- Email: [email protected]
Electron transport in II–VI compound semiconductors
J. Phys.
Chem.
Solids Vol. 45, No.
4, pp. 393-399,
@X-3697/84
1984
$3.00 + .oO
PergamonPress Ltd.
Printedin Great Britain.
ELECTRON TRANSPORT IN II-VI COMPOUND SEMICONDUCTORS D. Department
of Electronics
MUKHOPADHYAY
and Telecommunication 700-032,
Engineering, India
Jadavpur
University,
Calcutta
and D. P. Department
of Physics, (Received
BHATTACHARYAt
Jadavpur
University,
5 May 1983; accepted
Calcutta
700-032,
India
9 June 1983)
Abstract-The mobility characteristics of II-VI compound semiconductors have been investigated using a displaced Maxwellian model for the energy distribution of the free carriers and considering the combined
effects of acoustic, piezo-electric, ionised impurity and polar optical modes of scattering. The electric field dependence of the carrier mobility has been obtained at lattice temperatures of 77” K and 300°K. The effect of variation of the not too well known coupling constants on the characteristics has been observed. The variation of the low-field mobility with the lattice temperature and with the level of impurity concentration has also been obtained. The theoretical results agree quite satisfactorily with the available experimental data and with other theoretical works.
1. INTRODUCIION
These compounds have a larger band gap than the elemental or the III-V compound semiconductors and consequently are expected to have a larger breakdown field. The saturation drift velocity is also slightly higher in these compounds. Thus the highfield devices made from the II-VI compound semiconductors would have better characteristics and hence it is of interest to study the high-field characteristics of these compounds. We have presented in this paper the high-field mobility characteristics of the series of II-VI compound semiconductors: ZnO, ZnS, CdS, CdSe, CdTe at lattice temperatures of 77 and 300” K. Moreover, the characteristic variation of the low-field mobility with the lattice temperature has also been obtained for these semiconductors. To obtain the high-field characteristics, one has to solve the Boltzmann transport equation taking the various scattering mechanisms for the charge carriers into account. It is well known that in genera1 the mathematics involved in the problem of nonlinear electrical transport in semiconductors being very much complicated, one has to adopt simplifying assumptions in order to make the problem tractable. These necessary assumptions very often compromise with the physical validity of the theoretical results. To mention in particular, the scattering by optical phonons which are of the polar type in the II-VI compound semiconductors, is known to be predominant in these materials[&6] and a solution of the Boltzmann equation with predominant polar optical phonon scattering is beset with many difficulties. Hence one has to take recourse to the numerical techniques like the Monte Carlo technique, the
The problem of high electric-field properties in semiconductors has been extensively investigated for many years. This is mainly due to two reasons. First, more information on the physics of microscopic processes in these materials is obtained by studying the kinetic processes of the carrier ensemble in a state significantly perturbed from the state of thermodynamic equilibrium. Second, from a study of the high-field effects one can find new ways of better controlling the material properties by means of a variation of the external field strength, the knowledge of which is of great importance for their recent applications in solid state electronic devices. The theoretical as well as experimental studies of high field conductivity characteristics of semiconductors have been carried out in great details during the last two decades[l-31. Most of these studies have, however, been devoted to the elemental and the III-V compound semiconductors. II-VI compound semiconductors on the other hand have not received much attention. Though the acoustoelectric and optoelectronic properties of some of the II-VI compound semiconductors are well studied, not much work has been done on the high-field mobility characteristics of these semiconductors. This is mainly because of the technological difficulties in obtaining single crystals of desired purity required for the hot electron studies. II-VI compound semiconductors, however, should occupy a place of importance in high-field devices. ‘/‘On leave from: Department Kalyani,
Kalyani-741235,
of Physics,
University
of
India. 393
394
D.
MUKHOPADHYAYand D. P. BHATTACHARYA
method of path variables or the method of iterative integration [ l-31. All these methods however, require large computation time and it is difficult to study the variation of the high-field conductivity characteristics with the material parameters, some of which have not yet been determined from independent physical measurements in the II-VI compound semiconductors. A broad based analytical method for obtaining the high-field mobility characteristics would be thus useful, and more so for the study of the variation of the characteristics with the material parameters. If the carrier-carrier collisions are sufficiently frequent to randomise the energy and momentum distribution of the electrons, then the electron distribution function in the momentum space can be assumed to be in equilibrium at a temperature T, which depends upon the applied electric field and which is greater than the lattice temperature TL. Under this condition the distribution function becomes a drifted Maxwellian one. Using the displaced Maxwellian distribution function, the Boltzmann transport equation can be integrated to yield the energy and momentum conservation equations in which the average rate of momentum and energy loss is equal respectively to the momentum and energy gained from the applied electric field. The integrations can be carried out in a closed form if the distribution function is expanded in terms of the spherical harmonics and the series truncated after the first two terms[7]. This approximation is known as the diffusion approximation[8] and it amounts to assuming that the drift energy of the carriers is small compared with their thermal energy. Calculations show that for most of the semiconductors considered here this ratio is about 0.4 at 77°K in the low field region, and is still less at higher fields and at higher temperatures. The earlier calculations show that the retention of the drift energy terms in the distribution function does not significantly change the calculated characteristics in the III-V compound semiconductors even when this ratio is as high as 0.5[9]. It is to be noted however that for carrier-carrier scattering to dominate over the lattice-carrier scattering the required carrier concentration should be of the order of 10’6/cm3[7]. The hot electron studies are however performed in samples which have a carrier concentration about two orders of magnitude lower than this. For the displaced Maxwellian model to be valid what is required is that the carrier-carrier scattering should randomise the momentum distribution and this is achieved at a carrier concentration which is lower than that required for the predominance of the carrier-carrier scattering. The displaced Maxwellian model when applied to the study of the elemental and the III-V compound semiconductors is known to yield results which are in good agreement with the experimental results and with other numerical calculations even though the validity criterion is not strictly satisfied. This is presumably because of the above reason and also because of the inherent accuracy associated with the method of variation of
parameters[lO]. The II-VI compound semiconductors however, have a larger carrier concentration than the III-V compound semiconductors and the validity criterion for the predominant carrier-carrier scattering should be better satisfied in these materials. Therefore, though superior numerical methods are available, the displaced Maxwellian approach would be useful for studying the electrical transport in the II-VI compound semiconductors in particular.
