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Exact solutions of tightly bound electrons in a uniform electric field: perturbation theory
Volume 154, number 5,6
PHYSICS LETFERS A
8 April 1991
Exact solutions of tightly bound electrons in a uniform electric field: perturbation theory Xian-Geng Zhao Institute ofApplied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China Received 6 November 1990; accepted for publication 19 December 1990 Communicated by J. Flouquet
A two-band model for tightly bound electrons in infinite crystals under the action of a uniform electric field is investigated, in which the interband coupling is taken into account. For the case when the interband coupling is small, exact solutions for the energy spectrum and the wave functions are presented by using perturbation theory, by which it is shown quite rigorously that the energy spectrum is that oftwo interspaced Stark ladders.
It is known that in the case of a single-band model ofa one-dimensional crystal, the problem oftightly bound Bloch electrons under the influence of a uniform electric field has been solved easily [1,2]. However, in the case of a two-band model, to seek exact solutions for the energy spectrum and the wave functions is not easy except for the special case that the interband coupling is absent [3]. Taking into account the interband coupling, a two-band tight-binding model in infinite crystals was studied by Fukuyama, Ban and Fogedby [2] many years ago; though the exact results for the special case that two Bloch bands have the same nearest-neighbor hopping were found, the exact solutions for the general case have not been obtained to date. In this Letter, we make an attempt to investigate this problem. We find that for the case when the interband coupling is small, the problem can be solved exactly by using perturbation theory (PT). As result, explicit solutions for the energy spectrum and the wave functions are obtained. The Hamiltonian considered here for tightly bound electrons is given by Fukuyama, Ban and Fogedby [21 as H= —4~ata1+t
~
(aj~a,÷~ +aj~1ai)+4
+~~l(ata1+bj~b,)+V>
>~ bj~b~+1’~
(b7b1÷1+b~1b1)
(a,~b1+bta,),
(1)
where at (a1) and b j’ (b1) are the creation (annihilation) operators at lth Wannier site referring to the lower and upper bands respectively, —4 (4) is the site energy belonging to the lower (upper) band, I (t’) is the nearest-neighbor hopping matrix element according to the lower (upper) band, e = eaE0, where e, a, and E0 are the electronic charge, the lattice spacing, and the external electric fields, respectively. The interband coupling is described by V=eDE0, where D is the dipole matrix element between two atomic states at the same site. Here the matrix elements of the electric field between neighboring sites have also been neglected. And the weak interband coupling means that D/a << 1. As shown in ref. [21, by introducing —
a~= >~J,_~(2a)ai,
0375-9601/91/S 03.50 © 1991
(2a)
—
Elsevier Science Publishers B.V. (North-Holland)
275
Volume 154, number 5,6
PHYSICS LETTERS A
8 April 1991
fin_~Ji_n(2a’)bi, with
H=
(2b)
a=t/eaE0 and a’ =I’/eaE0, eq. (1) can be rewritten as ~
where e
(3)
~
,~
=
±4 + neaE0. This gives the eigenvalue equations (2a—2a’)fin.,
(4a)
(w—e~)fi~= V~J~_~(2a’ —2a)a~,
(4b)
(a—e;)an=V~Jn_n
where a is the energy and J~is the ordinary Bessel function. Using Fourier transforms of a~and /3~according to
fi(k)=
exp(—ink)a~,
(5a)
exp(—ink)fi~,
(5b)
where k is the (dimensionless) wavevector, eqs. (4a) and (4b) become (w+A—ieaEod/dk)a(k)= Vexp[ —2i(a—a’) sin k]fi(k),
(6a)
(w—4—ieaE0d/dk)fi(k)=Vexp[2i(a—a’)
(6b)
sink]a(k).
