# Interaction-Enhanced Superradiance of a Ryderg-Atom Array

Yiwen Han  CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei 230026, China  Haowei Li  CAS Key Laboratory of
Quantum Information, University of Science and Technology of China, Hefei
230026, China  Wei Yi<EMAIL_ADDRESS>CAS Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei 230026,
China  CAS Center For Excellence in Quantum Information and Quantum Physics,
Hefei 230026, China  Hefei National Laboratory, University of Science and
Technology of China, Hefei 230088, China

###### Abstract

We study the superradiant phase transition of an array of Rydberg atoms in a
dissipative microwave cavity. Under the interplay of the cavity field and the
long-range Rydberg interaction, the steady state of the system exhibits an
interaction-enhanced superradiance, with vanishing critical atom-cavity
coupling rates at a discrete set of interaction strengths. We find that, while
the phenomenon can be analytically understood in the case of a constant all-
to-all interaction, the enhanced superradiance persists under typical
experimental parameters with spatially dependent interactions, but at modified
critical interaction strengths. The diverging susceptibility at these critical
points is captured by emergent quantum Rabi models, each of which comprises a
pair of collective atomic states with different numbers of atomic excitations.
These collective states become degenerate at the critical interaction
strengths, resulting in a superradiant phase for an arbitrarily small atom-
cavity coupling.

_Introduction._ Combining strong long-range interaction, flexible spatial
configuration, and large polarizability, Rydberg-atom arrays are emerging as
an increasingly important platform for quantum simulation and computation [3,
4, 2, 1]. During the past decade, a vast range of intriguing many-body
phenomena have been simulated and investigated in Rydberg-atom arrays,
including magnetism and dynamics in quantum spin models [8, 6, 5, 7, 9],
symmetry protected topological phases [10], coherent excitation transfer [11,
12, 13], as well as emergent gauge field [14, 15, 16, 17] and many-body
localization [19, 18]. Furthermore, the recent demonstration of error
suppression using logical qubits on reconfigurable Rydberg-atom arrays
represents a seminal first step toward an era of error-corrected intermediate-
size quantum systems [20]. On the other hand, while the Rydberg states are
relatively long-lived, dissipation in Rydberg atoms is further tunable through
a variety of means, including laser-induced loss, microwave dressing, or
coupling to a cavity. As such, Rydberg atoms are also ideal for the study of
dissipative many-body dynamics [21, 22, 23, 24, 25, 26, 27]. A particularly
interesting configuration is the atom-cavity hybrid system involving Rydberg
states [28, 29], where the cavity-induced long-range interatomic interactions
and the back action of cavity dissipation conspire with the Rydberg
interactions, leading to exotic non-equilibrium dynamics.

For atoms coupled to a cavity, an outstanding phenomenon is the superradiant
phase transition, which can be traced back to the study of the Dicke model
half a century ago [30, 31]. In the Dicke model, an ensemble of two-level,
non-interacting atoms are coupled to a single-mode electromagnetic field,
corresponding to the atom-cavity coupling in the aforementioned hybrid system.
A superradiant phase transition occurs beyond a critical atom-cavity coupling
strength, wherein the cavity field becomes macroscopically populated [31, 32,
33, 34]. While the superradiant transition also arises in open quantum systems
[36, 37, 35], the advent of ultracold quantum gases further enriches its study
[38, 39, 40, 41, 42]. In particular, the transition was observed in a Bose-
Einstein condensate of atoms coupled to a dissipative cavity field, where,
driven by the collective light-matter interaction, the macroscopic population
of the steady-state cavity field is accompanied by a self-organized density
pattern in the condensate [43]. By contrast, a Fermi-surface-nesting-enhanced
superradiance was predicted and subsequently observed for degenerate Fermi
gases in a cavity [44]. In these studies, the interatomic interactions are
weak at best. Whether atomic interactions, particularly the long-range
interactions of Rydberg states, can have a significant impact on the steady-
state superradiant transition, remains to be explored.

Figure 1: Schematic illustration of Rydberg atoms in a microwave cavity. The
green stripes indicate the optical tweezers, used for trapping and arranging
the atom array. The dissipative cavity, characterized by a decay rate
$\kappa$, off-resonantly couples two Rydberg sates
$|s\rangle\leftrightarrow|p\rangle$. Microscopic illustration of the resonant
dipole-dipole interaction $V_{\text{dd}}$ is shown to the right.

In this work, we reveal a Rydberg-interaction-enhanced superradiance in an
array of atoms with microwave-cavity assisted Rydberg-state coupling. By
studying the steady-state superradiance of the atom-cavity hybrid system, we
show that the critical atom-cavity coupling rate for the superradiant
transition vanishes at a series of discrete interaction strengths, suggesting
divergent susceptibility at these critical points. Invoking a simplified model
with a constant all-to-all interaction, we derive an analytic expression for
the critical points, and find that the divergence of susceptibility thereof
coincides with the two-fold degeneracies of symmetric collective states in the
subspaces with $n$ and $n+1$ excitations, respectively. We then demonstrate,
through numerical simulations, that the interaction-enhanced superradiance
persists under typical experimental conditions, where both the Van der Waals
interaction and the spatial dependence of these interactions are considered.
Compared to the constant-interaction case, the critical interaction strengths
are modified, but a similar set of degenerate collective states also play a
key role. Notably, these collective states are no longer symmetric due to the
spatial dependence of the interaction potential. In either case, we show that
the enhanced superradiance near a critical point can be captured by an
emergent quantum Rabi model, which comprises the relevant near-degenerate
collective states.

_System configuration._ We consider a one dimensional Rydberg-atom array with
a lattice constant $R_{0}$, confined by optical tweezers in a single-mode
microwave cavity along the horizontal $x$ direction, as illustrated in Fig. 1.
For each atom, we consider two Rydberg states, denoted respectively as
$\left|s\right\rangle$ and $\left|p\right\rangle$. The Rydberg states are
coupled by the cavity mode characterized by frequency $\omega_{c}$ and
wavelength $\lambda_{c}$. In the case that $R_{0}$ is much smaller than
$\lambda_{c}$, the cavity field is identical for all atoms, leading to a
homogeneous atom-cavity coupling rate $\pazocal{G}$. The Rydberg atoms
interact through the resonant dipole interaction
$V\left(R_{jk}\right)=C_{3}/R_{jk}^{3}$, where $C_{3}$ is the dipole-dipole
interaction parameter, and $R_{jk}$ is the distance between the $j$ th and $k$
th atoms along the chain. Note that, although the Van der Waals interactions
become stronger than the dipole interaction at short interatomic distances,
they do not qualitatively modify the interaction-enhanced superradiance (see
later discussions). We therefore neglect the Van der Waals interactions for
now.

We start by studying the simplified case where the dipole-dipole interaction
is approximated by a constant all-to-all interaction
$V\left(R_{jk}\right)=V_{\text{dd}}$. The simplified atom-cavity coupling
Hamiltonian is a generalized Dicke model with dipole interactions (we take
$\hbar=1$) [45, 35, 46, 47]

$\hat{H}=\hat{H}_{\text{atom}}+\omega_{c}\hat{a}^{\dagger}\hat{a}+\sqrt{\frac{2}{N}}\pazocal{G}\sum_{j=1}^{N}\left(\hat{a}^{\dagger}+\hat{a}\right)\hat{\sigma}_{j}^{x},$
(1)

where the atomic Hamiltonian

$\hat{H}_{\text{atom}}=\frac{\omega_{a}}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}+\sum_{j<k}V_{\text{dd}}\left(\hat{\sigma}_{j}^{-}\hat{\sigma}_{k}^{+}+\text{H.c.}\right).$
(2)

Here $N$ and $\omega_{a}$ are the total atom number and the energy-level
difference between the two Rydberg states, respectively, $\hat{a}$ and
$\hat{a}^{\dagger}$ are the annihilation and creation operators of the cavity
photon, and $\hat{\sigma}_{j}^{\pm}=\hat{\sigma}_{j}^{x}\pm
i\hat{\sigma}_{j}^{y}$, and $\hat{\sigma}_{j}^{\beta}$ ($\beta=x,y,z$) are the
Pauli operators for the $j$ th atom, defined through
$\hat{\sigma}^{+}_{j}=|p\rangle\langle s|_{j}$,
$\hat{\sigma}^{-}_{j}=|s\rangle\langle p|_{j}$,
$\hat{\sigma}^{z}_{j}=|p\rangle\langle p|_{j}-|s\rangle\langle s|_{j}$, with
$|p\rangle_{j}$ and $|s\rangle_{j}$ indicating the Rydberg states of the $j$
th atom.

