Electron--vibration coupling constants in positively charged fullerene. home | group

April 10, 2001

Abstract

Introduction

Recent breakthrough experiments (Schön et al. 2000) have shown that a monolayer of C₆₀ can be positively field-doped, a goal hardly realized chemically so far (Datars and Ummat 1995). In that state, fullerene exhibits a higher resistivity than for negative doping, and becomes superconducting with critical temperatures that can exceed 50 K, about a factor 5 higher than the corresponding negative field-doped state. The general belief is that superconductivity in the fullerenes is related to a strong intra-molecular electron-phonon coupling, connected with the Jahn-Teller effect of the isolated ion (Antropov et al. 1993). Unlike the negative doping case, where both calculations (Antropov et al. 1993, Varma et al. 1991, Lannoo et al. 1991) and fits to data (Gunnarsson et al. 1995) exist, no quantitative evaluation of the actual Jahn-Teller coupling strengths is so far available for the positive fullerene ions.

In this work we undertake the task of determining the electron-vibration linear couplings for the H_u HOMO level of the C₆₀ molecule, along with that of spelling out some of the consequences for resistivity and for superconductivity. For that purpose we use density functional electronic structure calculations, yielding accurate molecular vibration frequencies and eigenvectors for the two a_g, six g_g, and eight h_g modes that couple linearly to the electronic H_u state. Knowing the form of the Jahn-Teller coupling matrices, we distort the molecule and extract the coupling constants from the calculated level shifts and splittings. As a parallel check, we repeat a similar calculation for the negative, electron doping case, where the a_g and h_g modes couple linearly to the T_1u LUMO level of the C₆₀ molecule. The couplings obtained for negatively doped C₆₀ are rather similar to those that can be found in the literature, and just represent a fresher, state-of-the-art theoretical determination. The hole-vibration couplings of positively doped C₆₀ are new, and can be put to use in a variety of manners, including predicting or explaining properties of molecular ions, such as photoemission (Brühwiler et al. 1997) and IR/Raman spectra. That is a task that we propose to consider in the near future.

The couplings obtained can also be used to determine the dimensionless electron-phonon coupling constant λ relevant for the superconductivity as well as for the vibron contribution to the high temperature T-linear resistivity in the crystalline phase. Comparing values for positive and negative doping we find that for positive doping λ is a factor 1.4 larger than for negative doping. These results provide a starting point for a discussion and comparison with the experimental findings.

This paper is organized as follows: the notation is set up in Sect. ham:sec; the ab-initio calculation and results for the molecular ion are described is Sect. dft:sec; in Sect. solid:sec we sketch the calculation of the resistivity in a band-degenerate case; Sect. superc:sec presents a formulation for superconductivity in that case; Sect. discussion:sec contains comparisons and discussion of experimental data.

The Jahn-Teller Hamiltonian

{ D^(2-) × D^(2-){}}_s = D⁽⁰⁺⁾ + D⁽²⁺⁾ + D⁽⁴⁺⁾

{ h_u × h_u {}}_s = a_g + h_g + (g_g + h_g) .

H = Σ_τ=ag,gg,hg Σ_i^n_modes(τ) [ H_harm^τ(h ω_{τ i},\vec P_{τ i},\vec Q_{τ i}) + H_e-v^τ (g_{τ i} h ω_{τ i},α_{τ i},\vec Q_{τ i}) ] .

H_e-v^τ(g h ω ,α,\vec Q)= ^{g h ω}/₂ Σ_{m µ ν} Q_m c⁺_µ c_-ν ^τC_µ,ν^m(α) .

\begin{array} c₀=d₀  \pmatrix{ c_m  c_-m } = ¹/ ₂ (

1	i
1	-i

) \pmatrix{ d_m  d_-m }, m=1,2 .

He-vag(g h ω,q)= g h ω/2 q Σµ ν d+µ dν Vagµν ,

\begin{array} Q₀  =  q₀ \ \ \ \ (h_g modes only)  \pmatrix{Q_m  Q_-m } = ^(-1)m/ ₂ (

1	i
1	-i

) \pmatrix{ q_m  q_-m } (m=1,2) .