2. THEORY
The Boltzmann
transport
equation
can be written
as
af =7x -I I at
fie,*
I
where the 1.h.s. is the rate of change of the distribution function due to the applied electric field F and 2f/atlj is the rate of change of the distribution function due to the jth collision mechanism. As has been explained earlier, if the carrier concentration is large enough to ensure momentum randomization due to electronelectron collision, the distribution function becomes displaced Maxwellian given by
fW =
(2
” exp[ nm *kBTJ312
-g]
(2)
where k is the electron wave vector, m * the isotropic effective mass of the electrons, T, the electron temperature and d is the displacement in the momentum space for the electron system. h = h/2n, h being the Planck’s constant and k, the Boltzmann constant. The carrier effective mass m * has been assumed to be isotropic. The II-VI compound semiconductors have larger band gap in comparison to the elemental or the III-V compound semiconductors, leading to a smaller value of the nonparabolicity factor. Though this simplifying assumption of a parabolic band structure is not strictly justified even for the II-VI compounds, it will be seen in what follows that the results obtained using the parabolic band structure quite agree with the experimental observations. The displacement d and the electron temperature T, may be obtained by solving the energy and momentum conservation conditions. The relevant balance condition for the monentum and energy can be written as
and
respectively, where E = h2k2/2m* is the carrier energy. In order to evaluate the integrals in (3) and (4)
Electron transport in II-VI compound semiconductors one must know the types of carrier collision mechanisms which dominantly limit the mobility value. The carrier scattering mechanisms important for the II-VI compound semiconductors are the scattering through acoustic phonons, both deformation potential and piezoelectric, scattering through polar optical phonons and the scattering with the ionised impurities. The intervalley scattering between the central and the higher lying satellite valleys should not play any significant role in the field range of our interest, because these valleys are considerably far removed from the central valley[l 11.. The scattering by the ionised impurities and piezoelectric phonons can be considered elastic and they do not contribute to the energy loss mechanism[l]. The integrals in (3) and (4) are evaluated in a closed form using the diffusion approximation which is true only when the scattering is elastic[l2]. In II-VI compound semiconductors polar optical phonon scattering which is strongly inelastic is predominant and the diffusion approximation would be valid in the presence of electric fields high enough so that the average carrier energy becomes much higher than the optical phonon energy. However, the other elastic collision processes which are always present randomizes the momentum distribution of the carriers and hence the diffusion approximation would not lead to much error even for lower values of the electric field[l3]. On substituting the values of @@)/&l, and 8f(k)/t%lj in (3) and (4) one can integrate term by term and obtains the following coupled equations F
_=v
d
. /?m *3’2h r8E,*E.*x”* . . h ‘+pc:
3nJ;;e
+
e’K_*E,‘l*x -‘I* “. . h 2~&g?l*
L + N,Z2e4Er~3/2x-3’2LBH 8c,,*q,*m*’
(5) 1
and m *W
8,,/5m**E,*E;i2x3/*(1
_ x -1)
Fd=p h2P
x&h3e
(6)
. Here E, = k,T,; x = TJT,; L, = ln( 1 + B) - B/( 1 + ~9); B = 24,rc~ * (kBTJ2/nh2e2; (exp(xJ
+ (exp(%) + 1%
$ (
- I)&
e >I
(exp(x,)
x, = es
--- .
- l)K,
(e )I;
395
$
K2m, the normalised electromechanical coupling constant, is given by K,,,* = P*/qc,pc,$ P being the pieozoelectric constant, p the density and c, the average acoustic velocity for the longitudinal mode. El is the deformation potential constant for the acoustic phonons, co is the free space permittivity. rcOand K, are respectively the static and the high frequency dielectric constants of the material. N, is the ionised impurity concentration and Z the degree of ionisation of the impurity centres, n the carrier concentration and Q9 the Debye temperature. & and K, are the modified Bessel functions of the second kind. Equations (5) and (6) can now be solved for F and d for different assumed values of the electron temperature T,, and hence one can find the mobility characteristics of a material. The parameters for different materials used in the numerical calculations are given in Table 1. 3. RESULTS AND DISCUSSIONS
Figure 1 shows the electric field dependence of the carrier mobility normalised to its zero-field value at a lattice temperature of 77°K taking Ni= 0. The nature of dependence is qualitatively the same for all the semiconducting compounds investigated here. For fields upto 0.5 kV/cm the mobility is more or less independent of the field strength, then except for ZnO it decreases in a nonlinear manner on increasing the field strength upto 3.0 kV/cm. On further increasing the field the mobility is found to fall almost linearly. For ZnO however, starting from the zero-field value, the mobility slowly increases with the electric field upto a field of 1.2 kV/cm. For higher fields the mobility at first decreases in a nonlinear manner for fields upto 4.5 kV/cm; beyond which the decrease is almost linear with the increase of the applied electric field. Further beyond the linear regime, at a field characteristic of the compound (lowest for CdTe; 7.0 kV/cm and highest for ZnO; 30 kV/cm) the mobility field characteristics again becomes nonlinear with a gradual decrease in the slope, tending towards a lower saturation. It is to be mentioned here that our calculation does not give any perceptible dependence of the carrier mobility on the strength of the electric field at a lattice temperature of 300” K, hence the hot-electron effects are not expected to be observed in the II-VI compound semiconductors at room temperature in the field range considered here. For CdTe, this result is in agreement with that of Butcher and Fawcett[l4] who made a theoretical study of the hot-electron transport in CdTe using the displaced Maxwellian model for the energy distribution of the free electrons but obtained the displacement d and the electron temperature T, by numerical solution of the re-
D. MUKHOPADHYAY and D. P. BHATTACHARYA
396
Table 1. Material parameters used for numerical calculations
all0
0.32
8.50
0.59
5.66
5.0
a51
0.074
10
ZnB
020
8.90
5-U
4.08
5.0
507
0.074
IO
OdTe
0.10
9.60
‘7.21
6.06
3.0
242
0.068
18
5.0
304
0.026
10
4.3
440
0,037
12.8
C&J
03
9.25
6.40
5.31
cas
025
9.19
5.24
4.80
Electric field
( kV/cm I
Fig. 1. Varation of the normalised mobility with the applied electric field at 77” K (with no impurity). (1) ZnO; (2) ZnS; (3) CdTe; (4) CdSe; (5) CdS.
