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Electrical properties of GaSb-InSb alloys
J. Phys. Chem. Solids
Pergamon
ELECTRICAL
Press 1960. Vol. 17, Nos. l/2, pp. 34-43.
PROPERTIES J. C. WOOLLEY
Department
of Physics,
(Received
12 April
Printed in Great Britain.
OF GaSb-InSb
and
C. M.
University
GILLETT
of Nottingham,
1960; revised
ALLOYS
12 May
England
1960)
Abstract-Reasonably homogeneous, polycrystalline, solid ingots of GaSb-InSb alloys have been produced by slow directional freezing techniques, and have been used to investigate the variation of electrical properties as a function of composition. Values of extrapolated thermal energy gap Es, electron mobility, and mobility ratio have been determined for some 25 specimens covering the complete composition range. Estimates have been made of the variation of effective mass values and, by comparison with optical energy gap values, of the energy gap temperature coefficient 8. From the variation of mobility with temperature, values of mobility temperature exponent x have been obtained. At the InSb-rich end of the composition range, the results indicate a smooth variation of the various parameters with composition, without any appreciable change in the band structure from that of InSb. The results at the GaSb-rich end of the range are complicated, firstly by the presence in GaSb of two conduction bands of only small energy separation, and secondly by an effect which extends over a considerable range of composition and is thought probably to be due to ordering in the alloys. As a result, correlation of the optical and electrical results is difficult in this range of composition.
prepared separately, by melting together under vacuum the appropriate amounts of the elements. The elements used were all of 99.999 per cent purity, and in the case of InSb the compound was further purified by normal zone refining methods. The alloys were produced by the slow directional freezing technique described previously,(12) and in order to obtain the required range of composition three ingots were used, these having approximate mean compositions of (A) 70 mol. per cent GaSb, 30 mol. per cent InSb (B) 30 mol. per cent GaSb, 70 mol. per cent InSb and (C) 10 mol. per cent GaSb, 90mol. per cent InSb respectively. Sections from various points on these ingots provided specimens covering practically the whole range of composition in the alloy system, i.e. from InSb to 94 mol. per cent GaSb, and the degree of homogeneity of these specimens was the same as that of the specimens used previously in the optical work.(i2) As indicated in that case, some inhomogeneity still remains however. To cover the range of composition from 94-100 mol. per cent GaSb, ingots of the appropriate compositions were made up and annealed at 520°C for 4 weeks. Some of these were rather less homogeneous than the specimens produced by directional freezing however and only one of them was used in the measurements described below. For the electrical measurements, specimens were cut from cross sections of the directionally frozen ingot, and for each composition measurements were made to determine the values of Hall coefficient RH and conductivity D as a function of temperature in the range
INTRODUCTION CONSIDERABLE work has been carried out to deter-
mine the available composition range of semiconductor alloys produced by solid solution beween two semiconducting compounds each having the zinc blende structure.(l*3~3~4*5)Information is now becoming available on the electrical properties of some of these alloys, but in many cases the data is limited. Electrical measurements have been reported on the following alloy systems: InAs-InP(Q GaAs-InAs, (3) CdTe-ZnTe, (5) CdTe-HgTr, (7) GaAs-Ga3Se3, (8) AlSb-GaSb,(s) InSb-In3Te3. (1s) In a recent paper, IVANOV-OMSKIIand KOLOMIET@) have reported electrical measurements on the equimolecular alloy of the GaSb-InSb system. The present authors have recently investigated the variation of optical energy gap as a function of composition for alloys of the GaSbInSb system.(13) The present paper reports an investigation of the electrical properties of these alloys. PREPARATION OF SPECIMENS AND METHODS Each
of the
OF MEASUREMENT
two compounds
concerned
was first 34
ELECTRICAL
PROPERTIES
OF
90-850°K. The specimens were cut to the approximate dimensions 7 x 1 x 1 mm. and were lightly etched in CP-4 etchant before attaching the electrical contacts. For the measurements below room temperature, the current electrodes and a copper-constantan thermocouple were soldered (using high purity indium solder) to the ends of the specimen, and the potential and Hall probes spot welded to the specimen in the normal positions. In the case of the high temperature measurements, the potential and Hall probes were again spot welded to the specimen but the current connections were made by pressure contacts, and the temperature determined by a Pt/PtRh thermocouple, the hot junction of which was held close to the specimen. Specimen currents of up to 60 mA were used, and all voltages were measured by means of a potentiometer accurate to 1pV. Magnetic fields of up to 6 kOe were provided by an electromagnet. During each set of observations at a given temperature, the temperature was held constant to +_ 1°C. CALCULATION
OF PARAMETER
GaSb-InSb
ALLOYS
then it is readily shown that
&max -=
(b- II2 4b
RH ext
where ~~~~~ is the maximum value attained by RH above the Hall inversion temperature. From equations (4), (5) and (6), th e variation of n can be determined as a function of temperature. Standard theory gives the equation
Fs = A(mn*m,*)3/2 exp - -$ i
= 1ogloA + ; loglO(mn*mp*) -
log~,‘;s
-
B
Eo
2.303 kT - ___ 2.303 k
(1)
Q = +&+p/b)
(2)
where for p-type material
P = n+pe
(3)
(7)
Hence the slope of the graph of logronp/T3vs. l/T gives Eo. Combining this value of Eo with the room temperature value of Es from optical work enables a value of the coefficient b to be estimated. Then assuming parabolic bands so that A = 4(2?rkT/hs)s, the intercept of the logrsnp/T3vs. l/T graph gives a value for (mn*ma*)1/2. If spherical energy surfaces and lattice scattering are assumed, then *
P nbz-p RHC_--e (nb+~)~
1
and assuming Ea = Eo+j?T in the temperature range considered, and that mn* and mp* are independent of T
VALUES
As shown below, the large majority of specimens used were p-type in the impurity range, and in the temperature range investigated each of these showed a transition from p-type to n-type behaviour as intrinsic effects began to predominate. Thus the simple single carrier formulae for RH and D could not be used and the more accurate equations had to be used to determine the variation of electron and hole densities (n and p) and of electron mobility PR as a function of temperature. When carriers of both signs are present, the standard equations for RH and o are
35
b=!?=
512
?.?_ CL?)
(
mn*
1
and hence mn* and mp* can be calculated. Although these assumptions can not be accurately applied to the present materials, the results can be used to give an indication of the probable variation of effective mass values. Finally from equations (2) and (3)
and b = t&p, pe being the resultant hole density due to impurities. Here the value of the Hall coefficient ratio r has been taken as unity. From equations (1) and (3) n=-
1
(b-1) ---
2e (b+l)
1
pe
+
b+l
-
-
RH
RESULTS
R Hext
the intrinsic
1 =
-
pee
AND
vs. log T
DISCUSSION
(a) Results of Hall measurements
(4) If it is assumed that at low temperature contribution is negligible so that
and assuming pn a T-2, a graph of logpn enables a value of x to be determined.
(5)
The variation of RH as a function of temperature for various typical specimens is shown (on a log scale) in Fig. 1 and the curves are seen to be those characteristic of materials with predominant p-type impurity. As the temperature is increased from the liquid oxygen temperature, some decrease in RH and hence increase in pe occurs, due
36
J.
