- Email: [email protected]
Peierls distortion in alloys of quasi-one-dimensional conductors
Solid State Communications, Vol. 33, PP. 951—954. Pergamon Press Ltd. 1980. Printed in Great Britain. PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS* C. Tannous and A. Caillé Departement de physique et Groupe de recherche sur les semiconducteurs Ct les diélectriques, Université de Sherbrooke, Québec, Canada J1K 2R1 (Received5Novemberl979byM.F. Collins) We have calculated the value of the mean field Peierls transition temperature in a quasi-one-dimensional binary A —B alloy as a function of the relative concentration. This was done in the framework of the single-dimer coherent potential approximation. Reductions in the transition temperature are predicted for isoelectronic constituants. THE PEIERLS TRANSITION (PT) results from an instabiity occuring in a half-filled band one-dimensional electron system coupled to the phonon field of the host lattice.t This transition happens at a low temperature a gap opens up in the electron band and the lattice exhibits a distortion with a wavenumber equal to twice the Fermi momentum [3]. It should be mentioned here that T~0is obtained theoretically within mean-field theory. The latter ignores critical fluctuations occuring in one-dimensional systems which strongly depress [4] the aforementioned temperature. Hence T~0is to be considered as a scale temperature below which the electrons and the ionic lattice are strongly coupled. This coupling leads to a one-dimensional distortion that gives rise to a diffuse X-ray scattering [5] The above considerations pertain to pure systems so that an important question may be posed: what happens to the transition when we induce disorder in the chain? A part of the answer to this is found in the recent experiments performed by Chiang et al. [6] who investigated the effect of controlled disorder on the transitions in TTF—TCNQ by irradiating the samples with a deuteronbeam. Besides, it is well-established that in KCP, the Br ions and the water molecules [71modify to a certain extent the physical properties of the linear chains of Pt and hence can be considered as impurities influencing the charge density wave state. The effects of disorder and impurities on the Peierls transition have been considered by many authors. Schuster [8] and Ono [91showed that, .
*
This research is partially supported by the National Research Council of Canada and le ministére de l’Education de Québec. Peierls transitions exist also in a non half-filled band [1] , however we will, be concerned only with the case of a half-filled band giving rise to a commensurate distortion of the lattice; the dimerization. Moreover we will consider a site description in contrast to a bond one [2]. 951
within the Born approximation, dilute non magnetic impurities influence the transition temperature and the energy gap of the Peierls phase in the same way as magnetic impurities change the corresponding quantities in a BCS superconductor. Sham and Patton [11] showed, by mapping the chain onto a 1 -D spin system where the impurities act as a random magnetic field, that a random distribution of impurities destroys the long-range order of the charge density wave state. It is to be noted also that Bulaevskii and Sadovskii [12] examined the effect of disorder on PT in the framework of the Lloyd model and the fragment model obtaining a strong reduction of the transition temperature. Working from the high ternperature phase, Sen and Varma [13] calculated the latter reduction by the method of the coherent potential approximation [14] (CPA). In this paper we present a microscopic theory based on a symmetry adapted representation of the chain in its low temperature phase, i.e. in the dimerized state. The impurity problem is solved in a scheme which we call the single dimer coherent potential approximation (SDCPA). The latter approximation permits us, with the help of proper approximations, to work with analytical expressions, recover all the results already obtained by the precedent theories, and find new interesting ones. It should be noted further, that the SDCPA is profoundly different from the cluster-CPA [15]. Nevertheless the method has the same limitations as the ordinary CPA, i.e. it works in principle for low concentrations and weak impurity potentials [141. We consider the following simple Hamiltonian in a Wannier representation H
