Optical Properties of Lattice Vibrationscmyles/Phys5335/Lectures/Optical Properties 3...Lectures on...

Lectures on Semiconductor Physics & Materials

2009 Lecture 16 1

Optical Properties of Lattice Vibrations

For a collection of classical charged Simple Harmonic Oscillators, the dielectric function is given by:

Where Ni is the number of oscillators with frequency ωi and e is the charge of the oscillators


2009 Lecture 16 2

Optical Properties of Lattice VibrationsFor optical phonons it is necessary to consider that the polarization is now in the form of a wave and also the effect of retardation.A wave has a wave vector k which determines whether the wave is transverse ( E┴k) or longitudinal (E||k)In the absence of free charges the medium has to satisfy the Gauss theorem: ∇•D=0 or ε(k•Eo)=0.If the EM wave is a plane wave described by:E=Eoexp[i(k•r-ωt)] then the equation ε(k•Eo)=0 can

be satisfied either by:ε=0 or (k•Eo)=0


2009 Lecture 16 3


When (k•Eo)=0 the EM wave is transverse and since P~E the polarization wave induced is also transverse.For a longitudinal wave (k•Eo)≠0 so ε=0. The frequency of this longitudinal wave will be denoted by ωL and it is given by the frequency when ε(ωL)=0


2009 Lecture 16 4


The classical expression for the dielectric function due to optical phonon with oscillation frequency ωT is:

To include the contribution to ε due to the valence electrons we will add a constant ε∞ to ε:

( )22

241)(ωω

πωε−

+=TMNQ

( )22

24)(ωω

πεωε−

+= ∞TMNQ


2009 Lecture 16 5


ε(ωL)=0 =>Solving this equation=>

Using this equation we can express ε in terms of ωT and ωL:

The longitudinal electric field of the longitudinal phonon is given by:

( ) 04)( 22

2=

−+= ∞

LTL

MNQ

ωωπεωε

(LST Relation)


2009 Lecture 16 6


In order to include retardation effect we will take the classical approach by going back to the Maxwell’s Equations and derive the wave equation in the medium:∇2E-(ε/c2)(∂E/∂t)2=0

Next substitute in the plane wave solution for E: E=Eoexp[i(k•r-ωt)], we obtain the photon dispersion:

εω 22 ⎟

⎠⎞

⎜⎝⎛=

ck


2009 Lecture 16 7

Optical Phonon PolaritonsSubstituting into this expression the optical phonons contribution to ε we obtain the polaritondispersion :


2009 Lecture 16 8

Coupled EM-Polarization Waves (Polariton)

Photon

Exciton

0

I

I

ωT

ωL

WAVEVECTOR

Two degenerate waves: photon and exciton

Any Interaction due to Her will split this degeneracy. The results are two “mixed waves” or polariton.

There are two branches to the polariton dispersion (upper branch and lower branch)

Lower Branch

Upper Branch


2009 Lecture 16 9

Exciton-Polariton

( )22

2)(4ωω

πεε−

+=XX

Xb m

eN

ωX = ωx(0)+[hk2/(2mx)]

22

2

2

222

2

2

22

)0()0(

)/(41

2)0(

)/(41

ωωω

επ

ωω

επωε

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+≈

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

XXX

xbX

XX

xbX

b

mk

meN

mk

meNkc

hh

Combine with

Exciton-Polariton Dispersion

A Exciton

CdS


2009 Lecture 16 10

Exciton-Polaritons Transmission in CdS

Wavenumber (cm )-120500 20600 20700 208000

1

2

3

4

ExperimentTheory

A Exciton

B Exciton

Experimental transmission Spectrum of CdS from Dagenais, M. and Sharfin, W. Phys. Rev. Lett. 58, 1776-1779 (1987). Oscillations due to interference between the two polariton branches


2009 Lecture 16 11

Absorption in the Polariton Picture

Polariton is a propagating wave in a medium. External wave is converted into a polariton inside the medium with a reflection and transmission coefficients.Absorption occurs when polaritons are scattered or disappear inside the medium (note the similarity between this case and the Landauer-Büttikerformalism for transport of charges)Since excitons and phonons are more strongly scattered, dissipation of polaritons is usually dominated by the polarization component of the polariton


2009 Lecture 16 12

Cavity PolaritonsPolaritons (as a form of coupled mode) can also exist in micro-cavitiesIn cavities the EM modes are confined in one or more directions but in most cases can propagate as a wave in at least one direction. When these cavity modes (either standing or guided waves) resonate with excitons in the medium, coupled EM-polarization modes, known as cavity polaritons, are formed.Cavity polaritons are important for understand the properties of a class of lasers known as vertically integrated cavity surface emitting lasers (VICSEL) which contain micro-cavities formed by Bragg reflectors.


2009 Lecture 16 13

PolaritonReflection

from a Micro-cavity

Reference: Phys. Rev. Lett. 90, 116401 (2003)


2009 Lecture 16 14

PolaritonReflection from a Micro-cavity


2009 Lecture 16 15

Theory of EmissionClassically emission of light is a common everyday experienceClassical theory: an oscillating dipole will radiate EM wave so the medium must be excited firstEmission excited by

light photoluminescenceElectrons electroluminescencHeating thermoluminescenceSound wave sonoluminescence

The semiclassical approach we have adopted cannot explain spontaneous emission since there is no EM field before emission making the interaction Hamiltonian between electron and EM field=0


2009 Lecture 16 16

Theory of EmissionThis problem is solved when we quantize the EM wave into photons.Probability of creating a photon is proportional to (1+N) where N is the photon occupancy: N=1/[exp(hω)/KbT-1]. The constant of proportionality is same as that for annihilation of a photon or absorption. Notice that even if N=0 there is still a non-zero probability of emitting a photon. Thus if the probability of absorbing a photon is given by BN, the probability for spontaneous emission is A while the probability for stimulated emission is AN. A and B are known as the Einstein’s A and B coefficients and are related to each other by the photon energy density. The reason is because the incident EM wave usually has a well-defined k but in emission the EM wave is emitted in all directions.


