Preparation and semiconducting properties of platinum antimonide

J. Phys. Chetn. Solids

Pergamon Press 1%8. Vol. 29, pp. 755-770.

PREPARATION

Printed in Great Britain.

AND SEMICONDUCTING PLATINUM ANTIMONIDEt

PROPERTIES

OF

R. A. REYNOLDS, M. J. BRAU and R. A. CHAPMAN Texas Instruments Incorporated, Dallas, Texas 75222, U.S.A. (Received 2 October 1967) Abstract- Single crystals of PtSb, having purities greater than any previously reported were prepared by horizontal zone refining and Czochralski methods. Extrinsic carrier concentrations near. 10IYcd and bole and electron mobilities at 50-60°K as high as 7,500 cm*/Vsec and 6,000 cm*/Vsec, respectively. were achieved. The thermal energy band gap, the intrinsic carrier concentration, the electron and hole mobilities and the mobility ratio of intrinsic PtSb, were derived from Hall coeficient and resistivity measurements in the mixed conduction range (77-3ooOK). The data give E,,O- O-11eV, n, = 1.32 x lo** T*‘*exp (0.110 eV/2kT), cup= 6.55 x lo@T-1’6Tandp,,/p,, = 0.50. Scattering by acoustic mode phonons limits the mobilities of electrons and holes in intrinsic PtSbl in the temperature range 77-300°K Infrared absorption measurements at =lO”K revealed that the band-to-band absorption threshold is =O.ll eV. The intrinsic absorption coefficient increases slowly above the threshold (a = 19 cm-’ at A = 8 CL),suggesting that the optical transition is either indirect or direct but electric-dipole forbidden. Free carrier absorption data at 77“K show that the conductivity masses are about 0.16 m&In for holes and O-31 m#* for electrons (where m, is the free electron mass and gl’* is an undetermined transport coefficient with a value between 1 and 1.8). The index of refraction at A = 16 p was found to be 5.5 f 0.2 from transmission interference fringe measurements. 1. INTRODUCTION

PtSb, IS A semiconducting compound having a previously reported energy gap of approximately O-075eV[ 11. It melts congruently at 1226°C [2] and crystallizes in the pyrite structure (symmetry T,, (m3)), which can be thought of as the NaCl structure with the Cl- ions replaced by anion pairs pointing in the (111) directions. Since the lead chalcogenides crystallize in the NaCl structure, one might expect the band structure and semiconducting properties of PtSb, to exhibit some similarities to those of PbS, PbSe and PbTe(3]. The PtSb, energy band structure and band parameters were studied by Damon, Miller and Sagar[l] using Hall coefficient, resistivity, Seebeck coefficient, magnetoresistance, piezoresistance, and magnetic susceptibility measurements. They suggested that the valence band tThis work was supported in part by the Air Force Avionics Laboratory, Wright-Patterson Air Force Base, Ohio, under contract AF 33 (615)-3218.

a multivalley structure with ellipsoidal constant energy surfaces located on the (100) direction in k-space, that Ego = 0.07 eV, and and m,* = 0.5 m. where ma* that mo* = O-7I?Z~ and m,* are the density-of-states masses in the valence and conduction bands respectively. Because of difficulties Damon et al. experienced in preparing homogeneous n-type samples, they could make no ’ definite conclusions concerning the nature of the conduction band. Furthermore, the rather high doping levels of their samples (p B 1.3 X 101*/cms) prevented them from obtaining data on the intrinsic Hall coefficient and resistivity below 300”K, and the analysis of data above 300°K was complicated due to the partial statistical degeneracy caused by the high carrier concentrations. We have measured the Hall coefficient, resistivity, and i.r. absorption coefficient of single crystal samples of zone refined PtSb+ The lower carrier concentrations of these samples compared to those of Damon et al. is

756

R. A. REYNOLDS,

M. J. BRAU

and in part&&r, the preparation of n-type samples having 1U1~/crn3 < n c f0%m5, have made it Possible to determine the Hall coefficient, resistivity, and mobility ratio of intrinsic PtSb,. in the temperature range 77300°K. From i.r. absorption measurements, the optical absorption threshold for band-toband transitions and the intrinsic absorption coefficient at photon energies near the band gap were determined. In addition, the hole and electron condu&ivity masses were estimated from the free carrier absorption coefheients. Our combined data show that Ee@= 0.11 eV, and suggest that ED has very little temperature dependence up to 300°K. The intrinsic absorption coefficient is small at photon energies near the band gap, suggesting that the optical transition is either indirect or electric-dipole forbidden. The conductivity mass of holes is smaller than that of electrons, and the intrinsic mobilities are determined by acoustic mode phonon scattering with dilt, = O+O which is independent of temperature. A preliminary report of these results has been previously pub~shed~4~. 2. MATERWSPREPARATION PtSb, melts at a temperature sufliciently low that zone-refining and CzochraIski techniques could be used to prepare high-purity single crystals. Polycrystalline ingots of PtSbz were prepared &om !B99+ per cent platinum sponge and antimony which contained no detectable impurities by emission spectrographic analysis. The hii purity antimony was obtained by zone refining 99+999 per sent antimony for a total of 40 zone passes at a rate of O-5in. per hr in a purified hydrogen atmosphere. Weighed quantities of the elements were sealed in quartz ampoules while under a pressure of approximately lo-* Torr, then reacted for several hrs in a rocking furnace held at 1250%. After sufhcient mixing, the melt was air-quenched to insure a homogeneous starting material. PurBcation and single crystal growth were

