# The rotating excitons in two-dimensional materials: Valley Zeeman effect and
chirality

Yu Cui  Tianjin Key Laboratory of Low Dimensional Materials Physics and
Preparing Technology, Department of Physics, School of Science, Tianjin
University, Tianjin 300354, China  Xin-Jun Ma  Research Team of Extreme
Condition Physics, College of Physics and Electronic Information, Inner
Mongolia Minzu University, Tongliao 028043, China  Jia-Pei Deng  Tianjin Key
Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, School of Science, Tianjin University, Tianjin 300354,
China  Shao-Juan Li  State Key Laboratory of Luminescence and Applications,
Changchun Institute of Optics, Fine Mechanics and Physics Chinese Academy of
Sciences, Changchun 130033, China  Ran-Bo Yang  Tianjin Key Laboratory of Low
Dimensional Materials Physics and Preparing Technology, Department of Physics,
School of Science, Tianjin University, Tianjin 300354, China  Zhi-Qing Li
Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing
Technology, Department of Physics, School of Science, Tianjin University,
Tianjin 300354, China  Zi-Wu Wang<EMAIL_ADDRESS>Tianjin Key Laboratory
of Low Dimensional Materials Physics and Preparing Technology, Department of
Physics, School of Science, Tianjin University, Tianjin 300354, China

###### Abstract

We propose the rotational dynamics of the intralayer and interlayer excitons
with their inherent momenta of inertia in the monolayer and bilayer transition
metal dichalcogenides, respectively, where the new chirality of exciton is
endowed by the rotational angular momentum, namely, the formations of left-
and right-handed excitons at the +K and -K valleys, respectively. We find that
angular momenta exchange between excitons and its surrounding phononic bath
result in the large fluctuation of the effective g-factor and the asymmetry of
valley Zeeman splitting observed in most recently experiments, both of which
sensitively depend on the magnetic moments provided by the phononic
environment. This rotating exciton model not only proposes a new controllable
knob in valleytronics, but opens the door to explore the angular momentum
exchange of the chiral quasiparticles with the many-body environment.

$Introduction.$—–A series of extraordinary optical and electric properties in
transition metal dichalcogenides (TMDs) are dominated by excitons in different
spin, valley and layer configurationsM. A. Green (2014, 2014, 2014, 2014,
2014, 2014). In particular, both intralayer excitons in monolayer and
interlayer excitons in bilayer structures are endowed with a valley degree of
freedom (valley pseudospin) at two inequivalent but energy-degenerate $\pm$K
valleysM. A. Green (2014, 2014). This valley pseudospin, like real spin, is
associated with valley magnetic moment, giving rise to exciton Zeeman
splitting in the presence of the external magnetic fieldM. A. Green (2014,
2014, 2014, 2014), which provides not only an attracted method of breaking the
valley degeneracy, but a powerful lever to exploit the fundamental physical
properties of the valley states, as well as for the development of new
approaches to valleytronic controlM. A. Green (2014, 2014).

Figure 1:  (a) Schematic diagrams for the left- and right-handed rotating
intralayer excitons in monolayer transition metal dichalcogenides (TMDs) at
the +K and -K valleys, respectively, where an out-of-plane rotational angular
momentum ${\hat{\bf{L}}_{z}}$ with opposite signs is formed. For a rotating
exciton, the electron and hole induce the opposite magnetic moments, but they
can’t cancel each other out due to the difference of effective masses between
them. Thus, a finite magnetic moment is produced, which could be equivalently
regarded as an effective charge $e^{*}=e(m_{h}-m_{e})/(m_{h}+m_{e})$ rotating
around a fixed axis with the radius $r$ (the relations between this effective
charge and the valley Zeeman term are given in supplemental materials Part
II). Similar to intralayer excitons, two rotating interlayer excitons in
bilayer TMDs are schemed in (b). $D$ is the interlayer distance between two
hosting layers.

The magnitude of this valley Zeeman splitting can be characterized by the
effective g-factor ($\textsl{g}_{eff}$) of exciton, whose predicted value is
-4 (or +4) including the contribution from the intra- and intercellular
orbital magnetic moments of the band structuresM. A. Green (2014, 2014).
However, there exists obviously disagreements between the theoretical
prediction and the experimental measurement. To explain this discrepancy,
several microscopic models have been proposed, e.g., the interplays among the
magnetic moment of the transition metal $d$${}_{x^{2}-y^{2}}$$\pm i$$d$xy
orbitalsM. A. Green (2014, 2014), the valley magnetic moment and the lattice
contribution stemming from Berry curvature should be consideredM. A. Green
(2014, 2014, 2014); the strain induces a hybridization of the direct and
indirect excitons, giving rise to the renormalization of $\textsl{g}_{eff}$E.
Yakaboylu (2017); instead of the local approximation, the distinct reduction
of $\textsl{g}_{eff}$ could be obtained when full Bloch states are included
based on the first-principles calculation in the Bethe-Salpeter equationE.
Yakaboylu (2017). For the larger $\textsl{g}_{eff}$ of interlayer exciton, the
mechanism that the brightening of forbidden optical transitions induced by the
moiré potential has also been suggestedM. A. Green (2014). Nevertheless, the
accuracy of the developed models is rather poor comparing with the
experimental data. The underlying physics for this factor remains a subject of
hot debate.

In this letter, we investigate the rotational motion of the intralayer and
interlayer excitons with their inherent momenta of inertia in monolayer and
bilayer TMDs, respectively. This rotational degree of freedom gives rise to an
additional orbital angular momentum in the out-of-plane due to the difference
of effective masses between electron and hole, adding a finite magnetic moment
that enhances the exciton Zeeman splitting upon interaction with an external
magnetic field. This picture enables us, on the one hand, to define the new
chirality of the exciton by the left- and right-handed rotations at the +K and
-K valleys, respectively, as shown in Fig. 1; on the other hand, to explore
angular momenta transfer between excitons and phononic bath. We find that
these values of $\textsl{g}_{eff}$ predicted by this model are in excellent
agreement with experimental measurements for both intralayer and interlayer
excitons. More importantly, the clockwise and anti-clockwise motions of the
phononic bath could be used to reveal an unexplained puzzle that the asymmetry
of valley splitting.

