# Magnetoelastic coupling and Grüneisen scaling in NdB4

R. Ohlendorf  Kirchhoff Institute for Physics, Heidelberg University, INF 227,
D-69120 Heidelberg, Germany<EMAIL_ADDRESS>S. Spachmann
Kirchhoff Institute for Physics, Heidelberg University, INF 227, D-69120
Heidelberg, Germany  L. Fischer  Kirchhoff Institute for Physics, Heidelberg
University, INF 227, D-69120 Heidelberg, Germany  K. Dey  Kirchhoff Institute
for Physics, Heidelberg University, INF 227, D-69120 Heidelberg, Germany  D.
Brunt  Department of Physics, University of Warwick, Coventry, CV4 7AL, United
Kingdom  G. Balakrishnan  Department of Physics, University of Warwick,
Coventry, CV4 7AL, United Kingdom  O. A. Petrenko  Department of Physics,
University of Warwick, Coventry, CV4 7AL, United Kingdom  R. Klingeler
Kirchhoff Institute for Physics, Heidelberg University, INF 227, D-69120
Heidelberg, Germany  Centre for Advanced Materials (CAM), Heidelberg
University, INF 225, D-69120 Heidelberg, Germany<EMAIL_ADDRESS>heidelberg.de

###### Abstract

We report high-resolution capacitance dilatometry studies on the uniaxial
length changes in a NdB4 single crystal. The evolution of magnetically ordered
phases below $T_{\rm N}$ = 17.2 K (commensurate antiferromagnetic phase,
cAFM), $T_{\rm IT}$ = 6.8 K (intermediate incommensurate phase, IT), and
$T_{\rm LT}$ = 4.8 K (low-temperature phase, LT) is associated with pronounced
anomalies in the thermal expansion coefficients. The data imply significant
magneto-elastic coupling and evidence of a structural phase transition at
$T_{\rm LT}$. While both cAFM and LT favor structural anisotropy $\delta$
between in-plane and out-of-plane length changes, it competes with the IT-type
of order, i.e., $\delta$ is suppressed in that phase. Notably, finite
anisotropy well above $T_{\rm N}$ indicates short-range correlations which
are, however, of neither cAFM, IT, nor LT-type. Grüneisen analysis of the
ratio of thermal expansion coefficient and specific heat enables the
derivation of uniaxial as well as hydrostatic pressure dependencies. While
$\alpha$/$c_{\rm p}$ evidences a single dominant energy scale in LT, our data
imply precursory fluctuations of a competing phase in IT and cAFM,
respectively. Our results suggest the presence of orbital degrees of freedom
competing with cAFM and successive evolution of a magnetically and orbitally
ordered ground state.

##  I Introduction

Geometric frustration provides a prime route to macroscopic degeneracy of
ground states, thereby suppressing or even completely eliminating static long-
range order Ramirez (1994). Often, when magnetic order eventually evolves,
competing interactions result in complex magnetic phase diagrams constituting
energetically similar spin configurations. Prototypical examples of
geometrically frustrated magnets are triangular, kagome, and pyrochlore
lattices Shimizu et al. (2003); Shen et al. (2016); Pratt et al. (2011); Lee
and Lee (2005); Yan et al. (2011); Han et al. (2012); Gingras and McClarty
(2014); Chalker and Eastmond (1992); Gardner et al. (2010). While many of
these systems realize quantum magnets, rare earth (R) borides, due to large
magnetic moment of the rare earth ions, are well suited for investigating
frustrated magnets in a classical limit. The family of tetraborides RB4 is one
particular example exhibiting magnetic frustration due to the rare earth ions
being arranged on the geometrically frustrated Shastry-Sutherland-lattice
(SSL) Shastry and Sutherland (1981); Gabani et al. (2020). Many of the RB4
compounds display a variety of competing phases at low magnetic fields Gabani
et al. (2020). As with all RB4 compounds featuring trivalent rare earth ions,
NdB4 reported here is metallic Johnson and Daane (1963). As such the RKKY
interaction may play an important role in addition to competing interactions
evolving on exchange paths formed by the SSL.

