GaN multiple quantum wells

GaN multiple quantum wells

ARTICLE IN PRESS Microelectronics Journal 40 (2009) 342–345 Contents lists available at ScienceDirect Microelectronics Journal journal homepage: www...

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ARTICLE IN PRESS Microelectronics Journal 40 (2009) 342–345

Contents lists available at ScienceDirect

Microelectronics Journal journal homepage: www.elsevier.com/locate/mejo

Microstructure analysis in strained-InGaN/GaN multiple quantum wells Huaping Lei a,b, Jun Chen c,, Xunya Jiang b, Ge´rard Nouet a a b c

´chal Juin, 14050 Caen, France Centre de Recherche sur les Ions, les Materiaux et la Photonique (CIMAP), 6 Boulevard Mare State Key Laboratory of Functional Materials for Informatics, SIMIT, CAS, Changning Road 865, Shanghai 200050, China ´te´s des Mate´riaux Nouveaux, Universite´ de Caen, IUT d’Alenc- on, 61250 Damigny, France Laboratoire de Recherche sur les Proprie

a r t i c l e in f o

a b s t r a c t

Available online 19 September 2008

The barrier thickness effect on the energy and microstructure properties of InGaN/GaN multiple quantum wells is investigated with Stillinger–Weber potential. The calculation indicates that the energy of a quantum well increases as the GaN barrier thickness rises, and that Ga–N and In–N bonds are shrunk with respect to those of random InGaN alloy. Moreover, a critical value of the barrier thickness exits. If the barrier thickness exceeds the critical value, the bond length of Ga–N in quantum wells reduces as a function of indium concentration. This singular behavior of Ga–N bond is analyzed with a force balance model. & 2008 Elsevier Ltd. All rights reserved.

PACS: 61.43.Bn 61.46.Bc 68.65.Fg Keywords: Computational simulation Multiple quantum wells InGaN Strain

1. Introduction

the properties of the tetrahedral semiconductors [8]

The narrow band gap of InN, 0.7 eV, has given a new impact to the wavelength range of the III-nitrides [1,2]. Thus, the InGaN heterostructures are very attractive for light-emitting diodes, laser diodes and solar cell devices working within a wide spectrum range. In reality, the light-emission mechanism of InGaN/GaN heterostructures is still controversial: the formation of In-rich clusters in InGaN quantum wells (QWs) due to Indium segregation should result in the localization of the excitons for the radiative recombination [3,4]. But recent experiment based on the three-dimensional atomic-probe technique indicates that no In-rich clusters exist in InGaN QWs [5]. The controversy about the existence of these clusters is still relevant by involving the effect of the electron beam damages in transmission electron microscopy (TEM) experiments [6,7] or equilibrium phase separation. Moreover, the large biaxial strain due to the lattice mismatch (10.7%) between GaN and InN remains a key parameter in InGaN/GaN heterostructures. Actually, there are still few works concerning the strain effect on the stability of wurtzite InGaN/GaN MQWs. In this paper, Stillinger–Weber (SW) potential is applied to investigate the energy and microstructure properties of InGaN/GaN MQWs with different barrier thicknesses.

Fð1; . . . ; NÞ ¼

2. Simulation method and model SW potential only considers the two- and three-body interactions of the nearest neighbor atoms and it is suitable to describe  Corresponding author.

E-mail address: [email protected] (J. Chen). 0026-2692/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.07.068

N X

f2 ði; jÞ þ

i;jðiojÞ

N X

f3 ði; j; kÞ.

