Journal of the Physical Society of Japan 83, 024715 (2014)
http://dx.doi.org/10.7566/JPSJ.83.024715

Evolution of the Impurity Band to Diamond-Like Valence Bandsin Boron Doped Diamond
Takashi Inushima1+, Yuichi Ota1,2, and Hiromu Shiomi321Department of Electronics, Tokai University, Hiratsuka, Kanagawa 259-1292, JapanTokyo Metropolitan Industrial Technology Research Institute, Koto, Tokyo 135-0064, Japan
3
Sumitomo Electric Industries, Ltd., Itami, Hyogo 664-0016, Japan(Received September 19, 2013; accepted December 18, 2013; published online January 30, 2014)

We present the absorption coefﬁcient and the refractive index of boron doped diamond having an impurity band at
0.07 eV above the valence band maximum and compare them with those obtained by ﬁrst principles calculation using a
C63B supercell model containing 1.57% boron. These optical constants are in good accordance with each other,indicating that the impurity band that forms at 2p excited states of impurity boron becomes top of the valence bands inmetallic condition. Based on this result we present a model of the evolution of boron atoms from isolated impurity toconstituent atoms in the boron doped diamond, where the valence electrons of boron become to have k dependence andform the top of the valence bands of the C63B supercell diamond.

1.

Introduction

Since the discovery of superconductivity in boron-doped
diamond,1) the distribution of impurity boron in diamondhas been a key issue for the understanding not only ofsuperconductivity but also of the physics of heavily doped
semiconductors. The physical properties of lightly doped
diamond are well understood by effective mass approximation.2) When the doping concentration NB exceeds a criticalvalue of Nc  3  1020 cm¹3, Mott transition occurs and the
diamond becomes a metal.3) It is understood that when acarrier concentration of a semiconductor is close to Nc , thecarriers are scattered by defects and the mean free path
becomes shorter, and ﬁnally Anderson localization isestablished.4) Actually the superconducting diamond is close
to the localization limit of kF l ’ 1 (kF is the Fermi
wavenumber and l is the carrier mean free path).5)
However, a study by an angle-resolved photoemission
spectroscopy (ARPES) revealed that the superconducting
diamond had clear solid diamond valence bands and its Fermi
energy shifted according to the increase of doping density.6)
This result is understood to show that the impurity states areabsent or located above the Fermi level.7) Interesting enough,the observed bands are reproduced well by a supercell-model
calculation where boron is a constituent atom.8,9) On theother hand, Shirakawa et al. claimed that boron should be
randomly distributed in diamond and form a disorderedpotential.10) They questioned the supercell model which
could not explain the continuous evolution of the impurity
states according to the increase of NB ; from the separate levelto the rigid band shift of the Fermi energy in the valenceband. But it should be stressed that the impurity band concept
applied in conventional semiconductors like Si and Ge does
not work in diamond.11)
In our previous papers we reported that when NB exceeds4  1018 cm¹3, the 2p wave functions of impurity boron
begin to overlap, and an impurity band forms not at the 1s
ground level but at the 2p excited states, which is located³0.07 eV above the valence band maximum. Simultaneously,
the Fermi level rises to this impurity band and the relaxation
time of the excited holes decreases drastically.12,13) Takeuchi
et al. obtained similar results of the impurity band structurewith the carrier concentration close to and below Nc using
a total photoelectron emission yield spectroscopy.14) Thisis a remarkable feature of diamond and it is important tounderstand how the impurity band evolves into solid bands
in boron doped diamond.
In this paper, we present the absorption coefﬁcient and therefractive index of boron doped diamond with an impurity
band, and show that they are in good accordance with thoseobtained by ﬁrst principles calculation using a supercell
model of C63B. Then we develop the idea presented in ourprevious works and show the boron concentration dependence of the Fermi level, and the impurity band evolution
caused by hybridization of boron atoms and carbon atoms.
2.

