#  Anomalous Circularly Polarized Light Emission induced by the Optical Berry
Curvature Dipole

Yizhou Liu  School of Physics Science and Engineering, Tongji University,
Shanghai 200092, China  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 76100, Israel  Binghai Yan
<EMAIL_ADDRESS>Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 76100, Israel

###### Abstract

The ability to selectively excite light with fixed handedness is crucial for
circularly polarized light emission. It is commonly believed that the
luminescent material chirality determines the emitted light handedness,
regardless of the light emitting direction. In this work, we propose an
anomalous circular polarized light emission (ACPLE) whose handedness actually
relies on the emission direction and current direction in electroluminescence.
In a solid semiconductor, the ACPLE originates in the band structure topology
characterized by the optical Berry curvature dipole. ACPLE exists in
inversion-symmetry breaking materials including chiral materials. We exemplify
the ACPLE by estimating the high circular polarization ratio in monolayer WS2.
In addition, the ACPLE can be further generalized to magnetic semiconductors
in which the optical Berry curvature plays a leading role instead. Our finding
reveals intriguing consequences of band topology in light emission and
promises optoelectric applications.

##  I Introduction

Circularly polarized (CP) light emission is important for various advanced
technologies including 3D display, optoelectrics, information storage, and
quantum spintronics [1, 2]. The CP light can be generated through optical
fibers composed by a linear polarizer followed by a quarter-wave plate or CP
light emission (CPLE) processes including photoluminescence (PL) and
electroluminescence (EL) in chiral organic luminophores [3, 4, 5, 6, 7, 8] and
chiral hybrid organic–inorganic perovskites [9, 10]. It is commonly believed
that the favored light handedness in both PL and EL are determined by the
material chirality but independent of the emitting direction. Recently, a
distinct CP EL was reported in the chiral polymers where the favored light
handedness depends on both the emitting direction and polymer chirality [11] .
Furthermore, the emission-direction-sensitive CPLE generates giant net
polarization ratio of CPLE by orders of magnitude. The anomalous CP light
emission (ACPLE) was interpreted by the orbital-momentum locking, a
topological nature of the chiral molecule electronic structure[12, 11].

Compared to chiral organic polymers, we are inspired to explore the ACPLE in
crystalline solids where the band structure topology [13, 14, 15, 16] can play
a significant role. For example, the Berry curvature and Berry connection[17]
is extensively considered in the content of light absorbance and the resultant
photocurrent, such as the photogalvanic effects [18, 19, 20, 21, 22, 23],
nonlinear anomalous Hall effect [24, 25, 26, 27] and the valley Hall effect
[28, 29].

In solids, giant CP EL was indeed observed in inversion-breaking
semiconductors, WSe2 [30] and WS2 [31] monolayers, where emitted light
handedness is switchable by reversing the current flow. It was rationalized by
the valley polarization and asymmetric band structures of electrons and holes.
Additionally, a magnetization switched CP PL was also reported in a magnetic
insulator CrI3 [32], which was interpreted by local molecular orbital
transitions of Cr3+ cations. From the theoretical aspect, the electronic
wavefunctions and topology should play an essential role there but remain
largely unexplored, although the band dispersion (e.g. band shift, and
triangular warping etc. [33]) was recently discussed in this content.

In this work, we formulate the ACPLE that exists in inversion-breaking solids
(including chiral solids) by the optical Berry curvature dipole. Different
from the ordinary CP EL, the favored light handedness depends both on the
direction of applied current and the light-emitting direction. Here, ACPLE can
be regarded as a reverse process of the circular photogalvanic effect where CP
light irradiation generates a dc current [19]. We demonstrate the ACPLE in the
strained WS2 monolayer. The CP EL in WS2 shows handedness switching by the
direction of applied strain, current flow, and emitting direction. In
addition, the ACPLE can also appear in magnetic solids, which originate in the
optical Berry curvature. Our theory rationalizes the current direction or
magnetization-switchable light handedness in previous experiments based on
CrI3 or WS2. Our predictions can be further verified by detecting the
emission-direction dependent handedness in present devices. The ACPLE promises
applications in exotic CP light-emitting diodes and ultrafast optoelectronic
information applications.

