1.3 angular momentum and spin · 2. sommerfeld theory 3. from atoms to solids 4. electronic...

19.1 QCMP Lent/Easter 2016

Quantum Condensed Matter Physics Lecture 19

David Ritchie

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

19.2

QCMP Course Contents

1. Classical models for electrons in solids

2. Sommerfeld theory

3. From atoms to solids

4. Electronic structure

5. Bandstructure of real materials

6. Experimental probes of the band structure

7. Semiconductors

8. Semiconductor devices

9. Electronic instabilities

Peierls transition, charge density waves, magnetism, local

magnetic moments, Curie Law, types of magnetic interactions

10. Fermi liquid theory

QCMP Lent/Easter 2016

Electronic instabilities • Crystal structure of solids much more complex

than expected

• Few solids are simple close packed structures

• e.g. Ga metal has several different phases as a

function of temperature and pressure

• Se crystallises in a structure which is an array of

spiral chains with 3 atoms per unit cell

• As, Sb, Bi, have puckered sheets where each

atom has 3 nearest neighbours

• Due to chemical bonding and balance of forces

• Fundamental principle of bonding – by placing

chemical potential in a gap, occupied states

lowered in energy (and unoccupied states go up)


Se

• Getting chemical potential to lie in a gap involves

the Brillouin zone boundary being in the right

place – at a momentum containing exactly the

correct number of states to account for all the

electrons in a solid

As

19.3

The Peierls Transition

• Consider a 1D chain of atoms

• Lattice constant , electron density chosen so the Fermi wavevector lies

in the middle of the band - a metal

• We can turn this metal into an insulator by applying an external potential

with periodicity where

• From previous lectures we know a periodic potential produces

Bragg scattering at a wavevector - a new Brillouin zone boundary

• If there is an energy gap induced on the Fermi surface

• Rather than applying an external potential we can get the same effect by

making a periodic lattice distortion (PLD) with the same periodicity

• So move the atom in the chain to a new position

• Assume the PLD amplitude is small

• We have already met this situation in the diatomic chain……

• If the atoms have a PLD with periodicity they will also produce a

new potential with the same period which is seen by the electrons

• The amplitude of the Fourier components is proportional to the displacement

and we can write with the electron-phonon coupling constant

QCMP Lent/Easter 2016 19.4

2 / Q 2 fQ k

0 cosV Qx/ 2Q

/ 2 fQ k

0 cos( )nR na u Qna

0u a

2 / Q

0Q QV g u Qg

afk

thn

The Peierls Transition

• Energy gap on the zone boundary is

• Hence an energy level at a momentum

just below is lowered by an energy

proportional to atomic displacement

- the unoccupied level just above is

raised by the same amount

• Period chosen to be so that a

band gap of is introduced at

exactly the chemical potential

• Overall there is an energy lowering as a

result of the PLD, the magnitude can be

calculated (see problem 4.4) by adding

up the energy changes of all occupied

states giving in the limit

where is a constant:

• The ln varies a bit faster than the

square, it is negative, the energy goes

down with the distortion QCMP Lent/Easter 2016 19.5

QV

0ufk

fk

2

0 0( / ) ln /elecE A u a u a

2 / 2 fk

02 Qg u

1D chain

1D chain with lattice modulation

0 / 1u aA

The Peierls Transition • So by an extension of the standard band structure result there is an

electronic charge modulation accompanying the periodic lattice distortion

• This is usually called a charge density wave (CDW)

• The result is just the electronic contribution to

the energy from states close to the Fermi surface

• We can model other interactions between atoms as springs and we can add

an elastic energy of the form giving a potential of the

form:


2

0 0( / ) ln /elecE A u a u a

2

0( / )ElasE K u a2 2( ) lnE x Ax x Bx

• This always has a minimum at

non-zero displacement

• The system lowers its energy by

distorting to produce a PLD and

CDW with a period determined

by the Fermi wave vector:

• This spontaneous lattice

distortion is a broken symmetry

phase transition named after its

discoverer Rudolf Peierls

2 / 2 fk

Charge density waves • Materials that are strongly anisotropic in electronic structures are prone to

spontaneous lattice formation and accompanying charge density wave


• Phase transitions occur on lowering T,

corresponding to onset of ordering

• Can be monitored by measuring Bragg

peaks in crystal structure using electron,

neutron or x-ray scattering

• Figures show electron diffraction images

from CDW in La0.29Ca0.71Mn03

• Top figure – real space shows short scale

atomic lattice with periodic modulation

• Bottom figure Fourier transform – widely

spaced bright peaks from small unit cell

• Less intense peaks from CDW

• Two periods not related since CDW

period determined by fermi surface size

and shape, depends on electron

concentration

Real space

K-space

Image from J Loudon, PA Midgeley, ND Mathur

Charge density waves

• The onset of a charge density wave can be seen in the phonon spectrum

• The coefficient of the quadratic term of energy as a function of displacement

gives the phonon stiffness for a mode of wavevector

• The onset of CDW is when the stiffness becomes zero (-ve below transition)

• At this point there is no restoring force and the phonon spectrum

shows a sharp minimum close to


2 fk

( )q2 fq k

• Figure shows Phonon dispersion

curves (measured by inelastic neutron

scattering at room T) for quasi 1D

organic compound TTF-TCNQ

(tetrathiofulvalene tetracyanoquinone)

• Dispersion curve is along the direction

of the chains in which there is a soft

phonon that turns into a periodic lattice

distortion at low temperature.

