- Email: [email protected]
The influence of exchange on the effective mass formalism
EVAN
0. KANE
General Electric Research Laboratory,
Schenectady,
New York
(RHx~~~~ 4 February 1958 ; revised 19 f”ebvlrary 1958) Absfra&--The &cc&e mass formalism of ABAM~ is generalized by the inclusion of exchange, The onfy difference is that the b%erband matrix elements in the eB&tive mass sums contain a term from the exchange operator in ad&&D to the usual term from the momentum operator. The exchange term is estimated theoretically arrd found to be significant. The interband matrix elements which cause optical absorption arise only from the momentnm operaior and not from the excba~ge crperaror. Hence a comparisun af matrix elements determined from absoiute q&al absorption with elements estimated from effective mass sums provides an experimental method of finding the exchange effect, in principle. For the cases investigated so hr, the exchange effect appears to lie within the uncertainties of the comparison. 1. INTRODUCTION
AND
CONCLUSIONS
THE effective mass formalism has been developed through the efforts of a number of authors.(l) A key feature of the formalism is that a great variety of experimental phenomena in the field of semiconductors can be explained in terms of a small number of constants, the effective masses. These efIective masses could, in principle, be calculated from first principles. At the present time, however, the greatest quantitative SWCC~SS~S of the theory have been in those substances for which cyclotron resonance has permitted an accurate experimental determination of the effective mass.(2) The influence of exchange on the effective mass theory was first treated by HEHUNG, However, in the later development of the theory the effect of exchange has generally been ignored. The present paper derives the effective mass theory with exchange,
f&ow&g
dOS&y
the
~~~~~0~~~~~
It
wilf
be
seen
that
2. EXCHANQE HAMILTONIAN STANT MAGNETIC
WITH FIELD
A
CON-
The Eamiftonian with the inclusion of exchange may be written symbolically
Of
introduction of exchange does not alter the formal appearance of the effective mass equation, but it does alter the interpretation of the ef3ective mass parameters. These parameters are no longer determined solely by the interband matrix elements of momentum, but contain a contribution from the exchange potential as well. This result indicates that attempts to deduce absolute values of optical absorption constants from effective mass measurements may P,mxs_~Q
be in error.(*) A crude estimate of the magnitude of the exchange term shows that it may be comparable with the momentum term. However, actual comparison between absolute optical absorption and effective mass suggests that exchange terms are not as large as n~omentum terms.+Q The present treatment, of course, makes the single particle ~F~~ox~mat~on. a recent paper by &xxN@f discusses the ef?ective mass approximation from a many-electron viewpoint which includes both exchange and correlation effects, At the present time KOHN'S theory has not been extended to include the effects of magnetic fields.
the
where V is the coulomb interaction and Z is the exchange interaction, In the usual r representatian, V is diagonal and Z is non-diagonal, having tha form
236
INFLUENCE
OF EXCHANGE
ON THE
EFFECTIVE
where the & are self-consistent electronic wave functions summed over all occupied states. In the same notation we should write V(rl, r2) =
=
= (3)
1 ’ T.
(5)
We observe the fact that e(i/fi)P*s is a displacement operator having the property e(t/fi)p.*#(r)
=
tJ(r+7).
(9) e(-ie/ch)(Al-A3.~Z(rl+7,r2+7)
(7)
where m runs over all degenerate states and smn is a unitary matrix. Then
* Strictly speaking these relations are valid only for the particular gauge chosen, namely A = fH x r. However, the Schroedinger equation resulting from the Hamiltonian (1) is known to be gauge-invariant, so that any physical deductions made with one choice of gauge will be valid for any gauge.
=
Z(rl,
r2).
Comparing equations (4) and (9), it is clear that Z(rl, r2) cannot be assumed to be the same as it was in the absence of a field. If we consider the operator W(rl, 12) defined by Z(rl, r2) = exp $A1-A2)
.