The RB4 family crystallizes in a tetragonal D${}^{5}_{4h}$-P4/mbm structure
Etourneau and Hagenmuller (1985). The boron sublattice is made up of octahedra
forming chains along the crystallographic $c$-direction. These chains are
connected via two boron atoms to form rings within the $ab$-plane. The rare
earth ions are located above and below the middle of these rings constituting
the rare earth sublattice. In NdB4, magnetism is due to Nd3+ ions which
electron configuration 4$f^{3}$ implies a $\prescript{4}{}{I}_{9/2}$ ground
state. Upon cooling, NdB4 shows three successive phase transitions signaling
the onset of commensurate antiferromagnetic (cAFM) order at $T_{\rm N}$ = 17.2
K, an incommensurate antiferromagnetic phase (IT) evolving at $T_{\rm IT}$ =
6.8 K and a low-temperature phase (LT) below $T_{\rm LT}$ = 4.8 K Brunt et al.
(2018); Watanuki et al. (2009). The former phase features an all-in-structure
of magnetic moments pointing into the squares formed by the Nd-ions Metoki et
al. (2018). In addition, the moments are slightly tilted towards the $c$-axis
Metoki et al. (2018). The pseudo-quartet ground state consisting of two
Kramers doublets carries an electric quadrupole moment giving rise to orbital
degrees of freedom Yamauchi et al. (2017). Here, we study the effects of long-
range order on the length changes in NdB4 by high-resolution capacitance
dilatometry. Despite the relevance of dilatometry for elucidating the coupling
of lattice, orbital, and magnetic degrees of freedom particularly in rare-
earth systems, such studies are yet missing for the whole family of
tetraborides RB4. We find pronounced thermal expansion anomalies at the
magnetic transitions as well as evidence for a structural transition at
$T_{\rm LT}$. In addition, precursing fluctuations well above $T_{\rm N}$
associated with negative in-plane thermal expansion indicate short-range
ordering in the paramagnetic phase which is, however, not associated with an
ordering phenomena evolving in either of the three low-temperature phases.
Analysis of the Grüneisen ratios and of structural anisotropy enables
elucidating the interplay of lattice, structure, and presumingly orbital
degrees of freedom.

##  II Experimental methods

Single crystals of NdB4 were grown by the optical floating-zone technique as
reported in detail in Ref. Brunt et al., 2019. The relative length changes
$dL_{i}/L_{i}$ along the crystallographic [001] and [110] directions (space
group 127), respectively, were studied on an oriented cuboid-shaped single
crystal of dimensions $1.476\times 1.478\times 1.880~{}$mm3. The measurements
were done by means of a three-terminal high-resolution capacitance dilatometer
Küchler et al. (2017) and the linear thermal expansion coefficients
${\alpha_{i}=1/L_{i}\cdot dL_{i}(T)/dT}$ were derived. Static magnetic
susceptibility $\chi=M/B$ was studied in a magnetic field of $B=0.1$ T in a
vibrating sample magnetometer (VSM) of a Magnetic Properties Measurement
System (MPMS3 SQUID magnetometer) by Quantum Design. Specific heat
measurements have been performed in a Quantum Design PPMS using a relaxation
method.

##  III Length changes upon evolution of magnetic order

Thermal expansion in NdB4 is positive at high temperatures and shows similar
behavior in-plane and out-of-plane (see Fig. 1). Upon cooling below $\simeq
50$ K, $L_{[001]}$ decreases significantly stronger as compared to
$L_{[110]}$. Distinct anomalies in the thermal expansion at $T_{\rm N}$ as
well as at $T_{\rm LT}$ and $T_{\rm IT}$ imply pronounced magnetoelastic
coupling.

Figure 1: Relative length changes $dL_{i}/L_{i}$ along the crystallographic
[001]- (black circles) and [110]-direction (red circles) $vs.$ temperature.
Insets show enlargement of the low temperature region. Left and right
ordinates are to scale but shifted in order to account for the difference at
200 K. Vertical dashed lines indicate the transition temperatures $T_{\rm N}$,
$T_{\rm IT}$, and $T_{\rm LT}$ towards the commensurate antiferromagnetic
phase (cAFM), the incommensurate IT-phase, and the low-temperature phase LT.
Figure 2: Uniaxial thermal expansion coefficients $\alpha_{i}$ along the
crystallographic (a) [001]- and (b) [110]-direction. Dashed lines indicate the
transition temperatures $T_{\rm LT}$, $T_{\rm IT}$, and $T_{\rm N}$ (see the
text).

