Optical Properties of Solids Flashcards
(40 cards)
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.