and R. A. CHAPMAN

achieved by moving a molten zone through a stationary ingot contained in a quartz. ampoule, The moiten zone was established either by direct-coupling with r.f., using a 1OkW r.f. generator at 5 MHz, or by using a graphite susceptor which was heated by the r-f. generator and in turn melted the PtSb,. Prior to use the graphite susceptor was coated with TIKOTE* silicon carbide to prevent oxidation at the elevated temperatures. The direct coupling process caused considerable turbulence and volatilization of antimony from the molten zone. 3ack-5lling the quartz ampoules with an inert gas and using ambient heaters at 700°C suppressed the volatilization of anti_ mony but some dissociation still occurred* so that the number of zone passes using the direct coupling method was limited to about five. Using the r.f. heated graphite susceptor rather than direct coupling to form the molt&n zone considerably reduced volatilization of antimony from the zone. However, the susceptur temperature necessary to establish a molten zone was approximately 14oO”C, and the resultant hii temperature of the quartz ampoule caused it to devitrify rapidly. A large numberof zone passes which may be necessary to attain high purity was not possible without re-encapsulating the ingot after only 10 to 20 zone passes. PtSbz also was grown by the Czochralski technique using seed crystals oriented either ( 111) or (100). Reacted material was placed in a quartz crucible contained in a graphite suseeptor which was heated by a 25 kW generator at 450 kHz. The PtSb, charge was melted in a dry nitrogen atmosphere, and held at a temperature just above the melting point while the crystal was pulled at a rate of 3 in. per hr, with the seed rotating at 12 rpm. Crystals of PtSba grown by the Czochralski technique usually contained a second phase in the last-to-freeze portion of the ingot. This second phase was identified by electron mieroprobe analysis as a Pt-Sb eutectic composed of *Re&temd Wemark.

PLATINUM

ANTIMONIDE

24 per cent antimony/76 per cent platinum by weight which corresponds to the eutectic composition of the phase diagram. The appearance of this Pt-rich second phase indicates that though antimony loss was not readily apparent, the melt becomes platinumrich during crystal growth. The usefulness of zone refining and Czochralski crystal growing techniques for the preparation of pure PtSb, were evaluated principally by measurement of the Hall coefficient and resistivity at 77°K. Detailed function of temperature measurements and the measurement procedure are presented in a following section. Several emission spectrographic analyses also were performed. Table 1 shows the electrical properties at 77°K of zone-refined single crystal PtSb, ingots. The samples listed are representative of the extrinsic carrier concentrations throughout most of each ingot except for a few cases in which the ingots contained both n- and p-type regions. Ingots 109 and 115 were grown from Pt-rich and Sb-rich material, the two starting compositions containing 2 mol-% excess Pt and 2 mol-% excess Sb respectively. All other ingots were prepared from a stoichiometric starting composition. It is apparent from the data of Table 1 that neither of these ingots have properties markedly different from the rest, suggesting that impurities rather

757

than stoichiometric defects are responsible for the extrinsic carriers. This conclusion is supported by the emission spectrographic analyses which always revealed the presence of a sufficient number of impurities to account for the extrinsic carriers (principally Si and Mg at levels ranging from O-1 to 1 ppm by weight). These impurities may have come from either the starting Pt (which was relatively impure) or from the quartz boat during the zone refining operation. Since all the ingots which were given a relatively large number of zone passes were partly or entirely n-type, the distribution coefficients for the donor impurities probably are larger than those for the acceptor impurities. Table 2 shows Hall coefficients and resistivities at 77°K of Czochralski-grown crystals of PtSb,. The carrier concentrations in zone refined crystals were generally a factor of 3-10 lower than in the Czochralski-grown crystals and no n-type material resulted from this latter method. The impurity content of pulled crystals was higher than zone refined crystals even when previously zone refined PtSb, was used as the charge in the Czochralski method. 3. HALL COEFFICIENT AND RESISTIYITY INTRINSIC AND EXTRINSIC PtSb,

The semiconducting properties of PtSb, were investigated through measurements of

Table 1. Electrical properties of zone-re$ned PtSb, measured at 77°K Crystal No. 1 23 27 33 98 105 109 115 122 140

Number of zone passes 53 12 33 1 1 6 6 7 23 78

l/R,e (ohm4cm) 7.1 3.6 1.7 2.0 1.0 6.4 2.9 1.2 1.5 7.9

x x x x x x x x x x

OF

lo-” lo+ 10-p lo-” 10-I lo-’ 10-l 10-p 10-l 10-1

(cmf;tH set) -1.3 -5.3 -1.1 +2*s +2*1 -1.3 +6*5 +2.3 -1.1 -3.2

x x x x x x x x x x

103 1P 101 103 10s 109 101 109 109 101

(cm+) -6.6 x -3.3 x -3.3 x +1.1 x +3*0x -7.4 x +3*3 x +2*2 x -3.9 x -2.5 x

10’S 10’8 10” 10” 10’6 IO” 10’6 10” 10’6 10”

758

R. A. REYNOLDS,

M. J. BRAU and R. A. CHAPMAN

Table 2. Electrical properties of Czochralski-grown measured at 77°K Cl-ptd No

Orientation

131 149 151 36

(111) (111) (100) (100)

x x x x

the Hall coefficient and resistivity between liquid helium temperatures and 300°K. The higher purity has made it possible to determine the intrinsic carrier concentration, the thermal band gap, and the electron and hole Hall mobilities as limited by lattic scattering in the temperature range 77-300°K. (a) Experimental procedure The electrical measurements were performed using a glass double dewar system and sample holder similar to that described by Mitchell and Putley [S], and an electromagnet which provided field strengths up to 6000 G. Temperatures were determined using a germanium thermometer below 30°K and a copper-constantan thermocouple above 30°K. The electrical measuring apparatus used was of the chopped d.c. type described by Dauphinee and Mooser[6]. The samples were cut ultrasonically from single crystal wafers of PtSb, having thicknesses of 0.5-l.Omm, and

l/&e (cmT3)

CLH (cm*/V set)

(ohAm) 5.6 8.6 8.6 9.4

PtSb,

lo+ lo+ lo+ IO-’

+1*7x +3.0x +3.0x +5.2 x

KY 1oJ 1oJ lo”

-l-6*6 x +2*3 x +2.4 x +1*3x

10” 10” 10” 10”

both the six-armed Hall bar and circular, fourcontact van der Pauw sample configurations were used. Most samples were etched in an HN03-HCl-Hz0 mixture [7] prior to applying pure indium soldered contacts. A few samples having freshly lapped surfaces also were measured and the results were identical to those obtained on the etched samples. The error in the electrical measurements is estimated to be 24 per cent and is largely due to uncertainty in the sample dimensions. The error in temperature measurement is estimated to be less than &0_3”K in all cases. (b) Results, analysis and discussion The properties at 77°K of the samples used in this investigation are listed in Table 3. Initial results revealed that in the mixed (intrinsic) conduction range the Hall coefficient R is positive due to the hole mobility being higher than the electron mobility. This study has emphasized the properties of samples

