# Bargmann Representation of Quantum Absorption Refrigerators

M. W. AlMasri1 and M. R. B. Wahiddin1,2
1Cybersecurity and Systems Research Unit, ISI-USIM,
Bandar Baru Nilai, 71800 Nilai, Negeri Sembilan, Malaysia.
2 Department of Computer Science, Kulliyah of ICT, IIUM,
P.O. Box 10, 50728, Kuala Lumpur, Malaysia.

###### Abstract

In this work, we solve the quantum absorption refrigerator analytically in the
space of holomorphic functions with Gaussian measure . Our approach simplifies
the calculations since for a given quantum system the coordinate
representation of any quantum state is always more complicated than its
corresponding expression written with respect to the phase-space coordinate
$z_{i}$. We finally discuss the computational complexity of the holomorphic
representation and compare it with the computational complexity of the
standard operator method and prove the efficiency of the holomorphic
representation in computing some tasks. Our treatment is applicable to all
quantum heat engines and refrigerators.
Keywords: Bargmann representation, Quantum Absorption Refrigerators,
Computational Complexity, Open Quantum Systems

##  1 Introduction

Thermodynamics is one of the oldest and well-established branches of physics
that sets boundaries to what can or cannot be achieved at macroscopic
level[1]. It also has deep connection with other branches of science such as
philosophy and computing theory[2, 3]. It was realized that large quantum
devices, such as masers and lasers, could be treated under the scope of
thermodynamics . The pioneering work of Scovil et al. has shown the
equivalence of the Carnot engine with three-level Maser [4, 5, 6]. This opened
the road for flourishing progress in the study of the connection between the
laws of thermodynamics and quantum devices [7, 8, 9, 10, 11, 12, 13, 14].

The interaction between any quantum mechanical system with environment may
lead to dissipation or loss of information contained in the system to its
environment. To obtain a complete description of any quantum mechanical
system, one needs to incorporate the effect of environment or baths to the
original system Hamiltonian. This was the philosophy behind investigating open
quantum systems [15, 16, 17]. One of the main tasks in this theory is to solve
the so-called Master equation of open quantum systems . Generally solving such
equation can be very complicated and requires many tedious calculations
especially in the non-Markovian case[18, 19].

In [20], Fock noticed the fact that one in principle express the raising and
lowering operators analytically in the complex $z$-plane as $z$ and
$\frac{\partial}{\partial z}$ respectively. In this case the wavefunction
should be written in term of the complex variable $z$ in a closed analytical
form not as a vector in the Hilbert space. For example, the wavefunctions of
the quantum harmonic oscillator in coordinate representation are given by
$\psi_{n}\left(x\right)=\frac{1}{\sqrt{2^{n}n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}}e^{-\frac{m\omega
x^{2}}{2\hbar}}H_{n}\left(\sqrt{\frac{m\omega}{\hbar}}x\right)$ where
$H_{n}\left(y\right)=\left(-1\right)^{n}e^{y^{2}}\frac{d^{n}}{dy^{n}}\left(e^{-y^{2}}\right)$
is the Hermite polynomials while in the Bargmann representation it is simply
$\psi_{n}\left(z\right)=\frac{z^{n}}{\sqrt{n!}}$ written with respect to the
phase space coordinate $z$ . The rigorous treatment of the holomorphic
function spaces described here was done by Segal and Bargmann in 60s. They
also provided a transform formula where one could obtain the coordinate
representation of wavefunctions from the monomials
$\psi_{n}\left(z\right)=\frac{z^{n}}{\sqrt{n!}}$ [21, 22]. The Bargmann
representation of some quantum mechanical system has been studied before such
as for Jaynes-Cummings model, thermal coherent states and WKB
approximation[23, 24, 25]. Very interestingly it was shown that Bargmann
representation gives better results in the case of WKB approximation comparing
with other known methods[25]. In [26], a generalization of the Segal-Bargmann
transform has been used in the context of canonical gravity and loop quantum
gravity.
For sake of simplicity we systematically discuss the theory of open quantum
systems in term of holomorphic functions. Moreover we solve the quantum
absorption refrigerators driven by a Gaussian noise as a model example in term
of analytical functions in the complex $z$-plane [27, 28]. Recently the
quantum absorption refrigerator was realized experimentally in the case of
trapped ions [29].

##  2 Bargmann representation of open quantum systems

Definition 1: The Bargmann space also known as the Fock-Bargmann or Segal-
Bargmann space, denoted by $\mathcal{H}L^{2}\left(\mathbb{C}^{n},\mu\right)$,
is the space of holomorphic functions with
$\mu\left(z\right)=(\pi)^{-n}e^{-|z|^{2}}$ and
$|z|^{2}=|z_{1}|^{2}+\dots+|z_{n}|^{2}$. Any entire analytic function
$f\left(z\right)$ in this space obeys the following square-integrability
condition [21, 22, 30, 31, 32, 33, 34]

$||f||^{2}:=\langle
f|f\rangle_{\mu}=\left(\pi\right)^{-n}\int_{\mathbb{C}^{n}}|f\left(z\right)|^{2}e^{-|z|^{2}}dz<\infty,$
(1)

where $dz$ is the $2n$-dimensional Lebesgue measure on $\mathbb{C}^{n}$.

Remark 1: The inner-product of any two analytic functions $f\left(z\right)$
and $g\left(z\right)$ satisfying the condition 1 is

$\langle
f|g\rangle_{\mu}=\left(\pi\right)^{-n}\int_{\mathbb{C}^{n}}\overline{f}\left(z\right)g\left(z\right)e^{-|z|^{2}}dz.$
(2)

Using the inner product defined in 2, we can prove that both $\overline{z}$
and $\frac{\partial}{\partial z}$ have the same effect i.e.
$\langle\frac{\partial f}{\partial z},g\rangle_{\mu}=\langle
f,zg\rangle_{\mu}$ [32].

Lemma 1: The Bargmann space $\mathcal{H}L^{2}\left(\mathbb{C}^{n},\mu\right)$
is a Hilbert space.
This observation was due to Bargmann in [21]. More generally, $L^{2}$ is the
only Hilbert space among the function spaces $L^{p}$[35].

According to Riesz representation theorem between any given Hilbert space and
its dual space there must exist a unique $f_{\alpha}$ such that

$f\left(\alpha\right)=\langle
f_{\alpha}\left(z\right)|f\left(z\right)\rangle_{\mu},$ (3)

where the function $f_{\alpha}$ is usually called a coherent state with
parameter $\alpha$ whereas the quantity $\overline{f_{\alpha}}$ is known as
the reproducing kernel [37, 38].

Lemma 2: The reproducing kernel in the Bargmann space is

$K\left(z,w\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}\frac{\overline{w}^{n}}{\sqrt{n!}}=\sum^{\infty}_{n=0}\frac{1}{n!}\left(z\overline{w}\right)^{n}=e^{z\overline{w}}.$
(4)

Lemma 3: The monomials $z^{n}/\sqrt{n!}$ or its alternate expression
$z^{n}/\sqrt{\Gamma\left(n+1\right)}$ form an orthonormal basis

$\int\frac{dz\;d\overline{z}}{\pi}\exp[-z\overline{z}]\;\overline{z}^{n}z^{m}=n!\;\delta_{mn}$
(5)

Thus the bosonic wave functions can be written as a uniformly convergent
series $f\left(z\right)=\sum_{n}c_{n}\frac{z^{n}}{\sqrt{n!}}$. The convergence
of this series in any compact domain of the complex $z$-plane is fixed by the
condition $\sum^{\infty}_{n}|c_{n}|^{2}=1$ [30, 36].

Lemma 4: In Bargmann representation, the raising and lowering operators can be
defined as $a^{\dagger}_{i}=z_{i}$ and $a_{i}=\frac{\partial}{\partial z_{i}}$
or $\overline{z}_{i}$.
This can be proven by straightforward calculations of the commutators,

$\displaystyle\left[\frac{\partial}{\partial z_{i}},z_{j}\right]=\delta_{ij},$
(6) $\displaystyle\left[\frac{\partial}{\partial
z_{i}},\frac{\partial}{\partial
z_{j}}\right]=\left[z_{i},z_{j}\right]=\left[\overline{z}_{i},\overline{z}_{j}\right]=0.$
(7)

With this construction we define the position and momentum operators in
natural units i.e. $\hbar=1$ as

$\displaystyle x_{j}=\frac{1}{2}\left(\frac{\partial}{\partial
z_{j}}+z_{j}\right),$ (8) $\displaystyle
p_{j}=\frac{1}{2i}\left(\frac{\partial}{\partial z_{j}}-z_{j}\right).$ (9)

Since the operators $x_{j}$ and $p_{j}$ satisfy the Weyl relations and act
irreducibly in the Bargmann space, one could in principle as consequence of
the Stone-von-Neumann theorem maps each position dependent quantity in the
Hilbert space $L^{2}\left(\mathbb{R}^{2n}\right)$ to its corresponding
holomorphic expression in the Bargmann space
$\mathcal{H}L^{2}\left(\mathbb{C}^{n},\mu\right)$[34]. This can be done in
virtue of the Segal-Bargmann transform given by [33, 34]

$\left(Af\right)\left(z\right)=\int_{\mathbb{R}^{n}}\exp\left[-\left(z\cdot
z-2\sqrt{2}z\cdot x+x\cdot x\right)\right]f\left(x\right)\;dx,$ (10)

As an example the harmonic oscillator Hamiltonian (up to a constant term)
$H_{0}=\hbar\omega a^{\dagger}a$ assumes the following form in the Bargmann
representation $H_{0}=\hbar\omega z\frac{d}{dz}$ [39]. Acting by $H_{0}$ to
the energy eigenstates we get ,

$H_{0}|n\rangle=\hbar\omega
z\frac{d}{dz}\frac{z^{n}}{\sqrt{n!}}=n\hbar\omega\frac{z^{n}}{\sqrt{n!}}=n\hbar\omega|n\rangle.$
(11)

Another interesting example from quantum optics is the cat-states which are
defined as the quantum superposition of two opposite-phase coherent states of
a single mode[4, 40, 41]. The even and odd cat states can be defined
respectively as [42]

$\displaystyle|\alpha_{+}\rangle=|\alpha\rangle+|-\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{2n}}{\sqrt{2n!}}|2n\rangle,$
(12)
$\displaystyle|\alpha_{-}\rangle=|\alpha\rangle-|-\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{2n+1}}{\sqrt{\left(2n+1\right)!}}|2n+1\rangle,$
(13)

where
$|\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}\left(a^{\dagger}\right)^{n}}{n!}|0\rangle$
in the Bosonic Fock number basis [41].

