Pressure-induced volumetric negative thermal expansion in CoZr 2 superconductor

We investigate the thermal expansion and superconducting properties of a CuAl2-type (tetragonal) superconductor CoZr2 under high pressures. We perform high-pressure synchrotron X-ray diffraction in a pressure range of 2.9 GPa<P<10.4 GPa and discover that CoZr2 exhibits volumetric negative thermal expansion under high pressures. Although the uniaxial positive thermal expansion (PTE) along the a-axis is observed under ambient pressure, that is suppressed by pressure, while the large uniaxial negative thermal expansion (NTE) along the c-axis is maintained under the pressure regime. As a result of a combination of the suppressed uniaxial PTE along the a-axis and uniaxial NTE along the c-axis, volumetric negative thermal expansion is achieved under high pressure in CoZr2. The mechanisms of volumetric NTE would be based on the flexible crystal structure caused by the soft Co-Co bond as seen in the iso-structural compound FeZr2, which exhibits uniaxial NTE along the c-axis. We also perform high-pressure electrical resistance measurements of CoZr2 to confirm the presence of superconductivity under the examined pressure regime in the range of 0.03 GPa<P<41.9 GPa. We confirm the presence of superconductivity under all pressures and observe dome-like shape pressure dependence of superconducting transition temperature. Because of the coexistence of two phenomena, which are volumetric NTE and superconductivity, in CoZr2 under high pressure, the coexistence would be achievable under ambient pressure by tuning chemical compositions after our present observation.


Introduction
Thermal expansion is one of the most significant characteristics of materials because it is connected to the crystal structure and electronic structure, which determines the physical properties of materials.In most cases, materials expand upon heating; this conventional property is called positive thermal expansion (PTE).In contrast, negative thermal expansion (NTE) is defined as contraction upon heating, and such NTE has been observed in various materials. [1,2,3,4]The mechanisms of NTE are diverse and correlated to the flexible crystal structure, [5,6] phase transition, [7] magnetic order-to-disorder transition, [8] and/or so on.NTE has been used to achieve zero thermal expansion in practical devices by making a composite using PTE and NTE materials.In superconducting devices, the heat cycle between working (very low) temperature and room temperature when turning off the device is a critical issue because the heat cycle degrades the surface and junction between different materials.If a superconductor with NTE is in a wide temperature range below room temperature, the heat-cycle problem will be improved.Isotropic and uniaxial NTE has been reported in various superconducting materials, such as single elements Nb [9,10] and Ta, [10] layered materials MgB2, [9,11] YBa2Cu3O7δ, [12] Bi2Sr2CaCu2O8+x, [13] PrFeAsO, [14] and Ba(Fe1−xCox)2As2 (x = 0.16, 0.23). [15]Those NTEs have been, however, observed in a limited temperature range, and volumetric NTE in a wide temperature range has not been achieved in any bulk superconducting materials.Recently, we observed uniaxial NTE with a wide temperature range in CuAl2-type (tetragonal) TrZr2 and TrZr3 (Tr: transition metal), which are superconductors. [16,17,18]In the TrZr2 system, we revealed that uniaxial NTE along the c-axis can be controlled by lattice-constant ratio, c/a, through chemical element substitution. [19,20]The anomalous bonding states related to uniaxial NTE along the c-axis in TrZr2 have been observed using X-ray absorption spectroscopy as well. [21]Moreover, Xu et al. revealed that FeZr2 exhibits giant uniaxial NTE along the c-axis, and they proposed that the soft Fe-Fe bond and flexible structure caused by optical phonons play an important role in the origin of uniaxial NTE along the c-axis. [22]Therefore, the crystal structural modification should be critical to the NTE phenomena in TrZr2.
Herein, we show the observation of volumetric NTE in CoZr2 under high pressure.At ambient pressure, CoZr2 shows superconductivity at Tc = 5.8 K (Tc: superconducting transition temperature), uniaxial PTE along the a-axis, and uniaxial NTE along the c-axis.The uniaxial PTE along the a-axis is suppressed by pressure, but the uniaxial NTE along the c-axis was not suppressed by pressure.As a consequence of competition between uniaxial PTE and NTE along the a-and c-axes, the volumetric NTE is realized.Since the coexistence of superconductivity and volumetric NTE in a wide temperature range is quite rare, we confirmed the presence of superconductivity in CoZr2 under high pressures by electrical resistance measurements.