H_e-v^τ(g h ω ,α,\vec q )= ^{g h ω}/₂ Σ_m q_m Σ_{µ ν} d⁺_µ d_ν V^{τ (m)}_{µ ν}(α) \ \ (τ=g_g,h_g) .

The simultaneous linear coupling to several modes will generally lead to a cumulative distortion and to an energy gain which is the sum of the individual energy gains. However, the form of the coupling (real interaction Hamiltonian) prevents the molecule to gain energy through both kinds of couplings. The system shall choose between a D₃ and a D₅ distortion, depending which one is energetically more convenient for given specific values of the couplings, vibration frequencies, and H_u orbital electronic filling. The calculation of the following section determines in particular which one of the two types of distortions prevails in C₆₀⁺.

Calculation of the couplings and results

Based on this electronic structure calculation, we used next density functional perturbation theory (Baroni et al. 1987) to compute three independent rows of the dynamical matrix. Icosahedral symmetry is then used to recover the full matrix, which determines the normal modes \vecξ_i,s and frequencies ω_i (Giannozzi and Baroni 1994) of the molecule. We obtained frequencies (see Table frequencies:tab) in good agreement with experiment (Prassides et al. 1991, Zhou et al. 1992), as well as with previous calculations (Giannozzi and Baroni 1994, Negri et al. 1988, Kohanoff et al. 1992).

Table 1

To determine the e-v couplings for the linearly coupled modes, we proceed subsequently to displace the atomic positions from the equilibrium position along each of the normal modes, choosing a suitable eigenvector combination in the linear space of each degenerate vibration. In particular for each h_g mode we selected the q₀ displacement, corresponding to the totally symmetric combination \vecξ_i,0 of the distortions \vecξ_i,s with respect to an (arbitrarily chosen) D₅ subgroup of the molecular symmetry group. The five initially degenerate (really, only nearly degenerate, owing to a weak cubic splitting due to the artificial supercell lattice) H_u Kohn-Sham eigenvalues split under this distortion with a pattern given by the eigenvalues of V^{h_g (0)}. We applied a displacement of the atomic positions along each of the eight normal-mode unit vectors, \vecξ_i,0 with a prefactor ranging from -0.1 to 0.1 Å. In Fig. split:fig we plot as an example the resulting energies for the sixth h_g mode. The pattern generated by V^{h_g (0)} should be 1+2+2 (a state separated by two pairs of twofold-degenerate states). The small residual splittings of these twofold degeneracies, due to the cubic crystal field and higher-than-linear couplings, give an estimate of the accuracy of the method. By standard linear fitting and comparison with Eq. (real interaction Hamiltonian), we obtained directly the linear dimensionless coupling coefficients g_hgi α_hgi collected in Table couplings:tab. We determined the sign of α_hgi by applying a distortion along \vecξ_i,1, and comparing the splitting of the HOMO with the eigenvalues of V^{h_g (1)}(± α_hgi). Following the same procedure we derived the couplings for the g_g modes, by applying here q_-1 distortions. The resulting linear coupling coefficients g_ggi are also collected in Table couplings:tab.

Table 2

For convenience, we also report in Table couplings:tab the amount of optimal JT distortion pertinent to each mode when the HOMO level is occupied by one electron/hole, and the corresponding energy lowering E_s for both D₅ and D₃ minima. Note in particular the large coupling associated to the lowest h_g mode, the corresponding distortion leading to an energy lowering practically equal to its quantum hω.

