# Spin-conserving Boltzmann theory for carriers and excitons in organic
semiconductors

Charlotte Bäcker  Institute of Theoretical Physics, Technische Universität
Dresden, 01062 Dresden, Germany  Linus Thummel  Imperial College London, South
Kensington Campus, London SW7 2AZ, United Kingdom  Institute of Theoretical
Physics, Technische Universität Dresden, 01062 Dresden, Germany  Carsten Timm
<EMAIL_ADDRESS>Institute of Theoretical Physics, Technische
Universität Dresden, 01062 Dresden, Germany  Würzburg-Dresden Cluster of
Excellence ct.qmat, Technische Universität Dresden, 01062 Dresden, Germany

(August 6, 2021)

###### Abstract

The rise of organic electronics calls for versatile modeling tools. In this
context, we develop a semiclassical Boltzmann theory that describes transport
and excitonic processes in crystalline organic semiconductors on equal
footing. The generation of singlet and triplet excitons out of the ground
state, their formation from free electrons and holes, the reverse processes,
as well as the fusion and fission of excitons are included. The corresponding
scattering integrals respect spin conservation, which requires matrix-valued
distribution functions. They also include fermionic and bosonic many-particle
effects such as Pauli blocking. We employ a multipole expansion of the
distribution functions, where quadrupolar terms turn out to be essential for
the triplet excitons. This work provides a basis for the modeling of organic
solar cells, in which excitonic processes are crucial for the performance.
Moreover, the theory is of general interest for transport and transitions of
multiple (quasi-) particle species carrying spin in nonequilibrium systems.

##  I Introduction

Organic semiconductors are promising materials for nanoelectronics and
photonics. Their physicochemical properties are easily adjustable by chemical
engineering, the production process is inexpensive, and they are advantageous
in situations where low weight or mechanical flexibility are desired For04 ;
Muc06 ; Kla06 ; BrA12 ; Sir14 ; RSP17 . Moreover, they are reaching
conductivities comparable to those of metals CFP77 ; BCF02 ; PMB04 ; JBP04 ;
KGM15 ; MaM17 .

However, the design of electronic properties requires a deeper theoretical
understanding of the charge-carrier-transport mechanisms. Several aspects make
this a challenging goal for organic materials: On the one hand, the lattice is
rather soft, i.e., the predominantly intermolecular phononic modes have low
energies. They are also relatively strongly coupled to the charge carriers.
This interaction is of short range with the largest contribution being the
coupling of the electronic occupation of a certain molecular orbital to
vibrations of the same molecule. This diagonal or _Holstein_ coupling Hol59
results in the formation of so-called small polarons Mah90 ; note.smallpolaron
. There is also a sizable coupling of intermolecular tunneling amplitudes to
vibrations, called nondiagonal or _Peierls_ coupling Gla63 ; GoC66 ; SSH79 ;
MSB17 ; FGB20 . For rubrene, Ordejón et al. OBP17  have found that the Peierls
coupling is indeed important. These couplings lead to strong polaronic effects
OBH09 ; OrR11 ; CiF12 ; FCM17 ; FGB20 ; HPO21 , which limit the carrier
mobility.

On the other hand, creation, decay, and recombination of excitons as well as
the fission of spin-singlet excitons into triplet excitons and the reverse
fusion process are crucial for organic photovoltaic and light-emitting devices
RWL07 ; WRL10 ; SmM10 ; Sil10 ; HRG12 ; MLH13 : Lifetimes of triplet excitons
can be six orders of magnitude longer than the ones of singlet excitons
because their transition into the ground state, i.e., the recombination
process, requires a spin flip Mik12 . Due to the spin exchange energy, the
singlet-exciton energy can be higher than twice the triplet-exciton energy
MoF15 . This enables singlet fission into two triplet excitons, which is spin
allowed. This fission process can be the key mechanism to induce transport of
excitation energy by triplet excitons over long distances, which is of major
importance for photovoltaic applications SmM10 ; Sil10 . Note that the
photogeneration in such devices is dominated by internal interfaces while the
exciton diffusion is a bulk effect. We return to this point in the
conclusions.

We aim at a unified description of transport and the aforementioned local
excitonic processes. A relevant property of organic materials is the typically
weak spin-orbit coupling, which ensures that spin is conserved to a good
approximation. The description of transport thus has to respect spin rotation
symmetry, in particular if singlet and triplet excitons are involved. We note
that any violation of spin conservation, i.e., any relaxation of spin, can
easily be incorporated by an additional scattering integral.

The excitons in typical devices are far from equilibrium. The quantum master
equation constitutes a powerful tool for the description of nonequilibrium
processes in quantum systems BrP02 . It is the equation of motion for the
reduced density matrix or statistical operator $\widehat{\rho}$ of an open
system. This equation is linear in the density matrix since it is derived from
the Schrödinger equation, which is linear in the state vector. Linearity is
necessary to preserve the unit trace of the density matrix. However, one can
relax this trace condition and interpret $\operatorname{Tr}\widehat{\rho}$ as
a concentration of quasiparticles. The matrix structure of $\widehat{\rho}$
then encodes the distribution of spin orientations, where $\widehat{\rho}$ is
a $(2s+1)\times(2s+1)$ matrix for spin-$s$ quasiparticles. Furthermore, the
equation of motion can then be nonlinear, for example describing generation
and decay processes. A generalized quantum master equation in this sense was
applied to local excitonic processes, such as fission and fusion, in organic
crystals as early as 1969 by Johnson and Merrifield JoM70 . Essentially the
same equation was proposed, under the name of _stochastic Liouville equation_
, in the context of quantum optics ScL67  and for the description of chemical
reactions involving spin EFL73 ; Hab76 ; StU89 . More recently, Schellekens et
al. SWK11  have applied it to excitonic processes in order to describe organic
magnetoresistance.

In case of the stochastic Liouville equation, the nonlinearity is given by a
$\widehat{\rho}$-independent source term JoM70 ; ScL67 ; SWK11 . The equation
takes the form

$\frac{\partial\widehat{\rho}}{\partial
t}=-\frac{i}{\hbar}\,\big{[}\widehat{H},\widehat{\rho}\big{]}-\frac{W}{2}\,\big{\\{}\widehat{P},\widehat{\rho}\big{\\}}+\widehat{\Gamma}.$
(1)

The first term on the right-hand side describes the unitary time evolution
determined by the Hamiltonian $\widehat{H}$. The second term implements decay
processes and is naturally linear in the density. Here $W$ is a decay rate and
$\widehat{P}$ is a projection operator onto the spin state from which a decay
is allowed by spin conservation. The specific anticommutator structure was
shown by Haberkorn Hab76  to ensure the positivity of $\widehat{\rho}$—since
the eigenvalues of $\widehat{\rho}$ are interpreted as concentrations they
must not become negative note.Haberkorn . The third term describes generation
processes and is of order zero in $\widehat{\rho}$ since the generation rate
is assumed to be independent of the density.

The interpretation of $\widehat{\rho}$ as a density also permits terms of
higher order. Consider, for example, a pair-annihilation process of the type
$A+A\to 0$. For spinless particles, $\widehat{\rho}$ just becomes the number
density $n$ and the process is naturally described by the equation

$\frac{\partial n}{\partial t}=-c\,n^{2}.$ (2)

This is of course well known in chemical kinetics McQ67 ; nonlinear .
Nonlinear terms also result from quantum statistics, which show that the rate
of a scattering process also depends on the occupation of the final state,
e.g., due to the Pauli principle.

The question arises what the corresponding equation looks like for particles
with spin. The right-hand side then contains a product of the matrix
$\widehat{\rho}$ with itself and the question is how this product must be
constructed so that spin is conserved. More complicated processes involving
multiple species, e.g., $A+B\to C$, can be described using multiple matrix-
valued densities and we have to construct spin-conserving terms out of
matrices of generally different sizes $(2s+1)\times(2s+1)$. This issue is
central for the description of excitonic processes. For spin-$1/2$
quasiparticles, i.e., electrons and holes, the matrix-valued densities are
$2\times 2$ matrices acting on their spin Hilbert space. For triplet excitons
with spin $1$, they are $3\times 3$ matrices, whereas for singlet excitons
they reduce to scalars.

We combine the spin-rotation-invariant description of local excitonic
processes with a spin-rotation-invariant description of carrier and exciton
transport. We are interested in bandlike transport in clean organic materials
at not very low temperatures. Characteristic quantum effects such as weak
localization and universal conductance fluctuations are thus expected to be
irrelevant and a semiclassical description is justified. Our goals thus
require the derivation of $\mathrm{SU}(2)$ spin-rotation-invariant Boltzmann-
type kinetic equations for the various quasiparticle species and of scattering
integrals describing transitions between them.

As noted above, polaronic effects are typically strong in organic materials
OBH09 ; OrR11 ; CiF12 ; FCM17 ; FGB20 ; HPO21 . The coupling to the phonons
causes the charge carriers and also excitons to dress as small polarons
note.smallpolaron . Unlike for inorganic materials, there is no well-developed
semiclassical transport theory for small polarons in organic semiconductors.
This is a promising topic for future research, where one obstacle will likely
be the inclusion of sizable Peierls coupling. Numerical results by Hutsch et
al. HPO21  indicate that the phononic modes can be usefully separated into
fast modes, which lead to polaron formation similar to inorganic materials,
and slow modes, which can be treated as quasistatic disorder. This suggests
that a semiclassical description should be possible, where the effect of the
slow modes is incorporated by scattering integrals. Since polaronic effects
are not at the focus of the present paper we assume that the polarons are
adiabatically connected to the undressed quasiparticles, i.e., that polaronic
effects only renormalize the parameters appearing in our description but do
not change its structure.

Spin-rotation invariance requires the Boltzmann theory to contain the matrix-
valued densities SmJ89 ; SXW97 ; Sim01 ; QiZ03 ; ZLZ04 ; PiT07 ; GSR10 ; PoN11
; Haj14 . We emphasize that it is not sufficient to use kinetic equations for
the densities in each spin channel, say spin-up and spin-down electrons, since
this is equivalent to assuming the matrix-valued densities to be diagonal in
the standard spin basis. This precludes the description of spin polarization
in the x and y directions. It is also clear that the scattering integrals are
significantly more complicated than for spinless particles since they depend
on matrix-valued functions of various dimensions and are themselves matrices.

Our program is of general interest beyond the field of organic electronics as
the principles developed here are useful for any system showing scattering of
and reactions between spin-carrying particles, for example in spintronics and
spin chemistry.

This paper is organized as follows. In Sec. II, we set up the spin-conversing
kinetic equations for charge carriers and excitons in uniform electric and
magnetic fields. The treatment of scattering and transitions between
quasiparticle species (excitonic processes) is then performed in Sec. III. The
theoretical framework is illustrated for a simple model in Sec. IV.
Conclusions are drawn in Sec. V.

##  II Spin-conserving kinetic equations

In this section, we present the Boltzmann transport equations for electrons,
holes, and singlet and triplet excitons including static, uniform electric and
magnetic fields. We concentrate on the drift terms conventionally appearing on
the left-hand side of the Boltzmann equation and leave the collision terms for
Sec. III.

Following Ref. QiZ03 , we write the general Boltzmann equation for the density
$\widehat{f}(\bm{r},\bm{k},t)$ as

$\displaystyle\frac{\partial}{\partial
t}\,\widehat{f}(\bm{r},\bm{k},t)+\bm{v}(\bm{k})\cdot\operatorname{\mbox{\boldmath$\nabla$}}_{\bm{r}}\widehat{f}(\bm{r},\bm{k},t)$
$\displaystyle\quad{}+\frac{q}{\hbar}\left(\bm{E}+\bm{v}(\bm{k})\times\bm{B}\right)\cdot\operatorname{\mbox{\boldmath$\nabla$}}_{\bm{k}}\widehat{f}(\bm{r},\bm{k},t)$
$\displaystyle\quad{}+\frac{i}{\hbar}\big{[}\widehat{H}(\bm{r},\bm{k},t),\widehat{f}(\bm{r},\bm{k},t)\big{]}=\widehat{I}[\widehat{f}],$
(3)

where $q\in\\{-e,0,e\\}$ is the charge of the quasiparticles under
consideration and the hat $\widehat{\bullet}$ denotes matrices acting on spin
space. The density $\widehat{f}(\bm{r},\bm{k},t)$ is a $(2s+1)\times(2s+1)$
matrix, where $s$ is the quasiparticle spin. Note that for $s>0$, the matrix
structure leads to an additional commutator term, which describes the dynamics
in spin space QiZ03 . For example, for spin-$1/2$ quasiparticles in a magnetic
field $\bm{B}$, the Hamiltonian $\widehat{H}$ contains a Zeeman term
proportional to $\bm{B}\cdot\widehat{\mbox{\boldmath$\sigma$}}$, where
$\widehat{\mbox{\boldmath$\sigma$}}$ is the vector of Pauli matrices
$\widehat{\sigma}_{1}$, $\widehat{\sigma}_{2}$, $\widehat{\sigma}_{3}$. We
suppress the arguments $\bm{r}$, $\bm{k}$, and $t$ from now on.
$\widehat{I}[\widehat{f}]$ contains all scattering integrals affecting the
distribution.

Since we will later consider nonlinear collision integrals, the normalization
of $\widehat{f}$ is important. We take the eigenvalues of $\widehat{f}$ to be
the occupation numbers per quantum state. For example, the density

$\widehat{f}_{e}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ (4)

for all $\bm{r}$ and $\bm{k}$ describes a filled electron band. This
convention will be useful for the description of Fermi and Bose statistics in
the collision integrals. As noted above, $\operatorname{Tr}\widehat{f}$ is not
restricted to unity and $\widehat{f}$ is not a density matrix in the sense of
a statistical operator SWK11 .

The velocity in Eq. (3) is given by the derivative of the Hamiltonian acting
on spin space with respect to momentum,

$\bm{v}=\frac{1}{\hbar}\,\frac{\partial\widehat{H}}{\partial\bm{k}},$ (5)

and thus, in principle, obtains a matrix structure. This structure is made
nontrivial by those terms in the Hamiltonian that contain both the momentum
and the spin, i.e., by spin-orbit coupling. Since we are primarily interested
in organic materials, which show weak spin-orbit coupling, we neglect them.
The velocity $\bm{v}$ then becomes proportional to the identity matrix and can
be treated as a scalar note.soc . The dispersion relation is the only relevant
momentum-dependent term in the Hamiltonian so that we can express $\bm{v}$ as
the group velocity

$\bm{v}=\frac{1}{\hbar}\bm{\nabla}_{\bm{k}}\epsilon(\bm{k})$ (6)

in terms of the quasiparticle dispersion $\epsilon(\bm{k})$.

###  II.1 Electrons

The charge carriers in organic semiconductors are electrons and holes. For
simplicity, we restrict ourselves to a model with one valence band and one
conduction band; the generalization to multiple bands is straightforward. The
spin $1/2$ of the electrons and holes is essential to understand the related
excitonic processes.

The identity matrix $\widehat{\sigma}_{0}$ and the three Pauli matrices
$\widehat{\sigma}_{x}$, $\widehat{\sigma}_{y}$, and $\widehat{\sigma}_{z}$
form a basis of the space of Hermitian $2\times 2$ matrices. Hence, for
spin-$1/2$ quasiparticles, the $2\times 2$ matrix $\widehat{f}$ can be
expanded as

$\widehat{f}=\frac{1}{2}\left(n\,\widehat{\sigma}_{0}+\bm{c}\cdot\widehat{\bm{\sigma}}\right).$
(7)

Then $n=\operatorname{Tr}\widehat{f}$ is the concentration of quasiparticles,
summed over spin orientations, and
$\bm{c}=\operatorname{Tr}\widehat{f}\,\widehat{\bm{\sigma}}$ is twice the spin
density (setting $\hbar=1$). Note that $\widehat{f}$ must be Hermitian to
ensure real densities. We will call relations of the form of Eq. (7)
_multipole expansions_.

We start with the description of electrons in the conduction band with the
Hamiltonian

$\widehat{H}_{e}=\widehat{H}_{e0}+\frac{1}{2}\,g_{\text{el}}\mu_{B}\bm{B}\cdot\widehat{\bm{\sigma}},$
(8)

with the electronic Landé factor $g_{\text{el}}$ and the Bohr magneton
$\mu_{B}$. Neglecting spin-orbit coupling, the bare electron Hamiltonian is
$\widehat{H}_{e0}=\epsilon_{e}(\bm{k})\,\widehat{\sigma}_{0}$, with the
dispersion $\epsilon_{e}(\bm{k})$. The Zeeman term remains the only part with
nontrivial matrix structure.

We decompose the electronic distribution function into the equilibrium
distribution $\widehat{{f}}_{e0}$ and a deviation $\widehat{{g}}_{e}$
according to

$\widehat{{f}}_{e}=\widehat{{f}}_{e0}+\widehat{{g}}_{e}.$ (9)

We assume the distribution function to be close to equilibrium because the
equilibrium distribution is perturbed only little by (a) a weak electric field
$\bm{E}$ and (b) decomposition of a low concentration of excitons. The weak
electric field also means that terms involving the product of $\bm{E}$ and
$\widehat{{g}}_{e}$ are small of higher order and can be neglected. This is
the assumption of linear response. It would be straightforward to drop this
approximation.

The multipole expansion of the deviation is

$\widehat{{g}}_{e}=\frac{1}{2}\left(\tilde{n}_{e}\widehat{\sigma}_{0}+\tilde{\bm{c}}_{e}\cdot\widehat{\bm{\sigma}}\right),$
(10)

similar to Eq. (7). In equilibrium, the distribution is spatially uniform
since the external fields and the scattering term are assumed not to depend on
position $\bm{r}$, and can thus be written as

$\widehat{{f}}_{e0}=\frac{1}{2}\left(n_{e0}\,\widehat{\sigma}_{0}+c_{e0}\bm{B}\cdot\widehat{\bm{\sigma}}\right).$
(11)

This yields

$\widehat{{f}}_{e}=\frac{1}{2}\,\big{[}(n_{e0}+\tilde{n}_{e})\,\widehat{\sigma}_{0}+(c_{e0}\bm{B}+\tilde{\bm{c}}_{e})\cdot\widehat{\bm{\sigma}}\big{]}.$
(12)

The equilibrium distribution can be derived starting from the specific case of
$\bm{B}\parallel\bm{e}_{3}$, where $\bm{e}_{3}$ is the unit vector in the $z$
direction. Here the Hamiltonian takes a diagonal form and so does the
equilibrium function. It is thus given by a linear combination of
$\widehat{\sigma}_{0}$ and $B\,\widehat{\sigma}_{z}$. The rotation to an
arbitrary direction of the magnetic field is then described by the
transformation
$B\,\widehat{\sigma}_{z}\rightarrow\bm{B}\cdot\widehat{\bm{\sigma}}$. The
induced magnetization is along $\bm{B}$ since we assume a magnetically
isotropic medium note.nonlinear . The equilibrium term $n_{e0}$ corresponds to
the Fermi-Dirac distribution of electrons with energy $\epsilon_{e}(\bm{k})$,
chemical potential $\mu$, and temperature $T$,

$n_{e0}(\bm{k})=\frac{1}{e^{(\epsilon_{e}(\bm{k})-\mu)/k_{B}T}+1}.$ (13)

In the linear-response approximation, the Boltzmann equation for the electrons
reads as

$\displaystyle\frac{\partial}{\partial
t}\,\widehat{{g}}_{e}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{e}\cdot\bm{\nabla}_{\bm{r}}\widehat{{g}}_{e}-\frac{e}{\hbar}\left(\bm{E}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}\widehat{{f}}_{e0}$
$\displaystyle\quad{}-\frac{e}{\hbar^{2}}\left(\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}\widehat{{g}}_{e}+\frac{i}{\hbar}\,\big{[}\widehat{H}_{e},\widehat{{f}}_{e0}\big{]}+\frac{i}{\hbar}\,\big{[}\widehat{H}_{e},\widehat{{g}}_{e}\big{]}$
$\displaystyle=2\widehat{I}_{e}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]},$
(14)

where the factor of 2 is due to the factor of $1/2$ in Eq. (7) and drops out
when the scattering term is also expanded. We now analyze the individual
expressions in the multipole expansion. The derivative

$\bm{\nabla}_{\bm{k}}\widehat{{f}}_{e0}=\frac{n_{e0}^{2}-n_{e0}}{2k_{B}T}\,\bm{\nabla}_{\bm{k}}\epsilon_{e}\widehat{\sigma}_{0}+\frac{1}{2}\,\bm{\nabla}_{\bm{k}}c_{e0}\bm{B}\cdot\widehat{\bm{\sigma}}$
(15)

can be used to simplify the expression

$\left(\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}\widehat{{f}}_{e0}=\frac{1}{2}\,(\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B})\cdot(\bm{\nabla}_{\bm{k}}c_{e0})\bm{B}\cdot\widehat{\bm{\sigma}}.$
(16)

The commutator terms are

$\displaystyle\frac{i}{\hbar}\big{[}\widehat{H}_{e},\widehat{f}_{e0}\big{]}$
$\displaystyle=0,$ (17)
$\displaystyle\frac{i}{\hbar}\big{[}\widehat{H}_{e},\widehat{g}_{e}\big{]}$
$\displaystyle=-\frac{g_{\text{el}}\mu_{B}}{2\hbar}\,(\bm{B}\times\bm{c}_{e})\cdot\widehat{\bm{\sigma}},$
(18)

where the latter expression describes the precession of the spin around the
direction of the magnetic field.

