Parallel multi-band k?p code for electronic ... - Semantic Scholar

PAPER

www.rsc.org/materials | Journal of Materials Chemistry

Parallel multi-band k?p code for electronic structure of zinc blend semiconductor quantum dots Stanko Tomic´,* Andrew G. Sunderland and Ian J. Bush Received 17th January 2006, Accepted 13th March 2006 First published as an Advance Article on the web 20th April 2006 DOI: 10.1039/b600701p We present a parallel implementation of the multi-bank k?p code ( ) for calculation of the electronic structure and optical properties of zinc blend structure semiconductor quantum dots. The electronic wave-functions are expanded in a plane wave basis set in a similar way to ab initio calculations. This approach allows one to express the strain tensor components, the piezoelectric field and the arbitrary shape of the embedded quantum dot in the form of coefficients in the Fourier transform, significantly simplifying the implementation. Most of the strain elements can be given in an analytical form, while very complicated quantum dot shapes can be modelled as a linear combination of the Fourier transform of several characteristic shapes: box, cylinder, cone etc. We show that the parallel implementation of the code scales very well up to 512 processors, giving us the memory and processor power to either include more bands, as in the dilute nitrogen quantum dot structures, or to perform calculations on bigger quantum dots/supercells structures keeping the same ‘‘cut-off’’ energy. The program performance is demonstrated on the pyramidal shape InAs/GaAs, dilute nitrogen InGaAsN, and recently emerged volcano-like InAs/GaAs quantum dot systems.

Computational Science and Engineering Department, CCLRC Daresbury Laboratory, Warrington, Cheshire, UK WA4 4AD. E-mail: [email protected]

when a material is grown on a substrate which is not lattice matched. The resulting strain produces coherently strained islands (quantum dots) on top of a two-dimensional wetting layer. The dot size, areal density, and optical properties depend on the growth parameters, such as growth temperature, growth rate, and group III over V ratio.13–15 The accurate prediction of the electronic properties of semiconductor QDs provides a direct relationship between their morphological and optical characteristics, and can reliably guide the design of the optimal structure, significantly reducing time, money and people effort. The solution of the multi-band Schro¨dinger equation with a complex 3D confinement potential influenced by the difference between the chemical potentials of the dot and surrounding material, inhomogeneous material strain and piezoelectric potential is a non-trivial task and requires immense computational power. The k?p and envelope function methods16–19 are still widely applied to study III–V semiconductor heterostructures.20–27 However, due to advances in high performance computing in recent years calculations which reflect the detailed atomistic nature have become feasible methods of nanostructure modelling despite the problem of size scaling as some power of the number of atoms in the system. These methods are based on the pseudopotential method,28,29 combined with advanced numerical algorithms30,31 In this article, we present a parallel implementation of the plane wave basis multi-band k?p code ( ) for calculation of the electronic structure and optical properties of the zinc blend semiconductor quantum dots. In the plane wave approach strain tensor elements, which are given in semi-analytical form,32 are uncorrelated and can be effectively parallelized at the moment that the complete Hamiltonian matrix is set-up for the direct diagonalization. Moreover, very complicated

This journal is ß The Royal Society of Chemistry 2006

J. Mater. Chem., 2006, 16, 1963–1972 | 1963

I. Introduction Semiconductor quantum dots (QD) are nanometer sized threedimensional (3D) objects made by embedding one semiconductor material into another.1,2 The usual semiconductor QD material systems are those from the III–V, IV, and II–VI groups of the periodic table. Since their size is smaller then the de Broglie wavelength of carriers, quantum effects are strongly manifest in QDs. The spatial confinement of electrons and holes along all three dimensions leads to a discrete energy spectrum and a delta-function atomic-like density of states, with sharp optical absorption lines.3 In that sense, a semiconductor QD can be recognized as an artificial solidstate atom. The energy spectrum of the QD depends on its size and material composition which can be controlled during the fabrication process. The ability to control systematically the quantum states, so called ‘‘band structure engineering’’,4 is essential for a variety of applications that range from semiconductor lasers,5 detectors and light emitting diodes, via spintronic devices on dynamic quantum dots,6 to quantum cryptography,7 quantum information processing and quantum computing.8–10 Most recently, semiconductor QDs have found applications in nanostructure–bioscience complexes,11 or as a scanning probe of DNA molecules.12 A major breakthrough in fabrication of the semiconductor QD was the discovery of the self-assembled Stranski– Krastanov growth mode. Self-assembled 3D island-like structures (quantum dots) nucleate spontaneously under certain conditions during molecular beam epitaxy (MBE)

quantum dot shapes, like the recently grown self-assembled quantum volcano, can be modelled as a linear combination of the Fourier transform of several characteristic primitives:32 box, cylinder, cone, etc. We show that the parallel implementation of the code scales well up to 512 processors, providing us with the memory and the processing power to either include more bands when necessary, as with quantum dot structures with a dilute amount of nitrogen substitutional impurities, or to perform calculations on bigger quantum dot/supercells structures keeping the same ‘‘cut-off’’ energy. The program scaling performance is demonstrated on the pyramidal shaped InAs/GaAs, while the dilute nitrogen InGaAsN quantum dot structure and recently emerged volcano-like InGaAs/GaAs quantum dot systems are discussed in terms of their electronic and optical properties.