The even and odd cat states defined in 12 and 13 can be rewritten in a closed
analytic form using the Bargmann representation as

$\displaystyle|\alpha_{+}\rangle=|\alpha\rangle+|-\alpha\rangle=2e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\left(\alpha
z\right)^{2n}}{2n!}$ (14)
$\displaystyle=2e^{-\frac{1}{2}|\alpha|^{2}}\cosh\left(\alpha\right)z=\sqrt{2\pi\alpha
z}\;e^{-\frac{1}{2}|\alpha|^{2}}I_{-1/2}\left(\alpha z\right),$
$\displaystyle|\alpha_{-}\rangle=|\alpha\rangle-|-\alpha\rangle=2e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\left(\alpha
z\right)^{2n+1}}{\left(2n+1\right)!}$ (15)
$\displaystyle=2e^{-\frac{1}{2}|\alpha|^{2}}\sinh\left(\alpha
z\right)=\sqrt{2\pi\alpha z}\;e^{-\frac{1}{2}|\alpha|^{2}}I_{1/2}\left(\alpha
z\right),$

since

$|\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\left(\alpha
z\right)^{n}}{n!}=e^{-\frac{1}{2}|\alpha|^{2}}e^{\alpha z},$ (16)

and $I_{1/2,-1/2}$ are the modified Bessel functions. Interestingly the
equations 14 and 15 are written with respect to complex variables $\alpha$ and
$z$ . As far as we are aware, these equations has not been given explicitly
before in the literature . It would be interesting to see if they will
reproduce all physical results found previously for the even and odd cat
states analytically.

In case of fermions, we can construct analytical representation on the
fermionic phase space $\mathbb{R}^{2d}$ by realizing the raising and lowering
operators in term of the Grassmann variable $\theta$ as

$\displaystyle b_{j}=\frac{\partial}{\partial\theta_{j}},$ (17) $\displaystyle
b^{\dagger}_{j}=\theta_{j},$ (18)

where $\theta_{j}=\frac{1}{2}\left(\xi_{2j-1}-i\xi_{2j}\right)$,
$\overline{\theta}_{j}=\frac{1}{2}\left(\xi_{2j-1}+i\xi_{2j}\right)$ with
$j=1,\dots d$ , $\left[\xi_{i},\xi_{k}\right]=2i\epsilon_{ikl}\xi_{k}$ and
$\left\\{\xi_{i},\xi_{k}\right\\}=0$ for $i\neq k$ and 2 for $i=k$. It is
important to note that this specific construction is when the fermionic phase
space is for spin-1/2 particles or in general any two-level systems and this
explains the 2 in front of the commutation relation between any two different
components of the vector $\xi$. For higher spins the generalization is not
difficult and one only need to generalize the Grassmann algebra for higher
spins and write the exact commutation and anti-commutation relations for each
case . The fermionic phase space is even since the complex structure should be
preserved. With this construction, the anticommutation relations are

$\left\\{\theta_{i},\theta_{j}\right\\}=\left\\{\overline{\theta}_{i},\overline{\theta}_{j}\right\\}=0,\quad\left\\{\overline{\theta}_{i},\theta_{j}\right\\}=\delta_{ij}.$
(19)

The Gaussian integral over complex Grassmann (anti-commuting) variables is

$\int d\overline{\theta}d\theta\;e^{-\overline{\theta}b\theta}=b,$ (20)

From this result we find

$\int
d\overline{\theta}d\theta\;\theta\overline{\theta}\;e^{-\overline{\theta}b\theta}=1.$
(21)

The basic integral rules for a single Grassmann variable $\theta$ are given by
the Berezin integrals defined as [43, 44]

$\displaystyle\int\theta d\theta=1,\;\;\;\int d\theta=1.$ (22)

Applying Berezin integrals over the most general one-variable Grassmann
polynomial $f\left(\theta\right)=a\theta+b$ gives

$\int f\left(\theta\right)d\theta=\int\left(a\theta+b\right)d\theta=a,$ (23)

where $a$ and $b$ are elements from $\mathbb{C}$. The orthogonal basis for a
single fermion are simply ${\bf 1}=|0\rangle$ and $\theta=|1\rangle$ since
$\theta^{2}=0$.

The total fermionic number operator for many-body fermionic system is

$N_{F}=\sum_{i}\theta_{i}\;\overline{\theta}_{i},$ (24)

and satisfies $N^{2}_{F_{i}}=N_{F_{i}}$.
By direct calculation we find for each $i$

$N^{2}_{F_{i}}=\theta_{i}\overline{\theta}_{i}\theta_{i}\overline{\theta}_{i}=\theta_{i}\left(1-\theta_{i}\overline{\theta}_{i}\right)\overline{\theta_{i}}=\theta_{i}\overline{\theta}_{i}=N_{F_{i}}$
(25)

since according to 19, we have $\theta_{i}^{2}=\overline{\theta}_{i}^{2}=0$.

The total Hamiltonian should be written as a function of $z_{i}$, $\theta_{i}$
and their complex conjugates i.e.
$H\left(z_{i},\overline{z}_{i},\theta_{i},\overline{\theta}_{i}\right)$ in
general. Moreover the corresponding bosonic eigenstates are written as a
polynomial of complex variable $z_{i}$

$|n_{1}\dots,n_{k}\rangle=\prod_{i=1}^{i=k}\frac{z^{n_{k}}}{\sqrt{n_{k}!}},$
(26)

and the fermionic eigenstates for each single fermion are 1 for the ground
state and $\theta$ for the excited state.

As a model example we consider the Jaynes-Cummings model that describes the
interaction between two-level atom and quantized field in an optical cavity.
The Jaynes-Cummings Hamiltonian up to a constant term in operator language
reads [4, 45]

$\hat{H}_{\mathrm{JC}}=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{eg}|e\rangle\langle
e|+\hbar
g_{c}\left(\hat{\sigma}_{+}\hat{a}+\hat{\sigma}_{-}\hat{a}^{\dagger}\right),$
(27)

where $\omega_{c}$ is the cavity frequency and $\omega_{eg}$ is the resonance
frequency of the transition between atomic sub-levels.
$\hat{a},\hat{a}^{\dagger}$ are the bosonic raising and lowering operators of
the cavity, $\hat{\sigma}_{+}=|e\rangle\langle
g|,\hat{\sigma}_{-}=|g\rangle\langle e|$ are the raising and lowering
operators of the atom. Moreover we can define the atomic inversion operator as
$\hat{\sigma}_{z}=|e\rangle\langle e|-|g\rangle\langle g|$. We re-write the
analytical version of the Jaynes-Cummings Hamiltonian as

$H_{\mathrm{JC}}=\hbar\omega_{c}z\overline{z}+\hbar\omega_{eg}\theta\overline{\theta}+\hbar
g_{c}\left(\theta\overline{z}+\overline{\theta}z\right),$ (28)

or equivalently

$H_{\mathrm{JC}}=\hbar\omega_{c}z\frac{\partial}{\partial
z}+\hbar\omega_{eg}\theta\frac{\partial}{\partial\theta}+\hbar
g_{c}\left(\theta\frac{\partial}{\partial
z}+\frac{\partial}{\partial\theta}z\right),$ (29)

where we identified the excited and ground states of the atom as
$|e\rangle=\theta$ and $|g\rangle=1$. We many define the number operator as

$N=\theta\overline{\theta}+z\overline{z}=\theta\frac{\partial}{\partial\theta}+z\frac{\partial}{\partial
z}.$ (30)

The eigenstates of the number operator commutes with the atom-field
Hamiltonian $[N,H_{\mathrm{JC}}]=0$. Thus it can be used as a basis of the
tensor product states $|e,n\rangle,|g,n\rangle,|e,n-1\rangle\dots$. However in
the analytical formulation we express these states as a product of monomials
not as a tensor product of state vectors. For example, the matrix element

$\displaystyle\langle g,n|\hat{H}_{\mathrm{JC}}|e,n-1\rangle=\hbar
g_{c}\langle g,n|\hat{a}^{\dagger}\hat{\sigma}_{-}|e,n-1\rangle+\hbar
g_{c}\langle g,n|\hat{a}\hat{\sigma}_{+}|e,n-1\rangle=\hbar\sqrt{n}g_{c},$
(31)

can be calculated analytically as

$\displaystyle\langle g,n|H_{\mathrm{JC}}|e,n-1\rangle=\hbar g_{c}\langle
g,n|z\overline{\theta}|e,n-1\rangle+\hbar g_{c}\langle
g,n|\overline{z}\theta|e,n-1\rangle$ (32) $\displaystyle=\hbar g_{c}\langle
g|\overline{\theta}|e\rangle\langle n|z|n-1\rangle_{\mu}+\hbar g_{c}\langle
g|\theta|e\rangle\langle n,\overline{z}|n-1\rangle_{\mu}$ (33)
$\displaystyle=\hbar g_{c}\langle g|\overline{\theta}|e\rangle\langle
n|z|n-1\rangle_{\mu}=\hbar g_{c}\langle
g|\overline{\theta}|e\rangle\int\frac{dzd\overline{z}}{\pi}\frac{\overline{z}^{n}}{\sqrt{n!}}e^{-|z|^{2}}z\frac{z^{n-1}}{\sqrt{\left(n-1\right)!}}$
(34) $\displaystyle=\hbar
g_{c}\int\frac{dzd\overline{z}}{\pi}\frac{\overline{z}^{n}}{\sqrt{n!}}e^{-|z|^{2}}\frac{z^{n}}{\sqrt{\left(n-1\right)!}}$
(35) $\displaystyle=\sqrt{n}\hbar
g_{c}\int\frac{dzd\overline{z}}{\pi}\frac{\overline{z}^{n}}{\sqrt{n!}}e^{-|z|^{2}}\frac{z^{n}}{\sqrt{n}!}=\sqrt{n}\hbar
g_{c},$ (36)

since $|e\rangle=\theta$, thus $\theta|e\rangle=\theta^{2}=0$, also
$\frac{\partial}{\partial z}|n-1\rangle=\sqrt{n-1}|n-2\rangle$ thus the
quantity $\langle g,n|\overline{z}\theta|e,n-1\rangle$= $\langle
g,n|\theta\frac{\partial}{\partial z}|e,n-1\rangle=0$ while $\langle
g|\overline{\theta}|e\rangle=1$ since
$\overline{\theta}|e\rangle=\overline{\theta}\theta={\bf 1}=|g\rangle$ and
$\langle g|g\rangle=1$ in an orthonormal basis. The previous calculations show
that one can separate the cavity states from the atomic level states safely
and express them as a product of analytical quantities so no need to worry
about the matrix dimensionality of the constructed space from tensor product
of the cavity and atomic states and other related issues. Finally we can
express the eigenstates of Jaynes-Cummings Hamiltonian as

$\displaystyle|n,+\rangle=\cos\left(\frac{\phi_{n}}{2}\right)|e,n-1\rangle+\sin\left(\frac{\phi_{n}}{2}\right)|g,n\rangle$
(37)
$\displaystyle=\cos\left(\frac{\phi_{n}}{2}\right)\theta\frac{z^{n-1}}{\sqrt{\left(n-1\right)!}}+\sin\left(\frac{\phi_{n}}{2}\right){\bf
1}\frac{z^{n}}{\sqrt{n!}},$
$\displaystyle|n,-\rangle=\cos\left(\frac{\phi_{n}}{2}\right)|g,n\rangle-\sin\left(\frac{\phi_{n}}{2}\right)|e,n-1\rangle$
(38) $\displaystyle=\cos\left(\frac{\phi_{n}}{2}\right){\bf
1}\frac{z^{n}}{\sqrt{n!}}-\sin\left(\frac{\phi_{n}}{2}\right)\theta\frac{z^{n-1}}{\sqrt{\left(n-1\right)!}},$

where ${\bf 1}$ is the fermionic ground state and obeys the Grassmann algebra
rules.

