Electron and hole conductivity in CuInS2

I.Pkys. Ckm. solids, 1976,Vol. 37,~~. 173-180. Pcrgamon Press. Printed inGnatBritain

ELECTRON AND HOLE CONDUCTIVITY IN CuInSzl D. C. LOOKand 3. C. MANTHURUTHIL Physics Department, University of Dayton, Dayton, OH 45469,U.S.A. (Receioed 11June 1975;accepted 15Juty 1975) Abstract-Single

crystals of CuInQ have been grown from the melt and annealed in In or S to produce good n- or

p-type conductivity, respectively. Two donor levels, one shallow and one deep (0.35eV), and one acceptor levd at ‘0*15eV

are identified. The hole-mobility data are best fitted with an effective mass m:== 1.3 m, which can be explained by simple, two band k‘.p’ theory if the valence band has appreciable d character. Above 300% the hole mobility falls rapidly, evidently due to multiband conduction and/or interband scattering between the nondegenerate and degenerate valence bands. The conduction band mobility appears to be dominated, in many samples, by large concentrations ( > 10” cm-‘) of native donors and acceptors, which are closely compensated.

2. EXPERIMENTAL CONSIDERATIONS

1. INTRODUCTION

The CuIr& crystals were grown from the melt by sealing stoichiometric quantities of Cu, In, and S in an evacuated quartz tube and heating to 1050, 1125, or I 175°Cfor about 1 day, and then cooling at a rate of about Z-S*Cfhr. Single crystals of approximate dimensions 5 mm x 5 mm x 1mm could easily be found, especially in the 1175°C growth. Good p-type behavior (P - l&cm, kp - 20 cm’/Vsec) could then be obtained by annealing at 550°C in a sulfur overpressure, and good n-type characteristics (p - l&cm, II. - 100-200 cm2/Vsec), by annealing in indium and excess CuIn& powder at 725-800°C. Typical room temperature electrical characteristics are summarized in Table 1. All temperaturedependent measurements discussed hereafter were performed on crystals grown at 117Y’C,since these were somewhat larger than the others. The electrical meas~ements were pe~ormed with a d.c. Hall-effect apparatus in conjunction with an He exchange-gas dewar, operating from 5*5-600°K. A fivecontact configuration was employed [51,with each voltage tWork performed at Aerospace Research Laboratories, Wright- lead guarded by an electrometer in a manner described Patterson Air Force Base, under Contract No. F33615-71-C-1877. elsewhere in detail[6]. Measurements accurate to a few The ternary I-III-VI compounds have recently received much attention because, unlike their binary II-VI analogues, they can often be made usefully both n- and p-type1 1,2]. The l~gest-b~dgap I-III-VI compound having this property is CuInSz (Eo= l-5 eV at room temperature) which has evoked technological interest both as an electroluminescent device[?i] (note that the bandgap is close to that of GaAs), and as a photovoltaic device with a high theoretical solar-cell efficiency[4]. In spite of the importance of its electrical properties there have been no studies, to our knowledge, of their temperature dependences, with subsequent identification of donor and acceptor energies, and hole and electron scattering mechanisms. We have attempted to fill this gap by producing both li- and p-type CuIn$, including some oriented single crystals, and then measu~ng the hole and electron concentrations and mobilities as functions of temperature.

Table 1. Room-temperature electrical characteristics of CuIr& crystals. All annealing times were about 60 hr Qrmh Tmou.ful* 10s0%

But Trutamt Dorl~ rS&

llZS°C

117s%

(8)

TTm

rGba) 4.6 I Id 1.7 x 103

sso% (5)

69

6SO"C(S)

78

UQM

me

4.9 f

ll&v.~ 240

4 or P (n-5 D

- 3.5 x $2

22

n * 1.6 r 1014

18

p - J.2x 10'3

ei

104

169

I3

0.4

II- s.4x 1016

p .rl x 102'

SSO'C(8)

4.8

700% (26)

2.1x IO4

725% (In)

1.1 *

104

160

14s% (1n)

1.6x 103

30

II- 1.2x 1$6 n.b,4xi$f a = 2.9 x 102S

17 0.7

rss% (1s)

24

60

77s% (IO)

