# Multiple magnetic interactions and large inverse magnetocaloric effect in
TbSi and TbSi0.6Ge0.4

Ajay Kumar<EMAIL_ADDRESS>Ames National Laboratory of the U.S. Department
of Energy, Iowa State University, Ames, Iowa 50011, USA  Prashant Singh  Ames
National Laboratory of the U.S. Department of Energy, Iowa State University,
Ames, Iowa 50011, USA  Andrew Doyle  Ames National Laboratory of the U.S.
Department of Energy, Iowa State University, Ames, Iowa 50011, USA  Deborah L.
Schlagel  Ames National Laboratory of the U.S. Department of Energy, Iowa
State University, Ames, Iowa 50011, USA  Yaroslav Mudryk  Ames National
Laboratory of the U.S. Department of Energy, Iowa State University, Ames, Iowa
50011, USA

###### Abstract

We present a comprehensive investigation of the electronic structure,
magnetization, specific heat, and crystallography of TbSi (FeB structure type)
and TbSi0.6Ge0.4 (CrB structure type) compounds. Both TbSi and TbSi0.6Ge0.4
exhibit two antiferromagnetic (AFM) transitions at T${}_{\rm N1}\approx$ 58 K
and 57 K, and T${}_{\rm N2}\approx$ 36 K and 44 K, respectively, along with an
onset of weak metamagnetic-like transition around 6 T between TN1 and TN2.
High-resolution specific heat (CP) measurements show the second- and first-
order nature of the magnetic transition at TN1 and TN2, respectively, for both
samples. However, in the case of TbSi, the low-temperature (LT) AFM to high-
temperature (HT) AFM transition takes place via an additional AFM phase at the
intermediate temperature (IT), where both LT to IT AFM and IT to HT AFM phase
transitions exhibit a first-order nature. Both TbSi and TbSi0.6Ge0.4 manifest
significant magnetic entropy changes ($\Delta S_{\rm M}$) of 9.6 and 11.6
J/kg-K, respectively, for $\Delta\mu_{0}H$=7 T, at TN2. The HT AFM phase of
TbSi0.6Ge0.4 is found to be more susceptible to the external magnetic field,
causing a significant broadening in the peaks of $\Delta S_{\rm M}$ curves at
higher magnetic fields. Temperature and field-dependent specific heat data
have been utilized to construct the complex H-T phase diagram of these
compounds. Furthermore, temperature-dependent x-ray diffraction measurements
demonstrate substantial magnetostriction and anisotropic thermal expansion of
the unit cell in both samples.

##  I  Introduction

Rare earth (RE)-Si-Ge compounds with varying stoichiometry have been a subject
of considerable interest within the scientific community for several decades
due to their intriguing properties, such as giant magnetocaloric effect
(GMCE), colossal magnetoresistance (CMR), metal-insulator transition (MIT),
Griffiths phase, kinetically arrested magnetostructural transitions, and
anomalous Hall effect, among others [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In
this family of materials, the RE5(SixGe1-x)4 compounds have garnered extensive
attention, especially after the discovery of GMCE near room temperature in
Gd5(Si2Ge2), a phenomenon associated with the peculiar first-order
magnetostructural phase transition at $\sim$276 K [1, 8, 12, 13]. The exchange
interactions between the RE ions in these compounds are indirect RKKY-type,
mediated by conduction electrons. The sign of these interactions is determined
by the distance between the rare earth ions, making them susceptible to
external perturbations such as temperature, hydrostatic pressure, chemical
pressure (doping), and magnetic field [7, 4, 14, 15]. A strong coupling
between the crystal structure and magnetic properties have been observed in
Gd5(SixGe1-x)4 pseudo-binary system. These compounds with $0<x\leq 0.2$ on
heating show a first-order transition from a low-temperature ferromagnetic
(FM) state having a Gd5Si4 [O(I)]-type crystal structure to an intermediate
antiferromagnetic (AFM) state with a Sm5Ge4 [O(II)]-type structure, and then
to a high-temperature paramagnetic (PM) phase having O(II)-type structure. In
these Ge-rich compounds the Curie temperature (TC) of FM-AFM transformation
increases and the Néel temperature (TN) remains nearly unchanged with
increasing $x$ [16, 17, 18]. For $0.2<x\leq 0.38$ there is a two-phase mixture
of O(II) and monoclinic structures at room temperature, and for $0.38<x\leq
0.5$, the system exhibits a first-order transformation from the same low-
temperature O(I)-type FM phase to a high-temperature PM phase with a
monoclinic crystal structure. Finally, Gd5(SixGe1-x)4 shows a second order
transition from FM to PM state with the same O(I)-type crystal structure in
both the magnetic phases for $0.5<x\leq 1$ [16, 17, 18].

The markedly different electronic structure of Tb3+ ($4f^{8}$; S=L=3, J=6), as
compared to Gd3+ ($4f^{7}$; J=S=7/2, L=0) ions having isotropic spin-only
magnetic moments, results in the strong directional character in Tb-based
compounds due to L$\neq$0 [3, 20, 2, 3, 4, 5, 6, 19, 21, 22]. Therefore, Gd is
often substituted with Tb ions due to their similar ionic radii and valence
state (both virtually trivalent) [6, 23, 19] to study the effects of single-
ion anisotropy on physical behavior of materials. For instance, Tb5Si2.2Ge1.8
shows a large anisotropic temperature and field dependent magnetization, and
hence magnetocaloric effect, where a GMCE is observed along the $a$(broad)-
and $c$ (sharp)-axes, but not along the $b$-axis [20, 21]. An anisotropic
electrical resistivity and magnetoresistance are also observed in the
Tb5Si2.2Ge1.8 with a CMR of 160% only obtained along the $a$-axis, whereas
$b$\- and $c$\- axes display a significantly smaller MR of around 8% and 5%,
respectively [3]. Similarly, the anisotropic magnetic exchange interactions
with positive Weiss temperature ($\theta_{\rm P}$) (FM interactions) along
$a$\- and $c$-axes, and negative (AFM coupling) along $b$-axis have been
observed in Tb5Ge4, unlike Gd5Ge4, where $\theta_{\rm P}$ remains positive
along all crystallographic directions [24, 25]. This change in magnetic
interactions in Tb5(SixGe1-x)4, despite similar RE-RE distances as compared to
Gd5(SixGe1-x)4 is attributed to the large orbital moment and hence directional
character of Tb3+ ions. Complex low-temperature phase diagrams have also been
observed in Sm5SixGe4-x [26], Ho5(SixGe1-x)4 [27], Yb5(SixGe1-x)4 [28], etc.,
by virtue of change in ionic radii, and electronic configurations of localized
4$f$ and itinerant conduction electrons, along the lanthanide series.

In this context, the systematic investigation of pseudo-binary compounds RE-
SixGe1-x adds a crucial piece of knowledge to our understanding and control of
the fundamental and functional properties of rare earth intermetallic
compounds [15, 29, 30, 31]. These compounds crystallize in either CrB-type
$Cmcm$ or closely related FeB-type $Pnma$ structure and display complex
composition-temperature ($x$-T) phase diagrams [32, 33, 34, 35, 36, 37]. In
HoSixGe1-x, the $x=$1 sample shows both CrB (35.5%)- and FeB (64.5%)- type
structures at room temperature, whereas other members adopt CrB-type only [35,
36]. The $x>0.2$ samples display two distinct AFM structures, where the low-
temperature (LT) lock-in phase with wave vector $q=(1/2,0,1/2)$ undergoes a
first-order phase transition (FOPT) into a high-temperature (HT)
incommensurate structure associated with $q=(q_{x}$, 0, $q_{z})$ at $T_{\rm
ic}$, and finally into PM state via a second-order phase transformation (SOPT)
at $T_{\rm N}$ [35]. Both $T_{\rm ic}$ and $T_{\rm N}$ increase with $x$ [35].
On the other hand, the magnetic structure of $x\leq 0.2$ samples remains
incommensurate down to 1.2 K [35]. The ErSixGe1-x shows even more complex
$x$-T phase diagram with no Lifshitz point, and two commensurate and two
incommensurate magnetic structures, which vary with both $x$ and T [37]. The
application of magnetic field (H) can provide an additional degree of freedom
to manipulate these $x$-T phase diagrams. For instance, a strong coupling
between localized 4$f$ moments and conduction electrons lead to anomalous
magnetic field-induced phase transitions in GdSi at 2.5, 3.8, and 19.5 T [15].
Feng et al. later reported the comprehensive H-T phase diagram of GdSi,
revealing four different types of AFM interactions in the sample, the
strengths and directions of which depend strongly on lattice directions [38].
Several experimental and theoretical investigations have aimed to understand
the metal-insulator transition in amorphous Gd-Si systems [39, 40, 9].
Moreover, large magnetic anisotropy and GMCE have been observed in single
crystals of PrSi and PrGe [29, 30].

P. Schobinger-Papamantellos $et$ $al.$ reported the crystal and magnetic
structure of TbSixGe1-x for $0\leq x\leq 1$ using temperature-dependent
neutron diffraction measurements [32, 33, 34]. All samples, except $x=1$,
crystallize in the orthorhombic CrB-type structure. The $0\leq x<0.6$ samples
exhibit LT planar AFM interactions (in $ac$ plane) with wave vector $q=$ (1/2,
0, 1/2), which undergo a HT uniaxial AFM (along $c$ axis) having $q=$ (0, 0,
1/2) (at Tic) via first-order transition, and finally experience a second-
order AFM to PM transition (at TN) [32, 33]. The $0.6>x>1$ samples, on the
other hand, show a transition from LT AFM with $q=$ (1/2, 0, 1/2) to a
sinusoidally modulated incommensurate AFM structure having q= (qx, 0, qz) at
Tic [33]. More importantly, at the critical doping of $x=$0.6 (TbSi0.6Ge0.4),
both commensurate and incommensurate magnetic structures coexist in equal
proportion in the HT phase region [33]. Unlike HoSixGe1-x, the TN in the case
of TbSixGe1-x remains almost invariant; however, Tic increases with $x$ [35,
33]. Further, TbSi ($x=1$) crystallizes in the FeB-type structure and displays
even more complex magnetic interactions. The low-temperature phase (below 36
K) is planar AFM (in the $ab$ plane) with $q=$ (0, 1/2, 0), which exhibits a
first-order transition into a three-dimensionally modulated incommensurate AFM
phase with $q=$ (qx, qy, qz) [34]. Interestingly, the incommensuration
persists predominantly up to 42 K only, and above this temperature, it tends
to commensurate with q $\approx$ (0, 1/2, 1/8) up to TN = 57 K, followed by a
second-order AFM to PM phase transition [34]. Therefore, it is vital to study
the the nature and behavior of this additional incommensurated magnetic
structure in case of the TbSi in this narrow temperature range. Further, the
detailed study of these multiple magnetic interactions and their evolution
with an external magnetic field is missing together with the theoretical
insights into the various unresolved magnetic phenomena in the RE-Si-Ge
systems. In particular, the response of the observed first-order transition to
applied magnetic field is important for understanding the potential for
magnetocaloric effect and other magnetoresponsive phenomena in the underlined
compounds [41, 42, 43].