II. Theoretical model A. 8-Band k?p Hamiltonian The conventional 8-band k?p Hamiltonian listed below can be applied directly to study most of the III–V zinc blend materials and related heterostructures. Here a variant of the 8-band Hamiltonian, Hk, is derived in angular momentum basis |J, mzT 1 1 ju1 T~ , z T~js; :T 2 2 3 3 i ju2 T~ , z T~ pffiffiffi ½jx; :Tzijy; :T 2 2 2 3 1 i ju3 T~ , z T~ pffiffiffi ½jx; ;Tzijy; ;T{2jz; :T 2 2 6 1 1 i ju4 T~ , z T~ pffiffiffi ½jx; ;Tzijy; ;Tzjz; :T 2 2 3 The set of Bloch basis states (|u5T 2 |u8T) are obtained under application of the time-reversal symmetry operator for zinc ^ J^ where sy is the spin Pauli matrix ^ d ~{isy C blend structures, T ˆ is the complex conjugation which flips the spin component, C operator, and Jˆ is the inversion about the midpoint between nearest neighbors ðs.{s, p.pÞ and reads as 1 1 ju5 T~ , { T~{js; ;T 2 2 3 3 i ju6 T~ , { T~{ pffiffiffi ½jx; ;T{ijy; ;T 2 2 2 3 1 i ju7 T~ , { T~z pffiffiffi ½jx; :T{ijy; :Tz2jz; ;T 2 2 6 1 1 i ju8 T~ , { T~z pffiffiffi ½jx; :T{ijy; :T{jz; ;T 2 2 3 In the absence of the external magnetic field, the set of states (|u5T 2 |u8T) are Kramer’s double degenerate to (|u1T 2 |u4T). The strain tensor in the QD is determined in analytical form as a solution of the Fourier transform of the lattice displacement Green’s function for the crystal with cubic symmetry convoluted with the forces (stress) existing on the surface of the QD.32 The strain tensor elements are used to form a Pikus–Bir Hamiltonian, 1964 | J. Mater. Chem., 2006, 16, 1963–1972

He, and this takes into account the crystal anisotropy, i.e. can = c11 2 c12 2 2c14 ? 0.32 From the strain tensor and spatially varying piezoelectric modulus e14, the piezoelectric potential, Vpz, is calculated and included in the final Hamiltonian. The 8 band k?p Hamiltonian includes k-dependent diagonal and off-diagonal matrix elements linking the conduction and valence basis states, and is given explicitly by: Huu Hul H~ zVpz I8|8 (1) Hlu Hll where Huu, Hul, Hlu, and Hll, are 4 6 4 submatrices, Vpz is the piezoelectric potential and I is the unitary matrix. The Hamiltonian sub-matrix linking upper basis states in eqn (1) is pffiffiffi pffiffiffi 0 1 { 3T 2U {U ECB p ffiffi ffi p ffiffi ffi B { 3T E 2S {S C B C pffiffiHH ffi pffiffiffi C (2) Huu ~B pffiffiffi @ 2U 2S ELH { 2Q A p ffiffi ffi {U {S { 2Q ESO while the Hamiltonian sub-matrix linking the lower set of the basis states in eqn (2) is complex conjugate Hll = Huu . The sub-matrices coupling the upper and lower set basis states are pffiffiffi 1 0 0 0 {T { 2T pffiffiffi B 0 0 {R { 2R C C B p ffiffiffi (3) Hul ~B C @ T R 0 3S A pffiffiffi pffiffiffi pffiffiffi 2T 2R { 3S 0 and Hlu = Hul. The diagonal elements are defined as: ECB = Ec0 + O EHH = Ev0 2 (P + Q) ELH = Ev0 2 (P 2 Q) ESO = Ev0 2 (P + Dso) The subscripts CB, HH, LH and SO stand for conduction, heavy hole, light hole and split-off band respectively, and Dso is the spin–orbital split-off energy. The unstrained conduction (Ec0) and valence (Ev0) band edges are all aligned relative to the average valence band of the dot or matrix material respectively.22,33 Other Hamiltonian matrix elements are given as a sum of kinetic terms (subscript k) and its strain counterpart (subscript e): O = Ok + Oe P = Pk + Pe Q = Qk + Qe R = Rk + Re S = Sk + Se T = Tk + Te U = Uk + Ue This journal is ß The Royal Society of Chemistry 2006

The kinetic terms are given by:

Ok ~

Pk ~

Qk ~

Rk ~

Sk ~

! B2 cc kx2 zky2 zkz2 2m0 ! B2 c1 kx2 zky2 zkz2 2m0 ! B2 c2 kx2 zky2 {2kz2 2m0 ! i B2 pffiffiffih 2 3 c2 kx {ky2 {2ic3 kx ky 2m0 ! B2 pffiffiffi 6c3 kx {iky kz 2m0

1 Tk ~ pffiffiffi P0 kx ziky 6 1 Uk ~ pffiffiffi P0 kz 3 Here kx, ky, and kz denote components of the wave vector along crystallographic directions [100], [010], and [001], respectively, P0 = 2i(B/m0)Ssc|px|xT (EP = 2m0P20 /B2) is the Kane matrix element, cc, c1, c2, and c3, are modified Luttinger parameters related to the original Luttinger parameters (superscript L) and remote bands as: cc = 1/mc 2 (EP/3)[2/Eg + 1/(Eg + 1/(Eg + Dso)], c1 = cL1 2 EP/(3Eg + Dso), and c2,3 = cL2;3 2 EP/(6Eg + 2Dso). The strain dependent terms in Hamiltonian eqn (1) are given by: Oe ~zac exx zeyy zezz Pe ~{av exx zeyy zezz bv exx zeyy zezz 2 pffiffiffi 3 bv exx {eyy zidv exy Re ~{ 2 dv Se ~{ pffiffiffi ezx {ieyz 2 X 1 Te ~ pffiffiffi P0 exj zieyj kj j 6 X 1 Ue ~ pffiffiffi P0 e k j zj j 3

