The magnetoresistance of electrons in InP and GaAs

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some of these speculations, more detailed cyclotron resonance information is desirable.

Acknowledgments-The author is indebted to R. K. WILLARDSONfor helpful comments, and to the Air Force Office of Scientific Research, Cambridge Research Center, and Wright Air Development Center for use of data in advance of publication. The kindness of Dr. J. F. H. CUSTERS, Diamond Research Lab., Industrial Distributors (1946) Ltd., Johannesburg, in supplying the II-B diamond is appreciated.

J. Phys. Chem. Solids

s.2

Pergamon

EFFECTS

511

REFERENCES

1. PETERSON E., SWANSON J., and TUCKER G. Bull.

Amw. Phys. Sot. 1, 117 (1956). 2. MCCLURE J. W. P/zys. Rev. 101, 1642 (1956). 3. GOLDBERGC., ADAMSE., and DAVIS R. Phys. Rev. 105, 865 (1957). 4. BEER A. C. and WILLARDSONR. K. Phys. Rev. 110, 1286 (1958). 5. BEER A., ARMSTRONGJ., and GREENBERG I. Phys. Rev. 107, 1506 (1957). 6. DEXTER R., ZEICW H., and LAX B. Phys. Ben 104, 637 (1956). 7. JOHNSONV. A. and WHITSELL W. J., Phys. Rev. 89. 941 (1953).

Press 1959. Vol. 8. pp. 511-515.

Printed in Great Britain

THE MAGNETORESISTANCE OF ELECTRONS InP AND GaAs*

IN

M. GLICKSMAN RCA Laboratories, AVAILABLE samples of single crystal indium phosphide and gallium arsenide contain appreciable concentration& 2, of impurities. It has been concluded from mobility observations that the purest indium phosphide containsc3) about 10la cmT3 impurities and the purest gallium arsenidec4) about 1017 cm-3 impurities. In spite of the high concentrations it is possible to deduce information about the anisotropy of the mobility (and indirectly the effective mass) and about the form of the scattering from galvanomagnetic measurements. Reported herein are observations made on single crystal slabs of n-type indium phosphide and gallium arsenide which indicate that (a) in both materials the bottom of the conduction band is spherically symmetric in k-space, and (b) in both materials the scattering is weakly dependent on the energy. Samples were prepared so that the magnetoresistance could be observed for the current parallel to a [l lo] direction or parallel to a [loo] direction. In a cubic crystal the first of these provides suffil This work was supported Force.

in part by the U.S.

Air

Princeton,

N. J.

cient measurements to deduce’s) the low-field magnetoresistance tensor, the second set serving as a check. It was found during the course of these investigations that the type and size of the side contacts could materially affect t6) the results. In particular, either the use of “large” area soldered side contacts or bridge-shape samples’7) introduces a contribution to the observed change in resistance in the magnetic field caused by a shorting out of the Hall voltage in the vicinity of the contacts. This effect is strongest when the magnetic field is in the plane of the contacts, and it is observed that the transverse magnetoresistance for crystals with such contacts is larger with the magnetic field in the plane of the contacts. The magnitude of this spurious magnetoresistance will depend on the mobility, and on the form of the contact shorting. In the case of bridge samples, a contribution of about O-3 x lo6 cm4/V2se? to Ap/p&P was observed for n-type germanium, indium phosphide, and gallium arsenide, all having mobilities of about 4000 cm2/V-sec. For germanium this corresponds to only 3-4 per cent of the bulk magnetoresistance; for indium phosphide and

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gallium arsenide it can be as high as 100 per cent. To minimize this spurious effect contacts were made using small diameter (0.002 in.) gold wires which were welded to the sides of the specimen. In the case of germanium samples, the spurious magnetoresistance was fcund to disappear (within experimental error) for this arrangement. Two approaches were used as a check on the spurious effect. One was to compare samples of the same crystal which had different contact arrangements. From these observations it was noted that in addition to an effect on the transverse magnetoresistance there is a much larger longitudinal magnetoresistance with the large area side contacts than when these are replaced by the small diameter

EFFECTS

Table 1 lists values for a number of n-type indium phosphide crystals. In all cases, H [OOl] refers to the magnetic field in the [OOl] direction and perpendicular to the plane of the contacts, whereas H [ilO] refers to the magnetic field in the [ilO] direction and in the plane of the contacts. It is clear from the results for the purest crystals (192) that there is a very small longitudinal magnetoresistance and a transverse magnetoresistance isotropic in the (110) plane. This is demonstrated in Figs. 1 and 2, which give the values of the magnetoresistance for one of the crystals 192 as a function of the magnetic field orientation. Fig. 3 yields a corroboratory check for another sample of the same crystal cut to give a current in the [loo] direction.

