PHYSICAL REVIEW B VOLUME 52, NUMBER 24 15 DECEMBER 1995-II

Low-temperature electron mobility studied by cyclotron resonance in ultrapure GaAs crystals

M. Kozhevnikov, B.M. Ashkinadze, E. Cohen, and Arza RonSolid State Institute, Technion Isra-el Institute of Technology, Haifa 32000, Israel

(Received 18 August 1995)

We report on a systematic study of the electron scattering processes in ultrapure GaAs crystals overthe temperature range 1.6—100 K by using the classical cyclotron resonance (CR) for a low density ofphotogenerated electrons ( n, —10' cm ). The highest electron mobility extracted from the CR data is

2.2X 10 cm /V sec at T =1.6 K. The electron scattering due to the acoustic-phonon piezoelectric po-tential is found to be the main intrinsic electron-phonon scattering mechanism in the temperature range1.6 & T & 30 K. Electron scattering by neutral impurities is shown to be significant even in these ultra-

pure GaAs crystals (unlike Ge and Si). The acoustic-phonon piezoelectric and deformation-potentialcoefficients as well as the impurity concentrations (-10' cm ) were obtained from the analysis of theelectron-scattering-rate dependence on temperature. A CR line narrowing is observed under increasingmicrowave power and is interpreted as due to a decrease in the acoustic-phonon piezoelectric scatteringe%ciency with electron heating. The electron e6'ective mass determined from the CR magnetic Geld ap-plied along [001] is m *= (0.0662+0.0002}mo.

I. INTRODUCTION

Carrier-scattering processes by imperfections play acrucial role in the transport phenomena in semiconduc-tors. For Ge and Si crystals there are detailed experi-ments and a good theoretical understanding of thedifferent scattering mechanisms for electrons andholes. '2

Despite the great progress in the growth and applica-tion of GaAs, the intrinsic electron-phonon scatteringmechanisms at low temperatures could not be investigat-ed experimentally, since the scattering of electrons by re-sidual impurities usually governs the mobility for T (50K.

One of the most powerful methods for the study ofcarrier-scattering mechanisms in semiconductors is theclassical cyclotron resonance (CR). In GaAs, CR wasstudied only in the quantum limit, where a high magnet-ic field and radiation frequency enabled the study of low-mobility GaAs samples. However, the scattering pro-cesses are different from those expected in the classicallimit.

Ultrapure epitaxial GaAs layers with a shallow impuri-ty concentration in the range 10' —10' cm can begrown by vapor-phase epitaxy (VPE) in a chlorine sys-tem. These samples exhibit a strong polariton photo-luminescence (PL) and a long electron momentum relax-ation time, resulting in an electron mobility of up to3 X 10 cm /V sec at 2 K. The availability of such ultra-pure samples enables the use of classical CR for studyingthe intrinsic electron-scattering processes at low tempera-tures.

The expression for the microwave (MW) power absorp-tion of linearly polarized radiation with frequency co andelectric-field strength E due to an electron rotating in themagnetic field B is given in the classical case by

1+(co +co, )rP(co)=on Ei

[I+(co —co, )r ] +4co2r

Here o.o= ne ~/I ' is the dc conductivity, ~ is momen-tum relaxation time, co, =eB/m*c is the cyclotron fre-quency, and m ' is the electron effective mass. If co,~) 1,the resonance occurs at co=co, and ~ can be obtainedfrom the full width of the CR line at half its maximum,AB, by the expression

co, i.=2B„/b, B, (2)

where B„ is the resonance magnetic field. ~ ' is the totalmomentum relaxation rate and it is expressed as a sum ofthe inverse-relaxation times for each of the scatteringmechanisms.

The momentum relaxation time, which is derived fromthe CR line shape, is the same as that determined from dcmeasurements, as long as the magnetic field is weak andthe cyclotron radius is large compared with the forcerange of the scattering center. This is the case of theclassical limit (fico( kT), wherein the scattering processesare not affected by the magnetic field.

This paper presents an experimental study of the classi-cal electron CR in photoexcited, ultrapure GaAs sam-ples. For the MW frequency used (35.6 GHz), the CRoccurs at a magnetic field of -0.085 T. Under these con-ditions, electron-scattering processes can be studied in ul-trapure GaAs crystals in the temperature range 1.6—100K. We compare the extracted low-temperature mobilityvalues for different GaAs samples grown by vapor-phaseepitaxy (VPE) and molecular-beam epitaxy (MBE) tech-niques. The analysis of the temperature-dependent mo-bility yields the relative significance of electron-phononand electron-impurity scattering processes.

