## Abstract

An electron–phonon theory for a quasi-one-dimensional band of electrons forming a moving incommensurate charge-density wave (CDW) state at zero temperature is presented. A useful analytic expression for the energy gap as a function of electron–phonon coupling is found. The gap is quite a bit larger than the well-known weak-coupling result, even for modest coupling, where the mean-field approximation should still be valid. It is also shown that the form of the Hamiltonian implies that there is no spread at all in the wavevector of the CDW in the vicinity of 2*k*_{F}. Comparisons of theoretical and experimental values of the energy gap and the lattice displacement amplitude are made for NbSe_{3}.

## 1. Introduction

The theory of the formation of charge-density waves (CDW) and the attendant periodic lattice displacement (PLD) through electron–phonon (el–phon) interaction in quasi-one-dimensional conductors has a long history, beginning with Fröhlich (1954) and Peierls (1955). Over the past three decades, there has been a wealth of data on many CDW systems accumulated, with NbSe_{3} probably being the most widely investigated (see Ong & Monceau 1977; Fleming *et al*. 1978; Fleming & Grimes 1979; Grüner & Zettl 1985; Ségransan *et al*. 1986; Grüner 1988, 1994; Ross *et al*. 1990; Monceau 2006; Lemay *et al*. 1999; Ringland *et al*. 2000; Danneau *et al*. 2002; Requardt *et al*. 2002; Schäfer *et al*. 2003; Sinchenko & Monceau 2003; Perucchi *et al*. 2004; Matsuura *et al*. 2006; see Grüner for a review). There have also been theoretical works addressing various aspects of the problem (see Allender *et al*. 1974; Lee *et al*. 1974; Rice 1975; Rice *et al*. 1976, 1979; Boriack & Overhauser 1977; McMillan 1977; Lee & Rice 1979; Wilson 1979; Horovitz 1986; Bjelis 1987; Tucker *et al*. 1988; Eremko 1992; McKenzie & Wilkins 1992; Buker 1999).

One impediment to a quantitative comparison of theory and experiment for the Fröhlich sliding mode in CDW systems is the use of the weak-coupling approximation. At a large coupling, it is known that the Fröhlich state does not occur; instead, the electrons form an array of localized polarons. According to Le Daeron & Aubry (1983), for the Holstein model, the polaron state should occur for the el–phon coupling *λ*_{H} greater than a critical value. It is assumed that also in the present model polarons would form at a higher coupling. However, in the lower range, nothing prevents the existence of the Fröhlich mode. This is the regime of interest in the present paper. As it turns out, even for el–phon coupling as low as *λ*=0.3, the correct energy gap is approximately twice as large as the weak-coupling prediction.

The present theory deals with an incommensurate CDW at zero temperature, but with no restriction on the strength of the electron–phonon interaction. A single electron band is allowed to interact with *n* distinct phonon bands. Excluded from the Hamiltonian are el–el interactions outside those affecting the band structure, i.e. those arising from the CDW. The el–phon Hamiltonian *H* is written as the sum of an electron part and a phonon part, such that the sums of the expectation values of the parts differ negligibly from the expectation value of the original *H*, for the state assumed. This energy is minimized, giving the zero-temperature ground state. Details of the model, and separation into electron and phonon parts, are discussed in §2. The electron part is discussed in §3 and the phonons in §4. Solution of the complete system for the static case comprises §5. Moving CDWs are discussed in §6, and the application to NbSe_{3} makes up §7. In §8, there is a demonstration justifying the restriction to just a single pair of phonon modes per phonon band in the PLD. The limits of validity are discussed in §9, and the application to other CDW systems is outlined in §10.

## 2. The model

Consider the Fröhlich Hamiltonian(2.1)Here, the *c*'s are electron operators; the *b*'s are phonon operators; and the *g*_{kqr} are the electron–phonon 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 travelling CDW and its accompanying PLD. It is assumed in particular that(2.2)Here, *g*_{r} is *g*_{kqr} evaluated at *k*=*k*_{F}, the Fermi wavevector and *q*=2*k*_{F}; *g*_{c}, presently unknown, and to be determined, is the electron energy gap. *b*_{r} refers to *b*_{Qr} and refers to *b*_{−Qr}. *Q*=2*k*_{F} throughout. In real space, this corresponds to a phonon wave with an amplitude 〈*y*〉 at position *x* of(2.3)Thus, the velocity of the wave will be *ω*/*Q*.

The Thomas–Fermi approximation for the *g*_{r} is assumed, for which (see Ashcroft & Mermin 1976)(2.4)Here, *k*_{e} is the electric constant, 9×10^{9} N m^{2} C^{−2}; and *Q*_{e} is the charge on the electron. The Thomas–Fermi wavevector *k*_{0} can be taken to be *k*_{0}≈*k*_{F}.

