Thermal, electrical and optical properties of (In,Ga)as alloys

J. Phys.

Chem. Solids

Pergamon

THERMAL,

Press 1959. Vol. 10. pp. 204-210.

ELECTRICAL OF

AND

(In,Ga)As

M. S. ABRAHAMS, R.C.A.

Research

Laboratory,

OPTICAL ALLOYS and F, D. ROSI

R. BRAUNSTEIN

David Sarnoff Research (Received

PROPERTIES

1.5 January

Center,

Princeton,

New Jersey

1959)

Abstract-The (In,Ga)As system exhibits complete solid miscibility. Homogeneous alloys were produced across the entire system by using the method of zone-levelling. Physical measurements show that the lattice thermal conductivity decreases markedly with alloying and exhibits a minimum value of about 0.05 Wdegrcm~’ at an alloying composition of about 50 per cent. The thermoelectric power varies linearly with the logarithm of the carrier concentration for non-degenerate crystals, and from these data it appears that the “density of states mass” is independent of alloying. The mobility of electrons decreases monotonically with increasing percentages, up to about 70 per cent of GaAs. The band gap varies continuously with composition, and a concave upward dependence is observed on going from InAs to GaAs.

1. IN~ODU~TION

properties.

In view of these results, it was thought that the system InAs-GaAs might also exhibit solid miscibility. This was considered possible, since a favorable size-factor exists for these two compounds, with due realization that size-factor alone is not a sufficient criterion. Complete solid

THE recent work of FOLBERTH(~)on alloy systems of III-V compounds has indicated that the GaA-GaP and InAs-InP systems exhibit complete solid solubility, and that by suitable alloying one can obtain a wide range of semiconducting 1

I

I

I

I

I

7

1200

1100 z E = too0 S 5 ; 900 S.S.

soo

7oY

,

to

I

I

t

,

20

30

40

50 Mol.

R

L

60 GaAs

FIG. 1. The system InAs-GaAs. 204

70

L

80

so

GOAS

THERMAL,

ELECTRICAL

AND

OPTICAL

miscibility in this system would be most desirable in view of the marked difference in the physical properties of the component compounds. Phase-equilibrium studies by VAN HOOK and LENKER@ have shown complete solid mis~ibiiity in the system InAs-GaAs. The phase diagram for this system, shown in Fig. 1, is characterized by a simple ascendent liquidus and solidus. The present work was undertaken to investigate the variation with composition of typical thermal, electrical and optical properties. The properties investigated were the lattice thermal conductivity, the Hall electron mobility and the band gap. In addition, the dependence of the thermoelectric power on carrier concentration was studied. 2. EXPERIMENTAL

PROCEDURE

(a) Preparation of alloys The alloys were prepared by reacting indium arsenide with a predetermined amount of gallium arsenide, either by gradient freezing or by zonelevelling.(s) The indium arsenide and gallium arsenide, which were used in preparing the final ingots, were obtained by reacting the elements in a graphite crucible which was contained in an evacuated silica tube. The crucible was heated to the melting point of the compound in a horizontal, resistance-type furnace and held at this temperature for about 1 hr. After this, the crucible was slowly cooled to room temperature with solidification proceeding from one end of the ingot to the other. The (In,Ga}As alloys were prepared by placing the desired amount of InAs and GaAs in a graphite crucible, and then sealing the crucible and its contents in a silica tube under a vacuum of about 10-Y mm Hg. A 1 per cent excess of arsenic was used in preparing these alloy ingots, as well as in the preparation of the components InAs and GaAs. The silica tube was long enough so that at all times there was a portion within the after-heater section of the furnace, which was controlled at 600°C. The after-heater served to maintain the arsenic vapor pressure at about 1 atm. In gradient freezing, the tube containing the crucible with the charge was maintained stationary while the furnace was moved from the front end of the ingot at constant speed. With this method, homogeneous sections of the ingots were produced

