quantum condensed matter physics...types of magnetic interactions; direct exchange, heisenberg...

1.1

Quantum Condensed Matter PhysicsLecture 1

David Ritchie

http://www.sp.phy.cam.ac.uk/drp2/homeQCMP Lent/Easter 2021

QCMP Lent/Easter 2021 1.2

Quantum Condensed Matter Physics: synopsis (1)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)Types of bonding; Van der Waals, ionic, covalent. Crystal structures. Reciprocal space, x-ray diffraction and Brillouin zones. Lattice dynamics and phonons; 1D monoatomic and diatomic chains, 3D crystals. Heat capacity due to lattice vibrations; Einstein and Debye models. Thermal conductivity of insulators. Electrons in a periodic potential; Bloch’s theorem. Nearly free electron approximation; plane waves and bandgaps. Tight binding approximation; linear combination of atomic orbitals, linear chain and three dimensions, two bands. Pseudopotentials. Band structure of real materials; properties of metals (aluminium and copper) and semiconductors.

Semi-classical model of electron dynamics in bands; Bloch oscillations, effective mass, density of states, electrons and holes in semiconductors


Quantum Condensed Matter Physics: synopsis (2)

3. Experimental probes of band structure (4L)Photon absorption; transition rates, experimental arrangement for absorption spectroscopy, direct and indirect semiconductors, excitons. Quantum oscillations; de Haas-Van Alphen effect in copper and strontium ruthenate. Photoemission; angle resolved photoemission spectroscopy (ARPES) in GaAs and strontium ruthenate. Tunnelling; scanning tunnelling microscopy. Cyclotron resonance.

Scattering in metals; Wiedemann-Franz law, theory of electrical and thermal transport, Matthiessen’s rule, emission and absorption of phonons. Experiments demonstrating electron-phonon and electron–electron scattering at low temperatures.


Quantum Condensed Matter Physics: synopsis (3)

4. Semiconductors and semiconductor devices (5L)Intrinsic semiconductors, law of mass action, doping in semiconductors, impurity ionisation, variation of carrier concentration and mobility with temperature - impurity and phonon scattering, Hall effect with two carrier types.

Metal to semiconductor contact. P-n junction; charge redistribution, band bending and equilibrium, balance of currents, voltage bias. Light emitting diodes; GaN, organic.

Photovoltaic solar cell; Shockley-Queisser limit, efficiencies, commercialisation. Field effect transistor; JFET, MOSFET. Microelectronics and the integrated circuit.

Band structure engineering; electron beam lithography, molecular beam epitaxy. Two-dimensional electron gas, Shubnikov-de Haas oscillations, quantum Hall effect, conductance quantisation in 1D. Single electron pumping and current quantisation, single and entangled-photon emission, quantum cascade laser.


Quantum Condensed Matter Physics: synopsis (4)5. Electronic instabilities (2L)The Peierls transition, charge density waves, magnetism, local magnetic moments, Curie Law. Types of magnetic interactions; direct exchange, Heisenberg hamiltonian, superexchange and insulating ferromagnets, band magnetism in metals, local moment magnetism in metals, indirect exchange, magnetic order and the Weiss exchange field.

6. Fermi Liquids (2L)Fermi liquid theory; the problem with the Fermi gas. Liquid Helium; specific heat and viscosity. Collective excitations, adiabatic continuity, total energy expansion for Landau Fermi liquid, energy dependence of quasiparticle scattering rate.

Quasiparticles and holes near the Fermi surface, quasiparticle spectral function, tuning of the quasiparticle interaction, heavy fermions, renormalised band picture for heavy fermions, quasiparticles detected by dHvA, tuning the quasiparticle interaction. CePd2Si2 ; heavy-fermion magnet to unconventional superconductor phase transitions.

