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Kp Method for Bulks Faculty of Physics, Modeling of Nanostructures University of Warsaw and Materials Jacek A. Majewski Lecture 12 – May 19, 2014 Summer Semester 2014 Continuous Methods for Modeling! Lecture ! Electronic Structure of Nanostructures! Modeling of Nanostructures and Materials !!Examples of nanostructures! !!From atomistic to continuum methods! !!k.p methods! !!Effective mass approximation! Jacek A. Majewski !!Envelope Function Theory! E-mail: [email protected] e-mail: [email protected] Nanotechnology – Low Dimensional Structures Modeling Nanostructures Quantum Quantum Quantum Wells Wires Dots A B Simple heterostructure 1! What about realistic nanostructures ? Examples of Nanostructures Inorganics Si(111)7!7 Surface TEM image of a InAs/GaAs dot 3D (bulks) : 1-10 atoms in the unit cell 2D (quantum wells): 10-100 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 GaN Organics HRTEM image: Nanotubes, DNA: 100-1000 atoms (or more) InGaN segregation of Indium in GaN/InGaN GaN Quantum Well Synthesis of colloidal nanocrystals Nanostructures: colloidal crystals! Injection of organometallic -!Crystal from sub-!m precursors spheres of PMMA (perpex) suspended in organic solvent; -! self-assembly when spheres density high enough; illuminated with white light Bragg‘s law different crystals different orientation Mixture of surfactants Heating mantle different ! 2! Hot topic (to come) – SEM of ZnTe nanowires grown by MBE on GaAs with Au nanocatalyst The curious world of nanowires Nanowire site control and branched NW structures: nanotrees and nanoforests Nanowire nanolasers Room temperature lasing action from chemically synthesized ZnO nanowires on sapphire substrate A SEM image http://www.nano.ftf.lth.se/ Schematic illustration One end of the nanowire is the epitaxial interface between the (A) Nanowires can be accurately positioned using lithographic methods such as EBL and NIL. (B) Subsequent seeding by aerosol deposition produces nanowire branches on an array as in panel (A). Shown here is a top view of sapphire such a ‘nanoforest’ where the branches grow in the <111>B crystal directions out from the stems. (C) Dark field STEM image and EDX line scan of an individual nanotree. An optically active heterosegment of GaAsP in GaP has been and ZnO, whereas the other end is the crystalline ZnO (0001) incorporated into the branches. plane Huang, M., Mao, S.S., et al., Science 292, 3! Detection of single viruses with NW-FET Controlled Growth and Structures of Molecular-Scale Silicon Nanowires Patolsky & Lieber, Materials Today, April 2005, p. 20 Patolsky et al., Proc. Natl. Acad. Sci. USA 101 (2004) 14017 (a)! TEM images of 3.8-nm SiNWs grown along the <110> direction (c) cross-sectional image (b) & (d) models based on Wulff construction Simultaneous conductance and optical data recorded for a Si nanowire device after the introduction of influenza A solution. Yue Wu et al., NANO LETTERS 4, 433 (2004) High Performance Silicon Nanowire Field Effect Transistors Heterostructured Nanowires