PROPERTIES

OF (In,Ga)As

ALLOYS

205

with compositions up to 30 molecular per cent of either compound. Growth rates of l-3 in./day were employed. Alloys with compositions greater than 30 per cent had a composition spread as large as 12 per cent. These composition variations were determined from lattice-constant measurements, using X-ray diffraction methods. In zone-levelling, a molten zone was passed from the front to the tail end of the ingot and then in the reverse direction. The overall ingot length was 8 in. and the length of the molten zone was about 1 in. In all cases the ingots were given two passes. Zoning rates of 5 and 10 in./day were used. The homogeneity of the alloys was strongly dependent on growth rate. The slower zoning rate produced alloys near the middle of the system with composition spreads of 4 per cent compared to the 12 per cent spread observed on gradient freezing. Homogeneity not only improves with decreased zoning rates, but also with decreased alloying at a constant zoning rate. This is to be expected in view of the occurrence of constitutional supercooling(4) on solidification. Alloys produced near the ends of the phase diagram exhibited composition spreads < 2 per cent when zone-levelled at 5 in./day. The ingots were polycrystafline with very-coarse grains elongated parallel to the growth direction There was no evidence of microcracks or porosity when grown in a graphite crucible. Chemically, the ingots contained 10-100 p.p.m. of spectrographically detectable impurities. The elements found in order of decreasing abundance were: Cu, Al, Si, Mg, Pb, Ag. (b) Physical measurements The electrical resistance was determined by voltage scanning. An a.c. current method was used in order to eliminate spurious thermoelectric c.m.f. due to Joule heating. The a.c. voltage drop across the crystal was scanned by a motor-driven potential probe, and was fed into a synchronous detector whose output was recorded on a d.c. recorder. Before each measurement, the apparatus was calibrated by switching the detector across a standard resistor. The therm\1 conductivity was determined by employing the technique of placing the specimen between a heater and a sink. The heater was

206

M.

S.

ABRAHAMS,

R.

BRAUNSTEIN

energized by passing an electric current through a resistance wire embedded in the heater. Copperconstantan thermocouples, the junctions of which were soldered into small holes located in the heater and sink, were used to measure the temperatures of the heater and sink. The diameters of the thermocouple wires were so chosen as to make the heat transfer along the thermocouples negligible. The reference junctions of the thermocouples were maintained at 0°C. measurements were carried out in a vacuum of about lo-4mm Hg. Details of this apparatus and that used to measure the electrical resistance are given in a previous paper.@) The thermal conductivity K was determined from the relation, K = (W/AT)(Z/A), (1) where W(watts) is the heat input to the heater, AT (“C) is the temperature difference between the heater and the sink, I (cm) is the length and A (cma) is the cross-sectional area of the sample with dimensions, 1 cm x 0.5 ems. All the results were corrected for radiation losses which amounted to 2-4 per cent of the measured value of K. The error due to the thermal contact resistance between the sample and the heater and sink was minimized by the use of a thin layer of silicone oil at the contact surfaces. This correction amounted to about O-2-0*4 per cent of the measured value of K. The thermoelectric power was measured in the thermal conductivity apparatus. The voltage (V) between the copper wires of the heater and sink thermocouples was measured, using the same specimens as that used in measuring K. The thermoelectric power is given by (2)

Q = V/AT,

AT has been defined above. The electron mobility was determined by measuring the Hall effect. The magnetic field intensity across the thickness of the specimen was equal to 2400 G, and a direct current of 10 mA was used along the length of the specimen. The specimen size was about I x 2 x 7 mm. Ohmic contacts were obtained in all cases by using indium dots between the specimen and the contact points of the apparatus. where

3. RESULTS

AND

DrSCUSSION

(a) Thermal conductivity The lattice thermal conductivity,

Kph, in extrinsic

and

D.

F.

ROSL

material is related to the measured conductivity, K, by the equation,

total thermal

(3)

K = Kph+ Kel,

-44f .42

-

.40 -38

-

.36 -34 ~32 -

0 Mol.% FIG.

GoAs

Variation of lattice thermal conductivity, with composition in the InAs-GaAs system.

2.

~ph,

where +r is the electronic component and is d.nectly related to the electrical conductivity, o, and the temperature, T, by the expression Kel

=

s(k/+-,.

(4) . The constant S has the numerical value of 2 for non-degenerate electrons and lattice scattering, 4 for non-degenerate electrons and impurity scattering, and ns/3 for degenerate electrons.@) Thus, if the test temperature and electrical conductivity are known, ~~1may be evaluated from equation (4) and Kph from equation (3).

