# Refrigeration by modified Otto cycles and modified swaps through generalized
measurements

Naghi Behzadi a
aDepartment of Theoretical Physics and Astrophysics,
University of Tabriz 5166614776, Tabriz, Iran

###### Abstract

We introduce two types of thermodynamic refrigeration cycles obtained through
modification of the Otto cycle refrigerator by a generalized measurement
channel. These refrigerators are corresponding to the activation of the
measurement-based stroke before (first type) and after (second type) the full
thermalization of the cooling medium by the cold reservoir in the related
familiar Otto cycle. We show that the coefficient of performance for the first
type modified refrigerator increases linearly in terms of measurement strength
parameter, beyond the classical cooling of the known Otto cycle refrigerator.
The second type interestingly introduces another autonomous refrigerator whose
supplying work is provided by a quantum engine induced by the measurement
channel along the modified cycle. By the considered measurement channel, we
also establish such modifications on the swap refrigerator. It is observed
that the thermodynamic properties of the obtained modified swap refrigerators
are the same as of the modified Otto cycle ones respectively.

Keywords: Modified Otto cycle refrigerator, Modified swap refrigerator,
Coefficient of performance, Generalized measurement, Measurement strength
parameter, Autonomous refrigerator

##  I. Introduction

The main landscape of quantum thermodynamics is concerned with how the effect
of quantum resources such as coherence, entanglement and quantum measurement
on the quality of work extraction, refrigeration, etc, in thermal devices can
be understood and applied [1-5]. As is well-known, quantum measurements are
able to alter the average energy and entropy of a quantum system, provided
that the measured observable does not commute with the Hamiltonian of that
system [6-10]. Thus they can play similar role as thermal reservoirs. The
measurement-driven devices which use energetic fluctuations generated by
quantum measurement, have recently been explored [5, 9-14].

On the other hand, in Ref. [15], a three-stroke quantum engine on the basis of
non-selective generalized measurement channels with adjustable measurement
strengths has been proposed and later, by employing nuclear magnetic resonance
(NMR) platform, it has been realized experimentally [16]. An interesting
advantage of the introduced engine in [15] which was also observed in the
experimental results reported in [16], is that this kind of quantum engine can
reach unit efficiency while also achieving maximum extracted power at the same
time with the fine-tuning of the measurement strengths parameter. The approach
of [15] then was developed to introduce another measurement-based devices such
as refrigerator and thermal accelerator according to a measurement setting
intensity [17].

The other feature of measurement channels introduced in [15] is that their
implementation running time is much smaller than any decoherence or
thermalization time scales [16]. This property along with the adjustable
measurement strength, encourage one to involve this type of measurement
channel, as a quantum resource, in the known thermodynamics cycles such as
Otto cycle and swap, as an additional stroke. Involving the thermodynamics
cycles with this additional stroke, called as modified cycles, may lead the
obtained devices to have surpassing properties relative to the traditional
ones.

In this work, we perform two types of modifications on the known Otto cycle
using a generalized measurement channel similar to the one introduced in [15],
as an additional stroke. As is well-known in the known Otto cycle, a quantum
system, particularly a two-level one considered as cooling medium (CM),
undergoes two isochoric strokes and two adiabatic ones. The modifications on
the Otto cycle refrigerator are obtained whenever the measurement-based stroke
is exploited before (first type) and after (second type) the full
thermalization of two-level system with the cold thermal reservoir. The first
type modification leads to a refrigerator whose coefficient of performance
(COP) increases in terms of measurement strength parameter beyond the case
related to the Otto cycle refrigerator, which is a genuine quantum effect. In
fact, for a given value of external work, the COP of modified Otto cycle
refrigerator goes beyond the Otto cycle one depending on the intensity of
measurement-based stroke. The second type modification interestingly leads to
another type of quantum autonomous refrigerator which by itself have been
subject of researches in last years [18-29]. In general, a refrigerator is
autonomous in the sense that it does not require any external sources of work
for cooling. The process which we are encountered in the obtained refrigerator
by the second modification, is a quantum effect because the required work for
the refrigeration is supplied by the extracted work from a three-stroke
quantum engine induced by the measurement-based stroke in the modified Otto
cycle. It is interesting to note that the efficiency of measurement-induced
quantum engine is the same as of an Otto cycle engine such that all of its
three strokes depend directly on the measurement strength parameter.

The modification scheme described on the Otto cycle refrigerator can be
applied easily on the two-stroke swap refrigerator. As the previous case, we
can obtain two types of three-stroke modified swap refrigerators depending on
that the measurement-based stroke activated before (first type) or after
(second type) the swap operation. These types of modifications lead to
refrigerators which have the same properties as of the discussed modified Otto
cycle refrigerators respectively.

##  II. Refrigeration by first type modified Otto cycle

In a four-stroke Otto cycle refrigerator, a given amount of work that is spent
by an external agent on the CM during two adiabatic strokes causes remove of
an amount of heat from the cold reservoir and in return flow of another amount
of heat into the hot reservoir during two isochoric strokes. In this section,
by considering the known Otto cycle which is in the refrigeration mode, we
involve a measurement apparatus into it before full thermalization of the CM
by the cold reservoir. We show that the obtained modified refrigerator makes
remove of more heat from the cold reservoir for the same amount of invested
work. To this aim, let us consider the CM is a two-level system with
Hamiltonian $H=\frac{\omega}{2}\sigma_{z}$ where $\sigma_{z}$ is the
z-component Pauli spin-$\frac{1}{2}$ operator and $\omega$ is the
corresponding transition frequency. In general, a measurement apparatus
(channel) makes generalized measurements on the CM described by completely
positive and trace-preserving (CPTP) maps [30]. To describe the first type
modified Otto cycle refrigerator, let us introduce a particular measurement on
the CM represented by the following measurement operators with adjustable
measurement strength [15], as follows

$\displaystyle M_{1}(\mathcal{\xi})=|0\rangle\langle
0|+\sqrt{1-\mathcal{\xi}}|1\rangle\langle 1|,\qquad
M_{2}(\mathcal{\xi})=\sqrt{\mathcal{\xi}}|0\rangle\langle 1|,$ (1)

where $\mathcal{\xi}$ is the measurement strength parameter. Obviously the
operators $M_{1}(\mathcal{\xi})$ and $M_{2}(\mathcal{\xi})$ satisfy
$\Sigma_{i}M_{i}(\mathcal{\xi})^{\dagger}M_{i}(\mathcal{\xi})=I$ which in
fact, is the essence of fundamental theorem of quantum measurement [30]. In
addition, the operators
$E_{1}(\xi):=M_{1}(\mathcal{\xi})^{\dagger}M_{1}(\mathcal{\xi})=|0\rangle\langle
0|+\left(1-\mathcal{\xi}\right)|1\rangle\langle 1|$ and
$E_{2}(\xi):=M_{2}(\mathcal{\xi})^{\dagger}M_{2}(\mathcal{\xi})=\mathcal{\xi}|1\rangle\langle
1|$ are the so-called POVM (positive operator valued measure) operators
corresponding to the measurement operators (1). It is clear that,
$0\leq\mathcal{\xi}\leq 1$, where $\mathcal{\xi}=0$ indicates that the system
has not been disturbed (measured) by the measurement process, and
$\mathcal{\xi}=1$ corresponds to performing strong or projective measurement
on the CM.

To achieve first modification on the Otto cycle refrigerator (Fig. 1), let us
assume that the CM, at the beginning, is in the thermal equilibrium with the
cold reservoir whose inverse temperature is $\beta_{c}$. In this time, the
Hamiltonian of the CM is denoted by $H_{c}=\frac{\omega_{c}}{2}\sigma_{z}$
where $\omega_{c}$ is the corresponding transition frequency in this
situation. So the corresponding Gibbs state of the CM is denoted by
$\rho_{c}=e^{-\beta_{c}H_{c}}/\mathcal{Z}_{c}$, where
$\mathcal{Z}_{c}=Tr[e^{-\beta_{c}H_{c}}]$ is the partition function. The mean
energy, in this step, is equal to
$E_{1}=Tr[H_{c}\rho_{c}]=-\frac{\omega_{c}}{2}\mathrm{tanh}\left(\beta_{c}\omega_{c}/2\right)$.
To describe the 1st stroke, we let the CM is isolated from the cold reservoir
and undergoes adiabatic unitary evolution in which only the transition
frequency (for simplicity and without lose of generality) changes from
$\omega_{c}$ to $\omega_{h}$ such that $\omega_{c}<\omega_{h}$. Therefore, the
mean energy after this stroke is obtained as
$E_{2}=Tr[H_{h}\rho_{c}]=-\frac{\omega_{h}}{2}\mathrm{tanh}\left(\beta_{c}\omega_{c}/2\right)$
where $H_{h}=\frac{\omega_{h}}{2}\sigma_{z}$.

The 2nd stroke is provided by full thermalization of the CM by a hot thermal
reservoir with inverse temperature $\beta_{h}$ $(\beta_{h}<\beta_{c})$. In
this time the corresponding Gibbs state is given by
$\rho_{h}=e^{-\beta_{h}H_{h}}/\mathcal{Z}_{h}$ where
$\mathcal{Z}_{h}=Tr[e^{-\beta_{h}H_{h}}]$ is the partition function. By this
stroke, the mean energy is changed from $E_{2}$ to $E_{3}$ which is equal to
$E_{3}=Tr[H_{h}\rho_{h}]=-\frac{\omega_{h}}{2}\mathrm{tanh}\left(\beta_{h}\omega_{h}/2\right)$.
In the 3rd stroke, which is similar to the first one, the CM is isolated from
the hot reservoir and undergoes adiabatically unitary evolution in which the
transition frequency is brought, in an isentropic way, to the original value
$\omega_{c}$. The corresponding mean energy becomes as
$E_{4}=Tr[H_{c}\rho_{h}]=-\frac{\omega_{c}}{2}\mathrm{tanh}\left(\beta_{h}\omega_{h}/2\right)$.

