December, 1994
by Masafumi Tamura

Acknowledgment

The present work has been conducted under the guidance of Professor
Haruo Kuroda, and has been completed under the supervision of Professor
Minoru Kinoshita. The author would like to express sincere gratitude for
their many valuable discussions, suggestions and encouragement.
The author is indebted to Professor Kyuya Yakushi (The Institute for
Molecular Science) for useful discussions and encouragement. The author
would like to acknowledge Dr. Hiroyuki Tajima for enlightening discussions and encouragement.
The author gratefully acknowledges Professor Hayao Kobayashi (Toho
University), Professor Akiko Kobayashi, Professor Reizo Kato for providing him with many attractive samples, useful comments and encouragement.
The author is grateful to Dr. Madoka Tokumoto (ElectrotechnicalLaboratory) for helping the magnetic measurements, valuable discussions
and encouragement. The author is also indebted to Professor James S.
Brooks (Boston University), Dr. Alka G. Swanson (AT&T Bell Laboratories), Professor C. C. Agosta (Clark University), Dr. S. T. Hannahs(Massachusetts Institute of Technology). Their many experimental efforts
have come to fruition of the first observation of the de Haas-van Alphen
oscillations in θ-(BEDT-TTF)2I3. The author would like to thank Dr.
Takehiko Mori (Tokyo Institute for Technology), Dr. Haruyoshi Aoki,
Dr. Shinya Uji and Dr. Taichi Terashima (National Research Institute for
Metals) for their excellent experiments and helpful discussions. The
author acknowledge his debt to Professor Koichi Kikuchi (Tokyo Metropolitan University) for providing the author with the interesting samples
of a salt of a non-symmetric donor, DMET.

Many thanks are given to all the members of the Laboratories ofProfessor Haruo Kuroda, Professor Minoru Kinoshita and Professor Reizo
Kato.

Contents

Chap. 1. General Introduction

1

1.1. Electronic Structures of Molecular Conductors

2

1.2. Tight-binding Band Model

5

1.3. Experimental Approach to the Band Structure

7

1.4. Optical Study of the Band Structure

91.5. The Purposes of This Study

13

1.6. The Contents of the Present Thesis

15

References

16

Chap. 2. Theoretical Aspects of the Optical Transitionsof Conduction Electrons19

2.1. Introduction

20

2.2. The Band Structure20

2.3. Intraband Transitions

23

2.4. Interband Transitions

27

References

30

Chap. 3. Measurement of Polarized Reflectance Spectrum
and Analysis of the Data33

3.1. Measurement of Polarized Reflectance Spectrum

34

3.2. Analysis of the Reflectance Spectrum

37

3.3. Derivation of Band Structure and Fermi Surface

38

3.4. A Case Study of the Relation between the Optical
Properties and Fermi Surface
References
Chap. 4. Electronic Structure of θ-(BEDT-TTF)2I3
4.1. Introduction

41
57
59
60

4.2. Experimental

63

4.3. Reflectance Spectra

65

4.4. Magnetic Properties in the Superconducting State80

4.5. Thermoelectric Power91

4.6. The de Haas-van Alphen (dHvA) Oscillations

97

4.7. The Subnikov-de Haas (SdH) Oscillations
and Small Fermi Surface

111

4.8. The Angle Dependent Magnetoresistance Oscillation(ADMRO)

121

4.9. Concluding Remarks

132

References

133

Chap. 5. Reflectance Spectra of κ-(BEDT-TTF)2I3

137

5.1. Introduction

138

5.2. Experimental

140

5.3. Results

141

5.4. Discussion

163

5.5. Concluding Remarks

166

References

167

Chap. 6. Reflectance Spectra of Me4N[Ni(dmit)2]2

171

6.1. Introduction

172

6.2. Experimental

175

6.3. Results and Discussion

176

6.4. Conclusion

183

References

184

Chap. 7. Reflectance Spectra of α-Et2Me2N[Ni(dmit)2]2

1857.1. Introduction

186

7.2. Experimental

190

7.3. Results and Discussion

190

7.4. Conclusion

199

References

200

Chap. 8. Reflectance Spectra of κ-(DMET)2AuBr2

201

8.1. Introduction

202

8.2. Experimental

205

8.3. Results

205

8.4. Discussion

214

8.5. Conclusion

216

References

216

Chap. 9. Concluding Remarks for the Present Thesis
References
Appendices

219
221
223

A. Disappearance of Interband transitions
due to Symmetry of Molecular Arrangement

224

B. Interplay of Interband Transitions and Electronmolecular-vibration (e-mv) Coupling

234

References

239

List of Publications

241

Chapter 1.

General Introduction

11.1. Electronic structures of Molecular Conductors

Molecular conductors (organic conductors or synthetic metals) haveafforded many research topics both in chemistry and physics for the last
40 years. The study originated from the pioneering work by Akamatu,
Inokuchi and Matsunaga in 1954 [1]. The most memorable is the discovery
of superconductivity in a series of quasi-one-dimensional conductors,
TMTSF salts [2,3], and in a variety of two-dimensional conductors, BEDTTTF salts [3]. Following them, some salts of non-symmetric organic
donors such as DMET [4-6] and those of the metal complexes, [Ni(dmit)2][7] and [Pd(dmit)2] [8], have been found to exhibit superconductivity. Sofar, various types of metallic systems have been identified. (For thechemical structures of TMTSF, BEDT-TTF, DMET and [M(dmit)2], seeFig. 1.1) The discovery of superconductivity in an alkaline metal salt of
C60 [9] is also striking in this field.
Not only superconductivity but also several peculiar physical properties such as CDW (charge density wave) and SDW (spin density wave),which are closely related to their low dimensionality, have been revealed
in molecular conductors. Such exotic features have activated many experimental and theoretical investigations on the electronic structures of thesesystems, both from chemical and physical sides. It has been widely
accepted that the dimensionality of the electronic structures plays a keyrole in the physical properties of these materials.
In the recent several years, application of sophisticated experimental
techniques at low temperatures, and sometimes under high magnetic fields,
have provided much information about the electronic structures of these2conductors (see Sec. 1.3). This makes it clear that their electronic band
structures can be basically understood within the framework of tightbinding band model based on the frontier molecular orbital, HOMO (thehighest occupied molecular orbital) or LUMO (the lowest unoccupied
molecular orbital). At the present time, this should be considered as themost significant characteristic of the electronic structures of the molecular
conductors.
Prior to the direct experimental evidence for the formation of band
structures, the tight-binding band model has been applied to elucidate thedimensionality of a molecular conductor from the structural viewpoint
mainly by chemists [10,11]. The basis of this approach is that the separationof the intramolecular energy levels is sufficiently larger than the intermolecular charge transfer energy. The estimation of the intermolecular transfer
integrals is the main part of the investigation in this framework. Usually,
they are obtained as intermolecular overlap integrals multiplied by anenergy factor, where the overlap integrals are calculated on the basis of
the extended Hückel method [12] and only the nearest neighbor ones are
considered. In this way, even rather involved features of the electronic
properties become describable in terms of the chemical ideas based on
the frontier orbitals and the transfer integrals between them. On the other
hand, in order to improve this framework and to develop a new design ofmolecular conductors, it is desired to estimate the intermolecular interactions experimentally.

3

4

1.2. Tight-binding Band Model

The most basic theory of metals is the electronic band theory basedon Bloch's theorem. In the theory, the one-electron states are describedby the Bloch wavefunctions, which is delocalized over the crystal and has
the periodicity in accord with that of the lattice. Fourier transformationof the Bloch function with respect to the lattice periodicity yields local
function localized on each unit cell. This localized function is called
Wannier function. The Bloch function is thus given as the linear combination of the Wannier functions.
The tight-binding band model is constructed by regarding the molecular orbitals (MOs) in the crystal as the Wannier functions. This meansthat the molecular identity still holds in the crystal. Diagonalization of
the one-electron Hamiltonian in the space of the MOs taking account of
the lattice periodicity yields the one-electron energy eigenvalues as functions of the wavevector, k. This is the energy band structure in thetight-binding framework. This treatment of the electronic state of thecrystal is analogous to the MO theory based on the linear combination of
atomic orbitals. The only difference is the consideration of the lattice
periodicity.
If only one MO on each molecule is taken into the basis set, thetreatment is in a complete analogy to the primitive Hückel theory of
π-MOs, in which only one 2p orbital is considered for each atom in aplanar aromatic compound. For simplicity, it is assumed that all the MOs
are equivalent in the following. The off-diagonal element of the oneelectron Hamiltonian between the MOs, | a > and | b >,5ta, b = < a | H | b >,(1.2.1)is called the transfer integral between | a > and | b >. The role of t isessentially the same as that of the resonance integrals in the primitive
Hückel MO theory. Similarly, only the nearest-neighbor ones are considered in the ordinary tight-binding band calculations. The transfer integrals
are usually approximated as the overlap integrals multiplied by an energy
factor, ε , that is,ta, b ≈ ε Sa, b ,(1.2.2)Sa, b = < a | b > ,(1.2.3)where,and ε is given by the Mulliken approximation,

ε < a | a > ≈ < a | H | a > = < b | H | b > ≈ ε < b | b >.

(1.2.4)The second equality comes from the equivalency of the MOs. With MOs
normalized, ε agrees with the energy of the MOs.
The extended Hückel method is the standard way to calculate theMOs in the molecular conductors. The band calculations start from theestimation of the overlap integrals between the MOs in the crystal calculatedby the extended Hückel method. Then, the transfer integrals are obtainedby use of Eqs. (1.2.2) and (1.2.4).

6

Within this framework, the metallic properties of molecular crystals
can be discussed in terms of the chemical intermolecular interactions. In
Chapter 2, the procedure of band calculations starting from a set of transfer
integrals is explained.

1.3. Experimental Approach to the Band Structure

Since most electronic properties of a metal are prescribed by the
Fermi surface, the study of the Fermi surface can be compared to thestudy of the frontier orbitals, which dominates the chemical properties ofa molecule. The band structure or the Fermi surface of a metallic material
can be studied by various types of experiments.
The electronic specific heat and the static spin susceptibility afford
the estimation of the density of states at the Fermi level. The coefficientof the T-linear term in specific heat, as well as the temperature independent
Pauli-like susceptibility, is proportional to the density of states.
The transport properties, including the Hall effect, thermoelectric
power and magnetoresistance give information on the band-filling, thecurvatures of band near the Fermi level, or the topology of the Fermi
surface. The electric resistivity, of course, provides significant informationon the band structure. Magnetic resonances (NMR and ESR) and (photo)electron spectroscopy (e.g., XPS, UPS and energy-loss spectroscopy)
are also known to work as useful tools to study the band structure, as well
as some microscopic features.
The de Haas-van Alphen (dHvA) effect and the Shubnikov-de Haas

7

(SdH) effect, the magneto-quantum oscillations, are the direct probe to
the Fermi surface [13]. These experiments are carried out under highmagnetic field at low temperature. Periodic oscillation of magnetization(dHvA) or resistance (SdH) as a function of inverse magnetic field can beobserved in such a condition. This is a result of the cyclotron motion ofcarriers along the Fermi surface. The frequency of the oscillation is
proportional to the extremal cross-sectional area of the Fermi surface
viewed along the field direction. From the temperature dependence ofthe oscillation amplitude, the cyclotron mass, a kind of effective mass ofthe carriers on the Fermi surface, is estimated. These techniques, which
are sometimes called the fermiology, are based completely on the natureof the Fermi surface. In other words, the information provided by thesemeasurements comprises purely of the features of the Fermi surface.
These experiments cannot be applied to one-dimensional conductors without closed Fermi surface.
There is a special technique which has been advanced in the field of
molecular conductors: the angle dependent magnetoresistance oscillation(ADMRO) [14-17]. This phenomenon is unique to the cylindrical Fermi
surface with small warping, which is characteristic of the quasi-twodimensional metals such as the BEDT-TTF salts [16,17]. The ADMROis characterized by the periodic oscillation of magnetoresistance as afunction of tan θ, where θ is the angle between the magnetic field and theaxis normal to the two-dimensional plane. From the period, the Fermi
wavenumber, i.e., the radius of the cylindrical Fermi surface can be derived[16].
Unlike the magneto-quantum oscillations and the ADMRO, the pre-8

viously mentioned experiments may include information on other features
related to phonons, defects, and so on. Sometimes, they may yield controversial results about the band structure. This caused the criticism on theapplicability of the tight-binding band model. However, the situation has
been changed by the observation of the SdH effect in κ-(BEDTTTF)2[Cu(NCS) 2] in 1988 [18], for the first time in molecular conductors.
Since then, the SdH or the dHvA experiment has been applied to many
two-dimensional molecular conductors [19]. This advance has greatly
contributed in establishing the validity of the tight-binding model. It isalso noteworthy that the observation of the magneto-quantum oscillations
requires considerably high quality of the sample. Therefore, the appearanceof the oscillations in a variety of molecular conductors indicates the highquality of the crystals of such molecular conductors.

1.4. Optical Study of the Band Structure

Except for the electron spectroscopy, the experimental methods mentioned in the preceding section are related to the ground state properties
or to low-lying states just above the ground state. In contrast to them, theoptical spectroscopy from the infrared to the visible range is the observationof the excitation of electrons by such photon energies. For this reason,
the optical properties generally reflect the global structure of the band
rather than the local structure near the Fermi surface. Consequently, theoptical spectrum over the infrared to visible regions is insensitive to thephenomena characterized by a low energy below about 0.05 eV, such as

9magnetism and superconductivity. In order to study such phenomena,
experiments in the far-infrared or microwave region are required.
Practically, the study of optical spectra of metals has the following
advantages.
(1) It is very easy to examine the dimensionality and anisotropy by
the optical experiment using polarized light, while it is difficult
to control the direction of current in a small single crystal in themeasurement of transport properties.
(2) The optical properties are not sensitive to internal cracks in the
crystal, while such cracks reduce the accuracy of the results ofthe transport measurements.
(3) The optical technique can be applied to a metal with many
defects or a non-metallic material, whereas the methods sensitive
to the Fermi surface cannot yield any information on such kindof materials.
(4) Materials with any dimensionality can be studied by the optical
spectroscopy.
(5) Neither high magnetic field nor very low temperature is requiredin the optical experiments.
The optical transitions of the conduction electrons can be divided
into two categories, intraband transitions and interband transitions. The
intraband transitions have the character of the collective motion of charge
carriers corresponding to the plasma oscillation, which is characteristic ofa metal. The intraband transitions are described by the Drude model ofdielectric function (complex ac dielectric constant),10ε (ω ) = ε C - ωP 2 / (ω 2 + i ω τ),(1.4.1)

where hω is the photon energy, τ is the relaxation time of the carriers,and ω P is the plasma frequency. The first term includes the constantcontributions from the higher excitations. From the plasma frequency,
the optical effective mass is estimated by the relation,
m* = 4 π n e 2 / ω P2 ,(1.4.2)where n is the number density of the carriers per unit volume. Thisnature of the intraband transition has been widely used to estimate theoptical effective mass, the number of the carriers and the relaxation timeof a metal. In view of the relation,

σdc = ω P2τ / 4π = n e 2τ / m*,(1.4.3)

it is clear that the intraband transitions are closely related to the dc conductivity. The integrated intensity of the intraband transitions, as well as
the dc conductivity, is determined by the character of the Fermi surface.
It should be remarked here that the optical effective mass should be
distinguished from the thermal effective mass related to the specific heatand from the cyclotron mass. These three masses agree with one another
only in the free-electron model. Generally, the three masses take different
values, particularly in a low-dimensional band described by the tightbinding model; only the orders of magnitudes are in agreement. These
are defined as the averages of the Fermi velocities by different ways of11integrations [20]. In addition, the optical mass is a tensor; the principal
values can be anisotropic with respect to the direction of light polarization.
And the cyclotron mass is defined for each closed cyclotron orbit. Caremust be taken, in comparing the "effective masses" obtained from different
experiments.
Outstanding examples illustrating the relation of the Fermi surface
and the intraband transitions can be found in recent literature. The one
concerns the metal-insulator transition in (DCNQI)2Cu system, where
DCNQI is N,N ′-dicyanoquinonediimine [21]. In this system, the intrabandtransitions disappear abruptly at the phase transition accompanied by a
drastic change in dimensionality. From this, it follows that the systemchanges from a three-dimensional metal into a one-dimensional insulator.
Another one deals with the temperature dependence of the reflectancespectra of κ-(BEDT-TTF)2Ag(CN)2·H2O [22]. Disappearance of a part of
the Fermi surface at low temperature was recognized as the suppressionof the intraband transitions in this case.
On the other hand, the interband transitions, optical transitions acrossthe band gaps, reflect the gap structure of the bands. By examining thefine structures of the spectral shape of interband transitions, the band
structures of inorganic crystals including many semiconductors have been
extensively investigated. However, the situation is somewhat different in
the organic molecular conductors. Because of considerably narrow widthof the conduction band, only a broad peak corresponding to the interband
transitions can be observed in the infrared spectra of the molecular conductors. It seems that the interband transitions afford only coarse featuresof the band structure. However, the interband transitions can probe the12molecular packing mode in the crystal, which is very useful informationin considering the band formation in view of the tight-binding model.
This characteristic arises from the fact that the interband transitions havean origin similar to that of the intermolecular excitations of a charge-transfer
complex, which is obvious in the framework of the tight-binding model.
In this model, the origin of the band gaps is attributed to the difference
between the transfer integrals, which are determined by the molecular
packing.
In the determination of the basic optical properties of metals, themeasurement of reflectivity is the most convenient and reliable technique.
The absorption spectroscopy is difficult, because most metallic materials
are not transparent. It is also difficult to measure the thickness of a small
single crystal sample, which results in inaccurate estimation of the opticalconstants. On the other hand, the measurement of reflectivity or reflectancespectra requires only smooth surface of the sample. In this study, polarizedreflectance spectra were measured by use of microspectrophotometers(spectrophotometers equipped with reflection microscopes and polarizers)
arranged especially for small single crystals of molecular conductors.
Cooling the sample down to low temperature is efficient to resolve different
types of transitions. For this purpose, low temperature measurements
were carried out.

1.5. The Purposes of This Study

This study mainly treats the determination of the band structures and13the Fermi surfaces of several two-dimensional conductors based on π-donoror π-acceptor molecules by use of the intensities of the intraband transitions
obtained from the polarized reflectance spectra. The compounds picked
up in this study possess considerably simple molecular arrangement, atleast approximately. This has made it possible to relate the intrabandtransitions to the transfer integrals describing the band structure. The
application of this method, developed in this study, is unique to themolecular conductors, which is describable by the simple tight-binding
band model. Nevertheless, it is expected to extend the applicability ofthe optical spectroscopy in this field.
The obtained results are then compared with those from the otherexperiments such as the magneto-quantum oscillations or the ADMRO,
in order to demonstrate the validity of the method. If both results are inagreement, it can be concluded that the band structure is well describedby the simple tight-binding model over the energy range of about 1 eV,
the highest excitation energy of conduction electrons within the band.
This information cannot be obtained from the other experiments.
The interband transitions play only a secondary role in this study.
In some cases, they are too weak to be treated quantitatively. Moreover,accurate theoretical calculations of the intensities of the interband transitions are considerably difficult, because they explicitly concern the excitedstates of a many-electron system. However, as mentioned in the preceding
section, the interband transitions give useful information on the moleculararrangement. The interband transition thus helps the determination of thetight-binding band structures.

14

1.6. The Contents of the Present Thesis

In Chapter 2, theory of the optical transitions of the conduction
electrons is treated in the framework of the tight-binding band model.
Chapter 3 summarizes the experimental techniques and the procedures ofanalysis of the optical data.
From Chapters 4 to 7, experimental results and discussion of theelectronic band structures and the shape of the Fermi surfaces of several
two-dimensional molecular conductors are given. Chapter 4 is devoted
to the study of the electronic structure of the θ-(BEDT-TTF)2I3. Since themolecular arrangement of this material has approximately high symmetry,θ-(BEDT-TTF)2I3 is considered to be suitable for the investigation fromdifferent points of view. Thereby, Chapter 4 includes not only the optical
study but also the studies of the magnetic properties, thermoelectric power,
the dHvA and SdH oscillations, and the ADMRO. Chapter 5 describes
the study of the reflectance spectra of a dimeric BEDT-TTF salt, κ-(BEDTTTF)2I3. Chapters 6 and 7 treat the studies of the reflectance spectra of
two [Ni(dmit)2] salts, Me4N[Ni(dmit)2]2 and α-Et2Me2N[Ni(dmit)2]2, respectively. Chapter 8 deals with the study of reflectance spectra of a
two-dimensional DMET salt, κ-(DMET)2AuBr2.
Concluding remarks for the present thesis are given in Chapter 9.
Appendices describe some useful properties of the interband transitions
in connection with the symmetry of the molecular arrangement.

15

References

1. H. Akamatu, H. Inokuchi and Y. Matsunaga: Nature 173 (1954) 168.
2. D. Jérome, A. Mazaud, M. Ribault and K. Bechgaard: J. Physique Lett. 41 (1980)
L95-98; K. Bechgaard: Mol. Cryst. liq. Cryst. 79 (1982) 1.
3. See for example, The Physics and Chemistry of Organic Superconductors, Ed. by
G. Saito and S. Kagoshima (Springer, Heidelberg, 1990); Organic Superconductors: Synthesis, Structure, Properties and Theory, Ed. by J. M. Williams, J. R.
Ferraro, R. J. Thorn, K. D. Carlson, U. Geiser, H. H. Wang, A. M. Kini and
M.-H. Whangbo (Prentice Hall, Englewood Cliffs, 1992).
4. K. Kikuchi, K. Murata, Y. Honda, T. Namiki, K. Saito, H. Anzai, K. Kobayashi,
T. Ishiguro and I. Ikemoto: J. Phys. Soc. Jpn. 56 (1987) 4241.
5. K. Murata, K. Kikuchi, T. Takahashi, K. Kobayashi, Y. Honda, K. Saito, K.
Kanoda, T. Tokiwa, H. Anzai, T. Ishiguro and I. Ikemoto: J. Mol. Elect. 4 (1988)
173.
6. G. C. Papavassiliou, G. A. Mousdis, J. S. Zambounis, A. Terzis, A. Hountas, B.
Hilti, C. W. Mayer and J. Pfeiffer: Synth. Metals 27 (1988) B379.
7. A. Kobayashi, H. Kim, Y. Sasaki, R. Kato, H. Kobayashi, S. Moriyama, Y.
Nishio, K. Kajita and W. Sasaki: Chem. Lett. 1819 (1987).
8. A. Kobayashi, H. Kobayashi, A. Miyamoto, R. Kato, R. A. Clark and A. E.
Underhill: Chem. Lett. 2163 (1991).
9. A. F. Hebard, M. J. Rosseinsky, R. C. Haddon, D. W. Murphy, S. H. Glarum, T. T.
M. Palstra, A. P. Ramirez and A. R. Kortan: Nature 350 (1991) 600.
10. T. Mori, A. Kobayashi, Y. Sasaki, H. Kobayashi, G. Saito and H. Inokuchi: Bull.
Chem. Soc. Jpn. 57 (1984) 627.
11. For example, see: M.-H. Whangbo, J. M. Williams, P. C. W. Leung, M. A. Beno,16

T. J. Emge, H. H. Wang, K. D. Carlson, G. W. Crabtree: J. Am. Chem. Soc. 107(1985) 5815.
12. R. Hoffmann: J. Chem. Phys. 39 (1963) 1397.
13. D. Shoenberg: Magnetic Oscillations in Metals (Cambridge Univ. press, Cambridge,1984).
14. M. V. Kartsovnik, P. A. Kononovich, V. N. Laukhin and I. F. Shchegolev: JETP
Lett. 48 (1988) 542.
15. K. Kajita, Y. Nishio, T. Takahashi, W. Sasaki, R. Kato, H. Kobayashi, A. Kobayashi
and Y. Iye: Solid State Commun. 70 (1989) 1189.
16. K. Yamaji: J. Phys. Soc. Jpn. 58 (1989) 1520.
17. R. Yagi, Y. Iye, T. Osada and S. Kagoshima: J. Phys. Soc. Jpn. 59 (1990) 3069.
18. K. Oshima, T. Mori, H. Inokuchi, H. Urayama, H. Yamochi and G. Saito: Phys.
Rev. B 38 (1988) 938.
19. For a review, see: J. Wosnitza: Int. J. Modern Phys. B 7 (1993) 2707.
20. For example, see: J. M. Ziman: Principles of the Theory of Solids (Cambridge
Univ. Press, Cambridge, 1972) 2nd Ed., Chaps. 8. and 9.
21. K. Yakushi, A. Ugawa, G. Ojima, T. Ida, H. Tajima, H. Kuroda, A. Kobayashi, R.
Kato and H. Kobayashi: Mol. Cryst. Liq. Cryst. 181 (1990) 217.
22. R. Masuda, H. Tajima, H. Kuroda, H. Mori, S. Tanaka, T. Mori and H. Inokuchi:
Synth. Metals 56 (1993) 2489.

17

18

Chapter 2.

Theoretical Aspects of the Optical Transitionsof Conduction Electrons192.1. Introduction

In this chapter, theoretical aspects of the optical transitions of the
conduction electrons are described. The calculations of the tight-binding
band are explained in Sec. 2.2. Section 2.3. deals with the intrabandtransitions on the basis of the relaxation time approximation of the Boltzmann equation. Finally, a brief treatment of the interband transitions isgiven in Sec. 2.4. It is recommended to refer to these descriptions tounderstand the analysis of the optical data in the following chapters.

2.2. The Band Structure

In the tight-binding framework, the band structure is calculated fromthe set of transfer integrals as follows. Using the second-quantization
expression, the one-electron Hamiltonian is simply expressed as,H=∑ ∑ t l m, l ′m ′ a†l m al ′m ′ ,l, l'(2.2.1)m, m'

where (l, m) specifies the m 'th MO in the l 'th unit cell. The summation
is taken over the crystal. The Fourier transform of a†l m with respect to thelattice periodicity is given by,a†l m = N –1/2 ∑ a†m, k exp(ik·Rl ) ,(2.2.2)kwhere N is the number of the unit cells in the crystal, and R l denotes the20position of the l 'th unit cell. Substitution with this converts Eq. (2.2.1)
into,
H = N –1 ∑∑ ∑ t l m, l ′m ′ exp[i(k·R l – k ′·R l ′)] a†m, k am′, k ′ . (2.2.3)

m, m ′ k, k′ l, l ′Taking account of the lattice periodicity, we have,∑ t l m, l ′m ′ exp[i(k · Rl – k ′ · R l ′)]l, l ′= ∑exp[i(k – k ′) · R l ′] ∑ t l m, l ′m ′ exp[ik · (Rl – Rl ′)]l′

l –l ′= ∑exp[i(k – k ′) · R l ′] ∑ t 0 m, l m ′ exp( ik · R l ) .
l′(2.2.4)l

The summation over l ′ results in∑ exp[i(k –k ′) · Rl ′] = N δ k, k ′ .

(2.2.5)l′Thus Eq. (2.2.3) is reduced into,
H= ∑m, m ′= ∑m, m ′∑ [∑ t 0 m, l m ′ exp(ik · Rl) a†m,k am,′k ]kl∑ hm,m ′(k) a†m,k am,′k ,kwhere,

21(2.2.6)

hm,m ′(k) = ∑ t 0 m, l m ′ exp(ik · Rl ) .

(2.2.7)lThis is an M × M matrix, M being the number of MOs in the unit cell.
Equation (2.2.6) has been already diagonalized with respect to k. The
major contribution in the summation over (l, m ′) in (2.2.6) comes fromthe nearest-neighbor sites of the (0,m) one. By diagonalizing hm,m ′(k), Eq.
(2.2.7), we obtain the eigenvalues, ε µ(k) (µ = 1, 2, ...M) for each k. Thatis,H = ∑ ε µ(k) a†µ, k aµ, k,(2.2.8)

µ, k

with,
aµ, k = ∑ Uµ,m(k) amk ,(2.2.9)m

where Uµ,m is the element of the unitary matrix to diagonalize hm,m ′(k).
The tight-binding band structure is thus given as εµ(k). From this, any
electronic properties related to the conduction band can be given in termsof µ (the branch index), k, and transfer integrals, as far as the one-electronband picture holds.

22

2.3. Intraband Transitions [1,2]

In the equilibrium state, the distribution of the electrons in a conduction
band is given by the Fermi distribution function, f0(ε k , T). Application ofan external electric field, E(r, t), perturbs the distribution into f(r, T) = f0
+ g(r, T). The Boltzmann equation for f(r,T) is given as,
df / dt = ∂f / ∂t + (vk – eE /h) · gradk f .

(2.3.1)

Assuming that the relaxation of g(r,T) to 0, as well as f(r,T) to f0, is asingle exponent type with the relaxation time, τ, we obtain a differential
equation for g(r,T),∂g / ∂t + vk · gradr g + g / τ = – eE · vk (∂f0 / ∂εk) .

(2.3.2)Since we concern only the linear response to the field, only the term
linear to E is retained. Suppose that both E and g are periodic in timeand space, i.e.,
E = Eq, ω exp[i(q · r – ω t)] ,(2.3.3)g = gq, ω exp[i(q · r – ω t)] .

(2.3.4)

Then, the solution of Eq. (2.3.2) is given in terms of gq, ω ,gq, ω (k) = – e τ Eq, ω · v k (– ∂f0 / ∂ε k) / [1+ τ (q · v k – iω )] . (2.3.5)

23

With this distribution function, any kind of the bulk collective responseto the field can be calculated.
In what follows, the principal value of the conductivity tensor is
calculated. The optical response can be expressed in terms of the complex
conductivity with respect to the electric field of the radiation. The current
density along the principal axis, x, is written in the form,

Jx = 2 (1/8π 3)∫ (– e v ) g d k .
3k(2.3.6)x

The factor 2 stems from the spin degrees of freedom. Substituting g in
Eq. (2.3.6) with Eqs. (2.3.4) and (2.3.5), we obtain(Jq, ω )x = (e 2/4π 3) (Eq, ω )x

∫ τ |v | [1+τ(q·v – iω )] (– ∂f / ∂ε ) d k .
2

k

-1

k

3

0

k

(2.3.7)Comparing this with the definition of the conductivity, σ ,Jq, ω = σ (q, ω ) Eq, ω ,(2.3.8)we have,∫

σx(q, ω) = (e 2/4π3) τ |vk | 2 [1+τ (q·vk – iω )] –1 (– ∂f0 / ∂εk) d 3k .
(2.3.9)This can be reduced into24∫σx(ω ) = (e 2/4π 3 h) [(vF )x2/|vF |] (1/τ – i ω ) –1 dS ,(2.3.10)F.S.

by use of the facts,ω = 2 π c′ | q | » | q·vk | ,(2.3.11)– ∂f0 / ∂ε k ≈ δ(ε – εF),(2.3.12)d 3k = (d ε / | gradk ε k |) dS ,(2.3.13)

where c ′ is the light velocity in the material, and kBT « εF is assumed.
The integration in Eq. (2.3.10) is taken over the Fermi surface. We
assume here that τ in Eq. (2.3.10) is independent of | k | and ω . Only theanisotropy of τ is phenomenologically taken account of. Thus we have,∫

4π σx(ω ) = (1 / τx – iω ) –1 (e 2 / 4π2 h) [(vF )x2 / |vF|] dS .

(2.3.14)F.S.

The assumptions made for τ is not a trivial one; it should be checked byexperiments in each case. Introduction of the plasma frequency, (ωP )x, by∫(ω P2)x = (e 2 / π 2 h) [(vF)x2 / |vF |] dS ,(2.3.15)

F.S.

leads to4π σx(ω ) = (ωP 2)x / (1 / τ – i ω ) ,25(2.3.16)or, in terms of the dielectric function,

ε x(ω ) –1 = (4π i /ω ) σ x(ω ) = – (ωP 2)x / ( ω 2 – i ω τ) . (2.3.17)
The final expression is equivalent to that of the Drude model. The
intensity of the intraband transitions, ω P2, is thus related to the band
structure near the Fermi surface. That is, the plasma frequency can becalculated from the band structure, εµ(k) in Eq. (2.2.8), using Eq. (2.3.15).
Application of Gauss' theorem to Eq. (2.3.15) yields∫(ωP 2)x = (e 2 / π2 h) (vF )x ux

dS

F.S.