Table 3. Transport properties at 77°K of the samples used in the study of intrinsic and extrinsic conduction in PtSbz (o&bcm)

Sample No. 140P 122 140N 105 27 15 23 109 136

+103 -1100 -260 -125 -19.0 -6.30 -1.14 -330 -300

1.76 5.80 8.28 748 1.70 7.98 2.91 2.75 2.00

x x x x x x x x x

lo-” 10-l 10-z 10-p 10-z 10-s 1O-3 lo-’ 10-l

(cmG

set)

5840 190 3140 1670 1120 790 392 1200 1500

PLATINUM which are n-type in the extrinsic range because a large amount of information could be gained from analysis of the temperature dependence of R and the resistivity p in the temperature region in which R changes sign. The mobilities listed in Table 3 essentially are one-carrier Hall mobilities except for sample 122, which had a sufficiently low extrinsic electron concentration that R is on the verge of changing sign at 77°K. The following results and discussion concerning these samples are divided into sections dealing with the mixed conduction range and the extrinsic conduction range respectively. (c) Properties of intrinsic PtSb, Samples 140N and 140P had the lowest total impurity contents of all samples studied, as shown by their higher mobilities compared to all the other samples in Table 3. The temperature dependencies of R and p for these two samples are shown in Fig. 1. The data for both samples merge to give pf and R1 for intrinsic PtSb, at temperatures above 125-150°K. There is no dependence of R upon magnetic field strength up to at least 6000 G except in sample 140N ‘at temperatures a few degrees above or below that at which R = 0. In order to calculate the intrinsic carrier concentration ni, the electron mobility wn, and the hole mobility pp from data such as that shown in Fig. 1, it is necessary to know accurately the extrinsic carrier concentration N,, and the electron-to-hole mobility ratio b. N,, can be determined from low temperature Hall measurements, and b can be calculated at the temperature of the positive maximum in R (for n-type samples) using the Hall effect data. As will be shown below, b = O-51 in sample 140N, and since this is near unity any small temperature dependence of b will strongly affect the calculated value of nf, pp and pn. Therefore the temperature dependencies of R and p were measured for a number of n-type samples having different uncompensated donor contents so that values of b could be obtained at several temperatures. These data are presented

ANTIMONIDE

759

102 -

= 3 8 “i

IO’

-

s

-

u?

-

IO0

-

1

_

‘r

v 140N.

6=41OG

14OP, .14OP I

B=426OG 6=666G I

1

I

I

6

IO

12

I4

l

c I 2

-

10-4

‘14ON,6*568OG

.

10-l

:

:

4

6

I6

103/T (‘I?) Fig. 1. Hall coefficient and resistivity vs. reciprocal temperature in the mixed conduction range for the n-type and p-type samples having the lowest total impurity contents.

in Figs. 2 and 3. Analysis of the data presented in Figs. l-3 to determine nl, pn and CL,,requires several simplifying assumptions concerning the semiconducting properties of PtSb,. It is assumed that the carrier concentrations are sufficiently low that nondegenerate statistics can be used, and that the ratio of the Hall mobility to the drift mobility is unity for both electrons and holes. The constant energy surfaces are assumed to be spherical, the densities-ofstates in both bands are assumed to be parabolic, and the effective mass ratios are assumed to be independent of both temperature and carrier concentration. Finally, only one set of valence band maxima and conduction band minima are assumed to be important in intrinsic conduction, and their energy band gap E0 is assumed to vary with temperature according to E, = E,O+PT

(1)

where E,O is the thermal (or 0°K) band gap of

760

R. A. REYNOLDS,

M. J. BRAU

and R. A. CHAPMAN

PtSb,, and p may be positive, negative or zero. Subsequent discussion wilI show that these assumptions allow thoroughly satisfactory inte~retation of the data up to at least 2WK, despite the complications to be expected, for example, due to the small value of E. and the less-than-full cubic symmetry of the crystal structure. Discrepancies above 200°K can be satisfactorily explained by taking into account the partial statistical degeneracy arisingdue to the high intrinsic carrier concentrations. The equations relating R and p to n, p, pp, and EL,,are 181 P-l=~=e(nf.h,+P&

(2)

and R _- a,,%,-I- cp2Rp+ u,~u,,~R+,R,(R,+ (cr,+~~)2+~~2a,2(R,+~,)2B2 i&T

PK-‘1

Fig. 2. Hail coefficients vs. reciprocal temperature for the n-type samples studied to determine the properties of intrinsic PtSb,.

I-

’

1

1

1

-I

’ i’

R,)B2

(3) where CTis the conductivity, B is the magnetic field strength, and the subscripts n and p indicate the electron and hole Hall coefficients and conductivities respectively. In order to solve for the four unknowns n, p, &n, and ppr equation (3) was written for a computer in terms of one variable &J using equation (2) and the additional relations

16’= N,,

(4)

b = ~~~~

(5)

k-p/ and

3

4 8

The four experimental quantities used in the solution of equations (2-5) were R, p, N,, = l/eRBr, and 6. b was calculated at the temperature of the positive maximum of R(R = R,,,) occurring in the plot of R vs. IIT for each n-type sample using the relation

4. Id2-

Kf32

R,,,IR,, 4

I

I

I

I

I

6

8

IO

12

14

16

1031TI’K-‘f Fig. 3. Resistivities vs. reciprocal temperature for the n-type samples studied to determine the properties of intrinsic PtSbs.

=

(b - 1)2

4b

-

(6)

Two methods were used to determine Q. The first was to calculate values of n and p at the temperature at which R = 0 using nb2 = P = (n-Ned

(7)

PLATINUM

and at the temperature at which R = R,,, using nb=p=(n-N.&. (8) nf was then calculated using ni2 = np.

