6 May 1996 & __ __ EB ELSEVIER

PHYSICS LETTERS A

Physics Letters A 2 I4 (I 996) 89-94

Isotope-shift exponent in the polaronic model of superconductors

Sourabh Banerjee a, A.N. Das b, D.K. Ray ’

a Depurtment of Physics. Visuu - Bhurati, Suntiniketm 731 235, India

b Saha Institute oj’Nuclrur Physics, I /AF. Bidhanrqur, Calcutta 700 064, India

’ Luhoratoire des Proprietes Mecmiques et Thermodynmniyues des Muter&u, CNRS. Uniuersite’ Paris Nod. F93430 Villrtctneuse,

Paris. France

Received 3 January 1996; accepted for publication 5 February 1996

Communicated by L.J. Sham

Abstract

We have studied the isotope-shift exponent ((~1 within the polaronic theory of superconductivity as functions of electron-phonon coupling and carrier concentration. cy is found to be consistently negative for all concentrations and electron-phonon couplings in contrast to experimental reports. Increasing Coulomb repulsion or use of a singular density of states pushes (Y to more negative values.

PACS: 74.20. - z; 74.20.Fg; 74.62. - c; 74.62.Dh

Keyworclst Isotope-shift exponent; Polaronic superconductivity

1. Introduction

The isotope-shift exponent ( (Y) of high-T, cuprates exhibit an anomalous behaviour with doping [l]. (Y is large and positive for underdoped systems where T, is low and attains a very small value at optimum doping where T, is maximum. In some samples of Bi-systems [2] negative values of LY in the over- doped region were reported. However, the general trend of (Y for most of the high-T, cuprates is that it is minimum at optimum doping and positive for the entire range of carrier concentration [l].

Recently, Alexandrov [3] and Alexandrov and Mott [4] suggested that the isotope-exponent be- haviour of high-T, cuprates may be explained within

the polaronic and bipolaronic models of supercon- ductivity. Using an approximate expression for T,,

which extrapolates between polaronic and bipola- ronic superconductivity, Alexandrov [3] studied the variation of T, and cr as a function of the electron- phonon (e-ph) coupling strength and obtained cr < 0 in the polaronic model but (Y > 0 for Bose condensa- tion of bipolarons.

The objective of the present work is to investigate the variation of LY with e-ph interaction and carrier concentration in the polaronic model without making any approximation to the equation for T,. The influ- ence of a van Hove singularity in the electron den- sity of states and Coulomb repulsion on a are also investigated.

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90 S. Bunerjee et d./ Physics Letters A 214 (19961 89-94

2. Model Hamiltonian and T,

The model polaronic Hamiltonian, relevant for

narrow-band systems and studying superconductiv- ity, is an extended Hubbard Hamiltonian [5,6]

H= -t, c c~cjq+U,~nifnil -Vxnin, 1.j.u i ij

- CWit (1) i

where I, = t exp( - g 2, is the reduced polaronic hop-

ping, r is the bare hopping and g denotes the e-ph coupling strength. U, is the effective on-site Coulomb repulsion and V is the inter-site attractive interaction

induced by the e-ph coupling. For large values of g, fP is very small and for WP -K V, bipolarons may

form (W, is the polaronic half-bandwidth). On the

other hand, for moderate values of g, Wp 2 V, po- larons are expected to be the charge carriers in the

system. Making the usual Hartree-Fock approximation to

the second and third terms of Eq. (1) and defining the order parameters

Ao=(c,Lc,t) (2)

and

A, = i(<ci, c/t > + (cjs c,t ))3 (3)

the Hamiltonian (Eq. (1)) may be written in the BCS form as

H= C&n,, + c+:t CL + h.c.), (4) ka k

where tk=ck--p, l k= -t,zy, and yk = (l/z)Cij exp(ikRij), Ri, being the nearest neigh- bour lattice vector. For a square lattice z = 4 and -yk = cos k, + cos k,. The superconducting gap pa- rameter corresponding to s-wave pairing is

A, = WeA, - zy,VA,. (5)

Using the Green function technique the supercon- ducting correlation is obtained as

<c:, c’, 1 > = - +nh( P&/2), (6) k

where E, = tk + A, and p = l/k,T. From the set \i, of coupled equations (5) and (6) for the order param- eters one obtains at T = T, the relations

1 = zVZ,( A,/A,) - &I, (7)

and

1 = zVI, - &I,( do/A,),

which leads to the equation for T, given by

(8)

(9)

where

I” = / ’ dN5)

-I

E: ta*( pEy?) 2(%-/h)

( 10)

for n = 0, 1, 2. er, pr and p,’ are a set of reduced variables, namely, energy, chemical potential and

l/k,T, in units of the polaronic half-bandwidth

WP = W, exp( -g’), (11)

W, (= zr) being the half-bandwidth of the bare electronic band. N(E,) is the electron density of states.