C.
WOOLLEY
and
to finite activation energy of impurities, but in most cases the curve levels out before the Hall inversion temperature is reached, and in each case the value of & in this range has been taken as
Fro 1. + l 0
C.
M.
GILLETT
per cent GaSb mean composition, was n-type along the whole length. The variation of pe with position along the length of ingot B is shown in Fig. 2, the variation of composition with position also being indicated. It is seen that the value of pe reaches a minimum of 1.7 x 1017/ems near the centre of the ingot and reaches values between 4 and 5 x 1017jcms at each end. The form of this curve is of interest as far as the behaviour of GaSb itself is concerned. It has been suggested that the p-type character usually observed with GaSb could be due to non-stoichiometry in the compound, but recently BOLTAKS and GUTORO@) have claimed that the result, in GaSb are due to the
Log&~ vs. togloT for various typical specimens 24 mol. per cent GaSb, 76 mol. per cent I&b 76 mol. per cent GaSb, 24 mol. per cent InSb G&b m0l
65 /
60 /
%
G&b
M4030 // I/'
IO 1 I
x) / I
InSb FIG.
Distancealong
ingot,
cm
FIG. 2. Variation of pa with position along ingot B (mean
composition 30 mol. per cent GaSb, 70 mol. per cent I&b). RH@~~ required in equations (5) and (6). These extrinsic range values of Rw showed that two of the three directionally frozen ingots, viz. A and B were p-type along the complete length of the ingot, while the third, C, which contained only 10 mol.
mol
% GoSb
G&b
3. Variation of mobility ratio 6 with composition.
presence of both donor and acceptor impurities with slightly different segregation coefficients. From the present results, it is seen that in the Iatter half of ingot B the value of pe rises from 1.7 to 4.8 x 1017,kms as the GaSb content falls from 30 to 10 mol. per cent. Since the InSb used for the alloy was n-type with less than 2x 10’s donors/cm3 and the GaSb was p-type with approximately 3 x 1017 acceptors/cma, it would appear impossible to explain the variation of pe along ingot B by non-stoichiomet~c behaviour. A plausible explanation can only be produced by postulating the presence of both p- and n-type impurities having segregation coefficients which are different one from the other and which vary along the length of the ingot as the composition varies. The variation of pe with position along ingot A and n, along ingot C further support this suggestion,
ELECTRICAL
PROPERTIES
From the results of RH vs. temperature on the P-type specimens a value of the mobility ratio b can be obtained immediately from equation (6). The use of specimens from ingots A and B thus enabled a value of b to be obtained for all alloy compositions except those in the range O-10 mol. per cent GaSb where only n-type specimens were obtained. The resulting variation of b with composition is shown in Fig. 3. It is seen that the value drops very rapidly from 85 for InSb, being 27 at 10 mol. per cent GaSb, but then stays reasonably constant up to approximately 75 mol.
IO%, FIG. 4. Loglsnp/Ts (a) (b) (c) (d) (e)
47 64 70 76 91
mol. mol. mol. mol. mol.
per per per per per
GaSb-InSb
37
ALLOYS
from the straight line at high temperatures. This is due to the onset of degenerate behaviour, for which case the equations in Section 3 do not apply. Sufficient results are available at lower temperatures for a value of Eo to be obtained. Secondly, graphs (d) and (e), which represent alloys of high GaSb content, are seen to divide into two parts, each being linear and hence giving a separate value of Eo. Thus for alloys of GaSb content higher than 70 mol. per cent, two values of Eo were obtained, a low temperature and a high temperature value, and also a transition temperature Tc. The values of Es are plotted against composition in Fig. 5 and
OK-1
vs. 10S/T for typical alloy specimens. cent cent cent cent cent
GaSb, GaSb, GaSb, GaSb, GaSb,
53 36 30 24 9
mol. mol. mol. mol. mol.
per per per per per
cent cent cent cent cent
InSb InSb InSb InSb InSb
cent GaSb after which it falls again to the value of 4.5 for GaSb. Using the data from the Hall effect measurements and equations (4), (5) and (6), values of npjT3 were calculated for each specimen and graphs of lognp/Ts vs. l/T plotted. Typical examples of these graphs are shown in Fig. 4. From equation (7) these should be straight lines with a slope dependent on E, and it is seen that the graphs are linear over a considerable range of temperature. Two points are illustrated by the graphs in Fig. 4. Firstly, graphs (a) and (b) for alloys of high InSb content, show some deviation
per
OF
Insi
ml
% GaSb
GaSb
FIG. 5. Variation of extrapolated energy gap Eo and transition temperature Tc with composition.
also the values of Tc over the range of composition where this is observed. On the question of the accuracy with which the two values of Eo and the value of Tc can be determined, it is clear that whatever the mechanism causing this effect, there will be some transition range around Tc. Hence there will be some uncertainty in the value of T,, and reasonable values of Ea will only be obtained from results at temperatures well away from the transition range. From a study of the experimental results, it is estimated that the error in Tc will not be greater than 20°C. The lower value of EO (l&l) can be obtained with reasonable accuracy (say
J.
38
C.
WOOLLEY
and
I: 0.02 eV) as measurements can be made at temperatures well below T,, but the upper value of Z&(Eos) will probably be less accurate as a large temperature range above Te cannot be used owing to the proximity of Tc to the melting point of the alloy. The variation of EO given above is to be compared with the corresponding variation of the optical energy gap EB previously published,(la) but repeated here for convenience in Fig. 6.
C.
M.
GILLETT
specimens of about 100 p thickness. Use of other conventions for interpreting optical data could give slightly different values for Eg, but should not greatly affect the shape of the Eg curve shown
‘0
GoSb
n-101 % GoSb
InSb
FIG. 7. Variation of energy gap temperature coefficient p with composition. in Fig. 6. Changes in the form of the Eg curve would result in corresponding changes in the g curve in Fig. 7, but it appears probable that whatever method is used to obtain EB from the optical
InSb
nwl % G&b
G&b
FIG. 6. Variation of room temperature optical energy gap Eg with composition.
If the values of &I
(Fig.
5) and the values of
Eq (Fig. 6) are compared, estimates of the value of a mean temperature coefficient j3 for the temperature range from room temperature to approximately 400°C can be made. The variation of p as a function of composition is shown in Fig. 7. Although the experimental scatter is large, it is seen that j3 has an approximately constant value of - 3 to - 4 x 10-4 eV/“C except in the composition range 65-95 mol. per cent GaSb where it rapidly changes and takes positive values at some compositions. The value of p obtained in this way will depend on the value assumed for Eg, which in turn depends on the method used to interpret the optical data. For polycrystalline alloys of the type used here, this interpretation is somewhat arbitrary and as indicated previouslycls) these particular values of EQ were obtained by determination of the apparent onset of transmission through
L
0
/
m
I nSb FIG.
8. Variation
/
40
mol
60
% GoSb
SO
100
G&b
of mn* and mP* with composition 0 mn* 0 -%*.
data, the peak in the /3 curve in the composition range 65-95 mol. per cent GaSb will still occur. It is hoped to carry out a more detailed investigation of the absorption edge of these alloys in the near future.