In this manuscript, we theoretically analyzed the structural stability and
experimentally performed detailed magnetization, specific heat, and
temperature-dependent x-ray diffraction (XRD) measurements on the two most
intriguing members of this series, namely the FeB-type TbSi and the CrB-type
TbSi0.6Ge0.4 samples. We observe a significant broadening in the magnetic
entropy change ($\Delta S_{\rm M}$) curves in the case of TbSi0.6Ge0.4 at
higher magnetic fields compared to TbSi. The high-resolution specific heat
measurements are used to uncover this phenomenon, which indicates the
evolution of additional field-induced AFM interactions in the HT AFM phase
region of this sample. Furthermore, we unravel an additional first-order
magnetic transition in the vicinity of the LT AFM phase of TbSi. Both samples
exhibit finite magnetostriction at both LT and HT AFM transitions and
demonstrate anisotropic thermal expansion of the unit cell in the magnetically
ordered as well as PM state. We construct complex H-T phase diagrams for TbSi
and TbSi0.6Ge0.4 using the temperature- and field-dependent specific heat
data.

##  II  Methodology

###  II.1 Theory

We employed density-functional theory (DFT) as implemented within Vienna ab
initio simulation package (VASP) [44] for electronic-structure calculation
based on the Perdew, Burke, and Ernzerhof (PBE) exchange-correlation
functionals [45]. The energy and Hellmann-Feynman force convergence criteria
for full relaxation (volume and atomic coordinates) are 10-7 eV and 10-6 eV/Å,
respectively. A kinetic-energy cutoff of 520 eV was used for the plane-wave
basis set. Spin-polarized calculations were performed due to the presence of
magnetic elements. To enforce the localization of the $f$-electrons, the
effective onsite Coulomb (U) and exchange (J) interaction parameters are set
to be 6.00 and 0.90 eV, respectively [46]. Monkhorst-Pack [47] k-mesh grids of
3×5×4 and 2×1×2 was used to sample the Brillouin zone in $Pnma$-TbSi and
$Cmcm$-TbSi0.6Ge0.4 phases, respectively, during relaxation, while the mesh-
size was doubled to accurately determine formation enthalpy and electronic-
structure properties. The partially ordered TbSi1-xGex ($x=$0–1) structures
were created using Alloy Theoretic Automated Toolkit (ATAT) [48] to assess
their thermodynamic (formation enthalpy) and structural (phonon) stability.
The formation energy (Eform=Etot-$\sum_{i}$niEi, i=Tb,Ge,Si) was used to
determine the phase stability (at 0 K) of TbSi1-xGex ($x=$0–1) intermetallics.
The Etot is total energy of disorder alloys while {ni, Ei} are number of atoms
of type “i” in their ground state unit cell, i.e., {Tb, Ge, Si}. The phonon
spectra and density of states are calculated using the finite displacement
method as implemented in Phonopy software. The density-functional perturbation
theory (DFPT) was used to construct force constant matrix needed to calculate
phonon dispersion [49].

###  II.2 Experimental

The TbSi and TbSi0.6Ge0.4 samples were prepared by arc melting stoichiometric
amounts of pure Tb, Si, and Ge elements in an argon (Ar) atmosphere. The
terbium metal was provided by the Materials Preparation Center of Ames
National Laboratory and had an overall purity better than 99.9 % with respect
to all elements, silicon was supplied by Alfa Aesar (99.9995 %), and germanium
by Meldform Metals ($\sim$99.99 %). Prior to the sample preparation, a Zr
getter was melted to minimize the presence of residual or impurity gases in
the chamber. Each sample was flipped and remelted 4-5 times to ensure the
homogeneity. The weight loss during the melting process was found to be less
than 1 wt. % for both samples. While the TbSi was found to be single phase
after melting, the TbSi0.6Ge0.4 sample required additional heat-treatment and
was post annealed at 900 °C for 7 days in an highly pure helium-filled quartz
tube. Room temperature x-ray powder diffraction (XRD) patterns were recorded
for both samples using a PANalytical X’Pert Pro diffractometer with Cu-Kα
($\lambda$ =1.5406 Å) radiation in the Bragg-Brentano geometry. Low-
temperature XRD data were collected in the temperature range of 10-300 K using
a Rigaku TTRAX powder diffractometer equipped with a rotating anode Mo Kα
source and a continuous helium flow cryostat [50]. The data were recorded in
cooling mode after stabilizing the sample at each temperature. The Rietveld
refinement of the XRD data was performed using the FullProf suite, employing a
pseudo Voigt peak shape and linear interpolation between background points.
The refinement process started with the 10 K data, followed by the Rietveld
refinement of the high-temperature data, where the lattice parameters, overall
B-factors, and atomic positions were the only varying parameters [51].
Temperature and magnetic field-dependent magnetization measurements were
conducted using a superconducting quantum interference device (SQUID) system
from Quantum Design, USA (model MPMS XL-7). The temperature dependence of
magnetization was measured using the following protocols: on warming in
applied magnetic field after zero-field cooling (ZFC), on cooling in applied
magnetic field (FCC), and on warming in applied magnetic field after cooling
in the same applied field (FCW). To record the field dependent magnetization
isotherms at different temperatures, we reduce the magnetic field from 70 kOe
to 500 Oe in linear mode and then from 500 Oe to 0 Oe in the oscillatory mode
before moving to next temperature (in heating mode) to avoid any remanence in
the samples, keeping their low coercivity ($\sim$10 Oe) in mind. The
temperature and field dependent specific heat measurements were performed
using a physical property measurement system (PPMS) from Quantum Design, Inc,
employing both standard relaxation technique as well as long heat pulse
method. Further details about the measurement protocols are provided along
with the discussion of the specific heat data.

##  III  Results and discussion

###  III.1 Structural stability

Figure 1: (a) The formation energy difference ($\Delta$Eform) between $Pnma$
and $Cmcm$ crystal structures for partially disordered TbSi1-xGex ($x=$0–1)
compounds calculated using DFT. (b, c) Phonon dispersion of TbSi and
TbSi0.6Ge0.4 in $Pnma$ and $Cmcm$ space group, respectively.

In order to understand the structural stability of pseudo-binary TbSi1-xGex
compounds, we plot the formation energy difference ($\Delta$Eform) between the
$Pnma$ (FeB-type) and $Cmcm$ (CrB-type) orthorhombic phases in Fig. 1(a) for
the $x=$0–1. The calculated Eform of the $x$=0 is -0.78 and -0.75 eV/atom, and
for $x=$1 is -0.57 and -0.88 eV/atom in the $Pnma$ and $Cmcm$ phases,
respectively, which indicate that TbSi ($x=$0) and TbGe ($x=$1) compounds
stabilize in the $Pnma$ and $Cmcm$ structures, respectively, consistent with
the reported neutron diffraction (ND) data [32, 33, 34]. The ND measurements
provide a lower limit of $x=$0.1 (10% Ge), sufficient to transform the TbSi
from the $Pnma$ to $Cmcm$ structure [33]. Therefore, in order to further
pinpoint the critical limit of Ge required for this structural transformation
in TbSi1-xGex, we calculate the $\Delta$Eform in a step of 2.5% in the
critical regime ($x=$0–0.1), as shown in Fig. 1 (a). Interestingly, the
$\Delta$Eform shows a crossover from $Pnma$ to $Cmcm$ orthorhombic phase at
2.5 at.% Ge, i.e., the $Pnma$ phase is stable only from 0 to 2.5 at. % Ge,
while $Cmcm$ phase is energetically more favourable for Ge $>$2.5 at.%. Now,
we will focus on the FeB-type ($Pnma$) TbSi and CrB-type ($Cmcm$) TbSi0.6Ge0.4
samples, explored in this study, to understand why Ge-doped TbSi0.6Ge0.4
adopts the $Cmcm$ structure instead of the parent crystal structure of binary
TbSi. We hypothesize that the origin of this change lies in local structural
features such as atomic distortion or displacement. Before delving deeper into
the discussion, for the clarity of readers, we want to emphasize that terbium,
being in an open $f$-shell structure with a $+3$ oxidation state (i.e.,
$4f^{8}$ electronic state), has its magnetic nature accounted for by placing
$f$-states into the valence in both thermodynamic and electronic structure
evaluations [52]. To study the local distortion, the TbSi0.6Ge0.4 composition
was fully relaxed in $Cmcm$ and $Pnma$ structures using DFT to obtain the
optimized atomic coordinates. The $Cmcm$ phase was kept at a fixed
experimental volume. For the $Pnma$-TbSi0.6Ge0.4 phase created using the
$Pnma$-TbSi crystal structure, we performed full lattice relaxation to obtain
the optimal volume and coordinates. Using the experimental crystal structure
as a reference, the average vector atomic displacement in the $Cmcm$ and
$Pnma$ phases of TbSi0.6Ge0.4 was calculated as 0.314 and 0.228, respectively.
The method to estimate atomic displacements in complex lattices has been
discussed in Ref. [53]. The increased vector atomic displacement in the $Cmcm$
phase suggests higher charge activity, i.e., higher distortion, improves the
phase stability, as the $Cmcm$ phase has higher crystal symmetry compared to
$Pnma$. The higher local distortion in the case of TbSi0.6Ge0.4 agrees with
experimental structural analysis discussed below, and is consistent with a
recent study where the degree of distortion was found to be correlated with
both bond strength and phase stability in disordered materials [53].