Inserting these expressions into the Boltzmann equation (14) gives

$\displaystyle\frac{\partial}{\partial
t}\,\tilde{n}_{e}\widehat{\sigma}_{0}+\frac{\partial}{\partial
t}\,\tilde{\bm{c}}_{e}\cdot\widehat{\bm{\sigma}}$
$\displaystyle\quad{}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{e}\cdot\left[\bm{\nabla}_{\bm{r}}\tilde{n}_{e}\widehat{\sigma}_{0}+\bm{\nabla}_{\bm{r}}(\tilde{\bm{c}}_{e}\cdot\widehat{\bm{\sigma}})\right]$
$\displaystyle\quad{}-\frac{e}{\hbar}\,\bm{E}\cdot\big{[}\bm{\nabla}_{\bm{k}}n_{e0}\widehat{\sigma}_{0}+\bm{\nabla}_{\bm{k}}c_{e0}\bm{B}\cdot\widehat{\bm{\sigma}}\big{]}$
$\displaystyle\quad{}-\frac{e}{\hbar^{2}}\left(\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}c_{e0}\bm{B}\cdot\widehat{\bm{\sigma}}$
$\displaystyle\quad{}-\frac{e}{\hbar^{2}}\left(\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\left[\bm{\nabla}_{\bm{k}}\tilde{n}_{e}\widehat{\sigma}_{0}+\bm{\nabla}_{\bm{k}}(\tilde{\bm{c}}_{e}\cdot\widehat{\bm{\sigma}})\right]$
$\displaystyle\quad{}-\frac{g_{\text{el}}\mu_{B}}{\hbar}\left(\bm{B}\times\bm{c}_{e}\right)\cdot\widehat{\bm{\sigma}}$
$\displaystyle=\widehat{I}_{e}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]}.$
(19)

It is useful to also expand the scattering term into multipole components, in
analogy to Eq. (7),

$\displaystyle\widehat{I}_{e}$
$\displaystyle=\frac{1}{2}\left(I^{n_{e}}\,\widehat{\sigma}_{0}+\bm{I}^{\bm{c}_{e}}\cdot\widehat{\bm{\sigma}}\right)$
$\displaystyle=\frac{1}{2}\left(I^{n_{e}}\,\widehat{\sigma}_{0}+I^{c_{e}^{x}}\,\widehat{\sigma}_{x}+I^{c_{e}^{y}}\,\widehat{\sigma}_{y}+I^{c_{e}^{z}}\,\widehat{\sigma}_{z}\right).$
(20)

From this, we can obtain equations for the coefficients of the basis matrices.
The notation can be shortened using the differential operator

$\mathcal{D}_{e}=\frac{\partial}{\partial
t}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{e}\cdot\bm{\nabla}_{\bm{r}}-\frac{e}{\hbar^{2}}\left(\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}.$
(21)

For the coefficient of $\widehat{\sigma}_{0}$, we obtain

$\mathcal{D}_{e}\tilde{n}_{e}-\frac{e}{\hbar}\,\bm{E}\cdot\bm{\nabla}_{\bm{k}}n_{e0}=I^{n_{e}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}]$
(22)

and for the coefficients of $\widehat{\sigma}_{i}$ with $i=1,2,3$,

$\displaystyle\mathcal{D}_{e}\tilde{c}_{ei}-\frac{e}{\hbar}\left(\bm{E}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{e}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}c_{e0}B_{i}$
$\displaystyle\quad{}-\frac{g_{\text{el}}\mu_{B}}{\hbar}\left(\bm{B}\times\tilde{\bm{c}}_{e}\right)_{i}=I^{c_{e}^{i}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}].$
(23)

###  II.2 Holes

The description of holes in the valence band is analogous. Missing electrons
close to the band top are described in terms of holes with positive mass and
positive charge. The Hamiltonian for the holes reads as

$\widehat{H}_{h}=\epsilon_{h}(\bm{k})\widehat{\sigma}_{0}-\frac{1}{2}\,g_{\textrm{ho}}\mu_{B}\bm{B}\cdot\widehat{\bm{\sigma}},$
(24)

where $g_{\mathrm{ho}}>0$ is the hole Landé factor.

The matrix-valued distribution function $\widehat{{f}}_{h}$ of the holes is
decomposed into the equilibrium distribution and a deviation,

$\widehat{{f}}_{h}=\widehat{{f}}_{h0}+\widehat{{g}}_{h}.$ (25)

The contributions are expanded according to

$\displaystyle\widehat{{f}}_{h0}$
$\displaystyle=\frac{1}{2}\left(n_{h0}\,\widehat{\sigma}_{0}+\bm{c}_{h0}\bm{B}\cdot\widehat{\bm{\sigma}}\right),$
(26) $\displaystyle\widehat{{g}}_{h}$
$\displaystyle=\frac{1}{2}\left(\tilde{n}_{h}\widehat{\sigma}_{0}+\tilde{\bm{c}}_{h}\cdot\widehat{\bm{\sigma}}\right).$
(27)

The kinetic equations for the multipole coefficients $\tilde{n}_{h}$ and
$\tilde{\bm{c}}_{h}$ can then be written down in analogy to the electronic
case by replacing subscripts “e” by “h”, $e$ by $-e$, and $g_{\mathrm{el}}$ by
$-g_{\mathrm{ho}}$. Defining the differential operator

$\mathcal{D}_{h}=\frac{\partial}{\partial
t}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{h}\cdot\bm{\nabla}_{\bm{r}}+\frac{e}{\hbar^{2}}\left(\bm{\nabla}_{\bm{k}}\epsilon_{h}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}},$
(28)

we obtain

$\displaystyle\mathcal{D}_{h}\tilde{n}_{h}+\frac{e}{\hbar}\,\bm{E}\cdot\bm{\nabla}_{\bm{k}}n_{h0}=I^{n_{h}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(29)
$\displaystyle\mathcal{D}_{h}\tilde{c}_{hi}+\frac{e}{\hbar}\left(\bm{E}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{h}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{k}}\bm{c}_{h0}B_{i}$
$\displaystyle\quad{}+\frac{g_{\text{ho}}\mu_{B}}{\hbar}\left(\bm{B}\times\tilde{\bm{c}}_{h}\right)_{i}=I^{c_{h}^{i}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}].$
(30)

###  II.3 Singlet excitons

Singlet excitons are spin-$0$ quasiparticles. Consequently, the density
$f_{s}$ and Hamiltonian $H_{s}$ are scalars so that the commutator term in Eq.
(3) vanishes. This simplifies the Boltzmann equation to

$\frac{\partial}{\partial
t}\,f_{s}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{s}\cdot\bm{\nabla}_{\bm{r}}f_{s}=I_{s}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]}.$
(31)

Again, we use the group velocity with regard to the dispersion relation of the
singlet excitons, $\epsilon_{s}(\bm{k})$. The description of excitons by a
band structure $\epsilon_{s}(\bm{k})$ and an associated velocity has a long
history Wan37 ; Dre56 ; Kan75  and has also been applied to organic
semiconductors Dav71 ; VrS03 ; NPT21 . The excitonic band structure in
pentacene has been studied experimentally by Schuster et al. SKB07 . One
should note that the band width and velocity of excitons can be larger than
the ones of the charge carriers since the motion of excitons can take place
through pure energy (Förster) transfer, without tunneling of electrons.

Similarly to the linear-response theory for electrons, we decompose the
distribution function into an equilibrium distribution $f_{s0}$ and a
deviation $g_{s}$,

$f_{s}=f_{s0}+g_{s}.$ (32)

However, here the deviation is not related to the electric field and is not
necessarily small since we also consider situations with optically excited
excitons far from equilibrium. We assume $f_{s0}$ to be uniform. In
equilibrium, excitons in semiconductors follow the Bose-Einstein distribution

$f_{0}(\epsilon)=\frac{1}{e^{(\epsilon-\mu)/k_{B}T}-1},$ (33)

where $\epsilon$ is the exciton energy and $\mu$ is their chemical potential
MoM00 . Inserting Eq. (32) into Eq. (31) gives

$\frac{\partial}{\partial
t}\,g_{s}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{s}\cdot\bm{\nabla}_{\bm{r}}g_{s}=I_{s}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]}.$
(34)

The scalar form of the Boltzmann equation for singlet excitons allows one to
obtain the solutions analytically. This requires specific assumptions for the
scattering term. In a simple case, we can neglect interactions with other
particle species and use the relaxation-time approximation,
$I_{s}[f_{s}]=-R_{s}\,(f_{s}-f_{s0})=-R_{s}\,g_{s}$, to describe the
scattering. $R_{s}$ is a possibly momentum-dependent relaxation rate. This
transforms the Boltzmann equation into a homogeneous partial differential
equation for $g_{s}$,

$\frac{\partial}{\partial
t}\,g_{s}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{s}\cdot\bm{\nabla}_{\bm{r}}g_{s}+R_{s}g_{s}=0.$
(35)

It is of first order and the coefficients are constant for fixed $\bm{k}$. The
analytical solution can be derived using Fourier transformation with respect
to $\bm{r}$. The result is

$g_{s}=g_{s}^{0}\,e^{-R_{s}t}\int\frac{d^{3}q}{(2\pi)^{3}}\>\exp\left(i\bm{q}\cdot\left[\bm{r}-\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{s}t\right]\right),$
(36)

with the integration constant $g_{s}^{0}$. This equation expresses an
exponential dampening of the deviation that is superimposed onto a ballistic
divergence in position space. This dynamics agrees with the expected behavior.
We can take this as a confirmation of the form of the Boltzmann equation
derived above.

###  II.4 Triplet excitons

Triplet excitons are spin-$1$ quasiparticles. Hence, their distribution
function $\widehat{{f}}_{t}$ is a $3\times 3$ matrix acting on the spin-$1$
Hilbert space and their Boltzmann equation takes the form

$\displaystyle\frac{\partial}{\partial
t}\,\widehat{{f}}_{t}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{t}\cdot\bm{\nabla}_{\bm{r}}\widehat{{f}}_{t}+\frac{i}{\hbar}\,\big{[}\widehat{H}_{t},\widehat{{f}}_{t}\big{]}=\widehat{I}_{t}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]},$
(37)

with the triplet-exciton dispersion $\epsilon_{t}(\bm{k})$. As before, we
decompose the distribution function into an equilibrium function and a
deviation, $\widehat{{f}}_{t}=\widehat{f}_{t0}+\widehat{g}_{t}$, and assume
$\widehat{f}_{t0}$ to be uniform. Inserting this into Eq. (37) yields

$\displaystyle\frac{\partial}{\partial
t}\,\widehat{g}_{t}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{t}\cdot\bm{\nabla}_{\bm{r}}\widehat{g}_{t}+\frac{i}{\hbar}\,\big{[}\widehat{H}_{t},\widehat{f}_{t0}\big{]}+\frac{i}{\hbar}\,\big{[}\widehat{H}_{t},\widehat{g}_{t}\big{]}$
$\displaystyle\quad=\widehat{I}_{t}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]},$
(38)

where the equilibrium commutator $[\widehat{H}_{t},\widehat{f}_{t0}]$ again
vanishes. We choose a basis of the space of Hermitian $3\times 3$ matrices in
a similar fashion as for the $2\times 2$ case. This includes

$\widehat{S}_{0}=\sqrt{\frac{2}{3}}\>\mathds{1}_{3}$ (39)

as the monopole term and the three-dimensional representation of
$\mathrm{SU}(2)$,

$\displaystyle\widehat{S}_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix},$ (40) $\displaystyle\widehat{S}_{2}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\ i&0&-i\\\
0&i&0\end{pmatrix},$ (41) $\displaystyle\widehat{S}_{3}$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\end{pmatrix},$ (42)

as dipole terms. The normalization condition for all basis matrices is
$\operatorname{Tr}\widehat{S}_{i}^{2}=2$. The remaining five matrices can be
interpreted as quadrupole terms and are arranged as a five-component vector
$\vec{{\widehat{Q}}}$. We use the Hermitian matrices AlH01

$\displaystyle\widehat{Q}_{1}$
$\displaystyle=\big{\\{}\widehat{S}_{2},\widehat{S}_{3}\big{\\}}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\
1&0&-1\\\ 0&-1&0\end{pmatrix},$ (43) $\displaystyle\widehat{Q}_{2}$
$\displaystyle=\big{\\{}\widehat{S}_{3},\widehat{S}_{1}\big{\\}}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&i\\\ 0&-i&0\end{pmatrix},$ (44) $\displaystyle\widehat{Q}_{3}$
$\displaystyle=\big{\\{}\widehat{S}_{1},\widehat{S}_{2}\big{\\}}=\begin{pmatrix}0&0&-i\\\
0&0&0\\\ i&0&0\end{pmatrix},$ (45) $\displaystyle\widehat{Q}_{4}$
$\displaystyle=\widehat{S}_{1}^{2}-\widehat{S}_{2}^{2}=\begin{pmatrix}0&0&1\\\
0&0&0\\\ 1&0&0\end{pmatrix},$ (46) $\displaystyle\widehat{Q}_{5}$
$\displaystyle=\sqrt{3}\,\big{(}\widehat{S}_{3}^{2}-\widehat{S}_{0}^{2}\big{)}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\
0&-2&0\\\ 0&0&1\end{pmatrix},$ (47)

which are also normalized such that $\operatorname{Tr}\widehat{Q}_{i}^{2}=2$.
The matrices $\widehat{Q}_{1}$, $\widehat{Q}_{2}$, and $\widehat{Q}_{3}$
transform as $t_{2g}$ under the cubic point group $O_{h}$, while
$\widehat{Q}_{4}$ and $\widehat{Q}_{5}$ correspond to $e_{g}$.

The multipole expansion of $\widehat{{f}}_{t}$ is

$\widehat{{f}}_{t}={n_{t}}\widehat{S}_{0}+\bm{c}_{t}\cdot\bm{\widehat{S}}+\vec{q}_{t}\cdot\vec{{\widehat{Q}}},$
(48)

with the coefficients ${n_{t}}\in\mathbb{R}$, $\bm{c}_{t}\in\mathbb{R}^{3}$,
$\vec{q}_{t}\in\mathbb{R}^{5}$. The related physical quantities are the number
density $\sqrt{6}\,{n_{t}}$, the spin density $\bm{c}_{t}$, and the quadrupole
density $\vec{q}_{t}$. This is a good place to also introduce the Cartesian
components of the quadrupole tensor operator,

$\widehat{G}_{ij}=\big{\\{}\widehat{S}_{i},\widehat{S}_{j}\big{\\}}-\frac{4}{3}\,\delta_{ij}\mathds{1}_{3},$
(49)

which form the symmetric and traceless matrix

$\underline{\widehat{Q}}=\begin{pmatrix}\widehat{Q}_{4}-\frac{1}{\sqrt{3}}\,\widehat{Q}_{5}&\widehat{Q}_{3}&\widehat{Q}_{2}\\\
\widehat{Q}_{3}&-\widehat{Q}_{4}-\frac{1}{\sqrt{3}}\,\widehat{Q}_{5}&\widehat{Q}_{1}\\\
\widehat{Q}_{2}&\widehat{Q}_{1}&\frac{2}{\sqrt{3}}\,\widehat{Q}_{5}\end{pmatrix}.$
(50)

Importantly, this matrix acts on the space of Cartesian spin components and
each component is a $3\times 3$ matrix on the Hilbert space of a spin of
length $1$. Using the Cartesian form, the quadrupolar term in the distribution
function $\widehat{{f}}_{t}$ can be rewritten as

$\vec{q}_{t}\cdot\vec{{\widehat{Q}}}=\frac{1}{2}\operatorname{Tr}_{\mathrm{Cart}}\underline{q}{}_{t}\,\underline{\widehat{Q}},$
(51)

where the trace is over Cartesian components and

$\underline{q}{}_{t}=\begin{pmatrix}{q_{t4}}-\frac{1}{\sqrt{3}}\,{q_{t5}}&{q_{t3}}&{q_{t2}}\\\
{q_{t3}}&-{q_{t4}}-\frac{1}{\sqrt{3}}\,{q_{t5}}&{q_{t1}}\\\
{q_{t2}}&{q_{t1}}&\frac{2}{\sqrt{3}}\,{q_{t5}}\end{pmatrix}.$ (52)

Next, we motivate the form of the Boltzmann equation for these physical
quantities. The Hamiltonian for the triplet excitons reads as

$\widehat{H}_{t}=\epsilon_{t}\mathds{1}_{3}+g_{\mathrm{tr}}\mu_{B}\bm{B}\cdot\bm{\widehat{S}},$
(53)

where $\epsilon_{t}(\bm{k})$ is the dispersion in the absence of a magnetic
field (see the comments on excitonic band structures in Sec. II.3) and
$g_{\mathrm{tr}}$ is the Landé factor of the triplet excitons, which is
assumed to be scalar. A triplet exciton is of course charge neutral but since
it generically consists of an electron and a hole with different Landé factors
its total magnetic moment is nonzero. Additional terms linear in quadrupole
matrices ${\widehat{Q}}_{i}$ have been proposed in the literature Mer71 ;
BaW16 ; Sal19 . This magneto-crystalline anisotropy is due to spin-orbit
coupling and is therefore weak in organic semiconductors. We omit these terms
for simplicity but implementing them in the Hamiltonian is straightforward.

If the external magnetic field is aligned along the $z$ axis, the Zeeman term
simplifies to $g_{\mathrm{tr}}\mu_{B}B\widehat{S}_{3}$ with
$B=\left|\bm{B}\right|$. The Hamiltonian is diagonal in this case. We use this
specific situation to derive the form of the equilibrium distribution function
$\widehat{f}_{t0}$. The equilibrium distribution is the Bose-Einstein
distribution and can immediately be evaluated for the diagonal components. The
resulting $\widehat{f}_{t0}$ has a diagonal matrix structure as well,

$\widehat{f}_{t0}={n_{t0}}\widehat{S}_{0}+c_{t0}^{z}B\widehat{S}_{3}.$ (54)

A rotation yields the general description for an arbitrary direction of
$\bm{B}$. The matrix $\widehat{S}_{3}$ generalizes to $\bm{\widehat{S}}$
projected onto the direction of $\bm{B}$. Thus, the general form of the
equilibrium distribution function is given by

$\widehat{f}_{t0}={n_{t0}}\widehat{S}_{0}+{c}_{t0}\bm{B}\cdot\bm{\widehat{S}}.$
(55)

The deviation $\widehat{g}_{t}$ is of the most general form of the multipole
expansion,

$\widehat{g}_{t}=\tilde{n}_{t}\widehat{S}_{0}+\tilde{\bm{c}}_{t}\cdot\bm{\widehat{S}}+\tilde{\vec{q}}_{t}\cdot\vec{{\widehat{Q}}}.$
(56)

The velocity is given by a derivative of the Hamiltonian of triplet excitons
in analogy to Eq. (5) and thus is a $3\times 3$ matrix. Since we assume spin-
orbit coupling to be negligible, we can approximate it by a scalar,
$\bm{v}=({1}/{\hbar})\bm{\nabla}_{\bm{k}}\epsilon_{t}$.

