Spectroscopic ellipsometry from 10 to 700 K

1 Introduction

Spectroscopic ellipsometry [1], [2], [3] is an optical reflection technique based on the polarization of electromagnetic waves. It is most commonly used to determine the thicknesses of transparent thin layers on thick single-side polished substrates (such as SiO₂ on Si) in research and development labs or for manufacturing control, for example in semiconductor wafer manufacturing [4] or in the deposition of optical coatings. Ellipsometry can also measure the refractive index of transparent bulk materials with moderate accuracy or determine the real and imaginary parts of the complex refractive index (including the absorption and extinction coefficients) of highly absorbing bulk materials with an absorption coefficient larger than about 1000 cm⁻¹. Depending on the complexity of an optical layer stack, it may also be possible to determine optical constants and thicknesses of thin layers [3].

Beyond these technology applications, spectroscopic ellipsometry is also a materials science tool for the study of materials and their energy spectra. Let’s assume for the moment that our material is described by a Hamiltonian H ₀ with many eigenstates of non-interacting energy levels. Some of these occupied energy levels can interact with light by absorbing a photon, which results in an excited state. Such energy levels form the initial and final states of allowed transitions with energy E _j ⁰, which we call oscillators. They result in peaks or other structures in the dielectric function ϵ(ω), where ω is the angular frequency. Other occupied energy levels cannot absorb a photon. They are related to forbidden transitions. For example, infrared-active phonons can absorb infrared photons, while Raman-active or silent phonons cannot.

For many materials the oscillators E _j ⁰ can be separated into three groups [5, 6]: (1) Oscillators with E _j ⁰ = 0 result from the absorption of light by free electrons and holes (if any). This is the Drude response of metals and doped semiconductors. (2) Lattice absorption in insulators and semiconductors occurs in the far- and mid-infrared spectral region up to about 1500 cm⁻¹ or 186 meV. (3) Electronic transitions usually have energies of 500 meV or higher. They can be observed in the near-infrared, visible, and ultraviolet spectral regions. Therefore, many materials have a region between 200 and 500 meV, where ϵ(ω) is real and nearly constant. The value of ϵ(ω) in this region is called the high-frequency dielectric constant ϵ_∞. (We should note that some materials like InSb or α-tin have very small band gaps and therefore this spectral separation of oscillators may not always be possible). With this classification of oscillators in mind, we can separate the dielectric function into three terms [7]

representing the Drude, lattice, and electronic absorption.

So far, we have assumed that the oscillators E _j ⁰ are non-interacting. They arise from the eigenstates of the unperturbed Hamiltonian H ₀, which is independent of temperature. The oscillators at E _j ⁰ are therefore δ-functions with a very small linewidth. We now turn on the interactions between the energy levels or with other quasiparticles and assume that they can be described by an interaction Hamiltonian H’(T). The complete Hamiltonian is therefore [8]

For example, the resistivity of a metal is limited by scattering of electrons with the lattice. Similar electron–phonon interactions broaden the final and/or initial states of electronic transitions. Another interaction is the anharmonic decay of optical phonons into acoustic phonons. Equation (2) is therefore the principal motivation for temperature-dependent ellipsometry. As we vary the temperature, we can study the eigenstates associated with the perturbed Hamiltonian H(T). By combining measurements over a broad temperature range, we can subtract the effects of the perturbation H′(T) and determine the energy levels of the unperturbed Hamiltonian H ₀, for example for comparison with ab initio lattice dynamics or band structure calculations. Additionally, the variations of the energy levels with temperature provide information about the nature of the interactions of electrons and holes with other elementary excitations such as phonons, magnons, etc.

The temperature-dependent interaction Hamiltonian H′(T) adds a complex self-energy Σ(T) to the unperturbed oscillator energies E _j ⁰. The oscillator energies observed in the experimental dielectric functions are therefore [8]

The real part Re[Σ(T)] is often (but not always) negative and usually leads to a redshift of oscillators with increasing temperature. The imaginary part Im[Σ(T)] is always positive and equals the homogeneous Lorentzian lifetime broadening of an oscillator. (For purposes of this review, we restrict ourselves to highly crystalline materials and ignore the inhomogeneous broadenings often found, for example due to impurities, extended defects, grain boundaries, variations of composition, etc.).

The self-energy Σ(T) does not vanish at low temperatures. For example, due to the zero-point energy of a quantum mechanical oscillator, the amplitudes of lattice vibrations do not vanish at T = 0. Therefore, we expect to find a red-shift and a broadening of transitions even at very low temperatures. There may also be temperature-independent self-energy contributions, for example due to alloy scattering in random alloys such as Si–Ge solid solutions. Nevertheless, low-temperature measurements usually yield the sharpest peaks with the smallest broadenings. This can help resolve closely spaced oscillators.

The self-energy Σ(T) usually varies quadratically at low temperatures and then changes linearly at higher temperatures. Therefore, much information can be gained from measurements between 80 and 700 K, if liquid helium is not available for experiments. Measurements above room temperature up to 700 K or higher are very important to understand the origin of Σ(T).

In our discussion of Eqs. (2) and (3), we have neglected that the unperturbed eigenstates E _j ⁰ of H ₀ depend on the lattice constant and therefore also show temperature dependence due to thermal expansion (TE). For optical phonons, this temperature dependence due to thermal expansion is given by [9]

where E _j ⁰ is the unperturbed energy at T = 0, ΔE _j ^TE(T) the shift of this eigenstate due to thermal expansion, α _l(T) the linear coefficient of thermal expansion (which depends on temperature), and γ the Grüneisen parameter of this phonon mode. A similar expression applies to the thermal expansion contribution to the shifts of band gaps, which involves the product of the bulk modulus and the pressure dependence of the band gap at constant temperature instead of the Grüneisen parameter [10]. The shift due to thermal expansion in Eq. (4) is usually smaller than the real part of the self-energy Σ(T).