•Course material will be useful for several part III courses.•Printed overheads & problem sheets provided.•All available in pdf on web: http://www.sp.phy.cam.ac.uk/drp2/home


Books

1. Band Theory and Electronic Properties of Solids, Singleton J (OUP 2008)

2. Optical properties of Solids, Fox M (2nd edn OUP 2010)

3. The Oxford Solid State Basics, Simon S H (OUP 2013)

4. Introduction to Solid State Physics, Kittel C (8th edn Wiley 1996)

5. Solid State Physics, Ashcroft N W and Mermin N D, (Holt, Rinehart and Winston 1976)


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)


Optical properties of insulators • Response to high frequency electric field in electromagnetic waves.

• Wavelength long compared to interatomic spacing.

• Classical picture – Lorentz dipole oscillator model

• Model atoms as nucleus + electron cloud.

• Applied electric field causes displacement of electron cloud,

• Assume restoring force is proportional to displacement.

No EM wave

electron cloud

nucleus

EM wave – electron cloud oscillates about nucleus displacement u.

+u-u

12

tf

=0t =

• Electron cloud behaves as damped harmonic oscillator

• ωT natural frequency, determined by force constant and mass; γ damping rate (no model for this yet…)

• Consider oscillating electric field , which induces oscillating displacement

• Resulting dipole moment per atom at angular frequency ωT : • Polarisation = dipole moment per unit volume

• From and the equation of motion we obtain for the polarisability:

• Frequency dependence of 𝜒𝜒𝜔𝜔 typical of harmonic oscillator


Optical properties of insulators

2Tmu m u m u qEγ ω+ + =

( ) i tE t E e ωω

−=( ) i tu t u e ω

ω−=

p quω ω=

0P Eω ω ωχ=

2 2

2 2 2 20 0( ) ( )T T

N q qnV m i m iωχ ω ω ωγ ω ω ωγ

= =− − − −

/P p N V npω ω ω= =

• Explains why different materials have very different static permittivities• Reflectivity between media of different permittivities: , • Power reflection coefficient given by:• In a solid polarization fields of other atoms can alter resonant frequency

Optical properties of insulators• Permittivity from model:• Figure shows typical

frequency dependence• Analogy with damped SHO

tells us that power absorbed by electron cloud is determined by )Im(𝜖𝜖𝜔𝜔 :

• Simple way to think of absorption lines in optical spectra

• At low frequencies:


2102 Im( )P Eω ωω= | |

1 2

1 2r −

+=

2

2 20

1 1( )T

qnm iω ωχ ω ω ωγ

= + = +− −

( ) 2 200 1 Tnq mω ω→ = +

2R r=

• Absorption coefficient and refractive index of sodium gas

• Atom density is • is the off resonance

refractive index• Absorption due strongest

hyperfine component of the D2 line at so

• Linewidth hence

• Very narrow…..


αn

7 31 10 m−×

Example – atomic absorption line

Taken from Optical properties of solids by M Fox

0n

589nmλ =145.1 10 Hzν = ×100MHzν∆ ≈

7/ 2 10ν ν −∆ = ×

Lorentz oscillator model: connection to quantum mechanics


1( ) ( )b a b aE E i E Eωχ ω δ ω−∝ − − + − −

T b aE Eω = −

• This classical oscillator model cannot be the whole story – works as a phenomenological description of optical response function

• For two sharp levels, from time-dependent perturbation theory

• Where the imaginary part is the transition rate

• This can also be written

where is very small

12b aE E iω ω γχ − − − /∝

γ

• As energy levels broaden into bands and increases, this expression becomes similar to Lorentz model close to resonance where

• We multiply by on top and bottom and approximate: hence:

γ

2 2 2 2

2( ) 2

T T

T T Ti iωω ω ωχ

ω ω ω ω γ ω ω ωγ+

∝− − + / − −

Tω ω+ 2T Tω ω ω+ ≈

• Lorentz oscillator model, superposition of spectra • May have a number of allowed transitions at energies• Usually high frequency – terahertz to ultraviolet • Resulting frequency-dependent permittivity from adding responses

associated with each transition:


Optical properties of insulators

1 2......T Tω ω

( ) 1 ( )iω χ ω= +∑

Perm

ittiv

ity

QCMP Lent/Easter 2021

Optical properties of insulators – comparison with experiments

( )( )