The band structure is linearized about the Fermi level. States important for the formation of a CDW state, i.e. states giving non-zero , lie in a region of energy (−*E*_{c}, *E*_{c}) around the Fermi level. Possible residual parts of the Fermi surface (FS), giving rise to free carriers as discussed, e.g. by Rice *et al*. (1979), are not included. The band is divided into four regions 1, 2, 3 and 4 (figure 1).

A (hole) state at wavevector *k* in region 1 interacts with the (electron) state at wavevector *k*+*Q* in region 2. This is called type 1 pairing. Also state |*k*〉 in region 3 interacts with |*k*+*Q*〉 in region 4. This is called type 2 pairing. Thus, *c*_{2} is an electron destruction operator at wavevector *k* in region 2, and is an electron creation operator at wavevector *k*+*Q* in region 1. Summations over wavevectors *k* are replaced by integrals , where is the density of states at the Fermi level and *E*_{c} is the cut-off energy.

## 3. The electrons

The Hamiltonian for the electrons is(3.1)

The ground-state energy for a type 1 pair of states that an energy *ϵ* above and below the FS works out to be (see Buker 1999)(3.2)For type 2 pairing (*ω*→−*ω*), in equation (3.2). Integrating the above two energies from *ϵ*=0 to *E*_{c}, and multiplying by the density of states (/2 for each type), gives the total electronic energy(3.3)

## 4. The phonons

The form of the electron expectation values allows one to write(4.1)Here, the is still-to-be-determined phonon-coupling constants. The *b*_{r}-phonon part of the Hamiltonian then becomes(4.2)Solving the Schrödinger equation for a phonon state |*ϕ*_{r}(*t*)〉 obeying *H*_{r}, one finds the expectation value of *H*_{r} at time *t*, while |*ϕ*_{r}(*t*)〉 is in a solution state, to be(4.3)with the subscript 0 on 〈*b*^{†}*b*〉_{0} referring to the initial value of the expectation value, and where(4.4)Similar equations hold for the phonons. When minimizing the total energy, the term oscillating in time (the last term on the r.h.s. of equation (4.3)) can be disregarded, since it averages to zero. Writing the phonon parts of the energy in terms of *g*_{c}, so that only one minimization variable needs to be considered, is straightforward for the term , since is easily expressible in electron quantities from equation (4.1). However, also this term can be neglected, since the solution arrived at without its inclusion gives *g*_{c} as being of the same order of magnitude as *E*_{c}, for which case the term is only approximately 0.1 per cent as large as the first term on the r.h.s. of equation (4.3); see appendix A for details. Thus, the phonon energy reduces to approximately(4.5)From equation (2.2), minimizing the energy at fixed *g*_{c} means that the values of |〈*b*_{r}〉| and are all equal, and that the phonon state |*ϕ*_{r}〉_{0} is an eigenstate of *b*_{r}. One has(4.6)The phonon energy becomes(4.7)Within the Thomas–Fermi approximation, the usual dimensionless el–phon coupling is independent of the phonon band *r*. However, for present purposes, it is convenient to introduce an effective el–phon coupling(4.8)This expression generates quite a large variation over different CDW systems in the effective coupling, and hence in the energy gaps and lattice distortions. From the expression, one sees that if one phonon frequency were much higher than the rest, then *λ*_{eff} would reduce to approximately *λ*, while at the other extreme, if all *n* frequencies were about equal, then *λ*_{eff}≈*nλ*. The total phonon energy becomes(4.9)

## 5. Solution for the energy gap and lattice displacement amplitude

Specialized to the static case (*ω*=0), minimization of the total energy *E*_{el}+*E*_{phon} with respect to *g*_{c} leads directly to an expression for *λ*_{eff} in terms of (5.1)The more natural inverse graph, together with the weak-coupling graph , is plotted in figure 2. As is clear from the graph, the predicted CDW gap is quite a lot larger than the weak-coupling result, even for fairly small *λ*_{eff}. Thus, to evaluate the energy gap from first principles, once *λ* is known, it is only necessary to know the width of the electron band and the phonon energies at wavevector 2*k*_{F}.

Specialized to a one-dimensional tube of length *Na* and cross-sectional area *a*_{2}*a*_{3}, the Thomas–Fermi approximation for *λ* gives(5.2)where for convenience the characteristic electron velocity has been inserted. In calculating *λ* for a one-dimensional system, the factor introducing the greatest uncertainty is the effective cross-sectional area *a*_{2}*a*_{3} of the one-dimensional tube for the electron band. A band confined to a narrow tube will have a larger *λ* than one confined to a wider one, proportional to (1/area).