The paper is laid out as follows: In Sec. II, the experi-mental setup, the samples used, and the experimental re-sults are presented. An analysis and a comparison of thetheoretical predictions with the experimental results aregiven in Sec. III. In Sec. IV, we discuss our findings andconclude the paper.

0163-1829/95/52(24)/17165(7)/$06. 00 17 165 1995 The American Physical Society

1'7 166 KOZHEVNIKOV, ASHKINADZE, COHEN, AND RON 52

II. EXPERIMENTAL PROCEDURE AND RESULTS

Ultrapure samples grown by the VPE and MBEmethods with layer widths 5 —10 pm are studied. A de-tailed analysis will be presented for three typical GaAssamples denoted A (VPE), 8 (VPE), and C (MBE). Theydiffer in the background impurity concentrations. Thesample is placed at the antinode of the MW electric fieldin an 8-mm waveguide which is short circuited by aplunger at one end. The MW radiation rejected from throm ep unger was directed onto a diode detector by a circula-tor. The waveguide is immersed either in liquid He or incold He gas so that the temperature is varied in the rangeT=1.6—100 K. The sample is illuminated by eitherabove-gap or below-gap laser light (from a He-Ne or atunable dye laser) through a pinhole in the waveguide.The light intensity II was low so that the photogeneratedfree-electron density was of the order of n & 10'n, cm ata photoexcitation level of 10 mW/cm . The excitationwas modulated at a frequency varying in the ran e10—400 Hz.

in e range

A stabilized 35.6-GHz Gunn diode was used as theMW source. An electrically controlled attenuator al-lowed us to vary the incident MW power, PM~, continu-ously so that the power incident on the sample was in therange 10 —10 mW. The MW radiation measured by thediode detector was modulated by the photoexcitation ofthe sample and the modulation signal was proportional tothe MW power absorbed by the photoexcited free elec-trons. The external dc magnetic field 8 was appliedparallel to the [001] direction and was swept over therange 0—0.25 T. The MW electric field was perpendicu-lar to B, and the absorbed MW power was measured as afunction of scanned magnetic field. The aforementionedII and PMw values were set at the minimal possiblevalues so that they do not affect the CR line shape.

The CR line shape has been measured under photoex-citation by a dye laser at the exciton resonance,EL = 1.515 eV, and also by a He-Ne laser at El = 1.96 eV.T ical CRyp' traces at various temperatures are presentedin Fig. 1, all normalized to the same peak value (sam le

~ ~

samp eL

= 1.96 eV). Similar traces are observed for the oth-er samples and for EL =1.515 eV. The CR line isbroadened as the lattice temperature T increases. Thetemperature ependence of the inverse relaxation timn ime,

( ), extracted from the CR lines [by using Eq. (1)], isshown in Fig. 2 for the three samples. The r '(T) curvesare different for each GaAs sample in the ran e1.6(T+3( 0 K (due to diFerent impurity concentrations),

p e in e range

but are the same for T) 30 K.We observed that the CR line is modified as PM~ in-

creases. The CR linewidth first decreases with increasingPMw (for example, up to I'Mw —1 pW for sample A) andthen broadens again with further power increase. Thisbehavior of the CR line shape for sample A is demon-strated in the inset of Fig. 3. A similar behavior is alsoobserved for the other samples having a lower mobility,but the range of PMw where the CR linewidth varies is

i erent, as shown in Fig. 3 for two samples. The CRlinewidth decrease in the terms of ~ ' is also marked bythe arrows in Fig. 2 for all three samples (at T-2 K).

Bulk GaAs (sample A)E„=1.96eV, !L=10mW/crn, PMw=0. 1pW

CD

o

oM

G5

DCD

0o

CL

I

IC

'I~ I

I

II

II

I

II

II

II

t

I

I

I

I

I

\

\

I

1

\

T= 2K10K-----40K- --

0.04 0.06I

0.08

B (Tesla)0.10 0.12

FIG. 1. Photoinduced CR traces of ultrapure GaAs observedat three lattice temperatures.

