Dressel.Gruner.Electrodynamics of Solids.2003

8	Part one: Introductory remarks

Broken symmetry states are described in their simple form using second quantized formalism.

General books and monographs

A.A. Abrikosov, Fundamentals of the Theory of Metals (North-Holland, Amsterdam, 1988)

M. Born and E. Wolf, Principles of Optics, 6th edition (Cambridge University Press, Cambridge, 1999)

J. Callaway, Quantum Theory of Solids, 2nd edition (Academic Press, New York, 1991)

R.G. Chambers, Electrons in Metals and Semiconductors (Chapman and Hall, London, 1990)

W.A. Harrison, Solid State Theory (McGraw-Hill, New York, 1970)

H. Haug and S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 3rd edition (World Scientific, Singapore, 1994)

J.D. Jackson, Classical Electrodynamics, 2nd edition (John Wiley & Sons, New York, 1975)

C. Kittel, Quantum Theory of Solids, 2nd edition (John Wiley & Sons, New York, 1987)

L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edition (Butterworth-Heinemann, Oxford, 1984)

I.M. Lifshitz, M.Ya. Azbel’, and M.I. Kaganov, Electron Theory of Metals (Consultants Bureau, New York, 1973)

G.D. Mahan, Many-Particle Physics, 2nd edition (Plenum Press, New York, 1990) D. Pines, Elementary Excitations in Solids (Addison-Wesley, Reading, MA, 1963)

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

J.R. Schrieffer, Theory of Superconductivity, 3rd edition (W.A. Benjamin, New York, 1983)

F.Stern, Elementary Theory of the Optical Properties of Solids, in: Solid State Physics

15, edited by F. Seitz and D. Turnbull (Academic Press, New York, 1963), p. 299

J.Tauc (ed.), The Optical Properties of Solids, Proceedings of the International School of Physics ‘Enrico Fermi’ 34 (Academic Press, New York, 1966)

F. Wooten, Optical Properties of Solids (Academic Press, San Diego, CA, 1972)

P.Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer-Verlag, Berlin, 1996)

J.M. Ziman, Principles of the Theory of Solids, 2nd edition (Cambridge University Press, London, 1972)

2

The interaction of radiation with matter

Optics, as defined in this book, is concerned with the interaction of electromagnetic radiation with matter. The theoretical description of the phenomena and the analysis of the experimental results are based on Maxwell’s equations and on their solution for time-varying electric and magnetic fields. The optical properties of solids have been the subject of extensive treatises [Sok67, Ste63, Woo72]; most of these focus on the parameters which are accessible with conventional optical methods using light in the infrared, visible, and ultraviolet spectral range. The approach taken here is more general and includes the discussion of those aspects of the interaction of electromagnetic waves with matter which are particularly relevant to experiments conducted at lower frequencies, typically in the millimeter wave and microwave spectral range, but also for radio frequencies.

After introducing Maxwell’s equations, we present the time dependent solution of the equations leading to wave propagation. In order to describe modifications of the fields in the presence of matter, the material parameters which characterize the medium have to be introduced: the conductivity and the dielectric constant. In the following step, we define the optical constants which characterize the propagation and dissipation of electromagnetic waves in the medium: the refractive index and the impedance. Next, phenomena which occur at the interface of free space and matter (or in general between two media with different optical constants) are described. This discussion eventually leads to the introduction of the optical parameters which are accessible to experiment: the optical reflectivity and transmission.

2.1 Maxwell’s equations for time-varying fields

To present a common basis of notation and parameter definition, we want to recall briefly some well known relations from classical electrodynamics. Before we consider the interaction of light with matter, we assume no matter to be present.

9

10	2 The interaction of radiation with matter

2.1.1 Solution of Maxwell’s equations in a vacuum

The interaction of electromagnetic radiation with matter is fully described by Maxwell’s equations. In the case of a vacuum the four relevant equations are

× E(r, t) +

× B(r, t) −

	1 ∂B(r, t)			=	0		,
	c		∂t
· B(r, t) = 0							,
	1 ∂E(r, t)			=	4π		J(r, t) ,
	c		∂t			c

· E(r, t) = 4πρ(r, t) .

