The factorized plasmon-phonon polariton description of the infrared dielectric function is generalized to include an additional factor to account for the effects of interband electronic transitions. This new formalism is superior to the usual Drude–Lorentz summation of independent oscillators, especially in materials with large transverse-longitudinal optical phonon splittings, multiple infrared-active phonon modes, or high concentrations of free carriers, if a broadband description of the dielectric function from the far-infrared to the vacuum-ultraviolet spectral region is desired. After a careful comparison of both approaches, the factorized description is applied to the dielectric function of undoped and doped semiconductors (GaAs, GaSb, and InAs) and metal oxides from 0.03 to 9.0 eV. Specifically, the authors find that both descriptions of the far-infrared dielectric function yield the same carrier density and mobility, at least for a single species of carriers. To achieve valid results for moderately high doping concentrations, measurements to lower energies would be helpful.
I. INTRODUCTION
The infrared optical spectra of semiconductors obtained from Fourier-transform infrared (FTIR) ellipsometry measurements contain rich information about their free carrier and lattice vibrational properties, such as the plasma frequency, carrier density, mobility, and energies and broadenings of transverse and longitudinal optical (LO) phonons.1 There have been many discussions in the literature whether the dielectric function should be written as a Drude–Lorentz sum of the contributions of various elementary excitations (such as plasmons, phonons, polaritons, excitons, etc.) or as a Berreman–Unterwald product of these terms. In this paper, we introduce a broadband factorized description of the dielectric function, which can be used from the far-infrared to the vacuum-ultraviolet spectral region, and apply it to undoped bulk cubic GaAs, wurtzite ZnO, and other materials. We also show that the factorized (Kukharskii) description of the infrared dielectric function for doped GaAs yields a similar electron concentration and mobility as the more commonly applied Drude–Lorentz model, if experimental errors and our limited experimental range (0.031–6.5 eV) are properly taken into account.
II. MODEL DIELECTRIC FUNCTIONS
A. Drude–Lorentz model (sum)
Following Helmholtz,2 Kettler,3 and Drude,4–6 one can write the dielectric function versus angular frequency as a sum
where the constant 1 is the contribution of the vacuum, the first term is
the susceptibility of free carriers, the second term is
the susceptibility of transverse optical (TO) phonons, and the last term is
the susceptibility of bound carriers due to interband optical transitions.
The justification for this summation (1) is the electromagnetic superposition principle: we assume that the polarization fields of various charges under the influence of the external electric field of the light source can be added, because they are independent of each other, ignoring interactions between charges.
As suggested by Drude,4,6 we allow more than one species of free carriers with an unscreened (angular) plasma frequency7
Drude scattering rate , carrier density , and effective mass to contribute to the dielectric function. is the electronic charge, is the vacuum permeability, and is the free electron mass. Usually, just a small number of free carrier species (often one or two) are sufficient to describe , such as electrons and holes, light and heavy holes, electrons in different conduction band valleys, - and -electrons, or bulk and surface electrons. For -type semiconductors with a single occupied conduction band valley or for metals with a simple spherical Fermi surface, only one term should be sufficient, while more terms might be needed for more complex Fermi surfaces of metals.
In the lattice absorption term (3), is the (angular) TO phonon frequency with scattering rate and dimensionless oscillator strength . Since electromagnetic waves are transverse, only TO phonons (not LO phonons) lead to a pole in the lattice susceptibility (3).
Similarly, in the interband absorption term (4) due to bound carriers, is the (angular) frequency of the transition, is its scattering rate,8 and is its dimensionless oscillator strength. The summation in Eq. (3) runs over all infrared-active phonon modes in the crystal (usually a small number, much less than three times the number of atoms in the primitive unit cell), but additional modes may be required due to higher-order phonon absorption or impurity-related vibrational modes.1
The summation in Eq. (4) in principle runs over all -vectors in the Brillouin zone and all possible combinations of interband transitions. Therefore, the interband contribution is usually replaced by a summation
over a much smaller number of Kramers–Kronig-consistent general oscillator functions , which might include Lorentzians with complex (or even negative) amplitudes, Gaussians, Tauc–Lorentz, or Cody–Lorentz lineshapes, or the Herzinger–Johs parametric oscillator model.9
Writing the dielectric function as a sum of Lorentzians or other lineshapes as in Eq. (1) implies that the various contributions are independent and that there is no cross-talk (interaction) between different transitions. Therefore, these models only use one broadening parameter for each term, in each denominator.