The dynamical stability of TbSi and TbSi0.6Ge0.4 compounds is further
confirmed via phonon dispersion curves, as shown in Figs. 1(b, c),
respectively. The dispersion shows no obvious imaginary phonon modes for the
two experimentally synthesized phases. The low-frequency contribution in TbSi
comes from mixed Tb/Si atoms, while the Ge-doped case shows phonon softening.
The low, mid, and high-frequency contributions in TbSi0.6Ge0.4 come from Tb,
Ge, and Si, respectively [see Fig. 1(c)]. This suggests that Tb atoms make the
main contribution to the low-energy phonons, which brings electron-phonon
coupling at low temperature and drives interesting magnetic characteristics.
Our argument is also consistent with previous reports, where adding a high Z
element was found to soften the phonons [54, 55]. In Fig. 1(c), the Ge doping
changes the Tb-Si interaction, which weakly softens the phonon modes as
clearly visible through the reduced acoustic mode frequency in the 25 to 50
THz range. The Ge doping also creates Ge-Si interactions that lead to the
spread of overall optical phonon modes, as seen by high-energy phonons, which
arise mainly from increased Ge/Si interaction. This also indicates that the
phonon free energies of TbSi0.6Ge0.4 are smaller than those of TbSi due to the
monotonicity of free energy with respect to the phonon frequency, which leads
to phonon softening and stabilizes this intermetallic compound in the $Cmcm$
phase. To further affirm our thermodynamic stability of the $Cmcm$ phase in
TbSi0.6Ge0.4, we present the phonon spectra of $Pnma$-TbSi0.6Ge0.4 in Fig. 1
of [56], which shows large imaginary phonons at $\Gamma$ points, indicative of
structural instability. The origin of this energy instability of the $Pnma$
structure for TbSi0.6Ge0.4 lies in the local charge rearrangement at the Tb-
site. We found that Tb $f$-states show significant charge gain (-0.204e),
resulting in more $f$-states at the Fermi level, and hence local charge
instability. On the other hand, Tb $d$-states, which play a key role in phase
stability through significant $d$-$f$ exchange in these compounds, weaken due
to noteworthy charge loss from Tb-5$d$ (-0.501e) on Ge substitution when
compared to the $Cmcm$ phase of TbSi0.6Ge0.4.

###  III.2 Room-temperature crystallography

Figure 2: The Rietveld refinement of room temperature XRD data of (a) TbSi and
(b) TbSi0.6Ge0.4 samples measured using Cu Kα radiation, where the observed
data points, simulated curves, Bragg positions, and the difference in observed
and calculated data are shown by open red circles, a continuous black line,
vertical green bars, and a continuous blue line, respectively. (c, d) The 3D
illustration of the FeB-type ($Pnma$ space group) and CrB-type ($Cmcm$ space
group) crystal structures of TbSi and TbSi0.6Ge0.4, respectively.

The results of the Rietveld refinement of the room temperature x-ray powder
diffraction patterns of TbSi and TbSi0.6Ge0.4 samples are presented in Figs.
2(a) and (b), respectively. The data confirm that substitution of Ge at Si
site changes the crystal structure from FeB-type ($Pnma$ space group) in TbSi
to CrB-type ($Cmcm$ space group) in TbSi0.6Ge0.4, consistent with our
theoretical simulations (discussed above) and Refs. [32, 33, 34]. The
extracted room temperature lattice parameters are $a$= 7.918 Å and 4.288 Å,
$b$= 3.835 Å and 10.615 Å, and $c$= 5.703 Å and 3.875 Å for TbSi and
TbSi0.6Ge0.4, respectively. The three dimensional crystal structure plots of
the TbSi and TbSi0.6Ge0.4 compounds are presented in Figs. 2 (c) and (d),
respectively, with their common axis of symmetry, i.e., the $b$-axis of TbSi
and the $c$-axis of TbSi0.6Ge0.4, shown out of the plane of the paper. Each
unit cell [represented by the dashed cuboids in Figs. 2 (c) and (d)] contains
the eight atoms (four formula units) in both the cases. In the TbSi (FeB-type
structure), both Tb and Si atoms occupy 4$c$ ($x$, 1/4, $z$) position, where
$x$=0.180 and 0.014, and $z$= 0.616 and 0.132, respectively; whereas in the
TbSi0.6Ge0.4 (CrB-type structure), all atoms occupy the 4$c$ (0, $y$, 1/4)
position with $y$=0.141 and 0.418 for Tb and Si/Ge, respectively, at room
temperature.

###  III.3 Magnetic properties and magnetocaloric effect

In order to understand the influence of Ge substitution on the magnetic
properties, temperature-dependent magnetization data have been recorded
between 2–350 K for both TbSi and TbSi0.6Ge0.4 in ZFC, FCW, and FCC modes at
100 Oe, as presented in Figs. 3 (a) and (b), respectively. Both samples show
two well-resolved antiferromagnetic transitions around 58$\pm$1 K (called TN1)
and 39$\pm$1 K (called TN2) for TbSi, and 57$\pm$1 K and 45$\pm$1 K for
TbSi0.6Ge0.4 [see Figs. 3(a, b)]. It is interesting to note that an increase
in the Si concentration in TbSixGe1-x samples was found to stabilize the low-
temperature (LT) planar AFM (increase in TN2) for $x\leqslant$0.9, having a
CrB-type crystal structure [33]. However, a reduction in TN2 in the case of
TbSi having FeB-type structure, as compared to TbSi0.6Ge0.4, indicates that
the transformation in crystal structure also influences the strength of LT AFM
in these samples. Although, no significant change in the strength of the high-
temperature (HT) AFM (TN1) is found between TbSi and TbSi0.6Ge0.4, analogous
to the other members of the TbSixGe1-x series [33]. Moreover, we observe an
abrupt enhancement in the magnetic moment in the case of TbSi between 35-39 K
(in the vicinity of TN2), as indicated by the dashed arrows in Fig. 3(a),
suggesting a possible change in the magnetic structure of TbSi in this narrow
temperature range. The neutron diffraction measurements also show a drastic
change in the wave vector of TbSi from q=(0, 1/2, 0) to a three-dimensionally
modulated AFM structure with q=(qx, qy, qz) in the 36-39 K temperature range
[34]. We will discuss this point in more detail while discussing the high-
resolution specific heat data in the next section. Furthermore, the ND
measurements reported the first-order nature of the LT to HT AFM transition in
both of these samples [33, 34]. However, no notable hysteresis has been
observed in the magnetization data recorded in the heating and cooling modes
(ZFC/FCW and FCC) in the entire temperature range for both samples. A larger
temperature step (1 K) compared to the thermal hysteresis ($<$0.5 K, discussed
below) along with the steep increase in magnetization at TN2 may be
responsible for this. The inverse magnetic susceptibility of the FCC curves is
plotted on the right axis of Figs. 3(a) and (b) for the TbSi and TbSi0.6Ge0.4
samples, respectively. We fit the 1/$\chi$ vs. T curves at various magnetic
fields [see Fig. 2 of [56] and the solid black lines in Figs. 3(a, b) at 100
Oe] for both samples, using the Curie-Weiss law: $\chi$=C/(T-$\theta_{\rm
p}$), where C and $\theta_{\rm p}$ are the Curie-Weiss constant and the
paramagnetic Curie temperature, respectively, which gives an average value of
$\theta_{\rm p}$= -7 K and -4 K, and effective magnetic moment ($\mu_{\rm
eff.}$=$\sqrt{8\rm C}$)= 10.11 $\mu_{B}$/f.u. and 10.12 $\mu_{B}$/f.u. for
TbSi and TbSi0.6Ge0.4, respectively. The almost field-independent values of
the $\mu_{\rm eff.}$ [see Figs. 2(g, h) of [56]] are slightly higher than the
theoretical value for the free Tb3+ ion with 7F6 ground term (9.72 $\mu_{B}$)
for both samples, indicating the small but finite contribution of the
itinerant moments from the conduction electrons in these samples. Furthermore,
the negative values of $\theta_{\rm p}$’s confirm the AFM interactions in both
samples. However, the values of $\theta_{\rm p}$ are significantly smaller
than the TN1 for both samples, which is also observed in Refs. [32, 33]. This
can be attributed to the deviation in the free Tb3+ states due to the crystal
field (CF) and hence non-negligible temperature-independent paramagnetic (TIP)
susceptibility term either from transition within these CF split states or
from the core electrons [58]. Furthermore, we observe a small but consistent
enhancement in $\theta_{\rm p}$ with an increase in the magnetic field for
both samples [see Figs. 2(e, f) of [56]], suggesting the possibility of short-
range magnetic correlations in these samples in the paramagnetic state.

Figure 3: Temperature-dependent direct (left axis) and inverse (right axis) dc
magnetic susceptibility for (a) TbSi and (b) TbSi0.6Ge0.4 compounds measured
at 100 Oe in ZFC, FCW, and FCC modes. Solid black lines represent the fit of
the FCC curves from 105 to 350 K using the Curie-Weiss law. Dashed arrows in
(a) represent the abrupt enhancement in the susceptibility of TbSi between 35
to 39 K. The field-dependent magnetization curves of (c) TbSi and (d)
TbSi0.6Ge0.4 samples, recorded at 2 K and 50 K, where insets represent the
dM/dH versus H plot for one -70 kOe to +70 kOe M-H cycle at 50 K.

The field-dependent magnetization (M-H) curves have been recorded at 2 K (in
LT AFM phase) and 50 K (in HT AFM phase) for TbSi and TbSi0.6Ge0.4, as shown
in Figs. 3(c) and (d), respectively. Interestingly, the M-H curves at 2 K show
linear behavior for both samples, whereas those at 50 K display a notable
convex nature at higher magnetic fields, suggesting the emergence of a
metamagnetic-like transition in both samples in the HT AFM region. In order to
clearly present this, the dM/dH versus H curves have been plotted in the
insets of Figs. 3(c) and (d) for one full M-H cycle (-70 kOe to +70 kOe). A
significant increase in the slope of the M-H curves up to around 60 kOe is
likely due to the ongoing spin-reorientation process [59, 60]. The dM/dH
decreases with further increase in H for H$>$60 kOe for both samples. The
magnetization at 70 kOe and 50 K is $\sim$2.5 $\mu_{B}$/f.u. for both samples,
which is around 28% of the theoretical saturation value (M${}_{\rm S}=$ gJJ =
9 $\mu_{B}$) for Tb3+ ions, indicating that the magnetic state is still
predominantly AFM. A metamagnetic transition with a critical field of 65 kOe
has also been observed in DySi with 80% FM alignment of the Dy3+ moments at 90
kOe and 5 K [61]. The linear behavior of the M-H curves below TN2 on the other
hand suggests stronger AFM interactions (in response to the external magnetic
field) in the LT phase of both samples. Here, it is worth mentioning that both
samples have planar AFM structure in the LT phase region, which is
commensurate with the crystal lattice; whereas in the HT region, TbSi has
sinusoidally modulated incommensurate magnetic structure [q=(qx, q$y$, q$z$],
and TbSi0.6Ge0.4 has 50% commensurated [q=(0, 0, 1/2)] and 50% incommensurated
[q=(qx, 0, qz)] magnetic phases [33, 34]. This suggests that the modulation of
the LT commensurated magnetic structures in the HT phase region decreases the
strength of the AFM coupling in both samples. Moreover, both samples show
negligible hysteresis in the M-H loop at 2 K as well as 50 K, which rules out
the presence of any ground-state anisotropic FM correlations and/or domains.

