Review of Analytic Properties of Dielectric Function in Solid State Physics
The lecture covers the analytic properties of the dielectric function, focusing on Kramers-Kronig transforms. Key equations and the practical evaluation of relations are discussed. Numerical methods for singular integrals and the use of Mathematica for Kramers-Kronig transforms are highlighted.
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Presentation Transcript
PHY 752 Solid State Physics 11-11:50 AM MWF Olin 107 Plan for Lecture 36: Review Comment on Kramers-Kronig transforms Some equations worth knowing Course assessment forms 4/24/2015 PHY 752 Spring 2015 -- Lecture 36 1
4/24/2015 PHY 752 Spring 2015 -- Lecture 36 2
4/24/2015 PHY 752 Spring 2015 -- Lecture 36 3
Review topic analytic properties of dielectric function = + Dielectric f unction ( ) ( ) ( ) i R I for ( ) can be sh own to be an alyic for 0 z z Kramers-Kronig transform for dielectric function: ( ) 0 ( ) 0 1 ' 1 = d 1 ' R I P - 1 ' ( ) with ( ) ; 1 ' 1 = d ' I R P - = ' 0 0 ( ) ( ) ( ) ( ) = R R I I 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 4
Practical evaluation of Kramers-Kronig relation ( ) = with ( ) 0 1 ' 1 = 1 ' R I P d - 1 ' 0 ( ) 0 ( ) ; 1 ' 1 d ' I R P - = ' 0 ( ) ( ) ( ) ( ) = R R I I ( ) ( ) ( ) = ( ) = Let R I 1 2 0 0 ( ') ( ') 1 2 ( ) 1 = = 2 2 P d P d 1 2 2 0 2 ( ') 1 ( ') 1 1 ( ) = = 1 1 P d P d 2 2 2 0 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 5
Practical evaluation of Kramers-Kronig relation ( ') 1 ( ) 1 = 2 P d 1 0 ( ') ( ') 1 + = 2 2 P d d 0 ( ') ( ') + 1 + = 2 2 P d d 0 0 Singular integral can be evaluated numerically: 0 0 W ( ') ( ) ( ') ( ') W = + ( )ln + 2 2 2 2 P d P d d 2 W 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 6
Evaluation of singular integral numerically: W ( ') ( ') ( ) ( ') W = + ( )ln + 2 2 2 2 P d P d d 2 0 0 W 2( ) ( ) ( ') 2 2 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 7
Evaluation of Kramers Kronig transform using Mathematica (with help from Professor Cook) 2( ) 1( ) 1 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 8
Another example 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 9
Some equations worth remembering -- 04/24/2015 PHY 712 Spring 2015 -- Lecture 36 10
4/24/2015 PHY 752 Spring 2015 -- Lecture 36 11
Bravais lattice vectors: Atomic basis vectors: x = + a a3 + a y z a 1 2 3 a a a a a1 a2 Reciprocal lattice ( a b a a m odulo 2 ) Distance between diffracting planes a j k a = ( ) 1 b i = d i j k hkl + + b b h k l = b a Note that i j i j 1 2 3 4/24/2015 PHY 752 Spring 2015 -- Lecture 36 12
Bragg diffraction incident beam defracted beam Condition for constructive interference: 2 sin hkl d n = kscat In terms of wave vectors = k k 2 sin kinc k 2 d 2 n = 2 s n i hkl 4/24/2015 PHY 752 Spring 2015 -- Lecture 36 13
Single particle wavefunction in a periodic system Bloch wave: k Eigenfunctions of the periodic Hamiltonian are Bloch states with eigenvalues Enkand electron velocity periodic function ( ) r ( ) r = k r i e u k n n 1 E k nk Wannier representation of electronic states -- continued Wannier function in lattice cell , associated with band is given by: V d k e Note that : = TT r T r T Comment: Wannier functions are not unique since the the Bloch function may be multiplied by a k-dependent phase, which may generate a different function Wn(r-T). T n = k T 3 i r T ( ) r W ( ) ( ) k n n 3 2 ( ) ( ') W W ' ' n n nn 4/24/2015 PHY 752 Spring 2015 -- Lecture 36 14
Understanding band structures --- Example of LiFePO4 and FePO4 Electronic structures of FePO4, LiFePO4, and related materials Ping Tang and N. A. W. Holzwarth -- Phys. Rev. B68, 165107 (2003) 4/24/2015 PHY 752 Spring 2015 -- Lecture 36 15
Partial densities of states FePO4 LiFePO4 4/24/2015 PHY 752 Spring 2015 -- Lecture 36 16
4/24/2015 PHY 752 Spring 2015 -- Lecture 36 17
4/24/2015 PHY 752 Spring 2015 -- Lecture 36 18
