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The electrical properties and growth conditions of CdS crystals
THE ELECTRICAL PROPERTIES AND GROWTH CONDITIONS OF CdS CRYSTALS M. A. SUBHAN, Department
of Applied
(Received
Physics
26 April
M. N. ISLAM and Electronics.
and J. WOODS University
I97 I ; in reuisedform
of Durham,
6 Ju/.v
England
I97 I )
Abstract-Large crystals of cadmium sulphide have been grown in various partial pressures of cadmium and sulphur vapour. The concentration and ionisation energy of the shallow donors in the different crystals have been determined by measuring the Hall coefficient and electrical conductivity of semiconducting samples, and by comparing the magnitudes of the drift and Hall mobilities in photoconducting (semi-insulating) samples. The variation of the donor and acceptor content can be explained in broad terms in relation to the parameters of crystal growth. Hall measurements indicate that the ionisation energy of the shallow donors is 0.021 k0402eV. The drift mobility experiment suggests that two shallow traps are present with energy depths of 0.015 and 0.037 eV. The variation of Hall mobility with temperature over the range 30-300°K can be explained in terms of polar optical mode. piezoelectric and ionised impurity scattering. The best value of effective mass obtained was 0.19m.
acceptor whereas I2 is associated with the recombination of an exciton bound to a neutral donor. Finally, although the variation of the Hall mobility with temperature has been discussed previously by several authors[4-71 a short discussion of the dominant scattering processes in CdS is included. Theoretical curves taking account of optical mode, piezoelectric and ionised impurity scattering have been fitted to the experimental data using the effective mass, m*, as an adjustable parameter. The best fit was obtained with a value of m” = 0.19m.
1. INTRODUCTION
of the work reported in this paper was to measure the donor concentration, compensation ratio and carrier mobility in crystals of cadmium sulphide which had been grown in various partial pressures of either cadmiuni or sulphur vapour. To this end the electrical conductivity and the Hall coefficient of semiconducting samples were measured over the temperature range from 10 to 293°K. An estimate of the donor concentration of high resistivity photoconducting samples was obtained by comparing the values of the drift and Hall mobilities between room temperature and 85°K. The drift mobility was determined by observing the critical field for departure from Ohm’s law which occurs when the drift velocity of the electrons equals the velocity of sound in the sample and acoustoelectric amplification of thermal phonons begins[ I, 21. With one or two samples we have also tried to compare the relative intensities of the I, and I:! bound exciton lines [3], which are observed in photoluminescence at helium temperatures. with the measured donor and acceptor concentrations. This is of interest because the I, line is attributed to recombination of an exciton bound to a neutral THE
OBJECT
2. CRYSTAL
GROWTH
The samples investigated were cut from boules grown in this laboratory by the method described by Clark and Woods[8]. Briefly the method consists of subliming CdS in a sealed silica tube down a temperature gradient. The CdS condenses at the conical tip of the silica growth capsule, and as the crystal grows the capsule is pulled slowly through the furnace. The capsule is connected via a narrow orifice to a reservoir which contains either cadmium or sulphur. and which is held at a fixed temperature throughout the growth schedule. In 229
230
M. A. SUBHAN.
M. N. ISLAM
this way the crystals can be grown in known pressures of cadmium and sulphur vapour. To prepare the charge for the growth capsule BDH Ltd. ‘Optran’ grade CdS was sublimed and recrystallised in a preliminary process in which argon was passed over a silica boat containing the starting material. With this technique a mass of rods, platelets and dendritic crystals were obtained on a silica liner. These crystals were then used as the charge in the growth capsule. During growth of a boule the charge was held at a temperature of 1150°C while the crystal grew at 1125°C. Reasonably perfect crystalline boules could be grown in partial pressures of cadmium up to about 400 torr and in pressures of sulphur up to about 3 atm. At the extremes of pressures the sublimation ratio was reduced so that the boules produced were less than 1 cm long. This compares with the 3-5 cm length obtained under the most favourable conditions, (the so-called pmin condition) where the vapour had the true stoichiometric proportions. 3. EXPERIMENTAL
The Hall and current saturation measurements were made on bars with approximate dimensions 10 X 2 X 1 mm3. For the current saturation experiments the bars were cut with the c-axis either parallel or perpendicular to the long dimension; With the semiconducting crystals where Hall measurements only were made, the bars had no particular orientation. The bars were cut from the boules using a reciprocating tungsten wire saw, they were then ground and polished with diamond paste down to 1 pm size. After this they were etched in chromic acid and tested for uniformity of resistivity by depositing a row of indium dots along the length of one of the larger faces. The non-uniform samples were discarded. The indium was removed from the remaining specimens with a further etch. Either five or six indium contacts were then applied to enable conventional d.c. Hall measurements to be made.
and J. WOODS
The sample was mounted on the copper block of a metal, helium cryostat with an exchange space between the block and the helium container. Conventional d.c. circuitry was used to measure the Hall coefficient and the electrical conductivity. When required the sample could be illuminated with band gap illumination which was obtained from a tungsten source filtered by a wide band-pass interference filter centred at 0.52 pm, and a 1.O cm path length of 10 per cent solution of copper sulphate. The magnetic flux density employed was variable up to 4.0 K G. In the current saturation experiment the current-voltage characteristics were measured using 20 ps pulses at a repetition frequency of 25 Hz. 4. EXPERIMENTAL
RESULTS
4.1 Carrier concentration in semiconducting samples The variation of the free carrier concentration. n. with temperature is reflected in the variation of the Hall coefficient RI,. For conduction in a single band n = r/R”e. where r is a constant which depends on the scattering process. For the present purpose it is sufficient to assume that r does not differ appreciably from unity. The curves in Fig. 1 illustrate the variation of the Hall coefficient with temperature for a number of semiconducting samples. At room temperature the carrier concentrations, calculated from n = l/R,,e, varied from 4-O x 1015 to 4.0 x 10’” cm-3. These concentrations remained almost constant down to 200°K as indicated by the small changes in RH. Below 200°K the electrons froze out and R, increased with decreasing temperature. With the two samples where the measurements were extended down tol4”K the Hall coefficient began to saturate in the vicinity of 30°K and remained constant as the temperature was reduced further. This effect is associated with the onset of impurity conduction through the donor levels and has been described previously by Crandall[9] and by Toyotomi and Morigaki [ lo].
THE
ELECTRICAL
PROPERTIES
0
Fig.
I.
AND
200
100
Hall
GROWTH
340
n(N.4+ n) (N,,-NJ-nn)
=Texp
400
coefficient as a function of semiconducting stiples
Assuming that the crystals contain a single set of donors of concentration ND cm3 with an ionisation energy ED, and a single set of compensating acceptors of density N,., cm-” the carrier concentration, n. is given by the expression (-E,/kT)
(1)
where N, is the effective density of states in the conduction band.
CONDITIONS
sod
inveke of CdS.
OF CdS CRYSTALS
231
600
temperature
for
5
Values of N,, N.4 and ED were obtained by fitting equation (1) to the experimental curves assuming n = l/R,e and using the knowledge of the magnitude of ND-N,., from the saturated concentration at room temperature. It was assumed that n < (ND - NA) and n < N,., and values of Nc were calculated on the assumption that the density of states effective mass was 0+2m. The values of ED, (ND-N,,), ND and N, obtained are shown in Table 1 together with the growth conditions. The values of N, and N,., obtained from the
Table I. Values of donor ionisation energy and donor and acceptor concentration obtainedfrom Hall data on 5 crystnls grown at 1125°C from a charge held at 1150°C Sample 167 107 148 183 182
Excess in reservoir Cd Cd Cd Cd s
Reservoir T”C ‘650 550 450 50 250
E,eV 0.029 0.02 I 0.021 0.020 0.020
ND cm-J 2.55 x 10’6 I.73 x 10’6 5.88 x IO’b 8.31 x lOI 7.4 x 10’5
N., cm-3 I .47 x 10’5 8.25 x lOI I.51 x IO’6 3.52 x IO’” 3.4 x 10’5
ND-N., 2’40 9.10x 4.37 4.79 4.0
x 10’6 IO’S x 10’6 x 10’5 x IO’S
NDINA 17.3 2.03 3.89 2.36 2.18
IJl, 480 50
232
M. A. SUBHAN.
M. N. ISLAM
andJ.
