BAND STRUCTURE AND SCATTERING MECHANISMS IN LEAD CHALCOGENIDES FROM TRANSPORT PHENOMENA

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BAND STRUCTURE AND SCATTERING

MECHANISMS IN LEAD CHALCOGENIDES FROM TRANSPORT PHENOMENA

Yu. Ravich

To cite this version:

Yu. Ravich. BAND STRUCTURE AND SCATTERING MECHANISMS IN LEAD CHALCO- GENIDES FROM TRANSPORT PHENOMENA. Journal de Physique Colloques, 1968, 29 (C4), pp.C4-114-C4-124. �10.1051/jphyscol:1968416�. �jpa-00213620�

JOURNAL DE PHYSIQUE Colloque C 4, supplkment au no 11-12, Tome 29, Novembre-Dkcembre 1968, page C 4 - 114

BAND STRUCTURE AND SCATTERING MECHANISMS IN LEAD CHALCOGENIDES FROM TRANSPORT PHENOMENA

Yu. I. RAVICH

Institute of Semiconductors, Academy of Sciences of the USSR, Leningrad, USSR

R6sumB. - Cet article porte sur la variation avec la tempkrature de la masse effective et des mkcanismes de diffusion dans PbTe, PbSe et PbS. Les mkthodes d'ktude de la non-parabolicite basBes sur la mesure du pouvoir thermoklectrique dans un fort champ magnetique et sur le coeffi- cient de Nernst-Ettingshausen (et d'autres coefficients cinktiques) font I'objet de discussion.

Les rksultats obtenus par ces mkthodes sont montrks et discutks. Les mesures de non-parabolicitk sont utiliskes pour dkfinir la variation en tempkrature de la masse effective et des parametres caractkrisant la diffusion. La masse effective varie avec la temperature proportionnellement ⁱⁱ la bande interdite.

Un certain nombre de risultats sur les effets thermoelectriques et thermomagnktiques suggerent des processus de diffusion de porteurs fortement inklastiques dans les sels de plomb. La plus grande partie de l'inklasticitk s'explique par des collisions electron-klectron et, pour une part moindre, par la diffusion sur les phonons optiques. La diffusion des phonons optiques affecte essentiellement la mobilite, de la meme manikre que la diffusion acoustique.

Abstract. - 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-Ettingshau- sen 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 tempera- ture 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 nonpara- bolicity. Further the scattering mechanisms among them nonelastic are considered which form the mobi- lity, 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 thermoma- gnetic 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 ope- rator 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

methods. In accordance with kp-perturbation theory [2] these data are sufficient to conclude the conside- rable non-parabolicity of conduction and valence bands in lead chalcogenides.

Nevertheless the thorough investigation of non- parabolicity 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 effec- tive 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 longitu- dinal components rn? and mt of effective mass separa- tely. 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 relati- vistic one by replacing mo with effective mass m* for the direction in question and light velocity c with ( ~ , / 2 m *)'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 trans- verse direction and leads to parabolic dispersion law for longitudinal direction, with longitudinal masses of electrons and holes allowing to differ. Energy sur- face 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 elec- trons 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 chalcoge- nides. 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 spin- orbital mixing wave-functions [9-111. Such mixing takes place in lead chalcogenides.

C 4 - 116 Y U . 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, there- fore 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 remo- ving 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 three- band 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 momen- tum P and P' in the given direction, but these conduc- tion 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 E 2 , we get the Kane model with effective gap

The non-parabolicity is still more for larger energies.

I f k - c o

The strong non-parabolicity which is not described with Kane model has been observed under Nernst- Ettingshausen effect investigation (see part 3).

The energy dependence of transverse and longitu- dinal effective masses is generally speaking different for the model represented in figure I. The measurement of Shubnikoff- de Haas effect [7] has discovered the longitudinal effective mass in PbSe and PbS to vary with energy approximately by such a way as the trans- verse one to do. The energy dependence of longitudinal effective mass in PbTe is alternatively weaker than that of transverse effective mass. This dependence is characterized with effective gap which is about two times more than the appropriate gap for transverse component. It means that Kane model is more fit in PbSe and PbS than Cohen one, but in PbTe some intermediate case is realized.

