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ARTICLE IN PRESS Optik xxx (2012) xxx–xxx

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Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots H. Kaviani a , A. Asgari a,b,∗ a b

Research Institute for Applied Physics, University of Tabriz, Tabriz 51665-163, Iran School of Electrical, Electronic and Computer Engineering, The University of Western Australia, Crawley, WA 6009, Australia

a r t i c l e

i n f o

Article history: Received 5 September 2011 Accepted 11 January 2012 Keywords: Nonlinear optic InGaN QDs Self-focusing effects

a b s t r a c t The third-order nonlinear optical properties in wurtzite InGaN/GaN pyramid and truncated-pyramid quantum dots are studied, and the oscillator strength, third-order nonlinear optical susceptibility and self-focusing effects are analyzed theoretically taken into account the strong built-in electric ﬁeld effect due to the piezoelectric and spontaneous polarization in nitride materials. The numerical results clearly show that the quantum dot (QD) size of InGaN/GaN have a signiﬁcant inﬂuence on the nonlinear optical properties of wurtzite InGaN/GaN quantum dots. Furthermore, the self-focusing effect increases with decrease in size of QDs. © 2012 Elsevier GmbH. All rights reserved.

1. Introduction In recent years, III-nitride alloys have been the wide research subjects due to practical applications in the ﬁeld of optoelectronic devices, high thermal conductivity, high electron-saturated drift velocity and small dielectric constant [1,2]. These materials have direct wide band gaps from 0.7 eV to 3.42 eV at room temperature, therefore, they are so useful in blue light emitting diodes industrial [3–5]. III–V nitride materials are founded in two different type structures: (i) wurtzite (WZ) [6] and, (ii) zinc-blend (ZB) [7]. In wurtzite structures, electronic states and optical properties are highly affected by the built-in electric ﬁeld due to spontaneous (Ps ) and piezoelectric (Pz ) polarizations. The magnitude of the built-in electric ﬁeld is estimated to be in the order of MV/cm. These properties do not exist in ZB structures, because they have high crystal symmetry [8–10]. Among III-nitride quantum dots (QDs) are already acknowledged as quantum nanostructures with high potentials in optoelectronic ﬁeld; for instance, in light emitting diodes (LEDs), laser diodes (LDs), optical memories and single electron transistors [11–13]. In this nanometer scale semiconductors, the charge carriers (electrons and holes) are conﬁned in all three dimensions [14,15] and due to the modiﬁcation in the density of states, these quantum nanostructures are expected to exhibit enhanced optical nonlinearities and enhanced electro-optic effects. Indeed many of these optical nonlinear properties associated with intersubband transitions those due to large dipole transition and very large oscil-

∗ Corresponding author at: Research Institute for Applied Physics, University of Tabriz, Tabriz 51665-163, Iran. Tel.: +98 411 339 3005; fax: +98 411 334 7050. E-mail address: [email protected] (A. Asgari).

lator strength, these optical intraband nonlinear are large too. As the intrasubband dipole length extend over the QDs which are in nanometer ranges [16,17]. Compare to bulk semiconductors, QD semiconductors have larger third-order nonlinear susceptibility [18,19]. One of the most interesting effect associated with third order susceptibility (3 ) is the self-focusing. It is typical type of nonlinear wave propagation that depends critically on the transverse proﬁle of the beam. Self-focusing of the light is the process in which an intense beam of light modiﬁes the optical properties of a material medium in such a manner that the beam is caused to come to a focus within or outside the material [20]. In this paper in order to understand the optical nonlinearity in QDs and its application as self-focusing effects, ﬁrst we assume two different shapes of wurtzite Inx Ga1−x N QDs (pyramid and truncated pyramid), then we calculated the Schrödinger equation in presence of the built-in polarization electric ﬁeld in the framework of the envelope function, and the effective mass theory. The third-order nonlinear susceptibility of the taken QDs as function of DQ’s size is investigated. Finally, the effects of QD size and shape on selffocusing effects have been analyzed.