Comparison of SiNW FET transport parameters with those for state-of-the-art planar MOSFETs show that “SiNWs have the potential to exceed substantially conventional devices, and thus could be coaxial longitudinal ideal building blocks for future heterostructured heterostructured nanoelectronics.” nanowire nanowire Yi Cui, et al. NANO LETTERS 3, 149 (2003) 4! Heterostructured Nanowires Transmission electron microscopy images of a GaN/AlGaN two Si/SiGe Methods core–sheath nanowire superlattice nanowires for Nanostructure Modeling Atomistic methods for modeling of nanostructures Density Functional Theory (DFT) One particle density determines the ground state energy of the system for arbitrary external potential ! ! ! Ab initio methods (up to few hundred atoms) E [ ! ] = d 3r!( r )! ( r ) + F [ ! ] ! ext E[!00]E= ground state energy ground state density Total energy Kinetic Exchange Correlation functional energy energy energy ! ! ! ! = ! ! + ! + ! + ! + ! E[ ] ! dr ext (r) (r) TS[ ] U[ ] Ex[ ] Ec[ ] External Classic Coulomb unknown!!! energy energy 5! Atomistic methods for modeling of Tight-Binding Hamiltonian nanostructures †† H = ! #i!cci! i! + ! ti! ,j" cci! j" Ab initio methods (up to few hundred atoms) !i !i,"j Semiempirical methods (up to 1M atoms) creation & anihilation operators Tight-Binding Methods 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 Atomistic methods for modeling of Atomistic vs. Continuous Methods nanostructures Microscopic approaches can be applied to calculate properties of realistic nanostructures 1e+06 Ab initio methods (up to few hundred atoms) Continuous 100 000 methods Semiempirical methods (up to 1M atoms) 10 000 Tight-Binding 1 000 Pseudo- (Empirical Pseudopotential) 100 potential Tight-Binding Methods Number of atoms 10 Ab initio 1 Continuum Methods 0 2 4 6 8 10 12 14 R (nm) (e.g., effective mass approximation) 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. 6! Electron in an external field ! ! pˆ 2 ! ! $ ! ! # +V (r ) + U(r )&! (r ) = !" (r ) "# 2m %& Continuum theory- Periodic potential of crystal Non-periodic external potential Strongly varying on atomic scale Slowly varying on atomic scale Envelope Function Theory Band structure of Germanium 5 "3 "1 #15 Band Structure $5 ! ! L3 $ 2 ‘ "1 U( r ) = 0 ! (k) #2 ‘ !1 n L1 $1 0 "3 L ‘ $5 3 # ‘ " 25 !4 L1 1 Energy [eV] Ge $ 2 ‘ -5 L 2‘ "1 $ 1 !1 #1 Wave vector k Electron in an external field Envelope Function Theory – ! # pˆ 2 ! !!$ ! Effective Mass Equation % +Vr()+ Ur ()&! () r= "! () r '% 2m (& J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955). ! Periodic potential of crystal Non-periodic external potential ! ! [!(!i") + U(r ) ! !]F (r ) = 0 (EME) Strongly varying on atomic scale Slowly varying on atomic scale n Which external fields ? ! e2 " Shallow impurities, e.g., donors Ur()= " ! EME does not couple different bands ! !!!! ! ||r ! ! ! "! Magnetic field B, B = curlA= ! " A ! (r ) = F (r )u (r ) "! Heterostructures, Quantum Wells, Quantum wires, Q. Dots n n0 “True” Periodic cbb Envelope wavefunction Bloch Function Function GaAs GaAlAs GaAlAs GaAs GaAlAs Special case of constant (or zero) external potential Does equation that involves the effective mass and a slowly varying ! ! ! ! ! ! U(r ) = 0 F (r ) = exp(ik ! r ) ! (r ) Bloch function function exist ?" pˆ 2 !!