The Interaction of Radiation and Matter: Semiclassical
Theory (cont.)
VII Semiconductor
Photonics (pdf)[1]
A. PRELIMINARIES: SEMICONDUCTOR BACKGROUND[1]
For a brief overview of this material (with movies) see
the webpage entitled
Self Study Materials
on Solid State Electronics 1
THE CRYSTAL HAMILTONIAN
For an assembly of atoms the classical energy is the sum of
the following:

the kinetic energy of the nuclei;

the potential energy of the nuclei in one another's electrostatic field;

the kinetic energy of the electrons;

the potential energy of the electrons in the field of the nuclei;

the potential energy of the electrons in one another's electrostatic field;

the magnetic energy associated with spin and orbital variables.
Dividing the electrons into core and valence electrons and leaving out
magnetic effects leads to the following expression for the crystal Hamiltonian:

[ VII0 ] 
where a and b
label
the ions, land mlabel the electrons,
is the momentum, M is an ionic mass, m is the mass of an
electron,
is the interionic potential, and
is the valenceelectronion potential.
The quantum mechanics of the assembly is treated to a good
approximation by taking the total wavefunction of the system as the product

[ VII1 ] 
where
is the wavefunction of all the ions and
is the wavefunction of all the electrons at the instantaneous ionic positions.
The Schrödinger equation is then written

[ VII2 ] 
where the total Hamiltonian is parsed into two independent
components 
viz.

[ VII3a ] 
.

[ VII3b ] 
The essential assumption of the adiabatic approximationis
that the bracketed term in Equation [ VII2] is negligible and that the
global problem may be treated as two independent problems  viz.

[ VII4a ] 

[ VII4b ] 
As a further refinement, the electron problem must be parsed
once more as
where
defines the problem of the many electron system interacting with the static
ionic lattice and
incorporates the effects of the electronphonon interaction.
LATTICE VIBRATIONS
For a brief overview of lattice vibrations (with movies)
see the webpage entitled
Self Study
Materials on Solid State Electronics 2
BLOCH ELECTRONS  Solutions of the Schrödinger equation for a single
electron or quasiparticle moving in a periodic lattice
If the electronelectron interaction is averaged out, deviations
from the average may be treated perturbations  i.e., we make the
replacement

[ VII5 ] 
where
leads to a constant replusive electronic energy component and
is a small fluctuating electronelectron interaction. If we neglect these
fluctuations, each electron interacts independently with a screened lattice
potential. In this approximation, the electronic wavefunction can be expressed
as

[ VII6 ] 
with the requirement that the occupation of oneelectron states
is in accordance with Pauli exclusion principle. Thus, each the wavefunction
for each electrons satisfies a Schrödinger equation in the form

[ VII7a ] 
or simplifying the notation

[ VII7b ] 
SCHEMATIC PERIODIC LATTICE POTENTIAL
The lattice potential far from the surface of the crystal has
the property that, for all lattice vectors

[ VII8 ] 
Bloch's (Floquet's) theorem: (See Periodic
Boundary Conditions and Bloch's Theorem)
The function

[ VII9 ] 
where
is a function with the same spatial periodicity as, is an eigenfunction
of Equation [ VII7 ].
Proof of theorem:
First rewrite Equation [ VII7b ] as

[ VII7b' ] 
Shifting the origin by a lattice vector, we again rewrite Equation
[ VII7b ] as

[ VII7b" ] 
In light of periodicity expressed in Equation [ VII8 ], it
follows that

[ VII10 ] 
Thus, any linear combination of the
possible eigenfunctions
is a valid eigenstate of the energy!
In particular, let us choose the combination

[ VII11 ] 
where
is, for the moment, taken to be an arbitrary complex vector. Since
the summation includes all possible lattice vectors it must be a periodic
function of
with the period of the lattice and may be identified with
of Equation
[ VII9 ].[2] If we impose cyclical
boundary conditions on ,
we see that, of necessity,
is a real vector! QED
In light of Bloch's theorem, Equation [ VII7b ] may be rewrite
in the form .

[ VII12 ] 
Since the reciprocal lattice vectors of crystal are defined[3]
so that ,
any function written in the form
has the spatial periodicity of the lattice. Conversely, we
are allowed to expand
and
in Fourier series  viz.

