Opt Quant ElectronDOI 10.1007/s11082-013-9806-x

How important is nonlinear piezoelectricity in wurtziteGaN/InN/GaN quantum disk-in-nanowire LEDstructures?

Krishna Yalavarthi · Vinay Chimalgi · Shaikh Ahmed

Received: 13 June 2013 / Accepted: 10 October 2013© Springer Science+Business Media New York 2013

Abstract Numerical investigation is carried out to compare the effects of linear and non-linear piezoelectric fields on the electronic and optical properties of recently-proposedwurtzite GaN/InN/GaN quantum disk-in-wire LED structures. The computational frame-work employs a combination of fully atomistic valence force-field molecular mechanics and10-band sp3s*-SO tight-binding electronic band-structure models, and accurately capturesthe interplay between the long-range electro-mechanical fields and the quantum atomicity inthe device. In particular, to model piezoelectricity in the wurtzite lattice, four different polar-ization models (based on the experimental and ab initio coefficients) have been considered inincreased order of accuracy. In contrast to recent studies on thin-film quantum well structures,simulation results obtained in this work show that the nonlinear (second-order) piezoelec-tric contribution has insignificant effects on the overall electronic and optical properties inreduced-dimensionality (nanoscale) disk-in-wire LED structures.

Keywords InGaN · Disk-in-wire LED · Non-linear piezoelectricity · Optical anisotropy ·Atomistic simulation

1 Introduction

Recently, optical emitters using InGaN nanostructures (quantum dots and nanowires) haveattracted much attention for applications in lasers, solid-state lighting, solar cells, consumerdisplays, as well as diagnostic medicine and biological imaging. Compared to conventionalbulk and quantum well (QW) structures, nanostructured devices offer several benefits as fol-low (Li et al. 2011; Merrill et al. 2012): (1) Strain in nanostructures is, to a large extent, relaxedand, therefore, threading dislocations can be smaller leading to increased quantum efficiency;(2) The fact that the concentration of strain-induced defects is small in nanostructures allows

K. Yalavarthi · V. Chimalgi · S. Ahmed (B)Department of Electrical and Computer Engineering, Southern Illinois University at Carbondale,1230 Lincoln Drive, Carbondale, IL 62901, USAe-mail: ahmed@siu.edu

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the use of higher indium content and more design freedom in bandgap engineering in thedevice, which potentially could lead to full-spectrum LEDs (as well as solar cells); and (3)Nanostructures used in the active region of optical devices provide improved electron con-finement (due to strongly peaked energy dependence of density-of-states) and thus highertemperature stability of threshold current and luminescence.

Most InGaN nanostructures have been realized from 0-D quantum dots (Ke et al. 2006;Kalden et al. 2010; Jarjour et al. 2008) and 1-D nanowires/nanorods (Li et al. 2011; Luet al. 2011). For example, in Lu et al. (2011), ultra-small single InGaN disk-in-wire LEDswere fabricated and shown to have potential for applications in near-field, subwavelengthphotolithography by controlling the exposure time and light intensity at chosen wavelengths.Also, recently, it has been suggested that short-wavelength single-photon emitters in III-nitrides would be useful for free-space quantum cryptography (Saniya et al. 2013). The greatmajorities of these devices crystallize in the thermodynamically stable wurtzite symmetryand are grown along the polar [0001] direction. Since the heteroepitaxy of InN on GaNinvolves a lattice mismatch of ∼11 %, these structures generally exhibit atomically inhomo-geneous and long-range polarization fields (on the order of MV/cm). In our recent work onInGaN disk-in-wire LEDs (Merrill et al. 2012), we have shown that, while relaxed activelayers translate to reduced quantum confined Stark effect (QCSE), true atomistic symme-try coupled with quantum mechanical size quantization effects in nanostructures result in:(1) Unconventional characteristics in the electronic structure related to non-degeneracy inthe excited P states and rotation (symmetry lowering) in the wavefunctions; and (2) A giantgrowth-plane optical polarization anisotropy. Both of these effects are strong function of InNdisk thickness and lead to overall degradation in internal quantum efficiency of LEDs madefrom these nanostructures. Nevertheless, in contrast to the frequently used first-order (linear)model, the piezoelectric polarization is generally a non-linear function of strain. Recently,it has been shown (Pal et al. 2011) that the piezoelectric polarization in strained InN/GaNquantum well devices (with 1-D quantum confinement) has strong contribution from thesecond-order effects that have so far been neglected. However, to the best of our knowledge,there is no theoretical and/or experimental report on the origin and effects of second-orderpiezoelectric polarization in nitride nanostructures (with 2-D and/or 3-D quantum confine-ment). In this work, our purpose is to computationally investigate the effects of linear andnonlinear piezoelectric fields on the electronic and optical properties of realistically-sizedwurtzite GaN/InN/GaN disk-in-wire nanostructures.

