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 ! + ( ) * = = exp[ik0 " r ]{ (k ! k0 ) + (k ! k0 ) " p + # j (k0 )}u j (k0, r ) ' Anj (k ) # V (r )&exp[i(k k0 ) r ] j (k0, r ) 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 theory = ! " k0 0 0 00 ! ! HH= 0 + H' Hn0 = Enn ! Expansion parameter: s = k ! k0 Hn= E n ! ! ! ! n Perturbation theory for non-degenerate states ! ! ! ! 2 ! ! 2 " ! " 2 2 " (s # pnj )(s # p jn ) 0 (1) (2) ! (k ) = ! (k ) + (k " k ) # p + (k " k ) + ! ! + EEE= + + E+ ! n n 0 m 0 nn 2m 0 2 % ! (k ) " ! (k ) nnn n m j$n n 0 j 0 |m0 | Hn '| 02 | 00 0 Energy to second order in the expansion parameter Enn= E+ nHn| '| + # + ! ! EE00" ! ! ! ! " ! p ! ! mn! nm u (k, r ) = u (k , r ) + s ! ! jn ! u (k , r ) +… 00 n n 0 m % " (k ) # " (k ) j 0 m| Hn '| j$n n 0 j 0 nn= 00+ m+ ! # 00 mn! EEnm" Periodic wave function to the first order in expansion parameter

k.p Method – Effective Mass Tensor k.p Method – Effective Mass Tensor ! ! 2 2 2 # $ # $ Taylor! series! ! ! ! 2 ! ! ! ! ! " (k ) " " pnj p jn + pnj p jn 1 ( ! n(k0 ) n 0 ! ! ! (k ) = ! (k ) + ("! (k )) #(k $ k ) + (k $ k ) (k $ k ) + = %#$ + ( n n 0 n 0 0 ' 0 % 0 & !k !k m m2 " (k ) & " (k ) 2 % ,& (k% (k& # $ j'n n 0 j 0 ! ! ! ! ! ! ! ! 2 ! ! 2 " ! " 2 2 " (s # pnj )(s # p jn ) ' ( ' ( ! (k ) = ! (k ) + (k " k ) # p + (k " k ) + ! ! + ! m $ 1 pnj p jn + pnj p jn n n 0 0 nn 0 2 % ! ! m – free electron mass m 2m m ! (k ) " ! (k ) # & = )'( + - j$n n 0 j 0 m* m * (k ) + * (k ) ! " " ! " " " n % '( j,n n 0 j 0 s ![ p + (k + k )] m nn 2 0 It is always possible to diagonalize the reciprocal effective Band n has extremum in k mass tensor by proper choice of the coordinate system. ! ! 0 ! ! If the extremum point k is a general point in the BZ, the choice of the axes !" n(k ) !" n(k ) p +"k = 0 0 ! = ! = 0 nn 0 depends on the details of the dynamics, that is, on the crystal potential. !k ! ! !s ! ! ! k =k0 s =0 = ! = pnn 0 k0 0 e.g., GaAs ! ! If the extremum occurs at a symmetry point or along an axis of symmetry, ! " ! pnn 0 k0 0 e.g., Si the axes may be partially determined by symmetry. Let us introduce second " In cubic crystals, for minima along [100], [110], [111], the symmetry axis rank tensor ! m $ m )2 * (k ) must be a principal axis. = n Reciprocal Effective # & 2 " m *% ! )k' )k( " " If k=0 is the extremum, the surface of constant energy in a cubic '( k =k Mass Tensor 0 crystal must be spherical

