2.1 Ellipsometry

The IRSE parameters Ψ and Δ are defined by the complex ratio of the p- and s-polarized reflectance coefficients r _p and r _s , respectively Reference Roeseler[13]

and depend on the angle of incidence Φ _a , the thickness d of each layer, and the dielectric functions ε _j of all materials from the heterostructure. Ellipsometry is an indirect technique and model calculations are needed to extract information from individual constituents. Nonlinear regression algorithms are used to vary physically significant model parameters until measured and calculated spectra match as closely as possible. Parametric dielectric function models can greatly reduce the number of free parameters during data analysis. Details and issues of IR ellipsometry data analysis have been extensively discussed elsewhere, and will not be repeated here. (See references Reference Roeseler[13], Reference Schubert, Kasic, Tiwald, Woollam, Härle and Scholz[17], Reference Schubert, Tiwald and Herzinger[21], Reference Kasic, Schubert, Tiwald, Woollam, Einfeldt and Hommel[22], and references therein.) The model approach for the infrared response of polar semiconductors with free carriers used in the present work is the same as that discussed in Ref. Reference Schubert, Kasic, Tiwald, Off and Kuhn[18]. The so-called four-parameter semi-quantum model was used in Ref. Reference Schubert, Kasic, Tiwald, Off and Kuhn[18] where ω _TOi , ω _LOi , γ _TOi , and γ _LOi are the transverse optical (TO), and longitudinal optical LO-phonon frequencies and broadening parameters, respectively. The free-carrier contribution can be parameterized through the carrier concentration n, the carrier effective mass m*, and the carrier mobility μ (See Equations 2 and 3 in Ref. Reference Schubert, Kasic, Tiwald, Off and Kuhn[18].) The high-frequency dielectric constant is ε _∞. Eqs.(2)-(4) in Ref. Reference Schubert, Kasic, Tiwald, Off and Kuhn[18] referred to a uniaxial material. However, in the present work the optical response parallel and perpendicular to the sample surface is treated as isotropic despite the tetragonal distortion of the GaN _x As_1−x SL sublayers due to the tensile strain within the pseudomorphically grown heterostructures. Because we have not observed deviation from the cubic lattice response, we will not consider the uniaxial perturbation of the GaN _x As_1−x SL sublayers parallel to the strain direction. We also did not observe anharmonicity of the lattice resonances, and we have set Γ = γ _TO = γ _LO throughout Reference Kukharskii[23].

2.2 Surface Polaritons

Surface polaritons (SP) are excitation states of transverse magnetic (TM) character at the boundary of two media whose dielectric functions fulfill certain conditions Reference Roeseler[13], Reference Agranovich and Mills[24]. The polariton wave vector is greater than the corresponding vector in the ambient medium. Normally, SP excitation by a simple reflection experiment is not possible, and gratings have to be ruled onto the surface, or light needs to be coupled in upon total reflection at the base of a high index prism Reference Agranovich and Mills[24]. However, it was pointed out by Röseler that under certain circumstances excitation of SP modes can be observed by IRSE at the interface between a polar and a metallic material Reference Roeseler[13]. There, the fact that the refractive index N = √ε in the polar medium becomes less than unity near the LO phonon frequency was discussed as the condition at which excitation of the SP mode at the metal interface can happen. The polar film primarily plays the role of the low-index gap for a prism setup, and the “prism” index of refraction is that of the ambient air.

The dispersion relation for SP modes at the boundary between two media, i.e., the condition for TM modes, follows directly from Maxwell, and can be written as follows Reference Agranovich and Mills[24]

with

being real and positive for a TM wave which is evanescent on both sides of the interface. ε ₁ and ε ₂ are the dielectric functions of film and substrate, respectively, and k _x = N _a sin (Φ _a ) is the x-component of the incident wave vector (angle of incidence Φ _a ; ambient index of refraction N_a ). The condition for the existence of a “true” SP follows immediately: One of the media must have a negative dielectric function in order to satisfy the equation above Reference Roeseler[13] Reference Agranovich and Mills[24]. This condition is fulfilled for the Berreman polariton (BP) observed in thin dielectric films attached to a metal surface, and was described in Berreman's original paper Reference Berreman[14].