ELECTRICAL
PROPERTIES
As indicated in Section 3, when ,9 is known equations (7) and (8) allow the values of m,* and mp* to be estimated provided spherical energy surfaces etc. are assumed. While these assumptions may not be accurately true here, it is of interest to observe the form of the variation of these parameters with composition, and these are shown in Fig. 8. Again a continuous trend in the values from InSb to GaSb is observed except for fluctuations in the composition range of anomalous behaviour. (b) Discussion of Hall measurement results In Fig. 5 the value of Es shows a practically linear variation with composition up to approximately 65 mol. per cent GaSb but at this point the graph splits into two parts. From a consideration of the work of SAGAF@) the two values of Es for GaSb itself can be attributed to two different minima in the conduction bands. SAGARhas suggested that at room temperature GaSb has its lowest conduction band minimum at the centre of the Brillouin zone but that a further set of minima occur at a slightly higher energy and along the [ll l] directions in K space. The energy separation of the minima at room temperature is estimated to be approximately 0.08 eV with a rate of change with temperature of &)
M - 3 x 1O-4 eV/“C.
This interpretation is confirmed by the work of KEYES and POLLAK~~). The present work indicates that the reduction in energy separation with temperature continues at higher temperatures and that the [ 11 l] minima fall in energy below the (000) minimum at temperatures above 450°C. Assuming from Fig. 6 a value of 0.72 eV as the room temperature value of _7&for GaSb, the values of Ea from Fig. 5 can be used to determine the values of the temperature coefficients fi for the two types of minima proposed, assuming that the two have the same energy at 450°C. The resulting values of fi are -3 x 10-4 eV/OC for the low temperature Ea and -7 x 10-J eV/“C for the high temperature Eo, giving a difference (A/3 3 dldt (AE)) of - 4 x 10-d eV/“C in reasonable agreement with the value given by SAGAR.The room temperature energy separation of the minima calculated
OF
GaSb-InSb
ALLOYS
39
on this basis (O-17 eV) is rather different from SAGAR’Svalue, however. When comparing these values, it should be borne in mind that in obtaining the value of 0.17 eV, it has been assumed that the variation of energy gap with temperature can be treated as strictly linear over the whole range of temperature from room to 600°C. While it is true that such a linear variation is a valid approximation over a limited range of temperature, e.g. 250”C-450°C, a typical range over which electrical measurements were made to obtain Eo, and hence that the measured EO values are truly characteristic of the electrical behaviour in that range of temperature, it is by no means certain that this assumption of linear behaviour can be applied over much larger temperature ranges.(lT) This fact, in addition to the inaccuracy in the value of Eos, could easily explain the discrepancy in the estimated values for the energy separation of the band minima of GaSb at room temperature. As a further check that the change in Ea was not in fact caused by some structural effect such as a phase change, powder photographs were taken of GaSb at 560°C. No change could be detected in the high temperature photographs as compared with similar photographs at room temperature except the usual small change in Bragg angle of the lines due to thermal expansion. This observed change allowed a calculation to be made of the mean linear expansion coefficient of GaSb over this range, and a value of 7+ 1 x lo-s/“C was obtained. When the GaSb-InSb alloys are considered it is of interest to see whether the two values of Eo can again be attributed to different conduction band minima as in the case of GaSb. For various reasons this does not appear to possible. One point is that the two Eo curves appear to coalesce at about 65 mol. per cent GaSb and that for alloys of lower GaSb content only one value of EO is observed, a behaviour which would appear very unlikely if two band minima were being observed. A second point comes from the variation of the transition temperature Tc with composition (Fig. 5). It is seen that as InSb is added to GaSb the value of Tc increases to 520°C at 96 mol. per cent GaSb, but at 94 mol. per cent GaSb the observed value of Tc has fallen to 420°C and as further InSb is added it rises to a maximum and then falls
40
J.
C.
WOOLLEY
and
again. Such a variation would seem to indicate that the values of Tc are being determined by more than one process. Initially, in order to explain the energy gap values, a band change similar to that in the Ge-Si alloys was considered, i.e. a variation of the relative energies of different band minima with composition, so that room temperature values would indicate one band minimum in one composition range and another band minimum at other compositions. This appears to be ruled out, if as seems to be generally accepted,(l5,rs) the room temperature band structures of GaSb and InSb are of the same type with the lowest minimum in the conduction band occurring at k = 0. Moreover, it would not be possible to explain the form of the room temperature optical energy gap results (Fig. 6) in terms of such a band change. Thus to explain the variation of Eo and Tc with composition over the alloy range 65-95 mol. per cent GaSb, some effect must be postulated other than the change of lowest band minimum with composition as seen with Ge-Si alloys or the change of lowest band minimum with temperature as shown by GaSb and alloys within a few per cent of this composition, as neither of these effects would appear to be satisfactory over the alloy range 65-95 mol. per cent GaSb. One explanation which is suggested by various experimental results is that of ordering of the gallium and indium atoms in the lattice. In such a case, up to 94 mol. per cent GaSb, the value Eel would apply to the minimum band gap in the ordered structure, and the value EOZto the minimum band gap in the disordered structure, Tc representing the ordering temperature. The following points can be quoted as supporting this : (a) The form of the Tc vs. composition curve between 70 and 94 mol. per cent GaSb is typical of the variation of ordering temperature with composition. (b) The form of the low temperature EO curve in this composition range is similar to that observed in systems where ordering occurs.(ls) (c) If the graph of lattice parameter with composition for this alloy system(a) is investigated as accurately as is possible with the present alloys, it is found to show small deviations from the Vegard line, but to intersect that line at an approximate composition of 75 mol. per cent
C.
M.
GILLETT
GaSb. It has been shown@@ that such an intersection can indicate the possibility of ordering in the vicinity of that composition. (d) The form of the solidus of the alloy system(21) indicates that the solid solution is not ideal in the composition range considered. To check this suggestion, specimens in this range of composition have been annealed at various temperatures above and below the possible ordering temperature given by T,, and then X-rayed in the normal way. No definite origins of ordering have been observed, although this could be partly due to the poor X-ray photographs obtained, because of some degree of inhomogeneity in the specimens used, which with the present method of preparation cannot be avoided. This inhomogeneity could be important, for because of the low diffusion rates observed in these alloys,(a) the degree of long range order might be relatively small, and so any superlattice lines would be weak and diffuse. It is of interest to note that BRAUNSTEIN et aZ.(22) have postulated a short range ordering in Ge-Si alloys in order to explain optical absorption data. It is hoped to carry out further work on this question of ordering when better alloy specimens are available. Thus it is seen from the above discussion that if ordering is postulated to explain the electrical results, this can only apply in the approximate composition range 65-94 mol. per cent GaSb, while the results for 96 mol. per cent GaSb and GaSb itself must be explained in terms of a band change. Thus the behaviour in the range 95-100 mol. per cent GaSb could be a more complicated form. Again, production of alloys in this range has proved somewhat difficult, but it is hoped to investigate this range more thoroughly with better alloy specimens. (c) Conductivity measurement In addition to Hall effect, the variation of conductivity with temperature was also observed for each alloy specimen. Typical curves of log u against l/T are shown in Fig. 9. These show the usual division into an intrinsic and extrinsic range. In order to compare the values for various alloys of different impurity content, the values of log u in the intrinsic range have been extrapolated to room temperature to give an effective room temperature intrinsic conductivity. The variation
ELECTRICAL
PROPERTIES
OF
GaSb-InSb
41
ALLOYS
type of carrier TV= CJRHhas been used. As the high InSb content alloys are n-type, the room temperature results for these give pa, while with the large majority of the alloys, the extrinsic range results give a room temperature value for pp. For comparison purposes, these values of pLp have been multiplied by the corresponding value of b, and Fig. 11 shows the resultant variation of pn with composition, the results from the two types of material being shown differently. It is seen that pn falls rapidly from the value for pure InSb, but levels out in the middle of the composition range at a value of the order 104 ems/V set, and then at high GaSb content falls again
0.5
’ 2
FIG. 9. (a) (b) (c) (d)
Loglou
10 47 76 94
mol. mol. mol. mol.