[ VII13a ] 

[ VII14a ] 
where

[ VII13b ] 
and

[ VII14b ] 
Substituting these expressions into Equation [ VII12 ] and
equating various Fourier components, we obtain the following infinite set
of algebraic equations:

[ VII15 ] 
which can, in principle, be solved for
and
. If we were to turn off the lattice potential off to effectuate the
empty
lattice approximation, the eigenfunctions and eigenvalues
should, obviously, be those of a free electron and given by

[ VII16 ] 
and

[ VII17 ] 
1D EMPTY LATTICE MODEL
extended zone scheme

reduced zone scheme

With gradual restoration of the lattice potential, the wave
function is gradually transformed from a plane wave, Equation [ VII16
], to a Bloch function, Equation [ VII9 ]. In general, the value of
is well defined since it does not change from its original value and the
plane wave merely becomes modulated by the function .
Clearly, it is also possible to write the Bloch wave function in the form

[ VII18 ] 
and since both
and
are spatially periodic, so is their product. Thus, the wave function can
be considered as obeying the Bloch theorem with any empty lattice wave
vector .
Accordingly, the theorem allows for two alternative modes of classifying
solutions  i.e. a solution may be specified with the original
value of
(extended zone scheme) or with that value "reduced" by a reciprocal lattice
vector plus a band index (reduced zone scheme).
In the nearly free electronapproximation we use
the
functions as a basis set and take the periodic potential
as a small static perturbation. Using Equations [ A6c ] and [ A7b ] of
this set of lecture notes (at the end of the Section II, Review of Basic
Quantum Mechanics: Dynamic Behavior of Quantum Systems) we obtain

[ VII19 ] 

[ VII20 ] 
In light of Equation [ VII13a ] we may write

[ VII21 ] 
so that Equations [ VII19 ] and [ VII20 ] become

[ VII22 ] 

[ VII23 ] 
Clearly, the strongest departure from free electron behavior
occurs when the denominator vanishes  i.e. when

[ VII24 ] 
Equation[ VII24 ] is the famous Bragg
scattering condition or the condition which defines the boundaries
of the Brillouin zones. At the zone edge for a particular pair of degenerate
free electron states the appropriate pair of equation from the set in Equation
[ VII19 ] may be approximated

[ VII25a ] 
or
.

[ VII25b ] 
1D NEARLY FREE ELECTRON MODEL
extended zone scheme

reduced zone scheme


These considerations then provide the context for the examination
of
real
band structures
For more on band structures go to: Bandgap
Engineering
For an introduction to pn junction physics go to: Properties
of PNJunctions
B. OPTICAL PROPERTIES OF BULK (3D) SEMICONDUCTOR
FREECARRIER THEORY
For most of our considerations a parabolicband
model provides a reasonable picture of optical properties. In
this model we assume that near a band edge the single particle energy of
the electron measured with respect to the edge is given by

[ VII26a ] 
where the effective mass
is a measure of the inverse curvature of the band.[4]
Introducing the notion of a deficiency of electrons in the valence band

i.e. holes  as positive charge carriers, the parabolicband
model is reinterpreted as, respectively, the quasiparticle energies of
an electron in the conduction band and a hole in the valence band

[ VII26b ] 

[ VII26c ] 
where
and
accounts for aggregate dynamics of all the electron in the valence minus
the single empty state.[5] Of course, the single particle
picture discussed thus far, does not include the critically important effects
of the interelectronic Coulomb interactions and, in particular, carriercarrierscattering.
In what is usually called freecarrier theory, it
is assumed that carriercarrier scattering causes a rapid (relaxation time
less than 0.1 picoseconds) "thermalization" of excited conduction band
electrons (and valence band holes) and, consequently,it is assumed that
carriers within a band are in
quasiequilibriumwith
energies distributed according to a FermiDirac distribution.[6],
[7]
Accordingly, the carrier density for a given band (i.e.
for the conduction band and
for the valence band) is determined by the condition

