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Unformatted text preview: Chapter 11: Dielectric Properties of Materials Lindhardt May 8, 2002 Contents 1 Classical Dielectric Response of Materials 2 1.1 Conditions on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Kramers Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . 6 2 Absorption of E and M radiation 8 2.1 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Reflectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Model Dielectric Response . . . . . . . . . . . . . . . . . . . . . . . . 13 3 The Freeelectron gas 17 4 Excitons 19 1 Electromagnetic fields are essential probes of material prop erties IR absorption Spectroscopy The interaction of the field and material may be described ei ther classically or Quantum mechanically. We will first do the former. 1 Classical Dielectric Response of Materials Classically, materials are characterized by their dielectric re sponse of either the bound or free charge. Both are described by Maxwells equations E = 1 c B t , H = 4 c j + 1 c D t (1) and Ohms law j = E . (2) Both effects may be combined into an effective dielectric con stant , which we will now show. For an isotropic medium, we 2 x << < G (k, ) 2245 ( ) k < E(x,t) 2245 E(x ,t) x e f8e5 x Figure 1: If the average excursion of the electron is small compared to the wavelength of the radiation < x > , then we may ignore the wavevector dependence of the radiation so that ( k , ) ( ) . have D ( ) = ( ) E ( ) (3) where E ( t ) = R de it E ( ) H ( t ) = Z de it H ( ) (4) D ( t ) = R de it D ( ) B ( t ) = Z de it B ( ) (5) E ( ) = E * ( ) E ( t ) < (6) Then H = 4 c j + 1 c D t 3 Z de it H ( ) = 4 c Z de it j ( ) + 1 c t Z de it D ( ) (7) Z de it H ( ) 4 c j ( ) 1 c ( i ) D ( ) = 0 (8) H = 4 c j i c D ( ) = 4 c E  i c E =  i c E  c 4 c = i c E 4 c E  c 4 i c = 4 c E (9) Thus we could either define an effective conductivity =  i 4 which takes into account dielectric effects, or an effective dielectric constant = + i 4 , which accounts for conduction. 1.1 Conditions on From the reality of D ( t ) and E ( t ), one has that E (+ ) = E * ( ) and D ( ) = D * ( ), hence for D = E , ( ) = * ( ) (10) 4 Additional constraints are obtained from causality D ( t ) = Z d ( ) E ( ) e it = Z d ( ) e it Z dt 2 e it E ( t ) = 1 2 Z dtd ( ( ) 1 + 1) E ( t ) e i ( t t ) (11) then we make the substitution ( )  1 4 (12) D ( t ) = 2 R dtd ( ) + 1 4 E ( t ) e i ( t t ) D ( t ) =...
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 Fall '11
 Electrodynamics

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