WOODS
Hall measurements have been compared with the results of edge emission studies made at liquid helium temperatures, see Orr, Clark and Woods[ 1 I]. The intensities of the I, (A = 4867A) and the I, (A = 4889 A) exciton emission components excited by 3650 A radiation had previously been measured for a number of CdS crystals grown in this laboratory. including Nos. 167 and 107. I, and I2 lines are attributed to the recombination of excitons bound to neutral acceptors and donors respectively [3]. Their relative intensities should give therefore an approximate indication of the relative concentrations of acceptors and donors. The values of I,/l, for crystals 167 and 107 are recorded in Table 1. The agreement between the ratio of IJf, and ND/N, for the two crystals is reasonably satisfactory. 4.2 The Hull mobilit>
lb5
1 0
The variation of the electrical conductivity. u. with temperature of the five crystals referred to is shown in Fig. 2. The Hall mobilities. pIr, of these crystals. (p,! = R,,c) are shown in Fig. 3 together with some measurements on a degenerate sample No. 239 which contained about I OLxcm-:’ chlorine donors. With decreasing temperature the Hall mobility for the three crystals where the measurements were extended over a sufficient range reached a maximum in the vicinity of 50°K. Previous workers have suggested that the dominant lattice scattering mechanisms in CdS are polar optical mode and piezoelectric scattering[5-71. Accordingly we have attempted to fit the experimental curves to calculated curves taking ionised impurity scattering into account in addition to the processes mentioned. It seems reasonable to ignore deformation potential scattering of acoustic modes since a calculation based on the well known expression for mobility limited by this process: i.e. p = 3.2 x 10-S pu,2 (m/m* )5/?lT:j/4E;~
(2)
1
1
1 200
1
1
1
1 400
1
1
1
‘Y\ ,.
1 600
1
1
1 I 800
lo’/ T (T in OK) -
Fig.
2. The electrical conductivity temperature for 5 semiconducting
as a function of inverse samples of CdS.
leads to a value of 4.5 x lO’cm’V-‘s-l at 300°K. which is much larger than the highest measured value of abaut 400. In equation (2) we have taken E,,. the deformation potential of the conduction band. to be 2.67eV. The longitudinal wave velocity. c,. has the value 4.4 x lO”cm. s-’ and p. the density of CdS is 4.82. (Incidentally Devlin has performed the same calculation and obtained an even higher, but erroneous. value for the mobility limited by acoustic mode scattering). The fitting of the experimental and calculated mobility curves will be illustrated with reference to the results obtained from sample No. 183. The three scattering processes considered. i.e. polar optical mode. piezoelectric and ionised impurity, would each lead to a value of mobility of pop. ,LL,,~and pi if each were operative alone. When more than one
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
CONDITIONS
233
OF CdS CRYSTALS
has been discussed by Frohlich[ 121 and by Howarth and Sondheimer [ 131. If the polaron coupling constant Q < I, Howarth and Sondheimer have shown that for non-degenerate semiconductors 4-
’
.z
xx
Fou = G. :x ’ I- .
II
I 7
1
I.3
’
’
I.5
’
’
I.7
’
’
Log T (T I” OK) Fig.
3. The
temperature variation semiconducting
’
I.9
’
’
2.1
’
2.3
’
’
2.5
-
of the CdS.
e 8W M” . 36
zlE
(ez- I)
(3)
where z = hw,/kT (w, is the angular frequency of the longitudinal optical phonons), and I,!J(z) = 1 for L 4 I and G(Z) = 3/8 (mz,)“’ when z + I. The polaron coupling constant (Yis
1,81.6.
1
Hall
mobility
of
scattering mechanism is operative the computation of the resultant mobility is a matter of considerable mathematical complexity which is beyond the scope of this work. For this reason the resultant mobility has been calculated following the not uncommon practice of adding the mobilities reciprocally. i.e. /..-I = CL,,,,-’+ I*,>:-’ + pi-l. This procedure is based on the assumption that the total collision probability is the sum of the individual probabilities for each process considered separately from the others. Polar optical mode scattering would be expected to dominate the mobility of crystals such as CdS with an appreciable component of ionic binding. Optical mode scattering
With CdS this leads to a value of a = 0.8 if the static and optical dielectric constants E,~and E, are taken to be 10.33 and 5.24 respectively and if the angular frequency of the phonon w, is taken to be 5.75 x IO’:’ set-i. The effective mass has been assumed to be 0.2m. Since CdS has no inversion symmetry a piezoelectric polarisation is developed by the strain associated with acoustic mode vibrations. The theory of piezoelectric scattering in CdS has been discussed by Hutson [ 141 who writes the piezoelectric mobility as /& = A($..‘g!)“”
(5)
where A is a factor determined by the piezoelectric and dielectric constants. A = 1.44E, c (K’)i{II:. lllOdl3
(6)
Here K is the electromechanical coupling factor. Hutson obtained a value of A = 4.0 X IO9 by taking a suitable average of K’. Since the results show that our crystals were partially compensated the effects of ionised impurity scattering must also be considered. The mobility associated with this
234
M. A. SUBHAN.
M. N. ISLAM
type of scattering has been calculated the Brooks-Herring[ 151 formula. pi = 3.2 x
using
(m,n,*)*~2CSP Z’N,
1015
I log 11.3 x lO’V%,
5 (11
/Il.
(7)
Here N, is the concentration of ionised centres, and Z their electronic change. The value of N, at a particular temperature was obtained by putting N, = 2N, + n. As stated previously the resultant mobility was calculated by adding the reciprocals of the three components. The effective mass was used as an adjustable parameter as was the factor A. The best fit with crystal 183 is shown in Fig. 4. where the broken line was calculated assuming m* = 0.19m and A = 3.0X lo?. A good fit with the experimental points was obtained down to 30°K. Good fits were in fact obtained for all the crystals
and J. WOODS
examined with the values of the adjustable parameters m” and A listed in Table 2. Note that although with one slight exception the best fit was obtained with the same value of the effective mass, i.e. m* = O.l9m, the values of A ranged between 1.3 x 10’ and 4.0 x lo?. Table 2. Values of effective mass and the constant A chosen to give the bestjt betirveen the experimental and calculated values qf mobility Crystal
m*lm
167 107 148 183 182
0.19 0.19 0.19 0.19 0.18
5. CURRENT
“‘“3 -
4.2
4.0-
t c2 =>” . “5 E -3 4
3-a3.63.4-
.
.
-
3.2
p where
I.3
I.5 Log
4.
.\ \\
3.0-
1.1
Fig.
.\ :\
The
1.7 T
calculated
2.1
2.3
2.5
(T III OK) -
and crystal
I.9
measured No. 183.
