# Magnetoelastic Coupling and Phases in the Skyrmion Lattice Magnet Gd2PdSi3
Discovered by High-resolution Dilatometry

S. Spachmann1<EMAIL_ADDRESS>A. Elghandour1 M.
Frontzek2,4 W. Löser3 R. Klingeler1,4 1Kirchhoff Institute for Physics,
Heidelberg University,Germany  2Oak Ridge National Laboratory, Oak Ridge, USA
3Leibniz Institute for Solid State and Materials Research (IFW), Dresden,
Germany  4Institute of Solid State and Materials Physics, Dresden University
of Technology, Germany  4Centre for Advanced Materials (CAM), Heidelberg
University, Germany

(August 27, 2024)

###### Abstract

We report detailed thermodynamic studies on high-quality single crystals of
the centrosymmetric skyrmion-hosting intermetallic Gd2PdSi3 by means of high-
resolution capacitance dilatometry in fields up to 15 T which are complemented
by specific heat and magnetization studies. Our dilatometric measurements show
magnetoelastic effects associated with antiferromagnetic order at
$T_{\mathrm{N1}}$ = 22.3 K and $T_{\mathrm{N2}}$ = 19.7 K, as well as strong
field effects in an applied magnetic field of 15 T up to 200 K (150 K) for
$B\parallel c$ ($B\parallel a$*, i.e. $B\perp c$). The data allow us to
complete the magnetic phase diagram, including a new feature at $T^{*}\approx
12$ K below which a new degree of freedom becomes relevant. For the first
time, the magnetic B vs. T phase diagram for the $a$*-axis is also reported.
Grüneisen analysis shows the onset of magnetic contributions around 60 K,
i.e., well above $T_{\mathrm{N1}}$. Uniaxial pressure dependencies of opposite
sign, $-1.3$ K/GPa and $0.3$ K/GPa, are extracted for the out-of-plane and in-
plane directions at $T_{\mathrm{N1}}$. For $T^{*}$ we obtain
${\partial}T^{*}/{\partial}p_{\mathrm{c}}=1.4$ K/GPa. In particular we
elucidate thermodynamic properties of the recently discovered skyrmion lattice
phase and show that it is strongly enhanced by uniaxial pressure.

##  I Introduction

Ternary intermetallic compounds of the type R2TX3 ($R$ = rare earth, $T$ =
transition metal, $X$ = element of main groups III to V) Hoffmann and Pöttgen
(2001); Pan et al. (2013) have been investigated extensively over the past
decades, due to their variety of intriguing electronic properties ranging from
superconductivity Majumdar and Sampathkumaran (2001), giant magnetoresistance
(GMR) Majumdar et al. (2000, 2001); Paulose et al. (2003), ferromagnetism Cao
et al. (2010) and incommensurate spin structures Kurumaji et al. (2019);
Hirschberger et al. (2020a), phenomena related to Kondo physics and heavy
fermions Majumdar et al. (1999); Saha et al. (2000); Majumdar et al. (2002),
to non-Fermi-liquid Majumdar et al. (1999) and spin-glass behavior Kaczorowski
and Noel (1993); Tien et al. (1997); Li et al. (1999, 2003). This is
particularly evident in the title material Gd2PdSi3 where a skyrmion lattice
phase featuring giant topological Hall and Nernst effect was discovered
recently. Saha et al. (1999); Kurumaji et al. (2019); Hirschberger et al.
(2020b)

Most members of the R2PdSi3 family of ternary silicides crystallize in a
highly symmetric AlB2-derived hexagonal structure (space group P6/mmm) with
triangular lattice layers of $R^{3+}$ magnetic sites sandwiching honeycomb
nets of Pd/Si sites. While the Pd and Si ions were originally believed to be
distributed statistically Szytuła et al. (1999), an X-ray and neutron
diffraction study by Tang et al. showed for Ho2PdSi3 that these ions actually
order into a superstructure along both in- and out-of-plane directions, while
the overall centrosymmetry of the structure is retained Tang et al. (2011).
This leads to two nonequivalent sites for the $R^{3+}$ ions, which has been
shown to affect the magnetism in an applied magnetic field for Er2PdSi3 Tang
et al. (2010). While no structural phase transition has been detected for R =
Gd, Tb, Dy, Ho, Er and Tm, most R2PdSi3 compounds show long-range magnetic
order at low temperatures Frontzek (2009); Smidman et al. (2019); Mukherjee et
al. (2011).

These various ordering phenomena are driven by a delicate interplay of
indirect exchange coupling mediated by the conduction electrons, i.e., the
Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction, spin-orbit coupling and the
influence of crystal field (CF) effects. The Gd3+ ions in Gd2PdSi3, however,
with a half-filled $4f$ shell, have vanishing orbital momentum (J $\approx$ S
$=7/2$) and are not influenced by crystal field effects. Magnetic order,
therefore, arises from the RKKY interaction and dipole-dipole interactions.
Gd2PdSi3 exhibits two successive phase transitions around $T_{\mathrm{N}}$ =
21 K Hirschberger et al. (2020a) and was found to exhibit a skyrmion lattice
(SkL) phase of Bloch-type skyrmions in low magnetic fields applied along the
$c$-axis Kurumaji et al. (2019). A number of incommensurate spin structures
both in zero-field as well as in higher applied magnetic fields have been
identified Hirschberger et al. (2020a) and the phase diagram in fields up to 9
T has been established through resistance and magnetization measurements as
well as resonant X-ray scattering Frontzek (2009); Kurumaji et al. (2019);
Hirschberger et al. (2020a); Zhang et al. (2020). Single crystal X-ray and
neutron diffraction measurements yielded lattice parameters at 300 K (2 K) of
$a$ = 4.079 Å (4.066 Å) and $c$ = 4.098 Å (4.091 Å), i.e. $\Delta a/a$ =
$3.2\cdot 10^{-3}$ and $\Delta c/c$ = $1.7\cdot 10^{-3}$. Tang et al.
(2011),FN (4)

Except for these measurements of the lattice parameters, however, there is at
present no study on magnetoelastic effects in Gd2PdSi3. Therefore, with a
particular focus on the skyrmion lattice phase, we report detailed
dilatometric studies of Gd2PdSi3 in a wide range of temperatures and magnetic
fields. Our thermal expansion and magnetostriction data show pronounced
magnetoelastic coupling and field effects extending up to temperatures of 150
K and above. Moreover, we uncover yet unreported phases and an anomaly in
zero-field which appears well below the Néel transitions at $T_{\mathrm{N1}}$
= 22.3(5) K and $T_{\mathrm{N2}}$ = 19.7(5) K, thereby evidencing competing
interactions already in zero-field. We update the magnetic phase diagram for
$B\parallel c$-axis, present for the first time the phase diagram for
$B\parallel a$*-axis, and discuss in detail the thermodynamic properties for
$B\parallel c$. Our results in particular elucidate the skyrmion lattice phase
and we show that it is enhanced by uniaxial pressure.

##  II Experimental Methods

Single crystals of Gd2PdSi3 have been grown by the optical floating-zone
method as reported in Ref. [Mazilu, 2006; Xu et al., 2011] and were previously
studied by AC susceptibility, neutron diffraction Frontzek (2009), and angle-
resolved photoemission spectroscopy Inosov et al. (2009). The magnetization
was studied in the temperature regime from 1.8 K to 300 K in magnetic fields
up to 7 T in a Magnetic Properties Measurement System (MPMS3, Quantum Design)
and up to 14 T in a Physical Properties Measurement System (PPMS, Quantum
Design) using the Vibrating Sample Magnetometry (VSM) option. Specific heat
measurements were performed on a PPMS-14 using a relaxation method on single
crystals of $m=20.79$ mg (2 K-300 K) and 13.49 mg (0.15 K-3 K). High-
resolution dilatometry measurements were performed by means of a three-
terminal high-resolution capacitance dilatometer in a home-built setup placed
inside a Variable Temperature Insert (VTI) of an Oxford magnet system Küchler
et al. (2012); Werner et al. (2017). With this dilatometer, the relative
length changes $dL_{i}/L_{i}$ along the crystallographic $c$ and $a$*
directions, respectively, were studied on an oriented cuboid-shaped single
crystal of dimensions $2.480\times 1.300\times 1.459\leavevmode\nobreak\ $mm3.
Measurements were performed at temperatures between 2 K and 300 K in magnetic
fields up to 15 T, applied along the direction of the measured length changes,
and the linear thermal expansion coefficients $\alpha_{i}=1/L_{i}\cdot
dL_{i}(T)/dT$ were derived. In addition, the field-induced length changes
$dL_{i}(B_{i})$ were measured at various fixed temperatures between 1.7 K and
200 K in magnetic fields up to 15 T. The longitudinal magnetostriction
coefficient $\lambda_{i}\leavevmode\nobreak\ =\leavevmode\nobreak\
1/L_{i}\leavevmode\nobreak\ \cdot\leavevmode\nobreak\ dL_{i}(B_{i})/dB_{i}$
was derived from $dL_{i}(B_{i})$.

