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Physica E 27 (2005) 221–226

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Piezoelectric field-dependent optical nonlinearities induced byinterband transition in InGaN/GaN quantum well

Junjie Li, Liming Liu, Duanzheng Yao�

Department of Physics, Wuhan University, Wuhan 430072, China

Received 2 November 2004; accepted 17 November 2004

Availabale online 23 January 2005

Abstract

The binding energy of GaN/InGaN quantum well (QW) has been studied by taking the strain-induced piezoelectric

and spontaneous polarization effects into account. The variational calculations are presented for the ground exciton

state in the quantum wells, and the third-order susceptibilities, as functions of In content x; well width w; and pumpphoton energy _o; have also been analyzed. In addition, the results are compared with the data of previous work andsome conclusions are given.

r 2005 Elsevier B.V. All rights reserved.

PACS: 78.67.D; 42.65.A; 77.65.L

Keywords: Optical properties of quantum well; Optical susceptibility; Piezoelectric field

1. Introduction

The physical properties of superlattices andsingle quantum well have attracted much attentionin the recent literature [1–5]. Especially, theelectronic and optical properties of the GaN/InGaN quantum well (QW) have received increas-ing interest for its particular characteristics [6,7].Many scientists have studied experiments anddeveloped theories for the exciton bindingenergy of the electron–hole pair and susceptibil-ities in a heterostructure. But many calculations

e front matter r 2005 Elsevier B.V. All rights reserve

yse.2004.11.011

ng author. Fax: +8627 87654569.

ss: dzyao@whu.edu.cn (D. Yao).

have been presented under infinite or finitesquare QW and there were few calculations aboutthe susceptibilities which arise from the bandtransition.In this paper, with the variational method and

effective mass approximation, the variation ofbinding energy under the presumption of triangu-lar QW is presented, and compared with that ofsquare wells. In addition, after our analyzingbriefly the hole subband wave function and thezone-center valence band Bloch state under a de-dipole moment approximation, it has been shownthat the third-order susceptibilities of GaN/InGaNQW with different well width and In content varyas a function of pump photo energy. Finally, the

d.

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J. Li et al. / Physica E 27 (2005) 221–226222

calculation results have been analyzed and someconclusions have been drawn.

2. Theory

Under the effective mass approximation, ne-glecting the couplings between heavy hole, lighthole and spinning-coupled splitting terms is a goodapproximation, especially when the carrier densityis low, when the electron and hole dispersionmostly concentrate on G point, and when thecenter of x–y mass momentum kk ¼ 0: With thesetwo approximations, the exciton wave functionFðrÞ must satisfy following Hamilton equation [8]:

�r2r

2 /�1

2me

q2

qz2e�

1

2mlh; hh; so

q2

qz2h

"

þ V eðzeÞ þ VhðzhÞ �e2

�r

#FðrÞ ¼ EbFðrÞ; ð1Þ

r ¼ jre � rhj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðze � zhÞ

2þ r2

q;

where V e and Vh are the potential functions inGaN/InGaN QW, r is the distance between theelectron and the hole in x–y plane, m is the reducedmasses of light, heavy hole or spinning-coupledsplitting terms within QW grown along the z-axis,me is the effective mass of electron, mlh; hh; so isthe effective mass of light hole, heavy holeor spinning-coupled splitting terms. Within theframework of variational calculation in whichsystem energy is required to be minimized, the trialwave function FðrÞ could be chosen as FðrÞ ¼jeðreÞjhðrhÞfðrÞ [9], where jeðrÞ;jhðrÞ; are, respec-tively electron wave function and hole wavefunction, which satisfy electron and hole Hamilto-nian formula, and fðrÞ is a function of r:

2.1. Electron, hole and exciton wave function

The electron bound-state energies and theconduction envelop functions of the QW formedby a heterostructure of GaN/InGaN are conven-tionally obtained by solving the following time-independent effective mass Eq. (2), in which themass discontinuity at the heterojunction is ne-glected since the difference between the barrier and

well is small [10]:

�1

2me

q2

qz2eþ V eðzeÞ

� �jemðzeÞ ¼ EemjemðzeÞ; (2)

where

V e ¼

eFwze for jzejoL=2;