The transport effects and some other ones (for exam- ple the infrared reflection) depend on some averaged effective mass. The energy dependences of averaged effective mass for both models differ not so much ; therefore both these models have being used with equal success to acount for experimental data for PbTe. For instance Dixon and Riedl[12] used Cohen model for PbTe, Kaydanoff, Chernik et al. [13] used Kane model for PbTe of n-and p-type.

2. THE DATA OBTAINED FROM T. E. P. MEASUREMENT IN STRONG MAGNETIC FIELD. - One of the methods of bands edges structure investigation is the t. e. p. mea- surement in strong magnetic field (in nonquantum sense). This method was used by Dubrowskaja and the author [14] to research nonparabolicity of conduc- tion band in PbTe, PbSe and PbS and of valence band in PbTe. The t. e. p. and Hall-effect measurements in strong magnetic field were carried out at the nytrogen temperature in PbTe with electron concen- tration n = 2 x 10'' - 1 x loz0 ~ m - ~ , in PbSe with n = 1,9 x 10'' - 3,8 x loi9 ~ m - ~ , in PbS with n ⁼ 2,8 x 10" - 5,4 x 10'' ~ m - ~ , in PbTe with hole concentration p = 5 x 1017 - 4 x 1019 ~ m - ~ .

Almost all the specimens were degenerate at this temperature. For example, the figure 3 represents a, in p-and n-PbTe.

FIG. 3. - T. e. p. in the strong magnetic field for degenerate samples of PbTe at 77 ^{O K} [14].

The limiting value of t. e. p. in strong magnetic field does not depend on scattering mechanism. The relations to come will show that a, measurement allows to find the density of states in the band as the function of energy p(E) and the Fermi-level as the function of the concentration [(n). This does need to attract no models of non-parabolicity. At the strong degeneration the a, is equal

a, = n2 k; Tp(n)

3 en ⁽⁸⁾

FIG. 4. - The density of states against the energy in the conduction band for PbTe, PbSe and PbS at 77 OK as obtained by t. e. p. measurement in strong magnetic field [14].

parameters E,, and mzo (mz0 = N''~ in:r3 -

the density of states effective mass at band edge, N- the number of ellipsoids) allows to achieve the agreement with both the first and second models.

In the table I the values obtained are listed, as well as the energy gap E, defined by Mitchell et al. [15]

from magnetooptic measurements. The values mfo obtained for two models coincide fairly. The magnitude E,, for conduction band is near E, in the case of Kane model while for Cohen model it is a little less than E,.

This formula allows to find the density of states p C,aV

at the Fermi-level I: corresponding to the concentration n. For finding the Fermi-energy one uses the relation

[,

and the value obtained is not sensitive to the model choosed ( ~ e t a i l s in [14]).

The Fermi-level dependence on concentration and the density of states on energy obtained for conduction band in PbTe, PbSe and PbS at 77 OK are shown at figures 4 and 5. The curve p(E) differs greatly from

that represented by the parabolic model.

I . . . . I 1 . , . . I

The plots obtained were compared to dependences 5@* u0'9 5;O" +/Ow

resulting from two above simple models of non-para- FIG. 5. -The Fermi-level against the concentration for bolicity. It occured that the appropriate choice of n-type of PbTe, PbSe and PbS at 77 ^OK as obtained from am [14].

C 4 - 118 YU. I. RAVICH

Band parameters obtained from t. e. p. -

measurements in the strong magnetic field at 77 OK [14]

&olmo

- ^Egi, - ^eV E, 1151, e v -

n-PbTe 0,12 0,19 (Kane) O,11 (Cohen)

p-PbTe 0,13 0,15 (Kane) 0,19

0,08 (Cohen)

n-PbSe 0,12 0,17 (Kane) 0,16

n-PbS 0,22 0,29 (Kane) 0,28

Further we shall only use the Kane model as the both models are approximate, the Cohen model is not better grounded, but the Kane one is more convenient for practical calculations.

