Lecture 2: Dispersion in Materials. 5 nm

Transcription

1 Lecture : Disersion in Materials 5 nm

2 What Haened in the Previous Lecture? Maxwell s Equations Bold face letters are vectors! B D = ρ f B = E = t Curl Equations lead to E P E = με + μ t t Linear, Homogeneous, and Isotroic Media P = ε χ E (under certain conditions) D H = + J t Wave Equation n E ( r, t) E( r, t) = c t Solutions: EM waves α E( zt, ) = Re E( z, ) ex iβz z+ it where β = kn ' and α = kn '' In real life: Resonse of matter (P) is not instantaneous + (, t) = ε dt' x( t t' ) E( r, t' ) P r Phase roagation χ ' = χ'( ) n' = n' ( ) χ '' χ''( ) = n'' = n'' ( ) absortion

3 Today: Microscoic Origin -Resonse of Matter Origin frequency deendence of χ in real materials Lorentz model (harmonic oscillator model) Insulators (Lattice absortion, color centers ) Semiconductors (Energy bands, Urbach tail, excitons ) Metals (AC conductivity, Plasma oscillations, interband transitions ) Real and imaginary art of χ are linked Kramers-Kronig But first.. When should I work with χ, ε, or n? They all seem to describe the otical roerties of materials!

4 n and n vs χ and χ vs ε and ε All airs (n and n, χ and χ, ε and ε ) describe the same hysics For some roblems one set is referable for others another n and n used when discussing wave roagation α E( zt, ) = Re E( z, ) ex iβz z+ it where β = kn ' and α = kn '' Phase roagation absortion χ and χ used when discussing microscoic origin of otical effects ε and ε As we will see today Inter relationshis Examle: n and ε From n = ε r n' + in'' = ε ' + iε '' r r ( ) ( ) ε r ' = n' n'' ε '' = nn ' '' r and n ' = n '' = ( ) ( ) ε ' + ε '' + ε ' r r r ( ) ( ) ε ' + ε '' ε ' r r r

5 Behavior of bound electrons in an electromagnetic field Otical roerties of insulators are determined by bound electrons Lorentz model Charges in a material are treated as harmonic oscillators m a = F + F + F el E, Local Daming Sring d r dr m + mγ + Cr = ee ex L i t dt dt Guess a solution of the form: Linear Dielectric Resonse of Matter ( ) = ex i t ( ) The electric diole moment of this system is: d d m + mγ + C = e E ex L i t dt dt d ; = i ex dt m imγ + C = e Ee L (one oscillator) ( ) = e ( it) r d dt E L C, γ r + ; ex = Solve for (E L ) Nucleus e -, m = er ( it)

6 Atomic Polarizability Determination of atomic olarizability Last slide: m imγ + C = e E L C e iγ L + = E (Divide by m) m m Define as (turns out to be the resonance ) = e 1 E iγ m L Atomic olarizability (in SI units) Define atomic olarizability: ( ) α e 1 = ε ε m iγ E L Resonance frequency Daming term

7 Characteristics of the Atomic Polarizability Resonse of matter (P) is not instantaneous -deendent resonse e 1 Atomic olarizability: α ( ) = Aex iθ ( ) ε = ε m iγ = E L Amlitude A = e ε m ( ) 1 + γ 1/ Amlitude smaller γ Phase lag of α with E: 18 θ = tan 1 γ Phase lag 9 smaller γ

8 Relation Atomic Polarizability (α) and χ: cases Case 1: Rarified media (.. gasses) Diole moment of one atom, : Polarization vector: Occurs in Maxwell s equation.. ( ) = εα E L P 1 ε = = α = ε Nα V E E sum over all atoms E-field hoton Density L L V α ( ) = e 1 ε m iγ P = Ne 1 L m iγ E = E Microscoic origin suscetibility: Plasma frequency defined as: ( ε χ ) ( ) χ Ne ε m = = χ( ) L Ne 1 ε m iγ = iγ

9 Remember: ε and n follow directly from χ Frequency deendence ε Relation of ε to χ : ε = 1+ χ = 1+ iγ ε ' + iε '' = 1 + χ ' + iχ '' = 1+ iγ ε ( ) ( ) ε ' = 1 + χ ' = 1+ ( ) ε '' = χ '' = ( ) ( ) + ( ) γ γ + γ χ ' = χ =

10 Proagation of EM-waves: Need n and n Relation between n and ε n = ε ε ( ) ( ) ( ) ε r ' = n' n'' ε '' = nn ' '' r n ( ) << : High n low v h =c/n : Strong deendence v h n ' n '' n ' = 1 Large absortion (~ n ) >> : n = 1 v h =c

11 χ = k Realistic Rarefied Media Realistic atoms have many resonances Resonances occur due to motion of the atoms (low ) and electrons (high ) Ne k 1 ε m iγ k Where N k is the density of the electrons/atoms with a resonance at k Examle of a realistic deendence of n and n α= k n Molecule rotation Atomic vibration Electron excitation n n > 1 indicates resence high oscillators 1 3 Log ()

12 Back to Relation Atomic Polarizability (α) and χ: Case : Solids Solid i E-field? Atom feels field from: 1) Incident light beam ) Induced diolar field from other atoms, i Local field: E = E + E L I Local field Field without matter Induced diolar field from all the other atoms

13 Local field Local field: Electric Suscetibility of a Solid E = E + E L I Local field Field without matter All the other atoms Induced diolar field Examle: For cubic symmetry: E = L E + P 3ε Polarization of a solid E I = P 3ε (Similar relations can be derived for any solid) (Solid state Phys. Books, e.g. Kittel, Ashcroft..) Solid consists of atom tye at a concentration N = ε N α = ε P N α + = ε N α + P P E E E N α 3 3 L ε 1 P 1 N α = ε N α 3 E P χ = = ε 1 E 1 3 N α N α

14 Clausius-Mossotti Relation Polarization of a solid Suscetibility: Limit of low atomic concentration:.or weak olarizability: retty good for gasses and glasses P χ = = ε 1 E χ 1 3 N α N α N α I II Clausius-Mossotti By definition: ε = 1+ χ Rearranging I gives ε 1 1 = ε + 3ε N α III Conclusion: Dielectric roerties of solids related to atomic olarizability This is very general!!