2 Simulation model

The overall simulation strategy is divided into four coupled phases: (1) Geometry construc-tion; (2) Atomistic structural (strain) relaxation; (3) Computing polarization fields; and (4)Determining the electronic structure and optical transitions. The purpose of the geometricconstructor is to create (from a basis set) the nanostructure having wurtzite symmetry andstore the atomistic details (atom type, coordinates, nearest neighbors, surface passivation,and computation model) into the memory of the computer. Initially, the atom positions inthe entire computational domain (including those of InN) are fixed to GaN lattice constant.Then, the atom positions are relaxed and the resulting strain (mechanical) fields are calculatedemploying an atomistic valence force-field (VFF) method using the Keating potentials. Inthis approach, the total elastic energy of the sample is computed as a sum of bond-stretchingand bond-bending contributions from each atom. The equilibrium atom positions are foundby minimizing the total elastic energy of the system. However, piezoelectricity is neglected

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Table 1 Piezoelectric coefficients used in the study

GaN InN

e15 (C/m2) e31(C/m2) e33(C/m2) e15(C/m2) e31(C/m2) e33(C/m2)

Expt. /Bulk(Winkelnkemperet al. 2006)

0.326 −0.527 0.895 0.264 −0.484 1.06

First-order(ab ini tio)(Pal et al. 2011)

−0.57 −0.55 1.05 −0.65 −0.55 1.07

e311 (C/m2) e333 (C/m2) e133 (C/m2) e311 (C/m2) e333 (C/m2) e133 (C/m2)

Second-order(ab ini tio)(Pal et al. 2011)

6.185 −8.090 1.543 5.151 −6.680 1.280

in this step. The strain simulations fix the atom positions on the bottom plane to the GaNlattice constant and open boundary conditions on the top surface. The strain parameters usedin this work are taken from Saito and Arakawa (2002) and validated through the calculationof Poisson ratio of the underlying materials (0.2743 for GaN, and 0.2798 for InN). The smallthermal strain contribution is neglected and hence was not accounted for while computingthe overall polarization. Next, the calculation of the internal electrostatic fields is carried out.The overall polarization P in a typical wurtzite semiconductor is given by P = PSP + PPZ,where PSP spontaneous polarization (pyroelectricity) and PPZ is the strain-induced piezo-electric polarization. The spontaneous polarization is strain-independent and arises fromthe fundamental asymmetry of the crystal structure. The piezoelectric polarization PPZ isobtained from the diagonal and shear components of the anisotropic atomistic strain fields(Pal et al. 2011; Bernardini and Fiorentini 2002). To do this, four different models for thepiezoelectric coefficient, based on Pal et al. (2011) and Winkelnkemper et al. (2006), havebeen considered: (1) Linear approximation using bulk/experimental values; (2) first-orderapproximation using microscopically-determined values; (3) second-order approximationusing microscopically-determined values; and (4) A combination of first-order and second-order effects using microscopically-determined values. The polarization constants used forthe (three independent) models are summarized in Table 1. The piezoelectric charge densityis derived by taking divergence of the polarization. To do this, the simulation domain isdivided into grids/cells by rectangular meshes. The polarization of each grid is computed bytaking an average of atomic polarization within each cell. A finite difference approach is thenused to calculate the charge density by taking divergence of the grid polarization. Finally, theinduced potential is determined by solving the 3-D Poisson equation on an atomistic grid.Next, the single-particle energies and wave functions are calculated using empirical nearest-neighbor 10-band sp3s∗-SO tight-binding model. For this purpose, we have augmented theopen source NEMO 3-D tool and used the core computational framework available therein.Detail description of this package can be found in Klimeck et al. (2007) and Ahmed et al.(2010). Here, the polarization induced potential is incorporated in the Hamiltonian as an exter-nal potential when needed. The computational domain for the calculation of the electronicstructure assumes a closed boundary condition with passivated dangling bonds. The calcu-lated valence-band maxima or highest occupied molecular orbital (HOMO), conduction-bandminima or lowest unoccupied molecular orbital (LUMO), and the bandgap depend on theconfining potential within the disk region. The tight-binding parameters for GaN and InN