I
I
I
I
4
6
6
IO
Id/T,
OK-’
-1 12
vs. 103/T for typical alloy specimens
per per per per
cent cent cent cent
GaSb, 90 mol. per cent GaSb, 53 mol. per cent GaSb, 24 mol. per cent GaSb, 6 mol. per cent
InSb InSb InSb InSb
of this value as a function of composition is shown in Fig. 10. The measured values of o were used to give the corresponding values of mobility, and in the extrinsic range the simple Hall mobility for one
t 00 InSb
mol
% GaSb
GaSb
FIG. 11. Variation of room temperature electron mobility with composition 0 from P-type alloys 0 from n-type alloys.
-31 0
InSb
I 20
40
mol
I 60
% G&b
I 60
I CO
G&b
FIG. 10. Variation of intrinsic conductivity (extrapolated to room temperature) with composition.
to the GaSb value. This form of curve is somewhat different from that which might be expected from simple alloy scattering theory, which would predict a minimum near 50 mol. per cent GaSb. However various other factors will influence the form of this curve, e.g. (i) the increase in effective mass as the GaSb content of the alloys is increased, (ii) the interband scattering which can be expected to occur with GaSb and alloys close to this composition, (iii) the effect on alloy scattering of any possible ordering in the alloys. Thus the simple curve showing a minimum at 50 per cent is not to be expected. For the P-type alloys, p, lies in the range 8 x 1016 to 4.5 x 1017/ems, while the values of RH for the n-type specimens give values of ne
42
J.
C.
WOOLLEY
and
in the range 6 x 1015 to 3 x lOls/cmS. These latter specimens are probably compensated however, with a total impurity content comparable with that of the p-type material. Thus the mobility values quoted correspond to material of relatively high impurity content, It is probable however that with alloys of this type, the predominant scattering mechanism at room temperature is not impurity scattering but alloy scattering. This is considered further below.
,
C.
M.
GILLETT
plotted against composition in Fig, 13. There is considerable experimental scatter on the points, but a smooth curve can reasonably be drawn through them. This value of xl falls from 1.5 for InSb as GaSb is added, but over a range from 20-65 mol. per cent GaSb lies within the limits 04 to 1.0. This is to be compared with the values of x between 0.7 and O-85, obtained by GLICKS~(23) for Ge-Si alloys, and the theoretical value of x = O-5 for alloy scattering given by NormHEIM(~~) and BRoOKsfz5f. It appears from this, that as indicated above, at these temperatures and in this range of composition the predominant scattering effect is that for alloy scattering. For alloys of GaSb content greater than 65 mol. per cent the
3.25
I 0 InSb
‘ho 10 47 70 86 91
mol. mol. mol. mol. mol.
per per per per per
cent cent cent cent cent
G&b, GaSb, GaSb, GaSb, GaSb,
T 90 53 30 14 9
40
60
mol % G&b
80
00 GOSb
Frc. 13. Variation of mobility temperature exponent x1 with composition (assuming that p CCTzl* at lower temperatures).
FIN. 12. Loglop* vs. log T for typical alloy specimens (a) (b) (c) (d) (e)
I
I 20
mol. mol. mol. mol. mol.
per per per per per
cent cent cent cent cent
InSb InSb InSb InSb InSb
To obtain values of pn for temperatures greater than that corresponding to the peak in the R- vs. T curve, equation (9) has been used, and for each alloy the variation of log pla against log T determined. Typical results are shown in Fig. 12. In a large number of cases the log plz vs. log T graph divides into two linear sections of different slope with a small transition region between the two. Applying the relation f2-a T-X to these results, two values of x can be obtained for each specimen. The low temperature values of x (~1) are shown
value of xl rises sharply, attaining values of 3.2 in the range 90-95 mol. per cent GaSb. Here the effects discussed previously appear to affect the scattering processes, but insufficient data is available for this to be discussed. The high temperature values of x (~2) are less definite, occurring near to the maximum temperature of measurement. Up to 60 mol. per cent GaSb however the value lies between 1-5 and 2-O and would appear to correspond to a predominence of lattice scattering at the higher temperatures. For compositions with more than 60 mol. per cent GaSb, larger values of xs are obtained rising to a peak value of 5.8 between 90 and 95 mol. per cent GaSb. In this range, the scattering processes will be complicated by the effects discussed above, and little can be said about the values of xs at this
ELECTRICAL
PROPERTIES
stage. The temperature range in which the transition from x1 to x2 occurs shows some correlation with the values of TC given in Fig. 5, but the effect is more complicated than the transition observed in the graphs giving I&J.
Acknowledgements-The authors are indebted to Professor L. F. BATES for the facilities of his laboratory. The work described forms part of an investigation carried out for the Admiralty. REFERENCES I. FOLBERTH 0. G., Z. Naturf. lOa, 502 (1955). 2. WOOLLEY J. C. and SMITH B. A., Proc. Phys. Sot. Lond. 72, 214 (1958). 3. WOOLLEY J. C. and SMITH B. A., Proc. Phys. Sot. Lond. 72, 867 (1958). 4. HERMAN F., GLICKSMANM. and PARMENTERR. H., Progress in Semiconductors Vol. II p. 17. Heywood, London (1957). 5. KOLOMIETS B. T. and MALKOVAA. A., Zh. tekh.fiz., Mask. 28, 1162 (1958). 6. ABRAHAMSM. S., BRAUNSTEINR. and Rosr F. D., J. Phys. Chem. Solids 10,204 (1959). 7. LAWSON W. D., NIELSEN S., PUTLEY E. H. and YOUNG A. S., J. Phys. Chews. Solids 9, 325 (1959).
