# Axis thermal expansion switching in transition-metal zirconides $Tr$Zr2 by
tuning the $c/a$ ratio

Hiroto Arima  Department of Physics, Tokyo Metropolitan University, Hachioji-
shi, Tokyo, 192-0397, Japan  Md. Riad Kasem  Department of Physics, Tokyo
Metropolitan University, Hachioji-shi, Tokyo, 192-0397, Japan  Yoshikazu
Mizuguchi<EMAIL_ADDRESS>Department of Physics, Tokyo Metropolitan
University, Hachioji-shi, Tokyo, 192-0397, Japan

###### Abstract

The development of materials with a preferred thermal expansion is essential
to achieve negative or zero thermal expansion. Recently, CuAl2-type CoZr2 has
been reported to exhibit $c$-axis negative thermal expansion over a wide
temperature range. In this study, we show the temperature evolution of lattice
constants for NiZr2, (Co,Rh,Ir)Zr2, and (Co,Ni,Cu,Rh,Ir)Zr2 with a CuAl2-type
structure (space group #140). We revealed that the lattice constants $a$ and
$c$ of NiZr2 expand on warming, which is positive thermal expansion.
Conversely, (Co,Rh,Ir)Zr2 and (Co,Ni,Cu,Rh,Ir)Zr2 exhibit anisotropic thermal
expansion with a $c$-axis contraction indicating negative thermal expansion.
The plot of $c$-axis thermal expansion amplitude as a function of the ratio of
lattice constants $c$ and $a$ ($c/a$) for the examined compounds and related
$Tr$Zr2 suggests that the $c$-axis thermal expansion can be controlled by
tuning $c/a$ through the $Tr$-site substitution. The results provide a design
guideline for negative thermal expansion of the $c$-axis in $Tr$Zr2, achieving
zero thermal expansion in a single compound.

##  I Introduction

Materials with zero thermal expansion (ZTE) that require ultra-precision
positive thermal expansion (PTE) and negative thermal expansion (NTE)
materials are a key material for application. Heat is known to cause thermal
expansion in almost all compounds, whereas some compounds exhibit shrinkage on
warming. NTE compounds have been developed based on phase transitions, (such
as structuralJohn S.O Evans and T.A Mary (2000); T.G. Amos and A.W. Sleight
(2001), ferroelectricD.K. Agrawal and A. Halliyal and J. Belsick (1988);
Agrawal et al. (1987), and magnetic transitionsW.F. Schlosser and G.M. Graham
and P.P.M. Meincke (1971); S.A. Nikitin and A.M. Tishin and M.D. Kuz’min and
Yu.I. Spichkin (1991)), intermetallic charge transfersYamada et al. (2013);
Nishikubo et al. (2019), and valence crossoversrsQiao et al. (2021); Azuma et
al. (2007). Although these compounds exhibit large NTE, it appears only at
specific temperatures, such as the boundary of the phase transition or
crossover temperature. Conversely, PTE is observed over a wide temperature
range. ZrW2O8 has been extensively studied because it shows isotropic NTE over
a broad temperature range from 0.3 K to 1000 KMartinek and Hummel (1968); Mary
et al. (1996).This NTE is assumed to be caused by the vibration of two rigid
units, a ZrO6 octahedron and WO4 tetrahedron, whose vibration mode is known as
rigid unit modeGiddy et al. (1993); A.K.A. Pryde and K.D. Hammonds and M.T.
Dove and V. Heine and J.D. Gale and M.C. Warren (1996); Ernst et al. (1998);
Dove (1997).

