- Email: [email protected]
On the variation of energy gaps in semiconducting alloy systems
Solid State Communications, Vol. 13, pp. 1397—1400, 1973.
Pergamon Press.
Printed in Great Britain
ON THE VARIATION OF ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS KS. Song Physics Department, University of Ottawa, Ottawa, Canada (Received 7 June 1973 by A.A. Maradudin)
The departure from linearity of energy gaps in a semiconducting alloy system is studied within the framework of virtual crystal approximation. Exact expressions for the slope of energy variation as function of composition x at the two ends x = 0 and x = I are derived. These can be evaluated theoretically with energy band calculation of the two pure materilas for a system. We discuss here several features of the slope.
INTRODUCTION
SLOPES AT x =0,1 WITHIN THE VCA
THE VARIATION of energy gaps in alloys systems of zinc-blend type semiconductors has been studied extensively.1 Usually they show a more or less pronounced departure from linear variation as function of cornposition x between the two pure materials. This socalled bowing effect, which is almost always downward, has been studied by several authors.23 The usual approach in treating this problem is to study the energies in the virtual crystal approximation (VCA) and consider the configurationally averaged fluctuation of potential as a perturbation. In the previous works the contribution of these two parts to the bowing was not satisfactorily handled due to either the simplicity of the model2 or the nature of the model employed.3
The valence electron wave functions in the semi. conductors we are interested in are very entended in the region between the atoms and the local fluctuation of potential in an alloy will not affect the energy levels drastically. This is the reason why the VCA is reasonable for these materials. Onodera and Toyozawa2 have shown that in these materials with large bandwidth the energy bands belong to the so-called amalgamation type upon alloying. It is also reasonable, then, to treat the fluctuation by the perturbation method. The virtual crystal potential is given by the following expression; V(x)
=
xV 0(a(x)) + (I —x) V~(a(x))
(I)
It is hence desirable to be able to evaluate exactly at least one of the two contributions to the bowing. where VA(a(x)) and VB(z(x)) are the potentials of the pure materials A and B respectively with their lattice parameter equal to a(x). The lattice parameter a changes usually as a function of composition x following Vegard’s law. The influence of the change of a is explicitly included in this study. The Hamiltonian in the VCA can now be written as;
In this note we present expressions for the slope of the energy gap E,(x) at x = 0 and x = I which are exact within the framework ofVCA. The slopes at x = 0 and x = 1 can be calculated with reliableband calculation of the two pure materials. We examine how these slopes deviate from the straight line between the pure materials. The effect of fluctuation is more difficult to study quantitatively. We will discuss briefly the effect of this on the VCA results.
2
H (x) = P /2,n + V(x) 2 2/2tn + V = P 8(a(x)) + x EVA (a(x)) Va~a(x))] 2/2m+ VA(a(x))+(l —x)[VB(a(x))— VA P ~a~x))] (2’) —
1397
1398
ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS
We now propose to calculate the energy levels of H°(x)near x = 0 (and x = 1) exactly to the first order in x (and x’ I x). As there is an obvious symmetry betweenx=Oandx= 1 points,we present the result
Vol. 13, No.9
Here a typical element after differentiation in x looks like;
—
only near x =0 point (pure B material). We asswne that the eigenvalue problem with V~(a(x))is solved; 212m+ VB(a(x))I ‘I’a,(a(x)) = [P (3) EBI (a(x))
ijl~,
(a (x))
H~,—E
+ V11((x))+x
=
~-~-‘—
dx
~
dx (6’) Call the energy level we studyE’ which goes toEBl as x -+0. Then putting x =0 and E = E 81 into equation (6) we obtain;
The eigen functions ~‘~~‘s constitute a complete set and we expand the unknown wave., function of the VCA hamiltonian (2) in this basis. As the virtual crystal hamiltonian (2) has the same symmetry elements and the same lattice paramerer sa V8(a(x)), we need only function of the same symmetry in the expansion of the unknown wave function, of a given symmetry.
dx
+ V11 (0)
—
dx
dx
(E~2 E’) (EB3 —
—
E’)...