2009 Lecture 16 17

Einstein Theory of Stimulated EmissionEinstein did not believe in Quantum Mechanics but he was able to derive the QM result before the development of QM.Einstein used Boltzmann’s theory of statistical mechanics and Planck’s radiation laws to argue that an electron with two levels cannot be in thermal equilibrium with a radiation field without stimulated emissionThe reason is: rate of absorption is proportional to intensity of light. If emission is entirely due to spontaneous emission, its rate is independent of intensity. By increasing the intensity one can make the excited state population larger than the ground state population violation of Boltzmann’s result that the excited state population is smaller than the ground state population by the factor : exp[-ΔE/KbT].


2009 Lecture 16 18

Einstein’s A & B coefficients

Let |n> and |m> represent 2 non-degenerate levels with En>Em. The rate for absorption (transition from |m> to |n>) for unit of incident EM energy density is Bmn and is equal to the rate for the reverse process (stimulated emission) Bnm. The rate for spontaneous emission is given by Anm and since spontaneous emission is spread over all directions the emission rate per unit EM energy density=Anm/ρ(ν) where ρ(ν) is energy density of the EM wave with frequency between ν and ν+Δν and is given by: Np(hν) (8πν2) (n/c)3.


2009 Lecture 16 19

Einstein’s A & B coefficients

Using the Principle of Detailed Balance Einstein obtained: Bmn = Bnm and Anm =(8πhν3) (n/c)3 BnmFrom Einstein’s result the total rate of emission is:Anm + Bnm ρ= Anm(1+ρBnm /Anm)

= Anm(1+Np)The rate of absorption is Bnm ρ= Anm ΝpΤhis is exactly the same result as obtained by

QM!


2009 Lecture 16 20

Emission Processes in Semiconductors

Based on Einstein’s result we expect the emission probability Rcan be determined from the absorption coefficient α (Roosbroek-Shockley relation):

Note that the indirect edge is almost as strong as the direct edge in emission because of the Boltzmannfactor

Ge300 K

k10

10

10

10

12

11

10

9

0.62 0.70 0.80 0.90 1.00

( )( )v vP ρ

hν (eV)

10 -5

10 -4

10 -3

10 -2


2009 Lecture 16 21

Photoluminescence Processes in Semiconductors

PL involves 3 distinct steps:

Real Excitation of e-h pair via absorptionRelaxation of e-h to lowest energy states (favored by the Boltzmann factor) and equilibrium with phononsEmission via radiativerecombination of e-hpairs


2009 Lecture 16 22

Emission Processes in Semiconductors

In pure semiconductors emission is intrinsic and dominated by recombination of free excitons at low T and conduction band-to-valence band transition at high TIn extrinsic semiconductors emission is dominated by defects and impurities:

Excitons bound to donors, acceptors or neutral centersFree-to-bound transitions such as donor=>valence bandDonor-acceptor pair (DAP) transitions

Since the Boltzmann factor tends to favor low energy states emission is a very sensitive probe of defect and impurity states


2009 Lecture 16 23

Free-to-bound transitions in Doped Semiconductors and Mott transition

Electrons in Shallow Donors have Bohr radii typically of the order of tens of lattice constants (or ~ several nm to 10 nm). When their concentration is high enough these electron wave function will overlap and electrons from one donor can hop to another. In another word the semiconductor becomes metallic and the discrete impurity levels will form bands, known as impurity bands. This transition from an insulating to a metallic state is known as Mott Transition. It is a classic example of a many-body effect and quantum phase transition (a transition which occurs even at T=0).


2009 Lecture 16 24

Free-to-bound transitions in Doped Semiconductors

The emission spectra of Zn-doped p-type GaAs as a function of doping concentration shows 2 many-body effects:Band gap shrinkage(or renormalization)Formation of a Fermi sea of electrons

1.25 X 10 20

1.3 X 10 19

3.7 X 10 17

1.9 X 10 18

1.40 1.50(eV)Photon energy

Fermi Level


2009 Lecture 16 25

DAP transitionsIn a compensated semiconductor there are both donors and acceptorsRecombination of electrons at a donor with a hole at an acceptor exhibits also a final state interaction:

Do+Ao=>hω+ D++A-

Since the donor and acceptor becomes charged in the final state there is a Coulomb attraction between them. As a result the emitted photon energy hω is given by:hω =Eg-ED-EA+e2/(εoR)Where R= distance between donor and acceptor


2009 Lecture 16 26

DAP transitions

Since the donors and acceptors can only sit on specific lattice sites R is discrete leading to sharp DAP linesThe position of the DAP lines depends on the lattice constant and on whether the donor and acceptor sit on the same sublattice (Type 1) or different sublattices (Type 2)


2009 Lecture 16 27

Type I DAP transitionsNumbers label the Shells counting from either D or A

Notice how the series converge towards lower energy!


2009 Lecture 16 28

Type II DAP transitions


2009 Lecture 16 29

DAP transitionsCurve C Coulombinteraction aloneCurve C+vdW Coulomb interaction plus van derWaals interaction between D and A pair in the initial stateParameters obtained from these fits: ε and (ED+EA)DAP transitions provide the most accurate determination of defect energies and separation

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