# Pressure-Induced Phase Transformations of Quasi-2D Sr3Hf2O7

M. C. B. Barbosa  IFIMUP, Institute of Physics for Advanced Materials,
Nanotechnology and Photonics, Department of Physics and Astronomy, Faculty of
Sciences, University of Porto, Rua do Campo Alegre, 687, 4169-007 Porto,
Portugal  E. Lora da Silva  IFIMUP, Institute of Physics for Advanced
Materials, Nanotechnology and Photonics, Department of Physics and Astronomy,
Faculty of Sciences, University of Porto, Rua do Campo Alegre, 687, 4169-007
Porto, Portugal<EMAIL_ADDRESS>P. Neenu Lekshmi  IFIMUP, Institute
of Physics for Advanced Materials, Nanotechnology and Photonics, Department of
Physics and Astronomy, Faculty of Sciences, University of Porto, Rua do Campo
Alegre, 687, 4169-007 Porto, Portugal  M. L. Marcondes  L. V. C. Assali  H. M.
Petrilli  Universidade de São Paulo, Instituto de Física, Rua do Matao 1371,
05508-090, São Paulo, SP, Brazil  A. M. L. Lopes  J. P. Araújo  IFIMUP,
Institute of Physics for Advanced Materials, Nanotechnology and Photonics,
Department of Physics and Astronomy, Faculty of Sciences, University of Porto,
Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

###### Abstract

We present an ab-initio study of the quasi-2D layered perovskite Sr3Hf2O7
compound, performed within the framework of the Density Functional Theory and
lattice dynamics analysis. At high temperatures, this compound takes a I4/mmm
centrosymmetric structure (S.G. n. 139); as the temperature is lowered, the
symmetry is broken into other intermediate polymorphs before reaching the
ground state structure, which is the Cmc21 ferroelectric phase (S.G. n. 36).
One of these intermediate polymorphs is the Ccce structural phase (S.G. n.
68). Additionally, we have probed the C2/c system (S.G n. 15), which was
obtained by following the atomic displacements corresponding to the
eigenvectors of the imaginary frequency mode localized at the
$\mathbf{\Gamma}$-point of the Ccce phase. By observing the enthalpies at low
pressures, we found that the Cmc21 phase is thermodynamically the most stable.
Our results show that the I4/mmm and C2/c phases never stabilize in the 0-20
GPa range of pressure values. On the other hand, the Ccce phase becomes
energetically more stable at around 17 GPa, surpassing the Cmc21 structure. By
considering the effect of entropy and the constant-volume free energies, we
observe that the Cmc21 polymorph is energetically the most stable phase at low
temperature; however, at 350 K the Ccce system becomes the most stable. By
probing the volume-dependent free energies at 19 GPa, we see that Ccce is
always the most stable phase between the two structures and also throughout
the studied temperature range. When analyzing the phonon dispersion
frequencies, we conclude that the Ccce system becomes dynamically stable only
around 19-20 GPa, and that the Cmc21 phase, is metastable up to 30 GPa.

###### keywords:

American Chemical Society, LaTeX

High Performance Computing Chair, University of Évora, Largo dos Colegiais 2,
7004-516 Évora Pressure,DFT,Lattice Dynamics,QHA

##  1 Introduction

Rational design and the discovery of new materials are essential towards the
advancement of future technological innovations. Perovskite structures (with
general chemical formula ABX3, where A and B are typically large and small
cations, respectively, and X is an anion) are one of the most versatile
classes of materials, with a variety of fascinating characteristics such as
magnetism, superconductivity, thermoelectricity, electrocatalysis,
ferroelectricity, and optical properties. 1, 2 While the greater part of the
known perovskite oxides are centrosymmetric, a small fraction of these
crystallize as non-centrosymmetric (polar perovskites).3 The polar phase
enables the emergence of ferroelectricity, piezoelectricity and nonlinear
optical effects which are crucial to modern electronic devices, such as
sensors, actuators, tunnel junctions for non-volatile memory, transducers for
medical imaging, and solar cells.1, 2, 4, 5, 6

Among perovskites, naturally layered perovskite oxides, such as quasi-2D
Ruddlesden-Popper (RP) and Dion-Jacobson (DJ) structures, have particular
interest because of their recently discovered mechanism for ferroelectricity,
termed as "hybrid improper ferroelectricity" (HIF). 7, 8 Structures with HIF
exhibit two non-polar structural distortion modes, which lead to a third polar
distortion mode, and are known as the trilinear coupling, resulting in a
macroscopic spontaneous polarization.7, 9 For perovskite structures which
undergo distortions towards the Pnma space group (i.e. double-perovskite
structures), the HIF phenomenon leads to an antiferro-distortive displacement
of the $A$-site cations and the adjacent layer displacement in opposite
directions.10, 11 The displacements of the adjacent layers cancel each other
for ABO3 single perovskites, leading to non-polar phases. However, naturally
layered quasi-2D perovskites, with disconnected octahedral layers and having
two inequivalent crystallographic A-sites, exhibit macroscopic layer
polarization, 12, 13 which are oriented oppositely with slightly different
intensities. In the RP structured naturally layered oxides, with the
(AO)(ABO3)n stoichiometry form, the $n$-layers of the ABO3 perovskite slabs
are stacked between the AO rock-salt layers, with a relative shift of the
neighboring perovskite slabs by a ($\frac{1}{2}$,$\frac{1}{2}$) translation.
Increasing the value of $n$ from 1 to $\infty$ it is possible to tune the
structural dimensionality of these perovskites from quasi-2D to 3D. At high
temperatures, each of the $n$ layers consists of vertex sharing oxygen
octahedra with a B-ion sitting at its center. 7, 9 As temperature decreases,
these systems can undergo structural phase transitions due to reorientation of
oxygen octahedra, leading to HIF. The HIF is also predicted to occur in oxide
superlattices. 14 In addition to HIF, the octahedral reorientations in layered
perovskites are of continuing interest because these effects also trigger many
interesting electronic properties, such as high-Tc superconductivity, colossal
magnetoresistance, metal-insulator transitions, and magnetic ordering.15

Recent theoretical works from Liu et al. 16 and da Silva et al.17 confirmed
that Sr3Hf2O7 (SHO), a $n$=2 quasi-2D RP layered perovskite system, with a
ferroelectric ground state Cmc21 (S.G. n. 36), is similar to the previously
reported HIF systems, such as Ca2Mn3O7 and Ca2Ti3O7, in Ref. 15. SHO is also
characterized by a high-temperature I4/mmm (S.G. n. 139) centrosymmetric
paraelectric structure. Previous studies showed that by applying group
theoretical analysis it was possible to probe potential intermediate
polymorphs that form as the temperature is lowered, enabling a transition
pathway as I4/mmm $\rightarrow$ Cmcm (S.G. n. 63) $\rightarrow$ Cmca (S.G. n.
64) $\rightarrow$ Cmc21 (S. G. n. 36).7, 17 The Ccce (S.G. n. 68) structural
phase is a possible polymorph, however it is not evidenced as being an
intermediate structure, occurring for these structural RP symmetries,
according to group theoretical analysis - a condensation of the X${}_{1}^{-}$
zone boundary mode would have to occur upon symmetry breaking of the
centrosymmetric system. It was also found by computing the phonon dispersion
curves of the Ccce phase,17 that the system is dynamically unstable at 0 K and
0 GPa, evidencing negative phonon modes localized at two of the high-symmetry
points of the Brillouin-zone (BZ): $\mathbf{\Gamma}$\- and Y-points. Therefore
the Ccce polymorph does not have the potential to crystallize at room
conditions. We must note that such a phase has been experimentally evidenced
as an intermediate phase for increasing temperatures in systems with similar
stoichiometry, i.e. Ca3(Mn/Ti)2O7, 18, 19 thus making it plausible for this
polymorph to form through a first-order transition for the SHO system as well.

Only recently have high-pressure (HP) studies been devoted to understand the
properties of the perovskite systems and the respective transition pathways.
Pressure is an important thermodynamic variable, for it enables a precise
control over the interatomic distances and hence the atomic interactions, in
turn driving phase-transitions. Almost all materials will undergo several
phase-transitions under compression, thus generating new polymorphs with
promising properties, different to those properties of materials under room
conditions. It has been found that factors which are known to control the ABO3
perovskites pressure response, such as the A- and B-site cation formal
charges, the tolerance factor, and the B-site chemical environments, also
affect the pressure response of the layered perovskites.20 Therefore, the
interest in applying pressure to probe the structural, electronic and
vibrational properties to such structures.

