AlGaN quantum wells caused by valence band mixing effect

Physics Letters A 372 (2008) 1691–1694 www.elsevier.com/locate/pla

Giant optical anisotropy in R-plane GaN/AlGaN quantum wells caused by valence band mixing effect Chun-Nan Chen a,∗ , Wei-Long Su b , Jih-Chen Chiang c , Ikai Lo c , Wan-Tsang Wang c , Meng-En Lee d a Department of Physics, Tamkang University, 151 Ying-chuan road, Tamsui, Taipei County, Taiwan 251, Taiwan, ROC b Department of Electronic Engineering, Lee-Ming Institute of Technology, Taishan, Taipei County, Taiwan 24305, Taiwan, ROC c Department of Physics and Center for Nanoscience and Nanotechnology, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC d Department of Physics, National Kaohsiung Normal University, Yanchao Township, Kaohsiung Country, Taiwan, ROC

Received 2 August 2007; received in revised form 23 September 2007; accepted 6 October 2007 Available online 11 October 2007 Communicated by R. Wu

Abstract ¯ This study investigates the optical anisotropy spectrum in the R-plane (i.e., the [1012]-oriented layer plane) of GaN/Al0.2 Ga0.8 N quantum wells of different widths. The optical matrix elements in the wurtzite quantum wells are calculated using the k · p finite difference scheme. The ¯ calculations show that the valence band mixing effect produces giant in-plane optical anisotropy in [1012]-oriented GaN/Al0.2 Ga0.8 N quantum wells with a narrow width. The nature of the in-plane optical anisotropy is found to be dependent on the well width. Specifically, it is found that the anisotropy changes from x  -polarization to y  -polarization as the well width increases. © 2007 Elsevier B.V. All rights reserved. PACS: 68.65.Fg; 78.55.Cr; 78.67.De Keywords: R-plane; Optical anisotropy; Valence band mixing; Optical matrix element

GaN-based semiconductors have received increasing attention in the past decade as a result of their unique properties and potential applications in the electronics and optoelectronics fields [1–5]. Recent advances in crystal growth techniques now enable the fabrication of high-quality III-nitride based semiconductor heterostructures on substrates with orientations other than the conventional [0001] direction [6,7]. The optical and electronic properties of such semiconductor heterostructures are quite different from those of semiconductor crystals grown on conventionally orientated substrates. For example, Rau et al. [8] and Sun et al. [9] identified the existence of giant in-plane optical anisotropy (over 90%) in non-polar III-nitride based wurtzite quantum wells, while Sharma et al. [10]. observed weaker optical anisotropy (∼32%) * Corresponding author.

E-mail addresses: [email protected] (C.-N. Chen), [email protected] (J.-C. Chiang). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.10.021

in semi-polar III-nitride based wurtzite quantum wells. This study employs an arbitrarily-oriented [hkil] Hamiltonian potential matrix to conduct a comprehensive investigation into inplane optical anisotropy in semi-polar wurtzite quantum wells. The optical anisotropy spectrum is calculated for semi-polar ¯ [1012]-oriented GaN/Al0.2 Ga0.8 N quantum wells of different widths. It is shown that the valence band mixing (VBM) ef¯ fect results in giant optical anisotropy in thin [1012]-oriented ¯ GaN/Al0.2 Ga0.8 N quantum wells. In a case of [1012]-oriented well plane, quantum well structures will have the internal field due to the piezoelectric and spontaneous polarizations. However, for highlighting the VBM effect we omitting the internal field and strain effects in the calculation. The physical origin of the giant in-plane optical anisotropy (over 90%) observed in non-polar quantum wells is discussed elsewhere, and is therefore not considered here. The current optical transition calculations are based on the formalism introduced by Lew Yan Voon and Ram-Mohan [11] and are solved using the k · p finite differ-

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¯ Fig. 1. (a) Valence band structures of [1012]-oriented GaN crystal for Δ2 = Δ3 = 0 (solid lines). In comparison with the solid lines, the dashed lines with open squared symbols show the valence band structures when the coupling between the |X  - and |Z  -states is not included (i.e., assuming N2 kx kz = 0). ¯ (b) Valence band structures of [1012]-oriented GaN crystal for Δ2 = Δ3 = 0 (dashed lines) and Δ2 = Δ3 = 0 (solid lines).

