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Density of states and localization length in weakly disordered peierls chains
4602
Synthetic Metals, 55-57 (1993) 4602-4607
DENSITY OF STATES AND LOCALIZATION LENGTH IN WEAKLY DISORDERED PEIERLS CHAINS
J. MERTSCHING Max-Planck-Arbeitsgruppe Halbleitertheorie, Hausvogteiplatz 5-7, O-1086 Berlin (FRG)
ABSTRACT The electron density of states and localization length are calculated for weakly disordered dimerized tight-binding chains. Pure bond disorder causes a singular density of states for small bond gaps, and a singular localization length for zero bond gap, whereas larger bond gaps are transformed into pseudogaps. A site gap for diatomic chains is preserved under bond disorder. Site disorder destroys any gap and does not create singularities in the density of states or localization length. These results are compared with those obtained from the continuum approximation with Gaussian disorder, which is not strictly justified for short-ranged disorder potentials. However, the continuum results are valid for weakly disordered chains provided the bond and site disorders in the chain correspond to bond-gap and combined site and site-gap disorders in the continuum model, respectively. KEYWORDS: Disordered Peierls chains.
INTRODUCTION Disordered chains are of great interest and have been studied by a large number of authors. Dyson [1] exactly calculated the vibration spectrum of specially disordered normal chains. His results are directly relevant for the electron density of states of a tight-binding chain with some special bond disorder, and yield a divergent density of states at the central energy E = 0. Normal tight-binding chains with site disorder have been studied by Economou and Papatriantafillou [3], and more analytically for weak disorder and small energies by Kappus and Wegner [4] and Derrida and Gardner [5]. They obtained a slightly enhanced density of states with finite localization lengths having a moderate maximum for E = 0. The disordered dimerized Peierls chain has been studied by Ovchinnikov and Erikhman [6], Hayn and Fischbeck [7, 8], and Wolf and Fesser [9] who invoked the continuum approximation [10], and assumed the disorder to be Gaussian. The continuum approximation, however, is not strictly justified for strongly localized disorder potentials. Therefore, we avoid this approximation and calculate the density of states and localization length for diatomic Peierls chains. Comparing our results with those obtained from the continuum approximation with Gaussian disorder [8, 9], we find agreement provided the parameters are chosen in some non-obvious manner.
0379-6779/93/$6.00
© 1993- Elsevier Sequoia. All rights reserved
4603
TIGHT-BINDING CItAIN AND THE CONTINUUM APPROXIMATION Let us consider a tight-binding chain with the eigenvalue equation -- t u _ l ~ u _
1 -~- (6. u -- E ) ~ ) u - ~ul/)u+l :
0
(1)
where the site energies and the square of the bond energies are
The site gap A . for diatomic chains and the bond gap Ab due to the Peierls distortion give rise + A~ in the ordered chain. The site and bond disorder terms to a total gap of halfwidth ~ ( ~ and ~b~ are assumed to be uncorrelated random variables with the average properties < &. > = 0 ,
< &~&,., > = 71~< ~2 > ~;.,(%.,
(i = .% b)
(3)
where r/~ and r/b are the relative portions of site and bond disorder, respectively (7/~+ 71~= 1). The continuum approximation [10] would be feasible if
(~ = ~ + (-1)~(/x~ + £ ~ ) ,
¢=t+L+~(- 1
l)~(Ab 4- Zxb.)
(4)
where the site, site-gap, bond, and bond-gap disorder terms ~., A.., [.,/~b. are slowly varying functions of the lattice position x = ua (a lattice constant). This condition, however, does not hold for the strongly localized random disorder potentials in (2).