Experimental Details

The home-epitaxial ﬁlms used in the experiments weregrown by the microwave-enhanced chemical vapor deposition method on an off-oriented Ib synthesized diamondsurface, which was (001) with 7° off-angle toward h100i. H2and CH4 were used as source gases with a ratio of CH4/H2 of
2%. B2H6 (10 ppm diluted by H2) was used as a doping gas.
The ratio of B/C in the reaction gas phase was used to
identify the samples, which was twice the value of thatused in the previous papers, where the ratio of B2H6/CH4
was used.15–17) The boron concentration determined by asecondary ion mass spectroscopy was 1 to 10  1019 cm¹3in the 10,000 ppm B/C sample.18) The boron-doped layer
thickness was 1.6 µm. There was a clear change ofconductivity from the valence band to the variable range
hopping (VRH) when NB ’ 4  1018 cm¹3.16) From theconductivity and Hall measurements, the B/C = 10,000
ppm sample had NB ¼ 2  1019 cm3,12) which was one orderof magnitude smaller than Nc .
Temperature dependence of the absorption and reﬂection
spectra was measured using a synchrotron beam at UVSOR
II of Institute for Molecular Science. In the infrared region
BL-6B was used together with a DTGS detector.19) In visible
and UV regions the reﬂection spectrum was obtained by
BL-7B, where a 3 m focal length Perkin-Elmer monochrometer and a silicon photodiode were employed at 8° to thenormal incident conﬁguration. The resolution was 0.3 meV at
4 eV.

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J. Phys. Soc. Jpn. 83, 024715 (2014)

T. Inushima et al.

Fig. 1. (Color online) Temperature dependence of the absorption spectraof boron doped diamond with (B/C = 5000 ppm) and without (B/C =1000 ppm) an impurity band.

3.

Experimental Results

Temperature dependence of the boron induced
absorption
Figure 1 shows a comparison of the temperature dependence of the absorption spectra of boron doped diamondwith and without an impurity band. The spectra of the
B/C = 1000 ppm sample have isolated sharp Gaussian shapeabsorptions at 0.305 and 0.347 eV, which are the carriertransitions from the boron 1s ground state to the ﬁrst and thesecond 2p excited states, respectively.11,20) When the temperature decreases, the peaks become sharper and the intensity
almost doubles. However, the phonon sidebands observed
at 0.46 and 0.51 eV, and the band continuum at 0.37 eV do
not show any temperature dependence. The temperaturedependence of the peak intensity and the broadening is
explained by the temperature dependence of the lifetime due
to acoustic phonon interactions.11)
When an impurity band exists as in the B/C = 5000 ppm
sample, the absorption spectrum has a Lorentzian shape
governed by the single absorption at 0.35 eV, and ﬁne
structure is not observed. This is because the Fermi level rises
to the 2p excited states and the relaxation time of the excited
carriers decreases drastically.13) As the temperature decreases,
the absorption coefﬁcient at the peak increases by 20% and ashoulder at ³0.3 eV becomes obvious, which is caused by theincrease of the absorption component at 0.305 eV. Such weaktemperature dependence of the absorption spectra suggeststhat despite some boron remaining as isolated impurity, mostboron forms the impurity band.
The shape of the absorption spectra observed at ³0.16 eV
in Fig. 1 is almost the same as what was assigned as defectinduced-ﬁrst-order (DIFO) phonon absorption.2,11) However,
the reported DIFO absorption coefﬁcient is ³1 cm¹1, and it is
difﬁcult to detect it in our sample thickness. Moreover, as isseen in the B/C = 5000 ppm sample, when the sample has animpurity band, DIFO absorption coefﬁcient becomes larger
than 1000 cm¹1 and depends strongly on temperature.
Figure 2 shows detailed DIFO absorption spectra at 11 Kof three samples with an impurity band. Though the DIFO
spectrum shows a one-phonon excitation of the bulk diamondin the Brillouin zone, the absorption coefﬁcient at 0.16 eV is
proportional to the boron concentration. It is reported that due
to a partial loss of translational symmetry, the DIFO spectrum
contains a sharp absorption peak at the Raman active opticalFig. 2. (Color online) Temperature dependence of the absorption spectraof boron doped diamond with an impurity band. Inset shows the microRaman spectra of the B/C = 10,000 and 1000 ppm samples obtained using
633 nm excitation energy under backscattering conﬁguration at room
temperature. !0 is the Raman active phonon energy.