ence scheme [12,13]. In the calculations, the directions of the ¯ ¯ 10] ¯ x-, y- and z-axes are assumed to lie along the [1010], [12 and [0001] directions, respectively. Furthermore, in the calculations the three unit vectors in the prime coordinates (x  , y  , z ) ˆ by setting the azimuthal angle φ = 0◦ are given by yˆ  = y, zˆ  = zˆ cos θ + xˆ sin θ and xˆ  = xˆ cos θ − zˆ sin θ , respectively, where θ is the polar angle and zˆ  is the growth direction. Note that the polar angles θ = 0◦ , 43.2◦ and 90◦ correspond to the ¯ and [1010] ¯ growth directions, respectively. zˆ  = [0001], [1012] Fig. 1(a) shows the GaN valence band structures along the ¯ ¯ [1010]and [1012]-directions without taking the spin–orbit interaction into consideration (i.e., Δ2 = Δ3 = 0). In this fig¯ ure, the growth direction is chosen to be [1012], and hence     ¯ and in the prime coordinate system (x , y , z ), zˆ = [1012] ¯   kz //[1012]. It is observed that the valence bands along the ¯ direction are pure |Y - (labeled Y ), pure |Z- (labeled Z) [1010] and pure |X- (labeled X) states, respectively. These pure states arise because the off-diagonal terms (i.e., N1 kx ky , N2 kx kz and N2 ky kz ) in the 3×3 valence band Hamiltonian given in Eq. (26) ¯ direction [14]. of Ref. [14] are all equal to zero in the [1010] ¯ direction, the off-diagonal term, i.e., However, in the [1012] N2 kx kz , is not equal to zero, and hence mixing occurs between the |X- and |Z-bands. Prior to mixing, as shown by the dashed lines with open squared symbol, the |X-band (labeled X) and the |Z-band (labeled Z) have a lower energy than the |Y -band (labeled Y ). However, after mixing (please refer to the solid  lines), the upper X–Z mixing band (labeled XX  Z  ) has a higher energy than the |Y -band (labeled Y  ) when the wave vector kz is greater than ∼0.07 (1/Å), because the coupling term, i.e., N2 kx kz , increases rapidly with increasing kz . In this figure, the

¯ Fig. 2. In-plane (i.e., x  –y  plane) valence subband structures of [1012]-oriented GaN/Al0.2 Ga0.8 N quantum well with 15 Å well width.   upper (lower) X–Z mixing band is labeled as XX  Z  (ZX  Z  ) because it is not only an |X-like (|Z-like) X–Z mixing band, but also an |X  -like (|Z  -like) X  –Z  mixing band. Additionally, the |Y -band is labeled as Y  because the |Y - and |Y  -bands are identical. Clearly, the X–Z valence band mixing leads to a  ¯ crossover of the Y  and XX  Z  bands in the [1012] direction. The dashed lines in Fig. 1(b) show the GaN valence band ¯ ¯ structures along the [1010]and [1012]-directions when the spin–orbital interaction is taken into consideration (i.e., Δ2 = Δ3 = 0). Here for convenience, the valence bands are labeled (from top to bottom) as the heavy-hole (HH) band, the lighthole (LH) band, and the crystal-field (CH) band, respectively.  It is seen that strong mixing between the Y  and XX  Z  bands near their crossing point causes the HH (LH) band along the ¯ [1012]-direction to change from a |Y  -like (|X  -like) state to   an |X -like (|Y -like) state as the wave vector kz increases. Additionally, it is also seen that the |X  -like HH and |Y  like LH bands are widely separated when kz is larger than ∼0.1 (1/Å). In this study, these phenomena are referred to as the valence band mixing (VBM) effect. As discussed later, the VBM effect implies that strong in-plane x  -polarization ¯ anisotropy exists in a [1012]-oriented unstrained quantum well with a narrow well width (L) (e.g., L < 30 Å). In the calculation, we took the valence band splitting energies as Δ1 = 16 meV, Δ2 = 4 meV, and Δ3 = 4 meV, and A parameters as A1 = −6.56(h¯ 2 /2m0 ), A2 = −0.91(h¯ 2 /2m0 ), A3 = 5.65(h¯ 2 /2m0 ), A4 = −2.83(h¯ 2 /2m0 ), A5 = −3.13(h¯ 2 /2m0 ), and A6 = −4.86(h¯ 2 /2m0 ) [14]. Fig. 2 shows the in-plane (i.e., x  –y  plane) valence sub¯ band structures of a [1012]-oriented GaN/Al0.2 Ga0.8 N quantum well with a well width of L = 15 Å. Reading from top to bottom, the valence subbands correspond to the first heavy-hole (HH1) band, the first light-hole (LH1) band, the second heavyhole (HH2) band, and the first crystal-field (CH1) band, respec-