Adding up the JT energy gain of the individual modes, we estimate the total classical potential energy lowering in C₆₀. The D₅ minima gain E_s=71 meV, while the D₃ minima gain only E_s=22 meV (the contribution of the a_g modes -- 2 meV -- being included in both cases). It is therefore apparent that the C₆₀⁺ ion will choose, at least within linear coupling, the D₅ distortion. As was shown in (Manini and De Los Rios 2000), the possibility of a switch to a nondegenerate A_u dynamical JT GS occurs, for large coupling strength g 6, only under the condition that the D₃ minima are energetically lower or equal to the D₅ minima. This settles finally the issue of the dynamical JT GS symmetry of this molecular ion: it is a regular Berry-phase vibronic state of symmetry H_u, like the parent electronic state (Manini and De Los Rios 2000). No level crossing to a nondegenerate A_u state is predicted to occur for C₆₀⁺.

Electronic charge density distribution of the hole in C60+ at the D5 JT minimum along a q0 distortion of any hg mode. Out of the six equivalent ones, this particular minimum is defined by the D5 axis drawn as a bold ``spit'' piercing the ball through two pentagons. The contours are drawn at 30% of the maximum density. }

A static JT coupling resolves the degeneracy of the H_u level: in a distorted configuration it is possible to distinguish individual levels within the HOMO, energy-wise. Figure bandHu5:fig depicts the square modulus of the hole wavefunction for a D₅ minimum. The probability density appears to be localized on an equatorial conjugated band, where the poles are the opposite pentagons centered around the D₅ axis we chose among the six possible ones.

As a check, with the same method used above to calculate the hole-vibration couplings of the HOMO (Fig. split:fig), we also computed the electron-vibration couplings of the LUMO and obtained the values in Table minus:tab. The total static JT potential energy lowering is 41 meV, of which 3 meV due to the a_g modes, and 38 meV due to the h_g modes. These values are generally in line with those calculated by previous authors (Antropov et al. 1993, Varma et al. 1991, Lannoo et al. 1991), although there are some differences in the details. Error bars in theoretical determinations of JT couplings of fullerene have proven surprisingly large, possibly reflecting and amplifying errors in the vibrational eigenvectors. Of course as is well known, somewhat larger energy gains are obtained when the true dynamical JT problem, including a full quantum treatment of the vibrons is considered (Auerbach et al. 1994, Manini et al. 1994). We shall leave this calculation in C₆₀ⁿ⁺ for future work.

Table 3

E_s^a_g(n)=n² E_s^a_g(1),

which can become as large as 179 meV for n=10 holes. Consider now the nontrivial JT part of the coupling, that to g_g and h_g modes. For n=2 electrons/holes (spin singlet configuration) in the HOMO orbital, the distortions simply become twice as large as in the n=1 case, with JT energy gains which are four times larger. Even though n=1,2 electrons/holes take advantage only of the D₅ stabilization energy, additional electrons/holes can benefit from the extra HOMO splitting induced by the h_g(r=2) plus g_g coupling. To test this, we relaxed completely the molecular structure in the 64-dimensional space of the 8 h_g and g_g modes, and determined the minima of the total potential energy, filling the five BO sheets as drawn in the insets of Fig. eclass_n:fig. For simplicity we only considered at this stage low-spin configurations, generally the most favored by JT. The resulting energy gains E_s^g_g+h_g(n) are reported in Fig. eclass_n:fig. The energy lowering is maximum for n=4 (and n=6): it is as large as 401 meV, compared to a modest E_s^h_g(n=2,4)=153 meV in C₆₀^-. Similarly to what happens in the T_1u LUMO case (see Fig. eclass_n:fig and Manini et al. 1994), the half-filled configuration n=5 is slightly unfavorable (by JT energetics) with respect to the neighboring n=4 and n=6 states. We find that the contributions of the g_g modes, strictly zero for n=1,2, are small (~ 1 meV) but nonzero in the 3 n 7 configurations. This indicates that for such large fillings the many-modes JT system (Manini and Tosatti 1998) can take some advantage also of the ``losing'' g_g + h_g(r=2) part of the coupling (favoring D₃ minima for n=1,2). The insets of Fig. eclass_n:fig, indicate that the largest displacement of a single level in the H_u HOMO, about 0.28 eV, is realized for n=2. Finally, we have particle-hole symmetry E_s^g_g+h_g(n)=E_s^g_g+h_g(10-n), at an opposite minimum distortion \vec q_s(n)=-\vec q_s(10-n). For the special case n=5 this means that the configuration drawn in the inset and the one obtained reflecting the 5 levels through zero give both the same optimal energy (at opposite distortions).

