Electrons in Crystals

Chris J. Pickard Electrons in Crystals

• The electrons in a crystal experience a potential with the periodicity of the Bravais lattice: U(r + R) = U(r)

• The scale of the periodicity is of the order of the de Broglie wavelength of an electron — 1A—˚ so we must use Quantum Mechanics

• Of course, the periodicity is an idealisation: impurities, defects, Cartoon of electrons (blue) in motion thermal vibrations, ﬁnite size eﬀects The Periodic Potential

• In principle, we are faced with a many electron problem

• But we can make a lot of progress using the independent electron approximation

• We investigate the properties of the Schr¨odinger equation for a single electron: h¯2 2 HΨ = (−2m∇ + U(r))Ψ = EΨ A 1D periodic crystalline potential with U(r + R) = U(r) Bloch’s Theorem

• Independent electrons which obey the one electron Schr¨odinger equation for ik·r Ψnk(r) = e unk(r) a periodic potential are called Bloch electrons and obey Bloch’s theorem

unk(r + R) = unk(r) • Bloch’s theorem can be written in two equivalent forms

ik·R Ψnk(r + R) = e Ψnk(r) Proof of Bloch’s Theorem

Consider the translation operator:

TRf(r) = f(r + R)

It forms a commuting set for all R and H:

TRHΨ(r) = H(r + R)Ψ(r + R) = H(r)Ψ(r + R) = HTRΨ(r)

TRH = HTR

TRTR0 = TR0TR = TR+R0 The eigenstates of H are simultaneous eigenstates of all TR:

HΨ(r) = EΨ(r)

TRΨ(r) = c(R)Ψ(r) The properties of TR imply a relationship between the eigenvalues:

0 TRTR0Ψ(r) = c(R)TR0Ψ(r) = c(R)c(R )Ψ(r)

0 TRTR0Ψ(r) = TR+R0Ψ(r) = c(R + R )Ψ(r) and so: c(R)c(R0) = c(R + R0) If ai are the primitive lattice vectors, we can always write:

2πixi c(ai) = e For an arbitrary Bravais lattice vector:

R = n1a1 + n2a2 + n3a3

and so, considering repeated applications of Tai:

n1 n2 n3 ik·R c(R) = c(a1) c(a2) c(a3) = e where bi · aj = 2πδij and k = x1b1 + x2b2 + x3b3 We arrive at the second form of Bloch’s Theorem:

ik·R TRΨ(r) = Ψ(r + R) = c(R)Ψ(r) = e Ψ(r) Born-von Karman Boundary Conditions

Apply periodic BCs to a commensurate The allowed Bloch wave vectors are: supercell: 3 mi k = X bi Ψ(r + N a ) = Ψ(r), i = 1, 2, 3 Ni i i i=1 and then Bloch’s theorem: and the volume ∆k of k-space per k:

3 iNik·ai 1 (2π) Ψnk(r + Niai) = e Ψnk(r) ∆k = b · (b × b ) = N 1 2 3 V iN k·a 2πiN x e i i = e i i = 1 The number of allowed k-vectors in the which means that xi = mi/Ni primitive cell of the reciprocal lattice is equal to the number of cells in the crystal The First Brillouin Zone

• The wave vector k can always be conﬁned to the ﬁrst Brillouin zone (or any primitive cell of the reciprocal L lattice)

• Any k0 not in the ﬁrst Brillouin zone K W X can be written as: k0 = k + K, where k is in the ﬁrst Brillouin zone and eiK·R = 1

• The labels K,L,W,X and Γ are high The ﬁrst FCC Brillouin zone symmetry points in the Brillouin zone Band Structure

• For a given k there many solutions to the Schr¨odinger equation: Hkuk = Ekuk(r), uk(r) = uk(r + R)

• The boundary condition ensure that there are many (labelled n) discretely spaced eigenvalues

L • The Hamiltonian depends on k as K W X a parameter, and so the eigenvalues W L Γ X W K vary continuously with wave vector for FCC free electron bandstructure a given n. Hence, they are bands Crystal Momentum