Thermal expansion under high pressure
We show the schematic images of the thermal expansions of a structural analogue NiZr2 and CoZr2 (under ambient pressure and high pressure) in Figure 1.These compounds have a tetragonal CuAl2-type crystal structure (space group: I4/mcm).NiZr2 shows PTE both along the a-and c-axes, and thus, the coefficient of volumetric thermal expansion β is positive (Figure 1(a)). [19,20]In the tetragonal crystal structure, β can be calculated to the following equation: where αa and αc are the coefficients of linear thermal expansion along a-and c-axes, respectively.
The crystal structure remains tetragonal CuAl2-type up to P = 10.4GPa.We observe a shift of the 002 and 220 peaks toward the higher angle side by applying pressure as shown in Figures 2(b) and 2(c).The shifts of the peaks result in decreasing lattice constants a and c. Figure 2(d) shows the pressure dependence of the lattice constants.Lattice constants normalized by a value at ambient pressure (a0 and c0) are shown in Figure 2(e).The a-axis is stiffer than the c-axis under high pressure.This implies that the crystal structure of CoZr2 along the c-axis direction is more flexible to pressure.The same trend of a and c against pressure was also observed in laboratory experiments, which were performed using another DAC with a Mo-Kα radiation (see Figure S2 in the Supporting information).As mentioned in the introduction part, FeZr2, which is an iso-structural compound with CoZr2 and NiZr2, exhibits giant uniaxial NTE along the caxis. [22]They revealed that the strong Fe3dz 2 -Fe3dz 2 interaction can play an important role in stabilizing large c/a in the CuAl2-type crystal structure and contributes to the soft Fe-Fe bond, which provides a large contraction space along the c-axis.Furthermore, the optical phonons with a phonon energy of several meV make the flexible structure in FeZr2, which leads to giant uniaxial NTE along the c-axis.These flexible characteristics of crystal structure would be common to CoZr2 because it has the same crystal structure with a similar c/a ratio as FeZr2 and shows large uniaxial NTE along the c-axis.Figure 2(f) shows the pressure dependence of lattice volume V of CoZr2.The solid line is the fit to the 3rd-order Birch-Murnaghan formula as expressed following equation: [23]  = 3 2  0 {(  0  ) where V0, B0, and  0 ′ are the volume at ambient pressure, bulk modulus, and 1st-order pressure derivative of B0, respectively.As the fitting result, we obtain B0 = 100 ± 6 GPa and  0 ′ = 6 ± 2.
The obtained B0 value is close to the computational calculation results, B0 = 134.81 [24]and 129 GPa [25] .We note here that the 3rd-order Birch-Murnaghan formula assumes the cubic structure, thus, the obtained B0 value can deviate from the actual value.Next, we show the results of the thermal expansion of CoZr2 at P = 2.9 GPa as an example of the high-pressure data set.For all the experiments under pressures, we evaluated the fluctuation of applied pressures because the thermal expansion is easily affected by pressure changes.The data shown in this paper has been carefully taken with the manner.Figure 3(a) shows the HP-SXRD patterns at P = 2.9 GPa under temperatures ranging from T = 303 K to 453 K with an increment of 10 K.There was no crystal structural transition between the temperature region under P = 2.9 GPa, which was commonly confirmed in other all applied pressures (see Figure S3 in the Supporting information).The absence of crystal structural transition at ambient pressure was confirmed in the wide temperature (7 K < T < 572 K) in Ref. 16.As the temperature increased, a clear shift of the 002 peak toward the higher angle side is observed as shown in Figure 3(b).However, the 220 peak position is almost the same with pressure as shown in Figure 3(c).The robustness of the a-axis to pressure seen from the 220 peak results in a small value of αa = 9 ± 1 μK -1 , which is clearly smaller than the value at ambient pressure (see Figure S1 in the Supporting information).Figures 3(d) and 3(e) are the temperature dependence of a and c at P = 2.9 GPa.Even under pressure, the large c-axis NTE is present with αc = -24 ± 1 μK -1 , which is almost the same as that observed under ambient pressure.From Equation (1), we obtain β = -6 ± 2 μK -1 , suggesting the pressure-induced volumetric NTE in CoZr2.Figure 3(f) shows the temperature dependence of V at P = 2.9 GPa.We see the volume slightly contracted as the temperature increased.We summarize the pressure dependence of αa, αc, and β in Figure 4.When the pressure is applied to CoZr2, the uniaxial PTE along the a-axis is suppressed; thus, αa under high pressure is lower than at ambient pressure (Figure 4(a)).In contrast, even under high-pressure conditions, the uniaxial NTE along the c-axis is not suppressed; therefore, αc is almost independent of pressure (Figure 4

