Hall coefficients and resistivities ofsingle crystals: The presenceof a mobility edge in correlated twodimensional electron system
Cite as: Journal of Applied Physics 92, 5214 (2002); https://doi.org/10.1063/1.1509842
Submitted: 09 April 2002 . Accepted: 05 August 2002 . Published Online: 21 October 2002
Katsuhiko Inagaki, and Satoshi Tanda

Journal of Applied Physics 92, 5214 (2002); https://doi.org/10.1063/1.1509842© 2002 American Institute of Physics.

92, 5214JOURNAL OF APPLIED PHYSICSVOLUME 92, NUMBER 9

1 NOVEMBER 2002

Hall coefficients and resistivities of Bi2¿ x Sr2À x CuOy single crystals:
The presence of a mobility edge in correlated two-dimensional electron
systemKatsuhiko Inagaki and Satoshi Tanda
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan

共Received 9 April 2002; accepted 5 August 2002兲
Hall coefficients and resistivities of a series of Bi2⫹x Sr2⫺x CuOy single crystals were measured toinvestigate the relation between the carrier concentration n and the resistivity-minimum temperature
T min . The T min was found to obey a simple linear relation: T min ⬀ nc⫺n, where n c ⯝4.2
⫻1021 cm⫺3 is a critical carrier concentration. A model of two-dimensional electron systems with amobility edge was introduced to explain this linear relation, and the presence of a mobility edge in
two-dimensional copper-oxide-based metals was suggested. © 2002 American Institute of Physics.
关DOI: 10.1063/1.1509842兴

I. INTRODUCTION

centrations. Accurate control of geometry by use of theexcimer-laser cutting technique allows us to evaluate the resistivity and the carrier concentration of each crystal and toinvestigate the evolution of the resistivity as a function of thecarrier concentration. We found that the temperature of theresistivity-minimum T min obeys a simple relation to the carrier
concentration
n: T min⬀nc⫺n,where
n c ⯝4.2⫻1021 cm⫺3 is a critical carrier concentration. This relationsuggests the presence of a mobility edge in the twodimensional CuO2 planes of the superconducting cuprates.

Recent studies have revealed that a new kind of twodimensional 共2D兲 metallic phase can exist, according to theresults of the dynamical scaling theory.1,2 Such 2D metallic
phases have been experimentally observed: Longitudinal resistivity in the quantum Hall system varies from infinity
共quantum Hall insulator兲 to zero 共quantum Hall liquid兲 by thetuning of the applied field, and a metal phase emerges between the two;3 and superconductivity of a molecular-beamepitaxy-grown Nd2⫺x Cex CuO4⫺y crystal is suppressed by
the applied field, and a metal phase is observed just at thesuperconductor-insulator 共SI兲 transition.4 Similar results arealso obtained in the SI transition of ultrathin films ofbismuth.5 In addition, a different type of 2D metal-insulatortransition at zero field has been proposed to explain the experimental results in metal-oxide-semiconductor field-effecttransistors.6 These theoretical and experimental developments exhibit a striking contrast to the well-known nature ofconventional noninteracting 2D systems, in which all electron states are localized and there is no 2D metallic state.7
These discussions seem analogous to those of minimum conductivity, first proposed by Mott,8 and later clearly denied by
the scaling theory of noninteracting 2D electron gas.7
High-T c superconducting cuprates provide a stage forthe investigation of a 2D electron system with strong correlation. The normal state of cuprates can be explained as a 2D
doped Mott insulator. The CuO2 plane of the cuprates show
metallic behavior when carriers are doped. Since the word‘‘metallic’’ here implies the electron system at finite temperatures exhibits a positive temperature coefficient of resistivity
共dR/dT⬎0兲, it is important to deduce the behavior of theelectron system at zero temperature from the data obtained at
finite temperatures. Bi2⫹x Sr2⫺x CuOy is an ideal material to
study a 2D electron system in the cuprates because of lowsuperconducting transition temperature 共T c ⬃8 K) even at
the optimum carrier concentration and its highly anisotropic
electric conduction 共␴ ab / ␴ c ⬎104 ). 9 In this article, we provide experimental results for resistivity and Hall coefficientof Bi2⫹x Sr2⫺x CuOy single crystals of different carrier con0021-8979/2002/92(9)/5214/4/$19.00