DENSITY OF STATES AND LOCALIZATION LENGTH We assume the energies E,t, As,Ab >_ 0 since the density of states and tile localization length do not depend on their sign. The quantities T~ = --t.~./~b.+l fulfil the recursion relations T~ t~ (,5) Let F+(T) and F_(T) be the normalized distribution functions of T~ for even and odd lattice sites, respectively, in an infinitely long chain. These functions satisfy the integral equations ¢oo
(t ± ~Ab) 2 2
F~CT) =< J_ ~ ( T - E T A ~ - t ~ - T
t,~b 2 ~
~'r'~ dT' >
'~+T~+ J
((i)
We confine ourselves to weak disorder and small energies, i.e. < (~ ><<< ~2 ><( 1 (k > 2) and E, As, Ab << t. Expanding (6) in first order with respect to E,A~,Ab, and < (2 >, and using the transformation T = t tan ~
( - - i < ~ < -~) '
F~(T) =
cos~ ~(~)
(7)
we obtain the differential equations 0 } ZF(~b sin 2~,+ T{sin qb COS 2 ~ - - ~ {sin 4 ~ - ~(3+cos .1 4~p)~-~ 4~y+ ~1 ( 1 - c o s 4 ~ ) ~0) ] w ± ( ~ ) = C
(s)
4604
where
E t<~2>
e - - - ,
~i=
Ai
(9)
t<~2>
and C is an integration constant. The integrated density of states D(E) per lattice site is the number of sign changes of ~b~, i. e. the average number of positive ~. Thus 1 t~r/2
D(E) = -~ Jo
1 {w+(~) + wm(~)} d~ = 2(1+ < ~2 > C)
(10)
and the density of states becomes N(E) = OD(E)/aE = (1/2t)OC/Oe. The localization length is determined from the mean increase of the amplitudes ~b~ [11] in one sublattice: L
=
[~12 ~b - 2 a < In I~b~-' ~/)u+ 1 [ > - 1 = - 2 a [3-~/2 l n l t a n ~ [ { w + ( ~ o ) + w _ ( ~ ) } d ~ + - f <
~
>]-1
(11)
The differential equations (8) may be solved by Fourier expansion
~(~)
=
(~l)"c.~
~'"~ ,
n=-oo
co =
-,
7r
e_,,
=
02)
c;,
which leads to a set of linear equations i [e+ ~n(1 +2y,)]c, + ~5,+i/~b c,-i
- 8(1-2~,)[(n-1)c.-2+(n+l)c.+2l
+ ~c.+a
= C6=,o
(i3) to be numerically solved for the Fourier coefficients c, (n > 0). The integration constant C is then given by (13) with n = 0. From (10) and (11) we finally obtain
La
b r ( D - 2) 1 - < ~22 > [~1 - ie - ~r(i5. + 5b)c, - ¼ _(1 - 2rb)c2 ]
(14)
The Fourier series (12) well converges except for pure bond disorder (r/. = 0), when
N-
1
27r2 ~ 0 e
0_ HO)r2, ~2/7-Z5~)1_2 ' 28b~ V ~
L-
<
4a x OH~[(2x) ,~2 > [ReH-!l)~2x~ 0 ~ ] ~ = ~ 25b~
1-'
(15)
where H denotes the Hankel function.
NUMERICAL RESULTS. COMPARISON WITH THE CONTINUUM APPROXIMATION Fig. 1 shows the results for pure bond disorder (7, = 0) and a finite site gap (5, = ¼). The density of states vanishes for e < 5,, i. e. the site gap persists in the presence of bond disorder. At the band edge e --+ 6,, the density of states diverges for 5b < I and vanishes for ~b > ½, whereas the localization length diverges only for 6b = 0 and becomes L = 2at/Ab independent of the disorder strength for 5b > 0. In the case of finite site disorder (r/8 > 0), the density of states and the localization length do not become singular, as shown in Fig. 2. Site disorder destroys any gap and creates smallenergy states which for sufficiently large gap parameters are more localized than the band states with larger energies. In those cases where res~!'~ ~ the continuum approximation
4605
a
Nt 0.3
0.2 L
/
!