9! k.p Method – Effective Mass Tensor Effective Mass - A Two-Band Model ' 2 Some insight into the nature of the results ! $ | p | 1 m 2 nj ! !* ! = 1 + ! ! ! -!refers now to one 1 = c p = p = p pp! = # * & + 01 10 3 " m % m j*n ( (k ) ) ( (k ) of the principal axes ! (0) = E n '' n 0 j 0 1 GAP mp2 2 mp2 2 EGAP = 1 ! = 1 + ! ! m* 3mE m* 3mE The interaction with the lower lying levels !0(0)= 0 0 g 1 g ! j (k0 ) < ! n(k0 ) tends to decrease an effective mass 0 = v ! ! ! (k ) > ! (k ) 2 CdS The interaction with the higher states j 0 n 0 2 p AlSb E << 0.2 AlAs tends to increase an effective mass Let us assume g ZnTe m GaP For cubic semiconductors with minimum of the conduction band This occurs, for example, CdSe ZnSe ! in ! point, the band energy in the neighborhood is ! 2 2 in GaSb, InAs, InSb " k 2 CdTe ! n(k ) = ! n(0) + mp2 2m* ! 0.1 InP Physical meaning of the effective mass ? n * m1 3mEg GaAs * m1 3m GaSb ! E Effective mass m*/m In the presence of external fields, 2 g InAs Dynamics m 2 p InSb the crystalline electrons behave as 0 of particles Effective mass proportional 0 1.0 2.0 3.0 particles with effective mass m*. to the energy gap Energy gap [eV]

k.P – Method – Band Degeneracies ! k.P – Method – Band Degeneracies The state of interest in k is degenerate ! 0 Löwdin perturbation theory The perturbation will remove degeneracy, (0) (1) (2) k HHnn= + H n+ H n= 0 at least in some directions ab( ab ) ( ab ) ( ab ) Going from a point of higher symmetry g 0 00 0 m !na|HH '|! mc! mc | '|! nb Valence band top of the energy band split = " 000# + ! |H '|! + n ab na nb & & 00 cubic semiconductors It is necessary to use degenerate m$ nc=1 " nm% " perturbation theory The essential idea of this procedure is to separate the states considered 0 00 e.g., L.I. Schiff, Quantum Mechanics H0!na= " n! na ag= 1,2,3,! , n Schwabl, Quantenmechanik in the perturbation calculation into two sets: one set involves a small number of strongly coupled states whose aH0 + H'0 b$ E#!" = interactions is treated exactly (gn), n First order perturbation Hab --- gg n! n --- Matrix the second set, with more states, contains those states that are well removed in energy from the first set ! 2 " " ! " " " In our case, perturbation is H ' = (k 2 ! k 2 ) + (k ! k ) " p ! 2m 0 m 0 For simplicity k = 0 One can also treat in this way situation in which some bands, although 0 2 " " ! 2 ! " not quite degenerate, approach each other closely. Valence band maximum H ' = k + k ! p Then treatment of such bands as non-degenerate does not make sense in most of semiconductors 2m m

10! k.P – Method – Band Degeneracies k.P – Method – Band Degeneracies ! ! pnm = u0 | pˆ | u0 Special case: valence band in ! ab,! {, x yz ,} nv! ab na mb n ! ! ! ! H is 3! 3 matrix 3 degenerated bands: vx,, vy vz ! 2 ! ! gm nm mn ab n " 2 " ! nn (k # pac )(k # pcb ) H = ! (k )" + k " + k # p + ! ! $ xx#111 xy# xz # % ab n 0 ab ab ab % % ()()()mmmµ! µ! µ! 2m m m&n c=1 ! (k ) $ ! (k ) & ' n 0 m 0 ˆ ! 2 33 ! ! Hˆ = " (0)+ kkmmm&()()()yx#111 yy# yz # ' k ! pnn = 0 nv* * µ µ! µ! µ! Band minimum in 0 ab 2m µ=11v= & ' nm mn & zx#111 zy # zz # ' ! 2 3 gm [ k ( p ) ][ k ( p ) ] ()()()mmmµ! µ! µ! n " 2 # µ µ µ ac #$ $ $ cb ( ) Hab = ! n(k0 )" ab + " ab # kµ + # # ! ! For cubic semiconductors, there are only three different matrix elements 2m µ =1 m&n c=1 ! (k ) % ! (k ) n 0 m 0 gm vm mv 111 2 ()()ppz zc z cz nm mn ! 2 3 3 3 gm ( p ) ( p ) L = = = = 1 + $ $ n " 2 µ ac $ cb xx yy zz m H = ! (k ) + " k + k k ! ! mmmxx yy zz m" nc=1 !n(0)# ! m (0) ab n 0 2m ab # µ # # µ v # # g vm mv µ =1 µ =1v=1 m&n c=1 ! n(k0 ) % ! m (k0 ) 111111 2 m ()()pp M ======1 + $ $ x zc x cz ! 2 3 3 mmmmmmyy zz xx xx yy zz m ! (0)# ! (0) n ! kµ kv xx yy zz yy zz xx m" nc=1 v m Hab = !n (k0 )!ab + ! ! g vm mv 2m mab 111111 2 m (pp )() µ=1v=1 µ! N ======1 + x xc z cz nm mn xy yz zx yx zy xz $ $ gm mmmm mm m " = ! (0)# ! (0) 1 2 ( pµ )ac ( p! )cb xy yz zx yx zy xz m nc 1 n m = ! ! + ! ! ab ab µ" m " " mµ! m#n c=1 !n (k0 ) ! !m(k0 ) L, M, N – Dresselhaus parameters

