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Strong electron correlations in κ-(BEDT-TTF)2X salts
PHYSICA ELSEVIER
Physica C 268 (1996)205-215
Strong electron correlations in K-(BEDT-TTF)2X salts Valery Ivanov * ,1, Kazushi Kanoda Institutefor Molecular Science, Nishigonaka 38, Myodaiji-cho, Okazaki, 444, Japan Received 5 June 1996
Abstract
The K-ETzX compounds are classified as strongly correlated electron systems. The tight-binding approach for correlated electrons is applied to the K-ET2X family. The electronic spectrum and density of states are calculated. The singlet superconducting pairing of different symmetries is studied based on the symmetry of the lattice model of ET2-1ayer, and the superconducting coupling constants are calculated. The comparison of the theoretical speculations with available experimental data is made for normal and superconducting phases in K-ET2X. Keywords: Coulomb interactions; Band structure; Anisotropic superconductor; Organic superconductor
1. Introduction
The discovery of metallic conductivity in organic condensed matter [1] and synthesis of the first organic superconductor [2] stimulated research for new conducting and superconducting organic materials, which is becoming an important branch of fundamental science and applications. A decade after the BEDT-TTF [bis(ethylenedithio)-tetrathiafulvalene] molecule (abbreviated as ET hereafter) was synthesized [3], a breakthrough occurred in K-ET2 X superconducting salts with the highest superconducting temperature, Tc, of greater than 10 K [4,5]. In recent years, systematic NMR, electrodynamic, thermodynamic and transport measurements under pressure (see detailed reviews [6,7]) have provided
* Corresponding author. Fax: +81 564 54 2254. IOn leave from: N.S. Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences, Leninskii Prospekt 31, 117 907 Moscow, Russia.
an opportunity to obtain unified knowledge on the K-phase family of ET-based compounds with a variety of metallic, insulating, magnetic and superconducting properties, in spite of the existing structural similarity. The crystal structure consists of dimer cation pairs, ETa-, arranged orthogonally to each other in the conducting layers, which are separated by the insulating anion sheets, X - , with a layer periodicity of about 15 A. The structural basic unit in the ET layer includes two dimers 2ET 2. The charge density profile of the hole within the ET is rich around the S and central C sites. However, C - S and C--C vibrational phonon modes do not influence the origin of the superconducting condensate according to the isotopic shift, ATc, measurements in 34S and 13C=~3C substituted ET-superconductors (see Ref. [6]). So, the role of the central C = C and C - S stretching motions and the intramolecular BCS-like electron-phonon mechanism may not be crucial. This implication is consistent with the absence of the Hebel-Slichter peak in the
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o
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NMR relaxation rate [8-10]. The electronic band calculation based on the extended Hfickel method gives the intradimer carrier hopping as tra ~ 0.25 eV and the interdimer one as ter ~ 0.1 eV in the ET-conduction plane. (The tra corresponds to bl and ter to b2, p or p' in Ref. [11].) The low values of carrier hoppings, t~r and tra, originate from the small overlap of molecular orbitals due to the Van der Waals nature of the molecular interaction, and mean that carriers prefer to stay at ET or ET 2 rather than to travel. It leads to enhancement of intraET and intraET 2 electron-electron repulsions, /JET ( ~ 1 eV) and UEV2, respectively. The authors of Refs. [6,7,12-14] have noted the importance of these magnitudes for organic materials based on ET, TMTSF (tetramethyltetraselenafulvalene) and BETS [bis(ethylenedithio)-tetraselenafulvalene]. For K-ETzX, it was estimated at UEV2 ~ 2 tr~ ~ 0.5 eV [7]. A similar value can be roughly derived from the dimer size, R, and the dielectric permeability, e - 6, [15]: UET2 ~ e Z / g R ~ 0.5 eV. Therefore, for the K-ETzX, one should consider explicitly that electronic properties are influenced by the electron correlations with energy scales of UEV ~ 1 eV, tra ~ 0.25 eV and t~r ~ 0.1 eV. Here we have set the goal to clarify the manner in which the internal degree of freedom in the ET 2 plane influences the normal electronic structure and the superconducting condensate, without going into the discussion on the nature of the superconducting pairing mechanism. The latter has obviously been a subject of several conjectures in narrow-energy band organic materials [6,16] and particularly in K-ET2 X superconductors [7], but is still an open question. Kino and Fukuyama discussed the ground state of K-ET2 X in the light of the electron correlation effect in the framework of the Hartree-Fock approximation [17]. In the present paper, the K-ETzX salts are firstly modeled as a correlated electron system with doubly degenerate sites in the ET z layer, where ET 2 dimers are considered as entities with two degenerate energy levels. We briefly discuss the mathematics of the generalized Hubbard-Okubo operators applicable to such a model. Then the tight-binding method previously developed for correlated electrons is applied beyond the Hartree-Fock approximation to the calculation of the electronic structure in the dimerized ET 2 layer. The electronic density of states is
derived and lines of its logarithmic divergencies are obtained. Based on the electronic band symmetry, the anisotropic superconducting pairing interaction is constructed for K-ET2X superconductors and coupling constants for singlet superconducting pairings of different symmetries, dx2_y2 and extended s*, are evaluated. The comparison of the present computations with the available experimental data is made for normal and superconducting K-ET2X compounds.
2. The X-operator machinery and the correlated narrowing of the energy band The strong electron-electron correlations suppress charge fluctuations not only inside the dimer but also in the correlated K-ET2X band near the top of the band filling, and therefore the long range interaction of electrons with interdimer lattice phonons is screened. It leads to reduction of the electron-phonon coupling. Here we neglect the role of phonons in the problem. The intradimer hopping, tra , is small compared to the intraET electron-electron correlations: tra/UEv << 1. Therefore, each dimer, ET 2 = ETaET b, can be represented by a site with two degenerated energy levels of orbitals, a and b. So, for some problems, the dimerized ET 2 layer in K-ET2X can be represented by a lattice of sites with degenerated energy levels, namely the doubly degenerated Hubbard model. According to the tight-binding method developed for correlated electrons [18], one needs to prepare a ground state and at least one nearest exited state. Since one hole spreads over a dimer/site, ETa', with three electrons, it is easier to deal with in the hole representation than in the electron one. We employ the former for hole concentration, n, around 1. Let us denote each of the 16 multielectron eigenstates in a dimer, ET 2 --- ETaETb, as ( a b ) . Then the one-particle ground state is formed by 4-fold degenerated states, ( 1' 0), ( $ 0), (01" ), (0 ~ ), each with population, nl, differentiated by spin projection and hole position in (ETaETb) +. The nearest exited states are the (00) state (the neutral dimer) with a population n 0, corresponding to the neutral dimer, and the triplet states. Due to the many-body interactions, the
V. Ivanov, K. Kanoda / Physica C 268 (1996) 205-215
2-particle triplet states, (o-o-) and ( a ? b $ + a ~, b 1 ' ) / ~ - , the next excited singlet one, S = ( a l ' b $ - a + b 1')/v/2, and the Hubbard doublets, ( 2 0 ) = o ' %+ % ± and ( 0 2 ) = o'b+bf, have higher energies than the empty bole state of (00) [18]. In the correlated electron picture, one has to use the generalized Hubbard-Okubo [19,20] operators Xan [18] instead of the Fermi operators. Each projection operator Xan = I B ) ( A I transforms the multielectron dimer state A to B. The basis of the X-operators is complete and orthogonal. That is why one-electron Fermi operators can be expanded in a series of basis X-operators in the following general form:
%= ~,(BI%IA)X~=
Y'.g~X,,,
A,B
et
b~= ~ ( B I b ~ I A ) X ~ = Y'.g~Xlj, A,B
,8
where (XA~)+ = Xa
with product
XAB ® Xc° =
207
{a+%} = {b+b~} = ~2a(XAa) (orthogonality of the normalized X-operator basis). Applying it to Eq. (1) and considering 4-fold degeneracy of the ground state, one can get the following system of equations for the site occupations of n, and no:
({a+a~,}}=({b+bo.}} = E ( X a a } = 1: A 4n I + n O =
1
(n~)=(a+ad)=(b+b~>:
n.=n/4.
All two-particle exited states have higher eigen energies than the ground one-particle state and the polar empty one, and therefore we do not take into account their exponentially negligible Boltzmarm populations in the second equation for number of carriers (n,,) = n / 4 (we consider a completely disordered paramagnetic state) of the above system, Eq. (2'). The so called correlation factor [18,21,22] is given
a,,cXL
as
Here ot and ~ numerate the intraET2 transitions and the essential expansion coefficients g~, (genealogical coefficients [18]) should be evaluated for all possible intradimer transitions between the ground state A = ( ? 0 ) , (~, 0), (01' ), (0 $ ) and all excited states B. For the problem under consideration, we get
f = ({X0o Xo~0}) = (X0~ > + ( X $ 0 )
ao.= E M , x,=X°°o + ( 1 / I / ~ ) ( X °w + o ' X ° ~ )
(2')
~rO
O0
~rO
= n o + n, = ( 4 - 3 n ) / 4 .
(3)
It reflects the filling of the correlated energy band by the carrier number per dimer, n. The correlation factor f is essential for the X-operator temperature Green's function:
G~t3( rt, tat') = - ( T X , ( rt) X_o( ta t') ).
+ x£" +
bo. = F_,g Xg
=
°,
oo
o + o-x
°)
o
- Xo~° - tr X°~.
