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Transcript
Optical properties of semiconductors
Today’s topics: interaction of light with electronic states in semiconductors
did this (C9S9)
did this (C9S11)
Fundamentals of Optical Science Spring 2008 - Class 12
slide 1
Schrödinger equation
To describe interaction of light with electrons, need to describe electron states
Schrödinger equation describes behavior of matter in terms of the wave function 
 2 2








V
x

x
,
t

i

x, t 


2
t
 2m x

Hamiltonian, giving the energy density of the wave function
Note that a ‘high curvature’ corresponds to a high energy (as for light waves) and
that the time dependence scales with the energy of the wave function
In free space and choosing V=0 we find ‘probability waves’ of the form
 
ik r
 k (r )  Ae
where the probability of finding matter at position x scales with ||2 or ×*
Compare: probability of detecting light scales with |E(x,t)|2 or E×E*
Fundamentals of Optical Science Spring 2008 - Class 12
slide 2
Momentum and energy
Classical relations:
Energy
E = ½ mv2
Momentum
p = mv = (2mE)
Quantum mechanical:
 2 2








V
x

x
,
t

i

x, t 


2
t
 2m x

Energy
 2 k e2
E V 
2m
Momentum
p = (2mE) =
or
 2 ke2
E
2m
 
ik r
 k (r )  Ae
for V=0
ke
We will see that the photon wave vector (and corresponding momentum k phot )
is often small compared to electron wave vector (e of the order of an atomic spacing)
Fundamentals of Optical Science Spring 2008 - Class 12
slide 3
Valence and conduction states
Inside a solid, electrons propagate in periodic potential: V(r) = V(r + a)
E low  bound solutions with low probability in between atoms
valence electrons
E high  propagating solutions with significant probability in between atoms
conduction electrons
Fundamentals of Optical Science Spring 2008 - Class 12
slide 4
Energy vs. wave vector
Shorter electron wavelength corresponds to higher energy:
 2 ke2
E
2m
Conduction electrons in crystal (shown for ‘very weak binding potential’)
If a = 1Å
E  150 eV
Energy of an electron with 1 A wavelength  150 eV
Energy of a photon with 1 A wavelength  12 keV
Fundamentals of Optical Science Spring 2008 - Class 12
slide 5
Reduced zone scheme vs. extended zone scheme
Functional form of solutions to Schrödinger equation can be written as Bloch wave:
 k ( r )  u k ( r )e

