MgZnO quantum wells

MgZnO quantum wells

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Tunable built-in electric field and optical properties in wurtzite ZnO/MgZnO quantum wells Congxin Xia a,∗ , Heng Zhang a , Jiao An a , Shuyi Wei a , Yu Jia b a b

Department of Physics, Henan Normal University, Xinxiang, Henan 453007, China International Laboratory for Quantum Functional Materials of Henan, Zhengzhou University, Zhengzhou, Henan 450001, China

a r t i c l e

i n f o

Article history: Received 15 January 2014 Received in revised form 11 May 2014 Accepted 17 May 2014 Available online xxxx Communicated by R. Wu Keywords: Built-in electric fields Optical properties ZnO/MgZnO QWs

a b s t r a c t Based on the effective-mass approximation, the built-in electric field, exciton states and optical properties are investigated theoretically in the wurtzite (WZ) ZnO/MgZnO quantum wells (QWs). Numerical results show that the built-in electric field, exciton states and optical properties can be tuned effectively by the MgZnO barrier layer in the QWs. When the MgZnO barrier width is decreased, the built-in electric field F w and the interband emission wavelength are decreased. However, the exciton binding energy and the electron–hole recombination rate increase with increasing the MgZnO barrier width. In particular, the electron–hole recombination rate approaches zero when the QWs structural parameters are large in the WZ ZnO/MgZnO QWs. © 2014 Elsevier B.V. All rights reserved.

1. Introduction In recent years, wurtzite (WZ) ZnO/MgZnO quantum heterostructures have attracted much attention due to novel potential applications in the blue and ultraviolet spectral optoelectronic devices. In order to understand the optoelectronic mechanism of WZ ZnO/MgZnO quantum heterostructures, different research groups have investigated the optical properties of the WZ ZnO/MgZnO quantum wells (QWs) [1–5]. Makino et al. investigated experimentally the charged exciton states in the WZ ZnO/MgZnO QWs [6]. The effects of exciton transport on luminescence efficiency were also investigated in the WZ ZnO/MgZnO QWs [7]. In addition, with the development of experiment technology, the WZ ZnO/MgZnO QWs-based optoelectronic devices, such as the UV light-emitting diodes (LEDs) and laser diodes (LDs), have been fabricated successfully [8–13]. However, experimental and theoretical studies have shown that the strong built-in electric fields are induced by the spontaneous and piezoelectric polarizations in the WZ ZnO/MgZnO QWs [14–16]. Moreover, the strong built-in electric fields degrade the performance of the WZ ZnO/MgZnO QWs-based optoelectronic devices, which are similar as the cases of WZ GaN-based QWs [17–19]. In order to improve further the performance of WZ ZnO/ MgZnO QWs-based optoelectronic devices, it is imperative to further understand the influence of the built-in electric field on prop-

*

Corresponding author. Tel.: +86 371 67739336; fax: +86 371 67767758. E-mail address: [email protected] (C. Xia).

http://dx.doi.org/10.1016/j.physleta.2014.05.015 0375-9601/© 2014 Elsevier B.V. All rights reserved.

erties of the WZ ZnO/MgZnO QWs. In the past years, different research groups have carried out some valuable experimental and theoretical studies. Laumer et al. have grown WZ ZnO/MgZnO QWs with graded MgZnO barriers [20]. The non-polar WZ ZnO/MgZnO QWs structures were also suggested to eliminate the piezoelectric polarization effects [21,22]. The optical gain and related optical properties were studied in the WZ ZnO/MgZnO QWs and related optoelectronic devices [23,24]. The exciton states and optical properties were also studied in the WZ ZnO/MgZnO QWs [16,25,26]. However, to our knowledge, the MgZnO barrier layer effects on the built-in electric field and optical properties have been not investigated in the WZ ZnO/MgZnO QWs to date. Thus, in this work, we will investigate theoretically exciton states and optical properties in the WZ ZnO/MgZnO QWs by means of the variational methods, considering different structural parameters of MgZnO barrier layer. Numerical results show that the built-in electric field, exciton states and optical properties can be tuned effectively by the MgZnO barrier layer in the WZ ZnO/MgZnO QWs. These results may be helpful to design the high-performance WZ ZnO QWsbased optoelectronic devices. 2. Built-in electric fields in the WZ ZnO/MgZnO QWs Fig. 1 presents the diagram of the WZ ZnO/Mgx Zn1−x O QWs theoretical model, in which the origin is taken at the center of the QWs and the z axis is defined to be the growth direction. In order to understand the influences of MgZnO barrier layer on optical properties of the WZ ZnO/MgZnO QWs, it is necessary to investigate firstly the influences of the barrier layer width on the

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Fig. 1. A diagram of the WZ ZnO/Mgx Zn1−x O QWs with ZnO well width L w and MgZnO barrier layer size L b , in which the origin is taken at the center of the QWs and the z axis is defined to be the growth direction.