##  III Experimental Results

###  III.1 Evolution of magnetic order at $B=0$

Uniaxial thermal expansion and specific heat show pronounced anomalies around
20 K which are associated with the onset of long-range magnetic order (Fig.
1). Close inspection of the anomalies indicates the proximity of not only one,
but two phase transitions around $T_{\mathrm{N}}$. While the anomaly at
$T_{\mathrm{N1}}$ = 22.3(5) K is seen as a jump in $\alpha_{i}$, the anomaly
at $T_{\mathrm{N2}}$ = 19.7(5) K (20.3(5) K for the $a$*-axis) is expressed as
a peak. This observation confirms the results by Hirschberger et al. of two
consecutive phase transitions in zero field Hirschberger et al. (2020a).
Beyond these two known transitions, however, our data display a third anomaly
indicative of a phase transition which has not been reported for single
crystals of Gd2PdSi3. This anomaly, marked by $T^{*}\approx 13$ K in the inset
of Fig. 1(b), is visible as a broad jump for both directions in
$\alpha_{\mathrm{i}}$ which extends between 10 K and 15 K and will be
discussed in more detail below.

Figure 1: (a) Relative length changes $dL_{i}/L_{i}$ along the $c$ and
$a$*-crystallographic directions. Directions w.r.t. the Gd lattice are
indicated in the inset. (b) Associated linear thermal expansion coefficients
$\alpha_{i}$ as well as the calculated 1/3 volume expansion
$\alpha_{\mathrm{V}}$. The inset displays the difference
$\alpha_{\mathrm{a^{*},mag}}-\alpha_{\mathrm{c,mag}}$ of the background
corrected thermal expansion coefficients
$\alpha_{\mathrm{i,mag}}=\alpha_{i}-\alpha_{\mathrm{ph}}$. Vertical dashed
lines indicate two phase transitions and the dashed-dotted line (inset) is a
guide to the eye highlighting a feature around $T^{*}$. (c) Specific heat
$c_{\mathrm{p}}$/T (black markers). Inset: Low-temperature regime plotted as
$c_{\mathrm{p}}$/T vs. $T^{2}$. The solid red line shows a fit with
$c_{\mathrm{p}}$ $=\gamma T+\beta T^{3}$. Dashed curves in all graphs mark the
non-magnetic background as explained in the text.

The specific heat data show the three observed features at $T_{\mathrm{N1}}$,
$T_{\mathrm{N2}}$ and $T^{*}$, too (Fig. 1(c)). The shape of the anomalies in
$c_{\mathrm{p}}$ is analogous to their shape in $\alpha_{\mathrm{a^{*}}}$. At
very low temperatures below about 400 mK a quasi-linear behavior of
$c_{\mathrm{p}}$/T vs. $T^{2}$ is found (see the inset of Fig. 1(c)) reminding
of similar observations in Lu2PdSi3 and Ce2PdSi3. Cao et al. (2013); Saha et
al. (2000) In this temperature regime, the data are described well by a linear
and a cubic term, i.e., $c_{\mathrm{p}}$/T $=\gamma+\beta T^{2}$. The quasi-
linear term is described by an effective Sommerfeld coefficient $\gamma=52(5)$
mJ/(mol K2). This parameter is in between the values obtained for Lu2PdSi3
(6.9 mJ/(mol K2)) and Ce2PdSi3 (108 mJ/(mol K2)), the latter being discussed
as evidence of heavy-fermion behaviour. Saha et al. (2000) Whereas phonons can
be neglected in this temperature regime, the coefficient $\beta=1.25(3)$
J/(mol K4) is rather large and reflects the contribution of low-energy
antiferromagnetic excitations.

The dashed lines in Fig. 1 show the phononic and electronic contributions to
the relative length changes, thermal expansion coefficients, and specific
heat. In order to obtain these contributions, the specific heat of the non-
magnetic analog Lu2PdSi3 as reported by Cao et al. Cao et al. (2013) was
fitted by phononic Debye and Einstein terms, as well as an electronic term,
according to

$c_{p}^{el,ph}=\gamma
T+n_{\mathrm{D}}D\left(\frac{T}{\Theta_{D}}\right)+n_{\mathrm{E}}E\left(\frac{T}{\Theta_{E}}\right)$
(1)

where $\gamma$ is the Sommerfeld coefficient, $n_{\mathrm{D}}$ and
$n_{\mathrm{E}}$ are constants, $D(T/\Theta_{\mathrm{D}})$ and
$E(T/\Theta_{\mathrm{E}})$ are the Debye and Einstein functions with the Debye
and Einstein temperatures $\Theta_{\mathrm{D}}$ and $\Theta_{\mathrm{E}}$. The
fit to the Lu2PdSi3 specific heat data yields $\Theta_{\mathrm{D}}$ = 213 K,
$\Theta_{\mathrm{E}}$ = 454 K, with $n_{\mathrm{D}}=3.69$ and
$n_{\mathrm{E}}=1.98$. $\gamma$ was fixed to the value reported by Cao et al.
of 6.93 mJ/(mol K2). Compared to $\Theta_{\mathrm{D}}=191$ K by Cao et al.,
extracted from the low temperature regime, our value is slightly larger.

Scaling the Debye and Einstein temperatures by the different masses of Lu and
Gd we obtain a scaling factor Tari (2003) of
$\Theta_{\mathrm{D,LPS}}/\Theta_{\mathrm{D,GPS}}=0.962$. The specific heat and
thermal expansion of Gd2PdSi3 were thus fitted with fixed
$\Theta_{\mathrm{D}}=222$ K $=\Theta_{\mathrm{D,LPS}}/0.962$ and
correspondingly $\Theta_{\mathrm{E}}=471$ K. For the fit to the specific heat,
$\gamma=52$ mJ/(mol K2) was also fixed. For the thermal expansion the
electronic contribution was negligibly small and therefore omitted, i.e. it
was fitted by

$\alpha^{ph}=n_{\mathrm{D}}D\left(\frac{T}{\Theta_{D}}\right)+n_{\mathrm{E}}E\left(\frac{T}{\Theta_{E}}\right)$
(2)

with parameters $n_{\mathrm{D}}$ and $n_{\mathrm{E}}$. The phononic
contributions to $dL_{i}/L_{i}$ in Fig. 1(a) were obtained by integrating the
background obtained for the respective $\alpha_{\mathrm{i}}$.

Subtracting the electronic and phononic backgrounds from the specific heat and
thermal expansion coefficients yields their respective magnetic contributions
which extend up to about 60 K. This agrees with the temperature regime where
the magnetization exhibits a non-linear field dependence up to 7 T (see Fig.
S1). From $c_{\mathrm{p,mag}}$/T the changes in magnetic entropy,
$S_{\mathrm{mag}}$, above 150 mK are calculated. We obtain a constant $\Delta
S_{\mathrm{mag}}(T>150\leavevmode\nobreak\ \mathrm{mK})=31.3$ J/(mol K) above
60 K, which is 90% of the full expected magnetic entropy of 2$R\ln{8}=34.6$
J/(mol K), where $R$ is the universal gas constant.

Returning to the thermal expansion data, we see that the anomalies in the
thermal expansion coefficients, at $T_{\mathrm{N1}}$ and $T_{\mathrm{N2}}$,
are of opposite sign for the $c$\- and $a$*-axis, indicating opposite pressure
dependencies ${\partial}T_{\mathrm{N}i}/{\partial}p_{\mathrm{c}}<$ 0 and
${\partial}T_{\mathrm{N}i}/{\partial}p_{\mathrm{a^{*}}}>$ 0\. The volume
thermal expansion also indicates a negative hydrostatic pressure dependence
${\partial}T_{\mathrm{N}i}/{\partial}p<$ 0 for both antiferromagnetic
transitions.

The Grüneisen ratio of the thermal expansion coefficient and the specific heat
is a valuable quantity to determine the relevant energy scales driving the
system and to quantify its pressure dependencies. In the presence of one
dominant energy scale $\epsilon$, this ratio is independent of temperature and
enables the determination of the pressure dependence of $\epsilon$, i.e.
Gegenwart (2016); Klingeler et al. (2006),

$\Gamma_{i}=\frac{\alpha_{i}}{c_{\mathrm{p}}}=\frac{1}{TV_{\mathrm{m}}}\frac{\partial
S/\partial p_{i}}{\partial S/\partial
T}=\frac{1}{V_{\mathrm{m}}}\frac{\partial\ln\epsilon}{{\partial}p_{i}}.$ (3)

Here, $V_{\mathrm{m}}$ is the molar volume and the index $i$ indicates a
linear direction or the volume. At $T_{\mathrm{N}}$, Eq. 3 converts to
$\Gamma$ =
($T_{\mathrm{N}}$$V_{m}$)${}^{-1}\cdot\partial$$T_{\mathrm{N}}$/$\partial p$.
Comparing the magnetic contributions $\alpha_{\mathrm{i,mag}}$ and
$c_{\mathrm{p,mag}}$ hence allows to identify temperature regimes where the
Grüneisen relation implies only one dominant energy scale while appropriate
scaling enables to read off the respective parameter
$\Gamma_{\mathrm{i,mag}}$. As shown in Fig. 2, the overall behavior of
$\alpha_{\mathrm{i,mag}}$ and $c_{\mathrm{p,mag}}$ is similar except for a
distinct jump in $\alpha_{\mathrm{i,mag}}$ at $\sim 12$ K which is much less
pronounced in the magnetic specific heat. In both cases, magnetic
contributions start to evolve around 60 K.

