MTLE-6120: Advanced Electronic Properties of Materials Fermi theory - - PowerPoint PPT Presentation

MTLE-6120: Advanced Electronic Properties of Materials Fermi theory of metals

Contents:

◮ Fermi energy and related properties ◮ Electronic heat capacity ◮ Fermi surfaces ◮ Quantum drude model

Reading:

◮ Kasap: 4.6 - 4.7, 4.11

1

Band theory (vs. free electrons)

◮ Band energies E = En(

k) with complex dependence (vs. E = 2k2/(2m))

◮ Group velocity

vn( k) = ∇

kE/ (vs.

v = k/m)

◮ Effective mass tensor ¯

m∗

n(

k) = 2 ∇

k∇ kEn(

k) −1 (vs. m∗ = m)

◮ Density of states g(E) = n

• 2d

k (2π)3 δ(E − En(

k)) (vs. g(E) =

√ E 2π2

√

2m

• 3

)

◮ Gaps in energy, usually at high-symmetry points in Brillouin zone such as

• k = 0 or zone boundaries (vs. all E > 0 allowed)

◮ Metals if HOMO = LUMO and semiconductor/insulator if not

(vs. no gap ⇒ always metallic)

2

Fermi statistics

◮ At temperature T and chemical potential µ, each electronic state of energy

E has average occupation f(E) = 1 1 + exp E−µ

kBT ◮ In contrast, classical occupation exp µ−E kBT

1 µ-2kBT µ µ+2kBT E f(E) Fermi Classical

3

Electron number

◮ Number of states per energy per volume = g(E) ◮ Average occupation per state of energy E at temperature T = f(E) ◮ Average number of electrons per volume at temperature T, n =

• dEg(E)f(E)

◮ Classically, for a free electron gas

n = ∞ dE √ E 2π2 √ 2m

• 3

exp µ − E kBT = 1 2π2 √ 2m

• 3

exp µ kBT ∞ dEE1/2 exp −E kBT = 1 2π2 √ 2m

• 3

exp µ kBT Γ(3/2)

√π/2

(kBT)3/2 = 1 4

• 2mkBT/π
• 3

exp µ kBT

4

Chemical potential: classical

◮ Classically, at finite temperature T,

n = 1 4

• 2mkBT/π
• 3

exp µ kBT

◮ Electron number density given, chemical potential varies with temperature

µ(T) = −kBT ln   1 4n

• 2mkBT/π
• 3



◮ Classically, µ decreases with T as ∼ −T ln T, with µ → 0 as T → 0 ◮ This is the correct behavior for gases, but not electrons!

5

Electron number: quantum at T = 0

◮ Fermi function f(E) = 1/

• 1 + exp E−µ

kBT

• → 1 for E < µ− few kBT

and f(E) → 0 for E > µ+ few kBT.

◮ Therefore, for T → 0, f(E) → Θ(µ − E) ≡

• 1,

E < µ 0, E > µ

◮ Number of electrons at T = 0 is

n = µ dEg(E) (general) = µ dE √ E 2π2 √ 2m

• 3

(free electrons) = 1 3π2 √2mµ

• 3

⇒ µ = 2 2m(3π2n)2/3, a non-zero constant

6

Electron number: finite T corrections

For constant n, let’s find the change in µ for small changes in T from T = 0, n =

• dEg(E)f(E)

⇒ 0 = ∂n ∂T =

• dEg(E)

∂f ∂T + ∂f ∂µ · ∂µ ∂T

• =
• dEg(E)
• ∂

∂T

• 1

1 + exp E−µ

kBT

• + ∂

∂µ

• 1

1 + exp E−µ

kBT

• · ∂µ

∂T

• =
• dEg(E)

exp E−µ

kBT

• 1 + exp E−µ

kBT

2 E − µ kBT 2 + 1 kBT · ∂µ ∂T

• =
• dEg(E)

1 4kBT cosh2 E−µ

2kBT

• sharp peak at E = µ

E − µ T + ∂µ ∂T

• =
• dE g(µ) + g′(µ)(E − µ) + · · ·

4kBT cosh2 E−µ

2kBT

E − µ T + ∂µ ∂T

• 7

Electron number: finite T corrections (contd.)

