phys. stat. sol. (b) 218, 83 (2000)

Subject classification: 71.30.+h; 72.20.Ee; 72.60.+g; S8.16

Universal Behaviour in the Variable Range HoppingRegime of Copper Ternary Compounds

M. Iqbal (a), J. Galibert1) (a), S.M. Wasim (b), and P. Bocaranda (b)

(a) Laboratoire de Physique de la MatieÁre CondenseÂe et Service National des ChampsMagneÂtiques PulseÂs, I.N.S.A. 135, Avenue de Rangueil, F-31077 Toulouse Cedex, France

(b) Centro de Estudios de Semiconductores, Facultad de Ciencias,Universidad de los Andes, VE-5101 Merida, Venezuela

(Received September 7, 1999)

Mott type variable range hopping conduction (VRH) below 50 K has been observed in severalsamples of CuInTe2. This set of data and those published earlier on CuInSe2 and Si:B are used tocheck the validity over an extended range of T/TX towards high temperature of the scalingexpression of the form r = r0 exp [Af(T/TX)] as proposed by Aharony et al. It is confirmed thatall data including that of CdSe fall on a universal curve. From the knowledge of the parameters Aand TX, one can get a better estimate of the critical temperature Tc in each sample, related to thecrossover from Mott to Efros-Shklovskii type conduction.

Introduction Electrical conduction on the insulating side of the metal±insulator transi-tion (MIT), characterized by n1=3

c aH � 0.25 [1], is governed by the variable range hop-ping (VRH) mechanism. This occurs at low temperatures when the electrical resistivityfollows the relation r = r0 exp (T0/T)s, where T0 is the localization temperature. In thecase of a three-dimensional system s = 1/4 if the conduction is of Mott type with non-vanishing constant density of states (DOS) at the Fermi level [2]. However, at muchlower temperatures or concentrations of impurities (N � nc), as proposed by Efros andShklovskii [3] (ES), s = 1/2 if the «Coulomb gap» appears in the DOS due to longrange electron±electron interactions between the localized states. Mott type VRH con-duction in semiconductors of group IV (Si, Ge) [4, 5], III±V (GaAs, InP) [6 to 8], II±VI(CdSe [9], CdTe) [10] and I-III±VI2 (CuInSe2) [11] is reported in the literature. Acrossover from Mott to ES type in n-CdSe [9] by lowering the temperature and fromboth Mott to ES type in n-InP [7], and from ES to Mott type in n-Ge [12] by theapplication of high magnetic fields has also been observed.

Theoretical Background A few years ago, from dimensional analysis, Aharony et al.[13] have proposed that the temperature dependence of the hopping resistivity can beexpressed by a scaling expression of the form r � r0 exp �A f �T=TX��, where f �T=TX�is a universal function and A, TX and r0 are sample-dependent constants. An explicitform for f �T=TX� is derived by optimizing the exponential in the probabilitygij � g0 exp �ÿ2rij=xÿ eij kBT��

for an electron to hop at a distance rij between localizedstates with an energy eij. This energy eij is combined to be the sum of a1=N�EF� r3

ij and

M. Iqbal et al.: Variable Range Hopping Regime of Copper Ternary Compounds 83

1) Author for correspondence: Tel.: +33-(0)5-61-55-99-63; Fax: +33-(0)5-61-55-99-50;e-mail: galibert@insa-tlse.fr

a2e2=j2rij for Mott and ES type hopping, respectively. N(EF) is the density of states atthe Fermi level and j is the dielectric constant.

The optimal hopping distance is a function of TX � a22e4N�EF� x24kBa1j2

and

A � 8=36ja1

a2e2N�EF� x2

" #.

The crossover universal function, where x = T/TX, is written as

f �x� �1� �1� x�1=2 ÿ 1

x��1� x�1=2 ÿ 1�1=2

:

Under this condition, in the asymptotic resistivity limits, one can get TMott = A4TX

and TES = 4.5 A2TX, where T0 is replaced by TMott and TES for Mott (s = 1/4) andES (s = 1/2) type of conduction, respectively, in the expression ln (r/r0) = (T0/T)s.

Thus, with the knowledge of A and TX one can get easily an estimation of TMott andTES even if the experimental resistivity shows only one type of conduction. On theother hand, the model proposed by Aharony et al. permits to establish the range of thecritical temperature where the crossover from Mott to ES type conduction occurs. Thistemperature Tc lies between 4TX and 200TX.