Figure 2:  Magnetic contributions to the thermal expansion coefficient (left
axis) and specific heat (right axis) for (a) the $a$*-axis and (b) the
$c$-axis after subtracting phononic and electronic contributions as described
in the text. FN (1)

Despite the overall similar behavior, there are differences at higher
temperatures, too. While the $a$*-axis shows a nearly perfect overlap between
$\alpha_{\mathrm{i,mag}}$ and $c_{\mathrm{p,mag}}$ down to 14 K as shown in
Fig. 2(a), we only observe a very good agreement around $T_{\mathrm{N1}}$ and
$T_{\mathrm{N2}}$ for the $c$-axis, in a range from 17 K to about 23 K. We
also note that below $\sim$14 K our results indicate the failure of Grüneisen
scaling rather than the presence of just a different scaling parameter.

Our data, however, clearly imply the presence of a single dominant energy
scale at and around the magnetic ordering temperatures $T_{\mathrm{N1}}$ and
$T_{\mathrm{N2}}$. The obtained Grüneisen parameters amount to
$\Gamma_{\mathrm{c,mag}}=-91(13)\cdot 10^{-8}$ mol/J and
$\Gamma_{\mathrm{a^{*},mag}}=22(3)\cdot 10^{-8}$ mol/J. From these values,
moderate pressure dependencies are derived, i.e., we obtain negative pressure
dependencies $\partial T_{\mathrm{N1}}/\partial p_{\mathrm{c}}=-1.3(2)$ K/GPa
and $\partial T_{\mathrm{N2}}/\partial p_{\mathrm{c}}=-1.4(2)$ K/GPa for
uniaxial pressure applied along the $c$-axis. The uniaxial pressure
dependencies for $p\parallel a$*-axis are positive and more than a factor of
four smaller, i.e., 0.31(5) K/GPa for $T_{\mathrm{N1}}$ and 0.34(5) K/GPa for
$T_{\mathrm{N2}}$.

While our data hence evidence that the ordering phenomena at $T_{\mathrm{N1}}$
and $T_{\mathrm{N2}}$ are governed by the same energy scale, an additional
energy scale becomes relevant upon further cooling, around $T^{*}$, as proven
by the failure of Grüneisen scaling (cf. Fig. 2). Closer inspection of the
associated anomalies implies not only a broad jump-like increase in the
thermal expansion coefficients but also a less pronounced anomaly in
$c_{\mathrm{p}}$ which is visible much more clearly in the
$c_{\mathrm{p}}$/$T$ data in Fig. 1c. In an attempt to deduce the anomaly size
associated with the respective features we obtain $\Delta
c_{\mathrm{p}}^{*}\approx 2.7(5)$ J/(mol K),
$\Delta\alpha_{\mathrm{V}}^{*}\approx 1.8\cdot 10^{-6}$/K,
$\Delta\alpha_{\mathrm{c}}^{*}\approx 4.1(6)\cdot 10^{-6}$/K,
$\Delta\alpha_{\mathrm{a^{*}}}^{*}\approx-1.0(3)\cdot 10^{-6}$/K. The changes
in magnetization around T∗ are very small for both axes and could not be seen
in the isothermal magnetization $M(B)$. However, temperature sweeps of the
magnetization in static field evidence a jump in ${\partial}\chi/{\partial}T$,
which is visible for $B\geq 0.2$ T (0.25 T) for $B\parallel c$ ($B\parallel
a$*) (see SI, Fig. S10(a)). At 0.2 T the jump height amounts to 5.6(1.4)$\cdot
10^{-3}\mu_{\mathrm{B}}$/(f.u. K). Further values are listed in Tab. S7.

###  III.2 Thermal Expansion at B $\neq$ 0 and Magnetostriction

The effect of high magnetic fields on the thermal expansion and specific heat
is shown in Fig. 3. A number of observations can be made: (1) The sharp
features indicating phase transitions are absent at $B=15$ T. (2) Significant
entropy is shifted to higher temperatures and, at $B=15$ T, significant field
effects are visible at least up to 150 K in all shown quantities, in
particular for $\alpha_{\mathrm{c}}$ even up to about 200 K. (3)
Magnetostriction from 0 T to 15 T is positive (negative) for the $c$-axis
($a$*-axis), and (4) the temperature region of negative thermal expansion of
the $c$-axis extends up to about 65 K at 15 T, compared to 38 K in zero-field.
Note, that the magnetostriction data fully agree to the thermal expansion data
at $B\neq 0$ T as shown by the (green) vertical lines in the inset of Fig. 3).

Figure 3:  The effect of high fields on (a) the thermal expansion coefficient
($B$ = 15 T) and (b) the specific heat $c_{\mathrm{p}}$/T ($B$ = 14 T) as
compared to zero-field measurements. The inset in (a) shows the relative
length changes. Vertical green bars indicate magnetostriction data from 0 T to
15 T at several temperatures. Figure 4:  Magnetostriction coefficients
$\lambda_{i}$ ($i$ = $c$, $a$*) and isothermal magnetic susceptibility,
$\partial M/\partial B$, at temperatures around 2 K, for $B\parallel c$ (a-b)
and $B\parallel a$* (c-d). Note the logarithmic scale in (b) and (c). Insets
in (a) and (d) show the relative length changes
$dL_{\mathrm{i}}(B)/L_{\mathrm{i}}$. Triangles and empty circles mark
anomalies in (a-d). Red markers and lines represent up-sweeps, black ones
down-sweeps.

Magnetostriction data at $T\approx 2$ K shown in Fig. 4(a) and (d) further
confirm strong magnetoelastic coupling and in addition clearly show the field-
induced phase transitions. For comparison the isothermal magnetic
susceptibility $\chi(B)=\partial M(B)/\partial B$ is also presented on the
same field scale for both directions (Fig. 4(b-c)). Considering the data for
$B\parallel c$, four anomalies can be identified (Fig. 4(a)): Up to 3.5 T,
there are two sharp peaks in $\lambda_{\mathrm{c}}$ signalling jumps in
$dL_{\mathrm{c}}(B)$ with only small field-hysteresis, followed by a broad
peak with a large hysteresis of $\sim$0.8 T. The size of the anomalies for up-
and down-sweep differs strongly. All three anomalies indicate discontinuous
phase transitions. Corresponding anomalies and hystereses are also visible in
the magnetic susceptibility. In addition, there is a broad downward jump in
$\lambda_{\mathrm{c}}$ at around 9 T, above which magnetostriction becomes
virtually zero which is also reflected by small $\chi$, i.e., rather full
alignment of magnetic moments in field (please note the logarithmic scale in
Fig. 4(b) and (c)). The overall region where hysteresis is visible extends
from about 6 T down to the lowest fields (see the inset in Fig. 4(a)) but no
remanent magnetostriction is visible which would indicate irreversible changes
in the sample, e.g., through domain effects. Four features are also visible in
$\lambda_{\mathrm{a^{*}}}$ for $B\parallel a$* (Fig. 4(d)). These anomalies
are smaller in magnitude, much broader and less well-defined than for the
$c$-axis. Similar to the findings for $\lambda_{\mathrm{c}}(B\parallel c)$,
there is a jump at higher fields, at about 7.3 T, but here of opposite sign.
Again, it signals a continuous transition to the saturated phase of vanishing
magnetostriction. In contrast to $B\parallel c$, the magnetostriction
measurements $dL_{\mathrm{a^{*}}}$($B\parallel a$*) feature pronounced
remanent magnetostriction below 5 K, i.e., non-zero overall length changes
after sweeping the field from 0 T to 15 T and back to 0 T. At 1.8 K, this
amounts to $(\Delta L/L)_{\mathrm{rem}}=1.4\cdot 10^{-5}$. We attribute this
observation to the irreversible effects of structural or magnetic domains as
seen, e.g., in CoCl2 Kalita et al. (2000), NiCl2 Kalita et al. (2002) and
NiTiO3 Dey et al. (2021). Such irreversible domain effects seem to be absent
in the measurements along the $c$-axis. The transition between a multidomain
and single domain state may thus be fully reversible for $B\parallel c$. We
conclude that hysteresis found for $B\parallel a$* below 3.5 T is both due to
the discontinuous nature of the phase transitions and domain effects.