V e0 þ eFwL=2 for ze4L=2;

V e0 � eFwL=2 for zeoL=2;

8><>:

in which L is the layer thickness, Voe is the depthof the QW, which is determined by the differencebetween the band gaps of GaN and InGaN andscaled by the conduction band offset usually takenas 0.6, e is the magnitude of the electron chargeand Fw is the sum of PZ fields in the QW, which isexpressed as [11]

Fw ¼ �Pwpz þ Pwsp � Pbsp

�0�r;

where superscripts w, b represent, respectively thewell and barrier, Pwsp;P

wpz;P

bsp; are the spontaneous

and strain-induced piezoelectric polarizations ofInGaN and the spontaneous polarization of GaN,respectively. The strain-induced piezoelectric po-larization Pwsp can be expressed by [11]

Pwpz ¼ 2a � a0

a0e31 � e33

c13

c33

� �;

where a is the lattice constant of buffer (substrate),and a0 is the lattice constant of well or barrier,c13; c33 and e31; e33 are elastic and piezoelectricconstants, respectively.In the QW, if we assume X ¼ �½2me=e2F2w_

2�1=3

ðE � eFwzeÞ; the variable ze in Eq. (2) will betransformed to a dimensionless coordinate X :The Schrodinger equation will be expressed as

d2

dX 2jemðX Þ � XjemðX Þ ¼ 0;

and then the solution jemðX Þ will be a combina-tion of Airy functions: jemðX Þ ¼ aAiðX Þ þ

bBiðX Þ; where a; b are constants and AiðZÞ;BiðZÞ

are Airy functions. Furthermore, with the con-tinuity at the edges of QW and the unity of wavefunction, the energy eigenvalue and eigenfunctionof electron can be obtained.In the same way, except that the valence band

offset is usually taken as 0.4, we can also solve the

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J. Li et al. / Physica E 27 (2005) 221–226 223

corresponding hole Hamilton equation and thenobtain the energy eigenvalue and eigenfunction ofvalence hole.Such an exciton model taking into account the

piezoelectric and spontaneous polarizations isillustrated in Fig. 1.

2.2. The third-order susceptibilities

When the carrier density is low, carrier disper-sion mostly concentrate on kk ¼ 0; and then thethird-order susceptibility wð3Þ can be expressed asfollows, [12]:

w3ð�2o1 þ o2;o1;o1;�o2Þ

¼�2iM4N

�0½i_ðo0 � 2o1 þ o2Þ þ _g?�½i_ðo2 � o1Þ þ _gk�

1

i_ðo0 � o1Þ þ _g?þ

1

i_ðo2 � o0Þ þ _g?

� �;

ð3Þ

where g?; gk are the transverse relaxation constantsand amount to 300 fs, and N is the number densityof carriers. The transition frequency o0; dipoletransition matrix element M can be defined by [13]

o0 ¼Ec � Ev

_and M ¼ er;

rmcv ¼

Zhcjf cðzÞrm f vðzÞjvidz

¼ hcjrmjvi

Zf cðzÞ f vðzÞdz;

rm¼1; 2; 3 ¼ x; y; z:

Fig. 1. Conduction and valence band profile of GaN/InGaN

QW, in consideration of strain effects.

Here f cðzÞ; f vðzÞ are the envelop functions ofconduction and valence bands which can be givenby the methods mentioned above, jci is thewave function of conduction electron and jni isthe wave function of valence hole, which is thecombination of jn1i; jn2i; jn3i; jn4i; jn5i; jn6iðjn1i ¼j3=2; 3=2i ¼ jðx þ iyÞ "i=

ffiffiffi2

p; jn2i ¼ j3=2; 1=2i ¼

ijx þ iyÞ #i=ffiffiffi6

p� i

ffiffiffiffiffiffiffiffi2=3

pjz "i; jn3i ¼ j1=2; 1=2i ¼

jðx þ iyÞ #i=ffiffiffi3

pþ jz "i=

ffiffiffi3

p; jn4i ¼ j1=2;�1=2i ¼

�ijðx � iyÞ "i=ffiffiffi3

pþ ijz #i=

ffiffiffi3

p; jn5i ¼ j3=2;�1=2i

¼ jðx � iyÞ "i=ffiffiffi6

pþ

ffiffiffiffiffiffiffiffi2=3

pjz #i; jn6i ¼ j3=2;�3=2i

¼ ijðx � iyÞ #i=ffiffiffi2

pÞ: It is also known that [14]

