Valence subband coupling effect on polarization of spontaneous emissions from Al-rich AlGaN/AlN Quantum Wells

doi:10.1364/OE.20.027384

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Valence subband coupling effect on polarization of spontaneous emissions from Al-rich AlGaN/AlN Quantum Wells

Huimin Lu, Tongjun Yu, Gangcheng Yuan, Chuanyu Jia, Genxiang Chen, and Guoyi Zhang »View Author Affiliations

Optics Express, Vol. 20, Issue 25, pp. 27384-27392 (2012)
http://dx.doi.org/10.1364/OE.20.027384

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▸ Article Outline
– Abstract
1. Introduction
2. Theoretical modeling and simulation
3. Simulation results and discussion
4. Conclusion
– References and links

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Abstract

The optical polarization properties of Al-rich AlGaN/AlN quantum wells (QWs) were investigated using the theoretical model based on the k⋅p method. Numerical results show that there is valence subband coupling which can influence the peak emission wavelength and emission intensity for TE and TM polarization components from Al-rich AlGaN/AlN QWs. Especially the valence subband coupling could be strong enough when CH1 is close to HH1 and LH1 subbands to modulate the critical Al content switching dominant emissions from TE to TM polarization. It is believed that the valence subband coupling may give important influence on polarization properties of spontaneous emissions and should be considered in designing high efficiency AlGaN-based ultraviolet (UV) LEDs.

1. Introduction

AlGaN alloys, especially Al-rich AlGaN alloys are promising semiconductors for the production of light-emitting devices in the spectral range from ultraviolet (UV) to deep-UV. However, it is demonstrated that the emission efficiency of these UV emitters decreases dramatically with the increase of Al content [1

H. S. Kim, R. A. Mair, J. Li, J. Y. Lin, and H. X. Jiang, “Time-resolved photoluminescence studies of AlGaN alloys,” Appl. Phys. Lett. 76(10), 1252–1254 (2000). [CrossRef]

–3

H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “231-261 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on AlN multilayer buffers grown by ammonia pulse-flow method on sapphire,” Appl. Phys. Lett. 91(7), 071901 (2007). [CrossRef]

]. Other than great difficulties in crystal growth of high quality Al-rich AlGaN, unusual optical polarization properties of AlGaN-based UV emitters have been attributed to the degradation of luminescence properties. It has been found that the intensity of polarized emission perpendicular to c-direction (TE polarization) decreases dramatically as Al content increase [4

K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett. 84(25), 5264–5266 (2004). [CrossRef]

–6

T. Kolbe, A. Knauer, C. Chua, Z. Yang, S. Einfeldt, P. Vogt, N. M. Johnson, M. Weyers, and M. Kneissl, “Optical polarization characteristics of ultraviolet (In)(Al)GaN multiple quantum well light emitting diodes,” Appl. Phys. Lett. 97(17), 171105 (2010). [CrossRef]

]. As a result, the surface emission from c-plane of Al-rich AlGaN-based emitters is very weak, and emission efficiency is low as about 1% or less [7

H. Kawanishi, M. Senuma, M. Yamamoto, E. Niikura, and T. Nukui, “Extremely weak surface emission from (0001) c-plane AlGaN multiple quantum well structure in deep-ultraviolet spectral region,” Appl. Phys. Lett. 89(8), 081121 (2006). [CrossRef]

, 8

H. Hirayama, S. Fujikawa, N. Noguchi, J. Norimatsu, T. Takano, K. Tsubaki, and N. Kamata, “222-282 nm AlGaN and InAlGaN-based deep-UV LEDs fabricated on high-quality AlN on sapphire,” Phys. Status Solidi A 206(6), 1176–1182 (2009). [CrossRef]

]. Obtaining Al-rich AlGaN-based emitters with dominant TE polarization is in great demand and has attracted much attention [9

S. H. Park, “Optical gain characteristics of non-polar Al-rich AlGaN/AlN quantum well structure,” J. Appl. Phys. 110(6), 063105 (2011). [CrossRef]

–16

R. G. Banal, M. Funato, and Y. Kawakami, “Extremely high internal quantum efficiencies from AlGaN/AlN quantum wells emitting in the deep ultraviolet spectral region,” Appl. Phys. Lett. 99(1), 011902 (2011). [CrossRef]

For wurtzite III-nitride material with spin-orbit interaction, the valence bands are usually labeled as heavy hole (HH) band, light hole (LH) band and crystal-field split-off (CH) band. It is generally accepted that emission light is TE polarized when transitions are from conduction band to HH (C-HH) and LH band (C-LH), while TM polarized light comes from the transitions to CH band (C-CH) [17

S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett. 68(12), 1657–1659 (1996). [CrossRef]

]. Then, the order of Brillouin zone center

Γ_{9}

and

Γ_{7}^{2}

corresponding to HH and CH bands determines the polarization properties of emission light. However, Nam et al. [4

K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett. 84(25), 5264–5266 (2004). [CrossRef]

] and Banal et al. [18

R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B 79(12), 121308 (2009). [CrossRef]

] reported that there was no obvious emission wavelength peak shift between TE and TM polarization components in AlGaN alloys and AlGaN/AlN quantum wells (QWs), which could hardly be understood with the existence of energy gap between CH and HH subbands. Furthermore, Hirayama et al. [19

H. Hirayama, N. Noguchi, and N. Kamata, “222nm deep-ultraviolet AlGaN quantum well light-emitting diode with vertical emission properties,” Appl. Phys. Express 3(3), 032102 (2010). [CrossRef]

] have revealed that TE polarized emission is dominant in AlGaN/AlN QWs with 83% Al content although the energy difference between

Γ_{9}

and

Γ_{7}^{2}

almost equals to zero. Therefore, detailed investigation for clarifying the physical mechanism of optical polarization in AlGaN/AlN QWs is required. This will also be beneficial to realize the polarization control or the polarization switch of UV light emissions from AlGaN/AlN QWs.