THERMAL,

ELECTRICAL

AND

OPTICAL

PROPERTIES

OF

(In,Ga)As

ALLOYS

207

mass” and the kinetic-energy terms does not vary with alloying. Since the kinetic-energy term depends upon the type of scattering and, hence, n, 1ACT/can be expected to be constant in view of the high carrier concentration of the measured samples. Therefore, the dominant scattering mechanism in these samples may be regarded as one of impurity

In Fig. 2, the lattice thermal conductivity at 300°K is plotted as a function of composition. The measured values of electrical conductivity indicate that Kel is negligible for all the crystals except the one containing 36 per cent InAs. For the degenerate alloy containing 36 per cent InAs, Kel is equal to 13 per cent of the total thermal conductivity. It is apparent from Fig. 2 that Kph decreases rapidly with alloying and achieves a minimum of 0.047 Wdeg-lcm-1 corresponding to an alloy composition of N 50 per cent. This decrease in Kph is most likely due to scattering of phonons(7) due to lattice strains resulting from the solid+olution alloying and, perhaps, from a mass effect due to the difference in weight of the gallium and indium atoms. The effectiveness of a small alloying addition on lowering the thermal conductivity of either InAs or GaAs is remarkable. A 10 per cent addition of either constituent into the respective pure compound lowers the thermal conductivity of the lattice by about 70 per cent. (b) Thermoelectric pewer The thermoelectric power@-1s) for a nondegenerate semiconductor with a single sign of carrier predominating may be expressed as: Q = k86.2

In

4.70 x 10’5 ?l

+iln--- miN+!!Y+; T

m

+

ln7-

1* (5)

Here, Q is the thermoelectric power in FV deg-I, and n is the carrier concentration per ems HERRING@) refers to the quantity, m(N), as the “den:;ity of states mass.” ~AcT[ represents the average kinetic energy of the electrons relative to the band edge. From equation (S), at constant temperature Q may be expected to’ vary as - In 11 with a slope equal to 86.2 PV deg-l, if the “density of states mass” and kinetic energy term are constant. In Fig. 3, Q is plotted against the logarithm of n at 300°K for a series of alloys ranging in composition from 0 to 100 molecular per cent InAs. Since all the points corresponding to the nc n-degenerate samples fall along a straight line having the required slope, it appears that the sum of the contribution due to the “density of states

scattering. This is substantiated by the fact that a calculation of the kinetic-energy term, 1ACTI/kT, from equation (5) yields the value of 3.1, which is in good agreement with the value reported by SPITZER and H.XRM(~~) for pure ionized-impurity scattering. This calculation was carried out from the data on pure InAs, using the optical value reported by SPITZER and FAN of 0.03 for m(N)/m. Thus, from the apparent constancy of ]AcT[ with alloying, it would appear that rn(N)/rnis invariant with alloying. (c) Electron mob&ties Ideally, the electron mobility, p, should be plotted as a function of composition at constant carrier concentration. Unfortunately, the available

208

M.

S.

ABRAHAMS,

R.

BRAUNSTEIN

data do not permit this to be done. In Fig. 4, however, several points have been chosen which are not widely different with respect to their carrier concentration. It can be seen that with decreasing percentages of InAs, the Hall electron mobility drops almost linearly down to about 30 per cent 2c

le

16

14

12 F i IC WE " : -0 ;"

6

4

2

I

0

I

I

IO 20

FIG. 4. Variation

InAs.

I

,

I

I

1

30 40 so 60 70 HOI. % ca35

I

I

80

90

of mobility with composition InAs-GaAp system.

IOC

in the

This

is to be compared to the work of a monotonic decrease in the mobility of InAs on alloying with InP. FOLBERTH,~~) who found

(d) Energy band gap The variation in band gap with composition in the InAs-GaAs system is shown in Fig. 5. These data were obtained from infrared transmission measurements made on optically polished, 20-mil-

and

F.

D.

ROSI

thick samples, employing CaFe optics; the usual sample-in-sample-out technique was used. A typical transmission curve is reproduced in Fig. 6. It was found that for samples relatively homogeneous in composition the transmission curves were parallel in the steep portions, and the band gap was arbitrarily taken as the energy at half the maximum transmission. In samples with large inhomogeneities in composition, the absorption edges were more diffuse. In no case did X-ray analysis and annealing studies indicate the presence of a second phase. It is interesting to note that it is now possible to produce material having any desired band gap in the range from 0.35 to 2,24 eV by solid-solution alloying. The range 0.35-1.35 eV is covered by the InAs-GaAs system, while the range 1.35-2.24 eV is covered by the GaAs-GaP system.(l) The concave upward dependence of the band gap on composition for InAs-GaAs, shown in Fig. 5, is to be contrasted with similar studies on other alloy systems. In the InAs-InP system,(r) a linear dependence of band gap on composition is observed, while in the GaAs-GaP system(l) a coricave downward dependence is observed. In the Ge-Si system,(ls) a distinct discontinuity of the rate of change of band gap versus composition is observed, and this indicates that the band edges shift to different positions in momentum space.(14) From the results in the InAs-GaAs system, no definite break in the slope is indicated, and consequently the band edges in this system probably remain essentially at the same position in K-space. This conclusion is consistent with the results of the thermoelectric-power measurements, where it appears that the density of states effective mass does not vary with alloying. The effective masses of electrons in GaAs and InAs are N 0.043m(rs) and O.O3m,(lr) respectively, as determined from optical measurements. Since it appears that such small effective masses are characteristic of conduction-band minima that lie at the center of the Brillouin zone,(ra) the lowest conduction-band minima are probably those with K = 0 for all compositions of InAs and GaAs. 4. SUMMARY (u) Some properties

