Optical Gain in Semiconductors
As shown in Figure 2-10, parabolic band structure approximation is used for conduction and valence band E-K relationship. The density of states for the combined system involving transition between E2 and E1 can be written as:
2
where the reduced effective mass is:
* emission photon energy.
mc* m*v
Assumed:
(1) The Einstein coefficient, B, is the same for semiconductor,
(2) Photon energy density is ,where nρ(ν)dν=np ⋅hν p is the photon density, (3) Density of states is expressed as Nr(ν),
The transition rate between E2 and E1 can be written as:
Stimulated absorption: R1−>2=B12ρ(ν)dν⋅Nr(ν)⋅[fv(E1)(1− fc(E2))] (2-32) Stimulated emission: R2−>1=B21ρ(ν)dν ⋅Nr(ν)⋅[fc(E2)(1− fv(E1))] (2-33) The net stimulated emission rate is:
) zero, the transparent condition occurs. Thus, the condition for net stimulated emission
is:
The gain is defined as the ratio of net power emitted per unit volume over the power crossing per unit area. As shown in Figure 2-11,.an incident light pass the distance of dz and gains the amount of power ∆I. The speed of light is vg, where dz = vg*∆t. The power of light can be express as: I(ν) = [ρ(ν)dν]*vg. The optical gain coefficient is written as:
) After the introduction to the concept of gain in a relative macroscopic point of view, the detail behavior and interaction between light and atomic system require quantum mechanical analysis. If we consider an atom with two-level energy system, the electron under light interaction has the Hamiltonian expressed by:
(2-38)
where A is the vector potential. The above equation can be expanded as:
' The term, H’, can be viewed as a time-dependent perturbation to the original Hamiltonian, Ho. This perturbation term is the driving force for transitions between the conduction and valence bands. From solutions of time-dependent Schrodinger’s
equation, the transition rate for semiconductors can be obtained as: Equation (2-40) is known as Fermi’s Golden Rules. Compared with equation (2-33) and (2-40), and substituted Einstein coefficient, B, with matrix element, |H’21|, the gain expression can be rewritten as:
)
The matrix element, |H’21|, determines the strength of interaction between two states. In semiconductors, ϕ2 and ϕ1 in equation (2-41) are expressed as:
ϕ1 = F1(r)*uv(r) for valence band, (2-43) ϕ2 = F2(r)*uc(r) for conduction band, (2-44) where:
(1) uv(r) and uc(r) are Bloch functions of parabolic potential with atomic scale, (2) F1(r) and F2(r) are envelope functions of macroscopic potential, satisfying
Schrodinger’s equation in such as quantum wells, quantum dots. The bulk, quantum-well, and quantum-wire envelope functions take the following form:
The H’ can be written as: Overlap of Bloch
function
Envelope function overlap integral
where |MT| is known as transition matrix element. Since photon energy density is nph*hν , and electromagnetic wave energy density = 1/2*nr2ξo|E|2, the Ao can be
And the equation (2-42) can be rewritten as:
) Optical Gain in Quantum Well Structures
We specially pay attention to the optical gain for quantum well structures, since
the active mediums for all the devices in this study have quantum wells (QW). If we assume the potential confinement is along the z direction, the envelope function overlap integral can be expressed as:
∫
Due to orthogonality between the quantum-well wave-function solutions, the overlap integral in equation (2-53) reduces to the following rule for sub-band transitions:v cn
F n
F2 | 1 |2 ,
|< > ≅δ (2-54)
This means that transitions can only occur between quantum-well sub-bands which have the same quantum number, nc = nv. These are referred to as allowed transitions.
Transitions between sub-bands with dissimilar quantum numbers are forbidden transitions. Both are illustrated in Figure 2-12.
Except for the envelope function overlap integral; the other term in transition matrix element is the overlap of Bloch function, which is also known as momentum matrix element |M|2: The momentum matrix element, which is polarization dependent, determines the transition probability between conduction band and valence band. To further define the Bloch functions of the various energy bands, the corresponding atomic orbitals have to be taken into account. The Bloch function, us, corresponding to the isotropic s atomic orbital in conduction band remains the same. However, the Bloch functions ux, uy and uz corresponding to three p atomic orbitals for valence bands: px, py and pz, interact with each other along with the spin up and down. Using the kp theory, the modified valence bands are shown in Figure 2-13. The three valence bands are
commonly known as the heavy-hole (HH), light-hole (LH), and split-off hole (SO) bands. Since the constant |M|2 can be determined experimentally, Table 2-2 has listed the reported values for several important materials. Table 2-3 summaries the results for bulk and quantum-well materials for either transverse electric (TE: electric field in the quantum-well plane) or transverse magnetic (TM: electric field perpendicular to quantum-well plane) polarizations.