By setting a(k)=a(k)exp[—i(a+4)k/eaE0]
,
fi(k)=b(k)exp[—i(w—4)k/eaE0]
,
(7)
eqs. (6a) and (6b) become -~-a(k)=i---~—exp{—2i[(a—a’)sink—4k/eaE0]}b(k), dk eaE0
(8a)
-~b(k)=i—-~—exp{2i[(a—a’)sin k—4k/eaEo]}a(k). d eaE0
(8b)
By introducing (9) G(k)=X(k)o~+Y(k)a~,
(10)
X(k)=——--~’-—cos{2[(a—a’)sink—4k/eaE0]}, eaE0
(11)
Y(k)=— ~ eaE0
(12)
where o~and ~ (as well as a~used below) are the Pauli matrices, whose explicit forms are [4] 276
Volume 154, number 5,6
/0 ~X~\1
l’~
PHYSICS LETTERS A
(0 ~‘~i
o)’
—i\
o)’
~
(1
8 April 1991
O’\ —1)’
(13)
eqs. (8a) and (8b) can be rewritten as ~A(k)=—iG(k)A(k)
(14)
or equivalently A(k)=A(0)_iJdkiG(ki)A(ki).
(15)
From (lO)—(12), it is easily shown that
IG(k)I={[X(k)]2+[Y(k)12}”2=IVIeaEoI=DIa.
(16)
For the case of weak interband coupling, we have D/a~z<1, i.e., IG(k) I <<1. This leads to Jdk 1G(k1)A(k1)~<
(17)
Therefore eq. (15) can be solved by using PT. As result, we obtain ~ u(m)(k,0)A(o),
(18)
m=O
where U~°~(k,0)=l
(19)
,
U(m)(k, o)~(_i)mJdk1
O(k)=l,
k>0,
=0,
k<0.
J
dk2
...
J
dkm 6(k1
k2)e(k2
k3)...O(km_i
km)G(ki)G(k2)...G(km),
(20)
(21)
It is easily shown that G(kl)G(k2)...G(km)X(klk2...km)+iY(kIk2...km)az, r~zX(kik2...km)ax+Y(kik2...km)ay,
ifm=21, ifm=21+l,
(22)
X(kik2...km)X(kik2...km_i)X(km)+Y(kik2...km_i)Y(km)
(m~2),
(23)
Y(kik2...km)X(kik2...km_i)Y(km)Y(kik2...km_i)X(km)
(m~>2).
(24)
Defining U~m)(k,0)=(_l)mJdkiJdk2...Jdk2mO(kj_k2)O(k2_k3)...O(k2m_i_k2m)X(kik2...k2m),
(25)
277
Volume 154, number 5,6
PHYSICS LETTERS A k
k
8 April 1991
k
Uc2m)(k,o)=(_l)m$dki$dk2...$dk2mo(k~_k2)o(k2_k3)...o(k2m_i_k2m)y(kik2...k2m), 0
0 k
(26)
0 k
k
U~m+1)(k,O)=(_l)m+UJdkiJdk2...$dk2m+iO(ki_k2)O(k2_k3)...O(k2m_k2m+i)X(kik2...k2m+i), o
o
0
k
k
k
(27) U~m+1)(k,O)=(_l)~~÷1$dkiJdk2...Jdk2m÷iO(ki_k2)O(k2_k3)...O(k2m_k2m+i)Y(kik2...k2m±i), o
o
0
(28) U~(k,0)=l
U~”(k,0)=0,
,
(29)
find
we
~ u~”~(k,0)=~ U~m)(k,0)+io~~ Uym)(k,0)+io~ ~ ,n=0
,n=O
,n=O
u~~(k,0)
,n=O
+io~ ~ UVm÷l)(k,0).