OF
GaSb-InSb
ALLOYS
43
8. NULEDOV D. N. and FELTIN’SH I. A., Sooiet PhysicsSolid State 1, 510 (1959-60). 9. BURDIYAN I. I. and KOLOMIETS B. T., Sosiet Physics-Solid State 1,1067 (1959-60). IO. WOOLLEY J. C., GILL!ZIT C. M. and EVANS J. A., J. Phys. Chem. Solids 16, 138 (1960). B. T., Soviet 1. IVANOV-OMSKIIV. I. and KOLOMIFZTS Physics-Solid State 1. 512 (1959-60). 12. WOO~LEY J. C., EVANS j. A. ‘and GI~LETT C. M., Proc. Phvs. Sot. Lond. 74. 244 (19591. 13. DETWILER D. P., Pkys. Rev: 97, 1‘575 (1955). 14. BOLTAKSB. I. and GUTOROVYu. A., Soviet PhysicsSolid State 1, 930 (1959-60). 15. SACARA., Phys. Rev. 117,93 (1960). 16. KEYES R. W. and POLLACK M., Westinghouse Research Labs. Scientific Paper, 6-40602-3-P5 (1959). 17. ANTONCIKE., Czech. J. Phys. 5, 449 (1955). 18. KANE E. O., J. Phys. Ckem. Solids 1, 249 (1957). 19. WOOLLEY J. C. and RAY B., J. Phys. Chem. Solids 15, 27 (1960). 20. APTEKARA. I. and FINKELSHTEINB., Zh. eksp. teor. fiz. 21, 900 (1951). 21. WOOLLEY J. C. and LEES D. G., J. Less Common Metals 1, 192 _ (1959). ~_ 22. BRAUNSTEIN K., MOORE A. R. and HERMAN F., Phys. Rew. 109, 695 (1958). 23. GLICKSMANM., Phys. Reo. 111, 125 (1958). 24. NORDHEIML., Ann. Phys. 9,607 (1931). 25. BROOKS H., Adv. in Electronics and Electron Physics, 7, 85 (1955).
Pergamon
ELECTRICAL
Press 1960. Vol. 17, Nos. l/2, pp. 34-43.
PROPERTIES J. C. WOOLLEY
Department
of Physics,
(Received
12 April
Printed in Great Britain.
OF GaSb-InSb
and
C. M.
University
GILLETT
of Nottingham,
1960; revised
ALLOYS
12 May
England
1960)
Abstract-Reasonably homogeneous, polycrystalline, solid ingots of GaSb-InSb alloys have been produced by slow directional freezing techniques, and have been used to investigate the variation of electrical properties as a function of composition. Values of extrapolated thermal energy gap Es, electron mobility, and mobility ratio have been determined for some 25 specimens covering the complete composition range. Estimates have been made of the variation of effective mass values and, by comparison with optical energy gap values, of the energy gap temperature coefficient 8. From the variation of mobility with temperature, values of mobility temperature exponent x have been obtained. At the InSb-rich end of the composition range, the results indicate a smooth variation of the various parameters with composition, without any appreciable change in the band structure from that of InSb. The results at the GaSb-rich end of the range are complicated, firstly by the presence in GaSb of two conduction bands of only small energy separation, and secondly by an effect which extends over a considerable range of composition and is thought probably to be due to ordering in the alloys. As a result, correlation of the optical and electrical results is difficult in this range of composition.
prepared separately, by melting together under vacuum the appropriate amounts of the elements. The elements used were all of 99.999 per cent purity, and in the case of InSb the compound was further purified by normal zone refining methods. The alloys were produced by the slow directional freezing technique described previously,(12) and in order to obtain the required range of composition three ingots were used, these having approximate mean compositions of (A) 70 mol. per cent GaSb, 30 mol. per cent InSb (B) 30 mol. per cent GaSb, 70 mol. per cent InSb and (C) 10 mol. per cent GaSb, 90mol. per cent InSb respectively. Sections from various points on these ingots provided specimens covering practically the whole range of composition in the alloy system, i.e. from InSb to 94 mol. per cent GaSb, and the degree of homogeneity of these specimens was the same as that of the specimens used previously in the optical work.(i2) As indicated in that case, some inhomogeneity still remains however. To cover the range of composition from 94-100 mol. per cent GaSb, ingots of the appropriate compositions were made up and annealed at 520°C for 4 weeks. Some of these were rather less homogeneous than the specimens produced by directional freezing however and only one of them was used in the measurements described below. For the electrical measurements, specimens were cut from cross sections of the directionally frozen ingot, and for each composition measurements were made to determine the values of Hall coefficient RH and conductivity D as a function of temperature in the range
INTRODUCTION CONSIDERABLE work has been carried out to deter-
mine the available composition range of semiconductor alloys produced by solid solution beween two semiconducting compounds each having the zinc blende structure.(l*3~3~4*5)Information is now becoming available on the electrical properties of some of these alloys, but in many cases the data is limited. Electrical measurements have been reported on the following alloy systems: InAs-InP(Q GaAs-InAs, (3) CdTe-ZnTe, (5) CdTe-HgTr, (7) GaAs-Ga3Se3, (8) AlSb-GaSb,(s) InSb-In3Te3. (1s) In a recent paper, IVANOV-OMSKIIand KOLOMIET@) have reported electrical measurements on the equimolecular alloy of the GaSb-InSb system. The present authors have recently investigated the variation of optical energy gap as a function of composition for alloys of the GaSbInSb system.(13) The present paper reports an investigation of the electrical properties of these alloys. PREPARATION OF SPECIMENS AND METHODS Each
of the
OF MEASUREMENT
two compounds
concerned
was first 34
ELECTRICAL
PROPERTIES
OF
90-850°K. The specimens were cut to the approximate dimensions 7 x 1 x 1 mm. and were lightly etched in CP-4 etchant before attaching the electrical contacts. For the measurements below room temperature, the current electrodes and a copper-constantan thermocouple were soldered (using high purity indium solder) to the ends of the specimen, and the potential and Hall probes spot welded to the specimen in the normal positions. In the case of the high temperature measurements, the potential and Hall probes were again spot welded to the specimen but the current connections were made by pressure contacts, and the temperature determined by a Pt/PtRh thermocouple, the hot junction of which was held close to the specimen. Specimen currents of up to 60 mA were used, and all voltages were measured by means of a potentiometer accurate to 1pV. Magnetic fields of up to 6 kOe were provided by an electromagnet. During each set of observations at a given temperature, the temperature was held constant to +_ 1°C. CALCULATION
OF PARAMETER
GaSb-InSb
ALLOYS
then it is readily shown that
&max -=
(b- II2 4b
RH ext
where ~~~~~ is the maximum value attained by RH above the Hall inversion temperature. From equations (4), (5) and (6), th e variation of n can be determined as a function of temperature. Standard theory gives the equation
Fs = A(mn*m,*)3/2 exp - -$ i
= 1ogloA + ; loglO(mn*mp*) -
log~,‘;s
-
B
Eo
2.303 kT - ___ 2.303 k
(1)
Q = +&+p/b)
(2)
where for p-type material
P = n+pe
(3)
(7)
Hence the slope of the graph of logronp/T3vs. l/T gives Eo. Combining this value of Eo with the room temperature value of Es from optical work enables a value of the coefficient b to be estimated. Then assuming parabolic bands so that A = 4(2?rkT/hs)s, the intercept of the logrsnp/T3vs. l/T graph gives a value for (mn*ma*)1/2. If spherical energy surfaces and lattice scattering are assumed, then *
P nbz-p RHC_--e (nb+~)~
1
and assuming Ea = Eo+j?T in the temperature range considered, and that mn* and mp* are independent of T
VALUES
As shown below, the large majority of specimens used were p-type in the impurity range, and in the temperature range investigated each of these showed a transition from p-type to n-type behaviour as intrinsic effects began to predominate. Thus the simple single carrier formulae for RH and D could not be used and the more accurate equations had to be used to determine the variation of electron and hole densities (n and p) and of electron mobility PR as a function of temperature. When carriers of both signs are present, the standard equations for RH and o are
35
b=!?=
512
?.?_ CL?)