C.-N. Chen et al. / Physics Letters A 372 (2008) 1691–1694

Fig. 3. Squared optical matrix elements of in-plane (i.e., x  –y  plane) optical ¯ transitions for [1012]-oriented GaN/Al0.2 Ga0.8 N quantum well with 15 Å well width.

tively. It is observed that near the Γ -point, the HH1 band is a strongly |X  -like symmetric (|X  S -like) state, the LH1 band is a strongly |Y  -like symmetric (|Y  S -like) state, the HH2 band is a strongly |X -like asymmetric (|X  A -like) state, and the CH1 band is a strongly |Z  -like symmetric (|Z  S -like) state. Meanwhile, in the kx  direction, the Γ -point effective mass of the HH2 band is negative and the Γ -point effective mass of the HH1 band is lighter than that of the LH1 band. As a result, strong 2-band mixings of (X  S –Y  S ) and (X  S –X  A ) occur near kx  = 0.03 (1/Å) and 0.07 (1/Å), respectively. Consequently, as kx  increases, the HH1 band changes from an |X  S -like state to a |Y  S -like state, while the LH1 band transits from a |Y  S -like state, through an |X  S -state, to an |X  A -like state. In the present study, this phenomenon is referred to as the quantum-well valence-band mixing (QWVBM) effect. Similarly, the QWVBM effect leads to 2-band mixing of X  A –Y  S near ky  = 0.07 (1/Å). As ky  increases, the LH1 band therefore changes from a |Y  S -like state to an |X  A -like state. However, it is observed that the HH1 band remains in an |X  S -like state. Fig. 3 shows |Mα |2c1–v1 as a function of the in-plane (i.e., ¯ the x  –y  plane) wave vector for a [1012]-oriented GaN/ Al0.2 Ga0.8 N quantum well with a well width of L = 15 Å, where |Mα |2c1–v1 denotes the squared optical matrix elements for an α-polarization (α = x  and y  ) transition from the first conduction subband (C1) to v1 valence subbands (where v1 = HH1, LH1 and CH1, respectively). Note that |Mα |2C1–CH1 (α = x  , y  ) is not shown in Fig. 3, mainly because the energy of the CH1 band is far less than that of either the HH1 band or the LH1 band and therefore it does not contribute to the optical properties of the GaN/Al0.2 Ga0.8 N quantum well. Near the zone center, the strongly |X S -like HH1 (|Y  S -like LH1) subband is a weak X  S –Y  S mixing band, and hence |Mx  |2C1–HH1 (|My  |2C1–LH1 ) is much larger than |My  |2C1–HH1 (|Mx  |2C1–LH1 ). This indicates that the weak X  S –Y  S mixing causes the photoluminescence generated by the C1–HH1 (C1–LH1) transition to be strongly polarized along the x  -axis (y  -axis). Certainly, the X  S –Y  S mixing becomes weaker as the energy separation between the HH1 and

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Fig. 4. Well-width dependence of optical anisotropy for C1–HH1 transition in ¯ GaN/Al0.2 Ga0.8 N quantum wells as function of substrate orientation (101l) with l = 0–∞ (polar angle θ = 90◦ –0◦ and azimuthal angle φ = 0◦ ).

