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A variational green function approach to the study of a vacancy and a substitutional impurity
Physica 45 (1970) 575-586
A VARIATIONAL
o North-Holland Publishing Co., Amsterdam
GREEN
OF A VACANCY
FUNCTION
APPROACH
AND A SUBSTITUTIONAL
P. GOOSSENS*
TO THE STUDY IMPURITY
and P. PHARISEAU
Laboratorium VOOY Kristallografie en Studie van de Vaste Stof, Rijksuniversiteit Gent, Belgit? Received 20 May 1969
Synopsis In this paper, the variational Green function method of Kohn-Rostoker, which is one of the most frequently used approaches in the energy band theory of perfect crystals, has been extended to a crystal containing a strongly localized perturbation. A vacancy and a substitutional impurity are discussed. The main difficulties arise from the determination of the appropriate Green function. The method discussed in this paper has the advantage of being applicable for practical calculations.
1. Introdzlction. So far, the study of the impurity states associated with a strongly localized point-perturbation, has been based upon formal arguments. Indeed, in the most important works?-s), the wave function, describing the perturbed system, is expanded in a complete set of known basis functions, the coefficients of which are determined by an infinite number of linear and homogeneous algebraic equations. In order for these theories to be useful for practical applications, plausible approximations have to be introduced, so that the secular equations may be reduced to a limited number. The method using the Wannier functionss-6) is the most currently used procedure. However, in actual cases, the calculation of these functions is a very difficult mathematical problem79 s s), since the Wannier functions are defined as the Fourier transforms of the Bloch waves. To avoid these complications we have developed a totally different mathematical technique for dealing with strongly localized impurity states. In this paper we have extended the variational Green function theory of Kohn and Rostokerla), which in the energy band theory of perfect crystals has already proved to be very efficient for practical applications. Here we have restricted ourselves to the study of a vacancy and a substitutional impurity. The more complicated problem of an interstitial impurity will be treated in a subsequent paper. * Aspirant of the “Nationaal
Fonds voor Wetenschappelijk 575
Onderzoek”, Belgium.
P. GOOSSENS AND>P. PHARISEAU
576
Fig. 1. Muffin-tin model of a substitutional impurity with b > a.
2. Determination of the variational functional. In order to take full advantage of the variational Green function techniquelo), we confine ourselves to a crystal field which may be described by a potential function of the muffin-tin type. Further we will ignore the effects produced by the relaxation of the atoms surrounding the imperfection. In fig. 1 this model is illustrated for a particular case of a substitutional impurity. The one-electron Schrijdinger equation, of an electron moving in a crystal with a strongly localized point defect, may be written as: (A + E -
V(r) -
V,(r)) Q(r)
=
0,
(1)
in which the function V(r) represents the potential corresponding unperturbed crystal field, and V,(r) is the localized perturbation Hence V(r) is of the form:
V(r) = C v,(r - q), i
II
Vu(P)= Vu(P)
(P
vu(p) = 0
(P2
< 4;
On the other hand the function 1. in the case of a vacancy =
-va(r
-
ri);
(2)
.,
where rf is a lattice vector and the functioti”v&3) is a so-called potential corresponding to a host crystal atom, defined by:
V,(r)
to the term.
(3)
a). V,(r)
atomic
’ ,_\,” ki given as:
‘.-I
(4)
GREEN FUNCTION APPROACH TO SUBSTITUTIONAL 2.
IMPURITY
577
in the case of a substitutional impurity V,(r) = ?@,(r -
ra) - &~(r- ri).
(5)
these formulas the vector ~2indicates the position of the point defect. We notice that in both cases ra is a lattice vector. Further the function Wb represents the atomic potential associated with the impurity atom, defined as: In
wb@) = w&‘)
(P <
b);
wb(p)
(P 2
4.
=
0
(6)
In order to transform the partial differential equation (1) into an equivalent integral equation we introduce the Green function G(rIr’) which satisfies the partial differential equation : (A + E -
V(r)) G(r(r’) = d(r - r’)
(7)
and represents an outgoing wave at infinity. Assuming the normalized Bloch waves (2x)-* !P*(k, r) to be known, this Green function may be immediately given as: G(rI’+‘) = 1-Z (2X)3 R
s
dk
y’,(k, r) !JJ;(k, r’) E - E,(k)
(8)
’
B.Z.
where the sum over the bandindex n is taken over all permitted energy bands, and the integration over the wave vector k is carried out over the first Brillouin zone. As we are concerned with the study of the extra energy states, the partial differential equation may be converted into the homogeneous integral equation : Q(r)
=
j
dp’G(r
I ri
+
P’)
J,‘s(Q
+
P’)
@@a
+
P’).
(9)
P’
In this expression the integral is taken over the sphere with centre in the defect-point and with a radius equal to a quantity c which in the various cases is defined by: 1. case of a vacancy and a substitional impurity with a 2 b c = a;
(10)
2. case of a substitutional,impurity with a < b c = b.