To obtain the approximate expression for T, used by Alexandrov [3], Eq. (9) is recast in a slightly different form as

1 = ZVJ, + 2zV/_L,J, + (ZV/_Lf - CIJJ,

+ zvu,( J,J, -JF),

where

(12)

/ ’ J, = de,N(e,) x+(E~-/L$-

--I

xtti( piyj, (13)

with n = 0, 1, 2. In the weak coupling limit (W, z++ zV ) J, reduces to a simple form

Jo= --& ln(l.l4P:d]). (14) P

The values of the integrals J, and J, are much smaller than J, in the weak coupling limit. Assum-

S. Banerjee et al./ Physics Letters A 214 (1996) 89-94 91

ing J, = 0 = J,, an approximate expression for T, is obtained as

T”pp’ = l.l4W,/-;lf_exp c (15)

This expression was used in Ref. [3] to derive the

isotope-shift exponent a.

3. Isotope-shift exponent

To derive an expression for the isotope-shift ex- ponent (a = -8 In T,/a In M) one has to differen-

tiate Eq. (9) with respect to M. The relevant factors which depend on the isotopic mass (M) are W,, and p,‘. The e-ph coupling strength varies with M ac-

cording to g* = & 131, correspondingly WP de- creases with increasing isotopic mass (this view has been supported by a recent experiment where an increase in the effective mass of the charge carriers on heavier isotopic substitution has been observed [7]). An expression for (Y is obtained as

2 1 &L ]-_

i

I* - 211, + I*&

2 I p,’ s, - 2zs, + Ps, ’ (‘6)

where

I= CbI,

1 +lJ& (17)

The In’s are defined as in Eq. (10) and S,, S, and S, are defined in a general way as

with n = 0, 1, 2. In the absence of Coulomb repul- sion (U, = 0) the expression for cx reduces to a very simple form

a=;(+&). (19)

The expression of (Y corresponding to TcapP’ (Eq. (15)) is

ET2 aaappr=_ ]_ ( 25

2 I ZV/.Lf - u, . (20)

The transition temperature CT’,> and the isotope-

shift exponent (a) are calculated using a square DOS

N(q) = &-, I E, I I 1 P

and a DOS with a van Hove singularity

(21)

0.20953 N( Er> = y--- lnl4/E,l, IE,I Il. (22)

P

4. Results and discussions

As stated in the objective our emphasis lies in the study of T, and (Y within the framework of a polaronic model as a function of g* and pr and a comparison with the results, obtained from an ap- proximated formula, is made. Our choice of parame- ters is guided by the following observations: for high-T, oxides W, - l-2 eV [8] and the effective mass of the charge carriers, m* - (5-lO)m, [9]. The

variation (of T, and CY> with respect to g2 is studied for W, = 12000 K and V = 2000 K while for the variation with respect to pF the choice has been Wp = 2000 K, V- 500-2000 K and g2 = 2.3. It may be noted, however, that the value of the isotope-shift exponent depends on the ratio of V/W, rather than on their independent values.

700 -

600-

500-

y LOO-

+” - 300-

200-

loo-

0 I , I [ I , , I , , ,

0 1 2 3 fs 5 6

g2

Fig. 1. Variation of T, with g* for W, = 12000 K and V = ZOCKI

K. Solid curves represent exact T, from Eq. (9). Dashed and

dot-dashed curves are for approximate T, from Eq. (15). UC = 0

for all curves.

92 S. Bunerjee et (II./ Physics Letrers A 214 (19961 89-94

In Fig. 1 we have compared the variation of T, with g2 from the exact and approximate equations, in the absence of on-site repulsion, for two values of pr (0.4 and 0.6). T, obtained from exact calculations increases with increasing g2, i.e. with decreasing polaronic bandwidth, and saturates to a value of

V/3, independent of F~, for large values of g*. However for very narrow polaronic bands (i.e. large

g2> the weak-coupling BCS-type approach becomes invalid. In fact, bipolarons may form in this limit.

