Dept of Phys

M.C. Chang

Energy bands (Nearly-free electron model)

NFE model is good for Na, K, Al… etc, in which the lattice potential is only a small perturbation to the electron sea.

• Electron diffraction and energy gap • Bloch theorem • The central equation • Empty-lattice approximation • Tight-binding model (see Chap 9 ) History of band theory (Ref: chap 4 of 半導體的故事 , by 李雅明 )

• 1928 – Bloch theory Both are students of • 1929 – Peierls Heisenberg • nearly-free electron model, diffraction and energy gap • effective mass • filled band does not conduct, hole • umklapp process

• 1930 – Kroenig-Penny model • 1931 – Wilson explains // Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Bloch recalled, The main problem was to explain how the electrons could sneak by all the in a metal so as to avoid a of the order of atomic distances . Such a distance was much too short to explain the observed resistances, which even demanded that the mean free path become longer and longer with decreasing temperature.

By straight Fourier analysis I found to my delight that the wave differed from the plane wave of free electrons only by a periodic modulation. This was so simple that I didn't think it could be much of a discovery, but when I showed it to Heisenberg he said right away: "That's it!" ik⋅ r ψ k()r= e ur k ()

Hoddeson L – Out of the Crystal Maze, p.107 Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Nearly-free electron model Free electron plane wave 2π ψ (re )=ik⋅ r , kn = ... etc k x x L • Consider 1-dim case, when we turn on a lattice potential with period a, the electron wave will be Bragg reflected when k=±π/a, which forms two different types of standing wave . (Peierls, 1930)

π  π ixaπ /− ixa / 2 x ˆ G ψ + =c() e + e = cos  , k⋅ G = a a  2 2 π x  ψ =ce() π ixaπ / − e− ixa / = i sin  . − a a  • Density distribution of the two standing waves Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

• These 2 standing waves have different electrostatic energies. This is the origin of the energy gap. If potential V(x)=Vcos(2 πx/a), then Note: Kittel use a potential energy U (=eV) Eg =∫ dx()ρ ρ− − + Vx( ) 0 a 2 2 =∫ dxe( − )| ψ()ψ − ()| x − | + ()| xVx () 0 2ea π π 2 π xxx  =−V∫ dxcos sin2 −= cos 2  eV a0  a aa  • Lattice effect on free electron’s energy dispersion ψ −

ψ +

Electron’s group velocity is zero near the boundary of the 1st BZ (because of the standing wave ). π π Q: where are the energy gaps when V(x)=V 1 cos(2 x/a)+V2 cos(4 x/a)? Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

A solvable model in 1-dim: The Kroenig-Penny model (1930) (not a bad model for superlattice)

Delta functions

a • Electron energy dispersion calculated from the Schrödinger eq.

Energy bands (Kittel, p.168) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Bloch theorem (1928)

The electron states in a periodic potential is of the form ψ ()r= eik⋅ r ur () k k , where uk(r+R )= uk(r) is a cell-periodic function .

• A simple proof for 1-dim: |Ψ(x)|2 is the same in each unit cell. Ψ(x+a )=C Ψ(x) Consider periodic BC, ψ ψ ψ (xNa+ ) = CN () x = () x s  →=Cexp 2π is  , = 0,1,2⋯ ,() N − 1 N  → ψψ (xa+ ) = C () x s Eigenstate of = eika ψ ( x ), k ≡ 2π translation op Na ikx write ψ k (x )= eu k ( x )

then uk ( x +a ) = u k ( x )

Similar proof applies to higher dimension. Energy gap Bloch theorem The central eq. Empty lattice approx. TBM important • Schrödinger eq for ψ: Lattice potential p2 H ψψ =ε , H = + U (r ) k k k 2m Energy ik⋅ r eigenstate ψ k()()r = e u k r

uk(x) depends on the form of the periodic lattice potential .

10 23 times less effort than • Schrödinger eq for u: the original Schrodinger eq. ɶ Hku ( ) k= ε k u k within one unit cell ɶ −ikr ⋅ i k ⋅ r ← where H( k ) ≡ e He Effective Hamiltonian for u k(r) ℏ2 ∇  2 = +k + U () r 2m i  Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Allowed values of k are determined by the B.C. important Periodic B.C. ψ ψ k(rNa+i i ) = k ( ri ), =1,2,3 (3-dim case) ik⋅ Ni a i →e =1, ∀ i N 3 N2 →kNa ⋅ii = 2π m ii ,m∈ Z , ∀ i

