why study the optical properties of solids?where m;qare phenomenologically-introduced mass and...

Why study the optical properties of solids?

Jimmy Qin

29 August 2019

Recently, I was reading the book by Yu and Cardona called Fundamentals of Semiconductors:Physics and Material Properties. The core of the book is a detailed and expansive treatment ofthe optical properties of semiconductors.

The entirety of this field of study could be summarized in just two short questions.

1. What is ε(ω), the dielectric constant? This is absorption.

2. What is I(ω), the intensity of radiative emission? This is emission.

I found it hard to organize all the information in my head because there are so many processesto know about. First, I will summarize the different processes that occur in both absorption andemission, which of course are microscopic opposites. Then, I will elaborate on why we would liketo know about ε(ω) and I(ω), two seemingly elementary quantities.

I included lots of nice pictures to reduce the amount of math (and also increase the intuition!).

Here are some references:

• Yu and Cardona, Fundamentals of Semiconductors: Physics and Material Properties. (Springer,2010)

• Pelant and Valenta, Luminescence Spectroscopy of Semiconductors. (Oxford, 2012)

• Dresselhaus, Solid state physics part II: Optical properties of solids. (MIT 6.732, 2001)web.mit.edu/course/6/6.732/www/6.732-pt2.pdf

• Stoneham, Theory of Defects in Solids: Electronic Structure of Defects in Insulators andSemiconductors. (Oxford, 2001)

Absorption

Absorption is the process by which the material annihilates photons, or absorbs light.

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web.mit.edu/course/6/6.732/www/6.732-pt2.pdf

Jimmy Qin Why optical properties of solids?

Contributions to absorption

• Band-to-band absorption: An electron near the top of the valence band absorbs a photon ofenergy ~ω > Eg and gets excited to the conduction band. This is the dominant process in

solar cells, of course.

The cutoff Eg suggests that εi(ω) = 0 for ~ω < Eg. Here, εi is the imaginary part of thedielectric constant, which controls absorption. In this sense, Eg is called the fundamentalabsorption edge. It can be direct or indirect depending on whether the bandgap is direct(GaAs) or indirect (Si). In the indirect case, the absorption is typically accompanied by asimultaneous process with a phonon, since the momentum of the photon is negligible (seeNelson, Physics of Solar Cells). Pictures of the fundamental absorption edge are below:

The fundamental absorption edge is really not so fundamental after all. In the case of excitonabsorption, there can be absorption for ~ω > Eg because exciton processes do not care aboutthe bandgap.

What do the measurements tell us about the material? In this case, measurementstell us about (1) the bandgap Eg (2) whether the bandgap is direct or indirect, in the sensethat if you drive T → 0 the phonon population should go to zero and indirect bandgaps wouldgive a “smearing” of the fundamental absorption edge (3) the occupation of the conductionand valence bands, based on the strength of the absorption edge.

• Excitonic absorption: An exciton is a bound state of a carrier electron and a (carrier) hole.The absorption of light γ can happen with an exciton X in two general ways:

γ → X and γ +X → X.

Generally, energy is only dissipated if the interaction of excitons with phonons is very large.(Otherwise, energy tends to get traded continuously between photons and excitons. The

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coupled process of EM and excitonic polarization Pexciton is called an exciton-polariton.There are different kinds of polaritons, depending on where the macroscopic polarizationcomes from.)

Consider the first process γ → X. In this case, absorption merely depends on the matrixelement

〈X|Hint|0〉 =1√N

∑rk

eik·rφnlm(r)〈ck|Vdip|vk〉.

Here, Vdip is roughly the A · p dipole interaction and ck means conduction electron withmomentum k, etc. φnlm is the “hydrogenic” wavefunction of the exciton (there is also awavefunction describing the translational motion of the center of mass). In this process, thephoton must have energy equal to

~ω = E0 − EX .

I think E0 is greater than EX because the electron should be attracted to the hole, i.e.excitons are energetically favorable. Because the exciton is like a hydrogen atom, EX isquantized. Therefore, ω for the excitonic creation processes also tends to be quantized. Thiscan show up experimentally:

The second process γ + X → X can be thought of as a photon-induced transition betweenthe energy levels ni and nf of the exciton. In this case,

~ω = Enf− Eni

.

