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:
     [ VII-0 ]
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 valence-electron-ion potential.
The quantum mechanics of the assembly is treated to a good approximation by taking the total wavefunction of the system as the product
     [ VII-1 ]
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
     [ VII-2 ]
where the total Hamiltonian is parsed into two independent components -- viz.
     [ VII-3a ]

.

     [ VII-3b ]
The essential assumption of the adiabatic approximationis that the bracketed term in Equation [ VII-2] is negligible and that the global problem may be treated as two independent problems -- viz.
     [ VII-4a ]

 
     [ VII-4b ]
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 electron-phonon 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 quasi-particle moving in a periodic lattice

If the electron-electron interaction is averaged out, deviations from the average may be treated perturbations -- i.e., we make the replacement
     [ VII-5 ]
where  leads to a constant replusive electronic energy component and  is a small fluctuating electron-electron 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
     [ VII-6 ]
with the requirement that the occupation of one-electron states is in accordance with Pauli exclusion principle. Thus, each the wavefunction for each electrons satisfies a Schrödinger equation in the form
      [ VII-7a ]
or simplifying the notation
     [ VII-7b ]

SCHEMATIC PERIODIC LATTICE POTENTIAL


The lattice potential far from the surface of the crystal has the property that, for all lattice vectors 
     [ VII-8 ]
Bloch's (Floquet's) theorem: (See Periodic Boundary Conditions and Bloch's Theorem)

The function

     [ VII-9 ]
where  is a function with the same spatial periodicity as, is an eigenfunction of Equation [ VII-7 ].
Proof of theorem:
First rewrite Equation [ VII-7b ] as
     [ VII-7b' ]
Shifting the origin by a lattice vector, we again rewrite Equation [ VII-7b ] as
     [ VII-7b" ]
In light of periodicity expressed in Equation [ VII-8 ], it follows that
     [ VII-10 ]
Thus, any linear combination of the possible eigenfunctions  is a valid eigenstate of the energy! In particular, let us choose the combination
     [ VII-11 ]
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
[ VII-9 ].[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 [ VII-7b ] may be rewrite in the form .
     [ VII-12 ]
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.
     [ VII-13a ]

 
     [ VII-14a ]
where
     [ VII-13b ]
and
     [ VII-14b ]
Substituting these expressions into Equation [ VII-12 ] and equating various Fourier components, we obtain the following infinite set of algebraic equations:
     [ VII-15 ]
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
     [ VII-16 ]
and
     [ VII-17 ]

 
 
 
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 [ VII-16 ], to a Bloch function, Equation [ VII-9 ]. 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
     [ VII-18 ]
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 [ A-6c ] and [ A-7b ] 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
     [ VII-19 ]

 
     [ VII-20 ]
In light of Equation [ VII-13a ] we may write
     [ VII-21 ]
so that Equations [ VII-19 ] and [ VII-20 ] become
     [ VII-22 ]

 
     [ VII-23 ]
Clearly, the strongest departure from free electron behavior occurs when the denominator vanishes -- i.e. when
     [ VII-24 ]
Equation[ VII-24 ] 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 [ VII-19 ] may be approximated
     [ VII-25a ]
or
 .
      [ VII-25b ]

 
1D NEARLY FREE ELECTRON MODEL
extended zone scheme
reduced zone scheme
These considerations then provide the context for the examination of real band structures
(Source
For more on band structures go to: Bandgap Engineering

For an introduction to pn junction physics go to: Properties of PN-Junctions

B. OPTICAL PROPERTIES OF BULK (3-D) SEMICONDUCTOR

FREE-CARRIER THEORY
For most of our considerations a parabolic-band 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
     [ VII-26a ]
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 parabolic-band model is re-interpreted as, respectively, the quasiparticle energies of an electron in the conduction band and a hole in the valence band
     [ VII-26b ]

 
     [ VII-26c ]
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 inter-electronic Coulomb interactions and, in particular, carrier-carrierscattering. In what is usually called free-carrier theory, it is assumed that carrier-carrier 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 Fermi-Dirac 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
     [ VII-27 ]

 
where  is the carrier quasichemical potential (or imref  quasi-Fermi energy see junction dynamics).[8]