3. THE DATA OBTAINED FROM MEASUREMENTS OF

NERNST-ETTINGSHAUSEN COEFFICIENT AND OTHER TRANSPORT EFFECTS.

Necessity of the strong magnetic field condition being fullfilled interferes with the use of this method at high temperatures and large carrier concentrations, when the mobility is low. In this case the other method is more fit which is based on Nernst-Ettingshausen effect in the weak magnetic field, t. e. p. and Hall- mobility measurements. This method was employed by Kaydanoff, Chernik et al. [13] under investigation of non-parabolicity in lead chalcogenides.

If the energy surfaces are ellipsoidal and anisotropy coefficient does not depend on energy one can find such combination of Nernst-Ettingshausen coefficient Q, Hall coefficient R, t. e. p. a and conductivity a in degenerate samples which does not depend on scattering mechanism if scattering is elastic. This combination gives density - of - states effective mass at the Fermi level :

where m>is defined by (2). The relation for definition rn; is the following :

The analysis of experimental data has shown the dependence rn; on the concentration to correspond well with Kane model in PbTe of n- and p-type as well as in PbSe at relatively small energies (down to

= 0.2 eV). The Kane model parameters at 120 OK are listed in the Table 11. In general they correspond well to results of employment of a, - measurement

method and to other data but sometimes one can observe the appreciable divergence, for example of the value Egi in p-PbTe. The reason of that is unknown by now. The error of a, - method may be related to unsufficient fullfillment of the strong magnetic field condition whereas the error of method of Nernst- Ettingshausen coefficient measurement may be related to the assumption of carrier scattering elasticity.

It will follow that scattering in lead chalcogenides at about 100 OK temperatures is unelastic. However the estimates made for some particular cases enable us to hope that the corrections of the scattering nonelasticity to formula (1 1) are not too large.

Band parameters obtained from the measurements of Nernst-Ettingshausen coefficient at 120 OK [I31

(The Kane model case)

At large concentrations (about loz0 cmL3) in PbSe Kaydanoff and Chernik have established the nonpara- bolicity to be stronger indeed than in Kane model. If we replace the right-hand side of (1) by

then it is necessary to adopt such parameters :

in order to fit the experiment in n-PbSe at 120 OK.

Such a strong non-parabolicity was discussed above (see part 1) in connection with six-band model. The dependence of m: on E, obtained for n-PbSe, is represented in figure 6.

4. INFLUENCE OF NON-PARABOLICITY ON TRANSPORT PHENOMENA. - The above data show that non-para- bolicity in lead chalcogenides is rather large, so that effective masses vary a few times within investigated energy range. The non-parabolicity manifests itself as directly through an effective mass dependence on energy as indirectly by relaxation time.

Because an effective mass and relaxation time appears in expressions for transport coefficients in combination

rim* [16], the energy dependence of effective mass may lead to the same effects as energy dependence of

FIG. 6. -The effective mass dependence on concentration at the Fermi-level for PbSe at 77 ^OK as obtained by Nernst- Ettingshausen and other transport coefficients measurement [13].

The deviation from Kane model is seen.

relaxation time may do. In particular, Efimova et al.

[17] have shown that non-parabolicity may lead to appreciable temperature dependence of Hall-coeffi- cient in the heavy-doped samples of n-PbTe, due to the variation of statistic factor in the Hall-coefficient formulas. There is also influence of non-parabolicity on Lorentz-number in the region of relatively high temperatures and concentrations [18].

Further the non-parabolicity changes the energy dependence of relaxation time due to increase of den- sity of states. It also leads to the change of temperature dependence of transport coefficients 117, 181. The value r = a ^{In z/a} In E can noticably decrease because of increasing of density of states [19].

