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Absorption

Macroscopic electrodynamics

What is light?

Electromagnetic radiation

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of

electric and magnetic fields.

Light

This diagram shows a plane linearly polarized wave propagating from left to right. The electric field is in

a vertical plane and the magnetic field in a horizontal plane.

From Wikipedia, the free encyclopaedia.

Maxwell`s equations

• Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.

• Maxwell's equations have two major variants. The "microscopic" set of Maxwell's equations uses total charge and total current including the difficult-to-calculate atomic level charges and currents in materials. The "macroscopic" set of Maxwell's equations defines two new auxiliary fields that can sidestep having to know these 'atomic' sized charges and currents.

Maxwell`s equations

• Name

• Gauss's law

• Gauss's law for magnetism

• Maxwell–Faraday equation (Faraday's law of induction)

• Ampère's circuital law (with Maxwell's correction)

• Constitutive relations

• In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between displacement field D and E, and the magnetic H-field H and B. These equations specify the response of bound charge and current to the applied fields and are called constitutive relations.

• where P is the polarization field and M is the magnetization field which are defined in terms of microscopic bound charges and bound current respectively

Fourier transform

• The Fourier transform is a mathematical operation with many applications in physics and engineering that expresses a mathematical function of time as a function of frequency, known as its frequency spectrum; Fourier's theorem guarantees that this can always be done.

• The function of time is often called the time domain representation, and the frequency spectrum the frequency domain representation. The inverse Fourier transform expresses a frequency domain function in the time domain. Each value of the function is usually expressed as a complex number (called complex amplitude) that can be interpreted as a magnitude and a phase component. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces.

Fourier transform

• There are several common conventions for defining the Fourier transform of an integrable function ƒ : R → C (Kaiser 1994). This article will use the definition:

• for every real number ξ.

• When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz).

Fourier transform

• Under suitable conditions, ƒ can be reconstructed from by the inverse transform:

• for every real number x.

• For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ.

• The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.

Electrodynamics

• In dielectric medium an external eletromagnetic wave with electrical field vector E(r,t)=E(q,w) exp i(qr-wt), where q is wave vector and w frequency, will induced polarization vector P which is related to applied field via second-rank tensor cij called as the electric susceptibility tensor.

Electrodynamics

• Pi(r`,t`)=Sj Integral cij(r,r`t,t`)Ej(r,t)dr dt (1) • Time is homogeneous, we assume that the space is also

homogeneous to avoid complication such a local field corrections. With this assumption (1) ccan be simplified to

• Pi(r`,t`)=Sj Integral cij(Ir-r`I, It-t`I)Ej(r,t)dr dt (2)

• From the convolution theorem (2) can be expressed in terms of the Fourier transforms of P, c, E

• Pi(q,w)=cij(q,w) Ej(q,w) (3)

Electrodynamics

• Note: Pi(r`,t`), cij(r,r`,t,t`), Ej(r,t) are real, but their Fourier transformation can be complex.

• The fact that cij(r,r`,t,t`) is real implies that cij(q,w)=cij*(-q,w)

• For comparison with experiment can be more convenient to define another complex second-rank tensor dielectric tensor eij which is defined by

• Di(q,w)=eij(q,w) Ej(q,w)

• Where Di(q,w) is the Fourier transform of the electric displacement D(r,t) defined by

Electrodynamics

• D(r,t)=E(r,t)+4pP(r,t)

• From their definition cij(q,w) and eij(q,w) are related by: eij(q,w)=1+4p cij(q,w),

There are important relations:

eij(-q,w)= eij*(q,w)

eij(q,w)= eji*(-q,w)

Approximations

• In the most cases the wavelength of the light is much larger than the lattice constan or other relevant dimensions.

• As results the magnitude of photon wave vector q can be assumed to be zero and we can abbreviate e(q,w)—e(w) which known as dielectric function.

• eij and cij depend on the crystal symmetry. For isotropic medium and cubic crystals eij has only three identical diagonals elements and we can replace eij(w) tensor by scalar e(w). e(w) can be write as e(w)=er(w)+iei(w)

Refractive index

• The macroscopic optical properties of an isotropic medium can be also characterized by a complex refractive index n~. The real part n is called refractive index, the imaginary part k is known as extinction index or coefficient.

