9. Fermi Surfaces and Metals

• Construction of Fermi Surfaces

• Electron Orbits, Hole Orbits, and Open Orbits

• Calculation of Energy Bands

• Experimental Methods in Fermi Surface Studies

Fermi Surface : surface of ε =εF in k-space

Separates filled & unfilled states at T = 0.

Close to a sphere in extended zone scheme.

Looks horrible in reduced zone scheme.

2nd zone nearly half-filled

Reduced Zone Scheme

Reduced Zone Scheme: k 1st BZ.

k is outside 1st BZ.k = k + G is inside.

ie u k r

k kr r i ie e u k r G r

k r

ie u k rk kr r

k r

iu e u G r

k kr r

Both ie G r u k r& are lattice-periodic.

So is

→

is a Bloch function

1-D Free Electrons / Empty Lattice

2

2 2 21 2 / /2m

k k k k

2 2/ /Constant

2m

k

Reduced zone scheme:εk is multi-valued function of k.Each branch of εk forms an energy band

Bloch functions need band index:

in ne u k r

k kr r 1st BZk

2

2

2nk nk Gm

inC e k G r

G

k GPWE

22

2k km

Periodic Zone Scheme

εk single-valued

εk multi-valuedεnk single-valued

εnk = εnk+G periodic

2 cos cos cosx y zk a k a k a k

E.g., s.c. lattice, TBA

Construction of Fermi Surfaces

Zone boundary:22 0 G Gk

3rd zone: periodic zone scheme

Harrison construction of free electron Fermi surfaces

Points lying within at least n spheres are in the nth zone.

Nearly free electrons:Energy gaps near zone boundaries → Fermi surface edges “rounded”.Fermi surfaces & zone boundaries are always orthogonal.

Electron Orbits, Hole Orbits, and Open Orbits

Electrons in static B field move on intersect of plane B & Fermi surface.

d e

dt k

kB

q f v B

Nearly filled corners:

P.Z.S.

P.Z.S.

Simple cubicTBM

Calculation of Energy Bands

• Tight Binding Method for Energy Bands

• Wigner-Seitz Method

• Cohesive Energy

• Pseudopotential Methods

Tight Binding Method for Energy Bands

2 neutral H atoms

Ground state of H2 Excited state of H2

1s band of 20 H atoms ring.

TBM / LCAO approximation Good for valence bands, less so for conduction bands.

, ,

lij l j

l j

e c

k Rk r k r R ρ α = s, p, d, …

ψk is a Bloch function since

j runs over the basis atoms

1li

m m ll

eN

k Rk r R r R R 1

m ni in

n

e eN

k R k R r R mie k R

k r

1st order energy: H k k k 3

,

1*l mi

m ll m

e d HN

k R R r r R r R

Bravais lattice , s-orbital only: 1li

ll

eN

k Rk r r R

3

,

1*l mi

m ll m

e d HN

k R R r r R R r 3

,

1*ni

nn m

e d HN

k R r r R r

3 *nin

n

e d H k R r r R r

Keep only on site & nearest neighbor terms:

3 *d H r r r 31*d H r r R r

. .

l

l

i

n n

e

k Rk

R

0/

0

2 1 aRy ea

For 2 H atoms ρ apart:

2

0 20.53a A

m e

4

213.6

2

meRy eV

Simple cubic lattice: 6 n.n. at

1,0,0 , 0,1,0 , 0,0,1aρ

y yx x z zi k a i k ai k a i k a i k a i k ae e e e e e k

2 cos cos cosx y zk a k a k a

Band width = 12

= surface. 1 e per unit cell.periodic zone scheme

2

2*

2m

a

2 N orbitals in B.Z.

. .

l

l

i

n n

e

k Rk

R

Fcc lattice: 12 n.n. at 1, 1,0 , 1,0, 1 , 0, 1, 12

a ρ

/2 /2 /2 /2, ,x y x y x y x yi k k a i k k a i k k a i k k a

e e e e x y y z x z y x k

1 12 cos cos

2 2

1 1cos cos

2 21 1

cos cos2 2

x y x y

y z y z

z x z x

k k a k k a

k k a k k a

k k a k k a

Band width = 24

1 1 1 1 1 14 cos cos cos cos cos cos

2 2 2 2 2 2x y y z z xk a k a k a k a k a k a

2 surface

. .