ik r
can describe high energy electrons (short waves) in terms of
phase difference between adjacent atoms (unit cells) described by ‘k’, and
shape of wave function within unit cell, described by function uk(r)
Behavior within unit cell gives
rise to energy bands
We can describe electron
behavior with wave vectors
that lie in the first Brillouin zone
1st Brillouin zone
Fundamentals of Optical Science Spring 2008 - Class 12
slide 6
Effective mass
In case of larger periodic potential, ‘repulsion’ at the zone boundary:
E
Bands no longer periodic, but
approximately parabolic near k=0
Can describe energy vs. k as before,
but use effective mass m*:
 2 ke2
E
2m *
Note: high curvature  m* must be small
-2/a -/a
0
/a
parabola = low effective mass’
2/a‘sharp
3/a
k
Key point: even though the electron energies are large, the energy differences
caused by ‘neighbor electron interactions’ are on the order of eV’s
Fundamentals of Optical Science Spring 2008 - Class 12
slide 7
Important note: things normally not that simple
Different direction of the wavevector: electron sees different ‘periodicity of crystal’
Example from Ashcroft and Mermin (p. 161): free electrons in FCC lattice
Letters on bottom axis represent directions and magnitudes of k
Fundamentals of Optical Science Spring 2008 - Class 12
slide 8
Inter-band absorption (direct gap)
Finite amount of electrons results in filled and empty states
In semiconductor, highest-energy filled states are in the
E states are in the conduction band
Lowest unoccupied
valence band
conduction electrons
empty
Egap
/a -/a
0
/a
valence electrons
filled
2/a 3/a k
Energy difference between valence and conduction band
is called the bandgap of the semiconductor
Material is called a semiconductor if Egap < 4eV, and insulator if Egap > 4eV
Fundamentals of Optical Science Spring 2008 - Class 12
slide 9
Energy levels in real semiconductors
In three dimensions, the dispersion relation depends on direction in the crystal
source: Optical
Properties of
Semiconductor
Nanocrystals,
Gaponenko
Fundamentals of Optical Science Spring 2008 - Class 12
slide 10
Summary of previous slides
Single atoms are ‘surrounded by’ bound electrons
In solids: electrons in neighboring atoms can interact:
 electron levels are modified, resulting in energy bands
In semiconductors the highest-energy band that is ‘filled’ (occupied by electrons)
is the valence band, and the lowest unoccupied states are in the conduction band
Coming slides:
strongest optical response if electron transitions can be induced
E
E
conduction
states
E
(usually empty)
empty
Egap
k
a -/a
0
/a
2/a 3/a
filled
Fundamentals of Optical Science Spring 2008 - Class 12
k
k
valence
states
(usually filled)
slide 11
Inter-band absorption in direct gap semiconductors
Direct-gap semiconductor: highest occupied and lowest unoccupied state occur at k=0
Light can induce electronic transitions if
energy and momentum are conserved:
E
Efinal – Einitial = Ephot and k = k phot  0
(Photon: long wavelength compared
to atomic spacing  kphot « /a )
k
Direct gap semiconductors
Photons with E < Egap have insufficient energy to ‘kick a valence electron
into a conduction state’  absorption starts at Ephot = Egap
These band-band absorptions have the usual implications for n and 
(recall Kramers-Kronig relations)
Fundamentals of Optical Science Spring 2008 - Class 12
slide 12
Interband absorption in indirect gap semiconductors
Indirect-gap semiconductor: highest occupied and lowest unoccupied state have k≠0
Direct transitions possible for k0
 strong direct interband absorption
occurs at E > Egap
Egap
Other possibility: momentum and
energy can be conserved by photon
absorption and simultaneous absorption or
emission of a phonon:
Indirect transitions possible with
‘assistance of a phonon’
Shown here are optically induced transitions
Egap
- during phonon emission
a phonon is generated in the process
- during phonon absorption
a phonon is generated in the process
Fundamentals of Optical Science Spring 2008 - Class 12
slide 13
Excitons
Excitons are combined electron-hole states:
A free electron and a free hole (empty electronic state in the valence band)
exert Coulomb force on each other:
hydrogen-like bound states possible: excitonic states
n=3
n=2
n=1
e
Coulomb
force
E
Eb is the exciton
binding energy =
h
energy released upon
exciton formation, or
Eb
k
energy required for
exciton breakup
Wave functions of electron and hole look similar to free electron and free hole
Note: exciton can move through crystal, i.e. not bound to specific atom!
Fundamentals of Optical Science Spring 2008 - Class 12
slide 14
Excitonic absorption
Light can excite an electron from the valence band and generate an exciton
at energies slightly below the bandgap
 see absorption at Ephot = Egap – Eb (absorption slightly below Egap)
n=3
n=2
n=1
e
Coulomb
force
E
h
Eb
k
Exciton binding energy on the order of a few meV
Thermal energy at room temperature: kT ~ 25 meV
 exciton rapidly dissociates at room temperature
 absorption lines broaden / disappear for higher temperatures
Fundamentals of Optical Science Spring 2008 - Class 12
slide 15
Optical transitions related to dopant atoms
Ga: 3 valence electrons
Si: 4 valence electrons
As: 5 valence electrons
Fundamentals of Optical Science Spring 2008 - Class 12
slide 16
Donor levels
Substitute Si atom with As atom (impurity atom in the Si lattice):
weakly bound extra valence electron
Low T
Low T: donors neutral, electron weakly bound
lew energy light can excite donor electron in to conduciton band
Binding energy Ed similar to kT at room temperature (‘RT’):
At room temperature the bound electron is quickly released
 impurity mostly ionized at RT : Arsenic is a donor in Si
RT
At RT such transitions are typically too broad to observe
Fundamentals of Optical Science Spring 2008 - Class 12
slide 17
Acceptor levels
Substitute Si atom with Ga atom : empty electronic state just above the Si
valence band: at finite temperature, Si valence electron may fill acceptor level
 location of unoccupied valence state (hole) can orbit the charged Ga dopant
‘hole’ =
available
electron
state
Binding energy Ea similar to kT at room temperature (‘RT’):
At room temperature the hole can leave the dopant,
producing a ‘free charge’
Fundamentals of Optical Science Spring 2008 - Class 12
slide 18
Infrared absorption due to dopants
Dopant binding energies low: donor level related absorptions invisible at RT,
but observable at low temperatures
Example: direct valence band → acceptor level absorption in boron doped Si
Transition at ~40 meV  absorption at 30 m : infrared
Fundamentals of Optical Science Spring 2008 - Class 12
slide 19
Dopant related transitions
Possible dopant related transitions:
Typically visible at low T, but not clearly observable at RT
Fundamentals of Optical Science Spring 2008 - Class 12
slide 20
Free carrier absorption (1/2)
At RT, predominant dopant related absorption is free carrier absorption
in which a photon excites an electron into a higher lying state
Example: p-type semiconductors: filled states in the conduction band:
optical transitions possible at Ephot < Egap !
Free electrons:
absorption typically indirect
phonon-assisted transition
Free holes can make direct transitions
form the heavy-hole band
to the light-hole band
 holes cause stronger free carrier
absorption than electrons
Fundamentals of Optical Science Spring 2008 - Class 12
slide 21
Free carrier absorption (2/2)
Free electron absorption can be described by the Drude model
Dopant levels in semiconductors range from ~1014 - 1018 /cm3
which is ~108 – 106 lower than free electron densities in metals
Plasma frequency of doped semiconductors 104 - 103 lower than of metals: IR
At frequencies above plasma frequency,εr and  described by
 p2
 r ' ( )  1  2 ,

 p2  p2 
 r " ( )  3  3


 p2  2
 ( )   "( ) 

2
c
c
c 2p

Electron FCA up for lower energies
Free hole absorption less well defined
Fundamentals of Optical Science Spring 2008 - Class 12
slide 22