In this work, the influence of valence subband coupling on polarization of spontaneous emissions from Al-rich AlGaN/GaN QWs was analyzed and discussed. Band structure and envelope wavefunctions of Al-rich AlGaN/AlN QWs were calculated using the theoretical model based on the k⋅p method [20

S. L. Chuang and C. S. Chang, “k⋅p method for strained wurtzite semiconductors,” Phys. Rev. B 54(4), 2491–2504 (1996). [CrossRef]

–23

H. Zhao, R. A. Arif, Y. K. Ee, and N. Tansu, “Self-consistent analysis of strain-compensated InGaN–AlGaN quantum wells for lasers and light-emitting diodes,” IEEE J. Quantum Electron. 45(1), 66–78 (2009). [CrossRef]

]. The simulation result demonstrated that there is obvious valence subband coupling at k_t = 0 and k_t ≠ 0, which can influence the momentum matrix element and spontaneous emission spectrum. This results in TE polarized emission from C-CH1 transitions and TM component from C-HH1 and C-LH1 transitions due to the valence subband coupling. When HH1 and CH1 subband are approaching each other, such coupling effects even become very strong and influence the critical Al content for polarization switching between dominant TE and TM emissions. It is believed that valence subband coupling may change the polarization properties of spontaneous emissions and should be considered in designing high efficiency AlGaN-based UV LEDs.

2. Theoretical modeling and simulation

For wurtzite III-nitride material the conduction subband energies

E_{n}^{c} (k_{t})

and associated envelope wavefunctions

φ_{n} (z; k_{t})

are determined by solving the effective mass equation

[\frac{ℏ^{2}}{2} (\frac{k_{t}^{2}}{m_{e}^{t}} + \frac{k_{z}^{2}}{m_{e}^{z}}) + E_{c}^{0} (z) + P_{c ε} (z) + P_{E} (z)] φ_{n} (z; k_{t}) = E_{n}^{c} (k_{t}) φ_{n} (z; k_{t})

(1)

where

E_{c}^{0} (z)

is the unstrained conduction band edge,

P_{c ε} (z)

is the hydrostatic energy shift in the conduction band induced by mechanical stresses and

P_{E} (z)

stands for an additional Hamiltonian term due to the spontaneous (SP) and piezoelectric (PZ) polarization electrostatic fields [24

F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56(16), R10024–R10027 (1997). [CrossRef]

, 25

V. Fiorentini, F. Bernardini, F. Della Sala, A. Di Carlo, and P. Lugli, “Effects of macroscopic polarization in III-V nitride multiple quantum wells,” Phys. Rev. B 60(12), 8849–8858 (1999). [CrossRef]

]. The carrier screening effect is ignored for the low carrier density in order to simplify calculation in this work. The 6 × 6 effective-mass Hamiltonian for the valence band can be block-diagonalized and the subband structure

E_{m}^{U} (k_{t})

can be determined by solving the following effective mass equation for the upper Hamiltonian

[H_{3 \times 3}^{U} (k_{t}; k_{z} = - i \frac{\partial}{\partial z}) + I_{3 \times 3} (E_{v}^{0} (z) + P_{E} (z))] \cdot [\begin{matrix} g_{m}^{(1)} (k_{t}; z) \\ g_{m}^{(2)} (k_{t}; z) \\ g_{m}^{(3)} (k_{t}; z) \end{matrix}] = E_{m}^{U} (k_{t}) [\begin{matrix} g_{m}^{(1)} (k_{t}; z) \\ g_{m}^{(2)} (k_{t}; z) \\ g_{m}^{(3)} (k_{t}; z) \end{matrix}]

(2)

where

E_{v}^{0} (z)

is the unstrained valence band edge and

g_{m}^{(j)} (z; k_{t})

are the envelope wavefunctions of the mth valence subbands. The bases

| 1 〉

| 2 〉

and

| 3 〉

for the upper Hamiltonian corresponding to the HH, LH and CH subbands, respectively, are defined as [21

S. L. Chuang, “Optical gain of strained wurtzite GaN quantum-well lasers,” IEEE J. Quantum Electron. 32(10), 1791–1800 (1996). [CrossRef]

]

{\begin{cases} | 1 〉 = α^{*} \frac{- 1}{\sqrt{2}} | (X + i Y) ↑ 〉 + α \frac{1}{\sqrt{2}} | (X - i Y) ↓ 〉 \\ | 2 〉 = β \frac{1}{\sqrt{2}} | (X - i Y) ↑ 〉 + β^{*} \frac{- 1}{\sqrt{2}} | (X + i Y) ↓ 〉 \\ | 3 〉 = β^{*} | Z ↑ 〉 + β | Z ↓ 〉 \end{cases}

(3)

where α and β are the parameters related to azimuthal angle in the k_x-k_y plane. That is, when valence subbands do not couple each other, the wavefunctions of HH, LH and CH subbands should only have

| 1 〉

| 2 〉

and

| 3 〉

components, respectively.

Using the calculated envelope wavefunctions, momentum matrix elements

{| {(M_{e})}_{n m} (k_{t}) |}^{2}

for TE polarization (

\hat{e} = \hat{x} or \hat{y} ⊥ c

axis) and TM polarization (

\hat{e} = \hat{z} ∥ c

axis) can be expressed as

{\begin{cases} {| {(M_{x})}_{n m} (k_{t}) |}^{2} = {| {(M_{y})}_{n m} (k_{t}) |}^{2} = \frac{1}{4} ({| \int_{- \infty}^{\infty} d z φ_{n}^{*} 〈 S | p_{x} | X 〉 g_{m}^{(1)} |}^{2} + {| \int_{- \infty}^{\infty} d z φ_{n}^{*} 〈 S | p_{x} | X 〉 g_{m}^{(2)} |}^{2}) \\ {| {(M_{z})}_{n m} (k_{t}) |}^{2} = \frac{1}{2} {| \int_{- \infty}^{\infty} d z φ_{n}^{*} 〈 S | p_{z} | Z 〉 g_{m}^{(3)} |}^{2} \end{cases}

(4)

where

〈 S | p_{x} | X 〉

and

〈 S | p_{z} | Z 〉

are the corresponding band-edge momentum matrix elements. Bases

| 1 〉

and

| 2 〉

are the mixing states of

| X 〉

and

| Y 〉

, and contribute to TE polarization of light. The inclusion of

| Z 〉

state correlates base

| 3 〉

with TM polarization as shown in Eq. (3).