(9)

The second method was to use equation (9) and the values of n and p obtained from the computer calculated solution of equations (2-5) at every measurement temperature for every sample appearing in Figs. l-3. Table 4 lists the values of b, the temperatures at which R = 0 and R = R,a,, and the values of ni at these temperatures for the six n-type samples concerned. b is temperature independent and has a value of 0.50 & O-03 for all samples studied except No. 122, for which b = O-62. The temperature at which R = 0 for sample 140N is about 1.0-f -5°K lower at 4 10 G than it is at 5900 G, which is consistent with b = O-50. The constant value of b for all but sample 122 can be understood[9] in terms of the relative magnitudes of lattice scattering and ionized impurity scattering at the temperature at which R = R,,, for each sample. At this temperature pn and pp for all samples except No. 122 are determined principally by acoustic mode phonon lattice scattering (discussed below) so that [lo] b=

(10)

(%)512@

ANTIMONIDE

761

which is independent of temperature. Here m,* and mp* are the conductivity masses of electrons and holes and E, and EP are the deformation potentials for the conduction and valence bands. In sample No. 122, however, there is an appreciable amount of ionized impurity scattering because the temperature at which R = R,,, is lower and the sample is very closely compensated (n Q Nd and NJ. For ionized impurity scattering [ 1 l] b

(11)

which, if E, = E,, gives a larger value of b since m,*lm,* must then be less than unity. The value of m,*/m,* calculated using equation (10) and assuming E,, = Ep is O-76 (for b = O-50), and this is in reasonable agreement with the ratio of O-5-0.6 suggested by the free carrier absorption measurements presented later in this paper. The small amount of ionized impurity scattering in samples other than No. 122 is most likely due to the relatively large dielectric constant (K(h) = 30 at A = 16 Jo and KStallcwill be larger) calculated from infrared measurements presented later. The values of log (nr/T312)as tabulated from the data in Table 4 are plotted versus 1OYT in Fig. 4. Also included are the values of log (nr/T312)obtained from the computer calculated solutions to equations (2-5) using the data from samples 140N and 140P and b = O-50.

Table 4. The extrinsic carrier concentrations, temperatures at which R = 0 and R = R,,, and the mobility ratio and intrinsic carrier concentrations calculated from the Hall coeficient data presented in Figs. 1-3. All data were taken at B = 4,000-6,000G except 140N*for which B = 410 G Sample 122 14ON 140N* 105 27 15 23

T(R

N,,.

(cm+) -5.68 -240 -240 -5.00 -3.30 -9.93 -5.48

x x x x x x x

10’5 10’6 10’6 10’6 10” 10” lO’8

= 0)

fJ = P”/PP

(“W

0.62 0.51 0.52 0.52 0.49 0.47 0.49

81.1 97.1 %.O 104.8 138.9 166.0 224

n,(R = 0)

(cm+) 5.66 x 1.63 x 1.50 x 3-54 x 2.14 x 5.97 x 3.53 x

10’5 lOI 10’8 10’6 10” 10’7 10’8

T(R

(“K)

n,(R = R,.,) (cm-3)

92.6 107.5 106.5 118.1 157.5 194.5 274

1.16 x 3.45 x 340 x 7.47 x 4.63 x 1.28 x 7.52 x

= Rmx)

lOa 10’6 10’6 10’6 10” lOI lO’8

R. A, REYNOLDS,

762 \ ’ - ‘\ ’

1

\P

,ol5_

1

I

. SAMPLE SAMPLE

I

.M. J. BRAU 1

14OP FOR b =O.SO 140N I 0 CALCULATED VALUES AT _ R’08R=RyAXFOR _ _ SAMPLES IN TABLE 4

I

and R. A. CHAPMAN 2.72 = ~~~c*)3,rexp

(2)

(13)

l

‘$

where m,* and mc* are the densities-of-states masses for the valence and conduction bands . 0 (including band extrema degeneracy factors), --TEXT EON. 12 !Q m. is the free electron rest mass, and p is the temperature coefficient of the band gap as defined by equation (1). If p = 0, then the geometric mean of m,* and m,* from equation (13) is 1.94 m,. The corresponding parameters for Si and Ge are 0*80m, and 0.45 m. respectively [ 121. Since the densities-of-states masses of PtSb, have been suggested[2] to be comparable to or slightly larger than those of Si and Ge, this rather close agreement in the magnitude of the mean density-of-states suggests that exp (-/3/2k) 5 2 and that p is small (1.5 < p < 0). I2 14 I6 6 8 IO 2 4 At temperatures below 90”K, Fig. 4 shows 103/T (‘K-I) that the calculated values of nJT3’* deviate Fig. 4. Log (Q/T~‘~) vs. reciprocal temperature for PtSb,. from the extrapolation of equation (12). This deviation is caused by small errors in the value The two sets of data are in excellent agreeof N,, which nevertheless became comparable ment thoughout the entire temperature range. to nt for T s 9O”K, and by the change in the as ionized The computer calculated values of log (ni/T3’*) value of b at low temperatures impurity scattering becomes more important. using the data of the other n-type samples Equation (12) for ni is believed to be applicable also are in excellent agreement with that shown in Fig. 4, but have been omitted for down to at least 77”K, and probably to temperatures considerably lower since the thermal clarity’s sake. Between 90 and 200°K the data and optical band gaps at 0°K coincide. of Fig. 4 is well represented by At temperatures above about 200°K. Fig. 4 shows that the calculated values of ni/T3’* ni = 2.72 A exp deviate upwards from the extrapolation of equation (12). and this is believed to be due to errors introduced by the use of nondegenerate where A = 4.84 X lOI T3’*/oK3/*cm3.Equation statistics. In order to calculate ni using the (12) is also shown in Fig. 4 and is a least squares fit of the ni data between 90 and 200°K proper statistics, it is necessary to know p, m,* and m,*. As an approximation we assume appearing in Table 4. If E, varies linearly with temperature then equation (12) gives Ego = that since /3 is small it can be set equal to zero, so that from equation (13) (m,*m,*)mo2 = O-110 eV. Since this is nearly equal to the value 1.94. We further assume that mV*/mc* = mp*l of 0.104-O. 113 eV obtained for the band-toband i.r. absorption threshold at lOoK, the m,* which is - 0.8 for b = O-50 and E, = E,, or -0.6 according to the free carrier absorpassumption of a linear variation of Eg with tion measurements. These approximate paratemperature is appropriate. The pre-exponential numerical factor of meters and the tabulated values [ 131 of the Fermi integrals Fi (q) were used to solve 2.72 in equation (12) represents the following: .