∫

= (e 2 / π 2 h) div [(vF)x ux] f0(ε k) d 3k
= (e 2 / π 2 h 2)

∫ (∂ ε / ∂k ) f (ε ) d k ,
2

2

x

3

0

k

(2.3.18)

where ux is the unit vector in the x-direction. For a parabolic band givenasε = – h 2[kx2 /2m*x + ky 2 /2m*y +kz2 /2m*z ] ,(2.3.19)

Eq. (2.3.18) reduces into a simple form,(ω P2)x = 4 π n e 2/m*x ,
where
26

(2.3.20)∫n = (1 / 4π 2) f0(ε k) d 3k ,(2.3.21)is the number density of the electrons.
In the above discussion, the intraband transitions are regarded as thetransverse collective motion of the conduction electrons. Such kind oftreatment is based closely on the theory of random-phase approximation
for the Fermi liquid. However, the assumption for τ to be independent of| k | and ω is a subtle problem. If there is a relaxation process important
only at low frequencies, the relation between the optical properties andthe Fermi surface, Eq. (2.3.15), may be modified. This can be detected as
the discrepancy between the band structure deduced from the opticalproperties and that concluded from the transport properties. Such a "dynamical" effect implies the deviation of the band structure just near the
Fermi surface from the global one over the range of the photon energy.

2.4. Interband Transitions [3-5]In the framework of the one-electron band theory, the interband
transitions can be regarded as one-electron excitations, which is in contrast
to the case of the intraband transitions. For such a process, the probabilityof the optical transition can be readily obtained. Suppose the Hamiltonian
for the tight-binding model, Eq. (2.2.1),H=∑ ∑ t l m,l ′m ′ a†l m al ′m ′ ,l, l '

m, m'

27(2.4.1)

And the x-component of the electric dipole moment operator is written in
the form,

Px = e∑ (Rl, m )x a†l m al m ,(2.4.2)l, mwhere Rl, m is the position of the m 'th molecular site in the l 'th unit cell.
From the standard first-order perturbation theory with the dipole approximation, the transition probability per unit time interval is obtained as,
w = Ex2 ∑ < i | P x | f >< f | P x | i > (π / 2h) δ(hω – hωf, i) , (2.4.3)
i, f

where i and f denotes the initial and final states, respectively. The energiesof the initial and final states are εi and ε f , respectively, and the transitionenergy is then, hω f, i = εf – ε i . The power loss of light due to theabsorption can be expressed in terms of the transition probability, that is,
p =h ω w .

(2.4.4)On the other hand, the power loss is related to conductivity as,
p = σ x E x2 V / 2 ,(2.4.5)where V is the volume of the crystal. Comparison of Eq. (2.4.5) with Eq.
(2.4.4) yields
28

σx = (2 / V E x2) hω w .

(2.4.6)Consequently, from Eqs. (2.4.3) and (2.4.6), we obtain,

σx(ω ) = (π / V h) ∑ ω f, i < i | P x | f > < f | Px | i > δ(ω – ωf, i ),i, f(2.4.7)
When the δ-function is replaced by a Lorentzian line-shape function withthe width of γ, Eq. (2.4.7) is rewritten in the form,

σx(ω ) = (2i / V h) ∑ ω f,i < i | P x | f >< f | Px | i > ω /( ω 2– ωf, i + i γω ) .
i, f(2.4.8)The commutator of H and P x is,[H , Px] = e ∑ ∑ t l m,l ′m ′ a†l m al ′m ′ (Rl, m– R l ′, m′)x (a†l m al ′m ′ – a†l ′m ′ al m).
l, l ′ m, m ′(2.4.9)The matrix element of [H, Px] is,< i | [H, P x] | f > = (εi – ε f) < i | Px | f > .
Using this, Eq. (2.4.8) is converted into,σx(ω ) = (2 / iV h 3) ∑ < i | [H, P x] | f > < f | [H, P x] | i >
i, f

29

(2.4.10)(ω / ω×f, i) / ( ω 2– ω f, i + i γω ) .
(2.4.11)which is calculable in the framework of the tight-binding model by use of
Eq. (2.4.9). In the actual calculations, the pair of states, (i, f), is specified
by the pair, (µ, k) and ( µ ′, k), the occupied and vacant one-electronstates, respectively. The matrix element, then, reduces into that between
the one-electron states (µ, k) and (µ ′, k) and the summation is taken overthe pairs of the branch indices (µ, µ ′) and all k in the first Brillouin zone.
This formulation is based simply on the one-electron picture; highly
excited one-electron states, in which the many-body effect may play anessential role even in the metallic state, are concerned. As regards thequantitative features, the result may be modified, even if the band pictureholds for low-lying states. However, it is still useful in elucidating thebasic origin and character of the transition.

References

1. J. M. Ziman: Principles of the Theory of Solids (Cambridge Univ. Press, Cambridge,1972) 2nd Ed., Chaps. 7. and 8.
2. N. W. Ashcroft and N. D. Mermin: Solid State Physics (Saunders College, Philadelphia, 1976).
3. H. Tajima: Doctorate Thesis, The Univ. of Tokyo, 1986.

4. H. Tajima, K. Yakushi, H. Kuroda and G. Saito: Solid State Commun. 56 (1985)

30

159.
5. Useful discussions by Mr. T. Naito (Toho Univ.) on the derivation of Eq. (2.4.7) isgratefully appreciated.

31

32

Chapter 3.

Measurement of Polarized Reflectance Spectrum
and Analysis of the Data33

3.1. Measurement of Polarized Reflectance Spectrum

As mentioned in Sec. 1.4., the polarized reflectance spectroscopy is
the most efficient tool to examine the basic optical properties of a metallic
material. The standard experimental configuration used for a single crystalof conductors is the normal reflectance measurement, where the incident
light is normal to the crystal face. The absolute value of reflectivity canbe determined as a function of the wavenumber and polarization of lightby comparing the intensity of the light reflected from the sample facewith that from a standard mirror.
With regard to the application to the organic π-molecular conductors,
the main difficulty lies in the smallness of their single crystals. The
typical dimension of the most developed crystal face is about 1 × 1 mm 2or less. This problem can be solved by combined use of a microscope
and a spectrophotometer, the microspectrophotometry
A mid-infrared reflection microspectrophotometer made by Dr. H.
Tajima (Univ. of Tokyo), JASCO MIR-300, has been used in this study.
The main experimental techniques for the room temperature and low
temperature measurements of the infrared reflectance spectra have been
established by him, and the details are described in his thesis [1,2]. The
spectrometer is originally equipped with a semiconductor detector of InSb
(1800-5400 cm –1) and HgCdTe (720-1800 cm –1) by Infrared Associates
Inc. The optical study in Chap. 4. was carried out with this combination.
The performance of the apparatus has been improved by the introductionof a liquid helium cooled composite Ge bolometer by Infrared Laboratories
Inc. This has extended the spectral to 450-6000 cm –1 and has shown to34

afford better signal-to-noise ratio over the mid-infrared region, during
this study. The experiments in Chaps. 5-8 were done with this improved
version. For the low temperature measurements, a liquid helium flow
type cryostat, CF 104A by Oxford Instruments Limited, was used.
In the near-infrared and visible regions (4200-25000 cm –1), a modified
version of a reflection microspectrophotometer, Olympus MMSP, hasbeen used. The low temperature measurements in these regions werecarried out by use of a closed cycle helium refrigerator, CTI SPECTRIM.
Since the focal length of the objective lenses in the microscope varies
with wavenumber, the height of the sample has to be adjusted at eachwavenumber. This process has been automated by Dr. T. Ida (HimejiInstitute of Technology). Since the optical system used for the mid-infrared
region contains no lens, the adjustment of sample height is not requiredin the mid-infrared measurements.
The alternate measurement of the reflectance signals of a sample
and a standard mirror at each wavenumber is employed in order to minimize
the effects of drifting of the atmosphere (H2O and CO2 in air), the source
light intensity and the detector sensitivity. This was done by use ofcomputer-controlled movable sample stages. The light was chopped,
monochromatized, polarized and then introduced in the microscopes
equipped with half mirrors separating the incident and reflected rays.
The light intensity data were collected by lock-in amplifiers and transferredto the computers. The reflectivity values were calculated from the ratioof the sample signal to the mirror signal. As the standard mirror, a silver
mirror is used for the mid-infrared measurement, and a Si plate for thenear-infrared and visible regions.

35

In every case in this study, the crystal face on which the spectra
were measured has rather high symmetry; it includes at least one uniqueaxis. In such a case, the principal directions of reflectance should agreewith the crystal axes, because the reflectivity, as well as the conductivity,
is a tensor quantity. This was checked in every measurement by comparing
the X-ray photographs with the reflectance results obtained by rotating
the polarizer.
In the low temperature measurements, a single crystal sample wasmounted on a small Si plate placed on the rotating center of a goniometerhead for each cryostat attached on the cold head of the cryostat. The
attachment of the sample and the Si plate was made by Apiezon L grease.
The crystal face was aligned so as to retain the normal incidence of lightat room temperature. The goniometer head was covered with a radiation
shield and a vacuum chamber. Both have windows of KBr (mid-infrared)or quartz (near-infrared and visible), through which the ray can pass. The
window and the sample face were not kept in parallel, in order to avoid
the light reflected from the window. Evacuation inside the vacuum chambershould be followed by readjustment of the position of the cryostat. In thecooling, the temperature of the goniometer head was monitored by
Au(Fe)/Chromel thermocouple.
The alternate measurement was applied also to the low temperaturecase. In the mid-infrared apparatus, a movable mirror was installed above
the cryostat to switch the ray focus from the sample in the cryostat to a
standard mirror. The effect of the KBr windows was corrected by measuring
the data for the standard Ag mirror placed in the cryostat instead of asample at room temperature. In the near-infrared and visible measurements,36the position of the cryostat was automatically adjustable. Thereby, thealternate measurement was done by changing the ray focus from thesample on the cold head to a Si plate outside by moving the cryostat. The
effect of the quartz windows was also corrected by measuring the room
temperature reflectance of a Si plate or the sample.

3.2. Analysis of the Reflectance Spectrum

In this study, the most important thing is the estimation of the intensityof the intraband transitions. Since the observed spectra contain otherspectral features such as the interband transitions, it is required to separate
the intraband contribution from the other ones.
A curve-fitting method was applied to extract the intraband contributions from the reflectance spectra. As the model function, the Drude-Lorentz
model [3] was used. It is written, in terms of the dielectric function, as

ε (ω ) = ε C – ω P2 / ( ω 2 + iω / τ) + ∑ fj / [(ω j 2–ω2) – i γj ω ] . (3.2.1)
j

The reflectivity is related to the dielectric function by the equation,
R = [1 + | ε | – √2(| ε | + Re(ε )) ] / [1 + | ε | + √2(| ε | + Re( ε)) ] . (3.2.2)
The first term in the right-hand side of Eq.(3.2.1), εC, stands for theconstant contribution from the higher energy transitions outside the experimental spectral region. The second term, the Drude term, gives thecontribution of the intraband transition. The square of the Drude plasma37frequency, ωP 2, is the integrated intensity of the intraband transitions, andτ is the relaxation time of the charge carriers. The summation describes
the contributions of other transitions: the interband transitions, the e-mv
coupling modes. The intensity, position and spectral width of each termin the summation, are given as fj , ω j and γj , respectively. Strictly
speaking, they do not follow the Lorentzian profile. However, it is foundin every case in this study that these contributions are well approximatedas a Lorentzian or as a superposition of a few Lorentzian profiles, at leastat sufficiently low temperature. Thus, this Drude-Lorentz fitting afforded
satisfactory results for the estimation of the intraband plasma frequencies.
As is clear from Eq. (3.2.2), the reflectance spectrum is not thesuperposition of different transitions. This makes it difficult to choose
the Lorentzian terms suitable for the curve-fitting. In stead of the reflectancespectra, the conductivity spectra can be used for the choice. The conductivity spectra, in which the contributions of different transitions are more
resolved and the superposition of them is visible, were obtained by the
Kramers-Kronig transformation [4] of the reflectance spectra. The use of
conductivity spectra enables us to find out a deviation from the Drude
behavior easily.

3.3. Derivation of Band Structure and Fermi Surface

In Sec. 2.3., a general expression for the intensity of the intrabandtransitions is derived. That is,38∫(ω P2)x = (e 2 /π2 h) [(vF )x2 / |vF| ] dS ,(3.3.1)F.S.

In a two-dimensional case, this can be reduced into a very simple form.
Let the xy-plane be the two-dimensional plane. It follows immediatelythat (ωP 2)z = 0 because (vF)z is zero at any k. The transformations,dS = |vF| dkFy dk Fz / |(v F)x| = |vF | dkFx dkFz / |(vF )y| ,(3.3.2)yield the set of equations,(ω P2)x = (e 2 /π2 h) (2π / z)

∫ |(v ) | dk
F x

Fy

,

(3.3.3),(3.3.4)

F.S.

(ω P2)y = (e 2 /π2 h) (2π / z)

∫ |(v ) | dk
F y

Fx

F.S.

where z denotes the lattice period along the z-direction. These equations
well describe the anisotropic shape of a two-dimensional Fermi surface
and the anisotropy of the plasma frequency. Since the Fermi velocity
vector is perpendicular to the Fermi surface at each kF, the major contribution
comes from the portion of the Fermi surface nearly perpendicular to thelight polarization.
If the band structure is describable only by two independent transfer
integrals, t1 and t2, they can be directly determined form the plasma
frequencies by Eqs. (3.3.3) and (3.3.4). In the actual calculations, theknowledge of the Fermi level is required to settle the Fermi surface. The
Fermi energy can be determined from the band filling. That is, the39

following equation should be satisfied,∫∫F SBZ = | kFx | dkFy = | kFy | dkFx ,(3.3.5)where F is the band filling expected from the stoichiometry. The integralsin (3.3.5) stand for the half of the area inside the Fermi surface. Bysolving Eqs. (3.3.3)-(3.3.5) simultaneously, the three band parameters t1,
t2 and E F can be derived from the three experimental data, ω Px, ω Py andthe stoichiometry. The integrals in Eqs. (3.3.3)-(3.3.5) were numericallyevaluated in a computational program to find out the optimal values of t1,
t2 and EF , which was developed in this study.
From the band structure and Fermi surface determined as above, thecross-sectional area of each cyclotron orbit and the extremal values of kFx
and k Fy can be readily calculated. These parameters are referred to in thecomparison of the optical results with those of the magneto-quantum
oscillations and the ADMRO. The cyclotron mass was also evaluatedfollowing the definition [5],m*c = (h 2 / 2π) (dS / dE) ; E = E F,

(3.3.6)

where S is the corresponding cross-sectional area. For a two-dimensional
system, it is noteworthy that the evaluation of the cyclotron mass is thesame as that of the density of states.

40

3.4. A Case Study of the Relation between the Optical Properties
and Fermi Surface

In order to illustrate how the Fermi surface determines the opticalproperties and other physical properties, a case study was carried out fora model of a realistic two-dimensional molecular arrangement shown in
Fig. 3.1. The plasma frequencies, the cross-sectional areas, the cyclotron
masses and the extremal values of the Fermi wavenumbers were calculated
as functions of the transfer integrals, assuming 1/2- or 3/4-filling of the
conduction band. The computational method is outlined in the preceding
section.
In the following, the lattice parameters normalized to unity are used.
The band structure of the model is given by,E(kx, ky) = 2tx coskx ± 2 (ty2 + 2ty ty′cos k y + ty′ 2 )1 / 2 cos (kx / 2), (3.4.1)
When ty = ty′, this reduces intoE(kx, ky) = 2tx cos kx ± 4 ty cos(kx / 2) cos (ky / 2) .

(3.4.2)

There are the degeneracies between the two branches at the zone boundary(kx = π or k y = π) in Eq. (3.4.2), while it is removed at the k y = π line in
Eq. (3.4.1). As ty =ty′= 0 is approached, the system becomes onedimensional, while the system remains two-dimensional even at tx = 0.
The physical properties were calculated for the 1/2- or 3/4-filledband of Eq. (3.4.2), as functions of the ratio, tx /ty or ty /tx , where tx +ty =41

1 eV. The results are displayed in Figs. 3.2-3.6. for the 1/2-filled caseand in Figs. 3.7-3.11 for the 3/4-filled case.
In Figs. 3.2 and 3.7, the ratio of the plasma frequencies for thepolarization E || x and E || y is plotted. The effect of the dimensionality
is obvious. By use of these diagrams, we can determine the ratio of thetransfer integrals from the observed anisotropy of the plasma frequency.
Since the plasma frequency is proportional to x /√V (for the E || x
polarization) or to y /√V (for E || y), where V is the cell volume, theplasma frequency should be normalized with respect to these factors
before use.
The absolute values of the transfer integrals in eV units can beobtained by comparing the observed values of the plasma frequency withthe ones plotted in Figs. 3.3 and 3.8, which are normalized for the lattice
parameters and for tT1 / 2 = (tx +ty) 1 / 2 and are measured in Å 1 / 2 eV 1 / 2
units. Suppose that tx +ty = tT eV in our case and that we have the valueof the observed plasma frequency of ωP obs eV Å 1 / 2 normalized for thelattice parameters. Using the diagrams, we can find the value for tx +ty =1 eV, ωP calc Å 1 / 2 eV 1 / 2, from the relevant transfer integral ratio. Then, we
obtain tT = (ωPobs/ωPcalc)2, because the plasma frequency is proportional to
the square root of the transfer integrals. From the sum and ratio of thetwo, the value of each transfer integral can be determined.
The fractional value of the cross-sectional area for each cyclotron
orbit is shown in Figs. 3.4 and 3.9, as a function of the transfer integral
ratio. The topology of the Fermi surface varies with the ratio as schematically shown in Fig. 3.12, where the notation of the orbits is also shown.
Since the β-orbit has the constant value (1.0 for the 1/2-filled case and420.5 for 3/4-filled case), it is omitted in the diagrams. Strictly speaking,the cyclotron motion around the α-orbit should not be observed for thisband structure. Therefore, the present results should be regarded as thosewith considerably small difference between ty and ty′.
Figures 3.5 and 3.10 show the cyclotron mass for each orbit for tx +ty= 1 eV. We can obtain the cyclotron mass values in our case, tx +ty = tTeV, by dividing those in the diagrams by x y tT. In the 1/2-filled case, themass for the β-orbit is about twice as large as that for the α-orbit over awide range of the transfer integral ratio.
Figures 3.6 and 3.11 display the radii of the cyclotron orbits. These
values are useful not only in drawing the approximate shape of the Fermi
surface but also in predicting the ADMRO periods.
In order to see the effect of the removal of degeneracy, some quantities
are calculated for the 1/2-filled band of Eq. (3.4.1) as functions of therelative difference, 2(ty – ty′)/(ty + ty′). The values of the transfer integrals,
tx = 0.048 eV and ty + ty′ = 0.104 eV are used. Figure 3.13 shows theplasma frequencies. It is found that the intensity of intraband transitions
for E || y decreases more rapidly than that for E || x. The cyclotron
masses are shown in Fig. 3.14, where the magnetic breakdown is assumed
for the β-orbit. Compared with the plasma frequencies, they seem to beless sensitive to the removal of degeneracy in this model.

43

ty
ty
tx
y

tx

t y′
t y′

x

Fig. 3.1. A realistic model of the intermolecular interactions in two-dimensional
molecular conductors. All the molecules are equivalent. Each molecule
is surrounded by six nearest neighbor ones. Solid lines indicates theunit cell.

441(x/y)(ω py /ω px )

0.8

(a)

0.6

1/2-Filled

0.4
0.2
0

0

0.2

0.4

tx / ty

0.6

0.8

1

0.6

0.8

1

1
(b)

(x/y)( ω py /ω px )0.8

1/2-Filled

0.6
0.4
0.2
0

0

0.2

0.4
t y / tx

Fig. 3.2. Anisotropy of plasma frequency plotted against the transfer integral
ratio, for the 1/2-filled model band with ty = ty´. (a) for tx < ty,and (b) for tx > ty.

45

(V1/2 / x t T 1/2 ) ω px / Å1/2 eV1/2

15

10

10

5

1/2-filled
0

0

0.2

0.4

0.6

0.8

tx /ty

1

150

0

50
40

100

30
20

(b)

50

10

1/2-filled
0

0

0.2

0.4

0.6

0.8

1

(V1/2/ y t T 1/2) ω py / Å1/2 eV1/2(V1/2 / x t T 1/2 ) ω px / Å1/2eV1/2(a)5(V1/2 / y t T 1/2) ω py / Å1/2 eV1/2

150

ty /t xFig. 3.3. Plasma frequencies normalized for the lattice constants and for totaltransfer integral, tT = tx + ty, is plotted against the transfer integral
ratio, for the 1/2-filled model band with ty = ty´. (a) for tx < ty,and (b) for tx > ty.

46

0.25
(a)

S / SBZ

0.2

1/2-filled
0.15
α

0.1
0.05
0

0

0.2

0.4

tx /t y

0.6

0.8

1

0.4

S / SBZ

0.3

α

0.2

γ
(b)

0.1
0

1/2-filled

0

0.2

0.4

ty /t x

0.6

0.8

1

Fig. 3.4. Fractional cross-sectional area of each cyclotron orbit plotted against
the transfer integral ratio, for the 1/2-filled model band with ty = ty´.
(a) for tx < ty, and (b) for tx > ty. For the notation of the orbits, seeFig. 3.12.

47

x y tT (m*/me) / Å 2 eV

50
(a)

40

1/2-filled

β

30
20

α

10
0

0

0.2

0.4 t /t 0.6
x

0.8

1

y

x y tT (m*/me) / Å 2 eV

50
(b)

40

1/2-filled

β

30
20

0

α

γ

10
0

0.2

0.4

ty /tx

0.6

0.8

1

Fig. 3.5. Cyclotron mass of each orbit normalized for the lattice constants andfor total transfer integral, , tT = tx + ty, is plotted against the transfer
integral ratio, for the 1/2-filled model band with ty = ty´. (a) fortx < ty, and (b) for tx > ty. For the notation of the orbits, see Fig. 3.12.

48

1.5
(a)

1/2 Filled
kFy β

1
kF / π

kFx β

kFx α

0.5

kFy α
0

0

0.2

0.4

t x / ty

0.6

0.8

1

0.8

1

1.5
(b)

1/2-Filled
kFyα

kF / π

1
kFxβ
0.5

kFxα
kFxγ

0

0

0.2

0.4

0.6
t y / tx

Fig. 3.6. Fractional radii of each cyclotron orbit plotted against the transfer
integral ratio, for the 1/2-filled model band with ty = ty´. (a) fortx < ty, and (b) for tx > ty. For the notation of the orbits, see Fig. 3.12.

491(x/y)(ω py /ω px )

0.8
0.6
(a)

0.4

3/4-filled

0.2
0

0

0.2

0.4

tx / ty

0.6

0.8

1

0.6

0.8

1

(x/y)( ω py /ω px )

1
0.8

(b)

0.6

3/4-filled

0.4
0.2
0

0

0.2

0.4

t y / tx

Fig. 3.7. Anisotropy of plasma frequency plotted against the transfer integral
ratio, for the 3/4-filled model band with ty = ty´. (a) for tx < ty,and (b) for tx > ty.

50

(V1/2 / x t T 1/2 ) ω px / Å1/2 eV1/2

15

10

10

5

3/4-filled
0

0

0.2

0.4

tx /ty

0.6

0.8

1

20

0

10

15
10

5
(b)

5
0

3/4-filled
0

0.2

0.4

0.6

0.8

1

(V1/2/ y t T 1/2) ω py / Å1/2 eV1/2(V1/2 / x t T 1/2 ) ω px / Å1/2eV 1/2(a)5(V1/2 / y t T 1/2) ω py / Å1/2 eV1/2

150

ty /txFig. 3.8. Plasma frequencies normalized for the lattice constants and for totaltransfer integral, tT = tx + ty, is plotted against the transfer integral
ratio, for the 3/4-filled model band with ty = ty´. (a) for tx < ty,and (b) for tx > ty.

51

0.01
0.008
S / SBZ(a)
0.006
3/4-filled
0.004

α

0.002
0
0

0.2

0.4

t x /t y

0.6

0.8

1

0.2
(b)
S / SBZ

0.15
3/4-filled
0.1
γ

α

0.05
0

0

0.2

0.4

t y /t x

0.6

0.8

1

Fig. 3.9. Fractional cross-sectional area of each cyclotron orbit plotted against
the transfer integral ratio, for the 3/4-filled model band with ty = ty´.
(a) for tx < ty, and (b) for tx > ty. For the notation of the orbits, seeFig. 3.12.
52

x y tT (m*/me) / Å 2 eV

20
β

15
10
(a)
5

3/4-filled
α

0
0

0.2

0.4 t /t 0.6
x

0.8

1

y

x y tT (m*/me ) / Å 2 eV

50
40

(b)

30

3/4-filled
β

20
γ

10
0

0

α
0.2

0.4

ty /tx

0.6

0.8

1

Fig. 3.10. Cyclotron mass of each orbit normalized for the lattice constants andfor total transfer integral, , tT = tx + ty, is plotted against the transfer
integral ratio, for the 3/4-filled model band with ty = ty´. (a) fortx < ty, and (b) for tx > ty. For the notation of the orbits, see Fig. 3.12.
53

2
(a)

3/4-Filled

1.5
kF / π

kFy β
1
kFx β

0.5

kFx α

kFyα
0

0

0.2

0.4

tx / ty

0.6

0.8

1

2
(b)

kF / π1.5

3/4-Filled
kFyβ

1
kFxγ

kFyα

kFxβ

0.5
kFxα
0

0

0.2

0.4

ty / tx

0.6

0.8

1

Fig. 3.11. Fractional radii of each cyclotron orbit plotted against the transfer
integral ratio, for the 3/4-filled model band with ty = ty´. (a) fortx < ty, and (b) for tx > ty. For the notation of the orbits, see Fig. 3.12.

54

Fig. 3.12. Schematic display of the Fermi surface topology. Notation of thecyclotron orbits and their radii are also shown. The topology changes
to right, with increasing tx /ty.

55

5
1/2-filled

ω p / eV

4

E // x3
2

E // y

1
0

0

0.5

1
1.5
2(t y -t y ')/(t y +t y')2

Fig. 3.13. Effect of symmetry lowering (ty > ty ´) on the plasma frequencies in thecase of V = 1 Å3, x = y = 1 Å, tx = 0.048 eV and ty + ty ´ = 0.104 eV.

500
1/2-filled

m*/m e

400
300

β

200
100
00

α
0.5

1
1.5
2(t y -t y ')/(t y +t y')2

Fig. 3.14. Effect of symmetry lowering (ty > ty ´) on the cyclotron mass in thecase of V = 1 Å3, x = y = 1 Å, tx = 0.048 eV and ty + ty ´ = 0.104 eV.

56

References

1. H. Tajima: Doctorate Thesis, The Univ. of Tokyo, 1986.
2. H. Tajima, K. Yakushi, H. Kuroda, G. Saito and H. Inokuchi: Solid State Commun.
49 (1984) 769.
3. J. M. Ziman: Principles of the Theory of Solids (Cambridge Univ. Press, Cambridge,1972) 2nd Ed., Chap. 8.
4. R. K. Ahrenkiel: J. Opt. Soc. Am. 61 (1971) 1651.
5. J. M. Ziman: Principles of the Theory of Solids (Cambridge Univ. Press, Cambridge,1972) 2nd Ed., Chap. 9.

57

58

Chapter 4.
Electronic Structure of θ-(BEDT-TTF)2I3

59

4.1. Introduction
A quasi-two-dimensional (2D) molecular conductor θ-(BEDTTTF)2I3, where BEDT-TTF is bis(ethylenedithio)tetrathiafulvalene, exhibits superconducting transition at 3.6 K under ambient pressure [1], whichis the first discovery of an organic superconductor in Japan (1986). The
arrangement of BEDT-TTF molecules in the donor sheets of this salt
(Fig. 4.1.1) [2] contains no distinct dimerization, but has approximately
high symmetry which affords a considerably simple band structure with alarge 2D free-electron-like Fermi surface [2]. For this reason, this salt
occupies a unique position among the molecular conductors; onedimensional stacking or dimerization is the most popular structural motif
in the molecular conductors.
However, the situation becomes somewhat complicated, when the
doubled periodicity stemming from the anion arrangement [3] is taken
account of. First, most crystals are twinned [2-4], which may yield
ambiguity in the experimental study of this salt. Second, the band structure
calculated on the basis of the detailed crystal structure gives complex
topology of the Fermi surface [4]. Therefore, experimental examinationsof the electronic structure of this salt have been desired.
The optical experiment [5] was first applied to the band structure ofthis salt (Sec. 4.3). Unlike other metallic BEDT-TTF salts, θ-(BEDTTTF)2I3 exhibited Drude-like spectra even at room temperature. It was
concluded from this that the optical properties of this salt are basicallyunderstandable on the basis of the simple band structure emerging fromthe highly symmetric molecular arrangement. Unlike the calculated ones60

Fig. 4.1.1.

The arrangement of BEDT-TTF in the conduction layer of θ-(BEDT-TTF)2I3.
The solid and broken lines indicate the orthorhombic and monoclinic unit cells,
respectively.

61[2,4], the shape of the Fermi surface predicted from the optical results is a
simple elliptic one [5]. Following this, the de Haas-van Alphen (dHvA)
oscillations were observed in this salt [6]. A clear saw-tooth-like waveform,
which is characteristic of a nearly ideal 2D system, was directly detected
for the first time [6]. The topology of the Fermi surface deduced fromthe dHvA results [6] have supported the prediction by the optical study[5]. The detailed analysis of the dHvA oscillations [7], which takes
account of the magnetic breakdown effect, have confirmed the geometryof the Fermi surface (Sec. 4.6).
Another significant finding in this salt is the angle dependent magnetoresistance oscillation (ADMRO); this phenomenon was found first in
this salt by Kajita el at. [8], though it was found also independently in
β-(BEDT-TTF)2IBr2 [9,10]. The origin of the ADMRO has been theoretically explained by considering a quasi-2D cylindrical Fermi surface withsmall warping [11,12]. However, the interpretation of the ADMRO in
θ-(BEDT-TTF)2I3 in terms of a realistic band structure has not been reported
yet, although the ADMRO is expected to provide valuable informationon the Fermi surface geometry. For this reason, the ADMRO in this salt
is reexamined and interpreted [13] (Sec. 4.7).
In this way, the study of θ-(BEDT-TTF)2I3 well illustrates how theband structure of a molecular conductor is explored. It also demonstrates
how the optical study is efficient. The main purpose of this chapter is thedescription of these aspects. The optical study is given in Sec. 4.3.
Section 4.4. deals with the magnetic properties in the superconducting
state, as a characterization of the superconductivity. Section 4.5. treats
the thermoelectric power measurement, which provides an aspect from a62different viewpoint. In Secs. 4.6 and 4.7, the dHvA results and theSubnikov-de Haas (SdH) results [14] are described, respectively. In thelatter, the recent finding of an extraordinarily small three-dimensional
Fermi surface [14] is also presented. Finally, the reexamination of the
ADMRO is reported in Sec. 4.8.