• The normal incident reflection coefficient or reflectance R of semiifinite isotropic medium in vacuum is given by:

• R=|(n~-1)/(n~+1)|2

Absorption coefficient

• When light is absorbed in passing through a medium from a point r1 to another point r2, the absorption coefficient a of the medium is defined by:

• I(r2)=I(r1) exp(-a |r2-r1|)

• I(r) is the light intensity at r.

• The absorption coefficient a is related to k by

a=4pk/l0

l0 is wavelength in vacuum.

The complex refractive index n~ is related to e(w) by e(w)=(n~)2

Kramers Kronig Relations

• Optical constant (function) n, k, er, ei R, F (phase angel of reflection) are not independent and they can be determined from the others. If we know spectral dependence of an one in all spectral region, we can calculate other one for constant frequency.

Kramers Kronig Relations

Microscopic theory of absorption coefficient

• We will used semi-classical approach to derive the Hamiltonian describing the interaction between an external electric field and electrons in semiconductor crystal.

• The electric field is treated classically while the electrons are described by QM (Bloch) wave function.

• This approach is not fully rigorous as fully QM treatment in which electromagnetic waves are quantized into photons.

Microscopic theory of absorption coefficient

• We start with unperturbed one-electron Hamiltonian H0=p2/2m +V(r)

• To describe the electromagnetic field we introduce a vector potential A(r,t) and scalar potential F(r,t)

The choice of these potentials is notunique due to gauge invariance, for simplicity we will choose the Coulomb gauge in which F=0 and div A=0. In this gauge E and B are defined by E=-1/c dA/dt and B=div x A

Microscopic theory of absorption coefficient

• The classical Hamiltonian of charge Q in the presence of external electromagnetic field can be obtained from the free-particle Hamiltonian by replacing the momentum p by p-(QA/c), for electron Q=-e

• H=1/2m [p+eA/c]2 + V(r) (5)

note that p does not commute with A

Microscopic theory of absorption coefficient

• We substitute to equation (5)

1/2m (p+eA/c)2=p2/2m+(e/mc) A p +(e/mc) p A +e2A2/2mc using the definition of p (momentum) as the operator (h/i) div, we can express the term (p A) f(r)=A(h/i divf) + (h/i divA) f(r) from the definition of A we know that divA=0 for the purpose of calculating linear optical properties we can neglect the term e2A2/2mc

Microscopic theory of absorption coefficient

• H= H0+ e/mc A p

• HeR=e/mc A p this term describes the interaction between the radiation and Bloch electron

• HeR depends on the gauge , another form of HeR commonly used in literature is: HeR=(-e) r E

These Hamiltonians are equivalent in the limit that momentum q is small. It corresponds to electric dipole approximation.

Microscopic theory of absorption coefficient

• We want to calculate absorption coefficient

• This product of transition probability of electrons from valence band to conduction band and density of states and distribution function.

• 1. Density of states we know: N(E)dE=(1/2ph`3) (2mef)

3/2 E1/2 dE

• 2. Distribution function is 1 at zero K.

• 3. Matrix element of the transition has to be determined.

Matrix element

• We assume that A is weak enough that we can applied time perturbation theory to calculate the transition probability per unit volume R from an electron in the valence band state |v> with energy Ev and wavevector kv to the conduction band state |c> with corresponding energy Ec and wavevector kc. To do this we need evaluate matrix element:

• |<c|HeR|v>|2= (e/mc)2 |<c|A p|v>|2

Matrix element

• We can write the potential vector A as A e, A is amplitude and e is unit vector parallel to A. In terms of the amplitude of the incident electric field E(q,w) the amplitude A can be rewrite as: A= -E/2q{exp[(i(qr-wt)] a c.c. }, where c.c. is complex conjugate.

• The calculation of the matrix element involves integration over time and space.

