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Hydrogenic impurity states in wurtzite GaN/AlGaN coupled quantum dots Congxin Xia a,∗ , Fengchun Jiang b , Shuyi Wei a a Department of Physics, Henan Normal University, Xinxiang, 453007, China b Department of Technology and Physics, Zhengzhou University of Light Industry, Zhengzhou, 450002, China

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Article history: Received 22 January 2008 Received in revised form 12 March 2008 Accepted 13 March 2008 Available online 2 May 2008 Keywords: Quantum dots Hydrogenic donor impurity Built-in electric fields

a b s t r a c t The ground-state binding energy of a hydrogenic donor impurity in wurtzite (WZ) GaN/AlGaN coupled quantum dots (QDs) is calculated by means of a variational method, considering the strong built-in electric fields caused by the piezoelectricity and spontaneous polarizations. The strong built-in electric fields induce an asymmetrical distribution of the ground-state binding energy with respect to the center of the coupled QDs. If the impurity is located at the low dot, the ground-state binding energy is insensitive to the interdot barrier width of WZ GaN/AlGaN coupled QDs. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, wurtzite (WZ) GaN/AlGaN quantum heterostructures have attracted much attention for conspicuous applications in electronic and optoelectronic devices. It has been found that the built-in electric fields are caused by the piezoelectricity and spontaneous polarizations in WZ GaN/AlGaN quantum heterostructures [1–3]. The magnitude of the built-in electric field is known as of the order of MV/cm. The strong built-in electric field effects on exciton states and optical properties of WZ GaN/AlGaN quantum heterostructures have also been investigated by experimental and theoretical works [1–6]. The study of hydrogenic impurity is one of the main problems in semiconductor low-dimensional systems because the presence of impurity in nanostructures influences greatly the electronic mobility and their optical properties [7]. So the investigation of properties related to the impurity is not only of fundamental interest, but is also of major importance in optoelectronic device applications (high electron mobility transistors, infrared photodetectors or emitters, etc) [7,8]. In the past years, there

∗ Corresponding author. Tel.: +86 0373 3325 151. E-mail address: [email protected] (C. Xia). 0749-6036/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2008.03.004

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Fig. 1. A diagram of the symmetric cylindrical WZ GaN coupled QDs structure with radius R, surrounded by two large energy gap materials Aly Ga1−y N (y < x) in the radial direction and Alx Ga1−x N in the z-direction. The heterointerfaces are located at z = z0 , z1 , z2 and z3 , respectively. In this paper, we take the dot height H = z1 − z0 = z3 − z2 and the interdot barrier width L = z2 − z1 .

have been many theoretical works on the binding energy of the hydrogenic impurity in quantum dots (QDs) [9–15]. These studies show that the binding energy in nanoscopic systems depends upon materials and geometry and impurity position. However, none of these calculations included electric fields larger than 300 KV/cm. To the knowledge of authors, few theoretical and experimental works have been done on the hydrogenic impurity states in WZ GaN/AlGaN quantum heterostructures. Quite recently, we have investigated theoretically the strong built-in electric field effects on the hydrogenic donor impurity states in WZ GaN/AlGaN single quantum dot (QD) [16]. However, multiple QDs rather than single QD are adopted in the commonly used GaN-based optoelectronic devices, such as highbrightness blue/green light-emitting diodes (LEDs) and laser diodes (LDs) [17]. In the multiple QDs sample, the electron can be localized in different QDs. Thus, it is interesting to investigate the hydrogenic impurity states in WZ multiple GaN QDs sample. In this paper, we will investigate theoretically the ground-state binding energy of a hydrogenic donor impurity in WZ GaN/AlGaN coupled QDs. 2. Theoretical model According to previous theoretical studies on WZ GaN coupled QDs [6], for simplicity, we consider the symmetric cylindrical WZ GaN/AlGaN coupled QDs structure (see Fig. 1). Within the framework of effective-mass approximation, the Hamiltonian for a hydrogenic donor impurity in the cylindrical WZ GaN/AlGaN coupled QDs can be written as