0.1
/
i
0.0
///
i ,"1
// i
1 !
I
I "l
i
L
i
1
i
i
s
i
L
~
L
I
I
t
b
15
L<~2>
a
10
......_---"y.0.0
0.5
.0
1.5
2.0
£
2.5
Fig. 1. (a) Density of states and (b) localization length of bond disordered (71, = 0) diatomic Peierls chains as a function of energy for 6, = 0.25, 3b =0 ( . ), 0.5 (- - -), and 1 ( . . . . ).
4606
O. 2
0
~ .......; ; :::::-:=--~=~. . . . . . . . . .
Nt
0"151
/ y'~'SS ~ - . . . . . . . .
..--.:=_:
// 0.10
/:'
0.(95I/'//' (9.0
8 L<~ 2 > a
(9.5
....
I .0
I .5
2.0
2.5
................
.i:J,l.
7 6 5 4
///
3
-=:::/.:
.....
2 1 0
0.0
t
t
[
I
I
O.B
I
I
I
I
I
1.0
I
I
6°
t
I
I
1.5
I
t
I
t
I
2.~
I
t
I
I
2.5
Fig. 2. (a) Density of states and (b) localization length of site disordered dimerized chains as a function of energy for r/, = 1, = 1, 6b = 0 ( ), r/s = 1, 6, = 6 b = 0.7071 ( - - - - ) , T/,= 1, g , = O , 6 b = 1 ( - - - ) , a n d 7 / , = 0 . 5 , 6 , = 1 , 6 b = O ( . . . . ).
4607
with Gaussian disorder are available [8, 9] we find agreement with our results, provided the parameters of the continuum model are chosen to be
t~=O,
< fk~,>
=
,lbt 2 <~2 >
(16)
Thus the site disorder in the chain corresponds to equal site and site-gap disorder in the continuum model, whereas the bond disorder corresponds to the continuum bond-gap disorder ~xb,. The continuum approximation may fail in the case of strong disorder, although the calculations are done for any strength of the Gaussian disorder.
SHORTCOMINGS In our investigation we have assumed some type of stochastic disorder, whereas it should be determined selfconsistently from the electron-lattice interaction as in the model of Su, Schrieffer, and Heeger [12]. Such selfconsistent calculations have been done only for impurities in finite chains [13] which cannot reveal singularities in the density of states, but are difficult to perform for infinite chains. Bond disorder arises in vibrating chains at finite temperatures. It is not quite clear, however, whether the density of states may really diverge in this case. A more detailed paper is submitted to phys. stat. sol. (b).
ACKNOWLEDGEMENTS I am indebted to Dr. R. Zimmermann, Dr. H. Puff, and Dr. H.-W. Streitwolf for interesting discussions and their critical comments.
REFERENCES
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
F. J. Dyson, Phys. Rev., 92 (1953) 1331. T. P. Eggarter and R. Riedinger, Phys. Rev. B, 18 (1978) 569. E. N. Economou and C. Papatriantafillou, Solid State Comm.,ll (1972) 197. M. Kappus and F. Wegner, Z. Physik B - Cond. Matter, 45 (1981) 15. B. Derrida and E. Gardner, J. Physique, 45 (1984) 1283. A. A. Ovchinnikov and N. S. Erikhman, Zh. eksper, teor. Fiz., 73 (1977) 650. R. Hayn and H. J. Fischbeck, Z. Phys. B - Cond. Matter, 76 (1989) 33. H. J. Fischbeck and R. Hayn, phys. stat. sol. (b), 158 (1990) 565. M. Wolf and K. Fesser, to appear in Annalen der Physik. H. Takayama, Y. R. Lin-Liu, and K. Maki, Phys. Rev. B, 21 (1980) 2388. [11] R. E. Borland, Proc. Roy. Soc. A, 274 (1963) 529. [12] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B, 22 (1980) 2099. [13] K. Harigaya and A. Terai, Phys. Rev. B, 44 (1991) 7835.