These values of ω are also quantized. Again, we can see them only if the interaction betweenthe phonons and excitons are large, such that the energy from the light ultimately goes tothe phonons and gets dissipated as heat.

What do the measurements tell us about the material? The measurements can tellus about the energy levels of the defects, and the strength of the absorption peaks can tellus about the density of defect states.

• (Optical) phonon absorption: Recall there are two kinds of phonons, acoustic (low-energy)and optical (higher energy). The simplest models of these phonon modes are due to Debyeand Einstein, respectively.

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Optical phonons interact with light (hence the name). This is sometimes called latticeabsorption. The energy levels of phonons are continuous, rather than discrete as in theexcitonic case, although for high quantum number n the excitonic energy levels can be treatedas roughly continuous. Hence, lattice absorption lends itself to description by classicaldifferential equation,

M u = −Mω2Tu +QE

where M,Q are phenomenologically-introduced mass and charge of the phonon dispersionand phonon-light interaction, respectively. The solution of D = εE is called Sellmeier’sequation and is continuous in ω,

εphonon(ω) = 1 +∑i

NiQ2i

ε0Mi(ω2i − ω2)

.

This is an approximate result. It can be improved by remembering that the phonon macro-scopic polarization Pphonon also induces an electric field, so in fact we should solve the coupledMaxwell wave equations inside the medium before obtaining the dielectric constant. This iscalled the phonon-polariton absorption.

Of course, this is merely approximate, and no real material would have ε(ω) blow up atω = ωi, although there would be a meaningful peak. This sickness can be cured phenomeno-logically by introducing a damping term −γu into the equation of motion above, see Peatrossand Ware Physics of Light and Optics. The result for the real and imaginary components ofε(ω) for a single harmonic oscillator at frequency ωT is:

What do the measurements tell us about the material? We can learn about thephonon dispersion relation and the temperature-dependence of the phonon occupations (i.e.the phonon density of states). This can be especially helpful for optical phonons in thecontext of the Einstein model https://en.wikipedia.org/wiki/Einstein_solid, whereall the optical phonons have the same frequency.

• Carrier electron absorption: A typical bandstructure is kind of complicated. In this bulletpoint, we use the word “band” in the sense that there are lots of bands for valence orconduction electrons to occupy. In other words, use “band” in the sense of the picturebelow.

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https://en.wikipedia.org/wiki/Einstein_solid


There are two important kinds of processes here. One is intraband absorption, whichis when a carrier electron absorbs a photon and goes to a different spot in the same band.Another is interband absorption, which is when a carrier electron absorbs a photon andgoes to a different band. In the case that the two bands involved are separated by less thanEg, this can produce a transition ~ω < Eg.

That was easy. Let’s just answer a few more questions.

1. Why don’t we consider the valence electrons? Typically there are not many emptystates for valence electrons, so there is nowhere for the valence electron to jump to if itis excited by a photon, except to the conduction band.

2. Why is this process smaller than the band-to-band absorption? This is because thereare not many electrons in the conduction band, especially at low temperatures, T .

The calculation of ε(ω) for conduction electrons can be done approximately with the Drudemodel. Typically we think of Drude model as a tool to use for metals, but it can be usedfor any material in which there are free electrons. The difference is that metals have lotsof free electrons and semiconductors have only a very low density of free electrons, so theDrude contribution is not as important in the case of semiconductors.

What do the measurements tell us about the material? We can learn about thedetailed bandstructure of the electronic bands and also learn about any symmetries presentin the bandstructure, based on selection rules for the photon absorption.

• Absorption by shallow impurities: We know that the bound states of impurities and elec-trons (or holes) attracted to them can be treated as approximately hydrogenic. Hence, theabsorption of light by shallow impurities is something like

~ω = Ec − Eimpurityn ,

where Ec is the conduction band energy and Eimpurityn is the nth energy level of the hydrogenic

impurity. Alternatively, we could jump between energy levels of the impurity:

~ω = Eimpuritynf

− Eimpurityni

.

And finally, we could jump from the valence band to an impurity state.

~ω = Eimpurityn − Ev .

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At low temperature, the absorption spectrum is quite sharp.

What do the measurements tell us about the material? We can learn how shallow(or deep) the impurities are, and learn about the impurity energy levels. We can also learnabout how many impurities are in the material, i.e. how “clean” the sample is.