The promised modification for the known Otto cycle refrigerator is introduced
by effect of the 4th stroke on the CM in which it is left to interact with a
measurement apparatus such that its effect is corresponding to the effect of a
non-selective generalized measurement on the CM as
$\mathcal{E}(\rho_{h})=\Sigma_{i}M_{i}(\xi)\rho_{h}M_{i}^{\dagger}(\xi)$ where
$M_{i}(\xi)$s are the measurement operators defined in Eq. (1). Therefore the
mean energy takes the following value as

$\displaystyle
E_{5}=Tr[H_{c}\rho_{{}_{h}}(\mathcal{\xi})]=-\frac{\omega_{c}}{2}\mathrm{tanh}\left(\beta_{h}\omega_{h}/2\right)-\mathcal{\xi}\omega_{c}\frac{e^{-\beta_{h}\omega_{h}/2}}{\mathcal{Z}_{h}},$
(2)

where

$\displaystyle\mathcal{E}(\rho_{h}):=\rho_{{}_{h}}(\mathcal{\xi})=\frac{1}{\mathcal{Z}_{h}}\left[(1-\mathcal{\xi})e^{{-\beta_{h}\omega_{h}/2}}|1\rangle\langle
1|+(e^{{\beta_{h}\omega_{h}/2}}+\mathcal{\xi}e^{{-\beta_{h}\omega_{h}/2}})|0\rangle\langle
0|\right].$ (3)

It is clear that $\rho_{{}_{h}}(\mathcal{\xi})$ is a non-equilibrium passive
state with no induced ergotropy [31], by the measurement apparatus.
Consequently, the cycle is completed by 5th stroke which is the thermalization
of the CM by the cold reservoir. By considering the condition
$\beta_{c}\omega_{c}<\beta_{h}\omega_{h}$, the CM throughout the modified Otto
cycle exchanges following works and heats with work and heat sources as well
as with the measurement apparatus as

$\displaystyle 1\mathrm{st}\quad\mathrm{stroke}:\quad
W_{1}=E_{2}-E_{1}=\left(\frac{\omega_{c}-\omega_{h}}{2}\right)\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right),$
(4) $\displaystyle 2\mathrm{nd}\quad\mathrm{stroke}:\quad
Q_{h}=E_{3}-E_{2}=\frac{\omega_{h}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)\right],$
(5) $\displaystyle 3\mathrm{rd}\quad\mathrm{stroke}:\quad
W_{2}=E_{4}-E_{3}=\left(\frac{\omega_{h}-\omega_{c}}{2}\right)\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right),$
(6) $\displaystyle 4\mathrm{th}\quad\mathrm{stroke}:\quad
Q(\xi)=E_{5}-E_{4}=-\mathcal{\xi}\omega_{c}\frac{e^{-\beta_{h}\omega_{h}/2}}{\mathcal{Z}_{h}},$
(7) $\displaystyle 5\mathrm{th}\quad\mathrm{stroke}:\quad
Q_{c}(\xi)=E_{1}-E_{5}=Q_{c}-Q(\xi),$ (8)

where

$\displaystyle
Q_{c}=\frac{\omega_{c}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right].$
(9)

$Q_{h}<0$ ($Q(\xi)<0$) is the amount of heat (quantum heat [5, 15]) flowed
from the CM into the hot reservoir (measurement apparatus) while
$Q_{c}(\xi)>0$ is as heat flowed from the cold reservoir into the CM (Fig. 1).
Also the work $W_{1}<0$ is extracted from the CM while $W_{2}>0$ is invested
on it as depicted in Fig. 1. Clearly in the absence the measurement process
$(\xi=0)$ then the received heat by the CM from the cold reservoir is reduced
to $Q_{c}$ which is corresponding to the case of Otto cycle refrigerator. The
COP for the corresponding modified refrigerator is obtained as follows

$\displaystyle\mathrm{COP}(\xi)=\frac{Q_{c}(\xi)}{W_{in}}=\left(\frac{\omega_{h}}{\omega_{c}}-1\right)^{-1}\left(1+\frac{2\mathcal{\xi}}{\mathcal{Z}_{h}}\frac{e^{-\beta_{h}\omega_{h}/2}}{\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right]}\right),$
(10)

where

$\displaystyle
W_{\mathrm{in}}=W_{1}+W_{2}=\left(\frac{\omega_{h}-\omega_{c}}{2}\right)\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right],$
(11)

is the required work $(W_{\mathrm{in}}>0)$ for the refrigeration which should
be invested on the CM by an external agent to enforce flow of heat from the
cold reservoir into the hot one and the measurement apparatus. It is clear
that when the measurement strength parameter becomes zero, i.e
$\mathcal{\xi}=0$, then the COP in Eq. (10), leads to the one of the familiar
Otto cycle refrigerator.

Fig. 2, shows the COP for the modified refrigerator in terms of measurement
strength parameter $\xi$, for various amounts of invested work. In fact, for
each values of invested works, the COP is always a linearly increasing
function of $\xi$, beyond the one of the familiar Otto cycle, such that the
rate of its increment is proportional inversely with the values of the
invested works. Therefore, the involution of generalized measurement (1) into
the Otto cycle, as a quantum resource, leads to the supremacy of the
refrigeration process by the introduced (first type) modified Otto cycle with
respect to the classical cooling of the Otto cycle refrigerator. On the other
hand, the cooling process characterized by the amount of removal heat from the
cold reservoir, i.e. $Q_{c}(\xi)$ (see Eq. (9)), is also an increasing
function of $\xi$ (Fig. 3), however, it is interesting to note that as
$\xi\rightarrow 1$, which is corresponding to the strong measurement case, the
cooling process, apart from the values of the invested work, reaches to a
unique value.

##  III. Refrigeration by second type modified Otto cycle

Let us consider the other type of modified Otto cycle obtained by activation
of the measurement channel after the full thermalization of the CM by the cold
reservoir, as shown in Fig. 4. The 1st stroke causes flow of quantum heat from
the two-level system into the measurement apparatus calculated as follows

$\displaystyle
Q(\xi)=E_{2}-E_{1}=-\xi\omega_{c}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}}<0,$
(12)

where $E_{1}=Tr[H_{c}\rho_{c}]$, as the previous case, is the mean energy at
the beginning of the 1st stroke and

$\displaystyle
E_{2}=Tr[H_{c}\rho_{c}(\xi)]=-\frac{\omega_{c}}{2}\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-\xi\omega_{c}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}},$
(13)

is at the end of it, with

$\displaystyle\rho_{c}(\xi)=\frac{1}{\mathcal{Z}_{c}}\left[(1-\xi)e^{{-\beta_{c}\omega_{c}/2}}|1\rangle\langle
1|+(e^{{\beta_{c}\omega_{c}/2}}+\xi
e^{{-\beta_{c}\omega_{c}/2}})|0\rangle\langle 0|\right].$ (14)

Obviously the non-equilibrium state (14), is a passive state without
ergotropy. The 2nd stroke causes evolution of the CM unitarily and
adiabatically such that only its transition frequency is changed from
$\omega_{c}$ to $\omega_{h}$ $(\omega_{c}<\omega_{h})$. Therefore, the mean
energy at the end of this stroke becomes

$\displaystyle
E_{3}=Tr[H_{h}\rho_{c}(\xi)]=-\frac{\omega_{h}}{2}\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-\mathcal{\xi}\omega_{h}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}},$
(15)

where, as previously, $H_{h}=\frac{\omega_{h}}{2}\sigma_{z}$. The extracted
work from the CM through this stroke is written as

$\displaystyle\mathcal{W}(\xi)=E_{3}-E_{2}=W_{1}+W(\xi),$ (16)

where

$\displaystyle
W_{1}=\left(\frac{\omega_{c}-\omega_{h}}{2}\right)\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right),$
(17)

and

$\displaystyle
W(\xi)=\mathcal{\xi}(\omega_{c}-\omega_{h})\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}}.$
(18)

$\mathcal{W}(\xi)$ is always negative because $W_{1}$ and $W(\xi)$ are always
negative. The expression (18), as a part of the extracted work in Eq. (16),
originates only from the measurement process such that
$\mathcal{W}(\xi=0)=W_{1}$. On the other hand, by the 3rd stroke, our two-
level system is brought into contact with the hot thermal reservoir and during
this process the amount of exchanged heat becomes as below

$\displaystyle Q_{h}(\xi)=E_{4}-E_{3}=Q_{h}+Q^{\prime}(\xi),$ (19)

where

$\displaystyle
Q_{h}=\frac{\omega_{h}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)\right],$
(20) $\displaystyle
Q^{\prime}(\xi)=\xi\omega_{h}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}}>0.$
(21)

and $E_{4}=Tr[H_{h}\rho_{h}]$. By considering the refrigeration condition for
the modified Otto cycle, i.e. $\beta_{c}\omega_{c}<\beta_{h}\omega_{h}$, we
have $Q_{h}<0$. Thus by this stroke, there is flow of heat $Q_{h}$ from the CM
into the hot reservoir and at the same time, flow of quantum heat
$Q^{\prime}(\xi)$ from the hot reservoir into the CM, which is only originated
from the effect of quantum measurement exploited in the 1st stroke. In the 4th
stroke, the Hamiltonian of the CM is returned to its original form, i.e. only
its transition frequency is changed from $\omega_{h}$ to $\omega_{c}$, by a
unitary and adiabatic process. The invested work, by this stroke, on the CM is
calculated as

$\displaystyle
W_{2}=E_{5}-E_{4}=\left(\frac{\omega_{h}-\omega_{c}}{2}\right)\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right),$
(22)

where $W_{2}>0$ and $E_{5}=Tr[H_{c}\rho_{h}]$. Ultimately, the CM is brought
again into contact with the cold reservoir and the cycle is completed by this
(5th) stroke so the amount of heat flowed from the cold reservoir into the CM
is obtained as follows

$\displaystyle
Q_{c}=E_{1}-E_{5}=\frac{\omega_{c}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right],$
(23)

where $Q_{c}>0$. The net work which should be invested by an external agent on
the CM, throughout the modified cycle, is

$\displaystyle
W_{ex}(\xi)=W_{in}+W(\xi)=\left(\frac{\omega_{h}-\omega_{c}}{2}\right)\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-2\xi\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}}\right],$
(24)

where $W_{ex}(\xi)\geq 0$. On the other hand,
$W_{in}=-(Q_{h}+Q_{c})=W_{ex}(\xi)-W(\xi)$ is the amount of work required to
invest on the CM in order to remove the heat $Q_{c}$ from the cold reservoir
and transfer the heat $Q_{h}$ into the hot reservoir. It is explicitly
independent from the measurement-based stroke whereas $W_{ex}(\xi)$ and
$W(\xi)$ are not. The COP for the refrigeration on the basis of modified Otto
cycle is

$\displaystyle\mathrm{COP}(\xi)=\frac{Q_{c}}{W_{ex}(\xi)-W(\xi)},$ (25)

which is also independent from the measurement-based stroke too, and as is
evident, it is the same as COP of familiar Otto cycle refrigerator. So what
does measurement do really? To answer to this question, let us at first
consider the case that the measurement process (1) has not been activated,
i.e. $\xi=0$. In this case $W(\xi=0)=0$ so $W_{in}=W_{ex}(\xi=0)$, thus the
required work for the refrigeration is provided only by an external agent.