Application of the above results to real C₆₀ⁿ⁺ must await the inclusion of electron-electron Coulomb repulsion. Coulomb interactions will generally compete with JT coupling and favor high-spin configurations, which, in turn, are generally less favorable for JT. (For example for n=2 the triplet configurations has a JT gain of 100 meV instead of 277 meV for the singlet.) We shall return to a more detailed description of C₆₀ⁿ⁺ ions in later work.

Solid-State Transport

e\vec{v}_kµ· \vec{E} ^n0(ε_kµ)/ _εkµ = ( ^n_k,µ/ _t)_coll.

δ n_k,µ = - ^n0(ε_kµ)/ _εkµ e \vec{v}_kµ· \vec{E} τ_kµ,

Since T>> Max_i(hω_i), after inserting (M:relaxation time approximation) into (M:collision term final) and using the Boltzmann equation, we find that

1/τkµ	=	2π/h1/2 V Σp Σi,τ,m,ν (gτ ihωτ i/2)2 | Vτ(m)νµ(ατ i)|2 kB T/hωτ i {\phantom{Σab} .
	& (1-\vec{v/<SUB>pν·\vec{n}} {\vec{v}kµ·\vec{n}}) [ δ(εkµ -εpν - hωτ i) + δ(εkµ -εpν + hωτ i)\right]},

If we approximately take ε_kµ=ε_k we find, since the electron-vibron matrices are symmetrical,

1/τk	=	π/h Σi,τ,m, (gτ ihωτ i/2)2   Tr[\left(Vτ)2]/d   kB T/hωτ i {\phantom{Σab} .
	& 1/VΣp (1-\vec{v/<SUB>p·\vec{n}} {\vec{v}k·\vec{n}}) [ δ(εk -εp - hωτ i) + δ(εk -εp + hωτ i)\right]} .

\tildeλ = Σ_i,τ,m ^{g_{τ i}2 hω_{τ i}}/₄   ^{Tr [\left(Vτ)2]}/_d ,

λ = N₁(ε_F)   \tildeλ ,

By using Eq. (M:lambdatransport) and the calculated e-v coupling parameters of Table couplings:tab, we find for holes in the H_u HOMO-derived band

\tilde λ₊ 0.277 eV.

This value can be compared with that calculated similarly, using our couplings of Table minus:tab for electrons in the T_1u band

\tildeλ_- 0.197 eV.

The dimensionless electron-vibron coupling which governs the transport properties, λ of Eqs. (M:lambdatransport,lambdadef:eq), is not in point of principle coincident with the parameter λ determining the superconducting properties. The latter must be determined by solving the Migdal-Eliashberg equation with the retarded interaction mediated by the vibrons plus the electron-electron Coulomb repulsion. However, a simple estimate of the order of magnitude of λ can be obtained by taking the unretarded limit, and imposing a Debye cutoff to the electron energies. We assume the electronic band operators to be related to the molecular creation and annihilation operators through the unitary (orthogonal) transformation c_nkσ = U^-1_nµ(k) d_{µ k σ}, n being the band index, σ the spin. Let us define a matrix W by

W_µν = Σ_{τ,i,\widehat m} ^{g_{τ i}2 hω_{τ i}}/₄ (V^{τ(\widehat m)}_µν)² = Σ_τ W^τ_µν Σ_i=1^{n_m(τ) g_{τ i}2 hω_{τ i}}/₄,

W_nk,mp = Σ_{τ,i,\widehat m} Σ_{µ1,µ2,µ3,µ4} ^{g_{τ i}2 hω_{τ i}}/₄ U^-1_nµ1(k) V^{τ(\widehat m)}_µ1µ2 U_{µ2 m}(-p) U_{µ3 m}(p) V^{τ(\widehat m)}_µ4µ3 U^-1_nµ4(-k),