Importantly, we consider an off-resonant coupling between the microwave cavity
field and the inter-Rydberg-state transition
$|s\rangle\leftrightarrow|p\rangle$, so that the counter-rotating terms are
retained in Eq. (1). Further, since the dependence of the Hamiltonian on the
atomic degrees of freedom is fully accounted for by the collective spin
operators $\hat{S}^{\beta}=\sum_{j}\hat{\sigma}_{j}^{\beta}/2$ and
$\hat{S}^{\pm}=\sum_{j}\hat{\sigma}_{j}^{\pm}$, we can rewrite Hamiltonian (1)
as

$\hat{H}=\omega_{a}\hat{S}^{z}+V_{\text{dd}}\hat{S}^{+}\hat{S}^{-}+\omega_{c}\hat{a}^{\dagger}\hat{a}+2\sqrt{\frac{2}{N}}\pazocal{G}\left(\hat{a}^{\dagger}+\hat{a}\right)\hat{S}^{x}.$
(3)

In a realistic system, dissipation is inevitable through its coupling to the
environment. For our setup, the lifetime of the Rydberg states are enhanced as
the cavity mode suppresses the effects of the blackbody radiation on the
Rydberg states. We further assume a negligible spontaneous decay rate from the
Rydberg states $|p\rangle$ and $|s\rangle$ to the electronic ground state, so
that the dominant dissipation channel is the cavity decay. The dynamics of the
atom-cavity hybrid system is then governed by the Lindblad master equation

$\dot{\hat{\rho}}=-i\left[\hat{H},\hat{\rho}\right]+\kappa\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}\right),$
(4)

where $\hat{\rho}$ is the density matrix containing both the atomic and
cavity-photon degrees of freedom, and $\kappa$ is the cavity decay rate.

_Interaction-enhanced superradiance._ We focus on the superradiant transition
in the steady state of the atom-cavity hybrid system. Adopting a mean-field
approximation on the cavity field, we replace $\hat{a}$ with the expectation
value $\alpha=\langle\hat{a}\rangle$, which, under the stationary condition
$\partial\alpha/\partial t=0$, is given by

$\alpha=\frac{-i2\sqrt{\frac{2}{N}}\pazocal{G}}{i\omega_{c}+\kappa}\left\langle\hat{S}^{x}\right\rangle.$
(5)

The steady state is solved self-consistently by numerically diagonalizing
Hamiltonian (3) while enforcing Eq. (5).

Figure 2: (a) The steady-state cavity photon number as a function of the atom-
cavity coupling rate for different interaction strengths. Solid lines
represent results under the mean-field approximation, while dash-dotted lines
are obtained by evolving the quantum master equation (4) for sufficiently long
times, starting from the initial direct-product state $\left|ss\dots
s\right\rangle\otimes\left|0\right\rangle$ of the atom-cavity hybrid system.
Here $|0\rangle$ is the vacuum state of cavity photons. Different colors
correspond to different interaction strengths. (b)(c) Steady-state phase
diagrams, characterized by the cavity photon number and the collective atom
correlation $\hat{S}_{x}$, respectively. Here “SR” stands for the superradiant
phase, and “NP” stands for the normal phase. (d) Impact of the cavity
dissipation on the critical atom-cavity coupling rate of the superradiant
transition. Different colors and line shapes correspond to different cavity
decay rates. For all calculations, we take $\omega_{a}$ as the unit of energy,
with other parameters given by $N=6$, $\omega_{c}=0.75$, and $\kappa=0.25$
[for (a)(b)(c)].

In Fig. 2(a), we show the numerically calculated steady-state cavity photon
number $\langle\hat{a}^{\dagger}\hat{a}\rangle=|\alpha|^{2}$ (solid curves)
with increasing atom-cavity coupling strength $\pazocal{G}$, for an array of
$N=6$ atoms. Under most of the interaction strengths, as well as in the non-
interacting case, a superradiant transition can be identified at a critical
$\pazocal{G}_{c}$, across which the cavity field starts to become finite.
Importantly, the sign and strength of the Rydberg interaction $V_{\text{dd}}$
have a significant impact on the superradiant transition. As a dramatic
example, at $V_{\text{dd}}=-1/3$, the critical $\pazocal{G}_{c}$ vanishes—the
system becomes superradiant even for an arbitrarily small atom-cavity coupling
rate. The mean-field results above are consistent with those from a full-
quantum calculation [see dashed curves in Fig. 2(a)], for which we evolve the
Lindblad equation (4) for a sufficiently long time so that the system is close
to the steady state. While the superradiant phase transitions from the full-
quantum treatment are not as sharp compared to the mean-field results, at the
critical value of $V_{\text{dd}}=-1/3$, both calculations indicate a vanishing
$\pazocal{G}_{c}$.

The overall interaction dependence of the superradiant transition is more
transparent in Fig. 2(b)(c), where the color contour respectively indicates
the steady-state cavity photon number and the collective atom correlation
$\langle\hat{S}_{x}\rangle$, on the $1/V_{\text{dd}}$–$\pazocal{G}$ plane.
Consistent with the collective nature of superradiance, the macroscopic
population of the cavity field is always accompanied by the emergence of
collective atomic correlations. Furthermore, we find that repulsive
interactions ($V_{\text{dd}}>0$) tend to suppress the transition, leading to
an ever increasing $\pazocal{G}_{c}$ for larger $V_{\text{dd}}$. By contrast,
under attractive interactions ($V_{\text{dd}}<0$), $\pazocal{G}_{c}$ exhibits
an undulating pattern. It vanishes at some discrete integer values of
$1/V^{c}_{\text{dd}}$, indicating interaction-enhanced superradiance, as first
observed in Fig. 2(a). Furthermore, while the critical coupling strength
$\pazocal{G}_{c}$ generally depends on the cavity-decay rate $\kappa$, the
critical interaction strengths $V^{c}_{\text{dd}}$ where $\pazocal{G}_{c}$
vanishes are independent of the cavity decay [see Fig. 2(d)]. While such
independence has potential significance for engineering superradiant
transitions in bad cavities, it also strongly suggests an underlying mechanism
hinged upon the atomic correlations.

_An emergent quantum Rabi model._ In the standard Dicke model, the
superradiant transition can be analyzed by dividing the Hilbert space into
subspaces with different numbers of atomic excitations, and the transition is
found to be dominated by symmetric states in each subspace [49, 48]. Of
particular importance are the ground state and the first excited state, the
former with no atomic excitations and the latter a symmetric superposition of
all instances of a single excitation. The critical atom-cavity coupling rate
of the superradiant transition is related to the energy difference between
these two states. Now the addition of Rydberg interactions would modify the
spectral outlook of the collective atomic excitations, giving rise to not only
level shifts but also degeneracies in the low-lying collective states. As we
show below, it is this interaction-induced low-energy degeneracy that is
directly responsible for the interaction-enhanced superradiance.

