# Quantification of electronic and magnetoelastic mechanisms of first-order
magnetic phase transitions from first principles: application to caloric
effects in La(FexSi1-x)13

Eduardo Mendive Tapia  Departament de Física de la Matèria Condensada,
Facultat de Física, Universitat de Barcelona, Martí i Franquès 1, E-08028
Barcelona, Catalonia Author to whom any correspondance should be addressed.
Email<EMAIL_ADDRESS>Christopher E. Patrick  Department of
Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
Tilmann Hickel  Department of Computational Materials Design, Max-Planck-
Institut für Eisenforschung, 40237 Düsseldorf, Germany BAM Federal Institute
for Materials Research and Testing, 12489 Berlin, Germany Jörg Neugebauer
Department of Computational Materials Design, Max-Planck-Institut für
Eisenforschung, 40237 Düsseldorf, Germany Julie B. Staunton  Department of
Physics, University of Warwick, Coventry CV4 7AL, United Kingdom

###### Abstract

La(FexSi1-x)13 and derived quaternary compounds are well-known for their
giant, tunable, magneto- and barocaloric responses around a first-order
paramagnetic-ferromagnetic transition near room temperature with low
hysteresis. Remarkably, such a transition shows a large spontaneous volume
change together with itinerant electron metamagnetic features. While
magnetovolume effects are well-established mechanisms driving first-order
transitions, purely electronic sources have a long, subtle history and remain
poorly understood. Here we apply a disordered local moment picture to quantify
electronic and magnetoelastic effects at finite temperature in La(FexSi1-x)13
from first-principles. We obtain results in very good agreement with
experiment and demonstrate that the magnetoelastic coupling, rather than
purely electronic mechanisms, drives the first-order character and causes at
the same time a huge electronic entropy contribution to the caloric response.

###### Contents

  1. 1 Introduction
  2. 2 Caloric effects and ab initio origin of first-order magnetic phase transitions
    1. 2.1 Fluctuating ’spins’ and the Disordered Local Moment picture
      1. 2.1.1 Statistical mechanics of slowly varying local moment orientations
      2. 2.1.2 Computation of caloric effects
      3. 2.1.3 Computationally efficient ensemble averaging of local moment configurations: A mean-field approximation
    2. 2.2 The magnetic Gibbs free energy: First-order magnetic phase transitions
      1. 2.2.1 Electronic and/or magnetoelastic mechanisms
  3. 3 The first-order magnetic transition and calculation of caloric effects in La(Fe1-xSix)13 from ab initio theory
    1. 3.1 Construction of the Gibbs free energy: Weiss fields against magnetic order parameters and lattice changes
    2. 3.2 Magnetocaloric and barocaloric effects
    3. 3.3 On the origin and nature of the transition and the role of itinerant-electron metamagnetism
  4. 4 Conclusions

##  1 Introduction

Large magnetocaloric and barocaloric effects can come from dramatic changes to
the magnetic order of a material when it is subjected to an external magnetic
field and/or other mechanical stimuli. Giant caloric responses for use in
solid-state refrigeration can occur around discontinuous (first-order)
magnetic phase transitions. A well-understood and extensively observed
mechanism driving such sharp transitions is a strong coupling of magnetism
with the lattice volume, known as magnetovolume coupling [1, 2]. There are
some materials, however, that exhibit first-order magnetic phase transitions
even though their magnetoelastic coupling is weak or even negligible, as
reported, for example, in Eu2In [3] and Mn3NiN [4, 5]. The fundamental cause
here derives from a response of the complex glue of electrons to the state of
magnetic order, which in turn causes a strong feedback on the magnetic
interactions and establishes an electronic origin for the magnetic
discontinuity.

Purely electronic mechanisms driving first-order transitions have a long and
subtle history and there are deeply insightful works such as that on itinerant
electron metamagnetism by Wohlfarth and Rhodes [6] and, more recently, on
electronic effects in La(FexSi1-x)13 compounds by Fujita et al [7, 8].
La(FexSi1-x)13 systems, and other derived quaternary compounds, are very
promising for applications, being made of relatively abundant, cheap, and non-
toxic elements. In particular, the materials have attracted huge attention
from the caloric refrigeration research community owing to the giant
magnetocaloric and barocaloric effects associated with a first order
paramagnetic-ferromagnetic transition near room temperature accompanied by a
small hysteresis [8, 9, 10, 11, 12]. This famous magnetic materials class
nonetheless is complicated since the changes of the lattice spacings that
occur at the transition are rather large. Apparently there is a magnetoelastic
coupling which is also playing an important and possibly pivotal role which is
intimately connected with the changes to the itinerant electronic structure.
In this work we describe a predictive first principles approach, based on the
disordered local moment (DLM) picture [13], that provides both electronic and
magnetoelastic contributions to first-order magnetic phase transitions, to
examine the subtleties in La(FexSi1-x)13. We obtain a good agreement with
experiment regarding both the nature of the transition and also the consequent
caloric responses. We quantify both electronic and magnetoelastic effects,
their synergy and demonstrate that the first-order character of the transition
can be said to be driven by the latter while a large proportion of the caloric
effect has an electronic source.

This work is organised as follows. In section 2 we describe our theoretical
framework for magnetic phase transitions and caloric effects. Section 2.2 is
devoted to an explanation of how it accounts for the first principles Gibbs
free energy of magnetic materials and describes the origin of first-order
magnetic phase transitions. In section 3 we present results obtained for a
specific La(FexSi1-x)13 compound. Our conclusions are given in the final
section 4.

##  2 Caloric effects and ab initio origin of first-order magnetic phase
transitions

###  2.1 Fluctuating ’spins’ and the Disordered Local Moment picture

Figure 1:  (a) Representation of a ferromagnetic state at zero temperature,
described with a value of the magnetic order parameter $m_{n}=1$ at all
magnetic sites, see Eq. (7). Panels (b) and (c) show single snapshots of
thermally excited local moment configurations for increasing values of the
temperature, respectively. Panel (c) shows the fully disordered local moment
state where $m_{n}=0$ at all magnetic sites establishes above the Curie
transition temperature $T_{c}$, while the local moment orientations in (b)
form a partially disordered ferromagnetic state below $T_{c}$ ($0<m<1$).

The formation of local magnetic moments in a crystalline solid is a process
involving spin-correlated electronic interactions which occur over very short
(femto-second) timescales, $\tau_{\text{elec}}$, of the electrons hopping from
atomic site to atomic site in the lattice. The central tenet of the Disordered
Local Moment (DLM) theory is to assume that the orientations of these moments
are slowly varying degrees of freedom evolving on a much longer time-scale
$\hbar/E_{SW}\gg\tau_{\text{form}}\gg\tau_{\text{elec}}$, where $E_{SW}$ is a
typical spin wave energy and $\tau_{\text{form}}$ is an intermediate time-
scale for the formation of the magnetic moment. Time averages over
$\tau_{\text{form}}$, therefore, result in a temporarily broken ergodicity
labelled by a collection of unit vectors identifying the local moment
orientations at each atomic site $n$, $\\{\hat{\textbf{e}}_{n}\\}$, which
emerge from the many electron interacting system [13]. If this assumption
holds, the spin-polarisation of the electronic structure in Density Functional
Theory calculations can be specified by $\\{\hat{\textbf{e}}_{n}\\}$ such that
the magnetic moment density, $\bm{\mu}(\textbf{r},\\{\hat{e}_{n}\\})$, is
constrained to satisfy

$\int_{V_{n}}d^{3}\textbf{r}\bm{\mu}(\textbf{r};\\{\hat{\textbf{e}}_{n}\\})=\mu_{n}(\\{\hat{\textbf{e}}_{n}\\})\hat{\textbf{e}}_{n},$
(1)

i.e. the orientation of the spin polarization within the volume $V_{n}$
centred at site $n$ is constrained to be along $\hat{e}_{n}$, $\mu_{n}$ being
its resulting magnitude which can have a dependence on the orientations of the
local moments surrounding it. Figure 1 illustrates this central concept of DLM
theory for a ferromagnet by showing different snapshots of constrained local
moment configurations at zero and higher temperatures. Hence, a magnetic
Hamiltonian can be established through the dependence of the material’s total
energy on the local moment orientations,