k.P – Method – Band Degeneracies Degenerated Valence Band of Cubic Semiconductors Degenerated Valence Band of Cubic Semiconductors ! Dispersion relations along ! " line k ![k ,0,0] ˆˆµ! x ˆˆ " 2 # H= D kkµ ! !vx(0) + Lk 0 0 ˆ $ % Hˆ = $ 0! (0) + Mk 2 0 % " 2 22 # vx Lkx+ M() k y+ k z Nkxy k Nkxz k $ 2 % 2 $ % $ 0 0!vx (0) + Mk % ˆ ! $ 2 22 % & ' HI= !v(0) + Nkxy k Lky+ M() k x+ k z Nkyz k 2 2m $ % ! 2 hh (2) $ 2 22% !hh= ! v (0) + Mkx Heavy hole Nkxz k Nkyz k Lkz+ M() k x+ k y 2m & ' 2 lh ! 2 (1) !lh= ! v (0) + Lkx Light hole ! ! " ˆ 2m line det(Hˆ ! EI )= 0 These equations can be solved analytically !! T. Manku & A. Nathan, J. Appl. Phys. 73, 1205 (1993) ! J. Dijkstra, J. Appl. Phys. 81, 1259 (1997) Generally ! v ( k ) are dependent on the direction of k   hh Pretty complicated task Warped bands lh L, M, N – Dresselhaus parameters SIMPLE: Find solutions along a symmetry line, ! obtained from fits to experimental e.g., k ![k ,0,0] x ! " line data

11! Role of spin in the band structure Spin-orbit coupling If one includes electron spin into the band structure obtained from In the case of Dirac equation Blochs theorem is still valid, ! spin independent effective Hamiltonian ˆ 2 ! since it depends only on the translational invariance of the Hamiltonian ˆ p ! ! ! ! ! the degeneracy of all bands will be doubled H = +V (r ) ! ! ! Four component 2m ! (k, r ) = exp(ik " r )u (k, r ) un(k, r ) (1) (2) n n Periodic spinor el el Bands around SPIN Systems that require full Dirac equation - Very heavy metals, point hh e.g., Uranium ! (2) hh (4) In semiconductors it is sufficient to consider only large component lh ! lh (1) ! (2) of the Dirac spinor #! $ Without spin With spin " % & ' Two-component spinor 0 Equation of electron with spin – Dirac equation for a single particle ( ) in periodic potential ! ! ! Foldy-Wouthuysen-Transformation Equation for ! !ˆ ˆ 2 ! ! ! F. Schwabl, Advanced Quantum Mechanics [!c"ˆ # p ! $mc +V (r )]% (k, r ) = E (k )% (k, r ) ! = c = 1 n n n ! ! Quantenmechanik für Fortgeschrittene ˆ 2 ! ˆ 4 ! ! ! # 0 ! $ "I 0 # p p 1 2 1 ˆ ˆ ˆ i !ˆ = 01 0 "i 10 {m + +V (r ) ! + " V (r ) + # $ "V % p}& = E& "i = % & $ % " # # $ " # 2m 3 2 2 '! i 0 ( '0 &I ( ! x = $ % ! y = % & ! z = $ % 8m 8m 4m ! ! &10' 'i 0 ( '01& ( ! (k, r ) Four-component Pauli matrices Darwin term n Relativistic mass correction Spin-orbit coupling spinor