4.2. Experimental
The single crystals of θ-(BEDT-TTF)2I3 used for the optical experiments were provided by Prof. Hayao Kobayashi (Toho Univ.), Prof.
Akiko Kobayashi and Prof. Reizo Kato (Univ. of Tokyo). The crystals
were grown by electrochemical oxidation of BEDT-TTF in a tetrahydrofuran solution containing 95:5 mixture of (n-C4H9)4NI3 and (n-C4H9)4NAuI2
as supporting electrolyte around 20 C [15]. The single crystals used in
the other studies (Secs. 4.4-8) were prepared by the author in a similar
way. The content of Au in the crystals was usually so small that this salt
is considered to be essentially the I3 salt. In fact, it was found in thepresent study that the addition of (n-C4H9)4NAuI2 is not necessary in the
crystal growth. Electrocrystallization without (n-C4H9)4NAuI2 using Ptplates or wires as the electrodes afforded the single crystals with satisfactorily high quality even under air. It took about a weak to obtain the
crystals with a constant current of 1 µA applied. However, it was also
found that the result of the crystallization strongly depends on the cellused. In most cases, the product was α-(BEDT-TTF)2I3.
Two nomenclatures of the crystal axes are used: the one is based on
the averaged orthorhombic structure with the space group of Pnma [2],63and the other is based on the monoclinic structure with the space group of
P21/c [3,4], which explicitly takes account of the doubled periodicity andthe twinned lattice. The relation between the two [3] is, am = 2c o, bm = aoand cm = bo – co, where the subscripts, m and o, mean the monoclinic andorthorhombic lattices, respectively. The 2D plane is the aoco-plane in theorthorhombic lattice. Since the present study concerns mainly theorthorhombic structure, the notation based on it is used in Secs. 4.3-4.5,
though Secs. 4.7 and 4.8 are written in terms of the monoclinic notation.
The crystal axes were identified by X-ray diffraction or by reflectanceanisotropy in all the experiments.
Polarized reflectance spectra (Sec. 4.3) were measured on the (010)
crystal face for the light polarizations parallel to the a- and c-axes. These
are the principal directions within the 2D plane. The spectra were measured
by the method described in Sec. 3.1, at 295, 200, 120, 75 and 16 K. The
spectrum for the polarization parallel to the b-axis (perpendicular to theplane) was also measured at 295 K on the (001) crystal face.
Static magnetization measurements were carried out by use of an
SHE SQUID magnetometer model 905. The correction for the sample
holder was made only when the magnetization of the sample is about assmall as that of the holder.
Thermoelectric power (Sec. 4.5) was measured by Prof. TakehikoMori (Tokyo Institute for Technology) in The Institute for Molecular
Science. The measurement was made along the a- and c-axes. The
results are little affected by whether gold foil as the thermal contact isused or not.
The dHvA measurements (Sec. 4.6) were carried out by Prof. J. S.

64

Brooks, Dr. A. G. Swanson (Boston Univ.), Prof. C. C. Agosta (Clark
Univ.), S. T. Hannahs (Massachusetts Institute of Technology) and Dr.
Madoka Tokumoto (Electrotechnical Laboratory) with the assistance by
the staff of the Francis Bitter National Magnet Laboratory at Massachusetts
Institute of Technology [16]. For the measurements, the single crystal
sample was installed in a 3He cryostat designed to operate in high fieldmagnets at the Francis Bitter National Magnet Laboratory. The magnetization of the sample was measured up to about 23 T by Brooks' method
using the small-sample force magnetometer [17]. Transverse magnetoresistance was also measured by the standard four-probe method with gold
contacts using a lock-in ac detection.
The SdH and ADMRO experiments were performed by Dr. Haruyoshi
Aoki, Dr. Shinya Uji and Dr. Taichi Terashima (National Research Insitutefor Metals). Electric contacts were made by gold paste.

The sample

used showed no superconductivity down to 0.05 K. The residual resistivity
ratio, ρ(280 K) / ρ(4.2 K), of the sample amounted to about 1000. The
SdH signals were recorded up to 13.8 T with electric current in the
direction perpendicular to the 2D plane by using the field modulation
technique. The ADMRO was also measured with this current direction.

4.3. Reflectance Spectra

4.3.1. Spectral Features
Figure 4.3.1 shows the polarized reflectance spectra measured atroom temperature. Both the || a and || c spectra exhibit Drude-like dispersions in the infrared region showing the plasma edge around 4000-6000

65

Fig. 4.3.1. Polarized reflectance spectra measured at room temperature.

66

cm –1. In the cases of the BEDT-TTF salts investigated before [18-22],
the shapes of the infrared reflectance spectra were not Drude-like, at leastat room temperature. To our knowledge, θ-(BEDT-TTF)2I3 is the firstexample of the BEDT-TTF salt whose room temperature spectra exhibit
Drude-like infrared dispersion for the two independent light polarizations
within the two-dimensional molecular sheet of BEDT-TTF.
As already mentioned, the reflectance spectra of metallic BEDT-TTF
salts usually show a non-Drude-like infrared dispersion. To elucidate theorigin of these dispersions, a systematic investigation of reflectance spectra
on a series of BEDT-TTF salts has been carried out [18-22]. In thesestudies, the probabilities of the optical interband transitions from thelower filled bands to the upper vacant (or partially filled) bands havebeen calculated in the framework of the one-electron band model. In thecases of the BEDT-TTF salts, these bands are formed from the HOMOsof BEDT-TTFs. It has been shown that the observed infrared dispersions
are primarily associated with the interband transitions and have relatively
small contribution of the intraband transitions [18-22].
Similar calculations based on the orthorhombic BEDT-TTF latticeof θ-(BEDT-TTF)2I3 yield no contribution of the interband transitions.
This is consistent with the rule given in Appendix A.: The BEDT-TTF
layer has glide symmetry along the a-axis and two-fold screw symmetry
along the c-axis, and contains two BEDT-TTFs in the two-dimensional
unit cell. This well explains the absence of dominant contribution of theinterband transitions. The infrared dispersions of θ-(BEDT-TTF)2I3 arethus attributed basically to the intraband transitions.
The broad peak near 20 × 10 3 cm –1 in the || c spectrum can be67

assigned to the intramolecular excitation of I3–, which is expected to becompletely polarized in the direction of the long axis of this linear anion.
The || b reflectance spectrum measured on the (001) crystal face is shownin the inset of Fig. 4.3.1. The prominent dispersion near 10 × 10 3 cm –1 isattributable to the intramolecular excitation of the cation radical of BEDTTTF, since the transition moment of this dispersion is almost parallel tothe long axis of the BEDT-TTFs. The excitation energy was evaluated tobe 1.22 eV by the curve-fitting analysis assuming a Lorentz oscillator.
The spectroscopic study on BPDT-TTF (bis(propylenedithio)tetrathiafulvalene) radical salts also supports this assignment [23]. Incidentally, thefact that there is no indication of in the || b spectrum for the presence of areflectivity minimum around 5000 cm –1 associated with the plasma edge,
indicates that the delocalization of charge carriers along the b-axis isstrongly interrupted by the layers of the counter anions sandwiched by
the BEDT-TTF molecular sheets.
Figure 4.3.2 shows the reflectance spectra of θ-(BEDT-TTF)2I3 measured at several different temperatures. On lowering temperature, theplasma edge becomes sharper accompanying the increase in reflectivity
in the low wavenumber region.
Although the spectra shown are Drude-like, it is not possible toreproduce exactly the observed reflectance curves only with the Drude
model. Figure 4.3.3 shows the conductivity spectra in the infrared region
obtained from the Kramers-Kronig analysis of the reflectance data. Small
broad shoulders around 2500 cm –1, which is indicated by the arrows in
Fig. 4.3.3, appear on the Drude tails. These shoulders seem to be theones associated with the interband transitions which used to appear in the68

Fig. 4.3.2.

Temperature dependence of the reflectrance spectra.

panels show the E //a and E // c spectra, respectively.
Drude-Lorentz model.

69

The upper and lower

Solid curves show the fit to theFig. 4.3.3.

Conductivity spectra at various temperatures.

absorption due to the interband transitions.

70

The arrows indicate the weak

spectra of the BEDT-TTF salts in the infrared region. If the arrangementof the BEDT-TTFs is exactly describable by the orthorhombic lattice, theinterband transition should be absent, as mentioned above. The appearanceof the weak interband transitions indicates the slight deviation of thearrangement from the orthorhombic structure.
According to the X-ray analysis of the crystal structure [3], the real
structure is described by the monoclinic lattice. The glide mirror and thescrew symmetries are no longer hold within the molecular layer of thismonoclinic structure. The unit cell is doubled along the c-axis. Therefore,
the symmetry discussion based on the orthorhombic lattice is an approximate one; weak interband transitions can be observed. Nevertheless, thefact that the interband contributions are very small supports that the consideration based on the orthorhombic lattice is a good approximation.
The small hump appearing around 1100 cm –1 in the || a and || cspectra (Figs. 4.3.1 and 4.3.2) is attributable to the totally-symmetricvibrational mode of BEDT-TTF coupled with the conduction electrons,
because its position and linewidth agree with the corresponding peakobserved in the spectra of other BEDT-TTF salts [24]. This electronmolecular-vibration (e-mv) coupling peak arises by borrowing the intensityof the interband transitions. The appearance of the weak e-mv peak inthe spectra of θ-(BEDT-TTF)2I3 also supports the presence of the weakinterband transitions in this salt.

4.3.2. Curve-fitting Analysis and Optical Effective Mass
In order to separate the contribution of the intraband transitions andthose of the interband transitions and the intramolecular transitions, the71

curve-fitting method based on the following Drude-Lorentz dielectric function,ε (ω ) = ε C – ω P2 / ( ω 2+iω / τ ) + ∑ f j / [(ω j2 –ω 2) – iγ jω ].
j(4.3.1)was used. The first term stands for the dielectric constant stemming fromthe higher energy excitations outside the experimental spectral region, thesecond term is the Drude term representing the intraband transitions, andthe third term describes the interband transitions (j = 1) and the local
excitation of I3– anion (j = 2). The results of the curve-fitting were shownby the solid curves in Fig. 4.3.2, and the obtained parameters are summarizedin Table 4.3.I, together with the dc conductivity values deduced from thismodel. The local excitation of I3– near 20000 cm –1 is explicitly included
for the analysis of the || c spectrum at room temperature, but it is not for
the low temperature spectra, which is limited below 4200 cm –1. However,the local excitation is implicitly included in the first term, ε C , because thecontribution of the third term for this excitation is almost constant below
4200 cm –1. The reflectivity is related to dielectric function by the equation,
R = [1 + | ε | –√2(| ε | + Re(ε)) ] / [1+| ε | + √2(| ε | + Re(ε)) ] .
(4.3.2)In terms of a two-dimensional parabolic band model (a nearly-free-electron
model), the anisotropic effective mass of the carriers can be derived fromthe plasma frequency for each polarization. The band model is expressed

72

as,

ε = – h 2 [ka2 /2m*a + kc 2 /2m*c ] ,(4.3.3)where m*a and m*c are the a- and c-components of the effective mass
tensor. The plasma frequency is given by,(ω P2)x = 4 π n e 2/m*x , x = a and c(4.3.4)

From the plasma frequency data at 16 K, the effective masses are calculated
to be m*a = 1.5 me and m*c = 3.0 me .

Table. 4.3.I. The Drude parameters obtained by the Drude-Lorentz fit of
the reflectance data.

Polarization T / K
// a

// c

295
200
120
16
295
200
120
75
16

εc

ωp / eV τ – 1 / eV σ dcopt / S cm – 1 a)

3.55 0.92
3.49 0.97
3.62 1.04
3.42 1.05
2.52 0.62
5.59 b) 0.71
5.68 b) 0.72
5.76 b) 0.72
5.98 b) 0.73

0.16
0.087
0.068
0.042
0.12
0.074
0.049
0.039
0.022

730
1440
2130
3550
440
900
1400
1910
3290

a) The values obtained by extrapolating the Drude conductivity tozero wavenumber.
b) These values are larger than the room temperature one, because
they involve the contribution of the local excitation of anions.
See Text.

73

4.3.3. Tight-binding Band Structure Derived from Optical Data
According to the tight-binding model taking account of HOMOs of
BEDT-TTFs, the energy band of θ-(BEDT-TTF)2I3 is expressed as,E(ka, k c) = 2tc cos(ckc ) ± 4 ta cos(aka / 2) cos (ck c / 2) ,(4.3.5)where the transfer integrals, ta and tc , are defined as indicated in Fig.
4.1.1. Because of the high symmetry of the molecular arrangement in theassumed orthorhombic BEDT-TTF sublattice, the band structure is described only by three parameters, EF , ta and tc , where EF is the Fermi
energy. These parameters can be related to the three experimental data,(ω P)a and (ω P)c , and the number of holes,

i.e. , the plasma frequencies,

Nh, which is estimated from the crystal structure and the stoichiometry.
The Fermi energy is determined implicitly by,

k Fc

2

Nh = (2 / 4π ) (4 /a)∫ cos {[E – 2t cos(ck )] / 4 t cos (ck / 2)} dk ,
–1

F

c

c

a

c

c

0

(4.3.6)

where Nh is the number of holes per unit area of the (010) plane (theconduction plane), and kFc denotes the maximum value of the c-componentof the Fermi wave vector. The plasma frequencies are related to the
Fermi velocities by,
k Fc∫(π 2h / e 2)(ωP2)a = [2π /(b / 2)] | vFa(kc) | dkc ,
–k Fc
74

(4.3.7)

k Fa∫(π 2h / e 2)(ωP2)c = [2π /(b / 2)] | vFc(ka) | dka ,–k Fa(4.3.8)where kFa is the maximum value of the a-component of the Fermi wave
vector, v Fa and v Fc denote the a- and c-components of the Fermi velocity,respectively, and b is the lattice constant perpendicular to the (010) plane.
Note that the unit cell of θ-(BEDT-TTF)2I3 includes two equivalent conduction layers. The Fermi velocity is defined as,vF = h –1 [gradk E(ka , kc)]; E = E F.

(4.3.9)The parameters, kFa and kFc are related to the Fermi energy by,
EF = 2tc + 4 ta cos(ak Fa / 2),
EF = 2tc cos(ckFc) + 4 ta cos (ck Fc / 2) ,(4.3.10)(4.3.11)

By numerically solving the simultaneous equations, (4.3.6)-(4.3.11), a setof solution, EF , ta , tc, kFa and kFc were obtained. The results are shown in
Table 4.3.II. For example, at 16 K, the parameters are E F = 0.12 eV, ta =0.080 eV, tc = 0.046 eV, k Fa = 0.30 Å –1 and k Fc = 0.42 Å –1 . Figure 4.3.4
shows the energy band structure and the Fermi surface calculated by useof these parameters. This is the first example of BEDT-TTF salt where
the transfer integrals of the tight-binding band model are directly inferredfrom the optical data.

75

Fig. 4.3.4.

The tight-binding band structure and the Fermi surface estimated from thereflectance spectra at 16 K.
76

Table 4.3.II. The tight-binding band parameters estimated from the optical data.

T/Kta / eV tc / eV EF / eV εF / eV a) kFa / Å–1 kFc / Å–1

200
120
16

0.069
0.078
0.080

0.045
0.043
0.046

0.10
0.11
0.12

0.26
0.28
0.29

0.30
0.29
0.30

0.41
0.42
0.42

a) The Fermi energy measured from the top of the conduction band, i.e.,εF = 2 tc + 4 ta – EF.

Fig. 4.3.5.

The shape of the Fermi surface calculated from the crystal structure by means of

the tight-binding approximation.

Left: for the orthorhombic lattic [2].

monoclinic lattice [4].

77

Right: for theThe band structure and the shape of the Fermi surface thus obtained
are somewhat different from those calculated from the crystal structure[2,4] (Fig. 4.3.5). In the calculations of the band structure from thecrystal structure, the transfer integrals (t 's) are assumed to be proportional
to the overlap integrals (S's) between the HOMOs of BEDT-TTFs. The
present optical study has yielded ta / tc = 1.7,

whereas the calculated

ratio of the overlap integrals is Sa / Sc = 0.66 [2]. This difference, whichis the origin of the difference between the experimental and calculatedband structures, is not negligible. In order to establish the band structureof θ-(BEDT-TTF)2I3, it is desired to study it by further experiments.
It should be also remarked here again that the present results are
based on the orthorhombic average structure. The doubled periodicity
along the c-axis in the monoclinic structure yields the Brillouin zone
folded at the mid point of the ΓZ line. This folding results in the closed
pocket area around the new zone boundary. The area of this pocket
amounts to 14 % of that of the first Brillouin zone. Furthermore, thelowered symmetry gives rise to removal of degeneracy on the new zone
boundary, which is not included in the present band model. The mid-infrared
optical properties are insensitive to the expected small gap structures of
the Fermi surface. Therefore, an experiment sensitive to the detailed
structure of the Fermi surface can afford results slightly different fromthe present ones. However, the origin of such difference should be understood in this way. The essential point of the present conclusion is that the
Fermi surface is an elliptic one elongated along the c-direction.

78

4.3.4. Temperature Dependence of the Spectra
The temperature dependence of the spectra is caused mainly by thetemperature dependence of the relaxation time, τ, as noticed from Table
4.3.I. From the Fermi velocity values and τ, the mean free path of thecarriers is estimated as,

Λ = vF τ,(4.3.12)Within the (010) plane, the vF values of about 2 × 10 7 cm s –1, are derived
from the band structure obtained as above. The mean free path within the(010) plane amounts to about 40 Å at 16 K, larger than the lattice constants.
On the other hand, the corresponding values at room temperature areshorter than 10 Å, comparable or smaller than the lattice constants. Thisimplies the breakdown of the simple picture of metal. Another contradictory
point concerns the dc conductivities deduced from the optical data (Table
4.3.I). The agreement between them and the values obtained directly by
the dc measurement is satisfactory at low temperatures. However, theagreement becomes remarkably worse as temperature increases; at room
temperature the former are about 400 - 700 S cm –1 , whereas the latter isless than 100 S cm –1. This fact suggests the possibility of anomalous
non-Drude dispersion at lower wavenumbers outside the experimentally
accessible region at high temperatures. The extraordinarily short mean
free paths at high temperatures indicates that the carriers are strongly
scattered by some thermally excited modes in the crystal. The physicalpicture has not been established for such a situation yet. Nevertheless,this is expected to be the origin of the anomaly. At low temperatures, the79concluded electronic structure seems to be free from such complication.
However, it is desired to test it by different experiments.

4.4. Magnetic Properties in the Superconducting State

Temperature dependence of the shielding and Meissner effect andanisotropy of the magnetization curve are investigated to characterize thesuperconductivity in θ-(BEDT-TTF)2I3. In the following, the sample
dependence is reported first. Next, the anisotropy of the magnetization
curve is discussed.

4.4.1. Sample Dependent Superconductivity
The magnetic behavior in the superconducting state is found to varyfrom sample to sample. This should be related to the sample dependent
behavior of the resistivity [25,26]. The sample-dependence of magnetization was examined after cooling the sample down to 1.8 K in zero field.
After the cooling, a magnetic field up to 10 G was applied along theb-axis in order to probe the diamagnetism. The samples could be divided
into two groups: Group I exhibited a diamagnetic shielding of 0.3 emu/g
at 10 G and Group II exhibited a very small shielding, 1/1000 of thevalue of the Group I. Among the 12 crystals examined, two were foundto be of the Group I.
Figures 4.4.1 and 4.4.2 show the temperature dependence of static
magnetization of the Group I and II samples, respectively. In this measurement, the sample was first cooled down to 1.8 K in zero field, then80Fig. 4.4.1.

Temperature dependence of the shielding and Meissner effect of the Group I

sample (#13-5 in Table 4.4.I) in the magnetic field of 10 G applied along the b-axis.
Correction for the background is not made.

81

Fig. 4.4.2.

Typical temperature dependence of the shielding and Meissner effect of theGroup II sample in the magnetic field of 10 G applied along b-axis.

Since the zero point ofmagnetization (ordinate) is ambiguous in this case, it was taken as the magnetization value at
5K.

Compare the scale of magnetization with that in Fig. 4.4.1.

82

the magnetic field of 10 G was applied, and the diamagnetic shielding
signal was measured on the warming process up to 5 K. Again thesample was cooled down to 1.8 K in the same field in order to measure
the Meissner signal. As shown in Figs. 2 and 3, the diamagnetism of theGroup I samples was much larger than that of the Group II samples in thelower temperature. The diamagnetic shielding decreased with increasing
temperature up to about 3.5 K. Above this temperature, the Group I
sample shows a small and constant signal due to the Pauli paramagnetism,
while the magnetization signals of the Group II sample were too small to
distinguish from background noises above 3 K. This behavior is not in
conflict with the reported temperature dependence of the resistivity atzero field, because the applied field (10 G) slightly exceeds the lower
critical field even at 1.8 K as described below.
On the cooling process in the magnetic field, the Group I sample
exhibited a diamagnetism due to the Meissner effect which gradually
increased on cooling down to 1.8 K. The Meissner signal, which was
defined as the ratio of the Meissner expulsion to the shielding signal, wasabout 1 above 3 K, while it reduced with decreasing temperature. It was
found to be about 0.1 at 1.8 K for the Group I sample. The Group II
sample showed a similar Meissner behavior. However, the Meissner
signal was estimate to be 5 times as large as that of the Group I sample at
1.8 K. Since the shielding signal of the Group II sample is remarkably
small for its size, the large Meissner signal never involves large superconducting volume.
Judging only from the Meissner signal value of about 0.1, the superconductivity in θ-(BEDT-TTF)2I3 seems to be a bulk one. In this case,83the shielding effect is suppressed by insufficient development of supercurrent path loop on the crystal surface as observed in the Group II samples.
Since the penetrating flux tends to be trapped mainly in the defect domain
in the Group II crystal, the Meissner to shielding ratio could be held at
rather larger value (0.5) than that of the Group I sample (0.1). Anyway in
the Group II crystals, the fractional volume of the superconducting part issignificantly suppressed. The reduction of the superconducting volume
can result in the appearance of the residual resistivity [25,26].

4.4.2. Anisotropy of the Lower Critical Field (H c1)
The anisotropy of magnetization curve of the Group I sample up to
40 or 60 G was investigated at 1.8 K. Figures 4.4.3 and 4.4.4 show theinitial magnetization and the magnetization curves obtained by applying
magnetic field along the crystal axis, b, a and c. The b-axis is normal tothe conduction layer, and the a- and c-axes are the in-plane directions.
Strong irreversibility in the magnetization curves found for all the field
directions is attributable to the flux trapping, which is a characteristicfeature of the non-ideal type II superconductor.
The initial magnetization for H || b was from ten to twenty times aslarge as those for the fields along the in-plane directions, H || a and H ||c. The magnetization curve for H || b has a definite diamagnetic maximum,
whereas the magnetization is approximately constant or gradually decreasing in the high field region for H || a and H || c. This difference indicatesthat the upper critical field, Hc2, has smaller value for H || b than that for
H || a or H || c. This observation agrees with the result of resistivity
measurements [26], in which the values of Hc2 for H || b was found to be84

Fig. 4.4.3. Initial magnetization of the Group I sample (#13-5 in Table 4.4.I) at 1.8 K for thethree field directions, b, a, and c. The solid lines are the least square fit of the first
several data points to linearity, indicating the initial slope.
85

Fig. 4.4.4 Magnetization curves of the Group I sample (#13-5 in Table 4.4.I) at 1.8 K for thethree field directions, b, a and c.

86

larger than that for an in-plane field by a factor of about 30. Furthermore,
a strong flux trapping may also prevent the field from penetrating into thesample in the higher field along the ac-plane.
We define Hc1obs as the field where the first deviation from linearity
in the magnetization curve of the shielding signal occurs. In fact, we
required that Hc1 obs should satisfy the following two conditions: i) at Hc1obs
the diamagnetic signal reduces to the value less than 98% of that extrapolated
linearly from the least square fit of the first 4 or 5 data points, and ii) at
the fields higher than Hc1 obs the signals show no recovery to the extrapolated
values. The ambiguity in estimating Hc1 obs was less than ± 1 G for H || band ± 0.2 G for H || a and H || c. The Hc1 obs values thus estimated for thethree field directions are compiled in Table I, together with the sizes ofthe samples used. At a glance, they do not seem to be very anisotropic;
the deviation from linearity starts at the field of about 2 G for all the
directions. However, they could be underestimated values of the lower
critical field, Hc1 , because of the demagnetization effect. The samples
used in our experiments were rather thin plates, so that the effective field
felt around the sample is much enhanced owing to the demagnetization,
especially for the field normal to the plane. Since the shapes of oursamples were far from ellipsoids, the magnetic field felt near the sample
surface should be inhomogeneous. Thus it is difficult to carry out an
accurate correction for the demagnetization effect. However, we havecarried out a rough estimation by approximating the sample shape as anellipsoid touching internally to crystal faces in order to obtain the demagnetization factors. The enhancement of the effective field is written as87

Heff = Happ / (1–D) = f Happ ,(4.4.1)

where D (0 ≤ D ≤ 1) is the demagnetization factor and f is the enhancement
factor. The ellipsoid approximation gives an enhancement factor of about
5 for the field normal to the plane, while the factors are near 1 for thein-plane fields. This estimation is considered to reflect the lower limits ofthe enhancement factors. The values of Hc1's, estimated roughly in thisway, are also listed in Table 4.4.I, together with the enhancement factors
calculated from the demagnetization coefficients for the ellipsoids [11].
Another way to evaluate the anisotropy of Hc1 is to use the initial
magnetization slope. Without the demagnetization effect, the slopes of
the curves for the three field directions should be equal to each other inthe linear region at low field. Even if there is the demagnetization effect,
we could still obtain the same slopes using Heff instead of Happ. Wechanged the scale of each field so that the three curves have the same
slope in the linear region. The result is shown in Fig. 4.4.5, where thedemagnetization factor for H || c is assumed to be equal to that calculatedby the ellipsoid approximation. This method gives the anisotropic Hc1values, Hc1, b = 40 G, Hc1, a = 1.4 G, and H c1, c = 2.0 G. Since the field
applied perpendicular to the plane exhibits strong inhomogeneity, thedifference between Hc1, b values estimated by the two methods is not
surprising. The Hc1, b value given by the ellipsoid approximation is clearly
underestimated.
Thus, anisotropic Hc1 values were estimated for this material, whichis compatible with the anisotropy of Hc2 reported previously. Similar
anisotropic behavior was found in other organic superconductors,

88

Table 4.4.I. The lower critical filed, the demagnetization factor and the size of thesample for each field direction.

Sample Direction Size /mm Hc1obs / G

Hc1 / G a)

Df

Hc1 / G

#13-5

17

0.77
0.15
0.08
0.82
0.09
0.08

4.35
1.18
1.09
5.56
1.10
1.09

40

#2-1

b
a
c
b
a
c

0.16
0.74
1.10
0.13
0.94
1.05

4.0
1.2
1.8
2.0
1.0
1.5

1.4
2.0
11.1
1.1
1.6

b)

1.4
2.0

a) The lower critical fields obtained by the ellipsoid approximation.
b) The lower critical fields evaluated by adjusting the initial magnetization slope.

Fig. 4.4.5 Initial magnetization of the Group I sample (#13-5 in Table 4.4.I) at 1.8 K,
plotted versus H eff.

For the H //c data (filled squares),

H eff is calculated by the ellipsoid

approximation, while those for the H //a (open circles) and H //b (filled circles) data areadjusted so that the initial slopes match with that of the H //c curve.

Note that the initial

magnetization is about 10 % of that of the perfect diamagnetism (broken lines).
3

The sample

volume was calculated from the density, 2.25 g / cm , and the crystal weight, 0.33 mg.

89

(TMTSF)2ClO4 [28,29], β-(BEDT-TTF) 2I3 [30], β-(BEDT-TTF) 2IBr2[31], β-(BEDT-TTF) 2AuI2 [31], and κ-(BEDT-TTF) 2[Cu(NCS)2] [32].
In the framework of the Ginzburg-Landau (GL) theory, Hc1 and Hc2
are related to GL parameter, κ, and the thermodynamical critical field,
Hc, by the equations,

κ = (1/2) 1 / 2 (ln κ + 0.497) Hc / Hc1,

(4.4.2)

κ = (1/2) 1 / 2 Hc2 /Hc .

(4.4.3)The parameter, κ, is defined as

κ=λ /ξ,(4.4.4)

where λ is the penetration depth and ξ is the coherence length. The
observed anisotropy of Hc1, as well as of Hc2 , is probably ascribable to theanisotropy of ξ, which is largest in the ac-plane [26]. The estimated H c1
values are about 40 G for H || b ( by the initial slope scaling method ) andfrom 1.1 to 2 G for the in-plane fields. The ratio, 20 to 36, is consistent
with that of 1/H c2, about 30.
Although the in-plane anisotropy of Hc2 has not been known, theanisotropic optical effective mass was estimated as [6],m*a = 1.5 me and m*c = 3.0 me ,(4.4.5)where me denotes the electron mass. It was elucidated that the nearly-

90free-electron model with parabolic dispersion can well reproduce the tightbinding band structure of this material [6]. Using the former model, we
obtain the ratio,

κc / κa = vFa / vFc = (m*a / m*c ) 1 / 2 = 0.7 ,(4.4.6)

where vFa and vFc are the a- and c-components of the Fermi velocity,respectively. On the other hand, to the limit of large κ, we can calculate
the ratio, κc / κa, from Eq.(4.4.2) as,

κc / κa = (Hc1 )a / (Hc1 )c .

(4.4.7)

By use of the Hc1 values obtained in this work, the ratio, κc / κa , isestimated to be 0.7. This is consistent with the result of Eq.(4.4.6).

4.5. Thermoelectric PowerIn Sec. 4.3, the structures of the conduction band and the Fermi
surface of θ-(BEDT-TTF)2I3 have been derived from the optical properties
by assuming the orthorhombic averaged structure. However, strictly speaking, the assumed structure, and thus the derived band structure, does notexactly hold for a real crystal of θ-(BEDT-TTF)2I3, because the true
structure is the monoclinic one with lowered symmetry of the moleculararrangement. Since the effect of this difference is small, as noted by the
weakness of the interband transitions, the global shape of the Fermi surface

91

derived from the optical data is reliable. In other words, the opticalexperiments probe only the global features and are insensitive to local
small structures of the Fermi surface. For more detailed examination of
the electronic structure of this salt, it is necessary to do other experiment
which is sensitive to small structures.
Thermoelectric power (TEP) measurement is one of such tools [33,34].
As well as dc conductivity measurement, it reflects the lowest energy
excitation (< kBT) around the Fermi level. Moreover, its sign and amplitude,
which may be anisotropic and temperature dependent, reflect the curvatureof the energy band around the Fermi surface. In some cases, TEP information has been shown to be useful to investigate the Fermi surface of
BEDT-TTF salts. Thus, it is expected that TEP measurement would
reveal a new feature of the structure of the Fermi surface of θ-(BEDTTTF)2I3, which was not detected in the optical study.
Figure 4.5.1 shows the TEP of θ-(BEDT-TTF)2I3 measured along
the two directions in the conduction plane, a and c. For both directions,
TEP is considerably small as compared with other BEDT-TTF salts.
Along the a-axis, TEP is negative and very small; it is almost zero atroom temperature. The TEP makes a broad shoulder and minimum around
50 K and 200 K, respectively. Along the c-axis, TEP is positive, with aprominent peak at about 20 K, and gradually decreases with increasing
temperature up to about 220 K. These features are qualitatively similar to
those of a series of BEDT-TTF salts, (BEDT-TTF)2MHg(SCN)4 (M=K,
Rb and NH4), except for the considerably suppressed values of TEP and
for the minimum at 220 K along the c-axis. The peak along the c-axis
should be ascribed to the phonon contribution, such as the phonon drug

92

Fig. 4.5.1.

Thermoelectric power of θ-(BEDT-TTF)2I3.

The broken curves stand for the

calculations with the following parameter sets:(1): t a = 0.090 eV, t a′ = 0.120 eV and t c = 0.055 eV.
(2): t a = 0.045 eV, t a′ = 0.060 eV and t c = 0.030 eV.
For the definition of the transfer integrals, see Fig. 4.5.2.

