OSE5312 Slides Class Zz - Summary for Final - 2up

8/13/2019 OSE5312 Slides Class Zz - Summary for Final - 2up

1/40OSE5312 Spring 2013 - Summary for Final Exam

College of Optics & Photonics, University of Central Florida www.creol.ucf.e

Light Matter Interaction Spring 2013 Summary for Final slide 1

Final exam Tuesday April 30, 4pm-6.50pm, Room 102/103

Closed book exam, covers all topics discussed in class


Broad outline

- Electric fields accelerate charge- Accelerating charge causes light re-radiation, complex index, absorption

- If we understand how many charges move by how much, can predict opt. prop.

Different models used to understand charge motion:

Classical

- Lorentz model bound charges: atoms, insulators, molecules, ~semic.

- Lorentz model (nonlinear), bound ch.: same,(2)() , (3)(), - Lorentz model (atom/ion motion): vibration in molecules, polar solids

- Debye model (molecular reorientation) molecular liquids

- Drude model free charges: metals, doped semiconductors

Quantum mechanical

- Schrdinger equation: mass has wave-like character

Time independent S. Eq.: e- in well, Energy Eigenstates

Limited basis set method: e- in well in DC field: vs. 0Time dependent pert. theory: e- in oscillating E (), Fermi golden rule Schrdinger eq. for atom cores: rotating and vibrating molecules, ()Electrons in periodic potential: semiconductors, energy bands, gap ()


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OSE5312 Spring 2013 - Summary for Final Exam



Broad outline

1. Maxwells equations

realize that any polarization response leads to changes in light propagation

derive wave equation,

time dependent polarization response complex susceptibility e()gave expressions for n, , , R, vg if you know , can calculate opt. prop.


Course recap / context

Assumed medium is homogeneous, linear, isotropic, has some polarization P

1

Ignored presence of atoms, assumed some dipole moment per unit volume P

Induced dipoles modify light propagation refractive index, absorption coefficient


3/40




Maxwells Equations including explicit current sources

Inside materials these quantities contain separate charge and current contributions

1. diverging E fields relate to charges (bound and free)

2. diverging magnetic flux density doesnt exist

3. rotating E fields are related to changing magnetic fields

4. rotating B relates to changing E and several currents

free current density magnetization current density polarization current density

Which of these three current densities have we described using the Lorentz model?

Scalar wave equation


Permittivity and permeability

permittivity

permeability

electrical susceptibility (~ -100-100)- related to the polarization of the medium

magnetic susceptibility (~10-5)- magnetic response of the medium

dielectric constant- response of medium + response of vacuum

Frequency dependent polarization:

with


4/40




Complex refractive index

Complex susceptibility complex index n+i, complex r= r+ir = 1+i2

Relation between and r: = r (assuming magnetic response negligible)

22

22

)1(

)1(

n

nR

er 1)( 00

c

2)(

')(21

rrn ')( 21 rr

Permittivity:

Refractive index from dielectric function:

Dielectric function from refractive index : )()(2'' nr )()('22 nr

Related optical properties:

Absorption coefficient

Reflectance


1. The phase velocity the speed at which a phase front moves through a material:

Phase velocity: In vacuum:

2. The refractive index n(w) is the reduction of the phase velocity compared to c: .

Dispersion of light inside media

Additional properties of plane wave solutions to Maxwells Equations:

with n the refractive index n(1+)3. Wave vector 4. Wavelength 5. Group velocity

with

dk

dvg

Why approximate?

IMPORTANT! These are the kinds of things you are asked to calculate on exams


5/40




Useful formulas

Reflection coefficient from medium 1 to medium 2 under normal incidence

Reflection coefficient under normal incidence from air on planar surface

1

1

Transmission through absorbing slab, ignoring multiple internal reflections

1


Broad outline

2. Kramers Kronig relations

Causal nature of (t) () is related to () and vice versa If you know (), can calculate n(), and from there R, vg


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Using reality condition OR Cauchy, found Kramers-Kronig relations linking

- real refractive index n to an absorption spectrum

- real susceptibility X to a spectrum of the imaginary susceptibility X

- a phase shift upon reflection to a reflectivity spectrum R

Summary - Kramers Kronig relations

Memorize these, and be able to construct a plot based on a provided X or spectrum


Broad outline

3. Lorentz model

Classical approach: Assume electrons bound to atom with spring constant

Calculate electron displacement using classical equation of motion

Gives dipole moment per atom, convert to P (dipole per unite volume) vs. field

Gives expression for if you know , can calculate opt. prop.