Figure 4 shows the photoluminescence spectra in theexciton region. Since the measured PI. intensity in therange of donor-acceptor pair recombination (not shown)is 200 times weaker than that of the excitons for all threesamples, the impurity concentration is genuinely ul-tralow.

Several authors have extracted the momentum relaxa-tion time from the optically detected CR (ODCR). Using

Bulk GaAs

EL——

O(Dio

CO

CO

0.5 I I ~ I I

10

Calculation:Total fit

neutral impurity contribution

100

Temperature (K)

FIG. 2. The temperature dependence of the electron inverserelaxation time for three samples. The values were obtainedfrom CR traces (as in Fig. 1) by using Eq. (1). The model-Atting

[Eq. (3)] curves are shown by solid lines. The decrease of rthat is due to electron heating (by increased PM~ at T-2 K ismarked by the arrows.

17 16752 LOW-TEMPERATURE ELECTRON MOBILITY STUDIED BY. . .

~ ~Bulk GaAs

21 5 5e IL=10mW/cm, 7=2K

Bulk GaAs

EL-—1.96eV, IL-—10IW/cm, T=2=2K

o 1.0-

—~—sample—o—sample

0~o—o0

~ S~ W=E~ro—o

CR traces (sample A)

CR15ev)

0OO~~

A

D

~oQo~

0 4 I ~ ~ ~I

~ ~ ~ I I I I ' I I

10" 10

0.06

s ~ a ~ I ~ ~/

~ ~ ~ s ~ I ~ t

10 10PM~ (mW)

1

0.08B (Tesla)

10'

I

0.10

10

nce of ~ ' on the microwave power.FIG. 3. he dependence o ~ onThe inset shows t eh CR line shape traces for samp e adifferent PM~ values.

~ 0

0~~6$

OE

CL

CJ0)0

0~O

h d the hotoluminescence quenching underthis met o, e p

ic field. Typica 1 ODCR traces are shown in Figs. 5 a orsample A) and 5(b) (for sample C), monitored a p-lariton line (1.515 eV) under photoexcitation at EL =1.96

arison, the CR line shapes are also shown.It is seen that the line shape of t e

Bulk GaAsX

0 . 0

0.04I

0.06I

0.08

B (Tesla)

I

0.10 0.12

n the ODCR and CR traces.FIG. 5. A comparison between t eed monitoring the exciton-The ODCR data were observed m

polariton PL line (1.515 eV).

from thatoft e w eh CR h n both are observed under theture MW power, and photoex-same conditions (temperature,

citation intensity).

III. ANALYSIS

M~ ~C

C5

V)

Q)

CL

1.512 1.513I I

1.514 1.515Photon energy (eV)

I

1.516

nce s ectra in the exciton regionFIG. 4. The photoluminescence spec' '

ionb) B and (c(observed witd th P =0) for samples (a) A, (

t d of free-electron scattenngG A b means of CR mea-

r a s stematic stu y o

kT fi ). I hrried out in a s y m

nl in the uantum limit ( & co .surements, on y in ethe efficiency of electron scattenng y

'pb im urities iscase, the e ciency

the electron-phonon scattering'

ms could be studied in samples with impuri y'' Th t 1 ltconcentration otterin rates (mea-indicate ad that the electron-phonon scattering

p 10 T) could be ex-sured under magnetic pnetic fields up tod b the classical limit expressions (8~

hat the theoretical prediction in thequantum imi i eq 1 't differs significantly from that or e c

nisms at B—+0it. The electron-scattering mechanism

unavailabihty of very pure samp es.r resent studies extend the electron

a'

l l' it. This allows us to obtainGaAs to the classica imi .lv f th main electron-scatteringthe relativive contributions o e m

'

s in the temperature rangeanisms in GaAs crysta s in emec an'd fit the experimental results2—100 K. The model use to

es that the total momentum relaxation rate is givenassumes t at t e ot times for each contrib-f the inverse-re axation imey a sum o

17 168 KOZHEVNIKOV, ASHKINADZE, COHEN, AND RON 52

uting electron-scattering mechanism,

7 '(T)=r„+r,; '+rp, '+rgb' . (3)

Here ~„and ~,.;' are the electron scattering rates by

neutral and ionized impurities, respectively; r, ' and ~d'

are the acoustic-phonon scattering rates via thepiezoelectric and the deformation potentials, respectively.The contribution of the electron scattering by polar opti-cal phonons via the Frohlich interaction takes over onlyat the upper edge of our temperature range, and there-fore, it was not included in the fitting procedure. Thetheoretical expressions for each process, ' '" as well asthe parameters extracted from the fitting are summarizedin Table I.

