Physica B 246—247 (1998) 282—285

Coulomb gap shrinkage in compensated Si:(P, B)in high magnetic fields

M. Iqbal!, J. Galibert!,*, J. Leotin!, S. Askenazy!, S. Waffenschmidt", J. Wosnitza",H. v. Lohneysen"

! Laboratoire de Physique de la Matie% re Condense&e et Service National des Champs Magne& tiques Pulse& s, I.N.S.A.,F-31077 Toulouse Cedex, France

" Physikalisches Institut and SFB 195, Universita( t Karlsruhe, D-76128 Karlsruhe, Germany

Abstract

Magnetoresistance measurements on heavily doped but still insulating n-type Si : (P, B) are reported for the temper-ature range 1.6 to 300 K and in magnetic fields up to 35 T. In zero magnetic field, the samples investigated (carrierconcentration 1.5, 2.5 and 2.95]1018 cm~3, compensation ratio &0.65) show Efros—Shklovskii (ES) variable rangehopping (VRH), o"o

0exp[(¹

0/¹)p] with p"1

2. For the sample with 2.95]1018 cm~3, the ES hopping conduction

remains unchanged in the low-field regime (up to 8 T). In addition, the observed B2 dependence of ln[o(B)] in this fieldrange is in good agreement with the expected behavior, with the proportionality constant depending on temperature asK

sJT ~3@2. However, in high fields ('25 T) p changes to 1

3as expected for Mott VRH with a finite density of states at the

Fermi level, indicating that the Coulomb gap becomes ineffective in large fields. This surprising result was previouslyfound for Ge : As. ( 1998 Elsevier Science B.V. All rights reserved.

Keywords: Metal—insulator transition; High field magnetoresistance; Variable range hopping; Coulomb gap

1. Introduction

In disordered compensated semiconductors, typi-cal conduction mechanisms are based on thermallyactivated charge carrier hopping between localizedstates. Variable range hopping (VRH) was identi-fied by Mott [1] and displays a temperature-depen-dent resistivity varying like o"o

0exp(¹

0/¹)s. The

*Corresponding author. Fax: #33-5-61-55-99-50; e-mail:galibert@insa-tlse.fr.

exponent s was found equal to 14

for three-dimen-sional systems. VRH involves low hopping energiesand usually occurs at low temperature. Efros andShklovskii (ES) [2] realized later that the assumption of a constant density of states (DOS) in thevicinity of the Fermi level does not hold whenlong-range electron—electron interactions give riseto a Coulomb gap at the Fermi level. The conse-quence is the change of the exponent to s"1

2.

A Coulomb gap was recently observed directly inSi : B by electron tunneling spectroscopy [3]. To-day, based on a large set of experimental data, it is

0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reservedPII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 9 1 6 - 2

generally understood that Mott’s law is expected atimpurity concentrations just below the metal insu-lator transition (MIT) while the ES regime shouldoccur at much lower concentrations.

Only recently, experimental studies have beendealing with the crossover between Mott and ESVRH (see, for example, Ref. [4]). Most of the stud-ies reported a crossover from Mott to ES VRH(s"1

4to s"1

2) when the system becomes ‘more

insulating’. For example, Hornung et al. [5] foundthis situation with decreasing impurity concentra-tion in Si:P. The crossover had been earlier ob-served by Zhang et al. [6] when lowering thetemperature down to the milliKelvin range. A sim-ilar behavior has also been realized [7,8] by shrink-ing the electron wave functions with application ofa magnetic field. Hence, the system is driven moretowards insulating behavior, i. e., it behaves as if thecarrier concentration was decreased. However, theopposite crossover from ES to Mott VRH withincreasing field was found in Ge : As by Shlimak etal. [9]. They put forward the existence of a criticaltemperature ¹

#below which the Coulomb gap is

effective. Because this temperature is decreased be-low the experimental temperature ¹ when the mag-netic field rises, a crossover towards a Mott regimeoccurs.

In the present work dealing with compensatedphosphorus-doped silicon, such opposite crossoverdriven also by magnetic field was found in Si:(P, B)measured in the temperature range 3—40 K in fieldsup to 35 T. The samples in the insulating regimewere the same as investigated earlier [10].

Evidence for a magnetic-field driven crossoverresults from both magnetoresistance signatures atfixed temperature as well as temperature signaturesat fixed magnetic field. Both behaviors, well estab-lished on theoretical and experimental grounds arediscussed in Ref. [11] for low and high magneticfields, respectively. The critical field B

#dividing the

low-field range from the high-field range is given by

Bc"

+c

e[N(E

F)k

B¹]1@4(a*

H)~5@4,

where a*H

is the effective Bohr radius in zero mag-netic field, N(E

F) the DOS at the Fermi level, and

the other constants have their usual meaning.

From these equations, one should expectln[o(B)/o(0)] to vary as B2 and B1@3 below andabove the critical field B

#in the case of Mott-type

VRH conduction. The temperature dependences areexpected to be ¹~3@4 and ¹~1@3, respectively. In thecase of ES-type VRH conduction o should exhibita field dependence of B2 and B1@5 below and aboveB#. The corresponding temperature dependences

are expected to be ¹~3@2 and ¹~3@5, respectively.