Magnetoelastic effects associated with the various ordering phenomena are
particularly evident if the thermal expansion coefficients are considered,
which display pronounced anomalies at $T_{\rm N}$ = 17.2 K, $T_{\rm IT}$ = 6.8
K, and $T_{\rm LT}$ = 4.8 K (see Figs. 2 and 3a). The onset of long-range
antiferromagnetic order at $T_{\rm N}$ is confirmed by previous magnetization
and neutron diffraction data Watanuki et al. (2009); Brunt et al. (2018);
Metoki et al. (2018), while two distinct antiferromagnetic phases develop at
$T_{\rm LT}$ and $T_{\rm IT}$ Metoki et al. (2018). At $T_{\rm N}$ and $T_{\rm
IT}$, the thermal expansion data imply $\lambda$-shaped anomalies typical for
continuous phase transitions. Anomalies in the volume thermal expansion
coefficient $\alpha_{v}=2\alpha_{\perp c}+\alpha_{\parallel c}$ show similar
features (Fig. 3a). All anomalies at $T_{\rm N}$ imply the decrease in length
upon onset of magnetic order, i.e., there is positive uniaxial pressure
dependence $dT_{N}/dp_{i}>0$ for both $i=$ [001] and [110]. Accordingly,
hydrostatic pressure dependence of $T_{\rm N}$ is positive, too.

In contrast, anomalies in the uniaxial thermal expansion coefficients exhibit
opposite signs at $T_{\rm IT}$, i.e., $dT_{\rm IT}/dp_{\parallel c}<0$ and
$dT_{\rm IT}/dp_{\perp c}>0$ (Fig. 2). The resulting volume response (Fig. 3a)
exhibits a positive sign of anomaly and hence positive hydrostatic pressure
dependence. At $T_{\rm LT}$ there are jumps $\Delta L_{i}$ in the observed
length changes for both directions $i$, which imply discontinuous evolution of
the LT phase. The corresponding jumps in $L_{i}(T)$ are negative upon cooling
(see the insets of Fig. 1) which implies $dT_{\rm LT}/dp_{i}>0$ in both cases.
Quantitatively, at $T_{\rm LT}$, we observe ${\Delta L_{\rm[110]}=-2.0(2)\cdot
10^{-6}}$ and ${\Delta L_{\rm[001]}=-5.0(2)\cdot 10^{-6}}$.

In order to quantitatively evaluate the effect of magnetic order and crystal
field on thermal expansion, phonon contributions are estimated from specific
heat data on LuB4 [Novikov et al., 2014] by assuming Grüneisen scaling of the
phonon heat capacity. Specifically, the phonon specific heat of LuB4 is given
in Novikov et al. (2014) as

$\displaystyle c_{p}^{ph}=$ $\displaystyle
a_{1}D\left(\frac{T}{\Theta_{D1}}\right)+a_{2}D\left(\frac{T}{\Theta_{D2}}\right)+a_{3}E\left(\frac{T}{\Theta_{E1}}\right)$
$\displaystyle+a_{4}E\left(\frac{T}{\Theta_{E2}}\right).$ (1)

Here, $a_{i}$ are parameters, and $D$ and $E$ denote Debye and Einstein
functions with Debye and Einstein temperatures as given in Table 1. The same
function was fitted to the uniaxial thermal expansion coefficients
$\alpha_{\parallel c}$ and $\alpha_{\perp c}$, above 65 K, using the same
Debye and Einstein temperatures and allowing the parameters $a_{i}$ to vary.
We note that, in LuB4, a linear term for the electronic part is very small
[Novikov et al., 2014] and has been omitted here. The volumetric thermal
expansion coefficient obtained by $\alpha_{v}=2\alpha_{\perp
c}+\alpha_{\parallel c}$ was fitted in the same way, except that $a_{2}$ was
set to the value obtained from the fit to $\alpha_{\parallel c}$. From these
fits the phononic Grüneisen parameters $\gamma=\alpha/c_{p}$ are determined
for both uniaxial thermal expansion coefficients and for the volume as listed
in Table 1.

