Dressel.Gruner.Electrodynamics of Solids.2003

field D, and the rate of electronic energy density absorbed per unit volume is given by the product

and we find that Im{1/ ˆ} describes the energy loss associated with the electrons moving in the medium. It is consequently called the loss function of solids and is the basic parameter measured by electron loss spectroscopy.5 From the dispersion relations (3.2.17) of the inverse dielectric function, we obtain the sum rule

which can also be derived using rigorous quantum mechanical arguments [Mah90]. Furthermore from Eq. (3.2.17a) we find the following relations:

where Z0 = 4π/c, because limω→0 ω XS(ω) = 0 for all conducting material [Bra74].

Not surprisingly, we can also establish sum rules for other optical parameters.

5 If we evaluate the energy and momentum transfer per unit time in a scattering experiment by the Born approximation, we obtain a generalized loss function W (ω) ≈ (ω/8π ) Im $1/ ˆ(ω)%. It is closely related to the structure factor or dynamic form factor S(ω), which is the Fourier transform of the density–density correlation [Pin63, Pla73]. It allows for a direct comparison of scattering experiments with the longitudinal dielectric function:

For example the sum rule for the components of the complex refractive index, n(ω) and k(ω), are

For the reflectivity, similar arguments yield:

These formulations of the sum rule do not express any new physics but may be particularly useful in certain cases.

In [Smi85] sum rules to higher powers of the optical parameters are discussed; these converge faster with frequency and thus can be used in a limited frequency range but are in general not utilized in the analysis of the experimental results. There are also dispersion relations and sum rules investigated for the cases of non-normal incidence [Ber67, Roe65], ellipsometry [Hal73, Ina79, Que74], and transmission measurements [Abe66, Neu72, Nil68, Ver68].

References

[Abe66] F. Abeles` and M.L. Theye,` Surf. Sci. 5, 325 (1966) [Ber67] D.W. Berreman, Appl. Opt. 6, 1519 (1967)

[Bod45] H.W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, Princeton, 1945)

[Bra74] G. Brandli,¨ Phys. Rev. B 9, 342 (1974)

[Hal73] G.M. Hale, W.E. Holland, and M.R. Querry, Appl. Opt. 12 48 (1973)

[Ina79] T. Inagaki, H. Kuwata, and A. Ueda, Phys. Rev. B 19, 2400 (1979); Surf. Sci. 96, 54 (1980)

[Kra26] H.A. Kramers, Nature 117, 775 (1926); H.A. Kramers, in: Estratto dagli Atti del Congr. Int. Fis., Vol. 2 (Zanichelli, Bologna, 1927), p. 545; Collected Scientific Papers (North-Holland, Amsterdam, 1956).

[Kro26] R. de L. Kronig, J. Opt. Soc. Am. 12, 547 (1926); Ned. Tjidschr. Natuurk. 9, 402 (1942)

[Lan80] L.D. Landau and E.M. Lifshitz, Statistical Physics, 3rd edition (Pergamon Press, London, 1980)

[Lin54] J. Lindhard, Dan. Mat. Fys. Medd. 28, no. 8 (1954)

[Mac56] J.R. Macdonald and M.K. Brachman, Rev. Mod. Phys. 28, 393 (1956)

[Mah90] G.D. Mahan, Many-Particle Physics, 2nd edition (Plenum Press, New York,1990)

[Mar67] P.C. Martin, Phys. Rev. 161, 143 (1967)

[Neu72] J.D. Neufeld and G. Andermann, J. Opt. Soc. Am. 62, 1156 (1972) [Nil68] P.-O. Nilson, Appl. Optics 7, 435 (1968)

[Nye57] J.F. Nye, Physical Properties of Crystals (Clarendon, Oxford, 1957)

[Pin63] D. Pines, Elementary Excitations in Solids (Addison-Wesley, Reading, MA, 1963)

[Pin66] D. Pines and P. Noizieres,` The Theory of Quantum Liquids, Vol. 1 (Addison-Wesley, Reading, MA, 1966)

[Pla73] P.M. Platzman and P.A. Wolff, Waves and Interactions in Solid State Plasmas

(Academic Press, New York, 1973)