Re

Im2

n

nκ

≈

≈

For weakly absorbing medium with

• Fused silica glass SiO2• Expected general

characteristics observed• Transparent in visible light• Strong absorption peaks in

infrared and ultraviolet• IR peaks due to vibrations of

SiO2 molecules• UV absorption background

across 10eV bandgap• UV absorption peaks caused

by inner core electron transitions in Si and O Visible

light

Infra-red Ultra-violet X-rays

n κOptical properties of Solids, M Fox 1.14

• Temporal broadening of pulse of spectralwidth in a dispersive medium length

• Lorentz model gives above one absorption line and below next absorption line

• Choosing we minimise dispersion


Optical properties of Solids, M Fox

1/ pf tδ ≈

2

2

d nLc dλτ λ

λ∆ = ∆

2 2d / d 0n λ <2 2d / d 0n λ >

2 2d / d 0n λ =

• SiO2 refractive index in more detail• n increases with frequency – ‘normal

dispersion’ caused by tails of absorption peaks in IR and UV

• Can be used to separate colours of light using a prism

• Short light pulses of duration have spread of frequencies

• Different velocities of frequency components causes problems in high speed optical fibre communications

1.3 mλ µ• For SiO2 this is at – a preferred optical fibre wavelength

pt

n

λ∆

Dispersion of light

L

Drude Model

• Assume we have a gas of electrons free to move between positive ion cores

• These electrons are only scattered by the ion cores

• Between collisions with the ion cores the electrons do not interact with each other.

• Collisions are instantaneous resulting in a change of electron velocity.

• The probability an electron has a collision in unit time is , the scattering rate.

• Electrons achieve thermal equilibrium with their surroundings only through collisions.


1τ −

taken from wikipedia

Drude Model - relaxation time approximation

• The current density J due to electrons of number density n, mass m, of average velocity v and momentum p is given by:

• Consider the evolution of p in time under the action of an external force

• Probability of a collision during is where is the average time between collisions.

• Probability of no collision during is• For electrons that have not collided momentum increases:

• So the contribution to the average momentum during for electrons that have not collided is:


nenem

= − = −J v p

tδ /tδ τ τ

tδ 1 /tδ τ−

tδ( )tf

2( ) ( )t t tδ δ δ= +Οp f

2( ) (1 / )( ( ) ( ) ( ) )t t t t t t O tδ δ τ δ δ+ = − + +p p f

tδ

Drude Model - relaxation time approximation• Electrons which have collided are a fraction of the total.• The momentum they will have acquired since colliding (where their

momentum was randomised) is• So contribution to average momentum for electrons which have collided is

of order - small• For momentum as before:

• If we can rearrange this to give

• Hence the collisions produce frictional damping.• Apply to electrical conductivity defined by , assume steady

state so: and the force on an electron is • From above• Hence


( )t tδf

/tδ τ

2( )tδ

2( ) (1 / )( ( ) ( ) ( ) )t t t t t t O tδ δ τ δ δ+ = − + +p p f0tδ →

d ( ) ( ) ( )d

t t tt τ

= − +p p f

σ=J Ed ( )

d 0tt =p

2( )( ) t m net ene m

τστ τ

= = − = − ⇒ =pf J E

( )t e= −f Eσ

nem= −J p

Summary of Lecture 1

• Introduction to course and recommended text books• Lorentz oscillator model for optical absorption in solids• Comparison with atomic absorption• Comparison of Lorentz model with experimental results• The Drude model of electron motion in solids• The relaxation time approximation, scattering and electrical

conductivity




The End

http://www.sp.phy.cam.ac.uk/drp2/home



David Ritchie

http://www.sp.phy.cam.ac.uk/drp2/home

Drude model

• From above:

• In magnetic and electric fields: and• Hence the same equation in different forms:


( )q= + ×f E v B

( )

( )

( )

( )

2

2

d ( ) ( )d

d ( ) ( )d

d ( ) ( )d

d ( ) ( )d

t t qtt t q

t mt t nqt mt t nq qt m m

τ

τ

τ

τ

= − + + ×

= − + + ×

= − + + ×

= − + + ×

p p E v B

v v E v B

j j E v B

j j E j B

nq=j v

d ( ) ( ) ( )d

t t tt τ

= − +p p f

Drude model: Frequency dependant conductivity

• From above

• If B=0: and if

• Then

• And given we have

• Where defines the Plasma frequency.