=
,,,
~
~
+ ~
~
ijo
+ ~} +
~ jao
g1c~Cj0 [b~o,+ bia]
+
~ wj~,{b~b1~ ja
(1)
952
PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS Vol. 33, No.9
where i,j are lattice sites, a is the spin. The c and b are respectively electrons and phonons operators. The ~ are the local electronic energy levels, w~,are the local phonon frequencies (a is a phonon branch) and g~is the electron—phonon coupling constant. For simplicity the kinetic energy-part is assured to be translationally invariant t~1= r(R~ ~), whereas c~,Wj~and g~are site
point to follow the forniatism developed by Weinkauf and Zittarz [17] in their CPA treatment of superconducting alloys and it is noteworthy to mention that their theory has been seriously confirmed by vanous experiments [18]. Furthermore, their theory was extended in order to include off-diagonal disorder by
—
dependent and random. Note that equation (1) has already been used by Strassler et al. [16] for the study of optical properties of KTCNQ. At this point, let us grow the atoms two by two so that our chain is now formed of two-atoms clusters, as if we had anticipated the state of dimerization. Consequently we define new quantities for: electron operators (c1 ~, c1, 2), atomic levels (~1’ j 2), phonon frequencies (w1, i, w~,2) and coupling constants ~g, 1, g, 2) where i is the cluster index and 1, 2 are the indices of each atom in the cluster. Now we use the mean-field approximation [3] in(represented order to introduce site-dependent static distortion by i~ the 2 for a given cluster i) in the system. Hence the meanfield Hamiltonian is written as [after neglecting the spin and considering only one branch of phonons] Hmf
Dubovskii [191who succeeded in obtaining a nealry perfect agreement between theory and experiment for the composition dependence of the superconducting transition temperature T~of the V—Ta alloy system. The equations of motion are now written in a condensed matrix form: A ~ E G(w) = 1 D C(w) ~th Wi
2~k,
A~k(w)
=
C~k(w)
=
e,~ —
6k
~,
1)~ik,
+
2gk 2~k 2)6ik
2
(4)
‘
~
=
—
t(ökl + ~
DIk
=
—
t(~~j+6k,
~+~)•
~j, i Cj,
=
+~
1CI, 1
+
+
~,
fl
If we average the Green’s function over the ensemble of atomic configurations [17] (denoted by a bar over the quantity) we obtain:
i, 2C1, 2C~,2
+ ~
~
2 [~
2
+ ~j
A(w) —2~ +
(w (w
~
~
g1,
+~
+ ~,
1
C1, 1C~,
A
1i,
g~,2’-
E
-
G=l
+
2C~,2C1 2(2)
t~c~c~,2+c~2c~+1,1+~2~,1
D with
(5)
C(w)
[A(w)]Ik
=
5~k
+ c~+1,~c~,2} where we define the site dependent distortions by:
[t.~(w)] ik = where i (w),
[~ 2
—
e2(w)
—
2g2~2(w)]
(w) are self-energies, 4
1(w), z~2(w) are
self-distortions (or self-gaps) and g1, g2 are effective 1 A
~i1
=
— — j2 —
phonon coupling constants. Note that and D remain unchanged after the averaging since weEare dealing only
C’i I
+
W~,2
~-.,
~,
\CjiCj2/d ‘
‘.
‘
where ( >~means a thermodynamic average in the deformed state. The dynamics of the system for a special atomic configuration {j,a, ~-‘.~i,a’ g~ a = 1, 2} is given by the configuration dependent propagator matrix: (s,) ¶~‘~/(~) G~~(w) = ~‘~j(W) ~‘~(w) ~
~
with the definition~’~~(w) = ((Cia, C~n))~. (The latm mdices are for the clusters whereas the greek ones denote the sites in a given cluster.) The Green’s function obtained in this form is written in what we call the “Dimer representation”. It is advantageous at this
I
with diagonal disorder. Following Weinkauf and Zittarz [17] we calculate now the matrix
A(w) — A (w)
—
W
=
G~
G’
=
0
0 -
which is diagonal in the dimer representation, whereas in the site representation of [17] it is not. Now we obtain the Tmatrix with the help of: T = W(l —~W)~ (6) and we only take the contribution T~° due to. a single . dimer so that the CPA self-consistency condition is .
wri en as. ~(j) = 0.