If we are only interested in the infrared portion of the dielectric function spectrum, we can define the high-frequency dielectric constant
This quantity describes the contribution of the vacuum and the electronic interband transitions to the static dielectric constant . Experimentally, one obtains for insulators by measurements at frequencies above the region of lattice absorption (thus the subscript ) but far below the band gap. The infrared dielectric function then becomes
where we have introduced the screened (angular) plasma frequency
For , Eq. (8) shows that in the absence of free carriers, the amplitudes describe the contribution of lattice absorption to the static dielectric constant since12
For a single phonon absorption band, we can use the Lyddane–Sachs–Teller (LST) relation13
to calculate the LO phonon frequency
(Kurosawa14 and Barker12 generalized the LST relation for cubic materials with multiple phonons and Schubert15 for anisotropic crystals.)
In the presence of free carriers, the dielectric function (1) diverges at low frequencies. It is convenient to introduce the complex optical conductivity
which cancels the divergence of the Drude term and therefore remains finite at low frequencies. We can then identify the quantity
with the electrical low-frequency conductivity. For the specific case of the Drude–Lorentz model (8), we find
where is the Drude collision time.
B. Kukharskii model (product)
Berreman and Unterwald16 take a completely different approach in their description of the dielectric function. Without making physical assumptions about the line shape of oscillators, they start with the mathematical fact that the dielectric function, like any analytic function in the complex plane, is completely determined by its zeroes and poles and therefore can be written as a quotient of two polynomials. Since approaches unity as the angular frequency goes to infinity, the number of poles must be equal to the number of zeroes and the highest-order polynomial coefficients in the numerator and denominator must be equal. Considering also the symmetry to ensure that the time-dependent dielectric displacement remains real, poles and zeroes come in pairs and those not located on the imaginary axis must be symmetric relative to the imaginary axis. This results in the functional form17
which was frequently applied to model the infrared reflectance of insulators.18 For insulators with many phonon modes or for large TO/LO splittings, it often gives a better description than the Drude–Lorentz model of independent oscillators.19,20
Since the dielectric function and its inverse (called the loss function) obey causality (i.e., the polarization response follows the applied electric field), both zeroes and poles must be located below the real axis in the complex plane, which is equivalent to the condition that all scattering rates and must be positive. (We assume a time-dependence for the electromagnetic wave. The other choice for the time-dependence leads to complex conjugate equations with poles and zeroes above the real axis, see Barker).12
To understand the physical significance of the zeroes and poles in Eq. (17), it is instructive to place the various factors into three groups
Berreman and Unterwald16 already recognized that the Drude response of free carriers can be described by
which corresponds to one pole at the origin and another one at . We chose the subscript after Kukharskii, who first applied Eq. (19) to describe the reflectance of doped GaAs.21,22 The zeroes in Eq. (19) are related to the lower longitudinal plasmon-phonon polaritons (LP).1,23,24 In the absence of free carriers, the LP angular frequency vanishes and the Drude factor (19) becomes unity.
The second factor
describes the dielectric response of infrared lattice absorption. The poles are related to TO phonons, while the zeroes are the upper longitudinal plasmon-phonon polaritons (UP). In the absence of free carriers, the UP modes are the LO phonons. They are pushed toward higher energy by the interaction with longitudinal plasmon oscillations of free carriers.23,24 (Since the plasmon oscillations are longitudinal, they interact only with the LO, but not with the TO phonons.) Additional factors may be attached to describe higher-order phonon absorption or impurity-related absorption.
For a single plasmon-phonon polariton mode, the lower and upper polariton frequencies are related to the screened plasma frequency and the LO frequency by22,23
The third factor
can be expressed as a sum similar to Eq. (4)
with the oscillator strength
if we pretend that all broadenings are small and thus neglect the coupling between different interband transitions. (The presence of broadenings justifies complex Lorentzian amplitudes.) More conveniently, we write this factor (24) as a sum of general oscillators
just like in the Drude–Lorentz case (6). We use the same definition (7) for . If we are only interested in the dielectric function of doped insulators well below the band gap, this allows us to write
which is known as Kukharskii’s equation.21,22
It is not straightforward to break up this product into a sum of contributions of different species of carriers to the DC conductivity, but for a single carrier species, we can write using Eq. (21)
which is exactly the same expression as in the Drude–Lorentz case (16). The Drude and Kukharskii scattering rates are therefore the same and we can omit this distinction.
III. EXPERIMENTAL PROCEDURE
In the infrared spectral region, we acquired the ellipsometric angles and as a function of angular frequency on a J. A. Woollam FTIR variable angle of incidence spectroscopic ellipsometer (FTIR-VASE) from 0.031 to 0.600 eV with a resolution of , usually at five equally spaced angles of incidence from to . We performed two-zone measurements with two polarizer angles (), two analyzer angles ( and ) and a rotating compensator (15 spectra per revolution, 20 FTIR scans per spectrum). We also acquired and from 0.50 to 6.60 eV with 0.01 eV steps at the same angles of incidence on a J.A. Woollam VASE ellipsometer equipped with a computer-controlled Berek wave plate compensator.