An evaluation of the lattice distortion is a little more complicated, involving also the nature of the phonons and the masses associated with them. However, a characteristic maximum lattice displacement *u*_{r} for the phonon mode can be found using(5.3)Here, *m*_{r} is the mass associated with the *r*th mode. For the present situation, from equation (4.6), this becomes(5.4)Specializing again to a one-dimensional tube, one obtains(5.5)Here, *a*_{0} is the Bohr radius (≈0.529×10^{−10} m) and *m*_{e} is the mass of the electron.

## 6. Moving CDWs

For the case of non-zero CDW speed (*ω*≠0), the general expression for *λ*_{eff} can be expanded in powers of to get(6.1)where and where *f*_{0}(*x*) comes from the zero-velocity equation (5.1). There is a resulting small increase in the energy gap compared with the static case, which for the representative point *x*=1 (i.e. *g*_{c}=*E*_{c}) amounts to a change Δ*g*_{c} of(6.2)This is a very small change for attainable CDW velocities, e.g. for *V*_{CDW}=1 m s^{−1}, one has of order 10^{−6} eV giving *v*^{2}≈10^{−10}, so that Δ*g*_{c}≈10^{−12}*g*_{c}.

There is a question of whether a CDW ever does slide at a constant velocity through the lattice. Even apart from impurity pinning, there is to be expected a jumpiness in the CDW motion arising from the voltage along the sample needed, for whatever reason, to have the CDW move. If the sample is a linear segment of length *L*, say, and the CDW is assumed to be sliding along, then due to the lumpiness of the CDW, there will be a fluctuating amount of electronic charge in each chain of the sample, oscillating by about ±*Q*_{e}. The applied voltage would act to accelerate this net charge, causing a jerky motion. If the CDW speed was greater, the effect would be less, since the voltage would have less time to act on the net charge on the chain. To estimate the size of this effect, consider the linear segment classically with a charge *Q*_{e} and mass *M*=*Nm*_{e} moving at speed *V*_{CDW} in an electric field . The field will be allowed to act for a time . Then the fractional change in the velocity of the CDW is(6.3)For typical values *L*≈1 cm, so *N*≈10^{7}; ≈1 V cm^{−1}; and *λ*_{CDW}≈10^{−9} m, the expression becomes(6.4)Thus, at *V*_{CDW}≈10^{−3} m s^{−1}, there would be fairly large fluctuations, while at the higher speed *V*_{CDW}≈1 m s^{−1}, the fluctuations due to this cause would be relatively small. A smoothly sliding state is also more easily obtained with a longer sample. This cause of velocity fluctuations should be eliminated if the sample was ring shaped, as in some recent resistance experiments by Matsuura *et al*. (2006) on NbSe_{3}.

## 7. Application to NbSe_{3}

The linearized band approximation inevitably brings in some uncertainty when calculating the energy gap *g*_{c}. However, good bounds can be placed on *g*_{c} by appropriate choices of an effective Fermi velocity *v*_{F} and band cut-off *E*_{c}. A minimum for *g*_{c} can be obtained by taking *v*_{F} to be the value actually measured, and *E*_{c} to be just . For this choice of *E*_{c}, all of the electron states are closer to the FS than the ones supposed by the linear approximation, making the actual *g*_{c} higher than this estimate. Also a maximum bound for *g*_{c} can be arrived at by taking *v*_{F} equal to one-half of its measured value, and also *E*_{c} equal to one-half of the previous value, i.e. . This corresponds to a linear band going directly from the FS to the bottom of the band, for which with a uniform band mass, the actual electron states will be further from the FS than those of the linear approximation.

For NbSe_{3}, *k*_{F}*a*≈*π*/4; *v*_{F}≈1.4×10^{−5} m s^{−1}; *a*=3.48×10^{−10} m (see Grüner 1988); the full tube cross-sectional area for the unit cell is (*a*_{2}*a*_{3})_{full}=1.48×10^{−18} m^{2}. There are six chains per unit cell, so the effective area can be estimated at approximately (1/6)(*a*_{2}, *a*_{3})_{full}.

The above values in equation (5.2) give *λ*=0.79. Modelling the system as having one phonon frequency much higher than the rest gives *λ*_{eff}≈*λ*. From the graph in figure 2, this leads to energy gap values *g*_{c,min}=0.13 eV and *g*_{c,max}=0.15 eV. These close bounds reflect the approximation made in linearizing the band. Values outside the range could occur through re-estimation of the effective cross-sectional area of the band. Sinchenko & Monceau (2003) found the gap experimentally to be approximately 130 meV, so the agreement is good.