$\mathcal{H}_{\text{mag}}=\mathcal{H}_{\text{mag,int}}(\\{\hat{\textbf{e}}_{n}\\},\omega)-\textbf{B}\cdot\sum_{n}\mu_{n}\hat{\textbf{e}}_{n},$
(2)

which defines $\mathcal{H}_{\text{mag,int}}$ as the Hamiltonian in the absence
of an external magnetic field. The ‘spins’ that emerge from itinerant electron
metallic systems such as CoMnSi, Fe2P etc are thus the sets of unit vectors
specifying the orientations of the local moments, the
$\hat{\textbf{e}}_{n}$’s. Importantly, the interactions among them can
strongly depend on the crystal structure. In this work we focus on the effect
of lattice volume changes produced spontaneously and also by the application
of a hydrostatic pressure. For convenience we describe them relative to the
paramagnetic limit,

$\omega=\frac{V-V_{\text{PM}}}{V_{\text{PM}}},$ (3)

where $V_{\text{PM}}$ is the unit cell volume obtained from a relaxation in
the paramagnetic state. The nature of the Hamiltonian
$\mathcal{H}_{\text{mag,int}}$ can, therefore, be quite complicated and the
sampling of the local moment configurations and the statistical mechanics
needs to be carefully carried out, which we address in the following.

####  2.1.1 Statistical mechanics of slowly varying local moment orientations

The description of temperature-dependent properties of materials in terms of
the atom-scale magnetic degrees of freedom is provided by
$\mathcal{H}_{\text{mag}}$ through the corresponding calculation of the
partition function for the local moment orientations in the canonical
ensemble,

$\mathcal{Z}_{\text{mag}}=\operatorname{Tr}_{\text{mag}}\exp\left[-\beta\mathcal{H}_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\right]=\int\prod_{n}\left[d\hat{\textbf{e}}_{n}\right]\exp\left[-\beta\mathcal{H}_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\right].$
(4)

Here $\beta=\frac{1}{k_{\text{B}}T}$ is the Boltzmann factor, and
$\operatorname{Tr}_{\text{mag}}$ is the trace over the magnetic degrees of
freedom, which takes the form of a product of integrals owing to the
continuous nature of each of the $\\{\hat{\textbf{e}}_{n}\\}$. The Boltzmann
probability distribution associated with the thermal fluctuations of these
local moment orientations is given by,

$P_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)=\frac{\exp\left[-\beta\mathcal{H}_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\right]}{\mathcal{Z}_{\text{mag}}},$
(5)

which leads to the exact magnetic Gibbs free energy of the material by
carrying out the appropriate ensemble averages,

$\begin{split}\mathcal{G}_{\text{mag}}&=-k_{\text{B}}T\ln\mathcal{Z_{\text{mag}}}=\int\prod_{n}\left[d\hat{\textbf{e}}_{n}\right]P_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\Big{[}\mathcal{H}_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)+k_{\text{B}}T\ln
P_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\Big{]}\\\
&=U_{\text{mag}}-TS_{\text{mag}}.\end{split}$ (6)

A central task of the DLM approach is to perform the averages over the local
moment orientations of different quantities of interest. For example,
$U_{\text{mag}}=\langle\mathcal{H}_{\text{mag}}\rangle_{P_{\text{mag}}}$ is
the internal magnetic energy, the single-site average of
$\hat{\textbf{e}}_{n}$ is

$\textbf{m}_{n}=\langle\hat{\textbf{e}}_{n}\rangle_{P_{\text{mag}}}=\int\prod_{n}\left[d\hat{\textbf{e}}_{n}\right]P_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\hat{\textbf{e}}_{n},$
(7)

i.e. a magnetic order parameter at site $n$, and the associated magnetic
entropy in Eq. (6) is given by

$S_{\text{mag}}=-k_{\text{B}}\left\langle\ln
P_{\text{mag}}\right\rangle_{P_{\text{mag}}}=-k_{\text{B}}\int\prod_{n}\left[d\hat{\textbf{e}}_{n}\right]P_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)\ln[P_{\text{mag}}(\\{\hat{\textbf{e}}_{n}\\},\omega)].$
(8)

The local moment orientations $\\{\hat{\textbf{e}}_{n}\\}$ emerge from the
many-electron interactions and as such they have an electronic origin. The DLM
approximation splits the trace over all the electronic degrees of freedom
$\text{Tr}_{\text{elec}}$ into the following two pieces [13], the first
capturing relatively slowly varying electronic degrees of freedom i.e. the
local moment orientations,

$\text{Tr}_{\text{elec}}\approx\text{Tr}_{\\{\hat{e}_{n}\\}}\text{Tr}_{\text{rest}},$
(9)

and $\text{Tr}_{\text{rest}}$ traces over the remaining faster ones. Eq. (9)
thus reflects a time-scale separation. Such a partition of the trace into
sequential parts, $\text{Tr}_{\text{rest}}$ being performed first for
constrained values of $\\{\hat{\textbf{e}}_{n}\\}$, implies that
$\mathcal{H}_{\text{mag}}$ is in fact the Gibbs free energy of the faster
electronic degrees of freedom,

$U_{\text{mag}}=\langle\mathcal{H}_{\text{mag}}\rangle_{P_{\text{mag}}}\equiv\langle\mathcal{G}_{\text{elec}}\rangle_{P_{\text{mag}}}=\langle\bar{E}\rangle_{P_{\text{mag}}}-TS_{\text{elec}},$
(10)

where $\bar{E}$ is the DFT-based total energy averaged over the faster
electronic degrees of freedom, and $S_{\text{elec}}$ is the associated
electronic entropy that has been also averaged over with respect to the local
moment configurations$\\{\hat{\textbf{e}}_{n}\\}$.

####  2.1.2 Computation of caloric effects

It is a common practice to write the total entropy of the material in terms of
the following three different parts,

$S_{\text{tot}}=S_{\text{vib}}+S_{\text{mag}}+S_{\text{elec}},$ (11)

associated with the fluctuations of vibrational, magnetic, and the remaining
electronic degrees of freedom. The description of caloric effects essentially
involves the calculation of entropy changes, both individually and exchanged
between these three different components, and of their response to the
application of external stimuli. The trace-separation made in Eq. (11) carries
fundamental implications. A careful analysis in relation to Eqs. (9) and (10)
shows that Eq. (11) applies to materials where time-scale separations between
the corresponding sets of degrees of freedom takes place. Particle-hole
excitations caused by thermal fluctuations are described by weighting the
Kohn-Sham single electron energies of SDFT by the Fermi-Dirac distribution,
$f(E;T)=\left(\exp[(E-\nu)/k_{\text{B}}T]+1\right)^{-1}$, $\nu$ being the
chemical potential (Fermi energy $E_{\text{F}}$ at $T=0$K). The corresponding
electronic entropy in Eq. (10) can then be calculated as

$S_{\text{elec}}(\\{\hat{\textbf{m}}_{n}\\},\omega;T)=\left\langle\int\frac{c}{T}dT\right\rangle_{P_{\text{mag}}}\approx\frac{\pi^{2}}{3}k_{\text{B}}^{2}T\left\langle
D(E_{\text{F}})\right\rangle_{P_{\text{mag}}},$ (12)

where the heat capacity, $c$, is the one associated with the internal energy
of the single-electron states and $D(E_{\text{F}})$ is the density of states
at the Fermi energy. Although the exact integral form can be formally
computed, the approximate expression on the right hand side, derived from a
Sommerfeld expansion, is a convenient numerical expression which is easily
accessible from DFT calculations. Most importantly, owing to the ensemble
averages over the local moment orientations of the density of states at
different lattice structures, weighted by $P_{\text{mag}}$, $S_{\text{elec}}$
depends on the magnetic order parameters $\\{\textbf{m}_{n}\\}$ introduced in
Eq. (7) and on the relative volume change $\omega$ [see Eq. (3)].

