- Email: [email protected]

On the fluctuation-induced superconductivity pseudogap N. Kristoffel b

a,)

¨ , T. Ord

b

a Institute of Physics, UniÕersity of Tartu, Riia 142, EE2400 Tartu, Estonia Institute of Theoretical Physics, UniÕersity of Tartu, Riia 142, EE2400 Tartu, Estonia

Received 2 December 1997; accepted 20 January 1998

Abstract The influence of spatially correlated order parameter fluctuations on the elementary excitations in the normal phase of a superconductor has been investigated. As a result of fluctuations, a pseudogap on the Fermi surface is present. The magnitude of the pseudogap decreases with temperature rising and it is essentially scaled with the parameter a s e FrkTc . This explains why the pseudogap has been observed in high-Tc systems and not in the conventional superconductors and its suppression in the overdoped high-Tc systems. q 1998 Elsevier Science B.V. PACS: 74.20 Keywords: Superconductivity; Pseudogap; Fluctuation

1. Introduction One of the new properties that has been heaved in sight in connection with superconductors possessing high transition temperatures ŽTc . is the pseudogap opening Žremainder. at T ) Tc , e.g., w1–7x. It concerns the onset of the superconducting correlations in the normal state and consists in a suppression of spectral weight of electronic excitations near the Fermi surface. In photoemission, one observes this phenomenon for a considerable interval of temperatures over Tc . Magnetic and heat measurements lead to the same result. In the superconducting state ŽT - Tc ., there is a well-established Žanisotropic. gap in the spectrum, )

Corresponding author. Fax: q372-7-383033; e-mail: [email protected]

which plays the role of superconductivity order parameter reflecting the long-range coherence. Over Tc , this gap does not close totally. This normal-state pseudogap is related to the superconducting gap below Tc , and is a normal-state precursor of the true gap. The pseudogap has the same symmetry as the true one. The high-Tc cuprate superconductors and related materials need a special doping treatment to start and optimize the desirable properties. The pseudogap seems to be a property of underdoped materials becoming suppressed with passing the optimally doped region. As the source of the pseudogap feature the pairing field Žorder parameter. fluctuations, or even the direct precursor, pair formation according to various superconductivity driving mechanisms has been identified in theoretical papers, e.g., Refs. w7–14x.

0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 0 4 9 - 5

¨ r Physica C 298 (1998) 37–40 N. Kristoffel, T. Ord

38

The first contribution to this problem seems to be the paper w15x on the fluctuation effects at a Peierls transition. Fluctuation effects are of general nature. Correspondingly, one can expect the pseudogap feature to be a general phenomenon. In this relation, two questions arise: Ži. why has the pseudogap not been notified in conventional low-Tc superconductors? Žii. what is the reason of suppression of pseudogap if the overdoped region is reached in high-Tc superconductors? We hope to demonstrate in this note that the pseudogap is a fluctuation-driven general feature in superconductors, and the two questions mentioned are naturally clarified by taking account of the role of the scaling parameter a s e Frk B Tc Ž e F is the Fermi energy. of the problem. A very simple model is used. We note here that an analogous w16x fluctuationinduced gap renormalization exists for vibronic ferroelectrics w17x explaining the observed nonlinear dielectric gap dependence on temperature in the paraphase Žalso the ‘excitonic insulator’ must be mentioned.. In this case, the electronic order parameter Žanomalous average. is of the electron-hole ‘excitonic’ type, whereas for superconductor, it is of pair creation Ždestruction. type.

of the fluctuating order parameter DŽ™ r . in the free energy functional for the system with the superconducting phase transition Fs

½

a

b

H

5

Ž 3.

,

Ž 4.

with a s 2 ru , u s

bs

cs

T y Tc Tc

7z Ž 3 . r 4p 2 Ž k B Tc . 7z Ž 3 . r 8 dp 2

2

,

Ž 5. 2

"n F

ž / k B Tc

.

Ž 6.

Here Tc designates the superconducting phase transition temperature in the mean-field approximation, r is the electron states density at the Fermi energy of the d-dimensional electron system with parabolic spectrum and Õ F is the Fermi velocity; z Ž3. f 1.2. The Fourier transform of the one particle two-time Green’s function for the system described by Eq. Ž1. satisfies the equation ™

X E y e Ž k . G Ž a k ≠ ; aq k ≠.

2. The model We will describe the superconductor by the Hamiltonian

d k kX

X D™ qD™ q )

™™

s

2p

q

Ý

™

™ ™X

q, q

E q e k y™ q

ž

/

™

q™ q ™ a™ q h.c. 4 , H s Ý e Ž k . aq k s a k s q Ý D™ q aykqq x k≠ ™

qX ™ ™X =G Ž a™ky ™ qqq ≠ ; ak ≠ . .