$The$ $Hamiltonian$ $of$ $the$ $rotating$ $exciton.$—–We propose an exciton as
an electric dipole with the rotational motion before its recombination schemed
in Fig. 1. For an exciton in two-dimensional material, its rotational degree
of freedom confers upon an out-of-plane orbital angular momentum with opposite
signs depending on the rotational directionsE. Yakaboylu (2017, 2017); M. A.
Green (2014). Meanwhile, the rotating exciton is immersed into the phononic
bath, leading to the exchange of angular momenta between them. In the presence
of an external magnetic field, the total Hamiltonian is given byR. Schmidt
(2015, 2016); E. Yakaboylu (2017, 2017)

$\displaystyle\mathcal{\hat{H}}=\hbar\xi_{0}{{\hat{\bf{L}}_{z}^{2}}}+{\mu_{B}}{\textsl{g}_{\kappa}}\hat{\bf{B}}\cdot{\hat{\bf{L}}_{z}}-{\mu_{B}}{\textsl{g}_{0}}\hat{\bf{B}}\cdot{\hat{\bf{L}}^{0}_{z}}$
$\displaystyle+\sum\limits_{q\lambda}{\overline{\hbar{\omega}}_{\nu}}\hat{b}_{q\lambda}^{\dagger}{\hat{b}_{q\lambda}}+\sum\limits_{q\lambda}{{\mathrm{V}}}_{\lambda}(q,r)\left[{{e^{i\lambda\hat{\varphi}}}{{\hat{b}}_{q\lambda}}+{e^{-i\lambda\hat{\varphi}}}\hat{b}_{q\lambda}^{\dagger}}\right],$

where the first term is the kinetic energy of a rotating exciton with the
rotational constant $\xi_{0}=\hbar/({{2I}})$, $I=\eta r^{2}$ is the moment of
inertia depending on the reduced mass $\eta^{-1}=m_{e}^{-1}+m_{h}^{-1}$ and
the relative distance $r$ between the electron and the hole; $m_{e}$ ($m_{h}$)
is the electron (hole) effective mass;
${\hat{L}_{z}}=-i\partial/\partial\varphi$ is the angular momentum operator
with the eigenvalues $l_{z}=0,\pm 1,\pm 2\ldots$ and eigenenergies
$E_{l_{z}}=\hbar\xi_{0}l_{z}^{2}$E. Yakaboylu (2017), where the sign $\pm$
represents two opposite directions of this angular momentum in the out-of-
plane of two-dimensional materials, corresponding to the left- and right-
handed excitons at the +K and -K, respectively, and representing a new
chirality of exciton, which is also corroborate with the formation of valley
excitons at the +K and -K valleys upon absorption of left- and right-handed
circularly polarized lightM. A. Green (2014, 2014). Just this angular momentum
endows a magnetic moment that results in the interaction between the rotating
exciton and the external magnetic field, thus giving rise to the additional
valley Zeeman effect, which can be described by the second term
${\mu_{B}}{\textsl{g}_{\kappa}}\hat{\bf{B}}\cdot{\hat{\bf{L}}_{z}}$ (the
detailed derivation for it in supplemental materials Part I) with
$\mu_{B}=e\hbar/(2m_{0}c)$ is the Bohr magneton and
$\textsl{g}_{\kappa}={m_{0}}({m_{h}}-{m_{e}})/(m_{e}m_{h})$ is the
gyromagnetic ratio for the rotating exciton ($m_{0}$ is the free electron
mass) and $\hat{\bf{B}}$ is the vector of magnetic field. In other word, the
magnetic moment of the rotating exciton is equivalent to a single particle
rotating around the fixed axis with the radius $r$ and the effective charge
$e^{*}=e(m_{h}-m_{e})/(m_{h}+m_{e})$; see the supplemental materials Part II
for details. We will see, in fact, this magnetic moment is very tiny for
intralayer exciton because of the small difference of effective masses between
electron and hole in TMDs. For interlayer exciton, however, it plays the key
role to modulate the valley Zeeman splitting because of the tunability of the
ratio of effective masses between electron and hole, hosting two dividual
layers, respectively. The third term represents the intrinsic exciton valley
splitting with the Lande factor $\textsl{g}_{0}=1$, which mainly attributes to
the contribution of $d$-orbital of transitional-metal atom with the magnetic
quantum $\mathfrak{L}_{z}^{0}=+2$ at the +K valley and
$\mathfrak{L}_{z}^{0}=-2$ at the -K valley, yield the ideal values
$\textsl{g}_{eff}^{0}=-4$, which has been predicted by several theoretical
models and proved by some experimentsM. A. Green (2014, 2014). The fourth term
describes the kinetic energy of the phononic bath with
$\overline{\hbar{\omega}}_{\nu}=\sqrt{\hbar\omega_{\nu}^{j}\hbar\omega_{\nu}^{j^{\prime}}}$
being the phonon energy depending on the hosting layer index $j$
$(j^{\prime})$ for hole (electron). For the intralayer exciton, the same layer
index $j=j^{\prime}$ is fulfilled. For the interlayer exciton in bilayer TMDs,
the electron and hole reside in different layers, so $j=j^{\prime}$ and $j\neq
j^{\prime}$ correspond to the homobilayer and heterobilayer structures,
respectively. Here ${q}=|{\bf q}|$ is the scalar representation of the phonon
wave vector, satisfying the relation $\sum\nolimits_{q}{\equiv\int{dq}}$.
$\lambda$ represents the phonon angular momentum. The corresponding creation
and annihilation operators, $\hat{b}_{\bf{q}}^{\dagger}$ and
${\hat{b}}_{\bf{q}}$, are expressed in polar coordinates,
$\hat{b}_{q\lambda}^{\dagger}$ and ${\hat{b}_{q\lambda}}$M. A. Green (2014);
E. Yakaboylu (2017). The last term describes the interaction between the
rotating exciton and phonons. The angular momentum-dependent coupling strength
$\mathrm{V}_{\lambda}(q,r)$ is given by