Table 1: Parameters of phonon background of thermal expansion. $\Theta_{\rm D}$ and $\Theta_{\rm E}$ are Debye- and Einstein-temperatures describing a phonon background of specific heat in LuB4 from Ref. Novikov et al., 2014. The parameters $\gamma$ are the uniaxial and volume phononic Grüneisen parameters. |  |  $\gamma$ (mol/MJ) |
---|---|---|---
$\Theta_{\rm D1}=1140$ K |  $\gamma_{\rm D1}^{\parallel c}$=0.028 |  $\gamma_{\rm D1}^{\perp c}$=0.019 |  $\gamma_{\rm D1}^{v}$=0.069
$\Theta_{\rm D2}=211$ K |  $\gamma_{\rm D2}^{\parallel c}$=0.317 |  $\gamma_{\rm D2}^{\perp c}$=0 |  $\gamma_{\rm D2}^{v}$=0.317
$\Theta_{\rm E1}=130$ K |  $\gamma_{\rm E1}^{\parallel c}$=0 |  $\gamma_{\rm E1}^{\perp c}$=0 |  $\gamma_{\rm E1}^{v}$=0
$\Theta_{\rm E2}=190$ K |  $\gamma_{\rm E2}^{\parallel c}$=0.356 |  $\gamma_{\rm E2}^{\perp c}$=0.680 |  $\gamma_{\rm E2}^{v}$=1.694

The result of this estimate shown by the dashed line in Fig. 1 implies
anisotropic non-phononic length changes up to about 65 K while for higher
temperatures the experimental data are well described by the fits (see Fig.
1).

Figure 3: (a) Volumetric thermal expansion coefficient corrected by the phonon
contribution. Inset: Uncorrected volumetric thermal expansion coefficient and
phonon fit to the high temperature data (see the text). (b) Corresponding
background corrected specific heat, and (c) Fisher’s specific heat
$\partial(\chi_{i}T)/\partial T$ for $B=0.1$ T applied along the
crystallographic [100]- (black circles) and [110]-direction (red circles),
respectively. Filled (open) circles indicate measurements upon heating
(cooling). Inset: Corresponding magnetic susceptibility $\chi_{[001]}$
highlighting hysteresis at $T_{\rm LT}$. Vertical dashed lines indicate
$T_{\rm LT}$, $T_{\rm IT}$, and $T_{\rm N}$.

Figure 3a shows the volumetric thermal expansion coefficient $\alpha_{\rm
v}^{\prime}$ corrected by the phonon contribution. The inset shows the bare
experimental data $\alpha_{\rm v}$ and the phonon fit derived as described
above. In the whole temperature regime under study, $\alpha_{\rm v}^{\prime}$
is positive and in particular – as described above – all anomalies are
positive indicating positive hydrostatic pressure dependencies $dT_{i}/dp>0$
of all transition temperatures. In addition, Fig. 3b displays the background
corrected specific heat $c_{\rm p}^{\prime}$ obtained by analogously
subtracting the phonon entropy changes. For comparison, we also present
Fisher’s specific heat Fisher (1962) $c_{\rm p}^{\rm
m}\propto\partial(\chi_{i}T)/\partial T$ for magnetic fields applied $B||c$
and $B\perp c$ (Fig. 3c) which provides an estimate of magnetic entropy
changes. Analogous to $\alpha$, $c_{\rm p}^{\prime}$ and $c_{\rm p}^{\rm m}$
display $\lambda$-shaped and discontinuous anomalies at the respective
ordering temperatures. Note that in general, $c_{\rm p}^{\rm m}$ resembles
$c_{\rm p}^{\prime}$ very well, which strongly confirms the procedure applied
for background correction. The fact that there is no visible anomaly in
$\partial(\chi_{c}T)/\partial T$ at $T_{\rm N}$ implies the easy axis being
$\perp c$ in the commensurate antiferromagnetic phase (cAFM) which agrees to
recent neutron data Yamauchi et al. (2017); Metoki et al. (2018). In addition,
the magnetic entropy changes are well detected by $\chi_{\|c}$ in the IT and
LT phases, suggesting significant out-of-plane spin components. We also note a
small but finite hysteresis between cooling and heating in the magnetic
susceptibility associated with the discontinuity at $T_{\rm LT}$ (see the
inset of Fig. 3c).

