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doi:10.1016/j.ph

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Physica E 28 (2005) 412–418

www.elsevier.com/locate/physe

Parameter-dependent resonant third-order susceptibilitycontributed by inter-band transitions in InxGa1�xN=GaN

quantum wells

Fei Gao, Guiguang Xiong�

Department of Physics, Wuhan University, Wuhan 430072, China

Received 15 April 2005; accepted 26 April 2005

Available online 11 July 2005

Abstract

The resonance enhancement of the third-order nonlinear optical susceptibility wð3Þ for degenerated four-wave mixing,due to the transition between valence band and conduction band, in InGaN/GaN multi-quantum wells have been

calculated. The band structures of valence bands and conduction bands and the wave functions are calculated by the

theory of effective mass. The contributions of the subbands of holes (heavy holes, light holes and the spin-orbit split-off

bands) to wð3Þ in the different directions are discussed. The correlations between wð3Þ in the different directions and thewidth of the quantum wells, and the constituents of the quantum wells are obtained.

r 2005 Elsevier B.V. All rights reserved.

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

Keywords: InGaN/GaN; Quantum wells; Optical susceptibility; Effective mass; Inter-band transition

1. Introduction

In recent years, the theoretical and experimentalstudies on the semiconductor quantum-confinedstructure materials are active, in which electrons,holes and excitons are confined in one or moredimensions. The discrete energy levels in quantum-

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

yse.2005.04.007

ng author. Fax: +8627 68752569.

ss: ggxiong@whu.edu.cn (G. Xiong).

confined structures lead to many unique physicalproperties, such as electrical structures, transpor-tation properties and optical properties. GaN-based semiconductor materials have good proper-ties of wide energy gap, high thermal conductivity,high electron saturated drift velocity and smalldielectric constants. People have given abroadattention on them [1–5]. With the development ofMBE, MOCVD and HVPE, the devices with goodperformance based GaN quantum well structureshave been manufactured. Scientists have paid

d.

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F. Gao, G. Xiong / Physica E 28 (2005) 412–418 413

more attention to the mechanism of luminescence.Compared to bulk materials, the nonlinear opticaleffects are stronger. As one means of coherentdynamics research, the calculation of the third-order nonlinear optical susceptibility ðwð3ÞÞ is veryimportant. Many studies on the transition inconduction bands of quantum wells have beenreported [6–8]. To LED and LD, the transitionbetween valence and conduction bands is moreimportant to the transition in conduction bands.The energy bands and the wave functions ofelectrons and holes are involved in the calculationsof the transition between valence and conductionbands. Since the spin-orbit split-off energy issimilar to the energy difference between subbandsin valence bands, its contribution to the transitionbetween valence and conduction bands cannot beneglected. This is different from that of GaAs-based quantum wells [9,10]. In this paper, the wð3Þ

for degenerated four-wave mixing of the InGaN/GaN quantum wells near the resonance frequencyat the directions of paralleling and vertical to the z-axis has been studied and the spectra of wð3Þ withdifferent width of the wells and the concentrationof In have been obtained.

2. Structure and formalism

The structure of InGaN/GaN multi-quantumwells are shown in Fig. 1. Because of the doped In,the energy gap of InGaN is smaller than GaN and

Fig. 1. The structure of InGaN/GaN multi-quantum wells.

the potential wells are formed. The potentialbarrier between two neighboring wells is wideenough so that the wave functions in the wellscannot overlap.With the approximation of effective mass

[11,12], the wave function near G point in Brillouinzone has the following form:

CðrÞ ¼X

n

UnðrÞF nðrÞ, (1)

where FnðrÞ is the slowly varying envelope functionand UnðrÞ is the Bloch function. For the electronsin conduction bands, U iðrÞ ¼ Cj i, Cj i here is theground state wave function of electrons. In thecase of valence bands, UnðrÞ is corresponding tosix band-edge wave functions of holes in valencebands vnj i ðn ¼ 1; 2; 3; . . . ; 6Þ. The terms of heavyholes, light holes and spin-orbit split-off withdifferent spin are included. Solving the effectivemass equation, the energy levels and the wavefunctions of the electrons and holes can beobtained.The effective mass equation of electrons is

_2k2

2m�e

þ V ðzÞ

� �F ðkÞ ¼ EF ðkÞ, (2)

where F ðkÞ is the envelope function of theconduction bands and m�

e is the effective mass ofthe electrons. Since the electrons are only confinedin the direction of z, there is no confinement to theelectrons in the directions of x and y. The envelopefunction F ðkÞ can be written as

