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«H.J. Schulz Laboratoire de Physique des Solides∗, Universit´ Paris-Sud, 91405 Orsay, France e arXiv:cond-mat/9301007v2 17 Mar 1993 Abstract A ...»

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Wigner Crystal in One Dimension

H.J. Schulz

Laboratoire de Physique des Solides∗, Universit´ Paris-Sud, 91405 Orsay, France


arXiv:cond-mat/9301007v2 17 Mar 1993


A one–dimensional gas of electrons interacting with long–range Coulomb

forces (V (r) ≈ 1/r) is investigated. The excitation spectrum consists of separate collective charge and spin modes, with the charge excitation energies

in agreement with RPA calculations. For arbitrarily weak Coulomb repulsion density correlations at wavevector 4kF decay extremely slowly and are best described as those of a one–dimensional Wigner crystal. Pinning of the Wigner crystal then leads to the nonlinear transport properties characteristic of CDW. The results allow a consistent interpretation of the plasmon and spin excitations observed in one–dimensional semiconductor structures, and suggest an interpretation of some of the observed features in terms of “spinons”.

A possible explanation for nonlinear transport phenomena is given.

71.45.-d, 72.15.Nj, 71.28.+d Typeset using REVTEX The properties of models of one–dimensional interacting electrons have been studied in great detail. Examples are the so–called “g–ology” model of fermions moving in a continuum [1,2], or the one–dimensional Hubbard model [3,1,4]. In these models, one usually assumes short–ranged (effective) electron–electron interactions. The so–called “Luttinger liquid” [5] behaviour in this type of models is characterized by separation between spin and charge degrees of freedom and by power–law correlation functions, with interaction–dependent exponents. Short–range interactions are a reasonable assumption for applications e.g. to quasi–one–dimensional conductors, where screeening between adjacent chains leads to effectively short range interactions within one chain [6]. However, the situation can be quite different if an isolated system of electrons moving in one dimension is considered. There then is no interchain screening, and the true long–range character of the Coulomb forces (V (r) = e2 /r) needs to be taken into account. This appears to be the case, e.g., in certain one–dimensional semiconductor structures, where the effects of one–dimensional Coulomb forces have indeed been observed [7].

The purpose of the present paper is to investigate the effects of the long–range Coulomb interaction in a one–dimensional model, using the bosonization method [1,2]. This allows in particular a rather straightforward and asymptotically (for low energies and wavenumbers) exact description of excitation spectra and correlation functions. The main conclusion is that the long–range force, even if it is very weak, leads to a state characterized by quasi– long–range order much closer to a one–dimensional Wigner crystal [8] than to an electron liquid. The calculations presented here provide a rather simple microscopic description of the Wigner crystal, a problem that in higher dimesnions has been difficult to treat by many–body techniques.

I will start by considering the particular case of one–dimensional electrons with a linear

–  –  –

Here a† (b† ) creates a right– (left–) moving electron with momentum k and spin projection k,s k,s s, vF is the Fermi velocity. In the interaction term ρq = ρa,q + ρb,q is the Fourier component of the total particle density, and V (q) is the Fourier transform of the interaction potential.

In strictly one dimension, a 1/r Coulomb interaction does not have a Fourier transform because of the divergence for r → 0, however, in a system of finite transverse dimension √ d, the singularity is cut off a r ≈ d [9]. Using the approximate form V (r) = e2 / r 2 + d2, one has V (q) = 2e2 K0 (qd). Finally, the backward scattering term Hbs describes processes where particle go from the right– to the left–moving branch and vice versa. This involves a nonsingular interaction matrix element at q ≈ 2kF, called g1.

