Theory of the Gain Spectrum of Semiconductor Lasers

Chapter 5: Second Quantized State Lasing of Single-Quantum-Well

6.6 Theory of the Gain Spectrum of Semiconductor Lasers

of the electron and hole state is made possible, corresponding to an imaginary momentum in the two-dimensional space. Rigorous k-selection is still requested for simplicity.

the spatial size of the electromagnetic mode. The preferred transverse structure in quantum-well lasers is a single-mode, optical wave-guide. This electromagnetic mode has a spatial extent that is large compared to the size of the quantum well.

In most cases, it is desirable to have the quantum well in the center of the optical mode. Consequently, considering the intensity of the electromagnetic mode to be constant over the region where the contributing eigenfunctions of the carriers are significantly large is a very good approximation Therefore, the modal gain does not depend on the width of the quantum well as long as its width is small compared to the width of the optical mode. A quantized state in a narrow quantum well has a high local gain over a small transverse region. In a wider quantum well, which has a lower local gain over a wider transverse region, the effective modal gain per quantized state is unchanged. Consequently, within this approximation, only the two-dimensional density is significant. The value of the three-dimensional density does not enter and is therefore of no concern.

The derivation of the gain is based on relating the density of subband transi- tions of quantum-well structures to the photon density. Thus the gain spectrum is derived independent of the line shape function of the transition. This approxima- tion is valid because the effective density of states is slowly varying with respect to the uncertainty energy associated with the interband transition rate {see Section 6.4). Furthermore, the carrier density is related to the current density required to maintain it via the spontaneous transitions. When the simplifying assumption of isotropic material is made, both relations can be derived directly for the Einstein relations.

The basic two-level system was treated by Einstein [25]. The dependence on the index of refraction, n, has been added. Using coefficients for the spontaneous

(A) and stimulated (B) transition rates and relating them to the fundamental dipole matrix element of the transition, the following equations are obtained:

2 · n^{3 •} Efrana - =

71" • t,,2 • c3

A

B (6.9)

(6.10) and consequently,

lµl

²

B - ---=-

- 2 · eo · n^{2 •} h ² (6.11)

where the square of the absolute of the dipole matrix element is

(6.12) which is a constant in isotropic material and applies to the spontaneous as well a~ to the ~tlro111ated tra.?1Sitionso ... -'\n equivalent qua..."1.tity, the oscillator strength, is occasionally quoted:

f =

^{2 ·}

\µ\2

e² • m · Etrcma

(6.13) In the following, the modal gain of quantum-well lasers is derived directly from the Einstein coefficient by relating the energy density of the transitions to the photon density. In this approach, the inverse dependence on the width of the optical mode emerges naturally, and it is evident that introducing a confinement factor is not necessary. The stimulated rate is given by the B coefficient times the photon density in photon frequency and volume.

d⁴ Ephoton field

p = - - - -

dv · d³V (6.14)

On the other hand, the density of states is a density in energy and area. A factor

211"·n.wEtrg.u provides the scaling to the number of photons, #photon, per energy and

op

area, where Wop is the characteristic width of the electromagnetic mode.

The definition of the gain coefficient is the relative increase in a variable inten- sity, upon propagation. This definition can be rewritten and a photon density per energy and area introduced in the rightmost factor in Equation {6.14}, here called Pn for short, can be substituted.

I dl n dl n dPn

g = - · - - - · - = - - · - -

I dz c · I dt c · Pn dt (6.15)

The rate of increase of the photon density can be re-expressed as the product of a density of transitions (a density per energy and area) Sn, and the transition rate 1/rstirn, leading to the last expression in Equation {6.15}.

By using I/rstirn

=

^{p · B} and (6.11} and substituting Sn via Equat£on (6.6}

and Pn via Equation {6.14) in Equation (6.15}, the following formula emerges:

me ·

1µ1

^{2 •} EtranJJ · Mef f

g= 3

n • eo • h · c · W (6.16)

In Equation (6.16),the gain of the mode is given. As outlined, this two-level model is applied to all pairs of electron and hole states. For each considered transition energy all contributions to the gain are gathered. Rather than introduce integrals over the electron and hole energy individually and delta functions to pick the val- ues consistent with the rigorous k-selection rule, constraints on the energies are specified, and no dummy integration is mentioned. Here the broadening, as intro- duced for the effective density of states, appears as well as the transverse overlap factor and the inversion factor. The set of quasi-Fermi energies for electrons and holes, EFerrnie and EFerrnih, is considered to obey the charge neutrality condition so that there is only one free parameter for the gain spectrum of a given transverse structure, the strength of the pumping:

E C^{l Ne Ni. 2 Q j [} M

. • ~ ~ µ, · ver nv · ^red

Gam(E)

=

W (E). n(E) L., L., mcq-m. mu>-m,.

op i=l i=l 1

+

e .ll.E 1

+

^{e .ll.l!!: (6.17)}

where Wop(E) is the effective width of the optical mode at the photon energy E.

