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Metal-insulator transitions and electron-phonon interactions in organic conductors
Synthetic Metals, 55-57 (1993) 4660-4665
4660
METAL-INSULATOR TRANSITIONS AND ELECTRON-PHONON INTERACTIONS IN ORGANIC CONDUCTORS
KIM-CHAU UNG and S. MAZUMDAR Physics Department, University of Arizona, Tucson, AZ 85721, U.S.A. D.K. CAMPBELL and H.Q. LIN Physics Department, University of Illinois, Urbana-Champalgn, IL 61801, U.S.A.
ABSTRACT Commensurate 1/4 and 1/3-filled bands in which the electrons are coupled to both intersite phonons and intramolecular vibrations are investigated. Unlike in the 1/2-filled band, the bond order wave and the charge density wave coexist and interact cooperatively for all values of the two electron-phonon coupling constants. In spite of the coexistence of the two kinds of density waves, solitons have charges that axe rational fractions. The relevance of the cooperative interaction in non-half-filled bands to the metal-insulator transitions in segregated stack charge-transfer solids is discussed.
INTRODUCTION Conducting segregated stack organic charge-transfer solids with less than half-filled bands undergo metal-insulator transitions at low temperatures that are driven by both electron-phonon and electron-electron interactions. Electron-moleculax vibration (e-mv) couplings play a strong role in the metal-insulator transitions in these molecular solids [1], leading to intrasite charge density waves (CDW). Experimental evidence also exist for periodic modulations of the intermoleculax distances [2], and therefore, bond order waves (BOW), that arise from intersite electron-phonon (e-ph) couplings. Indeed, the two kinds of ordering seem to coexist in real materials. In the present paper we focus on commensurate but non-half-filled one-dimensional bands in the presence of both kinds of electron-phonon couplings. The competition between the BOW and the CDW has been investigated only within the halffilled band. For noninteracting electrons, the two orderings are competing, and the BOW and the CDW coexist in a very narrow region [3]. This result changes only slightly when electron-electron interactions are included [4]. Within this narrow coexistence region, solitons with irrational charge are found [3]. We show in the present paper that this result is peculiar to the half-filled band. For commensuaxbility > 2, the BOW and the CDW always coexist within one-electron theory and interact cooperatively. Thus in the organic conductors, which presumably have weak e-ph couplings, the e-mv coupling can be the driving force for the BOW. We also investigate solitons in these systems,
0379-6779/93/$6.00
© 1 9 9 3 - Elsevier Sequoia. All rights reserved
4661
and show that in spite of the coexistence of the B O W and the CDW, solitons have charges that are rational fractions [5,6] for commensurability 3 and 4. It is well-known that for nonzero Coulomb interactions, the half-filled band is unique as a Mott insulator.
Our work indicates that even in
the absence of the electron-electron interaction, but with two different kinds of electron-phonon couplings, the behavior of the half-filled band is different from other band fillings.
T I i E O R E T I C A L MODEL The theoretical model we investigate is written as H = - E
+ + ~v~c+~c~ } {[t - o~(un+l - Un)][C+aCn+l,a d- ¢n+l,aCna]
n'~ 31_ 1 E [ / ( I ( U n + I n
(l) -- Ztn)2 ql- g2v
2]
where c+a creates an electron of spin a on the nth site, t is the nearest neighbor hopping integral, a and ~ are the e-ph and the e-my couplings, u , is the displacement of the nth molecular unit along the chain axis, v~ is an internal molecular distortion, and K t and K2 are the corresponding elastic constants. We choose bandfillings of 1/4 and 1/3, and are interested in the ground state broken symmetry as well as the nature of kink solutions. We emphasize that Eq. (1) is quite different from the model for M-x chains that have been studied by Gammel et al. [7].
