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GaN quantum dot
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Current Applied Physics 8 (2008) 153–158 www.elsevier.com/locate/cap www.kps.or.kr
Exciton states in zinc-blende InGaN/GaN quantum dot Congxin Xia b
a,*
, Fengchn Jiang b, Shuyi Wei
a
a Department of Physics, Henan Normal University, Xinxiang 453007, China Department of Technology and Physics, Zhengzhou University of Light Industry, Zhengzhou 450002, China
Received 14 February 2007; received in revised form 24 May 2007; accepted 4 July 2007 Available online 24 July 2007
Abstract Within the framework of effective-mass approximation, exciton states confined in zinc-blende(ZB) InGaN/GaN quantum dot(QD) are investigated by means of a variational approach, considering finite band offsets. The ground-state exciton binding energy and the interband emission energy are investigated as functions of QD structural parameters in detail. Numerical results show clearly that both the QD size and In content of InGaN have a significant influence on the exciton states and interband optical transitions in the ZB InGaN/GaN QD. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.35.y; 71.15.m Keywords: Exciton; InGaN; Quantum dot
1. Introduction In recent years, the wide-band-gap group-III nitrides based semiconductor heterostructures InGaN/GaN have attracted much attention due to conspicuous device applications in electronics and optoelectronics, such as highbrightness blue/green light-emitting diodes (LEDs) and laser diodes (LDs) [1]. They are usually grown in the thermodynamic stable configuration with hexagonal(wurtzite) crystal structure and in a metastable modification with cubic(zinc-blende)structure [2]. It is found that the electronic and optical properties of wurtzite(WZ) InGaN/ GaN heterostructures are affected by the strong built-in electric field induced by the piezoelectricity and spontaneous polarizations [3–7]. The magnitude of the strong built-in electric field is estimated to be in the order of MV/cm [3–7]. However, spontaneous polarization fields do not exist in the zinc-blende(ZB) nitrides due to the higher crystal symmetry, and piezoelectric fields are negligible due to the (0 0 1) growth direction of epitaxial layers *
Corresponding author. E-mail address: [email protected] (C. Xia).
1567-1739/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2007.07.001
[8–11]. Thus, one of the distinguished physical properties of ZB III-nitride quantum heterostructures is the absence of the built-in electric field. Since the band gap of ZB IIInitrides is lower than that of their hexagonal counterparts, 510 nm light emission can be obtained from ZB InGaN layers with reasonably lower In content. This may be an advantage due to the well-known difficulties in the growth of InGaN with high In content [8,12]. Therefore, ZB InGaN/GaN quantum heterostructures have attracted increasing attention because they are expected to have possible advantages for optoelectronic applications. Some experimental results show that many nanometer-scale InN-rich clusters are formed within the ZB InGaN active layer in the ZB InGaN/GaN quantum heterostructures [2,8–15]. These experiments indicate that the InN-rich clusters have cubic quantum dot(QD) or quantum disk shape. The ground-state transition energy as a function of QD size was calculated in a cubic QD [9]. In this paper, we model the InN-rich clusters using a cylindrical ZB InGaN/GaN QD. The reason for the choice of the cylindrical QD geometry lies in the fact that the structures of this type are relatively more convenient for fabricating devices than others [16]. On the other hand, it has been reported that In surface
154
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