he; jjrmjrn; ji ¼_

Ec � Ev

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEgðEg � DÞ2m�

e Eg þ23D

� �s

djjdm; n;

j ¼ 32;12;�

12;�

32;

where D is the spin–orbit split-off, and Eg is theband gap between conduction and valence band.

3. Result and discussion

For TM polarization of the incidence along zaxis, the band transition of 1 s exciton will happenonly between light hole subband and conductionsubband. Therefore, we confined our calculationto the light hole. The reduced mass m and theeffective mass of light hole in Eq. (1) are calculatedas [15]

1

/¼1

meþ1

m0ðg1 � g2Þ;

1

mlh¼1

m0ðg1 þ 2g2Þ;

where m0 is the mass of static electron, and g1; g2;are Luttinger parameters.

3.1. Exciton binding energy

Based on the theoretical framework outlined inSection 2, the function fðrÞ is assumed to be 1 sstate fðrÞ ¼

ffiffiffiffiffiffiffiffi2=p

pð1=aÞe�r=a where a is a varia-

tional parameter to minimize the exciton energy.In addition, when the In content is given, the sumof PZ fields Fw is calculated, and then by solvingHamilton Eq. (2), the electron energy eigenvalue.

Eem and corresponding conduction wavefunction jemðreÞ which is a combination ofAiry function are given. Also, the hole energy

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J. Li et al. / Physica E 27 (2005) 221–226224

eigenvalue Ehn and corresponding valence wavefunction jhnðrhÞ are obtained in the same way.With these wave functions, trial wave function isexpressed as

FðrÞ ¼

ffiffiffi2

p

r1

ajemðreÞjhnðrhÞe

�r=a:

The parameter a and the binding energy Eb aredetermined by importing FðrÞ into the followingequation and conducting a variational calculation:

Eb ¼ Eem þ Ehn

þmin FðrÞ �1

4p�e2

r

� ���������F rð Þ

� �� �:

Fig. 2. Binding energy as a function of In content x in

InxGa1�xN layer with well width 5 nm.

Table 1

Material parameters for GaN, InN, and InxGa1�xN [11,16]

GaN

Band gap 3.4

Lattice constant a ð (AÞ 3.189

Piezoelectric constant �0.20

Elastic constant (1011dyn/cm2)(C13Þ 11.4

C33 38.1

Spontaneous polarization (C/m2) �0.029

Effective mass, me 0.2

Effective mass, mlh 0.2

Luttinger parameters g1; g2; g3Spin–orbit split-off DStatic dielectric constant � 10.4

The binding energy of exciton which is confined inGaN/InGaN triangular QW is presented as afunction of In content in this section, andcompared with that of square QW (Fig. 2). InFig. 2, triangular quantum well (TQW) representsthe condition taking into account the piezoelectricand spontaneous polarization field and squarequantum well (SQW) represents the condition nottaking them into account.As can be seen from Fig. 2 above, the binding

energy of square QW is much higher than that oftriangular QW at any In content from 0.15 to 0.30,and with the increase of In content, the bindingenergy of triangular QW reduces faster. So it canbe concluded that the strain-induced piezoelectricand spontaneous polarization have a huge effecton the exciton binding energy. It is because whenthe strain-induced piezoelectric and spontaneouspolarization effects are taken into account, thepotential in QW reduces and then the possibleenergy eigenvalue reduces, too. The ground energyof the electron reduces and the energy of the holeincreases, which mean that the band gap decreases,so that the exciton binding energy of triangularQW reduces. The material parameters used in ourcalculations are summarized in Table 1.