Nonradiative Transitions
Nonradiative transition is relatively important when considering the overall carrier recombination process. Three major types of nonradiative transitions are depicted in Figure 2-14. The first type of nonradiative recombination happens when existing an energy level in the middle of the gap, which serves to trap an electron from the conduction band temporarily before releasing it to the valence band. Defects in the lattice structure are one source of traps. The recombination rate, also referred as Schockley-Read-Hall recombination, takes the form:
e where Ni is the intrinsic carrier concentration, τe is the time required to capture an electron from the conduction band assuming all traps are empty, τh is the time required to capture a hole from the valence band assuming all traps are full, and N*
and P* are the electron and hole densities that would exist if the Fermi level was aligned with the energy level of the trap. For the laser applications, equation (2-56) can be simplified with the high-level injection regime:
e
The second type of nonradiative recombination in Figure 2-13 depicts electrons recombining via surface states of the crystal. The surface recombination rate under
a 32
high level injection in the active region can be expressed as:
(2-58) where as is the exposed surface area, V is the volume of the active region, and vs is the surface recombination velocity. Surface recombination is most damaging when the exposed surface-to-volume ratio is large. In addition, devices when make use of regrowth technique can suffer from poor interfaces and hence high interface recombination. Surface recombination is also material dependent. The recombination velocity of short-wavelength GaAs system is one order greater than that of long-wavelength quaternary InGaAsP system.
The last type of nonradiative recombination depicted in Figure 2-14 is basically a collision between two electrons, which knocks one electron down to the valence band and the other to a higher energy state in the conduction band. An analogous collision can occur between two holes in the HH band and either SO or LH band. The above three types of collision are refereed to as Auger processed. In laser applications with high injection level, the Auger recombination rate can be expressed as:
(2-59) CN3
RA=
where C is a generic experimentally determined Auger coefficient. In long wavelength InGaAsP materials, the Auger coefficient is one order lager than GaAs systems since the smaller band-gap in InGaAsP materials enhances the probability of momentum conservation. The reduced material dimensionality, such as quantum well, appears to reduce the Auger process due to the modification of band structures.
Another possible method of minimizing Auger recombination is to use strained materials in active layers.
Optical Gain in Strained Quantum Wells
Strained QWs use a material, which has different native lattice constant than the
surrounding lattice constant. As shown in Figure 2-15, if the QWs native lattice constant is larger than the surrounding lattice constant, the QW lattice compress in the plane, and the lattice is said to be under compressive strain. If the opposite is true, the QW is under tensile strain. However, in any lattice-mismatched system, it is important to realize that there is a critical thickness beyond which the strained lattice will begin to revert back to its native state, causing high densities of lattice defects.
For typical applications, this critical thickness is on the order of a few hundred angstroms, thus limiting the number of strained QWs in active layers.
Because the energy gap of a semiconductor is related to its lattice spacing, distortions in the crystal lattice should lead to alterations in the bandgap of the strained layer. There are two types of modifications. The first effect produces an upward shift in the conduction band as well as a downward shift in both valence bands, increasing the overall bandgap by an amount, H (which is positive for compressive strain and negative for tensile strain). The H indicates that this shift originate from the hydrostatic component of the strain. The second important effect separates the HH and LH bands, each being pushed in opposite directions from the center by an amount, S. The S indicates that this shift originates from the shear component of the strain. Figure 2-16 illustrates the energy shifts of the bands for biaxial strains. No only the energy shift, the band curvatures will be modified due to the strain effect. For a quantum-structure such as an In1-xGaxAs layer sandwiched between InP barriers, the band structures are shown in Figure 2-17 for (a) a compressive strain (x < 0.468), (b) no strain (x = 0.468), and (c) a tensile strain (x >
0.468) [18]. The left-hand side shows the quantum-well band structures in real space vs. position along the growth (z) direction. The right-hand side shows the quantized subband dispersions in momentum space along the parallel (kx) direction in the plane
of the layer. These dispersion curves show the modification of the effective masses or the densities of states due to both the quantization and strain effects.
The above characteristics provide some advantages in using strained materials over unstrained materials. First, the bandgap can be adjusted to obtain certain emission wavelength. Next, the reduction in hole masses leads to lower threshold lasing and lower Auger recombination rate. Then, the applied strain can allow laser emission with tailored polarization. Finally, the built-in strain may suppress defect migration into the active region. More detailed demonstrations of the strain effect will be given in chapter 4.