# Pressure-Tuning Superconductivity in Noncentrosymmetric Topological Materials ZrRuAs

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## Abstract

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_{c}~ 7.74 K, while a large upper critical field ~ 13.03 T is obtained for ZrRuAs, which is comparable to the weak-coupling Pauli limit. The resistivity of ZrRuAs exhibits a non-monotonic evolution with increasing pressure. The superconducting transition temperature T

_{c}increases with applied pressure and reaches a maximum value of 7.93 K at 2.1 GPa, followed by a decrease. The nontrivial topology is robust and persists up to the high-pressure regime. Considering both robust superconductivity and intriguing topology in this material, our results could contribute to studies of the interplay between topological electronic states and superconductivity.

## 1. Introduction

_{2}P-type hexagonal structure (h-phase), and the MgZn

_{2}-type hexagonal structure (h’-phase). Both ZrRuAs and HfRuP belong to Fe

_{2}P-type structures and show intrinsic noncentrosymmetric superconductivity with relatively high transition temperatures. Recently the first-principles calculations predicted that both ZrRuAs and HfRuP host nontrivial bulk topology [8]. When ignoring the spin-orbit coupling (SOC), ZrRuAs/HfRuP possess two nodal rings slightly above the Fermi energy (E

_{F}) in the k

_{z}= 0 planes. However, when considering the SOC, they enter either a Weyl semimetal (WSM) phase (e.g., HfRuP) due to the lack of inversion symmetry or a topological crystalline insulating (TCI) phase (e.g., ZrRuAs) with trivial Fu-Kane Z

_{2}indices but nontrivial mirror Chern numbers. Combined with noncentrosymmetric superconductivity, the nontrivial topology of the normal state in ZrRuAs/HfRuP may generate unconventional superconductivity in both the bulk and surfaces.

_{c}increases with applied pressure and reaches a maximum value of 7.93 K at 2.1 GPa, then decreases with a dome-like behavior. We find that the application of pressure does not qualitatively change the electronic and topological nature of the two systems until the high-pressure regime, based on our ab initio band structure calculations.

## 2. Experimental Detail

^{−5}eV were adopted for relaxations of structures and calculations of electronic band structures. The spin-orbital coupling was considered self-consistent in this work. Wannier charge centers were calculated by Z2Pack [17] through tight binding models based on the maximally localized Wannier functions as obtained through the VASP2WANNIER90 interfaces in a non-self-consistent calculation.

## 3. Results and Discussion

_{2}P-type structure diffraction pattern is used to prove the same crystal structure.

_{xx}) of ZrRuAs are measured with the dc current along the c-axis direction. Under zero field, ZrRuAs exhibits a typical metallic behavior with residual resistivity of ρ

_{0}= 0.125 mΩ cm. At low temperatures, a sharp drop in ρ(T) to zero was observed, suggesting the onset of a superconducting transition. The inset of Figure 2a shows an enlargement of the superconducting transition. The transition temperature T

_{c}at 7.44 K in our single crystal is lower than the previous report in polycrystalline samples. The transition widths ΔT

_{c}are approximately 0.81 K, implying fairly good sample quality. The bulk superconductivity of ZrRuAs was further confirmed by the large diamagnetic signals and the saturation trend with the decreasing temperature. As shown in Figure 2b, the rapid drop of the zero-field-cooling data denotes the onset of superconductivity, consistent with the temperature of zero resistivity. The difference between zero-field-cooling and field-cooling curves indicates the typical behavior of type-II superconductors. We conducted resistivity measurements in the vicinity of T

_{c}for various external magnetic fields. As can be seen in Figure 2c, the resistivity drops shift to a lower temperature with an increasing magnetic field and keep a superconducting state even in the field of 9 T, indicating the high upper critical field (H

_{c}

_{2}). We extract the field (H) dependence of T

_{c}for ZrRuAs and plot the H(T

_{c}) in Figure 2d. We also tried to use the Ginzburg-Landau formula to fit the data [18]:

_{c2}= 13.03 T, which is comparable with the weak coupling Pauli limit value of 1.84 T

_{c}= 13.69 T. According to the relationship between H

_{c2}and the coherence length ξ, namely, H

_{c}

_{2}= Φ

_{0}/(2πξ

^{2}), where Φ

_{0}= 2.07 × 10

^{−15}Wb is the flux quantum, the derived ξ

_{GL}(0) was 5.03 nm.

