Double quantum wells (DQWs) have attracted attention due to their potential applications in low threshold lasers,^[1,2] photoswitches,^[3] and phase modulators^[4] since last two decades. Compared with symmetric DQWs, the carrier levels and distribution in asymmetric double quantum wells (ADQWs)^[3] are more complex due to the asymmetric quantum confinement. As a ZnO involved wide gap semiconductors, the excitons and their related polaritons became observable at room temperature.^[5,6] It is worthy to discuss the exciton related phenomena in ADQWs since they are the basic structures of devices.

It was well known that the stable structure of ZnO is wurtzite one, within the incorporation of Mg the structure of ternary mixed crystals (TMCs) Zn_xMg_{1 − x}O will change from a single wurtzite one to a mixed phase with both of wurtzite and zinc-blende in the range of 0.37 < x < 0.62.^[7] For zinc-blende, only piezoelectric polarization (PP) contributes to built-in electric fields (BEFs), whereas the BEFs are formed by both of the PP and spontaneous polarization (SP) for wurtzite.^[8,9] On the other hand, the physical parameters such as carrier effective masses, elastic and piezoelectric constants and band gaps, etc. are also different for the two kinds of structures. Thus, the exciton states show an obvious difference due to a sharp distinction of these parameters between wurtzite and zinc-blende. As a result, there are different transitions in wurtzite and zinc-blende structures respectively between the same exciton energy levels and the OACs should be studied necessarily in mixed phases.^[10–13]

In 2010, Zippel et al.^[14] fabricated ZnO/Zn_{1 − x}Mg_xO single QWs to show experimentally the modulations of the size and Mg components for exciton transitions. A clear blue shift of the photoluminescence peak with decreasing well width was observed. Additionally, the exciton transition energy increases with increasing x, whereas the authors just considered a single wurtzite of Zn_xMg_{1 − x}O. Later, Segawa et al.^[15] manufactured ten periods of ZnO/ZnMgO multiple QWs by pulsed laser deposition, and measured the optical absorption spectra to give the absorption peaks of exciton transitions. However, the explicit exciton states corresponding to the absorption peaks were neglected, and even without discussion of the modulation of exciton optical absorption by Mg components. More recently, Stachowiczet et al.^[16] demonstrated ZnO/Zn_{1 − x}Mg_xO ADQWs of high quality by molecular beam epitaxy technique, and the photoluminescence spectra of excitons with different components were obtained. The coupling between the wells by tunneling of charge carries from a narrow well to the wider well was found. In 2011, Yu et al.^[17] investigated theoretically the OACs of ZnO/ZnMgO quantum wells by the fractional-dimensional space approach with consideration of exciton effects, and indicated that the peaks move to a lower energy. In 2017, Gu et al.^[18] calculated the OACs of electron interband transitions in wurtzite ZnO/ZnMgO quantum wells in consideration of built-in electric fields (BEFs) and the Hartree potential, and the effects of size and TMCs of OACs are also given. Recently, Song et al.^[19] investigated optical absorption of exciton transitions in a single QW by introducing a mixed phase model.

In the present research, a finite difference method and a variational procedure were used to compute the exciton states in Zn_{1 − xl}Mg_xlO/ZnO/Zn_{1 − xc}Mg_xcO/ZnO/Zn_{1 − xr}Mg_xrO ADQWs, within a wider x range by considering the influence of BEFs and the Hartree potential. Compared with the existing theoretical works, this method can obtain energy levels of excitronic subband more efficiently. The OACs of exciton interstate transition in ADQWs are calculated by the density matrix method. In this process, we use the weight model to discuss the influence of Zn_xMg_{1 − x}O (0.37 < x < 0.62) mixed phases on the OACs, which was neglected by other authors. The size and TMCs effects of OACs were investigated in the case of the whole composition. Up to now, there are few experimental reports about the OACs of exciton interstates transition in ADQWs with mixed phases, so the exciton interband emission is calculated for a simple structure. The part of our results is compared with the experiment to verify the feasibility of this method. The paper is organized as follows: the theory and model used in our works are described in Section 2. Section 3 gives the computed results and discussions. Finally, the conclusions are shown in Section 4.

