Modelowanie Nanostruktur. On theoretical concepts for modeling nanostructures the messages to take home. Examples of Nanostructures

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1 Chair of Condensed Matter Physics stitute of Theoretical Physics Faculty of Physics, University of Warsaw Examples of Nanostructures Si(111)7 7 Surface TEM image of a As/GaAs dot Semester Zimowy 2011/2012 Wykład Modelowanie Nanostruktur Jacek A. Majewski Jacek.Majewski@fuw.edu.pl GaN GaN GaN HRTEM image: segregation of dium in GaN/GaN Quantum Well Modelowanie Nanostruktur, 2011/2012 Jacek A. Majewski Nanotechnology Low Dimensional Structures Lecture 12 January 3rd, 2012 On theoretical concepts for modeling nanostructures the messages to take home Quantum Wells Quantum Wires Quantum Dots Jacek.Majewski@fuw.edu.pl A B Simple heterostructure 1

2 Density of states Envelope Function Theory Effective Mass Equation J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955). Continuum theory- Envelope Function Theory [ ( i) U( r ) ] F ( r ) 0 EME does not couple different bands n ( r ) F ( r ) u ( r ) True wavefunction n Envelope Function n0 Periodic Bloch Function (EME) Special case of constant (or zero) external potential Ur ( ) 0 F ( r ) exp( ik r ) n ( r ) Uz ( ) F ( r ) exp[ i( k x k y)] F ( z) n x y n Bloch function Electron in an external field Electron States in Quantum Dots ˆ 2 Periodic potential of crystal Strongly varying on atomic scale p V ( r ) U( r ) ( r ) ( r ) 2m Band Structure U( r ) 0 n( k ) Non-periodic external potential Slowly varying on atomic scale Energy [ev] Band structure of Germanium L3 L1 L 3 L1 L Ge A B A Self-organized quantum dots Electrons confined in all directions U( x, y, z) F (,, ) (,, ) (,, ) (,, ) 2 * n x y z U x y z Fn x y z EnFn x y z m x y z Density of states for zero dimensional (0D) electrons (artificial atoms) (0 D G ) ( E) ( E E ) Wave vector k E1 E2 E3 E 4 E 2

3 Calculation of the strain tensor 1 Elastic E C ijkl ( x )ε ij ( x )ε kl ( x )dx 2 Minimization of elastic gives the strain distribution ε kl ( x ) It corresponds to ζ C ε ij ijkl kl ζ ij x i 0 GaN Hook s Law Strain Map AlGaN SiN Dot shape and piezoelectric charges Piezoelectric charges Localization of electron and hole wavefunction (for GaN/AlGaN HEMT ) No light emission Efficient light emission 3D nano-device simulator - nextnano 3 Simulator for 3D semiconductor nano-structures: Si/Ge and III-V materials Flexible structures & geometries Fully quantum mechanical Equilibrium & nonequilibrium Calculation of electronic structure : 8-band kp-schrödinger+poisson equation Global strain minimization Piezo- and pyroelectric charges Exciton energies, optical matrix elements,... Calculation of current only close to equilibrium with new approach How good is effective mass aprox.? E 1 AlGaAs GaAs E C Effective mass d Exact d [nm] Atomistic details sometimes matter! E 1 3

4 Farsightedness (hyperopia) of the Standard k p Model Alex Zunger, phys. stat. sol. (a) 190, 467 (2002) The use of a small number of bands in conventional k p treatment of nanostructures leads to farsightedness (hyperopia), detailed atomistic symmetry is not seen, only the global landscape symmetry is noted, the real symmetry is confused with a higher symmetry. Number of important symmetry-mandated physical couplings are unwittingly set to zero These are often introduced, after-the-fact, by hand, via an ansatz. atomistic theories of nanostructures, the physically correct symmetry is naturally forced upon us by the structure itself. Atomistic methods for modeling of nanostructures Ab initio methods (up to few hundred atoms) Semiempirical methods (up to 1M atoms) Empirical Pseudopotential Tight-Binding Methods What about realistic nanostructures? organics 3D (bulks) : 1-10 atoms in the unit cell 2D (quantum wells): atoms in the unit cell 1D (quantum wires): 1 K-10 K atoms in the unit cell 0D (quantum dots): 100K-1000 K atoms in the unit cell Ab initio = Density Functional Theory Based Methods Organics Nanotubes, DNA: atoms (or more) 4