The following notation is used in Table I: N& is theneutral impurity density concentration, Xl is the totalconcentration of ionized donors and acceptors, T, is theelectron temperature, T is the lattice temperature, c. is thedielectric constant, co is the permittivity of free space, Eis the electromechanical coupling constant, E, is thedeformation-potential constant, p is the crystal density,and s is an averaged velocity of sound. '

It is seen from Table I that the neutral impurityscattering rate is temperature independent. The scatter-ing of electrons by singly ionized impurities is describedby the Conwell-Weisskopf equation' and the inverse re-laxation time is proportional to T . For the acoustic-phonon piezoelectric scattering, ~ ' is proportional toT' and for the acoustic-phonon deformation-potentialscattering, to T ~ (at thermodynamic equilibrium,T=T, ). Note that the electron temperature may bedifferent from that of the lattice as a result of electronheating by the MW electric field.

At low temperatures ( T ~ 30 K), the observed values of

7' are different for the three samples, while at higher

temperatures they are essentially equal (Fig. 2). In thetemperature range 50—100 K, the experimental depen-dence is ~ ' —T for all three samples and, therefore, itis due to the acoustic-phono n deformation-potentialscattering. For T& 100 K, the ~ ' dependence is dom-inated by the polar optical-phonon scattering. ' Figure 2shows that the r '(T) for the purest sample A can be ap-proximated by a T' dependence in the range 4 ~ T & 50K, except for the temperature range 10—20 K (it will bediscussed below). It means that the main scatteringmechanism is due to the acoustic-phonon piezoelectricpotential (cf. Table I).

For T (10 K, the r '(T) is independent of T for sam-ple C and only weakly so for sample B. Thus, the scatter-ing rate in this temperature range is dominated byneutral-impurity scattering. This can be expected forsparsely compensated crystals with a low impurity con-tent, since under photoexcitation ionized impurities arecompletely neutralized and the ionization rate is extreme-ly small for T&10 K. Then the different ~ ' values foreach sample reAect the decrease in the impurity concen-tration. For T&30 K, the scattering by impurities isinsignificant, and the intrinsic phonon scattering becomesdominant.

In the temperature range 10—20 K, an additional in-crease of the scattering rate is observed for all three sam-ples. This is due to the fact that as the temperature in-creases, the neutral impurities are ionized and free-electron scattering by ionized impurities becomes impor-tant (r;;((r„;). As the temperature increases further, theionized-impurity contribution decreases as T . Wenote that a similar, less pronounced scattering-rate in-crease was observed in the quantum limit case.

TABLE I. Summary of theoretical expressions for each scattering process and the parameters extracted from the fitting pro-cedures.

Type of scattering

Acoustic-phonon

piezoelectric potential

Acoustic-phonon

deformation potential

rpe

dp

3+m +k~e~+2T

16+2776 KE,pT

3E T(m *'k T )'

2v'2M ps

This work

1/rp, =5X10 T' sec

(K'= 3.S9 X 10-')

1/rd =4X10 T' secdp

(E, =7.64 eV)

Parameters from references

1/r, =3.9X 10 T' sec ' (12)

1/rp, =5.7X 10 T' sec ' (13)K'=4.4X10-' (14)

1/rd =2.8X10 T secdp

(E& =7 eV) (13)

Ionized impurities1 e4Nr

1nP128+2vrm *(ceo) (k~ T, )

212''ccpkg Te

where P=1+e NI

E; =5.8 meV

N =1 SX10' cm (A)2.2 X 10' cm (8)

3.0 X 10' cm (C)

E; =5.8 meV (3)

E; =5.30—5.52 meV (13)

Neutral impurities1 = 80mcc, pA'

=aN& where a =f1 l m*e a =3.8X10 cm'sec ' (A) a =3.5X10 cm'sec ' (10)

=5.SX10 cm sec ' (B)=7.3X10 cm sec ' (C)

(m*=0.066m, v=12.5)

52 LOW-TEMPERATURE ELECTRON MOBILITY STUDIED BY. . . 17 169

The temperature-dependent densities of the ionizedand neutral impurities, XI and N~ =No NI respective-ly (No is the total impurity concentration), can be es-timated as follows. From the PL data (Fig. 4), we seethat the acceptor concentration N~ is very small in allthree samples, and hence we assume that Xo =XD, whereND is the donor concentration. Then, we may write'

10—

C)

C)

0.410

Temperature (K)

100

FICz. 6. The temperature dependence of each scattering rategiven in Table I with parameters corresponding to sample B.Also shown are the experimental scattering rate and the totalmodel fitting using Eq. (3) for the same sample.