A recent experimental study on Sr3Sn2O7 compound evidences a sequence of
pressure-induced phase transitions as Cmc21 $\rightarrow$ Pbcn (S.G. n. 60)
$\rightarrow$ Ccce $\rightarrow$ I4/mmm, at room temperature.21 Such
transition pathway would match the sequence of temperature-dependent
structural transitions observed for this compound between 77 and 1000 K. For
the Ca3M2O7-based oxides (M = Ti, Mn),22 ferroelectric switching is inhibited
by the irregular domains induced by the intermediate Ccce phase, and the
switching may be realized when the intermediate phase is suppressed by
applying chemical pressure. Moreover, it has been evidenced that the O
octahedra tilt and rotation modes of Ca3Ti2O7 undergo softening when
hydrostatic pressure or heating is considered.23, 24 This behavior under
temperature and pressure reveals the softness of the antiphase tilt, which
indicates the importance of the partially occupied $d$-orbitals of the
transition metal ions, since these determine the stability of the oxygen
octahedra distortion.23, 24 Moreover, through Raman measurements, it was
observed that under isotropic pressure, the polyhedra tilts of the Ca3Mn2O7
compound can be suppressed within the low pressure regime (1.4-2.3 GPa).25 On
the other hand, through DFT calculations (0 K) performed on the Ca3Ti2O7
compound, it was recently found by computing the enthalpies for several
pressure values, up to 20 GPa, that in fact the Ccce phase can become
energetically more favorable than Cmc21, with the transition occurring
slightly below 15 GPa.20

The goal of this work is to inspect, through first-principles calculations,
the evolution of the structural properties, the energetic and dynamic
stabilities of the Cmc21 and Ccce structural phases of the SHO compound for
increasing values of applied hydrostatic pressure. Further, enthalpy results
on the I4/mmm and C2/c structural phases have also been computed.

To probe the energetic (thermodynamic) stability, we evaluate the enthalpy
difference curves of the Ccce, C2/c, and I4/mmm structural phases, relative to
the ground state Cmc21 phase at 0 GPa, for hydrostatic pressures up to 20 GPa.
We also analyze the free energies, where the zero-point energy is considered,
in order to infer the energetic stability trend between the Ccce and Cmc21
structures. It is worth mentioning that we have considered not only the
harmonic approximation (HA), but also the quasi-harmonic approximation (QHA)
at constant pressure of 19 GPa. Upon these results, the dynamic stability was
probed by analyzing the phonon dispersion curves of the two energetically
lowest structural phases - Ccce and Cmc21 \- at relevant pressure values.

##  2 Theoretical Methodology

Density Functional Theory (DFT) 26, 27 calculations were performed through the
use of the Quantum Espresso (QE) code. 28, 29, 30 For the exchange-correlation
(xc) functional, the semi-local generalized-gradient approximation with the
Perdew-Burke-Ernzerhof revised for solids (PBEsol) 31, 32 was used.

Firstly, variable-cell relaxation calculations were performed for each
structural phase (Cmc21, Ccce, C2/c, and I4/mmm) of the SHO compound and for
each applied hydrostatic pressure value, ranging from 0 GPa up to 20 GPa. In
these structural relaxations, both the atomic positions and the unit-cell
parameters were allowed to change, by constraining the system to the targeted
pressure value. The Projector Augmented Wave (PAW) 33 datasets were used to
treat semi-core electronic states, with the Sr$\,[4s^{2}\,4p^{6}\,5s^{2}]$,
Hf$\,[5s^{2}\,5p^{6}\,5d^{2}\,6s^{2}]$, and O$\,[2s^{2}\,2p^{4}]$ electrons
being included in the valence shell. Additionally, the variable cell-shape
relaxation, considering the damped Beeman ionic dynamics and the Wentzcovitch
extended Lagrangian for the cell dynamics, 34 was performed with a plane-wave
kinetic-cutoff of 70 Ry. The electronic Brillouin-Zone (BZ) was sampled with a
$\mathbf{\Gamma}$-centered Monkhorst-Pack mesh 35 and defined with 6 $\times$
6 $\times$ 12 subdivisions for the Ccce and C2/c phases; 12 $\times$ 12
$\times$ 6 subdivisions for the ground-state Cmc21 system, and 14 $\times$ 14
$\times$ 14 subdivisions for the I4/mmm phase.

The enthalpy curves, $H$, were then calculated by interpolating the $V-p$ and
$E-V$ curves with a 3rd order Birch-Murnagham equation 36, 37 to obtain the
relation $H=E+pV$, where $E$ is the total electronic energy of the system, $p$
is the hydrostatic pressure to which the system is subjected to, and $V$ is
the volume per formula unit. The calculation of the relative enthalpy curves,
with respect to the lowest enthalpy at 0 GPa of the Cmc21 phase, allows the
analysis of the evolution of the thermodynamic stability of the several
structural phases, relative to one another, with increasing values of applied
pressure (up to 20 GPa).

For the (harmonic) lattice-dynamics calculations, the supercell finite-
displacement method was considered, with the Phonopy software package, 38
where QE is used as the second-order force-constant calculator. The supercells
employed to compute the phonon dispersion spectra were a 2 $\times$ 2 $\times$
2 expansion of the primitive-cell. Phonon calculations were performed for the
Ccce structure at several pressure values above which the system is
energetically more favorable than the Cmc21 phase. Respective calculations
were considered in order to probe the dynamical stabilities of the Ccce phase.
Phonon frequencies calculations were also considered for the Cmc21 system at
pressure values between 20 and 30 GPa.

In the harmonic model, the equilibrium distance among atoms is not temperature
dependent. The anharmonic effects needed to account for thermal expansion can
be introduced by the QHA, in which the thermal expansion of the crystal
lattice is obtained from the volume dependence of the phonon frequencies. 39,
40 To perform a QHA calculation, the phonon frequencies are computed for a
range of expansions and compressions about the 0 K equilibrium volume, and the
constant-volume free energy, for each configuration, is then evaluated as a
function of temperature. From this approach, the equilibrium volume, bulk
modulus, and Gibbs free energy can be obtained at several temperature values
by fitting the free energy as a function of volume to the Vinet equation of
state (EoS).41, 39, 38 We have performed QHA calculations for the two
structural phases of interest: Cmc21 and Ccce. The EoS fit was computed for a
constant pressure of 19 GPa and for a temperature range up to 500 K (more
details on the EoS fitting is discussed in Supp. Information).

##  3 Results and Discussion

###  3.1 Sr3Hf2O7 Polymorphs

Since the environment around the Sr atoms is inequivalent, the general formula
of the SHO system can be better written as (AHfO3)2A′O, where ${\rm
A}^{\prime}={\rm Sr}_{1}$ and ${\rm A}={\rm Sr}_{2}$. Just as other RP
structures, the system results from the inter-growth of rock-salt (R) and
perovskite (P) blocks. The P blocks are composed of two layers of HfO6
octahedra along the $a$-axis, sharing the O corner ions. The A′-site cations
occupy a nine-fold coordination site (R block), while the A-site cations have
a coordination number of twelve (cuboctahedral in P block).

The high-symmetry structure is centrosymmetric and belongs to the I4/mmm space
group (S. G. n. 139).17 By decreasing the temperature, lower symmetry
structural phases may be generated by inducing tiltings and/or rotations of
the O octahedra cages. In Ref. 17 it was theoretically shown that second-order
phase transitions may occur, towards lower structural phases, to the Cmcm or
Cmca space groups, from which the phonon instability modes of these two phases
may direct the final transition to Cmc21 space group. Interestingly enough, it
was experimentally observed that other related RP systems, such as Ca3Mn2O7
and Ca3Ti2O7 compounds, a structural transition path from low-temperature
Cmc21, intermediate-temperature Ccce (S.G. n. 68) and high-temperature I4/mmm
occurs.18 Since the Ccce phase is not related by group-symmetry analysis to
the I4/mmm $\rightarrow$ Cmc21 transition, since there is no single mode
connecting the phases, we infer that this structural phase may occur as a
first-order phase transition. Such a transition can occur by applying an
external perturbation, such as pressure, in order to induce the condensation
of the X${}_{1}^{-}$ zone boundary mode from the centrosymmetric system.17

Figure 1: Representation of the Cmc21, Ccce, C2/c, and I4/mmm structural
phases of Sr3Hf2O7. The red, green, and gold spheres represent, respectively,
the oxygen, strontium, and hafnium ions. The VESTA visualization software was
used to plot the figures. 42

We show in Fig. 1 the four structural phases we have studied in this
investigation. These are:

  1. 1.