(11)
Following the computation‘technique developed by Kohn and Rostoker 10) the integral equation (9) may be replaced by the equivalent variational principle : i w
= 0,
Y,: (12)
P. GOOSSENS AND P. PHARISEAU
578
with : r = lim J dp@*(ra + P) &(Q E-+fO p
+ P) *
+ p In + P’) L(ri
+ P’) @(rt -I- ~‘1)~ (13)
in which @*(t-a + p) is to be varied. The main interest of introducing this variational procedure is, that using a trial function with error of the order E, the error on the calculated energy values will be of the order ~2 (see ref. 10). Before computing explicitly the variational fonctional r, we bring eq. (13) into a simpler form. Using the wave equation (1) and the partial differential equation (7), the volume integrals in (13) may be transformed into surface integrals. So, we get
*
@(ri + p’)
In this expression G(n + plri
WP
1~')
w
-
G(PI P')
a@p-i + p’) aft
we also made use of the simplifying
*
(14
property: (15)
4 p’) = G(plp’)
of the Green function. Remark, that in the further computation of I’, it is obvious to choose the trial function as an expansion in spherical harmonics. So in the computation of the surface integrals it will also be advantageous, to expand the Green function in spherical harmonics. In the variational functional r (14)) 3. Exfiansion of the Greelz function. the Green function is wanted in the neighbourhood of the points p - p’, for which this function is singular. So we have to look for an appropriate expression of G(p 1p’), Which enables us to deal properly with these singularities. As to this problem, the different cases in which c is equal to a or equal to b need to be treated separately. Since the atomic potential function va(p) has Case 1: p-(p’
=
[
22 + 1 -yg-
(I -
spherical /ml)!
(I + bl) ! 1
harmonics *
Y&p)
Pimf(cos 0) eiq,
m-e defined as:
GREEN FUNCTION APPROACH TO SUBSTITUTIONAL
where the quantities
0 and q represent
IMPURITY
579
the polar angles of the vector p and
P/“’ is an associated
Legendre polynomial. The unknown expansion coefficients gl,, z’m’ in (16) will be determined in the following way. Substitution of the expression (16) into the partial differential equation (7), gives, after introducing spherical coordinates and using the properties of the spherical harmonics, a differential equation in gl,, lfrn* itself: (f
$(P”+j+ +
‘(’; ‘) gZm,Z'm'(P9P') >
E - zta(p) -
S(P L-
- P) dlZf drnrn#.
P2
(18)
If we assume that two linearly independent solutions of the homogeneous differential equation corresponding to (18) are known, the functions gl,, lcrn* can be determined by applying the method of Lagrange. Let Rl(p) and Sl(p) be two such functions and suppose Rz(p) to be regular in the origin. The function glm, I’m’ will be given by
s
( dzlg:p
glm,z~m~(p,p) =
dP WP)
-
S(P -
P’) +
+CdZZ’2’S dp Wp)
S(P -
Cl(P')
Rz(p) + >
(19)
SZ(P).
P’) + CZ(P’)
B
In this equation 01,B, Cr(p’) and Cs(p’) are integration constant w is equal to: w = p2[Rz,
constants.
Szla,
/l(P)
y
notation: dh (PI dp
-/z(p)
(
Since the Green function a > p, p’;
B <
> p=Cl
(21)
.
is regular in the origin, (Yand @ may be chosen as: P, P’.
(24
Taking into account that G(pIp’) is a symmetric function using (‘19) and (22), the expansion (16) can be written as: p
the
(20)
in which we used the shorthand [fit fzla =
Further
of p and p’ and
a; G(p 1~‘) =
+
If E
SZlfdmrn8 W
Rz(P)
F 5
Bzm, z*rn’RzCp) Rz,(P’)
kzm(P) Y&l
SZ~P’) >
(P’).
+
(23)
P. GOOSSENS AND P. PHARISEAU
580
where the quantities j3znz,1’ms are constants, which may easily be computed using an alternative expansion of the Green function. Indeed, in the region p < a, the normalized Bloch functions (2x)-* YY,(k, r) occurring in the expression (8) for the Green function, can be expanded in spherical harmonics : ~
yn(k P) = ii l=O
2
Czm&(k p) Yzm(p).
(24)
m=-1
In this formula the function
RF(k, p) is a solution of the radial equation:
+
G(P) -
which is regular in the origin. Thus substitution
)
En(k) R?(k P) = 0
(25)
of (24) in (8) gives
p, p’ < a; 1
G(PIP’) = -YZ (2n)3
n
s
dkCZEZ* 1 m
I’m’
B.Z.
.
ct,c;,Jq(k, P) RWc P') E - En(k)
Using this expression are defined as:
YZm(P)yGm’(P’)-
of the Green function,
the constants
(26) /l~~,l’~* in (23)
p
1 Blm, l’m’
=(2~)~
1 h(p)
Rv(P’)
X
dk CrmCim,R?(k* P) RF(k,
11 s
P’)
E - En(k)
-
B.Z.
