Detectors
Goal: Convert photons to an electronic signal (apologies to
photography...)
with as little accompanying noise as possible
ideally at the quantum limit enforced by the photons.
with as much conversion efficiency as possible
1 photon yields 1 electron (or ideally a bunch of electrons)
Primary Detection Methods
Bulk thermal response (bolometry)
incident radiation chages the temperature of the detector
electrical resistance changes with temperature
Conversion of photons to ''free'' electrons
quantum response
photoelectric or solid state detection
Coherent detection
sense wave nature (phase) of the photons
primarily through heterodyning to lower frequencies
Electron response -- free electrons / carriers
A free electron is a detectable electron
an electron can be free in space -- photoelectric effect
or it can be ''free'' within a crystal lattice -- solid state detection
The Photoelectric Effect
Metals are characterized by a work function which
determines the energy difference between the
highest energy state for an electron within the metal
and the energy of an electron in free space.
A photon with energy in excess of this work function will liberate
a free, detectable, electron -- the photoelectric effect
Heated metals will emit free electrons -- those with thermal
energy in excess of the material's work function -- thermionic
emission
via a Boltzmann law.
The Photoelectric Effect
Photomultipliers are based
on the cascade amplification
of individual electrons
liberated by the photoelectric
effect
Work functions for metals
are typically a few electron
volts
1 eV = 1240 nm
http://laxmi.nuc.ucla.edu:8248/M248_99/autorad/Scint/pmt.htmll
The Photoelectric Effect
Photomultipliers are based
on the cascade amplification
of individual electrons
liberated by the photoelectric
effect
Work functions for metals
are typically a few electron
volts
1 eV = 1240 nm
Photocathodes can be
engineered to have
sensitivity out to 1.5 um
(obviously not using pure
metals...)
http://hyperphysics.phy-astr.gsu.edu/hbase/tables/photoelec.html
The Photoelectric Effect
Shortcomings of photomultipliers
poor wavelength coverage
(<1.5um)
poor quantum efficiency (<20%
conversion of photons to electrons)
thermally emitted electrons -particularly for long-wavelength
devices.
large single-detector area
(overcome with devices like microchannel plates)
One big advantage -- photon counting
Modern detection systems use semiconductor detectors which
mimic the photoelectric effect in the solid state.
Photons create “free” electrons within the confines of the crystal
lattice.
Discussion requires an understanding of conductivity, or lack
thereof, in crystalline materials. Detectors
Electronic Energy Levels in Solids
Electrons are fermions and must obey exclusion rules.
At the atomic level, electrons fill available energy states up to the
valence levels... 1s, 2s, 2p, 3s, 3p, 3d, ... with two spin states in
each energy state.
In a solid, proximity leads to splitting of the degenerate valence
energy levels due to interference with neighboring atoms.
Lower energy levels remain tightly bound to their atoms. Higher
(possibly unoccupied) levels become states of the entire crystal
structure.
Idealized Electrical Conduction
Ignoring the lattice sites for the moment, “free” electrons in a
metal (or semiconductor) are particles in a box.
Momentum increases linearly with quantum state, energy
quadratically.
As particles are dropped into the box they fill the lowest energy
states.
Although the mean energy of particles is large (~Ef) the net
momentum is zero.
Application of an electrical potential can elevate electrons into
higher energy states -- conduction.
Electrical Conduction
The small disequilibrium between the right and left halves of
the parabola represents electronic conduction.
Most electrons in the material are still in momentum states which
cancel each other.
Only a limited number of electrons at the top of the Fermi sea
participate in conduction.
Electrical Conduction
In real materials the lattice sites produce a periodic potential.
When the electron DeBroglie wavelength becomes resonant
with the lattice spacing significant scattering occurs.
The dispersion relation exhibits breaks at these boundarys
providing an alternative view of the nature of conduction
“bands”.
These breaks are caused by the mutual interference of the
wavefunctions being reflected at each potential boundary.
Electronic Energy Levels in Solids
Electrons in the higher unbound states form a degenerate Fermi
gas.
The bulk of the lower unbound states are completely filled with
electrons (imagine dropping fermions into the classical infinite
square well).