The ground-state structural phase at room conditions, evidencing a polar
symmetry, Cmc21 (S.G. n. 36). This system phase is related to the
centrosymmetric I4/mmm structure through group-subgroup relations, by the
condensation of the X${}_{2}^{+}$ and the X${}_{3}^{-}$ modes, which
respectively lower the high-symmetry to the Cmcm and Cmca phases.17

  2. 2.

The Ccce structural phase (S.G. n. 68), which has experimentally been observed
in other similar RP compounds;18 however such a phase has not been evidenced
in SHO by analyzing the group-subgroup relations from the I4/mmm towards the
polar Cmc21 symmetry, and as described in Ref. 17. Respective polymorph is
induced through the condensation of the X${}_{1}^{-}$ mode, which does not
occur spontaneously for the SHO system. Such a distortion mode would have to
be induced by applying an external perturbation (i.e. hydrostatic pressure) to
break the symmetry from I4/mmm towards Ccce. We must note that there is no
group-subgroup relation (by considering single mode analysis) between the Ccce
phase and the Cmc21 phase.17

  3. 3.

The C2/c system (S.G. n. 15) was obtained by mapping out the anharmonic
potential energy surfaces by following the eigenvectors associated with the
soft (imaginary) phonon mode, localized at the $\mathbf{\Gamma}$-point, of the
Ccce structure (a more detailed description has been added in Supp.
Information). From this analysis it is possible to obtain a lower energy
structure, corresponding to the minima of the potential energy surface.

  4. 4.

And finally, the high-symmetry I4/mmm structural phase.

All three structural polymorphs are related to the high-symmetry, high-
temperature, I4/mmm phase through tiltings and/or rotations of the O
octahedra. When external perturbation is applied, symmetry-breaking occurs,
thus lowering the energy of the system.

###  3.2 Energetic Stability

To probe whether the structural phases of Fig. 1 can become energetically
competitive as a function of hydrostatic pressure, the enthalpy energies, for
pressures up to 20 GPa, have been calculated. These results are presented in
Fig. 2, where the relative enthalpy energies, $\Delta H$, with respect to that
of the structure with lowest enthalpy at 0 GPa (the Cmc21 phase) are defined.

Figure 2: (a) Relative enthalpy curves of the I4/mmm, Ccce, C2/c, and Cmc21
structural phases as a function of pressure, with respect to the structure
with the lowest energy value at ambient pressure (Cmc21). The cross marker
indicates the pressure value at which the Ccce structural phase becomes
energetically stable. (b) Schematics of the most plausible transition pathways
that Sr3Hf2O7 can undergo, and based also on Ref. 17.

By analyzing Fig. 2, we observe that at 0 GPa, the orthorhombic Cmc21 phase is
energetically the most stable, as already discussed in literature.17 The C2/c
is slightly higher in energy than Cmc21 ($\Delta$H=$\sim$0.02 eV) at 0 GPa,
however starts increasing the relative enthalpy as a function of pressure up
until $\sim$10 GPa. After 12 GPa the energetics of this system starts
decreasing, however never becomes energetically stable throughout the studied
pressure range. It is noteworthy of mentioning, that the energetic tendency to
decrease for increasing pressures, evidences the possibility of the C2/c
system stabilizing for pressures higher than 20 GPa. At 20 GPa, we observe a
enthalpy difference of the C2/c system with respect to Cmc21 of $\Delta$H=0.15
eV.

At room conditions, the Ccce phase is, in terms of enthalpy, the second
highest system with respect to the ground-state structure. However, for
increasing pressure we observe a considerable decrease in energy. At $\sim 5$
GPa this system will be competing energetically with the C2/c structure, thus
becoming more stable than the latter, and above 17.12 GPa the Ccce phase
becomes energetically the most favorable among the four studied systems.

The high-symmetry I4/mmm system shows quite elevated energy at 0 GPa, which is
expected since it is a high-temperature phase. At around 10 GPa, we observe
that the I4/mmm structural phase starts decreasing the enthalpy and, at 20 GPa
the energetic difference with respect to the Cmc21 phase is around 0.52 eV. In
comparison to the most stable phase at 20 GPa, the Ccce structure, the energy
difference between the two phases is of $\sim 0.60$ eV. Once again, we
envisage the possibility of the high-symmetry I4/mmm system to stabilize at
much higher pressures (> 30 GPa), than those considered in the present work.

The enthalpy calculations were performed at 0 K, without taking into
consideration the contributions to the free energy from the vibrations of the
solids. In order to probe whether these effects could alter the energy
ordering between the Cmc21 and Ccce phases of Sr3Hf2O7, we have performed
lattice-dynamics calculations on the equilibrium and compressed structures to
evaluate the constant-volume Hemholtz (F) and constant-pressure Gibbs (G) free
energies.

Within the harmonic approximation, the Helmholtz free energy as a function of
temperature can be obtained from the phonon frequencies and the lattice energy
of the equilibrium structure. Temperature effects can be included through the
Helmholtz free energy, which by introducing a transformation from the constant
volume function, we obtain the thermal properties at constant pressure, and
thus the Gibbs free energy. 40, 43 By increasing temperature, the volume
dependence of the phonon free energy changes, which in turn results in
different equilibrium volumes for different temperatures. This is regarded as
thermal expansion in the QHA.

The temperature dependence of the relative Helmholtz, $\Delta$F (0 GPa), and
Gibbs, $\Delta$G (19 GPa), free energies of the Cmc21 and Ccce phases are
shown in Fig. 3.

Figure 3: The harmonic (constant-volume/Helmoltz; red) and quasi-harmonic
(constant-pressure at 19 GPa/Gibbs; blue) free energies of the Ccce phase
relative to Cmc21 (black) as a function of temperature.

We observe from the Helmoltz free energy (Fig. 3; red dashed line) that the
Ccce structure becomes energetically more stable than the Cmc21 system (Fig.
3; black solid line), surpassing the latter at 370 K. This observation leads
to the conclusion that the Ccce may also be induced through temperature
effects, with the transition temperature being quite close to room
temperature. At 0 K and 0 GPa, the free-energy difference between both phases
is 38 meV, which is significantly lower than the relative enthalpy obtained
without considering the entropy effects ($\sim$220 meV).

Since the Ccce structure becomes dynamically stable around 19-20 GPa (more
detailed information in Subsec. 3.3), and that the Cmc21 structure is still
stable up until 30 GPa (Subsec. 3.3), we have calculated the Gibbs free energy
at 19 GPa (Fig. 3; blue dashed-dotted line) to probe the evolution of the free
energies of the two phases of interest. We observe that Ccce is
thermodynamically the most stable phase throughout the studied temperature
range (up to 500 K). At 0 K the free energy of the Ccce phase differs from the
Cmc21 system with a relative value of $\Delta$E$\sim$25 meV; as temperature
increases this energy difference also grows further apart ($\Delta$E=54 meV at
500 K). When comparing $\Delta$G with $\Delta$H (Fig. 2) for the two systems
at 0 K and 19 GPa, we observe that the enthalpy difference is $\sim$50 meV,
which is in much closer agreement to when comparing the relative enthalpy with
$\Delta$F (for 0 GPa and 0 K).