(1)
In the basis of the conventional Fermi operators, % and b~, the main dimer Hamiltonian includes fouroperator electron correlation terms. With respect to the X-operators, however, this Harniltonian is linear a s HO=~,I~lC~I6EKX ~, where all essential intradimer correlations are involved in eigenstates {er}. Average values of the diagonalized Xrr HubbardOkubo operators are equal to the Boltzmann populations of the corresponding eigenstates, e.g. ( X g ) = n o,
( Xffo°> = ( X o ,o,,, ) = n , .
(2)
For all of the 16 eigenstates of the dimer, the completeness relation for the X-operators yields 1 =
Below we take into account only the ground state (1'0), ( ~ 0 ) , (01"), (0,L) and the nearest excited polar state (00) with populations n 0, following from the system of Eqs. (2'). At the Matsubara fermion frequencies, for the dimer Hamiltonian H 0, the zeroth order dimer Green's function describes the intradimer transitions between the ground state and the nearest excited state and has the form n A -'1.-n B
c ° ( to°) =
- i to. + e s
-
,Ba
n o +n~ = -i
ton + ¢ 0 ~ - %0
.
(4)
where ~ denotes intradimer transition from A to B: a - - a ( ~ ) . For our problem, % 0 = 0 and e0~ = 6,,o(-- - / z ) is the 4-fold degenerate energy of the one-particle dimer ground state (/x is the chemical
208
V. Ivanov, K. Kanoda / Physica C 268 (1996)205-215
potential) and the zeroth order Green's function for E T 2 is f
o oo (o9.)
~[~o1"
-iw.-
ix
ix : ¼ ( 5 n - 4)w,
= O(Oooo)(o9.)
with correlation factor given by (3). The tunnel Hamiltonian is represented via all possible products of the X-operators in Eq. (1) for neighbouring dimers: ntunnel =
E a ,/~ ,p
~r0
f -- i o9n -- /z + f t e r ( P
)
=Coo (p, o9.).
(5)
0 ie,
Beyond the Hartree-Fock approximation, the spectrum of the one-particle excitations follows from poles of this Green's function by the analytic continuation i o9, ~ ~?+ i 8:
sc~'b=fie~(p) - ix= ¼(a- 3n)t~( p) - ix.
(6)
One can get the chemical potential from the following equation for the number of holes per site: 1
no, = i n
=f~"P,~°nei'S ([G° P 'o,0 L° =
(8)
which reflects the correlated narrowing of the energy band: at n = 1 (the K-ET2X salt case) we conclude that ix = w/4 (quarter band filling, because the chemical potential origin is the bottom of the energy band).
g°g~ter(p)Xa(p)X#(P)"
The tcr(p) refers to the bonding branch of the spectrum of uncorrelated hole carriers in the ET2-1ayer and will be calculated in the next section. In a perturbative approach, the interdimer hopping, t¢~(p), of hole with momentum p plays the role of a perturbation and the first order pertubative Green's function for the paramagnetic ET 2 layer is expressed as
Goo(P , t o ) =
carriers, one obtains from Eq. (7) a dependence of the chemical potential on carrier concentration as,
O9n)+G°°(P'o ~ o9,)] (7)
p
where O(x) is the Heviside function. In the righthand side of Eq. (7), the multiplier 2 is due to the a-b degeneracy of the spectrum of Eq. (6). Assuming a rectangular (fiat weighted) density of states with a half-bandwidth w, for noninteracting
3. Tight-binding electronic structure in ET2-1ayer model In spite of the seeming similarity of the layered structure in the K-ET2X and HTSC materials [6], their electronic band structures have a crucial dissimilarity. For the K-ET2X family, the internal degrees of freedom in the ET2-1ayers are important: dimeric nature of ET-molecules and two ET dimers per basic unit. A square lattice is assumed for ET dimers for simplicity, while the real arrangement has rhombic distortion. For the sake of clarity, all of the hopping integrals along lattice diagonals are reasonably supposed to be the same and are defined by t~ r. In reality, they are slightly split due to the non-symmetry of the dimer for some salts (see Ref. [11]). The t~r is an azimuthal hopping integral, which is, e.g. in the b direction of the crystal in Ref. [11] and is set along the y axis in this paper. The t~r and t~r are = 0.1 eV according to Ref. [ 11 ]. Based on the knowledge of the hopping integrals, the uncorrelated tight-binding calculations were used in Ref. [23] to analyze the optical reflectivity of tc-ET2I 3 and in Ref. [24] to analyze the quantum magnetic oscillations in K-ET2Cu(NCS)2 under pressure. The secular equation for the bonding branches of the uncorrelated hole spectrum sp of elementary excitations (antibonding branches of the electron kind) is, I(t ra - ee)6; i + Hij I = 0.
(9)
The matrix elements, Hij = (~i [ Hint 1~->, of the interdimer transfer Hamiltonian H ~n, are expressed via the Bloch functions ~i of the ith doubly degen-
V. lvanov, K. Kanoda / Physica C 268 (1996) 205-215
crate energy state of the dimer pair in the basic unit as,
209
Using the analytical expressions for tp, tp//2 t~ r ~ Sp:i: = COS py ~t- 2 C O S ( p y / / 2 ) C O S ( p x / / 2 )
Hu.22 = t~r{cos Py
= cos py _+/3(cos py) cos( p J 2 ) ,
+ 3[ 1 + cos p, + cos px + cos( p r + px)]} . e~[sin - py HI2 = , ~1/2
+ sin p~ + s i n ( p y + p x ) ] = H z , *.
Here we assumed an equal magnitude for the interdimer hopping integrals, and all momenta are given in reduced units: I Px I = [ Py I = 7r on the respective Brillouine zone boundaries. From Eq. (9), one obtains a pair of hole bonding spectrum branches [23]
where /3(x)=~]2(1 + x ) [c.f. Eq. (10)], one can calculate the total carrier density of states, with 6 the dimensionless energy in the range of - 1 _< e < 3 in the antibonding band as follows:
1
r2~
-(2~.)2j0
r2~
dpyJo dpx
as ~p = t ra + tp,
1 f,1
tp = 2t~r[cos py + 2 c o s ( p y / 2 ) cos( Px/2)].
7r 2 J - I d y ~/(1 _ y 2 ) [ [ 3 2 ( y ) -
(10) This derivation from the secular equation (9) is justified if the energy difference 2ira between the bonding and antibonding pairs exceeds the energy splitting in each bonding a n d / o r antihonding energy branch. Applying the tight-binding method for correlated electrons [18] to the dimeric structure of the ET 2 layer with a finite magnitude of tra/UET after a canonical transformation to natural bonding and antibonding orbitals, we get a pair of bonding branches and a pair of antibonding branches of the spectrum for correlated holes in conducting ET z layers as follows [25]: ~t, = +ira + (1 + 4tra/UzT)ft, -- IX.
O[/3(y)-e+y]
i
4
~2"2¢~-4e +3
x(jm
16 2q~-+6 + 3
J (12)
The above is divergent at e ~ - 1 , around which p(6) is of the form
p(e)-~
(11)
It is seen that in the limit of t~/UET ~ O, the upper branch of the spectrum of Eq, (11) corresponds to the dispersion of the doubly degenerate sites/dimers given by Eq. (6). In Eq. (11), the bonding and antibonding branches originate from dimerization. The splitting of each bonding or antibonding band in two branches occurs due to the K-ETzX crystal structure having a dimer pair per unit cell. Among the six electrons in one basic unit, 2 E T f , four of them occupy the two bonding bands and two electrons occupy the antibonding bands of Eq. (11). The uncorrelated branches of Eq. (10) reproduce the essentials of the first prinsiples calculation [26], tightbinding method [27] or numerical calculation [17] of the K-ET2X band structure.
y)2]
3
4 In---2,n.z42v~ + 3 e+ 1
3 4 -ln-27r 2 ~+ 1 '
where K ( x ) is the complete elliptic integral of the first kind. Note that / 3 ( y ) = 1 in Eq. (12) for the square lattice with a dispersion of e = % / 2 t = c o s py + COS Px or
= % / 2 t = cos py + cos ( p x / 2 ) and in this case DOS reduces to elliptic integrals with a prefactor of I/~r 2 and a modulus of V/1 - ( e / 2 ) 2 in the corresponding energy regions of -2<~<2 or - 1 < ~ < 2 . In comparison with HTSC cuprates, the main dissimilarity of the energy spectrum, Eqs. (10), (11) and (6), for x-ETaX is caused by the azimuthal
210
V. lvanov, K. Kanoda / Physica C 268 ( 1 9 9 6 ) 2 0 5 - 2 1 5
angular momentum in the ET 2 plane. For the paramagnetic state of cuprates, such components are very small. In Eqs. (10), (11) and (6), this anisotropy originates from the arrangement of the ET-molecules, which adds azimuthal transfers to the modelled transfer of square-lattice symmetry. It is interesting to note that the hopping integral asymmetry for the hole dispersion in the antiferromagnetic highTc cuprate background is governed by the symmetry of an antiferromagnetic Brillouin zone: a single hole prefers to jump to the next nearest neighbouring site of a square lattice rather than to the nearest neighbour; the nearest neighbour hopping is suppressed comparatively to the next nearest one [28].