‘.

PLATINUM ANTIMONIDE n( = iV,F,,, (r)) = N,Fllz (-e-r))

(14)

763

resistivity data of 140N and 14OP in the intrinsic range up to 227°K gives

where 3 = E&T (reduced Fermi level)), and E= E&T. We obtain ni = 8 X 10%m3 at p* = 76 X lo-5 exp ~~~~v)= (17) 300°K and this is changed little as long as -1.0 < /3 < 0 and 0.5 < m,*lm,* < 2-O.The The expressions for the hole and electron mObilitieS ;Ve then calculations based on ChSSiCd SQitiStiCS gh3 nf = 1.2 X 101%m9 and the extrapolation of equation (12) to 3OO*Kgives nf = 7-9 X 10X8f (18) ~~=4*15 X 108T-3Mexp~1~) cmS. The consistency of the approximate parameters used was tested by calculating R and at 300°K using the proper statistics and b = p,, = 2.07 X 108T-3’2exp (“‘;p”). (19) 0.50. We calculate R = -i-0.20 cmYcoulomb $7 WhetwiS the eXpeIiIUentd VdUe iS i-o-l85 or temperatures above 100°K the term exp cm%. flt*S’XfT) can be approximated by 1-58X There are two reasons why the use of classi- T-O’wso that or, can be expressed as cal statistics (and especially the use of equap, = 6.55 x I@’T-“*57. (20) tion (12)) does not lead to more serious error The occurrence of the near T-3/2dependence When calculating Q above 200°K. The first is that p is small so that even at 300°K E, = 0.1 of lu, and fin over a wide range of temperature eV = 4 kT. The second is that m,*lm,*is near is interpreted to show that acoustic mode Unity SO that the Fermi level in intrinsic PtSb2 lattice scattering limits the free carrier m&Ii.. is near the middle of the band gap and thus is t*ies in sufficiently pure PtSb,. The apparent separated from either band edge by about absence of any polar mode phonon scattering 2 kT. is most likely due to the relatively large value It is not likely that nf is increasing more of the dielectric constant of PtSb,, The smalf rapidly with increasing-T for T > 200°K due deviatiun of the temperature dependence of t0 eXCita~OIl Of h&hlSk CEkITkrS t0 other cLpand ILnfrom the theoreti& T-l*5 behavior energy band extrema or to the onset of a rapid may be due to calculational error introduced decrease of Eb with increasing 1”. These two b y assuming equality of Hall and drift mobilipossibilities would require that p decrease ties, or (as in PbTe3) a small concentration or temperature dependence of q* more rapidly with increasing T for T > 2WK, and m,*. F’igure 5 is a plot of equations (I 8) and ( 19) but this is not the case ex~~men~ly (Figs. 1 and 3). and of the computer calculated values of p@ The mobilities of electrons and holes in and p,, for samples 140N and 14OP for b = Ptgbz as limited by phonon scattering at O-50. Below 200°K there is excellent agreetemperatures below about 200°K can be ment between the analytical expressions and obtained from the calculated values, and the data join I smoothly with the mobility of h&es in the (15) extrinsic range of sample 14OP, and with the “=enlpJl+b) mobility of electrons in the extrinsic range of and (16) sample 140N. Above 200°K the calculated fin = bee, values of @I,and yR fall considerably below the by using the known value of b, equation ( 12) extrapolations of equations (18) and (I 9). This for ni, and the experimental values of pr deviation occurs in the same temperature obtained from samples 140N (T > 115°K)and range as does that in the plot of log (nllPi2) 140P (T > 150°K). A least squares fit of the vs. l/T (Fig. 4) and is believed to be due to the

R. A. REYNOLDS,

M. J. BRAU

103

102 T (OK)

Fig. 5. Electron and hole mobilities of intrinsic PtSb, in the temperature range 1OO-300°K.

use of nondegenerate statistics in the calculations. An alternative explanation is that a new scattering mechanism becomes important above 2OO”K, but this is unlikely because b is unchanged up to 274°K (sample 23, Table 4). Also, there is no discontinuity in the temperature dependence of p above 200°K. Such a discontinuity would be expected if the temperature dependencies of pn and p” changed above 200°K. We believe, therefore, that equations (18) and (19) for pp and p,, are valid up to 300°K. Assuming that equations (18) and (19) are valid between 200 and 3OO”K, they can be combined with the experimental values of the intrinsic resistivity to calculate nj in this temperature range. At 300°K we obtain nj= 7.7 X 10%ma which is in excellent agreement with the value of nj (8 x lOl*/cms) calculated using the proper statistics, and with the value obtained by extrapolation of equation (12) (7.9X lO%ms). We believe that for T > 200°K there are fewer uncertainties in the extrapolation of equations (18) and (19) for

and R. A. CHAPMAN

CL,,and CLI,than in the extrapolation of equation (12) for nj. Therefore, in the temperature range 2000K < T < 300”K, nj can be calculated most accurately with equation (15) in conjunction with the experimental values of Pj and b and equation (18) for cup Comparison of the present data with those of Damon, Miller and Sagar[l] reveals that their p-type samples were essentially intrinsic above 200-25O”K, and that the magnitudes of R and p found in the two investigations agree well with one another in the temperature range 200-300°K. At 500°K and above, however, Damon et al. found that R is negative in all samples, inferring that for T 2 500°K b > 1-O; The change in the value of b at high temperatures could be due to the onset of a new scattering mechanism (e.g. optical mode phonon scattering, carrier-carrier scattering, or intervalley scattering), to excitation of free carriers into other conduction and/or valence band extrema, or to the effective mass parameters becoming dependent on temperature or carrier concentration. The resistivity data of Damon et al. show a change in the temperature dependence at about 350-400”K, suggesting that one or more of the above possibilities do indeed occur. Also, the complex temperature dependence of R found by Damon et al. in their n-type sample in the temperature range 400-700°K could be due to the onset of the temperature dependence of b rather than to sample inhomogeneity as suggested by them. The values of nj, pp and p, for pure, intrinsic PtSb, at 77°K and 300°K are summarized in Table 5. The mobilities have been calculated at both temperatures using equations (18) and (19). nj was calculated at 77°K using equation Table 5. Calculated intrinsic carrier concentration and electron and hole mobilities of pure PtSb, at 300 and 77°K Quantity