spective conservation equations. They however predicted the onset of the hot-electron effects in CdTe for fields greater than 10 kV/cm at a lattice temperature of 300” K. As expected, the quantitative agreement of our results for fields below 10 kV/cm with those obtained by Butcher and Fawcett is not very good because of different values of the material parameters used in our calculations. Borsari and Jacoboni [15] theoretically investigated the high-field transport in CdTe by the Monte Carlo technique and reported definite nonlinear field dependence of the drift velocity of the carriers at 300” K in the range of field from IO kVjcm to 20 kV/cm. At fields below that range the drift velocity increases linearly with the electric field, whereas for fields above that range the drift velocity falls almost linearly with the increase of fields upto 50 kV/cm and it is in this range of electric field where their theory also predicts field-independent carrier mobility. On comparing our results for CdTe with the experimental observations of Canali et al. [ 161one can see that our results agree quite satisfactorily with their ex~rimentaI results. For example where our theory predicts drift velocities of 1.45 x IO’ and 1.73 x 107cm/sec at fields 7.0 and 8.5 kV/cm re-
spectively, their experimental observations give 1.5 x IO’ and 2.0 x 10’ cmjsec for the drift velocity at the above fields respectively. In the case of CdS our results are qualitatively in good agreement with those of Guha[ 121, who also obtained the high-field mobility characteristics in the displaced Maxwellian model. That our results for normalised mobility at different field strengths are consistently lower at all fields in comparison with the values obtained by Guha is attribute to the fact that the value of rcOwhich is chosen here is lower than that assumed by G&a. Figure 2 shows the dependence of the low-field mobility on the lattice temperature under the same condition when the effect of the ionised impurity scattering is not taken into account. It may be mentioned here that for CdSe the ionised impurity scattering plays insignificant role for impurity concentration upto 10’“/cm3[6] except at temperatufT below 77” K; whereas in the case of ZnO the same is true at lattice temperatures above 150” K upto the same level of impurity con~ntrat~ontl7~. It may be noted here that the general nature of the lattice temperature dependence of the low-field mo-
Electron transport in II-VI compound semiconductors 500400m III
200 I
3377 II
1000/ T,(
K
I
I
20
50
-’ )
Fig. 2. Variation of the low field mobility with the inverse of lattice temperature (with no impurity). (1) ZnO; (2) ZnS; (3) CdTe; (4) CdSe; (5) CdS.
Impwty
397
bility is also the same for all the compounds studied here. The low-field mobility decreases with the rise of lattice temperature. In the low temperature region the nonequilibrium carriers in CdSe possess highest mobility whereas in the room temperature regime the carrier mobility in CdTe is the highest. Moreover, the carrier mobility in CdS is usually lower than that in CdTe except in the range of temperature from 25” K to 100” K. It may be mentioned here that the values of the low-field mobility in ZnO at different lattice temperatures as measured and calculated by Hutson[4] agree well with the values obtained by the above theory. The calculated values for the low-field mobility in CdS at different lattice temperatures are found to be in good agreement with the ex~~mental values of Saitoh [ 181. For example, where our calculation yields the values 4.5 x IO3and 3.3 x lO*cm/sec for the drift velocity at 77” K and 300” K respectively, Saitoh’s results were 5 x lo3 and 3 x lo* cmjsec at these temperatures respectively. It is well known that the ionised impu~ty scattering becomes of very little significance for electrical conduction as the electric field is increased[6, 15, 171. Hence we have chosen to plot the variation of the low-field mobility with the impurity concentration in Fig. 3. It is to be observed that even the low-field mobility is almost independent of the impurity concentration in the sample at a lattice temperature of 300” K. At 77” K however, as expected, the low-field mobility decreases with the increase of the impurity
concentmtion
(cm-3)
Fig, 3. Variation of the low field mobility with the impurity concentration deformation potential coupling constant of 1OeV. -, 77°K: ---------, (3) CdTe; (4) CdSe; (5) CdS.
at 77°K and 300°K for 300”; (1) ZnO; (2) ZnS;
D.