Only the electrons at the top of the Fermi sea can hope to
change their energy/momentum states
Electronic Energy Levels in Conductors
An alternative approach:
At large separations, electronic orbitals have “atomic”
characteristics.
As atomic separation decreases these degenerate states must split
under the interacting potential of all of the nuclei in the crystal.
The ensemble of split energy levels is a “band” which may be
–
full, partially filled and/or overlapping with other bands
Electrons which have immediately adjacent energy states can
change state and thus “conduct”
Electronic Energy Levels in Conductors
Sodium, a metal, has a single 3s valence band electron.
the 3s state is only half filled since any atom can have a second electron in
this state with opposite spin.
In crystalline sodium (recall any chunk of metal is an assemblage of
crystalline domains) the 3s state is shared by the entire crystal.
There are 2N (twice the number of atoms) 3s states in the crystal and only N
electrons.
Conduction is “easy” since a valence electron sees a variety of nearby open
energy/momentum states.
Each atom brings a fixed
number of states and a fixed
number of electrons. It is
natural that bands
sometimes end up exactly
full.
Electronic Energy Levels in Conductors
Copper has a single 4s electron making it a natural conductor.
It also has an electronic configuration where the 4s band
overlaps the 3d band at the crystalline interatomic spacing.
Overlap between bands also provides access to infinitesimally
different energy states, permitting conduction (e.g. Magnesium –
which has a filled 3s state is still an electrical conductor).
Note that there are about
10²² atoms in a fist sized
chunk of metal.
Each atom contributes a
couple of 4s energy
states.
The energy states span a
couple of eV.
The degenerate
conduction band energy
levels are about 10¯²² eV
apart.
Electronic Energy Levels in Insulators
Insulators have filled energy bands which do not overlap with
adjacent energy bands for the interatomic equilibrium spacing.
There is no such thing as a
“semiconductor”!
At T=0 a material will either have
overlapping energy states and is a
conductor, or it will have a
“bandgap” above a completely filled
energy state and be an insulator.
Semiconductivity (if that is a word) is
a manifestation of Boltzmann factors
at finite temperature – kT vs. the
“bandgap”.
Band Filling and Band Interactions
Semiconductors
Semiconductors can consist of pure
elemental materials or alloys of
different elements.
In either case, materials with
complementary filled valence shells
are likely semiconductors.
Carbon (diamond) is a “semiconductor”
with an energy gap of 5.33 eV (0.23um)l
http://pearl1.lanl.gov/periodic/default.htm
Semiconductors
At T=0K, the world contains only conductors and insulators.
Above 0K, electrons at the top of the Fermi sea can be excited to
higher energy states if the states are sufficiently ( ~kT ) close.
Small bandgap materials are thus semiconductors with marginal
electrical conductivity at room temperature due to thermally excited
carriers.
The conductivity of metals improves at low
temperatures. The conductivity of
semiconductors declines.
Semiconductor Conductivity
The population of electrons (holes) in the conduction (valence)
band in an intrinsic semiconductor depends on temperature.
The electrons obey Fermi-Dirac statistics, but given the large
size of the bandgap vs. kT, occupancy in the conduction band
follows an apparent Boltzmann law.
E bandgap= 2∗ E−E f 
Room Temperature Electrical Conductivity
in Crystalline Silicon
At 300K, kT = 0.026 eV ( 1 eV = 11,600K)
Silicon's bandgap is 1.11 eV
The electron effective mass is 1.1 m
–
Electron density
1.3×1010 e cm−3
Room Temperature Electrical Conductivity
in Crystalline Silicon
1.3×10 e cm
10
−3
The resistance of a material is given by
R =
l
A
ohm cm−1
–
where l is the length traversed by the current
and A is the resistor cross section.  is the
“conductivity” (which represents the current
density induced by an electric field, E).
–
Conductivity can be expressed in terms of the
material carrier density if one knows the
electron mobility, , which is the proportionality
between carrier drift velocity and the electric
field.
J =  E = nq ⟨v⟩
 = nq
⟨v⟩
E
= nq 
a one centimeter cubic piece of silicon should have a
room temperature resistance of ...
At 77K this resistance falls by a factor of
−1.1
e
−1.1
2∗0.026
/ e
2∗0.007
= e
−21.1
−82.4
/e
= e
−61.3
= 2.4×10
−27