F ðkÞ ¼ f ðkx; kyÞf ðkzÞ. (3)

Setting kz ¼ �iðq=qzÞ, the effective mass functionsof electrons paralleling to and vertical to thedirection of z can be rewritten as

�_2

2m�e

d2

dz2f ðzÞ þ V ðzÞf ðzÞ ¼ Ezf ðzÞ, (4)

_2ðk2x þ k2yÞ

2m�e

f ðkx; kyÞ ¼ E==f ðkx; kyÞ. (5)

Setting the energy level at the top of the valenceband as zero, the wave function and the energylevel at G point for the electrons at the bottom of

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F. Gao, G. Xiong / Physica E 28 (2005) 412–418414

the conduction bands are

f nðkx; ky; zÞ ¼1ffiffiffiffiffiffiffiffiffiffiffi

LxLy

p exp½iðkxx þ kyyÞ

þ

ffiffiffiffi2

L

rsin

npz

Lð6Þ

E ¼ Eg þ_2ðk2x þ k2yÞ

2m�e

þ_2n2p2

2m�eL2, (7)

where Lx and Ly are the constants of normal-ization, which denote the length of quantum wellsin the direction of x and y, respectively, and Eg isthe energy gap.For the holes, the Bloch function is corresponding

to six band-edge functions of holes. So the envelopefunction of valence band can be expanded as 6� 6Luttinger–Kohn Hamiltonian matrix [13,14]:

H0ðkx; ky; kzÞ ¼

Aþ C iffiffi2

p C �iffiffiffi2

pB B 0

C� A� � iffiffi2

p ðAþ � A�Þ iffiffi32

qC 0 b

� iffiffi2

p C� iffiffi2

p ðAþ � A�Þ �Dþ 12ðAþ þ A�Þ 0 i

ffiffi32

qC i

ffiffiffi2

pB

iffiffiffi2

pB� �i

ffiffi32

qC� 0 �Dþ 1

2ðAþ þ A�Þ

iffiffi2

p ðAþ � A�Þiffiffi2

p C

B� 0 �iffiffi32

qC� � iffiffi

2p ðAþ � A�Þ A� �C

0 B� �iffiffiffi2

pB� � iffiffi

2p C� �C� Aþ

26666666666666666664

37777777777777777775

. ð8Þ

D here denotes the split of spin-orbit coupling.Aþ,A�, B, C have the following forms:

A� ¼ �_2

2m0½ðk2x þ k2yÞðg1 � g2Þ þ k2zðg1 2g2Þ,

B ¼ �

ffiffiffi3

p_2

2m0½g2ðk

2x � k2yÞ � i2g3kxky,

C ¼

ffiffiffi3

p_2

m0g3ðikx þ kyÞkz, ð9Þ

where g1; g2; g3 are Luttinger parameters. Similar tothe case of electrons, the envelope functions of holes

can be rewritten as

Fvðkx; ky; zÞ ¼1ffiffiffiffiffiffiffiffiffiffiffi

LxLy

p eiðkxxþkyyÞf vðzÞ. (10)

The effective mass equation for the valenceband is

H0ðkx; ky; zÞ½f mðzÞ ¼ Eðkx; ky; zÞ½f mðzÞ. (11)

The suffix m here denotes the terms of heavy holes,light holes and the spin-orbit split-off with theopposite spins. Near the top of valence band, f mðzÞ

can be expanded as

f mðzÞ ¼

ffiffiffiffi2

L

r XN

n¼1

cmn sin

npz

L. (12)

Combining (11) and (12), we can obtain that

½H½Cm ¼ E½Cm, (13)

where [H] is a 6N � 6N matrix obtained from (8).

qqz

� �mn

¼nð1� dmnÞ

L

1� cosðm þ nÞpm þ n

�

þ1� cosðm � nÞp

m � n

�, ð14Þ

q2

qz2

� �mn

¼ �mpL

2dmn.

