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Energy calculation of electrons in solid solutions of isovalent substitution
Solid State Communications,Vol. 19, pp. 1133-1135, 1976.
Pergamon Press.
Printed in Great Britain
ENERGY CALCULATION OF ELECTRONS IN SOLID SOLUTIONS OF ISOVALENT SUBSTITUTION Yu.A. Bratashevskii Physico-Technical Institute, Academy of Sciences of the Ukr. SSR, 340048 Donetsk, U.S.S.R.
(Received 4 March 1976 By E.A. Kaner) In the given paper the dependence of the electron energy on the concentration of isoelectronic impurity in solid solutions is discussed. The algorithm of calculation of the electron energy in substitutional isoelectronic solutions with the prescribed error of the approximation theory is found. THE ELECTRON ENERGY change at the solid solution formation may be expressed in the sum of contributions of deformation and impurity potentials: 1
AE =
+ A E U,
(1)
T
where (dE/dP)T and (-- d In V/dP)T are the isothermic coefficients of pressure and compressibility, A V / V is the relative change of the crystal volume, AEtr is the contribution of the impurity potential including the short-range part of local perturbation of the impurity U and the small addition connected with the dilatational field of deformation in the vicinity of an impurity site z which we have neglected for the purpose of simplifying calculations. Besides, we shall consider solutions which do not form localized states. In the linear approximation by the composition the electron energy in a two-band model is represented in terms of the expansion by cumulants of the self-energy part 2; (x; c) averaged over the impurity distribution of Green's one-electron function G. 3 For large impurity concentrations c it is convenient to use the representation Z (x; c) in the form of the nonterminating continued fraction: 4
(x; c) = UIc + xA(x; c)l,
where
A(x;c) =
approximation is determined by the inequality: . . . . .
K-~n
where K is the exact value of the fraction, P./Qn is the proper fraction of the n-order, Pn and Q. are its numerator and denominator, respectively. Let us rewrite (2) in the form: Z ( x ; c ) = G-' [(GU)c + (GU')(UG)f,(x;c)].
n(n -- 1)c(1 - - c ) f . ( x ; c ) -- 1 --n(1 -- 2c)x + x 2f~+l(x;c) ,n>~2; ix} = I U G I < I . As it is known from the theory of continued fractions, the best approximation is the representation of a continued fraction in the form of the proper fraction. 5.s Thereby, for the convergent continued fractions [bo, a,,/b= ] *~with the positive terms of links of the continued fraction (a,, > 0, b , > 0) the absolute error of the
(4)
The terms of the links of the continued fraction in brackets are equal:
a, = (GU)(UG)c(1 -- c)
b, = 1 -- (1 -- 2c)x
an = x2n(n -- 1)c(1 -- c), n i> 2
b , = 1 -- n(1 -- 2c)x
(5)
When the inequality is realized it is not difficult to show by the known criteria 8 the convergence of the stated fraction and the applicability of the error estimation to it in the form of the inequality (3). For the first approximation we have:
Zl(x;c)
(2)
c(1 - c) 1 - ( 1 - 2 c ) x +x=A(x;c)
(3)
en Qn_,
Q---~=
U
[c
XC(1--c))~]
+ 1 --(1 ----~c
"
(6)
The condition (5) does not confine the possibility of calculation of Z(x; c) for the whole range of values xc. In fact, in this case one has to take advantage of the symmetry property of the link terms of the continued fraction considered with respect to simultaneous change of signs at x and (1 -- 2c). For this it is necessary to make substitution of the variables e = 1 -- c and O = -- U, which is equivalent to mutual substitution of the ends of the same series of the solid solutions studied to have the inequality (5) always satisfied. For the calculation of matrix elements Z I (x; c) in a two-band model we make substitution of U = (1 -- 2c)Uin (6):
1133
ELECTRONS IN ISOVALENT SUBSTITUTION
1134
Z l ( x ; c ) = cUq
c(1 - - c ) 1 - - 2--------cUTG,
(7)
where T is the matrix satisfying the known integral equation: r = V' + U'GT.
