Abstract—The objective of this paper is to study the (com-

peting) effects of various internal (built-in) fields on the elec-

tronic structure and optical properties of wurtzite InN/GaN

quantum dot light emitting diodes (LEDs). A multiscale

approach has been employed where: 1) the atomistic NEMO 3-

D tool is used to calculate the strain distribution and one-

particle electronic states, and 2) using the NEMO 3-D outputs,

the Synopsys TCAD tool is then used to determine the terminal

electrical and optical properties of the device. A list of the main

findings is as follows: 1) Internal fields are long-ranged and

their proper treatment demands simulation of realistically

extended structures with millions of atoms; 2) Electronic struc-

tures show unconventional characteristics related to level shifts,

non-degeneracy in the excited P states, and rotation (symmetry

breaking) of the wavefunctions; and 3) Internal fields in these

nanostructured LEDs lead to strong suppression in the

interband optical transitions (near the center of the Brillouin

zone) and the conversion efficiency.

I. INTRODUCTION

N recent years, light-emitting diodes (LEDs) based on

wide-bandgap semiconductors have drawn much attention

for applications in lasers, fiber-optic communications, solid-

state lighting, solar cells, sensors, digital photography and

imaging, consumer displays, and diagnostic medicine. In the

last decade, GaN and its related alloys, especially InGaN,

have been viewed as the most promising materials for appli-

cations in light emitting optoelectronic devices because of

several potential advantages, such as, wide range of emis-

sion frequencies (near-infrared to near-ultraviolet), high

efficiency, low power consumption, fast switching time,

high color gamut, and low sensitivity to ionizing radiation

[1][2]. The impacted markets include energy, digital infor-

mation, healthcare, manufacturing, communications, and

security. For example, nitride-based solid-state lighting

(SSL) devices promise to replace conventional light sources,

with impressive economic and environmental savings. By

the year 2025, using SSL devices, worldwide electricity

consumption for lighting could be cut by more than 30%,

sparing the atmosphere ~28 million metric tons of carbon

emission annually [3]. SSL devices have the potential of

being over 90% efficient, are vibration and shock resistant,

and exceptionally long-lived (>50,000 hours). LEDs also

offer controllability of their spectral power distribution, spa-

tial distribution, color temperature, temporal modulation,

Manuscript received June 15, 2011. This work is supported by

ORAU/ORNL High-Performance Computing Grant 2009 and partially by

SIUC ORDA Seed Grant 2010.

The authors are with the Department of Electrical and Computer

Engineering, Southern Illinois University at Carbondale, IL 62901 USA

(phone: 618-453-7630; fax: 618-453-7972; email: ahmed@siu.edu).

and polarization properties. Such smart light sources can

adjust to specific environments and requirements.

An important step to design InGaN-based LEDs is the re-

alization of efficient active layers through appropriate

bandgap engineering. Multi-quantum well (MQW) low- in-

dium-content LEDs (less strain effects) are the current state-

of-the-art, offering mainly short-wavelength (blue and

green) emissions. Very recently, optical devices using

InGaN nanostructures have attracted much attention due

mainly to the following reasons: 1) In nanostructures, relax-

ation of elastic strain at free surfaces allows the accommo-

dation of a broader range of lattice mismatch and band-

lineups than is possible in conventional bulk and thin-film

quantum well heterostructures, and, therefore, threading

dislocations can be less in these nanostructures; 2)

Nanostructures, having large-indium-content and less strain

induced effects, can be used in full-solar-spectrum LEDs and

solar cells; and 3) Furthermore, nanostructures used in the

active region of optical devices provide better electron con-

finement (due to strongly peaked energy dependence of den-

sity of states) and thus a higher temperature stability of the

threshold current and the luminescence (enhanced radiative

recombination) than quantum wells. Most InGaN

nanostructures have been realized from 0-D quantum dots

[4][5] and some from 1-D nanowires/nanorods [6]. Since the

heteroepitaxy of InN on GaN involves a lattice mismatch up

to ~11%, a form of Stranski-Krastanov mode can be used for

growing InN on GaN by molecular beam epitaxy (MBE).