G R O U N D STATE B R O K E N S Y M M E T R I E S 1/4-filled band For weak couplings, we can write u,~ and vn in the usual forms involving first harmonics of the
2kF disortion only, viz., u,~ = uo cos(2kFna -- 0), vn = vo cos(2kFna -- ¢), where k F = r / 4 a , a being the lattice constant. We have not included the mean intramolecular displacement in our definition of vn, as this is merely a constant that would cancel if a term containing the chemical potential is included in Eq. (1). Consistent with our weak coupling assumption, we have not included tile second harmonic 4kF component of the displacements, and indeed our configuration space calculation (see below) shows that this is tiny compared to the 2kF-component. We now make the standard transformation from site to band orbitals. The electronic energies at each wave vector k are obtained as solutions to the equation
A4 _ A2k(4t2 + 4a2u2 + fl2v~o) + A k [ S t c ~ u o v o s i n ( ¢ _ 8)] + g~l 4v0(14 _ c o s 4 ¢ ) + 2 c ~ 4 U 4 o ( l _ c o s 4 8 ) + sin 2 2k[4t 4 - 8t2c~2u~ + 2a4u~(1 + cos 4R)] + 2(afluovo) 2 sin 20 s i n 2 ¢ = 0 .
(2)
For fi = 0, Eq. (2) is identical to that in Ref. [6]. The total energy ET is then obtained from
ET = - 2 E
Ak + Eet,stic
(3)
k6occ where ET is to be minimized with respect to both 0 and ¢.
The energy minima occur for
(0, ¢) = (0, 7r/2), (7r/2, r ) , ( r , 37r/2) and (37r/2, 27r), respectively. The strict phase relationship indicates coexistence of the B O W and the CDW. The phase relationship is shown in Fig. 1, where we show schematically the bond superalternation pattern and the intrasite charge densities. The BOW drives a superalternation with bond lengths a + ~, a + ~, a - ~, a - ~. The charge densities are
4662
Fig. 1. The bond superalternation pattern and the charge densitites for the l/4-filled band with nonzero e-ph and e-mv coupling constants. The lengths of the vertical bars on the atoms axe measures of the local charge densities.
b + c,/~,p - ~,/~, with the largest charge density occurring on the site that lies at the vertex of the two strong bonds. In order to prove the cooperative interaction between the BOW and the CDW we have solved the Hamiltoniaa in configuration space, by adopting the self-consistent-field (SCF) technique of Ono et al. [6], which involves solving for un and vn in configuration space. Instead of presenting the results in terms of un and vn we show our results for the bond order order parameter b, and the charge density order parameter Pn, given by b,~ : b + bo cos( 2kFna - 9b ) + ( - 1 ) n b ,
(4a)
p~ = # + p, cos(2kFna - Cp) + ( - 1 ) " p ,
(4b)
where the last terms on the right-hand side are the 4kF-components. In all cases the 4kF-components were found to be tiny justifying the assumption made in the k-space calculation above. In Figs. 2(a) and (b) we have plotted the amplitudes b0 and p0 of the 2kF BOW and the CDW against A~ and A~, respectively, where A~ = f12/I(2t and A~ : a ~ / K l t . Notice that b0 is expected to increase with a, but it also increases with /3, and is nonzero even when a = 0 but B ~t 0. Similarly, the CDW exists even when fl is zero, and increases with ~. These results prove that the two density waves interact in a cooperative manner.
•4
.........
I'''''""]'""''"I
'~ . . . . . . . I ' " ' ' " " I
.........
.6
I.........
.5
b03~
~
.4
Po .3
.2 .1 .1 .0
~''~ . . . . . . . ; . . . . . . . . .[. . . ÷ ' " 5 i . . . . . . . . . .I . . . . . . . . . . .I . . . . . . . . . . .I . . . .