At last, the non-parabolicity is connected with the dependence of modulating part of Bloch-function on wave-vector which leads to the change of matrix elements of carrier interaction with phonons, impu- rities etc. It is more difficult to take into account this comparatively small but still appreciable effect than the influence of the non-parabolicity on density-of- states. The calculation of this effect was produced by Ehrenreich [20] and Korenblit and Sherstobitoff [21]

for InSb-type semiconductors.

All the listed various manifestations of the non- parabolicity can often compensate each other in part.

This accounts for the relative success of previous attempts of interpreting many experimental data without taking the non-parabolicity into account.

For example when calculating the temperature depen- dence of t. e. p. the increase of effective mass with energy is compensated by the decrease of scattering parameter r. Still we need take the non-parabolicity into account for obtaining right values of effective mass, its temperature dependence, scattering parame- ters etc.

Let us mention of one interesting manifestation of non-parabolicity, namely in the magneto-tunnel effect. p-n junction is placed in the strong magnetic field which is transverse to the current. Aronoff and Picus [22] have established that both the energy corresponding to drift-velocity and the energy got by electron at Larmour-orbit are comparisable to the energy gap E,. Therefore the electron spectrum non- parabolicity plays here the principal role. As the elec- tron motion in two-bands model is described by the quasi-relativistic equation the authors [22] used the methods developed for Dirac equation. The theory has come to a good agreement with experimental data of Rediker and Calawa [23] for PbTe.

11. The temperature dependence of effective mass. - The dependence of effective mass on temperature is due to thermal expansion of a lattice and due to elec- tron interaction with phonons.

In accordance with Ehrenreich 1241 the temperature dependence of an effective mass which is due to ther- mal expansion is connected in Kane model with correspondent temperature dependence of the energy gap. For relatively small changes of effective mass at the band bottom we have :

The calculation produced by author [25] for InSb- type semiconductors gave the formula (12) also for the temperature dependence of effective mass which was due to the interaction with phonons (*).

Therefore the formula (12) signifies the total tempe- rature change of an effective mass in the Kane model.

In lead chalcogenides the Kane model is the approxi- mate one, so we can expect the relation (12) to hold approximately.

It follows from (12) that the temperature dependence

(*) In the paper [25] it was established that the temperature dependence of effective mass due to the interaction with the acoustical phonons is connected with the dependence of modu- lating part of Bloch-function o n the wave-vector. One of the results is also that the polar interaction with optical oscillations affects weakly electron spectrum even in such semiconductors in which it dominates in carrier scattering.

C 4 - 1 2 0 YU. I. RAVICH of effective mass is stronger in semiconductors with

the narrow energy gap where the non-parabolicity is also large. The temperature dependence of averagzd effective mass can be caused by thermal distribution of electrons over non-parabolic band. Thus under investigation of temperature dependence of effective mass it is necessary to analyse the experimental data with taking the non-parabolicity into account. This was produced by Kaydanoff et al. [I31 and by Efi- mova et al. [17] under such investigation of n-PbTe.

In the paper [13] the temperature dependence of an density-of-states effective mass m:o near the band bottom within the temperature range 120-300 OK was obtained by means of the Nernst-Ettingshausen effect and other effects measurements. In the paper [17] just the same was obtained from t. e. p. measurements at the temperatures 300-900°K. The both papers contain the result : m:o -- E,.

The temperature variations of an effective mass being due to thermal expansion and interaction with phonons can be separated with attracting the data of an effective mass variation under pressure. As shown by Averkin et al. [Ill, approximately a half a tempe- rature change of effective mass (and energy gap) is due to thermal expansion while another half is due to interaction with phonons.

111. Scattering mechanisms. - 1. SCATTERING

MECHANISMS FORMING THE MOBILITY. - At the tern- peratures higher about 100 OK the interaction with acoustical phonons (Moyzhes et al. [lo]) is the dominant mechanism of carriers scatte- ring in lead chalcogenides, while at low (helium) temperatures the scattering by the core impurity potential is one (Stylbans et al. [26]). The interaction with optical phonons was rarely attracted to account for experimental data on lead chalcogenides though some papers contain the conclusion that the optical phonons play the appreciable role together with a acoustical phonons [13].

