- Email: [email protected]
Generating function method for two-band electrons in a uniform electric field
Physics Letters A 163 (1992) North-Holland
219-222
PHYSICS
LETTERS
A
Generating function method for two-band electrons in a uniform electric field Wei-mou
Zheng
Institute of Theoretical Physics, Academia Sinica, Beijing. China Received 9 December I99 I; accepted Communicated by J. Rouquet
for publication
15 January
I992
A generating function method is developed to treat the problem of two-band electrons in a uniform electric field. The method is rather straightforward, and shows easily the existence of two interspaced Stark ladders and a relation between them. A perturbation solution is obtained.
In general, the application of an electric field lifts the degeneracy between local electronic levels, and results in Wannier-Stark localization of the electronic states [ 11. There has been a long-standing interest in understanding this localization [ 2,3 1. Fukuyama, Bari and Fogedby have proposed a two-band tight binding model of one-dimensional crystals in a uniform electric field. The Hamiltonian of the model is
H=-A,=z
m
I~)-<~I-+U-s)i~~
(l~)-(~+ll-+ll+l)-(~l)+d,=~ co
Ib+(~l+ 00
(1)
-17
where II) _ and 1f) + are the normalized local atomic states at site f referring to the lower and upper bands, and + A and t + 6 are the site energies and the nearest-neighbor hopping matrix elements of the two bands. We denote the scaled external field by t = eaE and the interband coupling by v= eDE, where - e, a, E and D are respectively the electronic charge, the lattice spacing, the external field and the dipole matrix element between the two atomic states at the same site referring to the two bands. Without loss of generality we set t= 1, i.e. take c as the unit of energy. In the case of v=O the model reduces to the single-band model, which is solvable. Another solvable case is the symmetric two-band model with 6~0 [2]. Recently, by taking the single-band wave functions as the basis for the Hamiltonian, a perturbation solution to the model ( 1) for a small interband coupling has been discussed by Zhao [ 41. Here we shall apply the generating function method to the model, and, as an example also interesting to physics, discuss the perturbation solution for small 6. Assume that the eigenstate of Hamiltonian ( 1) is
Iw>= ,=z Hlo)
(@/lb- +nlb+) 00
=~I~)
037%9601/92/$
3
(2)
> 05.00 0 1992 Elsevier Science Publishers
(3)
B.V. All rights reserved.
219
Volume 163, number 3
PHYSICS LETTERS A
16 March 1992
where 0; and q/; are the amplitudes of the atomic states [ l ) _ and l l) +. respectively. From eqs. ( 1 ) - ( 3 ) we can derive the coupled equations for 0; and ~,;:
(l-A)Ol+(t-6)(O;+:+O; ~)+t'gtl=to0;,
(l+A)qJl+(t+~)(~';+~+q.'; ~ ) + t ' 0 / = o ~ .
(4t
Let us introduce the following two generating functions, ,)c, [= -,~-
I=
:x
In terms of F ( ~ ) and G(~), eq. (4) may be written as
~F'-AF+ (t-fi)(~+ I/~)F+vG=toF,
~G' +AG+ ( t + c~) (~+ I/~)G+z,F=(oG.
(6
which may further be written in the concise vector form ~U' +1 [ t ( ~ + 1 / ~ ) - t o ] -
[ A + 6 ( ~ + l / l ) ] a - + ; ' ~ r , I,U=O,
(7
where the vector U(~) and the Pauli matrices (including o~,.used below) are
(F(~)~
6,. = (0l
U(~) = \ G ( ~ ) / '
10)'
¢7,.= ( 0 i - ; ) ,
a_ =(10 "
_0)
(8> "
From eqs. (6) we can directly draw two important conclusions. We denote the eigenfunctions for the eigenvalue to by Fo)(~) and G,,~(~), and assume that, as a function of & to = co (fi). The first conclusion is: If to(d) is an eigenvalue, then so is cO= - t o ( - 6 ) . The corresponding eigenfunctions are
F,,,(~:c~)=-G,,)(-1/~; -fi),
G
(9)
where/~o(~; fi) and G
t:,o+,,(~)=~"F,,,(~),
G,~+, (~) = ~G~,,(~).
