Dressel.Gruner.Electrodynamics of Solids.2003

the average

dSF ; vF

this average being distinctively different from the average sampled by σˆ (ω) given above. Therefore the thermal bandmass is, as a rule, different from the bandmass determined by optical experiments, the optical mass.

12.2 Effects of interactions and disorder

The optical experiments summarized in the previous section provide powerful evidence that the main assumptions which lie behind the Drude–Sommerfeld model are appropriate: for simple metals we can neglect the interaction of the electrons with lattice vibrations and with other electrons. This picture has to be modified by both interaction and disorder effects. We expect these to occur: (i) when the energy scale which describes the strength of the electron–phonon or electron–electron interactions and (ii) when – for disordered metals – the overall strength of the random potentials, are comparable to the kinetic energy of the electrons. This depends on many factors: electron–electron interactions are important for narrow bands, electron–phonon interactions are expected to be strong for soft lattices; while the strength of the random potential can be modified by alloying and/or by removing the underlying lattice periodicity, by making an amorphous solid.

12.2.1 Impurity effects

Most of the optical effects associated with impurities in metals can be absorbed into an impurity induced relaxation rate 1/τimp which affects the response in conjunction with the relaxation rate as determined by phonons 1/τP. This, together with Matthiessen’s rule, leads to the relaxation rate 1/τ = 1/τimp + 1/τP which enters into the relevant expressions of the Drude model. In addition to an increased relaxation rate, the plasma frequency can also be influenced by impurities which are non-isoelectric with the host matrix; for small impurity concentrations this modification is expected to be small. While the above effects can be incorporated into a Drude model, more significant are the effects due to impurities with unoccupied d or f orbitals. In this case the (originally) localized orbitals at energy position Ed (or Ef ) away from the Fermi level are, due to interaction with the broadened conduction band; this broadening is described by a transition matrix element Vsd between the localized orbitals and the conduction band. These states then acquire a finite width W – estimated through a simple golden-rule argument which gives W = 2π |Vsd |2 D(EF) with D(EF) the density of s states at the Fermi

materials, and optical studies have greatly contributed to the field by elucidating the important aspects of these interactions. In general the optical properties are treated by extending the formalism we have used in Part 1 in its simple form. This leads to a frequency dependent relaxation rate and mass (the two quantities are related by the Kramers–Kronig transformation), which in turn can be extracted from the measured optical parameters. In some cases the frequency dependencies can also be calculated both for electron–electron and electron–phonon interactions. This will not be done here; we confine ourselves to the main aspects of this fascinating field.

Without specifying the underlying mechanism, we assume that the relaxation

rate ˆ which appears in the Drude response of metals is frequency dependent:

relaxation rate in Eq. (5.1.4), the model referred to is the extended Drude model. If we define the dimensionless quantity λ(ω) = − 2(ω)/ω, then the complex conductivity can be written as

It is clear that this has the same form as the simple Drude expression (5.1.4) but with a renormalized and frequency dependent scattering rate

which approaches the constant 1/τ as λ → 0. The parameter which demonstrates this renormalization can be written in terms of a renormalized, enhanced mass m /mb = 1 + λ(ω), which then leads to another form of the conductivity:

By rearranging this equation, we can write down expressions for 1(ω) and m (ω) in terms of σ1(ω) and σ2(ω) as follows:

Kronig relation.

There are several reasons for such a frequency dependent relaxation rate: both electron–electron and electron–phonon interactions (together with more exotic mechanisms, such as scattering on spin fluctuations) may lead to a renormalized and frequency dependent scattering process. Such renormalization is not effective at high frequencies, but is confined to low frequencies; typically below the relevant phonon frequency for electron–phonon interactions or below an effective, reduced bandwidth in the case of electron–electron interactions. Hence the frequency dependent relaxation rate – and via Kramers–Kronig relations also a frequency dependent effective mass – approaches for high frequencies the unrenormalized, frequency independent value.

Electron–electron interactions leading to an enhanced thermodynamic mass are known to occur in various intermetallic compounds, usually referred to as heavy fermions. The thermodynamic properties are that of a Fermi liquid, but with renormalized coefficients [Gre91, Ott87, Ste84]. Within the framework of a Fermiliquid description, the Sommerfeld coefficient γ , which accounts for the electronic contribution to specific heat C(T ) = γ T + β T 3 at low temperatures where the phonon contribution can be ignored, is given by

where Vm is the molar volume, kF is the Fermi wavevector, m is the effective mass, and N is the number of (conduction) electrons per unit volume. The same formalism gives the magnetic susceptibility χm as

which has the same form as for a non-interacting Fermi glass, but with m replacing the free-electron mass m. As for non-interacting electrons, the so-called Wilson ratio γ (0)/χm(0) is independent of the mass enhancement. The enhanced χm and γ imply a large effective mass, hence the name heavy fermions. Because of the periodic lattice, the dc resistivity ρdc of metals is expected to vanish at T = 0. The finite values observed at zero temperature are attributed to lattice imperfections and impurities. At low temperatures, the temperature dependence of the resistivity is best described by

