Lectures on Semiconductor. Physics & Materials. 2009 Lecture 16. 1. Optical
Properties of Lattice Vibrations. ○ For a collection of classical charged Simple.
Optical Properties of Lattice Vibrations z For a collection of classical charged Simple Harmonic Oscillators, the dielectric function is given by:
z Where Ni is the number of oscillators with frequency ωi and e is the charge of the oscillators Lectures on Semiconductor Physics & Materials
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Optical Properties of Lattice Vibrations z For optical phonons it is necessary to consider that the polarization is now in the form of a wave and also the effect of retardation. z A wave has a wave vector k which determines whether the wave is transverse ( E┴k) or longitudinal (E||k) z In the absence of free charges the medium has to satisfy the Gauss theorem: ∇•D=0 or ε(k•Eo)=0. z 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 Lectures on Semiconductor Physics & Materials
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Optical Properties of Lattice Vibrations zWhen (k•Eo)=0 the EM wave is transverse and since P~E the polarization wave induced is also transverse. zFor 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 Lectures on Semiconductor Physics & Materials
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Optical Properties of Lattice Vibrations z The classical expression for the dielectric function due to optical phonon with oscillation frequency ωT is: 4πNQ 2 ε (ω ) = 1 + M ωT 2 − ω 2 z To include the contribution to ε due to the valence electrons we will add a constant ε∞ to ε:
(
ε (ω ) = ε ∞ +
Lectures on Semiconductor Physics & Materials
)
4πNQ 2
(
2
M ωT − ω 2
2009 Lecture 16
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Optical Properties of Lattice Vibrations 4πNQ 2
z ε(ωL)=0 => ε (ω L ) = ε ∞ + M (ω 2 − ω 2 ) = 0 T L z Solving this equation=>
z Using this equation we can express ε in terms of ωT and ωL: (LST Relation) z The longitudinal electric field of the longitudinal phonon is given by: Lectures on Semiconductor Physics & Materials
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Optical Properties of Lattice Vibrations z 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: 2 ⎛ω ⎞ 2 k =⎜ ⎟ ε ⎝c⎠ Lectures on Semiconductor Physics & Materials
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Optical Phonon Polaritons zSubstituting into this expression the optical phonons contribution to ε we obtain the polariton dispersion :
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Coupled EM-Polarization Waves (Polariton) I
Upper Branch
Photon
ωL
Lower Branch I
Two degenerate waves: photon and exciton Exciton
ωT
0 WAVEVECTOR Lectures on Semiconductor Physics & Materials
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) 2009 Lecture 16
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 Lectures on Semiconductor Physics & Materials
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Absorption in the Polariton Picture z Polariton is a propagating wave in a medium. External wave is converted into a polariton inside the medium with a reflection and transmission coefficients. z Absorption occurs when polaritons are scattered or disappear inside the medium (note the similarity between this case and the Landauer-Büttiker formalism for transport of charges) z Since excitons and phonons are more strongly scattered, dissipation of polaritons is usually dominated by the polarization component of the polariton Lectures on Semiconductor Physics & Materials
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Cavity Polaritons z Polaritons (as a form of coupled mode) can also exist in micro-cavities z In 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. z 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.
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Polariton Reflection from a Microcavity
Reference: Phys. Rev. Lett. 90, 116401 (2003) Lectures on Semiconductor Physics & Materials
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Polariton Reflection from a Micro-cavity
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Theory of Emission z Classically emission of light is a common everyday experience z Classical theory: an oscillating dipole will radiate EM wave so the medium must be excited first z Emission excited by { lightÙphotoluminescence { ElectronsÙelectroluminescenc { HeatingÙthermoluminescence { Sound waveÙsonoluminescence
z 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 Lectures on Semiconductor Physics & Materials
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Theory of Emission z This problem is solved when we quantize the EM wave into photons. z 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 nonzero probability of emitting a photon. z 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. Lectures on Semiconductor Physics & Materials
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Einstein Theory of Stimulated Emission z Einstein did not believe in Quantum Mechanics but he was able to derive the QM result before the development of QM. z 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 emission z The 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 populationÙ violation of Boltzmann’s result that the excited state population is smaller than the ground state population by the factor : exp[-ΔE/KbT]. Lectures on Semiconductor Physics & Materials
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Einstein’s A & B coefficients z 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. z 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. Lectures on Semiconductor Physics & Materials
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Einstein’s A & B coefficients z Using the Principle of Detailed Balance Einstein obtained: Bmn = Bnm and Anm =(8πhν3) (n/c)3 Bnm z From 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 Νp Τhis is exactly the same result as obtained by QM! Lectures on Semiconductor Physics & Materials
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Emission Processes in Semiconductors z Based on Einstein’s result we expect the emission probability R can be determined from the absorption coefficient α (Roosbroek-Shockley relation):
z Note that the indirect edge is almost as strong as the direct edge in emission because of the Boltzmann factor Lectures on Semiconductor Physics & Materials
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Ge 300 K 10 -2
10 12 P (v )ρ (v )
k
1011
10 -3
10
10
10 -4
10 9
10 -5
0.62
0.70
0.80
0.90
1.00
hν (eV) 20
Photoluminescence Processes in Semiconductors z PL involves 3 distinct steps: {Real Excitation of e-h pair via absorption {Relaxation of e-h to lowest energy states (favored by the Boltzmann factor) and equilibrium with phonons {Emission via radiative recombination of e-h pairs Lectures on Semiconductor Physics & Materials
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Emission Processes in Semiconductors z 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 T z In extrinsic semiconductors emission is dominated by defects and impurities: { Excitons bound to donors, acceptors or neutral centers { Free-to-bound transitions such as donor=>valence band { Donor-acceptor pair (DAP) transitions
z Since the Boltzmann factor tends to favor low energy states emission is a very sensitive probe of defect and impurity states
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Free-to-bound transitions in Doped Semiconductors and Mott transition
z Electrons in Shallow Donors have Bohr radii typically of the order of tens of lattice constants (or ~ several nm to 10 nm). z 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). Lectures on Semiconductor Physics & Materials
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Free-to-bound transitions in Doped Semiconductors z The emission spectra of Zn-doped p-type GaAs as a function of doping concentration shows 2 many-body effects: z Band gap shrinkage (or renormalization) z Formation of a Fermi sea of electrons
1.9 X 10 18
1.25 X 10
3.7 X 10 17
20
1.3 X 10 19
1.40
1.50
Photon energy
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Fermi Level
(eV)
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DAP transitions z In a compensated semiconductor there are both donors and acceptors z Recombination of electrons at a donor with a hole at an acceptor exhibits also a final state interaction: Do+Ao=>hω+ D++Az 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: z hω =Eg-ED-EA+e2/(εoR) z Where R= distance between donor and acceptor Lectures on Semiconductor Physics & Materials
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DAP transitions zSince the donors and acceptors can only sit on specific lattice sites R is discrete leading to sharp DAP lines zThe 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) Lectures on Semiconductor Physics & Materials
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Type I DAP transitions Numbers label the Shells counting from either D or A Notice how the series converge towards lower energy!
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Type II DAP transitions
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DAP transitions z Curve CÙCoulomb interaction alone z Curve C+vdWÙ Coulomb interaction plus van der Waals interaction between D and A pair in the initial state z Parameters obtained from these fits: ε and (ED+EA) z DAP transitions provide the most accurate determination of defect energies and separation Lectures on Semiconductor Physics & Materials