At present time one can ascertain the role of optical phonons in carrier scattering by the calculation, because all involved parameters are defined by the independent methods, and thus without taking any adopted parameters into account. For example, the relaxation time is given by [27]

at the temperature much more the Debye one for the longitudinal optical phonons (T >> 8 = &o/k,).

At lower temperatures one cannot introduce the

relaxation time and the expressions for mobility have the more complicated form /27] involving also the frequency of the longitudinal optical phonons o.

The calculation for this case is generally produced by the variational method. Further it is necessary to take nonparabolicity and screening of polar interaction by free carriers into account. The care of the nonparabolicity was taken in the form of the dependence of effective mass on energy. Screening by free carriers decreases a 1

+

(r, is the screening radius in the medium with E,, q is the phonon wave-vector). In particular, at T >> ⁸

this leads to appearance of factor

in formula (13) for 117.

There is a number of effects besides above ones neglected under calculations which results will be given. Firstly, the dependence of modulating part of Bloch-function on k in the nonparabolicity region modifies the matrix element of the carrier interaction with phonons [20, 211. Secondly, screening effect leads to change of the longitudinal optical frequency according to [28] as well as influencing directly on the matrix element, so that

( o , is the transverse optical frequency, ^8, and E, - the static and high-frequency dielectric constant).

This effect is especially large in lead chalcogenides because of large difference between o and a,, and

E , (*). At last the dynamical character of screening

effect needs also the consideration 1281. The calcula- tions taking care of these effects are produced at present but previous estimates enable to hope that the corrections to the results which follow will not be considerable and will not essentially change the conclu- sions concerning the role of optical phonons in the scattering.

In figure 7 the mobility in degenerate samples of n-PbTe is represented at the temperature 77 OK. The experimental points are taken from different papers

(*) The dispersion law of longitudinal optical phonons a t

EO 9 em is of the form : wz

-

⁺

FIG. 7. - The electron mobility for n-PbTe at 77 OK.

1. - The curve plotted over experimental points corresponding to the samples with the highest mobility for a given concentra- tion. 2. -The curve obtained after exclusion of scattering by the impurities with use of data on the mobility at 4,2OK, 3. - The theoretically obtained curve for the mobility which is due to scattering by optical phonons. 4. - The mobility due to scattering by acoustical phonons; this curve is obtained from the curves 2 and 3.

[13, 14, 29, 301. The curve 1 is plotted over points correspondent to the sampIes with the highest mobi- lity. The curve 2 is obtained from the curve 1 and from the mobility at 4 OK and represents the mobility due to scattering by phonons. The calculated curve 3 represents the mobility due to scattering by optical phonons. The curve 4 is obtained by means of compa- rison the curves 2 and 3 and gives the mobility due to scattering by acoustical phonons. As shown from the figure that optical phonons affect strongly the mobility at relatively small concentrations (2 x 10'' - 10'' ~ m - ~ ) .

By comparising the curve 3 in figure 7 with the curve in figure 4 for PbTe one can make sure that the product of the relaxation time for acoustical phonons by density of states z,, p is almost independent on energy in the region of relatively large energy. This product is reversely proportional to the square of matrix element of carrier interaction with acoustical phonons.

2. NONELASTIC SCATTERING. - The research of thermoelectric and thermomagnetic properties has established that carrier scattering in lead salts is

non-elastic, i. e. one can not introduce the relaxation time, which is the same for all effects. For instance, the electric current relaxation is only due to collisions which lead to the relatively large wave-vector change, whereas the heat current relaxation with the absence of the current in degenerate semiconductors may be produced also by processes which do not almost change the wave-vector but the energy change is comparisable to ko T.