For a given open quantum systems, we connect the system with one or more
reservoirs (baths). In the case of system coupled to a single bath we have the
Hamiltonian [15]

$H=H_{\mathrm{s}}+H_{\mathrm{b}}+H_{\mathrm{sb}},$ (39)

where $H_{\mathrm{s}}$ is the system Hamiltonian, $H_{\mathrm{b}}$ is the bath
Hamiltonian and $H_{\mathrm{sb}}$ is the system-bath interaction term. In the
Markovian approximation in which the time derivative of the operator depends
on the operator itself not its histories, the evolution of any observer $O$
such as the density matrix $\rho_{\mathrm{sb}}$ is given by the
Gorini–Kossakowski– Sudarshan–Lindblad (GKS-L) equation [46, 47, 48]

$\frac{d}{dt}O=-\frac{i}{\hbar}\left[H_{\mathrm{s}},O\right]+\mathcal{L}_{D}\left(O\right),$
(40)

where $\mathcal{L}_{D}$ is the dissipative part or dissipator given by [49]

$\mathcal{L}_{D}\left(O\right)=\sum_{n}\gamma_{n}\left(L_{n}O\overline{L}_{n}-\frac{1}{2}\left\\{\overline{L}_{n}L_{n},O\right\\}\right),\quad\gamma_{n}\geq
0,$ (41)

and $L_{n}$ are the Lindblad jump operators and $\overline{L}_{n}$ are their
complex conjugates.
Let $\\{\Lambda_{s}|s\geq 0\\}$ be a quantum dynamical semigroup and let
$\rho^{0}\in T\left(\mathcal{H}L^{2}\left(\mathbb{C}^{n},\mu\right)\right)$ be
an $\Lambda_{s}$-invariant state where
$T\left(\mathcal{H}L^{2}\left(\mathbb{C}^{n},\mu\right)\right)$ is the Banach
space of trace class operators, then the associated entropy production
$\sigma$ relative to $\rho^{0}$ is [50]

$\sigma(\rho)=-\frac{d}{dt}S\left(\Lambda_{s}\rho|\rho^{0}\right)|_{s=0}.$
(42)

As a final comment in this section, the holomorphic representation works also
in the non-Markovian approximation and in any quantum mechanical system
indeed.

##  3 Bargmann representation solution of quantum absorption refrigerator

The Hamiltonian of quantum absorption refrigerator is composed of three
interacting oscillators [12]

$\displaystyle H_{\mathrm{s}}=H_{0}+H_{\mathrm{int}},$ (43) $\displaystyle
H_{0}=\hbar\omega_{\mathrm{h}}z_{\mathrm{h}}\frac{\partial}{\partial
z_{\mathrm{h}}}+\hbar\omega_{\mathrm{c}}z_{c}\frac{\partial}{\partial
z_{c}}+\hbar\omega_{\mathrm{w}}z_{\mathrm{w}}\frac{\partial}{\partial
z_{\mathrm{w}}}=\sum_{\mathrm{i}}\hbar\omega_{\mathrm{i}}z_{\mathrm{i}}\frac{\partial}{\partial
z_{\mathrm{i}}},$ (44) $\displaystyle
H_{\mathrm{int}}=\hbar\omega_{\mathrm{int}}\left(z_{\mathrm{h}}\frac{\partial}{\partial
z_{\mathrm{c}}}\frac{\partial}{\partial
z_{\mathrm{w}}}+z_{\mathrm{c}}z_{\mathrm{w}}\frac{\partial}{\partial
z_{\mathrm{h}}}\right),$ (45)

where $\mathrm{i}=\mathrm{h},\mathrm{c},\mathrm{w}$ denotes the hot,cold and
work reservoirs. The unperturbed energy eigenstates for the quantum absorption
refrigerator in the Bargmann representation are
$|n_{\mathrm{h}},n_{\mathrm{c}},n_{\mathrm{w}}\rangle^{(0)}=\frac{1}{\sqrt{n_{\mathrm{h}}!n_{\mathrm{c}}!n_{\mathrm{w}}!}}z^{n_{\mathrm{h}}}_{\mathrm{h}}z^{n_{\mathrm{c}}}_{\mathrm{c}}z^{n_{\mathrm{w}}}_{\mathrm{w}}$.
Under steady-state conditions, the first and second laws of thermodynamics
should be satisfied, namely the following relations holds

$\displaystyle\mathcal{J}_{\mathrm{h}}+\mathcal{J}_{\mathrm{c}}+\mathcal{P}=0,$
(46)
$\displaystyle-\frac{\mathcal{J}_{\mathrm{h}}}{T_{\mathrm{h}}}-\frac{\mathcal{J}_{\mathrm{c}}}{T_{\mathrm{c}}}-\frac{\mathcal{P}}{T_{\mathrm{w}}}\geq
0,$ (47)

where the first relation represents the conservation of energy( first law of
thermodynamics) while the second inequality states that the sum of entropies
is equal or greater than 0(second law of thermodynamics, positive production
of entropy in the universe).
Quantum mechanically, we may re-write the Hamiltonian 43 as

$H=H_{0}+\lambda H_{\mathrm{int}}.$ (48)

Assuming the energy spectrum to be non-degenerate, we apply the holomorphic
perturbation theory where we expand the energy eigenstates in term of phase-
space coordinates $\left\\{z_{\mathrm{i}}\right\\}$ not
$\left\\{\psi(x_{\mathrm{i}})\right\\}$. The energy levels and eigenstates of
the perturbed Hamiltonian are given by time-independent Schrödinger equation

$(H_{0}+\lambda H_{\mathrm{int}})|n\rangle=E_{n}|n\rangle.$ (49)

If we assume the perturbation to be extremely small, we may expand the energy
levels and eigenstates as power series

$\displaystyle E_{n}|n\rangle=E^{(0)}_{n}+\lambda
E^{(1)}_{n}+\lambda^{2}E^{(2)}_{n}+\dots$ (50)
$\displaystyle|n\rangle=|n^{(0)}\rangle+\lambda|n^{(1)}\rangle+\lambda^{2}|n^{(2)}\rangle+\dots$
(51)

The first-order energy shift in the quantum absorption refrigerator is

$\displaystyle E^{(1)}_{n}=\langle
m^{(0)}|H_{\mathrm{int}}|n^{(0)}\rangle_{\mu}$ (52)
$\displaystyle=\hbar\omega_{\mathrm{int}}\big{(}\;\delta_{m_{\mathrm{h}},n_{\mathrm{h}+1}}\delta_{m_{\mathrm{c}},n_{\mathrm{c}-1}}\delta_{m_{\mathrm{w}},n_{\mathrm{w}-1}}\sqrt{(n_{\mathrm{h}}+1)n_{\mathrm{c}}n_{\mathrm{w}}}$
(53)
$\displaystyle+\delta_{m_{\mathrm{h}},n_{\mathrm{h}-1}}\delta_{m_{\mathrm{c}},n_{\mathrm{c}+1}}\delta_{m_{\mathrm{w}},n_{\mathrm{w}+1}}\sqrt{n_{\mathrm{h}}(n_{\mathrm{c}}+1)(n_{\mathrm{w}}+1)}\big{)}$

and the first-order energy eigenstates are

$\displaystyle|n^{1}\rangle=\sum_{m\neq n}\frac{\langle
m^{(0)}|H_{\mathrm{int}}|n^{(0)}\rangle_{\mu}}{E^{(0)}_{n}-E^{(0)}_{m}}|m^{(0)}\rangle$
(54)

where
$|m^{(0)}\rangle=\frac{1}{\sqrt{m_{\mathrm{h}}!m_{\mathrm{c}}!m_{\mathrm{w}}!}}z^{m_{\mathrm{h}}}_{\mathrm{h}}z^{m_{\mathrm{c}}}_{\mathrm{c}}z^{m_{\mathrm{w}}}_{\mathrm{w}}$
is the energy eigenstates of index
$m=(m_{\mathrm{h}},m_{\mathrm{c}},m_{\mathrm{w}})$ and
$E^{(0)}_{n}-E^{(0)}_{m}=\hbar\omega_{\mathrm{int}}(n-m)=\hbar\omega_{\mathrm{int}}(n_{\mathrm{h}}+n_{\mathrm{c}}+n_{\mathrm{w}}-m_{h}-m_{\mathrm{c}}-m_{\mathrm{w}})$.
By plugging the expressions of $\langle
m^{(0)}|H_{\mathrm{int}}|n^{(0)}\rangle_{\mu}$ and $E^{(0)}_{n}-E^{(0)}_{m}$
in 54 , we found the first-order correction of the energy eigenstates to be
independent of $\hbar\omega_{\mathrm{int}}$.