4. Superconducting pairing in the ETz-layer model As seen in the spectra of Eqs. (10), (11) and (6) derived for the present model of correlated carriers in K-ET2X, the band structure effects are important when the Fermi surface is near the Brillouin zone (this is the case of the K-ET2X family) where the influence of the crystal potential is strong. In the tight-binding approach, the symmetry of the Bloch functions, Er~b(r-R)e ip'r, and the symmetry of the Fermi surface manifest themselves in the symmetry of the superconducting order parameter A(p) [29,30], which represents the correlation function of the Cooper pairs as A ( p ) ~ (C~.pC~._p). Neglecting the retardation effects the latter becomes App, = E e x p [ i ( p ' - r ' - p .
r)]
rr'
× f ~b*(r-R)A( R)J/(r'-R) dR = ) ~ e x p [ i ( p ' - p ) . r]
r
x f A ( R ) I ¢ ( r - R ) I 2 dR It is seen from here that at constant energy gap [localization in real space of A(R)] the superconducting order parameter A(p) has the same symmetry as the Fermi surface. Dispersions of elementary excitations (10), (11) and (6) provide an anisotropic Fermi surface unlike the sphere cited in the BCS theory. A complicated Fermi surface does not pro-
vide the conventional isotropic s-wave pairing, i.e. either constant pairing interaction or a constant energy gap as Anderson had mentioned in earlier publications [31]. For the unconventional pairing, the symmetry of A(p) is lower than that of the Fermi surface. The pairing interaction Vpp, in the effective interaction Hamiltonian, 2'
Hint =
E
V ( q ) :p + q , o ' C p+' - q , ~ C p ' , ~ C p , o
",
p,p ,q,o" ,o"
and the superconducting order parameter, Apt,,, depend on the momenta difference of the pairing fermions, namely V ( p - p') and A(p - p ' ) . Here we do not attempt to specify the origin of the pairing interaction in K-ET2X superconductors. Due to particle correlation, on-site pairing should be ineffective. In the reference system of the square lattice, the general form of an effective pairing interaction for the nearest neighbours can be rewritten as the following identity:
Y'. cos(p,~-p~')
=2V
a=x,y
= 2V(I[(cos l[~x+COS py)(COS
+(cos
cos p,)(cos
P'x
+COS /~'y)
,-cos p,,)]
+ (sin py sin fly + sin Px sin fix)}.
(13)
Eliashberg had noted that the theory of superconductivity need not be based on an uncorrelated band spectrum and Green's functions of uncorrelated electrons [32]. We can start from the correlated Green's functions given by Eq. (5) to apply the formalism [33,34]. In the effective interaction Hamiltonian Hint, the Fermi operators should be expressed via X-operators (see Section 2). In the Born approximation, the amplitude of fermion-fermion scattering, F(p, p') = V(p-p'), is defined by Eq. (13). Then for the Cooper channel in the reference system of colliding fermions with opposite spin projections, the anomalous self-energy can be written as '
to,p'
t,
v
, ),
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V. lvanov, K. Kanoda/ Physica C 268 (1996) 205-215
G,o(P) =
~P
(14)
where the carrier dispersion law ~:, is determined mainly by the lower branch of the bonding hole spectrum (the upper antibonding branch of the electronic spectrum) of Eq. (11). In the reference system where the y-axis is along the azimuthal direction, it is rewritten in dimensionless units as g / / 2 ft~ r - oo
= cos Pr + 2 I cos(py/2) cos(px/2) = COS py + fl(cos
py)
I
cos(Px/2),
where fl(x) is defined in Eq. (12) and f is the correlation factor defined by Eq. (3). From now on, the ¢~ will be measured from the chemical potential p. and we will neglect the difference in interdimer transfer integrals, ter. Using the identity, oo
1
1
x
rn=0 y" x 2 + [(2n + 1)Trr] 2 = 2-7 tanh --,2T
order to solve Eq. (15), it is necessary to decompose the attractive interaction V(p - p ' ) and order parameter A(p) into contributions from all possible irreducible representations of the point group symmetry, C4v, of the square lattice under consideration. The symmetry of the representation with the largest Tc is the case according to the BCS type description of superconductivity in Refs. [35,36]. The identity of Eq. (13) is no more than the expansion of V ( / ~ - i f ) in a series with the eigenfunctions r/(/~) of the C4v irreducible representations. The first and second terms in the square brackets are products of basis functions of a one-dimensional irreducible representation A l (s*-wave pairing) and B 1 (dx2_y2-Wave pairing), respectively, and the last two terms include basis functions of the two-dimensional representation E (triplet p-wave pairing). Our reference system is rotated by an angle of ~'/4 from the "tilde" reference system. Therefore the carrier momenta in Eq. (13) are connected with the used dimensionless momenta as P x = ½( Px-- Py),
instead of Eq. (14), one can get the general equation of the standart BCS theory as
P y = ½( Px + Py),
and the basis functions are reduced to r/,(p) = cos(Px/2) cos(py/2),
~,(p) = E V ( p - p ' ) ~ , ( p ' ) p'
r h ( p ) = sin(px/2) sin(py/2), r/3.4( p ) = sin(px/2) cos(py/2)
×
2~/~(p')2 + ~2(p,)
dp'. + sin(py/2) cos(Px/2). (15)
Here the anomalous self-energy part, ~(p), has a character of the superconducting order parameter
a(p).
In K-ET2X superconductors, several experiments imply that anisotropic singlet d-type pairing with nodes in the superconducting order parameter such as Ao(/~)~ cos /~x- cos /~y (so called dx~_y~ pairing) or some other pairing (d~y) occurs at the Fermi surface. But they are not inconsistent with anisotropic singlet s* (extended s) pairing with nodes of the form A(/~) ~ cos /~ + cos /~y without a sign change, or minima of the gap on the Fermi surface, but in the same direction in the Brillouin zone as in the case of d-pairing. The problem now is to determine the type of Cooper pairing in our model. In
(16)
Putting the expansions V(p -p') = 2VE4= lr/i(p) rli(p') and A ( p ) = E4= lAirli(p), into Eq. (15), we can reduce it to the following system of algebraic equations for T~ in terms of orthogonality of the basis functions {r/(p)}:
4
tanh( ~p//2 Tc )
p;j= l
2 sop
A,=g i ~_.
Ajrli(p)rlj(p),
(17)
where gi = 2V and the summation is over momenta p in the ET2-Plane. The system of equations (17) has solutions of Tc when the following 4 × 4 determinant vanishes:
3ij-gY'~ P
tanh( p/2 ) 2g
[
Oi(p)rlj(P) I = 0 .
(18)
V. lvanoo, K. KamMa / Physica C 268 (1996) 205-215
212
The essential part of this secular equation is E tanh( ¢ / 2 r ~ ) ~ ( P ) nJ(~') p
2~p
f=
= -= de
tanh( (e)/2Tc) 2~:(e)
-, ¼ ( t + 1 ) ( 3 - t ) ,
( t + 1) 3
/
+ --K(q)
- 2
t--1
Fi4(e)'
F,j(e) = E FiT(el, Ot ~
q)
2t
Vt2--1
× [//(½~-, ¼ ( t + 1 ) ( 3 - t ) ,
where
~/(9-t2)/8)
,+3
++
+--K(~(9-t2)/8
F,7(e)
(20)
t-1
= E
(e-
Fa(s) = F22(e )
p
-(2 -r)3£ dpyf_ dx exp[ix(e-cos 1
-
[ ×[(3-t2)//(½~
".,r
1 (t-l)2( = F s . ( e ) + 87r2 ~
p,)]
-½(t-
× L ~ dpx exp[-ix~Ci(cos p y ) c o s ½p~] X Tli( P ) n j ( P ) ,
= rll(e)
1 8Ir 2
(t--
1) 2
~/t
1)(t + 3)
-4//(½~-,
and fl(x) is defined in Section 3 [c.f. Eq. (12)].
Because of oddness of the products, Th(p)Tij(p)(1 -- 6,), with respect to py, all of the nondiagonal matrix elements in Eq. (18) turn out to be zero and therefore superconducting pairing of different symmetries cannot be mixed. Here we will consider the realistic singlet d- and s*-pairing with nodes on the Fermi surface, because the Knight shift measurements suggest only a singlet form of pairing in K-ET2X superconductors [8-10]. The integrations in Eqs. (19) are performed over the variables Px, x, sequentially and then over momentum projection py. On the intermediate steps in the calculations, integrations with Bessel functions (after integration over variable p~) and Chebyshev polynomials (after integration over x) are useful. The final forms are expressed via elliptic integrals as follows: F~.(e)
2 t - 1 ] K(q)
× H(½7r, (t + 1)3(3 - t)/16t, q)
(19)
5. Coupling constants for different symmetries of singlet Cooper pairing
(t+l/2
¼(t+ 1 ) ( 3 - t ) ,
q)),
(21)
where K(k) and H ( ~1r , n, k) are the complete elliptic integrals of the first and the third kind, respectively, t = ~ 3 and one of elliptic integral modulus in Eqs. (20) and (21) is q = ~/1 - (t - l)3(t + 3 ) / 1 6 t . Then, the superconducting transition temperature for each irreducible representation is defined by .
l=gj %: -
-
de
2~:(e)
Fs'd(e)
to c
gf~(%) d~ tanh(~/2Tc ~'d)
,
(22) where % is a cut-off energy parameter and the derivative a~/ae = f is the correlation factor defined by Eq. (3) as evaluated by Eq. (6). We assumed, as usual, that the coupling constant g is nonvanishing in an energy range of width 2 % near the Fermi energy. Applying the logarithmic approximation to Eq. (22), we obtain Tc as Tcs~ .d = ~1tOc exp( - 1/A~.d) with coupling constant As.,d
(23) =
4gFs*.d(e = 0).