With the physical understanding above, we define the states
$\\{|\psi_{n}\rangle\\}$, which are the symmetric superpositions of all
possible $n$-atom excitations, with

$|\psi_{n}\rangle=\frac{1}{\sqrt{C_{N}^{n}}}\sum_{\\{j\dots k\\}}|s\dots
p_{j}\dots p_{k}\dots s\rangle,$ (6)

where $\\{j\dots k\\}$ in the summation denotes all possible combinations of
$n$ excitation locations. In the absence of the cavity field, the
corresponding eigenenergies for these collective states are
$\omega_{n}=-\left(\frac{N}{2}-n\right)\omega_{a}+n\left(N-n\right)V_{\text{dd}}$.
As illustrated in Fig. 3(a), with increasing attractive interaction
$V_{\text{dd}}$, the energies of some symmetric excited states shift downward
significantly, and the ground state becomes degenerate with at a discrete set
of interaction strengths

$V^{c}_{\text{dd}}(n)=-\frac{\omega_{a}}{N-\left(2n+1\right)},$ (7)

where $n\leq N/2$. More specifically, at each of the critical interaction
strength $V^{c}_{\text{dd}}(n)$, the two-fold degenerate ground-state subspace
is spanned by $|\psi_{n}\rangle$ and $|\psi_{n+1}\rangle$. It follows that the
superradiant transition near $V^{c}_{\text{dd}}(n)$ is dominated by the two
near-degenerate states $|\psi_{n}\rangle$ and $|\psi_{n+1}\rangle$. To
quantitatively confirm this point, we divide the overall range of
$1/V_{\text{dd}}$ into different regions, centered at the critical points with
vanishing $\pazocal{G}_{c}$. These regions are separated by the vertical
dotted lines in Fig. 3(b). As the background of Fig. 3(b), we show the color
contour of the overlap $P^{\left(n,n+1\right)}$ between the steady state
(denoted as $|\psi_{s}\rangle$) and the ground-state manifold at the critical
$V^{c}_{\text{dd}}$ of the corresponding region. For instance, we consider
$\\{|\psi_{0}\rangle,|\psi_{1}\rangle\\}$ in region I,
$\\{|\psi_{1}\rangle,|\psi_{2}\rangle\\}$ in II,
$\\{|\psi_{2}\rangle,|\psi_{3}\rangle\\}$ in III, and
$P^{\left(n,n+1\right)}=\left|\langle\psi_{n}|\psi_{s}\rangle\right|^{2}+\left|\langle\psi_{n+1}|\psi_{s}\rangle\right|^{2}$
in the corresponding region. Apparently, $P^{\left(n,n+1\right)}$ is close to
unity in the broad vicinity of the superradiant phase transition.

The observation above inspires us to examine an emergent quantum Rabi model

$\displaystyle\hat{H}_{\text{eff}}^{\left(n,n+1\right)}=$
$\displaystyle\frac{\Delta_{n}}{2}\left(\left|\psi_{n+1}\right\rangle\left\langle\psi_{n+1}\right|-\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right|\right)+\omega_{c}\hat{a}^{\dagger}\hat{a}$
$\displaystyle+\sqrt{\frac{2}{N}}\pazocal{G}\eta_{n}\left(\hat{a}^{\dagger}+\hat{a}\right)\left(\left|\psi_{n+1}\right\rangle\left\langle\psi_{n}\right|+\text{H.c.}\right),$
(8)

where $\Delta_{n}=\omega_{n+1}-\omega_{n}$ is the energy-level difference
between the lowest-lying states $|\psi_{n}\rangle$ and $|\psi_{n+1}\rangle$,
and
$\eta_{n}=\left\langle\psi_{n+1}\right|2\hat{S}_{x}\left|\psi_{n}\right\rangle=\sqrt{\left(N-n\right)\left(n+1\right)}$
is the modifier for the atom-cavity coupling in the symmetric-state subspace.
The corresponding superradiant tranition point for the quantum Rabi model is
analytically given as [50]

$\pazocal{G}_{c}=\frac{1}{2\eta_{n}}\sqrt{\frac{N\left|\varDelta_{n}\right|}{2}\frac{\omega_{c}^{2}+\kappa^{2}}{\omega_{c}}}.$
(9)

Naturally, when the states $|\psi_{n}\rangle$ and $|\psi_{n+1}\rangle$ become
degenerate at the critical interaction strengths, we have $\Delta_{n}=0$ and
$\pazocal{G}_{c}=0$, meaning a divergent susceptibility and superradiance for
an arbitrarily small atom-cavity coupling.

Figure 3: (a) Schematic illustration of the energy shift and degeneracies of
different symmetric states (bold lines) with different interaction strengths.
The left subplot illustrates the energy levels of the Dicke model with
$V_{\text{dd}}=0$. From bottom to top, the colored bold lines correspond to
subspaces with $n=0,1,2,3$, respectively, where collective states in the same
subspace have the same energy. The subplot on the right illustrates the energy
levels at the critical interaction strengths $1/V^{c}_{\text{dd}}=-5,-3,-1$
(left, middle, right), respectively. Here the bold lines indicate the energy
levels of the symmetric states, their colors indicate different excitation
numbers, consistent with those in the case of $V_{\text{dd}}=0$ (left
subplot). The thin grey lines denote non-symmetric collective states. (b) The
critical atom-cavity coupling rate for the superradiant transition with
varying interaction strengths, calculated from the effective quantum Rabi
models (dashed lines) and the full Hamiltonian (solid lines), respectively
(see main text for detailed discussion). The three regions (marked with I, II,
and III), separated by the vertical dotted lines at $1/V_{\text{dd}}=-4,-2$,
are centered at the critical interaction strengths
$1/V^{c}_{\text{dd}}=-5,-3,-1$. The color contour in the background represents
the overlap $P^{\left(n,n+1\right)}$ between the steady state and the
degenerate ground-state manifold at the central critical interaction strength
$V^{c}_{\text{dd}}$ of the corresponding region. Other parameters and the unit
of energy are the same as those in Fig. 2(b)(c). Figure 4: (a) Energies
$\omega_{n}$ of the low-lying collective states $|\psi_{n}\rangle$ within the
$n$-excitation subspace. Locations of degeneracy are highlighted with in
dashed yellow circles. (b) The critical atom-cavity coupling rate for the
superradiant transition with varying interaction strengths, is calculated from
the effective quantum Rabi models (dashed lines) and the full Hamiltonian
(solid lines), respectively. The three regions (marked with I, II, III,),
separated by the vertical dotted lines, correspond to regions where different
collective states dominate. For our calculations here, we take typical
experimental parameters [52]. Specifically, we take the Rydberg states
$|60S_{1/2},m_{j}=1/2\rangle$ and $|60P_{1/2},m_{j}=-1/2\rangle$ of 87Rb as
$|s\rangle$ and $|p\rangle$, respectively. Accordingly,
$C_{3}/\hbar\omega_{a}=-0.57~{}\upmu\text{m}^{3}$,
$C_{6}^{pp}/\hbar\omega_{a}=-11.48~{}\upmu\text{m}^{6}$,
$C_{6}^{ss}/\hbar\omega_{a}=51.10~{}\upmu\text{m}^{6}$,
$C_{6}^{sp}/\hbar\omega_{a}=-1.00~{}\upmu\text{m}^{6}$, and $N=6$,
$\omega_{c}/\omega_{a}=0.75$, $\kappa/\omega_{a}=0.25$, with
$\omega_{a}=16.7~{}\text{GHz}$.

To further illustrate the utility of the emergent quantum Rabi model (8), we
examine the effectiveness of the model across the critical points. Again, we
apply quantum Rabi models associated with the central critical point of each
region. Specifically, we apply $\hat{H}_{\text{eff}}^{\left(0,1\right)}$ in
region I, $\hat{H}_{\text{eff}}^{\left(1,2\right)}$ in region II, and
$\hat{H}_{\text{eff}}^{\left(2,3\right)}$ in region III. Following Eq. (9),
the superradiant transition points are given by the dashed curves in Fig.
3(b). The phase boundaries from the quantum Rabi models agree well with those
from the full Hamiltonian (3), when the interaction parameter is close to
$V_{\text{dd}}^{c}$. The agreement deteriorates near the region boundaries,
where we expect high-lying symmetric excitations to play a non-negligible
role.