Fig. 3. The built-in electric fields F b inside the barrier layer (a) and F w inside the well layer (b) as a function of the barrier width L b in the WZ ZnO/Mgx Zn1−x O QWs with the well widths L w = 3 nm, considering different Mg concentrations x. The solid, dashed and dotted lines are for x = 0.05, 0.10 and 0.15, respectively.

Fig. 2. The built-in electric fields F b inside the barrier layer (a) and F w inside the well layer (b) as a function of the barrier width L b in the WZ ZnO/Mg0.2 Zn0.8 O QWs with different well widths. The solid, dashed and dotted lines are for well widths L w = 2, 4 and 6 nm, respectively.

built-in electric fields in the QWs. According to previous theoretical studies on the built-in electric field in WZ GaN-based periodic heterostructures [27,28], the magnitudes of the built-in electric fields F w inside the well layer and F b inside the barrier layer in the WZ ZnO/MgZnO QWs can be expressed as

Fw = −

2( P w − P b ) L b 2ε0 ε w L b + ε0 εb L w

,

| z| ≤

Lw

(1)

2

and

Fb = −

( P w − P b )L w 2ε0 ε w L b + ε0 εb L w

,

Lw 2

< | z| ≤ L b +

Lw 2

(2)

where the indexes w and b indicate the ZnO well layer and MgZnO barrier layer, respectively. L w (b) is the width of well (barrier) layer, P w (b) is the total polarization and ε w (b) is the electronic dielectric constant. The direction of the built-in electric field inside the barrier layer (well layer) is the same (opposite) to the growth direction of QWs. In this work, the used material parameters of ZnO and Mgx Zn1−x O are taken from Ref. [24]. In Fig. 2, the magnitudes of the built-in electric fields F b inside the MgZnO barrier layer and F w inside the ZnO well layer are investigated as a function of the barrier width L b in the WZ ZnO/Mg0.2 Zn0.8 O QWs considering different well widths L w . From Fig. 2(a), one can see that the built-in electric field F b decreases when the barrier width increases (L b < 10 nm), and then becomes insensible to the variation of the barrier width. However, Fig. 2(b) also shows that the magnitude of the built-in electric field F w increases obviously when the barrier width increases (L b < 10 nm), and then becomes insensible to the variation of the barrier width. These results are very interesting and indicate that the built-in electric fields can be tuned effectively by MgZnO barrier layer size

in the QWs. We can understand these behaviors by the following extreme barrier width L b case, namely, L b → ∞. According to Eq. (1), one can see easily that when the barrier width L b is infinite, the built-in electric field F b approaches to zero, while the built-in electric field F w is independent of the variation of the barrier width. In addition, when the barrier width approaches to infinite (L b → ∞), the equilibrium lattice constant (aeq ) will be equal to the lattice constant of MgZnO barrier layer (ab ), thus the strain tensors of barrier and well layers approaches to zero and constant, respectively. In addition, Fig. 2 also shows that when the well width increases, the built-in electric field F b increases; while the built-in electric field F w decreases in the QWs. The reason is that the piezoelectric polarizations of ZnO well layer (MgZnO barrier layer) are decreased (increased) when the well width is increased in the QWs. In order to further understand the influences of Mg composition x on the built-in electric fields in the WZ ZnO/Mgx Zn1−x O QWs, in Fig. 3, we calculate the built-in electric fields F b and F w as a function of the MgZnO barrier width L b in the WZ ZnO/Mgx Zn1−x O QWs considering different Mg composition x. Fig. 3 shows that the built-in electric fields F b and F w are insensible to the variation of the barrier width L b when the barrier width L b > 10 nm for different Mg composition x. Moreover, Fig. 3 also shows that the magnitudes of the built-in electric field F w increase monotonously when Mg composition increases. The reason is that spontaneous and piezoelectric polarizations are increased when Mg composition is increased in the WZ ZnO/Mgx Zn1−x O QWs. These calculated results are in agreement with previous experimental and theoretical studies [29,30]. 3. Exciton states and optical properties of WZ ZnO/MgZnO QWs 3.1. Theory model Within the framework of the effective-mass approximation, the Hamiltonian of the exciton confined in the WZ ZnO/Mgx Zn1−x O QWs can be written in cylindrical coordinate as