In the absorption refrigerator , the noise substitutes the work bath and its
contact leading to the following interaction term

$H_{\mathrm{int}}=g(t)\left(z_{\mathrm{h}}\frac{\partial}{\partial
z_{\mathrm{c}}}+z_{\mathrm{c}}\frac{\partial}{\partial
z_{\mathrm{h}}}\right)=g(t)X_{1},$ (55)

where $g(t)$ is the stochastic noise field with zero mean $\langle
g(t)\rangle=0$ and delta time correlation $\langle
g(t)g(t^{\prime})\rangle=2\eta\;\delta\left(t-t^{\prime}\right)$ and $X_{1}$
is the generator of swap operation between the two oscillators and belongs to
the $\mathrm{SU(2)}$ Lie algebra together with the following operators

$\displaystyle X_{2}=i\left(z_{\mathrm{h}}\frac{\partial}{\partial
z_{\mathrm{c}}}-z_{\mathrm{c}}\frac{\partial}{\partial
z_{\mathrm{h}}}\right),$ (56) $\displaystyle
X_{3}=\left(z_{\mathrm{h}}\frac{\partial}{\partial
z_{\mathrm{h}}}-z_{\mathrm{c}}\frac{\partial}{\partial
z_{\mathrm{c}}}\right),$ (57) $\displaystyle
N=\left(z_{\mathrm{h}}\frac{\partial}{\partial
z_{\mathrm{h}}}+z_{\mathrm{c}}\frac{\partial}{\partial
z_{\mathrm{c}}}\right),$ (58)

forming a closed set of equations. The Heisenberg equation for an arbitrary
time-independent analytic function
$O(z_{\mathrm{i}},\overline{z}_{\mathrm{i}})$ reads

$\frac{d}{dt}O=-\frac{i}{\hbar}[H_{\mathrm{s}},O]+\mathcal{L}_{\mathrm{n}}(O)+\mathcal{L}_{\mathrm{h}}(O)+\mathcal{L}_{\mathrm{c}}(O),$
(59)

Where the noise dissipator for Gaussian noise is
$\mathcal{L}_{\mathrm{n}}(O)=-\eta[X_{1},[X_{1},O]]$ [27]. For simplicity we
assume the heat baths to be uncorrelated between themselves and also
uncorrelated with the driving noise. In the Bargmann representation, the hot
and cold dissipators are

$\displaystyle\mathcal{L}_{\mathrm{h}}(O)=\gamma_{\mathrm{h}}\left(\overline{n}_{\mathrm{h}}+1\right)\;\left(z_{\mathrm{h}}O\overline{z}_{\mathrm{h}}-\frac{1}{2}\\{z_{\mathrm{h}}\overline{z}_{\mathrm{h}},O\\}\right)$
(60)
$\displaystyle+\gamma_{\mathrm{h}}\overline{n}_{\mathrm{h}}\;\left(\overline{z}_{\mathrm{h}}Oz_{\mathrm{h}}-\frac{1}{2}\\{\overline{z}_{\mathrm{h}}z_{\mathrm{h}},O\\}\right),$
$\displaystyle\mathcal{L}_{\mathrm{c}}\left(O\right)=\gamma_{\mathrm{c}}\left(\overline{n}_{\mathrm{c}}+1\right)\;\left(z_{\mathrm{c}}O\overline{z}_{\mathrm{c}}-\frac{1}{2}\\{z_{\mathrm{c}}\overline{z}_{\mathrm{c}},O\\}\right)$
(61)
$\displaystyle+\gamma_{\mathrm{c}}\overline{n}_{\mathrm{c}}\;\left(\overline{z}_{\mathrm{c}}Oz_{\mathrm{c}}-\frac{1}{2}\\{\overline{z}_{\mathrm{c}}z_{\mathrm{c}},O\\}\right),$

For a vanishing stochastic driving field, these equations guide separately the
oscillators of hot and cold baths to thermal equilibrium provided that
$\overline{n}_{i}=[e^{-\hbar\omega_{\mathrm{i}}/k_{\mathrm{B}}T_{\mathrm{i}}}-1]^{-1}$
where $\mathrm{i}=\mathrm{h},\mathrm{c}$. To obtain the cooling current
$\mathcal{J}_{\mathrm{c}}=\langle\mathcal{L}_{\mathrm{c}}(\hbar\omega_{c}\;z_{c}\frac{\partial}{\partial
z_{c}})\rangle$, we search for the stationary solutions of $X_{3}$ and $N$ we
find

$\mathcal{J}_{\mathrm{c}}=\hbar\omega_{\mathrm{c}}\frac{\overline{n}_{\mathrm{c}}-\overline{n}_{\mathrm{h}}}{(2\eta)^{-1}+\gamma_{\mathrm{c}}^{-1}+\gamma_{\mathrm{h}}^{-1}}.$
(62)

Obviously cooling occurs when
$\overline{n}_{\mathrm{c}}-\overline{n}_{\mathrm{h}}>0$. The coefficient of
performance $\left(\mathrm{COP}\right)$ is

$\mathrm{COP}=\frac{\mathcal{J}_{\mathrm{c}}}{\mathcal{J}_{\mathrm{n}}}=\frac{\omega_{\mathrm{c}}}{\omega_{\mathrm{h}}-\omega_{\mathrm{c}}}\leq\frac{T_{\mathrm{c}}}{T_{\mathrm{c}}-T_{\mathrm{h}}}.$
(63)

In this section, we have considered Levy and Kosloff model driven by Gaussian
noise for the quantum refrigerator, however it is possible to consider the
Poisson noise case [27]or more enhanced models presented in [51].

##  4 The computational complexity

Bargmann representation of quantum absorption refrigerators and in principle
any quantum heat engine has some advantages comparing with the standard
treatment based on the operators $a$ and $a^{\dagger}$ that act in Hilbert
space $L^{2}(\mathbb{R}^{d})$. To put this into context we compute the
computational complexity of both pictures for a specific example. By
definition, the computational complexity is the time consumed by a multitape
Turing machine in performing computational tasks[52]. The coordinate
representation of any wavefunction in the bosonic hot or cold reservoir is
given by a Hermite polynomial of degree $n$ multiplied by exponential function
of Gaussian signature up to some numerical constants proportional with the
ground-state length scale $\xi=\sqrt{\hbar/2m\omega}$. However in Bargmann
representation the wavefunctions of the bosonic hot or cold reservoirs are
simply monomials of $z$ with power $n$. As an example consider the ground-
state of one specific state in hot or cold reservoir, it has simply the
formula $\psi_{0}(x)=(\frac{m\omega}{\pi\hbar})^{1/4}e^{-m\omega
x^{2}/2\hbar}$ with computational cost of order $O(N^{5/2})$ using schoolbook
multiplication algorithms and of order $O(N^{2.085})$ using Karatsuba
multiplication algorithm where $N$ denotes the input’s digits number. This
complexity is simplification of the problem since we considered the quantities
such as $m,\omega,\hbar,x^{2}$ and other possible combinations to be given
previous to calculations and thus have complexity of order $O(1)$ [52].
However it is very large comparing with the computational cost of the ground-
state in Bargmann representation which is of order $O(1)$. The situation
becomes very drastically complicated considering higher excited states.

Another interesting example which shows the advantage of the developed
formalism in this work appears in the calculations of quasiprobability
distributions in phase space [4, 54]. More concretely, the Husimi-Kano $Q$
representation has a simpler form in the Bargmann representation comparing
with its form in the phase space $\mathbb{R}^{d}$, and can be written formally
in natural units for normalized state $||\psi||=1$ as
$H_{\psi}=(2\pi)^{-d}|A\psi|^{2}(\frac{x-ip}{\sqrt{2}})e^{-|z|^{2}/2}$ [34,
53]. This simplification is mainly because instead of representing $Q$ in term
of the conjugate coordinates $X$ and $P$ in the phase space , we simply unify
the treatment using the phase space coordinate $z$ only. Thus Bargmann space
can be regarded as a natural home for the Husimi-Kano $Q$ representation and
other quasi-probability distributions in the phase space[54]. In the context
of heat engines and refrigerators, the Husimi-Kano $Q$-representation has been
used in the calculation of the quantum synchronization since the
synchronization measure $S$ is the integral of $Q$ up to a numerical
constant[55, 56, 57]. Thus it is legitimate to claim that quantum
synchronization formalism simplifies using Bargmann representation and this in
turn reduces the computational complexity of the problem.

##  5 Conclusion

In this work, we discussed the analytical theory of open quantum systems using
Bargmann representation of the bosonic raising and lowering operators
systematically. We also provided similar procedure for fermions in term of the
anti-commuting Grassmann variables $\left\\{\theta_{i}\right\\}$ and their
partial derivatives. This construction is useful in two counts first it allows
us to exploit the whole theory of analytical functions and all its techniques
throughout the computation of open system characterizations such as the
dissipators. Moreover it appears to be conceptually easier to understand than
the standard canonical approach based on the raising and lowering operators
$a$ and $a^{\dagger}$ especially for the bosonic case where there is no upper
bound on the number of excited states a particle can take.
More precisely, we have considered the quantum absorption refrigerator driven
by a Gaussian noise as a model example. However, the holomorphic
representation is applicable in all heat engines and refrigerators. We
discussed the computational complexity, the time required by a multitape
Turing machine to perform specific tasks, associated with both standard and
Bargmann representation for excitations in bosonic heat baths. We found that
working within Bargmann representation has less computational complexity
comparing with the standard algebraic or analytical methods in coordinate
representation. Another advantage comes from the fact that in holomorphic
picture, we use a phase coordinate $z$ instead of the canonical variables $X$
and $P$. This fact simplifies the computation of quasiprobability
distributions defined normally in phase space such as the Husimi-Kano $Q$
distributions and this might have impact on the numerical studies of quantum
synchronization in heat engines and in principle for any quantum system.

Acknowledgment
We are grateful to USIM for support. MRW acknowledges the support of IIUM
research grant IRF19-037-0037.

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# Structural-modulation-driven Curie-temperature enhancement in Cr-doped
SrRuO3

Pooja<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Kanpur, Kanpur 208016, India  Bikash Saha  Solid State Physics Division,
Bhabha Atomic Research Centre, Mumbai, 400085 India.  Nitesh Choudhary
Department of Polymer and Process Engineering, Indian Institute of Technology
Roorkee, Saharanpur Campus, 247001, U.P., India  Pradip K.Maji  Department of
Polymer and Process Engineering, Indian Institute of Technology Roorkee,
Saharanpur Campus, 247001, U.P., India  A. K. Bera  Solid State Physics
Division, Bhabha Atomic Research Centre, Mumbai, 400085 India.  S. M. Yusuf
Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai, 400085
India.  Chanchal Sow<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Kanpur, Kanpur 208016, India

###### Abstract

Strongly correlated system with competing ground states are often poised close
to the quantum critical point. External perturbations such as pressure,
strain, electric field, and chemical doping can stabilise its ground state
with exotic physical properties. Cr-doping is the lone exception which
enhances the Curie-temperature in one of such correlated system SrRuO3. To
find the origin of $T_{C}$ enhancement, we investigate temperature-dependent
structure, spectroscopic, magnetic and magnetotransport properties in
SrRu1-xCrxO3. Cr-doping squeezes the unit cell volume which effectively
enhances the stretching octahedral distortion by nearly five times than pure
SrRuO3. The Curie temperature increment by $\sim$ 22 K for x = 0.15 is found
to be intertwined with the structural-modulation. Temperature-dependent
Neutron diffraction analysis indicate that the unit cell volume minima
coincide exactly with the enhanced ferromagnetic ordering ($\sim$ 190 K).
Further analysis reveals that the effect of Cr-doping not only freezes the
octahedral tilt below 100 K but also suppresses the complex magnetism
responsible for exchange bias and topological hall effect in SrRuO3. The
spectroscopic measurements find a reduction of itinerancy of d-electrons with
Cr-doping. The magnetotransport measurements portray an evolution from
itinerant to localised ferromagnetism.

##  I Introduction

Figure 1: (color online) Schematic representation of 10$\times$10$\times$1
cell of SrRu1-xCrxO3. Here only the Ru (light blue balls) and oxygen (light
red balls) atoms are shown for clarity. The Cr atoms (shown in red and black
balls) are randomly distributed in the Ru matrix. The magnetic moments of Ru4+
(S=1) and Cr3+ (S=3/2) are shown with blue and red arrows respectively, and
are ferromagnetically aligned along the b-axis. Cr6+ (S=0) are shown as black
balls that are non-magnetic and act as a disorder. In such a system, there are
two dominant exchanges: the blue encircled region has pure Ru-Ru FM exchange,
and the Red region has Ru-Cr FM exchange. The grey region (contains the non-
magnetic Cr) is mainly involved in reducing the range of exchange interaction
whereas the red region enhances the strength of the exchange interaction than
SRO.