0 = ∂n ∂T =

• dE g(µ) + g′(µ)(E − µ) + · · ·

4kBT cosh2 E−µ

2kBT

E − µ T + ∂µ ∂T

• ≡ 1

T (I1g(µ) + I2g′(µ)) + ∂µ ∂T (I0g(µ) + I1g′(µ)) where In ≡

• dE

(E − µ)n 4kBT cosh2 E−µ

2kBT

=      1, n = 0 0, n = 1 (πkBT)2/3, n = 2 Therefore ∂µ ∂T = − I2g′(µ) I0Tg(µ) = −(πkB)2g′(µ)T 3g(µ) ⇒ µ(T) = µ(0)−(πkBT)2g′(µ) 6g(µ) For g(E) ∝ √ E, g′(µ)/g(µ) = 1/(2µ) ⇒ µ(T) = µ(0)

• 1 − (πkBT)2

12µ(0)2

• 8

Electron chemical potential: typical numbers

◮ Sodium: BCC 4.23 ˚

A, 1 valence electron/atom ⇒ n ≈ 2.6 × 1028 m-3

◮ Aluminum: FCC 4.05 ˚

A, 3 valence electrons/atom ⇒ n ≈ 1.8 × 1029 m-3

◮ µ(0) = 2 2m(3π2n)2/3 ≈ 3.2 eV for Na, and ≈ 12 eV for Al ◮ In comparison, kBT ≈ 0.026 eV at 300 K and ≈ 0.26 eV at 3000 K

(> Tmelt)

◮ Therefore µ(T) = µ(0)

• 1 − (πkBT )2

12µ(0)2

• is

essentially constant over relevant T range!

◮ Zero temperature chemical potential µ(0) ≡ EF , Fermi energy

9

Properties at the Fermi energy

◮ Fermi energy EF separates occupied states and unoccupied states at T = 0 ◮ For free electrons, EF = µ0 = 2 2m(3π2n)2/3 ◮ With band structure E = En(

k), Fermi surface ≡ set of k with E = EF

◮ For free electrons E(

k) = 2k2/2m, the Fermi surface is a sphere of radius kF = √2mEF / = (3π2n)1/3

◮ Fermi velocity vF = average magnitude of group velocity on Fermi surface ◮ For free electrons, vF = kF /m ◮ Many electronic properties of metals determined by Fermi properties alone

(exclusively a function of electron density for free electrons)

◮ Fermi-energy density of states g(EF ) ◮ For free electrons, g(ǫF ) = √EF 2π2

√

2m

• 3

=

3n 2EF

10

Electronic heat capacity: classical

Average energy in free electron gas of density n: CV ≡ dU dT = d dT

• dEEg(E)f(E)

= d dT

• dEEg0

√ Ee(µ−E)/(kBT ) = d dT g0eµ/(kBT )

• dEE3/2e−E/(kBT )

= d dT g0eµ/(kBT )Γ(5/2)(kBT)5/2 n ≡

• dEg(E)f(E) = g0eµ/(kBT )Γ(3/2)(kBT)3/2

⇒ CV = d dT

• nΓ(5/2)(kBT)5/2

Γ(3/2)(kBT)3/2

• = 3

2nkB A constant which looks familiar because equipartition theorem!

11

Electronic heat capacity: quantum

CV =

• dEEg(E)

∂f(E) ∂T + ∂f(E) ∂µ · dµ dT

• (just extra E than in n)

=

• dE

Eg(E) 4kBT cosh2 E−µ

2kBT

E − µ T + dµ dT

• =
• dE µg(µ) + (µg(µ))′(E − µ) + · · ·

4kBT cosh2 E−µ

2kBT

E − µ T − I2g′(µ) I0Tg(µ)

• = µg(µ)

I1 T − I0 I2g′(µ) I0Tg(µ)

• + (µg(µ))′

I2 T − I1 I2g′(µ) I0Tg(µ)

• + · · ·

(In defined in finite T corrections to µ derivation) = 0 − µg′(µ)I2 T + (µg(µ))′ I2 T − 0 + · · · (since I1 = 0) = g(µ)I2 T + · · · = g(EF )π2k2

BT

3 + O(T 2) (general g(Ef)) = 3 2nkB π2kBT 3EF (free-electron g(EF ))

12

Electronic heat capacity: comparison

◮ Classical: CV = 3 2nkB (equipartition) ◮ Quantum: CV = 3 2nkB π2kBT 3EF

(equipartition)

◮ Quantum mechanical result reduced by factor ∼ kBT/EF because only

electrons near Fermi energy ‘participate’

◮ Same reason for relative constancy of µ in quantum case ◮ Electrons in metal behave classically only when kBT ∼ EF , which is

∼ 3 × 104 K for Na and ∼ 1.4 × 105 K for Al, i.e. never!

