Apart from understanding the temperature dependence of material gain and electromagnetic
cavities, the understanding of the temperature dependence of the Purcell factor
*F*_{p} and the spontaneous emission factor β is
equally important, largely motivated by the quest for energy-efficient operations in nano-scale
devices. While the concept of thresholdless operation continues to be a subject of debate [25, 50],
the modulation and efficiency improvements enabled by wavelength-scale cavities, which are directly
related to the modification of the spontaneous emission (Purcell effect) in these cavities, is
fairly well understood [30–32]. For example, with proper design, the cavity of a
sub-wavelength laser may be designed such that most of the spontaneous emission is channeled into
the lasing mode. In so doing, unwanted emission into non-lasing modes is mitigated, and the
below-threshold efficiency is limited only by non-radiative recombination. In nano-scale lasers,
enhanced emission together with a reduced number of cavity modes relative to large lasers can have
significant effects, especially on sub-threshold behavior. These effects are generally desirable, as
they tend to increase the utilization of spontaneous emission into the lasing mode, thus lowering
the lasing threshold and increasing β. If the desired cavity mode has the highest Purcell
factor amongst all cavity modes, a high β laser can be realized even in a multi-mode cavity.
With this design goal in mind, it is important to accurately evaluate the
*F*_{p} of all cavity modes, taking into account the emitter
environment and the semiconductor gain material properties. While the original Purcell effect
evaluation was for radio frequency micro-cavities, the formal treatment of Purcell effect specific
to nano-scale devices wasn’t presented until the work of [26, 32, 51]. Tailored for
nanolasers, these studies provide insight into the fundamental physics in the nano-cavities,
although the temperature dependence of *F*_{p} was not
included.

For a cavity of quality factor Q, active region volume
*V*_{a} and refractive index *n*, emitting at
the free-space wavelength *λ*, the Purcell factor
*F*_{p} describes the rate of spontaneous emission relative
to the emission in bulk. In the simplified expression of a two-level system and a single cavity mode
that are on resonance and spatially overlapping, *F*_{p}
usually takes the form

$${F}_{p}=\frac{3}{4{\pi}^{2}}\frac{Q}{{V}_{a}}{\mathrm{(}\frac{\lambda}{n}\mathrm{)}}^{3}.\text{\hspace{1em}(1)}$$(1)

This simplified form is the one most commonly seen and used in the literature [52] to quantify
the cavity-enhanced or -inhibited rate of spontaneous emission, relative to the emission in
freespace. However, using the emitter-field-reservoir model in the quantum theory of damping,
effects ignored in the commonly used Purcell factor expression can be captured. The result revealed
that, this expression is far from accurate in semiconductor nanolasers with bulk or MQW gain
media.

We apply the results from the non-relativistic QED treatment of 2-level systems to a 3-level
laser, in which emitters are pumped from the ground state ∣1> to an excited state
∣3> and quickly decay from state |3> to a lower state |2>;
the lasing transition is between states |2> and |1> [26]. Semiconductor
lasers in particular are frequently modeled in this manner, even though their underlying physics
differs: state |2> describes the condition where a conduction band state is occupied
and the valence band state of the same crystal momentum is vacant, while state |1>
describes the condition when the conduction band state is vacant and the valence band state is
occupied [53]. To describe such a system, we construct a basic model similar to that in ([54].
§9) and [55]. We suppose each emitter to interact with all modes of the cavity, but ignore
direct interaction among emitters. The cavity modes, on their part, undergo damping as a result of
loss at the cavity boundaries, and we model the loss as a thermal reservoir.

Loss at the cavity boundary, such as loss in a metallic mirror, or loss of energy through the
mirror and its eventual conversion to heat at some remote point in space, generally satisfies the
assumptions of a reservoir model: it is weak interaction with a large stochastic system that is
disordered and does not retain memory of past interactions. Further, this reservoir is passive, as
it does not return energy to the mode. Rather, it drains the mode energy over time and is commonly
known as the zero temperature condition. The Hamiltonian describing each single emitter in this
system can be expressed as

$$\widehat{H}={\widehat{H}}_{A}+{\widehat{H}}_{F}+{\widehat{H}}_{AF}+{\widehat{H}}_{R}+{\widehat{H}}_{FR}\text{\hspace{1em}(2)}$$(2)

where $${\widehat{H}}_{A},\text{\hspace{0.17em}}{\widehat{H}}_{F}\text{\hspace{0.17em}and\hspace{0.17em}}{\widehat{H}}_{R}$$ are the emitter, field and reservoir Hamiltonian, respectively. $${\widehat{H}}_{AF}$$ denotes interaction between the emitter and the field modes, while $${\widehat{H}}_{FR}$$ denotes interaction between the field modes and the reservoir.