While the underlying assumption of DLM theory is that the local moment
orientations vary much more slowly than the other electronic motions, one can
use the Born-Oppenheimer approximation to justify that the vibrations of the
heavy nuclei at positions $\\{\textbf{R}_{n}\\}$ evolve on a still slower
time-scale. Indeed, Eq. (11) assumes an additional time-scale separation
between $\\{\hat{\textbf{e}}_{n}\\}$ and $\\{\textbf{R}_{n}\\}$. This
adiabatic approximation can be used to obtain the vibrational entropy
$S_{\text{vib}}$ as a function of the averages of the faster degrees of
freedom [14, 15], calculable within harmonic and quasi-harmonic approximations
[16, 17]. A simpler Debye model can also be employed [18],

$S_{\text{vib}}=k_{\text{B}}\left[-3\ln\left(1-e^{-\frac{\theta_{D}}{T}}\right)+12\left(\frac{\theta_{D}}{T}\right)^{3}\int_{0}^{\frac{\theta_{D}}{T}}\frac{x^{3}}{e^{x}-1}dx\right],$
(13)

where $\theta_{D}$ is the Debye temperature obtainable from experiment or
other ab initio sources [19]. Results shown in this work for the calculation
of caloric effects have been obtained by making use of Eqs. (8), (12), and
(13), $S_{\text{elec}}$ and $S_{\text{mag}}$ principally depending on
$\\{\textbf{m}_{n}\\}$ and $\omega$. To this end, the Gibbs free energy is
computed as a function of $\\{\textbf{m}_{n}\\}$ and $\omega$ by carrying out
the averages in Eq. (6) (see section 2.1.3) for different lattice constants.
It is then minimized for different values of the temperature and external
applied stimuli. We consider the effect of an external magnetic field, B, and
hystrostatic pressure, $p$. Isothermal entropy changes can then be directly
obtained by applying Eq. (11),

$\begin{split}&\Delta
S_{\text{iso}}(\textbf{B}_{0}\rightarrow\textbf{B}_{f};T,p)=S_{\text{tot}}(T,\textbf{B}_{f},p)-S_{\text{tot}}(T,\textbf{B}_{0},p),\\\
&\Delta S_{\text{iso}}(p_{0}\rightarrow
p_{f};T,\textbf{B})=S_{\text{tot}}(T,\textbf{B},p_{f})-S_{\text{tot}}(T,\textbf{B},p_{0}),\end{split}$
(14)

for magnetocaloric and mechanocaloric effects, respectively. Here subscripts
$0$ and $f$ indicate initial and final values of the applied fields. On the
other hand, adiabatic temperature changes are obtained by solving the
equations

$\begin{split}&S_{\text{tot}}(T,\textbf{B}_{0},p)=S_{\text{tot}}(T+\Delta
T_{\text{ad}}(\textbf{B}_{0}\rightarrow\textbf{B}_{f};T,p),\textbf{B}_{f},p),\\\
&S_{\text{tot}}(T,\textbf{B},p_{0})=S_{\text{tot}}(T+\Delta
T_{\text{ad}}(p_{0}\rightarrow
p_{f};T,\textbf{B}),\textbf{B},p_{f}),\end{split}$ (15)

where $\Delta T_{\text{ad}}(\textbf{B}_{0}\rightarrow\textbf{B}_{f};T,p)$ and
$\Delta T_{\text{ad}}(p_{0}\rightarrow p_{f};T,\textbf{B})$ are the
corresponding thermal responses.

The magnetocaloric effect associated with the second-order phase transition
between the paramagnetic and ferromagnetic states in pure gadolinium was one
of the first cases for which DLM theory was successfully applied in the
context of magnetic refrigeration. The transition temperature $T_{c}$ as well
as isothermal entropy and adiabatic temperature changes were computed in very
good agreement with experiment [20, 21]. Gadolinium’s heavier rare earth
element next neighbour, dysprosium, presents a much more complex magnetic
phase diagram containing continuous and discontinuous transitions to
helimagnetic and fan phases [22, 23]. DLM theory has demonstrated that the
general features of this magnetic phase diagram and consequent magnetocaloric
effects have their origin in the feedback on the magnetic interactions among
the local moments caused by the response of the mediating valence electrons to
different thermal states of magnetic order [24]. Such a purely electronic
mechanism is implicit in any metallic system. It underlies the presence of
higher than pairwise, multisite, magnetic correlations in the Gibbs free
energy, which can generate first order magnetic phase transitions in other
materials as shown, for example, in the ordered FeRh alloy [25], Mn-based
antiperovskites [4, 5] and the Eu2In compound [3] that exhibits a near
hysteresis-free first-order transition. In the next sub-section we address the
details of how DFT calculations can be used to perform the ensemble local
moment orientational averages at the heart of the first principles description
of these magnetic phase transitions and magneto caloric effects, the
elaboration of multisite correlations and expand on recent computational
findings.

####  2.1.3 Computationally efficient ensemble averaging of local moment
configurations: A mean-field approximation

Owing to its complex electronic origins in metallic materials,
$\mathcal{H}_{\text{mag}}$, introduced in Eq. (2), can have a very complicated
dependence on the local moment orientations. A tractable solution is
accessible by working with a trial, solvable, Hamiltonian [13]

$\mathcal{H}_{\text{tr}}=-\sum_{n}\textbf{h}_{n}\cdot\hat{\textbf{e}}_{n},$
(16)

where $\\{\textbf{h}_{n}\\}$ are single-site molecular fields establishing a
mean-field theory for the magnetic interactions, sometimes referred to as
Weiss fields. This approach continues by invoking the Peierls-Feynman
inequality [26, 27], also known as the Gibbs-Bogoliubov inequality [28, 16],
which provides an upper-bound of the magnetic Gibbs free energy [25, 29]:

$\mathcal{G}_{\text{mag,u}}=\langle\mathcal{H}_{\text{mag,int}}\rangle_{\text{tr}}-\textbf{B}\cdot\sum_{n}\mu_{n}\textbf{m}_{n}-TS_{\text{mag}}\geq\mathcal{G}_{\text{mag}}.$
(17)

Here the ensemble averages $\langle\dots\rangle_{\text{tr}}$ are instead
carried out with respect to a Boltzmann probability associated with the trial
Hamiltonian,

$P_{\text{tr}}(\\{\hat{\textbf{e}}_{n}\\})=\prod_{n}P_{n}(\hat{\textbf{e}}_{n}),$
(18)

where

$P_{n}(\hat{\textbf{e}}_{n})=\frac{1}{\mathcal{Z}_{\text{tr}}}\exp\left[\beta\textbf{h}_{n}\cdot\hat{\textbf{e}}_{n}\right],$
(19)

are single-site probabilities and

$\mathcal{Z}_{\text{tr}}=\prod_{n}\int
d\hat{\textbf{e}}_{n}\exp\left[\beta\textbf{h}_{n}\cdot\hat{\textbf{e}}_{n}\right]=\prod_{n}4\pi\frac{\sinh(\beta
h_{n})}{\beta h_{n}}$ (20)

is the corresponding partition function. The local order parameters and
magnetic entropy [see Eqs. (7) and (8)] therefore take the form of

