1. Introduction

Wurtzite Al_xGa_{1- x}N alloys are an attractive candidate for realizing ultraviolet (UV) light emitters and high-power high-frequency electronic devices, because they have large bandgap energies ranging from 3.43 eV (x = 0) to 6.01 eV (x = 1) and they principally are a hard material. Recently, a 210-nm electroluminescence (EL) has been demonstrated for c-plane AlN p-i-n homojunction light-emitting-diodes (LEDs) [1]. However, the external quantum efficiency (EQE) was as low as 10^-6% at 300 K, which was extremely lower than that of conventional blue or violet InGaN LEDs being ~84.3% for a 444 nm InGaN LED [2]. To improve overall performance of AlGaN-based deep UV (DUV) LEDs, many researchers have been trying to improve AlGaN quality and to optimize the multiple-quantum-well (MQW) design. As a consequence, the EQE values of 6.7% for a 345 nm LED [3], over 3% for 244-280 nm LEDs [4], and 0.003% for a 222 nm LED [5] have been achieved. Obviously, the values are still insufficient for practical use.

Currently, above mentioned UV LEDs are grown along the c-axis. In these structures, spontaneous and piezoelectric polarization discontinuity at the heterointerfaces induces the electric fields in the structure, which are parallel to the growth direction. These electric fields separate the electron (e) and hole (h) wavefunctions to the opposite interfaces of quantum wells (QWs), resulting in low optical efficiency and the forward-current-induced blueshift of the c-plane LEDs. These effects are called as quantum-confined Stark effects (QCSEs) [6]. To avoid QCSEs in (Al, In, Ga)N heterostructures, epitaxial growths on off-polar orientations such as (110) a-plane, and (100) m-plane (nonpolar planes) and (112), (201), and (10) planes (semipolar planes) have been attracting attentions [7–13]. In 2006, m-plane InGaN LEDs, whose EQEs were 3.1% at 435 nm [14] and 38.9% at 405 nm [15], have been demonstrated using the low threading-dislocation (TD) density, free-standing (FS) GaN substrates [16], which were sliced from a sub-cm-thick c-plane FS-GaN grown by halide vapor phase epitaxy (HVPE). Schematic drawing of a c-plane FS-GaN boule grown on a c-plane Al₂O₃ substrate and a sliced m-plane FS-GaN is shown in Fig. 1(a) . Subsequently, longer wavelength, 468 nm m-plane InGaN LEDs with EQEs higher than 15% have been reported [17]. However, drastic decrease in the output power cannot be suppressed for the m-plane InGaN LEDs whose peak wavelength is longer than 470 nm. The reason for this is often attributed to the degradation of crystal quality, which is caused by the low growth temperatures required for achieving high InN mole fraction InGaN. To the best of our knowledge, nonpolar DUV LEDs have not been reported yet.

 

Fig. 1 (a) Schematic drawing of a c-plane FS-GaN boule grown on a c-plane Al₂O₃ substrate by HVPE and a sliced m-plane FS-GaN. (b) X-ray rocking curves for the (100) diffraction of the m-plane FS-GaN. The x-rays were irradiated along the c-axis or a-axis, as shown in panel (c). (d) Schematic diagram of the notations of three axes.

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Possible reason for the low EQEs of DUV LEDs is the increase of TD and point defect densities [18] with increasing AlN mole fraction x. Indeed, high quality GaN and AlGaN alloys of reduced nonaradiative recombination center (NRC) concentration are indispensable to realize high efficiency UV LEDs. In addition to the extended defects such as TDs and stacking faults (SFs), there exist microscopic ones such as point defects, complexes, and impurities (O, C, and Si) in (Al, In, Ga)N crystals. From the results of the time-resolved photoluminescence (TRPL) and positron annihilation measurements, the origin of NRCs have been proposed to be certain point-defect complexes containing cation vacancies (V_III-X) [18]. Quite recently, radiative lifetime (τ_R) of AlN has been shown to increase with increasing point defect and impurity concentrations [19,20], meaning that point defects play an important role in determining the recombination dynamics.