The difference between effective relaxation times is followed by the appreciable deviation of Lorentz number L from its value Lo for elastic scattering which is equal to universal constant - 7cL (k0/e)' for strong

3

degeneration. The most direct method of defining the electron thermal conductivity of semiconduc- tors is the total thermal conductivity measurement in magnetic field H. In the limit of strong magnetic field (in a nonquantum sense) the electron part of thermal conductivity disappears, so that the difference between the two values of thermal conductivity (one for the H = 0 and another for H -+ co) gives the electron thermoconductivity K. This method was employed by Shalyt and Muzhdaba [31] for definition L/L, in lead salts. The results obtained are listed in the table 111 and represented in figure 8.

L differs appreciably from Lo at a temperature about 100 OK with L/L, tending to unity with a tem- perature decrease down to helium value.

The other method permitting to define to Lorentz number is based on the analysis of total thermoconduc- tivity dependence on temperature and concentration.

Smirnow et al. [32] have secured Lorentz number in PbTe, PbSe and solid solutions PbTe-PbSe with help of this method at the temperature of 77 OKand higher.

The values L/Lo obtained at 77 OK are in a satisfactory agreement with the results of thermoconductivity measurement in magnetic field with L tending to Lo under temperature increasing up to 300° (fig. 8).

By way of treatment of experimental data on concen- tration and temperature dependence of mobility u and t. e. p. a (the method is described in [19]) the values characterizing the carrier are obtained. It occured that the-magnitude rbefined formally from concentration dependence of mobility (r,) did not coinside with that obtained from t. e. p. (r,). The values r, and r, at 77 OK and at concentration

are listed in the table 111. The difference between r, and r, is as much appreciate as between L and Lo.

YU. I. RAVICH

%.

FIG. 8. - The Lorentz-number dependence on temperature for n-PbTe. 1 and 2 - the data for n = 5 x 1018 cm-3 and n = 1,3 x 1019 cm-3 respectively as obtained from thermal conductivity measurement in the strong magnetic field [31]. 3 and 4 - the data for n = 6 x 1018 cm-3 and n =9 X 1018 cm-3 as obtained by the analysis of total thermoconductivity dependence on temperature and concentration 1321 ; 5 - the result of calculation carrier-carrier collisions at n = 5 x 1018 cm-3.

The points related to the samples with near concentrations are bound with lines.

The values of L/Lo, r, and r, in lead salts at T = 77 OK and carrier concentration about 2 x 1018 ~ r n - ~ Calculated results

Experimental data

Mate- rial

- n-PbTe p-PbTe n-PbSe p-PbSe n-PbS

LILO C311 ra [I91 r" [I91 from x(H) from a from u

- - -

076 - 0,3 - 0,5

0,55 - 0,25 - 0,45

- 0,I - 0,4 074

0,45

The value of r, differs noticably from that of r, as well at concentrations about lo1' - lo2' ~ m - ~ .

At last the experimental data on the Nernst-Etting- shausen effect 113, 331 are also the evidence of scatter- ing nonelasticity. In particular, the mobility uQ obtained from the Nernst-Ettingshausen coefficient at r = - 0,5 is 1,3-2,O times less than the Hall-mobi- lity zx, in the non-degenerate samples of p-PbTe at 200-300 O K 1331.

LILo LILO LIL,

(elastic

+

+

scattering) ing) gether)

- - -

0,6 0,s 075

07 5 0,6 076 07 5

The processes of interaction with optical phonons as well as carrier-carrier collisions and intervalley transitions may result in- nonelastic scattering in lead chalcogenides. The latter mechanism appears to be a little efficient, as the transitions between extrema are forbidden by the selection rules in these semiconduc- tors.

The polar interaction with optical phonons is possible as a result of particular ionicity of the lattice.

As shown by the theoretical estimates for PbTe this interaction exerts some influence not only on the mobility, but also other transport coefficients. Howe- ver it is insufficient to explain the above experimental data concerning the scattering nonelasticity (at least in PbTe). For example, the theoretical value of L/Lo is equal to 0,8 in n-PbTe at 77 OK and

if the acoustical and optical scattering is taken into account. The results of the Nernst-mobility calcu- lation are represented in figure 9 (the plot 2) for nondegenerate sample of p-PbTe at 200-300 OK. The experimental data on scattering nonelasticity were explained by Moyzhes and the author [19] with help of adopting carrier-carrier scattering.

