Lecture 8 – Light-Matter Interaction Part 2 Basic excitation and coupling

EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku Schedule for the rest of the semester

„ Introduction to light-matter interaction (1/26):

„ How to determine ε(r)?

„ The relationship to basic excitations. „ Basic excitations and measurement of ε(r). (1/31) „ Structure dependence of ε(r) overview (2/2) „ Surface effects (2/7 & 2/9):

„ Surface EM wave

„ Surface polaritons

„ Size dependence „ Case studies (2/14 – 2/21):

„ Quantum wells, wires, and dots

„ Nanophotonics in microscopy

„ Nanophotonics in plasmonics „ Dispersion engineering (2/23 – 3/9):

„ Material dispersion

„ Waveguide dispersion (photonic crystals)

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 2 Last time

„ We learned:

„ To determine ε(r), we need to study how the microscopic interaction between atoms/electrons with the light.

„ This interaction is similar to coupling of two SHO’s.

„ The only details we need to know are the interaction near resonances of basic excitations.

„ The rest of the information needed to complete the calculation of ε(r) is through the Kramers-Kronig relation.

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 3 Today

„ Basic excitations by photons Æ polaritons

„ Plasmons (last time)

„ Phonons

„ Excitons, biexcitons, etc.

„ Measurement of ε(r)

Ref: D. L. Mills and E. Burstein, “Polaritons,” Rep. Prog. Phys., 37 (1974) 817. P. Y. Yu and M. Cardona, Fundamentals of Semiconductors, 2nd ed., Springer-Verlag (1999) chapters 6 and 7.

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 4 Review of the concept of polaritons

ω εω εω ε „ In a dielectric medium: ⎡⎤ ⎢⎥ω2 Nq2 ( ) 1pn where 2 nn =−0 ⎢⎥∑ 22 pn ≡ ⎢⎥n basic ()ω −+ωγω0n0nnim ε excitations ⎣⎦⎢⎥εεor oscillators 2ω ωωp γω = ∞ − 0 22 ()−+0 i where the index denotes the n-th kind of basic excitation or SHO.

„ QM analogue:

2 2 2 nq exˆ ωp =→ ε00m ε

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 5 Transverse and longitudinal vibrations

Longitudinally vibrated SHO’s

Vertically vibrated SHO’s

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 6 Transverse and longitudinal polaritons For homogeneous media: K ∇⋅D =0 KKK ⇒∇⋅()εεEkE =00 ⇒ ⋅ = K K ⇒⋅kE =0 or = 0 Normally EM waveε couples only to transverse SHO’s unless the dielectric constant vanishes.

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 7 Longitudinal polaritons

k 2 ε ==2 0 ωµ0 EM wave can couple to the longitudinal vibration.

Photons ω In free space

22 ω0 + ωp

ω0

k

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 8 Phonons M m

a vn un optical ⎧munnnnn=−αα()() uv −−+11 − uv − ω ⎨ ⎩Mvnnnnn=−αα()() vu −−+11 − vu − By periodicity:

⎪⎧uun =−exp[ inkat (2ω )] ⎨ acoustic ⎩⎪vvn =+−exp[] inkat (2( 1) ) ωα ω ⎧ 2 ⎡⎤ika− ika ⎪⎣−=mu ve() +− e2 u ⎦ ⇒ ⎨ ωα() k −=Mv2 ⎡⎤ ueeika +−− ika 2 v π / a ⎩⎪ ⎣⎦

ωα α 2 1/2 2 ⎛⎞⎛⎞mM++−⎡ mM2(1 cos ka )⎤ ⇒=⎜⎟⎜⎟ ±⎢ − ⎥ ⎝⎠⎝⎠mM⎣⎢ mM mM ⎦⎥

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 9 Orders of magnitude

7 „ At optical frequencies, k=ω/c~10 . 10 „ For typical crystal lattice, π/a~10 .

„ Only optical phonons couple to the light. k ≈ 0 For optical branch: u M ≈− Can generate the dipole moment v m For acoustic branch: u ≈ 1 v

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 10 Examples of phonon dispersion curves

silicon GaAs

Taken from P. Yu and M. Cardona.

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 11 Dispersion curve for phonon polaritons

Free-space-photon-like

Photons In free space ω

22 ω0 +=ωωpLO

ω0 = ωTO

k Photon-like but with strong phonon influence

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 12 Raman processes

„ When light is not at the infrared frequency, phonons can still participate in the inelastic processes with light Æ Raman processes.

„ The scattered light has a frequency shift w.r.t to the incident light due to its energy lost (or gain) to phonons.

„ Energy lost: Stokes process Energy gain: Anti-Stokes process

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 13 Excitons (two-level system)

el The Coulomb interaction b/w electron and hole makes the exciton. Exciton is like a hydrogen atom. hole

Exciton absorption

Coulomb enhancement

Eg E

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 14 Hot carriers relaxation processes

Carrier capture

Phase relaxation T2~100fs - ps k Thermalization

Recombination (T1~ns)

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 15 Size dependence (quantum confinement)

g(E) = Density of states

dot wire bulk sheet

3D 2D 1D 0D g(E) g(E) g(E) g(E)

Eg E Eg E Eg E Eg E

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 16 Exciton absorption in low-dim structures

Exciton binding energy: Exciton absorption: 23dd 2D: 2D: EEB,1nBn=== 4 ,1

1D:

1D:

a0=exciton Bohr radius~100A

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 17 Excitons (three-level system)

If we consider the polariton corresponding to exciton 1:

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 18 Electromagnetically induced transparency

1 α ωp= ω31 3

ω = ω s 21 n

2

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 19 Measurement of ε(r) - ellipsometry

ellipsometry

Sensitive to: Need to know underlying 1. Film thickness composition of materials. 2. Surface roughness 3. Anisotropy

EECS 598-002 Nanophotonics and Nanoscale Fabrication by P.C.Ku 20