Now, in the presence of measurement-based stroke, i.e. $\xi\neq 0$, it is
observed a measurement-induced quantum engine whose operation along the
modified cycle consists of absorption of heat $Q^{\prime}(\xi)$ from the hot
reservoir and transfer a part of it, i.e. $Q(\xi)$, into the measurement
apparatus along with extraction of work $W(\xi)$ such that
$W(\xi)=-(Q^{\prime}(\xi)+Q(\xi))$ (Fig. 4). Evidently, the efficiency of this
quantum engine is the same as of the familiar Otto cycle heat engine. The
extracted work can be regarded (by a suitable feedback) as a part of required
work for the refrigeration so the remainder part is supplied by an external
agent ($W_{in}=W_{ex}(\xi)-W_{ex}(\xi)$). Furthermore, since $W_{ex}(\xi)$ is
a linearly decreasing function of $\xi$, then it becomes zero for the
following value of measurement strength parameter as

$\displaystyle\xi_{c}=\mathcal{Z}_{c}\frac{e^{\beta_{c}\omega_{c}/2}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right].$
(26)

i.e. $W_{ex}(\xi_{c})=0$, thus $W_{in}=-W_{ex}(\xi_{c})$ (note that
$\xi_{c}<1$). The last relation explicitly shows that the required work for
the refrigeration is only supplied by the measurement-induced quantum engine.
So the second type of the modified Otto cycle presents an autonomous
refrigerator whose required work for the refrigeration is provided only by the
measurement-induced quantum engine in the cycle. Trivially, for
$\xi_{c}<\xi\leq 1$, the measurement-based quantum engine not only provides
the required work for the refrigerator but also can gives the work
$W_{ex}(\xi)<0$ into another work source.

There are also other types of modification through entering the measurement
based-stroke into the familiar Otto cycle other than the cases discussed
previously. In fact, these modifications are not effective because they do not
lead to any remarkable improvement on the outputs of obtained cycles in
comparison to Known Otto cycle.

##  IV. Refrigeration by modified swaps

In the previous sections, by using the generalized measurement (1), we
obtained two types of refrigerators on the basis of modifying the Otto cycle
refrigerator. Again, in this section, by considering the generalized
measurement (1), we perform such modifications on the two-stroke swap
refrigerator which in turn lead to the same refrigeration advantages obtained
in the previous section. Let us consider two two-level systems with
Hamiltonians $H_{i}=\frac{\omega_{i}}{2}\sigma_{z}^{i}$ ($i=h,c$) with
$\omega_{c}<\omega_{h}$, each of which is in thermal contact with thermal
reservoirs whose inverse temperatures are $\beta_{h}$ and $\beta_{c}$
$(\beta_{h}<\beta_{c})$ respectively. So in the thermal equilibrium, the
corresponding Gibbs states are $\rho_{h}$ and $\rho_{c}$ respectively (Sec.
II).

To describe the first type modified swap (Fig. 5), let us consider that at the
beginning of the 1st stroke, each subsystem (two-level system) has been
isolated from its own reservoir such that they are, as a whole, described by
the tensor product $\rho=\rho_{h}\otimes\rho_{c}$. Now consider that this
bipartite system is exchanged by the following swap operator

$\displaystyle U=|11\rangle\langle 11|+|10\rangle\langle 01|+|01\rangle\langle
10|+|00\rangle\langle 00|,$ (27)

which is an unitary operator. Since the entropy of the bipartite system is
left unchanged under this operation, so the amount of change in mean energy is
interpreted as work as follows

$\displaystyle\begin{split}&W_{in}=Tr\left(HU\rho
U^{\dagger}\right)-Tr\left(H\rho\right)\\\
&=\left(\frac{\omega_{h}-\omega_{c}}{2}\right)\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right].\end{split}$
(28)

where $H=H_{h}+H_{c}$ and $U\rho U^{\dagger}=\rho_{c}\otimes\rho_{h}$. The
invested work in Eq. (28) is maximal [32], which is the same as of Eq. (11) in
Sec. II. In the 2nd stroke, the subsystem with Hamiltonian $H_{c}$ and density
operator $\rho_{h}$ is affected by the generalized measurement (1), so the
state of the whole system after the measurement is left as
$(I\otimes\mathcal{E})U\rho U^{\dagger}=\rho_{c}\otimes\mathcal{E}(\rho_{h})$
where $\mathcal{E}(\rho_{h})=\Sigma_{i}M_{i}(\xi)\rho_{h}M_{i}^{\dagger}(\xi)$
and $M_{i}(\xi)$s are the measurement operators defined in Eq. (1). By this
stroke, the amount of quantum heat flowed from this subsystem into the
measurement apparatus is

$\displaystyle
Q(\xi)=Tr(H\rho_{c}\otimes\mathcal{E}(\rho_{h}))-Tr(H\rho_{c}\otimes\rho_{h})=-\mathcal{\xi}\omega_{c}\frac{e^{-\beta_{h}\omega_{h}/2}}{\mathcal{Z}_{h}},$
(29)

which is the same as Eq. (7), in Sec. II. The 3rd stroke is provided by
returning each subsystem into contact with its own reservoir and the cycle
(modified swap) is completed. Hence the exchanged heat between the each
subsystem with its respective reservoir is obtained as below

$\displaystyle
Q_{h}=Tr(H_{h}\rho_{h})-Tr(H_{h}\rho_{c})=\frac{\omega_{h}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)\right],$
(30)

and

$\displaystyle
Q_{c}(\xi)=Tr(H_{c}\rho_{c})-Tr(H_{c}\mathcal{E}(\rho_{h}))=Q_{c}-Q(\xi),$
(31)

where $Q_{h}$, $Q(\xi)$ and $Q_{c}$ are the same as of Eqs. (5), (7) and (9)
in Sec. II, respectively (so is $Q_{c}(\xi)$ as Eq. (8)). It is noted that in
the refrigerator mode ($\beta_{c}\omega_{c}<\beta_{h}\omega_{h}$), we have
$W_{in}>0$, $Q_{h}<0$ and $Q_{c}>0$. By this results, the corresponding COP
becomes as

$\displaystyle\mathrm{COP}(\xi)=\left(\frac{\omega_{h}}{\omega_{c}}-1\right)^{-1}\left(1+\frac{2\mathcal{\xi}}{\mathcal{Z}_{h}}\frac{e^{-\beta_{h}\omega_{h}/2}}{\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right]}\right),$
(32)

which is the same as the case of first type modified Otto cycle refrigerator
in Eq. (10).

At the end, let us introduce the second type modified swap refrigerator by
using the generalized measurement (1). Fig. 6, shows the details of second
type modified swap refrigerator. As the previous case, it is assumed that at
the beginning of the 1st stroke, each subsystem has been isolated from its own
reservoir such that they are described by the tensor product
$\rho=\rho_{h}\otimes\rho_{c}$. In the 1st stroke, the subsystem which was in
the thermal contact with the cold reservoir is affected by the measurement
process (1) such that the state of the whole system becomes
$\rho^{\prime}=\rho_{h}\otimes\mathcal{E}(\rho_{c})$. The amount of quantum
heat extracted from the considered subsystem by the measurement apparatus is

$\displaystyle
Q(\xi)=Tr\left(H\rho^{\prime}\right)-Tr\left(H\rho\right)=-\mathcal{\xi}\omega_{c}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}},$
(33)

which is the same as of Eq. (12) in Sec. III and that $H=H_{h}+H_{c}$. The 2nd
stroke is provided by exploiting the swap operator (27), which is
corresponding to the work stroke. The amount of work which should be invested
by an external agent in order to have refrigeration can be regarded as

$\displaystyle\begin{split}&W_{ex}(\xi)=Tr\left(HU\rho^{\prime}U^{\dagger}\right)-Tr\left(H\rho^{\prime}\right)\\\
&=W_{in}+W(\xi)=\left(\frac{\omega_{h}-\omega_{c}}{2}\right)\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-2\mathcal{\xi}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}}\right],\end{split}$
(34)

which is the same as of Eq. (24) in Sec. III. As discussed in the previous
section, $W_{ex}(\xi)\geq 0$, $W(\xi)\leq 0$ depend only on the measurement
strength parameter in the 1st stroke and $W_{in}(\xi)=W_{ex}(\xi)-W(\xi)$ is
the amount of required work which should be invested on the CM in order to
have refrigeration after completion of each cycle. In the 3rd stroke, the
subsystems are returned to contact with their own reservoirs in order to
recovering the respective original state of each subsystem so the cycle is
completed. The heat exchanged between each subsystem and the respective
thermal reservoir is obtained easily as