398
MUKHOPADHYAYand D. P. BHATTACHARYA
0”
5
2
03
5
2
0’
EleCtrlc
field
( kV/cm
)
Fig. 4. Variation of mobility
with electric field for various combinations of the deformation potential and the electromechanical coupling constant at 77” K. Curves l-3 are for ZnS, cures 4-6 are for CdSe and curves 7-8 are for CdS. (1) E, = 7 eV; (2) E, = 10 eV; (3) E, = 30 eV, EMCC = 0.074 for curves 1-3; (4)E, = 10eV, EMCC = 0.026; (5) E, = 18eV, &WCC = 0.026; (6) E, = 18eV, .&WCC= 0.035; (7)
E, = 12.8eV, &WCC = 0.037; (8) E, = 18eV, EMCC = 0.0455.
concentration. The decrease is slow for the impurity concentration upto 0.5 x 10”/cm3 and the slope becomes more and more steep as the concentration is increased. In the absence of any known experimental data under the identical conditions we cannot check our results. But our results are qualitatively in agreement with the experimental results of Finlayson et a1.[19] who actually studied the electrical conduction in CdSe samples having considerably high impurity concentrations. It may be observed that the calculations carried out here involve a large number of physical parameters. The values of all these parameters are not equally well known. The knowledge of the electron-phonon coupling constants being the least perfect [Z] compared to that of all the other parameters involved in the calculation, it is interesting to investigate the effect of the change of these constants upon the mobility-field characteristics of the samples. Hence in Fig. 4 the electric field dependence of the mobility has been plotted for various combinations of the values of the deformation potential and the electromechanical coupling constant at a lattice temperature of 77” K. Since a higher value of a coupling constant means a greater coupling between the electron-phonon systems leading to more frequent carrier coilision, the carrier mobility in ail these ~rnpo~ds at a particular field is found to be lower for higher values of either of these coupling constants. The effect of the change of the coupling constants on the mobility-field characteristics is prominent upto a field of about 3.0 kV/cm.
At higher fields the variation of the field dependence of the mobility becomes progressively insensitive to the variation in these coupling parameters. At a lattice temperature of 300” K however, our calculation does not give any perceptible effect of the change of ‘the coupling constants on the field dependence of the carrier mobility in the range of electric field considered here. It may’ be said in the context of the above details that though a comparison of our results with the experimental data wherever available would prompt the estimation of the numerical values for the coupling constants, their independent physical measurements are to be hoped for. 4. CONCLUSIONS
The mobility characteristics of the nonequilibrium carriers in a series of II-VI compound semiconductors are studied theoretically in the displaced Maxwellian model taking into account the different scattering mechanisms. In the absence of the experimental results for some of the compounds under identical conditions as assumed in our investigation, we cannot check our results for all the II-VI compounds. However, our results for the compounds studied here except for ZnS, are found to agree wit’? the available ex~rimental results; though the quantitative agreement
with the experimental
always very satisfactory. agreement
with
other
data is not
Our results are also in good theoretical
thereby that the mobility
results,
characteristics
indicating
are not much
Electron
transport
in II-VI
the exact form of the energy distribution function of the carriers under high-field condition. Finally it should be mentioned that our calculation is based on simplifying assumptions of a uniform homogeneous and perfectly crystalline semiconducting sample having discrete impurity states and spherical constant energy surfaces in k space. These assumptions may not be satisfied fully by the experimental samples and hence it is difficult to obtain experimental results which can be satisfactorily compared with those of the present calculation. Moreover, in compounds like CdS and ZnO having a large piezoelectric coefficient it is difficult to determine the true variation of the carrier mobility with the electric field because the current density in such compounds at high electric field is greatly modified by the acoustoelectric current which flows in a direction opposite to the normal drift current [20, 2 11.Hence experiment is to be performed with pulses having width smaller than the build-up time of the acoustoelectric domain in the sample, so that the effect of the acoustoelectric current can be eliminated and the theoretical results for the mobility characteristics at high electric field can be directly compared with the experimental data[l2]. Apart from these, the technology for obtaining single crystals of the II-VI compounds having desired purity is not known to have attained perfection. Hence it is clear that more experiments on the mobility characteristics of the II-VI compounds under varying physical conditions of the sample need be performed before a thorough survey of the success of any theoretical model can be assessed. sensitive
to
F’CS Vol. 45, No. 4--c
compound
semiconductors
399
REFERENCES 1. Conwell E. M.. High Field Transport in Semiconductors. Academic Press, New York (1967). 2. Nag B. R., Theory of Electrical Transport in Semiconductors. Pergamon Press, Oxford (1972). 3. Nag B. R., Electron Transport in Compound Semiconductors. Springer-Verlag, Berlin (1980): 4. Hutson A. R.. J. Phvs. Chem. Soliak 8. 967 (1959). I. A., Phys. Status Sdlidi. 5. Finlayson D. M. and-Johnson (b) 71, 395 (1975). D., Indian J. Phys. 53A, 528 (1979). 6. Mukhopadhyay I. Stratton R., Proc. Roy. Sot. (A) 246, 406 (1958). 8. Stenflo L., Proc. IEEE 54, 1970 (1966). D. and Nag B. R., Indian J. Pure Appl. 9. Mukhopadhyay Phys. 7, 616 (1969). D., Phys. Status Solidi. (b) 49, K41 10. Mukhopadhyay (1972). 11. Herman F., Kortum R. L., Kuglin C. C. and Shay J. L., Znt. Conf. II-VI Semiconducting Compounds, Rhode Island (Edited by D. G. Thomas), p. 503. Bell Telephone Laboratories (1967). 12. Guha S., Phys. Rev. B2, 4971 (1970). 13. Conwell E. M. and Vassel M. O., Phys. Rev. 166, 797 (1968). 14. Butcher P. N. and Fawcett W., Proc. Phys. Sot. 86, 1205 (1965). 15. Borsari V. and Jacoboni C., Phys. Status Solidi. (b) 54, 649 (1972). 16. Canali C., Martini M., Ottaviani G. and Zanio K. R., Phys. Rev. B4, 422 (1971). D. and Bhattacharya D. P., Indian J. 17. Mukhopadhyay Pure Appl. Phys. 17, 45 (1979). 18. Saitoh M., J. Phys. Sot. Jap. 21, 2540 (1966). D. M., Irvine J. and Peterkin L. S., Phil. 19. Finlayson Mug. B39, 253 (1979). 20. Smith R. W., Phys. Rev. Lett. 9, 87 (1962). 21. Hutson A. R., McFee J. H. and White D. L., Phys. Rev. Lett. 7, 237 (1961).
