S. BHAUMIK and C. K. SARKAR: Energy Loss Rate of Hot Electrons 617

phys. stat. sol. (b) 187, 617 (1995)

Subject classification: 72.10; 71.25; S8.15

Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta')

Energy Loss Rate of Hot Electrons in n-Hg,Cd, -,Te at High Temperatures in the Presence of a Quantizing Magnetic Field

BY S. BHAUMIK and C. K. SARKAR

The energy loss rate of hot electrons due to longitudinal polar optical phonons in narrow-gap semiconductors is calculated for the longitudinal configuration with classical and quantum screening for various lattice temperatures. The results are used to examine the influence of quantum screening on the energy loss rate. A relative comparison of the effects of quantum and classical screening on the energy loss rate due to optical phonons is made. The energy loss rate without screening is compared with the energy loss rate with classical and quantum screening.

1. Introduction

Hot electron transport in narrow-gap semiconductors such as Hg,Cd, -,Te has drawn a considerable attention in the last few years. These materials carry great technological interest in infrared devices. Recently, Nimtz and Stadler [l] and Yongping et al. [2] have reported experimental results on the transport properties of hot electrons in n-Hg,Cd, -,Te at low temperatures in the presence of a quantizing magnetic field.

The experimental energy loss rates in n-Hg,Cd, - ,Te at low temperatures have been analyzed extensively by the present authors [3] assuming the acoustic phonon scattering to be the most dominant energy loss mechanism at low temperatures.

The magnetic field quantization modifies the density of states in semiconductors and also alters the free carrier screening. It becomes a magnetic field dependent anisotropic parameter different from the classical Debye screening [4]. The effect of modified screening arising due to magnetic quantization is found to produce a different electric field dependence of the hot electron parameter compared to the classical screening. In the analysis of the energy loss rate at low temperatures, the inclusion of modified screening (also called quantum screening) is found to have a significant effect on the energy loss rate calculated assuming the screened electron acoustic phonon interaction. This result has prompted us to study the effect of quantum screening on the energy loss rate in narrow-gap semiconductors due to longitudinal polar optical phonon scattering for the longitudinal electric and quantizing magnetic fields.

In the present paper, a theoretical model has been developed to investigate the energy loss rate of hot electrons in n-Hg,,,Cd,,,Te in the presence of a quantizing magnetic field assuming a displaced Maxwellian distribution for carriers in longitudinal configuration. This assumption can be justified due to the dominance of electron-electron scattering in

') Calcutta 700032, India.

618 S. BHAUMIK and C. K. SARKAR

the presence of a quantizing magnetic field [5, 61. The electrons are assumed to populate the lowest two Landau subbands.

In the present model, the energy loss rate due to longitudinal polar optical phonon scattering in longitudinal configuration is calculated assuming intra- and inter-level scattering limited to the lowest two Landau subbands n = 0 and 1 [7]. The spin splitting of the levels has been neglected since the inclusion of the spin splitting will make the calculation much more complicated due to inter-level scattering within the levels at n = 0 and n = 1.

The theoretical results of the energy loss rate due to polar optical phonon scattering are presented with and without screening. A comparison of the results with quantum screening has been made with the results obtained with classical screening and without screening. The other important features of the model are the incorporation of band non-parabolicity and Landau level broadening due to electron-impurity interactions.

Finally, the energy loss rate due to polar optical phonon scattering has been compared with the energy loss rate due to low temperature acoustic phonon scattering and their relative importance in the temperature range has been examined.

2. Theory

When a non-parabolic semiconductor is subjected to a quantizing magnetic z-direction, the energy dispersion relation of an electron in the n-th sublevel can as P I

h2kk,2 E 2mtV 2

E = _ _ + " ( a , - l) ,

field B in be written

(1)

where k , is the z-component of the electron wave vector, E , is the band-gap energy, a, is the non-parabolicity factor given by

where w, = eB/m* is the cyclotron frequency, m* the band-edge effective mass, m,*V the average effective mass obtained by averaging over occupied Landau subbands given by [9]

1 1 hw,

e x p ( 5 ) - I]]?

where K , is given by

with

m*

m0

- 1

.=[I+;] and y = - - ,

where m0 is the rest mass of the electron and A the spin-orbit energy.

Energy Loss Rate of Hot Electrons in n-HgxCd, -,Te at High Temperatures 619

The energy loss rate expression is given as [lo]

- Avv,(q) f ( ~ v * , T,) 11 - f ( ~ v 9 Te)I) 9 (4)

where oq is the phonon frequency for wave vector q, T, the electron temperature, E, and E, .

are the electron energies in the Landau states v and v', respectively, 1C,12 is the electron-phonon coupling constant, and E,,, Avv, represent the emission and absorption terms, respectively.