10)
Thus, generally, the spectrum consists of two interspaced Stark ladders. When fi=0, by direct integration we have the solution to eq. (7): U(~) = ~ o exp[ - t ( ~ - 1/~) exp[ (A~: - vcr,) In ~] U( 1 )
=~¢°exp[-t(~-l/~)][cosh(121n~)+(Aa~-b~) s i n h ( p l n ~ ) ] U ( 1 ) ,
II)
w h e r e / ~ - x / A Z + v 2, A=A/lz, ~=v/lt, and we have made use of the formula 2
2
exp (xn.er) = c o s h x+n.t~ sinh x ,
12 )
with n being a unit vector. From the definition of the generating functions (5) it is obvious that U(@2~') = (~(~). Particularly, setting ~= 1, we have U(e2~i)=U(1) ,
(13)
which, by using eq. ( 11 ), means that
U( 1 ) = 2 ~[cos 0 + i ( z l a : - ~ a ~ ) sin 0] U( 1 ) .
14
where 2 - e - 2 ~ i ' ° and 0 = 2~p. For a non-trivial U( 1 ) to exist, vanishing of the determinant is required: cos 0 + i A sin 0 - 2 -i~sin 0
~,:.0
- ig sin 0
cosO-iAsinO-2
=0,
15
Volume 163,number 3
PHYSICS LETTERSA
16 March 1992
which gives eigenvalues o g ~ = n + A ~ z + v z,
(16)
and the corresponding eigenvectors (17) From eq. ( 11 ) we then find U(~) = ~ e x p [ - t ( ~ - 1/~)] U -+( 1 ) .
(18)
Finally, by noticing the generating function for the Bessel functions J~(~): exP[½t(~-l/()]=
~
~%(t),
(19)
from eqs. (5) we have Ot = gJn+t(-2t),
~ = (A+ 1 )J,+t( - 2 t ) .
(20)
Let us investigate a more interesting case when the parameter of asymmetry 6 ¢ 0. Because of the non-commutability of az and ax eq. (7) now possesses no simple solution. We introduce the operator W= - t ( ~ - 1 / ~ ) +
(o)+Aaz-Vax) In ~.
(21)
Using the formula exp(½i0n-~r)cr exp( -
½iOn.a) = n (n-~r) - n X (nX~r) cos 0+nX~r sin 0,
(22)
we have
#z=e-WazeW=~j(Czu-2)G-if)S2uG+ (ff2+~2C2u)az,
(23)
where
S . - ½(~"-~-"),
C . - ½(~"+~-").
(24)
It can then be verified that the transformation
U=eWu
(25)
converts eq. (7) to the equation for u,
u' =d(1-t-~-2)a~u ,
(26)
which may be solved by the standard perturbation scheme. Thus, we have to the first order in 6, ¢ ¢
u(~)=u(1)+O ~ (l+x-2)~z(X)U(X) d x ~ u ( 1 ) + 6 f (l+x-2)#z(X)U(1) dx. 1
(27)
1
By direct integration we further obtain that
u(~)=u(~ )+tj[~(dt~x+~t7z)(S2~+l+S2u-i)+~S'~(Jt7z-l)ax)-if)(~2~+1+~2'~-l-~)t7y]u(~ ) ~
(28)
where
g~-S./v,
C.-C~/v,
/ i = 4 U / ( 4 / t 2 - 1) .
(29) 221
Volume 163, number 3
PHYSICS LETTERS A
16 March t 992
The eigenvalues can then be determined from condition ( 13 ) in a similar way to that for the case of 5 = 0 . We obtain the equation for u( ! ): u(l ) = 2 - - ' [cos =2
~[cos
O+i(2a:-fJa~)
O+i(~a: -?'a.)
sin 0]{1
sin
-i~(1-22fiS)
sin0
sin 20-izT(cos 2 0 - 1 )a,'] }u( 1 )
O+ 2if¢&J(2a.y+ f)a:) sin O]u( 1 )
. ~{cosO+i(2+2fj2fl5) sinO =z- ~
+Afi[iO(3a,.+~a:)
- i ~ ( l - 2 A f i S ) s i n 0 "~u(l) cos0-i(2+2/~2fiS) sin0) "
(30)
The eigenvalue equation can be derived from the condition for the vanishing coefficient determinant, which reads 22-2.t cos 0 + 1 = O ( ~ z) .