_Realistic interaction potentials._ Under typical experimental conditions, two
types of interactions coexist that differ in character: (i) the resonant
dipole-dipole interactions considered above; (ii) the off-resonant Van der
Waals interactions between the Rydberg states [10, 51, 52, 53]. The two types
of interactions also differ in their spatial dependence, with the former $\sim
1/R_{jk}^{3}$ and the latter $\sim 1/R_{jk}^{6}$. As a result, the full atomic
Hamiltonian is given by

$\hat{H}_{\text{atom}}=\frac{\omega_{a}}{2}\sum_{j=1}^{N}\hat{\sigma}_{j}^{z}+\sum_{j<k}\frac{V_{\text{dd}}}{\left|j-k\right|^{3}}\left(\hat{\sigma}_{j}^{-}\hat{\sigma}_{k}^{+}+\text{H.c.}\right)+\hat{H}_{\text{VdW}},$
(10)

where $V_{\text{dd}}=C_{3}/R_{0}^{3}$, and

$\displaystyle\hat{H}_{\text{VdW}}=$
$\displaystyle\sum_{j<k}\left[\frac{V_{pp}}{\left|j-k\right|^{6}}\hat{P}_{j}^{\uparrow}\hat{P}_{k}^{\uparrow}+\frac{V_{ss}}{\left|j-k\right|^{6}}\hat{P}_{j}^{\downarrow}\hat{P}_{k}^{\downarrow}\right.$
(11)
$\displaystyle+\left.\frac{V_{sp}}{\left|j-k\right|^{6}}\left(\hat{P}_{j}^{\uparrow}\hat{P}_{k}^{\downarrow}+\hat{P}_{j}^{\downarrow}\hat{P}_{k}^{\uparrow}\right)\right].$
(12)

Here the Van der Waals interactions are characterized by
$V_{pp}=C_{6}^{pp}/R_{0}^{6}$, $V_{ss}=C_{6}^{ss}/R_{0}^{6}$, and
$V_{sp}=C_{6}^{sp}/R_{0}^{6}$, with the projectors
$\hat{P}_{j}^{\uparrow/\downarrow}=\left(\hat{\sigma}_{j}^{z}\pm 1\right)/2$.
In the following, we use the interatomic distance $R_{0}$ to tune the
different interaction terms in a consistent manner.

In this case, the homogeneity of the configuration is broken, and the
symmetric-state picture above no longer applies. Nevertheless, we find that
the interaction-enhanced superradiance persists, with vanishing
$\pazocal{G}_{c}$ at discrete values of $R_{0}$, which is explicitly
demonstrated in Fig. 4. The phenomenon is closely related to the lowest-energy
states in each $n$-excitation subspace. We define these states as
$|\psi_{n}\rangle$, and plot their energies in Fig. 4(a). Interestingly, the
locations of degenerate low-lying collective states coincide with those of
vanishing $\pazocal{G}_{c}$. This strongly suggests the applicability of the
emergent quantum Rabi models (8) near these critical points (with updated
coefficients $\Delta_{n}$ and $\eta_{n}$). In Fig. 4(b), we show the resultant
phase boundaries under Eq. (9) as dashed curves, which are in good agreement
with the full mean-field calculations near the critical points (purple solid).
Apparently, the interaction-enhanced superradiance here is also due to the
near degeneracy of the low-lying collective states in the atomic sector.

_Discussion._ We show that interatomic interactions can dramatically enhance
the superradiant phase transition for Rydberg atoms in a dissipative cavity.
Specifically, near a discrete set of interaction strengths, the susceptibility
of the system diverges, and the steady state becomes superradiant for an
arbitrarily small atom-cavity coupling rate. The phenomenon is attributed to
the interaction-induced degeneracy of low-lying collective atomic excitations,
which allow for the application of emergent quantum Rabi models near these
critical points.

For future studies, it would be interesting to explore the impact of
interaction on the superradiant transitions for Rydberg arrays in higher
dimensions. Understanding such processes can prove useful for the ongoing
study of quantum computation and simulation with neutral-atom arrays.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant No. 12374479) and the Innovation Program for Quantum Science and
Technology (Grant No. 2021ZD0301200).

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## Supplemental Material for “Interaction-Enhanced Superradiance of a Ryderg-
atom array”

##  I Symmetric state subspace

For superradiance, states with symmetric atomic excitations play a crucial
role. To see this, we first define a symmetric subspace composed of all
symmetric states $\\{|\psi_{n}\rangle\\}$, and project the Hamiltonian into
the subspace to obtain $\hat{H}_{\text{eff}}=\hat{P}\hat{H}\hat{P}$, with
$\hat{P}=\sum_{n=1}^{N}\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right|$.
The effective Hamiltonian can be written as

$\hat{H}^{[0,N_{c}]}_{\text{eff}}=\sum_{n=1}^{N_{c}}\omega_{n}\left(\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right|\right)+\omega_{c}\hat{a}^{\dagger}\hat{a}+\sqrt{\frac{2}{N}}\pazocal{G}\left(\hat{a}^{\dagger}+\hat{a}\right)\sum_{n=0}^{N_{c}-1}\eta_{n}\left(\left|\psi_{n+1}\right\rangle\left\langle\psi_{n}\right|+\text{H.c.}\right),$
(S.1)

where $\omega_{n}$ is the energy of the state $|\psi_{n}\rangle$, and
$\eta_{n}=\left\langle\psi_{n+1}\right|2\hat{S}_{x}\left|\psi_{n}\right\rangle=\sqrt{\left(N-n\right)\left(n+1\right)}$
is the modifier for the atom-cavity coupling in the symmetric-state subspace.
The superscript of the Hamiltonian $[0,N_{c}]$ indicates that the subspace is
cutoff at $N_{c}$ atomic excitations.

We then numerically confirm that the effective Hamiltonian can fully
characterize the superradiant phase transition on the mean-field level.
Specifically, as shown in Fig. S1(a)(b), we calculate the steady-state cavity
photon number as a function of the atom-cavity coupling rate, with increasing
$N_{c}$, that is, by sequentially enlarging the symmetric-state subspace to
include more symmetric atomic excitations. As we consider more symmetric
states, the results gradually converge. When we include all symmetric states
with $N_{c}=N$, the obtained results are completely consistent with those
obtained using the full Hamiltonian. In Fig. S1(c), we reproduce the results
of Fig. 2 of the main text, using the effective Hamiltonian (S.1) with
$N_{c}=N$.

Figure S1: The steady-state cavity photon number with increasing atom-cavity
coupling rate for (a) $V_{\text{dd}}=0$ and (b) $V_{\text{dd}}=-1/5$. The
solid curves with different colors represent calculation results using an
increasing number of symmetric states [with increasing $N_{c}$ in the model
(S.1)]. The green dashed curves represent the results from the full
Hamiltonian. (c) The steady-state phase diagram obtained from
$\hat{H}^{[0,N]}_{\text{eff}}$ under the mean-field approximation. All
parameters are the same as those in Fig. 3 of the main text.

The effective Hamiltonian (S.1) is useful for calculations of systems with
large atom numbers. Therein, the Hilbert-space dimension $d$ grows
exponentially with the number of atoms $N$. As the atom number increases, the
computational scale becomes significant, whether dealing with the full quantum
master-equation evolution or applying the mean-field approximation. However,
the effective model $\hat{H}^{[0,N]}_{\text{eff}}$ substantially reduces the
dimensionality to $d\propto N+1$, thus greatly reduces computational cost. As
an illustrative example, in Fig. S2, we calculate the steady-state phase
diagram for $N=20$.

Figure S2: Steady-state phase diagram of the generalized Dicke model with
dipole-dipole interactions for $N=20$ atoms.

##  II System with only spatially dependent dipole-dipole interactions.

When the spatial dependence of the dipole interactions is taken into account,
the homogeneity of the configuration is broken, and the symmetric-state
picture above no longer applies. Nevertheless, we find that the interaction-
enhanced superradiance persists, with vanishing $\pazocal{G}_{c}$ at discrete
values of $V_{\text{dd}}$ that are shifted with respect to Eq. (7) of the main
text. This is explicitly demonstrated in Fig. S3, where we take
$V\left(R_{jk}\right)=C_{3}/R_{jk}^{3}=V_{\text{dd}}/|j-k|^{3}$.
Interestingly, near the critical points where $\pazocal{G}_{c}=0$, the
emergent quantum Rabi models still apply, though a different set of collective
states must be considered. Specifically, we redefine $|\psi_{n}\rangle$ as the
lowest-energy state in the subspace with $n$ atom excitations, and update
$\Delta_{n}$ and $\eta_{n}$ in the quantum Rabi model [Eq. (8) of the main
text] accordingly. The resultant phase boundaries according to Eq. (9) of the
main text are shown as dashed curves in Fig. S3, which are in good agreement
with the full mean-field calculations near the critical points (purple solid).

