Dressel.Gruner.Electrodynamics of Solids.2003

10−1 eV or smaller. The gap in the superconducting state of aluminum for example is 0.3 meV; in contrast this gap in high temperature superconductors or in materials with charge density wave ground states can reach energies of 100 meV. Similarly, relatively small energy scales of the order of 10 meV characterize impurity states in semiconductors and also lattice vibrations.

Electron–phonon and electron–electron interactions also lead to reduced bandwidth (while keeping the character of states unchanged), and this reduction can be substantial if these interactions are large. This is the case in particular for strong electron–electron interactions; in the so-called heavy fermion materials, for example, the typical bandwidth can be of the order of 1 meV or smaller. The response of collective modes, such as a pinned density wave, lies below this spectral range; charge excitations near phase transitions and in a glassy state extend (practically) to zero energy.

In addition to these energy scales, the frequencies which are related to the relaxation process can vary widely, and the inverse scattering time 1/τ ranges from 1015 s−1 for a metal with a strong scattering process (leading to mean free paths of the order of one lattice constant) to 1012 s−1 in clean metals at low temperatures.

8.2 Response to be explored

In the case of a linear response to an electromagnetic field with a sinusoidal time variation, the response of only one individual frequency is detected, and all the other spectral components are suppressed, either on the side of the radiation source, or on the detection side. Since we consider only elastic light scattering, the frequency will not be changed upon interaction; also the response does not contain higher harmonics. The electric field E(ω) is described by

The current response measured at that single frequency ω is split into an in-phase and an out-of-phase component

which is then combined by the complex conductivity σˆ (ω) at this frequency:

J(ω) = J0 exp{−iωt} = [σ1(ω) + iσ2(ω)]E0 exp{−iωt} = σˆ (ω)E(ω) , (8.2.2)

using the complex notation of frequency dependent material parameters. In other words, the complex notation is used to indicate the phase shift which might occur

between stimulus (electric field) and response (current density). A full evaluation of the response which leads to both components of the optical conductivity obviously requires the measurement of two optical parameters; parameters which are examined by experiment and which are related – through Maxwell’s equations in a medium – to the complex conductivity. At low frequencies, as a rule, both the phase and amplitude of the optical response are measured – such as the resistance and capacitance, the reflected amplitude and phase, or the surface resistance and surface reactance, respectively. Alternatively, if only one parameter is observed, such as the absorbed power, experiments in a broad frequency range have to be conducted in order to perform the Kramers–Kronig analysis (see Section 3.2) for the determination of the complex conductivity. Since a single method does not cover the entire range, it is often necessary to combine the results obtained by a variety of different techniques. The main problem here is the extrapolation to low frequencies (ω → 0) and to above the measurement range (ω → ∞).

There are other issues which have to be resolved by a particular experiment. Assume that we apply a slowly varying time dependent electric field to a specimen with contacts applied at the ends. The purpose of the contacts is to allow the electric charge to flow in and out of the specimen and thus to prevent charges building up at the boundary. The wavelength λ = c/ f for small frequencies f , say below the microwave range, is significantly larger than the typical specimen dimensions. The wavevector dependence of the problem can under such circumstances be neglected. There is a time lag between the electrical field and the current, expressed by the complex conductivity σˆ (ω) = σ1(ω) + iσ2(ω). With increasing frequencies ω, two things happen. The alternating electric field may be screened by currents induced in the material, and the current flows in a surface layer. This layer is typically determined by the parameter called the skin depth δ0 = c(2π ωσ1)−1/2. For typical metals at room temperature the skin depth at a frequency of 1010 Hz is of the order of 10−5 m. Whenever this length scale δ0 is smaller than the dimension of the specimen, the skin effect has to be taken into account when the complex conductivity is evaluated. On increasing the frequency further, another point also becomes important: the wavelength of the electromagnetic field becomes comparable to the dimensions of the specimens, this typically occurs in the millimeter wave spectral range. Well above these frequencies, entirely different measurement concepts are applied. It is then assumed that in the two dimensions perpendicular to the direction of wave propagation the specimen spreads over an area which is large compared to the wavelength, and the discussion is based on the solution of Maxwell’s equations for plane waves and for an infinite plane boundary.

