Solid State Communications, Vol. 43, No. 5, pp. 375-378, 1982. 0038-1098/82/290375-04S03.00/0 Printed in Great Britain. Pergamon Press Ltd.

PEIERLS AND SPIN-PEIERLS PHASE TRANSITIONS IN STRUCTURALLY UNSTABLE QUASI-ONE-DIMENSIONAL SOLIDS

Y. IApine

Ddpartement de Physique, Universitd de Montrdal, C.P. 6128, Montrdal, Qudbec, Canada, H3C 3J7

(Received 29 December 1981 by R. Barrie)

The Peierls and spin-Peierls phase transitions are studied in solids in which a structural instability is already present. It is found that the presence of this intrinsic mode can increase considerably the critical temperature. For small values of the critical temperature, the transition is of the BCS-type, like the Peierls (or spin-Peierls) phase transition, but with an effective electron (or spin)-phonon coupling constant renormalized by the anharmonicity and by the instability of the phonon. Numerical results are also presented for larger critical temperatures. Then the BCS behaviour is no longer observed.

1. INTRODUCTION

THE PEIERLS DISTORTION is predicted to occur in quasi-one-dimensional crystals wi~h a half-tidied electron band, due to the electron-phonon coupling [ 1-3]. Similarly, in a quasi-one-dimensional spin system, with one unpaired spin for each unit cell, coupled through an Heisenberg or X Y antiferromagnetic interaction, a sp in- Peierls dimerization is predicted to occur [4-7] . In these systems, it is generally assumed as a first approxi- mation that the dimerization occurs along the stacking axis of planar molecules. However, it has been observed, specially in the spin-Peierls case, that the dimerization could occur in a different direction characterized by a precursive soft mode [6]. This pre-existence of a soft mode is characteristic of the phase transitions in T T F - MBDT (M = Cu, Au) and in MEM (TCNQ)2 [6]. The q-vector of this soft mode, in these systems, defines the dimerization axis. It has been argued that it was the three dimensional character of the soft mode that made the mean field treatment in these systems valid [6].

In this paper, I want to investigate the Peierls dis- tortion mechanism in a crystal where a softening phonon is present. In other words, an inherent lattice instability will be added to the Peierls-type instability. In the case of the Peierls instability, the electronic con- tribution will be described through the formalism of Rice and Strassler [ 1 ] and the lattice instability through the mean field formalism of Thomas [8] and of Pytte and Feder [9, 10]. In the spin-Peierls case, the formal- ism of Pytte [4] will be used to describe the sp in- phonon coupling.

2. MODEL CALCULATIONS

I will consider a quasi-one-dimensional solid for which the phase transition implies a displacement of the molecules that corresponds to an acoustic phonon at the comer of the Brillouin zone. The large organic molecules are thus considered as rigid. The magnitude of this dis- placement with respect to the high temperature phase will be --- Q, the sign alternating for two successive molecules. Q is then the normal coordinate (expressed in the Fourier space at the wavevector qF = zr/a) corre- sponding to the soft mode that will characterize the dimerization [8]. Here, a is the intersite distance of the uniform lattice in the direction of the soft mode. The Hamiltonian for the normal coordinates of the acoustic molecular vibration at site l (Qz) and for their coupling to the electrons can now be written, for the molecules of a chain, in the direction of dimerization [ I--4] (only the coordinates in the direction of dimerization are considered):

H = HLA x + HEL + HEL__LA T (1)

where

= Y. + 5 - + -g Z v,,, Q,Q,. l f

HZL = ~ eha~ak k

~I12

HEL_LA T = " ~

375

376 PEIERLS AND SPIN-PEIERLS PHASE TRANSITIONS

Qq is the Fourier transform of Qz, Pz is the momentum conjugate to Ql, ek = eF COS ka and a~, ak are the elec- tron second quantization operators. Pq (= Z @÷,~ap) is the electron density and g(q) is the electron-phonon coupling constant. In this Hamiltonian, the possibility of an intrinsic soft mode is included in/ /Lax and the possibility of a Peierls transition is included in HErrLAT-

If lit.At is treated in mean field theory [10] and HEI.,-LA T is diagonalized [1 ], the Helmholtz free energy can be written for the mode that becomes soft (only Qq=,r/,, = +- Q is kept and g(q = ~r/a) is writteng):

CO 2 3' F = _ y ( Q 2 + o) + ~.(Q4 + 60Q2 + 3aa)_~VoQ2

!