We must note that the (quasi-)harmonic approximation will not account for the
influence of the soft modes of the Ccce system on the free energy, a
deficiency which is amplified by the sensitivity of the volume to the free
energy. Therefore the transition temperature might be slightly underestimated
than expected.40 Since at 19 GPa, neither the two analyzed phases evidence
imaginary modes, we conclude that the QHA free energies would provide a much
reasonable description of the free energy behavior between the two systems.
Overall, and by comparing the free energies of Fig. 3 to what was obtained for
the relative enthalpy plots of Fig. 2, we observe comparable behaviors between
the energetics of both polymorphs: at 0 GPa and low temperatures the Cmc21
system is more stable than Ccce; whereas with pressure (19-20 GPa) the Ccce is
more stable than Cmc21, even for finite temperatures.

###  3.3 Dynamical Stability

Energetic stability is a necessary, but not a sufficient, condition for a
structural phase to be synthetically accessible. Another condition that should
be analyzed is the dynamical stability of the system, which requires the study
of the phonon spectra. If imaginary frequencies emerge (usually represented by
negative frequencies along the phonon dispersion curves), such a feature would
indicate that the system is at a transient state, undergoing a phase
transition and thus cannot be kinetically stable at the given temperature
and/or pressure conditions.44, 45, 46, 47, 48, 49

Thus being, we have proceeded to calculating the phonon structure for the Ccce
phase at several pressure values. These values were considered at: room
pressure (and taken from Ref. 17 for comparison); and at values above which
the energetically stability of the Ccce phase occurs - 12, 14, 16, 18, and 20
GPa, as shown in Fig. 4.

Figure 4: Phonon dispersion curves for the Ccce structural phase of the SHO
system for several pressure values: 0 GPa (taken from Ref. 17), 12 GPa, 14
GPa, 16 GPa, 18 GPa, and 20 GPa.

As illustrated in Fig. 4, the phonon dispersion curves of the Ccce structural
phase at 0 GPa (retrieved from Ref. 17) show negative phonon modes localized
at two of the high-symmetry points of the BZ: $\mathbf{\Gamma}$\- and
Y-points. We can therefore infer that such a structural phase is not
kinetically stable at ambient pressure. However, we observe that as the
applied pressure increases, the frequencies of the negative modes shift
towards positive values. The $\mathbf{\Gamma}$-point soft-mode hardens more
rapidly than the Y-point: at 18 GPa only a mild instability is observed at the
latter point, since at $\mathbf{\Gamma}$ the soft optical mode already
evidences positive frequencies. However, it is only at 20 GPa that we arrive
at a structure that is both thermodynamically and dynamically stable, with the
Y phonon mode evidencing positive values as well.

Figure 5: Phonon dispersion curves for the Cmc21 structural phase of the SHO
system for high pressures: 20 GPa, 24 GPa, 28 GPa, and 30 GPa.

We also show, in Fig. 5, the phonon dispersion curves of the Cmc21 system, at
several pressure values, and we observe that the structure is dynamically
stable still at 20 GPa, thus coexisting with Ccce phase, as a metastable
compound. It is noteworthy of mentioning that around the high-symmetry
S-point, the lowest frequency branch starts to decrease towards a soft mode,
which may be an indication that for higher pressure values a structural
transition will occur due to the instability at this q-point. Indeed, by
increasing the pressure to 30 GPa it is observed that this soft-mode becomes
unstable, thus rendering this structure kinetically unviable above this
pressure value.

##  4 Conclusions

In the present work we have performed ab-initio calculations to study the
thermodynamical stability of the room temperature Cmc21 phase and the high-
temperature I4/mmm structure; as well as different structural phases of
Sr3Hf2O7, which are not related to the aristotype polymorph through group-
subgroup relations, i.e., are a result of first order transitions. The latter
were the Ccce and C2/c structural phases.

It was observed that, at room conditions, the Cmc21 phase is the most
energetically stable, as expected; the C2/c phase being 0.05 eV higher in
energy than the ground-state; followed by the Ccce system with an energy
difference of $\sim 0.18$ eV with respect to Cmc21; and finally the I4/mmm
polymorph with the highest energy difference of $\sim$0.41 eV in relation to
the ground state phase. We observe that the enthalpy of Ccce decreases as
pressure increases, and at $\sim 17$ GPa, it surpasses that of the Cmc21
structure, becoming energetically the most stable system of all four studied
compounds.

To inspect in detail the two lowest energetic compounds, namely the Ccce and
Cmc21 phases, we have calculated the free energies (where entropy is accounted
for) and the phonon dispersion curves to analyze their respective dynamical
stability. From the constant-volume free energies, we observe that the Cmc21
system is energetically the most stable compound below 370 K. Above this
temperature, the Ccce polymorph has its free energy lowered, thus becoming
more stable than Cmc21 phase. By probing the volume-temperature dependent free
energies at 19 GPa, we observe that the Ccce is always the most stable phase
when compared to Cmc21, throughout the studied temperature range, up to 500 K.
When observing the phonon dispersion spectra, we conclude that the Ccce
structure only becomes dynamically stable at 20 GPa, which is at a higher
pressure range than when the energetic transition occurs from Cmc21. The Cmc21
phase, on the other hand, is dynamically stable up to 30 GPa, when an
imaginary mode at the high-symmetry S-point emerges. These results suggest
that both phases may coexist energetically at pressure values from $\sim 19$
GPa up to $\sim 30$ GPa; at this pressure interval the polar Cmc21 structure
is a metastable system.

From the present calculations we conclude that the quasi-2D layered
Ruddlesden-Popper perovskite Sr3Hf2O7 compound has a stable phase at room
conditions which is a ferroelectric ground state system with Cmc21 symmetry.
By applying hydrostatic pressure this phase may undergo a transition to the
Ccce structure, for it becomes thermodynamically (at $\sim 17$ GPa) and
dynamically stable (at 20 GPa).

This research was supported by the Network of Extreme Conditions Laboratories
(NECL), POCI-01-0145-FEDER-029454 (FLIP), POCI-01-0145-FEDER-032527
(vdW4device), UIDB/04968/2020 (IFIMUP - Financiamento Plurianual da Unidade de
I&D), funded by the Portuguese Foundation for Science and Technology (FCT) and
co-financed by NORTE 2020, through the programme Portugal 2020 and FEDER.
Authors also thank the financial support of the FAPESP projects 2018/07760-4
and 2019/07661-9, and CNPq projects 314884/2021-1 and 311373/2018-6. E.L.d.S.
acknowledges the CEEC individual fellowship 5th edition with Reference
2022.00082.CEECIND financed by FCT; and the High Performance Computing Chair -
a R&D infrastructure (based at the University of Évora; PI: M. Avillez),
endorsed by Hewlett Packard Enterprise, and involving a consortium of higher
education institutions, research centers, enterprises, and public/private
organizations. The authors acknowledge PRACE for awarding access to the Fenix
Infrastructure resources, which are partially funded from the European Union’s
Horizon 2020 research and innovation programme through the ICEI project under
the grant agreement No. 800858 (technical support provided by CINECA).
Additionally, the authors acknowledge the national computer resources from FCT
CPCA/A2/7087/2020 at MACC - Bob, and CPCA/A2/7263/2020 and CPCA/A1/431509/2021
(2ª Edição) at Univ. Évora - Oblivion, respectively.

##  5 Supplementary Material

###  5.1 Evolution of the lattice parameters with applied pressure

In Fig. 6 we show the lattice parameters of the four SHO studied systems as a
function of pressure in the 0-20 GPa range, namely the Cmc21, the C2/c, the
Ccce, and the I4/mmm phases.

Figure 6: Evolution of the calculated lattice parameters ($a$, $b$, and $c$)
for the four of the considered structures of the SHO system with increasing
hydrostatic pressure.

###  5.2 Potential-Energy Surface

By analyzing the Ccce phonon dispersion curves of SHO at 0 GPa (Fig. 7, top,
left) we observe two unstable modes localized at the high-symmetry
$\mathbf{\Gamma}$-point. Considering the frequency mode with higher degree of
instability (larger negative frequency), we have mapped out the anharmonic
potential energy surfaces by following the eigenvectors associated with this
instability/displacement. From this analysis, it is possible to obtain a lower
energy structure, corresponding to the minima of the potential energy surface.
This displacement involves the O-cage distortions and the Sr displacements at
the unit-cell, and minimizing the the energy down to -30.97 meV (Q=1), by
lowering the symmetry to C2/c, when compared to the Ccce phase at Q=0 (Fig.
7).