300°K

n,, cmvS pp, cmP/V set pm,cmg/V set

7.7 x 10’8 8.3 X 1oZ 4-2 X IOP

77°K 2.2 x 1O’J 7.3 x lff 3.7 x 1w

PLATINUM

ANTIMONIDE

765

(12), and at 300°K using the experimental value of ot and equations (18) and (19) for pn and pp. (d) Extrinsic conduction in PtSb, The Hall coefficients, resistivities, and /i4ON I/f -I mobilities at temperatures below 77°K are shown in Figs. 6-8 respectively for three n-type samples and one p-type sample. The temperature dependence of R and p show that at sufficiently low temperatures impurity conduction dominates the transport properties of these samples. The p-type sample shows practically no freezeout before the onset of metallic impurity band conduction. The three n-type samples show enough freezeout before impurity conduction becomes dominant so that an estimate of the donor ionization energy Ed can be made. Plots of RT312 vs. l/T have 0 0.10 020 0.30 slopes of 3-4meV for samples 109 and 136 IIT (‘K) and of 2-3 meV for sample 140N. The relative mobilities of these three samples suggest that Fig. 7. Resistivity in the extrinsic conduction range for several samples of PtSb%. 109 and 136 are closely compensated (Nd = ZV,)so that the slope of RT3’2 vs. l/T should be equal to Ed. If it is assumed that 140N is not closely compensated, then its slope of RT312 vs. l/T is equal to E,j2, which is consistent

1

o\,

L 0

I

9.76x10’ AT 2.13%

14OP o---o~ I

I

I

0.20

0.10 I/T

I

I

4 I

0.30

(*K-‘1

Fig. 6. Hall coefficient in the extrinsic conduction range for several samples of PtSb,.

I

IO

100

TPK)

Fig. 8. Hall mobility in the extrinsic conduction range for several samples of PtSb,.

766

R.

A. REYNOLDS,

M. J. BRAU and R. A. CHAPMAN

with the estimate of Ed obtained from 109 and 136. The small value of Ed is a consequence of the relatively large value of the static dielectric constant of PtSbz (K,,, > 30). The binding energy, Ear of a hydrogenic acceptor should be about *Ed if mh *5:tm,. The mobility data in Fig. 8 show that at sufficiently low temperatures (impurity conduction region) the mobility is independent of temperature which suggests that a metallic rather than a hopping type of impurity conduction is dominant. This type of behavior also was observed in sample 122 which had the lowest extrinsic electron concentration of all samples studied (5.68 X 1015/cm9). The dominance of metallic impurity band conduction at low temperatures is due to overlap of the (presumably) hydrogen-like impurity wavefunctions, whose Bohr radius is expected to be >53 a for Kstati, > 30 and m*lmo = O-3. According to the theory of Mott and Twose [14], the composition required to achieve metallic conduction for a0 3 53 A would be G 6 X 1016/cm3 which is in reasonable agreement with the data. Between approximately 10 and 40°K the mobilities appearing in Fig. 8 are limited p~ncip~ly by ionized impurity scattering. Since neither hlstatic nor (Nd + IV,) is known, it is not possible to make a quantitative estimate of m*/mo by fitting the data to the Brooks-Herring formula{ 111. However, the relative magnitudes of mn* and mp* can be estimated from the data of samples 140P and 14ON if it is assumed that they are not closely compensated (i.e. n(or p) = Nd + N,). Then cc14oplcl1~~= [m,*lm,*11’2 N,( 14ON)IN,, ( 14OP), and using the mobility data at 20°K we obtain m,*Im,* < 1, in agreement with the free carrier absorption data and the analysis of the mobility ratio in the mixed conduction range. 4. PHOTOA~~ON

The relatively low free-carrier concentration achieved by horizontal zone-refining PtSbt suggested that band-to-band photo-

absorption could be observed in this material. Photoabsorption studies on samples held near 10 and 77°K did provide this expected result. the conductivity effective Furthermore, masses could be estimated from the free carrier absorption observed in the 77°K data. Table 6 shows the sample thickness and 77°K carrier concentration and mobility derived from measurements of R and p on the thinned optical samples. All the p-type samples studied were cut from the same crystat and had similar electrical properties (Hall data were not available for the 167 micron sample). The samples were prepared for the i.r. studies by grinding and polishing with 3200~grit silicon carbide followed by O-3 ,u dia. alumina. Finally, the samples were etched with a 1: 1: 1 solution of HNO,: HCl: H@ [71. Table

6. PtSbz snmpies used measurements

Salllple

Tqe

thickness, (microns)

7?K

1067 540 167 565

P P P n

Carrier concentration at 77°K. (cme8)

in the i.r.

Mobility at 77°K. (cmW see)

+1*4x 10” +3-l x 10’7 ==+2

x 10”

-9.8 x 1O1’

2.5 x 103 1.8 x 10+x -2

x 10+3

7.5 x 10+*

Two different methods were utilized to derive the photoabsorption coefficient, LY,from the i.r. transmittance, T, of the samples. For a sample with parallel faces separated by a thickness t along the optical path [ 15 1 7=

(1 -p)2e-“f I - p2e-2a’

(21)

where p is the reflectivity. Initially, the reflectivity was not known accurately so that the 10°K data were analyzed using a technique less sensitive to the value of p. These measurements were performed by comparing the transmittances 71 and 72 of two samples of differing thicknesses tl and t2 but with identical ff.

PLATINUM

71172 =

eu(“-W{

(1 -p*e-“*‘2)/(

ANTIMONIDE

1 -p*e-“1)).

