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nextnano A large software project for the simulation of 3D nanometer semiconductor structures Walter Schottky Institut, TU München Current team: T. Zibold, T. Andlauer, R. Morschl, K. Smith, T. Kubis, P. Vogl Training and support: S. Birner Thanks to many contributors worIdwide: Vienna, Rome, ASU,... www.nextnano.de nextnano overview Goal: Provide quick global insight into basic physical properties of mesoscopic semiconductor structures Simulation software for 3D semiconductor nanostructures Si/Ge and III-V materials, Nitrides, alloys, zb and wz Flexible structures and geometries Quantum mechanical electronic structure Equilibrium properties and carrier transport Typically 5-10 downloads/day worldwide Calculation of electronic structure 8-band k.p-Schrödinger (+LDA) and Poisson equation Global strain minimization Piezoelectric, pyroelectric charges, deformation potentials Exciton energies and optical matrix elements Magnetic field and spin effects ISFET: Surface reactions @ semicond./electrolyte interfaces Calculation of charge current Quantum-drift-diffusion method : DD eq's + quantum densities Ballistic current through open systems: Contact Block Reduction Full quantum transport with scattering: NEGF for quasi-1D Examples of nanostructures 1000 nm In |1 Out |1 In |0 2DEG GaAs Out |0 300 nm Mach-Zehnderinterferometer z=4 nm Self-assembled quantum dot array y=500 nm x=500 nm InGaAs collector InAs wetting layers InP barrier InGaAs emitter Individual QD with e+h wave functions Quantum dotresonant tunneling diode Silicon 25 nm Triple-gate FET nextnano Program flow Input: structure, options Database: material parameters INITIALIZATION Bulk band structures, strain, def. pot's and piezo/pyro POISSON EQUATION Determination of potential CURRENT EQUATION QDD: Determination of quasi-Fermi levels Many-band kp SCHRÖDINGER (LDA) EQUATION Determination of wave functions and bound states OUTPUT MATRIX ELEMENTS WAVE FUNCTIONS OPTICAL SPECTRA,... Numerical principles and techniques Use state-of-the-art sparse linear systems solvers, CG methods for N > 106 Example: eigenvalue problems Hu=lu Problem: H huge (N > 106) + sparse, need only few l Solution: Subspace projection method Power method: u0 H u0 H2 u0 ... <u|H|u >= lmax ARPACK: very robust, degenerate evals, but fairly slow. Electronic structure principles and techniques Multiband k.p envelope function approach Based on "patching up" bulk Hamiltonians to build Hamiltonian for mesoscopic structures, is efficient and sufficiently accurate Method has built-in ambiguities that can lead to ghost states, spikes in density,... Spatial discretization can lead to instabilities and wrong oscillatory solutions Spikes Y 12 (S=0) Y21 Ghost states Have eliminated artifacts in k.p+envelope function theory by careful treatment of far-band contributions using operator orderings that are manifestly self-adjoint employing upwinding scheme for discretizing derivatives Multivalued operator ordering k·p HVol(k) = Ec iPk -i P k Ev+Lk2 P(x) /x or HVol() : /x P(x) or (P(x) /x + /x P(x)) Ordering unclear because of position dependent parameter P Different orderings yield Hermitian Ham., but cause contradictory boundary conditions (Non-self-adjointness) Solution of Problem: Self-adjoint H HVol(x) = Ref: B. A. Foreman, Phys. Rev. B 56, R12748 (1997) Ec P /x - /x P Ev- /x L /x Electronic structure principles and techniques Example: Eliminating oscillatory solutions Hbulk(k) H(): Discretization of 1. derivates is not unique. F(n) = F(n+1) - F(n-1) is compatible with n-1 n+1 n Solution: F(n) = [F(n) - F(n1)] excludes unphysical oscillatory solutions ForwardDifferencing H= BackwardDifferencing Ref: Andlauer et al, to be publ. equivalent to upwinding scheme Electronic structure principles and techniques Problem: How to solve Schrodinger equation for nanodevice in B-field? Vector potential A(x) diverges with x Discretized version of H violates gauge invariance arbitrary results (-i + A(x))2 H= +V(x) 2m Solution:* Invariance under gauge transformation is violated if f = lim f(x+e) - f(x) e e0 x y Define U(x,y) = exp( -i A(z) dz ) x Define D = 1e [f(x+e) - U(x+e,x) f(x)] Use Hamiltonian H = guarantees local gauge invariance D2 +V(x) 2m This Hamiltonian is gauge invariant and suitable for discretizing the Schrodinger equation in magnetic fields Works for any multi-band, relativistic k.p Hamilonian for nanostructures *) Morschl et al, to be publ. Carrier transport in nextnano Quantum drift-diffusion (QDD) equations: j (x) n(x)EF (x) 0 2 E (x)-Ei n(x) i (x) f F i k BT WKB-type approach, suitable for diffusive transport near equilibrium Good for barrier-limited transport Misses quantum resonances and interference effects Mamaluy, Sabathil, V., PRB 05 Contact block reduction-method (CBR): Efficient method to calculate strictly ballistic transport through open device with arbitrary number of leads