Tight-Binding Approximation. Faculty of Physics UW

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1 Tight-Binding Approximation Faculty of Physics UW

2 The band theory of solids

3 Periodic potential Bloch theorem φ n,k Ԧr = u n,k Ԧr eik Ԧr Bloch wave, Bloch function Bloch amplitude, Bloch envelope The solution of the one-electron Schrödinger equation for a periodic potential has a form of modulated plane wave: u n,k Ԧr = u n,k Ԧr + R We introduced coefficient n for different solutions corresponding to the same k (index). kvector is an element of the first Brillouin zone. u n,k Ԧr = ԦG C k ԦG ei ԦG Ԧr

4 Periodic potential Bloch theorem Proof: Translation operator T R T R f Ԧr = f Ԧr + R Perodic potential of the crystal lattice: T R V Ԧr = V Ԧr + R Hamiltonian with periodic potential T R H Ԧr ψ Ԧr = H Ԧr + R ψ Ԧr + R = H Ԧr ψ Ԧr + R = H Ԧr T R ψ Ԧr T R T R ψ Ԧr = ψ Ԧr + R + R = T R T R ψ Ԧr operators are commutative! Eigenfunctions ψ n,k Ԧr of the translation operator T R : T R ψ n,k Ԧr = C R ψ n,k Ԧr = e if R ψ n,k Ԧr C R 2 = 1 where f R + R = f R + f R f 0 = 0 f R = kr 6/5/2017 4

5 Periodic potential Bloch theorem Proof: Translation operator T R T R f Ԧr = f Ԧr + R Eigenfunctions ψ n,k Ԧr of the operator T R T R ψ Ԧr = C R ψ n,k Ԧr = e ikr ψ n,k Ԧr We denote our eigenfunction ψ n,k Ԧr where n distinguishes the different functions of the same k. Let us define: u n,k = ψ n,k ik Ԧr Ԧr e periodic function T R u n,k = T R ψ n,k Ԧr e ik Ԧr = e ikr ψ n,k Ԧr e ik Ԧr+R = ψ n,k Ԧr e ik Ԧr = u n,k Thus: ψ n,k Ԧr = u n,k eik Ԧr The eigenstates of the electron in a periodic potential are characterized by two quantum numbers k and n k wave vector n describes the energy bands (for a moment!) 6/5/2017 5

6 Periodic potential The nearly free-electron approximation k band band a x k x band 2π a π a a x π a 2π a 4π a 6π a 8π a

7 The electronic band structure It is convenient to present the energies only in the 1st Brillouin zone. The electron state in the solid state is given by the wave vector of the 1st Brillouin zone, band number and a spin

8 The electronic band structure W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics

9 The electronic band structure W. R. Fahrner (Editor) Nanotechnology and Nanoelectronics

10 Tight-Binding Approximation We describe the crystal electrons in terms of a linear superposition of atomic eigenfunctions (LCAO Linear Combination of Atomic Orbitals) H = H A + V : H A Ԧr R n φ j Ԧr R n = E j φ j Ԧr R n Equation for the free atoms that form the crystal H A = ħ2 2m Δ + V A Ԧr R n j-th state e Atom in position R n H = H A + V = ħ2 2m Δ + V A Ԧr R n + m n V A Ԧr R m Perturbation: the influence of atoms in the neighborhood of R m : V Ԧr R n = m n V A Ԧr R m Good for valence band of covalent crystals, d-orbital bands etc

11 Tight-Binding Approximation Approximate solution in the form of the Bloch function: Check: Φ j,k Ԧr = a n φ j Ԧr R n = exp i kr n φ j Ԧr R n n n Φ j,k+ ԦG Ԧr = Φ j,k Ԧr Φ j,k Ԧr + T = exp i kt Φ j,k Ԧr Energies determined by the variational method: Φ j,k Ԧr H Φ j,k Ԧr E k Φ j,k Ԧr Φ j,k Ԧr Expression Φ j,k Ԧr Φ j,k Ԧr = exp ik R n R m න φ j Ԧr R m φ j Ԧr R n dv n,m can be easily simplify assuming a small overlap of wave functions for n m Φ j,k Ԧr Φ j,k Ԧr = න φ j Ԧr R n φ j Ԧr R n dv = n