In order to optimize nonpolar Al_xGa_{1- x}N LED structures, the oscillator strengths (f) of three interband optical transitions in ideal, defect-free ones must be quantified as functions of light polarization direction and strains. With respect to the effect of anisotropic strain, optical polarization properties of (Al, In, Ga)N films and QWs of various orientations have been investigated [21–31]. Among them, Bhattacharyya et al. [31] have calculated the light polarization characteristics of the transitions involving three separate valence bands for m-plane Al_xGa_{1- x}N films on GaN, which suffer from anisotropic tensile stresses. They have estimated the anticrossing AlN mole fraction x = 0.10, and predicted that the lowest energy transition of Al_xGa_{1- x}N is predominantly polarization parallel to the substrate normal.

In this paper, the results of polarized CL measurement on m-plane Al_xGa_{1- x}N epilayers grown on the m-plane FS-GaN substrate are shown. All of the epilayers grown by ammonia-source molecular beam epitaxy (NH₃-MBE) and metalorganic vapor phase epitaxy (MOVPE) suffer from in-plane anisotropic tensile stresses. The results are quantitatively explained by calculating the transition energies and oscillator strengths of excitonic transitions involving three separate valence bands, as functions of in-plane strains. For the calculation, Bir-Pikus Hamiltonian [32] was used without fitting parameters.

2.Experimental details and results

Samples investigated are approximately 100 to 400-nm-thick m-plane Al_xGa_{1- x}N epilayers grown on an 1-μm-thick GaN homoepitaxial layer, which was grown on the m-plane FS-GaN substrates [16]. The surface of as-received substrates was inspected by atomic-force microscopy (AFM) and found to have smooth morphology with monolayer atomic steps [16,33,34]. However, the x-ray rocking curves (XRCs) exhibited multiple-peak or asymmetric line shapes [33], as shown in Fig. 1(b). This multiple grain structure principally originates from the bowing of the original c-plane FS-GaN. Because the state-of-the-art m-plane FS-GaN [16] is prepared by slicing nearly 1-cm-thick c-plane FS-GaN grown in Ga-polar [0001] direction, the c-plane tilt (including the wafer bowing) and twist mosaics of the initial crystal are transferred to blurring of the c-axis and the greater m-plane tilt mosaic along the a-axis, respectively, as shown schematically in Fig. 1(a). For instance, the values of full width at half-maximum (FWHM) of XRCs for the present substrate were Δω_mc = 76 and Δω_ma = 110 arcsec for the (100) diffraction along <0001> and <110> azimuths, respectively, [Fig. 1(b)] and Δω_r = 100 arcsec for the (102) diffraction. The TD and stacking fault (SF) densities were lower than 5 × 10⁶ cm⁻² and 1 × 10³ cm⁻¹, respectively [33,34]. The NH₃-MBE of m-plane Al_xGa_{1- x}N epilayers (0≤x≤0.70) were carried out at 870-970 °C using metallic Ga (7N) and Al (6N) sources. The beam-equivalent-pressures of metallic Ga and NH₃ were (2.1-4.0) × 10⁻⁵ and (1.3-4.1) × 10⁻² Pa, respectively. The growth details are found in Ref. 35. A 150-nm-thick Al_0.73Ga_0.27N and approximately 2.1-μm-thick AlN epilayers were grown at 1120 °C on the same m-plane FS-GaN substrates by MOVPE. The reactor pressure was 2.02 × 10⁴ Pa. Trimethylaluminum, triethylgallium, and NH₃ were used as the precursors.