Figure 1:  Schematics of the (a) ordinary CPLE and (b)-(e) ACPLE,
respectively. (a) Ordinary CP PL in chiral materials. The favored light
handedness depends solely on the chirality of the luminescent material but
independent of the emitting direction. (b)-(e) The ACPLE induced by finite
current in chiral or inversion-symmetry-breaking achiral materials sandwiched
by PN junctions. The favored light handedness depends on the emitting
direction and also reverses with the direction of current $I$. For chiral
materials the dirrection of ACPLE can be either parallel or perpendicular to
the direction of current [(b)-(e)], but for inversion-symmetry-breaking
achiral materials, the ACPLE can be only perpendicular to the current
[(b)-(c)].

##  II Theory of ACPLE

According to Fermi’s golden rule, the optical emission rate of left- and
right- handed CP light is determined by

$\begin{split}I_{L/R}=\frac{2\pi}{\hbar}\sum_{c,v,\bm{k}}|\langle
v\bm{k}|H^{\prime}|c\bm{k}\rangle|^{2}f_{c\bm{k}}(1-f_{v\bm{k}})\delta(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}-\hbar\omega)\end{split}$
(1)

where $|c\bm{k}\rangle$ ($|v\bm{k}\rangle$) refers to the Bloch state of
conduction (valance) band, $\varepsilon_{c\bm{k}}$ ($\varepsilon_{v\bm{k}}$)
is the conduction (valance) band energy, $f_{c\bm{k}}$ is the electron
occupation probability in the cunduction band and $1-f_{v\bm{k}}$ refers to
the hole occupation in the valance band, and
$H^{\prime}=-e\bm{E}\cdot\bm{r}-\bm{m}\cdot\bm{B}$ is the light-matter
interaction Hamiltonian with $e\bm{r}$ and $\bm{m}$ being the electric and
magnetic moments, respectively, $\bm{E}$ and $\bm{B}$ being the electric and
magnetic fields of light, respectively, and $\hbar\omega$ is the photon
energy. For CP light with wave vector $\bm{q}$, the electric and magnetic
fields are $\bm{E}=\frac{E_{0}}{\sqrt{2}}(\hat{\bm{e}}_{1}\pm
i\hat{\bm{e}}_{2})$ and $\bm{B}=\frac{1}{c_{0}}\hat{\bm{q}}\times\bm{E}$ with
$\hat{\bm{e}}_{2}$ and $\hat{\bm{e}}_{2}$ being the two unit vectors
perpendicular to $\bm{q}$, $c_{0}$ is speed of light, and
$\hat{\bm{q}}=\bm{q}/|\bm{q}|$ is the unit vector of $\bm{q}$.

The difference of generated CP light emission can be calculated as

$\begin{split}I_{L}&-I_{R}=-\frac{2\pi
e^{2}}{\hbar}|E_{0}|^{2}\sum_{c,v,\bm{k}}f_{c\bm{k}}(1-f_{v\bm{k}})\delta(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}-\hbar\omega)\\\
&\left[\frac{2}{ec_{0}}\textrm{Im}(r^{vc}_{1}m^{cv}_{1}+r^{vc}_{2}m^{cv}_{2})+\bm{\Omega}^{vc}\cdot\hat{\bm{q}}+O(m^{2})\right],\end{split}$
(2)

where $r^{vc}_{i}=\langle c\bm{k}|\bm{r}\cdot\hat{\bm{e}}_{i}|v\bm{k}\rangle$
and $m^{cv}_{i}=\langle c\bm{k}|\bm{m}\cdot\hat{\bm{e}}_{i}|v\bm{k}\rangle$
($i=1,2$) refer to the electric and magnetic transition dipoles within the
polarization plane of $\hat{\bm{e}}_{1}$ and $\hat{\bm{e}}_{2}$.
$\bm{\Omega}^{vc}=-i\bm{r}^{vc}\times\bm{r}^{cv}$ is the interband Berry
curvature [34, 35] whose expression is given by

