Transverse magnetoresistance and hall effect in N-type InSb

J. Phys. Cha.

Solids

Pergamon Press 1959. Vol. 9. pp. 119-128.

TRANSVERSE

Printed in Great Britain.

MAGNETORESISTANCE EFFECT

IN N-TYPE

R. T. BATE, R. K. WILLARDSON

AND

HALL

InSb*

and A. C. BEER

Battelle Memorial Institute, Columbus, Ohio (Received

14 July 1958)

Abstract-Data on the transverse magnetoresistance and Hall effect are given for fixed temperatures between 50 and 200”K, covering the range from the weak-magnetic-field region to the strongfield quantum limit in InSb samples of high mobility and sufficient purity that classical statistics are applicable. At weak fields, results are in semiquantitative agreement with the predictions of theory based on a simple mixed-scattering mechanism and a spherical conduction band. At the highest fields, the transverse-magnetoresistance data approach the H/T behavior expected on the basis of the quantum treatment of ARGYRE~ and ADAMS (Phys. Rev. 104, 900 (1956)), and the result in the quantum limit for the purest specimen shows good agreement with the theory for reasonable values of the electron effective mass.

1. INTRODUCTION recent years, analyses of the magnetic-field dependence of galvanomagnetic effects in germanium and silicon have contributed significantly to an understanding of the influence of band structure and crystalline imperfections on conductivity.(l*s) This method of investigation can also be helpful in studies of many other semiconducting materials. Indium antimonide is particularly well suited for such investigations because of its very small electron effective mass and consequent high electron mobility. Since galvanomagnetic coefficients depend on the dimensionless parameter p”H, where $J is the carrier drift mobility in electromagnetic units and H is the magnetic field intensity, one can observe in indium antimonide at moderate magnetic fields those effects which would require much higher fields in other semiconductors. FREDERIKSE and HOSLER(~)have made an extensive study of galvanomagnetic effects in n-type InSb between 1 and 78°K for samples with excess donor concentrations in the range 9 x1014 to IN

*This work was supported by the U.S. Air Force through the Air Force Office of Scientific Research of the Air Research and Development Command.

7 x 1015 cm-s. At 78’K, the electron concentration for which the Fermi level is 2kT from the edge of the conduction band is about 7 x 1014 cm-s, so that, strictly speaking, galvanomagnetic data on InSb with electron concentrations greater than this should be analyzed using Fermi-Dirac statistics for temperatures as low as that of liquid nitrogen. The behavior of the galvanomagnetic coefficients for such material may be quite complex, since the Fermi energy depends on the magnetic field, and the variation can be appreciable at large fields.(s*‘n Also, ionized-impurity scattering is probably the major scattering mechanism in such material below 78’K. In order to simplify analysis, it is therefore desirable to obtain data on samples of higher purity in a temperature range where classical statistics apply and impurity scattering is not predominant. High-field measurements are of particular interest in InSb because of the quantum effects in strong magnetic fields. As was originally pointed out by LANDAU,(~) the motion of a conduction electron is quantized in a strong magnetic field. The effects of this quantization on the galvanomagnetic coefficients should be most pronounced at low temperatures and high fields, such that WT > 1 and Aw/kT > 1 (when classical statistics apply, and where w is the cyclotron resonance 119

120

R. T.

BATE,

R. K. WILLARDSON

frequency and r is the relaxation time), since the quantized energy levels will be widely separated and the motion of the electrons will be characterized by small quantum numbers. The condition for which fiw/kT> 1, so that all carriers are in the lowest quantum state in the magnetic field, has been called the “quantum limit” by l%GYRES and ~~s.(s~ On the other hand, when fiw/kT $ 1, the motion is characterized by large quantum numbers and, by virtue of the correspondence principle, a quantum treatment should give the same results as the semiclassical treatment, which ignores the quantization. For high-purity n-type InSb at 77’K, the condition ~7 > 1 is satisfied for magnetic fields above 200 G, and the quantity kw/kT, which is approximately equivalent to 0.01 H/T, becomes unity at about 8000 G. Thus, n-type InSb is an ideal material for the study of quantum effects. 2. METHOD OF ANALYSIS The analysis of the data is carried out using the simplest possible model. At low fields, ionizedimpurity scattering is introduced following the treatment by JOHNSON and WHIT~ELL,(?~ which assumes that the Rutherford scattering and acoustical-mode lattice scattering (or, more generally, a scattering mechanism for the second process characterized by a relaxation time proportional to s-i) are the dominant mechanisms, that the energy surfaces are spherical, that quantum effects are unimportant, and that classical statistics apply. The high-geld data are analyzed by means of the quantum procedure developed by EGOS, which assumes spherical energy surfaces and acoustical-mode lattice scattering only. The treatment of JOHNSONand WHITESELL yields the following expressions for the Hall coefficient and transverse magnetoresistance