3.1

Fig. 3. (Color online) Reﬂection spectra of the B/C = 5000 ppm diamondsample. The n is plotted using Eq. (1).

phonon energy of !0 ¼ 0:165 eV.2) However, this phonon
energy corresponds to the upper sharp cut-off of the DIFO
absorption in Fig. 2.
A strong disorder induced broadening of the opticalphonon of heavily boron-doped diamond is reported in the
Raman spectra.21) However, as is shown in the inset of Fig. 2,
the micro-Raman spectrum of the optical phonon of theinvestigated samples does not show any disorder induced
broadening. Hence we conclude that the absorption shown in
Fig. 2 has not been induced by the defect, but that it has
become infra-red active absorption because of the occurrenceof the impurity band.
3.2 Refractive index of the impurity band
As measuring the refractive index by an ellipsometry iscomplex, we measured them using the reﬂection spectrum,
which is shown in Fig. 3 for the B/C = 5000 ppm sample.
Sinusoidal oscillation is observed in the energy range from
1.2 to 2.4 eV. In other energy region there is no structure.
For a nearly non-absorbing boron doped diamond ﬁlm witha refractive index n on a diamond substrate with index
ns , the reﬂectivity R at normal incidence is expressed asfollows;

024715-2

R¼

2
2
r10
þ r21
þ 2r10 r21 cos ;2
2 þ 2r r cos 
1 þ r10 r21
10 21

ð1Þ

©2014 The Physical Society of Japan

J. Phys. Soc. Jpn. 83, 024715 (2014)

T. Inushima et al.

where, r10 ¼ ð1  n Þ=ð1 þ n Þ and r21 ¼ ðn  ns Þ=ðn þ ns Þ are the Fresnel coefﬁcients between air and theﬁlm, and the ﬁlm and the substrate, respectively.  ¼
4 dn = is the optical path difference at wavelength ­,and d is the ﬁlm thickness. When n  1, the maxima in thefringe corresponds to  ¼ 2m, where m is an integer.
To estimate n using Eq. (1), the absolute value of m ischosen so as to ﬁt n below the bulk diamond value ofns ¼ 2:46. Then we set m ¼ 3 at the maxima of the fringe at³1.5 eV, where n exceeds 1. Thus n in Fig. 3 is obtainedusing the positions indicated in the ﬁgure. The estimated n
gives R  0:15 in the investigated region.
In the energy region below 1.5 eV where the signal is weakand noisy, the contribution of the absorption is important andEq. (1) is not applicable. Bustarret et al. reported on therefractive index of a 1:2  1021 cm¹3 boron doped diamond.21) They presented a concave dependence of the refractive index against energy, from 1.7 at 1 eV to 2.5 at 5 eV,
which is similar to the energy dependence shown in Fig. 3.
4.

Fig. 4. (Color online) Comparison of the refractive index (n ) of intrinsic(C64) and 1.57% boron doped (C63B) diamonds. The inset shows thecomparison of the n in the total energy range, where the main panel region
is indicated by an arrow.

First Principles Calculation

The electronic structure and the dielectric constants of
1.57% boron doped diamond (C63B) were calculated by ﬁrst
principles technique. We employed a super-cell model, a64-atom supercell constructed of 2  2  2 simple cubic
8-atom cells. One carbon atom was substituted by one boron
atom. The lattice parameters were determined so as tominimize the total energy of the crystal, and the obtained
lattice constant was 3.546 Å, which was in good agreementwith 3.567 Å of typical diamond. We employed the density
functional theory (DFT) provided in the code ADVANCE/
PHASE.22) Ultrasoft pseudopotentials were used, with thePerdew–Burke–Ernzerhof (PBE) generalized gradient approximation for the exchange–correlation functional. The
plane-wave basis set with a 25 hartree cutoff energy was
taken and a mesh of 8  8  8 Monkhorst–Pack k-points
sampling was used for the integration of the Brillouin zone.
Electronic dielectric function 2 was calculated using a
position operator r as follows;Z
e2 X
2 ¼ 2jhc ðkÞju  rjv ðkÞij2 ðEck  Evk  h !Þ dk; ð2Þ c,vwhere u is the polarization vector of the incident light, h ! isthe energy difference between the conduction (c) and valence(v) bands, Eck and Evk represent the Kohn–Sham energy level ofthe band at the wave vector k obtained using wave functions
c ðkÞ and v ðkÞ. The integration was done by LinearTetrahedron method,23) and the summation was made for all
the combinations of v and c. The transition moments werecorrected using the method of Kageshima and Shiraishi.24)
The real part 1 was calculated from the 2 using the
Kramers–Kronig relation, and with the combination of 1 and2 , we obtained the refractive index n . Figure 4 shows then of C63B together with that of the supercell C64 calculatedusing the same parameters. There is a strong anomalous
dispersion at ³0.5 eV caused by boron impurity, which is the
major difference of n between C64 and C63B as is shown in
the inset of Fig. 4. Due to this anomalous dispersion, thecalculated n is smaller than 2.46, which is what is observedin the bulk diamond and C64 below 2.5 eV. n becomes
smaller than 1 below 0.8 eV, and has a convex dependence on