Kinetic equations for the coefficients can be obtained but the derivation is
made complicated by quadrupolar terms. The details are relegated to Appendix
A. Defining the differential operator

$\mathcal{D}_{t}=\frac{\partial}{\partial
t}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{t}\cdot\bm{\nabla}_{\bm{r}}$
(57)

and expanding the scattering integral into multipoles, we find the coupled
equations

$\displaystyle\mathcal{D}_{t}\tilde{n}_{t}$
$\displaystyle=I^{n_{t}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(58) $\displaystyle\mathcal{D}_{t}\tilde{c}_{t}^{x}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\big{(}B_{2}\tilde{c}_{t}^{z}-B_{3}\tilde{c}_{t}^{y}\big{)}+I^{c_{t}^{x}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(59) $\displaystyle\mathcal{D}_{t}\tilde{c}_{t}^{y}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\big{(}B_{3}\tilde{c}_{t}^{x}-B_{1}\tilde{c}_{t}^{z}\big{)}+I^{c_{t}^{y}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(60) $\displaystyle\mathcal{D}_{t}\tilde{c}_{t}^{z}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\big{(}B_{1}\tilde{c}_{t}^{y}-B_{2}\tilde{c}_{t}^{x}\big{)}+I^{c_{t}^{z}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(61) $\displaystyle\mathcal{D}_{t}\tilde{q}_{t1}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\bm{B}\cdot\begin{pmatrix}-2\tilde{q}_{t4}-2\sqrt{3}\,\tilde{q}_{t5}\\\
-\tilde{q}_{t3}\\\ \tilde{q}_{t2}\\\ \end{pmatrix}$
$\displaystyle\quad{}+I^{q_{t1}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(62) $\displaystyle\mathcal{D}_{t}\tilde{q}_{t2}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\bm{B}\cdot\begin{pmatrix}\tilde{q}_{t3}\\\
-2\tilde{q}_{t4}+2\sqrt{3}\,\tilde{q}_{t5}\\\ -\tilde{q}_{t1}\\\
\end{pmatrix}$
$\displaystyle\quad{}+I^{q_{t2}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(63) $\displaystyle\mathcal{D}_{t}\tilde{q}_{t3}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\bm{B}\cdot\begin{pmatrix}-\tilde{q}_{t2}\\\
\tilde{q}_{t3}\\\ 4\tilde{q}_{t4}\\\
\end{pmatrix}+I^{q_{t3}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(64) $\displaystyle\mathcal{D}_{t}\tilde{q}_{t4}$
$\displaystyle=\frac{g_{\mathrm{tr}}\mu_{B}}{\hbar}\,\bm{B}\cdot\begin{pmatrix}\tilde{q}_{t1}\\\
\tilde{q}_{t2}\\\ -2\tilde{q}_{t3}\\\
\end{pmatrix}+I^{q_{t4}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}],$
(65) $\displaystyle\mathcal{D}_{t}\tilde{q}_{t5}$
$\displaystyle=-\frac{g_{\mathrm{tr}}\mu_{B}}{\sqrt{3}\hbar}\,\bm{B}\cdot\begin{pmatrix}-3\tilde{q}_{t1}\\\
3\tilde{q}_{t2}\\\ 0\\\
\end{pmatrix}+I^{q_{t5}}[f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}].\\!$
(66)

This complicated system cannot be simplified without further knowledge about
the scattering terms. A detailed investigation of these integrals is presented
in the following section.

##  III Spin-conserving scattering integrals

Table 1: Overview of scattering and transition processes treated explicitly in this paper. Process | Description | Reference
---|---|---
$e\to e$ | disorder scattering of electrons | Sec. III (introduction)
$s\to s$ | disorder scattering of singlet excitons | Sec. III (introduction)
$|0\rangle\to s$ | generation of singlet exciton | Sec. III.1
$s\to|0\rangle$ | decay of singlet exciton | Sec. III.1
$e+h\to s$ | binding of electron and hole in singlet state | Sec. III.2
$s\to e+h$ | unbinding of electron and hole in singlet state | Sec. III.2
$e+h\to t$ | binding of electron and hole in triplet state | Sec. III.3
$t\to e+h$ | unbinding of electron and hole in triplet state | Sec. III.3
$2s\to s$ | fusion of singlet excitons | Sec. III.4
$s\to 2s$ | fission of singlet exciton into singlet excitons | Sec. III.4
$t+s\to t$ | absorption of singlet exciton by triplet exciton | Sec. III.5
$2t\to s$ | fusion of triplet excitons into singlet exciton | Sec. III.6
$s\to 2t$ | fission of singlet exciton into triplet excitons | Sec. III.6
$e+t\to e$ | absorption of triplet exciton by electron | Sec. III.7
$2t\to t$ | fusion of triplet excitons into triplet exciton | Sec. III.8

In this section, we construct spin-conserving scattering integrals for the
Boltzmann equations for electrons, holes, and excitons. These scattering
integrals should not only describe collisions between quasiparticles and
scattering off disorder but, importantly, also generation, decay, and
transitions between excitons. An overview of the processes studied here is
given in Table 1. The selection and order of these processes as well as the
presentation in this section are partly pedagogical. We progress from
conceptually simpler to more complicated cases, where essentially every
subsection introduces an additional aspect, usually having to do with
conservation of spin or indistinguishability of quasiparticles. Additional
processes can be treated in an analogous manner, e.g., the reverse of some of
the processes in Table 1 and processes for holes instead of electrons. Which
processes are important of course depends on specific materials. Disorder
scattering is expected to be always present. Bipolar semiconductors in which
the carrier energies are sufficiently high in comparison to the exciton
binding energies will also show electron-hole binding and unbinding,
$e+h\leftrightarrow s$ or $e+h\leftrightarrow t$. For systems with
photogenerated singlet excitons, e.g., photovoltaic devices, the fission $s\to
2t$ and the fusion $2t\to s$ are crucial, as noted in Sec. I. Note that the
derivations in this section do not require any quasiparticle species to be
close to equilibrium.

Scattering off disorder is described by collision integrals that are linear in
the distribution function Zim72 ; AsM76 . Possanner and Negulescu PoN11  have
studied such terms for both spin-conserving and spin-flip scattering, from a
mathematical point of view. El Hajj Haj14  has studied spin-diffusion models
derived from the resulting linear matrix Boltzmann equation in detail.

Our main interest is in the description of transitions between quasiparticle
species. The required collision integrals are necessarily nonlinear. Quantum
statistics imply that the transition rates depend on the occupation of the
final states, contrary to what is assumed in the stochastic Liouville
equation. For fermions, complete occupation of the final state prevents a
process due to the Pauli principle, whereas for bosons, a large occupation of
the final state enhances the rate. In the context of excitonic processes, this
has been studied by Bisquert Bis08 , albeit not using an
$\mathrm{SU}(2)$-invariant formalism.

It is useful to first outline the principles behind the construction of the
collision integrals. Generation, decay, and transitions between excitons
involve quasiparticles on different energy levels. This is reminiscent of
laser theory, in which transition rates are derived by considering the
occupation of the initial and final states. However, unlike in laser theory,
these transition rates depend on momentum. Thus, they take a form similar to
the basic scattering term in Boltzmann theory,

$\displaystyle I(\bm{k})$
$\displaystyle=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W(\bm{k}^{\prime},\bm{k})\,f(\bm{k}^{\prime})\,[1-f(\bm{k})]$
$\displaystyle\quad{}-\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W(\bm{k},\bm{k}^{\prime})\,f(\bm{k})\,[1-f(\bm{k}^{\prime})],$
(67)

where $f$ is a distribution function. The two integrals describe in-scattering
and out-scattering, respectively. The processes considered here are more
complicated in a number of ways. First, our Boltzmann equations contain
matrix-valued density functions. The idea is to derive the matrix form from
the special case of z-polarized particles by means of a rotation into a
general direction, like discussed for the kinetic term in Sec. II.

Second, the initial and final states can consist of more than one particle. We
assume the initial state $A$ to contain particles of species
$a_{1},a_{2},\ldots$ (some of which may be the same) and the final state $B$
to contain particles of species $b_{1},b_{2},\ldots$ (some of which may be the
same). The integrand of the scattering integral is then proportional to

$N_{a_{1}}N_{a_{2}}\cdots(1\pm N_{b_{1}})(1\pm N_{b_{2}})\cdots,$ (68)

where $N_{a_{1}}$ etc. are the occupation numbers of states and the sign in
factors $1\pm N_{b}$ is $+$ ($-$) if $b$ is bosonic (fermionic). Considering
the limit $N_{b}\to 0$ yields the classical part of the process. For low
concentrations of quasiparticles in the final state, in particular low exciton
densities, this case is relevant for applications.

In the following, we will progress from simple to more complicated cases. To
fix the notation, we first discuss the essentially trivial case of elastic
scattering of electrons off nonmagnetic impurities. It will be beneficial to
write the matrix-valued distribution function of the electrons in components,

$\widehat{f}_{e}=\begin{pmatrix}f_{e\uparrow\uparrow}&f_{e\uparrow\downarrow}\\\
f_{e\downarrow\uparrow}&f_{e\downarrow\downarrow}\end{pmatrix}.$ (69)

Since nonmagnetic scattering does not flip the spin, the matrix-valued
collision integral appearing in the Boltzmann equation for $\widehat{f}_{e}$
is diagonal,

$\widehat{I}^{\widehat{f}_{e}}_{e\to e}=\begin{pmatrix}I^{e\uparrow}_{e\to
e}&0\\\ 0&I^{e\downarrow}_{e\to e}\end{pmatrix}.$ (70)

The collision integral for spin-up electrons can be written as Zim72 ; AsM76

$\displaystyle I^{e\uparrow}_{e\to e}(\bm{k})$
$\displaystyle=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W_{e^{\prime}\to
e}(\bm{k}^{\prime},\bm{k})\,f_{e\uparrow}(\bm{k}^{\prime})\,[1-f_{e\uparrow}(\bm{k})]$
$\displaystyle\quad{}-\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W_{e\to
e^{\prime}}(\bm{k},\bm{k}^{\prime})\,f_{e\uparrow}(\bm{k})\,[1-f_{e\uparrow}(\bm{k}^{\prime})]$
(71)

and analogously for spin-down electrons. Detailed balance for elastic
scattering implies that $W_{e^{\prime}\to e}(\bm{k}^{\prime},\bm{k})=W_{e\to
e^{\prime}}(\bm{k},\bm{k}^{\prime})$ and thus

$I^{e\uparrow}_{e\to
e}(\bm{k})=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W_{e\to
e^{\prime}}(\bm{k},\bm{k}^{\prime})\,[f_{e\uparrow}(\bm{k}^{\prime})-f_{e\uparrow}(\bm{k})]$
(72)

and analogously for spin down.

In the following, we employ a short-hand notation that suppresses (a)
integrals over momenta $\bm{k}^{\prime}$, $\bm{k}^{\prime\prime}$, … and (b)
momentum arguments. Functions depending on $\bm{k}$, $\bm{k}^{\prime}$, and
$\bm{k}^{\prime\prime}$ are denoted by no, one, and two primes, respectively.
Hence, we write the collision integrals for elastic impurity scattering as

$\displaystyle I^{e\uparrow}_{e\to e}$ $\displaystyle=W_{e\to
e^{\prime}}\,(f_{e\uparrow}^{\prime}-f_{e\uparrow}),$ (73) $\displaystyle
I^{e\downarrow}_{e\to e}$ $\displaystyle=W_{e\to
e^{\prime}}\,(f_{e\downarrow}^{\prime}-f_{e\downarrow}).$ (74)

From Eq. (20), we then obtain the collision integrals for the electron number
density,

$I^{n_{e}}_{e\to e}=I^{e\uparrow}_{e\to e}+I^{e\downarrow}_{e\to e}=W_{e\to
e^{\prime}}\,(n_{e}^{\prime}-n_{e}),$ (75)

and for the density of the z-component of spin,

$I^{c_{e}^{z}}_{e\to e}=I^{e\uparrow}_{e\to e}-I^{e\downarrow}_{e\to
e}=W_{e\to e^{\prime}}\,(c_{e}^{z\prime}-c_{e}^{z}).$ (76)

The crucial next step is to reconstruct the $\mathrm{SU}(2)$-invariant form of
the collision integral for the spin by rotating all quantities into an
arbitrary direction. In the present case, this is simple. We recognize that
the left-hand side and both summands on the right-hand side are z-components
of vectors. The $\mathrm{SU}(2)$-invariant form is thus

$\bm{I}^{\mathbf{c}_{e}}_{e\to e}=W_{e\to
e^{\prime}}\,(\bm{c}_{e}^{\prime}-\bm{c}_{e}).$ (77)

We can now use the multipole expansions in Eqs. (7) and (20) to express the
collision rate in terms of the matrix densities as

$\widehat{I}^{\widehat{f}_{e}}_{e\to e}=W_{e\to
e^{\prime}}\,(\widehat{f}_{e}^{\prime}-\widehat{f}_{e}).$ (78)

The case of holes is of course analogous.

The result for excitons is also analogous: For example for singlet excitons,
the collision integral is

$\displaystyle I^{s}_{s\to
s}(\bm{k})=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W_{s^{\prime}\to
s}(\bm{k}^{\prime},\bm{k})\,f_{s}(\bm{k}^{\prime})\,[1+f_{s}(\bm{k})]$
$\displaystyle\qquad{}-\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,W_{s\to
s^{\prime}}(\bm{k},\bm{k}^{\prime})\,f_{s}(\bm{k})\,[1+f_{s}(\bm{k}^{\prime})],$
(79)

where we have used that excitons are bosons. The second-order terms again
cancel and we obtain

$I^{s}_{s\to s}=W_{s\to s^{\prime}}\,(f_{s}^{\prime}-f_{s}).$ (80)

###  III.1 Generation and decay of singlet excitons

Singlet excitons are described by the scalar distribution function $f_{s}$, so
$\mathrm{SU}(2)$ invariance is automatically maintained. The transition rate
describing the generation of singlet excitons only depends on the occupation
of the final state. Since singlet excitons are bosons the resulting transition
rate reads as

$I_{\ket{0}\rightarrow s}^{s}=\widetilde{W}_{\ket{0}\rightarrow
s}\left(1+f_{s}\right).$ (81)

Here $\ket{0}\rightarrow s$ is a short-hand way to denote that the singlet
exciton is excited starting from the ground state with all single-electron
states below (above) the Fermi energy occupied (empty). This is of course not
a spontaneously occurring process but requires coupling to the radiation
field, which is suppressed in this notation. We use the symbol
$\widetilde{W}_{\ket{0}\rightarrow s}$ with a tilde for the rate; this is an
effective rate with properties of the radiation field such as occupation
numbers of photonic states absorbed. For the decay of a singlet exciton into
the Fermi sea, we analogously write

$I_{s\rightarrow\ket{0}}^{s}=-\widetilde{W}_{s\rightarrow\ket{0}}f_{s}.$ (82)

These transition rates can be added to the right-hand side of the Boltzmann
equation (31) for the singlet excitons. The corresponding processes for
triplet excitons are of course described analogously.

###  III.2 Binding and unbinding of electrons and holes in a singlet state

The transition between unbound electron-hole pairs and singlet excitons
involves three different particle species, which results in a description from
three different perspectives. We discuss these in turn.

####  III.2.1 Point of view of the singlet exciton

In the case of z-polarized particles, there are only two different
orientations of the electron and hole spins. To form a singlet exciton, a
spin-up electron has to be paired with a spin-down hole or vice versa. Hence,
the transition rate takes the form

$I_{e+h\rightarrow s}^{s}=W_{e+h\rightarrow
s}\,\frac{f^{\prime}_{e\uparrow}f^{\prime\prime\prime}_{h\downarrow}+f^{\prime}_{e\downarrow}f^{\prime\prime\prime}_{h\uparrow}}{2}\,(1+f_{s}).$
(83)

Here functions depending on the momenta
$\bm{k}^{\prime\prime}=\bm{k}+\bm{k}^{\prime}$ and
$\bm{k}^{\prime\prime\prime}=\bm{k}-\bm{k}^{\prime}$ are described by two and
three primes, respectively. Note that the momentum arguments of the electron
and the hole distribution functions could be interchanged as all possible
combinations are covered by integrating over $\bm{k}^{\prime}$. The origin of
the factor $1/2$ is that the electron-hole states $\ket{\uparrow\downarrow}$
and $\ket{\downarrow\uparrow}$ can be written as
$\ket{\uparrow\downarrow}=(\ket{\psi_{s}}+\ket{\psi_{t0}})/\sqrt{2}$ and
$\ket{\downarrow\uparrow}=(-\ket{\psi_{s}}+\ket{\psi_{t0}})/\sqrt{2}$,
respectively, where
$\ket{\psi_{s}}=(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2}$
is the singlet state and
$\ket{\psi_{t0}}=(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})/\sqrt{2}$
is the $m=0$ triplet state. Hence, the probability of, say,
$\ket{\uparrow\downarrow}$ being in the singlet state is $1/2$.

For a spin along the $z$ axis, only the diagonal elements of the distribution
function $\widehat{f}_{e}$ in Eq. (69) are nonzero and can be expressed in
terms of each other. Rewriting Eq. (83) in terms of the coefficients of the
multipole expansion in Eq. (7) then reads as

$I_{e+h\rightarrow s}^{s}=W_{e+h\rightarrow
s}\,\frac{n^{\prime}_{e}n^{\prime\prime\prime}_{h}-c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}}{4}\,(1+f_{s}).$
(84)

Now it is possible to generalize this transition rate to arbitrary spin
directions as there is a unique $\mathrm{SU}(2)$-invariant expression that
reduces to Eq. (84) for z-polarized spins:

$I_{e+h\rightarrow s}^{s}=W_{e+h\rightarrow
s}\,\frac{n^{\prime}_{e}n^{\prime\prime\prime}_{h}-\bm{c}^{\prime}_{e}\cdot\bm{c}^{\prime\prime\prime}_{h}}{4}\,(1+f_{s}).$
(85)

This transition rate $I_{e+h\rightarrow s}^{s}\equiv I_{e+h\to s}^{f_{s}}$
appears in the Boltzmann equation (31) for the scalar distribution function
$f_{s}$ of singlet excitons and is thus itself a scalar. It can also be
expressed in a basis-independent form in terms of matrix-valued distribution
functions as note.anticomm

$I_{e+h\to s}^{f_{s}}=\frac{W_{e+h\to
s}}{2}\,\operatorname{Tr}\big{\\{}\widehat{P}_{s},\widehat{f}^{\prime}_{e}\otimes\widehat{f}^{\prime\prime\prime}_{h}\big{\\}}\,(1+f_{s}),$
(86)

as can be shown by inserting the multipole expansions of the electron and hole
distribution functions. Here $\\{\bullet,\bullet\\}$ is the anticommutator and
$\widehat{P}_{s}$ is the projection operator onto the spin-singlet subspace of
the product Hilbert space of the electron and hole spins. This operator can be
written as

$\displaystyle\widehat{P}_{s}$
$\displaystyle=\frac{\mathds{1}_{4}}{4}-\widehat{\bm{S}}_{e}\cdot\widehat{\bm{S}}_{h}$
$\displaystyle=\frac{1}{4}\left(\widehat{\sigma}_{0}\otimes\widehat{\sigma}_{0}-\widehat{\sigma}_{x}\otimes\widehat{\sigma}_{x}-\widehat{\sigma}_{y}\otimes\widehat{\sigma}_{y}-\widehat{\sigma}_{z}\otimes\widehat{\sigma}_{z}\right),$
(87)

where $\widehat{\bm{S}}_{e}$ and $\widehat{\bm{S}}_{h}$ are the spin operators
of the electron and the hole, respectively. It expresses the fact that an
electron and a hole can only form a singlet exciton if they are in a relative
spin-singlet state. The structure of the momentum-dependent transition rate is
reminiscent of the Haberkorn approach JoM70 ; EFL73 ; Hab76 . As mentioned in
Sec. I, the anticommutator structure ensures positivity of the density matrix
on the electron-hole product space. The trace in Eq. (86) sums over all
contributions allowed by spin conservation and leads to a scalar transition
term.

