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A Landau theory of the Peierls-BCS transition
Volume
45A, number
5
PHYSICS
A LANDAU
THEORY
22 October
LETTERS
OF THE PEIERLS-BCS
1973
TRANSITION?
L. GUNTHER* Department
of Physics, Tufts University, Medford, Mass. 02155, Received
6 August
USA
1973
We introduce a Landau type theory of the Peierls instability-superconductor transition involving two complex order parameters. Under a particular inequality condition among expansion parameters, the theory predicts that the above the critical region and as (TE-T)-3’2 below the critical resuperfluid conductivity should vary as (T-Tc)-3’2 gion.
Recently, the theoretical and experimental search for high temperature superconductivity has been uplifted by the discovery at the University of Pennsylvania [l] that TTF-TCNQ exhibits paraconductor-like behavior [2] but that just above the temperature Tc at which the high temperature behavior of the superfluid conductivity would lead one to expect the system to become a superconductor, the system becomes an insulator. Bardeen [3] and Patton and Sham [4] have attempted to ascribe this paraconductivity to the “surf-riding” mechanism of Frohlich [S] while Gutfreund et al. [6] entertain the possibility that BCS [7] pairing is responsible. In this paper, we assume that BCS pairing is operative. Our work is a take-off on the paper of Bychkov et al. [8] who, as far as this author knows, were the first to note that the two instabilities of the one-dimensional Fermion system - the Peierls instability [9] and the BCS pairing instability - might, if independent, be resolved by a phase transition at approximately the same temperature?‘. In fact, because both instabilities are resolved by a gap in the electron energy spectrum, there is a competition between the two instabilities. That is, the resolution of one of the instabilities, accompanied by its corresponding phase transition, weakens the second instability and can prevent the occurrence of the second type of phase transition. We assume that the free energy can be expanded in a series involving two complex order parameters, $I and ?Research partially supported by National Science Foundation Grant Number GP-16025. *Most of this work was carried out while the author was on Sabbatical at the Departments of Physics at the Technion and at Tel Aviv University, Israel.
$, corresponding to K and A, respectively, of ref. [8], which correspond to the density variation of Peierls state and the pair wave function of the superconducting state, respectively. Thus, F(#, ti, T) = al Ml2 + b, lGi4/2 + clld#/dx12 +uZl$12 +b2111/14/2+c2id~ldx12 +dl~121J/12
(1)
where x is the position along a molecular chain, al = al (T-Tp), a2 = ii2(T-T,), and a,, a2, b,, b2, cl, c2 and d are approximated as positive constants independent of temperature. In line with the experimental results of ref. [ 11, we will assume that Tp > T,. Letting q50and Go denote the mean field values (assumed real) which minimize the free energy, we obtain the following results: above Tp, both r$o 2an_dtie vanish. Just below Tp, Go still vanishes but Go -al/b,, leading to an effective parameter a2eff = 2 q +d$o. In line with the experimental results of ref. [ 1] , we assume that al dTp/ii2 b 1 T, is several times unity so that 4, = 0 down to absolute zero. According to ref. [2], we expect the paraconductivity to be proportional to (T-Tc)-3/2 above the critical region - in agreement with experiment and proportional to (T,*-T)-3/2 below the critical region, where T,* -(ii2Tc-aldTp/b,)/(;i,-ii,d/b1), a result not yet checked by experiment. The full theory also predicts the possibility of two peaks in the specific heat, a result which can distinguish this theory from a theory involving a single order parameter. Detailed results will be reported elsewhere. t 'Under the approximations of ref. [8], they are resolved at exactly the same temperature. 367
Volume
45A, number
5
PHYSICS
The author is indebted to numerous people for stimulating and helpful discussions, most particularly S. Alexander, D. Bergman, G. Deutscher, 0. Entin, H. Gutfreund, Y. Imry, C. Kuper and J. Rudnick.
References [I] L.B. Coleman et al., Solid State Comm. 12 (1973) [2] R.A. Ferrell, J. Low Temp. Phys. 1 (1969) 241. [3] J. Bardeen, Solid State Comm. 13 (1973) 357.
36%
1125.
LETTERS
22 October
1973
[4] B.R. Patton and L.J. Sham, Phys. Rev. Lett. 31 (1073) 631. ]5] H. Frohlich, Proc. Roy. Sot. London A223 (1954) 296; C.G. Kuper, Proc. Roy. Sot. London A227 (1955’1 214. [6] H. Gutfreund, B. Horovits and M. Weger, J. Phya. C: Solid State Phys., to be published. [7] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [8] Yu.A. Bychkov, L.P. Gorkov and J.E. Dzyaloshinskiy. JETP 50 (1966) 738; Soviet Phys. JETP 23 (1966) 489. [9] R.E. Peierls, Quantum theory of solids (Oxford Univerrity Press. London 1953) p. 108.