$\displaystyle\begin{split}&Q_{h}(\xi)=Tr\left(H_{h}\rho_{h}\right)-Tr\left(H_{h}\rho^{\prime\prime}_{h}\right)\\\
&=Q_{h}+Q^{\prime}(\xi)=\frac{\omega_{h}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)+2\mathcal{\xi}\frac{e^{-\beta_{c}\omega_{c}/2}}{\mathcal{Z}_{c}}\right],\end{split}$
(35)

and

$\displaystyle\begin{split}&Q_{c}=Tr\left(H_{c}\rho_{c}\right)-Tr\left(H_{c}\rho^{\prime\prime}_{c}\right)\\\
&=\frac{\omega_{c}}{2}\left[\mathrm{tanh}\left(\frac{\beta_{h}\omega_{h}}{2}\right)-\mathrm{tanh}\left(\frac{\beta_{c}\omega_{c}}{2}\right)\right],\end{split}$
(36)

where $\rho^{\prime\prime}_{h}=Tr_{c}(U\rho^{\prime}U^{\dagger})$ and
$\rho^{\prime\prime}_{c}=Tr_{h}(U\rho^{\prime}U^{\dagger})$. Remarkably Eqs.
(34) and (35) are the same as Eqs. (19) and (23) in Sec III, respectively. In
fact, $Q_{h}(\xi)$ is the heat exchanged between the subsystem with
Hamiltonian $H_{h}$ and the hot reservoir while the $Q_{c}$ is the heat
exchanged between the subsystem with Hamiltonian $H_{c}$ and the cold
reservoir. Thus the second type modified swap refrigerator, as the second type
modified Otto cycle one, causes removal of heat $Q_{c}$ from the cold
reservoir and transfer heat $Q_{h}$ into the hot reservoir by investing work
$W_{in}$. As is evident from the Eq. (34), this work is supplied by an
external agent as well as by a measurement-induced quantum engine, i.e.
$W_{in}=W_{ex}(\xi)-W(\xi)$. The induced quantum engine by the measurement
process, in turns, absorbs quantum heat $Q^{\prime}_{h}(\xi)$ from the hot
reservoir and gives quantum heat $Q_{h}(\xi)$ into the measurement apparatus
and so its advantage is that the work $W(\xi)$ is extracted. By using the
results of last section, in the case $\xi=\xi_{c}$, the refrigerator is only
supplied by the measurement-induced quantum engine (through suitable feedback)
so there is no need to work supply by an external agent. Thus we obtain the
other type of autonomous refrigerator constructed by the second type modified
swap.

##  V. Conclusions

In this work, we provided a modification scheme to present the effect of a
generalized measurement channel such as one defined in Eq. (1), on the
thermodynamics cycles such as Otto cycle and swap when they are in the
corresponding refrigerator modes. To this aim, we performed two types of
modifications on the familiar Otto cycle refrigerator each of which led to the
corresponding other refrigerators. As elucidated, for the first one, the COP
depends linearly on the measurement strength parameter, so it goes beyond the
one of the Otto cycle refrigerator which in turns is a pure quantum mechanical
effect. Thus, in comparison to the Otto cycle refrigerator, this type of
refrigerator can have potential of interests when one consider it from a
finite-time practical realization point of view [33-35]. Since the
implementation running time of measurement channel (1) is much smaller than
any decoherence or thermalization time scales [16], so the total running time
of the modified refrigerator seems approximately to be the same as of the Otto
cycle one. Thus any finite-time protocol for realization of an Otto cycle
refrigerator may lead to be more advantageous for the first type modified
refrigerator which can be the subject of future research.

In the second modification, it was obtained an autonomous refrigerator which
does not require to any external work source such that the required work for
refrigeration process is supplied by a measurement-induced quantum engine
whose operation cycle is inside the considered modified Otto cycle. The COP
for the obtained refrigerator is the same of the Otto cycle refrigerator so is
the efficiency of its supplying engine as of the Otto cycle engine. All three
strokes of supplying engine depend directly on the intensity of generalized
measurement channel so this measurement-induced engine is purely a quantum
engine. From the practical point of view, addressing the finite-time
realization of such refrigerator would be also interesting.

Also it was tried to established such modifications on the two-stroke swap
refrigerator using the same generalized measurement (1). The first
modification led to a three-stroke refrigerator whose COP is exactly the same
as of the first type modified Otto cycle refrigerator. In the same way, the
second modification led to the other three-stroke autonomous refrigerator
whose supplying work is provided by a three-stroke quantum engine induced by
the measurement process in the same refrigeration cycle.

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Fig. 1. First type modified Otto cycle refrigerator. The five-stroke
refrigeration cycle is obtained through inserting measurement based-stroke
before full thermalization of the CM by the cold reservoir in the known Otto
cycle. In the 4th stroke, the measurement channel, as a quantum resource,
absorbs heat $Q(\xi)$ (blue dashed arrow) from the CM. Thus, the measurement
channel in the 4th stroke causes additional flow of heat from the cold
reservoir into the CM in the 5th stroke, i.e. $Q(\xi)$ (red dashed arrow),
beyond the classical cooling of the Otto cycle refrigerator.

Figure 1:

Fig. 2. The COP for the first type modified Otto cycle refrigerator (Eq. (9)),
in terms of measurement strength parameter, for some given invested works (Eq.
(11)). The parameters $\omega_{h}=10$, $\omega_{c}=2$ and $\beta_{c}=0.4$ have
been fixed. For $\beta_{h}=0.25$, $\beta_{h}=0.19$, $\beta_{h}=0.16$ and
$\beta_{h}=0.14$, the invested works are $W_{in}=1.87$, $W_{in}=1.43$,
$W_{in}=1.13$ and $W_{in}=0.89$ respectively and the corresponding
coefficients of performance are denoted by dotted-dashed black, dashed blue,
dotted red and solid green lines respectively.

Figure 2:

Fig. 3. The amount of removal heat from the cold reservoir according to Eq.
(8), in terms of measurement strength parameter. The solid green line is
corresponding to the invested work $W=0.89$ (Eq. (11)), the dotted red line
with $W=1.13$, the dashed blue line with $W=1.43$ and the dotted-dashed black
line with $W=1.89$. In the strong measurement limit ($\xi\rightarrow 1$), the
presence of measurement channel makes to have the most removal of heat from
the cold reservoir with the least amount of invested work.

Figure 3:

Fig. 4. Second type modified Otto cycle refrigerator. This five-stroke
refrigeration cycle is obtained through inserting the measurement based-stroke
after full thermalization of the CM by the cold reservoir in the known Otto
cycle. The second type modified Otto cycle, in one hand, provides
refrigeration in the same way as the Otto cycle refrigerator, and on the other
hand, gives a measurement-induced quantum engine whose efficiency is the same
as of the Otto cycle engine. The required work for the refrigeration is
$W_{in}=W_{1}+W_{2}$ which causes remove of heat $Q_{c}$ (red solid arrow)
from the cold reservoir and flow of heat $Q_{h}$ (blue solid arrow) into the
hot reservoir, whereas the solid yellow and green arrows are corresponding to
the works $W_{1}$ (extracted) and $W_{2}$ (invested) respectively. The work
$W_{in}$ is provided by an external agent, i.e. $W_{ex}(\xi)$, as well as by
the measurement-induced engine, i.e. $W(\xi)$ (yellow dashed arrow) through
suitable feedback, such that $W_{in}=W_{ex}(\xi)-W(\xi)$. The extracted work
from the quantum engine is the consequence of absorption of heat
$Q^{\prime}(\xi)$ (red dashed arrow) from the hot reservoir and flow of heat
$Q(\xi)$ (blue dashed arrow) into the measurement apparatus. In the limit
$\xi\rightarrow\xi_{c}$, $W_{ex}(\xi_{c})=0$, i.e. the modified refrigerator
has no need to work supplying by an external agent. In this situation the
modified Otto cycle leads to an autonomous refrigerator whose invested work is
only provided by the work extracted from the quantum engine such that
$W_{in}=-W(\xi_{c})$.

Figure 4:

Fig. 5. First type modified swap refrigerator. (a) At the beginning, each
subsystem has been isolated from its own reservoir then the state of bipartite
system is $\rho_{h}\otimes\rho_{c}$. (b) The 1st stroke is related to
application of swap operator on the bipartite system which makes the cold
qubit becomes colder and the hot one becomes hotter. This requires investment
of work $W_{in}$ (green solid arrow) on the CM, supplied by an external agent.
(c) The effect of measurement process (1), as the 2nd stroke, on the colder
qubit causes flow of quantum heat $Q(\xi)$ (blue dashed arrow) from CM into
the measurement apparatus. (d) In the 3rd stroke, each subsystem is brought
into contact with its own reservoir so the CM absorbs heat $Q_{c}$ and
$Q(\xi)$ (red solid and dashed arrows respectively) from the cold reservoir
and the heat $Q_{h}$ is flowed from the CM into the hot reservoir.

(a) (b)
(c) (d)

Figure 5:

Fig. 6. Second type modified swap refrigerator. (a) At the beginning, each
subsystem has been isolated from its own reservoir so that the state of
bipartite system is $\rho_{h}\otimes\rho_{c}$. (b) The subsystem with state
$\rho_{c}$ is affected by the generalized measurement defined by (1), so by
this (1st) stroke the quantum heat $Q(\xi)$ (blue dashed arrow) is transfered
from the mentioned subsystem into the measurement apparatus. (c) The 2nd
stroke consists of operation of swap operator which exchanges the subsystems
with each other. By this stroke, the required work for the refrigeration, i.e.
$W_{in}$ (green solid arrow), is supplied by the measurement-induced quantum
engine whose extracted work is $W(\xi)$ (red dashed arrow), as well as by an
external agent, i.e. $W_{ex}(\xi)$, such that $W_{in}=W_{ex}(\xi)-W(\xi)$ and
$0\leq\xi<\xi_{c}$ (see Eq. (34)). At the end of this stroke, the cold qubit
becomes colder and the hot one becomes hotter. (d) The 3rd stroke is provided
by bringing the subsystems into contact with their respective reservoirs.
During this stroke, the heat $Q_{c}$ (red solid arrow) is transfered from cold
reservoir into the CM and the heat $Q_{h}$ (blue solid arrow)is transfered
from the CM into the hot reservoir, whereas the heat $Q^{\prime}(\xi)$ (red
dashed arrow) is absorbed from the hot reservoir by the CM. Clearly
$W_{in}=-(Q_{h}+Q_{c})$ and $W(\xi)=-(Q^{\prime}(\xi)+Q(\xi))$. In the case
$\xi=\xi_{c}$, the modified swap provides an autonomous refrigerator where the
required work for the refrigeration is only supplied by the quantum engine
induced by the measurement channel in the same cycle.