(
mn*
1
and hence mn* and mp* can be calculated. Although these assumptions can not be accurately applied to the present materials, the results can be used to give an indication of the probable variation of effective mass values. Finally from equations (2) and (3)
and b = t&p, pe being the resultant hole density due to impurities. Here the value of the Hall coefficient ratio r has been taken as unity. From equations (1) and (3) n=-
1
(b-1) ---
2e (b+l)
1
pe
+
b+l
-
-
RH
RESULTS
R Hext
the intrinsic
1 =
-
pee
AND
vs. log T
DISCUSSION
(a) Results of Hall measurements
(4) If it is assumed that at low temperature contribution is negligible so that
and assuming pn a T-2, a graph of logpn enables a value of x to be determined.
(5)
The variation of RH as a function of temperature for various typical specimens is shown (on a log scale) in Fig. 1 and the curves are seen to be those characteristic of materials with predominant p-type impurity. As the temperature is increased from the liquid oxygen temperature, some decrease in RH and hence increase in pe occurs, due
36
J.
C.
WOOLLEY
and
to finite activation energy of impurities, but in most cases the curve levels out before the Hall inversion temperature is reached, and in each case the value of & in this range has been taken as
Fro 1. + l 0
C.
M.
GILLETT
per cent GaSb mean composition, was n-type along the whole length. The variation of pe with position along the length of ingot B is shown in Fig. 2, the variation of composition with position also being indicated. It is seen that the value of pe reaches a minimum of 1.7 x 1017/ems near the centre of the ingot and reaches values between 4 and 5 x 1017jcms at each end. The form of this curve is of interest as far as the behaviour of GaSb itself is concerned. It has been suggested that the p-type character usually observed with GaSb could be due to non-stoichiometry in the compound, but recently BOLTAKS and GUTORO@) have claimed that the result, in GaSb are due to the
Log&~ vs. togloT for various typical specimens 24 mol. per cent GaSb, 76 mol. per cent I&b 76 mol. per cent GaSb, 24 mol. per cent InSb G&b m0l
65 /
60 /
%
G&b
M4030 // I/'
IO 1 I
x) / I
InSb FIG.
Distancealong
ingot,
cm
FIG. 2. Variation of pa with position along ingot B (mean
composition 30 mol. per cent GaSb, 70 mol. per cent I&b). RH@~~ required in equations (5) and (6). These extrinsic range values of Rw showed that two of the three directionally frozen ingots, viz. A and B were p-type along the complete length of the ingot, while the third, C, which contained only 10 mol.
mol
% GoSb
G&b
3. Variation of mobility ratio 6 with composition.
presence of both donor and acceptor impurities with slightly different segregation coefficients. From the present results, it is seen that in the Iatter half of ingot B the value of pe rises from 1.7 to 4.8 x 1017,kms as the GaSb content falls from 30 to 10 mol. per cent. Since the InSb used for the alloy was n-type with less than 2x 10’s donors/cm3 and the GaSb was p-type with approximately 3 x 1017 acceptors/cma, it would appear impossible to explain the variation of pe along ingot B by non-stoichiomet~c behaviour. A plausible explanation can only be produced by postulating the presence of both p- and n-type impurities having segregation coefficients which are different one from the other and which vary along the length of the ingot as the composition varies. The variation of pe with position along ingot A and n, along ingot C further support this suggestion,
ELECTRICAL
PROPERTIES
From the results of RH vs. temperature on the P-type specimens a value of the mobility ratio b can be obtained immediately from equation (6). The use of specimens from ingots A and B thus enabled a value of b to be obtained for all alloy compositions except those in the range O-10 mol. per cent GaSb where only n-type specimens were obtained. The resulting variation of b with composition is shown in Fig. 3. It is seen that the value drops very rapidly from 85 for InSb, being 27 at 10 mol. per cent GaSb, but then stays reasonably constant up to approximately 75 mol.
IO%, FIG. 4. Loglsnp/Ts (a) (b) (c) (d) (e)
47 64 70 76 91
mol. mol. mol. mol. mol.
per per per per per
GaSb-InSb
37
ALLOYS
from the straight line at high temperatures. This is due to the onset of degenerate behaviour, for which case the equations in Section 3 do not apply. Sufficient results are available at lower temperatures for a value of Eo to be obtained. Secondly, graphs (d) and (e), which represent alloys of high GaSb content, are seen to divide into two parts, each being linear and hence giving a separate value of Eo. Thus for alloys of GaSb content higher than 70 mol. per cent, two values of Eo were obtained, a low temperature and a high temperature value, and also a transition temperature Tc. The values of Es are plotted against composition in Fig. 5 and
OK-1
vs. 10S/T for typical alloy specimens. cent cent cent cent cent
GaSb, GaSb, GaSb, GaSb, GaSb,
53 36 30 24 9
mol. mol. mol. mol. mol.
per per per per per
cent cent cent cent cent
InSb InSb InSb InSb InSb
cent GaSb after which it falls again to the value of 4.5 for GaSb. Using the data from the Hall effect measurements and equations (4), (5) and (6), values of npjT3 were calculated for each specimen and graphs of lognp/Ts vs. l/T plotted. Typical examples of these graphs are shown in Fig. 4. From equation (7) these should be straight lines with a slope dependent on E, and it is seen that the graphs are linear over a considerable range of temperature. Two points are illustrated by the graphs in Fig. 4. Firstly, graphs (a) and (b) for alloys of high InSb content, show some deviation
per
OF
Insi
ml
% GaSb
GaSb
FIG. 5. Variation of extrapolated energy gap Eo and transition temperature Tc with composition.
also the values of Tc over the range of composition where this is observed. On the question of the accuracy with which the two values of Eo and the value of Tc can be determined, it is clear that whatever the mechanism causing this effect, there will be some transition range around Tc. Hence there will be some uncertainty in the value of T,, and reasonable values of Ea will only be obtained from results at temperatures well away from the transition range. From a study of the experimental results, it is estimated that the error in Tc will not be greater than 20°C. The lower value of EO (l&l) can be obtained with reasonable accuracy (say
J.
38
C.
WOOLLEY
and
I: 0.02 eV) as measurements can be made at temperatures well below T,, but the upper value of Z&(Eos) will probably be less accurate as a large temperature range above Te cannot be used owing to the proximity of Tc to the melting point of the alloy. The variation of EO given above is to be compared with the corresponding variation of the optical energy gap EB previously published,(la) but repeated here for convenience in Fig. 6.
C.
M.