Superconducting properties
We measured the electrical resistance (R) of CoZr2 under high pressures (0.03 GPa < P < 41.9 GPa) to confirm the presence of superconductivity.Figures 5(a) and 5(b) show the temperature dependence of R under pressures.As temperature decreases, R decreases with a negative curvature, which is a trend commonly seen in d-electron superconductors. [26,27] t low temperatures, the R drops to zero at Tc under all applied pressures.The dome-shaped pressure dependence of Tc is observed as shown in Figure 5(c).The Tc taken from the R data at ambient pressure (P = 0.03 GPa) is 5.8 K, consistent with the value taken from magnetic susceptibility measurement at ambient pressure (see Figure S5 in the Supporting information).As pressure increases, the Tc increases up to P =17.9 GPa and reaches 6.5 K, but the trend of Tc changed at P > 17.9 GPa; Tc decreases with pressure at higher pressures.In a study on a single crystal of CoZr2, the Tc reached 9.5 K at P = 8 GPa, [26] which is higher than the highest Tc obtained in this study with polycrystalline CoZr2.The discrepancy of the highest Tc values may be due to the difference in the reactions of Tc to generated pressure caused by the experimental conditions: pressure cells and sample type (single or poly crystals).In the low-temperature region where Tc < T ≪ ΘD (Debye temperature), the R could be fitted to the power-law relation: where R0 is the residual resistance, A is the numerical temperature-independent coefficient, and n is a component depending on the carrier scattering mechanisms.We used R at 10 K < T < 30 K under pressures in the fitting to power-law relation, which yielded n ~ 3 for all applied pressures as shown in Figure S6 (see Supporting information) The T 3 dependence on lowtemperature R can be explained with a phonon-assisted s-d electron scattering model. [28]The R of compounds composed of d-block elements is empirically known that it could be fitted to the Parallel-resistor model [27,29] developed by Wiesmann et al. [30] In the model, the R is described as the following equation: where Rsat is the saturated R at high temperatures.Fisk and Webb found that, at high temperatures, the R of strong-coupled superconducting transition-metal compounds such as Nb3Sn and Nb3Sb saturates at a certain value that corresponds to an electron mean free path of the order of the interatomic spacing in the compound. [31]Rideal is composed of residual electrical resistance R0 and phonon-assisted s-d electron scattering term as the following expression: where C is the numerical temperature-independent coefficient.We used the R data at 10 K < T < 300 K for fitting to the Parallel-resistor model and evaluated the ΘD values as a function of applied pressure as shown in Figure 5(d).The calculated ΘD is 292 K at ambient pressure (P = 0.03 GPa), which slightly deviates from the value ΘD = 260 K obtained by another experimental result using specific heat measurement. [32]As mentioned above, we observed the dome-shaped pressure dependence of Tc (Figure 5(c)).
In conventional weak-coupling electron-phonon (BCS) superconductors, Tc can be expressed by the following equation: [33] where N(0) is the electronic density of states at the Fermi energy, and U is the effective Coulomb interaction constant.According to Equation ( 6), the Tc is mostly controlled by ΘD and N(0).The ΘD gradually increases upon applying pressure (Figure 5(d)), contributing to an enhancement of Tc.In contrast, N(0) usually decreases under high-pressure conditions because of the expansion of bandwidth, [34] contributing to the suppression of Tc.Therefore, the competition of the contributions of ΘD and N(0) to pairing would cause the dome-shaped pressure dependence of Tc.The dome-shaped pressure dependence of Tc is observed in other superconductors such as CaSb2, [35] CeV3Sb5, [36] AuTe2, [37] and Cd2Re2O7 [38] .The possible causes of causing discontinuation or dome shaped pressure dependence of Tc are proposed as crystal structural transition or Lifshitz transitions.
ΘD is related to the elastic properties of the material, especially the stiffness. [39]As well known, diamond or crystals with a diamond-type structure whose large ΘD exhibits a small linear thermal expansion coefficient. [40,41]Therefore, the trend of ΘD against applied pressure as shown in Figure 5(d) is consistent with the decrease of αa under pressure.The pressure effect on αc would be negligible because of the Co-Co soft bond like the Fe-Fe soft bond seen in FeZr2. [22]

Electrical structure
Here, we show the total electronic density of states (DOS) and partial DOS in Figure 6.The

Sample preparation:
A polycrystalline sample of CoZr2 was prepared using a Co rod (99.98 %, Nilaco) and Zr plates (99.2 %, Nilaco) by the Arc melting method.The sample chamber was filled with Ar gas after three times gas replacements.We synthesized the sample on a water-cooled Cu stage and turned it over several times when each melting step for homogenization.