(7)
Vol. 33, No.9 PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS
953
This is what we call the single-dimer coherentpotential approximation (SDCPA). Equation (7) gives us, in principle, a means for calculating the self-energies e~ (ci.,), ~2 (w) and the self-gaps ~ (w), ~2 (w). In order to study the dependence of T~on the different parameters of the alloy we linearize equation (7) with respect to 2’ (w) and ~2 (W). Furthermore we use a Lorentzian density of states since it permits us to work with analytical expressions while the nature of the PT in the quasi ID conductor is not qualitatively altered by this choice. Here we specify that a full discussion of this approximation, its range of validity and a comparison with other model density of states will be published elsewhere [20], so that we confme ourselves in this
.I~.
paper to the above mentioned approximations. Consequently the linearized gap equation is obtained:
D(B — CA). The normalized electron phonon coupling constants and bandwidths are taken respectively to be: XA = 0.4, XB = 0.33, ~F AID = 0.9, ~F 8/D = 1.0. One
~‘
~‘
0 0.2
~
8
07 8-04
0
0.5
X
.0
Fig. 1. Normalized transition temperature vs concentration x in A~B1-x alloy, for different values of ~, i.e.
sees quite clearly that wiih the increase of & the curve depresses until it reaches zero for critical concentrations. 4g~ Im ~ Dw~j3d.~z) >
L~(z) =
____________
z,~— e~z~)
~
iir
(2n+l)— ‘3
=
(8)
where = irN(o) [N(o)substance] is the density states at the Fermi D level in the pure is theofcharacteristic bandwidth of the Lorentzian [17], 13 = 1 IkBT is the inverse temperature (kB = Boltzmann constant), and e(z) and L~(z)are the self-energy and self-gap of the effective crystal undergoing the effective dimerization with g1 and w~as site-dependent effective coupling constant and effective phonon frequency. Moreover the following functions are given by: 17(z) =
F°(z) =
sgn Im Z
—
[z
—
e(z) +
e(z)]
(i/D)i~(z) 2
d~(z)=
[F°(Z)]
+
2
n(z)
—
with e~the site energy levels, and z representing the energy. At this stage we look at the gap equation (8) in the limit6F(3CCF 1 where j3~,is the temperature the half bandwidth [3]inverse ; after critical the frequency and summation we obtain: ~‘
“~ ~“
log~—J \lr/
--
~)
~/‘(
=
v ~og ~
ir
—
~
\
where ~~(x)is the digamma function [21],
V
=
Dc? D2e? +
/
I D2? + 1
+
C
27r))
(10)
and /3~,4 are the inverse transition temperature and half-bandwidth of the pure substance. Besides 2e?I g~/g~Il —D 2efl w~/w 0[1 + D where g 0 and W0 are the coupling constant and phonon frequency of the pure substance. It is noteworthy to mention that the frequency summation had to be restricted to energies < 2e~.in order to recover the exact numerical coefficient of the gap equation of Rice and Strassler [3] in the pure case. Moreover one notes that equation (10) for the determination of the critical temperature differs formally from the equation obtained by Bulaevskii and Sadovskii [12] in the Uoyd model only by the factor v. The same comment applies to the comparison of equation (10) with the superconducting critical temperature equation in presence of magnetic impurities [10]. Now in order to get the dependence of the critical temperature in an A~—B1 -x alloy as a function of the concentrationx we proceed [17] by fixing the distance ~and [ö B D(B — CA)] between the bands ofCB) thehas pure substance whereas the sum D(CA + toAbe determined from the condition that the Fermi level of =
the alloyofisstates density located function at zeroone energy. obtains Integrating for x: the x
=
tg~(DEB) tg’ (DEB) — tg~(DCA)
(11)
since case we have taken one electron per Atreated and B in atoms. (The of different valences a forthcoming paper [20].) The systemwill of be equations (10) and (11) permits us to determine the transition temperature
954
PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS Vol. 33, No.9
as a function of x, once the normalized coupling constants
x.