In the present paper, continuing our earlier studies of CuInSe2, see e.g. [11, 14], wereport on the temperature dependence of the resistivity in several single crystal samplesof p-CuInTe2, another member of the I±III±VI2 family of chalcopyrite semiconductors,below 100 K when the electrical conduction is dominated by the charge carriers in theimpurity band. As in the case of CuInSe2, hopping conduction of Mott type that startsat temperature as high as 50 K is observed in p-CuInTe2. Since hopping occurs at rela-tively lower temperatures, this allows us to check the validity of the universal functionproposed by Aharony et al. at high temperatures.

Experimental Methods Single crystals of CuInTe2 were cut in the form of a parallele-piped from a relatively homogeneous ingot that was grown by a new technique consist-ing on the tellurization of a stoichiometric mixture of Cu and In in the liquid phasearound 1100 �C. This is very similar to that employed by us earlier for the growth ofCuInSe2 [15].

The Hall coefficient RH measured between 1.6 and 300 K for samples CIT#W2 andCIT#W3 exhibits between the two extreme temperatures, a maximum around 35 and50 K. This confirms that electrical conduction in both samples below these temperaturesis dominated by the charge carriers in the impurity band. From the Mott criteriump1=3

c a0 = 0.25, the critical concentration pc is estimated to be 3.4 � 1019 cm±±3 with theeffective hole mass m*h = 0.78m0 [16] and the low frequency dielectric constant j = 11.3[17]. This indicates that the two samples fall well on the insulating side of the MITsince the residual doping concentration NA ±± ND is 7.5 � 1017 and 4.4 � 1017 cm±±3 forCIT#W2 and CIT#W3, respectively; the compensation ratio ND/NA lies between 10%and 20%. Hence, at low temperatures, and due to the low carrier concentration of oursamples, conduction by the VRH mechanism is to be expected.

Results and Discussion The plot of logarithm of the resistivity r for samples CIT#W2and CIT#W3 (not shown here, see Iqbal et al. [18]), and of CIS1 and CIS2 of n-type

84 M. Iqbal et al.

CuInSe2 published earlier by us [11] as a function of T±±1/4, shows, as expected, an al-most linear dependence. From the intercept and the gradient, the parameters T0 and r0

can be calculated. However, to get a better estimate of these parameters, two differentapproaches are found in the literature. The first, employed by us earlier [19], and Fin-layson [20], consist in varying the value s and then fit the experimental data to theVRH law r � r0 exp �T0=T�s with r0 and T0 as adjustable parameters. The best value ofs is obtained from minimizing the percentage deviation which is expressed as

% dev � 1n

� �Pni�1

100rfr0 exp �T0=Ti�s ÿ rig1=2

� �;

where ri and Ti are the experimental values.Capoen [21] and Zhang et al. [9] have followed another approach used by Zabrods-

kii [22] in the case of n-Ge. By defining the parameter w�T� � lnd ln �s�d ln �T�� �

, we get

from the hopping generic expression: w(T) = ln (s) + s ln (T0) ±± s ln (T). Thus, from aplot of w(T) against ln (T), the exponent s and the corresponding hopping temperatureranges are defined.

In Fig. 1, using the second approach, we plot w(T) as a function of ln (T) calculatedfrom the r versus T data of samples CIT#W2, CIT#W3, CIS1 and CIS2. Two tempera-ture regions having different slopes are observed. At lower temperature the data canbe fitted to a straight line with the slope s very close to 0.25. However, in this kind oflocal analysis, a slight dispersion of the calculated points w(T) is probably due to the

fast change of the resistance valueswith respect to an infinitesimal tem-perature change. At higher tempera-tures the slope is around unity. Thisconfirms that below 50 K the conduc-tion is of Mott type, whereas at highertemperatures it is due to the activationof charge carriers. The values of s andthose of T0 calculated by this methodfor samples CIT#W2, CIT#W3, CIS1and CIS2 are given in Table 1, togetherwith the values deduced from the meth-od of percentage deviation. With theknowledge of T0, as given in Table 1,the ratio Rhop/x � 0.4(T0/T)1/4 betweenthe optimal hopping length Rhop andthe localization length x is calculated.