• Figure also shows many non-

dispersive optical modes

H A Mook and CR Watson Phys Rev

Lett 36, 801 (1976)

Magnetism • Defined as capacity of materials to change local magnetic field

• Not possible for classical systems in thermal equilibrium to be magnetic:

Bohr-van Leeuwen theorem – see Feynman lectures volume 2

• All magnetic phenomena rooted in protection of orbital or spin angular

moment afforded by quantum mechanics

• Distinguish between materials which are diamagnetic, paramagnetic or

magnetically ordered

• Diamagnetic materials - magnetism induced is opposite to applied field, a

relative permeability and magnetic susceptibility

• Due to motion of charged quantum mechanical particles, a weak effect,

negligible in many materials exhibiting paramagnetism or magnetic ordering

Examples: Bismuth (χ = -16.6x10-5), copper (χ = -1x10-5), water (-9.1x10-6),

superconductors (-1) ‘perfect’ diamagnets


1 1 0

• Diamagnets can be levitated and held in a stable

equilibrium in a strong magnetic with no power

consumption, total (magnetic+gravitational) energy

having a minimum.

• Experiment – high water content allow a frog to be

levitated in a 16T field M V Berry and A K Geim Eur J Phys 18, 307 (1997)

Magnetism

• Paramagnetic materials - magnetism induced is in

direction of applied field

• Observed in systems with partially filled atomic

shells or unpaired electrons. Orbital and spin

angular momentum give rise to paramagnetic

response significantly greater than any diamagnetic

effect due to paired electrons.

• Permeability Magnetic susceptibility

• Paramagnetic effects quite small

• Examples: Oxygen, sodium (χ = 7.2x10-6),

aluminium (χ = 2.2x10-5), calcium, uranium.


1 0 1 610 10

Motion of liquid oxygen in

a magnetic field

wikipedia

• As temperature is reduced in a paramagnet, below a critical value

magnetic dipoles may be ordered

• Ferromagnetism – dipoles aligned giving a permanent magnetic field

• Antiferromagnetism – dipoles anti-aligned

• Magnetic ordering due to quantum mechanical exchange interaction -

magnetic dipole-dipole interaction much smaller

Local magnetic moments

• Consider a local magnetic moment of magnitude in a small

applied magnetic field

• Dipole energy is given by

• Probability of finding dipole pointing in a particular direction at temperature

• The average moment

• The magnetic susceptibility is then given by and is isotropic

so

• We can express as

• Since for


m m0H

0( )E m m H

T

( ) 2( ) / , ( )d , 1/E

Bp e Z Z p k T

m

m

m m m

02 21( )d dp e

Z

m H

m m

m m m m m m

d

d( ) i

jij T m

H

( ) ( )ij ijT T

( )ij T

0 02 2

2

0 0

d 1 d 1 dZ(e )d e d

d d Z d

ii i

j j j

i j i j i j

Z

m H m H

m m

mm m m m

H H H

m m m m m m

0

dZ1 1Z d

0ii

Hm 0H

Local magnetic moments – Curie Law

• In the limit and

• If rather than a single moment we have

moments in volume the associated

susceptibility is:

• This is known as the Curie law of

susceptibility, note the inverse

temperature dependence.

• Figure shows plotted against

temperature measured for paramagnetic

ions in a gadolinium salt

Gd(C2H3SO4).9H2O

• The straight line is the Curie law


00, 1e

m H

H22 2 2 21 1

3 3x y zm m m m

NV

2

0

1 1

3 B

N

V k T

1

L C Jackson and H Kamerlingh Omnes

From Kittel 19.12

Types of magnetic interactions

• In many materials a finite magnetism is often observed in absence of

magnetic field

• Must be produced by interactions coupling to electron magnetic moment

• First idea – moments couple through dipole magnetic fields

• Interaction energy of two magnetic dipoles is of order

• Using a magnetic moment of order a Bohr magnetron:

• Where is the fine structure constant

• At typical atomic separations of 2nm this is about

corresponding to a temperature of less than 1K

• This is far too small to explain ordering magnetic temperatures for Co -

1388K, Fe - 1043K, Ni - 627K

• The real explanation revolves around the symmetry of wavefunctions and

the Pauli exclusion principle, the large energies are due to the Coulomb

interaction between electrons


2 34om r

32

2

3

1Ryd

4 2

o Bohrdipolar

aeU

m r r

1 137

54 10 eV

Summary of Lecture 19


• Peirels transition

• Charge density waves

• Magnetism

• Local magnetic moments

• Curie Law

• Types of magnetic interactions

19.15 QCMP Lent/Easter 2016

Quantum Condensed Matter Physics Lecture 19

The End

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1.3 angular momentum and spin · 2. sommerfeld theory 3. from atoms to solids 4. electronic...

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