(30)
in=0
From (7), (9) and (18), we have /exp(—iAk/eaE0) 0
(a(k)\ A(0)
=
0 exp(iAk/eaE0)
)
~ U(m)(k,0)A(0),
(31)
m=O
(a(0)\ /a(0)\ b(0))~fi(0))~
(32)
Note that from (5a) and (5b), a(k+2x)=a(k) and ,I3(k+2it)=/3(k). Therefore, without any loss of generality, a ( 2x) = a (0) and /J(2x) = /1(0). This leads to the following equation,
I
~fi(0))=exP(_iX2~w/eaEo)(
exp(—iX2itA/eaE0) 0
0 exp(ix2x4/eaEo))m~oU(m)(2x,0)(fl(0)). (33)
The eigenvalue equation determined by eq. (33) is [Iexp(—ix2it4/eaE0
0
det[~ 0 exp(ix2x4/eaEo),n~o um(27t,O)_exp(ix2xw/eaEQ)]=O, with solutions (see the appendix)
(34)
eaE0 (n=integer),
(35)
m~(2~, 0)]. 27tA~ \eaE Ø(2ir, 0)=cos_1[cos(-~~-~ ~ U5?”° (2x, 0)+sin(~—~_)~uV 0, ,n=O
(36)
where
Corresponding to w,~,the solutions for a±(0) and /1~(0) are 278
Volume 154, number 5,6
PHYSICS LETTERS A 2
+~,)UVm~)(2~,0)
1 Il \ I a±(0)=__[~~ U~m~~)(2x~0)) x[l_cos(2+4)) eaE0 /
~ U~m)(2~,O)_sin(2 m=O
no
2
+4)) eaE0
8 April 1991 1/2
U~m)(2m,O)],
(37)
m=O
no
fi±(0)=( ~ U~m±l)(27t0)+i ~ U~m~)(2x 0) \m=O
m=O
x[exp(i2+4))_(~ U~m)(27t,0)+i ~ UVm)(2E,0))]a±(0). eaE0 m=O m=O
(38)
it is straightforward to check the orthogonality conditions, i.e., a±(0)12+I/1±(0)12=1
,
a~(0)a~(0)+fi’~(0)fi~(0)=0.
(39)
It is clearly seen from (35) that the energy spectrum is that oftwo interspaced Stark ladders, which is consistent with many theoretical results about the existence of Wannier—Stark localization in solids for the case of charged particles (or Bloch electrons) under the influence of uniform electric fields [5—71. In ref. [2], two such interspaced ladder energy structures were also found in the special case that two Bloch bands had the same nearest-neighbor hopping, i.e., t=t’ or a=a’. Obviously, compared to the results of ref. [2], our conclusion in this Letter has a quite general meaning not only for the special case t = I’ or a = a’. In principle, our results (31), (32), (35)—(39) can exactly hold for the case of weak interband coupling. However, this means that one needs to calculate infinite integrals, which, obviously, is impossible. Hence in practice, we have to make further approximations up to the orders needed. For example, as the zero order of PT, we get from (29) and (36) that ~
U~m)(2x,0)~U5,~)(2x,0)=l,
m=O
~ UVm)(2n,0)~Uc2)(2~,0)=0.
(40)
,n~O
This leads to Ø(2x,0)=2itA/eaE0,
(41)
therefore o4=neaE0±A.
(42)
For other orders of PT, to calculate the energy spectrum is straightforward but tedious. Finally, we should like to indicate that since the wave functions have been obtained in this Letter, it is possible to calculate other physical quantities up to any order of PT. It is a pleasure to thank Professors S.G. Chen, X.W. Zhang and W.X. Zhang for valuable conversations.
Appendix Substituting (30) (taking k=27t) into (34), we get
279
Volume 154, number 5,6
PHYSICS LETTERS A
exP(ix -~~)_2exp(ix -~)[cos(-~ ‘no
~
\2
U~~(2~r 0)
U~m)(2,t,0)+sin(~-~)~ UV°°(2it,0)] \2
,.
+1\,n=o ~ UVm)(2x
I
8 April 1991
0)
I
/
+1
\2
~
U$~m~-I)(2~ 0)
\m=o
/
1 +(
I
\2
~
U~m±l)(2x0)
\m=O
=0.