+(dEB2
dE’ + V22(O) 1)(E —E
(EBI =
~
C(x)1, I,IIBI (a(x))
83—E’)...
(4) (7)
The coefficients C(x)~,are determined by solving the following secular determinant;
II H~, E 6.•,~ —
=
(5)
0
where H,,
=
1(0) = E If we realize that E 8~and EBI’S are all different (except for some very unlikely situation where two or more states of same symmetry are degenerate), the above equation reduces to;
EB,((x)) 61, + x
(5’)
V~((x)) =
f I/,e~(a(x))[VA(a(x))
—
+ V11(0)—
VB(a(x))J ijlaj(a(x)) dv
=
0
0,
hence
0
(8) As we are interested in the variation of E in first order of x, we differentiate the above secular determinant (5) in x. The result is a sum of determinants each of which is obtained after one of the lines is differentiated in x, the rest of the lines remaining unchanged: H~1—E’
H~2
H21
H22 —E
..
+
H11 —E
H12
H~1
H~2—E
..
dE —
dEBI =
dx
—
dx
+ V11 (0)
This is the slope at x = 0 of the energy level which goes to ~ for pure B material. Experimental measurements usually study the variation of interband energy gaps,instead of that of individual energy levels. It is thus necessary to follow the variation of two optically connected energy bands and take the difference between them. The corresponding expression of the slope of energy gap variation is easily obtained and is as follows:
(6) ~-
dx
=~—~1 +V(0)
dx
0
(9)
ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS
Vol. 13, No.9 where =
dx
~I
j
—
dx Jo
dx
1399
rewritten expression: equations (10—11). As the actual slope will always be compared to the straight line between the two pure materials, we examine the three correction terms of equations (10—11).
I 10
=
V~(0) V,(0)
=
< 4i~(as)lVA(aB)— V3(a3)J ‘1’3~(aa))
—
(*BV(aB
—
)I VA (a3)
—
V3(a3) I !PBV(aB)> (9’)
The above equation can be slightly rewritten in such a way that different terms can be more readily analyzed. + ~(°) dx =
one. It represents the gap variation of the other pure
.~.f
.
da
(i) the first term (dE/da da/dx) represents the variation of energy gap of pure material A (or B) as its lattice constant is changed upon alloying with B (orA). This quantity can be calculated theoretically as it has been the case in some band calculations. More easily this can be obtained from experimental measurements of pressure coefficicns1of the energy gap, together with the compressibility of the pure material. Usually the direct gaps of zinc-blend type materials 4 (ii) The third termisiscompressed. very analogous to the first increase as the crystal
+ < EA~(aBPB EBg
material latticeseen constant is being modified alloying. as It its is easily that the two terms (firstupon and
—
dx~
0 =
~
+
.~
dxIo
da
EAg(aB))B
+ [EA~(aB) EAg] + [EAU —
EA,(aB)]
E3g]
(10)
As is clear from equations (9’)—(lO) (EAg(aa))B is the expectation value for the energy gap of A calculated with wave function of B at lattice constant of B.EA,(aB) is the energy gap of A at lattice constant of B. Similar expression is obtained for x = I point;
dx
=
da
dx
I
I
+
V,(1)
1
+ [(E,8~(a~))A
—
+ [EB,(aA) —E3,] + [EB,~EA,J
E3~(aA~ (II)
The theoretical slope dEg/dx at x = U and x = I can be evaluated using expressions given in equation (9’). To this end we need band calculations of pure materialsA and B at two different lattice constants. Such an attempt is being undertaken. This will allow us to separate the effect of the virtual crystal potential from that of potential fluctuation in the observed sowing of gap variation. In the present note we analyze briefly the influence of the different terms as given in