Hall
mobility
of
SATURATION MOBILITY
A 3.5 3.2 I.3 3.0 4.0
x x x x x
AND
IO’ lo’ IO lo’ IO’
DRIm
Although all five of the crystals listed in Table 1 were semiconducting with resistivities between 1 and 10 fl cm at 300°K. many crystals grown with any cadmium reservoir temperature below about 350°C. and most crystals grown with sulphur reservoir temperatures above 250°C were semi-insulating with room temperature resistivities ranging from 1Ox to 10’” fl cm. Those with resistivities at the lower end of this range, i.e. IO’-10” 0 cm. were usually highly photoconductive so that illumination with tungsten light of 100 ft. candles intensity reduced the resistivity to about 1O3R cm. The higher resistivity samples were usually less photosensitive. Since we were unable to make satisfactory Hall measurements on samples in the dark with resistivities in excess of some lo7 fl cm., the activation energy of the donors and their concentration was determined by comparing the Hall and drift mobilities of illuminated samples over a range of temperatures from
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
CONDITIONS
OF CdS CRYSTALS
235
85 to 300°K. The method is in essence that According to Uchida et al. when WT > 1, previously described by Moore and Smith [2]. the drift mobility ,& = bf&,. The saturated Briefly, provided the resistivity of CdS is current density, J,,,, is then obtained by reless than lo” R cm.. the current voltage characplacing$, in (8) by bJ;,, so that teristics measured under pulsed conditions show a departure from Ohm’s law (saturation) JSat = tzeu,/bf;, = J,,,,/b = (n + n,) eu,/b. (9) at a critical field E, of the order IO:’ Vcm-‘. The saturation is caused by the acoustoelecIn our crystals the condition WT < I obtains tric interaction of the drifting electrons with at 300°K. but at 77°K T has increased so that amplified acoustic waves, the source of which WT 9 I. thus the above considerations are is either the thermal phonons in the crystal important. or the shock wave produced by the leading If we assume a simple model with one edge of the voltage pulse. discrete set of traps of density N, at a depth, Saturation occurs when the electron drift E. below the conduction band then it is easily velocity, u+ equals the velocity of sound in shown that the sample. Then u,! = p,,E,. = u,~, so that a measure of E, allows the drift mobility p.d to I. pH =pH{ 1 + N/N&Xp (E/kr)}-’ f”PH = & be calculated, provided v, is known. In the small signal theory of acoustoelectric (10) amplification the effect of electron traps has and from the considerations outlined above been taken into account by assuming that a fraction. 5 of the bunched space charge is .t&t = /-b/b = /-dJsaJJsato) . (11) mobile, while the fraction (1 -f) is trapped. This theory was extended by Uchida et al. Thus combining (10) and ( 11) [ 161 and applied by Moore and Smith to measure the depth and concentration of the ~~tr/~dJsato/Jsal) - 11 = N/N, em (EIkT) . traps. (12) The part played by the traps depends on the trapping time. 7, which is a function of tempThe object of the experiment therefore is to erature. If T is short compared with the period measure the Hall and drift mobilities pH and of the acoustic wave. i.e. 2.rrlw, the trapped & over a range of temperatUreS and to record electrons communicate rapidly with the the saturated current densities, so that Ln conduction band and can be bunched in the {[PH/P~(Js~~Js,~) - 11T3’*l can be plotted piezoelectric field in phase with the bunched against I/T. A straight line should be obtained, free electrons. In this eventfis a real number from the slope and intercept of which, values J;,. In contrast when OT > I a phase difference of E and N can be calculated. is established between the free and bound Although some ten photoconducting space charge waves and f becomes complex. samples have been examined the results Uchida et al. showed that when WT > I. reported in this section refer to one particular and that when wr < I, ud = sample, No. 78, grown with a cadmium reserU
236
M. A. SUBHAN.
M. N. ISLAM
In all the samples studied saturation effects could be observed in the’ current voltage curves provided the illumination was sufficiently intense to reduce the resistivity to a value less than 10” R cm. The three sets of curves in Fig. 5 show the Hall and drift mobilities of sample 78 under illumination with light of intensity 18. 6 and 1.8ft. candles. At the highest intensity the resistivity at 300°K was 3.46 x lo? fl cm. J,,, remained constant from 300°K down to about 200°K. whereafter it decreased. In the three curves in Fig. 6 Ln { (pHIpdUSa,o/JSa,) - I ) T3”l is plotted against l/T for each of the three intensities of light employed. The three dashed line curves shown in Fig. 6 are composed of two straight line segments. each of which has substantially the same slope regardless of the intensity of illumination used. It is concluded that the drift mobility was limited by the action of two sets of traps. The slopes and intercepts of the pairs of lines in Fig. 6 show that the two traps had depths of 0.015
and J. WOODS
and 0.037 eV. at concentrations of 1.35 X I 017 and 3.14 x IOlfi cm-3 respectively. Since the saturated current density J,,, fell below the room temperature value JSato at temperatures below about 200°K it is reasonable to assume that WT becomes of the order of unity at this temperature. Following the arguments outlined by Moore and Smith and assuming that the correct value to choose for w is that of the angular frequency for maximum acoustoelectric gain we calculate the trap relaxation time associated with the shallower trap to be 1.87 X IO-“‘s. In its turn this leads to an estimate of the capture cross section of the 0.015 eV trap of 5 X lo-“‘cm’. This is of the order of magnitude to be expected for a singly charged trapping centre. It is interesting to note that Photo Hall measurements of the type described by Bube and Mac Donald [ 181 indicate that crystal No. 78 contained two sets of deeper traps with ionisation energies of 0.28 and 0.36eV. The relaxation times of electrons in these
Fig. 5. The Hall and drift mobilities. p,, and gd. of a photoconducting sample No. 78 at three different light intensities (a) I.8 ft. candles, (b) 6 ft. candles. (c) I8 ft. candles.
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
Fig. 6. The results shown in Fig. 5 plotted according to equation t I?) to obtain trap concentrations and ionisation energies (a) 1.8 ft. candles. (b) 6 ft. candles. (c) 18 ft. candles.
traps would be so high that these electrons would not be able to bunch in the piezoelectric field. This explains why a comparison of drift and Hall mobilities measured in the manner described here can reveal information about shallow traps only. 6. DISCUSSION
6.1 Transport properties
The initial object of the work was to try and determine the donor and acceptor concentrations in semiconducting samples of CdS. and the concentration of shallow traps in photoconducting, semi-insulating samples. and to examine how these concentrations varied with the conditions under which the crystals were grown. Before commenting on this
CONDITIONS
OF CdS CRYSTALS
237
aspect of the work it is desirable to compare our results with those of other workers. Although the Hall coefficient and electrical conductivity of CdS have been investigated several times previously there are relatively few estimates of the ionisation energy of shallow donors to be found in the literature. Kroger et al. reported a value of ED of 0.02 eV at a concentration of IO” crne3 but most of their samples showed impurity banding effects. Piper and Halsted[S] found a donor level at 0.032 eV in samples with a concentration of 5 x IO’jcm-” which is sufficiently dilute to allow interaction effects to be ignored. Woodbury found a value of 0.024eV at concentrations of 10’” cm-3 (see [7]) whereas 1takura and Toyada [ 191 interpreted their results as indicating the presence of two shallow donors at 0.014 and 0.007eV. The smaller value is almost certainly suspect, as it was measured near the regime of impurity conduction at low temperatures. Our results on the semiconducting samples (with the exception of one sample) indicate a donor ionisation energy of 0.021 2 0.002 eV. The lowest donor concentration of 7.4 x 1O’j cm+ was not very different from Piper and Halsted’s infinite dilution of 5 x 10i5cm-“. The comparison of the drift and Hall mobilities of the photoconducting sample indicates the presence of two shallow traps with ionisation energies of 0.015 and 0.037 eV. It is reasonable to assume that the shallower of these. which had a measured concentration of I .35 x 10” cm+, was identical with the shallow donors found in the semi-conducting specimens. For comparison it should be recorded that Moore and Smith using the drift mobility technique found a trap at 0.017 eV in a semiconducting sample of CdS and a trap at 0.025 eV in a photoconducting specimen. It is interesting to speculate whether in fact the various samples examined by the different authors contained different donor centres. although all the results quoted above are for unintentionally doped crystals. For example, it may be that Piper and Halsted’s shallow