Figure 7: Double-well potential-energy surface (bottom plot) for the phonon
instabilities associated with the high-symmetry $\mathbf{\Gamma}$-point
negative mode of the Ccce phase [top, left figure (results of the phonon
dispersion are from Ref. 17)]. The minima of the potential-energy surface
correspond to the energetically favorable structural phase at 0 GPa, the C/2c
polymorph, where the respective phonon modes are also shown (top, right plot).
The normal-mode coordinates Q have been normalized so that the minima are
located at Q=$\pm$1 and the energy differences are those calculated in a unit-
cell.

For these calculations we used the open-source MODEMAP package.43, 50 A
sequence of displaced structures is generated by displacing along the phonon
eigenvectors over a range of amplitudes of the normal-mode coordinate Q. The
total energies of the “frozen phonon” structures are then evaluated from
single-point DFT calculations, after which the E(Q) curves will be fitted to a
polynomial function, with the number of terms depending on the form of the
potential well.

By computing the phonon dispersion structure of the energetically favorable
system, the C2/c, we observe that it is dynamically unstable at room
conditions (Fig. 7, top, right). A negative mode appears at the Y2 q-point,
which is related to the Y mode from the Ccce structure.

###  5.3 Electronic Band Structure of the Ccce and Cmc21 Structural Phases

The electronic band dispersions at pressure values above which the dynamical
stability has been confirmed for the Ccce structure is presented in Fig. 8.
These values were chosen to be 20 GPa, 24 GPa, 28 GPa, and 30 GPa. We may
infer that as pressure increases, a subtle decrease of the band-gap occurs,
from 4.25 eV (at 20 GPa) towards 4.18 eV (at 30 GPa). All structures display
an indirect band-gap, where the valence band maximum (VBM) is located at the
$\mathbf{\Gamma}$-point, and the conduction band minimum (CBM) is slightly
shifted away from the $\mathbf{\Gamma}$-point and along the
$\mathbf{\Gamma}$-Z segment. It should be noted, however, that for the Ccce
structural phase (at all considered pressure values), the difference between
direct and indirect band-gaps is small (less than $\sim 0.1$ eV) and within
the DFT error margin. Thus being, we may infer that most optical absorption
will be direct.

Figure 8: Electronic band structures of the Ccce structural phase of the SHO
system at several applied pressure values: 20 GPa, 24 GPa, 28 GPa, and 30 GPa.

The same procedure has been adopted for the Cmc21 phase, and we observe an
opposite trend with respect to the Ccce system: a tendency to increase the
band-gap energy for increasing pressures. The Cmc21 structure evidences also
an indirect band-gap, except at 12 GPa. The location of the VBM varies for
different pressure values; however it is noteworthy of mentioning that the
extremal points throughout the BZ are very close in energy, and therefore are
within the DFT error margin. While for 0 GPa, the VBM is located at the G0-T
high-symmetry segment, at 12 GPa the VBM is along $\mathbf{\Gamma}$-Z. For
larger pressures, the VBM is locked along the Z-B0 high-symmetry segment. The
CBM seems to be more stable under different pressure values, remaining located
at the $\mathbf{\Gamma}$-point. The gap increases from 4.27 eV (at 0 GPa) to
4.52 eV (at 28 GPa).

Figure 9: Electronic band structures of the Cmc21 structural phase of the SHO
system at different applied pressure values: 0 GPa (taken from Ref. 17, 12
GPa, 14 GPa, 20 GPa, 24 GPa, and 28 GPa.

We have found in literature, and from experimental data of the parent
perovskite systems, that at room temperature these may yield band-gap values
around $\sim$ $6.1$ eV;51, 52, 53 however, theoretical DFT calculations yield
lower values [as expected due to the well-known problem of the asymptotic
behavior when using (semi-)local functionals]. From photoluminescence
measurements, it was observed two visible emission lines close to 420 and 500
nm in AZrO3 and AHfO3 (A = Ca, Sr, and Ba), with respective spectral weights
being sensitive to the local distortions.51 Moreover, calculations for the
high-symmetry and high-temperature, cubic Pm$\overline{3}$m structural phase
of the parent systems returned band-gaps ranging from $3.7$ eV to $5.79$ eV.
54, 55, 56, 57, 58 These values were obtained through DFT calculations, by
employing LDA and GGA for the xc functional. For room temperature and the Pnma
structural phase, the calculations resulted in a $4.18$ eV band-gap width. 59

Tuning and/or engineering of the band-gaps towards lower values for double-
layered RP systems could in fact be possible, either through cation mutations
and/or replacing O by other chalcogen elements (i.e. S, Se), since these may
alter the Fermi energy accordingly, by changing the position of the CBM and/or
VBM, respectively. This study is however out of the scope of the present work,
and therefore we will not enter into further discussion.

###  5.4 Electronic Band Structure of Cmc21 with Spin-Orbit Coupling Effects

We compare in Fig. 10 the electronic band dispersion of the Cmc21 system when
taking the Spin-Orbit Coupling (SoC) effects into account, and when these
interactions are not considered in the calculations. We see that at 0 GPa the
energy splitting of the CBM is only 0.062 eV, with an increase of the band
energy splitting occurring at 20 GPa with 0.126 eV.

Figure 10: Fully relativistic electronic band structure (solid red line) of
the conduction band minimum, compared to the system without considering the
spin-orbit interaction (purple dashed line) of the polar Cmc21 structure at 0
GPa (top) and 20 GPa (bottom).

On the other hand, for the VBM, we observe that the position of the band
maximum changes. For room pressure (0 GPa), the VBM goes from the high-
symmetry segment G0-T, when we do not consider SoC, to a direct character,
when SoC is included, with the VBM aligning with the CBM at the same high-
symmetry point, along the segment $\mathbf{\Gamma}$-Z. When pressure is
applied with 20 GPa, we also observe a change of position of the VBM from the
Z-B0 high-symmetry segment to $\mathbf{\Gamma}$-Z, although remaining with an
indirect character, as to what occurs without relativistic effects. We must
state that the changes of the VBM position when considering, and not
considering, SoC, are related to the fact that the valence band extrema along
the BZ are very close to the Fermi energy, having energy maxima very similar
to each other, and therefore these changes are quite subtle.

From these calculations we do not observe any major effects of the electronic
properties when fully relativistic calculations are considered, and most
importantly these calculations do not evidence any momentum-dependent band
spin splitting for the studied polar structural phase.

###  5.5 Quasi-Harmonic Approximation

Since at 19 GPa the Cmc21 and Ccce polymorphs are both dynamically stable, we
have probed the QHA derived-properties of the two systems at this pressure
range. Moreover, at this pressure value, we obtain good fittings of the free-
energy equations of state for both the studied phases, as can be observed in
Fig. 11. Since at low pressures the Ccce phase evidences several imaginary
frequencies (more detailed information in Subsec. 3.3), we found anomalies in
the low pressure free-energy equations of state, which were only resolved when
fixing the pressure close to 19 GPa.

Figure 11: Helmoltz free-energy equation-of-state fits for the two Sr3Hf2O7
polymorphs at different lattice temperatures: Cmc21 (left), and Ccce (right),
for a pressure value of 19 GPa and maximum temperatures up until 500 K. Note
that the free energies are given as relative energies with respect to the
lowest energy-volume point at 500 K.

The structural properties, as a function of temperature, obtained from the QHA
are shown for each phase in Fig. 12. These include the temperature-dependent
volume (per formula unit), $V(T)$ and the volumetric thermal-expansion
coefficient, $\alpha_{V}(T)$.

Figure 12: Volume per formula unit, $V(T)$ (left), and volumetric-expansion
coefficient, $\alpha_{\textrm{V}}(T)$ (right), for the two Sr3Hf2O7
polymorphs: Cmc21 (black), and Ccce (red), for a pressure value of 19 GPa and
maximum temperatures until 500 K.