-
allrfirnrn’ Iv
SZ(p’)
As a consequence of the symmetry property constants are connected through the relation Blm, l’m’ =
81p-m’,
(27)
Rl(p’)’
1-m.
of the Green function
these
(28)
So the formula (23) gives the desired expansion of the Green function inside the sphere with a radius equal to a. Case 2: a < p < p’ ( b. The Green function will be determined in this region, starting from the known expansion (23) and taking into account the continuity properties of this function. First of all G(p [ p’) is computed in the region defined by the inequalities p < a < p’ < c. Inside this region the Green function satisfies, respectively, as a function Ibf p and of p’ the homo-
GREEN
FUNCTION
geneous differential (4
+ E -
(4
APPROACH
TO SUBSTITUTIONAL
IMPURITY
581
equations :
dp))
+ E) ‘+I
G(p I P’) = 0;
(29)
P’) = 0.
(30)
Thus we may write: p
1P’) f
=
?
E
F
2
(wm,
Wm, t*rn*&(p)
z*mgRz(p)
‘%‘(KP’))
il+P’)
Ylm(P)
+
Y&~(p’)-
(31)
The functions i&z) and nr(x) are the spherical Bessel- and spherical Neumann functions of order 1, the quantity K being defined as: K=&??
E20;
,=i+E
E
(32)
The unknown constants prm, r’7n’ and vrm,r*m* in (31) are computed in function of the constants /3lm,r’m’, expressing the continuity of G(p 1p’) and i3G(p 1p’)/i3p’ on the surface of the sphere p’ = a. Hence, taking into account (23) and (31) we get: /Jlrn, l’m’ =
Ka2
Blm, z~rn*[Rl*, wla
(33)
+
(34) On the other hand, in the region a < p I the inhomogeneous equation : (A + E) G(P~P’) = S(P Thus,
p‘ < b the Green function satisfies
P’).
G(p 1p’) can be expanded
(35)
as
a
1P’)
=
F
$
$ +
z,(fltrn)
Z’rn’il(KP)
$-
(EZm,Z’m’
Kh!dmm’)
+
TZm, z’m’%(KP)
iz’(KP’)
+
czrn, Ztm@l(Kp)
fiZf(KP’))
iZ(KP)
iV(KP’)
-k
flZ’(Kp’)
f
+
yirn(P)
Y&*(/f),
(36)
in which the expansion coefficients Orm,r*rn’, Elm,rsm*, qrm,r’m’ and crm r’m* will be determined as functions of the constants plm, z’~ and vrm, r’m’, using the continuity properties of the Green function upon the sphere p = a. Thus taking into account the formulas (31) and (36) for the Green function
P. GOOSSENS AND P. PHARISEAU
582
and using the relations (33) and (34) we finally get these constants as functions of the constants &nt,l~m~
(37)
+
hg6mrnJ
W
3
[SZ,nila ;
(38)
(39)
We conclude that in the actual cases, the difficult problem of determining the Green function G(p 1p’) is reduced to the computation of the constants Btrn,l*rn*-
4. The extra energystates atid corresfiondingeigenfwctions. We first need to compute explicitly the variational functional r defined in (14). This implies the choice of an appropriate trial function in the various considered cases of point impurities and the use of one of the expansions (23) or (36) of the Green function. The application of the variational principle enables us to compute the extra energy states and corresponding eigenfunctions. A. Case of a vacancy. Taking into account the definition (lo), the integrations in the formula (14) are to be carried out respectively over the surfaces with a radius a - 2~ and a - E.In this region the potential function V(r) - V,(r) in th e wave equation (1) is equal to zero. So it is obvious to use a trial function of the form: p <
a;
@(rg + p) =,i
m ;=,dtm iz(Kp)_Ylm(P)~
(41)
in which ZP represents a finite number. Substitution of (23) and (41) into
GREEN
FUNCTION
APPROACH
TO SUBSTITUTIONAL
583
IMPURITY
(14) gives
The variational principle (12) prescribes that the coefficients &, in (40) must be equated in such a way that the functional I’ in (42) remains stationary under variation of d;Y. This requirement gives rise to a set of (ZP + 1)2 linear and homogeneous algebraic equations in the unknown coefficients dzm:
The exclusion of the trivial zero solution, leads to the determinantal
equation :
(44 which determines the possible extra energy levels of the system. The eigenfunctions corresponding to these energy states, may be computed using the formulas (41) and (9). After calculation of the constants dz, from the set (42), the expansion (41) gives @(rc + p) inside the sphere p = a. For arguments lying outside this region, the integral equation (9) can be transformed into the simpler form: p >
a;
aw, + ~7 ap,
WP IP') -
ap,
p’=a-6
Taking into account the expansions
(8), (24) and (41) we get the formula:
P > a;
(2;j3 ,$mi’_plm C dk Y& ‘) 12s E -E,(k)
@(r$+ p) = -___
G,X(k),
ida
(46)
B.Z.
which gives the eigenfunctions of the perturbed problem as functions of the Bloch waves. Henceforth we shall no longer be concerned with the wave functions, as their knowledge is of lesser importance than that of the extra energy levels, and as in the other cases they may be computed in a totally* analogous way. B. Case of a substitutional impurity with b = a. In this case the trial functions can be chosen as: P <
b;
@(n
+
P)
=
2 Z=O
;: m=-1
&J-~(p)
Yz,(P)>
(47)
P. GOOSSENS
584
AND
P. PHARISEAU
in which Tz@) is an in the origin regular solution of the second order differential equation :
(-+$(pZ$)+~E-rvl(p) - I’l; +(P) =0.