Qe ~{

where eij are the strain tensor components, ac and av are the conduction and valence band hydrostatic deformation potential of the host material, and bv and dv are the shear deformation potentials along the [001] and [111] directions of the host material respectively. B. 10-Band k?p Hamiltonian In the dilute nitrogen containing quantum dot structures the strong interaction between the N resonant states and the conduction band edge34 means that the conventional 8-band This journal is ß The Royal Society of Chemistry 2006

k?p method cannot be applied directly to (In, Ga)AsN and related heterostructures. To describe the electronic structure of InyGa12yAs12xNx and related alloys, two s-like (spin-degenerate) nitrogen-related bands needto be added35–37 N 1 1 1 1 , z T~s ; :T and , { T~{sN ; ;T to the usual two 2 2 2 2 conduction and six valence bands in the conventional 8-band model in eqn (1), so producing the 10 band k?p Hamiltonian. Several studies support that the energy of the N resonant state and its coupling to the CBE vary with In composition in InyGa12yAs12xNx.38–40 We assume the N resonant level to vary with In composition y as EN0(eV) = 1.65 2 0.18y, where the zero of energy is taken at the GaAs valence band maximum. The host material unperturbed conduction band energy is assumed to vary with N composition x as Ec0(y) 2ax, with a (in eV) = 1.55 2 0.14y, while the matrix element linking the N state and host material CBE is presumed to vary with N composition x and In pffiffiffi composition y as39,41 VNc0 ðeVÞ~{ð2:45{1:17yÞ x. From the comparison with the hydrostatic pressure experiment on the quantum well structures we estimate the nitrogen level deformation potentials to vary with indium concentration y as aN # av + (0.21 2 0.08y)(ac 2 av),42,43 where ac and av are the conduction and valence band deformation potential of the InyGa12yAs host material respectively. C. Plane wave implementation of k?p hamiltonian The quantum dot breaks the translational symmetry along all three Cartesian directions implying operator replacement kv A 2ih/hv in eqn (1), where v = x, y, z. To solve the multiband system of Schro¨dinger equations, eqn (1), for an embedded quantum dot into a supercell of the volume V = Lx 6 Ly 6 Lz, a plane wave (PW) methodology is implemented. In the PW representation the eigenvalues (En) and coefficients (Ank ) of the nth eigenvector, (|n,rT 5 SkAnk |krT); are linked by the relation X

X

hi,j ðq, kÞz~upz ðq, kÞI Ank ~En Ank

k

(4)

k

where the Fourier transform of the Hamiltonian matrix elements Hi,j(r) is given by hi,j ðq, kÞ~

1 SqrjHi,j ðrÞjkrT V

(5)

All the elements in the Hamiltonian matrix, eqn (1), can be expressed as a linear combination of three different kinetic ð0Þ ð1Þ ð2Þ related terms: Hi;j = f(r), Hi;j = f(r)ki and Hi;j = f(r)kikj, and ðe0Þ ðe1Þ two different strain related terms: Hi;j = f(r)ei,j and Hi;j = f(r)eijkl, where the material parameters spatially varying function is of the form f(r) = fqdxqd(r) + fmx[1 2xqd(r)] The quantum dot characteristic function xqd and characteristic function of the surrounding matrix material xmx are related as xmx(r) = 1 2 xqd(r). Since by definition xqd(r) equals 1 in the QD region and 0 outside, it can be weighted to account for spatial varying of the atomic species inside QD, i.e., spatial variation of the strain.44,45 After proper symmetrization46,47 of the f(r)ki = [kif(r) + f(r)ki]/2 and f(r)kikj = [kif(r)kj + kjf(r)ki]/2 J. Mater. Chem., 2006, 16, 1963–1972 | 1965

operators, the Hermiticity of the Hamiltonian Fourier transform is restored, and characteristic elements are: hð0Þ ðq, kÞ~f qd dk,q z

ð2pÞ3 Df ~xmx ðq{kÞ V

1 hð1Þ ðq,kÞ~ ðqi zki Þhð0Þ ðq,kÞ 2 hð2Þ ðq,kÞ~

hðe0Þ ðq,kÞ~

1 ki qj zkj qi hð0Þ ðq,kÞ 2

ð2pÞ3 qd ð2pÞ6 f ~eij ðq{kÞz 2 Df F ½xmx F eij V V 1 hðe1Þ ðq,kÞ~ ðql zkl Þhðe0Þ ðq,kÞ 2

(6)

(7)

(8)

(9)

(10)

IV. Results and discussion A. Parallelization performance

The convolution of the matrix oblique and the elastic constants Fourier transforms is P xmx ðq{k{k0 Þ~eij ðk0 Þ, and Df = f mx 2 F ½xmx F eij ~ k0 ~ f qd. Under the crystal anisotropy assumption (can ? 0) of the cubic symmetry unit cell, the strain term in inverse space reads:

ð11Þ

where ea = (amx 2 aqd)/aqd is the relative mismatch of the lattice constants of the QD and matrix material, c11, c12 and xqd ðkÞ c44 are elastic constants for the matrix material, and ~ is the Fourier transform of the characteristic QD shape function. The piezoelectric potential ~upz in the PW expansion is given by

eyz ðq{kÞzc:p: eqd 14 ðqx {kx Þ~ ~upz ðq, kÞ~{2i qd { ðq{kÞ2 er (12)

e14 ðqx {kx ÞF ½xmx F eyz zc:p: 2iD er ðq{kÞ2 where the symbol y denotes Fourier transform, and c.p. stands for cyclic product. The number of plane waves per Cartesian dimension n is 2nn + 1 leading the overall rank of the Hamiltonian matrix in the inverse space, eqn (4), to be Nk = Nband(2nx + 1) (2ny + 1) (2nz + 1), where Nband is the number of bands taken in the real space Hamiltonian, eqn (1). Nband equals 8 in the 8-band k?p model and 10 in the 10-band k?p model.