.00
.09
.18
.27~.36 P
.45
.54
.63
.0 .00
.16
.32
X
.48
.64
.80
a
Fig. 2(a). The amplitude of the BOW in the 1/4-filled band vs the dimensionless e-mv coupling constant ( A :An = 0; • : An = 0.05; * : A ~ = 0.2; • : A~ = 0.45). Fig. 2(b). The amplitude of the CDW in the 1/4-filled band vs the dimensionless e-ph coupling constant ( A : A~ = 0; * : )~ = 0.i; * : A~ = 0.4; • : A~ = 0.9).
4663
1/3-filled b a n d T h e functional forms for un and vn are still the same. T h e electronic H a m i l t o n i a n now is a 3 x 3 m a t r i x whose eigenvalues are obtained from
A~ - Ak(3t 2 + 7 a Uo ~- 7P Vo) +
¢-
¢) + 2cos
- i r a Uo -
+ •v° [9aZu2o cos(¢ 4- 20) + 3 V ~ tano sin(0 - ¢)] = 0
(5)
Once again the total energy, electronic plus elastic, is found as a function of O and 0. T h e s e results are s u m m a r i z e d in Fig. 3, where the phase relationship again indicates coexistence. A peak in the charge density occurs on the site which is at t h e vertex of two strong bonds.
Fig. 3. T h e b o n d s u p e r a l t e r n a t i o n p a t t e r n and the charge densities for the 1/3-filled band. T h e n o t a t i o n s are t h e s a m e as in Fig. 1. As with t h e 1/4-filled band we have calculated b0 and P0 while varying c~ a n d ft. T h e s e results are shown in Figs 4(a) and (b), and are found to be similar to those in Figs. 2(a) and (b) respectively. While we have not been able to find a general proof, we believe t h a t a cooperative interaction is true for all bandfillings other t h a n 1/2. W i t h increasing c o m m e n s u r a b i l t y , one gradually reaches the i n c o m m e n s u r a t e limit, where no distinction can be m a d e between the B O W and t h e C D W .
.4
.........
i .........
1 .........
T .........
I .........
I .........
.6
; .........
......
I .........
i .........
I .........
I .........
(b)
.5
!
.3 .4
bo
Po
.2
.3 .2
.! .1
J .0
,,-~ ......
.00
.; ........
.09
.,.,',,,,I
.18
.........
.27
t .........
.36
X~
Fig. 4(a). T h e a m p l i t u d e c o n s t a n t ( A : A a = 0; • : the C D W in t h e 1/3-filled 0.1; , : A~ = 0.4; • : , ~ =
i ..................
.45
.54
.63
.0 .00
.16
.32
X
.48
.64
.80
of t h e B O W in the 1/3-filled b a n d vs t h e dimensionless e-mv coupling Aa = 0.05; * : A s = 0.2; • : A,~ = 0.45). Fig. 4(b). T h e a m p l i t u d e of b a n d vs the dimensionless e-ph coupling c o n s t a n t ( A : Af~ = 0; • : An = 0.9).
4664
Kink Excitations In t h e case of t h e 1/2-filled b a n d , t h e narrow coexistence region containing b o t h t h e B O W and t h e C D W h a s soliton excitations t h a t have irrational charge. Similar results have been obtained [7] for t h e (AB):c polymer, where, however, t h e coexistence region is broader. In t h e case of t h e 1/4- and 1/3-filled b a n d s , solitons have fractional charge when only one kind of density wave is present [6]. Clearly it is of interest to d e t e r m i n e soliton charges here. T h e soliton excitations are studied using t h e S C F technique of O n o et al. [6]. In order to determine w h e t h e r kinks have irrationM charge, we have studied t h e two-kink cases. For charges to be irrational, t h e excess charges on the two kinks m u s t be unequal. We define s m o o t h e d charge densities/~n as (6a)
~,~ = ¼ [0.5p,,_2 + p ~ - i + p~ + p.+~ + 0.5p.+2] Pa
=
1
[,On_l
+
Pn
+
(65)
/On+i]
for t h e 1/4-filled a n d 1/3-filled b a n d s respectively. T h e s m o o t h e d densities of t h e excess electrons are
*p(n) -- ~ - 1/2 and t~p(n) = ~,~ - 2/3. T h e self-consistent calculations were done within periodic b o u n d a r y condition, with 120 electrons a n d 160 electrons respectively on 238 sites, t h e r e b y giving two charged kinks in b o t h cases. For comparison, t h e calculations were done also for t h e (AB)~:
.07
........
i ......