High-resolution x-ray diffraction (XRD) measurements were carried out using a four-crystal monochromator and an analyzer crystal (Bruker D8). A system with a one-dimensional detector array was used to obtain the x-ray reciprocal space mapping (X-RSM) images. The m-plane Al_xGa_{1- x}N epilayers of low x (0<x≤0.32) were confirmed by X-RSM to grow coherently on the GaN base layers. Other high AlN mole fraction epilayers (x≥0.35) were partially or nearly fully relaxed. Due to the tensile strain, the epilayers of x≥0.35 had macroscopic surface cracks. Representative X-RSM images for the pseudomorphic Al_0.25Ga_0.75N and mostly relaxed Al_0.70Ga_0.30N epilayers grown by NH₃-MBE are displayed in Fig. 2 . In each panel, the ideal location for strain-free AlN is shown by a closed (red) circle. We note that critical layer thicknesses calculated by the model given by People and Bean [35] are 202 nm for x = 0.25 and 31 nm for x = 0.52, taking in-plane anisotropic lattice and thermal-expansion mismatches into account. Because both the pseudomorphic and relaxed films suffered from anisotropic strains, x values were calculated from the out-of-plane (m-axis) and in-plane (c- and a-axes) lattice parameters, which were obtained from a 2θ-ω scan for the (100) XRD and X-RSM results for (201) and (120) diffractions, respectively (see Fig. 2, for example). The relation ${\epsilon }_{X_3{X}_3}=\left(-C_{12}{\epsilon }_{X_{}{X}_1}-C_{13}{\epsilon }_{X_2{X}_2}\right)/C_{11}$ was used, where ${\epsilon }_{X_3{X}_3}$, ${\epsilon }_{X_1{X}_1}$and ${\epsilon }_{X_2{X}_2}$ are the strains along m-, a-, and c-axes and $C_{ij}$are the elastic stiffness constants of Al_xGa_{1- x}N alloys were assumed to obey [36] Vegard’s law. In-plane tensile strains increased with x for the pseudomorphic (coherently grown) films, as shown in Fig. 3(a) . Conversely, the strains were gradually relaxed by the partial relaxation for x>0.32. For the quantitative discussion of polarization properties, we define the notations for the three axes: X ₁ (perpendicular to the c-axis in the growth plane); X ₂ (parallel to the c-axis in the growth plane); and X ₃ (normal to the growth plane), as shown in Fig. 1(d).

 

Fig. 2 Representative X-RSM images for the pseudomorhic m-plane Al_0.25Ga_0.75N epilayer grown on the m-plane FS-GaN taken in the vicinity of (a) (130) and (b) (201) diffraction spots. The X-RSM images for partially lattice-relaxed Al_0.70Ga_0.30N epilayers taken for (c) (120) and (d) (201) diffractions. Both the epilayers were grown by NH₃-MBE. The closed circle in each panel shows the location of strain-free AlN, for comparison.

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Fig. 3 (a) FWHM values for the XRCs (Δω_mc, Δω_ma, and Δω_r) of m-plane Al_xGa_{1- x}N epilayers grown by NH₃-MBE and MOVPE. (b) Strain components ${\epsilon }_{X_1{X}_1},$ ${\epsilon }_{X_2{X}_2},$ and ${\epsilon }_{X_3{X}_3}$ of the m-plane Al_xGa_{1- x}N films as a function of AlN mole fraction x.

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The surface of the MOVPE epilayers exhibits well-aligned 0.26-nm-high monolayer atomic step lines. In contrast, the MBE epilayers exhibit surface striations parallel to the c-axis. We note that the striations have already been formed during the underlayer GaN growth [33]. Therefore, different from the MOVPE case, it seems difficult for NH₃-MBE [33,37] to eliminate such striations, even for the binary GaN growth. The reason for this may be an inappropriate surface preparation or insufficient migration of Ga adatoms due to lower growth temperature (T_g) in comparison with MOVPE, because NH₃-MBE is a surface-sensitive growth method. Indeed, the depth of the striations in the Al_xGa_{1- x}N films was deeper than the GaN case, where Al-containing materials need higher T_g to ensure sufficient surface migration. Here, we note that the in-plane polar direction of the epilayers was confirmed by convergent-beam electron diffraction (CBED) measurement to be the same as the substrate.

Despite the presence of such surface striations, all the m-plane Al_xGa_{1- x}N films grown by NH₃-MBE exhibit a single (100) XRD peak, similar to the case for the MOVPE films. The results mean that both the MOVPE and MBE epilayers did not show noticeable phase separation or compositional ordering along the m-axis. However, similar to FS-GaN, XRCs for the pseudomorphic Al_xGa_{1- x}N films exhibit a multiple-peak or asymmetric line shape. Because this is also the case for both underlying GaN homoepitaxial films [34,38] and the substrate [16], the multidomain structure (bowing) of the original c-plane FS-GaN must be the major reason [see Fig. 1(a)]. In contrast, relaxed Al_xGa_{1- x}N films exhibit broad but single-peaked XRCs. In those films, the multiple domain fine structure seems to be hidden due to the broadness of the line shape.