• For Bloch electrons k is not p, and known as crystal momentum proportional to electronic momentum • The dynamical signiﬁcance of ¯hk ¯h ¯h ik·r ∇Ψnk = ∇(e unk(r)) is revealed by considering electrons i i response to applied electromagnetic ¯h = ¯hkΨ + eik·r ∇u ﬁelds nk i nk • A quantum number characteristic • The Ψnk are not momentum of the translational symmetry of eigenstates the periodic potential, as p is characteristic of the full translational • However, ¯hk is a natural extension of symmetry of free space Velocity and Effective Mass

• The velocity of an electron at k in band n is given by the gradient of the band and the inverse eﬀective mass is given by the curvature

• The velocity operator is: v = dr/dt = (1/i¯h)[r, H] = p/m = ¯h∇/im

V=hk/m L

K W X • Electrons in a perfect crystal move at V=0 a constant mean velocity W L Γ X W K Density of States

• Many electronic properties are q = R dEg(E)Q(E) weighted sums over the electronic levels of the form: • The density of states of a band is: Q = 2 Qn(k) Pnk g (E) = dk δ(E − E (k)) which is an integral in a large crystal: n R 4π3 n q = 2 dk Q (k) Pn R (2π)3 n • It can be written as a surface integral: • Often Qn(k) depends only on n and dS 1 gn(E) = RSn(E) 4π3 |∇En(k)| k through En(k), and the density of states g(E) = Pn gn(E) is a useful with Sn(E) a surface of constant construct: energy Van Hove Singularities

• Because En(k) is periodic in • In 1D this results in a divergence of reciprocal space, and for each n the density of states itself bounded from above an below, and diﬀerentiable everywhere there must • In 3D the divergence is integrable, and be k for which |∇E| = 0 results in discontinuities in dgn/dE • Thus, the integrand in the expression for gn(E) diverges • These are the van Hove singularities Metals and Insulators

• Fill the electronic states, lowest energy ﬁrst across the whole ﬁrst Brillouin zone (so that each level is counted only once), until all the electons are accomodated

• If there is a gap between the highest occupied state and the lowest unoccupied the crystal is an insulator (and a called a semiconductor if the gap is close to kBT ) The Fermi Surface

• If there is no gap then the crystal is a metal

• There will be a surface in k-space separating occupied from unoccupied levels: this is known as the Fermi surface and may consist of several branches. It determines the transport and optical properties of the metal The Fermi Surface

Li K • If there is no gap then the crystal is a metal

• There will be a surface in k-space separating occupied from unoccupied Al Be levels: this is known as the Fermi surface and may consist of several branches. It determines the transport and optical properties of the metal

Zn Y Electrons in a Realistic Crystal Potential

• In principle, it remains only to solve the Schr¨odinger’s equation for the Bloch wavefunctions

• This might be viewed as a job of pure numerics – aside from the choice of U(r)

• But we can do better than brute force – and with greater understanding The Nearly Free Electron Approximation

• Possibly suprisingly, the electronic structure of some metals is considered to arise from a weak periodic perturbation of the free electron gas

• This is due to the combined eﬀects of the Pauli exclusion principle and O(U) { screening

k • The perturbation has diﬀerent results, K K 1 depending whether the free electron Deﬁning near degeneracy states are nearly degenerate or not The Nearly Free Electron Approximation

E • If there is near degeneracy, the perturbation can be linear in the potential

• Symmetries and structure factor

2|U k| eﬀects can eliminate the splitting – returns with spin-orbit coupling k 1 0 2K K Splitting at a single Bragg plane • The bandstructure can be plotted in • If there is no degeneracy, the the reduced or extended or repeated zone schemes pertubation is second order in UK Free electron and a Real Bandstructure

L L

K K W X W X

W L Γ X W KW L Γ X W K The free electron and DFT bandstructure of FCC Aluminium The Tight Binding Approximation

