Survey of Materials. Lecture 3

Electronic structure of materials

Andriy Zhugayevych

October 5, 2017

Outline

• Quantum mechanics and Schrodinger equation

• Electronic structure of atoms and molecules

• Electronic structure of crystals

• Atomic motions

• Total energy

• Electronic properties

• Defects

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Why do we need to know Quantum Mechanics

• mechanical

• thermal

• at macroscale

• electronic

• chemical

• anything at nanoscale

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Pragmatic approach to Quantum MechanicsLet’s start with a single particle

Particle position r(t) → probability distribution |ψ(r , t)|2, wherethe wave-function ψ is the solution of Schrodinger equation:

i~∂ψ

∂t= Hψ ≡ − ~2

2m∆ψ + U(r , t)ψ

Any observable can be calculated as follows:

A = 〈ψ|A|ψ〉 ≡∫ψ(r , t) (Aψ) (r , t) dV

For example, average position:

r(t) =

∫r |ψ(r , t)|2 dV

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Example: particle in uniform field

Initial conditions (a particle at r0 with velocity ~k/m):

ψ(r , 0) = C exp

[ikr − (r − r0)2

a2

]If U(r , t) = −Fr then

ψ(r , t) ∼ exp

i(k +F t

~

)r −

(r − r0 − ~kt

m −F t2

2m

)2

a2 + i2~tm

=⇒ classical dynamics + gaussian broadening

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Example: simple model of electron transfer

Two-site Holstein model:

H =

(g x VV −g x

)+

Mx2

2+

kx2

2

Evolution: real space and coefficients5 / 22

Stationary Schrodinger equation

If the Hamiltonian is time-independent then the evolution can bewritten explicitly:

ψ(t) =∑n

cnψne−i En~ t , cn = 〈ψn|ψ(0)〉,

here (En, ψn) are eigenvalues and eigenfunctions of the stationarySchrodinger equation:

Hψ = Eψ

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Examples

potential ψn En

free particle eikr ~2k2

2m k ∈ R3

potential box sin πnxa

π2~2n2

2ma2 n = 1,∞

oscillator Hn(ξ)e−ξ2/2 ~ω

(n + 1

2

)n = 0,∞

Coulomb r lL2l+1n−l−1

(2rn

)e−

rnYlm(θ, φ) − α

2an2 n ∈ N,

l < n, |m| 6 l

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Practical considerations: basis setPlane waves or atomic orbitals

Wikipedia

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Combination of atomic orbitals: examples

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Many particle systems: fermions

Slater determinant – basis for many-body systems:

Ψ(ξ1, . . . , ξN) =1√N!

∣∣∣∣∣∣∣∣ψ1(ξ1) ψ1(ξ2) . . . ψ1(ξN)ψ2(ξ1) ψ2(ξ2) . . . ψ2(ξN). . . . . . . . . . . .

ψN(ξ1) ψN(ξ2) . . . ψN(ξN)

∣∣∣∣∣∣∣∣where ψi is i-th orbital and ξj is coordinate+spin of j-th electron.

Methods:

• Hartree–Fock (HF) – take single Slater determinate

• DFT – the same but modify energy functional

• post-HF – expand in basis of finite excitations

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One-electron orbitals

• molecular orbitals (MO) – eigenfunctions of one-electronHamiltonian (HF/DFT)

• localized molecular orbitals (LMO) – a rotation of MOslocalizing each orbital in space

• natural orbitals (NO) – eigenfuntions of one-electron densitymatrix ρ1e

Ψ (ξ; η) = N∫

Ψ(ξ, ζ2, . . . , ζN)Ψ(η, ζ2, . . . , ζN)dζ

• natural transition orbitals (NTO) – the same for transitiondensity matrix ρ1e

ΨΦ(ξ; η) = N∫

Ψ(ξ, ζ2, . . . , ζN)Φ(η, ζ2, . . . , ζN)dζ

Explore examples here

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More examples: MO vs NO

HOMO LUMO

ground statenh = 2ne = 0

hole NO electron NO

cation/anionnh/e = 1

∆n2 = .07/.06

singlet excitonnh = 1 + .12ne = 1 − .12

triplet excitonnh = 1 + .17ne = 1 − .18

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More examples: NO vs NTO

hole NTO/NO electron NTO/NO

singlet excitonnh = 1 + .12ne = 1 − .12

singlet transitionnh/e = 1 ± .17

triplet excitonnh = 1 + .17ne = 1 − .18

triplet transitionnh/e = 1 ± .25

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Strong correlations: Extended Hubbard modelPopulation analysis: ground state, hole, exciton; U/V = 2/1 vs 16/4

Methods: Uniform electron gas approximation for simple metals (LDA)

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Extended Hubbard model: density waves1e density matrix (diagonal & subdiagonal): ground state & hole; U/V = 2/1 vs 16/4

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Quantum mechanics of crystals

Bloch’s theorem for one-electron wave-function:

ψnk(r) = un(r)eikr

where un is periodic, n enumerates electronic bands, k is thewave-vector “periodic” in the reciprocal space define by vectors

bi = eijk2π

υ(aj × ak)

υ = a1 · (a2 × a3) is unit cell volume

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Example 1: Kronig–Penney model

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Example 2: Huckel model of trans-polyacetylene

H =

. . .

0 −t1 0−t1 0 −t2

0 −t2 0. . .

ψn = c1,2e

ikn, n ∈ Z, |k | 6 π/2

E (k) = ±√

t21 + t2

2 + 2t1t2 cos 2k

Ebandgap = 2|t1−t2|, Ebandwidth = 2(t1+t2)

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Example 2: Total π-electron energy of trans-polyacetylene

Total electronic energy E = 2π

∫ π/20 E (k)dk = −3.34

4x supercell −3.30

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Example 3: Silicon crystal

Band structure of Si

L

zk z

Γ

k x

ky

W

VΣ

3X

Λ

∆

KM

Q

1X

U

XS

Σ1

©bilbao crystallographic serverhttp://www.cryst.ehu.esBrillouin zone for Fm-3m

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Silicon crystal: charge density

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Summary and Resources

See summary here

• Wikipedia

• Bilbao Crystallographic Server

A few textbooks out of many:

• C Kittel, Introduction to Solid State Physics (2005)

• N W Ashcroft, N D Mermin, Solid state physics (1976)

Visualization software:

• Jmol

• Vesta

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