2 exp I (p ED )I—kii T I + 1

where p is the temperature-dependent Fermi level' andEL, is the energy of the 1s donor state. For extremely lowphotoexcitation intensities, the photoexcited-electronconcentration is far less than that of donors, and there-fore, the steady-state Fermi level is approximated by itsequilibrium value. Setting this value into Eq. (3), we ob-tain the expression for the ionized-donor concentration,

21VD

1+(1+8ND/NcexpIE,

/k/GATI

)'/z

where E; =E —ED is the ionization energy of donorsand Nc is the conduction-band density of states (forGaAs, NC=8. 36X10 T cm ).

We fit the experimentally observed r '(T) by using alinear combination of all the scattering rates [Eq. (3)].The contribution of each of the electron-scattering mech-anisms to the total fitting is demonstrated in Fig. 6 forsample B. The fitting curves for all three samples understudy are shown in Fig. 2. All three demonstrate verygood agreement between the experimental data and themodel calculations. From this fitting procedure, the in-teraction coefficients and other parameters (appearing inTable I} are obtained. The parameters determined fromthe fitting for sample 8 were used without change in the

fitting procedure for the other samples. These values arein good agreement with the theoretical predictions forthe acoustic-phonon piezoelectric scattering and for theacoustic-phonon deformation-potential scattering (seeTable I}.

The donor concentration for each of the samples wasestimated from the calculated contribution of ionized-impurity scattering to the fitted r '(T) in the tempera-ture range 10—20 K. The concentration values, takingE, =5.8 meV, are given in Table I. Then, these XDvalues are used in order to determine the coe%cient a inErginsoy's formula, using the v„value obtained fromthe fitting procedure. This derived value of a is differentfor each one of the samples and exceeds slightly thatgiven by Erginsoy's formula (cf. Table I}. The discrepan-cy may be explained, on the one hand, by a low degree ofcompensation in our samples, which is not taken into ac-count here, or, on the other hand, by the theoretical pre-diction that Erginsoy's formula is not fully applicable tocompound semiconductors.

The estimated trend of the impurity concentrations ob-tained from the CR measurements is consistent with theexperimental PL spectra (Fig. 4). The PL spectrum ofsample A displays a most intense PL line that is due tothe exciton polaritons (X). In sample 8, the donor-boundexciton PL intensity exceeds that of the free exciton. Theexciton lines in sample C are much broader. The absenceof an acceptor-bound exciton line for samples A and 8shows that they have a very low compensation level.

Consider now the effect of free-electron heating by theMW field on the electron-scattering mechanisms. AsPMw increases, the free-electron energy increases so thatT, )T." Figure 3 shows that the CR linewidth de-creases initially with PMw and then increases. Underhigh PMw, a distortion of the CR line shape is observedalong with the line broadening. At T=2 K, the onlyelectron-scattering processes are those due to neutral im-purities and acoustic phonons via the piezoelectric poten-tial. For example, for the purest A sample, these contri-butions are approximately equal. Under electron heating,the neutral-impurity-scattering rate remains unchangedwhile that due to the acoustic-phonon piezoelectricscattering decreases as T, '/ (cf. Table I). Consequently,we expect a decrease of the total scattering rate r '(T)approximately to the neutral-impurity-scattering limit.Still at higher PMw, the CR line shape is affected by vari-ous nonlinear electron heating processes that may includeimpurity ionization (these will be reported elsewhere). InFig. 2, the CR linewidth narrowing due to electron heat-ing measured for each sample is marked by an arrow (atT=2 K). One can see that, indeed, r '(T) approachesthe predicted neutral-impurity-limit value. Note that thePM~ range wherein the CR line narrowing occurs isshifted to a higher power for the samples with larger im-purity concentrations, namely, from 1 pW for sample 2to 10 pW for sample C (Fig. 3). This is a result of thehigher MW absorption rate in the sample with the highermobility ( /I ).

