Correlation effects and superconductivity in doped fullerite

PHYSICA

Physica B 186-188 (1993) 1071-1073 North-Holland

Correlation effects and superconductivity in d o p e d fullerite Valery A. Ivanov N.S. Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences, Moscow, Russian Federation The proposed AxC60model involves strong Hubbard energy of alkali electrons in the narrow molecular LUMO band (lowest unoccupied molecular orbital) of C6o. By invoking a kinematical high-Tc mechanism, we obtained an expression for Tc(X) and found the superconductivity gap Ao which satisfies the inequality 2Ao/Tc> 3.53.

1. Introduction The discovery of 3D superconductivity in doped fullerites AxC6o [1-4] raised the need for a description of its nature and superconductivity features. It follows from magnetic readings in KxC6o [5] that superconductivity is formed by a narrow band about 0.04 eV wide (for 2Ao/Tc = 3.53) which means large localization of and strong correlations between electron wavefunctions. The first quantum-chemical estimates [6-8] showed that C6o has a large H O M O - L U M O gap of about 2 eV. Following the tight-binding method for correlated electrons [9], we can show that threefold degeneracy of the t~u band in the compound AxC60 can be eliminated in this way: E , ( p ) = (E,, u - Eh° ) -- 2 f t z T ( p ) , Ez,3(p) = (Equ - Eh.) + tfzT( p) ,

(1)

where z is the number of nearest neighbours, 3'(P) is the structure factor of the FCC lattice, f is the correlation factor [9] dependent on the population of correlated levels by alkali A-electrons, and t is the hopping integral of electrons between alkali neighbours. The electronic structure (1) of noninteracting electrons is defined by the Slater-Koster method as f = 1 [10].

and a double-degenerate narrow band. Let us assume a wide band filled with one alkali electron below the Fermi level. If we denote interband and intraband hopping integrals as h and t, respectively, the Hamiltonian for strongly correlated alkali electrons of the degenerate narrow band will be H =

Z (i,/)a,A=

+

""~°g°'~ tqXia ja 1,2

Z

As is evident from the simplest tight-binding estimates (1), electron tunnelling in AxC6o splits the L U M O band of C6o into a nondegenerate wide band Correspondence to: V.A. Ivanov, N.S. Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences, 31 Leninsky Prospekt, GSP-1 Moscow 117907, Russian Federation.

x..06- - Z,,,,

<2)

i

where X pq is the Hubbard operator and ix is the chemical potential. The most recent electron energy loss spectroscopy findings have confirmed the existence of separate wide and narrow bands in doped fullerites [12,13]. The interband effective hopping integral h between alkali A-ions can be linked to the electron-phonon interaction of the C6o ion [14]. Contrary to some studies [15], the Hubbard energy U >> t, h is essential for the suggested model. In a loopless approximation, the inverse Green function in line with model (1) can be written as

A=I A=2

2. A model of the electronic structure of doped fullerite

¢rO


- ito. - ix +tp f

h.

hp

/

- ito. - ix + tpJ '

f

(3) As we analytically extend ito, ~ to + i6, we arrive at single-particle energies of electrons ~ = f ( t , ± h,) - ix = f(1 ± b)tp - ix,

(4)

which form energy bands in the normal phase of AxC6o. In this model of AxC6o, one electron is in the wide band, while the other x - 1 electrons are equally

0921-4526/93/$06.00 ¢~) 1993- Elsevier Science Publishers B.V. All rights reserved

V.A. Ivanov / Correlation effects and superconductivity in AxC6o

1072

distributed inside four-spin sub-bands of two degenerate correlated bands (4). Therefore, the correlation factors in eqs. (3) a n d (4) are

homogeneous Bethe-Salpeter equations:

ro~(p)G2(-p)G2(p)r~'b(p)

F ab= T ~_~

(8)

p,to,A,A'

f=l-

(X++) =1

x-1 ~ -

~

5-x

(l
Taking into account eq. (5), in rectangular DOS we arrive at: = 3w( 1 _ b 2 ) ( x _ 7 ) .

for vertex parts in the Cooper channel, where G~ a' (p) is determined by expression (3), Y'02= - h e - h e = Y~' and F ~ ' = - t p - tp = F0z2. Taking into account eq. (6), the logarithmic approximation of eq. (8) suggests that superconductive instability of the initial paramagnetic phase occurs at

