Solid State Communications, Vol. 60, No. 4, pp. 407-410, 1986. 0038-1098/86 $3.00 + .00 Printed in Great Britain. Pergamon Journals Ltd.

NON-CRITICAL BEHAVIOUR OF THE HALL-COEFFICIENT AT THE MOBILITY EDGE

E. Tousson and Z. Ovadyahu*

Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, 84120, Israel

(Received 24 June 1986 by S. Alexander)

Measurements of the Hall-coefficient, R~ for In2 03-x samples with K v l = 0.03 to ~ 8.5 are reported. In this range of disorder, the low-temperature conductivity varies over 4 decades• Ru however, exhibits "classical" met- allic behaviour down to 1.4 K.

THE BEHAVIOUR OF the Hall coefficient, R H at the metal-insulator-transition (IVlIT) has been the subject of several recent investigations [1-3]• According to the scaling theory of Shapiro and Abrahams, R~ is expected to retain its "metallic" character as the system crosses the MIT. Namely, R H is predicted to be temperature independent up to (and possibly beyond) the critical degree of disorder needed to bring about the MIT.

In this letter we report on measurements of the Hall-coefficient for In20~_ x samples that are near the metal insulator transition. The experimental results demonstrate that R H, measured at a finite scale, is a continuous function at the MIT in agreement with the scaling theory [1 ].

The In20a_x crystalline samples used in this study were ~ 2000 A thick and their carrier densities (inferred from Rg measured at room temperature), ranged between 1019 cm -a to 6" 1019 cm -a. In the following, different samples are labelled by their room-temperature conduc- tivity, trRr or K e l values. + Samples with KFI "~ 0.03 to K v l ~ 8.5" were studied in this research. This range is wide enough to include conducting as well as insulating samples (i.e., samples with o( T -+ O) > 0 and o( T ~ O) = 0 respectively• The low temperature conductivities of these samples spanned almost 4 orders of magnitude• Our main result is that RH remains virtually constant throughout this series of samples and in the entire temperature range studied (1.4 to 300 K).

The necessity of keeping the electric and magnetic fields sufficiently small for a meaningful comparison of such studies with the theory, makes a phase-sensitive technique mandatory. Our measurements were made in a 4He glass dewar. A split-coil electromagnet (wired on a non-magnetic bore) was employed to generate an alter- nating field of ~ 160 Oe (r.m.s) and frequency 13Hz for the Hall-effect experiment. A d.c. current was main- tained along the sample and the a.c. transverse voltage

*Present address: Racah Institute, The Hebrew Univer- sity, Jerusalem.

2 2/3 1/3 5 / 3 +(K~.I was taken as: (31r) h(RH) ORT/e )

was monitored by a PAR-124 lock.in amplifier. The latter was phase-synchronized to 90 ° of the voltage induced in a small pick-up coil mounted on the (non- conducting) sample-probe. The Hall-voltage, V H was derived by properly summing the final readings obtained for each of the two opposite current directions. In all cases considered below, it was established that VH is linear with both, bias-current and magnetic field.

a(T) curves of typical conducting and insulating samples are given in Figs• 1 and 2 respectively. For the former group o(T) is empirically found to follow: o(T) = Oo + Ao T x with x = 0.3-0.6• The best fit parameters for several such samples (taking x = 0.5) are depicted in Fig. 3. A dependence of this form may reflect Coulomb correlations for which theory predicts [4].

o = Oo + C(~ - 2F)D -1'2 T 1/2, (1)

where F, is the screening parameter, D is the diffusion

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T ( K ) Fig. 1. Conductivity vs temperatures for In: O a_x samples with KFI values (top to bottom) of: 8.48, 3.99, 2.43, 2.17, 1.92.

407

408 NON-CRITICAL BEHAVIOUR OF THE HALL-COEFFICIENT Vol. 60, No. 4

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T ( K ) Fig. 2• Conductivity vs temperature for samples with Kel values (top to bottom) of: 0.50, 0•33, 0.19, 0.11, 0.10, 0.03. The conductivity decreases with decreasing temperature at least as fast as logarithmically which, by extrapolation, gives in each case zero-conductivity as T ~ 0 .

4 C T 120 ,,¢- E i.o T

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Fig. 3. Best-fit parameters (Oo and Ao, depicted by circles and squares respectively) for the 4 bottom samples of Fig. 1 as a function of disorder• The solid line is given by: Oo = (ORT--48) °"82. OM denotes Mott's minimum- metallic conductivity = 0.03e 2/ha.

coefficient and C is a numerical constant. Alternatively, weak-localization [5] may account for the observed o(T). According to the latter:

o(T) = Ae 2/1~ + Be 2/hLi,, (2)

where ~ is the localization length, Lin = ( D T i n ) 1/2 , the inelastic-diffusion length and A and B are numerical constants• Thus, if Lin ~ T -1/2 , (2) could explain the data of Fig. 1. However, there are difficulties with reconciling the values of the parameter Ao with either theoretical model• Equation (1) predicts Ao cc on~ ~ in

08

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T ( K )

Fig. 4. The Hall coefficient as a function of temperature for typical samples in the liquid He range. The inset depicts Rn at room temperature vs KF 1 (log-log scale).

disagreement with the data (Fig. 3). This is in addition to the fact that F in In2 O3_x is of the order [6] of ", 0.9 and the correction to the conductivity might be expected to have the opposite sign. Equation (2) can account for the data if Lin is assumed to be fairly independent of on~, (in addition to scaling like T-l/z). This apparently means that the relevant inelastic-mean- free-time, Tin follows Tin ~ o~/T . Previous [7] inde- pendent studies did indicate such a possibility. The mechanism responsible for such an inelastic scattering rate remains, however, obscure. We conclude that neither model can, at present, fully account for the observed o(r).

