ARTICLE IN PRESS

Physica E 41 (2008) 245–253

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

A study in optical properties of AlGaN/GaN pyramid and prism-shapequantum dots

H. Rasooli Saghai a, A. Asgari b,�, H. Baghban Asghari Nejad c, A. Rostami c

a Science and Research Branch, Islamic Azad University (IAU), Tehran, Iranb Photonics-Electronics Group, Research Institute for Applied Physics, University of Tabriz, Tabriz 51665-163, Iranc Photonics and Nanocrystals Research Laboratory (PNRL), Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 51664, Iran

a r t i c l e i n f o

Article history:

Received 18 February 2008

Received in revised form

13 July 2008

Accepted 14 July 2008Available online 9 September 2008

PACS:

78.20.Jq

78.67.De

78.67.Hc

73.21.Fg

73.21.La

Keywords:

Third-order susceptibility

Absorption coefficient

Pyramid quantum dot

Prism quantum dot

77/$ - see front matter & 2008 Elsevier B.V. A

016/j.physe.2008.07.009

esponding author. Tel.: +98 4113393007; fax

ail address: asgari@tabrizu.ac.ir (A. Asgari).

a b s t r a c t

A three-dimensional numerical method for investigating electrical and optical properties of pyramid

and prism-shape AlGaN/GaN quantum dots (QDs) has been developed. The effect of QD structure

parameter such as height and base area on energy levels, dipole matrix element, absorption coefficient,

and third-order susceptibilities (quadratic electro-optic effect and third harmonic generation (THG)

parameters) are studied. Our calculation results show the decreasing of the energy levels, resonant

frequency and absorption peak, and the increasing of the quadratic electro-optic effect and THG

by increasing the QD height and base area. Also with increasing the QD height and base area a red

shift occurs.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Intersubband transitions (ISBT) in quantum dots (QDs) andnanocrystal structures are subjects of interest for both funda-mental physics study and development of infrared optoelectronicdevices. Compared with two-dimensional quantum well struc-tures, the intersubband absorption in zero-dimensional QDstructures has advantages in optical applications due to theirsharp delta-like density of states, reduced intersubband relaxationtimes [1,2]. ISBT in QD structures have also attracted considerableresearch attention in recent years due to the large values of thedipole transition matrix elements to achieve both linear andnonlinear optical properties, extremely large oscillator strengthsand relatively narrow linewidths [3–7].

The interstate transition as well as the energy level spectrumdepends on the chemical identity, crystal structure of thenanocrystals and their size and shape. Many II–VI and III–Vsemiconductors have been prepared as insulating nanocrystals by

ll rights reserved.

: +98 4113347050.

colloidal chemistry. Nucleation and growth in solution leads to(nearly) spherical crystals—unless the growth conditions arecarefully manipulated. In contrast, deposition on a surface(molecular beam epitaxy and electrodeposition) typically yieldsnon-spherical structures. The question is to how far the energylevel spectrum of QD is affected by the shape and size of thesestructures. However, as is clear from recent optical work oninsulating quantum rods, relatively small deviations from sphe-rical symmetry can have dramatic effects on the optical propertiesof semiconductor nanocrystals [8].

The shape of the QD can be significantly modified duringregrowth or postgrowth annealing or by applying complex growthsequences [9]. The QDs may assume the shape of pyramids [10] orflat circular lenses [11]. The deposition of a thin layer of onematerial on top of a substrate where there is a large difference inlattice constants, can lead to the formation of pyramidal-shapedQD [12]. A typical example for pyramidal QD is an InAs pyramidQD embedded in a cubic GaAs matrix [13,14].

