Temperature dependence of the infrared dielectric function and the direct bandgap of InSb from 80 to 725 K

Indium antimonide (InSb) is a III–V compound semiconductor with the zinc blende crystal structure.^1–3 Of all III–V semiconductors, it has the smallest bandgap^4,5 (only 185 meV at 300 K) and, therefore, the smallest electron mass⁴ (0.014$m0$⁠) and the strongest nonparabolicity effects.^1,6 It also has the largest spin-orbit splittings^1,4 and the smallest optical phonon energies.⁷ The valence band maximum and the conduction band minimum are at the $Γ$-point of the Brillouin zone. The satellite valleys in the conduction band are at least 500 meV above the $Γ$-minimum and do not need to be considered for the interpretation of our experiments.^8,9

The properties of InSb, especially the intrinsic carrier concentration,^3,10–15 the band structure,^8,16 and band filling, due to doping^17–19 (Burstein–Moss shift) were studied extensively from the 1950s to the 1980s. It was noted that InSb was the only elemental or III–V semiconductor where degenerate carrier statistics with Fermi–Dirac distribution functions had to be considered.²⁰ For most other semiconductors with larger bandgaps, such as Ge or GaAs, nondegenerate (Maxwell–Boltzmann) statistics can be applied, which has led to the textbook theories of intrinsic semiconductors.^21,22 The absorption coefficient and the refractive index near the direct bandgap were determined using transmission and interference measurements^23–25 and from spectroscopic ellipsometry.²⁶ It is difficult to determine the absorption coefficient from transmission measurements alone, because the reflection losses depend on the refractive index, which may not be known with sufficient precision to calculate the absorption coefficient from the transmittance. This difficulty can be overcome by measuring the transmission of several samples with different thicknesses.^27,28

Temperature-dependent ellipsometry measurements of InSb were performed in the region of interband transitions (above the direct bandgap) by Logothetidis et al.²⁹ as well as by others.³⁰ The infrared dielectric function of InSb near the direct bandgap was measured by Schaefer et al.²⁶ and explained with a theoretical model. There are several experimental reports and calculations^9,31,32 of the temperature dependence of the direct bandgap up to room temperature¹³ using transmission measurements,^28,33 photoluminescence,^34,35 piezoreflectance,³⁶ and two-photon Hall response.³⁷ The only data available above room temperature seem to be the transmission data by Liu and Maan,²⁸ which, unfortunately, are not compatible with our ellipsometry results at the highest temperatures. While transmission measurements determine small absorption coefficients (⁠$20−300cm−1$⁠) of the Urbach tail below the band gap, ellipsometry is more sensitive to large absorption coefficients $>1000cm−1$ near and above the bandgap.

Open questions to be addressed by our work are the following: (1) How does the direct bandgap vary with temperature between room temperature and the melting point at 800 K? (2) How does the intrinsic carrier concentration depend on temperature?¹⁵ and (3) How does this affect the optical absorption?³⁸ (4) What is the role of excitons in optical absorption?³⁹ and (5) How do excitons get screened^40,41 above 300 K due to thermally excited electron-hole pairs? (6) What is the temperature dependence of the electron⁴² and hole¹⁵ effective masses?⁴³ (7) Are the effective masses determined by the unrenormalized bandgap (including thermal expansion) or by the renormalized bandgap due to deformation-potential electron phonon interaction?^15,43,44 (8) Does InSb become a semimetal (a semiconductor with zero bandgap) or a topological insulator at high temperatures?³¹ (9) How do the longitudinal optical phonons couple to the thermally excited carriers?²⁸ 

Our findings are relevant not only for InSb, but also for all small-gap semiconductors considered for mid-wave infrared detectors, such as $Ge1−ySny$ alloys.

The pseudodielectric function in the region of the direct bandgap of a commercially obtained undoped bulk InSb sample with (100) surface orientation was measured between 80 and 725 K using a J. A. Woollam Fourier-transform infrared variable angle spectroscopic ellipsometer (FTIR-VASE) equipped with a Lakeshore ST-400 ultrahigh vacuum (UHV) cryostat and diamond windows as described elsewhere.⁴⁵ As received, the back surface of the sample was rough and we did not attempt to roughen it further. Reflections from the back surface led to depolarization effects and artifacts in the data at the longest wavelengths, especially at low temperatures, as described below. The sample was cooled with liquid nitrogen and heated with a resistor. The temperature was measured with two type-E (NiCr-CuNi) thermocouples, one attached to the sample (more accurate) and the other located near the heater (more stable control). Our InSb substrate melted at a thermocouple reading of 750 K, lower than the melting point of InSb (800 K).⁴⁶ It is, therefore, likely that the sample temperature (above 300 K) is somewhat higher than the reading of the thermocouple attached to the sample. This temperature difference is smaller at room temperature and increases to about 50 K near the InSb melting point.