2. Results and discussion

The carrier concentrations of the investigatedn—Si:(P, B) samples as determined from the Hallconstant are: Sid1 with N"2.95]1018 cm~3,Sid2 with N"2.5]1018 cm~3 and Sid3 withN"1.50]1018 cm~3. These values are below thecritical concentration of the MIT for this compen-sation ratio, N

#"4.94]1018 cm~3 [12]. The com-

pensation ratio is K+0.65.Ohmic contacts were established on the samples

with gold leads spot-welded to the sample. Mag-netoresistance measurements were carried out inpulsed magnetic field, at the S.N.C.M.P. of Tou-louse, in a temperature range between 1.6 and300 K, and up to 34 T using the standard four-probes AC technique. The dissipated power intothe sample was maintained at a level of less thana few nW, especially in the lowest temperaturerange.

The zero-field resistivity o(¹, B"0) plotted asa function of ¹~1@2 was reported in Ref. [10].Evidence of ES regime in the full range 3.5—40 Kwas found in contrast to the temperature drivencrossover towards the Mott regime found inGe : As [9].

Fig. 1 shows the resistivity on a logarithmic scaleas a function of ¹~1@2 for different magnetic fieldsfor Sid1. For fields up to 14 T lno varies linearlywith ¹~1@2 in the temperature range from 3 to40 K. At high magnetic fields, deviations from thelinearity are observed. In order to determine theexponent describing the temperature dependence,s was varied to minimize the least-squares devi-ation s2"R[ln(o

%91)!ln(o

#!-#)]2. As an example,

the result of such an analysis for Sid1 is shown inFig. 2.

M. Iqbal et al. / Physica B 246—247 (1998) 282—285 283

Fig. 1. Logarithmic dependence of the resistivity as a functionof ¹~1@2 at fixed magnetic fields.

Fig. 2. Percentage of deviation obtained in fitting the experi-mental resistivity to various exponents s at fixed magnetic fields.

Fig. 3. Plot of the logarithm of the slope K4obtained from the

dependence ln[o(B)/o(0)]"K4B2 in the low-field regime as

a function of ln(¹) showing that K4J¹~3@2.

A sharp minimum with s"0.52 close to thetheoretical ES value s"1

2occurs at zero magnetic

field. The ES regime remains unchanged up to 14 T.In addition, we find ln(K

s)+!1.57 ln(¹) (Fig. 3) in

close agreement with the typical coefficient ofthe low-field ES regime, equal to !1.5. Uponincreasing the magnetic field, the exponents shifttowards lower values and lie between 0.28 and 0.35,i.e., not far from the theoretical value s"1

3for the

Mott-type VRH. In the high-field regime, above26 T, the Mott regime is clearly established fromfirst, the linear plot of the resistivity in logarithmicscale as a function of ¹~1@3 (Fig. 4), second, fromthe linear dependence of the above slopes withmagnetic field (insert of Fig. 4) as expected fromTable 1 [11].

The crossover from ES to Mott VRH was ex-plained by Shlimak et al. [9] in terms of a criticaltemperature ¹

#below which the Coulomb gap

becomes effective. Whereas in the case of Ge : Asthis temperature falls within the experimental rangeat zero field, in the present experiment the criticaltemperature ¹

#at zero field is likely to be above

40 K. Due to free carrier excitation into the con-duction band, the Mott regime therefore seems notaccessible because a direct transition from domi-nant ES VRH to activated conduction might occurwith increasing ¹. However, should ¹

#decrease

with magnetic field as suggested by Shlimak et al.[9] and fall into the present experimental range

284 M. Iqbal et al. / Physica B 246—247 (1998) 282—285

Fig. 4. ln[o(T,B)] plotted as a function of ¹~1@3 in the high-fieldrange. The inset shows the linearity of the characteristic temper-ature ¹

0with the field.

Table 1Expected behavior for Mott and Efros—Shklovskii (ES) variable-range hopping-in magnetic fields

Regime ¹ and B laws Mott ES

Low magneticfield

ln[o(B)/o(0)]"K

4B2

K4J¹~3@4 K

4J¹~3@2

High magneticfield

ln[(o(B)/o(0)]"[¹

0(B)/¹]4

s"1/3 s"3/5

¹0JB ¹

0JB1@3

3—40 K, one may expect the magnetic field to drivethe crossover towards the Mott regime. It shouldbe possible to determine the key parameters includ-ing the DOS at the Fermi level in the Mott regime

from the transition between low-field and high-field regime as in Ref. [13] in order to reacha quantitative estimate of ¹

#(B) using the Shlimak

model. This work is in progress together with thefull analysis of samples 2 and 3 which exhibit alsothe crossover.

In conclusion, we have confirmed in theSi : (P, B) system the VRH crossover from ES toMott regime previously observed in Ge : As.

We thank D.F. Holcomb, Cornell University, forthe Si : (P, B) samples. This work was supported bythe DFG through SFB 195.

References

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M. Iqbal et al. / Physica B 246—247 (1998) 282—285 285