5 Density Functional Theory (DFT) One particle density determines the ground state of the system for arbitrary external potential E[ρ] d 3 r ρ( r )υ ext ( r ) F [ ρ] Total functional E[ρ ] E 0 0 ground state density ground state E[ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ] ext S x c External Kinetic Classic Coulomb Exchange Correlation unknown!!! υ ( r ) υ ( r ) υ ( r ) υ ( r ) θ ( r ) ε θ ( r ) 2 2 ext H x c i i i 2m The Kohn-Sham Method Kohn-Sham equations with local potential δu ρ( r') υ H ( r ) dr' δρ r r' υ ( r ) e ext 2 s δe x [ ρ] δρ Z rη R s,n s n N ρ( r ) θ* i ( r )θ i( r ) i1 δe c [ ρ] δρ These equation are nonlinear and must be solved iteratively (self-consistently) Density Functional Theory (DFT) Approximations to the exchange & correlation E[ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ] ext S x c unknown!!! Local Density Approximation (LDA) LDA hom Exc [ ] dr( r ) xc ( ( r )) Generalized Gradient Approximation (GGA) GGA xc E [ ] dr f ( ( r ), ( r )) xc The Kohn-Sham Method Total Energy & Forces 2 N * 2 E[ρ] drθ i ( r ) θ i( r ) U [ ρ] E x [ ρ] E c [ ρ] drυ ext( r )ρ( r ) 2m i 1 total electronic in the field of ions 1 ZαZβ Eion 2 α,β Rα R β Energy of ions Total E ({ R }) E ({ R }) E ({ R }) tot α el α ion α Forces on ions Equilibrium: αf 0 Dynamics: MαRα F α α F α E R tot α Rα E tot 5

6 Solution of the Kohn-Sham Equations - Methods Expansion of the Kohn-Sham orbitals in a basis { ( r k )} ( r ) c ( n, k ) ( r ) nk k Plane waves and pseudopotential χ ( r ) exp( ig r ) G Tight-Binding methods Linear combination of atomic orbitals (LCAO) Discretization in real space χ i( r ) δ( r r i ) Local in space suitable for transport DFT for silicon nanostructures Silicon nanoparticles (clusters, dots) optoelectronic materials on silicon basis biosensors to detect biological and chemical warfare agents Si H 2 H replaced by O 71 Si atoms passivated by hydrogens Electrons are in the center of the dot O Dramatic change of the optical properties (wavelength) of the silicon nanostructure G. Gali & F. Gygi, Lawrence Livermore National Laboratory Tight-Binding Formalism i n n i, i ( r ) c ( r ) index of orbital αi iα index of atom αi βj n n n n n iα jβ H ε iα iα t iα jβ αi,βj iα, jβ { χ αi }orthogonal set ε θ H θ ( c )*c iα H jβ Tight binding Hamiltonian on-site of functions NOT ATOMIC ORBITALS! hoping TB-parameters 6

7 αi Tight-Binding Hamiltonian iα iα iα iα, jβ iα jβ αi,βj H ε c c t c c creation & anihilation operators On-site energies are not atomic eigenenergies They include on average the effects of neighbors Problem: Transferability E.g., Si in diamond lattice (4 nearest neighbors) & in fcc lattice (12 nearest neighbors) Dependence of the hopping energies on the distance between atoms Why Tight-Binding? Allows us to describe the band structure over the entire Brillouin zone Relaxes all the approximations of Envelope Function approaches Allows us to describe thin layer perturbation (few Å) Describes correctly band mixing Gives atomic details The computational cost is low It is a real space approach Molecular dynamics Scalability (from empirical to ab-initio) Scalability of TB approaches Empirical Tight-Binding Hamiltonian matrix elements are obtained by comparison of calculated quantities with experiments or ab initio results. Very efficient, poor transferability. The sp 3 s* Hamiltonian [Vogl et al. J. Phys. Chem Sol. 44, 365 (1983)] order to reproduce both valence and conduction band of covalently bounded semiconductors a s* orbital is introduced to account for high orbitals (d, f etc.) Semi-Empirical Hartree-Fock Methods used in the chemistry context (INDO, PM3 etc.). Medium transferability. Density Functional based Tight-Binding (DFTB, FIREBAL, SIESTA) DFT local basis approaches provide transferable and accurate interaction potentials. The numerical efficiency of the method allows for molecular dynamics simulations in large super cells, containing several hundreds of atoms. 7