segregation effects occur during the growth of InGaN ternary alloys [11,12,17–19]. The In segregation effects induce a blue-shift of the optical transitions and a decrease of the oscillator strength in InGaN/GaN quantum heterostructures [18]. However, recent studies show that some improved experiment methods can relieve the In segregation effects in ZB InGaN/GaN quantum heterostructures [17]. Therefore, we neglect the In surface segregation effects on exciton states and optical properties of ZB InGaN/GaN QD in this paper. To the knowledge of the authors, few theoretical investigation on exciton states in ZB InGaN/ GaN QD were published. In this paper, we investigate variationally the ground-state exciton binding energy and the interband emission energy of ZB InGaN/GaN QD. This paper is organized as follows. In Section 2, a theoretical model used to describe exciton states confined in ZB InGaN/GaN QD is outlined. Numerical results for the ground-state exciton binding energy and the interband emission energy are given and discussed in Section 3. Finally, we summarize the main conclusions obtained in this paper in Section 4. 2. Theoretical model For simplicity, let us now consider an exciton confined in the cylindrical ZB InGaN/GaN QD. The cylindrical QD is specified by its radius R and height h (see Fig. 1). Within the framework of effective-mass approximation, the Hamiltonian of the exciton can be written as, b ex ¼ H be þ H b h þ Eg H
e2 ; ! 4pe0ej r e ! r hj
ð1Þ
b h ) is the electron (hole) Hamiltonian, Eg is the b e(H where H band gap energy of the zinc-blende InxGa1xN material,
y In Ga y
In Ga x
1-x
1-y
N
N
x
R
! r h ) is the position vector of the electron (hole), e is r e (! the absolute value of the electron charge, e0 is the permittivity of free space, and e is the effective mean relative dielectric constant of the embedding material between the electron and hole. Throughout the present paper, we choose the conduction band bottom and the valence band top of the ZB InGaN layer at the origin z = 0 as the reference energy level for the electron in the conduction band and for the hole in the valence band, respectively. The Hamiltonian of the electron (hole) in the cylindrical coordinates reads, " ! # 2 2 2 h 1 o o 1 o o bj ¼ H qj þ þ 2 þ V ðqj Þ þ V ðzj Þ: 2mj qj oqj oqj qj ou2j oz2j ð2Þ
where the subscript j = e or h denotes the electron or hole. V(zj) and V(qj) are, respectively, the electron (hole) z-direction and in-plane confinement potential due to the band offset(Qj) in our ZB InGaN/GaN QD. ( 0; jzj j < h2 ; ð3Þ V ðzj Þ ¼ Qj ½Eg ðGaN Þ Eg ðInx Ga1x N Þ; jzj j > h2 ; ( 0; qj 6 R; V ðqj Þ ¼ Qj ½Eg ðIny Ga1y N Þ Eg ðInx Ga1x N Þ; qj > R: ð4Þ Here, we adopt the variational approach to estimate the ground-state exciton binding energy and interband emission energy. According to previous work [6,7,20], the following trial wave function is chosen in our calculations, 2 2 Uex ð! r e; ! r h Þ ¼ f ðqe Þhðze Þf ðqh Þhðzh Þeareh ebzeh ;
where the ground-state radial wave function f(qj) of the electron (hole) can be obtained using the Bessel function J0 and the modified Bessel function K0. The z-axis wave function h(zj) of the electron (hole) can obtained using linear combinations of analytical functions sin(n) and 2 2 cos(n)(dot), or exp(n)(barrier). r2eh ¼ ðxe xh Þ þ ðy e y h Þ and zeh = ze zh. a and b are variational parameters. The ground-state exciton wave function and energy in the ZB InGaN/GaN QD can be determined by minimizing the total energy, Eex ¼ min
In Ga
GaN
y
In Ga x
1-y
1-x
N
N
GaN
R
0
h 2
z h 2
Fig. 1. A diagram of a cylindrical ZB InxGa1xN/GaN QD with height h and radius R, surrounded by two large energy gap materials ZB InyGa1yN (y < x) in the radial direction and GaN in the z-direction.
ð5Þ
a;b
b ex jUex i hUex j H : hUex jUex i
ð6Þ
The exciton binding energy Eb and the interband optical transition energy Eph associated with the exciton can be defined as follows, Eb ¼ Ee þ Eh þ Eg Eex Eph ¼ Ee þ Eh þ Eg Eb ;
ð7Þ ð8Þ
where Ee(Eh) is the electron (hole) confinement energy in the ZB InGaN/GaN QD. In order to investigate the influence of the QD confinement potential on the electron-hole spatial separation, we
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
a
90 85 80
b
E (meV)
define the in-plane distance reh and the distance zeh in the z direction as 1 hUex ð! r e; ! r h Þ j q2eh j Uex ð! r e; ! r h Þi 2 reh ¼ ; ð9Þ r e; ! r h Þ j Uex ð! r e; ! r hÞ > Uex ð! !12 hUex ð! r e; ! r h Þ j ðze zh Þ2 j Uex ð! r e; ! r h Þi zeh ¼ : ð10Þ r e; ! r h Þ j Uex ð! r e; ! r hÞ > Uex ð!
