magnetic effect, we consider an initial state with a distribution of _triplet_
excitons with momenta close to $\pm k_{0}$ and helical spin polarization: the
spin is polarized parallel (antiparallel) to $x$ for $k>0$ ($k<0$). Except for
the spin polarization, this distribution is similar to the one obtained by
singlet fission in the previous examples. The occupation number $\rho_{t}$ of
triplet excitons, shown in Fig. 4(a), is independent of the applied magnetic
field. The x component of the spin polarization given by

$s_{t}^{x}(x,t)=\int\frac{dk}{2\pi}\,\frac{1}{2}\operatorname{Tr}\widehat{S}_{1}\widehat{f}_{t}(k,x,t)$
(210)

and plotted in Fig. 4(b) shows two effects: First, due to the helical spin
polarization of the initial distribution, triplet excitons moving to the right
(left) have positive (negative) spin. Second, their spin precesses about the
magnetic field, i.e., the z direction, with a frequency given by the Larmor
frequency $\omega_{L}$ in Eq. (193).

##  V Summary and conclusions

The understanding of transport of charge carriers and excitons in organic
semiconductors is crucial for applications. Semiclassical Boltzmann theory is
a valuable tool for this since it is formulated in terms of physically
intuitive quantities and processes, thereby simplifying the interpretation. In
this paper, we have addressed the spin-conserving charge transport and
scattering in a semiclassical framework, including the creation, decay,
recombination, fission, and fusion of excitons, which are crucial for organic
solar cells. However, our results are of more general interest for any system
showing scattering of and reactions between spin-carrying particles, for
example in spintronics and spin chemistry.

To start with, we have set up kinetic equations for charge carriers as well as
for singlet and triplet excitons in uniform electric and magnetic fields,
taking care to preserve $\mathrm{SU}(2)$ spin-rotation invariance. The case of
triplet excitons is the most interesting in that these spin-$1$ quasiparticles
permit and require a _quadrupolar_ contribution to their distribution
function. We have obtained coupled equations for the concentration, spin-
density, and quadrupole-density components. The generalization to nonuniform
systems is conceptually straightforward.

In the second step, we have constructed $\mathrm{SU}(2)$-invariant transition
terms or scattering integrals describing disorder scattering, singlet-exciton
generation, binding of electrons and holes into singlet and into triplet
excitons, and exciton fusion, as well as the reverse processes. The occupation
of final states enhances a process for bosons and suppresses it for fermions.
These quantum effects are naturally included and lead to higher-order terms in
the scattering integrals, which are relevant for large occupation numbers. It
proved useful to employ two different representations of distribution
functions, namely a multipole expansion and a description as matrix-valued
functions acting on the spin Hilbert space. For the simpler processes, the
multipole representation allows one to directly write down the
$\mathrm{SU}(2)$-invariant terms based on the transition terms under the
(symmetry-breaking) assumption of all moments being parallel or antiparallel
to the $z$ axis. The results can be used to describe real materials when
realistic models for the bare transition rates $W$ are available. Moreover,
our work serves as a guide on how to construct scattering integrals for
complex transitions involving several quasiparticles with spin.

Future directions following from this work are quite clear. First, it will be
necessary to derive models for the bare transition rates $W$ for real
materials and link the equations to observables, in order to obtain
quantitative descriptions. For the application to photovoltaic devices, it is
important to note that the generation of excitons takes place at internal
interfaces in organic blends, while the diffusion of (triplet) excitons is a
bulk effect. Our framework offers two approaches to this problem: On long
length scales relative to the typical scale of the phases in the blend, coarse
graining leads to an effective-medium description with an effective exciton-
generation rate. On the other hand, on shorter length scales—but still long
compared to the single-molecule scale—the spatial structure of the phases can
be modeled. The semiclassical approach is in principle suitable for this but
of course relies on a good structural model.

Second, additional processes are relevant in certain materials. For example,
bound states of two electrons and one hole or vice versa, i.e., trions, can
exist Lam58 ; Gla20 . The formation and decay of trions can easily be
described in our framework. Third, we have here assumed that polaronic
effects, which are strong in organic materials, can be treated simply be
renormalizing model parameters for charge carriers and excitons. This is
certainly simplistic, in particular due to the broad distribution of
timescales of phonons in organic materials. The connection of transport and
excitonic processes with polaronic effects in a semiclassical framework is
thus an important goal for the future.

###### Acknowledgements.

The authors thank A. Knoll, K. Leo, F. Ortmann, and S. Reineke for stimulating
discussions. C. T. acknowledges financial support by the Deutsche
Forschungsgemeinschaft through Collaborative Research Center SFB 1143, project
A4, and the Cluster of Excellence on Complexity and Topology in Quantum Matter
ct.qmat (EXC 2147).

##  Appendix A Kinetic equation for triplet excitons

In this Appendix, we derive kinetic equations for the concentration, spin, and
quadrupolar coefficients of the nonequilibrium triplet-exciton distribution
function in Eq. (56). We first introduce an alternative way to express the
quadrupole term that simplifies calculations. Using the Cartesian quadrupole
tensor operator defined by Eq. (50), the distribution function can be written
as

$\widehat{{f}}_{t}={n_{t}}\widehat{S}_{0}+\bm{c}_{t}\cdot\bm{\widehat{S}}+\bm{a}\cdot\underline{\widehat{Q}}\bm{b}.$
(211)

The decomposition contains 10 real coefficients but has only nine degrees of
freedom, so we choose $|\bm{a}|=1$ as an additional condition. We call
relations such as Eq. (211) matrix expansions. Analogously, the deviation is
written as

$\widehat{g}_{t}=\tilde{n}_{t}\widehat{S}_{0}+\tilde{\bm{c}}_{t}\cdot\bm{\widehat{S}}+\tilde{\bm{a}}\cdot\underline{\widehat{Q}}\tilde{\bm{b}},$
(212)

with $|\tilde{\bm{a}}|=1$.