Δ_k,n = ¹/_2VΣ_p,m W_nk,mp ^Δ_p,m/_Ep,m tanh(β^E_p,m/₂),

where E_p,m=((ε_pm-µ₀)² + Δ_p,m²), µ₀ being the chemical potential. The critical temperature is obtained by solving the eigenvalue equation δ_nmδ_kp - ¹/_2VΣ_p,m W_nk,mp ¹/_|εpm-µ0| tanh(β_c^{|ε_pm-µ₀|}/₂) = 0. In general, the BCS gap equation (M:BCSeq) leads to interference between the Cooper pairs belonging to different bands. That, in turn, increases the critical temperature relative to a situation in which the pairs do not interfere. We may therefore foresee two opposite limits of strongly interfering and of non interfering pairs which, respectively, over and underestimate the effective coupling strength λ.

If we assume that averages over the Fermi surface do not depend on the band indices (interfering pairs, corresponding, for example, to the choice of Lannoo et al. 1991), then we can replace W with W, and we find that the critical temperature is determined by the maximum eigenvalue of the matrix W in Eq. (M:W). Under this assumption, the superconducting λ is determined through

λ = N₁(ε_F)Maxeigenvalue(W),

For our H × (a+g+h) e-ph coupling the matrices W^τ are:

W^g_g=¹/₂₄ \pmatrix{ 8  1  6  1  8  1  8  6  8  1  6  6  0  6  6  1  8  6  8  1  8  1  6  1  8  }

Whg	=	1/60 \pmatrix{ 16	14	9	14	7
14	16	9	7	14
9	9	24	9	9
14	7	9	16	14
7	14	9	14	16
}
	&+cos(2 α)/30 \pmatrix{ -2	2	-3	2	1
2	-2	-3	1	2
-3	-3	12	-3	-3
2	1	-3	-2	2
1	2	-3	2	-2
}
	&+sin(2 α)/4 5 \pmatrix{ 0	0	-1	0	1
0	0	1	-1	0
-1	1	0	1	-1
0	-1	1	0	0
1	0	-1	0	0
} ,

while W^a_g is trivially the unit matrix. We note that, even though the matrices W^τ, for different τ's, do not commute, the eigenvector (1,1,1,1,1)/ 5 is an eigenstate of each of them. Moreover, for each τ this eigenstate provides the largest eigenvalue

Maxeigenvalue(W^τ)=Σ_n W^τ_mn=1

(for any m and τ). This means that the totally-symmetric paired state, delocalized over all the five H_u orbitals or bands, is favored by the couplings to all modes. It also implies that, contrary to molecular JT, the couplings to all modes cooperate evenly to this superconducting state, and contribute additively to λ. Totally equivalent (even if at first sight apparently different) results were derived for the d=3 case (K₃C₆₀) in Refs. (Lannoo et al. 1991, Rice et al. 1991). We note, however, that the claim that orbital degeneracy enhances the superconducting λ through a factor d (Rice et al. 1991) is not really justified, as one must at the same time reduce the density of states from total to single-band, a factor 1/d smaller. We also note that

1= Σ_n W^τ_mn= ^{Σ_mn Wτ_mn}/_d= ^{Σ_mn Vτ_mnVτ_nm}/_d= ^{Σ_m [(Vτ)2]_mm}/_d = ^Tr[(Vτ)2]/_d ,

which shows the identity of the \tildeλ computed for superconductivity to the one obtained for transport in Eq. (M:lambdatransport).

If, in the opposite limit, the pairs did not interfere between different bands, the effective λ would be reduced by a factor d. Although we cannot identify a physical situation corresponding to this limit, we can assume that a general case will be intermediate between the limits (interfering/not-interfering pairs). For simplicity, we will stick here to the interfering limit.