Recently, transition-metal zirconides CoZr2 and alloyed $Tr$Zr2 (Tr: Fe, Co,
Ni, Rh, Ir) have been reported as a new family of NTE compoundsMizuguchi et
al. (2022); Kasem et al. (2022). Figure 1 shows the crystal structure of
$Tr$Zr2, which has a body-centered tetragonal structure (CuAl2 type, space
group #140). In CoZr2, neutral powder diffraction (NPD) and X-ray diffraction
(XRD) at low and high temperatures, respectively, revealed anisotropic thermal
expansion of the $a$\- and $c$-axesMizuguchi et al. (2022).Additionally,
heating induces the expansion and contraction of lattice constants $a$ (PTE)
and $c$(NTE), respectively. Furthermore, the $c$-axis NTE is observed from 7 K
to 573 K. Similar wide temperature range anisotropic thermal expansion has
been reported for (Fe,Co,Ni)Zr2, (Fe,Co,Ni,Rh,Ir)Zr2, and (Fe,Co,Ni,Cu,Ga)Zr2,
which were synthesized based on the concept of high-entropy alloys (HEA). In
this study, we synthesized NiZr2, (Co,Rh,Ge)Zr2, and (Co,Ni,Cu,Rh,Ir)Zr2 and
analyzed their temperature dependence of lattice constants by XRD experiment
to explore NTE compounds with a CuAl2-type structure in space group #140.

Figure 1: (Color online) A schematic image of the tetragonal crystal structure
of $Tr$Zr2.

##  II Method

The polycrystalline samples of NiZr2, (Co,Rh,Ir)Zr2, and (Co,Ni,Cu,Rh,Ir)Zr2
were synthesized by the arc melting method in an Ar gas environment on a
water-cooled copper hearth. Powders of pure Ni (99.9 %), Co (99 %), Rh (99.9
%), Ge (99.99 %), Cu (99.9 %), and Ir (99.9 %) were pelletized and melted with
Zr plates (99.2 %). During arc melting, the ingot was turned over and remelted
three times to homogenize the samples. Energy-dispersive X-ray spectroscopy
(EDX, SwiftED, Oxford) equipped with a scanning electron microscope (TM3030,
Hitachi High-Tech) was used to determine the actual composition of the samples
(Co,Rh,Ir)Zr2 and (Co,Ni,Cu,Rh,Ir)Zr2. The configurational entropy of mixing
($\Delta S_{\rm{mix}}$) at the $Tr$ site was calculated from the actual
composition of the $Tr$ site using the formula $\Delta S_{\rm{mix}}=-R\Sigma
c_{i}\ln c_{i}$, where $R$ is the gas constant and $c_{i}$ is the mole
fraction of component $i$. The crystal structure in high temperature was
investigated by employing a powder XRD technique using the Cu-K$\rm{\alpha}$
radiation by the $\theta$-$2\theta$ method (Miniflex-600 RIGAKU) equipped with
a heating temperature attachment (BTS 500). The Rietveld refinement of the XRD
data was performed using the RIETAN-FP packageIzumi and Momma (2007). A
schematic image was obtained by VESTAMomma and Izumi (2011).

##  III Results and discussion

The actual composition of the $Tr$ site estimated by EDX for (Co,Rh,Ir)Zr2 and
(Co,Ni,Cu,Rh,Ir)Zr2 were Co0.28Rh0.43Ir0.29Zr2 and
Co0.19Ni0.10Cu0.08Rh0.33Ir0.31Zr2, respectively. Using these calculations, the
composition of Zr was fixed at 2. In this study, we abbreviated
Co0.28Rh0.43Ir0.29Zr2 as (Co,Rh,Ir)Zr2 and Co0.19Ni0.10Cu0.08Rh0.33Ir0.31Zr2
as (Co,Ni,Cu,Rh,Ir)Zr2. Entropies of mixing of (Co,Rh,Ir)Zr2 and
(Co,Ni,Cu,Rh,Ir)Zr2 were 1.08$R$ and 1.50$R$, respectively.

Figure 2 (a) shows the XRD pattern and Rietveld refinement results for NiZr2
at $T=$293 K. The obtained pattern was refined using the CuAl2-type (space
group: #140) structural model with an impurity phase of NiZr (9 %). The
reliability factor was $R_{\rm{wp}}$= 4.1 %, and the goodness-of-fit indicator
was $S$ = 1.3. Figure 2 (b) illustrates the XRD patterns at high temperatures,
the profiles at each temperature are similar, which confirms no structural
transition.