The linear energy–momentum relation makes the model (1) exactly solvable using standard bosonization methods. Moreover, at least for weak Coulomb interactions, when states near the Fermi energy play the major role, linearizing the spectrum is not expected to change the physics drastically, and one therefore expects that the model (1) correctly represents the low–energy physics even for more realistic bandstructures. The model can be easily solved introducing the phase fields

–  –  –

where ν = ρ, σ, and ρr (p) (σr (p)) are the usual charge (spin) density operators for right– (r = +) and left–(r = −) going fermions. The Hamiltonian then decomposes into commuting parts for the charge and spin degrees of freedom. The charge part takes the simple quadratic form

–  –  –

˜ where V (q) = V (q)/(πvF ). The long–wavelength form, ωρ (q) ≈ |q 2 ln q|1/2, agrees with RPA calculations [9,10], however, the effect of g1, which is a short–range exchange contribution, are usually neglected in those calculations.

The spin part of the Hamiltonian does not involve the long–range part of the interaction and only depends on the backward scattering amplitude g1. For repulsive interaction, the long–wavelength spin excitations then are described by a Hamiltonian similar to the first term in (3), giving rise to collective spin oscillations with ωσ (q) = uσ |q|, and spin wave ˜2 velocity uσ = vF 1 − g1. Together with the charge oscillations (4), these excitations are the complete spectrum of the model.

The bosonization method makes the calculation of correlation functions rather straightforward. Here, the charge–charge correlations are of particular interest. Using the expression

–  –  –

where A1,2 are interaction dependent constants, and only the most slowly decaying Fourier components are exhibited. The most interesting point here is the extremely slow decay (much slower than any power law!) of the 4kF component, showing an incipient charge density wave at wavevector 4kF (instead of the usual 2kF of the Peierls instability). This slow decay should be compared with the case of short–range interactions, where the 2kF and 4kF components deacay as with the power laws x−1−Kρ and x−4Kρ, respectively, with an interaction–dependent constant Kρ [1,2,4]. The 4kF oscillation period is exactly the average interparticle spacing, i.e. the structure is that expected for a one–dimensional Wigner crystal. Of course, because of the one–dimensional nature of the model, there is no true long–range order, however, the extremely slow decay of the 4kF oscillation would produce strong quasi–Bragg peaks in a scattering experiment. It is worthwhile to point out that this 4kF contribution arises even if the Coulomb interaction is extremely weak and depends only on the long–range character of the interaction. On the other hand, any 2kF scattering is considerably weaker, due to the 1/x prefactor in (7) which has its origin in the contribution of spin fluctuations.

Other correlation functions are easily obtained. For example, the spin–spin correlations are

–  –  –

where there is no 4kF component. On the other hand, correlation functions that involve operators changing the total number of particles (e.g. the single particle Green’s function) decay like exp[−cst.(ln x)−3/2 ], i.e. faster than any power law. This in particular means that the momentum distribution function nk and all its derivatives are continuous at kF, and there is only an essential singularity at kF. The calculations are also straightforwardly generalized to finite frequency and temperature [1,2], however the rather complicated formulae are not of immediate interest here.

The presence of metallic screening changes the above behaviour: a finite screening length ξs would lead to a saturation of V (q) for q → 0 at 2e2 ln(ξs /d). One than would have, for x ξs, power–law decay of the type discussed above for short–range interactions, with √ Kρ ≈ 1/ ln ξs. On the other hand, if the interaction potential decays more slowly than 1/r (a rather hypothetical case), the integral (6) remains finite for x → ∞, and therefore there then is real long–range order of the Wigner crystal type.

It is instructive to compare the above result (7), obtained in the limit of weak Coulomb interactions, with the case of strong repulsion (or, equivalently, heavy particles). The conguration of minimum potential energy is one of a chain of equidistant particles with lattice constant a, and quantum effects are expected to lead only to small oscillations in the distances between particles. The Hamiltoniam then is

–  –  –

Noticing that kF = π/(2a), one observes that the results (7) and (10) are (for g1 = 0) identical as far as the long–distance asymptotics are concerned, including the constants in the exponentials. Eq. (7) was obtained in the weak interaction limit, whereas (10) applies for strong Coulomb forces. Similarly, the small–q limit of the charge excitation energies is identical. We thus are lead to the rather remarkable conclusion that the long–distance behaviour of correlation functions is independent of the strength of the Coulomb repulsion, provided the interaction is truly long–ranged.