(48)
Taking into account (lo), the variational functional has to be computed again using the formula (23). Hence the results may be directly deduced from the preceding case. Replacing jr by Tz in (44) we get the secular equation :
(4% It is interesting to mention that Dupreeii) previously investigated the crystal model considered here. This author was concerned mainly with the study of the energy levels inside the allowed energy bands. The condition he obtains for the existence of a localized energy state may easily be proved to be equivalent to the determinental equation (49), if the number ZP approaches + 90. C. Case of a substitutional with b > a. In view of (ll), we now have to consider the expansion (36) of the Green function. Thus, using (47), (36), (14) and (12) the supplementary energy states are now determined by the condition :
D. Case of a substitutional impurity with b < a. In this case, we have to find an accurate expression for the wave function, in the region just outside the muffin-tin sphere associated with the impurity atom. Taking into account the continuity of @(ri + p) and SD(rd + p)/+ on the surface p = b, and using the expansion (47) for the wave function where we take 2, -+ + co, inside this sphere, we may write that: b
@trZ-k P)
=,go mg (
:Zm il(KP)
-
PI, jllb LTl
nYtllb
nZ(KP)
ydP)p
(51)
I
where elm represents a new set of expansion constants. In the choice of the trial function, this expansion is broken off after a finite number of terms. So, using this trial function and the expansion (23) of the Green function the
GREEN
FUNCTION
APPROACH
TO SUBSTITUTIONAL
IMPURITY
585
secular equation will be given by:
B~rn,lprn*
det
o =,
(52)
+ which defines the extra energy levels of the considered
system.
5. Discussion and con&&on. The dependence of the extra energy levels upon the different parameters of the crystal lattice, may apparently be read from the different obtained secular equations. First of all we remark that the atomic potential function zIa, associated with a host crystal atom, appears in the quantities RJ, Sl and @cm,r#ml.As may be deduced from (27) these last constants also depend upon the crystal structure. The parameters of the imperfection enter implicitly in the function Tz and explicitly through the constant b. Further we observe that the vector rg, indicating the position of the point defect does not enter into the results. Thus, by means of this variational Green function technique, the difficult problem of solving the Schrodinger equation has been converted into the simpler one of solving a limited number of linear algebraic equations. Moreover, contrary to the existing theories using the Wannier functions43 596) the variational Green function method seems to be very powerful for practical application. Indeed, it is now possible to use directly the Bloch-waves in the computation of the results, instead of first passing to the Fourier transforms of these functions. In practice this means firstly, a considerable gain in calculation time and an increase in correctness of the results. In the evaluation of the secular equations, the functions RI, Sz and TZ may be obtained by numerical integration. The calculation of the constants Bzrn,~1~’ is more complicated. These are defined namely as a sum in which all the existing allowed energy bands have to be considered, in practice it will thus be impossible to determine exactly these quantities. Fortunately, as appears from the definition (27), the terms corresponding to the bands lying nearest to the examined extra energy level play a predominant role. So that approximately only these terms have to be considered. To illustrate this more visually, we have examined numerically the convergence of the Green function method, by applying it to a simple linear modelre). From the numerical data, obtained in this paper, we may conclude that the Green function method gives very accurate results, under the condition that in the sums over the bandindex a sufficient number of terms are considered. Another interesting feature of this method is, that the proposed energy band approximation may be proved to be equivalent with the band approximation, introduced in the generalized Koster-Slater theorys).
586
GREEN
FUNCTION
APPROACH
TO SUBSTITUTIONAL
IMPURITY
Acknowledgement. We wish to thank Professor Dr. W. Dekeyser for his continuous interest and discussions.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
Adams, E. N., J. them. Phys. 21 (1953) 2013. Luttinger, J. M. and Kohn, W., Phys. Rev. 97 (1955) 869. Slater, J. C., Phys. Rev. 76 (1949) 1592. Koster, G. F., Phys. Rev. 95 (1954) 1436. Koster, G. F. and Slater, J. C., Phys. Rev. 95 (1954) 1167. Koster, G. F. and Slater, J. C., Phys. Rev. 96 (1954) 1208. Herman, F., Rev. modern Phys., 30 (1958) 102. Callaway, J. and Hughes, A. J., Phys. Rev. 156 (1967) 860. Callaway, J. and Hughes, A. J.. Phys. Rev. 164 (1967) 1043. Kohn, W. and Rostoker, N., Phys. Rev. 94 (1954) 1111. Dupree, T. H., Ann. Phys. 15 (1961) 63. Goossens, P. and Phariseau, P., Physica 45 (1970) 587.