$\Omega^{vc}_{a}=-i\epsilon_{abd}r^{vc}_{b}r^{cv}_{d}=-\hbar^{2}\epsilon_{abd}\textrm{Im}\frac{\langle
v\bm{k}|v_{b}|c\bm{k}\rangle\langle
c\bm{k}|v_{d}|v\bm{k}\rangle}{(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}})^{2}}$
(3)

where $a,b,d=x,y,z$ refer to Cartesian corrdinates, $\epsilon_{abd}$ refers to
the rank-3 antisymmetric Levi-Civita tensor and $v_{b}$, $v_{d}$ refer to the
velocity operator. The first term on the second line of Eq. (2) gives rise to
the ordinary CPLE or natural circular dichroism which has been extensively
investigated in nonmagnetic chiral molecules and polymers [36, 5] as shown in
Fig. 1(a).

The second term in Eq. (2) ($\bm{\Omega}^{vc}\cdot\hat{\bm{q}}$) gives rise to
the ACPLE. This anomalous term is usually neglected before because the total
Berry curvature vanishes in nonmagnetic materials such as organic molecules
and polymers. It is not surprising that this term becomes significant in
magnetic materials. Here, the magnetic order determines the sign of
$\bm{\Omega}^{vc}$ and thereafter the sign of CPLE polarization. Furthermore,
$\bm{\Omega}^{vc}\cdot\hat{\bm{q}}$ clearly indicates the emission-direction
($\hat{\bm{q}}$) dependence of CPLE, which was not awared before. Because the
nonequilibrium current flow breaks time-reversal symmetry (TRS) even in
nonmagnetic materials, the ACPLE term is also significant in CP EL. For a
nonmagnetic system with TRS, the Berry curvature satisfies
$\bm{\Omega}^{vc}(\bm{k})=-\bm{\Omega}^{vc}(-\bm{k})$ which makes the
anomalous term cancel with each other between $\bm{k}$ and $-\bm{k}$ at
thermodynamic equilibrium. Therefore there are two cases the anomalous term
can take effects: (i) The TRS is broken e.g. by spontaneous magnetization
$\bm{M}$ in magnetic materials (e.g. CrI3 monolayer); (ii) the system is in
nonequilibrium state with the occupation probability satisfying
$f_{c\bm{k}}\neq f_{c-\bm{k}}$ and $f_{v\bm{k}}\neq f_{v-\bm{k}}$ driving by
an external electrical current i.e. anomslous CP EL as shown in Fig. 1(b)-(c).
Switching the current direction reverses the induced polarization.
Particularly the anomalous term explicitly depends on the light-emitting
direction $\hat{\bm{q}}$ resulting in the direction-dependent CP EL signals.

Optical Berry curvature dipole. In nonmagnetic materials, the system is driven
out of thermal equilibrium by an external electrical dc current. The
nonequilibrium distribution functions $f_{c\bm{k}}$ and $f_{v\bm{k}}$ can be
determined by the Boltzmann equation under relaxation time approximation,

$\begin{split}f_{\alpha\bm{k}}=f^{0}_{\alpha\bm{k}}+\tau\frac{e}{\hbar}\bm{\mathcal{E}}\cdot\bm{\nabla_{k}}f^{0}_{\alpha\bm{k}}+O(\tau^{2})\end{split}$
(4)

where $\alpha=c,v$ and
$f^{0}_{\alpha\bm{k}}=\left[e^{(\varepsilon_{\alpha\bm{k}}-\mu)/k_{B}T}+1\right]^{-1}$
refers to the equilibrium Fermi distribution function, $\tau$ is the
relaxation time, and $\bm{\mathcal{E}}$ is the external dc electric field.
Substitute Eq. (4) into Eq. (2), the anomalous CP EL (second term of Eq. (2))
can be calculated as,