Synthetic Metals, 55-57 (1993) 4602-4607
DENSITY OF STATES AND LOCALIZATION LENGTH IN WEAKLY DISORDERED PEIERLS CHAINS
J. MERTSCHING Max-Planck-Arbeitsgruppe Halbleitertheorie, Hausvogteiplatz 5-7, O-1086 Berlin (FRG)
ABSTRACT The electron density of states and localization length are calculated for weakly disordered dimerized tight-binding chains. Pure bond disorder causes a singular density of states for small bond gaps, and a singular localization length for zero bond gap, whereas larger bond gaps are transformed into pseudogaps. A site gap for diatomic chains is preserved under bond disorder. Site disorder destroys any gap and does not create singularities in the density of states or localization length. These results are compared with those obtained from the continuum approximation with Gaussian disorder, which is not strictly justified for short-ranged disorder potentials. However, the continuum results are valid for weakly disordered chains provided the bond and site disorders in the chain correspond to bond-gap and combined site and site-gap disorders in the continuum model, respectively. KEYWORDS: Disordered Peierls chains.
INTRODUCTION Disordered chains are of great interest and have been studied by a large number of authors. Dyson [1] exactly calculated the vibration spectrum of specially disordered normal chains. His results are directly relevant for the electron density of states of a tight-binding chain with some special bond disorder, and yield a divergent density of states at the central energy E = 0. Normal tight-binding chains with site disorder have been studied by Economou and Papatriantafillou [3], and more analytically for weak disorder and small energies by Kappus and Wegner [4] and Derrida and Gardner [5]. They obtained a slightly enhanced density of states with finite localization lengths having a moderate maximum for E = 0. The disordered dimerized Peierls chain has been studied by Ovchinnikov and Erikhman [6], Hayn and Fischbeck [7, 8], and Wolf and Fesser [9] who invoked the continuum approximation [10], and assumed the disorder to be Gaussian. The continuum approximation, however, is not strictly justified for strongly localized disorder potentials. Therefore, we avoid this approximation and calculate the density of states and localization length for diatomic Peierls chains. Comparing our results with those obtained from the continuum approximation with Gaussian disorder [8, 9], we find agreement provided the parameters are chosen in some non-obvious manner.
0379-6779/93/$6.00
© 1993- Elsevier Sequoia. All rights reserved
4603
TIGHT-BINDING CItAIN AND THE CONTINUUM APPROXIMATION Let us consider a tight-binding chain with the eigenvalue equation -- t u _ l ~ u _
1 -~- (6. u -- E ) ~ ) u - ~ul/)u+l :
0
(1)
where the site energies and the square of the bond energies are
The site gap A . for diatomic chains and the bond gap Ab due to the Peierls distortion give rise + A~ in the ordered chain. The site and bond disorder terms to a total gap of halfwidth ~ ( ~ and ~b~ are assumed to be uncorrelated random variables with the average properties < &. > = 0 ,
< &~&,., > = 71~< ~2 > ~;.,(%.,
(i = .% b)
(3)
where r/~ and r/b are the relative portions of site and bond disorder, respectively (7/~+ 71~= 1). The continuum approximation [10] would be feasible if
(~ = ~ + (-1)~(/x~ + £ ~ ) ,
¢=t+L+~(- 1
l)~(Ab 4- Zxb.)
(4)
where the site, site-gap, bond, and bond-gap disorder terms ~., A.., [.,/~b. are slowly varying functions of the lattice position x = ua (a lattice constant). This condition, however, does not hold for the strongly localized random disorder potentials in (2).