• At low frequencies we get

• Where is defined as the carrier mobility


2d ( ) ( )d

t t nqt mτ+ =

j j E ( ) ( )exp , expi t i tω ωω ω= − = −j j E E

σ=j E

( )2d ( ) ( )

dt t nq qt m mτ

= − + + ×j j E j B

( ) ( ) ( )2

2 21

1 1nqm

nq nqim im i

ω ωω ω ω

ττ ωωττ ω

−−

− = ⇒ = =−−

E Ej E j

( ) ( )

220

1 1pnq

m i iω

ε ω ττσωτ ωτ

= =− −

2

0 0,nq q nqm mτ τσ µ σ µ= = ⇒ =1ωτ

22

0p

nqm

ω =

µ

Optical properties of metals: connection to ac conductivity

• Current density (velocity x density x charge) • Polarisation Hence, for the conduction electrons• Adding in the polarisation of the core electrons as - the background

polarisability we get:• At angular frequency ω, substituting:

• We get• Hence

• This relates the imaginary part of the permittivity to the real part of the frequency dependant conductivity.

• Combining this with our expression from above we obtain:


nq=j un q=P u c =P j

0χ∞= +P j E χ∞

0exp( ), exp( ), exp( ),i t i t i tω ω ω ω ω ωω ω ω χ= − = − = − =E E j j P P P E0 0 0( ) ( )i iω ω ω ω ω ω ω ωω χ χ ω χ χ σ∞ ∞= − − = − − =j E E E E

0 00

( ), ( ) ii i ωω ω ω ω ω

σσ ω χ χ σ ωω∞ ∞ ∞= − − = − − ⇒ = +

( )

2 2

21 /p pii iω

ω τ ωω ωτ ω ω τ∞ ∞= + = −

− +

( ) 120 1p iωσ ε ω τ ωτ −= −

• Use Lorentz oscillator model for bound electrons giving background• Use Drude model for conduction electrons, zero restoring force

• diverges as so metals are highly reflecting at low frequency• Peak in at low ω, due to enhanced absorption, ‘Drude peak’:• crosses zero at and approaches 1 at a high frequency so metals

become transparent in the ultraviolet.

Optical properties of metals


0Tω⇒ =

Im( )ω( )ω| | 0ω →

( )ω Pω∗

2

2 20

2

20

2 22

20

, 0,( )

( )

,( )

TT

pp

nqm i

nqm i

nqi m

ω

ω

χ ωω ω ωγ

ω ωγ

ωω

ω ωγ

∞

∞

= =− −

⇒ = −+

= − =+

∞

Drude Peak Resonance due to core electrons

Optical properties of metals• Reflectivity at the interface between two media using the Drude model• If permeability is unchanged at interface: • Power reflection coefficient

• Plateau in R at low frequency, related to conductivity of material • at high frequency – blue line on figure• If background permittivity - due to polarisability of core electrons is

significant, R can go to zero at finite frequency – red lineQCMP Lent/Easter 2021 2.7

1 2

1 2

r−

=+

2R r=

4R ω−∝∞

Pow

er re

flect

ion

coef

ficie

nt R

4( / )pω ω∝

1∞ =

5∞ =

Experimental reflectivity of Aluminium

• Experimental Reflectivity of Aluminium as a function of photon energy - green curve, (solid)

• Reflectivity above 80% for visible region of spectrum - aluminium coating is used for commercial mirrors

• Plasma frequency in ultra-violet• Both theory curves assume we

have• Red (dashed) no damping


15.8eVpω =

158.0 10 sτ −= ×

1.00.80.60.40.20.0

Ref

lect

iviti

y

0 5 10 15 20Energy (eV)

Experimental data

Theory

Reflectivity of Aluminium

Data from Ehrenreich et al (1963)