Wrl,
r2>, (10)
equation
(9) shows that we have u/(rl+T,
(6)
Further p-(e/c)A commutes with p+(e/c)A and p * T commutes with A - T. Using these facts, it is easily seen that S commutes with the first two terms of equation (l).* It can then be shown that it is self-consistent to assume that S commutes with Z. For if S commutes with H Sh = 2 h&n m
h(r$,h*(r2),
z
szs-l=
’
where T is any lattice vector. We now show that in the presence of a uniform magnetic field, 2 acquires a symmetry different from that given by equation (4). Consider the unitary operator S = & /JL){ p-k/c)A
hd*ln*m(n)*~*(rz)
since the summation over n contains all degenerate states. Also S commutes with r12. Hence S commutes with Z, which is the requirement of consistency. This result may be written
(4)
.
c c m
In the absence of a magnetic field, the potential and exchange terms both have lattice periodicity. That is
Z(r1+7, r2+7) = Z(r1, r2)
237
FORMALISM
I.m,n
W$(rl--2)
W1+7) = V(n)
MASS
r2+$
=
W(rl,
r2).
(11)
W(q, 12) and Z(rl, 12) become identical in the absence of a magnetic field. The presence of a small perturbation should not strongly affect the operator Z(rl, rs), because any perturbation which mixes only the filled states will leave Z invariant. A small change in Z will result from a small amount of band mixing. Equation (10) shows that a magnetic field cannot be regarded as a small perturbation in the ordinary sense no matter how small the field may be. This difficulty arises because (Hxr)/Z may become arbitrarily large. In a mathematical sense, infinite band mixing results. The definition of W’(rl, t-2)in equation (10) essentially extracts this gaugedependence and leaves a factor which may be assumed unchanged for small fields, in accord with the physical expectation that the field has only a very slight effect on the electron density. 3.
EFFECTIVE
MASS FORMALISM CHANGE
WITH
EX-
In treating the influence of exchange on the effective mass formalism, we follow closely the
238
EVAN
0.
method introduced by AmMS.(l~ This method is centered around the Bioch function representation of the Hamiltonian. The Bloch functions &k(r) = &%,k(r) diagonalize the Hamiltonian in the absence -of a magnetic field &AL@)
(12)
= E?@+&(r)
HO= (p2/2m)+ V+ W. In the presence of a magnetic tonian may be written
(13)
KANE
Al)Al\nS(r) gives the following expressions for the impoutant operators in the Bloch representation r=<+X (k’n’j?Jkn) (k’n’JX/kn)
= &,*V#G--k’) = &/,(k)G(k-k’) (18)
(k’n’JpJkn > = p,,,(k)?@--k’) (k’n’~Hojkn) = S,,,G(k-k’)&(k).
field, the Hamil-
(14) The brace ( Ii signifies the anticommutator, written in this case only for the sake of symmetry. The convention of summing on repeated indices is used throughout. We now expand the exponential in equation (10) in powers of A to give the result
ADAMS has also shown that the diagonal component XaS is independent of k and is zero in lattices possessing inversion symmetry. We need the following commutator relations
[y’, [y“, HollGz=
(19)
GE,
- dE+
In deriving the effective mass equations, we are interested in the region about a band maximum or ~nimum. For simplicity, let us assume the extremum to be at k = 0. If the band is nondegenerate, we can write Equation (15) may also be written notation as
in operator
It is easily verified by the use of equations f13) and (16) that equation (14) can be written in the form H= Ho+
(A/1, {A “, [rfi, [r “,Ho]]-))++
.... (17)
If HO does not include exchange, all commutators higher than the second vanish.
,
(20)
where the principal axes of the ellipse are taken as co-ordinate axes, and the rngn are effective mass constants. The technique for diagonalizing the Hamiltonian of equation (17) is to make a perturbation expansion removing interb~d matrix efements in successively higher orders in the magnetic field. An off-diagonal matrix element removed in a given order contributes to diagonal elements in higher orders. The intraband off-diagonal elements associated with the differential operator 5 cannot be removed in this way, since the expansion would not converge. The removal of interband matrix elements to a given order therefore results in a differential operator for each band, the bands being made independent to the desired order ofx, the magnetic field. The further diagonalization is then accomplished by solving the differential
INFLUENCE
OF
EXCHANGE
ON
THE
equation or by using some other appropriate technique. The-usual effective mass equation is derived to be correct to the first order in the magnetic field.* To eliminate higher-order terms from consideration, we consider all operators to be expanded in powers of k; k should then be considered to be of order .X1/a and ajak to be of order $P-l/a. We must then retain the following terms in equation (17):
-HS [S,HoI]-= gJ’(Afe, Using the relations given by equations
A+g, (
EFFECTIVE
MASS
H’ = e~Hf?-+ = H+[S,
FORMALISM
HI-+-&, JL:
239
[S, HI]-+
.. . (22)
[S, &I-
= (
;
1
{A%), [r/l, ~olkV]+n#n’.