/
(A.!)
Note that from (33), we have
Ia(0)12+I/1(0)12=(a*(0) =
(a*(0)
fi*(0))(~) tm~(2m,0))(~).
fi*(0))(~ U(m)(2x, 0))(~
(A.2)
U
When using (30) again, eq. (A.2) becomes Ia(0) 12+1/1(0) 12=[( ~ U~m)(2m,0))+( ~ Uym(2x, 0)) tnO
~
,nO
U~~(2~ 0))+(
,n=O
~
U~m+I)(2x 0))][Ia(0)12+Ifi(0)12].
(A.3)
m=O
This gives ‘no
\2
~ \m=o
U~?~?~(2~ 0)1 I
‘no
+(
\2
~ U~m)(21t 0))
\,n=o
I
/
\2
‘no
+1\,n=o ~ U~m+I)(2x 0)1 +( ~ U~m~(2it 0)1 I \m=o I
Substituting (A.4) into (A. 1), and by introducing cos[Ø(2x,0)]=cos(-~-) ~ U~2m)(2x,0)+sin(~~-~_) ~ U~(2~,0),
=1
.
(A.4)
(A.5)
eaE 0
eaE0
pn=O
pn=O
we obtain exp{i[2mw/eaE0~Ø(2it,0)]}=l
.
This leads to (35).
References [1] [2] [3] [4] [5] [6] [7]
E.O. Kane, J. Phys. Chem. Solids 12 (1959)181. H. Fukuyama, R.A. Ban and H.C. Fogedby, Phys. Rev. B 8 (1973) 5579. P. Feuer, Phys. Rev. 88 (1952) 92. E. Merzbacher, Quantum mechanics (Wiley, New York, 1970) p. 270. G.H. Wannier, Phys. Rev. 181 (1969) 1364. D. Emin and C.F. Hart, Phys. Rev. B 36 (1987) 7353. C.F. Hart and D. Emin, Phys. Rev. B 37 (1988) 6100.
280
(A.6)
PHYSICS LETFERS A
8 April 1991
Exact solutions of tightly bound electrons in a uniform electric field: perturbation theory Xian-Geng Zhao Institute ofApplied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China Received 6 November 1990; accepted for publication 19 December 1990 Communicated by J. Flouquet
A two-band model for tightly bound electrons in infinite crystals under the action of a uniform electric field is investigated, in which the interband coupling is taken into account. For the case when the interband coupling is small, exact solutions for the energy spectrum and the wave functions are presented by using perturbation theory, by which it is shown quite rigorously that the energy spectrum is that oftwo interspaced Stark ladders.
It is known that in the case of a single-band model ofa one-dimensional crystal, the problem oftightly bound Bloch electrons under the influence of a uniform electric field has been solved easily [1,2]. However, in the case of a two-band model, to seek exact solutions for the energy spectrum and the wave functions is not easy except for the special case that the interband coupling is absent [3]. Taking into account the interband coupling, a two-band tight-binding model in infinite crystals was studied by Fukuyama, Ban and Fogedby [2] many years ago; though the exact results for the special case that two Bloch bands have the same nearest-neighbor hopping were found, the exact solutions for the general case have not been obtained to date. In this Letter, we make an attempt to investigate this problem. We find that for the case when the interband coupling is small, the problem can be solved exactly by using perturbation theory (PT). As result, explicit solutions for the energy spectrum and the wave functions are obtained. The Hamiltonian considered here for tightly bound electrons is given by Fukuyama, Ban and Fogedby [21 as H= —4~ata1+t