and A. C. BEER

where R is the Hall coefficient, p is the resistivity, ~~0 and pIs are the carrier drift mobilities for the respective cases of scattering only by lattice vibrations and by ionized impurities in the absence of a magnetic field. The integrals K(& y) and .Z@, r) have recently been evaluated over a wide range of values of j3 and y by BEERet ai. (8) Analysis of the expressions reveals that the Hall coefficient approaches the field-independent value 1jne at strong fields regardless of the value of j3, and it approaches a p-dependent value at weak fields. The transverse magnetoresistance, Ap/ps,is proportional to Ws at low fields, and becomes fieldindependent at high fields, but is dependent on p at all fields. EGG has taken into account the quantization of the electron motion to calculate the strongfield magnetoresistance and Hall coefficient of a semiconductor with spherical energy surfaces for which acoustical-mode lattice scattering is the only important scattering mechanism. His results, which apply as long as WT> 1 and tiw/kT > 1, can be simplified when classical statistics apply and when kT7 3 k to yield :

:

PH

-Y3

PO

2 rr ( ~--,“,

fw F

>

(logr-1-0*577)

,“x !@&logf-l-0.577),

where where and &&Bis the Bohr magneton. For high-purity InSb at 80”K, l’ 21 10-s, so that the predicted strong-field Hall coefficient agrees with the classical

TRANSVERSE

MAGNETORESISTANCE

AND

value l/ne to within about O-1 per cent for this case. On the other hand, the behavior of the transverse magnetoresistance is quite different, the magnetoresistance being roughly proportional to H/T at high fields and low temperatures (as long as classical statistics apply), instead of saturating as the classical theory predicts. Several objections to the above model can be raised, and it would be well at this point to examine the limitations. KANlX@) has calculated the structure of the conduction band in InSb and has found that the relation between energy and wave number is nonparabolic for sufficiently large energies above the bottom of the band. In our case, however, this consideration causes no difficulty because of the small carrier concentrations and low temperatures of measurements. Hence, the assumption of spherical energy surfaces appears quite well justified. It is interesting to note that in germanium the warped heavy mass band is responsible for a minimum in the magnetic-field variation of the Hall coefficient(s) similar to that observed in InSb (see Fig. 2). Recently EHRENREICH(~~) has shown that polar scattering is the dominsnt scattering mechanism in InSb above 200°K. Since the Debye temperature for the polar modes in InSb is approximately 290°K we would expect them to be somewhat less important at low temperatures, but a rough extrapolation of EHRENREICH’Scalculations shows that T&e

HALL

EFFECT

IN N-TYPE

InSb

121

the mobility contribution from polar-mode scattering still predominates over that from acoustical modes at 80°K. In addition, HARRISON has pointed out that piezoelectric scattering should be important in InSb at low temperatures. Therefore, a truly satisfactory treatment should incorporate both of these scattering mechanisms. This would be quite complicated, especially in the case of polar scattering, inasmuch as one cannot define a relaxation time in the temperature range of interest. In view of these complications, it is obvious that the analysis adopted here can be expected to give only a semiquantitative description of the situation, and that any magnitude quoted for lattice mobility must be considered as only an approximation to the true mobility. However, as will be apparent, the treatment presented here does indicate the importance of considering a mixed scattering mechanism rather than a single process. This conclusion is based on the fact that our development appears to account better for the experimental data than does a treatment where the relaxation time is determined only by polar scattering or by Rutherford scattering. 3. ~~~~ PROCEDURE The specimens used in this investigation were highpurity single crystals of InSb cut from ingots which had been given many zone-refining passes. Ultrasonic cutting was used to fabricate “bridge” samples. Current and potential leads and thermocouples were welded

1. Electric characteristics of n-type

InSb samples

Z-Z!=

Sample

Temperature (OR)

Hall coefficient,

Resistivity,

Mobility,

R mm (cm”/C)

P (Cl-cm)

ILo (cm*/V-set)

Impurityscattering parameter, j? = ~(PL’)/&)

Excess-donor concentration, ND-NA

(cm-“)

Estimated ionizedimpurity concentration, Nr(est.) (cm-s)

_-

1

195 77 55

=8*8x =8.0x =8*0x

2

195 77 55

3

195 77 55 =

103 10” lo*

6.0 x lo-% 1-70x 10-r 1.56 x lo-’