767

mission through each sample was measured using a copper-doped germanium photoconductive detector mounted in the same dewar. The experimental apparatus and procedure have been previously described[l6]. Figure 9 shows the photoabsorption coefficient, cr, to the l/2-power plotted versus photon energy hv. The middle curve given by the filled-circles is (Al’*for the 10°K runs. The lower curve given by the open-circle data points shows the data after a wavelength independent correction of 1.4 cm-l has been subtracted from the experimental data, i.e. (fY- l-4)“*. A similar plot of (ahv) I’* vs. photon energy also is linear from a threshold at 0.104eV up to 0~17eV. This latter fit is given by

(22) The expression in the braces (p appears here only) approaches 1 as (Yapproaches either 0 or ~0, and in general (Ycalculated using equation (22) is less sentitive to p than that calculated using (21). Fortunately, a reasonably accurate value of p could be derived from transmission interference fringes near 16 Jo observed at 10°K using the 167 p thick sample of PtSb,. The spacing of these fringes indicated that the index of refraction, n, in this wavelength region is 5.5 f O-2 and the reflectivity is 0.48 (= (n-1)2/(n+1)2). This value of p was used in equation (22), in which case the expression in the brackets was as large as 1.2 for intermediate values of (Y.Since all absorption coefficients observed were less than 100 cm-‘, one would expect only small changes in n with wavelength in the region studied. In each run, two PtSb, samples of different thicknesses were mounted on the bottom of a Cryoflask* liquid-helium dewar providing a sample temperature near lO”K, and the trans-

&y=

103(hv-O~104)2eVcm-1

where hv is in eV. Other fractional power plots show that there is a range of exponents which provide fits to the data with small changes in the infrared threshold; for instance 3 x 102(hv-0~113)3/2eVcm-1

ahv=

WAVELENGTH*(MICRONS) 7 I

6 I

IO I

12 I

6 I

.

.

1

T

6-

E v

z! -8

4

2-

Lt.OS

a

-.X$3

:

I 0.10

t

y

b . . . . b”

.

.

cpB

.H J@ I 0.12

(24)

provides a fit to the data which covers a slightly more narrow range of photon energies than that provided by equation (23). The ex-

*Registeredtrademark.

IO

(23)

I 0.14

PHOTON

I 0.16

ENERGY

I 0.16

I 0.20

0.22

(cV)-,

Fig. 9. Square root of the intrinsic absorption coefficient of PtSb, vs. photon energy. Data represented by the different symbols are explained in the text.

768

R. A. REYNOLDS,

M. J. BRAU and R. A. CHAPMAN

ponents 1 and 3 (a plotted vs. hv and a plotted vs. (h~)“~) do not provide good fits to the data. The carrier absorption measurements were done at 77°K using the 1067 and 540 CLthick p-type samples and the 565 p n-type sample. Since the free carrier absorption varied from sample to sample, equation (22) could not be used; equation (21) with p = O-48 was used in all the 77°K measurements. Figure 10 shows the absorption coefficient so derived for these samples as a function of wavelength. A dependence a a A2 is expected for free carrier absorption within a parabolic band[ 171; best fits to the data for this theory are shown as solid lines in this figure. To obtain the intrinsic band-to-band absorption at 77”K, the free carrier absorption was extrapolated to shorter wavelengths and subtracted from the total absorption. The 77°K intrinsic photoabsorption so derived is shown by the triangular data points in Fig. 9. Subtraction of 4 cm-’ from the 77°K data points places the corrected a1j2(77”K) within experimental error of the corrected a1j2( 10°K) shown by the open circles. The difference in the value of a in the

TO 60 50 40

intrinsic range at 10 and 77°K may be due in part to the use of the less accurate equation (21) to analyze the 77°K data. The intrinsic absorption for the n-type sample was slightly larger and is not shown in Fig. 9. Since the i.r. absorption threshold energy for the data of Fig. 9 corresponds to the band gap energy deduced from the transport measurements, a definitely is representative of band-to-band transitions despite its small magnitude. Without reference to the proposed band structure of PtSb, [18], the nature of the band-to-band transition can be interpreted in A dependence (ah) a several ways. is predicted for electric-dipole(hv-Ep forbidden band-to-band direct transitions [ 193 where EG is the minimum direct band gap. At low temperatures, a dependence (C&V)= (hv - EC - Ep)2 can be obtained under restricted conditionst for electric-dipole-allowed band-to-band indirect transitions where EG is tThe more rigorous expression for indirect transitions with the emission of phonons has been simplified by assuming that all direct energy gaps are much larger than hv.

PI Sbz T-77 lK .

i40-

.

/

/

/

WAVELENGTH

P i

”

(MICRONS)

+

Fig. 10. Absorption coefficient vs. photon energy showing both the free carrier absorption (A > 15~) and intrinsic absorption (A <: 12~).

PLATINUM

the minimum indirect gap and Ep is the energy of the phonon emitted to conserve crystal momentum. The maximum phonon energy in PtSb, is not known, but it should be relatively small because of the large mass of both Pt and Sb; for instance, the maximum phonon energy in PbTe is 12 meV [20]. At 77”K, phononabsorption-assisted indirect photoabsorption could be present on this model and may account for the increased intrinsic absorption at this temperature. Emtage[l8] has predicted that both the valence and conduction bands of PtSb, are derived from platinum d-orbitals with an additional higher-energy s-like conduction band. On this theory, all band-to-band transitions near the absorption threshold must be since electric-dipole-forbidden Ai#+l. Emtage further predicted an indirect gap of 0.08 eV followed by a direct gap of 0.091 eV, both of which are near (but not at) the center of the Brillouin zone. Thus, to be in qualitative agreement with this band calculation, the photoabsorption of Fig. 9 must be fit by ahv = A(~v--E~)~‘~ and interpreted as direct forbidden transitions with a direct band gap of 0.11 eV and an (unobRrved) indirect forbidden band gap at a somwhat lower energy. However, angular momentum selection rules may break down sufficiently to permit indirect allowed transitions to dominate in spite of the predictions of the band structure calculations. Because of the slow rise of absorption coefficient above threshold, it can be firmly stated that the threshold is not due to directallowed transitions. Furthermore, the mobilities of electrons and holes in PtSb, are considerably higher than is typical of d-band semiconductors, suggesting that the bands of PtSb, contain considerable contributions from s and p orbitals neglected in the model of Emtage. However, a rigorous determination of the nature of the band-to-band absorption can not be made at present because of the approximations necessary in the derivation of the simplified formula used to predict the wave-

ANTIMONIDE

769

length dependence of band-to-band absorption for the various mechanisms. Free carrier photoabsorption in a parabolic band can be predicted as a function of wavelength using [ 171

a

=

e3 NAo2 4de,mo2c3n p

(25) where e, l0, m0 and c have their usual meanings, n is the index of refraction, N is the carrier concentration, A,,the vacuum wavelength, p the carrier mobility, and m* the conductivity effective mass. g is the ratio of the resistivity at infinite magnetic field to that at zero magnetic field. Using known values of constants and n = 5.5 a = 9.55x 10-14$f g(s)2

m-l.