2 Experimental details

2.2 Ultrahigh vacuum chamber design for heating and cooling

Many researchers have realized that only an ultrahigh vacuum (UHV) chamber can consistently protect samples from condensation at low temperatures and oxidation at high temperatures. A typical example is given in Figure 1, which shows a Lakeshore Janis ST-400 cryostat mounted on a J. A. Woollam Co. vertical variable angle of incidence spectroscopic ellipsometer (VASE). The bottomless exterior stainless steel vacuum envelope has four windows mounted at about 0° and 180° for transmission measurements and at about ±70° for reflection and ellipsometry measurements. The windows are misaligned slightly to avoid multiple reflections. To achieve good vacuum base pressure (as low as 10⁻⁸ Torr at room temperature), the vacuum envelope is mounted directly on top of a small air-cooled turbopump (Agilent TwisTorr 84 FS), which is backed by a roughing pump. The whole assembly is attached to the Woollam VASE instrument, just like other sample stages. The same assembly (with different windows) can also be attached to a Woollam Fourier-transform infrared (FTIR) VASE instrument. With some modifications, cryostat measurements are also possible in the nitrogen-purged vacuum ultraviolet (VUV) VASE instrument [13], [14], [15]. Below 80 K, the pressure usually falls below 10⁻⁸ Torr, since the sample holder acts as a cryopump. At 800 K, the pressure may rise up to 10⁻⁶ Torr or even higher, as the sample outgases. Care must be taken not to let the pressure rise too much. A high pressure when heating the sample indicates that the sample may disintegrate. This may lead to thin layers being deposited on the inside of the vacuum windows, which reduces their transparency and may also disturb the polarization. The vacuum shroud contains an electrical UHV feedthrough to measure the sample temperature with a thermocouple, Si diode, or Pt resistor.

Figure 1:

The Lakeshore Janis ST-400 ultrahigh vacuum cryostat is shown attached to a J. A. Woollam Co. vertical variable angle of incidence spectroscopic ellipsometer (VASE).

Inside the vacuum shroud rests a cold finger insert made from copper with high thermal conductivity. It contains a cryogen reservoir, which is filled with liquid helium or nitrogen for cooling. The top of the insert is heated to prevent the UHV seal between the insert and the stainless steel vacuum shroud from freezing (which may cause vacuum leaks). Sample temperatures near 10 K can be achieved with helium and about 80 K with nitrogen. The evaporating cryogen can be captured at the vent port or allowed to escape into the ambient. The cryogen is usually supplied from a Dewar through an evacuated transfer arm. If a recirculating helium cooler is used instead, one has to worry about vibrations being transferred to the optical setup, which may increase the noise in the ellipsometry measurement. This will require some testing and perhaps isolation measures.

Heating of the sample is achieved with a 50 Ω resistor located just under the cryogen reservoir. To reduce corrosion of the cold finger insert at high temperatures, the vent port must be evacuated with a roughing pump while the cryogen intake is plugged with a rubber stopper. Even with this precaution, corrosion is not avoidable in the long run. The heater or thermocouple may fail or vacuum leaks may develop. The cold finger should therefore be treated as a consumable, unfortunately, for measurements at the highest temperatures. Additional details, photos, and drawings can be found at the manufacturer’s web site [16].

2.3 Mounting of samples

Below the cryogen reservoir, a two-inch long semicylindrical copper sample holder with a one-inch diameter is attached. Slotted through-holes allow a slight shift of the sample holder along the surface normal to accommodate different sample thicknesses. How to mount the sample on this sample holder deserves some discussion, especially for small samples. The sample must have good thermal contact with the sample holder to establish thermal equilibrium. This requirement suggests the use of a paste based on silver or carbon nanoparticles. However, this paste may outgas at higher temperature and may get redeposited on the windows, which is detrimental, or on other surfaces. Furthermore, it is important (especially for infrared measurements, where the copper sample holder has a high reflectivity) that the ellipsometry beam only hits the sample, not the sample holder. If light reflected by the sample holder hits the detector, then the ellipsometry measurements yield false results. Usually, this can be recognized by a higher than normal depolarization signal. To avoid such sample holder reflections, one may cover the exposed sample holder (areas not covered by the sample) with a silver or carbon-based paint, which scatters the light rather than reflecting it. Again, this paste may outgas at high temperatures. It is even better to have multiple sample holders available, to match the width of the sample with that of the sample holder. The highly reflective copper sample holder may also increase the backside reflections of transparent samples. Their back side will have to be roughened even more than for measurements of samples in ambient at room temperature on less reflective sample holders.

For our purposes and for sufficiently large and mechanically stable samples, we found it best to mount the sample mechanically (without any paste) with horizontal stainless steel strips that push the sample against the sample holder, see Figure 2. The sample should have the full width of the sample holder and the beam should not hit the stainless steel strips, unless they are sufficiently rough or covered with a nonreflective paste. The steel strips are attached to the sample holder with spring-loaded screws. Different sample heights can be accommodated by a copper sample holder with several pairs of screw holes. To achieve the best vacuum, the screw holes should go through the entire sample holder. If that is not feasible, then vented screws must be used.

Figure 2:

A Si sample mounted on the copper sample holder with horizontal stainless steel strips.