ˆ = Hˆ0 − H

e2

4πε0 ε¯ r

,

(1)

with Hˆ0 = −

h¯ 2

2m∗

"

1 ∂

ρ ∂ρ

ρ

∂ 1 ∂2 ∂2 + 2 2 − 2 + V (ρ, z) + V (z), ∂ρ ρ ∂ϕ ∂z

#

(2)

p where r = (x − xi )2 + (y − yi )2 + (z − zi )2 is the distance between the electron and the impurity site. x(xi ), y(yi ) and z(zi ) are the coordinates of the electron(impurity) in the coupled QDs, e is the absolute

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value of the electron charge, ε0 is the permittivity of free space, and ε¯ is the effective mean relative dielectric constant of the embedding material, m∗ is the electron effective mass. The electron confinement potential V (ρ, z) due to the band offset in the WZ GaN/AlGaN coupled QDs structure is given by V (ρ), z0 < z < z1 or z2 < z < z3 , V (ρ, z) = (3) vII , otherwise, 0, ρj ≤ R, V (ρ) = (4) vI ,

ρ > R.

The static electric potential V (z) induced by the built-in electric fields can be written as 0, z ≤ z0 , GaN (z − z0 ), z0 < z < z1 , −eF V (z) =

−[eF GaN z1 − eF AlGaN (z − z1 )], z1 ≤ z ≤ z2 , −eF GaN (z − z3 ), z2 < z < z3 , 0, z ≥ z3 ,

(5)

here the strength of the built-in electric fields F GaN in the GaN dot layer and the F AlGaN in the AlGaN interdot barrier layer caused by the spontaneous and piezoelectric polarizations in the WZ GaN/AlGaN coupled QDs are expressed as follows [1,3]: (PGaN + PGaN − PAlGaN )L SP PE SP (6) F GaN = − , GaN L) ε0 (2εAlGaN H + ε e e and F

AlGaN

(PGaN + PGaN − PAlGaN )H SP PE SP = 2 , ε0 (2εAlGaN H + εGaN L) e e

(7)

GaN GaN AlGaN where PSP , PPE and PSP are the spontaneous and piezoelectric polarizations of GaN and the spontaneous polarization of AlGaN, respectively. According to Refs. [16,18–20], the trial wave function may be written as

ψ = f (ρ)h(z)e−αρei −βzei , 2

2

(8)

where wave functions f (ρ) and h(z) describe the motions of the electron in in-plane and z-direction in the coupled QDs, respectively. The α and β are variational parameters. ρ2ei = (x − xi )2 + (y − yi )2 and zei2 = (z − zi )2 . We would like to point out that the two-parameter variational wave function is a reasonable approximation for studying the hydrogenic impurity states in the WZ GaN/AlGaN coupled QDs [16,18–20]. The ground-state energy of a hydrogenic donor impurity in the WZ GaN/AlGaN coupled QDs can be determined by E = min α,β

hψ|Hˆ |ψi . hψ|ψi

(9)

The ground-state binding energy Eb of a hydrogenic donor impurity is given by Eb = E0 − E,

(10)

where E0 is the ground-state energy for the Hamiltonian of Eq. (2). The z-direction distance zei between the electron and the impurity can be written as zei =

hψ | (ze − zi )2 | ψi hψ | ψi

!1 2

.

(11)

124

C. Xia et al. / Superlattices and Microstructures 44 (2008) 121–126

Fig. 2. The ground-state binding energy Eb as a function of the impurity position zi along the growth direction in the WZ GaN/Al0.15 Ga0.85 N coupled QDs with the interdot barrier width L = 2 nm, radius R = 5 nm and height H = 3 nm, surrounded by Al0.02 Ga0.98 N material in the radial direction.

Fig. 3. Conduction band edges of the WZ GaN/Al0.15 Ga0.85 N coupled quantum wells and the related electron wave function.