The above expression (4) reduces to the following form after some simplifications:

where q2 = q: + qt , TL is the lattice temperature, no the carrier concentration, and n and n' are the two sublevels representing the Landau index of lower and upper sublevels, respectively.

After some manipulations (5) reduces to the form

For polar optical phonon scattering,

where e is the electronic charge, wo the POP frequency, and E, are, respectively, the optical and static permittivities, and q. represents the Debye screening term which is simply given as q: = 1;' = noe2/(&,cokBK), where AD is the Debye screening length for classical screening and no the electron concentration. In the classical limit, qs is independent of the magnetic field. However, the presence of a quantizing magnetic field modifies the screening term qs which can be expressed as [4]

620 S. BHAUMIK and C. K. SARKAR

(-x)" = c

n = O 2"(2n + l)!!. In order to obtain the energy loss rate due to optical phonon scattering, IC,12 given by

The integrations over qL and q, are then performed making the following change in (7) is substituted in (6) and appropriate qs is also substituted for different screening.

variables:

After some algebraic simplification, we then get

p = Po0 + Po1 2 (8)

where Po, is the energy loss rate due to transitions between the levels at n = 0,

and Pol represents the transition between levels n = 0 and 1 and is given by

x ( a,+-+ B V aAPv + as))} 2 4

Here we have made the following substitutions:

where 1 = (h/eB)'/' is the Landau radius for the lowest Landau subband. The average value of u is taken as 1/4 [ll].

Energy Loss Rate of Hot Electrons in n-Hg,Cd,_xTe at High Temperatures 62 1

3. Results

The energy loss rate of hot electrons due to polar optical phonon scattering as a function of electron temperature has been calculated for n-Hgo,,Cdo,,Te for lattice temperatures TL = 10 and 20 K at B = 4 T. The results for TL = 10 and 20 K are shown in Fig. 1 and 2, respectively. The calculations have been done at B = 4 T since certain parameters used for the present calculations were reported at B = 4 T [3, 121.

Results of other magnetic fields can be obtained in a straightforward manner. The energy loss rates due to polar optical phonon scattering for TL = 20 K at B = 7 T has also been shown in Fig. 3. The composition x = 0.8 is chosen since most of the measurements are reported for this composition [12].

It can be seen from Fig. 1 that the energy loss rate is higher for the case without screening compared to the energy loss rate calculated including the quantum and classical screening. This is due to the fact that the inclusion of the screening reduces the scattering matrix element causing less scattering and results in a lower energy loss rate compared to the case of scattering without screening. However, the loss rate is higher for the quantum screening compared to the classical screening due to the enhancement of the scattering rate of electrons causing more electron energy dissipation due to quantum screening compared to the classical screening. The enhancement of the scattering rate is partly due to the confinement effect caused by the quantum screening.

E = 4 r $=IOK

Fig. 1. Variation of the energy loss rate with electron temperature at TL = 10 K and B = 4 T. Curves A, B, C represent energy loss rates with- out screening, with quantum screen- ing, and with classical screening, re- spectively

622 S. BHAUMIK and C. K. SARKAR

Fig. 2. Variation of the energy loss rate with electron temperature at TL = 20K and B = 4 T. Curves A represent the intra- level scattering for n = 0. Curves B repre- sent the total scattering involving transi- tion between levels n = 0 and 1 and intra- level scattering for n = 0. The continuous curves represents classical screening, while the dashed ones the quantum screening

6

A

/6 -

0 I I I I

25 28 32 36 40 g 00 -

A relative comparison of the energy loss rates due to acoustic phonon scattering (Pa,.) and polar optical phonon scattering (PpOp) has also been made for various lattice temperatures without screening and with quantum screening. This has been shown in Table 1 for T, = 4.2 and 10 K. It can be seen from the table that the energy loss rate due to polar optical phonon scattering is dominating at higher temperatures while the acoustic phonon scattering dominates at lower temperatures due to the low phonon energy. It can also be found that the energy loss rate due to polar optical phonon scattering is less dominant

Fig. 3. Variation of the energy loss rate with electron temperature at TL = 20 K and B = 7T. CurvesA represent the intra-level scattering at n = 0. Curves B represent the total scattering involving transition between levels n = 0 and 1 and intra-level scattering for n = 0. The con- tinuous curves represent classical scree- ning, while the dashed ones the quantum screening