155
75 70 65 60 55 50
3. Numerical results and discussion
b
45 2.45
Eph (eV)
2.40 2.35 2.30 2.25 2.20
c
2.2 2.0 1.8
zeh (nm)
We have calculated the ground-state exciton binding energy Eb and the interband emission energy Eph as functions of QD structural parameters, such as the dot height h, radius R and In content of the InGaN QD. For simplicity, we ignore the effect of the complicated valence band structure of InGaN and GaN and consider only the heavy-hole exciton states. The material parameters used in our calculations are taken from Refs. [9,15,21]. The ZB GaN and InxGa1xN layer band gap energy is Eg = 3.22 eV and Eg = [3.22(1 x) + 1.9x 1.4x(1 x)] eV, respectively. The electron me ¼ 0:19m0 and the hole mh ¼ 0:86m0 for ZB GaN effective mass. The electron me ¼ ½0:19ð1 xÞ þ 0:10xm0 and the hole mh ¼ ½0:86ð1 xÞ þ 0:84xm0 for ZB InxGa1xN effective mass. m0 is the free space electron mass. The band offset is assumed to be 70:30 [22]. In Fig. 2, the ground-state exciton binding energy Eb and the interband emission energy Eph are investigated as a function of height h for the ZB InGaN/GaN QD. Fig. 2a shows that the ground-state exciton binding energy Eb is reduced when the dot height h is increased. This is because that the relative distance zeh between the electron and hole is increased when h is increased [see Fig. 2c]. The Coulomb interaction between the electron and hole is reduced. The exciton binding energy is an indication of the electron-hole Coulomb interaction. So the Eb reduces when h increases. We can also see from Fig. 2b that the interband emission energy Eph is reduced if h is increased. The reason is that the confinement energies of the electron and hole are decreased when h is increased. Thus, the effective band gap of the ZB InGaN QD material is reduced when dot height h is increased. The ground-state exciton binding energy Eb and the interband emission energy Eph as a function of radius R of ZB InGaN/GaN QD are shown in Fig. 3. We can see from Fig. 3a that the ground-state exciton binding energy Eb is reduced when R is increased. This is mainly because that the electron-hole in-plane relative distance reh is increased when R is increased [see Fig. 3c], which reduces the Coulomb interaction between the electron and hole. Thus, the Eb is reduced when the R is increased. Fig. 3b shows the interband emission energy Eph is decreased monotonically if radius R is increased. This is due to the reduction of the the in-plane confinement energies of the electron and hole when R increases.
1.6 1.4 1.2 1.0 0.8
2
3
4
5 h(nm)
6
7
8
Fig. 2. The ground-state exciton binding energy Eb (a), the interband emission energy Eph (b) and the distance zeh (c) as a function of the height h of the ZB In0.5Ga0.5N/GaN QD with radius R = 6 nm, surrounded by the ZB In0.02Ga0.08N material in the radial direction.
In order to understand the influence of In content on exciton state and optical properties in the ZB InGaN/ GaN QD, we further investigate the ground-state exciton binding energy Eb and the interband emission energy Eph as a function of In content x of the InxGa1xN in Fig. 4. We can see from Fig. 4a that the ground-state exciton binding energy Eb increases if x increases. This is because the electron and hole wave functions are more strongly localized inside the QD. The Coulomb interaction between the electron and hole is enhanced when x is increased. Thus, the ground-state exciton binding energy Eb is increased when x is increased. Fig. 4b also shows that the interband emission energy Eph is decreased when x is increased. This
156
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
a
a
100
64 63 62
Eb (meV)
E b (meV)
90 80 70
61 60 59 58
60
57 56
50
b
55
b
2.40
c
2.6 E ph (eV)
Eph (eV)
2.36
2.32
2.4
2.28
2.2
2.24
2.0 0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
3.6 Fig. 4. The ground-state exciton binding energy Eb (a) and the interband emission energy Eph (b) as a function of the In content x of the ZB InxGa1xN/GaN QD with height h = 5 nm and radius R = 6 nm surrounded by the ZB In0.02Ga0.08N material in the radial direction.
3.3 reh (nm)
2.8
3.0 2.7 2.4 2.1 1.8 2
3
4
5 R(nm)
6
7
8
Fig. 3. The ground-state exciton binding energy Eb (a), the interband emission energy Eph (b) and the in-plane distance reh (c) as a function of the radius R of the ZB In0.5Ga0.5N/GaN QD with height h = 5 nm, surrounded by the ZB In0.02Ga0.08N material in the radial direction.
is due to the reduction of the effective energy gap of ZB InxGa1xN material when x is increased. This behavior is in agreement with the experimental measurements [9,11]. We can see from Figs. 2–4 that the ground-state exciton binding energy Eb is 50 meV around. The large exciton binding energy paves the way for an intense exciton emission at room-temperature(RT) because the RT thermal energy is 25 meV about. We can expect that some optoelectronics based on ZB InGaN/GaN QD, such as LDs or LEDs, can be revealed at RT or at even higher working temperature when the QD height h 6 8 nm (or radius R 6 8 nm) and it has the higher In content in the ZB
InGaN active layer. In order to study the quantum dots(QDs) interaction influence on the exciton binding energy, we further calculate the ground-state exciton binding energy Eb as a function of the interdot barrier layer thickness of the ZB InGaN/GaN coupled QDs in Fig. 6. The coupled QDs model is also given in Fig. 5. We can see from Fig. 6 that the ground-state exciton binding energy Eb has a minimum in the ZB InGaN/GaN coupled QDs. This non-monotonic behavior is mainly due to the influence of the interdot barrier layer on the distribution of the electron (hole) wave function. When the interdot barrier layer thickness Lb is small(Lb < 2.2 nm), it is easier for the electron(hole) wave function to penetrate into the interdot barrier layer. The relative distance zeh between the electron and hole is increased when Lb is increased. So the exciton binding energy decreases with increasing Lb. When the interdot barrier layer thickness Lb increases to 2.2 nm, the electron(hole) wave function has almost zero in the middle of the interdot barrier layer. Therefore the ground-state exciton binding energy as a function of the interdot barrier layer thickness is expected to go through a minimum. When the interdot barrier layer thickness Lb increases continuously, the coupling between the QDs is weak and the exciton behaves like in isolated QD. So the ground-state exciton binding energy Eb increases and tends
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
ground-state exciton binding energy Eb and the interband emission energy Eph are decreased monotonically when QD size is increased. The ground-state exciton binding energy Eb is increased when In content x of the InxGa1xN is increased. The interband emission energy Eph is decreased if In content x is increased. The QDs interaction has also been considered simply. The ground-state exciton binding energy has a minimum when the interdot barrier layer thickness increases. We hope that our numerical results can stimulate further experimental investigations of ZB InGaN/GaN QD, as well as device applications of group-III nitrides.