~
(aj~a,÷~ +aj~1ai)+4
+~~l(ata1+bj~b,)+V>
>~ bj~b~+1’~
(b7b1÷1+b~1b1)
(a,~b1+bta,),
(1)
where at (a1) and b j’ (b1) are the creation (annihilation) operators at lth Wannier site referring to the lower and upper bands respectively, —4 (4) is the site energy belonging to the lower (upper) band, I (t’) is the nearest-neighbor hopping matrix element according to the lower (upper) band, e = eaE0, where e, a, and E0 are the electronic charge, the lattice spacing, and the external electric fields, respectively. The interband coupling is described by V=eDE0, where D is the dipole matrix element between two atomic states at the same site. Here the matrix elements of the electric field between neighboring sites have also been neglected. And the weak interband coupling means that D/a << 1. As shown in ref. [21, by introducing —
a~= >~J,_~(2a)ai,
0375-9601/91/S 03.50 © 1991
(2a)
—
Elsevier Science Publishers B.V. (North-Holland)
275
Volume 154, number 5,6
PHYSICS LETTERS A
8 April 1991
fin_~Ji_n(2a’)bi, with
H=
(2b)
a=t/eaE0 and a’ =I’/eaE0, eq. (1) can be rewritten as ~
where e
(3)
~
,~
=
±4 + neaE0. This gives the eigenvalue equations (2a—2a’)fin.,
(4a)
(w—e~)fi~= V~J~_~(2a’ —2a)a~,
(4b)
(a—e;)an=V~Jn_n
where a is the energy and J~is the ordinary Bessel function. Using Fourier transforms of a~and /3~according to
fi(k)=
exp(—ink)a~,
(5a)
exp(—ink)fi~,
(5b)
where k is the (dimensionless) wavevector, eqs. (4a) and (4b) become (w+A—ieaEod/dk)a(k)= Vexp[ —2i(a—a’) sin k]fi(k),
(6a)
(w—4—ieaE0d/dk)fi(k)=Vexp[2i(a—a’)
(6b)
sink]a(k).
By setting a(k)=a(k)exp[—i(a+4)k/eaE0]
,
fi(k)=b(k)exp[—i(w—4)k/eaE0]
,
(7)
eqs. (6a) and (6b) become -~-a(k)=i---~—exp{—2i[(a—a’)sink—4k/eaE0]}b(k), dk eaE0
(8a)
-~b(k)=i—-~—exp{2i[(a—a’)sin k—4k/eaEo]}a(k). d eaE0
(8b)
By introducing (9) G(k)=X(k)o~+Y(k)a~,
(10)
X(k)=——--~’-—cos{2[(a—a’)sink—4k/eaE0]}, eaE0
(11)
Y(k)=— ~ eaE0
(12)
where o~and ~ (as well as a~used below) are the Pauli matrices, whose explicit forms are [4] 276
Volume 154, number 5,6
/0 ~X~\1
l’~
PHYSICS LETTERS A
(0 ~‘~i
o)’
—i\
o)’
~
(1
8 April 1991
O’\ —1)’
(13)
eqs. (8a) and (8b) can be rewritten as ~A(k)=—iG(k)A(k)
(14)
or equivalently A(k)=A(0)_iJdkiG(ki)A(ki).
(15)
From (lO)—(12), it is easily shown that
IG(k)I={[X(k)]2+[Y(k)12}”2=IVIeaEoI=DIa.
(16)
For the case of weak interband coupling, we have D/a~z<1, i.e., IG(k) I <<1. This leads to Jdk 1G(k1)A(k1)~<
(17)
Therefore eq. (15) can be solved by using PT. As result, we obtain ~ u(m)(k,0)A(o),
(18)
m=O
where U~°~(k,0)=l
(19)
,
U(m)(k, o)~(_i)mJdk1
O(k)=l,
k>0,
=0,
k<0.
J
dk2
...
J
dkm 6(k1
k2)e(k2
k3)...O(km_i
km)G(ki)G(k2)...G(km),
(20)
(21)
It is easily shown that G(kl)G(k2)...G(km)X(klk2...km)+iY(kIk2...km)az, r~zX(kik2...km)ax+Y(kik2...km)ay,
ifm=21, ifm=21+l,
(22)
X(kik2...km)X(kik2...km_i)X(km)+Y(kik2...km_i)Y(km)
(m~2),
(23)
Y(kik2...km)X(kik2...km_i)Y(km)Y(kik2...km_i)X(km)
(m~>2).