13

130

p - 7.9 x 1016 I,'4rul I4 n = 3.7 x 10'2

174

D. C. LOOKand J. C.

percent could be performed on samples with resistances =SlO”fi and the associated time constant was less than 10 sec. The highest resistances measured in the present study were only about 1O9R.The magnetic field strength used was about 18kG, and this insured a low-field condition (@ 4 1) for all measurements. The contacts were pure indium soldered with an ultrasonic iron and these were generally ohmic within a few percent over several orders of magnitude of current in both polarities. Point current contacts were used to avoid shorting of the Hall voltage since, in most cases, the length-to-width ratio of the samples was less than three [5]. The data are presented in Figs. l-4. The p-type sample used for the data of Figs. 1 and 2 was originally a single crystal of approximate dimensions 4 mm X2 mm X 0.6 mm, with the c-axis oriented perpendicular to the long (4 mm) side and lying in the big (4mm X2 mm) plane. Measurements on this crystal, with ZlC, are denoted by circles in Fig. 1. Then this crystal was cut in half and measurements were taken with both ZlC and ZllCon one of these small crystals (now of dimensions 2 mm x 2 mm x 0.6 mm). The ZLC data are shown as squares in Fig. 1 and are within experimental error of the Zlz data on the bigger sample. The p vs T data for ZlC, shown in Fig. 2, were almost identical for both samples and thus are plotted as single points. The ZllC measurements were made twice on the small sample, with the old contacts removed and new contacts soldered on between data runs. The results of these two runs differed by about 40%, and this is attributed to the geometrical difficulties in putting contacts on a small sample. The average ~11was about 50% larger than kLIbut this result must be considered tentative until a larger crystal is obtained, with f parallel to the long edge.

MANTHURUTHIL

T(“K)

t

I

I

0

IO"

i

I1

I

I

'"'"2 3

4

5

6 IO’

7

9

9

IO

II 12

TA’K-‘I

Fig. 2. The hole density vs inverse temperature for a single crystal of CuInS2.The solid line is a theoretical fit derived from the usual single-acceptor model. 200r -

IOO-

80 60 40

2 3 E ” 2o 2 10 8-

600

600 400 300 -

6-

200 -

4-

2.

100;j. $ 60-

II

$ 60E _ 0 p 40a 30 -

W

20

40

60 60100

200

400 600

J

TPKI

Fig. 3. The electron mobilities vs temperature for several CuInS, crystals annealed at different temperatures in In. The solid lines are a best fit of the experimental points.

20 -

IOB-

50

100

200

300400 600

WKI Fig. 1. The hole mobility vs temperature for a single crystal of Cub& with II?. The solid lines result from a theoretical analysis: seetextfor explanation.

The precision (error bars) in the measurements can be taken as about 5% for wp> 10cm’/Vsec, i.e. below 300”K, and about 20% for measurements above 350°K where the Hall-voltage signal-to-noise ratio is becoming small. The Hall factor (r = epRH) is taken as unity in calculating Z.L~ and p since it is within 20% of unity for the lattice scattering mechanisms, and ionized-impurity scattering is only important at the very lowest temperatures. Later, in

175

Electronand hole conductivity in CuIr& T(“K)

Id’

0

5

IO

If

20

f03T/~oK-t)

Fig. 4. The electron densities vs inverse temperature for the samplesillustratedin Fig. 3. The solid lines are a best fit of the experimental points. comparing theory with experiment, it will be seen that other possible errors are larger than these. The data for four representative n-type samples are presented in Figs. 3 and 4. As can be seen from these figures and from Table 1, neither P,, nor n, at a given tempera~re, varies monotonica~y with annealing temperature, although p does. This could be due to poor furnace regulation, differences in the initial condition of the crystals, dependences upon the placement of the components (crystal, In, and powder) in the annealing tube, or other reasons. At any rate, it was possible to produce nearly degenerate n-type CuInS by heating in equilibrium In vapor at 800°C. The large positive temperature coefficient of the electron mobility, in some samples, will be discussed later. 3. ANALYSESOF DATA