93effect, rather than to the electronic band origin. The TEP values are sosmall that it is difficult to distinguish exactly the contributions of phononsand electrons at low temperatures. In the following, the TEP values
above about 100 K are mainly considered.
It possible to simulate the temperature dependence of TEP on thebasis of the tight-binding band structure [33]. According to the Boltzmann
equation [35], TEP (S) of a metal is expressed as

3

Si j = (1 / eT) ∑ (K0 – 1) i j K 1, k j (i, j, k = x, y, z) ,(4.5.1)k=1

K0, i j = (τ /4π 3h)∫∫ (v ) (v ) (– ∂f / ∂ε) (dS /v ) dε ,K1, i j = ( τ /4π 3h)

F i

F j

0

(4.5.2)F∫∫ (v ) (v ) (ε – ε )(– ∂f / ∂ε) (dS /v ) dε ,
F i

F j

F

0

F

(4.5.3)where f0 is the Fermi distribution function, εF is the Fermi energy, and vF
and (vF)i are the Fermi velocity and its component, respectively. The
quantities, S, K 0 and K1 are tensors. In the simulations presented here, theintegrals over the Fermi surface were numerically calculated by dividing
the first Brillouin zone into 160 × 160 meshes.
A simulation based on the band structure derived from the optical
study yields positive TEP both for a- and c-axis, which is in contradiction
to the observation. This band structure is derived by assuming the orthorhombic lattice. In this assumed structure, the energy band is described
only by two transfer integrals, ta and tc . The Fermi surface is determined94

so as to satisfy the requirement of 3/4-filling of this band. The charge
carriers are considered to be holes for this 3/4-filled band. This is theorigin of the positive sign of the simulated TEP.
It is thus evident that the band structure is more complicated one
than assumed previously. The effect of lowered symmetry should betaken account of. For example, a simple model given in Fig. 4.5.2 canapproximate this feature. In Fig. 4.5.1, the curves calculated from twoparameter sets for this model are depicted. Qualitative agreement withthe observation, i.e., the sign for each direction and the small values, isobtained. This model can describe the removal of degeneracy on the XMzone boundary. This affords the positive curvature of the upper bandnear the XM boundary, so that this portion of the Fermi surface retains
the electron character (Fig. 4.5.3). The origin of the negative TEP along
the c-axis can be understood in this way.
The curvature of the band near the zone boundary is a subtle problem.
This is because of the cos(ak a / 2) dependence of the band dispersion forthe basic orthorhombic packing, where the curvature is exactly zero at theboundary. This situation is similar to those in κ-(BEDT-TTF) 2[Cu(NCS)2][33] and (BEDT-TTF) 2MHg(SCN)4 [34]. The small values of TEP canalso be understood in this framework, while the considerably wide band
width of this salt also responsible for the suppression of TEP. It is also
noticed that the TEP values are prescribed not only by the amplitudes oftransfer integrals but also by the differences between them. Therefore,
further improvement of the fit between the calculation and observation isdifficult. For instance, the different two parameter sets yields similar
temperature dependence of TEP.

95

tc
c

ta

t a′

ta

t a′
a

tc

0

Fig. 4.5.2. The model molecular arrangementt assumed in the simulation

Energyof TEP

EFΓ

X

Fig. 4.5.3. Schematic display of the effect of the symmetry lowering t(a >t a′ )on the band structure near the zone boundary. Thick and brokenlines are with and without the symmetry lowering, respectively.

96

The present TEP results have revealed that the effect of lowered
symmetry, i.e., the removal of degeneracy, plays a crucial role in thetransport phenomena in θ-(BEDT-TTF)2I3. This affords a viewpoint different from that of the optical study.

4.6. The de Haas-van Alphen (dHvA) Oscillations
The first observation of the "saw-tooth" dHvA oscillation in θ-(BEDTTTF)2I3 has shown that a satisfactory clear 2D electronic system is realized
in this salt [6]. The topology of the Fermi surface (FS) has been deduced
from the interpretation of the dHvA oscillations in terms of the magnetic
breakdown (MB) effect, the effect of tunnelling across the gap at Fermi
surface under high magnetic field [6]. This point is examined in moredetail in this section. Moreover, the dHvA results, including the cyclotron
mass, are compared with the band structure inferred from the optical
study (Sec. 4.3)

4.6.1. Analysis of the dHvA Oscillations and Magnetic Breakdown EffectFigure 4.6.1 shows the magnetization as a function of magnetic fieldat the selected temperatures [16]. The data was recorded for the crystal
other than the one previously reported [6,36-38]. A significant difference
between these two samples is the appearance of the nodes at about 12, 15
and 20 T in the data in Fig. 4.6.1. The Fourier transform of the data
displayed in Fig. 4.6.2. shows that these nodes result from the beat of the97

Fig. 4.6.1.

dHvA oscillations of θ-(BEDT-TTF)2I3 as a function of field atselected temperatures.

The curves for T=0.89 K and T=1.67 K

are shifted up for clarity. Twinning effect is visible as the nodes
at about 12, 15 and 20 T.

Fig. 4.6.2.

Fourier transform of the 0.89 K data in Fig. 4.6.1.
around the higher frequencies.
98

The inset is the close-up

two close frequency components at fβ′=4440 T and f β″=4494 T. The
coexistence of such two fast oscillations, corresponding to approximately
100 % of the first Brillouin zone (BZ), in an exactly single crystal contradicts
to the basic chemical architecture of the salt. As mentioned above,θ-(BEDT-TTF)2I3 often crystallizes in a twinning form. Therefore, it isvery likely to think of the beat as a result of the twinning in the crystal
examined. If this is the case, the nodes due to this beat should beobserved at the field, Hn, satisfying,cos[π(f β″– f β′) / Hn] = 0,(n = 0, 1, 2,....)(4.6.1)(f β″– f β′) / Hn = n + 1/2.

(n = 0, 1, 2,....)(4.6.2)or,Corresponding to n = 3 and 4, the fields Hn= 15.4 and 12.0 T are respectively
obtained, which is consistent with the observation. The oscillation without
such beat was able to be observed for the other crystal [6,36-38] (Fig.
4.6.3). On the other hand, beat behavior may come from small warpingof a single cylindrical FS. The beat having such origin should give nodes
at the field, H ′ n, satisfying,cos[π(f β″– f β′) / H ′ n – π/4] = 0, (n = 0, 1, 2,....)
or,99(4.6.1′)

(f β″– f β′) / H ′ n = n + 3 / 4,(n = 0, 1, 2,....)(4.6.2′)where f β′ and f β″ correspond to the minimum and maximum of the crosssectional area of the warped FS, respectively. For n=3 and 4, we obtain
H ′n = 14.4 and 11.4 T, respectively. These values are appreciably lower
than those observed for θ-(BEDT-TTF) 2I3. Thus we conclude that theobserved behavior is composed of the contributions from two domains indifferent orientations; each domain exhibits the fast and slow oscillations,f α and f β at about 800 T and 4000 T, respectively.
We derive the cyclotron mass values from the data showing twinningeffect (Figs. 4.6.1 and 4.6.2) below, because only they are available forthe analysis of the temperature dependence. The inset of Fig. 3 displays
the Fourier transform of the magnetization data without a remarkablebeat, showing the fast oscillation at fβ = 4290 T. Assuming that the field
direction was parallel to the principal axis in this measurement, we canestimate the inclination of each twinning domain with respect to the fieldfrom the deviation of fβ′ and fβ″ from fβ . The angles between the principal
axes of the domains and the field direction are 15 and 17 , corresponding
to fβ ′ and fβ ″, respectively. This misorientation may yield at most 6 % oferror in our mass analysis for a cylindrical FS.
In Fig. 4.6.4, the mass-plot, the plot of A / T versus T, is shown,
where A is the amplitude of the Fourier component at each frequency.
From the slope of each plot, the cyclotron masses are evaluated: mβ = 3.6
me for the fast oscillations, and mα = 2.0 me for the slow oscillation, areobtained.

100

Fig. 4.6.3.

dHvA oscillations and its Fourie tranform (inset) of θ-(BEDTTTF)2I3 showing no distinct beat.

Fig. 4.6.4.

The mass-plot, i.e., the plot of A / T versus T, where A is the amplitude of theFourier component at each frequency.

101

Fig. 4.6.5. Top: Field dependence of the oscillation amplitude. Bottom: The Dingle plot of
the data in Fig. 4.6.3.

The solid lines stand for the calculations considering themagnetic breakdown effect.

102

Field dependence of the oscillation amplitude is also analyzed interms of the Lifshitz-Kosevich (LK) formula [39]. In the presence of themagnetic breakdown (MB) effect, the tunnelling probability is approximately expressed as,

P = exp(– HB /H),

(4.6.3)where HB denotes the breakdown field. We consider here the tunnelling
junctions between the closed pocket (α) and the pair of open portion.
Such a situation modifies the original LK formula for a single orbit. The
results can be expressed as [40],Sβ= Mβ(H) H 1/2 sinh[K(mβ / me )T / H]
∝ P 4 exp[– K(m β / me )TDβ / H],(4.6.4)and,Sα = Mα(H) H 1/2 sinh[K(mα /me )T / H]
∝ (1 – P) 2 exp[– K(mα/me)TDα / H],(4.6.5)where K = 2π 2meck B /e h = 14.7 T / K, M(H) is the amplitude of theoscillation, and TD is the Dingle temperature; the subscripts, α and β,
denote the slow and fast oscillations, respectively. By using mα and mβ
evaluated as above, TDα, TDβ and HB can be estimated from the slope of103

log(S) versus 1 / H plots. Figure 4.6.5 shows such a plot of the data
given in Fig. 4.6.3. Too small amplitude of the oscillations below 15 T
prevents us from evaluating the parameters precisely. Nevertheless, theparameter set TDα = 2 K, TDβ = 0.8 K, and HB = 15 T seems to reproduce
the data well, as depicted by the solid curves in Fig. 4.6.5. The overall
field dependence is thus explained by the MB effect.
Another evidence for the MB effect is the overall field dependenceof the resistivity [16]. Figure 4.6.6 shows the transverse magnetoresistanceof θ-(BEDT-TTF)2I3 at 0.7 K for the field perpendicular to the basal
plane. SdH oscillation appears above about 15 T. Superconductivity was
not observed for this sample. The low field behavior is consistent withthat reported in Ref.8. Noteworthy is the maximum around 6 T.
By taking account of the MB effect, the decrease in resistance above
Hmax= 6 T can be qualitatively explained as follows. In the low field
limit, H « HB, where the MB effect is negligible, the resistivity is dominatedby the electron scattering. The resistivity should increase quadratically
until the MB effect becomes appreciable. In the high field limit, thedominant term in the resistivity can be calculated by means of a path
integral method including the possibility of the magnetic breakdown.
This contribution is written as [40-42],
R = (H / n e c )8(1 – P) / πP,(4.6.6)

where n is the number density of the carriers. As H → ∞, this term takes
the form 1 + HB / 2H, so that the resistivity decreases with increasing
field. Hence, a maximum of resistivity should appear at an intermediate104Fig. 4.6.6. Transverse magnetoresistance of θ-(BEDT-TTF)2I3 at 0.7 K.

The

field was applied perpendicular to the two-dimensional plane.

Fig. 4.6.7. The Fermi surface topology concluded in this study, together with the notation of
the cyclotron orbits, α and β. The directions, x and y, corresponds to the b and aaxes in the monoclinic lattice defined in Ref. 3, respectively.

105

field, Hmax, which depends on HB and the scattering time. Such behavior
has been found also for (BEDT-TTF)2KHg(SCN)4 [43], which has FS
topology similar to that of θ-(BEDT-TTF) 2I3. According to the semiclassical model assuming a field independent scattering time, Hmax lies between
HB / 3 and H B / 10 [41]. The present results, HB =15 T and H max= 6 T, canbe qualitatively explained by this scheme.

4.6.2. Fermi Surface and Band Structure
The intermolecular transfer interactions in the donor sheets of
θ-(BEDT-TTF)2I3 can be categorized into two groups; there are three
face-to-face ones and four face-to-side ones [4]. The difference withineach group is related to the structural modulation on the uniform packingof BEDT-TTFs arising from I3– ion arrangement. Since this modulation
is expected to be small, we first consider only two types of transfer
integrals, tf-f and tf-s , standing for the two groups in describing the band
structure. (On the basis of the notation in Sec. 4.3, tf-f and tf-s correspondto tc and ta, respectively.) This simplification enables us to treat the
Fermi surface geometry and other physical properties exactly. In thispicture, the open and closed portions of FS stick together at the MB
junctions. The geometry of the FS is determined only by the ratio of thetransfer integrals, tf-f / tf-s.
On the basis of such approximation, the infrared reflectance spectraof this salt were first analyzed and the band structure was deduced [5](Sec. 4.3). The ratio of the transfer integrals, tf-f / tf-s, found to reproduce
the optical results, is 0.59 [5]. From the reported band parameters [5], thecyclotron mass corresponding to the orbit on each portion of FS can be106

calculated by use of the following equations,
mc = (h 2 / 2π) (∂A c / ∂ε ); ε = εF

(4.6.7)(∂Ac / ∂ε ) = SBZ ρc(ε); ε = εF

(4.6.8)and,

where mc and A c are the mass and cross sectional areas for the cyclotron
orbit c, SBZ the area of the first BZ, and ρc (εF) density of states per 2D unit
cell per spin for the portion of FS corresponding to the orbit c. It is also
possible to estimate the cross sectional areas. The cyclotron masses
calculated in this way are, mβ opt=2.2 me and mαopt =1.0 me, with the crosssectional areas, A β and A α, 100 % (exact) and 14 % of SBZ , respectively.
Here, c = β and c = α denote the whole FS and the closed portion,
respectively (Fig. 4.6.7).
Equations (7) and (8) indicate that the cyclotron mass for a 2D
system is proportional to the (partial) density of states of the relevant
portion of the Fermi surface. It follows from this that the larger cyclotron
mass value for the fast oscillation arising from the MB effect is thesimple sum of the partial densities of states of the portions to be coupledeach other by the breakdown; it is not an average along the coupledorbit. Therefore, for a coupled large orbit composed of a closed pocket
and open portions, we usually obtain cyclotron mass value larger thanthat for the small pocket only. The larger value of the coupled orbit
mass, as in the present case, is not surprising and does not necessarily

107

involve the larger mass for the open portions. We have to be careful incomparing the cyclotron masses for different types of orbits. The enhanced
mass for the open portions of the FS larger than that for the coupled orbit[45] cannot be concluded.
The geometry of the FS concerns the two ratios, mβ /m α and A α / Aβ .
The values, mβ/mα = 2.2 and Aα / Aβ = 0.14, are inferred from the opticalresults, while the present dHvA results give A α / A β = fα / fβ = 0.18 and mβ /mα = 1.8. This comparison supports the picture shown in Fig. 7, although
there remains discrepancy which probably comes from the simplification
introduced in the analysis of the optical data. The MB effect clearlyshows opening of a small gap at the junction of FS. This gap should giverise to interband optical transitions. However, the transition energy, i.e.,
the gap width, is so small that this interband contribution cannot be well
separated from the intraband one in the observed spectra. In Ref. 5,
therefore, the band parameters were estimated from the plasma frequencies
including the low energy interband contributions. Since the FS is stillconnected along the x-direction in Fig. 4.6.7 even in the presence of thegap, such overestimation of the plasma frequency is expected mainly forthe y-direction data. This means that k Fyβ in Fig. 4.6.7 is relatively underestimated, which results in the underestimation of mα and Aα in the analysisof the optical data. The discrepancy between the optical and the dHvA
results can be qualitatively explained in this way.
On the other hand, the values of cyclotron mass, mβ = 3.6 me deduced
from the dHvA oscillations is larger than that calculated from the opticalresults by the factor of about 1.6. In order to remove this discrepancy,
the transfer integrals smaller than the reported ones [5] by the factor of108

0.6 should be used. Such difference cannot be ascribable only to experimental error. It is also difficult to explain this by taking account of thedifference in the transfer integrals within each group. Mass enhancement
near the Fermi level due to many-body effect, such as electron-phonon
interaction, is thereby suggested. The optical spectra probe the global
structure of the conduction band rather than the detailed features near the
Fermi level, so that the optical mass is insensitive to the many-bodyeffect on the quasiparticle energy near the Fermi level.
Next, let us direct our attention to the gap affording the MB effect.
The breakdown field, H B, is approximately related to the energy gap near
the junction, εg, by the equation [40,45],
HB = (π / 4) (c /e h) (εg2 / vx vy ),(4.6.9)where vx and vy are the components of the Fermi velocity at the junction
parallel and perpendicular to the zone boundary, respectively. The simplified tight-binding model with the parameters inferred from the opticalresults [5] yields vx = 1.5 × 10 7 cm/s and vy =1.1 × 10 7 cm/s. From this,
together with HB = 15 T, it follows that εg=1.5 × 10 – 2 eV. If the mass
enhancement near the Fermi level is taken into account, the gap shouldbe about the half of this value. A more popular formula [40],
HB ≈ (mα c / e h) (εg2 /εF),(4.6.10)with the cyclotron mass, mα=3.6 me , results in the same order of magnitude,i.e., εg ≈ 1 × 10 – 2 eV. The corresponding spacing between the open and109the closed portions of the FS at the junction is as small as about 2 % of
π / b, where b is the lattice constant along the x-direction. This justifies
the application of the simplified band model to approximate estimation of
the masses and εF .
In the framework of the tight-binding band model, the origin of the
gap at the zone boundary is the difference of transfer integrals within thegroups, which reflects the small deviations from the uniform packing of
BEDT-TTFs. In the present case, it seems difficult to predict such asmall energy structure accurately enough. Nevertheless, the empiricaltight-binding band calculations can provide a reasonable estimation of
the order of magnitude of the gap [4]. Another problem which exists inthe present case is the failure in prediction of the topology of FS by thecalculations [4]. Even in the presence of the deviations from the uniform
packing, the geometry of FS is basically determined by the ratio of thegroup averages of the transfer integrals, <tf-f> / <tf-s >. Since the modes
and characters of the intermolecular contacts are entirely different between
the groups, attention should be paid in comparing tf-f with tf-s. The calculations [4] yielded <tf-f> / <tf-s> = 1.7, while the optical and the dHvA
experiments support <tf-f> / <tf-s> = 0.59. The former one is sensitive tothe details in the parameterization in the extended Hückel molecular orbital
calculations, and can be suppressed down to about 0.2 by taking accountof the contributions of sulfur 3d orbitals [46]. Therefore, the contradiction
between the experiments and the calculations is expected to be removed,
when moderate weight of the sulfur 3d contributions is adopted. In spiteof such arbitrariness in the empirical parameterization, it is evident that instudying the electronic structures of the molecular conductors the frame-110

work of the tight-binding band model works well with the help of experiments, as demonstrated in this study.

4.6.3. Summary
From the analysis of the dHvA oscillations in θ-(BEDT-TTF)2I3, thecyclotron masses have been evaluated to be 2.0 me and 3.6 m e , for theslow and fast oscillations, about 800 T and 4000 T. The latter has beenshown to be as a consequence of the magnetic breakdown effect whichconnects the closed and open portions of the Fermi surface. These two
portions are separated by the gap of 1.5 × 10 – 2 eV at the junction. These
results are consistent with the topology of the Fermi surface deduced
from the tight-binding band model with the parameters inferred from theoptical measurement. It is also proposed that the contribution of thesulfur 3d orbitals should be taken account of in the extended Hückel
calculations of the HOMO of BEDT-TTF.

4.7. The Subnikov-de Haas (SdH) Oscillations
and Small Fermi Surface

In this section, the notation of the crystal axes follows that of themonoclinic system.
A typical SdH signal observed in a high magnetic field range isshown in Fig. 4.7.2 (a). The Fourier spectrum shows the two fundamental
frequencies marked by the symbols, α and β, and their combination
frequencies. The α and β oscillations are attributed to the closed orbit

111

Fig. 4.7.1. The Fermi surface of θ-(BEDT-TTF)2I3 concluded from the optical and the dHvA
results in the framework of the tight-binding model.

112

Fig. 4.7.2.

(a) Typical SdH oscillations in a high magnetic field range. The inset shows thecorresponding Fourier spectrum.

(b) Frequencies of α and β as a function of the field angle

measured from the c*-axis. The lines show the 1 / cosθ fits.

113

around the zone boundary and the magnetic breakdown orbit, respectively(Fig. 4.7.1). The angle dependence of their frequencies is summarized inFig. 4.7.2 (b). The angle dependence fits well with the formula, f = f0 /cos θ, both for H || ac* and for H || bc*, where θ is the angle between themagnetic field and the c*-axis. The 1/cos θ dependence indicates that
the Fermi surfaces responsible for the α and β oscillations are almost
perfect cylinders, as expected for the two-dimensionality of the electronic
structure of θ-(BEDT-TTF)2I3. The frequency, f0, is 779 T for α and4234 T for β. The corresponding cross-sectional areas are 19 % and 102% of the area of the first Brillouin zone viewed along the c*-axis for α
and β, respectively. From the temperature dependence of the amplitudes
between 0.05 and 0.6 K, the cyclotron masses of the α and β orbits are
estimated to be (1.8 ± 0.1) me and (3.5 ± 0.1) me , respectively. The
frequencies and the cyclotron masses are in good agreement with thedHvA results [6,7] given in the preceding section. The cross-sectional
area of the orbit α, 19 %, is also closed to that predicted from the optical
study, 14 % [5]. The angle dependence, the frequencies, and the cyclotron
mass have definitely established the topology and the shape of the twodimensional Fermi surface of θ-(BEDT-TTF)2I3.
New SdH oscillations are found in a low magnetic field range below
about 3 T. Figure 4.7.3 shows the SdH oscillations together with their
Fourier spectra for several field directions. The angle dependence of thefrequencies is summarized in Fig. 4.7.4, where the frequencies are tentatively classified into branches, γ (γ ′,γ ″), δ and ε. We have examined theangle dependence from about – 40 to 90

for each of θa , θb, and φ, andconfirmed the symmetry with respect to c* and b axes.

114

Fig. 4.7.3.

(a) SdH oscillations in a low field range.

115(b) Corresponding Fourier spectra.

Fig. 4.7.4. Angle dependence of the new series of SdH frequencies.

116

The γ oscillation is observed for field directions around H || c*. For
H || c*, the frequency is about 6 T and the corresponding cross-sectional
area is only 0.15 % of the first Brillouin zone viewed along the c*-axis.
As the field is tilted towards the b-axis, the frequency of γ decreases andthe γ oscillation disappears above θb ≥ 31 . When the field is tilted fromthe c*-axis towards the a-axis up to θa = 20 , the frequency slightly
increases. The three data points for θa ≥ 33 , which are denoted by thesymbol, (γ), in Fig. 4.7.4, seem to be related to the γ branch, though SdH
oscillation was not detected for θ a = 26 nor θa = 30 . On the other
hand, another branch marked by γ ′ appears for θa ≥ 20 . Its frequency
increases rapidly with θ a , approaching that of γ. The peaks appearing in
the Fourier spectra for θa = 45 and 50

are relatively broad. Accordingly,

the two branches, γ and γ ′, may join and be connected with γ ″.
The δ oscillation is remarkable; it is visible for the magnetic fields
parallel to the two-dimensional conducting plane (the ab plane) in a wide
angle range. A Clear SdH oscillation observed for observed for H || b isshown in Fig. 4.7.3. With the field rotating within the ab-plane, we could
observe the δ oscillation up to φ = 60 . Its amplitude decreases as thefield direction approaches the a-axis. The frequency of δ gradually increases
as φ increases.

When the field is rotated from the b-axis towards thec*-axis, the frequency increases as well.
The cyclotron effective masses were determined from the amplitudes
in the Fourier spectra between 0.05 and 1.2 K for γ (H || c*) and δ (H ||b).
Figure 4.7.5(a) shows a mass-plot for the γ oscillation (H || c*). The
cyclotron mass can be estimated to be (0.05 ± 0.02) me for γ, though thedata points are scattered because of the small amplitudes.

117

A veryFig. 4.7.5.

The SdH amplitudes are plotted against temperature for (a) γ (H //c *) and (b) δ(H // b).

The lines are those calculated with the Lifshitz-Kosevich formulausing the cyclotron mass values designated in the figure.
118

small mass value of (0.014 ± 0.002) me is found for the δ oscillation, as
shown in Fig. 4.7.5(b).
The band structures of BEDT-TTF-based conductors have been considered to be highly two-dimensional, as expected from their typical layered
structures. The two-dimensional band structures are calculated on thebasis of the tight-binding approximation in terms of the HOMOs of BEDTTTF molecules. Usually, only the transfer integrals within one BEDT-TTF
layer are taken into consideration. For several BEDT-TTF salts, thedHvA or SdH experiments have indeed revealed the 1/cos θ dependenceof the frequencies, indicating the presence of almost cylindrical Fermi
surfaces as evidence for the two-dimensionality [47]. The α and β branches
are clearly explained within this framework. They come from the large
cylindrical Fermi surface, as is demonstrated by the 1/cos θ dependence(Fig. 4.7.2(b)). They can be regarded as originating from the HOMOs of
BEDT-TTFs.
On the other hand, the interlayer interactions have been believed toplay only a minor role in determining the band structures; they have beentreated as much smaller (10– 3) transfer interactions between the BEDT-TTF
layers only leading to slight corrugation of the cylindrical Fermi surfaces.
The angle-dependent magnetoresistance oscillation observed for several
BEDT-TTF salts has been thought to arise from such slight warping of
the Fermi surface [9-12].
The above framework does not necessarily exclude the possibility ofa very small closed Fermi surface; it can be formed around the zone
boundary, if a slightly corrugated Fermi surface touches the boundary.
However, this cannot explain the new SdH oscillations, γ, δ and ε. First,

119the frequency of γ decreases as the field is tilted from the c*-axis towards
the b-axis. This implies that there exists a considerable energy dispersion
along the c*-axis at least around the γ cyclotron orbit. That is, if we thinkof an ellipsoidal shape of Fermi surface for the γ oscillation, it must be
elongated along the b-axis. Such a shape of the Fermi surface cannot be
expected from the presumed small interlayer interactions. Second, thecyclotron mass of the δ orbit for H || b is very small (0.014 me ). Within
an effective mass band model (parabolic band), the observed mass isexpressed as (mc * ma) 1 / 2, where mc * and ma are the c*- and a-componentsof the effective mass tensor, respectively. Thus, not only mc * but also mashould be small. It is difficult to explain this only by the small interlayerinteractions.
Accordingly, the conclusions can be summarized as follows. Two
different types of the Fermi surface coexist in θ-(BEDT-TTF)2I3. The
first group consists of the large quasi-two-dimensional Fermi surfaces
originating from the HOMOs of BEDT-TTFs, corresponding to the α andβ oscillations. The second group, related to the γ, γ′, γ″, δ and ε oscillations,comprises the very small Fermi surfaces exhibiting a pronounced dispersion
even for the direction perpendicular to the BEDT-TTF layer. They cannot
be understood within the conventional tight-binding framework based
only on the HOMOs of BEDT-TTFs. In order to interpret the present
results, further improvement of the band structure calculations, whichtakes account of the other molecular orbitals of BEDT-TTFs or anions,seems necessary. Thus, this SdH study has established the HOMO-based
band structure on one hand, and revealed the new features outside thesimple picture on the other hand.

120

4.8. The Angle Dependent Magnetoresistance Oscillation(ADMRO)As shown first by Yamaji [11], the period of the ADMRO peaks canbe related to the Fermi wavenumber kF, by
kF = π / d ∆ tan θ ,(4.8.1)

where d is the periodicity in the direction perpendicular to the twodimensional plane, and θ is the angle between the magnetic field direction
and the normal direction. Therefore, the ADMRO is expected to provide
detailed information on the geometry of the Fermi surface. Althoughθ-(BEDT-TTF)2I3 is one of the first material in which the ADMRO is
observed [8], the interpretation of the oscillation period based on a realistic
band structure has not been done yet. Nowadays, the Fermi surface of
this salt has been well understood by the optical (Sec. 4.3) [5], dHvA
(Sec. 4.4) [6,7] and SdH (Sec. 4.5) [14] studies. In order to establish therelation between the ADMRO and the Fermi surface geometry in thissalt, the ADMRO is reexamined here. The notation of the crystal axes
based on the monoclinic lattice is referred to again in this section.
Figure 4.8.1 shows the resistance measured along the c*-axis withthe magnetic field rotated in the ac*-plane, where θ is the angle between
the c*-axis and the field direction. Two oscillations with short and long
period can be distinguished. The short-period oscillation with sharp peaks
is prominent for small values of tan θ (| tan θ | < ca. 5). The values oftanθ at the peaks are plotted against the peak number in Fig. 4.8.2 (a),121

Fig. 4.8.1.

The ADMRO for H // ac *.
againsttan θ.

The

Top: Plot against the angle θ.

short-period oscillation

atBottom: Plot

smallvalues of|tanθ | and the long-period oscillation at large values of | tan θ | are evident.
The broken line shows the envelope function which smoothly connects thepeaks due to the short-period oscillation.
positions due to the long-period oscillation.

122

The ticks indicate the peakFig. 4.8.2.

The plot of tan θ versus peak number for H // ac *.
oscillation.

(b) The long-period oscillation.

123(a) The short period

where the peaks for tan θ > 0 (tan θ < 0) are numbered with positive(negative) integers from the one closest to tan θ = 0. The period, ∆ tan θ,
is determined to be 0.42, which is in good agreement with the value
reported by Kajita et al. [8]. On the other hand, a long-period oscillation,
which was not reported so far, becomes evident for large values of tan θ.
This finding may be ascribed to the quite high quality of the crystal (RRR
= 1200, where RRR is the residual resistivity ratio, ρ(280 K) / ρ(4.2 K))
and to the fine angle-step (0.3 ) around θ = 90

used in recording theresistance. The long-period oscillation is visible as an oscillation of theenvelop function for small values of tan θ; the height of the peaks due tothe short-period oscillation periodically oscillates as is shown by thedotted curve in Fig. 4.8.1. The peak positions are independent of themagnetic field for the long-period oscillations, as is the case for the
short-period one. The values of tan θ at the peaks, including those of theenvelop function, are plotted against the peak number in Fig. 4.8.2 (b).
The data points lie on the straight line, from which the period, ∆ tan θ =1.6, is obtained.
The resistance for the magnetic field rotating in the bc*-plane isplotted against tan θ in Fig. 4.8.3. In this case, the oscillatory features are
considerably weak, and therefore the second derivative, d2R / dθ 2, is also
shown to identify the peaks. The values of tan θ at the peaks are plottedagainst the peak number in Fig. 4.8.4. The peak positions lie on thestraight line with ∆ tan θ = 0.68 and 0.97. The discrepancy between the
periods for tan θ > 0 and tan θ < 0 does not seem intrinsic. The long-period
oscillation seems to overcome the short-period one at large values of|tanθ |.

124

Fig. 4.8.3.

The ADMRO for H // bc *.

Top: Plot against the angle, θ.

Bottom: Plotagainst tan θ, where the most upper curve shows the second derivative at H = 13
T. The ticks indicate the peak positions due to the long-period oscillation.

125

Fig. 4.8.4. The plot of tan θ versus peak number for H // bc *.