2. Theoretical model To model the device, two pyramid and truncated-pyramid shaped InGaN wurtzite QDs embedded in GaN material are assumed. The proposed structure has been shown in Fig. 1. In order to study the electronic structures, different methods have been experienced [21–24]. The single band method is used

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Please cite this article in press as: H. Kaviani, A. Asgari, Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2012.01.012

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Fig. 1. The proposed pyramid (a) and truncated-pyramid (b) shaped InGaN QD within GaN material.

in this study. In the framework of the envelope function, and the effective mass theory, the Hamiltonian can be written as [22]: 2

H=

−¯h 1 ∇ ∗ ∇ + V (x, y, z) 2 m (x, y, z)

(1)

In which m* is the electron effective mass and is given by:

∗

m (x, y, z) = and

V (x, y, z) =

m∗InGaN m∗GaN

in QD , in barrier

0 EC (EV )

(2)

inside QD , else

(3)

where EC (EV ) is the conduction and valence bands discontinuity [25]:

EC = 0.7 m × 6.13 + (1 − m) × 3.42 − m(1 − m) − Eg0

eV,

(4)

where m notiﬁes Al molar fraction in the Inm Ga1−m NQD. As the most optoelectronic system needs an applied electric ﬁeld to operate and III-nitride based structures also has a strong built-in electric ﬁeld, one has to take into account the total ﬁelds effect in the Hamiltonian: 2

H=

→→ −¯h 1 ∇ ∗ ∇ + v(x, y, z) + e F r 2 m (x, y, z)

(5)

→

where F denotes both the external and built-in electric ﬁelds. It should be mentioned that III-nitrides in the wurtzite phase have a strong spontaneous and piezoelectric polarization. The abrupt variation of the polarization at the interfaces gives rise to large polarization sheet charges which creates the built-in electric ﬁeld. Therefore, the optical properties of wurtzite AlGaN/GaN QDs are affected by the 3D conﬁnement electrons and the strong built-in electric ﬁeld. This causes the simulation of the systems extremely challenging task. The built-in electric ﬁeld which applied in the equations is [26]: Fd =

br − P d ) Lbr (Ptot tot ε0 (Ld εbr + Lbr εd )

Fig. 2. A schematic QD nonlinear medium for self-focusing effect.

(6)

Fig. 3. The wavefunctions of the ground state and ﬁrst excited state in InGaN pyramid, and truncated pyramid QDs. The wavefunctions are m−2 unit.

where εbr (d) is the relative dielectric constant of the barrier (dot), br/d Ptot is the total polarization and Lbr/d is the width of the barrier and height of the dot. br/d

br/d

br/d

Ptot = Ppiezo + Psp

(7)

The piezoelectric polarization includes: one part induced by the lattice mismatch (ms), and the other caused by br/d br/d br/d br/d thermal strain (ts): Ppiezo = Pms + Pts , where Pms = 2(e31 −

d = −3.2 × 10−4 c/m2 [26]. e e33 c13 /c33 ) (a − a0 )/a and Pts 31 and e33 are the piezoelectric coefﬁcients, c31 and c33 are elastic constants, and ‘a’ is the lattice constant. All other material parameters can be found in [27]. To solve the Schrödinger equation, assuming that the wave functions are expanded in terms of the normalized plane waves [22]: nx,ny,nz (x, y, z)

=

1

Lx Ly Lz

anx,ny,nz exp i(knx x + kny y + knz z)

nx,ny,nz

(8) where knx = kx + nx Kx , kny = ky + ny Ky , knz = kz + nz Kzs and Kx = 2/Lx , Ky = 2/Ly , Kz = 2/Lz . Lx , Ly and Lz are lengths of the unit cell along the x, y and z directions. nx , ny and nz are the number of plane waves along the x, y and z directions respectively. As reported in [28], the attraction of the normalized plane wave approach is the fact that there is no need to explicitly match the wave function, across the boundary of the barrier and QD. Hence this method is easy to apply to an arbitrary conﬁning potential problem. As more plane waves are taken, more accurate results are achieved. We used thirteen normalized plane waves in each direction to form the Hamiltonian matrix (i.e. nx , ny and nz from −6 to 6) and we formed 2197 × 2197 matrix. It was found that using more than 13 normalized plane waves in each direction takes signiﬁcantly long computational time and only about 1 meV more accurate energy eigenvalues. By substituting the Eq. (8) in Schrödinger equation, eigenfunctions and eigenvalues are calculated. To study the nonlinear optical absorption coefﬁcient and refractive index, one has to know the third-order nonlinear susceptibility