3.1 Hofe mobility The I-III-VI compounds are the ternary analogues of the binary II-VI compounds and thus we might expect CuInSl to behave like CdS or ZnS]7J. However, the bandgap is much smaller (1.55 eV at Z’K) than in either of these compounds, and is, in fact, about the same as in GaAs. Since the band parameters, especially the effective masses, are largely determined by the bandgap, and the mobilities determined, in turn, by the effective masses, we might expect the mobilities in CuInSs to be close to those of GaAs (,u. - 8000, gP - 400 cm’/Vsec at room temperature) rather than CdS or ZnS (CL.- 300, pP - 20 cm’/Vsec). However, the mobilities are indeed about the same as those of the latter, and we will have to explain this. We will consider the hole scattering from the fo~owing sources: the acoustical Iattice modes, the non-polar

optical modes, the polar optical modes, the piezoelectric potential, ionized impurities, and neutral impurities. As has been pointed out recently by a number of authors, the usual formulae for the mobilities resulting from these sources are derived for a non-degenerate s-like conduction band and cannot simply be extended to the p-like valence bands by using the hole effective mass and valence-band deformation potential [8- 101. In fact, one must consider (1) the overlap integral in the scattering matrix element, which is unity for a sure s-like state, (2) interband scattering, and (3) differing hole densities and masses in the different bands. The overlap integrals have been shown to be quite close to 0.5 for all the aforementioned lattice scattering mechanisms [l&10] and are probably about the same for the impurity scattering. (Actually the neutral impurities are never important, and the ionized impurities are only important below 120°K.) Thus, we will need to multiply all the standard mobility formulae by about a factor two. The interband scattering, and other degenerate band effects, can be neglected well below room temperature if the valence-band mode1 put forward by Shay et al. [ 1l] is correct. They deduce from their electroreflect~ce spectra that the spin-orbit split valence band is above the degenerate light- and heavy-hole bands, a situation that is rarely encountered (two other examples are CuCl and, possibly, ZnO). The reason, they believe, is that the negative spin-orbit splitting of the Cu 3d-bands slightly dominates the positive contribution from the S 3p-bands; i.e. the valence bands have mixed p and d character. The splitting of the nondegenerate upper band and degenerate lower bands is about 0~02eV. Thus, much below room temperature, we need not be concerned about the effects of other bands. This ~-function mixing may also affect the overlap factor but, without further i~ormation, we will not consider this possibility. The acoustical-mode-scattering mobility is given by [S], 2 @oc= j$j

(8a)‘neh4pu2 3EZ,,m:5/*(kT)“*cmZIVsec

(>

(1)

where the 2 in the numerator is due to the overlap and the 300 in the denominator is to convert to practical units (cm*/Vsec) when all the parameters are entered in cgs units. Here, p is the density, II the longitudinal velocity of sound, E,, the valence-bu~d deformation potential, and the other symbols have their usual meanings. We will not worry about anisotropic effects, here or later, even though we will eventually compare with data for talc-axis. The reason is that preliminary investigations, mentioned earlier, have shown that Pr is only about 50% larger than ILL,and other possible errors in the parameters for IL,, and the other contributions to p are probably larger than this. Wiley and DiDomenico[l2] have combined the acoustical-mode contribution with the non-polar opticalmode contribution by properly averaging the relaxation times (i.e. p = (TU”)/(U~)where T-’ = 7:: + Q-&)and then nume~c~ly solving the resulting integral. They find a combined mobility, panpotgiven by

176

D. C.

LOOK and

where 0 is the optical-phonon characteristic temperature, and v = (EnpO IE., I’, where EnpOis the non-polar-optical deformation potential [ 131.An analytical approximation to S(0, q, T), good to within a few percent over a wide range of 0, q, and T, is[12] s(e, 1, T) 1: (1 + (BIT)$!Il[exp (e/T)- DII-’

(3)

where H and D are constants given for each value of u. Wiley and DiDomenico have found that eqn (2) (multiplied by an interband term near unity) satisfactorily explains the hole mobilities in many III-V compounds even though the polar-optical contribution (neglecting overlap) has been considered to be dominant in many past studies of holes in these compounds[l4]. Wiley has further shown that E,, = 5-6 eV in several III-V compounds, and that 77= 4 in these compounds, as well as Si and Ge [ 151.We will also assume that u = 4 in Cub&, but E,, is probably a little larger than 5 or 6 eV. A rough ruleof-thumb estimate [ 161 is that E,, (valence band) = E,,,(conduction band) - 6 eV. Rode[ 171 finds that E.,(cond. band) = 14.5 eV for both CdS and ZnS, so we will take E,,(valence band) = 8.5 eV for CuIn$. The density, derived from the constituent molecular weights and lattice constants, is P = 4.74 gm/cm”[l8]. In the present approximation we will take simple averages (between CdS and ZnS)[l7] of the other parameters in eqn (1): 8 = 470°K and u = 4.97 x lo5 cmlsec. Also for 7) = 4, we must use H = 1.34, and D = 0.914[12]. Then