Both data sets, hence, imply a series of four phase transitions in magnetic
field, at $T\approx 2$ K, which is also corroborated by magnetization studies
(also see Fig. S7) and agrees to the recently published phase diagram for
$B\parallel c$. Hirschberger et al. (2020a) Following the notations in Refs.
[Hirschberger et al., 2020a; Kurumaji et al., 2019] for the phases appearing
for $B\parallel c$, we label the respective phases as IC-1, A, IC-2, DP, and
field induced ferromagnetic (fiFM) phase, with IC-1/IC-2 being characterized
by incommensurate spin configurations, A by the formation of a skyrmion
lattice (SkL), and DP by the depinning of the direction of magnetic moments
(see also the phase diagram in Fig. 6). We note, however, that while the
magnetostriction data evidence field-driven structural changes, domain effects
may obscure the actual phase transitions up to the field and temperature
regions at which a single domain state is achieved. In particular, broad peaks
in the magnetostriction coefficients as seen in $\lambda_{\mathrm{a^{*}}}$
(Fig. 4(b), S4(e) and (f)) do not necessarily indicate the actual phase
boundaries, but the peak positions may differ from those found in the
magnetization studies, as shown by a phenomenological model by Kalita et al.
Kalita et al. (2000). Therefore, for the further thermodynamic analysis of the
phase boundaries as well as the construction of the phase diagram, for
$B\parallel a$* we will only consider anomalies in the magnetostriction which
can directly be linked to anomalies in isothermal magnetization.

Figure 5:  Thermal expansion coefficients (a,c) and static magnetic
susceptibility $\chi=M/B$ (b), respectively its derivative
$\partial\chi/\partial T$ (d), for (a-b) $B\parallel c$ and (c-d) $B\parallel
a$*. Curves are offset vertically by (a) 1.6$\cdot 10^{-6}$/K, (b) 0.5 erg/(G2
mol), (c) 1$\cdot 10^{-6}$/K, and (d) 0.2 erg/(G2 mol K), respectively, for
better visibility. 0.75 T to 3 T data in (d) are multiplied by a constant as
indicated in the legend and 3 T data is offset by 1.5 erg/(G2 mol K) instead
of 1.4 erg/(G2 mol K). ”FCW” indicates a field-cooled-warming measurement. FCW
Empty circles mark temperature positions of the anomalies as extracted for the
phase diagram.

In order to obtain the phase boundaries, we have performed thorough
magnetostriction and isothermal magnetization studies at various fixed
temperatures as well as corresponding temperature sweeps at fixed magnetic
field (see Figs. S4, S5, S7). FN (2) This is demonstrated in Fig. 5, where the
thermal expansion coefficients in low fields up to 3 T and the corresponding
magnetization data are presented. For $B\parallel c$ (Fig. 5(a) and (b)), the
evolution of two different phase boundaries can be traced straightforwardly.
Specifically, applying small fields yields a suppression of $T_{\mathrm{N2}}$
while the jump at $T_{\mathrm{N1}}$ becomes more distinguished. In increasing
field $T_{\mathrm{N1}}$ is also suppressed to lower temperature, but to a
smaller extent (Fig. 5a). Above 0.3 T the peak assigned to $T_{\mathrm{N2}}$
at zero-field changes its shape, signaling the transition from the previously
reported skyrmion lattice A phase to the IC-2 phase. It is associated with a
jump $\Delta\chi_{\mathrm{c}}^{\mathrm{A}}$ in the static susceptibility from
one $T$-independent value to another (Fig. 5(b)). At 0.4 T, a second feature
corresponding to the transition from the A phase to the IC-1 phase is visible
in both $\alpha_{\mathrm{c}}$ and $\chi_{\mathrm{c}}$, while above 0.9 T (1 T
for $\chi_{\mathrm{c}}$) all features below $T_{\mathrm{N1}}$ are gone.
Quantitatively, $\Delta\chi_{\mathrm{c}}^{\mathrm{A}}$ gradually decreases
from 0.91 $\mu_{\mathrm{B}}$/Gd3+ at 0.9 T and 8.6 K to 0.36
$\mu_{\mathrm{B}}$/Gd3+ at 0.4 T and 16.4 K. The pronounced jump in
$\alpha_{\mathrm{c}}$ at $T_{\mathrm{N1}}$ corresponds to a kink in the static
susceptibility, i.e., a jump in its derivative (see SI, Fig. S7).

As mentioned before, anomalies seen for measurements along the $a$*-axis are
in general much weaker and less well-defined than for the $c$-axis.
Furthermore, the evolution of anomalies in the thermal expansion and the
static susceptibility along the $a$*-axis is even more complex than for
$B\parallel c$ (Fig. 5(c) and (d)). In zero-field, the anomaly at $T^{*}$ is
also visible (see Fig. 1, as well as SI, Fig. S10(d) and can be traced up to
0.4 T in ${\partial}\chi/{\partial}T$. Also, a jump in
$\alpha_{\mathrm{a^{*}}}$ evolving from $T_{\mathrm{N1}}$ can be followed to
lower temperatures for increasing fields, corresponding to a jump in
$\partial\chi/\partial T$. Above 0.6 T this jump splits into two jumps,
uncovering an additional phase between the IC-2 and fiFM phases, while we only
see one broad jump in $\alpha_{\mathrm{a^{*}}}$.

##  IV Discussion

From our detailed dilatometric and thermodynamic data we construct the phase
diagrams for the $c$\- and $a$*-axes in Fig. 6. While the general features for
$B\parallel c$ confirm previous results Kurumaji et al. (2019); Hirschberger
et al. (2020a); Zhang et al. (2020), our data evidence two phases in zero-
field which were previously unknown: (1) Our isothermal magnetization data
between 19 K and 22 K (SI, Fig. S7) clearly indicate that the IC-2 phase does
not extend to zero field, but there is a separate pocket closed off by a phase
boundary extending from the edge of the A(SkL) phase to $T_{\mathrm{N1}}$ =
22.3 K. We label this new phase IC-3, since incommensurate spin structures
were previously reported for this temperature regime Kurumaji et al. (2019).
(2) Furthermore, the phase boundary at $T^{*}$ splits the IC-1 phase into IC-1
and IC-1’ (Fig. 1(b) inset). The yet unreported phase diagram for $B\parallel
a$* in general shows a similar behavior, with the critical fields of the IC-1’
and IC-4 phases at lowest temperatures assuming higher values than IC-1’ and
the A phase for the $c$-axis. The IC-4 phase appearing for $B\parallel a$*
(see Fig. 6b) reminds of the A(SkL) phase for $B\parallel c$, however, it was
shown previously by angle-dependent resistivity measurements in the
$a$*-$c$-plane at 2 K that it does not connect to the A(SkL)
phase.Hirschberger et al. (2019) The magnetic structure of this phase needs to
be clarified by diffraction studies. One major difference between the phase
diagrams is seen for $B\parallel a$*, where the IC-2 phase is not directly
adjacent to the field-induced FM phase, but there is an additional phase
(labelled B) in between (see Fig. 5(c) and (d)). The B phase is bordered by
two jumps both in $c_{\mathrm{p}}$ as well as in ${\partial}\chi/{\partial}T$.

Figure 6:  Phase diagrams for (a) $B\parallel c$ and (b) $B\parallel a$*
constructed from different experimental techniques as indicated in the
legends. The shaded areas show two strong hysteresis regimes. The
abbreviations for the phases are: paramagnetic (PM), incommensurate magnetic
orders (IC-x), antiferromagnetic (A,B), antiferromagnetic skyrmion lattice
(SkL), and depinned phase (DP).

From the anomalies in the thermal expansion, specific heat and magnetization
at the phase boundaries we calculated the uniaxial pressure and field
dependence of the critical temperatures and critical fields, as well as the
entropy changes at the phase boundaries for $B\parallel c$-axis. Considering
$T^{*}(B)$ marking a continuous phase transition, the associated jumps in
specific heat ($\Delta$$c_{\mathrm{p}}$), magnetic susceptibility
($\Delta(\partial M/\partial B)$), and thermal expansion coefficient
($\Delta\alpha$) are connected with the magnetic field and pressure
dependencies of $T^{*}$ by the Ehrenfest-type relations (see e.g. Ref. [Barron
and White, 1999; Klingeler et al., 2005])

$\left.\left(\frac{{\partial}T^{*}}{{\partial}B_{i}}\right)\right|_{p}=-T^{*}\frac{\Delta(\left.\frac{\partial
M_{i}}{\partial T})\right|_{B}}{\Delta c_{p,B}}$ (4)

and

$\left.\left(\frac{{\partial}T^{*}}{{\partial}p_{i}}\right)\right|_{B}=T^{*}V_{\mathrm{m}}\frac{\Delta\alpha_{i}}{\Delta
c_{\mathrm{p}}}.$ (5)