It is useful to restore the momentum arguments and integrals at this point.
For the process $e+h\to s$ from the point of view of the exciton, the outer
momentum $\bm{k}$ is the one of the exciton, the momentum $\bm{k}^{\prime}$ of
the electron needs to be integrated over and the momentum
$\bm{k}^{\prime\prime\prime}=\bm{k}-\bm{k}^{\prime}$ of the hole is then fixed
by momentum conservation. This leads to

$\displaystyle I_{e+h\to s}^{f_{s}}(\bm{k})$
$\displaystyle=\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,\frac{W_{e+h\to
s}(\bm{k}^{\prime},\bm{k}-\bm{k}^{\prime})}{2}$
$\displaystyle\quad{}\times\operatorname{Tr}\big{\\{}\widehat{P}_{s},\widehat{{f}}_{e}(\bm{k}^{\prime})\otimes\widehat{{f}}_{h}(\bm{k}-\bm{k}^{\prime})\big{\\}}\,\big{[}1+f_{s}(\bm{k})\big{]}.$
(88)

The breaking of singlet excitons into unbound electrons and holes can be
described in the same way. For this reverse process only the sign and the
occupation numbers have to be changed. Now the final state is fermionic, which
yields

$I_{s\rightarrow e+h}^{s}=-W_{s\rightarrow
e+h}\,\frac{(2-n^{\prime}_{e})(2-n^{\prime\prime\prime}_{h})-\bm{c}^{\prime}_{e}\cdot\bm{c}^{\prime\prime\prime}_{h}}{4}\,f_{s}$
(89)

in the multipole form and

$\displaystyle I_{s\rightarrow e+h}^{f_{s}}$ $\displaystyle=-\frac{W_{s\to
e+h}}{2}$
$\displaystyle\quad{}\times\operatorname{Tr}\Big{\\{}\widehat{P}_{s},\big{(}\mathds{1}_{2}-\widehat{f}^{\prime}_{e}\big{)}\otimes\big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime\prime}_{h}\big{)}\Big{\\}}\,f_{s}$
(90)

as the basis-independent expression; compare Eqs. (7) and (20).

####  III.2.2 Point of view of the electron

We now consider the perspective of the electron (the hole case is of course
analogous). The electron’s momentum is the momentum argument $\bm{k}$ of the
solution of the Boltzmann equation. The transition rates for spin-up and spin-
down electrons read as

$\displaystyle I_{e+h\rightarrow s}^{e\uparrow}$
$\displaystyle=-W_{e+h\rightarrow
s}\,\frac{f_{e\uparrow}f^{\prime}_{h\downarrow}}{2}\,(1+f^{\prime\prime}_{s}),$
(91) $\displaystyle I_{e+h\rightarrow s}^{e\downarrow}$
$\displaystyle=-W_{e+h\rightarrow
s}\,\frac{f_{e\downarrow}f^{\prime}_{h\uparrow}}{2}\,(1+f^{\prime\prime}_{s}),$
(92)

respectively. The transition rate for the electron number density is the sum
of these two transition rates, while the transition rate of the density of the
$z$-component of the spin is their difference. After generalizing to an
arbitrary polarization direction, we obtain

$I_{e+h\rightarrow s}^{n_{e}}=-W_{e+h\rightarrow
s}\,\frac{n_{e}n^{\prime}_{h}-\bm{c}_{e}\cdot\bm{c}^{\prime}_{h}}{4}\,(1+f^{\prime\prime}_{s})$
(93)

and

$\bm{I}_{e+h\rightarrow s}^{\bm{c}_{e}}=-W_{e+h\rightarrow
s}\,\frac{\bm{c}_{e}n^{\prime}_{h}-\bm{c}^{\prime}_{h}n_{e}}{4}\,(1+f^{\prime\prime}_{s}).$
(94)

The transition rate of the electron number density is the negative of the
transition rate from the point of view of the singlet exciton given in Eq.
(85). According to Eqs. (7) and (20), the matrix-valued transition term then
reads as

$\displaystyle\widehat{I}^{\widehat{f}_{e}}_{e+h\to s}$
$\displaystyle=\frac{1}{2}\,\big{(}I_{e+h\rightarrow
s}^{n_{e}}\widehat{\sigma}_{0}+\bm{I}_{e+h\rightarrow
s}^{\bm{c}_{e}}\cdot\widehat{\bm{\sigma}}\big{)}$
$\displaystyle=-\frac{W_{e+h\to
s}}{2}\,\operatorname{Tr}_{h}\big{\\{}\widehat{P}_{s},\widehat{{f}}_{e}\otimes\widehat{f}^{\prime}_{h}\big{\\}}\,(1+f^{\prime\prime}_{s}),$
(95)

where $\operatorname{Tr}_{h}$ is the trace over the hole sector. To restore
the momenta and integrals, we note that the outer momentum $\bm{k}$ is the one
of the electron, the hole momentum $\bm{k}^{\prime}$ is integrated over, and
the exciton momentum $\bm{k}^{\prime\prime}=\bm{k}+\bm{k}^{\prime}$ is then
fixed. The rate reads as

$\displaystyle\widehat{I}^{\widehat{f}_{e}}_{e+h\to s}(\bm{k})$
$\displaystyle=-\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\,\frac{W_{e+h\to
s}(\bm{k},\bm{k}^{\prime})}{2}$ $\displaystyle\leavevmode\nobreak\
{}\times\operatorname{Tr}_{h}\big{\\{}\widehat{P}_{s},\widehat{{f}}_{e}(\bm{k})\otimes\widehat{{f}}_{h}(\bm{k}^{\prime})\big{\\}}\,\big{[}1+f_{s}(\bm{k}+\bm{k}^{\prime})\big{]}.$
(96)

These transition rates can be compared to the transition rates from the point
of view of the singlet exciton. This yields

${-}I_{e+h\rightarrow s}^{s}=I_{e+h\rightarrow s}^{n_{e}}=I_{e+h\rightarrow
s}^{n_{h}},$ (97)

as the creation of one singlet exciton results in the annihilation of a free
electron and a free hole. Furthermore, the spin transition rates satisfy

$\bm{I}_{e+h\rightarrow s}^{\bm{c}_{e}}+\bm{I}_{e+h\rightarrow
s}^{\bm{c}_{h}}=0,$ (98)

which shows that spin is conserved, as the singlet exciton does not posses an
overall spin.

The reverse unbinding process can be described analogously. It can also be
read off from Eq. (95), essentially by reversing the sign and interchanging
occupied and empty states. The resulting rate reads as

$\widehat{I}^{\widehat{f}_{e}}_{s\to e+h}=\frac{W_{s\to
e+h}}{2}\,\operatorname{Tr}_{h}\Big{\\{}\widehat{P}_{s},\big{(}\mathds{1}_{2}-\widehat{{f}}_{e}\big{)}\otimes\big{(}\mathds{1}_{2}-\widehat{f}^{\prime}_{h}\big{)}\Big{\\}}\,f^{\prime\prime}_{s}.$
(99)

###  III.3 Binding and unbinding of electrons and holes in a triplet state

Like for electrons and holes, it is beneficial to write the matrix-valued
distribution function of the triplet excitons in components,

$\widehat{{f}}_{t}=\begin{pmatrix}f_{t,11}&f_{t,10}&f_{t,1\bar{1}}\\\
f_{t,01}&f_{t,00}&f_{t,0\bar{1}}\\\
f_{t,\bar{1}1}&f_{t,\bar{1}0}&f_{t,\bar{1}\bar{1}}\end{pmatrix}.$ (100)

The indices of the matrix elements denote the magnetic quantum number of the
triplet exciton, where we use $\bar{1}=-1$. For a spin along the $z$ axis,
only the diagonal elements of the matrix are nonzero so that in the multipole
representation it takes the form

$\widehat{{f}}_{t}=\begin{pmatrix}f_{t,1}&0&0\\\ 0&f_{t,0}&0\\\
0&0&f_{t,\bar{1}}\end{pmatrix}={n_{t}}\widehat{S}_{0}+{c_{t}^{z}}\widehat{S}_{3}+{q_{t5}}{\widehat{Q}}_{5};$
(101)

cf. Eq. (48). This allows us to express the elements of the density matrix in
terms of the diagonal elements of the multipole expansion and vice versa,

$\displaystyle{n_{t}}$
$\displaystyle=\frac{f_{t,1}+f_{t,0}+f_{t,\bar{1}}}{\sqrt{6}},$ (102)
$\displaystyle{c_{t}^{z}}$ $\displaystyle=\frac{f_{t,1}-f_{t,\bar{1}}}{2},$
(103) $\displaystyle{q_{t5}}$
$\displaystyle=\frac{f_{t,1}-2f_{t,0}+f_{t,\bar{1}}}{2\sqrt{3}}.$ (104)

####  III.3.1 Point of view of the triplet exciton

The transition term for the triplet-exciton distribution function
$\widehat{f}_{t}$ also has to be a $3\times 3$ matrix on the spin-$1$ Hilbert
space. For the spin along the $z$ axis, spin conservation implies that we only
need to consider the transition rates of the diagonal components, which read
as

$\displaystyle I_{e+h\rightarrow t}^{t_{1}}$ $\displaystyle=W_{e+h\rightarrow
t}\,f^{\prime}_{e\uparrow}f^{\prime\prime\prime}_{h\uparrow}\,(1+f_{t,1}),$
(105) $\displaystyle I_{e+h\rightarrow t}^{t_{0}}$
$\displaystyle=W_{e+h\rightarrow
t}\,\frac{f^{\prime}_{e\uparrow}f^{\prime\prime\prime}_{h\downarrow}+f^{\prime}_{e\downarrow}f^{\prime\prime\prime}_{h\uparrow}}{2}\,(1+f_{t,0}),$
(106) $\displaystyle I_{e+h\rightarrow t}^{t_{\bar{1}}}$
$\displaystyle=W_{e+h\rightarrow
t}\,f^{\prime}_{e\downarrow}f^{\prime\prime\prime}_{h\downarrow}\,(1+f_{t,\bar{1}}).$
(107)

Recall that three primes denote a dependence on
$\bm{k}^{\prime\prime\prime}=\bm{k}-\bm{k}^{\prime}$. The next step is again
to construct the $\mathrm{SU}(2)$-invariant transition rates by rotating the
spin into a general direction. In analogy to the multipole expansion of the
distribution function in Eq. (48), we expand the transition term as

$\widehat{I}^{\widehat{f}_{t}}_{e+h\rightarrow t}=I^{n_{t}}_{e+h\rightarrow
t}\widehat{S}_{0}+\bm{I}^{\bm{c}_{t}}_{e+h\rightarrow
t}\cdot\bm{\widehat{S}}+\vec{I}^{\,\vec{q}_{t}}_{e+h\rightarrow
t}\cdot\vec{{\widehat{Q}}}.$ (108)

For the spin along the $z$ direction, this yields

$\displaystyle I_{e+h\to t}^{{n_{t}}}$ $\displaystyle=\frac{I_{e+h\rightarrow
t}^{t_{1}}+I_{e+h\rightarrow t}^{t_{0}}+I_{e+h\rightarrow
t}^{t_{\bar{1}}}}{\sqrt{6}}$ $\displaystyle=\frac{W_{e+h\rightarrow
t}}{24}\,\big{(}3\sqrt{6}\,n_{e}^{\prime}n_{h}^{\prime\prime\prime}+\sqrt{6}\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}$
$\displaystyle\quad{}+6\,n_{e}^{\prime}n_{h}^{\prime\prime\prime}n_{t}+2\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}n_{t}+2\sqrt{6}\,c_{e}^{z\prime}n_{h}^{\prime\prime\prime}c_{t}^{z}$
$\displaystyle\quad{}+2\sqrt{6}\,n_{e}^{\prime}c_{h}^{z\prime\prime\prime}c_{t}^{z}+4\sqrt{2}\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}q_{t5}\big{)}$
(109)

and analogously

$\displaystyle I_{e+h\to t}^{{c_{t}^{z}}}$
$\displaystyle=\frac{W_{e+h\rightarrow
t}}{12}\,\big{(}3\,n_{e}^{\prime}c_{h}^{z\prime\prime\prime}+3\,c_{e}^{z\prime}n_{h}^{\prime\prime\prime}$
$\displaystyle\quad{}+3\,n_{e}^{\prime}n_{h}^{\prime\prime\prime}c_{t}^{z}+\sqrt{6}\,n_{e}^{\prime}c_{h}^{z\prime\prime\prime}n_{t}+\sqrt{6}\,c_{e}^{z\prime}n_{h}^{\prime\prime\prime}n_{t}$
$\displaystyle\quad{}+3\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}c_{t}^{z}+\sqrt{3}\,n_{e}^{\prime}c_{h}^{z\prime\prime\prime}q_{t5}+\sqrt{3}\,c_{e}^{z\prime}n_{h}^{\prime\prime\prime}q_{t5}\big{)},$
(110) $\displaystyle I_{e+h\to t}^{{q_{t5}}}$
$\displaystyle=\frac{W_{e+h\rightarrow
t}}{12\sqrt{3}}\,\big{(}6\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}$
$\displaystyle\quad{}+3\sqrt{3}\,n_{e}^{\prime}n_{h}^{\prime\prime\prime}q_{t5}+3\,n_{e}^{\prime}c_{h}^{z\prime\prime\prime}c_{t}^{z}+3\,c_{e}^{z\prime}n_{h}^{\prime\prime\prime}c_{t}^{z}$
$\displaystyle\quad{}+2\sqrt{6}\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}n_{t}-\sqrt{3}\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}q_{t5}\big{)}.$
(111)

In these equations, the terms of second order in the coefficients represent
the classical part since they do not depend on the occupation of the final
state. The terms of third order are the quantum corrections due to the Bose-
Einstein statistics of the excitons.

Unlike for the previous examples, the $\mathrm{SU}(2)$-invariant
generalization of Eqs. (109)–(111) is not obvious since these transition rates
contain the quadrupole coefficient ${q_{t5}}$. The transformation of the
coefficient vector $\vec{q}_{t}$ under rotations is not clear. On the other
hand, the Cartesian quadrupole density $\underline{q}{}_{t}$ defined in Eq.
(52) transforms like a matrix under rotations. To make use of this, we have to
identify the factors multiplying ${q_{t5}}$ as components of Cartesian
matrices and write Eqs. (109)–(111) as components of proper matrix products.

This can be done by considering the two-particle state of an electron and a
hole. As the unbound electron-hole pair can be of triplet character, it
generally has a quadrupole moment. The projection operator

$\widehat{P}_{t}=\mathds{1}_{4}-\widehat{P}_{s}=\frac{3\mathds{1}_{4}}{4}+\widehat{\bm{S}}_{e}\cdot\widehat{\bm{S}}_{h}$
(112)

projects onto the triplet subspace, in analogy to the singlet projection
operator $\widehat{P}_{s}$ defined in Eq. (87). Since this subspace is three
dimensional the projected density matrix
$\widehat{P}_{t}\big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{f}^{\prime\prime\prime}_{h}\big{)}\widehat{P}_{t}$
can be written as a $3\times 3$ matrix with respect to a suitable basis of
this subspace. To be consistent, we choose the canonical spin-$1$ basis
$\\{|{\uparrow_{e}\uparrow_{h}}\rangle,(|{\uparrow_{e}\downarrow_{h}}\rangle+|{\downarrow_{e}\uparrow_{h}}\rangle)/\sqrt{2},|{\downarrow_{e}\downarrow_{h}}\rangle\\}$.
The notation $\left[\bullet\right]_{3}$ refers to the corresponding matrix.
The density matrix for an electron-hole pair in a triplet state is thus
written as

$\widehat{{f}}_{eh}=\Big{[}\widehat{P}_{t}\,\big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{f}^{\prime\prime\prime}_{h}\big{)}\,\widehat{P}_{t}\Big{]}_{3}.$
(113)

The resulting matrix $\widehat{{f}}_{eh}$ is an operator on the Hilbert space
of spin-1 particles and can thus be expressed in terms of the nine basis
matrices of the multipole expansion. For a spin along the $z$ axis, only three
coefficients are nonzero,
$\widehat{{f}}_{eh}=n_{eh}\widehat{S}_{0}+c_{eh}^{z}\widehat{S}_{3}+q_{eh5}{\widehat{Q}}_{5}$,
where

$\displaystyle n_{eh}$
$\displaystyle=\frac{c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}+3n^{\prime}_{e}n^{\prime\prime\prime}_{h}}{4\sqrt{6}},$
(114) $\displaystyle c_{eh}^{z}$
$\displaystyle=\frac{c_{e}^{z\prime}n^{\prime\prime\prime}_{h}+c_{h}^{z\prime\prime\prime}n^{\prime}_{e}}{4},$
(115) $\displaystyle q_{eh5}$
$\displaystyle=\frac{\sqrt{3}\,c_{e}^{z\prime}c_{h}^{z\prime\prime\prime}}{6}.$
(116)

As electron and hole together have the overall momentum $\bm{k}$ due to
momentum conservation, $\widehat{{f}}_{eh}$ and the corresponding multipole
coefficients do not carry any primes. By eliminating the variables
$n^{\prime}_{e}$, $n^{\prime\prime\prime}_{h}$, $c_{e}^{z\prime}$, and
$c_{h}^{z\prime\prime\prime}$ from Eqs. (109)–(111) and (114)–(116), we obtain

$\displaystyle I_{e+h\to t}^{{n_{t}}}$ $\displaystyle=\frac{W_{e+h\rightarrow
t}}{3}\,\big{(}3n_{eh}+\sqrt{6}\,n_{eh}n_{t}+\sqrt{6}\,c_{eh}^{z}c_{t}^{z}$
$\displaystyle\quad{}+\sqrt{6}\,q_{eh5}q_{t5}\big{)},$ (117) $\displaystyle
I_{e+h\to t}^{{c_{t}^{z}}}$ $\displaystyle=\frac{W_{e+h\rightarrow
t}}{3}\,\big{(}3c_{eh}^{z}+\sqrt{6}\,n_{eh}c_{t}^{z}+\sqrt{3}\,q_{eh5}c_{t}^{z}$
$\displaystyle\quad{}+\sqrt{6}\,c_{eh}^{z}n_{t}+\sqrt{3}\,q_{t5}c_{eh}^{z}\big{)},$
(118) $\displaystyle I_{e+h\to t}^{{q_{t5}}}$
$\displaystyle=\frac{W_{e+h\rightarrow
t}}{3}\,\big{(}3q_{eh5}+\sqrt{6}\,q_{eh5}n_{t}+\sqrt{6}\,n_{eh}q_{t5}$
$\displaystyle\quad{}+\sqrt{3}\,c_{eh}^{z}c_{t}^{z}-\sqrt{3}\,q_{eh5}q_{t5}\big{)}.$
(119)

These relations allow us to construct the $\mathrm{SU}(2)$-invariant
generalizations without ambiguities. For terms containing quadrupole moments,
Eq. (52) should be noted. The transition rate of the triplet exciton number
density is

$\displaystyle I_{e+h\rightarrow t}^{{n_{t}}}$
$\displaystyle=\frac{W_{e+h\rightarrow
t}}{6}\,\big{(}6\,n_{eh}+2\sqrt{6}\,n_{eh}{n_{t}}+2\sqrt{6}\,\bm{c}_{eh}\cdot\bm{c}_{t}$
$\displaystyle\quad{}+\sqrt{6}\,\operatorname{Tr}\underline{q}{}_{eh}\underline{q}{}_{t}\big{)},$
(120)

where the classical part is proportional to $n_{eh}$. For the last term, the
identity
$\operatorname{Tr}\underline{q}{}_{eh}\underline{q}{}_{t}=2\vec{q}_{eh}\cdot\vec{q}_{t}$
has been used. The spin transition rate becomes

$\displaystyle\bm{I}_{e+h\rightarrow t}^{\bm{c}_{t}}$
$\displaystyle=\frac{W_{e+h\rightarrow
t}}{6}\,\big{(}6\,\bm{c}_{eh}+2\sqrt{6}\,n_{eh}\bm{c}_{t}+3\,\underline{q}{}_{eh}\bm{c}_{t}$
$\displaystyle\quad{}+2\sqrt{6}\,\bm{c}_{eh}{n_{t}}+3\,\underline{q}{}_{t}\bm{c}_{eh}\big{)},$
(121)

where the classical part is proportional to $\bm{c}_{eh}$. The transition rate
of the quadrupole part has to be traceless and symmetric in order to ensure
that the full triplet-exciton transition rate is Hermitian. This requires the
subtraction of the diagonal parts of matrices obtained by a naive restoration
of $\mathrm{SU}(2)$ symmetry and yields

$\displaystyle\underline{I}_{e+h\rightarrow t}^{\underline{q}{}_{t}}$
$\displaystyle=W_{e+h\rightarrow
t}\,\bigg{[}\underline{q}{}_{eh}+\frac{\sqrt{6}}{3}\,\underline{q}{}_{eh}{n_{t}}+\frac{\sqrt{6}}{3}\,n_{eh}\underline{q}{}_{t}$
$\displaystyle\quad{}+\frac{1}{2}\,(\bm{c}_{t}\otimes\bm{c}_{eh}+\bm{c}_{eh}\otimes\bm{c}_{t})-\frac{1}{3}\,(\bm{c}_{eh}\cdot\bm{c}_{t})\,\mathds{1}_{3}$
$\displaystyle\quad{}-\frac{1}{2}\,\big{(}\underline{q}{}_{eh}\underline{q}{}_{t}+\underline{q}{}_{eh}\underline{q}{}_{t}\big{)}+\frac{1}{3}\,\big{(}\operatorname{Tr}\underline{q}{}_{eh}\underline{q}{}_{t}\big{)}\,\mathds{1}_{3}\bigg{]},$
(122)

with the classical part being proportional to $\underline{q}{}_{eh}$.