Figure 2: (Color online) (a) The XRD pattern and Rietveld refinement results
for NiZr2 at $T$=293 K. The numbers indicate Miller indices. The asterisk
indicates an impurity peak. The red dots and green solid line are experimental
data and fitted line, respectively. The blue line is a residual curve. The
green ticks are the positions of the Bragg peaks for NiZr2 and NiZr. (b) XRD
patterns at $T$= 303 , 330 , 369 , 410 , 450 , 490 , 530, and 570 K.

Table 1 lists the determined crystal structure parameters for NiZr2,
(Co,Rh,Ir)Zr2, and (Co,Ni,Cu,Rh,Ir)2; we summarized the lattice constant and
volume. The labels of atomic distances $Tr$-Zr, $Tr$-$Tr$, and Zr-Zr, and the
angle of Zr-$Tr$-Zr are shown in Figure.1. The XRD patterns of (Co,Rh,Ir)Zr2
and (Co,Ni,Cu,Rh,Ir)Zr2 at room temperature and high temperatures are shown in
the Supporting Information (Figure S1). Compounds with shorter $c$-axis
lengths had smaller $Tr$-$Tr$ distances, directly reflecting the lattice
constant $c$. Zr-Zr(ii) and $Tr$-$Tr$(ii), which relate to the lattice
constant $a$, became smaller as $a$ decreased. Because the lengths of the
$a$\- and $c$-axes contribute to Zr-Zr(i) and $Tr$-Zr, it is difficult to
determine the factors they depend on. However, Zr-Zr(i) and $Tr$-Zr generally
tend to be longer when the ratio of the lattice constants $a$ and $c$ ($c/a$)
is large. The Zr-$Tr$-Zr angle increased with larger lattice constant $c$ and
smaller $a$, that is with smaller $c/a$. The Zr-$Tr$-Zr angle was large for
NiZr2 with a relatively small $c/a$.

Table 1: Crystal structure parameters at room temperature and entropy of
mixing for $Tr=$ Co, Ni, (Co,Rh,Ir), (Co,Ni,Cu,Rh,Ir).

$Tr=$ | Co (Ref.17) | Ni | (Co,Rh,Ir) | (Co,Ni,Cu,Rh,Ir)
---|---|---|---|---
Space group |  $I$4/$mcm$ (#140)
$a$ (Å) | 6.3584 | 6.4932 | 6.4797 | 6.4892
$c$ (Å) | 5.5002 | 5.2673 | 5.5952 | 5.5191
$c/a$ | 0.8667 | 0.8111 | 0.8634 | 0.8505
$V$ (Å3) | 222.37 | 222.08 | 234.92 | 232.40
$Tr$-Zr (Å) | 2.7293 | 2.7599 | 2.7825 | 2.7761
$Tr$-$Tr$ (i) (Å) | 2.7501 | 2.6337 | 2.7976 | 2.7595
$Tr$-$Tr$ (ii) (Å) | 4.4961 | 4.5914 | 4.5815 | 4.5886
Zr-Zr (i) (Å) | 3.0954 | 3.0640 | 3.1585 | 3.1259
Zr-Zr (ii) (Å) | 3.334 | 3.4302 | 3.4017 | 3.4067
Zr-$Tr$-Zr (deg) | 119.49 | 123.00 | 119.64 | 120.39
$\Delta S_{\rm{mix}}$ | 0 | 0 | 1.08$R$ | 1.50$R$