$\displaystyle{{V}}_{\lambda}(q,r)=\sqrt{\frac{q}{{2\pi}}}\left[\mathcal{M}_{h}^{j}{{J_{\lambda}}\left({-{\beta_{1}}qr}\right)-\mathcal{M}_{e}^{j^{\prime}}{J_{\lambda}}\left({{\beta_{2}}qr}\right)}\right],$
(2)

depending on the microscopic details of two-body interaction
${\mathcal{M}^{j(j^{\prime})}_{h(e)}}=[{{{e^{2}}\alpha
Z_{0}\hbar{\omega^{j(j^{\prime})}_{\nu}}}/({{2\mathbb{A}{\varepsilon_{0}}}})}]^{1/2}$
between hole (electron) and the $\nu$th branch of phonon modeM. A. Green
(2014, 2014), where $Z_{0}$ is the monolayer thickness, $\mathbb{A}$ is the
quantization area in the monolayer materials, and $\varepsilon_{0}$ is the
permittivity of vacuum. $\alpha$ denotes the strength of the rotating exciton
coupled with its surrounding phononic bath and is regarded as a changeable
parameter in the large scale, because of the following reasons: (i) different
branches of phonon modes will give the coupling contribution together; (ii)
the strength could be tuned significantly by the dielectric environment of the
monolayer and bilayer structures, e.g., the various choices of the substrate
and encapsulation materials, and the structural parameters, e.g., the internal
distances between the hosting monolayer and the substratesM. A. Green (2014,
2014); E. Yakaboylu (2017); (iii) the moiré potential induced by stacking
angle not only gives the new phonon modes, but enhances the exciton-phonon
coupling remarkably due to the strong quantum confinement effectE. Yakaboylu
(2017, 2017). ${J_{\lambda}}(-{\beta_{1}}qr)$ and
${J_{\lambda}}({\beta_{2}}qr)$ are the Bessel functions of the first kind,
where $\beta_{1}=m_{h}/m$ and $\beta_{2}=m_{e}/m$ for hole and electron,
respectively ($m$ is the total mass.).

To show the significant influence of phonon angular momenta on the rotating
particle, a variational ansatz for an exciton rotating in the phononic bath is
introduced based on the angulon model developed by Schmidt and LemeshkoR.
Schmidt (2015); M. A. Green (2014):

$\displaystyle\left|\Psi\right\rangle=\sqrt{C}\left|j_{z}\right\rangle\left|0\right\rangle+\sum\limits_{q\lambda}{{\beta_{q\lambda}}}\left|{j_{z}-\lambda}\right\rangle\hat{b}_{q\lambda}^{\dagger}\left|0\right\rangle,$
(3)

where ${\hat{J}_{z}}={\hat{L}_{z}}+{\hat{\Lambda}_{z}}$ is the total angular
momentum operator of the system with the eigenvalues of $j_{z}=0,\pm 1,\pm
2,\ldots$, and ${\hat{\Lambda}_{z}}$ is the collective angular momentum
operator of the phononic bath. $\sqrt{C}$ and ${\beta_{q\lambda}}$ are the
variational parameters with the normalization condition
$\left|C\right|+\sum\nolimits_{q\lambda}{{{\left|{{\beta_{q\lambda}}}\right|}^{2}}}=1$,
and $\left|0\right\rangle$ represents the vacuum of phonons. Here, the angular
momentum conservation satisfies $l_{z}+\lambda=j_{z}$. By minimizing of the
function $\left\langle\Psi\right|\mathcal{\hat{H}}-E\left|\Psi\right\rangle$
with respect to the parameters $\sqrt{C}^{*}$ and ${\beta^{*}_{q\lambda}}$, we
get the valley-dependence eigenenergies of the system (the detailed
mathematical processes are given in the supplemental materials Part III)

$\displaystyle E_{\pm\mathrm{K}}$ $\displaystyle=$
$\displaystyle{\hbar\xi_{0}}{l_{z}^{2}}+{\mu_{B}}\textsl{g}^{*}_{\kappa}Bl_{z}^{\pm\mathrm{K}}-{\mu_{B}}\textsl{g}_{0}B\mathfrak{L}_{z}^{0,{\pm\mathrm{K}}}$
(4)
$\displaystyle+{{\overline{\hbar\omega}}_{\nu}}-\sum\nolimits^{(1)}_{l_{z}}{(E)},$

with

$\displaystyle{\rm{g}}_{\kappa}^{*}=\left[1+\sum\limits_{q\lambda}{\frac{{{\lambda{\left|{{V_{\lambda}}(q,r)}\right|}^{2}}}}{{{l_{z}}{{\left({\hbar{\xi_{0}}j_{z}^{2}-{{\overline{\hbar\omega}}_{\nu}}-\hbar{\xi_{0}}l_{z}^{2}}\right)}^{2}}}}}\right]{\rm{g}}_{\kappa},$
$\displaystyle\sum\nolimits^{(1)}_{l_{z}}{(E)}=\sum\limits_{q\lambda}{\frac{{{{\left|{{V_{\lambda}}(q,r)}\right|}^{2}}}}{{{\hbar\xi_{0}}{j_{z}^{2}}-{{\overline{\hbar\omega}}_{\nu}}-{\hbar\xi_{0}}{l_{z}^{2}}}}}.$

Obviously, the valley Zeeman splitting is determined by the second and third
terms in Eq. (4), where the energy level diagram showing the contribution to
the splitting is schemed in Fig. S1 in supplemental materialsE. Yakaboylu
(2017). The magnitude of the energy splitting with valley dependence could be
rewritten as
$E^{S}_{\pm\mathrm{K}}={\mu_{B}}\textsl{g}^{*}_{\kappa}Bl_{z}^{\pm\mathrm{K}}-{\mu_{B}}\textsl{g}_{0}B\mathfrak{L}_{z}^{0,{\pm\mathrm{K}}}$.
In the following sections, the lowest rotational states
$l_{z}^{+\mathrm{K}}=+1$ ($l_{z}^{-\mathrm{K}}=-1$) for the +K (-K) valley is
adopted. The impact of phonon magnetic moment on the valley splitting is
reflected by the renormalization of ${\rm{g}}_{\kappa}^{*}$ in the second
term. Consequently, the effective ${\textsl{g}}-$factor of exciton could be
obtained by
${\textsl{g}}_{eff}=(E^{S}_{+\mathrm{K}}-E^{S}_{-\mathrm{K}})/(\mu_{B}B)$. The
last term is the self-energy of the rotating exciton due to the angulon
effect, which, in fact, only gives the tiny contribution to the valley
splitting, but plays the crucial role in inducing the fine structure of the
spectroscopy for the rotating particles in the many-body environment. Here,
the first-order approximation of this term is adopted (see the supplemental
materials Part III).

