Quantum Condensed Matter Physics · QCMP Lent/Easter 2019 3.2 Quantum Condensed Matter Physics 1....

3.1

Quantum Condensed Matter PhysicsLecture 3

David Ritchie

http://www.sp.phy.cam.ac.uk/drp2/homeQCMP Lent/Easter 2019

QCMP Lent/Easter 2019 3.2

Quantum Condensed Matter Physics

1. Classical and Semi-classical models for electrons in solids (3L)Lorentz dipole oscillator, optical properties of insulators. Drude model and optical properties of metals, plasma oscillations. Semi-classical approach to electron transport in electric and magnetic fields, the Hall effect. Sommerfeld model, density of states, specific heat of; electrons in metals, liquid 3He/4He mixtures. Screening and the Thomas-Fermi approximation. 2. Electrons and phonons in periodic solids (6L)3. Experimental probes of band structure (4L)4. Semiconductors and semiconductor devices (5L)5. Electronic instabilities (2L)6. Fermi Liquids (2L)

Sommerfeld Model – density of states• Free electron gas - Schrodinger equation:• Introduce eigenstates

satisfying periodic boundary conditions etc.• Allowed values of momentum are discrete: where 𝑛𝑛𝑥𝑥,𝑛𝑛𝑦𝑦,𝑛𝑛𝑧𝑧 are positive or negative integers


2 22 ( ) ( )m Eψ ψ− ∇ =r r

22( ) exp( ), / 2A i E mψ = ⋅ =k kr k r k

( ) ( )x L y z x y zψ ψ+ , , = , ,2 ( )x y zL n ,n ,nπ=k

• At zero temperature fill up Fermi sphere to the Fermi energy

• Each triplet of quantum numbers corresponds to 2 states – electron spin degeneracy

• volume in k-space

FE

3(2 )Lπ /

2 1 3(3 )Fk nπ /=

Sommerfeld Model – density of states• Number of occupied states in Fermi sphere:

• Hence if then• Also

• Density of states

• Hence

• Factor of 2 for spin degeneracy

• Often is given per unit volume so V disappears. QCMP Lent/Easter 2019 3.4

3

3

4 32(2 )

FkNLπ

π/

= ⋅/

2 1 3(3 )Fk nπ /=3/n N V N L= = /

2

3

Volume of shell in k space 4( ) 2 2Volume of k space per state (2 )

k dkg E dEV

ππ

−= ⋅ = ⋅

− /12

23 2 2 2

2( ) 2 4(2 )

V dk V m mEg E kdE

ππ π

= =

( )g E

2 2 2 2 2 3 23/ 2 (3 ) / 2 ln( ) ln( )+const

d 2 d d 3( )3 d 2

F F F

FF

F F F

E k m n m E nE n n ng EE n E E

π /= = ⇒ =

⇒ = ⇒ = =

Sommerfeld Model – electronic specific heat• Occupancy of states in thermal equilibrium – Fermi distribution:

• Chemical potential• Number density of particles• Energy density• At room temperature For metals a few eV hence

• From above

• Since the Fermi function is a step function is sharply peaked at the chemical potential

• Contributions to the specific heat only come from states within of the chemical potential, with each state having specific heat and we can guess that


( )1( )

1BE k Tf Ee µ− /=

+0 FT Eµ= ⇒ = 1 ( ) di

in N V f E g(E)f(E) E

V= / = =∑ ∫du Eg(E)f(E) E= ∫

( )( ) dv vf Ec u T Eg E E

T∂

= ∂ / ∂ | =∂∫

0.025Bk T eV≈B Fk T E

FE ≈

( )f ET

∂∂

Bk T

Bv B

F

k Tc n kE

≈Bk

Sommerfeld Model – electronic specific heat• To calculate this more accurately…. We take the density of

states as a constant so

• changing variables

• The number of particles is conserved so

• The first term in the square brackets is odd, is even so• To the same level of accuracy:


( ) ( )( ) d ( ) dvf E f Ec Eg E E g E E E

T T∂ ∂

= ≈∂ ∂∫ ∫

2

1( )( 1)

x

B xB

f e xx E k TT e T k T T

µµ ∂ ∂

= − / ⇒ = × + ∂ + ∂

2

( ) 10 ( ) d ( ) d( 1)

x

F F B xB

dn f E e xg E E g E k T xdT T e T k T T

µ∞

−∞

∂ ∂= = = × + ∂ + ∂

∫ ∫2( 1)

x

xe

e + 0Tµ∂∂

2

2 22 2

2

( )( ) d ( ) ( ) d( 1)

( ) d ( )( 1) 3

x

v F F B B x

x

F B B Fx

f E e xc g E E E g E k T k T x xT e T

x eg E k T x k Tg Ee

µ

π

∞

−∞

∞

−∞

∂= = +

∂ +

= =+

∫ ∫

∫

Sommerfeld Model – electronic specific heat

• Since we can write

• This result is of the same form as the equation above obtained from a simple argument but with a different prefactor - as opposed to 1.