Figure S3: Critical atom-cavity coupling rate as a function of the Rydberg
interaction strength $V_{\text{dd}}$. The grey solid curve represents results
from a constant all-to-all interaction. The purple solid curve represents
results from the full Hamiltonian with spatially dependent dipole-dipole
interaction, while the grey dashed curve represents results from the quantum
Rabi models in the corresponding region. The spatially dependent interaction
is given by $V\left(R_{jk}\right)=C_{3}/R_{jk}^{3}=V_{\text{dd}}/|j-k|^{3}$.
Parameters and the unit of the energy are the same as those in Fig. 3 of the
main text.

##  III System with only Van der Waals interactions

In the main text, for each atom, we consider a pair of Rydberg states. Here we
focus on a more common Rydberg simulator with one ground state and a Rydberg
state. The atomic Hamiltonian takes the following form

$\hat{H}_{\text{atom}}=\omega_{a}\sum_{j=1}^{N}\frac{\hat{\sigma}_{j}^{z}}{2}+\sum_{j<k}V_{pp}\frac{1+\hat{\sigma}_{j}^{z}}{2}\frac{1+\hat{\sigma}_{k}^{z}}{2},$
(S.2)

where we consider an all-to-all Van der Waals interaction characterized by
$V_{pp}$. The energies of the collective atomic states with $n$ excitations
are then

$\omega_{n}=-\left(\frac{N}{2}-n\right)\omega_{a}+C_{n}^{2}V_{pp}.$ (S.3)

In Fig. S4(a), we show the energy gap $\Delta E$ between the lowest two
collective states. When the energy gap is significant, a large atom-cavity
coupling rate is required for the superradiant transition to happen. It is
clear that no matter how the interaction strength is adjusted, the ground-
state degeneracy consisting of collective states $|\psi_{n}\rangle$ and
$|\psi_{n+1}\rangle$ cannot be achieved, unlike in the resonant-dipole case.
Fig. S4 shows the calculated phase diagram of this model. The Van der Waals
interaction alone appears to suppress the superradiance.

Figure S4: Superradiance under the Van der Waals interactions. (a) The energy
gap $\Delta E$ between the ground state and the first excited state as a
function of the interaction strength $V_{pp}$. (b)(c) The steady-state phase
diagram of the system, characterized by the cavity photon number (b), and the
atomic correlation $\left\langle\hat{S}_{x}\right\rangle$ (c), respectively.
Other parameters for the calculations are the same as those of Fig. 3 in the
main text.

##  IV Multi-fold degeneracies under realistic interaction potentials.

Under realistic interaction potentials, accidental multi-fold degeneracies of
the low-lying collective states can also arise. For instance, under the
parameters of Fig. 4 in the main text, a three-fold degeneracy emerges in the
atomic ground state for $R_{0}\sim 1.48\,\mu$m, as circled out in Fig. S5(a).
The three degenerate states are the lowest-energy collective states in the
$n=4,5,6$-excitation subspace, respectively. The degeneracy also leads to a
diverging susceptibility for the superradiant transition. Remarkably, we can
reproduce the critical atom-cavity coupling near the three-fold degenerate
point, by constructing the emergent Rabi model. The result is shown in Fig.
S5(b), where the lowest-energy collective states $|\psi_{n=5}\rangle$ and
$|\psi_{n=6}\rangle$ are used to the left of the three-fold degenerate point,
and the states $|\psi_{n=4}\rangle$ and $|\psi_{n=5}\rangle$ are used to the
right.

Figure S5: (a) Energies $\omega_{n}$ of the lowest-energy collective states
$|\psi_{n}\rangle$ within the $n$-excitation subspace. (b) The critical atom-
cavity coupling rate for the superradiant transition with varying interaction
strengths, calculated from the effective quantum Rabi models (dashed lines)
and the full Hamiltonian (solid lines), respectively. Other parameters are the
same as those of Fig. 4 in the main text.


# First principle investigation of hydrogen behavior in M doped Cu2O (M $=$
Na, Li and Ti)

A. Larabi${}^{\textsf{\footnotesize{\mbox{?}}}}$, A.
Mahmoudi${}^{\textsf{\footnotesize{\mbox{?}}}}$, M.
Mebarki${}^{\textsf{\footnotesize{\mbox{?}}}}$, M.
Dergal${}^{\textsf{\footnotesize{\mbox{?}}}}$ Corresponding author: A. Larabi,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received October 30, 2018, in final form March 4, 2019)

###### Abstract

We study the hydrogen effect on the electronic, magnetic and optical
properties of Cu2O in presence of different dopants (Na, Li and Ti). The
electronic properties calculations show that hydrogen changes the conductivity
of Cu2O from p to n-type. The results show that interstitial hydrogen atom
prefers to locate in the tetrahedral site in Cu2O system and it decreases the
band gap value of the later. The Na or Li doping Cu2O preserves the p-type
conductivity of Cu2O, while hydrogen is the source of n-type conductivity in
Na or Li doped Cu2O systems. Ti doping increases the band gap value of Cu2O
and makes it an n-type semiconductor. Hydrogen increases the optical
transmittance of M doped Cu2O.

Key words: Cu2O, hydrogen, density functional theory

PACS: 70.

###### Abstract

Äîñëäæóòüñÿ âïëèâ âîäíþ íà åëåêòðîíí, ìàãíòí òà îïòèí âëàñòèâîñò Cu2O ó
ïðèñóòíîñò ðçíèõ ëåãóþèõ äîìøîê (Na, Li Ti). Ðîçðàõóíêè åëåêòðîííèõ
âëàñòèâîñòåé ïîêàçóþòü, ùî âîäåíü çìíþ ïðîâäíñòü Cu2O ç p äî n-òèïó.
Ðåçóëüòàòè ïîêàçóþòü, ùî ìæâóçëîâèé àòîì âîäíþ âîë ðîçòàøóâàòèñÿ â òåòðàåäðèíé
äëÿíö â ñèñòåì Cu2O çìåíøó øèðèíó çàáîðîíåíî¿ çîíè îñòàííüî¿. Cu2O, ëåãîâàíèé
Na àáî Li çáåðãà p-òèï ïðîâäíîñò Cu2O, â òîé àñ ÿê âîäåíü äæåðåëîì n-òèïó
ïðîâäíîñò â ñèñòåìàõ Cu2O, ëåãîâàíèõ Na àáî Li. Ti ëåãóâàííÿ çáëüøó øèðèíó
çàáîðîíåíî¿ çîíè Cu2O ðîáèòü ¿¿ íàïâïðîâäíèêîì n-òèïó. Âîäåíü ïäâèùó êîåôöíò
îïòèíîãî ïðîïóñêàííÿ Ì-ëåãîâàíîãî Cu2O.

Ключов слова: Cu2O, âîäåíü, òåîðÿ ôóíêöîíàëó ãóñòèíè

##  1 Introduction

Owing to high optical absorption coefficient, non-toxicity, abundancy and
inexpensive layer formation, copper oxide (CuxO) is appealing to a renewed and
rising interest. It could have a potential role in fabricating semiconductor
devices, among others, for nanoelectronic, spintronic and photovoltaic
applications [1, 2, 3]. We know copper oxide as a p-type semiconductor in two
stable forms, cupric oxide (CuO) and cuprous oxide (Cu2O) and this latter
enables higher conversion efficiency when used as an absorber material for
solar cell applications [4, 5]. Moreover, the possibility of synthesizing
nanoparticles enables excellent properties and opens the way to a new class of
innovative and more efficient devices [6]. The ability to control the
conductivity type can be important for the synthesizing of Cu2O thin films
paired with other p-type materials. The challenge of p-type doping of Cu2O
must take hydrogen into account, since it is known to passivate any acceptors
in solar cell absorbers [7] and reverse the polarity of conductivity in other
materials [8]. Hydrogen is present in all semiconductors, including oxides,
and is highly reactive, forming a complex with most impurities and defects in
the lattice. Hydrogen is a potential source of n-type conductivity [9, 10,
11]. The capability of hydrogen to remove or stimulate the electrical and
optical activity of impurities makes it sometimes a helpful, sometimes a
disagreeable agent, in either case. However, a theoretical understanding of
the principal interactions is very important. The diversity and complexity of
hydrogen states, inside the lattice and in traps, has made their
characterization impossible by experimental methods alone; the complementary
role of theory is proved to be essential [12]. The earlier calculations
suggested that interstitial hydrogen prefers to locate in the bond center site
(BC) in CdTe and ZnO lattices [13, 14, 15, 16]. It could locate in the
tetrahedral site in other oxides such SnO2 and TiO2 [17, 18]. Theoretical
prediction and spectroscopy confirmed that hydrogen in semiconductors may be a
cause of n-type conductivity; it forms shallow-donor states in the zinc oxide
(ZnO) [13, 19, 20, 21]. Peacock and Robertson [22] showed that hydrogen acts
as a shallow donor in many oxides, while deep in the silicates, SiO2 and
Al2O3. Scanlon and Watson [23] studied the behavior of hydrogen in p-type
Cu2O; they showed that hydrogen could weaken the performance of Cu2O solar
cell devices. It is important, however, to study the H atom influence on
different properties of Cu2O and M doped Cu2O (M= Na, Li, Ti). As the
fabrication process of solar cell devices uses different kind of gas, the
knowledge of the hydrogen effect is important in choosing the gas in the
annealing process.