The elimination of the off-diagonal term gives a diagonal term in the second order of S which is of order X.
P, ~Olnn*[~ “,Ho]&+ [Yy,Ho];?v[Yl”,H&n
+
?a,*n.
(&--En,)
(23)
(19), we can write H’ as follows: (24)
(k’n’lHlkn) +
= EnSnn2@--k’)+ {A%),
Equation
I
P, Hol,}+Sw+
{A’V)nw>
(24) can be written
H’ = t
P, Ho]&}+n#n’+
1
c21)
even more simply as
E,(kfi+$/(()),
(25)
which is easily shown to be true to second order in k 1‘f we use the symmetrical expansion of En
{Afw, {A%)> +
{A%),
(26)
En(k) =En(~)+k+$)o+tk~kv(;)o
P‘, P’, Ho]l;~)}+S,w+ [y“, Hol&v}+~#n’. i
The first four terms on the right-hand side of equation (21) are already diagonal in bands. All quantities in equation (21) are to be evaluated to lowest order in k. The last term is non-diagonal in bands and of order 21’2 according to our prescription. This term may be eliminated by the transformation * Terms in Hz are required for the calculation of the magnetic susceptibility. To develop a theory to order Hz, the change in the operators Y and W with magnetic field would have to be included.
and substitute kr+(e/ch)Ap( 5) directly into this expansion. Equation (25) shows explicitly that the formal nature of the effective mass equation is not changed by the inclusion of exchange. This is clearly a consequence of the fact that equation (17) could be written in a form which did not depend on whether exchange was included or not. Equations (19) give the following expressions for the effective mass, rnpv fi2 -zzzz rnry
lPE
-( ) dkpdk”
o
(27)
240
EVAN
fi2
-
mpy
= - [YP, p, Ho]]& -
w,
0.
KANE
ffol&&“,ffo],%+[r“,Ho]Gv[r~, Ho],,), Zn’ (h---w)
[rp, Ho]- = (ih/m)p+
p,
[TV,Ho]]- = - $/ty+[rP,
From these equations we see that exchange contributes to the effective mass through [rp, w];,,, and [YW, [YY, W]];n. 4. OPTICAL
[W, w]-
ABSORPTION
One of the results of effective mass theory which is sensitive to exchange is the comparison of optical absorption constants and effective masses.@) The Hamiltonian in the presence of a light wave is given by equation (1). A is the vector potential of the light which may be taken to be Aoe-iR’r+ + Ao*ebk*r. Unlike the case of the constant magnetic field, the vector potential does not become arbitrarily large for large values of r. Hence A may be considered as a small perturbation which will not affect the exchange operator Z(rl, t-2). The perturbation H, may be written
(29)
[IV, W]]-.
(30)
conduction bands in germanium@) and indium antimonide.(4) In this work exchange was neglected and higher bands of proper symmetry were assumed to be unimportant in the sum over bands of equation (28). The uncertainty in the comparison is such that it can only be claimed that exchange is less important than the ordinary momentum contribution to the effective mass. We can obtain a very crude estimate of the size of [YP, WI- which shows that it is not negligible compared to tpplrn. We approximate the exchange charge density -eL’t,bt(r$,h~*(r2) by a constant i
equal to the average charge density of electrons of parallel spin for ~12 < Y~Jand equal to zero for ~12 > ~0. The value of ro is given by -ep
4x 3y03P = l’
Hl-LA.p, mc
(31)
since A0 * k = 0. The absolute value of the optical absorption, therefore, gives the momentum matrix element. In the case of germanium and indium antimonide, the closely adjacent conduction and valence bands contribute a large share of the effective mass sum in equation (28). Reasonable agreement with the cyclotron masses has been found with the use of the optical matrix element between valence and
(k’, .‘I [YY,WI-l k, n > = +e2p
(32)
so that the total exchange charge is equal to -e. This approximation is based on arguments given by sLATERc7) and should be reasonably accurate. We further assume that the exchange hole is small compared to the size of the unit cell and to the scale of variation of the functions U,(Q). This approximation is clearly very bad, but it should give an order-of-magnitude estimate. With this approximation we can write u(r2) = u(r1) + (rz-r1). vu(rl) over the region of the exchange hole. We may then write
(y1P-yZP)~k,n.*(r1)(r1-r2)
fS
(28)
’
~#kn(rl) drld(rl--2)
Iu--f-21
(33)
INFLUENCE
OF
EXCHANGE
ON
THE
The integral of rs-rlis over a sphere ~12 < TO. The integral is easily performed to give Iy~,cyl=+[~yO](q.P’~.*
(34)
The importance of exchange in this approximation depends only on the electron density. For germanium the factor mesre/4tia has the value O-65. -, which indicates that the two terms may be comparable. With the same approximations as made above, the term [YP, [YY, WI]- has the value e2ro [YB,
[Y“,
w-J]- = - --&. 4
This term then, gives a contribution to the reciprocal effective mass of +0*65 reciprocal elec* Note added in proof: The author wishes to thank Dr. JAMESC. PHILLIPSfor correcting a sign in equations (33) and (34).