238
M. A. SUBHAN.
M. N. ISLAM
donor with ED = 0.032 eV is similar to our deeper donor in the photoconducting sample. Much of the previous work on the transport properties of CdS, which is well summarised by the article by Devlin[7], has been concerned with the intrinsic scattering mechanisms. Piper and Halsted described the temperature variation of the Hall mobility of their best crystal using a combination of polar optical and piezoelectric scattering. They obtained a good fit down to 70°K using a value of the effective mass of 0.16 eV. Devlin used a variational method to combine the effects of the two scattering mechanisms (thus avoiding the assumption that the mobilities add reciprocally). His experimental results were limited to nitrogen temperatures. He obtained a good fit down to 80°K with m* = 0.20m. Our results on crystal No. I83 show good agreement down to 30°K when ionised impurity scattering is also taken into account. and indicate a value of m* of 0.19m. This agrees well with previously published work. At temperatures below 30°K the experimental and calculated values of mobility in sample No. 183 began to diverge. Further, reasonable fits for samples 167 and I82 could not be extended below 70°K. The reason for this is to be sought in the increasing importance of conduction through the donor levels as the temperature is reduced below 70°K. Thus at the lowest temperatures, conduction takes place through the conduction band and via the donor levels (‘hopping’). It is difficult to assess the contribution of each of these but clearly the onset of the impurity conduction process will cause the measured mobility to decrease more rapidly than considerations of pop, ppl and pi would predict. Before leaving the subject of scattering processes it should be pointed out that in order to obtain good agreement with our experimental results it has been necessary to choose values of the factor A in Hutson’s piezoelectric scattering equation which lie between the limits of 1.3 x IO? and 4.0 x lo?. The value of A calculated by Hutson was
and J. WOODS
4.0 x 109. There are several reasons for supposing that the magnitude of A may vary from sample to sample. For example the piezoelectric constants used by Hutson were obtained from measurements on samples annealed in sulphur vapour and which in consequence were highly insulating. It is possible that the value of the electromechanical coupling constant is different in samples with higher conductivities because of .the screening of the piezoelectric field by free carriers. However it must be admitted that the present results offer no evidence to support such a possibility. Hutson also assumed that the elastic and dielectric properties of the crystal were isotropic. The true value of A may obviously depend on the crystallographic orientation of the sample. Finally c-axis inversion, which has been observed in thin films[20], may occur in crystals, and the quantity A would vary if single crystals contain anti-phase boundaries. 6.2 Donor content and crystal gro~r~tll
When CdS and either sulphur or cadmium are maintained at various temperatures in a sealed tube, each of these components will make a contribution to the vapour pressure in the system. CdS dissociates according to the reaction 2CdS ,$G= 2Cd,+ S,,.
(13)
In a tube containing CdS only, the partial pressures are given by the equation p’;,, . Psz = KrdS
(14)
KCdS is the dissociation constant. At any temperature there will be a minimum total pressure (p& which can be shown to occur when pea = 2p,, = (2KC,s)“‘1.
(15)
Under these conditions the vapour will have stoichiometric proportions. On heating, the
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
CdS charge will vapourise and attempt to establish the Pmin condition throughout the enclosure. When a reservoir containing cadmium or sulphur is added the equilibrium will be disturbed provided the vapour pressure of Cd or S? at the reservoir temperature exceeds the dissociation pressure of the corresponding component at the charge temperature. When the charge is held at 1150°C the Pmin condition requires a cadmium partial pressure of 80 torr and a sulphur pressure of 40 torr. These conditions are obtained for reservoir temperatures of TCd - 600°C or 300°C. Thus crystals grown with a TS cadmium reservoir at 600°C should have similar properties to crystals grown with sulphur reservoirs at 300°C. This is borne out by the measured properties of crystals Nos. I07 and 182 which have substantially similar concentrations of donors and compensation ratios (see Table 1). We have found by experience that it is only possible to grow boules of appreciable size when the partial pressure in the growth chamber is equal or less than the dissociation pressure which establishes the pmi,, condition. Thus when either TCd exceeds 600°C or T, exceeds 3OO”C, transport from the charge to the growing crystal is reduced and small boules only are produced. The extent to which this occurs depends upon the temperature gradient in the growth chamber (i.e. the relative temperatures of charge and boule), but in our experiments we attempted to maintain a fixed gradient. Hardly any growth at all occurred for cadmium reservoir temperatures above 700°C and sulphur temperatures above 400°C. The highest sulphur temperatures led to insulating crystals with such low photoconductivity that it has not as yet been possible to use the current saturation method to determine their shallow donor content. In contrast, crystal 167 is typical of boules grown in pressures of cadmium which exceed the pmln condition. It is interesting to note that the Hall measurements show that although the donor content is only increased
CONDITIONS
OF CdS CRYSTALS
239
slightly relative to that of sample 107, the major effect of increasing the cadmium pressure is the reduction of the acceptor content. With either TCd < 600°C or T, < 3OO”C, when the reservoir pressures are less than the dissociation pressures, the dissociation products will constantly diffuse to the reservoir and the pressures in the growth chamber are difficult to calculate. To prevent excessive diffusion the connection between the growth chamber and the reservoir was via a very narrow orifice. In general with cadmium reservoir temperatures exceeding 35O”C, semiconducting crystals were produced, but with cadmium reservoirs at lower temperatures, or with T, < 250°C the electrical properties of the crystal were unpredictable, and were sometimes semiconducting and sometimes semiinsulating. Crystal I83 is an example of a semiconducting sample, with properties not very different from those expected for a sample grown under Pmin conditions. In contrast crystal 78 grown with TCd = 350°C was photoconducting with a high donor content, I.35 x 1017cm-3, but this crystal clearly contained an almost equal large concentration of acceptors. Current saturation measurements on other samples which are unreliable because of the inhomogeneity of the crystals tend however to support the conclusion that photoconducting samples contain concentrations of donors and acceptors of about 10” cm-3. REFERENCES I. SMITH R. W..Phys. Rev. Left. 9,527 (1962). 7. MOORE A. R. and SMITH R. W.. Phys. Reo. MA, 1’50 (1965). 3. THOMAS D. G. and HOPFIELD J. J., Phys. Rev. Lef~.7,316(196l);Phys.Reu.12.8,2135(1962). 4. KROGER E. A.. VINK H. J. and VOLGER J., Phillips Res. Repr. lo,39 (1955). 5. PIPER W. W. and HALSTED R. E., Proc. Int. Conf. on Semiconductor Physics, Prague 1960, rx 1046. Academic Press, New York (1961). 6. iOOK D. and DEXTER R. N., Phys. Rev. 129,198O (1963). 7. DEVLIN S. S., Physics and Chemistry of II-VI Compounds, p. 552. (Edited by Aven and Prener) N. Holland (1967).
240
M. A. SUBHAN.
M. N. ISLAM
8. CLARK L. and WOODS J.. J. Crysr. Growth 3.4. 127 (1968). 9. CRANDALL R. S.. Phys. Rev. 169.577 (1968). IO. TOYOTOMI S. and MORIGAKI K.. J. phys. Sot. Japan 25,807 ( 1968). I I. ORR D. S., CLARK L. and WOODS J..J. Phys. D. (Appl. Phys.) 1,1609 (I 968). 12. FROHLICH H.,Adu. Phys. 3.325 (1954). 13. HOWARTH D. J. and SONDHEIMER E. A., Proc.R.Soc.219.53(1953). 14. HUTSONA.R..J.appl.Phys.32.2287(1961).
and J. WOODS
15. BROOKS H..Adv. Electron. Phys. 7. I56 (1955). 16. UCHIDA I., ISHIGURO T.. SASAKI Y. and SUZUKI T..J. phys. Soc.Japanl9.674 (1964). 17. ISLAM M. N. and WOODS J.. J. Phys. D. (Appl. Phvs.)3,1197 (1970). 18. BABE R. H. and MACDONALD H. E..Phys. Rev. 121.473 ( 1961). 19. ITAKURA M. and TOYODA H.. J. phys. Sot. Japan 18, I 50 ( 1963). 20. ROZYANYI G. A. and FOSTER N. F.. J. appl. Phys. 38. 5 172 (I 967 ).