Our work uses the first-principles calculations. We study the effect of
hydrogen on the electronic, magnetic and optical properties of Cu2O and
Cu2(1-x)MxO. To find the favoured position of the hydrogen atom in the systems
studied, we calculated the formation energies of different configurations
proposed. In addition, we try to discuss the influence of hydrogen and
different dopants on physical properties of M doped Cu2O. We also predict and
give an easy, low-cost, and scalable strategy to prepare the Cu2O composite on
heterostructure solar cells, hydrogen storage and/or production.

##  2 Computational details

We used ab initio total-energy and molecular-dynamics program VASP (Vienna ab
initio simulation program) developed at the Fakultät für Physik of the
Universität Wien [24, 25], to do all calculations. For treatment of the
exchange and correlation energies, we used the generalized gradient
approximation (GGA) [26] with projector-augmented wave (PAW) [25, 27] pseudo-
potentials. Brillouin zone integrals are converged with a 450 eV plane-wave
cut-off, and a $2\times 2\times 2$ Monkhorst-Pack k-point mesh suffices to
make sure that energy convergence takes place for supercell. These
calculations showed a discrepancy within 10-7 eV. We performed a relaxation
with the standard conjugated gradient algorithm. We used the supercell
approach to simulate the H doped Cu2O or (Cu2, M)O system. The optimized
lattice constant of Cu2O, $a=4.26$ Å, is in good agreement with the
experimental value of $a=4.27$ Å [28, 29] and theoretical values $a=4.18$ and
4.20 Å [30, 31].

In these calculations, we positioned the M atoms at the cation sites (Cu). We
performed the H defect formation energy calculations on $3\times 3\times 3$
supercell, containing 162 atoms, using the calculated lattice constant with
the defect at different positions in the cell. The supercell size is necessary
to allow for a detailed study of many different dopant geometric structures.
For calculations of the electronic properties of H doped (Cu2, M)O, we suppose
that M atom leads to the formation of $3\times 3\times 3$ supercell with
chemical composition Cu2(107)MO54, with one atom of hydrogen at the
interstitial site. All calculations were spin polarized. We can determine
optical properties using the complex dielectric function
$\varepsilon(\omega)=\varepsilon_{1}(\omega)+\mathrm{i}\varepsilon_{2}(\omega)$
[32]. The imaginary part of the dielectric function $\varepsilon_{2}(\omega)$
was calculated from the momentum matrix elements between the occupied and
unoccupied wave functions [33]:

$\varepsilon_{2}(\hbar\omega)=\frac{2e^{2}\piup}{\Omega\varepsilon_{0}}\sum_{\mathbf{k},V,C}\left|\left\langle\Psi_{\mathbf{k}}^{C}|\widehat{\mathbf{u}}\cdot\mathbf{r}|\Psi_{\mathbf{k}}^{V}\right\rangle\right|^{2}\delta\big{(}E_{\mathbf{k}}^{C}-E_{\mathbf{k}}^{V}-\hbar\omega\big{)}.$
(2.1)

Here, $\Omega,C,V,\mathbf{k},\omega,\widehat{\mathbf{u}}$ are the volume of
cell, conduction band, valence band, $\mathbf{k}$ point, photon frequency,
external field vector, respectively, while $\Psi_{\mathbf{k}}^{C}$ and
$\Psi_{\mathbf{k}}^{V}$ represent the eigenstates. We can obtain the real part
of the dielectric function, $\varepsilon_{1}$ from $\varepsilon_{2}$ by the
Kramer–Kronig relationship [34]. The corresponding absorption coefficient was
obtained using the following equation:

$\alpha(\omega)=\sqrt{2}\Big{(}\frac{\omega}{c}\Big{)}\Big{[}\sqrt{\varepsilon_{1}^{2}(\omega)-\varepsilon_{2}^{2}(\omega)}-\varepsilon_{1}(\omega)\Big{]}^{\frac{1}{2}}.$
(2.2)

##  3 Results and discussion

###  3.1 Hydrogen in Cu2O

In Cu2O lattice we can attribute different substitutional and interstitial
positions to hydrogen impurity. Figure 1 illustrates all positions where
copper, oxygen and hydrogen atoms are, respectively, represented by blue, red
and green spheres. In this figure, we notice four possible sites for the
hydrogen impurity, T (Tetrahedral site), BC (Bond Center site), C (Cation
substitutional site) and A (Anion substitutional site).

Figure 1: (Colour online) Location of H impurity in Cu2O lattice. Cu atoms in
blue color, O atoms in red and H atoms are in green.

To find the preferred site of H in Cu2O, we calculated the formation energies
for all hydrogen positions. We show the obtained results in table 1. The
references [35, 36] give the formation energy for hydrogen insertion in Cu2O
as follows:

$E_{\text{f}}=E_{\text{Cu}_{2}\text{O}+\text{H}}-E_{\text{Cu}_{2}\text{O}}-E_{\text{tot}}^{\text{H}}+qE_{\text{F}}\,.$
(3.1)

In case of substitution, the formation energy associated is:

$E_{\text{f}}=E_{\text{Cu}_{2}\text{O}+\text{H}}-E_{\text{Cu}_{2}\text{O}}-E_{\text{tot}}^{\text{H}}+E_{\text{tot}}^{\text{Cu}}+qE_{\text{F}}\,,$
(3.2)
$E_{\text{f}}=E_{\text{Cu}_{2}\text{O}+\text{H}}-E_{\text{Cu}_{2}\text{O}}-E_{\text{tot}}^{\text{H}}+E_{\text{tot}}^{\text{O}}+qE_{\text{F}}\,.$
(3.3)

$E_{\text{Cu}_{2}\text{O}+\text{H}}$ is the total energy of supercell with one
hydrogen atom, $E_{\text{Cu}_{2}\text{O}}$ is the total energy of supercell
without defects $E_{\text{tot}}^{\text{H}}$ , $E_{\text{tot}}^{\text{Cu}}$ and
$E_{\text{tot}}^{\text{O}}$ are the total energies of an isolated hydrogen,
cuprous and oxygen atoms, respectively, in the ground state. The last term in
the formation energy ($q$) accounts because H+ donates an electron $(q=-1)$,
and H- accepts an electron $(q=+1)$. $E_{\text{F}}$ is the Fermi level energy.
If the formation energy is positive $(E_{\text{f}}>0)$, the implantation of
the hydrogen atom into the Cu2O lattice is energetically unfavorable.

One can see from table 1 that the lowest formation energy is that of the T
site with $-0.77$ eV. These values show that hydrogen prefers to localize in
the tetrahedral site in Cu2O lattice, which confirms earlier results [23, 37].
Figure 2 shows different configurations of hydrogen in Cu2O relaxed lattice.
We can see that with insertion in the tetrahedral site, anionic site and
cationic site, the hydrogen position does not change, while with insertion in
Cu-O bond center, the hydrogen atom takes another position in the Cu2O
lattice.