• Blue (Dotted) with value deduced from DC conductivity values - slightly better fit

• Two unexplained features in experimental results: (1) reflectivity is smaller than predicted (2) small dip in reflectivity around

• Both explained by considering interband absorption rates 1.5eV

15.8eVpω =

Optical properties of Solids, M Fox

Optical properties of metals: plasma oscillations• Part IB electromagnetism – “Plasma Oscillations”• electrons moving in a positively charged environment - model for metal. • Consider probing a slab of material by applying an oscillating field D:

• In metals and usually so peaks and • More generally as at we get Plasma Oscillations at a

frequency defined by • Polarisation causes build-up of surface charge, which generates the

restoring force driving the oscillations. The electrons slosh back and forth.


0pω ≈

2 20/p ne mω =

1∞ =1/2

pω ω −∞=

( )10 0

2

2

21

2 2

1

( )

1

p

p

ii

i

ω ω

ω

ω

ωω ωγ

ω ωγω ω ωγ

−

∞

−∞

= = + ⇒ = −

= −+

+≈ ⇒ =

− +

D E E P P D

1pω γ τ −= 𝜖𝜖𝜔𝜔−1

0ω →

Optical properties of metals: plasma oscillations and electron energy loss spectroscopy in Ge and Si

• Incoming electron at wavevector 𝑞𝑞, energy ℏ𝜔𝜔. Outgoing electron at wavevector 𝑘𝑘, energy ℏ𝜈𝜈. Plasmon generated with energy ℏ 𝜔𝜔 − 𝜈𝜈

• Response to oscillating applied D field given by

• Resonance at plasma frequency width 𝜏𝜏−1.

21

2 2p

iiω

ω ωγω ω ωγ

− +=

− +

22

0p

nem

ω = 2.10

EELs

From optical absorption


Transport in electric and magnetic fields

• Previously

• If B is parallel to z-axis, taking components

• Steady state

• Hall effect: current confined to x-axis by transverse field due to charge build up.

• Hall coefficient gives carrier density and sign of charge

• If then carrier ‘mobility’ • Mobility defined as velocity of charge carrier per unit electric field.• since• Cyclotron frequency defined as QCMP Lent/Easter 2021 2.11

1dd

1dd

( ) ( )

( ) ( )

qy y xt m

qx x yt m

j nqE Bj

j nqE Bj

τ

τ

−

−

+ = −

+ = +dd 0t =

j

( ), ( )q qy y x x x ym mj nqE Bj j nqE Bjτ τ= − = +

eBc cm Bω ω τ µ= ⇒ =

( 0)yj =

1yH

x

ER

j B nq= =

( )2d ( ) ( )

dt t nq qt m mτ

= − + + ×j j E j B

x x

x x

j vq qx xm nqE E mj nqEτ τ µ= ⇒ = = =

2nqm nqτσ σ µ= ⇒ = =j E

0B =

(E )y

Hall effect in metals• First measured 1879 E H Hall• From the last slide• Drude theory predicts is

independent of and • In metals however it is found that

does vary with magnetic field as well as temperature and sample purity

• Measuring pure samples at low temperatures and high magnetic fields (1T) limiting values are obtained.

• Comparison made between observed and predicted number of free electrons per atom for a range of metals

• Theory seems to work quite well for alkali metals & noble metals but not for the others - new theories needed!


1/HR nq=HR

B τHR

Metal Valence -1/RhneLi 1 0.8Na 1 1.2K 1 1.1Cu 1 1.5Ag 1 1.3Au 1 1.5Be 2 -0.2Mg 2 -0.4In 3 -0.3Al 3 -0.3

0.01 0.1 1.0 10 100 1000

1/ hR ne−

Bωτ µ=

0.33−

aluminiumR Lück, Phys Stat. Sol. 18, 49(1966)

Scanning Hall Probe Microscopy• First report Chang et al Appl Phys Lett 61, 1974 (1992)• Hall sensor mounted on scanning system to measure local magnetic field• Magnetic field sensitivity spatial resolution • Now commercial product from Attocube, Nanomagnetics, Magcam……


510 T− 0.35 mµ

Problems with Drude Theory• Drude model predicts the electronic head capacity to be from equipartition

of energy independent of T.• Measured heat capacity falls far below that expected from equipartition

theorem and is temperature dependent.