13

Electronic density of states: Al free electron

10 20 30

• 10
• 5

5 10 E-EF [eV] g(E) [eV-1nm-3] All states Occupied states (T = 300 K) Occupied states (T = 3000 K) Occupied states (T = 30000 K) ◮ Parabolic DOS, EF ≈ 11.8 eV: now plotted relative to E − EF ◮ Heat stored by moving electrons ∼ kBT below Fermi level by ∼ kBT ◮ Therefore, U ∝ T 2 and CV ∝ T ◮ Only narrow window around Fermi level participates even at Tmelt ◮ Resembles Maxwell-Boltzmann (classical) distribution only for kBT ∼ EF

14

Electronic density of states: real metals

DOS resembles free electrons for an energy window around Fermi level for best conducting metals (also the plasmonic metals)

0.1 0.2 0.3 0.4 DOS [1029 eV-1 m-3]

a) Al

PBEsol+U (this work) Lin et al. 2008 free electron 0.5 1 1.5 2 2.5

b) Ag

0.5 1 1.5

• 10
• 5

5 10 DOS [1029 eV-1 m-3]

ε-εF [eV]

c) Au

1 2 3

• 10
• 5

5 10

ε-εF [eV]

d) Cu

• Phys. Rev. B 91, 075120 (2016)

15

Electronic heat capacity: real metals

Linear heat capacity till kBT accesses difference from free electron model

2 4 6 8

Ce [105 J/m3K]

a) Al

• Eq. 3 (this work)

Lin et al. 2008 Sommerfeld 4 8 12

b) Ag

4 8 12 2 4 6 8

Ce [105 J/m3K] Te [103 K]

c) Au

5 10 15 20 2 4 6 8

Te [103 K]

d) Cu

• Phys. Rev. B 91, 075120 (2016)

16

Fermi surfaces: real metals

Fermi surface somewhat spherical for best conducting metals

ACS Nano 10, 957 (2016)

17

Fermi surface: density of states

◮ Consider arbitrary shaped equi-energy surfaces in k-space ◮ (For E = EF , that would be the Fermi surface) ◮ Let A(E) be the area in k-space of this surface (with elements dA) ◮ Number of states between E and E + dE is

dN = 2 (2π)3

• dAdk

= 2 (2π)3

• dA dk

dE dE = 2 (2π)3

• dA

1 v(k)dE = 2 (2π)3 A(E)dE 1 ¯ v(E) ≡ g(E)dE ⇒ g(E) = 2A(E) (2π)3¯ v(E)

18

Fermi surface: electronic current

◮ Apply electric field

E

◮ Force on electrons −e

E

◮ Average change in momentum −e

Eτ

◮ Average change in

k is K = −e

Eτ

• ◮ Effectively field displaces Fermi surface by

K

◮ Current without field must be zero

• j =
• d

k 2 (2π)3 Θ(EF − E)(−e v( k)) = 0

◮ Current only carried by difference between old and new Fermi surfaces! ◮ Normal displacement of Fermi surface ˆ

v · K, so

• j =

2 (2π)3

• dA(ˆ

v · K)(−e v)

19

Fermi surface: electronic conductivity

• j =

2 (2π)3

• dA(ˆ

v · K)(−e v) = 2 (2π)3 A(EF )(−e¯ v(EF ))ˆ v(ˆ v · K) = 2A(EF ) (2π)3 (−e¯ v(EF ))

• K

3 = (¯ v(EF )g(EF ))(−e¯ v(EF ))−e Eτ 3 = e2v2

F g(EF )τ

3

• E

⇒ σ = e2v2

F g(EF )τ

3

◮ Note that ˆ

v(ˆ v · K) = K/3 is for isotropic Fermi surface

◮ In general case

j is not parallel to K ⇒ σ is a tensor

20

Fermi surface conductivity vs. Drude model

◮ Fermi surface result:

σ = e2v2

F g(EF )τ

3

◮ For free electron model with Fermi wave-vector kF :

σ = 1 3e2 kF m 2 √EF 2π2 √ 2m

• 3

τ = 1 3e2 kF m 2 √2mEF / 2π2 √ 2m

• 2

τ = 1 3e2 2k2

F

m2 kF 2π2 2m 2 τ = e2τ m k3

F

3π2 = ne2τ m which is exactly the Drude model result!

◮ Conceptual difference: in Drude model all electrons contribute to σ ◮ In Fermi theory, only electrons near the Fermi surface do!

21