Large optical intraband nonlinearities are expected to occurin semiconductor QD since the intraband dipole lengths extendover the QD size and are in nano-meter range [15]. For instant,quadratic electro-optic effects (QEOE) and electro-absorption (EA)

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Table 1Material parameters used in calculations for the AlGaN/GaN QD

AlxGa1�xN parameters Unit Value

Electron effective mass (m*) m 0.252x+0.228

H.R. Saghai et al. / Physica E 41 (2008) 245–253246

process in InGaN/GaN cylinder QD have been investigated, wherethe third-order susceptibility dispersion behaviors of directcurrent are obtained [16]. Also it is found that the increasing ofthe QD height and radius increase the magnitudes of the quadraticelectro-optic susceptibility real part and the imaginary part of theEA susceptibility as well increase the resonant frequency and shiftit to the lower energy region. The obtained susceptibility is about10�16–10�14 m2/V2. Another example is well-width-dependentthird-order optical nonlinearities of a ZnS/CdSe cylindrical QD-quantum well (QDQW) in Ref. [17], where the eigenenergies andwave functions of electrons in QDQW have been calculated underthe effective mass approximation by solving a three-dimensionalnonlinear Schrodinger equation and by means of compact densitymatrix method. The reported values for the third-order nonlinearsusceptibilities are around 10�15–10�14 m2/V2.

AlGaN alloy and associated AlGaN/GaN heterostructures arethe subjects of intense research interest because of great potentialfor fabrication of high electron mobility transistor and for futureapplications in industries requiring high power and/or high-temperature microwave field [18]. Besides, the ISBT in GaN-basedheterostructures have been recently the topic of extensiveresearches for their advantages. Ultrafast optical modulation anda broad wavelength range are available in these structures. AlsoAlGaN withstands high-power optical excitation and high-temperature operation [19].

In order to investigate the characteristics of the QDs and theeffects of QD shape and size on the optical and electricalproperties, an appropriate model is needed. On the other hand,the modeling of the QDs is still lacking; mainly hampered by thecomplex three-dimensional nature of the QDs system. So,theoretical descriptions of the QDs are limited to one- or two-dimensional approximation [20]. Therefore, a full three-dimensionalmodel is required to provide a full description of the QD structure.The 3D-finite difference model is one of the most popularmethods to solve the Schrodinger equation for the QDs.

In the present article, the model derivations to calculate theelectrical and optical properties of pyramid and prism QD incooperate with 3D-finite difference model to solve 3D Schrodingerequation are presented.

0

Band gap (Eg(x)) eV 6.13x+(1�x)�3.42�x(1�x)

Band offset (DEC(x)) eV 0.7� [Eg(x)–Eg(0)]

Typical relaxation constant (_G) meV 0.3

Barrier density of carriers m�3 1�1024

Density of carriers inside the QD m�3 1�1021

Relative dielectric constant (er) � 8.5x+10.4(1�x)

2. Model derivation

In order to calculate the eigenvalues, wave functions andrelated optical and electrical quantities, the effective mass

ab

h

Fig. 1. Potential distributions of (a) pyramid q

equation has been solved numerically. It has been started byconsidering the Schrodinger equation in the slowly varyingenvelope approximation in three dimensions as [13]

�h2

2m�ir

2x;y;z þ Viðx; y; zÞ

)cðx; y; zÞ ¼ Ecðx; y; zÞ

((1)

where

m�i ¼m�w for inside of the dot

m�b for the barrier region

(

are the effective mass, Vi(x, y, z) and c(x, y, z) are the overallpotential distribution and the slowly varying envelope in differentregions, respectively. The potential profile has been denoted inFig. 1. The effective mass Schrodinger equation is solved using3D-finite difference method. The main advantage of utilizing thefinite difference method relies in the simplicity of investigatingthe effect of different parameters on the considered equation. Forinstance, one can investigate the effects of external field, strain,carrier exchange potential, and polarization field as a correctionpotential to diagonal potential matrix.

The Fermi energy level is obtained as [21]Z þ1�1

NþD dv ¼

Z þ1�1

nðx; y; zÞdv

¼

Z þ1�1

2X

k

ckðx; y; zÞ�� ��2f1þ exp½ðEk � EF�=KBTg�1 dv

(2)

where N+D, Ek, EF, kB, T are doping concentration in the barrier

region, energy levels, Fermi energy level, Boltzmann constant, andtemperature, respectively. To determine precisely the Fermienergy level, all occupied and partially occupied levels insidethe dot region should be included.

uantum dot and (b) prism quantum dot.