As long as coherent growth was maintained, Δω_mc of the Al_xGa_{1- x}N epilayers are the same as the substrate, regardless of the presence of striations, as shown in Fig. 3(b). The result is similar to the case with pseudomorphic m-pane In_xGa_{1- x}N (x≤0.14) epilayers [38] grown on the m-plane FS-GaN prepared by the same provider (Mitsubishi Chemical Holdings Group). However, Δω_ma and Δω_r immediately increased for 0<x≤0.25, presumably because the twist mosaic along the c-axis of the initial c-plane FS-GaN was exaggerated by the lattice and thermal-expansion mismatches between AlGaN and GaN during surface-sensitive NH₃-MBE. This must be another origin for the deeper striations in the Al_xGa_{1- x}N films, in comparison with GaN films. The broader horizontal width of the (120) X-RSM spot the Al_0.25Ga_0.75N epilayer [Fig. 2(a)] also reflects the result. For the Al_0.32Ga_0.68N film, Δω values were larger than those for x≤0.25, reflecting the increase in misfit dislocation densities.

Steady-state CL was excited with an electron beam operated at 3.0 kV. The probe current density was 1.0 × 10⁻² A/cm² at sample. The emission was dispersed by a 30-cm-focal-length grating monochromator, and detected using a multi-channel charge-coupled device. A Glan-Thompson prism polarizer was used for the polarized CL measurement. Polarized near-band-edge (NBE) CL spectra at 12 K of the m-plane Al_xGa_{1- x}N films are shown as a function of x in Fig. 4(a) . The intensities are normalized to that of stronger polarization direction (X ₁ or X ₂) for each x. The entire spectra shifted to the higher energy with increasing x, although some of them exhibited double emission peaks. As shown, the light polarization direction altered from X ₁ to X ₂ between x = 0.25 and 0.32. The value of polarization ratio ρ was defined as$\left(I_{X_1}-I_{X_2}\right)/\left(I_{X_1}+I_{X_2}\right)$, where $I_{X_2}$ and $I_{X_1}$ are the spectrally-integrated CL intensities of the NBE emission. The values are plotted by closed circles in Fig. 4(b).

 

Fig. 4 (a) Polarized CL spectra at 12 K of m-plane Al_xGa_{1- x}N epilayers grown on the m-plane FS-GaN substrates. (b) Polarization ratios, which are defined as $\left(I_{X_1}-I_{X_2}\right)/\left(I_{X_1}+I_{X_2}\right),$ of the Al_xGa_{1- x}N films as a function of AlN mole fraction x. Corresponding values calculated using the relative oscillator strengths are also shown. The films of x≤0.70 were grown by NH₃-MBE and x≥0.73 were grown by MOVPE.

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3. Theoretical analysis

To quantitatively explain the experimental findings, the energies and oscillator strengths of the interband transitions involving three separate valence bands (VBs) are calculated using the Bir-Pikus Hamiltonian [32], taking the anisotropic strains into account. At the Γ point, the states at the conduction band minimum (CBM) have an atomic s orbital with wavefunctions of |S〉 symmetry. The three valence band maximum (VBM) states have atomic p orbitals with wavefunctions of a combination of |X〉, |Y〉, and |Z〉 symmetries. For simplicity, excitonic effects are neglected in the calculation. The Hamiltonian for the strain dependence of VB is given by the following 6 × 6 matrix$H=\left[\begin{array}{cccccc}F& 0& -H^{\ast }& 0& K^{\ast }& 0\\ 0& G& \Delta & -H^{\ast }& 0& K^{\ast }\\ -H& \Delta & \lambda & 0& I^{\ast }& 0\\ 0& -H& 0& \lambda & \Delta & I^{\ast }\\ K& 0& I& \Delta & G& 0\\ 0& K& 0& I& 0& F\end{array}\right],$ where$F={\Delta }_1+{\Delta }_2+\lambda +\theta ,$ $G={\Delta }_1-{\Delta }_2+\lambda +\theta ,$ $H=i\left(A_6{k}_z{k}_{+}+A_7{k}_{+}+D_6{\epsilon }_{z+}\right),$ $I=i\left(A_6{k}_z{k}_{+}-A_7{k}_{+}+D_6{\epsilon }_{z+}\right),$ $K=A_5{k}_{+}^2+D_5{\epsilon }_{+},$ $\lambda =A_1{k}_z^2+A_2{k}_{\perp }^2+D_1{\epsilon }_{zz}+D_2\left({\epsilon }_{xx}+{\epsilon }_{yy}\right),$ $\theta =A_3{k}_z^2+A_4{k}_{\perp }^2+D_3{\epsilon }_{zz}+D_4\left({\epsilon }_{xx}+{\epsilon }_{yy}\right),$ $\Delta =\sqrt{2}{\Delta }_3,$ $k_{\perp }^2=k_x^2+k_y^2,$ $k_{+}=k_x+i{k}_y,$ ${\epsilon }_{z+}={\epsilon }_{xz}+i{\epsilon }_{yz},$ and