(2.1.1a)

(2.1.1b)

(2.1.1c)

(2.1.1d)

E and B are the electric field strength and the magnetic induction, respectively; c = 2.997 924 58 × 108 m s−1 is the velocity of light in free space. The current density J and the charge density ρ used in this set of equations refer to the total quantities including both the external and induced currents and charge densities; their various components will be discussed in Section 2.2. All quantities are assumed to be spatial, r, and time, t, dependent as indicated by (r, t). To make the equations more concise, we often do not explicitly include these dependences. Following the notation of most classical books in this field, the equations are written in Gaussian units (cgs), where E and B have the same units.1

The differential equations (2.1.1a) and (2.1.1b) are satisfied by a vector potential A(r, t) and a scalar potential (r, t) with

	B = × A				(2.1.2)
and
1			∂A
E +				= − .	(2.1.3)
	c		∂t

The first equation expresses the fact that B is an axial vector and can be expressed as the rotation of a vector field. If the vector potential vanishes (A = 0) or if A is time independent, the electric field is conservative: the electric field E is given by the gradient of a potential, as seen in Eq. (2.1.3). Substituting the above expressions into Ampere’s` law (2.1.1c) and employing the general vector identity

× ( × A) = − 2A + ( · A)

(2.1.4)

gives the following equation for the vector potential A:

2A −

1 ∂2A

4π

1

∂

= −

J +

+ ( · A)

;

(2.1.5)

c2

∂t2

c

c

∂t

a characteristic wave equation combining the second time derivative and the second

1For a discussion of the conversion to rational SI units (mks), see for example [Bec64, Jac75]. See also Tables G.1 and G.3.

2.1 Maxwell’s equations for time-varying fields

11

spatial derivative. As this equation connects the current to the scalar and vector potentials, we can find a corresponding relationship between the charge density and the potentials. Substituting Eq. (2.1.3) into Coulomb’s law (2.1.1d) yields

· E = 4πρ = − 2		1 ∂
	−				( · A) .
		c		∂t
Using the Coulomb gauge2
· A = 0 ,						(2.1.6)

the last term in this equation vanishes; the remaining part yields Poisson’s equation

2 = −4πρ ,

(2.1.7)

expressing the fact that the scalar potential (r, t) is solely determined by the charge distribution ρ(r, t). From Eq. (2.1.1c) and by using the definition of the vector potential, we obtain (in the case of static fields) a similar relation for the vector potential

4π
2A = − c	J ,	(2.1.8)

connecting only A(r, t) to the current density J(r, t). Employing the vector identity· ( × B) = 0, we can combine the derivatives of Eqs (2.1.1c) and (2.1.1d) to obtain the continuity equation for electric charge

∂ρ	= − · J ,	(2.1.9)
∂t

expressing the fact that the time evolution of the charge at any position is related to a current at the same location.

Equation (2.1.7) and Eq. (2.1.9) have been obtained by combining Maxwell’s equations in the absence of matter, without making any assumptions about a particular time or spatial form of the fields. In the following we consider a harmonic time and spatial dependence of the fields and waves. A monochromatic plane

2

Another common choice of gauge is the Lorentz convention · A

1 ∂

= 0, which gives symmetric wave

+ c ∂t

equations for the scalar and vector potentials:

2 −

1 ∂2

= −4πρ ,

c2

∂t2

2A −

1 ∂2A

4π

= −

J ;

c2

∂t2

c

or – in the case of superconductors – the London gauge, assuming 2 = 0. For more details on the properties of various gauges and the selection of an appropriate one, see for example [Jac75, Por92]. Our choice restricts us to non-relativistic electrodynamics; relativistic effects, however, can safely be neglected throughout the book.

12	2 The interaction of radiation with matter
electric wave of frequency		f = ω/2π traveling in a certain direction (given by
the wavevector q) can then be written as
	E(r, t) =	E01 sin{q · r − ωt} + E02 cos{q · r − ωt}
	=	E03 sin{q · r − ωt + ψ } ,	(2.1.10)

where E0i (i = 1, 2, 3) describe the maximum amplitude; but it is more convenient to write the electric field E(r, t) as a complex quantity

E(r, t) = E0 exp{i(q · r − ωt)} .