V. lvanov, K. Kanoda / Physica C 268 (1996) 205-215
Here the Fs*,d are defined by Eqs. (20) and (21). The numerical multiplier in the coupling constant is due to the correlation factor discussed above: I/f= 4. It should be noted that for conventional s-wave pairing with basis functions rli(p) = 1 in the irreducible representation, A l, Eq. (19) reduces to Eq. (12). In this case, the coupling constant of isotropic s-pairing is defined by the normal electronic density of states Eq. (12) at the Fermi surface ( e = 0) and one
can
get
4gp(e
As =
4
= 0) = (4g/Tr2~/-3-)
K(½1/2 + v~- ) = 0.87g(K(½!/2-+ ~ ) -~ 2.77 according to a numerical evaluation) for the same coupling constant g. Using Eqs. (20) and (21), the coupling constants are evaluated as follows: 2+v~A~.
- -
×
2-f3-
As
2
[
H
2'
rr 2
2'2
+ ~ g
g,
(24)
with / / ( r r / 2 , ~ / 2 , ¢-5-/2)= 1.50 and K ( ¢ 3 - / 2 ) = 2.16, i.e. As. = 1.16g,
ad = ¼(1 + ¢ 5 ) a , + a,. 2[~-/3-~ -7r V-/-32- - V~-//( 'n" ) ' 2 +42~ - ' ¢-2+2 ¢ 5
+4//
2'
2 '
2
g
1-I(~r/2, (2 + v/-3")/4, 1/2 + ~ / 2 ) H(rr/2, vr3/2, ~ + ~ - / 2 ) = 1.84,
with
(25)
= 1.80 and
i.e. Ad = 1.53g. So, the evaluated ratio of anisotropic coupling constants is Ad/a ~. = 1.32. For nonphonon superconducting pairings, the cutoff energy parameter % in Eq. (23) is determined by the electronic structure. Assuming that the cut-off energy parameter is the half-bandwidth, i.e. 4t~r ~ 0.4 eV, and the coupling constant is g = 0.17, we can estimate the superconducting transition temperature Tc at = 12 K for d-wave pairing in the weak cou-
213
piing limit from Eq. (23). This value is a reasonable magnitude for K-ET2X superconductors. The Tc of anisotropic s*-wave pairing is smaller by half an order of magnitude.
6. C o n c l u d i n g r e m a r k s
From the spectrum of Eq. (11) we found for correlated carriers in the K-ET2 X model the chemical potential as /~ = ( 1 / 4 - 3tra/UET)W, that means quarter narrowing of the noncorrelated energy bands in the strongly correlated limit, tra/uEX = 0 (c.f. Eq. (8) at n = 1). So, the actual bandwidth should be considerably smaller due to the particle correlations; in the normal paramagnetic phase, the correlated holes move in the narrowed band of 0 . 8 / 4 ~ 0.2 eV width. This correlated reduction of the energy band is essential for band structure [23] and may be involved in the optical [27] and cyclotron [24] effective masses. For a conventional square lattice, the electron dispersion is ~ = 2t(cos p y + COS p ~ ) and the DOS is given by the expression K((1- (~/4t)2)/ (2~-21) with logarithmic divergences only along boundaries of the Brillouine zone: e = + 4 t . Unlike this case, Eq. (12) for the K-ET2X lattice has an energy-dependent prefactor and has a logarithmic divergence in energy at the point of t~/2ter = -1 (in case of A ter = 0), i.e. along the boundaries M(rr, "n') - Z(O, "n') - ( - rr, rr), ( - "n', - rr) ( r r , - r r) and along the diagonals M(rr, r r ) F(0, 0) - ( - ~, - 7r) and (Tr, 7r) - ( - ~r, - ~-) of the Brillouine zone. The positions of the logarithmic divergence are in agreement with the results of the first-principles calculation [26]. Fortunately, these divergencies are located well outside the Fermi level and the assumption of a flat weighted density of states is justified. Referring to magnitudes of the interdimer hopping i n t e g r a l , ter = 0.1 e V , Eq. (12) yields the density of 4
states at the Fermi level as p(p,) = (l/2"n'2Vl'3ter)K (I/2 + ~ / 2 ) = 1.05 (eV spin)-l (c.f. Section 5). Note that this value is in agreement with the value p -- 1 states/eV spin ET, evaluated in Ref. [37] for noninteracting electrons in metallic K-ET2 X salts.
214
V. Ioanov, K. Kanoda / Physica C 268 (1996) 205-215
For the superconductivity we used a model with a correlated energy scale of UET ~ 1 eV, not with the hopping energy scale tra, ter ~ 10 -1 eV. The anisotropy of the symmetry of the derived electronic transfer, Eqs. (6), (10) and (11), is involved in anisotropy and type of the superconducting order parameter. The irreducible representation B l possesses the highest coupling constant and Tc among various pairings with possible symmetries. So, the superconducting dx2_y~-wave pairing [see Eq. (13)] is preferable. Our treatment of the superconductivity gives the possibility of nodes in the gap parameter. In this case, the superconducting order parameter has line nodes along the Px and py axes, and four inequivalent nodes on the K-ET2X Fermi surface. For s *-wave pairing of the irreducible representation A l with the basis functions tll(p)=cos(px/2) cos(py/2), also four inequivalent nodes exist on the Fermi surface. The present study may favour some kind of nonphonon superconducting mechanism, e.g. the attractive pairing can be induced by antiparamagnon excitations [38]. As a final conclusion on pairing, however, it is useful to compare not only the Tc but also energies of different pairing symmetries (c.f. Refs. [35,39]). In addition, we cannot exclude the possibility that the actual distortion of the square lattice in the ET2-1ayers may lead to a contribution of rhombic components in their pairing symmetry. Admixture of different pairing components occurs for tetragonal distortions of the square lattice in high-Tc cuprates
[40]. The interdimer hopping te~ increases under pressure. As can be seen from Eq. (23) (after replacing the coupling parameter g by a dimension magnitude g/2t~) it leads to a decrease of the coupling cons t a n t Ad and Tc in the proposed model of K - E T E X superconducting salts. The opposite tendency occurs with the increase of the lattice parameters or expansion of the basic unit. These are in agreement with empirical conclusions. The electron-electron correlations manifest themselves in narrowing of the normal energy band and in the appearance of the correlated factor f < 1 in Eqs. (23), and may enhance T¢ and the condensation energy in comparison with the uncorrelated case. The present treatment is reasonable for the description of the low energy experiments such as
NMR [8-10] and electronic specific heat [41] in the superconducting state. The present study can be useful and applicable to other organic materials with layer structure, e.g. BETS compounds [42].
Acknowledgements The authours thank H. Fukuyama for providing a preprint, Y. Maruyama and S. Nakajima for helpful discussions and T. Kasuya for useful critical comments. The authors acknowledge K. Yakushi and N.M. Plakida for valuable advice and are thankful to T. Iwatsuki for her help. This work was supported by a Grand-in-Aid for Scientific Research, Nos. 06452064 and 06243102 (Priority Area "Novel Electronic States in Molecular Conductors") from the Ministry of Education, Science, Sports and Culture of Japan. V.I. acknowledges the support of the Ministry of Education, Science, Sports and Culture of Japan during a stay at the Institute for Molecular Science, Okazaki.
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V. Ivanov, K. Kanoda / P hysica C 268 (1996) 205-215 [15] American Institute of Physics Handbook, ed. D.E. Gray (McGraw-Hill, New York, 1972) pp. 5-128. [16] T. Ishiguro and K. Yamaji, Organic Superconductors, Solid State Science, Vol. 88 (Springer, Berlin, 1990); J.M. Williams et al., Organic Superconductors (Prentice Hall, Englewood Cliffs, 1992). [17] H. Kino and H. Fukuyama, J. Phys. Soc. Jpn. 64 (1995) 2726. [18] V.A. Ivanov in: Studies on High-Tc Superconductors, Vol. 11, ed. A. Narlikar (Nova Science, New York, 1993) p. 331; J. Phys. Condens. Mat. 6 (1994) 2065. [19] J. Hubbard, Proc. R. Soc. A 285 (1965) 512. [20] S. Okubo, Prog. Theor. Phys. 27 (1962) 949. [21] S. Nakajima, M. Sato and Y. Murayama, J. Phys. Soc. Jpn. 61 (1991) 2333; Y. Murayama, S. Nakajima, J. Phys. Soc. Jpn. 61 (1991) 4265. [22] Y. Murayama, V.A. Ivanov and S. Nakajima, Phys. Lett. A 211 (1996) 181. [23] M. Tamura, H. Tajima, K. Yakushi, H. Kuroda, A. Kobayashi, R. Kato and H. Kobayashi, J. Phys. Soc. Jpn. 60 (1991) 3861; M. Tamura, thesis, University of Tokyo (1994). [24] J. Caulfield, W. Lubczynski, F.L. Pratt et al., J. Phys.: Condens. Mat. 6 (1994) 2911. [25] V.A. Ivanov, to be published. [26] Y.-N. Xu, W.Y. Ching, Y.C. Jean et al., Phys. Rev. B 52 (1995) 12 946. [27] A. Ugawa, G. Ojima, K. Yakushi and H. Kuroda, Phys. Rev. B 38 (1988) 5122.