II. EXPERIMENTThe Bi2⫹x Sr2⫺x CuOy single crystals used in this study
were synthesized by the self-flux method. We changed theratio of starting powders Bi2 O3 and SrCO3 , so that we could
control Bi/Sr substitution. A composite powder consisting of
Bi2 O3 , SrCO3 , and CuO was placed in an MgO crucible,
and the mixture was heated in a furnace at 1000 °C for 4 h.
The furnace was cooled at the rate of 2 °C/h. All processes
were carried out under standard atmosphere. The resulting
crystals were black and slabshaped.
The obtained crystals were cut into the shape of the Hallbar by repeating step-by-step laser drilling. The laser system
consists of a Kr-F excimer laser source 共wavelength 248
nm兲, an attenuator, a lens 共f⫽1:300兲, and a computercontrolled x-y stage. The beam was focused to a 50
⫻50 ␮ m2 spot. The sample was fixed on the x-y stage, which
can be positioned with an accuracy of 2 ␮m. Shortly we will
discuss why such accuracy is required for this study.
Bi2⫹x Sr2⫺x CuOy crystals of 10 to 200 ␮m thickness wereable to be pierced by 50 to 1000 laser pulses. The typical sizeof the sample is 2.7 mm in length, and 0.5 mm in width. The
sample has two potential terminals and two Hall terminals,
the size of which is 0.3⫻0.4 mm2 . Sample thickness t wascalculated from the sample area and density. Weight of thesample was measured by a microbalance 共Mettler M3兲 up to
the order of 0.1 ␮g. The typical thickness was ⬃10 ␮m.
The resistivity of the samples was measured by the standard four probe method with a current of less than 100 ␮A.
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J. Appl. Phys., Vol. 92, No. 9, 1 November 2002

K. Inagaki and S. Tanda

5215

The Hall voltage and the voltage drop across the sample wasmonitored with a nanovoltmeter 共Keithley 182兲. In the Hallmeasurement, magnetic fields up to 4 T were applied perpendicular to the CuO2 plane. We measured the Hall voltages ofboth directions in order to eliminate the offset voltage at zero
field. We used a battery-operated current generator to minimize the line noise induced by the power lines.
It is important to evaluate if the geometry of the Hallbars
have the accuracy required for this study. Magnetoresistancewill appear on Hall terminals, if the terminals are not positioned on the same equipotential line. When the Hall terminals are misaligned by ⌬l, a voltage drop across the Hallterminals occurs
V⫽I ␳

⌬l
,
tw

共1兲

where ␳ is resistivity, t is thickness, and w is width. Magnetoresistance effect may change ␳ in Eq. 共1兲 as much as a few
percent. Since the Hall voltage is represented as V H⫽R HIB/t, we compared it to the voltage drop across theHall terminals
⌬V
␳ ⌬l
.
⫽
V H R HB w

共2兲In cuprates, the typical resistivity ␳ ⬃10⫺6 ⍀m, and the Hall
coefficient R H⬃10⫺9 C⫺1 m3 . To make ⌬V/V H comparable,
thus ⌬l/wB⬃10⫺3 (tesla) ⫺1 is required. Assuming a magnetic field of several teslas, and a sample width w ⬃ 1 mm,
one should align the Hall terminals within ⌬l⬍10⫺3 mm tosatisfy the above conditions. We attained this accuracy bymeans of the laser cutting technique described above. We
also estimated ⌬l by the voltage drop across the Hall probes
at zero magnetic field. The electrically estimated misalignment of ⌬l was, however, the order of 10⫺2 mm. Typically,⌬V/V H⬃100 was obtained.
III. RESULTS