We mainly discuss the renormalization of ${\textsl{g}}-$factors for the
rotating intralayer and interlayer excitons in four typical monolayer TMDs and
their bilayers structures, where phonon angular momenta $\lambda=-1$ and
$\lambda=+1$ are selected in the calculation processesE. Yakaboylu (2017),
representing the clockwise and anti-clockwise rotations of the phononic bath
respecting to the motion of exciton, respectively. The adopted values of other
parameters for these materials are shown in Table S1 and Table S2 in the
supplemental materialsE. Yakaboylu (2017). In addition, we only consider the
A-type excitons in this work and assume that B-type excitons have the similar
behavior.

$Exciton$ $\textsl{g}_{eff}$ $in$ $monolayer$ TMDs.—– Fig. 2(a) shows the
Zeeman splitting of intralayer excitons with the magnetic field for four well-
known monolayer TMDs, where the precondition is assumed that the orientation
of the phonon angular moment is always keeping opposite respecting to the
right- and left-handed rotating excitons at the +K and -K valleys,
respectively. Comparing to the normal valley Zeeman splitting (purple solid
lines for $\textsl{g}_{eff}=-4$)M. A. Green (2014, 2014), the slopes of energy
shift decrease obviously under the influence of the transfer of phonon angular
momentum, which also directly show the existence of phonon magnetic moment,
and even the corresponding magnitude could be evaluated quantitatively.

Figure 2:  (a) Energy shift of intralayer exciton at +K and -K valleys without
(solid lines) and with (dash lines) the transfer of phonon angular momentum in
the presence of the magnetic field at $\alpha=0.08$. (b) Energy shifts of
exciton with the contribution of the phonon angular momentum $\lambda=+1$ at
the +K and -K valleys for the monolayer WS2 at $\alpha=0.024$, in which
experimental data (pink dots) are reproduced from Ref. 20. (c) The
renormalization of $\textsl{g}-$factors with the positive (the upper-limit
$\textsl{g}^{p}_{eff}$) and negative (the lower-limit $\textsl{g}^{n}_{eff}$)
contribution of phonon magnetic moments as a function of the coupling constant
$\alpha$ in monolayer MoS2, where these solid dots represents the values of
$\textsl{g}_{eff}$ obtained in experiments.

In addition, the slopes of energy shift for the W-based monolayers are flatter
than Mo-based ones. This behavior could be attributed to two reasons: the
stronger strength of the exciton-phonon coupling and the bigger value of the
gyromagnetic ratio ${\rm{g}}_{\kappa}$ stemming from the larger difference of
effective masses between electron and hole in W-based monolayer (see the Table
S1), both of which lead to the more correction to exciton magnetic moment.

Another abnormal phenomenon that the asymmetrical distribution of Zeeman
shifts at two valleys in the monolayer TMDs has been widely observed by
earlier experimentsE. Yakaboylu (2017); M. A. Green (2014). To explain it from
the microscopic aspect, several advanced theoretical models based on the
first-principles calculation have been given, e.g., Caruso $et$ $al.$ proposed
the out-of-plane component of the orbital angular momentum of an exciton is
expressed in terms of the Bethe-Salpeter equationE. Yakaboylu (2017), Deilmann
$et$ $al.$ developed a new approach merging the contribution of full Bloch
states to the excitonic magnetic momentsM. A. Green (2014). However, the
effective simulations for the experimental measurements are still lacking.
Fig. 2(b) shows the valley Zeeman shift of the rotating exciton as a function
of the magnetic field in monolayer WS2 (solid lines), along with magneto-
reflectance spectroscopy data from Ref. 20 (pink dots), where phonon angular
momentum $\lambda=+1$ at both the +K and -K valleys is assumed. Namely, the
direction of phonon angular momentum is the same to the exciton angular
momentum at the +K valley, but reversely at the -K valley, as illustrated in
the insets. One can see that the experimental data are very excellently fitted
with the negative and positive magnetic moments provided by phonon angular
momentum at the +K and -K valley excitons, respectively. The quantum state
$\lambda=-1$ also has the similar effect. This implies that (i) the rotation
of valley exciton is not just hindered by the phononic bath, and it also can
get the assistance from the surrounding environment; (ii) the magnitude of
Zeeman shift could be varied in the large scale due to the phononic bath
endowing the positive and negative contribution to the exciton magnetic moment
in each valley, which may provide a potential explanation for the large
fluctuation of $\textsl{g}_{eff}$ in experiments. To show the latter effect
clearly, we present the renormalization of $\textsl{g}_{eff}$ as a function of
the coupling constant for the monolayer MoS2 in Fig. 2(c). According to the
positive and negative contribution to the exciton magnetic moment, the upper-
($\textsl{g}^{p}_{eff}$) and lower-limits ($\textsl{g}^{n}_{eff}$) of the
renormalization of $\textsl{g}_{eff}$ are presented, in which some values of
$\textsl{g}_{eff}$ measured by recent experiments are also illustratedM. A.
Green (2014, 2014, 2014); E. Yakaboylu (2017, 2017, 2017, 2017, 2017). It is
obviously show that the large fluctuation of $\textsl{g}_{eff}$ could be
covered successfully with increasing the coupling strength. The same results
are obtained for other monolayer TMDs in Fig. S3.

From 2015, Schmidt and Lemeshko introduced a new quansiparticle—angulon,
describing the quantum impurity rotating in phononic bathR. Schmidt (2015),
such as molecules merging into superfluid helium droplets. They pointed out
the angulon induces the rotational fine structures due to the angular momenta
transfer between the rotor and the phonons. These transfer processes reflected
by the rotational Lamb shift in spectra have been proved in experimentsE.
Yakaboylu (2017, 2017, 2017). However, the distinguishment between the
clockwise and anti-clockwise motions for phononic bath is still a challenge
task. The model of the rotating exciton in these two-dimensional TMDs systems
presents an ideal platform to overcome this problem by analyzing the
renormalization of the $\textsl{g}-$factor of the valley exciton.

Figure 3:  (a) $\textsl{g}_{\kappa}$ for interlayer excitons in sixteen TMDs
bilayers based on the difference of effective masses between electron and
hole. (b) The correction of $\textsl{g}^{0}_{eff}=-4\textsl{g}_{0}$ by the
2$\textsl{g}_{\kappa}$ for interlayer excitons in sixteen TMDs bilayers. The
renormalization of $\textsl{g}_{eff}$ for the rotating interlayer excitons
with the positive (c) and negative (d) phonon magnetic moments in these
bilayer structures at $\alpha=0.13$, $D=1.5$ nm. (e) The impact of the
effective mass ratios between electron and hole on the $\textsl{g}_{eff}$ with
and without the contribution of phonon magnetic moments. (f)
$\textsl{g}_{eff}$ depends on the internal distance with the positive (the
upper-limit $\textsl{g}^{p}_{eff}$) and negative (the lower-limit
$\textsl{g}^{n}_{eff}$) phonon magnetic moments in MoSe2/WSe2 heterostructure
at $\alpha=0.35$.