12 Tight-Binding Approximation Approximate solution in the form of the Bloch function: Check: Φ j,k Ԧr = a n φ j Ԧr R n = exp i kr n φ j Ԧr R n n n Φ j,k+ ԦG Ԧr = Φ j,k Ԧr Φ j,k Ԧr + T = exp i kt Φ j,k Ԧr Energies determined by the variational method: Φ j,k Ԧr H Φ j,k Ԧr E k Φ j,k Ԧr Φ j,k Ԧr E k 1 N Φ j,k Ԧr H Φ j,k Ԧr = = exp ik R n R m න φ j Ԧr R m E j + V Ԧr R n φ j Ԧr R n dv n,m

13 Tight-Binding Approximation Approximate solution in the form of the Bloch function: Check: Φ j,k Ԧr = a n φ j Ԧr R n = exp i kr n φ j Ԧr R n n n Φ j,k+ ԦG Ԧr = Φ j,k Ԧr Φ j,k Ԧr + T = exp i kt Φ j,k Ԧr Energies determined by the variational method: Φ j,k Ԧr H Φ j,k Ԧr E k Φ j,k Ԧr Φ j,k Ԧr E k 1 N Φ j,k Ԧr H Φ j,k Ԧr = Only the vicinity of R_n = exp ik R n R m න φ j Ԧr R m E j + V Ԧr R n φ j Ԧr R n dv n,m Only diagonal terms R n = R m in E j

14 Tight-Binding Approximation E k 1 N Φ j,k Ԧr H Φ j,k Ԧr = Only the vicinity of R_n = exp ik R n R m න φ j Ԧr R m E j + V Ԧr R n φ j Ԧr R n dv n,m Only diagonal terms R n = R m in E j When the atomic states φ j Ԧr R n are spherically symmetric (s-states), then overlap integrals depend only on the distance between atoms: E n k E j A j B j exp ik R n R m m A j = න φ j Ԧr R n V Ԧr R n φ j Ԧr R n dv B j = න φ j Ԧr R m V Ԧr R n φ j Ԧr R n dv Restricted to only the nearest neighbours of R n

15 Tight-Binding Approximation When the atomic states φ j Ԧr R n are spherically symmetric (s-states), then overlap integrals depend only on the distance between atoms: E n k E j A j B j exp ik R n R m A j = න φ j Ԧr R n V Ԧr R n φ j Ԧr R n dv m B j = න φ j Ԧr R m V Ԧr R n φ j Ԧr R n dv The result of the summation depends on the symmetry of the lattice: For sc structure: R n R m = ±a, 0,0 ; 0, ±a, 0 ; 0,0, ±a ; E n k E j A j 2B j cos k x a + cos k y a + cos k z a For bcc structure : Forfcc structure : E n k E j A j 8B j cos k xa 2 E n k E j A j 4B j cos k xa 2 cos k ya 2 cos k ya 2 cos k za 2 + c. p

16 B j = න φ j H. Ibach, H. Lüth, Solid-State Physics Ԧr R m V Ԧr R n φ j Ԧr R n dv Tight-Binding Approximation For sc structure: R n R m = ±a, 0,0 ; 0, ±a, 0 ; 0,0, ±a ; E n k E j A j 2B j cos k x a + cos k y a + cos k z a

17 B j = න φ j H. Ibach, H. Lüth, Solid-State Physics Ԧr R m V Ԧr R n φ j Ԧr R n dv Tight-Binding Approximation For sc structure: R n R m = ±a, 0,0 ; 0, ±a, 0 ; 0,0, ±a ; E n k E j A j 2B j cos k x a + cos k y a + cos k z a

18 The electronic band structure bands Expanding E n k = E n ħ2 k 2 2m near extremum, e.g. k = 0: Landolt-Boernstein

19 The electronic band structure Expanding E n k = E n ħ2 k 2 2m near extremum, e.g. k = 0: Landolt-Boernstein

20 Tight-Binding Approximation Linear dispersion relation in graphene: Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, The Band Theory of Graphite, Physical Review 71, 622 (1947).] gives: E k = ± γ cos k ya cos k ya 2 cos k x 3a 2 ħcǁ k k i

Tight-Binding Approximation Linear dispersion relation in graphene: Tight binding approach with the only nearest neighbors interaction [P. R.