A different type of solution for TM waves with wavevector along the interface can be read from the equation above when both ε ₁ and ε ₂ are positive, but one κ is imaginary and positive, and the other κ is imaginary and negative. We will call this type of TM wave a pseudo surface polariton (PSP) because the wave is not evanescent on both sides of the interface, but presents a power flow transport along the interface similar to that of a “true” SP. This situation is actually often observed (although not referred to in the manner just described) because the Berreman-effect in dielectric films attached to dielectric materials (e.g., a free-standing semiconductor material film) belongs to this second type of TM wave solution. (For the free standing film near its LO frequency neither side of the film interfaces possesses negative dielectric function values. More details of this discussion will be given somewhere else Reference Schubert, Hofmann, Tiwald and Sik[25].)

The samples investigated here consist in principle of two identical polar materials except for the concentration of free carriers. For now we will treat the films deposited on the doped substrate as one single GaAs film, which combines all layers including the GaN _x As_1−x layers. Note that the Ga-N related contributions to the GaN _x As_1−x IR dielectric functions are very small, and can be omitted in the meantime. The dielectric functions ε ₁ (the undoped GaAs film) and ε ₂ (the doped GaAs substrate) differ then by the Drude term contribution to ε ₂ only. These subtle differences lead to well-defined branches of PSP modes, which are related to the occurrence of the coupled plasmon-phonon bulk modes within the substrate (ε ₂). We therefore refer to the PSP modes as surface-plasmon-phonon induced pseudo polaritons. Figure 1 presents the solution of Equation 2 for positive and imaginary κ ₁ but negative and imaginary κ ₂ as a function of the substrate free carrier concentration n for Φ _a = 70°. For simplicity, and only for now, we assume no broadening (Γ₁ = Γ₂ = 0, μ ₂ = +∞). We find two branches (PSP⁺, PSP⁻) of PSP modes at spectral positions where the substratematerial has an index of refraction of less than 1. The PSP modes follow closely those of the longitudinal-optical coupled plasmon-phonon modes (LPP) in polar semiconductors with free carriers Reference Mitra and Palik[26], Reference Kukharskii[23]. However, the PSP frequencies are slightly larger than the LPP modes, and depend on the angle of incidence. The inset in Figure 1 shows the wavenumber differences between LPP and PSP modes for both branches as a function of n.

Figure 1. Dispersion of pseudo surface polariton modes at the interface doped / undoped GaAs as a function of the free-carrier concentration n for k_x = sin(70°). The two branches (PSP⁺, PSP⁻) follow closely those of the longitudinal-optical coupled plasmon-phonon bulk modes (LPP⁺, LPP⁻). The horizontal lines indicate the TO and LO frequencies of GaAs. The inset displays the difference between the PSP and LPP modes (PSP⁺-LPP⁺: right axis, PSP⁻-LPP⁻: left axis). Parameters used for calculations are ω _LO =291.7 cm⁻¹, ω _TO = 267.8 cm⁻¹, ε _∞ = 11.7, m* = 0.063m_e . No broadening is assumed here for simplicity.

The BP is also present in our sample situation because the BP is one solution of Equation 2. ε ₂ is negative near the GaAs-LO frequency in the substrate because of the free-carrier coupling, and the combined GaAs film has a positive ε ₁. κ ₁ and κ ₂ are real and positive and the BP is a “true” polariton in the case observed here. The occurrence of the BP was discussed and interpreted in several other publications (See, e.g., Refs. Reference Roeseler[13], Reference Humlicek[15], Reference Schubert, Rheinländer, Franke, Neumann, Tiwald, Woollam, Hahn and Richter[27], Reference Schubert, Hofmann, Tiwald and Sik[25]), and will not be further addressed here.