Then the spontaneous emission rates of TE and TM polarizations can be given as

{\begin{cases} r_{s p}^{T E} (ℏ ω) = C \frac{2}{L_{z}} \sum_{n, m} \int \frac{k_{t} d k_{t}}{2 π} \cdot ({| {(M_{x})}_{n m} (k_{t}) |}^{2} + {| {(M_{y})}_{n m} (k_{t}) |}^{2}) \cdot \frac{Γ / (2 π) f_{n}^{c} (k_{t}) (1 - f_{m}^{v} (k_{t}))}{{(E_{n m}^{c v} (k_{t}) - h ω)}^{2} + {(Γ / 2)}^{2}} \\ r_{s p}^{T M} (ℏ ω) = C \frac{2}{L_{z}} \sum_{n, m} \int \frac{k_{t} d k_{t}}{2 π} \cdot {| {(M_{z})}_{n m} (k_{t}) |}^{2} \cdot \frac{Γ / (2 π) f_{n}^{c} (k_{t}) (1 - f_{m}^{v} (k_{t}))}{{(E_{n m}^{c v} (k_{t}) - h ω)}^{2} + {(Γ / 2)}^{2}} \end{cases}

(5)

where L_z is the well width and C is a coefficient independent from QW structure.

f_{n}^{c} (k_{t})

and

f_{m}^{v} (k_{t})

are Fermi-Dirac distributions for the electrons in the conduction and valence subbands. The half linewidth Г of Lorentzian function is taken as 6.58meV for simplicity in this work. In the above equations, the expression

\frac{2}{V} \sum_{k_{t}} = \frac{1}{π L_{z}} \int_{0}^{\infty} k_{t} d k_{t}

is used in the calculation of emission spontaneous spectra. Other expression

\frac{2}{V} \sum_{k_{t}} = \int_{0}^{\infty} ρ_{r}^{2 D} d E {}_{t}

also is applied for calculating emission intensity varying k_t. The parameters for wurtzite GaN and AlN used in this work are taken from Refs [26

J. Piprek, Semiconductor Optoelectronic Devices (Academic Press, 2003).

–30

H. M. Lu and G. X. Chen, “Design strategies for mitigating the influence of polarization effects on GaN-based multiple quantum well light-emitting diodes,” J. Appl. Phys. 109(9), 093102 (2011). [CrossRef]

3. Simulation results and discussion

Previously, it is presented that the light emission with TE polarization of AlGaN-based QWs profits from compressive strain and thin well width [18

R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B 79(12), 121308 (2009). [CrossRef]

, 31

A. A. Yamaguchi, “Valence band engineering for remarkable enhancement of surface emission in AlGaN deep-ultraviolet light emitting diodes,” Phys. Status Solidi C 5(6), 2364–2366 (2008). [CrossRef]

]. Therefore, for our simulation model, the QW consisted of a 1.5nm thick AlGaN well and a 10 nm thick AlN barrier. Figure 1 gives the valence subband structure and labels the subband order for AlGaN/AlN QWs with Al content of 81%, 83% and 90%, respectively. As shown in Fig. 1, the HH1 subband is the first valence subband for Al_0.81Ga_0.19N/AlN and Al_0.83Ga_0.17N/AlN QWs, but for Al_0.9Ga_0.1N/AlN QW the CH1 subband becomes the first subband. It also can be seen from Figs. 1(a) and 1(c) that there is an energy difference of more than 30meV between HH1 and CH1 subbands in Al_0.81Ga_0.19N/AlN QW at Brillouin zone center, and only CH1 and CH2 subbands are confined in Al_0.9Ga_0.1N/AlN QW. When Al content is 83% in Fig. 1(b), CH1 subband locates between HH1 and LH1 subbands and becomes close to HH1 subband at zone center. This could correlate Al_0.83Ga_0.17N/AlN QW with its spectacular optical properties of polarization switching from dominant TE to TM light emission [18

R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B 79(12), 121308 (2009). [CrossRef]

Fig. 1 The valence subband structures of (a) Al_0.81Ga_0.19N/AlN QW, (b) Al_0.83Ga_0.17N/AlN QW and (c) Al_0.9Ga_0.1N/AlN QW.

Because the Al_0.83Ga_0.17N/AlN QW has special valence subband structure, the envelope wavefunctions of the three topmost valence subbands are calculated and given in Fig. 2 . For clarity, Fig. 2 only plots a part of the envelope wavefunctions near well region from 6nm to 16nm. The envelop wavefunction for CH is not well confined in the well layer for the reason that effective mass of CH is much smaller than that of HH and LH for GaN-based material. It is indicated from Figs. 2(a)–2(c) that the CH1 subband is decoupled with the HH1 subband but strongly couples with LH1 subbands at k_t = 0. That is, the wavefunction of HH1 subband only has

| 1 〉

component at k_t = 0 and that of LH1 and CH1 subbands include both

| 2 〉

and

| 3 〉

components. As a result, when the C-LH1 and C-CH1 transitions occur at zone center, the emission light should be characterized with both TE and TM polarizations. Furthermore, HH1, CH1 and LH1 subbands are coupled with each other at k_t = 1nm⁻¹ as described in Figs. 2(d)–2(f), which results in the emission light with both TE and TM polarizations for C-HH1, C-CH1 and C-LH1 transitions. Therefore, compared with weak coupling of the HH1, LH1 and CH1 subbands in InGaN/GaN QWs [32

C. Jia, T. Yu, R. Tao, X. Hu, Z. Yang, Z. Qin, Z. Chen, and G. Zhang, “Spontaneous luminescence polarizations of wurtzite InGaN/GaN quantum wells,” Appl. Phys. Lett. 93(17), 171114 (2008). [CrossRef]

], the influence of valence subband coupling on the optical polarization of AlGaN/AlN QWs is much greater and should not be ignored.

Fig. 2 Envelope wavefunctions of the three topmost valence subband for Al_0.83Ga_0.17N/AlN QW at (a)-(c) k_t = 0 nm⁻¹ and (d)-(f) k_t = 1.0 nm⁻¹.

Using the calculated envelope wavefunctions, the momentum matrix elements in the vicinity of the center zone from k_t = 0 to k_t = 1nm⁻¹ for the Al_0.83Ga_0.17N/AlN QW are obtained in Figs. 3(a) –3(c). It is demonstrated that the momentum matrix elements can be influenced seriously by valence subband coupling especially at k_t ≠ 0. As expected from the above discussions on envelope wavefunctions, at k_t = 0 the momentum matrix elements for C1-CH1 and C1-LH1 transitions have both TE and TM components, and away from zone center all the C1-HH1, C1-CH1 and C1-LH1 transitions contribute to both TE and TM polarization components. If only valence subband order is considered, the optical polarization can be determined by momentum matrix elements of topmost subband at k_t = 0 [10

A. A. Yamaguchi, “Theoretical investigation of optical polarization properties in Al-rich AlGaN quantum wells with various substrate orientations,” Appl. Phys. Lett. 96(15), 151911 (2010). [CrossRef]

]. However, as shown in Figs. 3(a)–3(c), the momentum matrix elements are very different at k_t = 0 and k_t ≠ 0 due to the valence subband coupling. So, the influence of the transitions at k_t ≠ 0 should be discussed in more exactly analyzing optical polarization properties.

