- Email: [email protected]
GaN quantum dots: Geometry dependence and competing effects on the electronic structure
Physica E 43 (2011) 1235–1239
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Internal fields in InN/GaN quantum dots: Geometry dependence and competing effects on the electronic structure K. Yalavarthi, V. Gaddipati, S. Ahmed n Department of Electrical and Computer Engineering, Southern Illinois University at Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901, USA
a r t i c l e i n f o
abstract
Article history: Received 26 November 2010 Received in revised form 20 January 2011 Accepted 16 February 2011 Available online 23 February 2011
Self-assembled InN quantum dots grown on lattice-mismatched GaN substrates are subject to internal structural and electrostatic fields originating mainly from: (1) the fundamental crystal atomicity and the interface discontinuity between two dissimilar materials, (2) atomistic strain, (3) piezoelectricity, and (4) spontaneous polarization (pyroelectricity). In this paper, using the multimillion-atom NEMO 3-D simulator, we study the origin and effects of these four competing internal fields on the electronic structure of self-assembled InN/GaN quantum dots having three different geometries, namely, box, dome, and pyramid. It is shown that internal electrostatic fields in InN/GaN quantum dots are longranged (demanding simulations using millions of atoms) and lead to a global shift in the one-particle energy states, significant modifications in the valence bandstructure (pronounced carrier localization), anisotropy and twofold degeneracy in the conduction band P level, formation of mixed excited bound states, and polarized optical transitions near the Brillouin zone center. & 2011 Elsevier B.V. All rights reserved.
1. Introduction In the last decade, GaN and its related alloys especially InGaN have been the subject of intense experimental and theoretical research due mainly to their wide range of emission frequencies, high stability against defects, and potential for applications in various optoelectronic, solid-state lighting, and high-mobility electronic devices [1,2]. 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). This finding gives rise to the possibility of growing InN quantum dots (QDs) on GaN substrates. 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 islands [3–5]. Relaxed InN islands with controllable size and density can be formed by changing the growth parameters (such as temperature) in either MBE or metalorganic chemical vapor deposition (MOCVD) [6]. Relaxation of elastic strain at free surfaces in semiconductor dots (and nanowires) allows the accommodation of a broader range of lattice mismatch and band-lineups in coherent nanostructures than is possible in conventional bulk and thin-film (quantum well) heterostructures, and, therefore, threading dislocations can be all but nonexistent
n
Corresponding author. Tel.: +1 618 453 7630; fax: + 1 618 453 7972. E-mail address: [email protected] (S. Ahmed).
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.02.007
in quantum dots (and nanowires). Furthermore, QDs used in the active region of optical devices provide better electron confinement (due to strongly peaked energy dependence of density of states) and thus a higher temperature stability of the threshold current and the luminescence than quantum wells. It is also clear that high-quality bulk GaN is an ideal substrate material for nitride nanostructures. Pure GaN crystal is five times more thermally conductive than sapphire, and optically transparent at visible and near-UV wavelengths [7]. Very recently, July 2010 issue of the IEEE Spectrum covers the story of a small polish company establishing a huge technical edge by manufacturing nearly perfect 2 in. crystal of GaN [8]. Knowledge of the electronic bandstructure of nanostructures is the first and an essential step towards the understanding of the optical performance (luminescence) and reliable device design. 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 [9,10]. These observations suggest the existence of certain symmetry-lowering mechanisms (structural and electrostatic fields) in these low dimensional nanostructures. While 0-D QDs promise better performance, only very few and recent experimental results exist concerning the photoluminescence (PL) and electroluminescence (EL) of nitride QDs in the visible spectral region [5,11,12], and experiments revealing polarization anisotropy in InN QDs are rare. Similar to the 2-D QW structures, the optical properties of the QDs are expected, to a large extent, to be
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K. Yalavarthi et al. / Physica E 43 (2011) 1235–1239
determined by an intricate interplay between the structural and the electronic properties, and (since not yet been fully assessed experimentally) demand detailed theoretical investigations. In this paper, we study the electronic bandstructures of wurtzite InN/GaN quantum dots having three different geometries, namely, box, dome, and pyramid. The main objectives are two-fold—(1) to explore the nature and quantify the role of crystal atomicity, strain fields, piezoelectric, and pyroelectric potentials in determining the energy spectrum and the wavefunctions, and (2) to address a group of related phenomena including shift in the energy state, symmetry-lowering, and non-degeneracy in the first excited state, strong band-mixing in the overall conduction band electronic states, and strongly suppressed and optically anisotropic interband transitions. We have also demonstrated the importance of full 3-D atomistic material representation and the need for using realistically extended substrate and cap layers (multimillion-atom modeling) in the study of electronic structure of these reduced-dimensional QDs.
