Optical Properties of Solids Flashcards
Reflectivity/Transmissivity
Reflectivity R = reflected power / incident power
Transmissivity T = transmitted power / incident power
Conseration of energy R + T = 1 (or if absorption occuring R + T - A = 1 )
R = |~n-1/~n+1|^2
=((n-1)^2+k^2)/((n+1)^2+k^2)
Refractive index
~n = n + ik n = c/v, k is extinction coefficient originating from absorption in the medium.
k = k_r + ik_i
n = ck_r/omega k = ck_i/omega
Beer’s Law
I = I0 e^(-alpha x) alpha = 2k_i
Crystalline materials
Display translational periodicity which falls into 32 point groups. Neumann’s principle relates symmetry of observed physical properties to that of the crystal:
Any macroscopic physical property must have at least the symmetry of the crystal structure.
Optical Anisotropy
Lifting of degeneracies
Refractive index depends on diretion (along which crystallographic axis propagation or polarisation is oriented)
Free atoms are spherically symmetric but this can be broken by eg Zeeman.
Band structure
Electronic states form continuous bands rather than the discrete states seen in free atoms: optical absorption and emission can occur over a range of energies defined by band width.
Simplest treatment of plasma excitations in a solid
1) treat free electrons as uniform gas held in place by the background
2) displace electrons with respect to background - treat as single harmonic oscillator, natural frquency = plasma frequency
w_p = root(ne^2/mE0)
Lorenz model: atomic oscillators
Treat electrons in atoms as dipole oscillators
w0 = root(K_s/mu)
where 1/mu = 1/m0+1/mN~1/m0 and thus mu = m0
Resonant freq. of core electrons, valence electrons, phonons
Core electrons - x-ray
Valence electrons - UV/vis
Phonons - IR
Dispersion
GVD
Variation of n with wavelength in medium arising naturally from Lorentz model leads to dispersion: light of different wavelength refracted to different degree.
Normal dispersion: n increases with freq.
Anomalous dispersion: n decreases with freq, ocurs near resonance lines.
Dispersion leads to Fourier components of wave pulse spreading.
GVD = group velocity dispersion - in Si optical figbres, GVD zero for 1.3 micro m, short pulses can be transmitted with little temporal spreading
Measuring Optical Coefficients (3)
Can use Snell’s law
Or find critical angle for total internal reflection from interface between prism and material (in transparency region)
Above fundamental edge absorption is large and materials become opaque. Possible to measure attenuation of beam through thin sample and then get k. Measurement of real or imaginary part of RI or dielectric function is enough to get full knowledge of coefficient.
Strong absorption means they suffer badly with surface contamination - use ultra high-vacuum conditions.
~E components
E1 = 1 - (w_p^2 tau^2)/(1+w^2tau^2) E2 = (w_p^2tau)/(w(1+w^2tau^2))
~E_r = E1 + E2 (these are all epsilons)
low freq wtau «_space;1 so E2»_space; E1
Plasma freq. in doped semiconductor
w_p^2 = Ne^2 / Eopt E0 m*
Dispersion relations for transverse and longitudinal fields
Transverse: c^2k^2 = w^2 - w_p^2 - modification of dispersion relation of EM waves is bc they couple strongly with free-electron plasma and form volume plasmon polariton.
L: w = w_p - supports dispersionless longitudinal modes of frequency. (light in vacuum uses this).These single freq. oscillations correspond to plasmons with E = hbar w_p
Surface plasmons
Introducing free surface breaks translational symmetry of a solid - new modes of excitation. SP are compressions and rarefractions of charge strictly localised at 2D surface.
WHy are elements Ge Si and C semiconductors with outer shell configuration s2p2?
Energetically favourable to promote an electron from s to p to form four sp3 hybrid orbitals. Energy cost of s->p promotion overcome by forming bonds to four nearest neighbours instead of 2.
Bonding/antibonding
Can see how this arises by considering wavefunction spossible when combining H atoms - symmetric is bonding and vice versa.
Angles in bonding
Mixing orbits changes angle between hybrides. Atomic p orbitals are 90 degrees apart, two sp3 hybrids are 109.5, sp2 are 120 and sp are 180.
angle between two hybrids sp^m and sp^n are cos theta = -1/(m.n)^0.5
Direct and indirect
Direct takes one photon with band gap energy hbar w. Indirect is ‘off-centre’ so requires phonon too - thus less likely to happen.