III. The model quantum dot The model QD is a pyramidal shaped dot with a base length, b, of 20 lattice constants (11.3 nm) on top of a one monolayer wetting layer (WL), with a height of h = b/2. Since the strain anisotropy and piezoelectric field are included the symmetry of the supercell is reduced to C2v. The assumed temperature is T = 0 K. The material parameters for the unperturbed InAs and GaAs crystals are taken from ref. 22, while the lattice 1966 | J. Mater. Chem., 2006, 16, 1963–1972

constant, elastic constants and deformation potentials of zinc blend GaN and InN are taken from ref. 48. The transition energies and dipole matrix elements given by our 8-band k?p model of the InAs/GaAs reference QD are in very good agreement with those obtained from a k?p model with a Keating–Martin valence force field (VFF) model of the strain which ignores Cauchy violation of elastic constants.22 Further, the Coulomb energies between the first two states in the conduction and valence band are also in good agreement with those obtained by an atomistic method based on empirical pseudopotentials29 (here systematically smaller for 0.5–1.1 meV). The error in the calculated e0 2 h0 transition energy is estimated to be ,3 meV.

All performance results described here were run on the HPCx system located at CCLRC Daresbury Laboratory in the UK.49 The system consists of 96 IBM Power 5 p5-575 SMP nodes connected via the IBM high performance switch (HPS). Each node comprises of 16 1.5 GHz processors with 32 GBytes of shared memory. The Fortran compiler used is IBM’s xlf version 9.1 and the code was compiled with optimisation settings in 64-bit mode and linked to SCALAPACK (Scalable Linear Algebra Package),50 BLACS (Basic Linear Algebra Communication Subprograms),50 ESSL (IBM Engineering and Scientific Software Library)51 (this includes optimised BLAS routines),50 and MPI (Massage Passing Interface)50 libraries. The Hamiltonian matrix is generated in parallel as part of a 2D block-cyclic distribution.52 Although this method of data distribution can appear relatively complex it ensures very good load-balancing across the processor grid and it allows the dense matrix algorithms in SCALAPACK to take advantage of highly optimised Level 3 BLAS routines. Moreover, utility routines which facilitate the set-up and use of this distribution are widely available to code developers. The initial stage of the parallelisation involves the definition of a 2D processor grid via the BLACS routine BLACS_GRIDINIT. Calls to the utility routines NUMROC and DESCINIT define the basic descriptor array which will be passed to the SCALAPACK routines in order to provide relevant details of the parallelization. The local elements of the distributed Hamiltonian matrix h(ir, ic) are then formed on each processor using the SCALAPACK utility routines INDXG2P, INDXG2L to determine data locality. Pseudocode for this procedure is shown in the Appendix. As it is expected that the eigensolve of the Hamiltonian will represent the major computational cost for this calculation the performance of two alternative SCALAPACK parallel matrix diagonalization routines, PZHEEVX and PZHEEVD, has been investigated on HPCx. PZHEEVX is an ‘‘expert’’ driver that uses bisection and inverse iteration algorithms53 in order to obtain all or a subset of the eigenpairs. PZHEEVD uses a divide-and-conquer algorithm54 that is generally highly efficient, but functionality is restricted to finding all the eigenpairs of the matrix (PZHEEVD is therefore said to use a ‘‘simple’’ driver). Since the physics of interest only depends upon a few This journal is ß The Royal Society of Chemistry 2006

Fig. 1 Speed-up curves on the HPCx parallel supercomputer for two different ranks of Hamiltonian matrix in the k space: the solid symbols correspond to Nk = 37800 and open symbols correspond to Nk = 25688. The square symbols represent the time taken generating the distributed Hamiltonian matrix. The circle symbols represent the time taken calculating eigenpairs from the whole spectrum by the simple driver routine PZHEEVD, and triangles represent the time taken calculating a subset of eigenpairs within the energy band (Emx CBM 4 Emx VBM ) by the expert routine (PZHEEVX).

states near the band gap this requirement to calculate all eigenpairs is potentially a performance overhead, despite the high efficiency of the divide-and-conquer algorithm. The parallel performance results for the two methods on HPCx are shown in Fig. 1. For the PZHEEVD-based calculations, the timings from two problem sizes are given here, involving the generation and diagonalization of Hamiltonian matrices of order Nk = 25688 [8-band k?p code with (nx, ny, nz) = (6, 6, 9) (open symbols)] and Nk = 37800 [8-band k?p code with (nx, ny, nz) = (7, 7, 10) (solid symbols)]. For the PZHEEVXbased calculations the timings from the 25688 ordered system are shown (the Hamiltonian generation time for each calculation is independent of the chosen eigensolver). Usually the quantum dot modelling requires only a few eigenenergies and eigenfunctions around the energy gap of the QD material. Unless the folded spectrum algorithm is implemented31 those energies of interest lie in the middle of the spectrum and are not known beforehand. The best guess would be to restrict the required eigenpairs to the energy interval between the conduction band minimum and the valence band maximum of the matrix material. In that range we have found 128 eigenpairs for the 25688 order problem. Furthermore, if we narrow the energy bandwidth of interest in an ad-hoc manner by ¡80 meV in the valence and conduction band respectively, to get rid of the states in the wetting layer, this leaves just 22 states to calculate. Despite reducing the problem in this way, the time saving is a mere 2 seconds on 256 processors, suggesting that the first few eigenpairs are the most expensive to find. However, there remains the potential for significant computational savings when using the expert driver PZHEEVX. The extent of this saving across different sized processor arrays is given by the difference between the open circles and This journal is ß The Royal Society of Chemistry 2006