½Cm ¼ ðCHH";CLH";CSO";CSO#;CLH#;CHH#ÞT,

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Ci ¼ ðCi1;C

i2;C

i3; . . . ;C

iNÞT. Thus, the effective

mass equation of holes is rewritten as a 6N � 6Nsecular equation. Given enough N, the wavefunction of holes near the top of valence band canbe got from (12), and E are the energies of subbandsof holes.According to the selection rules of the transi-

tion between valence and conduction bands,the transition between valence band and con-duction subband can exist only when Dn ¼ 0.Since the envelope function varies slowly, theelement of transition matrix, whose quantumnumber is n, is

hrpin ¼

ZC�

c ðrÞrpCvðrÞd3r

¼X6i¼1

hcjrpjvii

Zf �

c ðzÞfivðzÞdz

¼X6i¼1

hcjrpjviiCin. ð15Þ

Note that [15]

hcjrpjvii ¼ he; j0jrpjrq; ji

¼_

Ec � Ev

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEgðEg þ DÞ2m�

e ðEg þ23DÞ

sdj;j0dp;q. ð16Þ

In (16), rp;q¼1;2;3 ¼ x; y; z, dj;j0dp;q denotes thatthere is no transition between different directionsand different spins. The band-edge wave functionsof holes ðj3=2;�3=2i; j3=2;�1=2i; j1=2;�1=2iÞare shown in Ref. [9]. Take them into (16), andsetting

_

Ec � Ev

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEgðEg þ DÞ2m�

e ðEg þ23DÞ

s¼ O,

we can obtain the matrix element of dipoletransition between the nth subband ofvalence band and the nth subband of conductionband.

xe";v ¼C1nffiffiffi2

p �iC4nffiffiffi3

p þC5nffiffiffi6

p

� �O,

xe#;v ¼iC2nffiffiffi6

p þC3nffiffiffi3

p þiC6nffiffiffi2

p

� �O,

ye";v ¼iC1nffiffiffi2

p �C4nffiffiffi3

p �iC5nffiffiffi6

p

� �O,

ye#;v ¼ �C2nffiffiffi6

p þiC3nffiffiffi3

p þC6nffiffiffi2

p

� �O,

ze";v ¼ �i

ffiffiffi2

3

rC2n þ

C3nffiffiffi3

p

!O,

ze#;v ¼iC4nffiffiffi3

p þ

ffiffiffi2

3

rC5n

!O. ð17Þ

It can be noted from (17) that the dipole matrixis symmetrical in the directions of x and y.It indicates that the quantum wells are iso-tropic in the direction vertical to the z-axis.The electrons and holes are confined bythe potential field of quantum wells in thedirection of z. The polarizations in the directionsparalleling and vertical to the z-axis are quitedifferent. We calculated wð3Þðo� oþ oÞ in bothdirections.For the degenerated four-wave mixing of three

resonance between valence and conduction bands,there is [16]

wð3Þðo;o;o;oÞ

¼e4N

_3�0

Xv¼HH;LH;SO

hcjrmjvihvjrmjc0ihc0jrmjv

0i

�hv0jrmjciðr0c � r0vÞðo� ocv þ itcvÞ�1

�fðitvv0 Þ�1½ðo� oc0v0 � itc0v0 Þ

�1

� ðo� oc0v þ itc0vÞ�1

þ ðitcc0 Þ�1½ðo� oc0v0 � itc0v0 Þ

�1

� ðo� ocv0 þ itcv0 Þ�1g, ð18Þ

where N is the electron density in quantumwells, �0 the permittivity of vacuum, e theelectronic charge. c, c0 and v, v0 denote thestates of electrons and holes with different spins.r0c and r0v are the quasi-Fermi energies, r0c ¼

0:15 eV and r0v ¼ 0:05 eV. tcc and tvv are therelaxation time of the electrons and the holes(i.e., the reciprocal of the energy level line width ofthe electrons and the holes), respectively. tcc ¼

tvv ¼ t; tcv ¼ ðtcc þ tvvÞ=2 and t�1 ¼ 10�13 s. oij ¼

ðEi � EjÞ=_ is the energy separation betweensubbands i and j.

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3. Numerical results and discussion

The formulas of the energy levels and the wavefunctions of the valence and conduction bands ofquantum wells near G point in Brillouin zone havebeen given above. The parameters of GaN andInN at room temperature are shown in Table 1.The parameters of the band structure calculations

for III–V compound semiconductors are given inRef. [17]. The energy gap of the ternary semicon-ductors can be obtained from the following formula:

EgðA1�xBxÞ ¼ ð1� xÞEgðAÞ þ xEgðBÞ � xð1� xÞC.