The FTIR-VASE resolution was set to $16cm−1$ (2 meV), which is smaller than the $E0$ broadening at the lowest temperatures (4 meV, as shown below). Room temperature data in air were also taken from 0.5 to 6.5 eV on a J. A. Woollam VASE ellipsometer for comparison with the literature.^29,47 All measurements were performed at $70°$ angle of incidence.

Before the measurements, the InSb sample was cleaned ultrasonically, first in water for 15 min and then in isopropanol for 15 min. We did not use the harsh chemicals (bromine solution in methanol, hydrochloric acid diluted with methanol) as suggested by Aspnes and Studna.⁴⁷ The pseudodielectric function of InSb measured at room temperature in air from 0.5 to 6.5 eV before and after this wet clean is shown in Fig. S1.⁶¹ By comparison with the optical constants of Ref. 47 for InSb (especially the maximum of $⟨ϵ2⟩$ at the $E2$ critical point near 3.7 eV) and Refs. 48 and 49 for its native oxide, the surface layer thickness in air was found to be 36 Å before cleaning and 29 Å after wet cleaning.

After the wet clean, the InSb sample was mounted in the UHV cryostat, which was then pumped down to a base pressure below $10−8$ Torr.⁴⁵ We tried several methods of mounting the sample.⁴⁵ Our best results were obtained by clamping the sample to the copper sample holder with stainless steel strips and screws, without the use of any adhesives, silver paint, etc. The sample was then heated at $450°C$ for more than 12 h to remove volatile surface layers such as water (known as a degas) and allowed to cool back down to room temperature. The pseudodielectric function of InSb at 300 K was then acquired on the FTIR-VASE instrument inside the cryostat under UHV conditions. Window corrections⁴⁵ were applied automatically by our commercial data acquisition software.

To correct the pseudodielectric function and obtain an estimate for the dielectric function of InSb, the thickness and optical constants of the oxide must be known. We assume a constant value of $ϵ∞=3.8$ for the native oxide on InSb in the infrared spectral region.⁴⁸ Since the oxide thickness is small, the errors introduced by lattice absorption of oxygen-related vibrational modes in the native oxide will be small.

To determine the oxide thickness, we use the Jellison-Sales method for transparent glasses,⁵⁰ which is based on the premise that the imaginary part of the dielectric function for an insulator must be exactly zero in the transparent region below the bandgap (not positive, not negative). If one assumes an oxide thickness that is too small, the resulting dielectric function of InSb will be positive. If the oxide thickness is assumed too large, then the corrected dielectric function will be negative. Using this premise, we were able to determine that the thickness of the native oxide is $25±5$ Å. The accuracy of this method is determined by the systematic and statistical errors of the ellipsometric angle $Δ$ in the transparent region, including the window correction, which affects the ellipsometric angle $Δ$⁠. This method does not work for conductors (such as InSb at high temperatures) since the free carrier absorption obscures the transparent region below the bandgap. See Fig. S3 in the supplementary material for more information.⁶¹ 

After this sample preparation, the InSb substrate was cooled down to 80 K and FTIR-VASE spectra were taken up to 750 K in steps of approximately 25 K. The resulting pseudodielectric functions are shown in Fig. S4 in the supplementary material.⁶¹ Important trends can already be observed in these $⟨ϵ⟩$ spectra, as described in the supplementary material.⁶¹ At the lowest temperatures, there are incoherent reflections from the back side of the sample (and the sample holder), which lead to artifacts in the pseudodielectric function and strong depolarization effects below 0.1 eV. At higher temperatures, the transmission of the substrate decreases due to free carrier absorption and the artifacts at long wavelengths disappear.

Our previous studies on GaP⁵¹ and Ge⁵² showed that the oxide thickness can vary by 20–30 Å when performing a temperature series. We ignore these variations for this work because small fluctuations in the oxide thickness do not have a significant impact on the temperature dependence of the bandgap, which is the primary focus of our work. There is no indication in the pseudoabsorption below the bandgap that oxide thickness variations with temperature were significant.

The dielectric function $ϵ$⁠, calculated from $⟨ϵ⟩$ assuming a constant oxide thickness of 25 Å, is shown in Fig. 1. We observe a number of trends: (1) The $E0$ bandgap, seen as the onset of absorption and as a peak of $ϵ1$⁠, decreases from the lowest temperatures (purple) to room temperature (blue), remains constant near 500 K (yellow), and then increases again at the highest temperatures (red). (2) The broadening increases monotonically with temperature. (3) Above room temperature, $ϵ1$ diverges toward $−∞$ at the lowest photon energies, while $ϵ2$ diverges towards $+∞$⁠. This indicates free carrier absorption, consistent with a Drude model, at elevated temperatures. (4) There is a 30% decrease in the magnitude of $ϵ2$ at high energies (near 0.5 eV) as the temperature increases from 77 to 700 K.