$\textbf{m}_{n}=\langle\hat{\textbf{e}}_{n}\rangle_{\text{tr}}=\int
d\hat{\textbf{e}}_{n}P_{n}(\hat{\textbf{e}}_{n})\hat{\textbf{e}}_{n}=\left[\frac{-1}{\beta
h_{n}}+\coth(\beta h_{n})\right]\hat{\textbf{h}}_{n},$ (21)

which is the Langevin function, and

$S_{\text{mag}}=\sum_{n}S_{n}=-k_{\text{B}}\sum_{n}\int
d\hat{\textbf{e}}_{n}P_{n}(\hat{\textbf{e}}_{n})\ln
P_{n}(\hat{\textbf{e}}_{n})=k_{\text{B}}\sum_{n}\left[1+\ln\left(4\pi\frac{\sinh(\beta
h_{n})}{\beta h_{n}}\right)-\beta h_{n}\coth(\beta h_{n})\right],$ (22)

respectively. One can write the corresponding equation of state in this mean-
field model by taking the derivative of Eq. (17) and solving
$\frac{\partial\mathcal{G}_{\text{mag,u}}}{\partial\textbf{m}_{n}}=\textbf{0}$.
This results in

$\left\\{\textbf{h}_{n}=\textbf{h}_{n\text{,int}}+\mu_{n}\textbf{B}\right\\},$
(23)

where

$\textbf{h}_{n,\text{int}}=-\frac{\partial\langle\mathcal{H}_{\text{mag,int}}\rangle_{\text{tr}}}{\partial\textbf{m}_{n}}$
(24)

is the magnetic exchange contribution to the total molecular field and
$\mu_{n}$ is the average magnitude of a local moment on a site. Increasing
$\beta h_{n}$ to very large values means that $\theta_{n}$, the angle between
$\hat{\textbf{e}}_{n}$ and $\textbf{h}_{n}$ is statistically close to zero,
which corresponds to the values of the order parameter approaching unity and a
low state of magnetic entropy at this site. This is the absolute zero Kelvin
limit, while in the high temperature, disordered, limit $m_{n}=0$ and the
magnetic entropy takes its maximum, $S_{n}=k_{\text{B}}\ln(4\pi)$, i.e., the
paramagnetic state forms.

The mean-field theory presented above sets up a computationally tractable
scheme to calculate the magnetic Gibbs free energy in Eq. (17) by reducing the
thermally fluctuating local moment configurations to a numerically affordable
and meaningful magnetic phase space given by $P_{\text{tr}}$. The
corresponding ensemble averages over the local moment orientations with
respect to $P_{\text{tr}}$ can be carried out following two different
approaches. The first is based on the so-called coherent potential
approximation (CPA) [30, 31, 32], implementable within the formalism of DFT
based on multiple-scattering theory known as the Korringa-Kohn-Rostoker (KKR)
electronic structure method [33, 34]. The CPA reproduces the material
properties on the average and directly provides the Weiss fields as given in
Eq. (24) [13]. Typically self-consistent DLM-DFT calculations are carried out
for the paramagnetic state, $T>T_{c}$, or a fully ordered magnetic state
appropriate to zero temperature, to form effective potentials and magnetic
fields from the local charge and magnetisation densities. Further information
on this computational method is described elsewhere [35] and with this
implementation DLM-DFT has been successfully applied for a number of years to
study temperature-dependent magnetic anisotropy, e.g. [32, 36], magnetic phase
transitions and diagrams of different sorts, e.g. [37, 25, 38, 24, 39], and
(multi-)caloric effects [40, 3, 4]. The second approach averages the total
energy over $N_{\text{MC}}$ supercell calculations containing local magnetic
moments that are magnetically constrained to follow $P_{\text{tr}}$ [5]. In
this cited work we have demonstrated that $N_{\text{MC}}$ is a computationally
accessible small number even for affordable supercell sizes containing a few
hundreds of atoms. It can be implemented in a wide range of DFT codes not
restricted to KKR-based ones, and can describe the coupling of magnetism with
the crystal structure and consequent lattice relaxations as function of
magnetic temperature more accurately.

###  2.2 The magnetic Gibbs free energy: First-order magnetic phase
transitions

One can generally write the average of the magnetic Hamiltonian as

$U_{\text{mag}}=\langle\mathcal{H}_{\text{mag,int}}\rangle_{\text{tr}}=U^{(0)}-\frac{1}{2}\sum_{nn^{\prime}}U^{(2)}_{nn^{\prime}}\textbf{m}_{n}\cdot\textbf{m}_{n^{\prime}}-\frac{1}{4}\sum_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}}U^{(4)}_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}}(\textbf{m}_{n}\cdot\textbf{m}_{n^{\prime}})(\textbf{m}_{n^{\prime\prime}}\cdot\textbf{m}_{n^{\prime\prime\prime}})-\dots,$
(25)

where
$\\{U^{(2)}_{nn^{\prime}},U^{(4)}_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}},\dots\\}$
are second and higher order coefficients of a Ginzburg-Landau-type expansion
in terms of the magnetic order parameters $\\{\textbf{m}_{n}\\}$. The higher
order coefficients describe the effect of multisite magnetic interactions on
the stability and competition of different magnetic phases at finite
temperature [24]. Terms arising from magnetic anisotropy have not been
considered for simplicity. Most importantly, the sizes and signs of these
internal energy coefficients are characteristic of each magnetic material
under study. An expression for the Weiss field directly follows by taking the
first order derivative to Eq. (25)

$\textbf{h}_{\text{n,int}}=\sum_{n^{\prime}}U^{(2)}_{nn^{\prime}}\cdot\textbf{m}_{n^{\prime}}+\sum_{n^{\prime}n^{\prime\prime}n^{\prime\prime\prime}}U^{(4)}_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}}(\textbf{m}_{n^{\prime\prime}}\cdot\textbf{m}_{n^{\prime\prime\prime}})\textbf{m}_{n^{\prime}}+\dots$
(26)

which can be directly computed. The other components of the magnetic Gibbs
free energy and their dependence on $\\{\textbf{m}_{n}\\}$, i.e. the magnetic
entropy and the coupling to an applied magnetic field, take a known analytical
form in this mean-field model [see Eqs. (17) and (22)].
$\mathcal{G}_{\text{mag,u}}(\\{\textbf{m}_{n}\\})$ can therefore be computed,
either using the KKR-CPA or supercell approaches, by performing the following
steps

  1. 1.

A set of values
$\\{\textbf{m}_{n}\\}=\\{\textbf{m}_{1},\dots,\textbf{m}_{N}\\}$ is chosen,
where $N$ is the number of magnetic sites in the magnetic primitive unit cell
or the number of magnetically constrained sites in the supercell,
respectively.

  2. 2.

$P_{\text{tr}}(\\{\hat{\textbf{e}}_{n}\\})$ is obtained from
$\\{\textbf{m}_{n}\\}$ using equations (18) and (21) via mapping through
$\\{\beta\textbf{h}_{n}\\}$.

  3. 3.

Information on $U_{\text{mag}}(\\{\textbf{m}_{n}\\})$ is obtained either by
computing its first derivative (CPA approach) or its value directly (supercell
approach).

  4. 4.

Steps 1.-3. are repeated for different sets of values of
$\\{\textbf{m}_{n}\\}$ describing the magnetic states of interest.

At this point such ab initio data for many sets of $\\{\textbf{m}_{n}\\}$ are
fit via a linear regression to output a minimal set of internal energy
coefficients containing the
$\\{U^{(2)}_{nn^{\prime}},U^{(4)}_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}},\dots\\}$.
This procedure provides the values for a reduced number of magnetic
coefficients describing particular magnetic states of interest [25, 29, 5].
The additional dependence of $U_{\text{mag}}$ on crystal deformation can be
calculated by performing the four steps above for different lattice parameters
and distortions, which results in the internal energy coefficients as
functions of the crystal structure. Since the internal magnetic energy
$U_{\text{mag}}$ is the only material-dependent term of the Gibbs free energy,
the latter is directly given by Eq. (17) once all the terms and dependencies
of $U_{\text{mag}}$ have been obtained following the procedure described here.