Matrix element

• Integration over time leads to law of energy conservation: ~d(Ec(kc)-Ev(kv)-hw)

• From the periodicity and summation over all lattice vectors Rj we obtain law of momentum conservation: d(q-kc+kv)

• Now we have to integrate only over unit cell Integral u*ckcexp [i(q-kc+kv) r] p uvkv dr` is simplified to

Integral u*ckv+q p uvkv dr` ; if q is small we can expande it into Taylor series: uckv+q= uckv+q duckv/dk+ q is small, we can use only the first term and this is electric dipole matrix element, dipole approximation.

If the electric dipole is zero, matrix elements gives rise to electric quadrupole and magnetic dipole tansition.

Matrix element

The same procedure can be used for k, when k is small (near the gap of semiconductor).

<u*ck|ep|uvk>=<u*c0|ep|uv0>+k d/dk <u*ck|ep|uvk>+….

Matrix element |<u*c0|ep|uv0>|2 does not depend on k is constant.

Absorption coefficient follows the density of states.

Absorption coefficient of direct semiconductors

Absorption coefficient of direct semiconductors

• 1.

a= |pcv(0)|2 (2mr/h2)3/2 (hw-Eg)1/2

dipole allowed transition of the first order

• 2. Matrix element |<u*c0|ep|uv0>|2 is zero a= |pcv(0)|2 (2mr/h2)3/2 (hw-Eg)

3/2 dipole transition of the second order or direct forbidden transition

Matrix element |<u*c0|ep|uv0>|2 is not zero

Absorption coefficient of direct semiconductors

Absorption coefficient of indirect semiconductors

Absorption coefficient of indirect semiconductors

• Main differences three particles (electron, photon and phonon)

• 1. Laws of conservation

• A) Energy : Ev+hw+-hW=Ec

• B) Momentum: kc=kv+qphot+qphon

Absorption coefficient of indirect semiconductors

• 1. Matrix element |<u*c0|ep|uv0>|2 is not zero a~ A (hw-Eg

ind+-hW)2

• 2. Matrix element |<u*c0|ep|uv0>|2 is zero a~ A (hw-Eg

ind+-hW)3

A contains distribution function of phonons and matrix element.

Absorption coefficient of indirect semiconductors

Absorption coefficient of indirect semiconductors

Absorption coefficient of indirect semiconductors

Absorption coefficient

How to explain peaks on the curves? By excitons.

Excitons

• Exciton is a pair of electron and hole which is bound by Coulomb interaction.

• Exciton can be solved as hydrogen atom, with different mass.

• En(K)=Eg-EX/n2 +Ekin, where EX=(mr/m0)/e2 EH, EH=13.6 eV

Excitons

Absorption coefficient of exitons

• Direct semiconductors

• 1. S=0 Sz=0 singlet state a~1/n3 d(hw-Eg+EX/n2)

• 2. S=1 Sz=-1, 0, 1 triplet state a~n2-1/n5 d(hw-Eg+EX/n2)

Absorption coefficient of exitons

Absorption coefficient of exitons

Absorption coefficient of exitons

• Indirect semiconductors

• 1. S=0 Sz=0 singlet state a~1/n3 (hw-Eg

ind+EX/n2+-hW)1/2

• 2. S=1 Sz=-1, 0, 1 triplet state a~n2-1/n5 (hw-Eg

ind+EX/n2+-hW)3/2

Absorption coefficient of exitons in Ge

Notes

• 1. We studied only transition hw>=Eg

• 2.The conservation of momentum is possible not only interaction with phonons, but also by scattering on impurities.

Two photons absorption

• We studied one photon absorption.

• Is possible two photons absorption for photons with energy < Eg?

• This process was predicted by M. Göppert-Mayer in 1931.

• This is based on the interaction of light with matter in semi classical approach, in the second order of perturbation theory, because in the first order the this contribution is zero.

Two photons absorption

• In OPA the matrix element in dipole approximation is |<c|ep|v>|2

• In TPA the matrix element in dipole approximation is: |Ss{(<c|e1p|s> <s|e2p|v>)/(Es-Ev- hw2))+ + (<c|e2p|s> <s|e1p|v>)/(Es-Ev- hw1))}|2 d(Ec-Ev-hw1-hw2)

Two photons absorption

• In this case the absorption coefficient depends on light intensity

• a(2)=A I