Our first objective is to determine the bandgap $E0$ as a function of temperature. To achieve this, one typically plots the square of the absorption coefficient as a function of photon energy (Tauc plot) and then finds the bandgap by extrapolating to zero.⁵ We could only find a linear region in such Tauc plots below room temperature and, therefore, chose not to use this method.

Instead, we described the pseudodielectric function of InSb with the Herzinger–Johs parametric semiconductor model⁵³ and varied the parameters of the direct bandgap as a function of temperature until a good fit was achieved. This model has been shown to work well for the direct gap of Ge over a broad range of temperatures.⁵² In particular, the direct bandgaps of Ge determined from the parametric semiconductor model agreed well with those obtained using second-derivative and reciprocal space methods, see Fig. 6 in Ref. 52.

At elevated temperatures, a Drude term⁵ 

was added to the parametric semiconductor dielectric function to describe absorption by free electrons, where $ℏω$ is the photon energy, $ωP$ is the (angular) plasma frequency, $n$ is the free electron density, $e$ is the electronic charge, $m∗m0$ is the effective electron mass (⁠$m∗=0.014$⁠, which we kept independent of temperature), $ϵ0$ is the permittivity of vacuum, and $γ$ is the electron scattering rate. (The absorption by free holes was neglected due to their large effective mass.) The mobility can be calculated from Ref. 54, $μ=e/γm0m∗$⁠.

Examples of such fits are shown in Fig. 2. The agreement between data and fit is usually quite good. It may be possible to achieve an even better agreement with the lineshapes for screened excitons proposed by Tanguy,^39,41 which will be discussed in more detail below. We recently applied the excitonic (Tanguy) line shapes to the direct bandgap of Ge, with good success.⁵⁵ In particular, the broadenings for InSb may be somewhat smaller than those obtained from a parametric semiconductor fit. The asymmetry of the experimental line shape leads to a slight over-estimation of the bandgap values by a few meV.

The direct bandgaps and broadenings of InSb as a function of temperature obtained from the parametric semiconductor fit are shown in Fig. 3. As already observed by inspection of the dielectric functions in Fig. 1, the bandgap decreases up to 450 K, remains approximately constant up to 600 K, and then increases again at the highest temperatures by about 50–70 meV. Figure 3 also shows the broadening parameter of the direct bandgap, which increases more or less monotonically with temperature. The broadenings of $E0$ in InSb are at least five times larger than in Ge⁵⁵ but comparable to GaP⁵⁶ and GaAs.⁵⁷ The last data point at 725 K is very close to the melting point of our sample.

Our bandgaps agree reasonably well with prior results^28,37 at low temperatures. There are some differences at higher temperatures that could be explained by errors in the temperature measurement on the order of 50 K, which is not uncommon at such high temperatures.

Up to 450 K, the dependence of the bandgap can be described with a Bose–Einstein model,^29,45

where $EB$ is the unrenormalized bandgap (in the absence of electron-phonon interactions), $T$ is the temperature, $aB$ is the strength of the electron-phonon interaction, $Ω$ is an effective phonon energy, and $kB$ is the Boltzmann constant. Parameters obtained from our experimental data are shown in Table I and are in good agreement with the literature.⁹ 

By fitting the data shown in Fig. 1 with Eq. (1), we were also able to determine the carrier concentration $n$ and the mobility $μ$⁠, which are shown in Fig. 4. Strictly speaking, Eq. (1) yields a plasma frequency and a scattering rate. This can be converted into a carrier density and mobility, if the effective mass $m∗$ is known. Through $k→⋅p→$-theory, $m∗$ depends on the bandgap. One should also include the carrier-density dependence of $m∗$ due to the large nonparabolicity of the conduction band.⁴² Such considerations are beyond the scope of the present article and we keep the effective mass constant at $m∗=0.014$ for free electrons in InSb. The Drude contributions of holes are much smaller (due to the large mass of heavy holes and the low concentration of light holes) and have been neglected.