2. Models and software used It is clear that, at nanoscale, modeling approaches based on a continuum representation (such as effective mass [13], and kp [14]) 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 CNn, whereas pyramidal dots are assumed to have C4n symmetry. In a recent effort on modeling In1 xGaxN quantum dots using the kp approach [14], it was found that the envelope S function reproduces the symmetry of the confining potential, the excited P and D states are energetically degenerate and optically isotropic—a group of observations that clearly suppresses the true fundamental atomistic symmetry of the underlying crystal and thus overestimates the quantum efficiency of the light emitters in these quantum dots. On the other side, various ab initio atomistic materials science methods (fundamental manyelectron correlated methods based on perturbation theory, quantum Monte Carlo method, or GW approach) offer intellectual appeal, but can only predict masses and bandgaps for very small systems (around 100 atoms). Thus, for quantum dot simulations, the simulation domain requiring multimillion atoms prevent the use of ab initio methods. Empirical methods (Pseudopotentials [15] and Tight-Binding [16–18]), which eliminate enough unnecessary details of core electrons, but are finely tuned to describe the atomistically dependent behavior of valence and conduction electrons, are attractive in realistically sized nanodevice (containing millions of atoms) simulations. Tight-binding 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. This study has been performed through atomistic simulations using the Nanoelectronic Modeling tool NEMO 3-D. Description of this package can be found in Ref. [19]. Based on the atomistic valence-force field (VFF) method for the calculations of strain fields and a variety of tight-binding models (s, sp3s*, sp3d5s*) optimized with genetic algorithms to match experimental and theoretical electronic structure data, NEMO 3-D enables the computation of atomistic (non-linear) strain and piezoelectric field for over 64 million atoms and of electronic structure for over 52 million atoms, corresponding to volumes of 110 and 101 nm3, respectively. Excellent parallel scaling up to 8192 cores has been
demonstrated [17]. We are not aware of any other semiconductor device simulation code that can simulate such large number of atoms. NEMO 3-D includes spin in its fundamental atomistic tight-binding representation. Effects of interaction with external electromagnetic fields are also included [20–22]. Recent work on multimillion-atom simulations demonstrates the capability of NEMO 3-D to model a large variety of relevant, realistically sized nanoelectronic devices [17] including bulk materials, quantum dots, quantum wires, quantum wells, and nanocrystals. The overall polarization P in a typical Wurtzite semiconductor is given by P¼PPZ + PSP, where PPZ is the strain-induced piezoelectric polarization and PSP is the spontaneous polarization (pyroelectricity). The magnitude of the piezoelectric field depends on strain, piezoelectric constants, and (importantly) on device geometry. Previous theoretical investigations on the effects of internal fields in III-nitride structures have mainly focused on 2-D quantum wells and either neglected the strain fields (in cladding layers) [23,24], or used a linear harmonic strain model [14]. In clear contrast, in our work, the piezoelectric polarization PPZ is obtained from the diagonal and shear components of the anisotropic atomistic strain fields, which is calculated using a VFF Keating model [19]. In Ref. [25] the piezoelectric coupling constants in III–V nitrides were calculated from ab initio studies using the Berry-phase approach and found to be up to ten times larger than in conventional III–V and II–VI semiconductor compounds. 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 non-zero component only along the polar [0 0 0 1] (growth) direction. The polarization constants used in this study are taken from Ref. [14], whereas the tight-binding parameters and strain constants are taken from Ref. [16] and the small thermal strain contribution is neglected. Also, it is a key point to notice that, PPZ may have (due to its strain dependence) the same or the opposite 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 rectangular 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 density 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.
3. Simulation results Fig. 1 shows the simulated quantum dots with box, dome, and pyramid geometries. The InN QDs grown in the [0 0 0 1] direction and embedded in a GaN substrate used in this study have (unless otherwise stated) diameter/base length, d 10.1 nm and height, z
c
y x b
Delectronic Dstrain
BOX
h s
DOME
PYRAMID
Fig. 1. Simulated InN/GaN quantum dots on a thin InN wetting layer. Two major computational domains are also shown. Delec: central smaller domain for electronic structure (quantum) calculation, and Dstrain: outer domain for strain calculation. In the figure: s is the substrate height, c is the cap layer thickness, h is the dot height, and d is the dot diameter/base length as appropriate.