3.1 Results and Discussion

Figure 2 presents experimental (symbols) and calculated (solid lines) Ψ spectra from all three samples (Φ _a = 70°). Presentation of further angle-of-incidence data is omitted here for clarity. Vertical lines indicate the GaAs ω _TO and ω _LO frequencies. The best-fit calculations were performed considering the layered structure of the samples including the SL sequence. The dielectric functions of the constituents were calculated using the dielectric function model and the Drude approximation mentioned above. For the GaN _x As_1−x SL sublayers we included an additional harmonic oscillator to account for the Ga-N sublattice vibration (ω _TO2, ω _LO2, Γ₂). For the Te-doped n-type GaAs substrates we assumed an effective mass parameter of 0.063 free electron mass units, and we obtained an optical mobility parameter of μ = 3020 cm²/Vs. The logarithms of the substrate free-carrier concentrations (in units of cm⁻³) obtained from the IRSE data analysis where 17.425 ± 0.004, 17.395 ± 0.005 and 17.490 ± 0.003 for samples with x = 0.9, 1.3 and 3.3%, respectively. We found that the GaAs phonon frequencies are the same for all sample constituents, even for the GaN _x As_1−x SL sublayers, except for the broadening parameters (Γ =1.8, 2.5 and 3.9 cm⁻¹ for x = 0.9%, 1.3% and 3.3%, respectively) of the GaN _x As_1−x sublayers, which increase with increasing x. The increased number of dislocation and defects within the GaN _x As_1−x SL sublayers may explain the latter. Because the nitrogen incorporation within the GaN _x As_1−x sublayers is very small, the Ga-N vibration has a small amplitude (i.e., [ω _LO2 − ω _TO2]/ω _TO2 << 1) whereas the Ga-As resonance is almost unchanged. The best-fit calculations shown through Figures 2, 3, 4 and 5 are obtained from the best-fit parameters given in Tables 1 and 2. The best-fit GaAs model parameters are ω _LO =291.7 cm⁻¹, ω _TO =267.8 cm⁻¹, and ε _∞= 11.7. The high-frequency dielectric constant for the GaN _x As_1−x SL sublayers were found as ε _∞=10.0.

Figure 2. Experimental data (symbols), and best-fit calculation (solid lines) of Ψ _x As_1−x -SL heterostructures (See Table 1). Vertical lines indicate the GaAs ω _TO (TO₁) and ω _LO (LO₁) frequencies. The Berreman-polariton effect above the GaAs LO₁ frequency causes distinct resonance features in all samples. The upper-branch PSP⁺ mode is excited near 306cm⁻¹ between the n-type GaAs substrate and the undoped GaAs buffer layer / SL heterostructure. See also Figure 3. The angle of incidence is Φ _a = 70°. Spectra are to scale, but shifted for convenience by 20° each. The PSP⁺ resonance becomes washed out for higher nitrogen contents due to screening of the incident electromagnetic field components within the GaN _x As_1−x -SL sublayers by free carriers, which concentrate within the GaN _x As_1−x -SL sublayers. At TO₂, the Ga-N sublattice vibration is detected within the IRSE spectra. See also Figure 4.

Figure 3. Index of refraction N and extinction coefficient k (ε = [N + ik]²) of the n-type GaAs substrate (dotted lines), the undoped GaAs buffer layer and SL sublayers (solid lines), and GaN_0.009As_0.991 (dashed line) for example. The position of the PSP⁺ mode is indicated as the frequency at which the substrate refractive index becomes less than 1. The GaN_0.009As_0.991 optical spectra reveal the Ga-N sublattice resonance at TO₂ (vertical line).

Figure 4. Enlarged section of Figure 2 within the Ga-N sublattice resonance frequency (TO₂) range. (Spectrum for x=0.9% is shifted by −1° for convenience.) The feature labeled 2LO₁ is assigned as the forbidden but disorder-activated second harmonic of the GaAs LO frequency in sample GaNAs019.

Figure 5. GaN _x As_1−x relative TO-LO splitting f and carrier concentration n versus x obtained from IRSE data analysis (See Table 2; Lines are drawn to guide the eye.).

The Ga-N sublattice vibration resonance, observed within the IR-SE data at ~70 cm⁻¹, and labeled as “TO₂” in Figures 2 and 3, provides sensitivity to the nitrogen concentration within the GaN _x As_1−x sublayers, which is further discussed below. The excitation of the BP (Berreman polariton) causes the dip within all data sets near the GaAs ω _LO frequency, and provides sensitivity to the buffer-layer and GaAs SL sublayer thicknesses and phonon frequencies. We also observe a sharp resonance structure near ω ~306 cm⁻¹. This resonance, labeled by PSP⁺ in Figures 2 and 3, is related to the occurrence of the upper-branch PSP TM mode between the GaAs substrate and the combined GaAs buffer layer/GaAs/GaN _x As_1−x SL heterostructure film.