Chem.
Solids Vol. 45, No.
4, pp. 393-399,
@X-3697/84
1984
$3.00 + .oO
PergamonPress Ltd.
Printedin Great Britain.
ELECTRON TRANSPORT IN II-VI COMPOUND SEMICONDUCTORS D. Department
of Electronics
MUKHOPADHYAY
and Telecommunication 700-032,
Engineering, India
Jadavpur
University,
Calcutta
and D. P. Department
of Physics, (Received
BHATTACHARYAt
Jadavpur
University,
5 May 1983; accepted
Calcutta
700-032,
India
9 June 1983)
Abstract-The mobility characteristics of II-VI compound semiconductors have been investigated using a displaced Maxwellian model for the energy distribution of the free carriers and considering the combined
effects of acoustic, piezo-electric, ionised impurity and polar optical modes of scattering. The electric field dependence of the carrier mobility has been obtained at lattice temperatures of 77” K and 300°K. The effect of variation of the not too well known coupling constants on the characteristics has been observed. The variation of the low-field mobility with the lattice temperature and with the level of impurity concentration has also been obtained. The theoretical results agree quite satisfactorily with the available experimental data and with other theoretical works.
1. INTRODUCIION
These compounds have a larger band gap than the elemental or the III-V compound semiconductors and consequently are expected to have a larger breakdown field. The saturation drift velocity is also slightly higher in these compounds. Thus the highfield devices made from the II-VI compound semiconductors would have better characteristics and hence it is of interest to study the high-field characteristics of these compounds. We have presented in this paper the high-field mobility characteristics of the series of II-VI compound semiconductors: ZnO, ZnS, CdS, CdSe, CdTe at lattice temperatures of 77 and 300” K. Moreover, the characteristic variation of the low-field mobility with the lattice temperature has also been obtained for these semiconductors. To obtain the high-field characteristics, one has to solve the Boltzmann transport equation taking the various scattering mechanisms for the charge carriers into account. It is well known that in genera1 the mathematics involved in the problem of nonlinear electrical transport in semiconductors being very much complicated, one has to adopt simplifying assumptions in order to make the problem tractable. These necessary assumptions very often compromise with the physical validity of the theoretical results. To mention in particular, the scattering by optical phonons which are of the polar type in the II-VI compound semiconductors, is known to be predominant in these materials[&6] and a solution of the Boltzmann equation with predominant polar optical phonon scattering is beset with many difficulties. Hence one has to take recourse to the numerical techniques like the Monte Carlo technique, the
The problem of high electric-field properties in semiconductors has been extensively investigated for many years. This is mainly due to two reasons. First, more information on the physics of microscopic processes in these materials is obtained by studying the kinetic processes of the carrier ensemble in a state significantly perturbed from the state of thermodynamic equilibrium. Second, from a study of the high-field effects one can find new ways of better controlling the material properties by means of a variation of the external field strength, the knowledge of which is of great importance for their recent applications in solid state electronic devices. The theoretical as well as experimental studies of high field conductivity characteristics of semiconductors have been carried out in great details during the last two decades[l-31. Most of these studies have, however, been devoted to the elemental and the III-V compound semiconductors. II-VI compound semiconductors on the other hand have not received much attention. Though the acoustoelectric and optoelectronic properties of some of the II-VI compound semiconductors are well studied, not much work has been done on the high-field mobility characteristics of these semiconductors. This is mainly because of the technological difficulties in obtaining single crystals of desired purity required for the hot electron studies. II-VI compound semiconductors, however, should occupy a place of importance in high-field devices. ‘/‘On leave from: Department Kalyani,
Kalyani-741235,
of Physics,
University
of
India. 393
394
D.
MUKHOPADHYAYand D. P. BHATTACHARYA
method of path variables or the method of iterative integration [ l-31. All these methods however, require large computation time and it is difficult to study the variation of the high-field conductivity characteristics with the material parameters, some of which have not yet been determined from independent physical measurements in the II-VI compound semiconductors. A broad based analytical method for obtaining the high-field mobility characteristics would be thus useful, and more so for the study of the variation of the characteristics with the material parameters. If the carrier-carrier collisions are sufficiently frequent to randomise the energy and momentum distribution of the electrons, then the electron distribution function in the momentum space can be assumed to be in equilibrium at a temperature T, which depends upon the applied electric field and which is greater than the lattice temperature TL. Under this condition the distribution function becomes a drifted Maxwellian one. Using the displaced Maxwellian distribution function, the Boltzmann transport equation can be integrated to yield the energy and momentum conservation equations in which the average rate of momentum and energy loss is equal respectively to the momentum and energy gained from the applied electric field. The integrations can be carried out in a closed form if the distribution function is expanded in terms of the spherical harmonics and the series truncated after the first two terms[7]. This approximation is known as the diffusion approximation[8] and it amounts to assuming that the drift energy of the carriers is small compared with their thermal energy. Calculations show that for most of the semiconductors considered here this ratio is about 0.4 at 77°K in the low field region, and is still less at higher fields and at higher temperatures. The earlier calculations show that the retention of the drift energy terms in the distribution function does not significantly change the calculated characteristics in the III-V compound semiconductors even when this ratio is as high as 0.5[9]. It is to be noted however that for carrier-carrier scattering to dominate over the lattice-carrier scattering the required carrier concentration should be of the order of 10’6/cm3[7]. The hot electron studies are however performed in samples which have a carrier concentration about two orders of magnitude lower than this. For the displaced Maxwellian model to be valid what is required is that the carrier-carrier scattering should randomise the momentum distribution and this is achieved at a carrier concentration which is lower than that required for the predominance of the carrier-carrier scattering. The displaced Maxwellian model when applied to the study of the elemental and the III-V compound semiconductors is known to yield results which are in good agreement with the experimental results and with other numerical calculations even though the validity criterion is not strictly satisfied. This is presumably because of the above reason and also because of the inherent accuracy associated with the method of variation of
parameters[lO]. The II-VI compound semiconductors however, have a larger carrier concentration than the III-V compound semiconductors and the validity criterion for the predominant carrier-carrier scattering should be better satisfied in these materials. Therefore, though superior numerical methods are available, the displaced Maxwellian approach would be useful for studying the electrical transport in the II-VI compound semiconductors in particular.