In summary, is there any enhancement of superconductivity due to orbital degeneracy? We can still identify one possible source for that, namely Coulomb pseudopotential. In fact, we note that, although a large λ due to tunneling of the Cooper pairs between different orbitals/bands, can be seen as orbital degeneracy enhancing the effective λ to the highest eigenvalue of W, there is no corresponding enhancement of the repulsive Coulomb pseudo-potential µ^*, at least within the Migdal-Eliashberg theory. The reason is that the main contribution to the Coulomb pseudo-potential is a charge-charge repulsion which does not include tunneling processes between different bands, and being band-diagonal it does not get enhanced. In conclusion, in the above restricted sense, orbital degeneracy may in principle favor superconductivity.

Discussion

In order to connect with important solid state properties including transport and superconductivity we have formulated a theory of the Boltzmann relaxation time, and of the BCS-type pairing, suitable for an orbitally degenerate multiband case with Jahn Teller coupling. This confirms that the same parameter λ determines both transport and superconducting properties of the multiband degenerate solid. As we previously observed, not all the computed g_i (Table couplings:tab) are small parameters, and thus weak-coupling BCS theory is strictly not applicable for C₆₀ⁿ⁺. However, the overall λ is still moderate. Our calculation neglects couplings to acoustic phonons and librations, which in principle should also contribute to e-ph scattering. In addition, similarly to Lannoo et al.'s calculation (Lannoo et al. 1991), we assume that the dispersion of the HOMO band and of the optical phonons has a negligible effect on the integrated value of λ. (This assumption was tested and proved correct in Ref. (Antropov et al. 1993) for the couplings to the LUMO band.) With all these approximations, Eq. (M:lambda) should provide a semi-quantitative estimate of the total e-ph scattering.

Assuming conservatively a total average density of states of ~ 10/0.6 eV~17 states eV^-1 for the HOMO band (i.e. a single-band density of states N₁(0) = 1.7 states eV^-1 per band per spin), our calculated effective dimensionless λ₊ for hole superconductivity in C₆₀ is in conclusion about λ₊~0.47. The Coulomb pseudopotential µ^* is not available yet, but possibly in the same range of values as for negative C₆₀ [µ^*~ 0.2÷ 0.3 (Gunnarsson et al. 1995, Gunnarsson and Zwicknagl 1992)]. With this value of λ₊, weak coupling would predict T_c ~ 1.14 hω_D k_B^-1 exp[-1/(λ₊ -µ^*)]~ 40 K for µ^*= 0.2, and T_c~ 5 K for µ^*= 0.3 (assuming a typical phonon energy ω_D of about 1500 K). This seems of the correct order of magnitude, although somewhat on the low side, in comparison with T_c= 52 K found experimentally. However, it is difficult to justify weak coupling in this case.

The corresponding value λ_- which we obtain for electrons in the T_1u orbitals is, assuming the same bandwidth of 0.6 eV, thus again N₁(0) = 1.7 states eV^-1 per band per spin for the T_1u band, λ_-~ 0.33. The factor λ₊/λ_-=1.4 of holes relative to electrons is in qualitative agreement with a larger T_c of the former. Assuming the same typical phonon frequency, BCS would predict here T_c~ 0.8 K for µ^*= 0.2, and T_c~ 0 K for µ^*= 0.3. That is obviously way smaller than the observed superconducting T_c= 10 K found experimentally in the field emission transistor (FET) experiment, let alone the higher values found in the fullerides.

Coming to transport, the measured resistivities for holes are larger than those of electrons, and this also agrees with a larger λ value. Calculation of the T-linear high temperature resistivity ρ = ^{λ_tr T}/_4πωp² would however predict a moderately larger value for hole- than for electron-doped C₆₀, at least assuming (somewhat arbitrarily) the same plasma frequencies for the same carrier densities. Quantitatively, Batlogg's FET data (Schön et al. 2000) differ strongly from this expectation. They, first of all, indicate a nonlinear temperature dependence, closer to T²; secondly, they show values about 5 times larger for holes than for electrons. While there are second order processes (see Appendix) that would indeed yield a T² resistivity at low temperatures, we do not believe that they may explain the discrepancy here. Zettl and coworkers (Vareka and Zettl 1994) proposed that the apparent T² in the electron resistivity is an effect of thermal expansion, and showed that a linear T increase is recovered at constant volume, for negative doping.