(31)
Thus, the spectra are approximately still ~o=n_+#, and the correction to them is at least of the second order in 6. However, the correction to the eigenvectors u + ( 1 ) is of the first order. From eq. (30) we find that ( ~-22~fl5 "~ u t (1)=_ 2 _ 14-2/~2/25] "
(32)
Finally, after a tedious calculation, from eqs. (25) and (28) we obtain U~ (~) = g " e x p [ - t ( ~ -
T UT25~[ / l ( 222j +j ) f_l,(]2S#t) +- ' 5 (2+_ 1 2 2 /12) C ] S,,1- ) 5~fi(2_+ 1 ) C, ) , 1/~) ] \(u-512fi(2-T-2_+l+5v2fi_5122(l+_2)
(33)
which gives % = ~ [ l - 5 ~ f i ( 2 g l ) ] J . + A - 2 t ) - 5 g \ 2 / ~ T l (2_+_1
q/¢=(2++_l+Sf~z12)J,,+,(-2t)+5( 2~
22)
J,,+/+,(-2t)-St ~'( '~_ + 1
+22)
J,,~ ,
-2t).
l 22(1_+2)) J ~ + , + , ( - 2 1 ) + 5 ( 77777+22(1_+2) .,.a. /-)2 )J,,+~_,(-2t). (,34)
Noticing that the relation 2 2 + 3_+1 -
~2=1 results in
-2_+ 1 v
(35)
one can verify that solutions (33) satisfy relation (9), as expected. In the above we have developed the generating function method to deal with the problem of two-band electrons in a uniform electric field. The method is rather straightforward, and shows easily the existence of two interspaced Stark ladders and the relation between them o)= - ¢ o ( - 5 ) in terms of the parameter 5 describing the a s y m m e t r y of the two bands. Furthermore, we have obtained a perturbation solution of the lowest order in 5. References
[ 1] G.H. Wannier, Elements of solid state theory (Cambridge Univ. Press, Cambridge, 1959 ). [2] H. Fukuyama, R. Bari and H. Fogedby, Phys. Rev. B 8 (1973) 5579. [3] J.B. Krieger and G.J. Iafrate, Phys. Rev. B 33 (1986) 5494; D. Emin and C.F. Hart, Phys. Rev. B 36 (1987) 7353: C.F. Hart and D. Emin, Phys. Rev. B 37 (1988) 6100. [4] X.-G. Zhao, Phys. Lett. A 154 ( 1991 ) 275. 222
219-222
PHYSICS
LETTERS
A
Generating function method for two-band electrons in a uniform electric field Wei-mou
Zheng
Institute of Theoretical Physics, Academia Sinica, Beijing. China Received 9 December I99 I; accepted Communicated by J. Rouquet
for publication
15 January
I992
A generating function method is developed to treat the problem of two-band electrons in a uniform electric field. The method is rather straightforward, and shows easily the existence of two interspaced Stark ladders and a relation between them. A perturbation solution is obtained.
In general, the application of an electric field lifts the degeneracy between local electronic levels, and results in Wannier-Stark localization of the electronic states [ 11. There has been a long-standing interest in understanding this localization [ 2,3 1. Fukuyama, Bari and Fogedby have proposed a two-band tight binding model of one-dimensional crystals in a uniform electric field. The Hamiltonian of the model is
H=-A,=z
m
I~)-<~I-+U-s)i~~
(l~)-(~+ll-+ll+l)-(~l)+d,=~ co
Ib+(~l+ 00
(1)
-17
where II) _ and 1f) + are the normalized local atomic states at site f referring to the lower and upper bands, and + A and t + 6 are the site energies and the nearest-neighbor hopping matrix elements of the two bands. We denote the scaled external field by t = eaE and the interband coupling by v= eDE, where - e, a, E and D are respectively the electronic charge, the lattice spacing, the external field and the dipole matrix element between the two atomic states at the same site referring to the two bands. Without loss of generality we set t= 1, i.e. take c as the unit of energy. In the case of v=O the model reduces to the single-band model, which is solvable. Another solvable case is the symmetric two-band model with 6~0 [2]. Recently, by taking the single-band wave functions as the basis for the Hamiltonian, a perturbation solution to the model ( 1) for a small interband coupling has been discussed by Zhao [ 41. Here we shall apply the generating function method to the model, and, as an example also interesting to physics, discuss the perturbation solution for small 6. Assume that the eigenstate of Hamiltonian ( 1) is
Iw>= ,=z Hlo)
(@/lb- +nlb+) 00
=~I~)
037%9601/92/$
3
(2)
> 05.00 0 1992 Elsevier Science Publishers
(3)
B.V. All rights reserved.