The temperature-induced volume changes in SHO are dependent on the stretching
of the Hf-O bonds and the tilting of the HfO6 octahedra. As the Cmc21 and Ccce
structures have a similar HfO6 octahedral framework, one may infer that the
octahedral cages remain largely ideal (i.e., undistorted), and so the thermal
expansion is accommodated primarily by changes in the volume of the cage.40 We
observe that the volume expansion (Fig. 12, left) of the Ccce polymorph has a
larger increase than Cmc21 when the temperature raises, which makes the former
system a more compressible structure. Moreover, the thermal expansion of the
Ccce phase (Fig. 12, right) is found to be higher than that of the lower-
symmetry phase, for increasing temperatures, however showing a similar trend
at temperatures above 50 K. For the low temperature regime, the expansion
coefficients of the Cmc21 system decreases slightly to negative values,
indicating negative thermal expansion (NTE), i.e. a reduction in volume upon
heating. The NTE can be associated to several microscopic mechanisms, however
for this specific system to ferroelectricity, since this is in fact a polar
phase.60 Above 50 K, a rapid increase in the expansion coefficients are
evidenced for Cmc21 until $\sim$150 GPa, followed by a slower increase up
until 500 K. Such a behavior is also noticed for Ccce, although with a much
steeper increase between the interval of 50 K and 150 K.

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# Switching off microcavity polariton condensate near the exceptional point

Yao Li  Tianjin Key Laboratory of Molecular Optoelectronic Science, Institute
of Molecular Plus, School of Science, Tianjin University, Tianjin 300072,
China  Xuekai Ma<EMAIL_ADDRESS>Department of Physics and Center for
Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn,
Warburger Strasse 100, 33098 Paderborn, Germany  Zaharias Hatzopoulos
Institute of Electronic Structure and Laser (IESL), Foundation for Research
and Technology-Hellas (FORTH), Heraklion 71110, Greece  Pavlos Savvidis
School of Science,Westlake University, 18 Shilongshan Road, Hangzhou 310024,
Zhejiang Province, China  Institute of Natural Sciences, Westlake Institute
for Advanced Study, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province,
China  Institute of Electronic Structure and Laser (IESL), Foundation for
Research and Technology-Hellas (FORTH), Heraklion 71110, Greece  Department of
Nanophotonics and Metamaterials, ITMO University, St. Petersburg 197101,
Russia  Stefan Schumacher  Department of Physics and Center for
Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn,
Warburger Strasse 100, 33098 Paderborn, Germany  Wyant College of Optical
Sciences, University of Arizona, Tucson, AZ 85721, USA  Tingge Gao
<EMAIL_ADDRESS>Tianjin Key Laboratory of Molecular Optoelectronic
Science, Institute of Molecular Plus, School of Science, Tianjin University,
Tianjin 300072, China

###### Abstract

Gain and loss modulation are ubiquitous in nature. An exceptional point arises
when both the eigenvectors and eigenvalues coalesce, which in a physical
system can be achieved by engineering the gain and loss coefficients, leading
to a wide variety of counter-intuitive phenomena. In this work we demonstrate
the existence of an exceptional point in an exciton polariton condensate in a
double-well potential. Remarkably, near the exceptional point, the polariton
condensate localized in one potential well can be switched off by an
additional optical excitation in the other well with very low (far below
threshold) laser power which surprisingly induces additional loss into the
system. Increasing the power of the additional laser leads to a situation in
which gain dominates in both wells again, such that the polaritons re-condense
with almost the same density in the two potential wells. Our results offer a
simple way to optically manipulate the polariton condensation process in a
double-well potential structure. Extending such configuration to complex
potential well lattices offers exciting prospects to explore high-order
exceptional points and non-Hermitian topological photonics in a non-
equilibrium many-body system.

Effective Hamiltonians of non-Hermitian nature play a crucial role in our
understanding of a plethora of modern physical systems [1, 2]. This is in
spite of the underlying common principle that in quantum mechanical systems
Hermiticity is required to keep a real-valued spectrum of eigenvalues. Complex
eigenvalues generally appear in systems governed by non-Hermitian
Hamiltonians. However, Bender et.al. [3] observed that under one special
condition systems can still sustain a real-valued energy spectrum. That is if
the Hamiltonian commutes with the PT operator, where P and T are the parity
and time-reversal operator, respectively. In recent years photonic platforms
are widely employed to study PT symmetry in view of the fact that the
refractive index distribution can be judiciously engineered, that is, the real
part of the refractive index can be tailored to be an even function whereas
the imaginary part remains an odd function such that PT symmetry can be
fulfilled. A striking phase transition can occur at a point in parameter space
(the symmetry breaking point or exceptional point) where the real eigenvalues
become complex-valued. This transition can be induced by modulation of the
real or imaginary components of the external potential in the effective
Hamiltonian. Near this exceptional point, a large variety of counter-intuitive
phenomena has been reported, for example, PT symmetric synthetic lattices [4]
or microcavities [5], single mode [6, 7] and vortex lasing [8] in a ring
cavity, unidirectional light propagation in a waveguide [9], suppression and
revival of lasing due to loss in coupled whispering-gallery-mode microcavities
[10], improved sensitivity in single or coupled resonators [11, 12], optical
isolation in PT symmetric microcavities [13], formation of an exceptional ring
in a photonic crystal [14], and an enhanced linewidth of a phonon laser mode
at the exceptional point [15]. In contrast to regular laser operation, a non-
Hermitian system can behave surprisingly near the exceptional point where
changes in gain can shut down or re-excite lasing as demonstrated in coupled
resonators [16, 17, 18]. Approaching the exceptional points in these works is
achieved using asymmetrical pumping schemes.

Exciton polaritons form as hybrid particles from excitons in a quantum well
that strongly couple with the photon mode of a planar optical resonator. In an
excitation regime where polaritons behave like bosonic quasi-particles, they
can experience condensation and show spontaneous macroscopic coherence under
non-resonant excitation [20, 21, 22], as shown in Fig. 1(a). Thanks to the
spontaneous decay of photons from the microcavity, polariton systems are
inherently driven-dissipative in nature and as such provide an excellent
platform to study PT symmetry [23] and non-Hermitian physics. In our previous
works [24, 25] we demonstrated that the polariton condensate wavefunction and
energy distribution can be strongly modified by the gain and loss coefficients
in an optical billiard. Also in an exciton polariton system in coupled
microresonators the influence of pump asymmetry was explored and it was
demonstrated that relaxation kinetics plays a crucial role in the condensation
process [19]. Polariton dimers also offer a platform to study polariton-
polariton nonlinear interactions with Josephson oscillations, quantum self
trapping [26], interplay of interference and nonlinearity [27], and
bistability [28]. However, the existence of an exceptional point and its role
for the polariton condensation was not demonstrated or discussed in such
structures.

Figure 1: Excitation configuration. Schematics of polariton condensation
process in the planar microcavity (a) and the structure of a built-in double-
well potential (b) which is pumped by two independent non-resonant laser
beams.

In the present work we utilize a particularly simple and robust double-well
potential structure with slightly asymmetric sites as sketched in [Fig.
1(b)](the details of the sample can be seen in the experimental section). We
show that for non-resonant optical excitation polariton condensates can be
loaded into the double-well in a controlled manner , and demonstrate that an
exceptional point can be realized and explored in detail varying the optical
excitation parameters. In our experiment, we use two independent non-resonant
pumping lasers to separately excite the two potential wells and find that the
polariton condensation in one potential well can be altered dramatically and
even shut down completely by only controlling the pump intensity on the
adjacent potential well (in this case the double-well potential system as a
whole thus falls below the condensation threshold). The reason is that varying
the pump intensity on one of the potential wells changes the coupling of them,
giving rise to the appearance of an exceptional point which results in
substantial redistribution of the modes and reduced effective gain for
polaritons in the coupled wells. With further increasing the pump intensity on
the adjacent well, the polariton dimer starts condensation again as the system
is tuned away from the exceptional point and bifurcates into two modes,
antibonding and bonding. Our results show that the exceptional point can be
easily tailored in such kind of macroscopic quantum system and our work can be
systematically extended to 1D or 2D lattices to investigate anomalous edge
modes based on polariton condensates [29, 30].