(1)

i;j;kðiojokÞ

The two-body interaction term is

f2 ði; jÞ ¼ f 2 ðrij =sÞ,

(2)

with ( f 2 ðrÞ ¼

AðB=r 4  1Þ exp½1=ðr  aÞ;

roa;

0;

r4a

(3)

and the three-body interaction term is *

*

*

f3 ði; j; kÞ ¼ f 3 ð r i =s; r j =s; r k =sÞ; *

*

*

f 3 ð r i ; r j ; r k Þ ¼ hðr ij ; r jk ; yijk Þ þ hðr ji ; r ik ; yjik Þ þ hðr jk ; r ki ; yjki Þ,

(4) (5)

hijk ðr ij ; r jk ; yijk Þ ¼ l exp½g=ðr ij  aÞ þ g=ðr jk  aÞðcos yijk þ 1=3Þ2 .

(6)

e and s are energy and length units, respectively. a is the cut-off * * distance. y(i, j, k) is the angle formed by the r i;j and r j;k vectors. A, B, l and g are the bond-strength factors. The modified SW potential parameters for Ga–N and In–N are presented in Table 1. In order to validate the SW parameters, the crystallographic parameters and elastic constants of GaN and InN are calculated and compared with the experimental and first-principle calculation values [9–11]. The values calculated with SW potential are in fair agreement with the experimental and first-principle calculation data (Table 2). The periodic boundary condition and Verlet algorithm are used. The size of the supercell along [1 2¯ 1 0] and

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343

Table 1 SW potential parameters for Ga–N and In–N

Ga–N In–N

A

B

l

a

g

e (eV)

s (A˚)

7.718 7.755

0.694 0.699

28.5 18.5

1.8 1.8

1.2 1.2

2.265 1.993

1.700 1.879

Table 2 Crystallographic parameters (A˚) and elastic constants (GPa)

a

Bond length

Lattice constant

C11

C12

C13

C33

C44

B

GaN

Expt. Present

1.949 1.949

3.189 3.183

390 394

145 133

106 109

398 419

105 106

210 212

InN

Cal. b Present

2.156 2.156

3.538 3.521

220 206

120 109

91 101

249 214

36 41

125 139

a b

Ref. [10]. Ref. [11].

Fig. 1. (Color online) Schematic diagram of InGaN/GaN MQWs along [1 2¯ 1 0] direction (red, white and blue circles represent Ga, In and nitrogen atoms, respectively).

[1 0 1¯ 0] is 40a  24O3a. The width of InGaN QWs is typically taken as 4 monolayers (MLs) and In atoms randomly distribute inside (Fig. 1). In the first stage of this analysis, no GaN barriers are considered, and the whole system is equivalent to a pure random InGaN alloy. Secondly, the heterostructure is formed with the insertion of 8, 12, 16 and 20 MLs GaN (2 MLs ¼ 1cGaN) as barriers regularly spaced along the c-axis. The atom number in the supercell is in the order of magnitude 104.

3. Results and discussion As shown in Fig. 2, the deformation energy of QWs in InGaN/ GaN heterostructure is calculated as a function of the barrier thickness for different Indium concentration. The formation of random InGaN alloy (0 MLs) results in the energy increase from 1.4 to 3.1 meV/A˚3 when Indium concentration changes from 10% to 40%. With the effect of the barriers, the deformation energy of InGaN QWs in InGaN/GaN heterostructure is higher than that of the alloy with the same Indium concentration. For example, when the GaN barrier is 8 MLs thick, the energy is higher 0.007 meV/A˚3 (0.5%) for In0.1Ga0.9N QWs and 1.671 meV/A˚3 (53%) for In0.4Ga0.6N QWs than that of the InGaN alloys with the corresponding Indium

Fig. 2. (Color online) Deformation energy of InGaN QWs with different Indium concentration as a function of barrier thickness. 0 MLs indicates the random InGaN alloy.