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ˆ exc = E g + Hˆ ez + Hˆ hz − H −

e2 4πεε0 r

h¯ 2



1 ∂ 1 ∂2 ∂2 + + 2 2 2 2μ ∂ ρ ρ ∂ ρ ρ ∂φ

3



(3)

, 



where E g is the band gap of ZnO, H ez ( H hz ) is the z-direction electron (hole) Hamiltonian. μ is the reduced mass in the plane. ρ and φ are the relative coordinates of the electron and hole, re spectively. In addition, r = ρ 2 + ( ze − zh )2 is the electron–hole distance, e is the absolute value of the electron charge, ε0 is the dielectric constant of vacuum and ε is the effective mean relative dielectric constant. The z-direction Hamiltonian of the electron (hole) in the cylindrical coordinates is given by

ˆ jz = − H

h¯ 2

2m∗ j

∂2 + V (z j ) ± e F z j , ∂ z2j

(4)

where the index j is e or h, m∗j is the effective mass of the electron (hole), V ( z j ) is the electron (hole) confinement potential due to the band offset in the WZ ZnO/Mgx Zn1−x O QWs, F is the strength of the built-in electric field, which is opposite to the z axis. The sign −(+) is for the electron (hole). Considering the correlation of the electron–hole relative motion in the WZ ZnO/Mgx Zn1−x O QWs, the trial wavefunction of exciton in the QWs can be chosen as [31,32]

Φexc (re , rh ) = f e ( ze ) f h ( zh ) exp(−β r ),

(5) 



where f e ( ze ) and f h ( zh ) are the eigenfunctions of H ez and H hz , respectively. In addition, β is the variational parameter. The ground-state exciton energy can be investigated by minimizing the expectation value of the Hamiltonian with the trial wavefunction

E exc = min β

Φexc | Hˆ exc |Φexc  . Φexc |Φexc 

(6)

The ground-state exciton binding energy E b , the interband optical transition energy E ph and the interband emission wavelength λ can also be defined as follows:

E b = E g + E e + E h − E exc , E ph = E g + E e + E h − E b ,

λ=

hc E ph

,

(7)

where E e(h) is the electron (hole) confinement energy in the WZ ZnO/Mgx Zn1−x O QWs, which can be obtained from the groundstate energies of Eq. (4). In addition, the electron–hole recombination rate A can be given by

A=

  | dre drh f e ( ze ) f h ( zh )|2 . Φexc |Φexc 

Fig. 4. The ground-state exciton binding energy E b , the electron–hole recombination rate A and the interband emission wavelength λ are investigated as a function of the barrier width L b in the WZ ZnO/Mg0.15 Zn0.85 O QWs. The solid, dashed and dotted lines are for the well width L w = 2.0, 4.0 and 6.0 nm, respectively.

(8)

3.2. Numerical results and discussions We have calculated the ground-state exciton binding energy E b , the electron–hole recombination rate A and the interband emission wavelength λ as functions of structural parameters, such as the MgZnO barrier layer width L b , ZnO well width L w and Mg composition x in the WZ ZnO/Mgx Zn1−x O QWs. In Fig. 4, the ground-state exciton binding energy E b , the electron–hole recombination rate A and the interband emission wavelength λ are investigated as a function of the MgZnO barrier