##  I results

## Theory of non-hermitian degeneracy

In theory, the Hamiltonian of a two-level non-Hermitian system like a coupled
polariton dimer trapped in two potential wells 1 and 2 can be expressed as
follows:

Figure 2: Eigenenergies of a two-level non-Hermitian Hamiltonian near an
exceptional point. (a) Real and (b) imaginary parts of the eigenenergy in a
two dimensional parameter space, where $\Delta\gamma$ =
$\varGamma_{1}-\varGamma_{2}$, $\Delta E$= $E_{1}-E_{2}$. Relations between
the (c) real and (d) imaginary parts of the eigenenergy with $\Delta\gamma$
along the route $\Delta{E}=0$ highlighted in (a) and (b). The arrow indicates
the decrease of $\Delta{\gamma}$. The exceptional point appears at
$\Delta{\gamma}=J$.

$H=\left[\begin{array}[]{ccc}E_{1}+i\varGamma_{1}&J/2\\\
J^{*}/2&E_{2}+i\varGamma_{2}\\\ \end{array}\right]$ (1)

where $E_{1}$($\varGamma_{1}$) and $E_{2}$($\varGamma_{2}$) correspond to the
real (imaginary) part of the polariton energies in the two potential wells, J
is the coupling strength between the two wells. The eigenenergies of the
Hamiltonian are
$E_{\pm}=(E_{1}+E_{2}+i\varGamma_{1}+i\varGamma_{2})/2\pm\sqrt{(J^{2}+[E_{1}-E_{2}+i\varGamma_{1}-i\varGamma_{2}]^{2})}/2$.
The dependence of the real and imaginary parts of the eigenenergies on
$\Delta{E}=E_{1}-E_{2}$ and $\Delta{\gamma}=\Gamma_{1}-\Gamma_{2}$ are shown
in Figs. 2(a) and 2(b). We introduce asymmetric pumping of the polariton dimer
by sequentially exciting the two potential wells: the potential well 1 is
excited above the threshold, then the pumping power of the potential well 2 is
increased from zero. Before the system approaches the exceptional point with
the difference of the gain level of the two potential wells $\Delta\gamma>J$
(the potential well 1 has a larger gain than 2) along the route at
$\Delta{E}=0$ shown in Figs. 2(a) and 2(b), one of the two eigenstates of the
above non-Hermitian Hamiltonian lies above the condensation threshold with
large gain [see the lower branch in Fig. 2(d)] and another is below the
threshold with large loss rate [see the upper branch in Fig. 2(d)] [16]. When
the pump power on the other potential well increases, the difference of the
gain level diminishes as shown in Fig. 2(b) and (d) until $\Delta\gamma=J$,
thus approaches the exceptional point [18], where
$J^{2}+[E_{1}-E_{2}+i\varGamma_{1}-i\varGamma_{2}]^{2}=0$. In this case, the
polaritons feel a larger effective loss when the gain/loss ratio in the
coupled wells approaches a critical value, which can leads to the already
condensed polariton being switched off. As a consequence, the polariton dimer
is below the threshold and the emission intensity is reduced greatly. When
$\Delta{\gamma}$ decreases further, the system is pulled away from the
exceptional point and the gain/loss contrast is reduced, thus the real parts
of the eigenvalues of the system are repelled away from each other [see the
bifurcation in Fig. 2(c)] and finally the polariton dimer can condensate
again. The switching off of the polariton dimer can also occur when
${E_{1}}\neq{E_{2}}$ where the anticrossing of the eigenenergy of the system
will appear [17, 16, 18].

## Experimental realization

In the experiment, we use a microcavity which contains 12 GaAs quantum wells
in the cavity with the Rabi splitting of around 9.2 meV and is cooled down at
around 4 K. The two potential wells with the distance of around 5 $\mu$m [see
the sketch in Fig. 1 (b)] are formed unintentionally during the growth process
due to some defects which induce loss into the system. Such kind of potential
wells can also be tailored with microstructuring mesas into the planar
microcavity. From the measurement taken under low pumping power at the
potential wells and planar microcavity (shown in the Supplemental Materials),
we find the potential depth of the two potential wells are 1.46 meV and 1.30
meV, respectively, as demonstrated in Fig. S1 (b) and (c). This double-well is
chosen which allows for realizing equal energy distribution in the two
potential sites with deep well above threshold and shallow well below
threshold, and polaritons in the system experience large gain level
difference. All the experiments are taken using a CW laser (wavelength: 750
nm) and the laser beam is chopped with a mechanical chopper with the duty
circle of 5% to reduce heating.

Figure 3: Switching off exciton polariton condensates. (a) The emission
intensity of the polariton molecule as the function of the pumping intensity
of the potential well 2 with the potential well 1 being continuously excited.
The real space distribution of polaritons is taken at the pump densities of
0.006MW/cm2 (b), 0.048MW/cm2 (c), and 0.115MW/cm2 (d) of the second laser
beam. (e-g) Time-integrated density profiles of the polariton condensates from
numerical simulations at $P_{1}=$12 $\mu$m-2 ps-1 and different $P_{2}$: (e)
$P_{2}=$ 0 $\mu$m-2 ps-1, (f) $P_{2}=$ 5 $\mu$m-2 ps-1, and (g) $P_{2}=$ 15
$\mu$m-2 ps-1. Note that the density profiles in the experiment (b-d) and in
the simulation (e-g) have different orientations.

Firstly we focus the first laser beam onto the potential well 1 with the spot
size of around 2 $\mu$m. At the pump density of around 0.45 MW/cm2 ($\sim
1.2P_{th}^{1}$, where $P_{th}^{1}$ is the condensation threshold of the
potential well 1), the polaritons condense and are mainly located in the
driven potential well 1 [see Fig. 3 (b)] with the energy of 1.5056 eV (the
data showing the polariton condensation can be seen in Fig. S2 of the SM).
Under quasi-resonant pumping, similar localization of exciton polaritons in a
photonic dimer due to quantum self localization was observed in [26], and
polaritons can be localized or delocalized in a photonic dimer [28] due to the
interplay between the nonlinear interactions and interference when the laser
energy is tuned across the antibonding/bonding modes.

In the following, the second potential well 2 is excited using another laser
beam with the same wavelength and spot size, during which the pumping flux of
the potential well 1 is fixed at around 1.2 $P_{th}^{1}$. We monitor the
emission of the polariton dimer by gradually increasing the pump density of
the second laser beam from zero to above the threshold $P_{th}^{2}$ (here
$P_{th}^{2}=$0.4 MW/cm2 is the threshold of the condensation in the potential
well 2). Surprisingly, the intensity of the polariton dimer decreases
dramatically to nearly zero (Fig. 3 (a) and (c)) then increases again with the
symmetric emission pattern appearing across the coupled potential wells, as
shown in Fig. 3 (d). It is worth noting that the above results in Fig. 3 are
reversible, i.e., when we gradually reduce the power of the second laser, the
polariton dimer shows the same behavior at the same pump power.

To check whether an exceptional point exists or not in the polariton dimer, we
measure both the energy levels (real and imaginary parts) and the real space
distribution of different polariton modes as the power of the second laser
beam varies. Thanks to the coupling and the potential depth difference in the
system, the polariton double-well potential can be tuned to the vicinity of
the exceptional point by varying the pump power of the second laser beam. When
the pump density of the potential well 2 is smaller than 0.032 MW/cm2, there
are two states whose energy share the same real part but different imaginary
parts, as shown in Fig. 4. The total emission of the polariton dimer is mainly
located in the potential well 1. At higher pumping density of the potential
well 2, the polariton condensate is switched off and the coupled wells are
below the threshold, which can be seen from the dispersion taken in the
switching off regime in Fig. S3 of the SM, thus the emitted light intensity is
greatly reduced. When the pump power density of the potential well 2 is larger
than 0.115 MW/cm2, the emission intensity of the polariton dimer increases
sharply and we can find two states: the antibonding mode and bonding mode, as
shown in the inserts in Fig. 4 (a). The intensity of the mode with larger
energy is much larger than another state thus dominates the total emission
pattern (Fig. 3 (d)). The energy difference between these two states increases
with the power of the second laser beam (Fig. 4 (a)) and the intensity of the
lower-energy state grows faster. At the same time, the linewidths of the two
modes are around the same [Fig. 4 (b)]. Hence, the clear bifurcations of the
eigenenergies of the coupled condensates shown in Fig. 4 are consistent with
the theoretical prediction in Fig. 2 (c,d), evidencing the existence of an
exceptional point. In this scenario, the pump power of the second laser beam
(the gain level difference between the two potential wells) acts as one
parameter ($\Delta\gamma$) in the two dimensional parameter space in Fig. 2,
which switches off the polariton condensate due to the substantial
modification of the condensate wavefunction near the exceptional point. Under
higher power of the second laser, the system approaches symmetric pumping, the
emission of the polariton dimer becomes mainly localized at each potential
site. In the coupled classical laser systems, we note that the appearance of
two modes after switching off the coupled lasers or not also depends on the
modal cross-saturation or spatial hole burning effect [16].