y In Ga y
In Ga x
1-x
1-y
N
N
x
R
In Ga
GaN
y
In Ga x
1-y
1-x
N
In Ga
GaN
y
N
In Ga x
1-y 1-x
N
157
GaN
N
z
Acknowledgement
z
z
0
1
z
z
2
3
Fig. 5. A diagram of a cylindrical ZB InxGa1xN/GaN coupled QDs with radius R, surrounded by two large energy gap materials InyGa1yN (y < x) in the radial direction and GaN 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 layer thickness Lb = z2 z1.
43.5
E (meV) b
43.0 42.5 42.0 41.5 41.0 40.5
1
2
3 L (nm) b
4
5
Fig. 6. The ground-state exciton binding energy Eb as a function of the interdot barrier layer thickness Lb in the ZB In0.5Ga0.5N/GaN coupled QDs with radius R = 6 nm, height h = 5 nm, surrounded by In0.02Ga0.98N material in the radial direction.
towards a constant value. We can see from Fig. 6 that the QDs interaction decreases the exciton binding energy. Thus, we can expect that some optoelectronics based on ZB InGaN/GaN single QD can be revealed at RT or at even higher working temperature of LDs or LEDs. 4. Conclusions In conclusion, we have investigated exciton states confined in the ZB InGaN/GaN QD by means of a variational approach, with the framework of effective-mass approximation. Numerical results clearly show that the groundstate exciton binding energy and the interband emission energy depend on QD structural parameters, such as height h, radius R and In content x of the InxGa1xN. The
This work was supported by the National Natural Science Foundation of China under Grant No. 60476047, by the Natural Science Foundation of the Education Bureau of Henan Province, China under Grant No. 2004140004. References [1] S. Nakamura, S.F. Chichibu, Introduction to Nitride Semiconductor Blue Lasers and Light Emitting Diodes, Taylor and Francis, London, 2000. [2] S.F. Chichibu, M. Sugiyama, T. Onuma, T. Kitamura, H. Nakanishi, T. Kuroda, A. Tackeuchi, T. Sota, Y. Ishida, H. Okumura, Appl. Phys. Lett. 79 (2001) 319. [3] S.F. Chichibu, M. Sugiyama, T. Kuroda, A. Tackeuchi, T. Sota, S.P. Denbaars, S. Nakamura, Y. Ishida, H. Okumura, Appl. Phys. Lett. 79 (2001) 3600. [4] P. Ruterana, S. Kret, A. Vivet, G. Maciejewski, P. Dluzewski, J. Appl. Phys. 91 (2002) 8979. [5] R.A. Taylora, J.W. Robinsona, J.H. Ricea, A. Jarjoura, J.D. Smitha, R.A. Oliverb, G.A.D. Briggsb, M.J. Kappersc, C.J. Humphreysc, Y. Arakawad, Physica E 21 (2004) 285. [6] J.J. Shi, Z.Z. Gan, J. Appl. Phys. 94 (2003) 407. [7] J.J. Shi, C.X. Xia, S.Y. Wei, Z.X. Liu, J. Appl. Phys. 97 (2005) 083705. [8] S.F. Chichibu, T. Onuma, T. Sota, S.P. Denbaars, S. Nakamura, T. Kitamura, Y. Ishida, H. Okumura, J. Appl. Phys. 93 (2003) 2051. [9] O. Husberg, A. Khartchenko, D.J. As, H. Vogelsang, T. Frey, D. Schikora, K. Lischka, O.C. Noriega, A. Tabata, J.R. Leite, Appl. Phys. Lett. 79 (2001) 1243. [10] O. Husberg, A. Khartchenko, H. Vogelsang, D.J. As, K. Lischka, O.C. Noriega, A. Tabata, L.M.R. Scolfaro, J.R. Leite, Physica E 13 (2002) 1090. [11] K. Lischka, J. Crystal Growth 231 (2001) 415. [12] L. Shunfeng, J. Schormann, A. Pawlis, D.J. As, K. Lischka, Microelectron. J. 36 (2005) 963. [13] D.G.P. Salazar, S.F. Li, F. Cerdeira, E.A. Meneses, J.R. Leite, L.M.R. Scolfaro, D.J. As, K. Lischka, J. Crystal Growth 284 (2005) 379. [14] S.P. Lepkowski, J.A. Majewski, Solid State Commun. 