Lecture 6
6. The Tight-Binding Approximation
… or from Bonds to Bands
Basic concepts in quantum chemistry – LCAO and molecular orbital theory The tight binding model of solids – bands in 1, 2, and 3 dimensions
References: 1. Marder, Chapters 8, pp. 194-200 2. Kittel, Chapter 9, pp.244-265 3. Ashcroft and Mermin, Chapter 8 4. R. Hoffmann, “Solids and Surfaces: A chemists view of bonding in extended structures" VCH, 1988, pp 1-55, 65-78. 5. P.A. Cox, “The Electronic Structure and Chemistry of Solids”, Oxford, 1987, Chpts. 1, 2(skim), 3 (esp. 45-62), and 4 (esp.Lecture 79-88). 5 1
Bonds to Bands
• Forces in solids – Covalent (e.g., Si, C …) – Ionic (e.g., NaCl, MgO …) – Metallic (e.g., Cu, Na, Al …)
– Molecular (weak) (e.g., N 2, benzene)
• Forces between atoms (chemistry) – Covalent – Polar covalent – Ionic – Weak (London/dispersion, dipole)
Lecture 5 2
But for most problems we use same approximation methods:
Two main approaches in electronic structure calculations:
– Build up from atom: atomic orbitals + … – Infinite solid down: plane waves + …
Two methods used in quantum mechanics are variational and perturbation theory methods
Perturbation theory : H = H o + λ H1 + λ2 H2+ … oΨ o o Ψ o H n = En n (often known)
Lecture 6 3
1 Variational Method
Variational Method Use matrix notations Ψ = | Ψ> Η| Ψ> = Ε| Ψ> (time indep. Sch. Eqn.) For stationary stats, if Ψ is normalized, well-behaved function, it can be shown that
< Ψ *| Η | Ψ> = < Ψ *| Ε| Ψ> = Ε <Ψ *| Ψ> ← (variational integral)
⇒ Minimize variational integral to get ground state Ψs Ψ |* H | Ψ E = ← (overlap integral) Ψ |* Ψ
Lecture 6 4
Linear Combination of Atomic Orbitals (LCAO)
Best to start to solve Sch. eq. by choosing good ψ’s! ψ = φ ∑cn n n φ Any ψ can be expanded from set of orthonormal functions n (satisfying appropriate boundary conditions)
φ In quantum physics/chemistry often use hydrogen-like “atomic orbitals” n (s, p, d…) to study molecules (molecular ψ’s)
This approach is called LCAO – linear combination of atomic orbitals
Lecture 6 5
LCAO
General problem is to minimize variational integral by finding coefficients cn that make variational integral stationary:
∂c n = !0 ∂E HΨ = EΨ ⇒ φ = φ H ∑cn n E∑cn n n n − φ = (H E)∑cn n 0 n φ − φ = ∑cn (H n E n ) 0 n φ * φ * − φ * φ * = ∑cn (∫ m H n dV ∫ mE n dV ) 0 n []− δ = φ ∑cn H mn E mn 0 if mn orthonorma l, if not n c []H − ES = 0 ∑ n mn mn Lecture 6 6 n
2 + Molecular system H 2
+ • Most simple molecular system H 2 (molecular ion with one electron)
2 2 2 2 2 ˆ = − h ∇2 − e − e + e + 2e + H elec ... 2m rIA rIB rAB r12
kin Elect.-nuc. nucl repulsion fro H 2 electronic or attract. rep. larger systems
• For trial functions choose 2 simple 1s atomic orbitals ψ = φ + φ i cAi ISA cBi ISB = φ * φ = H AA ISA | H | ISA H BB = φ * φ = H AB ISB | H | ISA H BA = φ * φ = S AB ISB ISA SBA = S AA 1 Lecture 6 7
Bonds to Bands
• Determinant becomes: H − E H − ES AA AB AB = 0 − − H AB ES AB H BB E − − − − 2 = (H AA E)( H BB E) (H AB ES AB ) 0 − 2 2 + − + 2 − 2 = 1( S AB )E 2( H AB 2H AA )E (H AA H AB ) 0 • Solve quadratic eqs and get 2 solutions:
H − H S + H − H S Energy diagram: E = AA AB AB AB AA AB + sol' n − + 1( S AB )( 1 S AB ) E2 H − H S − H + H S E = AA AB AB AB AA AB − sol' n β − + 1( S AB )( 1 S AB ) H + H E = AA AB (+) α 1 + 1 S AB H − H E1 E = AA AB (-) 1 2 − ψ = (φ +φ ) 1 S AB 1 2 ISA ISB 1 ψ = (φ −φ ) Lecture 6 2 2 ISA ISB 8
Definitions
ψ • 1 is called bonding orbital ψ • 2 is called anti-bonding orbital
• Remember: each orbital can have 2 electrons • For 2+ electron system must add e-e repulsion + screening of core effective potential e.g., Hydrogen
Lecture 6 9
3 Nitrogen
Lecture 6 10
Correlation Diagrams
Homonuclear diatomics Heteronuclear diatomics
Lecture 6 11
Some symmetry considerations
• Strength of bond due to: - overlap of wavefunctions in space - inv. prop. to energy difference of non-interacting orbitals - electron + core repulsions
Lecture 6 12
4 Extended Hückel Theory - EHT
• Quantum chemical approximation technique (semi-empirical) • Study only valence (i.e., bonding ) orbitals • Solve the problem with LCAO variational method, but using ideas from perturbation theory
• The initial state orbitals are atomic Slater type orbitals – STOs
−5r − = n 1 n0 θ ϕ X i Ar exp Ylm ( , )
• The initial state energies, Hii are taken from experiments or some other calculations
Lecture 6 13
Extended Hückel Theory - EHT
• Intuitive idea is that the strength of a bond, as represented by the off- diagonal matrix elements Hij, should be proportional to the extent of overlap (Sij) and the mean energy of the interacting orbitals(Hii+Hjj)/2
H + H H = K ii jj S ij 2 ij • K is an empirical parameter between 1 and 2 • Problems:
-Hij is not exact - no charge self consistency
-Hij increases with overlap causing system to show minimum energy when atoms collapse - no core-core repulsion