• Correct description: degenerate Fermi gas (cf. Stat. Phys. course). • Interpret velocity in Drude model as drift velocity, averaged over many

particles. Individual electrons actually travel at up to 1% of speed of light!


u

32 kel BC n=

Drude Model – is it valid?• The Drude model is very crude (devised only 3 years after discovery of

electron by J J Thomson in 1897!)– assumes electrons have random (or 0) momentum after a collision– assumes scatterers are all the positive ions (but mean free path can be

much longer than atomic spacing)– no concept of QM, Pauli exclusion principle, fermions, …

• We should instead consider a Fermi sphere of electrons occupying all the states in k-space up to the Fermi energy EF– only two electrons in each state (two spins)– only electrons near Fermi surface can gain energy or scatter– Fermi-Dirac function gives probability of finding electron in a state– distribution function is modified by electric field, which provides

momentum, so there are more electrons moving in one direction than in the opposite one


ff

Drude model – its replacement• Describe using Boltzmann transport formalism

(Part III major option AQCMP)– scattering causes total momentum to decay, same results

• Various types of scattering in metals (and semiconductors):– phonons (dominate at room temperature T)– impurities and lattice defects (important at low T)– other electrons (surprisingly unimportant; total momentum

conserved so no effect on current)– we will see that the ions themselves have no effect provided that

they are in a periodic lattice (band structure)



• The Drude model – frequency dependant conductivity• Optical properties of metals• Experimental reflectivity of aluminium• Plasma oscillations, electron energy loss spectroscopy• Hall effect from Drude model• Hall effect in metals• Applications of Hall effect – scanning Hall probe microscopy• Problems with Drude theory


2.18


The End


3.1


David Ritchie


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 2021 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 2021

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 2021


• 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


The End


4.1


David Ritchie




1. Classical and Semi-classical models for electrons in solids (3L)2. Electrons and phonons in periodic solids (6L)Types of bonding; Van der Waals, ionic, covalent. Crystal structures. Reciprocal space, x-ray diffraction and Brillouin zones. Lattice dynamics and phonons; 1D monoatomic and diatomic chains, 3D crystals. Heat capacity due to lattice vibrations; Einstein and Debye models. Thermal conductivity of insulators. Electrons in a periodic potential; Bloch’s theorem. Nearly free electron approximation; plane waves and bandgaps. Tight binding approximation; linear combination of atomic orbitals, linear chain and three dimensions, two bands. Pseudopotentials.. ……….

3. Experimental probes of band structure (4L)4. Semiconductors and semiconductor devices (5L)5. Electronic instabilities (2L)6. Fermi Liquids (2L)

4.2

From atoms to solids – the binding of crystals• Cohesion due to interaction between electrons and nuclei giving rise to

effective interaction potential between atoms• Several different types of bonds – firstly Van der Waals• Example: Inert gases – filled electron shells, large ionization energies• Electron configuration in solid similar to that in separated atoms• Atoms neutral – interaction weak, due to Van der Waals interaction• Consider atom as oscillator – electrons fluctuate around nucleus• Zero point fluctuations cause dipole moment , a second atom at distance

experiences induced electric field • This field induces a dipole at the second atom where

is the atomic polarizability• The second atom induces an electric field at the first

• The energy of the system changes by:• Induced interaction and is always attractive • Note - Energy depends on and not which is zeroQCMP Lent/Easter 2021 4.3

1p R3

1 /E p R∝3

2 1 /p p Rα∝ α

3 61 2 1E p R p Rα∝ / ∝ / .

2 61 1 1U p E p Rα∆ = − ⋅ ∝ − / .