(2.1.11)

We should keep in mind, however, that only the real part of the electric field is a meaningful quantity. We explicitly indicate the complex nature of E and the possible phase factors only if they are of interest to the discussion. A few notes are in order: the electric field is a vector, and therefore its direction as well as its value is of importance. As we shall discuss in Section 3.1 in more detail, we distinguish longitudinal and transverse components with respect to the direction of propagation; any transverse field polarization can be obtained as the sum of two orthogonal transverse components. If the ratio of both is constant, linearly polarized fields result, otherwise an elliptical or circular polarization can be obtained. Similar considerations to those presented here for the electric field apply for the magnetic induction B(r, t) and other quantities. The sign in the exponent is chosen such that the wave travels along the +r direction.3 We assume that all fields and sources can be decomposed into a complete continuous set of such plane waves:

	1		∞
E(r, t) =			−∞ E(q, ω) exp{i(q · r − ωt)} dω dq ,	(2.1.12)
	(2π )4

and the four-dimensional Fourier transform of the electric field strength E(r, t) is

∞
E(q, ω) = −∞ E(r, t) exp{−i(q · r − ωt)} dt dr .	(2.1.13)

Analogous equations and transformations apply to the magnetic induction B. Some basic properties of the Fourier transformation are discussed in Appendix A.1. Equation (2.1.12), which assumes plane waves, requires that a wavevector q and a frequency ω can be well defined. The wavelength λ = 2π/|q| should be much smaller than the relevant dimensions of the problem, and therefore the finite sample size is neglected; also the period of the radiation should be much shorter than the typical time scale over which this radiation is applied. We assume, for the moment, that the spatial dependence of the radiation always remains that of a plane wave.

3For exp{i(q · r + ωt)} and exp{−i(q · r + ωt)} the wave travels to the left (−r), while exp{i(q · r − ωt)} and exp{−i(q · r − ωt)} describe right moving waves (+r). With the convention exp{−iωt} the optical constantsˆ and Nˆ have positive imaginary parts, as we shall find out in Section 2.2. The notation exp{iωt} is also common and leads to Nˆ = n − ik and ˆ = 1 − i 2.

2.1 Maxwell’s equations for time-varying fields

13

Using the assumption of harmonic waves of the form (2.1.12), the spatial and time derivatives of the electric field strength E(r, t) can be calculated as

× E(r, t) = iq × E(r, t)
· E(r, t)			=	iq · E(r, t)	(2.1.14)
	∂E(r, t)		=	−iωE(r, t) ,
	∂t

with the same relations found for the magnetic induction B(r, t). The spatial and time periodicity of the radiation can now be utilized to write Maxwell’s equations in Fourier transformed form:

ω

q × E(q, ω) − c B(q, ω)

q · B(q, ω)

iq × B(q, ω) + iωc E(q, ω) iq · E(q, ω)

=	0		,	(2.1.15a)
=	0		,	(2.1.15b)
=		4π	J(q, ω) ,	(2.1.15c)
		c
=	4πρ(q, ω) .			(2.1.15d)

The vector and scalar potentials, A and , can also be converted to Fourier space, and from Eqs (2.1.2), (2.1.3), and (2.1.7) we then obtain

B(q, ω)

E(q, ω)

q2 (q, ω)

=	iq × A(q, ω) ,
		ω
= i			A(q, ω) − iq (q, ω) ,
		c
=	4πρ(q, ω) .

(2.1.16)

(2.1.17)

(2.1.18)

As mentioned above, the assumption of harmonic waves is not a restriction for most cases; since the transformation to Fourier space turns out to be particularly convenient, we have utilized these relations heavily.