Works for materials with sharp optical transitions,

and approximately for materials with broad transitions (well below absorption)

Atoms, molecules

Insulators (below transition)

Semiconductors (well below bandgap)

D-band contribution to metal properties (sharp transitions within metal)

Example questions : dopants in transparent hosts, solids made up of Lorentz resonators


7/40





Built classical model of isolated atoms, completely ignored quantum effects

1

Assumed optical response was due to electron motion r() in harmonic potentialAssume that all atoms in a solid (or gas) act independently (no coupling effects)

(Added anharmonic binding potential to classical model)


Equation of motion

Equation of motion: totFma

Illuminate atom:

Charge displacement

Average electron position: tr

All forces acting on the electronsrestorefrictionelectricalt FFFF

Understand that our Lorentz model represents displacement of the electron cloud (!), and

position r(t) is the central position of this cloud


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Equation of motion

Equation of motion

)()()()( 2

02

2

tEetrmt

trm

t

trm

erf FFFam

or

Note: gamma here is in s-1 whereas all omegas are radians / second

Also note: is not defined as FWHM. In some cases the FWHM is equal to the value of

Illuminate atom:

Charge displacement


Lorentz model: complex susceptibility

Polarization in terms of susceptibility

)()()( ENP

im

Ne22

00

2 1)(

Polarization in terms of polarizability (=E)

which with

im

e

)(

1)(

22

0

2

gives

where the prefactor is related to the plasma frequency p throughm

Nep

0

22

Assumptions:

- restoring force linear in r

- damping linear in dr/dt

- averaged/macroscopic fields and local fields equal

- dipoles all point along the applied field (isotropic)

0

)()(

N

memorize

memorizememorize


9/40




Lorentz model examples - narrow resonance

0 = 4 p = 8 = 0.3

reflectance from air at normal incidence

22

22

)1(

)1(

n

nR0 2 4 6 8 10 12

40

20

0

20

Re ( )

Im ( )

0 2 4 6 8 10 120

2

4

6

n( )

( )

2 ( )

p

0 2 4 6 8 10 12

0

0.5

1

R( )

e 1)( 0

c

2)(

understand these trends

memorize

(or derive)

')(21

rrn

')(21

rr


Lorentz model examples - narrow resonance

Types of questions:

- Show index spectrum, calculate density of atoms

(e.g. one resonance know 0, know n(0), can find N)

- Show absorption spectrum of doped host material,

calculate gamma based on peak absorption and Ndopant,

calculate Ndopant if gamma is given

Show transmission spectrum (1-R)^2 e-z ,

find n(0)

Note - I may ask for an absorption cross-section based on transmission(see next slide)


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Absorption cross-section

Fraction of absorbed light in thin slab given by Nz

Write in terms of differential equation:

This will result in a solution of the form

Comparing this to we find

or

Absorption cross section

- if light passes within area it is absorbed by the ion / atom

Note: I can ask for absorption cross-section of a Lorentz resonator!


Optical properties of insulators at visible frequencies

Now that we know the concept of band structure: revisit insulators.

Insulators: electronic prop. also described by Schodinger equation in period lattice

BUT: strong binding of valence electrons leads to relatively flat VB

Example: calculated LiF band structure => well defined EVB , sharp onset of

http://www.crystal.unito.it/mssc2006_cd/tutorials/defects/defects_tut.html

DOS

VB

- - -___ n

LiFLiF


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Overview

4. Drude model

Same as Lorentz model, but assuming zero restoring forceMetals: large electron concentration, no restoring force large charge motion

Looks like extremely strong Lorentz resonator with zero res. frequency

Optical propeties described by TA RT

Large reflection below p, skin depth

Note: doped semiconductors with large free electron or free hole concentration:

Also partly described by Drude model


Refractive index vs. frequency Gold

Metals (free charges) can be described as oscillator with zero resonance frequency :- high absorption near resonance (at low freq)

- anomalous dispersion and n


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Color of metals look at frequency dependent reflectance

Ag and R

Cu: effect of d-electrons insufficient to reach r=0Reduced reflectance at short (blue) wavelengthsreddish color

Cu and R

Ag

Ag

Cu

Cu


Band structure of metals?