1s • Now, let’s think of the electronic structure as being a modiﬁcation of

1s that of the isolated constituent atoms 2s 2p 3s 2p • This is a good picture if the overlap 3s of the atomic wavefunctions is small

2s • Clearly, this is not the case for metallic Sodium electronic wavefunctions – sodium – it is a nearly free electron separated by 3.7A˚ metal The Tight Binding Approximation

• To calculate the corrections, consider: H = Hat + ∆U(r) rφ

• The general form of the wavefunction ik·R r R is: ψ(r) = PR e φ( − ) where ∆U(r) r r φ( ) = Pn bnψn( )

• There is a strong hybridisation and splitting of levels close to each other in When rφ(r) is large, ∆U(r) is small energy – recall the nearly free electron and vice versa model The Tight Binding s-Band

ik·R Consider a single s-band: |ψki = PR e |ψsi

hψs|∆U|ψki (Ek − Es)hψs|ψki = hψs|∆U|ψki ⇒ Ek = Es + hψs|ψki

Ignoring the devations from unity of the denominator, summing over the nearest neighbours only, and using the inversion symmetry of the potential:

k R ∗ r R r R r Ek = Es + hψs|∆U|ψsi + Pnn cos( · ) R ψs ( )∆U( )ψs( − )d The Tight Binding s-Band

R X M

X R M Γ RX R M Γ R Sodium in a Simple Cubic cell of side 3.7A:˚ Ek = α − β(cos akx + cos aky + cos akz) Valence Band Wavefunctions

1s • All electonic structure methods must face up to a truth: the valence

1s wavefunctions are diﬃcult to describe 2s 2p in any one way 3s 2p

• They are smooth in the interstitial 3s regions, and oscillatory in the core

2s regions

• This is because the valence Sodium electronic wavefunctions – wavefunctions must be orthogonal to separated by 3.7A˚ the core states Electronic Structure Methods

• The Cellular method

• The Augmented Plane-wave method The Cellular potential (APW)

• The Green’s Function method (KKR)

• The Orthogonalised Plane-wave method (OPW) The Muﬃn Tin potential The Pseudopotential Approach

1s • The pseudopotential was originally thought of as the weaker eﬀective

1s potential due to the orthogonality of 2s 2p the valence to the core wavefunctions 3s 2p

• This concept provided a justiﬁcation 3s and theoretical framework for the nearly free electron approximation 2s

• The true power of the method unleashed by much later ab initio formulations The Pseudopotential Scheme

Ψ (r) ps • Do an all electron atomic calculation

Ψ rc ae (r) • Choose a core radius, and pseudise

Vps(r) the atomic wavefunctions in the channels of interest

• The simplest schemes conserve the norm but this can be relaxed

Vae(r) • Invert the Schr¨odinger equation to ﬁnd the pseudopotential The Pseudopotential Scheme

Ψ (r) ps • The resulting pseudopotential will reproduce the reference eigenstates Ψ (r) rc ae by construction, and is nonlocal in general

Vps(r) • Transferability is of great importance – the potentials should be tested extensively

• There may be ghost states

Vae(r) Projector Augmented Waves

• The Projector Augmented Wave following: 1 ˜ approach puts pseudopotential theory T = + PR,n[|φR,ni − |φR,ni]hp˜R,n| on a firm theoretical footing • The projectors are defined so that: • It defines an operator T which hp˜ |φ˜ 0 i = δ 0δ maps pseudowavefunctions to the all R,n R ,m R,R n,m electron wavefunctions: |Ψi = T |Ψ˜ i • Pseudo operators can be defined: • This operator can be chosen to be the O˜ = T +OT

• The PAW scheme can be applied to the all electron Hamiltonian H

add and subract V − V loc which is localised:

˜ 1p2 loc r nl H = 2 + V ( ) + PR VR

• An arbitary choice for the local potential V loc(r) is made and the nonlocal potential is of the form:

nl R VR = Pn,m |p˜R,nian,mhp˜R,m|