Please cite this article in press as: H. Kaviani, A. Asgari, Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2012.01.012

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190

2.44

Oscillator Strenght

Transition Wavelenght(μm )

H. Kaviani, A. Asgari / Optik xxx (2012) xxx–xxx

2.40 2.36 2.32

P-QD (h=4 nm)

180

170

160

P-QD (h= 4nm) TP-QD (h=3 nm)

TP-QD (h=3 nm)

2.28 6.5

7.0

7.5

8.0

8.5

150

9.0

6.4

6.8

7.2

7.6

Fig. 4. Transition wavelength of InGaN pyramid (P), and truncated pyramid (TP) QDs versus the base-length.

-12

-2

2

3 0 h=3, b=7 nm

-3

, b=8 nm

3

-4 0.18

0.20

0.52

0.54

0.56

, b=9 nm

-6 0.14

0.16

0.50

0.52

0.54

4

-12

b2 2

Im(X )[(THG) (m /v )] x10

0

2

2

0 -2

h=4, b=6.5 nm , b=7 nm , b=7.5 nm

-4 -6

3

-12

2 2 3

0.18

Pump Photon Energy(eV)

a2 Im(X )[(THG)(m /V )] x10

9.2

6

Pump Photon Energy (eV)

-8 0.17

0.18

0.52

0.54

-4 h=3, b=7 nm , b=8 nm

-8

, b=9 nm

-12

0.56

0.16

Pump Photon Energy(eV)

0.18

0.50

0.52

0.54

Pump Photon Energy(eV )

b3 1.2

h=3, b=7 nm , b=8 nm , b=9 nm

0.9

2

2

0.6

2

h=4, b=6.5 nm , b=7 nm , b=7.5 nm

abs(X )[(THG)(m /V )]x10

0.8

-11

a3 -11

8.8

2

0

3

2

2

Re(Ξ )[(THG) (m /V )] x10

b=6.5 nm 7 nm 7.5 nm, h=4 nm

2

Re(X )[(THG) (m /V )] x10

4

0.16

2

8.4

Fig. 5. Oscillator strength of InGaN pyramid (P), and truncated pyramid (TP) versus base-length.

b1

-12

a1

abs(X )[(THG)(m /V )]x10

8.0

b (nm)

b(nm)

0.4

0.3

3

3

0.2

0.6

0.0 0.16

0.18

0.20

0.51

0.54

Pump Photon Energy (eV)

0.57

0.0 0.14

0.16

0.18

0.48 0.50 0.52 0.54

Pump Photon Energy(eV)

Fig. 6. Third-order susceptibility of THG (m2 /V2 ) versus pump photon energy for InGaN pyramid, (b) truncated pyramid QDs.

Please cite this article in press as: H. Kaviani, A. Asgari, Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2012.01.012

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4

b1 Re(X )[(QEOE)(m /V )] x10

, b=7 nm

2

1

, b=7.5 nm

2

h=4, b=6.5 nm

0

3

3

-1 -2 0.40

3 h=3, b=7 nm

-12

2

2

2

Re(X )[(QEOE)(m /V )] x10

-12

a1

0.45

0.50

0.55

0.60

0.65

2

0 -1 -2 -3 0.40

h=4, b=6.5 nm

2

-1

, b=7 nm , b=7.5 nm

-2

-4 0.40

0.45

0.50

0.55

0.60

-2 -3 -4

h=3, b=7 nm , b=8 nm

-5

, b=9 nm 0.45

h=4, b=6.5 nm nm

, b=7.5 nm

0.2

3

0.1 0.0 0.40

0.45

0.50

0.55

0.60

0.55

0.60

0.65

6.0 h=3, b=7 nm 5.0

, b=8 nm , b=9 nm

2

, b=7

2

0.3

-12

b3 0.4

0.50

Pump Photon Eergy(eV)

abs(X )[(QEOE) (m /V )] x10

-11

0.65

-1

-6 0.40

0.65

2

2

0.60

3

-3

a3

3

0.55

0

-12 2

Im(X )[(QEOE)(m /V )] x10

-12

0

Pump Photon Energy(eV)

abs(X )[(QEOE)(m /V )] x10

0.50

b2

2

2

0.45

Pump Photon Energy(eV )

a2

3

, b=9 nm

1

Pump Photon Energy(eV )