~,l
.........
i .........
i .........
i .........
.06 .05 .04 dp .03 .02 .ol .oo
............ £ 1 ..... 40
o
80
120 site
160
200
240
n .030
.025 :. ........ , ......... , ......... , ........ ~......... , .........
(b .025
.020
.020 6p .015
.015 5p .010
.010 .005 .000
.005 0
40
80
t20 160 site n
200
240
.000
4O
ao
120 t60 SlLe n
200
240
Fig. 5. T h e excess charge densities on the solitons for the two-kink case for (a) t h e (AB)x polymer, (b) t h e 1/4-filled b a n d ( A = site 4n + 1,o = site 4n + 2 , * = site 4n + 3, It = site 4n)) a n d (c) t h e 1/3-filled b a n d (& = site 3n + 1 , . = site 3n + 2 , * = site 3n). In (b) and (c) t h e overall charge densities are obtained by s u m m i n g over all sites.
4665
polymer with two kinks, by adding alternating site energies to Eq. (1) while removing the e-mv coupling. In Fig. 5(a) we show the excess charges for the half-filled
(AB)~: system,
for a site energy
difference of 0.1 in units of t. Note that the charges on the kinks are very different, indicating irrational charges. In Fig. 5(b) we have plotted the excess charges for the 1/4-filled band with two kinks. The overall charge is obtained by summing over the 4nth, (4n + 1)th, (4n + 2 ) t h and (4n + 3)th sites for each kink. It should be obvious that the overall charges of the two kinks axe equal. The same is true for the the two kinks in the 1/3-filled band in Fig. 5(c). We therefore conclude that in spite of the coexistence of the BOW a~ld the CDW, the charges on the kinks are rational fractions.
APPLICATION TO ORGANIC CONDUCTORS The most important conclusion of our work is that the BOW is always accompanied by the CDW and vice versa. In the case of organic conductors, intermoleculax distances along the stack axes are 3 - 3.5A, only slightly smaller than the sum of the van der Waals' radii. Thus the e-ph coupling should be rather weak. In spite of this, considerable modulations of intermoleculax distances are common below the metal-insulator transitions. Our work is able to explain this. The e-my coupling can drive both a CDW and a BOW, and the latter leads to a modulation of the intermolecular distances.
REFERENCES 1 M.J. Rice, Phys. Rev. Lett., 37 (1976) 36; R. Bozio and C. Pecile, in L. Alc~cer (ed.), The Physics and Chemistry of Low Dimensional Solids, Reidel, Dordrecht, 1980, pp. 165-186. 2 S. Huizinga et ai., Phys. Rev. B, 25 (1982) 1717. 3 S. Kivelson, Phys. Rev. B, 28 (1983) 2653 ; W.P. Sn, Solid St. Commun., 48 (1983) 479. 4 S. Mazumdax and D.K. Campbell, Phys. Rev. Lett., 55 (1985) 2067; A. Painelli and A. Girlando, Phys. Rev. B, 45 (1992) 8913. 5 W.P. Su, Phys. Scripta, 32 (1985) 34. 6 Y. Ono, Y. Ohfuti and A. Terai, J. Phys. Soc. (Japan), 54 (1985) 2641; Y. Ohfuti and Y. Ono, ibid., 55 (1986) 4323. 7 J.T. Gammel et al., Phys. Rev. B, 45 (1992) 6408. 8 E.J. Mele and M.J. Rice, Phys. Rev. Lett., 49 (1982) 1455; D.K. Campbell, Phys. Rev. Lett., 50 (1983) 865.