(26)

This theory can be only approximately applied to PtSb, because of the possibly complex nature of the valence and conduction bands. Furthermore, the value of g has not been measured. Nevertheless, this analysis when applied to the data of Fig. 10 will yield relative sizes of the conductivity effective masses of the conduction and valence bands. For the 1067 p p-type, the 540 p p-type, and the 565 p n-type PtSb, samples, this analysis yields values of g”2(m*/mo) of 0.14, 0.18 and 0.31 respectively. Thus, the electron conductivity effective mass, is roughly twice as large as the hole conductivity effective mass, and since g will have values between I.13 (acoustic mode phonon scattering) and 3.39 (ionized impurity scattering), the data suggest 0.17 < m,*lmo < O-30, and 0.34 < m,*lmo < O-57. That the electron effective mass is greater than the hole effective mass is also suggested by the fact that in intrinsic material, the hole mobility is larger than the electron mobility. Analysis of the free carrier absorption data predicts plasma frequencies corresponding to wavelengths in the 100-200 p region for these samples at 77°K. At room temperature, the

770

R. A. REYNOLDS,

M. J. BRAU

intrinsic carrier ~n~n~on of 7-7 X lO’*fcc predicts a hole-plasma frequency corresponding to a 26 p wavelength. Reflectivity measurements performed at room temperature displayed a reflectivity minimum at 2 1 k and a wavelength dependence characteristic of free carrier absorption[21] with a plasma-oscillation wavelength near this value. These reflectivity data have not yet been analyzed in detail because of the increased reflectivity at short wavelengths due to band-to-band photoabso~tion.

Ac~wfedgements-me authors wish to thank several associates for assistance. H. E. Jarman and D. Thompson grew the PtSb, crystals. D. R. Powell wrote the pro8ram for analysis of the Hall coefficient and resistivity data in the mixed conduction range. Tbe i.r. absorption measurements were performed by W. G. Hutchison (10°K data) and J. D. Parker (77°K data and room temperature reflectivity). The authors wish to thank R. E. Johnson and S. R. Borrello for useful discussions and suggestions concerning this work. REFERENCES 1. DAMON

2. 3.

5. SUMMARY

The thermal (or 0°K) band gap of PtSb, determined from Hall coefficient and resistivity data is 0.11 eV and has little dependence on temperature up to 300°K. The band-toband optical absorption threshold at = 10°K is O-11 eV in agreement with that expected on the basis of the electrical properties. The inQinsic absorption is very small for photon energies near. O-11-0*20 eV, suggesting that the optical transition is either indirect and dielectric dipole allowed, or direct and electricdipole-forbidden. The mobilities of electrons and holes in pure PtSb, are limited by acoustic mode phonon scattering for which the mobility ratio b = p,,/pp is 0.50, independent of temperature. The electron conductivity mass is about twice that of the hole mass based on free carrier absorption results. On the basis of the large dielectric constant, the hydrogenie io~~tion energy is at least as small as 4 meV for donors and 2 meV for acceptors. Consequently the wavefunctions of neutral impurities overlap sufliciently for carrier concentrations above about 101*/cm3 that metallic impurity band conduction dominates carrier transport at low temperatures.

and R. A. CHAPMAN

8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

D. H., MILLER R. C. and SAGAR A., Phys. Reu. 138,636 ( 1964). HANSEN M., Constitution of Binary Alloys, p. 1139. McGraw-Hill, New Y&k (1958). PUTLEY E. H. In ~ute~a~s Used in Semicandu~tor Lkuices (Edited by C. A. Hogarth), p. 71. Interscience. New York 119651. BRAU’M. J.,J. efe&och>m Sot. 113,215~ (1966). MITCHELL W. H. and PUTLEY E. H. Rev. scient. Instrum. 36,134 (1959). DAUPHINEE T. M. and MOOSER E., Rev. scient. Instrum. 26,660 (1955). MILLER R. C., DAMON D. H. and SAGAR A., J. appl. Phys. 35,3582 (1964). PUTLEY E. H., Hall Effect and Related Phenomena, pp. 106-122. Butterwortbs, London (1960). HOWARTH D. J., JONES R. H. and PUTLEY E. H., PFbC. Phys. Sot. ?OB, 124 (1957). SHOCKLEY w.. Electrons and Holefin Semiconductors, Chap. 11. Van Nostrand, Princeton (1950). BROOKS H. In Advances in Ele~?roni&s and Electron Physics {Edited by L. Mat-ton), Vol. 7. Academic Press, New York(1955). GEBALLE T. H. In Semiconductors (Edited by N. B. Hammy),p. 331. Reinhold, New York(1959). BLAKEMORE J. S., Semiconductor Statistics, Appendices A-C. Pergamon, New York (1962). MOTT N. F. and TWOSE W. D., Su~pl. __ Phil. Man. 10,107 (1961). SMITH R. A.. Semiconductors. o. 304. Universitv Press, Cambridge (1959). ’CHAPMAN R. A. and HUTCHINSON W. G., Phys. Reu. 157,6 1S (1967). SMITH R. A. lot. c&p. 219. EMTAGE P. R., Phys. Rev. 138,246 (1965). SMITH, R. A., lot. cit. pp. 196-197, 204, and 209. This text contains the original references on band-toband absbrption. HALL R. N. and RACE’ITE J. N.,J. appi. Phys. 32, 2078 (1961). FAN H. Y. In Semiconductors and Semimetals, Optical Properties of 111-V Compounds, Vol. III, Chap. 9, p. 406. Academic Press, New York (1967).