(24)
Defining U~m)(k,0)=(_l)mJdkiJdk2...Jdk2mO(kj_k2)O(k2_k3)...O(k2m_i_k2m)X(kik2...k2m),
(25)
277
Volume 154, number 5,6
PHYSICS LETTERS A k
k
8 April 1991
k
Uc2m)(k,o)=(_l)m$dki$dk2...$dk2mo(k~_k2)o(k2_k3)...o(k2m_i_k2m)y(kik2...k2m), 0
0 k
(26)
0 k
k
U~m+1)(k,O)=(_l)m+UJdkiJdk2...$dk2m+iO(ki_k2)O(k2_k3)...O(k2m_k2m+i)X(kik2...k2m+i), o
o
0
k
k
k
(27) U~m+1)(k,O)=(_l)~~÷1$dkiJdk2...Jdk2m÷iO(ki_k2)O(k2_k3)...O(k2m_k2m+i)Y(kik2...k2m±i), o
o
0
(28) U~(k,0)=l
U~”(k,0)=0,
,
(29)
find
we
~ u~”~(k,0)=~ U~m)(k,0)+io~~ Uym)(k,0)+io~ ~ ,n=0
,n=O
,n=O
u~~(k,0)
,n=O
+io~ ~ UVm÷l)(k,0).
(30)
in=0
From (7), (9) and (18), we have /exp(—iAk/eaE0) 0
(a(k)\ A(0)
=
0 exp(iAk/eaE0)
)
~ U(m)(k,0)A(0),
(31)
m=O
(a(0)\ /a(0)\ b(0))~fi(0))~
(32)
Note that from (5a) and (5b), a(k+2x)=a(k) and ,I3(k+2it)=/3(k). Therefore, without any loss of generality, a ( 2x) = a (0) and /J(2x) = /1(0). This leads to the following equation,
I
~fi(0))=exP(_iX2~w/eaEo)(
exp(—iX2itA/eaE0) 0
0 exp(ix2x4/eaEo))m~oU(m)(2x,0)(fl(0)). (33)
The eigenvalue equation determined by eq. (33) is [Iexp(—ix2it4/eaE0
0
det[~ 0 exp(ix2x4/eaEo),n~o um(27t,O)_exp(ix2xw/eaEQ)]=O, with solutions (see the appendix)
(34)
eaE0 (n=integer),
(35)
m~(2~, 0)]. 27tA~ \eaE Ø(2ir, 0)=cos_1[cos(-~~-~ ~ U5?”° (2x, 0)+sin(~—~_)~uV 0, ,n=O
(36)
where
Corresponding to w,~,the solutions for a±(0) and /1~(0) are 278
Volume 154, number 5,6
PHYSICS LETTERS A 2
+~,)UVm~)(2~,0)
1 Il \ I a±(0)=__[~~ U~m~~)(2x~0)) x[l_cos(2+4)) eaE0 /
~ U~m)(2~,O)_sin(2 m=O
no
2
+4)) eaE0
8 April 1991 1/2
U~m)(2m,O)],
(37)
m=O
no
fi±(0)=( ~ U~m±l)(27t0)+i ~ U~m~)(2x 0) \m=O
m=O
x[exp(i2+4))_(~ U~m)(27t,0)+i ~ UVm)(2E,0))]a±(0). eaE0 m=O m=O
(38)
it is straightforward to check the orthogonality conditions, i.e., a±(0)12+I/1±(0)12=1
,
a~(0)a~(0)+fi’~(0)fi~(0)=0.