GILLETT
specimens of about 100 p thickness. Use of other conventions for interpreting optical data could give slightly different values for Eg, but should not greatly affect the shape of the Eg curve shown
‘0
GoSb
n-101 % GoSb
InSb
FIG. 7. Variation of energy gap temperature coefficient p with composition. in Fig. 6. Changes in the form of the Eg curve would result in corresponding changes in the g curve in Fig. 7, but it appears probable that whatever method is used to obtain EB from the optical
InSb
nwl % G&b
G&b
FIG. 6. Variation of room temperature optical energy gap Eg with composition.
If the values of &I
(Fig.
5) and the values of
Eq (Fig. 6) are compared, estimates of the value of a mean temperature coefficient j3 for the temperature range from room temperature to approximately 400°C can be made. The variation of p as a function of composition is shown in Fig. 7. Although the experimental scatter is large, it is seen that j3 has an approximately constant value of - 3 to - 4 x 10-4 eV/“C except in the composition range 65-95 mol. per cent GaSb where it rapidly changes and takes positive values at some compositions. The value of p obtained in this way will depend on the value assumed for Eg, which in turn depends on the method used to interpret the optical data. For polycrystalline alloys of the type used here, this interpretation is somewhat arbitrary and as indicated previouslycls) these particular values of EQ were obtained by determination of the apparent onset of transmission through
L
0
/
m
I nSb FIG.
8. Variation
/
40
mol
60
% GoSb
SO
100
G&b
of mn* and mP* with composition 0 mn* 0 -%*.
data, the peak in the /3 curve in the composition range 65-95 mol. per cent GaSb will still occur. It is hoped to carry out a more detailed investigation of the absorption edge of these alloys in the near future.
ELECTRICAL
PROPERTIES
As indicated in Section 3, when ,9 is known equations (7) and (8) allow the values of m,* and mp* to be estimated provided spherical energy surfaces etc. are assumed. While these assumptions may not be accurately true here, it is of interest to observe the form of the variation of these parameters with composition, and these are shown in Fig. 8. Again a continuous trend in the values from InSb to GaSb is observed except for fluctuations in the composition range of anomalous behaviour. (b) Discussion of Hall measurement results In Fig. 5 the value of Es shows a practically linear variation with composition up to approximately 65 mol. per cent GaSb but at this point the graph splits into two parts. From a consideration of the work of SAGAF@) the two values of Es for GaSb itself can be attributed to two different minima in the conduction bands. SAGARhas suggested that at room temperature GaSb has its lowest conduction band minimum at the centre of the Brillouin zone but that a further set of minima occur at a slightly higher energy and along the [ll l] directions in K space. The energy separation of the minima at room temperature is estimated to be approximately 0.08 eV with a rate of change with temperature of &)
M - 3 x 1O-4 eV/“C.
This interpretation is confirmed by the work of KEYES and POLLAK~~). The present work indicates that the reduction in energy separation with temperature continues at higher temperatures and that the [ 11 l] minima fall in energy below the (000) minimum at temperatures above 450°C. Assuming from Fig. 6 a value of 0.72 eV as the room temperature value of _7&for GaSb, the values of Ea from Fig. 5 can be used to determine the values of the temperature coefficients fi for the two types of minima proposed, assuming that the two have the same energy at 450°C. The resulting values of fi are -3 x 10-4 eV/OC for the low temperature Ea and -7 x 10-J eV/“C for the high temperature Eo, giving a difference (A/3 3 dldt (AE)) of - 4 x 10-d eV/“C in reasonable agreement with the value given by SAGAR.The room temperature energy separation of the minima calculated
OF
GaSb-InSb
ALLOYS
39
on this basis (O-17 eV) is rather different from SAGAR’Svalue, however. When comparing these values, it should be borne in mind that in obtaining the value of 0.17 eV, it has been assumed that the variation of energy gap with temperature can be treated as strictly linear over the whole range of temperature from room to 600°C. While it is true that such a linear variation is a valid approximation over a limited range of temperature, e.g. 250”C-450°C, a typical range over which electrical measurements were made to obtain Eo, and hence that the measured EO values are truly characteristic of the electrical behaviour in that range of temperature, it is by no means certain that this assumption of linear behaviour can be applied over much larger temperature ranges.(lT) This fact, in addition to the inaccuracy in the value of Eos, could easily explain the discrepancy in the estimated values for the energy separation of the band minima of GaSb at room temperature. As a further check that the change in Ea was not in fact caused by some structural effect such as a phase change, powder photographs were taken of GaSb at 560°C. No change could be detected in the high temperature photographs as compared with similar photographs at room temperature except the usual small change in Bragg angle of the lines due to thermal expansion. This observed change allowed a calculation to be made of the mean linear expansion coefficient of GaSb over this range, and a value of 7+ 1 x lo-s/“C was obtained. When the GaSb-InSb alloys are considered it is of interest to see whether the two values of Eo can again be attributed to different conduction band minima as in the case of GaSb. For various reasons this does not appear to possible. One point is that the two Eo curves appear to coalesce at about 65 mol. per cent GaSb and that for alloys of lower GaSb content only one value of EO is observed, a behaviour which would appear very unlikely if two band minima were being observed. A second point comes from the variation of the transition temperature Tc with composition (Fig. 5). It is seen that as InSb is added to GaSb the value of Tc increases to 520°C at 96 mol. per cent GaSb, but at 94 mol. per cent GaSb the observed value of Tc has fallen to 420°C and as further InSb is added it rises to a maximum and then falls
40
J.
C.
WOOLLEY
and
again. Such a variation would seem to indicate that the values of Tc are being determined by more than one process. Initially, in order to explain the energy gap values, a band change similar to that in the Ge-Si alloys was considered, i.e. a variation of the relative energies of different band minima with composition, so that room temperature values would indicate one band minimum in one composition range and another band minimum at other compositions. This appears to be ruled out, if as seems to be generally accepted,(l5,rs) the room temperature band structures of GaSb and InSb are of the same type with the lowest minimum in the conduction band occurring at k = 0. Moreover, it would not be possible to explain the form of the room temperature optical energy gap results (Fig. 6) in terms of such a band change. Thus to explain the variation of Eo and Tc with composition over the alloy range 65-95 mol. per cent GaSb, some effect must be postulated other than the change of lowest band minimum with composition as seen with Ge-Si alloys or the change of lowest band minimum with temperature as shown by GaSb and alloys within a few per cent of this composition, as neither of these effects would appear to be satisfactory over the alloy range 65-95 mol. per cent GaSb. One explanation which is suggested by various experimental results is that of ordering of the gallium and indium atoms in the lattice. In such a case, up to 94 mol. per cent GaSb, the value Eel would apply to the minimum band gap in the ordered structure, and the value EOZto the minimum band gap in the disordered structure, Tc representing the ordering temperature. The following points can be quoted as supporting this : (a) The form of the Tc vs. composition curve between 70 and 94 mol. per cent GaSb is typical of the variation of ordering temperature with composition. (b) The form of the low temperature EO curve in this composition range is similar to that observed in systems where ordering occurs.(ls) (c) If the graph of lattice parameter with composition for this alloy system(a) is investigated as accurately as is possible with the present alloys, it is found to show small deviations from the Vegard line, but to intersect that line at an approximate composition of 75 mol. per cent
C.