21 Tight-Binding Approximation Linear dispersion relation in graphene: Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, The Band Theory of Graphite, Physical Review 71, 622 (1947).] gives: E k = ± γ cos k ya cos k ya 2 cos k x 3a 2 ħcǁ k k i

22 Tight-Binding Approximation Linear dispersion relation in graphene: Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, The Band Theory of Graphite, Physical Review 71, 622 (1947).] gives: E k = ± γ cos k ya cos k ya 2 cos k x 3a 2 ħcǁ k k i

23 Tight-Binding Approximation Linear dispersion relation in graphene: Tight binding approach with the only nearest neighbors interaction [P. R. Wallace, The Band Theory of Graphite, Physical Review 71, 622 (1947).] gives: E k = ± γ cos k ya cos k ya 2 cos k x 3a 2 ħcǁ k k i The number of states in the volume of πk 2 : N E = 2 2π 2 πk2 = 2 2π 2 π k k i 2 = 2 2π 2 π E ħcǁ 2 ρ E = N E E = E π ħcǁ

24 H. Ibach, H. Lüth, Solid-State Physics Tight-Binding Approximation The existence of the band structure arising from the discrete energy levels of isolated atoms due to the interaction between them. We can classify the electronic states as belonging to the electronic shells s, p, d etc. The existence of a forbidden gap is not tied to the periodicity of the lattice! Amorphous materials can also display a band gap. If a crystal with a primitive cubic lattice contains N atoms and thus N primitive unit cells, then an atomic energy level E i of the free atom will split into N states (due to the interaction with the rest of N 1 atoms). Each band can be occupied by 2N electrons

25 H. Ibach, H. Lüth, Solid-State Physics Tight-Binding Approximation If a crystal with a primitive cubic lattice contains N atoms and thus N primitive unit cells, then an atomic energy level E i of the free atom will split into N states (due to the interaction with the rest of N 1 atoms). Each band can be occupied by 2N electrons. An odd number of electrons per cell (metal) An even number of electrons per cell (non-metal) An even number of electrons per cell but overlapping bands (metals of the II group, e.g. Be next slide!)

26 H. Ibach, H. Lüth, Solid-State Physics Tight-Binding Approximation The states can mix: for instance sp 3 hybridization. C, Si, Ge Be

27 Tight-Binding Approximation

28 Tight-Binding Approximation Michał Baj

29 Fermi surfaces of metals Cu Ashcroft, Mermin

30 Fermi surfaces of metals

31 Fermi surfaces of metals Michał Baj

32 Tight-Binding Approximation

33 Szmulowicz, F., Segall, B.: Phys. Rev. B21, 5628 (1980). Tight-Binding Approximation Michał Baj

34 Szmulowicz, F., Segall, B.: Phys. Rev. B21, 5628 (1980). Tight-Binding Approximation Density of states like for free electrons! Michał Baj

35 Tight-Binding Approximation Cu: [1s 2 2s 2 2p 6 3s 2 3p 6 ] 3d 10 4s 1 [Ar] 3d 10 4s 1 d-band Michał Baj

36 Ni: [1s 2 2s 2 2p 6 3s 2 3p 6 ] 3d 9 4s 1 [Ar] 3d 9 4s 1 ferromagnetic ordering, exchange interaction, different energies for different spins, two different densities of states for two spins and Δ Stoner gap

37 Semiconductors Semiconductors of the group IV (Si, Ge) diamond structure Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) zinc-blend or wurtzite structure Semiconductors AIVBVI (np. SnTe, PbSe) NaCl structure Si Indirect bandgap, Eg = 1,1 ev Conduction band minima in Δ point, constant energy surfaces ellipsoids (6 pieces), m =0,92 m0, m =0,19 m0, The maximum of the valence band in Γ point, mlh=0,153 m0, mhh=0,537 m0, mso=0,234 m0, Δso= 0,043 ev

38 Semiconductors Semiconductors of the group IV (Si, Ge) diamond structure Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) zinc-blend or wurtzite structure Semiconductors AIVBVI (np. SnTe, PbSe) NaCl structure Ge Indirect bandgap, Eg = 0,66 ev The maximum of the valence band in Γ point, mlh=0,04 m0, mhh=0,3 m0, mso=0,09 m0, Δso= 0,29 ev Conduction band minima in Δ point, constant energy surfaces ellipsoids (8 pieces), m =1,6 m0, m =0,08 m0,

