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Optical Spectroscopy
• Study of the interaction of matter and light (radiated energy).• A key experimental technique use to probe all states of matter (atoms,
gas, plasma, solids, liquids etc).• Gives direct information about electronic structure of a system.• Can be used to explore relative composition of a material (i.e. relative
concentration of a known compound in solution).• Widely used in industry and quality assurance.• Key technique in semiconductor research.
SummaryAbsorption and emission of radiationTransition ratesSelection rules for optical transitionsDirect vs indirect semiconductorsMeasurement of absorptionMeasurement of emission and quantum yieldFluorescence decay lifetime
For more information, see “Optical Properties of Solids” by Prof. Mark Fox on which this lecture is loosely based.
• In quantum systems, we have interaction of a stationary-state (e.g. an atom) with an oscillating state – a photon.
• Coupling of the states is strongest when the energy difference between the two states matches the energy of the photon, e.g.
• Atoms can either absorb or emit this energy when they jump between states.
E1
E2
E1
E2
Absorption Emission
Transition rates• Transition rates (W12) between two states can be calculated
using the wavefunction of the initial and final states through Fermi’s golden rule.
• Here the M12 is the matrix element for the transition between bands of levels. The matrix element can written as
where H is the perturbation that causes the transition (the interaction between the atom and the photon). The strongest interaction process is called the “Electric dipole interaction”.
• g(hn) is the photon density of states. This is defined as the number of photon states per unit volume that fall into the energy range E + dE, where E = hn
• Here, the perturbation is expressed by
H = -p. ewhere p is the dipole moment of the electron and e is the electric field applied.
Because of this, such transitions are called ‘Electric Dipole’, or ‘Dipole-Allowed’ transitions.
Selection rules for Electronic transitions.
Electric dipole transitions can only happen if a number of rules about initial and final states are obeyed.
These are rules about the quantum numbers of the initial and final states.
For an electron in a system with quantum numbers l, m, ms, the selection rules are
(1)Parity of initial and final states must be different(2)Dm = -1, 0 or +1(3)Dl = +/- 1(4)Dms = 0
The photon carries away one unit of angular momentum, so the total angular momentum of the atom must change by one unit in the transition.
From single emitters to electronic bands.• Atoms in a solid are packed close-together, so outer orbitals
interact strongly together.• This broadens discrete levels to bands. Electronic bands retain
of atomic character of states.• Transitions between bands allowed if they are allowed by
selection rules. • Absorption allowed over continuous range of wavelengths.• Such materials have sizable optical effects, making them useful
for device technology.
GaAsGaAs crystal structure Interatomic separation
E
Luminescence from a semiconductor
Excited states
Ground states
Inject electrons
Inject holes
tRtNR
Relaxation Rapid relaxation and thermalisationApplies for electrons and holes.
Recombination of electrons andholes can result in luminescence.
This often different from absorption.
Radiative and non-radiative processes compete.
Quantifying luminescence efficiency
Luminescence quantum efficiency (F) can be calculated using
tR = radiative rate, tNR = non-radiative rate
Have,
where A = 1 / tR.
If tR << tNR, then F ~ 1.0 and all energy comes out as light.
If tR >> tNR, then F ~ 0 and the energy is lost internally as heat.
Interband Semiconductor Luminescence
Direct-gap materials
Conduction band
Valence band
electrons
holes
hnEg
Photons emitted whenelectrons at the bottomof the conduction band recombinewith holes at the top of the valence band.
Since momentum of the photonis negligible compared to thatof the electron, e and h have same k-vector.
Energy of luminescence closeto energy-gap.
Examples: GaAs, GaN, GaInP
k
E
Generating Photoluminescence
• Excite a direct bandgap semiconductor with a photon whose energy is greater than the energy-gap.
• Photons are absorbed, raising electrons into the CB and holes into the VB.
• Electrons loose energy very quickly by emitting phonons. Each step corresponds to the emission of a phonon (~ 100 fs). Energy and momentum is conserved in this process.
• Relaxation process much faster than radiative emission, so electrons collect at bottom of CB before recombining radiatively.
electrons
holes
hnEg
k = 0
E
Photoluminescence at low carrier density
holes
k = 0
E E
Density of states
electrons
At low carrier densities, the occupation ofelectrons and holes have a Boltzmann distribution
PL emission above energy-gapfalls off exponentially due to Boltzmann factor.
From “Optical Properties of Solids” by Mark Fox
Interband Semiconductor absorption
Conduction band
Valence band
hnEg
k
E
Transitions are possible at aWide range of energy (wavelengths) from the Valence band to the Conduction band.
GaN: Absorption and luminescence
Absorption and emissionare not the same!From “Optical Properties of Solids” by Mark Fox
Perovskites can be direct bandgap semiconductors
Saidaminov et al Nature Communications 6, 7586 (2015)
Mott et al Nature Communications 6 (2015) 7026
Indirect gap semiconductors
Conduction band
Valence band
electrons
holes
hnEg
Phonon Conduction band minimumand valence band maximumare at different point in Brillouin zone.Conservation of momentum requiresthat a phonon is either emitter or absorbedwhen the photon is emitted.
This represents a ‘second-order’ process,giving it a low probability.
Radiative lifetime is therefore slow, andcompetition with non-radiative processesmakes luminescence weak.Examples: Si, Ge
k
E
Quantifying absorbance
Sample
Io(l) I(l)
d
I0 and I are the power perunit area of the radiation incidentand transmitted.
s is the attenuation cross-section ofthe absorber
N is the number density of the absorbers
Absorbance is defined as
Very often, when measuring a thin film, we are actually interested in S – the attenuation coefficient, whereUnits for S are often expressed as cm-1.
Measurement of optical absorption
Basic principle - split white light into component wavelengths using a dispersive element (e.g. a grating or prism). Measure how efficiently the different wavelengths are absorbed.
Components inside an absorption spectrometer
Light-source
Optical grating on rotating turret
Entrance slit
Exit slit
Sample
Detector
Scan ‘probe’ wavelengthand measure transmittedsignal
Measuring optical absorption using a CCD spectrometer.
Shine a white light through a sampleDetermine the wavelengths transmitted
Measuring photoluminescence emissionThis is considerably easier if the sample under test isan efficient emitter (i.e. F ~ 1.0)
Emission dynamics
• Emission from an excited state comes via the process of spontaneous emission
• If upper level has a population N at time t, the radiative emission rate is given by
• This can be solved to show
where tR is the radiative lifetime of the transition.
The actual luminescence intensity I(hn) can be dependent on the emission rate, and the relative probability that the upper level is
occupied and the lower level is unoccupied.
‘Allowed’ transitions often have radiative lifetimes of ~ 1 ns.
Measuring fluorescence decay lifetime• Sample excited with a very short light-pulse.• Emission recorded as a function of time.• Lasers with pulses 100 fs to 1 ps now commonplace.• Time-resolutions of 100 ps achievable with photomultiplier tubes. Resolutions of 1
ps possible with a ‘streak-camera’.
Pulsed laser
DetectorSample
Mirror
Lenses
Fluorescence dynamics
Gives direct information about carrier relaxation, diffusion and recombination mechanisms.
Often a series of processes going on, so practically need to fit decay lifetimes to a series of exponential decay functions.
Summary• Absorption and PL spectra allow the
electronic states within a material to be directly accessed.
• The properties of many semiconductor materials have similarities, but also distinct differences, depending on atomic / molecular structure.
• Reviewed a number of key techniques of use in routine spectroscopy labs.