+ ~ga,(n + I/2) -- ~- [(1 + n) In (1 + n) - -n In n ]

2 rtlZa ~'~ ! dk In 2 cosh (3Ek/2

where

1 o = ~ ctgh/392s/2

g2~ = co = + 33'(o + Q=)

1 t/ -~-

1 + exp (/~2,)

Et~ = (e~- cosaka + Aa) t/2

2 g 2

(2)

A 2 = r reFTkcoaQ 2 ~. = _ NhrreFoo

Vo is the Fourier transform of V n, evaluated at q = rr/a. The dimerization curve for Q as a function of tempera- ture is obtained from equation (2) by a minimization with respect to Q:

I"

Q [Vo -- co 2 -- 37o _3'Q2 + 6V;kC02 /

~x~ ~ d k t a ~ ' k / 2 ] = 0 (3)

0

If Vo = 7 = 0, equation (3) reduces to the gap equation of Rice and Strassler [1]. IfX = 0, it reduces to the equation of the displacement of the molecules as a func- tion of temperature in a structural phase transition [8, 9]. Equations (2) and (3) have been written in a sys- tem of units such that the displacement and the energies are dimensionless [1,8, 9].

For a spin-Peierls phase transition, the integral term of equation (2) must be replaced by [4]

r r /2a

rr(32 f dk ln 2 coshl~E~/2 + p2j4 PJ2 (4) 0

Vol. 43, No. 5

with

E~ = (A 2 + (p:J"-- A2) cos 2 ka) t/2

and

A = 2g'Qp

where J is the exchange parameter and g" is the spin- phonon coupling constant as defined by Pytte in [4]. The equation of Q as a function of temperature is then equation (3) with the integral term replaced by:

rr/za tanh (/tgJ,/2) sin2ka P J X S w 2 i dk ( 5 )

o El,

where

4gS~p ~k s -- J~¢.o 2

and p is a slowly varying function of temperature that can be taken as equal to 1.64 [4, 11]. E~ is the energy of excitation of a quasi-fermion, the elementary excitation of the spin system.

The temperature dependence of the frequency of the soft mode renormalized by the electron (or spin)- phonon interaction is obtained from the formalism of structural phase transitions [8-10] , for T larger than T e and at q = Tr/a. It is

~rlaa tanh/3Ek/2 ~22 = w2--Vo+ 37a--eFXW 2 j" dk Ek (6)

0

for the Peierls case. In the spin-Peierls case, it is

~2 = co2_ Vo+ 370--pJXSco 2

'r/c2~ tanh (/1E~/2) sin2ka

x j ~ El, ' (7) O

in the Pytte approximation.

3. ANALYTICAL AND NUMERICAL RESULTS

Inspection of equation (3) reveals that two terms depend on temperature: the term proportional to a and the integral term. If the integral term which is propor- tional to the electron-phonon coupling is small, equation (3) describes a pure structural phase transition. If, however the term in o is small or independent of the temperature, the phase transition will be of the Peierls- type. This is the case when the transition temperature is small. Then an analytical expression can be derived for the critical temperature Tc by observing that the gap equation (equation (3)) looks like that of a pure Peierls transition with X replaced by Xeff:

Vol. 43, No. 5

hco 2 h ~ :

heir G32 -- Vo + 3"la + 7Q: co2 - 11"o + 3"/o'

i f Q , ~ 1. This gives for the critica! temperature (when

kTe "~ eF and hf2,):

kTe = 2.28eF exp-- lc°2 + 3"t° -- V°

= 2.28eF exp -- (1/here) (9)

with

1 cr = 2(¢o 2 + 37a)1/2

a is independent o f the temperature in this limit. In the same limit, the gap at zero temperature is given by:

A(0) = 4e Fexp(-~-~lf) for [QI < 1 (10)

The behaviour o f the gap as a function o f temperature is thus the same as for the Peierls transition: BCS-type for kTe "~ eF. From equation (6), in this limit, the fre- quency of the renormalized soft mode can also be found. It is:

~22 = hco 2 In (T/Te), for T > T e. (11)

In the intermediate case, when the term in o varies appreciably with the temperature, the phase transition has a mixed character and the dimerization curve must be obtained from equation (3), without approximation. In the extreme case where there is no e lec t ron-phonon interactions we have a structural transition for which Olfly the lattice is involved. From equation (3), with the h-term equal to zero, the critical temperature is:

kTe = 2 tanh-1 (12)

This expression and the corresponding gap equation (equation (3) with h = 0) are not of the BCS-type.

For the spin-Peierls case, hse~ is defined as:

h 'oJ : h ~ = (13)

co 2 -- 1Io + 37o

The critical temperature is [6, 11 ] :

kTe = 0.83pJ exp (-- 1 / h ~ ) , (14)

i fkTe '~ J and if h s is not too small. The frequency of the renormalized soft mode is now:

i22 = hSw 2 In(T/Te). (15)

Intermediate cases, when both aspects o f the tran- sition are present (soft mode and e lec t ron-phonon interaction) or when Telis not small, must be treated

PEIERLS AND SPIN-PEIERLS PHASE TRANSITIONS

(8)

o"° ~ - " - ~ 5 t

: \

.4 .3 c d

.2

C .5 T

Fig. 1. Molecular displacements as a function of tem- perature for Vo = 2.1, ~' = co 2 = eF = 1. (a) h = 0, (b) h = 0.01, (c) h = 0.05, (d) h = 0.1, (e) h = 0.5. Insert: Critical temperature as a function h.

377

1.0(

.75 - - ~

.5C

b o~° 0

I'

.5

f

) I 2 V .