Figure 1 shows that the temperature dependence of theresistivity in Bi2⫹x Sr2⫺x CuOy single crystals with x⫽0.0
共hereafter A兲, 0.2 共B兲, 0.3 共C兲, and 0.4 共D and E兲. Data for asample showing a superconducting transition is also shown
as a reference 共S兲. It is shown that the resistivity depends
systematically on the Bi/Sr ratio of the samples. Sample A
showed a superconducting transition at 8 K, whereas B, C,
D, and E showed no superconducting transition down to 1.8
K. We found that the SI transition lies between samples A
and B in the observed Bi2⫹x Sr2⫺x CuOy system. We calculated the sheet resistance per CuO2 plane, taking the separation between CuO2 planes as 12 Å. 10 The sheet resistance of
1 k⍀ corresponds to the resistivity 1.2⫻10⫺4 ⍀ cm. It was
proposed that associated with the superconductor-insulatortransition is a universal critical resistance h/4e 2 (6.45 k⍀). 1
The critical sheet resistance obtained here was about two
times this value.
Sample A exhibits ln T upturn in the resistivity in thetemperature regime from superconducting transition temperature to 50 K. In the weak localized regime, the conductivity obeys the following logarithmic form:7

FIG. 1. Resistivities of the samples with different Bi/Sr ratio. The sheet
resistance of 1 k⍀ corresponds to the resistivity 1.2⫻10⫺4 ⍀ cm. The residual resistivity becomes higher when a significant amount of Sr is substituted by Bi.

␴ 共 T 兲 ⫽ ␴ 0 ⫹ ␣ p 共 e 2 /2␲ 2 ប 兲 ln T,

共3兲

where ␣ is a parameter of the order of unity, and p is determined as ␶ i ⬀T ⫺p . We obtained ␴ 0 ⫽9.24⫻10⫺5 ⍀ ⫺1 and␣ p(e 2 /2␲ 2 ប)⫽7.17⫻10⫺6 ⍀ ⫺1 , hence, ␣p⫽0.57. Consequently, it is shown that the behavior of sample A is described in terms of the weak localization. The possibility of
the resistivity upturn being due to magnetic impurities 共theKondo effect兲 should also be evaluated. Several authors including us have independently reported anisotropic magnetoresistance in Bi2 Sr2 CuOy .11,12 The anisotropy of the magnetoresistance is a reliable indicator for Anderson
localization. In contrast, for the Kondo effect the magnetoresistance is purely isotropic. We shall therefore, interpret theresistivity upturn on the basis of the Anderson localization.
We evaluated the disorder of the crystals from the residual resistivity. The randomness of a system is characterized by the metallic parameter k Fᐉ, where k F is the Fermi
wave vector and ᐉ is an elastic mean free path. The metallic
parameter k Fᐉ is related to the residual sheet resistance R 0
1/R 0 ⫽ 共 e 2 /h 兲 k Fᐉ.

共4兲

We obtained k Fᐉ⫽4.0 for sample A, 2.0 for B, 1.6 for C, 1.4
for D, and 1.1 for E. This result suggests that the Bi/Sr substitution effectively enhances system disorder. If much Bi/Sr
substitution were carried out up to the Ioffe-Regel limit
k Fᐉ⬃1, the carriers could not extend over the sample due todisorder. However, since k Fᐉ⬎1 for the all substitutions, oursystem remains metallic.
Figure 2 shows the carrier concentration of samples A,
B, C, D, and E, defined as n⫽1/R He, where R H is the Hall
coefficient. This treatment is justified by applying a singlecarrier model as the first approximation. We found the following two characteristics in the Hall coefficients: the carrier
concentration became smaller when Sr sites were substituted
by excess Bi atoms; and the Hall coefficients of each sample
were temperature independent. The observed temperature independent Hall coefficient is consistent with the Anderson
localization picture.13 The carrier concentrations of thesamples are 3.6⫾0.4, 2.6⫾0.3, 2.0⫾0.2, 1.1⫾0.1, and 0.9

5216

J. Appl. Phys., Vol. 92, No. 9, 1 November 2002

K. Inagaki and S. Tanda

FIG. 2. Carrier concentration of samples A, B, C, D, and E, determined asn⫽1/R He, where R H is the Hall coefficient. The carrier concentration becomes smaller when a significant amount of Sr is substituted by Bi. It isshown that the Hall coefficient of each sample is independent of temperature.

⫺3⫾0.1⫻10 cm for A, B, C, D, and E, respectively. Theseare equivalent to the carrier number per each Cu site as
0.25⫾0.03, 0.18⫾0.02, 0.14⫾0.01, 0.08⫾0.01, and 0.06
⫾0.01.10
21

We now consider two-dimensional electron systems witha mobility edge. The Fermi energy E F is proportional to thecarrier concentration, that is,
E F⬀k F2⬀n.