Mounting the sample in the correct position can be tricky, especially for small samples. It is better not to seal the cryostat until the correct position of the sample has been verified with a test measurement. Given all these considerations, the ideal sample size for commercial instrumentation is probably about 20 by 20 mm²; the smallest sample size is about 10 by 10 mm². It is easier to measure small samples on the VASE because it can be seen clearly where the white light beam hits the sample. On the FTIR-VASE, the infrared beam is not visible; on the VUV-VASE, there is no optical access to see the beam. In both cases, small samples are harder to measure. Some researchers have built custom sample holders to measure small samples at the expense of intensity.

2.6 Black-body radiation

At elevated temperatures, the sample holder and the gold-plated heat shield can produce significant amounts of black-body radiation. Modern ellipsometers measure the reflected intensity for each configuration of the polarizing elements with and without the incident light hitting the sample, either by employing a shutter or by chopping the incident beam. This allows to subtract the dark currents in the detection system and intensity from spurious light sources, such as ambient room light or black-body radiation, but only within the linearity limits of the detector and the associated electronics. At the highest temperatures, the black-body radiation can saturate the detector used in ellipsometry experiments. This is true particularly for near-infrared VASE measurements using an InGaAs detector. A Si photodiode or a photomultiplier (used at larger photon energies) have a larger range of linearity and are affected less than the InGaAs detector at lower energies (where black-body radiation is stronger). It may also be possible to use customized detection electronics, targeted to allow a higher total light intensity without saturating the circuit by reducing the overall amplification factor of the circuit. Using a photomultiplier tube, ellipsometry measurements of Si between 2.0 and 4.3 eV up to 1200 K have been reported [12].

The effect of black-body radiation is shown in Figure 3 based on work described in the supplementary material of ref. [27]. For high-temperature VASE measurements, we therefore place a small carefully aligned aperture just outside the exit window of the cryostat. The aperture is just larger than the reflected beam and therefore reduces the overall black-body radiation. This procedure was not needed for FTIR-VASE measurements at high temperature [28].

Figure 3:

Pseudodielectric function of Ge at 583 K from 0.5 to 1.0 eV with and without an aperture (iris) placed between the sample and the detector to reduce the effects of black-body radiation. Data from ref. [27].

2.7 Surface effects in UHV during heating and cooling

For precise ellipsometry measurements on bulk materials, surface layers (including surface roughness, native oxides, and adsorbed overlayers such as water) must be minimized as much as possible. This can be achieved with ex situ as well as in situ techniques. Wet cleaning techniques have been described for many semiconductors to reduce the native oxide thickness while keeping the sample in a flow of dry nitrogen gas during the ellipsometry measurement [29]. For oxides, a wet clean using organic solvents followed by UV ozone clean can be used [30, 31]. This UV ozone clean is also effective to remove water from metal surfaces [32]. For bulk Ge and Ge–Sn alloys, we developed a hybrid technique involving a wet ultrasonic clean in isopropanol and water followed by UV ozone clean [27]. The effectiveness of each clean is confirmed with an ellipsometry measurement before mounting the sample in the cryostat for temperature-dependent measurements. The cleanest surface will have the lowest/highest value of the ellipsometric angle Δ (for an incidence angle lower/larger than the Brewster angle) below the band gap (for semiconductors and insulators) and the highest value of 〈ϵ ₂〉 at the E ₂ critical point (for semiconductors) [29, 33]. For insulators, the cryostat windows and surface overlayers both affect primarily the ellipsometric angle Δ. Therefore, it can be difficult to distinguish between window and surface effects in a measurement.

After the best possible ex situ preclean for the material, the sample is immediately loaded in the UHV chamber, but the pumps are not turned on yet. An ellipsometry measurement (with air still in the cryostat) should yield the same pseudodielectric function as the measurement just before the sample was loaded in the cryostat. If the results are different and cannot be explained with a gradual growth of surface overlayers, then this indicates an issue with the parameters describing the window effects. The window calibration may have to be repeated using a SiO₂ on Si calibration sample. After this control measurement, the pumps are turned on. It will take about 24 h until the base pressure of 10⁻⁸ Torr is reached. In most cases, the sample is then heated to 800 K overnight to outgas any residual adsorbed surface layers, followed by a cooldown of the sample to room temperature and another control measurement.

An example of this process for bulk GaP is shown in Figure 4 from the supplemental material in ref. [28]. We report the surface layer thickness in our experiments, modeled as a 50/50 mixture of GaP and voids within the Bruggeman effective medium approximation. We call this the apparent surface layer thickness, because it might be affected by systematic errors from the cryostat windows. An initial measurement of the GaP sample in air (without windows) finds a surface layer thickness of 38 Å, shown by the green data point. The sample is then mounted in the cryostat followed by pumping to UHV conditions. This reduces the surface layer thickness to 29 Å, as shown by the blue data point. This reduction might be due evaporation of some of the overlayer, but it could also be due to errors caused by our windows. We then heat the sample to 800 K for several hours and let it cool down to room temperature overnight. This reduces the surface layer thickness to 9 Å, as shown by the red data point. Most likely, this value of 9 Å represents the residual surface roughness, after all adsorbed contaminants have evaporated. We now start our temperature series at 80 K, where the surface layer thickness is 19 Å. We gradually increase the temperature to 275 K and measure the pseudo-dielectric function at each step for several hours. This increases the surface layer thickness to 48 Å, as ice forms on the sample. At the next temperature (300 K), the surface layer thickness is only 18 Å, because most of the ice has evaporated. As we heat the sample to 325 K, the surface layer thickness is 8 Å and it gradually is reduced to zero as we heat towards 700 K. It is not likely that the surface layer thickness will actually vanish at 700 K. It is more likely that our data are affected by experimental errors due to the diamond windows. These windows are not getting warm, even at the highest sample temperatures. Therefore, the errors due to the windows are likely independent of temperature. It is likely that we underestimate the surface layer thickness in the cryostat by about 10 Å, due to the retardance of the diamond windows. This discussion emphasizes that one must be careful when interpreting changes of optical constants with temperature. A careful preparation of the sample is required to minimize surface overlayers. It can be very helpful to heat the sample for several hours to reduce airborne molecular contamination, if this does not change the properties of the sample. It is inevitable for ice to form on the sample below room temperature, even under the best (10⁻⁸ Torr) vacuum conditions achievable in our setup.