Interband Absorption
Process of absortpion in which electrons are excited from a (part) filled to a (part) empty band
Fermi’s Golden Rule
Wi->f = (2pi/h)|M|^2g(hbarw)
M =
Selection Rules
Parity Changes
l delta l = +- 1
delta m = + 1 circular: sigma +
m delta m = - 1 circular: sigma -
delta m = 0 linear || z
delta m = +- 1 linear || (x,y)
s delta m = 0
m_s delta m_s = 0
Nanophotonics
Subwavelength manipulation of light. Still issue of coupling since plasmon-polariton dispersion falls below light line
Phonons
Two branches: optical and acoustic. Acoustic branch intersects light line only at w = 0 thus cannot be excited by incident light.
Finite freq. at q = 0 for optic modes means optical branch intersects light line, allowing excitation provided bonds are polar or ionc, modes are transverse, selection rules satisfied.
Polaron
Combo of lattice distortion and electron is known as polaron.
Bands (from top to bottom)
excited
heavy hole
light hole
split off (starting at -delta so)
Excitons
An interband transition at the edge of a SC or insulator will for a hole in VB and electron in CB. Coulomb interaction can ead to increased rates of optical transition and even to a bound electron-hole pair - exciton. Only form if electron and hole group velocities are the same or they will not move as a pair - only met when CB/VB have same gradient.
2 types: Mott-Wannier (or free) exciton and Frenkel (or tightly bound) exciton. Frenkel easier to observe.
Can only observe free excitons at:
Low temperatures (rule of thum, their binding energy is < kbT) In very pure samples, since dopants can release carriers which screen e-h interaction and charged impurities generate electric fields which can ionize excitons.
Mott-Wannier excitons
Interactions between excitons becomes significant when their average separtion is equal to their diameter, at Mott density: Nmott ~ 1/(4/3 . pi rn^3). Easily achieved with focused laser beam.
Effects that occur when Nmott is approached
Exciton gas may dissociate into e-h plasma due to collisions - broadening exciton linewidth and reducing absorption
Biexcitons may form - bound exciton pairs
In Si/Ge excitons can condense to form liquid - e-h droplet, which manifests as broad feature at lower energy
Bose-Einstein Condensates of excitons may form (observation is controversial)
Frenkel Excitons
Highly localised - considered as excited states of atoms or molecules which can propagate by hopping - strong coupling to lattice - strongly bound.
Observed in insulators, molecular crystals
Luminescense
Light may be absorbed by electronic transitions, may also be emitted via decay of excited states - luminescense
Photoluminescense - re-emission of light after absorption of photon of higher energy
Electroluminescense: emission of light resulting from passing an electric current through material
Photoluminescense at low rho and high T/ high rho low T
Low rho high T - electrons and holes can be described by classical statistics. Luminescense spectrum should rsie sharply at Eg and decay exponentially with decay constant kbT
High rho have to use Fermi-Dirac statistics and electrons and holes are said to be degenerate
Low T - take limit of T = 0 at which al states below Fermi level of e/h are full and all above are empty.
Electroluminescense
Consider LED. Consists of p-n junction in which both regions are heavily doped to produce distributions of holes in p region and electrons in n region.
Molecular Materials
Characterised by weak (usually vdw) bonds. As result, electronic structure is only weakly pertubed by formation of solid. Electronic/vibrational modes are therefore localised. Organic molecules form most important class of materials from opto-electronic standpoint and we therefore focus on these.
Optical properties of molecules divided into three regions
Far IR (wl > 100 micro m) - rotational
IR ( wl 1 - 100 micro m) - vibrational
UV-vis (wl < 1 micro m) electronic (vibronic) excitations
Coupling of electronic/vibrational degrees of freedom leads to concept of vibronic transition. Understood by Franck-Condon principle.
LUMO/HOMO
Lowest Unnoccupied Molecular Orbital
Highest Occupied Molecular Orbital
A molecule absorbing EM may have electron promoted from HOMO to LUMO. Since nuclei are heaver than e we assume degrees of freedom are decoupled - Born Oppenheimer
Franck Condon
Two atoms (??) go through cycle of absorbing relaxing, emitting and relaxing again and back round
Cubic crystals, uniaxial and biaxial
Cubic crystals are isotropic
Uniaxial have single optical axis
Biaxial cyrstals have two optical axes.
Birefringence
Incident unpolarized light on birefringent material has emerging light polarised in opposite directions: E ray and O ray.