45A, number
5
PHYSICS
A LANDAU
THEORY
22 October
LETTERS
OF THE PEIERLS-BCS
1973
TRANSITION?
L. GUNTHER* Department
of Physics, Tufts University, Medford, Mass. 02155, Received
6 August
USA
1973
We introduce a Landau type theory of the Peierls instability-superconductor transition involving two complex order parameters. Under a particular inequality condition among expansion parameters, the theory predicts that the above the critical region and as (TE-T)-3’2 below the critical resuperfluid conductivity should vary as (T-Tc)-3’2 gion.
Recently, the theoretical and experimental search for high temperature superconductivity has been uplifted by the discovery at the University of Pennsylvania [l] that TTF-TCNQ exhibits paraconductor-like behavior [2] but that just above the temperature Tc at which the high temperature behavior of the superfluid conductivity would lead one to expect the system to become a superconductor, the system becomes an insulator. Bardeen [3] and Patton and Sham [4] have attempted to ascribe this paraconductivity to the “surf-riding” mechanism of Frohlich [S] while Gutfreund et al. [6] entertain the possibility that BCS [7] pairing is responsible. In this paper, we assume that BCS pairing is operative. Our work is a take-off on the paper of Bychkov et al. [8] who, as far as this author knows, were the first to note that the two instabilities of the one-dimensional Fermion system - the Peierls instability [9] and the BCS pairing instability - might, if independent, be resolved by a phase transition at approximately the same temperature?‘. In fact, because both instabilities are resolved by a gap in the electron energy spectrum, there is a competition between the two instabilities. That is, the resolution of one of the instabilities, accompanied by its corresponding phase transition, weakens the second instability and can prevent the occurrence of the second type of phase transition. We assume that the free energy can be expanded in a series involving two complex order parameters, $I and ?Research partially supported by National Science Foundation Grant Number GP-16025. *Most of this work was carried out while the author was on Sabbatical at the Departments of Physics at the Technion and at Tel Aviv University, Israel.
$, corresponding to K and A, respectively, of ref. [8], which correspond to the density variation of Peierls state and the pair wave function of the superconducting state, respectively. Thus, F(#, ti, T) = al Ml2 + b, lGi4/2 + clld#/dx12 +uZl$12 +b2111/14/2+c2id~ldx12 +dl~121J/12
(1)
where x is the position along a molecular chain, al = al (T-Tp), a2 = ii2(T-T,), and a,, a2, b,, b2, cl, c2 and d are approximated as positive constants independent of temperature. In line with the experimental results of ref. [ 11, we will assume that Tp > T,. Letting q50and Go denote the mean field values (assumed real) which minimize the free energy, we obtain the following results: above Tp, both r$o 2an_dtie vanish. Just below Tp, Go still vanishes but Go -al/b,, leading to an effective parameter a2eff = 2 q +d$o. In line with the experimental results of ref. [ 1] , we assume that al dTp/ii2 b 1 T, is several times unity so that 4, = 0 down to absolute zero. According to ref. [2], we expect the paraconductivity to be proportional to (T-Tc)-3/2 above the critical region - in agreement with experiment and proportional to (T,*-T)-3/2 below the critical region, where T,* -(ii2Tc-aldTp/b,)/(;i,-ii,d/b1), a result not yet checked by experiment. The full theory also predicts the possibility of two peaks in the specific heat, a result which can distinguish this theory from a theory involving a single order parameter. Detailed results will be reported elsewhere. t 'Under the approximations of ref. [8], they are resolved at exactly the same temperature. 367
Volume
45A, number
5
PHYSICS
The author is indebted to numerous people for stimulating and helpful discussions, most particularly S. Alexander, D. Bergman, G. Deutscher, 0. Entin, H. Gutfreund, Y. Imry, C. Kuper and J. Rudnick.
References [I] L.B. Coleman et al., Solid State Comm. 12 (1973) [2] R.A. Ferrell, J. Low Temp. Phys. 1 (1969) 241. [3] J. Bardeen, Solid State Comm. 13 (1973) 357.
36%
1125.
LETTERS
22 October
1973
[4] B.R. Patton and L.J. Sham, Phys. Rev. Lett. 31 (1073) 631. ]5] H. Frohlich, Proc. Roy. Sot. London A223 (1954) 296; C.G. Kuper, Proc. Roy. Sot. London A227 (1955’1 214. [6] H. Gutfreund, B. Horovits and M. Weger, J. Phya. C: Solid State Phys., to be published. [7] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [8] Yu.A. Bychkov, L.P. Gorkov and J.E. Dzyaloshinskiy. JETP 50 (1966) 738; Soviet Phys. JETP 23 (1966) 489. [9] R.E. Peierls, Quantum theory of solids (Oxford Univerrity Press. London 1953) p. 108.