(39)
It is clearly seen from (35) that the energy spectrum is that oftwo interspaced Stark ladders, which is consistent with many theoretical results about the existence of Wannier—Stark localization in solids for the case of charged particles (or Bloch electrons) under the influence of uniform electric fields [5—71. In ref. [2], two such interspaced ladder energy structures were also found in the special case that two Bloch bands had the same nearest-neighbor hopping, i.e., t=t’ or a=a’. Obviously, compared to the results of ref. [2], our conclusion in this Letter has a quite general meaning not only for the special case t = I’ or a = a’. In principle, our results (31), (32), (35)—(39) can exactly hold for the case of weak interband coupling. However, this means that one needs to calculate infinite integrals, which, obviously, is impossible. Hence in practice, we have to make further approximations up to the orders needed. For example, as the zero order of PT, we get from (29) and (36) that ~
U~m)(2x,0)~U5,~)(2x,0)=l,
m=O
~ UVm)(2n,0)~Uc2)(2~,0)=0.
(40)
,n~O
This leads to Ø(2x,0)=2itA/eaE0,
(41)
therefore o4=neaE0±A.
(42)
For other orders of PT, to calculate the energy spectrum is straightforward but tedious. Finally, we should like to indicate that since the wave functions have been obtained in this Letter, it is possible to calculate other physical quantities up to any order of PT. It is a pleasure to thank Professors S.G. Chen, X.W. Zhang and W.X. Zhang for valuable conversations.
Appendix Substituting (30) (taking k=27t) into (34), we get
279
Volume 154, number 5,6
PHYSICS LETTERS A
exP(ix -~~)_2exp(ix -~)[cos(-~ ‘no
~
\2
U~~(2~r 0)
U~m)(2,t,0)+sin(~-~)~ UV°°(2it,0)] \2
,.
+1\,n=o ~ UVm)(2x
I
8 April 1991
0)
I
/
+1
\2
~
U$~m~-I)(2~ 0)
\m=o
/
1 +(
I
\2
~
U~m±l)(2x0)
\m=O
=0.
/
(A.!)
Note that from (33), we have
Ia(0)12+I/1(0)12=(a*(0) =
(a*(0)
fi*(0))(~) tm~(2m,0))(~).
fi*(0))(~ U(m)(2x, 0))(~
(A.2)
U
When using (30) again, eq. (A.2) becomes Ia(0) 12+1/1(0) 12=[( ~ U~m)(2m,0))+( ~ Uym(2x, 0)) tnO
~
,nO
U~~(2~ 0))+(
,n=O
~
U~m+I)(2x 0))][Ia(0)12+Ifi(0)12].
(A.3)
m=O
This gives ‘no
\2
~ \m=o
U~?~?~(2~ 0)1 I
‘no
+(
\2
~ U~m)(21t 0))
\,n=o
I
/
\2
‘no
+1\,n=o ~ U~m+I)(2x 0)1 +( ~ U~m~(2it 0)1 I \m=o I
Substituting (A.4) into (A. 1), and by introducing cos[Ø(2x,0)]=cos(-~-) ~ U~2m)(2x,0)+sin(~~-~_) ~ U~(2~,0),
=1
.
(A.4)
(A.5)
eaE 0
eaE0
pn=O
pn=O
we obtain exp{i[2mw/eaE0~Ø(2it,0)]}=l
.
This leads to (35).
References [1] [2] [3] [4] [5] [6] [7]
E.O. Kane, J. Phys. Chem. Solids 12 (1959)181. H. Fukuyama, R.A. Ban and H.C. Fogedby, Phys. Rev. B 8 (1973) 5579. P. Feuer, Phys. Rev. 88 (1952) 92. E. Merzbacher, Quantum mechanics (Wiley, New York, 1970) p. 270. G.H. Wannier, Phys. Rev. 181 (1969) 1364. D. Emin and C.F. Hart, Phys. Rev. B 36 (1987) 7353. C.F. Hart and D. Emin, Phys. Rev. B 37 (1988) 6100.
280
(A.6)