Quantum materials with tunable multi-functionalities is an attractive area of
research due to its intriguing magnetic, electronic and optical
characteristics [1, 2, 3]. Such diverse physical properties bring vast
possibilities for applications in microelectronic, magnetic and spintronic
devices [4]. Tuning the relative strength of spin-orbit coupling ($\lambda$)
and on-site Coulomb interaction ($U$) from low to high, various quantum phases
have emerged [5]. In particular, ruthenates with moderate strength of
$\lambda$/$U$ become interesting candidates as they lie near the crossover
region. The layered Ruthenates, especially in Ruddlesden-Popper (RP) series:
(Sr/Ca)1+nRunO3n+1, where $n$ (1 $<$ $n$ $<$ $\infty$) is the number of Ru-O
layers, exhibit a drastic change of physical properties with a change in $n$.
In the Ca-based RP series, a robust antiferromagnetic (AFM) insulating state
(Mott Insulator [6]) is stabilized for $n$ = 1, which gradually decreases with
increasing $n$. Thus the magnetic state of metallic CaRuO3 (end member) is
critically poised among AFM, ferromagnetic (FM) and paramagnetic (PM) [7, 8].
Sr2RuO4 ($n$ = 1) is a nonmagnetic metal down to 2 K and becomes a $p$-type
superconductor below 1.5 K [9]. For $n$ $>$ 2, the ferromagnetic metallic
state is stabilized in Sr-based RP series and the curie temperature is
enhanced with $n$. The end member SrRuO3 ($n$ = $\infty$) is an itinerant
ferromagnet below $T_{C}$ $\sim$ 166 K [7]. SrRuO3 identifies with distorted
orthorhombic ($Pnma$) perovskite structure with sizeable $U$ (= 3-3.5 eV [10])
and $\lambda$ (= 40-80 meV [11]). In addition, SrRuO3 also possesses strong
hybridization between Ru-4$d$ and O-2$p$ orbitals. These features facilitate a
diverse range of intriguing physical characteristics in SrRuO3 [12, 13, 14,
15, 16]. The recent finding of topological Hall effect and magnetic skyrmions
in SrRuO3 thin-films arising due to the interfacial strain has gained
considerable interest [17, 18, 19, 20].

In bulk systems the external perturbations such as pressure, temperature,
internal strain by chemical doping, and electric field change the physical
properties drastically. For example, the crystal structure undergoes
perovskite ($Pbnm$) to post-perovskite ($Cmcm$) structure accompanied by a 1.9
% volume collapse with application of high pressure ( $\sim$ 40 GPa ) [21],
which also alters the magnetism by decreasing the Ru-O bond lengths [22]. On
the other hand, temperature also drives two structural phase transitions from
orthorhombic to tetragonal (at 800 K) and from tetragonal to cubic (at 975 K)
[23]. Chemical doping provides an additional method for tuning the magnetic
exchange by internally deforming the structure. The size of the cation can be
chosen higher or lower than the Ru depending on the requirement of unit cell
volume expansion or contraction [24]. This also affects the orthorhombicity
and distortion in RuO6 octahedra thereby a significant change in the magnetism
[25, 26, 27]. In particular SrRu1-xMxO3, where M (= Zn2+, Mn3+, Co2+, Ni2+ and
Ti4+) is transition metal ions (TMI) suppresses the $T_{C}$ down to 100 K [28,
29]. Doping Cr is an exceptional case (among TMI) that enhances the $T_{C}$ of
SrRuO3 up to $\sim$ 190 K [30, 31]. The enhancement of $T_{C}$ ($\sim$ 210 K)
is also noticed in Pb-doped SrRuO3, the origin is found to be structural
distortion [31]. However, the origin of increased $T_{C}$ with Cr-doping in
SrRuO3 is still not clear. A schematic picture of magnetic moments with
possible exchange interactions in SrRu1-xCrxO3 is depicted in Fig-1.
Experimental reports suggest the mixed valence state of ruthenium, which
creates Ru4+/5+-O-Cr4+/3+ minority double exchange interaction and enhances
the ordering temperature [30, 32, 33]. While the First-principles calculation
(using LSDA, LSDA+U and GGA+U) explain the origin of the increased TC in
SrRu1-xCrxO3 system in terms of Cr4+ \- Ru4+ hybridization [34] and p-d
coupling [35].

In order to find a clear picture of such TC enhancement, we present a detailed
and systematic structural, spectroscopic, magnetic, transport, and microscopic
magnetic state by Neutron diffraction studies on single phase SrRu1-xCrxO3 (0
$<$ x $<$ 0.15) samples. Our study established a strong connection between
structural modulation with the increased $T_{C}$ in Cr-doped SrRuO3.

##  II Sample preparation and experimental details-

High quality, SrRu1-xCrxO3 (0 $<$ x $<$ 0.15) (SRCO) samples are synthesized
via solid-state reaction method. Powders of SrCO3 ($>$ 99.95$\%$), RuO2 ($>$
99.9$\%$), and Cr2O3 ($>$ 99.9%) are used as ingredients. The mixture is
pelletized and sintered for 48 hours at 900 - 1400 ∘C with multiple
intermediate grindings. RuO2 is volatile as it sublimates at 1200∘C. To
compensate, the extra 5 weight % of RuO2 is added to preserve the
stoichiometric ratio. The atomic percentage of Ru and Cr is examined by the
electron probe microscopy analyzer (EPMA) equipped with JXA-8230; and JEOL (as
tabulated in Table- 1). The Phase purity and crystallinity of all samples are
checked by using a high-resolution X-ray powder diffractometer (PAN analytical
Empyrean Cu K$\alpha$).

Table 1: EPMA results: Atomic percentage of Ru% and Cr% with the nominal ( xnom) and determined (xdet) value of x in SrRu1-xCrxO3 samples. Sample | xnom | xdet | Ru% | Cr%
---|---|---|---|---
SRO | 0 | 0 | 102.0 | 0
SRCO5 | 0.05 | 0.07 | 87.9 | 7.06
SRCO10 | 0.10 | 0.11 | 90.32 | 11.51
SRCO15 | 0.15 | 0.15 | 86.52 | 15.86

The valence states of Cr, Ru, and Sr are verified by PHI Versa Probe-II X-ray
photoelectron spectroscopy (XPS) consisting of a monochromatic Al K$\alpha$
(h$\nu$ = 1486.6 eV) source. The magnetic and transport measurements are
conducted between 2-300 K temperature and 0-6 T of applied magnetic field
range using commercial quantum design MPMS (MPMS-XL) and PPMS respectively.
Temperature-dependent (5-300 K) Neutron diffraction (ND) measurements are
carried out on a high-quality SRCO15 polycrystalline sample using a powder
neutron diffractometer, PD-I, at Dhruva Research Reactor, Mumbai (India). The
powder sample is filled in a cylindrical vanadium can and attached to the cold
finger of the He-4 gas-based closed-cycle refrigerator.

##  III Calculation details-

The spin polarised density of states (DOSs) calculation is carried out on
SrRuO3 by using generalised gradient approximation (GGA) and GGA+U method with
Quantum Espresso [36]. The Hubbard U term which is the Coulomb repulsion
between on-site electrons, is considered to understand the effect of electron
correlation on DOSs. We considered the conventional cell (Z = 4) of SrRuO3
consisting of 20 atoms with lattice parameters obtained from Rietveld-refined
XRD-data for SRO and SRCO15 (without any structural relaxation calculation).
The calculation is performed with 4$\times$4$\times$4 K-points, by using
Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional [37]. The ultra
soft pseudo potentials (USPP), and a basis set for valence electrons
consisting of plane waves with cut-off energy 25 Rydberg (340 eV) are used for
self-consistent calculation. To understand the electron correlation effect on
DOS the value of $U$ is taken 3.5 eV [35] for Ru atom.

##  IV Results and discussion-

###  IV.1 Structural analysis-

Figure 2: (color online) Rietveld refined X-Ray diffraction pattern of (a) SRO
and (b) SRCO15. The open black circles and red solid lines are observed and
calculated patterns respectively and the blue line shows the difference
between observed and calculated patterns, whereas green vertical bars are
allowed Bragg peak positions. (c) The orthorhombic crystal structure of SRCO15
along with (Ru/Cr)O6 octahedral arrangement. The crystal structure is drawn
using VESTA visualizing program [38], from the output file after rietveld
refinements. Schematics of (Ru/Cr)O6 octahedra to demonstrate the octahedral
tilt and various oxygen atoms in the (d) $bc$-plane and (e) $ac$-plane. The
black solid line represents the unit cell. Variation of (f) lattice constants,
(g) unit cell volume (left y-axis) and $\Delta$ (right-y axis) is shown upon
Cr doping.

Figure-2(a) and (b) displays the experimentally measured and calculated
Rietveld-refined (using FULLPROF [39]) X-ray diffraction patterns of the SRO
and SRCO15 respectively. Rietveld analysis reveals that SRCO samples have an
orthorhombic crystal symmetry ($a$ $\neq$ $b$ $\neq$ $c$, $\alpha$ = $\beta$ =
$\gamma$ = 90) with space group $Pnma$ (62) [32, 40]. The Cr atom occupies the
same Wyckoff-4$a$ site as the Ru atom. The schematic crystal structure is
shown in Fig-2(c). The crystal structure consists of corner-sharing identical
(Ru/Cr)O6 octahedra with tilt along all three directions as shown in Fig-2(d,
e). Fig-2(f) shows the variation of lattice constant with respect to Cr-
content. At 15% Cr-doping the lattice constants $a$, $b$, and $c$ decrease by
$\sim$ 0.13%, 0.24%, and 0.22% respectively. As a result, the overall $\sim$
0.54% decrease in unit cell volume is shown in Fig-2(g). The decrease in the
unit cell volume is attributed to the smaller cationic radius of Cr6+(0.044
nm)or Cr3+(0.061 nm) (confirmed by XPS) compared to Ru4+(0.062 nm). The
refined crystal structure reveals three distinct octahedral bond lengths
namely Ru-O11 (basal), Ru-O12 (basal), and Ru-O2 (apical) with values 1.976,
2.0596 $\AA$ and 1.929 respectively in case of SRCO15. The unequal bond
lengths give rise to the octahedral distortion [41]. The octahedral stretching
distortions ($\Delta$) and angular distortions ($\Sigma$) can be calculated
using the equations:

$\Delta=\frac{1}{6}\Sigma^{6}_{i=1}\Big{[}\frac{|d_{i}-d_{mean}|}{d_{mean}}\Big{]}^{2}$
(1) $\Sigma=\frac{1}{12}\Sigma^{12}_{i=1}|\phi_{i}-90|$ (2)

respectively, where $d_{mean}$ and $d_{i}$ denote the average and $i^{th}$
Ru/Cr-O bond length (there is a total of 6 different Ru-O lengths in a RuO6
octahedra) respectively. The angle $\phi_{i}$ is considered as $i^{th}$
O-(Ru/Cr)-O bond angles in the (Ru/Cr)O6 octahedra. The stretching distortion
increases as we increase the Cr-doping in SRO. At 15% Cr-doping, the value of
$\Sigma$ is increased by 22%, whereas $\Delta$ is enhanced by nearly 5 times
compared to parent compound SRO. Such increased distortion due to Cr-doping
affects the superexchange/double-exchange interaction between Ru moments. In
this regard, theoretical work by L. Wang et al. suggests for low Cr-doped
samples (x$<$ 0.25), Cr3+ couples ferromagnetically with Ru4+ moments which is
responsible for enhancement of the $T_{C}$ [35]. The calculated value of
$\Sigma$, $\Delta$ and derived lattice parameters of SRCO15 in comparison with
SRO at room temperature are tabulated in Table-2.