Figure 4: The magnetization isotherms recorded from 0 to 70 kOe in the
indicated temperature regimes for (a–d) TbSi and (e–h) TbSi0.6Ge0.4.

Further, the magnetization isotherms have been recorded from 0 to 70 kOe
across the transition temperatures, TN1 and TN2, and are shown in Figs. 4(a–d)
for TbSi and 4(e–h) for TbSi0.6Ge0.4. It can be clearly observed that the
virgin curves are sufficiently straight below 30 K for both samples [see Figs.
4(a, e)], indicating rigid coupling in the LT AFM phase in both samples. The
curves begin to display a convex shape above 30 K, which becomes more
pronounced as the temperature approaches 50 K, as shown in Fig. 4(b) for TbSi.
However, the convex character of the virgin isotherm at 31 K in the case of
TbSi0.6Ge0.4 is weaker compared to that of TbSi [see Figs. 4(b, f)], which
confirms the better stability of the LT planar AFM in the former. Moreover,
close to TN1, i.e., above 50 K, both samples show the “S-like” shape of the
virgin curves [see Figs. 4(c, g)], which is a typical signature of the
metamagnetic transition [62]. This affirms the weaker AFM coupling in these
samples in the HT phase region due to finite incommensuration in the magnetic
structure for T${}_{\rm N2}<T<$ TN1, discussed above [33, 34]. Above 50 K, the
magnetization increases with the temperature for low magnetic fields but
decreases monotonically at 70 kOe for both samples, as shown in Figs. 4(c, g),
which indicates the shift in the TN1 to the lower temperature with an increase
in the magnetic field. Figures 4(d, h) show the virgin curves for both samples
deep in the paramagnetic region (T$\geqslant$80 K), where the curves exhibit a
slight concave nature even up to around 100 K. Thus, the presence of short-
range magnetic correlations is possible in these compounds. Recently, short-
range magnetic interactions have been also detected in other AFM materials
using Raman spectroscopy up to temperatures as high as $\sim$3TN [63].

Figure 5: Temperature dependent magnetization at different magnetic fields
extracted from the isothermal curves for (a) TbSi and (b) TbSi0.6Ge0.4. Black
and red dashed arrows represent the shift in TN1 and TN2, respectively, with
increase in magnetic field. (c, d) The temperature dependent magnetic entropy
change ($\Delta$SM) for different $\Delta\mu_{0}$H in the vicinity of TN1 and
TN2 for both the samples. Insets show the field dependent $\Delta S_{\rm
M}^{max}$ at TN2, where black solid curves represent the best fit using the
power law.

The temperature-dependent magnetization (M-T) curves derived from these
magnetization isotherms are plotted in Figs. 5(a, b) for TbSi and
TbSi0.6Ge0.4, respectively. It can be clearly observed that both transition
temperatures, TN1 and TN2, shift to lower values with an increase in the
magnetic field, as indicated by black and red dashed arrows, respectively.
However, this effect is much more prominent for TN1 compared to TN2 [see Figs.
5 (a, b) and Figs. 2 (a, b) of [56] for more clarity], supporting weaker AFM
coupling in the HT phase. In fact, we observe a significant broadening and
hence a plateau-like feature at TN1 in the case of TbSi for H$>$5 T [see Fig.
5(a)]. This effect is more clearly evident in the high-field specific heat
measurements, discussed below. Moreover, in the case of TbSi, the transition
at TN2 remains sharp and distinctly visible even at 7 T, whereas it becomes
broad and nearly merges with TN1 in the case of TbSi0.6Ge0.4 [see Figs. 5 (a,
b)]. This is due to the evolution of additional field-induced magnetic
interactions between TN1 and TN2 in the latter, which we discuss below in
detail.

The change in magnetic entropy ($\Delta$SM) is a crucial parameter for
investigating the magnetocaloric effect (MCE) and can also serve as a
sensitive tool to probe the interplay of complex magnetic interactions and
their evolution with temperature and magnetic field in magnetically ordered
compounds [60, 62, 65, 64]. Therefore, $\Delta$SM is calculated as a function
of temperature for different $\Delta$H values using the following Maxwell’s
thermodynamic relation [65].

$\displaystyle\Delta S_{M}(\Delta H,T)=\int_{0}^{H}\bigg{(}\frac{\partial
M(T,H)}{\partial T}\bigg{)}_{H}dH$ (1)

Table 1: The comparison of the transition temperatures (TC/TN), $\Delta$S${}_{\rm M}^{max}$, and TEC(10) of various Tb-based compounds with TbSi and TbSi0.6Ge0.4 for the given values of change in the magnetic field ($\Delta\mu_{0}$H). Sample |  TC/TN (K) |  $\Delta\mu_{0}$H(T) |  $\Delta$S${}_{\rm M}^{max}$ (J/Kg-K) |  TEC(10) (J/Kg-K) |  Reference
---|---|---|---|---|---
TbSi |  35 |  7 |  9.6 |  5.4 |  present study
TbSi0.6Ge0.4 |  44 |  7 |  11.6 |  9.3 |  present study
TbCr2Si2C |  16.6 |  7 |  -16.4 |  15.79 |  [70]
TbMn2Ge2 |  95 |  7$||c$ |  -24.02 |  21.39 |  [22]
Tb2Dy4FeSb2 |  182 |  5 |  -7.72 |  7.45 |  [71]
Tb4CoIn |  50 |  7 |  -7.2 |  7.1 |  [41]
GdTbDyHo |  198 |  5 |  -8.2 |  8.11 |  [64]
Tb1.4Dy0.6In |  155 |  5 |  -5.97 |  5.9 |  [42]
Tb2Co0.8Si3.2 |  58 |  7 |  -3.32 |  3.2 |  [72]
TbCo2 |  233 |  5 |  -5.7 |  5.4 |  [43]
Gd0.2Tb0.2Dy0.2Ho0.2Er0.2Ni2 |  32 |  5 |  -14.3 |  13.8 |  [73]

The $\Delta$SM(T) curves for the TbSi and TbSi0.6Ge0.4 samples are depicted in
Figs. 5(c) and (d), respectively. A clear transition from the inverse
magnetocaloric effect (IMCE) (positive $\Delta$SM) to the conventional
magnetocaloric effect (CMCE) (negative $\Delta$SM) is observed at $T_{\rm
N1}\approx$ 58 K (for $\Delta\mu_{0}$H = 0.5 T) for both samples, as expected
for the AFM to PM transition [66]. This crossover temperature decreases with
the increase in applied magnetic field as a consequence of a field-induced
shift in $T_{\rm N1}$, as discussed above. Moreover, we observe a stronger
peak in the $\Delta S_{\rm M}$ curves around 36 K (for $\Delta\mu_{0}$H = 0.5
T) corresponding to the LT AFM transition ($T_{\rm N2}$) in the case of the
TbSi sample [see Fig. 5(c)]. This large IMCE, with the value of $\Delta S_{\rm
M}$ = 9.6 J/kg-K at $\Delta\mu_{0}$H = 7 T, indicates the possible use of TbSi
as a heat sink to improve the efficiency of magnetic refrigerators [67].
Interestingly, a significant broadening in the $\Delta S_{\rm M}$ peak of
TbSi0.6Ge0.4 has been observed at higher magnetic fields, where the two peaks
corresponding to LT and HT AFM phases almost merge into each other at
$\Delta\mu_{0}$H = 7 T [see Fig. 5(d)]. This is due to the evolution of
additional maxima in the $\Delta S_{\rm M}$ curves of TbSi0.6Ge0.4 between
$T_{\rm N1}$ and $T_{\rm N2}$ at higher fields. The presence of maxima in the
$\Delta S_{\rm M}$ curves, i.e., positive entropy change (IMCE), at higher
fields indicates the evolution of field-induced additional AFM coupling(s) in
the HT phase region of this sample. This is more clearly evident from the
behavior of specific heat data at higher magnetic fields, discussed below. The
maximum value of $\Delta S_{\rm M}$ ($\Delta S_{\rm M}^{max}$) at
$\Delta\mu_{0}$H = 7 T increases to 11.6 J/kg-K in TbSi0.6Ge0.4, which along
with the significant broadening in the $\Delta S_{\rm M}$ peak compared to
pure TbSi [see Figs. 5(c, d)], makes the TbSi0.6Ge0.4 sample more suitable for
magnetic refrigerator applications. We note that the maximum MCE of our
compounds is smaller compared to other well-known materials in this
temperature range, such as Eu2In [68], but the plateau-like temperature
profile of the TbSi0.6Ge0.4 compound can make it enticing for engineers
looking for materials with inverse MCE for hydrogen liquefaction. For
instance, a rough approximation of a single peak corresponding to IMCE in the
case of TbSi0.6Ge0.4 gives a moderate relative cooling power (RCP) of 125 J/kg
for $\Delta\mu_{0}$H = 7 T, where RCP is defined as RCP = $\Delta S_{M}^{\rm
max}\delta_{\rm FWHM}$, where $\delta_{\rm FWHM}$ represents the full width at
half maximum of the peak. Further, we calculated the temperature-averaged
entropy change (TEC), defined by Griffith et al. [69], as