2
=
2N(0)g,
(i
=
A B)
6. 7.
Wi
and the respective half-bandwidths CF ~(i = A, B) are given. The results are displayed in Fig. 1, and one sees quite clearly that for small ~, T~is nearly linear in X whereas for larger values, T~is strongly depressed even going to zero for critical concentrations. Similar qualitative behaviour [22] has been observed for the metal-insulator transition temperature of (TSCF)X(TTF)I -x TCNQ. In conclusion, we have studied in this paper the effect of alloying on the mean field Peierls transition temperature in the framework of SDCPA which is valid only for low relative concentrations and weak impurity diffusion potential. Moreover we performed the calculations within a simple model density of states. In order to improve on the above treatment one should take into account the off-diagonal disorder and a more realistic band structure. Acknowledgement The authors are grateful to Professor M.J. Zuckermann for very enlightening discussions. —
8. 9. 10. 11 12. 13. 14 15. 16. 17. 18. 19. 20.
REFERENCES 1. 2. 3. 4. 5.
Y. Ono,J. Phys. Soc. Japan 41, 817 (1976);M.C. Leung,Phys. Rev. Bil, 4272 (1975). A. Madhukar, Client Phys. Lett. 27606 (1974). M.J. Rice & S. Strässler,Solid State Commun. 13, 125 (1973). P.A. Lee, T.M. Rice & P.W. Anderson, Phys. Rev, Lett. 31, 462 (1973). A. Braude, A. Lindegaard-Andersen, K; Carneiro &
21. 22.
A. Underhill, Synthetic Metals 1, 35 (1979) and references therein. C.K. Chiang, M.J. Cohen, P.R. Newman & A.J. Heeger,Phys. Rev. B16, 5163 (1977). J.C. Inkson, Solid State Commun. 14, 1229 (1974); T. Takahashi, H. Akagawa, H. Doi & H. Nagasawa, Solid State Commun. 23, 809 (1977) and references therein. H.G. Schuster, Solid State Commun. 14, 127 (1974) Y. Ono, Solid State Commun. 28, 763 (1978). A.A. Abrikosov & L.P. Gorkov, Soy. Phys. JETP 12, 1243 (1961). L.J. Sham & B.R. Patton, Phys. Rev. B13, 3151 (1976) L.N. Bulaevskii & M.V. Sadovskii, Soy. Phys. Solid State 16, 743 (1974). P.N. Sen & C.M. Varma, Solid State Commun. 15, 1905Elliott, (1974).J.A. Krumhansl & P.L. Leath, Rev. R.J. Mod. Phys. 46, 465 (1974) and references therein. W.H. Butler, Phys~Rev. B8, 4499 (1973). M.J. Rice, N.O. Lipari & S. Strässler, Phys. Rev. Lett. 39,1359 (1977). A. Weinkauf & J. Zittarz, J. Low Temp. Phys. 18, 229 (1975). A. Weinkauf & J. Zittarz, Solid State Commun. 14, 365 (1974). Soy. Phys. Solid State 20, 1798 O.A. Dubovskii, (1978). C. Tannous, A. Caillé & M.J. Zuckenmann (to be published). M. Abramowitz & l.A. Stegun, Handbook of Mathematical Functions,Dover, New York (1965). T.D. Schultz & R.A., Craven, Highly Conducting One-dimensional Solids (Edited by J.T. Devreese. R.P. Evrard & V.E. van Doren), p. 147. Plenum Press, New York (1979); T.D. Schultz, Quasi Onedimensional Conductors, Vol. I, Dubrovnik 1978, (Edited by S. Barisic, A. Bjelis, J.R. Cooper & B. Leontic), p. 31. Springer-Verlag, Berlin (1979).