Using the molar volume $V_{\mathrm{m}}=7.06\cdot 10^{-5}$ m3/mol (see Ref.
[Frontzek, 2009]) as well as the anomaly values presented above and in Tab. S7
– for their extraction from the experimental data see the supplement, Fig. S10
– we obtain a moderate uniaxial pressure dependence of
${\partial}T^{*}/{\partial}p_{\mathrm{c}}=-1.4(3)$ K/GPa. In a field of 0.2 T,
the field dependence is very small and amounts to only
${\partial}T^{*}/{\partial}B_{\mathrm{c}}=-37(13)$ mK/T. At 0.3 T,
${\partial}T^{*}/{\partial}B_{\mathrm{c}}$ rises to $-0.28(18)$ K/T. These
results demonstrate that the IC-1 phase is stabilized both under pressure and
applied field at the expense of the IC-1’ phase. Also, from these values we
can calculate the expected jump in $\partial M/\partial B$ ($\Delta\chi$), at
$T^{*}(B)$ via

$\left.\left(\frac{{\partial}T^{*}}{{\partial}B}\right)\right|_{p}=-\frac{\Delta\left(\frac{{\partial}M}{{\partial}B}\right)|_{p,T}}{\Delta\left(\frac{{\partial}M}{{\partial}T}\right)|_{p,B}}.$
(6)

At 0.2 T this yields $\Delta\chi=-2\cdot 10^{-4}\mu_{\mathrm{B}}$/(f.u. T),
which is well below the resolution limit of our experiment, explaining why our
isothermal magnetization studies do not show anomalies at $T^{*}$ (see SI,
Fig. S10). Note, that in an early report on Gd2PdSi3 by Mallik et al. the
authors detected a jump in the effective local field $|$B${}_{\mathrm{eff}}|$
at 15 K by Mössbauer spectroscopy and attributed it to a lower ordering
temperature of one of the two Gd sites in Gd2PdSi3. This transition was not
detected in any of the later reports on single crystalline samples, but our
results presented in this work clearly confirm its presence.

The phase boundaries between the A(SkL), IC-2, and DP phases are of
discontinuous nature, exhibiting jumps $\Delta(dL_{i}/L_{i})$ in the length
changes and $\Delta M$ in the magnetization (i.e., $\Delta m$ in magnetic
moment). Therefore, the Clapeyron equations Barron and White (1999)

$\displaystyle\left.\left(\frac{{\partial}T_{c}}{{\partial}p_{i}}\right)\right|_{B}$
$\displaystyle=V_{\mathrm{m}}\frac{\frac{{\Delta}L_{i}}{L_{i}}}{\Delta S}$ (7)
$\displaystyle\left.\left(\frac{{\partial}T_{c}}{{\partial}B_{i}}\right)\right|_{p}$
$\displaystyle=-\frac{\Delta m_{i}}{\Delta S}=-\frac{\Delta(M_{i}\cdot
V)}{\Delta S}$ (8)
$\displaystyle\left.\left(\frac{{\partial}B_{c}}{{\partial}p_{i}}\right)\right|_{T}$
$\displaystyle=V_{\mathrm{m}}\frac{\frac{{\Delta}L_{i}}{L_{i}}}{\Delta m_{i}}$
(9)

apply for the pressure and field dependence of the respective critical
temperatures $T_{\mathrm{c}}$ and critical fields $B_{\mathrm{c}}$. Hence, the
observed slopes ${\partial}T_{c}/{\partial}B$ (see Tab. S1–S4) and the jumps
$\Delta M$ allow us to obtain the associated entropy changes $\Delta S$.

For the transition from the incommensurate IC-1’ phase to the skyrmion lattice
A phase, a jump $\Delta M=0.99(10)\mu_{\mathrm{B}}/$Gd accompanies the
$c$-axis contraction of $\Delta dL_{\mathrm{c}}/L_{\mathrm{c}}=-10.8(1.1)\cdot
10^{-6}$ at 4.3 K. Applying the above-mentioned thermodynamic relations yields
small entropy changes on the order of $\Delta S_{\mathrm{calc}}=125(13)$
mJ/mol K and a negative uniaxial pressure dependence of
${\partial}T_{\mathrm{c}}/{\partial}p_{i}=-6.1(9)$ K/GPa. At higher
temperatures these values decrease down to $110(11)$ mJ/mol K and $-1.5(2)$
K/GPa at 16 K (see Tab. S1). At the transition from the skyrmion lattice A
phase to the IC-2 phase the $c$-axis also contracts, but these contractions
are much smaller ($\Delta dL_{\mathrm{c}}/L_{\mathrm{c}}=-2.4(3)\cdot 10^{-6}$
at 4.3 K) while the jumps in magnetization again roughly correspond to one
Bohr magneton per Gd ion ($\Delta M=1.01(11)\mu_{\mathrm{B}}/$Gd at 4.3 K).
Accordingly, this phase boundary shows much smaller pressure dependence, i.e.,
${\partial}T_{\mathrm{c}}/{\partial}p_{i}=-0.47(7)$ K/GPa at 4.3 K (see Tab.
S2). At the same time, the steeper slope of the phase boundary
$B_{\mathrm{c}}(T)$ implies larger changes in entropy of $360(40)$ mJ/mol K at
4.3 K which increases to almost $600(60)$ mJ/mol K at 16 K. We note that the
analysis of anomalies from temperature instead of field sweeps confirms these
values (Tab. S3). Looking at higher fields, the slope of the phase boundary
from IC-2 to the depinned phase (DP) is very small, changing from a small
negative slope below 6 K to a small positive slope above. Considering the
measured anomalies $\Delta M=0.08(4)\mu_{\mathrm{B}}/$Gd at 4.3 K, this yields
negligible associated entropy changes (Tab. S4). In contrast, there are
pronounced lattice effects ($\Delta dL_{\mathrm{c}}/L_{\mathrm{c}}=-17(2)\cdot
10^{-6}$ at 1.77 K) yielding a very large pressure dependence for the phase
boundary IC-2 $\rightarrow$ DP.

| $T$ | $B_{c}$ | $\Delta(dL/L)$ | $\Delta m$ | ${\partial}B_{\mathrm{c}}/{\partial}T$ | $\Delta S$ | ${\partial}T_{\mathrm{c}}/{\partial}p_{\mathrm{i}}$ | ${\partial}B_{\mathrm{c}}/{\partial}p_{i}$
---|---|---|---|---|---|---|---|---
Transition | (K) | (T) | (10-6) | ($\mu_{\mathrm{B}}$/Gd) | (T/K) | (mJ/mol K) | (K/GPa) | (mT/GPa)
IC-1 $\rightarrow$ A(SkL) | 4.3$\pm$0.1 | 0.49$\pm$0.02 | –10.8$\pm$1.1 | 1.0$\pm$0.1 | –0.01 | 125$\pm$13 | –6.1$\pm$0.9 | –50$\pm$50
A(SkL) $\rightarrow$ IC-2 | 4.3$\pm$0.1 | 1.04$\pm$0.03 | –2.4$\pm$0.3 | 1.0$\pm$0.1 | –0.03 | 360$\pm$40 | –0.47$\pm$0.07 | –15$\pm$2
Table 1: Relevant quantities and anomaly sizes at the phase boundaries of the
skyrmion lattice phase, at $T=4.3$ K, and in magnetic fields $B\parallel c$
which have been either directly extracted from the experimental data or were
obtained by using thermodynamic relations as given in the text.

The results of the thermodynamic analyses are shown in table I as well in
tables S1 to S4 in the supplement. In particular, our analysis evidences
pronounced negative uniaxial pressure dependencies for all three phase
transitions between IC-1/IC-1’, A(SkL), IC-2 and DP at low temperatures. This
implies that the IC-1’/IC-1, A(SkL) and IC-2 phases are all destabilized by
pressure along the $c$-axis with respect to the higher temperature phases,
i.e., the field-induced FM phase – and the paramagnetic phase at low fields –
is stabilized.