Fig. 3 (a-c) Momentum matrix elements versus k_t and (d-f) spontaneous emission rate versus k_t for the conduction band to the three topmost valence subband transition in Al_0.83Ga_0.17N/AlN QW.

Considering that momentum matrix element directly determines the transition rate from conduction to valence subbands, spontaneous emission intensities of TE and TM components varying with k_t for C1-HH1, C1-CH1 and C1-LH1 transitions are calculated at carrier density of 1.0 × 10¹⁸cm⁻³ and given in Figs. 3(d)–3(f). It is indicated by solid lines that dominant TE component comes from not only the C1-HH1 and C-LH1 transitions near k_t = 0, but also C1-CH1 transitions away from zone center. While, the main TM polarization component originates also from both C1-HH1 and C1-LH1 transitions away from zone center and the C1-CH1 transition near k_t = 0 shown by dashed lines. As a result, the corresponding emission peak wavelength of TE polarization component should be close to that of TM component. Furthermore, the ratio of TE component to TM component in emission light should be influenced by the valence subband coupling for AlGaN/AlN QW.

Figure 4 give the contribution proportions of C1-HH1, C1-LH1 and C1-CH1 transitions to total emission of AlGaN/AlN QWs with different Al contents considering the valence subband coupling. The valence subband orders are also labeled. In Fig. 4(b) for Al_0.83Ga_0.17N/AlN QW, the proportion of C1-CH1 transition in TE polarization emission even exceeds the value in TM emission due to valence subband coupling. As a result, the total TE polarization emission is much stronger than TM component for AlGaN/AlN QW with 83% Al content although the energy level of

Γ_{9}

almost is equal to that of

Γ_{7}^{2}

. Then, it can be deduced that the critical Al content for polarization switching between dominant TE and TM emissions should be changed by valence subband coupling. When the valence subband coupling decreases as CH1 subband moves away from HH1 or LH1 subbands in Al_0.81Ga_0.19N/AlN and Al_0.9Ga_0.1N/AlN QWs, TE component from C1-CH1 transition and TM component from C1-HH1 and C1-LH1 transitions decrease as shown in Figs. 4(a) and 4(c).

Fig. 4 Spontaneous emission proportion of transitions from conduction band to the three top valence subbands for (a) Al_0.81Ga_0.19N/AlN, (b) Al_0.83Ga_0.17N/AlN and (c) Al_0.9Ga_0.1N/AlN QWs.

The spontaneous emission spectra of AlGaN/AlN QWs with different Al contents are calculated and compared with experimental result from other groups [18

R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B 79(12), 121308 (2009). [CrossRef]

, 19

H. Hirayama, N. Noguchi, and N. Kamata, “222nm deep-ultraviolet AlGaN quantum well light-emitting diode with vertical emission properties,” Appl. Phys. Express 3(3), 032102 (2010). [CrossRef]

, 33

H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “226-273 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on multilayer AlN buffers on sapphire,” Phys. Status Solidi C 5(9), 2969–2971 (2008). [CrossRef]

]. As seen in Fig. 5 , the spontaneous emission spectra of Al_0.83Ga_0.17N/AlN QW have dominant TE component and there is not emission peak difference between TE and TM components. For Al_0.81Ga_0.19N/AlN and Al_0.9Ga_0.1N/AlN QWs, the shift of emission peak for TE and TM emissions is less than 0.8nm due to the subband coupling although CH1 is over 30meV away from HH1 subband. In addition, TE polarization emission is dominant in Al_0.81Ga_0.19N/AlN QW and TM polarization emission is dominant in Al_0.9Ga_0.1N/AlN QW. The agreement of unobvious peak shift and polarization composition of TE and TM polarized components in our calculations with experimental results surely becomes the evidence of valence subband coupling effect on the optical polarization properties of Al-rich AlGaN/AlN QWs.

Fig. 5 The spontaneous emission spectra of Al_0.81Ga_0.19N/AlN, Al_0.83Ga_0.17N/AlN and Al_0.9Ga_0.1N/AlN QWs.

It is also interesting that the critical Al composition for tuning TE and TM polarizations may be influenced by such valence band coupling. In Fig. 6 , the critical Al contents considering only valence subband order and including valence subband coupling are compared for different well widths from 1.0nm to 4.0nm. Several experimental results taken from the papers of Banal et al. [18

R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B 79(12), 121308 (2009). [CrossRef]

] and Hirayama et al. [19

H. Hirayama, N. Noguchi, and N. Kamata, “222nm deep-ultraviolet AlGaN quantum well light-emitting diode with vertical emission properties,” Appl. Phys. Express 3(3), 032102 (2010). [CrossRef]

, 33

] are cited as references. It is clear that the critical Al content deduced from valence subband order slightly decrease as well width decreases. However, when valence subband coupling also is considered, the polarization switch point shifts towards higher Al content with the decrease of well width. The critical Al contents in well layer should be included in a region between that for TE and TM polarization emission, the shaded parts in Fig. 6, and our calculated results under the consideration with influence of valence subband coupling fit well with tendency of experimental results. Such changes of optical polarization property with well width are of combination of quantum confinement, piezoelectric field effect and valence subband coupling.

Fig. 6 Calculated Al contents for polarization switching between TE and TM polarization emission of AlGaN/AlN QWs with and without valence subband coupling.

4. Conclusion

In summary, the optical polarization properties of Al-rich AlGaN/AlN QWs were analyzed using the theoretical model based on the k⋅p method. Numerical results demonstrated that there is a coupling between the CH1 and LH1 subbands at k_t = 0, but HH1, CH1 and LH1 subbands are coupled with each other at k_t ≠ 0, which can change emission light properties with TE and TM polarizations. Therefore, the emission peak wavelength difference and emission intensity of TE and TM components are determined by C-HH, C-LH and C-CH transition at k_t = 0 and k_t ≠ 0 duo to the valence subband coupling. Especially, the effect of valence subband coupling on the optical polarization properties for AlGaN/AlN QWs is very obvious when the Al content close to the critical content for polarization switching. It is concluded that valence subband coupling can change the polarization properties of spontaneous emissions and should be considered in designing high efficiency AlGaN-based UV LEDs.

