1. Introduction

Gallium nitride and related alloys are attractive materials for various high-temperature high-power electronic applications Reference Pearton and Ren[1] Reference Nakamura and Fasol[2] Reference Nakamura, Senoh, Nagahama, Iwasa, Yamada, Matsushita, Kiyoku and Sugimoto[3] Reference Amano, Kito, Hiramatsu and Akasaki[4]. Advances in the technology of low temperature epitaxial growth allow one to grow GaN layers of 2 - 10 nm thickness with the smoothness of hetero-interfaces at the order of one atomic layer Reference Nakamura, Senoh, Nagahama, Iwasa, Yamada, Matsushita, Kiyoku and Sugimoto[3] Reference Amano, Kito, Hiramatsu and Akasaki[4]. When the layer thickness is comparable with the de-Broglie wavelength, the electron and hole energy spectrum depends on the layer thickness due to quantum confinement effects. These effects open new possibilities to control the wavelength of the emitted light through the variation of the quantum well width.

In spite of a great deal of interest in these materials, many features of strained GaN layers, such as the influence of strains on the electron and hole effective masses, longitudinal-transverse exciton splitting, etc., are poorly explored. The study of the impact of strain upon material characteristics is important since the growth of GaN layers on sapphire and 6H-SiC substrates is a common practice due to the lack of suitable GaN substrates. Strains in GaN layers are caused by large lattice and thermal mismatches between GaN and the substrate materials.

In this work, we investigate optical properties of GaN layers grown on sapphire substrates and compare optical parameters of these strained layers with the data published on non-stressed material.

2. Experimental

The GaN layers used in our experiments were grown by low-pressure MOCVD on (0001) c-plane sapphire using trimethylgallium (TMGa) and ammonia (NH₃) as source materials Reference Hubbard, Pavlidis, Valiaev, Stevens-Kalceff and Tiginyanu[5]. The carrier gas was Pd-cell purified hydrogen (H₂). Heating was accomplished by RF induction of the graphite susceptor. A buffer layer of about 25-nm thick GaN was first grown at 510 °C. Subsequently n-GaN layers were grown at 1100 °C. The concentration of free electrons in the top n-GaN layer was 1.7 × 10¹⁷ cm⁻³.

Photoreflectivity and absorption spectra were measured using the light from a halogen lamp. The white light reflected from the sample (in reflectivity experiments) or passed through the sample (in absorption experiments) was analysed through a double spectrometer with 1200 lines/mm gratings assuring a linear dispersion of 0.8 nm/mm. The signal from an FEU-106 photomultiplier with SbKNaCs photocathode working in a photon counting mode was introduced in an IBM computer via the IEEE-488 interface for further data processing. The spectral resolution was better than 0.5 meV. The samples were mounted on the cold station of a LTS-22-C-330 workhorse-type optical cryogenic system.

3. Results and discussion

It is known from numerous experiments on Cu₂O and II-VI compounds that exact values of the exciton bound energy (R) and bandgap (E_g) are determined from the energy position of n = 2, n = 3, etc. exciton excited states. The errors in calculating the R and E_g values are higher in the case when only the n = 1 and n = 2 states are observed, since the screening of the Coulomb electrostatical potential influences the energy position of the n = 1 line. The strains inherent to GaN layers grown on sapphire or SiC substrates due to large lattice and thermal mismatches between GaN and these materials strongly influence the energy position of exciton lines Reference Skromme, Zhao, Goldenberg, Kong, Leonard, Bulman, Abernathy and Pearton[6] Reference Skromme[7] Reference Skromme, Jayapalan, Vaudo and Phanse[8] Reference Jayapalan, Skromme, Vaudo and Phanse[9]. According to our measurements and literature data the shift of the ground (n = 1) exciton states can reach several tens of meV, while the shift of this state due to the screening of the Coulomb electrostatical potential is several meV (an order of magnitude less). Apart from that, according to the literature data Reference Steube, Reimann, Frohlich and Clarke[10] Reference A Shikanai, Azuhata, Sota, Chichibu and Kuramata[11] Reference Edwards, Yoo, Bremser, Weeks, Nam, Davis, Lui, Stall, Horton, Perkins, Kuech and Aspnes[12], strains lead to a shift of all exciton lines, but practically do not influence the exciton binding energy.