K. Yalavarthi et al. / Physica E 43 (2011) 1235–1239
h 5.6 nm, and are positioned on an InN wetting layer of one atomic-layer thickness. The simulation of strain is carried out in the large computational box, while the electronic structure computation is restricted to the smaller inner domain. All the strain simulations fix the atom positions on the bottom plane to the GaN lattice constant, assume periodic boundary conditions in the lateral dimensions, and open boundary conditions on the top surface. The strain parameters used in this work were validated through the calculation of Poisson ratio of the bulk materials. The inner electronic box assumes a closed boundary condition with passivated dangling bonds. Fig. 2 shows the topmost valence (HOMO) and first four conduction band wavefunctions (projected on the X–Y plane) for the quantum dots without strain relaxation. Here, both the InN dot and the GaN barrier assume the lattice positions of perfect wurtzite GaN. The topmost valence (HOMO) state in all three quantum dots has orbital S-character retaining the geometric symmetry of the dots. The first electronic state is S-like, while the next three states are P-like and the split (non-degeneracy) in these levels originate from the crystal fields alone. Note that the magnitude of the split (defined as DP¼E010–E100) in the P level is largest in a box ( 45.294 meV) and minimum in a dome ( 1.476 meV). Also, the anisotropy in the P states assumes different orientations—for box and pyramid dots, the first P state is oriented along the [0 1 0] direction and the second along [1 0 0] direction, while the converse occurs in a dome. It is clear that the fundamental crystal atomicity and the interfaces (between the dot material InN and the barrier material GaN) lower the geometric shape symmetry even in the absence of strain relaxation. Therefore, the interface plane creates a short-range interfacial potential and cannot be treated as a reflection plane. Next, we introduce atomistic strain relaxation in our calculations using the VFF method with the Keating potential. In this approach, the total elastic energy of the sample is computed as a sum of bond-stretching and bond-bending contributions from each atom. The equilibrium atomic positions are found by minimizing the total elastic energy of the system [17]. However, piezoelectricity is neglected in this step. The total elastic energy in the VFF approach has only one global minimum, and its functional form in atomic coordinates is quartic. The conjugate gradient minimization algorithm in this case is well-behaved and stable. Strain modifies the effective confinement volume in the device, distorts the atomic bonds in length and angles, and hence modulates the confined states. From our calculations, atomistic strain
BOX
E =2.25 eV
DOME
E = 2.29 eV
E –E
E –E
PYRAMID
was found to be anisotropic and long-ranged penetrating deep (20 nm) into both the substrate and the cap layers stressing the need for using realistically extended substrate and cap layers (multimillion-atom modeling) in the study of electronic structure of these reduced-dimensional QDs. Fig. 3 shows the wavefunction distributions for the topmost valence band and first 4 (four) conduction band electronic states in a 2-D projection. Noticeable are the deformed LUMO (electronic) states, and the pronounced optical anisotropy and non-degeneracy in the P levels. Strain introduces uniform orientational pressure (adds negative potential) in all three quantum dots with DP to be largest in a pyramid and minimum in a dome. Also, strain relaxation causes blue-shift in the conduction band electronic states and results in strongly displaced HOMO (hole) wavefunctions. These observations will have significant implications on the optical polarization and performance of devices based on these nanostructures. In pseudomorphically grown heterostructures, the presence of non-zero atomistic stress tensors results in a deformation in the crystal lattice and leads to a combination of piezoelectric and pyroelectric field, which has been incorporated in the Hamiltonian as an external potential (within a non-selfconsistent approximation). The resulting potential distributions along the growth direction are shown in Fig. 4 for all three quantum dots. One can see that the potential (both the piezoelectric and pyroelectric), in accordance with the dot volume, has the largest magnitude in a box, and is minimum in a pyramid. The pyroelectric potential is significantly larger ( 5 times) than the piezoelectric counterpart and tends to oppose the latter. 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. 5 shows the topmost valence band and first 4 (four) conduction band wavefunctions for all three quantum dots including the strain relaxation and the piezoelectric potential. The piezoelectric potential introduces a global red-shift in the energy spectrum and opposes the strain induced field (without any significant modifications in the wavefunction orientations) in the box and pyramid dots. Fig. 6 shows the same wavefunctions for all three quantum dots including the combined effects of all four types of internal fields, namely, interface, strain, piezoelectricity, and pyroelectricity in the calculations. The pyroelectric potential was found to be large enough to create band mixing and strong wavefunction anisotropy in the conduction band energy landscape. Fig. 7 shows the interband optical transition rates between ground hole (HOMO) and ground electronic states (LUMO) in all
BOX
E = 2.68 eV
DOME
E = 2.77 eV
PYRAMID
E = 2.82 eV
= -45.294 meV
= 1.476 meV
E = 2.4 eV
E –E
1237
= -1.626 meV
Fig. 2. Topmost valence (HOMO) and first four conduction band wavefunctions due to fundamental crystal and interfacial symmetry. Noticeable are the split and the anisotropy in the P level. Number of atoms simulated: 1.78 million (strain domain) and 0.8 million (electronic domain).
E –E
= -48.75 meV
E –E
= -2.79 meV
E –E
= -56.61 meV
Fig. 3. Topmost valence (HOMO) and first four conduction band wavefunctions due to combined effects of atomicity and strain relaxation. Noticeable are the strongly displaced HOMO (hole) wavefunctions, deformed LUMO states, and split and the anisotropy in the P level.