The objective is the evaluation of the two components of the optical conductivity or alternatively the dielectric constant – the parameters which characterize the medium – and to connect the experimental observations to the behavior proposed

by theory. The quantities which describe the modification of the electromagnetic wave in the presence of the specimen under study are, for example, the power reflected off or transmitted through a sample of finite thickness, and also the change of the phase upon transmission and reflection. The task at hand is either to measure two optical parameters – such as the refractive index and absorption coefficient, or the surface resistance and surface reactance, or the amplitude and phase of the reflected electromagnetic radiation. Although this is often feasible at low frequencies where, because of the large wavelength, spatial sampling of the radiation is possible, the objective, however, is difficult to meet at shorter wavelengths (and consequently at higher frequencies) and therefore another method is commonly used: one evaluates a single parameter, such as the reflected or transmitted power, and relies on the Kramers–Kronig relations to evaluate two components of the complex conductivity. Note, however, that the Kramers–Kronig relations are nonlocal in frequency, and therefore the parameter has to be measured over a broad frequency range, or reasonable (but often not fully justified) approximations have to be made for extrapolations to high or to low frequencies.

8.3 Sources

The spectroscopies which are discussed here depend fundamentally on the characteristics of the electromagnetic radiation utilized; consequently a discussion of some of the properties of the sources are in order. As we have seen, the electromagnetic spectrum of interest to condensed matter physics extends over many orders of magnitude in frequency. This can only be covered by utilizing a large variety of different sources and detectors; their principles, specifications, and applicable range are the subject of a number of handbooks, monographs, and articles (see the Further reading section at the end of this chapter).

We distinguish between four different principles of generating electromagnetic waves (Fig. 8.1). At low frequencies mainly solid state electronic circuits are used; they are monochromatic and often tunable over an appreciably wide range. Above the gigahertz frequency range, electron beams are modulated to utilize the interaction of charge and electric field to create electromagnetic waves from accelerated electrons. Thermal radiation (black-body radiation) creates a broad spectrum, and according to Planck’s law typical sources have their peak intensity in the infrared. Transition between atomic levels is used in lasers and discharge lamps. The radiation sources might operate in a continuous manner or deliver pulses as short as a few femtoseconds.

In recent years significant efforts have been made to extend the spectral range of synchrotron radiation down to the far infrared and thus to have a powerful broadband source available for solid state applications. Also free-electron lasers

Fig. 8.1. Ranges of the electromagnetic spectrum in which the different radiation sources are applicable. At low frequencies solid state devices such as Gunn oscillators or IMPATT diodes are used. Up to about 2 THz, coherent monochromatic sources are available which generate the radiation by modulation of an electron beam (e.g. clystron, magnetron, backward wave oscillator). White light sources deliver a broad but incoherent spectrum from the far-infrared up to the ultraviolet; however, at both ends of the range the intensity falls off dramatically according to Planck’s law. In the infrared, visible, and ultraviolet spectral ranges, lasers are utilized; in some cases, the lasers are tunable.

are about to become common in solid state spectroscopy since they deliver coherent but tunable radiation, and also short pulses.

For sources of electromagnetic radiation the bandwidth is an important parameter. Radiation sources produce either a broad spectrum (usually with a frequency dependent intensity), or they are limited (ideally) to a single frequency (a very narrow bandwidth) which in some cases can be tuned. A quantitative way of distinction is the coherence of the radiation, which is defined as a constant phase relation between two beams [Ber64, Mar82]. Time coherence, which has to be discriminated from the spatial coherence, is linked to the monochromaticity of the radiation, since only light which is limited to a single frequency and radiates over an infinite period of time is fully time coherent. Thus in reality any radiation