$\begin{split}I_{L}-I_{R}=&\frac{2\pi
e^{3}\tau}{\hbar^{2}}f_{B}(\hbar\omega)\mathcal{E}_{a}D_{ab}(\hbar\omega)\hat{q}_{b}+O(\tau^{2}),\\\
D_{ab}(\hbar\omega)=&\sum_{c,v,\bm{k}}(f^{0}_{v\bm{k}}-f^{0}_{c\bm{k}})\partial_{k_{a}}\Omega^{vc}_{b}(\bm{k},\omega)\\\
\approx&\sum_{c,v,\bm{k}}\partial_{k_{a}}\Omega^{vc}_{b}(\bm{k},\omega)\end{split}$
(5)

The identity
$f^{0}_{c\bm{k}}(1-f^{0}_{v\bm{k}})=f_{B}(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}})(f^{0}_{v\bm{k}}-f^{0}_{c\bm{k}})$
has been used in the derivation, and the last line of Eq. (5) is valid for
large-gap semiconductors or insulators which are often used for optoelectronic
materials. Here $f_{B}(\hbar\omega)=(e^{\hbar\omega/k_{B}T}-1)^{-1}$ is the
Bose distribution function. The optical Berry curvature refers to Berry
curvature at the optical transition energy $\hbar\omega$,

$\Omega^{vc}_{b}(\bm{k},\omega)=\hat{\bm{e}}_{b}\cdot\bm{\Omega}^{vc}(\bm{k})\delta(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}-\hbar\omega)$
(6)

which represent the $b$-th component. The $D_{ab}(\hbar\omega)$, as expressed
on the last line of Eq. (5), represents the sum of the optical Berry curvature
dipole (OBCD). It is a rank-2 pseudo tensor i.e. it is even under TRS but odd
under inversion, so that the anomalous CP EL in Eq. (5) can only occur in
inversion-symmetry-breaking materials including chiral materials. In addition
to the material-specific OBCD, the anomalous CP EL is also linear to the
external driving field $\bm{\mathcal{E}}$ which rationalize previous
experimental results about electrical switchable CP EL [30].

What was uncovered in previous experiments is that the CP EL also depends on
the emission direction $\hat{\bm{q}}$. In optoelectronic literature, a
dimensionless dissymmetry factor $g=(I_{L}-I_{R})/\frac{1}{2}(I_{L}+I_{R})$ is
often used to describe the circular polarization ratio. The dissymmetry factor
for the anomalous CP EL is

$\begin{split}g_{EL}=&\frac{2e\tau}{\hbar}\frac{\mathcal{E}_{a}D_{ab}\hat{q}_{b}}{I_{0}}+O(\tau^{2}),\end{split}$
(7)

where
$I_{0}=\sum_{c,v,\bm{k}}\left(|r^{vc}_{1}|^{2}+|r^{vc}_{2}|^{2}\right)\delta(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}-\hbar\omega)$
is the ordinary absorption magnitude. According to Eq. (7), the dissymmetry
factor can be enhanced by improving the relaxation time $\tau$ and increasing
electric field $\bm{\mathcal{E}}$ besides optimizing OBCD.

In the context of the nonlienar anomalous Hall effect [24], the intra-band
Berry curvature dipole was discussed and requires that the electric field
$\mathcal{E}_{a}$ is orthogonal to $\Omega_{b}$, i.e., $a\neq b$. In contrast,
the ACPLE exists for both $a\neq b$ and $a=b$ in Eq. (5). If $a=b$, the
material should be chiral because either inversion symmetry or mirror symmetry
forces $D_{aa}$ to vanish. If $a\neq b$, the material only requires to break
the inversion symmetry, which also applies for chiral materials. In the
device, $a\neq b$ ($a=b$) means that light emission direction $\hat{q}_{b}$
can be normal [see Figs.1 (b) & (c)] [parallel (see Figs. 1 (d) & (e))] to the
current flow.

##  III Anomalous CP EL in inversion asymmetric materials

Among the various inversion asymmetric materials found so far, the transition
metal dichalcogenides (TMDs) are of particular interest because of the low-
energy Dirac physics at Brillouin zone corners $K$ and $K^{\prime}$ which
exhibit valley-contrasting CP excitations processes. Zhang et al. first
demonstrated the electrically switchable CP EL in a WSe2 monolayer transistor
[30] and this effect was attributed to the triangular warping effect of the
band structure [30, 33]. Recently Pu et. al. have experimentally demonstrated
the room-temperature chiral light-emitting diode in strained WS2 monolayer
[31]. In the following we illustrate the role of OBCD in the anomalous CP EL
of strained WS2 monolayer.

