Pyroelectricity 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 rank 1: pyroelectric effect, pyromagnetic effect, electrocaloric effect, magnetocaloric effect Rank 2 Tensors Electric susceptibility Dielectric constant Magnetic susceptibility Thermal expansion Electrical conductivity Thermal conductivity Seebeck effect Peltier effect Polarization tensor Pi ij E j Px xx xy xz E x Py yx yy yz E y Pz zx zy zz E z Transforming P and E by a crystal symmetry must leave the polarization tensor unchanged UP UE U1 UP U 1 UE = U-1U If rotation by 180 about the z axis is a generating matrix, 1 0 0 1 0 0 xx xy xz 1 1 U 0 1 0 U U U U 0 1 0 yx yy yz 0 0 1 0 0 1 zx zy zz xz = yz = zx = zy = 0 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 Nonlinear optics Period doubling crystals no inversion symmetry P1 E 2 E2 3 E 3 2 3 G1 G Pi E j EE jk EEij2 EEE ijk 2 1 cost 2 1 cos(2 t ) 806 nm light : lithium iodate (LiIO3) 860 nm light : potassium niobate (KNbO3) 980 nm light : KNbO3 1064 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 Piezoelectricity average position + G is Pk E average position - k P 2G average position + k d ijk not ij E k ij average position - Rank 4 Tensors Stiffness tensor Compliance tensor Piezoconductivity Electrostriction Magnetostriction How the Seebeck effect depends on stress How the electric susceptibility depends on stress How the magnetic susceptibility depends on stress Nonlinear electric susceptibility Nonlinear magnetic susceptibility Electrostriction 2 P ij G k d ijk ij E k E kij piezoelectricity Electrostriction Symmetric and asymmetric tensors 2 2 GP G Pj k kj jk EEjk E j EE kj E k Symmetric Asymmetric electric susceptibility Seebeck effect magnetic susceptibility Peltier effect electrical conductivity piezoconductivity thermal conductivity stiffness tensor Structural phase transitions Some materials make a transition from one crystal structure to another. Two allotropes of tin: gray tin (a-Sn) is stable at temperatures below 13.2°C and white tin (b-Sn) is stable above. The phase with the lowest free energy prevails. (White tin can be stabilized below 13.2 C by adding impurities.) F U TS Structural phase transition in Sn semiconductor diamond crystal structure metal tetragonal crystal structure Structural phase transition in Sn a Sn metal b Sn 2 D( EF ) 2 s kTB semiconductor: electrons make a 3 negligible contribution to the entropy 2 ** 3 / 4 Eg 3 / 2 E g s mmexp kTk 5 , 2 3 eh B B 2 2kTB T .