131 (2004) 763. [15] V. Lemos, E. Silveira, J.R. Leite, A. Tabata, L.M.R. Scolfaro, T. Frey, D.J. As, D. Schikora, K. Lichka, Phys. Rev. Lett. 17 (2000) 3666. [16] R. Charrour, M. Bouhassoune, M. Fliyou, D. Bria, A. Nougaoui, J. Phys.: Condens. Matter 12 (2000) 4817. [17] L. Shunfeng, J. Schormann, J.As. Donat, K. Lischka, Appl. Phys. Lett. 90 (2007) 071903. [18] A. Dussaigne, B. Damilano, N. Grandjean, J. Massies, J. Crystal Growth 251 (2003) 471.
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[19] V. Potin, E. Hahn, A. Rosenauer, D. Gerthsen, B. Kuhn, F. Scholz, A. Dussaigne, B. Damilano, N. Grandjean, J. Crystal Growth 262 (2004) 145. [20] S.L. Goff, B. Ste´be´, Phys. Rev. B 47 (1993) 1383.
[21] E.W.S. Caetano, V.N. Freire, G.A. Farias, J. Crystal Growth 246 (2002) 341. [22] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815.
Current Applied Physics 8 (2008) 153–158 www.elsevier.com/locate/cap www.kps.or.kr
Exciton states in zinc-blende InGaN/GaN quantum dot Congxin Xia b
a,*
, Fengchn Jiang b, Shuyi Wei
a
a Department of Physics, Henan Normal University, Xinxiang 453007, China Department of Technology and Physics, Zhengzhou University of Light Industry, Zhengzhou 450002, China
Received 14 February 2007; received in revised form 24 May 2007; accepted 4 July 2007 Available online 24 July 2007
Abstract Within the framework of effective-mass approximation, exciton states confined in zinc-blende(ZB) InGaN/GaN quantum dot(QD) are investigated by means of a variational approach, considering finite band offsets. The ground-state exciton binding energy and the interband emission energy are investigated as functions of QD structural parameters in detail. Numerical results show clearly that both the QD size and In content of InGaN have a significant influence on the exciton states and interband optical transitions in the ZB InGaN/GaN QD. Ó 2007 Elsevier B.V. All rights reserved. PACS: 71.35.y; 71.15.m Keywords: Exciton; InGaN; Quantum dot
1. Introduction In recent years, the wide-band-gap group-III nitrides based semiconductor heterostructures InGaN/GaN have attracted much attention due to conspicuous device applications in electronics and optoelectronics, such as highbrightness blue/green light-emitting diodes (LEDs) and laser diodes (LDs) [1]. They are usually grown in the thermodynamic stable configuration with hexagonal(wurtzite) crystal structure and in a metastable modification with cubic(zinc-blende)structure [2]. It is found that the electronic and optical properties of wurtzite(WZ) InGaN/ GaN heterostructures are affected by the strong built-in electric field induced by the piezoelectricity and spontaneous polarizations [3–7]. The magnitude of the strong built-in electric field is estimated to be in the order of MV/cm [3–7]. However, spontaneous polarization fields do not exist in the zinc-blende(ZB) nitrides due to the higher crystal symmetry, and piezoelectric fields are negligible due to the (0 0 1) growth direction of epitaxial layers *
Corresponding author. E-mail address: [email protected] (C. Xia).