$Exciton$ $\textsl{g}_{eff}$ $in$ $bilayer$ TMDs.—–Comparing with the
intralayer excitons, the interlayer excitons have more tunabilities stemming
from the separation of the hosting layers for electron and hole as well as the
modulation of internal distance between two hosting layers. For instance, four
typical monolayer TMDs could be stacked into sixteen bilayer structures,
including four homostructures and twelve heterostructures. Based on the
difference of effective masses between electron and hole (see Table S1 in the
supplemental materials), these values of the gyromagnetic ratios
($\textsl{g}_{\kappa}$) for sixteen rotating excitons and their correction to
the ideal value $\textsl{g}^{0}_{eff}=4$ are listed in Figs. 3(a) and (b),
respectively. We can see that $\textsl{g}_{\kappa}$ varies remarkably with
increasing the ratios of effective masses between electrons and holes, and
thus results in the larger variation of $\textsl{g}_{eff}$ in Fig. 3(b), which
presents a very practical way to control the valley Zeeman splitting of
interlayer exciton by different ratios of effective masses of electron-hole
pairs. Furthermore, the renormalization effect of the angular momentum
transfer of the phononic bath on $\textsl{g}_{eff}$ are shown in Figs. 3(c)
and (d) with the positive and negative magnetic moment contribution at
$\alpha=0.13$, respectively. One can see that phonon angular momenta give the
smaller correction to $\textsl{g}_{eff}$ comparing with the ratio of effective
mass. Besides these monolayer TMDs materials, various of heterostructures
could be composed of by the huge members of two-dimensional material family
for interlayer excitons, ensuring the almost continuously changing ratios of
effective mass for electron-hole pairs. This allows us to obtain much larger
variation for the $\textsl{g}_{eff}$ as plotted in Fig. 3(e). Moreover, the
positive and negative correction of phonon magnetic moments will be enhanced
clearly with the increasing this ratio. These results suggest that the
difference between electron and hole plays a predominate role to adjust the
$\textsl{g}-$factor of interlayer exciton in two-dimensional van der Waals
heterostructures.

Another controllable parameter for the interlayer exciton is the internal
distance $D$ between two hosting layers that related to the rotational
constant $\xi_{0}$ (see the supplemental materials Part II). The interlayer
distance dependence of $\textsl{g}_{eff}$ is given in Fig. 3(f) for MoSe2/WSe2
bilayer with the positive ($\textsl{g}^{p}_{eff}$) and negative
($\textsl{g}^{n}_{eff}$) contribution of phonon magnetic moments. We find the
renormalization of $\textsl{g}_{eff}$ varies in the range of $-4.5\sim-9.5$
with the increasing of $D$, covering most experimental data (solid dots)
illustrated in Fig. 3(f)M. A. Green (2014, 2014); E. Yakaboylu (2017, 2017,
2017). This behavior can be attributed to the enhanced transfer of phonon
angular momentum with $D$, resulting in the large fluctuation of
$\textsl{g}_{\kappa}^{*}$. Hence, the interlayer distance is also a key
parameter for the modulation of the rotational interlayer exciton. But this
role has been neglected in most experiments, which may explain why different
values of $\textsl{g}_{eff}$ were obtained for the same bilayer structuresM.
A. Green (2014, 2014); E. Yakaboylu (2017, 2017, 2017).

Recently, the chiral phonon has been predicted by the theoretical studies and
proved by some experiments in two-dimensional TMDs structuresE. Yakaboylu
(2017, 2017, 2017, 2017, 2017, 2017), which plays an important role for the
interconversion between the dark and bright excitons due to the required
momentum conservation. Here, the chirality of the rotational angular momentum
constitutes an inherent degree of freedom of valley excitons that induces the
coupling to other degree of freedom, e.g., spin and orbital angular momentum,
and the external perturbationsE. Yakaboylu (2017). In turn, the chirality of
other degrees of freedom could be reflected by the exchange of angular momenta
between them. On the other hand, the larger fluctuation of $\textsl{g}_{eff}$
for trion has been observed in recent experimentsE. Yakaboylu (2017, 2017,
2017). Using the rotating exciton model, we present a potential explanation
that an extra electron (or hole) in a rotating trion will induce the bigger
magnetic moment, enhancing the valley Zeeman effect. Moreover, the transfer of
phonon angular momentum results in the fine-structures of trions.

$Conclusion.$—–In summary, we propose the rotational motion of valley excitons
in monolayer TMDs and their bilayer structures, where the new chirality of
exciton is defined by the rotational angular momentum, which could be
manifested by the valley Zeeman splitting. Furthermore, the rotating excitons
induce the transfer of phonon angular momentum, resulting in the
renormalization effect of exciton $\textsl{g}-$factors, and indicating the
chirality of the phonon and phonon magnetic moment. These results also provide
important enlightenments for revealing the underlying physics of the optical
properties of valley excitons in two-dimensional materials.

This work was supported by National Natural Science Foundation of China (Grant
Nos. 11674241, 62022081, 61974099 and 12174283).

## References

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# NGC 7538 IRS 1 - an ionized jet powered by accretion

Göran Sandell  SOFIA-USRA, NASA Ames Research Center, MS N211-3, Moffett
Field, CA 94035, U.S.A<EMAIL_ADDRESS>W. M. Goss  National Radio
Astronomy Observatory, P.O. Box O, Socorro, NM 87801, U.S.A.  Melvyn Wright
Radio Astronomy Laboratory, University of California, Berkeley 601 Campbell
Hall, Berkeley, CA 94720, U.S.A.  Stuartt Corder11affiliation: Jansky Fellow,
NRAO Joint ALMA Observatory, Av Apoquindo 3650 Piso 18, Las Condes, Santiago,
Chile