It is worth noting that the o(T) data [8] on Nb:Si show similar features to these discussed here for In2 Oa-x samples. Hertel et al. [8] have interpreted their results and the observation of a disorder independent Ao using McMillan's theory [9] (which purports to include both localization and interaction effects)• Other possible explanations for the apparent "insensitivity" of Ao to disorder will be discussed elsewhere.

For samples with KFI < 1, the experimental o(T) (Fig. 2) could not be fitted to any simple law except for the most resistive sample that follows o ( T ) ~ e x p [--(To~T) x/4 ] with To ~ 3 K , a form consistent with Mott's varible-range hopping law [10].

The behaviour of RH as a function of temperature in the liquid He range is illustrated in Fig. 4 for typical samples from each range of KFI studied (i.e., KFI > 1 and KFI< 1). For each sample Rt-t was measured at room-temperature and for most, also a't 77K. Our £mdings, summarized in Fig. 5 are next compared with current interaction and localization theories.

Interaction theory [11] predicts that RH will change with temperature along ~,i,tla the change .in conductivity• It is expected that 6RH[RH =2~o[o, namely, the change in RH should be twice that of o. This expectation is not borne out by our experiments.

Vol. 60, No. 4

50

t ~

O.5

NON-CRITICAL BEHAVIOUR OF THE HALL-COEFFICIENT

0

0

0 0

. . . . . . . . . _i°_?_ . . . . T I o

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I I 0.01 0.1 I I0

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Fig. 5. The variation of RH and the longitudinal resis- tivity, p, for the studied samples in the temperature range 1.4-300 K. Pl.4/PR.T. (empty circles)is the ratio of # at 1.4 K to/9 at room-temperature. RH[RH(R.T.) is the range of RH values measured normalized to its value at room temperature.

Focusing attention on the Kel > 1 regime (for which the theory applies), we find e.g., that for the samples with KFl = 8.48, 3.99, 2.43, 2.17 and 1.92 the conductivity increases by 2.8, 8, 10, 26 and 39% respectively (between 1.4 and 77 K). In the same temperature range, RH shows no systematic dependence. The maximum variation in the RH(T) data in this range amounts to less than 0.4, 1.5, 6, 0.5 and 2% respectively. This observation is consistent with earlier claims [12] that interaction effects are not dominant in the magneto-transport pro- perties of In 2 O3_x samples.

Shapiro and Abrahams [1 ] have recently developed a scaling theory for the Hall effect in disordered metals. Their results suggest that RH remains constant as a function of scale as the mobility-edge is approached from the metallic side. The relevant length scale in our Hall experiment is essentially dictated by Lin(T). For conducting In20a-x films Lin is ~ 50 A at ~ 300 K and

2000 A at ~ 1 K. The perturbation introduced by H = 160 Oe to the longitudinalconductivity usually amounted to less than 0.01% (Aa(H) being always positive). The magnetic length, LH = (c~eH) 1/2, associated with this field is ~ 1800A. One then expects that the field will not severely limit the scale of the experiment for tem- peratures above ~ 1K. The observation of a temperature independent R~ for samples that are in the vicinity of the MIT seems, therefore, to be in agreement with the prediction of [1] (though, for reasons given below, only in a limited sense).

Our findings may seem to extend the "classical" result: RH(T)= constant, independent of disorder, to

samples that are apparently insulating, but measured at finite temperatures. We note, however, that in these samples the relevant length could be considerably shorter than that of the conducting samples at the same temper- ature. Also, it is quite possible that even in the most resistive sample a significant number of carriers are excited to states above the mobility edge. It may then be necessary to view these R~(T) data as characteristic of effectively conducting samples (with highly reduced mobilities) rather than as insulators. It would be of interest to investigate the behaviour of R~t(T) in such samples at very low temperatures where hopping- conductivity should become the evident mechanism. For conducting samples, it is recognized that any finite temperature measurement places a limit on how close can one meaningfully approach the transition. At the same time, for measurements below ~ 1 K the magnetic field used may severely limit the scale otherwise set by the temperature. For the issues raised above, a temper- ature of ~ 1.4 K may be considered as quite small. The fermi-energy, EF in In2 Oa-x is of the order of 2000 K. In terms of the reduced temperature, KB T/EF, 1.4 K is tantamount to a ~ 50 mk measurement on a system such as Ge:Sb where EF ~ 50 K at the transition.

In summary, we have measured the Hall coefficient for In2 O3-x samples near the MIT using extremely low magnetic fields and reduced temperatures down to

5" 10 -4. Our results demonstrate that a considerable reduction in the system conductivity can be obtained even in the range where the temperature coefficient of a is positive (and increases rapidly with disorder) while RH remains virtually constant. Intuitively speaking, these results show that in this system the reduced conductivity brought about by the spatial disorder is a mobility effect, consistent with the one electron picture underlying the localization theory.

Acknowledgements - We acknowledge discussions with B. Shapiro, M. Kaveh, D.E. Prober and G. Deutscher. This research is partially supported by a grant adminis- tered by the Fund for Basic Research of the Israel Academy for Sciences and Humanities. One of us (E.T.) gratefully acknowledges support from the Wolf Foun- dation.

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4!0 NON-CRITICAL BEHAVIOUR OF THE HALL-COEFFICIENT Vol. 60, No. 4

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