The parameters D_j (j = 1 to 6) denote the deformation potential constants for the valence band and A_j (j = 1 to 7) are Luttinger parameters, and ε_lm and k_l (l, m = X ₁, X ₂, X ₃) are the strain and wavevector components, respectively. Here we assume that non-diagonal elements of the strain tensor are zero. Δ₁ = Δ_cr is the crystal field splitting, while 3Δ₂ = 3Δ₃ = Δ_so are the spin-orbit splitting under quasi-cubic approximation. The basis functions of Bir-Pikus Hamiltonian are $\left(1/\sqrt{2}\right)|X+iY,\alpha 〉,$ $\left(1/\sqrt{2}\right)|X+iY,\beta 〉,$ $\left(1+\sqrt{2}\right)|X-iY,\alpha 〉,$ $\left(1/\sqrt{2}\right)|X-iY,\beta 〉,$, $|Z,\alpha 〉$ and $|Z,\beta 〉$. Here |α〉 and |β〉 denote the spin-wave functions corresponding to up spin and down spin, respectively. The method described here is universal, and Bhattacharyya et al. [31] have also used the same approach for calculating the electronic states of m-plane (Al, In, Ga)N alloys.

The exciton transition energies are obtained from the band energies and exciton binding energy:$E_j=E^{\ast }+E^c-E_j^v-E_{ex}^b,$ where E* = E_g + Δ₁ + Δ₂. The parameters E_g, E^c, E^v_j, and E^b_ex are the band gap energy, the CBM energy, the VBM energies, and the exciton binding energy, respectively. The E^b_ex values for the　A-, B-, and C-transition were set identical to 26 meV [39] for GaN and 51.3 meV for AlN [40].

The oscillator strength components for the transitions are obtained from momentum matrix elements ${\left|〈{\Psi }^{CB}|p_l|{\Psi }^{VB}〉\right|}^2$ with l = x, y, and z. Here, $〈{\Psi }^{CB}|=〈S|$ and $|〈{\Psi }^{VB}|〉=a_1|X〉+a_2|Y〉+a_3|Z〉$ represent the orbital parts of the CB and VB basis functions, respectively. The coefficients a_j are obtained by determining the eigenvectors of Hamiltonian. The relative values of ${\left|〈S|p_x|X〉\right|}^2,$ ${\left|〈S|p_y|Y〉\right|}^2,$ and ${\left|〈S|p_Z|X〉\right|}^2$ are set unity under the quasi-cubic approximation, ${\displaystyle {\sum }_{i=1}^3{f}_{i,\beta }=1}.$ The calculations stated herein are carried out exclusively at k = 0, meaning that the 6 × 6 matrix was effectively treated as 3 × 3.

For the practical calculation on Al_xGa_{1- x}N alloy films, the material parameters of end-point compounds, namely GaN and AlN, are taken from the literature, as shown in Table 1 . The parameters for the alloys are assumed to obey the Vegard's law, and the bowing parameter for the bandgap energy of strain-free Al_xGa_{1- x}N is chosen as 0.82 eV [41]. We use energy notations E ₁, E ₂, and E ₃ hereafter, because the crystal symmetry of the Al_xGa_{1- x}N films suffering from anisotropic stresses is no longer $C_{6v}^4.$