It is also useful to express the triplet-exciton transition rate in a basis-
independent form. In contrast to the formation of a singlet exciton, the final
state is now also described by a matrix-valued distribution function. The
resulting transition rate takes the form

$\widehat{I}^{\widehat{f}_{t}}_{e+h\rightarrow t}=\frac{W_{e+h\to
t}}{2}\,\Big{\\{}\Big{[}\widehat{P}_{t}\,\big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{f}^{\prime\prime\prime}_{h}\big{)}\,\widehat{P}_{t}\Big{]}_{3},\mathds{1}_{3}+\widehat{{f}}_{t}\Big{\\}},$
(123)

where $\widehat{{f}}_{eh}$ as defined in Eq. (113) appears. For the reverse
process $t\to e+h$, the matrix

$\widehat{{f}}_{1-eh}=\widehat{P}_{t}\,\Big{[}\big{(}\mathds{1}_{2}-\widehat{f}^{\prime}_{e}\big{)}\otimes\big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime\prime}_{h}\big{)}\Big{]}\,\widehat{P}_{t}$
(124)

occurs instead, which results in a similar structure. In particular, the
basis-independent transition rate from the point of view of the triplet
excitons reads as

$\displaystyle\widehat{I}^{\widehat{f}_{t}}_{t\to e+h}$
$\displaystyle=-\frac{W_{t\to e+h}}{2}$
$\displaystyle\quad{}\times\Big{\\{}\Big{[}\widehat{P}_{t}\,\Big{[}\big{(}\mathds{1}_{2}-\widehat{f}^{\prime}_{e}\big{)}\otimes\big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime\prime}_{h}\big{)}\Big{]}\,\widehat{P}_{t}\Big{]}_{3},\widehat{{f}}_{t}\Big{\\}},$
(125)

from which the rates for the number, spin, and quadrupole densities can be
inferred.

####  III.3.2 Point of view of electron and hole

In the following, we discuss the process $e+h\to t$ from the point of view of
the electrons; the results for the holes are analogous. The transition rates
of the different spin orientations for z-polarized electrons,

$\displaystyle I_{e+h\rightarrow t}^{e\uparrow}$
$\displaystyle=-W_{e+h\rightarrow
t}\,\bigg{[}f_{e\uparrow}f^{\prime}_{h\uparrow}\,(1+f^{\prime\prime}_{t,1})$
$\displaystyle\quad{}+\frac{f_{e\uparrow}f^{\prime}_{h\downarrow}}{2}\,(1+f^{\prime\prime}_{t,0})\bigg{]},$
(126) $\displaystyle I_{e+h\rightarrow t}^{e\downarrow}$
$\displaystyle=-W_{e+h\rightarrow
t}\,\bigg{[}f_{e\downarrow}f^{\prime}_{h\downarrow}\,(1+f^{\prime\prime}_{t,\bar{1}})$
$\displaystyle\quad{}+\frac{f_{e\downarrow}f^{\prime}_{h\uparrow}}{2}\,(1+f^{\prime\prime}_{t,0})\bigg{]},$
(127)

can be combined to obtain the transition rates for the electron number and
spin densities in analogy to Eqs. (75) and (76). The resulting expressions are
not symmetric in the coefficients of $\widehat{{f}}_{e}$ and
$\widehat{f}^{\prime}_{h}$ for the simple reason that we are writing down
transition rates for the electrons, not for the holes. The rates are not
sufficient to read off the $\mathrm{SU}(2)$-invariant form, as we will see
shortly.

The basis-independent form of the transition rate is particularly useful for
this reason. In analogy to the previous cases, the transition rate in terms of
density matrices can be written as

$\displaystyle\widehat{I}_{e+h\to t}^{\widehat{{f}}_{e}}$
$\displaystyle=-\frac{W_{e+h\to
t}}{2}\operatorname{Tr}_{h}\bigg{[}\left(\mathds{1}_{3}+\widehat{f}^{\prime\prime}_{t}\right)_{4}\widehat{P}_{t}\left(\widehat{{f}}_{e}\otimes\widehat{f}^{\prime}_{h}\right)$
$\displaystyle\quad{}+\left(\widehat{{f}}_{e}\otimes\widehat{f}^{\prime}_{h}\right)\widehat{P}_{t}\left(\mathds{1}_{3}+\widehat{f}^{\prime\prime}_{t}\right)_{4}\bigg{]}.$
(128)

The notation $\left(\bullet\right)_{4}$ means that a $3\times 3$ matrix on the
triplet subspace is extended to the full two-particle space by adding a
$1\times 1$ null matrix on the singlet subspace. The matrix is then
transformed into the product basis of electron and hole spins,
$\\{\ket{\uparrow_{e}}\ket{\uparrow_{h}},\ket{\uparrow_{e}}\ket{\downarrow_{h}},\ket{\downarrow_{e}}\ket{\uparrow_{h}},\ket{\downarrow_{e}}\ket{\downarrow_{h}}\\}$.
The partial trace over the hole sector is defined by

$\operatorname{Tr}_{h}\bullet=\bra{\uparrow_{h}}\bullet\ket{\uparrow_{h}}+\bra{\downarrow_{h}}\bullet\ket{\downarrow_{h}}.$
(129)

The resulting $\mathrm{SU}(2)$-invariant transition rates given in the
multipole expansion are thus

$\displaystyle I_{e+h\rightarrow t}^{n_{e}}$
$\displaystyle=-\frac{W_{e+h\rightarrow
t}}{12}\,\big{[}9n_{e}n^{\prime}_{h}+3\bm{c}_{e}\cdot\bm{c}^{\prime}_{h}+3\sqrt{6}\,n_{e}n^{\prime}_{h}n^{\prime\prime}_{t}$
$\displaystyle\quad{}+6n_{e}\bm{c}^{\prime}_{h}\cdot\bm{c}^{\prime\prime}_{t}+6n^{\prime}_{h}\bm{c}_{e}\cdot\bm{c}^{\prime\prime}_{t}+\sqrt{6}\,n^{\prime\prime}_{t}\bm{c}_{e}\cdot\bm{c}^{\prime}_{h}$
$\displaystyle\quad{}+6\bm{c}_{e}\cdot\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}^{\prime}_{h}\big{]}$ (130)

for the electron number density and

$\displaystyle\bm{I}_{e+h\rightarrow t}^{\bm{c}_{e}}$
$\displaystyle=-\frac{W_{e+h\rightarrow
t}}{12}\,\big{[}9\bm{c}_{e}n^{\prime}_{h}+3n_{e}\bm{c}^{\prime}_{h}$
$\displaystyle\quad{}+3\sqrt{6}\,\bm{c}_{e}n^{\prime}_{h}n^{\prime\prime}_{t}+\sqrt{6}\,n_{e}\bm{c}^{\prime}_{h}n^{\prime\prime}_{t}+6n_{e}n^{\prime}_{h}\bm{c}^{\prime\prime}_{t}$
$\displaystyle\quad{}+6\bm{c}_{e}\left(\bm{c}^{\prime}_{h}\cdot\bm{c}^{\prime\prime}_{t}\right)+6n_{e}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}^{\prime}_{h}\big{]}$ (131)

for the spin density. This rate contains a product of three spin densities.
The corresponding term is proportional to
$c_{e}^{z}c_{h}^{z\prime}c_{t}^{z\prime\prime}$ in the limit of z-polarized
spins. It is thus clear that this limit does not allow to infer the full
$\mathrm{SU}(2)$-invariant expression (although one might guess correctly) so
that the basis-independent expression (128) is required.

The results for the holes are analogous. We thus find that the rates satisfy
the consistency relations

$\displaystyle-\sqrt{6}\,I_{e+h\to t}^{{n_{t}}}$ $\displaystyle=I_{e+h\to
t}^{n_{e}}=I_{e+h\to t}^{n_{h}},$ (132) $\displaystyle-4\bm{I}_{e+h\to
t}^{\bm{c}_{t}}\,$ $\displaystyle=\bm{I}_{e+h\to
t}^{\bm{c}_{e}}+\bm{I}_{e+h\to t}^{\bm{c}_{h}}.$ (133)

Note that the factors $\sqrt{6}$ and $4$ in these two equations stem from the
different normalization conditions concerning electrons and triplet excitons.
The reverse process can be treated similarly.

###  III.4 Fusion and fission of singlet excitons

In preparation for the important transitions between a singlet exciton and two
triplet excitons, $s\leftrightarrow 2t$, we first consider the simpler process
$s\leftrightarrow 2s$. In contrast to the processes investigated before, the
fusion and fission of singlet excitons involves the same particle species in
the initial and final states. Furthermore, the two singlet excitons on the
same side of the reaction have to be treated as indistinguishable bosons.

Starting with the fusion, this process can either create or fuse a singlet
exciton with the outer momentum $\bm{k}$. This leads to in-scattering and out-
scattering terms in the transition rate, which can be written as

$I_{2s\to s}^{f_{s}}=W^{\prime\prime\prime}_{2s\to
s}f_{s}^{\prime\prime\prime}f_{s}^{\prime}\left(1+f_{s}\right)-2W_{2s\to
s}f_{s}f_{s}^{\prime}\left(1+f_{s}^{\prime\prime}\right).$ (134)

Here the short-hand notations $W\equiv W(\bm{k}^{\prime},\bm{k})$ and
$W^{\prime\prime\prime}\equiv W(\bm{k}^{\prime},\bm{k}-\bm{k}^{\prime})\equiv
W(\bm{k}^{\prime},\bm{k}^{\prime\prime\prime})$ are used. The factor $2$ in
the out-scattering term results from the two equal contributions of
indistinguishable singlet excitons in the initial state. Conversely, the
fission of one singlet exciton into two singlet excitons is described by

$\displaystyle I_{s\to 2s}^{f_{s}}$ $\displaystyle=2W_{s\to
2s}\left(1+f_{s}\right)\left(1+f_{s}^{\prime}\right)f_{s}^{\prime\prime}$
$\displaystyle\quad{}-W^{\prime\prime\prime}_{s\to
2s}\left(1+f_{s}^{\prime\prime\prime}\right)\left(1+f_{s}^{\prime}\right)f_{s}.$
(135)

Note that due to the spin-singlet character of all involved quasiparticles,
$\mathrm{SU}(2)$ symmetry is trivially satisfied.

###  III.5 Absorption of a singlet exciton by a triplet exciton

In contrast to the previous process, the particles in the initial state are
distinguishable. In the limit of $z$ polarization, this yields, from the point
of view of the singlet exciton,

$\displaystyle I_{t+s\to t}^{s}$ $\displaystyle=-W_{t+s\to
t}\,\big{[}f^{\prime}_{t,1}(1+f^{\prime\prime}_{t,1})+f^{\prime}_{t,0}(1+f^{\prime\prime}_{t,0})$
$\displaystyle\quad{}+f^{\prime}_{t,\bar{1}}(1+f^{\prime\prime}_{t,\bar{1}})\big{]}f_{s}.$
(136)

In terms of the multipole expansion this becomes

$\displaystyle I_{t+s\to t}^{s}$ $\displaystyle=-W_{t+s\to
t}\,\big{(}\sqrt{6}\,n^{\prime}_{t}$
$\displaystyle\quad{}+2n^{\prime}_{t}n^{\prime\prime}_{t}+2\bm{c}^{\prime}_{t}\cdot\bm{c}^{\prime\prime}_{t}+2\vec{q}^{\>\prime}_{t}\cdot\vec{q}^{\>\prime\prime}_{t}\big{)}f_{s}.$
(137)

The singlet exciton is described by a scalar distribution function so that the
basis-independent transition rate is

$I_{t+s\to t}^{s}=-\frac{W_{t+s\to
t}}{2}\,\operatorname{Tr}\Big{\\{}\widehat{f}^{\prime}_{t},\mathds{1}_{3}+\widehat{f}^{\prime\prime}_{t}\Big{\\}}\,f_{s}.$
(138)

The transition term from the perspective of the triplet exciton features in-
scattering and out-scattering terms as the triplet exciton is present on both
sides of the reaction. The transition rates for the different spin
orientations are

$\displaystyle I_{t+s\to t}^{t_{1}}$
$\displaystyle=W^{\prime\prime\prime}_{t+s\to
t}f^{\prime}_{t,1}f_{s}^{\prime\prime\prime}\left(1+f_{t,1}\right)$
$\displaystyle\quad{}-W_{t+s\to
t}f_{t,1}f_{s}^{\prime}\left(1+f^{\prime\prime}_{t,1}\right),$ (139)
$\displaystyle I_{t+s\to t}^{t_{0}}$
$\displaystyle=W^{\prime\prime\prime}_{t+s\to
t}f^{\prime}_{t,0}f_{s}^{\prime\prime\prime}\left(1+f_{t,0}\right)$
$\displaystyle\quad{}-W_{t+s\to
t}f_{t,0}f_{s}^{\prime}\left(1+f^{\prime\prime}_{t,0}\right),$ (140)
$\displaystyle I_{t+s\to t}^{t_{\bar{1}}}$
$\displaystyle=W^{\prime\prime\prime}_{t+s\to
t}f^{\prime}_{t,\bar{1}}f_{s}^{\prime\prime\prime}\left(1+f_{t,\bar{1}}\right)$
$\displaystyle\quad{}-W_{t+s\to
t}f_{t,\bar{1}}f_{s}^{\prime}\left(1+f^{\prime\prime}_{t,\bar{1}}\right).$
(141)

Since the (positive) in-scattering and the (negative) out-scattering
contributions have the same structure it is sufficient to investigate one
type. For the out-scattering contributions to the $\mathrm{SU}(2)$-invariant
transition rates of the multipole coefficients we obtain

$\displaystyle I_{t+s\to t}^{{n_{t}},\,\mathrm{out}}$
$\displaystyle=-\frac{W_{t+s\to t}}{\sqrt{6}}\,\big{(}\sqrt{6}\,n_{t}$
$\displaystyle\quad{}+2n_{t}n^{\prime\prime}_{t}+2\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{t}+\operatorname{Tr}\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\big{)}f_{s}^{\prime},$ (142) $\displaystyle\bm{I}_{t+s\to
t}^{\bm{c}_{t},\,\mathrm{out}}$ $\displaystyle=-\frac{W_{t+s\to
t}}{6}\,\big{(}6\bm{c}_{t}+2\sqrt{6}\,n_{t}\bm{c}^{\prime\prime}_{t}+2\sqrt{6}\,\bm{c}_{t}n^{\prime\prime}_{t}$
$\displaystyle\quad{}+3\underline{q}{}_{t}\bm{c}^{\prime\prime}_{t}+3\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}_{t}\big{)}f_{s}^{\prime},$ (143)
$\displaystyle\underline{I}_{t+s\to t}^{\underline{q}{}_{t},\,\mathrm{out}}$
$\displaystyle=-W_{t+s\to
t}\,\bigg{[}\underline{q}{}_{t}+\frac{\sqrt{6}}{3}\,\big{(}n_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+\underline{q}{}_{t}n^{\prime\prime}_{t}\big{)}$
$\displaystyle\quad{}+\frac{1}{2}\,\big{(}\bm{c}_{t}\otimes\bm{c}^{\prime\prime}_{t}+\bm{c}^{\prime\prime}_{t}\otimes\bm{c}_{t}\big{)}-\frac{1}{3}\,\bm{c}^{\prime\prime}_{t}\cdot\bm{c}_{t}\mathds{1}_{3}$
$\displaystyle\quad{}-\frac{1}{2}\,\big{(}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}{}_{t}+\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\big{)}+\frac{1}{6}\,\operatorname{Tr}{\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt}\mathds{1}_{3}\bigg{]}f_{s}^{\prime},$ (144)

where the quadrupole transition rate is again symmetric and traceless. In a
basis-independent form, these relations can be summarized as

$\displaystyle I_{t+s\to t}^{\widehat{{f}}_{t}}$
$\displaystyle=\frac{W^{\prime\prime\prime}_{t+s\to
t}}{2}\big{\\{}\widehat{f}^{\prime}_{t},\mathds{1}_{3}+\widehat{{f}}_{t}\big{\\}}f_{s}^{\prime\prime\prime}$
$\displaystyle\quad{}-\frac{W_{t+s\to
t}}{2}\big{\\{}\widehat{{f}}_{t},\mathds{1}_{3}+\widehat{f}^{\prime\prime}_{t}\big{\\}}f_{s}^{\prime}.$
(145)

The treatment of singlet-exciton emission is analogous.

###  III.6 Fusion and fission involving two triplet excitons and one singlet
exciton

The important fusion of two triplet excitons forming a singlet exciton StU89
is similar to the fusion of electron and hole into a singlet exciton. In
contrast to the latter process, the two particles forming the singlet exciton
are indistinguishable bosons.