In particular, our data for $B\parallel c$ provide further information on the
skyrmion phase. Both the onset of the SkL phase from incommensurate magnetic
order IC-1/IC-1’ and its transition into the incommensurate IC-2 phase are
discontinuous in nature. In both cases, transitions are associated with the
increase of magnetization by about 1 $\mu_{\mathrm{B}}/$Gd. Rather flat phase
boundaries in the magnetic phase diagram are indicative of comparably small
entropy changes. Our quantitative analysis evidences that the evolution of the
SkL phase, depending on the temperature, yields an entropy gain of $\Delta
S\approx 100-150$ mJ/(mol K) while the entropy jumps at the transition out of
the skyrmion phase by 300-600 mJ/(mol K). These values are by far larger than
for the chiral magnet MnSi where latent heat at the phase boundaries only
amount to a few mJ/(mol K). Bauer et al. (2013) Uniaxial pressure along the
$c$-axis significantly enhances the SkL phase as seen by the uniaxial pressure
dependencies of the transition temperatures. Specifically, at 4 K there is a
rapid decrease of the IC-1/IC-1’ $\rightarrow$ A(SkL) transition temperature
$\partial T_{\mathrm{in}}/\partial p_{c}\approx-6$ K/GPa, leading to an
expansion of the A(SkL) phase towards lower temperatures under pressure. At
the same time the temperature of the A(SkL) $\rightarrow$ IC-2 transition,
i.e. exiting the SkL phase towards higher temperatures, changes by only
$\partial T_{\mathrm{out}}/\partial p_{c}\approx-0.5$ K/GPa (see tables I and
II). Enhancement of skyrmion lattice phases under pressure is also observed in
other materials. In the insulating skyrmion system Cu2OSeO3, Levatić et al.
report a dramatic enhancement of the skymion pocket under pressure by about 8
K at 0.6 GPa. Levatić et al. (2016) While in Gd2PdSi3 the SkL phase appears at
lower temperatures, our results ($\partial T_{\mathrm{in}}/\partial
p_{c}-\partial T_{\mathrm{out}}/\partial p_{c}$) imply about half of this
effect. We also note similar findings to the ones reported at hand in the
chiral magnet MnSi Chacon et al. (2015); Nii et al. (2015) where uniaxial
pressure along [001] yields a rapid decrease of the onset temperature of the
skyrmion phase while the high temperature phase boundary shows a much smaller
pressure dependence. FN (3) Recent theoretical studies by Hayami et al.
investigated the influence of single-ion anisotropy on the formation and
stability of the skyrmion lattice phase. They show that easy-axis anisotropies
stabilize magnetic-field-induced skyrmion crystals in frustrated magnetsHayami
et al. (2016) and easy-axis (easy-plane) anisotropy substantially increases
(decreases) the stable field-range for a Skyrmion lattice. These findings
suggest, that the pressure dependencies stabilizing the skyrmion lattice phase
in Gd2PdSi3 may originate from small distortions in the local environment of
Gd leading to an increase of the weak magnetic anisotropy of the Gd moments.

The transition from the depinned phase to the field-induced ferromagnetic
phase is of a continuous type. It exhibits a jump in the magnetostriction
coefficient $\Delta\lambda=-4.7(5)\cdot 10^{-5}/T$ (at 1.77 K) and in the
derivative of the magnetization $\Delta\partial M/\partial
B=-0.159(16)\mu_{\mathrm{B}}$/(T Gd) (T = 1.9 K). Using an Ehrenfest relation,
the uniaxial pressure dependence of the critical field can be expressed as
$dB_{\mathrm{c}}/dp_{i}=\Delta\lambda/\Delta(dM/dB)$, which yields
$dB_{\mathrm{c}}/dp_{c}=1.9(5)$ T/GPa at 1.77 K, i.e. the depinned phase is
stabilized under uniaxial pressure $p_{c}$. Similarly, using the anomaly
values listed in Tab. S5, we find uniaxial pressure dependencies of the
critical field between $-0.65(16)$ T/GPa (10 K) and $-0.50(8)$ T/GPa (14 K) at
the continuous transition IC-2 to fiFM.

While Gd2PdSi3 shows only moderate frustration, magnetic entropy and length
changes are observed up to about 2.7 $T_{\mathrm{N1}}$ (60 K), thereby
implying the evolution of short range magnetic order in this temperature
regime. Effects of fluctuations above $T_{\mathrm{N}}$ in Gd2PdSi3 were
observed before in resistivity measurements. Measurements on polycrystalline
samples show a well-defined minimum around 45 K Mallik et al. (1998) which was
also confirmed in single crystals Saha et al. (1999). A theoretical
explanation of this behavior based on the RKKY-interaction in combination with
frustration was given by Wang et al.Wang et al. (2016). Grüneisen scaling
suggests that these precursor fluctuations are of the IC-1/IC-3 type. Both
ordering phenomena are driven by the same dominating energy scale which
differs from the one driving IC-1’. As expected for a Gd3+-system,
magnetoelastic coupling is moderate. It is, hence, somehow surprising that
magnetostriction is large at high temperatures and displays pronounced effects
up to 200 K. In addition, despite linear field dependence of the
magnetization, magnetostriction does not follow a $B^{2}$-law below 200 K (see
Fig. S6) as would be expected from the relation
$dL_{i}/L_{i}=-1/2V\partial\chi_{i}/\partial p_{i}B^{2}$ in the paramagnetic
regime [Johannsen et al., 2005]. Tentatively, magnetostriction above 100 K
implies negative uniaxial pressure dependence, $\partial\chi/\partial
p_{c}<0$, of the magnetic susceptibility while $\partial\chi/\partial
p_{\mathrm{a^{*}}}>0$. This observation suggests that antiferromagnetic
exchange interactions are strengthened by uniaxial pressure along the $c$-axis
and weakened upon application of $p\parallel a$*. Notably, however, the long-
range magnetic ordering temperatures do not follow this trend as $\partial
T_{\mathrm{N}}/\partial p_{c}<0$ which further highlights the complex nature
of magnetism in Gd2PdSi3.

##  V Conclusions

For the first time, high-resolution dilatometry was used to study the
interplay between magnetism and the lattice of single crystalline Gd2PdSi3.
Strong magnetoelastic coupling and field effects up to high temperatures are
found. Pronounced anomalies in the thermal expansion, magnetostriction and
magnetization allow us to obtain the magnetic phase diagram. This yields in
particular several novel phases for $B\parallel c$ while the B vs. T phase
diagram for $B\parallel a$* has not yet been reported at all in the
literature. Grüneisen analysis shows the onset of magnetic contributions well
above $T_{\mathrm{N1}}$, and the pressure dependencies of ordering phenomena
are obtained. In particular, we find that uniaxial pressure strongly enhances
the skyrmion lattice phase.

## Acknowledgements

We thank I. Mazilu and Y. Xu for support in the crystal growth. We acknowledge
financial support by BMBF via the project SpinFun (13XP5088) and by Deutsche
Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy
EXC2181/1-390900948 (the Heidelberg STRUCTURES Excellence Cluster), through
project KL 1824/13-1, and within the SFB 463.

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## Supplementary Material:
Magnetoelastic Coupling and Phases in the Skyrmion Lattice Magnet Gd2PdSi3
Discovered by High-resolution Dilatometry