Figure 4:

Evolution of the apparent roughness layer thickness on GaP during a temperature-dependent ellipsometry measurement. Data from ref. [28].

A similar discussion of the surface layer thickness during a temperature-dependent ellipsometry measurement for bulk Ge can be found in the supplemental material of ref. [27], see Figure 5. Below room temperature, the apparent native oxide thickness increases gradually as the measurements progress, since ice from the residual pressure in the UHV chamber forms on the sample. Above room temperature, the ice desorbs and the surface layer thickness decreases. Above 500 K, the surface layer increases due to thermal oxidation of Ge [34].

Figure 5:

Thickness of the native oxide on Ge as a function of temperature. Data from ref. [27].

Drastic differences of the pseudodielectric function at room temperature before and after heating in the UHV chamber can also be found for bulk nickel [35, 36]. Since surface overlayers on Ni desorb near the Curie temperature at around 600 K, it is easy to confuse surface effects with changes in the magnetic structure of Ni [36].

Surface oxidation can be even more pronounced in nitrogen-purged ellipsometry chambers at very high temperatures [12].

3 Temperature dependence of the refractive index of insulators

Ellipsometry is not the most accurate technique to determine the refractive index of bulk transparent materials. Much higher accuracy can be achieved with minimum-deviation prism methods, but such setups are rare [45], [46], [47]. Interferometric techniques [48, 49] are also competitive. Nevertheless, ellipsometry offers a convenient method to determine refractive indices over a broad spectral and temperature range.

For insulators and semiconductors without free carriers, the dielectric function in the infrared spectral region can be written as a sum of Lorentzian oscillators [7]

where the sum runs over all infrared-active (transverse optical, TO) phonons with amplitude A _i , angular frequency ω _i and broadening γ _i . The high-frequency dielectric constant introduced earlier can be expressed as [8, 50]

where E _u is the unscreened plasma frequency of the valence electrons [28] and the Penn gap E _Penn represents an average separation of the valence and conduction band. For many semiconductors, E _Penn is approximately the same as the energy of the interband critical point E ₂ or the band gap at the symmetry point X in the face-centered cubic Brillouin zone.

The temperature dependence of the high-frequency dielectric constant is therefore given as [28, 48]

The first term due to volume expansion with the linear thermal expansion coefficient α _l is usually negative. The second term, which describes the pure temperature effect, is usually (but not always) positive, because the Penn gap (E ₂ energy) usually decreases with increasing temperature. Both terms are similar in magnitude and cancel partially.

For spinel (MgAl₂O₄), a temperature coefficient dϵ₁/dT = 3.2 × 10⁻⁵/K was found at 2 eV [51]. A similar value was determined for LaAlO₃ [52]. These values are smaller than the thermal expansion term alone (−10⁻⁴/K) because of the partial cancellation from the second term. dϵ ₁/dT may even change its sign [52] because of the temperature dependence of the various parameters in Eq. (7).

The temperature dependence of ϵ _∞ for GaP [28] is shown in Figure 6. At room temperature, the rate of change of ϵ _∞ is about 4.5 × 10⁻⁴/K, which compares favorably with the prediction of 6.4 × 10⁻⁴/K given by Eq. (7). Using the Lyddane–Sachs–Teller relation

where ω _TO and ω _LO are the angular frequencies of the transverse and longitudinal optical phonons in a zinc blende crystal, it is also possible to calculate the static dielectric constant ϵ _s from the high-frequency one. This result is also shown in Figure 6. Such results are typical. A similar value of dϵ _∞/dT = 1.6 × 10⁻⁴/K for the ordinary optical constants was found for wurtzite ZnO [53]. Note that dϵ/dT = 2ndn/dT. Other examples for silicon and germanium, where ϵ _s = ϵ _∞, are given in Refs. [12, 54].

Figure 6:

Temperature dependence of the (a) high-frequency and (b) static dielectric constant of GaP. Results from the literature at 300 K are also shown. Reprinted from ref. [28] with the permission of AIP Publishing.

In low-symmetry monoclinic Ga₂O₃ crystals, the high-frequency dielectric function ϵ _∞ is a tensor with four different elements, including one off-diagonal element. Using Mueller matrix ellipsometry of two Ga₂O₃ single crystals with different surface orientations, Mock et al. [55] could determine the temperature dependence of all four tensor elements of ϵ _∞ from extrapolating the dielectric tensor to low frequencies. They also determined the temperature dependence of the energies of six different critical points, which are required to fit the dielectric tensor of Ga₂O₃.

It is rather unusual, if ϵ ₁ or the refractive index n decrease with increasing temperature, but it is possible due to phase transitions or in the vicinity of interband optical transitions. This occurs, for example, in the mineral delafossite CuFeO₂, where ϵ ₁ at 1 eV decreases with increasing temperatures below 50 K due to magnetic phase transitions at 11 K and 16 K [56]. More commonly known is the temperature dependence of the static dielectric constant of ceramics such as BaTiO₃ used as capacitor materials, where the crystal symmetry is lowered as the temperature decreases [57].