exhibits only partial coherence as it originates from an atomic transition between levels with finite lifetime leading to a broadening. Monochromatism is determined by the bandwidth of the power spectrum, while the ability to form an interference pattern measures the time coherence. If light which originates from one source is split into two arms, with electric fields E1 and E2, of which one can be delayed by the time period τ – for instance by moving a Michelson interferometer out of balance by the distance δ = τ /2c as displayed in Fig. 10.6 – the intensity of superposition is found to vary from point to point between maxima, which exceed the sum of the intensities in the beam, and minima, which may be zero; this fact is called interference. For the quantitative description a complex auto-coherence function is defined

where E denotes the complex conjugate of the electric field; the period of the sinusoidal wave with frequency f = ω/2π is defined by T = 1/ f . The degree of coherence is then given by γ (τ ) = (τ )/ (0), and we call |γ (τ )| = 1 totally coherent, |γ (τ )| ≤ 1 partially coherent, and γ (τ ) = 0 totally incoherent. By definition, the coherence time τc is reached when |γ (τ )| = 1/e; the coherence length lc is then defined as lc = cτc, where c is the speed of light. lc can be depicted as the average distance over which the phase of a wave is constant. For a coherent source it should be at least one hundred times the period of the oscillation; the wavepackages are then in phase over a distance more than one hundred times the wavelength. If the radiation is not strictly monochromatic, the coherence decreases rapidly as τc increases. A finite bandwidth f reduces the coherence length to lc = c/2 f . In general, the finite bandwidth is the limitation of coherence for most radiation sources.

The spatial coherence, on the other hand, refers to a spatially extended light source. The coherence of light which originates from two points of the source decreases as the distance δ between these positions increases. In analogy to the time coherence, we can define the auto-correlation function (δ) and the degree of coherence γ (δ). Spatial coherence is measured by the interference fringes of light coming through two diaphragms placed in front of the radiation source. The absence of coherence becomes especially important for arc lamps, but also for some lasers with large beam diameters.

8.4 Detectors

Electromagnetic radiation is in general detected by its interaction with matter. The most common principles on which devices which measure radiation are based are

Fig. 8.2. Operating range of detectors. Semiconductor devices, such as Schottky diodes, can be used well into the gigahertz range of frequency. Thermal detectors, such as Golay cells and bolometers, operate up to a few terahertz; the infrared range is covered by pyroelectric detectors. Photomultipliers are extremely sensitive detectors in the visible and ultraviolet spectral ranges.

the photoelectric effect and the thermal effect (heating); less important are luminescence and photochemical reactions. In the first category, photons of frequency ω excite carriers across a gap Eg if h¯ ω > Eg and thus the conductivity increases. The second class is characterized by a change in certain properties of the material due to an increase in temperature arising from the absorption of radiation; for example, the resistivity decreases (semiconductor bolometer) or the material expands (Golay cell). Most detectors are time averaging and thus probe the beam intensity, but non-integrating detectors are also used for measuring the power of the radiation; the time constant for the response can be as small as nanoseconds. Details of the different detector principles and their advantages are discussed in a large number of books [Den86, Der84, Kin78, Key80, Rog00, Wil70]. The various detectors commonly used for optical studies are displayed in Fig. 8.2.

Grating spectroscopy

Fourier transform spectroscopy

Fabry–Perot

resonator

Terahertz time domain spectroscopy

Resonant

cavity

Microstripline

resonator

Time domain spectroscopy

Frequency domain spectroscopy

Energy (eV)

Fig. 8.3. Commonly employed methods of exploring the electrodynamics of metals in a wide spectral range. Also shown are the various energy units used. RF refers to radio frequencies; mw to microwaves; FIR and IR mean far-infrared and infrared, respectively; vis stands for visible, and UV stands for ultraviolet.

8.5 Overview of relevant techniques

The broad range of energy scales we encounter in solids implies that various techniques which are effective at vastly different parts of the electromagnetic spectrum are all of importance, and in fact have to be combined if a full account of the physics under question is attempted. This explains the variety of methods which are

used, the variety of the hardware, the means of propagation of the electromagnetic radiation, and the relevant optical parameters which are measured.