Acknowledgments

This work is supported by the National Key Basic Research Program of China under Grant Nos 2011CB301900 and 2012CB619306, the National High-Tech Research and Development Program of China under Grant No 2011AA03A103, and the National Natural Science Foundation of China under Grant Nos 61076012, 61076013, 60976009, 61204054 and 61275052.

References and links

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4.	K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett. 84(25), 5264–5266 (2004). [CrossRef]
5.	H. Kawanishi, M. Senuma, and T. Nukui, “Anisotropic polarization characteristics of lasing and spontaneous surface and edge emissions from deep-ultraviolet (λ≈240 nm) AlGaN multiple-quantum-well lasers,” Appl. Phys. Lett. 89(4), 041126 (2006). [CrossRef]
6.	T. Kolbe, A. Knauer, C. Chua, Z. Yang, S. Einfeldt, P. Vogt, N. M. Johnson, M. Weyers, and M. Kneissl, “Optical polarization characteristics of ultraviolet (In)(Al)GaN multiple quantum well light emitting diodes,” Appl. Phys. Lett. 97(17), 171105 (2010). [CrossRef]
7.	H. Kawanishi, M. Senuma, M. Yamamoto, E. Niikura, and T. Nukui, “Extremely weak surface emission from (0001) c-plane AlGaN multiple quantum well structure in deep-ultraviolet spectral region,” Appl. Phys. Lett. 89(8), 081121 (2006). [CrossRef]
8.	H. Hirayama, S. Fujikawa, N. Noguchi, J. Norimatsu, T. Takano, K. Tsubaki, and N. Kamata, “222-282 nm AlGaN and InAlGaN-based deep-UV LEDs fabricated on high-quality AlN on sapphire,” Phys. Status Solidi A 206(6), 1176–1182 (2009). [CrossRef]
9.	S. H. Park, “Optical gain characteristics of non-polar Al-rich AlGaN/AlN quantum well structure,” J. Appl. Phys. 110(6), 063105 (2011). [CrossRef]
10.	A. A. Yamaguchi, “Theoretical investigation of optical polarization properties in Al-rich AlGaN quantum wells with various substrate orientations,” Appl. Phys. Lett. 96(15), 151911 (2010). [CrossRef]
11.	P. Y. Dang and Y. R. Wu, “Optical polarization anisotropy of tensile strained InGaN/AlInN quantum wells for TM mode lasers,” J. Appl. Phys. 108(8), 083108 (2010). [CrossRef]
12.	T. K. Sharma, D. Naveh, and E. Towe, “Strain-driven light-polarization switching in deep ultraviolet nitride emitters,” Phys. Rev. B 84(3), 035305 (2011). [CrossRef]
13.	D. Fu, R. Zhang, B. Wang, Z. Zhang, B. Liu, Z. Xie, X. Xiu, H. Lu, Y. Zheng, and G. Edwards, “Modification of the valence band structures of polar and nonpolar plane wurtzite-GaN by anisotropic strain,” J. Appl. Phys. 106(2), 023714 (2009). [CrossRef]
14.	J. Zhang, H. Zhao, and N. Tansu, “Large optical gain AlGaN-delta-GaN quantum wells laser active regions in mid- and deep-ultraviolet spectral regimes,” Appl. Phys. Lett. 98(17), 171111 (2011). [CrossRef]
15.	K. Kojima, A. A. Yamaguchi, M. Funato, Y. Kawakami, and S. Noda, “Impact of nonpolar AlGaN quantum wells on deep ultraviolet laser diodes,” J. Appl. Phys. 110(4), 043115 (2011). [CrossRef]
16.	R. G. Banal, M. Funato, and Y. Kawakami, “Extremely high internal quantum efficiencies from AlGaN/AlN quantum wells emitting in the deep ultraviolet spectral region,” Appl. Phys. Lett. 99(1), 011902 (2011). [CrossRef]
17.	S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett. 68(12), 1657–1659 (1996). [CrossRef]
18.	R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B 79(12), 121308 (2009). [CrossRef]
19.	H. Hirayama, N. Noguchi, and N. Kamata, “222nm deep-ultraviolet AlGaN quantum well light-emitting diode with vertical emission properties,” Appl. Phys. Express 3(3), 032102 (2010). [CrossRef]
20.	S. L. Chuang and C. S. Chang, “k⋅p method for strained wurtzite semiconductors,” Phys. Rev. B 54(4), 2491–2504 (1996). [CrossRef]
21.	S. L. Chuang, “Optical gain of strained wurtzite GaN quantum-well lasers,” IEEE J. Quantum Electron. 32(10), 1791–1800 (1996). [CrossRef]
22.	R. A. Arif, H. Zhao, Y. K. Ee, and N. Tansu, “Spontaneous emission and characteristics of staggered InGaN quantum-well light-emitting diodes,” IEEE J. Quantum Electron. 44(6), 573–580 (2008). [CrossRef]
23.	H. Zhao, R. A. Arif, Y. K. Ee, and N. Tansu, “Self-consistent analysis of strain-compensated InGaN–AlGaN quantum wells for lasers and light-emitting diodes,” IEEE J. Quantum Electron. 45(1), 66–78 (2009). [CrossRef]
24.	F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56(16), R10024–R10027 (1997). [CrossRef]
25.	V. Fiorentini, F. Bernardini, F. Della Sala, A. Di Carlo, and P. Lugli, “Effects of macroscopic polarization in III-V nitride multiple quantum wells,” Phys. Rev. B 60(12), 8849–8858 (1999). [CrossRef]
26.	J. Piprek, Semiconductor Optoelectronic Devices (Academic Press, 2003).
27.	I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys. 94(6), 3675–3696 (2003). [CrossRef]
28.	P. Rinke, M. Schefﬂer, A. Qteish, M. Winkelnkemper, D. Bimberg, and J. Neugebauer, “Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory,” Appl. Phys. Lett. 89(16), 161919 (2006). [CrossRef]
29.	J. Bhattacharyya, S. Ghosh, and H. T. Grahn, “Are AlN and GaN substrates useful for the growth of non-polar nitride films for UV emission? The oscillator strength perspective,” Phys. Status Solidi B 246(6), 1184–1187 (2009). [CrossRef]
30.	H. M. Lu and G. X. Chen, “Design strategies for mitigating the influence of polarization effects on GaN-based multiple quantum well light-emitting diodes,” J. Appl. Phys. 109(9), 093102 (2011). [CrossRef]
31.	A. A. Yamaguchi, “Valence band engineering for remarkable enhancement of surface emission in AlGaN deep-ultraviolet light emitting diodes,” Phys. Status Solidi C 5(6), 2364–2366 (2008). [CrossRef]
32.	C. Jia, T. Yu, R. Tao, X. Hu, Z. Yang, Z. Qin, Z. Chen, and G. Zhang, “Spontaneous luminescence polarizations of wurtzite InGaN/GaN quantum wells,” Appl. Phys. Lett. 93(17), 171114 (2008). [CrossRef]
33.	H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “226-273 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on multilayer AlN buffers on sapphire,” Phys. Status Solidi C 5(9), 2969–2971 (2008). [CrossRef]