4 Temperature dependence of band gaps and broadenings

The temperature dependence of band gaps and broadenings has been measured for many different semiconductor materials [8]. As an example, we show the dielectric function of germanium near the direct band gap E ₀ between 80 and 450 K in Figure 7. Its imaginary part ϵ ₂ shows a step-like absorption edge, which is accompanied by a discrete exciton peak at low temperatures and broadened at higher temperatures. The corresponding peak of ϵ ₁ is Kramers–Kronig consistent.

Figure 7:

Temperature dependence of the (a) real and (b) imaginary parts of the dielectric function of Ge near the direct band gap E ₀ and (c, d) their second derivatives with respect to photon energy calculated using an extended Gaussian (EG) numerical filter. The lines show fits to the data using the Elliott–Tanguy oscillator. Reprinted from ref. [54] with the permission of AIP Publishing.

For the precise determination of the band gap and broadening at each temperature, one calculates the second derivatives of the spectra with respect to photon energy using sophisticated numerical methods, which remove the nonresonant background from higher-energy interband transitions [27, 54]. These derivatives are then fitted with analytical oscillator lineshapes, for example those introduced by Aspnes [58, 59] for interband critical points with parabolic bands of the form

where A, ϕ, E _g , and Γ are the amplitude, phase angle, energy, and Lorentzian broadening of the critical point and B is the nonresonant background from other critical points. ℏ is the reduced Planck’s constant. This lineshape does not work well for the direct band gap E ₀ of Ge, which cannot be described as a Lorentzian peak (μ = 1) or a square-root onset of absorption for direct transitions (μ = −0.5). Instead, we have used the Elliott–Tanguy oscillator lineshape to describe the dielectric function [54, 60], which includes the excitonic Sommerfeld enhancement of band-to-band transitions. Results are shown by the lines in Figure 7.

The only adjustable parameters to achieve the fits shown in Figure 7 are the energy and broadening of the direct gap (and two Sellmeier parameters for the nonresonant contributions to ϵ ₁), because the peak amplitude can be calculated from the momentum matrix element within k → ⋅ p → theory [54]. Results are shown by symbols in Figure 8. Energies and broadenings vary quadratically at low temperatures and linearly at higher temperatures. Empirically, this is described by a Bose–Einstein model

and a similar equation for the broadenings [40]. E ₀ is the transition energy derived from eigenstates of the unperturbed Hamiltonian H ₀ in Eq. (2), which are perturbed through the interaction with another quasiparticle (usually a phonon or magnon) with energy Ω. E _b is the coupling strength and k the Boltzmann constant. The second term in Eq. (10) represents the real part of the complex self-energy Σ introduced in Eq. (3). This model neglects the redshift of band gaps due to thermal expansion of the crystal, which is usually smaller than the scattering shift described by Eq. (10).

Figure 8:

Temperature dependence of the energy and Lorentzian broadening of the direct gap of Ge (symbols) fitted with a Bose–Einstein model given by Eq. (10). See ref. [54] for more details.

As a second example, the dielectric function of Si between 2.0 and 4.3 eV at temperatures up to 1123 K is shown in Figure 9. It is dominated by the critical points E ₁ around 3.4 eV and E ₂ around 4.2 eV, which can be fitted with Eq. (9) to determine the energies and broadenings. These show a similar behavior as the direct gap of Ge, see Figure 8, and can also be described with a Bose–Einstein model (10).

Figure 9:

Temperature dependence of the dielectric function of undoped Si from 298 to 1123 K. Reprinted from ref. [12] with the permission of AIP Publishing.

The precise numerical procedure to fit critical point spectra can have a significant impact on the resulting band gaps and broadenings [61]. For example, fitting the dielectric function or its first, second, or third derivatives can yield different results [12]. Since the excitonic phase angle ϕ and the energy E _g are strongly correlated, it is possible to obtain an invalid temperature dependence of the energy E _g if ϕ is treated as a temperature-dependent parameter [12]. On the other hand, the excitonic phase angle ϕ should depend on temperature, since the excitonic Coulomb force between the electron and the hole decreases with increasing temperature [40]. Experimentally, window effects and surface layers will also impact the fitted phase angle ϕ and therefore can slightly affect the energies. This problem has been solved for the direct band gap of Ge (see Figure 8) where the introduction of temperature-dependent amplitudes and phase angles could be avoided, since the excitonic effects were included explicitly with the Elliott–Tanguy model [54].

The optical spectra of wurtzite compounds like ZnO near the direct band gap are more complicated to fit, because the crystal field splitting and the small spin–orbit splitting result in three excitons with several exciton-phonon replicas. Once these effects are properly taken into account, the temperature dependence of the band gap energy and broadening of wurtzite ZnO [53] resembles the results for Ge shown in Figure 8.

To understand the optical spectra and band structure of wide band-gap materials, ellipsometry measurements in the vacuum-ultraviolet spectral range are required. Results up to 8.5 eV below room temperature have been reported for In₂O₃ [13], Ga₂O₃ [14], and CuI [15]. As shown in Figure 10, CuI shows the typical E ₀, E ₁, E ₀’, and E ₂ critical points of a compound semiconductor, but shifted to higher energies. The broadenings and energies of these critical points follow Eq. (10). The excitonic enhancement of the E ₀ exciton is quite pronounced at 10 K and decreases at higher temperatures [15].

Figure 10:

Temperature dependence of the dielectric function of a thin CuI layer on sapphire from 10 to 300 K. The spectra are shifted vertically for clarify. The inset shows ϵ ₂ near the excitonic E ₀ resonance. Reprinted from ref. [15] with the permission of AIP Publishing.