Now that we know band structure, revisit metals:

Electron states in metals also described by Schrdinger equation in periodic structure

metals also display electronic band structure

Difference with insulators and semiconductors: Fermi level within a band

low-energy transitions possible, DC conduction possible

Gold band structure

From thesis Pina Romaniello (page 94, online PDF)

SR=scalar relativistic, SO=including spin-orbit effects


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Overview

5. Quantum mechanics

(Matter and therefore) electron motion is wave-like in nature

Need wave equation to describe behavior of charges

QM model of charge position and motion QM description of ()



Started modeling electron motion quantum mechanically using square wells

1

Limited basis set method: DC charge displacement dipole moment, (0)Time dependent perturbation: dipole moment (), transition rates ()

All electrons / quantum systems assumed to behave independently, no coupling


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Time-independent Schrodinger equation

If p=mv is the electron momentum, then p2/2m = mv2 is the kinetic energy

write wave equation in terms of kinetic energy = total energy E - potential energy V

Where V is the position dependent potential energy of the electron

(warning: V is not the electric potential ! Notation potentially confusing.. )

Alternative notation:

This is the Time-independent Schrdinger equation.

Left part is the energy operator, called the Hamiltonian H

light: affected by n, electrons: affected by V


Confined electron modes in infinite potential well

Depending on n, wavefunctions have different symmetry

Note: in sketch energy of n indicated by dotted line, and shown as solid line

Lowest order solution (n=1)

is symmetric (even), whereas n=2

solution is antisymmetric (odd)

(unfortunately the odd solutions are

represented by even numbers n)

Terminology:

1 has even parity and

2 has odd parityNote increasing

level spacing

Solutions are called Eigensolutions or Eigenfunctions, and the

allowed energies are called Eigenvalues

Remember shape of

allowed modes


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Corresponding wavefunctions of finite well

First three Eigenmodes in finite well with depth of 8 energy units

Second even mode

First odd mode

First even mode

[ Slightly lower energy than E1 ]

Note:

- at higher energy, wavefunction enters further into high-V region

- Energy level spacing increases as energy goes up, just as in inf. well case

- Finite number of bound modes


Linear applied field

Finite well with small applied field : perturbed system

Ground state will deform, and have slightly different energy

Wavefunction will still look similar to n , but not exactly

1 is no longer an Eigenfunction ofthe perturbed system

Question: how can we find solutions to our perturbed system?

Looking for solutions to H = E (E=energy) with the Hamiltonian now given by

[ Warning: here Ez is electric field magnitude ]

Concept of a perturbed Hamiltonian, notion of a perturbed wavefunction

Realize that weak perturbation new Eigenfunctions, look similar to unperturbed n


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Modified ground state in infinite well with applied electric field

Example: new lowest energy Eigenfunction has c1 = 0.985, c2=0.174, c3=0.013

Corresponding to

Field has pulled wavefunction

to the left as expected

Exact solution (solid line) lies

close to result using limited

basis set (dashed line)

The dotted line indicates

first order perturbation theory,

which we will cover next

Energies close to inf. square well:

compared to 1, 4, and 9

Understand physical result of perturbing field


Pulsed perturbation with oscillatory field

To calculate time dependent polarization in response to oscillatory EM wave:

(note: electric field amplitude here is 2E0)

Describe Hp(t) as with

To see how our system (e.g. atom) responds to perturbation:

- start with unperturbed system in one state (e.g. ground state)

- turn on perturbation at t=0

- turn off perturbation at t=t0

Integrate amplitude in another state (e.g. 1st excited state) over time

Assume that amplitude in initial state does not change much (low power limit)