of Applied
(Received
Physics
26 April
M. N. ISLAM and Electronics.
and J. WOODS University
I97 I ; in reuisedform
of Durham,
6 Ju/.v
England
I97 I )
Abstract-Large crystals of cadmium sulphide have been grown in various partial pressures of cadmium and sulphur vapour. The concentration and ionisation energy of the shallow donors in the different crystals have been determined by measuring the Hall coefficient and electrical conductivity of semiconducting samples, and by comparing the magnitudes of the drift and Hall mobilities in photoconducting (semi-insulating) samples. The variation of the donor and acceptor content can be explained in broad terms in relation to the parameters of crystal growth. Hall measurements indicate that the ionisation energy of the shallow donors is 0.021 k0402eV. The drift mobility experiment suggests that two shallow traps are present with energy depths of 0.015 and 0.037 eV. The variation of Hall mobility with temperature over the range 30-300°K can be explained in terms of polar optical mode. piezoelectric and ionised impurity scattering. The best value of effective mass obtained was 0.19m.
acceptor whereas I2 is associated with the recombination of an exciton bound to a neutral donor. Finally, although the variation of the Hall mobility with temperature has been discussed previously by several authors[4-71 a short discussion of the dominant scattering processes in CdS is included. Theoretical curves taking account of optical mode, piezoelectric and ionised impurity scattering have been fitted to the experimental data using the effective mass, m*, as an adjustable parameter. The best fit was obtained with a value of m” = 0.19m.
1. INTRODUCTION
of the work reported in this paper was to measure the donor concentration, compensation ratio and carrier mobility in crystals of cadmium sulphide which had been grown in various partial pressures of either cadmiuni or sulphur vapour. To this end the electrical conductivity and the Hall coefficient of semiconducting samples were measured over the temperature range from 10 to 293°K. An estimate of the donor concentration of high resistivity photoconducting samples was obtained by comparing the values of the drift and Hall mobilities between room temperature and 85°K. The drift mobility was determined by observing the critical field for departure from Ohm’s law which occurs when the drift velocity of the electrons equals the velocity of sound in the sample and acoustoelectric amplification of thermal phonons begins[ I, 21. With one or two samples we have also tried to compare the relative intensities of the I, and I:! bound exciton lines [3], which are observed in photoluminescence at helium temperatures. with the measured donor and acceptor concentrations. This is of interest because the I, line is attributed to recombination of an exciton bound to a neutral THE
OBJECT
2. CRYSTAL
GROWTH
The samples investigated were cut from boules grown in this laboratory by the method described by Clark and Woods[8]. Briefly the method consists of subliming CdS in a sealed silica tube down a temperature gradient. The CdS condenses at the conical tip of the silica growth capsule, and as the crystal grows the capsule is pulled slowly through the furnace. The capsule is connected via a narrow orifice to a reservoir which contains either cadmium or sulphur. and which is held at a fixed temperature throughout the growth schedule. In 229
230
M. A. SUBHAN.
M. N. ISLAM
this way the crystals can be grown in known pressures of cadmium and sulphur vapour. To prepare the charge for the growth capsule BDH Ltd. ‘Optran’ grade CdS was sublimed and recrystallised in a preliminary process in which argon was passed over a silica boat containing the starting material. With this technique a mass of rods, platelets and dendritic crystals were obtained on a silica liner. These crystals were then used as the charge in the growth capsule. During growth of a boule the charge was held at a temperature of 1150°C while the crystal grew at 1125°C. Reasonably perfect crystalline boules could be grown in partial pressures of cadmium up to about 400 torr and in pressures of sulphur up to about 3 atm. At the extremes of pressures the sublimation ratio was reduced so that the boules produced were less than 1 cm long. This compares with the 3-5 cm length obtained under the most favourable conditions, (the so-called pmin condition) where the vapour had the true stoichiometric proportions. 3. EXPERIMENTAL
The Hall and current saturation measurements were made on bars with approximate dimensions 10 X 2 X 1 mm3. For the current saturation experiments the bars were cut with the c-axis either parallel or perpendicular to the long dimension; With the semiconducting crystals where Hall measurements only were made, the bars had no particular orientation. The bars were cut from the boules using a reciprocating tungsten wire saw, they were then ground and polished with diamond paste down to 1 pm size. After this they were etched in chromic acid and tested for uniformity of resistivity by depositing a row of indium dots along the length of one of the larger faces. The non-uniform samples were discarded. The indium was removed from the remaining specimens with a further etch. Either five or six indium contacts were then applied to enable conventional d.c. Hall measurements to be made.
and J. WOODS
The sample was mounted on the copper block of a metal, helium cryostat with an exchange space between the block and the helium container. Conventional d.c. circuitry was used to measure the Hall coefficient and the electrical conductivity. When required the sample could be illuminated with band gap illumination which was obtained from a tungsten source filtered by a wide band-pass interference filter centred at 0.52 pm, and a 1.O cm path length of 10 per cent solution of copper sulphate. The magnetic flux density employed was variable up to 4.0 K G. In the current saturation experiment the current-voltage characteristics were measured using 20 ps pulses at a repetition frequency of 25 Hz. 4. EXPERIMENTAL
RESULTS
4.1 Carrier concentration in semiconducting samples The variation of the free carrier concentration. n. with temperature is reflected in the variation of the Hall coefficient RI,. For conduction in a single band n = r/R”e. where r is a constant which depends on the scattering process. For the present purpose it is sufficient to assume that r does not differ appreciably from unity. The curves in Fig. 1 illustrate the variation of the Hall coefficient with temperature for a number of semiconducting samples. At room temperature the carrier concentrations, calculated from n = l/R,,e, varied from 4-O x 1015 to 4.0 x 10’” cm-3. These concentrations remained almost constant down to 200°K as indicated by the small changes in RH. Below 200°K the electrons froze out and R, increased with decreasing temperature. With the two samples where the measurements were extended down tol4”K the Hall coefficient began to saturate in the vicinity of 30°K and remained constant as the temperature was reduced further. This effect is associated with the onset of impurity conduction through the donor levels and has been described previously by Crandall[9] and by Toyotomi and Morigaki [ lo].
THE
ELECTRICAL
PROPERTIES
0
Fig.
I.
AND
200
100
Hall
GROWTH
340
n(N.4+ n) (N,,-NJ-nn)
=Texp
400
coefficient as a function of semiconducting stiples
Assuming that the crystals contain a single set of donors of concentration ND cm3 with an ionisation energy ED, and a single set of compensating acceptors of density N,., cm-” the carrier concentration, n. is given by the expression (-E,/kT)
(1)
where N, is the effective density of states in the conduction band.
CONDITIONS
sod
inveke of CdS.
OF CdS CRYSTALS
231
600
temperature
for
5
Values of N,, N.4 and ED were obtained by fitting equation (1) to the experimental curves assuming n = l/R,e and using the knowledge of the magnitude of ND-N,., from the saturated concentration at room temperature. It was assumed that n < (ND - NA) and n < N,., and values of Nc were calculated on the assumption that the density of states effective mass was 0+2m. The values of ED, (ND-N,,), ND and N, obtained are shown in Table 1 together with the growth conditions. The values of N, and N,., obtained from the
Table I. Values of donor ionisation energy and donor and acceptor concentration obtainedfrom Hall data on 5 crystnls grown at 1125°C from a charge held at 1150°C Sample 167 107 148 183 182
Excess in reservoir Cd Cd Cd Cd s
Reservoir T”C ‘650 550 450 50 250
E,eV 0.029 0.02 I 0.021 0.020 0.020
ND cm-J 2.55 x 10’6 I.73 x 10’6 5.88 x IO’b 8.31 x lOI 7.4 x 10’5
N., cm-3 I .47 x 10’5 8.25 x lOI I.51 x IO’6 3.52 x IO’” 3.4 x 10’5
ND-N., 2’40 9.10x 4.37 4.79 4.0
x 10’6 IO’S x 10’6 x 10’5 x IO’S
NDINA 17.3 2.03 3.89 2.36 2.18
IJl, 480 50
232
M. A. SUBHAN.
M. N. ISLAM
andJ.