l

l

i

n n

e

k Rk

R

Band Structure

Wigner-Seitz Method

Bloch function: ie u kk

rk r r

Schrodinger eq.:

ie u i u kk k k

rr k r

2 ie i i u u i u u k

k k k k krr k k k

2 22ie u i u u kk k k

r k k

22 2

22 212

2 2 2ie u

m m m rk

k

pk pk p

2

2ie u

m

kk

r k p

2

2U u u

m

k k k

p k

a a a A A A

j ji

k xx

k r

j i jk ik k ie i u u kk k

r k

For k = 0, we have2

0 0 02U u u

m

pu0 is periodic in R l .

is a Bloch function;can serve as an approximate solution of the Schrodinger eq. for k 0.

0ie u k r

k r r

0ie u k r

k r r 0 0 0H u u

2 2

0 0 02H u u

m m k k

kk p

202

iH e U um

krk p k

r 2

0

2

0ie u

m m

rk kpk

0 0 1u u Prob 8

Wigner-Seitz result for 3s electrons in Na.

Wigner-Seitz B.C.: d /d r = 0 at cell boundaries.

Table 3.9, p.70 ionic r = 1.91A r0 of primitive cell = 2.08A n.n. r = 1.86A is constant over 7/8 vol of cell.

Cohesive Energy

linear chain

Na

5.15 eV for free atom.0 ~ 8.2 eV for u0 . +2.7 eV for k at zone boundary.

Table 6.1, p.139: F ~ 3.1 eV.K.E. ~ 0.6 F ~ 1.9 eV.

~ 8.2+1.9 ~ 6.3 eV

Cohesive energy ~ 5.15 +6.3 ~ 1.1 eVexp: 1.13 eV

Pseudopotential Methods

Conduction electron ψ plane wave like except near core region.Reason: ψ must be orthogonal to core electron atomic-like wave functions.Pseudopotential: replace core with effective potential that gives true ψ outside core.

Empty core model for Na(see Chap 10)

Rc = 1.66 a0 .U ~ –50.4 ~ 200 Ups at r = 0.15

2

0 c

ps

c

r R

U forer Rr

With Thomas-Fermi screening.

Typical reciprocal space Ups

Empirical Pseudopotential Method

Cohen

2

3ps FU G 0 (see Chap 14)

Experimental Methods in Fermi Surface Studies

• Quantization of Orbits in a Magnetic Field

• De Haas-van Alphen Effect

• Extremal Orbits

• Fermi Surface of Copper

• Example: Fermi Surface of Gold

• Magnetic Breakdown

Experimental methods for determining Fermi surfaces:• Magnetoresistance• Anomalous skin effect• Cyclotron resonance• Magneto-acoustic geometric effects• Shubnikov-de Haas effect• de Haas-van Alphen effect

Experimental methods for determining momentum distributions:• Positron annihilation• Compton scattering• Kohn effect

Metal in uniform B field → 1/B periodicity

Quantization of Orbits in a Magnetic Field

kin field p p p q

c k A

Bohr-Sommerfeld quantization rule: d n h p r

Phase corrector γ = ½ for free electrons

q = –e for electrons

d q d

dt c dt

k rB → 0 0

q

c k k r r B B = const

0d d k r k k r qd

c r B r

qd

c B r r

2qd

c B S

2q

c

0

qd

c r r B r

q qd d

c c A r A S

q

c

qd

c p r n h

electronn D

hcn n

q 7 24.14 10D

h cGauss cm

e

Flux quantization Dirac flux quantum

q

c k r B →

cr k

qB

For Δr B :

Let A = Area in r-space, S = Area in k-space. →2

cA S

qB

Hence2

c S

q B

B A 2 cn

q

2n

qS n B

c

Area of orbit in k-space is quantized

1

1 1 2

n n

qS

B B c

If 1n nS S S then

1 2 qconst

B cS

Properties that depend on S are periodic functions of 1/B.

