Dressel.Gruner.Electrodynamics of Solids.2003

Wavevector q

Fig. 12.4. The interaction of high velocity electrons with excitations of a threedimensional free-electron gas. For vb > vF plasmons are excited, while for vb < vF single-particle excitations are responsible for the electron energy loss.

beam of high energy electrons is passed through a thin film, and the energy loss (and momentum transfer) of the electrons is measured. This loss is described by the so-called loss function

Here ˆ(q, ω) refers to the wavevector and frequency dependent longitudinal dielectric constant. As discussed in Section 3.1, this can be different from the transverse dielectric constant, sampled by an optical experiment. Both single-particle and collective plasmon excitations contribute to the loss; which of these contributions is important depends on the velocity vb of the electron beam with respect to the Fermi velocity of the electron gas vF. The situation is shown in Fig. 12.4. If vb is large and exceeds the Fermi velocity vF, single-particle excitations cannot occur, and the exchange of energy between the electron beam and the electron gas is possible only by creating plasma excitations; here we can use the expression of ˆ(q, ω) obtained

Table 12.1. Plasma frequencies of simple metals, as obtained from the onset of transparency, from electron energy loss (EEL), and from theory [Kit63, Rae80].

The values are given in energy h¯ ωp or in wavenumber νp = ωp/2π c.

A typical result, obtained for aluminum at high electron beam velocities, is shown in Fig. 12.5. The value of the plasma frequency is obtained from the spacing of the peaks at E = 15.0 eV. Assuming three electrons per atom and free-electron mass leads to a plasma frequency ωp/2π c = 1.3 × 105 cm−1, corresponding to 16 eV, again showing excellent agreement between theory and experiment. This value also compares well with what is obtained from reflectivity, the data for aluminum being displayed in Fig. 12.6. The plasma edge is not particularly sharp, and this can be interpreted as damping; however, the notoriously bad surface characteristics of aluminum may be responsible for this feature of the reflectivity data. Nevertheless, the drop in R occurs around 1.2 × 105 cm−1, in full agreement with the value derived from the electron energy loss spectroscopy. In principle, the loss function can be calculated from the dielectric constant as evaluated from the optical experiments, and may be compared with the loss measured directly with electron energy loss spectroscopy. This has been done for certain semiconductors, but not for metals.

Some plasma frequency values obtained from both reflectivity and electron energy loss spectroscopy are collected in Table 12.1, together with the values calculated assuming free electrons, with the number of electrons per atom as given in the table. The excellent agreement between theory and experiment is perhaps one of the most powerful arguments for applying the free-electron theory to metals, where the bandwidth, i.e. the kinetic energy of electrons is large and exceeds all other energy scales of the problem.

12.1.2 The anomalous skin effect

The prevailing notion that we have relied on in the previous sections is the local response to the applied electromagnetic field, the assumption that the current

at a particular position is determined by the electric field at the same position only, and hence that the conductivity is independent of the position at which it is examined. Of course this does not mean that the currents and fields are independent of position, as the exponential decay of both J and E at the surface of a conducting medium – the examination of which leads to the normal skin effect described by Eq. (2.3.16) – so clearly demonstrates. The various consequences of this wavevector independent response are well known and were discussed in the previous section. The local response also leads, via Eq. (5.1.18), to a surface impedance Zˆ S = RS + iXS where – in the Hagen–Rubens regime – the components RS and XS are proportional to ω1/2 and RS = −XS. This is all confirmed by experiments on various simple metals, where the mean free path is not extremely large. This approximation progressively breaks down if the mean free path of the electrons becomes longer, and in the limit where exceeds the skin depth δ0 the non-local response has to be taken into account. Utilizing the Chambers formula (5.2.27) to examine what happens near to the surface of a metal for which> δ0 leads to the so-called anomalous skin effect, and the fundamental expression is given by Eq. (5.2.32). Both the normal and the anomalous skin effect have been derived in the Hagen–Rubens regime ωτ < 1, although it is straightforward to develop appropriate expressions in the opposite, so-called relaxation, regime (see Appendix E).

Let us estimate where the gradual transition from normal to anomalous skin effect occurs if we cool down a good metal such as copper. At room temperature the dc conductivity is typically 1×105 −1 cm−1, and the number of carriers (assuming that each copper atom donates approximately one electron into the conduction

with the same parameters at 1 GHz, for instance (the upper end of the radio frequency spectral region), is 20 000 A˚ . As the skin depth is much larger than the mean free path, copper at this frequency is in the normal skin effect regime at room temperature. On cooling, the conductivity increases and consequently the mean free path increases while the skin depth decreases. At liquid nitrogen temperature the resistivity is about one order of magnitude larger than at room temperature, and estimations similar to those given above lead to ≈ 10 000 A˚ and δ0 = 7000 A,˚ placing the material in the anomalous limit. There must be therefore a smooth crossover from the normal to the anomalous regime at relatively high temperatures. This has indeed been found by Pippard [Pip57, Pip60], and the experimental results are displayed in Fig. 12.7. For small conductivities, RS−2

Fig. 12.8. Section of an anisotropic Fermi surface in the xz plane, with the shaded region representing electrons which are affected by the electric field in the δ0 limit.