We note that even if, by assumption, a given emitter does not directly interact with other
emitters, the field modes still interact with all emitters present, rather than only with a single
emitter. This interaction is not included in the Hamiltonian in Eq. (2), either explicitly or as
part of the reservoir. In a solid-state system where an emitter interacts with the field, and the
field interacts with a thermal reservoir, the cavity Purcell factor
*F*_{p} is defined as the ratio of spontaneous emission in a
cavity to that in bulk material of effective index *n*_{r},
with no cavity [56, 57]. The spontaneous emission probability in the bulk material, $${P}_{2\to 1,|0\dots 0\u3009}^{material},$$ takes the same form as in free space, except that
*ε*_{0} is replaced by the permittivity of the medium $${\epsilon}_{r}={n}_{r}^{2}{\epsilon}_{0}$$ and that *c* is scaled down by the refractive index
*n*_{r}. It is expressed as

$$\begin{array}{c}{P}_{2\to 1,|0\dots 0\u3009}^{material}\approx {\displaystyle \int \frac{{\omega}_{21}^{3}}{3\pi \overline{h}{\epsilon}_{r}{\mathrm{(}c/{n}_{r}\mathrm{)}}^{3}}{\tau}_{\text{coll}}|{\wp}_{12}\mathrm{(}{\omega}_{21}\mathrm{)}{|}^{2}D\mathrm{(}{\omega}_{21}\mathrm{)}d{\omega}_{21}}\\ \approx {\displaystyle \int \frac{{\overline{\omega}}_{21}^{3}}{3\pi \overline{h}{\epsilon}_{r}{\mathrm{(}c/{n}_{r}\mathrm{)}}^{3}}{\tau}_{\text{coll}}|{\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}{|}^{2}}\end{array}\text{\hspace{1em}(3)}$$(3)

where *ω*_{21} is the mode resonant frequency, $${\wp}_{12}\mathrm{(}{\omega}_{21}\mathrm{)}$$ is the dipole matrix element, and
*D*(*ω*_{21}) characterizes the inhomogeneity of the
system. The intraband collision time, *τ*_{coll}, is the average time
between carrier-carrier and carrier-phonon collisions, and decreases with increasing temperature
[58]. In the second line of Eq. (3), we evaluate $${\omega}_{21}^{3}$$ and $${\wp}_{12}\mathrm{(}{\omega}_{21}\mathrm{)}$$ at the center frequency $${\overline{\omega}}_{21}$$ of the inhomogeneous broadening spectrum
*D*(*ω*_{21}) and pull them out of the integration,
because these quantities vary relatively little over the homogenous broadening range.

In a damped cavity, the mode interacts with the reservoir. Provided that equilibrium between the
mode and the reservoir is reached, we obtain the photonemission probability in steady-state,

$$\begin{array}{c}{P}_{2\to 1,equilibrium}^{cav}={\displaystyle \sum _{k}\frac{{\omega}_{k}}{\overline{h}}\mathrm{(}\overline{n}\mathrm{(}{\omega}_{k}\mathrm{)}+1\mathrm{)}}\\ {\displaystyle \int |{\wp}_{12}\mathrm{(}{\omega}_{21}\mathrm{)}\cdot {e}_{k}\mathrm{(}{r}_{e}\mathrm{)}{|}^{2}D\mathrm{(}{\omega}_{21}\mathrm{)}}\\ {\displaystyle \int {L}_{k}\mathrm{(}\omega \text{-}{\omega}_{k}\mathrm{)}R\mathrm{(}\omega \text{-}{\omega}_{21},{\tau}_{\text{coll}}\mathrm{)}d\omega d{\omega}_{21}}\end{array}\text{\hspace{1em}(4)}$$(4)

where *R*(*ω*-*ω*_{21},
*τ*_{coll}) is the homogeneous broadening function and depends on
*τ*_{coll}. Viewed as a function of *ω*,
*R*(*ω*) peaks at *ω*_{21}, has a
width on the order of 1/*τ*_{coll}, and satisfies $$\int R\mathrm{(}\omega \text{-}{\omega}_{21},{\tau}_{\text{coll}}\mathrm{)}d\omega}=2\pi \cdot {\tau}_{\text{coll}$$ [59]. The Lorentzian
*L*_{k}(*ω*-*ω*_{k})
in Eq. (4) is expressed as