The crystal structure of unstrained WS2 monolayer has a 3-fold rotation
symmetry $C_{3z}$, under which the OBCD $D_{ab}$ ($a=x,y$, and $b=z$) vanishes
for any $\omega$. Therefore any finite Berry dipole requires the breaking of
$C_{3z}$ which can be realized by an externally applied uniaxial strain field
$u_{xx}$ and $u_{yy}$. Figures 2 shows the calculated band structure, OBCD,
and $g_{EL}$ of WS2 monolayer under different strain conditions based on
first-principles methods (see Methods section for details). As is shown there,
the calculated OBCD and $g_{EL}$ vanish for the unstrained case but become
nonzero with finite $u_{xx}$ or $u_{yy}$ which is consistent with the symmetry
analysis.

On the other hand the strain-induced OBCD and $g_{EL}$ are of opposite sign
between $u_{xx}$ and $u_{yy}$ as shown in Figs. 2(c)-(d) which enables strain
engineering of CP EL in WS2 monolayer. According to the semiclassical theory,
the applied strain can induce an effective gauge potential [37, 38]

$\bm{A}^{\text{eff}}=\tau_{z}\frac{\beta}{a_{0}}(-2u_{xy},u_{xx}-u_{yy})$ (8)

where $a_{0}$ is the lattice constant, $\beta$ is a parameter proportional to
the electron-phonon coupling strength, and $u_{ab}$ ($a,b=x,y$) refers to the
applied external strain tensor. $\tau_{z}=\pm 1$ is the valley index
indicating $K$ and $K^{\prime}$ valleys, respectively, reflecting the fact
that the induced effective gauge potential are of opposite signs between two
valleys, which is a manifestation of the TRS. According to Eq. (8) the gauge
potentials induced by different uniaxial strains $u_{xx}$ and $u_{yy}$ are of
the opposite signs which moves the conduction and valance bands along opposite
directions as is shown in Figs. 2(a)-(b). As a result the induced OBCD and
$g_{EL}$ are of opposite sign for nonzero $u_{xx}$ and $u_{yy}$ as is shown in
Figs. 2(c)-(d). The calculated value of $g_{EL}$ depends on magnetude of the
applied field $\mathcal{E}$ and quality of sample (i.e. relaxation time
$\tau$). For moderate parameters $|\bm{\mathcal{E}}|=10^{4}$ V/m and
$\tau=\hbar/\eta$ with $\eta=0.1$ eV the calculated maximum $g_{EL}$ is 0.1 at
around 3.8 eV which is comparable to the experimental value of polarization
rate between 9% and 24% [31]. The $g_{EL}$ can be further enhanced by using
larger electrical field or improving the sample quality according to Eq. (7).
However, it should be noted that $g_{EL}$ cannot become infinitely large with
increasing $|\bm{\mathcal{E}}|$ and $\tau$ because it is bounded as
$|g_{EL}|\leq 2$ by its definition. This is because the relaxation time
approximation in Eq. (4) is valid only under moderate $|\bm{\mathcal{E}}|$ and
$\tau$ which can be justified by the fact that the values of occupations
$f_{\alpha\bm{k}}$ and $f^{0}_{\alpha\bm{k}}$ must be bounded by $0\leq
f_{\alpha\bm{k}},f^{0}_{\alpha\bm{k}}\leq 1$ so that
$\tau\frac{e}{\hbar}\left|\bm{\mathcal{E}}\cdot\bm{\nabla_{k}}f^{0}_{\alpha\bm{k}}\right|=|f_{\alpha\bm{k}}-f^{0}_{\alpha\bm{k}}|\leq
1$. For significantly large values of $|\bm{\mathcal{E}}|$ and $\tau$, the
higher-order terms in $O(\tau^{2})$ and the achiral magnetic dipole
contributions $O(m^{2})$ of Eq. (2) must be included [39]. The ACPLE in WS2
monolayer exhibit rich switchable properties controlled by the direction of
applied electrical current, strain, and emitting directions which is favorable
for future optoelectronic device applications.