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are taken from Cocoletzi et al. (2005). Finally, the spontaneous optical emission (absorption)rate is calculated by Schubert (2008):

R(ki , k f , E

) = 2π

h̄

q2 A2

4m20

∣∣∣ �Mcv

(ki , k f

) · n̂∣∣∣2 × δ

[Ev

(k f

) − Ec (ki )+ E] × F, (1)

where, �Mcv is the momentum matrix calculated from the overlap of HOMO and LUMOwavefunctions and F is the product of Fermi factors for electrons and holes:

�Mcv(ki , k f

) =∮

�

d3rψ∗c,ki(r)

h̄

j�∇ψv,k f (r), (2)

F = fc (Ec (ki ))× fv(Ev

(k f

)). (3)

For F = 1, emission (absorption) rate solely depends on the momentum matrix, whereabsorption and emission lose their meaning and are termed as transition rates. Note that fornon-periodic finite-sized nanostructures to calculate �Mcv , one needs to integrate in the entiredomain.

3 Results and discussion

Figure 1 shows the simulated GaN/InN/GaN disk-in-wire structure. As shown in the bottompanel of this figure, the physical structure (which includes the GaN nanowire and the buriedInN quantum disk) being simulated has a rhombus cross-section, and the device consistsof hexagonal inner cells in the wurtzite crystal system. In NEMO 3-D, the dimensions ofrhombohedron are defined as the lengths along the wurtzite primitive vectors. The GaNnanowire is grown in the [0001] direction (c-axis) having a base length of 14 nm and a heightof 100 nm. The InN quantum disk is buried in the GaN wire and has a base length of 10 nm anda height of 3 nm. In the strain calculation, initially, the atom positions in the entire simulateddomain are fixed to GaN lattice constant (top left panel in Fig. 1). During relaxation, theequilibrium (relaxed) atom positions are found by minimizing the total elastic energy of thesystem (top right panel in Fig. 1). On the other hand, the electronic bandstructure calculation(diagonalization of the Hamiltonian) assumes a closed boundary condition with passivateddangling bonds.

In the strain calculation, the equilibrium/relaxed atomic positions (as shown in the rightpanel of Fig. 1) are determined by minimizing the total elastic energy of the system. The totalelastic energy in the VFF approach has only one global minimum, and its functional form inatomic coordinates is quartic. The conjugate gradient minimization algorithm in this case wasfound to be well-behaved and stable. As shown in the 1-D strain profiles in Fig. 2, atomisticstrain was found to be long-ranged (penetrating ∼15 nm into the substrate and the cap layers)stressing the need for using realistically-extended structures (multimillion-atom modeling)in modeling electronic structure of these nano-heterostructures. As expected in these reduceddimensionality disk-in-wire structures, εxx and εyy were found to be compressive within theInN disk region and tensile in the surrounding GaN material matrix, while εzz exhibits anopposite trend, that is, tensile within the InN disk region and compressive in the surroundingGaN material matrix. Both the hydrostatic strain