$$\begin{array}{c}{L}_{k}\mathrm{(}\omega \text{-}{\omega}_{k}\mathrm{)}\equiv \frac{1}{\pi}\frac{\frac{1}{2}{C}_{k}}{{\mathrm{(}\frac{1}{2}{C}_{k}\mathrm{)}}^{2}+{\mathrm{(}\omega \text{-}{\omega}_{k}\mathrm{)}}^{2}}\\ =\frac{2}{\pi}\cdot \frac{Q}{{\omega}_{k}}\frac{{\mathrm{(}\frac{1}{2}\Delta {\omega}_{k}\mathrm{)}}^{2}}{{\mathrm{(}\frac{1}{2}\Delta {\omega}_{k}\mathrm{)}}^{2}+{\mathrm{(}\omega \text{-}{\omega}_{k}\mathrm{)}}^{2}},\text{\hspace{0.17em}where\hspace{0.17em}}{C}_{k}=\Delta {\omega}_{k},\end{array}\text{\hspace{1em}(5)}$$(5)

and the quality factor is defined as
*Q*≡*ω*_{k}/Δ*ω*_{k}.
From the discussion in Section 2, the quality factor is a strong function of temperature due to the
dependence of the full-width-at-half-maximum (FWHM),
Δ*ω*_{k}, on the imaginary part of the
metal-cladding permittivity. This is especially true for well-confined modes where the
*Q* is dominated by absorptive, rather than radiation, losses. The convolution in Eq.
(4) determines the emission probability in a cavity for an inhomogeneously broadened ensemble of
emitters, when the mode-reservoir equilibrium has been reached. The effect of the reservoir on the
emission probability is described by
*L*_{k}(*ω*-*ω*_{k}),
whose spectral property is described by Eq. (5).

The Purcell factor *F*_{p} is then

$$\begin{array}{c}{F}_{p}\equiv \frac{{P}_{2\to 1,\text{\hspace{0.17em}}\text{equilibrium}}^{cav}}{{P}_{2\to 1,|0\dots 0\u3009}^{material}}\approx {\displaystyle \sum _{k}\frac{3\pi {\epsilon}_{r}{\mathrm{(}c/{n}_{r}\mathrm{)}}^{3}}{{\tau}_{\text{coll}}}\frac{{\omega}_{k}}{{\overline{\omega}}_{21}^{3}}\frac{|{\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}\cdot {e}_{k}\mathrm{(}{r}_{e}\mathrm{)}{|}^{2}}{|{\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}{|}^{2}}}\\ {\displaystyle \int D\mathrm{(}{\omega}_{21}\mathrm{)}}{\displaystyle \int {L}_{k}}\mathrm{(}\omega \text{-}{\omega}_{k}\mathrm{)}R\mathrm{(}\omega \text{-}{\omega}_{21},\text{\hspace{0.17em}}{\tau}_{\text{coll}}\mathrm{)}d\omega d{\omega}_{21}.\end{array}\text{\hspace{1em}(6)}$$(6)

The emission probability in Eq. (4), and hence the Purcell factor in Eq. (6), depends on the
location **r**_{e} of the emitter. More precisely, it depends on
the normalized mode field at the location of the emitter
**e**_{k}(**r**_{e}),
as well as on the orientation of the emitter’s dipole moment matrix element $${\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}$$ relative to the field. If the emitters are randomly oriented and uniformly
distributed over an active region of volume *V*_{a}, the
quantity $$|{\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}\cdot {e}_{k}\mathrm{(}{r}_{e}\mathrm{)}{|}^{2}$$ is replaced by its average over all locations and orientations.

$$|{\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}\cdot {e}_{k}\mathrm{(}{r}_{e}\mathrm{)}{|}^{2}\to \frac{1}{3}|{\wp}_{12}\mathrm{(}{\overline{\omega}}_{21}\mathrm{)}{|}^{2}\frac{1}{{V}_{a}}{\displaystyle \int {}_{{V}_{a}}}|{e}_{k}\mathrm{(}r\mathrm{)}{|}^{2}{d}^{3}r\text{\hspace{1em}(7)}$$(7)

where the coefficient 1/3 accounts for the random emitter orientation.