Recently Goldoni et al. measured by EELS the plasma frequency in K₃C₆₀. They found it slowly decreasing with temperature, its width growing approximately quadratically with T. These data support the view that the T² resistivity is directly related with a T^-2 decrease of relaxation time, most likely linked with lattice expansion. It seems plausible that a similar physics could apply to holes too, in which case the predicted constant-pressure relaxation-time drop with temperature would also be non-linear, and quantitatively larger than the electron case. If, on the other hand, it became possible to obtain the constant-volume inverse relaxation time and resistivity, then, assuming the same plasma frequency ω_p, its increase should be linear with T with a slope 1.4 times larger than that of negatively-charged C₆₀. This conjecture must await experimental test.

Acknowledgements

We are indebted to B. Batlogg, O. Gunnarsson and G. Santoro for useful discussions. This work was partly supported by the European Union, contract ERBFMRXCT970155 (TMR Fulprop), and by MURST COFIN99.

Appendix: Quadratic Resistivity

Within Fermi's golden rule, the vibron contribution to the electrical resistivity is exponentially decreasing if temperature is much smaller than the vibron frequencies. However, the above result is not true any more if higher order corrections are taken into account. Indeed, at second order, the electron-vibron coupling generates an effective electron-electron interaction. Since the electrons involved lie on a shell of width T around the Fermi energy, the vibron-originated electron-electron interaction

Vel-el(ω)	=	- Στ,i,m gτ i2hωτ i/4 ωτ i2/ωτ i2 - ω2
	&1/VΣkpqΣµνγβΣσ,σ' Vτ(m)µν(α) Vτ(m)γβ(α) d+µ,σ,k+q d+γ,σ',p dβ,σ',p+q dν,σ,k ,

1/τ	~	(kB T)2/5h N1(εF)3 Σi,j,τ,τ',m,m' (gτ i2 hωτ i/4) (gτ' j2 hωτ' j/4)
	& { 5/4 [Tr(Vτ(m)Vτ'(m'))\right]2 - Tr( Vτ(m)Vτ'(m')Vτ(m)Vτ'(m')) } .

