EFFECT OF ELECTRON-PHONON INTERACTIONS ON RAMAN LINE AT FERROMAGNETIC ORDERING
L. A. Falkovsky*
Landau Institute for Theoretical Physics 142432, Chernogolovka, Moscow Region, Rissia
Verechagin Institute of the High Pressure Physics 142190, Troitsk, Moscow, Rissia
Received April 3, 2014
The theory of Raman scattering in half-metals by optical phonons interacting with conduction electrons is developed. We evaluate the effect of electron-phonon interactions at ferromagnetic ordering in terms of the Boltzmann equation for carriers. The chemical potential is found to decrease as the temperature decreases. Both the linewidth and frequency shift exhibit a dependence on temperature.
DOI: 10.7868/S0044451014090260
1. INTRODUCTION
Recently, the Raman scattering in the half-metallic C0S2 was studied fl] in a wide temperature region. The u) = 400 cm"1 Raman line, observed previously at room temperature in Refs. [2,3], demonstrates a particular behavior near the ferromagnetic transition at Tc = 122 Iv. The unusual large Raman linewidth and shift of the order of 10 cm"1 were observed. The reflectivity singularities of C0S2 were explained in Ref. [4] by the temperature variation of the electronic structure. Another example of electron phonon interactions is adduced in Ref. [5] in order to explain the phonon singularity at the F point in graphene. The electron phonon interactions should also be considered in interpreting the observed Raman scattering around the Curie temperature.
Thermal broadening of phonon lines in Raman scattering is usually described in terms of the three-phonon anharmonicity, i.e., by the decay of an optical phonon with a frequency ui into two phonons. The simplest case where the final state has two acoustic phonons from one branch (the Klemens channel) was theoretically studied by Klemens [6], who obtained the temperature dependence of the Raman linewidth. The corresponding line shift was considered in Refs. [7,8]. This theory was compared in Refs. [7 9] with experimental data for Si,
* E-mail: falk'flitp.ac.ru
Ge, C, and a:-Sn. A model was also considered with the final-state phonons from different branches. It was found that anharnionic interactions of the forth order should be taken at high temperatures T > 300 Iv into account.
The situation is more complicated in substances with magnetic ordering. The interaction of phonons with niagnons in antiferroniagnets was discussed in review article [10] and more recently in the analysis of thermal conductivity [11], the spin Seebeck effect [12, 13], high-temperature superconductivity [14], and optical spectra [15]. The niagnon phonon interaction results in the niagnon damping [16], but 110 effect for phonons was observed. The influence of antiferro-magnetic ordering is considered in Ref. [17], where only the line shift was calculated. Damping of the optical phonons was found [18] to become large in the rare-earth Gd and Tb below the Curie temperature, achieving a value of 15 cm"1, which is much greater than the three-phonon interaction effect.
A contradiction is known to exist in the Migdal theory [19] of electron phonon interaction. O11 one hand, Migdal showed that the vertex corrections for acoustic phonons are small by the adiabatic parameter s/m/M, where in and M are the respective electron and ion masses (the "Migdal theorem"). The theory correctly-described the electron lifetime and rcnormalization of the Fermi velocity vp- But 011 the other hand, the theory resulted in a strong rcnormalization of the sound velocity = ,s(l — 2A)1/2, where A is the dimcnsionlcss
16 >K9T<E>, libiu. 3 (9)
657
coupling constant. For a sufficiently largo electron plionon coupling constant A —¥ 1/2, the plionon frequency approaches zero, marking an instability point of the system. Instead, one would intuitively expect the plionon ronormalization to be weak along with the adiabatic parameter.
This discrepancy was resolved by Brovman and Iva-gan [20] almost a decade later (see also [21]). They found that there are two terms in the second-order perturbation theory that compensate each other and produce a result small by the adiabatic parameter. Namely, in calculating the plionon self-energy function Il(ui,k) with the help of the diagram technique, one should eliminate an adiabatic contribution of the Fröhlich model by subtracting Il(ui,k) — 11(0, k).
