Modelowanie Nanostruktur

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1 Chair of Condensed Matter Physics Institute of Theoretical Physics Faculty of Physics, Universityof Warsaw Modelowanie Nanostruktur, 2012/2013 Jacek A. Majewski Semester Zimowy 2012/2013 Wykad Modelowanie Nanostruktur Jacek A. Majewski Wykad 3 9 X 2012 Metody ci"ge w modelowaniu nanostruktur Metoda masy efektywnej dla studni kwantowej Jacek.Majewski@fuw.edu.pl Nanotechnology Low Dimensional Structures What about realistic nanostructures? Inorganics 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 Quantum Wells Quantum Wires Quantum Dots 0D (quantum dots): 100K-1000 K atoms in the unit cell A Simple heterostructure Organics Nanotubes, DNA: atoms (or more) 1

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8 Synthesis of colloidal nanocrystals Nanostructures: colloidal crystals Injection of organometallic precursors -Crystal from sub-m spheres of PMMA (perpex) suspended in organic solvent; - self-assembly when spheres density high enough; Mixture of surfactants Heating mantle Atomistic vs. Continuous Methods Microscopic approaches can be applied to calculate properties of realistic nanostructures Number of atoms 1e Tight-inding 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. Atomistic methods for modeling of nanostructures Ab initio methods (up to few hundred atoms) Semiempirical methods (up to 1M atoms) (Empirical Pseudopotential) Tight-inding Methods Continuum Methods (e.g., effective mass approximation) 2

9 Electron in an external field Continuum theory- Envelope Function Theory Periodic potential of crystal Strongly varying on atomic scale U( r ) = 0 ˆp 2 2m +V ( r ) +U( r $ # )& "# %& ( r ) = "( r ) and Structure n ( k) Non-periodic external potential Slowly varying on atomic scale Energy [ev] and structure of Germanium L3 L1 L 3 L1 L 2 " 3 " 1 " 3 " 1 " 1 " 1 Ge # 15 # 2 # 25 #1 $ 5 $ 2 $1 $ 5 $ 2 $ Wave vector k Electron in an external field ˆp 2 2m +V ( r ) +U( r $ # )& "# %& ( r ) = "( r ) Periodic potential of crystal Non-periodic external potential Strongly varying on atomic scale Slowly varying on atomic scale Which external fields? Shallow impurities, e.g., donors U( r ) = e2 r Magnetic field, = curl A = " A Heterostructures, Quantum Wells, Quantum wires, Q. Dots cbb GaAs GaAlAs GaAlAs GaAs GaAlAs Does equation that involves the effective mass and a slowly varying function exist? $ ˆp2 # 2m * +U( r ) & F( r ) = F( r ) " % F( r ) =? Envelope Function Theory Effective Mass Equation J. M. Luttinger & W. Kohn, Phys. Rev. 97, 869 (1955). [(i ") +U( r ) ]F n ( r ) = 0 EME does not couple different bands True wavefunction ( r ) = F n ( r )u n0 ( r ) Envelope Function Special case of constant (or zero) external potential U( r ) = 0 F n ( r ) = exp(i k r ) U( z) Periodic loch Function F n ( r ) = exp[i(k x x + k y y)]f n (z) (EME) ( r ) loch function 3