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BOX
E = 2.62 eV
DOME
E = 2.65 eV
PYRAMID
E = 2.69 eV
E
–E
= -26.94 meV
E
–E
= -31.5 meV
E
–E
= -27.05 meV
Fig. 6. Topmost valence band (HOMO) and first four conduction band wavefunctions including all four (4) competing internal fields originating from interfacial effects, strain, piezoelectricity, and pyroelectricity.
Fig. 4. Induced piezoelectric and pyroelectric potential distributions along the z (growth) direction. Note the spread/penetration in the surrounding material matrix.
BOX
E = 2.29 eV
DOME
E = 2.38 eV
PYRAMID
E = 2.28 eV
E –E
= -29.04 meV
E –E
= -17.06 meV
Fig. 7. Interband optical transition rates between ground hole (HOMO) and ground electronic states (LUMO) in all three quantum dots.
E –E
= -19.57 meV
three quantum dots revealing significant suppression and strong polarization anisotropy (peak occurring at angles greater than zero) due to spatial irregularity (displacement) in the wavefunctions originating from the combined effects of all four internal fields. The true atomistic symmetry of the quantum dots, thus, influences the electronic bandstructure and in general the strengths of the optical transitions differ for different geometry. The transition rates were found to be inversely proportional to volume of QD with values maximum in the pyramid and minimum for the box structures.
piezoelectric and pyroelectric potentials are found to be long-ranged and penetrating deep ( 20 nm) inside both the substrate and the cap layers. This stresses the need for using realistically extended substrate and cap layers containing at least 3 million atoms in the theoretical study of electronic structure of these reduced-dimensional QDs. In contrast to the interfacial symmetry, strain is found to have a general (uniform) tendency to orient the electronic wavefunctions along the [0 1 0] direction and further lowers the symmetry of the system under study. The induced piezoelectric and pyroelectric potentials are significantly large (tens of meV in some cases), opposing, and anisotropic in the QD planes. All four types of internal fields introduce a global shift and a band mixing in the energy spectrum, and lead to significant suppression and strong polarization anisotropy in the interband optical transitions.
4. Conclusion
Acknowledgments
Atomistic simulations using the NEMO 3-D tool have been carried out to study the influence of internal electrostatic fields in wurtzite InN/GaN quantum dots having three different geometries, namely, box, dome, and pyramid, all having a diameter/base length of 10.1 nm and a height of 5.6 nm. Atomistic strain and the resulting
This work was supported by the ORAU/ORNL High-Performance Computing Grant 2009. Computational resources supported by the National Science Foundation under Grant no. 0855221 and the Rosen Center for Advanced Computing (RCAC) at Purdue University are also acknowledged. The development of
Fig. 5. Topmost valence band (HOMO) and first four conduction band wavefunctions including interfacial effects, strain, and piezoelectricity.
K. Yalavarthi et al. / Physica E 43 (2011) 1235–1239
the NEMO 3-D tool involved a large number of individuals at JPL and Purdue University, whose work has been cited. Shaikh Ahmed would like to thank Gerhard Klimeck and Muhammad Usman at Purdue University for useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
F.A. Ponce, D.P. Bour, Nature 386 (1997) 351. H. Morkoc-, S.N. Mohammad, Science 267 (1995) 51. S. Ruffenach, B. Maleyre, O. Briot, B. Gil, Phys. Status Solidi (c) 2 (2005) 826. W. Ke, C. Fu, C. Chen, L. Lee, C. Ku, W. Chou, W.-H. Chang, M. Lee, W. Chen, W. Lin, Appl. Phys. Lett. 88 (2006) 191913. J. Kalden, C. Tessarek, K. Sebald, S. Figge, C. Kruse, D. Hommel, J. Gutowski, Nanotechnology 21 (2010) 015204. H. Wang, D. Jiang, J. Zhu, D. Zhao, Z. Liu, Y. Wang, S. Zhang, H. Yang, J. Phys. D 42 (2009) 145410. X.A. Cao, S.D. Arthur, Appl. Phys. Lett. 85 (2004) 3971. R. Stevenson, IEEE Spectrum 47 (2010) 40. J. Bhattacharyya, S. Ghosh, M.R. Gokhale, B.M. Arora, H. Lu, W.J. Schaff, Appl. Phys. Lett. 89 (2006) 151910. P. Waltereit, O. Brandt, A. Trampert, H.T. Grahn, J. Menniger, M. Ramsteiner, M. Reiche, K.H. Ploog, Nature 406 (2000) 865.