####  III.6.1 Point of view of the singlet exciton

Considering the Clebsch-Gordan coefficients, the transition rate in terms of
diagonal elements of the distribution function is

$I_{2t\to s}^{s}=\frac{W^{\prime\prime\prime}_{2t\to
s}}{3}\,\big{(}f^{\prime}_{t,1}f^{\prime\prime\prime}_{t,\bar{1}}+f^{\prime}_{t,0}f^{\prime\prime\prime}_{t,0}+f^{\prime}_{t,\bar{1}}f^{\prime\prime\prime}_{t,1}\big{)}(1+f_{s}).$
(146)

Written in the multipole expansion and generalized to the
$\mathrm{SU}(2)$-invariant form this yields

$I_{2t\to s}^{s}=\frac{2W^{\prime\prime\prime}_{2t\to
s}}{3}\,\big{(}n^{\prime}_{t}n^{\prime\prime\prime}_{t}-\bm{c}^{\prime}_{t}\cdot\bm{c}^{\prime\prime\prime}_{t}+\vec{q}^{\>\prime}_{t}\cdot\vec{q}^{\>\prime\prime\prime}_{t}\big{)}(1+f_{s}),$
(147)

where the relation
$\operatorname{Tr}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime\prime}_{t}=2\vec{q}^{\>\prime}_{t}\cdot\vec{q}^{\>\prime\prime\prime}_{t}$
has been used. The only differences compared to the creation of singlet
excitons from free charge carriers are the prefactor and the additional
quadrupole term. The basis-independent form is

$I_{2t\to s}^{f_{s}}=\frac{W^{\prime\prime\prime}_{2t\to
s}}{2}\,\operatorname{Tr}\big{\\{}\widehat{P}_{s},\widehat{f}^{\prime}_{t}\otimes\widehat{f}^{\prime\prime\prime}_{t}\big{\\}}\left(1+f_{s}\right),$
(148)

which is analogous to the basis-independent form in the process $e+h\to s$;
see Eq. (86). Here $\widehat{P}_{s}=|0_{2t}\rangle\langle 0_{2t}|$ projects
the two-triplet-exciton state onto the singlet sector, where

$|0_{2t}\rangle=\frac{1}{\sqrt{3}}\left(|1\bar{1}\rangle-|00\rangle+|\bar{1}1\rangle\right)$
(149)

is the singlet state of the two triplet excitons.

The rate for the reverse fission process can be obtained analogously or be
read off from Eq. (148),

$I_{s\to 2t}^{f_{s}}=-\frac{W^{\prime\prime\prime}_{s\to
2t}}{2}\,\operatorname{Tr}\bigg{\\{}\widehat{P}_{s},\Big{(}\mathds{1}_{3}+\widehat{f}^{\prime}_{t}\Big{)}\otimes\Big{(}\mathds{1}_{3}+\widehat{f}^{\prime\prime\prime}_{t}\Big{)}\bigg{\\}}f_{s}.$
(150)

####  III.6.2 Point of view of the triplet exciton

For the diagonal elements of the distribution function, we obtain the three
transition rates

$\displaystyle I_{2t\to s}^{t_{1}}$ $\displaystyle=-\frac{2W_{2t\to
s}}{3}\,f_{t,1}f^{\prime}_{t,\bar{1}}\left(1+f_{s}^{\prime\prime}\right),$
(151) $\displaystyle I_{2t\to s}^{t_{0}}$ $\displaystyle=-\frac{2W_{2t\to
s}}{3}\,f_{t,0}f^{\prime}_{t,0}\left(1+f_{s}^{\prime\prime}\right),$ (152)
$\displaystyle I_{2t\to s}^{t_{\bar{1}}}$ $\displaystyle=-\frac{2W_{2t\to
s}}{3}\,f_{t,\bar{1}}f^{\prime}_{t,1}\left(1+f_{s}^{\prime\prime}\right),$
(153)

where the factor $2$ is due to the indistinguishable triplet excitons in the
initial state. The transition rate referring to the triplet-exciton number
density is

$I_{2t\to s}^{{n_{t}}}=-\frac{2\sqrt{6}\,W_{2t\to
s}}{9}\,\big{(}n_{t}n^{\prime}_{t}-\bm{c}_{t}\cdot\bm{c}^{\prime}_{t}+\vec{q}_{t}\cdot\vec{q}^{\>\prime}_{t}\big{)}(1+f_{s}^{\prime\prime}).$
(154)

Since the actual density is $\sqrt{6}\,n_{t}$, where $n_{t}$ is the expansion
coefficient in Eq. (48), the transition rate for the actual density is twice
the negative transition rate for the singlet-exciton density in Eq. (147). For
the spin transition rate we obtain

$\displaystyle\bm{I}_{2t\to s}^{\bm{c}_{t}}$ $\displaystyle=-\frac{W_{2t\to
s}}{3}\,\bigg{(}{-}\frac{2\sqrt{6}}{3}\,n_{t}\bm{c}^{\prime}_{t}+\frac{2\sqrt{6}}{3}\,n^{\prime}_{t}\bm{c}_{t}$
$\displaystyle\quad{}-\underline{q}{}_{t}\bm{c}^{\prime}_{t}+\underline{q}^{\prime}{}_{t}\bm{c}_{t}\bigg{)}\left(1+f_{s}^{\prime\prime}\right)$
(155)

and the traceless and symmetric quadrupole transition rate is

$\displaystyle\underline{I}_{2t\to s}^{\underline{q}{}_{t}}$
$\displaystyle=-\frac{W_{2t\to
s}}{3}\,\bigg{[}{-}\big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}+\underline{q}^{\prime}{}_{t}\underline{q}{}_{t}\big{)}+\frac{2}{3}\operatorname{Tr}\big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\big{)}\,\mathds{1}_{3}$
$\displaystyle\quad{}-\big{(}\bm{c}_{t}\otimes\bm{c}^{\prime}_{t}+\bm{c}^{\prime}_{t}\otimes\bm{c}_{t}\big{)}+\frac{2}{3}\,\big{(}\bm{c}_{t}\cdot\bm{c}^{\prime}_{t}\big{)}\,\mathds{1}_{3}$
$\displaystyle\quad{}+\frac{2\sqrt{6}}{3}\,\big{(}\underline{q}{}_{t}n^{\prime}_{t}+n_{t}\underline{q}^{\prime}{}_{t}\big{)}\bigg{]}\left(1+f_{s}^{\prime\prime}\right).$
(156)

The basis-independent transition rate then assumes the form

$I_{2t\to s}^{\widehat{{f}}_{t}}=-W_{2t\to
s}\operatorname{Tr}_{t^{\prime}}\Big{\\{}\widehat{P}_{s},\widehat{{f}}_{t}\otimes\widehat{f}^{\prime}_{t}\Big{\\}}\left(1+f_{s}^{\prime\prime}\right),$
(157)

where $\operatorname{Tr}_{t^{\prime}}$ is the partial trace over the sector of
the triplet exciton with momentum $\bm{k}^{\prime}$. The reverse fission
process is then described by

$I_{s\to 2t}^{\widehat{{f}}_{t}}=W_{s\to
2t}\operatorname{Tr}_{t^{\prime}}\bigg{\\{}\widehat{P}_{s},\Big{(}\mathds{1}_{3}+\widehat{{f}}_{t}\Big{)}\otimes\Big{(}\mathds{1}_{3}+\widehat{f}^{\prime}_{t}\Big{)}\bigg{\\}}f_{s}^{\prime\prime}.$
(158)

###  III.7 Absorption of a triplet exciton by an electron

The process $e+t\leftrightarrow e$ is more complicated than the previous cases
because both particle species carry spin, in contrast to, e.g.,
$t+s\leftrightarrow t$. For the process $e+t\leftrightarrow e$ to be possible,
the total spin of the electron and triplet exciton on the left-hand side must
be $1/2$, i.e., they must form a doublet. The two states are

$\displaystyle|\psi^{d}_{\uparrow}\rangle$
$\displaystyle=\frac{\big{|}\frac{1}{2}\big{\rangle}|0\rangle-\sqrt{2}\,\big{|}{-}\frac{1}{2}\big{\rangle}|1\rangle}{\sqrt{3}},$
(159) $\displaystyle|\psi^{d}_{\downarrow}\rangle$
$\displaystyle=\frac{\sqrt{2}\,\big{|}\frac{1}{2}\big{\rangle}|{-}1\rangle-\big{|}{-}\frac{1}{2}\big{\rangle}|0\rangle}{\sqrt{3}}.$
(160)

These relations allow us to read off the numerical prefactors in the rates.

####  III.7.1 Point of view of the triplet excitons

In the limit of $z$ polarization, the three transition rates describing
triplet excitons are

$\displaystyle I_{e+t\to e}^{t_{1}}$ $\displaystyle=-\frac{2W_{e+t\to
e}}{3}\,f^{\prime}_{e\downarrow}f_{t,1}(1-f^{\prime\prime}_{e\uparrow}),$
(161) $\displaystyle I_{e+t\to e}^{t_{0}}$ $\displaystyle=-\frac{W_{e+t\to
e}}{3}\,\big{[}f^{\prime}_{e\uparrow}f_{t,0}(1-f^{\prime\prime}_{e\uparrow})+f^{\prime}_{e\downarrow}f_{t,0}(1-f^{\prime\prime}_{e\downarrow})\big{]},$
(162) $\displaystyle I_{e+t\to e}^{t_{\bar{1}}}$
$\displaystyle=-\frac{2W_{e+t\to
e}}{3}\,f^{\prime}_{e\uparrow}f_{t,\bar{1}}(1-f^{\prime\prime}_{e\downarrow}).$
(163)

The $\mathrm{SU}(2)$-invariant transition rates in terms of multipole
coefficients are obtained in analogy to the previous cases. For the triplet-
exciton number density we find

$\displaystyle I_{e+t\to e}^{{n_{t}}}$ $\displaystyle=-\frac{W_{e+t\to
e}}{18}\,\big{(}6{n_{t}}n^{\prime}_{e}-2\sqrt{6}\,\bm{c}_{t}\cdot\bm{c}^{\prime}_{e}$
$\displaystyle\quad{}-3n^{\prime}_{e}n^{\prime\prime}_{e}{n_{t}}+{n_{t}}\bm{c}^{\prime}_{e}\cdot\bm{c}^{\prime\prime}_{e}$
$\displaystyle\quad{}+\sqrt{6}\,\bm{c}_{t}\cdot\bm{c}^{\prime}_{e}n^{\prime\prime}_{e}-\sqrt{6}\,\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{e}n^{\prime}_{e}+\sqrt{6}\,\bm{c}^{\prime}_{e}\cdot\underline{q}{}_{t}\bm{c}^{\prime\prime}_{e}\big{)},$
(164)

for the spin density

$\displaystyle\bm{I}_{e+t\to e}^{\bm{c}_{t}}$ $\displaystyle=-\frac{W_{e+t\to
e}}{18}\,\bigg{[}6\bm{c}_{t}n^{\prime}_{e}-2\sqrt{6}\,{n_{t}}\bm{c}^{\prime}_{e}-3\underline{q}{}_{t}\bm{c}^{\prime}_{e}$
$\displaystyle\quad{}+\sqrt{6}\,{n_{t}}\,(\bm{c}^{\prime}_{e}n^{\prime\prime}_{e}-n^{\prime}_{e}\bm{c}^{\prime\prime}_{e})$
$\displaystyle\quad{}+\frac{3}{2}\,(\bm{c}_{t}\cdot\bm{c}^{\prime}_{e})\bm{c}^{\prime\prime}_{e}+\frac{3}{2}\,(\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{e})\bm{c}^{\prime}_{e}$
$\displaystyle\quad{}-3\bm{c}_{t}n^{\prime}_{e}n^{\prime\prime}_{e}+\frac{3}{2}\,n^{\prime\prime}_{e}\underline{q}{}_{t}\bm{c}^{\prime}_{e}-\frac{3}{2}\,n^{\prime}_{e}\underline{q}{}_{t}\bm{c}^{\prime\prime}_{e}\bigg{]},$
(165)

and for the symmetric traceless quadrupole density

$\displaystyle\underline{I}_{e+t\to e}^{\underline{q}{}_{t}}=-\frac{W_{e+t\to
e}}{18}$
$\displaystyle{}\times\bigg{(}{-}3\,\bigg{[}\bm{c}^{\prime}_{e}\otimes\bm{c}_{t}+\bm{c}_{t}\otimes\bm{c}^{\prime}_{e}-\frac{2}{3}\,(\bm{c}^{\prime}_{e}\cdot\bm{c}_{t})\,\mathds{1}_{3}\bigg{]}+6n^{\prime}_{e}\underline{q}{}_{t}$
$\displaystyle\quad{}-3n^{\prime}_{e}n^{\prime\prime}_{e}\underline{q}{}_{t}$
$\displaystyle\quad{}+\sqrt{6}\bigg{[}\bm{c}^{\prime}_{e}\otimes\bm{c}^{\prime\prime}_{e}+\bm{c}^{\prime\prime}_{e}\otimes\bm{c}^{\prime}_{e}-\frac{2}{3}\,(\bm{c}^{\prime}_{e}\cdot\bm{c}^{\prime\prime}_{e})\,\mathds{1}_{3}\bigg{]}{n_{t}}$
$\displaystyle\quad{}+\frac{3}{2}\,\bigg{[}\bm{c}^{\prime}_{e}\otimes\bm{c}_{t}+\bm{c}_{t}\otimes\bm{c}^{\prime}_{e}-\frac{2}{3}\,(\bm{c}^{\prime}_{e}\cdot\bm{c}_{t})\,\mathds{1}_{3}\bigg{]}n^{\prime\prime}_{e}$
$\displaystyle\quad{}-\frac{3}{2}\,\bigg{[}\bm{c}_{t}\otimes\bm{c}^{\prime\prime}_{e}+\bm{c}^{\prime\prime}_{e}\otimes\bm{c}_{t}-\frac{2}{3}\,(\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{e})\,\mathds{1}_{3}\bigg{]}n^{\prime}_{e}$
$\displaystyle\quad{}+3\bigg{[}(\bm{c}^{\prime}_{e}\cdot\bm{c}^{\prime\prime}_{e})\underline{q}{}_{t}+\frac{1}{2}\,\bm{c}^{\prime}_{e}\otimes(\underline{q}{}_{t}\bm{c}^{\prime\prime}_{e})+\frac{1}{2}\,(\underline{q}{}_{t}\bm{c}^{\prime\prime}_{e})\otimes\bm{c}^{\prime}_{e}$
$\displaystyle\qquad{}-\frac{1}{2}\,\bm{c}^{\prime\prime}_{e}\otimes(\underline{q}{}_{t}\bm{c}^{\prime}_{e})-\frac{1}{2}\,(\underline{q}{}_{t}\bm{c}^{\prime}_{e})\otimes\bm{c}^{\prime\prime}_{e}$
$\displaystyle\qquad{}-(\bm{c}^{\prime}_{e}\otimes\bm{c}^{\prime\prime}_{e})\underline{q}{}_{t}-\underline{q}{}_{t}(\bm{c}^{\prime\prime}_{e}\otimes\bm{c}^{\prime}_{e})+\frac{2}{3}\,\bm{c}^{\prime}_{e}\cdot\underline{q}{}_{t}\bm{c}^{\prime\prime}_{e}\,\mathds{1}_{3}\bigg{]}\bigg{)}.$
(166)

Like for the process $e+h\to t$, one might guess at this
$\mathrm{SU}(2)$-invariant form based on the limit of z-polarized spins but
cannot infer it rigorously. The problem lies in the terms containing three
$\mathbf{c}$ vectors and in the terms with two $\mathbf{c}$ vectors and the
triplet-exciton quadrupole tensor $\underline{q}{}_{t}$. The above expressions
have thus been checked by comparing them with the basis-independent expression
describing this process, which takes the form

$\displaystyle I_{e+t\to e}^{\widehat{{f}}_{t}}$
$\displaystyle=-\frac{W_{e+t\to
e}}{2}\,\operatorname{Tr}_{e}\bigg{[}\Big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime}_{e}\Big{)}_{6}\widehat{P}_{d}\Big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{{f}}_{t}\Big{)}$
$\displaystyle\quad{}+\Big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{{f}}_{t}\Big{)}\widehat{P}_{d}\Big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime}_{e}\Big{)}_{6}\bigg{]},$
(167)

where $\widehat{P}_{d}$ is the projection operator onto the doublet subspace.
This is a $6\times 6$ matrix. The notation $\left(\bullet\right)_{6}$ means
that a $2\times 2$ matrix on the doublet subspace is extended to the full two-
particle space by adding a $4\times 4$ null matrix on the quartet subspace.
The matrix is then transformed into the product basis for electron and
triplet-exciton spin to be consistent with
$\widehat{{f}}_{e}\otimes\widehat{{f}}_{t}$.

####  III.7.2 Point of view of the electron

The perspective of the electron leads to in-scattering and out-scattering
terms. The resulting transition rates for the two spin orientations are

$\displaystyle I_{e+t\to e}^{e\uparrow}=-\frac{W_{e+t\to e}}{3}$
$\displaystyle\quad{}\times\big{[}2f_{e\uparrow}f^{\prime}_{t,\bar{1}}(1-f^{\prime\prime}_{e\downarrow})+f_{e\uparrow}f^{\prime}_{t,0}(1-f^{\prime\prime}_{e\uparrow})\big{]}$
$\displaystyle{}+\frac{W^{\prime\prime\prime}_{e+t\to
e}}{3}\,\big{[}f^{\prime}_{e\uparrow}f^{\prime\prime\prime}_{t,0}(1-f_{e\uparrow})+2f^{\prime}_{e\downarrow}f^{\prime\prime\prime}_{t,1}(1-f_{e\uparrow})\big{]},$
(168) $\displaystyle I_{e+t\to e}^{e\downarrow}=-\frac{W_{e+t\to e}}{3}$
$\displaystyle\quad{}\times\big{[}2f_{e\downarrow}f^{\prime}_{t,1}(1-f^{\prime\prime}_{e\uparrow})+f_{e\downarrow}f^{\prime}_{t,0}(1-f^{\prime\prime}_{e\downarrow})\big{]}$
$\displaystyle{}+\frac{W^{\prime\prime\prime}_{e+t\to
e}}{3}\,\big{[}f^{\prime}_{e\downarrow}f^{\prime\prime\prime}_{t,0}(1-f_{e\downarrow})+2f^{\prime}_{e\uparrow}f^{\prime\prime\prime}_{t,\bar{1}}(1-f_{e\downarrow})\big{]},$
(169)

where the Clebsch-Gordan coefficients have been used. As the in-scattering and
out-scattering terms have the same structure it is sufficient to investigate
out-scattering. For the multipole transition rates this yields

$\displaystyle I_{e+t\to e}^{n_{e},\,\mathrm{out}}$
$\displaystyle=-\frac{W_{e+t\to
e}}{18}\,\big{(}6\sqrt{6}\,n_{e}n^{\prime}_{t}-12\bm{c}_{e}\cdot\bm{c}^{\prime}_{t}$
$\displaystyle\quad{}-3\sqrt{6}\,n_{e}n^{\prime}_{t}n^{\prime\prime}_{e}-6n_{e}\bm{c}^{\prime\prime}_{e}\cdot\bm{c}^{\prime}_{t}+\sqrt{6}\,n^{\prime}_{t}\bm{c}_{e}\cdot\bm{c}^{\prime\prime}_{e}$
$\displaystyle\quad{}+6n^{\prime\prime}_{e}\bm{c}_{e}\cdot\bm{c}^{\prime}_{t}+6\bm{c}_{e}\cdot\underline{q}^{\prime}{}_{t}\bm{c}^{\prime\prime}_{e}\big{)},$
(170) $\displaystyle\bm{I}_{e+t\to e}^{\bm{c}_{t},\,\mathrm{out}}$
$\displaystyle=-\frac{W_{e+t\to
e}}{18}\,\bigg{[}12n_{e}\bm{c}^{\prime}_{t}-6\sqrt{6}\,\bm{c}_{e}n^{\prime}_{t}$
$\displaystyle\quad{}-\sqrt{6}\,n_{e}n^{\prime}_{t}\bm{c}^{\prime\prime}_{e}-6n_{e}\bm{c}^{\prime}_{t}n^{\prime\prime}_{e}+3\sqrt{6}\,\bm{c}_{e}n^{\prime}_{t}n^{\prime\prime}_{e}$
$\displaystyle\quad{}+6\bm{c}_{e}(\bm{c}^{\prime}_{t}\cdot\bm{c}^{\prime\prime}_{e})-\frac{3}{2}\,n_{e}\underline{q}^{\prime}{}_{t}\bm{c}^{\prime\prime}_{e}\bigg{]},$
(171)

which can be combined to obtain the full transition rate from the point of
view of the electron.