¯ LH1 bands increases. In a thin [1012]-oriented wurtzite GaN unstrained quantum well (L < 30 Å), the VBM effect results in a large energy difference between the |X  S -like HH1 band and the |Y  S -like LH1 band near the zone center, and this implies the presence of strong in-plane x  -polarization anisotropy. In Fig. 3, as kx  increases toward the 2-band (X  S –Y  S ) mixing region (kx  ≈ 0.03 (1/Å)), the QWVBM (X  S –Y  S mixing) effect causes |Mx  |2C1–HH1 and |My  |2C1–LH1 to fall rapidly to zero and |My  |2C1–HH1 to increase to its maximum value. Similarly, the QWVBM effect causes |My  |2C1–LH1 to vanish for ky  > 0.07 (1/Å) and |Mx  |2C1–LH1 to assume a single-peak structure with a peak near kx  = 0.03 (1/Å). Fig. 4 shows the in-plane optical anisotropies at the zone center (i.e., ρ(kx  = ky  = 0)) of the z -oriented GaN/ Al0.2 Ga0.8 N quantum wells. Note that the in-plane optical anisotropy, ρ, is defined as ρ=

|Mx  |2C1–HH1 − |My  |2C1–HH1 |Mx  |2C1–HH1 + |My  |2C1–HH1

.

It can be seen that the largest optical anisotropy (y  -polariza¯ tion) occurs in the [1010]-oriented quantum wells, while sub ¯ sidiary anisotropy (x -polarization) appears in the thin [1012]oriented quantum wells (L < 30 Å). The strong anisotropy in ¯ the thin [1012]-oriented quantum wells is x  -polarized and is the result of the VBM effect. However, the strong anisotropy in ¯ the [1010]-quantum wells is y  -polarized and is caused by the crystal field effect (which will be discussed elsewhere). Most ¯ interestingly, in the [1012]-oriented quantum wells, it is observed that as the well width increases and reaches a critical value of LC (LC ≈ 30 Å), the in-plane anisotropy changes from x  -polarization to y  -polarization. Similar calculated results have been already reported in Ref. [15]. Again, this phenomenon is caused by the VBM effect (i.e., the HH band shown in Fig. 1 changes from an |X  -like state to a |Y  -like state as the wave vector kz decreases). The critical value, LC , decreases (increases) with increasing compressive (tensile) strain since compressive (tensile) strain enlarges (reduces) the crystal field split energy and therefore weakens (enhances) the cou-

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pling between the |X- and |Z-bands (refer to Fig. 1). Clearly, when the compressive (tensile) strain is sufficiently high, only y  -polarization (x  -polarization) anisotropy exists. Finally, inplane optical anisotropy is not found in the [0001]-oriented quantum wells because this particular quantum well belongs to the D6h high-symmetry point group. The results presented in this study are in good agreement with the experimental findings reported in the literature. For example, Rau et al. [8] and Sun et al. [9] identified the presence of giant in-plane optical anisotropy (over 90%) in non-polar III-nitride based wurtzite quantum wells, while Sharma et al. [10] observed weaker optical anisotropy (∼32%) in semi-polar III-nitride based wurtzite quantum wells. In conclusion, this study has calculated the optical transition of GaN/Al0.2 Ga0.8 N wurtzite quantum wells of different well widths using the k · p finite difference scheme. The results have shown that giant in-plane optical anisotropy exists in the ¯ R-plane (i.e., [1012]-oriented layer plane) of GaN/Al0.2 Ga0.8 N quantum wells with a narrow width as a result of the valence ¯ band mixing effect. The anisotropy in [1012]-oriented quantum wells is width dependent, i.e., the in-plane anisotropy changes from x  -polarization to y  -polarization as the well width increases. The results presented in this study are in good agreement with the experimental findings reported in the literature and provide valuable guidelines for the design of polarization stabilization devices based on polarization control or selectivity.

Acknowledgements The current authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan, ROC, under Grant No. NSC-94-2112-M-269-001. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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