215
[28] N.M. Plakida, P. Horsch, A. Liechtenstein et al., Preprint ICTP, IC/95/157. [29] B.T. Geilikman and V.Z. Kresin, Sov. Phys. - JETP 13 (1961) 677; Sov. Phys. - Solid State 5 (1964) 2605. [30] V.L. Pokrovsky, Soy. Phys. - JETP 13 (1961) 447; V.L. Pokrovsky and M.S. Ryvkin, Sov. Phys. - JETP 16 (1963) 67. [31] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26; Phys. Rev. Lett. 3 (1959) 325. [32] G.M. Eliashberg et al., JETP Lett. 59 (1994) 441. [33] L.P. Gor'kov, Sov. Phys. - JETP 7 (1958) 505. [34] Y. Nambu, Phys. Rev. 112 (1958) 812. [35] K. Miyake, T. Matsuura, H. Jichu and Y. Nagaoka, Prog. Theor. Phys. 72 (1984) 1063. [36] R. Fehrenbacher and M.R. Norman, Phys. Rev. Lett. 74 (1995) 3884. [37] G. Saito, A. Otsuka et al., Mol. Cryst. Liq. Crystals (1996) to be published. [38] H. Won and K. Maki, Physica C 235-240 (1994) 1689; preprint. [39] Zs. Szabo, Zs. Gulacsi, submitted to Phys. Rev. B. [40] O.T. Valls and M.T. Beal-Monod, Phys. Rev. B 51 (1995) 8438; K. Maki and M.T. Beal-Monod Phys. Lett. A 208 (1995) 3650; H. Kim and E.J. Nicol, Phys. Rev. B 52 i1995-II) 13 576. [41] Y. Nakazawa and K. Kanoda, submitted to Phys. Rev. Lett. [42] H. Kobayashi et al., J. Amer. Chem. Soc. 118 (1996) 368.
Physica C 268 (1996)205-215
Strong electron correlations in K-(BEDT-TTF)2X salts Valery Ivanov * ,1, Kazushi Kanoda Institutefor Molecular Science, Nishigonaka 38, Myodaiji-cho, Okazaki, 444, Japan Received 5 June 1996
Abstract
The K-ETzX compounds are classified as strongly correlated electron systems. The tight-binding approach for correlated electrons is applied to the K-ET2X family. The electronic spectrum and density of states are calculated. The singlet superconducting pairing of different symmetries is studied based on the symmetry of the lattice model of ET2-1ayer, and the superconducting coupling constants are calculated. The comparison of the theoretical speculations with available experimental data is made for normal and superconducting phases in K-ET2X. Keywords: Coulomb interactions; Band structure; Anisotropic superconductor; Organic superconductor
1. Introduction
The discovery of metallic conductivity in organic condensed matter [1] and synthesis of the first organic superconductor [2] stimulated research for new conducting and superconducting organic materials, which is becoming an important branch of fundamental science and applications. A decade after the BEDT-TTF [bis(ethylenedithio)-tetrathiafulvalene] molecule (abbreviated as ET hereafter) was synthesized [3], a breakthrough occurred in K-ET2 X superconducting salts with the highest superconducting temperature, Tc, of greater than 10 K [4,5]. In recent years, systematic NMR, electrodynamic, thermodynamic and transport measurements under pressure (see detailed reviews [6,7]) have provided
* Corresponding author. Fax: +81 564 54 2254. IOn leave from: N.S. Kurnakov Institute of General and Inorganic Chemistry of the Russian Academy of Sciences, Leninskii Prospekt 31, 117 907 Moscow, Russia.
an opportunity to obtain unified knowledge on the K-phase family of ET-based compounds with a variety of metallic, insulating, magnetic and superconducting properties, in spite of the existing structural similarity. The crystal structure consists of dimer cation pairs, ETa-, arranged orthogonally to each other in the conducting layers, which are separated by the insulating anion sheets, X - , with a layer periodicity of about 15 A. The structural basic unit in the ET layer includes two dimers 2ET 2. The charge density profile of the hole within the ET is rich around the S and central C sites. However, C - S and C--C vibrational phonon modes do not influence the origin of the superconducting condensate according to the isotopic shift, ATc, measurements in 34S and 13C=~3C substituted ET-superconductors (see Ref. [6]). So, the role of the central C = C and C - S stretching motions and the intramolecular BCS-like electron-phonon mechanism may not be crucial. This implication is consistent with the absence of the Hebel-Slichter peak in the
0921-4534/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved PH S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 4 0 6 - 6
o
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v. lvanov, K. Kam)da/ Physica C 268 (1996) 205-215
NMR relaxation rate [8-10]. The electronic band calculation based on the extended Hfickel method gives the intradimer carrier hopping as tra ~ 0.25 eV and the interdimer one as ter ~ 0.1 eV in the ET-conduction plane. (The tra corresponds to bl and ter to b2, p or p' in Ref. [11].) The low values of carrier hoppings, t~r and tra, originate from the small overlap of molecular orbitals due to the Van der Waals nature of the molecular interaction, and mean that carriers prefer to stay at ET or ET 2 rather than to travel. It leads to enhancement of intraET and intraET 2 electron-electron repulsions, /JET ( ~ 1 eV) and UEV2, respectively. The authors of Refs. [6,7,12-14] have noted the importance of these magnitudes for organic materials based on ET, TMTSF (tetramethyltetraselenafulvalene) and BETS [bis(ethylenedithio)-tetraselenafulvalene]. For K-ETzX, it was estimated at UEV2 ~ 2 tr~ ~ 0.5 eV [7]. A similar value can be roughly derived from the dimer size, R, and the dielectric permeability, e - 6, [15]: UET2 ~ e Z / g R ~ 0.5 eV. Therefore, for the K-ETzX, one should consider explicitly that electronic properties are influenced by the electron correlations with energy scales of UEV ~ 1 eV, tra ~ 0.25 eV and t~r ~ 0.1 eV. Here we have set the goal to clarify the manner in which the internal degree of freedom in the ET 2 plane influences the normal electronic structure and the superconducting condensate, without going into the discussion on the nature of the superconducting pairing mechanism. The latter has obviously been a subject of several conjectures in narrow-energy band organic materials [6,16] and particularly in K-ET2 X superconductors [7], but is still an open question. Kino and Fukuyama discussed the ground state of K-ET2 X in the light of the electron correlation effect in the framework of the Hartree-Fock approximation [17]. In the present paper, the K-ETzX salts are firstly modeled as a correlated electron system with doubly degenerate sites in the ET z layer, where ET 2 dimers are considered as entities with two degenerate energy levels. We briefly discuss the mathematics of the generalized Hubbard-Okubo operators applicable to such a model. Then the tight-binding method previously developed for correlated electrons is applied beyond the Hartree-Fock approximation to the calculation of the electronic structure in the dimerized ET 2 layer. The electronic density of states is
derived and lines of its logarithmic divergencies are obtained. Based on the electronic band symmetry, the anisotropic superconducting pairing interaction is constructed for K-ET2X superconductors and coupling constants for singlet superconducting pairings of different symmetries, dx2_y2 and extended s*, are evaluated. The comparison of the present computations with the available experimental data is made for normal and superconducting K-ET2X compounds.
2. The X-operator machinery and the correlated narrowing of the energy band The strong electron-electron correlations suppress charge fluctuations not only inside the dimer but also in the correlated K-ET2X band near the top of the band filling, and therefore the long range interaction of electrons with interdimer lattice phonons is screened. It leads to reduction of the electron-phonon coupling. Here we neglect the role of phonons in the problem. The intradimer hopping, tra , is small compared to the intraET electron-electron correlations: tra/UEv << 1. Therefore, each dimer, ET 2 = ETaET b, can be represented by a site with two degenerated energy levels of orbitals, a and b. So, for some problems, the dimerized ET 2 layer in K-ET2X can be represented by a lattice of sites with degenerated energy levels, namely the doubly degenerated Hubbard model. According to the tight-binding method developed for correlated electrons [18], one needs to prepare a ground state and at least one nearest exited state. Since one hole spreads over a dimer/site, ETa', with three electrons, it is easier to deal with in the hole representation than in the electron one. We employ the former for hole concentration, n, around 1. Let us denote each of the 16 multielectron eigenstates in a dimer, ET 2 --- ETaETb, as ( a b ) . Then the one-particle ground state is formed by 4-fold degenerated states, ( 1' 0), ( $ 0), (01" ), (0 ~ ), each with population, nl, differentiated by spin projection and hole position in (ETaETb) +. The nearest exited states are the (00) state (the neutral dimer) with a population n 0, corresponding to the neutral dimer, and the triplet states. Due to the many-body interactions, the
V. Ivanov, K. Kanoda / Physica C 268 (1996) 205-215
2-particle triplet states, (o-o-) and ( a ? b $ + a ~, b 1 ' ) / ~ - , the next excited singlet one, S = ( a l ' b $ - a + b 1')/v/2, and the Hubbard doublets, ( 2 0 ) = o ' %+ % ± and ( 0 2 ) = o'b+bf, have higher energies than the empty bole state of (00) [18]. In the correlated electron picture, one has to use the generalized Hubbard-Okubo [19,20] operators Xan [18] instead of the Fermi operators. Each projection operator Xan = I B ) ( A I transforms the multielectron dimer state A to B. The basis of the X-operators is complete and orthogonal. That is why one-electron Fermi operators can be expanded in a series of basis X-operators in the following general form:
%= ~,(BI%IA)X~=
Y'.g~X,,,
A,B
et
b~= ~ ( B I b ~ I A ) X ~ = Y'.g~Xlj, A,B
,8
where (XA~)+ = Xa
with product
XAB ® Xc° =
207
{a+%} = {b+b~} = ~2a(XAa) (orthogonality of the normalized X-operator basis). Applying it to Eq. (1) and considering 4-fold degeneracy of the ground state, one can get the following system of equations for the site occupations of n, and no:
({a+a~,}}=({b+bo.}} = E ( X a a } = 1: A 4n I + n O =
1
(n~)=(a+ad)=(b+b~>:
n.=n/4.