OCIS Codes
(230.3670) Optical devices : Light-emitting diodes
(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

ToC Category:
Optoelectronics

History
Original Manuscript: August 31, 2012
Revised Manuscript: October 26, 2012
Manuscript Accepted: November 14, 2012
Published: November 26, 2012

Citation
Huimin Lu, Tongjun Yu, Gangcheng Yuan, Chuanyu Jia, Genxiang Chen, and Guoyi Zhang, "Valence subband coupling effect on polarization of spontaneous emissions from Al-rich AlGaN/AlN Quantum Wells," Opt. Express 20, 27384-27392 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-25-27384

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References

H. S. Kim, R. A. Mair, J. Li, J. Y. Lin, and H. X. Jiang, “Time-resolved photoluminescence studies of AlGaN alloys,” Appl. Phys. Lett.76(10), 1252–1254 (2000). [CrossRef]
V. Adivarahan, W. H. Sun, A. Chitnis, M. Shatalov, S. Wu, H. P. Maruska, and M. Asif Khan, “Visual system-response functions and estimating reflectance,” Appl. Phys. Lett.85(12), 2175–2177 (2004). [CrossRef]
H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “231-261 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on AlN multilayer buffers grown by ammonia pulse-flow method on sapphire,” Appl. Phys. Lett.91(7), 071901 (2007). [CrossRef]
K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett.84(25), 5264–5266 (2004). [CrossRef]
H. Kawanishi, M. Senuma, and T. Nukui, “Anisotropic polarization characteristics of lasing and spontaneous surface and edge emissions from deep-ultraviolet (λ≈240 nm) AlGaN multiple-quantum-well lasers,” Appl. Phys. Lett.89(4), 041126 (2006). [CrossRef]
T. Kolbe, A. Knauer, C. Chua, Z. Yang, S. Einfeldt, P. Vogt, N. M. Johnson, M. Weyers, and M. Kneissl, “Optical polarization characteristics of ultraviolet (In)(Al)GaN multiple quantum well light emitting diodes,” Appl. Phys. Lett.97(17), 171105 (2010). [CrossRef]
H. Kawanishi, M. Senuma, M. Yamamoto, E. Niikura, and T. Nukui, “Extremely weak surface emission from (0001) c-plane AlGaN multiple quantum well structure in deep-ultraviolet spectral region,” Appl. Phys. Lett.89(8), 081121 (2006). [CrossRef]
H. Hirayama, S. Fujikawa, N. Noguchi, J. Norimatsu, T. Takano, K. Tsubaki, and N. Kamata, “222-282 nm AlGaN and InAlGaN-based deep-UV LEDs fabricated on high-quality AlN on sapphire,” Phys. Status Solidi A206(6), 1176–1182 (2009). [CrossRef]
S. H. Park, “Optical gain characteristics of non-polar Al-rich AlGaN/AlN quantum well structure,” J. Appl. Phys.110(6), 063105 (2011). [CrossRef]
A. A. Yamaguchi, “Theoretical investigation of optical polarization properties in Al-rich AlGaN quantum wells with various substrate orientations,” Appl. Phys. Lett.96(15), 151911 (2010). [CrossRef]
P. Y. Dang and Y. R. Wu, “Optical polarization anisotropy of tensile strained InGaN/AlInN quantum wells for TM mode lasers,” J. Appl. Phys.108(8), 083108 (2010). [CrossRef]
T. K. Sharma, D. Naveh, and E. Towe, “Strain-driven light-polarization switching in deep ultraviolet nitride emitters,” Phys. Rev. B84(3), 035305 (2011). [CrossRef]
D. Fu, R. Zhang, B. Wang, Z. Zhang, B. Liu, Z. Xie, X. Xiu, H. Lu, Y. Zheng, and G. Edwards, “Modification of the valence band structures of polar and nonpolar plane wurtzite-GaN by anisotropic strain,” J. Appl. Phys.106(2), 023714 (2009). [CrossRef]
J. Zhang, H. Zhao, and N. Tansu, “Large optical gain AlGaN-delta-GaN quantum wells laser active regions in mid- and deep-ultraviolet spectral regimes,” Appl. Phys. Lett.98(17), 171111 (2011). [CrossRef]
K. Kojima, A. A. Yamaguchi, M. Funato, Y. Kawakami, and S. Noda, “Impact of nonpolar AlGaN quantum wells on deep ultraviolet laser diodes,” J. Appl. Phys.110(4), 043115 (2011). [CrossRef]
R. G. Banal, M. Funato, and Y. Kawakami, “Extremely high internal quantum efficiencies from AlGaN/AlN quantum wells emitting in the deep ultraviolet spectral region,” Appl. Phys. Lett.99(1), 011902 (2011). [CrossRef]
S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett.68(12), 1657–1659 (1996). [CrossRef]
R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B79(12), 121308 (2009). [CrossRef]
H. Hirayama, N. Noguchi, and N. Kamata, “222nm deep-ultraviolet AlGaN quantum well light-emitting diode with vertical emission properties,” Appl. Phys. Express3(3), 032102 (2010). [CrossRef]
S. L. Chuang and C. S. Chang, “k⋅p method for strained wurtzite semiconductors,” Phys. Rev. B54(4), 2491–2504 (1996). [CrossRef]
S. L. Chuang, “Optical gain of strained wurtzite GaN quantum-well lasers,” IEEE J. Quantum Electron.32(10), 1791–1800 (1996). [CrossRef]
R. A. Arif, H. Zhao, Y. K. Ee, and N. Tansu, “Spontaneous emission and characteristics of staggered InGaN quantum-well light-emitting diodes,” IEEE J. Quantum Electron.44(6), 573–580 (2008). [CrossRef]
H. Zhao, R. A. Arif, Y. K. Ee, and N. Tansu, “Self-consistent analysis of strain-compensated InGaN–AlGaN quantum wells for lasers and light-emitting diodes,” IEEE J. Quantum Electron.45(1), 66–78 (2009). [CrossRef]
F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B56(16), R10024–R10027 (1997). [CrossRef]
V. Fiorentini, F. Bernardini, F. Della Sala, A. Di Carlo, and P. Lugli, “Effects of macroscopic polarization in III-V nitride multiple quantum wells,” Phys. Rev. B60(12), 8849–8858 (1999). [CrossRef]
J. Piprek, Semiconductor Optoelectronic Devices (Academic Press, 2003).
I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys.94(6), 3675–3696 (2003). [CrossRef]
P. Rinke, M. Schefﬂer, A. Qteish, M. Winkelnkemper, D. Bimberg, and J. Neugebauer, “Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory,” Appl. Phys. Lett.89(16), 161919 (2006). [CrossRef]
J. Bhattacharyya, S. Ghosh, and H. T. Grahn, “Are AlN and GaN substrates useful for the growth of non-polar nitride films for UV emission? The oscillator strength perspective,” Phys. Status Solidi B246(6), 1184–1187 (2009). [CrossRef]
H. M. Lu and G. X. Chen, “Design strategies for mitigating the influence of polarization effects on GaN-based multiple quantum well light-emitting diodes,” J. Appl. Phys.109(9), 093102 (2011). [CrossRef]
A. A. Yamaguchi, “Valence band engineering for remarkable enhancement of surface emission in AlGaN deep-ultraviolet light emitting diodes,” Phys. Status Solidi C5(6), 2364–2366 (2008). [CrossRef]
C. Jia, T. Yu, R. Tao, X. Hu, Z. Yang, Z. Qin, Z. Chen, and G. Zhang, “Spontaneous luminescence polarizations of wurtzite InGaN/GaN quantum wells,” Appl. Phys. Lett.93(17), 171114 (2008). [CrossRef]
H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “226-273 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on multilayer AlN buffers on sapphire,” Phys. Status Solidi C5(9), 2969–2971 (2008). [CrossRef]