8 The sp 3 d 5 s* Hamiltonian Tight-binding for quantum wells (1) Many parameters, but works quite well! Schroedinger equation m ; k // H E k // 0 Matrix form M H mm l, lm C lm EC l where M is the number of interacting planes For M=1 we have: H l,l l+1 l l-1 H l,l+1 H l,l-1 H = but boundary conditions? z y x LCAO wavefunction k // Tight-binding for quantum wells (1) For a quantum well and similar 2D heterojunctions, the symmetry is broken in one direction, thus the Bloch theorem cannot apply in this direction. We define the Bloch sums in the m-th atomic plane m k planes, m orbitals, ; // 1 N m R// e C m k m k // ; m ik // R// // R this case the number of C s is related to the number of planes z md R R y m m d R// m R // x Tight-binding for quantum wells (3) Boundary conditions Periodic After P planes the structure repeats itself. Suitable for superlattices Finite chain Open boundary conditions H= After P planes the structure ends. Suitable for quantum wells. After P planes there is a semiinfinite crystal. Suitable for current calculations. H= BULK P P BULK P P 8

9 Where do we put the atoms? To describe the electronic and optical properties of a nanostructure we need to know where the atoms are. 1) We know a priori where the atoms are (for example X-ray informations) 2) We need to calculate the atomic positions Quantum calculation (DFTB, SIESTA) Continum theory elasticity theory Valence Force Field (Keating model & extensions) Aditional Repulsive terms added to TB Hamiltonian T-B molecular dynamics Tight-Binding Formalism Dependence of the hopping integrals on atomic distance Calculations for systems with distorted lattice The dependence of the hopping integrals on the inter-atomic distance Harrison s ~d -2 dependence Exponential dependence t t0 exp( βr ) t αβ ( R ij ) t ( r ) f ( R αβ 0 ij ) n nc nc r r r 0 exp n 0 r r f ( r ) r 1 rc rc 2 3 c0 c 1( r r 1 ) c 2( r r 2 ) c 3( r r 2 ) r r1 C. Xu et al., J. Phys. Condens. Matter 4, 6047 (1992) Tight-Binding Formalism The Total Energy Tight-Binding Formalism Dependence of the hopping integrals on atomic distance Band structure Repulsive Charge transfer The total repulsive contains ion ion repulsion, exchange correlation, and accounts for the double counting of electron electron interactions in the bandstructure term. The last term imposes an penalty on large inter-atomic charge transfers Charge transfer Typical U ~ 1eV LDA ( ) and tight-binding ( ) band structures for GaAs in the zinc-blende structure for two different bond lengths 9

10 contact Tight-Binding Formalism The Total Energy Boundary conditions for transport The binding (E coh ), repulsive (E rep ), and band-structure (E bs ) for GaAs in the zinc-blende structure, as functions of the interatomic distance. The transport problem is: contact active region where symmetry is lost + contact regions (semi-infinite bulk) active region contact Open-boundary conditions can be treated within several schemes: Transfer matrix Green Functions These schemes are well suited for localized orbital approach like TB Tight-Binding Formalism Parametrization of the repulsive term Using the interpolated hopping integrals, the tightbinding band-structure can be obtained for any geometry and inter-atomic distance. We then define the repulsive as the difference between the exact binding, obtained using ab initio calculations, and the tight-binding band-structure Several crystallographic phases of a material are usually used Structure independent parametrization of the repulsive terms Device description: Green Function Approach - H L G E 1 L LD Lead D H D R g DL L LD g DR R RD H D Molecular region r a L, R i L, R L, R L RD H R Lead R + Retarded (r) and advanced (a) Green functions are defined as follow (matrix notation) H L H DR H 0 With these definitions and considering that r, a E ii HG I Let us write H and G in a block form LD D RD 0 DR H R G G G G L DL RL G G G LD D RD GLR G DR GR ii H g I E L, R L, R defines the Green function g L,R of the semi-infinite lead (which can be easily calculated) Self- T tr G G L r D R a D Transmission coefficient 2e G 2 T Conductance 10