(see chap 2) m1 m 2 m3 N1 → k= b1+ b 2+ b 3 N1 N 2 N 3 b b b  1 b⋅( b × b ) 3 1 2 3 1 2 3 = ∆=⋅k ×  = bbb1 ⋅×( 2 3 ) 3 N N1 N 2 N 3  N ∆ k

N= NN1 2 N 3 V Vol of 1 st BZ i aa⋅( × a ) == v , Vol of a k-point 1 2 3 N (2π ) 3 i b⋅( b × b ) = , 1 2 3 v (2π ) 3 i ∆3k = , as in the free-electron case. V Therefore, there are N k-points in a BZ (a unit cell in reciprocal lattice), where N = total number of primitive unit cells in the crystal. Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Counting in r-space and k-space

r-space k-space

a crystal with PBC:

G R

N points here

N points here

• N unit cells • Infinite reciprocal lattice points (N lattice points x 1 atom/point) • N k-points in 1 st BZ • If each atom contributes q • N k-points in an energy band conduction electrons, then Nq electrons in total. Q: what if there are p atoms per lattice point? Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Difference between conductor and insulator (Wilson, 1931 )

• There are N k-points in an energy band, each k-point can be occupied by two electrons ( up and down). ∴ each energy band has 2N “seats” for electrons. conductor E • If a has odd number of valence electron per primitive cell, then the energy band is half-filled (conductor). ε For example, all alkali are conductors. F k • If evena solid number has even of electronsnumber of per valence primitive electron cell, th peren primitivethere are cell, 2 thenpossibilities: the energy (a) noband energy can eitheroverlap be or insulator (b) energy or metal overlap.. E.g., alkali-earth elements can be conductor or insulator.

Insulator (later) conductor E E

k k Filled band partially-filled bands Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Band theory of

energy

Insulator metal semiconductor (n-type) (p-type) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Some mathematics before moving on

• Orthogonal relation

iG(− G ') x 1D ∫ dx e= a δG, G ' cell

3iG (− G ') ⋅ r See A+M, App. D 3D ∫ dre= v δG, G ' cell

 Vδ for discrete k  k, k ' 3ik (− k ') ⋅ r note: ∫ d r e =  3 all ()2π δ ()k− k ' for continuous k

• Fourier decomposition If f( r) has lattice translation symmetry, that is, f( r)=f( r+R) for any lattice vector R, then it can be expanded as, = iG⋅ r fr( ) ∑ ef G , where G is the reciprocal lattice vector. G (see Chap 2) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

How do we determine uk(r) from lattice potential U(r)?

2 ℏ  Schrödinger equation ()p+ k  +Urur() () = ε ur () 2m  k k k   Keypoint: go to k-space to avoid derivatives Fourier transform: UrR(+= ) Ur (), urRk ( += ) ur k () 1. the lattice potential Ur( ) = Ue iG⋅ r ∑ G G=2 πn/a G

2. the wave function = −iG ⋅ r k=2 πn/L urk()∑ CGe k () G unkown ℏ2k 2 Schröd. eq. in k-space εε 0 − + = ε 0 ≡ ()kG− k CG k( )∑ UG'− G C k (G ') 0, k G' 2m

aka. the central eq. Kittel λ uses k Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Matrix form of the central eq . (in 1D) G=ng (g ≡2π/a)

⋱ ⋮  0  εε kgk+2− U g U 234 g U g U g Ck (− 2 g )   0    U−g εε kgk + − UUU g2 g 3 g Ck (− g )   UU ε0 − UU  C (0)  = 0  −2g − gkkg2 g  k  0 for a  UUU−3g − 2 g −− gkgk εε − U − g  Ck ( g )  particular k  0     U−432g U − g U − g U −− gkgk εε 2 − Ck (2 g )  ⋱ ⋮

• For a given k, there are many eigen-values εnk , with eigen-vectors Cnk .

ε • The eigenvalues n(k) determine the energy bands .

• The eigenvectors { Cnk (G), ∀G} determine the Bloch states :

−iGx uxnk()= ∑ CGe nk () G Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Example: U(x) = 2U cos2 πx/a π π = U exp (2 ix/a )+U exp (-2 ix/a ) (Ug=U -g=U )

⋱ ⋮  0  εε k+2 g− k U 0 0 0 Ck (− 2 g )   0     U εε k+ g− k U 0 0   Ck (− g )   0 U ε0 − U 0   C (0)  = 0  k k  k 0    0 0 U εε k− g− k U   Ck ( g )   0     0 0 0 U εε k−2 g− k   Ck (2 g )  ⋱ ⋮ An infinite matrix.