J. C. MANTHURUTHIL CuInS (a = 11),whereas E=has not, to our knowledge. In cases where EOappears by itself we will use e. = 11. We will not dwell long on the piezoelectric-potential scattering[21], and neutral-impurity scattering[22], since their contributions are unimportant here. We simply note that pp = 4.6 x 104/T”2mz3’*,and pN = 1.31 x 102’mg/No, where No is the pumber of neutral impurity centers. We will see later than PN > 18,000cm*/Vsec. Finally, the ionized impurity contribution to the mobility will be assumed to be given by the BrooksHerring formula[5], again multiplied by a factor two:

T3’2 = 7.96 x 10”

Nrm :““f(x)

cm’/V set

(7)

where N, is the ionized-impurity density and f(x) = ln(ltx)-x/(1+x). Here .X=

6eorn ~(/cT)~ ITmV2 = 1.435 x 10 -

ne2h2p

P

(8)

The ionized-impurity concentration, N,, is calculated by assuming that the maximum in P vs T is given by ELI= pL = 2~, where PL is the total lattice mobility. This is valid (in Matthiessen’s approximation) if b is symmetrical about this maximum, which is approximately true (/LT- T3”, PL - Te3”). Then, N, = 3.2 x 10” cm-‘, and /.LI= 2.49

X [exp (470/T) - 0.914]}-’cm*/Vsec

(4) where

(In eqn (4), and also in the other “working” equations hereafter, m * is given in units of the free electron mass.) Since the II-VI compounds are in general more polar than the III-V’s, we might expect PpO to be more dominant. Indeed, the important factor e,+/(eo - E,), where e. and E, are, respectively, the low- and high-frequency dielectric constants, is about five times smaller in CdS than GaAs[l7,19]. The polar-optical mode contribution [S, 201 is

X(exp B/T - 1)cm’/Vsec

(5)

where, again, the 2 in the numerator is due to the overlap integral, and x(0/T) is a slowly varying function of I’, which may be approximated, in the range 120-300”K, by x(B/T) = 0.48 exp (0:18 e/T). Although our computations extend slightly outside this range, the resulting error is not at all serious. Again, by using an average eOe~/(eO - em)= 13.7 (13.9 for CdS, 13.4 for ZnS)[l7], we get pLpO = -$& T”*e*s’r[exp(470/T) - 1]cm’/Vsec. (6) It should be noted that e0 has been determined [l l] for

x =@357m:T”*exp(1681/T).

(10)

In eqn (10) we have used the low-temperature fit to the p vs T data (Fig. 2) valid below 2SO”K, where PI is important. The total mobility is now calculated using Matthiessen’s approximation, CL _I = P &, t b;i + p I’, which is probably good to within 20%[23]. The only undetermined parameter left in eqns (4), (6)-(8) is m z. By using m $ = I .3 we get a good fit to the experimental data below 300°K as shown in Fig. 1.The mobility above 300°Kbegins to decline more rapidly, but this is probably due to interband effects, as will be discussed later. It should be noted that the largest contribution to p comes from kanpowhich varies as m *“‘. A factor two change in P..,,~ would only change the calculated rn: by 30%. 3.2 Hole density The hole concentration, below 3OO”K,is given quite well by the usual formula for thermal excitation from a single acceptor level [5]; 1

p =z(4‘N~)

I+

1 I (11)

4(N,.,- ND)~ ‘I*_ 1 (~+No)>

where NA and ND are the concentrations of acceptors and

111

Electronand hole conductivity in CuInS, donors, respectively, and

N" qt=-exp(-&/kT) gu

*3/Z

=

5g, 7”” exp (-

4.825 x 1O'5

At low temperatures (p Q ND,

NA -ND)

E, /kT). we

(12)

have

P=

x exp ( - EA/kT)

(13)

