Band Structure and Scattering Mechanisms in Lead Chalcogenides from Transport Phenomena

The non-parabolicity, temperature dependence of an effective mass and scattering mechanisms in PbTe, PbSe, PbS are considered. The methods of investigation of non-parabolicity based upon measurement of thermoelectric power in strong magnetic fields and Nernst-Ettingshausen coefficient (together with other kinetic coefficients) are discussed. The results given by these methods are shown and discussed. The data of non-parabolicity are used to define the temperature dependence of an effective mass and parameters characterizing the scattering. The effective mass varies with temperature proportionally to the energy gap. A number of experimentaI results on thermoelectric and therrnomagnetic effects point out a considerable non-elasticity of carrier scattering in lead chalcogenides. In the main the non-elasticity is explained by carrier-carrier collisions and to a less extent it is due to scattering by optical phonons. The scattering by optical phonons affects essentially the mobility in the same manner as acoustical scattering. The problem of carrier scattering mechanisms in lead chalcogenides can not be considered as a settled one by now. The generally accepted view that all available experimental data a t not too low temperature are explained by acoustical scattering contradicts the whole number of experimental data recently obtained. In particular the scattering in these semiconductors occurs to be essentially non-elastic. The results of experimental data analysis show that a general picture of scattering mechanisms is rather complicated indeed. This picture can not be got without researching many transport effects as a whole, with reliable values of scattering parameters being able to define only after thorough investigation of band non-parabolicity and temperature dependence of an effective mass. This paper contains the conclusions on some details of band structure and scattering mechanisms in lead chalcogenides recently obtained by means of transport phenomena investigation. At first the non-parabolicity of conduction and valence bands is considered. Then we deal with temperature dependence on an effective mass for defining which we use the data on nonparabolicity. Further the scattering mechanisms among them nonelastic are considered which form the mobility, heat conductivity and other transport coefficients. In particular the assumption about essential role of carrier-carrier scattering is attracted to explain the experimental data on thermoelectric and thermomagnetic phenomena. I. The band non-parabolicity. 1. THE MODELS OF NON-PARABOLICITY. The Scanlon's research [I] of optical properties of lead salts has shown that extrema of valence and conduction bands are at the same point of k-space, that energy gap is relatively narrow and the matrix element of momentum operator between these extrema states is non zero. These conclusions were supported afterwards by many other Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1968416 BAND STRUCTURE AND SCATTERING MECHANISMS C 4 115 methods. In accordance with kp-perturbation theory [2] these data are sufficient to conclude the considerable non-parabolicity of conduction and valence bands in lead chalcogenides. Nevertheless the thorough investigation of nonparabolicity was undertaken only for last years, because the interpretation of numerous experimental data connected with transport phenomena needs the detailed information on non-parabolicity. The simplest model of non-parabolicity takes place when the dispersion law near band edges is fully determined by interaction of bands separated with energy gap. In this case the effective mass value and its energy dependence is defined by the energy gap E, and by matrix element of momentum operator between status corresponding to band edges. The carrier effective mass occurs to be much less than the free electron mass m,. Allgaier's investigation of magnetoresistence [3] and other investigations have shown that extrema of valence and conduction bands are at L-point, and thus we need to consider the transverse and longitudinal components rn? and mt of effective mass separately. If both masses are defined by the interaction of conduction and valence bands the dispersion law for electrons and holes is We shall call this model the Kane model because the dependence E(k) to any direction coincides with the dependence for conduction band in InSb 121. Let us note the dispersion law to have quasirelativistic form in this case, i. e. it can be obtained from relativistic one by replacing mo with effective mass m* for the direction in question and light velocity c with (~ , /2m *)'I2. The model (I) is characterized by identity of electron and hole effective masses, the smallness of transverse and longitudinal effective masses as compared with m,, ellipsoidality of energy surfaces and independence of anisotropy coefficient on energy. If we introduce an effective mass for given direction with the relation then the energy dependence of this mass is The other model of non-parabolicity considered by Cohen [4] takes place when the transverse effective mass is defined by interaction of the nearest bands while the longitudinal one is defined by the interaction with far placed bands as well as by contribution of free electron mass. In this case the dispersion law is This model coincides with the Kane one for transverse direction and leads to parabolic dispersion law for longitudinal direction, with longitudinal masses of electrons and holes allowing to differ. Energy surface are nonellipsoidal in Cohen model and their form depends on energy. The theoretical calculations carried out by different methods [5-81 lead to the conclusion that in lead chalcogenides near the energy gap there are 6 relatively close bands (fig. 1). Though effective masses of electrons and holes are in general defined by the interaction FIG. 1. -The arrangement of bands at the point L in lead chalcogenides. The nonzero matrix elements of longitudinal and transverse components of the momentum operator are shown with the arrows. of lower conduction band and upper valence band (*) other bands also contribute to effective mass. Therefore the above simple models can only approximately describe the structure of band edges in lead chalcogenides. The models parameters must be adopted in such a way that choosed model would correspond to the experiment as fair as possible. For example the (*) Let us note that the states at the point L may be connected simultaneoulsy by the transverse and longitudinal components of momentum operator only under condition of strong spinorbital mixing wave-functions [9-111. Such mixing takes place in lead chalcogenides. C 4 116 YU. I. RAVICH effective interaction gap E,, may differ from the real energy gap E,. Let us consider the lower conduction band. Its bottom is described with an odd wave-function, therefore at the point it is connected with three valence bands but not with other conduction bands. At first sight the nonparabolicity of lower conduction band (as well as of upper valence band) seems for this reason to be weaker than in the two-band model (1) (when one expands in powers E the right-hand side of the relationship replacing (I), it should be substitued Egi instead of E, in the quadratic term with E,, being more than E,). It is not the case in fact. When removing from the L-point the wave function of conduction band consists of the odd and even parts, thus it appears the possibility of interaction with upper conduction bands. To understand the mutual interaction of the bands with the same parity better we consider the threeband model (fig. 2), reserving the main features of FIG. 2. The three-band model of nonparabolicity. six-band model. Two conduction and one valence bands are connected with matrix elements of momentum P and P' in the given direction, but these conduction bands are not connected with each other. The dispersion law to the given direction is defined by formula : At the relatively small E, when retaining the terms up to E2, we get the Kane model with effective gap The non-parabolicity is still more for larger energies.