(19)

In the case of InGaN, C ¼ 3 eV. The effectivemass parameter and Luttinger parameter can beobtained from

PðA1�xBxÞ ¼ ð1� xÞPA þ xPB. (20)

Fig. 2. The energy of the holes as a function of k*vertical to the

z-direction in In0.05Ga0.95N quantum wells.

Table 1

Material parameters for numerical calculations

GaN InN

Eg (eV) 3.299 1.94

D (eV) 0.017 0.006

m�e 0.15 0.12

g1 2.67 3.72

g2 0.75 1.26

g3 1.10 1.63

With these parameters, setting that the width ofthe wells is 7.0 nm and the concentration of In is0.05, we calculated the valence band structure(Fig. 2). The axis y denotes the energy and theenergy at the top of valence band is set to zero.The axis x denotes k

*vertical to the z-direction. The

carriers in the quantum wells can move freelywithout confinement vertical to the z-direction. Theyare confined in the direction paralleling to the z-axisand the energy levels appear to be discrete. It can benoted from (7) that the motions of electrons in thedirections of x, y and z are independent. The energylevels have the shape of parabola. Comparing withthe electrons, the motions of holes in the directionsof x, y and z are coupled. All the subbands areformed by the heavy holes, light holes and the spin-orbit split-off bands. In Fig. 2, the overlap of theenergy levels occurs because of the coupling ofsubbands.At room temperature, the electrons and holes in

the InGaN/GaN quantum wells are distributedmainly near the bottom of conduction band andthe top of valence band, respectively. Most ofcarriers are near the state of n ¼ 1, k

*¼ 0.

According the rules of selection, the transitionbetween valence and conduction bands can bepredigested as the transition between HH1, LH1and SO1 valence subbands and conduction sub-bands. We can obtain wð3Þ at the directions parallel-ing and vertical to the z-axis when the degeneratedfour-wave mixing occurs. wð3Þ spectra in both

Fig. 3. The third-order nonlinear optical susceptibility in the

directions paralleling and vertical to z-axis.

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Fig. 4. The third-order nonlinear optical susceptibilities in the

directions vertical to z-axis as the well width changes from 8.0

to 6.0 nm (a), and in the directions paralleling to z-axis as the

well width changes from 8.0 to 6.0 nm (b).

F. Gao, G. Xiong / Physica E 28 (2005) 412–418 417

directions are shown in Fig. 3, wð3Þ vertical to the z-direction has three peaks at 3.151, 3.158 and3.193 eV. The peak values of wð3Þ are 2:42� 10�9,2:21� 10�9 and 7:83� 10�11 esu. The locations ofthe three peaks are corresponding to the energydifferences of HH1, LH1 and SO1 to the firstsubband of electrons, respectively. In the z-direction,wð3Þ has only two peaks with the value of 1:76�10�10 esu at 3.158eV and 5:25� 10�9 esu at 3.193eV.The location of the peaks are corresponding to

the energy difference between LH1, SO1 and thefirst subband of electrons. It can be noted from(17) and (19) that the value of wð3Þ depends on theelements of the transition matrix and the locationof the peak of wð3Þ depends on the energy differencebetween subbands. Meanwhile, the contribution ofsubbands to wð3Þ is determined by the elements oftransition matrix. In the calculations for theGaAs-based quantum wells [9,10], the contribu-tion of spin-orbit split-off subband to wð3Þ can beneglected since the energy of spin-orbit split-off isrelatively large (D � 0:25 eV). It is very different inGaN-based quantum wells, the spin-orbit split-offbands should be included since they are not farfrom the other two bands in InGaN/GaN so thattheir contribution to wð3Þ is very important.According to (17), the transition between conduc-tion band and HH1 subband is only related to thewð3Þ vertical to the z-direction and the transitionbetween conduction band and LH1, SO1 subbandis related to wð3Þ in two directions. At roomtemperature, light holes mainly contribute to wð3Þ

vertical to the z-direction and the spin-orbit split-off mainly contributes to wð3Þ paralleling to the z-direction. This numerical result is also shown inFig. 3. The peak value of wð3Þ vertical to the z-direction is 1 order larger than the peak value ofwð3Þ paralleling to the z-direction at 3.158 eV, andthe peak value of wð3Þ paralleling to the z-directionsis 2 orders larger than the peak value of wð3Þ

vertical to the z-direction at 3.193 eV.To understand the correlations between wð3Þ and

the structure and the constituents of quantumwells, we performed a series of calculations on wð3Þ.Firstly, setting the concentration of In as 0.05, andvarying the width of the wells from 60 to 80 A, aset of wð3Þ spectra with different well-width areobtained, shown in Fig. 4. Then, setting the width