ARTICLE IN PRESS

1.16

1.15

1.14

1.13

1.12

1.11

30 40 50

1.5

1

0.5

2000 4000 6000 8000

1.3

1.2

1.1

1

0.9

0.8

3000 4000

1.25

1.2

1.15

1.1

1.05

20 40 60 80

1.1

1

0.9

0.84000 6000

1.1

1

0.9

0.82500 3000 3500

Height (h) [A°] Base Area (s = ab) [A°2] Base Area (s = ab) [A°2]

a = 50 A°

a = 50 A°

a = b

a = b

Dip

ole

Mat

rix E

lem

ent [

nm]

Fig. 3. Dipole transition matrix element of pyramid QD vs. (a) height (base area ¼ 80 A�80 A), (b) square base (height ¼ 60 A), (c) rectangular base (height ¼ 60 A), and

prism QD vs. (d) height (base area ¼ 80 A�80 A), (e) square base (height ¼ 60 A), (f) rectangular base (height ¼ 60 A).

400

350

300

250

200

15030 40 50

E0E1

E0E1

E0E1

500

400

300

200

2000 4000 6000 8000

a = b

500

400

300

200

3000 4000

a = 50 A°

500

400

300

200

10020 40 60 80

Height (h) [A°]

400

300

200

4000 6000

400

350

300

250

2003000 3500 4000

Base Area (S = ab) [A°2] Base Area (S = ab) [A°2]

a = ba = 60 A°

E2

E2

E0E1E0

E1

E0E1

Ene

rgy

[meV

]

Fig. 2. Energy states of pyramid QD vs. (a) height (base area ¼ 80 A�80 A), (b) square base (height ¼ 40 A), (c) rectangular base (height ¼ 40 A), and prism QD vs. (d) height

(base area ¼ 80 A�80 A), (e) square base (height ¼ 40 A), (f) rectangular base (height ¼ 40 A).

H.R. Saghai et al. / Physica E 41 (2008) 245–253 247

ARTICLE IN PRESS

3.5

3

2.5

2

1.5

1

0.5

00.105 0.11 0.115 0.12

6

5

4

3

2

1

00.1 0.12 0.14 0.16 0.18

6

5

4

3

2

1

00.08 0.1 0.12 0.14 0.16 0.18

Pump Photon Energy [eV]

Abs

orpt

ion

Coe

ffici

ent [

1/cm

]

x 106 x 106 x 106

h = 30 A° a = b = 50 A°

90 A°80 A°70 A°

b = 60 A°b = a = 50 A°

50 A°40 A°

80 A°70 A°60 A°

Fig. 5. Absorption coefficient of pyramid QD vs. pump photon energy for different (a) height (base area ¼ 80 A�80 A), (b) square base (height ¼ 40 A), (c) rectangular base

(height ¼ 40 A) sizes.

0.542

0.54

0.538

0.536

0.534

0.532

0.53

0.528

0.526

0.524

0.52230 35 40 45 50

Ferm

i-Diff

eren

ce

0.6

0.58

0.56

0.54

0.52

0.5

0.48

0.4620 40 60 80

Height [A]

Fig. 4. Fermi difference vs. height for (a) pyramid and (b) prism structures (base area ¼ 80 A�80 A).

H.R. Saghai et al. / Physica E 41 (2008) 245–253248

ARTICLE IN PRESS

2.5

2

1.5

1

0.5

0.105 0.11 0.115 0.12

4

3.5

3

2.5

2

1.5

1

0.5

0.1 0.12 0.14 0.16 0.18

6

5

4

3

2

1

00.1 0.12 0.14 0.16 0.18

Pump Photon Energy [eV]

abs

(XQ

EO

E) [

m2 /

V2 ]

x 10-15 x 10-15 x 10-15

50 A°40 A°

h = 30 A°

90 A°80 A°70 A°60 A°

a = b = 50 A°

90 A°80 A°70 A°

b = 60 A°b = a = 50 A°

Fig. 7. Third-order susceptibility of quadratic electro-optic effect (QEOE) for pyramid QD [(m2/V2)] vs. pump photon energy for different (a) height (base

area ¼ 80 A�80 A), (b) square base (height ¼ 40 A), (c) rectangular base (height ¼ 40 A) sizes.

