Institute of Solid State PhysicsTechnische Universität Graz

22. Crystal Physics

Jan. 7, 2018

Hall effect / Nerst effect

Institute of Solid State PhysicsTechnische Universität Graz

Crystal Physics

Crystal physics explains what effects the symmetries of the crystal have on observable quantities.

An Introduction to Crystal Physics Ervin Hartmannhttp://ww1.iucr.org/comm/cteach/pamphlets/18/index.html

International Tables for Crystallographyhttp://it.iucr.org/

Kittel chapter 3: elastic strain

Strain

A distortion of a material is described by the strain matrix

ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )ˆ ˆ ˆ(1 )

xx xy xz

yx yy yz

zx zy zz

x x y z

y x y z

z x y z

x̂

ŷ

ẑ

x̂

ŷ

ẑ

Stress

9 forces describe the stress

Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz

Xx is a force applied in the x-direction to the plane normal to x

Xy is a sheer force applied in the x-direction to the plane normal to y

yx z

x y z

yx z

x y z

yx z

x y z

XX XA A A

YY YA A A

ZZ ZA A A

stress tensor:

Stress is force/m2

Xx Xy

Xz

x

y

z

Stress and Strain

The stress - strain relationship is described by a rank 4 stiffness tensor. The inverse of the stiffness tensor is the compliance tensor.

ij ijkl kls

ij ijkl klc

Einstein convention: sum over repeated indices.

xx xxxx xx xxxy xy xxxz xz xxyx yx xxyy yy

xxyz yz xxzx zx xxzy zy xxzz zz

s s s s s

s s s s

Statistical Physics

Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N).

ij ij k K l ldU TdS d E dP H dM

ij K lij k l

U U U UdU dS d dP dMS P M

The normal modes must be solved for in the presence of electric and magnetic fields.

Internal energy in an electric field

In an electric field, if the dipole moment is changed, the change of the energy is,

U E P

Using Einstein notation

This is part of the total derivative of U

ij ij k K l ldU TdS d E dP H dM

k kdU E dP

kk

UEP

Statistical Physics

Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P, , N).

Cannonical ensemble: At constant temperature, make a Legendre transformation to the Helmholtz free energy.

F = U - TS

F(V, T, N, M, P, )

Make a Legendre transformation to the Gibbs potential G(T, H, E, )

ij ij k K l lG U TS E P H M

ij K lij k l

ij ij k K l l

U U U UdU dS d dP dMS P M

dU TdS d E dP H dM

ij ijV

Helmholtz free energyCannonical ensemble: At constant temperature, make a Legendre transformation to the Helmholtz free energy.

F = U - TS

F(T, N, M, P, )

dF = dU - TdS - SdT

i i ij ij k k l ldF SdT dN d E dP H dM

, , ,N M P

FST

, , , , j ii

i T M P N

FN

, , ,

ijij N M P T

F

, , ,k

k N M T

FEP

, , ,

ll N T P

FHM

i ij k li ij K l

F F F F FdF dT dN d dP dMT N P M

Gibbs free energy

, , ,

ijij T E H

G

, , ,k

k T H

GPE

, , ,

ll T E

GMH

i i ij ij k k l ldG SdT N d d P dE M dH

i ij k li ij k l

G G G G GdG dT d d dE dHT E H

i i ij ij k K l lG U TS N E P H M

i i ij ij k K l ldU TdS dN d E dP H dM

, , ,E H

GST

, , ,

ii T E H

GN

G(T, , H, E, )

http://www.iucr.org/education/pamphlets/18

Direct and reciprocal effects (Maxwell relations)

Useful to check for errors in experiments or calculations

simultaneously ferroelectric and ferromagnetic

Multiferroics

BiFeO3

If two magnetic sublattices have different charge, changing the magnetic field can change the polarization and changing the electric field can change the magnetization.