A striking result is the discrepancy of temperature dependencies between the resistivity, increasing by several orders, and the Hall coefficient, being almost constant. This
discrepancy between the resistivity and the Hall coefficient
has been observed in various superconducting cuprates, suchas ion-beam-damaged YBCO films,14 BiSr共Ca, Y兲CuO single
crystals,15 polycrystalline BiSr共Ca, Pr兲CuO,16 and BiSrCuO
single crystals.11,17 We focus on the temperatures at resistivity minima T min to understand the discrepancy. Figure 3 is a
close up of the resistivity variation with temperature in theregion of T min for different carrier concentrations n determined from the Hall coefficients. The temperature T min systematically increases as carrier concentration n decreases.
A few authors have discussed the physical interpretationof the T min in terms of localization theories: the temperaturewhen the inelastic scattering length becomes comparable tothe localization length,18 and the energy difference between
the Fermi level and the mobility edge.16
We analyze the experimental results, making use of theidea that T min represents the energy difference between the
Fermi level and the mobility edge.16 The resistivity has ametal-like temperature dependence (d ␳ /dT⬎0) above T min ,whereas it shows insulating properties below T min . The insulating sample 关d ␳ /dT(T→0)⬍0] can also exhibit metallic
behavior at finite temperatures for which the thermal energyk BT exceeds the difference between the mobility edge E c
and the Fermi level E F . Therefore, the temperature of resistivity minimum is given by the following formula:16
共5兲

共6兲

Hence, Eqs. 共5兲 and 共6兲 lead to a relation between the T min
and n,T min⬀n c ⫺n,

IV. DISCUSSIONS

k B T min⬃ 兩 E c ⫺E F兩 .

FIG. 3. Resistivity minima of samples A, B, C, D, and E. The temperaturesof the resistivity minima depend on the carrier concentration.

共7兲

where n c is a critical carrier concentration, if the mobility
edge E c is not sample dependent.
To see if the relation 共7兲 is satisfied, we plot T min as afunction of n in Fig. 4. It is shown that the relation 共7兲 fits
our data well, and the critical carrier concentration is obtained as n c ⯝4.2⫻1021 cm⫺3 . This value corresponds to
5.0⫻1014 cm⫺2 共or 0.35 per Cu site兲 in terms of twodimensional electron systems.
The model of two-dimensional electron systems with amobility edge describes our experimental results of the temperature dependences of the resistivities and Hall coefficientsof Bi2⫹x Sr2⫺x CuOy single crystals. By extrapolating our re-

FIG. 4. Relation between the resistivity-minimum temperature T min and thecarrier concentration n.

J. Appl. Phys., Vol. 92, No. 9, 1 November 2002

sults to T min→0 K, we predict that a metal-insulator transition should take place in Bi2⫹x Sr2⫺x CuOy when E F is equal
to E c at n⫽n c , and a two-dimensional metallic system willappear at the phase transition point. We believe further investigation to determine the critical exponent will reveal thedetails of this noble type of metal-insulator transition, by
lower temperature studies of the cuprates whose carrier concentrations lay in the neighborhood of n⫽n c .
V. CONCLUSIONSIn conclusion, we present the experimental results forresistivity and Hall coefficient of Bi2⫹x Sr2⫺x CuOy single
crystals of different carrier concentrations. We found that thetemperature of the resistivity-minimum T min obeys a simple
relation to the carrier concentration n: T min⬀nc⫺n, wheren c ⯝4.2⫻1021 cm⫺3 is a critical carrier concentration. This
relation suggests the presence of a mobility edge in twodimensional CuO2 planes of the superconducting cuprates.
ACKNOWLEDGMENTS

We are grateful to K. Yamaya and T. Nakayama for fruitful discussions, and K. Sajiki and T. Ishibashi for experimental support. One of the authors 共S. T.兲 thanks Sir N. F. Mottfor his encouragement. This work is partly supported by theJapan Society for the Promotion of Science and Kurata
Foundation.

K. Inagaki and S. Tanda

5217

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