${\rm TEC}(\Delta T_{\rm lift})=\frac{1}{\Delta T_{\rm
lift}}\stackunder{max}{\scriptsize Tmid}\left\\{\int\limits_{T_{\rm
mid-\frac{\Delta T_{\rm lift}}{2}}}^{T_{\rm mid+\frac{\Delta T_{\rm
lift}}{2}}}\Delta S_{M}(T)_{\Delta H,T}dT\right\\}$ (2)

where, $\Delta T_{\rm lift}$ represents the temperature range within which the
material exhibits its optimal response to $\Delta$H, while Tmid denotes the
temperature at which ${\rm TEC}(\Delta T_{\rm lift})$ reaches its maximum
value. This value is determined by sweeping over different temperature ranges
for a given $\Delta T_{\rm lift}$. We obtained a reasonably high value of
TEC(10) = 9.3 J/kg-K for TbSi0.6Ge0.4 at Tmid = 46 K, compared to TEC(10) =
5.4 J/kg-K for TbSi at Tmid = 37 K, both at $\Delta\mu_{0}$H = 7 T, owing to
the relatively broad transition in the former. These $\Delta S_{\rm M}^{max}$
and TEC(10) values are moderate compared to other Tb-based compounds listed in
Table I. However, such values, along with positive $\Delta S_{\rm M}$ (IMCE)
and negligible thermal hysteresis, make these samples particularly
significant, considering the low magnetic moment and sharp nature of AFM
transitions, resulting in small values of $\Delta S_{\rm M}$ and TEC(10),
respectively, in materials exhibiting IMCE. Furthermore, the values of $\Delta
S_{\rm M}^{max}$ at TN2 monotonically increase with the increase in
$\Delta\mu_{0}$H up to 7 T for both samples. The field dependence of $\Delta
S_{\rm M}^{max}$ is depicted in the insets of Fig. 5(c, d) for TbSi and
TbSi0.6Ge0.4, respectively. The best fit of the $\Delta
S_{M}^{max}$($\Delta$H) curves using the power law $\Delta S_{\rm
M}^{max}\propto$Hϕ is illustrated by the solid black curves, yielding
$\phi=$1.93(2) and 2.36(4) for TbSi and TbSi0.6Ge0.4, respectively. The nearly
quadratic H dependence of $\Delta S_{\rm M}^{max}$ and the rigidity of the LT
AFM phase in response to the external field suggest that the giant
magnetocaloric effect can be achieved in these samples at the higher magnetic
fields [74]. For instance, extrapolation of the $\Delta S_{\rm
M}^{max}$($\Delta$H) curves gives $\Delta S_{\rm M}^{max}$ = 36.1 and 58.6
J/kg-K at $\Delta\mu_{0}$H = 14 T for TbSi and TbSi0.6Ge0.4, respectively.

###  III.4 Specific heat

The substantial IMCE and TEC(10) at TN2 and the noteworthy enhancement in the
latter with the Ge substitution, make it imperative to investigate the nature
of this LT AFM transition in these samples. Therefore, we record the zero-
field specific heat data from 2 to 200 K, as depicted in Figs. 6(a, b) for
TbSi and TbSi0.6Ge0.4, respectively. Two well-defined peaks have been observed
at TN2 $\approx$ 35 K and 43 K, and TN1 $\approx$ 57 K and 56 K for the TbSi
and TbSi0.6Ge0.4 samples, respectively, which is consistent with the LT and HT
AFM orderings observed in the magnetization measurements. We observe an
asymmetric $\lambda$-like peak at TN1 for both both TbSi and TbSi0.6Ge0.4,
suggesting the second-order nature of this HT AFM to PM transition in these
samples [75]. On the other hand, the peak at TN2 is delta-like, sharp and
symmetric in nature, indicating the first-order nature of the LT AFM to HT AFM
phase transition [76]. The specific heat at 200 K at is around 45 J/mole-K for
both samples, which is slightly lower than the classical Dulong-Petit limit,
i.e., 3$n$R=49.9 J/mole-K, where $n$ and R are the number of atoms per formula
unit (two in this case) and gas constant, respectively, as shown by horizontal
dashed lines in Figs. 6(a, b). We estimate the lattice and electronic
contribution in the specific heat curves by fitting the CP(0, T) curves from
115-200 K using the Debye model with a single Debye temperature, and an
electronic term, $\gamma$T, as given by [77]

$C_{V}(T)=\gamma
T+9nR\left(\frac{T}{\theta_{D}}\right)^{3}\int_{0}^{\theta_{D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)}dx,$
(3)

Figure 6: The zero-field temperature-dependent specific heat curves measured
using the standard relaxation technique for (a) TbSi and (b) TbSi0.6Ge0.4
samples. The solid black curves represent the best fit obtained using the sum
of electronic and Debye lattice specific heat from 115-200 K, with dashed
curves indicating their extrapolation down to $\sim$2 K. The horizontal dashed
lines denote the classical Dulong-Petit limit. (c, d) The temperature-
dependent magnetic specific heat (Cmag)/T (on the left axis) and magnetic
entropy (Smag) (on the right axis) for the TbSi and TbSi0.6Ge0.4 samples,
respectively. Dashed lines represent the theoretical value of Smag, Rln13, for
the free Tb3+ ion.  Figure 7: (a–f) The temperature dependent specific heat
curves of TbSi measured using heat pulse method in the vicinity of TN2 in both
heating and cooling modes at different magnetic fields from 0–5 T. Arrows in
(b) mark the additional transition (IT AFM) observed in the high resolution CP
data.  Figure 8: (a–h) The temperature dependent specific heat curves of TbSi
in the vicinity of TN2 in both heating and cooling modes for $\mu_{0}$H=
5.25–7 T. Arrows in (h) represent the evolution of addition features with
increase in the magnetic field.

where $\theta_{D}$, and $\gamma$ are the Debye temperature and Sommerfeld
coefficient, respectively. The solid black curves in Figs. 6(a) and (b) depict
the best fit using the Debye function, which gives $\theta_{D}$ = 333$\pm$3 K
and 296$\pm$2 K, and $\gamma$ = 3.4(6) and 2.1(3) mJ/mole-K2 for the TbSi and
TbSi0.6Ge0.4 samples, respectively. We extrapolate these curves down to
$\sim$2 K, as indicated by the black dashed curves in Figs. 6(a, b), to
extract the lattice and electronic contribution across the entire temperature
range under study. The magnetic entropy (Smag) is then calculated for both
samples using the formula Smag(T) = $\int\frac{C_{mag}}{T}$ dT, where Cmag
represents the magnetic specific heat obtained by subtracting the sum of
lattice and electronic specific heat (dashed lines) from the CP(0, T) curves.
The temperature-dependent Cmag/T and Smag are shown on the left and right axes
of Figs. 6(c) and (d) for the TbSi and TbSi0.6Ge0.4 samples, respectively. The
Smag exhibits a saturation value of around 20 J/mole-K for both samples, close
to the theoretical value of Rln(2J+1) = Rln13 = 21.33 J/mole-K for a free Tb3+
ion (J=6), as illustrated by the dashed lines in Figs. 6(c) and (d). However,
a precise estimation of the lattice specific heat using non-magnetic
isostructural reference samples is important for a more reliable estimation of
Smag.

Further, we record the temperature-dependent specific heat curves at different
magnetic fields [CP(H, T)] for both TbSi and TbSi0.6Ge0.4 samples. We observe
that the low-temperature peak (at TN2) remains sharp and almost symmetric up
to 7 T [see Figs. 3(a,b) of [56]], which confirms the first-order nature of
the LT to HT AFM transition in both samples. Moreover, the TbSi displays an
additional sharp peak in the vicinity of the LT AFM, as shown by the arrow in
Fig. 3(a) of [56]. Here, it is important to mention that the specific heat
measurements performed using the conventional thermal-relaxation technique
assume that CP remains constant, and have the same values during heating and
cooling cycles for a given heat pulse. However, for a very sharp (delta-like)
FOPT, as in the present case at TN2, both of these assumptions become
unreliable, leading to considerable inaccuracies in the measured values of CP
in the vicinity of FOPT [68, 78, 79]. Therefore, a point-by-point time-
dependent analysis of both heating and cooling curves, after giving a long
heat pulse which can essentially capture the whole phase transition, is more
appropriate in the present case [79]. In Figs. 7(a–f) and 8(a–h), we show the
specific heat curves of TbSi measured using this method in the vicinity of LT
AFM in both heating and cooling cycles at different applied magnetic fields.
To record the data at each field, the sample was first heated up to 100 K and
then cooled down to the desired temperature in the absence of magnetic field,
in order to remove the possibility of any thermal and/or magnetic remanence.
The temperature and magnetic field were then stabilized for 20 minutes before
applying a heat pulse for 500 secs. A clear shift of $\sim$0.3 K in TN2
measured in heating and cooling modes can be observed at all the magnetic
fields, which confirm the first-order nature of this LT AFM to HT AFM phase
transition in TbSi. This small thermal hysteresis was overshadowed due to the
larger temperature step (1 K) used in the magnetization measurements performed
in heating and cooling modes, discussed above. This demonstrates the potential
of the CP measurements performed using the heat pulse method in resolving the
fairly sharp features in the vicinity of FOPT. Furthermore, the peak value of
CP significantly increases compared to that observed using the standard
relaxation technique [see Figs. 6(a) and 7(a)], as the instantaneous value of
CP in the former is free from both of the assumptions described above [68,
78]. More importantly, now we clearly see an additional somewhat broad but
almost symmetric feature above TN2 in both heating and cooling modes,
represented as TN3 in Fig. 7(b). In the absence of magnetic field, this
feature has almost half intensity but significantly higher thermal hysteresis
($\sim$2 K) as compared to the LT AFM transition, indicating the first-order
nature of this transition as well. Here, it is important to reemphasize that
ND measurements performed on the TbSi showed the planar (collinear) nature of
the LT AFM phase below 36 K [q=(0, 1/2, 0)], where the magnetic moments lie in
the $xy$-plane, making an angle of 66$\pm$20 with the $a$-axis [34]. This LT
planar AFM was found to transformed into a three dimensionally modulated
incommensurate HT AFM phase with q=(qx, qy, qz) [34]. However, the most
striking modulation in the magnetic wave vector was observed between 36-39 K,
and above 42 K magnetic structure is almost commensurate with the crystal
lattice with q$\approx$(0, 1/2, 1/8) [34]. The presence of additional peak in
the intermediate temperature (IT) range ($\sim$36-39 K on heating) in the
CP(0, T) curves clearly indicates the existence of a distinct magnetic phase
between LT and HT AFM phases of TbSi. This peak shifts to the lower
temperature with increase in the magnetic field [see Figs. 4(a, c) and 5(a, c)
of [56]], which indicates the AFM nature of this phase. Therefore, in TbSi,
the LT planar AFM transformed into an IT AFM via a FOPT, which then
transformed into HT AFM through another FOPT and finally into PM state with
second order phase transformation (SOPT). In fact, an abrupt variation in the
position, line width, and the integrated intensity of (1, 1/2, 0) reflection
in the ND data reported between 36-39 K in Ref. [34] along with a thermal
hysteresis of 2 K in TbSi is most likely to be associated with this IT AFM
phase. In order to further confirm the nature of the IT to HT AFM phase
transition, we apply a short heat pulse only up to $\sim$37 K (below TN3) and
interestingly, no peak corresponding to IT AFM was observed in the cooling
curve (see Fig. 6 of [56]), which confirms the first-order nature of this
transition. Note that a slightly higher background of the heating curves at
the higher temperatures in Figs. 7 and 8 originates from the partial breakdown
of the ideal adiabatic condition at the end of the long heating pulse. The
field induced shift in TN3 is higher than TN2, which causes an increase in the
strength of IT AFM peak with the magnetic field, especially for $\mu_{0}$H$>$5
T, due to their overlap with the LT AFM peaks. This effect is more prominent
in the cooling curves owing to their lesser separation ($\sim$1.7 K at 0 T)
between LT and IT AFM peaks as compared to the heating curves ($\sim$2.5 K at
0 T) [see Figs. 8 (a–h]. However, both transitions remain sharp even up to 7
T, where both heating and cooling curves confine within a narrow temperature
range of $\sim$3 K, contrary to TbSi0.6Ge0.4, which shows significant
broadening at the higher fields (discussed below). Moreover, the LT AFM peaks
measured in the heating mode show an asymmetry towards higher temperatures for
$\mu_{0}$H$>$5 T and a clear evolution of an additional feature for
$\mu_{0}$H$>$6 T [see Figs. 8(a–h)], as shown by the solid red arrow in Fig.
8(h). However, due to their sharp and intense nature, this feature is visible
in the cooling curves only for $\mu_{0}$H$>$6 T [see Figs. 8 (e–h)]. Further,
a weak feature is also deconvoluted below TN2 (between $\sim$33–34 K) in both
heating and cooling curves for $\mu_{0}$H$>$5 T, as represented by the dashed
blue arrow in Fig. 8(h) [see Fig. 5(b, d) of [56] for clarity]. Both of these
field-induced features also shift to lower temperatures with an increase in
the magnetic field, indicating their AFM nature. More importantly, both of
these features evolve at the expense of the LT AFM peak [see Figs. 8(a–h)] and
show a thermal hysteresis of $\sim$0.3 K. This indicates the field-induced
modulation of the LT AFM phase into different AFM configurations having small
formation energy differences as their origin [38]. We call these low- and
high-temperature field-induced features split from the LT AFM as (LT AFM)S1
and (LT AFM)S2, respectively. Furthermore, we calculate the magnetic entropy
and adiabatic temperature change ($\Delta$Tad) using these CP(H, T) curves for
TbSi [shown in Figs. 7(a–d) of [56]], which show good agreement (in
$\Delta$SM) with those calculated from the magnetization measurements,
discussed above.