1567-1739/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2007.07.001
[8–11]. Thus, one of the distinguished physical properties of ZB III-nitride quantum heterostructures is the absence of the built-in electric field. Since the band gap of ZB IIInitrides is lower than that of their hexagonal counterparts, 510 nm light emission can be obtained from ZB InGaN layers with reasonably lower In content. This may be an advantage due to the well-known difficulties in the growth of InGaN with high In content [8,12]. Therefore, ZB InGaN/GaN quantum heterostructures have attracted increasing attention because they are expected to have possible advantages for optoelectronic applications. Some experimental results show that many nanometer-scale InN-rich clusters are formed within the ZB InGaN active layer in the ZB InGaN/GaN quantum heterostructures [2,8–15]. These experiments indicate that the InN-rich clusters have cubic quantum dot(QD) or quantum disk shape. The ground-state transition energy as a function of QD size was calculated in a cubic QD [9]. In this paper, we model the InN-rich clusters using a cylindrical ZB InGaN/GaN QD. The reason for the choice of the cylindrical QD geometry lies in the fact that the structures of this type are relatively more convenient for fabricating devices than others [16]. On the other hand, it has been reported that In surface
154
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
segregation effects occur during the growth of InGaN ternary alloys [11,12,17–19]. The In segregation effects induce a blue-shift of the optical transitions and a decrease of the oscillator strength in InGaN/GaN quantum heterostructures [18]. However, recent studies show that some improved experiment methods can relieve the In segregation effects in ZB InGaN/GaN quantum heterostructures [17]. Therefore, we neglect the In surface segregation effects on exciton states and optical properties of ZB InGaN/GaN QD in this paper. To the knowledge of the authors, few theoretical investigation on exciton states in ZB InGaN/ GaN QD were published. In this paper, we investigate variationally the ground-state exciton binding energy and the interband emission energy of ZB InGaN/GaN QD. This paper is organized as follows. In Section 2, a theoretical model used to describe exciton states confined in ZB InGaN/GaN QD is outlined. Numerical results for the ground-state exciton binding energy and the interband emission energy are given and discussed in Section 3. Finally, we summarize the main conclusions obtained in this paper in Section 4. 2. Theoretical model For simplicity, let us now consider an exciton confined in the cylindrical ZB InGaN/GaN QD. The cylindrical QD is specified by its radius R and height h (see Fig. 1). Within the framework of effective-mass approximation, the Hamiltonian of the exciton can be written as, b ex ¼ H be þ H b h þ Eg H
e2 ; ! 4pe0ej r e ! r hj
ð1Þ
b h ) is the electron (hole) Hamiltonian, Eg is the b e(H where H band gap energy of the zinc-blende InxGa1xN material,
y In Ga y
In Ga x
1-x
1-y
N
N
x
R
! r h ) is the position vector of the electron (hole), e is r e (! the absolute value of the electron charge, e0 is the permittivity of free space, and e is the effective mean relative dielectric constant of the embedding material between the electron and hole. Throughout the present paper, we choose the conduction band bottom and the valence band top of the ZB InGaN layer at the origin z = 0 as the reference energy level for the electron in the conduction band and for the hole in the valence band, respectively. The Hamiltonian of the electron (hole) in the cylindrical coordinates reads, " ! # 2 2 2 h 1 o o 1 o o bj ¼ H qj þ þ 2 þ V ðqj Þ þ V ðzj Þ: 2mj qj oqj oqj qj ou2j oz2j ð2Þ
where the subscript j = e or h denotes the electron or hole. V(zj) and V(qj) are, respectively, the electron (hole) z-direction and in-plane confinement potential due to the band offset(Qj) in our ZB InGaN/GaN QD. ( 0; jzj j < h2 ; ð3Þ V ðzj Þ ¼ Qj ½Eg ðGaN Þ Eg ðInx Ga1x N Þ; jzj j > h2 ; ( 0; qj 6 R; V ðqj Þ ¼ Qj ½Eg ðIny Ga1y N Þ Eg ðInx Ga1x N Þ; qj > R: ð4Þ Here, we adopt the variational approach to estimate the ground-state exciton binding energy and interband emission energy. According to previous work [6,7,20], the following trial wave function is chosen in our calculations, 2 2 Uex ð! r e; ! r h Þ ¼ f ðqe Þhðze Þf ðqh Þhðzh Þeareh ebzeh ;
where the ground-state radial wave function f(qj) of the electron (hole) can be obtained using the Bessel function J0 and the modified Bessel function K0. The z-axis wave function h(zj) of the electron (hole) can obtained using linear combinations of analytical functions sin(n) and 2 2 cos(n)(dot), or exp(n)(barrier). r2eh ¼ ðxe xh Þ þ ðy e y h Þ and zeh = ze zh. a and b are variational parameters. The ground-state exciton wave function and energy in the ZB InGaN/GaN QD can be determined by minimizing the total energy, Eex ¼ min
In Ga
GaN
y
In Ga x
1-y
1-x
N
N
GaN
R
0
h 2
z h 2
Fig. 1. A diagram of a cylindrical ZB InxGa1xN/GaN QD with height h and radius R, surrounded by two large energy gap materials ZB InyGa1yN (y < x) in the radial direction and GaN in the z-direction.