The overlap factor and the inversion factor are, respectively:

! ^+11-

²

Overl

= I _

^D. t,L,(i)* · t,b(j)dxl

2

(6.18)

-1 (6.19)

And the relative reduced effective mass is

Mred

=

^{_ l _ _ _} 1 ^{l _}

I (6.20)

M(i) T M(i)

There is a very special relation between Ee, Eh, and E implied, which depends on i and

i-

First, the excess energy over the bandgap energy is the sum of the energy in the conduction band and the energy in the valence band:

(6.21) Second, the excess energy over the bandgap energy is split in such a way that the rigorous k-selection rule for the electron eigenstate i, and the hole eigenstate

i

applies:

E - Eez - E(i) - E(j) E( ")

e - • 1 1

+

^g

M(i). (M(i)

+

M(.i))

(6.22) Eh

=

~e; -( E~i) - EV'))

+

E(j)

M J ' M(i)

+

M(i)

(6.23) Finally, C' includes the basic constants, the rest mass of the electron me, the vacuum permittivity eo, h Planck's constant divided by 211", the speed of light in vacuum c and in the matrix element µ. The matrix element is based on the value µ₀

=

5.2 · 10-²⁹ msA as given in Reference [26], but is scaled according to Reference [ 27] as follows:

2 2 (1.424eV)2

xlm-

¹

µ

=

^{µO. . l}

ETrana Mo -1 (6.24)

where 1.424 eV and Mo

=

0.0667 are the reference values of the transition energy and effective, relative electron mass. The constant is

C ' -_- ^me ₃

eo · h • c (6.25)

so that C' •

µ5

⁼ 7.913 • 10¹⁷ J-¹ = 0.127

ev-

^1• This model involves a consider- able amount of computation, which can be reduced about eightfold by considering symmetrical structures exclusively. In many applications the maximum of the gain is the only important part of the gain spectrum, but this value is wanted as a func- tion of carrier density or quasi-Fermi energy. In these cases the spectral range to be evaluated can be reduced considerably; however, great care must be taken to ensure that the absolute maximum of the gain is found in cases of high excitation

(see second quantized-state lasing of a single-quantum-well laser, Chapter ,I.).

While the gain spectrum is the response to probing the system with light, that 1s, the stimulated emission, the consideration of the spontaneous emission gives the radiating component to the current density required to maintain the gain at steady state. The radiating component seems to be the dominant contribution to the threshold current of high-quality material semiconductor lasers if the leakage current is low.

The calculation is based on the two-level system: A definite electron in the conduction band and a definite hole in the valence band with an equal in-plane k-vector have a spontaneous lifetime of

7r. eo. h,4. cs

r,vpont

=

₂

µ •n

1

(6.26)

For Etrayu = 1.424 eV and n = 3.4, this computes to Tapont = 0.85 ns; the actual value is reduced by dividing through the occupation factors. The total rate of

transitions per area multiplied by the charge of the electron is the current density.

The current density is calculated by integrating the inverse spontaneous lifetime multiplied by the density of transitions (per area and energy) over the transition energies, Etrans. The density of the transitions is determined by the density of possible transitions, as given from the density of states, multiplied by the probability of having the electron and hole state individually occupied and the weighting factor for these states, as introduced for the density of effective states:

(6.27) where the occupation factor for the transition is

1 (6.28)

Again, C" includes the basic constants and the matrix element and is

(6.29) so that

µ5 ·

^C"

=

^{1.219 · 1083}

^m.f

_{1 ,}

=

8.03 · 10³

cm.lev

^{4 •} The integrand in Equatfon

6. 27 scales with the photon energy-dependent index of refraction n as well as with the reduced effective mass {6.20}. Equations {6.21} to {6.29} are used to specify E1

and Eh.

6. 7 Conclusion.

A model to compute gain spectra for quantum-well lasers is presented. In con- trast to previous models, the derivation is specifically for quantum-well lasers and is based directly on the Einstein coefficients. The conventionally used Lorentzian- broadening of the transition is analyzed and found not to be theoretically justified.

A heuristic smearing of the density of states is introduced to define the effective

density of states, which is then used as the base of the intraband transitions. The used density of states is the density in area, not volume, because the distribution in the transverse direction is found to be irrelevant to the gain spectrum. The model provides a natural transition to wider structures; therefore, it can be used directly to calculate double heterostructures.