Adivarahan, V.
Arif, R. A.
Asif Khan, M.
Banal, R. G.
Bernardini, F.
Bhattacharyya, J.
Bimberg, D.
Chang, C. S.
Chen, G. X.
Chen, Z.
Chitnis, A.
Chua, C.
Chuang, S. L.
Dang, P. Y.
Della Sala, F.
Di Carlo, A.
Edwards, G.
Ee, Y. K.
Einfeldt, S.
Fiorentini, V.
Fu, D.
Fujikawa, S.
Funato, M.
Ghosh, S.
Grahn, H. T.
Hirayama, H.
Hu, X.
Jia, C.
Jiang, H. X.
Johnson, N. M.
Kamata, N.
Kawakami, Y.
Kawanishi, H.
Kim, H. S.
Knauer, A.
Kneissl, M.
Kojima, K.
Kolbe, T.
Li, J.
Lin, J. Y.
Liu, B.
Lu, H.
Lu, H. M.
Lugli, P.
Mair, R. A.
Maruska, H. P.
Meyer, J. R.
Nakarmi, M. L.
Nam, K. B.
Naveh, D.
Neugebauer, J.
Niikura, E.
Noda, S.
Noguchi, N.
Norimatsu, J.
Nukui, T.
Ohashi, T.
Park, S. H.
Qin, Z.
Qteish, A.
Rinke, P.
Schef?er, M.
Senuma, M.
Sharma, T. K.
Shatalov, M.
Sun, W. H.
Takano, T.
Tansu, N.
Tao, R.
Towe, E.
Tsubaki, K.
Vanderbilt, D.
Vogt, P.
Vurgaftman, I.
Wang, B.
Weyers, M.
Winkelnkemper, M.
Wu, S.
Wu, Y. R.
Xie, Z.
Xiu, X.
Yamaguchi, A. A.
Yamamoto, M.
Yang, Z.
Yatabe, T.
Yu, T.
Zhang, G.
Zhang, J.
Zhang, R.
Zhang, Z.
Zhao, H.
Zheng, Y.

Appl. Phys. Express

H. Hirayama, N. Noguchi, and N. Kamata, “222nm deep-ultraviolet AlGaN quantum well light-emitting diode with vertical emission properties,” Appl. Phys. Express3(3), 032102 (2010). [CrossRef]

Appl. Phys. Lett.