All two-particle exited states have higher eigen energies than the ground one-particle state and the polar empty one, and therefore we do not take into account their exponentially negligible Boltzmarm populations in the second equation for number of carriers (n,,) = n / 4 (we consider a completely disordered paramagnetic state) of the above system, Eq. (2'). The so called correlation factor [18,21,22] is given
a,,cXL
as
Here ot and ~ numerate the intraET2 transitions and the essential expansion coefficients g~, (genealogical coefficients [18]) should be evaluated for all possible intradimer transitions between the ground state A = ( ? 0 ) , (~, 0), (01' ), (0 $ ) and all excited states B. For the problem under consideration, we get
f = ({X0o Xo~0}) = (X0~ > + ( X $ 0 )
ao.= E M , x,=X°°o + ( 1 / I / ~ ) ( X °w + o ' X ° ~ )
(2')
~rO
O0
~rO
= n o + n, = ( 4 - 3 n ) / 4 .
(3)
It reflects the filling of the correlated energy band by the carrier number per dimer, n. The correlation factor f is essential for the X-operator temperature Green's function:
G~t3( rt, tat') = - ( T X , ( rt) X_o( ta t') ).
+ x£" +
bo. = F_,g Xg
=
°,
oo
o + o-x
°)
o
- Xo~° - tr X°~.
(1)
In the basis of the conventional Fermi operators, % and b~, the main dimer Hamiltonian includes fouroperator electron correlation terms. With respect to the X-operators, however, this Harniltonian is linear a s HO=~,I~lC~I6EKX ~, where all essential intradimer correlations are involved in eigenstates {er}. Average values of the diagonalized Xrr HubbardOkubo operators are equal to the Boltzmann populations of the corresponding eigenstates, e.g. ( X g ) = n o,
( Xffo°> = ( X o ,o,,, ) = n , .
(2)
For all of the 16 eigenstates of the dimer, the completeness relation for the X-operators yields 1 =
Below we take into account only the ground state (1'0), ( ~ 0 ) , (01"), (0,L) and the nearest excited polar state (00) with populations n 0, following from the system of Eqs. (2'). At the Matsubara fermion frequencies, for the dimer Hamiltonian H 0, the zeroth order dimer Green's function describes the intradimer transitions between the ground state and the nearest excited state and has the form n A -'1.-n B
c ° ( to°) =
- i to. + e s
-
,Ba
n o +n~ = -i
ton + ¢ 0 ~ - %0
.
(4)
where ~ denotes intradimer transition from A to B: a - - a ( ~ ) . For our problem, % 0 = 0 and e0~ = 6,,o(-- - / z ) is the 4-fold degenerate energy of the one-particle dimer ground state (/x is the chemical
208
V. Ivanov, K. Kanoda / Physica C 268 (1996)205-215
potential) and the zeroth order Green's function for E T 2 is f
o oo (o9.)
~[~o1"
-iw.-
ix
ix : ¼ ( 5 n - 4)w,
= O(Oooo)(o9.)
with correlation factor given by (3). The tunnel Hamiltonian is represented via all possible products of the X-operators in Eq. (1) for neighbouring dimers: ntunnel =
E a ,/~ ,p
~r0
f -- i o9n -- /z + f t e r ( P
)
=Coo (p, o9.).
(5)
0 ie,
Beyond the Hartree-Fock approximation, the spectrum of the one-particle excitations follows from poles of this Green's function by the analytic continuation i o9, ~ ~?+ i 8:
sc~'b=fie~(p) - ix= ¼(a- 3n)t~( p) - ix.
(6)
One can get the chemical potential from the following equation for the number of holes per site: 1
no, = i n
=f~"P,~°nei'S ([G° P 'o,0 L° =
(8)
which reflects the correlated narrowing of the energy band: at n = 1 (the K-ET2X salt case) we conclude that ix = w/4 (quarter band filling, because the chemical potential origin is the bottom of the energy band).
g°g~ter(p)Xa(p)X#(P)"
The tcr(p) refers to the bonding branch of the spectrum of uncorrelated hole carriers in the ET2-1ayer and will be calculated in the next section. In a perturbative approach, the interdimer hopping, t¢~(p), of hole with momentum p plays the role of a perturbation and the first order pertubative Green's function for the paramagnetic ET 2 layer is expressed as
Goo(P , t o ) =
carriers, one obtains from Eq. (7) a dependence of the chemical potential on carrier concentration as,
O9n)+G°°(P'o ~ o9,)] (7)
p
where O(x) is the Heviside function. In the righthand side of Eq. (7), the multiplier 2 is due to the a-b degeneracy of the spectrum of Eq. (6). Assuming a rectangular (fiat weighted) density of states with a half-bandwidth w, for noninteracting
3. Tight-binding electronic structure in ET2-1ayer model In spite of the seeming similarity of the layered structure in the K-ET2X and HTSC materials [6], their electronic band structures have a crucial dissimilarity. For the K-ET2X family, the internal degrees of freedom in the ET2-1ayers are important: dimeric nature of ET-molecules and two ET dimers per basic unit. A square lattice is assumed for ET dimers for simplicity, while the real arrangement has rhombic distortion. For the sake of clarity, all of the hopping integrals along lattice diagonals are reasonably supposed to be the same and are defined by t~ r. In reality, they are slightly split due to the non-symmetry of the dimer for some salts (see Ref. [11]). The t~r is an azimuthal hopping integral, which is, e.g. in the b direction of the crystal in Ref. [11] and is set along the y axis in this paper. The t~r and t~r are = 0.1 eV according to Ref. [ 11 ]. Based on the knowledge of the hopping integrals, the uncorrelated tight-binding calculations were used in Ref. [23] to analyze the optical reflectivity of tc-ET2I 3 and in Ref. [24] to analyze the quantum magnetic oscillations in K-ET2Cu(NCS)2 under pressure. The secular equation for the bonding branches of the uncorrelated hole spectrum sp of elementary excitations (antibonding branches of the electron kind) is, I(t ra - ee)6; i + Hij I = 0.
(9)
The matrix elements, Hij = (~i [ Hint 1~->, of the interdimer transfer Hamiltonian H ~n, are expressed via the Bloch functions ~i of the ith doubly degen-
V. lvanov, K. Kanoda / Physica C 268 (1996) 205-215
crate energy state of the dimer pair in the basic unit as,
209
Using the analytical expressions for tp, tp//2 t~ r ~ Sp:i: = COS py ~t- 2 C O S ( p y / / 2 ) C O S ( p x / / 2 )
Hu.22 = t~r{cos Py
= cos py _+/3(cos py) cos( p J 2 ) ,
+ 3[ 1 + cos p, + cos px + cos( p r + px)]} . e~[sin - py HI2 = , ~1/2
+ sin p~ + s i n ( p y + p x ) ] = H z , *.
Here we assumed an equal magnitude for the interdimer hopping integrals, and all momenta are given in reduced units: I Px I = [ Py I = 7r on the respective Brillouine zone boundaries. From Eq. (9), one obtains a pair of hole bonding spectrum branches [23]
where /3(x)=~]2(1 + x ) [c.f. Eq. (10)], one can calculate the total carrier density of states, with 6 the dimensionless energy in the range of - 1 _< e < 3 in the antibonding band as follows:
1
r2~
-(2~.)2j0
r2~
dpyJo dpx
as ~p = t ra + tp,
1 f,1
tp = 2t~r[cos py + 2 c o s ( p y / 2 ) cos( Px/2)].
7r 2 J - I d y ~/(1 _ y 2 ) [ [ 3 2 ( y ) -
(10) This derivation from the secular equation (9) is justified if the energy difference 2ira between the bonding and antibonding pairs exceeds the energy splitting in each bonding a n d / o r antihonding energy branch. Applying the tight-binding method for correlated electrons [18] to the dimeric structure of the ET 2 layer with a finite magnitude of tra/UET after a canonical transformation to natural bonding and antibonding orbitals, we get a pair of bonding branches and a pair of antibonding branches of the spectrum for correlated holes in conducting ET z layers as follows [25]: ~t, = +ira + (1 + 4tra/UzT)ft, -- IX.