We will calculate the resulting amplitude in the excited state after the pulse

Functional form of time dependent perturbation due to incident field


17/40




Polarization under steady state illumination

Polarization

with

Which with leads to prediction of susceptibility

Question: Describe meaning of a3(1)

Be prepared to calculate an index

based on provided transition matrix element

Oscillatory perturbing potential find polarization from time dependent pert. theory:


Fermis Golden rule

- if the Eigenfunctions of a system are known, complete, and orthonormal

- if we know the form of the perturbation (e.g. ~linear in space for plane wave)

we can calculate susceptibility and absorption of a large collection of these systems

Quantum mechanical susceptibility for a system initially in ground state 1

2

Transition rates (Fermis Golden Rule):

Absorption coefficient: or

Optical cross-section stimulated em/abs

4

Dipole matrix element

Joint density of states per atom / quantum system

Volume per absorber (here: V-1 = Ne = e- per unit volume)

4

Joint density of states per volume


18/40




Blackbody model

Hot objects radiate due to thermal motion of charges (atom cores and electrons)

In an idealized object (perfect blackbody radiator), the emission spectrum is adirect measure of energy present in the system

Perfect blackbody: reflectivity = transmission = 0 emissivity = 1

Model: aperture in large box that is at some equilibrium temperature

[Based on Verdeyen, Chapter 7]


Allowed modes

Like e- in semiconductors, optical modes in finite rectangular volume described bygrid of allowed k-vectors count allowed k-values find photon density of states

kx

ky

m=2, p=1m=3, p=1

m=4, p=1

m=1, p=1

m=2, p=2m=4, p=2

Result: states in frequency range

d per unit volume given by p()d =


19/40




Bose-Einstein distribution function: /(valid for Bosons indistinguishable particles with integer spin)

Analytical solution for blackbody radiation

Photons are Bosons, thermal population (probability to be occupied) given by

Energy density of blackbody emitter at a given frequency :

This is the Planck blackbody radiation formula

number of modes energy of photon

average number of photons in mode with energy h

Spectrum = modes x energy per photon x times photons per mode


Blackbody Radiation

Short wavelength behavior:

Result of quantum nature of light

mode density thermal population

Long wavelength behavior:

Result of the wave nature of light

(density of states / modes)

photon energy


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Spontaneous emission

Excited atoms can spontaneously relax to a lower lying energy level:

Energy level diagram:

spsp

NAN

dt

dN

22

2

Rate equation


Absorption (stimulated absorption)

Electromagnetic field induces electronic transition from level 1 to level 2

Energy conservation requires phonon energy


)(1212 BN

dt

dN

abs

h

IN

dt

dN

abs

)(1212

or

Rate equations:Understand applicability:

atom in broad spectrum, or atom in narrowband light field)


21/40




Stimulated emission

Electromagnetic wave can induce transition from excited state to lower lying level


Rate equations

)(2122 BN

dt

dN

st

h

IN

dt

dN

st

)(2122

or


Rate equations spontaneous emission

Suppose you can bring atoms in the excited state by some energy input

look at time dependence of N2 after the energy input is turned off at t=0:

spt

spsp

eNtNN

dt

dN

/

2222 0

Note that the N2 drops to 1/e of its original value when t=sp.

We have solved our first rate equation to calculate the time dependent

concentration of excited atoms (congratulations!)


22/40




Simplified rate equations for three-level system

The simplified rate equations for this quasi three level system become

Steady state solution can be found by setting dN2/dt = 0

This can easily be shown to result in

Question: can this system provide population inversion? (N2 > N1, or N21 > 0)

Be able to solve a simple

steady state rate equation


Overview

6. Vibrations

Charged atoms can move under influence of E

Large mass low frequency resonances

Molecules: eigenmodes of Harmonic oscillator, well defined absorption energy

Polar solids: phonon dispersion relation: strong interaction with light at TAbsorption peak, reflection band, high index at low freq.