Recent studies have shown that the strain between InN and

GaN can be relieved by misfit dislocations at the hetero-

interface after the deposition of the first few InN bilayers

and before the formation of InN QDs. Relaxed InN islands

with controllable size and density can be formed [7] by

changing the growth parameters (such as temperature) in

either MBE or metalorganic chemical vapour deposition

(MOCVD).

Knowledge of the electronic bandstructure of nanostruc-

tures is the first and an essential step towards the under-

standing of the optical performance (luminescence) and

reliable device design. The great majorities of InN/GaN QDs

crystallize in the thermodynamically stable configuration

with a wurtzite crystal structure and are grown along the

polar [0001] direction [8]. These structures exhibit large

spontaneous and strain-induced polarization. These effects

lead to a large internal electrostatic field, which is unique to

III-nitride heterostructures and has a significant effect on the

electronic and optical properties of the LEDs. The magni-

tude of the electrostatic built-in field has been estimated to

be on the order of MV/cm [8]. Such fields spatially separate

Multiscale Modeling of Wurtzite InN/GaN Quantum Dot LEDs

Krishna Yalavarthi, Sasi Sundaresan, and Shaikh Ahmed, Member, IEEE

I

2011 11th IEEE International Conference on NanotechnologyPortland MarriottAugust 15-18, 2011, Portland, Oregon, USA

978-1-4577-1515-0/11/$26.00 ©2011 IEEE 881

the electrons and holes, which leads to a reduction in the

optical transition rate (oscillator strength) and enhanced ra-

diative lifetimes. Hexagonal group-III nitride 2-D quantum

well (QW) heterostructures have experimentally been shown

to demonstrate polarized transitions in quantized electron

and hole states and non-degeneracy in the first excited state

in various spectroscopic analyses [9]. While 0-D QDs (and

1-D nanowires) promise better performance, only very few

experimental results exist concerning the photoluminescence

(PL) and electroluminescence (EL) of these III-N

nanostructures in the visible spectral region [5][6][10]. Simi-

lar to the 2-D QW structures, the optical properties of the

QDs are expected, to a large extent, to be determined by an

intricate interplay between the structural and the electronic

properties, and (since not yet been fully assessed experi-

mentally) demand detailed theoretical investigations.

In this paper, we study the electronic bandstructure and

optical properties of wurtzite InN/GaN quantum dots. The

main objectives are three-fold—1) To explore the origin and

nature of various built-in fields including crystal atomicity,

strain fields, piezoelectric, and pyroelectric potentials; 2) To

quantify the role of these internal fields on the electronic

bandstructure in terms of shift in energy levels, split (non-

degeneracy) in the excited P states, anisotropy in the wave-

functions, and strongly suppressed optical transitions, and 3)

Finally, demonstrate how the atomistically-calculated optical

transition rates can be integrated into a commercial LED

TCAD tool (Synopsys) to calculate the terminal electrical (I-

V) and optical properties of reduced-dimensional LEDs.

II. SIMULATION MODELS

As described in the previous section, efficiency and opti-

cal performance of nanostructured LEDs is determined by an

intricate interplay of internal fields, electronic bandstructure

effects, and dynamics of charge and phonon transport phe-

nomena. Therefore, any efforts of modeling these devices

involve a multiphysics problem and tackle a large number of

identified hurdles of scientific uncertainty. To address these

needs, in this work, we have employed a multiscale simula-

tion approach, which essentially bridges the gap between

contemporary continuum and ab initio models and have the

capability of handling realistically-sized devices containing

more than 100 million atoms! The simulation strategy is

divided into different computational phases spanning from

the atomistic structure of the constituting elements to the

electron transport and optical coupling and is depicted in

Figure 1. The Figure also shows the various length and time

scales and the associated observables and how one passes

between them, and the codes/tools used.