Together with the in-scattering terms the basis-independent form is then

$\displaystyle I_{e+t\to e}^{\widehat{{f}}_{e}}$
$\displaystyle=\frac{W^{\prime\prime\prime}_{e+t\to
e}}{2}\,\operatorname{Tr}_{t^{\prime\prime\prime}}\bigg{[}\Big{(}\mathds{1}_{2}-\widehat{{f}}_{e}\Big{)}_{6}\widehat{P}_{d}\Big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{f}^{\prime\prime\prime}_{t}\Big{)}$
$\displaystyle\quad{}+\Big{(}\widehat{f}^{\prime}_{e}\otimes\widehat{f}^{\prime\prime\prime}_{t}\Big{)}\widehat{P}_{d}\Big{(}\mathds{1}_{2}-\widehat{{f}}_{e}\Big{)}_{6}\bigg{]}$
$\displaystyle\quad{}-\frac{W_{e+t\to
e}}{2}\,\operatorname{Tr}_{t^{\prime}}\bigg{[}\Big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime}_{e}\Big{)}_{6}\widehat{P}_{d}\Big{(}\widehat{{f}}_{e}\otimes\widehat{f}^{\prime}_{t}\Big{)}$
$\displaystyle\quad{}+\Big{(}\widehat{{f}}_{e}\otimes\widehat{f}^{\prime}_{t}\Big{)}\widehat{P}_{d}\Big{(}\mathds{1}_{2}-\widehat{f}^{\prime\prime}_{e}\Big{)}_{6}\bigg{]}.$
(172)

The absorption and emission of a triplet exciton by a hole is analogous.

###  III.8 Fusion of triplet excitons into a triplet exciton

The fusion of two triplet excitons into a triplet exciton StU89  is the most
complex process considered here. In the case of $z$ polarization, the
transition terms of the components of the density matrix read as

$\displaystyle I_{2t\to t}^{t_{1}}=W^{\prime\prime\prime}_{2t\to
t}\,\frac{f^{\prime\prime\prime}_{t,1}f^{\prime}_{t,0}}{2}\,(1+f_{t,1})$
$\displaystyle\quad{}-2W_{2t\to
t}\bigg{[}\frac{f_{t,1}f^{\prime}_{t,\bar{1}}}{2}\,(1+f^{\prime\prime}_{t,0})+\frac{f_{t,1}f^{\prime}_{t,0}}{2}\,(1+f^{\prime\prime}_{t,1})\bigg{]},$
(173) $\displaystyle I_{2t\to t}^{t_{0}}=W^{\prime\prime\prime}_{2t\to
t}\,\frac{f^{\prime}_{t,1}f^{\prime\prime\prime}_{t,\bar{1}}}{2}\,(1+f_{t,0})$
$\displaystyle\quad{}-2W_{2t\to
t}\bigg{[}\frac{f^{\prime}_{t,1}f_{t,0}}{2}\,(1+f^{\prime\prime}_{t,1})+\frac{f^{\prime}_{t,\bar{1}}f_{t,0}}{2}\,(1+f^{\prime\prime}_{t,\bar{1}})\bigg{]},$
(174) $\displaystyle I_{2t\to t}^{t_{\bar{1}}}=W^{\prime\prime\prime}_{2t\to
t}\,\frac{f^{\prime\prime\prime}_{t,\bar{1}}f^{\prime}_{t,0}}{2}\,(1+f_{t,\bar{1}})$
$\displaystyle\quad{}-2W_{2t\to
t}\bigg{[}\frac{f^{\prime}_{t,1}f_{t,\bar{1}}}{2}\,(1+f^{\prime\prime}_{t,0})+\frac{f_{t,\bar{1}}f^{\prime}_{t,0}}{2}\,(1+f^{\prime\prime}_{t,\bar{1}})\bigg{]}.$
(175)

In this process, the in-scattering and out-scattering terms differ in
structure. This makes it necessary to investigate both terms separately.

####  III.8.1 In-scattering for $2t\to t$

Combining Eqs. (III.8)–(175), we can obtain the transition rates expressed in
the multipole expansion. Their generalization to the
$\mathrm{SU}(2)$-invariant form is ambiguous at first glance. However, the
$\mathrm{SU}(2)$-invariant form can be obtained in analogy to the formation of
triplet excitons out of free charge carriers. We construct the density matrix
$\widehat{f}^{\prime}_{t}\otimes\widehat{f}^{\prime\prime\prime}_{t}$ on the
product space of the two triplet-exciton spins and project it onto the triplet
subspace. The resulting $3\times 3$ matrix, which will be referred to as
$\widehat{f}_{tt}$, can be expressed in terms of coefficients of the nine
basis matrices $\widehat{S}_{i}$, $i=1,2,3$, and $\widehat{Q}_{j}$,
$j=1,2,3,4,5$. For a spin along the $z$ axis, only three coefficients do not
vanish:

$\displaystyle n_{tt}$
$\displaystyle=\frac{2\sqrt{6}\,n^{\prime}_{t}n^{\prime\prime\prime}_{t}-\sqrt{6}\,c_{t}^{z\prime}c_{t}^{z\prime\prime\prime}-\sqrt{6}\,q^{\prime}_{t5}q^{\prime\prime\prime}_{t5}}{6},$
(176) $\displaystyle c_{tt}^{z}$
$\displaystyle=\frac{\sqrt{6}\,c_{t}^{z\prime}n^{\prime\prime\prime}_{t}-2\sqrt{3}\,c_{t}^{z\prime}q^{\prime\prime\prime}_{t5}+\sqrt{6}\,c_{t}^{z\prime\prime\prime}n^{\prime}_{t}-2\sqrt{3}\,c_{t}^{z\prime\prime\prime}q^{\prime}_{t5}}{6},$
(177) $\displaystyle q_{tt5}$
$\displaystyle=\frac{2\sqrt{2}\,c_{t}^{z\prime}c_{t}^{z\prime\prime\prime}-2n^{\prime}_{t}q^{\prime\prime\prime}_{t5}-2n^{\prime\prime\prime}_{t}q^{\prime}_{t5}-2\sqrt{2}\,q^{\prime}_{t5}q^{\prime\prime\prime}_{t5}}{6}.$
(178)

Combining these equations and using Eqs. (III.8)–(175), we can unambiguously
generalize the rates to their $\mathrm{SU}(2)$-invariant form:

$\displaystyle I_{2t\to t}^{{n_{t}},\,\mathrm{in}}$
$\displaystyle=W^{\prime\prime\prime}_{2t\to
t}\bigg{[}{n}_{tt}+\frac{\sqrt{6}}{3}\,n_{t}{n}_{tt}+\frac{\sqrt{6}}{3}\,\bm{c}_{t}\cdot\bm{c}_{tt}$
$\displaystyle\quad{}+\frac{\sqrt{6}}{3}\,\vec{q}_{t}\cdot\vec{q}_{tt}\bigg{]},$
(179) $\displaystyle\bm{I}_{2t\to t}^{\bm{c}_{t},\,\mathrm{in}}$
$\displaystyle=W^{\prime\prime\prime}_{2t\to
t}\bigg{[}\bm{c}_{tt}+\frac{\sqrt{6}}{3}\,{n}_{tt}\bm{c}^{\prime}_{t}+\frac{\sqrt{6}}{3}\,\bm{c}_{tt}n^{\prime}_{t}$
$\displaystyle\quad{}+\frac{1}{2}\,\underline{q}{}_{tt}\bm{c}^{\prime}_{t}+\frac{1}{2}\,\underline{q}^{\prime}{}_{t}\bm{c}_{tt}\bigg{]},$
(180) $\displaystyle\underline{I}_{2t\to
t}^{\underline{q}{}_{t},\,\mathrm{in}}$
$\displaystyle=W^{\prime\prime\prime}_{2t\to
t}\bigg{[}\underline{q}{}_{tt}+\frac{\sqrt{6}}{3}\,{n}_{tt}\underline{q}{}_{t}+\frac{\sqrt{6}}{3}\,n_{t}\underline{q}{}_{tt}$
$\displaystyle\quad{}-\frac{1}{2}\,\underline{q}{}_{tt}\underline{q}{}_{t}-\frac{1}{2}\,\underline{q}{}_{t}\underline{q}{}_{tt}+\frac{1}{3}\,\operatorname{Tr}\big{(}\underline{q}{}_{tt}\underline{q}{}_{t}\big{)}\mathds{1}_{3}$
$\displaystyle\quad{}-\frac{1}{3}\,(\bm{c}_{tt}\cdot\bm{c}_{t})\mathds{1}_{3}+\frac{1}{2}\,(\bm{c}_{tt}\otimes\bm{c}_{t}+\bm{c}_{t}\otimes\bm{c}_{tt})\bigg{]}.$
(181)

The basis-independent in-scattering term reads as

$\displaystyle I_{2t\to t}^{\widehat{{f}}_{t},\,\mathrm{in}}$
$\displaystyle=\frac{W^{\prime\prime\prime}_{2t\to
t}}{2}\,\operatorname{Tr}_{t^{\prime\prime\prime}}\bigg{[}\Big{(}\mathds{1}_{3}+\widehat{{f}}_{t}\Big{)}_{9}\widehat{P}_{t}\Big{(}\widehat{f}^{\prime}_{t}\otimes\widehat{f}^{\prime\prime\prime}_{t}\Big{)}$
$\displaystyle\quad{}+\Big{(}\widehat{f}^{\prime}_{t}\otimes\widehat{f}^{\prime\prime\prime}_{t}\Big{)}\widehat{P}_{t}\Big{(}\mathds{1}_{3}+\widehat{{f}}_{t}\Big{)}_{9}\bigg{]}.$
(182)

The notation $(\bullet)_{9}$ is analogous to $(\bullet)_{4}$ in Sec. III.3.2
and to $(\bullet)_{6}$ in Sec. III.7.1.

####  III.8.2 Out-scattering for $2t\to t$

For out-scattering, the strategy employed for in-scattering does not work
since the rates for the case of $z$ polarization are not symmetric in the two
incoming triplet excitons at the outer momentum $\bm{k}$ and the running
momentum $\bm{k}^{\prime}$. Hence, the projected density matrix on the product
space, $\widehat{f}_{tt}$, is not useful. The situation is analogous to the
process $e+h\to t$ from the point of view of the electron or hole; see Sec.
III.3.2. Like for $e+h\to t$, the matrix-valued transition rate in terms of
projection operators and density matrices

$\displaystyle I_{2t\to t}^{\widehat{{f}}_{t},\,\mathrm{out}}$
$\displaystyle=-W_{2t\to
t}\,\operatorname{Tr}_{t^{\prime}}\bigg{[}\Big{(}\mathds{1}_{3}+\widehat{f}^{\prime\prime}_{t}\Big{)}_{9}\widehat{P}_{t}\Big{(}\widehat{{f}}_{t}\otimes\widehat{f}^{\prime}_{t}\Big{)}$
$\displaystyle\quad{}+\Big{(}\widehat{{f}}_{t}\otimes\widehat{f}^{\prime}_{t}\Big{)}\widehat{P}_{t}\Big{(}\mathds{1}_{3}+\widehat{f}^{\prime\prime}_{t}\Big{)}_{9}\bigg{]}$
(183)

can be used to read off the correct $\mathrm{SU}(2)$-invariant multipole
transition rates. For the triplet exciton number density this yields

$\displaystyle I_{2t\to t}^{{n_{t}},\,\mathrm{out}}$
$\displaystyle=-\frac{W_{2t\to
t}}{3}\,\bigg{[}2\sqrt{6}\,n_{t}n^{\prime}_{t}-\sqrt{6}\,\bm{c}_{t}\cdot\bm{c}^{\prime}_{t}-\frac{\sqrt{6}}{2}\,\operatorname{Tr}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}$
$\displaystyle\quad{}+4n_{t}n^{\prime}_{t}n^{\prime\prime}_{t}+2\big{(}n_{t}\bm{c}^{\prime}_{t}\cdot\bm{c}^{\prime\prime}_{t}+n^{\prime}_{t}\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{t}+n^{\prime\prime}_{t}\bm{c}_{t}\cdot\bm{c}^{\prime}_{t}\big{)}-\Big{(}n_{t}\operatorname{Tr}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+n^{\prime}_{t}\operatorname{Tr}\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+n^{\prime\prime}_{t}\operatorname{Tr}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\Big{)}$
$\displaystyle\quad{}-\sqrt{6}\,\Big{(}\bm{c}_{t}\cdot\underline{q}^{\prime}{}_{t}\bm{c}^{\prime\prime}_{t}+\bm{c}^{\prime}_{t}\cdot\underline{q}{}_{t}\bm{c}^{\prime\prime}_{t}+\bm{c}_{t}\cdot\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}^{\prime}_{t}\Big{)}-\frac{\sqrt{6}}{2}\,\Big{(}\operatorname{Tr}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+\operatorname{Tr}\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}^{\prime}{}_{t}\Big{)}\bigg{]}.$ (184)

The spin transition rate becomes

$\displaystyle\bm{I}_{2t\to t}^{\bm{c}_{t},\,\mathrm{out}}$
$\displaystyle=-\frac{W_{2t\to
t}}{6}\,\Big{[}2\sqrt{6}\,(-2n_{t}\bm{c}^{\prime}_{t}+n^{\prime}_{t}\bm{c}_{t})-3\Big{(}\underline{q}{}_{t}\bm{c}^{\prime}_{t}+\underline{q}^{\prime}{}_{t}\bm{c}_{t}\Big{)}-4\big{(}n_{t}n^{\prime\prime}_{t}\bm{c}^{\prime}_{t}-n_{t}n^{\prime}_{t}\bm{c}^{\prime\prime}_{t}-2n_{t}n^{\prime\prime}_{t}\bm{c}^{\prime}_{t}\big{)}$
$\displaystyle\quad{}-\sqrt{6}\,\Big{(}n^{\prime}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}_{t}-n^{\prime}_{t}\underline{q}{}_{t}\bm{c}^{\prime\prime}_{t}+n^{\prime\prime}_{t}\underline{q}^{\prime}{}_{t}\bm{c}_{t}+n^{\prime\prime}_{t}\underline{q}{}_{t}\bm{c}^{\prime}_{t}+2n_{t}\underline{q}^{\prime}{}_{t}\bm{c}^{\prime\prime}_{t}-2n_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}^{\prime}_{t}\Big{)}$
$\displaystyle\quad{}-3\big{(}\bm{c}^{\prime}_{t}(\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{t})+\bm{c}^{\prime\prime}_{t}(\bm{c}^{\prime}_{t}\cdot\bm{c}_{t})-2\bm{c}_{t}(\bm{c}^{\prime}_{t}\cdot\bm{c}^{\prime\prime}_{t})\big{)}-3\Big{(}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}_{t}+\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}^{\prime}{}_{t}\bm{c}_{t}+\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\bm{c}^{\prime\prime}_{t}-\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}_{t}\Big{)}\Big{]}.$ (185)

The quadrupole part reads as

$\displaystyle\underline{I}_{2t\to t}^{\underline{q}{}_{t},\,\mathrm{out}}$
$\displaystyle=-\frac{W_{2t\to
t}}{6}\,\bigg{(}2\sqrt{6}\,\big{(}-n_{t}\underline{q}^{\prime}{}_{t}+2n^{\prime}_{t}\underline{q}{}_{t})-3\big{[}\bm{c}_{t}\otimes\bm{c}^{\prime}_{t}+\bm{c}^{\prime}_{t}\otimes\bm{c}_{t}-2(\bm{c}^{\prime}_{t}\cdot\bm{c}_{t})\mathds{1}_{3}\big{]}-3\big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}+\underline{q}^{\prime}{}_{t}\underline{q}{}_{t}\big{)}-2\operatorname{Tr}\big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\big{)}\mathds{1}_{3}$
$\displaystyle\quad{}-4\Big{(}n_{t}n^{\prime}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+n_{t}n^{\prime\prime}_{t}\underline{q}^{\prime}{}_{t}-2n^{\prime}_{t}n^{\prime\prime}_{t}\underline{q}{}_{t}\Big{)}$
$\displaystyle\quad{}-\sqrt{6}\,\big{[}2n_{t}(\bm{c}^{\prime}_{t}\otimes\bm{c}^{\prime\prime}_{t}+\bm{c}^{\prime\prime}_{t}\otimes\bm{c}^{\prime}_{t})-n^{\prime}_{t}(\bm{c}_{t}\otimes\bm{c}^{\prime\prime}_{t}+\bm{c}^{\prime\prime}_{t}\otimes\bm{c}_{t})+n^{\prime\prime}_{t}(\bm{c}_{t}\otimes\bm{c}^{\prime}_{t}+\bm{c}^{\prime}_{t}\otimes\bm{c}_{t})\big{]}$
$\displaystyle\quad{}+\frac{2\sqrt{6}}{3}\,\Big{[}2n_{t}(\bm{c}^{\prime}_{t}\cdot\bm{c}^{\prime\prime}_{t})-n^{\prime}_{t}(\bm{c}_{t}\cdot\bm{c}^{\prime\prime}_{t})+n^{\prime\prime}_{t}(\bm{c}_{t}\cdot\bm{c}^{\prime}_{t})\Big{]}\mathds{1}_{3}$
$\displaystyle\quad{}-\sqrt{6}\,\Big{[}2n_{t}\Big{(}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}^{\prime}{}_{t}\Big{)}-n^{\prime}_{t}\Big{(}\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}{}_{t}\Big{)}-n^{\prime\prime}_{t}\Big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}+\underline{q}^{\prime}{}_{t}\underline{q}{}_{t}\Big{)}\Big{]}$
$\displaystyle\quad{}+\frac{2\sqrt{6}}{3}\,\Big{[}2n_{t}\operatorname{Tr}\big{(}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\big{)}-n^{\prime}_{t}\operatorname{Tr}\big{(}\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\big{)}-n^{\prime\prime}_{t}\operatorname{Tr}\big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\big{)}\Big{]}\mathds{1}_{3}$
$\displaystyle\quad{}-3\Big{[}\big{(}\bm{c}_{t}\otimes\bm{c}^{\prime\prime}_{t}\big{)}\underline{q}^{\prime}{}_{t}+\underline{q}^{\prime}{}_{t}\big{(}\bm{c}^{\prime\prime}_{t}\otimes\bm{c}_{t}\big{)}-\big{(}\bm{c}_{t}\otimes\bm{c}^{\prime}_{t}\big{)}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt-\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\big{(}\bm{c}^{\prime}_{t}\otimes\bm{c}_{t}\big{)}-\big{(}\bm{c}^{\prime}_{t}\otimes\bm{c}^{\prime\prime}_{t}\big{)}\underline{q}{}_{t}-\underline{q}{}_{t}\big{(}\bm{c}^{\prime\prime}_{t}\otimes\bm{c}^{\prime}_{t}\big{)}$
$\displaystyle\qquad{}-\big{(}\bm{c}^{\prime\prime}_{t}\otimes\bm{c}^{\prime}_{t}\big{)}\underline{q}{}_{t}-\underline{q}{}_{t}\big{(}\bm{c}^{\prime}_{t}\otimes\bm{c}^{\prime\prime}_{t}\big{)}\Big{]}-2\Big{(}\bm{c}_{t}\cdot\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\bm{c}^{\prime}_{t}-\bm{c}_{t}\cdot\underline{q}^{\prime}{}_{t}\bm{c}^{\prime\prime}_{t}+\bm{c}^{\prime\prime}_{t}\cdot\underline{q}{}_{t}\bm{c}^{\prime}_{t}\Big{)}\mathds{1}_{3}$
$\displaystyle\quad{}+3\Big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt+\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}^{\prime}{}_{t}\underline{q}{}_{t}+\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}{}_{t}+\underline{q}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\underline{q}^{\prime}{}_{t}\Big{)}-4\operatorname{Tr}\Big{(}\underline{q}{}_{t}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\Big{)}\mathds{1}_{3}-6\operatorname{Tr}\Big{(}\underline{q}^{\prime}{}_{t}\underline{q}^{\prime\prime}{}_{t}\hskip
1.49994pt\Big{)}\underline{q}{}_{t}\bigg{)}.$ (186)

The derivation of the rates for the spin and for the quadrupole moment again
require the basis-independent form in Eq. (183).