1.46 x lo5 4.9 x105 5.1 x105

0.3 4.5 12

=9*1 x 103 =6.2x lo* =6*2x lo*

6.0 x lo-” 1.38 x 10-i 1.20 x IO-’

1.51 x 106 4.6 x10& 5.2 x105

=7*.5x =3-7x =3*7x

4.8 6.0 5.0

1.57 x 106 6.2 x106 7.4 x106

10s 104 104

x lo-* x lo-* x10-r

7.8 x lOXa

8 x 10’”

0.3 3 8

1.0x

10’4

5x10’*

0.3 1 25

1.7x

10’4

1.7 x 10’4

122

R.

T.

BATE,

R.

K.

WILLARDSON

to the samples, which were then sealed in Pyrex tubes in a helium atmosphere. Voltage measurements were made with an L & N K-2 potentiometer and an L & N White double potentiometer. Measurements of the Hall coefficient were done at fixed temperatures for magnetic fields in the range 2020,000 G. Fields of 50 G and below were obtained from Helmholtx coils. The constant-temperature baths and the associated temperatures were: dry ice and acetone, 195°K; liquid nitrogen, 77°K; pumping of vapor from liquid nitrogen containing oxygen, approximately 55°K. In taking the measurements, the sample current and magnetic field were reversed independently and the readings averaged to eliminate thermomagnetic and thermoelectric voltages.

Fro.

and

A.

4. RESULTS

C.

BEER

AND

DISCUSSION

(a) Mobility data and determination of degree of impurity scattering Table 1 lists the Hall coefficients, resistivities, and mobilities of three samples at the fixed temperatures employed. The mobilities were obtained by assumingthat IRlmin = l/m, where IRl,i,is the absolute value of the Hall coefficient measured at the magnetic field for which it is minimum (see Fig. 1). Before the galvanomagnetic data can be interpreted, it is necessary to determine the appropriate values of 8,

1. Magnetic-field dependence of the Hall coefficient (Sample 1, No-N4 = 7.8 x 1Ors cm-*).

in n-type

InSb

TRANSVERSE

MAGNETORESISTANCE

AND

II,

HALL

EFFECT

IN

N-TYPE

InSb

123

gauss

FIG. 2. Comparison of experimental and theoretical magnetic-field dependence of the Hall coefficient (Sample 1, No--N_4 = 7.8 X IOrs cm-a). the impurity-scattering parameter. A self-consistent way of doing this is as follows : according to the JOHNSONand WHITELSELI. treatment,(‘) which is based on the simplified mixed-scattering mechanism previously discussed, the mobility at a temperature Tl in the extrinsic region is

Thus, if k” is the mobility at another temperature the extrinsic range, then: P1° -= ct2e

-

TI

( Tz )

TBin

--3/2 K(bT~-3) K(bT2-3)’

an equation which may be solved graphically for b when the ratios pro/ho and TJTs are known. Then, j?( N& T) = b T-3. This procedure, which employs only the ratios of measured mobilities at different temperatures, avoids the use of absolute magnitudes of total impurity content and lattice mobilities required when a direct calculation is made of pr,//.cr. The method thus eliminates errors due to compensation effects and those resulting from measurements of sample geometry. Results are summa rixed in Table 1.

In calculating the /3values in the intrinsic region, hole conduction and electron-hole scattering must also be taken into account. Since the effective mass of holes in InSb is much greater than that of the electrons, electron-hole scattering can, in first approximation, be treated like ionized-impurity scattering, taking into account the temperaturedependence of the hole concentration. Thus, the chief effect of electron-hole scattering is to modify the temperature-dependence of p in the intrinsic region, and this modification has been taken into account in the calculation of /3at 195°K. As a consequence of the large mobility ratio in InSb, the effect of hole conduction on the conductivity and on the galvanomagnetic coefficients below 1000 G is quite small in the intrinsic region. However, the effects of the holes on the strongfield galvanomagnetic effects may be significant, and these will be discussed. @) The Hall coe@ikimt Fig. 1 shows the magnetic-field dependence, at

124

R.

T.

BATE,

R.

K.

WILLARDSON

and

A.

1.70

C.