(a) (b)
(c) (d)

Figure 6:


# Strain effect on the ground-state structure of Sr2SnO4 Ruddlesden-Popper
oxides

Hwanhui Yun  Department of Chemical Engineering and Materials Science,
University of Minnesota  Dominique Gautreau  Department of Chemical
Engineering and Materials Science, University of Minnesota  School of Physics
and Astronomy, University of Minnesota  K. Andre Mkhoyan  Department of
Chemical Engineering and Materials Science, University of Minnesota  Turan
Birol<EMAIL_ADDRESS>Department of Chemical Engineering and Materials
Science, University of Minnesota

###### Abstract

Ruddlesden-Popper (RP) oxides (An+1BnO3n+1) comprised of perovskite (ABO3)n
slabs can host a wider variety of structural distortions than their perovskite
counterparts. This makes accurate structural determination of RP oxides more
challenging. In this study, we investigate the structural phase diagram of
$n=1$ RP Sr2SnO4, one of alkaline earth stannates that are promising for opto-
electronic applications by using group theory-based symmetry analysis and
first-principles calculations. We explore the symmetry breaking effects of
different dynamical instabilities, predict the energies of phases they lead
to, and take into account different (biaxial strain and hydrostatic pressure)
boundary conditions. We also address the effect of structural changes on the
electronic structure and find that compressive biaxial strain drives Sr2SnO4
into a regime with wider bandgap and lower electron effective mass.

##  I Introduction

The ABO3 perovskites constitute one of the most studied structural family of
oxides, thanks to the plethora of interesting phenomena they host. The
perovskite crystal structure is very flexible and can host a very large number
of combinations of different metal cations on the A and B sites. This
flexibility is in large part due to subtle crystal structural distortions
(atomic displacements,) which don’t break the connectivity of the corner
sharing network of oxygen octahedra, but modify the coordination of the
cations to compensate for under- or over-bonding. In fact, the majority of
perovskite oxides have so-called oxygen octahedral rotations that reduce the
cubic symmetry of the crystal structure. The cubic, high-symmetry perovskite
structure is an exception observed in a small number of compounds, rather than
the prototypical examples [1].

In addition to structural distortions and local defects, many ABO3 perovskites
also form extended defects such as 2-dimensional stacking faults to
accommodate nonstoichiometry [2]. Furthermore, double-AO stacking faults are
often observed to order as well and give rise to the so-called Ruddlesden-
Popper (RP) structures with the chemical formula An+1BnO3n+1 [3]. The nth
member of the RP series consists of n-unit-cell-thick perovskite slabs with an
additional AO rock-salt layer between the slabs [4, 5] (Fig. 1a). The network
of corner-sharing BO6 octahedra in the perovskite structure is broken by the
extra AO layer, and hence the structure consists of relatively isolated
2-dimensional slabs stacked on top of each other in a body-centered tetragonal
fashion. As a result, various material properties can be tuned by the
thickness of the perovskite slabs, as $n$ goes from $n=1$ in A2BO4 to infinity
in ABO3 [6, 7, 8]. As in the ABO3 perovskites, the perovskite slabs in RP
oxides exhibit various structural distortions from their high-symmetry
structure including octahedral tiltings, octahedral distortions, and cation
displacements that may give rise to ferroelectricity [9, 10, 11, 12]. In
addition, when subjected to external stimuli such as temperature, pressure, or
strain, RP oxides can undergo even more complex structural modifications,
which subsequently affects materials’ properties [13, 14].

Figure 1: (a) Atomic models of RP oxides (An+1BnO3n+1). (b) Illustration
showing the AO rock-salt layer and perovskite slabs in RP oxides. Alignment
between neighboring perovskite slabs 1 and 2 is viewed from the out-of-plane
($[001]_{T}$) direction in a box. The subscript $T$ is tetragonal. The
conventional unit cell is indicated with solid lines.

In this article, we present a comprehensive study of the crystal structure of
the $n=1$ RP Sr2SnO4, which is a cousin of the perovskite SrSnO3. SrSnO3 has
been actively studied as an ultra-wide bandgap material and has a nontrivial
structural phase diagram as a function of temperature and strain [15, 16, 17,
18, 19]. Sr2SnO4 is one of two experimentally synthesized RPs from SrSnO3,
($n=1,2$ Srn+1SnnO3n+1)[20, 21] and is positioned at the end of the RP series
(farthest from SrSnO3, Fig. 1a), thus, understanding its properties is
essential to comprehend characteristics of RP series in comparison with
perovskite. Sr2SnO4 also has a wide bandgap and, when doped, can be utilized
as a phosphor due to its luminescence properties such as long-lasting
phosphorescence [22, 23], mechanoluminescence [24], and thermoluminescence
[25]. Ceramics of Sr2SnO4 are synthesized with various methods, and
optical/dielectric characterizations have been performed by multiple groups
[26, 27, 28]. Yet, interestingly, the crystalline structure of Sr2SnO4, the
most fundamental characteristic of the material, is still disputed. Neutron
diffraction experiments performed by Green et al. point to a tetragonal
structure with a space group of $P4_{2}/ncm$ (#138) at $T=12$ K, and an
orthorhombic $Bmab$ (#64) phase at $T=295$ K [20]. On the other hand, Fu et
al. report that Sr2SnO4 has an orthorhombic $Pccn$ (#56) structure at $T=4$ K
and undergoes phase transitions of $Pccn\rightarrow Bmab\rightarrow I4/mmm$
(#139) phases with increasing temperature [29, 30]. The reported structures
have different SnO6 oxygen octahedral rotation patterns in addition to
rotation angles. While some of these patterns and the resulting space groups,
such as $P4_{2}/ncm$, are also reported in other $n=1$ RP compounds, Sr2SnO4
is the only stoichiometric RP for which the space group $Pccn$ is reported to
our best knowledge [10]. Additional disagreements exist in the reported
structural transition temperatures, which might imply that the structure is
highly sensitive to extrinsic effects [28].

In order to resolve the disagreements in the experimentally reported crystal
structures and assess the possible sensitivity of optical and electronic
properties to boundary conditions such as strain, we perform a first-
principles and symmetry-based analysis of Sr2SnO4. The first-principles
density functional theory (DFT), in combination with the standard local
density approximation or generalized gradient approximations, is expected to
accurately predict the crystal structure and lattice energetics of this
compound, since it is a band insulator with a rather simple electronic
structure. The symmetry analysis is based on application of group theory,
which involves labeling possible ionic displacements (i.e. normal modes) in
crystals by the irreducible representations (irreps) of the space groups and
is commonly used to describe the octahedral rotations in perovskites and
related systems [31, 32]. For example, out-of-phase and in-phase octahedral
rotation patterns in perovskite oxides can be described as atomic
displacements associated with the three-dimensional irreps $R_{4}^{+}$ and
$M_{3}^{+}$ of the cubic perovskite structure with space group $Pm\bar{3}m$
(#221) [31]. Likewise, the octahedral rotation pattern in the $n=1$ RP Sr2SnO4
structure can be depicted as lattice distortions mediated by irreps of the
high-symmetry body-centered tetragonal $I4/mmm$ (#139) structure. We identify
the unstable structural modes (imaginary frequency phonon modes) in the
$I4/mmm$ structure by calculating the phonon dispersions, and then determine
the irreps that they transform as accordingly. This enables finding possible
meta-stable low-symmetry structures using group theory and predicting their
energies by relaxing the crystal structures using DFT.

Next, we explore the biaxial strain and hydrostatic pressure effects on the
structure and symmetry also by employing the group-theoretical analysis and
DFT. Biaxial strain, imposed on thin films by lattice mismatched substrates,
can lead to significant changes in the crystal and electronic structure of
crystalline materials [33, 34, 17]. In addition to changing the crystal
structure, biaxial strain may also have a significant effect on the electronic
properties of materials. For example, the bandgap of a large number of oxide
and halide perovskites are predicted to be strongly strain dependent [35, 36,
37], strain induced magnetic and coupled metal-insulator transitions coexist
in SrMoO3 [38, 39], and the correlation strength in correlated metallic SrVO3
and CaVO3 are found to be strain tunable through the changes in the crystal
structure [40]. The stannate RPs are wide bandgap semiconductors, and hence an
electronic phase transition is not likely under strain, but changes in their
bandgap and effective mass can nevertheless be significant enough to affect
their optical and electronic properties. Hence, we study the impact of the
strain-driven structural changes on the electronic properties by calculating
the electronic band structures of Sr2SnO4 with different symmetry crystal
structures. Lastly, the effect of hydrostatic pressure on the Sr2SnO4
structure is investigated and compared with the biaxial strain effects.

##  II Methods

DFT calculations were performed by using the Vienna ab-initio simulation
package (VASP) [41, 42] with the projector-augmented wave approach [43] and
Perdew-Burke-Ernzerhof (PBE)-sol functional [44]. $10^{-6}$ eV was used as the
convergence threshold in the self-consistent field cycle for the electronic
wavefunctions. The energy cut-off for the plane wave basis set was set to 550
eV. The Brillouin zone of the tetragonal $I4/mmm$ structure ($4.068\times
4.068\times 12.560$ Å) was sampled with a $12\times 12\times 6$ Monkhorst-Pack
k-point grid, which was scaled down accordingly for larger supercells used for
distorted crystal structures. As the convergence criterion for ionic
relaxations, a force per atom of $5\times 10^{-4}$ eV/Å was used. The energy
and volume of optimized structures are presented per formula unit (f.u.) of
Sr2SnO4. Phonon calculations were carried out from a $2\times 2\times 2$
primitive supercell of the $I4/mmm$ structure of Sr2SnO4 using the frozen
phonon approach. Dispersion curves were computed with the Phonopy software
(v2.9.1) interfaced with VASP [45]. Non-analytical term correction was
incorporated with dielectric tensor and Born effective charges calculated
using density functional perturbation theory as implemented in VASP [46]. The
atomic structures corresponding to individual modes were constructed using the
eigen-displacements, and irreducible representations (irreps) of each mode
were determined by using ISODISTORT [47]. Construction of distorted structures
adapting normal modes with different order parameters and directions was also
conducted using ISODISTORT. The Bilbao Crystallographic Server was used as a
reference for group theory tables [48]. Crystal structure visualization was
carried out using the VESTA software [49]. Simulation data are available at
the Data Repository for the University of Minnesota [50].

