Influence of quantum well design on light polarization switching in AlGaN ultraviolet emitters

AlGaN-based deep ultraviolet (UV) light-emitting diodes (LEDs) and lasers are demonstrated to be ideal candidates for a variety of applications such as water/air purification, sterilization and high-density optical recording.¹ However, many hindrances remain in AlGaN-based emitters to realize higher devices external quantum efficiency (⁠$ηEQE$⁠),^1,2 especially the limited light extraction efficiency (⁠$ηLEE$⁠).³ It is well-known that low $ηLEE$ from UV LEDs is attributed to the high refractive index of AlGaN materials (or sapphire) and the light absorption from the p-electrode.¹ In addition, light polarization is also a significant contributing factor to limit the UV photons emitting from the device surface or substrate.¹ Specifically, most of transverse-electric (TE)-polarized light is emitted at small angles respect to the c-axis while transverse-magnetic (TM)-polarized light travels at large angles. Those photons emitted at large angle will be trapped and absorbed in the LED structure more easily, which results in a low $ηLEE$⁠. For AlGaN quantum wells (QWs), the emitted photons switch from TE-polarization to difficult-escaped TM-polarization as the Al-composition increases because of the remarkable difference of the crystal-field splitting energies (Δ_CR) between AlN (-169 meV) and GaN (10 meV) which consequently changes the order of valence subbands.⁴ Note that the light polarization change in the AlGaN QW does not only limit the $ηLEE$⁠, but also strongly influence spontaneous emission rate and internal quantum efficiency as well.⁵ Hence, it is essential to investigate and predict the critical Al-content at which the switching from TE to TM occurs for efficient deep-UV active region design. Several research groups have reported different critical Al-compositions^5–7 for TE/TM switching, which are influenced by many factors such as strain state of the AlGaN layer^7,8 and quantum confinement.^6,7,9–12 However, the physical mechanisms related to the change of the TE/TM crossover point are still unclear. For instance, R. G. Banal et.al. showed that the light polarization switched from TM to TE as the QW thickness decreased from 13 nm to 1.3 nm in Al_0.82Ga_0.18N/AlN QWs.⁷ Meanwhile, a similar trend was also observed from Al_0.65Ga_0.35N/AlN system.¹⁰ However, a dominant TE-polarized emission was reported from 2-nm Al_0.66Ga_0.34N/4.8-nm Al_0.66Ga_0.34N QWs¹³ while a TM-polarized emission was observed from 1-nm Al_0.66Ga_0.34N/4.1-nm Al_0.83Ga_0.17N QWs,¹⁴ which is inconsistent with the previous polarization switching behavior as the QW thickness changes. Therefore, it is very important to understand the physics of AlGaN QW design on the influence of optical polarization switching point, which has not been reported yet.

In this work, we theoretically investigated the light polarization crossover Al-contents with different AlGaN QW designs by using the 6-band k·p model.¹⁵ The valence band orders from AlGaN/AlN QWs were studied by the band structure calculations. Specifically, the critical Al-contents at which heavy hole (HH) and crystal-field split-off (CH) valence subbands crossover occurred were investigated. Both TE-polarized and TM-polarized spontaneous emission recombination rates per unit volume (R_sp) were studied for different QW thicknesses by taking into account strain effect, internal polarization fields and carrier screening effect.¹⁵ Furthermore, the Al-contents for TE-R_sp/TM-R_sp switching points were plotted and compared to the valence subbands crossover Al-contents. To explain the difference between the two crossover points, two AlGaN QWs with different well thicknesses (1.5 nm and 3 nm) were further investigated. In addition, this work summarized reported optical polarization from AlGaN QWs and categorized by QW thicknesses. A similar critical Al-content trend was observed from the reported values, as compared to the calculations, which indicates the importance of this study for high-efficiency UV LEDs designs.