(8)
Matrix elements of the potential by virtue of its locality do not depend on the quasi-momentum, a Moreover, as in the isoelectronic series one-electron Green's functions G°(E) = ~r G°(K' E) of the components A and B of the binary solution differ slightly from each other, we represent the averaged Green's function of the solid solution in the form of linear combination: Ga = (1 - - c ) G A + cG~,
(9)
where ct is the band index. Then the matrix elements 2;1 (x; c) are determined from the equations: ~,tm,(ElC ) -= cU
'+
c(1 - c )
U,~T~,(E, c)V,,,(e, c) /~=1
2
c) =
+ E
(10)
V'~.~(c)G,(E, c)r,~,(e, ~).
7=1
The contribution of the local perturbation of the potential AEu(c) necessary for the calculation of the change of the energy gap (overlap) AE(c) at the formation of solid solution has the form:
Vol. 19, No. 11
tudes (dE/dP)r and [-- (d In V)[dP] r in the solution series change slightly. In a general case the magnitudes Uu, U22 and UnU21 necessary for the calculation may be defined from the equation (1) with the help of any two known experimental values AE(c) within the solution series. In the cases when the contribution of the interband transitions may be neglected, the number of prescribed experimental values AE(c) reduces to one. Finally, when the electron mass is far less than the hole mass, and, therefore, we may neglect Un as compared to/-/22, we may calculate AE(c) without experimental data of solution parameters. In this case the only necessary matrix element U22 for the calculation is found from (1) at c = 1. The Green's function of the solution components G ° (E) may be evaluated as G ° --~2/AEa where AEa is the width of the corresponding band. The relative error of the calculation arising in the approximation in the form of the fraction of the first, second, and third orders amounts to 1/3, 0.125, and 4 x 10 -2 , respectively. The suggested algorithm of the calculation of the energy gaps in solid solutions also allows one to calculate such band parameters as the effective mass and g-factor of free carriers. For this one can use the known relations of models of ideal crystals which proved to be sufficiently reliable, for example, the Kane's model, where the theory parameters are the values of the width of the forbidden band Eg, spin-orbital splitting of bands A, and matrix element of the momentum P after the
AEu(c) = Re [ ~ n (c) -- ~22 (c)] = c(Ull -- Us 2)
(11)
+ c(1 -- c) Gl U~ll -- G2 U~22 + (G 1 -- G2)U12 U21 -- (1 -- 2c)G 1G2 (Ull -- U22)(Ull U22 -- U12 U21 ) [1 -- (1 -- 2c)U11Gl ] [1 -- (1 -- 2c)U22 G2 ] -- (1 -- 2c) 2 GI G2 U12 U21 The magnitude AE(c) is calculated after the substitution (11) into (1) knowing lattice parameters, band structures [GaA(E), G~(E), E(0), E(1)] and the pressure, and compressibility coefficients of the initial components of the solution A and B, and it is assumed that the magni-
substitution of Eg and A calculated by the above method into these relations within the solution series. The P value, as it is known, does not change practically within the isovalent series of solutions, and it is considered known by the end components of the solution.
REFERENCES 1.
BRATASHEVSKII Yu.A., Solid State Commun. 13, 1405 (1973).
2.
ZAKHAROV A.Yu., Fizika i Takhnika Poluprovodnikov 9,425 (1975).
3.
BRATASHEVSKII Yu.A., ZAKHAROV A.Yu. & IVANCHENKO Yu.M., Solid State Commun. 15, 1777 (1974).
4.
JONESAWA F., Progr. Theor. Phys. 40, 734 (1968).
5.
KHINCHIN A.Ya., "Zepnyi drobi", Gosizdat Fiz. Mat. Let., M., (1961).
6.
KHOVANSKII A.N., "Priblizhenie zepnych drobei i ich obobtschenie k boprosam priblitschennogo analiza", Gosuzdat tekhniko-teoreticheskoi literatury, M. (1956).
Vol. 19, No. 11
ELECTRONSINISOVALENTSUBSTITUTION
1135
B HacToH~e~ CTaTBe 0OCyx~aeTCH SaBEC~MOCTB sHepr~ SneKTp0HOB OT EOHIIeHTpa~ ~SosneHTpOHH0~ n p ~ e c ~
B TBep~Nx
pacT~opax, floayyeH anrop~Tu pac~eTa ~Heprzz aneKTp0Ha B paCTBOpaX MSOB~eETp0HHOPO sauet~eH~H c sa~OHHO~ HoPpelTTHOCTBD aHHpoNczMa~
Te0pz~.