##  IV Discussion

Figure 4: Grüneisen parameters $\gamma_{i}=\alpha_{i}^{\prime}$/$c_{\rm
p}^{\prime}$ where $i$ denotes the two directions [001] (a) and [110] (b) as
well as the volume (c). Dashed lines show the transition temperatures $T_{\rm
N}$, $T_{\rm IT}$, and $T_{\rm LT}$.

Comparing the anomalous length and entropy changes provides quantitative
information on coupling between spin and structure. In the presence of one
dominant energy scale $\epsilon$, the ratio of the thermal expansion
coefficient and specific heat is independent on temperature and enables the
determination of the pressure dependence of $\epsilon$ by evaluating the
volume or uniaxial Grüneisen parameter Gegenwart (2016); Klingeler et al.
(2006)

$\gamma_{i}=\frac{\alpha_{i}}{c_{\rm p}}=\frac{1}{V_{\rm
m}}\left.\frac{\partial\ln\epsilon}{dp_{i}}\right|_{T}.$ (2)

Here, $V_{\rm m}$ is the molar volume and the index $i$ indicates a uniaxial
direction or the volume. Exploiting the non-phononic contributions
$\alpha_{\rm[110]}^{\prime}$, $\alpha_{\rm[001]}^{\prime}$, $\alpha_{\rm
v}^{\prime}$, and $c_{\rm p}^{\prime}$ yields the Grüneisen parameters in Fig.
4. The experimentally obtained Grüneisen ratios distinctly change at the
different phase boundaries thereby signaling clear changes of magnetoelastic
coupling. At $T_{\rm N}$ we observe jumps in $\gamma_{\rm v}$ and
$\gamma_{\rm\perp c}$ but no clear anomaly appears in $\gamma_{\rm c}$. This
absence of a clear anomaly corresponds to the absence of an anomaly in the
magnetic susceptibility $\chi_{\rm[001]}$ at $T_{\rm N}$. Within the cAFM
phase, both the uniaxial Grüneisen parameters $\gamma_{\parallel c}$ and
$\gamma_{\perp c}$ as well as the volume Grüneisen parameter $\gamma_{v}$ are
rather constant but begin to change a few Kelvin above $T_{\rm IT}$. We
conclude that critical fluctuations start to evolve at around 10 K, associated
with structural changes of opposite uniaxial pressure dependence. The presence
of opposite signs of anomalies tends to volume conservation at the phase
transition and is also a feature of the thermal expansion anomalies at $T_{\rm
IT}$ (see Fig. 2). We hence conclude that fluctuations below 10 K are of the
IT-type. Notably, the changes of Grüneisen scaling in the cAFM phase
correspond to the evolution of the purely magnetic (1 0 0) reflection which
gradually evolves below around 10 K and sharply increases at $T_{\rm LT}$
Metoki et al. (2018). Microscopically, the fluctuation regime in cAFM seems to
be associated with the rotation of magnetic moments out of the $ab$-plane as
the out-of-plane angle is found to evolve below ca. 10 K as a secondary order
parameter. In the IT-phase, all parameters $\gamma_{i}$ increase upon cooling
which we again attribute to precursing fluctuations – here of the LT-type as
corroborated by the equal signs of associated pressure dependencies –
superimposed to the response of the IT-type of order. Lastly there is a (non-
volume conserving) jump at the transition into the LT-phase which features
constant values of $\gamma_{i}$.

Our analysis yields the $\gamma$ values summarized in Table 2. Using the
Ehrenfest relation for the continuous phase transitions at $T_{\rm N}$ and
$T_{\rm IT}$, the obtained values of $\gamma$ yield the pressure dependencies
of the ordering temperatures at vanishing pressure, i.e., $dT_{\rm
N/IT}/dp_{i}=T_{\rm N/IT}V_{\rm m}\gamma_{i}$. Klingeler et al. (2006) Here,
the pressure dependence of $T_{\rm IT}$ is estimated by the value of the
Grüneisen parameter directly below the phase transition. For the discontinuous
phase transition at $T_{\rm LT}$, the pressure dependence is governed by the
respective discontinuities in length, $\Delta L_{i,{\rm LT}}/L_{i}$ and in
entropy, $\Delta S_{\rm LT}$, via the Clausius-Clapeyron equation:

$\frac{dT_{\rm LT}}{dp_{i}}=V_{m}\frac{\Delta L_{i,{\rm LT}}/L_{i}}{\Delta
S_{\rm LT}},$ (3)

with the molar volume $V_{m}=3.22\cdot 10^{-5}$ m3 Brunt et al. (2019).
Analysis of the experimental specific heat data yields $\Delta S_{\rm
LT}=0.13(3)$ J/(mol K). We note a general experimental uncertainty due to the
relaxation method used for determining the heat capacity at a discontinuous
phase transition. From the field dependence ${dT_{\rm LT}/dB\approx 0.4(1)}$
K/T obtained around 1 T and the jump in magnetization $\Delta M_{\rm
LT}\approx 85(10)$ erg/(mol$\cdot$G) we obtain $\Delta S_{\rm LT}=(dT_{\rm
LT}/dB)^{-1}\cdot\Delta M_{\rm LT}\approx 0.21(8)$ J/(mol K) which is hence
used for the Clausius-Clapeyron analysis. The results for all phase
transitions are shown in Table 2.

Table 2: Pressure dependence of ordering temperatures and Grüneisen parameter for the respective phases for uniaxial pressure along the [001]- and the [110]-direction as well as for hydrostatic pressure. The asterisks mark averaged values for the regimes where $\gamma$ varies. Parameters for the cAFM phase are obtained above 10 K. | $dT/dp_{\rm c}$ | $\gamma_{\rm c}$ | $dT/dp_{\rm\perp c}$ | $\gamma_{\rm\perp c}$ | $dT/dp$ | $\gamma_{\rm v}$
---|---|---|---|---|---|---
| (K/GPa) | (mol/MJ) | (K/GPa) | (mol/MJ) | (K/GPa) | (mol/MJ)
$T_{\rm N}$ | 0.38(4) | 0.69(7) | 0.034(8) | 0.061(14) | 0.44(19) | 0.80(3)
$T_{\rm IT}$ | -0.15* | -0.69* | 0.11* | 0.52* | 0.04* | 0.18*
$T_{\rm LT}$ | 0.8(3) | 1.8(2) | 0.3(1) | 1.0(1) | 1.3(6) | 3.8(1)
Figure 5: Temperature dependence of the relative distortion of the uniaxial
length changes $\delta=(dL_{[110]}-dL_{[001]})$/$(dL_{[110]}+dL_{[001]})$. The
inset shows an enlargement of the low-temperature behavior. Arrows indicate
the transition temperatures of competing magnetic phases.

The thermal expansion data do not show clear signatures associated with the CF
levels which have been detected at 2.9, 12.5 and 17.7 meV Brunt (2017).
Excitation of these CF levels would imply Schottky contributions to the
thermal expansion in the paramagnetic phase centered around 60 K and 82 K,
respectively, while a potential Schottky peak at around 14 K associated with
the lowest CF level might be masked and shifted due to the presence of long-
range magnetic order. The absence of clear anomalies above $T_{\rm N}$ is,
however, evident from Fig. 1 and the inset of Fig. 3a. Below $T_{\rm N}$,
there is no clear signature of CF levels either, which might have been also
expected in the experimental Grüneisen parameters in Fig. 4 Werner et al.
(2019). As the magnitude and the shape of the Schottky anomalies in the
thermal expansion coefficients are completely determined by the energy gaps
$\Delta_{i}$ and their pressure dependencies $\partial\ln{\Delta}/\partial
p_{i}$ (see Eq. 2), we conclude rather small pressure dependencies of all
$\Delta_{i}$ for hydrostatic pressure and uniaxial pressure along [110] and
[001] (cf., e.g., Berggold et al. (2007)).

In order to highlight the difference of in-plane and out-of-plane behavior, a
structural anisotropy parameter
$\delta=(dL_{[110]}-dL_{[001]})$/$(dL_{[110]}+dL_{[001]})$ is presented in
Fig. 5. It represents the difference of in-plane and out-of-plane length
changes, assuming finite values below $T^{\ast}\sim 65$ K and smoothly
increasing cubically upon cooling to $T_{\rm N}$. Kinks at $T_{\rm N}$ and
$T_{\rm IT}$ as well as a jump at $T_{\rm LT}$ indicate the magnetic phase
transition. While a rather conventional increase towards saturation is
observed in cAFM, the distortion actually decreases in IT but eventually
linearly increases in LT down to lowest accessible temperatures. The jump in
length at $T_{\rm LT}$ implies that there is a combined structural-magnetic
phase transition associated with abrupt shrinking of both $L_{\rm[001]}$ and
$L_{\rm[110]}$, a jump-like increase of the structural anisotropy parameter
$\delta$, and evolution of the LT-type of spin configuration Brunt (2017);
Metoki et al. (2018).