The n = 1 lineshape in high quality crystals is determined by exciton polaritons sensitive to the longitudinal-transverse (ω_LT) splitting and the damping parameter (γ). The dielectric function in the region of exciton resonance is written as:

The reflectivity from an isotropic crystal in the case of normal incident light is determined by:

where

, , ε_b is the background dielectric constant, t is the dead-layer thickness, k is the exciton wave-vector, n ₁ and n ₂ are longitudinal and transverse components of the refractive index determined from:

An oscillating structure due to the Fabré-Perot interference is observed in the reflectivity and absorption spectra of GaN layers in the energy region E < 3.46 eV. The spectral dependence of the refractive index at photon energies approaching the exciton energy can be drawn from this interference pattern (Figure 1). At photon energies E > 3.47 eV the interference structure in spectra taken from a GaN layer of d = 1.5 μm thickness is practically absent. Instead, evident excitonic features are observed in this spectral range (Figure 2). The maxima in the reflectivity spectrum at 3.4870 and 3.4970 eV are due to the n = 1 states of A and B excitons, respectively. The weak feature at 3.5220 eV is attributed to the n = 1 state of the C exciton. A weak band at 3.5062 eV (associated with n = 2 state of the A exciton) as well as two weak features at 3.5120 and 3.5170 eV (associated with n = 2 state of the B exciton) are observed at the long-wavelength side of the n = 1 state of the C exciton. A value of 25.2 meV is deduced for the A exciton binding energy from the energy position of the n = 1 and n = 2 states. As mentioned above, the ground state energy is affected by the screening of the effective interaction between electrons and holes. Recently, it was shown that anisotropy, intersubband coupling and polaron corrections should be also taken into account when estimating ground and first excited exciton states in GaN Reference Rodina, Dietrich, Goldner, Eckey, Hoffmann, Efros and Rosen[13]. Therefore, the calculation of the effective Rydberg from the experimentally observed separation between the n = 1 and the n = 2 exciton states in the simplest hydrogen-like model is only a first approximation.

Figure 1. The spectral dependence of the refractive index calculated from the interference pattern in the reflectivity and absorption spectra of a GaN layer at 10 K and 300 K.

Figure 2. Experimental (solid curve) and calculated (dashed curve) spectra of 1.5 μm thick GaN layer. T = 10 K.

The simulation of the reflectivity contours for the n = 1 resonance of the A and B excitons in terms of a model taking into account the spatial dispersion, the presence of a dead-layer and the additional Pekar boundary conditions is illustrated by the dotted-line in Figure 2. The best fit to the experimental spectrum is achieved using ε _b = 9.3 and the following exciton parameters: transverse exciton energy E₀ ^A = 3.4870 eV, E₀ ^B = 3.497 eV; longitudinal transverse splitting E_LT ^A = 1.5 meV, E_LT ^B = 2.2 meV; damping parameter γ^A = 3.6 meV, γ^B = 2.2 meV; and exciton mass m^A = 1.2 m₀, m^B = 0.65 m₀.

One should note that the values obtained for the damping parameters of A and B excitons are too high. This could be a consequence of the strains our layers experience. According to the literature data, the damping parameter in strained GaN layers is higher than that inherent to non-stressed layers Reference Skromme, Jayapalan, Vaudo and Phanse[8] Reference Jayapalan, Skromme, Vaudo and Phanse[9] Reference Kornitzer, Ebner, Grehl, Thonke, Sauer, Kirchner, Schwegler, Kamp, Leszczynski, Grzegory and Porowski[14]. In order to check if the obtained values are not a mistake of simulation, we have applied our method for the calculation of spectra presented in Ref. Reference Kornitzer, Ebner, Grehl, Thonke, Sauer, Kirchner, Schwegler, Kamp, Leszczynski, Grzegory and Porowski[14] for non-stressed GaN layers (Figure 3). We have obtained the following parameters as a result of simulation: ε_b = 9.9, E₀ ^A = 3.4770 eV, E_LT ^A = 1.25 meV, m^A = 1.2 m₀, γ^A = 0.001 meV for the A exciton and E₀ ^B = 3.4814 eV, E_LT ^B = 1.15 meV, m^B = 0.45 m₀, γ^B = 0.02 meV for the B exciton. These values are in excellent agreement with those deduced in Ref. Reference Kornitzer, Ebner, Grehl, Thonke, Sauer, Kirchner, Schwegler, Kamp, Leszczynski, Grzegory and Porowski[14].

Figure 3. Experimental spectrum reproduced from Ref. Reference Kornitzer, Ebner, Grehl, Thonke, Sauer, Kirchner, Schwegler, Kamp, Leszczynski, Grzegory and Porowski[14] for a non-stressed GaN layer (dots) and the result of calculations (solid curve).