M.
GILLETT
GaSb. It has been shown@@ that such an intersection can indicate the possibility of ordering in the vicinity of that composition. (d) The form of the solidus of the alloy system(21) indicates that the solid solution is not ideal in the composition range considered. To check this suggestion, specimens in this range of composition have been annealed at various temperatures above and below the possible ordering temperature given by T,, and then X-rayed in the normal way. No definite origins of ordering have been observed, although this could be partly due to the poor X-ray photographs obtained, because of some degree of inhomogeneity in the specimens used, which with the present method of preparation cannot be avoided. This inhomogeneity could be important, for because of the low diffusion rates observed in these alloys,(a) the degree of long range order might be relatively small, and so any superlattice lines would be weak and diffuse. It is of interest to note that BRAUNSTEIN et aZ.(22) have postulated a short range ordering in Ge-Si alloys in order to explain optical absorption data. It is hoped to carry out further work on this question of ordering when better alloy specimens are available. Thus it is seen from the above discussion that if ordering is postulated to explain the electrical results, this can only apply in the approximate composition range 65-94 mol. per cent GaSb, while the results for 96 mol. per cent GaSb and GaSb itself must be explained in terms of a band change. Thus the behaviour in the range 95-100 mol. per cent GaSb could be a more complicated form. Again, production of alloys in this range has proved somewhat difficult, but it is hoped to investigate this range more thoroughly with better alloy specimens. (c) Conductivity measurement In addition to Hall effect, the variation of conductivity with temperature was also observed for each alloy specimen. Typical curves of log u against l/T are shown in Fig. 9. These show the usual division into an intrinsic and extrinsic range. In order to compare the values for various alloys of different impurity content, the values of log u in the intrinsic range have been extrapolated to room temperature to give an effective room temperature intrinsic conductivity. The variation
ELECTRICAL
PROPERTIES
OF
GaSb-InSb
41
ALLOYS
type of carrier TV= CJRHhas been used. As the high InSb content alloys are n-type, the room temperature results for these give pa, while with the large majority of the alloys, the extrinsic range results give a room temperature value for pp. For comparison purposes, these values of pLp have been multiplied by the corresponding value of b, and Fig. 11 shows the resultant variation of pn with composition, the results from the two types of material being shown differently. It is seen that pn falls rapidly from the value for pure InSb, but levels out in the middle of the composition range at a value of the order 104 ems/V set, and then at high GaSb content falls again
0.5
’ 2
FIG. 9. (a) (b) (c) (d)
Loglou
10 47 76 94
mol. mol. mol. mol.
I
I
I
I
4
6
6
IO
Id/T,
OK-’
-1 12
vs. 103/T for typical alloy specimens
per per per per
cent cent cent cent
GaSb, 90 mol. per cent GaSb, 53 mol. per cent GaSb, 24 mol. per cent GaSb, 6 mol. per cent
InSb InSb InSb InSb
of this value as a function of composition is shown in Fig. 10. The measured values of o were used to give the corresponding values of mobility, and in the extrinsic range the simple Hall mobility for one
t 00 InSb
mol
% GaSb
GaSb
FIG. 11. Variation of room temperature electron mobility with composition 0 from P-type alloys 0 from n-type alloys.
-31 0
InSb
I 20
40
mol
I 60
% G&b
I 60
I CO
G&b
FIG. 10. Variation of intrinsic conductivity (extrapolated to room temperature) with composition.
to the GaSb value. This form of curve is somewhat different from that which might be expected from simple alloy scattering theory, which would predict a minimum near 50 mol. per cent GaSb. However various other factors will influence the form of this curve, e.g. (i) the increase in effective mass as the GaSb content of the alloys is increased, (ii) the interband scattering which can be expected to occur with GaSb and alloys close to this composition, (iii) the effect on alloy scattering of any possible ordering in the alloys. Thus the simple curve showing a minimum at 50 per cent is not to be expected. For the P-type alloys, p, lies in the range 8 x 1016 to 4.5 x 1017/ems, while the values of RH for the n-type specimens give values of ne
42
J.
C.
WOOLLEY
and
in the range 6 x 1015 to 3 x lOls/cmS. These latter specimens are probably compensated however, with a total impurity content comparable with that of the p-type material. Thus the mobility values quoted correspond to material of relatively high impurity content, It is probable however that with alloys of this type, the predominant scattering mechanism at room temperature is not impurity scattering but alloy scattering. This is considered further below.
,
C.
M.
GILLETT
plotted against composition in Fig, 13. There is considerable experimental scatter on the points, but a smooth curve can reasonably be drawn through them. This value of xl falls from 1.5 for InSb as GaSb is added, but over a range from 20-65 mol. per cent GaSb lies within the limits 04 to 1.0. This is to be compared with the values of x between 0.7 and O-85, obtained by GLICKS~(23) for Ge-Si alloys, and the theoretical value of x = O-5 for alloy scattering given by NormHEIM(~~) and BRoOKsfz5f. It appears from this, that as indicated above, at these temperatures and in this range of composition the predominant scattering effect is that for alloy scattering. For alloys of GaSb content greater than 65 mol. per cent the
3.25
I 0 InSb
‘ho 10 47 70 86 91
mol. mol. mol. mol. mol.
per per per per per
cent cent cent cent cent
G&b, GaSb, GaSb, GaSb, GaSb,
T 90 53 30 14 9
40
60
mol % G&b
80
00 GOSb
Frc. 13. Variation of mobility temperature exponent x1 with composition (assuming that p CCTzl* at lower temperatures).
FIN. 12. Loglop* vs. log T for typical alloy specimens (a) (b) (c) (d) (e)
I
I 20
mol. mol. mol. mol. mol.
per per per per per
cent cent cent cent cent
InSb InSb InSb InSb InSb
To obtain values of pn for temperatures greater than that corresponding to the peak in the R- vs. T curve, equation (9) has been used, and for each alloy the variation of log pla against log T determined. Typical results are shown in Fig. 12. In a large number of cases the log plz vs. log T graph divides into two linear sections of different slope with a small transition region between the two. Applying the relation f2-a T-X to these results, two values of x can be obtained for each specimen. The low temperature values of x (~1) are shown
value of xl rises sharply, attaining values of 3.2 in the range 90-95 mol. per cent GaSb. Here the effects discussed previously appear to affect the scattering processes, but insufficient data is available for this to be discussed. The high temperature values of x (~2) are less definite, occurring near to the maximum temperature of measurement. Up to 60 mol. per cent GaSb however the value lies between 1-5 and 2-O and would appear to correspond to a predominence of lattice scattering at the higher temperatures. For compositions with more than 60 mol. per cent GaSb, larger values of xs are obtained rising to a peak value of 5.8 between 90 and 95 mol. per cent GaSb. In this range, the scattering processes will be complicated by the effects discussed above, and little can be said about the values of xs at this
ELECTRICAL
PROPERTIES
stage. The temperature range in which the transition from x1 to x2 occurs shows some correlation with the values of TC given in Fig. 5, but the effect is more complicated than the transition observed in the graphs giving I&J.
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GaSb-InSb
ALLOYS
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