3. Numerical results and discussion We have calculated the ground-state binding energy Eb of a hydrogenic donor impurity as functions of the impurity positions zi along the growth direction and the interdot barrier width L of WZ GaN/AlGaN coupled QDs. All material parameters used in the present paper are the same as in Ref. [16]. In Fig. 2, the ground-state binding energy Eb is investigated as a function of the impurity position zi in the WZ GaN/AlGaN coupled QDs. Fig. 2 shows that the ground-state binding energy Eb has a maximum when the impurity is located at the center of the low dot. The reason is that the electron wave function is localized mainly inside the low dot due to the existence of the strong built-in electric fields in the WZ GaN/AlGaN coupled QDs (see Fig. 3). This is different from the distribution of the electron wave function in the WZ GaN/AlGaN coupled QDs when the strong built-in electric fields are ignored. From the above results, we can find that the strong built-in electric fields and the impurity positions have a significant influence on the ground-state binding energy in the WZ GaN/AlGaN coupled QDs. The ground-state binding energy Eb as a function of the interdot barrier width L of WZ GaN/AlGaN coupled QDs with different impurity positions zi is given in Fig. 4. We can see from Fig. 4 that the ground-state binding energy Eb decreases with increasing the interdot barrier width L when the impurity is localized inside the interdot barrier and the upper dot (Curves D, E, F and G). This is because the distance zei between the electron and the impurity is increased when the interdot barrier width L is increased in these cases (See curves D, E, F and G in Fig. 5). Thus, the Coulomb interaction between the electron and the impurity is reduced when the interdot barrier width L is increased. Fig. 4 also shows that the ground-state binding energy Eb is larger when the impurity is located at the low dot than that

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Fig. 4. The ground-state binding energy Eb as a function of the interdot barrier width L of the WZ GaN/Al0.15 Ga0.85 N coupled QDs with radius R = 5 nm and height H = 3 nm, surrounded by Al0.02 Ga0.98 N material in the radial direction. The curves A, B, C, D, E, F and G are for the impurity positions zi = z0 ,

z0 +z1 z +z z +z , z1 , 1 2 , z2 , 2 3

2

2

2

and z3 , respectively.

Fig. 5. The distance zei between the electron and the impurity as a function of the interdot barrier width L of the WZ GaN/Al0.15 Ga0.85 N coupled QDs with radius R = 5 nm and height H = 3 nm, surrounded by Al0.02 Ga0.98 N material in the radial direction. The curves A, B, C, D, E, F and G have the same meaning as in Fig. 4.

at the upper dot and interdot barrier layer (Curves A, B and C). The reason is that the electron wave function is located mainly inside the low dot when the strong built-in electric fields are ignored in the WZ GaN/AlGaN coupled QDs. In particular, we also find from Fig. 4 that the ground-state binding energy Eb is insensitive to the interdot barrier width L in the WZ GaN/AlGaN coupled QDs. This is because the distance zei between the electron and the impurity is insensitive to the interdot barrier width (see curves A, B and C in Fig. 5). 4. Conclusions In conclusion, we have calculated the ground-state binding energy of a hydrogenic donor impurity located anywhere along the growth axis in the symmetric cylindrical WZ GaN/AlGaN coupled QDs. Numerical results show that the ground-state binding energy is highly dependent on the impurity positions and the interdot barrier width. The strong built-in electric fields induce an asymmetrical distribution of the ground-state binding energy with respect to the center of the WZ GaN/AlGaN coupled QDs. In particular, we find that the ground-state binding energy is insensitive to the interdot barrier width if the impurity is located at the low dot in the WZ GaN/AlGaN coupled QDs. These results may be of interest for technological purpose, as it could involve a source of control some

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impurity-related properties in these systems. We also hope that our calculation results can stimulate further experimental and theoretical investigations on the impurity states of group-III nitrides. Acknowledgments The authors thank Professor Yuan Ping Feng for useful discussion and comments on the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

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