3.2. The third-order susceptibilities related to

degenerated four waves mixing

In Eq. (3), if we assume o1 ¼ o2 the third-ordersusceptibility wð3Þ will become to be the description

InN InxGa1�xN

1.9 3:4ð1� xÞ þ 1:9x � 3:2xð1� xÞ

3.548 3:189ð1� xÞ þ 3:548x

�0.23 �0:20ð1� xÞ � 0:23x9.4 11:4ð1� xÞ þ 9:4x

20.0 38:1ð1� xÞ þ 20:0x�0.032 �0:029ð1� xÞ � 0:032x

0.2

0.2

3.03, 0.80, 1.26

17

10.4

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Fig. 3. jwð3Þð�o;o;o;�oÞj of TQW and SQW versus the pump

photoenergy for different In content x: (a) x ¼ 0:15; (b) x ¼

0:20; (c) x ¼ 0:30:

J. Li et al. / Physica E 27 (2005) 221–226 225

of degenerated four waves mixing and if o1 ¼�o2; Eq. (3) will give the third-order suscepibilitiesfor third harmonic generation (THG) of QW. Inthis letter, only the degenerated four-wave mixing(DFWM) is in our consideration. For GaN/InGaN QW, we only consider the condition inwhich polarization is along z-axis. Using the wavefunctions of electron and hole which are obtainedthrough the methods mentioned in the previoussections, we got the third-order susceptibility wð3Þ

for degenerated four-wave mixing with various Incontent, and compared these data of TQW (takingthe strain-induced piezoelectric and spontaneouspolarization effects into account) with that ofSQW (not taking those into account), as shown inFig. 3.In Fig. 3, we displayed jwð3Þj as a function of

pump photo energy _o for different In content x

in a 5 nm GaN/InGaN quantum well. We caneasily find that when the piezoelectric and sponta-neous polarizations are taken into account, theresonant peak shifts to lower energy region. Thiscan be explained with the same reason as thereduction of the exciton binding energy. Inaddition, the susceptibility of TQW is muchsmaller than that of SQW under the samecondition, and different from square QW wherethe third-order susceptibility is almost invariablewhen the In content increases, the susceptibility inthe triangular QW decreases faster. These are dueto the presence of the strain-induced piezoelectricand spontaneous polarization effects: the electronand hole wave function tend to separate due totheir opposite charge in the piezoelectric andspontaneous polarization field, then the overlapis smaller than that in SQW, and with the increaseof In content, this overlap becomes much less.In addition, the third-order susceptibilities

of four different width QWs are also calculated,and depicted in Fig. 4. The peak values of jwð3Þjat pump photo energy (2.661, 2.505, 2.3524,2.2000 eV) amount to well width W ¼ ð3; 4; 5;6 nm), respectively. That means the nonlinearsusceptibilities reduce and that the resonantpeak shifts to lower energy when the wellwidth increases. These also can be explainedto the effect of the strain-induced piezoelectricand spontaneous polarization: when QW width

increases, reduction of the electron energyeigenvalue and increase of the hole energyeigenvalue lead to the transition frequency de-crease. The reduction of the overlap betweenelectron and hole wave functions leads to thereduction of susceptibilities.

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Fig. 4. jwð3Þð�o;o;o;�oÞj of TQW as a function of pump

photoenergy for different well width in GaN/In0:2Ga0:8N QW.

J. Li et al. / Physica E 27 (2005) 221–226226

4. Conclusion

In this paper, we have calculated the bindingenergy by using the variational method. We alsocalculated third-order nonlinear optical suscept-ibilities of GaN/InGaN QW taking into accountinfluences of strain-induced piezoelectric andspontaneous polarization with some approxima-tions. The third-order susceptibilities for differentwell width and In content are given, too.Opposite to the condition of intersubband

transition discussed in Refs. [10,17], the presenceof the strain-induced piezoelectric field duringband transition decreases the overlap of subbandwave functions and therefore reduces the suscept-ibilities of intersubband transition. Since thestrain-induced piezoelectric and spontaneous po-larization effects are inevitable in many situations,

we can improve the third-order susceptibilities byadding an electronic field opposite to the strain-induced piezoelectric field.Our calculation is confined to the light hole, the

same work can be done to the heavy hole, andmore complex condition will appear when thecouplings between light hole, heavy hole, andspinning-coupled splitting terms are considered.

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