of the InAs-GaAs system, which exhibits complete solid solubility, have been examined.

THERMAL,

ELECTRICAL

1'40

AND

OPTICAL

PROPERTIES

OF (In,Ga)As

1

1

I

1

I

I

1

I 20

30

I 40

I 50

I GO

I 70

I 90

ALLOYS

209

1'30 1.20 I.10 I.00 ;: *go 2 ,**?I0 -70 *GO -

306

IO

Mol.

r

ICO

‘7. GaAr

FIG. 5. Band gap versus composition in the InAs-GaAs

r.4

I 90

system.

(d) The thermoelectric power varies linearly with the logarithm of the carrier concentration for nondegenerate crystals. It appears that the “density of states mass” is independent of alloying. (e) The Hall electron mobility decreases monotonically with increasing percentages, up to about 70 per cent, of GaAs. (f) The band gap varies continuously with composition for the entire system. A concave upward dependence is observed on going from InAs to GaAs.

FIG. 6. Transmission versus photon energy for an alloy containing 11 per cent GaAs. Band gap = 040 eV.

(b) The process of zone-levelling has produced homogeneous alloys across the entire system. Ilomegeneity is favored by slow zoning rates. (c) The lattice thermal conductivity, q,h, decreases markedly with alloying and exhibits a minimum at an alloying composition of N 50 per cent.

Acknowledgements-The authors would like to express their appreciation to Mr. P. D. DO~NEY for assisting in the preparation of the alloys; to Mr. D. A. KRAMER for assisting in the physical measurements; to Mr. H. H. WHITAKEF~for performing the spectrographic analyses; to Dr. J. WHITE and Mr. W. C. ROTH for performing the X-ray diffraction studies; and to Messrs. H. J. VAN HOOK and E. S. LENKER for permission to reproduce the InAsGaAs phase diagram.

REFERENCES 1. FOLBERTH 0. G., Z. Naturforsch. lOa,502 (1955). 2. VAN HOOK H. J., and LENKER E. S., The System InAs-GaAs. In course of publication. 3. PFANN W. G., Trans. Amer. Inst. Min (Met&) Engrs. 194, 747 (1952). 4. RU~TER J. W. and CHALMERSB., Canad. J. Phys. 31, 15 (1953).

210

M.

S. ABRAHAMS,

R.

BRAUNSTEXN

and

F.

D.

ROSI

5. Rosr F. D., ABELE~B., and JENSENR. V., Materials for Tk~~oelec~yic Refrigeration, J. Phys. Chem.

10. HERRINGC., Pkys. Rev. 96, 1163 (1954). 11. SPITTERL. Jr. and HXRIWR., Pkys. Rev. Ss, 977

So&& 10, 191 (1959). 6. BLATT F. J., Solid State Physics (Ed. F. SEITZ and

(1953). 12. SPITZEHW. G. and FAN H. Y., Pkys. Rev. 106,882 (1957). 13. JOHNSONE. R. and CHRISTIANS. M., Pkys. Rev. 95, 560 (1954). 14. HERMANF., Pkys. Rev. 95, 847 (1954). 15. BARCUS L. C., PERLMUTTER A., and CALLAWAYJ., f Phys. Rev. 111, 167 (1958).

D. TURNBULL)Vol. 4, p. 221. Academic Press, Inc., New York (1957). 7. KLEMEKSP. G., Handbuch der Pkysik Vol. 14, p. 198. Springer-Verlag, Berlin (19.56). 8. JOHNSONV. A. and LARK-H•ROVITZK., Pkys. Rev.

92, 226 (1953). 9. YAMA~HITAJ., J. Pkys. Sot. Japan 4,310 (1949).