third) contribute in opposite direction and as they have the samesmall. order of magnitude. the net effect may be relatively (iii) The second correction term of equations (10—11) is one which can only be evaluated theoretically We will present here arguments which indicate that this term will more likely be negative. Variational methods in quantum mechanics tells us that if a state “i” is the ground state of the system (lowest state of a given symmetry in our case), the expectation value (EAI(aBP.B is always higher than the exact eigenvalue EA1(aB). If “I” is an excited state, the proposed variational function ~8j should be orthogonal io all lower states in order that the same statement holds. For a valence band this argument probably applies, as the core states are usually very deep (except when there is a nearby d-band) and little overlap will result between core and valence wave functions. This holds both at x = 0 and x = I. It follows that the second correction term will then tend to make the valence band variation bow upward. A conduction band, however, has several lower states (valence bands) of the same symmetry in the near vicinity. It is then unlikely that the conduction band variation will show large bowing due to the second term of equations (10—Il). Analogous considerations can be made to study the spin-orbit splitting of the valence band in a system of alloys. Expressions for the slope of splitting similar to equation (9) are then obtained. There is a departure
1400
ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS
from simple linearity even within the VCA, in contrast 5 to assumptions made by some workers. The slope evaluated from above expressions cannot be directly compared to the experimental data, as the effect is difficult to evaluate quantitatively. General effects of such local fluctuation havefunction been studied by 2 by Green’s method Onodera and Toyozawa with a simple model Hamiltonian. Parmenter6 treated this effect by perturbation method with free electron model. Van Vechten and Bergstresser3 employed the dielectric model of Phillips to evaluate this effect Generally speaking the valence band maximum will be pushed upward,while the conduction band minimum will be pushed downward,giving a downward bowing to the energy gap. All the interband enervy gaps show downward bowing without exception, as far as we are aware. For a given alloy system E~
I.
Vol. 13, No. 9
(F15 F’15) gap show a bowing which is very much smaller than forE 0 (F15 F’1 ).~‘This might indicate that the s-like conduction band (F’1) is more strongly affected by fluctuation than p-like (F~)conduction band, which is plausible in view of the localized nature of the potential fluctuation. With the expressions of slope 0 and x = Itowhich we derived for the VCA, at it xis =now possible evaluate theoretically the bowing due to the effect of the virtual crystal. Such an attempt is underway. It will also be of considerable interest to measure, by photoemission studies or soft X.ray emission studies for example, the variation of valence band maximum as a function of composition for some of the alloy systems. This will allow a more direct comparison with some of our conclusions, in particular the different behaviour of the two bands across the gap. ~
,
REFERENCES ThOMPSON A.G., CARDONA M., SHAKLEE K.L. and WOOLLEY J.C.,Phvs. Rev. 146,601 (1966). WOLLEY J.C.,ThOMAS MB. and THOMPSON A.G.,Can. J. Phi’s. 46,157(1968).
2. 3.
ONODERA Y. and TOYOZAWA Y.,J. Phys. Soc. Japan 24, 341 (1968). VAN VECHTEN J .A. and BERGSTRESSER T.K., P/n’s. Rev. 1,3351(1970).
4.
NEUBERGER M., Handbook ofElectronic Materials Vol. 2, Plenum Press, New York (1971).
5.
VAN VECHTEN J.A., BEROLO 0. and WOOLLEY J.C.,Phvs. Rev. Lett. 29, 1400 (1972).
6. 7.
PARMENTER R.H.,Phys. Rev. 97,587 (1955). WOOLLEY J.C., Private communication.
La variation des gaps d’énergie dans un système d’alliage semiconducteur est étudiée dans le cadre de l’approximation du cristal virtuel. L’expression exacte de la pente d’énergie en fonction de la composition x est donnée pour x = 0 et x = 1. Ces quantites peuvent ëtre évaluées théoriquement avec les calculs de bandes des deux matériaux conslituant l’alliage. Nous discutons de quelques conclusions qui sont tirées de ces expressions de la pente.