It is clear that, at nanoscale, electronic structure modeling

approaches based on a continuum representation (such as

effective mass, and k•p) are invalid. Continuum models

assume the symmetry of the nanostructure to be that of its

overall geometric shape. For example, in quantum dot

simulations using continuum models, dome-shaped dots are

assumed to have continuous cylindrical symmetry C∞ν,

whereas pyramidal dots are assumed to have C4ν symmetry.

In a recent effort on modeling In1-xGaxN quantum dots using

k•p approach [11], it was found that the envelope S function

reproduces the symmetry of the confining potential, the ex-

cited P and D states are energetically degenerate and opti-

cally isotropic—a group of observations that clearly sup-

presses the true fundamental atomistic symmetry of the

underlying crystal and thus overestimates the quantum

efficiency of the light emitters in these quantum dots.

For computing the atomistic strain distribution and the

electronic structure (energy eigenvalues and

wavefunctions), we have used the extended version of open

source NEMO 3-D tool. Detail description of this package

can be found in Ref. [12][13][14]. Using a variety of tight-

binding models (s, sp3s*, sp3d5s*) that are optimized with a

genetic algorithm tool (PGApack), NEMO 3-D currently

enables the computation of electronic structure for over 52

million atoms, corresponding to a volume of (101nm)3.

Tight-binding (as opposed to other Empirical methods such

as pseudopotentials [15]) is a local basis representation,

which naturally deals with finite device sizes, alloy-disorder

and hetero-interfaces and it results in very sparse matrices.

The requirements of storage and processor communication

are therefore minimal compared to pseudopotential

implementations and perform extremely well on inexpensive

Linux clusters. For the calculation of atomistic (non-linear)

strain relaxation, NEMO 3-D employs the atomistic valence-

force field (VFF) with strain-dependent Keating potentials

and can handle over 100 million atoms corresponding to a

volumes of (125nm)3! This versatile software currently

allows the calculation of single-particle electronic states and

optical response of various semiconductor structures

including bulk materials, quantum dots, impurities, quantum

wires, quantum wells and nanocrystals. NEMO 3-D includes

spin in its fundamental atomistic tight binding rep-

resentation. Effects of interaction with external

electromagnetic fields are also included. Excellent parallel

scaling up to 8192 cores on various TOP500 HPC machines

has been demonstrated with NEMO 3-D.

The overall polarization P in a typical wurtzite semicon-

ductor is given by P = PPZ + PSP, where PPZ is the strain-

induced piezoelectric polarization and PSP is the spontaneous

Fig. 1. The integrated multiscale simulation platform used in this work.

882

polarization (pyroelectricity). Previous theoretical investiga-

tions on piezoelectric effects in III-nitride structures have

mainly focused on 2-D quantum wells and either neglected

the strain fields (in cladding layers), or used a linear har-

monic strain model [11]. In clear contrast, in our work, the

piezoelectric polarization PPZ is obtained from the diagonal

and shear components of the anisotropic atomistic strain

fields. Also, in contrast to zincblende semiconductors, in III-

V nitride based devices the spontaneous polarization is an

unavoidable source of large electric fields even in lattice-

matched (unstrained) systems. The spontaneous polarization

is dependent on the material and has nonzero component

only along the polar [0001] (growth) direction. The polari-

zation constants used in this study are taken from Ref. [11],

whereas the tight binding parameters and strain constants are

taken from Ref. [16] and the small thermal strain contribu-

tion is neglected. Also, it is a key point to notice that, PPZ

may have (due to its strain dependence) the same or the op-

posite sign with respect to the fixed PSP depending on the

epitaxial relations. The polarization induced charge density

is derived by taking divergence of the polarization. To do

this, we divide the simulation domain into cells by rectan-

gular meshes. Each cell contains four cations. The

polarization of each grid is computed by taking an average

of atomic (cations) polarization within each cell. A finite

difference approach is then used to calculate the charge den-

sity by taking divergence of the grid polarization. Finally,

the induced potential is determined by the solution of the 3-

D Poisson equation on an atomistic grid (using an in-house

PETSc-based [17] parallel full 3-D Poisson solver).