[ VII27 ] 
where
is the carrier quasichemical potential (or
imref
quasiFermi energy see junction
dynamics).[8]
In the discussion of Equation [ IV5 ] of this set of lecture, it was
argued that the 3D density of states in
space is given by

[ VII28 ] 
which translates (within a given band) into a density of states
per unit energy per unit volume of

[ VII29] 
Hence, for a 3D semiconductor, Equation [ VII27 ] becomes

[ VII30 ] 
OPTICAL MATRIX ELEMENTS
Since optical interactions in a semiconductor are essential
distributed,
it is probably more precise and appropriate in this case to write the interaction
Hamiltonian as [9]

[ VII31 ] 
Thus, the critical optical matrix element [10]
between a state in the valence band and one in the conduction band is

[ VII32a ] 
where the so called transition matrix element is
given by

[ VII32b ] 
Using the expansion for
in Equation [ VII12a ] the matrix element becomes

[ VII33 ] 
However,
so that

[ VII34 ] 
With the parsing
and using the defining relationship ,
the volume integral may be expressed

[ VII35 ] 
The summation
unless

[ VII36 ] 
 i.e. essentially only
vertical transitions are allowed!
Therefore, .

[ VII37a ] 
For a given polarization of the applied field, the transition
matrix can be written

[ VII37b ] 
where the polarization factors are of order one and typical
values of
are given in the following table:[11]
Material system


Ga As 
28.8 
Al_{x} Ga_{1x} As (x < 0.3) 
29.83 + 2.85 x 
Inx Ga_{1x} As 
28.8  6.6 x 
In P 
19.7 
In_{1x} Ga_{x} As_{y}
P_{1y} (x = 0.47 y) 
19.7 + 5.6 y 
Of course, the radiation must also satisfy the energy conservation condition

[ VII38a ] 
where

[ VII38b ] 
"FREECARRIER THEORY" OF OPTICAL PROCESSESS
The bottom lineis that in the freecarrier
theory of optical interactions, the effective Hamiltonian
for the carriers 
i.e. "free" particle kinetic energy plus electromagnetic
interaction  is separable into a series of dependent
terms.[12] Thus, if we neglect
correlations in the treatment of optical properties, we need only consider
the following manifold of states:
Equation [ VII37a ] tells us that the absorption or emission
of a single photon connect only the two states in the manifold which can
be identified as and where
the first number in the ket specifies electron occupancy of a state
with momentum
and the second hole occupancy of a state with momentum .
Since no
correlations are involved in the free carrier theory, the density operator
of the complete system may be expressed as a product of component density
operators

[ VII39 ] 
so that the Schrödinger equation of motion for the density
operator is also separable  viz. .

[ VII39 ] 
C. INJECTION LASER THEORY
See pictures of Injection
Laser Configurations
A fairly satisfactory model of lasing (and other optical processes)
in semiconductors may be obtain by adaptation of the twolevel, semiclassical
discussed earlier. In light of the discussion in the previous section,
we adapt the twolevel theory by making the identifications

[ VII40 ] 
Given the free carrier effective Hamiltonian discussed
above and the equation of motion expressed in Equation [ VII31 ], we can
write

[ VII41a ] 

[ VII41b ] 

[ VII41c ] 
which tells us, once again, that
is driven by
and vice versa. There is, however, an important new element to be
considered in this problem to wit, as we have illustrated above, we are,
in fact, actually dealing with a fourlevel system which
includes of the states
and
as well as the pseudo "a" and "b" states. Fortunately, inclusion of these
states does not unduly complicate the analysis since, as discussed above,
electronelectron scattering induces a rapid relaxation or "thermalization"
of the probability of finding a given state occupied. After thermalization,
the probability of finding a state with a particular momentum value in
a given band is provided by a FermiDirac distribution referenced to a
quasiFermi energy or
imref appropriate to that band. To obtain
an expression for
we note that[1]

[ VII42 ] 
so that

[ VII43 ] 
where
is the probability of finding a electron with momentum
independent of whether or not there is a hole with momentum
and
is the corresponding probability for a hole. Adapting Equations
[ VI13ac ] from this set of lecture notes, we may write

[ VII44a ] 

[ VII44b ] 