2. THEORY
The Boltzmann
transport
equation
can be written
as
af =7x -I I at
fie,*
I
where the 1.h.s. is the rate of change of the distribution function due to the applied electric field F and 2f/atlj is the rate of change of the distribution function due to the jth collision mechanism. As has been explained earlier, if the carrier concentration is large enough to ensure momentum randomization due to electronelectron collision, the distribution function becomes displaced Maxwellian given by
fW =
(2
” exp[ nm *kBTJ312
-g]
(2)
where k is the electron wave vector, m * the isotropic effective mass of the electrons, T, the electron temperature and d is the displacement in the momentum space for the electron system. h = h/2n, h being the Planck’s constant and k, the Boltzmann constant. The carrier effective mass m * has been assumed to be isotropic. The II-VI compound semiconductors have larger band gap in comparison to the elemental or the III-V compound semiconductors, leading to a smaller value of the nonparabolicity factor. Though this simplifying assumption of a parabolic band structure is not strictly justified even for the II-VI compounds, it will be seen in what follows that the results obtained using the parabolic band structure quite agree with the experimental observations. The displacement d and the electron temperature T, may be obtained by solving the energy and momentum conservation conditions. The relevant balance condition for the monentum and energy can be written as
and
respectively, where E = h2k2/2m* is the carrier energy. In order to evaluate the integrals in (3) and (4)
Electron transport in II-VI compound semiconductors one must know the types of carrier collision mechanisms which dominantly limit the mobility value. The carrier scattering mechanisms important for the II-VI compound semiconductors are the scattering through acoustic phonons, both deformation potential and piezoelectric, scattering through polar optical phonons and the scattering with the ionised impurities. The intervalley scattering between the central and the higher lying satellite valleys should not play any significant role in the field range of our interest, because these valleys are considerably far removed from the central valley[l 11.. The scattering by the ionised impurities and piezoelectric phonons can be considered elastic and they do not contribute to the energy loss mechanism[l]. The integrals in (3) and (4) are evaluated in a closed form using the diffusion approximation which is true only when the scattering is elastic[l2]. In II-VI compound semiconductors polar optical phonon scattering which is strongly inelastic is predominant and the diffusion approximation would be valid in the presence of electric fields high enough so that the average carrier energy becomes much higher than the optical phonon energy. However, the other elastic collision processes which are always present randomizes the momentum distribution of the carriers and hence the diffusion approximation would not lead to much error even for lower values of the electric field[l3]. On substituting the values of @@)/&l, and 8f(k)/t%lj in (3) and (4) one can integrate term by term and obtains the following coupled equations F
_=v
d
. /?m *3’2h r8E,*E.*x”* . . h ‘+pc:
3nJ;;e
+
e’K_*E,‘l*x -‘I* “. . h 2~&g?l*
L + N,Z2e4Er~3/2x-3’2LBH 8c,,*q,*m*’
(5) 1
and m *W
8,,/5m**E,*E;i2x3/*(1
_ x -1)
Fd=p h2P
x&h3e
(6)
. Here E, = k,T,; x = TJT,; L, = ln( 1 + B) - B/( 1 + ~9); B = 24,rc~ * (kBTJ2/nh2e2; (exp(xJ
+ (exp(%) + 1%
$ (
- I)&
e >I
(exp(x,)
x, = es
--- .
- l)K,
(e )I;
395
$
K2m, the normalised electromechanical coupling constant, is given by K,,,* = P*/qc,pc,$ P being the pieozoelectric constant, p the density and c, the average acoustic velocity for the longitudinal mode. El is the deformation potential constant for the acoustic phonons, co is the free space permittivity. rcOand K, are respectively the static and the high frequency dielectric constants of the material. N, is the ionised impurity concentration and Z the degree of ionisation of the impurity centres, n the carrier concentration and Q9 the Debye temperature. & and K, are the modified Bessel functions of the second kind. Equations (5) and (6) can now be solved for F and d for different assumed values of the electron temperature T,, and hence one can find the mobility characteristics of a material. The parameters for different materials used in the numerical calculations are given in Table 1. 3. RESULTS AND DISCUSSIONS
Figure 1 shows the electric field dependence of the carrier mobility normalised to its zero-field value at a lattice temperature of 77°K taking Ni= 0. The nature of dependence is qualitatively the same for all the semiconducting compounds investigated here. For fields upto 0.5 kV/cm the mobility is more or less independent of the field strength, then except for ZnO it decreases in a nonlinear manner on increasing the field strength upto 3.0 kV/cm. On further increasing the field the mobility is found to fall almost linearly. For ZnO however, starting from the zero-field value, the mobility slowly increases with the electric field upto a field of 1.2 kV/cm. For higher fields the mobility at first decreases in a nonlinear manner for fields upto 4.5 kV/cm; beyond which the decrease is almost linear with the increase of the applied electric field. Further beyond the linear regime, at a field characteristic of the compound (lowest for CdTe; 7.0 kV/cm and highest for ZnO; 30 kV/cm) the mobility field characteristics again becomes nonlinear with a gradual decrease in the slope, tending towards a lower saturation. It is to be mentioned here that our calculation does not give any perceptible dependence of the carrier mobility on the strength of the electric field at a lattice temperature of 300” K, hence the hot-electron effects are not expected to be observed in the II-VI compound semiconductors at room temperature in the field range considered here. For CdTe, this result is in agreement with that of Butcher and Fawcett[l4] who made a theoretical study of the hot-electron transport in CdTe using the displaced Maxwellian model for the energy distribution of the free electrons but obtained the displacement d and the electron temperature T, by numerical solution of the re-
D. MUKHOPADHYAY and D. P. BHATTACHARYA
396
Table 1. Material parameters used for numerical calculations
all0
0.32
8.50
0.59
5.66
5.0
a51
0.074
10
ZnB
020
8.90
5-U
4.08
5.0
507
0.074
IO
OdTe
0.10
9.60
‘7.21
6.06
3.0
242
0.068
18
5.0
304
0.026
10
4.3
440
0,037
12.8
C&J
03
9.25
6.40
5.31
cas
025
9.19
5.24
4.80
Electric field
( kV/cm I
Fig. 1. Varation of the normalised mobility with the applied electric field at 77” K (with no impurity). (1) ZnO; (2) ZnS; (3) CdTe; (4) CdSe; (5) CdS.