 
 

In the discussion of Equation [ IV-5 ] of this set of lecture, it was argued that the 3-D density of states in  space is given by

     [ VII-28 ]
which translates (within a given band) into a density of states per unit energy per unit volume of
     [ VII-29]
Hence, for a 3D semiconductor, Equation [ VII-27 ] becomes
     [ VII-30 ]

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]
     [ VII-31 ]
Thus, the critical optical matrix element [10] between a state in the valence band and one in the conduction band is
     [ VII-32a ]
where the so called transition matrix element is given by
     [ VII-32b ]
Using the expansion for  in Equation [ VII-12a ] the matrix element becomes
     [ VII-33 ]
However,  so that
     [ VII-34 ]
With the parsing  and using the defining relationship , the volume integral may be expressed
     [ VII-35 ]
The summation   unless
     [ VII-36 ]
-- i.e. essentially only vertical transitions are allowed! Therefore, .
     [ VII-37a ]
For a given polarization of the applied field, the transition matrix can be written
     [ VII-37b ]
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
Alx Ga1-x As  (x < 0.3) 29.83 + 2.85 x 
Inx Ga1-x  As 28.8 - 6.6 x 
In P  19.7 
In1-x Gax  Asy  P1-y  (x = 0.47 y)  19.7 + 5.6 y 

Of course, the radiation must also satisfy the energy conservation condition

     [ VII-38a ]
where
     [ VII-38b ]
"FREE-CARRIER THEORY" OF OPTICAL PROCESSESS
The bottom lineis that in the free-carrier 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 [ VII-37a ] 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
     [ VII-39 ]
so that the Schrödinger equation of motion for the density operator is also separable -- viz. .
     [ VII-39 ]
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 two-level, semiclassical discussed earlier. In light of the discussion in the previous section, we adapt the two-level theory by making the identifications
     [ VII--40 ]
Given the free carrier effective Hamiltonian discussed above and the equation of motion expressed in Equation [ VII-31 ], we can write
      [ VII--41a ]

 
     [ VII--41b ]

 
     [ VII--41c ]
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 four-level 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, electron-electron 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 Fermi-Dirac distribution referenced to a quasi-Fermi energy or imref appropriate to that band. To obtain an expression for  we note that[1]
     [ VII--42 ]
so that
     [ VII--43 ]
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 [ VI-13a-c ] from this set of lecture notes, we may write
     [ VII--44a ]
     [ VII--44b ]
     [ VII--44c ]
where  represents the pumping rate due to carrier injection and  effect of carrier-carrier scattering.  and  are, respectively, phenomenological representations of nonradiative decay and radiative recombination (spontaneous emission), respectively, . The probability difference (gain factor)
     [ VII--45 ]
 
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 equilibriumFermi-Dirac distributions -- viz.

     [ VII--46 ]

 
BULK LASER ENERGY DIAGRAM

 
Thus, the absorption factoris given by
     [ VII--47 ]
and the spontaneous emissionfactoris given by
 
    [ VII--48 ]

where  and  .[15]
 


Therefore, for population inversion
which is a very stringent condition!

 
Adapting Equation  [ VI-26a ]  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
     [ VII--49 ]
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.,


which yields a single electron  energy 

and the density of paired states which is a constant or a staircase.

 
 
2D (QW) LASER ENERGY DIAGRAM
2D (QW) Gain Variation with Carrier Concentration

Gain in a 2D semiconductor with
= 1.06, 1.08, and 1.10

http://www.shef.ac.uk/uni/academic/D-H/eee/cf/newslett/laser-5.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 many-body Coulomb interactions among the valence electrons.

.

[6] See a Derivation of the Fermi-Dirac distribution function at http://ece-www.colorado.edu/~bart/book/fermi-dirac_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 one-half.

[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 first-order perturbation theory tells us that the radiation-induced 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, Semiconductor-Laser Physics, Springer-Verlag (1994) -- that


where and are, respectively, electron and hole operatorsis 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