39 Semiconductors Semiconductors of the group IV (Si, Ge) diamond structure Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) zinc-blend or wurtzite structure Semiconductors AIVBVI (np. SnTe, PbSe) NaCl structure GaAs Direct bandgap, Eg = 1,42 ev The maximum of the valence band in Γ point, mlh=0,076 m0, mhh=0,5 m0, mso=0,145 m0, Δso= 0,34 ev The minimum of the conduction band in Γ point, constant energy surfaces spheres, mc=0,065 m

40 Semiconductors Semiconductors of the group IV (Si, Ge) diamond structure Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) zinc-blend or wurtzite structure Semiconductors AIVBVI (np. SnTe, PbSe) NaCl structure a-sn Diamond structure Zero energy gap, Eg = 0 ev (inverted bandstructure) The maximum of the valence band in Γ point, mv=0,195 m0, mv2=0,058 m0, Δso=0,8 ev The minimum of the conduction band in Γ point, constant energy surfaces spheres, mc=0,024 m

41 Semiconductors Semiconductors of the group IV (Si, Ge) diamond structure Semiconductors AIIIBV (e.g. GaAs, GaN) i AIIBVI (np. ZnTe, CdSe) zinc-blend or wurtzite structure Semiconductors AIVBVI (np. SnTe, PbSe) NaCl structure PbSe Direct bandgap in L-point, Eg = 0,28 ev The maximum of the valence band in L point, constant energy surfaces ellipsoids, m =0,068 m0, m =0,034 m0, The minimum of the conduction band in L point, constant energy surfaces ellipsoids m =0,07 m0, m =0,04 m0,

42 H. Ibach, H. Lüth, Solid-State Physics Photoemission spectroscopy ħω = φ + E kin + E b work function potential barrier

43 H. Ibach, H. Lüth, Solid-State Physics Photoemission spectroscopy ħω = φ + E kin + E b work function potential barrier

44 Photoemission spectroscopy Phys. Rev. B 71, (2005)

45 Photoemission spectroscopy Phys. Rev. B 71, (2005)

46 Semiconductor heterostructures

47 Semiconductor heterostructures Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application

48 Bandgap engineering How can we change the heterostructure band structure: selecting a material (eg., GaAs / AlAs) controlling the composition controlling the stress

49 Heterostruktury półprzewodnikowe Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application

50 Bandgap engineering Valence band offset (Anderson s rule) Valence band offset: (powinowactwo) 50

51 Su-Huai Wei, Computational Materials Science, 30, (2004) Bandgap engineering Valence band offset (Anderson s rule) (powinowactwo) Valence band offset:

52 Walukiewicz Physica B (2001) Bandgap engineering Valence band offset

53 Bandgap engineering How can we change the heterostructure band structure: selecting a material (eg., GaAs / AlAs) controlling the composition controlling the stress Vegard s law: the empirical heuristic that the lattice parameter of a solid solution of two constituents is approximately equal to a rule of mixtures of the two constituents' (A and B) lattice parameters at the same temperature: a = a A 1 x + a B x

54 Z Dridi et al. Semicond. Sci. Technol. 18 No 9 (September 2003) Bandgap engineering How can we change the heterostructure band structure: selecting a material (eg., GaAs / AlAs) controlling the composition controlling the stress Vegard s law: Relationship to band gaps of binary compound : E = E A 1 x + E B x bx(1 x) b so-called bowing (curvature) of the energy gap

55 A wave packet S. Kryszewski Gdańsk

56 A wave packet Note that is a fourrier transform of Gaussian packet: S. Kryszewski Gdańsk

57 A wave packet Gaussian packet: S. Kryszewski Gdańsk

58 A wave packet Derivation: S. Kryszewski Gdańsk

59 A wave packet S. Kryszewski Gdańsk

60 A wave packet Ordering expression:

61 A wave packet Ordering expression: S. Kryszewski Gdańsk

62 Pakiet falowy

63 A wave packet Δx Δp ħ 2 m e = kg h = Js = ev s ħ = J s = ev s e = C

64 A wave packet

65 A wave packet Mass particle (electron) and massless (photon) foton elektron

Bonding in solids The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together.

Bonding in solids The interaction of electrons in neighboring atoms of a solid serves the very important function of holding the crystal together. For example Nacl In the Nacl lattice, each Na atom is