126

In order to examine the present results by using Eq. (4.8.1), we haveto know the fundamental period of the energy band structure along thec*-direction in the k-space, 2π / d. By applying the Bloch theorem withrespect to this direction, the wavefunction can be expanded in a Fourier
series,Ψk(r) = ∑ φk, G (x, y) exp[– i(k c + G) z],(4.8.2)G

where x, y and z are the components of r in the real space, and G is thec*-component of the reciprocal lattice vector. The matrix element of the
Hamiltonian of the tight-binding band model is written as,< Kz | H | K ′z > = ∑ <Kz | χm > < χm | H | χm ′ > < χm ′ | K ′z >m, m ′= ∑ <Kz | χm > < χm ′ |K ′z > tm, m ′ ,(4.8.3)m, m ′

in terms of the representation based on plane waves propagating along
the z-direction, where | χm > is the molecular orbital, tm, m ′ is the transfer
integral between the molecular orbitals | χm > and | χm ′ > and| Kz > = (2L) –1 / 2 exp(– iKz z) ,(4.8.4)

L being the length of the crystal along the z-direction.
Let us consider the symmetry operation of the crystal, α, defined by127

α: r → ρα r + τα ,(4.8.5)where ρα and τ α are the rotation and translation part of the operation, α.
Operating αon χm and χm ′ , we have,tα m, α m ′ = tm, m ′ ,(4.8.6)

because the Hamiltonian is not affected by the symmetry operation. Therefore, Eq. (4.8.3) can be written as,< Kz | H | K ′z> = ∑ tm, m ′ ∑ <Kz | αn χm> < αn χm ′ | K ′z > ,(m , m ′ )n(4.8.7)
where (m , m ′ ) runs over the pairs in which m cannot be related to m ′ by
αn, with n running up to a very large integer.
There is a glide mirror symmetry along the c-axis in the monoclinic
structure of θ-(BEDT-TTF)2I3. Regarding this as the operation, α, i.e., τα= c / 2, we obtain,<K z | χ [αnm ] > ==∫ exp(iK ·z) χ (ρ α r – n c / 2) d r
–1

z

3

m

∫ exp[iK · (z + n c sin β / 2)] χ (r) d r
3zm= exp(in Kz · c sin β / 2) <K z | χm >, (4.8.8)

128

taking account of the fact that ρα (π-rotation around the z-axis) does notaffect the z-component. Accordingly, the matrix element (4.8.7) is converted into,<K z | H | K ′z> = ∑ exp[i n (Kz – K ′z) c sin β / 2]n× ∑ tm, m ′ <K z | χm > < χm ′ | K ′z >.

(4.8.9)(m , m′ )The factor,∑ exp[i n (K z – K ′z) c sin β / 2] ,(4.8.10)nis the Laue function appearing in the X-ray diffraction theory. From thenature of the Laue function, it follows that <K z | H | K′z> becomes negligible
unless there exists an integer, I, satisfying,K′z = Kz + 4πI / c sin β .

(4.8.11)

Therefore, only the terms with
G = 4πI / c sin β ,(4.8.12)is required in the expansion (4.8.2). This indicates that the fundamental
period along the c*-direction in the k-space is 4π / c sin β.
With this knowledge, the observed periods of ADMRO can be related129

to the Fermi wavenumbers by Eq. (4.8.1); the interlayer distance, d ,should be c sin β / 2 = 16.93 Å. From the periods, ∆ tan θ, of theshort-period oscillation, the Fermi wavenumbers are estimated to be 0.44
Å –1 and 0.27 Å –1 along the a- and b-axes, respectively. First of all, theshort-period oscillation is ascribable to the large magnetic breakdown
orbit, β, because the Fermi wavenumber along the a-axis points outside
the first Brillouin zone (a* / 2 = 0.32 Å –1 ). The both Fermi wavenumbers,i.e., the long and short radii of the elliptic Fermi surface, are in excellent
agreement with those predicted from the optical study.
Accordingly, it is natural to attribute the long-period oscillation to
the closed orbit, α, around the zone boundary. Using the observed periods,∆ tan θ = 1.6 and 0.97, the Fermi wavenumbers are evaluated to be 0.12
Å –1 and 0.19 Å –1 along the a- and b-axes, respectively. Again, thesevalues are in good agreement with the size of the orbit α deduced fromthe Fermi surface determined by the optical study.
Recently, another study of the angle dependent magnetoresistanceoscillation has been carried out [48]. In this study, the short-period
oscillation is examined for tilting of the field from the c*-axis towards
various directions in the ab-plane. From the obtained period as a functionof the tilting direction, the shape of the magnetic breakdown orbit, β, hasbeen inferred as shown in Fig. 4.8.6 [48]. This shape is in satisfactory
agreement with the results above.
Thus, the angle dependent magnetoresistance oscillation in θ-(BEDTTTF)2I3 has been clearly explained on the basis of the Fermi surface
concluded from the optical study.

130

Table 4.8.I. The Fermi wavenumbers, k F.

ADMRO
Optical Study [5]∆ tan θkF /Å– 1 kF /Å – 1

Orbit
α
β

// a
// b
// a
// b

1.6
0.97
0.42
0.68

0.12
0.19
0.44
0.27

0.10
0.20
0.42
0.30

Fig. 4.8.5. The Fermi surface of θ-(BEDT-TTF)2I3.

Fig. 4.8.6.

The geometry of the Fermi surface derived from the ADMRO with various tilting
way of field [48].
1314.9. Concluding Remarks
The geometry of the 2D Fermi surface of θ-(BEDT-TTF)2I3 hasbeen almost entirely established. As regards the Fermi surface geometry,the simple tight-binding band model inferred from the optical data isfound to be satisfactorily accurate. The discrepancy between the calculated
and experimental band structures has been attributed to the neglect of thesulfur 3d contribution in the calculations of the HOMO of BEDT-TTF.
This conclusion has been provided by the dHvA, SdH and ADMRO
experiments, which also clarify that this salt affords an almost ideal 2D
electron system showing magnetic breakdown effect.
However, the cyclotron mass values predicted from the optical data
do not well agree with those obtained from the dHvA and SdH results.
This may be an indication of enhancement of the density of states justnear the Fermi level due to the many-body effect [49-51]. In this connection,
attention should be directed to the discrepancy between the observed dc
conductivity and that estimated from the optical data at high temperatures.
On the other hand, the TEP results indicate that the simplified bandmodel fails to give the local curvatures near the Fermi surface. From theviewpoint of the optical properties, this corresponds to the appearance of
weak interband transitions, which are not allowed in the simplified bandmodel. In order to explain these features, it is desired to improve theoretical
tools.
The finding of the extraordinarily small three-dimensional Fermi
surface by the SdH experiments is striking: it seems impossible to explain
this within the HOMO based tight-binding model. Contribution of another132degree of freedom is suggested.

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6. M. Tokumoto, A. G. Swanson, J. S. Brooks, M. Tamura, H. Tajima and H.
Kuroda: Solid State Commun. 75 (1990) 439.
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Brooks, C. C. Agosta and S. T. Hannahs: J. Phys. Soc. Jpn. 63 (1994) 615.
8. K. Kajita, Y. Nishio, T. Takahashi, W. Sasaki, R. Kato, H. Kobayashi, A. Kobayashi
and Y. Iye: Solid State Commun. 70 (1989) 1189.
9. M. V. Kartsovnik, P. A. Kononovich, V. N. Laukhin and I F. Shchegolev: JETP
Lett. 48 (1988) 541.
10. M. V. Kartsovnik, V. N. Laukhin, S. I. Pesotskii, I. F. Shchegolev and V. M.
Yakovenko: J. Phys. I. Paris 2 (1991) 89.
11. K. Yamaji: J. Phys. Soc. Jpn. 58 (1989) 1520.

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12. R. Yagi, Y. Iye, T. Osada and S. Kagoshima: J. Phys. Soc. Jpn. 59 (1990) 3069.
13. T. Terashima, S. Uji, H. Aoki, M. Tamura, M. Kinoshita and M. Tokumoto:
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Chem. Lett. 957 (1986).
16. A. G. Swanson: Ph. D. Thesis, Boston University, 1991.
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159.
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H. Kobayashi and R. Kato: Synth. Metals 24 (1988) 301.
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Lett. 957 (1986).
26. K. Kajita, Y. Nishio, S. Moriyama, R. Kato, A. Kobayashi, H. Kobayashi and W.
Sasaki: Jpn. J. Appl. Phys. 26 Suppl. 26-3 (1987) 1363; Solid State Commun. 64

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30. H. Schwenk, C. P. Heidmann, F. Gross, E. Hess, K. Andres, D. Schweizer and H.
J. Keller: Phys. Rev. B 31 (1985) 3138.
31. H. Scwenk, S. S. P. Parkin, V. Y. Lee and R. L. Greene: Phys. Rev. B 34 (1986)
3156.
32. M. Tokumoto, H. Anzai, K. Takahashi, K. Murata, N. Kinoshita and T. Ishiguro:
Synth. Metals 27 (1988) A305.
33. T. Mori and H. Inokuchi: J. Phys. Soc. Jpn. 57 (1988) 3674.
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Jpn. 59 (1990) 2624.
35. J. M. Ziman: Principle of the Theory of Solids (Cambridge Univ. Press, Cambridge,1972) 2nd. ed., Secs. 7.7 and 7.9.
36. M. Tokumoto, N. Kinoshita, H. Anzai, A. G. Swanson, J. S. Brooks, S. T.
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Yakushi: Synth. Metals 42 (1991) 2459.
37. M. Tokumoto, A. G. Swanson, J. S. Brooks, C. C. Agosta, S. T. Hannahs, N.
Kinoshita, H. Anzai, M. Tamura, H. Tajima, H. Kuroda and J. R. Anderson: in
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47. To our knowledge, only one exception is the case of β-(BEDT-TTF)2 IBr2 given
in Ref. 10, where decrease in the small SdH frequency with the tilting of fieldfrom the normal direction has been reported.
48. S. J. Klepper, G. J. Athas, J. S. Brooks, M. Tokumoto, M. Tamura and M.
Kinoshita: Synth. metals, in press.
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Kurmoo and P. Day: J. Phys. Condens. Metter 6 (1994) 2911.
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A. J. Perenboom, M. Kurmoo and P. Day: Synth. Metals 56 (1993) 2566.
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M. Kurmoo and P. Day: Synth. Metals, in press.

136

Chapter 5.
Reflectance Spectra of κ-(BEDT-TTF)2I 3

137

5.1. Introduction
The κ-form of di[bis(ethylenedithio)tetrathiafulvalene] triiodide,κ-(BEDT-TTF)2I3, exhibits superconducting transition at 3.6 K under ambient pressure [1]. So far, other κ-type BEDT-TTF salts, such as [Cu(NCS)2]
salt [2], [CuN(CN)2Br] salt [3], were found to be ambient pressure superconductors. The transition temperatures (Tc) of the [Cu(NCS)2] and[CuN(CN)2Br] salts exceed 10 K, which being the highest ones among
the organic superconductors. In this connection, the κ-type moleculararrangement is believed to have a key to the higher Tc; extensive studies
have been made on these κ-type salts. The κ-type structure is characterized
by a two-dimensional arrangement of the dimeric units of donor molecules[4], as shown in Fig. 5.1.
The reflectance spectra of metallic BEDT-TTF salts have been investigated, in order to elucidate their electronic structures [5-15]. The dispersion of the reflectance spectra of the BEDT-TTF salts often deviates fromthe Drude-type one in the infrared region. With the aid of the calculationsof the optical transition probabilities based on the one-electron theory [7],the origin of the non-Drude dispersion has been pointed out to be theinterband transitions between the split band arising from non-uniform
arrangements of BEDT-TTF molecules. In this connection, κ-(BEDTTTF)2I3 looks to be a suitable example to examine whether the opticalproperties of metallic BEDT-TTF salts are understandable within theframework of the conventional band model, since the κ-salt has a typicaldimeric structure.

138

Fig. 5.1. Left: Arrangement of BEDT-TTF molecules in κ-(BEDT-TTF)2I3 viewed along thelong-axis of BEDT-TTF, and the definition of the transfer integrals.
Orientaion of BEDT-TTFs within the bc-plane.
139

Right:

In this study, reflectance spectra of κ-(BEDT-TTF)2I3, in the infraredand visible region (500-25000 cm –1) at 295 K and 15 K, are presented.
In 5.3.1, a distinct correlation between the appearance of the non-Drude
anomaly and the dimeric arrangement of BEDT-TTF molecules is describedin detail. The curve-fitting analysis of the observed spectra is explainedin 5.3.2. Strong two-dimensionality of the optical effective mass is elucidated. In 5.3.4, the inter-dimer transfer integrals describing the HOMO
bands of κ-(BEDT-TTF)2I3 are evaluated by use of the band scheme
based on the dimer model. The four inter-molecular transfer integrals arealso estimated, and the optical conductivity spectra are calculated using
these integrals.

5.2. Experimental
The single crystals of κ-(BEDT-TTF)2I3 were provided by Prof. H.
Kobayashi (Toho Univ.), Prof. A. Kobayashi and Prof. R. Kato (Univ. ofTokyo). The single crystals were obtained by electrochemical oxidationof BEDT-TTF in a tetrahydrofuran solution containing mixed electrolyte,(n-C4H9)4NX (X=I3 and AuI2, or IBr2 and AuI2) [16]. The content of Au
and Br of the samples used were less than the detectable limit of electronprobe micro-analyzer (EPMA) [17]. The crystal axes of the samples
were identified by X-ray diffraction. Reflectance Spectra were measured
on the (100) crystal face, which is the most developed one, following themethod described in Sec. 3.1. Details of the apparatus used in this work
were described elsewhere [5]. A composite bolometer (Infrared Lab-140

oratories Inc.) was used in the measurement below 2000 cm –1. The
principal axes of reflectance in the bc-plane were confirmed to be the band c-axes by measuring the change in reflectance at 2500 cm –1 onrotating the light polarizer.
Since there were many scratches on the crystal surface, we had toselect out a small area free from scratches by use of a microspectrophotometer with a highly magnifying objective lens in the visible and nearinfrared region. However, we can not use such a procedure in the infrared
region. Thus it was not possible to determine directly an absolute reflectivity
in this region. Consequently, a calibration was carried out by comparing
the reflected lights from the sample surface with that from the gold-coatedsample surface.

5.3. Results

5.3.1 Reflectance Spectra
Figure 5.2 shows the reflectance spectra of κ-(BEDT-TTF)2I3 at 295
K and 15 K. The intra-molecular transition of I3– anion appeared around23000 cm –1 for the polarization direction parallel to the c-axis, along
which the linear I3– anions are oriented. The peak position shifts tohigher wavenumber on lowering the temperature. This transition wasobserved also for other (BEDT-TTF)2I3 salts [7-10,12,13]. A hump was
found near 10000 cm –1 in the || c spectrum, but no such dispersion
appears in the || b spectrum. Since the long axes of BEDT-TTF molecules
are almost perpendicular to b-axis in κ-(BEDT-TTF)2I3, this dispersion is141

Fig. 5.2. Reflectance spectra of κ-(BEDT-TTF)2I3.

The solid lines are the fit to the Drude-

Lorentz model. The inset shows the reflectance spectra of β-(BEDT-TTF)2I3.

142

Fig. 5.3. Conductivity spectra of κ-(BEDT-TTF)2I3.

143

assigned to the intra-molecular transition of BEDT-TTF+ radical. This
assignment is supported by our studies on θ-(BEDT-TTF)2I3 [13] andBPDT-TTF salts [17] (BPDT-TTF denotes bis(propylenedithio)tetrathiafulvalene). The curve-fitting procedure described below gives the peak
position for this transition as 11200 cm –1 at 295 K and 11300 cm –1 at 15
K.
Let us concentrate our attention on the electronic transition observedin the infrared region. As Fig. 5.2 shows, the dispersion curves for thetwo different polarization directions are similar to each other, suggesting
the strongly two-dimensional character of this salt. Although both || band || c spectra indicate the plasma-edge-like minima around 5000 cm –1 ,the line shapes do not follow the Drude behavior below 4000 cm –1 . The
reflectance curves exhibit broad shoulders around 3000 cm –1 and prominentvibronic structures at about 1100 cm –1 . The latter can be assigned to themolecular vibration coupled with the intermolecular transition. Anothervibronic transition was found around 800 cm–1. A similar spectral feature
has been reported for the || [110] spectrum of β-(BEDT-TTF)2I3 at room
temperature [7], which is depicted in the inset of Fig. 5.2.
In many BEDT-TTF salts, BEDT-TTF molecules form twodimensional networks as regards the electron transfer interactions. Generally this network is not uniform but sometimes has a dimeric unit. For
example, in the donor sheets of β-(BEDT-TTF)2I3, BEDT-TTF molecules
form dimers along the [110] axis [19]. The network structure of κ-salt isthe most typical case of dimerization. A strong intra-dimer interaction
splits the conduction band formed from HOMO's of BEDT-TTF into two
branches, between which the dimerization gap open [20]. The lower

144

branch has a bonding character of the dimer, while the upper branch has
an anti-bonding character. Thus, the optical transition between thesebranches is infrared active and polarized along the dimerization direction,i.e., the direction connecting the centers of molecules in the dimer.
As a consequence of this interband transition, the infrared dispersion
is expected to be a superposition of the two contributions, i.e., the Drude-like
dispersion due to the intraband transitions and the contribution of theinterband transitions. If this is the case, the temperature dependence ofthe spectra should give insight into the presence of these two contributions.
At room temperature, the Drude-like dispersion is broad and obscured by
the electron scattering near the Fermi energy. On lowering the temperature,
the scattering rate becomes saliently suppressed, and decreases more andmore. This yields large negative values of the real part of dielectric
function at lower wavenumbers and a distinct plasma-edge in the reflectancecurve. On the other hand, the line shape of the interband transitions isbasically ruled by the band dispersion, thus being rather insensitive to thetemperature than that of the intraband transitions. Consequently, thespectra are expected to approach a Drude-like dispersion on lowering thetemperature. Such a temperature dependence is in fact observed for
κ-(BEDT-TTF)2I3, as shown in Fig. 5.2. A similar result was also reported
for β-(BEDT-TTF)2I3 (Fig. 5.2 inset) [8]. The conductivity spectra obtainedby means of the Kramers-Kronig analysis of the reflectance data at 295 K
and 15 K are shown in Fig. 5.3. For both the || b and || c polarizations, thecontribution of the intraband transitions centered at zero wavenumber
becomes fairly sharp at 15 K. On the other hand, the absorption bandaround 2500 cm –1 corresponding to the interband transitions shows an145intensity decrease and a slight red-shift at 15 K. These facts support theview that the intra- and interband transitions coexist in the infrared region.
Now, let us compare the reflectance spectra of different forms of(BEDT-TTF)2I3 to see the relationship between the structure and opticalproperty. In β-(BEDT-TTF)2I3, BEDT-TTF molecules form a dimerized
stack and all the dimers are oriented in parallel [19,20]. The prominent
conductivity peak appears only in the || [110] spectra. It has been proposedthat the inter-band transition across the dimerization gap is responsible
for this prominent peak by comparing with the calculation of conductivity
spectra [7]. On the other hand, there are two different dimerization axesin the κ-salt, which are almost perpendicular to each other. As waspointed out before, the infrared peak appears along two polarization directions in the bc-plane. In the average structure of θ-(BEDT-TTF)2I3, all
the BEDT-TTF molecules are crystallographically equivalent, forming auniform two-dimensional network [21,22]. The molecular arrangementin the κ-salt [4] is connected with the average structure of θ-salt [22], by
replacing each BEDT-TTF with a BEDT-TTF dimer. The infrared propertyof the θ-salt is Drude-like as expected for a two-dimensional metal [9,13].
The calculation of conductivity spectra also showed that there would be
no contribution from the interband transitions. This is because the interband
transition is forbidden by the high crystal symmetry. (See also Appendix
A.)
As mentioned above, the prominent peak in the infrared region is
ascribed to the inter-band transition across the dimerization gap. The lowtemperature spectra also supported this interpretation. This infrared peakalways appears only along the dimerization direction: It appears in two146

directions in the κ-salt, in one direction in the β-salt, and disappears inany directions in the θ-salt. One of the simplest way to explain thisstrong correlation between the structures and the spectra is to ascribe thisinfrared peak to the one associated with the interband transitions acrossthe dimerization gap based on the tight-binding band scheme [7,12,15].
Let us make here a brief comment on the relation between the electronmolecular vibration (e-mv) coupling and the interband transition. The
e-mv coupling have been investigated mainly for one-dimensional organic
conductors [23,24]. Strong e-mv peaks appear in the spectra of the β-salt[7,8] and κ-salt, whereas only weak e-mv peaks were observed for theθ-salt [13]. This suggests a correlation between the e-mv coupling andthe interband transition; the intensity and the dichroism of the e-mv mode
seem to reflect those of the interband transition. These modes, usuallyobserved around 1200 cm –1 for BEDT-TTF salts, correspond to the totallysymmetric (breathing) intramolecular vibrational modes with the characterof C=C stretching [25,26]. In a free molecule these modes are infraredinactive. However, they appear in the infrared spectra by borrowing theintensity of the interband transitions. On the analogy of the theory for a
dimer chain [23,24], the inter-band transition across the dimerization gap,
referring to the electronic charge oscillation between the two molecules
within the dimer, can couple to the asymmetric combination of the totallysymmetric vibrations of the two molecules. Consequently, it appears as
peaks observed at frequencies slightly lower than those of the original
C=C stretching modes.
From an experimental viewpoint, it can be pointed out that the observation of the e-mv modes affords a probe to the properties of the interband

147transitions; the high intensity of the e-mv peak should indicate the presenceof strong interband transitions in the same light polarization. This consideration is consistent with the experimental results for several BEDT-TTF
salts [5-15]. It works as a convenient index, because the e-mv coupling
modes are easier to distinguish than the interband transition itself; Theinterband transitions have a broad-band dispersion, which is often indistinguishable from that of the intraband transitions. (See also Appendix
B., about the discussion presented here.)
Let us apply the above consideration to the cases of the β-, θ-, andκ-(BEDT-TTF)2I3. The appearance of strong e-mv peaks in the β- [7,9]and the κ-salts reflects large intensities of the interband transitions inthese salts. In spite of the Drude-like behavior, weak e-mv modes wereobserved in the θ-salt [13], suggesting the existence of weak interband
transitions. In fact there are small shoulders in the conductivity spectra
around 2500 cm –1 [13]. This can be interpreted as a result of the loweringof symmetry in the real structure of the θ-salt [27]. The present investigation
is limited to qualitative one because of low resolution of our spectra.
Detailed experimental features of the vibronic structures have been reported
for some BEDT-TTF salts [25,26,28-31].

5.3.2 Curve-Fitting Procedure
The intraband transitions contain the information concerning thedispersion of the conduction band, while the interband transitions reflect
the band splitting due to the dimeric intermolecular interaction. We
estimated the band parameters by analyzing the intraband transitions separated from the interband transitions. The separation was carried out by a148

curve-fitting by use of the Drude-Lorentz model,

3 or 4ε (ω ) = ε C – ω / ( ω + iω / τ) + ∑ f j / [(ω j2 – ω2) – iγ j ω ] .
2
P

2

(5.1)

j

The reflectivity is related to the dielectric function by the equation,
R = [1 + | ε | – √2(| ε | + Re(ε )) ] / [1 + | ε | + √2(| ε | + Re( ε)) ] .
(5.2)

The first term in the right-hand side of Eq.(5.1) stands for the constantcontribution from the higher energy transitions outside the experimental
spectral region. The second term, the Drude term, gives the contributionof the intraband transitions. The third term describes the contribution ofother transitions: the interband transitions, the e-mv coupling modes andthe intramolecular electronic transitions. The line shape of the interband
transitions is empirically expressed as a superposition of two Lorentzian
oscillators, since the shape of the interband transitions shown in Figs. 5.2
and 5.3 could not be reproduced well by a single Lorentzian. Instead ofincluding the contribution of the transitions beyond 20000 cm –1 explicitly,
we limited the fitting region below 15000 cm –1, so that they were implicitly
included in the first term, εc. The curve-fitting results were shown as thesolid curves in Fig. 5.2, and the obtained parameters were compiled inTable 5.I.

149

Table 5.I. Curve-fitting results.

Polarization T / K εc ω p / eV τ / eV – 1Ωp / eV a) m*/m e σ dcopt / Scm – 1 b)

// b
// c

295
15
295
15

4.0
4.0
3.9
3.6

0.63
0.74
0.76
0.82

2.7
22
3.2
16

0.93
0.98
0.95
0.96

4.1
3.0
2.9
2.4

150
1600
250
1500

a) Ωp2 = ω p2 + ∑ j fj , where the summation is taken over the e-mv mode and theinterband transitions.
b) The values obtained by extrapolating the Drude conductivity to zero wavenumber.

5.3.3 Optical Effective Mass
To obtain the information on the conduction band structure from theoptical data, the effective mass model is widely used as a simple descriptionof the band dispersion. In this model the structure of the two-dimensional
conduction band is expressed as a parabola around the Γ point, that is,ε = – h 2 [kb2 / 2m*b + kc 2 / 2m*c ] ,(5.3)

where m*b and m*c are the principal values of the two-dimensional effective
mass tensor defined in the bc plane. The effective mass values are estimatedfrom the intraband plasma frequencies, ωP 's, by using Eq.(5.4),
m*x = 4 π n e 2 /m*x(ω P2)x , x = b and c

150

(5.4)

where n and e denote the number density of the holes per unit volume
and the electron charge, respectively. The effective mass values arecalculated to be m*b = 3.0 me and m*c = 2.4 me, when the plasma frequency
values obtained from the 15 K spectra, (ω P)b = 0.74 eV and (ω P)c = 0.82
eV, together with n = 1.185 × 10 21 cm –1, are used. Small temperaturedependence of the lattice constant and the hole density are neglected
here. The obtained ratio, m*b / m*c = 1.3, is very small among the
known organic conductors, a similar value being known for κ-(BEDTTTF)2[Cu(NCS) 2] [11]. For β- and θ-(BEDT-TTF)2I3, the corresponding
effective mass ratios are estimated to be 3.5 and 2.0, respectively [8,13].
This elucidates the strong two-dimensionality of the electronic structureof κ-(BEDT-TTF)2I3. The obtained results are shown in Table 5.I.
5.3.4 Analysis by Tight-Binding Band Model
The effective mass model is simple and applicable to any metallic
material. However, this model gives little information on the intermolecularinteraction. Moreover, when the Fermi level is far from the edge of the
conduction band, this model is no longer a good approximation. In thisrespect, it is worthwhile to examine the analysis based on the tight-binding
approximation. This approximation is formulated in terms of transfer
integrals between the frontier molecular orbitals of the nearest-neighbormolecules calculated by the extended Hückel method [32-34]. This approach has been successfully applied on several BEDT-TTF salts [68,13,32-38]. The recent self-consistent-field band calculations suggestthat the molecular orbital identities of BEDT-TTF are maintained in the
crystal electronic structure of β-(BEDT-TTF)2X [39].

151Fig. 5.5. Effective interdimer interactions in κ-(BEDT-TTF)2I3.

152

For the sake of estimating the tight-binding band parameters fromthe optical data, we have developed a dimer-based treatment using a
perturbation theory. The overlap integrals between the HOMO's of theneighboring molecules are given as Sb1 = 10.92 × 10 – 2, Sb2 = 2.46 × 10 – 3,
Sp = 5.05 × 10 – 3, and Sq = -1.85 × 10 – 3 for κ-(BEDT-TTF)2I3 [4]. The
intradimer overlap, tb1, is considerably larger than the others, indicating
the dimeric structure of this salt. According to the calculation by thetight-binding approximation, the higher two branches and the lower two
are separated by the dimerization gap, as shown in Fig. 5.4. If the
dimerization is sufficiently strong, i.e., the intradimer transfer integral
sufficiently excels the interdimer ones, we can start from the unperturbed
dimer state with two energy levels, the bonding (lower) and the anti-bonding(higher) levels of the dimer. Due to the perturbation of interdimer interaction, the two levels form two bands, keeping the dimerization gap between
them. Indeed, the energy band structure shown in Fig. 5.4 has thischaracter.
The one-electron Hamiltonian, which is diagonalized to give tightbinding band energies, is divided into two terms as follows:H(tb1, tb2, tp, tq; k) = H0(tb1; k) + H1(tb2, tp, tq; k).

(5.5)

Since there are four molecules in the unit cell, the Hamiltonian is 4 × 4
matrix. We first solve the diagonalization problem for the unperturbed
Hamiltonian, H 0, and find the eigenvalues,
E10 = E20 = tb1 and E 30 = E40 = –tb1.

153

(5.6)

E10 and E20 correspond to the bonding levels and E30 and E 40 to two
anti-bonding levels of the dimer. These levels are independent of wavevector, k, since H0 contains no interdimer interaction. The degeneracy comesfrom the presence of the two crystallographically equivalent dimers in theunit cell. The next step is to calculate the matrix elements of H1, (H1)i, j =<Ψ i0 | H1 |Ψj0 >, where Ψi 0 stands for the Bloch sum of the HOMO's of themolecules, i. Following this, we obtain the first-order correction energy,

δE, by solving the secular equation,
det [(H1)i, j – δE δi, j ] = 0 ,(5.7)

for each pair of the doubly degenerating states, (1, 2) and (3, 4). The
resultant energies to the first-order, E 1 = E 0+δE , are written in the
explicit forms,

E11(k) = E 10 + tb2 cos(b k) – 2 | tp+tq | cos(b k / 2) cos(c k / 2),
E21(k) = E 20 + tb2 cos(b k) +2 | tp+tq |cos(b k / 2) cos(c k / 2),
E31(k) = E 30 – tb2 cos(b k) – 2 | tp–tq | cos(bk / 2) cos(c k / 2),

E41(k) = E 40 – tb2 cos(b k) + 2 | tp–tq | cos(bk / 2) cos(c k / 2) .
(5.8)

The obtained energy dispersion is similar to that of θ-(BEDT-TTF)2I3(the average structure), which is given by,154

E(k) = 2 tc cos(c k) ± 4 |tb | cos(ck/ 2) cos(ak / 2) .

(5.9)

Since the arrangement of BEDT-TTF dimers in the κ-structure has
the same symmetry as that of the molecules in the θ-structure, the similarity
between Eqs.(5.8) and (5.9) is natural. In this approximation, as a resultof the interdimer interactions, tb2, tp and tq, the doubly degenerate bonding
levels of the two dimers in the unit cell form the pair of branches, E 11 (k)and E21(k), and the anti-bonding levels form the other pair, E31(k) andE41(k). In other words, we regard the dimers as supermolecules (Fig.5.5).
This approximation is good, when the energy separation E31(k) –E21(k)sufficiently exceeds E 31(k) –E41(k) and E11(k) –E21(k). As Fig.5.4 shows,
this condition is actually satisfied around the Fermi surface.
Taking account of this perturbation theory, we can estimate thevalues of effective interdimer transfer integrals from the Drude plasma
frequencies. The plasma frequencies, (ω P )b and (ω P)c are the function of
the Fermi velocities. Since the Fermi level crosses the upper two branches,
E31 and E41, we can explicitly relate the two plasma frequencies and thenumber of states beyond the Fermi level per unit area, Nh, to the threeparameters, tb = -tb2 / 2, tc = | tp-tq | / 2 and the Fermi energy, E F, by thefollowing equations,∫(5.10)∫(5.11)(π2h / e 2) (ωP 2)b = (2π / a sin β) | (vF)b | dkc ,(π2h / e 2) (ωP 2)c = (2π / a sin β) | (v F)c | dkb ,Nh = (2/4π2)∫ | k (k ) | dk ,
Fc

155

b

b

(5.12)

where (v F)b and (vF)c denote the b- and c-components of the Fermi velocity,respectively. The components of the Fermi wavenumber, k Fb and kFc,
satisfy E(k Fb, k Fc) = E F. These three integrals are taken over the twodimensional Fermi surface in the bc-plane. As have been done in the caseof θ-(BEDT-TTF)2I3 [13], it is possible to solve these three equations
simultaneously to determine tb, tc and EF for given values of (ω P)b , (ω P)cand N h. By using (ω P)b = 0.74 eV, ( ωP )c = 0.82 eV, a = 16.387 Å, sin(β)=
0.948, and Nh = 1.841 × 10 14 cm – 2, we obtained the result: tb = 0.034, tc =0.054 and E F = – 0.037 eV. Thus, the effective band parameters havebeen determined for the upper half of the HOMO band of κ-(BEDT-TTF)2I3.
In this "θ-model", the band structure of κ-salt is modeled on that of
θ-salt.
Next the procedure to estimate the full parameter set (tb1, tb2, tp, tq) of
the exact κ-band is described. The intradimer transfer integral, tb1, isclosely related to the excitation energy of the inter-band transition. Since
the line shape of the conductivity spectra (Fig.5.3) is broad, it is difficult
to specify the accurate excitation energy values of the inter-band transitions.
Nevertheless, we assumed that,tb1 ≈ – 0.15 eV ,(5.13)

because this value gives preferable peak positions in the calculation of
the inter-band transitions described below (Fig.9). There is no experimental
information on the values of tp and tq, so that we assumed the ratio, tp / tq,based on the calculations of the overlap integrals as Eq.(5.14):156

tp / tq = – 5.05 / 1.85 ≈ – 2.7 .

