## Abstract

A theory for a quasi-one-dimensional incommensurate charge-density wave state arising from electron–phonon (el–phon) interaction connecting electron states in two different bands is presented. An expression for the fundamental component of the energy gap as a function of the effective el–phon coupling, valid for all coupling strengths, has been found. For a single band, the expression simplifies to a reciprocal hyperbolic-sine dependence on the reciprocal effective coupling. The effective coupling, although simply related to the *n* assumed phonon-band frequencies, is not generally expressible as a sum of independent functions of these frequencies. The theory is applied to tetrathiofulvalinium–tetracyanoquinodimethane and to potassium blue bronze.

## 1. Introduction

Since the first description of charge-density waves (CDWs) in solids in the 1950s (Fröhlich 1954; Peierls 1955), there have been numerous advances on the theory of their formation, including those described in the following works: Overhauser (1971), Lee *et al.* (1973, 1974), Allender *et al.* (1974), McMillan (1975, 1976), Rice (1975, 1976), Rice *et al.* (1976, 1979), Wilson (1979), Mertsching & Fischbeck (1981), Monceau (1985), Grüner & Zettl (1985), Horovitz (1986), Bjelis (1987), Grüner (1988, 1994), Tucker *et al.* (1988), Degiorgi *et al.* (1991, 1995), Eremko (1992), McKenzie & Wilkins (1992), McKenzie (1995), Mozos *et al.* (2002), Sing *et al.* (2003) and Perroni *et al.* (2004).

There are a large number of CDW compounds. Accompanying the electron wave is a periodic lattice displacement (PLD) of the background ions. Given the band structure above the transition temperature, it is a fair game to predict the ensuing CDW properties of the ground state. The goal is to make quantitative predictions for some real materials.

The Hamiltonian is broken into electron and phonon parts, which are separately solved for their ground-state energies, in terms of a single parameter, the electron energy gap *g*_{c}. The total energy is then minimized with respect to this parameter, which yields a fairly simple analytic expression for the gap in terms of an effective electron–phonon (el–phon) coupling *λ*_{eff}.

In a previous treatment (Buker 2009), the CDW–PLD system was solved in the simple context of a single electron band. The present work extends the theory to the common case where the CDW arises from two bands.

Derivation of the two-band gap equation is discussed in §2. Specialization to the single-band case is made in §3. Possible refinements are discussed in §4, and a sketch of how electron–electron (el–el) interactions would enter is made in §5. The application to tetrathiofulvalinium–tetracyanoquinodimethane (TTF–TCNQ) and K-blue bronze comprise §§6 and 7.

## 2. The two-band gap equation

Consider the Fröhlich Hamiltonian2.1Here the *c*’s are electron operators, the *b*’s are phonon operators and the *g*_{kqr}’s are the el–phon coupling constants; *r* labels the phonon bands, running from 1 to *n*. The sum over *k* is meant to imply also a sum over electron spin. Attention is narrowed to a state with a CDW and its accompanying PLD. It is assumed in particular that2.2

Let *k*_{F1} and *k*_{F2} be the Fermi wavevectors of the two bands, and let *Q*=*k*_{F1}+*k*_{F2} be the wavevector of the CDW. Then, *g*_{r} is *g*_{kqr} evaluated at *k*=*k*_{F1} and *q*=*Q*; *g*_{c}, presently unknown, and to be determined, is the fundamental component of the electron energy gap. *b*_{r} refers to *b*_{Qr} and refers to *b*_{−Qr}. In real space, this corresponds to a phonon wave with an amplitude 〈*y*〉 at position *x* of2.3The Thomas–Fermi approximation for the *g*_{r} is assumed, for which (see Ashcroft & Mermin 1976)2.4Here, *k*_{e} is the electric constant, 9×10^{9} Nm^{2} C^{−2}, and *Q*_{e} is the charge on the electron. *k*_{0} will be called the screening wavevector.

The band structure is linearized about the Fermi level. Consider the two bands shown in figure 1. Choose one of the bands to be band 1. States separated by the fixed wavevector *Q* are paired. State |1〉 in band 1 at an energy *ϵ* above the Fermi level is paired with state |2〉 in band 2 at an energy *pϵ* below the Fermi level. The parameter *p* is the ratio of the dispersion at the Fermi level of band 1 to that of band 2. Similarly, state |3〉 is paired with state |4〉. *E*_{c1} is the energy cutoff for the pairing of |1〉 and |2〉; and *E*_{c2} is the cutoff for the pairing of |3〉 and |4〉. Summations over wavevectors in the Hamiltonian are replaced by integrals , where 𝒟 is the density of states at the Fermi level and *E*_{c} is the cutoff.

The part of the Hamiltonian concerning states |1〉,|2〉 is2.5Diagonalizing, one finds the ground-state energy *E*_{12} to be2.6Integrating *E*_{12} over the energies *ε* yields the total electronic energy for this part of the band as2.7where2.8and2.9The pairings at of the |1〉,|2〉 pairing may be accommodated by taking 𝒟_{1} to be the full density of states of band 1. Similarly, the electron energy *E*_{el2} corresponding to pairs |3〉,|4〉 is the same expression as *E*_{el1}, except that the energy cutoff is now *E*_{c2}.

Just as for the single-band case (see Buker 2009), the phonon energy is2.10where the effective coupling *λ*_{eff} is given by2.11and where the dimensionless el–phon coupling *λ* is given by2.12

Minimization of the total energy, *E*_{el}+*E*_{phon}, with respect to *g*_{c} leads now directly to the basic two-band gap equation2.13where *x*_{1}=(2*g*_{c}/(1+*p*))(1/*E*_{c1}) and *x*_{2}=(2*g*_{c}/(1+*p*))(1/*E*_{c2}).

