Peak Emission Wavelength Tuning for Light Emitting Diodes and Lasers for Quantum Well by varying the Composition of the Delta well

Saumya Biswas, Ifana Mahbub School of Engineering and Computer Science

Independent University Bangladesh Dhaka, Bangladesh

biswas.saumya@yahoo.com

Md. Shofiqul Islam, Saugata Biswas Bangladesh University of Engineering & Technology

Dhaka, Bangladesh shofiqul@eee.buet.ac.bd

Abstract— Four InGaN-delta NGaIn yy 1 Quantum Wells

(QW) with four different values of y (1, 0.9, 0.8, 0.7) were investigated through the numerical solution of a k.p Hamiltonian with the Finite Difference Method (FDM). The spontaneous emission spectra and optical gain spectra of the four structures have been found to have a definite pattern where increasing the value of y gradually shifted the spectra toward the higher energy regions. The electron–hole ground state overlaps were not found to change by a great amount. Increasing y brought about a predictable shift of the spectra toward the higher energy regions at the expense of electron-hole wave function overlap.

Keywords—delta; InN; InGaN; gainspectra

I. INTRODUCTION

The III Nitride based Quantum Wells (QW) are widely used in optoelectronic device applications. They cover a wide range of wavelength from Ultra Violet to Infra Red and the physics for their design is well developed [1],[2]. Delta InN wells were introduced as a scheme to increase the electron-hole wave function overlap and increase the radiative recombination efficiency for Light Emitting Diodes (LED) [3]. In case of lasers the delta wells have been employed to increase the band mixing and improve device characteristics [4]. The emission wave length of these devices depends critically on the barrier width, height, well width and strain. A deep delta well increases the electron-hole wave function overlap substantially. Decreasing the depth of the delta well will result in a decrease in the electron hole wave function overlap and consequently the radiative recombination efficiency. But the variation of the depth of the well by changing the composition of the delta well can lend itself to design purposes.

II. THEORICAL FORMALISM AND NUMERICAL SCHEME

A. Single Band Effective Mass Equation For the Conduction BandThe single band effective mass equation in the Envelope

Function Approximation (EFA) that needs to be solved for the wave functions and energy levels is the following: [5]

)()(2

),( 0222

zPzEmk

mk

kkH ccze

zte

tzt

c

(1)

where )(0 zEc is the potential energy profile for electron in

the heterostructure, )(zPc is the strain induced band edge

shift, )(zm te and )(zm z

e are the position dependent transverse and longitudinal effective masses respectively.

B. Six Band Block Diagonalized Effective Mass Equation for Holes The valence band structure of the bulk material can be

calculated by numerically solving the following eigenvalue equation of 6 coupled differential equations ignoring the conduction band-valence band coupling effects.

0)()(det kEkH vij

vij (2)

)(kH vij is a 66 matrix with the following Block

diagonalized form for the basis of zone-center Bloch functions presented in [5].

)(00)(

)(33

3366 kH

kHkH

L

Uv

(3)For heterostructures we have to solve the conduction and

valence band effective mass equation with

tt ik zik z

(4)

C. Electron and Hole Wavefunctions for the QW The wavefunction of the nth conduction sub-band can be

written as [5]

Independent University Bangladesh. (Sponsor)

NGadeltaInInGaN yy 1

207

,)()(

.

, SzA

ez n

rikc

kn

tt

t

(5)The hole wave function obtained from the upper

Hamiltonian for sub-band m can be written as [5]

3);(

2);(1);();(

)3(

)2()1(.

tm

tmtm

rik

tUm

kzg

kzgkzgA

ekztt

(6)For the lower Hamiltonian the expression is [5]

3);(

2);(1);();(

)6(

)5()4(.