Table 1. Magnetoresistance of n-type InP Crystal

51 54 T43 T43 192 192 192” T5” T43’ T43”

I

1

___ Hall mobility (cmz/V-set)

3150 3150 2800 4300 8700 4200 23400 4600 2750 4000 6900

_-._____ Electron concentration (cm-“)

1.4 1.2 6 4.3

x x x x

6

x lOI5

_-

10” 10” lo= 1016

6 x 1Ol5 4.6 x 10” 5 x 101”

Temperature (“K)

Magnetoresistance bl~oH=

(10” cm4/V2-sec2) I [llO] I [llOJ H[llO] H[OOl] 0.31 0.55 0.03 0.13 0.37 1.15 0.20 0.66 0.6 8.2 0.05 1.3 34.0 0.4 1.11 0.02 0.14 0.02 0.02 0.28 0.09 6.2

289 290 290 293 77 294 77 292 293 292 77

I [llO] H[TlO] 0.26 0.10 1.22 0.25 6.6 1.5 31.0 1.09 0.10 0.21 5.8

I I * The side contacts were 0.002 in. diameter gold wires welded to the crystal.

welded wires. The other type of check was made using a sample with the current in the [loo] direction and the magnetic field in the perpendicular (100) plane. In this case the bulk magnetoresistance should be independent of the direction of the magnetic field in the (100) plane. This was found to be the case only when the contacts were small diameter welded wires. Included in the tables of values are measurements involving both forms of contacts, with those crystals with welded wires starred. These latter values are the most reliable. The spurious contributions to the unstarred group vary because of the many types of contacts used-small area pressure contacts, soldered contacts of different areas and bridge samples with well-defined areas.

Although crystal 192 shows this isotropy, it is clear that more impure specimens (with larger electron concentration) show some anisotropy. In the conventional notationt5) three coefficients b, c, and d may be defined by AP POH2

=

b+(I- V+ IZHZ

T I2Hi’ d F’

The last three columns of Table 1 refer respectively to measurements of b +c + &d, b, and 6 + id. The form of anisotropy shown by the observations gives a negative value for d, although the effect is small. From the measurements in indium phosphide, it is concluded that the mobility tensor of er/m* is isotropic for crystals with electron concentrations

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INDIUM

PHOSPHIDE

CRYSTAL

513

EFFECTS

192H

I [II01 H = IO 000 H[001]

OERSTEDS

H [IlO]

4

H [ooi]

+

+

0012 -

30

0

60

120

150

160

8tD:REESI

FIG. 1. The magnetoresistance of n-type InP crystal 192 as a function of the angle the magnetic field makes with the [OOl] direction in the (ilO) plane. The temperature was 292°K.

the order of 1W6 cm-3 or less, and shows a small anisotropy of a [loo] typef8y g, for crystals with larger electron concentrations. The first result indicates that the effective mass at the bottom of the conduction band is a scalar. The second may be due to an anisotropic scattering process entering or to a possible anisotropic second conduction band quite close to the lowest spherically symmetric one.“O) The latter suggestion seems more reason-

I

INDIUM

PHOSPHIDE

H 0011

0012-

‘,

CRYSTAL

192l-l

I [I I oJ H = IO 000

+ 0016 -

AP

able, and there is other evidence for the presence of such a band (with a heavier mass) from thermoelectric and optical measurements.(ll) From the form of the anisotropy, this second state may be similar to the state of symmetry A which is the lowest conduction band in silicon.‘12’ Table 2 lists a similar series of measurements for crystals of n-type gallium arsenide. The best measurements (on crystals 61 and M12) indicate a

H

_

_

"

0

OERSTEDS

[ii01 4

"

0

-

a

OO

I

I

30

60

I

120

I

150

8( DiiREESI

FIG. 2. The magnetoresistance of n-type InP crystal 192 as a function of the angle the magnetic field makes with the [OOI] direction in the (110) plane. The temperature was 292°K. LL

180

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INDIUM

PHOSPHIDE

EFFECTS

CRYSYAL

I92

1 I

I [IO01 H = IO 000

OQl6

H tooil

OERSTEDS

4

H [IO01

f

I

t

8 (DEGREES) FIG. 3. The

magnetoresistance of n-type InP crystal 192 as a function of the angle the magnetic field makes with the [OOI] direction in the (010) plane. The temperature was 292°K.

small longitud~al ma~etor~sist~ce in the [llO] and in the [loo] directions and an isotropic transverse magnetoresistance. In this case, too, it is concluded that the mobility tensor er/m* is isotropic, with the effective mass m* thus a scalar. The anisotropies shown for the other crystals in Table 2 are all of the magnitude and type due to the contacts and are ascribed as spurious. The isotropic effective mass is in agreement with calculations reported by CALLAWAY. For both indium phosphide and gallium arsenide the magnetoresistance is considerably less than the @28 times the square of the Hall mobility

expected(r@ for lattice scattering with a relaxation time inversely proportional to the carrier velocity. However the scattering in these III-V compounds is likely of the type believed dominant in InSb,(r6) the electrons being most strongly scattered by polar optical modes with a resulting effective relaxation time(lQ weakly dependent on the energy. This would reduce the magnetoresistance, in qualitative agreement with the observations reported. It is necessary to reduce the impurity scattering effect or correct for it with confidence in order to give a more quantitative comparison of theory and experiment.