• But… EHT does usually show qualitative correct trends in an easily interpretable manner Lecture 6 14
Orbitals and Bands in 1D
1. Equally spaced H atoms 2. Isomorphic p-system of a non-bond-alternating delocalized polyene 3. Chain of Pt II square-planar complexes
Lecture 6 15
5 Lecture 6 16
Band Dispersion or band width
The band structure of a chain of hydrogen atoms spaced 3, 2, and 1 Å apart. The energy of an isolated H atom is - 13.6eV
Lecture 6 17
Density of States (DOS) Concept
• E(k) curve, the band structure, has a simple cosine curve shape • DOS(E) is proportional to the inverse of the slope of E(K) vs k; ⇒ the flatter the band, the greater the density of states at that energy
Lecture 6 18
6 s vs p orbitals
For s orbitals, the interaction integral is negative (lowering energy) Total energy span from -2β to +2 β ⇒ total width is 4 β, which is proportional to the degree of interacting between neighbouring atoms General trend : strongly overlapping orbitals give large values of β, and wide bands, where as contracted atomic orbitals that overlap poorly give rise to narrow bands
Energy as a function of k for bands of s Linear combination p−σ orbitals for k = 0 and pσ orbitals in a linear chain (antibonding) and k= ±π/a (bonding) Lecture 6 19
Examples
(a) E(k) curve showing the allowed k values for a chain with N = 8 atoms; (b) Orbital energies for eight-atom chain, showing clustering at the top and bottom of the band; (c) Density of states for a chain with very large N
X-ray photoelectron spectrum of the
long chain alkane C36 H74 , showing the density of states in the 2 s band. (J.J.Pireau et.al.,Phys.Rev.A, 14 (1976), Lecture 6 2133.) 20
Acronyms of current experimental methods for band structure studies
AES Auger electron spectroscopy Photon (electron) in, electron out, inner shells EELS Electron energy loss spectroscopy Electron in/out; Conduction electrons EXAFS Extended x-ray absorption fine structure Photon in; filled bands IPS Inverse Photoemission Spectroscopy Electron in, photon out; Empty Levels STM(S) Scanning Tunneling Microscopy Electrons in, conduction and valence (Spectroscopy) bands Vis(UV) abs Vis./ UV absorption Photon in; band gap, defects
UPS Ultraviolet photoelectron spectroscopy Photon in, electron out, filled bands XPS X-ray photoelectron spectroscopy Photon in, electron out, filled bands XAS X-ray absorption Photon in, empty levels
Kittel, p. 328 - modified Lecture 6 21
7 Spectroscopic techniques
(a) optical absorption in the visible /UV range; (b) photoelectron spectroscopy; (c) inverse photoelectron spectroscopy; (d) X-ray absorption; (e) X-ray emission. Electrons with energies above the vacuum level can enter or leave the solid; in techniques (b) and (c) the scale shows the kinetic energy measured in a vacuum outside the solid. Lecture 6 22
Crystal Orbital Overlap Population (COOP)
• Better than DOS for determining extent of bonding and anti-bonding interactions
Lecture 6 23
p and d orbitals in linear chain
p and d orbitals (in addition to s) – for linear chain
Lecture 6 24
8 A polymer - linear array of PtH 4 molecules
• Monomer-monomer separation ~ 3 Å
• The major overlap between d z2 and pz orbitals
Lecture 6 25
2- Band structure and DOS for PtH 4
The DOS curves are broadened so that the two-peaked shape of the xy peak in the DOS is not resolved
Lecture 6 26
Buckling of chain
Buckling of 1D chain to minimize E (periodicity changes - doubles!)
Lecture 6 27
9 Energy minimization for 1D chain - Peierls instability
Solid-state chemistry analog of Jahn-Teller effect
Lecture 6 28
Peierls Distortion
• Qualitative electronic structure and energy minimization of cyclobutadiene ⇒ Symmetry breaking
Lecture 6 29
LCAO theory for 2D array of s orbitals
• The simplest possible 2D crystal consists of atoms of the same kind arranged on a square lattice, with spacing a. ψ = χ • In the LCAO approach, we need to find crystal orbitals in the form: ∑cr,s r,s r,s where the atomic orbital coefficients cr,s are determined by the periodicity of the = + lattice cr,s exp(irk xa isk ya)
(a) The band structure and (b) DOS of a square lattice of LectureH atoms 6 30
10 2D array of p orbitals
• pz orbitals will give a band structure similar to that of the s orbitals (topology of the interaction of these orbitals is similar)
Each crystal orbital can be characterized by σ or π bonding present
E.g., at Γ the x and y combinations are σ antibonding and π bonding
Lecture 6 31
Qualitative Band Structure
Schematic band structure of a planar square lattice of atoms bearing ns and np orbitals The s and p levels have a large separation that the s and p band do not overlap
Lecture 6 32
The electronic structure of graphite
•Pz orbitals for different wave-vectors in graphite.
Lecture 6 33
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