WOODS
Hall measurements have been compared with the results of edge emission studies made at liquid helium temperatures, see Orr, Clark and Woods[ 1 I]. The intensities of the I, (A = 4867A) and the I, (A = 4889 A) exciton emission components excited by 3650 A radiation had previously been measured for a number of CdS crystals grown in this laboratory. including Nos. 167 and 107. I, and I2 lines are attributed to the recombination of excitons bound to neutral acceptors and donors respectively [3]. Their relative intensities should give therefore an approximate indication of the relative concentrations of acceptors and donors. The values of I,/l, for crystals 167 and 107 are recorded in Table 1. The agreement between the ratio of IJf, and ND/N, for the two crystals is reasonably satisfactory. 4.2 The Hull mobilit>
lb5
1 0
The variation of the electrical conductivity. u. with temperature of the five crystals referred to is shown in Fig. 2. The Hall mobilities. pIr, of these crystals. (p,! = R,,c) are shown in Fig. 3 together with some measurements on a degenerate sample No. 239 which contained about I OLxcm-:’ chlorine donors. With decreasing temperature the Hall mobility for the three crystals where the measurements were extended over a sufficient range reached a maximum in the vicinity of 50°K. Previous workers have suggested that the dominant lattice scattering mechanisms in CdS are polar optical mode and piezoelectric scattering[5-71. Accordingly we have attempted to fit the experimental curves to calculated curves taking ionised impurity scattering into account in addition to the processes mentioned. It seems reasonable to ignore deformation potential scattering of acoustic modes since a calculation based on the well known expression for mobility limited by this process: i.e. p = 3.2 x 10-S pu,2 (m/m* )5/?lT:j/4E;~
(2)
1
1
1 200
1
1
1
1 400
1
1
1
‘Y\ ,.
1 600
1
1
1 I 800
lo’/ T (T in OK) -
Fig.
2. The electrical conductivity temperature for 5 semiconducting
as a function of inverse samples of CdS.
leads to a value of 4.5 x lO’cm’V-‘s-l at 300°K. which is much larger than the highest measured value of abaut 400. In equation (2) we have taken E,,. the deformation potential of the conduction band. to be 2.67eV. The longitudinal wave velocity. c,. has the value 4.4 x lO”cm. s-’ and p. the density of CdS is 4.82. (Incidentally Devlin has performed the same calculation and obtained an even higher, but erroneous. value for the mobility limited by acoustic mode scattering). The fitting of the experimental and calculated mobility curves will be illustrated with reference to the results obtained from sample No. 183. The three scattering processes considered. i.e. polar optical mode. piezoelectric and ionised impurity, would each lead to a value of mobility of pop. ,LL,,~and pi if each were operative alone. When more than one
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
CONDITIONS
233
OF CdS CRYSTALS
has been discussed by Frohlich[ 121 and by Howarth and Sondheimer [ 131. If the polaron coupling constant Q < I, Howarth and Sondheimer have shown that for non-degenerate semiconductors 4-
’
.z
xx
Fou = G. :x ’ I- .
II
I 7
1
I.3
’
’
I.5
’
’
I.7
’
’
Log T (T I” OK) Fig.
3. The
temperature variation semiconducting
’
I.9
’
’
2.1
’
2.3
’
’
2.5
-
of the CdS.
e 8W M” . 36
zlE
(ez- I)
(3)
where z = hw,/kT (w, is the angular frequency of the longitudinal optical phonons), and I,!J(z) = 1 for L 4 I and G(Z) = 3/8 (mz,)“’ when z + I. The polaron coupling constant (Yis
1,81.6.
1
Hall
mobility
of
scattering mechanism is operative the computation of the resultant mobility is a matter of considerable mathematical complexity which is beyond the scope of this work. For this reason the resultant mobility has been calculated following the not uncommon practice of adding the mobilities reciprocally. i.e. /..-I = CL,,,,-’+ I*,>:-’ + pi-l. This procedure is based on the assumption that the total collision probability is the sum of the individual probabilities for each process considered separately from the others. Polar optical mode scattering would be expected to dominate the mobility of crystals such as CdS with an appreciable component of ionic binding. Optical mode scattering
With CdS this leads to a value of a = 0.8 if the static and optical dielectric constants E,~and E, are taken to be 10.33 and 5.24 respectively and if the angular frequency of the phonon w, is taken to be 5.75 x IO’:’ set-i. The effective mass has been assumed to be 0.2m. Since CdS has no inversion symmetry a piezoelectric polarisation is developed by the strain associated with acoustic mode vibrations. The theory of piezoelectric scattering in CdS has been discussed by Hutson [ 141 who writes the piezoelectric mobility as /& = A($..‘g!)“”
(5)
where A is a factor determined by the piezoelectric and dielectric constants. A = 1.44E, c (K’)i{II:. lllOdl3
(6)
Here K is the electromechanical coupling factor. Hutson obtained a value of A = 4.0 X IO9 by taking a suitable average of K’. Since the results show that our crystals were partially compensated the effects of ionised impurity scattering must also be considered. The mobility associated with this
234
M. A. SUBHAN.
M. N. ISLAM
type of scattering has been calculated the Brooks-Herring[ 151 formula. pi = 3.2 x
using
(m,n,*)*~2CSP Z’N,
1015
I log 11.3 x lO’V%,
5 (11
/Il.
(7)
Here N, is the concentration of ionised centres, and Z their electronic change. The value of N, at a particular temperature was obtained by putting N, = 2N, + n. As stated previously the resultant mobility was calculated by adding the reciprocals of the three components. The effective mass was used as an adjustable parameter as was the factor A. The best fit with crystal 183 is shown in Fig. 4. where the broken line was calculated assuming m* = 0.19m and A = 3.0X lo?. A good fit with the experimental points was obtained down to 30°K. Good fits were in fact obtained for all the crystals
and J. WOODS
examined with the values of the adjustable parameters m” and A listed in Table 2. Note that although with one slight exception the best fit was obtained with the same value of the effective mass, i.e. m* = O.l9m, the values of A ranged between 1.3 x 10’ and 4.0 x lo?. Table 2. Values of effective mass and the constant A chosen to give the bestjt betirveen the experimental and calculated values qf mobility Crystal
m*lm
167 107 148 183 182
0.19 0.19 0.19 0.19 0.18
5. CURRENT
“‘“3 -
4.2
4.0-
t c2 =>” . “5 E -3 4
3-a3.63.4-
.
.
-
3.2
p where
I.3
I.5 Log
4.
.\ \\
3.0-
1.1
Fig.
.\ :\
The
1.7 T
calculated
2.1
2.3
2.5
(T III OK) -
and crystal
I.9
measured No. 183.