Numerous studies on exciton spectroscopy have shown that the energy position of the transverse exciton excited states are determined much more accurately from the absorption spectra. Figure 4 presents the reflectivity and absorption spectra of a GaN layer with d = 0.5 μm thickness. The simulation of the reflectivity contour of this layer by application of the method discussed above gives the following parameters: ε_b = 9.3, E₀ ^A = 3.4853 eV, E_LT ^A = 1. 5 meV, m^A = 1.2 m₀, γ^A = 3.3 meV for the A exciton and E₀ ^B = 3.4950 eV, E_LT ^B = 2.2 meV, m^B = 0.65 m₀, γ^B = 2.0 meV for the B exciton. One can observe that the parameters obtained for the 0.5 μm thick layer are very close to those of the 1.5 μm thick layer. This means that these layers are equally stressed in spite of the existence of an intermediate buffer layer. While the influence of biaxial strain on the energies of excitons in GaN layers is well documented Reference Gil, Pankove and Moustakas[15], experimental data on the stress-induced variation of the longitudinal-transverse splitting and exciton masses are practically absent. Using the data presented in Figure 18 of Ref. Reference Gil, Pankove and Moustakas[15], we estimated the strain in our layers to be of about 10 kbar. Comparing the values of E_LT deduced above for stressed and non-stressed layers, one can conclude that the LO-TO splitting of the A-exciton is practically unaffected by the stress, while this parameter increases from 1.25 meV to 2.2 meV for the B-exciton when the stress increases from 0 to 10 kbar. These observations are in fairly good agreement with the results of calculations presented in Figure 24 of ref. Reference Gil, Pankove and Moustakas[15]. On the other hand, the m^A exciton mass in our layers coincides with that of non-stressed layers, while the m^B exciton mass is increased by a factor of 1.4. These data are consistent with the results of theoretical analysis of GaN layers subjected to biaxial strain Reference A Shikanai, Azuhata, Sota, Chichibu and Kuramata[11]. On the basis of the obtained m^A and m^B exciton masses and the known electron effective mass (m_å*= 0.22-0.24m₀) Reference Skromme, Zhao, Goldenberg, Kong, Leonard, Bulman, Abernathy and Pearton[6] Reference Skromme[7] Reference Skromme, Jayapalan, Vaudo and Phanse[8] Reference Jayapalan, Skromme, Vaudo and Phanse[9] one can calculate the M*_v1= (1±0.1)m₀ and M*_v2= 0.39m₀ effective masses of the valence bands split due to the crystal field and spin-orbital interaction.

Figure 4. The reflectivity (dashed curve) and absorption (solid curve) spectra of a 0.5 μm thick GaN layer. T = 10 K.

A maximum at 3.4945 eV corresponding to the ω_tBtransversal B exciton frequency can be seen in the absorption spectrum. Two other maxima at 3.4910 and 3.4865 eV are observed on the long-wavelength side of this band. Taking into account the C_6v symmetry of the wurtzite GaN, one can conclude that Γ₅ and Γ₆ excitons are formed as a result of interaction between a hole from Γ₉ valence band and an electron from Γ₇ conduction band. The Γ₅ excitons are active in σ polarization, while Γ₆ excitons are inactive in dipole approximation Reference Dingle, Sell, Stokowski and Ilegems[16]. The weak feature at 3.4865 eV in the absorption spectrum differs by 1.2 meV from the

(3.4853 eV) determined from the reflectivity spectrum. This feature is either due to the Γ₅ exciton or due to the Γ₆ exciton appearing as result of breakdown of the selection rules caused by the biaxial strain. In the last case, the 1.2 meV value characterizes the exciton splitting due to the short-range exchange interaction.

The B exciton series is formed by the Γ₇ valence band and the Γ₇ conduction band. A Γ₁ exciton active in π polarization, a Γ₅ exciton active in σ polarization, and a Γ₂ exciton forbidden in both π and σ polarizations are formed as a result of this interaction. Taking this into account one can suppose that the absorption peak at 3.4945 eV is related to the transverse Γ₅(ω_t) exciton. The absorption peak at 3.4910 eV is due to the transverse Γ₁(ω_t) exciton. According to theoretical evaluations Reference Dingle, Sell, Stokowski and Ilegems[16], the intensity of the Γ₁ exciton in π polarization is lower than that of Γ₅ exciton in σ polarization. We suppose that the weak features at 3.5115 and 3.5153 eV are caused by the n = 2 excited state of the Γ₁ and Γ₅ excitons, respectively. In such a case, from the ground and the first excited exciton states one can deduce the binding energy for the Γ₁ and Γ₅ excitons: 27.1 and 27.6 meV respectively.

The n = 1 line of the C exciton forms a doublet at 3.5220 and 3.5250 eV which is supposed to be related to the Γ₁ and Γ₅ excitons, respectively. The weak feature at 3.5415 eV can be attributed to the n = 2 state of the C exciton. The binding energy of the C exciton deduced from this assignment equals 26.0 meV.

Three absorption bands at 3.5829, 3.5939 and 3.6110 eV were evidenced in the high-energy spectral range (Figure 5). We attribute these bands to transitions from Γ₉, Γ₇, and Γ₇ valence bands to the second E_c2 band of Γ₃ symmetry. The splitting of the Γ₁(E_c1) and Γ₃(E_c2) bands at k = 0 is estimated then to be around 100 meV, which is in rather good correlation with theoretical prediction Reference Bloom, Harbeke, Meier and Ortenburger[17] Reference Bougrov, Levinshtein, Rumyantsev, Zubrilov, Levinshtein, Rumyantsev and Shur[18].

Figure 5. The absorption spectrum of a 0.5 μm thick GaN layer measured in the high-energy spectral range. T = 10 K.