where ρ0 is the residual resistivity due to impurities, the T 2 contribution is a consequence of electron–electron scattering, and A is proportional to both γ 2 and χm2 . This relation holds for a broad variety of materials spanning a wide range of mass enhancements [Dre97, Miy89]. In most cases, the dc resistivity has a broad maximum at a temperature Tcoh, which is often referred to as the

for heavy fermion behavior [Awa93], is displayed at two different temperatures. The temperature below which electron–electron interactions are effective and lead to a coherent, heavy fermion state is approximately Tcoh ≈ 3 K in this material. At T = 10 K, above the coherence temperature, σ1(ω) is that of a simple metal with a relaxation rate γ = 1/(2π cτ ) = 1000 cm−1 and νp = ωp/(2π c) = 30 000 cm−1. This value of the plasma frequency is in agreement with the calculated ωp as

Therefore at this temperature CeAl3 behaves as a simple metal, with no interaction induced renormalization. In contrast, at T = 1.2 K, well in the coherent regime, one observes a narrow resonance in σ1(ω) and the data joins the 10 K data at approximately 3 cm−1, i.e. at a frequency ωc corresponding to Tcoh. In order to analyze this behavior, and to extract quantities which are related to the enhanced effective mass m , we can assume that the narrow feature is that of a renormalized Drude response, with a renormalized (and likely frequency dependent, but at first approximation assumed to be constant) relaxation rate and mass. Spectral weight arguments are utilized to evaluate this renormalized mass. In order to do so we can evaluate the following integrals:

which is directly related to the unrenormalized bandmass; ωc is the usual cutoff frequency just below interband transitions (see Section 3.2.2). In Eq. (12.2.9a), ωc /2π c = 3 cm−1 is the frequency above which the data at 10 K and at 1.2 K are indistinguishable.5 This is also the frequency where the renormalization becomes ineffective as discussed above. From the resulting ratio A0/ A , we obtain m /mb = 450 ± 50, a value comparable with the one estimated from the thermodynamic quantities [Awa93].

Consequently, at low temperatures, where electron–electron interactions are effective, the optical properties can be described in terms of a renormalized Drude response

with the renormalization simply incorporated into a renormalized mass. Note that

5Such division of the spectral response is somewhat arbitrary, and a fit to an extended Drude response with a frequency dependent scattering rate and mass is certainly more appropriate.

326 12 Metals

the dc conductivity – while larger at 1.2 K than at 10 K due to the usual temperature dependence observed in any metal – is not enhanced; what is enhanced is the relaxation time τ and mass m . The ratio τ /m , however, is not modified by the electron–electron interactions – at least in the limit where τ is determined by impurity scattering.

The analysis described above has been performed for a wide range of materials with differing strength of electron–electron interactions – and thus with different enhancements of the effective mass. This enhancement has been evaluated using the thermodynamic quantities, such as specific heat and magnetic susceptibility, with both given by Eqs (12.2.6) and (12.2.7). The mass has also been evaluated from the reduced spectral weight, and in Fig. 12.14 the two types – thermodynamic and electrodynamic – of masses are compared. Here the specific heat coefficient, γ , is plotted; as neither N nor kF vary much from material to material, from Eq. (12.2.6) it follows that γ is approximately proportional to the mass enhancement. We find that thermodynamic and optical mass enhancements go hand in hand, leading to the notion that in these materials electron–electron interactions lead to renormalized Fermi liquids, to first approximation.

Of course, the above description is an oversimplification, in light of what was said before. The resistivity is strongly temperature dependent at low temperatures, reflecting electron–electron interactions. As approximately kBT /EF electrons are

whereas the effective mass is not frequency dependent in lowest order. At low temperatures the dc resistivity

is proportional to the square of the temperature (see also Eq. (12.2.8)), as expected for a (renormalized) Fermi liquid. Using both components of the conductivity as evaluated by experiments over a broad spectral range, the frequency dependence of both γ = 1/(2π cτ ) and m can be extracted by utilizing Eqs (12.2.5); these are shown in the insert of Fig. 12.13. The leading frequency dependence of 1/τ is indeed ω2 as expected for a Fermi liquid, and this is followed at higher frequencies by a crossover for both 1/τ and m to a high frequency region where these parameters assume their unrenormalized value. Thus, just like the temperature

Fig. 12.14. Enhanced mass m relative to the free-electron mass m plotted versus the Sommerfeld coefficient γ obtained from the low temperature specific heat C/ T for several heavy fermion compounds. The solid line represents m /m0 γ (0). Also included are the parameters for a simple metal, sodium, and for several transition metals (after [Deg99]).

driven destruction of the correlated heavy fermion state, renormalization effects are ineffective at high frequencies.

Electron–phonon interactions also lead to an enhancement of the thermodynamic mass, as evidenced by the moderately enhanced low temperature specific heat C/ T . The influence of these interactions on the optical properties of metals has been considered at length. The leading term is the emission of phonons in a metal; this so-called Holstein process involves the absorption of photons by simultaneously emitting a phonon and scattering an electron. The process obviously depends on the phonon spectrum, which we describe as the density of states F(ωP) and by a weight factor αP2 which is the electron–phonon matrix element. With these