Hall
mobility
of
SATURATION MOBILITY
A 3.5 3.2 I.3 3.0 4.0
x x x x x
AND
IO’ lo’ IO lo’ IO’
DRIm
Although all five of the crystals listed in Table 1 were semiconducting with resistivities between 1 and 10 fl cm at 300°K. many crystals grown with any cadmium reservoir temperature below about 350°C. and most crystals grown with sulphur reservoir temperatures above 250°C were semi-insulating with room temperature resistivities ranging from 1Ox to 10’” fl cm. Those with resistivities at the lower end of this range, i.e. IO’-10” 0 cm. were usually highly photoconductive so that illumination with tungsten light of 100 ft. candles intensity reduced the resistivity to about 1O3R cm. The higher resistivity samples were usually less photosensitive. Since we were unable to make satisfactory Hall measurements on samples in the dark with resistivities in excess of some lo7 fl cm., the activation energy of the donors and their concentration was determined by comparing the Hall and drift mobilities of illuminated samples over a range of temperatures from
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
CONDITIONS
OF CdS CRYSTALS
235
85 to 300°K. The method is in essence that According to Uchida et al. when WT > 1, previously described by Moore and Smith [2]. the drift mobility ,& = bf&,. The saturated Briefly, provided the resistivity of CdS is current density, J,,,, is then obtained by reless than lo” R cm.. the current voltage characplacing$, in (8) by bJ;,, so that teristics measured under pulsed conditions show a departure from Ohm’s law (saturation) JSat = tzeu,/bf;, = J,,,,/b = (n + n,) eu,/b. (9) at a critical field E, of the order IO:’ Vcm-‘. The saturation is caused by the acoustoelecIn our crystals the condition WT < I obtains tric interaction of the drifting electrons with at 300°K. but at 77°K T has increased so that amplified acoustic waves, the source of which WT 9 I. thus the above considerations are is either the thermal phonons in the crystal important. or the shock wave produced by the leading If we assume a simple model with one edge of the voltage pulse. discrete set of traps of density N, at a depth, Saturation occurs when the electron drift E. below the conduction band then it is easily velocity, u+ equals the velocity of sound in shown that the sample. Then u,! = p,,E,. = u,~, so that a measure of E, allows the drift mobility p.d to I. pH =pH{ 1 + N/N&Xp (E/kr)}-’ f”PH = & be calculated, provided v, is known. In the small signal theory of acoustoelectric (10) amplification the effect of electron traps has and from the considerations outlined above been taken into account by assuming that a fraction. 5 of the bunched space charge is .t&t = /-b/b = /-dJsaJJsato) . (11) mobile, while the fraction (1 -f) is trapped. This theory was extended by Uchida et al. Thus combining (10) and ( 11) [ 161 and applied by Moore and Smith to measure the depth and concentration of the ~~tr/~dJsato/Jsal) - 11 = N/N, em (EIkT) . traps. (12) The part played by the traps depends on the trapping time. 7, which is a function of tempThe object of the experiment therefore is to erature. If T is short compared with the period measure the Hall and drift mobilities pH and of the acoustic wave. i.e. 2.rrlw, the trapped & over a range of temperatUreS and to record electrons communicate rapidly with the the saturated current densities, so that Ln conduction band and can be bunched in the {[PH/P~(Js~~Js,~) - 11T3’*l can be plotted piezoelectric field in phase with the bunched against I/T. A straight line should be obtained, free electrons. In this eventfis a real number from the slope and intercept of which, values J;,. In contrast when OT > I a phase difference of E and N can be calculated. is established between the free and bound Although some ten photoconducting space charge waves and f becomes complex. samples have been examined the results Uchida et al. showed that when WT > I. reported in this section refer to one particular and that when wr < I, ud = sample, No. 78, grown with a cadmium reserU
236
M. A. SUBHAN.
M. N. ISLAM
In all the samples studied saturation effects could be observed in the’ current voltage curves provided the illumination was sufficiently intense to reduce the resistivity to a value less than 10” R cm. The three sets of curves in Fig. 5 show the Hall and drift mobilities of sample 78 under illumination with light of intensity 18. 6 and 1.8ft. candles. At the highest intensity the resistivity at 300°K was 3.46 x lo? fl cm. J,,, remained constant from 300°K down to about 200°K. whereafter it decreased. In the three curves in Fig. 6 Ln { (pHIpdUSa,o/JSa,) - I ) T3”l is plotted against l/T for each of the three intensities of light employed. The three dashed line curves shown in Fig. 6 are composed of two straight line segments. each of which has substantially the same slope regardless of the intensity of illumination used. It is concluded that the drift mobility was limited by the action of two sets of traps. The slopes and intercepts of the pairs of lines in Fig. 6 show that the two traps had depths of 0.015
and J. WOODS
and 0.037 eV. at concentrations of 1.35 X I 017 and 3.14 x IOlfi cm-3 respectively. Since the saturated current density J,,, fell below the room temperature value JSato at temperatures below about 200°K it is reasonable to assume that WT becomes of the order of unity at this temperature. Following the arguments outlined by Moore and Smith and assuming that the correct value to choose for w is that of the angular frequency for maximum acoustoelectric gain we calculate the trap relaxation time associated with the shallower trap to be 1.87 X IO-“‘s. In its turn this leads to an estimate of the capture cross section of the 0.015 eV trap of 5 X lo-“‘cm’. This is of the order of magnitude to be expected for a singly charged trapping centre. It is interesting to note that Photo Hall measurements of the type described by Bube and Mac Donald [ 181 indicate that crystal No. 78 contained two sets of deeper traps with ionisation energies of 0.28 and 0.36eV. The relaxation times of electrons in these
Fig. 5. The Hall and drift mobilities. p,, and gd. of a photoconducting sample No. 78 at three different light intensities (a) I.8 ft. candles, (b) 6 ft. candles. (c) I8 ft. candles.
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
Fig. 6. The results shown in Fig. 5 plotted according to equation t I?) to obtain trap concentrations and ionisation energies (a) 1.8 ft. candles. (b) 6 ft. candles. (c) 18 ft. candles.
traps would be so high that these electrons would not be able to bunch in the piezoelectric field. This explains why a comparison of drift and Hall mobilities measured in the manner described here can reveal information about shallow traps only. 6. DISCUSSION
6.1 Transport properties
The initial object of the work was to try and determine the donor and acceptor concentrations in semiconducting samples of CdS. and the concentration of shallow traps in photoconducting, semi-insulating samples. and to examine how these concentrations varied with the conditions under which the crystals were grown. Before commenting on this
CONDITIONS
OF CdS CRYSTALS
237
aspect of the work it is desirable to compare our results with those of other workers. Although the Hall coefficient and electrical conductivity of CdS have been investigated several times previously there are relatively few estimates of the ionisation energy of shallow donors to be found in the literature. Kroger et al. reported a value of ED of 0.02 eV at a concentration of IO” crne3 but most of their samples showed impurity banding effects. Piper and Halsted[S] found a donor level at 0.032 eV in samples with a concentration of 5 x IO’jcm-” which is sufficiently dilute to allow interaction effects to be ignored. Woodbury found a value of 0.024eV at concentrations of 10’” cm-3 (see [7]) whereas 1takura and Toyada [ 191 interpreted their results as indicating the presence of two shallow donors at 0.014 and 0.007eV. The smaller value is almost certainly suspect, as it was measured near the regime of impurity conduction at low temperatures. Our results on the semiconducting samples (with the exception of one sample) indicate a donor ionisation energy of 0.021 2 0.002 eV. The lowest donor concentration of 7.4 x 1O’j cm+ was not very different from Piper and Halsted’s infinite dilution of 5 x 10i5cm-“. The comparison of the drift and Hall mobilities of the photoconducting sample indicates the presence of two shallow traps with ionisation energies of 0.015 and 0.037 eV. It is reasonable to assume that the shallower of these. which had a measured concentration of I .35 x 10” cm+, was identical with the shallow donors found in the semi-conducting specimens. For comparison it should be recorded that Moore and Smith using the drift mobility technique found a trap at 0.017 eV in a semiconducting sample of CdS and a trap at 0.025 eV in a photoconducting specimen. It is interesting to speculate whether in fact the various samples examined by the different authors contained different donor centres. although all the results quoted above are for unintentionally doped crystals. For example, it may be that Piper and Halsted’s shallow