On the other hand, the spontaneous optical emis-

sion/absorption rate is given by [18]:

where, Pcv is the momentum matrix and depends on the light-

polarization. P is probability of hole occupation and electron

vacancy, and for P = 1, absorption rate solely depends on the

momentum matrix, where absorption and emission lose their

meaning and we use a term of transition rate. Note that for

nanostructures with finite size, to calculate Pcv, one needs to

integrate with respect to the entire domain. To dermine the

terminal electrical and optical properties, we have used

Synopsys’s 3 core TCAD tools namely, Sentaurus Structure

Editor, Sentaurus Device, and Tecplot or Inspect. Sentaurus

Device includes [18] models for the comprehensive simula-

tion of LEDs, which solves drift-diffusion or hydrodynamic

transport equations for the carriers, the Schrödinger equation

for gain in the active optical and optical rate equations, and

the Helmholtz equations self-consistently in the quasi-sta-

tionary and transient modes. Photon recycling is another

important model used to predict the light trapping in the

device by total internal reflection.

III. SIMULATION RESULTS

Figure 2 shows the simulated wurtzite InN/GaN quantum

dot (which is used in the core active region in the LED). The

QD is grown in the [0001] direction (c-axis) on GaN buff-

ered substrate (actual LED substrate may be SiC or

sapphire), has diameter, d~11 nm and height, h~5 nm, and is

positioned on a one atomic-layer thick InN wetting layer.

The simulation of strain is carried out in the large computa-

tional box, while the electronic structure is restricted to the

(smaller) inner domain. All the strain simulations fix the

atom positions on the bottom plane to the GaN lattice con-

stant, assume periodic boundary conditions in the lateral

dimensions, and open boundary conditions on the top sur-

face. The strain parameters used in this work were validated

through the calculation of Poisson ratio of the bulk materials

(0.2743 for GaN, and 0.2798 for InN). The inner electronic

box assumes a closed boundary condition with passivated

dangling bonds.

In the strain calculation, the equilibrium atomic positions

are determined by minimizing the total elastic energy of the

system. The total elastic energy in the VFF approach has

only one global minimum, and its functional form in atomic

coordinates is quartic. From our calculations, as shown in

Figure 3, atomistic strain was found to be long-ranged (pen-

etrating ~20 nm into the substrate and the cap layers)

stressing the need for using realistically-extended structures

(multimillion-atom modeling) in modeling electronic struc-

ture of these QDs. The conjugate gradient minimization

algorithm in this case was found to be well-behaved and

stable. The biaxial strain (εxx+ εyy-2εzz) was found to be

negative in the lower part of the quantum dot.

The net polarization and polarization-induced potential is

calculated using a parallel full 3-D Poisson solver. Both the

piezoelectric and the pyroelectric potentials are found to be

significantly large (tens of meV), anisotropic in the lateral

and vertical planes, and long-ranged (Figure 4). The pyroe-

lectric potential is significantly larger and asymmetric and

tends to oppose the piezoelectric counterpart. This also

suggests that for an appropriate choice of alloy composition

and quantum dot size/geometry, spontaneous and

piezoelectric fields may be caused to cancel out!

Fig. 2. Left: Dome shaped wurtzite InN quantum dot on a thin (one atomic

layer) wetting layer. Right: Atom distribution (reduced view). Delec: central

smaller domain for electronic structure (quantum) calculation, and Dstrain:

outer domain for strain calculation. In the figure: s is the substrate height ~

30 nm, c is the cap layer thickness ~10 nm, h is the dot height ~5 nm, and d

is the dot diameter ~11 nm.