Fig. 5. (Color online) Comparison of the absorption coefﬁcient of intrinsic(C64) and 1.57% boron doped (C63B) diamonds. The inset shows thecomparison in the total energy range, where the main panel region isindicated by an arrow.

the increase of energy, which is in accordance with what wehave observed in Fig. 3.
The absorption coefﬁcient is calculated using 1 and 2 ,which is shown in Fig. 5. There is a sharp and strong
absorption of 1:7  105 cm¹1 at ³0.5 eV, which is one orderof magnitude larger than that shown in Fig. 1. Electricallydetermined boron concentration of the B/C = 5,000 ppm
sample is two orders of magnitude smaller than that of C63B.
As is shown in the inset of Fig. 5, the absorption inducedby boron is observed at 0.5 eV above the valence band
maximum (VBM) as a separate peak. This is because thethreefold degeneracy of the VBM is held in this calculation.
Detailed VBM of C63B is shown in Fig. 6(A). The band has ahole pocket and the Fermi level is 0.18 eV below the VBM.
The bands labeled a, b, and c are not split at ¥ point becauseof the symmetry, and there are threefold degenerate bands d,e, and f, which newly appear due to the boron involvement.
These features are similar to those of the C127B supercell
model calculated by Lee and Pickett.9)
The bands that contribute to the absorption at 0.5 eV are
determined by computing the separate absorption coefﬁcients. They are calculated using Eq. (2) and shown in
Fig. 6(B), where the initial valence and the ﬁnal conduction
bands are speciﬁed in the legends. Since the absorption
coefﬁcient shown in Fig. 6(B) is calculated without taking
the combinations of c and v in Eq. (2), there is a small

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J. Phys. Soc. Jpn. 83, 024715 (2014)

T. Inushima et al.

Fig. 6. (Color online) (A) Valence band structure of the C63B supercell.
(B) Absorption coefﬁcients of the optical transitions between the bands
labeled in (A). The transition direction is indicated by the arrows in thelegend.

discrepancy between the sum of the separately calculated
absorption coefﬁcients and that shown in Fig. 5. However it
is obvious that the absorption is caused by b ! a and c ! btransitions, which are the intraband transitions between the
topmost valence bands of C63B.
Periodicity of boron atoms is absent in the diamond where
the boron impurities are randomly distributed. Focusing on
this point, Lee and Pickett studied the band structure ofmetallic diamond calculated using a supercell model andcoherent potential approximation (CPA), and concluded that
the CPA and supercell results were similar at the valenceband structure near the Fermi level.9) Hence it is expectedthat the metallic boron-doped diamond has a sharp and strong
intraband absorption, which causes the big change of therefractive index shown in Fig. 4.
5.

Discussion

When we compare the optical constants of the boron doped
diamond with an impurity band and those of a C63B supercell
obtained by the ﬁrst principles calculation, it is obvious thatthe calculated values will be experimentally obtained at oneorder of magnitude larger NB . In other words, a boron doped
diamond with an impurity band is in a CxB supercell
condition. That is, when the NB is small, boron has aseparated energy level expressed by the Bloch wavefunctionsof the valence band maxima. However, when the 2p
wavefunctions of impurity boron begin to overlap at Nc1 ¼
4  1018 cm¹3 (B/C = 2000 ppm), an impurity band appears
at the 2p excited level. Simultaneously, the Fermi energy
shifts to the 2p excited level, which indicates that the diamondwith the impurity band has a metallic character. Then, thetopmost valence band structure is determined by the valence
electrons of boron as is expected in the supercell condition,and the separate impurity states fade into the valence band.

Fig. 7. (Color online) Boron concentration (NB ) dependence of the
impurity band structure in diamond. Band A forms at the boron groundstates. EF is the Fermi level position. Nc1 is the critical density for the changeof the conduction. Nc is the Mott transition density. The dashed lines are thedensity of states formed by boron wave functions.