##  III Results and Discussion

###  III.1 Crystal structure of bulk Sr2SnO4

####  III.1.1 Experimentally Observed Structures

At high temperature, Sr2SnO4 has the high-symmetry reference RP structure with
the space group $I4/mmm$ (#139) [30]. In this structure, there are isolated
perovskite slabs that are single-octahedron-thick, and are half-unit-cell
shifted along the $[110]_{T}$ direction with respect to their neighbors, as
shown in Fig. 1b. (Throughout this manuscript, all directions and planes are
indexed using the axes of the conventional unit cell of the high-symmetry
body-centered tetragonal structure.) All of the experimentally reported
crystal structures at room temperature have structural distortions that break
translational symmetry of the $I4/mmm$ RP structure and give rise to new Bragg
peaks at the $X$ point of the Brillouin zone[20, 29, 30, 51]. These
distortions are due to oxygen octahedral rotations (or tilts) which are
commonly observed and are classified in detail in multiple $n=1$ RP oxides
[52, 53]. The three experimentally observed space groups ($Pccn$, $Bmab$, and
$P4_{2}/ncm$) are all subgroups of the high-symmetry $I4/mmm$ space group of
the RP structure, which itself is observed above 573 K [30, 29, 20]. These
three structures can be obtained by the same octahedral rotation mode, which
transforms as the $X_{3}^{+}$ irrep, and differ only in the direction of
2-dimensional order parameter: $X_{3}^{+}(a;a)$, $X_{3}^{+}(a;0)$, and
$X_{3}^{+}(a;b)$ reduce the symmetry to $P4_{2}/ncm$, $Bmab$ (or equivalently
$Cmca$), and $Pccn$ respectively. The tetragonal space group $P4_{2}/ncm$
($X_{3}^{+}(a;a)$) has out-of-phase oxygen octahedral tilts only along a
single in-plane Cartesian axis, but this direction is alternating on
neighboring perovskites slabs. In other words, it corresponds to an
alternating $a^{-}b^{0}c^{0}$ and $a^{0}b^{-}c^{0}$ rotation pattern in the
Glazer notation. The base-centered orthorhombic $Bmab$ ($X_{3}^{+}(a;0)$) is
obtained by out-of-phase tilts along a $[110]_{T}$ direction, which is
$a^{-}a^{-}c^{0}$ in the Glazer notation. The experimentally observed $Pccn$
structure, on the other hand, involves rotations around low-symmetry
directions on the $a$-$b$ plane. The atomic model is illustrated in
Supplemental Material (SM) Fig. S1 [54].

Figure 2: (a) Phonon dispersion diagram of $I4/mmm$ Sr2SnO4. Unstable $X$
modes with imaginary frequencies are marked. The band path is visualized in SM
Fig. S2. (b,c,d) Atomic models of Sr2SnO4 structure transformed with the three
unstable $X$ modes. (b) $X_{3}^{+}(a;0)$ mode-implemented structure exhibiting
OOT. (c) $X_{2}^{+}$ mode-implemented structure showing OOR. (d) Atomic models
describing different alignments between neighboring perovskite slabs in the
$X_{3}^{+}$ and $X_{4}^{+}$ irreps-associated structures. When the structures
are viewed from the out-of-plane ($[001]_{T}$) direction, and the layer 1 is
fixed, the configurations of the layer 2 in $X_{3}^{+}$ and $X_{4}^{+}$ irreps
are dissimilar. The direction of the atomic displacements of $O_{SrO}$ due to
OOT are indicated with the arrows.

####  III.1.2 Lattice dynamics of Sr2SnO4

To investigate the zero temperature ground-state (or the lowest energy)
crystal structure of Sr2SnO4, we perform first-principles lattice dynamics
calculations in the $I4/mmm$ structure and identify lattice instabilities that
may freeze in and reduce the symmetry, as well as the energy. The phonon
dispersion is shown in Fig. 2a. Phonons with imaginary frequencies, which
indicate lattice instabilities, are present at the $X$, $P$, and $N$ k-points
of the Brillouin zone. Since all the structures proposed based on experimental
data are associated with the $X_{3}^{+}$ irrep, we focus our analysis on the
unstable modes at the $X$-point only, but consider all three unstable
$X$-modes. These modes break the translational symmetry in the same way, and
hence lead to primary Bragg spots at the same positions but with different
intensities in diffraction experiments. While there are different systematic
absences due to nonsymmorphic symmetries in these space groups, some of the
allowed peaks may be hard to detect due to the low structure factor of oxygen
atoms. As a result, differentiating the different structures the $X$ modes
lead to might be experimentally challenging, and thus all of them need to be
considered to theoretically predict the ground state structure. In the SM
section 3, we also provide a partial analysis of the unstable $N$ and
$P$-modes. This analysis shows that they lead to monoclinic structures without
larger energy gains than the $X$-modes, and therefore, are not critical for
the present discussion.

We identified the three unstable modes at the $X$-point with the $X_{3}^{+}$,
$X_{2}^{+}$, and $X_{4}^{+}$ irreps of the $I4/mmm$ space group. All of these
irreps correspond to SnO6 octahedral rotations about different
crystallographic axes as well as associated Sr displacements. The $X_{3}^{+}$
and $X_{4}^{+}$ irreps both correspond to out-of-phase oxygen octahedral tilts
(OOT) about the in-plane axes ($[100]_{T}$ and $[010]_{T}$) but with different
phases along the out-of-plane direction, while the $X_{2}^{+}$ irrep
corresponds to SnO6 oxygen octahedral rotations (OOR) about the out-of-plane
($[001]_{T}$) axis. These modes are illustrated in Fig. 2b-c. All three $X$
irreps are 2-dimensional, and hence have three possible order parameter
directions (OPD)($(a;0)$, $(a;a)$, and $(a;b)$) that lead to different space
groups.

The oxygen octahedral rotation modes in RP’s are different from their
counterparts in bulk perovskites in two important aspects. The first
difference is that while the octahedral rotations can induce (improper)
displacements of A-site cations in perovskites, there is no coupling between
rotations and A-site displacements at the harmonic ($2^{nd}$) order, and
hence, the octahedral rotation phonon eigenvectors of perovskites don’t
include any A-site displacements. However, in RP’s, the phonon eigenvectors
include small but nonzero A-site displacements as well. For example, the
$X_{3}^{+}(a;0)$ mode includes atomic displacements of Sr in addition to OSrO
(apical O in the SrO$(001)$ layer), and O${}_{SnO_{2}}$ (O in the
SnO${}_{2}(001)$ layer). The other difference is due to the broken
connectivity of the octahedral network and the shift of the neighboring
perovskite layers. Unlike in perovskites, where there is only one symmetry
allowed way to have an in-phase (out-of-phase) octahedral rotation along a
certain axis, there is more than one degree of freedom in the RP’s that
correspond to an in-phase (out-of-phase) octahedral rotation along an axis.
For example, both the $X_{3}^{+}$ and $X_{4}^{+}$ irreps correspond to out-of-
phase OOT around in-plane axes. However, the alignment between neighboring
perovskite slabs of the $X_{4}^{+}$ displacement pattern is different from
that of the $X_{3}^{+}$ mode, which accounts for their disparate symmetries
and stabilities. When the relative in-plane shift between the neighboring
perovskite slabs, e.g. layers 1 and 2 in Fig. 2b and d, is parallel to the
octahedral rotation axis, i.e. perpendicular to the in-plane displacement
direction of OSrO (white arrows in Fig. 2d), a structure with a
$X_{3}^{+}(a;0)$ mode is produced, and when the shift is perpendicular to the
rotation axis, i.e. parallel to the OSrO displacements, a $X_{4}^{+}(a;0)$
mode is generated. See SM section 4 for details. Despite their similarities,
the $X_{3}^{+}$ mode is more unstable than the $X_{4}^{+}$ mode as can be seen
in Fig. 2a, implying that a structure incorporating the $X_{3}^{+}$ distortion
pattern reduces the structural energy more at quadratic order. We address why
this may be from a bond-valence point of view in the following sections.

####  III.1.3 Order parameter directions

In the previous section, we compared the relative strengths of lattice
instabilities. All three of these instabilities are 2-dimensional, and higher
order terms in the Landau free energy leads to different energy gains for
different OPDs. In this subsection, we consider these different order
paramater directions to map out the energy surface, and predict the lowest
energy structure. For each of the three 2-dimensional irreps, there are three
distinct phases, corresponding to different values of OPDs, $a$ and $b$. For
example, as discussed earlier, freezing in the $X_{3}^{+}(a;0)$ mode in the
$I4/mmm$ structure leads to a $Bmab$ phase with OOT around the $[110]_{T}$
axis, i.e. equal OOT around $[100]_{T}$ and $[010]_{T}$ axes. In contrast, an
$X_{3}^{+}(a;b)$ distortion leads to a $Pccn$ phase exhibiting OOT with
dissimilar tilts around $[100]_{T}$ and $[010]_{T}$, and an $X_{3}^{+}(a;a)$
leads to the tetragonal $P4_{2}/ncm$ where the octahedral rotation axes
alternate between $[100]_{T}$ and $[010]_{T}$ in consecutive perovskite
layers. (These distortions are illustrated in detail in SM Fig. S5.)
Similarly, there are three phases each for the $X_{2}^{+}$ and $X_{4}^{+}$
modes, corresponding to distinct OPDs, which are shown in SM Fig. S6.

To optimize structures for each $X$ mode, we calculate the energy of
structures with different $(a;b)$ values on a grid for different irreps from
DFT. Fig. 3a shows the energy map of a Sr2SnO4 structure distorted with an
$X_{3}^{+}$ mode as a function of amplitudes (Q) along the two order parameter
components. The phonon eigenvectors are used to determine the displacement
patterns for each order parameter component; in other words, the ratio between
the three types of atomic displacements
(QSr:Q${}_{O_{SrO}}$:Q${}_{O_{SnO_{2}}}$) is fixed to the values obtained from
phonon calculation. We find that for $X_{3}^{+}$, the minimum energy is
obtained for $a=b$ (space group $P4_{2}/ncm$) whereas for $X_{2}^{+}$, the
energy is lowest when only one of the two order parameter components is
nonzero (space group $Bmab$). For $X_{4}^{+}$ (SM Fig. S10), the minimum
energy is also obtained for $a=b$ (space group $P4_{2}/nnm$). Of all these
structures, the lowest energy is obtained for $X_{3}^{+}(a;a)$, and thus, the
lowest energy structure in the absence of strain is $P4_{2}/ncm$.