Figure 2: Anomalous CP EL in strained WS2 monolayer. (a)-(b) Evolution of band
structures of WS2 monolayers under unidirectional strain $u_{xx}$ and
$u_{yy}$, respectively. (c)-(d) Calculated OBCD and $g_{EL}$ for different
strains. Under uniaxial strain $u_{xx}$ and $u_{yy}$, $D_{xz}(\omega)$
vanishes because of the combined mirror-time-reversal symmetry
$M_{y}\mathcal{T}$. For $g_{EL}$, parameters $|\bm{\mathcal{E}}|=10^{4}$ V/m
and $\tau=6.6$ fs are used in the calculations.

##  IV Anomalous CP PL in magnetic materials

CrI3 is a van der Waals magnetic material with (anti-) ferromagnetic
(interlayer) intralayer coupling [40, 41]. Its monolayer exhibit in-plane
ferromagnetism which breaks the TRS. The crystal structure has inversion
symmetry which prohibit the ordinary term in Eq. (2) so that only the
anomalous term contributes to the CP luminescence.

We calculate the anomalous CP PL in the ferromangetic CrI3 monolayer. The
anomalous CP PL is determined by the OBC $\Omega(\bm{k},\omega)$ which is the
product of interband Berry curvature $\Omega^{vc}_{\bm{k}}$ and the joint
density of states (JDOS)
$g^{cv}_{j}(\bm{k},\omega)=\delta(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}-\hbar\omega)$.
Figure 2(a) shows the electronic band structure of CrI3 monolayer along high-
symmetry lines (HSLs) with a direct band gap of 0.85 eV at $\Gamma$ point.
Derived from the band structure and Bloch wave functions along the HSLs, we
plotted the optical excitation energy spectrum
$\hbar\omega=\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}$ together with
$\bm{k}$-resolved optical Berry curvature
$\Omega(\bm{k},\omega)=\sum_{c,v}\Omega^{vc}_{z}(\bm{k},\omega)$ along high-
symmetry lines (HSLs) in Fig. 2(b). The interband Berry curvature generally
decreases with increasing $\omega$ because
$\Omega^{cv}_{\bm{k}}\propto\frac{1}{(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}})^{2}}\propto\frac{1}{\omega^{2}}$,
however, the JDOS $g^{cv}_{j}(\omega)$ generally increase with $\omega$
because of the increasing number of channels. As a result, there exist a
maximum value of the optical Berry curvature $\Omega(\omega)$ as function of
$\omega$ as is shown in the red line of Fig. 2(c). We also calculated the
dissymmetry factor for the CP PL

$g_{PL}=2\times\frac{I_{L}-I_{R}}{I_{L}+I_{R}}=\frac{2\Omega(\omega)}{I_{0}}$
(9)

as shown in the blue line of Fig. 2(c), where
$\Omega(\omega)=\sum_{c,v,\bm{k}}\Omega^{vc}_{z}(\bm{k},\omega)$ refers to the
total OBC. It should be noted that the anomalous CP PL in Eq. (9) is
determined only by band structure properties in contrast to the anomalous CP
EL which also depends on relaxation time and external electrical field. Figure
2(d) shows the $\bm{k}$-resolved distribution of the optical Berry curvature
$\Omega(\bm{k},\omega)$ at the peak value $\omega=1.14$ eV of the curve of
optical Berry curvature $\Omega(\omega)$, which indicates the most prominent
contribution from the highest valance band and the second lowest conduction
band. The calculated maximum $g_{PL}$ of CrI3 monolayer approaches about 0.42
at $\omega\approx 1.14$ eV which matches the order of magnitude with the
experimental $g\approx 1$ at $\omega=1.15$ eV [32]. Here, reversing the
magnetization lead to opposite $\Omega(\omega)$ and thus opposite $g_{PL}$.