tm

tmtm

rik

tLm

kzg

kzgkzgA

ekztt

(7) D. Calculation of Strain and Piezoelectric Polarization

spP and zpP ,

are respectively the spontaneous and piezoelectric polarizations of the layer. In our QW the growth axis is c(0001). The NGaIn xx 1 layer is grown on GaN substrate and have a lattice constant very close to that of GaN. The well is biaxially strained. The amount of strain can help us design the barrier height for the quantum well. For a strained layer wurtzite crystal grown pseudomorphically along the (0001) (z axis) the strain tensor has only the following non-vanishing diagonal elements:

aaa

yyxx0 xxzz C

C

33

1320zxyzxy

(8)

E. Internal Electric Field Fb and Fw are the internal electric fields in the barrier and

well respectively. These electric fields are the consequence of the polarization difference between the two layers. The electric fields are related to the difference in the sum of spontaneous and piezoelectric polarization in the two layers through the following relationship:

bwwb

bwwbbwwb LL

LPPF ,

0

,,,

)(

(9)

Where bwP , is the total polarization (spontaneous and

piezoelectric) in the well (barrier). bwL , is the length of the well (barrier).

zpsp

bw PPP ,, (10)

xxPZ CC

CCdP )2(233

213

121131 (11)

31d is one of the piezoelectric moduli.

F. The Finite Difference Method The coupled differential equation set was solved using the

Finite Difference Method (FDM) with a uniform grid [5]. The differential coefficients at any point were expanded in terms of the preceding and succeeding grid points with the following coefficients:

)(2

)()(

)(2

)()()(

)(2

)()(

)()(

121

211

121

2

2

iii

iiii

iii

zzzz

zgz

zAzA

zgz

zAzAzA

zgz

zAzA

zgzA

zzgzA

ii

(12)

)(4

)()()(

4)()(

)()(21)(

11

11

iii

iii

zzzz

zgz

zBzBzg

zzBzB

zBg

zgzB

zgzB

ii

(13)

This Finite Difference scheme ensures the Hermitian properties of the Hamiltonian and the enforcement of the boundary conditions.

G. Interband Momentum Matrix Elements The expressions for the interband Momentum Matrix

Elements for TE and TM polarization are the following [6]: TE- polarization ( ce axis): For U

2)2(2)1(

22

4)( mnmn

xtnmx gg

XpSkM

(14)For L

2)5(2)4(

22

4)( mnmn

xtnmx gg

XpSkM

(15)TM- polarization ( ze ˆˆ || c axis)

For U

2)3(

22

2)( mn

ztnmx g

ZpSkM

(16)For L

2)6(

22

2)( mn

ztnmx g

ZpSkM

(17)

208

H. Spontaneous Emission Rate The expressions for the spontaneous emission rate per unit

volume per energy interval at an optical energy is given by [6]

)()( 22

22esp

rsp g

cnr

(18)

22,

2

, , ,200

2

)())(()/))((1)((

)(.ˆ22

)(

tcv

nm

tmtc

n

tnmtt

LU mnzr

esp

kEkfkf

kMeddkkLmcn

qg

(19)

I. Optical Gain Spectra The relationship between material gain and the spontaneous

emission rate is [6]

]exp1)[()(Tk

FggB

esp

(20)Where

vc FFF (21) is the separation between the quasi Fermi levels for electrons and holes.

III. RESULTS AND DISCUSSION

Our structures showed eligibility to be employed in LEDs and lasers both at certain carrier injection levels. The value of y could tune the peak emission wavelength from 690 nm to 720 nm. The emission spectra are sensitive to the internal electric field and consequently on the barrier width as well as the well width. So the calculated peak emission wavelengths are specific to layer widths used. In all four structures the layer widths were the following: 2.8 nm GaN/1 nm InGaN/ 0.4 nm delta well/ 1 nm InGaN/2.8 nm GaN. The necessary k.p parameters were taken from [2].

Table 1 presents the data that illustrates how the composition of the delta well can tune the peak emission wavelength.

TABLE I.