For III-nitride materials in the wurtzite structure, the electronic and optical properties are strongly determined by the band structures at the Brillouin zone center (⁠$Γ$ point). Due to the crystal-field splitting and the spin-orbit splitting (Δ_SO) from the nature of the wurtzite crystal structure, the top of the valence band splits into HH, light hole (LH) and CH subbands. The HH (⁠$Γ$₉) band and the LH (⁠$Γ$₇) band have atomic p_x- and p_y-like character while the CH (⁠$Γ$₇) band is governed by atomic p_z-like state. An analytical expression of the energy splitting in a strained Al_xGa_1-xN layer was derived by Chuang et al. and given by:^16,17

where D_i are the deformation potentials, C_ij are the elastic stiffness constant and ε_zz is the strain tensor element along the growth direction (c-axis). So the HH and CH subbands switch (⁠$Γ$₉ = $Γ$₇) only occurs at ∆′ = 0, corresponds to the critical Al composition. Assuming a linear relation between the AlN and the GaN parameters, the critical Al-compositions were calculated as x = 0.044 for unstrained Al_xGa_1-xN and x = 0.60 for strained Al_xGa_1-xN on AlN template.⁷ Moreover, the critical Al composition can be further affected by the quantum confinement due to the lighter hole effective mass for CH subband in wurtzite AlN. In general, better quantum confinement pushes the CH subband downward and higher critical Al composition is required for HH/CH subbands switching.⁷ 

Theoretically, the light polarization properties from the active region can be predicted by the calculations of the TE-polarized (e = x) and TM-polarized (e = z) spontaneous emission rate, which are defined as follows based on the Fermi’s Golden rule:¹⁸ 

where $q$ is the electron charge, $nr$ and $Lω$ are the refractive index and thickness of the quantum well, c, $ϵ0$ and $m0$ are the velocity of light, permittivity and electron mass in free space. $γ=(0.1ps)−1$ is used in our calculation.¹⁵ $fnckt$ and $fσmvkt$ represent the Fermi-Dirac distributions for electrons in the conduction and valence bands, respectively. $Menmσkt$ is the momentum matrix element for transitions between the n^th conduction band state and the m^th valence band state, which is obtained by using the calculated envelop functions. Specifically, large electron-hole wavefunction overlap corresponds to large momentum matrix element. The total spontaneous emission rate per unit volume per unit energy interval (r^spon) and total R_sp are given by:¹⁸ 

Note that the total R_sp is strongly determined by both momentum-matrix element from n^th conduction band to m^th valence band, as well as the surface carrier densities at these two energy states. The transitions from conduction bands to HH subbands (C-HH) generate TE-polarized photons while the transitions to CH subbands (C-CH) mainly result in TM-polarized photons. For conduction bands to LH subbands transitions (C-LH), the light polarization is strongly influenced by the coupling between the LH subband and CH subband. Strong coupling between the two valence subbands leads to dominant TM-polarized emission from C-LH transition while weak coupling is preferable for the generations of TE-polarized photons. In addition, C-CH transition also generates significant amount of TE-polarized photons when CH and LH are strongly coupled.¹⁸ 

In this study, the band structures and spontaneous emission were calculated by a self-consistent 6-band k·p model with considering spontaneous and piezoelectric fields, strain effect and carrier screening effect, and the modeling details can be found in Ref. 15. Table I lists the material parameters for GaN and AlN used in this study, which are taken from Ref. 4. The carrier density and the QW barrier thickness are fixed at n = 5×10¹⁸ cm^-3 and 6 nm respectively throughout the calculations.

The critical Al-contents at which valence subbands switch in AlGaN/AlN QW were first studied at different QW thicknesses and plotted in Figure 1(a). Without polarization field, the crossover occurs at lower Al-contents for thicker QW due to the weaker quantum confinement as the QW thickness (d) increases. Specifically, the lighter CH hole will be pushed more downward by the quantum confinement, as compared to HH and LH. For the QWs thinner than ∼1.5 nm, although the better quantum confinement caused by the reduction of the QW thickness pushes the CH-related band downward, the critical Al-content shows a decline trend with the QW thickness decreases, which is attributed to the barrier height difference between CH and HH subbands. Specifically, for AlN and high Al-content AlGaN, the top valence band is CH (⁠$E20$⁠), as shown in figure 1 inset. And the barrier height of CH band (⁠$ΔE20$⁠) is smaller than that of HH band (⁠$ΔE10$⁠), which causes overall weaker quantum confinement for CH subband in thinner QW.