7

6

5

4

3

2

1

0

7

6

5

4

3

2

1

0

7

6

5

4

3

2

1

00.1 0.12 0.14 0.12 0.14 0.16 0.18 0.12 0.14 0.16 0.18

Pump Photon Energy [eV]

Abs

orpt

ion

Coe

ffici

ent [

1/cm

]

x 106 x 106 x 106

70 A°60 A°50 A°40 A°

h = 30 A°

80 A°70 A°60 A°

a = b = 50 A° b = a = 50 A°

80 A°70 A°

b = 60 A°

Fig. 6. Absorption coefficient of prism QD vs. pump photon energy for different (a) height (base area ¼ 80 A�80 A), (b) square base (height ¼ 60 A), (c) rectangular base

(height ¼ 60 A) sizes.

H.R. Saghai et al. / Physica E 41 (2008) 245–253 249

ARTICLE IN PRESS

2

1.5

1

0.5

00.11 0.12 0.13 0.14 0.15

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0.12 0.14 0.16 0.180

2

4

6

8

10

12

14

16

0.12 0.14 0.16 0.18Pump Photon Energy [eV]

abs

(XQ

EO

E) [

m2 /

V2 ]

x 10-15 x 10-15 x 10-16

70 A°60 A°50 A°40 A°

h = 30 A°

80 A°70 A°60 A°

a = b = 50 A°

70 A°80 A°

b = 60 A°b = a = 50 A°

Fig. 8. Third-order susceptibility of quadratic electro-optic effect (QEOE) for prism QD [(m2/V2)] vs. pump photon energy for different (a) height (base area ¼ 80 A�80 A),

(b) square base (height ¼ 60 A), (c) rectangular base (height ¼ 60 A) sizes.

54.5

43.5

32.5

21.5

10.5

00.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

Pump Photon Energy [eV]

abs

(XTH

G) [

m2 /

V2 ]

abs

(XTH

G) [

m2 /

V2 ]

abs

(XTH

G) [

m2 /

V2 ]

x 10-15 x 10-15

50 A°40 A°

h = 30 A°

90 A°80 A°70 A°60 A°

a = b = 50 A°

b = 70 A°

90 A°80 A°

b = 60 A°, a = 50 A°

4

3.5

3

2.5

2

1.5

1

0.5

00.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

Pump Photon Energy [eV]

x 10-15

14

12

10

8

6

4

2

00.04 0.06 0.08 0.1 0.12 0.14

Pump Photon Energy [eV]

Fig. 9. Third-order susceptibility of third harmonic generation (THG ) effect for pyramid QD [(m2/V2)] vs. pump photon energy for different (a) height (base

area ¼ 80 A�80 A), (b) square base (height ¼ 40 A), (c) rectangular base (height ¼ 40 A) sizes.

H.R. Saghai et al. / Physica E 41 (2008) 245–253250

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H.R. Saghai et al. / Physica E 41 (2008) 245–253 251

With solving Schrodinger equation and knowing the energylevels, corresponding wavefunctions, and electron densitydistribution, it is possible to calculate the different electricaland optical properties such as the dipole matrix element, thethird-order susceptibility, and the linear absorption coefficient[22–24].