FIG. 9. - The Hall - and Nernst - mobility for p-PbTe with hole concentration of 8 X 1017 cm-3 (experimental points [33] and theoretically obtained curves) ; 1. - the Hall-mobi-

8 16 qc

lity u~ = - oRc ; 2. - the Nernst-mobility us = - - I Q 1

3 n 3 nkn

when taking care of scattering by acoustical and optical-pho- nons ; 3. - the same for scattering by acoustical phonons and for carrier-carrier collisions ; 4. - the same when taking care of all the above three scattering mechanisms.

3. CARRIER-CARRIER SCATTERING. - The collisions between carriers can result in the significant change of heat current and thus can appreciably influence on the thermoconductivity, t. e. p. and thermomagnetic effects which occurs due to energy redistribution bet- ween carriers [27]. The efficiency of this process is rather large in lead chalcogenides in spite of the real absence of scattering by impurity Coulomb potential.

When rapidly moving the carriers are no time in order to polarize the lattice so that their mutual interaction is defined by the high-frequency dielectric constant

E, while an interaction with impurity ions is defined by the static one e0 which is an order more.

The variational principle was also employed for elucidating the role of carrier-carrier collisions. Their interaction was described by the screened Coulomb potential. Taking the care of the carrier-carrier colli- sions does not require adopting any fit parameters as well as the calculation of the optical oscillations role in the scattering does, For example, the thermal resistance W e , due to carrier-carrier collisions is equal to :

at v, kF 2 1 [34] in degenerate semiconductor. Here kF and v, are wave-number and the velocity at the Fermi-level, Wo is the thermal resistance related to elastic scattering of carriers and given by Wiedemann- Franz law.

The calculation according to (16) gave L/Lo

--

0,6 in all lead chalcogenides of n- andp-type when taking care of carrier-carrier scattering and elastic scattering for degenerate samples of PbTe, PbSe and PbS with n

=

The results of Nernst mobility calculation are drawn at the figure 9 for p-PbTe with hole concentration 8 x 10'' ~ m - ~ . This figure illustrates the contribution of each nonelastic scattering mechanism to Nernst mobility. As seen from that the carrier-carrier colli- sions can account for the difference between uQ and u, which is observed.

Conclusions. - The treatment of non-parabolicity gave the possibility to explain the experimental depen- dences of transport coefficients on the temperature and the carrier concentration, as well as to secure more exact values of an effective mass and parameters characterizing the scattering. The carrier-carrier colli- sions and optical scattering occurred to play a consi- derable role together with acoustical scattering. In particular, the nonelasticity of scattering is connected with these mechanisms. Taking care of these mecha-

C 4 - 124 YU. I. RAVICH

nisms has resulted in a good agreement of the theory with the number of experimental data for samples with carrier concentration about up to lo1' ~ m - ~ .

However the above mechanisms become unsuffi- ciently effective t o account for the observed scattering nonelasticity with help t o available formulae at large concentrations (about 10'' ~ m - ~ ) . The divergence of theory from experiment may clearly come from the non-perfection of our understanding the problem of electron-electron interaction. It should be remarked that the results of the account of carrier-carrier colli- sions, are very sensitive t o the form of interaction poten- tial for degenerate semiconductors (see r: in (16)).

One can hope the nonelastic scattering investigation to give some information concerning the character of carrier-carrier interaction in degenerate semiconduc- tors.

The following task is to interprete the large amount of experimental data on,transport phenomena from universal point of view.

~ e c a u s e taking care of the scattering by optical oscillations and of carrier-carrier collisions reauires adopting no fit parameters it is necessary to choose the only one parameter for all the effects, temperatures and carrier concentrations, namely the deformation potential constant (*).

The author is greatly indebted t o B. Y. Moyzhes, L. S. Stylbans, V. I. Kaydanoff, I. A. Chernick, I. N. Dubrovskaya, V. I. Tamarchenko, B. A. Efimova for their help with preparing this paper.

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