Structure

Results

yElectron-Hole wavefunction

overlap(%)

Peak Emission Wavelength( nm)

1 1 23.14 721.38

2 0.9 21.38 708.97

3 0.8 19.19 697.02

4 0.7 16.89 685.47

Table1. Electron-Hole wavefunction overlap and peak emission wavelength of the four structures

These structures may be used for Emitters in the UV range. We may be able to tailor their emission characteristics to our specific wavelength necessity. Ordinarily the peak emission wavelength depends mostly on the ground state electron and hole energy levels in a emitter made of nonpolar materials. Changing the well width may be the most suitable way of designing for maximizing emission in a specific wavelength. But in Nitride materials the internal electric field complicates the design procedure.

Fig.1. The conduction and valence band edge energies (right axis) and ground state moduli squared electron and hole wavefunction (left axis)

We found the emission spectra to shift toward the shorter wavelength regions with an increase in the value of y i.e. decrease in the depth of the delta well. The peak of the emission spectra decreased a little and the overlap between electron and hole wavefunction reduced slightly. But the change in the delta well composition lends itself to the output wavelength tuning in a very convenient way.

Fig.2. Spontaneous Emission Spectra for the four structures under carrier

injection levels 18105 , 18108 and 181010 3cm

We present the TE (Transverse Electric) and TM (Transverse Magnetic) optical spectra separately because it

1600 1700 1800 1900 2000

0.00E+000

1.00E+032

2.00E+032

3.00E+032

4.00E+032

Spon

tane

ous

emis

sion

spe

ctru

m (

s-1 e

V-1 c

m-3 )

photon energy (meV)

structure 1 structure 2 structure 3 structure 4

0 20 40 60 80

0.00

0.05

0.10 electron hole

distance ( angstroms)

-2

0

2

4 Ec Ev0

209

enables us to understand the relative contributions of forces at play better. Our structures had a dominant TE gain.

Fig.3. Optical gain for the four structures under the carrier injection levels 19105 and 19107 3cm

IV. CONCLUSION

For our particular well and barrier widths the peak emission wavelength tuning could be done very conveniently. The

structures were suitable for designing in a large range of wavelengths.

REFERENCES

[1] H. Zhao, G. Liu, J. Zhang, J. D. Poplawsky, V. Dierolf, and N. Tansu, “Approaches for high internal quantum efficiency green InGaN light-emitting diodes with large overlap quantum wells,” OPTICS EXPRESS, vol. 19, pp A991-A1007, July 2011

[2] H. Zhao, R. A. Arif, Y. Ee, and N. Tansu, “ Self-Consistent Analysis of Strain-Compensated InGaN-AlGaN Quantum Wells for Lasers and Light-Emitting diodes.” IEEE Journal of Quantum Electronics, vol. 45, pp 66-78, Jan. 2009.

[3] H. Zhao, G. Liu, N. Tansu, “ Analysis of InGaN -delta- InNquantum wells for light-emitting diodes,” Applied Physics Letter, vol. 97, pp 131114, October, 2010.

[4] J. Zhang, H. Zhao, N. Tansu, “ Large optical gain AlGaN -delta-GaN quantum well laser active regions in mid- and deep-ultraviolet spectral regimes,” Applied Physics Letter, vol. 98, pp 171111, April 2011.

[5] S. L. Chuang and C. S. Chang, “ A band-structure model of strained quantum-well wurtzite semiconductors,” Semicond. Sci. Technol., vol. 12, pp 252-263, 1997.R. Kajitani, K. Kawasaki and M. Takeuchi, “ Barrier-height and well-width dependence of photoluminescence from AlGaN-based quantum well structures for deep-UV emitters,” Materials Science and Engineering B, vol. 139, pp 186-191, 2007.

[6] S. L. Chuang, “Optical Gain of Strained Wurtzite GaN Quantum-Well Lasers,” IEEE Journal of Quantum Electronics, vol. 32, pp 1791-1800, October, 1996.

1500 2000

0

500

1000 Structure1, TE Structure1, TM Structure2, TE Structure2, TM Structure3, TE Structure3, TM Structure4, TE Structure4, TM

Opt

ical

Gai

n (c

m-1)

photon energy (meV)

210