Pergamon Press.
Printed in Great Britain
ENERGY CALCULATION OF ELECTRONS IN SOLID SOLUTIONS OF ISOVALENT SUBSTITUTION Yu.A. Bratashevskii Physico-Technical Institute, Academy of Sciences of the Ukr. SSR, 340048 Donetsk, U.S.S.R.
(Received 4 March 1976 By E.A. Kaner) In the given paper the dependence of the electron energy on the concentration of isoelectronic impurity in solid solutions is discussed. The algorithm of calculation of the electron energy in substitutional isoelectronic solutions with the prescribed error of the approximation theory is found. THE ELECTRON ENERGY change at the solid solution formation may be expressed in the sum of contributions of deformation and impurity potentials: 1
AE =
+ A E U,
(1)
T
where (dE/dP)T and (-- d In V/dP)T are the isothermic coefficients of pressure and compressibility, A V / V is the relative change of the crystal volume, AEtr is the contribution of the impurity potential including the short-range part of local perturbation of the impurity U and the small addition connected with the dilatational field of deformation in the vicinity of an impurity site z which we have neglected for the purpose of simplifying calculations. Besides, we shall consider solutions which do not form localized states. In the linear approximation by the composition the electron energy in a two-band model is represented in terms of the expansion by cumulants of the self-energy part 2; (x; c) averaged over the impurity distribution of Green's one-electron function G. 3 For large impurity concentrations c it is convenient to use the representation Z (x; c) in the form of the nonterminating continued fraction: 4
(x; c) = UIc + xA(x; c)l,
where
A(x;c) =
approximation is determined by the inequality: . . . . .
K-~n
where K is the exact value of the fraction, P./Qn is the proper fraction of the n-order, Pn and Q. are its numerator and denominator, respectively. Let us rewrite (2) in the form: Z ( x ; c ) = G-' [(GU)c + (GU')(UG)f,(x;c)].
n(n -- 1)c(1 - - c ) f . ( x ; c ) -- 1 --n(1 -- 2c)x + x 2f~+l(x;c) ,n>~2; ix} = I U G I < I . As it is known from the theory of continued fractions, the best approximation is the representation of a continued fraction in the form of the proper fraction. 5.s Thereby, for the convergent continued fractions [bo, a,,/b= ] *~with the positive terms of links of the continued fraction (a,, > 0, b , > 0) the absolute error of the
(4)
The terms of the links of the continued fraction in brackets are equal:
a, = (GU)(UG)c(1 -- c)
b, = 1 -- (1 -- 2c)x
an = x2n(n -- 1)c(1 -- c), n i> 2
b , = 1 -- n(1 -- 2c)x
(5)
When the inequality is realized it is not difficult to show by the known criteria 8 the convergence of the stated fraction and the applicability of the error estimation to it in the form of the inequality (3). For the first approximation we have:
Zl(x;c)
(2)
c(1 - c) 1 - ( 1 - 2 c ) x +x=A(x;c)
(3)
en Qn_,
Q---~=
U
[c
XC(1--c))~]
+ 1 --(1 ----~c
"
(6)
The condition (5) does not confine the possibility of calculation of Z(x; c) for the whole range of values xc. In fact, in this case one has to take advantage of the symmetry property of the link terms of the continued fraction considered with respect to simultaneous change of signs at x and (1 -- 2c). For this it is necessary to make substitution of the variables e = 1 -- c and O = -- U, which is equivalent to mutual substitution of the ends of the same series of the solid solutions studied to have the inequality (5) always satisfied. For the calculation of matrix elements Z I (x; c) in a two-band model we make substitution of U = (1 -- 2c)Uin (6):
1133
ELECTRONS IN ISOVALENT SUBSTITUTION
1134
Z l ( x ; c ) = cUq
c(1 - - c ) 1 - - 2--------cUTG,
(7)
where T is the matrix satisfying the known integral equation: r = V' + U'GT.