10 Electronic states in Quantum Wells Envelope Function Theory- Electrons in Quantum Wells Effect of Quantum Confinement on Electrons Let us consider an electron in the conduction band near point cbb Potential U( r )? GaAlAs GaAs GaAlAs U( " $ r ) = U(z) = c0 z GaAs # %$ c0 + E c z GaAlAs Growth direction (z direction ) U( r c ( k) = c0 + 2 k 2 ) is constant in the xy plane 2m * Effective Mass Equation for the Envelope Function F 2 " 2 2m * x y + 2 % $ # 2 z 2 ' F(x, y, z) +U(z)F(x, y, z) = EF(x, y, z) & Separation Ansatz F( x, yz, ) = Fx( xf ) y( yf ) z( z) # " 2 F x 2m * "x F 2 y F z + " 2 F y "y F 2 x F z + " 2 & % F z "z F 2 x F ( % y ( +U(z)F x F y F z = EF x F y F z $ ' 2 Effective Mass Equation of an Electron in a Quantum Well # 2 " 2 F x 2m * "x F 2 y F z + " 2 F y "y F 2 x F z + " 2 & % F z "z F 2 x F ( % y ( +U(z)F x F y F z = EF x F y F z $ ' E = Ex + Ey + Ez 2 2 F x 2m * x F 2 y F z = E x F x F y F z 2 2 F y 2m * y F 2 x F z = E y F x F y F z 2 " 2 F z 2m * "z F 2 x F y +U(z)F x F y F z = E z F x F y F z 2 2 F x 2m * x = E 2 x F x F x ~ e ik x x, Ex = 2 2m * k 2 x 2 2 F y 2m * y = E 2 y F y F y ~ e ik y y, E y = 2 2m * k 2 y 2 2 F zn +U(z)F 2m * z 2 zn = E zn F zn Conduction band states of a Quantum Well F k ( r ) = 1 A exp[i(k x x + k y y)] = 1 A exp(i k r ) F n, k ( r ) = F k ( r )F zn (z) = 1 A exp(i k r )F zn (z) E n ( k ) = 2 " k 2 2m * + E zn E zn Energy Energies of bound states in Quantum Well c0 + "Ec n=3 n=2 n=1 c0 Wave functions F zn (z) E zn E 2 E F zn +U(z)F 2m * z 2 zn = E zn F zn E n ( k ) Fzn ( z ) k E = E k 2m * E = E k 2m * 4

11 HgTe ZnTe CdSe and structure Engineering and lineups in semiconductors 0 Energy Gap [ev] ENERGY (ev) CdTe ZnSe ZnO ZnS CdS InSb GaSb AlSb InAs GaAs AlAs InP GaP AlP InN GaN AlN SiC SiGe Lattice constant The major goal of the fabrication of heterostructures is the controllable modification of the energy bands of carriers -12 and edges compile by: Van de Walle (UCS) Neugebauer (Padeborn), Nature 04 Envelope Function Theory- Electrons in Quantum Structures Various possible band-edge lineups in heterostructures Type-I Type-II (staggered) (misaligned) E c E c E cbb E cbb E gap c A gap E gap A E gap A A vbt A E gap cbb A E v E v E gap E v vbt vbt e.g., GaAs/GaAlAs GaN/SiC InAs/GaSb E v - Valence and Offset (VO) E c - Conduction band offset VO s can be only obtained either from experiment or ab-initio calculatio cbb and lineup in GaAs / GaAlAs Quantum Well with Al mole fraction equal 20% vbt 1.75 ev 1.52 ev GaAlAs GaAs 0.14 ev 0.09 ev Effective Mass Theory with Position Dependent Electron Effective Mass m * ( z) * m A * m z = 0 * * ma m z = 0 Graded structures 2 d 2 2m *(z) dz 2 IS NOT HERMITIAN 2 d 1 d Symetrization of the kinetic energy operator 2 dz m *(z) dz 2 2 d 1 d $ # &F(z) +U(z)F(z) = EF(z) dz " m *(z) dz % F( z) General form of the kinetic energy operator T [ m*( z)] pˆ " [ m*( z)] pˆ = [ m*( z)] with 2 + " = # 1 z z m * A 1 df( z) and m *( z) dz IS HERMITIAN ARE CONTINOUOS 5

12 Energy band diagram of a selectively doped AlGAAs/GaAs Heterostructure before (left) and after (right) charge transfer Doping in Semiconductor Low Dimensional Structures A A E g VACUUM LEVEL AlGaAs E c E g E v GaAs A E g l d 1 Positively charged region L A E v E F E g Negatively charged region A and - The electron affinities of material A & The Fermi level in the GaAlAs material is supposed to be pinned on the donor level. The narrow bandgap material GaAs is slightly p doped. (0) U( z) = U ( z) + e ( z) Effects of Doping on Electron States in Heterostructures E F E c (z) E F Unstable E c Charge transfer E 1 Thermal equilibrium Resulting electrostatic potential Fermi distribution function 2 ( r ) = 4e $ "( r " R A ) " ( r " R ' & # # D ) "# f " ( r ) 2 ) %& (acc) (don) () should be taken into account in the Effective Mass Equation ) 2 # " 2 2m * "x + " 2 2 "y + " 2 &, + % $ 2 "z 2 ( +U(x, y, z) e( r ). * + ' -. " ( r ) = E " ( r ) Electrostatic potential can be obtained from the averaged acceptor and donor concentrations 2 ( r ) = 4e $ & N A ( r ) " N D ( r ' ) "# f " ( r ) 2 ) %& () The self-consistent problem, so-called Schrödinger-Poisson problem To be continued 6