Figure 3 shows the temperature dependence of the lattice constants and volume
for NiZr2, (Co,Rh,Ir)Zr2, and (Co,Ni,Cu,Rh,Ir)Zr2. The lattice constant $a$
expanded with increasing temperature in all compounds, which is PTE. However,
the temperature dependence of the $c$-axis lengths varied depending on the
$Tr$ composition. On heating, the lattice constant $c$ increased in NiZr2, and
decreased in (Co,Rh,Ir)Zr2 and (Co,Ni,Cu,Rh)Zr2. The volumes of NiZr2 and
(Co,Ni,Cu,Rh,Ir)Zr2 monotonically increased with increasing temperature.
Although the volume of (Co,Rh,Ir)Zr2 increased in the macroscopic view, but it
decreased near room temperature. In order to reveal the origin of the
anisotropic and variable axis thermal expansion, we demonstrated the
temperature dependence of the $Tr$-Zr, $Tr$-$Tr$(i), $Tr$-$Tr$(ii), Zr-Zr(i),
and Zr-Zr(ii) distances and Zr-$Tr$-Zr angles for NiZr2 and (Co,Rh,Ir)Zr2,
(Co,Ni,Cu,Rh,Ir)Zr2 in the Supporting Information (Figure S2). In NiZr2, the
$Tr$-Zr-$Tr$ angle showed insignificant changes, and all bond lengths
increased with increasing temperature; these are the characteristics of PTE.
In (Co,Rh,Ir)Zr2, the bond length of $Tr$-Zr, $Tr$-$Tr$(ii), and Zr-Zr(ii)
increased with increasing temperature, whereas the $Tr$-$Tr$ (i) distance
decreased. Although the Zr-Zr(i) distance also decreased, its magnitude was
relatively small and became independent of temperature above 400 K. In
addition, the Zr-$Tr$-Zr angle increased with increasing temperature. The
temperature dependence of the bond length and bond angle of
(Co,Ni,Cu,Rh,Ir)Zr2 was similar to that of (Co,Rh,Ir)Zr2. The contraction of
the $c$-axis in (Co,Rh,Ir)Zr2 and (Co,Ni,Cu,Rh,Ir)Zr2 is attributed to the
Zr-$Tr$-Zr angle the increases while the length of Zr-Zr(i) remains at a
distance, and the $Tr$-$Tr$(i) distance is relatively decreased. This
mechanism is the same as that reported for CoZr2.

Figure 3: (Color online) Crystal structure parameters for (a)(b) NiZr2, (c)(d)
(Co,Rh,Ir)Zr2, and (e)(f) (Co,Ni,Cu,Rh,Ir)Zr2. Temperature dependence of
lattice constant $a$-, $c$-axes (left panels), and $V$ (right panels).

To compare the magnitude of change in the lattice constants, Figure 7 shows
the temperature dependence of the change ratio of lattice constants in the
$a$-axis ($\Delta a/a=1-a(300\,{\rm K})/a(T)$) and $c$-axis ($\Delta
c/c=1-c(300\,{\rm K})/c(T)$) normalized at room temperature of $Tr$Zr2 ($Tr$ =
Ni, Co, (Fe,Co,Ni), (Co,Rh,Ir), (Fe,Co,Ni,Rh,Ir), (Co,Ni,Cu,Rh,Ir),
(Fe,Co,Ni,Cu,Ga))Mizuguchi et al. (2022); Kasem et al. (2022). As mentioned
previously, the lattice constant $a$ increasesed with increasing temperature,
and the change ratio was similar regardless of compounds. In contrast, the
temperature dependence of the $c$-axis varied among different compounds. The
$c$-axis of (Fe,Co,Ni,Cu,Ga)Zr2 was almost temperature independent, while that
of NiZr2 exhibited PTE; other compounds exhibited $c$-axis NTE. The absolute
value of the slope differed among the $c$-axis NTEs compounds, and the slope
became steep in the following order: (Co,Ni,Cu,Rh,Ir)Zr2, (Co,Rh,Ir)Zr2,
(Fe,Co,Ni)Zr2, and CoZr2, (Fe,Co,Ni)Zr2, and (Co,Ni,Cu,Rh,Ir)Zr2.

Figure 4: (Color online) Temperature dependence of the change ratio of
(a)$a$-axis and (b)$c$-axis lengths scaled at room temperature in $Tr$Zr2
($Tr$=Ni, Co, (Fe,Co,Ni), (Co,Rh,Ir), (Fe,CoNi,Rh,Ir), (Co,Ni,Cu,Rh,Ir),
(Fe,Co,Ni,Cu,Ga)). Lattice parameters were obtained from Ref. 18 ($Tr$=Co,
(Fe,Co,Ni), (Co,Ni,Cu,Rh,Ir)) and Ref.19($Tr$=(Fe,Co,Ni,Cu,Ga)).