Since data from these two instruments were merged, small discrepancies can be noticed in the region of overlap, possibly due to slight misalignment. Most noticeably, the data taken with the FTIR ellipsometer are noisy above 0.5 eV.
IV. RESULTS AND DISCUSSION
A. Intrinsic and n-type GaAs, doped GaSb, and InAs
In Fig. 1, we show the pseudodielectric functions for a nominally undoped (intrinsic) and a Si-doped (n-type) GaAs substrate from 0.031 to 6.5 eV. Above 1 eV, we used tabulated optical constants for undoped and n-type GaAs and its native oxide taken from the literature.25–27 This allowed us to determine the native oxide thickness (40 and 26 Å, respectively). This fit is generally quite good although some discrepancies were found, most likely due to polishing damage near the surface and uncertainties in the optical constants of the native oxide.27
Real and imaginary parts of the pseudodielectric function for silicon-doped (top) and undoped (bottom) GaAs covered with native oxide. Data from two different instruments were merged. The insets show expanded views of the regions of lattice absorption. Symbols show experimental data, lines the best fit to Eq. (18) with parameters given in Table I.
Real and imaginary parts of the pseudodielectric function for silicon-doped (top) and undoped (bottom) GaAs covered with native oxide. Data from two different instruments were merged. The insets show expanded views of the regions of lattice absorption. Symbols show experimental data, lines the best fit to Eq. (18) with parameters given in Table I.
Screened plasma frequency , Drude broadening , high-frequency dielectric constant , carrier density , mobility , TO and LO phonon energies and and broadening , Kukharskii broadening , and lower and upper plasmon-polariton frequencies and and their broadenings and for undoped and n-type GaAs as well as n-type and p-type GaSb and InAs. For uniaxial undoped ZnO, values for the ordinary (o) and extraordinary (eo) parameters are listed separately. Quantities marked (f) were fixed during the fit, and those marked with an asterisk were taken from the literature. For each material, the top row (model DL) shows a fit with Eq. (8), where , , , , and are experimental values from ellipsometry data, whereas , , and were calculated using Eqs. (9), (16), and (12); the bottom row (model KK) shows experimental values , , , , , , , and determined from a fit to the ellipsometry data with Eq. (18), whereas , , , and were calculated using Eqs. (21), (22), (9), and (29), respectively. Calculated quantities are shown in bold. The broadenings shown in italics show strong parameter correlations and therefore are not reliable.
Sample . | Model . | . | . | . | . | . | . | . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
. | . | (meV) . | (meV) . | (1) . | . | (cm/V s) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . |
u-GaAs | DL | 12.4 | 4.2 | 10.8 | 4400 | 33.3 | 35.9 | 0.3 | ||||||
u-GaAs | KK | 13.0 | 10.8 | 45 000 | 33.2 | 35.8 | 0.3 | 0.4 | 11.9 | 36.2 | 0.6 | 0.3 | ||
n-GaAs | DL | 35.6 | 5.4 | 11.0 | 3400 | 33.3 | 35.6 | 0.3 | ||||||
n-GaAs | KK | 36.7 | 11.0 | 1600 | 33.0 | 35.0 | 0.2 | 11.3 | 29.2 | 41.5 | 4.9 | 4.3 | ||
n-GaSb | DL | 17 | 2 (f) | 14.6 | 1000 (f) | 27.78 | 28.89 | 0.3 (f) | ||||||
n-GaSb | KK | 10.9 | 14.6 | 1000 | 27.78 | 28.3 | 0.3 (f) | 2 | 10.7 | 28.4 | 1 | 1 (f) | ||
p-GaSb | DL | 33.3 | 32.8 | 14.0 | 120 | 27.78 | 28.89 | 0.3 (f) | ||||||
p-GaSb | KK | 28.9 | 14.0 | 600 | 27.78 | 28.6 | 0.3 (f) | 6.4 | 25.1 | 32.0 | 1 (f) | 11.7 | ||
InAs | KK | 17 | 12.2 | 50 000 | 27 | 30 | 0.3 (f) | 1 (f) | 15 | 31 | 1 (f) | 1 (f) | ||
ZnO (o) | KK | 3.73 | 50.7 | 1.2 | 73.2 | 1.1 | ||||||||
ZnO (eo) | KK | 3.81 | 46.8 | 1 (f) | 71.1 | 0.9 |
Sample . | Model . | . | . | . | . | . | . | . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
. | . | (meV) . | (meV) . | (1) . | . | (cm/V s) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . | (meV) . |