A. Thermal expansion of CoZr2 at ambient pressure
We measured High-temperature X-ray diffraction (XRD) patterns at ambient pressure as shown in Figure S1.The usual volumetric positive thermal expansion was observed.
μK -1 , respectively.Therefore, we obtain β = 24 ± 1 μK -1 (see FigureS1in the Supporting information).As discussed later, under high pressures, we find that the uniaxial PTE along the a-axis is suppressed, but uniaxial NTE along the c-axis is not suppressed; as a consequence, β could be a negative value, which means volumetric NTE is realized in CoZr2 under pressures (Figure1(c)).

Figure 1 .
Figure 1.Schematic images of thermal expansion (a) NiZr2 under ambient pressure, (b) CoZr2 under ambient pressure, and (c) CoZr2 under high pressure.

Figure 2 .
Figure 2. (a) HP-SXRD patterns at T = 303 K for CoZr2.(b,c) A shift of 002 and 220 peaks due to pressure.Pressure dependence of (d) lattice constants a and c, (e) a and c normalized by a value at ambient pressure, (f) lattice volume V.The solid line is fit to the 3rd-order Birch-Murnaghan formula.

Figure 3 .
Figure 3. (a) HP-SXRD patterns at P = 2.9 GPa for CoZr2.(b,c) A shift of 002 and 220 peaks due to heating.Temperature dependence of lattice constants (d) a, (e) c.The solid line is fit to the linear line.(f) lattice volume V.
(b)).Above the ambient pressure, the β value can be negative because the impact of uniaxial NTE along the c-axis on the volume exceeds the suppressed uniaxial PTE along the a-axis, resulting in volumetric NTE.The results of Rietveld refinement of CoZr2 under P = 2.9 GPa (303 K, 453 K) and P = 10.4GPa (303K, 403 K) are shown in FigureS4(see Supporting information).

Figure 4 .
Figure 4. Pressure dependence of the coefficient of linear thermal expansion (a) along the aaxis αa, (b) along the c-axis αc.(c) Pressure dependence of the coefficient of volumetric thermal expansion β.In the tetragonal crystal structure, β can be calculated to β = 2αa+αc.

Figure 5 .
Figure 5. (a,b) Temperature dependence of electrical resistance (R) under applied pressures.The solid lines are fit to the Parallel-resistor model.(c,d) Pressure dependence of superconducting transition temperature Tc and Debye temperature ΘD.
Figures 6(b) and 6(c) at P = 0 and 10 GPa, respectively.The total DOS near the EF mainly consists of Co-3d and Zr-4d orbitals.Figures 6(d) and 6(e) show the details of Co-3d partial DOS at P = 0 and 10 GPa, respectively.The most of the partial Co-3d DOS are located below the EF at both P = 0 and 10 GPa.However, the DOS for the Co-3dz 2 orbital presents above the EF, which is an unoccupied state.

Figure 7 (
Figure 7(a) shows the charge density isosurface of CoZr2 at P = 0, 3, 5, and 10 GPa.As pressure is applied, the charge density isosurface increases between Co-Co and Co-Zr bonds, resulting in a decrease in the bond lengths.The Co-Co and Co-Zr bond lengths are reversed at P = 10 GPa (Figure 7(b)).Figure 7(c) shows the pressure dependence of bond lengths normalized by a value at ambient pressure.The normalized Co-Co bond length becomes shorter than the normalized Co-Zr bond length under pressures, suggesting that CoZr2 flexibly contract along the c-axis as compared to the a-axis as shown in Figure 2(e).The change in the bonding state with increasing pressure would affect to the suppression of uniaxial PTE along the a-axis, which leads volumetric NTE.

Figure 7 .
Figure 7. (a) Charge density isosurface of CoZr2 at P = 0, 3, 5, and 10 GPa.The bond length comparison between Co-Zr and Co-Co is shown at each applied pressure.(b) Pressure dependence of Co-Co and Co-Zr bond lengths.(c) Bond lengths normalized by a value at ambient pressure.

Figure S1 .
Figure S1.(a) High-temperature XRD patterns at P = 0 GPa (ambient pressure) for CoZr2.(b,c) Shifts of 002 and 220 peaks due to heating.(d,e) Temperature dependence of lattice constants (d) a and (e) c.The solid line is the fit to the linear line.(f) Temperature dependence of lattice volume V.

Figure S2 .
Figure S2.Pressure dependences of (a) lattice constants a and c and (b) lattice constants normalized by the ambient pressure.These data were collected on another sample (different DAC from that shown in Fig. 2 in the main text) with a laboratory XRD.

Figure S6 .
Figure S6.(a) Temperature dependence of R near the superconducting transition.The solid line is fit to the Power-law relation (b) Pressure dependence of calculated exponent n.