To discuss the conditions of $c$-axis NTE in $Tr$Zr2 compounds, we
demonstrated the temperature dependence of the ratio of the $c$\- and $a$-axis
lattice constants ($c/a$) of $Tr$Zr2 ($Tr$ = Ni, Co, (Fe,Co,Ni), (Co,Rh,Ir),
(Fe,CoNi,Rh,Ir), (Co,Ni,Cu,Rh,Ir), and (Fe,Co,Ni,Cu,Ga)) in figure 5. In the
$c$-axis-NTE compounds, $c/a$ decreases with increasing temperature due to the
contraction and expansion of the lattice constants $c$ and $a$, respectively.
NiZ2 and (Fe,Co,Ni,Cu,Ga)Zr2 showed a smaller change in $c/a$ than other
compounds because the lattice constant $c$ exhibited PTE or almost ZTE,
respectively. In addition, room temperature $c/a$ values was found to be
larger for $c$-axis-NTE compounds than for PTE compounds.

Figure 5: (Color online) Temperature dependence of $c$/$a$ for $Tr$Zr2
($Tr$=Ni, Co, (Fe,Co,Ni), (Co,Rh,Ir), (Fe,CoNi,Rh,Ir), (Co,Ni,Cu,Rh,Ir),
(Fe,Co,Ni,Cu,Ga)). The value of $c$/$a$ is taken from Ref.18 ($Tr$=Co,
(Fe,Co,Ni), (Co,Ni,Cu,Rh,Ir)) and Ref.19($Tr$=(Fe,Co,Ni,Cu,Ga)).

We estimate the change ratio of the lattice constant $c$ as a function of
temperature ($\Delta c/\Delta T$); Figure 6 illustrates $c/a$ at room
temperature dependence for $\Delta c/\Delta T$. $\Delta c/\Delta T$ was
obtained from the slope evaluated in Figure 7(b). In this study, the whole
temperatures were approximated as linear and fitted to the results for each
compound. The $\Delta c/\Delta T$ of CoZr2, (Co,Rh,Ir)Zr2, (Fe,Co,Ni)Zr2,
(Co,Ni,Cu,Rh,Ir)Zr2, and (Fe,Co,Ni,Rh,Ir)Zr2 exhibiting NTE was negative,
while that of NiZr2 exhibiting PTE was positive. The $\Delta c/\Delta T$ of
(Fe,Co,Ni,Cu,Ga)Zr2 with a small temperature dependence of the lattice
constant $c$ was near zero. Figure 6 clearly demonstrates the correlation
between $\Delta c/\Delta T$ and $c/a$. $\Delta c/\Delta T$ increased as $c/a$
decreased, and the sign of the $\Delta c/\Delta T$ changed from negative to
positive, that is from NTE to PTE at a boundary of around $c/a\sim$ 0.83 by
tuning $c/a$. In this experiment, $\Delta S_{\rm{mix}}$ at the $Tr$ sites was
different, which possibly affected the temperature dependence of the lattice
constant $c$. This experiment does not exclude the possibility that the
difference in $\Delta S_{\rm{mix}}$ of the $Tr$ sites affected the electronic
state and caused anisotropic thermal expansion. However, we claim that the
relationship between $\Delta S_{\rm mix}$ and anisotropic thermal expansion is
insignificant because CoZr2 and NiZr2 with $\Delta S_{\rm{mix}}$=0 exhibit
contrasting thermal expansions while (Fe,Co,Ni,Rh,Ir)Zr2 ($\Delta
S_{\rm{mix}}$=1.60$R$)Kasem et al. (2022) and (Co,Ni,Cu,Rh,Ir)Zr2 ($\Delta
S_{\rm{mix}}$=1.50$R$) with relatively large $\Delta S_{\rm{mix}}$ demonstrate
different temperature dependence of lattice constant $c$. Notably, the
anisotropy of thermal expansion depends more on the $c/a$ ratio associated
with the Zr-$Tr$-Zr angle than on the type of $Tr$-site atoms. This phenomenon
indicates that the $c$-axis thermal expansion can be controlled by flexibly
substituting the $Tr$ sites in $Tr$Zr2. In the future, the temperature
dependence of the lattice parameters of various $Tr$Zr2, including HEA, should
be investigated to reveal the systematics of the anisotropy between the
crystal structure and thermal expansion. To clarify the origin of the
anisotropy of thermal expansion of the $c$-axis, it is important to elucidate
the local crystal structure by high-resolution XRD and NPD.

