Free electron theory of metals

Electronic Properties of Solids R.J. Nicholas Electronic Properties: Combination of : • • • • • • • Metals Semiconductors Insulators Paramagnets Di...
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Electronic Properties of Solids R.J. Nicholas Electronic Properties:

Combination of :

• • • • • • •

Metals Semiconductors Insulators Paramagnets Diamagnets Ferromagnets Superconductors

Crystal Structure Atomic Structure

Free electron theory of metals • Metals are good conductors (both electrical and thermal) • Electronic heat capacity has an additional (temperature dependent) contribution from the electrons. • Why are some materials metals and others not?

Simple approximation: treat electrons as free to move within the crystal

Metals – HT10 – RJ Nicholas

1

Free electron theory of metals • Alkali metals (K, Na, Rb) and Noble metals (Cu, Ag, Au) have filled shell + 1 outer s-electron. • Atomic s-electrons are delocalised due to overlap of outer orbits. • Crystal looks like positive ion cores of charge +e embedded in a sea of conduction electrons • Conduction electrons can interact with each other and ion cores but these interactions are weak because:

(1) Periodic crystal potential (ion cores) is orthogonal to conduction electrons - they are eigenstates of total Hamiltonian e.g. for Na conduct. electrons are 3s states, but cores are n=1 and n=2 atomic orbitals. (2) Electron-electron scattering is suppressed by Pauli exclusion principle. Assumptions: (i) ions are static - adiabatic approx. (ii) electrons are independent - do not interact. (iii) model interactions with ion cores by using an “effective mass” m* (iv) free electrons so we usually put m* = me

Metals – HT10 – RJ Nicholas

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Free Electron Model

L Put free electrons into a very wide potential well the same size as the crystal i.e. they are 'de-localised'

Free electron properties Free electron Hamiltonian has only kinetic energy operator: Free electrons are plane waves

= 2 ∂ 2ψ Eψ = − 2m ∂x 2

ψ = A e ± ikx

with: Momentum: i=

∂ψ = ± =k ψ ∂x

Metals – HT10 – RJ Nicholas

Energy: −

= 2 ∂ 2ψ =2k 2 ψ = 2m ∂x 2 2m

Group velocity: ∂ω 1 ∂E =k = = ∂k = ∂k m

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Free Electron Model – Periodic boundary conditions

L

L

Add a second piece of crystal the same size: The properties must be the same.

Density of states Calculate allowed values of k. Use periodic (Born-von Karman) boundary conditions: L = size of crystal

ψ ( x) = ψ ( x + L) ∴ e ikx = e ik ( x + L ) ∴ e ikL = 1 2π 4π , ± L L 2π ∴δk = L

∴ k = 0, ±

Density of allowed states in reciprocal (k-) space is: ΔK in 1 − D δ k

Metals – HT10 – RJ Nicholas

or

ΔVK in 3 − D δ k3

x 2 for spin states

4

Density of states (2)

States have energies ε to ε + dε

g (ε )d ε = g (k )dk = 4π k 2 dk × dk dε dε

∴ g (ε )d ε = g (k )

=2 k 2 ε = 2m ⎛ 2mε ⎞ k =⎜ 2 ⎟ ⎝ = ⎠

= 1

dk ⎛ 2m ⎞ =⎜ ⎟ dε ⎝ =2 ⎠

2

1

= 2

1 2ε

1

2

8π ⎛ 2π ⎞ ⎜ ⎟ ⎝ L ⎠

k

3

4π ⎛ 2π ⎞ ⎜ ⎟ ⎝ L ⎠

2 δ k3

⎛ 2m ⎞ ⎜ 2 ⎟ ⎝= ⎠

2

1



2mε ⎛ 2m ⎞ ⎜ ⎟ =2 ⎝ =2 ⎠

3

⎛ 2m ⎞ = 4π ⎜ 2 ⎟ ⎝h ⎠

3

2

ε

1

1

2

2

1

1



2

1

2

ε

1

2



dε × V

Fermi Energy Electrons are Fermions

N =



∫ g (ε ) f

F −D

(ε ) d ε

0

μ

at T = 0

N =

∫ g (ε ) d ε 0

μ at T = 0 is known as the Fermi Energy, EF

Metals – HT10 – RJ Nicholas

N 8π ⎛ 2mEF ⎞ n= = ⎜ ⎟ V 3 ⎝ h2 ⎠ ⎛ 3N ⎞ EF = ⎜ ⎟ ⎝ 8π V ⎠

2

3

3

2

h2 2m

5

Typical value for EF e.g. Sodium (monatomic) crystal structure: b.c.c.