Let us assume that an exciton is confined in a Zn_{1 − xl}Mg_xlO/ZnO/Zn_{1 − xc}Mg_xcO/ZnO/Zn_{1 − xr}Mg_xrO ADQW as shown in Fig. 1. Here, L_b1, L_b, and L_b2 are chosen as the thicknesses corresponding to the left, center, and right barriers. L_w1 and L_w2 are the wideness of left and right wells of the ADQWs, respectively. The growth direction of materials is chosen as the z axis whereas the perpendicular to this direction as the x–y plane. The Hamiltonian of the exciton in this ADQW can be expressed by

Fig. 1.

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	Fig. 1. The schematic diagram of a Zn1 − xlMgxlO/ZnO/Zn1 − xcMgxcO/ZnO/Zn1 − xrMgxrO ADQW.

In Eq. (1), H_e–h consists of two parts, the kinetic energy term of the electron–hole in the x–y plane and the Coulomb interaction between the electron and hole with charge e and −e, respectively.^[20,21] It can be given by

In Eq. (1), H(z_i) (i = e, h) are the parts of Hamiltonian for a single electron and hole in the z direction. They can be written as

Here E_g,ZnMgO is the band gap of Zn_{1 − x}Mg_xO and can be given by E_g,ZnMgO = xE_g,MgO − (1 − x)E_g,ZnO, where xr epresents the component of Mg. E_g,MgO and E_g,ZnO are the band gaps for MgO and ZnO alloys, respectively.

In Eq. (3), V_H,i is the Hartree potential induced by the distribution of free carriers, it satisfies Poisson equation^[22] for electron and hole

N_D is the doping concentration at temperature T.^[18] E_f,i is the Fermi energy and E_D the donor energy level. And k_B is the Boltzmann constant. In Eq. (5), N_2D (z_i) is the two-dimensional electron (hole) gas concentration in all conduction (valance) sub-band related to the distribution of electron (hole). And it satisfies^[23]

In Eq. (3), F_k is the BEF in layer k, and it can be expressed as follows:^[25]

For a zinc-blende structure, is just induced by the and can be written as^[26]

In Eq. (10), e_14,k is the piezoelectric coefficient. In Eq. (11), a_MgO and a_ZnMgO are the lattice constants of bulk materials MgO and ZnMgO, respectively. C_11,k, C_12,k, and C_44,k are elasticity constants for a zinc-blend. The lattice constant of a TMC can be interpolated linearly by binary compounds and satisfies a_ZnMgO = xa_MgO + (1 – x)a_ZnO.^[16]

Let us consider that exciton transition is only caused by the transition of the electron (hole) in the z direction mainly due to the confinement potential since the separation between the ground state and first excited state caused by Coulomb interaction is larger than the photon energy within infrared region, and the exciton is at the ground state in the x–y plane. So that its excited states can be written as^[27]

The OACs of exciton transitions can be written as^[28]

There are two phases of zinc-blende and wurtzite in Zn_{1 − x}Mg_xO for 0.37 < x < 0.62.^[10] We assume that the mixed phase can be fitted with a Gauss pacific distribution and the proportion of zinc-blende in the structure can be expressed as follows:^[19]

The parameters used in the numerical calculation are given in Table 1. In addition, in Eq. (6) N_D = 5 × 10¹⁹/cm³. In Eq. (7), E_D is adopted as 0.03 eV below the bottom of the conduction band. We choose η = 0.5 and δ = 0.25 in Eq. (18), and Γ_mn = 0.14 ps.^[14]

Table 1.

Table 1.

Table 1. Material parameters about ZnO and MgO of WZ and ZB used in the calculations. . Wurtzite Zinc-blende ZnO MgO ZnO MgO Lattice constant/nm aa 0.3219 0.3259 ab 0.4538 0.4436 Elasticity constants/GPa C11a 217 222 C11b 193 159 C12a 117 90 C12b 139 133.3 C13a 121 58 C44b 96 74 C33a 225 109 Piezoelectric coefficients/(c/m2) e31a –0.57 –0.38 e14b 0.69 0.78 e33a 1.34 2.26 Spontaneous polarizationa/(c/m2) Psp –0.022 –0.06 Bad gap/eV Eg 3.37c 6.34d Eg 3.27e 7.8f Effective mass (free electron mass m0) 0.24g 0.35f 0.233h 0.24i 0.78j 1.6f 1.578h 0.78i a Ref. [30]; b Ref. [31]; c Ref. [32]; d Ref. [33]; e Ref. [34]; f Ref. [35]; g Ref. [36]; h Ref. [19]; i Ref. [26]; j Ref. [37].

Table 1. Material parameters about ZnO and MgO of WZ and ZB used in the calculations. .