A VARIATIONAL
o North-Holland Publishing Co., Amsterdam
GREEN
OF A VACANCY
FUNCTION
APPROACH
AND A SUBSTITUTIONAL
P. GOOSSENS*
TO THE STUDY IMPURITY
and P. PHARISEAU
Laboratorium VOOY Kristallografie en Studie van de Vaste Stof, Rijksuniversiteit Gent, Belgit? Received 20 May 1969
Synopsis In this paper, the variational Green function method of Kohn-Rostoker, which is one of the most frequently used approaches in the energy band theory of perfect crystals, has been extended to a crystal containing a strongly localized perturbation. A vacancy and a substitutional impurity are discussed. The main difficulties arise from the determination of the appropriate Green function. The method discussed in this paper has the advantage of being applicable for practical calculations.
1. Introdzlction. So far, the study of the impurity states associated with a strongly localized point-perturbation, has been based upon formal arguments. Indeed, in the most important works?-s), the wave function, describing the perturbed system, is expanded in a complete set of known basis functions, the coefficients of which are determined by an infinite number of linear and homogeneous algebraic equations. In order for these theories to be useful for practical applications, plausible approximations have to be introduced, so that the secular equations may be reduced to a limited number. The method using the Wannier functionss-6) is the most currently used procedure. However, in actual cases, the calculation of these functions is a very difficult mathematical problem79 s s), since the Wannier functions are defined as the Fourier transforms of the Bloch waves. To avoid these complications we have developed a totally different mathematical technique for dealing with strongly localized impurity states. In this paper we have extended the variational Green function theory of Kohn and Rostokerla), which in the energy band theory of perfect crystals has already proved to be very efficient for practical applications. Here we have restricted ourselves to the study of a vacancy and a substitutional impurity. The more complicated problem of an interstitial impurity will be treated in a subsequent paper. * Aspirant of the “Nationaal
Fonds voor Wetenschappelijk 575
Onderzoek”, Belgium.
P. GOOSSENS AND>P. PHARISEAU
576
Fig. 1. Muffin-tin model of a substitutional impurity with b > a.
2. Determination of the variational functional. In order to take full advantage of the variational Green function techniquelo), we confine ourselves to a crystal field which may be described by a potential function of the muffin-tin type. Further we will ignore the effects produced by the relaxation of the atoms surrounding the imperfection. In fig. 1 this model is illustrated for a particular case of a substitutional impurity. The one-electron Schrijdinger equation, of an electron moving in a crystal with a strongly localized point defect, may be written as: (A + E -
V(r) -
V,(r)) Q(r)
=
0,
(1)
in which the function V(r) represents the potential corresponding unperturbed crystal field, and V,(r) is the localized perturbation Hence V(r) is of the form:
V(r) = C v,(r - q), i
II
Vu(P)= Vu(P)
(P
vu(p) = 0
(P2
< 4;
On the other hand the function 1. in the case of a vacancy =
-va(r
-
ri);
(2)
.,
where rf is a lattice vector and the functioti”v&3) is a so-called potential corresponding to a host crystal atom, defined by:
V,(r)
to the term.
(3)
a). V,(r)
atomic
’ ,_\,” ki given as:
‘.-I
(4)
GREEN FUNCTION APPROACH TO SUBSTITUTIONAL 2.
IMPURITY
577
in the case of a substitutional impurity V,(r) = ?@,(r -
ra) - &~(r- ri).
(5)
these formulas the vector ~2indicates the position of the point defect. We notice that in both cases ra is a lattice vector. Further the function Wb represents the atomic potential associated with the impurity atom, defined as: In
wb@) = w&‘)
(P <
b);
wb(p)
(P 2
4.
=
0
(6)
In order to transform the partial differential equation (1) into an equivalent integral equation we introduce the Green function G(rIr’) which satisfies the partial differential equation : (A + E -
V(r)) G(r(r’) = d(r - r’)
(7)
and represents an outgoing wave at infinity. Assuming the normalized Bloch waves (2x)-* !P*(k, r) to be known, this Green function may be immediately given as: G(rI’+‘) = 1-Z (2X)3 R
s
dk
y’,(k, r) !JJ;(k, r’) E - E,(k)
(8)
’
B.Z.
where the sum over the bandindex n is taken over all permitted energy bands, and the integration over the wave vector k is carried out over the first Brillouin zone. As we are concerned with the study of the extra energy states, the partial differential equation may be converted into the homogeneous integral equation : Q(r)
=
j
dp’G(r
I ri
+
P’)
J,‘s(Q
+
P’)
@@a
+
P’).
(9)
P’
In this expression the integral is taken over the sphere with centre in the defect-point and with a radius equal to a quantity c which in the various cases is defined by: 1. case of a vacancy and a substitional impurity with a 2 b c = a;
(10)
2. case of a substitutional,impurity with a < b c = b.