crystal basis: single Na atom

lattice points per conventional (cubic) unit cell: 2 conduction electrons per unit cell

2

∴ electrons per lattice point = 1 lattice constant (cube side) = a = 0.423 nm ∴ density of electrons n = N/V= 2/a3 = 2.6 x 1028 m-3 ∴ EF = 3.2 eV Fermi Temperature TF?

kBTF = EF

∴ TF = 24,000 K

Finite Temperatures and Heat Capacity

Fermi-Dirac distribution function fF-D = 1/(eE-μ/kBT + 1) electrons are excited by an energy ~ kBT Number of electrons is ≈ kBT g(EF) ∴ ΔE ≈ kB2T2 g(EF) ∴ CV = ΔE/ ΔT ≈ 2kB2T g(EF)

Metals – HT10 – RJ Nicholas

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∴ ln n = 3 ln E + const. 2 dn 3 dE ∴ = n 2 E dn 3 n = = g ( EF ) dE 2 EF

Previously we have n = AEF3/2

∴ C v = 3nk B

k BT EF

= 3nk B

T TF

∴ Heat Capacity is: (i) less than classical value by factor ~kBT/EF (ii) proportional to g(EF)

Is this significant? Lattice

Room Temperature Low Temperature

Electrons

3nat.kB

π2/2 nkB (kBT/EF)

12π4/5 nat.kB (T/ΘD)3

π2/2 nkB (kBT/EF)

C/T = βT2 + γ Debye term

Metals – HT10 – RJ Nicholas

free electron term

7

Rigorous derivation ∞

U = ∫ ε g (ε

)

∂f F − D ? ∂T

f F − D (ε ) d ε

0



∂U ∂f = ∫ ε g (ε ) dε ∂T ∂ T 0



2

= g ( EF ) k B T





− EF k BT

=

π2 3

f =

2 x

x e dx

(e

x

)

+ 1

2

+ δ

≈−∞

2

g ( EF ) k B T

1 ε − μ , x= e + 1 kT x

∂f − ex = ∂T ex + 1

(

)

δ ∝





− EF k BT

=

×

2

∂x ∂T

x e x dx

(e

x

)

+ 1

2

0 (why?)

Magnetic susceptibility • Susceptibility for a spin ½ particle is:

χ=

μ B2 μ0 kT

/ electron

• This is much bigger than is found experimentally - Why?

Metals – HT10 – RJ Nicholas

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Pauli paramagnetism Separate density of states for spin up and spin down, shifted in energy by ± ½gμBB (g=2) Imbalance of electron moments Δn Δn = ½ g(εF) × 2μBB giving a magnetization M M = μB Δn = μB2 g(εF) B and a susceptibility χ = M/H = μ0 μB2 g(εF) = 3nμ0 μB2 /2εF

k-space picture and the Fermi Surface T=0 states filled up to EF

= 2k 2 ∴ = EF 2m

Map of filled states in k-space = Fermi surface

∴ kF =

2mEF =2

N = 2×

4π k F3 3 ⎛⎜ 2π ⎞⎟ 3

or we can write:

⎝ L ⎠

∴ k F3 =

Metals – HT10 – RJ Nicholas

3π 2 N V

9

k-space picture and the Fermi Surface ∴

T=0 states filled up to EF

E

= 2k 2 = EF 2m

∴ kF =

2mEF =2

EF

k kF

How big is Fermi surface/sphere compared to Brillouin Zone? Simple cubic structure volume of Brillouin Zone = (2π/a)3 electron density n = 1/a3 volume of Fermi sphere = 4πkF3/3 = 4π3/a3 = half of one B.Z.

Metals – HT10 – RJ Nicholas

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Electron Transport - Electrical Conductivity Equation of motion: Force = rate of change of momentum =

∂k = − e (E + B × v ) ∂t

Apply electric field - electrons are accelerated to a steady state with a drift velocity vd - momentum is lost by scattering with an average momentum relaxation time τ ∴ momentum loss =

mvd

τ

∴ current j = nevd =

= −eE ne 2τ E m

ne 2τ ∴ conductivity σ = = neμ m

μ is mobility with: vd = μE

What happens in k-space? All electrons in k-space are accelerated by electric field: On average all electrons shifted by: δ k = − eEτ =

= δ k = Fδ t = − eE δ t E EF

k δk

Metals – HT10 – RJ Nicholas

kF

11

What happens in k-space? All electrons in k-space are accelerated by electric field:

= δ k = Fδ t = − eE δ t

On average all electrons shifted by:

δk = −

eEτ =

Fermi sphere is shifted in k-space by δk

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