The third-order susceptibility is

wð3Þð�2o1 þo2;o1;�o2Þ

¼�2iNq4jafgj

4

�0_3

�1

½iðo0 � 2o1 þo2Þ þG�½iðo2 �o1Þ þ G�

� �

�1

iðo0 �o1Þ þGþ

1

iðo2 �o0Þ þG

� �(3)

where G ¼ 1/t,o0 ¼ Ef–Eg/_, afg ¼ /cf|r|cgS, and N are therelaxation rate (inverse of relaxation time), transition frequency(resonance frequency between first excited and ground states),dipole transition matrix element, and carrier density, respectively.To calculate the third-order susceptibilities of the QEOE and THG,it has been taken into account that o1 ¼ 0, o2 ¼ –o ando1 ¼ –o2 ¼ o in Eq. (3), respectively. The linear absorptioncoefficient a(o) can be clearly calculated by computing through

7

6

5

4

3

2

1

00.040.050.060.070.080.09 0.1 0.110.120.130.14

Pump Photon Energy [eV]

abs

(XTH

G) [

m2 /

V2 ]

Pump Ph

abs

(XTH

G) [

m2 /

V2 ]

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00.04 0.06 0.08

x 10-15

x 10-15

70 A°60 A°50 A°40 A°

h = 30 A°

Fig. 10. Third-order susceptibility of third harmonic generation (THG) effect for

area ¼ 80 A�80 A), (b) square base (height ¼ 60 A), (c) rectangular base (height ¼ 60 A

the density matrix approach as [25–27]

aðoÞ ¼ 4poe2

Vo_c�0ffiffiffiffi�rp

Xi;j

dij

�� ��2 � ff ðEiÞ � f ðEjÞg �gij

g2ij þ ðo�oijÞ

2(4)

where o, e, c, e0, er, |dij| ¼ |/cj|r|ciS|, gij, oij are the photonfrequency, the electron charge, the speed of light, the permittivityof vacuum, the relative permittivity of semiconductor, dipoletransition matrix element, the relaxation rate, and the transitionfrequency, respectively. The expression {f(Ei)–f(Ej)} denotes theFermi energy distribution, difference of initial and final states.Also, in the equation the Lorentzian broadening is considered[26,28]. The material parameters of the proposed QD are given inTable 1 [29–31].

3. Results and discussion

The results in this paper characterize the electrical and opticalproperties of two shape (pyramid and prism) QDs structures. Inthe first part, the effects of structure shape and size on energylevels and matrix element has been studied and are shown inFigs. 2 and 3, respectively.

As illustrated from Fig. 2, with increasing QD structure heightand base area, the QD energy levels decrease in both the shapes

Pump Photon Energy [eV]

abs

(XTH

G) [

m2 /

V2 ]

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00.04 0.06 0.08 0.1 0.12 0.14 0.16

oton Energy [eV]0.1 0.12 0.14 0.16

x 10-15

a = b = 50 A°

80 A°70 A°60 A°

80 A°70 A°

b = 60 A°b = 50 A°, a = 50 A°

prism QD [(m2/V2)] vs. pump photon energy for different (a) height (base

) sizes.

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H.R. Saghai et al. / Physica E 41 (2008) 245–253252

where the decreasing rate is much higher for the prism shape(the volume of the prism QD is twice of the pyramid QD). Thedemonstrated decreasing of the energy levels is due to wideningof the well potential. To study the effects of base area, twodifferent cases have been considered: (i) a symmetric base (squarebase), and (ii) asymmetry base (rectangular base). In thesymmetric case, the first and second excited states are degener-ated which are split in asymmetric case. Also, it is shownthat with increasing the height and base area, the dipoletransition matrix element increases (Fig. 3). The increasing ofheight and base area, decrease the difference between the groundand first excited state energy levels and so enhance the over-lapping of the wave functions and lead to higher dipole matrixelement values.

Decreasing the ground state energy with height and base areaincreases its occupancy probability, so the Fermi differenceincreases. This behavior is clear in both pyramid and prismstructures (Fig. 4).