open upper triangles symbol lines in the graph. From Fig. 1, for the smaller problem size, computing 25688 eigenpairs with PZHEEVD takes 357 seconds on 256 HPCx processors, computing 128 eigenpairs with PZHEEVX on 256 processors takes 209 seconds (computing the 22 eigenpairs takes 207 seconds). Fig. 1 also demonstrates that the generation of the distributed Hamiltonian matrix and both eigensolver approaches scale very well with number of processors used on HPCx. Taking the overall times to solution for the order 25688 case, the PZHEEVD-based and PZHEEVX-based timings show a speed-up of 1.933 and 1.913 respectively when comparing runs on 32 processors and 256 processors (ideal speed-up is 2log2 256/32) while the 37800 order case shows a speed up of 1.933 when run on 64 and 512 processors (ideally 2log2 512/64). A SCALAPACK implementation of the new MRRR eigensolver algorithm, PDSYEVR, is currently scheduled for release in 2007.55 This routine is expected to feature an expert driver, highly efficient parallel performance and low memory overheads. It is anticipated that this algorithm will be ideally suited to this code and therefore may yield further advantages for performance. B. Dilute nitrogen InGaAsN QD Recent advances in growth techniques facilitate the fabrication of self-assembled QDs with a very small amount (less than 5%) of nitrogen substitutional impurities in the QD region or in the capping layer.56–62 With appropriate tailoring of the QD morphology this opens the possibility of GaAs-based optoelectronic devices emitting at 1.55 mm and beyond, considerably expanding the capabilities of the already well established GaAs technology. It has been found that replacing a small amount of the group V element by nitrogen in a III–V compound reduces the energy gap and increases the electron effective mass in the conduction band. This reduction of the energy gap (of about 0.1 eV per % of N for x , y0.05) occurs due to a band anti-crossing (BAC) interaction between the conduction band edge and a higher-lying nitrogen resonant band34 and dramatically changes the electronic structure, thus offering a new route to band structure engineering. From a band structure point of view the addition of indium into GaAs12xNx to form InyGa12yAs12xNx has two main effects on the conduction band structure. Firstly, the conduction band edge (CBE) shifts down in energy on an absolute scale with increasing y in InyGa12yAs.33 Secondly, because In has a larger atomic radius than Ga, there is a weaker overall lattice perturbation around an isolated N atom bonded to In neighbours, leading to a reduced BAC interaction. There is also a weaker distortion around N–N pairs and other cluster states,63 which, when bonded predominantly to In neighbours, consequently lie higher in energy than equivalent states in GaAsN.64 Improved agreement was shown between the electron effective mass predicted by the BAC model and that observed experimentally when there are no cluster states close by in energy with which the InGaAsN CBE can interact.41,42 This justifies the assumption that, away from cluster states, the BAC model provides an excellent description not just of the energy gap but also of the band dispersion. J. Mater. Chem., 2006, 16, 1963–1972 | 1967

Fig. 2 Projection of the electron ground state wave function onto their basis states. (a) N-free In0.7Ga0.3As/GaAs QD and (b) In0.7Ga0.3As0.96N0.04/GaAs QD. Surfaces are plotted at 25% (transparent) and 75% of the particular charge density. The integral probabilities of the respective envelopes after summation over both spins are: (a) 0.000/0.897/0.029/0.049/0.025 and (b) 0.317/0.579/0.030/0.051/0.023. Table 1 Transition energy between electron (e0) and hole (h0) ground state in eV, direct transition scaled matrix element I[100], Coulomb interaction energy Je0,h0 in meV, and character decomposition summed over both degenerate states QD

e0 2 h0 I[100] Je0,h0 |sNT / |scT / |hhT / |lhT / |soT

InAs 1.1094 0.107 27.2 0.000/0.891/0.031/0.053/0.025 In0.7Ga0.3As 1.2328 0.128 28.8 0.000/0.897/0.029/0.049/0.025 In0.7Ga0.3As0.96N0.04 1.0325 0.071 30.8 0.317/0.579/0.030/0.051/0.023

To achieve a longer wavelength of the fundamental e0 2 h0 optical transition than in the sample with pure InAs QD, we have studied a sample with 4% of N in In0.7Ga0.3As. The reduction of the In content in the dot from pure InAs to y = 70% significantly increases the energy gap of the dot material and reduces the hydrostatic strain. When 4% of nitrogen is added in the QD region the conduction band is strongly affected due to both the dominant effects of BAC interaction and a further strain reduction. The character decomposed ground state wave functions of the In0.7Ga0.3As/ GaAs and In0.7Ga0.3As0.96N0.04/GaAs QDs are given in Fig. 2. In In0.7Ga0.3As0.96N0.04/GaAs QD the electron ground state is composed predominantly of the s-like states which are 31.7% nitrogen in character, and 57.9% host material conduction band. The effect of increasing the nitrogen concentration

increases the CB offset by about y280 meV relative to the N-free and y = 70% QD, inducing more tightly confined electron states. The CB minimum in y = 70% and x = 4% QD sample is at E_ = 0.787 eV, reducing the optical transition energy to e0 2 h0 = 1.033 eV, which is y77 meV less than in pure InAs/GaAs QDs. In contrast to the N-free and y = 70% QD, the dipole matrix element along the [100] polarization plane drops by about 45%, and the polarization anisotropy is unchanged, see Table 1. To illustrate the influence of nitrogen on the carrier confinement in Fig. 3 we compare the charge densities of the first few CB and VB states in the y = 70% and N-free sample, against those in y = 70% and x = 4% QD. A stronger confinement of the first three states in the CB of x = 4% QD is noticeable. Since the amplitude of nitrogen character |sNT in dilute nitrogen heterostructures increases with increasing confinement energy,42 an improved localization of higher states, e1 and e2, in In0.7Ga0.3As0.96N0.04/GaAs QD is also visible. A slightly better confinement of the VB states in the N-free sample is attributed mainly to a stronger influence of strain. We briefly discuss the influence of the quantum size effect on the e0 2 h0 transition energy of QDs. Fig. 4 shows the variation of the e0 2 h0 transition energy in the

Fig. 3 The wave-function squared of the three lowest CB and two highest VB states in (top row) N-free In0.7Ga0.3As/GaAs QD and (bottom row) In0.7Ga0.3As0.96N0.04/GaAs QD. Isosurfaces are plotted at 25% (transparent) and 75% of the maximum charge density. A better electron confinement is visible in the sample with x = 4%.