23/40




EM interaction with polar di-atomic molecule

Electromagnetic waves can interact directly with molecular vibrations on

molecules with built-in dipoles

Assume slightly positive atom A and slightly negative atom B

coupling to rotation

coupling to vibration

E

E

Realize that charge separation is necessary to excite phonons with light

- Permanent dipole moment: far IR rotational absorption possible

- No permanent dipole, but dipole active vibrations: expect combined rot+vib trans.


EM interaction with tri-atomic molecule

E

E

Dipole inactive modes (due to symmetry of charge distribution)

Dipole active modes (dipole moment changes during vibration)

Understand what dipole active means

For this particular charge distribution, these modes cannot be directly excited

Requirement for activity: dipole moment needs to be changing during oscillation


24/40





Started taking into account atom cores (positive charge)

0

3 0 3

11

Consider field-induced alignment of polar molecules in liquid (Debye model)

Molecule-molecule interaction considered classically (statistical mechanics)


Molecular rotation in polar liquids low frequency susceptibility

Liquids: molecules can rotate, but many collisions between molecules

Polar molecules can be (partially) aligned by an applied field:

Field induced polarization dipolar molecules have a susceptibility

Amount of alignment depends on dipole moment and collisional reorientation

In gases: few collisions per second - sharp lines

In liquids: many collisions per second - mostly broad features

1108 1109 11010 11011 110121

1.2

1.4

n ( )

-

+

+

O

H

H

Microwave oven: ~122mm

Debye model

Index of water

http://www.philiplaven.com/p20.html

Good news! Your eyeball is transparent


25/40




Debye relaxation model

Time constant implies: takes finite time to change polarization.

Time dependence of polarization can be empirically described by :

)(3

2

0 tEkT

NpP

dt

Pd

Debye relaxation equation

Be prepared to give physically reasonableexplanations for the various features in the

refractive index of water



Considered sustained molecule vibrations classically and quantum mechanically

? ?

Ignored molecule-molecule interactions (assumed gas phase, dilute, few collisions)

Ignored quantum mechanical nature of electron states

Phenomenically considered electron-atom core interaction (Stokes shift)


26/40




Vibrations on a diatomic molecule 3/3

Mm

111

reduced mass

remember frequency of stretch mode

m m

1

2

K

Solve equation of motion:

andmK/22

Substitute into equation of motion

tm

KKxt

m

KKxt

m

Kx

m

Km

2cos

2cos

2cos

2221212gives

22122212122 xxKxKxKx : equal and opposite motion

)( 211 XXKXm

112122cos)( Xt

m

KxtX

222222cos)( Xt

m

KxtX

Note: unequal masses m and M would give with /2 K


Vibration spectrum 1/2

Bound atoms are in low energy state, atoms oscillate around equilibrium positions

Binding potential ~harmonic (parabola) for small r

/0 K

Quantum mechanics shows that a

harmonic oscillator can be excited

in discrete steps or vibrational quanta.

This results in equally spaced levels of

excitation with energies

Evib = (v + )0 , v = 0, 1, 2,

with v the vibrational quantum number and

0 the classical resonance frequency

Remember: for perfectly harmonic atomic binding potential absorption freq.

independent of number of phonons excited

Remember effect of anharmonicity


27/40




Rotational modes

Light can interact with rotations of polar molecules

Classically no restoring force no absorption resonances?

Wrong: see absorption spectrum for HCl gas

H+

Cl-

Energy in wave numbers (cm-1) = how many

waves fit on a cm. Question: =1um ? cm-1

Corresponding energy :wavenumbers / 8 E(meV)

Remember what these lines mean


Rotational modes 4/6

Example: H2 has hcB = 5.8 x 10-22 J, so lines at 2hcB spacing = 7.25 meV (~171 um)

Remember: thermal energy at room temperature ~25 meV

rotational levels thermally populated,so high values of J less likely than low J

Result: absorption lines weaker at high energy

()

0 2hcB 4hcB 6hcB 8hcB 10hcB 12hcB Energy

Note: H2 not dipole active,

but this analysis gives order

of magnitude of rotational

transition energies

Heavier molecules:

spacings < 7.25 meVwavelengths > 171 um

This represents the far infrared absorption spectrum of polar molecules

Remember: small masses fast rotation

remember typical frequency range, understand ~exponential decay of line strength


28/40




Interaction of light with vibrating molecules (simplified)