Figure 3: Anomslous CP PL in magnetic CrI3 monolayer. (a) Band sturcture of
CrI3 monolayer whose crystal structure and Brillouin zone are shown in the
insets. (b) Distribution of optical Berry curvature dipole
$\Omega(\bm{k},\omega)$ along hight-symmetry lines. (c) Integrated total
optical Berry curvature
$\Omega(\omega)=\sum_{\bm{k}}\Omega(\bm{k},\omega)=\sum_{c,v,\bm{k}}\Omega^{vc}_{\bm{k}}\delta(\varepsilon_{c\bm{k}}-\varepsilon_{v\bm{k}}-\hbar\omega)$
and dissymmetry factor $g_{PL}=2(I_{L}-I_{R})/(I_{L}+I_{R})$ as functions of
$\omega$. The peak value of $g_{PL}\sim 0.42$ at $\omega=1.14$ is comparable
to the experimental value of maximum $g$ value of around 1 at $\omega=1.15$ eV
[32]. (d) Momentum resolved optical Berry curvature distribution
$\Omega(\bm{k},\omega)$ at the peak value of $\omega=1.14$ eV.

##  V Discussion

Because of the intimate relation between Berry curvature (dipole) with the
linear (nonlinear) anomalous Hall effect (AHE) [17, 42, 24], the ACPLE
satisfies a sum rule,

$\begin{split}\sigma^{\textrm{AH}}=\frac{1}{\pi}\int^{+\infty}_{0}d\omega~{}\frac{I_{L}-I_{R}}{f_{B}(\hbar\omega)}\end{split}$
(10)

where $\sigma^{\textrm{AH}}$ refer to the anomalous Hall conductivity in the
plane perpendicular to the wave vector of light $\hat{\bm{q}}$. For 2D
semiconductors/insulators, the linear AHE conductivity is quantized in
multiples of $\frac{e^{2}}{h}$. Thus the integrated ACPLE spectrum give rise
to topologically quantized signals which awaits for future experimental
verification.

The emission-direction dependence of ACPLE is a characteristic feature
compared to the ordinary CPLE. In literature, CPLE is usually studied with the
exciton-coupling model [43, 44] which cannot rationalize the current-direction
and emission-direction dependence of ACPLE.

The emission-direction has a significant impact to enhance the polarization of
a LED device. Figure 4 shows the geometry of back-reflective LED. For the
ordinary CPLE, the emitted CP light is of the same handedness for both
directions. The handedness changed after back reflection(Fig. 4(a)). Therefore
the net polarization diminishes when both light beams mix together. On the
other hand, for the ACPLE, the handedness of the CP light for both directions
are opposite. After reflection, both light beams exhibit the same handedness,
avoiding the cancellation of polarization (Fig. 4(b)). Furthermore, the
current-direction dependence enables high-speed switching of the light
handedness by electric control, which essential for future optical quantum
technology.

##  VI Summary

In summary, we proposed the quantum theory of anomalous circular polarized
light emission based on the Berry curvature and Berry curvature dipole. Our
calculations on WS2 and CrI3 monolayers are consistent with recent
experimental results. We propose that experiments can further detect the
emission-direction dependence of the light handedness. Our findings deepen the
understanding of band topology induced exotic optical phenomena and promise
technology applications for high-speed, handedness-tunable optoelectrics.

Figure 4: Comparison between (a) ordinary CPL and (b) ACPLE in the presence of
reflective substrates. The net polarization of ordinary CPL is suppressed
while it is enhanced for the ACPLE.

##  VII Methods

First-principles calculations. Our first-principles calculations are based on
density functional theories as implemented in Vienna ab initio Simulation
Packages (VASP) [45]. The generalized gradient approximation of Perdew-Burke-
Ernzerhof type [46] has been used for the exchange-correlation functionals.
The tight-binding Hamiltonians of CrI3 and WS2 monolayers are constructed
using the code Wannier90 [47, 48, 49].

###### Acknowledgements.

We thank Daniel Kaplan and Hengxin Tan for helpful discussions. B.Y.
acknowledges the financial support by the European Research Council (ERC
Consolidator Grant “NonlinearTopo”, No. 815869) and the Minerva foundation
with funding from the Federal German Ministry for Education and Research.

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