Spin-orbit coupling Transformation properties of spinors !ˆ 2 ! ! ! % p ! " !ˆ !ˆ ( ! ! There are fundamental differences in the description of the effect of ' +V (r ) + ! "(#V $ p)*+ (k, r ) = , (k )+ (k, r ) rotations on spinor, as compared to a scalar or an ordinary vector. 2m 2 2 n n n & 4m c ) Spinor transforms under rotations according to the j=1/2 representation ! ! ! ! of the rotational group In the absence of the spin-orbit coupling ! = " # n(k, r ) n(k, r ) $ Two different quantum mechanical operators exist that correspond to the same physical transformations of points in space. "1# #0$ Spin functions (spinors) chosen !+ = $ % !" = % & The representations of the operators, rather than physical &0' '1( as eigenfunctions of !ˆz !ˆ transformations themselves, are required. Spin operator ! commutes with the non-relativistic Hamiltonian The components of a spinor under the rotation (about axis n Spin is a good quantum numer ! ! through an angle ! ) transform as !u " ! $ " (+)(k, r ) ' 1 ! & n ) ! u = # $ Case with spin-orbit interaction !n(k, r ) = ! u ' = Rˆ (!,n)u %u2 & & (#) ! ) ! ! ! " n (k, r ) ˆ ! ˆ ! ! !ˆ % ( R(! , n) = cos " i# $ nsin Rˆ (! , n) is unitary 2x2 matrix Spin operator ! does not commute with the Hamiltonian 2 2 or representation Spin is not a good quantum number " ˆ ! ˆ ! ˆ ! (1/2) ˆ ! State describing band n can be a mixture R(! + 2" , n) = # R(! , n) $ R(! , n) D (R) ! R(" , n) of spin up (+) and spin down (-) states The two rotation that are not regarded as distinct Removing of some degeneracies expected in the r-space are represented by different matrices !

12! Transformation properties of spinors Spin-splitting of the valence band Action of an operator in spinor space on spinor ! ! of cubic semiconductors ! $ (+) #1 ' ! (1/2) " n (k, R r ) P ! (k, r ) = D (R) & ! ) ! 15 3x3 2x2 R n & (#) #1! ) 4x4 2x2 " n (k, R r ) % ( ! " D(1/ 2) = ! + ! Note there are twice so many operators PR as Rs 15 8 7 We wish to determine the behavior of an electron state that belonged 6x6 Representations ()l of the double group to the representation ! before the spin was considered Valence band without Representation
 We form direct product ! (l )" D (1/ 2) spin-orbit interaction of the single group

The representations so obtained may be reducible as ! so hh the representations of the group of operators in spinor space ! 15 3 ! 8 hh ! ()l# Dn (1/2)= ()ii! () 2! lh $ Irreducible representations so !so "ì 3 lh ! 7 Physical meaning: the splitting of a degenerate state l by spin-orbit coupling into states of symmetry i so

Band structure of cubic semiconductors Spin-orbit splitting in semiconductors around the ! point Comparison of Theory (LDA) with Experiment el (1/ 2) el ! 16" D = ! ! 1 ! 6 (2) (2) Representations cb E of the double group (4) EGAP ! (1/ 2) so hh (6) ! 15 " D = ! 8+ ! 7 3 ! 8 hh hh 2! lh ! 15 so !so 3 lh lh (2)! 7 so so Band structure of a typical Non-relativistic zinc-blende semiconductor With relativistic effects, DFT gives very accurate predictions of spin-orbit splitting spin-orbit coupling included

