Peierls-Fröhlich model

In the previous section, we have seen that the electron gas is unstable under the perturbation of a periodic potential. Here, we introduce a microscopic model with an explicit electron-phonon

coupling that mimics the interaction with a lattice. Hence, this extended model also covers the structural change of the lattice that is associated with the Peierls transition.

Electron-phonon coupling has been studied extensively, both experimentally and theoretically, due to its ubiquity in condensed matter physics [60–62], in particular in the context of superconduc-tivity. In the discussion of charge-density waves, a frequently encountered model is the so-called FröhlichHamiltonian [33]:

H =H₀+H_el-ph. (1.7)

It consists of two parts, namely an unperturbed partH₀describing electron quasiparticles and lattice vibrations (phonons), and the interaction HamiltonianH_el-ph, considering a short-ranged electron-phonon coupling. For the derivation, we refer to the Refs. [26,55,63]. In the formalism of second quantization [64], the unperturbed part is given by

H₀=Õ

k

_kc_k⁺c_k +Õ

k

~ωQb⁺_Qb_Q, (1.8)

where the first term describes the electron quasiparticle gas by a sum of creation and annihilation operatorsc_k andc_k⁺, respectively, with dispersionk and wave vectork. For simplicity, we omit the spin and consider a single band. The second term covers the quantized harmonic vibrations of the lattice with the corresponding bosonic ladder operatorsb⁺_Qandb_Q of a phonon mode with energy ω_Qand wave vectorQ[65]. For convenience, only a single longitudinally polarized acoustic phonon branch is considered. With the given notation, the interaction Hamiltonian for lowest-order coupling (Born approximation and small displacements of atoms [66]) between electrons and phonons reads

H_el-ph=Õ where g_k,k⁰ is the electron-phonon coupling constant that describes the probability amplitude for scattering an electron with momentumkto a state with momentumk⁰= k±Qunder the simultaneous absorption (emission) of a phonon with momentum Q(−Q). The quantities M, N andV_k−k⁰ are the atomic mass, the atom density and the single atom potential in Fourier space, respectively.

The scattering can be visualized diagrammatically (see Fig. 1.3) and corresponds to the terms b⁺_−Qc_k+Qc_k andb_Qc⁺_k+Qc_k in the interaction Hamiltonian. The lattice displacement, in terms of the bosonic ladder operators, is given by

u(x)=Õ

Q

s

~

2N Mω_Q(b_Q+b⁺_−Q)e^iQx. (1.11)

Electron Phonon

Phonon emission Phonon absorption

Figure 1.3: First order scattering processes between electrons and phonons in the Peierls-Fröhlich model.

Adapted from Ref. [67].

1.1.3.1 Kohn Anomaly

Based on the Fröhlich Hamiltonian, we investigate the impact of electron-phonon interaction on the phonon dispersion relation. As an outcome, we will find a renormalized phonon dispersion at Q =2kF calledKohn anomaly. In three dimensions, the change in energy correction is relatively small and can be calculated via second-order perturbation theory [54], whereas in quasi-one-dimensional systems the correction to the energy can be significant resulting even in a vanishing phonon energy (giant Kohn anomaly) and a structural phase transition with afrozen-inCDW-coupled mode (see Fig. 1.4a). In the latter case, instead of low-order perturbation theory, a preferable theoretical description is a mean-field theory that treats the phonon system in the presence of a mean electronic density.

Following the derivation of the Kohn anomaly in Ref. [26], the essential idea is to determine the temporal evolution for the periodic lattice distortionu(x) that leads to a simple equation of an harmonic oscillator. This can be done by evaluating the relevant commutators of the phonon operatorsb_Qandb⁺_−Q, and leads to the following equation

d dt

₂

(b_Q+b⁺_Q)=− ωQ+ 2g²ω_Q

M~ χ(Q,T)

!

(b_Q+b⁺_Q), (1.12) wheregis again the electron-phonon coupling taken to be constant and independent of korQ. In the derivation, the electron densityn_Qemerging in the coupling term was replaced by its expectation valuehn_Qi (mean field) which is associated with the lattice deformation via the response function

χ. From equation1.12, we can extract a renormalized phonon frequency ω²_ren,Q=ω_Q² + 2g²ω_Q

M~

χ(Q,T). (1.13)

As discussed above, the one-dimensional electron gas is unstable against a perturbation with wavevector Q = 2k_F yielding a diverging susceptibility. Therefore, the phonon dispersion will

be strongly lowered, or softened, in the vicinity of this wavevector where an optic mode starts to condensate [68]. Inserting the temperature-dependent expression given in Equation 1.6, the renormalized phonon frequency in 1d is then given by

ω²_ren,2k

Figure1.4b shows the phonon dispersionω²_ren,2k

F relation for various temperatures as determined by equation1.14. At the transition temperatureT_cand in a one-dimensional system, the renormalized phonon frequency vanishes due to a diverging response function χ, and the system undergoes a structural phase transition, which is calledPeierls transition. In higher dimensions shown in Fig.