Interband critical points are observed not only in semiconductors, but also in metals. A recent example is the temperature dependence of the dielectric function of Ni from 77 to 770 K [62]. Within the framework of Eq. (1), the contributions of the free carriers are described with two Drude oscillators and interband transitions with four Lorentzians, the strongest one at 4.8 eV. Fitting the temperature dependent redshift of this main peak with Eq. (10) results in a boson energy Ω = 53 ± 3 meV. This value is larger than optical phonon energies of Ni (below 30 meV), but of the right magnitude for a magnon energy. We therefore conclude that the main absorption peak of Ni at 4.8 eV is strongly perturbed by magnons (quantized spin waves), which lead to a red shift of this peak with increasing temperature.

While the redshift of the main absorption peak of Ni at 4.8 eV with increasing temperature is similar to Si, Ge, and many other semiconductors, the temperature dependence of the broadening of this peak shows a very unusual behavior, see Figure 11. Instead of a gradual increase of the broadening with increasing temperature (as seen in Figure 8 for the direct band gap of Ge) we observe a decrease of the peak width, which seems to follow the normalized magnetization M/M ₀ below the Curie temperature T _C and remains constant above T _C . Our interpretation is that the main peak of Ni is a doublet below T _C arising from two bands in the vicinity of the L-point separated by the exchange splitting of the d-band [38], which is on the order of 0.38 eV. The exchange splitting vanishes above T _C and the two peaks merge into a single peak, which reduces the broadening as shown in Figure 11. The temperature dependence of the dielectric function of Ni displays several other interesting phenomena related to its magnetic properties, which are beyond the scope of this review [62].

Figure 11:

Broadening of the main peak of Ni at 4.8 eV (symbols) and normalized magnetization (line). Data from ref. [62].

5 Anharmonic phonon decay

Scattering with other particles (phonons or magnons) by electrons and holes participating in interband transitions affects the dielectric function in the near-infrared to UV spectral range, as discussed in Section 4. Similarly, the lattice absorption in insulators and semiconductors by long-wavelength infrared-active optical phonons also changes with temperature.

We select the wide band-gap semiconductor GaP as an example, see Figure 12. With increasing temperature, the transverse optical (TO) phonon peak of ϵ ₂ shifts towards lower energies and broadens. The peak amplitude also decreases, because [28]

Figure 12:

Temperature dependence of the dielectric function of GaP in the region of lattice absorption (dashed) in comparison with the literature (solid). Reprinted from ref. [28] with the permission of AIP Publishing.

The infrared dielectric function can be fitted with different oscillator models [7, 28] to determine the phonon parameters. The peak of ϵ ₂ occurs at ω _TO while the peak of the loss function − Im 1 ϵ determines ω _LO.

The temperature dependence of the TO and LO energies of GaP [28] is shown in Figure 13. Both phonon modes show a distinct redshift with increasing temperature, similar to the redshift of the direct band gap of Ge shown in Figure 8. The thermal expansion contribution to this redshift is small (dashed). The corresponding TO and LO broadenings increase with temperature [28], see Figure 14. Redshifts and broadenings of phonons have been studied for decades with Raman spectroscopy [9], but the use of FTIR ellipsometry for this purpose is new.

Figure 13:

Temperature dependence of (a) transverse and (b) longitudinal optical phonon energies of GaP (·) in comparison to a Bose–Einstein model given by Eq. (10) (solid). The thermal expansion contribution from Eq. (4) is shown by the dashed line. Raman shifts (×) from ref. [63] are shown for comparison. See ref. [28] for details.

Figure 14:

Broadenings of the transverse (·) and longitudinal (∘) optical phonons in GaP as a function of temperature. The lines show a fit with Eq. (10). Data from ref. [28].

The theory of these temperature effects on phonons is complicated [9, 63], but can be summarized as follows: An optical phonon has a finite lifetime on the order of a few 10⁻¹² s. Through an anharmonic perturbation of the Hamiltonian H ₀ with self-energy Σ, long-wavelength optical phonons can decay into two or more acoustic phonons. This leads to the observed shifts and broadenings. An ab initio calculation summing over all possible decay paths in the Brillouin zone has been performed by Debernardi [64, 65]. Empirically, we can use a Bose–Einstein expression (10) to describe the redshifts and broadenings, as shown by the solid lines in Figures 13 and 14.

6 Temperature dependence of the Drude response

The temperature dependence of the Drude contribution to the dielectric function in metals or doped semiconductors is determined by two parameters [7]: (1) the density of free carriers, which determines the unscreened plasma frequency E _P ² = ℏ ² ne ²/ϵ ₀ m ₀ m ^*, where ϵ ₀ is the permittivity of vacuum, m ₀ the free electron mass, and m ^* an effective mass related to the curvature of the band, and (2) their scattering rate (proportional to the Drude broadening), which is related to the mobility. There may also be secondary effects, such as the temperature dependence of the effective masses [54]. Electron–phonon (or hole–phonon) scattering is often the dominating broadening mechanism in semiconductors [8] and metals [66].

In a metal like Au or Ni, the density of free carriers is nearly constant, given by the density of states at the Fermi level, which only depends weakly on temperature, for example through thermal expansion [66] or small shifts of band energies with temperature. The same is true for a doped semiconductor, where the carrier density is given by the density of activated donors or acceptors (doping density), as long as the temperature is low enough to ignore the intrinsic carriers discussed in Section 7. For metals and doped semiconductors, the temperature dependence of the Drude term is therefore dominated by the broadening parameter, which increases with temperature, similar to the broadening of band gaps and phonons shown in Figures 8 and 14. This explains the increased resistivity (reduced conductivity) at high temperatures in such materials.

In a simple metal like Ag and Au, the Drude contribution to the dielectric function can be described with a single species of carriers, i.e., a single Drude term. At room temperature, the unscreened plasma energy of single-crystalline Au (Ag) is 8.5 (8.9) eV and the Drude broadening is 44 (18) meV [35, 66]. Temperature-dependent Drude parameters for different types of Ag films (single- and poly-crystalline) were recently reported, derived from fits of ellipsometry data taken from 450 to 800 nm [66].

