The inelastic electron – polar optical phonon scattering in HgTe
Malyk O.P.
Semiconductor Electronics Department
“Lviv Polytechnic” NationalUniversity
Bandera Street. 12, 79013, Lviv,
Ukraine
Abstract: -The model of inelastic electron scattering on polar optical phonons is proposed in which the scattering probability does not depend on macroscopic parameter – crystal permittivity. The reviewed model gives the good agreement between the theory and experiment in temperature range 77 - 300 К.
Key-Words: -Electronic Transport, Inelastic Scattering, Mercury Telluride
1 Introduction
The electron-polar optical phonon scattering was considered in relaxation time approximation in [1]. In [3, 4] this scattering mechanism was considered in view of inelastic character of scattering within the framework of a precise solution of the stationary Boltzmann equation. There was exhibited that the usage of standard model of electron - polar optical phonon scattering reduces to a disagreement between the theory and experiment in temperature range Т> 100 К. According to our opinion this model has following shortages: 1) the use of macroscopic parameter - permittivity- is not reasonable in microscopic processes; 2) the interaction potential of an electron with optical oscillations of a crystal lattice is long-range that contradicts the special relativity. The purpose of the present work is the build-up of such a model of scattering which at firstwould well match with experiment and secondly would not have the abovementioned shortages.
2 The model of the electron – polar optical phonon scattering
Let's consider adisplacement of j-th (j =1, 2)atomin a unit cell of a crystal with zinc blende structure under the influenceof optical oscillations [5]:
where G – a number of unit cells in a crystal volume; M = MHg + MTe - the mass of the unit cell; and - wave vector and angular frequency of -th branch of a crystal optical oscillations respectively () ; - polarizationvector of crystal oscillations; and - operators of phonons annihilation and birth respectively of -th branch with wave vector q ;
, () , - lattice constant; - unitary vectors along crystal principal axis.
Under optical oscillations in a unit cell a polarizationvector arises:
, (2)
where - the volume of the unit cell; e - elementary charge.
Using (1) and taking into account only the long wave ( ) oscillations one can obtain:
It must be noticed that the polarization vector is a function of discrete variables. To calculate the bound charge let's make following replacement of a partial derivative of a polarization vector on coordinates:
where for a unit cell of the zinc blende structure.
The similar relations can be written for partial derivatives and with .
Then Poisson equation for a scalar potential bound up with crystal oscillations becomes:
where the relation is used and only optical longitudinal vibrations are taken into account, - dielectric constant.
To solve the equation (5) let’s substitute the unit cell by an orb of effective radius the magnitude which lays within the limits from half of smaller diagonal up to half of greater diagonal of a unit cell (0.5 < γ < ). Magnitude γ =0.628 is picked so toadjust the theory with experiment. Spherically symmetric solution of a Poisson equation looks like:
. (6)
Then the interaction energy of an electron with polar optical oscillations of a lattice is determined from expression:
Let's mark that the potential (7) is short-range as it takes into account interaction of an electron only with one unit cell. To calculate the transition probability connected with electron – phonon interaction let’s write the wave function of the system “electron + phonons” in a form:
, (8)
where - crystal volume; - wave function of the system of independent harmonic oscillators.
Then the transition matrix element from interaction energy looks like:
The integration over the electron coordinates is carried out in the limits of unit cell and gives:
The calculation shows that the electron wave vector ( and s together with it) varies within the limits from 0 up to 10 9 м-1 at energy changing from 0 up to 10 кВТ (кВ – Boltzmann constant) at the temperature range 4.2 – 300 К .
Fig. 1. The dependence of the function from .As introduced at fig. 1 the dependence of the value from it is seen that at the indicated limits of varying of wave vector the following relation is well fulfilled:
The integration over the harmonic oscillators coordinates gives the factors and
(Nq–thenumber of phonons with a frequency at) for phonon annihilation and birth operators respectively. To calculate the sum over the vector q let’s do the following simplifications –1) taking into consideration quasi continuous character of varying of wave vector let’s pass from summation to integration over q ; 2) let's pass from an integration on a cube with a crossbar to an integration on an orb with effective radius :
Then we obtain for the sum the following expression:
As introduced at fig. 2 the dependence of the value from it is seen that the function can be approximated by the expression:
Fig.2. The dependence of the function from .After the calculations we can obtain the expression for electron transition probability connected with phonon absorption and radiation:
where - electron energy.
Fig. 3. ThetemperaturedependenceofelectronmobilityinHgTe : solidline –mixed scattering mechanism; 1 –intraband scattering; 2 –interband scattering. Experiment – [2].On the base of this for intra- and interband electron transitions the values figuring in a method of a precise solution of the stationary Boltzmann equation can be obtained [3,4] :
where - Kronecker delta ; - Fermi – Dirac function; - step function.
The calculation of the temperature dependence of electron mobility was made for acceptor concentration NA = 3 x 1015 сm-3 thus it is possible to neglect the contribution of heavy holes ( about 1%). At calculations the same scattering mechanisms as in [3,4] were taken into consideration. As it seen from Fig. 3 the theoretical curve well coincides with experimental data at the temperature range T >100 K. It testifies that the model, offered by us, adequately describes the electron-polar optical phononscattering process as against model introduced in [1]. From figure it is also seen that the basic scattering mechanism in the interval T > 100 K is intraband scattering on polar optical phonons. The contribution of the interband scattering is negligible and can be neglected.
3 Conclusion
The model of inelastic electron-polar optical phonon scattering in HgTe is designed which in the framework of a precise solution of the stationary Boltzmann equation well coincides with experiment at the temperature range T > 100 K.
References:
[1] W.Szymanska, T.Dietl, Electron scattering and transport phenomena in small-gap zinc-blende semiconductors, J. Phys. Chem. Solids, Vol.39, 1978, pp. 1025-1040.
[2] J. Dubowski, T. Dietl, W.Szymanska, Electron scattering in Cd xHg 1-xTe, J. Phys. Chem. Solids, Vol. 42, 1981, pp. 351-362.
[3] O.P.Malyk, Nonelastic electron scattering in mercury telluride. Ukr. Phys. Zhurn., Vol.47, 2002, pp. 842-845.
[4] O.P. Malyk, Nonelastic charge carrier scattering in mercury telluride, Acceptedto publication in Journal of Alloys and Compounds.
[5] V.L. Bonch-Bruevich, S.G. Kalashnikov, Physics of Semiconductors, Nauka, Moscow, 1977 (in Russian).