The ineffectiveness concept summarized in Section 5.2.5, and which led to the expression displayed above, can be extended to anisotropic Fermi surfaces. Such an anisotropic Fermi surface is shown in Fig. 12.8, where a cross-section in the xz plane is displayed for a particular value of the y component ky . The radius of curvature ρ(ky ) at any given point is wavevector dependent. The slices in Fig. 12.8 indicate electrons with velocities which lie within the angle δ0/ of the surface. The current is proportional to

Jx eEx τ vF dS . (12.1.14)

The surface dS over which the integration must be performed is given by this slice defined by a constant ky , and consequently we find

The integral over dky has to be taken around the line where the Fermi surface is cut by the xz plane. Inserting this effective conductivity, defined through Eq. (12.1.15) by Jx /Ex , into the expression of the anomalous skin effect, we find that the surface resistance

" −1 1/3

Although this is a qualitative argument, analytical results have been obtained for ellipsoidal Fermi surfaces by Reuter and Sondheimer [Reu48]. The importance of this result lies in the fact that the main contribution to the integral comes from regions of large curvature; these correspond to the flat regions of the Fermi surface. Therefore – at least in principle – the anisotropy of the Fermi surface can be mapped out by surface resistance measurements with the electromagnetic

Fig. 12.9. Surface resistance and surface reactance of tin measured at different orientations of the electric field with respect to the crystallographic axis [Pip50]. The normalized values of the real and imaginary parts of the surface impedance δr = (c/4π )2 RS/ω and δi = (c/4π )22XS/ω are plotted in the lower panel. The upper panel shows the ratio of the two components.

field pointing in different directions with respect to the main crystallographic axes. A typical result obtained for tin at low temperatures by Chambers [Cha52] is displayed in Fig. 12.9 with both the surface resistance and surface reactance displaying substantial anisotropy. One has to note, however, that the evaluation of the characteristics of the Fermi surface from such data is by no means straightforward, and other methods of studying the Fermi-surface phenomena, like Subnikov–de Haas oscillations, and the de Haas–van Alphen effect, or cyclotron resonance, have proven to be more useful.

12.1.3 Band structure and anisotropy effects

The Drude model, as it was used before, applies for an isotropic, three-dimensional medium where subtleties associated with band structure effects are fully neglected. Such effects enter into the various expressions of the conductivity in different

ways, and usually the Boltzmann equation in its variant forms where the electron velocities and the applied electric field appear in a vector form can be used to explore such band structure dependent phenomena.

A particularly straightforward modification occurs when the consequence of band structure effects can be absorbed into a dispersion relation which retains a parabolic form. If this is the case

i.e. the electron mass m is not the same as the bandmass mb. This parameter depends on the scattering of electrons on the periodic potential, and may be anisotropic. When the above expression applies, all features of the interband transitions remain unchanged, for example the plasma frequency is given by ωp = (4π N e2/mb)1/2, and, through Eq. (12.1.17), is dependent on the orientation of the electric field with respect to the crystallographic axes. Such effects are particularly important when the band structure is highly anisotropic; an example is displayed in Fig. 12.10. This material, (TMTSF)2ClO4, is an anisotropic metal, and band structure calculations suggest rather different single-particle transfer integrals in the two directions, with ta ≈ 200 meV and tb ≈ 20 meV. The small bandwidth also suggests that a tight binding approximation is appropriate. The resistivity is metallic in both directions, and its anisotropy ρb/ρa ≈ 102 is in full agreement with the anisotropic band structure as determined by the above transfer integrals. The metallic, but highly anisotropic, optical response leads to different plasma frequencies in the two directions; and the expression for ωp given above, together with the known carrier concentration, leads to bandmass values which, when interpreted in terms of a tight binding model, are in full agreement with the transfer integrals.

We encounter further complications if the approximation in terms of an effective bandmass as given above is not appropriate and we have to resort to the full Boltzmann equation as given by Eq. (5.2.16). The relevant integral which has to

where nE is the unit vector in the direction of the electric field E; through nE and vk it leads to a complex dependence on the band structure. As σ = N e2τ /m, one can define an effective mass

which, in general, will depend also on the orientation of the applied electromag-