Technische Universität Graz Institute of Solid State Physics 22. Crystal Physics Jan. 7, 2018
Hall effect / Nerst effect
Technische Universität Graz Institute of Solid State Physics Crystal Physics Crystal physics explains what effects the symmetries of the crystal have on observable quantities. An Introduction to Crystal Physics Ervin Hartmann http://ww1.iucr.org/comm/cteach/pamphlets/18/index.html International Tables for Crystallography http://it.iucr.org/ Kittel chapter 3: elastic strain
Strain A distortion of a material is described by the strain matrix x (1 ) xˆ yˆ zˆ xx xy xz y xˆ(1 ) yˆ zˆ yx yy yz z xˆ yˆ(1 ) zˆ zx zy zz ẑ ẑ ŷ ŷ ˆx ˆx
Stress z 9 forces describe the stress Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz Xz y x Xx Xy 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 stress tensor: Stress is force/m 2 X x X y X A A A x y z Yx Yy Y A A A x y z Z x Z y Z A A A x y z z z z
Stress and Strain s ij ijkl kl The stress - strain relationship is described by a rank 4 stiffness tensor. The inverse of the stiffness tensor is the compliance tensor. c ij ijkl kl Einstein convention: sum over repeated indices. s s s s s xx xxxx xx xxxy xy xxxz xz xxyx yx xxyy yy s s s s xxyz yz xxzx zx xxzy zy xxzz zz
Statistical Physics Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P,, N). U U U U du ds d dp dm S P M ij K l ij k l du TdS d E dp H dm ij ij k K l l 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, Using Einstein notation U EP du E dp k k This is part of the total derivative of U E k U P k du TdS d E dp H dm ij ij k K l l
Statistical Physics Microcannonical Ensemble: Internal energy is expressed in terms of extrinsic quantities U(S, M, P,, N). V U U U U du ds d dp dm S P M ij K l ij k l du TdS d E dp H dm ij ij ij k K l l ij 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, ) GUTS E P HM ij ij k K l l
Helmholtz free energy Cannonical ensemble: At constant temperature, make a Legendre transformation to the Helmholtz free energy. F = U - TS F(T, N, M, P, ) F F F F F df dt dn d dp dm T N P M i ij k l i ij K l df = du - TdS - SdT df SdT dn d E dp H dm i i ij ij k k l l S F T F F ij ij,,, i NMP,,, Ni TMP,,,, Nji NMPT E k F F l Pk NMT,,, M l NTP,,, H
Gibbs free energy G(T,, H, E, ) G U TS N E P H M i i ij ij k K l l du TdS dn d E dp H dm i i ij ij k K l l dg SdT N d d P de M dh i i ij ij k k l l G G G G G dg dt d d de dh T E H i ij k l i ij k l S G T, E, H, N i G i T, E, H, ij G ij T, E, H, P k G G M l Ek T,, H, l T,, E, H
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Direct and reciprocal effects (Maxwell relations) Useful to check for errors in experiments or calculations
Multiferroics simultaneously ferroelectric and ferromagnetic BiFeO 3 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://upload.wikimedia.org/wikipedia/commons/0/09/thermodynamic_map.svg Replace P and V with and
http://it.iucr.org 2006 edition available through TU Graz library
The properties of solids H Ze e ZZe 2 2 2 2 2 2 2 A A B i A i 2me A 2mA i, A 40riA ij 40rij AB40rAB electronic band structure E vs. k structure bond potentials phonon band structure vs. k density of states Boltzmann transport optical absorption density of states equilibrium properties c v, free energies, bulk modulus,... optical properties