• This calculation is the leading order term in an expansion in powers of .

• To next order the chemical potential is temperature dependent (see below) but because for metals we can usually ignore it.

• Examples: • Electron gas in solids – often much smaller than lattice specific heat• Liquid helium mixtures of 3He in 4He – near ideal Fermi gas


3( )2F

F

ng EE

=2 2 2 2

2 2 3( )3 3 2 2 2

Bv B F B B B

F F F

k Tn Tc k Tg E k T nk nkE E T

π π π π= = = =

2( )B Fk T E/

2 411 ( ) ( )3 2

BF B F

F

k TE O k T EE

πµ

= − + /

2 / 2π

B Fk T E

Specific Heat of mixtures of 3He and 4He


Helium dilution refrigerator

5mK

50mK

1.5K

0.7K

4K• Experimental procedure: • Cool helium mixtures to mK temperatures, • Isolate from surroundings• Input heat for given time• Measure temperature rise• Calculate specific heat at particular

temperatures and pressures

Polturak and Rosenbaum JLTP, 43, 477 (1981)

From wikipedia

Specific heat of mixtures of liquid 3He and 4He

3.9

( )2 2 2/32, 3

2 2F

v B FF B B

ETc nk T nT k mk

π π= = =

, ,vc T n FT

FT T

• From above

• Linear behaviour in Fermi gas regime

• So knowing we can calculate the Fermi temperature and the effective mass

• Effective mass 2.44 to 3.07 time bare 3He mass – due to interactions m

Polturak and Rosenbaum JLTP, 43, 477 (1981) QCMP Lent/Easter 2019

Screening and Thomas-Fermi approximation• Placing a positive charge in a metal will result in electrons moving around

to screen its potential resulting in zero electric field.• This is quite different from a dielectric where electrons are not able to

move freely and the potential is reduced by dielectric constant• In a classical picture electrons can move anywhere, but quantum

mechanics dictates this is not possible - an electron cannot sit right on top of a nucleus.

• In metals a balance is reached between minimising potential and kinetic energy, screening over a short but finite distance.

• We estimate the response of a free electron gas to a perturbing potential. )𝑉𝑉0(𝐫𝐫 is the electrostatic potential, )𝜌𝜌0(𝐫𝐫 the charge distribution.

• Consider the positive background charge to be homogeneous with the electron gas moving around - plasma or “Jellium” model and in this case

)𝜌𝜌0(𝐫𝐫 = 0 everywhere (this does not include the charges used to set up the perturbing potential).


2 00

0

( )( )V ρ∇ = −

rr

Screening and Thomas-Fermi approximation

• In the presence of a perturbing potential the electron charge density redistributes which changes the potential , . The changes are related by:


( )extV r0( ) ( ) ( )ρ ρ δρ= +r r r

0

( )2 ( )V δρδ∇ = − rr

22 ( ) ( )( ( ) ( )) ( ) ( )

2 exte V V Em

ψ δ ψ ψ− ∇ + − + =r r r r r

• We link the charge redistribution to the applied potential by assuming the perturbing potential shifts free electron energy levels – the same as assuming a spatially varying Fermi energy. This is the “Thomas-Fermi” approximation.

• The potential is the total produced by the added external charge and the induced “screening” charge , hence:

( )δρ r

tot extV V Vδ= +

( ) ( ) ( )0V V Vδ= +r r r

Screening and the Thomas-Fermi approximation• Assume the induced potential is slowly varying on the scale of the

Fermi wavelength so the energy eigenvalues are just shifted by potential as a function of position:

where has a free electron parabolic dispersion• Keeping the electron states filled up to a constant energy means we

adjust the local Fermi energy as measured from the bottom of the band so:


( ) ( ) ( )0 totE E eV, = −k r k r( )0E k 2 2

2km

2 Fkπ /

µ( )FE r

( ) ( )F totE eVµ = −r r• A small shift in the local Fermi

Energy leads to a change in the local electron number density, n.