In certain situations, the carrier distribution over
*V*_{a} may become non-uniform. For example, in MQW
structures, the carrier distributions in the well and barrier regions differ significantly. Even in
bulk semiconductors, the recombination of carriers may vary spatially, with the highest rates
occurring at field antinodes. This is the case if the recombination at field antinodes is so rapid
that diffusion of carriers from other parts of the active volume is not fast enough to avoid
depletion. Carrier depletion at field antinodes and subsequent diffusion from the nodes toward the
antinodes leads to the spatial inhomogeneity of the recombination. At room temperature, the
diffusion length of carriers in InGaAsP (i.e., average distance traveled before recombination) is on
the order of 1–2 μm [60]. The distance between the field node and antinode in visible
and near infra-red sub-wavelength semiconductor cavities, on the other hand, is usually <0.5
μm [19, 20]. Thus, the depletion regions would remain relatively depleted due to the finite
diffusion time. Under these circumstances, Eq. (7) should then be replaced by an appropriately
weighted average.

$$\begin{array}{c}{F}_{p}={\displaystyle \sum _{k}\frac{\pi {\mathrm{(}c/{n}_{r}\mathrm{)}}^{3}}{{\tau}_{\text{coll}}}\frac{{\omega}_{k}}{{\overline{\omega}}_{21}^{3}}\frac{1}{{V}_{a}}\left\{{\Gamma}_{k}\right\}{\displaystyle \int D\mathrm{(}{\omega}_{21}\mathrm{)}}}\\ {\displaystyle \int {L}_{k}\mathrm{(}\omega \text{-}{\omega}_{k}\mathrm{)}R\mathrm{(}\omega \text{-}{\omega}_{21},\text{\hspace{0.17em}}{\tau}_{\text{coll}}\mathrm{)}d\omega d{\omega}_{21}}\\ ={\displaystyle \sum _{k}{F}_{p}^{\mathrm{(}k\mathrm{)}}},\end{array}\text{\hspace{1em}(8)}$$(8)

where Γ_{k} is the energy confinement factor of mode
*k*. Equation (8) permits several observations. Firstly, the double integral in Eq.
(8) is the convolution of inhomogeneous broadening
*D*(*ω*_{21}), cavity Lorentzian
*L*_{k}(*ω*-*ω*_{k}),
and homogeneous broadening
*R*(*ω*-*ω*_{21},
*τ*_{coll}). It should be noted that although the homogenous
broadening function R(*ω*) and the inhomogeneous broadening function
D(*ω*) appear symmetrically in Eq. (8), they may in principle exhibit
different dynamics. In particular, rapid recombination of carriers near the mode frequency
*ω*_{k} may deplete the carrier population at that frequency faster
than it is replenished by intraband scattering (this phenomenon is known as “spectral hole
burning”). In such cases, it could be meaningful to disaggregate the integral in
*d**ω*_{21} in Eq. (8) and define separate Purcell
factors for carriers at different frequencies *ω*_{21} [61]. More
typically, however, especially at room temperatures, the intraband relaxation time
*τ*_{coll} ∼0.3 ps of InGaAsP is much shorter than
photonemission time, and the distribution of carriers D(*ω*_{21}) is
at all times the equilibrium distribution ([44]. Appendix 6). This equilibrium distribution closely
resembles the photoluminescence spectrum [62]. In semiconductor lasers utilizing bulk or MQW gain
material, it is the broadest of the three convolution factors in Eq. (8) and therefore dominates the
convolution. For InGaAsP at room temperature, the FWHM of
*D*(*ω*_{21}) and
*R*(*ω*-*ω*_{21},
*τ*_{coll}) are approximately 7×10^{13} rad/s and
6.7×10^{12} rad/s, respectively.
*D*(*ω*_{21}) dominates the convolution in Eq. (8) as
long as the cavity Q factor is above 19, which corresponds to a FWHM of 7×10^{13}
rad/s. For practical cavities, the Q factor will be significantly larger; thus diminishing the
contribution of
*L*_{k}(*ω*-*ω*_{k})
to the resulting Purcell factor. In fact,
*R*(*ω*-*ω*_{21},
*τ*_{coll}), alone, dominates
*L*_{k}(*ω*-*ω*_{k})
if the Q factor is greater than 200 [58, 63]. Consequently, in typical III-V semiconductor lasers
with MQW or bulk gain material, the cavity Q factor plays a negligible role in determining the
spontaneous emission rate and *F*_{p}. Further, while the
cavity lineshape broadens with temperature for well-confined cavity modes, the homogeneous lineshape
broadens as well. Secondly, *F*_{p} may be large in small
laser cavities due to its inverse proportionality to the active region volume
*V*_{a}. However,
*F*_{p} is actually inversely proportional to the effective
size of the mode,
*V*_{a}/Γ_{k}, where the
mode-gain overlap factor Γ_{k} is defined in Eq. (8) and describes
the spatial overlap between the mode and the active region. Thus, if the mode is poorly confined,
Γ_{k} <<1,
*F*_{p} will remain small, despite a small active
region.

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