We use our calculated m $ = 1.3 and choose g, = 2, since the upper valence band is non-degenerate; however, it is not clear that the two lower degenerate bands will not influence g,. A plot of p/T” vs l/T then gives E., = 0.145 eV and N., /ND = 2.1. From the mobility fit we found that, at low temperature, N, = 2ND t p = 2ND = thus, ND = 1.6 x 10” and NA = 3-2 x 10” cm-‘; 3.4 x 10” cm-‘. These parameters give a good fit below 3OO’K;above 300”,p begins to rise again, due perhaps to a deeper acceptor level. Below lOO”K,p begins to bend upward from the expected behavior, probably signalling the beginning of an impurity-conduction effect [24]. 3.3 Electron density and mobility A salient feature of the electron mobility, p. vs T, is the dominance of impurity scattering, even at room temperature, as shown in Fig. 3. Most of the n-type samples investigated fell into two broad categories: (1) those for which p. - T3’*in the impurity-scattering regime; and (2) those for which p. vs T was much steeper than this. For category (1) samples, it seems reasonable to apply the Brooks-Herring relationship, given by eqns (7) and (8), except for letting rn: + m !: and p --) n, and also omitting the overlap factor of 2, since the conduction band wave functions are s-like. Thus, by assuming that p, = 2~ at the p vs T maximum, and estimating rn? = 0.16 (to be discussed later) we can calculate Nr = 6 x 10” cm-’ for both the sample annealed at 800°C and the sample annealed at 755°C. For an n-type sample, Nr = 2N, t n, so that N* = 3 x 10” cm-j. It is evident that the compensation must be very high, since n e Nn, NA, and this fact may indeed explain the rapid temperature variation of the mobility of the category (2) samples[25]. If an initially degenerate semiconductor is increasingly compensated, the bottom of the conduction band becomes distorted and inhomogeneous in space, forming energy valleys and barriers. Eventually, with very close compensation, the electrons sit in “droplets” at the bottom of their respective potential wells [26].The conduction in this case must be “activiated” over the barriers between the potential wells, and consequently the measured mobility may be low and extremely temperature dependent. The Hall constant also becomes temperature dependent and thus one must interpret activation energies with caution. We believe, however, that the activation energy (0.35 eV) measured from the 725°C data (Fig. 4) arises from a true, deep donor level, for the following reasons: (1) it has been IPCSVOL.37NO.Z-D

measured in at least three different samples, prepared in different ways; and (2) the 0.35 eV slope does not change above 300°K where clearly, as seen in Fig. 3, the normal conduction-band lattice scattering has begun to take over. In this model then, the effect of the In anneal is to “uncompensate” the donors by reducing the acceptor density. If the acceptors were predominantly In vacancies, the process would be straightforward. However, bonding considerations seem to favor Cu vacancies[l] and X-ray studies have indeed shown that crystals grown from the melt are nonstoichiometric, favoring the In [ 181. Thus, it is possible that the reactions are somewhat complicated. We might note that a plot of nvs the inverse annealing temperature, for the samples annealed in In, has a least-squares slope of about 5.5 f l-5 eV; however, if our analysis above is correct, then this must not simply be interpreted as an enthalpy of formation for S vacancies. It should also be mentioned that a rapid (i.e. faster than T”‘) temperature variation of the hole mobility was also seen for T < 80°K; however, this effect was not studied as a function of annealing temperature. 4.DISCUSSION

4.1 Effective mass CuIn& crystallizes in the chalcopyrite structure, point group IL,, with a direct band gap at a = (O,O,0)[7]. Without spin-orbit coupling, the lowest conduction band transforms like I, (rotationally invariant) while the upper valence bands, under a crystal-field distortion, transform like I&, xy) and 15(x,y) separated by Acr,the crystal-field splitting[271. Then, there are I2 and IJ conduction bands above II, and another Is valence band below the upper I4 and Is valence bands. When spin is included, the double group must be used, adding Ia and I’,, with the following transformations: I, + 16; I2 + 16; I3 + I,; Ir + I,; and I5 + I6 t I',[28]. Unfortunately, little is known about the positions of the upper conduction bands and lower valence bands. An often successful procedure for finding the effective masses of II-VI, III-V, or group IV semiconductors is to use nondegenerate k . p perturbation theory on the lowest conduction band, and degenerate i. j theory on the upper valence bands [29]. If CuInS behaved like most of the known semiconductors, which have an s-like conduction band and p-like valence bands, the spin-orbitsplit I7 valence band (I,““) would lie below the other two (Is and I,) valence bands. However, Shay et al. [ 1l] have found from electroreflectance data that I,“’ lies above the other two, by about 0.02eV and they attribute this to strong mixing of the Cu 3d states into the S 3p states. A recent NMR study supports this conclusion[30]. (Note that I4 allows both pZ and d,, wavefunctions within its basis.) A further result of the electroreflectance study is that AC,= 0; i.e. the two bands below r7’O are nearly degenerate. This mixture of p- and d-like wavefunctions complicates the calculation of the relevant momentum matrix element. Furthermore, pL (transforming like I,) will connect r6 bands with both I6 and I7 bands and the same for I7 bands, while pt (transforming like Id) will connect