Table 2: Rietveld-refinement structural parameters of SRO and SRCO15 in comparison with the room-temperature ND results. Sample | a | b | c | V | Ru-O1-Ru | Ru-O2-Ru | $\Sigma$ | $\Delta$
---|---|---|---|---|---|---|---|---
| ($\AA$) | ($\AA$) | ($\AA$) | ($\AA^{3}$) | (∘) | (∘) |  |
SRO | 5.529 | 7.845 | 5.569 | 241.5 | 163.9 | 159.1 | 4.5 | 1.6$\times 10^{-4}$
SRCO15 | 5.522 | 7.826 | 5.557 | 240.2 | 163.5 | 157.9 | 5.5 | 9.1$\times 10^{-4}$

###  IV.2 X-ray photoelectrons spectroscopy-

Table 3: The Binding energy (eV) values of different constituent elements in SRO and SRCO15. Sample | Ru3$d_{5/2}$ | Ru3$d_{3/2}$ | Sr3$p_{1/2}$ | Cr2$p_{3/2}$ | Cr2$p_{1/2}$ | Sr3$d_{5/2}$ | Sr3$d_{3/2}$
---|---|---|---|---|---|---|---
| (eV) | (eV) | (eV) | (eV) | (eV) | (eV) | (eV)
SRO | 280.75 | 285.95 | 278.0 | - | - | 132.21 | 135.21
| 282.05 | 288.55 | - | - | - | 133.76 | 136.34
SRCO15 | 281.35 | 286.05 | 278.75 | 577.86 (Cr3+ | 587.92(Cr3+) | 132.39 | 134.94
| 282.6 | 288.5 | - | 581.54(Cr6+) | 590.69(Cr6+) | 133.64 | 136.34

The X-ray photoelectron spectroscopy (XPS) is employed to investigate the
valence states of various constituent elements and the density of state (DOS)
in the vicinity of Fermi energy (EF). The XPS survey scan identifies the
presence of Sr, Ru, Cr, and O in SRCO samples (not shown here). Fig-3(a)
depicts the Ru-3$d$ XPS core spectra of SRO and SRCO15 (normalised the
intensity by removing the Shirley background after fitting). The peak at
$\sim$ 278 eV is agreed as Sr3$p_{1/2}$ and the other two peaks in the Ru-3d
spectra corresponds to spin-orbit doublet (SOD): Ru 3$d_{5/2}$ and Ru
3$d_{3/2}$ with splitting energy $\Delta$E${}_{SO}^{Ru}$ $\sim$ 5 eV,
suggesting Ru4+ valence state [42] in all samples. Each Ru doublet has two
peaks labelled as screened ($s$) and unscreened ($u$). The $s$ peak is
generally present at $\sim$ 2 eV lower binding energy (B.E.) than $u$ peak, in
metallic systems due to the screening effect of core electrons [43]. The
presence of both peaks also indicates that the samples are metallic at room
temperature. It is important to note that in the case of SRCO15 the Ru core
spectra are shifted towards the higher binding energy. The shift is visible in
the zoomed-in-view of Ru 3$d_{5/2}$ peak in the inset of Fig-3(a). This
spectral shift arises due to the reduction of delocalized electrons [44] in
the system with increasing disorder via Cr-doping. The Sr-3d spectra (not
shown) have fitted with two SOD peaks Sr3d5/2 and Sr3d3/2 confirming the
presence of Sr2+ state. Fig-3(b) shows the Cr-2p core spectra of SRCO15. The
Cr SOD peaks labelled as Cr 2p1/2 and Cr 2p3/2 are separated by
$\Delta$E${}_{SO}^{Cr}$ $\sim$ 9.2 eV. These doublets are further split into
two peaks corresponding to the two different oxidation states as
Cr${}^{3+}3d^{3}_{t_{2g}}$ and Cr${}^{6+}3d^{0}_{t_{2g}}$ (splitting energy
$\sim$ 3.68 eV). The peak intensity ratio of Cr6+ and Cr3+ peaks is found to
be $\sim$ 2:1 (Fig-3(b)).

Figure 3: (color online) Core level XPS-spectrum of (a) Ru-3d in SRO and
SRCO15, (b) Cr-2p in SRCO15, where the open circle shows the raw data and the
solid lines are peak sum, and different coloured peaks represent the fittings
(using XPSPEAK-4.1 [45] ). All fitting peaks are labelled according to their
B.E. positions in all curves. (c) XPS valence band spectrum of SRO and SRCO15.
The inset figure in a, and c represent the zoomed-in-view of spectra. (d)
Theoretically calculated TDOS spectra for SRO with U = 0 (gray), U = 3.5 eV
(blue-line), and SRCO15 with U = 0 (red)

The overall valence state of Cr is estimated by the relation:
Cr${}^{6+}_{x-y}$Cr${}^{3+}_{y}$ = Cr${}^{4+}_{x}$. It maintains an overall 4+
oxidation state at the Ru site. For SRCO15, y $\sim$ 0.045, suggesting one-
third of Cr is in the 3+ state and two-thirds is in the 6+ oxidation state.
Thus overall charge neutrality equation for SRCO15:
Sr2+Ru${}^{4+}_{0.85}$Cr${}^{4.5+}_{0.15}$O${}^{2-}_{3+\delta}$, where
$\delta$ $\sim$ 0.04 (calculated from average charge valency of Oxygen [46]).
The B.E. values of Ru, Cr, and Sr of SRO and SRCO15 are tabulated in Table-3.
Fig-3(c) shows the comparative valence band spectra of SRO and SRCO15. The
peak centred $\sim$ 1.2 eV and 6.2 eV correspond to the Ru-4d/Cr-3d and O-2p
electrons respectively, indicating the strong hybridization between Ru/Cr and
oxygen [47]. The inset Fig-3(c) shows a zoomed-in-view of spectral region at
$\sim$ 1.2 eV, which is marked as feature-A (red) and B (green) ascribed as
coherent (delocalized) and incoherent (localized) electrons respectively. In
SRCO15 the spectral weight of feature-A near $E_{F}$ is shifted to feature-B,
suggesting the decrement of a delocalized electron with increasing Cr-doping
in SRO. Such shift of the delocalized electron mainly arises due to electron
correlation [48]. To understand the effect of electron correlation and
structure distortion in the DOS with Cr-doping, we theoretically calculated
the DOS. The total density of state (TDOS) for SrRuO3 is calculated with $U$ =
0, $U$ = 3.5 eV and with modified lattice parameters at 15% Cr-doping (with U
= 0) as shown in Fig-3(d). The calculated TDOS shows half- metallic state and
the bands between 2.5 eV to -9 eV energy containing Ru-4d, O-2p electronic
contributions consisting with the earlier reported results [35, 34]. The up-
spin TDOS spectra resemble the experimentally obtained valence band spectra.
It is crucial to note that in both cases: with finite $U$ and modified lattice
parameter (for SRCO15), the delocalised electron near the $E_{F}$ is decreased
and shifted towards the localised state. Thus the Cr-doping modifies the
crystal structure and induces the electron correlation in the system hence
reducing the number of itinerant electrons. This localised behaviour of charge
carriers is further understood through the magnetic and transport properties
of SRCO.

Table 4:  Magnetic parameters of SRO and SRCO15. Parametrs/ | $M_{5K}$ | $T_{C}$ | $H_{c}$ | $M_{s}$ | $A_{SW}$ | $B_{SE}$ | J$\times$kB | $T_{C}^{cal}$
---|---|---|---|---|---|---|---|---
Samples | (emu/g) | (K) | (kOe) | ($\mu_{B}$/f.u.) | ($K^{-3/2}$) | (K-2) |  | (K)
SRO | 17.6 | 166.2 | 3.30 | 1.42 | 1.33$\times$10-4 | 2.3$\times$10-6 | 28.98 | 158.6
SRCO15 | 13.4 | 188.5 | 2.88 | 1.05 | 1.22$\times$10-4 | 2.04$\times$10-6 | 30.7 | 167.9

###  IV.3 Magnetic measurements-

The structural distortion and the mixed valence state of Cr ion certainly
change the magnetic exchange interaction in SRCO. Fig-4(a) and (b) shows the
temperature-dependent FC-ZFC magnetization data of SRO and SRCO15
respectively. Both exhibit FM behaviour. The FM Curie-temperature ($T_{C}$) is
calculated by fitting the FC magnetisation data with the scaling law: $M$
$\approx$ ($T_{C}$ \- $T$)β, in the vicinity of the critical region. The value
of $T_{C}$ turns out to be 166 K and 188 K for SRO and SRCO15 respectively.
The critical exponent $\beta$ is $\sim$ 0.5 for both samples indicating mean
field type FM-interactions. Most importantly 22 K enhancement in the $T_{C}$
is noticed, whereas the FC magnetization at 5 K ($M_{5K}$) is reduced with Cr-
doping (see Table-4). At low temperatures, the thermally excited magnons
(following Bloch $T^{3/2}$ law), as well as Stoner excitations (following
$T^{2}$ law), reduce the magnetic moment. Hence the low temperature
magnetization data fitted with: $M(H,T)$ = $M(H,0)$[1 - $A_{SW}$ $T^{3/2}$ \-
$B$ ${}_{SE}T^{2}$], where $A_{SW}$ is a spin-wave stiffness constant and
$B_{SE}$ is Stoner excitation parameter. The value of $A_{SW}$ is related with
the exchange constant ($J$) between two Ru4+ atom as: $A_{SW}$ =
(0.0587/S)($k_{B}/2JS$)3/2 and with $T_{C}$ as: ($A_{SW}$ )-2/3= 2.42 $T_{C}$
[49]. The obtained fitting values of $A_{SW}$ and $B_{SE}$ decreases, $J$ and
TC increases with increasing Cr-doping in SRO (see Table-4). In Fig-4(c) the
exchange constant, $J\times k_{B}$ is plotted against the Cr-doping, which
shows a gradual increase and explains the $T_{C}$ enhancement.