spective conservation equations. They however predicted the onset of the hot-electron effects in CdTe for fields greater than 10 kV/cm at a lattice temperature of 300” K. As expected, the quantitative agreement of our results for fields below 10 kV/cm with those obtained by Butcher and Fawcett is not very good because of different values of the material parameters used in our calculations. Borsari and Jacoboni [15] theoretically investigated the high-field transport in CdTe by the Monte Carlo technique and reported definite nonlinear field dependence of the drift velocity of the carriers at 300” K in the range of field from IO kVjcm to 20 kV/cm. At fields below that range the drift velocity increases linearly with the electric field, whereas for fields above that range the drift velocity falls almost linearly with the increase of fields upto 50 kV/cm and it is in this range of electric field where their theory also predicts field-independent carrier mobility. On comparing our results for CdTe with the experimental observations of Canali et al. [ 161one can see that our results agree quite satisfactorily with their ex~rimentaI results. For example where our theory predicts drift velocities of 1.45 x IO’ and 1.73 x 107cm/sec at fields 7.0 and 8.5 kV/cm re-
spectively, their experimental observations give 1.5 x IO’ and 2.0 x 10’ cmjsec for the drift velocity at the above fields respectively. In the case of CdS our results are qualitatively in good agreement with those of Guha[ 121, who also obtained the high-field mobility characteristics in the displaced Maxwellian model. That our results for normalised mobility at different field strengths are consistently lower at all fields in comparison with the values obtained by Guha is attribute to the fact that the value of rcOwhich is chosen here is lower than that assumed by G&a. Figure 2 shows the dependence of the low-field mobility on the lattice temperature under the same condition when the effect of the ionised impurity scattering is not taken into account. It may be mentioned here that for CdSe the ionised impurity scattering plays insignificant role for impurity concentration upto 10’“/cm3[6] except at temperatufT below 77” K; whereas in the case of ZnO the same is true at lattice temperatures above 150” K upto the same level of impurity con~ntrat~ontl7~. It may be noted here that the general nature of the lattice temperature dependence of the low-field mo-
Electron transport in II-VI compound semiconductors 500400m III
200 I
3377 II
1000/ T,(
K
I
I
20
50
-’ )
Fig. 2. Variation of the low field mobility with the inverse of lattice temperature (with no impurity). (1) ZnO; (2) ZnS; (3) CdTe; (4) CdSe; (5) CdS.
Impwty
397
bility is also the same for all the compounds studied here. The low-field mobility decreases with the rise of lattice temperature. In the low temperature region the nonequilibrium carriers in CdSe possess highest mobility whereas in the room temperature regime the carrier mobility in CdTe is the highest. Moreover, the carrier mobility in CdS is usually lower than that in CdTe except in the range of temperature from 25” K to 100” K. It may be mentioned here that the values of the low-field mobility in ZnO at different lattice temperatures as measured and calculated by Hutson[4] agree well with the values obtained by the above theory. The calculated values for the low-field mobility in CdS at different lattice temperatures are found to be in good agreement with the ex~~mental values of Saitoh [ 181. For example, where our calculation yields the values 4.5 x IO3and 3.3 x lO*cm/sec for the drift velocity at 77” K and 300” K respectively, Saitoh’s results were 5 x lo3 and 3 x lo* cmjsec at these temperatures respectively. It is well known that the ionised impu~ty scattering becomes of very little significance for electrical conduction as the electric field is increased[6, 15, 171. Hence we have chosen to plot the variation of the low-field mobility with the impurity concentration in Fig. 3. It is to be observed that even the low-field mobility is almost independent of the impurity concentration in the sample at a lattice temperature of 300” K. At 77” K however, as expected, the low-field mobility decreases with the increase of the impurity
concentmtion
(cm-3)
Fig, 3. Variation of the low field mobility with the impurity concentration deformation potential coupling constant of 1OeV. -, 77°K: ---------, (3) CdTe; (4) CdSe; (5) CdS.
at 77°K and 300°K for 300”; (1) ZnO; (2) ZnS;
D.