Table 1. Optical Constants of Thin Films of Materials^a

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The calculated relative oscillator strengths for the three interband transitions in the m-plane GaN are shown as functions of ${\epsilon }_{X_1{X}_1}$ and ${\epsilon }_{X_2{X}_2}$ by gray-scale contour plots in Fig. 5 . For each E ₁, E ₂, and E ₃ transition, the measured strain coordinate (strain-free) is plotted by a closed circle on the panel exhibiting the predominant polarization direction. As shown, the calculated polarization directions are X ₁, X ₃, and X ₂ in order of decreasing electron energy. The result is consistent with previous studies on GaN [42,43]. Similar calculated results for the m-plane Al_0.03Ga_0.97N are given in Fig. 6 . For each E ₁, E ₂, and E ₃ transition, the measured strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)=\left(0.08\%,0.13\%\right)$ is plotted by a closed circle on the panel exhibiting the calculated predominant polarization direction. Different from the case for GaN, the calculated polarization directions are X ₃, X ₁, and X ₂ in order of decreasing electron energy. The result means that anisotropic strain induces a remarkable change in the electronic band structures. In the case of Al_0.70Ga_0.30N alloy with $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)=\left(0.79\%,0.35\%\right),$ the polarization directions are X ₃, X ₂, and X ₁ in order of decreasing electron energy, as shown in Fig. 7 . As revealed from Figs. 6 and 7, the polarization ordering of Al_0.70Ga_0.30N was different from Al_0.03Ga_0.97N, and the oscillator strengths of Al_0.70Ga_0.30N showed weaker contrast than those of GaN and Al_0.03Ga_0.97N. Similar results are found in the case of AlN, as shown in Fig. 8 . The reason for this will be explained later. Gil and Alemu [23] have reported a theoretical study on the electronic band structure of m-plane GaN under anisotropic biaxial strain. They predicted that E ₁ and E ₃ transitions were E⊥c (X ₁) polarized under large in-plane compressive strain. With respect to E ₃ transition, their result differs from our calculated result for AlGaN alloys suffering from biaxial compressive strain (data not shown in this paper, because our AlGaN films basically suffered from in-plane tensile strains). The discrepancy might arise from the fact that we did not consider the excitonic effects. However, it is likely that their notation was different [24] from ours so that their m-plane would correspond to a-plane in our case, which might be the cause for this discrepancy. Similar arguments have been given by Bhattacharyya et al [31].

 

Fig. 5 Relative oscillator strengths of E ₁, E ₂, and E ₃ transitions for the m-plane GaN film as functions of in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)\text{ .}$ Closed circles indicate the experimentally obtained in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)=\left(0.00\%,0.00\%\right)\text{,}$ which are plotted on the respective predominant polarization directions.

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Fig. 6 Relative oscillator strengths of E ₁, E ₂, and E ₃ transitions for the m-plane Al_0.03Ga_0.97N film as functions of in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)\text{ .}$ Closed circles indicate the experimentally obtained in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)=\left(0.08\%,0.13\%\right)\text{,}$ which are plotted on the respective predominant polarization directions.

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Fig. 7 Relative oscillator strengths of E ₁, E ₂, and E ₃ transitions for the m-plane Al_0.70Ga_0.30N film as functions of in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right).$ Closed circles indicate the experimentally obtained in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)=\left(0.79\%,0.35\%\right),$ which are plotted on the respective predominant polarization directions.

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Fig. 8 Relative oscillator strengths of E ₁, E ₂, and E ₃ transitions for the m-plane Al_.N film as functions of in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right).$ Closed circles indicate the experimentally obtained in-plane strain coordinate $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)=\left(0.25\%,1.96\%\right).$ They are plotted on the outside of the frameworks of respective predominant polarization directions.

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The three exciton transition energies calculated for m-plane GaN, Al_0.03Ga_0.97N, Al_0.70Ga_0.30N, and AlN films under isotropic in-plane strain $\left({\epsilon }_{X_1{X}_1}={\epsilon }_{X_2{X}_2}\right)$ are shown in Figs. 9(a) , 9(b), 9(c), and 9(d), respectively. In the case of GaN and Al_0.03Ga_0.97N, the VB anticrossing immediately takes place when in-plane biaxial tensile strain is introduced. On the contrary, the VB anticrossing gradually takes place with increasing the tensile strain for Al_0.70Ga_0.30N and AlN. The latter result means that corresponding VBs are strongly hybridized, which gives rise to much lower oscillator strength contrast for Al_0.70Ga_0.30N and AlN, as shown in Figs. 7 and 8. The energy differences between E ₁ and E ₂ bands for the GaN, Al_0.03Ga_0.97N, Al_0.70Ga_0.30N, and AlN films are shown as functions of ${\epsilon }_{X_1{X}_1}$ and ${\epsilon }_{X_2{X}_2}$ in Figs. 9(e), 9(f), 9(g), and 9(h), respectively, using contour lines. Similar to Figs. 5, 6, 7, and 8, the measured strain coordinates are plotted by closed black circles. The E ₂-E ₁ values, in the same order, are predicted to be 7.7 meV, 6.2 meV, 106 meV, and 142 meV.