Note that the proper coupling strength entity to use in the gap equation is *λ*_{eff}=*f*_{phon}*λ*, where the phonon factor *f*_{phon} is given by2.14*λ*_{eff} is not generally expressible as a sum of independent functions of the phonon frequencies.

The choice of which band to call band 1 leads to some variation in the parameters used above. To clarify this, imagine that two experimenters, call them Elliott and Finnegann, when faced with the same two-band system, happen to choose different bands for their band 1. Then, when Elliott measures *p*,𝒟_{1},*E*_{c1} and *E*_{c2}, Finnegann measures and . The two sets of quantities are related by and .

Also, *λ* and *λ*_{eff} are experimenter-dependent quantities. If Elliott refers to *λ*, then Finnegann will be referring to *λ*/*p*. To make each side of the gap equation invariant under band interchange, one can multiply the whole equation by the factor (1+*p*).

## 3. Specialization to a single band

Setting *p*=1, and noting that *λ*(1 band)=2*λ*(2 bands), then for a single band, the gap equation can be simplified to3.1where *x*=*g*_{c}/*E*_{c}. This can be solved for the gap to give3.2

This expression for the gap replaces the well-known weak-coupling result3.3

Considering the case of a single phonon band, so that *λ*_{eff}=*λ*, then the new expression with its sinh dependence instead of exponential dependence on 1/*λ*, results in a 2 per cent increase in the gap at *λ*=0.5 and a 16 per cent increase at *λ*=1.0.

The phonon factor, changing *λ* to *λ*_{eff}, carries vital information on the phonon modes to the gap equation.

## 4. Further refinements

The present method allows the inclusion of other interactions. If there are energies other than those so far considered, labelled as ‘Else’, then the total energy will be4.1Minimizing with respect to the gap, *g*_{c} leads to the general gap equation4.2

The major influences so far unaccounted for are the second harmonic interaction and the el–el interaction. Each of these interactions turns out to yield quite a large correction with use of the above equation. However, the additional terms depend relatively sensitively on the energies of states far away from the Fermi surface (FS). In the present model, the bands have been linearized about the Fermi level, so that these effects cannot here be accurately evaluated. For transparency, the form the el–el correction takes is briefly given in the next section.

## 5. Electron–electron interactions

The el–el interaction Hamiltonian may be written as (Kittel 1987)5.1Here, the *c*_{k}^{′} operators are plane-wave electron operators. Thus, the direct Coulomb energy associated with the fundamental component of the CDW is5.2

This time the summation over *k* is meant to also include spin. As an approximation, the *c*′ operators can be taken to be electron operators for the Bloch states of the band. Bloch-state corrections could be entertained once these are known more accurately. The exchange energy is smaller, vanishing in the infinite-volume limit.

## 6. Application to tetrathiofulvalinium–tetracyanoquinodimethane

For a comparison of theory and experiment, it is convenient to take the measured value of the gap and use the gap equation to find the expected coupling *λ*_{eff}. The comparison is then between this value and *f*_{phon}*λ*_{bare}, where in the Thomas–Fermi approximation with *k*_{0}=*Q*/2, from equation (2.12)6.1

For TTF–TCNQ, the gap determined experimentally is 2*g*_{c}≈40 meV, according to Sing *et al.* (2003). From the density-functional band structure of Sing *et al.* (2003), I find *E*_{c1}=0.36 eV, *E*_{c2}=0.3 eV and *p*=1.4. This yields *x*_{1}=0.046 and *x*_{2}=0.056. With these values, the basic gap equation (2.13) predicts *λ*_{eff}=0.16, referring to the shallow band.

For the cross-sectional area *a*_{2}*a*_{3} per chain, Allender *et al.* (1974), give 114×10^{−20} m^{2}. From Sing *et al.* (2003), I find *v*_{F1}=1.7×10^{5} m s^{−1}, referring to the shallow TCNQ band. With *Q*=0.485×10^{10} m^{−1}, one has, from equation (6.1), *λ*_{bare}=0.384. The phonon factor *f*_{phon} is at least unity, so that the coupling expected from the band structure is quite a lot greater than that found using the gap equation. However, as mentioned in §4, the second-harmonic and el–el corrections are large, and require, for their accuracy, energy bands known more accurately than a linear approximation at the FS.

## 7. Application to K-blue bronze

From a scanning tunnelling microscope experiment (Tanaka *et al.* 1993), the CDW gap is 2*g*_{c}=140±20 meV. From the density functional band structure of Mozos *et al.* (2002), I find *E*_{c1}=0.23 eV, *E*_{c2}=0.3 eV and the ratio *p* of the two dispersions at the Fermi level to be *p*=3.7. The gap equation (2.13) then yields *λ*_{eff}=0.41, referring to the shallow band. For comparison to experiments, where both bands are accounted for in the density of states, i.e. multiplying by the factor ((1+*p*)/*p*), the full el–phon coupling is *λ*_{eff,2 bands}=0.52.

Degiorgi *et al.* (1995) give the experimental coupling as *λ*_{exptl}=0.5.

The apparent agreement of theoretical and experimental values for the gap is, at this stage, somewhat illusory. For there is still the effect of el–el interactions to consider. Also the phonon factor, *f*_{phon} should be accounted for. But this is set aside until more information on the phonon modes is available.

## 8. Conclusion

An equation has been found, for quasi-one-dimensional systems when two electron bands are connected by the el–phon interaction, relating the CDW energy gap to band-structure data above the transition temperature. The relation is valid for all coupling strengths. When specialized to a single band, the theory yields a simple expression for the energy gap.

- Received February 10, 2010.
- Accepted June 1, 2010.

- This journal is © 2010 The Royal Society