Q
EFFECTIVE
MASS
FORMALISM
241
tron masses in germanium. According to the oversimplified model used, this contribution would be thesameforallbands* Acknowledgements-The author would like to thank Professor M. COHEN for bringing this problem to his attention. He is indebted to Dr. F. HAM and Professor G. DRESSELHAUS for several very helpful discussions.
REFERENCES 1. WANNIER G. H. Phys. Rev. 52, 191 (1937). ADAMS E. N.J. Ch&. Phys. 2i, 2013 (1953). LUTTINGER1. M. and KOHN W. Phvs. Rev. 97. 869
(1955).
-
G., KIP A. F., and KITTEL C. Phys. 2. DRESSELHAUS Rev. 98, 368 (195.5). 3 HERRINGC. Phys. Rev. 52, 361 (1937). 4: KANE E. 0. 3. Phys. Chem. Solids 1, 83 (1956); 1, 249 (1957). 5. KOHN W. Phys. Rev. 105, 509 (1957). 6. KANE E. 0. Unpublished work. 7. SLATERJ. C. Phys. Rev. 81, 385 (1951).
0. KANE
General Electric Research Laboratory,
Schenectady,
New York
(RHx~~~~ 4 February 1958 ; revised 19 f”ebvlrary 1958) Absfra&--The &cc&e mass formalism of ABAM~ is generalized by the inclusion of exchange, The onfy difference is that the b%erband matrix elements in the eB&tive mass sums contain a term from the exchange operator in ad&&D to the usual term from the momentum operator. The exchange term is estimated theoretically arrd found to be significant. The interband matrix elements which cause optical absorption arise only from the momentnm operaior and not from the excba~ge crperaror. Hence a comparisun af matrix elements determined from absoiute q&al absorption with elements estimated from effective mass sums provides an experimental method of finding the exchange effect, in principle. For the cases investigated so hr, the exchange effect appears to lie within the uncertainties of the comparison. 1. INTRODUCTION
AND
CONCLUSIONS
THE effective mass formalism has been developed through the efforts of a number of authors.(l) A key feature of the formalism is that a great variety of experimental phenomena in the field of semiconductors can be explained in terms of a small number of constants, the effective masses. These efIective masses could, in principle, be calculated from first principles. At the present time, however, the greatest quantitative SWCC~SS~S of the theory have been in those substances for which cyclotron resonance has permitted an accurate experimental determination of the effective mass.(2) The influence of exchange on the effective mass theory was first treated by HEHUNG, However, in the later development of the theory the effect of exchange has generally been ignored. The present paper derives the effective mass theory with exchange,
f&ow&g
dOS&y
the
~~~~~0~~~~~
It
wilf
be
seen
that
2. EXCHANQE HAMILTONIAN STANT MAGNETIC
WITH FIELD
A
CON-
The Eamiftonian with the inclusion of exchange may be written symbolically
Of
introduction of exchange does not alter the formal appearance of the effective mass equation, but it does alter the interpretation of the ef3ective mass parameters. These parameters are no longer determined solely by the interband matrix elements of momentum, but contain a contribution from the exchange potential as well. This result indicates that attempts to deduce absolute values of optical absorption constants from effective mass measurements may P,mxs_~Q
be in error.(*) A crude estimate of the magnitude of the exchange term shows that it may be comparable with the momentum term. However, actual comparison between absolute optical absorption and effective mass suggests that exchange terms are not as large as n~omentum terms.+Q The present treatment, of course, makes the single particle ~F~~ox~mat~on. a recent paper by &xxN@f discusses the ef?ective mass approximation from a many-electron viewpoint which includes both exchange and correlation effects, At the present time KOHN'S theory has not been extended to include the effects of magnetic fields.
