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; Guoyi Zhang

doi:10.1364/OE.20.027384

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 [1–3]. 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 [4–6]. 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, 8]. Obtaining Al-rich AlGaN-based emitters with dominant TE polarization is in great demand and has attracted much attention [9–16].

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]. 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] and Banal et al. [18] 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] 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 [20–23]. 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})

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, 25]. 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}]

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]

{\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}

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}

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}

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 [26–30].

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, 31]. 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].

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.

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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], 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.

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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]. 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.

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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.

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The spontaneous emission spectra of AlGaN/AlN QWs with different Al contents are calculated and compared with experimental result from other groups [18, 19, 33]. 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.

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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] and Hirayama et al. [19, 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.

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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.

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

Abstract

1. Introduction

2. Theoretical modeling and simulation

3. Simulation results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (5)

Optics Express