1968 | J. Mater. Chem., 2006, 16, 1963–1972


Fig. 4 Transition energy between QD electron and hole ground states as a function of dot size at the T = 0 K.

In0.7Ga0.3As0.96N0.04/GaAs QDs when the pyramid base length is changed from 20 to 36 GaAs lattice constants. The corresponding dependence in pure InAs/GaAs QDs is also given, for the purpose of comparison. For the biggest (b = 20.4 nm) QD with x = 4% the e0 2 h0 transition energy is estimated to be 0.894 eV (y1.39 mm) at T = 0 K, while its Coulomb energy is 15 meV. At room temperature this emission will move toward a longer wavelength region. However the shift will be smaller than in the bulk III–V compound because both the dilute nitride and the QD structure itself reduce the Eg temperature sensitivity. C. Quantum volcano A remarkable change in topology occurs when InGaAs quantum dots, grown by Stranski–Krastanov self-assembly, are partially covered by a thin layer of GaAs which destabilizes them. During an annealing step, the dots acquire a ‘‘vent’’-hole in their center and take on a volcano shape.65,66 The nanoislands rearrange themselves to form volcano-like islands with distinct, ring-like features. The self-assembled ring confines both electron and hole,67 not just the electron like

lithographically defined one, and therefore it is possible to load a self-assembled ring with a small number of electrons starting from zero occupancy.65 Particularly interesting are the magnetic properties of such quantum systems, which are related to the possibility of trapping magnetic flux in their interior. A perfect ring exhibits quantum interference effects in a magnetic field, resulting in Aharanov–Bohm-like oscillations in the magnetization as a function of applied magnetic field. There is evidence in self-assembled rings for the first Aharanov–Bohm oscillation from vertical tunnelling experiments.65 In Fourier space it is reasonably easy to represent quite complex relief volcano quantum dot topology.32,45 The volcano-like shape can be modelled as a linear combination of the Fourier transforms of several representative primitives. In the figures (Fig. 5) the volcano quantum dot is modelled as a crossover of several primitive shapes (solids): truncated cone (slope of the volcano) < cylinder (neck of the volcano) < box (wetting layer) > inverted truncated cone (volcano’s vent). Here we focus our attention on the quantum size and piezoelectric field effects in the quantum volcano. Two structures were examined, small (big), with the outer radius of the neck 100 nm (200 nm), height of the neck 2.5 nm (2.5 nm), outer radius of the volcano slope 150 nm (300 nm), height of the slope 2.5 nm (2.5 nm), upper radius of the slope to be equal to the outer radius of the neck, upper radius of the vent 70 nm (150 nm) and bottom radius of the vent 30 nm (80 nm). In Fig. 5, the hydrostatic component of the strain (upper row) and biaxial component of the strain (bottom row) is given for the two representative quantum volcanoes. The indium composition is assumed to be constant in the volcano material (InAs) while the surrounding matrix material is GaAs. Biaxial strain which basically shapes the piezoelectric potential in this kind of structure is located towards the outer part of the volcano slope and in the inner bottom part of the vent, see Fig. 5. This part of the piezo-field partly influences orientation of the wave functions. Comparing with the ring like shapes,68 where the biaxial strain is located inside the ring, there is a region in the vicinity of the neck–slope interface with

Fig. 5 Maps of the hydrostatic strain component (ehy = exx + eyy + ezz) (upper row) and the biaxial strain component [ebx = ezz 2 (exx + eyy)/2] (bottom row) in the xz plane through the centre of the QD. Left column correspond to the small quantum volcano and the right column correspond to the big quantum volcano. Dimensions are given in the main text.


J. Mater. Chem., 2006, 16, 1963–1972 | 1969

Fig. 6 Piezoelectric field in small quantum volcano. Dark and light surfaces are plotted at 240 meV and +40 meV respectively. Maximum piezoelectric field value is 90 meV.

very small or even no biaxial strain. This suggests that in the electrostatic sense a quantum volcano can be modelled as two close vertically coupled rings with different outer and inner diameters. Also it means that unlike well defined quantum rings with a square cross-section, there exists a significant piezo-potential (due to superposition of piezo-field from the slope and piezo-field from the neck) inside the contour of the quantum volcano, Fig. 6. Most probably this occurs in the region between the volcano neck and the volcano slope. This proposition is confirmed by detailed inspection of several electron wave functions, Fig. 7. It can be seen from Fig. 7 that, regardless of quantum volcano size, the electron wave functions (up to e4) are confined predominantly to the radius of the volcano neck. As in the case of quantum rings for the large volume and high indium concentration, the electron ground state is localized along the [11¯0] direction. For the small volcano-like dot case only a small portion of the ground state charge density (,10%) is extended around the whole rim.

V. Conclusions In summary, we have presented a parallel implementation of , for the the plane wave basis multi-band k?p code, calculation of the electronic structure and optical properties of zinc blend semiconductor quantum dots. Our analysis suggests that both the distributed Hamiltonian matrix generation and parallel diagonalization stages of the calculation scale very well up to 512 processors on the HPCx machine, . resulting in good overall parallel performance for This has entitled the code to an HPCx Silver Capability Incentive Award.49 The enlarged memory and processing power made available through parallelization allow us to include more electron bands where required, as in the case of dilute nitrogen quantum dots, and/or enable the modelling of quantum size objects of very complicated shape such as the quantum volcano. Theoretical analysis of ideal dilute nitrogen InGaAsN/ GaAs(N) QDs shows that the influence of nitrogen induces more confined states in the CB than in equivalent N-free QDs, reducing the energy of the fundamental optical transition. The better confinement in dilute nitrogen QD is due to both significantly reduced compressive strain, which was one of the major obstacles for a long-wavelength emission from InAs/GaAs QDs, and the BAC effect. These two effects play a crucial role in enabling significantly improved device operation 1970 | J. Mater. Chem., 2006, 16, 1963–1972