At low temperatures, the Drude response of free carriers is masked by depolarization effects (backside reflections), but at 275 K and above, there is a clear decrease in $ϵ1$ at low energies, see Fig. 2, which yields the plasma frequency and electron concentration. The increase in $ϵ2$ at low energies is damped by a factor of $γ/ω$ compared to $ϵ1$⁠, see Eq. (1), and only visible at 450 K and higher, see Fig. 4. The carrier density $n$ derived from our plasma frequency data with $m∗=0.014$ agrees well with a simple non-degenerate model (dashed) and an analysis of temperature-dependent Hall effect measurements (solid) near room temperature, see Fig. 4. At higher temperatures, we underestimate the electron concentration because of our assumption of a constant electron mass. If we correctly calculated the temperature-dependent mass, our electron concentration would likely be closer to the literature results.

Below 450 K, the scattering rate and mobility parameters in the Drude fit have very large error bars (as indicated by open symbols in Fig. 4) and should be ignored. Between 450 and 700 K, the mobility decreases from 100 000 to 20 000$cm2/Vs$⁠, consistent with Hall effect measurements.⁵⁸ 

In summary, we determined the infrared dielectric function of InSb in the region of the direct bandgap at temperatures from 77 to 725 K using Fourier-transform infrared spectroscopic ellipsometry. The native oxide thickness was determined to be $25±5$ Å using the Jellison–Sales method for transparent glasses and appropriate corrections were made to the experimental data.

We observe a decrease in the direct bandgap up to 450 K due to electron-phonon interactions and thermal expansion, which can be described with a Bose–Einstein model, similar to direct band-gaps in other semiconductors. The broadening increases with temperature. At the lowest photon energies (below 0.1 eV), there is a Drude tail, from which the free electron concentration and the electron mobility can be determined. We underestimate the electron concentration and overestimate the Fermi energy and the mobility because we assumed a constant effective mass and ignored the conduction band nonparabolicity in our analysis, see Fig. 1 in Ref. 6, but this does not change our conclusions qualitatively.

At the highest temperatures, the free carrier concentration due to thermally excited electron-hole pairs becomes very large and the Fermi level is above the conduction band minimum.⁶ This reduces the magnitude of $ϵ2$ due to band filling and screening of the excitonic (Sommerfeld) enhancement of absorption. Band filling also results in a thermal Burstein–Moss shift: The onset of absorption is larger than the bandgap by about 100 meV at 700 K due to band filling. Therefore, the observed bandgap increases again above 600 K.

A more complete model would attempt to describe $ϵ2$ of InSb with the absorption of screened excitons given by Tanguy,^38,39,41

$A$ and $R$ are the excitonic amplitude and binding energy, respectively, for transitions from the heavy hole to the electron band. (Transitions from the light hole and split-off hole bands could be added, but they are smaller.) The parameters $A$ and $R$ need to be corrected for nonparabolicity effects. Another correction may be needed to take into account the $k$-dependence of the momentum matrix element.^25,$E0$ is the bandgap, i.e., the energy difference between the conduction band minimum and the valence band maximum. $E0$ depends on temperature and on the carrier concentration due to many-body effects (band bap renormalization).^59,60 The first term in brackets describes the absorption of the $n$th discrete exciton, and the second term describes the absorption by the exciton continuum. $g$ is the Hulthen potential screening parameter,⁴⁰ which can be calculated from the Thomas–Fermi screening length. $H$ is the Heaviside unit step function and $k=(ℏω−E0)/R$⁠. The functions $fh$ and $fe$ describe band filling and are calculated from the degenerate Fermi–Dirac distribution functions in the valence and conduction bands.³⁸ The real part can be found by Kramers–Kronig transformation, after introducing Lorentzian broadenings of transitions.⁴¹ Modifications are also required to take into account the effects of nonparabolicity on free carrier absorption.

A numerical evaluation of this expression is beyond the scope of the present article and will be presented elsewhere.

This work was supported by the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-20-1-0135. J.R.L. and C.M.Z. acknowledge support from the New Mexico Alliance for Minority Participation (NMAMP).

The authors have no conflicts to disclose.

Melissa Rivero Arias: Data curation (lead); Formal analysis (lead); Investigation (lead); Visualization (lead); Writing – review & editing (supporting). Carlos A. Armenta: Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Supervision (equal); Validation (supporting). Carola Emminger: Data curation (supporting); Formal analysis (equal); Investigation (supporting); Software (lead); Supervision (equal); Validation (equal); Writing – review & editing (supporting). Cesy M. Zamarripa: Data curation (equal); Formal analysis (supporting); Investigation (equal); Visualization (equal). Nuwanjula S. Samarasingha: Supervision (equal); Validation (equal). Jaden R. Love: Investigation (supporting); Resources (equal). Sonam Yadav: Formal analysis (lead). Stefan Zollner: Conceptualization (lead); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (lead); Validation (lead); Visualization (supporting); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.