4660
METAL-INSULATOR TRANSITIONS AND ELECTRON-PHONON INTERACTIONS IN ORGANIC CONDUCTORS
KIM-CHAU UNG and S. MAZUMDAR Physics Department, University of Arizona, Tucson, AZ 85721, U.S.A. D.K. CAMPBELL and H.Q. LIN Physics Department, University of Illinois, Urbana-Champalgn, IL 61801, U.S.A.
ABSTRACT Commensurate 1/4 and 1/3-filled bands in which the electrons are coupled to both intersite phonons and intramolecular vibrations are investigated. Unlike in the 1/2-filled band, the bond order wave and the charge density wave coexist and interact cooperatively for all values of the two electron-phonon coupling constants. In spite of the coexistence of the two kinds of density waves, solitons have charges that axe rational fractions. The relevance of the cooperative interaction in non-half-filled bands to the metal-insulator transitions in segregated stack charge-transfer solids is discussed.
INTRODUCTION Conducting segregated stack organic charge-transfer solids with less than half-filled bands undergo metal-insulator transitions at low temperatures that are driven by both electron-phonon and electron-electron interactions. Electron-moleculax vibration (e-mv) couplings play a strong role in the metal-insulator transitions in these molecular solids [1], leading to intrasite charge density waves (CDW). Experimental evidence also exist for periodic modulations of the intermoleculax distances [2], and therefore, bond order waves (BOW), that arise from intersite electron-phonon (e-ph) couplings. Indeed, the two kinds of ordering seem to coexist in real materials. In the present paper we focus on commensurate but non-half-filled one-dimensional bands in the presence of both kinds of electron-phonon couplings. The competition between the BOW and the CDW has been investigated only within the halffilled band. For noninteracting electrons, the two orderings are competing, and the BOW and the CDW coexist in a very narrow region [3]. This result changes only slightly when electron-electron interactions are included [4]. Within this narrow coexistence region, solitons with irrational charge are found [3]. We show in the present paper that this result is peculiar to the half-filled band. For commensuaxbility > 2, the BOW and the CDW always coexist within one-electron theory and interact cooperatively. Thus in the organic conductors, which presumably have weak e-ph couplings, the e-mv coupling can be the driving force for the BOW. We also investigate solitons in these systems,
0379-6779/93/$6.00
© 1 9 9 3 - Elsevier Sequoia. All rights reserved
4661
and show that in spite of the coexistence of the B O W and the CDW, solitons have charges that are rational fractions [5,6] for commensurability 3 and 4. It is well-known that for nonzero Coulomb interactions, the half-filled band is unique as a Mott insulator.
Our work indicates that even in
the absence of the electron-electron interaction, but with two different kinds of electron-phonon couplings, the behavior of the half-filled band is different from other band fillings.
T I i E O R E T I C A L MODEL The theoretical model we investigate is written as H = - E
+ + ~v~c+~c~ } {[t - o~(un+l - Un)][C+aCn+l,a d- ¢n+l,aCna]
n'~ 31_ 1 E [ / ( I ( U n + I n
(l) -- Ztn)2 ql- g2v
2]
where c+a creates an electron of spin a on the nth site, t is the nearest neighbor hopping integral, a and ~ are the e-ph and the e-my couplings, u , is the displacement of the nth molecular unit along the chain axis, v~ is an internal molecular distortion, and K t and K2 are the corresponding elastic constants. We choose bandfillings of 1/4 and 1/3, and are interested in the ground state broken symmetry as well as the nature of kink solutions. We emphasize that Eq. (1) is quite different from the model for M-x chains that have been studied by Gammel et al. [7].