In order to show the influence of Hartree potential V_H,i on the total potentials, the potential profiles before and after the self-consistent computations and V_H,i are given for specified ADQW1 with well widths of L_w1 = 6 nm and L_w2 = 4 nm and the width of central barrier with L_b = 1.5 nm in Fig. 2.

Fig. 2.

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	Fig. 2. Total potential profiles before (a), after (b) self-consistent computations, and Hartree potential VH,i (c) for electrons (holes) in the specified ADQW1.

It can be seen from Figs. 2(a) and 2(b) that V_H,i bends the conduction and valance bands to lower (elevate) the energy levels of electrons (holes). Consequently, this effect strengths the binding of excitons. Figure 2(c) also indicates boundary condition for Eq. (5).

The eigen-wavefunctions and corresponding energy levels of the electron (hole) are computed for ADQW2 and ADQW1 mentioned in Section 2 by solving Eqs. (5) and (13) with parameters in Table 1. According to Eq. (15), the three energy levels of an exciton are calculated to be E₁(e₁h₁), E₂(e₁h₃), and E₃(e₂h₁), where e_i (h_i) presents an electron (hole) at state i. The OACs of exciton transitions between these energy levels are calculated based on the density matrix principle and the OACs for mixed phases and are obtained by the weight model interpolated by Eq. (19). In order to verify the feasibility of our theory, the exciton interband emission energy in ADQW2 is denoted by E_ph = E_e + E_h + E_g − E_b with E_e and E_h as the confinement energies of the electron and hole respectively, E_g as the band gap energy of ZnO, and E_b as the exciton binding energy. Our computed result agrees well with the experiment^[38] for a special asymmetric structure with equal height of wells and barriers and the corresponding OACs are presented in Fig. 3.

Fig. 3.

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	Fig. 3. The OACs of the exciton interband transition as a function of the incident photon energy in a Zn0.85Mg0.15O/ZnO/Zn0.85Mg0.15O/ZnO/Zn0.85Mg0.15O (9 nm/2 nm/2 nm/7 nm/9 nm) ADQW. The red dash line is the experimental result.[38] and the black solid line represents our theoretical result.

Figure 4 shows the optical absorption coefficients for the exciton transition of E₁ to E₃ with variation of L_b in the two kinds of ADQWs. The two kinds of peaks are given in Figs. 4(a) and 4(b) to present the results for ADQW1 and the ADQW2, respectively. For the two cases, the hole is in the right well, whereas the electron can stay in the two wells to form a direct exciton or an indirect exciton. It can be seen from Fig. 4, the variation of the optical absorption coefficients is insensitive to the variation of L_b for L_w1 = L_w2 in ADQW2 that indicates the exciton is direct. Whereas an obvious red shift appears in Fig. 4(a) when L_b increases from 1.7 nm for ADQW1. This phenomenon is due to the change of the excitons from the indirect type to the direct type. For a direct exciton, the electron is mainly distributed in the right well with a lower barrier potential because of the quantum confinement effects (QCEs). The formation of indirect excitons is that the electron tunnels into the left well within the influence of the BEFs when L_b < 1.7 nm, whereas the hole remains in the right well with the increase of L_b. It should be pointed out that the BEF has a stronger influence on the indirect exciton to separate the electron and hole. During the transition, the electron plays an important role. The energy level intervals of the electron in the left well with a higher barrier potential are larger than that in the right well with a lower barrier potential. Consequently the change of the exciton type results in a redshift of the optical absorption coefficients. When L_w1 > L_w2, the tunneling of the electron in the right well into the left well is easier under the strong quantum confinement effects as decreasing L_b to keep the exciton being indirect. As a result, it can be seen that the optical absorption coefficients only have a slightly red shift caused by the weakened BEFs in the left well with decreasing L_b for ADQW1 in Fig. 4(b).

Fig. 4.

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	Fig. 4. OACs of the exciton transition from E1 to E3 with variation of incident photon wavelength λ as a function of Lb for Zn1 − xlMgxlO/ZnO/Zn1 − xcMgxcO/ZnO/Zn1 − xrMgxrO ADQWs with Lb1 = 7 nm, Lb2 = 7 nm, xl = 0.6, xc = 0.4, and xr = 0.4 for (a) Lw1 = 4 nm and Lw2 = 4 nm; (b) Lw1 = 5 nm and Lw2 = 3 nm. The right images are for ADQW1, the left ones for ADQW2.