(11)
Following the computation‘technique developed by Kohn and Rostoker 10) the integral equation (9) may be replaced by the equivalent variational principle : i w
= 0,
Y,: (12)
P. GOOSSENS AND P. PHARISEAU
578
with : r = lim J dp@*(ra + P) &(Q E-+fO p
+ P) *
+ p In + P’) L(ri
+ P’) @(rt -I- ~‘1)~ (13)
in which @*(t-a + p) is to be varied. The main interest of introducing this variational procedure is, that using a trial function with error of the order E, the error on the calculated energy values will be of the order ~2 (see ref. 10). Before computing explicitly the variational fonctional r, we bring eq. (13) into a simpler form. Using the wave equation (1) and the partial differential equation (7), the volume integrals in (13) may be transformed into surface integrals. So, we get
*
@(ri + p’)
In this expression G(n + plri
WP
1~')
w
-
G(PI P')
a@p-i + p’) aft
we also made use of the simplifying
*
(14
property: (15)
4 p’) = G(plp’)
of the Green function. Remark, that in the further computation of I’, it is obvious to choose the trial function as an expansion in spherical harmonics. So in the computation of the surface integrals it will also be advantageous, to expand the Green function in spherical harmonics. In the variational functional r (14)) 3. Exfiansion of the Greelz function. the Green function is wanted in the neighbourhood of the points p - p’, for which this function is singular. So we have to look for an appropriate expression of G(p 1p’), Which enables us to deal properly with these singularities. As to this problem, the different cases in which c is equal to a or equal to b need to be treated separately. Since the atomic potential function va(p) has Case 1: p-(p’
=
[
22 + 1 -yg-
(I -
spherical /ml)!
(I + bl) ! 1
harmonics *
Y&p)
Pimf(cos 0) eiq,
m-e defined as:
GREEN FUNCTION APPROACH TO SUBSTITUTIONAL
where the quantities
0 and q represent
IMPURITY
579
the polar angles of the vector p and
P/“’ is an associated
Legendre polynomial. The unknown expansion coefficients gl,, z’m’ in (16) will be determined in the following way. Substitution of the expression (16) into the partial differential equation (7), gives, after introducing spherical coordinates and using the properties of the spherical harmonics, a differential equation in gl,, lfrn* itself: (f
$(P”+j+ +
‘(’; ‘) gZm,Z'm'(P9P') >
E - zta(p) -
S(P L-
- P) dlZf drnrn#.
P2
(18)
If we assume that two linearly independent solutions of the homogeneous differential equation corresponding to (18) are known, the functions gl,, lcrn* can be determined by applying the method of Lagrange. Let Rl(p) and Sl(p) be two such functions and suppose Rz(p) to be regular in the origin. The function glm, I’m’ will be given by
s
( dzlg:p
glm,z~m~(p,p) =
dP WP)
-
S(P -
P’) +
+CdZZ’2’S dp Wp)
S(P -
Cl(P')
Rz(p) + >
(19)
SZ(P).
P’) + CZ(P’)
B
In this equation 01,B, Cr(p’) and Cs(p’) are integration constant w is equal to: w = p2[Rz,
constants.
Szla,
/l(P)
y
notation: dh (PI dp
-/z(p)
(
Since the Green function a > p, p’;
B <
> p=Cl
(21)
.
is regular in the origin, (Yand @ may be chosen as: P, P’.
(24
Taking into account that G(pIp’) is a symmetric function using (‘19) and (22), the expansion (16) can be written as: p
the
(20)
in which we used the shorthand [fit fzla =
Further
of p and p’ and
a; G(p 1~‘) =
+
If E
SZlfdmrn8 W
Rz(P)
F 5
Bzm, z*rn’RzCp) Rz,(P’)
kzm(P) Y&l
SZ~P’) >
(P’).
+
(23)
P. GOOSSENS AND P. PHARISEAU
580
where the quantities j3znz,1’ms are constants, which may easily be computed using an alternative expansion of the Green function. Indeed, in the region p < a, the normalized Bloch functions (2x)-* YY,(k, r) occurring in the expression (8) for the Green function, can be expanded in spherical harmonics : ~
yn(k P) = ii l=O
2
Czm&(k p) Yzm(p).
(24)
m=-1
In this formula the function
RF(k, p) is a solution of the radial equation:
+
G(P) -
which is regular in the origin. Thus substitution
)
En(k) R?(k P) = 0
(25)
of (24) in (8) gives
p, p’ < a; 1
G(PIP’) = -YZ (2n)3
n
s
dkCZEZ* 1 m
I’m’
B.Z.
.
ct,c;,Jq(k, P) RWc P') E - En(k)
Using this expression are defined as:
YZm(P)yGm’(P’)-
of the Green function,
the constants
(26) /l~~,l’~* in (23)
p
1 Blm, l’m’
=(2~)~
1 h(p)
Rv(P’)
X
dk CrmCim,R?(k* P) RF(k,
11 s
P’)
E - En(k)
-
B.Z.