DENSITY OF STATES AND LOCALIZATION LENGTH We assume the energies E,t, As,Ab >_ 0 since the density of states and tile localization length do not depend on their sign. The quantities T~ = --t.~./~b.+l fulfil the recursion relations T~ t~ (,5) Let F+(T) and F_(T) be the normalized distribution functions of T~ for even and odd lattice sites, respectively, in an infinitely long chain. These functions satisfy the integral equations ¢oo
(t ± ~Ab) 2 2
F~CT) =< J_ ~ ( T - E T A ~ - t ~ - T
t,~b 2 ~
~'r'~ dT' >
'~+T~+ J
((i)
We confine ourselves to weak disorder and small energies, i.e. < (~ ><<< ~2 ><( 1 (k > 2) and E, As, Ab << t. Expanding (6) in first order with respect to E,A~,Ab, and < (2 >, and using the transformation T = t tan ~
( - - i < ~ < -~) '
F~(T) =
cos~ ~(~)
(7)
we obtain the differential equations 0 } ZF(~b sin 2~,+ T{sin qb COS 2 ~ - - ~ {sin 4 ~ - ~(3+cos .1 4~p)~-~ 4~y+ ~1 ( 1 - c o s 4 ~ ) ~0) ] w ± ( ~ ) = C
(s)
4604
where
E t<~2>
e - - - ,
~i=
Ai
(9)
t<~2>
and C is an integration constant. The integrated density of states D(E) per lattice site is the number of sign changes of ~b~, i. e. the average number of positive ~. Thus 1 t~r/2
D(E) = -~ Jo
1 {w+(~) + wm(~)} d~ = 2(1+ < ~2 > C)
(10)
and the density of states becomes N(E) = OD(E)/aE = (1/2t)OC/Oe. The localization length is determined from the mean increase of the amplitudes ~b~ [11] in one sublattice: L
=
[~12 ~b - 2 a < In I~b~-' ~/)u+ 1 [ > - 1 = - 2 a [3-~/2 l n l t a n ~ [ { w + ( ~ o ) + w _ ( ~ ) } d ~ + - f <
~
>]-1
(11)
The differential equations (8) may be solved by Fourier expansion
~(~)
=
(~l)"c.~
~'"~ ,
n=-oo
co =
-,
7r
e_,,
=
02)
c;,
which leads to a set of linear equations i [e+ ~n(1 +2y,)]c, + ~5,+i/~b c,-i
- 8(1-2~,)[(n-1)c.-2+(n+l)c.+2l
+ ~c.+a
= C6=,o
(i3) to be numerically solved for the Fourier coefficients c, (n > 0). The integration constant C is then given by (13) with n = 0. From (10) and (11) we finally obtain
La
b r ( D - 2) 1 - < ~22 > [~1 - ie - ~r(i5. + 5b)c, - ¼ _(1 - 2rb)c2 ]
(14)
The Fourier series (12) well converges except for pure bond disorder (r/. = 0), when
N-
1
27r2 ~ 0 e
0_ HO)r2, ~2/7-Z5~)1_2 ' 28b~ V ~
L-
<
4a x OH~[(2x) ,~2 > [ReH-!l)~2x~ 0 ~ ] ~ = ~ 25b~
1-'
(15)
where H denotes the Hankel function.
NUMERICAL RESULTS. COMPARISON WITH THE CONTINUUM APPROXIMATION Fig. 1 shows the results for pure bond disorder (7, = 0) and a finite site gap (5, = ¼). The density of states vanishes for e < 5,, i. e. the site gap persists in the presence of bond disorder. At the band edge e --+ 6,, the density of states diverges for 5b < I and vanishes for ~b > ½, whereas the localization length diverges only for 6b = 0 and becomes L = 2at/Ab independent of the disorder strength for 5b > 0. In the case of finite site disorder (r/8 > 0), the density of states and the localization length do not become singular, as shown in Fig. 2. Site disorder destroys any gap and creates smallenergy states which for sufficiently large gap parameters are more localized than the band states with larger energies. In those cases where res~!'~ ~ the continuum approximation
4605
a
Nt 0.3
0.2 L
/
!
0.1
/
i
0.0
///
i ,"1
// i
1 !