In this work we investigate the finite-temperature magnetic properties of
La(Fe1-xSix)13 compound. The crystal structure of this material is of the type
NaZn13, which corresponds to the space group $Fm\bar{3}c$. The internal atomic
positions used in the simulations are those ones observed experimentally [41]
scaled by the relevant lattice parameter. An important trait of the
La(Fe1-xSix)13 compounds is that the Fe atoms sit on two non-equivalent sites
corresponding to Wyckoff symbols $96i$ and $8b$. This means that the
ferromagnetic state is in fact described in terms of two different magnetic
order parameters, $m_{96i}$ and $m_{8b}$. The results shown in section 3 are
such that $m_{8b}$ is self-consistently obtained along with Weiss fields that
fully minimise the Gibbs free energy for a chosen set of values of $m\equiv
m_{96i}$, and hence a set of temperatures. To achieve this we have followed
the iterative procedure to compute full equilibrium Weiss fields as done in
previous works [42, 43, 36].

The internal magnetic energy can thus be expressed as
$U_{\text{mag}}(m_{96i},m_{8b};\omega)$, where $\omega$ is the relative volume
change that we introduced in Eq. (3). In principle, a larger number of DLM-DFT
calculations should be done to extract such a full dependence, which also
describes non-equilibrium states. However, the primitive unit cell of
La(Fe1-xSix)13 contains 24 and 2 Fe atoms at these Wyckoff positions,
respectively. A large contribution to the derivatives of $U_{\text{mag}}$
might, therefore, come from those made with respect to $m_{96i}$. In this work
we have found that this applies to La(Fe1-xSix)13 ($x=0.12$) and so have
considered the approximation that the internal magnetic energy is a function
of $m\equiv m_{96i}$ only. In order to validate this approximation we
performed again all the calculations presented in section 3 but setting
$m_{8b}=0$, i.e. constraining the local moment orientations on these sites to
be completely disordered. Following this route we obtained qualitatively
similar results. The observed quantitative differences were very small, the
transition remains first-order and magnitudes of caloric effects and
spontaneous volume changes were almost identical. A significant quantitative
difference was that the computed linear magnetovolume coupling of $U^{(2)}$
was 8% smaller. Nonetheless, the role of magnetic order developing on the Fe
atoms on the $8b$ sites deserves a more careful finite-temperature analysis
which we plan to perform in future investigations using our supercell DLM
approach.

####  2.2.1 Electronic and/or magnetoelastic mechanisms

We illustrate how the analysis of DLM-DFT outputs can be used to investigate
first-order phase transitions and quantify their origin by studying the
simplest case of a ferromagnet. Since the unit cell of a ferromagnetic phase
contains a single crystal position, the magnitudes of the magnetic local order
parameters are the same for every site in the lattice. In other words, a
single value $m=|\textbf{m}_{n}|$ is sufficient to specify the state of
ferromagnetic order. We remark that such a scenario also applies approximately
to La(Fe1-xSix)13, as explained in the former section, as well as to fully
compensated antiferromagnetic triangular states, as observed in Mn-based
antiperovskite materials [44, 4]. Inclusion of external forces and spontaneous
material responses result in alterations to the unit cell volume, expressed in
terms relative to the paramagnetic limit,
$\omega=\frac{V-V_{\text{PM}}}{V_{\text{PM}}}$ [see Eq. (3)]. In this
situation Eq. (25) in units of energy per atom becomes

$\frac{1}{N_{\text{at}}}U_{\text{mag}}=\frac{1}{N_{\text{at}}}\langle\mathcal{H}_{\text{mag,int}}\rangle_{\text{tr}}=U^{(0)}(\omega)-U^{(2)}(\omega)m^{2}-U^{(4)}(\omega)m^{4}-\text{h.o.},$
(27)

where $N_{\text{at}}$ is the number of atoms, and

$\begin{split}&U^{(2)}(\omega)=-\frac{1}{2N_{\text{at}}}\sum_{nn^{\prime}}U^{(2)}_{nn^{\prime}}(\omega)\cos\Delta\theta_{nn^{\prime}}\\\
&U^{(4)}(\omega)=-\frac{1}{4N_{\text{at}}}\sum_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}}U^{(4)}_{nn^{\prime}n^{\prime\prime}n^{\prime\prime\prime}}(\omega)\cos\Delta\theta_{nn^{\prime}}\cos\Delta\theta_{n^{\prime\prime}n^{\prime\prime\prime}}\end{split}$
(28)

are effective internal energy coefficients containing the effect of the local
moment correlations via $\cos\Delta\theta_{nn^{\prime}}$. These are relative
angles between the local order parameters at sites $n$ and $n^{\prime}$ whose
study can be used to extract more free energy coefficients [38, 24]. Second
and higher order of these coefficients describe the dependence of the single-
site Weiss fields against $m$ as given by Eq. (26),

$\textbf{h}_{n,\text{int}}=U^{(2)}(\omega)m+2U^{(4)}(\omega)m^{3}+4U^{(6)}(\omega)m^{5}+\cdots.$
(29)

The typical parabolic behavior of the energy in the paramagnetic limit
($m\rightarrow 0$) is captured by

$\lim_{m\rightarrow
0}U_{\text{mag}}=U^{(0)}(\omega)=U^{(0)}(V_{\text{PM}})+\frac{1}{2}\gamma
V_{\text{PM}}\omega^{2},$ (30)

where $\gamma$ is the inverse of the compressibility, or bulk modulus, in the
paramagnetic state. For simplicity, we also assume a magnetovolume coupling
that is linear in $\omega$. For second, fourth, and sixth order coefficients
this means that

$\begin{split}&U^{(2)}(\omega)\approx U^{(2)}_{0}+\alpha^{(2)}\omega,\\\
&U^{(4)}(\omega)\approx U^{(4)}_{0}+\alpha^{(4)}\omega,\\\
&U^{(6)}(\omega)\approx U^{(6)}_{0}+\alpha^{(6)}\omega.\end{split}$ (31)

The magnetic Gibbs free energy in Eq. (17) therefore is

$\begin{split}\frac{1}{N_{\text{at}}}\mathcal{G}_{\text{mag,u}}=&-\textbf{B}\cdot\mu\textbf{m}+pV_{\text{PM}}\omega-
TS_{\text{mag}}\\\ &+U^{(0)}(V_{\text{PM}})+\frac{1}{2}\gamma
V_{\text{PM}}\omega^{2}-\left(U^{(2)}_{0}+\alpha^{(2)}\omega\right)m^{2}-\left(U^{(4)}_{0}+\alpha^{(4)}\omega\right)m^{4}-\text{h.o.}.\end{split}$
(32)

This result is minimized with respect to $\omega$ to yield

$\omega=\frac{1}{V_{\text{PM}}\gamma}\left(\alpha^{(2)}m^{2}+\alpha^{(4)}m^{4}\right)-\frac{p}{\gamma}.$
(33)

The terms in the Gibbs free energy in Eq. (32) can be arranged into different
orders of $m$ by using a Taylor expansion for the magnetic entropy,

$S_{\text{mag}}=k_{\text{B}}\sum_{n}\left(\ln
4\pi-\frac{3}{2}m_{n}^{2}-\frac{9}{20}m_{n}^{4}-\cdots\right),$ (34)

and Eq. (33) to give

$\begin{split}&\frac{1}{N_{\text{at}}}\mathcal{G}_{\text{mag,u}}=-k_{\text{B}}T\ln
4\pi+U^{(0)}(V_{\text{PM}})-\frac{V_{\text{PM}}p^{2}}{\gamma}-\textbf{B}\cdot\mu\textbf{m}\\\
&-\left(U^{(2)}_{0}-\frac{\alpha^{(2)}p}{\gamma}-\frac{3}{2}k_{\text{B}}T\right)m^{2}-\left(U^{(4)}_{0}+\frac{\left[\alpha^{(2)}\right]^{2}}{2V_{\text{PM}}\gamma}-\frac{\alpha^{(4)}p}{\gamma}-\frac{9}{20}k_{\text{B}}T\right)m^{4}-\text{h.o.}\end{split}$
(35)