The lattice distortions depend more sensitively on the phonon characteristics than the energy gap does. Once these characteristics are known, then equation (5.5) can be used to calculate the distortions. As a preliminary estimate, for the acoustic mode at 7 meV, and associating the mass of a niobium atom (1.54×10^{−25} kg) with it, the maximum displacement would calculate out to be *u*≈0.29×10^{−10} m. The existence of the other modes greatly decreases this value.

## 8. Why there is no spread in *Q*

How sharply peaked at 2*k*_{F} is the CDW wavevector? One could imagine some spread in *Q* in the immediate vicinity of 2*k*_{F}. But as is now demonstrated, the focus on 2*k*_{F} is right sharp, with no neighbouring spread at all, beyond the discrete special wavevector. First, consider a pair of electron states (1, 2) at an energy *ϵ* above and below the FS coupled by the el–phon interaction. Since states close to the FS are the most important, work right at the FS and take *ϵ*≈0 (compared with the coupling). Then an effective Hamiltonian *H*_{12} for a single electron might be(8.1)Take *g* as real and positive. The ground-state energy for *H*_{12} is *E*_{gnd}=−*g*. Next, add in a pair of *k* states (5, 6) adjacent to the original (1, 2) pair (i.e. the very next to the discrete momentum eigenstates) (figure 3). Consider occupation of the four states by two electrons. Allow pairing at wavevectors slightly different from *Q*, i.e. also of states 1 and 6 (wavevector slightly less than *Q*); 5 and 2 (wavevector slightly greater than *Q*); and 5 and 6 (coupled again at wavevector *Q*). Allowing the additional pairings gives a Hamiltonian(8.2)The two-electron base states might be |12〉, |15〉, |16〉, |25〉 and |26〉. The determinant *D*=det(*H*_{1256}−*μI*) has the characteristic equation(8.3)for which the lowest root is *μ*=0. Thus the ground-state energy of *H*_{1256} is *E*_{gnd}=0, higher than (double) that of *H*_{12}. So having a spread in the wavevector *Q* of the CDW would only serve to increase the energy of the state. Therefore, the restriction of *Q* to being a single pair ±2*k*_{F} is not an approximation, rather it is a prediction to the el–phon Hamiltonian.

## 9. Limits of validity

The main approximation in the present description is the use of the Thomas–Fermi wavevector for screening of the el–phon interaction. This keeps the problem tractable, while incorporating at least a part of the many-body complexity into the theory, hopefully allowing transparent comparison of some facets of CDW systems.

Fortuitously, the uncertainty arising from linearizing of the bands can be definitely bounded, as described in the application to NbSe_{3}. In this typical case, the uncertainty in the energy gap due to linearization amounts to approximately ±7 per cent.

## 10. Other CDW systems

The present work is directed at the case of a single one-dimensional electron band. A similar framework could be used for other situations, giving somewhat different results for the equation relating the electron energy gap to the effective el–phon coupling. For example, for the case of several bands, the electron energy (equation (3.3)) would have a dependence on each of the band cut-offs. K-blue bronze falls into this category. In the case of nesting in two and three dimensions, a workable version of the present theory should also be able to be set up. The main electronic input would be the electron velocities *v*_{k} over the FS, replacing the band cut-off *E*_{c} used here.

## 11. Conclusion

A useful graph of the CDW energy gap versus the el–phon coupling *λ* has been derived. The theory brings a quantitative comparison of theory and experiment for the energy gap and lattice distortions of some CDW systems within reach.

## Relative sizes of the phonon energy terms

From equation (4.3), the non-oscillatory parts of the phonon Hamiltonian expectation values can be written as(A1)where and the adornment(A2)It will now be established that the adornment *A*_{r} is much smaller than *E*_{r}. First, consider *A*_{r}. From equation (4.1), one has(A3)Using the electron Hamiltonian *H*_{el}, one finds for the states |1〉, |3〉 at an energy *ϵ* above the FS and states |2〉, |4〉 at an energy *ϵ* below the FS that(A4)where(A5)Replacing the summation over wavevectors by , one finds(A6)where(A7)and where in turn and . Note that for *ω*=0 and *g*_{c}=*E*_{c}, one has *I*=0.88, with the log function tame, so that *I* can safely be taken to be about unity. Thus, with(A8)where volume=*Naa*_{2}*a*_{3} and , one has finally(A9)Turning now to the other term, *E*_{r}, one has(A10)which becomes(A11)Now and *ϵ*_{F} are roughly in the 1 eV region and , so that the factor differentiating *A*_{r} and *E*_{r} is just (1/40)(1/4*π*)^{2}, leaving *A*_{r} less than 0.1 per cent as large as *E*_{r}.

## Footnotes

- Received November 26, 2008.
- Accepted March 11, 2009.

- © 2009 The Royal Society