Scales with N2 rather than N3 Suitable for very short quantum devices close to resonance Non-equilibrium Green’s function method (NEGF): Kubis, V., subm. Full quantum transport with all relevant sccattering mechanisms Only for vertical transport (quasi-1D) Assessment of QDD Tunneling through thin barrier Fully SC-NEGF -0.05 Ballistic 20 30 40 Position [nm] *) Kubis et al, Poster #96 50 Fully SC-NEGF nextnano -0.10 0V -0.18 10 20 30 40 50 Position [nm] 0.00 10 -0.02 0 0.05 -0.10 n-GaAs (1018 cm-3) 0.06 n-Si V=0 nextnano InGaAs n-GaAs 60 Conduction band [eV] Conduction band [eV] 0.10 n-Si (1018 cm-3) SiO2 Carrier capture by quantum well Conduction band [eV] Comparison of QDD with fully self-consistent NEGF*) shows good agreement... close to equilibrium in situations where interference effects are weak 0.06 nextnano Fully SC-NEGF -0.02 -0.10 0.08 V -0.18 0 10 20 30 Position [nm] 40 50 1D Results: Optical absorption in SiGe QCL-structures Si/SiGe p-type Quantum Cascade Laser Structure Collector -0,70 HH1 Energy [eV] -0,80 Injector -0,90 -1,00 -1,10 SiGe Si -1,20 HH2 50 kV/cm -10 0 10 20 30 40 Distance [nm] QC structure: G. Dehlinger et al., Science 290, 2277 (2000) HH1 HH2 Exp: 125 meV Nextnano: 124.5 meV Lateral non-parabolicity plays important role for optical spectra: accurate 6-band kp-model necessary 2D Results: Equilibrium + QDD for Si DG-FET Effect of el-el interaction: exchange-correlation potential Local density functional theory adds VXC=VX+VC to VHartree -10 -20 y VX >> VC VX = a n1/3 = 50 meV for n=1020 cm-3 Exchange Correlation G D S G x -30 VSD = 0.05 V VSG = 0 V 2 10 -40 -50 Source Drain Hartree + XC Hartree -60 -1 -70 0 5 10 15 20 25 30 35 40 45 50 X axis [nm] Hartree Hartree+XC -0.02 -0.04 Potential [eV] Source-drain current 101 Current [Acm ] Potential energy [meV] 0 Gate Gate -0.06 -0.08 0 10 VSD=0.05 V -1 10 10-2 10-3 10-4 Gate current -0.10 -0.12 yS -0.14 G D G x -0.16 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Y axis [nm] 10-5 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 Gate Voltage VSG [V] 0.1 VXC has very large effect for small VSD B InP InAs Nanowire InP 53 nm 32 nm Effective g-factor 3D Results: Prediction of magnetic g-tensors in nanowires 10 8 nextnano g^ 6 nextnano 4 g// Experiment (Björk et al, 05) 2 8 Ground state density (front view) B=20 T B=0 T g// 10 12 14 16 18 20 Length [nm] Excellent agreement between calculated g-factor and experiment 3D results: Self-assembled buried QD exx GaAs exx exz GaAs exz InAs InAs Material strain 20 nm 20 nm - + + - Hole Piezoelectric polarization charge Hole Electron Efficient light emission Electron Modified exciton states No light emission 3D results: Quantum Dot Molecule Vertically stacked Data: QD InGaAs/GaAs vertically stacked QD Strain field (nextnano) exx WL 6 nm WL Electron wave functions: Electron & hole wave functions bonding antibonding Calculate exciton energies 3D Results: Neutral excitons in QD-Molecule Quantum coupling + strain + Coulomb interaction Large separation: direct and indirect excitons Small separation: el-dominated bonding and antibonding excitons Exciton Energy [eV] 1.29 antibonding 1.28 1.27 1.26 indirect Ex bonding Coulomb interaction [~20 meV] 1.25 direct Ex 1.24 2 4 6 8 Dot separation [nm] 10 3D Results: Anticrossing of direct + indirect states Energy [meV] indirect direct 1285 * Nextnano 1280 Experiment 1275 14 16 18 20 22 24 Applied Field [kV/cm] *) P.W. Fry et al, PRL 84, 733 (2000), G. Ortner et al., PRL 94, 157401 (2005) H. J. Krenner et al., PRL 94, 057402 (2005), G. Bester et al., cond-mat/0502184 3D results: ballistic transport: resonant tunneling through QD-molecule InGaAs/InP dots InGaAs collector InAs wetting layers J InP barrier 40 nm InGaAs emitter 250 x 10 200 150 Current [pA] 100 Exp. Theory 50 0 0.2 0.3 0.4 0.5 Voltage [V] Bryllert et al, APL 82 , 2655(2003) Ballistic current (CBR) Good agreement in resonance position Absolute value of J off by factor of 10 Current depends strongly on inter-dot distance and lateral misalignment Resonant tunneling through QD-molecule 30 25 20 15 10 5 0 7 Line width [meV] Line width [meV] The resonance line width ... 11 12 13 14 15 Distance between WLs [nm] ...decreases exponentially with increasing inter-dot distance 6 5 4 25 50 75 100 Overlap of base areas [%] ...and is proportional to overlap of base areas How to get nextnano? Software including source is free (nextnano) Online documentation is free Online registration is free Support, customized input files + on-site training available on request (by S. Birner) Some complex tutorial files (QCLs, MOSFETs) are not free www.nextnano.de Summary Nextnano provides base for physics of 1D, 2D, and 3D semiconductor nanostructures Handles equilibrium electronic structure, optics, magnetic fields Nonequilibrium: QDD approach, ballistic current, and NEGF Successful application to 2D+3D nano-MOS, QD molecules, excitons, magnetic field effects, QCL‘s,...