burstein shift excitons interband transitions in quantum wells...
TRANSCRIPT
NNSE 618 Lecture #15
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Lecture contents
• Burstein shift
• Excitons
• Interband transitions in quantum wells
• Quantum confined Stark effect
NNSE 618 Lecture #15
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Absorption edges in semiconductors
• Offset corresponds to bandgap
• Abs. coefficient is orders of
magnitude higher for direct
transitions
• Abs. coefficient roughly follows
density of states
NNSE 618 Lecture #15
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Burstein-Moss shift
• Shift of absorption edge in degenerate semiconductors
• Usually in direct n-type semiconductors with low effective mass
• Due to occupation of band energy states up to: , the edge shifts:
From Seeger, 1973
Absorption edge shift in doped n-InSb
Burstein edge in degenerate n-type semiconductor
eBn
e
Tkm
k4
2 *
22
**
22 11
2he
gmm
kE
NNSE 618 Lecture #15
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Wannier Excitons
r
erVi
2
)( Similar to hydrogen-like impurity: electron and hole
bound by screened coulomb interaction
Solution for discrete energy levels:
With reduced effective mass (electron and
hole orbiting around their center of mass):
Envelope function of the ground state
(hydrogen-like):
Bohr radius:
Free exciton can move in the crystal
as a quasiparticle with a mass
BB
a
r
a
rF exp1
)(213
0
20
2 m
emaB
22222
4 111
2 nmRy
n
eEex
Exciton Ry*
Ry* = 6 meV, aB = 100 A
**
111
he mm
For excitons in GaAs
(me*=0.07m: and = 12.6 ):
**he mmM
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Exciton absorption
Exciton absorption edge in GaAs
• Exciton absorption red-shifts the absorption edge by the exciton binding energy
• Exciton edge absorption is higher than for band absorption (Sommerfeld Enhancement)
• Exciton peaks at room temperature are difficult to resolve in most materials (notable exception -
quantum wells)
• Excitons in bound states are fragile e.g., broken by colliding with phonons (e.g., in a few hundred
femtoseconds).
• By the uncertainty principle, they must then have broad linewidth
From Harris, 2004
NNSE 618 Lecture #15
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Sommerfeld Enhancement
Lots of bound states near
the onset of continuum sum
together to give Sommerfeld
enhancement.
• Even without excitonic peaks, bandedge
absorption is enhanced due to Coulomb
interaction between electrons and holes
• The reason is an increased density of states of
excitons over the band edge
• This results in increasing of the absorption
coefficient at the band edge:
• Above the bandedge exciton contribution is due to
“mobile” excitons with nonzero wavevector k:
with
Absorption edge in direct band semiconductors
exex
Ry
n
dE
dnDOS
3
2
21
212
g
exfreegex
E
RyE
x
xeE
x
freegex
sinh
21
21
g
ex
E
Ryx
NNSE 618 Lecture #15
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Exciton absorption in “forbidden” direct band edges
Exciton absorption in Cu2O at 4 K
• “Forbidden” direct band-to-band
transition Cu2O due to even parity of
electron and hole wavefunctions
(momentum operator has odd parity)
• Higher order transition (quadruple
instead of dipole) and dipole transition
for non-zero k in confined states are
allowed
From Seeger , 1973
NNSE 618 Lecture #15
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Interband transitions in quantum wells
From Singh, 2003
• Calculated absorption spectrum of 100A
GaAs/Al0.3Ga0.7As without exciton effects
• Strong exciton effects are present
Absorption spectra of GaAs/Al0.3Ga0.7As and
In0.53Ga0.47As/n0.52Ga0.48As QWs
Alloy broadening
Heavy-hole exciton binding
energy as a function of well size
NNSE 618 Lecture #15
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Franz-Keldysh effect in GaAs
Modulation of interband transitions in bulk semiconductors:
Franz-Keldysh effect
From Seeger, 1973
• Concept of Franz-Keldysh effect:
solution for electron and hole
envelope wavefunctions with
constant field are Airy functions.
• Wavefunctions now "tunnel" into
the bandgap region allowing
overlap of electron and hole
wavefunctions even for photon
energies less than the bandgap
energy, hence allowing optical
absorption below the bandgap
energy.
Franz-Keldysh effect
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Absorption spectrum due to Franz-Keldysh effect
Franz-Keldysh effect
Franz -Keldysh effect is a central-force problem
with perturbation:
Airy function Ai(Z)
• Z>0: electron-hole energy < electric field potential
• Z<0: electron-hole energy > electric field potential, i.e.
above bandgap oscillation wavefunction
Absorption spectrum reduce to the familiar
square root energy when field 0
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• With applied field, electron and hole wavefunctions are
distorted (second order perturbation)
• The intersubband separation decreases with electric
field (dominant term)
• Binding energy of excitons decreases with field;
carriers are separated by the field (few meV effect)
Modulation of interband transitions in quantum wells
QW
no electric field
QW
in electric field
Calculated variation of ground state intersubband
transition in W= 100A GaAs/Al0.3Ga0.7As QW
2
422
2
)2(1
*1
15
24
1
WemE
NNSE 618 Lecture #15
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Electric field modulation of transmission spectra
of 100 A GaAs/AlGaAs QW at two polarizations
Modulation of intersubband transitions in quantum wells
• Absorption edge red-shifts with
electric field
• Exciton absorption strength
reduces with field because the
electron and hole wavefunctions
are separated by electric field
• Polarization rules apply due to
symmetry of electron-radiation
matrix elements
From Miller, 1986
fiferi pAmc
eH