1.4a, the dependence on χ(Q,T) is weaker and the phonon softening less prominent. For a weak electron-phonon couplingg, the renormalized phonon frequency therefore remains finite, and no phase transition occurs. Consequently, a Peierls state in quasi-2d materials is favored by a non-zero electron-phonon coupling and an efficient Fermi nesting.

1d

Phonon band structure ω(Q)

Nesting vector Q

Dimensionality Temperature dependence in 1d case

Nesting vector Q

Figure 1.4:Phonon softening. (a) Dimensionality dictates the strength of the Kohn anomaly. (b) AboveT_c, phonon softening is an indicator for the Kohn anomaly. Adapted from Ref. [26]

Furthermore, below the transition temperature, zero-energy 2k_F-phonons condensate in a macro-scopic number, motivating the definition of an order parameter∆that is based on the expectation value of the phonon operators. Borrowed from the theory of superfluidity [54], the expectation values behave as complex numbers, vanish aboveT_c, and have a finite value below. The complex order parameter∆is described by

∆=|∆|e^iϕ =g_2kF

Accordingly, the expectation value of the lattice displacementhu(x)i (PLD), an observable in

Finite values ofhb_±2kFi lead to atom displacements away from their equilibrium positions and a static PLD with a 2kF-periodicity. In other words, the presence of a PLD is a direct measure of the CDW-order in the system. The complex nature of the order parameter will lead to characteristic collective excitations in the system, shown below in the context of the Ginzburg-Landau theory.

1.1.3.2 Electronic Band Structure

We now switch our point of view and examine the electronic spectrum in the presence of a mean distortion field. The Fröhlich Hamiltonian in this phonon mean-field approximation takes the form

H=Õ In the nearly-free electron approximation, the Hamiltonian can be diagonalized via a canonical transformation [63] for the relevant 2kF-phonon modes. We omit the detailed derivation and state the result for the electronic dispersion [26,54]

E_k =_F +sign(k−kF) q

~²v_F²(k−kF)²+∆², (1.19) wherev_Fis the Fermi velocity. The dispersionE_kexhibits single-particle gaps at the modesk =±k_F of the size 2|∆| transforming the prior metal state into anPeierls insulatorif the condensate does not contribute to the electric conductivity. Consequently, the amplitude of the order parameter can be experimentally accessed, for example, via spectroscopic techniques measuring the band structure of the material and identifying CDW gaps.

Moreover, the charge densityρin the Peierls state can be determined utilizing the new ground state wave functions yielding

where ρ₀ is the constant electron density in the metallic state. The periodic form of the charge density motivates the namecharge-density wave, already introduced above.

Figure1.5shows the charge density modulation, the periodic lattice distortion and the electronic dispersion relation, in the metallic and the Peierls state for a one-dimensional chain. For illustration purposes, the band is half-filled which, however, represents a special case since the chain dimerizes with a periodicityλ_PLD=π/k_F =π/2a.

Generally, a system with two coexisting periodicities is classified ascommensurateor incom-mensuratecorresponding to a rational or irrational ratio of periodicities. In this example, the two periodicities are associated with the regular lattice of the chain and the distortion field. Hence, the dimerized chain is commensurate while it is incommensurate for an irrational filling. In the latter case, the total energy of the Peierls state is independent of the order parameter phaseϕ. For arbitrary values ofϕ, the charge-density wave can adiabatically go from one energy state to another and has the freedom toslidealong the chain resulting insliding modesof the CDW state. This notion orig-inally stems from magnetic and compositional incommensurate structures [68] which represent a different type of incommensurateness compared to the CDW-induced displacive character studied in this work [69,70]. The additional degree of freedom has important consequences for the collective excitation spectrum as well as for the electronic transport behavior [71].

a

b

k_F 0

a ρ(x)

2a normal state CDW state 0

-k_F

ΔΔ

-π/a π/a

Electron dispersion E(k)

Figure 1.5: CDW as an coupled object. (a) In the mean field of lattice modes, the electronic spectrum is altered resulting in CDW gaps at±kF. (b) The charge density (green) and atomic positions (black and grey dots) are periodically modulated. Adapted from Ref. [26].