O[/3(y)-e+y]
i
4
~2"2¢~-4e +3
x(jm
16 2q~-+6 + 3
J (12)
The above is divergent at e ~ - 1 , around which p(6) is of the form
p(e)-~
(11)
It is seen that in the limit of t~/UET ~ O, the upper branch of the spectrum of Eq, (11) corresponds to the dispersion of the doubly degenerate sites/dimers given by Eq. (6). In Eq. (11), the bonding and antibonding branches originate from dimerization. The splitting of each bonding or antibonding band in two branches occurs due to the K-ETzX crystal structure having a dimer pair per unit cell. Among the six electrons in one basic unit, 2 E T f , four of them occupy the two bonding bands and two electrons occupy the antibonding bands of Eq. (11). The uncorrelated branches of Eq. (10) reproduce the essentials of the first prinsiples calculation [26], tightbinding method [27] or numerical calculation [17] of the K-ET2X band structure.
y)2]
3
4 In---2,n.z42v~ + 3 e+ 1
3 4 -ln-27r 2 ~+ 1 '
where K ( x ) is the complete elliptic integral of the first kind. Note that / 3 ( y ) = 1 in Eq. (12) for the square lattice with a dispersion of e = % / 2 t = c o s py + COS Px or
= % / 2 t = cos py + cos ( p x / 2 ) and in this case DOS reduces to elliptic integrals with a prefactor of I/~r 2 and a modulus of V/1 - ( e / 2 ) 2 in the corresponding energy regions of -2<~<2 or - 1 < ~ < 2 . In comparison with HTSC cuprates, the main dissimilarity of the energy spectrum, Eqs. (10), (11) and (6), for x-ETaX is caused by the azimuthal
210
V. lvanov, K. Kanoda / Physica C 268 ( 1 9 9 6 ) 2 0 5 - 2 1 5
angular momentum in the ET 2 plane. For the paramagnetic state of cuprates, such components are very small. In Eqs. (10), (11) and (6), this anisotropy originates from the arrangement of the ET-molecules, which adds azimuthal transfers to the modelled transfer of square-lattice symmetry. It is interesting to note that the hopping integral asymmetry for the hole dispersion in the antiferromagnetic highTc cuprate background is governed by the symmetry of an antiferromagnetic Brillouin zone: a single hole prefers to jump to the next nearest neighbouring site of a square lattice rather than to the nearest neighbour; the nearest neighbour hopping is suppressed comparatively to the next nearest one [28].
4. Superconducting pairing in the ETz-layer model As seen in the spectra of Eqs. (10), (11) and (6) derived for the present model of correlated carriers in K-ET2X, the band structure effects are important when the Fermi surface is near the Brillouin zone (this is the case of the K-ET2X family) where the influence of the crystal potential is strong. In the tight-binding approach, the symmetry of the Bloch functions, Er~b(r-R)e ip'r, and the symmetry of the Fermi surface manifest themselves in the symmetry of the superconducting order parameter A(p) [29,30], which represents the correlation function of the Cooper pairs as A ( p ) ~ (C~.pC~._p). Neglecting the retardation effects the latter becomes App, = E e x p [ i ( p ' - r ' - p .
r)]
rr'
× f ~b*(r-R)A( R)J/(r'-R) dR = ) ~ e x p [ i ( p ' - p ) . r]
r
x f A ( R ) I ¢ ( r - R ) I 2 dR It is seen from here that at constant energy gap [localization in real space of A(R)] the superconducting order parameter A(p) has the same symmetry as the Fermi surface. Dispersions of elementary excitations (10), (11) and (6) provide an anisotropic Fermi surface unlike the sphere cited in the BCS theory. A complicated Fermi surface does not pro-
vide the conventional isotropic s-wave pairing, i.e. either constant pairing interaction or a constant energy gap as Anderson had mentioned in earlier publications [31]. For the unconventional pairing, the symmetry of A(p) is lower than that of the Fermi surface. The pairing interaction Vpp, in the effective interaction Hamiltonian, 2'
Hint =
E
V ( q ) :p + q , o ' C p+' - q , ~ C p ' , ~ C p , o
",
p,p ,q,o" ,o"
and the superconducting order parameter, Apt,,, depend on the momenta difference of the pairing fermions, namely V ( p - p') and A(p - p ' ) . Here we do not attempt to specify the origin of the pairing interaction in K-ET2X superconductors. Due to particle correlation, on-site pairing should be ineffective. In the reference system of the square lattice, the general form of an effective pairing interaction for the nearest neighbours can be rewritten as the following identity:
Y'. cos(p,~-p~')
=2V
a=x,y
= 2V(I[(cos l[~x+COS py)(COS
+(cos
cos p,)(cos
P'x
+COS /~'y)
,-cos p,,)]
+ (sin py sin fly + sin Px sin fix)}.
(13)
Eliashberg had noted that the theory of superconductivity need not be based on an uncorrelated band spectrum and Green's functions of uncorrelated electrons [32]. We can start from the correlated Green's functions given by Eq. (5) to apply the formalism [33,34]. In the effective interaction Hamiltonian Hint, the Fermi operators should be expressed via X-operators (see Section 2). In the Born approximation, the amplitude of fermion-fermion scattering, F(p, p') = V(p-p'), is defined by Eq. (13). Then for the Cooper channel in the reference system of colliding fermions with opposite spin projections, the anomalous self-energy can be written as '
to,p'
t,
v
, ),
211
V. lvanov, K. Kanoda/ Physica C 268 (1996) 205-215
G,o(P) =
~P
(14)
where the carrier dispersion law ~:, is determined mainly by the lower branch of the bonding hole spectrum (the upper antibonding branch of the electronic spectrum) of Eq. (11). In the reference system where the y-axis is along the azimuthal direction, it is rewritten in dimensionless units as g / / 2 ft~ r - oo
= cos Pr + 2 I cos(py/2) cos(px/2) = COS py + fl(cos
py)
I
cos(Px/2),
where fl(x) is defined in Eq. (12) and f is the correlation factor defined by Eq. (3). From now on, the ¢~ will be measured from the chemical potential p. and we will neglect the difference in interdimer transfer integrals, ter. Using the identity, oo
1
1
x
rn=0 y" x 2 + [(2n + 1)Trr] 2 = 2-7 tanh --,2T
order to solve Eq. (15), it is necessary to decompose the attractive interaction V(p - p ' ) and order parameter A(p) into contributions from all possible irreducible representations of the point group symmetry, C4v, of the square lattice under consideration. The symmetry of the representation with the largest Tc is the case according to the BCS type description of superconductivity in Refs. [35,36]. The identity of Eq. (13) is no more than the expansion of V ( / ~ - i f ) in a series with the eigenfunctions r/(/~) of the C4v irreducible representations. The first and second terms in the square brackets are products of basis functions of a one-dimensional irreducible representation A l (s*-wave pairing) and B 1 (dx2_y2-Wave pairing), respectively, and the last two terms include basis functions of the two-dimensional representation E (triplet p-wave pairing). Our reference system is rotated by an angle of ~'/4 from the "tilde" reference system. Therefore the carrier momenta in Eq. (13) are connected with the used dimensionless momenta as P x = ½( Px-- Py),
instead of Eq. (14), one can get the general equation of the standart BCS theory as
P y = ½( Px + Py),
and the basis functions are reduced to r/,(p) = cos(Px/2) cos(py/2),
~,(p) = E V ( p - p ' ) ~ , ( p ' ) p'
r h ( p ) = sin(px/2) sin(py/2), r/3.4( p ) = sin(px/2) cos(py/2)
×
2~/~(p')2 + ~2(p,)
dp'. + sin(py/2) cos(Px/2). (15)
Here the anomalous self-energy part, ~(p), has a character of the superconducting order parameter
a(p).
In K-ET2X superconductors, several experiments imply that anisotropic singlet d-type pairing with nodes in the superconducting order parameter such as Ao(/~)~ cos /~x- cos /~y (so called dx~_y~ pairing) or some other pairing (d~y) occurs at the Fermi surface. But they are not inconsistent with anisotropic singlet s* (extended s) pairing with nodes of the form A(/~) ~ cos /~ + cos /~y without a sign change, or minima of the gap on the Fermi surface, but in the same direction in the Brillouin zone as in the case of d-pairing. The problem now is to determine the type of Cooper pairing in our model. In
(16)
Putting the expansions V(p -p') = 2VE4= lr/i(p) rli(p') and A ( p ) = E4= lAirli(p), into Eq. (15), we can reduce it to the following system of algebraic equations for T~ in terms of orthogonality of the basis functions {r/(p)}:
4
tanh( ~p//2 Tc )
p;j= l
2 sop
A,=g i ~_.
Ajrli(p)rlj(p),
(17)
where gi = 2V and the summation is over momenta p in the ET2-Plane. The system of equations (17) has solutions of Tc when the following 4 × 4 determinant vanishes:
3ij-gY'~ P
tanh( p/2 ) 2g
[
Oi(p)rlj(P) I = 0 .
(18)
V. lvanoo, K. KamMa / Physica C 268 (1996) 205-215
212
The essential part of this secular equation is E tanh( ¢ / 2 r ~ ) ~ ( P ) nJ(~') p
2~p
f=
= -= de
tanh( (e)/2Tc) 2~:(e)
-, ¼ ( t + 1 ) ( 3 - t ) ,
( t + 1) 3
/
+ --K(q)
- 2
t--1
Fi4(e)'
F,j(e) = E FiT(el, Ot ~
q)
2t
Vt2--1
× [//(½~-, ¼ ( t + 1 ) ( 3 - t ) ,
where
~/(9-t2)/8)
,+3
++
+--K(~(9-t2)/8
F,7(e)
(20)
t-1
= E
(e-
Fa(s) = F22(e )
p
-(2 -r)3£ dpyf_ dx exp[ix(e-cos 1
-
[ ×[(3-t2)//(½~
".,r
1 (t-l)2( = F s . ( e ) + 87r2 ~
p,)]
-½(t-
× L ~ dpx exp[-ix~Ci(cos p y ) c o s ½p~] X Tli( P ) n j ( P ) ,
= rll(e)
1 8Ir 2
(t--
1) 2
~/t
1)(t + 3)
-4//(½~-,
and fl(x) is defined in Section 3 [c.f. Eq. (12)].