u-GaAs | DL | 12.4 | 4.2 | 10.8 | 4400 | 33.3 | 35.9 | 0.3 | ||||||
u-GaAs | KK | 13.0 | 10.8 | 45 000 | 33.2 | 35.8 | 0.3 | 0.4 | 11.9 | 36.2 | 0.6 | 0.3 | ||
n-GaAs | DL | 35.6 | 5.4 | 11.0 | 3400 | 33.3 | 35.6 | 0.3 | ||||||
n-GaAs | KK | 36.7 | 11.0 | 1600 | 33.0 | 35.0 | 0.2 | 11.3 | 29.2 | 41.5 | 4.9 | 4.3 | ||
n-GaSb | DL | 17 | 2 (f) | 14.6 | 1000 (f) | 27.78 | 28.89 | 0.3 (f) | ||||||
n-GaSb | KK | 10.9 | 14.6 | 1000 | 27.78 | 28.3 | 0.3 (f) | 2 | 10.7 | 28.4 | 1 | 1 (f) | ||
p-GaSb | DL | 33.3 | 32.8 | 14.0 | 120 | 27.78 | 28.89 | 0.3 (f) | ||||||
p-GaSb | KK | 28.9 | 14.0 | 600 | 27.78 | 28.6 | 0.3 (f) | 6.4 | 25.1 | 32.0 | 1 (f) | 11.7 | ||
InAs | KK | 17 | 12.2 | 50 000 | 27 | 30 | 0.3 (f) | 1 (f) | 15 | 31 | 1 (f) | 1 (f) | ||
ZnO (o) | KK | 3.73 | 50.7 | 1.2 | 73.2 | 1.1 | ||||||||
ZnO (eo) | KK | 3.81 | 46.8 | 1 (f) | 71.1 | 0.9 |
Using the tabulated optical constants for the electronic part of the dielectric function , we then fitted the remaining parameters (TO phonon and polariton energies and broadenings) in Eqs. (18) and (27), with results shown in Table I. was taken as the zero-energy limit of the tabulated dielectric functions.
Since the effects of plasmon-phonon polaritons on the optical constants can only be seen at the lowest photon energies, we also show the ellipsometric angle below 0.09 eV for both substrates in Fig. 2. For each sample, there are two regions called reststrahlen bands (shown in gray), where is close to (and the normal-incidence reflectance is high). One of these bands extends from zero to , while the other one extends from to . Our FTIR ellipsometer has good sensitivity to (which appears as a strong peak in the loss function shown in Fig. S1) and its broadening. is right at the edge of our experimental range but clearly visible in , see the insets in Fig. 1. The energy range of the lower plasmon-phonon polariton band, on the other hand, is too low to be measurable using our instrument. Nevertheless, we obtain reasonable values for all relevant parameters, see Table I. The only exceptions are the Kukharskii and LP broadenings, which are strongly correlated.
Ellipsometric angle at five angles of incidence for silicon-doped (top) and undoped (bottom) GaAs covered with native oxide in the region of plasmon-polariton absorption (gray). Symbols show experimental data, lines the best fit to Eq. (18) with parameters shown in Table I. The reststrahlen bands are shaded in gray.
Ellipsometric angle at five angles of incidence for silicon-doped (top) and undoped (bottom) GaAs covered with native oxide in the region of plasmon-polariton absorption (gray). Symbols show experimental data, lines the best fit to Eq. (18) with parameters shown in Table I. The reststrahlen bands are shaded in gray.
Using Eqs. (21) and (22), we calculated the plasma and LO phonon frequencies. From for undoped GaAs, we obtain a carrier density of , which should be considered an upper limit. For n-type GaAs, implies a carrier density of , which is within the range of doping densities specified by the supplier (5.5–). An effective electron mass of was used.
For comparison, we also fitted the same data shown in Fig. 2 with the Drude–Lorentz model (8). The Drude and TO phonon energies and broadenings are also shown in Table I. The LO energy, carrier density, and mobility were calculated from Eqs. (12), (9), and (16), respectively. The carrier densities and LO frequencies agree quite well between both models. Also, the optical mobility for undoped and doped GaAs obtained from the Drude–Lorentz fit agrees with the electrical mobility expected for the given carrier concentration.28
Unfortunately, the Kukharskii scattering rate disagrees with the Drude scattering rate , comparing Eqs. (16) and (29), and therefore the Kukharskii mobilities are not reliable, see Table I. As mentioned earlier, this is due to our limited spectral range and the correlations in the fit between and because of the larger number of broadening parameters in the Kukharskii model.