Eq. (35) can be used to analyse the character of a magnetic phase transition
from the paramagnetic state. If, in the absence of an external magnetic field
($\textbf{B}=\textbf{0}$), such a transition is second-order, i.e., the order
parameter changes continuously from $m=0$ to $m\neq 0$ by lowering the
temperature, the corresponding transition temperature can be obtained by
solving $\frac{\partial^{2}\mathcal{G}_{\text{mag,u}}}{\partial
m^{2}}|_{m=0}=0$. This gives

$T_{\text{tr,sec}}=\frac{2}{3k_{\text{B}}}\left(U^{(2)}_{0}-\frac{\alpha^{(2)}p}{\gamma}\right).$
(36)

On the other hand, the transition becomes discontinuous when

$\lim_{m\rightarrow 0}\frac{\partial^{2}\mathcal{G}_{\text{mag,u}}}{\partial
m^{2}}\Bigg{|}_{T=T_{\text{tr,sec}}}=0^{-}.$ (37)

A first-order character arises, therefore, when the overall fourth order
coefficient at $T=T_{\text{tr,sec}}$ is negative, which gives rise to the
following condition

$U^{(4)}_{0}+\frac{\left[\alpha^{(2)}\right]^{2}}{2V_{\text{PM}}\gamma}+\frac{p}{\gamma}\left(\frac{3}{10}\alpha^{(2)}-\alpha^{(4)}\right)>\frac{3}{10}U^{(2)}_{0}.$
(38)

Figure 2:  Second- (continuous) and first- (discontinuous) order magnetic
phase transitions obtained after minimising Eq. (35). (a) Transition
temperatures from the paramagnetic ($m=0$) to a (partially-)ordered ($m>0$)
magnetic state for different values of $U_{0}^{(4)}/U_{0}^{(2)}$ and
increasing values of the lowest order of the magnetovolume coupling
$\alpha^{(2)}$. Continuous and discontinuous lines correspond to second- and
first-order magnetic phase transitions, respectively. Representative values of
$U_{0}^{(2)}=190$ meV, $\alpha^{(4)}=0$, $\gamma=200$ GPa, and
$V_{\text{PM}}=11.3$Å3 have been used (approximate for bcc Fe [5]), and
$T_{\text{ref}}=\frac{2U_{0}^{(2)}}{3k_{\text{B}}}=1470$K is taken as a
reference temperature. (b) Magnetic order parameter against $T$ for different
values of the coefficients studied in panel (a). Here $\alpha^{(2)}=0$ for the
first four, non-solid, coloured lines, while $U_{0}^{(4)}/U_{0}^{(2)}=0$ and
$\alpha^{(2)}=1700$ meV apply to the continuous black curve. Lower panels show
the Gibbs free energy below, at, and above $T_{\text{tr}}$. Results in panel
(d) are also equivalent to those obtained using $U_{0}^{(4)}/U_{0}^{(2)}=0$
and $\alpha^{(2)}=1470$ meV.

Eq. (38) demonstrates that a magnetovolume coupling for the pairwise
correlations, $\alpha^{(2)}$, always contributes to enhance the first-order
character of the magnetic phase transition regardless of its sign. On the
other hand, $U^{(4)}_{0}$ has to be positive to do so. In the absence of
external stimuli, a purely electronic origin for a first-order transition,
i.e. when $\alpha^{(2)}=\alpha^{(4)}=0$, occurs if $U^{(4)}_{0}$ is larger
than 30% of the absolute magnitude of $U^{(2)}_{0}$. This is shown in Fig.
2(a), where the transition temperature is plotted for different values of
$U^{(2)}_{0}$ and $\alpha^{(2)}$. Here continuous/dashed lines indicate
second-/first- order magnetic phase transitions. These results are obtained
after examining the dependence of $m$ on temperature resulting from the
minimization of the Gibbs free energy $\mathcal{G}_{\text{mag,u}}$, which we
show in the vicinity of the transition temperature for some illustrative cases
in panels (b) and (c-e), respectively. Indeed, increasing values of
$U^{(2)}_{0}$ and $[\alpha^{(2)}]^{2}$ enhance the first-order character of
the transition. Interestingly, Eq. (38) also shows that the application of a
hydrostatic pressure can change the character of the transition depending on
the signs and relative sizes of the magnetovolume coefficients.

##  3 The first-order magnetic transition and calculation of caloric effects
in La(Fe1-xSix)13 from ab initio theory

La(Fe1-xSix)13 is one of the most important and widely studied magnetic
materials classes in the field of caloric refrigeration. There is strong
motivation to obtain a fundamental understanding of the first-order magnetic
phase transition which is heightened by several intriguing aspects and trends.
The Curie transition temperature, varying within the range $T_{c}=190-260$K,
decreases together with an enhancement of the first-order character by
reducing the content of Si ($x=0.6-0.12$), despite the fact that LaFe13 does
not crystallize in experiment [8]. The unit cell volume spontaneously
increases by a significant order of magnitude of $1\%$ when the transition is
crossed by lowering the temperature from the paramagnetic state, the so-called
negative thermal expansion (NTE) [8, 45]. An itinerant electron metamagnetic
origin was directly proposed as the principal mechanism driving the first-
order character, ascribed to the observation of how the density of states
responds to magnetic stimuli [8, 46]. On the other hand, it is reasonable to
consider the broadly-invoked magnetoelastic coupling as a source of the
discontinuity, which is supported by the substantial spontaneous volume change
at the transition. Here we apply DLM-DFT theory to the intermetallic
La(Fe0.88Si0.12)13 ($x=0.12$) in order to quantify both electronic and
magnetoelastic mechanisms, and we conclude that it is latter mechanism which
is driving its first-order character.

All the DLM-DFT calculations presented in this work were performed using the
MARMOT code [35], an implementation of the DLM picture within the Korringa-
Kohn-Rostoker electronic structure method that employs the CPA to simulate
different states of magnetic disorder at finite-temperature (see section
2.1.3). We used an atomic sphere approximation (ASA) together with a scalar-
relativistic scheme. The input potentials for MARMOT were generated to
simulate the paramagnetic limit. Here Fe sites are occupied by two magnetic
species distinguished by the orientation of their spin moments (‘up’ and
‘down’) with equal concentrations. The local spin density approximation (LSDA)
was chosen to treat exchange and correlation effects and the single-site
scattering problem was solved using an angular momentum cutoff
$l_{\text{max}}=3$. The Brillouin zone integration was carried out to very
high numerical accuracy, by setting MARMOT’s parameter tolint to $10^{-5}$.
The CPA was also employed to account for the chemical disorder of Si and Fe
atoms on the $96i$ sites [47], see section 2.2 for information on the crystal
structure.

###  3.1 Construction of the Gibbs free energy: Weiss fields against magnetic
order parameters and lattice changes

Figure 3:  (a) Weiss fields and (b) density of states at the Fermi energy
computed for La(Fe0.88Si0.12)13 as functions of the ferromagnetic local order
parameter ($m$). Results are shown for different values of the lattice
constant. Data points and curves in the graphs correspond to DLM-DFT data and
their linear regressions, respectively.  Figure 4:  (a,b,c) Internal energy
coefficients obtained from a linear regression performed to
$\textbf{h}_{n,\text{int}}$, shown in Fig. 3(a), against the volume of the
unit cell (per magnetic atom). Continuous curves correspond to the second
linear regression performed. Bottom panels (d,e,f) show a similar analysis
made for the density of states at the Fermi energy considering our DLM-DFT
data in Fig. 3(b) and Eq. (39).