On the other hand, Drude already recognized in 1900 that two Drude terms were required to achieve a good description of the dielectric function of Ni [67], [68], [69]. This idea has not always been well received, but one might perhaps think of s- and d-electrons in transition metals (or electrons and holes in doped semiconductors), which might have different carrier densities and scattering rates. It is more common in the literature to describe the free carrier contribution of metals with a single Drude term, where the scattering rate depends on the photon energy [35, 70]. For this review, we prefer the approach with a finite number of Drude terms, each with a carrier density, effective mass, and scattering rate that are independent of photon energy.

To study the temperature dependence of the Drude parameters in Ni, we measured its dielectric function from 0.1 to 6.0 eV between 77 and 770 K. Since the resonance frequency of free carriers vanishes, because there is no restoring force, one introduces the optical conductivity

which removes the singularity of ϵ(ω) at ω = 0. Results for the real part σ ₁, which corresponds to ϵ ₂, are shown in Figure 15. The temperature dependence of the main peak at 4.8 eV has already been discussed in Section 4.

Figure 15:

Real part of the optical conductivity of Ni from 0.1 to 6.0 eV between 77 and 770 K. Data from ref. [62].

Both Drude scattering rates for Ni are shown in Figure 16. The scattering rate for d-electrons is very large (about 2.9 eV) and nearly independent of temperature. This should be expected, since the strongly localized d-electrons do not contribute much to the electrical conductivity. The scattering rate for s-electrons is much smaller (44 meV at room temperature) and shows a significant increase with temperature as expected for a metal. The slope of the scattering rate for s-electrons changes at the Curie temperature T _C , since the scattering mechanisms are different in the ferromagnetic and paramagnetic phases [71]. This kink is also observed in electrical conductivity measurements of Ni and Ni–Pt alloys [72]. In the paramagnetic phase above T _C , s-electrons with both spin states can scatter into d-bands, because both d-bands are only partially occupied. On the other hand, in the ferromagnetic state below T _C , one d-band is completely full. Therefore, s-electrons with spin up can scatter into the d-bands, while those with spin down cannot. This reduces the scattering rate of s-electrons very significantly in the ferromagnetic phase [71].

Figure 16:

Temperature dependence of the Drude scattering rate for s- (▲) and d-electrons (■) of Ni between 77 and 770 K, determined from fitting the dielectric function. The Curie temperature T _C is shown by the vertical dashed line. Reprinted from ref. [62] with the permission of AIP Publishing.

The squares of the plasma frequencies E _P ² for s- and d-electrons in Ni, which are determined by the ratios of the carrier density to the effective mass, also vary somewhat with temperature, but this is difficult to quantify, because the two plasma frequencies are correlated as fitting parameters in the Drude–Lorentz model used to fit the data shown in Figure 15. We can only state that E _P ² equals 23 ± 1 eV² for the s-electrons and ranges from about 140 eV ² at low temperatures to 155 eV² at high temperatures for the d-electrons. It is helpful to know that ℏ ² e ²/ϵ ₀ m ₀ = 1.379 × 10⁻²¹ cm³ eV². For an effective mass m ^* = 3 for d-electrons, this plasma frequency corresponds to a free carrier density of about 3 × 10²³ cm⁻³. This means that 3.6 d-electrons per atom are near the Fermi level and can participate in electronic transport, an order of magnitude higher than what has been reported [73]. For s-electrons, we find a carrier concentration of 0.25 electrons per atom, which is reasonable [73]. Because of the uncertainty of the effective masses and their variation across the Brillouin zone [74], these carrier densities should only be considered rough estimates.

Another good example for the temperature dependence of the Drude response in complex metal oxides at very long wavelengths is La_2−x Sr _x CuO₄ [26].

7 Temperature-dependent intrinsic carrier concentrations

In an intrinsic semiconductor (without dopants), the free carrier density is a strong function of temperature, through thermal excitation of electrons across the gap into the conduction band, given by [75, 76]

for a direct semiconductor with a single conduction band valley, such as InSb, see Figure 17. m _e and m _h are the density-of-states effective masses for electrons and holes, respectively. The prefactor equals 4.84 × 10¹⁵ cm⁻³K^−3/2. Indirect materials like Ge have a more complex conduction band structure and several minima need to be considered [54]. Because of the smaller band gap of InSb compared to Ge, the intrinsic carrier concentration of InSb is much higher.

Figure 17:

Carrier density of intrinsic InSb (solid) and Ge (dashed) as a function of temperature.

Following Eq. (13), even pure semiconductors should display a Drude contribution to the dielectric function at sufficiently high temperatures. We are not aware of such results for any semiconductor, but narrow-gap compounds such as InSb or InAs would be interesting candidates.

To illustrate this point, we show preliminary results for the temperature dependence of the energies and broadenings of the direct band gap of InSb in Figure 18 [77]. The mid-infrared pseudo-dielectric function of InSb was measured from 80 to 700 K with 16 cm⁻¹ resolution in a UHV cryostat with diamond windows and corrected for a 2.5 nm thick native oxide with a constant ϵ = 3.8. The resulting dielectric function of InSb was fitted from 500 to 5000 cm⁻¹ with a parametric oscillator model similar to ref. [27]. The parameters in this model were the energy E ₀, broadening Γ, and amplitude of the direct band gap and poles in the IR and UV outside the measured spectral range.