However, taking into consideration the effect of both spontaneous polarization electric field and piezoelectric polarization electric field, the critical Al-content shows a dissimilar trend. The internal electrostatic field in each layer (j^th) of the structure was calculated as:¹⁵ 

where P is the total polarization, ε is the dielectric constant and l is the thickness of each layer. From the calculations, the absolute value of the electric field drops as the QW thickness increases, which results in the critical Al-content rises as the QW thickness increases. However the influence of the internal electric field on energy level positions in a quantum well is still under debate.

The spontaneous emission recombination rates for TE- and TM-polarizations were calculated and the critical Al-contents at which TE- and TM-polarized R_sp switching occurs are plotted as a function of QW thickness, as shown in figure 1(b). As the QW thickness increases, the R_sp critical Al-content increases at thinner QW (d < 1.5 nm) while drops when QW is thicker than 1.5 nm. The valence subbands crossover points were replotted in figure 1(b) for comparison. Note that the two critical Al-content curves are dissimilar. Specifically, the critical Al-content for valence subbands crossover is higher than the Al-content for TE/TM R_sp crossover in thinner QW (d < 2.5 nm) while that is lower than the Al-content for TE/TM R_sp crossover in thicker QW. To explore the physics underneath the critical Al-contents change, a thin AlGaN/AlN QW (1.5 nm) and a thick AlGaN/AlN QW (3 nm) with critical Al-compositions at which TE/TM R_sp are identical were further studied.

Figure 2(a) plots the valence band structure of the 1.5-nm Al_0.75Ga_0.25N QW. As shown in the figure, the ground state CH subband (CH1) occupies the highest energy state in this structure with a small energy separation to the ground state HH/LH subbands (HH1/LH1) since the Al-composition is slightly larger than the valence subbands crossover point (x = 0.73). As a result, the carriers populating all the three valence subbands without large differences in surface carrier densities (within one order of magnitude). Note that the HH subband locates in the middle of the CH and LH subbands, the coupling between these two subbands is weak. Consequently, most of the photons generated by the C-CH1 transition are TM-polarized while TE-polarized for the photons generated by the C-LH1 transition at $Γ$ point, as shown in figure 2(b). The TE-polarized matrix elements from C-HH1/C-LH1 transitions and the TM-polarized matrix element from C-CH1 transition have large and similar values due to the large electron-hole wavefunction overlaps in the thin QW. Although both the surface carrier density of CH1 subband and the corresponding matrix element from C-CH1 at $Γ$-point are larger than that of HH1 and LH1 subbands individually, the TM-polarized R_sp generated by the C-CH1 transition is identical to the total TE-polarized R_sp produced by both C-HH1 and C-LH1 transitions.

For 3-nm Al_0.73Ga_0.27N QW, the ordering of valence subbands from higher to lower energy is HH1, LH1, CH1; and the 1^st excitation state is HH subband (HH2). There is a strong coupling between the CH1 and LH1 subbands, which significantly influences the polarization of spontaneous emissions from AlGaN/AlN QWs.¹⁹ As a result, large TM-polarized matrix element can be realized from the C-LH1 transition, as compared to the TE-polarized value generated by the same transition. In addition, the matrix element value difference between C-CH1 and C-HH1/C-LH1 is more obvious for the thicker 3 nm QW, as compared to that of thinner QW, due to the more severe quantum confined stark effect (QCSE) in thicker QW. Specifically, the QCSE caused by the large electric fields push the electron and hole wavefunctions toward the edge of QW but the impacts can be mitigated for the lighter CH hole with wider hole wavefunction. As a result, although HH1 has the largest surface carrier density (one order of magnitude higher than CH1 subband), the total TE-polarized R_sp is identical to the TM-polarized R_sp, which is attributed to the large TM-polarized matrix element from C-CH1 transition and also the unneglectable portion from C-LH1 transition.