Fig. 7 The wave-function squared of the five lowest CB states in (left column) small quantum volcano dot and (right column) big quantum volcano QD. Isosurfaces are plotted at 10% (transparent) and 50% of the maximum charge density.

at room temperature and above. In conjunction with QD size variation, this effect can be of great utility for the design of devices emitting at longer wavelengths. Furthermore, in contrast to N-free QDs, dilute nitrogen QDs exhibit reduced dipole matrix elements and a larger Coulomb interaction energy. Our findings are in good agreement with the reported experimental results on similar structures.56,57,62 We conclude that possible routes toward 1.55 mm wavelength emission would be: further increase of the indium content in nitrogen to reach (x y 4%) QD; an increase of the QD size/volume ratio; or vertically aligned and coupled dilute nitrogen QDs. As with quantum rings, we have observed in the quantum volcano-like dots strong localization of the ground state wave function along the [11¯0] plane. However, in quantum volcano dots, we have observed much stronger confinement of the wave function in the potential pockets produced by the piezoelectric field in the volcano interior. Only for the small This journal is ß The Royal Society of Chemistry 2006

quantum volcano have we found that a small portion of the ground state charge density is spread around the whole rim.

Appendix In this appendix we present pseudo-code which describes the key part of the generation and distribution of the global Hamiltonian matrix h(ir, ic), eqn (5), over the 2D processor grid, ir_p 6 ic_p, each of which take a local part of the Hamiltonian h(ir_l, ic_l). In the part of pseudo-code below the terms ‘‘processor column’’ and ‘‘processor row’’ refer to the coordinates of the processors within this 2D grid. For clarity SCALAPACK block factors are omitted from the pseudo-code here. In the full code implementation these are taken into account in order to optimise performance for this parallel construct.

Acknowledgements The author is grateful to the Engineering and Physical Sciences Research Council (EPSRC) UK, and the CCP3 Consortium for supporting this work. We wish to thank B. G. Searle for useful discussions about graphical visualization.69

References 1 D. Bimberg, M. Grundman and N. N. Ledentsov, Quantum Dots Heterostructures, Wiley, New York, 1999. 2 Z. I. Alferov, Nobel Lecture, Rev. Mod. Phys., 2001, 73, 767. 3 M. Grundmann, J. Christen, N. N. Ledentsov, J. Bo¨ hrer, D. Bimberg, S. S. Ruvimov, P. Werner, U. Richter, U. Go¨sele, J. Heydenreich, V. M. Ustinov, A. Yu. Egorov, A. E. Zhukov, P. S. Kop’ev and Zh. I. Alferov, Phys. Rev. Lett., 1994, 74, 4043. 4 M. Grundmann, Nano-Optoelectronics: Concepts, Physics and Devices, Springer, Berlin, 2002. 5 V. M. Ustinov and A. E. Zhukov, Semicond. Sci. Technol., 2000, 15, R41. 6 J. A. H. Stotz, R. Hey, P. V. Santos and K. H. Ploog, Nat. Mater., 2005, 4, 585. 7 Z. Yuan, Science, 2002, 295, 102. 8 D. P. DiVincenzo, D. Bacon, J. Kempe, G. Burkard and K. B. Whaley, Nature, 2000, 408, 339. 9 M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern and A. Forchel, Science, 2001, 291, 451. 10 G. Bester, J. Shumway and A. Zunger, Phys. Rev. Lett., 2004, 93, 047401. 11 D. Alexon, H. Chen, M. Cho, M. Dutta, Y. Li, P. Shi, A. Raichura, D. Ramadurai, S. Parikh, M. A. Stroscio and M. Vasudev, J. Phys.: Condens. Matter, 2005, 17, R637. 12 C. Y. Zhang, H.-C. Yeh, M. T. Kuroki and T.-H. Wang, Nat. Mater., 2005, 4, 826.