G R O U N D STATE B R O K E N S Y M M E T R I E S 1/4-filled band For weak couplings, we can write u,~ and vn in the usual forms involving first harmonics of the
2kF disortion only, viz., u,~ = uo cos(2kFna -- 0), vn = vo cos(2kFna -- ¢), where k F = r / 4 a , a being the lattice constant. We have not included the mean intramolecular displacement in our definition of vn, as this is merely a constant that would cancel if a term containing the chemical potential is included in Eq. (1). Consistent with our weak coupling assumption, we have not included tile second harmonic 4kF component of the displacements, and indeed our configuration space calculation (see below) shows that this is tiny compared to the 2kF-component. We now make the standard transformation from site to band orbitals. The electronic energies at each wave vector k are obtained as solutions to the equation
A4 _ A2k(4t2 + 4a2u2 + fl2v~o) + A k [ S t c ~ u o v o s i n ( ¢ _ 8)] + g~l 4v0(14 _ c o s 4 ¢ ) + 2 c ~ 4 U 4 o ( l _ c o s 4 8 ) + sin 2 2k[4t 4 - 8t2c~2u~ + 2a4u~(1 + cos 4R)] + 2(afluovo) 2 sin 20 s i n 2 ¢ = 0 .
(2)
For fi = 0, Eq. (2) is identical to that in Ref. [6]. The total energy ET is then obtained from
ET = - 2 E
Ak + Eet,stic
(3)
k6occ where ET is to be minimized with respect to both 0 and ¢.
The energy minima occur for
(0, ¢) = (0, 7r/2), (7r/2, r ) , ( r , 37r/2) and (37r/2, 27r), respectively. The strict phase relationship indicates coexistence of the B O W and the CDW. The phase relationship is shown in Fig. 1, where we show schematically the bond superalternation pattern and the intrasite charge densities. The BOW drives a superalternation with bond lengths a + ~, a + ~, a - ~, a - ~. The charge densities are
4662
Fig. 1. The bond superalternation pattern and the charge densitites for the l/4-filled band with nonzero e-ph and e-mv coupling constants. The lengths of the vertical bars on the atoms axe measures of the local charge densities.
b + c,/~,p - ~,/~, with the largest charge density occurring on the site that lies at the vertex of the two strong bonds. In order to prove the cooperative interaction between the BOW and the CDW we have solved the Hamiltoniaa in configuration space, by adopting the self-consistent-field (SCF) technique of Ono et al. [6], which involves solving for un and vn in configuration space. Instead of presenting the results in terms of un and vn we show our results for the bond order order parameter b, and the charge density order parameter Pn, given by b,~ : b + bo cos( 2kFna - 9b ) + ( - 1 ) n b ,
(4a)
p~ = # + p, cos(2kFna - Cp) + ( - 1 ) " p ,
(4b)
where the last terms on the right-hand side are the 4kF-components. In all cases the 4kF-components were found to be tiny justifying the assumption made in the k-space calculation above. In Figs. 2(a) and (b) we have plotted the amplitudes b0 and p0 of the 2kF BOW and the CDW against A~ and A~, respectively, where A~ = f12/I(2t and A~ : a ~ / K l t . Notice that b0 is expected to increase with a, but it also increases with /3, and is nonzero even when a = 0 but B ~t 0. Similarly, the CDW exists even when fl is zero, and increases with ~. These results prove that the two density waves interact in a cooperative manner.
•4
.........
I'''''""]'""''"I
'~ . . . . . . . I ' " ' ' " " I
.........
.6
I.........
.5
b03~
~
.4
Po .3
.2 .1 .1 .0
~''~ . . . . . . . ; . . . . . . . . .[. . . ÷ ' " 5 i . . . . . . . . . .I . . . . . . . . . . .I . . . . . . . . . . .I . . . .
.00
.09
.18
.27~.36 P
.45
.54
.63
.0 .00
.16
.32
X
.48
.64
.80
a
Fig. 2(a). The amplitude of the BOW in the 1/4-filled band vs the dimensionless e-mv coupling constant ( A :An = 0; • : An = 0.05; * : A ~ = 0.2; • : A~ = 0.45). Fig. 2(b). The amplitude of the CDW in the 1/4-filled band vs the dimensionless e-ph coupling constant ( A : A~ = 0; * : )~ = 0.i; * : A~ = 0.4; • : A~ = 0.9).
