  Many-body electron-hole complexes in a semiconductor are important both from
a fundamental physics point of view and for practical device applications. A
three-body system of electrons (e) and holes (h) (2e1h, or 1e2h) in a two-band
semiconductor is commonly believed to be associated with two spectral peaks for
the exciton and trion (or charged exciton), respectively. But both the validity
of this understanding and the physical meaning of a trion or charged exciton
have not been thoroughly examined. From the physics point of view, there are
two different configurations, (e)(eh) or (eeh), which could be considered
charged exciton and trion, respectively. Here (...) represents an irreducible
cluster with respect to Coulomb interactions. In this paper, we consider these
issues related to the 2e1h three-body problem theoretically and experimentally
using monolayer MoTe2 as an example. Our theoretical tools involve the
three-body Bethe-Salpeter Equation (BSE) and the cluster expansion technique,
especially their correspondence. Experimentally, we measure the
photoluminescence spectrum on a gate-controlled monolayer MoTe2. We found two
spectral peaks that are 21 and 4 meV, respectively, below the exciton peak, in
contrast to the single "trion" peak from the conventional understanding. We
show that, while the three-body BSE in a two-band model can reproduce all
spectral features, the cluster-expansion technique shows that the two peaks
correspond to the charged exciton (e)(eh) and trion (eeh), respectively. In
other words, there is a spectral splitting due to the two different many-body
configurations. Furthermore, we find that the trion only exists in the
intervalley case, while the charged exciton exists both for the intervalley and
intravalley cases.