Pergamon Press.
Printed in Great Britain
ON THE VARIATION OF ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS KS. Song Physics Department, University of Ottawa, Ottawa, Canada (Received 7 June 1973 by A.A. Maradudin)
The departure from linearity of energy gaps in a semiconducting alloy system is studied within the framework of virtual crystal approximation. Exact expressions for the slope of energy variation as function of composition x at the two ends x = 0 and x = I are derived. These can be evaluated theoretically with energy band calculation of the two pure materilas for a system. We discuss here several features of the slope.
INTRODUCTION
SLOPES AT x =0,1 WITHIN THE VCA
THE VARIATION of energy gaps in alloys systems of zinc-blend type semiconductors has been studied extensively.1 Usually they show a more or less pronounced departure from linear variation as function of cornposition x between the two pure materials. This socalled bowing effect, which is almost always downward, has been studied by several authors.23 The usual approach in treating this problem is to study the energies in the virtual crystal approximation (VCA) and consider the configurationally averaged fluctuation of potential as a perturbation. In the previous works the contribution of these two parts to the bowing was not satisfactorily handled due to either the simplicity of the model2 or the nature of the model employed.3
The valence electron wave functions in the semi. conductors we are interested in are very entended in the region between the atoms and the local fluctuation of potential in an alloy will not affect the energy levels drastically. This is the reason why the VCA is reasonable for these materials. Onodera and Toyozawa2 have shown that in these materials with large bandwidth the energy bands belong to the so-called amalgamation type upon alloying. It is also reasonable, then, to treat the fluctuation by the perturbation method. The virtual crystal potential is given by the following expression; V(x)
=
xV 0(a(x)) + (I —x) V~(a(x))
(I)
It is hence desirable to be able to evaluate exactly at least one of the two contributions to the bowing. where VA(a(x)) and VB(z(x)) are the potentials of the pure materials A and B respectively with their lattice parameter equal to a(x). The lattice parameter a changes usually as a function of composition x following Vegard’s law. The influence of the change of a is explicitly included in this study. The Hamiltonian in the VCA can now be written as;
In this note we present expressions for the slope of the energy gap E,(x) at x = 0 and x = I which are exact within the framework ofVCA. The slopes at x = 0 and x = 1 can be calculated with reliableband calculation of the two pure materials. We examine how these slopes deviate from the straight line between the pure materials. The effect of fluctuation is more difficult to study quantitatively. We will discuss briefly the effect of this on the VCA results.
2
H (x) = P /2,n + V(x) 2 2/2tn + V = P 8(a(x)) + x EVA (a(x)) Va~a(x))] 2/2m+ VA(a(x))+(l —x)[VB(a(x))— VA P ~a~x))] (2’) —
1397
1398
ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS
We now propose to calculate the energy levels of H°(x)near x = 0 (and x = 1) exactly to the first order in x (and x’ I x). As there is an obvious symmetry betweenx=Oandx= 1 points,we present the result
Vol. 13, No.9
Here a typical element after differentiation in x looks like;
—
only near x =0 point (pure B material). We asswne that the eigenvalue problem with V~(a(x))is solved; 212m+ VB(a(x))I ‘I’a,(a(x)) = [P (3) EBI (a(x))
ijl~,
(a (x))
H~,—E
+ V11((x))+x
=
~-~-‘—
dx
~
dx (6’) Call the energy level we studyE’ which goes toEBl as x -+0. Then putting x =0 and E = E 81 into equation (6) we obtain;
The eigen functions ~‘~~‘s constitute a complete set and we expand the unknown wave., function of the VCA hamiltonian (2) in this basis. As the virtual crystal hamiltonian (2) has the same symmetry elements and the same lattice paramerer sa V8(a(x)), we need only function of the same symmetry in the expansion of the unknown wave function, of a given symmetry.
dx
+ V11 (0)
—
dx
dx
(E~2 E’) (EB3 —
—
E’)...