• Franz-Keldysh effect: If we apply a (quite strong) uniform electric field E to the sample, thenthe material is no longer translationally symmetric in that direction. The bandstructure willget a little messed up and the conduction and valence bands will kind of “bleed” into thebandgap; they will be described by Airy functions. As we know from the theory of classicaloptics, Airy functions are oscillatory and die out rather quickly. The oscillatory nature ofthe modified bandstructure turns up in the absorption spectrum, which can now be nonzerofor ~ω < Eg:

What do the measurements tell us about the material? Not sure. I do knowthat Franz-Keldysh effect is useful technologically; for example see https://en.wikipedia.org/wiki/Electro-absorption_modulator. “An electro-absorption modulator (EAM) is

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https://en.wikipedia.org/wiki/Electro-absorption_modulator

https://en.wikipedia.org/wiki/Electro-absorption_modulator


a semiconductor device which can be used for modulating the intensity of a laser beam viaan electric voltage.”

Tools in absorption calculations

• Due to the Kramers-Kronig relations

εr(ω)− 1 =2

πP∫ ∞0

dω′ω′εi(ω

′)

ω′2 − ω2and εi(ω) =

−2ω

πP∫ ∞0

dω′εr(ω

′)

ω′2 − ω2,

we can choose to measure either εr(ω) or εi(ω) and either choice gives equivalent information.

• The basic interaction parameter of matter with light is the interaction

Hint =−qm

A · p,

where p is the quantum-mechanical momentum (i.e. of the wavefunction). To learn moreabout it, see my notes “Appendix: Electric dipole approximation.”

• Absorption strength is described by the rate R(ω), where

R(ω) =2π

~(e|E(ω)|)2

(2mω)2

∑k

∣∣Pcv(k)∣∣2δ(εkc − εkv − ~ω).

Actually, the expression above is only for the band-to-band transition. Generally it couldtake the form

R(ω) =2π

~(e|E(ω)|)2

(2mω)2

∑k

∣∣Pfi(k)∣∣2δ(εkf − εki − ~ω).

Here,

Pfi(k) = 〈fk|e · p|ik〉 =

∫r,unit cell

u∗kf (r)(e · p)uki(r)

is the dipole matrix element. Generally, p, the momentum operator, is allowed to act oneither the bra or the ket.

A similar form of Fermi Golden Rule is applicable to all of the absorption processes describedearlier.

Emission

Emission means photons are produced by the sample. There are two broad categories of emission:

1. Luminescence: some electrons or excitons, etc. are already excited. Their return to theground state is accompanied by an emitted photon. Luminescence is also called radiativerecombination.

There are different ways to get the sample excited in the first place. These include electro-luminescence, thermoluminescence, photoluminescence, cathodoluminescence.

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2. Inelastic scattering: light entering the sample is scattered by time-dependent fluctuationsinside the sample. Time-dependent fluctuations can always be described by phonons, soequivalently the photons are scattered inelastically by phonons.

Raman scattering is scattering of photons by optical phonons. Brillouin scatteringis scattering of photons by acoustic phonons. In fact, microscopically the photons cannotinteract directly with the phonons; the interaction is mediated by a virtual electron-hole pair(i.e. exciton). I suppose it would be easy to derive an effective theory of phonon-photoninteraction.

Types of luminescence

• Band-to-band transition: The emission of light by a conduction electron which returns tothe valence band is called a band-to-band transition. They are the dominant process forsemiconductors at high-T . It is easy to show that the intensity of emitted photons at energy~ω is roughly

I(ω) ∼ Θ(~ω − Eg)(E − Eg)1/2e−~ω/kT .

Figure 1: The left edge is smeared out, compared to the sharp edge predicted theoretically

What do the measurements tell us about the material? This tells us about thebandgap and the population of carriers at varying temperature.

• Free-to-bound transition: At low-T , there are not many conduction electrons, so band-to-band transition gets suppressed. Instead, a new process called the free-to-bound transi-tion may occur. Consider a p-type semiconductor with acceptor-type impurity at energyEA above the valence band. The impurity is much more likely than the valence band to haveempty states, so any conduction electrons that are still alive will be likely to jump down andstick to the impurity. The peak is at

~ω = Eg − EA .

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Technically, a bound electron on an impurity has the hydrogenic energy levels, so thereshould be more than one peak.