-
allrfirnrn’ Iv
SZ(p’)
As a consequence of the symmetry property constants are connected through the relation Blm, l’m’ =
81p-m’,
(27)
Rl(p’)’
1-m.
of the Green function
these
(28)
So the formula (23) gives the desired expansion of the Green function inside the sphere with a radius equal to a. Case 2: a < p < p’ ( b. The Green function will be determined in this region, starting from the known expansion (23) and taking into account the continuity properties of this function. First of all G(p [ p’) is computed in the region defined by the inequalities p < a < p’ < c. Inside this region the Green function satisfies, respectively, as a function Ibf p and of p’ the homo-
GREEN
FUNCTION
geneous differential (4
+ E -
(4
APPROACH
TO SUBSTITUTIONAL
IMPURITY
581
equations :
dp))
+ E) ‘+I
G(p I P’) = 0;
(29)
P’) = 0.
(30)
Thus we may write: p
1P’) f
=
?
E
F
2
(wm,
Wm, t*rn*&(p)
z*mgRz(p)
‘%‘(KP’))
il+P’)
Ylm(P)
+
Y&~(p’)-
(31)
The functions i&z) and nr(x) are the spherical Bessel- and spherical Neumann functions of order 1, the quantity K being defined as: K=&??
E20;
,=i+E
E
(32)
The unknown constants prm, r’7n’ and vrm,r*m* in (31) are computed in function of the constants /3lm,r’m’, expressing the continuity of G(p 1p’) and i3G(p 1p’)/i3p’ on the surface of the sphere p’ = a. Hence, taking into account (23) and (31) we get: /Jlrn, l’m’ =
Ka2
Blm, z~rn*[Rl*, wla
(33)
+
(34) On the other hand, in the region a < p I the inhomogeneous equation : (A + E) G(P~P’) = S(P Thus,
p‘ < b the Green function satisfies
P’).
G(p 1p’) can be expanded
(35)
as
a
1P’)
=
F
$
$ +
z,(fltrn)
Z’rn’il(KP)
$-
(EZm,Z’m’
Kh!dmm’)
+
TZm, z’m’%(KP)
iz’(KP’)
+
czrn, Ztm@l(Kp)
fiZf(KP’))
iZ(KP)
iV(KP’)
-k
flZ’(Kp’)
f
+
yirn(P)
Y&*(/f),
(36)
in which the expansion coefficients Orm,r*rn’, Elm,rsm*, qrm,r’m’ and crm r’m* will be determined as functions of the constants plm, z’~ and vrm, r’m’, using the continuity properties of the Green function upon the sphere p = a. Thus taking into account the formulas (31) and (36) for the Green function
P. GOOSSENS AND P. PHARISEAU
582
and using the relations (33) and (34) we finally get these constants as functions of the constants &nt,l~m~
(37)
+
hg6mrnJ
W
3
[SZ,nila ;
(38)
(39)
We conclude that in the actual cases, the difficult problem of determining the Green function G(p 1p’) is reduced to the computation of the constants Btrn,l*rn*-
4. The extra energystates atid corresfiondingeigenfwctions. We first need to compute explicitly the variational functional r defined in (14). This implies the choice of an appropriate trial function in the various considered cases of point impurities and the use of one of the expansions (23) or (36) of the Green function. The application of the variational principle enables us to compute the extra energy states and corresponding eigenfunctions. A. Case of a vacancy. Taking into account the definition (lo), the integrations in the formula (14) are to be carried out respectively over the surfaces with a radius a - 2~ and a - E.In this region the potential function V(r) - V,(r) in th e wave equation (1) is equal to zero. So it is obvious to use a trial function of the form: p <
a;
@(rg + p) =,i
m ;=,dtm iz(Kp)_Ylm(P)~
(41)
in which ZP represents a finite number. Substitution of (23) and (41) into
GREEN
FUNCTION
APPROACH
TO SUBSTITUTIONAL
583
IMPURITY
(14) gives
The variational principle (12) prescribes that the coefficients &, in (40) must be equated in such a way that the functional I’ in (42) remains stationary under variation of d;Y. This requirement gives rise to a set of (ZP + 1)2 linear and homogeneous algebraic equations in the unknown coefficients dzm:
The exclusion of the trivial zero solution, leads to the determinantal
equation :
(44 which determines the possible extra energy levels of the system. The eigenfunctions corresponding to these energy states, may be computed using the formulas (41) and (9). After calculation of the constants dz, from the set (42), the expansion (41) gives @(rc + p) inside the sphere p = a. For arguments lying outside this region, the integral equation (9) can be transformed into the simpler form: p >
a;
aw, + ~7 ap,
WP IP') -
ap,
p’=a-6
Taking into account the expansions
(8), (24) and (41) we get the formula:
P > a;
(2;j3 ,$mi’_plm C dk Y& ‘) 12s E -E,(k)
@(r$+ p) = -___
G,X(k),
ida
(46)
B.Z.
which gives the eigenfunctions of the perturbed problem as functions of the Bloch waves. Henceforth we shall no longer be concerned with the wave functions, as their knowledge is of lesser importance than that of the extra energy levels, and as in the other cases they may be computed in a totally* analogous way. B. Case of a substitutional impurity with b = a. In this case the trial functions can be chosen as: P <
b;
@(n
+
P)
=
2 Z=O
;: m=-1
&J-~(p)
Yz,(P)>
(47)
P. GOOSSENS
584
AND
P. PHARISEAU
in which Tz@) is an in the origin regular solution of the second order differential equation :
(-+$(pZ$)+~E-rvl(p) - I’l; +(P) =0.