(5.14)

Then, to be searched is the set of parameters (tb1, tb2, tp, tq) which
satisfies the following conditions: i) tb1 = – 0.15 eV, ii) tp / tq ≈ – 2.7 eV,
and iii) the plasma frequencies fits with the observed ones. The mostreliable parameters are tb1 = – 0.15 (by assumption), tb2 = – 0.037, tp =–0.074, and tq = 0.027 eV. Figure 5.6 shows the band structure and the
Fermi surface drawn by the use of these parameters and those of the
θ-model described by tb and tc. In the θ-model the top of the band is 0.32
eV from the Fermi level, while in the exact band the corresponding value
is 0.24 eV. The obtained shape of the Fermi surface is substantially thesame with each other. These results are in agreement with the calculation[4] displayed in Fig. 5.4.
There are the relations to the first-order linking tb and tc with tb2, tp
and tq , i.e.,tb = –tb2 / 2 and tc = |tp – tq | / 2 .

(5.15)

Due to the contribution of tb1, which is neglected here, the effective
parameters, tb and tc , should be somewhat larger than those expected fromthese relations. In fact, we have,tb = 0.034 > –tb2 / 2 = 0.019 eV,(5.16)

tc = 0.054 > | tp – tq | / 2 = 0.051 eV.

(5.17)The density of states (DOS) for the tight-binding bands of κ- and157

Fig. 5.6.

Tight-binding band structure and the Fermi surface estimated from the reflectancespectra at 15 K.

Solid curves are by the "exact κ-band" and broken curves are by

the "θ-model".

158

Fig. 5.7.

Density of states of κ-(BEDT-TTF)2I3 (upper panel) and θ-(BEDT-TTF)2I3(lower panel) calculated from the tight-binding bands estimated from thereflectance spectra.

159

θ-(BEDT-TTF)2I3 is calculated. The transfer integrals for θ-(BEDT-TTF)2I3
are taken from Ref. 13. The obtained DOS curves are shown in Fig. 5.7.
The DOS values around the Fermi level of the two models for κ-(BEDTTTF)2I3 are in agreement with each other. Just below the Fermi level, asingular point appears, which comes from the degenerate states along the
Y-M line. Another singularity at the bottom of the θ-model band arises
from pseudo-one-dimensional character near the bottom of the θ-type
band structure. This singularity is artificial, because the perturbation
theory cannot be applied around this energy region. The DOS at the
Fermi level per formula unit for a single spin was calculated; ρ(EF )= 3.1
eV – 1 for κ-(BEDT-TTF)2I3 (from both models) and ρ(EF )= 2.2 eV – 1 forθ-(BEDT-TTF)2I3. The latter value match with that calculated from theeffective mass band model.
Two de Haas-van Alphen (dHvA) results have been reported for the
Fermi surface of κ-(BEDT-TTF)2I3 [40,41]. As explained in Chap. 3, thecross-sectional area and cyclotron mass for the cyclotron orbit relevant to
the dHvA effect can be calculated from the band structure. From thepresent band structure, the area of the closed pocket around the Z-point isestimated to be 14 % of that of the first Brillouin zone. The corresponding
value from the dHvA results by Oshima et al. is 20 % [40] and that by
Wosnitza et al. is 15 % [41]. These are close to each other. The cyclotron
masses for the pocket and for the whole Fermi surface are also calculated
to be 1.2 me and 2.7 m e, respectively, from the band structure determinedin this study. The latter is in accord with that reported by Oshima et al.
(2.7 me ) [40], while Wosnitza et al. have stated that it is 3.9 me [41].
Furthermore, we calculated the conductivity spectra in the framework160

of the point-charge approximation using the parameter set (tb1, tb2, tp, tq).
As the DOS calculation shows, there is a peak just below the Fermi levelin the DOS curve. The electrons here are thermally excited even at room
temperature, thus introducing vacant levels below E F. Therefore, theintensity of the interband transition from the lower band to the upperband is expected to increase. Consequently, the low energy portion of
the interband transition should grow on increasing temperature. To examine
this effect, we take account of the temperature dependence of the FermiDirac distribution function in the calculation. Small temperature dependence of chemical potential is neglected. The calculated spectra wereshown in Fig. 5.8 and the intensities were listed in Table 5.II. The
position of the main peak, about 2500 cm –1 in the || c spectra, and thegrowth of the low energy portion of the transition at room temperatureare reproduced by this calculation. However, the overall intensities of theinterband transition are less than the observed ones. The hump around3500 cm –1 in the || b spectra cannot be given by the calculation.

Table 5.II. Intensities of the interband transitions.

Polarization T / K

// b
// c

300
0
300
0

Intensity / eV 2
calc.
obs. a)
0.14
0.086
0.058
0.032

0.47 (295 K)
0.42 (15 K)
0.34 (295 K)
0.24 (15 K)a) ∑j fj , where the summation is taken over thee-mv mode and the interband transitions.

161Fig. 5.8. Conductivity spectra calculated by use of the parameters: tb1 = – 0.15, tb2 = – 0.037,
tp = – 0.074 and tq = 0.027 eV.

162

5.4. Discussion
The comparison of the reflectance spectra of κ-, β-, and θ-(BEDTTTF)2I3 shows that there is a definite correlation between the dimericarrangement of BEDT-TTF and the appearance of the non-Drude midinfrared absorption. In the light of the tight-binding band model, weinterpreted these mid-infrared features as the interband transitions acrossthe dimerization gap. This is consistent with the results for other BEDT-TTF
salts [5-15]. The calculations of the interband transitions of κ-(BEDTTTF)2I3 give preferable peak position. On the other hand, an interdimer
charge-transfer [26] assisted by the electron-electron Coulomb repulsion
cannot explain the correlation between the spectra and the crystal structure.
The intensities of the interband transitions are calculated, taking
account of the temperature dependence of the Fermi-Dirac distributionfunction. Although the qualitative tendency of the observed spectra was
reproduced by this calculation, the absolute values of the observed intensity
could not be explained. This discrepancy is conspicuous particularly atroom temperature. Similar result was found for β-(BEDT-TTF)2I3 [7].
The larger intensity of the interband transition requires a larger intradimertransfer integral or smaller interdimer ones. However, the larger intradimer
integral gives too high transition energies and the smaller interdimer
integrals yield too small intraband plasma frequencies. This is a dilemma.
Thus, in order to explain the large intensity of the interband transition, we
have to take other factors into consideration.
One of the possible reasons for the discrepancy is the inaccuracy of
our simple point-charge approximation. The point-charge approximation,163

in which a molecule is treated as a point located at the center of eachmolecule, is useful to predict the characteristics of the interband transition
dominated by the crystal symmetry. However, the inter-molecular dipole
moments calculated from the point-to-point vectors in this approximation
are not very precise in the real crystal.
Many-body effect may influence the infrared property. To the limitof large intradimer Coulomb repulsion, we have one hole localized oneach dimer. Interdimer charge-transfer transition across the (extended)
Hubbard gap occurs in this localized picture. However, the observationsof the Shubnikov-de Haas effect in κ-(BEDT-TTF) 2[Cu(NCS)2] [35], thetemperature dependence of thermoelectric power in β-(BEDT-TTF)2I3 andκ-(BEDT-TTF)2[Cu(NCS) 2] [37] and the de Haas-van Alphen effect inθ-(BEDT-TTF)2I3 [38] suggest that in the metallic BEDT-TTF salts the
one-electron picture is preferable than the localized picture, at least at
low temperatures. Thereby, we surmise that the qualitative structure of
the conduction band agrees with that predicted by the band calculation,even if the renormalization effect [42] reduces the band width.
In this picture, the intensity and the energy of the interband transition
are probably affected by the electron-electron scattering or the Coulomb
attraction within the electron-hole pair created by the excitation. Since
the metallic BEDT-TTF salts exhibit rather low dc conductivity, particularly
at room temperature, the screening effect is expected to be so insufficientthat the Coulomb interaction survives and coexists with the metallic conduction. Since the strong dimerization causes the system to approach thehalf-filled limit, the DOS around the Fermi level is sensitive to many-bodyeffect. As the present calculations show, the interband transition is sensitive164

to DOS enhancement. The hump around 3500 cm –1 on the || b spectra
may indicate additional transitions assisted by these effects. Even if the
many-body effect influences the infrared transition, it should be noticed
that the transition energy does not necessarily indicate the Coulomb potential energy as long as the one-electron picture holds. However, theeffect of the Coulomb interaction on the intensity of the inter-band transition
still remains unsettled. To make it clear, we need further investigations
on the conduction mechanism of these materials at high temperatures.
Very short mean free path in many organic conductors [13,43] at room
temperature is also possibly in connection with the infrared properties.
The superconducting transition temperature of κ-(BEDTTTF)2[Cu(NCS) 2], Tc = 10.4 K [2], is about three times as large as that of
κ-(BEDT-TTF)2I3, 3.6 K [1], though their donor sheet structures are quite
similar to each other. Moreover, the dc conductivity of the I3 salt is
metallic between room temperature and Tc [1,44], while the [Cu(NCS) 2]
salt exhibits a resistivity maximum [2] or shoulder [11] near 100 K.
Several studies have been made on the infrared properties of the [Cu(NCS)2]
salt [11,26,29,30,45-47]. The observed line shapes of the two salts resemble
each other; no evidence for a substantial difference in the band structure
is found in the shape of the observed spectra. However, the plasma
frequencies of the I3 salt are somewhat higher than those of the [Cu(NCS)2]
salt. The increase in the plasma frequency indicates increase in the band
width and decrease in the density of states. In fact, the DOS at the Fermi
level per formula unit for a single spin is estimated to be 3.1 eV – 1 for theI3 salt, while for the [Cu(NCS) 2] salt the corresponding value is deducedas 7.1eV – 1 [2] (from the static paramagnetic susceptibility at room tem-165perature) or 7.5 eV – 1 [11] (from reflectance spectra). Therefore, thehigher plasma frequencies of the I3 salt can be related to the lower Tc .
The fact that the values of the cyclotron mass of the I3 salt [40,41] aremuch smaller than those of the [Cu(NCS)2] salt [42] also supports thisinterpretation. A possible origin of the difference in the electronic structureis the interlayer electron transfer with the help of the donor-anion interaction[36].

5.5. Concluding Remarks

The band parameters describing the structure of the HOMO band of
κ-(BEDT-TTF)2I3 have been estimated from the Drude plasma frequencies
by the method based on the dimer approximation developed in this work.
The advantage of this method is that the procedure requires only the
intraband plasma frequencies and that the energy dispersion can be expressed by a simple and explicit formula. This approximation, which isformulated by use of simple perturbation theory, can be easily generalized
and should be applicable to other dimeric organic metals: other κ-type
BEDT-TTF salts, β-(BEDT-TTF)2X, κ-(DMET)2AuBr2 [48,49], κ-(MDTTTF)2AuI2 [50], and so on. It has been shown that the interband transitions
across the dimerization gap [12,15] are essential to the mid-infrared absorption observed for κ-(BEDT-TTF)2I3. We pointed out the possibilityof many-body effect on the intensity of the optical transition. Further
investigations will be necessary to elucidate the origin of the large intensityof the mid-infrared transitions.

166

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170

Chapter 6.

Reflectance Spectra of Me4N[Ni(dmit)2]2171

6.1. Introduction

The anion radical salt of [Ni(dmit)2], Me4N[Ni(dmit)2]2, is the firstexample of a metallic molecular conductor based on π-radical anion andclosed-shell cation [1,2]. The reported temperature dependence of dc
resistivity of this salt is somewhat complicated [1]; it depends on thecooling-heating cycles and is accompanied by a resistivity jump around
100 K with hysteresis or by non-metallic behavior at low temperatures.
These anomalies have been suggested to be a metal-semimetal transition
associated with the freezing of the rotational degrees of freedom of themethyl groups [1]. The anomalies have been found to be suppressed byapplication of pressure, and superconducting transition have been discovered above 3 kbar, for the first time in the closed-shell cation salt of[Ni(dmit)2] [1]. The superconducting transition temperature rises withincreasing pressure, up to 5 K at 7 kbar, which is in contrast to the case of
the salts of TTF derivatives [1].
In the crystal of Me4N[Ni(dmit)2]2, the conduction layers of [Ni(dmit)2]and cation sheets are arranged alternately along the c-axis (Fig. 6.1) [1].
Two types of conduction layers are distinguished; the one has [Ni(dmit)2]
stacks elongated in the [110] direction, and the other has the equivalent
stacks in the [110] direction. These two layers appear alternately along
the c-axis (Fig. 6.2). Two pairs of quasi-one-dimensional Fermi surfaces
are expected, each pair arising from each type of layer. Thus, the wholesystem has a two-dimensional electronic structure. In fact, tight-binding172

Fig. 6.2. Crystal structure of Me4N[Ni(dmit)2]2 [1].

173Fig. 6.3.

The tight-binding band structure and Fermi surface of Me4N[Ni(dmit)2]2 inferredfrom the intermolecular LUMO overlap calculations [1, 2]. The primitive lattice,(a ′, b ′, c ′), was adopted, where a ′ = (a + b) / 2, b ′ = –b and c ′= – c.
lines indicate the Fermi surface of the adjacent conduction layer.

174

Dottedband calculations based on the LUMOs of [Ni(dmit)2] results in suchfeatures (Fig. 6.3) [1]. Similar structures have been found also for α- andβ-Me 4N[Pd(dmit)2]2 [3,4], Me4N[Pt(dmit)2]2 [5], Cs[Pd(dmit)2]2 [6] andMe4As[Pd(dmit)2]2 [3].
The main purpose of this study is an experimental examination ofthis two-dimensional electronic band structure. The observed spectrum
should correspond to a superposition of that of the two types of layers.
However, it is possible to derive each part of the contribution, by taking
account of the fact that the two types of layers are equivalent. Furthermore,
introduction of a simplification of the molecular arrangement in the layers
affords an estimation of an approximate shape of the Fermi surface.

6.2. Experimental

The single crystal samples of Me4N[Ni(dmit)2]2 were provided by
Prof. H. Kobayashi (Toho Univ.), Prof. A. Kobayashi (Univ. of Tokyo)
and Prof. R. Kato (Univ. of Tokyo). The black plate single crystals wereobtained by galvanostatic electrolysis of an acetonitrile solution of
Me4N[Ni(dmit)2] and Me4NClO 4. Polarized reflectance spectra were measured on the (001) crystal face, by the methods described in Sec. 3.1,
with the light polarization, E || a and E || b, which are the principal axes
within the (001) plane. The crystal axes were identified by X-ray oscillation
photographs.

175

6.3. Results and Discussion

Figure 6.4 shows the reflectance spectra measured at room temperature, 150 K and 25 K. The two-dimensionality is obvious, as is expectedfrom the crystal structure. The reflectivities at low wavenumbers are
somewhat low, particularly for E || b. However, they are remarkably
enhanced with lowering temperature, in accord with the metallic behavior
as expected for this material. At low temperatures, the spectral shape isbasically metallic accompanied by the plasma edges around 4000 cm –1 inthe E || a spectra and 2000 cm –1 in the E || b spectra, though there aresmall deviations from the Drude type dispersion, that is, the humps around1100 and 1500 cm –1 in the E || a spectra and that around 3000 cm –1 in the
E || b spectra. The Drude-like dispersion suggests that this salt is essentiallymetallic even at ambient pressure in spite of the anomalies in behavior ofthe dc resistivity, at least from the viewpoint of the optical properties.
This is consistent with the suggestion that the anomalous behavior of thedc resistivity may be caused localization due to weak randomness such as
the rotation of the methyl groups.
The small humps indicate the presence of the contribution of theinterband transitions across band gaps. In fact, weak broad maxima
corresponding to the humps can be recognized in the conductivity spectra(Fig. 6.5) obtained by the Kramers-Kronig analysis of the reflectancespectra. Tails of the Drude contributions are evident at low temperatures.
On the other hand, peaks of conductivity appear below 1000 cm –1 in both176

Fig. 6.4.

Reflectance spectra of Me4N[Ni(dmit)2]2.
Drude-Lorentz model.

177

Solid curves shows the fit to theFig. 6.5. Conductivity spectra of Me4N[Ni(dmit)2]2.
178

E || a and E || b spectra at room temperature. However, they are due tothe inaccuracy of the extrapolation of the considerably low reflectivity
data towards zero wavenumber. The extrapolation is required for the
Kramers-Kronig analysis. But it is difficult for metals with low reflectivity,
because reflectivity should increase very rapidly towards zero wavenumber
outside the experimental region. So that, the peaks should be regarded as
artificial ones, and do not necessarily imply the gap formation at room
temperature.
In order to extract the contribution of the intraband transitions (theDrude terms), a curve-fitting analysis of the reflectance spectra was carriedout, on the basis of the Drude-Lorentz dielectric function,

1 or 2ε (ω ) = ε C – ω / ( ω + iω / τ ) + ∑ f j / [(ω j 2–ω2) – iγ j ω ] .
2
P

2

(6.1)

j=1

The first term in the right-hand side of Eq.(6.1), the Drude term, describes
the intraband transitions. The second term stands for the interband transitions corresponding to the conductivity maxima around 1600 cm –1 and4000 cm –1 in the E || a spectra and that around 4000 cm –1 in the E || bspectra. The reflectivity is related to dielectric function by the equation,
R = [1 + | ε | – √2(| ε | + Re(ε )) ] / [1 + | ε | + √2(| ε | + Re( ε)) ] .
(6.2)

The obtained parameters are summarized in Table 6.I. At room temperature,179

the fitting results are not reliable because of the too damped Drude contributions.

Table 6.I. Drude parameters obtained by theDrude-Lorentz fit of the reflectancespectraPolarization T / K εc ωp / eV τ – 1 / eV
// a
// b

150
25
150
25

3.1
2.8
4.1
4.1

0.75
0.69
0.41
0.41

0.083
0.040
0.094
0.038

The squares of the obtained plasma frequencies, ωp2, are the sums ofthe contributions of the two types of [Ni(dmit)2] layers with different
orientations. Since the a- and b-axes are the principal axes of the crystal
and the two types of layers are equivalent, the sum is simply the twice of

ωp2 of the one type of the layer. In this way, the observed values of ωp2
can be compared with those calculated from the band structure.
The calculations of inter-molecular overlap integrals of
Me4N[Ni(dmit)2]2 suggest the following features. The amplitudes of thetwo intrastack transfer integrals are almost equal to each other, and they
have opposite signs. Only one type of transfer integrals is predominantamong the interstack ones, referring to that marked by the symbol p in
Ref. 1. Thereby, the band structure of this salt is approximately described180

by the two transfer integrals, the intra- and interstack ones, t1 and t2,
respectively, as schematically illustrated in Fig. 6.6. This approximation,i.e., the assumption of no dimerization in a [Ni(dmit)2] stack, is also
supported by the weakness of the observed interband transitions.
The plasma frequencies can be calculated from the band structure by
the equations,

∫

(6.3)

∫

(6.4)

(ωp2) a = 2 (e2 / π2h) (π / c sin β) | (vF)a | dk Fb ,(ωp2)b = 2 (e 2 / π2h) (π / c sin β) | (vF)b | dkFa ,where the factor 2 stands for the sum over the two different [Ni(dmit)2]
layers, and (vF )a and (v F)b are the a- and b-components of the Fermi
velocity, vF, given by,
vF = h –1 grad k E(t1, t2; k a, kb) ; E = E F(6.5)

The band filling, which determines the Fermi energy E F, is set to be 1/4,
as inferred from the stoichiometry. By use of these equations, the optimal
set of t1 and t2 which reproduces the observed plasma frequencies was
searched. When the values of plasma frequencies at 25 K, (ωp)a = 0.69
eV and (ωp)b = 0.41 eV, together with the lattice parameters at 95 K, a =13.675, b = 6.476, c = 36.048 Å and β = 94.74 , are used, the results t1 =0.10 eV and t2 = 0.052 eV are obtained.
Figure 6.7 shows the band structure and Fermi surface estimated inthis way. The overall shapes of the band dispersion and the Fermi surface

181

Fig. 6.7.

The band structure and the Fermi surface inferred from the optical results at 25 K.
The primitive lattice, (a ′, b ′, c ′), was adopted, where a ′ = (a + b) / 2, b ′ = – b
and c ′= – c.
182are in agreement of those of the calculated ones (Fig. 6.3). The opticalresults give slightly larger corrugation of the quasi-one-dimensional Fermi
surface. This comes from the larger ratio t2/t1 = 0.52; the corresponding
ratio is about 0.3 according to the band calculations. This deviation may
stem partly from the simplification of the molecular arrangement introducedin the analysis of the optical data.
In the light of this band structure, the interband transitions appear
around 4000 cm –1 are ascribable to the transitions from around the bottomof the lower band to around the top of the upper band. Such transitions
for zigzag stacks, as in this material, should have larger intensities for thelight polarization perpendicular to the stacks, which is consistent with theobservation. The transition energy of them corresponds to the overall
width of the LUMO bands, which is slightly larger than 4t1.

6.4. Conclusion

The reflectance spectra of Me4N[Ni(dmit)2]2 suggest that this salt isessentially metallic even at low temperature, in spite of the non-metallic
behavior of the dc resistivity. The band structure and the Fermi surface
deduced from the Drude plasma frequencies are consistent with the tightbinding calculations.

183

References

1. A. Kobayashi, H. Kim, Y. Sasaki, R. Kato, H. Kobayashi, S. Moriyama, Y. Nishio,
K. Kajita and W. Sasaki: Chem. Lett. 1819 (1987).
2. H. Kim, A. Kobayashi, Y. Sasaki, R. Kato, and H. Kobayashi: Chem. lett. 1799
(1987).
3. A. Kobayashi, H. Kim, Y. Sasaki, K. Murata, R. Kato, H. Kobayashi: J. Chem.
Soc. Faraday Trans. 86 (1990) 361.
4. A. Kobayashi, R. Kato, R. A. Clark, A. E. Underhill, A. Miyamoto, K. Bun, T.
Naito and H. Kobayashi: Synth. Metals 55-57 (1993) 2927.
5. A. Kobayashi, A. Miyamoto, H. Kobayashi, R. A. Clark, A. E. Underhill: J. Mater.
Chem. 1 (1991) 827.
6. R. A. Clark and A. E. Underhill: Synth. Metals 27 (1988) B515.

184

Chapter 7.
Reflectance Spectra of α-Et2Me 2N[Ni(dmit)2]2

185

7.1. Introduction
The anion radical salt of [Ni(dmit)2], α-Et2Me2N[Ni(dmit)2]2, retains
metallic conductivity down to liquid helium temperature [1]. Three modifications, α-, β- and γ-types, have been hitherto identified in this compound[2]. The β-type is semiconducting. The α-type exhibits resistivity jumpat 240 K due to cracks of the crystals caused by the abrupt structuralphase transition, which has been revealed very recently [2]. The γ-type
crystal shows smooth decrease in resistivity down to low temperature [2].
The crystal of α-Et2Me2N[Ni(dmit)2]2 is composed of the conduction
layers of [Ni(dmit)2] and cation sheets, which are arranged alternately
along the a-axis [1]. In the [Ni(dmit)2] layers, all the molecular planes
are almost perpendicular to the c-axis and the molecules are packed in aface-to-face fashion. However, there is a special characteristic overlapping
mode in this crystal, which is called as the "spanning overlap" mode [1].
That is, a molecule overlaps with two molecules equivalently, in thismode (Fig. 7.1). As a result, molecular bridging array is formed. Anothersignificant overlapping mode is a conventional face-to-face one to form adimeric packing of [Ni(dmit)2]. The calculations of the intermolecular
overlap integrals [1] have shown that these two overlaps are dominant.
These interactions afford a honey-comb-like two-dimensional network.
According to the tight-binding band calculations [1], this molecular arrangement in fact yields two-dimensional Fermi surface (Fig. 7.2). The
shape of the Fermi surface is similar to those of the typical two-dimensional
BEDT-TTF salts with the κ-type structure. Therefore, this structure is an
important example providing a way to construct a two-dimensional structure186

Fig. 7.1. Top: Arrangement of [Ni(dmit)2] in the conduction layer of α-Et2Me2N[Ni(dmit)2]2[1].

Bottom: Spanning overlap mode, designated as p1 in the top.

187

Fig. 7.3.

The tight-binding band structure and Fermi surface of α-Et2Me2N[Ni(dmit)2]2
inferred from the intermolecular LUMO overlap calculations using the crystal
structure at 11 K [2].

The doubled periodicity along the b-axis is omitted forsimplicity.

188

only by the face-to-face overlapping most favored by planar π-molecules.
Similar molecular packing is also found for α-(C7H16N)[Ni(dmit) 2]2, where
C7H16N+ is piperidinium cation [3].
However, the recent low temperature X-ray study has revealed thatthe crystal structure below the phase transition at 240 K is more complicatedthan that at room temperature [2]. The phase transition is associated withan ordering of Et2Me2N + cations and changes in the locations of [Ni(dmit)2].
The low temperature structure includes two independent [Ni(dmit)2] layers
arranged alternately along the a-axis. Though the molecular arrangementin each layer still includes the spanning overlap modes and is similar tothat of the room temperature structure, the band structure and the Fermi
surface are modified into those shown in Fig. 7.3 [2]. This low-temperatureband structure can explain several features of the recent measurements ofthe magnetoresistance and the Shubnikov-de Haas effect [4,5].
The main purpose of this study is the experimental examination of
the two-dimensional electronic structure of this material and the estimationof the transfer integral corresponding to the spanning overlap. For thispurpose, the anisotropy of the intensities of intraband transitions can beused. The analysis of the optical properties may be involved due to thecomplicated low temperature structure. Nevertheless, it is interesting to
examine what information is provided by the optical measurements applied
to such a complicated situation.

189

7.2. Experimental
The single crystals of α-Et2Me2N[Ni(dmit)2]2 are provided by Prof.
H. Kobayashi (Toho Univ.), Prof. A. Kobayashi (Univ. of Tokyo) andProf. R. Kato (Univ. of Tokyo). Galvanostatic electrochemical oxidationof Et2Me2N[Ni(dmit)2], in an acetonitrile-acetone (1:1) solution containing
Et2Me2NClO4 as supporting electrolyte, yielded black plate single crystals.
Polarized reflectance spectra were measured on the (100) crystal face, by
the methods described in Sec. 3.1, with the light polarization, E || b and E|| c, which are the principal axes within the (100) plane. The crystal axes
were identified by X-ray oscillation photographs.

7.3. Results and Discussion

Figure 7.4 shows the reflectance spectra measured at room temperature, 200 K, 100 K and 25 K. The spectra clearly show the twodimensionality of this salt, with the plasma edges appearing around 2000
cm –1 for E || b and 4000 cm –1 for E || c. As is expected for metals, theplasma edges become sharper with decreasing temperature accompaniedby the increase in reflectivity at low wavenumbers. No significant change
was observed on the Drude-like spectral shape at low wavenumbers in
the cooling through the phase transition. However, non-Drude features
grew around 4000 cm –1 with lowering temperature, which can be seen as
small humps in the reflectance spectra.

190

Fig. 7.4. Reflectance spectra of α-Et2Me2N[Ni(dmit)2]2.
Drude-Lorentz model.

191

Solid curves shows the fit to theFig. 7.5. Conductivity spectra of α-Et2Me2N[Ni(dmit)2]2.

192

The deviation from the Drude behavior indicates the presence of thecontribution of the interband transitions across band gaps. In fact, broad
maxima corresponding to the humps can be recognized around 4000 cm –1
and around 7000 cm –1 for both polarizations in the conductivity spectra,
which were obtained by the Kramers-Kronig analysis of the reflectancespectra (Fig. 7.5). The contribution of the interband transitions is relatively
small, particularly at room temperature. This is in contrast to the spectralfeatures of the dimeric κ-type BEDT-TTF salts (see Chap. 5) [6], in spiteof the resemblance of the shape of the Fermi surface (Fig. 7.2). From this
it follows that the dimeric character in α-Et2Me2N[Ni(dmit)2]2 is not remarkable from the viewpoint of the electronic structure. In other words,
at least one of the interdimer transfer integrals, such as the spanning
overlap one, is comparable to the intradimer one. The increase in theintensities of the interband transitions with decreasing temperature, on
the other hand, suggests a gradual structural change such that modifies
the character of the electronic structure, and may be related to the relative
displacement of the [Ni(dmit)2] molecules shown by the low temperature
X-ray study [2].
In order to extract the contribution of the intraband transitions (theDrude terms), a curve-fitting analysis of the reflectance spectra was carriedout, on the basis of the Drude-Lorentz dielectric function,

1 or 2ε (ω ) = ε C – ω / ( ω + iω / τ ) + ∑ f j / [(ω j 2 – ω 2) – iγ jω ] .
2
P

2

(7.1)

j=1

The first term in the right-hand side of Eq.(7.1), the Drude term, describes

193

the intraband transitions. The second term stands for the interband transitions corresponding to the conductivity maxima. The reflectivity is related
to dielectric function by the equation,
R = [1 + | ε | – √2(| ε | + Re(ε )) ] / [1 + | ε | + √2(| ε | + Re( ε)) ] .
(7.2)

The obtained parameters are summarized in Table 7.I. The solid curves
in Fig. 7.4 stand for the calculated spectra based on these parameters.
The fitting results are much less reliable for the room temperature data
because of the overdamped (small τ) shape of the plasma edges.
If we consider only the two dominant transfer integrals, t1 and t2,
within the [Ni(dmit)2] layer, the band structure can be explicitly expressedas the form,E(kb, kc ) = ± [t1 2 + 4t2 2 cos 2(b ·kb / 2) ± 4t1t2 cos(c ·kc / 2) cos(b ·k b / 2)]1 / 2 ,(7.3)

where t1 is the intradimer transfer integral and t2 is that corresponds to thespanning overlap. For the lower branches, which concern the Fermi
surface, only the case of E(kb, k c) < 0 should be considered. The situation
described by this model is schematically illustrated in Fig. 7.6. This bandmodel brings about the degeneracy at the boundaries of the first Brillouinzone, in accord with the assumed high symmetry.
The squares of the obtained plasma frequencies, ωp2, can be calculated
from this band structure described by t1 and t2. That is to say,194

Table 7.I. Drude parameters obtained by the DrudeLorentz fit of the reflectance spectra

Polarization T / K εc ωp / eV τ – 1 / eV
// b

// c

295
100
25
295
100
25

4.0 0.51
4.2 0.52
4.1 0.52
1.2 a) 0.67
3.1 0.73
3.1 0.73

0.27
0.077
0.061
0.26
0.082
0.065

a) In the fitting of this spectrum, the near-infraredline-shape used is different from those of the othercases. This results in the small value of ε c.

t1
t2
t1

t2
t1

t2

t2
t1

Fig. 7.6. Schematic display of the two dominant intermolecular interactions in the
conduction layer of α-Et2Me2N[Ni(dmit)2]2. The spanning overlap
corresponds to t 2.

195

∫

(7.4)

∫

(7.5)

(ωp2) b = 2 (e2 / π2h) (2π / a sin β) | (vF )b | dkFc ,(ωp2) c = 2 (e2 / π2h) (2π / a sin β) | (v F)c | dkFb ,where (vF )b and (vF)c are the b- and c-components of the Fermi velocity,
vF, given asvF = h –1 grad k E(t1, t2; kb, kc ) ; E = EF .