[Que74] M.R. Querry and W.E. Holland, Appl. Opt. 13, 595 (1974) [Roe65] D.M. Roessler, Brit. J. Appl. Phys. 16, 1359 (1965)

[Smi85] D.Y. Smith, Dispersion Theory, Sum Rules, and Their Application to the Analysis of Optical Data, in: Handbook of Optical Constants of Solids, Vol. 1, edited by E.D. Palik (Academic Press, Orlando, FL, 1985), p. 35

[Tau66] J. Tauc, Optical Properties of Semiconductors, in: The Optical Properties of Solids, edited by J. Tauc, Proceedings of the International School of Physics ‘Enrico Fermi’ 34 (Academic Press, New York, 1966)

[Tol56] J.S. Toll, Phys. Rev. 104, 1760 (1956) [Vel61] B. Velicky,´ Czech. J. Phys. B 11, 787 (1961)

[Ver68] H.W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968)

Further reading

[Agr84] V.M. Agranovich and V.L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons, 2nd edition (Springer-Verlag, Berlin, 1984)

[But91] P.N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge, 1991)

[Gei68] J. Geiger, Elektronen und Festkorper¨ (Vieweg-Verlag, Braunschweig, 1968) [Gin90] V.L. Ginzburg, Phys. Rep. 194, 245 (1990)

[Kel89] L.V. Keldysh, D.A. Kirzhnitz, and A.A. Maradudin, The Dielectric Function of Condensed Systems (North-Holland, Amsterdam, 1989)

[Mil91] P.L. Mills, Nonlinear Optics (Springer-Verlag, Berlin, 1991) [Sas70] W.M. Saslow, Phys. Lett. A 33, 157 (1970)

[Smi81] D.Y. Smith and C.A. Manogue, J. Opt. Soc. Am. 71, 935 (1981)

the electromagnetic field is given by

Here we have neglected second order terms (e.g. terms proportional to A2). Note that in Eq. (4.1.4) p, A, and are quantum mechanical operators which observe the commutation relations. For the moment, however, we will treat both A and as classical fields.

As discussed in Section 3.1.1, in the Coulomb gauge the vector potential A is only related to the transverse response of the medium, while the scalar potential determines the longitudinal response to the applied electromagnetic fields. These two cases are treated separately. As usual, magnetic effects are neglected, and throughout this and subsequent chapters we assume that µ1 = 1 and µ2 = 0.

4.1.1 Transverse conductivity: the response to the vector potential

Next we derive an expression for the complex conductivity in terms of the current– current correlation function. The operator of the electrical current density is defined as

i=1

where vi is the velocity of the ith particle at position ri (and we have implicitly assumed the usual commutation rules). The velocity operator of an electron in the presence of an electromagnetic field is given by v = p/m + eA/mc; consequently the current density has two terms,

JT(r) = Jp(r) + Jd(r)

The second term follows from the fact that A(ri ) commutes with δ{r − ri }. The first term is called the paramagnetic and the second the diamagnetic current. As the vector potential A depends on the position, it will in general not commute with the momentum. However, since p = −ih¯ we obtain

Note, if Coulomb gauge ( · A = 0) is assumed, A and p commute.

Using the definition (4.1.5) of the electric current density, the interaction term

Eq. (4.1.4) for transverse fields (note that = 0) can be written as:

We have replaced the summation over the individual positions i by an integral, and we have assumed that the current is a continuous function of the position r. Note that the diamagnetic current term Jd leads to a term in the interaction Hamiltonian which is second order in A; thus it is not included if we restrict ourselves to interactions proportional to A. Our objective is to derive the wavevector and frequency dependent response, and in order to do so we define the spatial Fourier transforms (see Appendix A.1) of the current density operator J(q) and of the vector potential

A(q) as

Here denotes the volume element over which the integration is carried out; if just the unit volume is considered, is often neglected. It is straightforward to show that with these definitions the q dependent interaction term in first order perturbation becomes