*
This research is partially supported by the National Research Council of Canada and le ministére de l’Education de Québec. Peierls transitions exist also in a non half-filled band [1] , however we will, be concerned only with the case of a half-filled band giving rise to a commensurate distortion of the lattice; the dimerization. Moreover we will consider a site description in contrast to a bond one [2]. 951
within the Born approximation, dilute non magnetic impurities influence the transition temperature and the energy gap of the Peierls phase in the same way as magnetic impurities change the corresponding quantities in a BCS superconductor. Sham and Patton [11] showed, by mapping the chain onto a 1 -D spin system where the impurities act as a random magnetic field, that a random distribution of impurities destroys the long-range order of the charge density wave state. It is to be noted also that Bulaevskii and Sadovskii [12] examined the effect of disorder on PT in the framework of the Lloyd model and the fragment model obtaining a strong reduction of the transition temperature. Working from the high ternperature phase, Sen and Varma [13] calculated the latter reduction by the method of the coherent potential approximation [14] (CPA). In this paper we present a microscopic theory based on a symmetry adapted representation of the chain in its low temperature phase, i.e. in the dimerized state. The impurity problem is solved in a scheme which we call the single dimer coherent potential approximation (SDCPA). The latter approximation permits us, with the help of proper approximations, to work with analytical expressions, recover all the results already obtained by the precedent theories, and find new interesting ones. It should be noted further, that the SDCPA is profoundly different from the cluster-CPA [15]. Nevertheless the method has the same limitations as the ordinary CPA, i.e. it works in principle for low concentrations and weak impurity potentials [141. We consider the following simple Hamiltonian in a Wannier representation H
=
,,,
~
~
+ ~
~
ijo
+ ~} +
~ jao
g1c~Cj0 [b~o,+ bia]
+
~ wj~,{b~b1~ ja
(1)
952
PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS Vol. 33, No.9
where i,j are lattice sites, a is the spin. The c and b are respectively electrons and phonons operators. The ~ are the local electronic energy levels, w~,are the local phonon frequencies (a is a phonon branch) and g~is the electron—phonon coupling constant. For simplicity the kinetic energy-part is assured to be translationally invariant t~1= r(R~ ~), whereas c~,Wj~and g~are site
point to follow the forniatism developed by Weinkauf and Zittarz [17] in their CPA treatment of superconducting alloys and it is noteworthy to mention that their theory has been seriously confirmed by vanous experiments [18]. Furthermore, their theory was extended in order to include off-diagonal disorder by
—
dependent and random. Note that equation (1) has already been used by Strassler et al. [16] for the study of optical properties of KTCNQ. At this point, let us grow the atoms two by two so that our chain is now formed of two-atoms clusters, as if we had anticipated the state of dimerization. Consequently we define new quantities for: electron operators (c1 ~, c1, 2), atomic levels (~1’ j 2), phonon frequencies (w1, i, w~,2) and coupling constants ~g, 1, g, 2) where i is the cluster index and 1, 2 are the indices of each atom in the cluster. Now we use the mean-field approximation [3] in(represented order to introduce site-dependent static distortion by i~ the 2 for a given cluster i) in the system. Hence the meanfield Hamiltonian is written as [after neglecting the spin and considering only one branch of phonons] Hmf
Dubovskii [191who succeeded in obtaining a nealry perfect agreement between theory and experiment for the composition dependence of the superconducting transition temperature T~of the V—Ta alloy system. The equations of motion are now written in a condensed matrix form: A ~ E G(w) = 1 D C(w) ~th Wi
2~k,
A~k(w)
=
C~k(w)
=
e,~ —
6k
~,
1)~ik,
+
2gk 2~k 2)6ik
2
(4)
‘
~
=
—
t(ökl + ~
DIk
=
—
t(~~j+6k,
~+~)•
~j, i Cj,
=
+~
1CI, 1
+
+
~,
fl
If we average the Green’s function over the ensemble of atomic configurations [17] (denoted by a bar over the quantity) we obtain:
i, 2C1, 2C~,2
+ ~
~
2 [~
2
+ ~j
A(w) —2~ +
(w (w
~
~
g1,
+~
+ ~,
1
C1, 1C~,
A
1i,
g~,2’-
E
-
G=l
+
2C~,2C1 2(2)
t~c~c~,2+c~2c~+1,1+~2~,1
D with
(5)
C(w)
[A(w)]Ik
=
5~k
+ c~+1,~c~,2} where we define the site dependent distortions by:
[t.~(w)] ik = where i (w),
[~ 2
—
e2(w)
—
2g2~2(w)]
(w) are self-energies, 4
1(w), z~2(w) are
self-distortions (or self-gaps) and g1, g2 are effective 1 A
~i1
=
— — j2 —
phonon coupling constants. Note that and D remain unchanged after the averaging since weEare dealing only
C’i I
+
W~,2
~-.,
~,
\CjiCj2/d ‘
‘.