the
where V is the coulomb interaction and Z is the exchange interaction, In the usual r representatian, V is diagonal and Z is non-diagonal, having tha form
236
INFLUENCE
OF EXCHANGE
ON THE
EFFECTIVE
where the & are self-consistent electronic wave functions summed over all occupied states. In the same notation we should write V(rl, r2) =
=
= (3)
1 ’ T.
(5)
We observe the fact that e(i/fi)P*s is a displacement operator having the property e(t/fi)p.*#(r)
=
tJ(r+7).
(9) e(-ie/ch)(Al-A3.~Z(rl+7,r2+7)
(7)
where m runs over all degenerate states and smn is a unitary matrix. Then
* Strictly speaking these relations are valid only for the particular gauge chosen, namely A = fH x r. However, the Schroedinger equation resulting from the Hamiltonian (1) is known to be gauge-invariant, so that any physical deductions made with one choice of gauge will be valid for any gauge.
=
Z(rl,
r2).
Comparing equations (4) and (9), it is clear that Z(rl, r2) cannot be assumed to be the same as it was in the absence of a field. If we consider the operator W(rl, 12) defined by Z(rl, r2) = exp $A1-A2)
.
Wrl,
r2>, (10)
equation
(9) shows that we have u/(rl+T,
(6)
Further p-(e/c)A commutes with p+(e/c)A and p * T commutes with A - T. Using these facts, it is easily seen that S commutes with the first two terms of equation (l).* It can then be shown that it is self-consistent to assume that S commutes with Z. For if S commutes with H Sh = 2 h&n m
h(r$,h*(r2),
z
szs-l=
’
where T is any lattice vector. We now show that in the presence of a uniform magnetic field, 2 acquires a symmetry different from that given by equation (4). Consider the unitary operator S = & /JL){ p-k/c)A
hd*ln*m(n)*~*(rz)
since the summation over n contains all degenerate states. Also S commutes with r12. Hence S commutes with Z, which is the requirement of consistency. This result may be written
(4)
.
c c m
In the absence of a magnetic field, the potential and exchange terms both have lattice periodicity. That is
Z(r1+7, r2+7) = Z(r1, r2)
237
FORMALISM
I.m,n
W$(rl--2)
W1+7) = V(n)
MASS
r2+$
=
W(rl,
r2).
(11)
W(q, 12) and Z(rl, 12) become identical in the absence of a magnetic field. The presence of a small perturbation should not strongly affect the operator Z(rl, rs), because any perturbation which mixes only the filled states will leave Z invariant. A small change in Z will result from a small amount of band mixing. Equation (10) shows that a magnetic field cannot be regarded as a small perturbation in the ordinary sense no matter how small the field may be. This difficulty arises because (Hxr)/Z may become arbitrarily large. In a mathematical sense, infinite band mixing results. The definition of W’(rl, t-2)in equation (10) essentially extracts this gaugedependence and leaves a factor which may be assumed unchanged for small fields, in accord with the physical expectation that the field has only a very slight effect on the electron density. 3.
EFFECTIVE
MASS FORMALISM CHANGE
WITH
EX-
In treating the influence of exchange on the effective mass formalism, we follow closely the
238
EVAN
0.
method introduced by AmMS.(l~ This method is centered around the Bioch function representation of the Hamiltonian. The Bloch functions &k(r) = &%,k(r) diagonalize the Hamiltonian in the absence -of a magnetic field &AL@)
(12)
= E?@+&(r)
HO= (p2/2m)+ V+ W. In the presence of a magnetic tonian may be written
(13)
KANE
Al)Al\nS(r) gives the following expressions for the impoutant operators in the Bloch representation r=<+X (k’n’j?Jkn) (k’n’JX/kn)
= &,*V#G--k’) = &/,(k)G(k-k’) (18)
(k’n’JpJkn > = p,,,(k)?@--k’) (k’n’~Hojkn) = S,,,G(k-k’)&(k).