Next we derive the rate of absorption of electromagnetic radiation. As we have seen in Section 2.3.1, this absorption rate can also be written as P = σ1T(ET)2, in terms of the real part of the complex conductivity. If we equate the absorption which we obtain later in terms of the electric currents using this expression, it will then lead us to a formula for σ1 in terms of the q and ω dependent current densities. The imaginary part of σˆ (ω) is subsequently obtained by utilizing the Kramers–Kronig relation. We assume that the incident electromagnetic wave with wavevector q and frequency ω results in the scattering of an electron from one state to another with higher energy. Fermi’s golden rule is utilized: for one electron the number of transitions per unit time and per unit volume from the initial state |s to a final state |s of the system is

Here h¯ ωs and h¯ ωs correspond to the energy of the initial and final states of the system, respectively. The energy difference between the final and initial states h¯ ωs − h¯ ωs is positive for photon absorption and negative for emission. In this notation, |s and |s do not refer to the single-particle states only, but include all excitations of the electron system we consider. The average is not necessarily restricted to zero temperature, but is valid also at finite temperatures if the bracket is interpreted as a thermodynamic average. This is valid also for the expression we proceed to derive.

The matrix element for the transition is given by Eq. (4.1.11); substituting

s |HintT |s = − 1c s |JT(q)|s AT(q) into Eq. (4.1.12) leads to

where J (q) = J(−q). The summation over all occupied initial and all empty final states gives the total transfer rate per unit volume:

For the complete set of states #s |s s| = 1. In the Heisenberg representation exp{−iωs t}|s = exp {−iH0t/h¯ } |s , and then the time dependence is written as

where we have replaced s by s in order to simplify the notation.

For applied ac fields the equation relating the transverse electric field and the vector potential is given by ET = iωAT/c, leading to our final result: the absorbed power per unit volume expressed as a function of the electric field and the current– current correlation function:

With P = σ1T(ET)2 the conductivity per unit volume is simply given by the current–current correlation function, averaged over the states |s of our system in question

i.e. the Kubo formula for the q and ω dependent conductivity. Here the right hand side describes the fluctuations of the current in the ground state. The average value of the current is of course zero, and the conductivity depends on the time correlation between current operators, integrated over all times. The above relationship between the conductivity (i.e. the response to an external driving force) and current fluctuations is an example of the so-called fluctuation-dissipation theorem. The formula is one of the most utilized expressions in condensed matter physics, and it is used extensively, among other uses, also for the evaluation of the complex conductivity under a broad variety of circumstances, both for crystalline and noncrystalline solids. Although we have implied (by writing the absolved power as P = σ1T(ET)2) that the conductivity is a scalar quantity, it turns out that σ1T is in general a tensor for arbitrary crystal symmetry. For orthorhombic crystals the conductivity tensor reduces to a vector, with different magnitudes in the different crystallographic orientations, this difference being determined by the (anisotropic) form factor.

Now we derive a somewhat different expression for the conductivity for states for which Fermi statistics apply. At zero temperature the transition rate per unit volume between the states |s and |s is given by

explicitly include the q dependence since we are only interested in the q = 0 limit of the conductivity at this point. (Note that in this Hamiltonian we neglected the diamagnetic current term of JT.) The energies Es = h¯ ωs and Es = h¯ ωs correspond to the energy of the initial state |s and of the final state |s , respectively; the energy difference between these states is Es − Es = h¯ ωs s . In the T = 0 limit, the Fermi

equals unity and f (Es ) = 0, as all states below the Fermi level are occupied while the states above are empty. Integrating over all k vectors we obtain

This is the probability (per unit time and per unit volume) that the electromagnetic

As before, the conductivity σ1(ω) is related to the absorbed power through Ohm’s law. In the T = 0 limit, f (Es )[1 − f (Es )] = 1, and we obtain

This expression is appropriate for a transition between states |s and |s , and we assume that the number of transitions within an interval dω is Ns s (ω). We define the joint density of states for these transitions as Ds s (ω) = dNs s (ω)/dω:

where the factor of 2 refers to the different spin orientations. The conductivity is then given in terms of the joint density of states for both spin directions and of the transition probability as

This equation, often referred to as the Kubo–Greenwood formula, is most useful when interband transitions are important, with the states |s and |s belonging to different bands. We utilize this expression in Section 6.2 when the optical conductivity of band semiconductors is derived and also in Section 7.3 when the absorption of the electromagnetic radiation in superconductors is discussed.