Dielectric functions and carrier concentrations of Hg_1−xCd_xSe films determined by spectroscopic ellipsometry

There has been a plethora of research devoted to the development of materials suited for infrared (IR) applications in the past few decades.¹ Semiconductors such as Hg_1−xCd_xTe have been carefully investigated and developed in order to produce devices capable of operating at IR-wavelengths.² However, lattice-matched substrates used to grow Hg_1−xCd_xTe—typically Cd_1−xZn_xTe—tend to have constraints in terms of size, scalability, and cost. Although there has been a considerable effort placed on growing Hg_1−xCd_xTe on scalable substrates such as Si or GaAs, these efforts have been hampered due to high dislocation densities.^3,4 Due to these reasons, there has been a recent interest in probing alternate semiconductors, such as Hg_1−xCd_xSe, as possible candidates for IR-based applications.⁵ This is in large part due to the fact that Hg_1−xCd_xSe is nearly lattice matched to GaSb, which can serve as a large-area, high quality, III-V scalable substrate.⁶ While few groups have worked on the growth and characterization of Hg_1−xCd_xSe, focusing on the molecular beam epitaxy (MBE)-growth, and the subsequent analysis of these films through various methods, it is essential to explore their optical properties more comprehensively, if Hg_1−xCd_xSe alloys are to be used in IR-based devices.^7,8 Additionally, one other aspect that needs further attention is the fact that the characteristics of charge carriers in Hg_1−xCd_xSe are difficult to ascertain since some of the potential substrates used to grow these films, such as GaSb, are electrically conductive.⁹ Consequently, charge carriers from the substrates tend to obfuscate the carrier information derived for Hg_1−xCd_xSe layers using conventional methods such as single-field Hall measurements.⁵ 

The complex dielectric function (⁠$ϵ=ϵ1+iϵ2$⁠) of a semiconductor provides useful information regarding their electronic structure.¹⁰ Although there are several methods available to determine ϵ, spectroscopic ellipsometry is one of the more efficient methods, as it does not require one to perform a Kramers-Kronig transformation.^11–13 The ϵ₁ and ϵ₂ spectra can be used to determine the electronic transitions in the Brillouin zone which in turn allows one to formulate insights on the band structure of the semiconductor system.^14,15 By extending the ellipsometric measurements to the deep IR spectral regime (⁠$∼500 cm−1$⁠), one can also determine the carrier concentration of semiconductors, which is particularly significant in instances where semiconductor thin films are grown on substrates that are fairly conductive.^16–18 

In a previous publication, we reported the feasibility of obtaining the carrier characteristics using IR-ellipsometry for two samples grown on GaSb substrates.¹⁹ In this work, we have undertaken a more detailed study of the Hg_1−xCd_xSe alloy system, by extending spectroscopic ellipsometry to a wide spectral range (i.e., between $35 meV$ to $6 eV$⁠), and have explored Hg_1−xCd_xSe films grown on ZnTe/Si substrates, where the carrier characteristics obtained from ellipsometry can be compared directly with Hall results. Furthermore, the samples grown on ZnTe/Si substrates allow us to obtain the effective masses of the carriers, which is of great significance. Since our spectroscopic results extend to a fairly high energy regime, the ϵ spectra manifest the transitions due to band electrons. Thus, we are able to map how the fundamental band gap and the higher order electronic transitions of Hg_1−xCd_xSe changes as a function of Cd composition.

The Hg_1−xCd_xSe samples were grown by MBE on ZnTe/Si(112) and GaSb(112) substrates. The (211)B specific orientation of the substrates was chosen because it is established that alloys of Hg_1−xCd_xSe are less sensitive to microtwin formation in this specific direction.¹² All of the details related to the growth of the samples are included in Refs. 5 and 6. While none of the Hg_1−xCd_xSe samples are intentionally doped, all samples were heavily n-type due to impurities in the source material and native defects formed during growth. For the samples grown on (non-conducting) ZnTe/Si substrates, Hall measurements were performed to determine their carrier concentrations and mobilities.⁶ The alloy compositions of Hg_1−xCd_xSe films were measured via Fourier transform-IR (FTIR) measurements, utilizing a model for the energy band gap vs. composition developed by Summers and Broerman.²⁰ Alloy compositions determined by FTIR corresponded to those measured through energy-dispersive x-ray spectroscopy. Spectroscopic ellipsometry measurements were performed using a J. A. Woollam IR-ellipsometer, covering a spectral range from $2000 nm$ (⁠$5000 cm−1$⁠) to $40 000 nm$ (⁠$250 cm−1$⁠). The instruments had a spectral resolution of $2 cm−1$ and also the capability of performing measurements at multiple angles of incidence.²¹ For each Hg_1−xCd_xSe sample, ellipsometric spectra were obtained at three angles of incidence (i.e., 65°, 70°, and 75°). Prior to ellipsometric measurements, the backsides of the samples were roughened by a sand-blaster to avoid complications due to backside reflections.