Figure S1: Static magnetic susceptibility $\chi=M/B$ for (a) $B\parallel a$*
and (b) $B\parallel c$ (b). Figure S2:  (a-b) Thermal expansion measurements
up to 220 K for $B\parallel c$ in magnetic fields up to 15 T. (a) Thermal
expansion coefficient $\alpha_{\mathrm{c}}$. The inset shows the magnified low
temperature region with an offset of $1.2\cdot 10^{-6}$ 1/K. (b) Relative
length changes $dL_{\mathrm{c}}(T)/L_{\mathrm{c}}$. Vertical lines mark the
changes $\Delta L(B)/L$ derived from magnetostriction measurements. The data
at different fields are normalized by these changes at 4.3 K. (c-d) Thermal
expansion measurements up to 100 K for $B\parallel a$* in magnetic fields up
to 15 T. (c) Thermal expansion coefficient $\alpha_{\mathrm{a^{*}}}$ offset by
$2\cdot 10^{-7}$ 1/K. (d) Relative length changes
$dL_{\mathrm{a^{*}}}(T)/L_{\mathrm{a^{*}}}$. Vertical lines again mark the
changes $\Delta L(B)/L$ taken from magnetostriction measurements. The data at
different fields are offset and normalized by these changes at 2.2 K w.r.t.
the zero-field data. Figure S3:  Comparison of the scaled Lu2PdSi3 specific
heat $c_{\mathrm{p}}$/T measured by Cao et al.Cao et al. (2013) (red empty
circles) with the Debye and Einstein fit (blue line). Note that the Lu2PdSi3
data was interpolated and therefore contains more data points than the
published data. Figure S4:  Magnetostriction measurements at temperatures from
1.7 K to 25 K for $B\parallel c$ (a-c) and $B\parallel a$* (d-f): The relative
length change $dL(B)/L$ (a, d) and magnetostriction coefficient $\lambda_{i}$
(b, c, e, f) are shown. Solid lines in (a, d) represent up-sweeps, dashed
lines down-sweeps. Triangles pointing upwards in (d) mark up-sweeps, triangles
pointing downwards mark down-sweeps. Data in (a-c, insets also) and (e) are
shifted vertically for better visibility by: (a) different amounts, (b, c)
$6\cdot 10^{-6}$ 1/T, (e) $3.7\cdot 10^{-6}$ 1/T (3.5 K data is omitted).
Insets show magnifications of the low field regions. Figure S5:
Magnetostriction measurements at temperatures above $T_{\mathrm{N1}}$ for
$B\parallel c$ (a-b) and $B\parallel a$* (c-d): The relative length change
$dL(B)/L$ (a, c) and magnetostriction coefficient $\lambda_{i}$ (b, d) are
shown. Solid lines in (a, c) represent up-sweeps, dashed lines down-sweeps.
Triangles pointing upwards in (b, d) are up-sweeps, triangles pointing
downwards mark down-sweeps. Figure S6:  Relative length changes $dL/L$ plotted
vs. $B^{2}$. The uniaxial pressure dependence of the susceptibility is related
to the magnetostriction by the Maxwell relation
${\partial}(dL/L)/{\partial}B=-{\partial}M/{\partial}p$. For a paramagnetic
material with $M=\chi B$ we thus have
$dL/L=-\frac{1}{2}{\partial}\chi/{\partial}pB^{2}$, i.e.
${\partial}\chi/{\partial}p$ is proportional to the slope in the above plot.
Figure S7:  Measurements of the isothermal magnetization as well as the static
magnetic susceptibility $\chi(T)$ and their derivatives for $B\parallel c$
(a-d) and $B\parallel a$* (e-h). $\chi(T)$ measurements were performed in a
zero-field cooled manner unless stated otherwise (FCW: field-cooled warming,
FCC: field-cooled cooling). Data are offset for better visibility by (b) 1.2
$\mu_{\mathrm{B}}$/(T Gd), (c, inset) $4.5\cdot 10^{-6}$ m3/mol, (d) $-8\cdot
10^{-7}$ m3/(mol K), (f) 1.1 $\mu_{\mathrm{B}}$/(T Gd), (g, inset) $5\cdot
10^{-6}$ m3/mol, and (h) $-1.4\cdot 10^{-6}$ m3/(mol K) starting with the data
for $B$ = 0.10 T. Figure S8:  Close-up of the measurements of the magnetic
susceptibility ${\partial}M/{\partial}B$ for $B\parallel c$ (a) and
$B\parallel a$* (b) in the temperature regime of the IC-3 $\rightarrow$ IC-2
transition. Figure S9:  (a) Relative length changes
$dL_{\mathrm{c}}/L_{\mathrm{c}}$ and (b) magnetostriction coefficient
$\lambda_{\mathrm{c}}$ in the low-field regime at temperatures below and above
$T_{\mathrm{N1}}$ and $T_{\mathrm{N2}}$ for $B\parallel c$. Only up-sweeps are
shown. Figure S10:  Extraction of $T^{*}$ from thermal expansion (a, d),
${\partial}\chi/{\partial}T$ (b, e) and ${\partial}M/{\partial}B$ for
$B\parallel c$ (a-c) and $B\parallel a$* (d-f). Vertical lines mark the
position of $T^{*}$ for the different measurements. Data in (a) and (d) are
offset by $2\cdot 10^{-6}$ 1/K and $1\cdot 10^{-6}$ 1/K, respectively.
IC-1/IC-1’ $\rightarrow$ A(SkL) – $c$-axis: Relevant quantities, entropy
changes and uniaxial pressure dependencies
---
T (K) | Bc (T) |  $\Delta(dL/L)$ (10-6) |  $\Delta m$ ($\mu_{\mathrm{B}}$/Gd) |  ${\partial}B_{\mathrm{c}}/{\partial}T$ (T/K) |  $\Delta S_{\mathrm{calc}}$ (mJ/mol K) |  ${\partial}T_{\mathrm{c}}/{\partial}p_{\mathrm{i}}$ (K/GPa) |  ${\partial}B_{\mathrm{c}}/{\partial}p_{i}$ (T/GPa)
1.77 | 0.52$\pm$0.02 | –7.3$\pm$0.8 | 1.00$\pm$0.10 | –0.01135 | 126$\pm$13 | –4.1$\pm$0.6 | –0.046$\pm$0.007
4.3 | 0.49$\pm$0.02 | –10.8$\pm$1.1 | 0.99$\pm$0.10 | –0.01133 | 125$\pm$13 | –6.1$\pm$0.9 | –0.069$\pm$0.010
5.01 | 0.48$\pm$0.02 | –8.9$\pm$0.9 | 0.99$\pm$0.10 | –0.01151 | 127$\pm$13 | –4.9$\pm$0.7 | –0.056$\pm$0.008
7.94 | 0.45$\pm$0.02 | –8.4$\pm$0.9 | 0.95$\pm$0.10 | –0.01318 | 140$\pm$14 | –4.2$\pm$0.6 | –0.056$\pm$0.008
9.83 | 0.42$\pm$0.02 | –8.7$\pm$0.9 | 0.81$\pm$0.08 | –0.01499 | 135$\pm$14 | –4.6$\pm$0.7 | – 0.068$\pm$0.010
9.99 | 0.42$\pm$0.02 | –7.2$\pm$0.8 | 0.81$\pm$0.08 | –0.01516 | 137$\pm$14 | –3.7$\pm$0.6 | –0.056$\pm$0.008
12.02 | 0.38$\pm$0.02 | –7.0$\pm$0.7 | 0.75$\pm$0.08 | –0.01783 | 149$\pm$15 | –3.3$\pm$0.5 | –0.059$\pm$0.009
14.35 | 0.34$\pm$0.02 | –4.5$\pm$0.5 | 0.52$\pm$0.05 | –0.02260 | 131$\pm$14 | –2.4$\pm$0.4 | –0.055$\pm$0.008
16 | 0.30$\pm$0.02 | –2.4$\pm$0.3 | 0.40$\pm$0.04 | –0.02505 | 110$\pm$11 | –1.5$\pm$0.2 | –0.038$\pm$0.006
IC-3 $\rightarrow$ IC-2 – $c$-axis: Relevant quantities, entropy changes and
uniaxial pressure dependencies
18.39 | 0.30$\pm$0.04 | –9$\pm$2 | 0.24$\pm$0.03 | –0.1 | 270$\pm$40 | –2.3$\pm$0.6 | –0.23$\pm$0.06
20.27 | 0.23$\pm$0.04 | –3.8$\pm$0.4 | 0.21$\pm$0.05 | -0.04 | 94$\pm$30 | –2.9$\pm$0.8 | –0.12$\pm$0.03
Table S1: Relevant quantities and jumps, calculated changes of entropy, and
calculated pressure dependencies for the discontinuous transition in magnetic
field from IC-1/IC-1’ $\rightarrow$ A(SkL) and IC-3 $\rightarrow$ IC-2 for the
$c$-axis according to Eq. (7). The column of
${\partial}B_{\mathrm{c}}/{\partial}T$ was calculated by taking the derivative
of a polynomial fit to the values $B_{\mathrm{c}}(T)$ for IC-1/IC-1’
$\rightarrow$ A(SkL) and a linear fit (between between 18.39 K and 19 K as
well as between 19 K and 21 K) for IC-3 $\rightarrow$ IC-2. A(SkL)
$\rightarrow$ IC-2 – $c$-axis: Relevant quantities, entropy changes and
uniaxial pressure dependencies
---