The infra-red active phonon density of states in the 0.15 eV
region induced by the impurity band shown in Fig. 2 isanother evidence of the CxB supercell condition. The phonon
structure of the boron doped diamond was calculated using
a supercell model, and because of the boron involvement,
phonon softening, which plays an important role for theoccurrence of the superconductivity, is expected at thehighest frequency of the zone center optical phonons.25)
Actually the phonon softening was experimentally observedin superconducting diamond.26) Moreover, the observed
softening was much closer to that expected by supercell
model calculation than that by virtual crystal approximation.27) These results offer a scenario where the boron atoms
are involved in the carbon network, satisfying the CxB
supercell condition.
In previous papers we showed how boron impurity
evolved from isolated level to an impurity band.12,13) Now,
the ﬁrst principles calculation shows that the impurity band
continues to evolve into the solid valence band throughMott transition. Here, we present an idea of the evolution,
schematically shown in Fig. 7.
When NB is small, boron has the energy level at A, which
is 0.37 eV above the valence band maximum, and the 2p
excited level is isolated, which is shown in Fig. 7(a). The
Fermi level depends on the degree of compensation. When
NB is larger than the number of charged donors, the Fermi
level lies in band A (EF ). When the temperature is high,conduction is governed by the carriers excited from the Fermi
level to the valence band maximum.
The conduction mechanism drastically changes at Nc1 ,which is shown in Fig. 7(b). At Nc1 the average distance
between borons is µ4 nm, and the 2p wavefunction starts tooverlap that of the neighboring boron atoms, and EF rises tothe E2p position.13) Hence, the supercell condition is partially

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J. Phys. Soc. Jpn. 83, 024715 (2014)

T. Inushima et al.

satisﬁed and the 2p wavefunctions have k dependence. The
appearance of k dependence is indicated by the dashed line.
In our model Mott transition occurs when the Fermi level
reaches the top of the valence band at NB ¼ Nc ( 3  1020
cm¹3), which is shown in Fig. 7(c). The impurity band
formed by 2p wavefunctions merges into the bulk diamondband at Nc . The energy position of the top of the valenceband is not the same as that of the intrinsic one and it should
be determined by the supercell condition. At Nc , the 1s wave
functions of borons partially overlap and have k dependence,but the majority remains in isolated condition. Hence, theabsorption spectrum is governed by the transition between A
and the top of the valence band.
Figure 7(d) corresponds to the C63B supercell condition
and EF ¼ 0:18 eV. From a study by an X-ray absorption
and emission spectroscopy, it is reported that there is an ingap state about 0.3 eV above the top of the valence band in
the metallic and nonmetallic boron doped diamonds.28) We
understand that the in-gap state is, when NB is 0.1%, theimpurity band absorption shown in Fig. 1, and when NB is4.3%, it is the intra-band absorption shown in Fig. 5. This
assignment supports the idea of the impurity band structure
presented in Fig. 7. Moreover, the band structure of the C63B
supercell is almost the same as that of bulk diamond, which
explains the result given by a photoemission spectroscopy.6)
In Figs. 7(a) to 7(d) there is no drastic change of theabsorption spectra, because the transition including the intraband transition is governed mainly by boron wavefunctions.
In usual semiconductor physics, heavily doped impurity
disturbs the crystal ordering and the potential ﬂuctuationinduces Anderson localization.4) This scenario does notwork in the superconductivity of diamond. Experimentally,
it is reported from a nuclear magnetic resonance study ofsuperconducting diamond that, though boron is distributed
randomly in diamond, the boron that substitutes carbon in thehigh symmetry diamond structure produces effective carriers
for the superconductivity, while the boron in the lower local
symmetry does not.29) This result supports our model where
the doped boron works as a constituent atom in the bulk
having a CxB supercell and the potential disturbance is not
so strong. Moreover, the involvement of boron atoms in
diamond structure is advantageous to producing a strong
bond polarization and inducing a strong electron–phonon
interaction as has been observed in the superconductivity ofsemiconductor InN which also shows superconductivity near
the Mott transition.30)
6.

Summary

We compared the absorption coefﬁcient and the refractive
index of boron doped diamond having an impurity band at
0.07 eV above the valence band maximum and those obtainedby ﬁrst principles calculation using a C63B supercell model
containing 1.57% boron. From the comparison we presented
a model of the evolution of boron atoms from isolated
impurity to constituent atoms in the boron doped diamond,
where, when the boron concentration exceeds 4  1018 cm¹3,
the valence electrons of boron become to have k dependence,and they form the top of the valence bands of C63B supercell
diamond. The presented model explains the boron concentration dependence of the reported X-ray absorption andemission spectra.

Acknowledgments
This work was supported by the Joint Studies Program(2011) of the Institute for Molecular Science. The authors
are grateful to the Ministry of Education, Culture, Sports,
Science and Technology of Japan for the Grant-in-Aid for
Scientiﬁc Research No. 22560304.

+

inushima@keyaki.cc.u-tokai.ac.jp
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024715-5©2014 The Physical Society of Japan