™™

k,s

k,q

Ž 1. ™

where aq with the momentum k, k s creates an electron ™ ™ spin s and the energy e Ž k ., e Ž k F . s 0. The second term in Eq. Ž1. describes generation and annihilation of pairs with the total momentum ™ q by the fluctuating source D™ q . This quantity changes in the normal phase randomly so that - D™ q G 0. In fact, D™ q must be treated as the Fourier transform ™™

D q s Vy1 D Ž ™ r . eyi q r d™ r ™

H

Ž 2.

Ž 7.

In Eq. Ž7. we replace now the product of ran)X domly fluctuating quantities D ™ qD™ q by its average value in the Gaussian approximation. By means of Eq. Ž2. one has ²D ™ : ²< q < 2 :d™ X X qD™ q ) s D™ q™ q , ²< D™ <2:s q

js

c

ž / a

k BT y2

Vc j

qq2

,

Ž 8. Ž 9.

1r2

,

Ž 10 .

¨ r Physica C 298 (1998) 37–40 N. Kristoffel, T. Ord

where j is the correlation length Žthe pair size.. As a result, one finds G Ž a k ≠ ; aq k≠ . s

1 2p

In the reliable case Eg - 2 e F one has from Eq. Ž17.

Ž 11 .

Eg2 s y

with the mass operator

gs

k BT c

V

™

q

Ž 4e F A jy2 q A2jy4 .

2

1

Ý

2 1

q

g P Ž E. s

1

y1

™

Eye Ž k. yP Ž E.

™

E q e k y™ q

Ž jy2 q q 2 .

ž

,

Ž 12 .

/ Ž 13 .

Ž 4e F A jy2 q A2jy4 .

° ¢

s~y

Eg 0

a p

a q

p

3. The pseudogap feature We approximate now the electron band energy dispersion by a two-dimensional parabolic spectrum Ž A s " 2r2 m. ™

e Ž k . s Ak 2 y e F

Ž 14 .

with the constant density of states r s mr2p " 2 . Performing the ™ q-integration in Eq. Ž12., one obtains for the mass operator the expression g P Ž E. s . 1r2 ™

y2

2

ž E q e Ž k . y Aj /

q 4 A2 k 2jy2

Ž 15 . The Green’s function poles determine the elementary excitation energy spectrum of the system under consideration. With Eqs. Ž11. and Ž15., the corresponding equation is ™

™

Eye Ž k. ™

q4 e Ž k . q e F A jy2 ™

y2

2

ž E q e Ž k . y Aj / 1r2

sg .

Ž 16 .

™

For k s k F , it follows from Eq. Ž16. that E s Eg / 0, i.e., there exists a pseudogap Eg at the Fermi surface caused by fluctuating superconductivity order parameter amplitude. The pseudogap satisfies the equation Eg Eg2 q 2 Ž 2 e F y Eg . A jy2 q A2jy4

1r2

s g . Ž 17 .

2

q 4g 2

1r2

.

Ž 18 .

This result can be written also as Eg

.

39

uq

uq

2pu 2 7z Ž 3 .

2pu 2 7z Ž 3 .

ay1

y1

a

¶• ß

1r2 1r2

2

q Ž1qu .

2

,

Ž 19 . where Eg 0 s

k B Tc c

1r2

s

16p 3 Ž k B Tc . 7z Ž 3 . e F

3

1r2

Ž 20 .

is the pseudogap at T s Tc . Eq. Ž17. determining the temperature dependence of the pseudogap ŽT G Tc . contains one free parameter eF as . Ž 21 . k B Tc The value of a determines essentially the pseudogap behaviour, as depicted in Fig. 1. It is seen that the pseudogap decreases with increasing temperature and the fall-off of Eg is scaled by the parameter a . Thus, the increase of a suppresses the pseudogap compared to Eg0 . At the same time, the ratio Eg0re F ; ay3 r2 diminishes with a increasing. Consequently, smaller a values are more favourable to observe a pseudogap of fluctuative origin. In conventional superconductors a ; 10 3 % 10 4 and the pseudogap forms only in a very narrow temperature region above Tc . This circumstance explains why it has not been observed before the discovery of high-Tc systems. In cuprate superconductors with short coherence lengths a is essentially smaller and the pseudogap becomes observable. By going from slightly underdoped to overdoped compounds a increases. In slightly underdoped region, the increase of Tc with doping is slow and in

40

¨ r Physica C 298 (1998) 37–40 N. Kristoffel, T. Ord

Acknowledgements This work has been partly supported by the Estonian Science Foundation grant No. 1929.

References

Fig. 1. The dependence of the relative pseudogap Eg r Eg0 on temperature near Tc . Curve 1: a s10; 2: a s10 2 ; 3: a s10 3 ; 4: a s10 4 .

overdoped region Tc falls. At the same time, the Fermi energy rises continuously and quickly w18x. As a result, the pseudogap regime will be suppressed as observed. In such a way, the superconductivity fluctuations manifest themselves through a pseudogap opening before the phase transition happens. Precursor and residual dynamic fluctuation Žrelaxor.-type phenomena have been observed in connection with various phase transitions in the last time.

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