Figure 9: (a–f) The temperature dependent specific heat curves of TbSi0.6Ge0.4
in the vicinity of TN2 in both heating and cooling modes at different magnetic
fields from 0–5 T. Arrows in (d) and (e) show the splitting of the peaks into
a doublet and triplet at 3 T and 4 T magnetic fields, respectively.  Figure
10: (a–h) The temperature dependent specific heat curves of TbSi0.6Ge0.4 in
the vicinity of TN2 in both heating and cooling modes for $\mu_{0}$H= 5.25–7
T.  Figure 11: The temperature dependent specific heat of (a1–a8) TbSi and
(b1–b8) TbSi0.6Ge0.4 in the vicinity of TN1 at different magnetic fields.
Black bars and green arrows represent the position of TN1 and the evolution of
additional features for $\mu_{0}$H$\geqslant$5 T, respectively, in both the
samples.

The TbSi0.6Ge0.4 sample, on the other hand, has an equal proportion of
uniaxial AFM (collinear along the $c$-axis) with q=(0, 0, 1/2) and a two-
dimensionally modulated (sinusoidal) AFM with q=(qx, 0, qz) in the HT AFM
phase [33]. Furthermore, the LT planar AFM phase (in the $ac$ plane) can also
have the possibility of both collinear as well as canted alignment of the
magnetic moments [33], and hence is expected to display an even more complex
H-T phase diagram due to field-induced transformation between these different
possible magnetic structures. The temperature-dependent specific heat curves
of TbSi0.6Ge0.4 at the different magnetic fields from 0–5 T in the vicinity of
LT AFM are shown in Figs. 9(a–f) in both heating and cooling modes. A
hysteresis of around 0.4 K is observed in the CP(0, T) curves recorded in
heating and cooling modes, indicating the first-order nature of the LT to HT
AFM transition. However, we observe a shoulder-like feature towards the lower
temperature side of the peak in both heating and cooling curves of this sample
[see Fig. 9(a)]. The transformation of LT AFM into two different HT AFM phases
[commensurate with q=(0, 0, 1/2) and incommensurate with q=(qx, 0, q$z$) [33]]
may give rise to this shoulder-like feature in the CP(0, T) curves when one
phase transformation is favored over the other. Furthermore, we observe a
significant broadening in the LT AFM peak with an increase in the magnetic
field even at 2 T [see Fig. 9(c)]. Interestingly, with a further increase in
the magnetic field, this peak first splits into a doublet at $\mu_{0}$H=3 T
and then into a triplet for $\mu_{0}$H$\geqslant$4 T, as shown by arrows in
Fig. 9(d) and (e), respectively. All three peaks shift to lower temperatures
with an increase in the magnetic field [see Figs. 8(a, c) and 9(a, c) of
[56]], indicating their AFM nature. We define these low- and high-temperature
field-induced features as (LT AFM)S1 and (LT AFM)S2, respectively.
Importantly, the strength of (LT AFM)S1 and (LT AFM)S2 increases compared to
the main LT AFM peak with an increase in the magnetic field. The field-induced
increment in the latter is more prominent compared to the former. This effect
in the CP(H, T) curves of TbSi0.6Ge0.4 is more clearly observed at higher
fields for $\mu_{0}$H$>$5 T, as shown in Figs. 10(a–h). The strength of (LT
AFM)S1 dominates over the main LT AFM peak for $\mu_{0}$H$\geqslant$5.25 T.
Furthermore, the (LT AFM)S2 peak dominates over that of (LT AFM)S1 for
$\mu_{0}$H$\geqslant$6.5 T [see Figs. 10(f–h)]. These observations indicate
that with an increase in the magnetic field, the fraction of LT planar AFM
transforming into uniaxial or 2D modulated AFM at zero field [33] decreases,
and that into two new field-induced HT AFM structures increases. Although with
varying degrees, but all these LT to HT phase transitions coexist in
TbSi0.6Ge0.4 for $\mu_{0}$H=4-7 T. The coexistence of these multiple phase
transitions at the higher magnetic field in the case of TbSi0.6Ge0.4 causes
the observed significant broadening in the $\Delta$SM peaks (discussed above)
compared to the pure TbSi sample. The thermal hysteresis has been observed for
all these LT to HT AFM transitions, indicating their first-order nature.

Figure 12: The field dependent shift in the various AFM transition temperatures measured in heating and cooling modes for (a, b) TbSi and (c, d) TbSi0.6Ge0.4. The solid black curves represent the best fit using eq. 4.  Table 2: The best fit parameters extracted by fitting the field dependent AFM transition temperatures, extracted from the CP(H, T) curves, using eq. 4 for the TbSi and TbSi0.6Ge0.4 samples. |  |  TN0 (K) |  $\gamma$ |  $\mu_{0}$H0 (T)
---|---|---|---|---
|  HT AFM |  57.6(1) |  0.39(3) |  13.5(8)
|  IT AFM (heating) |  38.0(1) |  0.58(6) |  31(4)
|  IT AFM (cooling) |  36.9(1) |  0.64(7) |  35(5)
TbSi |  LT AFM (heating) |  35.5(1) |  0.61(4) |  50(2)
|  LT AFM (cooling) |  35.2(1) |  0.59(5) |  47(3)
|  (LT AFM)S1 (heating) |  35.1(4) |  0.50(2) |  30(1)
|  (LT AFM)S2 (heating) |  35.9(4) |  0.50(3) |  34(2)
|  HT AFM |  55.7(1) |  0.45(2) |  13.8(4)
|  LT AFM (heating) |  44.0(1) |  0.50(2) |  18(1)
|  LT AFM (cooling) |  43.6(1) |  0.50(2) |  17(1)
TbSi0.6Ge0.4 |  (LT AFM)S1 (heating) |  44.5(2) |  0.57(3) |  23(2)
|  (LT AFM)S1 (cooling) |  43.9(2) |  0.55(3) |  22(2)
|  (LT AFM)S2 (heating) |  43.9(2) |  0.40(2) |  19(2)
|  (LT AFM)S2 (cooling) |  43.4(3) |  0.40(3) |  18(2)

Further, we also record the specific heat curves of both samples in the
vicinity of the HT AFM to PM transition using the long heat pulse method, as
shown in Figs. 11(a1-a8) for TbSi and (b1-b8) for TbSi0.6Ge0.4, at different
magnetic fields in both heating and cooling modes. No hysteresis has been
observed in the CP data recorded in heating and cooling modes, which is in
agreement with the second-order nature of this transition in both samples.
Interestingly, both samples show the evolution of an additional feature above
TN1 for $\mu_{0}$H$\geqslant$5 T, as shown by solid green arrows in Figs.
11(a6-a8) and 11(b6-b8) for TbSi and TbSi0.6Ge0.4, respectively. The strength
of this feature increases, accompanied by a shift to lower temperature with an
increase in the magnetic field, causing a plateau-like behavior in the case of
TbSi at 7 T, similar to that observed in the M-T measurements, discussed
above. This indicates that the external magnetic field also stabilizes an AFM
structure having a higher transition temperature compared to the zero-field HT
AFM phase in both samples. This effect is more prominent in the case of
TbSi0.6Ge0.4, where the main HT AFM peak almost merges into the LT AFM at 7 T.
We use several long heat pulses at different temperatures to record the high-
resolution CP data in the wide temperature range in the case of TbSi0.6Ge0.4
for $\mu_{0}$H=0 and 4-7 T.