11 Conduction Band Edge [ev] dium fluctuations Tight-Binding method Applications Optical properties of GaN/GaN QWs Simulations of reactive collisions in biased CNT dium fluctuations are obtained from experimental results Tight-binding calculations with atoms dium fluctuations in GaN/Gan QWs Conduction Band Edge Profile Average strained 15% Pseudomorphic strain on GaN mev Depth [nm] Typical photoluminescence results exhibit two peaks 11

12 Number of atoms Green s Function + Molecular dynamics Carbon Nanotubes Molecular Dynamics simulations of a reactive collision of a biased nanotube (V=100mV) and benzene Current flowing in the nanotube calculated at each MD step v = 0.6 Å/ps Atomistic vs. Continuous Methods Microscopic approaches can be applied to calculate properties of realistic nanostructures 1e Tight-Binding Pseudo- 100 potential 10 Ab initio R (nm) Continuous methods Number of atoms in a spherical Si nanocrystal as a function of its radius R. Current limits of the main techniques for calculating electronic structure. Nanostructures commonly studied experimentally lie in the size range 2-15 nm. Time Dependent Current Conclusions TB Approaches A Current [ m A] R CN = 8 KW R CN-C6H4 = 10KW CNT without C 6 H 4 I = 20% C C A B B Time [ps] can range from empirical to ab-initio, avoid all the Envelope Function approximations, Green-Function or Transfer matrix techniques can be easily implemented to calculate electronic transport, molecular dynamics and current calculations can be coupled together. 12

13 Large scale modeling - Coarse-Graining Coarse-graining & effective approaches For large scale modeling, one may introduce alternative approaches using simplified coarse-grained models (lattice gas models) These models can be treated with the methods used commonly in statistical mechanics such as mean-field theory, the cluster variation method (CVM), Monte Carlo methods. Question: how to provide a link between atomistic calculations (ab initio, classical potentials) and the potential parameters suitable for coarse-grained models. What to do with large systems? Coarse-grained model for ions around DNA Polyelectrolyte problem: ions around DNA All-atom model Coarse-grained model Atomistic DFT calculations not really possible to sample distances A from DNA Na + Water molecule Na + 13

14 tegral charge Coarse-grained model for ions around DNA Lipid bilayer in water Na + Ions interacting with DNA by effective solvent-mediated potentials Different potentials for various parts of DNA No explicit water density profile and integral charge Li + Na + Na * K + * Cs PB r (Å) Picture contains atoms Minimal, reasonable piece of bilayer for atomistic calculations 72 lipids + at least 20 H 2 O per lipid = atoms Another example: Coarse-grained lipid model Hierarchy of Theoretical Approaches All-atom model 118 atoms Coarse-grained model 10 sites Device growth Time [s] Formation 10-3 of islands 10-6 Atomic motion 10-9 Atomic vibrations Monte Carlo Classical MD accelerated Classical MD Ab-itio MD DFT Level Set Continuum Methods size 1nm 1μm 1mm 1m length islands device circuit wafer 14

15 Simulation of growth processes Kinetic Monte Carlo (KMC) Kinetic Monte Carlo a tool for simulation of growth processes Modeling crystal growth with the KMC method allows one to cover experimentally relevant growth times and system sizes, since each event on the surface is just described by a single quantity the transition rate rather than by modeling the full reaction path including atomic geometries and energies Bridging of length and time scales From Molecular Dynamics to Kinetic Monte Carlo KMC Simulations: Effect of Nearest Neighbor Bond Energy E N KMC Simulations Large E N : Irreversible Growth Small E N : Compact Islands Experimental Data Au/Ru(100) Ni/Ni(100) Hwang et al., PRL 67 (1991) Kopatzki et al., Surf.Sci. 284 (1993) 15

16 Computational Materials Science: A Scientific Revolution about to Materialize Due to the complexity of materials systems, progress has necessarily proceeded either within the Bohr quadrant or Edison s quadrant Pasteur's Quadrant experiment and theory done on model systems research and development by trial and error Realistic simulation is the vehicle for moving materials research firmly into Pasteur's quadrant. Thank you! 16