P. Rinke, M. Schefﬂer, A. Qteish, M. Winkelnkemper, D. Bimberg, and J. Neugebauer, “Band gap and band parameters of InN and GaN from quasiparticle energy calculations based on exact-exchange density-functional theory,” Appl. Phys. Lett.89(16), 161919 (2006). [CrossRef]
C. Jia, T. Yu, R. Tao, X. Hu, Z. Yang, Z. Qin, Z. Chen, and G. Zhang, “Spontaneous luminescence polarizations of wurtzite InGaN/GaN quantum wells,” Appl. Phys. Lett.93(17), 171114 (2008). [CrossRef]
H. S. Kim, R. A. Mair, J. Li, J. Y. Lin, and H. X. Jiang, “Time-resolved photoluminescence studies of AlGaN alloys,” Appl. Phys. Lett.76(10), 1252–1254 (2000). [CrossRef]
V. Adivarahan, W. H. Sun, A. Chitnis, M. Shatalov, S. Wu, H. P. Maruska, and M. Asif Khan, “Visual system-response functions and estimating reflectance,” Appl. Phys. Lett.85(12), 2175–2177 (2004). [CrossRef]
H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “231-261 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on AlN multilayer buffers grown by ammonia pulse-flow method on sapphire,” Appl. Phys. Lett.91(7), 071901 (2007). [CrossRef]
K. B. Nam, J. Li, M. L. Nakarmi, J. Y. Lin, and H. X. Jiang, “Unique optical properties of AlGaN alloys and related ultraviolet emitters,” Appl. Phys. Lett.84(25), 5264–5266 (2004). [CrossRef]
H. Kawanishi, M. Senuma, and T. Nukui, “Anisotropic polarization characteristics of lasing and spontaneous surface and edge emissions from deep-ultraviolet (λ≈240 nm) AlGaN multiple-quantum-well lasers,” Appl. Phys. Lett.89(4), 041126 (2006). [CrossRef]
T. Kolbe, A. Knauer, C. Chua, Z. Yang, S. Einfeldt, P. Vogt, N. M. Johnson, M. Weyers, and M. Kneissl, “Optical polarization characteristics of ultraviolet (In)(Al)GaN multiple quantum well light emitting diodes,” Appl. Phys. Lett.97(17), 171105 (2010). [CrossRef]
H. Kawanishi, M. Senuma, M. Yamamoto, E. Niikura, and T. Nukui, “Extremely weak surface emission from (0001) c-plane AlGaN multiple quantum well structure in deep-ultraviolet spectral region,” Appl. Phys. Lett.89(8), 081121 (2006). [CrossRef]
A. A. Yamaguchi, “Theoretical investigation of optical polarization properties in Al-rich AlGaN quantum wells with various substrate orientations,” Appl. Phys. Lett.96(15), 151911 (2010). [CrossRef]
J. Zhang, H. Zhao, and N. Tansu, “Large optical gain AlGaN-delta-GaN quantum wells laser active regions in mid- and deep-ultraviolet spectral regimes,” Appl. Phys. Lett.98(17), 171111 (2011). [CrossRef]
R. G. Banal, M. Funato, and Y. Kawakami, “Extremely high internal quantum efficiencies from AlGaN/AlN quantum wells emitting in the deep ultraviolet spectral region,” Appl. Phys. Lett.99(1), 011902 (2011). [CrossRef]
S. L. Chuang and C. S. Chang, “Effective-mass Hamiltonian for strained wurtzite GaN and analytical solutions,” Appl. Phys. Lett.68(12), 1657–1659 (1996). [CrossRef]

IEEE J. Quantum Electron.

S. L. Chuang, “Optical gain of strained wurtzite GaN quantum-well lasers,” IEEE J. Quantum Electron.32(10), 1791–1800 (1996). [CrossRef]
R. A. Arif, H. Zhao, Y. K. Ee, and N. Tansu, “Spontaneous emission and characteristics of staggered InGaN quantum-well light-emitting diodes,” IEEE J. Quantum Electron.44(6), 573–580 (2008). [CrossRef]
H. Zhao, R. A. Arif, Y. K. Ee, and N. Tansu, “Self-consistent analysis of strain-compensated InGaN–AlGaN quantum wells for lasers and light-emitting diodes,” IEEE J. Quantum Electron.45(1), 66–78 (2009). [CrossRef]

J. Appl. Phys.

H. M. Lu and G. X. Chen, “Design strategies for mitigating the influence of polarization effects on GaN-based multiple quantum well light-emitting diodes,” J. Appl. Phys.109(9), 093102 (2011). [CrossRef]
I. Vurgaftman and J. R. Meyer, “Band parameters for nitrogen-containing semiconductors,” J. Appl. Phys.94(6), 3675–3696 (2003). [CrossRef]
K. Kojima, A. A. Yamaguchi, M. Funato, Y. Kawakami, and S. Noda, “Impact of nonpolar AlGaN quantum wells on deep ultraviolet laser diodes,” J. Appl. Phys.110(4), 043115 (2011). [CrossRef]
P. Y. Dang and Y. R. Wu, “Optical polarization anisotropy of tensile strained InGaN/AlInN quantum wells for TM mode lasers,” J. Appl. Phys.108(8), 083108 (2010). [CrossRef]
S. H. Park, “Optical gain characteristics of non-polar Al-rich AlGaN/AlN quantum well structure,” J. Appl. Phys.110(6), 063105 (2011). [CrossRef]
D. Fu, R. Zhang, B. Wang, Z. Zhang, B. Liu, Z. Xie, X. Xiu, H. Lu, Y. Zheng, and G. Edwards, “Modification of the valence band structures of polar and nonpolar plane wurtzite-GaN by anisotropic strain,” J. Appl. Phys.106(2), 023714 (2009). [CrossRef]

Phys. Rev. B

T. K. Sharma, D. Naveh, and E. Towe, “Strain-driven light-polarization switching in deep ultraviolet nitride emitters,” Phys. Rev. B84(3), 035305 (2011). [CrossRef]
R. G. Banal, M. Funato, and Y. Kawakami, “Optical anisotropy in [0001]-oriented AlxGa1−xN/AlN quantum wells (x>0.69),” Phys. Rev. B79(12), 121308 (2009). [CrossRef]
F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B56(16), R10024–R10027 (1997). [CrossRef]
V. Fiorentini, F. Bernardini, F. Della Sala, A. Di Carlo, and P. Lugli, “Effects of macroscopic polarization in III-V nitride multiple quantum wells,” Phys. Rev. B60(12), 8849–8858 (1999). [CrossRef]
S. L. Chuang and C. S. Chang, “k⋅p method for strained wurtzite semiconductors,” Phys. Rev. B54(4), 2491–2504 (1996). [CrossRef]

Phys. Status Solidi A

H. Hirayama, S. Fujikawa, N. Noguchi, J. Norimatsu, T. Takano, K. Tsubaki, and N. Kamata, “222-282 nm AlGaN and InAlGaN-based deep-UV LEDs fabricated on high-quality AlN on sapphire,” Phys. Status Solidi A206(6), 1176–1182 (2009). [CrossRef]

Phys. Status Solidi B

J. Bhattacharyya, S. Ghosh, and H. T. Grahn, “Are AlN and GaN substrates useful for the growth of non-polar nitride films for UV emission? The oscillator strength perspective,” Phys. Status Solidi B246(6), 1184–1187 (2009). [CrossRef]

Phys. Status Solidi C

A. A. Yamaguchi, “Valence band engineering for remarkable enhancement of surface emission in AlGaN deep-ultraviolet light emitting diodes,” Phys. Status Solidi C5(6), 2364–2366 (2008). [CrossRef]
H. Hirayama, T. Yatabe, N. Noguchi, T. Ohashi, and N. Kamata, “226-273 nm AlGaN deep-ultraviolet light-emitting diodes fabricated on multilayer AlN buffers on sapphire,” Phys. Status Solidi C5(9), 2969–2971 (2008). [CrossRef]

Other

J. Piprek, Semiconductor Optoelectronic Devices (Academic Press, 2003).

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