In order to quantitatively summarize these multiple magnetic interactions, we
plot the field dependence of different transition temperatures [TN(H)] for
TbSi and TbSi0.6Ge0.4 in Figs. 12(a, b) and (c, d), respectively. The precise
peak positions of different features have been determined by fitting the sum
of Voigt functions. The error bars represent the estimated upper limit of the
uncertainty, cumulatively from the peak fitting, temperature step size, and
instrumental resolution. We fit the TN(H) data using the following relation
[80]

$\displaystyle H=H_{0}\left(1-\frac{T_{\rm N}}{T_{\rm N0}}\right)^{\gamma},$
(4)

where H0 and TN0 represent the critical magnetic field required to shift TN at
0 K and the AFM transition temperature at zero field, respectively. The solid
black curves in Figs. 12(a–d) represent the best fit using the above equation,
and the best fit parameters are listed in Table II. It can be clearly observed
from Fig. 12(a) and (c) that the field-induced shift in the LT AFM is
significantly higher in the case of TbSi0.6Ge0.4 compared to TbSi, which is
also evident from the considerably lower values of H0 for the former in both
heating and cooling modes (see Table II). The difference between heating and
cooling curves remains almost independent of the applied magnetic field for
all the transitions in both samples. However, the separation of the field-
induced features (LT AFM)S1 and (LT AFM)S2 from LT AFM peaks increases with
the magnetic field for both samples. The higher overlapping of the (LT AFM)S1
and (LT AFM)S2 with LT AFM and IT AFM transitions, respectively, in the
cooling curves of TbSi makes us unable to precisely determine their positions
and hence they are not included in this H-T phase diagram. The field-dependent
shift indicates the predominating AFM nature of all the transitions. High-
resolution magnetization and/or transport measurements on the single-
crystalline samples [38] or the magnetic field-dependent ND measurements [81]
can be helpful to further understand the nature and origin of these multiple
field-induced magnetic phases in TbSi and TbSi0.6Ge0.4.

###  III.5 Low-temperature crystallography

Figure 13: (a–e) The Rietveld refinement of the XRD data of TbSi recorded
using the Mo K$\alpha$ radiation at selected temperatures. The temperature
dependence of the lattice parameters (f) $a$, (g) $b$, (h) $c$, (i) unit-cell
volume, and (h) overall $B$ parameter, extracted from the Rietveld refinement.
The vertical dashed lines represent the AFM transitions at 58 and 36 K, taken
from the heat capacity measurements (Table I). The solid curves in (f–i)
represent the best fit of the unit cell parameters from 70–300 K using the
Grüneisen approximation up to first order and the dashed curves represent the
extrapolation of the fitted data down to 10 K. The yellow lines in (i) are the
guide to eye to show the discontinuity in cell volume at TN2.

The temperature-dependent XRD measurements have been performed on both samples
from 10 to 300 K to understand the correlation between the crystal structure
and magnetic properties of these compounds and further uncover the nature of
their magnetic transitions. The selected Rietveld refinements of the powder
XRD data of the TbSi sample recorded using the Mo Kα radiation are shown in
Figs. 13(a–e). The refinement shows that the crystal structure of TbSi remains
FeB-type (orthorhombic symmetry, $Pnma$ space group) down to 10 K. However, we
observe a significant deviation in the lattice parameters near both magnetic
transition temperatures, as shown in Figs. 13(f–j). This indicates the
presence of strong magnetostriction in this sample. Note that a large
temperature step of 5 K across the magnetic transitions unable us to
distinguish the structural changes associated with the LT and IT AFM phases.
The lattice parameters $a$ and $b$ show a notable elongation and contraction
below TN1, as shown in Figs. 13(f, g), respectively. Interestingly, the $c$
parameter on the other hand shows no significant deviation from the
conventional temperature-induced cell contraction (phonons) at TN1. In the HT
AFM region of TbSi, the Tb moments lie in the $ab$ plane at an angle of 80o
with the $a$-axis associated with wave vector q$\approx$(0, 1/2, 1/8) [33] and
hence change in only $a$ and $b$ lattice parameters is in accordance with the
strong spin-lattice coupling in this sample. The lattice parameters $a$ and
$c$ show a change in their behavior below TN2/TN3, where the lattice parameter
$a$ remains almost invariant; however, the parameter $c$ shows an abrupt jump
(elongation) below this temperature. The ND measurements demonstrate the
first-order transformation from a commensurate magnetic structure with q = (0,
1/2, 0) to a three-dimensionally modulated phase having q = (qx, qy, qz), with
a most remarkable change only in the qz value from 0 to 0.126 between 36-39 K
[33]. Therefore, the observed first-order-like abrupt change in the $c$
lattice parameter in this temperate regime of TbSi is associated with the
drastic modulation in the magnetic wave vector in the $z$-direction,
indicating the sizable magnetostriction in the sample also at the order-order
transition. To quantify the observed spontaneous magnetostriction in TbSi, the
lattice parameters above the magnetic transition temperatures are fitted using
the Grüneisen function, given as [82, 51].

Figure 14: (a–e) The Rietveld refinement of the powder XRD data of
TbSi0.6Ge0.4 shown at some representative temperatures. The temperature
dependence of the lattice parameters (f) $a$, (g) $b$, (h) $c$, (i) cell
volume, and (j) overall $B$ parameter. The vertical dashed lines represent the
AFM transitions at 57 and 44 K, taken from the heat capacity measurements
(Table I). The solid curves in (f–i) represent the best fit of the unit cell
parameters from 70–300 K using the Grüneisen approximation and the dashed
curves represent the extrapolation of the fitted data down to 10 K.

$\epsilon(T)=\epsilon_{0}+K_{0}U(T),$ (5)

where $\epsilon_{0}$ is a lattice constant value at 0 K, $K_{0}$ is the
measure of the incompressibility of the sample (a constant), and $U(T)$ is
internal energy, which can be expressed using the Debye function as

$U(T)=9Nk_{B}T\left(\frac{T}{\theta_{D}}\right)^{3}\int_{0}^{\theta_{D}/T}\frac{x^{3}}{(e^{x}-1)}dx,$
(6)

where $N$ represents the effective number of atoms in a unit cell (eight in
the present case) and $k_{B}$ is the Boltzmann constant. The solid curves in
Figs. 13(f–i) represent the best fit of the lattice parameters from 70 to 300
K using the above equations, and the fitting parameters are given in Table
III. Interestingly, the $b$ parameter shows a remarkably different temperature
evolution not only in the magnetically ordered state below TN1 (discussed
above) but also in the paramagnetic state. The lattice parameter $b$ shows
almost linear temperature dependence between 70 and 300 K, whereas $a$ and $c$
parameters show significant curvature in this temperature range [see Figs.
13(f–h)], which is evident from the significantly lower value of the Debye
temperature, $\theta_{D}$, for the $b$-axis as compared to $a$ and $c$, given
in Table III. This highly anisotropic temperature dependence of the lattice
parameters even in the PM state of TbSi may arise from the presence of large
orbital angular momentum (L=3) and hence finite single-ion anisotropy of Tb3+
ions [20, 21]. The calculated deviations in the lattice parameters at 10 K
from the Grüneisen function along $a$, $b$, and $c$ directions are 0.04%,
0.1%, and 0.06%, respectively, as shown by the arrows in Figs. 13(f–h), where
the deviation $\Delta$a10K=100*(aexp-afit)/aexp. Note that despite the first-
order-like discontinuous change in the $c$ parameter, the overall relative
deviation is higher in the case of $b$. The unit cell volume shows a small but
notable change at TN2/TN3, as shown by a discontinuity in the yellow lines in
Fig. 13(i). The overall Debye-Waller factor ($B$) increases below TN1, as
shown in Fig. 13(j), indicating strong spin-lattice coupling in the sample.
However, the $B$ parameter starts increasing from $\sim$80 K, i.e., well above
the HT AFM region, which suggests the presence of short-range magnetic
correlations and their coupling with the lattice even in the PM region.

Table 3: Fit parameters extracted from the Grüneisen approximation of the lattice parameters of TbSi and TbSi0.6Ge0.4 samples from 70 to 300 K. |  a |  b |  c |  V
---|---|---|---|---
|  |  TbSi |  |
$\epsilon_{0}$ [Å(Å3)] |  7.9120(1) |  3.8261(6) |  5.6957(3) |  172.49(2)
$K_{0}$ [Å(Å3)/J] |  1.53 x 1017 |  1.29 x 1017 |  1.45 x 1017 |  1.47 x 1019
$\theta_{D}$ (K) |  350(7) |  90(8) |  310(5) |  320(6)
|  |  TbSi0.6Ge0.4 |  |
$\epsilon_{0}$ [Å(Å3)] |  4.2827(3) |  10.5940(2) |  3.8625(4) |  175.29(2)
$K_{0}$ [Å(Å3)/J] |  1.15 x 1017 |  4.92 x 1017 |  1.52 x 1017 |  1.59 x 1019
$\theta_{D}$ (K) |  715 (20) |  640(17) |  80(13) |  300(6)
Figure 15: (a, b) The temperature dependent first, second, and third nearest
neighbour Tb-Tb distances for TbSi and TbSi0.6Ge0.4 samples, respectively. The
number in the parenthesis represent the degeneracy of the respective
interatomic distances. The horizontal dashed lines are drawn only to show the
temperature variation in these Tb-Tb distances. (c) Average Tb-Tb distances as
a function of temperature for both the samples. (d, e) Temperature dependence
of neighbouring Tb-Si/Ge distances for TbSi and TbSi0.6Ge0.4 samples,
respectively. (f) Average Tb-Si/Ge distances for both the samples. (g, h) A 3D
representation of the Tb atoms surrounded by seven Si/Ge atoms in TbSi and
TbSi0.6Ge0.4 crystals, respectively. (i) Temperature dependent local
distortion parameter ($\Delta$) around Tb atoms for both the samples (see text
for more details).