25 27 29 31 33 35 37 39 41 6 (K) -

Energy Loss Rate of Hot Electrons in n-HgxCd,-,Te at High Temperatures 623

Table 1

T. = 4.2K 10 11 12 13 14

TL = 10K 12 14 16

6.2x 10-15

7.1 x 10-1~ 7.29 x 1 0 - 1 5

6.57 x 6 . 8 6 ~

1.765 x 2.984 x 3.865 x 10-l5

5.29 x lo-” 2.12 x 10-16

3.374x 1 . 0 6 6 ~ 10-15

9 . 0 4 ~ 1 0 - 1 5

1.016 x 10-15 8.99 x 1 0 - 1 5

4.459 x 10-14

pa, (W) ( 4

5.627 x lo-’’ 6 x 6 . 3 3 7 ~ 6.6 x lo-” 6.82 x lo - ’ ’

1.63 x 1 0 - 1 5

2.79 x 1 0 - 1 5

3.65 x 1 0 - 1 5

p,,, (W) (4

2.99 x lo-” 1.62 x 6 . 6 4 ~ 2 . 1 8 6 ~

6.32 x 10-l’ 6.03 x lo-’’ 3.17 x lO-I4

6 . 0 6 ~ 10-15

pa, (W) (qs)

6 x lo-‘’ 6 . 3 8 8 ~ 6.68 x lo-’’ 6.9 x lo-’’

1 . 7 1 9 ~ 2 . 9 1 9 ~

7.13 x 10-15

3.793 x 10-15

P,“, (W) (qs)

3.5 x 10- ” 1.88 x 10-lb 7.6 x

6.86 x

7.62 x

2.49 x 10-15

6.82 x 1 0 - 1 5

3.54 x 10-14

ns no screening, cs classical screening, qs quantum screening.

compared to acoustic phonon scattering when screening is incorporated in the calculation. Hence the importance of the polar optical phonon scattering at low lattice temperature is reduced due to the presence of scattering.

The effect of screening on the energy loss rate is found to decrease with increasing electron temperature. This suggests that screening is not important for optical phonon scattering which dominates at higher temperature compared to low temperature acoustic phonon scattering.

Fig. 2 and 3 show the contributions of the energy loss rate for inter- and intra-Landau level optical phonon scattering with quantum screening for TL = 20 K at B = 4 and 7 T, respectively. It can be seen that the total energy loss rate for the lowest two Landau subband occupations (n = 0 and n = 1) is slightly higher than the energy loss rate due to the lowest Landau level. This means that the inter-level contributions between the lowest two Landau sublevels are quite insignificant compared to the intra-level contribution to the total energy loss rate. This is partly due to the large separation of the Landau subbands in n-Hg,Cd, -,re (x = 0.8) compared to thermal energy (kBT,) and phonon energy (ho,).

Acknowledgement

One of the authors (S.B.) wishes to acknowledge the CSIR, New Delhi for financial support.

References [l] G. NIMTZ and J. P. STADLER, Physica (Utrecht) 134B, 359 (1985). [2] C. YOUNGPING, Z. GUOZHEN, G. YAQUIAN, G. SHAOLING, c. JIANXIANG, and T. DINGYUAN,

Semicond. Sci. Technol. 5, S304 (1990). [3] S. BHAUMIK, C. K. SARKAR, and K. SANTRA, phys. stat. sol. (b) 161, 329 (1990). [4] A. FORTINI, phys. stat. sol. (b) 125, 259 (1984). [5] E. M. CONWELL, Solid State Physics, Suppl. 9, Ed. F. SEITZ, D. TURNBULL, and H. EHRENRICH,

[6] B. R. NAG, Springer Series Solid State Sci. 11, 332 (1980). [7] R . V. POMORTSEV and G. I. KHARUS, Soviet Phys. - Solid State 9, 1150 (1967). [8] U. P. PHADKE and S. SHARMA, J. Phys. Chem. Solids 36, 1 (1975). [9] C. K. SARKAR, R. J. NICHOLAS, J. C. PORTAL, M. RAZEGHI, J. CHEVRIER, and J. MASSIES, J . Phys.

Academic Press, New York 1967 (p. 12).

C 18, 2667 (1985). [lo] G. BAUER, H. KAHLERT, and P. KOCEVAR Phys. Rev. B 11, 968 (1975). [Ill A. M. Z L ~ B I N and P. S. ZYRYANOV, Soviet Phys. - Solid State 14, 379 (1972). [I21 R. DORNHAUS and G. NIMTZ, Springer Tracts mod. Phys. 98, 119, (1983).

(Received September 1, 1993; in revisedform July 19, 1994)