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[11] A. Jarjour, R. Taylor, R. Oliver, M. Kappers, C. Humphreys, A. Tahraoui, Appl. Phys. Lett. 93 (2008) 233103. [12] M. Senes, K. Smith, T. Smeeton, S. Hooper, J. Heffernan, Phys. Rev. B 75 (2007) 045314. [13] M. Buongiorno Nardelli, K. Rapcewicz, J. Bernholc, Appl. Phys. Lett. 71 (1997) 3135. [14] M. Winkelnkemper, A. Schliwa, D. Bimberg, Phys. Rev. B 74 (2006) 155322. [15] Gabriel Bester, Alex Zunger, Phys. Rev. B 71 (2005) 045318. [16] J.M. Jancu, F. Bassani, F. Della Sala, R. Scholz, Appl. Phys. Lett. 81 (2002) 4838. [17] G. Klimeck, S. Ahmed, N. Kharche, H. Bae, S. Clark, B. Haley, S. Lee, M. Naumov, H. Ryu, F. Saied, M. Prada, M. Korkusinski, T.B. Boykin, IEEE Trans. Electron Dev. 54 (2007) 2079. [18] S. Ahmed, S. Islam, S. Mohammed, IEEE Trans. Electron Dev. 57 (2010) 164. [19] G. Klimeck, F. Oyafuso, T. Boykin, R. Bowen, P. von Allmen, Comput. Model. Eng. Sci. 3 (2002) 601. [20] M. Graf, P. Vogl, Phys. Rev. B 51 (1995) 4940. [21] T. Boykin, R. Bowen, G. Klimeck, Phys. Rev. B 63 (2001) 245314. [22] T. Boykin, P. Vogl, Phys. Rev. B 65 (2001) 035202. [23] V. Fiorentini, F. Bernardini, F. Della Sala, A. Di Carlo, P. Lugli, Phys. Rev. B 60 (1999) 8849–8858. [24] N. Baer, S. Schulz, P. Gartner, S. Schumacher, G. Czycholl, F. Jahnke, Phys. Rev. B 76 (2007) 075310. [25] F. Bernardini, V. Fiorentini, D. Vanderbilt, Phys. Rev. B 56 (1997) R10024.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Internal fields in InN/GaN quantum dots: Geometry dependence and competing effects on the electronic structure K. Yalavarthi, V. Gaddipati, S. Ahmed n Department of Electrical and Computer Engineering, Southern Illinois University at Carbondale, 1230 Lincoln Drive, Carbondale, IL 62901, USA
a r t i c l e i n f o
abstract
Article history: Received 26 November 2010 Received in revised form 20 January 2011 Accepted 16 February 2011 Available online 23 February 2011
Self-assembled InN quantum dots grown on lattice-mismatched GaN substrates are subject to internal structural and electrostatic fields originating mainly from: (1) the fundamental crystal atomicity and the interface discontinuity between two dissimilar materials, (2) atomistic strain, (3) piezoelectricity, and (4) spontaneous polarization (pyroelectricity). In this paper, using the multimillion-atom NEMO 3-D simulator, we study the origin and effects of these four competing internal fields on the electronic structure of self-assembled InN/GaN quantum dots having three different geometries, namely, box, dome, and pyramid. It is shown that internal electrostatic fields in InN/GaN quantum dots are longranged (demanding simulations using millions of atoms) and lead to a global shift in the one-particle energy states, significant modifications in the valence bandstructure (pronounced carrier localization), anisotropy and twofold degeneracy in the conduction band P level, formation of mixed excited bound states, and polarized optical transitions near the Brillouin zone center. & 2011 Elsevier B.V. All rights reserved.
1. Introduction In the last decade, GaN and its related alloys especially InGaN have been the subject of intense experimental and theoretical research due mainly to their wide range of emission frequencies, high stability against defects, and potential for applications in various optoelectronic, solid-state lighting, and high-mobility electronic devices [1,2]. 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). This finding gives rise to the possibility of growing InN quantum dots (QDs) on GaN substrates. 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 islands [3–5]. Relaxed InN islands with controllable size and density can be formed by changing the growth parameters (such as temperature) in either MBE or metalorganic chemical vapor deposition (MOCVD) [6]. Relaxation of elastic strain at free surfaces in semiconductor dots (and nanowires) allows the accommodation of a broader range of lattice mismatch and band-lineups in coherent nanostructures than is possible in conventional bulk and thin-film (quantum well) heterostructures, and, therefore, threading dislocations can be all but nonexistent
n
Corresponding author. Tel.: +1 618 453 7630; fax: + 1 618 453 7972. E-mail address: [email protected] (S. Ahmed).
