MTLE-6120: Advanced Electronic Properties of Materials Maxwells - - PowerPoint PPT Presentation

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

1

Electrostatics

◮ Coulomb’s law: electric field around a point charge q

• E(

r′) = q( r′ − r) 4πǫ0| r′ − r|3 = −∇

r′

• q

4πǫ0| r′ − r|

• φ(

r′)

• ◮ Gauss’s law (integral form):

ǫ0

• E · d

a = q

◮ Gauss’s law (differential form), for charge density ρ(

r): ǫ0∇ · E = ρ and ∇ × E = 0

• r equivalently, in terms of the potential:

−ǫ0∇2φ = ρ

2

Magnetostatics

◮ Biot-Savart law: magnetic field around a current I

• B(

r′) = µ0I dl × ( r′ − r) 4π| r′ − r|3

◮ Ampere’s law (integral form)

• B · d

l = µ0I

◮ Ampere’s law (differential form), for current density

j( r): 1 µ0 ∇ × B = j and ∇ · B = 0

x 3

Electromagnetic induction

◮ Faraday’s law for electromotive force:

EMF =

• dl ·

E( r′) = − d dt

• da ·

B( r)

• Flux Φ

◮ Differential form:

∇ × E = −∂ B ∂t

4

Equations so far

ǫ0∇ · E = ρ ∇ × E = −∂ B ∂t 1 µ0 ∇ × B = j ∇ · B = 0 Are these correct in general? Apply divergence to third equation: ∇ · j = 1 µ0 ∇ ·

• ∇ ×

B

• = 0

Divergence of current is the rate at which charge leaves a point (continuity equation i.e. charge conservation): ∇ · j = −∂ρ ∂t = − ∂ ∂t

• ǫ0∇ ·

E

• So how can we fix the equations?

5

Maxwell’s equations

ǫ0∇ · E = ρ ∇ × E = −∂ B ∂t 1 µ0 ∇ × B = j + ǫ0 ∂ E ∂t ∇ · B = 0 Apply divergence to third equation: ∇ · j + ∂(ǫ0∇ · E) ∂t = 1 µ0 ∇ ·

• ∇ ×

B

• = 0

⇒ ∇ · j + ∂ρ ∂t = 0 Now consistent with charge conservation.

6

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 = δqδxˆ x

◮ Polarization is the density of induced dipoles:

• P =

δ p δxδa = δq δa ˆ x

7

Bound charge due to polarization

◮ Charge density in infinitesimal chunk

ρb = δq1 − δq2 δaδx =

δq1 δa − δq2 δa

δx = Px(x) − Px(x + δx) δx = −∂Px ∂x

◮ Similarly accounting for y and z components:

ρb = −∇ · P

8

Current density due to polarization

◮ Charge crossing dotted surface in time δt:

δq = δq2 − δq1

◮ Corresponding current density:

jx = δq2 − δq1 δaδt =

δq2 δa − δq1 δa

δt = Px(t + δt) − Px(t) δt = ∂Px ∂t

◮ Polarization current density in general direction:

• jP = ∂

P ∂t

9

Materials in magnetic fields

◮ Charges circulate around magnetic field due to force q

v × B

◮ Magnetic dipole moment of infinitesimal current loop (per unit δz)

• δµ = 1

2

• r ×

dlδI = 1 2 (0 + δxδyˆ zδI + δyδxˆ zδI + 0) = δxδyˆ zδI

◮ Magnetization is density of induced magnetic dipoles:

• M =

δµ δxδy = ˆ zδI

(Per unit )

10

Bound current density due to magnetization

◮ Current within dotted element (per unit δz)

δIy = δI1 − δI2

◮ Corresponding current density:

jy = Iy δx = δI1 − δI2 δx = Mz(x) − Mz(x + δx) δx = −∂Mz ∂x

◮ Generalizing to all directions:

• jb = ∇ ×

M

(Per unit )