4663
1/3-filled b a n d T h e functional forms for un and vn are still the same. T h e electronic H a m i l t o n i a n now is a 3 x 3 m a t r i x whose eigenvalues are obtained from
A~ - Ak(3t 2 + 7 a Uo ~- 7P Vo) +
¢-
¢) + 2cos
- i r a Uo -
+ •v° [9aZu2o cos(¢ 4- 20) + 3 V ~ tano sin(0 - ¢)] = 0
(5)
Once again the total energy, electronic plus elastic, is found as a function of O and 0. T h e s e results are s u m m a r i z e d in Fig. 3, where the phase relationship again indicates coexistence. A peak in the charge density occurs on the site which is at t h e vertex of two strong bonds.
Fig. 3. T h e b o n d s u p e r a l t e r n a t i o n p a t t e r n and the charge densities for the 1/3-filled band. T h e n o t a t i o n s are t h e s a m e as in Fig. 1. As with t h e 1/4-filled band we have calculated b0 and P0 while varying c~ a n d ft. T h e s e results are shown in Figs 4(a) and (b), and are found to be similar to those in Figs. 2(a) and (b) respectively. While we have not been able to find a general proof, we believe t h a t a cooperative interaction is true for all bandfillings other t h a n 1/2. W i t h increasing c o m m e n s u r a b i l t y , one gradually reaches the i n c o m m e n s u r a t e limit, where no distinction can be m a d e between the B O W and t h e C D W .
.4
.........
i .........
1 .........
T .........
I .........
I .........
.6
; .........
......
I .........
i .........
I .........
I .........
(b)
.5
!
.3 .4
bo
Po
.2
.3 .2
.! .1
J .0
,,-~ ......
.00
.; ........
.09
.,.,',,,,I
.18
.........
.27
t .........
.36
X~
Fig. 4(a). T h e a m p l i t u d e c o n s t a n t ( A : A a = 0; • : the C D W in t h e 1/3-filled 0.1; , : A~ = 0.4; • : , ~ =
i ..................
.45
.54
.63
.0 .00
.16
.32
X
.48
.64
.80
of t h e B O W in the 1/3-filled b a n d vs t h e dimensionless e-mv coupling Aa = 0.05; * : A s = 0.2; • : A,~ = 0.45). Fig. 4(b). T h e a m p l i t u d e of b a n d vs the dimensionless e-ph coupling c o n s t a n t ( A : Af~ = 0; • : An = 0.9).
4664
Kink Excitations In t h e case of t h e 1/2-filled b a n d , t h e narrow coexistence region containing b o t h t h e B O W and t h e C D W h a s soliton excitations t h a t have irrational charge. Similar results have been obtained [7] for t h e (AB):c polymer, where, however, t h e coexistence region is broader. In t h e case of t h e 1/4- and 1/3-filled b a n d s , solitons have fractional charge when only one kind of density wave is present [6]. Clearly it is of interest to d e t e r m i n e soliton charges here. T h e soliton excitations are studied using t h e S C F technique of O n o et al. [6]. In order to determine w h e t h e r kinks have irrationM charge, we have studied t h e two-kink cases. For charges to be irrational, t h e excess charges on the two kinks m u s t be unequal. We define s m o o t h e d charge densities/~n as (6a)
~,~ = ¼ [0.5p,,_2 + p ~ - i + p~ + p.+~ + 0.5p.+2] Pa
=
1
[,On_l
+
Pn
+
(65)
/On+i]
for t h e 1/4-filled a n d 1/3-filled b a n d s respectively. T h e s m o o t h e d densities of t h e excess electrons are
*p(n) -- ~ - 1/2 and t~p(n) = ~,~ - 2/3. T h e self-consistent calculations were done within periodic b o u n d a r y condition, with 120 electrons a n d 160 electrons respectively on 238 sites, t h e r e b y giving two charged kinks in b o t h cases. For comparison, t h e calculations were done also for t h e (AB)~:
.07
........
i ......