References

1. Altmann, S. L., and Herzig, P., 1994, Point-Group Theory Table (Oxford Science Pub., Oxford).
2. Antropov, V. P., Gunnarsson, O., and Liechtenstein, A. I., 1993, Phys. Rev. B 48, 7651.
3. Auerbach, A., Manini, N., and Tosatti, E., 1994, Phys. Rev. B 49, 12998.
4. Baroni, S., Giannozzi, P., and Testa, A., 1987, Phys. Rev. Lett. 58, 1861; Dal Corso, A., Pasquarello, A., and Baldereschi, A., 1997, Phys. Rev. B 56, R11369; Baroni, S., de Gironcoli, S., Dal Corso, A., and Giannozzi, P., 2001, Rev. Mod. Phys. 73, in print.
5. Brühwiler, P., Maxwell, A. J., Balzer, P., Andersson, S., Arvanitis, D., Karlsson, L., and M\aa rtensson, N., 1997, Chem. Phys. Lett. 279, 85.
6. Butler, P. H., Point Group Symmetry Applications (Plenum, New York).
7. Ceulemans, A., and Fowler, P. W., 1990, J. Chem. Phys. 93, 1221.
8. Ceulemans, A., Fowler, P. W., and Vos, I., 1994, J. Chem. Phys. 100, 5491.
9. Datars, W. R., and Ummat, P. K., 1995, Solid State Commun. 94, 649.
10. De Los Rios, P., and Manini, N., 1997, in Recent Advances in the Chemistry, Physics of Fullerenes, Related Materials: Volume 5; in Recent Advances in the Chemistry and Physics of Fullerenes and Related Materials: Volume 5, edited by Kadish, K. M., and Ruoff, R. S. (The Electrochemical Society, Pennington), p. 468.
11. De Los Rios, P., Manini, N., and Tosatti, E., 1996, Phys. Rev. B 54, 7157.
12. The C pseudopotential is described in Favot, F., and Dal Corso, A., 1999, Phys. Rev. B 60, 11427.
13. Fowler, P. W., and Ceulemans, A., 1985, Mol. Phys. 54, 767.
14. Giannozzi, P., and Baroni, S., 1994, J. Chem. Phys. 100, 8537.
15. Goldoni, A., Sangaletti, L., Parmigiani, F., Comelli, G., and Paolucci, G., 2001, Phys. Rev. Lett., in print.
16. Gunnarsson, O., Handschuh, H., Bechthold, P. S., Kessler, B., Ganteför, G., and Eberhardt, W., 1995, Phys. Rev. Lett. 74, 1875; Gunnarsson, O., 1995, Phys. Rev. B 51, 3493 (1995).
17. Gunnarsson, O., and Zwicknagl, G., 1992, Phys. Rev. Lett. 69, 957-960.
18. Kohanoff, J., Andreoni, W., and Parrinello, M., 1992, Phys. Rev. B 46, 4371; Kohanoff, J., Andreoni, W., and Parrinello, M., 1992, Chem. Phys. Lett. 198, 472); Feuston, B. P., Andreoni, W., Parrinello, M., and Clementi, E., 1991, Phys. Rev. B 44, 4056.
19. Lannoo, M., Baraff, G. A., and Schlüter, M., 1991, Phys. Rev. B 44, 12106.
20. Manini, N., and De Los Rios, P., 1998, J. Phys.: Condens. Matter 10, 8485.
21. Manini, N., and De Los Rios, P., 2000, Phys. Rev. B 62, 29.
22. Manini, N., and Tosatti, E., 1998, Phys. Rev. B 58, 782.
23. Manini, N., Tosatti, E., and Auerbach, A., 1994, Phys. Rev. B 49, 13008.
24. Moate, C. P., Dunn, J. L., Bates, C. A., and Liu, Y. M., 1997, J. Phys.: Condens. Matter 9, 6049.
25. Moate, C. P., O'Brien, M. C. M., Dunn, J. L., Bates, C. A., Liu, Y. M., and Polinger, V. Z., 1996, Phys. Rev. Lett. 77, 4362.
26. Negri, F., Orlandi, G., and Zerbetto, F., 1988, Chem. Phys. Lett. 144, 31.
27. Prassides, K., Dennis, T. J. S., Hare, J. P., Tomkinson, J., Kroto, H. W., Taylor, R., and Walton, D. R. M., 1991, Chem. Phys. Lett. 187, 455:tHeReEnDsArEfErEnCeHeRe; Prassides, K., Christides, C., Rosseinsky, M. J., Tomkinson, J., Murphy, D. W., and Haddon, R. C., 1992, Europhys. Lett. 19, 629:tHeReEnDsArEfErEnCeHeRe; Jishi, R. A., and Dresselhaus, M. S., 1992, Phys. Rev. B 45, 2597; Mitch, M. G., Chase, S. J., and Lannin, J. S., 1992, Phys. Rev. B 46, 3696.
28. Rice, M. J., Choi, H. Y., and Wang, Y. R., 1991, Phys. Rev. B 44, 10414.
29. Schön, J. H., Kloc, Ch., Haddon, R. C., and Batlogg, B., 2000, Science 288, 656; Schön, J. H., Kloc, Ch., and Batlogg, B., 2000, Nature 408, 549.
30. Vanderbilt, D., 1990, Phys. Rev. B 41, 7892.
31. Vareka, W. A., and Zettl, A., 1994, Phys. Rev. Lett. 72, 4121.
32. Varma, C. M., Zaanen, J., and Raghavachari, K., 1991, Science 254, 989.
33. Zhou, P., Wang, K. A., Wang, Y., Eklund, P. C., Dresselhaus, M. S., Dresselhaus, G., and Jishi, R. A., 1992, Phys. Rev. B 46, 2595.

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