I I , 0.5 1.0 1.5 T

Fig. 2. Molecular displacements as a function of tem- perature for *o 2 = eF = ~' = 1.0 and k = 0.4. (a) Vo = 0, (b) Vo = 0.5, (c) Vo = 1.0, (d) Vo = 1.5, (e) Vo = 2.0, (f) Iio = 2.2. Insert: Critical temperature as a function of go.

378 PEIERLS AND SPIN-PEIERLS PHASE TRANSITIONS

numerically. In Fig. 1, I plotted Q as a function o f T for Vo = 2.1, co 2 = 3' = e F = 1 and X = 0, 0.01, 0.05, 0.1 and 0.5. In the insert, T e is plotted as a function o f X. It can be seen that very small values of e lec t ron-phonon coupling are sufficient to alter considerably the dimeriz- ation curve, particularly in the low temperature limit. Also the critical temperature increases as a function o f X. In Fig. 2 I plotted Q as a function of temperature for X = 0.4, 602 = 7 = eF = 1.0 and Vo = 0, 0.5, 1, 1.5, 2.0 and 2.2. In the insert, T c is plotted as a function o f Vo. V0 is a measure o f the intrinsic lattice instability. Note that without the e lec t ron-phonon interactions, for 3' = 6o2 = 1, a phase transition is possible only for Vo >- 2.048. In Fig. 2, with X = 0.4, it is observed that Q and T c increase with Vo and more rapidly for large Vo's than for smaller ones. The effect o f adding both mechanisms for the phase transition is thus to increase considerably the critical temperature. This can be more easily seen in the following example: for a purely struc- tural transition with co 2 = 7 = 1.0 and Vo = 2.1, Tc is 0.28. For a purely Peierls phase transition with to 2 = eF = 1.0 and 2, = 0.5, Tc is o f the same order o f mag- nitude: 0.41. If both mechanisms are combined ( co2 = 3' = eF = 1.0, Vo = 2.1 and 2, = 0.5), Tc is 1.0174. The critical temperature has more than doubled. For the spin-Peierls phase transition, comparable results would be obtained as a function o f J and 2,".

In this communication, two limiting cases were considered analytically, besides the numerical results. For the purely structural instability, it was found that the dimerization occurs only for values o f the intersite potential (Vo) larger than a minimum value Vo MII~. In the other limit, the purely Peierls (or spin-Peierls) instability, it was found that, as soon as an e lec t ron- phonon coupling is present, there exists a critical tem- perature, T c larger than zero at which a dimerization occurs. This explains why a very small e lec t ron-phonon coupling is sufficient to change considerably the dimeriz- ation curve o f the structural phase transition (Fig. 1) while much larger values of Vo are necessary to perturb the dimerization curve o f a purely Peierls transition

Vol. 43, No. 5

(Fig. 2). When both mechanisms are present, I found that, if kT e is small with respect to the characteristic energies o f the system, the dimerization curve obeys a BCS law, as in the pure Peierls case, but with the e lec t ron-phonon coupling constant renormalized to herr def'med in equations (8) and (13) for the Peierls and spin-Peierls cases respectively. The larger tempera- ture cases have been analysed numerically. Both systems, Peierls or spin-Peierls, behave similarly. In considering real systems, like T T F - M B D T (with M = Cu, Au or Pt) one can adjust the behaviour o f the intrinsic soft mode by adjusting V o and 3'- Vo measures the inherent lattice instability and 3' governs its behaviour as a function o f temperature. For the moment, numerical application to these crystals is not possible since the dependence o f the intrinsic soft mode frequency on temperature is not known.

Acknowledgements - This research was partially supported by the Natural Science and Engineering Research Council o f Canada and the Ministdre de l '6ducation du Qu6bec (F.C.A.C.). I would also like to thank N. Pigeon for help in the numerical calculations.

REFERENCES

1. M.J. Rice & S. Strassler, Solid State Commun. 13 ,125 (1973).

2. G.A. Toombs, Physics Reports 40, 181 (1978). 3. A.J. Berlinsky, Rep. Prog. Phys. 42, 1243 (1979). 4. E. Pytte, Phys. Rev. B10,4637(1974) . 5. A.I. Buzdin & L.N. Bulaevskii, Soy. Phys. Usp. 23,

409 (1980); Usp. Fiz. Nauk 131,495 (1980). 6. J.W. Bray, L.V. Interrante, I.S. Ja¢obs & J.C.

Bonner, to be published in Extended Linear Chain Compounds, Vol. III, Plenum Press.

7. M.C. Cross & D.S. Fisher, Phys. Rev. B I 9 , 4 0 2 (1979).

8. H. Thomas, Structural Phase Transitions and Soft Modes (Edited by Samuelsen et al.) p. 15, Universitetsforlaget, Oslo (1971).

9. J. Feder, Ref. [8], p. 171. 10. E. Pytte & J. Feder, Phys. Rev. 187, 1077 (1969). 11. Y. IApine, Can. J. Phys. 59, 1661 (1981).