In the second part, the effect of height and base area on theabsorption coefficient in terms of pump photon energy in twodifferent structures is presented. The absorption coefficient iscalculated for ground to the first excited state transition. It isshown that the increasing of the height and base area, generallydecreases the absorption peak (overall variations: from 7�106 to1.5�106 and from 7.2�106 to 3�106 for the pyramid and prismQD, respectively), and a red shift occurs in both structures (Figs. 5and 6). The observed red shift is due to the decreasing of thedifference between energy levels. In order to justify the variationof the absorption peak, according to Eq. (4), the competitionbetween resonant frequencies, o, dipole matrix element, and theFermi energy distribution difference should be considered. Thiscompetition may lead to increase or decrease of the absorptionpeak in the two considered structures. These results correspondwith the fact that the absorption peak weakens when the resonantfrequency shifts to lower energies (long wavelengths).

2

1

0

-1

-2

0.095 0.1 0.105 0.11 0.115Pump Photon

Pump Photon

Imag

(χ3 )

[m2 /

V2 ]

Rea

l (χ3 )

[m2 /

V2 ]

x 10-15

x 10-15

22

0

-1

-2

-3

0.095 0.1 0.105 0.11 0.115

XQE

Fig. 11. Real part, nonlinear reflection coefficient (n2), imaginary part, and nonlinear abs

photon energy for different heights (base area ¼ 80 A�80 A).

In the third part, the QEOE and THG susceptibilities forpyramid and prism QDs in terms of pump photon energy areinvestigated. Figs. 7 and 8 depict QEOE susceptibilities for the QDswith different heights and base area and with the pyramid and theprism shape, respectively. These figures illustrate that the peak ofthe QEOE (near o ¼ o0) increases and a red shift is observed withincreasing the height and the base area of the QDs. These effectsare due to the increase of the matrix element and the decrease ofthe energy levels difference, respectively. Our simulation resultshave agreement with the quantum size effect which implies thatQD with bigger size have stronger optical nonlinearity and theresonant peaks of that shift to lower frequencies [18].

Also, the third-order susceptibility of THG is illustrated inFigs. 9 and 10 for the pyramid and prism QDs, respectively. Asshown in the figure, the susceptibility has two peaks near o ¼o0=3 and o0 and by increasing the height and base area of theQDs, the peak of the susceptibility increases (the whole range ofvariations are: from 1�10�15 to 1.5�10�14 and from 0.5�10�15

to 8�10�15 for the pyramid QD and prism QD, respectively) and ared shift occurs. So, it is possible to manage the resonancefrequency and the amplitude of third-order susceptibilities bycontrolling the height and base area sizes in both structures.

However, one can express the nonlinear change of absorptioncoefficient and refractive index as the function of imaginaryand real parts of third-order nonlinear optical susceptibility,respectively:

a ¼ a0 þ a2I

a2 ¼3o

4n0cIm wð3Þ� �

8<: (5)

and

n ¼ n0 þ n2I

n2 ¼3

8n0Re wð3Þ� �

8<:

2

1

0

-1

-2

0.1 0.11 0.12 Energy [eV]

Energy [eV]

x 10-16

x 10-10

Ref

ract

ive

Inde

x [m

/V]

Abs

orpt

ion

[m/V

2 ]

1

0

-1

-2

0.04 0.06 0.08 0.1 0.12

50 A40 A

h = 30 A

OE

orption coefficient (a2) of third-order susceptibility (QEOE) in pyramid QD vs. pump

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H.R. Saghai et al. / Physica E 41 (2008) 245–253 253

where I is incident power density, a0 and n0 are linear absorptioncoefficient and refractive index, respectively, and c is the speed oflight. For instant, the above parameters are simulated for apyramid QD structure with base area of 80 A�80 A and height ofQD as a variation parameter (Fig. 11).

4. Conclusion

In this paper a three-dimensional numerical method ispresented to solve the Schrodinger equation with constanteffective mass in pyramid and prism-shape AlGaN/GaN QDstructures. The modeling results show that the increasing of theheight and base area in both structures decrease the energy levels,differences between ground and first excited energy states andabsorption peak, and also increase the dipole matrix element,QEOE and THG. Moreover a red shift occurs. It is concluded thatthe resonance frequency and the amplitude of absorption peakand third-order susceptibilities can be controlled by tailoring theheight and base area sizes in both QDs.

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