2.1.2 Wave equations in free space

In the absence of matter and in the absence of free current and external charge (J = 0 and ρ = 0), the combination of Faraday’s induction law (2.1.1a) and Coulomb’s law (2.1.1d) yields

1 ∂ ( × B) = 2E ; c ∂t

using Eq. (2.1.1c), we obtain the following relation for the electric field:

1 ∂2E	= 0 .
2E − c2 ∂t2		(2.1.19a)

This is a wave equation in its simplest form without dissipation or other complications; the second spatial derivative is equal to the second derivative in time with

14		2 The interaction of radiation with matter
	x, E	Electric field E

λ

z, q

y, B	Magnetic field B

Fig. 2.1. Linearly polarized electromagnetic radiation of wavelength λ propagating along the z direction. Assuming that the electric field E is along the x direction, the magnetic field B points in the y direction and the wavevector q is directed along the z axis.

the square of the velocity as proportionality factor. One possible solution of this differential equation is given by a harmonic wave E(r, t) = E0 exp{i(q · r − ωt)} as suggested in Eq. (2.1.11). Similarly, the value of the wavevector q is simply ω/c. The corresponding wave equation for the B field can easily be derived from Maxwell’s equations and has the same form:

1 ∂2B	= 0 .
2B − c2 ∂t2		(2.1.19b)

A linearly polarized electromagnetic wave is shown in Fig. 2.1, with the notations and conventions adopted in this book. According to Eqs (2.1.15a) and (2.1.15c) the three vectors E, B, and q are perpendicular to one another. The electric and magnetic fields are oriented in the x and y directions, respectively; the wave propagates along the z axis. Note that the electric and magnetic fields are oscillating in phase as seen from Eq. (2.1.15a). At any given r and t the magnetic field is zero when the electric field vanishes; because J = 0 in free space, we see from Eq. (2.1.15c) that E is also zero when there is no B field. Since the equation of an undamped wave contains only the second spatial and time derivatives, the choice of the particular sign in the exponent of Eq. (2.1.11) has no significance: leftand right-going waves satisfy the wave equation.

There is an energy associated with the creation of electric and magnetic fields; the time average density of the electric field energy is ue = 41π E 2t and of the magnetic field um = 41π B 2t . A traveling electromagnetic wave is associated with transport of electromagnetic energy. There will be no dissipation of energy (or attenuation of the waves) as long as we do not consider damping effects. The energy density, i.e. the energy in a unit volume, associated with a harmonic wave

	2.2 Propagation of electromagnetic waves in the medium					15
	is written as	1		1
	u =		E2 +		B2 .	(2.1.20)
		8π		8π

In the case of free space |E| = |B|, and the energies associated with the electric

and magnetic fields are equal (ue = um). Let us define the vector
S =	c
	4π E × B ,	(2.1.21)

called the Poynting vector, which is directed along the direction of propagation. It describes the energy transported, i.e. the flow of energy per unit area and per unit time, as can be easily seen by

du
· S = − dt	,	(2.1.22)

an expression of energy conservation. We will use these relations in Section 2.3.1 when the energy dissipated in the medium is discussed.

2.2 Propagation of electromagnetic waves in the medium

Because we are interested in the optical properties of solids, we now have to discuss the influence of matter on the wave propagation. First we will define some material parameters and see how they enter Maxwell’s equations.

2.2.1 Definitions of material parameters

The presence of a medium in electric and magnetic fields may lead to electric dipoles and magnetic moments, polarization charges, and induced current. Clearly the electric and magnetic fields will not be uniform within the material but fluctuate from point to point reflecting the periodicity of the atomic lattice. For wavelengths appreciably larger than the atomic spacing we can nevertheless consider an average value of the electric and magnetic fields. These fields, however, are different compared with the fields in vacuum; consequently the electric displacement D instead of the electric field strength E, and the magnetic induction B associated with the magnetic field strength H, are introduced to account for the modifications by the medium.

The total charge density ρ = ρtotal used in Coulomb’s law (2.1.1d) now has two components

ρtotal = ρext + ρpol ,

(2.2.1)

an external charge ρext added from outside and a contribution due to the spatially varying polarization

ρpol = − · P .