13 J. S. Lee, in Semiconductor Quantum Dots, ed. Y. Masumoto and T. Takagahara, Springer, Berlin–Heidelberg, 2002, ch. 1, pp. 1–75. 14 K. Hoshino, S. Kako and Y. Arakawa, Appl. Phys. Lett., 2004, 85, 1262. 15 D. Gao, A. Kaczynski and J. A. Jaszczaka, Appl. Phys. Lett., 2005, 85, 133102. 16 E. O. Kane, J. Phys. Chem. Solids, 1957, 1, 249. 17 E. O. Kane, Semicond. Semimet., 1966, 1, 75. 18 J. M. Luttinger and W. Kohn, Phys. Rev., 1955, 97, 869. 19 J. M. Luttinger, Phys. Rev., 1956, 102, 1030. 20 M. A. Cusack, P. R. Briddon and M. Jaros, Phys. Rev. B, 1996, 54, R2300. 21 C. Pryor, Phys. Rev. B, 1998, 57, 7190. 22 O. Stier, M. Grundmann and D. Bimberg, Phys. Rev. B, 1999, 59, 5688. 23 A. D. Andreev and E. P. O’Reilly, Phys. Rev. B, 2000, 62, 15851. 24 G. Liu and S. L. Chuang, Phys. Rev. B, 2002, 65, 165220. 25 V. Mlinar, M. Tadic´, B. Partoens and F. M. Peeters, Phys. Rev. B, 2005, 71, 205305. 26 W. Sheng, S.-J. Cheng and P. Hawrylak, Phys. Rev. B, 2005, 71, 035316. 27 S. Birner, nextnano3 package, http://www.nextnano.de/. 28 A. Zunger, MRS Bull., 1998, 23, 35. 29 L. W. Wang, J. Kim and A. Zunger, Phys. Rev. B, 1999, 59, 5678. 30 L. W. Wang and A. Zunger, Phys. Rev. B, 1999, 59, 15806. 31 A. Canning, L. W. Wang, A. Williamson and A. Zunger, J. Comput. Phys., 2000, 160, 29. 32 A. D. Andreev, J. R. Downes, D. A. Faux and E. P. O’Reilly, J. Appl. Phys., 1999, 86, 297. 33 C. G. Van de Walle, Phys. Rev. B, 1989, 39, 1871. 34 W. Shan, W. Walukiewicz, J. W. Ager, E. E. Haller, J. F. Geisz, D. J. Friedman, J. M. Olson and S. R. Kurtz, Phys. Rev. Lett., 1999, 82, 1221. 35 S. Tomic´, E. P. O’Reilly, R. Fehse, S. J. Sweeney, A. R. Adams, A. D. Andreev, S. A. Choulis, T. J. C. Hosea and H. Reichert, IEEE J. Sel. Top. Quantum Electron., 2003, 9, 1228. 36 S. Tomic´, 5th NUSOD’05 Conference, Berlin, Germany, ed. H.-J. Wunsche, J. Piprek, U. Bandelow and H. Wenzel, IEEE LEOS Press, NY, 2005, p. 3, TuA2. 37 S. Tomic´, Phys. Rev. B, 2006, 73, 125348. 38 P. J. Klar, H. Gru¨ning, J. Koch, S. Scha¨fer, K. Volz, W. Stolz, W. Heimbrodt, A. M. Saadi, A. Lindsay and E. P. O’Reilly, Phys. Rev. B, 2001, 64, 121203(R). 39 A. Lindsay and E. P. O’Reilly, Physica B, 2003, 340–342, 434. 40 M. Hetterich, A. Grau, A. Yu. Egorov and H. Riechert, J. Appl. Phys., 2003, 94, 1810. 41 S. Tomic´ and E. P. O’Reilly, Phys. Rev. B, 2005, 71, 233301. 42 S. Tomic´, E. P. O’Reilly, P. J. Klar, H. Gru¨ning, W. Heimbrodt, W. M. Chen and I. A. Buyanova, Phys. Rev. B, 2004, 69, 245305. 43 S. A. Choulis, T. J. C. Hosea, S. Tomic´, M. Kamal-Saadi, A. R. Adams, E. P. O’Reilly, B. A. Weinstein and P. J. Klar, Phys. Rev. B, 2002, 66, 165321. 44 J. A. Barker and E. P. O’Reilly, Phys. Rev. B, 2000, 61, 13840. 45 S. Tomic´, P. Howe, N. M. Harrison and T. S. Jones, J. Appl. Phys., 2006, 99, in press. 46 R. A. Morrow and K. R. Brownstein, Phys. Rev. B, 1984, 30, 678. 47 N. Vukmirovic´, D. Indjin, V. D. Jovanovic´, Z. Ikonic´ and P. Harrison, Phys. Rev. B, 2005, 72, 075356. 48 I. Vurgaftman and J. R. Meyer, J. Appl. Phys., 2003, 94, 3675. 49 http://www.hpcx.ac.uk. 50 http://www.netlib.org. 51 http://publib.boulder.ibm.com/clresctr/windows/public/esslbooks.html. 52 http://www.netlib.org/scalapack/html/fig/2dbsd.ps. 53 The calculation of specified eigenvectors by inverse iteration, contribution II/18, Handbook of Automatic Computation, SpringerVerlag, Berlin, 1971, vol. 2, pp. 418–439. 54 F. Tisseur and J. Dongarra, SIAM J. Sci. Comput., 1999, 20, 2223. 55 P. Bientinesi, I. S. Dhillon and R. A. van de Geijn, UT CS Technical Report TR-03026, 2003, http://www.cs.utexas.edu/users/ plapack/papers/pareig.ps. 56 C. W. Tu, in Physics and Applications of Dilute Nitrides, ed. I. A. Buyanova and W. M. Chen, Taylor & Francis Cooks, Inc., New York–London, 2004, ch. 10, pp. 281–307. 57 M. Sopanen, H. P. Xin and C. W. Tu, Appl. Phys. Lett., 2000, 76, 994.

J. Mater. Chem., 2006, 16, 1963–1972 | 1971

58 A. Nishikawa, Y. G. Hong and C. W. Tu, Phys. Status Solidi B, 2003, 240, 310. 59 A. Nishikawa, R. Katayama, K. Onabe, Y. G. Hong and C. W. Tu, J. Cryst. Growth, 2005, 278, 244. 60 R. Oshima and Y. Okada, Thin Solid Films, 2004, 464–465, 229. 61 K. C. Yew, S. F. Yoon and Z. Z. Sun, J. Cryst. Growth, 2004, 271, 8. 62 O. Schumann, S. Birner, M. Baudach, L. Geelhaar, H. Eisele, L. Ivanova, R. Timm, A. Lenz, S. K. Becker, M. Povolotskyi, M. Da¨hne, G. Abstreiter and H. Reichert, Phys. Rev. B, 2005, 71, 245316. 63 P. R. C. Kent and A. Zunger, Phys. Rev. Lett., 2001, 86, 2613.

1972 | J. Mater. Chem., 2006, 16, 1963–1972

64 A. Lindsay and E. P. O’Reilly, Phys. Rev. Lett., 2004, 93, 196402. 65 A. Lorke, R. J. Luyken, A. O. Govorov, J. P. Kotthaus, J. M. Garcia and P. M. Petroff, Phys. Rev. Lett., 2000, 84, 2223. 66 A. Lorke, R. Blossey, J. M. Garcia, M. Bichler and G. Abstreiter, Mater. Sci. Eng., B, 2002, 88, 225. 67 R. J. Warburton, C. Schaflein, D. Haft, F. Bickel, A. Lorke, K. Karrai, J. M. Garcia, W. Schoenfeld and P. M. Petroff, Nature, 2000, 405, 926. 68 J. A. Barker, R. J. Warburton and E. P. O’Reilly, Phys. Rev. B, 2004, 69, 035327. 69 B. G. Serle, Comput. Phys. Commun., 2001, 137, 25.