883

Next we calculate the electronic structure of the quantum

dot. Here, we quantify the contributions of inter-

face/atomistic symmetry (without strain), strain,

piezoelectricity, and pyroelectricity separately by calculating

the shift in the conduction band (CB) ground state, splitting

in the P level, and wavefunction orientations. When needed,

the influence of the piezoelectric and the pyroelectric fields

are incorporated in the Hamiltonian as an external potential

(within a non-self-consistent approximation). Figure 5 shows

the topmost valence (HOMO) and first four conduction band

wavefunctions (projected on the X-Y plane) for the quantum

dot. In the first row, where the effects of strain relaxation,

piezoelectricity, and pyroelectricity are all excluded, the

small split (non-degeneracy) in the P level (1.476 meV) is

due mainly to the atomistic interface and fundamental mate-

rial discontinuity in the underlying device structure. In the

second row, atomistic strain is included resulting in a -2.79

meV split and a flip in the P level. The first P state is ori-

ented along the [110] direction and the second along [110]

direction. In the third row, piezoelectricity is included on top

of strain, which, while retaining the polarization in the P

states, induces a split of -17.06 meV and results in a mixed

D band. In the fourth row, a combined effect of strain, pie-

zoelectricity and spontaneous polarization (PSP) is shown.

The inclusion of PSP results in a mixed ground CB state and

a flip in the P states, and induces an overall split of -31.5

meV. Importantly, while the lowest CB state (unoccupied

molecular orbital, LUMO) retains an S-character, the top-

most valence (HOMO) assumes an unconventional P-

character. Figure 6 shows the influence of all of the four

types of internal fields on the single-particle conduction

band ground states (left panel) and split in the P level (right

panel) in the quantum dot. One can see that: 1) strain, while

modifying the effective confinement volume, introduces a

pronounced blue shift in the conduction band ground state,

2) piezoelectricity causes a red shift, and finally 3) pyroe-

lectricity, while opposing the piezoelectric contribution,

introduces a blue shift and a strong P-split.

Figure 7 shows the polar plots of the interband optical

transition rates between ground hole (HOMO) and ground

electronic states (LUMO) in the quantum dot without strain

(top panel) and with strain field (bottom panel). The Figure

reveals significant suppression and strong polarization

anisotropy due to spatial irregularity (rotation) in the

wavefunctions. The true atomistic symmetry of the quantum

dots, thus, influences both the electronic bandstructure and

Fig. 6. Ground conduction band and split in P level in the QD including

interface effects (w/out strain), strain, piezoelectricity, and pyroelectricity.

0

0.5

1

1.5

2

2.5

3

W/O STRAIN PIEZO PYRO

CB

GR

OU

ND

EN

ERG

Y [

eV

]

-35

-30

-25

-20

-15

-10

-5

0

5

P-L

EVEL

SP

LIT

∆E P

[me

V]

W/O STRAIN PIEZO PYRO

Fig. 3. Atomistic strain along the growth ([0001]) direction through the

center of the QD. Strain is seen to penetrate deep into the substrate and the

cap layers. Also, noticeable is the gradient of strain inside the dot region.

Number of atoms simulated: 1.8 million (strain domain), 0.8 million

(electronic domain).

Fig. 4. Polarization induced potential along the z direction. Note the spread

of the potential in the substrate and the cap layers.

Fig. 5. Quantum dot wavefunctions due to 1) interface, 2) strain, 3)

piezoelectric, and 4) pyroelectric fields—all resulting in shift in energy

spectrum. Wave functions in InN dot showing deformed valence band,

conduction band P-level anisotropy and non-degeneracy, and formation of

mixed orbitals resulting from these competing fields.

E010 – E100 = 1.476 meVE0 = 2.29 eV

E010 – E100 = -2.79 meVE0 = 2.77 eV

E010 – E100 = -17.06 meVE0 = 2.38 eV

E010 – E100 = -31.5 meVE0 = 2.65 eV

(1)

(2)

(3)

(4)

884

the strengths of the optical transitions. The anisotropy ratios

for w/out and with strain cases were found to be 0.032 and

0.99 respectively (obtained from actual transition values).