Figure 3 shows the complex index of refraction N + ik = √ε of the n-type GaAs substrate, the GaAs buffer layer (which is identical to the GaAs SL sublayers), and the GaN_0.009As_0.991 layer (Sample GaNAs016, see Tables 1 and 2). The position at which the substrate index of refraction N is less than 1 is indicated by a vertical line, and labeled by PSP⁺. This spectral position matches exactly the frequency at which we observe the resonance feature within our IRSE data on all samples labeled by PSP⁺ in Figure 2. It further matches the condition mentioned above for existence of the unbound TM wave propagating along the substrate/film interface, and its frequency can be exactly located in Figure 1 for the GaAs substrate carrier concentration of log(n[cm⁻³]) = 17.425. As can be seen in Figure 2, the PSP⁺ resonance is sharp for the sample with x = 0.9%, and less pronounced for the sample with 1.3%. For the sample with x = 3.3% the PSP⁺ resonance is almost subsumed by the GaAs reststrahlen band, but still present as a weak shoulder on the high-energy side of the GaAs TO-LPP⁺ reflectivity band. This damping behavior is not due to the slight increase of the lattice resonance broadening within the GaAs/GaN _x As_1−x SL heterostructure (See Table 2). In order to successfully model the damping of the PSP⁺ feature we need to consider free-carrier contributions to the optical response of the GaN _x As_1−x SL sublayers. A simple explanation for the damping of the PSP feature is that free carriers within the GaN _x As_1−x SL sublayers screen the incident electromagnetic fields, which otherwise penetrate through the film into the substrate/film interface region. The carrier absorption within the GaN _x As_1−x SL sublayers therefore effectively suppresses the excitation of the PSP⁺ resonance. We obtain from our best-fit analysis that the carrier concentration parameter increases with increasing x (Table 2), in accordance with the observation of the PSP damping in Figure 2. For the GaN _x As_1−x sublayers we assumed the GaAs effective mass parameter. The optical mobility parameter of μ ~100 cm² / Vs obtained for the GaN _x As_1−x SL sublayers is much less than that for the Te-doped substrate. This may indicate holes as majority carriers, which are known to obey smaller mobility. However, from this optical experiment one cannot differentiate between n- or p-type conductivity. As discussed in Ref. Reference Schubert, Kasic, Tiwald, Off and Kuhn[18], the free-carrier related quantities derived from the IRSE experiments are the ratios m*/n, and 1/(μ n). Here we assumed the GaAs electron effective mass m*, and obtained n given in Table 2, and μ~100 cm²/Vs . Concentration and mobility would change accordingly if the hole effective mass parameter would be chosen for data analysis, but the calculated best-fit spectra as well as all other parameters would remain unchanged.

Figure 4 presents the enlarged section for the Ga-N sublattice resonance frequency (TO₂) within our IRSE data (Figure 2). The vertical line indicates the TO₂ resonance. The frequency shift of TO₂ versus x (see Tab. II) is negligible, and within its uncertainty limit of ± 0.5 cm⁻¹. On the other hand, the TO-LO splitting of the Ga-N resonance, f = (ω_TO-ω_LO )/ω_TO , i.e., the polar strength of the Ga-N phonon branch, increases with increasing x. We obtain that f increases linearly versus x. We also observe an increase in Γ₂ with x due to the increase in dislocation and disorder within the GaN _x As_1−x SL sublayers. Accordingly, the forbidden second harmonic of the GaAs LO frequency 2LO₁ is detected within the IR-SE data from the sample with x= 3.3 %. This observation is explained by the breakdown of selection rules in the GaN _x As_1−x sublayers due to the increase of strain-induced lattice disorder.

Figure 5 shows the dependences of the relative Ga-N resonance LO-TO splitting f and the GaN _x As_1−x sublayer carrier concentration n versus x. The linear dependence of f versus x obtained here can be used to test the nitrogen concentration in other GaN _x As_1−x epilayers. Note that the sensitivity of the IR-SE data to n is larger for higher concentration x because free-carrier detection limits exist for small concentration values Reference Schubert, Kasic, Tiwald, Off and Kuhn[18], Reference Kasic, Schubert, Tiwald, Woollam, Einfeldt and Hommel[22].