BEER 1

Expanmentol Theoretml

ro = [R/A,,,]H_O r. = nelRIt,t,O

GO’

I

0

Sample

A

Sample 2

I.50

lO-3

IO+

IO”

I

IO'

IO2

B(N1, 7’) FIG. 3. Dependence of the weak-field Half coefficient on the

impuri~-~~~~ring parameter fl(@ = 6~~‘~~~*).

three fixed temperatures, of the Hall coefficient for a typical specimen of high-purity n-type InSb. It will be noted that a minimum occurs in the highfield region, and the Hall coefficient has been arbitrarily normalized to unity at this minimum to facilitate comparison. (The values of R at the minimum are given in Table 1.) Note that there appears to be little evidence of the oscillatory or “sinuous” variation reported for samples with larger conduction-electron concentrations.@) In Fig. 2 the experimental data at 55°K are compared with calculated values. Although the appropriate value of ,6 is 12, a value of 10 was used, since K(/3, r) and L@, y) are tabulated(*) for this value. The lattice mobility used in determining y was found by applying the relation ~~0 = COCK to the mobility data on several samples. The agree-

ment is seen to be fairly good at low fields, but at higher fields the data exhibit a fine structure not accounted for by the theoretical curve. The increase in the absolute value of the extrinsic Hall coefficient at strong fields seems rather difficult to explain. As has already been pointed out, the warped heavy-mass valence band in germanium leads to such a behavior. Although the present case is somewhat different due to the absence of the band degeneracy, we are not inclined to rule out anisotropy as a point for consideration before directional effect studies are completed.* Another possible cause is the magnetically-induced * In this connection, GOLDand ROTH have shown how anisotropy introduced via combinations of ellipsoidal energy surfaces can also lead to minima in Rx as a function of magnetic field.

TRANSVERSE

MAGNETORESISTANCE

195%

20

AND

HALL

EFFECT

IN N-TYPE

InSb

125

(intrlnsc)

100

IOOOO

1000 H,gauss

FIG.

4.

Typical magnetoresistance data for n;type InSb (Sample 3, No -Na

impurity banding investigated by KEYES and SLADEK.(1s) For the measurements at 200”K, an increase in Hall coefficient at large fields is to be expected as a result of contributions from hole conduction. Fig. 3 shows a comparison of a theoretical plot of the normalized weak-field Hall coefficient as a function of /3, with experimental points for two samples. Note that the agreement is quite good, although it must be remembered that the data were rather arbitrarily normalized. It is significant, however, that the minimum occurs in the predicted range of p values. (c) Weak-field magnetoresistance Transverse magnetoresistance data for one sample of n-type InSb at three fixed temperatures are shown in Fig. 4. The transverse magnetoresistance increases with decreasing temperature, because of the increased mobility, and it is proportional to H2 at low fields. At high fields, the magnetoresistance is proportional to H in the

= 1.7 x 1014cm-“).

extrinsic region, whereas it increases more rapidly with H when intrinsic. Fig. 5 shows the data for 77°K replotted along with the theoretical curve for the appropriate value of /?.Although the agreement with the simple theory is fairly good at low fields, the magnetoresistance begins to deviate when WT becomes of the order of unity, and approaches a linear dependence on Hat high fields, whereas the semiclassical theory predicts saturation. This highfield behavior is evidently a quantum effect, but the initial deviation occurs at a rather small value of ho/kT to be attributed to quantization, and may result from some other mechanism as, for example, a delayed saturation due to polar scattering.(14) A plot of the weak-field magnetoresistance analogous to that of Fig. 3 for the weak-field Hall coefficient is shown in Fig. 6, where the parameter,

[

T3-

AP

pidf2

1

H-CO,

is plotted as a function of /3.The theoretical variation, assuming

R.

126

T.

BATE,

R.

FIG. 5. Comparison

/&(77”K)

K.

WILLARDSON

of experimental

= 1.2 x 10s=, __ V-set

is shown for comparison. Note that the theoretical low-field magnetoresistance is extremely sensitive to impurity scattering (and electron-hole scattering), the relative change in resistance having been reduced two orders of magnitude when p1 = 10~~. The agreement here is not so good as in the case of the Hall coefficient, although, in most cases, the deviation is less than 50 per cent. Note that there seems to be a correlation of the deviation from theory with the degree of compensation. (d) Strong-jield magnetoresistance Turning now to the strong-field region, the quantum treatment suggests a transverse magnetoresistance proportional to H at high fields, and the predicted high-field behavior is shown by the

and

magnetoresistance

A.

C.