Table 1: Formation energies for different sites of hydrogen in Cu2O lattice. Sites | H${}_{\text{T}}$ | H${}_{\text{BC}}$ | H${}_{\text{A}}$ | H${}_{\text{C}}$
---|---|---|---|---
$E_{\text{f}}$(eV) | – 0.77 | 5.64 | 0.15 | 0.41
Figure 2: (Colour online) H impurity in Cu2O lattice after relaxation. Cu
atoms in blue color, O atoms in red and H atoms are in green.

We plotted the total densities of state (TDOS) of pure Cu2O and a single
H-doped Cu2O in figure 3. The TDOS figures show that hydrogen causes a change
in the position of the Fermi level from near the valence band towards
conduction band, leading the system to n-type semiconductor characteristic;
this result agrees well with earlier works [23, 38]. In fact, this behavior of
hydrogen was found in other p-type oxide materials such as SnO2, TiO2 and ZnO
[38, 39, 40, 41]. The calculated band gap of Cu2O, $E_{\text{g}}$=0.55 eV, is
smaller than the experimental value (2.17 eV) [42] according to the DFT-GGA
calculations. The result agrees well with the earlier theoretical values
reported in the literature (0.6 eV [43, 44], 0.64 eV [45] and 0.7 eV [46]).
For hydrogen doped Cu2O, we notice a narrow energy gap of 0.09 eV of the
majority-spin electrons, while for the minority-spin electrons, the band gap
is 0.23 eV. The hydrogen in Cu2O decreases the band gap value of Cu2O.

Figure 3: (Colour online) Total density of states TDOS: (a) Pure Cu2O, (b) H
doped Cu2O.

To improve the physical properties of Cu2O for voltaic applications, we have
tried three different dopants: Sodium (Na), Lithium (Li) and Titanium (Ti). We
investigated the properties of M doped Cu2O, M= Na, Li and Ti, by substituting
one M atom for one Cu atom in the $3\times 3\times 3$ supercell Cu(108)O54.

In Cu2(1-x)MxO structure, we consider two configurations where interstitial
hydrogen atom is at the tetrahedral site. In the first one, a short distance
(S${}_{\text{D}}$) separates M and H while in the second one, a long distance
(L${}_{\text{D}}$) separates M and H atoms.

###  3.2 Na and H doped Cu2O

In the first case configuration (S${}_{\text{D}}$), the distance between Na
and H was 1.78 Å while after relaxation this distance becomes 2.38 Å. This
suggests that H atom relaxes to move away from the Na atom. For the second
configuration (L${}_{\text{D}}$) and before relaxation, the distance between
Na and H was 6.98 Å. After relaxation, this distance hardly moves to only
7.012 Å.

We show the total density of states of Na doped Cu2O in figure 4 (a). The
figure indicates that the Na doping decreases the band gap of Cu2O. The doping
of Cu2O by Na does not change the nature of his conductivity; the calculated
energy of the gap is 0.35 eV. Elfadill et al. [47] showed that Na doping Cu2O
is a p-type characteristic, which shows that our calculations are in good
agreement with experimental data. In addition, Minami et al. used Cu2O doped
Na as p-type in p-n heterojunction and/or homojunction solar cells [48, 49].

Figure 4: (Colour online) Total density of states TDOS: (a) Na-doped Cu2O, (b)
H-Na-doped Cu2O (S${}_{\text{D}}$), (c) H-Na-doped Cu2O (L${}_{\text{D}}$).

For hydrogen implantation in Na doped Cu2O, the calculated band gap is 0.47 eV
for a short distance. In the long distance case, we note two different band
gap values, 0.47 eV and 0.35 eV for the majority-spin and minority-spin
electron, respectively. From the DOS figures [figure 4 (b), figure 4 (c)], we
note that hydrogen changes the position of Fermi level ($E_{\text{F}}$) of Na
doped Cu2O, $E_{\text{F}}$ moves from the valence band to conduction band.
Thus, Na doped Cu2O becomes n-type semiconductor. The distance between Na and
H atoms does not influence the electronic properties of Na doped Cu2O, we
found the same band gap value in the two configurations.

###  3.3 Li and H doped Cu2O

For the short distance case, the distance between Li and H was 1.78 Å. After
the relaxation, the distance becomes 1.69 Å while in the long distance case,
the distance between Na and H was 5.27 Å. After the relaxation, the distance
becomes 5.35 Å.

Figure 5: (Colour online) Total density of states TDOS: (a) Li-doped Cu2O, (b)
H-Li-doped Cu2O (S${}_{\text{D}}$), (c) H-Li-doped Cu2O (L${}_{\text{D}}$).

Figure 5 (a) shows the total densities of states for Li-doped Cu2O. From Fermi
level position we note that the Li doped Cu2O is p-type characteristic. This
result agrees well with the experimental work of Kyung-Su Cho et al. [50]
where they used Li doped Cu2O as p-type in p-n heterojunction. The calculated
band gap of Li-doped Cu2O is 0.61 eV.

From the DOS figures [figure 5 (b), figure 5 (c)] of H-Li-doped, we see the
shift in the Fermi level position. The hydrogen implantation in Li-doped Cu2O
does not change the value of the band gap but it makes the Cu2O an n-type
semiconductor characteristic. The distance between Li and H does not influence
the band gap value of Li-doped Cu2O.

###  3.4 Ti and H doped Cu2O

For the first configuration (S${}_{\text{D}}$), the distance between Ti and H
is 1.8 Å. After the relaxation, the distance becomes 1.88 Å. In the second
case configuration (L${}_{\text{D}}$), the distance between Ti and H was 7.92
Å. After the relaxation, the distance becomes 7.9 Å.

Figure 6: (Colour online) Total density of states TDOS: (a) Ti-doped Cu2O, (b)
H-Ti-doped Cu2O (S${}_{\text{D}}$), (c) H-Ti-doped Cu2O (L${}_{\text{D}}$).

Figure 6 shows the total densities of states for Ti-doped Cu2O and H-Ti-doped
Cu2O. The calculated band gap of Ti-doped Cu2O is 0.73 eV and 0.86 eV for the
majority-spin and minority-spin electron, respectively. These values are
larger than that of the pure Cu2O (0.55 eV). From the DOS figures, we note
that Ti doping changes the position of Fermi level from near the valence band
towards the conduction band, leading the system to n-type semiconductor
characteristic. In the first case configuration, the hydrogen implantation can
decrease the band gap value of Ti doped Cu2O to 0.59 eV, while in the second
case configuration, hydrogen does not change the band gap value. In the two
configurations, it does not change the conductivity type of Ti doped Cu2O.

Table 2: Calculated band gap $E_{\text{g}}$(eV) and lattice parameter $a$(Å). |  $E_{\text{g}}$(eV) |  $a$(Å)
---|---|---
Cu2O | 0.55 | 4.26
H-Cu2O (spin up) | 0.46 | 4.28
H-Cu2O (spin dn) | 0.23 | –
Na-Cu2O | 0.35 | 4.26
H-Na-Cu2O (S${}_{\text{D}}$) | 0.47 | 4.27
H-Na-Cu2O (L${}_{\text{D}}$) (spin up) | 0.47 | 4.27
H-Na-Cu2O (L${}_{\text{D}}$) (spin dn) | 0.35 | –
Li-Cu2O | 0.61 | 4.12
H-Li-Cu2O (S${}_{\text{D}}$) | 0.61 | 4.11
H-Li-Cu2O (L${}_{\text{D}}$) | 0.61 | 4.11
Ti-Cu2O (spin up) | 0.73 | 4.11
Ti-Cu2O (spin dn) | 0.86 | –
H-Ti-Cu2O (S${}_{\text{D}}$) | 0.59 | 4.24
H-Ti-Cu2O(L${}_{\text{D}}$) | 0.86 | 4.12

Table 2 shows the calculated band energies ($E_{\text{g}}$) and lattice
parameters (a) for Cu2O and M doped Cu2O with or without hydrogen. From this
table, we can see that the band gap value of Na doping Cu2O makes it the best
of the three as a p-type dopant in Cu2O. Compared with the value of pure Cu2O,
the lattice parameter of Na doped Cu2O does not change while Li and Ti doping
should compress the lattice of Cu2O. For all cases, hydrogen does not change
the lattice of systems except with hydrogen implantation in the first
configuration (S${}_{\text{D}}$) in Ti doped Cu2O. In this configuration,
hydrogen should expand the lattice of this system from 4.11 to 4.24 Å.