###### Abstract

Analysis of high spatial resolution VLA images shows that the free-free
emission from NGC 7538 IRS 1 is dominated by a collimated ionized wind. We
have re-analyzed high angular resolution VLA archive data from 6 cm to 7 mm,
and measured separately the flux density from the compact bipolar core and the
extended (1${}^{\prime\prime}\mskip-7.6mu.\,$5 - 3′′) lobes. We find that the
flux density of the core is $\propto\nu^{\alpha}$, where $\nu$ is the
frequency and $\alpha$ is $\sim$ 0.7. The frequency dependence of the total
flux density is slightly steeper with $\alpha$ = 0.8. A massive optically
thick hypercompact core with a steep density gradient can explain this
frequency dependence, but it cannot explain the extremely broad recombination
line velocities observed in this source. Neither can it explain why the core
is bipolar rather than spherical, nor the observed decrease of 4% in the flux
density in less than 10 years. An ionized wind modulated by accretion is
expected to vary, because the accretion flow from the surrounding cloud will
vary over time. BIMA and CARMA continuum observations at 3 mm show that the
free-free emission still dominates at 3 mm. HCO+ J = $1\to 0$ observations
combined with FCRAO single dish data show a clear inverse P Cygni profile
towards IRS 1. These observations confirm that IRS 1 is heavily accreting with
an accretion rate $\sim$2 10-4 M⊙/yr.

###### Subject headings:

accretion, accretion disks – H II regions – stars: early-type – stars:
formation

##  1\. Introduction

NGC 7538 IRS 1 was first detected in the radio at 5 GHz by Martin (1973), who
found three compact radio sources at the SE edge of the large ($\sim$ 4′) H II
region NGC 7538, which is at a distance 2.65 kpc (Moscadelli et al., 2008).
The brightest of the three, source B, has later become known as IRS 1 (Wynn-
Williams, Becklin, & Neugebauer, 1974). The far-infrared luminosity of the
three sources is $\sim$1.9 105 L⊙, completely dominated by IRS 1(Hackwell,
Grasdalen & Gehrz, 1982). IRS 1 was first resolved with the VLA at 14.9 GHz by
Campbell (1984), who showed that it has a compact
($\sim$0${}^{\prime\prime}\mskip-7.6mu.\,$2) bipolar N-S core with faint
extended fan-shaped lobes, suggesting an ionized outflow. This is an extremely
well-studied source with numerous masers, a prominent molecular outflow and
extremely broad hydrogen recombination lines indicating substantial mass
motion of the ionized gas (Gaume et al., 1995; Sewilo et al., 2004; Keto et
al., 2008). Although it has been modeled as an ionized jet (Reynolds, 1986) or
a photo-ionized accretion disk (Lugo, Lizano & Garay, 2004), the source is
usually referred to as an Ultra Compact or Hyper Compact H II region with a
turnover at $\sim$ 100 GHz. Since we have carried extensive observations of
IRS 1 with BIMA and CARMA, the desire to determine the contribution from dust
emission at mm-wavelengths led us to investigate where the free-free emission
from IRS 1 becomes optically thin. Such a determination is not possible from
published data, because IRS 1 has been observed with a variety of VLA
configurations, some of which resolve out the extended emission and others may
only quote total flux. This prevented us from separately estimating the flux
for the compact bipolar core and the extended lobes. Furthermore the flux
density varies with time (Franco-Hernández & Rodríguez, 2004), which makes it
difficult to derive an accurate spectral index.

##  2\. VLA archive data

Figure 1.—  C-band (4.8 GHz) image of the IRS1 - 3 field. The beam size is
plotted in the bottom right corner of the image. The first contour is at
3-$\sigma$ level, the next contour 3 times higher and from there on we plot 7
logarithmic contours. An outline of the jet from IRS 1, plotted in red, shows
the optically thick inner part of the jet and where the jet expands and
becomes optically thin. Fig 2 zooms in on the inner optically thick part of
the jet and illustrates how the jet becomes smaller when we go up in frequency
due to the outer portion of the jet becoming optically thin.

We have retrieved and re-analyzed a few key observations of NGC 7538 IRS 1
from the VLA data archive. These are all long integration observations with
very high angular resolution, good uv-coverage, and high image fidelity. Table
1 gives the observing dates, synthesized beam width, sensitivity, the
integrated flux of the bipolar core, and the observed total flux. We included
a BnA array observation obtained by us at X-band (8.5 GHz)(Sandell, Goss &
Wright, 2005), because it is more contemporary with the high frequency data
sets and provides an additional data point at a frequency where there are no
other high angular resolution VLA data available. Fig 1 shows the IRS 1 - 3
field at 4.8 GHz, while Fig 2 shows the morphology of the compact bipolar core
as a function of frequency.

##  3\. Analysis of VLA data

We determined flux densities of the compact bipolar core with an accuracy of a
few percent. The results are given in Table 1. Table 1 also gives total flux
densities, obtained by integrating over the whole area where we detect
emission from IRS 1. At 4.9 GHz the total linear extent of IRS 1 is $\sim$
7′′, while it is $\sim$ 1${}^{\prime\prime}\mskip-7.6mu.\,$4 at 43.4 GHz. Even
though we can reliably determine the flux density of the bipolar core at 43.4
GHz, some of the faint extended emission is filtered out; therefore the total
flux is underestimated. At 4.9 GHz the bipolar core has a total length of
$\sim$ 1${}^{\prime\prime}\mskip-7.6mu.\,$1 FWHM (Full Width Half Maximum),
with a separation between the two peaks of
0${}^{\prime\prime}\mskip-7.6mu.\,$68, at 14.9 GHz the size is
0${}^{\prime\prime}\mskip-7.6mu.\,$44 with a peak separation of
0${}^{\prime\prime}\mskip-7.6mu.\,$24 (Table 1). At 43.4 GHz we can still fit
the core with a double Gaussian with a peak separation of $\sim$
0${}^{\prime\prime}\mskip-7.6mu.\,$12, while the linear size (length) has a
FWHM of 0${}^{\prime\prime}\mskip-7.6mu.\,$20\. Since one of our data sets (
X-band, 8.4 GHz) has insufficient resolution to resolve the two bipolar cores,
we plot the linear size (length) as a function of wavelength in Fig. 3,
although fitting to the lobe separation gives virtually identical results, but
with a larger uncertainty. We find the size to vary with frequency as
$\nu^{-0.8\pm 0.03}$, while the flux density increases with frequency as
$\nu^{0.7\pm 0.05}$. The frequency dependence of the total flux is more
difficult to estimate, since some VLA configurations are not very sensitive to
faint extended emission and will therefore underestimate the total flux. If we
chose data sets which give a reliable flux for IRS 2111The emission from IRS 2
is already optically thin at 4.8 GHz (C-band) with an integrated flux of 1440
mJy., which is a spherical H II region with a size of $\sim$ 8′′, we find that
the total flux from IRS 1 has a slightly steeper spectral index, 0.8 $\pm$
0.03 determined from data sets from 4.8 GHz to 49 GHz. The fit to the total
flux densities for IRS 1 is shown as a dotted line in Fig. 3. Such a shallow
frequency dependence would require a very steep density gradient in the
ionized gas, if the emission originates from an H II region ionized by a
central O-star, although it is not impossible, see e.g. Keto et al. (2008). A
high-density H II region with a steep density gradient should be spherical,
not bipolar with a dark central lane as we observe in IRS 1. Neither can a
steep density gradient model explain the extremely broad recombination lines
observed in IRS 1 (Gaume et al., 1995; Keto et al., 2008). All these
characteristics can be explained, if the emission originates in an ionized
stellar wind or jet (Reynolds, 1986). The inner part of the jet is optically
thick and further out, where the jet expands, the emission becomes optically
thin. At higher frequencies the outer part of the jet becomes optically thin
and therefore appears shorter and more collimated, exactly what we see in IRS
1, see Fig 2. For a uniformly expanding spherical wind, the size of the source
varies as a function of frequency as $\nu^{-0.7}$, while the flux density
density goes as $\nu^{0.6}$ (Panagia & Felli, 1975). Since we resolve the
emission from IRS 1, we know that it is not spherical, but instead it appears
to originate in an initially collimated bipolar jet (opening angle $\lesssim$
30∘) approximately aligned with the molecular outflow from IRS 1, see Section
4. For a collimated ionized jet, Reynolds (1986) showed that it can have a
spectral index anywhere between 2 and $-$0.1, depending on gradients in jet
width, velocity, degree of ionization, and temperature.