~,l
.........
i .........
i .........
i .........
.06 .05 .04 dp .03 .02 .ol .oo
............ £ 1 ..... 40
o
80
120 site
160
200
240
n .030
.025 :. ........ , ......... , ......... , ........ ~......... , .........
(b .025
.020
.020 6p .015
.015 5p .010
.010 .005 .000
.005 0
40
80
t20 160 site n
200
240
.000
4O
ao
120 t60 SlLe n
200
240
Fig. 5. T h e excess charge densities on the solitons for the two-kink case for (a) t h e (AB)x polymer, (b) t h e 1/4-filled b a n d ( A = site 4n + 1,o = site 4n + 2 , * = site 4n + 3, It = site 4n)) a n d (c) t h e 1/3-filled b a n d (& = site 3n + 1 , . = site 3n + 2 , * = site 3n). In (b) and (c) t h e overall charge densities are obtained by s u m m i n g over all sites.
4665
polymer with two kinks, by adding alternating site energies to Eq. (1) while removing the e-mv coupling. In Fig. 5(a) we show the excess charges for the half-filled
(AB)~: system,
for a site energy
difference of 0.1 in units of t. Note that the charges on the kinks are very different, indicating irrational charges. In Fig. 5(b) we have plotted the excess charges for the 1/4-filled band with two kinks. The overall charge is obtained by summing over the 4nth, (4n + 1)th, (4n + 2 ) t h and (4n + 3)th sites for each kink. It should be obvious that the overall charges of the two kinks axe equal. The same is true for the the two kinks in the 1/3-filled band in Fig. 5(c). We therefore conclude that in spite of the coexistence of the BOW a~ld the CDW, the charges on the kinks are rational fractions.
APPLICATION TO ORGANIC CONDUCTORS The most important conclusion of our work is that the BOW is always accompanied by the CDW and vice versa. In the case of organic conductors, intermoleculax distances along the stack axes are 3 - 3.5A, only slightly smaller than the sum of the van der Waals' radii. Thus the e-ph coupling should be rather weak. In spite of this, considerable modulations of intermoleculax distances are common below the metal-insulator transitions. Our work is able to explain this. The e-my coupling can drive both a CDW and a BOW, and the latter leads to a modulation of the intermolecular distances.
REFERENCES 1 M.J. Rice, Phys. Rev. Lett., 37 (1976) 36; R. Bozio and C. Pecile, in L. Alc~cer (ed.), The Physics and Chemistry of Low Dimensional Solids, Reidel, Dordrecht, 1980, pp. 165-186. 2 S. Huizinga et ai., Phys. Rev. B, 25 (1982) 1717. 3 S. Kivelson, Phys. Rev. B, 28 (1983) 2653 ; W.P. Sn, Solid St. Commun., 48 (1983) 479. 4 S. Mazumdax and D.K. Campbell, Phys. Rev. Lett., 55 (1985) 2067; A. Painelli and A. Girlando, Phys. Rev. B, 45 (1992) 8913. 5 W.P. Su, Phys. Scripta, 32 (1985) 34. 6 Y. Ono, Y. Ohfuti and A. Terai, J. Phys. Soc. (Japan), 54 (1985) 2641; Y. Ohfuti and Y. Ono, ibid., 55 (1986) 4323. 7 J.T. Gammel et al., Phys. Rev. B, 45 (1992) 6408. 8 E.J. Mele and M.J. Rice, Phys. Rev. Lett., 49 (1982) 1455; D.K. Campbell, Phys. Rev. Lett., 50 (1983) 865.