(εxx + εyy + εzz

)and the biaxial strain(

εxx + εyy − 2εzz)

profiles are found to be compressive within the disk and tensile in thesubstrate and cap layers. Figure 2b shows the 1-D off-diagonal strain distributions alongthe growth direction through the center of the structure. Here, εxz and εyz are found to be

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Fig. 1 Atomistic representationof a rhombohedral GaN/InN/GaNquantum disk-in-wire structureusing the open source Jmolmolecular viewer toolkit before(left) and after (right) strainrelaxation. Lower panel showsthe cross-section of the physicalstructure cut through the buriedInN quantum disk. Also shownare the directions of the axes xand y as well as the base lengthof the rhombi

InN

GaN

GaN

UNRELAXED RELAXED

notic

eabl

ech

ange

Ly~

24n

m

L x ~ 14nm

x

y

GaNInN

inequivalent and have maxima close to the center of the disk. The inequivalence (with respectto the phase and the amplitude) of the εxz and the εyz profiles results from the fact that thecross-section of the physical structure being simulated in this work is a rhombus (and nota square). As mentioned previously, the dimensions of rhombohedron are defined as thelengths along the wurtzite primitive vectors (preserving the internal hexagonal symmetry ofthe underlying crystal structure). In contrast, εxy is small and confined mainly within the diskregion. Figure 2c shows the two-dimensional radial distribution of the six strain componentsin the x–y plane halfway through the InN disk region. Diagonal strain, as expected, wasfound to be somewhat relaxed at the GaN/InN side interfaces, as depicted by thin black lineswithin the disk in the 2-D εxx plot.

Figure 3a shows the polarization-induced piezoelectric potential distribution for the fourmodels along the growth direction and through the center of the disk. The induced potentialis found to be: (1) long-ranged; (2) peaking at the GaN/InN interfaces, the magnitude being∼800 mV; and (3) almost isotropic with respect to the center of the disk. The magnitude ofthe peak polarization field within the disk, including the net effects, was determined to be∼6 MV/cm. More importantly, the second-order contribution tends to: (1) be significantlysmaller than the first-order counterpart, magnitude being ∼80 mV at the GaN/InN interfaces;(2) be almost negligible in GaN substrate and cap layers; and (3) aid the first order counterpartinside the InN disk region. Figure 3b shows the net 2-D potential distributions projected inthe x–z plane, featuring formation of dipoles in the growth direction.

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xx yy zz

xz yz xy

Red Compressive; Blue Tensile

(a)

(b)

(c)

Fig. 2 a Atomistic diagonal, biaxial, hydrostatic, and b off-diagonal/shear strain profiles along the growth([0001]) direction through the center of the structure. Strain is seen to penetrate deep (∼15 nm) into thesubstrate and the cap layers. c 2-D radial distribution of strain components in the x–y plane halfway throughthe InN disk (color: red indicates compressive strain, blue indicates tensile strain, black indicates zero strain).Number of atoms simulated: ∼1 million. (Color figure online)

Figure 4 shows the HOMO (valence band, EV) and the LUMO (conduction band, EC)

wavefunctions projected on the x–y plane in the InN quantum disk. Here, with the inclusion ofatomistic strain and electrostatic fields in the calculation the isotropic character (as seen in theunrelaxed but size-quantized lattice depicted as W/O) in the overall wavefunctions is almostlost. The valence band wavefunction, in particular, becomes highly localized. Also, the first-order field exerts a strong influence in localizing the conduction band (LUMO) wavefunction.The bottom (third) row in this figure lists the single-particle bandgap (EG = EC − EV ) ofthe device. The contribution of various internal fields in shifting the energy bandgap (�EG )