From the previous discussions on the 1.5-nm and 3-nm AlGaN QWs, the critical Al-contents difference between TE/TM-polarized R_sp crossover and HH/CH valence subbands crossover can be attributed to both QCSE and the band mixing effect. Figure 3 (red) plots the Al-content difference (∆x = R_sp crossover Al-content (x₁₎ – valence subbands crossover Al-content (x₂₎) as a function of QW thickness. The positive ∆x value represents the critical Al-content for R_sp crossover is larger than that for valence subbands crossover. Note that the Al-contents difference (∆x) switches from a positive value to a large negative value as the QW thickness increases. To study the influence of the QCSE on the critical Al-contents change, the structures with critical Al contents at which valence subbands crossover occurs were compared. Specifically, the momentum matrix elements ratio between C-CH1 and C-HH1 transitions at each valence subbands crossover point is calculated and plotted in figure 3 (black). An increasing trend is observed for the matrix element ratio as the QW thickness increases. This is because the thicker QW suffers from more severe QCSE, the electron-HH/LH wavefunction overlaps shrink significantly while the electron-CH wavefunction overlap slowly changes due to the wider CH wave function. Note that there is a negative correlation between the matrix element ratio and the critical Al-contents difference, as presented in figure 3. For thinner QWs, the momentum matrix elements from C-HH1 and C-CH1 are similar as well as the surface carrier densities for HH1 and CH1 subbands. Therefore, the TM-polarized R_sp generated by C-CH1 is comparable to the TE-polarized R_sp generated by C-HH1. However, the existence of C-LH1 transition determines to the total TE-R_sp > TM-R_sp at the valence crossover point and a slightly higher Al-content is required to equalize the TE- and TM-R_sp. For thicker QWs, the matrix element ratios are large due to the severe QCSE. Consequently, TM-R_sp >> TE-R_sp is observed at the valence subbands crossover point and hence the R_sp crossover occurs at lower Al-content.

To verify the predicted critical Al-contents for R_sp crossover, the reported light polarization properties from AlGaN-based UV emitters are summarized and are categorized by QW thickness,^{6,7,10,12–14,20–29} as shown in figure 4. The critical Al-contents at different QW thicknesses are estimated based on the reported polarization properties and have a highest value (x = 0.82) at d = 1.5 nm, which shows a similar trend with the calculated results. The variations from absolute values in reported values and the calculated results are mainly due to influencing factors such as the carrier density, barrier composition, temperature and strain state. Therefore, it is demonstrated that the calculations in this study will provide a useful guidance for knowing the polarization properties from active regions, which helps to optimize the light extraction efficiency and internal quantum efficiency by engineering the design of AlGaN QWs.

In summary, the effect of quantum well thickness on the optical polarization crossover point of AlGaN-based UV LEDs were theoretically investigated in this study. The valence subbands crossover Al-content shows a rising trend as the QW thickness increases taking into account the polarization electric field. A dissimilar behavior was observed for the TE- and TM-polarized R_sp crossover point as the QW changes, which can be attributed to the QCSE and the band mixing effect. Specifically, the critical Al-content for R_sp crossover has a maximum value at QW thickness of 1.5 nm and the trend is verified by reported polarization properties from AlGaN-based UV emitters. Note that device light extraction efficiency and internal quantum efficiency are strongly influenced by the light polarization properties. Therefore, this study would help us to predict the polarization properties from the device and optimize the active region design for high-efficiency DUV LEDs.

The authors would like to acknowledge the support by the Office of Naval Research under Award No. N00014-16-1-2524, and by the National Science Foundation under Award No. ECCS 1751675.