As the concentration of impurities increases, the impurity wavefunctions tend to overlap andthey form a band in the middle of the bandgap. Therefore the emission peak tends to getsmeared out at higher doping – see the figure below.

Figure 2: The floating numbers are the concentrations of impurities.

What do the measurements tell us about the material? The peaks tell us about theimpurity energy levels. The width of the peaks tell us about the density of impurities inthe sample. Additionally, comparison of the free-to-bound transition with the band-to-bandtransition tells us about the behavior of the material with varying temperature.

• Donor-acceptor pair transition: Consider a semiconductor at low temperature which has bothdonors D and acceptors A. Obviously the impurities can be ionized; the DAP transitionis

D + A→ ~ω +D+ + A−.

Here,

~ω = Eg − EA − ED +e2

4πεε0R,

where R is the distance between the donor and acceptor impurities, taken to be much greaterthan the lattice spacing R� a. Because the impurities tend to be located at lattice sites, Ris quantized. The corresponding spectrum has all kinds of interesting peaks depending onthe value of R:

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The large number of peaks is equivalent to saying there are is a large number of data pointswith which to calculate results. For example, the value EA +ED can be calculated to greatprecision. The lower the temperature, the sharper the peaks are.

What do the measurements tell us about the material? The large number of peaksgives us very accurate information about EA + ED. Also, the strengths of the peaks givesus information about how far apart the acceptors are from the donors. If there is some kindof spatial pattern in the doping (i.e. where the acceptors and donors tend to be, relative toeach other), it will show up in the peaks.

• Free exciton transition: The free exciton emission is when an exciton self-annihilates,

X → γ.

This reasoning will be wrong, but I will write it down to show what is incorrect.An exciton consists of a carrier electron and a hole which tango together. Compared tothe energy of a valence electron, the carrier electron has extra energy Eg. However, theattraction of the electron to the hole lowers the exciton energy by a hydrogenic energy level;hence the energy of the emitted photon is like

~ω = Eg −EX

n2.

This is just the microscopic reverse of the absorption spectrum.

Nope. Nature is more complicated than that. Actually, the excitons tend to thermalizerather quickly, but the thermalization gets much slower near the bottleneck region. Manyexcitons get trapped in this bottleneck region, and only then do they live long enough to self-annihilate and produce radiation. There are essentially two bottleneck regions: one on theUPB (upper polariton branch) and another on the LPB (lower polariton branch). Therefore,there are two emission peaks. Actually for very pure materials, these peaks tend to merge.

Figure 3: Two peaks; one is much weaker than the other.

The bottleneck region is where the dispersion relation gets very flat. In any region, the lossof energy via phonons must happen with phonons of small wavevector k. In the bottleneckregion, the loss of energy must proceed with acoustic phonons, since they have lower energy.

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Figure 4: Bottleneck is where dEdk

is small.

What do the measurements tell us about the material? For very pure materials, thetwo peaks tend to merge; this turns out to be because the defects mediate the existence ofthe gap energy or something weird like that. Therefore, this gives a highly sensitive test ofhow pure the sample is, i.e. whether it is contaminated with defects.

• Bound exciton transition: The bound exciton transition is also a kind of exciton self-annihilation, except this exciton has lower energy because it is bound to an impurity. ForWannier excitons of large radius aX , it is quite likely that there will be an impurity inthe region of volume a3X . Therefore at low temperature, the bound exciton transition issignificantly stronger than the free exciton transition.

The process is (for example, for excitons X bound to an ionized donor D+)

D+X → D+.

This emits light of frequency

~ω = (Eg − EX)− Ebind ,

where Ebind is the binding energy of the exciton to the defect. We see that the general ideaof ~ω = Eg − EX is applicable to the bound excitons but not to the free excitons. For freeexcitons, the exciton-polariton many-body effect is dominant. Here, the excitons are kindof stuck on a single site so we do not have to worry about the dispersion relation and stufflike that.

What do the measurements tell us about the material? Because a single Wannierexciton can see lots of defects, the bound exciton emission peak tends to be very strong.This is termed the giant oscillator strength. The strength of the bound exciton emissionpeak tells us about the concentration of impurities, relative to the Wannier radius aX .

Of course, this can also tell us about the exciton-defect binding energies.

