Interband and Intersubband optical transition

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Interband and Intersubband optical transition

• Illia Marderfeld

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Outline

• Optical absorption in quantum structures
• Fermis Golden Rule
• Differences in quantum structures (quantum well)
• Form of wavefunctions for "non-excitonic"quantum
  well absorption
• Subbands and densities of states
• Calculation of optical absorption matrix element
• Selection rules, Experimental geometry

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Optical absorption in quantum structures
We want to develop a set of equations to describe
the absorption of a photon in semiconductor
material. The electromagnetic field is a
quantized system (with a set of modes, each of
which is a harmonic oscillator). In absorption, a
photon is absorbed by the crystal and the energy
of the electromagnetic field is transferred to
the crystal. The initial state in the region of
interest in the crystal is Ei,while the final
state is Ef.
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What are the assumptions and approximations we
must consider?

• The electromagnetic field is perturbed by the
  electronic crystal.

• If the wavelength associated with a
  mono-energetic field is larger than the
  perturbing charge (like in an atom or quantum
  dot), then we can make the dipole approximation
  and assume there is no position dependence to the
  field (and solve just using the time-dependent
  field E(t)E0sin( ?t)).

• Otherwise, we assume, Bloch waves.

• We can assume the intensity of the field is
  large,so that changes in the photon number in
  each mode is small. Called semiclassical
  approximation (which we will make most of the
  time).

• This means we neglect the action of the charge
  back on to the field (back action).

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Fermis Golden Rule
In order to determine the probability or
amplitude of the absorption we must find the
overlap of the initial and final wavefunctions.
Instead of single initial and final states in
single-particle picture, we have in principle a
large density of final states - ?(k)
The probability of absorption or emission will
depend on the overlap and energy difference of
the initial and final state, and the density of
these states.
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The matter-field interaction Hamiltonian (I)
The H we discuss has been based on the quantum
mechanical form of the kinetic energy and a
scalar field.
But now instead of the scalar potential we have
the momentum of the particle interacting with a
vector field, the electromagnetic field
containing the photon.
The absorption process is an interaction between
the matter and the electromagnetic field.
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The matter-field interaction Hamiltonian (II)
Initial state photon in the vacuum field,
carrier in a lower energy, ground state.
Final state the vacuum field is emptied of one
photon, carrier in the an upper, excited state.
The kinetic energy of the carrier in the vector
field PeeA (often called the dressed state).
And, disregarding any scalar potential (V(x)),
In the weak-field regime,e2? A2 is neglectable.
The kinetic energy of the carrier in the absence
of the field.(H0)
Responsible for the interaction energy between
the carrier and field (i.e.,the absorption).(H)
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Differences in quantum structures case
Choice of the wavefunctions for the initial and
final states
Two different kinds of possibilities in quantum
structure
Transitions between the valence and conduction
bands
Transitions between the quantum-confined states
within a given band, so-called "intersubband
transitions
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Form of wavefunctions for "non-excitonic quantum
well absorption (quantum well)

• Start by neglecting any excitonic effects (and
  other Coulomb effects many particle effects)
• Treat the initial state as being some electron
  state corresponding to an electron in the valence
  band or some lower subband
• Treat the final state as an electron in the
  conduction band or a higher subband

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Wavefunctions of initial and final states
In this approximation,we will write the
wavefunction in the form Bloch form

• ?k(r)- represents the actual electron (or hole)
  wavefunction
• Bqw is a normalization constant
• u(r) is appropriate unit cell function
  (i.e.,repeats exactly every unit cell)
• ?qw(z) is the "envelope function calculated
  for the appropriate quantum well state
• kxy is a wavevector in the x-y plane
• rxy is a position in the x-y plane