Because of oddness of the products, Th(p)Tij(p)(1 -- 6,), with respect to py, all of the nondiagonal matrix elements in Eq. (18) turn out to be zero and therefore superconducting pairing of different symmetries cannot be mixed. Here we will consider the realistic singlet d- and s*-pairing with nodes on the Fermi surface, because the Knight shift measurements suggest only a singlet form of pairing in K-ET2X superconductors [8-10]. The integrations in Eqs. (19) are performed over the variables Px, x, sequentially and then over momentum projection py. On the intermediate steps in the calculations, integrations with Bessel functions (after integration over variable p~) and Chebyshev polynomials (after integration over x) are useful. The final forms are expressed via elliptic integrals as follows: F~.(e)
2 t - 1 ] K(q)
× H(½7r, (t + 1)3(3 - t)/16t, q)
(19)
5. Coupling constants for different symmetries of singlet Cooper pairing
(t+l/2
¼(t+ 1 ) ( 3 - t ) ,
q)),
(21)
where K(k) and H ( ~1r , n, k) are the complete elliptic integrals of the first and the third kind, respectively, t = ~ 3 and one of elliptic integral modulus in Eqs. (20) and (21) is q = ~/1 - (t - l)3(t + 3 ) / 1 6 t . Then, the superconducting transition temperature for each irreducible representation is defined by .
l=gj %: -
-
de
2~:(e)
Fs'd(e)
to c
gf~(%) d~ tanh(~/2Tc ~'d)
,
(22) where % is a cut-off energy parameter and the derivative a~/ae = f is the correlation factor defined by Eq. (3) as evaluated by Eq. (6). We assumed, as usual, that the coupling constant g is nonvanishing in an energy range of width 2 % near the Fermi energy. Applying the logarithmic approximation to Eq. (22), we obtain Tc as Tcs~ .d = ~1tOc exp( - 1/A~.d) with coupling constant As.,d
(23) =
4gFs*.d(e = 0).
V. lvanov, K. Kanoda / Physica C 268 (1996) 205-215
Here the Fs*,d are defined by Eqs. (20) and (21). The numerical multiplier in the coupling constant is due to the correlation factor discussed above: I/f= 4. It should be noted that for conventional s-wave pairing with basis functions rli(p) = 1 in the irreducible representation, A l, Eq. (19) reduces to Eq. (12). In this case, the coupling constant of isotropic s-pairing is defined by the normal electronic density of states Eq. (12) at the Fermi surface ( e = 0) and one
can
get
4gp(e
As =
4
= 0) = (4g/Tr2~/-3-)
K(½1/2 + v~- ) = 0.87g(K(½!/2-+ ~ ) -~ 2.77 according to a numerical evaluation) for the same coupling constant g. Using Eqs. (20) and (21), the coupling constants are evaluated as follows: 2+v~A~.
- -
×
2-f3-
As
2
[
H
2'
rr 2
2'2
+ ~ g
g,
(24)
with / / ( r r / 2 , ~ / 2 , ¢-5-/2)= 1.50 and K ( ¢ 3 - / 2 ) = 2.16, i.e. As. = 1.16g,
ad = ¼(1 + ¢ 5 ) a , + a,. 2[~-/3-~ -7r V-/-32- - V~-//( 'n" ) ' 2 +42~ - ' ¢-2+2 ¢ 5
+4//
2'
2 '
2
g
1-I(~r/2, (2 + v/-3")/4, 1/2 + ~ / 2 ) H(rr/2, vr3/2, ~ + ~ - / 2 ) = 1.84,
with
(25)
= 1.80 and
i.e. Ad = 1.53g. So, the evaluated ratio of anisotropic coupling constants is Ad/a ~. = 1.32. For nonphonon superconducting pairings, the cutoff energy parameter % in Eq. (23) is determined by the electronic structure. Assuming that the cut-off energy parameter is the half-bandwidth, i.e. 4t~r ~ 0.4 eV, and the coupling constant is g = 0.17, we can estimate the superconducting transition temperature Tc at = 12 K for d-wave pairing in the weak cou-
213
piing limit from Eq. (23). This value is a reasonable magnitude for K-ET2X superconductors. The Tc of anisotropic s*-wave pairing is smaller by half an order of magnitude.
6. C o n c l u d i n g r e m a r k s
From the spectrum of Eq. (11) we found for correlated carriers in the K-ET2 X model the chemical potential as /~ = ( 1 / 4 - 3tra/UET)W, that means quarter narrowing of the noncorrelated energy bands in the strongly correlated limit, tra/uEX = 0 (c.f. Eq. (8) at n = 1). So, the actual bandwidth should be considerably smaller due to the particle correlations; in the normal paramagnetic phase, the correlated holes move in the narrowed band of 0 . 8 / 4 ~ 0.2 eV width. This correlated reduction of the energy band is essential for band structure [23] and may be involved in the optical [27] and cyclotron [24] effective masses. For a conventional square lattice, the electron dispersion is ~ = 2t(cos p y + COS p ~ ) and the DOS is given by the expression K((1- (~/4t)2)/ (2~-21) with logarithmic divergences only along boundaries of the Brillouine zone: e = + 4 t . Unlike this case, Eq. (12) for the K-ET2X lattice has an energy-dependent prefactor and has a logarithmic divergence in energy at the point of t~/2ter = -1 (in case of A ter = 0), i.e. along the boundaries M(rr, "n') - Z(O, "n') - ( - rr, rr), ( - "n', - rr) ( r r , - r r) and along the diagonals M(rr, r r ) F(0, 0) - ( - ~, - 7r) and (Tr, 7r) - ( - ~r, - ~-) of the Brillouine zone. The positions of the logarithmic divergence are in agreement with the results of the first-principles calculation [26]. Fortunately, these divergencies are located well outside the Fermi level and the assumption of a flat weighted density of states is justified. Referring to magnitudes of the interdimer hopping i n t e g r a l , ter = 0.1 e V , Eq. (12) yields the density of 4
states at the Fermi level as p(p,) = (l/2"n'2Vl'3ter)K (I/2 + ~ / 2 ) = 1.05 (eV spin)-l (c.f. Section 5). Note that this value is in agreement with the value p -- 1 states/eV spin ET, evaluated in Ref. [37] for noninteracting electrons in metallic K-ET2 X salts.
214
V. Ioanov, K. Kanoda / Physica C 268 (1996) 205-215
For the superconductivity we used a model with a correlated energy scale of UET ~ 1 eV, not with the hopping energy scale tra, ter ~ 10 -1 eV. The anisotropy of the symmetry of the derived electronic transfer, Eqs. (6), (10) and (11), is involved in anisotropy and type of the superconducting order parameter. The irreducible representation B l possesses the highest coupling constant and Tc among various pairings with possible symmetries. So, the superconducting dx2_y~-wave pairing [see Eq. (13)] is preferable. Our treatment of the superconductivity gives the possibility of nodes in the gap parameter. In this case, the superconducting order parameter has line nodes along the Px and py axes, and four inequivalent nodes on the K-ET2X Fermi surface. For s *-wave pairing of the irreducible representation A l with the basis functions tll(p)=cos(px/2) cos(py/2), also four inequivalent nodes exist on the Fermi surface. The present study may favour some kind of nonphonon superconducting mechanism, e.g. the attractive pairing can be induced by antiparamagnon excitations [38]. As a final conclusion on pairing, however, it is useful to compare not only the Tc but also energies of different pairing symmetries (c.f. Refs. [35,39]). In addition, we cannot exclude the possibility that the actual distortion of the square lattice in the ET2-1ayers may lead to a contribution of rhombic components in their pairing symmetry. Admixture of different pairing components occurs for tetragonal distortions of the square lattice in high-Tc cuprates
[40]. The interdimer hopping te~ increases under pressure. As can be seen from Eq. (23) (after replacing the coupling parameter g by a dimension magnitude g/2t~) it leads to a decrease of the coupling cons t a n t Ad and Tc in the proposed model of K - E T E X superconducting salts. The opposite tendency occurs with the increase of the lattice parameters or expansion of the basic unit. These are in agreement with empirical conclusions. The electron-electron correlations manifest themselves in narrowing of the normal energy band and in the appearance of the correlated factor f < 1 in Eqs. (23), and may enhance T¢ and the condensation energy in comparison with the uncorrelated case. The present treatment is reasonable for the description of the low energy experiments such as
NMR [8-10] and electronic specific heat [41] in the superconducting state. The present study can be useful and applicable to other organic materials with layer structure, e.g. BETS compounds [42].
Acknowledgements The authours thank H. Fukuyama for providing a preprint, Y. Maruyama and S. Nakajima for helpful discussions and T. Kasuya for useful critical comments. The authors acknowledge K. Yakushi and N.M. Plakida for valuable advice and are thankful to T. Iwatsuki for her help. This work was supported by a Grand-in-Aid for Scientific Research, Nos. 06452064 and 06243102 (Priority Area "Novel Electronic States in Molecular Conductors") from the Ministry of Education, Science, Sports and Culture of Japan. V.I. acknowledges the support of the Ministry of Education, Science, Sports and Culture of Japan during a stay at the Institute for Molecular Science, Okazaki.
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