13! k.p Method with spin-orbit coupling k.p Method with spin-orbit coupling ! ! $ pˆ 2 ! ! ! ! ! 2k2 ! ! !ˆ ' ! ! ! ! ! $ ˆ 2 ! ! ! ' ! ! ! ! ! +V (r ) + !ˆ !("V # pˆ ) + + k ! ! u (k,r ) = ! (k )u (k,r ) p ! ˆ ˆ & 2 2 ) n n n & +V (r ) + ! !("V # p))!n(k,r ) = !n(k )!n(k,r ) %& 2m 4m c 2m m () & 2m 4m2c2 ) ˆ % ( H SO ! ! ! ! ! ! Main strategy of the! k.p Method Two-component Bloch spinor !n(k,r ) = exp(ik ! r )un(k,r ) ! ! Build unknown u n ( k , r ) out of known u n 0 ( r ) for wave vector! Hˆ k0 = 0 = ! !ˆ 2 What to do with SO ? $ p ! ! !ˆ !ˆ & +V (r ) + ! !("V # p) + Periodic part 2 2 1 Hˆ %& 2m 4m c Include SO into unperturbed Hamiltonian 2 2 ! ! 2 2 " 2 ! ! ! ! ˆ ! k ! ˆ ! k ! "ˆ ! !ˆ ' ! ! H ' = + k ! ! + + k ! p + ! !("V # k ))u (k,r ) = ! (k )u (k,r ) 2m m 2 2 n n n 2 2 ! 2m m 4m c () ! k ! ! ˆ Include H ˆ into perturbed Hamiltonian Hˆ ' = Hˆ + + k ! ! 2 SO ! ! SO 2m m !ˆ ! ! ! Hˆ u (r ) = ! (0)u (r ) ! = pˆ + (!ˆ ! "V ) Note! ! ! ! ! ! 0 n0 n n0 2 a !(b ! c) = c !(a ! b) ˆ 4mc H0 - is spin independent !ˆ 2 2 2 ! ! ! ! ! $ p ! ! !ˆ !ˆ ! k ! ˆ ' ! ! Use Löwdin perturbation theory including valence and conduction band & +V (r ) + ! !("V # p) + + k ! ! )u (k,r ) = ! (k )u (k,r ) 2 2 n n n states in the group A states and the rest of states in group B %& 2m 4m c 2m m ()

8 band k.p Method 8 band k.p Method - Terms without spin-orbit ˆ States of group A: s + s ! x + y + z + x ! y ! z ! " ˆ # {}ui0 ˆ $h44! 0 % hij (8! 8) = $ ˆ % Conduction band Valence band & 0 h44! ' !cc(0) " ! 0 !vv(0) " ! 0 s + x + y + z + B iH|ˆˆ '| b bH | '| j " 22 % ˆ ! k 2 Hˆ= iH| ˆˆ+ H '| j+ s + &!c0 + + A' k Bkyz k+ iPk x Bkxz k+ iPk y Bkxy k+ iPk z ' ij 0 # ! " ! & 2m ' b ib0 & 22 ' ! ! ! k ˆ & !v0 + + ' ˆ Bkk# iPk 2m Nkk''Nkk 8 x 8 Matrix det(Hij ! !i (k )) = 0 ! !i (k ) x + & yz x xy xz ' & 2 22 ' 2 2 ! ! Lk'()x+ Mk yy+ k ˆ ˆ ! k ! ˆ & ' Commonly employed simplifications in H ' = H + + k ! ! ˆ & 22 ' SO 2m m h44$ = ! k ! ! ! ! ! 2 2 & !v0 + + ' ˆ ! k! p ! k ! " ! Bkk# iPk Nkk''2m Nkk ! ! pˆ Hˆ ' = Hˆ + Hˆ Hˆ k! p = + k ! pˆ y + & xz y xy yz ' SO & Lk'()2+ Mk 22+ k ' 2m m & y xz ' ! ! ! ! k! p k! p & ! 22k ' ! ! B i | Hˆ | b b | Hˆ | j & ! + + ' ˆ k! p v0 Hˆ = i | Hˆ + Hˆ | j + + i | Hˆ | j z + & Bkkxy# iPk z Nkk''xz Nkkyz 2m ' ij 0 # ! " ! SO & 2 22' b i0 b0 (& Lk'()z+ Mk xy+ k )' 2 ! ˆ ˆ ! sp||xy b bp || z ˆ Contains part with operators ()HSO ij B = 2 P= ! i sp||x x hij independent on spin 2 # m m b (!cv00+ ! )/2" ! b 0