Figure 18:

Temperature dependence of the energies E (■) and broadenings Γ (·) of the direct band gap of InSb. The lines show a Bose–Einstein fit to Eq. (10) with parameters E ₀ = 267 meV, E _b  = 20.2 meV, and Ω/k = 175 K. Data from ref. [77].

Below room temperature, the band gap E ₀ follows Eq. (10) with reasonable parameters. Above room temperature, the data deviate from the model (dotted line). Between 300 and 600 K, E ₀ is nearly constant and then even increases at the highest temperature. Our InSb sample melted at a thermocouple temperature of 750 K, just below the reported melting point of InSb of 800 K. Close to the melting point, both E ₀ and Γ show an unexpected (anharmonic) behavior. We believe that thermally activated electron-hole pairs cause a blueshift of the band gap due a thermal Burstein–Moss shift [78]. This needs to be confirmed with detailed calculations.

8 Phase transitions in complex metal oxides

The literature contains many reports on the temperature dependence of the optical conductivity of materials, especially related to magnetic phase transitions, insulator to metal (Mott) transitions, and the superconducting transition of cuprates. Such results were usually obtained by near normal incidence reflectance followed by Kramers–Kronig transform [79]. In a reflectance measurement, the infrared beam can easily be focused on a small sample, while ellipsometry usually requires a much larger sample size. Therefore, ellipsometry studies of phase transitions are rare, since the size of single-crystal metal oxides is often quite small. On the other hand, ellipsometry directly measures the dielectric function of materials (at least in the absence of surface layers) and is not affected by the errors of a Kramers–Kronig transformation. Furthermore, infrared reflectance measurements can usually be carried out at lower photon energies than those accessible with commercial infrared ellipsometers. It can be useful to combine reflectance and ellipsometry measurements to obtain the dielectric function over a broad spectral range. Such an analysis has been performed to investigate the temperature dependence of the optical conductivity of nickelates [80].

As an example, the temperature dependence of the optical conductivity of a NdNiO₃ thin layer grown on SrTiO₃ is shown in Figure 19. Between 0.5 and 6.0 eV, the letters indicate several interband optical transitions between different molecular orbitals. In the mid-infrared range (0.02–0.08 eV), several sharp lattice absorption peaks can be seen. At even lower energies, the optical spectra are dominated by the Drude response. At temperatures below 100 K, the conductivity vanishes at low energies, indicating that the oxide is an insulator. Above 100 K, the optical conductivity is high at low energies and displays a clear Drude response indicating metallic character of the NdNiO₃. The insulator-to-metal transition occurs at about 100 K. The metallic response at the higher temperatures leads to a large broadening of the interband transitions A′ and B. The inset shows the temperature dependence of the plasma frequency, calculated from the optical sum rule [6] with different cut-off energies Ω. In the insulating regime below 100 K, the plasma frequency is low. It rises sharply at the transition temperature and then decreases gradually with increasing temperature. See ref. [80] for more detail.

Figure 19:

Temperature dependence of the optical conductivity of NdNiO₃ on SrTiO₃. The letters indicate interband optical transitions between different orbitals. The inset shows the temperature dependence of the plasma frequency. Note that only data above 60 meV were obtained by ellipsometry, while far-infrared data were derived from infrared reflectance. Reprinted with permission from ref. [80]. Copyright 2011 by the American Physical Society.

9 Phase-change materials

It is well known that the dielectric function of amorphous materials is very different from single- or poly-crystalline materials of the same composition. Crystalline materials show sharp critical points known as van Hove singularities, where unfilled and filled electron bands run parallel in the Brillouine zone, see Figure 9 [12, 29, 50]. Amorphous materials, on the other hand, show a single broad peak in the dielectric function, which can usually be described with a Tauc–Lorentz oscillator [1]. Silicon is a good example, which has been studied extensively in the amorphous state and after annealing at various temperatures [81, 82]. It is possible to determine the adiabatic amorphous to crystalline transition temperature by heating the sample in a UHV cryostat while performing spectroscopic ellipsometry measurements at preselected temperature intervals.

As an example, we show the temperature dependence of the dielectric function of carbon-doped Ge₂ Sb₂ Te₅ (GST) alloys [83] in Figure 20. Such GST alloys are commonly used in optical recording and non-volatile semiconductor memories or as tunable infrared optical filters [84]. They are crystalline at high temperatures and remain crystalline at room temperature if cooled slowly. Rapid cooling, however, leads to an amorphous state with a very different dielectric function.

Figure 20:

Temperature dependence of the dielectric function of Ge₂ Sb₂ Te₅:C. Data from ref. [83].

For this work, 750 nm of GST:C were deposited on Si wafers covered with a 400 nm thick thermal oxide. As deposited, the layers are amorphous with a band gap of about 1 eV, which is indicated by sharp interference fringes in the ellipsometric angles below the gap (not shown). The alloys remain amorphous when heated to 400 K and display a very broad peak of ϵ ₂ centered on 2.5 eV, nearly independent of temperature. At higher temperatures (425–675 K), the onset of absorption decreases below 0.5 eV, the interference fringes disappear, and the main absorption peak is much narrower, indicating a crystalline phase. When heated from 425 to 675 K, the absorption peak blueshifts from 1.0 to 1.5 eV and its integrated intensity increases. This is unusual, since peaks in ϵ ₂ usually redshift with increasing temperature, see Figure 15. At even higher temperatures (above 700 K), the layer evaporates and only the thermal oxide remains on the wafer. This example demonstrates the power of in situ ellipsometry for thermal process monitoring.

10 Summary

Temperature-dependent spectroscopic ellipsometry from the infrared to the ultraviolet spectral region inside a UHV cryostat is a powerful method to investigate the interactions of electronic and vibrational states with other quasiparticles. The required experimental techniques were described in detail and selected examples were presented.