B. Bulk undoped ZnO
According to Kukharskii,22 the factorized dielectric function (18) can also be applied to each diagonal component of the dielectric tensor of anisotropic materials with at least orthorhombic symmetry, where the dielectric properties can be described by a diagonal tensor in a coordinate system that is invariant with photon energy.
To illustrate this point, we analyze the ellipsometric angles and the pseudodielectric function of a bulk c-axis oriented ZnO substrate obtained commercially, see Fig. 3. To model these data, we use a uniaxial model for a bulk substrate with an ordinary and an extraordinary dielectric function, each described independently with the form given by Eq. (18). Since no free carrier effects are visible for this substrate, the Drude factor was set to unity. A surface roughness layer thickness (described with a 50/50 mixture of ZnO and voids using the Bruggeman effective medium approximation) of 21 Å was found from the magnitude of the pseudoabsorption below the band gap of about 3.1 eV. The electronic part of the ordinary dielectric function was described using two Tauc–Lorentz oscillators to account for the absorption of the main exciton triplet29–31 (not resolved) and the exciton-phonon complexes.29,32,33 At higher energies, we added two simplified Herzinger–Johs parametric oscillators and a pole at 11 eV. For our c-axis oriented ZnO substrate, the extraordinary dielectric function does not have a significant impact on the pseudodielectric function above the region of lattice absorption. We therefore assume that the electronic part of the extraordinary dielectric function is equal to that of the ordinary dielectric function, except for a rigid shift upward by 0.08, see Ref. 34. Our data are not sensitive to the complications described by Shokhovets et al.35 The infrared lattice absorption of ZnO is dominated by the () phonons in the ordinary (extraordinary) dielectric function, each of which is split into a TO/LO pair by the Fröhlich interaction.34
Real and imaginary parts of the pseudodielectric function for undoped c-axis oriented bulk ZnO with 21 Å surface roughness. Data from two different instruments were merged. Points show experimental data, lines the best fit to Eq. (18) with parameters in Table I.
Our experimental ellipsometry data for bulk ZnO along with the best fit using Eq. (18) in the low and high photon energy regions are shown in Figs. 3 and 4, respectively. The physical significance of various structures in the spectra has been explained elsewhere.34 Our main point here is to show that Eq. (18) gives an excellent description over the complete spectral range from 0.03 to 6.5 eV. Our fit only has one problem: we are unable to describe the exact energy where the pseudo-Brewster angle of the sample changes from 0 to . This energy depends on the precise value of the high-frequency dielectric constant . In our approach, is not a free parameter, but determined by the electronic part of the spectrum shown in Fig. 3. A slight mismatch of the data from the two instruments causes a small error in (on the order of 0.05), which is responsible for the error seen near 0.14 eV in Fig. 4.
Ellipsometric angles and for undoped c-axis oriented bulk ZnO with 21 Å surface roughness in the region of infrared lattice absorption. Points show experimental data, lines the best fit to Eq. (18) with parameters in Table I.
V. SUMMARY
Fifty years ago, modulation spectroscopy36 (to study the electronic band structure of materials) and vibrational spectroscopy1,37 (Raman and FTIR, especially) were distinctly different fields, with different approaches to describe experiments. The availability of modern commercial ellipsometry instruments covering the range from 0.03 to 9.0 eV requires a consistent broadband approach suitable for insulators, semiconductors, and metals. The Drude–Lorentz summation (1) meets the broadband requirement, but it is not suitable for interacting excitations, such as insulators with multiple phonons19,20 or doped semiconductors with coupled longitudinal phonon-plasmon polaritons.21,22 We therefore introduced a factorized formalism (17), a generalization of Kukharskii’s equation (27), which is appropriate to describe insulators, semiconductors, and metals over the complete spectral range from the mid-IR to the vacuum-UV. This product (17) is easily implemented in commercial software. Several examples were given in the main text and in the supplementary material.38
Specifically, we also presented an approach to calculate the carrier mobility of doped semiconductors from Kukharskii’s equation, if the effective mass of the carriers is known. We applied this approach to doped GaAs, InAs, and GaSb. Our results are reasonable, but the study of doped semiconductors could be more reliable if the lower spectral range of commercial FTIR ellipsometers could be extended to 0.01 eV.
ACKNOWLEDGMENTS
This paper is based, in part, on research sponsored by Air Force Research Laboratory (AFRL) (Agreement No. FA9453-18-2-0046). The work at NMSU was supported by the National Science Foundation (NSF) (No. DMR-1505172). The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory (AFRL) and/or the US Government.