178

D. C. Loon and J. C.

Is with I,, only, and vice versa. Thus, the upper conduction bands and lower valence bands will contribute to the effective masses and they will be anisotropic. However, for most known semi-conductors, the dominant contribution to the conduction-band mass, m$, is from the upper valence bands, while the dominant contribution to the spin-orbit-split mass, mt,( = m:), is from the lowest conduction band [29,3 11. In this approximation, neglecting A, (O-02eV) compared with the band gap Eo(l.53 eV), we can write

(19 where

Here, W, and Y, are, respectively, the conduction and valence-band wave functions, pX is the x-component of momentum, and, again, the m *‘s are in units of m,. We are assuming that (pz) = (py’)= (p:)and are ignoring anisotropic effects. It is known that P* is almost constant for most of the II-VI, III-V, and group IV compounds, and can be given as P2 * 23? 3 eV[29,31]. For these compounds, however, T, is nearly pure s-like, and ‘v,, p-like. For Cub& this is evidently not true for v,,, since d-functions mix in. In a tight-binding approximation, suppressing the spatial variation, we will assume

where LYis the fraction of p character and (I - cu), d character. Then, realizing that the momentum operator will not connect q\v, with qd (since AI = 2), we get P*+ (YP*in eqn (16). By using Eo = 1.53eV, P2 = 23 eV, and our measured value mt, = 1.3, we can calculate (Y= O-35 from eqn (15). This value of (Ymeans that the valence bands (at least the spin-orbit-split band) have roughly 65% d character. In the light of our approximations this does not disagree with the 45% d character found by Shay et al.11l] from the relationship A,, = a A, t (1 - cu)Aa. From eqn (14) we can now calculate m t = 0.16, close to the value for CdS. Thus, this simple model can explain why the measured mobilities are quite close to those of CdS, even though the band gap of CuIn& is much lower (I.5 eV compared to 2.5 eV). The reason is that although the lower band gap increases the strength of the interaction, the diminished matrix element reduces it. Further speculation on this matter must await a better knowledge of the band energies and wave functions.

4.2 Defect energies and ~de~ti~cutio~s From the p vs T data (Fig. 2) we measured an acceptor level at 0.145 eV above the valence band. This was the

MANTHURUTHIL

lowest acceptor energy identified in any of our p-type samples. It is interesting to calculate the “hydrogenic” acceptor level: EA = 13*62=0*146eV.

(18)

This agreement is, of course, entirely fortuitous since the Bohr radius for a level this deep is only about 4.5 A, and thus EOshould be diminished (closer to the vacuum value), and m: should also be diminished (closer to the free electron value). Furthermore, we would expect some decrease in the hydrogenic hole binding energy due to the coulombic attraction of other ionized acceptors, an increase due to the repulsion of ionized donors in the neighborhood, and a further decrease due to the screening by other free holes. Usually, the latter two effects are not important, and EA is described by E, = E,, - @VA”‘.For both shallow donors (with E,, -+Eo, N, +&)I321 and acceptors [33] in Ge it is found that @= 2.4 x lo-’ eV-cm. Yu et aL[34] also find this value of B to be consistent with measurements of EA in su~ur-~ne~ed CuGa&. If such a relationship holds in CuInSt, then Eao= 0.16eV. This may be compared with Eao(CuGaS2)= O-11 eV 1341for which the hydrogenic value is calculated to be about 0.13 eV. Thus, it appears that the calculated hydrogenic levels in both CuInS and CuGa$ are within about 20% of the measured values. In the n-type samples we have identified a deep (0.35 eV) level and further believe that a shallow level is necessary to explain the near degeneracy of some of the data. The shallow level may also be given by a hydrogenic approximation, calculated below by using our estimated value of nat = O-16: E. = 13~6$=0.018eV.