• And from above so we have:

( ) ( )V F F V F totn g E E eg E Vδ δ= =tot extV V Vδ= +

( )( )V F extn eg E V Vδ δ= +

Screening and the Thomas-Fermi approximation• Since the added potential and induced electron number density are small

we can use Poisson’s equation to write:

• We can calculate the induced potential and density response using Fourier transformation. Assume an oscillatory perturbing potential : and a resulting oscillatory induced potential: substituting into the equation above:

• Where we define the Thomas–Fermi wavevector:


2

0 0

( )( ) ( ( ) ( ))2

Vext

e g EeV n V Vδ δ δ∇ = = +Fr r r

( )ext extV V .= iq rq e( )V Vδ δ .= iq rq e

( )2 2

0

20 0

02

0

( )( ) ( ) ( ( ) ( ))

( ) ( ) / ( ) ( ) /

( ) /( ) ( ) ( )( ) /

2i i iV F

ext

2 2V F ext V F

2 2V F TF

ext ext2 2 2V F TF

e g EV e q V e V V e

V q e g E V e g E

e g E qV V Vq e g E q +q

δ δ δ

δ

δ

⋅ ⋅ ⋅∇ = − = +

⇒ + = −

⇒ = − = −+

q r q r q rq q q q

q q

q q q

122

0( )TF V Fq e g E

= /

Screening and the Thomas-Fermi approximation• The Thomas-Fermi wavevector , and given that for

the free electron gas:

• We obtain:

• where the Bohr radius isand the Wigner-Seitz radius, rs is defined by

• To find the induced electron number density – from above we have


122

0( )TF V Fq e g E

= /

222 1

2 2 1/20

1 4 2 95ÅF

TF FB s

kmeq ka rπ π

− .= = =

2

02

4 0 53ÅB mea π= .

3 1(4 3) Sr nπ −/ =

( ) 2 2

2 2 2 2

3/2 1/22122

( ) , ( )Fkm mV F V F Fmg E E E g E k

π π= = ⇒ =

( ) 2 20

20 0

( ) , ( ) ( ), ( )

( ) ( ) ( ) 1 ( )/ 1

2TF

V F ext ext TF V F2 2TF

2 2TF TF

ind ext ext2 2 2 2TF TF

qn eg E V V V V q e g Eq +q

q q qn n V Ve q +q e q q +

δ δ δ

δ

= + = − = /

⇒ = = − =

q q

q q q q

The Thomas-Fermi dielectric function• The wavevector dependent dielectric function relates the electric

displacement D to the electric field E by • given:

• Since from above

• Using

• And hence the “Thomas Fermi dielectric function” is given by:

• is the Thomas-Fermi screening length, for copper where the electron density we have .


( )q0 ( ) ( ) ( )=q E q D q

0( ) / , ( ) ( )ext tot ext totV V V V∇ = − ∇ = − ⇒∇ = ∇D q E q q

2

( ) ( ) ( ) ( )2TF

ext tot ext2 2 2 2TF TF

q qV V V Vq +q q +q

δ = − ⇒ =q q q q

( )( ) ( ) ( ) ( )tot

tot ext ext ext

V

V V V V V Vδ δ= + ⇒ = +q q q q

2( ) 12

TF TFqq

= +q 1

TFq−

22 38.5 10 cmn −= × 1/ 0.055nmTFq =

Thomas-Fermi screening• From last slide• For small (long distances)• Long range part of Coulomb potential

also so it is exactly cancelled• In real space if (Coulombic

and long range) then is the short range screened potential.

2( ) 1 /TF 2TFq q= +q

2TF q−∝q

2q−∝extV Q r= /

( ) ( ) TFq rV r Q r e−= /

(problem sheet 1 question 6)• The screened potential is known as the “Yukawa potential” in particle physics• Exponential factor reduces range of Coulomb potential – screened over

distances comparable to inter-particle spacing• Mobile electron gas highly effective at screening external charges.• Application to resistivity of alloys – atoms of Zn (valency 2) added

substitutionally to metallic copper, (valency 1) has an excess charge.• Foreign atom scatters conduction electrons with interaction given by

screened Coulomb potential – scattering contributes to increase in in resistivity, theory and experiment in agreement.

r0 5

Pote

ntia

l ene

rgy

-0.1

-0.2

-0.3

exp( )krr−

1r

screenedunscreened

In this graph screening length 1/k=1

3.16QCMP Lent/Easter 2019

Summary of Lecture 3

• The Sommerfeld model – electrons in a degenerate Fermi gas• Free electron gas in three dimensions • Fermi surface and density of states• Thermal properties of the Fermi gas – specific heat• Experimental measurements of specific heat in liquid helium.• Screening and the Thomas-Fermi approximation, • Thomas-Fermi wavevector and dielectric function• Effect of screening on a Coulomb potential


3.18

Quantum Condensed Matter PhysicsLecture 3

The End

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Quantum Condensed Matter Physics · QCMP Lent/Easter 2019 3.2 Quantum Condensed Matter Physics 1....

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