From the densities of states figures, we can see that hydrogen creates new
states in the band gap in all systems. Na and Li act as acceptor impurities in
Cu2O and hydrogen should passivate these acceptors. Ti is a source of n-type
conductivity and hydrogen could not passivate this donor in Cu2O.

In the cases of H doped Cu2O, Ti doped Cu2O and H-Na-co-doped Cu2O for the
second configuration (L${}_{\text{D}}$), the diference in band gap value of
majority-spin and minority-spin electrons shows anisotropy of the gap which
determines the elastic distribution between the hole states and electrons.

From these results, it seems that hydrogen can passivate the acceptors (Na,
Li) or donors (Ti) in Cu2O. Similar results have been found in experimental
works; Lu et al. [51] revealed that a treatment of N-doped Cu2O films in
hydrogen plasma could passivate the carrier traps. On the other hand, Ishizuka
et al. [52, 53] show that a treatment of N-doped Cu2O films in hydrogen plasma
leads to an increase of conductivity.

###  3.5 Magnetic and optical properties

We list the values of the total and local magnetic moments in table 3. With
hydrogen, we have the values of the total magnetic moments approximately equal
to 1 $\mu_{\text{B}}$ except with Ti-doped Cu2O, where it is almost zero. In
all systems, the Cu moments dominate the values of local magnetic moments with
a maximum value of 0.16 $\mu_{\text{B}}$ excluding the H-Ti-codoped Cu2O
system where local magnetic moments values are negative.

Table 3: Magnetic moments (in unit of $\mu_{\text{B}}$) of the supercell ($m_{\text{tot}}$), H atom ($m_{\text{H}}$), Na ($m_{\text{Na}}$), Li atom ($m_{\text{Li}}$), Ti atoms ($m_{\text{Ti}}$), Cu atom ($m_{\text{Cu}}$) and O atom ($m_{\text{O}}$). |  $m_{\text{tot}}$($\mu_{\text{B}}$) |  $m_{\text{H}}$($\mu_{\text{B}}$) |  $m_{\text{Na}}$($\mu_{\text{B}}$) |  $m_{\text{Li}}$($\mu_{\text{B}}$) |  $m_{\text{Ti}}$($\mu_{\text{B}}$) |  $m_{\text{Cu}}$($\mu_{\text{B}}$) |  $m_{\text{O}}$($\mu_{\text{B}}$)
---|---|---|---|---|---|---|---
Cu2O | 0.0 | – | – | – | – | 0.0 | 0.0
H-Cu2O | 0.9999 | 0.056 | – | – | – | 0.13 | 0.05
Na-Cu2O | 0.0 | – | 0.0 | – | – | 0.0 | 0.0
H-Na-Cu2O (S${}_{\text{D}}$) | 1.0 | 0.056 | 0.004 | – | – | 0.16 | 0.06
H-Na-Cu2O (L${}_{\text{D}}$) | 0.9999 | 0.058 | 0.0 | – | – | 0.12 | 0.05
Li-Cu2O | 0.0 | – | – | 0.0 | – | 0.0 | 0.0
H-Li-Cu2O (S${}_{\text{D}}$) | 1.0 | 0.06 | – | 0.013 | – | 0.15 | 0.06
H-Li-Cu2O (L${}_{\text{D}}$) | 0.9955 | 0.06 | – | 0.0 | – | 0.12 | 0.04
Ti-Cu2O | 1.0 | – | – | – | 0.75 | 0.024 | – 0.012
H-Ti-Cu2O (S${}_{\text{D}}$) | – 0.0006 | 0.0 | – | – | 0.05 | 0.02 | – 0.007
H-Ti-Cu2O(L${}_{\text{D}}$) | 0.0699 | – 0.07 | – | – | 0.74 | – 0.11 | – 0.05

As far as the principle of contamination of the samples by ferromagnetic
impurities can be discounted, the common factor in all cases is crystal
defects. Here, they have been established, but it is plausible that they are
point defects-atomic vacancies or interstitials. Examples of $d^{0}$
ferromagnetism have been reported in the literature [54].

The absorption coefficient $\alpha(\omega)$ is important to estimate the
optical properties of Cu2O and H-M-codoped Cu2O (M= Na, Li, Ti) used in
photovoltaic cells or as optoelectronic devices. A material with a low
absorption coefficient means more transmitted light. Cuprous oxide Cu2O is a
promising p-type TCO, but its relatively low band gap (2.1 eV) limits its
optical transmittance in the visible spectrum. In this paper, we aim at
increasing this value.

Figure 7 presents the absorption coefficient (of the order of 104 cm-1) versus
the wavelength (nm) of all configurations compared to the pure compound Cu2O
in the visible light region (400–900 nm) and UV region (250–400 nm). The
absorbance of pure Cu2O decreases from 1.92$\times 10^{5}$ cm-1 to 0.89$\times
10^{5}$ cm-1 in (250–400 nm) and in the visible region continues to decrease
to 0.086$\times 10^{5}$ at 650 nm but we observed two peaks at 530
(0.287$\times 10^{5}$ cm-1) and 800 nm (0.428$\times 10^{5}$ cm-1). However,
we noticed a small change when the M (Na, Li, Ti) atoms are at the cation
sites (Cu) in the pure Cu2O. We remark the same peaks when we doped with Na
with a small displacement of the second peak (from 800 to 650 nm). The Li or
Ti doped Cu2O gives the same result in visible and UV regions but without the
initial peak recorded in the pure system, and the absorbance increases between
650 and 750 nm. Furthermore, incorporation of Hydrogen reduces the absorbance
of pure Cu2O and Ti doped Cu2O. Hydrogen does not decrease the absorbance of
Na or Li-doped Cu2O.

Figure 7: (Colour online) The variation of the absorption coefficient of H
inserted in M doped Cu2O compared with pure one.

##  4 Conclusion

We studied the electronic properties of M-doped Cu2O (M= Na, Li, Ti) with and
without hydrogen by first-principles calculations. The results show that
hydrogen is most stable in the tetrahedral site with the low formation energy.
Interstitial hydrogen reduces the band gap value of Cu2O and it gives an
n-type conduction behavior. Adding hydrogen does not decrease the band gap
value of Na or Li-doped Cu2O but it changes the charges conductivity type from
p- to n-type. In these cases, the distance between H atom and Na or Li atom
does not affect the structural and electronic properties of Na or Li-doped
Cu2O systems. The Ti doping of Cu2O changes its conductivity type. Hydrogen
does not change the n-type characteristic of Ti doped Cu2O system (hydrogen
effect is negligible compared to that of Ti) but it decreases the magnetic
moments of this system. The distance between H and Ti atoms can influence the
structural, electronic and magnetic properties of Ti doped Cu2O. From these
results, we see that Na is the best p-type dopant in the Cu2O and it is
necessary to remove hydrogen in this material to achieve an increased
performance. The absorption shows that Cu2O is a promising optical material in
the visible region. In this paper, we established the effects of H, Ti, Li or
Na doping, which lead to a significant decrease of the absorption, i.e.,
increase the optical transmittance. The results of this work make it possible
to adapt Cu2O to several technological applications and to different
scientific fields. In the next stage of our research, we will study the
dopants concentration effects and we hope to confirm our results by
experimental investigations.

## Acknowledgements

This work was funded under the Algerian General Directorate of Scientific
Research and Technological Development (DGRSDT).

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Ukrainian 0 Ïåðøîïðèíöèïíå äîñëäæåííÿ ïîâåäíêè âîäíþ ó M-ëåãîâàíîìó Cu2O (M
$=$ Na, Li Ti) A. Ëÿðàá${}^{\textsf{\footnotesize{\mbox{?}}}}$, A.
Ìàõìóä${}^{\textsf{\footnotesize{\mbox{?}}}}$, M.
Ìåáàðê${}^{\textsf{\footnotesize{\mbox{?}}}}$, M.
Äåðãàë${}^{\textsf{\footnotesize{\mbox{?}}}}$

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${}^{\textsf{\footnotesize 2}}$ Âääëåííÿ âèâåííÿ òà ïðîãíîçóâàííÿ ìàòåðàëâ,
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