(2.2.2)

16	2 The interaction of radiation with matter

For a homogeneous polarization the positive and negative charges cancel everywhere inside the material, leading to no net charge ρpol in our limit at large enough wavelengths.

Let us assume that there is no external current present: Jext = 0. The total current density J = Jtotal entering Maxwell’s equation then consists of a contribution Jcond arising from the motion of electrons in the presence of an electric field and of a contribution Jbound arising from the redistribution of bound charges:

Jtotal = Jcond + Jbound .

(2.2.3)

Ohm’s law, i.e. current proportional to electric field E, is assumed to apply to this conduction current:

Jcond = σ1E .

(2.2.4)

σ1 is the conductivity of the material; since we limit ourselves to the linear response, σ1 does not depend on the electric field strength. We also assume it should not be changed by an external magnetic field, implying that we do not consider magnetoresistive effects. The time dependent polarization ∂P/∂t or spatially dependent magnetization × M contribute to what is called the displacement current

or bound current density Jbound.

Let us now include these terms explicitly into Maxwell’s equations (2.1.1). The first two equations do not change because they contain neither current nor charge densities. Upon substitution, Eq. (2.1.1c) becomes

1			∂E(r, t)			4π
× B(r, t) −					=			Jtotal(r, t)
	c			∂t			c			(r, t)
						4π			∂
					=			Jcond(r, t) +		P	+ c × M(r, t)
							c			∂t

and Eq. (2.1.1d) changes to

· E(r, t) = 4πρtotal(r, t) = 4π [ρext(r, t) − · P(r, t)] .

These reformulations of Maxwell’s equations suggest the definitions of the electric displacement D and magnetic field strength H. The electric field strength E and the electric displacement (or electric flux density) D are connected by the dielectric constant (or permittivity) 1:

D = 1E = (1 + 4π χe)E = E + 4π P ,

(2.2.5)

where χe is the dielectric susceptibility and P = χeE is the dipole moment density or polarization density. The dielectric constant 1 can be either positive or negative. Similarly, the magnetic field strength H is connected to the magnetic induction B

2.2 Propagation of electromagnetic waves in the medium	17
by the permeability µ1:
B = µ1H = (1 + 4π χm)H = H + 4π M ,	(2.2.6)

where χm is the magnetic susceptibility and M = χmH is the magnetic moment density, or magnetization. The quantities 1, χe, µ1, and χm which connect the fields are unitless. The magnetic susceptibility χm is typically four to five orders of magnitude smaller (except in the case of ferromagnetism) than the dielectric susceptibility χe, which is of the order of unity. For this reason the diaand para-magnetic properties can in general be neglected compared to the dielectric properties when electromagnetic waves pass through a medium. Throughout the book we shall not discuss the properties of magnetic materials and therefore we assume that µ1 = 1 (unless explicitly indicated otherwise), although we include it in some equations to make them more general. We assume furthermore that there are no magnetic losses and consequently that the imaginary part of the permeability is zero.

2.2.2 Maxwell’s equations in the presence of matter

With these definitions we can rewrite Maxwell’s equations (2.1.1) in the presence

of matter. The rearrangement of the terms and some substitution yield

× E +

1 ∂B

=

0

,

(2.2.7a)

c

∂t

· B

=

0

,

(2.2.7b)

1

∂D

4π

× H −

=

Jcond ,

(2.2.7c)

c

∂t

c

· D

=

4πρext .

(2.2.7d)

Using Eqs (2.2.5) and Ohm’s law (2.2.4) and recalling that there is no external current, Eq. (2.2.7c) can be written as

c × H = −iω 1E + 4π σ1E = −iω ˆE ,

where we have assumed a harmonic time dependence of the displacement term ∂D/∂t = −iωD, and we have defined the complex dielectric quantity4

ˆ = 1 + i	4π σ1	= 1 + i 2 .	(2.2.8)
	ω

By writing D = ˆE, the change in magnitude and the phase shift between the

4Whenever we need to distinguish complex quantities from real ones, we indicate complex quantities with a circumflex, and their real and imaginary parts by subscripts 1 and 2, except if they have well established names.