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Fig. 3 a Polarization-induced piezoelectric potential distribution for the four models along the growth direc-tion and through the center of the disk. b Potential distribution projected in the x–z plane

Fig. 4 HOMO and LUMO wavefunctions featuring strong localization and anisotropy due to quantum atom-icity, strain, and the piezoelectric fields

is illustrated more clearly (in an incremental order) in Fig. 5. The salient features one canextract from this figure are as follows: (1) When the strain in the InN layer is ignored (theW/O case), due only to quantum mechanical size quantization, there is a blue-shift in thebandgap of ∼422 meV; (2) When strain in included in the calculation, the energy bandgapincreases to 1.229 eV, which means that strain alone introduces a blue shift of ∼107 meV; (3)All of the electrostatic fields (pyroelectric and piezoelectric) lead to red-shift in the bandgapand pyroelectricity has the smallest and almost negligible effect on the bandgap; (4) The first-order piezoelectric field exhibits the largest impact (red-shift) on the bandgap (∼520 meV)due mainly to the QCSE; (5) The contribution from the second-order piezoelectric effect, ascompared to the first-order counterpart, is almost negligible (�EG ∼22 meV only); and (6)The use of the experimental (bulk) coefficient alone in the calculations, as compared to the net(first+second-order) piezoelectric effects, underestimates the change in the energy bandgap.

Figure 6 shows polar plot (as a function of polarization angle) of the interband opticaltransition strength between the HOMO and the LUMO states in the x–y (growth) plane within

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Fig. 5 Effects of size quantization, atomistic strain, spontaneous/pyroelectric polarization, and the four piezo-electric models on shifting the energy bandgap (EC − EV ) in an incremental order

Fig. 6 Polar plot (as a function of polarization angle) of interband optical transition strength between theHOMO (ground hole) and the LUMO (ground electron) states in the x–y plane in the InN quantum disk. Thedefinitions of the curves have the same meaning as in Fig. 5

the InN quantum disk. The following observations can be made from this figure: (1) Whilequantum size quantization has a large impact on the energy bandgap (color of optical emis-sion), the isotropy of the optical transition remains almost unchanged. The slight anisotropyoriginates mainly from the atomicity and the material discontinuity at the GaN/InN inter-faces; (2) Atomistic strain relaxation not only reduces the transition rate, but, to a large extent,lowers the symmetry of the optical emission, the transition rate being maximum along thehorizontal axis (0◦–180◦ line); (3) As expected, the pyroelectric and the second-order piezo-electric polarization have negligible effects on the optical emission characteristic; and (4)The first-order piezoelectric field reduces the transition rate and further lowers the symmetryin the optical emission. In contrast, the inclusion of the net piezoelectric polarization effectsin the calculations improves, to some extent, the isotropy in the optical emission.

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4 Conclusion

Using a combination of valence force-field molecular mechanics, 10-band sp3s∗-SO atom-istic tight-binding approach, and appropriate post-processing tools, we have studied the com-peting effects of linear and nonlinear piezoelectric fields on the electronic and optical prop-erties of a realistically-sized wurtzite GaN/InN/GaN disk-in-wire LED structure. We foundthat, while the use of a nanostructured active region in the LED results in somewhat relaxedlattice (due to the surface atoms), true atomistic symmetry coupled with quantum mechan-ical size quantization effects in nanostructures lead to strong suppression and anisotropyin the optical transitions. The second-order piezoelectric potential, while in phase with thefirst-order counterpart inside the thin InN disk region, was found to be almost an order ofmagnitude smaller than the latter and negligible in the GaN substrate and cap layers. Thefirst-order piezoelectric field, although leads to a large reduction in the transition rate, wasfound to overestimate the anisotropy in the optical emission. The inclusion of both the first-order and the second-order (that is, net) polarization effects improves, to some extent, theisotropy in the optical emission.