21p< > 2

1p< >

21p∝< >

From atoms to solids – Van der Waals bonding• If atoms move together so the electron charge distributions overlap they

repel each other• Repulsion due to electrostatic forces and the Pauli exclusion principle

which prevents electrons having the same quantum numbers• For example if we try to force 2 spin parallel electrons into the same 1s

state in H, one will go into the 2s state at a large energy cost• Calculations of repulsive interactions is complex

there is a short range (hard core) potential• Common empirical “Lennard-Jones” potential

• are constants depending on atoms involved• Except He, Inert gases form close-packed face-centered cubic solids with

high coordination nos. (10-12), low cohesive energies and melting points• He is special – due to zero point motion does not solidify even at absolute

zero, unless pressurised to 30 Bar!


12 612 6min min( ) 4 2R RV r

r r r rσ σε ε

= − = −

,ε σ

Van de Waals bonding for low-dimensional structures


Liu et al, Nature Materials 1, 16042 (2016)

From atoms to solids – Ionic bonding• Atoms with electronic configuration close to a filled shell will tend to gain or

lose electrons to fill the shell• Energy for reaction in gas phase is ionization energy• Energy for reaction in gas phase is electron affinity • To form ionic molecules it costs an energy but the electrostatic

potential energy between charges is reduced by a greater amount. • Electrostatic interaction for a diatomic crystal:

• Where is sum of all Coulomb energies between ions. • If the system is on a regular lattice with constant then we can write:

where the Madelung constant depends on structure NaClCsCl cubic ZnS or Zincblende

• We must add repulsive short range force, since ions are different sizes we get different energies - explains why NaCl has rocksalt structure and not CsCl structure

• Intermediate coordination numbers (6-8)QCMP Lent/Easter 2021 4.6

M M e+ −→ + IX e X− −+ → A

I A+

12electrostatic ij

i jU U= ∑∑

204ij ijU q Rπε= ± /

R2

0/ 4electrostatic MU q Rα πε= −( 1.7476)Mα =

( 1.6381)Mα =( 1.7627)Mα =

Crystal structures

QCMP Lent/Easter 2021 4.7From Kittel

From atoms to solids – Covalent bonding• Consider covalent bonding in hydrogen molecule – due to electron pair• Overlapping orbitals on neighbouring atoms hybridise• Hamiltonian symmetric about point between nuclei, hence eigenstates have

odd or even parity about this point• With a basis of atomic states where the nucleus is at R we get

two states, of odd and even parity

• is non zero and has a node between the nuclei • For an attractive potential we will have and the two electrons of

opposite spin will fill the lower ‘bonding’ state .• The ‘anti-bonding’ state will be separated from the bonding state by an

energy gap and will be unfilled


( )φ −r R

( ) ( ) ( )rψ φ φ± = − ± −a br R r Rψ+ ψ−

E E+ −<ψ +

gE E E− += −

aR bR

( )Va r ( )Vb r

φa φb

E−

E+

From atoms to solids – Covalent bonding• An example of degenerate perturbation theory• We calculate single-electron energy levels, ignore electron-electron

repulsion and use Dirac notation, so:

• Look for energy eigenfunction within subspace spanned by orthonormal basis functions:

• Apply Hamiltonian, and since and have the same eigenenergy:

• Left multiply by and we obtain two equations:

• From this, given we can obtain an eigenvector equation for the coefficients which only has a non trivial solution if the determinant is zero:


( ) ( )a bφ φ= − = −a br R r R

a bψ α β= +

E a b E a E bψ ψ α β α β= ⇒ + = +H H Ha b

a a a b E

b a b b E

α β α

α β β

+ =

+ =

H H

H H

a b

,α β

0aa ab

ab bb

H E HH H E∗

−=

−

ba abH H ∗=

From atoms to solids – Covalent bonding

• From previous slide:

• From this we can find the energy eigenvalues which are distributed around the average of and

• For we get covalent bonding

• If or then or and is irrelevant - we have ionic bonding

• See Problem Sheet 1 question 7


0aa ab

ab bb

H E HH H E∗

−=

−

aaH bbH1 22

2

2 2 2 2aa bb aa bb

abH H H HE EE H

/

+ −∆ ∆ = ± , = +

, 2aa bb abH H E H∆ / =| |

aa bbH H<< aa bbH H>> aaE H= bbE H= abH

Covalent and ionic semiconductors• With s electrons molecules are formed

which form a weakly bound molecular solid.