‘
where ( >~means a thermodynamic average in the deformed state. The dynamics of the system for a special atomic configuration {j,a, ~-‘.~i,a’ g~ a = 1, 2} is given by the configuration dependent propagator matrix: (s,) ¶~‘~/(~) G~~(w) = ~‘~j(W) ~‘~(w) ~
~
with the definition~’~~(w) = ((Cia, C~n))~. (The latm mdices are for the clusters whereas the greek ones denote the sites in a given cluster.) The Green’s function obtained in this form is written in what we call the “Dimer representation”. It is advantageous at this
I
with diagonal disorder. Following Weinkauf and Zittarz [17] we calculate now the matrix
A(w) — A (w)
—
W
=
G~
G’
=
0
0 -
which is diagonal in the dimer representation, whereas in the site representation of [17] it is not. Now we obtain the Tmatrix with the help of: T = W(l —~W)~ (6) and we only take the contribution T~° due to. a single . dimer so that the CPA self-consistency condition is .
wri en as. ~(j) = 0.
(7)
Vol. 33, No.9 PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS
953
This is what we call the single-dimer coherentpotential approximation (SDCPA). Equation (7) gives us, in principle, a means for calculating the self-energies e~ (ci.,), ~2 (w) and the self-gaps ~ (w), ~2 (w). In order to study the dependence of T~on the different parameters of the alloy we linearize equation (7) with respect to 2’ (w) and ~2 (W). Furthermore we use a Lorentzian density of states since it permits us to work with analytical expressions while the nature of the PT in the quasi ID conductor is not qualitatively altered by this choice. Here we specify that a full discussion of this approximation, its range of validity and a comparison with other model density of states will be published elsewhere [20], so that we confme ourselves in this
.I~.
paper to the above mentioned approximations. Consequently the linearized gap equation is obtained:
D(B — CA). The normalized electron phonon coupling constants and bandwidths are taken respectively to be: XA = 0.4, XB = 0.33, ~F AID = 0.9, ~F 8/D = 1.0. One
~‘
~‘
0 0.2
~
8
07 8-04
0
0.5
X
.0
Fig. 1. Normalized transition temperature vs concentration x in A~B1-x alloy, for different values of ~, i.e.