layer width L b in the WZ ZnO/Mg0.2 Zn0.8 O QWs considering different well width cases. It can be seen clearly from Figs. 4(a) and (b) that the exciton binding energy and the electron–hole recombination rate decrease initially, and then become insensitive to the variation of the barrier width. The main reason is that the built-in electric field F w increases initially and then remains insensitive to the variation of the barrier width with increasing the MgZnO barrier layer width in the QWs. In addition, Fig. 4(c) also shows that the MgZnO barrier layer size has an obvious influence on the interband emission wavelength. The reason is that the MgZnO barrier layer size affects the strength of the built-in electric fields inside the QWs, which modifies further the effective band gap and interband emission wavelength of the QWs. Moreover, Fig. 4 also shows that the influences of well width are obvious on the exciton binding energy, the electron–hole recombination rate and the interband emission wavelength in the QWs. In order to understand further the influences of well width on exciton states and optical properties in the WZ ZnO/MgZnO QWs, in Fig. 5, the ground-state exciton binding energy E b , the electron–hole recombination rate A and the interband emission wavelength λ are investigated as a function of the well width L w in the WZ ZnO/Mg0.2 Zn0.8 O QWs considering different MgZnO barrier layer sizes. Fig. 5(a) shows that the exciton binding energy has a maximum value with the variation of the well width for each considered case, which is due to well-known quantum size effects in finite barrier quantum structures. In addition, it is also clear that the exciton binding energy is decreased when the MgZnO barrier layer thickness is increased, which agrees with the results of Fig. 4. We can also see from Fig. 5(b) that the electron–hole recombination rate decreases with increasing the well width. The reason is that when the well width increases, quantum confinement effects are reduced in the QWs. In particular, Fig. 5(b) also shows that the electron–hole recombination rate is reduced to zero when the well width is large. Moreover, we can see from Fig. 5(c) that the interband emission wavelength increases as the well width increases, which is in agreement with previous experimental mea-

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Now we turn to investigate the effects of Mg composition on exciton states and optical properties in the WZ ZnO/Mgx Zn1−x O QWs considering different barrier layer cases. We can see from Figs. 6(a) and (b) that the exciton binding energy and the electron– hole recombination rate decrease when Mg composition increases in the QWs for each considered case. This behavior can be understood as follows. Although quantum confinement can be increased when Mg composition increases in the QWs, the built-in electric fields effects have much more obvious influences on electrons and holes states than quantum confinement effects. The builtin electric field increases with increasing Mg content in the WZ ZnO/MgZnO QWs. Therefore, the electron–hole Coulomb interaction and wavefunction overlap are decreased when Mg content is increased in the QWs. In addition, Fig. 6(c) also shows that the interband emission wavelength increases when Mg composition increases. The reason is that the built-in electric fields can reduce the effective band gap of the QWs. 4. Conclusions

Fig. 5. The ground-state exciton binding energy E b , the electron–hole recombination rate A and the interband emission wavelength λ are investigated as a function of the well width L w in the WZ ZnO/Mg0.2 Zn0.8 O QWs. The solid and dashed lines are for the barrier width L b = 2.0 and 6.0 nm, respectively.

In conclusion, we have investigated theoretically the builtin electric fields, the ground-state exciton binding energy, the electron–hole recombination rate and the interband emission wavelength in the WZ ZnO/MgZnO QWs. Numerical results show that when the MgZnO barrier layer width increases in the QWs, the built-in electric field F w inside the well layer increases, the exciton binding energy and electron–hole recombination rate decrease. Moreover, the built-in electric fields F b and exciton binding energy become insensible to the variation of the barrier width when the barrier layer size is large. The emission wavelength can also be modified from 350 to 400 nm ultraviolet range by the finite barrier width. In addition, the electron–hole recombination rate is reduced and the interband emission wavelength is increased when the well width is increased in the QWs. These results may be interesting for the technological purpose, as it could involve a source of control for exciton states and optical properties in the WZ ZnO/MgZnO QWs by tuning the barrier layer width and Mg composition. Acknowledgements This research was supported by the basic research projects with cutting-edge technology of Henan Province (No. 122300413208), and National Natural Science Foundation of China under grant No. 11274280. References

Fig. 6. The ground-state exciton binding energy E b , the electron–hole recombination rate A and the interband emission wavelength λ are investigated as a function of Mg composition in the WZ ZnO/Mgx Zn1−x O QWs. The solid and dashed lines are for the barrier width L b = 2.0 and 6.0 nm, respectively.

sures [33,34]. The reason is that the electrons and holes confinement energies decrease when the well width increases in the QWs. These results indicate that the well width effects should be considered for the high-performance WZ ZnO/MgZnO QWs-based LEDs and LDs.

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