Our experimental ODCR results show that themomentum relaxation rate, extracted from the ODCR

17 170 KOZHEVNIKOV, ASHKINADZE, COHEN, AND RON

linewidth, appears to be much larger than that deducedfrom the CR line shape, for the same PMw and IL (Fig.5). There is a basic difference between the CR andODCR methods: ODCR results from the hot electrontemperature dependence on the magnetic-field strengthand the hot electron effect on the recombination process-es. Since the PL intensity is not proportional to the elec-tron temperature, ' and, moreover, the mobility valuecannot be extracted from the dependence T, (B), ' theODCR line shape does not provide a reliable estimate ofthe carrier mobility [and hence of the r '(T)].

Finally, the very narrow electron CR line observed atT=1.6 K allows us to measure precisely the electroneffective mass. For a magnetic field applied along the[001] direction, it is m *= (0.0662+0.0002)m o. Thisvalue corresponds to the very bottom of the GaAs con-duction band, namely, to a mean electron energy of 0.14meV.

IV. SUMMARY

So far, the highest electron mobility value was reportedfor MBE-grown n-GaAs layers, with a peak mobility of-4X10 cm /V sec obtained at T=30 K. ' Our study ofelectron-scattering mechanisms shows that the electronmobility in the purest GaAs sample (AD=1.5X10'cm ) increases as the temperature decreases and reaches2.2X10 cm /Vsec at T=1.6 K. From the analysis of7 ( T) we found that this experimentally obtained mobil-ity value is only twice less than that expected for theacoustic-phonon piezoelectric scattering limit. In thecase of electron heating by the microwave field, when theelectron scattering via the acoustic-phonon piezoelectricpotential is greatly reduced (cf. Table I), the measuredmobility value increases up to 4.2 X 10 cm /V sec. Thus,we find that, in the ultrapure GaAs crystals studied here,the acoustic-phonon piezoelectric scattering is the mostsignificant process even for T& 10 K.

The difficulty to reach high electron mobility in GaAsis due to the high efficiency of impurity scattering relativeto the various phonon-assisted scattering mechanisms.Unlike the well-studied bulk Ge and Si crystals, wherethe electron-phonon scattering (due to the acoustic defor-

mation potential) dominates even down to T-2 K (atND —10' cm ), the relative contribution of electron-impurity scattering to the mobility in GaAs is higherthan that in Ge and Si by —100. This is a result of thelow value of the electron effective mass in GaAs. There-fore, the electron-impurity scattering is significant forT& 30 K even in high-quality samples and it was believedthat the electron-phonon scattering mechanisms couldnot be studied experimentally in the low-temperature lim-it. '

In conclusion, we presented a detailed study of theclassical electron cyclotron resonance and the tempera-ture dependence of the electron momentum relaxationrate r '(T) in ultrapure GaAs crystals. The results wereanalyzed by a combination of the various mechanisms ofelectron scattering. From the fit of r '(T), the electron-acoustic phonon interaction parameters as well as the im-purity concentrations were obtained. It was found thatthe electron scattering by acoustic-phonon piezoelectricpotential is the main momentum relaxation mechanism inthe purest samples at low temperatures. The CR lineshape narrowing resulting from free-electron heating bythe MW field is explained by the decrease of theacoustic-phonon piezoelectric scattering with electron en-ergy when the momentum relaxation rate decreases downto the neutral-impurity-scattering limit. It is shown that,unlike the CR measurements, the ODCR line shape doesnot provide a reliable estimate of the carrier mobilityvalue.

ACKNOWLEDGMENTS

We would like to thank Dr. V. V. Bel'kov and Dr. V.V. Rossin of the A. F. Ioffe Institute (St. Petersburg, Rus-sia) for their participation in the first stage of this studyand to Dr. V. Umansky of the Weizmann Institute(Rehovot, Israel) for supplying us with the MBE-grownsamples. The research at Technion was done in the Bar-bara and Norman Seiden Center for Advanced Opto-electronics. B.M.A. acknowledges the support of Fonda-tion Rich (France) and the Alexander Goldberg Fund ofthe Technion.

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LOW-TEMPERATURE ELECTRON MOBILITY STUDIED BY. . . 17 171

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