(6) T = w l--~4 b ~ / 2 ( x - 1 ) ( 3

3. K i n e m a t i c a l m e c h a n i s m of s u p e r c o n d u c t i v i t y

In the simplest strongly correlated U >> t Hubbard model with a small number of electrons ( n ' ~ 1), no bound states are formed in pair collisions, resulting in higher electron energies and kinematic repulsion of electrons at the bottom of the band. At the same time, in the upper part of the band at 2 / 3 < n < 1 [16], electrons interact at lower energies tp---~ft~ = (1 - n~ 2)t, and therefore they should be attracted. We c a n arrive at the energy of interaction from the fully correlated paramagnetic energy J"/x dn minus the kinetic energy 2 ~] f t , O(l~ - f t p ) = 2nw(n - 1 ) / ( 2 -

n).

P

In a rectangular DOS, this energy is 3 2 2/3 - n Ei, t = ~n 2 - n w,

(7)

that is, Eim < 0 at n > 2/3, and electrons are kinematically attracted. Thus, the Hubbard U, excluding a large number of lattice sites from translation motion of electrons at n < 1, causes the paired attraction of neighbouring electrons forming Cooper pairs. This t-proportional correlation effect is at the heart of the kinematical high-T~ mechanism [17-22], which is due to a change of sign in the Born amplitude of scattering of correlated electrons with opposite spins at the point n = 2/3 in the Hubbard model [23,24] and at x = 7/3 in our model (2) (c.f. eq. (6)). The existence of such effects in spin-wave scattering was first reported by Dyson [25]. Remarkably, the kinematical high-T c mechanism of the Hubbard model seems to produce results similar to those of the Hirsch mechanism [26,27].

4. S u p e r c o n d u c t i v i t y

in d o p e d fullerites

We can arrive at T~ using the formalism in ref. [28] under the condition of solvability of the system of

x exp

{

1

2b 4+3b

8/27

1 + b x - (--~3)J "

x)

(9)

We can arrive at the superconductive gap using equations similar to (8) for abnormal Green functions 1I - roOOl = 0 or 1

tp

a

- -- th - -

a

a

hi, a ~- th

I

2T

th

=0 1-

(10)

th

where a = g ~ + 27~. Disregarding the relaxation of correlated electrons, the superconducting gap ~p ( T = 0) = Ao can be defined similarly to T¢ (9) by the same interaction constant A 0 = (1 + b)(x - 7/3) 27/8, this being the reason why 2A/T¢ = 3.53. However, spinflip effects are important in a system of correlated electrons at T = 0, their relaxation times r ~ 1 / x T being linked to the spin susceptibility X. Therefore, parameters of the superconductive phase near T~ a n d the Tc value itself are determined by the renormalized interaction constant a = A o + ~o[½ + (2,nT¢,rs)-' ] - ~0(½).

(11)

Therefore, we finally arrive at 2// 2rr -----exp{ATc y

A0) > 3 . 5 3 .

(12)

5. C o n c l u s i o n s

Applying the kinematical high-T c mechanism to the

AxC6o model (2), we arrive at expression (9) for T c as a function of x for a narrow interval 7/3 < x < [3 - 2b/ (4t + 3b)]. In our model T max occurs at x = 2.93. By reducing the lattice cell size (either by inserting A~ = Rb2Cs--~Rb 3 [2]--~K 3 [1]--~Na2Cso.sRbo. 5, or by increasing the external pressure [29]), we see that the values of the ratio h/t = b and T c both decrease. Since we substitute C6o for Cv0 in superconductors

V.A. Ivanov / Correlation effects and superconductivity in AxC6o

(C60)l_y(C7o)yg 3 in eq. (9), we should also replace w---* w(1 - y) so that C6o has no tlu molecular bands. As a result, we obtain a linear decrease in T c together with an increase in y, which agrees perfectly with experimental findings [30]. According to eq. (12), the superconducting gap 2Ao/T c exceeds the BCS value because of the spin-flip scattering of correlated electrons. This work was performed u n d e r project no. 91112 ' H u b b a r d ' of the Russian State H i g h - T c Superconductivity Program.

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