Figure 4: (color online) (Left y-axis) Temperature-dependent FC and ZFC
magnetization of (a) SRO and (b) SRCO15 measured at 500 Oe, where solid red
(green) lines are FC-magnetization fitting near critical region (below 50 K).
(Right y-axis) inverse FC-susceptibility, where pink solid lines show the
Curie-Weiss linear fit. (c) Variation of $J\times k_{B}$ with Cr-doping. (d)
M-H hysteresis loops of SRCO samples at 2 K. (e) Comparative plot of M/Ms vs
field for SRO and SRCO15. (f) Variation of TC (right y-axis) and Ms (left
y-axis) at 60 kOe with different Cr concentrations.

Fig-4(d) shows the M-H hysteresis plot at 2 K for SRCO samples. The
magnetization increases up to a saturation magnetic moment with increasing the
magnetic field. For SRO the magnetic moment at 6 T (Ms) is $\sim$ 1.42
$\mu_{B}/f.u.$. This ordered moment is lower than the anticipated complete S =
1 moment (2 $\mu_{B}$/Ru), which may arise due to several reasons: (a)
itinerant ferromagnetic nature [50], (b) spin glass state at low-temperature
[16], and (c) large magnetocrystalline anisotropy (MCA). In case of SRCO15,
the resultant magnetic moment is reduced by $\sim$25% than the parent compound
SRO. In SRCO15 the origin of the reduced moment can be understood with the
presence of Cr6+ (S = 0). It is important to note that the total spin-only
moment is reduced (as 10% spin is getting zero). This is in agreement with the
fact that an FM-like hysteresis is observed with a non-saturating component
for all SRCO samples. The non-saturating behaviour in SRO is supported by
large MCA. In Cr-doped SRO, due to the randomness of exchange interactions:
[(a) FM with Ru4+-O-Ru4+, Ru4+-O-Cr3+ (b) PM Ru4+-O-Cr6+, and rare possibility
of (c) AFM Cr3+-O-Cr3+] a non-saturation of the magnetisation is expected.
Fig-4(e) shows the comparative plot of normalized magnetization (M/Ms) as a
function of field (0 - 6 T). It is evident from the plot that SRO possesses a
higher MCA than SRCO15. The origin of higher anisotropy in SRO compared to
SRCO15 is intriguing. Below 100 K various kinds of magnetic anomalies were
observed in bulk as well as in thin film SrRuO3. In bulk, SrRuO3 the spin
glass, memory effect, slow domain wall dynamics, anomalous magneto transport,
and Hall effect sign inversion have been observed while in thin film exchange
bias, Skyrmionic behaviour, topological Hall effect, etc. have been reported
[51, 17]. All these observations suggest the formation of a complex magnetic
order below 100 K in SRO. The origin of this “complex ferromagnetism” (CFM)
was found to be srtuctural-modulation driven [52]. The effect of Cr-doping
reduces the anisotropy field in SRO. Such reduction in the anisotropy field is
related to the reduction of CFM in SRCO15, which is discussed in the
magnetotransport section. Fig-4(f) shows the variation of $M_{s}$ (at 2 K, 6
T) as well as $T_{C}$ as a function of Cr-concentration. The value of Ms is
decreasing with increasing Cr-doping in SRO, whereas $T_{C}$ is increasing.
All obtained values of $T_{C}$, $A_{SW}$, $B_{SE}$, $J$, and $M_{FC}$(5 K) of
SRCO15 in comparison with SRO are tabulated in Table-4. To understand the
microscopic magnetism and evolution of magnetic structure the temperature-
dependent ND measurement is performed.

Figure 5: (color online) (a) Neutron diffraction patterns (Nuclear + magnetic)
recorded over 5 – 300 K temperature range. (b) Temperature evolution of
magnetic Bragg peak at 1.6 $\AA^{-1}$. (c) The magnetic structure of SRCO15.
(d) Temperature dependence of the ordered magnetic moment of SRCO15 (blue
hexagon data) at the Ru site. The dashed line is the magnetic exponent
fitting. Here SrRuO3 data (gray) [52], is plotted for comparison. (e - h)
Temperature evolution of the lattice constants, and unit cell volume of
SRCO15, where colored vertical dotted lines at 190 K (orange) and 100 K (blue)
represent onset temperature of ferromagnetic (FM) and complex ferromagnetic
(CFM) ordering respectively.

###  IV.4 Neutron diffraction-

Figure-5(a) shows the experimentally recorded and Rietveld-refined ND patterns
for the SRCO15 sample at various temperatures (5 – 300 K). The peaks at Q
$\sim$ 1.6 $\AA^{-1}$ (2$\theta$ = 16∘) in the neutron diffraction data
correspond to magnetic Bragg peaks (101) and (020). To get the pure magnetic
signal the nuclear background (PM-state) taken at 250 K is subtracted from the
low-temperature data (below the magnetic ordered state). Fig-5(b) shows such
background subtracted data for various temperatures. The intensity of the
magnetic peak (Q $\sim$ 1.6 $\AA^{-1}$) starts to appear below 190 K and
attains a maximum at 5 K. This reveals a ferromagnetic ordering with a
propagation vector, k = (0,0,0), as reported for the parent compound SrRuO3
[52]. To determine the symmetry of all the allowed magnetic structures of
SRCO15, representation analysis [53, 54, 55] is carried out using the BASIREPS
program [39]. The symmetry analysis for the space group $Pnma$ and the
propagation vector k = (0,0,0) reveal four possible non-zero irreducible
representations ($\Gamma_{1}$, $\Gamma_{3}$, $\Gamma_{5}$, and $\Gamma_{7}$)
(Table-5). Out of these four irreducible representations, the best agreement
of the experimentally observed to the Rietveld refined magnetic diffraction
pattern is achieved for $\Gamma_{5}$. Since, the experimental value of the
magnetic form factor for Ru4+ is not available in the literature, therefore
the magnetic form factor of Ru1+ is used to solve the magnetic refinement of
the diffraction patterns of SRCO15 (similar to the magnetic refinement for the
several other Ru4+ based compounds [56, 57]). The magnetic structure obtained
from the Rietveld analysis of the ND pattern is shown in Fig-5(c). From the
magnetic structure, it is evident that the moments are lying predominantly
along the $b$-axis. The component of the moments along the a and c maximize is
found to be negligible ($<$ 0.1 $\mu_{B}$/Ru4+). The refined ordered moment
value at 5 K is found to be; $M$ = 1.328 $\pm$ 0.1 $\mu_{B}$/Ru4+, closely
matches with the experimental ( Ms value of SRCO15 i.e. $\sim$ 1.1
$\mu_{B}$/Ru4+). The temperature dependence of the ordered magnetic moment of
SRCO15 and in comparison with SRO is shown in Fig-5(d). The fitting of the
power law equation, $\mu_{B}$(T) = A(TC-T)β, finds $T_{C}$ = 190 K and $\beta$
= 0.52 (closely agree with experimental value ($T_{C}$ = 188 K, and $\beta$ =
0.5). From ND it is evident that the Cr-doping results $\sim$ 22 K enhancement
of FM ordering temperature, however, the mean field-like behaviour remains the
same. It has to be noted that the ordered magnetic moment is considerably
lower than the spin-only moment 2 $\mu_{B}$/Ru4+.

Table 5: Basis vectors for the atoms of the magnetic sites 4b site of the orthorhombic space group $Pnma$, with the propagation vector, k = (0,0,0). | $\psi_{1}$ |  $\psi_{2}$ (-x+$\frac{1}{2}$, |  $\psi_{3}$ (-x, |  $\psi_{4}$ (x+$\frac{1}{2}$,
---|---|---|---|---
| (x, y, z) | -y, z+$\frac{1}{2}$) | y+$\frac{1}{2}$, -z) | -y+$\frac{1}{2}$, -z+$\frac{1}{2}$)
$\Gamma_{1}$ | (1, 1, 1) | (-1, -1, 1) | (-1, 1, -1) | (1, -1, -1)
$\Gamma_{3}$ | (1, 1, 1) | (-1, -1, 1) | (1, -1, 1) | (-1, 1, 1)
$\Gamma_{5}$ | (1, 1, 1) | (1, 1, -1) | (-1, 1, -1) | (-1, 1, 1)
$\Gamma_{7}$ | (1, 1, 1) | (1, 1, -1) | (1, -1, 1) | (1, -1, -1)

Now begin with the question of what is the origin of enhancement in $T_{C}$.
The answer lies within the evolution of structure parameters with temperature.
The lattice parameters are obtained from the Rietveld analysis of all ND
patterns. The variation of lattice parameters and the unit cell volume is
plotted against temperature as shown in Fig-5(e - h). A gradual contraction of
the lattice constants and unit cell volume has been observed with the decrease
in the temperature. The volume contraction below the ordering temperature can
be understood as a spontaneous magnetostriction effect that arises due to the
magnetic ordering of the atomic moments below $T_{C}$. The unit cell volume
resembles a minima at 190 K. A minima in unit cell volume represents a stable
configuration (minimum free energy of the system), when a magnetic ordering is
usually formed. Fig-6(a) shows the comparative volume plot of SRCO15 and
SrRuO3. In the case of SRO, an anomaly in the lattice parameters and unit cell
volume at the ferromagnetic ordering temperature (166 K). It is important to
note that 0.5 % volume contraction occurs only due to Cr doping and lowering
the temperature further enhances the contraction affecting the $\Sigma$ and
$\Delta$. Thus Cr doping results in a relative change in the crystal structure
with respect to temperature. Such change is visible in the form of a minimum
in volume and a peak in $\Delta$ near 190 K (22 K higher than the parent
compound SRO). Thus the origin of the $T_{C}$ enhancement is closely
associated with structural modulations due to both Cr-doping and temperature.
Further, in the case of SrRuO3, another minimum below 100 K ($\sim$ 75 K) is
noticed, which is responsible for the CFM in addition to the dominant FM
order. However, Cr-doping suppresses the formation of such CFM. This picture
is more clear in the transport section later.

Figure 6: (color online) The temperature variation of (a) unit cell volume and
(b) stretching distortion, (c) angular distortion of Ru/CrO6 octahedra and (d)
octahedral tilt angle ($\theta$) for SRCO15 and SRO compounds. The values are
normalized with respect to their reference values at 300 and 250 K for SRCO15
and SRO, respectively. The data for SRO are taken from [52, 58].