Figure 4: Demonstration of the exceptional point. The real (a) and imaginary
(b) part of the polariton energy of the coupled wells as the function of the
pumping intensity of the potential well 2. The inserted graphs in (a)
correspond to the real space imaging of the eigenmodes at several pumping
intensities where the bottom ones are multiplied by 3 times, 0.2 times, 0.5
times respectively. The dashed regions show the switching off region.

If the pump power of the potential well 1 is increased, the polariton
condensate can be partly switched off by repeating above experimental steps.
For example, if the pump density of the potential well 1 is kept at around 2
$P_{th}^{1}$, the second laser can reduce the emission intensity at potential
well 1 by about 40$\%$ at the pump density of 0.064 MW/cm2 then enhance the
intensity afterwards (see Fig. S5 in the SM). During this process, the
emission intensity evolves from asymmetric to symmetric pattern.

The coupled lasers can also be switched off when the frequencies of the two
resonators is different [16, 17, 18]. If we swap the two potential wells, that
is, the potential well 2 is pumped above its threshold firstly and the pumping
flux of potential well 1 is varied. In this configuration, the real parts of
the polariton energy of the two potential wells are not the same due to the
difference of the potential depths, however, the switching off polariton
condensate can still be observed (see Fig.S6 in the SM). It means that in this
case, the system experiences an anticrossing along a different route, instead
of $\Delta{E}=0$, in Fig. 2(a,b).

## Numerical analysis

We numerically mimic the dynamics of the polariton condensate in our
experiments by applying the extended Gross-Pitaevskii equation with loss and
gain [31]:

$\displaystyle i\hbar\frac{\partial\Psi(\mathbf{r},t)}{\partial t}$
$\displaystyle=\left[-\frac{\hbar^{2}}{2m_{\text{eff}}}\nabla_{\bot}^{2}-i\hbar\frac{\gamma_{\text{c}}}{2}+g_{\text{c}}|\Psi(\mathbf{r},t)|^{2}\right.$
(2)
$\displaystyle+\left.\left(g_{\text{r}}+i\hbar\frac{R}{2}\right)n(\mathbf{r},t)+V(\mathbf{r})\right]\Psi(\mathbf{r},t),$

Here, $\Psi(\mathbf{r},t)$ is the polariton wavefunction, $m_{\text{eff}}$
denotes the effective mass of polaritons in the vicinity of the bottom of the
lower polariton branch, $\gamma_{\text{c}}$ is the polariton loss rate,
$g_{\text{c}}$ represents the polariton-polariton interaction, and
$g_{\text{r}}$ represents the polariton-reservoir interaction. The reservoir
$n$ is incoherent and provides the gain for the condensate with a rate $R$. It
obeys the following equation of motion:

$\frac{\partial n(\mathbf{r},t)}{\partial
t}=\left[-\gamma_{r}-R|\Psi(\mathbf{r},t)|^{2}\right]n(\mathbf{r},t)+P(\mathbf{r})\,.$
(3)

Here, $\gamma_{\text{r}}$ is the loss rate of the reservoir and
$P(\mathbf{r})$ represents the two non-resonant pumps with Gaussian
distribution, i.e.,

$P(x,y)=P_{1}e^{-\frac{(x-x_{1})^{2}+y^{2}}{w^{2}}}+P_{2}e^{-\frac{(x-x_{2})^{2}+y^{2}}{w^{2}}},$
(4)

with pump intensity $P_{1}$ and $P_{2}$ and pump width $w$. The two pumps are
placed at $x_{1}$ and $x_{2}$, respectively. The double-well potential
$V(\mathbf{r})$ takes the following form:

$V(x,y)=V_{\text{1}}e^{-\left(\frac{(x-x_{1})^{2}+y^{2}}{W^{2}}\right)^{2}}+V_{\text{2}}e^{-\left(\frac{(x-x_{2})^{2}+y^{2}}{W^{2}}\right)^{2}}.$
(5)

Here, $V_{1}$ and $V_{2}$ are the depths of the potential well 1 and 2,
respectively, and $W$ indicates the size of each well. The parameters used for
numerical modeling are summarized in [32]. We fix the intensity of pump 1 that
excites potential well 1 with $P_{1}=12$ $\mu$m-2 ps-1 ($\sim
1.2~{}P_{th}^{1}$) and gradually increase the other pump intensity $P_{2}$ in
potential well 2, i.e., numerically repeating the experimental excitation
process presented in Fig. 3(a-d). The numerical results are shown in Fig.
3(e-g) (here the density profiles are integrated over time to match the
experimental measurements) and are in very good agreement with the
experimental results in Fig. 3(b-d).

## Discussion

We note that the exciton polariton condensate in the polariton dimer can not
be switched off under uniform pumping where the emission intensity
continuously increases with the pumping flux, as shown in Fig. S7 of the SM.
In this case the gain and loss coefficient of the two potential wells are
modulated simultaneously. In addition, as we also find in our numerical
simulations, the coupling between the two potential wells plays a critical
role in the switching off process, which can not be observed if we use two
tightly focused laser spots to excite a simple planar microcavity (see Fig. S8
in the SM).

To summarize, the counter-intuitive results reported in the present work are
rooted in the optically induced gain and loss modulation near an exceptional
point in a polariton dimer. We demonstrate that this can be used to
efficiently control the polariton condensation process and explore the role of
the exceptional point in detail. Switching off the polariton condensate with
increasing total pump power can also be used to investigate further the
bistability or multistability [33] in the non-Hermitian double-well potentials
if the detuning between the excitons and cavity mode are tuned to be more
positive. If the polariton condensate were loaded in multiple coupled
potential wells, high-order exceptional points could be explored where the
polariton energy is very sensitive against the perturbation[11, 12]. For the
future, we also envision that novel edge modes based on polariton condensates
can be created in 1D or 2D potential lattices when the gain and loss
coefficients are modulated similarly to the demonstration in the present work
[29, 30].

## Acknowledgments

T.G.thanks Li Ge for fruitful discussion and acknowledges the support from the
National Natural Science Foundation of China (grant No. $11874278$). The
Paderborn group acknowledges the Deutsche Forschungsgemeinschaft (DFG) through
the collaborative research center TRR142 (project A04, grant No. 231447078)
and Heisenberg program (grant No. 270619725). X.M. further acknowledges
support from the National Natural Science Foundation of China (grant No.
11804064). P.S thanks Project No. 041020100118 supported by Westlake
University and Program 2018R01002 supported by Leading Innovative and
Entrepreneur Team Introduction Program of Zhejiang. P.S. acknowledges
financial support from Russian Science Foundation Grant No. 19-72-20120 for
supporting MBE sample growth and from Greece Polisimulator project co-financed
by Greece and EU Regional Development Fund.

## Methods

The setup we used in the experiment is a hand-made momentum-space spectroscopy
system. The laser is divided into two identical beams through a 50:50 beam
splitter, and is focused onto the sample through the same objective (X 50, NA
0.42). The intensity of each laser can be adjusted continuously. The signal
emitted from the sample is collected by the objective and analyzed by a
spectrometer (Princeton instrument), from which we can get the energy-momentum
spectrum and energy-position spectrum of the exciton polaritons. The lens
closest to the spectrometer is installed on a one-dimensional motorized stage
(NewPort), and tomography can be performed (Fig. 4) by continuously moving the
translation stage from which we can get the energy-resolved real space
distribution of polaritons.

Author contributions. T.G. and X.M. conceived the project, Y.L. performed the
experiment and analyzed the results, X.M. performed the theoretical analysis
and numerical simulation, T.G. and X.M. prepared the manuscript with
contributions from S.S. and P.S., Z. H and P.S. fabricated the sample. All
authors discussed the results.

Competing interests. The authors declare no competing interests.

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