I
I "l
i
L
i
1
i
i
s
i
L
~
L
I
I
t
b
15
L<~2>
a
10
......_---"y.0.0
0.5
.0
1.5
2.0
£
2.5
Fig. 1. (a) Density of states and (b) localization length of bond disordered (71, = 0) diatomic Peierls chains as a function of energy for 6, = 0.25, 3b =0 ( . ), 0.5 (- - -), and 1 ( . . . . ).
4606
O. 2
0
~ .......; ; :::::-:=--~=~. . . . . . . . . .
Nt
0"151
/ y'~'SS ~ - . . . . . . . .
..--.:=_:
// 0.10
/:'
0.(95I/'//' (9.0
8 L<~ 2 > a
(9.5
....
I .0
I .5
2.0
2.5
................
.i:J,l.
7 6 5 4
///
3
-=:::/.:
.....
2 1 0
0.0
t
t
[
I
I
O.B
I
I
I
I
I
1.0
I
I
6°
t
I
I
1.5
I
t
I
t
I
2.~
I
t
I
I
2.5
Fig. 2. (a) Density of states and (b) localization length of site disordered dimerized chains as a function of energy for r/, = 1, = 1, 6b = 0 ( ), r/s = 1, 6, = 6 b = 0.7071 ( - - - - ) , T/,= 1, g , = O , 6 b = 1 ( - - - ) , a n d 7 / , = 0 . 5 , 6 , = 1 , 6 b = O ( . . . . ).
4607
with Gaussian disorder are available [8, 9] we find agreement with our results, provided the parameters of the continuum model are chosen to be
t~=O,
< fk~,>
=
,lbt 2 <~2 >
(16)
Thus the site disorder in the chain corresponds to equal site and site-gap disorder in the continuum model, whereas the bond disorder corresponds to the continuum bond-gap disorder ~xb,. The continuum approximation may fail in the case of strong disorder, although the calculations are done for any strength of the Gaussian disorder.
SHORTCOMINGS In our investigation we have assumed some type of stochastic disorder, whereas it should be determined selfconsistently from the electron-lattice interaction as in the model of Su, Schrieffer, and Heeger [12]. Such selfconsistent calculations have been done only for impurities in finite chains [13] which cannot reveal singularities in the density of states, but are difficult to perform for infinite chains. Bond disorder arises in vibrating chains at finite temperatures. It is not quite clear, however, whether the density of states may really diverge in this case. A more detailed paper is submitted to phys. stat. sol. (b).
ACKNOWLEDGEMENTS I am indebted to Dr. R. Zimmermann, Dr. H. Puff, and Dr. H.-W. Streitwolf for interesting discussions and their critical comments.
REFERENCES
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
F. J. Dyson, Phys. Rev., 92 (1953) 1331. T. P. Eggarter and R. Riedinger, Phys. Rev. B, 18 (1978) 569. E. N. Economou and C. Papatriantafillou, Solid State Comm.,ll (1972) 197. M. Kappus and F. Wegner, Z. Physik B - Cond. Matter, 45 (1981) 15. B. Derrida and E. Gardner, J. Physique, 45 (1984) 1283. A. A. Ovchinnikov and N. S. Erikhman, Zh. eksper, teor. Fiz., 73 (1977) 650. R. Hayn and H. J. Fischbeck, Z. Phys. B - Cond. Matter, 76 (1989) 33. H. J. Fischbeck and R. Hayn, phys. stat. sol. (b), 158 (1990) 565. M. Wolf and K. Fesser, to appear in Annalen der Physik. H. Takayama, Y. R. Lin-Liu, and K. Maki, Phys. Rev. B, 21 (1980) 2388. [11] R. E. Borland, Proc. Roy. Soc. A, 274 (1963) 529. [12] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. B, 22 (1980) 2099. [13] K. Harigaya and A. Terai, Phys. Rev. B, 44 (1991) 7835.