In Figs. 14(a-e), we show the refined powder XRD patterns of TbSi0.6Ge0.4 at
the same selected temperatures as for TbSi. The crystal structure of
TbSi0.6Ge0.4 also remains the same, viz., CrB-type (orthorhombic symmetry,
$Cmcm$ space group), in the whole studied temperature range down to 10 K. The
temperature dependence of the Rietveld refined lattice parameter of
TbSi0.6Ge0.4 is shown in Figs. 14(f–j). The lattice parameters $a$ and $c$,
shown in Figs. 14(g, i), respectively, exhibit elongation and contraction
below TN1. Notably, below TN2, this effect slightly decreases along the
$a$-axis but increases along the $c$-axis. The magnetic moments in the HT AFM
phase of TbSi0.6Ge0.4 lie along the $c$-axis as well as in the $ac$ plane in
equal proportion [33]. Therefore, a change in the $a$ and $c$ parameters is
attributed to the magnetostriction due to spin-lattice coupling in the sample.
However, it is important to note that the lattice parameter $c$ shows higher
magnetostriction in the LT AFM phase, despite the fact that all the Tb moments
lie only in the $ac$-plane in the LT phase of TbSi0.6Ge0.4 [33]. This shows
that the planar AFM with q=(1/2, 0, 1/2) in the LT phase produces more (less)
magnetostriction along the $c$ ($a$)-axis compared to the sinusoidally
modulated AFM having q=(qx, 0, qz) in the HT AFM phase region of TbSi0.6Ge0.4.
More strikingly, the lattice parameter $b$ in TbSi0.6Ge0.4 first shows
contraction below TN1 and then elongation below TN2, despite having no
magnetic moments lying along the $b$-axis in any of the magnetically ordered
phases. This indicates that any change in the $b$-parameter in the two AFM
regions is a consequence of change in the unit cell along the $a$ and
$c$-directions due to finite magnetoelastic coupling in the sample. The
fitting of high-temperature lattice parameters of TbSi0.6Ge0.4 from 70 to 300
K using equations 5 and 6 [shown by solid lines in Figs. 14(f–i)] gives
significantly lower (around one order of magnitude) Debye temperature for the
$c$ parameter compared to $a$ and $b$ (see Table III). Note that the lattice
parameter $b$ in $Pnma$ setting is equivalent to the $c$ parameter in the
$Cmcm$ symmetry [83, 84]. This explains the similar temperature-dependent
behavior of the lattice parameter $c$ in TbSi0.6Ge0.4 to the $b$ parameter in
TbSi and confirms the highly anisotropic thermal expansion of the lattice
parameters in both the samples in the magnetically ordered as well as
disordered (PM) states. Further, the unit cell volume of TbSi0.6Ge0.4, shown
in Fig. 14(i), increases compared to the TbSi sample in accordance with the
larger atomic size of the Ge atoms than Si [85]. However, interestingly, the
unit cell volume of TbSi0.6Ge0.4 displays no significant change at TN2, making
it an interesting case where the deviation in the unit cell volume associated
with the first-order magnetic transition is negligible. The compounds
exhibiting an abrupt change in magnetization at the first-order transition
accompanied by a small change in the cell volume are desirable for magnetic
refrigeration applications owing to their large value of $\Delta S_{\rm M}$
along with better cycle stability. Recently, RE2In (RE=Eu and Pr) compounds
were also found to display a similar behavior, with a volume change of as low
as $\sim$0.1% at their first-order FM$\rightarrow$PM transition [68, 86, 87].
The absence of any discontinuous change in the cell volume along with a
significant broadening in the magnetic transition at the higher field and
negligible thermal hysteresis make TbSi0.6Ge0.4 a possible candidate to use
for magnetocaloric applications. While in TbSi0.6Ge0.4 the change in magnetic
entropy is moderate, our work indicates that such almost anhysteretic first-
order transitions associated with small volume changes may be abundant in rare
earth compounds. We observed a change in the overall Debye-Waller factor at
both TN1 and TN2 in the case of TbSi0.6Ge0.4 as shown in Fig. 14(j), keeping
the large error associated with the B-factor in the Rietveld refinement in
mind.

Now we discuss the change in the local coordination environment of these
compounds with Ge substitution. In Figs. 15(a, b), the temperature dependence
of the first, second, and third nearest neighbor (NN) Tb-Tb distances is
plotted for TbSi and TbSi0.6Ge0.4, respectively, where the numbers in
parentheses show their respective degeneracies. Each Tb atom is coordinated
with eight different Tb atoms with varying distances in both $Pnma$ and $Cmcm$
structures. Interestingly, in the case of TbSi, the first NN Tb-Tb distance is
four-fold degenerate and the second NN distance is two-fold degenerate,
whereas the opposite trend is observed in TbSi0.6Ge0.4 [see Figs. 15(a, b)].
The third NN Tb-Tb bond distance is doubly degenerate in both cases. In fact,
this third NN Tb-Tb bond distance represents the unit cell parameters $b$ and
$c$, respectively, for TbSi and TbSi0.6Ge0.4 samples in their respective
crystal structures [83, 84]. The increase in the interatomic distance between
the Tb ions for both samples occurs on heating in accordance with general
lattice expansion [the horizontal dashed lines in Figs. 15(a, b) are the
baselines]. The temperature-dependent average Tb-Tb distances are plotted in
Fig. 15(c) for both compounds. An increase in the average Tb-Tb distances has
been observed with temperature and Ge substitution due to thermal and chemical
expansion of the unit cell, respectively. Each Tb atom is surrounded by seven
Si/Ge atoms in both samples. We note that the $Pnma$ space group has lower
crystal symmetry compared to $Cmcm$. Therefore, in the case of TbSi, we
observe a total of five different Tb-Si bond distances, with two sets of two
equal (1st and 2nd NN) and three unequal (3rd, 4th, and 5th NN) bonds, as
schematically represented in Fig. 15(g). Whereas, only three non-equivalent
Tb-Si/Ge bond distances were observed in TbSi0.6Ge0.4 with degeneracies of
four, one, and two for the first, second, and third NN, respectively, as
represented in Fig. 15(h). The temperature evolution of these Tb-Si/Ge bonds
for TbSi and TbSi0.6Ge0.4 is presented in Figs. 15(d) and (e), respectively.
Here, a number in the subscript of letter d represents the order of the NN
distance of the Tb atom, from the shortest to the longest. Due to increasing
lattice symmetry, d1 and d2 of TbSi merge and become d1 in the case of
TbSi0.6Ge0.4, whereas d4 and d5 become d3. The temperature evolution of the
average Tb-Si/Ge distance is presented in Fig. 15(f) for both samples, showing
the same increase with temperature and Ge doping as the Tb-Tb distances. In
order to further understand the change in the local coordination environment
around the Tb atoms, we calculate the local distortion parameter, $\Delta$,
defined as [82, 88].

$\Delta=\frac{1}{n}\sum_{n=1}^{n}\left(\frac{d_{n}-<d>}{<d>}\right)^{2},$ (7)

where n represents the number of Si/Ge atoms coordinated to Tb (seven in this
case), dn represents each Tb-Si/Ge bond distance, and $<d>$ is the average Tb-
Si/Ge distance. The temperature-dependent local distortion parameter around Tb
atoms (by Si/Ge atoms only) is presented in Fig. 15(i) for both samples. It is
interesting to note that the distortion parameter, $\Delta$, increases in
TbSi0.6Ge0.4 compared to the TbSi sample, despite the higher overall crystal
symmetry of the former. Thus, the change from the $Pnma$ structure in our
system towards the $Cmcm$ is coupled with the enhancement in the local
distortion within the unit cell. Interestingly, one may argue that the bond
distances obtained by Rietveld refinement are already averaged with respect to
Si/Ge distribution, and further averaging of atomic distances should blunt all
the details about the local structure. To the contrary, the $\Delta$ parameter
accurately identifies the possibility of lattice changes due to distortion
caused by chemical substitution, making it useful for future predictions.

##  IV  Conclusion

In summary, theoretical investigations reveal that a minimal 2.5% Ge doping is
sufficient to induce a structural transition from the FeB-type $Pnma$ lattice
to the CrB-type $Cmcm$ in TbSi1-xGex ($x=$0–1) compounds. This transition is
attributed to the increased local distortion resulting from Ge substitution,
favoring the emergence of the higher symmetry $Cmcm$ structure. Subsequent
experimental analyses, encompassing magnetization, specific heat, and
temperature-dependent X-ray diffraction (XRD), are conducted on TbSi and
TbSi0.6Ge0.4 samples. TbSi crystallizes in the FeB-type crystal structure
($Pnma$ space group), while TbSi0.6Ge0.4 adopts the CrB-type structure ($Cmcm$
space group). Both TbSi and TbSi0.6Ge0.4 samples exhibit two magnetic
transitions: a high-temperature (HT) antiferromagnetic (AFM) to paramagnetic
(PM) transition at T${}_{\rm N1}\approx$ 58 K and 57 K, and a low-temperature
(LT) AFM to HT AFM transition at T${}_{\rm N2}\approx$ 36 K and 44 K,
respectively. Detailed CP measurements manifest the second- and first-order
nature of these phase transitions, respectively, in both samples. We unravel
an additional intermediate temperature (IT) AFM phase between TN1 and TN2 in
the case of TbSi, where both LT to IT and IT to HT AFM phase transitions
exhibit first-order nature. Both samples display a metamagnetic-like behavior
with a critical magnetic field of around 6 T at 50 K, indicative of weak HT
AFM interactions under an external magnetic field. The TbSi and TbSi0.6Ge0.4
samples show a magnetic entropy change ($\Delta S_{\rm M}$) of 9.6 (and
$\Delta$T${}_{\rm ad}\approx$ 1.2 K) and 11.6 J/kg-K, respectively, at TN2 for
$\Delta\mu_{0}$H= 7 T. The TbSi0.6Ge0.4 sample displays a significant
broadening in the $\Delta$SM curves at higher magnetic fields, yielding a
reasonably large value of TEC(10)=9.3 J/kg-K as compared to TbSi (5.4 J/kg-K)
at $\Delta\mu_{0}$H=7 T. The TbSi0.6Ge0.4 and, to a lesser extent, TbSi
display field-induced additional AFM interactions in the HT phase region,
which are found to be responsible for the significant broadening in the
$\Delta S_{\rm M}$ curves of the former at higher magnetic fields. Moreover,
both TbSi and TbSi0.6Ge0.4 exhibit highly anisotropic thermal expansion
behavior in both AFM and PM states, accompanied by notable magnetostriction
and magnetoelastic coupling at the observed transition temperatures. Finally,
a detailed H-T phase diagram for both samples has been constructed utilizing
the CP(H, T) data, elucidating their complex magnetic behavior under varying
temperature and magnetic fields.

###  IV.1 Acknowledgments

This work was supported by the U.S. Department of Energy (DOE), Office of
Science, Basic Energy Sciences, Materials Science and Engineering Division.
The research was performed at Ames National Laboratory, which is operated for
the U.S. DOE by Iowa State University under contract # DE-AC02-07CH11358.

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