If polarizability depends on bond length, vibrating molecule develops (t) = (t) E(t)

configuration 2

configuration 1

harmonic driving field

orientation 1: large induced dipole moment

orientation 2: small induced dipole moment

rotating molecule: beating in dipole moment

spacing dependent polarization modulated response

Excitation with single wavelength can generate polarization response at multiple


Interaction of light with vibrational modes: Raman scattering

This interaction of light with vibrations and rotations is called Raman scattering:

inelastic interaction of light with vibrational/rotational modes

is

is

Stokes

Scattering

Anti-Stokes

Scattering

is

is

Stokes

Scattering

Anti-Stokes

Scattering

Stokes scattering: illumination at i produces light at lower frequency santi-Stokes scattering: illumination at i produces light at higher frequency s

Virtual state: state of molecule only while light is present(not an intrinsic state)

virtual state

from http://neon.otago.ac.nz/chemlect/chem306/pca/IR_Raman/page8.html

example spectrum

Be able to calculate vibration energy from Raman shift


29/40




Relaxation + configuration diagram

Example: high frequency excitation (UV-VIS) changes electron wavefunctions

which in turn affects the atom-atom binding energy (and thus the equilibrium spacing)

(a)

(b)

(c)

(d)(a)

(b)

(c)

(d)

(a)Frank-Condon principle:molecular reconfiguration much slower than electronic

transitions (related to Born-Oppenheimer approximation)

light comes in

excites electron

molecule reconfigures

electrons relax

molecule reconfigures (back)

Q = configuration

coordinate

Energy diagram:

E vs. configuration

Be able to describe steps leading to Stokes shift in molecular emission


Vibrations in solids - atomic chain model

m

1

K m

1

m mK K

a

Equation of motion can be rewritten as )2( 11 XXXKXm

kaiikz exexX 00

Phase and amplitude at atom

Look for solutions of the form eikz-it : propagating waves and correlated atom motion

with oscillation phase difference between neighboring atoms given by wave vector k

Remember the approach

Understand meaning of k

Note difference with Bloch waves (see semiconductor part):

- Phonons: wave amplitude only has meaning on discrete points in space

(around equilibrium positions NOT in between atoms)

- Bloch waves: (x) has physical meaning everywhere in crystal


30/40




Vibrations on an infinite chain

Coupled motion: kaikaikaikai eeeKxexm 2)1()1(002

2sin2

2sin2

2)cos(1

2

)cos(122

ka

m

Kka

m

Kka

m

K

kam

K

k

0

m

K2

- /a + /a

Edge of Brillouin zone: Neighboring atoms 180o out of phase

animation from http://en.wikipedia.org/wiki/Image:1D_normal_modes_%28280_kB%29.gif


Coming up:

Modeled atom-atom interaction classically: dipole active lattice vibrations

Describe solid as chain of mechanically coupled positive and negative atoms

Look at all possible mechanical waves, find small subset that is dipole active

Find polarization waves trends in susceptibility for polar solids


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Linear molecules with two types of atoms

Again, only consider nearest neighbor interactions

)v2(v

)2vv(

1

1

ssss

ssss

uuKm

uKuM

tiiska

s

tiiska

s eeuuee 00 ,vv

Note that in this case the unit cell of length a contains 2 atoms, in other words

the unit cell has an internal degree of freedom (this will become useful later)

Acceleration now depends on neighbor positions, but different for M and m

As before, substitute harmonic waves in the equations of motion,

but this time distinguishable atoms v and u can have different amplitude and phase:

s counts unit cells

understand unit cell (and realize that a large cell means /a has a smaller value)


Phonons in di-atomic (or poly atomic) solids

Discuss vibrations on molecules and in solids

Model dispersion of vibration modes (phonons) in a linear chain of atoms

Atom-atom coupling results in mechanical modes with well defined and k

Animations from: http://www.chembio.uoguelph.ca/educmat/chm729/Phonons/optmovie.htm

k

0 + /a

m

2K

2K

K2

)(

ck

transverse optical phonon

transverse acoustic phonon


32/40




Low-k amplitude solutions and resulting polarization

Define v in terms of u, and substitute in one of the simplified equations of motion:

22

0022

00

/,

/

TT

mqEv

MqEu

KT

2

Large mass small amplitudeOpposite charge opposite motionExcitation near T (natural frequency of optical k=0 branch) large amplitude

with

Lattice contribution to the polarization )( 00 vuNeP

Total polarization given by boundPPP

Assume for now that q is equal to the unit charge e

(atoms have opposite and equal charge -e and +e)

atomic / nuclear / phonon response

important at low (infrared) frequencies electronic response

becomes important at high

Number of atompairs or unit cells(dipole moment requires the 2 atoms here)

understand that knowing charge q and

atom positions u and v gives dipole moment


General shape of the dielectric function

Sketch of dielectric function (lossless phonon modes):

T L

(0)()

1

()electronic transitions

(difference scales with how many charges available, and how easy to get large amplitude)

negative high reflection

Remember how you can recognize T and L in spectra of n, , , R, and

at what wavelength they typically occur


33/40




Overview

7. Semiconductors

Valence electrons bound with binding energy < 4eV

Multiple valence electrons per atom and weak binding large index, large R

Electron excitation occurs into conduction band (continuum of allowed states)

Broad absorption bands

Predict from Fermi Golden Rule, demanding energy and momentum conservation

Idealized case: parabolic increase of absorption above band gap

Can find index from predicted together with Kramers-Kronig relations


Coming up:

Modeled solid as inifinite quantum mechanically coupled electron system

Ignore any motion of atom cores, consider coupling between adjacent atoms

Find all possible electronic states in periodic Coulomb potential

Assume matrix element known, use Fermi Golden Rule to find shape of ()


34/40




Schrdinger equation

To describe interaction of light with electrons, need to describe electron states

txt

itxxVxm

,,2 2

22

Schrdinger equation describes behavior of matter in terms of the wave function

Note that a high curvature corresponds to a high energy (as for light waves) and

that the time dependence scales with the energy of the wave function

In free space and choosing V=0 we find probability waves of the form

rki

k Aer

)(

Hamiltonian, giving the energy density of the wave function

where the probability of finding matter at position x scales with ||2 or *

Compare: probability of detecting light scales with |E(x,t)|2 or EE*


Effective mass

In case of periodic potential with finite potential fluctuations,level repulsion occurs at the zone boundary:

E

0 /a- /a

Bands no longer periodic, but

approximately parabolic near k=0

Can describe energy vs. k as before,

but use effective mass m*:

*2

22

m

kE e

Note: high curvature m* must be small

sharp parabola = low effective mass

Key point: even though the electron binding energies can be large,

the energy differences caused by interactions with neighbors are on the order of eVs

Remember relation of near-gap

band structure to effective mass


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Interband absorption in indirect gap semiconductors

Indirect-gap semiconductor: highest occupied and lowest unoccupied state have k0

Direct transitions possible for k0 strong direct interband absorption

occurs at E > Egap

Other possibility: momentum and

energy can be conserved by photon

absorption and simultaneous absorption or

emission of a phonon:

Indirect transitions possible with

assistance of a phonon

Shown here are optically induced transitions

- during phonon emissiona phonon is generated in the process

- during phonon absorption

a phonon is generated in the process

Egap

Egap


Free carrier absorption (1/2)

At RT, predominant dopant related absorption is free carrier absorption

in which a photon excites an electron into a higher lying state

Example: p-type semiconductors: filled states in the conduction band:

optical transitions possible at Ephot < Egap !

Note: free carrier absorption can be described by a Drude-like model with m the

effective electron or hole mass.