The reverse fission process $t\to 2t$ can be investigated in the same way. In
the out-scattering term, the introduction of $\widehat{f}_{1-tt}$ leads to the
same structure as in the in-scattering term for $2t\to t$.

##  IV Examples

In this section, we illustrate the theoretical framework by applying it to a
simple model. In order to reduce numerical complications, we consider a one-
dimensional toy model containing fission and fusion processes, disorder
scattering of singlet and triplet excitons, as well as relaxation of singlet
and triplet excitons. All bare rates $W$ are assumed to be independent of
momentum, except as dictated by scattering being elastic. The temperature is
assumed to be sufficiently low so that the equilibrium thermal population of
excitonic states is negligible. A magnetic field, if present, is taken to be
uniform, constant in time, and oriented along the z direction.

The dispersion relations of singlet and triplet excitons are written as

$\displaystyle\epsilon_{s}(k)$
$\displaystyle=\Delta\epsilon+\frac{\hbar^{2}k^{2}}{2m_{s}},$ (187)
$\displaystyle\epsilon_{t}(k,m)$
$\displaystyle=\frac{\hbar^{2}k^{2}}{2m_{t}}-g_{\mathrm{tr}}\mu_{B}Bm,$ (188)

respectively, where $m=1,0,-1$ specifies the spin state of triplet excitons
and

$\Delta\epsilon\equiv\frac{\hbar^{2}k_{0}^{2}}{2m_{t}}$ (189)

is the energy difference between singlet and triplet excitons, which defines a
momentum scale $k_{0}$.

Choosing our units such that $\hbar/m_{t}=1$ and defining the parameters

$\displaystyle\alpha$
$\displaystyle\equiv\frac{m_{t}}{m_{s}}=\frac{\hbar}{m_{s}},$ (190)
$\displaystyle\omega_{L}$
$\displaystyle\equiv\frac{1}{\hbar}\,g_{\mathrm{tr}}\mu_{B}B,$ (191)

the Boltzmann equation for the singlet-exciton distribution function
$f_{s}(x,k,t)$ can be written as

$\frac{\partial f_{s}}{\partial t}+\alpha k\,\frac{\partial f_{s}}{\partial
x}=I^{f_{s}},$ (192)

and the corresponding equation for the matrix-valued triplet-exciton
distribution function $\widehat{f}_{t}(x,k,t)$ as

$\frac{\partial\widehat{f}_{t}}{\partial
t}+k\,\frac{\partial\widehat{f}_{t}}{\partial
x}+i\omega_{L}\,[\widehat{S}_{3},\widehat{f}_{t}]=\widehat{I}^{\widehat{f}_{t}}.$
(193)

Here $I^{f_{s}}$ and $\widehat{I}^{\widehat{f}_{t}}$ are the full scattering
integrals, which contain the following contributions, where we only make the
momentum dependence explicit:

(a) Decay is discussed in Sec. III.1. It is described by the scattering
integrals

$\displaystyle I^{f_{s}}_{s\to\ket{0}}(k)$
$\displaystyle=-\widetilde{W}_{s\to\ket{0}}\,f_{s}(k)\equiv-\eta_{s0}\,f_{s}(k),$
(194) $\displaystyle\widehat{I}^{\widehat{f}_{t}}_{t\to\ket{0}}(k)$
$\displaystyle=-\widetilde{W}_{t\to\ket{0}}\,\widehat{f}_{t}(k)\equiv-\eta_{t0}\,\widehat{f}_{t}(k).$
(195)

x

(b) Disorder scattering is discussion in the introduction to Sec. III and is
described by

$\displaystyle I^{f_{s}}_{s\to s}$
$\displaystyle=\int\frac{dk^{\prime}}{2\pi}\,w_{s\to
s^{\prime}}\,\delta\bigg{(}\frac{\hbar^{2}(k^{\prime})^{2}}{2m_{s}}-\frac{\hbar^{2}k^{2}}{2m_{s}}\bigg{)}\,[f_{s}(k^{\prime})-f_{s}(k)]$
$\displaystyle=w_{s\to
s^{\prime}}\,\frac{m_{s}}{2\pi\hbar^{2}}\,\frac{f_{s}(-k)-f_{s}(k)}{|k|}$
$\displaystyle\equiv\eta^{\mathrm{dis}}_{s}\,\frac{f_{s}(-k)-f_{s}(k)}{|k|},$
(196) $\displaystyle\widehat{I}^{\widehat{f}_{t}}_{t\to t}$
$\displaystyle=\int\frac{dk^{\prime}}{2\pi}\,w_{t\to
t^{\prime}}\,\delta\bigg{(}\frac{\hbar^{2}(k^{\prime})^{2}}{2m_{t}}-\frac{\hbar^{2}k^{2}}{2m_{t}}\bigg{)}\,\Big{[}\widehat{f}_{t}(k^{\prime})-\widehat{f}_{t}(k)\Big{]}$
$\displaystyle=w_{t\to
t^{\prime}}\,\frac{m_{t}}{2\pi\hbar^{2}}\,\frac{\widehat{f}_{t}(-k)-\widehat{f}_{t}(k)}{|k|}$
$\displaystyle\equiv\eta^{\mathrm{dis}}_{t}\,\frac{\widehat{f}_{t}(-k)-\widehat{f}_{t}(k)}{|k|}.$
(197)

Note that the singlet-triplet energy difference $\Delta\epsilon$ and the
magnetic energy of triplet excitons cancel.

(c) Fusion is discussion in Sec. III.6. From the point of view of the singlet
excitons, fusion is described by

$\displaystyle I^{f_{s}}_{2t\to s}(k)$
$\displaystyle=\int\frac{dk^{\prime}}{2\pi}\,\frac{w_{2t\to
s}}{2}\,\delta\bigg{(}\frac{\hbar^{2}(k^{\prime})^{2}}{2m_{t}}+\frac{\hbar^{2}(k-k^{\prime})^{2}}{2m_{t}}-\Delta\epsilon-\frac{\hbar^{2}k^{2}}{2m_{s}}\bigg{)}\operatorname{Tr}\big{\\{}\widehat{P}_{s},\widehat{f}_{t}(k^{\prime})\otimes\widehat{f}_{t}(k-k^{\prime})\big{\\}}\,[1+f_{s}(k)]$
$\displaystyle=w_{2t\to
s}\,\frac{m_{t}}{4\pi\hbar^{2}}\,\frac{1}{\kappa}\,[1+f_{s}(k)]\bigg{(}\operatorname{Tr}\widehat{P}_{s}\bigg{[}\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\otimes\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}+\operatorname{Tr}\widehat{P}_{s}\bigg{[}\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\otimes\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\bigg{]}\bigg{)}$
$\displaystyle\equiv
w_{\mathrm{fusion}}\,\frac{1}{\kappa}\,[1+f_{s}(k)]\bigg{(}\operatorname{Tr}\widehat{P}_{s}\bigg{[}\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\otimes\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}+\operatorname{Tr}\widehat{P}_{s}\bigg{[}\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\otimes\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\bigg{]}\bigg{)},$
(198)

with

$\kappa\equiv\sqrt{\textstyle
k_{0}^{2}+\frac{m_{t}}{2m_{s}}\,k^{2}-\frac{k^{2}}{4}}=\sqrt{\textstyle
k_{0}^{2}+\frac{\alpha}{2}\,k^{2}-\frac{k^{2}}{4}},$ (199)

if $\kappa$ is real, otherwise the scattering integral vanishes. Note that the
magnetic energy of triplet excitons cancels. Introducing the unitary part of
the antiunitary time-reversal operator,

$\widehat{U}_{T}=e^{i\widehat{S}_{2}\pi}=\begin{pmatrix}0&0&1\\\ 0&-1&0\\\
1&0&0\end{pmatrix}=\widehat{U}_{T}^{\dagger},$ (200)

we can simplify the expression to

$I^{f_{s}}_{2t\to
s}(k)=\frac{2}{3}\,w_{\mathrm{fusion}}\,\frac{1}{\kappa}\,[1+f_{s}(k)]\operatorname{Tr}\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\widehat{U}_{T}\widehat{f}_{t}^{\,T}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\widehat{U}_{T}^{\dagger}.$
(201)

The corresponding scattering integral for the triplet excitons reads as

$\displaystyle\widehat{I}^{\widehat{f}_{t}}_{2t\to s}$
$\displaystyle=-\int\frac{dk^{\prime}}{2\pi}\,w_{2t\to
s}\,\delta\bigg{(}\frac{\hbar^{2}k^{2}}{2m_{t}}+\frac{\hbar^{2}(k^{\prime})^{2}}{2m_{t}}-\Delta\epsilon-\frac{\hbar^{2}(k+k^{\prime})^{2}}{2m_{s}}\bigg{)}\operatorname{Tr}_{t^{\prime}}\Big{\\{}\widehat{P}_{s},\widehat{f}_{t}(k)\otimes\widehat{f}_{t}(k^{\prime})\Big{\\}}\,[1+f_{s}(k+k^{\prime})]$
$\displaystyle=-\frac{2w_{\mathrm{fusion}}}{|1-\alpha|}\,\frac{1}{\kappa^{\prime}}\,\bigg{(}\operatorname{Tr}_{t^{\prime}}\bigg{\\{}\widehat{P}_{s},\widehat{f}_{t}(k)\otimes\widehat{f}_{t}\bigg{(}\frac{\alpha}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\bigg{\\}}\bigg{[}1+f_{s}\bigg{(}\frac{1}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\bigg{]}$
$\displaystyle\quad{}+\operatorname{Tr}_{t^{\prime}}\bigg{\\{}\widehat{P}_{s},\widehat{f}_{t}(k)\otimes\widehat{f}_{t}\bigg{(}\frac{\alpha}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\bigg{\\}}\bigg{[}1+f_{s}\bigg{(}\frac{1}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\bigg{]}\bigg{)},$
(202)

with

$\kappa^{\prime}\equiv\sqrt{\textstyle\frac{2}{1-\alpha}\,\Big{(}k_{0}^{2}+\frac{\alpha-1/2}{1-\alpha}\,k^{2}\Big{)}},$
(203)

if $\kappa^{\prime}$ is real, otherwise the scattering integral vanishes. With
some algebra, the expression can be rewritten as

$\displaystyle\widehat{I}^{\widehat{f}_{t}}_{2t\to s}$
$\displaystyle=-\frac{2}{3}\,\frac{w_{\mathrm{fusion}}}{|1-\alpha|}\,\frac{1}{\kappa^{\prime}}\,\bigg{(}\bigg{[}\widehat{f}_{t}(k)\widehat{U}_{T}\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\widehat{U}_{T}^{\dagger}+\widehat{U}_{T}\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\widehat{U}_{T}^{\dagger}\widehat{f}_{t}(k)\bigg{]}\bigg{[}1+f_{s}\bigg{(}\frac{1}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\bigg{]}$
$\displaystyle\quad{}+\bigg{[}\widehat{f}_{t}(k)\widehat{U}_{T}\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\widehat{U}_{T}^{\dagger}+\widehat{U}_{T}\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\widehat{U}_{T}^{\dagger}\widehat{f}_{t}(k)\bigg{]}\bigg{[}1+f_{s}\bigg{(}\frac{1}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\bigg{]}\bigg{)},$
(204)

which only contains $3\times 3$ instead of $9\times 9$ matrices.

(d) Fission is also discussed in Sec. III.6. From the point of view of the
singlet excitons, we have

$\displaystyle I^{f_{s}}_{s\to 2t}$
$\displaystyle=-\int\frac{dk^{\prime}}{2\pi}\,\frac{w_{s\to
2t}}{2}\,\delta\bigg{(}\frac{\hbar^{2}(k^{\prime})^{2}}{2m_{t}}+\frac{\hbar^{2}(k-k^{\prime})^{2}}{2m_{t}}-\Delta\epsilon-\frac{\hbar^{2}k^{2}}{2m_{s}}\bigg{)}\operatorname{Tr}\Big{\\{}\widehat{P}_{s},\big{[}\mathds{1}_{3}+\widehat{f}_{t}(k^{\prime})\big{]}\otimes\big{[}\mathds{1}_{3}+\widehat{f}_{t}(k-k^{\prime})\big{]}\Big{\\}}\,f_{s}(k)$
$\displaystyle=-w_{s\to
2t}\,\frac{m_{t}}{4\pi\hbar^{2}}\,\frac{1}{\kappa}\,f_{s}(k)\,\bigg{[}\operatorname{Tr}\widehat{P}_{s}\bigg{(}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\bigg{]}\otimes\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}\bigg{)}$
$\displaystyle\quad{}+\operatorname{Tr}\widehat{P}_{s}\bigg{(}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}\otimes\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\bigg{]}\bigg{)}\bigg{]}$
$\displaystyle\equiv-
w_{\mathrm{fission}}\frac{1}{\kappa}\,f_{s}(k)\,\bigg{[}\operatorname{Tr}\widehat{P}_{s}\bigg{(}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\otimes\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}\bigg{)}\bigg{]}$
$\displaystyle\quad{}+\operatorname{Tr}\widehat{P}_{s}\bigg{(}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}\otimes\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\bigg{]}\bigg{)}\bigg{]},$
(205)

with $\kappa$ given by Eq. (199), if $\kappa$ is real and zero otherwise. The
expression can be simplified to

$I^{f_{s}}_{s\to
2t}=-\frac{2}{3}\,w_{\mathrm{fission}}\,\frac{1}{\kappa}\,f_{s}(k)\,\operatorname{Tr}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}\bigg{(}\frac{k}{2}+\kappa\bigg{)}\bigg{]}\widehat{U}_{T}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}^{\,T}\bigg{(}\frac{k}{2}-\kappa\bigg{)}\bigg{]}\widehat{U}_{T}^{\dagger}.$
(206)

Analogously, we find

$\displaystyle\widehat{I}^{\widehat{f}_{t}}_{s\to 2t}$
$\displaystyle=\frac{2}{3}\,\frac{w_{\mathrm{fission}}}{|1-\alpha|}\,\frac{1}{\kappa^{\prime}}\,\bigg{[}\bigg{(}\big{[}\mathds{1}_{3}+\widehat{f}_{t}(k)\big{]}\widehat{U}_{T}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\bigg{]}\widehat{U}_{T}^{\dagger}+\hat{U}_{T}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}\bigg{]}\widehat{U}_{T}^{\dagger}\big{[}\mathds{1}_{3}+\widehat{f}_{t}(k)\big{]}\bigg{)}$
$\displaystyle\qquad{}\times
f_{s}\bigg{(}\frac{1}{1-\alpha}\,k+\kappa^{\prime}\bigg{)}$
$\displaystyle\quad{}+\bigg{(}\big{[}\mathds{1}_{3}+\widehat{f}_{t}(k)\big{]}\widehat{U}_{T}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\bigg{]}\widehat{U}_{T}^{\dagger}+\hat{U}_{T}\bigg{[}\mathds{1}_{3}+\widehat{f}_{t}^{\,T}\bigg{(}\frac{\alpha}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\bigg{]}\widehat{U}_{T}^{\dagger}\big{[}\mathds{1}_{3}+\widehat{f}_{t}(k)\big{]}\bigg{)}$
$\displaystyle\qquad{}\times
f_{s}\bigg{(}\frac{1}{1-\alpha}\,k-\kappa^{\prime}\bigg{)}\bigg{]},$ (207)

with $\kappa^{\prime}$ given by Eq. (203), if $\kappa^{\prime}$ is real and
zero otherwise.

One can now expand $\widehat{f}_{t}$ into multipoles using Eq. (48). This is
particularly useful if some of the multipoles vanish by symmetry. In any case,
the equations to be solved are coupled first-order nonlinear partial
differential equations for $f_{s}$ and $\widehat{f}_{t}$ as functions of time
$t$ and position $x$. The momentum $k$ acts as a parameter labeling different
coupled functions. None of the coefficients depend on $x$ or $t$.
Nevertheless, due to the nonlinearity of the equations, an analytical solution
seems out of reach. We therefore use numerical forward propagation of the
initial functions with discrete time steps and also discretize the $x$ and $k$
dependence.

Figure 1: Spatial distribution function $\rho_{s}(x,t)$ for singlet excitons
for the case without fission. There are no triplet excitons in this case. The
parameters are $\alpha=0.8$, $k_{0}=4$, $\sigma_{k}=1$, $\sigma_{x}=1$,
$\eta_{s0}=2$, and all other rates are set to zero. The discretization
intervals for momentum, space, and time are $\Delta k=0.1$, $\Delta x=0.1$,
and $\Delta t=2.5\times 10^{-3}$.

(a)
(b)

Figure 2: Spatial distribution function for (a) singlet and (b) triplet
excitons as a function of time for the case with fission but without fusion.
The parameters are $\eta_{t}^{\mathrm{dis}}=6$ and $w_{s\to 2t}=10$, the other
parameters are the same as in Fig. 1.

(a)
(b)

Figure 3: Spatial distribution function for (a) singlet and (b) triplet
excitons as a function of time for the case with fission and strong fusion.
The fusion rate is $w_{2t\to s}=400$, the other parameters are the same as in
Fig. 2.

As an example, we consider a localized distribution of singlet excitons,
photogenerated close to the energy threshold. The distribution is modeled by a
Gaussian function with mean zero and width $\sigma_{x}$ in real space
multiplied by a Gaussian function with mean zero and width $\sigma_{k}$ in
momentum space. To reflect some of the features of real systems, the singlet
excitons are assumed to decay rapidly. We consider the singlet-excition
distribution function in real space,

$\rho_{s}(x,t)=\int\frac{dk}{2\pi}\,f_{s}(k,x,t),$ (208)

and the distribution function of the occupation of triplet excitons in real
space,

$\rho_{t}(x,t)=\int\frac{dk}{2\pi}\,\operatorname{Tr}\widehat{f}_{t}(k,x,t).$
(209)

For reference, we plot the time evolution of the singlet-exciton distribution
function $\rho_{s}$ in Fig. 1 for the situation without fission. In this case,
the singlets decay rapidly and triplet excitons are not generated. Note that
the densities $\rho_{s}$ and $\rho_{t}$ in all plots are presented in
arbitrary units but on the same color scale.

If fission (but not fusion) is switched on the singlet excitons quickly
transition into triplet excitatons, as shown in Fig. 2(a). The triplet
excitons show predominant diffusive motion due to disorder scattering, see
Fig. 2(b). The cone from ballistically moving, i.e., not scattered, excitons
is visible as a weaker feature.

If we now also switch on fusion with a high bare rate we observe recovery of
singlet excitons and loss of triplet excitons, as expected, see Fig. 3.
However, the distribution of recreated singlet excitons is much narrower than
the one of the triplet excitons. We attribute this to the fact that for our
parameters two triplet excitons must collide head on in order to fuse, which
is more likely in the center of the triplet cloud. It is an interesting
question for the future to what extent such an effect is also present in real
three-dimensional systems.

(a)
(b)

Figure 4: Spatial distribution function of (a) the occupation number of
triplet excitons and (b) the x component of their spin polarization as
functions of time in an applied magnetic field in the z direction. The initial
distribution of triplet excitons has helical spin polarization with the spin
parallel (antiparallel) to $x$ for $k>0$ ($k<0$). The dimensionsless Larmor
frequency is $\omega_{L}=10$, the other parameters are the same as in Fig. 3.

In the previous examples, the spin of the triplet excitons does not matter.
The initial singlet distribution is of course nonmagnetic and there is no
mechanism that generates any spin polarization. To exhibit a characteristic