1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.02.007
in quantum dots (and nanowires). Furthermore, QDs used in the active region of optical devices provide better electron confinement (due to strongly peaked energy dependence of density of states) and thus a higher temperature stability of the threshold current and the luminescence than quantum wells. It is also clear that high-quality bulk GaN is an ideal substrate material for nitride nanostructures. Pure GaN crystal is five times more thermally conductive than sapphire, and optically transparent at visible and near-UV wavelengths [7]. Very recently, July 2010 issue of the IEEE Spectrum covers the story of a small polish company establishing a huge technical edge by manufacturing nearly perfect 2 in. crystal of GaN [8]. Knowledge of the electronic bandstructure of nanostructures is the first and an essential step towards the understanding of the optical performance (luminescence) and reliable device design. 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 [9,10]. These observations suggest the existence of certain symmetry-lowering mechanisms (structural and electrostatic fields) in these low dimensional nanostructures. While 0-D QDs promise better performance, only very few and recent experimental results exist concerning the photoluminescence (PL) and electroluminescence (EL) of nitride QDs in the visible spectral region [5,11,12], and experiments revealing polarization anisotropy in InN QDs are rare. Similar to the 2-D QW structures, the optical properties of the QDs are expected, to a large extent, to be
1236
K. Yalavarthi et al. / Physica E 43 (2011) 1235–1239
determined by an intricate interplay between the structural and the electronic properties, and (since not yet been fully assessed experimentally) demand detailed theoretical investigations. In this paper, we study the electronic bandstructures of wurtzite InN/GaN quantum dots having three different geometries, namely, box, dome, and pyramid. The main objectives are two-fold—(1) to explore the nature and quantify the role of crystal atomicity, strain fields, piezoelectric, and pyroelectric potentials in determining the energy spectrum and the wavefunctions, and (2) to address a group of related phenomena including shift in the energy state, symmetry-lowering, and non-degeneracy in the first excited state, strong band-mixing in the overall conduction band electronic states, and strongly suppressed and optically anisotropic interband transitions. We have also demonstrated the importance of full 3-D atomistic material representation and the need for using realistically extended substrate and cap layers (multimillion-atom modeling) in the study of electronic structure of these reduced-dimensional QDs.
2. Models and software used It is clear that, at nanoscale, modeling approaches based on a continuum representation (such as effective mass [13], and kp [14]) 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 CNn, whereas pyramidal dots are assumed to have C4n symmetry. In a recent effort on modeling In1 xGaxN quantum dots using the kp approach [14], it was found that the envelope S function reproduces the symmetry of the confining potential, the excited P and D states are energetically degenerate and optically isotropic—a group of observations that clearly suppresses the true fundamental atomistic symmetry of the underlying crystal and thus overestimates the quantum efficiency of the light emitters in these quantum dots. On the other side, various ab initio atomistic materials science methods (fundamental manyelectron correlated methods based on perturbation theory, quantum Monte Carlo method, or GW approach) offer intellectual appeal, but can only predict masses and bandgaps for very small systems (around 100 atoms). Thus, for quantum dot simulations, the simulation domain requiring multimillion atoms prevent the use of ab initio methods. Empirical methods (Pseudopotentials [15] and Tight-Binding [16–18]), which eliminate enough unnecessary details of core electrons, but are finely tuned to describe the atomistically dependent behavior of valence and conduction electrons, are attractive in realistically sized nanodevice (containing millions of atoms) simulations. Tight-binding 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. This study has been performed through atomistic simulations using the Nanoelectronic Modeling tool NEMO 3-D. Description of this package can be found in Ref. [19]. Based on the atomistic valence-force field (VFF) method for the calculations of strain fields and a variety of tight-binding models (s, sp3s*, sp3d5s*) optimized with genetic algorithms to match experimental and theoretical electronic structure data, NEMO 3-D enables the computation of atomistic (non-linear) strain and piezoelectric field for over 64 million atoms and of electronic structure for over 52 million atoms, corresponding to volumes of 110 and 101 nm3, respectively. Excellent parallel scaling up to 8192 cores has been
demonstrated [17]. We are not aware of any other semiconductor device simulation code that can simulate such large number of atoms. NEMO 3-D includes spin in its fundamental atomistic tight-binding representation. Effects of interaction with external electromagnetic fields are also included [20–22]. Recent work on multimillion-atom simulations demonstrates the capability of NEMO 3-D to model a large variety of relevant, realistically sized nanoelectronic devices [17] including bulk materials, quantum dots, quantum wires, quantum wells, and nanocrystals. The overall polarization P in a typical Wurtzite semiconductor is given by P¼PPZ + PSP, where PPZ is the strain-induced piezoelectric polarization and PSP is the spontaneous polarization (pyroelectricity). The magnitude of the piezoelectric field depends on strain, piezoelectric constants, and (importantly) on device geometry. Previous theoretical investigations on the effects of internal fields in III-nitride structures have mainly focused on 2-D quantum wells and either neglected the strain fields (in cladding layers) [23,24], or used a linear harmonic strain model [14]. In clear contrast, in our work, the piezoelectric polarization PPZ is obtained from the diagonal and shear components of the anisotropic atomistic strain fields, which is calculated using a VFF Keating model [19]. In Ref. [25] the piezoelectric coupling constants in III–V nitrides were calculated from ab initio studies using the Berry-phase approach and found to be up to ten times larger than in conventional III–V and II–VI semiconductor compounds. 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 non-zero component only along the polar [0 0 0 1] (growth) direction. The polarization constants used in this study are taken from Ref. [14], whereas the tight-binding parameters and strain constants are taken from Ref. [16] and the small thermal strain contribution is neglected. Also, it is a key point to notice that, PPZ may have (due to its strain dependence) the same or the opposite 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 rectangular 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 density 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.