In practice, it an be truncated according to the precision demanded. iGx important − TBM G k ∑ = ⋅ ? ik R ) () () (QHE, TI …) TI (QHE, nk nk = ux CGe 2g r r + = () () + = : rRe r ( ) ( ); + ( ') ( ) nk ,, , , ' nknkG nk nk Empty lattice approx. lattice Empty C + ψψ ψ ψ BZ is is enough BZ k G k C GG CG st g is a linear linear is a 3k 2k 1k ε ε ε C(-g)=1 C(0)=1 C(g)=1 C(0)=1 C(-g)=1 topological obstruction topological nk 0 BZ u Gx i , st ( ) info 1 info the in − 1 k k k ε iGx ∴ − ( The central eq. Thecentral = = nk ε x x () () = G n G + ∑ + , +G , n k = nkG nk n,k uψ xeux ψ ε () () , for a particular , for particular a -2g -g 0 nk nk Periodic gauge. gauge. Periodic of presence the in valid Not ≠ ( ) ux⇒ CGe U(x) Bloch theorem U(x)=0 • can that see central eq., the one From • U=0 when and eigen-states are eigen-energies the What • when combination of plane waves, with with waves, coefficients of plane combination • energy Bloch Energy gap Energy Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Approximation of the central equation

ik(− G ) x • The Bloch state ψ nk()x= ∑ CGe nk () G is a superposition of … exp[ i(k-g)x], exp[ ikx ], exp[ i(k+g )x] …

～ • If k 0, then the most significant component of ψ1k (x) is See Fig in exp[ ikx ] (little superposition from other plane waves). previous page

～ • If k g/2 , then the most significant components of ψ1k (x) and ψ2k (x) are exp[ i(k-g)x] and exp[ ikx ], others can be neglected.

Truncation: ⋱ ⋮  0  εε k+2 g− k U 0 0 0 Ck (− 2 g )   0     U εε k+ g− k U 0 0   Ck (− g )   0 U ε0 − U 0   C (0)  = 0  k k   k  0 0 U εε 0 − U C( g )  k− g k   k  cutoff for  0     0 0 0 U εε k−2 g− k   Ck (2 g )  k～g/2! ⋱ ⋮ Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Energy levels near zone boundary k ～ g/2  ε0 − U  C( k )  • Cut-off form of the central eq. k  = 0    0 U εε k− g −  C( k− g )  • Energy eigenvalues • Energy eigenstates

100 1 00 2 2 ikx i( k− g ) x ε ε ε (k )=() +±() − + () 2 U ψ ±k ()x= Cke ± () + Ck ± ( − ge ) ±2k kg − 2 k kg − g define kɶ = k − 2 ψ +,k ℏ2kɶ2 2ε 0  ɶ 0 g /2  then ε ε ± (k )= g/2 ± U + 1 ±  2m U 

parabola ψ −,k

cos(π x / a ) ⇒ ≃ ψ g (x ) ±, 2 sin()π x / a Agree with the diffraction scenario Energy gap Bloch theorem The central eq. Empty lattice approx. TBM 3 ways to plot the energy bands:

(k '∈ 1st BZ)

εε n, k ' → k (k= k ' ± ng ) 1st Brillouin zone

Sometimes it is convenient to repeat the domains of k

Kittel, p.225 (Chap 9)

Fig from Dr. Suzukis’ note (SUNY@Albany) Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Nearly-free-electron model in 2-dim (energy bands) • 0th order approx.: empty lattice (U( r)=0) • 1st order approx.: energy gap opened by Bragg reflection

k G Laue condition k⋅ Gɵ = 2 G/2

→ Bragg reflection whenever k hits the BZ boundary Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

“Empty lattice” in 2D 2D square lattice’s reciprocal lattice

• Free electron in vacuum:

ℏ2k 2 εk = 2m

• Free electron in empty lattice:

2 ℏ2 ()k′ + G εε k= k = n ′ 2m k= k′ + G k′∈1st BZ

M 2π/a • How to fold a parabolic “surface” back to the first BZ?

