1 MTLE-6120: Advanced Electronic Properties of Materials Maxwell’s equations in materials Contents: ◮ Electromagnetism in free space ◮ Material response to EM fields ◮ Electromagnetic waves Reading: ◮ Kasap: not discussed ◮ Griffiths EM: Chapter 7
2 Electrostatics ◮ Coulomb’s law: electric field around a point charge q r ′ − � q ( � r ) � q � � r ′ ) = E ( � r | 3 = −∇ � r ′ − � r ′ − � r ′ 4 πǫ 0 | � 4 πǫ 0 | � r | � �� � φ ( � r ′ ) ◮ Gauss’s law (integral form): � � ǫ 0 E · d � a = q ◮ Gauss’s law (differential form), for charge density ρ ( � r ) : ǫ 0 ∇ · � E = ρ and ∇ × � E = 0 or equivalently, in terms of the potential: − ǫ 0 ∇ 2 φ = ρ
3 Magnetostatics ◮ Biot-Savart law: magnetic field around a current I � µ 0 I � r ′ − � d l × ( � r ) x � B ( � r ′ ) = r ′ − � r | 3 4 π | � ◮ Ampere’s law (integral form) � B · d � � l = µ 0 I ◮ Ampere’s law (differential form), for current density � j ( � r ) : 1 ∇ × � B = � j and ∇ · � B = 0 µ 0
4 Electromagnetic induction ◮ Faraday’s law for electromotive force: � � r ′ ) = − d d l · � � d a · � � EMF = E ( � B ( � r ) d t � �� � Flux Φ ◮ Differential form: E = − ∂ � B ∇ × � ∂t
5 Equations so far Apply divergence to third equation: � � ǫ 0 ∇ · � j = 1 E = ρ ∇ · � ∇ × � ∇ · B = 0 µ 0 E = − ∂ � B ∇ × � ∂t Divergence of current is the rate at which 1 charge leaves a point (continuity equation ∇ × � B = � j µ 0 i.e. charge conservation): ∇ · � B = 0 j = − ∂ρ ∇ · � ∂t Are these correct in general? � � = − ∂ ǫ 0 ∇ · � E ∂t So how can we fix the equations?
6 Maxwell’s equations ǫ 0 ∇ · � E = ρ Apply divergence to third equation: E = − ∂ � B j + ∂ ( ǫ 0 ∇ · � ∇ × � E ) = 1 � � ∇ · � ∇ × � ∇ · B = 0 ∂t ∂t µ 0 ∂ � 1 E ∇ × � B = � j + ∂ρ j + ǫ 0 ⇒ ∇ · � ∂t = 0 µ 0 ∂t ∇ · � B = 0 Now consistent with charge conservation.
7 Materials in electric fields ◮ All materials composed of charges: electrons and nuclei ◮ Charges pulled along/opposite electric field with force q � E ◮ Charges separated in each infinitesimal chunk of matter ⇒ dipoles ◮ Induced dipole moment: δ� p = δqδx ˆ x ◮ Polarization is the density of induced dipoles: δxδa = δq δ� p � P = δa ˆ x
8 Bound charge due to polarization ◮ Charge density in infinitesimal chunk ρ b = δq 1 − δq 2 δaδx δq 1 δa − δq 2 δa = δx = P x ( x ) − P x ( x + δx ) δx = − ∂P x ∂x ◮ Similarly accounting for y and z components: ρ b = −∇ · � P
9 Current density due to polarization ◮ Charge crossing dotted surface in time δt : δq = δq 2 − δq 1 ◮ Corresponding current density: j x = δq 2 − δq 1 δaδt δq 2 δa − δq 1 δa = δt = P x ( t + δt ) − P x ( t ) δt = ∂P x ∂t ◮ Polarization current density in general direction: j P = ∂ � P � ∂t
10 Materials in magnetic fields v × � ◮ Charges circulate around magnetic field due to force q� B ◮ Magnetic dipole moment of infinitesimal current loop (per unit δz ) � δµ = 1 � r × � � d lδI 2 = 1 2 (0 + δxδy ˆ zδI + δyδx ˆ zδI + 0) = δxδy ˆ zδI ◮ Magnetization is density of induced magnetic dipoles: δµ � M = δxδy = ˆ zδI (Per unit )
11 Bound current density due to magnetization ◮ Current within dotted element (per unit δz ) δI y = δI 1 − δI 2 ◮ Corresponding current density: j y = I y δx = δI 1 − δI 2 δx = M z ( x ) − M z ( x + δx ) δx = − ∂M z ∂x (Per unit ) ◮ Generalizing to all directions: � j b = ∇ × � M
12 Material response summary ◮ Response to electric field � E is polarization � P ◮ Polarization corresponds to bound charge density ρ b = −∇ · � P and current density j P = ∂ � P � ∂t ◮ Response to magnetic field � B is magnetization � M ◮ Magnetization corresponds to bound current density j b = ∇ × � � M