Spectroscopic ellipsometry measures two parameters, $Ψ$ and Δ, which are related to the ratio of reflection coefficients by

where R_p is the complex reflection coefficient for light polarized parallel to the plane of incidence, and R_s is the coefficient for light polarized perpendicular to the plane of incidence.¹¹ In Fig. 1, we show $Ψ$ and Δ spectra obtained for a representative sample of Hg_1−xCd_xSe (x = 0.245) at an angle of incidence of $70°$⁠. The symbols correspond to the experimental data while the solid line is the fit obtained from our model, which we will discuss later. In order to show all of the important details of the spectra, we have chosen a log scale for the x-axis (i.e., the x-axis spans from 316 cm⁻¹= 3.92 meV to 56 234 cm⁻¹= 6.972 eV). In both $Ψ$ and Δ spectra, the oscillations at lower energies occur due to thin film interference, and the disappearance of these oscillations signals the advent of the fundamental band gap (i.e., critical point E₀).¹⁴ 

In order to obtain the thickness and the dielectric function, the ellipsometry data were fitted using a layered-model. We chose a four layer model for Hg_1−xCd_xSe samples grown on ZnTe/Si (Si-substrate, ZnTe-buffer, Hg_1−xCd_xSe film, and a surface-oxide-layer) and a three layer model for the Hg_1−xCd_xSe grown on GaSb, which did not have a buffer layer. The surface oxide layer was modeled using an effective medium approximation.¹⁰ Prior to measuring the Hg_1−xCd_xSe samples, we used two bare samples of ZnTe/Si and GaSb to recover their optical constants which were consistent with the literature values.^23,24 These were then used to model the entire structure, where the thickness and the dielectric function of Hg_1−xCd_xSe layer were adjusted to match the experimental data. The dielectric functions of Hg_1−xCd_xSe were modeled as follows:¹⁸ 

The first term on the right side of the equation represents the contributions from the critical points to ϵ, and in our case, all of the Hg_1−xCd_xSe layers were modeled as a collection of four critical points (i.e., E₀, E₁, $E1+Δ1$⁠, and E₂).²² While the critical point E₀ corresponds to the fundamental band gap, E₁, $E1+Δ1$⁠, and E₂ critical points are associated with the higher order electronic transitions in the Brillouin zone. The second term on the right side of the equation represents a Drude oscillator that incorporates the absorption due to free carriers. This oscillator is represented as follows:²⁵ 

where ρ, τ, ϵ₀, and $ℏ$ correspond to resistivity, scattering time, permittivity of free space, and the Planck constant divided by $2π$⁠, respectively. It is important to note that the amplitude and the broadening parameter of the Drude oscillator is related to ρ and τ, which is taken into account in the above description. Furthermore, ρ and τ are related to effective mass (m*), carrier concentration (N), and mobility (μ) through the relationship $ρ=m*Nq2τ=1qμN$⁠, where q corresponds to the charge of the electron. This suggests that if ρ and τ are determined by representing ϵ by a Drude oscillator in the low-energy regime, then one could determine either m* or N, provided that one of them is known.¹⁸ 

Consequently, ρ and τ of the Drude oscillator were adjusted to obtain a reliable fit to the $Ψ$ and Δ data, especially focusing on the lower energy region. Simultaneously, the thickness of the Hg_1−xCd_xSe layer was also adjusted to obtain a good fit. The Fabry-Perot-like oscillations conspicuous in the $Ψ$ and Δ spectra (see Fig. 1) enabled us to recover the proper thickness of the film via the layered-model as shown in Table I. After we obtained a fairly good fit in the lower energy regime, next the oscillators corresponding to the three critical points were incorporated into the model in order to obtain a fit over the entire energy region.²² In Figs. 2(a) and 2(b), we show the results of ϵ₁ and ϵ₂ determined from this fitting procedure, respectively, for three different samples of Hg_1−xCd_xSe. In addition, the solid lines on both $Ψ$ and Δ spectra, depicted in Fig. 1, correspond to the fit we obtained by modeling the sample as described above.

It is evident from the ϵ₂ spectrum in Fig. 2(b) that there seemed to be two distinct regions with high absorption; a fairly significant absorption at low end of the spectrum and another region above ∼3.25 on the x-axis (i.e., $∼1800 cm−1$= 223 meV). While the absorption at low energies is due to free carriers, the absorption in the second region is due to the critical points in Hg_1−xCd_xSe.²⁶ As noticeable in Fig. 2(b), the transition to the second region occurs at different energies, indicating a connection between the fundamental band gap and the Cd composition in Hg_1−xCd_xSe. This is consistent with the previous FTIR measurements.