219
Volume 163, number 3
PHYSICS LETTERS A
16 March 1992
where 0; and q/; are the amplitudes of the atomic states [ l ) _ and l l) +. respectively. From eqs. ( 1 ) - ( 3 ) we can derive the coupled equations for 0; and ~,;:
(l-A)Ol+(t-6)(O;+:+O; ~)+t'gtl=to0;,
(l+A)qJl+(t+~)(~';+~+q.'; ~ ) + t ' 0 / = o ~ .
(4t
Let us introduce the following two generating functions, ,)c, [= -,~-
I=
:x
In terms of F ( ~ ) and G(~), eq. (4) may be written as
~F'-AF+ (t-fi)(~+ I/~)F+vG=toF,
~G' +AG+ ( t + c~) (~+ I/~)G+z,F=(oG.
(6
which may further be written in the concise vector form ~U' +1 [ t ( ~ + 1 / ~ ) - t o ] -
[ A + 6 ( ~ + l / l ) ] a - + ; ' ~ r , I,U=O,
(7
where the vector U(~) and the Pauli matrices (including o~,.used below) are
(F(~)~
6,. = (0l
U(~) = \ G ( ~ ) / '
10)'
¢7,.= ( 0 i - ; ) ,
a_ =(10 "
_0)
(8> "
From eqs. (6) we can directly draw two important conclusions. We denote the eigenfunctions for the eigenvalue to by Fo)(~) and G,,~(~), and assume that, as a function of & to = co (fi). The first conclusion is: If to(d) is an eigenvalue, then so is cO= - t o ( - 6 ) . The corresponding eigenfunctions are
F,,,(~:c~)=-G,,)(-1/~; -fi),
G
(9)
where/~o(~; fi) and G
t:,o+,,(~)=~"F,,,(~),
G,~+, (~) = ~G~,,(~).
10)
Thus, generally, the spectrum consists of two interspaced Stark ladders. When fi=0, by direct integration we have the solution to eq. (7): U(~) = ~ o exp[ - t ( ~ - 1/~) exp[ (A~: - vcr,) In ~] U( 1 )
=~¢°exp[-t(~-l/~)][cosh(121n~)+(Aa~-b~) s i n h ( p l n ~ ) ] U ( 1 ) ,
II)
w h e r e / ~ - x / A Z + v 2, A=A/lz, ~=v/lt, and we have made use of the formula 2
2
exp (xn.er) = c o s h x+n.t~ sinh x ,
12 )
with n being a unit vector. From the definition of the generating functions (5) it is obvious that U(@2~') = (~(~). Particularly, setting ~= 1, we have U(e2~i)=U(1) ,
(13)
which, by using eq. ( 11 ), means that
U( 1 ) = 2 ~[cos 0 + i ( z l a : - ~ a ~ ) sin 0] U( 1 ) .
14
where 2 - e - 2 ~ i ' ° and 0 = 2~p. For a non-trivial U( 1 ) to exist, vanishing of the determinant is required: cos 0 + i A sin 0 - 2 -i~sin 0
~,:.0
- ig sin 0
cosO-iAsinO-2
=0,
15
Volume 163,number 3
PHYSICS LETTERSA
16 March 1992
which gives eigenvalues o g ~ = n + A ~ z + v z,
(16)
and the corresponding eigenvectors (17) From eq. ( 11 ) we then find U(~) = ~ e x p [ - t ( ~ - 1/~)] U -+( 1 ) .
(18)
Finally, by noticing the generating function for the Bessel functions J~(~): exP[½t(~-l/()]=
~
~%(t),
(19)
from eqs. (5) we have Ot = gJn+t(-2t),
~ = (A+ 1 )J,+t( - 2 t ) .
(20)
Let us investigate a more interesting case when the parameter of asymmetry 6 ¢ 0. Because of the non-commutability of az and ax eq. (7) now possesses no simple solution. We introduce the operator W= - t ( ~ - 1 / ~ ) +
(o)+Aaz-Vax) In ~.
(21)
Using the formula exp(½i0n-~r)cr exp( -
½iOn.a) = n (n-~r) - n X (nX~r) cos 0+nX~r sin 0,
(22)
we have
#z=e-WazeW=~j(Czu-2)G-if)S2uG+ (ff2+~2C2u)az,
(23)
where
S . - ½(~"-~-"),
C . - ½(~"+~-").