Figure 6: (Color online) Change ratio of the $c$-axis with respect to the
temperature ($\Delta c/\Delta T$) of $Tr$Zr2 ($Tr$=Ni, Co, (Fe,Co,Ni),
(Co,Rh,Ir), (Fe,CoNi,Rh,Ir), (Co,Ni,Cu,Rh,Ir), and (Fe,Co,Ni,Cu,Ga)) plotted
as a function of $c$/$a$. $\Delta c/\Delta T$ and $c/a$(@RT) values were
obtained from Ref. 18 ($Tr$=Co, (Fe,Co,Ni), (Co,Ni,Cu,Rh,Ir)) and Ref. 19
($Tr$=(Fe,Co,Ni,Cu,Ga)).

##  IV Conclusion

In conclusion, we synthesized transition-metal zirconides NiZr2,
(Co,Rh,Ir)Zr2, and (Co,Ni,Cu,Rh,Ir)Zr2 with a CuAl2-type structure (space
group #140) and analyzed their crystal structure by powder XRD at high
temperatures. With increasing temperature, the lattice constant $a$ of all
compounds exhibited expansion (PTE), while the temperature dependence of the
$c$-axis differed for the compounds. The lattice constant $c$ in NiZr2
demonstrated PTE, while NTE was observed for (Co,Rh,Ir)Zr2 and
(Co,Ni,Cu,Rh,Ir)Zr2. We found that the temperature dependence of the lattice
constant $c$ correlated with the ratio of the $c$-axis to the $a$-axis
($c/a$), and tuning $c/a$ with elemental substitution at the $Tr$ site
switched NTE and PTE. Although the origin of the anisotropic lattice
structural change is unclear, this study is expected to provide a guideline
for developing ZTE compounds.

The authors would like to thank M. Fujita and O. Miura for their support in
experiments and discussion. This work was partly supported by Grant-in-Aid for
Scientific Research (KAKENHI) (Proposal No. 21K18834 and 21H00151) and Tokyo
Government Advanced Research (H31-1).

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Supporting information
Hiroto Arima,1 Md. Riad Kasem,1 and Yoshikazu Mizuguchi1

1Department of Physics, Tokyo Metropolitan University, Hachioji-shi, Tokyo,
192-0397, Japan
(Dated: )

Figure 7 shows XRD patterns of (a)(Co,Rh,Ir)Zr2 and (Co,Ni,Cu,Rh,Ir)Zr2 at
high temperatures. For each compounds, the profiles for each temperature are
similar, that indicate is no structural transition and magnetic ordering.

Figure 8 shows a atomic distance of $Tr$-Zr, $Tr$-$Tr$(i), $Tr$-$Tr$(ii), Zr-
Zr(i), Zr-Zr(ii), and an angle of Zr-$Tr$-Zr for NiZr2, (Co,Rh,Ir)Zr2, and
(Co,Ni,Cu,Rh,Ir)Zr2.

Figure 7: (Color online) Powder X-ray diffraction patterns for (a)
(Co,Rh,Ir)Zr2, and (b) (Co,Ni,Cu,Rh,Ir)Zr2 in high temperatures. Figure 8:
(Color online) Temperature dependence for (a)(e)(i) $Tr$-Zr distance,
(b)(f)(j) $Tr$-$Tr$ distance, (c)(g)(k) Zr-Zr distance, and (d)(h)(i)
Zr-$Tr$-Zr angle for NiZr2 (upper panels), (Co,Rh,Ir)Zr2 (middle panels), and
(Co,Ni,Cu,Rh,Ir)Zr2 (lower panels.)