Fig. 3(a) shows the internal magnetic field $\textbf{h}_{n,\text{int}}$
computed for La(Fe0.88Si0.12)13, directly given by MARMOT [35], as function of
ferromagnetic local order parameter ($m$) for different lattice constants, see
Eq. (24). As illustrated in section 2.2.1, the analysis of how
$\textbf{h}_{n,\text{int}}$ depends on $m$ and the crystal structure provides
the Gibbs free energy of the material, which can be minimized at different
temperatures and external fields to obtain the equilibrium states.

In order to achieve this we have performed two consecutive linear regressions.
The first is done to $\textbf{h}_{n,\text{int}}$ against $m$ in order to
extract the internal energy coefficients
$\\{U^{(2)}(\omega),U^{(4)}(\omega),\dots\\}$ at different values of the
lattice constant [see Eq. (29)]. The latter relates to the volume relative to
the paramagnetic state, $\omega=\frac{V-V_{\text{PM}}}{V_{\text{PM}}}$, where
we have considered $V_{\text{PM}}=14.3$ Å3 per magnetic atom (total unit cell
volume of $V_{\text{PM}}=371.8$ Å3 with $a\approx 11.45$Å) as a reference
value in order to get a closer agreement with experiment. The corresponding
regression lines are shown as curves in Fig. 3 and the coefficients given in
the top panels of Fig. 4 as data points. We observe a pronounced linear
behaviour for the lowest, leading, second order term $U^{(2)}(\omega)$. This
means that there is a substantial magnetovolume coupling despite the fact that
$U^{(4)}$ and $U^{(6)}$ barely depend on $V$. The second linear regression is
applied to extract another set of magnetovolume coefficients, shown in table
1, which captures such a linear dependence as given in Eq. (31). Indeed,
$\alpha^{(2)}$ is the largest and most significant term.

$\alpha^{(2)}$ | $\alpha^{(4)}$ | $\alpha^{(6)}$
---|---|---
523 meV | -6 meV | -56 meV
Table 1: Ab initio computed magnetovolume coefficients obtained for
La(Fe0.88Si0.12)13 from the analysis of the Weiss fields at different values
of the magnetic order parameter and lattice constant.

In Fig. 3(b) and bottom panels of Fig. 4 we show a similar analysis for the
computed density of states at the Fermi energy, $D(E_{\text{F}})$. Such a
calculation is necessary in order to obtain the change of the electronic
entropy following Eq. (12), which will be used in the next section for the
calculation of magnetocaloric and barocaloric effects. We have found that an
expansion up to fourth order suffices to model the density of states at
$E_{\text{F}}$:

$D(E_{\text{F}})=d^{(0)}+d^{(2)}m^{2}+d^{(4)}m^{4}.$ (39)

As can be seen in Fig. 3(d,e,f), the second linear regression of these
coefficients against the volume,
$\\{d^{(0)}(\omega),d^{(2)}(\omega),d^{(4)}(\omega)\\}$, has required the
inclusion of a parabolic magnetovolume coupling.

While the gradient of $\textbf{h}_{n,\text{int}}$ in the paramagnetic limit
($m\rightarrow 0$, described by $U^{(2)}$) gives a major contribution to the
value of $T_{c}$ [see Eq. (36)], the deviation of this linear dependence at
higher values of $m$ caused by higher order interactions can generate a first-
order character. However, $U^{(4)}$ and $U^{(6)}$ are not only small but
negative. This directly implies that such a potential, electronic, source does
not contribute positively to the generation of a discontinuity at the
transition, as given by Eq. (38). On the other hand, we have found that
$\alpha^{(2)}$ is large enough to do so: $\frac{[\alpha^{(2)}]^{2}}{2\gamma
V_{\text{PM}}U^{(2)}_{0}}=0.66>\frac{3}{10}$, where $U^{(2)}_{0}=24\text{
meV}$, and an estimation of the value of the bulk modulus $\gamma=97$ GPa has
been directly taken from experiment [48, 49]. In other words, the first-order
phase transition in La(Fe0.88Si0.12)13, and consequent giant caloric effects,
which we address in the following, are driven by a strong magnetovolume
coupling.

###  3.2 Magnetocaloric and barocaloric effects

Figure 5:  Temperature dependence of (a) magnetic local order parameter and
(b) relative volume change, $\omega=\frac{V-V_{\text{PM}}}{V_{\text{PM}}}$,
computed for La(Fe0.88Si0.12)13 without external stimuli and applying an
external magnetic field $B=2$ T and a hydrostatic pressure $p=2.1$ kbar.
Corresponding magnetocaloric and barocaloric effects quantified by isothermal
entropy changes are shown in the right-hand-side panels (c) and (d),
respectively. Their total values are composed of magnetic and electronic
contributions.

We have extracted all the qualitatively and quantitatively relevant internal
energy coefficients, $\\{U^{(2)}(\omega),U^{(4)}(\omega),\dots\\}$, and their
dependence on volume, which provides
$U_{\text{mag}}(m,\omega)=-U^{(2)}(\omega)m^{2}-U^{(4)}(\omega)m^{4}-U^{(6)}(\omega)m^{6}$.
We can, therefore, now access the Gibbs free energy (per magnetic atom),

$\mathcal{G}_{\text{mag,u}}=U_{\text{mag}}-TS_{\text{mag}}+\frac{1}{2}\gamma
V_{\text{PM}}\omega^{2}-B\mu m+pV_{\text{PM}}\omega.$ (40)

Note that the DLM-DFT computational analysis that we have done here has
yielded more coefficients than those used in Eq. (32), since the latter
contained a reduced set for illustrative purposes. Panels (a) and (b) of Fig.
5 show the temperature dependence of $m$ and $\omega$ obtained after
minimising $\mathcal{G}_{\text{mag,u}}$ at different values of $T$ with
respect to these parameters. The effects of applying an external magnetic
field $B=2$T and a hydrostatic pressure $p=2.1$kbar are also plotted. Indeed,
we find a first-order phase transition from the paramagnetic to ferromagnetic
states. The computed Curie temperature is about $T_{c}=188$K and the
spontaneous volume change at the transition is approximately
$\Delta\omega=0.7$%, which are in good agreement with experimental
measurements.

Table 2 summarises a comparison of the results obtained with experiment. The
application of pressure $p$ reduces $T_{c}$ while enhancing the first-order
character, also following experimental trends. The value found is $\Delta
T_{c}/\Delta p\approx-10$ K/kbar, which is very large in accordance with the
strong magnetovolume coupling calculated earlier and matches very well the
experimental measurement of $[\Delta T_{c}/\Delta p]^{\text{exp}}\approx-9$
K/kbar [8]. In the right-hand-side panels of the same figure we show the
corresponding magnetocaloric and barocaloric effects quantified by the
isothermal entropy change. The total value of each of these quantities at the
transition is giant as measured experimentally, $|\Delta
S_{\text{tot}}|=15-20$ J/kgK, being conventional (negative) and inverse
(positive), respectively. An important observation is that a large component
of such responses comes from the electronic entropy, representing more than
50% of the total change. This is a consequence of the density of states at the
Fermi energy strongly decreasing/increasing in response to the lattice
expansion/compression caused when the transition is crossed by applying a
magnetic field/pressure, see Fig. 3(b) and Eq. (12). A further inspection of
Fig. 3(b) reveals that the effect coming from the change of $m$ at the
transition, i.e. $\Delta D(E_{\text{F}})/\Delta m$, changes sign depending on
the value of the lattice constant. Importantly, while the large negative
thermal expansion is tied to the strong magnetovolume coupling, the sharp
change of $m$ is independent of the fact that the origin of the first order
character of the transition is not electronic.

|  $T_{c}$ (K) |  $\frac{\Delta T_{c}}{\Delta p}$ $\left(\frac{\text{K}}{\text{kbar}}\right)$ |  $\Delta\omega$ (%) | MCE (J/kgK) | BCE (J/kgK) | Refs.
---|---|---|---|---|---|---
Theory | 188 | -10 | 0.7 | -10 | +17 | this work
Experiment | 195-208 | -9 | 1 | -14 | +9 | [45, 8, 10]
Table 2: Comparison of computed theoretical results for La(Fe0.88Si0.12)13
with experiment. The magnetocaloric (MCE) and barocaloric (BCE) effects are
quantified by those isothermal entropy changes observed at the Curie
transition temperature that are obtained by applying $B=2$ T and $p=2.1$ kbar,
respectively. The experimental value for the MCE taken from Ref. [45]
corresponds to a material with a slightly different composition in comparison
with the one studied in this work. The experimental value given for the BCE
has been taken from Ref. [10], which reports measurements made for a La-Fe-Si-
Co compound that exhibits a less pronounced first-order character with a
higher transition temperature.