3. Simulation results Fig. 1 shows the simulated quantum dots with box, dome, and pyramid geometries. The InN QDs grown in the [0 0 0 1] direction and embedded in a GaN substrate used in this study have (unless otherwise stated) diameter/base length, d 10.1 nm and height, z
c
y x b
Delectronic Dstrain
BOX
h s
DOME
PYRAMID
Fig. 1. Simulated InN/GaN quantum dots on a thin InN wetting layer. Two major computational domains are also shown. Delec: central smaller domain for electronic structure (quantum) calculation, and Dstrain: outer domain for strain calculation. In the figure: s is the substrate height, c is the cap layer thickness, h is the dot height, and d is the dot diameter/base length as appropriate.
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h 5.6 nm, and are positioned on an InN wetting layer of one atomic-layer thickness. The simulation of strain is carried out in the large computational box, while the electronic structure computation is restricted to the smaller inner domain. All the strain simulations fix the atom positions on the bottom plane to the GaN lattice constant, assume periodic boundary conditions in the lateral dimensions, and open boundary conditions on the top surface. The strain parameters used in this work were validated through the calculation of Poisson ratio of the bulk materials. The inner electronic box assumes a closed boundary condition with passivated dangling bonds. Fig. 2 shows the topmost valence (HOMO) and first four conduction band wavefunctions (projected on the X–Y plane) for the quantum dots without strain relaxation. Here, both the InN dot and the GaN barrier assume the lattice positions of perfect wurtzite GaN. The topmost valence (HOMO) state in all three quantum dots has orbital S-character retaining the geometric symmetry of the dots. The first electronic state is S-like, while the next three states are P-like and the split (non-degeneracy) in these levels originate from the crystal fields alone. Note that the magnitude of the split (defined as DP¼E010–E100) in the P level is largest in a box ( 45.294 meV) and minimum in a dome ( 1.476 meV). Also, the anisotropy in the P states assumes different orientations—for box and pyramid dots, the first P state is oriented along the [0 1 0] direction and the second along [1 0 0] direction, while the converse occurs in a dome. It is clear that the fundamental crystal atomicity and the interfaces (between the dot material InN and the barrier material GaN) lower the geometric shape symmetry even in the absence of strain relaxation. Therefore, the interface plane creates a short-range interfacial potential and cannot be treated as a reflection plane. Next, we introduce atomistic strain relaxation in our calculations using the VFF method with the Keating potential. In this approach, the total elastic energy of the sample is computed as a sum of bond-stretching and bond-bending contributions from each atom. The equilibrium atomic positions are found by minimizing the total elastic energy of the system [17]. However, piezoelectricity is neglected in this step. The total elastic energy in the VFF approach has only one global minimum, and its functional form in atomic coordinates is quartic. The conjugate gradient minimization algorithm in this case is well-behaved and stable. Strain modifies the effective confinement volume in the device, distorts the atomic bonds in length and angles, and hence modulates the confined states. From our calculations, atomistic strain
BOX
E =2.25 eV
DOME
E = 2.29 eV
E –E
E –E
PYRAMID
was found to be anisotropic and long-ranged penetrating deep (20 nm) into both the substrate and the cap layers stressing the need for using realistically extended substrate and cap layers (multimillion-atom modeling) in the study of electronic structure of these reduced-dimensional QDs. Fig. 3 shows the wavefunction distributions for the topmost valence band and first 4 (four) conduction band electronic states in a 2-D projection. Noticeable are the deformed LUMO (electronic) states, and the pronounced optical anisotropy and non-degeneracy in the P levels. Strain introduces uniform orientational pressure (adds negative potential) in all three quantum dots with DP to be largest in a pyramid and minimum in a dome. Also, strain relaxation causes blue-shift in the conduction band electronic states and results in strongly displaced HOMO (hole) wavefunctions. These observations will have significant implications on the optical polarization and performance of devices based on these nanostructures. In pseudomorphically grown heterostructures, the presence of non-zero atomistic stress tensors results in a deformation in the crystal lattice and leads to a combination of piezoelectric and pyroelectric field, which has been incorporated in the Hamiltonian as an external potential (within a non-selfconsistent approximation). The resulting potential distributions along the growth direction are shown in Fig. 4 for all three quantum dots. One can see that the potential (both the piezoelectric and pyroelectric), in accordance with the dot volume, has the largest magnitude in a box, and is minimum in a pyramid. The pyroelectric potential is significantly larger ( 5 times) than the piezoelectric counterpart and tends to oppose the latter. 