Calculating free energies Electronic component Phonon component
Groups Crystals can have symmetries: translation, rotation, reflection, inversion,... x 1 0 0 x y 0 cos sin y z 0 sin cosz 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 C 2 C 4 http://en.wikipedia.org/wiki/cyclic_group
Pyroelectricity i 2 G E T i Pyroelectricity is described by a rank 1 tensor Pi i T 1 0 0 x x x 0 1 0 y y y 0 0 1 z z 0 1 0 0 x x 0 0 1 0 y y 0 0 0 1 z z 0
Pyroelectricity Quartz, ZnO, LaTaO 3 example Turmalin: point group 3m for T = 1 C, E ~ 7 10 4 V/m Pyroelectrics have a spontaneous polarization. If it can be reversed by an electric field they are called Ferroelectrics (BaTiO 3 ) Pyroelectrics are at Joanneum research to make infrared detectors (to detect humans). 10 Pyroelectric crystal classes: 1, 2, m, mm2, 3, 3m, 4, 4mm, 6, 6mm
Rank 2 Tensors Electric susceptibility Dielectric constant Magnetic susceptibility Thermal expansion Electrical conductivity Thermal conductivity Seebeck effect Peltier effect
UP Electric susceptibility Px xx xy xz Ex P E y yx yy yz y P z zx zy zz E z P E i ij j Transforming P and E by a crystal symmetry must leave the susceptibility tensor unchanged UE 1 1 U UP U UE ij 2 G EiE = U -1 U j If rotation by 180 about the z axis is a symmetry, 1 0 0 U 0 1 0 U 0 0 1 1 1 0 0 U 0 1 0 U 0 0 1 xz = yz = zx = zy = 0 1 xx xy xz U yx yy yz zx zy zz
Cubic crystals All second rank tensors of cubic crystals reduce to constants 216: ZnS, GaAs, GaP, InAs 221: CsCl, cubic perovskite 225: Al, Cu, Ni, Ag, Pt, Au, Pb, NaCl 227: C, Si, Ge, spinel 229: Na, K, Cr, Fe, Nb, Mo, Ta
Rank 3 Tensors Piezoelectricity Piezomagnetism Hall effect Nerst effect Ettingshausen effect Nonlinear electrical susceptibility
Tensor notation We need a way to represent 3rd and 4th rank tensors in 2-d. 1 11 1 2 6 2 2 2 1 3 5 2 3 4 3 3 3 g rank 3 rank 4 36 g312 14 1123 g g
Piezoelectricity average position + is average position - P k G E k average position + not average position - 2 P k G ij Ek ij d ijk
Piezoelectricity (rank 3 tensor) AFM's, STM's Quartz crystal oscillators Surface acoustic wave generators Pressure sensors - Epcos Fuel injectors - Bosch Inkjet printers No inversion symmetry lead zirconate titanate (Pb[Zr x Ti 1 x ]O 3 0<x<1) more commonly known as PZT barium titanate (BaTiO 3 ) lead titanate (PbTiO 3 ) potassium niobate (KNbO 3 ) lithium niobate (LiNbO 3 ) lithium tantalate (LiTaO 3 ) sodium tungstate (Na 2 WO 3 ) Ba 2 NaNb 5 O 5 Pb 2 KNb 5 O 15 Piezoelectric crystal classes: 1, 2, m, 222, mm2, 4, -4, 422, 4mm, -42m, 3, 32, 3m, 6, -6, 622, 6mm, -62m, 23, -43m
Nonlinear optics Period doubling crystals no inversion symmetry 1 2 3 P E E E 2 3 2 3 G 1 G Pi Ej EjEk EE 2 EE E i j i j k t t cos 1 cos(2 ) 2 1 2 806 nm light : lithium iodate (LiIO 3 ) 860 nm light : potassium niobate (KNbO 3 ) 980 nm light : KNbO 3 1064 nm light : monopotassium phosphate (KH 2 PO 4, KDP), lithium triborate (LBO). 1300 nm light : gallium selenide (GaSe) 1319 nm light : KNbO 3, BBO, KDP, lithium niobate (LiNbO 3 ), LiIO 3
Birefringence (Doppelbrechung) Calcite Two indices of refraction http://en.wikipedia.org/wiki/birefringent k