Maxwell relations

Useful to check for errors in experiments or calculations

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http://it.iucr.org

2006 edition available throughTU Graz library

The properties of solids

electronic band structure E vs. k

2 22 2 22 2

, 0 0 02 2 4 4 4A A B

i Ai A i A i j A Be A iA ij AB

Z e Z Z eeHm m r r r

structurebond potentials

density of states

Boltzmann transport

equilibrium propertiescv, free energies, bulk modulus,...

optical absorption

optical properties

phonon band structure vs. k

density of states

Calculating free energies

Electronic component

Phonon component

Groups

Crystals can have symmetries: translation, rotation, reflection, inversion,...

1 0 00 cos sin0 sin cos

x xy yz z

Symmetries can be represented by matrices.All such matrices that bring the crystal into itself form the group of the crystal.

A, B G AB G

32 point groups (one point remains fixed during transformation)230 space groups

Cyclic groups

http://en.wikipedia.org/wiki/Cyclic_group

C2

C4

Pyroelectricity

1 0 00 1 00 0 1 0

x x x

y y y

z z

Pyroelectricity is described by a rank 1 tensor

ii

PT

1 0 0 00 1 0 00 0 1 0

x x

y y

z z

2

ii

GE T

Pyroelectricity

10 Pyroelectric crystal classes: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm

Pyroelectrics are at Joanneum research to make infrared detectors (to detect humans).

Quartz, ZnO, LaTaO3

Pyroelectrics have a spontaneous polarization. If it can be reversed by an electric field they are called Ferroelectrics (BaTiO3)

exampleTurmalin: point group 3m

for T = 1°C, E ~ 7 ·104 V/m

Rank 2 Tensors

Electric susceptibilityDielectric constantMagnetic susceptibilityThermal expansionElectrical conductivityThermal conductivitySeebeck effectPeltier effect

Electric susceptibility

x xx xy xz x

y yx yy yz y

z zx zy zz z

P EP EP E

Transforming P and E by a crystal symmetry must leave the susceptibility tensor unchanged

= U-1U

If rotation by 180 about the z axis is a symmetry, 1 0 0

0 1 00 0 1

U

1

1 0 00 1 00 0 1

U U

1xx xy xz

yx yy yz

zx zy zz

U U

xz = yz = zx = zy = 0

i ij jP E

1 1U UP U UE

UP UE

2

iji j

GE E

Cubic crystals

All second rank tensors of cubic crystals reduce to constants

216: ZnS, GaAs, GaP, InAs221: CsCl, cubic perovskite225: Al, Cu, Ni, Ag, Pt, Au, Pb, NaCl227: C, Si, Ge, spinel229: Na, K, Cr, Fe, Nb, Mo, Ta

Rank 3 Tensors

PiezoelectricityPiezomagnetismHall effectNerst effectEttingshausen effectNonlinear electrical susceptibility

Tensor notation

1 1 12 2 2

3 3 3

2 3 4

1 3 51 2 6

36 312g g 14 1123g grank 3 rank 4

We need a way to represent 3rd and 4th rank tensors in 2-d.

Piezoelectricity

average position + is average position -

average position + not average position -

kk

GPE

2k

ijkij k ij

P G dE

Piezoelectricity (rank 3 tensor)

AFM's, STM'sQuartz crystal oscillatorsSurface acoustic wave generatorsPressure sensors - EpcosFuel injectors - BoschInkjet printers

lead zirconate titanate (Pb[ZrxTi1−x]O3 0

Nonlinear optics

Period doubling crystals

no inversion symmetry

1 2 32 3P E E E 2 31

2i j j ki j i j k

G GP E E EE E E E E

806 nm light : lithium iodate (LiIO3)860 nm light : potassium niobate (KNbO3)980 nm light : KNbO31064 nm light : monopotassium phosphate (KH2PO4, KDP), lithium triborate (LBO).1300 nm light : gallium selenide (GaSe)1319 nm light : KNbO3, BBO, KDP, lithium niobate (LiNbO3), LiIO3

2 12cos 1 cos(2 )t t

Birefringence (Doppelbrechung)

http://en.wikipedia.org/wiki/Birefringent

Two indices of refractionCalcite

k