# ! ! n $ + Ur()% Fr ()= ! Fr () ! $ % Fr()= ? F (r ) = exp[i(k x + k y)]F (z) & 2*m ' Uz() n x y n 7! k.p Method- Luttinger-Kohn basis Bloch functions are orthogonal in the wave vector and band index ! ! ! ! ! ! ! d 3r" *(k, r )" (q, r ) = # # (k $ q) ! l n nl ! ! ! ! ! ! ! " ! ! ! d 3ru*(k, r )u (k, r ) = $ ! (k, r ) = exp[ik " r ]u (r ) ! l n 3 nl n n (2# ) ! ! ! ! ! ! ! d 3ru*(k, r )u (q, r ) " 0 k ! q ! l n for Luttinger-Kohn basis k.p method for bulks ! ! ! ! ! ! ! ! ! ! ! ! (k, r ) = exp[i(k " k ) # r ]$ (k , r ) = exp[ik # r ]u (k , r ) j 0 j 0 j 0 ! ! ! ! ! ! ! d 3r " *(k, r )" (q, r ) = # # (k $ q) ! l n nl ! ! ! ! ! ! ! Luttinger-Kohn basis 3 # * # = $ % ! " d k n(k, r ) n(k, r ') (r r ') is orthogonal and complete n Expansion of the unknown Bloch function in terms of known Luttinger-Kohn functions ! ! ! ! ! ! n(k, r ) = # Anj (k )" j (k, r ) j k.p Method - Derivation k.p Method - Derivation ! 2 " " ! " " " " " " { (k ! k )2 + (k ! k ) " p + # (k )}$ (k , r ) = ! ! ! ! ! ! 0 0 j 0 j 0 ! p2 ! $ ! ! ! ! ! 2m m ! n(k, r ) = # Anj (k )" j (k, r ) # +V (r )&' n(k, r ) = ( n(k )' n(k, r ) 2 " " " " " " " 2m j ! 2 ! " " " " % = { (k ! k ) + (k ! k ) " p + # (k )}exp[ik " r ]u (k , r ) = 2m 0 m 0 j 0 0 j 0 2 ! 2 " " " " " " " " " " ! ! p ! $ ! ! ! ! ! ! 2 2 ! A (k ) +V (r ) exp[i(k ( k ) ) r ]* (k , r ) = = exp[ik0 " r ]{ (k ! k0 ) + (k ! k0 ) " p + # j (k0 )}u j (k0, r ) ' nj # & 0 j 0 2m m j "2m % ! ! ! ! ! ! 2 ! ! ! ! ! ! ! ! " 2 2 " ! ! ! = + n(k )exp[i(k ( k0 ) ) r ]' Anj'(k )* j'(k0, r ) ! Anj (k ){ (k " k0 ) + (k " k0 ) # p + $ j (k0 )}u j (k0, r ) = p ! "i"# 2m m j' j ! ! ! ! ! ! u*(k , r ) = $ (k ) A (k )u (k , r ) Multiply both sides by l 0 ! ! ! ! ! ! ! ! ! ! ! ! n ! nj' j' 0 3! ! exp[i(k " k ) # r ]$ (k , r ) = exp[i(k " k ) # r ]{! + i(k " k )}$ (k , r ) j' Integrate over the unit cell d r 0 j 0 0 0 j 0 " ! ! " 2 ! ! " ! ! ! ! ! ! ! ! 2 $ ! ! ! 2 2 p ! ! ! ! {[" j (k0 ) + (k # k0 )]$ lj + (k # k0 ) % plj +}Anj (k ) = " n(k )Anl (k ) # +V (r )&exp[i(k ' k ) ( r ]) (k , r ) = 2m m 0 j 0 j "2m % k dot p ! ! ! " 2 ! ! " ! ! ! ! ! ! 3 ! ! 2 ! (2! ) 3! * ! !ˆ ! = exp[i(k ' k0 ) ( r ]{ (k ' k0 ) + (k ' k0 ) ( p + * j (k0 )}) j (k0, r ) p = d ru (k , r ) pu (k , r ) 2m m lj " # l 0 j 0 " Momentum matrix elements 8! k.p Method – Main Equation k.p Method – Non-degenerate band ! 2 ! ! ! ! ! ! ! " 2 2 " ! ! {[" j (k0 ) + (k # k0 )]$ lj + (k # k0 ) % plj +}Anj (k ) = " n(k )Anl (k ) Let us consider non-degenerate band and look j 2m m ! for band energies around extremum in k0 A (k ) e.g., conduction band minimum in GaAs System of homogeneous equations for expansion coefficients n!j 2 " " " " This equations couple all bands ! n(k ) ! ! " H ' = (k 2 ! k 2 ) + (k ! k ) " p 2m 0 m 0 ! What one needs is decoupling scheme, e.g., perturbation
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