398
MUKHOPADHYAYand D. P. BHATTACHARYA
0”
5
2
03
5
2
0’
EleCtrlc
field
( kV/cm
)
Fig. 4. Variation of mobility
with electric field for various combinations of the deformation potential and the electromechanical coupling constant at 77” K. Curves l-3 are for ZnS, cures 4-6 are for CdSe and curves 7-8 are for CdS. (1) E, = 7 eV; (2) E, = 10 eV; (3) E, = 30 eV, EMCC = 0.074 for curves 1-3; (4)E, = 10eV, EMCC = 0.026; (5) E, = 18eV, &WCC = 0.026; (6) E, = 18eV, .&WCC= 0.035; (7)
E, = 12.8eV, &WCC = 0.037; (8) E, = 18eV, EMCC = 0.0455.
concentration. The decrease is slow for the impurity concentration upto 0.5 x 10”/cm3 and the slope becomes more and more steep as the concentration is increased. In the absence of any known experimental data under the identical conditions we cannot check our results. But our results are qualitatively in agreement with the experimental results of Finlayson et a1.[19] who actually studied the electrical conduction in CdSe samples having considerably high impurity concentrations. It may be observed that the calculations carried out here involve a large number of physical parameters. The values of all these parameters are not equally well known. The knowledge of the electron-phonon coupling constants being the least perfect [Z] compared to that of all the other parameters involved in the calculation, it is interesting to investigate the effect of the change of these constants upon the mobility-field characteristics of the samples. Hence in Fig. 4 the electric field dependence of the mobility has been plotted for various combinations of the values of the deformation potential and the electromechanical coupling constant at a lattice temperature of 77” K. Since a higher value of a coupling constant means a greater coupling between the electron-phonon systems leading to more frequent carrier coilision, the carrier mobility in ail these ~rnpo~ds at a particular field is found to be lower for higher values of either of these coupling constants. The effect of the change of the coupling constants on the mobility-field characteristics is prominent upto a field of about 3.0 kV/cm.
At higher fields the variation of the field dependence of the mobility becomes progressively insensitive to the variation in these coupling parameters. At a lattice temperature of 300” K however, our calculation does not give any perceptible effect of the change of ‘the coupling constants on the field dependence of the carrier mobility in the range of electric field considered here. It may’ be said in the context of the above details that though a comparison of our results with the experimental data wherever available would prompt the estimation of the numerical values for the coupling constants, their independent physical measurements are to be hoped for. 4. CONCLUSIONS
The mobility characteristics of the nonequilibrium carriers in a series of II-VI compound semiconductors are studied theoretically in the displaced Maxwellian model taking into account the different scattering mechanisms. In the absence of the experimental results for some of the compounds under identical conditions as assumed in our investigation, we cannot check our results for all the II-VI compounds. However, our results for the compounds studied here except for ZnS, are found to agree wit’? the available ex~rimental results; though the quantitative agreement
with the experimental
always very satisfactory. agreement
with
other
data is not
Our results are also in good theoretical
thereby that the mobility
results,
characteristics
indicating
are not much
Electron
transport
in II-VI
the exact form of the energy distribution function of the carriers under high-field condition. Finally it should be mentioned that our calculation is based on simplifying assumptions of a uniform homogeneous and perfectly crystalline semiconducting sample having discrete impurity states and spherical constant energy surfaces in k space. These assumptions may not be satisfied fully by the experimental samples and hence it is difficult to obtain experimental results which can be satisfactorily compared with those of the present calculation. Moreover, in compounds like CdS and ZnO having a large piezoelectric coefficient it is difficult to determine the true variation of the carrier mobility with the electric field because the current density in such compounds at high electric field is greatly modified by the acoustoelectric current which flows in a direction opposite to the normal drift current [20, 2 11.Hence experiment is to be performed with pulses having width smaller than the build-up time of the acoustoelectric domain in the sample, so that the effect of the acoustoelectric current can be eliminated and the theoretical results for the mobility characteristics at high electric field can be directly compared with the experimental data[l2]. Apart from these, the technology for obtaining single crystals of the II-VI compounds having desired purity is not known to have attained perfection. Hence it is clear that more experiments on the mobility characteristics of the II-VI compounds under varying physical conditions of the sample need be performed before a thorough survey of the success of any theoretical model can be assessed. sensitive
to
F’CS Vol. 45, No. 4--c
compound
semiconductors
399
REFERENCES 1. Conwell E. M.. High Field Transport in Semiconductors. Academic Press, New York (1967). 2. Nag B. R., Theory of Electrical Transport in Semiconductors. Pergamon Press, Oxford (1972). 3. Nag B. R., Electron Transport in Compound Semiconductors. Springer-Verlag, Berlin (1980): 4. Hutson A. R.. J. Phvs. Chem. Soliak 8. 967 (1959). I. A., Phys. Status Sdlidi. 5. Finlayson D. M. and-Johnson (b) 71, 395 (1975). D., Indian J. Phys. 53A, 528 (1979). 6. Mukhopadhyay I. Stratton R., Proc. Roy. Sot. (A) 246, 406 (1958). 8. Stenflo L., Proc. IEEE 54, 1970 (1966). D. and Nag B. R., Indian J. Pure Appl. 9. Mukhopadhyay Phys. 7, 616 (1969). D., Phys. Status Solidi. (b) 49, K41 10. Mukhopadhyay (1972). 11. Herman F., Kortum R. L., Kuglin C. C. and Shay J. L., Znt. Conf. II-VI Semiconducting Compounds, Rhode Island (Edited by D. G. Thomas), p. 503. Bell Telephone Laboratories (1967). 12. Guha S., Phys. Rev. B2, 4971 (1970). 13. Conwell E. M. and Vassel M. O., Phys. Rev. 166, 797 (1968). 14. Butcher P. N. and Fawcett W., Proc. Phys. Sot. 86, 1205 (1965). 15. Borsari V. and Jacoboni C., Phys. Status Solidi. (b) 54, 649 (1972). 16. Canali C., Martini M., Ottaviani G. and Zanio K. R., Phys. Rev. B4, 422 (1971). D. and Bhattacharya D. P., Indian J. 17. Mukhopadhyay Pure Appl. Phys. 17, 45 (1979). 18. Saitoh M., J. Phys. Sot. Jap. 21, 2540 (1966). D. M., Irvine J. and Peterkin L. S., Phil. 19. Finlayson Mug. B39, 253 (1979). 20. Smith R. W., Phys. Rev. Lett. 9, 87 (1962). 21. Hutson A. R., McFee J. H. and White D. L., Phys. Rev. Lett. 7, 237 (1961).