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Normalization of quantum well wavefunctions
Choose to normalize ?qw(z) over the quantum well
structure,i.e.
Quntum well structure (z direction)
Integral is over the whole structure of the
quantum well, including the barrier material,
not only the thickness of the quantum well
material itself
In the case of a more complex structure, such as
a coupled quantum well or other structure with
many different layers, the integral is over the
whole structure of many layers.
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Subbands and densities of states
Simple picture of the states in a quantum well
eigenstates of a particle (electron or hole)
Particle in one of the states of the one
dimensional potential, with quantum number n, and
envelope wavefunction ?n(z), in the z direction.
Unconstrained "free", plane-wave motion in the
other two directions with wavevector kxy.
The motion in the plane of the layers
With the total energy
Inplane motiom
We have so-called "subbands", the bottom of each
subband has the energy En of the one-dimensional
quantum well problem.
In bulk crystal
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Subbands
Sketch of the first three subbands for one of the
particles in a quantum well
Solid regions merely emphasize that the subbands
are paraboloids of revolution
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Density of states in a subband
Need to understand explicitly how many states
there are available to an electron or hole in a
given energy range about some particular energy
of interest
Density of states (in energy)- DOS
Number of possible electron states per unit
energy per unit volume
Start by calculating how many states there are
per unit k
Formally impose periodic boundary conditions in
the x and y directions
Gives us allowed values of the wavevector in the
x direction, kx, spaced by 2?/Lx, where Lx is the
length of the crystal in the x direction
similarly, the allowed values of ky are spaced by
2?/Ly, with analogous definitions
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DOS in Q.W. case
For each value of kxy, there are two allowed
electron states, so the number of states in a
small area d2kxy of kxy-space is
Hence can define (kxy-space) density of states
per unit (real) area, g2D(kxy),
Number of states between energies Exy and
ExydExy,i.e.,g2D(Exy)dExy, is then the number of
states in kxy-space in the annular ring, of area
2?kxydkxy, between kxy and kxydkxy, where
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DOS in Q.W. case (II)
Using the parabolic relation
We have
Constant for all Exy gt0.
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Special case of "infinite Q.W. density of states
Simple relation between the density of states in
an "infinite"quantum well and the DOS in a
conventional bulk (3D)semiconductor
"3D"density of states is
Evaluate the DOS in a bulk semiconductor at the
energy that corresponds to the first confined
state of a quantum well of thickness Lz
Same as the density of states per unit volume
(rather than per unit area) of an
"infinite"quantum well, i.e.,dividing g2D by Lz.
The infinite Q.W DOS (per unit volume), it
touches the bulk DOS (per unit volume) at the
edge of the first step
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Comparison of bulk DOS with Q.W. DOS
Increase the thickness of the Q.W
Steps would get closer and closer together
But their corners would still touch the bulk
density of states
We would eventually not be able to distinguish
its density of states from that of the bulk
material.
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Optical absorption
Electrical field
The Vector potential
q propagation vector
Hamiltonian
Under this electric dipole approximation( q
photon wavevector as 2?/q1?m ) Hint reduced to
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Optical matrix elemen
In the case where the levels i? and f? are
partially or completely occupied, one has to
account for impossibility of allowing transition
from either an empty level or toward a filled one
.
The probability per unit time that an electron
makes a i? ? f? transition or, equivalently,
that a photon disappears is thus equal to
Constants
f(Ei,f) Fermi distribution
Under electric dipole approximation( q photon
wavevector as 2?/q1?m ) Hint reduced to
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Interband optical absorption
Choose valence band wavefunction as initial
state. Conduction band wavefunction as the final
state.
For convenience choose the x-direction for the
optical electric vector
Assume the slowly varying envelope approximation
where the?I,f (z)terms and the oscillating
exponentials are slowly varying
over a unit cell, so we can replace them by their
values in the middle of the unit cells.
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Interband optical absorption (II)
Where Rxy,m runs over all the unit cells in a
plane in the x-y direction,zi runs over all the
unit cells in the z-direction in the quantum well
structure.
Sum over Rxy,m averages to zero unless kxy,v
kxy,c ?n0 selection rule
i.e.,unless momentum is conserved in the x-y
direction
This would be the case for the z-direction as
well in bulk material.
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Calculation of interband optical absorption
The total transition rate in the quantum well
structure
Replace
2D "joint"density of states in energy (so,per
unit area)
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2D "joint"density of states in energy
?(q) is called the Heaviside step function
Where we are saying that the 2D DOS is zero for
energies less than the separation of the bands in
question, and constant for energies greater than
the subband separation.
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Selection rules
In an idealized "infinite"quantum well
all our eigenfunctions form a complete orthogonal
set
only transitions between bands with the same
quantum number (nv nc) are allowed
In a finite quantum well,
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Interband and Intersubband Absorption
Intersubband absorption
Interband absorption
Energy (meV)
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Intersubband optical transition
Vanishes because changing parity
Non-zero only in the z direction
Vanishes because orthogonality
Rxy,m runs over all the unit cells in a plane in
the x-y direction
in-plane momentum conservation
zi runs over all the unit cells in the
z-direction in the quantum well structure
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Polarization selection rules
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Experimental geometry
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Energy lost
The transition probability per unit time
(absorption)
The occupancy of the level ??
In a similar way for emission
Energy loss of EM field per unit time to the
transition
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The absorption coefficient
Energy density
Volume of the sample
Time average over period 2?/?
Absorption coefficient
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Orders of magnitude
Averaging
Continuity equation for the photon density
In the steady state
per photon
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Justification of separation of wavefunction
Form of the wavefunction
as a separation into a unit cell wavefunction
u(r) multiplied by some envelope functions is
justified first of all by
presumption that we have envelope functions that
are slowly-varying over a unit cell
separation of the envelope function as
Can formally be justified by considering the
Schrödinger equation for the envelope function
Difference here compared to the conventional bulk
semiconductor case is that we have the additional
"quantum well" potential V(z)
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Separation of wavefunction
May formally apply the usual product method of
separating variables, which leads to
and
solution
So the separation of variables is justified.
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Envelope functions
In bulk semiconductors,use Bloch form for
electron in perfect crystal
Can consider uk(r) as being unit-cell part of
wavefunction, and exp(ik?r) as being one example
of an "envelope function"
Resulting wavefunctions can be approximately
written in a similar form
Especially where the perturbation is not large
over a unit cell
Approximately separate wavefunction into
Envelope that is slowly varying over a unit
cell multiplying unit cell function
Called a "slowly-varying envelope function
approximation"