concentration. Meanwhile, the energy of QWs depends on the barrier thickness. The thicker barrier results in the higher deformation energy of QWs. But the barrier thickness effect is less pronounced for the low Indium concentration. As the barrier thickness increases from 0 to 20 MLs, the deformation energy increases 0.073 meV/A˚3 for In0.1Ga0.9N QWs but 2.578 meV/A˚3 for In0.4Ga0.6N QWs. The barrier thickness effect on the microstructure properties of QWs is investigated. As a function of Indium concentration, Ga–N and In–N bond lengths of MQWs with different barrier thicknesses are shown in Figs. 3(a) and (b), respectively. The inset within Fig. 3(a) is the variation of Ga–N bond length in the barrier region. Compared with those of the random InGaN alloy (dash lines in Figs. 3(a) and (b), respectively), Ga–N and In–N bonds are compressed in MQWs. And Ga–N bonds in the barriers are different from those in QW region. Taking In0.3Ga0.7N (2cInGaN)/ GaN(4cGaN) MQWs for example, the bond length of Ga–N in the barriers and QWs, respectively, reduces to 0.46% and 0.79%, and In–N bond length reduces to 0.89% compared with those of In0.3Ga0.7N alloy (1.967 A˚ for Ga–N and 2.117 A˚ for In–N). The compression of In–N bond in the strained InGaN has been validated through EXAFS experiments [12] (Fig. 3(b)). And both of Ga–N and In–N bonds are compressed more by the thicker barriers. Furthermore, when the barrier is thin, Ga–N and In–N bond lengths increase normally as a function of Indium concentration like those of InGaN alloy. However, as shown in Fig. 3(a), there is a critical value of the barrier thickness, above which the bond length of Ga–N in QWs behaves singularly and inversely decreases as Indium concentration increases. For the 4 MLs thick InGaN QWs, the critical barrier thickness is estimated to 6cGaN. Meanwhile, In–N bonds keep increasing as a function of Indium concentration. We change the width of InGaN QWs to 2 MLs and 6 MLs. The results indicate that the abnormality of Ga–N bonds still occurs, but the critical barrier thickness shifts to 3cGaN for 2 MLs and 9cGaN for 6 MLs thick InGaN QWs. The strain, defined as e(x) ¼ (a(x)a0(x))/a0(x) where a(x) and a0(x) are, respectively, the lattice constant of QWs and the alloy with the same Indium composition is introduced to describe the abnormality of Ga–N bonds. As shown in Fig. 4, independent of the width of the QWs, there is a critical value of the strain, over which Ga–N bonds decreases as a function of Indium concentration. Thus, the strain would be responsible for the abnormality of Ga–N bonds. The

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Fig. 3. (Color online) Variation of Ga–N and In–N bond lengths in InGaN/GaN MQWs with different barrier thicknesses as the function of Indium concentration. (a) Ga–N bonds. The inset shows the variation of Ga–N bond length in the barrier region and (b) In–N bonds.

Fig. 4. (Color online) Critical strain for the abnormality of Ga–N bonds. The negative sign means the compressive strain.

critical strain divides e(x) into the normality and abnormality regions for Ga–N bonds. The abnormal behavior of Ga–N bonds could be analyzed based on the force balance model. In this model, only the radial interaction is considered, namely the two-body term of SW potential f2(r). Taking 4 MLs thick InGaN QWs, for example, f2(r) of Ga–N and In–N bonds are shown in Fig. 5(a). The variation range of Ga–N and In–N bond lengths is indicated by the black and red shadow regions, respectively. The analysis to the force on * * Gallium and Indium atoms (F ¼ qf2 ðrÞ=q r ) is shown in Fig. 5(b). The arrows direct the increase of Indium concentration. When the barrier is thin (4cGaN), even though Ga–N and In–N bonds shrink with respect to InGaN alloy (as shown in Figs. 3(a) and (b)), In atoms provide enough space to relax Ga–N bonds since In atoms has a larger radius and In–N bond is softer than Ga–N (Table 2), which stretches Ga–N bond to be larger than the equivalent value r0(GaN). The force on Gallium is negative and Ga–N bonds are in the attractive state (block circles in Fig. 5(b)). With the increase of barrier thickness, InGaN QWs are compressed much more and Indium atoms cannot provide enough space for Ga–N bonds so that Ga–N bonds shrink. When the barrier is up to 6cGaN thick, the distance between Ga and N atoms decreases to r0(GaN) and the force is about zero (red squares in Fig. 5(b)). Ga–N