In passing, we note that the fact that we find local minima only for either
(i) when $a=b$, or (ii) when one of $a$, $b$ is zero, is consistent with a 6th
order Landau free energy expansion. Symmetry imposes strict conditions on the
allowed terms in the expansion of the free energy as a polynomial of the order
parameter components around the high-symmetry structure, which is the
undistorted $I4/mmm$ RP structure in this case [55]. All three unstable
$X$-point irreps $X_{i}^{+}$ ($i=1,~{}2,~{}3$) of Sr2SnO4 have the same
trivial Landau Free energy up to 6th order:

$\mathcal{F}(a,b)=\alpha(a^{2}+b^{2})+\beta_{4}(a^{2}+b^{2})^{2}+\gamma_{4}(a^{2}b^{2})\\\
+\beta_{6}(a^{2}+b^{2})^{3}+\gamma_{6}(a^{2}+b^{2})(a^{2}b^{2}),$ (1)

where the Greek letters are material specific coefficients that can be
obtained from DFT. The $(a^{2}+b^{2})$ terms do not depend on the relative
direction of the order parameter, and hence cannot differentiate between
$X_{i}^{+}(a;0)$, $X_{i}^{+}(a;b)$, and $X_{i}^{+}(a;a)$. The only terms that
depend on the direction of the order parameters are the $\gamma$ terms, which
have the same type of $\sim a^{2}b^{2}$ dependence. Depending on the signs of
$\gamma$, this expression is minimized either for $|a|=|b|\neq 0$ or for one
of the two components being zero. Thus, the space group $Pccn$, which is
obtained by the $X_{3}^{+}(a;b)$ OPD, requires higher order terms in the free
energy expansion to be important in determining the ground state crystal
structure. Our analysis (not shown) of the higher order Landau free energy
expansion for $X_{3}^{+}$ shows that the 8th order terms indeed allow
solutions with $|a|\neq|b|\neq 0$. Since in most crystalline materials a
lattice free energy expansion up to just 6th order is sufficient to determine
the correct low temperature crystal structure, we find that the space group
$Pccn$ is an unlikely ground state candidate for Sr2SnO4.

In principle, both strain-coupling and higher order terms can be important in
determining the symmetry of the lowest energy structure. In order to take
these effects into account completely, we perform relaxations of the crystal
structure of Sr2SnO4 using DFT, and allowing both the internal atomic
coordinates and the lattice vectors to relax. Since the DFT calculations
preserve the symmetry and the basin of attraction of different metastable
phases can be large, for each OPD we use multiple starting points with
$(a;0)$, $(a;a)$, and $(a;b)$. Note that these complete ionic relaxations
allow the QSr:Q${}_{O_{SrO}}$:Q${}_{O_{SnO_{2}}}$ ratio, as well as the strain
state of the lattice, to change. We could not stabilize a state with $a\neq b$
in any of these calculations. All calculations initiated with such a structure
relax to a higher symmetry structure with either $(a;a)$ or $(a;0)$. While
they do not make a qualitative difference in the ground state, we find the
strain effects to be quantitatively important: For example, when relaxed, the
$P4_{2}/ncm$ phase has in-plane and out-of-plane lattice constants change by
+0.83% and -1.38% compared to the undistorted $I4/mmm$ structure with the same
supercell size. All in all, the lowest energy structure is still $P4_{2}/ncm$
obtained by $X_{3}^{+}(a;a)$, and the minimum energy for structures with
$X_{2}^{+}$ and $X_{4}^{+}$ distortions are obtained at OPDs $X_{2}^{+}(a;0)$
and $X_{4}^{+}(a;a)$, respectively, as shown in Fig. 3c.

Figure 3: (a,b) Three-dimensional (3D) energy map of Sr2SnO4 structures
containing $X(a;b)$ irreps as a function of mode amplitudes of a and b. 3D
maps were constructed for the $X_{3}^{+}$ (a) and $X_{2}^{+}$ (b) irreps. The
minimum energy structure was further relaxed, and the optimized structures are
presented on the right. (c) The energy and (d) GII values of the relaxed
structures with different irrep modes.

While the $X_{3}^{+}(a;a)$ irrep gives the most stable structure, the other
unstable modes may also be present in the lowest energy structure in
combination with the $X_{3}^{+}$ mode. In order to check this possibility, we
also relaxed structures that have a superposition of $X_{3}^{+}$ and the
$X_{2}^{+}$ or $X_{4}^{+}$ modes. These structures all relax to a structure
containing only the $X_{3}^{+}(a;a)$ mode. As a further check, we also
performed structural relaxations of the experimentally-determined low-
temperature crystal structure with the space group $Pccn$ and the
$X_{3}^{+}(a;b)$ distortion [29]. This structure also relaxed to the same
final state with the $X_{3}^{+}(a;a)$ distortion. As a result, at the DFT
level, there is no reason to doubt that the lowest energy crystal structure of
Sr2SnO4 is obtained by an $X_{3}^{+}(a;a)$ distortion of the high-symmetry
$I4/mmm$ structure and has the space group $P4_{2}/ncm$.

####  III.1.4 Bond valence approach

To gain further insight into the structural stability of the different phases
in Sr2SnO4, we use the bond valence approach. The bond valence sum and the
related quantity of the Global Instability Index (GII) have been proven to be
effective tools to explain structural stability of crystals where modification
of the local bonding environment can explain structural transitions [1, 56,
57, 58, 59]. As an example, different rotation patterns in perovskites were
shown to be predicted by the optimization of bond valences more than two
decades ago [60]. The valence $V_{i}$ of an atom $i$ can be estimated by
summing the individual bond valences $v_{ij}$ that are calculated from the
bond length between the atoms:

$V_{i}=\sum_{j=1}v_{ij}$ (2)

and

$v_{ij}=\mathrm{exp}((R_{0}-R_{ij})/b),$ (3)

where $R_{0}$ is the empirical value of the expected bond distance, $R_{ij}$
is the actual bond distance between atoms $i$ and $j$, and $b=0.37$ Å is an
empirical constant [59, 56]. The observed valence of an atom $i$ typically
deviates from its formal valence ($V_{formal,Sr}$ = 2, $V_{formal,Sn}$ = 4)
and the discrepancy is referred to as a discrepancy factor,
$d_{i}=V_{i}-V_{formal}$. The discrepancy factor is an indicator of whether an
atom is under-($d_{i}<0$) or over-($d_{i}>0$)bonded. The root mean square of
all the discrepancy factors in a structure,

$GII=\sum_{i=1}^{N}\left(\frac{d_{i}^{2}}{N}\right)^{1/2},$ (4)

is referred to the GII. A small GII is an indicator of overall structural
stability.

In Fig. 3d, we show the GII values for the DFT-optimized structures of
Sr2SnO4. Whereas the original form of GII includes bond valences of all ions
in an examined structure, our careful analysis shows that, interestingly, the
stability of Sr2SnO4 is better-explained by GII using only cations; and thus,
we report cation-based GII. (See SM section 7 for the GII calculation
analysis.) The GII values of different phases follow the same trend as the DFT
energies, which indicates that the phase stability of Sr2SnO4 can be predicted
from the local bonding environment of cations. The bond valences of Sr (shown
in SM Fig. S7) make the dominant contribution to the GII and determine the
low-symmetry structure. This is in-line with the observation in perovskites
that the octahedral rotations are A-site driven instabilities, and serve to
optimize the bonding environment of the underbonded A-site cations.

###  III.2 Strain effects on the ground-state structure of Sr2SnO4

In this section, we evaluate the effect of biaxial strain on Sr2SnO4. We limit
our discussion to (001) grown thin films (i.e. structures where the
crystallographic $c$ axis is relaxed but the $a$ and $b$ lattice parameters
are fixed by a substrate), since this is the most commonly grown orientation
by, for example, molecular beam epitaxy method. (See, for example, [61, 12,
62, 17].)

As hinted from the distinct lattice parameter change upon relaxation of the
minimum energy structures in the energy maps, each irrep mode responds to
biaxial strain differently. In order to predict how the lattice instabilities
evolve, we relax the high-symmetry $I4/mmm$ structure with in-plane biaxial
strain ranging from -3 % to +3 % and calculate the phonon dispersions for each
of these strained structures. We define strain for a structure with in-plane
lattice parameter $a$ as $\eta=(a-a_{0})/a_{0}$, where $a_{0}$ is the lattice
parameter of the fully relaxed but undistorted ($I4/mmm$) structures. The
phonon dispersion curves presented in SM Fig. S9 indicate not only that the 3
unstable $X$ modes remain unstable under strain, but also that the strongest
instability remains to be at the $X$ point. The frequencies of the three
relevant $X$-modes are shown in Fig. 4a. Under compressive strain, the
$X_{3}^{+}$ and $X_{4}^{+}$ instabilities (related to OOT around the in-plane
axes) weaken while the $X_{2}^{+}$ becomes (associated with the out-of-plane
axis OOR) even more unstable. On the other hand, under tensile strain, the
$X_{3}^{+}$ and $X_{4}^{+}$ modes become slightly more unstable, while the
$X_{2}^{+}$ instability is suppressed, and its frequency becomes real at +3 %.

Figure 4: Strain effect on the $X$ modes. (a) Phonon frequency of the three
$X$ modes. (b) The relative energy of the $X_{3}^{+}$ modes with dissimilar
OPDs with respect to the $I4/mmm$ structure. The region where $X_{3}^{+}(a;b)$
is stabilized is magnified in the inset. (c) The relative energy of the
optimized $X$ modes. OPD of each structure is indicated via the symbol types:
closed circle-$X(a;0)$, open circle-$X(a;a)$, open circle with a center
dot-$(a;b)$.