Figure 5 shows the OACs of the exciton transition from E₂ to E₃ with variation of L_b when L_w1 = L_w2 and L_w1 > L_w2 for ADQW1 and ADQW2, respectively. As discussed for Fig. 4, it can be clearly seen that there is a transition between a direct and an indirect excitons near L_b ∼ 1.7 nm for ADQW1. The BEF in the central barrier is weakened gradually as increase of L_b, and the BEFs in the wells also increase. There three effects enhance the QCEs on the electron and hole in different wells to increase the intervals of exciton energy levels for ADQW2. As a result, both panels 5(a) and 5(b) show that OACs appear a blue shift with increasing L_b. It can be also seen from Fig. 5(a) that the OACs have an obvious blue shift after a red shift for ADQW1 contributed by the exciton-type transformation, which widens the excitonic energy levels intervals due to the stronger Coulomb interaction between electron and hole when N_b > 1.7 nm. Figure 5(b) shows that the transition energy decreases gradually as L_b increases in ADQW1 and is contrast to that in ADQW2 when L_w1 = L_w2. As increase of L_b for ADQW1 when L_w1 > L_w2, the BEFs in the two wells are stronger than that in the central barrier. The BEFs separate the electron and hole to weaken their Coulomb interaction. Consequently, the intervals of excitonic energy levels decrease and the OACs have a red shift. The two kinds of peaks merger together when L_b = 3 nm. In addition, the excitonic transition energy from E₁ to E₂ is also calculated. The result shows that OACs appear a redshift for the two kinds of ADQWs with increasing center barrier thickness. For the sake of shortness, we do not give the results in the paper.

Fig. 5.

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	Fig. 5. OACs of the exciton transition from E2 to E3 with variation of incident photon wavelength λ as a function of Lb for Zn1 − xlMgxlO/ZnO/Zn1 − xcMgxcO/ZnO/Zn1 − xrMgxrO ADQWs with Lb1 = 7 nm, Lb2 = 7 nm, xl = 0.6, xc = 0.4, and xr = 0.4, for (a) Lw1 = 4 nm and Lw2 = 4 nm; (b) Lw1 = 5 nm and Lw2 = 3 nm. Panel (b) is for ADQW2, panel (a) is for ADQW1.

Figure 6 depicts the wave functions of an electron at e₂ and a hole at h₁ in ADQW1. For comparison, the conduction and valance bands are also given. As shown in Fig. 6(a), it can be seen that the electron and hole are mainly in the right well corresponding to a direct exciton. Whereas the electron and hole are separately in different wells that reveals an indirect exciton. This effect is mainly from the BEFs to tilt the energy bands.

Fig. 6.

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	Fig. 6. Wave functions of an electron at e2 (green lines), a hole at h1 (red lines) for ADQW1 with Lb1 = 7 nm, Lb2 = 4 nm, xl = 0.6, xc = 0.4, and xr = 0.4 for (a) Lw1 = 4 nm, Lw2 = 4 nm, and (b) Lw1 = 5 nm, Lw2 = 3 nm.

Figure 7 shows the wave functions of direct and indirect excitons at ground state E₁ and excited state E₂ computed using Eq. (12) for both ADQW1 and ADQW2 with the fixed ρ as (3.16 nm for former and 2.24 nm for later). As shown in Figs. 7(a) and 7(b), the electron and hole distribute mainly in the right well corresponding to a direct exciton at E₁ for ADQW1, whereas the electron and hole are expelled to the opposite directions of the well due to the BEFs and asymmetric properties. It can be seen from Fig. 7(b) that the electron for an exciton at E₂ obviously tunnels into the center barrier and the left well. It can be also seen form Figs. 7(c) and 7(d), the electron and hole distribute in the same wells to form a direct exciton for ADQW2. It can be seen form Figs. 7(c) and 7(d) that the wave functions of exciton at E₁ and E₂ have a wider distribution due to the stronger BEFs. It should be pointed out that the electron in an exciton at E₁ and E₂ has a stronger tunneling effect in ADQW1 comparing with that in ADQW2 since the BEFs are weak for the former. In Figs. 7(e) and 7(f), the electron and hole distribute in the different wells corresponding to an indirect exciton at E₁ and E₂ for ADQW2.

Fig. 7.