BEER

with theory.

broken line in Fig. 5. However, this calculation was made for acoustical-mode lattice scattering only and for Aw/kT > 1; and neither of these conditions has been completely satisfied experimentally. It is interesting to note that the values of hw/kT for the maximum field of 20,000 G are 2.6 at 77’K and 3.5 at 55’K. To investigate the effects of impurity scattering and to illustrate the approach to the quantum limit, the dimensionless parameter PH _.-

PO

kT

I-H

has been plotted in Fig. 7 as a function of field for three samples at two different tures. The theory predicts that for the phonon scattering, this parameter should approximate value

magnetic temperaacoustichave the

TRANSVERSE

MAGNETORESISTANCE

AND

HALL

Ltlrnlt for pure lotttce scottermg Cfi =O)

r

Theoretical

EFFECT

0

pldt ossuming pL (7f0) = I.2 x d

IN

N-TYPE

I&b

127

Sxld’cm-”

-$&

FIG. 6. Dependence of the weak-field magnetoresistance on the impurity-scattering parameter /3(fi = 6p~‘/p)

;

?r

S(log

r-1-0*577),

an expression which is relatively independent of temperature between 50 and 80”K, and equals about 350 for m*/m of O-013.(15) Note that according to our data this parameter does seem to be approaching a constant value for each sample, but the value appears to be a function of the total ionizedimpurity concentration, with the result for the purest sample approaching most nearly the theoretical value for pure lattice scattering. Unfortunately, it is not possible at this time to continue the analysis further, because the quantum treatment has not been extended to encompass the case of mixed scattering.

predicted influence of impurity scattering and electron-hole scattering on the Hall coefficient and magnetoresistance. The existence of quantum effects for nondegenerate material in strong magnetic fields has been verified, and the data show that the strong-field magnetoresistance and Hall coefficient are in semiquantitative agreement with the quantum treatment. Ackmwledgements-The authors are gratefui to Dr. P. N, ARGYRES for sending us his manuscript before publi~tion, and to Dr. E. N. ADAMS, Jr., Dr. II. EHRENREICH, and Dr. C. &&BRING for heipful comments. Thanks are also due to Miss M. I?. TANNER, who produced some of the material, to G. L. KENDALL, who prepared the bridge-type samples, and to J. SCHROEDER of Ohio Semiconductors Inc., who supplied the highestpurity sample.

5. SUMMARY

The experimental data tend to support the hypothesis of nearly spherical energy surfaces in the conduction band of InSb, and to confirm the

Note added in proof: Experiments are now being carried out on specimens taken from pa&d crystals. Although this material is slightly less pure than that cut

128

R.

T.

BATE,

R.

K.

WILLARDSON

and

A.

--

C.

-77%

BEER

(2.6)

I I1 I 2600 a= Q.0 600

400

200

IO' H, gauss

FIG. 7. Strong-field

magnetoresistance

from zone-refined ingots (ND -NA - 5 X 1014, p,,ax - 310,000), etch-pit densities are negligible compared to the 104/cm2 count for the zone-refined material. The behavior of RH is substantially different from that shown in Fig. 1. After plastic bending, however, the difference disappears. It may be possible that dislocations exert a significant influence on the weak-field behavior of RH. This problem is now being investigated by JULESDUGA. REFERENCES ABEL= B. and MEIBOOMS., Phys. Reo. 95,31(1954). SHIBUYA M., Phys. Rev. 95, 1385 (1954). WILLARDSONR. K., HARMANT. C. and BEER A. C., Phys. Rev. 96 1512 (1954). BEERA. C. and WILLARDSONR. K., Phys. Reu. 110, 1286 (1958). FREDERIKSEH. P. R., and HOSLERW. R., Phys. Rm. 108,1136 (1957).

in n-type I&b.

4. fbGYRFS P. N., Phys. Rm. 109,111s (1958). 5. LANDAU L., 2. Phys. 64,629 (1930). 6. ARGYRE~P. N. and ADAMS E., Phys. Rev. 104, 900 (1956). 7. JOHNSONV. A. and WHITEYELLW. J., Phys. Rm. 89, 941 (1952). I. N., 8. BEERA. C., ARMSTRONGJ. A. and GREENBERG Phys. Rew. 107, 1506 (1957). 9. KANE E. E., J. Phys. Chem. Solids 1, 249 (1957). H., J. Phys. Chm. Solids 2,131 (1957). 10. EHRENREICH 11. HARRISON W. H., Phys. Rev. 100, 903 (1956). 12. KEYES R. K. and SLADEK R. J., J. Phys. Chm. Solids 1, 143 (1956). 13. GOLD L. and ROTH L. M., Phys. Rev. 107, 358 (1957). 14. LEWIS B. F. and SONDHEIMER E. H., PYOC. Roy. Sot. AZ27, 241 (1955). 15. DRESSELHAUSG., KIP A. F., KITTEL C. and WAGONERG., Phys. Rev. 98,556 (1955).