In the next phase, the output results (optical transition rate

and its degradation due to strain field) from NEMO 3-D are

integrated into the Synopsys’s TCAD tool to simulate the

terminal electrical and optical characteristics of an InN/GaN

quantum dot LED. Sentaurus structure editor is used to de-

fine the LED structure in a parameterized manner. The

structure consists of 40 nm thick sapphire (Al2O3), 4 nm

thick n-type GaN wide buffer layer, 18 nm thick n-type GaN

buffer layer, the quantum dot along with a 2 nm thick Al-

GaN window layer, which is followed by 5 nm thick p-type

GaN Cap layer and 2 nm thick p-type GaN FER (Fast

Electron Recombination) layer. In the following, we present

some of our recent simulation results using the integrated

platform. Figure 8 shows the active region filled with an

abundance of carriers (orange represents the holes and blue

represents the electrons). The right panel of this Figure

shows the energy band diagram of the LED at a drive volt-

age of 2.7 V (which is less than 2.95 V, the turn-on voltage

of the LED). The local spontaneous emission as a function

of energy is computed and shown in Figure 9. The local

spontaneous emission is given by [18]:

( ) ∑∫ | | ( )

( ) ( ) ( )

where,

D(E) is the reduced density of states,

| | is the overlap integral of quantum mechanical wave-

functions, and Pij is the polarization dependent factor of the

optical momentum matrix | | . These emission coeffi-

cients determine the rate of production of photons when

given the number of available quantum dot carriers at the

active vertex. ksp is the scaling factor for the optical matrix

| | which is determined from the NEMO 3-D calcula-

tions. Figure 10 shows the internal quantum efficiency (IQE)

of the device. The inset shows the I-V characteristics of the

device without and with the statin field. To model the optical

emission, Synopsys uses the raytracing method which ap-

proximates the optical intensity inside the device as well as

the amount of light that can be extracted from the device

making it computationally less complex. Figure 11 shows

the LED light output emission pattern. Raytracing does not

contain phase information, so it is not possible to compute

the far-field pattern for an LED structure. Instead, the

outgoing rays from the LED raytracing are used to produce

the radiation pattern.

Fig. 7. Polar plots of the interband optical transition rates between ground

hole (HOMO) and ground electronic states (LUMO) in the quantum dot

without strain (top panel) and with strain field (bottom panel).

Fig. 8. Space charge concentration (left) and the energy band diagram

(right) of the LED at bias voltage of 2.7 V (which is less than 2.95 V, the

turn-on voltage of the LED).

Fig. 9. Local spontaneous emission as a function of energy.

885

IV. CONCLUSION

A multiscale (spanning from the fundamental atomic

structure to the realistically-extended device) and mul-

tiphysics (molecular mechanics, quantum electronics

structure, transport, and optical coupling) approach has been

used to study the terminal properties of a nanostructured

InN/GaN quantum dot LED. Here, the atomistic NEMO 3-D

tool is used to calculate the strain distribution and one-parti-

cle electronic states, and the Synopsys TCAD tool is used to

determine the terminal electrical and optical properties of the

device. The internal fields are found to be long-ranged

(spreading ~16 nm in the substrate) and their proper treat-

ment demands simulation of realistically extended structures

with at least 2 million of atoms. Electronic structures show

unconventional characteristics related to shift in the energy

spectrum, mixed HOMO and LUMO bands, non-degeneracy

in the excited P states, and rotation (symmetry breaking) in

the wavefunctions. True atomistic symmetry due to the pres-

ence of the internal fields in these nanostructured InGaN

LEDs lead to strong suppressions in the interband optical

transitions (near the center of the Brillouin zone) and the

conversion efficiency.

ACKNOWLEDGMENT

This work was supported by the ORAU/ORNL High-Per-

formance Computing Grant 2009 and partially by SIUC

ORDA Seed Grant 2010. Computational resource supported

by the National Science Foundation under Grant No.

0855221 and access to Synopsys TCAD tools are also

acknowledged. Currently, the open source NEMO 3-D code

is being maintained by Gerhard Klimeck at Purdue

University. Discussion with Muhammad Usman and Hoon

Ryu is also acknowledged.

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886