sees quite clearly that wiih the increase of & the curve depresses until it reaches zero for critical concentrations. 4g~ Im ~ Dw~j3d.~z) >
L~(z) =
____________
z,~— e~z~)
~
iir
(2n+l)— ‘3
=
(8)
where = irN(o) [N(o)substance] is the density states at the Fermi D level in the pure is theofcharacteristic bandwidth of the Lorentzian [17], 13 = 1 IkBT is the inverse temperature (kB = Boltzmann constant), and e(z) and L~(z)are the self-energy and self-gap of the effective crystal undergoing the effective dimerization with g1 and w~as site-dependent effective coupling constant and effective phonon frequency. Moreover the following functions are given by: 17(z) =
F°(z) =
sgn Im Z
—
[z
—
e(z) +
e(z)]
(i/D)i~(z) 2
d~(z)=
[F°(Z)]
+
2
n(z)
—
with e~the site energy levels, and z representing the energy. At this stage we look at the gap equation (8) in the limit6F(3CCF 1 where j3~,is the temperature the half bandwidth [3]inverse ; after critical the frequency and summation we obtain: ~‘
“~ ~“
log~—J \lr/
--
~)
~/‘(
=
v ~og ~
ir
—
~
\
where ~~(x)is the digamma function [21],
V
=
Dc? D2e? +
/
I D2? + 1
+
C
27r))
(10)
and /3~,4 are the inverse transition temperature and half-bandwidth of the pure substance. Besides 2e?I g~/g~Il —D 2efl w~/w 0[1 + D where g 0 and W0 are the coupling constant and phonon frequency of the pure substance. It is noteworthy to mention that the frequency summation had to be restricted to energies < 2e~.in order to recover the exact numerical coefficient of the gap equation of Rice and Strassler [3] in the pure case. Moreover one notes that equation (10) for the determination of the critical temperature differs formally from the equation obtained by Bulaevskii and Sadovskii [12] in the Uoyd model only by the factor v. The same comment applies to the comparison of equation (10) with the superconducting critical temperature equation in presence of magnetic impurities [10]. Now in order to get the dependence of the critical temperature in an A~—B1 -x alloy as a function of the concentrationx we proceed [17] by fixing the distance ~and [ö B D(B — CA)] between the bands ofCB) thehas pure substance whereas the sum D(CA + toAbe determined from the condition that the Fermi level of =
the alloyofisstates density located function at zeroone energy. obtains Integrating for x: the x
=
tg~(DEB) tg’ (DEB) — tg~(DCA)
(11)
since case we have taken one electron per Atreated and B in atoms. (The of different valences a forthcoming paper [20].) The systemwill of be equations (10) and (11) permits us to determine the transition temperature
954
PEIERLS DISTORTION IN ALLOYS OF QUASI-ONE-DIMENSIONAL CONDUCTORS Vol. 33, No.9
as a function of x, once the normalized coupling constants
x.
2
=
2N(0)g,
(i
=
A B)
6. 7.
Wi
and the respective half-bandwidths CF ~(i = A, B) are given. The results are displayed in Fig. 1, and one sees quite clearly that for small ~, T~is nearly linear in X whereas for larger values, T~is strongly depressed even going to zero for critical concentrations. Similar qualitative behaviour [22] has been observed for the metal-insulator transition temperature of (TSCF)X(TTF)I -x TCNQ. In conclusion, we have studied in this paper the effect of alloying on the mean field Peierls transition temperature in the framework of SDCPA which is valid only for low relative concentrations and weak impurity diffusion potential. Moreover we performed the calculations within a simple model density of states. In order to improve on the above treatment one should take into account the off-diagonal disorder and a more realistic band structure. Acknowledgement The authors are grateful to Professor M.J. Zuckermann for very enlightening discussions. —
8. 9. 10. 11 12. 13. 14 15. 16. 17. 18. 19. 20.
REFERENCES 1. 2. 3. 4. 5.
Y. Ono,J. Phys. Soc. Japan 41, 817 (1976);M.C. Leung,Phys. Rev. Bil, 4272 (1975). A. Madhukar, Client Phys. Lett. 27606 (1974). M.J. Rice & S. Strässler,Solid State Commun. 13, 125 (1973). P.A. Lee, T.M. Rice & P.W. Anderson, Phys. Rev, Lett. 31, 462 (1973). A. Braude, A. Lindegaard-Andersen, K; Carneiro &
21. 22.
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