Variable Range Hopping Regime of Copper Ternary Compounds 85

Fig. 1. Reduced activation energy of conduc-tion w(T) versus ln (T) for CIT#W2,CIT#W3, CIS1 and CIS2

86 M. Iqbal et al.

Ta b l e 1Values of the VRH conduction parameters obtained by analyzing the electrical resistivitydata using two different methods; percentage deviation and Zabrodskii's approaches. DTrepresents the temperature range in which Mott's type of VRH conduction is observed

samples percentage deviation approach Zabrodskii's approach

DT (K) r0 (10±±3 Wcm) T0 (104 K) s T0 (104 K) s

CIT#W2 4.5 to 47 9.90 1.84 0.25 1.23 0.26CIT#W3 11 to 84 23.0 1.76 0.25 2.40 0.24CIS1 4 to 20 65 0.86 0.22 0.57 0.25CIS2 4 to 20 118 0.94 0.25 1.13 0.23

Fig. 2. ln [(r/r0)/A] versus ln (T/TX) for CIT#W2, CIT#W3, CIS1, CIS2, and CdSe. Inset: theoreti-cal fits for CIT#W2, CIT#W3, CIS1, CIS2 (see text)

One can verify that Rhop/x decreases with increasing temperature, as expected, butalways remains larger than unity.

To check whether our data of CIT#W2, CIT#W3, CIS1 and CIS2 fall on a universalcurve of ln �r=r0�=A versus ln �T=TX� proposed by Aharony et al. [13], the followingprocedure was used. With a given value of r0, the r versus T data were fitted to theequation ln �r=r0� � Af �T=TX� with A and TX as adjustable parameters. r0 obtainedfrom Zabrodskii's analysis was used as a guideline. This process was repeated with differ-ent values of r0 until a set of r0, A and TX was obtained. The best values of r0, A and TX

were taken that gave the least deviation between the theoretical and experimental values.In Fig. 2 we plot ln �r=r0�=A against ln �T=TX� for the data of samples CIT#W2,

CIT#W3, CIS1 and CIS2 with the values of r0, A and TX given in Table 2. In the samefigure we plot the data of sample #4 of CdSe [13] and the data of Si:B reported by Daiet al. [23]. With no reported values of r0 for both samples, a preliminary estimation isobtained using the percentage deviation method. With these values of r0 the data arefitted to Aharony's universal function with A and TX as variable parameters. The bestfit, thus obtained with the parameters given in Table 2, is also shown in Fig. 2. Forbetter perception and clarity, the theoretical fits to the data of samples CIS1, CIS2,CIT#W2, and CIT#W3 are also shown as insert in the same figure. Thus, we find, asexpected from the scaling model of Aharony et al. (but shown only for different sam-ples of the same semiconductor, CdSe), that the temperature dependence of the electri-cal resistivity in the VRH regime can be represented by a universal curve.

Furthermore, with the knowledge of A and TX the hopping characteristic tempera-tures TMott and TES can be calculated. These, in turn, can be used to estimate the criti-cal temperature Tc that corresponds to the transition from Mott to Efros and Shklovs-kii's type of VRH. The values of TMott, TES and Tc calculated for the semiconductorsstudied in the present work are also given in Table 2. The crossover from Mott to ESVRH below 2 K in CdSe and the absence of this transition in CuInSe2 and CuInTe2 arein agreement with the calculated values of Tc.

However, it must be noted that we used here the criterium of Rosenbaum [24], whoobtained Tc � 16T2

ES=TMott by equating the average hopping energy in the two regimeswhen the crossover occurs. To give crossover points for the conduction mechanism bytransforming between two regimes which do not have a common limit is quite question-able, since there must exist a more or less broad transition region. In that sense, Shli-mak et al. [12] use another criterium; at the temperature Tc, the width of the «optimal

Variable Range Hopping Regime of Copper Ternary Compounds 87

Ta b l e 2Parameters r0, A, TX obtained from Aharony's model for CIT#W2, CIT#W3, CIS1 andCIS2. The characteristic temperatures TMott, TES, and Tc are calculated for the studiedsamples

samples r0 (Wcm) A TX (K) T 0Mott (K) T 0ES (K) Tc(K)

CIT#W2 0.013 49.33 0.0018 10598.7 19.6 0.58CIT#W3 0.019 40.89 0.005 14033.2 37.77 1.63CIS1 0.065 50.04 0.00007 438.9 0.78 0.02CIS2 0.12 31.25 0.0008 772.9 3.56 0.26CdSe:In 1.174*) 6.95 0.011 25.26 2.36 3.53Si:B 0.013 7.2 0.028 76.85 6.67 9.26

*) The value of the resistance prefactor (in W).

band» D(T) of localized levels which are involved in the hopping conduction becomesequal to the half width of the Coulomb gap. This leads to Tc � 2:7T2

ES=TMott. The ap-proach of Shlimak means that the Coulomb gap is effective only for temperatures low-er than Tc, this comes from what he calls a «temperature induced smearing of the Cou-lomb gap» [25].