(7.6)

Half-filling over the lower two branches, i.e., 1/4-filling for the overall
LUMO band, is assumed so as to satisfy the requirement of stoichiometry.
This determines the Fermi energy EF . By use of these equations, theoptimal set of t1 and t2 which reproduce the observed plasma frequencies
were searched. For example, when the values, (ωp)b = 0.67 eV and (ωp)c= 0.52 eV, at room temperature are used together with the lattice constantsat room temperature, a = 38.95 Å, b = 6.494 Å, c = 13.835 Å and β =99.63 , the transfer integrals are estimated to be t1 = 0.09 eV and t2 =0.11 eV. As was expected from the spectral features, the intra- andinterdimer interactions are comparable to each other.
Before applying this method to the low temperature spectra, theeffect of the structural phase transition [2] should be considered. A
possible interpretation of the low temperature Fermi surface is as follows.
There are two independent [Ni(dmit)2] layers, between which only very
weak interlayer interactions are found. Each layer, having a similar
structural character, forms each two-dimensional band. However, the196difference in the position and width between these bands split the degenerate
Fermi surface into the larger and smaller ones. Moreover, the larger
Fermi surface changes its topology into the one open along the c-axis dueto the reconstruction around the Γ point, while the small one almost
retains its original topology. This situation is depicted in Fig. 7.7, where
the solid and dotted lines show the smaller closed Fermi surface and thelarger open one, respectively, and the doubled periodicity along the b-axis
is omitted for simplicity. The magnetoresistance and SdH results havebeen interpreted in terms of the small Fermi surface [5].
It seems quite natural to regard the observed optical properties at
low temperatures as the average of the large and small Fermi surface. If
this is the case, the analysis of the optical data should yield an intermediateshape of the Fermi surface between the two. By use of the values at 25
K, (ωp)b = 0.73 eV and (ωp)c = 0.52 eV, and the lattice constants at 11 K,
a = 37.499 Å, b = 6.444 Å, c = 13.718 Å and β = 92.69 , the transfer
integrals were estimated to be <t1> = 0.12 eV and <t2> = 0.10 eV fromthe same method as above, where the brackets denote the average over
the two different [Ni(dmit)2] layers. Figure 7.8 shows the band structure
and the Fermi surface based on these average values.
The validity of this estimation can be checked by the comparison
with the results of the SdH and magnetoresistance oscillations. The
cyclotron mass for the small Fermi surface connected by the magnetic
breakdown effect has been reported to be m*c = 4.3 ± 0.2 me , with thecorresponding cross-sectional area of 85.9 % of the first Brillouin zone
[4,5]. The cyclotron mass can be calculated also for the average Fermi
surface based on the optical results, by use of the relation,

197

Fig. 7.8. The average band structure and Fermi surface inferred from the optical data at 25 K.

198

<m*c> = (h 2 / 2π) (dS / dE) ;E = EF ,(7.7)

where S is the cross-sectional area. This yields <m*c> = 4.3 me, whichcoincides with the observed cyclotron mass for the smaller Fermi surface.
Note that the cross-sectional area is assumed to be 100 % in the analysisof the optical data. The optical results would give the cyclotron mass forthe smaller Fermi surface slightly less than <m*c> = 4.3 m e, because thesmaller Fermi surface corresponds to the wider band according to Fig.
7.7. Considering the simplicity of the approximation made for such acomplicated low temperature Fermi surface, this result does not seem to
be inconsistent with the observed cyclotron mass. The origin of thedifference may be attributed to the modified topology of the large Fermi
surface around the Γ point (Fig. 7.7).
Next, the value of the Fermi wavenumber inferred from the angle
dependent magnetoresistance oscillation is considered. For the diameterof the closed portion of the small Fermi surface around the Z point, 0.29
b* along the c-axis, is suggested by the magnetoresistance results [5].
The corresponding value for the average Fermi surface should be slightly
larger, and calculated to be 0.32 b* from the optical results. This result issatisfactory.

7.4. Conclusion

The present results can be regarded as the estimation of the averaged
geometry of the multiple Fermi surfaces emerging from the two independent199

[Ni(dmit)2] layers. The transfer integrals estimated from the low temperature data should be the average of the corresponding ones over the
independent layers. The present study well illustrates the nature of the
optical investigation of the Fermi surface using the intraband transitions.
The intraband transitions are less sensitive to the detailed features of the
Fermi surface as in the low temperature one in this case. Nevertheless,they efficiently reflect the global and overall geometry of the Fermi surface
and always afford the satisfactory estimation of the basic dimensionalityof a metal, whatever the detailed structure of the Fermi surface is.

References

1. R. Kato, H. Kobayashi, H. Kim, A. Kobayashi, Y. Sasaki, T. Mori, H. Inokuchi:
Chem. Lett. 865 (1988).
2. A. Kobayashi, T. Naito and H. Kobayashi: Phys. Rev. B, in press.
3. H. Kobayashi, R. Kato, A. Kobayashi: Synth. Metals 41-43 (1991) 2495.
4. H. Tajima, S. Ikeda, M. Inokuchi, A. Kobayashi, T. Ohta, T. Sasaki, N. Toyota, R.
Kato, H. Kobayashi and H. Kuroda: Solid State Commun. 88 (1993) 605.
5. H. Tajima, S. Ikeda, M. Inokuchi, A. Kobayashi, T. Ohta, T. Sasaki, N. Toyota, R.
Kato, H. Kobayashi and H. Kuroda: Synth. Metals, in press.
6. For example, see: M. Tamura, H. Tajima, K. Yakushi, H. Kuroda, A. Kobayashi,
R. Kato and H. Kobayashi: J. Phys. Soc. Jpn. 60 (1991) 3861.

200

Chapter 8.
Reflectance Spectra of κ-(DMET)2AuBr 2201

8.1. Introduction
An organic π-donor, DMET (dimethyl(ethylenedithio)diselenadithiafulvalene) is the hybrid molecule between TMTSF and BEDT-TTF as
shown in Fig. 8.1. The TMTSF salts are the first family of organic
superconductors [1,2]. They favor a quasi-one-dimensional structure andhave spin-density-wave phase at low temperature [2,3]. In contrast tothem, a variety of two-dimensional molecular pickings are found in the
BEDT-TTF family [4]. The DMET family salts were synthesized withthe expectation that they can fill the gap between the TMTSF and BEDTTTF families [5]. In fact, the DMET family includes both a quasi-onedimensional conductor similar to the TMTSF salts and a two-dimensional
conductor similar to the BEDT-TTF salts [6-12]. The DMET family has
afforded 7 superconductors, for the first time as radical salt based on a
non-symmetric molecule.
The salts, (DMET)2AuBr2, crystallizes at least in two different modifications [13,14]. The crystals of κ-(DMET)2AuBr2 can be categorized
into two groups, r1 and r2, by the difference in the electric properties (Fig.
8.2) [14]. Both show a resistivity maximum around 150 K (= Tmax).
Around this maximum, only the r1 crystals exhibit an "anti-hysteresis"[14]. The r2 crystal is an ambient pressure superconductor with thetransition temperature Tc = 1.9 K, while the r 1 ones are metallic down to
0.5 K without superconductivity [14]. In spite of these differences, thecrystal structures of the r1 and r2 samples are substantially the same. A
crystal of the r2 type is picked up in this study. The crystal structure is
the so-called κ-type one, with the space group P21/a, in which dimers of202

Fig. 8.2. Temperature dependence of electric resistivity of the three types of (DMET)2AuBr2.
The types, p1 and p2, belong to κ-(DMET)2AuBr2.

203

Fig. 8.3. Crystal structure of κ-(DMET) 2AuBr2 projected on to the ab-plane.

204

DMET molecules are arranged to form a two-dimensional packing within
the ab-plane (Fig. 8.3) [14]. In the dimer, molecules are related to eachother by the inversion symmetry.
In this study, the difference between the optical properties above
Tmax and those below Tmax is first focused on. Second, the intensities of
the intraband transitions at low temperature are analyzed in order toestimate the electronic structure. The results are compared with the dHvA
experiments. A comparison with the BEDT-TTF salts is also given.

8.2. Experimental
Single crystals of κ-(DMET)2AuBr2 were provided by Prof. Kikuchi(Tokyo Metropolitan Univ.). The crystals were grown by electrochemical
oxidation of DMET in chlorobenzene with (n-C4H 9)4NAuBr2 as a supporting
electrolyte [13]. The most developed crystal face is the (001) one, with arhombus-like shape. The reflectance spectra were measured on this face,with the light polarization E || a and E || b, i.e., parallel to the two
principal axes within the ab-plane. The crystal axes were identified by
X-ray oscillation photographs. The measurements of the spectra werecarried out by following the method described in Sec. 3.1.

8.3. Results

8.3.1. Spectral Aspects

205

Fig. 8.4. Reflectance spectra of κ-(DMET)2AuBr2.
206

Figure 8.4 shows the reflectance spectra measured at room temperature, 160 K, 80 K, 50 K and 20 K. The overall spectral shapes are
similar to those of the κ-type BEDT-TTF salts [15]; there are remarkabledeviations from the Drude behavior, i.e., the humps around 3000 cm –1
and the prominent peaks at about 1300 cm –1. As we have pointed out inthe cases of dimeric BEDT-TTF salts [15], this non-Drude behavior is
ascribable to the strong interband transitions stemming from the dimeric
structure of the donor molecule. The structures around 3000 cm –1 isattributable to the interband transitions, whose character is close to that of
the intradimer transitions. The peaks at about 1300 cm –1 are considered
to be due to the electron-molecular-vibration coupling corresponding to
C=C stretching modes. This can be induced by the interband transitions.
At room temperature and 160 K, that is, above Tmax, the reflectivity
is constant or slightly decreasing below 1000 cm –1 towards low wavenumber. This suggests that the intraband transitions are considerably suppressed. On the other hand, at 80 K, 50 K and 20 K, the reflectivity
increases below 1000 cm –1 towards low wavenumber, indicating the contribution of the intraband transitions. Figure 8.5 shows the conductivity
spectra obtained by the Kramers-Kronig analysis of the reflectance spectra.
Above Tmax, the conductivity seems to decrease rapidly towards zero whenzero wavenumber is approached, indicating the absence of the intrabandtransitions. Below Tmax, the conductivity value extrapolated to zero wavenumber does not seem to drop to zero. Thus, the optical properties aboveand below Tmax are in remarkable contrast. These features are consistent
with the dc resistivity behavior. This indicates that the change in theelectronic properties through the resistivity maximum concerns not only

207

Fig. 8.5. Conductivity spectra of κ-(DMET)2AuBr2.

208

the states just near the Fermi level but also those lying over the energy
width of, at least, 0.1 eV.
Another noticeable point is the splitting of the broad peaks of theinterband transitions below Tmax (Fig. 8.5). This is observed both for the
E || a and E || b spectra, and is evident also in the reflectance spectra (Fig.
8.4). The splitting width seems to increase with decreasing temperature.
A similar splitting was reported also for α-(BEDT-TTF)2I3 [16]. However,
the origin should be different, because the splitting in α-(BEDT-TTF)2I3
accompanies the transition to the low temperature insulating phase; thesplitting in κ-(DMET)2AuBr2 appears when the material becomes metallic.
Moreover, as is noticed by the first order character of the metal-insulatortransition of α-(BEDT-TTF)2I3, the appearance of the splitting can be
related to the structural change, whereas no structural phase transition has
been found in κ-(DMET)2AuBr2. Further discussion about this splitting
will be given below in connection with the non-metallic resistivity behavior
above Tmax.
8.3.2. Estimation of Band Structure
In order to extract the intensities of the interband transitions at low
temperature, a curve-fitting analysis based on the Drude-Lorentz model,
3 or 4ε (ω ) = ε C – ω / ( ω +iω / τ) + ∑ f j / [(ω j2 – ω2) – iγ j ω ] .
2
P

2

(8.1)

j=1

was carried out. The reflectivity is related to dielectric function by theequation,

209

Table 8.I. Drude parameters at 20 K obtained by theDrude-Lorentz fit of the reflectance spectra.

Polarization
// b
// a

εc

ωp / eVτ – 1 / eV

3.3
3.2

0.72
0.48

0.035
0.061

100
20 K

Reflectance / %80

E // b60
40
20
0E // a2
51
10
Wavenumber / 103 cm – 1

25

Fig. 8.6. Reflectance spectra at 20 K. Solid curves shows the fit to the DrudeLorentz model.

210

R = [1 + | ε | – √2(| ε | + Re(ε)) ] / [1 + | ε | + √2(| ε | + Re(ε)) ] .
(8.2)

The first term in the right-hand side of Eq.(8.1) stands for the constantcontribution from the higher energy transitions outside the experimental
spectral region. The second term, the Drude term, gives the contributionof the intraband transition. The third term describes the contribution ofother transitions: the interband transitions, the e-mv coupling modes. We
empirically expressed the interband transition as a superposition of two
Lorentzian oscillators to describe the split shape of the interband transition.
The relatively small contributions of the intraband transitions, as comparedwith those of the interband ones, result in less satisfactory convergence ofthe fitting, particularly for the 80 K and 50 K data. For the 20 K spectra,
the curve-fitting results were shown as the solid curves in Fig. 8.6, together
with the observed data points. The obtained parameters related to theintraband transitions were compiled in Table 8.I.
The same method which was applied to the case of κ-(BEDT-TTF)2I3(see Chap. 5) [15], is applicable again to the estimation of the band
structure of this material, because of the symmetry of the packing of
dimers. This method is based on the two effective interdimer transfer
integrals, ta and tb , which describe the band structure near the Fermi
surface. The definition of them is given in Fig. 8.7. In this method, theconduction bands are regarded as those formed from the anti-bonding
state of the dimer; the effective Hamiltonian, which is derived by treating
the interdimer transfer integrals as perturbation to the energy splitting due
to the strong intradimer interactions, is used to describe the band structure.

211

b

ta
tb

ta
a

0

Fig. 8.7. Schematic display of the effective interdimer interactions.

b*Energy / eV

0.2

Y

0.1

Γ

X

0
EF

-0.1
X

Γ

Y

Fig. 8.8. The band strcuture and the Fermi surface estimated fromthe optical data at 20 K.

212

a*

The band structure is expressed as,E(ka, k b) = 2 t b cos (b ·kb) ± 4 ta cos(a ·ka / 2) cos(b ·k b / 2) .

(8.3)

When the intraband plasma frequency values at 20 K given in Table 8.I
are used together with the lattice parameters at room temperature, a =11.542, b = 8.238, c = 16.168 Å and β = 90.99 , the two transfer
integrals are estimated to be ta = 0.024 eV and tb = 0.031 eV. The band
structure and the shape of the Fermi surface calculated by use of theseparameters are displayed in Fig. 8.8.
The results are compared with the those of the dHvA experiments[17], in order to check the validity. From the above band structure
inferred from the optical data, the values of the cyclotron mass and thecross-sectional area can be calculated both for the large and small cyclotron
orbits (marked by the α and β, in Fig. 8.8) detected in the dHvA experiments.
The cyclotron masses for the α and β orbits are calculated to be m*αopt =3.4 me and m*βopt = 5.6 me , respectively, from the optical results. The
corresponding values deduced from the dHvA results are, m*αdHvA = 3.8
me and m*β dHvA = 6.0 me [17]. The agreement is satisfactory. The crosssectional area for the β orbit should be 100 % of the first Brillouin zone,
in both results. The α orbit has the cross-sectional area of 26 % of thefirst Brillouin zone according to the dHvA optical results [17], while thecorresponding value from the optical results is 21 %, which is slightly
underestimated probably due to the simplification of the band structureusing the dimer-based model. However, it can be concluded that theband structure derived from the optical data is consistent with the dHvA

213

results upon the whole.
It should be noted that the values of the cyclotron mass inκ-(DMET)2AuBr2 are larger than those of κ-(BEDT-TTF)2I3. There areseveral reports which point out the correlation between the massenhancement and high Tc in a series of BEDT-TTF superconductors [18].
The mass-enhancement involves the enhancement of the density of statesat the Fermi level. As the origin of the mass-enhancement, the role of theelectron correlation has been suggested [18]. The cyclotron mass of
κ-(DMET)2AuBr2 is as heavy as those of a class of BEDT-TTF salts with
Tc = ca. 10 K. However, Tc of κ-(DMET)2AuBr2, 1.9 K, is lower thanthose of the 10 K class BEDT-TTF superconductors. The mass of
κ-(DMET)2AuBr2 is about twice as heavy as that of κ-(BEDT-TTF)2I3,
though Tc of the former is about half as low as that of the latter, 3.6 K.
From these facts, it follows that not only the mass (and density of states)
but also the nature of the molecule is responsible for Tc . This means thatthe argument on the correlation between the mass and Tc cannot be applied
over systems based on different molecules.

8.4. Discussion

Let us return to the temperature dependence of the spectral shape.
Usually the absence of the intraband transitions is evidence for the gap
opening around the Fermi level. On the other hand, from an ESR study,
it is concluded that the spin susceptibility is Pauli-like and insensitive totemperature above 50 K and that the temperature dependence of the ESR

214

line width well corresponds to that of the dc resistivity [19]. It is suggested
from the ESR results that the anomalous non-metallic behavior has adynamical origin such as some unusual molecular vibrations [19]. In
fact, a simple band picture of a metal cannot explain both the absence of
the intraband transitions and the Pauli-like susceptibility. It is, therefore,
likely to think that the intraband transitions disappear due to some dynamical
disorder above Tmax. In this picture, the coherence of the Bloch wavefunctions is considered to be destroyed by strong inelastic scattering due tothe dynamical disorder. Consequently, the mobility and the mean freepath of the charge carriers would be suppressed, without drastic change
in long-time average of the band structure. The conduction in such asituation should resemble to the hopping one. The seeming optical gap
width, below which the optical intraband transitions are suppressed, hasbeen found to amount to about 0.1 eV. Roughly speaking, this is ameasure of the characteristic energy of the disorder or of the inelastic
scattering. The order of magnitude seems to support the suggestion thatthe origin of the dynamical effect may be some molecular vibration [19].
In this connection, the broadening of X-ray diffraction peak aroundTmax is
also suggestive [20].
The splitting of the interband transitions is next focused on. Theinterband transitions occur mainly across the gaps between the lower
doubly split band and the highest branch. Therefore, the splitting in the
spectra observed below Tmax may reflect the splitting of the lower band.
If this is the case, the disappearance of the spectral splitting above Tmax
can be explained as follows. According to the uncertainty principle, thedynamical effect above Tmax removes the physical meaning of the detailed

215

band structure; only coarse-grained features with the scale of about 0.1
eV can remain. The splitting of the lower band is thus obscured, as wellas the spectral splitting.
It should be noted that even the other molecular conductors with noresistivity maximum are close to such a dynamically disordered situation.
This is based on the fact the mean free path of the charge carriers is very
short. However, an exact and explicit treatment of such a dynamicallydisordered state of metals has not been established yet. Thus, detailed
quantitative analysis of the anomalous behavior is not available.

8.5. Conclusion
The optical properties of κ-(DMET)2AuBr2 correlate with the anomalous resistivity behavior. The correlation appears not only in low wavenumber region but also the in the spectral shape in the mid-infrared
region. This suggests the dynamical origin of the non-metallic behaviorof dc resistivity at high temperature. The analysis of the intraband transitions
at low temperature yields the band structure and Fermi surface consistent
with the dHvA results.

References

1. D. Jérome, A. Mazaud, M. Ribault and K. Bechgaard: J. Physique Lett. 41 (1980)
L95-98.
2. K. Bechgaard: Mol. Cryst. Liq. Cryst. 79 (1982) 1.

216

3. D. Jérome and H. J. Schulz: Adv. Phys. 31 (1982) 299.
4. See for example, The Physics and Chemistry of Organic Superconductors, Ed. by
G. Saito and S. Kagoshima (Springer, Heidelberg, 1990); Organic Superconductors:
Synthesis, Structure, Properties and Theory, Ed. by J. M. Williams, J. R. Ferraro,
R. J. Thorn, K. D. Carlson, U. Geiser, H. H. Wang, A. M. Kini and M.-H.
Whangbo (Prentice Hall, Englewood Cliffs, 1992).
5. K. Kikuchi, I. Ikemoto and K. Kobayashi: Synth. Metals 19 (1986) 551.
6. I. Ikemoto, K. Kikuchi, K. Saito, K. Kanoda, T. Takahashi, K. Murata and K.
Kobayashi: Mol. Cryst. Liq. Cryst. 181 (1990) 185.
7. K. Kikuchi, M. Kikuchi, T. Namiki, K. Saito, I. Ikemoto, K. Murata, T. Ishiguro
and K. Kobayashi: Chem. Lett. 931 (1987).
8. Y. Nogami, M. Tanaka, S. Kagoshima, K. Kikuchi, K. Saito, I. Ikemoto and K.
Kobayashi: J. Phys. Soc. Jpn. 56 (1987) 3783.
9. K. Kanoda, T. Takahashi, T. Tokiwa, K. Kikuchi, K. Saito, I. Ikemoto and K.
Kobayashi: Phys. Rev. B 38 (1988) 39.
10. K. Kikuchi, K. Murata, Y. Honda, T. Namiki, K. Saito, T. Ishiguro, K. Kobayashi
and I. Ikemoto: J. Phys. Soc. Jpn. 56 (1987) 3436.
11. K. Murata, K. kikuchi, T. Takahashi, K. Kobayashi, Y. Honda, K. Saito, K.
Kanoda, T. Tokiwa, H. Anzai, T. Ishiguro and I. Ikemoto: J. Mol. Elect. 4 (1988)
173.
12. K. Kikuchi, K. Murata, Y. Honda, T. Namiki, K. Saito, H. Anzai, K. Kobayashi,
T. Ishiguro and I. Ikemoto: J. Phys. Soc. Jpn. 56 (1987) 4241.
13. K. Kikuchi, K. Murata, Y. Honda, T. Namiki, K. Saito, K. Kobayashi, T. Ishiguro
and I. Ikemoto: J. Phys. Soc. Jpn. 56 (1987) 2627.
14. K. Kikuchi, Y. Honda, Y. Ishikawa, K. Saito, I. Ikemoto, K. Murata, H. Anzai, T.
Ishiguro and K. Kobayashi: Solid State Commun. 66 (1988) 405.
15. For example, see: M. Tamura, H. Tajima, K. Yakushi, H. Kuroda, A. Kobayashi,

217

R. Kato and H. Kobayashi: J. Phys. Soc. Jpn. 60 (1991) 3861, and references
cited therein.
16. K. Yakushi, H. Kanbara, H. Tajima, H. Kuroda, G. Saito and T. Mori: Bull.
Chem. Soc. Jpn. 60 (1987) 4251.
17. K. Kanoda, K. Kato, A. Kawamoto, K. Oshima, T. Takahashi, K. Kikuchi, K.
Saito and I. Ikemoto: Synth. Metals 56 (1993) 2309.
18. For example, see: N. Toyota, E. W. Fenton, T. Sasaki and M. Tachiki: Solid State
Commun. 72 (1989) 859.
19. K. Kanoda, T. Takahashi, K. Kikuchi, K. Saito, Y. Honda, I. Ikemoto, K. Kobayashi,
K. Murata and H. Anzai: Solid State Commun. 69 (1989) 415.
20. S. Kagoshima and Y. Nogami: Synth. Metals 27 (1988) A299.

218

Chapter 9.

Concluding Remarks for the Present Thesis219

In this study, the intermolecular transfer integrals have been derived
from the optical data in the five organic π-molecular conductors. Thus, it
has been shown that the band structures and Fermi surfaces are essentially
understandable in the framework of the simple tight-binding model, although there remain some subtle discrepancies between the optical results
and the transport results. This is the first demonstration of the determinationof the Fermi surface by the optical experiments, which can be comparedwith the transport results. The findings of the materials having rather
high symmetries of the molecular arrangements have enabled this study.
It may be surprising that the conduction bands of these materials arecomposed only of one frontier molecular orbital on each molecule. Thishas made the analysis of intraband transitions very easy, in spite of thecomplicated structure of the materials. The strong correlation between
the interband transitions and the dimeric structure also supports this view(see Chap. 5 and Appendix A.). Except for a few examples such as[Pd(dmit)2] salts [1-3], the intramolecular electronic degrees of freedom
play only a minor role in the conduction. Only other kinds of flexibility,
such as molecular vibrations, can contribute to subtle features of theelectronic properties in most cases.
From this point of view, a molecule having degenerate frontier levels
or another molecular level very close to the frontier level is of greatinterest. The carbon cluster, C60, belongs to this class of molecules. The
high symmetry of the molecule is essential for such a situation. Suchkind of molecules have attracted attentions mainly from the magnetic
point of view, because the degeneracy of molecular levels should induce
the local magnetic interactions. However, the donor or acceptor molecule220with high symmetry would afford also some novel features in molecular
conductors, because the electronic flexibility of the molecule can directly
affect the conduction in such a case. This strategy would be also usefulin controlling the dimensionality and the electron correlation by the molecular degrees of freedom. In this connection, the finding of the verysmall three-dimensional Fermi surface in θ-(BEDT-TTF)2I3 [3] (Sec. 4.7)
is a suggestive result.

References

1. H. Tajima, T. Naito, M. Tamura, A. Kobayashi, H. Kuroda, R. Kato, H. Kobayashi,
R. A. Clark and A. E. Underhill: Solid State Commun. 79 (1991) 337.
2. A. E. Underhill, R. A. Clark, I. Marsden, M. Allan, R. H. Friend, H. Tajima, T.
Naito, M. Tamura, H. Kuroda, A. Kobayashi, H. Kobayashi, E. Canadell, S. Ravy
and J. P. Pouget: J. Phys. Condens. Matter 3 (1991) 933.
3. E. Canadell, S. Ravy, J. P. Pouget and L. Brossard: Solid State Commun. 75 (1990)
633.
4. T. Terashima, S. Uji, H. Aoki, M. Tamura, M. Kinoshita and M. Tokumoto: Solid
State Commun. 91 (1994) 595.

221

222

Appendices223

A. Disappearance of Interband Transition
due to Symmetry of Molecular Arrangement

In the following, a rule for disappearance of the optical interband
transition due to crystal symmetry is shown.

The conventional selection

rule for interband transtitions concerns the transition probability at a special
k-point. On the other hand, this rule states that the optical interband
transition disappears over the whole region in the first Brillouin zone, if
the light polarization and the crystal structure satisfy the following conditions:i) The crystal has an n-fold non-symmorphic symmetry operation
along the x-direction (principal axis).
ii) The light polarization is parallel to the x-direction.
iii) All the molecular orbitals (MOs) relevant to the conduction bands
are equivalent to each other.
iv) The number of the MOs relevant to the conduction bands in theunit cell is not more than n.
The non-symmorphic symmetry operation is defined as the symmetry
operation accompanied by the translation by 1 / n of the unit lattice
vector, where n = 2, 3, 4, or 6. This category comprises the screw andglide mirror operations. The condition iii) is satisfied, at least approximately, for wide range of the molecular conductors based on radical
salts. The tight-binding approximation is applied here; the MOs are theWannier functions for the conduction bands. The rule is analogous to theextinction rules in the diffraction crystallography, and illustrates the roleof the interband transition as a probe to the symmetry of molecular packing.

224

The interband optical spectra for the light polarized in the x-direction
with the frequency, ω, is described by the optical conductivity (see Sec.
2.4.),σ x (ω) = (2 / iV h 3)

∑ < j | [H, Px ] | j′ ><j′ | [H, Px ] | j >j, j ′

× f(E j) [1– f(Ej ′ )] (ω / ω j j ′) / (ω 2– ω j j ′ 2+iγ ω) ,(A.1)

where V is the crystal volume, j and j′ are the electronic states of thecrystal, andh ω j j ′ = (E j– Ej ′ ) ,(A.2)

is the energy of the transition from the state j to j'. The function f(E)stands for the Fermi distribution, and,H = ∑ t m m′ a †m a m′ ,

(A.3)

m, m′

and,
Px = e ∑ (R m) x a †m a m ,

(A.4)

m, m′

are the Hamiltonian and the x-component of the electronic dipole, respectively, where tm m′ is the transfer integral between the molecules m and m′,(R m) x the x-component of the position vector pointing the molecular sitem. And a† and a are the creation and annihilation operators of electron,225respectively. These are corresponding to the MOs on the sites m and m ′.
Here it is assumed that each molecule has only one MO (Cf. the conditioniii)). It follow that
[H, P x ] = e ∑ t m m′ (R m, m′ ) x a †m a m′ ,

(A.5)

m, m′

where,(R m, m′ ) x =(R m – R m′ ) x .

(A.6)

Since [H, Px ] is an anti-Hermitian, [H, Px ] j ′ j =0 when [H, Px ] j j ′ .
Suppose that the unit cell contains µΜ molecules. In the reducedzone scheme, this results in µΜ bands. The vertical optical transition
form the M 'th band to M ′ 'th band at wavevector k is prescribed by thematrix element,<Ψ M, k | X |Ψ M ′, k > ≡ < Ψ M, k | [H, P x ] |Ψ M ′, k > .

(A.7)

The wavefunction is given byµΜΨ M, k (r) = ∑ C M, µ, k ψ µ, k (r) ,

(A.8)

µ =1

ψ µ, k (r) = N –1 / 2 ∑ exp(– ik ·R l ) χ µ (r – R l ),
l

226

(A.9)

where N denotes the number of unit cells in the crystal, and R l the
position of the unit cell, l. The summations in Eqs. (A.8) and (A.9) aretaken over the unit cell and the crystal, respectively. The coefficient in
Eq. (A.8), C, is determined by the band calculations. It is assumed herethat all the MOs, χ µ , are equivalent to each other (Cf. the condition iii)).
Sinceexp(– i G · R l ) = 1 ,(A.10)

holds for any reciprocal lattice vector, G,ψ µ, k + G = ψ µ, k ,(A.11)

follows from Eq. (A.9). At the same time, we obtain,ψ µ, k + G = ∑ CM, µ, k + G ψ µ, k ,(A.12)µ

by substituting k with k + G. This means that CM, µ, k + G , as well as CM, µ, k ,is the eigenvector of the Hamiltonian in the (M, k) space. Therefore, we
can label each state by (G, k) as well as by (M, k). This is, namely, theextended zone scheme. The M 'th band in the reduced zone scheme
corresponds to the M 'th Brillouin zone in the extended zone scheme. We
have only to consider up to the µΜ'th zone, because M ≤ µΜ. By use ofCM, µ, k + G instead of CM, µ, k , Eq.(A.8) is expressed as,227Ψ k + G (r) = ∑ C M, µ, k + G ψ µ, k (r).

(A.13)µFrom now on, we are referring to the representation on the basis of
the plane waves propagating along the x-direction,| k > = exp(– i k · x),(A.14)

namely, k x-representation.
Application of Bloch's theorem to Ψ k + G (r) for the x-direction, thatis, the Fourier expansion of Ψ k + G (r) with respect to x, yields,

Ψ k + G (r) = ∑ φ (k + G, G ′x ; y, z) exp[– i(k x+Gx +G ′x ) · x]. (A.15)
G ′x
The matrix element of the Hamiltonian in the kx-representation is givenby,< Kx | H | K′x > = ∑ < Kx | m > < m | H | m ′ > < m ′ | K′ x >m, m′= ∑ < Kx | m > < m ′ | K′x > t m m′ ,(A.16)

m, m′

where | m > stands for the molecular orbital at site m, χ (r – R m), and thesummation is taken over all the molecular pairs in the crystal. It should
be noted that < Kx | m > is the Fourier transform of the molecular orbital,| m >.
Next let us examine the effect of a crystal symmetry operation, α.
Since all the MOs are equivalent to each other, | m ′ > may be obtained by228

operating a certain α on | m >:| m ′ > = | α n m >.