Table 2. Magnetoresistance Crystal

59 14 61 M12” Ml27

Hall mobility (cmz/v-sec)

4100 4400 3400 4000 3800

Electron concentration (cm+)

4 7 4 47 4

x x x x x

of n-type GaAs

AP/PR

(IO8 cm*/V2-se@

(OK)

29s 295 295 292 290 I

-

Magnetoresistance

Temperature

101’ 10’6 10’6 101’ 101’

_--

I[llO] HfllO] 0.21 O-15 0.02 0.16 0.02

I[llO] H[OOl] @30 0.39 0.43 0.61 0.27

-

1[110] H[ilO] 068 0.57 0.48 O-29 0.27 -

1) For this crystal only, the case I [IIO] H [OOl] corresponds to H in the plane of the contacts and I [I IO] H IrIO] to N perpendicular to the plane of the contacts. t The side contacts were O%lO2in. diameter gold wires welded to the crystal. NIeasurements on a sample with current in the [loo] direction gave the same results.

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Acknowledgments-1 should like to thank J. J. GANNON and R. V. VANNOZZIfor their aid in making the measurements, and R. GUIRE, J. R. WOOLSTON, and Dr. WEISER for supplying the crystals.

K.

REFERENCES WELKER H. and WEISS H. Solid State Physics Vol. 3, (editedby F.SeitzandD.Turnbull),p. 1. Academic Press, New York (1956). JENNY D. Proc. Inst. Radio Engrs. 46, 959 (1958). GLICICSMANM. and WEISER K. r. Electrochem Sot. 105, 728 (1958). WEISBERG L. R., WOOLSTONJ. R., and GLICKSMAN M. r. Appl. Phys. 29, 1514 (1958).

J. Phys.

Chem. Solids

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5. PEARSON G. L. and SUHL H. Phys. 6. 7. 8. 9. IO. Il. 12. 13. 14. 15. 16.

Rev. 83, 768 (1951). BROOM R. F. PYOC. Phys. Sot. 71, 500 (1958). GLICK.SMANM. Phys. Rev. 100, 1146 (1955). ABELES B. and MEIBOOMS., Phys. Rev. 95,31 (1954). SHIBUYA M. Phys. Rev. 95, 1385 (1954). GLICKSMAN M. Bull. Amer. Phys. Sot. 3, 120 (1958). NEWMAN R. Phys. Rev. 111, 15.18 (1958). HERMAN F. Phys. Rev. 95, 847 (1954). CALLAWAYJ. r. Electronics 2, 330 (1957). SEITZ F. Phys. Rev. 79, 372 (1950). EHRENREICHH. r. Phys. Chem. Solids 2, 131 (1957). HOWARTH D. J. and SONDHEIMERE. H. Proc. Roy. Sot. A 219, 53 (1953).

Press 1959. Vol. 8. pp. 515-518.

Printed in Great Britain

BAND CONDUCTION IN STRONG MAGNETIC FIELDS R. J. SLADEK

Westinghouse

Research Laboratories,

1. INTRODUCTION

mobility, t.~$, of electrons in magnetically induced impurity bands (1) in n-InSb has been determined as a function of temperature and magnetic field strength by means of resistivity and Hall effect measurements at liquid-helium temperatures. Field strengths up to 28,000 G were employed. At all temperatures PC decreases very rapidly with increasing magnetic field strength in general accord with the decrease of the overlap between electronic wave functions centered on neighboring donors. In most cases pt is a weakly activated function of temperature with the activation energy increasing with field strength. THE

2. EXPERIMENT Electrical conductivity and Hall effect data were obtained for three n-type InSb samples having between 1-O x 1014 and 3.3 x 1014 excess donors per cm3. The Hall data have been published recently.(l) Representative electrical conductivity data are presented in Fig. 1. The log of the conductivity, c at constant magnetic field is plotted vs. the

Pittsburgh

35, Pennsylvania

reciprocal of the absolute temperature. The magnetic field was perpendicular to the current direction as for Hall effect measurements. From Fig. 1 it can be seen that v decreases strongly as the temperature is decreased or the magnetic field is increased. 3. DISCUSSION Previously”) it was deduced from the Hall effect data that electrons fall from the conduction band into donor levels which have been split off from the conduction band by the strong magnetic field. Electrons in these donor levels have non-zero mobility, i.e. there is impurity band conduction. Resistivity measurements at one low temperature indicated that the impurity band mobility varied with magnetic field strength like an exchange integral in which gaussian-type donor wave functions taken from theory(2’ were used. Thus it was concluded that the impurity band conduction involved electron jumps between neighboring donor ions-the jump frequency being given by the exchange integral. Our present purpose is to determine w over a range of temperatures and field strengths. Since in