Acknowledgments This work was supported by The US National Science Foundation Grant No. EECS-1102192. The nanoHUB computational resources at Purdue University were used for part of this work.

References

Ahmed, S., Islam, S., Mohammed, S.: Electronic structure of InN/GaN quantum dots: multimillion atomtight-binding simulations. IEEE Trans. Electron Devices 57(1), 164–173 (2010)

Bernardini, F., Fiorentini, V.: First-principles calculation of the piezoelectric tensor �d of III–V nitrides. Appl.Phys. Lett. 80, 4145–4147 (2002)

Cocoletzi, H., Contreras, D.A., Arriaga, J.: Tight-binding studies of the electronic band structure of GaAlNand GaInN alloys. Appl. Phys. A 81, 1029–1033 (2005)

Deshpande, S., Heo, J., Das, A., Bhattacharya, P.: Electrically driven polarized single-photon emission froman InGaN quantum dot in a GaN nanowire. Nature Commun. 4(1675), 1–7 (2013)

Jarjour, A., Taylor, R., Oliver, R., Kappers, M., Humphreys, C., Tahraoui, A.: Electrically driven singleInGaN/GaN quantum dot emission. Appl. Phys. Lett. 93(233103), 1–3 (2008)

Kalden, J., Tessarek, C., Sebald, K., Figge, S., Kruse, C., Hommel, D., Gutowski, J.: Electroluminescencefrom a single InGaN quantum dot in the green spectral region up to 150 K. Nanotechnology 21(015204),1–4 (2010)

Ke, W., Fu, C., Chen, C., Lee, L., Ku, C., Chou, W., Chang, W.-H., Lee, M., Chen, W., Lin, W.: Photolumi-nescence properties of self-assembled InN dots embedded in GaN grown by metal organic vapor phaseepitaxy. Appl. Phys. Lett. 88(191913), 1–3 (2006)

Klimeck, G., Ahmed, S., Kharche, N., Bae, H., Clark, S., Haley, B., Lee, S., Naumov, M., Ryu, H., Saied,F., Prada, M., Korkusinski, M., Boykin, T.: Atomistic simulation of realistically sized nanodevices usingNEMO 3-D. IEEE Trans. Electron Devices 54(9), 2079–2099 (2007)

Li, Q., Westlake, K.R., Crawford, M.H., Lee, S.R., Koleske, D.D., Figiel, J.J., Cross, K.C., Fathololoumi, S.,Mi, Z., Wang, G.T.: Optical performance of top-down fabricated InGaN/GaN nanorod light emitting diodearrays. Opt. Express 19(25), 25528–25534 (2011)

Lu, Y., Lin, H., Chen, H., Yang, Y.-C., Gwo, S.: Single InGaN nanodisk light emitting diodes as full-colorsubwavelength light sources. Appl. Phys. Lett. 98(233101), 1–3 (2011)

Merrill, K., Yalavarthi, K., Ahmed, S.: Giant growth-plane optical anisotropy in wurtzite InN/GaN disk-in-wirestructures. Superlatt. Microstruct. 52, 949–961 (2012)

Pal, J., Tse, G., Haxha, V., Migliorato, M.A.: Second-order piezoelectricity in wurtzite III–N semiconductors.Phys. Rev. B 84(085211), 1–7 (2011)

Saito, T., Arakawa, Y.: Electronic structure of piezoelectric In0.2Ga0.8N quantum dots in GaN calculatedusing a tight-binding method. Phys. E Low-Dimens. Syst. Nanostruct. 15, 169–181 (2002)

Schubert, E.F.: Light-Emitting Diodes, 2nd edn. University Press, Cambridge (2008)Winkelnkemper, M., Schliwa, A., Bimberg, D.: Interrelation of structural and electronic properties in

Inx Ga1−x N/GaN quantum dots using an eight-band kp model. Phys. Rev. B 74(155322), 1–12 (2006)

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