T (K) | Bc (T) |  $\Delta(dL/L)$ (10-6) |  $\Delta m$ ($\mu_{\mathrm{B}}$/Gd) |  ${\partial}B_{\mathrm{c}}/{\partial}T$ (T/K) |  $\Delta S_{\mathrm{calc}}$ (mJ/mol K) |  $dT_{\mathrm{c}}/dp_{\mathrm{i}}$ (K/GPa) |  ${\partial}B_{\mathrm{c}}/{\partial}p_{i}$ (T/GPa)
1.77 | 1.11$\pm$0.03 | 0 | 0.95$\pm$0.10 | –0.028 | 300$\pm$30 | 0 | 0
4.3 | 1.04$\pm$0.03 | –2.4$\pm$0.3 | 1.01$\pm$0.11 | –0.031 | 360$\pm$40 | –0.47$\pm$0.07 | –0.015$\pm$0.002
5.01 | 1.01$\pm$0.02 | –1.4$\pm$0.2 | 1.00$\pm$0.10 | –0.033 | 370$\pm$40 | –0.27$\pm$0.04 | –0.009$\pm$0.002
7.94 | 0.91$\pm$0.02 | –2.5$\pm$0.3 | 0.99$\pm$0.10 | –0.041 | 450$\pm$50 | –0.40$\pm$0.06 | –0.016$\pm$0.003
9.83 | 0.82$\pm$0.02 | –5.1$\pm$0.5 | 0.98$\pm$0.10 | –0.048 | 520$\pm$60 | –0.69$\pm$0.10 | –0.033$\pm$0.005
9.99 | 0.82$\pm$0.02 | –2.9$\pm$0.3 | 0.98$\pm$0.10 | –0.048 | 530$\pm$60 | –0.39$\pm$0.06 | –0.019$\pm$0.003
12.02 | 0.7$\pm$0.02 | –5.9$\pm$0.6 | 0.93$\pm$0.10 | –0.057 | 590$\pm$60 | –0.70$\pm$0.10 | –0.040$\pm$0.006
14.35 | 0.56$\pm$0.02 | –5.9$\pm$0.6 | 0.75$\pm$0.08 | –0.069 | 580$\pm$60 | –0.72$\pm$0.11 | –0.050$\pm$0.007
16 | 0.45$\pm$0.02 | –4.0$\pm$0.4 | 0.66$\pm$0.07 | –0.079 | 580$\pm$60 | –0.49$\pm$0.07 | –0.039$\pm$0.006
18.39 | 0.23$\pm$0.02 |  |  | -0.095 |  |  |
Table S2: Relevant quantities and jumps, calculated changes of entropy, and
calculated pressure dependencies for the discontinuous transition in magnetic
field from A(SkL) to IC-2 for the $c$-axis according to Eq. (7). The column of
${\partial}B_{\mathrm{c}}/{\partial}T$ was calculated by taking the derivative
of a polynomial fit to the values $B_{\mathrm{c}}(T)$. A(SkL) $\rightarrow$
IC-2 – $c$-axis: Relevant quantities, entropy changes and uniaxial pressure
dependencies
---
$B$ (T) |  $T_{\mathrm{c}}$ (K) |  $\Delta$(dL/L) (10-6) |  $\Delta m$ ($\mu_{\mathrm{B}}$/Gd) |  ${\partial}T_{\mathrm{c}}/{\partial}B$ (K/T) |  $\Delta$Scalc (mJ/mol K) |  $dT_{\mathrm{c}}/dp_{\mathrm{i}}$ (K/GPa)
0 | 19.7$\pm$0.5 |  | no SkL |  |  |
0.1 | 19.3$\pm$0.3 |  | no SkL | –4.8 |  |
0.2 | 18.8$\pm$0.5 |  | no SkL | –8.1 |  |
0.3 | 17.9$\pm$0.7 |  | no SkL | –10.8 |  |
0.4 | 16.4$\pm$0.4 | –5.0$\pm$1.0 | 0.36$\pm$0.04 | –13.0 | 310$\pm$40 | –1.2$\pm$0.3
0.5 | 15.3$\pm$0.4 | –7.4$\pm$1.0 | 0.54$\pm$0.04 | –14.8 | 410$\pm$40 | –1.3$\pm$0.2
0.6 | 13.6$\pm$0.3 | –7.0$\pm$0.7 | 0.70$\pm$0.04 | –16.0 | 490$\pm$30 | –1.02$\pm$0.12
0.7 | 12.0$\pm$0.3 | –5.1$\pm$0.5 | 0.77$\pm$0.04 | –16.7 | 510$\pm$30 | –0.70$\pm$0.08
0.8 | 10.4$\pm$0.3 | –2.7$\pm$0.3 | 0.86$\pm$0.04 | –17.0 | 560$\pm$30 | –0.34$\pm$0.04
0.9 | 8.60$\pm$0.3 | –1.3$\pm$0.1 | 0.91$\pm$0.04 | –16.7 | 610$\pm$30 | –0.15$\pm$0.02
Table S3: Relevant quantities and jumps, calculated changes of entropy, and
calculated pressure dependencies for the discontinuous transition in
temperature from A(SkL) to IC-2 for the $c$-axis according to Eq. (7). The
column of ${\partial}T/{\partial}B$ was calculated by taking the derivative of
a polynomial fit to the values Tc(B), including the values from 0 T to 0.3 T.
IC-2 $\rightarrow$ DP – $c$-axis: Relevant quantities, entropy changes and
uniaxial pressure dependencies
---
T (K) | Bc (T) |  $\Delta(dL/L)$ (10-6) |  $\Delta m$ ($\mu_{\mathrm{B}}$/Gd) |  ${\partial}B_{\mathrm{c}}/{\partial}T$ (T/K) |  $\Delta S_{\mathrm{calc}}$ (mJ/mol K) |  $dT_{\mathrm{c}}/dp_{\mathrm{i}}$ (K/GPa) |  ${\partial}B_{\mathrm{c}}/{\partial}p_{i}$ (T/GPa)
1.77 | 3.51$\pm$0.05 | –17.1$\pm$1.7 | 0.08$\pm$0.03 | –0.04$\pm$0.04 | 16$\pm$16 | –80$\pm$80 | –1.4$\pm$0.6
4.3 | 3.44$\pm$0.05 | –12.2$\pm$2 | 0.08$\pm$0.04 | –0.014$\pm$0.014 | 7$\pm$7 | –130$\pm$130 | –0.9$\pm$0.4
5.01 | 3.44$\pm$0.10 | –11.8$\pm$0.4 | 0.07$\pm$0.03 | –0.008$\pm$0.008 | 3$\pm$3 | –250$\pm$250 | –1.0$\pm$0.6
7.94 | 3.45$\pm$0.15 | –6$\pm$3 | 0.044$\pm$0.018 | 0.018$\pm$0.009 | -4$\pm$3 | 100$\pm$80 | -0.9$\pm$0.6
Table S4:  Relevant quantities and jumps, calculated changes of entropy, and
calculated pressure dependencies for the discontinuous transition in magnetic
field from IC-2 to DP for the $c$-axis according to Eq. (7). The column of
${\partial}B_{\mathrm{c}}/{\partial}T$ was calculated by taking the derivative
of a polynomial fit to the values $B_{\mathrm{c}}(T)$. IC-2 $\rightarrow$ fiFM
– $c$-axis: Relevant quantities, entropy changes and uniaxial pressure
dependencies
---
T (K) | Bc (T) |  ${\Delta}\lambda$ (10-6/T) |  ${\Delta}({\partial}M/{\partial}B)$ ($\mu_{\mathrm{B}}$/(T Gd)) |  ${\partial}B_{\mathrm{c}}/{\partial}p_{\mathrm{i}}$ (T/GPa)
9.99 | 3.7$\pm$0.1 | 15$\pm$2 | –0.15$\pm$0.03 | -0.7$\pm$0.2
12.02 | 3.0$\pm$0.1 | 17$\pm$2 | –0.19$\pm$0.03 | -0.6$\pm$0.2
14.35 | 2.3$\pm$0.1 | 19$\pm$2 | –0.25$\pm$0.03 | -0.5$\pm$0.1
Table S5:  Relevant quantities and jumps and calculated field and pressure
dependencies for the continuous transition from IC-2 to the field-induced
ferromagnetic (fiFM) phase for the $c$-axis according to Eq. (4), (5). DP
$\rightarrow$ fiFM – $c$-axis: Relevant quantities, entropy changes and
uniaxial pressure dependencies
---
T (K) | Bc (T) |  ${\Delta}\lambda$ (10-5/T) |  ${\Delta}({\partial}M/{\partial}B)$ ($\mu_{\mathrm{B}}$/(T Gd)) |  ${\partial}B_{\mathrm{c}}/{\partial}p_{\mathrm{i}}$ (T/GPa)
1.77 | 8.8$\pm$0.5 | –4.7$\pm$1.0 | –0.159$\pm$0.018 | 1.9$\pm$0.5
4.3 | 7.94$\pm$0.4 | –1.9$\pm$1.0 | –0.094$\pm$0.009 | 1.3$\pm$0.7
5.01 | 6.6$\pm$0.4 | –1.4$\pm$1.0 | –0.074$\pm$0.018 | 1.2$\pm$0.9
7.94 | 6.48$\pm$0.6 | –0.7$\pm$0.4 | –0.065$\pm$0.018 | 0.6$\pm$0.4
Table S6:  Relevant quantities and jumps and calculated field and pressure
dependencies for the continuous transition from the depinned phase (DP) to the
field-induced ferromagnetic (fiFM) phase for the $c$-axis according to Eq.
(4), (5). IC-1 $\rightarrow$ IC-1’ – $c$-axis
---
B | Tc |  $\Delta$$\alpha_{\mathrm{c}}$ |  $\Delta$$c_{\mathrm{p}}$ | $\Delta({\partial}M/{\partial}T)$ | $dT_{\mathrm{c}}/dp_{i}$ | $dT_{\mathrm{c}}/dB_{i}$
(T) | (K) | (10-6) | (J/mol K) | (10${}^{-3}\mu_{\mathrm{B}}$/f.u. K) | (K/GPa) | (K/T)
0 | 12.8$\pm$0.7 | 4.1$\pm$0.5 | –2.7$\pm$0.5 |  | –1.4$\pm$0.3 |
0.2 | 12.15$\pm$0.5 | 2.7$\pm$0.4 | –1.8$\pm$0.4 | –5.6$\pm$1.4 | –1.3$\pm$0.4 | –0.037$\pm$0.013
0.25 | 11.45$\pm$0.5 |  |  | –11$\pm$4 |  |
0.3 | 10.6$\pm$0.5 | 0.9$\pm$0.3 | –0.6$\pm$0.3 | –16$\pm$6 | –1.1$\pm$0.7 | –0.28$\pm$0.18
0.35 | 8.4$\pm$0.5 |  |  | –20$\pm$7 |  |
Table S7:  Relevant quantities, jumps, and calculated field and pressure
dependencies for the continuous transition at T∗ from IC-1 to IC-1’ for the
$c$-axis according to Eq. (4), (5). Note that the $c_{\mathrm{p}}$ data was
only measured at 0 T. The jumps in field $B$ were scaled by the changes of the
jumps in $\alpha_{\mathrm{c}}$, i.e., $\Delta$$c_{\mathrm{p}}$(B) =
$c_{\mathrm{p}}$(0) $\cdot\leavevmode\nobreak\
\Delta\alpha$(B)/$\Delta\alpha$(0).