Free holes can make direct transitions

form the heavy-hole band

to the light-hole band

holes cause stronger free carrierabsorption than electrons


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Counting allowed k-vectors in 2D systems

Density of states: how many states in energy interval {E,E+dE}

Two-dimensional system:

- look at energy interval dE around

a given |k|

- Find corresponding k-interval dk

- Look at k-area k : area taken up byeach allowed wavevector point

- Divide area of 2kdk by k tofind number of k values within dE

Area of ring: 2k dk

Note: assumed parabolic and isotropic conduction band around k=0 analysisonly valid near the band edge, and for isotropic direct gap semiconductors


Density of states for electrons and holes

Setting energy of the top of the valence band

to E=0 gives D(E) in conduction band as:

21

23

22

*2

2

1)( ge EE

mE

21

23

22

*2

2

1)( E

mE LHLH

And for light and heavy holes:

21

23

22

*2

2

1)( E

mE HHHH

k

E

*2

)(22

e

g

m

kEkE

*2

)(22

hm

kkE

Be able to draw DOS for holes and electrons (in parabolic band assumption)


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Inter-band transitions - requirements

Light absorption from initial state (i) to final state (f) requires

- energy conservation :

- momentum conservation :

which due to small kphot becomes

Transitions from heavy holes to the conduction band:k

E

Ei

Ef

Ef= Ei+

kf= ki+ k

kf ki

g

e

cfHH

Vi Em

kEE

m

kEE

*

2222

2,

2

HHgVC

kEEE

2

22

HHeHH mm

111

*

The allowed k-dependent transition energy is thus given by: (for light holes replace HH by LH)

with

Joint density of states:

Initial energy final energy

2123

22

2

2

1)( gJ EEE

2

1

2

3

)( gE


Free carrier absorption (2/2)

Free charges? Use Drude model to approximate optical response

Carrier concentrations in semiconductors ~1014-1018 /cm3 (metals ~108106 higher)

Plasma frequency of doped semiconductors 104 - 103 lower than of metals: IR

3

2

2

2

)(",1)('

pr

p

r 2

2

2

2

)(")(p

p

ccc

Electron FCA up for lower energies

Free hole absorption less well defined

3

2

2

2

)(",)('

p

r

p

r 2

2

0)(")(pncnc

Drude model:

nhost + Drude:

Warning: in the second model,

r does not become zero at


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Additional electronic states: excitons

Thus far: e- bound to atoms (in valence band) or free (in conduction band)

Optical properties dominated by light-induced transitions from VB to CB

BUT: If electrons and holes are present in a semiconductor, Coulomb interaction

between + hole and - electron can lead to bound e-h states: excitons

Result: hydrogen-like bound states possible: excitonic states with E < Egap

e

hCoulomb

force

n=3

n=2

n=1

Note: exciton can move through crystal, i.e. not bound to specific atom!

Binding reduces energy

Eb : exciton binding energy =

energy released upon

formation of a ground-state

exciton (1s type orbit), or

energy required for breakupof 1s-like exciton into free e+hReal-space sketch with

E

k

Eb

k-space sketch of new energy levels


Excitonic absorption

Light can excite an electron from the valence band and generate an excitonat energies slightly below the bandgap

absorption observed at Ephot = Egap Eb (slightly below Egap)

Exciton binding energy on the order of a few meV (depends on semiconductor)

Thermal energy at room temperature: kT ~ 25 meV

exciton rapidly dissociates at room temperature, short lifetime = large absorption lines broaden / disappear at room temperature

E

k

Eb

GaAs

Note: T-dependent band gap

294 K

186 K

90 K21 K


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Dopant related transitions

Possible dopant-related transitions at low temperature:

Low temperature processes: sharp lines

High temperature (room temperature) : broad lines / shoulders

D + A

and high T

Donors only

Low T

Acceptors only

Low T

D + A

and high T


Optical absorption processes in semiconductors summary

Energy

k

Interband

absorption

*

22

02 e

Cm

kEE

*

22

2 HHm

kE

*

22

2 LHm

kE

FHA

FCA

Midgap states

Excitons / Donor states

Acceptor states

or:

CB

HH

LH

SO


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The usual tips

- Read carefully. If I ask for the dielectric function, dont give me n !

- Give units

- Use acceptable significant digits: if I provide r=9.1, dont tell me n=3.0166206

- show your work = spell out your logical steps

- Write something, anything for partial credit

Good luck studying!

OSE5312 Slides Class Zz - Summary for Final - 2up

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Transcript of OSE5312 Slides Class Zz - Summary for Final - 2up