Next, we evaluate the Boltzmann equation (II.4) in the matrix expansion,

$\displaystyle\frac{\partial}{\partial
t}\,\tilde{n}_{t}\widehat{S}_{0}+\frac{\partial}{\partial
t}\,\tilde{c}_{ti}\widehat{S}_{i}+\frac{\partial}{\partial
t}\,\big{(}\tilde{{a}}_{m}\tilde{{b}}_{n}\big{)}\,\widehat{Q}_{mn}$
$\displaystyle\quad{}+\frac{1}{\hbar}\,\bm{\nabla}_{\bm{k}}\epsilon_{t}\cdot\left[\bm{\nabla}_{\bm{r}}\tilde{n}_{t}\widehat{S}_{0}+\bm{\nabla}_{\bm{r}}\tilde{c}_{ti}\widehat{S}_{i}+\bm{\nabla}_{\bm{r}}\big{(}\tilde{{a}}_{m}\tilde{{b}}_{n}\big{)}\,\widehat{Q}_{mn}\right]$
$\displaystyle\quad{}+\frac{g_{\mathrm{tr}}\mu_{B}}{2\hbar}\,\varepsilon_{ijk}\tilde{{b}}_{j}\tilde{c}_{tk}\widehat{S}_{i}+\frac{g_{\mathrm{tr}}\mu_{B}}{2\hbar}\left(\bm{B}\times\tilde{\bm{a}}\right)_{m}\tilde{{b}}_{n}\widehat{Q}_{mn}$
$\displaystyle\quad{}+\frac{g_{\mathrm{tr}}\mu_{B}}{2\hbar}\,\big{(}\bm{B}\times\tilde{\bm{b}}\big{)}_{m}\,\tilde{{a}}_{n}\widehat{Q}_{mn}$
$\displaystyle=\widehat{I}_{t}\big{[}f_{s},\widehat{{f}}_{e},\widehat{{f}}_{h},\widehat{{f}}_{t}\big{]},$
(213)

where $\widehat{Q}_{mn}$ denote the components in Eq. (50) and
$\varepsilon_{ijk}$ is the Levi-Civita symbol. We employ the Einstein
summation convention. Equations for the number, spin, and quadrupole densities
can be obtained using the relations

$\displaystyle\big{[}\widehat{S}_{a},\widehat{S}_{b}\big{]}$
$\displaystyle=i\varepsilon_{abm}\widehat{S}_{m},$ (214) $\displaystyle
2\,\mathds{1}_{3}$
$\displaystyle=\widehat{S}_{1}^{2}+\widehat{S}_{2}^{2}+\widehat{S}_{3}^{2},$
(215) $\displaystyle\big{[}\widehat{Q}_{ab},\widehat{S}_{c}\big{]}$
$\displaystyle=i\varepsilon_{acm}\widehat{Q}_{mb}+i\varepsilon_{bcm}\widehat{Q}_{ma},$
(216) $\displaystyle\widehat{Q}_{ab}\widehat{S}_{c}$
$\displaystyle=\frac{i}{2}\,\big{(}\varepsilon_{bcm}\widehat{Q}_{am}+\varepsilon_{acm}\widehat{Q}_{bm}\big{)}$
$\displaystyle\quad{}+\frac{1}{2}\,\big{(}\delta_{ac}\widehat{S}_{b}+\delta_{cb}\widehat{S}_{a}\big{)}-\frac{1}{3}\,\delta_{ab}\widehat{S}_{c},$
(217)

with $a,b,c,m\in\\{1,2,3\\}$. After the Boltzmann equation has been solved
with regard to Eq. (211), we can convert the quadrupole coefficients to the
form of the multipole expansion in Eq. (48) by the transformation

$\displaystyle{q_{t1}}$ $\displaystyle=a_{2}b_{3}+a_{3}b_{2},$ (218)
$\displaystyle{q_{t2}}$ $\displaystyle=a_{3}b_{1}+a_{1}b_{3},$ (219)
$\displaystyle{q_{t3}}$ $\displaystyle=a_{1}b_{2}+a_{2}b_{1},$ (220)
$\displaystyle{q_{t4}}$ $\displaystyle=\frac{1}{2}(a_{1}b_{1}-a_{2}b_{2}),$
(221) $\displaystyle{q_{t5}}$
$\displaystyle=-\frac{\sqrt{3}}{6}(a_{1}b_{1}+a_{2}b_{2}-2a_{3}b_{3}).$ (222)

Analogously to the density matrix, we expand the scattering term as

$\widehat{I}_{t}=I^{n_{t}}\,\widehat{S}_{0}+\bm{I}^{\bm{c}_{t}}\cdot\bm{\widehat{S}}+\vec{I}^{\,\vec{q}}\cdot\vec{{\widehat{Q}}}.$
(223)

Using the differential operator $\mathcal{D}_{t}$ defined in Eq. (57), we
obtain Eqs. (58)–(66).

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