In Fig. 4b, we show the energy of structures with $X_{3}^{+}$ distortions. In
SM Fig. S10, 3D energy maps of the Sr2SnO4 structures at different strain
degree are also presented. Extending the compressive strain range to beyond
-3% shows that there is a transition from $X_{3}^{+}(a;a)$ ($P4_{2}/ncm$) to
$X_{3}^{+}(a;0)$ ($Bmab$). This transition can be understood to stem from a
strain-induced sign change of the sum of the $\beta_{2}$ term in Eq. 1 with
the higher order terms with the same dependence on $|a|/|b|$. As this term
crosses zero, higher order terms that favor different $|a|/|b|$ ratios become
apparently important, and the $X_{3}^{+}(a;b)$ ($Pccn$) phase has a lower
energy in a narrow strain range as shown in the inset of Fig. 4b. However,
this does not make $Pccn$ experimentally observable either, since under
compressive strain, the $X_{2}^{+}$ turns out to lead to a lower energy
structure, as seen from Fig. 4c where we present the DFT-predicted energies of
structures obtained from each of the $X$ point instabilities. The transition
from structures obtained by $X_{3}^{+}$ distortions under tensile strains to
structures with $X_{2}^{+}$ octahedral rotations can be understood as a
transition from a phase with octahedral tilts around in-plane axis to a phase
with octahedral rotations around the out-of-plane axis. Such transitions are
rather common in perovskite and perovskite-related structures and are
primarily driven by the fact that out-of-plane octahedral rotations increase
the in-plane B–O bond lengths, which becomes favorable under compressive
strain. This enforces a shorter than ideal B–O distance.

In passing, we note that the strain driven structural transitions observed in
Fig. 4c can also be predicted from the GII to a large extent, which means that
electrostatics and local bonding are the determining factor for phase
stability of Sr2SnO4 under strain as well. We show the details of strain
dependent GII calculations in SM Fig. S11c-d.

We finally consider the combinations of different $X$ point distortions to
examine if any phases which is not considered in Fig. 4c are stabilized by
strain. The resultant phase diagram, Fig. 5a, hosts a $Pbca$ phase that has
octahedral rotations around both in-plane and out-of-plane axis under moderate
compressive strain. The $Pbca$ phase is obtained by a superposition of
$X_{3}^{+}(a;0)$ and $X_{2}^{+}(b;0)$ distortions as illustrated in Fig. 5b.
Therefore, our result highlights that as the phase transition from
$P4_{2}/ncm$ to $Cmca$ takes place in Sr2SnO4 due to compressive biaxial
strain, the intermediate phase $Pbca$ can be stabilized in the strain range of
$-3\%\leq\eta\leq 0\%$. See SM Fig. S11 for stability comparison between the
$Pbca$ and $P4_{2}/ncm$ (ground-state structure) phases as a function of
strain.

Figure 5: Combination of the $X$ modes under the biaxial strain. (a) The
energy of structures mediated by the three $X$ irreps and by superposition of
two irreps ($X_{3}^{+}\oplus X_{2}^{+}$). (b) The atomic model of the
optimized structures under the biaxial strain of -1%. The structure is a
$Pbca$ phase, and its distortion is described with the reducible
representation of $X_{3}^{+}(a;0)\oplus X_{2}^{+}(b;0)$.

###  III.3 Strain effect on the electronic properties of Sr2SnO4

Perovskite stannates including SrSnO3 are known to exhibit strain-sensitive
electronic band structures, and strained thin films have been reported to have
bandgaps larger by $\sim$1 eV than bulk [63, 64, 65]. Given the similar
character of valence and conduction bands in Sr2SnO4, similar changes that
would affect its luminescence properties may be possible. The bottom of the
conduction band of Sr2SnO4 is comprised of Sn $5s$ states, which hybridize
with O $2p$ derived bands. Thus, the Sn-O bonding configurations such as Sn-O-
Sn bonding angles controlled by octahedral rotations directly influence the
bandgap and the electron effective mass. The electronic band diagram of the
ground-state structure at zero strain, the $P4_{2}/ncm$ phase containing the
$X_{3}^{+}(a;a)$ irrep, is shown in Fig. 6.

Figure 6: (a) The electronic band structure of the $P4_{2}/ncm$ (#138) phase
of Sr2SnO4 structure. The regions near the conduction band minimum (inside the
green boxes) were used for the effective mass calculation. (inset) A schematic
of the Brillouin zone of the $P4_{2}/ncm$. The band path used for the diagram
is illustrated with the blue lines.

The key feature of the conduction bands of perovskite stannates BaSnO3 and
SrSnO3, a single parabolic s-band at the $\Gamma$ point, is also present in
Sr2SnO4, and it is resistant to the structural distortions in all phases we
considered. However, different degrees and patterns of octahedral rotations
lead to different bandgaps, as shown in Fig. 7a. Following the lowest energy
structure (yellow) as a function of strain, it is evident that larger in-plane
lattice parameters reduce the bandgap by as much as $\sim$0.4 eV for 3%
strain. (Note that while the standard DFT+GGA calculations we perform
underestimate the bandgap, they usually capture the correct trends with
strain.) The same effect of tensile (compressive) biaxial strain decreasing
(increasing) the bandgap was also observed in SrSnO3 [64] and suggests that
strain engineering can be employed to achieve photoemission with different
wavelengths from this material.

Figure 7: Electronic properties of Sr2SnO4 with respect to biaxial strain. (a)
Bandgap. (b) Effective electron mass $m_{e}^{*}$ of ground-state structures
along the in-plane and out-of-plane directions. Two in-plane effective masses,
originating from the in-plane anisotropy in the orthorhombic phases, are
plotted as closed circles.

Next, we examine the electron effective mass of Sr2SnO4, which is an important
quantity for perovskite stannates as high mobility n-type semiconductors [66,
67, 19]. In Fig. 7b, the electron effective mass calculated in the lowest
energy crystal structure is presented as a function of biaxial strain. The
symmetry of the Sr2SnO4 structures lead to dissimilar effective masses along
the out-of-plane and the in-plane directions. (See SM section 12 for details.)
Under zero strain, while the in-plane effective mass is similar to that of
perovskite SrSnO3 ($\sim$0.4me)[68, 17], the out-of-plane effective mass is
higher by $\sim$30%, which is in line with the broken Sn-O network in this
direction. Interestingly, under large compressive strain, the effective mass
tensor of the $Pbca$ and $Cmca$ phases become closer to isotropic, and the
out-of-plane effective mass becomes comparable to the in-plane one. While
compressive strain leads to both SnO6 octahedral distortion, i.e. elongation
along the c-axis, and tilting, i.e. OOT to OOR, the reduction of the electron
effective mass indicates that the change of the octahedral tilting patterns
plays a major role in determining the effective mass of Sr2SnO4. As OOT
weakens under compressive strain, the out-of-plane direction O-Sn-O bonds are
aligned to the c-axis, and consequently, the effective mass along the
direction decreases. The isotropic and perovskite-comparable electron
effective mass, in combination with the enhanced bandgap, under compressive
strain renders Sr2SnO4 as a viable wide-bandgap semiconductor.

###  III.4 Hydrostatic pressure effect

Lastly, we consider the effect of hydrostatic pressure on the structural
phases of Sr2SnO4. Different symmetry structures are relaxed under fixed
stress boundary conditions, and the enthalpy $H$ ($H=E-PV$, where $E$ is total
energy, $P$ is pressure, and $V$ is a volume) of the optimized structures are
compared to examine their relative stability (see Fig. 8). Pressures up to 250
kBar are considered. The pressure response of Sr2SnO4 is found to be very
similar to its response to biaxial compressive strain, which also reduces the
unit cell volume. With increasing pressure, the structure with
$X_{3}^{+}(a;0)$ and $X_{4}^{+}(a;0)$ distortions become energetically less
favorable over the structures with $X_{3}^{+}(a;a)$ and $X_{4}^{+}(a;a)$
modes. Until $P$= 200 kBar, the structural ground state does not change, while
above this pressure the $Cmca$ structure hosting $X_{2}^{+}(a;0)$ distortions
becomes the lowest energy structure. We find that in contrast to biaxial
strain, neither a combination of multiple structural distortions, nor a
different direction of these three order parameters, are stabilized under
pressure.

Figure 8: The effect of pressure on the $X$ modes of Sr2SnO4. (Top) Relative
enthalpy of $X_{3}^{+}$ mode with different OPDs. (Bottom) Relative enthalpy
of $X$ mode-adapted structure with respect to the $I4/mmm$ structure. The
closed and open circles represent the order parameter configurations of
$(a;0)$ and $(a;a)$, respectively. Space group of the minimum energy structure
at different pressures is shown as the background color.

##  IV Conclusions

In this paper, we provide a comprehensive first-principles and group-
theoretical study of the crystal and electronic structures of $n=1$ RP
Sr2SnO4. Our main findings are that (i) the lowest energy structure at ambient
pressure is $P4_{2}/ncm$, and the next lowest energy metastable phase is
$Bmab$, which is consistent with the experimental observation of a phase
transition between these two phases, followed by a transition to the $I4/mmm$
phase with increasing temperature[20], (ii) the $Pccn$ phase is not even
metastable (except for a narrow range under large biaxial strain), and
therefore not likely to be present in a bulk material, (iii) biaxial strain
leads to both structural phase transitions and significant changes in the
bandgap and electron effective mass, which is relevant to both semiconducting
and optical properties of the material, (iv) structural phase transition to
$Cmca$ occurs above 200 kPa without necessarily producing the intermediate
$Pbca$ phase seen in the phase diagram under biaxial strain.

The RP structures bear both additional degrees of freedom (layering) and
structural complexity compared to ABO3 perovskites. This study provides an
example of theoretical determination of the RP oxide structure, analytical
understanding of stability, and phase transitions under different boundary
conditions and shows that even in the absence of additional complications
(such as Jahn-Teller distortions, magnetic transitions, etc.) the phase
diagrams can host multiple phases. The effect of these structural phases on
electronic properties such as the bandgap and the effective mass will enable
tuning of material properties and broadening of its applicability in opto-
electronic devices.

###### Acknowledgements.

This work was supported primarily by the National Science Foundation through
the University of Minnesota MRSEC under Award Number DMR-2011401. The authors
acknowledge the Minnesota Supercomputing Institute (MSI) at the University of
Minnesota for providing computational resources that contributed to the
research results reported within this paper.

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