• With p and d orbitals, bonds become directional such as sp3 in C, Si and Ge

• These hybrids point in tetrahedral directions and each atom donates 1 electron

• This forms an open tetrahedral network – “diamond structure”

(111),(111),(111),(111)

• In GaAs and cubic ZnS the total number of electrons satisfies “octet rule” • Same tetrahedral structure as diamond with alternating atoms - “zincblende” • Zincblende structure cohesion is partly covalent and partly ionic• Hexagonal crystal structure based on local tetrahedral network - “wurtzite”• Wurtzite structure favoured in more ionic systems.• With increasing ionicity we get : Group IV Ge (diamond), III-V GaAs (zincblende),

II-VI ZnS (zincblende or wurtzite), II-VI CdS (wurtzite), I-VII NaCl (rocksalt)

Tetrahedral bonding in diamond


Metals• Band forms from atomic states. Partially filled which

implies energy gain generalisation of covalent bond to ‘giant molecule’

• Electrons in band states delocalised, high conductivity. Bonding is isotropic, like van der Waals.

• Close packing. to maximise density while keeping atomic cores far apart: fcc or hcp. High coordination numbers

• Screening by conduction electrons. Screening length of order atomic spacing

• Within a row in periodic table: ion core potential grows, density increases, crossover to covalent semiconductors, then insulating molecular structures.

• Transition metals. d-electrons more localised, inside s- and p- orbitals. Are often spin-polarised hence magnetism in 3d elements. 4d and 5d orbitals overlap giving high binding energy (e.g. W melts at 3700K).


10 12≈ −

1Å

Crystal Lattices• An ideal crystal is an infinite repetition of structural unit in space• Repeating structure called the lattice• Group of atoms which is repeated is called the basis• The basis may be a single atom or very complex – such as a polymer

• Lattice is defined by primitive translation vectors:• An arbitrary lattice translation is:• The atomic arrangement looks the same from equivalent points in the unit

cell:

• The lattice formed is called a Bravais latticeQCMP Lent/Easter 2021 4.13

Lattice Basis Crystal

, 1 2 3i i = , ,a

integeri in n′ = + ∀∑ ii

r r a

i iin=∑T a

Wigner-Seitz cell

• The region in space around a particular lattice point closer to it than any other lattice point

• To construct – draw lines from a given lattice point to all of its neighbours.• Draw planes perpendicular with each line intersecting at the line midpoint.• The smallest volume enclosed is the Wigner-Seitz primitive unit cell

QCMP Lent/Easter 2021 4.14Bcc lattice, Wigner-Seitz cell fcc lattice, Wigner-Seitz cell

Space and point groups

• Symmetry operations which map lattice onto itself Space group• Map lattice onto itself, but keep one point fixed Point group. • Point group operations: reflections, inversions, rotations • Seven point groups for Bravais lattices = Seven crystal systems • cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, hexagonal. • Fourteen space groups for Bravais lattices • General crystal structures (Bravais lattice + basis): 32 point groups, 230

space groups.


The Bravais lattice types


See also p.170 Singleton

Index system for crystal planes

• Coordinates of three lattice points enough to define crystal plane• Label plane by coordinates where it cuts axes.

• denotes plane which cuts axes ator multiple thereof, so that


{ } { }1 2 3x y z= , ,ir a a a( )hkl 1 2 3, ,h k l/ / /a a a

integerxh yk zl= = =

• is the index of the plane• For set of planes equivalent by

symmetry use• e.g. For a cubic crystal has

equivalent symmetry planes:

• Note that overbar denotes negation

( )hkl

{ }{ }100

(100),(010),(010),(100),(010),(001)

1

4.18

Crystal plane indices

From Kittel



• The binding of crystals• Van der Waals• Ionic • Covalent • Metals• Crystal lattices• Wigner –Seitz cell• Space and Point groups• Index system for crystal planes


4.20


The End


quantum condensed matter physics...types of magnetic interactions; direct exchange, heisenberg...

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