(8)
Matrix elements of the potential by virtue of its locality do not depend on the quasi-momentum, a Moreover, as in the isoelectronic series one-electron Green's functions G°(E) = ~r G°(K' E) of the components A and B of the binary solution differ slightly from each other, we represent the averaged Green's function of the solid solution in the form of linear combination: Ga = (1 - - c ) G A + cG~,
(9)
where ct is the band index. Then the matrix elements 2;1 (x; c) are determined from the equations: ~,tm,(ElC ) -= cU
'+
c(1 - c )
U,~T~,(E, c)V,,,(e, c) /~=1
2
c) =
+ E
(10)
V'~.~(c)G,(E, c)r,~,(e, ~).
7=1
The contribution of the local perturbation of the potential AEu(c) necessary for the calculation of the change of the energy gap (overlap) AE(c) at the formation of solid solution has the form:
Vol. 19, No. 11
tudes (dE/dP)r and [-- (d In V)[dP] r in the solution series change slightly. In a general case the magnitudes Uu, U22 and UnU21 necessary for the calculation may be defined from the equation (1) with the help of any two known experimental values AE(c) within the solution series. In the cases when the contribution of the interband transitions may be neglected, the number of prescribed experimental values AE(c) reduces to one. Finally, when the electron mass is far less than the hole mass, and, therefore, we may neglect Un as compared to/-/22, we may calculate AE(c) without experimental data of solution parameters. In this case the only necessary matrix element U22 for the calculation is found from (1) at c = 1. The Green's function of the solution components G ° (E) may be evaluated as G ° --~2/AEa where AEa is the width of the corresponding band. The relative error of the calculation arising in the approximation in the form of the fraction of the first, second, and third orders amounts to 1/3, 0.125, and 4 x 10 -2 , respectively. The suggested algorithm of the calculation of the energy gaps in solid solutions also allows one to calculate such band parameters as the effective mass and g-factor of free carriers. For this one can use the known relations of models of ideal crystals which proved to be sufficiently reliable, for example, the Kane's model, where the theory parameters are the values of the width of the forbidden band Eg, spin-orbital splitting of bands A, and matrix element of the momentum P after the
AEu(c) = Re [ ~ n (c) -- ~22 (c)] = c(Ull -- Us 2)
(11)
+ c(1 -- c) Gl U~ll -- G2 U~22 + (G 1 -- G2)U12 U21 -- (1 -- 2c)G 1G2 (Ull -- U22)(Ull U22 -- U12 U21 ) [1 -- (1 -- 2c)U11Gl ] [1 -- (1 -- 2c)U22 G2 ] -- (1 -- 2c) 2 GI G2 U12 U21 The magnitude AE(c) is calculated after the substitution (11) into (1) knowing lattice parameters, band structures [GaA(E), G~(E), E(0), E(1)] and the pressure, and compressibility coefficients of the initial components of the solution A and B, and it is assumed that the magni-
substitution of Eg and A calculated by the above method into these relations within the solution series. The P value, as it is known, does not change practically within the isovalent series of solutions, and it is considered known by the end components of the solution.
REFERENCES 1.
BRATASHEVSKII Yu.A., Solid State Commun. 13, 1405 (1973).
2.
ZAKHAROV A.Yu., Fizika i Takhnika Poluprovodnikov 9,425 (1975).
3.
BRATASHEVSKII Yu.A., ZAKHAROV A.Yu. & IVANCHENKO Yu.M., Solid State Commun. 15, 1777 (1974).
4.
JONESAWA F., Progr. Theor. Phys. 40, 734 (1968).
5.
KHINCHIN A.Ya., "Zepnyi drobi", Gosizdat Fiz. Mat. Let., M., (1961).
6.
KHOVANSKII A.N., "Priblizhenie zepnych drobei i ich obobtschenie k boprosam priblitschennogo analiza", Gosuzdat tekhniko-teoreticheskoi literatury, M. (1956).
Vol. 19, No. 11
ELECTRONSINISOVALENTSUBSTITUTION
1135
B HacToH~e~ CTaTBe 0OCyx~aeTCH SaBEC~MOCTB sHepr~ SneKTp0HOB OT EOHIIeHTpa~ ~SosneHTpOHH0~ n p ~ e c ~
B TBep~Nx
pacT~opax, floayyeH anrop~Tu pac~eTa ~Heprzz aneKTp0Ha B paCTBOpaX MSOB~eETp0HHOPO sauet~eH~H c sa~OHHO~ HoPpelTTHOCTBD aHHpoNczMa~
Te0pz~.