Γ X Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Folded parabola along ΓX (reduced zone scheme)

• There would be energy gaps at BZ boundaries because of the Bragg reflection

• The folded parabola along ΓM is different M

Γ X

• Usually we plot only major directions. For 2D square lattice, they are ΓX, XM, M Γ

2π/a Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Empty Lattice in 3D

Simple cubic lattice

2 ℏ2 ()k′ + G εε k= k = n ′ 2m

k= k′ + G k′∈1st BZ Energy gap Bloch theorem The central eq. Empty lattice approx. TBM Empty FCC lattice 1st Brillouin zone:

Energy bands for empty FCC lattice along the Γ-X direction. Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Comparison with real band structure The energy bands for Actual band structure for “empty” FCC lattice copper (FCC, 3d 10 4s 1)

d bands

lampx.tugraz.at/~hadley/ss1/empty/empty.php From Dr. J. Yates’s ppt Energy gap Bloch theorem The central eq. Empty lattice approx. TBM ε DOS for a Bloch band nk

An energy shell

dS ε dk ⊥

Surface ε+d ε =const Surface ε=const

L  3 Dd(ε ε ) =   ∫ dk3 (per spin) 2π  Shell

3 d k= dSdkε ⊥

d εε = ∇ k dk ⊥

3 dε ∴d k = dS ε ∇k ε

3 If v = ∇ε = 0, then there is L  dS k k ⇒ D(ε ) =   ∫ ε 2π  "van Hove singularity " (1953) ∇k ε Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

For example, DOS of 1D energy bands

Van Hove singularity Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Origin of energy bands - an opposite view

Tight-binding model (see chap 9) Wiki: electronic band structure band electronic Wiki: Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Tight-binding method (Bloch, 1928)

Let am(r) be the eigenstate of an electron in the potential Uat (r) of an isolated atom.

at atomic Haratm()= ε mm ar () orbital

Consider a crystal with N atoms at lattice sites R, • A wave function with translation symmetry: (but still not an energy eigenstate) ϕ = − mk()r∑ dRarR k ()(m ) R 1 Linear combination of =eik⋅ R a ( r− R ) ∑ m atomic orbitals (LCAO) N R

1 Check: ϕ +=ik⋅ R ' −− mk ()rR∑ earRRm ((')) N R' 1 ikR⋅ ikR ⋅' − = ikR ⋅ ϕ =e∑ earRem ( ')mk ( r ) N R' Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Schrödinger equation Vlatt= V at + ∆ V p2 H= + V( r ) ()b= () a + () c 2m latt H εψ nk= nk nk

Energy eigenstate (Bloch state) n ϕ ψ nk()r= ∑ Cm mk () r m

One-band approximation ≃ ϕψ nk()r nk () r

Hunklinger, Festkörperphysik H ε ϕϕ nk= nk nk

Given an , the other εϕ ϕ H a= a nkn0 nk nk n 0 atoms is just a perturbation. Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

= + ∆ Define overlap integral H Hat V i α R= a a εϕ ϕ + ∆ = n( ) nR n 0 nkHVaat n0 nknk a n 0 off-diagonal matrix elements ⇒ −ik ⋅ R ∑ e aHanR at n 0 are neglected R +−ik ⋅ R ∆ Intra-atom ∑ e anR Va n0 R i an0∆ V a n 0 ≡ βn = ε e−ik ⋅ R a a nk ∑ nR n0 energy shift due to the potential R of neighboring atoms.

△V(r) Inter-atom 2 |a n0(r)| i anR ∆ V a n0 ≡ γ n (R ) V(r) Inter-atomic matrix element between nearby atoms γ ⇒ β α ε at−⋅ ik R+ + −⋅ ik R n∑e nn() R ∑ eR n () R R ≠0 α= ε −ik ⋅ R nk ∑ en ( R ) R Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

−ik ⋅ R α = α α =3 * − n (k ) ∑ n()Re ; n () R∫ drarRar nn ( )() R≠0 3 * βn=∫ dra n() r ∆ Va n () r −ik ⋅ R γ = γ γ =3 * − ∆ n (k ) ∑ n()Re ; n () R∫ drarn (R ) V an () r R≠0 α ε ⇒ β γ α ε at + ++ =  + nn1()k nn () knk  1() n k β γ + (k ) ⇒ atn n ≃ at β γε ε ε nk =+n nnn ++ (k ) 1+αn (k )

• If we keep only the NN integrals, then γ γ α(α (−=R ) ( R ); ( −= R ) ( R ) will be used )

γ n (k ) = 2∑ γ γ γ 0 cos(k⋅R) , 0 = n( R N N ) half of NN Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Square lattice, with only NN coupling solidstate.quantumtinkerer.tudelft.nl/lecture_6

at • 1D β γε nk= n+ n + n (k ) Energy contours = 2∑ γ 0 cos()kR +const. half of NN

= 2γ 0 cos ka

• 2D γε nk =20 ( coskax+ cos ka y )

• 3D γε nk =20 ( coskax+ cos ka y + cos ka z )

Density of states /

From Dr. P. Young’s at UCSC Energy gap Bloch theorem The central eq. Empty lattice approx. TBM

Tight-binding model vs nearly-free electron model • Covalent solid •

• d-electrons in • noble metal transition metals