###  3.3 On the origin and nature of the transition and the role of itinerant-
electron metamagnetism

Our calculations, reported in section 3.1, quantify mechanisms for the first-
order character of the paramagnetic-ferromagnetic phase transition in
La(FexSi1-x)13. We have found that the strong magnetovolume coupling is the
principal origin. On the other hand, those complex itinerant electron effects
which are entwined with the transverse fluctuations of the local moment
orientations, lead to significantly large high order coefficients $U^{(n>2)}$
in our theory. The negative signs of these coefficients, however, act to
oppose the occurrence of a first order discontinuity [see Eq. (38) and Fig.
4]. While seminal works carried out for La(FexSi1-x)13 and similar compounds
[8, 50, 51] focus on itinerant electron metamagnetism and claim a crucial role
for the coupling of the electronic structure with magnetic order, they also
acknowledge the great importance of large magnetoelastic effects and their
role in the first-order character of the transition.

In the context of itinerant electron metamagnetism, several authors have
investigated the effect of variations in the sizes $\mu_{n}$ of the local
moments, by means of fixed-spin-moment (FSM) calculations [52, 53, 54]. The
terms produced by such calculations, involving longitudinal local moment size
fluctuations, are fundamentally different from the $U^{(n>2)}$ internal energy
coefficients computed in our study based on transverse local moment
orientation fluctuations. The application of the FSM approach to
La(FexSi1-x)13 has revealed the emergence of several shallow minima in the
total energy as a function of the local moment magnitude in the ferromagnetic
state ($m=1$) when the lattice is contracted. Such a feature is proposed as a
potential factor for the low thermal hysteresis observed in experiment.

Related itinerant electron physics is captured in the DLM theory in cases
where the size of a local moment on a site is found to depend sensitively on
the orientations of the local moments on the sites surrounding it,
$\mu_{n}(\\{\hat{\textbf{e}}_{n}\\})$, and hence the extent and nature of
magnetic order. Indeed, such insights for La(FexSi1-x)13 could resonate with
some features found in early DFT and DLM-DFT studies of face-centered-cubic
iron [55, 56]. Here, as the lattice parameter is decreased, the metal’s $T=0$K
magnetic state changes from ferromagnetic with a large, $>$2$\mu_{B}$,
magnetisation per atom, to ferromagnetic with a smaller magnetisation, onto
antiferromagnetic and eventually non-magnetic. In the paramagnetic DLM state,
the size of the local moment on a Fe site, when averaged over all
configurations, collapses to zero for lattices with parameters below a
critical value.

Figure 6:  Computed local moment magnitudes at both $96i$ and $8b$ sites in
both (a) the disordered local moment (DLM, $m=0$) and (b) the ferromagnetic
states as functions of the unit cell volume. The paramagnetic volume in this
work, which expands when the ferromagnetic state stabilizes, is indicated by a
vertical dashed line.

We have investigated similar circumstances for La(Fe0.88Si0.12)13. In figure 6
we show the computed local moment magnitudes for both the paramagnetic state
as well as for the ferromagnetic phase plotted against the unit cell volume.
We observe that for the computed paramagnetic volume, $V_{\text{PM}}$, the
local moment of an iron atom at a $96i$ site in the DLM state is reasonably
large, $\mu_{\text{Fe},96i}=1.7\mu_{\text{B}}$, and in agreement with
photoemission spectroscopy measurements, $\mu_{\text{exp}}=1.5\mu_{\text{B}}$
[57]. When the first-order transition takes place, the lattice expands and
ferromagnetic order develops, both events causing an increase of the local
moment sizes. Our calculations fully include the magnetovolume effect but are
done throughout using self-consistent DLM potentials produced for the
paramagnetic state, $m=0$. In order to investigate the potential influence of
fully self-consistent calculations, and hence sensitivity of the moment sizes
to magnetic order and unit cell volume, we repeated the calculation of the
leading magnetovolume coefficient using potentials from the self-consistent
ferromagnetic calculations, $m=1$. The result obtained is $\alpha^{(2)}=415$
meV, which compares reasonably with the DLM outcome given in table 1 of $523$
meV and indicates that the effects of the local moments size variations are
not overly significant. Further investigations along these lines, however, are
needed to fully quantify these aspects.

##  4 Conclusions

Around a first-order, discontinuous, magnetic phase transition a material can
exhibit a large caloric response to the application of a small or moderate
external field which results from an abrupt change of the state of thermal
order of its local magnetic moments. Analysis shows that such a change
originates from a strong dependence of second order coefficients in the
material’s internal magnetic energy on attributes of the material’s crystal
structure and electronic glue. There is a consequent shaping of the material’s
Gibbs free energy with several competing minima and hence a sharp
discontinuity in the magnetic order parameters at the transition temperature.
Substantial magnetostructural or magnetovolume couplings are well understood
sources of first-order transitions in this regard [1]. Purely electronic
mechanisms, however, are at an earlier stage of study and have a rich
potential for new caloric effects [6, 7, 3].

In this work we have applied the disordered local moment (DLM) picture in
density functional theory (DFT), as a predictive, first-principles tool to
quantify the extent of and interplay between magneto-structural and electronic
sources to the origin of the famous first-order paramagnetic-ferromagnetic
phase transition in La(Fe1-xSix)13 for $x=0.12$. We have found that the
principal origin of such a transition and of the consequent giant
magnetocaloric and barocaloric effects is a strong magnetovolume coupling
while purely itinerant-electron mechanisms linked to transeverse fluctuations
act in opposition to the occurrence of the first-order character. The origin
of the transition is linked to large changes of the magnetic order parameter
driven by a sharp dependence of magnetic local moment interactions on the
value of the lattice constant. The substantial negative thermal expansion
computed here and observed in experiment mirrors this fact. The itinerant
electron structure, however, which is fundamental to the determining the
crystal structure as well as the local moments and magnetic properties,
adjusts strongly as the lattice changes and leads to the total isothermal
entropy changes with very large magnetic and electronic contributions. Our
work demonstrates that the coupling of the electronic structure in its
entirety with the lattice at finite temperature is pivotal for the magnetic
and caloric properties of La(Fe1-xSix)13. Further investigation of the effects
of substituting a small proportion of the Fe atoms with other elements from
this perspective show promise for improving and tuning the properties as well
as the understanding of this important materials class.

## Acknowledgments

We are grateful to A. Planes for fruitful discussions. E. M.-T. acknowledges
funding received from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No.
101025767, support from MCIN/AEI/10.13039/501100011033 (Spain) under Grant No.
PID2020-113549RB-I00/AEI, and computing time provided at the NHR Center
NHR4CES at RWTH Aachen University (project number p0020248). The latter is
funded by the Federal Ministry of Education and Research, and the state
governments participating on the basis of the resolutions of the GWK for
national high performance computing at universities. Presented results have
been achieved also using computing time granted by the supercomputer of the
Department of Computational Materials Design, operated by the Max Planck
Computing and Data Facility in Garching. This work is supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Projektnummers 405621160 and 405621217. J. B. S. acknowledges support from
EPSRC (UK) Grant No. EP/M028941/1 (PRETAMAG project).

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