Types of inelastic scattering

• Raman scattering: Raman scattering describes the scattering of light off of optical phonons.In the simplest macroscopic description, the phonons have energy ω0 and change the suscep-tibility to

χ = χ0 + χ1 cos(ω0t).

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If the incoming EM wave has frequency ω, then the outgoing wave will be like

P = E0

[χ0 cos(ωt) +

χ1

2cos((ω + ω0)t) +

χ1

2cos((ω − ω0)t)

].

Light is either scattered elastically (Rayleigh scattering), inelastically to lower energy(Stokes process), or inelastically to higher energy (anti-Stokes process). Even in elasticscattering, the photon is technically allowed to change direction. This is because to a firstapproximation, the photon carries no momentum.

The phonon frequency is equal to the difference between incident and scattered photonenergies. This is called the Raman shift.

Figure 5: TO = transverse optical; LO = longitudinal optical.

Microscopically, there is no direct photon-phonon interaction. Therefore, the Raman scat-tering process is mediated by virtual electron-hole pairs.

What do the measurements tell us about the material? Raman scattering can tellus about the vibrational and rotational modes of molecules, and is the preferred method bywhich chemists (and forensic scientists!) determine the identity of an unknown substance.Besides phonons, there are analogous Raman scattering processes off of many kinds of dy-namical excitations, such as plasmons, magnons, and superconducting gap excitations, soRaman scattering can tell us about those frequencies as well. Sometimes it is used to learnmore about old paintings and such. Lots of uses!

On the theoretical side, the absence of Raman scattering can tell us about the symmetries ofthe zone-center phonon mode. This can be described theoretically with the Raman tensor,which is described in the book by Yu and Cardona. Also, it can tell us about the phonondispersion near the zone-center, q ≈ 0.

• Brillouin scattering: Brillouin scattering is the scattering of light off of acoustic phononwaves. The main difference is the different dispersion relation,

ω(q) = vac|q|.

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Compared to an acoustic phonon, we can no longer say the photon has negligible momentum,in the sense that optical phonons have much higher energy than acoustic phonons. Thereforewe can determine the energy of the phonon ωphonon with the energy ω and scattering angleθ of the light. Ignoring the frequency-dependence of refractive index, approximately

ωphonon = vac2nω

csin

θ

2.

Figure 6: Brillouin scattering can also produce sharp Stokes peaks, if our detector is at afixed angle θ relative to the incoming light probe.

What do the measurements tell us about the material? Brillouin scattering isextraordinarily useful in determining elastic properties of materials. It can sense where fiberoptic cables are highly stressed, for example, and is the preferred way of determining thecomplete elastic tensor of a solid. Finally, it can tell us about vac or about the index ofrefraction, n(ω).

• Resonant Raman or Brillouin scattering: The scattering amplitudes in either Raman orBrillouin scattering are enhanced by orders of magnitude of the transition energy of a virtualprocess happens to correspond to some kind of transition energy for the electrons. (Recallthat the Raman and Brillouin scatterings are microscopically identical and proceed via theelectron-photon and electron-phonon scatterings.) These transitions energies could be thosecorresponding to bandgaps, energies between microscopic bands, or energies correspondingto the bound states of excitons. For more about resonances, see Weinberg Quantum Theoryof Fields, volume I.

What do the measurements tell us about the material? This can tell us about thebandstructure of the material, or about the energies of the excitons.

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Why ε(ω) and I(ω)?

From the notes by Dresselhaus at MIT:

The optical properties of solids provide an important tool for studying energy band structure,impurity levels, excitons, localized defects, lattice vibrations, and certain magnetic excita-tions. In such experiments, we measure some observable, such as reectivity, transmission,absorption, ellipsometry or light scattering; from these measurements we deduce the dielec-tric function ε(ω), the optical conductivity σ(ω), or the fundamental excitation frequencies.It is the frequency-dependent complex dielectric function ε(ω) or the complex conductivityσ(ω), which is directly related to the energy band structure of solids.

This illustrates the cycle between theory and experiment. To get absorption or emissionpeaks and their strengths and frequencies, we have to do experiment. To understand wherethey come from, we have to describe them with theory. If we believe that our theoretical de-scription is sufficient, then we can describe the microscopic properties of unknown materialsusing only the measurable and macroscopic parameters ε(ω) and I(ω).

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why study the optical properties of solids?where m;qare phenomenologically-introduced mass and...

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