(48)
Taking into account (lo), the variational functional has to be computed again using the formula (23). Hence the results may be directly deduced from the preceding case. Replacing jr by Tz in (44) we get the secular equation :
(4% It is interesting to mention that Dupreeii) previously investigated the crystal model considered here. This author was concerned mainly with the study of the energy levels inside the allowed energy bands. The condition he obtains for the existence of a localized energy state may easily be proved to be equivalent to the determinental equation (49), if the number ZP approaches + 90. C. Case of a substitutional with b > a. In view of (ll), we now have to consider the expansion (36) of the Green function. Thus, using (47), (36), (14) and (12) the supplementary energy states are now determined by the condition :
D. Case of a substitutional impurity with b < a. In this case, we have to find an accurate expression for the wave function, in the region just outside the muffin-tin sphere associated with the impurity atom. Taking into account the continuity of @(ri + p) and SD(rd + p)/+ on the surface p = b, and using the expansion (47) for the wave function where we take 2, -+ + co, inside this sphere, we may write that: b
@trZ-k P)
=,go mg (
:Zm il(KP)
-
PI, jllb LTl
nYtllb
nZ(KP)
ydP)p
(51)
I
where elm represents a new set of expansion constants. In the choice of the trial function, this expansion is broken off after a finite number of terms. So, using this trial function and the expansion (23) of the Green function the
GREEN
FUNCTION
APPROACH
TO SUBSTITUTIONAL
IMPURITY
585
secular equation will be given by:
B~rn,lprn*
det
o =,
(52)
+ which defines the extra energy levels of the considered
system.
5. Discussion and con&&on. The dependence of the extra energy levels upon the different parameters of the crystal lattice, may apparently be read from the different obtained secular equations. First of all we remark that the atomic potential function zIa, associated with a host crystal atom, appears in the quantities RJ, Sl and @cm,r#ml.As may be deduced from (27) these last constants also depend upon the crystal structure. The parameters of the imperfection enter implicitly in the function Tz and explicitly through the constant b. Further we observe that the vector rg, indicating the position of the point defect does not enter into the results. Thus, by means of this variational Green function technique, the difficult problem of solving the Schrodinger equation has been converted into the simpler one of solving a limited number of linear algebraic equations. Moreover, contrary to the existing theories using the Wannier functions43 596) the variational Green function method seems to be very powerful for practical application. Indeed, it is now possible to use directly the Bloch-waves in the computation of the results, instead of first passing to the Fourier transforms of these functions. In practice this means firstly, a considerable gain in calculation time and an increase in correctness of the results. In the evaluation of the secular equations, the functions RI, Sz and TZ may be obtained by numerical integration. The calculation of the constants Bzrn,~1~’ is more complicated. These are defined namely as a sum in which all the existing allowed energy bands have to be considered, in practice it will thus be impossible to determine exactly these quantities. Fortunately, as appears from the definition (27), the terms corresponding to the bands lying nearest to the examined extra energy level play a predominant role. So that approximately only these terms have to be considered. To illustrate this more visually, we have examined numerically the convergence of the Green function method, by applying it to a simple linear modelre). From the numerical data, obtained in this paper, we may conclude that the Green function method gives very accurate results, under the condition that in the sums over the bandindex a sufficient number of terms are considered. Another interesting feature of this method is, that the proposed energy band approximation may be proved to be equivalent with the band approximation, introduced in the generalized Koster-Slater theorys).
586
GREEN
FUNCTION
APPROACH
TO SUBSTITUTIONAL
IMPURITY
Acknowledgement. We wish to thank Professor Dr. W. Dekeyser for his continuous interest and discussions.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
Adams, E. N., J. them. Phys. 21 (1953) 2013. Luttinger, J. M. and Kohn, W., Phys. Rev. 97 (1955) 869. Slater, J. C., Phys. Rev. 76 (1949) 1592. Koster, G. F., Phys. Rev. 95 (1954) 1436. Koster, G. F. and Slater, J. C., Phys. Rev. 95 (1954) 1167. Koster, G. F. and Slater, J. C., Phys. Rev. 96 (1954) 1208. Herman, F., Rev. modern Phys., 30 (1958) 102. Callaway, J. and Hughes, A. J., Phys. Rev. 156 (1967) 860. Callaway, J. and Hughes, A. J.. Phys. Rev. 164 (1967) 1043. Kohn, W. and Rostoker, N., Phys. Rev. 94 (1954) 1111. Dupree, T. H., Ann. Phys. 15 (1961) 63. Goossens, P. and Phariseau, P., Physica 45 (1970) 587.