(19)

As discussed in Section 3.3 the n-type data are consistent with a model in which the as-grown (1175°C) crystals have ND and Na of approximately 3 x 10” cm--‘. Annealing in In then tends to “uncompensate” the present donors, rather than creating large numbers of new donors. From our sample annealed at 550°C in S, we found ND = 16 x 10”cm-” and NA = 3.4 x 10” cm-‘. However, as seen in Fig. 2, there appears to be a deeper acceptor level, which shows no signs of satiating at 4OO”K,and which may indeed have a density b 10” cmm3.If so, then a sulfur anneal may also simply “uncompensate” the present acceptors by reducing the donor density from about 10’8cm-3 to 10’7cm-3. A spark-source mass spectrographic analysis gives the following impurity concentrations for a crystal grown at 1175°C: 0.3 ppm Li; 10ppm Na; 0.6ppm Al; 3 ppm Si; 6 ppm Cl; 0.2 ppm Ca; 4 ppm Fe; and less than 20 ppm Zn. No other impurities were detected, although H, C, N, and 0 are not determinable by this technique. (It is possible that the Na and Cl came from inadvertently touching the sample prior to analysis.) At any rate, it appears that there are less than 100ppm of impurities and thus our as-grown samples must be dominated by native defects. The

179

Electronand holeconductivityin CuIr& shallow donors are possibly S vacancies, as is thought to be the case in CdS[35], and the “deep” acceptors are possibly Cu or In vacancies, with bonding considerations favoring the former. Furthermore, the deep (0.35 eV) donor and the 0.15 eV acceptor may also be native defects, although their concentrations are not high enough to completely rule out impurities. The Mott-transition concentrations [241for the donors and acceptors would be about 3 x 101’cm-’ and 2 x 10mcm-3, respectively, if mX = 0.16 and rn: = 1.3. Above these concentrations we would expect relatively free conduction for the electrons and holes. Our donor concentration is higher than this for the n-type samples, and, indeed, n is nearly temperature independent for the 800°C sample. The strong compensation modifies the expected behavior somewhat, however. 4.3 High-temperature hole mobility and density Above 300°K it is seen in Fig. 1 that the hole mobility begins to fall from the expected temperature dependence. This effect was seen in most of the p-type samples, even though the Hall-voltage signal-to-noise ratio was some: what poorer at these temperatures. For conduction in two-bands, the Hall mobility is given

by[51, pH’RU=

R,a,‘+ Rm2 ut +

u2

(20)

where band 1 is the spin-orbit split band, which we have assumed is dominant below 3OO”K,and band 2 is the combination of the two degenerate bands below band 1. Here we have again assumed unity Hall factors for all bands. If panpo is dominant in this temperature region, then pZ/pl=(mT/mT)S’Z where mt=1.3 and mf: will probably be close to the heavier of the two masses in the other bands. Also, p2/pI = (m T/mT)3’2exp (-ASO/kT), giving

This model would also predict that p(T) = pl( T)K’( T) from an analysis similar to that giving eqn (20), where K’(T) is equal to K(T) except with the denominator squared. Since the denominator is always greater than unity, we must have p(T) 5 p,(T) if K(T) G 1. However, from Fig. 2 we see that p(T) actually rises above pl( T) at high temperatures. This effect is probably due to a deeper acceptor level, as mentioned earlier. 5. SUMMARY CuInS crystals, grown from the melt at 1175”C,have very low mobility, evidently due to high concentrations (> lO”~rn-~) of native donors and acceptors which are closely compensated. However, anneals in S at 550°C or in In at 800°C produce good p - and n-type conductivity, respectively. An acceptor level at 0.15 eV and a donor level at O-35eV have been identified, but there also is a shallow donor level and probably a deeper acceptor level. The hole-mobility data are well fitted below 300°K with an effective mass m : 2 1.3 if all undetermined parameters in the mobility equations are taken as averages of those same parameters in CdS and ZnS, the II-VI analogues. This high value of m $ can be explained by using simple two-band E. fi perturbation theory if the valence bands have about 65% d character. However, interactions with other bands, which will also affect m$, have not been taken into account. Above 300°K pP begins to fall below the expected temperature dependence, evidently due to multiband hole conduction and interband scattering arising from a pair of degenerate bands about 0.02eV below the upper (spin-orbit split) band. Acknowledgement-We wish to thank Dennis Walters for performing the mass spectrographic analysis. REFERENCES

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=

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180

D. C. LOOKand J. C. MANTHURUTHIL

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