Approximated values of T,, on the other hand, show a peak at an intermediate value of g* = gi. Alexan-

drov [3] has pointed out that the approximate expres- sion for T, (Eq. (15)) is capable of describing the

Bose condensation of intersite bipolarons for g* > gz. Thus, the falling off in c on both sides of g, may be explained as follows: for g* < g,‘, WP 2 V and the superconductivity is polaronic while for g* > g,’ (W, < V) Bose condensation of bipolarons sets in and T, decreases due to an increase in the effective mass of the bipolarons with increasing g2.

In Fig. 2 the variation of exact values of a

(obtained from Eq. (19)) with g2 is shown for CL, = 0.8, 0.6 and 0.4. cr is negative for the entire range of g values and becomes zero as g --) 0 as well as for large values of g where T, saturates and dT,.d g 2 = 0. For intermediate values of g, the re- gion of interest for polaronic superconductivity, cy is negative and decreases (becoming more negative) with decreasing values of p,. For BCS-type pola- ronic superconductivity (Y is negative even for large g. However, as mentioned earlier that for large g superconductivity may be due to the Bose condensa- tion of intersite bipolarons with the transition tem- perature given by T, = W, exp( - g2)4(n> [3] which results in cy = g2/2. Thus a crossover from pola- ronic to bipolaronic superconductivity, as expected with increasing g values, should associate with a change in sign of cy as shown in Ref. [3].

For the study of the variation of T, and cy with pLr we confine our numerical exercises to the region of polaronic superconductivity i.e. V/W, 5 1.

In Fig. 3 the variation of T, as a function of p-Lr is shown for cl, = 0. The curves c. f and g are obtained from Eq. (9) with a square DOS for V = 2000, 1250 and 500 K respectively, keeping WP = 2000 K. Evi- dently, a decrease in V/W, results in a decrease in T. For V = W, = 2000 K we have also shown the T,

-3.51 I , I , , , I , , , I , 0.0 1 2 3 4 5 6

g2

Fig. 2. Variation of exact cy with gL for W, = 12OCQ K and

V = 2000 K. Curves a, b and c are for ~~ = 0.8, 0.6 and 0.4

respectively. cl, = 0 for all curves.

obtained from Eq. (9) using a DOS with a vHs (curve d) and that obtained from the approximate formula (Eq. (15)) of T, for a square DOS (curve a>. It is seen that the approximate formula predicts a much higher value of T, (for 0.55 < pu, < 0.951, whereas a vHs in DOS reduces the value of T, compared to that for a square DOS. Within the polaronic model, increase in isotopic-mass leads to a decrease in WP. In Fig. 3 we also show the effect of decreasing WP on exact and approximated values of

600

500

s LOO-

> 300-

200-

loo-

o- 0.0 0.2 0.4 0.6 0.8 1.0

Fig. 3. Variation of T, with CL, ( = p/W,,). Curves a, b: approxi-

mate T, with Wp = 2000 and 1800 K respectively and V = 2ooO

K. Curves c, e, f and g: exact T, for a square DOS. Wp = 2000 K

and V = 2000, I250 and 500 K for curves c, f and g respectively.

For curve e: Wp = 1800 K and V = 1250 K. Curve d: exact T, for

a DOS with a vHs for W,, = V = 2000 K.

S. Banerjee et al./ Physics Letters A 214 (19961 89-94 93

T, for a square DOS. It is seen that exact values of

T, increases with decreasing Wp for all values of pr while approximated T, increases with decreasing Wp

in one region of pu, and decreases in the other part. These results clearly indicate that a would change

sign when approximate expression of T, is used, while use of exact equation for T, would yield negative a for the entire range of CL, for polaronic

superconductivity. In Fig. 4 we plot the variation of (Y with the

chemical potential Pi. The exact values of (Y for a square DOS are shown by the solid curves for

different values of V/W, (= 1, 0.625, 0.25) and the dashed curve shows the variation of (Y, derived from the approximate formula of T,, for V/W, = 1. It is

seen that exact values of (Y are considerably negative and its magnitude increases with decreasing V/W,. Approximate values of (Y show a smooth variation from positive to negative values with decreasing Pi. Thus in the polaronic region of BCS-type supercon- ductivity use of the approximate expression for (Y is inadequate, since it predicts positive values in a region of pr where exact values of (Y are consis- tently negative.

In Fig. 5 we have shown the effect of Coulomb repulsion and a vHs in DOS on cr. Both Coulomb repulsion and the vHs in DOS shift the values of IX towards more negative values.