Fig. 5. (Color online) Force balance model. (a) Two-body term of Stillinger-Weber potential for Ga–N and In–N bonds. The black and red shadow regions indicate the change range of Ga–N and In–N bonds and (b) force on Gallium and Indium atoms in InGaN QWs with the different barrier thicknesses and Indium concentration.

bond length changes little as a function of Indium concentration. Compressed by the thicker barriers, Ga–N bonds shrink to be smaller than r0(GaN) and enter the repulsive state (green uptriangles and blue down-triangles in Fig. 5(b)). The additional In atoms enhance the compression since the force on In atoms is larger than that on Ga atoms as shown in Fig. 5(b). The Ga–N bond

ARTICLE IN PRESS H. Lei et al. / Microelectronics Journal 40 (2009) 342–345

length reduces as Indium concentration increases. On the contrary, In–N bond length is always smaller than the equivalent value r0(InN) and In–N bonds are in the repulsive state for any barrier thicknesses. Substituting Ga atoms, the additional Indium relaxes In–N bonds. So, In–N bond length increases as a function of Indium concentration. Therefore, the occurrence of the abnormality of Ga–N bonds could be attributed to the change from attractive to repulsive state.

4. Conclusion The barrier thickness effect on the deformation energy and microstructure properties of InGaN/GaN MQWs is investigated through Stillinger–Weber potential. The results indicate that the strain from the mismatch of the heterostructures increases the deformation energy of QWs, and that both of Ga–N and In–N bonds in the strained-InGaN QWs are compressed with respect to the alloy. An abnormal behavior of Ga–N bonds has been found and analyzed based on the force balance model. The change from the attractive to repulsive state would result in the abnormality of Ga–N bonds with the effect of the strain.

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Acknowledgment Acknowledge to the support of MRTN-CT-2004-005583 (PARSEM) and Eiffel grant. References [1] J. Wu, W. Walukiewicz, K.M. Yu, J.W. Ager III., E.E. Haller, H. Lu, W.J. Schaff, Y. Saito, Y. Nanishi, Appl. Phys. Lett. 80 (2002) 3967. [2] T. Matsuoka, H. Okamoto, M. Nakao, H. Harima, E. Kurimoto, Appl. Phys. Lett. 81 (2002) 1246. [3] D.M. Graham, A. Soltani-Vala, P. Dawson, M.J. Godfrey, T.M. Smeeton, J.S. Barnard, M.J. Kappers, C.J. Humphreys, E.J. Thrush, J. Appl. Phys. 97 (2005) 103508. [4] I.H. Ho, G.B. Stringfellow, Appl. Phys. Lett. 69 (1996) 2701. [5] M.J. Galtrey, R.A. Oliver, M.J. Kappers, C.J. Humphreys, D.J. Stokes, P.H. Clifton, A. Cerezo, Appl. Phys. Lett. 90 (2007) 061903. [6] T.M. Smeeton, M.J. Kappers, J.S. Barnard, M.E. Vickers, C.J. Humphreys, Appl. Phys. Lett. 83 (2003) 5419. [7] T.M. Smeeton, C.J. Humphreys, J.S. Barnard, M.J. Kappers, J. Mater. Sci. 41 (2006) 2729. [8] F.H. Stillinger, T.A. Weber, Phys. Rev. B 38 (1985) 1537. [9] H.P. Lei, X.Y. Jiang, J. Chen, P. Ruterana, G. Nouet, Appl. Phys. Lett. 90 (2007) 111901. [10] A. Polian, J. Appl. Phys. 79 (1996) 3343. [11] K. Kim, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 53 (1996) 16310. [12] K.P. O’Donnell, J.F.W. Mosselmans, R.W. Martin, S. Pereira, M.E. White, J. Phys.: Condens. Matter 13 (2001) 6977.