14! 8 band k.p Method – Spin-Orbit Terms 8 band k.p Method – Angular momentum ˆ Representation of Valence States sH! |SO |"! '0= ! "{,}+ # ! "{,xyz ,} ˆˆ 3! ! 3! !V ! !V ! Jm,J Jm,J | H0 + HSO |, Jm J ! = !i x+ | (!V " pˆ ) | z ! = ! x+ | " | z ! 2 2 y 2 2 4m c 4m c !z !x !x !z 33 1 ! ,() x+ + iy+ "v0 + 22 2 3 x + y + z + x ! y ! z ! ! v 3 1 21 ! x + , -(z+ + ( x# + iy# ) "v0 + # 0 i 00 0" 1$ 22 36 3 ! 8 % & y + "ii000 0 21 3 ,# 1 -(z# # ( x+ # iy+ ) " + ! % & 22 v0 3 z + ˆ ! % 0 001"i 0& 36 ˆ 1 H SO = " % & 3 1 ! x ! ,# - ()x# # iy# "v0 + 3 % 0 010"i 0& 22 2 3 y ! % 0 0ii 00 & 11 112! , (x# + iy# )+ () z+ "v0 # ! # " 22 3 v z ! % & 33 '%"1"i 00 0 0(& 1 12! ! 7 11,# (x+ # iy+ )# () z# " # 22 v0 3 This matrix can be diagonalized easily 33

8 band k.p Method The most popular form of the k.p Method Hamiltonian matrices in both bases used in the calculations 8 x 8 matrix easily handled numerically For analytical purposes one must take further simplifications E. O. Kane,
The k.p Method, Semiconductors and Semimetals, Envelope Function Theory – Vol. 1, eds. R. K. Willardson and A. C. Beer, (Academic Press, San Diego, 1966), p. 75. Effective Mass Equation One author – one notation 2m ! 1 = " (LM+ 2 )1" e.g., Luttinger parameters 3! 2 m ! 2 = " ()LM" 3! 2 m ! 3 = " N 3! 2

Crystal potential hidden in the parameters of the k.p matrix

15! Electron in an external field Envelope Function Theory !ˆ 2 J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955). ! p ! ! $ ! ! ! 2 ! pˆ ! ! $ ! ! # +V (r ) + U(r )&! (r ) = !" (r ) +V (r ) + U(r ) ! (r ) = !" (r ) # 2m & # & " % "# 2m %& Periodic potential of crystal Non-periodic external potential ! ! ! ! Ansatz: ! (r ) = dk ' A (k ')! ! (r ) " ! n' n'k ' Strongly varying on atomic scale Slowly varying on atomic scale n' BZ 2 Integration over the Brillouin Zone Which external fields ? ! e ! ! U(r ) = ! ! ! ik!r ! ! "! Shallow impurities,! e.g., donors! ! ! ! ! (r ) = e u (r ) ! | r | nk n0 Luttinger – Kohn functions k0 = 0 "! Magnetic field B, B = curlA = ! " A ! ! ! * ! (r ) 3 "! Heterostructures, Quantum Wells, Quantum wires, Q. Dots Multiply both sides of Eq. by nk " dr and integrate over the crystal volume ! cbb ! 2 ! ! ! " " ! [! + k 2]A (k ) + k " p A (k ) + GaAlAs n0 n ! nn' n' GaAs GaAlAs GaAs GaAlAs 2m n' m ! ! ! ! ! ! + = ! ! # dk ' nk |U(r ) | n'k ' An'(k ') An(k ) Does equation that involves the effective mass and a slowly varying n' BZ function exist ? !ˆ 2 ! p ! $ ! ! ! ! ! ! ! ! ! ! ! ! ! # + U(r )& F(r ) = !F(r ) F(r ) = ? nk |U(r ) | n'k ' = dr exp[i(k '!k )# r ]U(r )u* (r )u (r ) " 2m* % " n0 n'0