of the wells as 70 A, varying the concentration ofIn from 0.040 to 0.060, another set of wð3Þ spectrawith different concentration of In are obtained,shown in Fig. 5.It can be noted from Figs. 4 and 5 that, though

wð3Þ in the different directions are derived from thetransitions between different valence subbandsand conduction band, the trends of the locationsof the peaks and the peak values for wð3Þ in thedifferent directions are same when varying thewidth of the wells and the concentration of In.When the width of wells decreases, the location ofthe peak of wð3Þ occurs blue shift. Since the locationof the peak of wð3Þ is determined by the energydifference between conduction band and valencesubbands, the blue shift indicates the increase of

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Fig. 5. The third-order nonlinear optical susceptibilities in the

directions vertical to z-axis as the concentration of In changes

from 0.040 to 0.060 (a), and in the directions paralleling to z-

axis as the concentration of In changes from 0.040 to 0.060 (b).

F. Gao, G. Xiong / Physica E 28 (2005) 412–418418

this energy difference with the decrease of the wellwidth. The peak values of wð3Þ increases with thedecrease of the well width. When the concentrationof In increases, the location of the peak of wð3Þ

occurs red shift. It indicates the band gap ofquantum wells decreases. This leads to thedecrease of the energy difference between conduc-tion band and valence subbands and the increaseof the peak value of wð3Þ.In conclusion, we calculated the valence and

conduction band structures and wave functions ofGaInN/GaN quantum wells by the theory ofeffective mass. Under the approximation of nearresonance, wð3Þ owing to the degenerated four-wavemixing in the directions paralleling and vertical tothe z-axis are obtained. At room temperature, only

the transitions between conduction band andheavy-hole subbands and light-hole subbandscontribute to wð3Þ vertical to the z-direction, whilewð3Þ paralleling to the z-direction is determinedonly by the transition between conduction bandand spin-orbit split-off subbands. As the width ofquantum wells decreases, the band gap of quan-tum wells increases and it leads to the blue shift ofthe peak location of wð3Þ and the increase of peakvalue of wð3Þ. As the concentration of In increases,the band gap of quantum wells decreases. It leadsto the red shift of the peak location of wð3Þ and theincrease of the peak value of wð3Þ.

Acknowledgement

This work was financially supported by theNatural Science Foundation of Wuhan, China(Grant: 1320017010121).

References

[1] K. Tachibana, T. Someya, S. Ishida, Y. Arakawa, Appl.

Phys. Lett. 76 (2000) 3212.

[2] E. Martinez-Guerrero, F. Chabuel, B. Daudin, J.L.

Rouviere, H. Mariette, Appl. Phys. Lett. 81 (2002) 5117.

[3] Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939.

[4] A. Yariv, Appl. Phys. Lett. 52 (1988) 88.

[5] B. Shen, T. Someya, O. Moriwali, Y. Arakawa, Physica E

7 (2000) 939.

[6] O. Ambacher, B. Foutz, J. Smart, J.R. Shealy, N.G.

Weimasnn, K. Chu, M. Murphy, A.J. Sierakowski, W.J.

Scha5, L.F. Eastman, R. Dimitrov, A. Mitchell, M.

Stutzmann, J. Appl. Phys. 87 (2000) 334.

[7] F. Bernandini, V. Fiorentini, Phys. Rev. B 64 (2001) 5207.

[8] J. Liu, Y. Bai, G. Xiong, Physica E 23 (2004) 70.

[9] Y. Takahashi, A. Neogi, H. Kawaguchi, IEEE.

J. Quantum Electron. 34 (1998) 1660.

[10] Y. Takahashi, H. Kawaguchi, IEEE. J. Quantum Electron.

36 (2000) 864.

[11] J.M. Luttinger, W. Kohn, Phys. Rev. 97 (1955) 869.

[12] J.M. Luttinger, Phys. Rev. 102 (1955) 1030.

[13] J. Singh, Physics of Semiconductors and their Hetero-

structures, McGraw-Hill, New York, 1993.

[14] E.L. Ivchenko, G. Pikus, Superlattices and Other Hetero-

structures, Springer, Berlin, Germany, 1995.

[15] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249.

[16] Y.R. Shen, The Principles of Nonlinear Optics, Wiley,

New York, 1984.

[17] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl.

Phys. 89 (2001) 5815.