Conclusion In conclusion, it has been established that the temperature dependence ofthe electrical resistivity of semiconductors belonging to group IV, II±VI and I±III±VI2

that follows a variable range hopping conduction mechanism can be expressed by ascaling expression of the form ln �r=r0� � Af �T=TX� according to the model of Ahar-ony et al. The r versus T data of the various systems, when plotted as ln �r=r0�=Aagainst ln �T=TX�, fall on the same curve. The temperature range of validity of thisuniversal behaviour has been extended towards higher temperatures. In this high tem-perature range no deviation of the data from the theoretical curve is noted.

In addition, even if the exact analytical expression remains questionable, the criticaltemperature Tc, related to the crossover from Mott to ES type VRH conduction hasbeen derived from the knowledge of A and TX.

Acknowledgements This work was supported by grants from EEC (Contract #CI1-CT-92-0099VE) and the Franco±Venezuelian cooperation through CEFI and CONICIT.

References

[1] N.F. Mott, Phil. Mag. 6, 287 (1961).[2] N.F. Mott, J. Non-Cryst. Solids 1, 1 (1968).[3] A.L. Efros and B.I. Shklovskii, J. Phys. C 8, L49 (1975).[4] W.N. Shafarman and T.G. Castner, Phys. Rev. B 33, 3570 (1986).[5] F.R. Allen and C.J. Adkins, Phil. Mag. 26, 1027 (1972).[6] F. Tremblay, M. Pepper, D. Ritchie, D.C. Peacock, J.E.F. Frost, and G.A.C. Jones, Phys. Rev.

B 39, 8059 (1989).[7] G. Biskupski, H. Dubois, and A. Briggs, J. Phys. C 21, 333 (1988).[8] M. Benzaquen, D. Walsh, and K. Mazuruk, Phys. Rev. B 38, 10933 (1988).[9] Y. Zhang, P. Dai, M. Levy, and M.P. Sarachik, Phys. Rev. Lett. 64, 2687 (1990).

[10] N.V. Agrinskaya and A.N. Aleshin, Soviet Phys. ±± Solid State 31, 1996 (1989).[11] S.M. Wasim, L. Essaleh, J. Galibert, and J. LeÂotin, phys. stat. sol. (a) 144, 149 (1994).[12] I.S. Shlimak, M. Kaveh, M. Yosefin, M. Lea, and P. Fozooni, Phys. Rev. Lett. 68, 3076 (1992).[13] A. Aharony, Y. Zhang, and M.P. Sarachik, Phys. Rev. Lett. 68, 3900 (1992).[14] L. Essaleh, S.M. Wasim, J. Galibert, and J. LeÂotin, phys. stat. sol. (b) 189, 209 (1995).[15] M.A. Arsene, A. Albacete, F. Voillot, J.P. Peyrade, A. Barra, J. Galibert, S.M. Wasim, and

E. HernaÂndez, J. Cryst. Growth 158, 97 (1996).[16] Y. Riede, H. Neumann, H. Sobota, R. Tomlinson, D. Elliott, and L. Haworth, Solid State

Commun. 33, 557 (1980).[17] R. Marquez and C. RincoÂn, phys. stat. sol. (b) 191, 115 (1995).[18] M. Iqbal, 8th Internat. Conf. Hopping and Related Phenomena, Sept. 7 to 9, 1999, Murcia

(Spain); Ph. D. Thesis, Toulouse III (France), 1998.[19] L. Essaleh, J. Galibert, S.M. Wasim, E. HernaÂndez, and J. LeÂotin, Phys. Rev. B 50, 18040 (1994).[20] D.M. Finlayson and P.J. Mason, J. Phys. C 19, L299 (1986).[21] B. Capoen, Ph. D. Thesis, Lille, 1993.[22] A.G. Zabrodskii, Soviet Phys. ±± Semicond. 14, 670 (1980).[23] P. Dai, Y. Zhang, and M.P. Sarachik, Phys. Rev. Lett. 66, 1914 (1991).[24] R. Rosenbaum, Phys. Rev. B 44, 3599 (1991).[25] I.S. Shlimak, M. Kaveh, R. Ussyshkin, V. Ginodman, S.D. Baranovskii, P. Thomas, H. Vaupel,

and R.W. van der Heijden, Phys. Rev. Lett. 75, 4764 (1995).

88 M. Iqbal et al.: Variable Range Hopping Regime of Copper Ternary Compounds