ð5Þ
a;b
b ex jUex i hUex j H : hUex jUex i
ð6Þ
The exciton binding energy Eb and the interband optical transition energy Eph associated with the exciton can be defined as follows, Eb ¼ Ee þ Eh þ Eg Eex Eph ¼ Ee þ Eh þ Eg Eb ;
ð7Þ ð8Þ
where Ee(Eh) is the electron (hole) confinement energy in the ZB InGaN/GaN QD. In order to investigate the influence of the QD confinement potential on the electron-hole spatial separation, we
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
a
90 85 80
b
E (meV)
define the in-plane distance reh and the distance zeh in the z direction as 1 hUex ð! r e; ! r h Þ j q2eh j Uex ð! r e; ! r h Þi 2 reh ¼ ; ð9Þ r e; ! r h Þ j Uex ð! r e; ! r hÞ > Uex ð! !12 hUex ð! r e; ! r h Þ j ðze zh Þ2 j Uex ð! r e; ! r h Þi zeh ¼ : ð10Þ r e; ! r h Þ j Uex ð! r e; ! r hÞ > Uex ð!
155
75 70 65 60 55 50
3. Numerical results and discussion
b
45 2.45
Eph (eV)
2.40 2.35 2.30 2.25 2.20
c
2.2 2.0 1.8
zeh (nm)
We have calculated the ground-state exciton binding energy Eb and the interband emission energy Eph as functions of QD structural parameters, such as the dot height h, radius R and In content of the InGaN QD. For simplicity, we ignore the effect of the complicated valence band structure of InGaN and GaN and consider only the heavy-hole exciton states. The material parameters used in our calculations are taken from Refs. [9,15,21]. The ZB GaN and InxGa1xN layer band gap energy is Eg = 3.22 eV and Eg = [3.22(1 x) + 1.9x 1.4x(1 x)] eV, respectively. The electron me ¼ 0:19m0 and the hole mh ¼ 0:86m0 for ZB GaN effective mass. The electron me ¼ ½0:19ð1 xÞ þ 0:10xm0 and the hole mh ¼ ½0:86ð1 xÞ þ 0:84xm0 for ZB InxGa1xN effective mass. m0 is the free space electron mass. The band offset is assumed to be 70:30 [22]. In Fig. 2, the ground-state exciton binding energy Eb and the interband emission energy Eph are investigated as a function of height h for the ZB InGaN/GaN QD. Fig. 2a shows that the ground-state exciton binding energy Eb is reduced when the dot height h is increased. This is because that the relative distance zeh between the electron and hole is increased when h is increased [see Fig. 2c]. The Coulomb interaction between the electron and hole is reduced. The exciton binding energy is an indication of the electron-hole Coulomb interaction. So the Eb reduces when h increases. We can also see from Fig. 2b that the interband emission energy Eph is reduced if h is increased. The reason is that the confinement energies of the electron and hole are decreased when h is increased. Thus, the effective band gap of the ZB InGaN QD material is reduced when dot height h is increased. The ground-state exciton binding energy Eb and the interband emission energy Eph as a function of radius R of ZB InGaN/GaN QD are shown in Fig. 3. We can see from Fig. 3a that the ground-state exciton binding energy Eb is reduced when R is increased. This is mainly because that the electron-hole in-plane relative distance reh is increased when R is increased [see Fig. 3c], which reduces the Coulomb interaction between the electron and hole. Thus, the Eb is reduced when the R is increased. Fig. 3b shows the interband emission energy Eph is decreased monotonically if radius R is increased. This is due to the reduction of the the in-plane confinement energies of the electron and hole when R increases.
1.6 1.4 1.2 1.0 0.8
2
3
4
5 h(nm)
6
7
8
Fig. 2. The ground-state exciton binding energy Eb (a), the interband emission energy Eph (b) and the distance zeh (c) as a function of the height h of the ZB In0.5Ga0.5N/GaN QD with radius R = 6 nm, surrounded by the ZB In0.02Ga0.08N material in the radial direction.
In order to understand the influence of In content on exciton state and optical properties in the ZB InGaN/ GaN QD, we further investigate the ground-state exciton binding energy Eb and the interband emission energy Eph as a function of In content x of the InxGa1xN in Fig. 4. We can see from Fig. 4a that the ground-state exciton binding energy Eb increases if x increases. This is because the electron and hole wave functions are more strongly localized inside the QD. The Coulomb interaction between the electron and hole is enhanced when x is increased. Thus, the ground-state exciton binding energy Eb is increased when x is increased. Fig. 4b also shows that the interband emission energy Eph is decreased when x is increased. This
156
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
a
a
100
64 63 62
Eb (meV)
E b (meV)
90 80 70
61 60 59 58
60
57 56
50
b
55
b
2.40
c
2.6 E ph (eV)
Eph (eV)
2.36
2.32
2.4
2.28
2.2
2.24
2.0 0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
3.6 Fig. 4. The ground-state exciton binding energy Eb (a) and the interband emission energy Eph (b) as a function of the In content x of the ZB InxGa1xN/GaN QD with height h = 5 nm and radius R = 6 nm surrounded by the ZB In0.02Ga0.08N material in the radial direction.