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Fig. 7. Wave functions of a direct exciton at state E1 (a), at state E2 (b) for ADQW1, at state E1 (c), and at state E2 (d) for ADQW2 with Lb1 = 7 nm, Lw1 = 5 nm, Lb = 1.8 nm, Lw2 = 3 nm, Lb2 = 4 nm, xl = 0.6, xc = xr = 0.4, and that of an indirect exciton at state E1 (e), and at state E2 (f) for ADQW2 with Lb1 = 7 nm, Lw1 = Lw2 = Lb2 = 4 nm, Lb = 1.8 nm, xl = 0.6, xc = xr = 0.4. The unit a.u. is short for arbitrary units.

Figure 8 shows the OACs of the exciton transitions from E₁ to E₂, E₂ to E₃, and E₁ to E₃ with increasing L_w2, respectively. In Fig. 8(a), the energy level intervals of excitons decrease due to the weakening QCEs. Consequently, the OACs of the transition from E₁ to E₂ have a red shift with increasing L_w2 for both ADQW1 and ADQW2. Whereas the effect of the Coulomb interaction on the exciton transitions are more important when an energy level of exciton approaches another one. Such like the transition from E₂ to E₃ in Fig. 8(b), the trend of OACs is contrary to that of Fig. 8(a). The differences of average positions between an electron at e₁, e₂, and a hole at h₃, h₁ increase with increasing L_w2 since the BEFs and asymmetrical structure. The Coulomb interaction becomes weaker to reduce E₂ and E₃, thus the OACs have a blue shift for both of ADQW1 and ADQW2. For the transition from E₁ to E₃ in Fig. 8(c), the intervals of electron energy levels are widened due to the increasing BEF in the left well as increase of L_w2 to widen interval between E₁ and E₃. As a result, a blue shift appears for both of the ADQWs. At this moment, the electron plays a dominant role in the transition.

Fig. 8.

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	Fig. 8. OACs with the variation of incident photon wavelength λ as a function of Lw2 for Zn1 − xlMgxlO/ZnO/Zn1 − xcMgxcO/ZnO/Zn1 − xrMgxrO ADQWs with Lb1 = 4 nm, Lw1 = 5 nm, Lb2 = 7 nm, xl = 0.6, xc = 0.4, and xr = 0.4, for the transition from (a) E1 to E2, (b) E2 to E3, and (c) E1 to E3. The right images are for ADQW1, the left ones are for ADQW2.

Figure 9 shows the xl-dependent OACs for ADQW1 and ADQW2. It can be seen from Fig. 9(a), the distance between the peaks of ADQW1 and ADQW2 becomes bigger as xl increases. For ADQW2, the BEFs in the left barrier is strengthened gradually as increase of xl that is equivalent to narrowing L_b1 to enlarge the electron energy level intervals. Consequently, the exciton energy level intervals become larger to shift the OACs toward a smaller λ. Contrary to this phenomenon, the BEFs in the left barrier in ADQW1 are opposite to that in ADQW2, a red shift occurs in the former. In Fig. 9(b), the differences of average positions between electrons and holes increase speedily due to the strengthened BEFs in both wells, the Coulomb interaction are weaken and the intervals of excitonic energy levels decrease for ADQW2. Whereas the BEFs in both wells are weakened as increase of xl for ADQW1. Thus the OACs have a red shift and a blue one for ADQW2 and ADQW1, respectively. The peaks merge together at xl = 0.48. For the transition from E₁ to E₃ mainly comes from the contribution of the hole distributed in the right well, and the variation of xl has a slight influence on the hole to bring an unobvious change on the OACs for the two kinds of ADQWs in Fig. 9(c).

Fig. 9.

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	Fig. 9. OACs with the variation of incident photon wavelength λ as a function of xl for Zn1 − xlMgxlO/ZnO/Zn1 − xcMgxcO/ZnO/Zn1 − xrMgxrO ADQWs with Lb1 = 4 nm, Lw1 = 5 nm, Lb2 = 7 nm, Lw2 = 3 nm, xc = 0.4, and xr = 0.4, for the transition from (a) E1 to E2, (b) E2 to E3, and (c) E1 to E3.

Our computed results for OACs for exciton interstate transition show the following conclusions. The BEFs in ADQWs tilt the conduction and valance bands to influence the exciton energy levels and exciton types. Firstly, the exciton types occur a transformation when an electron at state e₂ participates a transition in ADQW2 for L_w1 = L_w2 to mutate OACs. Secondly, the electron-involved exciton transitions are more sensitive to the change of size and composition than that of hole-involved for ADQW2 and ADQW1. Finally, the peaks of OACs in the two kinds of QWs merge together under some special conditions and this is helpful to both of theory and experiment.