(A.17)

So that, Eq. (A.16) can be rewritten in the form,< Kx | H | K′x > = ∑ t m m′ ∑ < Kx | α n m > <α n m ′ | K′x > , (A.18)(m, m ′ )nwhere (m, m ′ ) runs over all the molecular pairs in which the molecules
are not related by α. The Fourier transform of an MO, | α n m >, is,∫< Kx | α n m > = exp(iKx x) χ m [ρα– n (r – n τ α)] dx ,(A.19)

where ρ α and τ α represent the rotation operator and translation vector ofα, respectively. The integration is taken over the crystal. For a nonsymmorphic operation with translation along the x-direction (Cf. the condition i)), ρ α does not affect the x-component, and τ α is parallel to thex-direction. Taking account of these facts, we obtain,∫< Kx | α n m > = exp[iKx (x + n τα)] χm (r) dx= exp (iKx n τα) < Kx | m > ,(A.20)

for a non-symmorphic operation, α.
Equation (A.18) thus becomes,< Kx | H | K′x >= ∑ t m m′ <Kx | m > < m ′ |K′ x> ∑ exp[i(Kx – K′x ) n τα] ,(m, m ′ )n(A.21)

229

The summation over n is equivalent to the Laue function appears in thetheory of X-ray diffraction, because n runs up to a very large integer
number. It is known that this function vanishes if Kx – K′x ≠ (2π /τα)I , for
any integer I .
Therefore, in the expansion (A.15), only the terms with
G ′x = 2πI / τα ≡ I gx ,(A.22)

is required for the evaluation of the matrix elements. The expansion(A.15) is rewritten in the form,Ψ k + G (r) = ∑ φ (k + G, Igx ; y, z) exp[–i(kx+Gx +Igx ) · x]. (A.23)
I

The matrix element (A.7) is, then, given by<Ψ M, k | X |ΨM ′, k > = <Ψk +G | X | Ψk + G ′ >=∑∫ ∫ φ *(k + G, Ig ; y, z) φ (k + G′, I′ g ; y, z) dy dzxx

I, I ′∫

× exp[i(kx+Gx + Igx ) · x] X exp[–i(kx+Gx ′+I′gx ) · x] dx ,(A.24)

where it is used that X concerns only the x-coordinate. The x-integration
is modified into,230

< k x +Gx+Ig x | X | k x+Gx ′+I ′g x > = ∑ < m | X | m ′>m, m ′

× < k x +Gx+Ig x | m > < m ′ | k x+Gx ′+I ′g x >× ∑ exp[i{Gx –Gx ′ + (I – I ′) g x }n τ α] ,(A.25)

nby an analogous treatment from Eq. (A.16) to (A.21). Again the Laue-likefactor becomes negligible unless there is an integer J satisfying
Gx – Gx ′ + (I – I ′)g x = J g x ,Gx – Gx ′ = J g x ,(A.26)

Here it should be reminded that both Gx and Gx′ are limited within theµΜ'th Brillouin zone, that is,| Gx – G x′ | < 2 π µΜ / ξ ,(A.27)

where ξ is the lattice constant in the x-direction. There is no such integer
J that satisfies (A.26), unless Gx =Gx ′. Hence, (A.25) and thus (A.24) are
negligible, if,µΜ ≤ ξ / τ α ,(A.28)

Gx ≠ Gx′ .

(A.29)

and231

This is equivalent to the condition iv). So that, the interband opticaltransition (M ≠ M ′, i.e., Gx ≠ Gx ′ .) for the x-polarized light disappears at
any k.
In what follows, several comments are given in connection with thisrule.
Let us consider the matrix element of the Hamiltonian (A.21) at thezone boundary, where both K x and K ′x are the x-components of reciprocallattice vectors. It is readily noticed that the matrix element may become
zero in the presence of non-symmorphic symmetry operation. This corresponds to the degeneracy of the bands at the zone boundary.
The rule obtained here implies an analogy to the extinction rules for
X-ray diffraction, i.e., the character of the Laue function. Both are thegeneralization of the law of conservation of the crystal momentum. For asystem with no non-symmorphic symmetry, the present rule reduces tothe simple law of conservation of crystal momentum, i.e., only the verticaltransitions are allowed. It is possible and useful to regard the interband
optical transition as the scattering of the electron Bloch wave by theradiation field. The analogy to the diffraction phenomena turns out to be
obvious in this way. For instance, the extinction of the (1 0 0) diffraction
corresponds to the disappearance of the interband transitions with Gx –Gx′ = 2π / ξ.
As anticipated from this analogy, it is possible to show a similar rule
for multiple primitive cell systems, such as body-centered and face-centeredlattices, because these systems have the symmetry operations with translations shorter than the lattice constants. However, if the band structure is
calculated for the corresponding primitive cell, such a rule is trivial.

232

It is interesting to rewrite the matrix element of the transition moment(A.7) in terms of the structure factor and the phase factor (the Laue
function). The calculation is straightforward and we obtain,<Ψ M, k | X |Ψ M ′, k > = ∑G″∑ < k + G + G ″ | X | k + G ′ + G ″′ >G ″′× φ *k +G ,G ″ φ k + G ′ , G ″′=∑∑ < m | X | m′>< k + G +G ″ | m > < m′ | k +G ′ +G ″′>m, m ′

G ″, G ″′× φ*k +G ,G ″ φ k + G ′ , G ″′
=

∑′
m≠m ′

∑ t m m′ (R m – R m′ ) xm′× ∑ < k + G + G ″ | m > < m′ | k +G ′ + G ″′> φ*k +G ,G ″ φ k + G ′ , G ″′G ″, G ″′× ∑ exp[i(G + G ″ – G ′– G ″′ ) · R l ].

(A.30)lwhere the Fourier coefficient φ k +G ,G ″ is defined by,Ψk + G (r) = ∑ φk +G ,G ″ exp[-i(k + G + G ″) · r].

(A.31)G″

Thus the matrix element can be expressed in terms of the factors
related to the intermolecular interactions (t), structure factors (Fouriertransforms) and the Laue function.
The present discussion is based on the assumption that the conduction
233

bands are formed from equivalent MOs. It is easily found that the interband
transition never disappears if two or more MOs are relevant to the conduction bands, as in [Pd(dmit)2] salts. Furthermore, the rule can be broken
by strong short-range electron correlation. In the presence of very large
on-site Coulomb repulsion, the state of a molecule distinctly depends on
the number of electrons on the molecule. Therefore, the equivalent MO
picture no longer holds, even if the MOs themselves are equivalent. In
other words, the present rule is useful to probe not only the symmetry but
also the composition of the conduction bands and the effect of strong
correlation.

B. Interplay of Interband Transitions
and Electron-molecular-vibration (e-mv) Coupling

In Sec. 5.3.1, the role of the e-mv coupling peaks as a probe to theinterband transitions has been mentioned from an experimental point ofview. Here, some general features of the e-mv coupling are briefly
discussed.
The origin of the e-mv coupling is the dependence of the relevantmolecular level on the molecular conformation. In other words, it is theinteraction between the occupation of the MO and the normal modes ofthe molecular vibration. A Hamiltonian describing this interaction is,
H e-mv = ∑ ∑ g v a†l, m a l, m (b†l, m, v + b l, m, v ) ,
l, m

v

234

(B.1)

where g v is the linear coupling constant for the normal mode, v, a†l, m a l, m
denotes the number of electrons on the m 'th molecule in the l 'th unit
cell, and the operator, b†l, m, v + b l, m, v is the second-quantization expression
for the normal mode coordinate. It is well known that the major contributions arise from the (nearly) totally symmetric modes, which are infrared
inactive. With the unperturbed Hamiltonian, He + H mv, the Hamiltonianof the system is given as, H = He + Hmv + He-mv, where He and Hmv standfor the unperturbed electronic and vibrational states, respectively.
Let us look at the role of the intraband transitions here, before
turning to that of the interband transitions. As mentioned in Sec. 2.3., theintraband transitions are the collective transverse motions of the electrons.
Such motions are accompanied by no change in distribution of electrons
over the molecules, because it corresponds to a longitudinal mode. In
other words, a continuous flow of electronic current does not involve anychange in charge density distribution, as far as the principal axis components
concern. Therefore, the intraband transitions do not contribute to thee-mv coupling, in principle. One the other hand, the distinction between
the transverse and longitudinal motions loses its physical meaning for theinterband transitions.
The e-mv coupling with regard to the intermolecular charge-transfer
excitations have been treated mainly on the basis of the dimer model [1],or similar local models [2]. Since dimeric packing is often found inmolecular crystals, it is convenient to refer to such a framework. However,from the viewpoint of the conduction band electrons, it is not necessarily
efficient to stick to the treatment based on an isolated dimer or to the
localized picture.

235

Generally speaking, the e-mv coupling should be regarded as theinterference between the continuum spectrum of the interband transitions
and the discrete one of the vibrational excitation. A general theory by
Fano [3] is available for such kind of problem, and this phenomenon is
known as the Fano resonance. For simplicity, we refer only to the case of
the interference between one discrete level and one continuum.
The discrete level is given in terms of the diagonal elements of Hmv,as,< φ | Hmv | φ > = Eφ .

(B.2)

The continuum spectrum of purely electronic excitations is described by
He, that is,<ψ E ″ | He | ψ E ′ > = E′ δ(E ″ – E′) .

(B.2)

where the continuum states, | ψ E ′ >, are normalized with respect to energy
as noted by the δ-function. The cross terms arise from He-mv:<ψ E ′ |He-mv | φ > = He-mv (E′) .

(B.3)

The eigenstates of the total Hamiltonian, H, should have the form,

Ψ E = aφ (E) +∫ b (E) ψ d E ′ .
E′

E′

(B.4)

where E is the corresponding eigenvalues and aφ and b E ′are the coefficients.

236

The solution of this eigenvalue problem is given by Fano [3]. The resultsare,aφ (E) = sin ∆ (E) / πHe-mv (E) ,(B.5)

b E′(E) = H e-mv (E′) sin ∆(E) P / π (E – E′)He-mv (E)– δ(E– E′) cos ∆ (E),

(B.6)

where P means that Cauchy's principal value should be taken when integrated, and,tan ∆ (E) = – π | H e-mv (E) | 2 / [E – E φ + F(E)] ,F(E) = P

∫ |H

e-mv

(E′) | 2 dE′ / (E – E ′) .

(B.7)

(B.8)

Let H ′ be the interaction between the system and the radiation field.
The transition moment matrix corresponding the transition from the initialstate, |i >, to | Ψ E> is,<Ψ E | H′ | i > = aφ *(E) < φ | H′ | i >∫+ b E ′∗(E) <ψ E ′ | H′ | i >dE ′. (B.9)
The first term vanishes, because the direct transition to | φ > is notallowed. From the second term and Eq. (B.6), we obtain,237

<Ψ E | H′ | i > = [sin ∆(E) /π He-mv*(E)]×P∫ < φ |He-mv

| ψ E ′ ><ψ E ′ | H′ | i > dE′/(E – E′)– cos ∆(E) <ψE | H′ | i > .

(B.10)The second term corresponds to the interband transitions to the purely
electronic excited states | ψE >. On the other hand, the first term describes
the transitions to the vibrational state, | φ >, which are forbidden in thecase of no perturbation. Clearly, this stems form the presence of He-mv,
and corresponds to the appearance of the e-mv peaks in the optical spectrum.
The formulation of the transition moment taking account of the band
structure is a too involved subject to be treated here explicitly in detail.
Rather, two significant characteristics emerging from the above discussion
are pointed out in the following. First, the divergence of tan ∆ (E) occurs
at the energy which satisfies, E = Eφ – F(E). Since F(E) can be regarded
as independent of E over a sufficiently narrow range, the shift of resonanceenergy is approximately F(E). According to the expression of F(E), Eq.
(B.8), this indicates that the shift of an e-mv peak is nearly proportional
to the square of the coupling constant, g 2, which is consistent with thedimer case [4]. Second, the first term in (B.10) is linear to <ψ E ′ | H′ | i>.
This means the linear dependence of the intensity of the e-mv peaks onthat of the interband transitions. This character enables us to judge theintensity of the interband transitions by that of the e-mv peaks, as pointed
out in Sec. 5.3.1.

238

References

1. M. J. Rice: Solid State Commun. 31 (1979) 93.
2. A. Painelli and A. Girlando: J. Chem. Phys. 84 (1986) 5655.
3. U. Fano: Phys. Rev. 124 (1961) 1866.
4. M. Meneghetti and C. Pecile: J. Chem. Phys. 84 (1986) 4149.

239

240

List of Publications

2411. Temperature Dependence of the Reflectance Spectra of BEDT-TTF Polyhalides.
K. Yakushi, H. Tajima, H. Kanbara, M. Tamura, H. Kuroda, G. Saito, H. Kobayashi,
R. Kato and A. Kobayashi
Physica 143 B(1-3), 463-467 (1986).
2. Optical Study on Bis(propylenedithio)tetrathiafulvalenium (BPDT-TTF) Salts.
K. Yakushi, H. Tajima, T. Ida, M. Tamura, H. Hayashi, H. Kuroda, A. Kobayashi,
R. Kato and H. Kobayashi
Synth. Metals, 24(4), 301-309 (1988).
3. Temperature Dependence of the Polarized Reflectance Spectra of the θ-type ofBis(ethylenedithio)tetrathiafulvalenium Triiodide θ-(BEDT-TTF)2I3: Estimationof Band Parameters.
M. Tamura, K. Yakushi, H. Kuroda, A. Kobayashi, R. Kato and H. Kobayashi
J. Phys. Soc. Jpn., 57(9), 3239-3247 (1988).
4. Reflectance Spectra of β-, θ-, κ-(BEDT-TTF)2I3, and β´´- and β´-(BEDT-TTF)2ICl2:
Relation between the Inter-band Transition and the Dimeric Structure.
H. Kuroda, K. Yakushi, H. Tajima, A. Ugawa, M. Tamura, Y. Okawa and A.
Kobayashi
Synth. Metals, 27(1-2), A491-A498 (1988).
5. New Molecular Conductors, α- and β-(EDT-TTF)[Ni(dmit)2]. Metal with
Anomalous Resistivity Maximum vs. Semiconductor with Mixed Stack.
R. Kato, H. Kobayashi, A. Kobayashi, T. Naito, M. Tamura, H. Tajima and H.
KurodaChem. Lett., 1989(10), 1839-1842.
6. The Reflectance Spectra of (BEDT-TTF)5Hg3Br11 and (BEDT-TTF)HgBr3. The
Estimation of Effective On-Site Coulomb Interaction.
H. Tajima, M. Tamura, H. Kuroda, T. Mori, H. Inokuchi
Bull. Chem. Soc. Jpn., 63(2), 538-543 (1990).
7. Magnetization and Meissner Effect of Organic Superconductor θ-(BEDT-TTF)2I3.
M. Tamura, H. Tajima, H. Kuroda and M. Tokumoto
J. Phys. Soc. Jpn., 59(5), 1753-1759 (1990).
8. Reflectance Spectra of [M(dmit)2] Salts.
H. Tajima, M. Tamura, T. Naito, A. Kobayashi, H. Kuroda, R. Kato, H. Kobayashi,
R. A. Clark and A. E. Underhill
Mol. Cryst. Liq. Cryst., 181, 233-242 (1990).
9. Structural and Electronic Properties of Cs[Pd(dmit)2]2.
A. E. Underhill, R. A. Clark, I. Marsden, M. Allan, R. H. Friend, H. Tajima, T.
Naito, M. Tamura, H. Kuroda, A. Kobayashi, H. Kobayashi, E. Canadell, S.
Ravy and J. P. Pouget
J. Phys. Condens. Matter, 3(8), 933-954 (1991).
10. Fermi Level Inversion in Strongly Dimerized [Pd(dmit)2] Salts.
H. Tajima, T. Naito, M. Tamura, A. Kobayashi, H. Kuroda, R. Kato, H. Kobayashi,

242

R. A. Clark and A. E. Underhill
Solid State Commun., 79(4), 337-341 (1991).
11. Fermi Surface Study of Two-dimensional Organic Metals (BEDT-TTF)2X.
M. Tokumoto, A. G. Swanson, J. S. Brooks, C. C. Agosta, S. T. Hannahs, N.
Kinoshita, H. Anzai, M. Tamura, H. Tajima, H. Kuroda, A. Ugawa and K.
Yakushi
Synth. Metals, 42(3), 2459-2466 (1991).
12. Optical Study on [M(dmit)2] Salts.
H. Tajima, T. Naito, M. Tamura, A. Takahashi, S. Toyoda, A. Kobayashi, H.
Kuroda, R. Kato, H. Kobayashi, R. A. Clark and A. E. Underhill
Synth. Metals, 42(3), 2417-2421 (1991).
13. Reflectance Spectra of Some Two-dimensional Organic Metals based on BEDTTTF and [Ni(dmit)2].
M. Tamura, R. Masuda, T. Naito, H. Tajima, H. Kuroda, A. Kobayashi, K.
Yakushi, R. Kato, H. Kobayashi, M. Tokumoto, N. Kinoshita and H. Anzai
Synth. Metals, 42(3), 2499-2502 (1991).
14. An Organic Radical Ferromagnet.
M. Kinoshita, P. Turek, M. Tamura, K. Nozawa, D. Shiomi, Y. Nakazawa, M.
Ishikawa, M. Takahashi, K. Awaga, T. Inabe and Y. Maruyama
Chem. Lett., 1991(7), 1225-1228.
15. Discovery of a Quasi-1D Organic Ferromagnet, p-NPNN.
M. Takahashi, P. Turek, Y. Nakazawa, M. Tamura, K. Nozawa, D. Shiomi, M.
Ishikawa and M. Kinoshita
Phys. Rev. Lett., 67(6), 746-748 (1991); Erratum 69, 1290
16. Bulk Ferromagnetism in the β-Phase Crystal of the p-Nitrophenyl Nitronyl Nitroxied
Radical.
M. Tamura, Y. Nakazawa, D. Shiomi, K. Nozawa, Y. Hosokoshi, M. Ishikawa,M. Takahashi and M. Kinoshita
Chem. Phys. Lett., 186(4,5), 401-404 (1991).
17. Reflectance Spectra of κ-(BEDT-TTF)2I3: Electronic Structure of Dimeric BEDTTTF Salts.
M. Tamura, H. Tajima, K. Yakushi, H. Kuroda, A. Kobayashi, R. Kato and H.
Kobayashi
J. Phys. Soc. Jpn., 60(11), 3861-3873 (1991).
18. Low-temperature Magnetic Properties of the Ferromagnetic Organic Radical, pNitrophenyl Nitronyl Nitroxide.
Y. Nakazawa, M. Tamura, N. Shirakawa, D. Shiomi, M. Takahashi, M. Kinoshita
and M. Ishikawa
Phys. Rev. B, 46(14), 8906-8914 (1992).
19. Ferromagnetic Intermolecular Interaction in an Organic Radical 3-Quinolyl
Nitronyl Nitroxide (3-QNNN).

243

T. Sugano, M. Tamura, M. Kinoshita, Y. Sakai and Y. Ohashi
Chem. Phys. Lett., 200(3), 235-240 (1992).
20. Magnetic Properties of an Organic Biradical, m-BNN: m-Phenylene Bis(α-Nitronyl
Nitroxide).
D. Shiomi, M. Tamura, H. Sawa, R. Kato and M. Kinoshita
J. Phys. Soc. Jpn., 62(1), 289-300 (1993).
21. Growth of Large Isomeric YBa2Cu3Ox Single Crystals from Coexisting Region ofSolid with Melt in Y2Cu3 Crucibles.
H. Asaoka, H. Takei, Y. Iye, M. Tamura, M. Kinoshita and H. Takeya
Jpn. J. Appl. Phys. Part 1, 32(3A), 1091-1096 (1993).
22. Magnetic Properties and Crystal Structures of Some Nitronyl Nitroxide Derivatives.
K. Nozawa, Y. Hosokoshi, D. Shiomi, M. Tamura, N. Iwasawa, H. Sawa, R.
Kato and M. Kinoshita
Synth. Metals, 56(2,3), 3323-3326 (1993).
23. Ferro- and Antiferromagnetic Intermolecular Interactions in the Aromatic
Heterocyclic Derivatives of α-Nitronyl Nitroxide.
T. Sugano, M. Tamura and M. Kinoshita
Synth. Metals, 56(2,3), 3305-3310 (1993).
24. Novel Magnetic Interactions in Organic Polyradical Crystals Based on Nitronyl
Nitroxide.
D. Shiomi, M. Tamura, H. Sawa, R. Kato and M. Kinoshita
Synth. Metals, 56(2,3), 3279-3284 (1993).
25. Searching for Spontaneous Magnetic Order in an Organic Ferromagnet. µSR
Studies of β-Phase p-NPNN.
L. P. Le, A. Keren, G. M. Luke, W. D. Wu, Y. J. Uemura, M. Tamura, M.
Ishikawa and M. Kinoshita
Chem. Phys. Lett., 206(1-4), 405-408 (1993).
26. Preparation and Physical Properties of an Alloyed (DMe-DCNQI)2Cu with Fully
Deuterated DMe-DCNQI (DMe-DCNQI=2,5-Dimethyl-N,N'-dicyanoquinonediimine).
R. Kato, H. Sawa, S. Aonuma, M. Tamura, M. Kinoshita and H. Kobayashi
Solid State Commun., 85(10), 831-835 (1993).
27. Weak Ferromagnetism and Magnetic Anisotropy in Cu Salt of Fully Deuterated
DMe-DCNQI, (DMe-DCNQI-d8)2Cu.
M. Tamura, H. Sawa, S. Aonuma, R. Kato, M. Kinoshita and H. Kobayashi
J. Phys. Soc. Jpn., 62(5), 1470-1473 (1993).
28. Novel Electronic States of Partially Deuterated (DMe-DCNQI)2Cu.
H. Sawa, M. Tamura, S. Aonuma, R. Kato, M. Kinoshita and H. Kobayashi
J. Phys. Soc. Jpn., 62(7), 2224-2228 (1993).
29. Metallic and Superconducting Salts Based on an Unsymmetrical π-Donor
Dimethyl(ethylenedithio)tetraselenafulvalene (DMET-TSeF).

244

R. Kato, S. Aonuma, Y. Okano, H. Sawa, M. Tamura, M. Kinoshita, K. Oshima,
A. Kobayashi, K. Bun and H. Kobayashi
Synth. Metals, 61(3), 199-206 (1993).
30. Magnetic Quantum Oscillations in Low-dimensional Organic Conductors (BEDTTTF)2X.
M. Tokumoto, A. G. Swanson, J. S. Brooks, C. C. Agosta, S. T. Hannahs, N.
Kinoshita, H. Anzai, M. Tamura, H. Tajima, H. Kuroda, A. Ugawa and K.
YakushiPhysica B 184 (1-4), 508-512 (1993).
31. Magnetic Energy Gap in the Crystal of an Organic Biradical, m-BNN: m-Phenylene
Bis(α-Nitrophenyl Nitroxide).
D. Shiomi, H. A. Katori, T. Goto, H. Sawa, R. Kato, M. Tamura and M. Kinoshita
Mol. Cryst. Liq. Cryst., 232(1), 109-116 (1993).
32. Ferro- and Antiferro-magnetic Interactions and Crystal Structures of Some Radicals
Based on Nitronyl Nitroxide.
T. Sugano, M. Tamura, T. Goto, R. Kato, M. Kinoshita, Y. Sakai, Y. Ohashi, M.
Kurmoo and P. Day
Mol. Cryst. Liq. Cryst., 232(1), 61-68 (1993).
33. Magnetic Properties of Phenyl Nitronyl Nitroxides.
M. Tamura, D. Shiomi, Y. Hosokoshi, N. Iwasawa, K. Nozawa, M. Kinoshita, H.
Sawa and R. Kato
Mol. Cryst. Liq. Cryst., 232(1), 45-52 (1993).
34. Growth, Structure and Properties of Y2Cu2O5 Single Crystal.
W.-J. Jang, M. Hasegawa, T.R. Zhao, H. Takei, M. Tamura and M. Kinoshita
J. Crystal Growth, 141(1,2), 153-158 (1994).
35. Analysis of de Haas-van Alphen Oscillations and Band Structure of an Organic
Superconductor, θ-(BEDT-TTF)2I3.
M. Tamura, H. Kuroda, S. Uji, H. Aoki, M. Tokumoto, A. G. Swanson, J. S.
Brooks, C. C. Agosta and S. T. Hannahs
J. Phys. Soc. Jpn., 63(2), 615-622 (1994).
36. Weak Ferromagnetism in (DBr-DCNQI)2Cu (DBr-DCNQI=2,5-Dibromo-N,N'dicyanoquinonediimine).
M. Tamura, H. Sawa, Y. Kashimura, S. Aonuma and M. Kinoshita
J. Phys. Soc. Jpn., 63(2), 425-428 (1994).
37. Magnetic Study of Metal-Insulator-Metal Transitions in (DMe-DCNQI-α,α´d2)2Cu.
M. Tamura, H. Sawa, S. Aonuma, R. Kato and M. Kinoshita
J. Phys. Soc. Jpn., 63(2), 429-432 (1994)
38. Magnetic Structure of the 4,4,4´,4´,5,5,5´,5´-Octamethyl-2,2´-m-phenylenebis(4,5dihydroimidazol-1-oxyl 3-Oxide) Biradical: Quantum Spin Effect of S=1 Species
Associated with Structral Change.

245

D. Shiomi, M. Tamura, H. Aruga Katori, T. Goto, A. Hayashi, Y. Ueda, H.
Sawa, R. Kato and M. Kinoshita
J. Mater. Chem., 4(6), 1915-1920 (1994).
39. Three-dimensional Fermi Surface in θ-(BEDT-TTF)2I3.
T. Terashima, S. Uji, H. Aoki, M. Tamura, M. Kinoshita and M. Tokumoto
Solid State Commun., 91(8), 595-598 (1994).
40. Magnetic Properties and Crystal Structure of the p-Fluorophenyl Nitronyl Nitroxide
Radical Crystal: Ferromagnetic Intermolecular Interactions leading to a Three
dimensional Network of Ground Triplet Dimeric Molecules.
Y. Hosokoshi, M. Tamura, M. Kinoshita, H. Sawa, R. Kato, Y. Fujiwara and Y.
Ueda
J. Mater. Chem., 4(8), 1219-1226 (1994).
41. Organic Radical Crystals, α-Nitronyl Nitroxide Family: High-field Magnetization
Study
Y. Hosokoshi, M. Tamura, D. Shiomi, N. Iwasawa, K. Nozawa, M. Kinoshita, H.
Aruga Katori and T. Goto
Physica B, 201, 497-499 (1994).
42. Two-dimensional Ferromagnetic Intermolecular Interactions in Crystals of the
p-Cyanophenyl Nitronyl Nitroxide Radical.
Y. Hosokoshi, M. Tamura, H. Sawa, R. Kato and M. Kinoshita
J. Mater. Chem., in press.
43. Coexistence of One- and Three-dimensional Fermi Surfaces and Heavy Cyclotron
Mass in Molecular Conductor (DMe-DCNQI)2Cu.
S. Uji, T. Terashima, H. Aoki, J. S. Brooks, R. Kato, H. Sawa, S. Aonuma, M.
Tamura and M. Kinoshita
Phys. Rev. B, in press.
44. Magnetic Properties and Crystal Structures of 2-Hydro and 2-Halo Nitronyl
Nitroxide Radical Crystals.
Y. Hosokoshi, M. Tamura, K. Nozawa, S. Suzuki, M. Kinoshita, H. Sawa and R.
Kato
Synth. Metals, in press.
45. Ferromagnetic Resonance and Nonlinear Absorption in p-NPNN.
K. Oshima, H. Kawanoue, Y. Haibara, H. Yamazaki, K. Awaga, M. Tamura andM. Kinoshita
Synth Metals, in press.
46. Fermi Surface and Cyclotron Mass in (DMe-DCNQI)2Cu System.
S. Uji, T. Terashima, H. Aoki, J. S. Brooks, R. Kato, H. Sawa, S. Aonuma, M.
Tamura and M. Kinoshita
Synth. Metals, in press.
47. Charge Transfer and Reentrant Phenomenon in Deuterated (DMe-DCNQI)2Cu.
H. Sawa, M. Tamura, S. Aonuma, M. Kinoshita and R. Kato

246

Synth. Metals, in press.
48. Crystal Structure and Physical Properties of Molecular Metal (DI-DCNQI)2Cu(DI-DCNQI=2,5-Diiodo-N,N'-dicyanoquinonediimine).
R. Kato, Y. Kashimura, H. Sawa, S. Aonuma, M. Tamura, M. Kinoshita, H.
Takahashi and N. Môri
Synth. Metals, in press.
49. Magnetic Properties of DCNQI-Cu Systems.
M. Tamura, H. Sawa, Y. Kashimura, S. Aonuma, R. Kato, M. Kinoshita and H.
Kobayashi
Synth. Metals, in press.
50. The Shubnikov-de Haas Oscillations and a Small Closed Orbit in θ-(BEDT-TTF)2I3.
T. Terashima, S. Uji, H. Aoki, M. Tamura, M. Kinoshita and M. Tokumoto
Synth. Metals, in press.
51. Reexamination of Angle Dependent Magnetoresistance Oscillation in θ-(BEDTTTF)2I3.
T. Terashima, S. Uji, H. Aoki, M. Tamura, M. Kinoshita and M. Tokumoto
Synth. Metals, in press.
52. Angular Magnetoresistance Oscillations and Magnetic Break-down in θ-(BEDTTTF)2I3.
S. J. Klepper, G. J. Athas, J. S. Brooks, M. Tokumoto, T. Kinoshita, M. Tamura
and M. Kinoshita
Synth. Metals, in press.
53. Coexistence of Superconductivity with CDW (or SDW): Ambient-pressure
Superconductivity Based on [Ni(dmit)2], α-EDT-TTF[Ni(dmit)2]
H. Tajima, M. Inokuchi, S. Ikeda, M. Arifuku, T. Naito, M. Tamura, T. Ohta, A.
Kobayashi, R. Kato, H. Kobayashi and H. Kuroda
Synth. Metals, in press.
54. Charge Transfer Controlled Phase Transition in a Molecular Conductor, (DMeDCNQI)2Cu – Doping Effect –.
H. Sawa, M. Tamura, S. Aonuma, M. Kinoshita and R. Kato
J. Phys. Soc. Jpn.., in press.
55. Fermi Surface and Absence of Additional Mass Enhancement near the Insulating
Phase in (DMe-DCNQI)2Cu.
S. Uji, T. Terashima, H. Aoki, R. Kato, H. Sawa, S. Aonuma, M. Tamura and M.
Kinoshita
Solid State Commun., in press.
56. Magnetism of the β-Phase p-Nitrophenyl Nitronyl Nitroxide Crystal.
K. Takeda, K. Konishi, M. Tamura and M. Kinoshita
Mol. Cryst. Liq. Cryst., in press.
57. (DCNQI)2Cu: Variety of Magnetism Arising from pπ-d Interactions.
M. Tamura, H. Sawa, Y. Kashimura, S. Aonuma, R. Kato and M. Kinoshita

247

Mol. Cryst. Liq. Cryst., in press.
58. Magnetic Interactions in the Crystals of α- and β-Phases of 2-Hydro Nitronyl
Nitroxide and Related Compounds.
Y. Hosokoshi, M. Tamura, K. Nozawa, S. Suzuki, H. Sawa, R. Kato and M.
Kinoshita
Mol. Cryst. Liq. Cryst., in press.
59. Ferromagnetic Resonance in p-NPNN below 1 K.
K. Oshima, Y. Haibara, H. Yamazaki, K. Awaga, M. Tamura and M. Kinoshita
Mol. Cryst. Liq. Cryst., in press.
60. Enhanced Magnetic Susceptibility of (DI-DCNQI)2Cu.
M. Tamura, Y. Kashimura, H. Sawa, S. Aonuma, R. Kato and M. Kinoshita
Solid State Commun., 93 (1995) 585-588.

248