Figure 2.—  VLA images at 4.86, 14.94, 22.37, & 43.37 GHz showing the bipolar core surrounding IRS 1. Beam sizes are given in Table 1. The C-band image (Fig. 1) shows faint extended lobes extending several arcseconds to the north and south of IRS 1. These images show that the core shrinks in size with increasing frequency. Table 1Observational parameters and results of VLA archive data of IRS 1  Frequency | Synthesized beam | rms | Size | flux density | flux density | Observing dates
---|---|---|---|---|---|---
|  |  | bipolar core | bipolar core | total |
[GHz] |  ′′ $\times$ ′′ pa = ∘ | [mJy beam-1] |  ′′ $\times$ ′′ pa = ∘ | [mJy] | [mJy] |
4.86 | 0.43 $\times$ 0.37 $+$05.4∘ | 0.300 | 1.06 $\times$ 0.44 $-$9.6 | 94.3 $\pm$ 4.8 |  00122.5 | 1984:1123
8.46 | 1.21 $\times$ 0.47 $-$00.5∘ | 0.050 | 0.76 $\times$ 0.30 $-$10.9 | 135.7 $\pm$ 13.6aaInsufficient spatial resolution, blends in with IRS 2. Unreliable total flux. |  00148.1aaInsufficient spatial resolution, blends in with IRS 2. Unreliable total flux. | 2003:1014
14.94 | 0.14 $\times$ 0.12 $+$47.6∘ | 0.017 | 0.44 $\times$ 0.20 $-$0.3 | 185.9 $\pm$ 5.6 |  00248.7 | 2006:0226,0512bbAverage of two data-sets at different epochs during the same year.
14.91 | 0.14 $\times$ 0.10 $+$13.7∘ | 0.081 | 0.50 $\times$ 0.20 $-$1.2 | 165.2 $\pm$ 13.3 |  00291.0 | 1994:1123
22.37 | 0.08 $\times$ 0.08 $-$43.0∘ | 0.017 | 0.34 $\times$ 0.17 $+$3.1 | 286.5 $\pm$ 11.5 |  00370.0 | 1994:1123
43.37 | 0.14 $\times$ 0.12 $+$14.5∘ | 0.170 | 0.20 $\times$ 0.14 $-$1.0 | 429.7 $\pm$ 12.9 |  00473.7 | 2006:0706,0914bbAverage of two data-sets at different epochs during the same year.
Figure 3.—  Top panel: A least squares power-law fit to the length of the
optically thick jet as a function of frequency $\nu$ gives $\theta$(arcsec) =
4.13 $\nu^{-0.8}$ where $\nu$ is in GHz. Bottom panel: Least squares fit to
VLA data (open squares) of the compact bipolar core surrounding IRS1 plotted
as a solid line, and to total fluxes (open triangles) plotted with a dotted
line. In the figure we also show mm data points obtained with BIMA and CARMA
as well as interferometer data from the literature, but they are not used in
any of the fits. This plot clearly demonstrates that the free-free emission
still dominates the emission in high angular resolution interferometer data
even at 1.3 mm.

##  4\. Outflow and accretion

Although the molecular outflow from IRS 1 in early studies was found to be
quite compact and going from SE to NW (Scoville et al., 1986), we find that
the molecular outflow is very extended ($>$ 4′) and approximately aligned with
the free-free jet from analyzing large mosaics obtained with CARMA in 12CO and
13CO J = $1\to 0$, and HCO+ J = $1\to 0$. For HCO+ we filled in missing zero
spacing with fully sampled single dish maps obtained with FCRAO. A thorough
discussion of the molecular outflow is beyond the scope of this paper, and
will be discussed in a forthcoming paper (Corder et al. 2009, in prep). Here
we therefore summarize some of the main results. The CARMA mosaics show that
the outflow is very large and has a position angle (pa) of $\sim$ $-$20∘,
which is similar to the orientation of the collimated inner part of the
ionized jet driven by IRS 1 on smaller angular scales. The outflow extends
several arcminutes to the north and starts as a wide angle limb brightened
flow. To the south the outflow is difficult to trace, because of several other
outflows in the giant molecular core in which IRS 1 is embedded, but appears
more collimated with a linear extent of $\sim$ 2′. Analysis of Spitzer IRAC
archive data show that the outflow may be even larger to the south. The 5.8
and 8.0 $\mu$m IRAC images show a jet-like feature projecting back towards IRS
1, extending to $\sim$ 3.5′ (2.6 pc) from IRS 1, which is much further out
than what was covered by the CARMA mosaics (Corder, 2008). However, the CARMA
observations also confirm that IRS 1 is still heavily accreting. In HCO+ J =
$1\to 0$ we observe a clear inverse P Cygni profile towards the strong
continuum emission from IRS 1 (Fig 4). The absorption is all red-shifted and
has two velocity components, which could mean that the accretion activity is
varying with time (episodic). From these data Corder (2008) estimates $\sim$
6.8 M⊙ of infalling gas. With the assumption that the accretion time is
similar to the free-fall time, $\sim$ 30,000 yr, Corder finds an accretion
rate $\sim$ 2 10-4 M⊙/yr, which will block most of the uv photons from the
central O-star, allowing them to escape only at the polar regions.