The methods which have been utilized are of course too numerous to review. Some of the most commonly used experimental techniques are displayed in Fig. 8.3 together with the electromagnetic spectrum, as measured by different units. All these units have significance and Table G.4, p. 464, may be helpful. The s−1 or hertz scale is the natural unit for the radiation with a sinusoidal time variation; attention has to be paid not to confuse frequency f and angular frequency ω = 2π f . The energy associated with the angular frequency is h¯ ω; the unit commonly used is electron-volts. The temperature scale is in units for which kBT is important, as it establishes a correspondence between temperature driven and electromagnetic radiation induced charge response. Often this response is fundamentally different in the so-called quantum limit kBT < h¯ ω as opposed to the kBT > h¯ ω classical limit. Finally, the wavelength λ of the electromagnetic radiation is important, not least because it indicates its relevance with respect to typical sample dimensions.

The various methods applied in different ranges of the electromagnetic spectrum have much in common, in particular as far as the principles of light propagation and the overall measurement configurations are concerned. Interferometric techniques work equally well in the infrared and in the microwave spectral ranges – although the hardware is vastly different. Resonant techniques have been also employed in different spectral ranges – the advantages and disadvantages of these techniques do not depend on the frequency of the electromagnetic radiation.

Finally, one should keep in mind that the optical spectroscopy covered in this book is but one of the techniques which examine the charge excitations of solids; complementary techniques, such as electron energy loss spectroscopy, photoemission, or Raman and Brillouin scattering, are also widely used. The response functions which are examined are different for different methods, and often a comparison of information offered by the different experimental results is required for a full characterization of the charge excitations of solids.

References

[Ber64] M.J. Beran and G.B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, NJ, 1964)

[Den86] P.N.J. Dennis, Photodetectors (Plenum Press, New York, 1986)

[Der84] E.L. Dereniak and D.G. Crowe, Optical Radiation Detectors (J. Wiley, New York, 1984)

[Kin78] Detection of Optical and Infrared Radiation, edited by R.H. Kingston, Springer Series in Optical Sciences 10 (Springer-Verlag, Berlin, 1978)

[Key80] Optical and Infrared Detectors, edited by R.J. Keyes, 2nd edition, Topics in Applied Physics 19 (Springer-Verlag, Berlin, 1980)

[Mar82] A.S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982) [Rog00] A. Rogalski, Infrared Detectors (Gordon and Breach, Amsterdam, 2000)

[Wil70] Semiconductors and Semimetals, 5, edited by R.K. Willardson and A.C. Beer (Academic Press, New York, 1970)

Further reading

[Gra92] R.F. Graf, The Modern Oscillator Circuit Encyclopedia (Tab Books, Blue Ridge Summit, PA, 1992)

[Hec92] J. Hecht, The Laser Guidebook, 2nd edition (McGraw-Hill, New York, 1992)

[Hol92] E. Holzman, Solid-State Microwave Power Oscillator Design (Artech House, Boston, MA, 1992)

[Hen89] B. Henderson, Optical Spectroscopy of Inorganic Solids (Clarendon Press, Oxford, 1989)

[Kin95] R.H. Kingston, Optical Sources, Detectors, and Systems (Academic Press, San Diego, CA, 1995)

[Kuz98] H. Kuzmany, Solid-State Spectroscopy (Springer-Verlag, Berlin, 1998)

[Pei99] K.-E. Peiponen, E.M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis

and Optical Spectroscopy, Springer Tracts in Modern Physics 147 (Springer-Verlag, Berlin, 1999)

[Sch90] Dye Lasers, edited by P. Schafer,¨ 3rd edition, Topics in Applied Physics 1 (Springer-Verlag, Berlin, 1990)

[Sch00] W. Schmidt, Optische Spektroskopie, 2nd edition (Wiley-VCH, Weinheim, 2000)

[Yng91] S.Y. Yngversson, Microwave Semiconductor Devices (Kluwer Academic Publisher, New York, 1991)