Im(X )[(QEOE)(m /V )] x10

, b=8 nm

4.0 3.0 2.0 1.0 0.0 0.40

0.65

Pump Photon Energy (eV)

0.44

0.48

0.52

0.56

0.60

0.64

Pump Photon Energy (eV )

Fig. 7. Third-order susceptibility of QEOE (m2 /V2 ) versus pump photon energy for InGaN pyramid, (b) truncated pyramid QDs.

which can be obtained after eigenvalues and eigenfunctions calculation.

˛ = ˛0 + ˛2 I 3ω , Im (3 ) ˛2 = 4n0 c

excited states ( f ), is given by:

3

(−2ω1 + ω2 ; ω1 , −ω2 ) = − (9)

×

and

⎧ ⎨

n = n0 + n2 I 122 3 , ⎩ n2 = n2 c Re ( ) 0

× (10)

where I is incident power density, ˛0 and n0 are linear absorption coefﬁcient and refractive index, respectively [17]. The third-order nonlinear susceptibility for two energy levels, ground ( g ) and ﬁrst

2

2Ne4 ˛fg ε0h ¯

3

1 [i(ω0 − 2ω1 + ω2) + ][i(ω2 − ω1 ) + ] 1 1 + i(ω0 − ω1 ) + i(ω2 − ω0 ) +

,

(11)

g , e and N are where = 1/ , ω0 = (Ef − Eg )/¯h, ˛fg = f r the relaxation rate (inverse of relaxation time), transition frequency (resonance frequency between ﬁrst excited and ground states), dipole transition matrix element, electron charge and density of electron in QDs, respectively. In the transition matrix element, “r” has been set along the polarization of the incident light.

Please cite this article in press as: H. Kaviani, A. Asgari, Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2012.01.012

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4.0

Inm Ga1−m N parameters

Unit

Value

Electron effective mass m* Relative dielectric constant (εr ) Density of carriers (N) Piezoelectric constant (e31 ) Piezoelectric constant (e33 ) Elastic constant (c13 ) Elastic constant (c33 ) Lattice constant (a) Spontaneous polarization (Psp )

m0 – m−3 c/m2 c/m2 GPa GPa A˚

0.12m + 0.15(1−m) 14.6m + 10.3(1−m) 5 × 1024 −0.57m−0.49(1−m) 0.97m + 0.79(1−m) 94m + 106(1−m) 200m + 398(1−m) 3.548m + 3.189(1−m) −0.032m−0.029(1−m)

2

c/m

For calculating the third-order susceptibilities of quadratic electro-optic effect (QEOE) and third-harmonic-generation (THG), we take ω1 = 0, ω2 = −ω and ω1 = −ω2 = ω, respectively [16–19,29]. Knowing the nonlinear behavior of the absorption coefﬁcient and refractive index, one can study the intensity-dependent refractive index phenomena such as self-focusing effects, which is illustrated in Fig. 2. This process can occur when a beam of light having a nonuniform transverse intensity distribution propagates through a material in which n2 is positive. Under these conditions, the material effectively acts as a positive lens, which causes the rays to curve towards each other. This process is of great practical importance because the intensity at the focal spot of the self-focused beam is usually sufﬁciently large to lead to optical damage of the material. The self focusing given by: 2n0 ω02 0

1 P/Pcr

,

(12)