field, the Hamil-
(14) The brace ( Ii signifies the anticommutator, written in this case only for the sake of symmetry. The convention of summing on repeated indices is used throughout. We now expand the exponential in equation (10) in powers of A to give the result
ADAMS has also shown that the diagonal component XaS is independent of k and is zero in lattices possessing inversion symmetry. We need the following commutator relations
[y’, [y“, HollGz=
(19)
GE,
- dE+
In deriving the effective mass equations, we are interested in the region about a band maximum or ~nimum. For simplicity, let us assume the extremum to be at k = 0. If the band is nondegenerate, we can write Equation (15) may also be written notation as
in operator
It is easily verified by the use of equations f13) and (16) that equation (14) can be written in the form H= Ho+
(A/1, {A “, [rfi, [r “,Ho]]-))++
.... (17)
If HO does not include exchange, all commutators higher than the second vanish.
,
(20)
where the principal axes of the ellipse are taken as co-ordinate axes, and the rngn are effective mass constants. The technique for diagonalizing the Hamiltonian of equation (17) is to make a perturbation expansion removing interb~d matrix efements in successively higher orders in the magnetic field. An off-diagonal matrix element removed in a given order contributes to diagonal elements in higher orders. The intraband off-diagonal elements associated with the differential operator 5 cannot be removed in this way, since the expansion would not converge. The removal of interband matrix elements to a given order therefore results in a differential operator for each band, the bands being made independent to the desired order ofx, the magnetic field. The further diagonalization is then accomplished by solving the differential
INFLUENCE
OF
EXCHANGE
ON
THE
equation or by using some other appropriate technique. The-usual effective mass equation is derived to be correct to the first order in the magnetic field.* To eliminate higher-order terms from consideration, we consider all operators to be expanded in powers of k; k should then be considered to be of order .X1/a and ajak to be of order $P-l/a. We must then retain the following terms in equation (17):
-HS [S,HoI]-= gJ’(Afe, Using the relations given by equations
A+g, (
EFFECTIVE
MASS
H’ = e~Hf?-+ = H+[S,
FORMALISM
HI-+-&, JL:
239
[S, HI]-+
.. . (22)
[S, &I-
= (
;
1
{A%), [r/l, ~olkV]+n#n’.
The elimination of the off-diagonal term gives a diagonal term in the second order of S which is of order X.
P, ~Olnn*[~ “,Ho]&+ [Yy,Ho];?v[Yl”,H&n
+
?a,*n.
(&--En,)
(23)
(19), we can write H’ as follows: (24)
(k’n’lHlkn) +
= EnSnn2@--k’)+ {A%),
Equation
I
P, Hol,}+Sw+
{A’V)nw>
(24) can be written
H’ = t
P, Ho]&}+n#n’+
1
c21)
even more simply as
E,(kfi+$/(()),
(25)
which is easily shown to be true to second order in k 1‘f we use the symmetrical expansion of En
{Afw, {A%)> +
{A%),
(26)
En(k) =En(~)+k+$)o+tk~kv(;)o
P‘, P’, Ho]l;~)}+S,w+ [y“, Hol&v}+~#n’. i
The first four terms on the right-hand side of equation (21) are already diagonal in bands. All quantities in equation (21) are to be evaluated to lowest order in k. The last term is non-diagonal in bands and of order 21’2 according to our prescription. This term may be eliminated by the transformation * Terms in Hz are required for the calculation of the magnetic susceptibility. To develop a theory to order Hz, the change in the operators Y and W with magnetic field would have to be included.
and substitute kr+(e/ch)Ap( 5) directly into this expansion. Equation (25) shows explicitly that the formal nature of the effective mass equation is not changed by the inclusion of exchange. This is clearly a consequence of the fact that equation (17) could be written in a form which did not depend on whether exchange was included or not. Equations (19) give the following expressions for the effective mass, rnpv fi2 -zzzz rnry
lPE
-( ) dkpdk”
o
(27)
240
EVAN
fi2
-
mpy
= - [YP, p, Ho]]& -
w,
0.