Envelope Function Theory Envelope Function Theory – Condition ! ! ! nk |U(r ) | n'k ' ! 0 nn' ! for the external potential When for ! ! ! ! ! ! 3 nn' then the coefficients of different bands are decoupled. nk |U(r ) | n'k ' = (2! ) " Bm U(k !k '+ Gm ) ! m u (r ) -- are periodic functions n0 For m! 00 Bnn' ! for nn! ' ! ! ! ! m Orthogonality of periodic u* (r )u (r ) = Bnn' exp["iG ! r ] Fourier series Bloch functions n0 n'0 ! m m For m= 00 Bnn' = for nn! ' m m Fourier coefficients ! ! ! ! ! ! ! ! 1 ! iG !r ! ! 1 nk |U(r ) | n'k ' ! 0 for nn' ! !U(k ! k '+ G ) ! 0 for m ! 0 Bnn' = dre m u* (r )u (r ) Bnn' = ! m m ! " n0 n'0 0'3 nn 0 ! (2" ) It means – only the first Fourier component of the potential U 0 ! ! ! ! ! ! ! ! ! ! " ! = nn' i(k ' k Gm ) r is of any importance ! nk |U(r ) | n'k ' $ Bm # dre U(r ) m ! U(r ) NOT !! ! 1 ! ! ! ! U(k ) = dre!ik"rU(r ) 3 # Fourier coefficients of the potential U ! (2! ) Potential U ( r ) should change slowly in space ! ! ! ! ! ! ! ! ! ! U(k ! k '+ G ) ! ! ! ! ! nk |U(r ) | n'k ' = (2! )3 Bnn'U(k !k '+ G ) m " m m ! ! ! 1 ! nk |U(r ) | n'k ' = U(k ! k ')! nn' m U(k ! k ')

16! Envelope Function Theory – Effective Mass Equation Envelope Function Theory- Transformation of the effective in Momentum Space ! 2 ! ! ! " ! ! ! ! ! ! ! mass equation in momentum space into r-space [! + k 2]A (k ) + k ! p A (k ) + dk 'U(k !k ')A (k ') = ! A (k ) n0 2m n ! m nn' n' " n n ! ! ! ! ! ! ! n' BZ ! + ! = ! n(k )Bn(k ) " dk 'U(k k ')Bn(k ') Bn(k ) This equation still couples different bands BZ " ! ! 2k 2 S ()new! S S ! (k ) = Canonical transformation A= eB Hˆˆ= e He For isotropic effective mass n * ! ! 2mn New coefficients " k ! p This equation reminds the Schrödinger equation in momentum space S ! nn' ! 1 " Case for non-degenerate bands m EGAP (here ! k is quasi-momentum, not momentum !!) ! ! ! ! ! % ! 2k 2 ! 2 (k ! p )(k ! p )( ! ! ! ! ! ! ! + + nn'' n''n B (k ) + dk 'U(k "k ')B (k ') = ! B (k ) ' n0 2 $ * n + n n Transformation into r - space & 2m m ! " ! ) ! ! ! n''#n n0 n''0 BZ ik!r Multiply both sides of the equation by e and integrate ! dk ( 2 3 ! $ + ! ! ! ! ! ! BZ * ! 1 - ! ! ! ! ! + # & k k B (k ) + dk 'U(k .k ')B (k ') = ! B (k ) ! ik!r * n0 2 ' # * & µ ! - n / n n F (r ) = dke B (k ) Envelope function µ,! m BZ Define function n n ) " µ! % n , ! Effective Mass BZ ! ! ! ! ! ! ! ! ONLY BZ !! ! (k )B (k ) + dk 'U(k !k ')B (k ') = ! B (k ) Equation n n " n n Write ! (k ) = ! + ! k k +… in Momentum n n0 ! µ" µ ! BZ µ ! Space ,