2electron

2D

Bn

2

electron

2

D

electron

Dn

2

electron

2

DS

2

electron 2D SB

electron 2D kB

De Haas-van Alphen Effect dHvA effect: M of a pure metal at low T in strong B is a periodic function of 1/B.

1

2n n

eBS S S

c

2-D e-gas:PW in (B) dir.

# of states in each Landau level = (spin neglected)

22

2

eB LD

c

B2

2

eL

c

22

D

B

22

DA

2eL

hc

2

D

L

2

D

L

A

DDA

B

B = 0

Allowed levels

22

D

SA

2

D

BLD

22

D

B

B

2D

h c c

e e

1

2n cE n

*c

e B

m c See Landau & Lifshitz, “QM: Non-Rel Theory”, §112.

B 0

10T TE E B 2 30T T TE B E E B

1 2 3B B B

1

2n cE n

*c

e B

m c

For the sake of clarity, n of the occupied states in the circle diagrams is 1 less than that in the level diagrams.

Number of e = 48

D = 16 D = 19 D = 24

Critical field (No partially filled level at T = 0):

50ss B N

s = highest completely filled level

2

0.5D

L

2

D

BLD

B

Black lines are plots of n = s ρ B,n = N = 50 at B = Bs .

Red lines are plots of n = s N / ( N / ρ B ),n = N = 50 at N / ρ B = s .

100N

1

2n cE n

*c

e B

m c

1

1

2

s

T f cn

E D n

2

D

BLD

B

1 11

2 2cD s s s

Total energy in fully occupied levels:

21

2 cD s

Total energy in partially occupied level s + 1: 1

2T p cE s N sD

21 1

2 2T T f T p cE E E Ds s N sD 1

2 1 12 c N s D s s

21

2 c

ND

D

→2

2 *T T f

e NE E

m c

for1 1

s

sB B N

for

2

2

2 *

e NB

m c B

221

2 1 12 c

N D Ds s s

D N N

22

2 1 12 *

e N D Ds s s

m c N N

2

1 12 * s s s s

e N B B B B

m c B B B B

2

1

1 1 12 * s s

e N B B

m c B B

for

1

1 1 1

s sB B B

1

1 1 1

s sB B B

11s s s

D B

N s B s B s B

where

/s

B s

B N B

TE

B

2

1

1 1

2 *ss s

e NB

m c B B

2

1

1 1 12 *T

s s

e N B BE

m c B B

for

1

1 1 1

s sB B B

2

1 1

1 11 1

2 * s s s s

e N B B

m c B B B B

2

1 1

1 1 2

2 * s s s s

e N B

m c B B B B

2 *

e N

m c

2

11

1 1

2 *ss s

e NB

m c B B

2 *

e N

m c

2

1

1 1

s s D

L

B B N N

where 22

DS

S = extremal area of Fermi surface B

Section AA is extremal.Its contribution dominates due to phase cancellation effect.

Fermi Surface of Copper

Cu / Au

Monovalent fcc metal: n = 4 / a3 1/323Fk n 1/3212

a

4.90

a

Shortest distance across BZ = distance between hexagonal faces 23

a

2 Fk10.88

a

Band gap at zone boundaries → band energy there lowered → necks

Distance between square faces 12.57/a : necking not expected

Example: Fermi Surface of GolddHvA in Au with B // [110]: Dogbone

μ has period 210–9 gauss–1 for most directions →

22

1/D

SB

16 24.8 10 cm

Table 6.1: 8 11.2 10Fk cm → 16 24.5 10S cm

Period along [111] is 610–8 gauss–1 → 15 21.6 10S cm → neck

Dogbone area ~ 0.4 of belly area

Multiply-connected hole surface of Mg in bands 1 & 2

Magnetic Breakdown

~ free electron-like

→breakdown

Change of connectivity

Affected quantities ~ sensitive to connectivity : magnetoresistance

2c F gE

E.g., hcp metals with zero (small if spin-orbit effect included) gap at hexagonal zone boundaryMg: Eg ~ 10–3 eV, εF ~ 10 eV, breakdown :

510c eV 310B G