chap11(1) - Chapter 11: Dielectric Properties of Materials...

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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 Free-electron 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 wave-vector 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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chap11(1) - Chapter 11: Dielectric Properties of Materials...

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