KANE
ffol&&“,ffo],%+[r“,Ho]Gv[r~, Ho],,), Zn’ (h---w)
[rp, Ho]- = (ih/m)p+
p,
[TV,Ho]]- = - $/ty+[rP,
From these equations we see that exchange contributes to the effective mass through [rp, w];,,, and [YW, [YY, W]];n. 4. OPTICAL
[W, w]-
ABSORPTION
One of the results of effective mass theory which is sensitive to exchange is the comparison of optical absorption constants and effective masses.@) The Hamiltonian in the presence of a light wave is given by equation (1). A is the vector potential of the light which may be taken to be Aoe-iR’r+ + Ao*ebk*r. Unlike the case of the constant magnetic field, the vector potential does not become arbitrarily large for large values of r. Hence A may be considered as a small perturbation which will not affect the exchange operator Z(rl, t-2). The perturbation H, may be written
(29)
[IV, W]]-.
(30)
conduction bands in germanium@) and indium antimonide.(4) In this work exchange was neglected and higher bands of proper symmetry were assumed to be unimportant in the sum over bands of equation (28). The uncertainty in the comparison is such that it can only be claimed that exchange is less important than the ordinary momentum contribution to the effective mass. We can obtain a very crude estimate of the size of [YP, WI- which shows that it is not negligible compared to tpplrn. We approximate the exchange charge density -eL’t,bt(r$,h~*(r2) by a constant i
equal to the average charge density of electrons of parallel spin for ~12 < Y~Jand equal to zero for ~12 > ~0. The value of ro is given by -ep
4x 3y03P = l’
Hl-LA.p, mc
(31)
since A0 * k = 0. The absolute value of the optical absorption, therefore, gives the momentum matrix element. In the case of germanium and indium antimonide, the closely adjacent conduction and valence bands contribute a large share of the effective mass sum in equation (28). Reasonable agreement with the cyclotron masses has been found with the use of the optical matrix element between valence and
(k’, .‘I [YY,WI-l k, n > = +e2p
(32)
so that the total exchange charge is equal to -e. This approximation is based on arguments given by sLATERc7) and should be reasonably accurate. We further assume that the exchange hole is small compared to the size of the unit cell and to the scale of variation of the functions U,(Q). This approximation is clearly very bad, but it should give an order-of-magnitude estimate. With this approximation we can write u(r2) = u(r1) + (rz-r1). vu(rl) over the region of the exchange hole. We may then write
(y1P-yZP)~k,n.*(r1)(r1-r2)
fS
(28)
’
~#kn(rl) drld(rl--2)
Iu--f-21
(33)
INFLUENCE
OF
EXCHANGE
ON
THE
The integral of rs-rlis over a sphere ~12 < TO. The integral is easily performed to give Iy~,cyl=+[~yO](q.P’~.*
(34)
The importance of exchange in this approximation depends only on the electron density. For germanium the factor mesre/4tia has the value O-65. -, which indicates that the two terms may be comparable. With the same approximations as made above, the term [YP, [YY, WI]- has the value e2ro [YB,
[Y“,
w-J]- = - --&. 4
This term then, gives a contribution to the reciprocal effective mass of +0*65 reciprocal elec* Note added in proof: The author wishes to thank Dr. JAMESC. PHILLIPSfor correcting a sign in equations (33) and (34).
Q
EFFECTIVE
MASS
FORMALISM
241
tron masses in germanium. According to the oversimplified model used, this contribution would be thesameforallbands* Acknowledgements-The author would like to thank Professor M. COHEN for bringing this problem to his attention. He is indebted to Dr. F. HAM and Professor G. DRESSELHAUS for several very helpful discussions.
REFERENCES 1. WANNIER G. H. Phys. Rev. 52, 191 (1937). ADAMS E. N.J. Ch&. Phys. 2i, 2013 (1953). LUTTINGER1. M. and KOHN W. Phvs. Rev. 97. 869
(1955).
-
G., KIP A. F., and KITTEL C. Phys. 2. DRESSELHAUS Rev. 98, 368 (195.5). 3 HERRINGC. Phys. Rev. 52, 361 (1937). 4: KANE E. 0. 3. Phys. Chem. Solids 1, 83 (1956); 1, 249 (1957). 5. KOHN W. Phys. Rev. 105, 509 (1957). 6. KANE E. 0. Unpublished work. 7. SLATERJ. C. Phys. Rev. 81, 385 (1951).