[ VII44c ] 
where
represents the pumping rate due to carrier injection and
effect of carriercarrier scattering.
and
are, respectively, phenomenological representations of nonradiative decay
and radiative recombination (spontaneous emission), respectively,
. The probability difference (gain factor)

[ VII45 ] 
is the critical factor in the analysis of stimulated
processes 
i.e. for inversion .
This gain factor may be written as
 i.e. the population inversion is proportional to the probability
difference of an electron in corresponding states in the conduction
and valence bands.
Alternatively and more usefully, we see that it varies directly
with the product of
the spontaneous emission factor 

and the so called absorption factor 

In the spirit of the free carrier model
and in the unsaturated
limit the's
are given by the quasi equilibriumFermiDirac distributions 
viz.

[ VII46 ] 
BULK LASER ENERGY DIAGRAM

Thus, the absorption factoris
given by

[ VII47 ] 
and the spontaneous emissionfactoris
given by

[ VII48 ] 
where
and
.[15]
Therefore, for population
inversion
which is a very
stringent condition!
Adapting Equation
[ VI26a ] from earlier in this set of lecture notes,
we may see that the small signal gain in a semiconductor, may be expressed
in the form

[ VII49 ] 
In bulk material
the density of paired states varies as the square root of the energy.
Therefore, we may draw the following gain curves:
Gain in a bulk semiconductor with
= 1.00, 1.02, 1.04, 1.06, 1.08, and 1.10
QUANTUM CONFINEMENT
For reference see Quantum
Well Structures and the Quantum Well Laser
A Typical Confinement Structure

(click for full
size picture)

The quantum mechanics may easily solved on the assumption of
an idealized quantum well  i.e.,
2D (QW) LASER ENERGY DIAGRAM

http://www.shef.ac.uk/uni/academic/DH/eee/cf/newslett/laser5.html
http://adelaide.dcs.hull.ac.uk/AP/theory/pretty_pictures/Eb_SQW.html
[1] This discussion draws heavily on B. K.
Ridley, Quantum Processes in Semiconductors
(3rd
edition), Clarendon Press (1993).
[2] We keep for the record
[3] The reciprocal lattice vectors:
where
[4] That is, the effective mass is given by
and, thus, in the conduction band the effective mass of electrons
is positive and in the valence band it is negative  i.e.
and
.
[5] In particular,
since
must of necessity include the manybody Coulomb interactions among the
valence electrons.
.
[6] See a Derivation of the FermiDirac distribution
function at http://ecewww.colorado.edu/~bart/book/fermidirac_derivation.htm
[7] See the applets Fermi Level vs. Carrier Concentration
and Doping of Donor and Acceptor Impurities
(at http://www.acsu.buffalo.edu/~wie/applet/fermi/fermi.html)
and Fermi Level, Fermi Function and Electron Occupancy of Localized
Energy States
(at http://www.acsu.buffalo.edu/~wie/applet/fermi/functionAndStates/functionAndState.html).
[8] As are the electron and
hole energies , the imrefs are measured with respect to the appropriate
band edge  viz.
and
Thus, if the band does contain enough carriers to populate any state
with a probability greater than onehalf.
[9] In the next set of notes entitled The
Interaction of Radiation and Matter: Quantum Theory we develop the
following expression for the nonrelativistic Hamiltonian of a single
charged particle:
where
is the canonical conjugate momentum of the charged particle.
[10] Fermi Golden Rule and firstorder perturbation
theory tells us that the radiationinduced transition rate is given
by
[11] From Diode Lasers and Photonic Integrated Circuits
by Larry A. Coldren and Scott W. Corzine, Wiley (1995)
[12] In particular, it can be shown  see Weng W.
Chow, Stephan W. Koch and Murray. Sargent III, SemiconductorLaser Physics,
SpringerVerlag (1994)  that
where and are,
respectively, electron and hole operators. is
the dipole matrix element between vertical states in the valence and conduction
bands.
[13] That is, there must be either 0 or 1 electrons
and 0 or 1 holes in the given state.
[14]
[15] For plotting purposes it is useful to write
the complete
gain factor as
where , , ,
and .
.
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This page was prepared and is maintained
by R. Victor Jones, jones@deas.harvard.edu
Last updated April 4, 2000