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. 5 shows the topmost valence band and first 4 (four) conduction band wavefunctions for all three quantum dots including the strain relaxation and the piezoelectric potential. The piezoelectric potential introduces a global red-shift in the energy spectrum and opposes the strain induced field (without any significant modifications in the wavefunction orientations) in the box and pyramid dots. Fig. 6 shows the same wavefunctions for all three quantum dots including the combined effects of all four types of internal fields, namely, interface, strain, piezoelectricity, and pyroelectricity in the calculations. The pyroelectric potential was found to be large enough to create band mixing and strong wavefunction anisotropy in the conduction band energy landscape. Fig. 7 shows the interband optical transition rates between ground hole (HOMO) and ground electronic states (LUMO) in all
BOX
E = 2.68 eV
DOME
E = 2.77 eV
PYRAMID
E = 2.82 eV
= -45.294 meV
= 1.476 meV
E = 2.4 eV
E –E
1237
= -1.626 meV
Fig. 2. Topmost valence (HOMO) and first four conduction band wavefunctions due to fundamental crystal and interfacial symmetry. Noticeable are the split and the anisotropy in the P level. Number of atoms simulated: 1.78 million (strain domain) and 0.8 million (electronic domain).
E –E
= -48.75 meV
E –E
= -2.79 meV
E –E
= -56.61 meV
Fig. 3. Topmost valence (HOMO) and first four conduction band wavefunctions due to combined effects of atomicity and strain relaxation. Noticeable are the strongly displaced HOMO (hole) wavefunctions, deformed LUMO states, and split and the anisotropy in the P level.
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BOX
E = 2.62 eV
DOME
E = 2.65 eV
PYRAMID
E = 2.69 eV
E
–E
= -26.94 meV
E
–E
= -31.5 meV
E
–E
= -27.05 meV
Fig. 6. Topmost valence band (HOMO) and first four conduction band wavefunctions including all four (4) competing internal fields originating from interfacial effects, strain, piezoelectricity, and pyroelectricity.
Fig. 4. Induced piezoelectric and pyroelectric potential distributions along the z (growth) direction. Note the spread/penetration in the surrounding material matrix.
BOX
E = 2.29 eV
DOME
E = 2.38 eV
PYRAMID
E = 2.28 eV
E –E
= -29.04 meV
E –E
= -17.06 meV
Fig. 7. Interband optical transition rates between ground hole (HOMO) and ground electronic states (LUMO) in all three quantum dots.
E –E
= -19.57 meV
three quantum dots revealing significant suppression and strong polarization anisotropy (peak occurring at angles greater than zero) due to spatial irregularity (displacement) in the wavefunctions originating from the combined effects of all four internal fields. The true atomistic symmetry of the quantum dots, thus, influences the electronic bandstructure and in general the strengths of the optical transitions differ for different geometry. The transition rates were found to be inversely proportional to volume of QD with values maximum in the pyramid and minimum for the box structures.
piezoelectric and pyroelectric potentials are found to be long-ranged and penetrating deep ( 20 nm) inside both the substrate and the cap layers. This stresses the need for using realistically extended substrate and cap layers containing at least 3 million atoms in the theoretical study of electronic structure of these reduced-dimensional QDs. In contrast to the interfacial symmetry, strain is found to have a general (uniform) tendency to orient the electronic wavefunctions along the [0 1 0] direction and further lowers the symmetry of the system under study. The induced piezoelectric and pyroelectric potentials are significantly large (tens of meV in some cases), opposing, and anisotropic in the QD planes. All four types of internal fields introduce a global shift and a band mixing in the energy spectrum, and lead to significant suppression and strong polarization anisotropy in the interband optical transitions.
4. Conclusion
Acknowledgments
Atomistic simulations using the NEMO 3-D tool have been carried out to study the influence of internal electrostatic fields in wurtzite InN/GaN quantum dots having three different geometries, namely, box, dome, and pyramid, all having a diameter/base length of 10.1 nm and a height of 5.6 nm. Atomistic strain and the resulting
This work was supported by the ORAU/ORNL High-Performance Computing Grant 2009. Computational resources supported by the National Science Foundation under Grant no. 0855221 and the Rosen Center for Advanced Computing (RCAC) at Purdue University are also acknowledged. The development of
Fig. 5. Topmost valence band (HOMO) and first four conduction band wavefunctions including interfacial effects, strain, and piezoelectricity.
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