Figure 4.—  Inverse P Cygni profile in HCO+ J = $1\to 0$ observed with CARMA
supplemented with FCRAO single dish data (Corder, 2008). Note that the
spectrum is extracted from a continuum subtracted spectral line cube, hence
the flux density appears to go below zero. The angular resolution is
4${}^{\prime\prime}\mskip-7.6mu.\,$5\. The vertical dotted line shows the
systemic velocity of IRS 1.

##  5\. Millimeter data - where is the accretion disk?

In Fig. 3 we also plotted flux densities at 3 and 1 mm from the literature
(Lugo, Lizano & Garay, 2004), supplemented with our own results at 3 mm from
BIMA (Wright et al. 2009, in prep) and CARMA (Corder, 2008). All the observed
flux densities at 3 mm or even at 1.3 mm can be explained by free-free
emission, with at most a marginal excess from dust emission. High angular
resolution CARMA continuum observations at 91.4 and 108.1 GHz confirm what is
predicted from the fit to total flux densities in Fig 3, i.e. that the free-
free emission dominates at 3 mm. These CARMA observations resolve IRS 1 with a
size of 0${}^{\prime\prime}\mskip-7.6mu.\,$5
$\times\leq$0${}^{\prime\prime}\mskip-7.6mu.\,$3 pa $-$1∘ and
0${}^{\prime\prime}\mskip-7.6mu.\,$4
$\times\leq$0${}^{\prime\prime}\mskip-7.6mu.\,$1 pa 20∘ at 94 and 109 GHz,
respectively, i.e. the 3 mm emission is aligned with the free-free emission,
and not with an accretion disk, which is expected to be perpendicular to the
jet. Since we detect a collimated free-free jet and since there is a strong
accretion flow towards IRS 1 (Section 4), IRS 1 must be surrounded by an
accretion disk. The morphology of the free-free emission suggests that the
disk should be almost edge-on, and centered halfway between the northern and
the southern peak of the optically thick inner part of the free-free jet.
However, if such an edge-on accretion disk is thin, i.e. the height of the
disk is small relative to its diameter, it may not provide much surface area,
and is therefore difficult to detect at 3 mm. At 3 mm IRS 1 and IRS 2 are very
strong in the free-free, and even the free-free emission from IRS 3 cannot be
ignored, which makes it very hard to detect dust emission from the accretion
disk. Since the spectral index for dust is $\sim$ 3 - 3.5, the dust emission
may start to dominate at frequencies above 300 GHz, observable only by SMA,
but even at 1.3 mm we may have a much better chance to detect the accretion
disk. There is some evidence for such a disk in 450 and 350 $\mu$m continuum
images obtained at JCMT (Sandell & Sievers, 2004). Until higher spatial
resolution interferometer images are available, it is not clear whether this
emission originates in a disk, from the surrounding in-falling envelope, or
from a superposition of several nearby sources.

##  6\. Discussion and Conclusions

Free-free emission from ionized jets is very common in low mass protostars
(Class 0 and Class I sources), which all appear to drive molecular outflows.
This situation is especially true for young stars, which often have rather
well collimated outflows (Rodríguez, Anglada & Curiel, 1999). Jets may also be
common in young early B-type high mass stars (Gibb & Hoare, 2007), although
they have not been studied as well as low-mass protostars, because they are
short lived, more distant and harder to identify. Although there have not been
any detections of jets in young O stars, i.e. stars with a luminosity of $>$
105 L⊙, such objects are likely to exist. Broad radio recombination line (RRL)
objects, many of which are classified as Ultra Compact or Hyper Compact H II
regions (Jaffe & Martín-Pintado, 1999; Kurtz & Franco, 2002; Sewilo et al.,
2004; Keto et al., 2008) offer a logical starting point, because they show
evidence for substantial mass motions, which would be readily explained if the
recombination line emission originates in a jet. Several of them also show
evidence for accretion. In the sample analyzed by Jaffe & Martín-Pintado
(1999), which partly overlaps with the sources discussed by Gibb & Hoare
(2007), they find four sources (including IRS 1) with bipolar morphology. All
appear to be dominated by wind ionized emission, although not necessarily jet
driven winds. K 3-50 A, however, at a distance of 8.7 kpc (De Pree et al.,
1994), has a luminosity of a mid-O star, drives an ionized bipolar outflow (De
Pree et al., 1994), and has a spectral index of 0.5 in the the frequency range
5 - 15 GHz (Jaffe & Martín-Pintado, 1999). K 3-50 A is therefore another
example of an O-star, where the free-free emission appears to be coming from
an ionized jet. Detailed studies of broad RRL objects will undoubtedly
discover more examples of jet-ionized H II regions. Such objects are likely to
drive molecular outflows, excite masers and show evidence for strong
accretion.

To summarize: We have shown that the emission from IRS 1 is completely
dominated by a collimated ionized jet. The jet scenario also readily explains
why the free-free emission is variable, because the accretion rate from the
surrounding clumpy molecular cloud will vary with time. It also explains why
IRS 1 is an extreme broad RRL object. Since most Hyper Compact H II regions
are broad RRL objects, and show rising free-free emission with similar
spectral index to IRS 1, other sources now classified as Hyper Compact H II
regions, may also be similar to IRS 1.

The National Radio Astronomy Observatory (NRAO) is a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc. The BIMA array was operated by the Universities of
California (Berkeley), Illinois, and Maryland with support from the National
Science Foundation. Support for CARMA construction was derived from the states
of California, Illinois, and Maryland, the Gordon and Betty Moore Foundation,
the Kenneth T. and Eileen L. Norris Foundation, the Associates of the
California Institute of Technology, and the National Science Foundation.
Ongoing CARMA development and operations are supported by the National Science
Foundation under a cooperative agreement, and by the CARMA partner
universities.

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