(24)
It can then be verified that the transformation
U=eWu
(25)
converts eq. (7) to the equation for u,
u' =d(1-t-~-2)a~u ,
(26)
which may be solved by the standard perturbation scheme. Thus, we have to the first order in 6, ¢ ¢
u(~)=u(1)+O ~ (l+x-2)~z(X)U(X) d x ~ u ( 1 ) + 6 f (l+x-2)#z(X)U(1) dx. 1
(27)
1
By direct integration we further obtain that
u(~)=u(~ )+tj[~(dt~x+~t7z)(S2~+l+S2u-i)+~S'~(Jt7z-l)ax)-if)(~2~+1+~2'~-l-~)t7y]u(~ ) ~
(28)
where
g~-S./v,
C.-C~/v,
/ i = 4 U / ( 4 / t 2 - 1) .
(29) 221
Volume 163, number 3
PHYSICS LETTERS A
16 March t 992
The eigenvalues can then be determined from condition ( 13 ) in a similar way to that for the case of 5 = 0 . We obtain the equation for u( ! ): u(l ) = 2 - - ' [cos =2
~[cos
O+i(2a:-fJa~)
O+i(~a: -?'a.)
sin 0]{1
sin
-i~(1-22fiS)
sin0
sin 20-izT(cos 2 0 - 1 )a,'] }u( 1 )
O+ 2if¢&J(2a.y+ f)a:) sin O]u( 1 )
. ~{cosO+i(2+2fj2fl5) sinO =z- ~
+Afi[iO(3a,.+~a:)
- i ~ ( l - 2 A f i S ) s i n 0 "~u(l) cos0-i(2+2/~2fiS) sin0) "
(30)
The eigenvalue equation can be derived from the condition for the vanishing coefficient determinant, which reads 22-2.t cos 0 + 1 = O ( ~ z) .
(31)
Thus, the spectra are approximately still ~o=n_+#, and the correction to them is at least of the second order in 6. However, the correction to the eigenvectors u + ( 1 ) is of the first order. From eq. (30) we find that ( ~-22~fl5 "~ u t (1)=_ 2 _ 14-2/~2/25] "
(32)
Finally, after a tedious calculation, from eqs. (25) and (28) we obtain U~ (~) = g " e x p [ - t ( ~ -
T UT25~[ / l ( 222j +j ) f_l,(]2S#t) +- ' 5 (2+_ 1 2 2 /12) C ] S,,1- ) 5~fi(2_+ 1 ) C, ) , 1/~) ] \(u-512fi(2-T-2_+l+5v2fi_5122(l+_2)
(33)
which gives % = ~ [ l - 5 ~ f i ( 2 g l ) ] J . + A - 2 t ) - 5 g \ 2 / ~ T l (2_+_1
q/¢=(2++_l+Sf~z12)J,,+,(-2t)+5( 2~
22)
J,,+/+,(-2t)-St ~'( '~_ + 1
+22)
J,,~ ,
-2t).
l 22(1_+2)) J ~ + , + , ( - 2 1 ) + 5 ( 77777+22(1_+2) .,.a. /-)2 )J,,+~_,(-2t). (,34)
Noticing that the relation 2 2 + 3_+1 -
~2=1 results in
-2_+ 1 v
(35)
one can verify that solutions (33) satisfy relation (9), as expected. In the above we have developed the generating function method to deal with the problem of two-band electrons in a uniform electric field. The method is rather straightforward, and shows easily the existence of two interspaced Stark ladders and the relation between them o)= - ¢ o ( - 5 ) in terms of the parameter 5 describing the a s y m m e t r y of the two bands. Furthermore, we have obtained a perturbation solution of the lowest order in 5. References
[ 1] G.H. Wannier, Elements of solid state theory (Cambridge Univ. Press, Cambridge, 1959 ). [2] H. Fukuyama, R. Bari and H. Fogedby, Phys. Rev. B 8 (1973) 5579. [3] J.B. Krieger and G.J. Iafrate, Phys. Rev. B 33 (1986) 5494; D. Emin and C.F. Hart, Phys. Rev. B 36 (1987) 7353: C.F. Hart and D. Emin, Phys. Rev. B 37 (1988) 6100. [4] X.-G. Zhao, Phys. Lett. A 154 ( 1991 ) 275. 222