13 Maxwell’s equations including material response ◮ Fields produced by external ‘free’ charges and currents ( ρ f and � j f ) as well as bound ones induced in the materials ∂ � 1 E ǫ 0 ∇ · � ∇ × � B = ( � j f + � j b + � E = ( ρ f + ρ b ) j P ) + ǫ 0 µ 0 ∂t E = − ∂ � B ∇ × � ∇ · � B = 0 ∂t ◮ Rewrite bound quantities in terms of polarization and magnetization ∂ � 1 E ǫ 0 ∇ · � ∇ × � B = ( � j f + � j b + � E = ( ρ f + ρ b ) j P ) + ǫ 0 µ 0 ∂t M + ∂ � ∂ � 1 P E ǫ 0 ∇ · � E = ρ f − ∇ · � ∇ × � j f + ∇ × � B = � P ∂t + ǫ 0 µ 0 ∂t � � � j f + ∂ ( ǫ 0 � E + � B P ) ∇ · ( ǫ 0 � E + � − � = � P ) = ρ f ∇ × M µ 0 ∂t
14 Maxwell’s equations in media ◮ Define fields D ≡ ǫ 0 � � E + � P � B � − � H ≡ M µ 0 ◮ Yields equations with free charge and current densities as the sources: ∇ · � D = ρ f E = − ∂ � B ∇ × � ∂t j f + ∂ � D ∇ × � H = � ∂t ∇ · � B = 0
15 Constitutive relations ◮ Material determines how � P (and hence � D ) depends on � E ◮ Material determines how � M (and hence � H ) depends on � B ◮ Simplest case: linear isotropic dielectric P = χ e ǫ 0 � � M = χ m � � E H D = (1 + χ e ) ǫ 0 � � B = (1 + χ m ) µ 0 � � E H ǫ = (1 + χ e ) ǫ 0 µ = (1 + χ m ) µ 0 ◮ Anisotropic dielectric: � χ e · ǫ 0 � P = ¯ E with susceptibility tensor ¯ χ e ◮ Nonlinear dielectric: � P = χ e ( E ) ǫ 0 � E
16 Ohm’s law ◮ Response of metals to constant electric fields given by � j = σ � E with electrical conductivity σ ◮ But what is the corresponding constitutive relation � P ( � E ) ? ◮ The current is actually a polarization current � j P = σ � E ◮ Remember � j P = ∂ � P/∂t , so � � d t� j P = � P 0 + tσ � P = E ◮ Important: the response is not instantaneous in general ◮ It can depend on the history i.e. is non-local in time
17 Frequency domain ◮ For linear materials, convenient to work in frequency domain where all quantities have time dependence f ( t ) ≡ fe − iωt with angular frequency ω ◮ Maxwell’s equations take the form ∇ · � D = ρ f ∇ × � E = iω � B ∇ × � H = � j f − iω � D ∇ · � B = 0 ◮ For Ohmic metal (with frequency-dependent conductivity σ ( ω ) ): σ ( ω ) � j P ≡ ∂ � P/∂t = − iω � E = � P P = iσ ( ω ) � E ⇒ � ω ⇒ ǫ ( ω ) = ǫ 0 + iσ ( ω ) ω
18 Electromagnetic waves ◮ Linear response of materials described very generally by ǫ ( ω ) and µ ( ω ) ◮ Maxwell’s equations in the absence of free charges and currents ∇ · ( ǫ ( ω ) � E ) = 0 ∇ × � E = iω � B � B µ ( ω ) = − iω ( ǫ ( ω ) � ∇ × E ) ∇ · � B = 0 ◮ Substitute second equation in curl of third equation: ∇ × ( ∇ × � B ) = − iωǫ ( ω ) ∇ × � E = ω 2 ǫ ( ω ) � B µ ( ω ) −∇ 2 � B = ω 2 ǫ ( ω ) µ ( ω ) � using ∇ · � B B = 0
19 Electromagnetic wave equation ◮ Write Maxwell’s equation in linear media with no free charge or current as: v 2 ( ω ) ∇ 2 � B = − ω 2 � B v 2 ( ω ) ∇ 2 � E = − ω 2 � E � where v ( ω ) ≡ 1 / ǫ ( ω ) µ ( ω ) ◮ First consider 1D version, v 2 ∂ 2 f ∂x 2 = − ω 2 f . General solution: f ( x ) = A exp( ikx ) + B exp( − ikx ) with k = ω/v ◮ General solution for v 2 ∇ 2 f = − ω 2 f : � k exp( i� f ( � r ) = A � k · � r ) � k with | � k | = ω/v
20 Electromagnetic wave speed ◮ Going back to EM waves and to time domain: � k 0 exp( i� � � E ( � r, t ) = E � k · � r − iωt ) � k � � � k 0 exp( i� B ( � r, t ) = B � k · � r − iωt ) � k with E 0 ⊥ B 0 ⊥ � k to satisfy Maxwell’s equations ◮ For each � k , two linearly-independent choices for � E (polarizations) � ◮ Wave speed is v ( ω ) = ω/ | � k | = 1 / ǫ ( ω ) µ ( ω ) ◮ In vacuum, ǫ ( ω ) = ǫ 0 and µ ( ω ) = µ 0 , so speed of light c = 1 / √ ǫ 0 µ 0 ◮ In materials, speed of light usually specified by refractive index � c ǫ ( ω ) µ ( ω ) n ( ω ) ≡ v ( ω ) = ǫ 0 µ 0
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