where 0 P and Pcr are the vacuum wavelength of the laser radiation, the power carried by the beam, and critical power, respectively [30]. A schematic QD nonlinear medium for self-focusing effect has been shown in Fig. 2. 3. Results and discussion The considered two pyramid and truncated-pyramid shaped InGaN wurtzite QDs which are embedded in GaN material, have a squared (b × b) base and h height. The base is in the x–y plane. In the calculation, the QD size varies with the b and h variation, and also we considered the refractive index (n0 = 3.37), in molar (m = 0.25), beam reduce (ω0 = 5 m), 0 = 2 nm, P = 20 W. The other parameters used in the calculation are summarized in Table 1. In Fig. 3, the wavefunctions of the ground and ﬁrst excited states for two typical InGaN QDs (pyramid and truncated pyramid) have been shown. The truncated-pyramid QD has a height, h = 3 nm, and the base, b = 8 nm, and the pyramid QD has a height, h = 4 nm, and the base, b = 7 nm. By scaling down the QDs, reducing the base length of the QD, the energy level distance between the electron ground and ﬁrst excited states in conduction band becomes larger. Therefore, the transition energy wavelength decreases. The transition energy wavelength as a function of QDs’ base length for both QDs is shown in Fig. 4. The oscillator strength is given by: Fos =

2m∗ h ¯

2

f εr

(Ef − Eg )

g

2 ,

(13)

where ε is the unit vector for the photon polarization [31]. Fig. 5 shows the size dependency for oscillator strength that was associated with intraband transitions in the conduction band of ∼InGaN/GaN quantum dots. By increasing the base length, the oscillator strength increases, because the dipole transition matrix elements increase with enhancement of base length. Although the

Self- Focusing (cm )

Table 1 Material parameters for Inm Ga1−m N [27].

Zsf =

5

P-QD (h=4 nm)

3.5

TP-QD (h=3 nm)

3.0 2.5 2.0 1.5

6.5

7.0

7.5

8.0

8.5

9.0

b (nm) Fig. 8. The self-focusing length of InGaN pyramid (P) and truncated pyramid (TP) QDs versus QD base-length.

energy level separations decrease however this lead to increase the overlapping in wavefunctions. In the following we study the effects of QD size on third order susceptibility (3 ). Our calculation results show that the large QDs have larger optical nonlinearity, and with increase in QD size, the magnitude of the resonance peak in 3 raises and also trend to lower energy. This subject is illustrated in Figs. 6 and 7 clearly. In Fig. 6 the variation of THG has been shown. It shows that THG has two peaks in near ω = ω0 , ω = ω0 /3 and these two peaks have red shift as base length increases. Also, Fig. 7 demonstrate that QEOE as function of photon energy with different QD size. It has one peak near ω = ω0 and this peak has a red shift as the base length increases. Finally, the effects of QD size and shape on self focusing have been studied. Fig. 8 shows the size dependency for the length of self focusing (Zsf ) in InGaN/GaN QDs. As depicted in the ﬁgure, for the both shapes (pyramid QDs and truncated-pyramid QDs), the length of self focusing decreases with increase in QDs’ base length. Because the length of self focusing increases with increase in the critical power and the critical power increases with decrease in optical nonlinearity. Therefore, the small QDs are very good nonlinear medium for the self-focusing effects. Also, as shown in Fig. 8, for small InGaN QDs, the truncated-pyramid shaped QDs have more self-focusing effects than pyramid shape QDs. But, for QDs with long base length, the effects of self focusing in pyramid QDs are higher than the truncated-pyramid QDs. Because the pyramid shaped InGaN QDs have larger optical nonlinearity than the truncated-pyramid shaped InGaN QDs at long base lengths.

4. Conclusion In this paper, we have considred two shapes (pyramid and truncated pyramid) of Inx Ga1−x N QDs with different sizes, and calculated the size-dependent electronic and nonlinear optical properties such as quantum energy levels, wavefunction, transition wavelength, oscillator strength, third order susceptibility (THG, QEOE) and self focusing in presence of the built-in electric ﬁeld. The calculated results show that the increase in the QDs’ size (not bigger than the size which the dots’ optical properties belong to the region of the bulk material), the transition wavelength, oscillator strength and also third order susceptibility(THG, QEOE) increase, moreover the resonances peak trend to low frequencies (red shift). Also, for both pyramid QDs and truncated-pyramid QDs, the length of self focusing decreases with increase in the QDs’ base length.

Please cite this article in press as: H. Kaviani, A. Asgari, Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2012.01.012

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Please cite this article in press as: H. Kaviani, A. Asgari, Investigation of self-focusing effects in wurtzite InGaN/GaN quantum dots, Optik - Int. J. Light Electron Opt. (2012), doi:10.1016/j.ijleo.2012.01.012

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