The data in Fig. 5 imply that commensurate magnetic order in HT and LT
promotes structural anisotropy while in contrast $\delta$ competes with
incommensurate magnetic order in IT. This competition is also evidenced by the
Grüneisen parameters which, in particular, display a sign change of
$\gamma_{\perp c}$ at $T_{\rm IT}$ and $T_{\rm LT}$. We conclude the presence
and interplay of several degrees of freedom. It is straightforward to
attribute this observation to the magnetic and orbital degrees of freedom of
the system. This leads to the scenario that the LT phase features combined
magnetic and orbital order which evolution is associated with the qualitative
change from a volume-conserving behaviour in the IT phase to a non-volume-
conserving one below $T_{\rm LT}$. Note, that only the magnetically
incommensurate IT phase and the associated transition exhibit opposite
uniaxial pressure dependencies in-plane and out-of-plane. In addition, the IT
phase also shows a small but distinct non-volume-conserving temperature
dependence of the Grüneisen parameters (see Fig. 4). This again indicates the
presence of several degrees of freedom and may be interpreted as signature of
a slowly evolving long- or short-range orbital order. Accordingly, the
fluctuation regime below 10 K may indicate the presence of short-range orbital
order competing with cAFM-type of magnetism.

Finally, we note finite values of $\delta$ up to about 65 K, i.e., the
evolution of anisotropy in the length changes as also seen in Fig. 1,
indicating short-range fluctuations in this temperature regime. As visible in
the inset of Fig. 1, changes in $\delta$ above $T_{\rm N}$ are associated with
anomalous in-plane expansion which yields a negative thermal expansion
coefficient $\alpha_{\rm[110]}$ (cf. Fig. 2). As both magnetic and quadrupolar
degrees of freedom are strongly coupled to the lattice, this phenomenon may be
in general related to fluctuations of either ordering phenomena. It is however
not directly associated with fluctuations of any of the long-range ordering
phenomena actually observed at low temperatures so that its nature remains
unclear. Specifically, the uniaxial Grüneisen parameter $\gamma_{\perp c}$
displays a sign change at $T_{\rm N}$ which implies opposite pressure
dependencies $\partial\ln{\epsilon}/\partial p_{c}$ above and below $T_{\rm
N}$, thereby excluding the short-range fluctuations being of cAFM nature. An
analogous argument holds for the LT-type order. We also exclude that
precursing anisotropy above $T_{\rm N}$ is associated with order of the IT-
type as $\delta$ is suppressed and $L_{[110]}$ considerably shrinks in this
phase.

##  V Summary

In summary, high-resolution capacitance dilatometry studies on the uniaxial
length changes in a single crystal enable to elucidate the interplay of spin
and quadrupolar order in NdB4. The data imply significant magneto-elastic
coupling and a structural phase transition at $T_{\rm LT}$. While cAFM and LT
favor structural anisotropy between in-plane and out-of-plane length changes,
${\delta=(dL_{[110]}-dL_{[001]})/(dL_{[110]}+dL_{[001]})}$ competes with order
in the IT phase. Notably, finite anisotropy well above $T_{\rm N}$ indicates
short-range correlations which are however of neither cAFM, IT, nor LT-type.
Grüneisen analysis is used to derive the uniaxial as well as the hydrostatic
pressure dependencies. While the ratio $\alpha$/$c_{\rm p}$ features a single
dominant energy scale in LT, the data implies precursory fluctuations of a
competing phase in IT and cAFM. Our results suggest the presence of orbital
degrees of freedom competing with cAFM and evolving towards the magnetically
and orbitally ordered ground state.

###### Acknowledgements.

Partial support by Deutsche Forschungsgemeinschaft (DFG) via Project KL
1824/13-1 and by BMBF via the SpinFun project (13XP5088) is gratefully
acknowledged. SP acknowledges fellowship by the HGSFP. Work at the University
of Warwick was funded by EPSRC (UK) through Grant EP/T005963/1.

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