238
M. A. SUBHAN.
M. N. ISLAM
donor with ED = 0.032 eV is similar to our deeper donor in the photoconducting sample. Much of the previous work on the transport properties of CdS, which is well summarised by the article by Devlin[7], has been concerned with the intrinsic scattering mechanisms. Piper and Halsted described the temperature variation of the Hall mobility of their best crystal using a combination of polar optical and piezoelectric scattering. They obtained a good fit down to 70°K using a value of the effective mass of 0.16 eV. Devlin used a variational method to combine the effects of the two scattering mechanisms (thus avoiding the assumption that the mobilities add reciprocally). His experimental results were limited to nitrogen temperatures. He obtained a good fit down to 80°K with m* = 0.20m. Our results on crystal No. I83 show good agreement down to 30°K when ionised impurity scattering is also taken into account. and indicate a value of m* of 0.19m. This agrees well with previously published work. At temperatures below 30°K the experimental and calculated values of mobility in sample No. 183 began to diverge. Further, reasonable fits for samples 167 and I82 could not be extended below 70°K. The reason for this is to be sought in the increasing importance of conduction through the donor levels as the temperature is reduced below 70°K. Thus at the lowest temperatures, conduction takes place through the conduction band and via the donor levels (‘hopping’). It is difficult to assess the contribution of each of these but clearly the onset of the impurity conduction process will cause the measured mobility to decrease more rapidly than considerations of pop, ppl and pi would predict. Before leaving the subject of scattering processes it should be pointed out that in order to obtain good agreement with our experimental results it has been necessary to choose values of the factor A in Hutson’s piezoelectric scattering equation which lie between the limits of 1.3 x IO? and 4.0 x lo?. The value of A calculated by Hutson was
and J. WOODS
4.0 x 109. There are several reasons for supposing that the magnitude of A may vary from sample to sample. For example the piezoelectric constants used by Hutson were obtained from measurements on samples annealed in sulphur vapour and which in consequence were highly insulating. It is possible that the value of the electromechanical coupling constant is different in samples with higher conductivities because of .the screening of the piezoelectric field by free carriers. However it must be admitted that the present results offer no evidence to support such a possibility. Hutson also assumed that the elastic and dielectric properties of the crystal were isotropic. The true value of A may obviously depend on the crystallographic orientation of the sample. Finally c-axis inversion, which has been observed in thin films[20], may occur in crystals, and the quantity A would vary if single crystals contain anti-phase boundaries. 6.2 Donor content and crystal gro~r~tll
When CdS and either sulphur or cadmium are maintained at various temperatures in a sealed tube, each of these components will make a contribution to the vapour pressure in the system. CdS dissociates according to the reaction 2CdS ,$G= 2Cd,+ S,,.
(13)
In a tube containing CdS only, the partial pressures are given by the equation p’;,, . Psz = KrdS
(14)
KCdS is the dissociation constant. At any temperature there will be a minimum total pressure (p& which can be shown to occur when pea = 2p,, = (2KC,s)“‘1.
(15)
Under these conditions the vapour will have stoichiometric proportions. On heating, the
THE
ELECTRICAL
PROPERTIES
AND
GROWTH
CdS charge will vapourise and attempt to establish the Pmin condition throughout the enclosure. When a reservoir containing cadmium or sulphur is added the equilibrium will be disturbed provided the vapour pressure of Cd or S? at the reservoir temperature exceeds the dissociation pressure of the corresponding component at the charge temperature. When the charge is held at 1150°C the Pmin condition requires a cadmium partial pressure of 80 torr and a sulphur pressure of 40 torr. These conditions are obtained for reservoir temperatures of TCd - 600°C or 300°C. Thus crystals grown with a TS cadmium reservoir at 600°C should have similar properties to crystals grown with sulphur reservoirs at 300°C. This is borne out by the measured properties of crystals Nos. I07 and 182 which have substantially similar concentrations of donors and compensation ratios (see Table 1). We have found by experience that it is only possible to grow boules of appreciable size when the partial pressure in the growth chamber is equal or less than the dissociation pressure which establishes the pmi,, condition. Thus when either TCd exceeds 600°C or T, exceeds 3OO”C, transport from the charge to the growing crystal is reduced and small boules only are produced. The extent to which this occurs depends upon the temperature gradient in the growth chamber (i.e. the relative temperatures of charge and boule), but in our experiments we attempted to maintain a fixed gradient. Hardly any growth at all occurred for cadmium reservoir temperatures above 700°C and sulphur temperatures above 400°C. The highest sulphur temperatures led to insulating crystals with such low photoconductivity that it has not as yet been possible to use the current saturation method to determine their shallow donor content. In contrast, crystal 167 is typical of boules grown in pressures of cadmium which exceed the pmln condition. It is interesting to note that the Hall measurements show that although the donor content is only increased
CONDITIONS
OF CdS CRYSTALS
239
slightly relative to that of sample 107, the major effect of increasing the cadmium pressure is the reduction of the acceptor content. With either TCd < 600°C or T, < 3OO”C, when the reservoir pressures are less than the dissociation pressures, the dissociation products will constantly diffuse to the reservoir and the pressures in the growth chamber are difficult to calculate. To prevent excessive diffusion the connection between the growth chamber and the reservoir was via a very narrow orifice. In general with cadmium reservoir temperatures exceeding 35O”C, semiconducting crystals were produced, but with cadmium reservoirs at lower temperatures, or with T, < 250°C the electrical properties of the crystal were unpredictable, and were sometimes semiconducting and sometimes semiinsulating. Crystal I83 is an example of a semiconducting sample, with properties not very different from those expected for a sample grown under Pmin conditions. In contrast crystal 78 grown with TCd = 350°C was photoconducting with a high donor content, I.35 x 1017cm-3, but this crystal clearly contained an almost equal large concentration of acceptors. Current saturation measurements on other samples which are unreliable because of the inhomogeneity of the crystals tend however to support the conclusion that photoconducting samples contain concentrations of donors and acceptors of about 10” cm-3. REFERENCES I. SMITH R. W..Phys. Rev. Left. 9,527 (1962). 7. MOORE A. R. and SMITH R. W.. Phys. Reo. MA, 1’50 (1965). 3. THOMAS D. G. and HOPFIELD J. J., Phys. Rev. Lef~.7,316(196l);Phys.Reu.12.8,2135(1962). 4. KROGER E. A.. VINK H. J. and VOLGER J., Phillips Res. Repr. lo,39 (1955). 5. PIPER W. W. and HALSTED R. E., Proc. Int. Conf. on Semiconductor Physics, Prague 1960, rx 1046. Academic Press, New York (1961). 6. iOOK D. and DEXTER R. N., Phys. Rev. 129,198O (1963). 7. DEVLIN S. S., Physics and Chemistry of II-VI Compounds, p. 552. (Edited by Aven and Prener) N. Holland (1967).
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M. N. ISLAM
8. CLARK L. and WOODS J.. J. Crysr. Growth 3.4. 127 (1968). 9. CRANDALL R. S.. Phys. Rev. 169.577 (1968). IO. TOYOTOMI S. and MORIGAKI K.. J. phys. Sot. Japan 25,807 ( 1968). I I. ORR D. S., CLARK L. and WOODS J..J. Phys. D. (Appl. Phys.) 1,1609 (I 968). 12. FROHLICH H.,Adu. Phys. 3.325 (1954). 13. HOWARTH D. J. and SONDHEIMER E. A., Proc.R.Soc.219.53(1953). 14. HUTSONA.R..J.appl.Phys.32.2287(1961).
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15. BROOKS H..Adv. Electron. Phys. 7. I56 (1955). 16. UCHIDA I., ISHIGURO T.. SASAKI Y. and SUZUKI T..J. phys. Soc.Japanl9.674 (1964). 17. ISLAM M. N. and WOODS J.. J. Phys. D. (Appl. Phvs.)3,1197 (1970). 18. BABE R. H. and MACDONALD H. E..Phys. Rev. 121.473 ( 1961). 19. ITAKURA M. and TOYODA H.. J. phys. Sot. Japan 18, I 50 ( 1963). 20. ROZYANYI G. A. and FOSTER N. F.. J. appl. Phys. 38. 5 172 (I 967 ).