+(dEB2
dE’ + V22(O) 1)(E —E
(EBI =
~
C(x)1, I,IIBI (a(x))
83—E’)...
(4) (7)
The coefficients C(x)~,are determined by solving the following secular determinant;
II H~, E 6.•,~ —
=
(5)
0
where H,,
=
1(0) = E If we realize that E 8~and EBI’S are all different (except for some very unlikely situation where two or more states of same symmetry are degenerate), the above equation reduces to;
EB,((x)) 61, + x
(5’)
V~((x)) =
f I/,e~(a(x))[VA(a(x))
—
+ V11(0)—
VB(a(x))J ijlaj(a(x)) dv
=
0
0,
hence
0
(8) As we are interested in the variation of E in first order of x, we differentiate the above secular determinant (5) in x. The result is a sum of determinants each of which is obtained after one of the lines is differentiated in x, the rest of the lines remaining unchanged: H~1—E’
H~2
H21
H22 —E
..
+
H11 —E
H12
H~1
H~2—E
..
dE —
dEBI =
dx
—
dx
+ V11 (0)
This is the slope at x = 0 of the energy level which goes to ~ for pure B material. Experimental measurements usually study the variation of interband energy gaps,instead of that of individual energy levels. It is thus necessary to follow the variation of two optically connected energy bands and take the difference between them. The corresponding expression of the slope of energy gap variation is easily obtained and is as follows:
(6) ~-
dx
=~—~1 +V(0)
dx
0
(9)
ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS
Vol. 13, No.9 where =
dx
~I
j
—
dx Jo
dx
1399
rewritten expression: equations (10—11). As the actual slope will always be compared to the straight line between the two pure materials, we examine the three correction terms of equations (10—11).
I 10
=
V~(0) V,(0)
=
< 4i~(as)lVA(aB)— V3(a3)J ‘1’3~(aa))
—
(*BV(aB
—
)I VA (a3)
—
V3(a3) I !PBV(aB)> (9’)
The above equation can be slightly rewritten in such a way that different terms can be more readily analyzed. + ~(°) dx =
one. It represents the gap variation of the other pure
.~.f
.
da
(i) the first term (dE/da da/dx) represents the variation of energy gap of pure material A (or B) as its lattice constant is changed upon alloying with B (orA). This quantity can be calculated theoretically as it has been the case in some band calculations. More easily this can be obtained from experimental measurements of pressure coefficicns1of the energy gap, together with the compressibility of the pure material. Usually the direct gaps of zinc-blend type materials 4 (ii) The third termisiscompressed. very analogous to the first increase as the crystal
+ < EA~(aBPB EBg
material latticeseen constant is being modified alloying. as It its is easily that the two terms (firstupon and
—
dx~
0 =
~
+
.~
dxIo
da
EAg(aB))B
+ [EA~(aB) EAg] + [EAU —
EA,(aB)]
E3g]
(10)
As is clear from equations (9’)—(lO) (EAg(aa))B is the expectation value for the energy gap of A calculated with wave function of B at lattice constant of B.EA,(aB) is the energy gap of A at lattice constant of B. Similar expression is obtained for x = I point;
dx
=
da
dx
I
I
+
V,(1)
1
+ [(E,8~(a~))A
—
+ [EB,(aA) —E3,] + [EB,~EA,J
E3~(aA~ (II)
The theoretical slope dEg/dx at x = U and x = I can be evaluated using expressions given in equation (9’). To this end we need band calculations of pure materialsA and B at two different lattice constants. Such an attempt is being undertaken. This will allow us to separate the effect of the virtual crystal potential from that of potential fluctuation in the observed sowing of gap variation. In the present note we analyze briefly the influence of the different terms as given in