In recent experiments, one–dimensional structures with two partially filled subbands have been investigated [7]. If only the long–range part of the Coulomb interaction is considered, the appropriate generalization of the model (3) to that case is described by the Hamiltonian

–  –  –

where vi is the Fermi velocity of band i, and φρ,i, Πρ,i are the charge fields of band i. In the long–wavelength limit the charge oscillation eigenmodes have energies

–  –  –

part of (10) (q = 4(kF,0 + kF,1)) is again the one that corresponds to Wigner–crystal type ordering, e.g. the electrons order approximately equidistantly. In fact, this type of ordering is determined only by the ω+ mode, whereas all other Fourier componets contain contributions from the ω− mode, which lead to power law decay.

To compare the present results with experiment [7] one can first notice that, provided that 2kF d 1 and including the background dielectric screening, one has g1 0.2, and ˜ consequently to within a few percent uσ = vF, i.e. the triplet spin mode (“SDE”) is expected at vF q, as experimentally observed. Further, in the experimentally accessible range q

0.2kF the plasmon energies found here are indistinguishable from RPA results, and thus the present results provide a good fit to the experimental plasmon dispersion.

More difficult to explain is the extra feature which has been interpreted as an electron– hole continuum (“SPE”) [7]: in fact in the present model with its linear electron dispersion relation, there is no such continuum (and it would not exist in an RPA calculation either).

However, the model offers an alternate possibility: together with the triplet spin mode, there is also a singlet mode [11]. The existence of the singlet mode is a consequence of spin– charge separation in one–dimensional fermion systems, and in particular it is degenerate with the triplet mode. This mode can be found e.g. in energy density correlation functions (as opposed to the spin mode, which appears as a pole of the spin density correlation function), and therefore also is expected to be seen in the polarized Raman spectra. This interpretation requires the SDE and SPE features to appear at the same energy, which seems to be consistent with the results published in ref. [7]. It is noteworhty that, if correct, this interpretation would mean that these results constitute the first direct spectroscopic evidence for the existence of individual spin–1/2 objects (“spinons”): the existence of degenerate triplet and singlet mode implies that they are both build up from non–interacting spin–1/2 excitations.

One might argue that the existence of the particle–hole continuum is due to effects of band curvature, which is neglected in the present model. RPA calculations including band curvature certainly predict both a plasmon and a particle–hole continuum [9,10]. However, within RPA the total spectral weight for the continuum is about two orders of magnitude smaller than that of the plasmon, whereas in ref. [7] plasmon and SPE have comparable weight. Moreover, in exactly solved one–dimensional models like the Hubbard model [3], one finds both a plasmon–like collective mode and the singlet mode discussed above, but no separate particle–hole continuum [12]. There thus seems to be little theoretical evidence in favor of an interpretation of the SPE feature in terms of a particle–hole continuum.

The nearly long–range Wigner crystal type order should have important consequences for transport properties: in fact, in the presence of disorder, a classical charge density wave (which has real long–range order at T = 0) becomes disordered [13], with a “pinning length” describing the decay of spatial correlations given by

–  –  –

where n is the density of impurities, and V the Fourier component of the impurity potential at the wavevector of the CDW (4kF in our case). Inclusion of quantum effects in systems with short range interaction only leads to corrections to the exponent 1/3 in (11) [14].

Following the same arguments, I expect (11) to be valid for the Coulomb system too (up to logarithmic corrections), i.e. as far as low–frequency phenomena are concerned, the system of electrons interacting with Coulomb forces behaves like a classical charge density wave.

In particular, all the unusual dynamical properties associated with nonlinear transport in CDW systems should also occur in the one–dimensional electron system.

At finite temperature, thermal agitation can become sufficiently strong to depin a CDW.

In the present case, this is expected to happen when the thermal correlation length, ξT, in the absence of impurities, given by ωρ (1/ξT ) ≈ T, becomes shorter than ξpin.

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