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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
http
://up
load
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ia.o
rg/w
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edia
/com
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09/T
herm
odyn
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.svgReplace P and V with and
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