Further, the structural analysis is done from the variation of bond lengths,
bond angles, and octahedral tilt. Fig-6(b) and (c) depicts the normalized
stretching and angular distortions of SRCO15 (SrRuO3), with respect to the
value at 300 K (250 K), as a function of $T/T_{C}$. In the vicinity of FM
ordering the relative stretching distortion ($\Delta$/$\Delta_{300K}$) shows a
peak for both SRO and SRCO15. Such peak in $\Delta$ neat $T_{C}$ is expected.
Below 100 K in SRCO15, $\Delta$ shows an increasing trend whereas SrRuO3
exhibits a peak-like feature (in the locality of CFM). While the relative
angular distortion ($\Sigma$/$\Sigma_{300K}$) show an increasing trend down to
FM ordering, and then it freezes down to $\sim$ 150 K. Lowering the
temperature results decrease in $\Sigma$/$\Sigma_{300K}$ down to 100 K and
then again freezes. Fig-6(d) displays the relative inter-octahedral tilt
($\theta$/$\theta_{300K}$) against the $T/T_{C}$. The octahedral tilt is
calculated from the Ru-(O1/O2)-Ru bond angles as: tilt =
$\frac{180-\angle{(Ru-(O1/O2)-Ru)}}{2}$ (angle $\theta$ schematically shown in
Fig-2(d)). With lowering the temperature, the octahedral tilt along the apical
plane initially increases by 2.5% and gets frozen in the vicinity of $T_{C}$
followed by a reduction (while along the basal plane, it is increased) of 2.5%
down to 100 K. Further lowering temperature neither affects the apical nor
basal tilt. Such freezing in the angular distortion and octahedral tilt
indicates the formation of an ordered phase (in this case it is FM). To get
more insight into magnetotransport study on various SRCO samples is carried
out up to magnetic field 5 T.

Figure 7: (color online) (a) The temperature-dependent normalised electrical
resistivity ($\rho/\rho_{250K}$) of SRCO samples at 0 T (coloured data points)
and 5 T (solid green line). (b) Variation of $\rho_{2K}$, $\rho_{300K}$ (left
y-axis), and MR% at 2 K, 50 K, and 166 K (right y-axis) with increasing Cr
concentrations. (c) Resistivity fitting of SRCO5 sample with WL-model. (d)
G-VRH models fitting of resistivity in SRCO10 and SRCO15. (e) $MR$
($\frac{\rho(5T)-\rho(0)}{\rho(0)}$ $\times$100)% vs temperature data;
vertical dotted lines represent the onset of CFM and FM. (f) MR% vs magnetic
field at 2 K; inset shows MR in the vicinity of FM ordering for SRCO samples.

###  IV.5 Transport measurements-

Figure-7(a) displays the temperature-dependent normalized resistivity
($\rho/\rho_{250}$ K). The parent compound SRO exhibits the metallic behaviour
down to 2 K (with residual resistivity ratio (RRR) $\sim$ 20, with a slope
change at $T_{C}$ (166 K). A gradual increase in resistivity with Cr-doping
indicates the evolution of non-metallic behaviour. Fig-7(b) depicts the
variation of $\rho_{300K}$ and $\rho_{2K}$ with respect to Cr doping. The two
and four orders of magnitude increase in the $\rho_{300K}$ and $\rho_{2K}$
respectively, is estimated at 15% Cr doping in SRO. The increase in
resistivity as well as the low-temperature resistivity upturn arises due to
the disorder and electron correlation effect induced by Cr doping. The role of
disorder and electron correlation on electrical transport can be understood by
appropriate fitting of the resistivity data with various models. The parent
compound shows FL to Non-FL behaviour $\sim$ 45 K (fitting not shown) as
reported earlier [59, 52]. A resistivity minima is noticed at $T_{min}$ $\sim$
117 K for x=0.05. In an itinerant electron system with such low Cr doping
induces weak localization (WL) [60]. On the other hand, electron-electron
interaction (EEI) in such system also drives low-temperature resistivity
upturn [61]. In correlated metals with low disorder, WL [60] and EEI [61]
drive the insulating state at low temperatures. In such case the resistivity
upturn can be fitted using the following relation [62]:

$\rho=\frac{1}{1/\rho_{0}+aT^{p/2}+bT^{1/2}}+cT^{2}.$ (3)

Here $\rho_{0}$ refers to the residual resistivity, the term $aT^{p/2}$ and
$bT^{1/2}$ represents WL and EEI effect respectively. The value of $a$ and $b$
gives an idea about the strength of WL and EEI respectively. In the WL term,
$p$ depends on the scattering rate of the electron’s dephasing mechanism and
dimensionality of the system [61]. The value of $p$ indicates the nature of WL
to be electron-electron (p=2) or electron-phonon (p=3) driven. The $cT^{2}$
term in Eq-3 represents EEI scattering at high-temperatures. Fig-7(c) shows
the resistivity fitting of SRCO5 using Eq-3. The fitting parameters of
resistivity at 0 and 5 T are tabulated in Table-6. The value of $p$ turns out
to be 2.98. Similar value of $p$ has been reported for Ba and Ti doped SrRuO3
as well [62], indicates that the WL is mainly driven by the electron-phonon
interaction. However, it is evident from the value of $b$ ($\sim$ 200$a$, see
Table-6) that EEI dominates in the overall conduction process. For x $>$ 0.05
the electron transport follows the modified Mott-Variable Range Hopping also
known as Greaves-VRH (G-VRH) model [63, 64], where the resistivity can be
expressed as

$\rho(T)=A\sqrt{T}\exp\Big{[}\Big{(}\frac{T_{G}}{T}\Big{)}^{1/4}\Big{]}$ (4)

where $T_{G}$ is Greaves characteristic temperature and $A$ is pre factor.
Fig-7(d) shows the G-VRH model fitting in a high-temperature regime for SRCO10
($T_{G}$ = 8.7$\times$103 K) and SRCO15 ($T_{G}$ = 1.7$\times$104 K). The
characteristic temperature $T_{G}$ is inversely proportional to the
localization length of conduction electron [64, 62]. With Cr-doping the
increased value of $T_{G}$ indicates that the localization length of
conduction electron decreases, which suggests the system is getting localised
with increasing the Cr-doping.

Further to study the magnetic contribution in transport the magnetoresistance
measurement is performed. Fig-7(e) depicts the temperature dependent
magnetoresistance ($MR\%$ = $\frac{\rho(H)-\rho(0)}{\rho(0)}$ $\times$100), at
5 T in SRCO. Two negative peaks in MR data are observed for the parent
compound SRO. The first peak arises owing to the spin fluctuation from PM to
FM phase transition at $\sim$ 166 K. The second peak $\sim$ 50 K is attributed
to the CFM. With small Cr-doping (x = 0.05), the MR peak at FM order is
shifted to a higher temperature ($\sim$ 190 K), and a second peak at 50 K is
suppressed. However, both of these negative peaks almost disappear in the case
of SRCO10 and SRCO15. Interestingly for SRO a crossover from negative MR to
positive MR is noticed near 10 K. Such crossover in MR behavior is also
reflected from the R-H data measured at various temperatures as shown in
Fig-7(f). For SRO a large non-saturating positive linear MR is observed at 2K.
However Cr-doped SRCO samples exhibit a negative MR. The negative MR for
ferromagnetic metallic systems can be explained by using spin fluctuation
theory. While the origin of positive MR in bulk SrRuO3 is not known. Although,
it can be attributed to CFM in SRO. It has to be noted that, the 5 T field is
not sufficient to align the complex magnetic structure, which can effectively
increase the spin disorder scattering and hence the positive MR at low
temperatures [65, 66]. However the positive MR in SrRuO3 thin films has
quantum origin such as weak antilocalisation due to large spin-orbit coupling,
orbital effect, and Weyl fermionic nature [67, 68, 69]. The hysteretic nature
in MR is observed for all SRCO samples. This hysteretic nature increases with
increasing Cr doping. The value of MR is reduced to $\sim$ -2% for SRCO15 at 2
K as shown in Fig-7(b) (right y-axis). At 150 K the hysteresis in MR decreases
and a weak negative MR ($\sim$ -0.5%) is noticed in the case of SRCO samples
(inset Fig-7(f)). Whereas large negative MR (-2%) is observed for the parent
compound (corresponds to an FM ordering at $\sim$ 166 K). The temperature-
dependent MR response of SRCO is nicely correlated with the temperature-
dependent structural evolution obtained from ND data. In a nutshell, Cr
doping: (i) distorts the crystal structure, (ii) dilutes the total number of
magnetic ions and (iii) increases the disorder in SRO. These three facts
effectively install a non-metallic state with reduced ferromagnetic strength.
This study indicates that the structural modulation with temperature enhances
the TC.

Table 6: Fitting parameters of SRCO5 resistivity at 0 and 5T. | $\rho_{0}$ |  $a$ ($\Omega$cm-1 | $p$ |  $b$ ($\Omega$cm-1 |  $c$ ($\Omega$cm
---|---|---|---|---|---
| ($\Omega$cm) | K-p/2) |  | K-1/2) | K-2)
0T | 0.121 | 0.0015 | 2.98 | 0.32 | 8.9$\times$ $10^{-7}$
5T | 0.119 | 0.0014 | 2.92 | 0.34 | 7.9$\times$ $10^{-7}$

##  V CONCLUSION-

Here we observe enhanced Curie-temperature by 22 K in Cr doped SrRuO3 and
installs non-metallic behaviour as the delocalised Ru-4d orbitals are replaced
by localised Cr-3d. The structural analysis in SrRu1-xCrxO3 (0 $<$ x $<$ 0.15)
finds orthorhombic ($Pnma$) crystal structures with squeezed volume. The
stretching distortions of the RuO6 octahedra increase up to five times at x =
0.15. XPS core level spectra find mixed valance state of Cr (as Cr3+ and Cr6+)
and the valence band spectra exhibit spectral weight shifting of delocalised
electron towards the localised state. The low-temperature M(T) fit indicates a
larger $J$ value for SRCO than SRO supporting higher Curie temperature. ND
refinements of SRCO15 find that the magnetic moment is aligned along the
$b$-axis with $M_{s}$ = 1.3 $\mu_{B}$/Ru4+ which closely matches with the
experimental value of the bulk magnetic moment at 2 K, 6 T (1.1
$\mu_{B}$/f.u.). Temperature-dependent ND analysis reveals a structural
modulation which is closely associated with the enhanced $T_{C}$. The unit
cell volume minima of SRCO15 shifts by 22 K compared to SRO, exactly mimics
the $T_{C}$ shift. In transport, the resistivity upturn for x = 0.05 and the
non-metallic behaviour for x $>$ 0.05 is fitted in terms of disorder and
localization picture. The magnetotransport data reveals a gradual suppression
of spin scattering near FM as well as CFM ordering and an evolution of a non-
metallic state with Cr-doping. Thus, the itinerant-localised duality of the
d-electron becomes crucial in deciding the fate of the electronic as well as
magnetic character.

## ACKNOWLEDGMENTS-

The authors acknowledge SRG SERB Grants (SRG-2019-001104, CRG-2022-005726,
EEQ-2022-000883), India, and Initiation Grant (IITK-2019-037), IIT Kanpur, for
financial support. Special Thanks to S. Yonezawa (Kyoto University, Japan) for
the magnetic and magnetotransport measurements.

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