As mentioned previously, the energies of the critical points were obtained by modeling the experimental $Ψ$ and Δ spectra with the layered structure. In Fig. 3, we plot E₀, E₁, and $E1+Δ1$ critical point energies as a function of Cd composition for several samples of Hg_1−xCd_xSe. Since the critical point E₂ manifests near the high-energy end of the experimental spectral range (see Fig. 2), we were unable to deduce its characteristics with confidence. In Fig. 3, the dashed-line corresponds to the results obtained from Ref. 20 for E₀, while the solid lines through E₁ and $E1+Δ1$ critical points correspond to results taken from Ref. 8. It is evident that E₀, E₁, and $E1+Δ1$ critical points blue shift as more Cd is incorporated into the lattice. As for E₀, since the fundamental band gap of CdSe is larger than HgSe, it is apparent that the E₀ transition will increase as more Cd is incorporated into the lattice. The values obtained for E₀ from the current work seemed to be slightly higher than the results reported in Ref. 20, possibly due to the fact that the assignment of the fundamental band gap is slightly different in the two methods.^12,20 It is important to note that the blue shift in E₀ is not due to Burstein-Moss effect as the samples used for this study are not heavily doped.^27,28

While we used three critical points to model the absorption of Hg_1−xCd_xSe layers, the transition $E1+Δ1$⁠, which generally occurs close to the transition E₁, seemed to merge with the latter for most samples. These broadening effects are probably due to the presence of impurities, where carriers are able to undergo indirect transitions as a result of their scattering with impurities.²⁹ This is the reason why there are only two data points for $E1+Δ1$⁠. The blue shift in both E₁ and $E1+Δ1$ are consistent with the previous results, obtained for bulk samples.⁸ 

Now, we turn our attention to the characteristics of the Drude oscillator. In order to show the dramatic effect of carrier concentration on the low-energy regime of the dielectric function, in Fig. 4, we plot ϵ₂ for four samples of Hg_1−xCd_xSe with varying carrier concentrations. We note that all four samples shown in Fig. 4 have the different alloy compositions. The magnitude of the Drude oscillator increases sharply as N increases, evident by the increase in ϵ₂ at lower energies. As discussed before, there is an intimate connection between the oscillator parameters (i.e., amplitude and broadening) and m*, N, and μ of a crystal. For the Hg_1−xCd_xSe samples grown on ZnTe/Si, since N and μ were determined by Hall measurements, we were able to determine m* for each of these samples. The values of m* and N are listed in Table I for samples grown on ZnTe/Si (i.e., samples A through F). Similar to other ternary families, it is likely that both the alloy composition (x) and N influence the value of m*. Since we cannot decouple the contributions of x and N on m*, we can only conclude that there seemed to be a general trend towards higher values of m* as N is increased. Furthermore, we note that the influence of N on m* seemed to be more dominant than the effect x on m*, at least for $N>1017 cm−3$⁠. Once greater control over doping is achieved for Hg_1−xCd_xSe, further experiments are needed to determine if this is true for lower values of N. In comparison with Hg_1−xCd_xTe, we note that the m* values determined for Hg_1−xCd_xSe are slightly higher.³⁰ 

Once we can estimate a lower and an upper bound for m* from the values obtained for Hg_1−xCd_xSe grown on ZnTe/Si, we can use those values to calculate N for Hg_1−xCd_xSe samples grown on GaSb substrates. We show these values for samples G, H, and I in Table I. The uncertainty is calculated by evaluating N for the lower and the upper bound values of m*. Incidentally, while samples B and H have similar Cd-compositions, the reason for the large discrepancy between their carrier concentrations is probably due to the different growth conditions. Note that by examining a series of samples with varying alloy composition (x) and carrier concentrations, one could clearly establish a relationship between m*, x, and N. This relationship can be subsequently used to obtain a more accurate value for N through a single ellipsometric measurement.

Using spectroscopic ellipsometry, we obtained the dielectric functions of a series of Hg_1−xCd_xSe films grown on both ZnTe/Si and GaSb substrates, covering a wide spectral range. The fundamental band gap (i.e., E₀) blue shifts as a function of the Cd composition and the two other critical points (i.e., E₁ and $E1+Δ1$⁠) also show a similar result corroborated by the previous studies. The magnitude of the Drude oscillator, representing the free carrier absorption, increases as a function of the carrier concentration. The effective masses were used to determine the carrier concentrations of Hg_1−xCd_xSe samples grown on GaSb substrates, which is a conducting substrate. Using ellipsometry to determine the carrier concentration of films grown on conducting substrates is of great benefit as conventional single-field Hall measurements are inadequate to probe such films.

The work at Kenyon was funded by the National Science Foundation DMR-1207169 grant. Support for the growth and study of the Hg_1−xCd_xSe samples was provided by the U.S. Army Research Laboratory and the U.S. Army Research Office under Contract/Grant Nos. W911NF-10-2-0103, W911NF-10-1-0335, and W911NF-12-2-0019.