For high-T, oxides CY is positive and large in the

l-

-l-

1 -3-

a _/---

.- , b

. / c

/ /

d

/ /

-7 :n;;

/ I

-5- / I

I

0.2 0.L 0.6

pr

0.8 1.0

Fig. 4. Plot of a versus CL, for a square DOS. Curve a: approxi-

mate a for Wp = V = 2COO K. Curves b, c and d: exact LY for

Wp = 2000 K and V = 2000, 1250 and 500 K respectively.

+h 1.0

Pr Fig. 5. Plot of a versus fir for Wp = V = 2000 K for different

values of C/,. Solid curves are for a square DOS: a (U, = 0) and b

((/, = 1000 K). Dashed (UC = 0) and dot-dashed ((i, = 1000 K)

curves are for a DOS with a vHs.

underdoped region and decreases with increasing carrier density. In most of the high-T, samples, a! is minimum (and positive) near optimum doping. Our study shows that (Y is negative for polaronic BCS-

type superconductivity for the entire range of doping and becomes more negative with the introduction of Coulomb repulsion or the presence of a vHs in the DOS. Positive values of Q may be obtained within the polaronic theory for large values of g, where the superconductivity is due to the bose condensation of bipolarons [3]. A transition from bipolarons to a polaronic state should occur at a high or intermediate value of g and be reflected in LY through a smooth transition from positive to negative values. For most high-T, oxide systems LY is positive and increases beyond optimum doping where polaronic supercon- ductivity is expected to be present. But the approxi- mated calculations [3,4] based on the polaron theory as well as our more exact calculations give (Y to be always negative.

It may be mentioned that the Hamiltonian (Eq. (1)) also yields the d-wave solution for the supercon- ducting order parameter. The s-wave solution is sta- ble when the chemical potential lies near the bottom or top of the band, whereas the d-wave solution is stable for a wide range of intermediate values of pr [5,6]. For d-wave pairing also T, increases with decreasing bandwidth provided the pairing interac-

94 S. Banrrjee et al./ Physics Letters A 214 f 1996) 89-94

tion remains constant. Within the polaronic model the isotopic substitution changes the bandwidth only whereas the pairing interaction remains unaltered.

Thus the isotope-shift exponent within the polaronic model should be negative even for d-wave pairing.

It is, therefore, difficult to reconcile how the polaronic mechanism of superconductivity in the

present form can explain the observed behavior of the isotope-shift exponent in high-T, oxide systems.

This is in contrast to the success of the polaronic theory in explaining the optical conducitivity experi-

ments [lo]. The observed (Y > 0 in high-T, systems suggests that the superconductivity in high-T sys- tems may be bipolaronic [3,4], however the increase

of LY beyond optimum doping, as observed in high-T, systems [l], is difficult to be understood within the bipolaronic theory.

Acknowledgement

S.B. acknowledges the financial support of the University Grants Commission, India. D.K.R. is thankful to Professor AS. Alexandrov of I.R.C., Cambridge for clarifications regarding the influence

of polarons and bipolarons on the isotope-shift expo- nent in high-ir, superconductors.

References

111

121

131 [41

[51

[61

[71 [81

bl

II01

J.P. Frank, in: Physical properties of high-temperature su-

perconductivity, Vol. 4, ed. D.M. Ginsberg (World Scien-

tific, Singapore, 1994) p. 189.

H.J. Bomemann, D.E. Morris and H.B. Liu, Physica C I82

(1991) 132.

AS. Alexandrov, Phys. Rev. B 46 (1992) 14932.

AS. Alexandrov and N.F. Mott, Rep. Prog. Phys. 57

(1994) 1197.

R. Micnas, J. Ranninger and S. Robaszkiewicz, Rev. Mod.

Phys. 62 (1990) 113.

D.K. Ray, J. Konior, A.M. Oles and A.N. Das, Phys. Rev. B

43 (1991) 5606;

A.N. Das, J. Konior, D.K. Ray and A.M. Oles, Phys. Rev. B

44 (1991) 7680.

G. Zhao and D.E. Morris, Phys. Rev. B 51 (199.5) 16487.

K.C. Hass, in: Solid state physics, Vol. 42, eds. H. Ehren-

reich and D. Tumbull (Academic Press, London, 1989) p.

213.

G.A. Thomas et al., Phys. Rev. Lett. 61 (1988) 1313;

V.Z. Kresin and S.A. Wolf Phys. Rev. B 41 (1990) 4278.

AS. Alexandrov, V.V. Kabanov and D.K. Ray, Physica C

224 (1994) 247, and references therein; Phys. Rev. B 49

(1994) 9915.