The interaction of electrons with optical phonons was considered first. A splitting of the optical plionon into two branches at finite wavonumbors k was predicted by Engelsberg and Schrioffer [22] within Migdal's many-body approach for dispersion-less phonons. Later, Ipatova and Subashiov [23] calculated the optical plionon attenuation in the col-lisionloss limit and pointed out that the Brovman Kagan ronormalization should be carried out for optical phonons in order to obtain the correct plionon ronormalization. In Ref. [24], Alexandrov and Schrioffer corrected the calculational error in Ref. [22] and argued that no splitting was in fact found. Instead, they predicted an extremely strong dispersion of optical phonons, u'ft = u>o + Xv'pk2/3u>o, due to the coupling to electrons. The large plionon dispersion is a typical result of Migdal's theory [25] using the Frölich Hamiltonian. No such dispersion has 6V6F been observed experimentally. The usual dispersion of optical phonons in metals has the order of the sound velocity. Reizer [26] stressed the importance of taking screening effects into account. Papers [24, 26] are limited to the case where both electron and plionon systems are collisionless. Moreover, only the plionon renormaliza-tion was considered, with no results available for the attenuation of optical phonons.
A semiclassical approach, which is different from the many-body technique and is based on the Boltz-niann equation and the elasticity theory equations was developed by Akhiezer, Silin, Gurevich, Kontorovich, and many others (we refer the reader to review [27]). This approach was compared with various experiments, such as attenuation of sound waves, the effects of strong magnetic fields, crystal anisotropy, and sample surfaces on sound attenuation, and so on. It can be applied to the problem of the electron optical-phonon interaction [28] as well.
In the previous paper [29], we developed a quantum theory for the optical-phonon attenuation and shift induced by the interband electron transitions and tuned with a temperature variation. Here, we consider the optical plionon ronormalization as a result of the electron plionon interaction taking ferromagnetic ordering into account. We argue that the reasonable plionon damping and shift can be obtained using the semiclassical Boltzmann equation for electrons and the equation of motion for phonons coupled by the deformation potential.
2. ELECTRON-PHONON INTERACTIONS AT FERROMAGNETIC ORDERING
We assume that the electron bands in C0S2 have the shape shown in Fig. 1. The ferromagnetic ordering results in a spin splitting /¡bHc of the unfilled half-metallic band,
r(p) =
jr_ 2 m*
IibHc
l(p) =
jr_ 2 m*
lißHc (1)
in the effective Weiss field Hc. As the temperature decreases, the magnetization, determined in the mean-field approximation as
m
(2)
T < Tt,
T > Tc
Fig. 1. Proposed band scheme for two-electron spin projections
appears according to experimental data in C0S2 at approximately Tc = 122 Iv, and the spin splitting is proportional to the magnetization.
We write the interaction of electrons with the optical phonon i/j as the deformation potential
= (3)
where AT ~ 1/u3 is the number of cells in unit volume and a is the interatomic distance. For the acoustic phonon electron interaction, we should substitute the strain tensor u.y instead of the displacement in order to satisfy the translation symmetry of the lattice.
The Boltzmann equation for the nonequilibrium part of the distribution function /(p) has the form
[—¿(u> — k • v) + r-1]/(p) =
= _^[PV.E-/-u;Mj0(p)], (4)
where /0 is the equilibrium distribution function. In Boltzmann equation (4), we omit the spin index ,s that determines all the electron parameters. The electron collision frequency r_1 takes the collisions with impurities and phonons into account. The collision frequency-is calculated for C0S2 in the Debye model with the temperature To = 500 Iv [30]. It follows from Eq. (4) that the condition
(Q) = 0
must be satisfied for the current continuity equation to hold; here, the brackets denote averaging over the Fermi surface for temperatures TCff.
In the ferromagnetic phase, as the temperature changes, the carriers overflow from one spin state to another, but the total number of carriers
remains constant. This condition determines the chemical potential and the concentration of carries with spin up and spin down, shown in Fig. 2. All figures correspond here and in what follows to the carrier concentration AT = 1021 cm"3 in the considered band with the chemical potential //. = 0.36 oV above the Curie temperature.
We write the equation of motion for the phonon mode in the form
/2 2\ QEi 1 9Hinl
7VT
1.0
0.8
0.6
0.4
0.2
0 50 100 150 200 250
T, K
Fig. 2. Calculated temperature dependence of the carrier concentration for spin up Nf and spin down (relative to the total concentration at temperatures above the ferromagnetic ordering temperature), and the dependence of the chemical potential //
where M is the reduced ion mass of the cell, Q is the charge corresponding to the optical vibration, and u>o is the frequency of the considered mode. Here, the last term represents the electron phonon interaction. Using Boltzmann equation (4), we rewrite this term as
1 ^ll,,,! _ "i M diii ~ MN x
x v f ^nt(p) f_dfo\ <Pp . iOT+i V 9e J (2TT/Ï)3
The term with the electric field in the Boltzmann equation disappears in integrating over p due to the velocityinversion v —v. The term with the wave vector k has to be omitted for a Raman phonon, because the vector k is determined in this
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