Envelope Function Theory- Transformation of the effective Envelope Function Theory- Transformation of the effective mass equation in momentum space into r-space mass equation in momentum space into r-space ! ! ! ! ! ! ! ! ! ! ! ! ! ! ik"r !n(k )Bn(k ) + dk 'U(k !k ')Bn(k ') = ! Bn(k ) = ! = " (b) ## dk 'dkU(k k ')e Bn(k ') BZ BZ (a) ! ! ! ! ! ! ! ! 1 ! ! i(k '!k )"r ' ik"r ! (b) ! ! ! ! ! ! ! ! ! = dk 'dk dr 'U(r ')e e Bn(k ') = = ik"r ! = ! + ! ik"r = 3 ## # (a) ! dke n(k )Bn(k ) n0Fn(r ) # µ" ! dkkµ k! e Bn(k ) (2! ) BZ BZ µ,! BZ $ ! ! ! ! ' $ ! ! ! ! ' % ( ! ! ! ! ! ! ik '"r ' 1 ik"(r !r ') ! 1 $ % 1 $ ( ik"r = dr 'U(r ') dk 'e B (k ') dke = ! F (r ) + ! ' * dke B (k ) = # & # n ) & 3 # ) n0 n # µ" i $x ' i $x * ! n (2! ) µ,! & µ ) & v ) BZ % BZ ( % BZ ( ! ! ! ! % 1 $ ( % 1 $ ( ! F (r ') !(r ! r ') = ! F (r ) + ! F (r ) = n n0 n # µ" ' * ' * n ! ! ! µ,! & i $xµ ) & i $xv ) ! ! 1 ! !(r ! r ') = dkeik"(r !r ') In the full k-space 3 # ! ! ! ! + % 1 $ ( % 1 $ ( . ! (2! ) !(r ! r ') = ! (r ! r ') = -! + ! 0 F (r ) BZ n0 # µ" ' * ' * n ! - & i $xµ ) & i $xv ) 0 ! ! ! ! ! 3! ! , µ,! / ! !(r )d 3r = 1 ! !(r ! r ') f (r ')d r ' ! f (r ) If f ( r ) is smooth 1 ! ! 1 ! (a) = ! ( !)F (r ) ! ! ! ! ! ! ! n n kµ " (b) = dr 'U(r ')F (r ')!(r " r ') # U(r )F (r ) i ix! µ ! n n

17! Envelope Function Theory- Effective Mass Equation Envelope Function Theory- Degenerate Bands ! Matrices obtained from k.p method, e.g., 8 band k.p method ! ! 3 33 [!(!i") + U(r ) ! !]F (r ) = 0 (EME) ˆ (0) (1)µ (2)µ! n Hab = Dab+ " D ab kµ + " " Dab kkµ v µ=1 µ=11! = ! ! ! ! ! ! ! ! ! ! A (k ) ! B (k ) ! = ik!r = n n (r ) " # dkBn(k )e un0(r ) " Fn(r )un0(r ) Periodic potential hidden in the parameters of the Hamiltonian matrix n BZ n The effect of non-periodic external potential can be described by EME does not couple different bands a system of differential equations for the envelope functions ! ! ! s 3 3 ! (r ) = F (r )u (r ) (2)µ! n n0 ![ ! ! Dab ("i#µ )("i#! ) + b=1 µ=1! =1 True Periodic Envelope 3 wavefunction Bloch Function (1)µ (0) ! ! ! Function + ! Dab ("i#µ ) + Dab + U(r )!ab]Fb(r ) = !Fb(r ) µ=1 Special case of constant (or zero) external potential ! s ! ! ! ! ! ! ! Wave function ! (r ) = F (r )u (r ) F (r ) = exp(ik ! r ) Bloch function ! b b0 U(r ) = 0 n ! (r ) b=1 ! Basis theory for studies of low dimensional systems Uz() Fn(r ) = exp[i(kx x + ky y)]Fn(z)

Envelope Function Theory - Applications

a) Magnetic field ! ! e ! ! pˆ ! pˆ ! A(r ) Minimal coupling principle for full Hamiltonian c e ! k ! "i! ! A (r ) In effective Hamiltonian µ µ c µ Non-degenerate case - conduction band electrons ! e ! ! ! [!(!i! " A(r )) ! !]F (r ) = 0 Landau levels Thank you ! c n Degenerate case of valence band s µ! e ! e ! ! ! #[Dab (!i"µ ! Aµ (r ))(!i"! ! Av (r ))]Fb(r ) = !Fb(r ) b=1 c c

b) Donors in semiconductors NEXT LECTURE c) Low dimensional semiconductor structures

18!