3.3 reh (nm)
2.8
3.0 2.7 2.4 2.1 1.8 2
3
4
5 R(nm)
6
7
8
Fig. 3. The ground-state exciton binding energy Eb (a), the interband emission energy Eph (b) and the in-plane distance reh (c) as a function of the radius R of the ZB In0.5Ga0.5N/GaN QD with height h = 5 nm, surrounded by the ZB In0.02Ga0.08N material in the radial direction.
is due to the reduction of the effective energy gap of ZB InxGa1xN material when x is increased. This behavior is in agreement with the experimental measurements [9,11]. We can see from Figs. 2–4 that the ground-state exciton binding energy Eb is 50 meV around. The large exciton binding energy paves the way for an intense exciton emission at room-temperature(RT) because the RT thermal energy is 25 meV about. We can expect that some optoelectronics based on ZB InGaN/GaN QD, such as LDs or LEDs, can be revealed at RT or at even higher working temperature when the QD height h 6 8 nm (or radius R 6 8 nm) and it has the higher In content in the ZB
InGaN active layer. In order to study the quantum dots(QDs) interaction influence on the exciton binding energy, we further calculate the ground-state exciton binding energy Eb as a function of the interdot barrier layer thickness of the ZB InGaN/GaN coupled QDs in Fig. 6. The coupled QDs model is also given in Fig. 5. We can see from Fig. 6 that the ground-state exciton binding energy Eb has a minimum in the ZB InGaN/GaN coupled QDs. This non-monotonic behavior is mainly due to the influence of the interdot barrier layer on the distribution of the electron (hole) wave function. When the interdot barrier layer thickness Lb is small(Lb < 2.2 nm), it is easier for the electron(hole) wave function to penetrate into the interdot barrier layer. The relative distance zeh between the electron and hole is increased when Lb is increased. So the exciton binding energy decreases with increasing Lb. When the interdot barrier layer thickness Lb increases to 2.2 nm, the electron(hole) wave function has almost zero in the middle of the interdot barrier layer. Therefore the ground-state exciton binding energy as a function of the interdot barrier layer thickness is expected to go through a minimum. When the interdot barrier layer thickness Lb increases continuously, the coupling between the QDs is weak and the exciton behaves like in isolated QD. So the ground-state exciton binding energy Eb increases and tends
C. Xia et al. / Current Applied Physics 8 (2008) 153–158
ground-state exciton binding energy Eb and the interband emission energy Eph are decreased monotonically when QD size is increased. The ground-state exciton binding energy Eb is increased when In content x of the InxGa1xN is increased. The interband emission energy Eph is decreased if In content x is increased. The QDs interaction has also been considered simply. The ground-state exciton binding energy has a minimum when the interdot barrier layer thickness increases. We hope that our numerical results can stimulate further experimental investigations of ZB InGaN/GaN QD, as well as device applications of group-III nitrides.
y In Ga y
In Ga x
1-x
1-y
N
N
x
R
In Ga
GaN
y
In Ga x
1-y
1-x
N
In Ga
GaN
y
N
In Ga x
1-y 1-x
N
157
GaN
N
z
Acknowledgement
z
z
0
1
z
z
2
3
Fig. 5. A diagram of a cylindrical ZB InxGa1xN/GaN coupled QDs with radius R, surrounded by two large energy gap materials InyGa1yN (y < x) in the radial direction and GaN 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 layer thickness Lb = z2 z1.
43.5
E (meV) b
43.0 42.5 42.0 41.5 41.0 40.5
1
2
3 L (nm) b
4
5
Fig. 6. The ground-state exciton binding energy Eb as a function of the interdot barrier layer thickness Lb in the ZB In0.5Ga0.5N/GaN coupled QDs with radius R = 6 nm, height h = 5 nm, surrounded by In0.02Ga0.98N material in the radial direction.
towards a constant value. We can see from Fig. 6 that the QDs interaction decreases the exciton binding energy. Thus, we can expect that some optoelectronics based on ZB InGaN/GaN single QD can be revealed at RT or at even higher working temperature of LDs or LEDs. 4. Conclusions In conclusion, we have investigated exciton states confined in the ZB InGaN/GaN QD by means of a variational approach, with the framework of effective-mass approximation. Numerical results clearly show that the groundstate exciton binding energy and the interband emission energy depend on QD structural parameters, such as height h, radius R and In content x of the InxGa1xN. The
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