third) contribute in opposite direction and as they have the samesmall. order of magnitude. the net effect may be relatively (iii) The second correction term of equations (10—11) is one which can only be evaluated theoretically We will present here arguments which indicate that this term will more likely be negative. Variational methods in quantum mechanics tells us that if a state “i” is the ground state of the system (lowest state of a given symmetry in our case), the expectation value (EAI(aBP.B is always higher than the exact eigenvalue EA1(aB). If “I” is an excited state, the proposed variational function ~8j should be orthogonal io all lower states in order that the same statement holds. For a valence band this argument probably applies, as the core states are usually very deep (except when there is a nearby d-band) and little overlap will result between core and valence wave functions. This holds both at x = 0 and x = I. It follows that the second correction term will then tend to make the valence band variation bow upward. A conduction band, however, has several lower states (valence bands) of the same symmetry in the near vicinity. It is then unlikely that the conduction band variation will show large bowing due to the second term of equations (10—Il). Analogous considerations can be made to study the spin-orbit splitting of the valence band in a system of alloys. Expressions for the slope of splitting similar to equation (9) are then obtained. There is a departure
1400
ENERGY GAPS IN SEMICONDUCTING ALLOY SYSTEMS
from simple linearity even within the VCA, in contrast 5 to assumptions made by some workers. The slope evaluated from above expressions cannot be directly compared to the experimental data, as the effect is difficult to evaluate quantitatively. General effects of such local fluctuation havefunction been studied by 2 by Green’s method Onodera and Toyozawa with a simple model Hamiltonian. Parmenter6 treated this effect by perturbation method with free electron model. Van Vechten and Bergstresser3 employed the dielectric model of Phillips to evaluate this effect Generally speaking the valence band maximum will be pushed upward,while the conduction band minimum will be pushed downward,giving a downward bowing to the energy gap. All the interband enervy gaps show downward bowing without exception, as far as we are aware. For a given alloy system E~
I.
Vol. 13, No. 9
(F15 F’15) gap show a bowing which is very much smaller than forE 0 (F15 F’1 ).~‘This might indicate that the s-like conduction band (F’1) is more strongly affected by fluctuation than p-like (F~)conduction band, which is plausible in view of the localized nature of the potential fluctuation. With the expressions of slope 0 and x = Itowhich we derived for the VCA, at it xis =now possible evaluate theoretically the bowing due to the effect of the virtual crystal. Such an attempt is underway. It will also be of considerable interest to measure, by photoemission studies or soft X.ray emission studies for example, the variation of valence band maximum as a function of composition for some of the alloy systems. This will allow a more direct comparison with some of our conclusions, in particular the different behaviour of the two bands across the gap. ~
,
REFERENCES ThOMPSON A.G., CARDONA M., SHAKLEE K.L. and WOOLLEY J.C.,Phvs. Rev. 146,601 (1966). WOLLEY J.C.,ThOMAS MB. and THOMPSON A.G.,Can. J. Phi’s. 46,157(1968).
2. 3.
ONODERA Y. and TOYOZAWA Y.,J. Phys. Soc. Japan 24, 341 (1968). VAN VECHTEN J .A. and BERGSTRESSER T.K., P/n’s. Rev. 1,3351(1970).
4.
NEUBERGER M., Handbook ofElectronic Materials Vol. 2, Plenum Press, New York (1971).
5.
VAN VECHTEN J.A., BEROLO 0. and WOOLLEY J.C.,Phvs. Rev. Lett. 29, 1400 (1972).
6. 7.
PARMENTER R.H.,Phys. Rev. 97,587 (1955). WOOLLEY J.C., Private communication.
La variation des gaps d’énergie dans un système d’alliage semiconducteur est étudiée dans le cadre de l’approximation du cristal virtuel. L’expression exacte de la pente d’énergie en fonction de la composition x est donnée pour x = 0 et x = 1. Ces quantites peuvent ëtre évaluées théoriquement avec les calculs de bandes des deux matériaux conslituant l’alliage. Nous discutons de quelques conclusions qui sont tirées de ces expressions de la pente.