 

Fig. 9 Calculated E ₁, E ₂, and E ₃ exciton transition energies for the m-plane (a) GaN, (b) Al_0.03Ga_0.97N, (c) Al_0.70Ga_0.30N, and (d) AlN films. The energy difference between E ₂ and E ₁, (E ₂-E ₁), as functions of in-plane strains $\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)$ for the m-plane (e) GaN, (f) Al_0.03Ga_0.97N, (g) Al_0.70Ga_0.30N, and (h) AlN films. Closed circles indicate respective in-plane strains.

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Table 2 summarizes the polarization directions for the transitions and E ₂-E ₁ values calculated for m-plane Al_xGa_{1- x}N suffering from the experimentally obtained strain values. As shown, E ₁ transition is X ₃-polarized regardless of x. The result means that E ₁ (exciton) emission is essentially undetectable from the surface normal. Apart from E ₁, the polarization directions alter from X ₁ to X ₂ for E ₂ emission (X ₃ to X ₂ for E ₃ emission) between x = 0.25 and 0.32. Assuming that the experimentally observed CL peaks principally originate from E ₂ and E ₃ transitions, the calculated prediction is consistent with the experimental results, as shown in Fig. 4(b). In Fig. 4(b), ρ values of m-plane Al_xGa_{1- x}N films calculated using the oscillator strengths for the measured$\left({\epsilon }_{X_1{X}_1},{\epsilon }_{X_2{X}_2}\right)$ coordinates are plotted as a function of x by open circles. As shown, the experimental data nearly agree with the calculated ones, except rather low ρ values for x≤0.25. The low ρ values may be due to the light depolarization caused by high density surface striations along the c-axis, which had been disclosed using atomic force microscopy observation [33]. We must note in Fig. 4(a) that overall CL intensity for the samples of x≥0.58 is much weaker than that for x≤0.32. From Table 2, it is obvious that E ₂-E ₁ increases with x. Therefore, Boltzmann distribution gives rise to very low hole populations in E ₂ and E ₃ bands in comparison with E ₁ band for high x samples. This may be one of the reasons for the low CL intensities at low temperature, where nonradiative recombination channels are in principle frozen.

Table 2. Calculated polarization directions for E₁, E₂, and E₃ transitions and energy differences between E₁ and E₂ band (E₂-E₁).

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Finally, calculated E ₂ transition energies (closed squares), CL peak energies (open circles), and their energy differences (ΔE) for the m-plane Al_xGa_{1- x}N films are plotted as a function of x in Fig. 10 . The ΔE value, which is similar to the Stokes-type shift, ranges between 3 and 577 meV for the alloys. These values are slightly larger than those reported for c-plane AlGaN films grown by MOVPE (100 ~250 meV) [40], indicating the difficulties in growing homogeneous, excellent quality m-plane AlGaN epilayers. Leastwise, current m-plane Al_xGa_{1- x}N films exhibit UV emission peaks between 360 nm (x = 0) and 210 nm (x = 1).

 

Fig. 10 Calculated E ₂ transition energies (closed squares), measured CL peak energies (open circles), and their energy differences (closed diamonds) for the m-plane Al_xGa_{1- x}N films as a function of x.

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4. Conclusion

Interband optical polarization characteristics of UV-light-emitting m-plane Al_xGa_{1- x}N alloy films grown on the m-plane FS-GaN substrates were interpreted by means of polarized CL measurements and theoretical calculations. The predominant light-polarization direction of the emission peak for the films suffering from in-plane anisotropic tensile stresses was shown to alter from E⊥c (X ₁) to E//c (X ₂) between x = 0.25 and 0.32. Theoretical analysis of the oscillator strengths of interband transitions and exciton transition energies was carried out using Bir-Pikus Hamiltonian, taking the anisotropic strain into account. The calculation predicted that the lowest energy transition (E ₁) is X ₃-polarized regardless of x, meaning that edge-emitting LED configuration is preferred for E ₁ exciton emission. For surface-emitting configuration, the calculated polarization direction altered from X ₁ (E⊥c) to X ₂ (E//c) for E ₂ transition and X ₃ to X ₂ for E ₃ transition between x = 0.25 and 0.32. The variations of polarization ratio and overall CL intensity as a function of x were quantitatively reproduced through the calculation. These achievements may open the way of designing UV LEDs using nonpolar AlGaN alloys.