11

Material response summary

◮ Response to electric field

E is polarization P

◮ Polarization corresponds to bound charge density

ρb = −∇ · P and current density

• jP = ∂

P ∂t

◮ Response to magnetic field

B is magnetization M

◮ Magnetization corresponds to bound current density

• jb = ∇ ×

M

12

Maxwell’s equations including material response

◮ Fields produced by external ‘free’ charges and currents (ρf and

jf) as well as bound ones induced in the materials ǫ0∇ · E = (ρf + ρb) 1 µ0 ∇ × B = ( jf + jb + jP ) + ǫ0 ∂ E ∂t ∇ × E = −∂ B ∂t ∇ · B = 0

◮ Rewrite bound quantities in terms of polarization and magnetization

ǫ0∇ · E = (ρf + ρb) 1 µ0 ∇ × B = ( jf + jb + jP ) + ǫ0 ∂ E ∂t ǫ0∇ · E = ρf − ∇ · P 1 µ0 ∇ × B = jf + ∇ × M + ∂ P ∂t + ǫ0 ∂ E ∂t ∇ · (ǫ0 E + P) = ρf ∇ × B µ0 − M

• =

jf + ∂(ǫ0 E + P) ∂t

13

Maxwell’s equations in media

◮ Define fields

• D ≡ ǫ0

E + P

• H ≡
• B

µ0 − M

◮ Yields equations with free charge and current densities as the sources:

∇ · D = ρf ∇ × E = −∂ B ∂t ∇ × H = jf + ∂ D ∂t ∇ · B = 0

14

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 = χeǫ0

E

• M = χm

H

• D = (1 + χe)ǫ0

E

• B = (1 + χm)µ0

H ǫ = (1 + χe)ǫ0 µ = (1 + χm)µ0

◮ Anisotropic dielectric:

P = ¯ χe · ǫ0 E with susceptibility tensor ¯ χe

◮ Nonlinear dielectric:

P = χe(E)ǫ0 E

15

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

jP = σ E

◮ Remember

jP = ∂ P/∂t, so

• P =
• dt

jP = P0 + tσ E

◮ Important: the response is not instantaneous in general ◮ It can depend on the history i.e. is non-local in time

16

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 = ρf ∇ × E = iω B ∇ × H = jf − iω D ∇ · B = 0

◮ For Ohmic metal (with frequency-dependent conductivity σ(ω)):

σ(ω) E = jP ≡ ∂ P/∂t = −iω P ⇒ P = iσ(ω) E ω ⇒ ǫ(ω) = ǫ0 + iσ(ω) ω

17

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) µ(ω) = −iωǫ(ω)∇ × E = ω2ǫ(ω) B −∇2 B = ω2ǫ(ω)µ(ω) B using ∇ · B = 0

18

Electromagnetic wave equation

◮ Write Maxwell’s equation in linear media with no free charge or current as:

v2(ω)∇2 B = −ω2 B v2(ω)∇2 E = −ω2 E where v(ω) ≡ 1/

• ǫ(ω)µ(ω)

◮ First consider 1D version, v2 ∂2f ∂x2 = −ω2f. General solution:

f(x) = A exp(ikx) + B exp(−ikx) with k = ω/v

◮ General solution for v2∇2f = −ω2f:

f( r) =

• k

A

k exp(i

k · r) with | k| = ω/v

19

Electromagnetic wave speed

◮ Going back to EM waves and to time domain:

• E(

r, t) =

• k
• E

k0 exp(i

k · r − iωt)

• B(

r, t) =

• k
• B

k0 exp(i

k · r − iωt) with E0 ⊥ B0 ⊥ 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) = ǫ0 and µ(ω) = µ0, so speed of light c = 1/√ǫ0µ0 ◮ In materials, speed of light usually specified by refractive index

n(ω) ≡ c v(ω) =

• ǫ(ω)µ(ω)

ǫ0µ0

20