_{c}

_{1}, the field-dependent magnetization M(H) of ZrRuAs was measured at various temperatures up to T

_{c}using a ZFC protocol. Some representative M(H) curves are shown in Figure 2e. For each temperature, H

_{c}

_{1}was determined as the value where M(H) deviates from linearity (dotted line) and summarized in Figure 2f. A linear fit gives μ

_{0}H

_{c}

_{1}(0) = 51.85 ± 0.6 Oe. Using the formula [18]:

_{300K}) of ZrRuAs exhibits a non-monotonic evolution with increasing pressure. Over the whole temperature range, the ρ

_{300K}is first suppressed with applied pressure and reaches a minimum value at about 17.0 GPa, then displays a moderate increase with further increasing pressure until 29.9 GPa. A similar evolution of ρ

_{300K}is observed for HfRuP in Figure 3b; the lowest ρ

_{300K}occur at 18.5 GPa and turn into an increase as the pressure increase until 32.3 GPa. In Figure 3c, it is found that the superconducting transition temperature (T

_{c}) of ZrRuAs increases from ∼7.74 K to a maximum of ~7.93 K at 2 GPa. Beyond this pressure, T

_{c}decreases slowly, showing a dome-like behavior. A similar evolution is observed for HfRuP, as shown in Figure 3d; a maximum T

_{c}of 6.50 K is attained at P = 2.8 GPa, then T

_{c}fluctuates as the pressure increase to 9.8 GPa and eventually drops rapidly until 32.3 GPa.

_{300K}and T

_{c}s with the pressure of ZrRuAs and HfRuP; the two materials show the same variation trend under exerted pressure. The high T

_{c}under various pressures of HfRuP can last longer than ZrRuAs, but once the T

_{c}breaks into a fast drop, the slopes of HfRuP and ZrRuAs are similar. The T

_{c}-Pressure phase diagram indicates that the TT’X family could have the same dome-like shape. It is worth mentioning that both ZrRuAs and HfRuP show the difference of imposed pressure between corresponding the max T

_{c}and the minimum ρ

_{c}. Although there are inevitable errors in the measurement of the absolute value of resistivity root in the thickness of the gasket, the tendency of resistivity under pressure is reliable.

_{z}= 0 plane between M-K without SOC. Further, including SOC, the nodal lines are fully gapped, leading to a continuous gap between valance and conduction bands. As explained later, the gapping of the nodal lines gives nontrivial mirror Chern numbers on k

_{z}= 0, k

_{y}= 0, and other equivalent mirror planes.

_{z}= 0, π, k

_{y}= 0, and the two equivalent ones relate to k

_{y}= 0 by ${\widehat{C}}_{3}$. On the mirror plane, mirror operators take definite eigenvalues +i, −i. Because mirror operations belong to the little group, the Hamiltonian on the mirror planes becomes block diagonal with eigenvalues of mirror operators +i, −i. With time-reversal symmetry, the Chern number in any mirror plane is always zero. However, we can define a nonzero Chern number ${C}_{+i}$, ${C}_{-i}$ separately for each subspace. In the analogy of the spin Chern number, the mirror Chern number is defined as ${C}_{M}=\left({C}_{+i}-{C}_{-i}\right)/2$ [29]. On mirror symmetric planes, time-reversal symmetry transforms eigenstates with opposite momentum in different subspaces into each other, which makes them a Kramer pair. Similar to the time-reversal invariant ${Z}_{2}$, mirror Chern numbers can be obtained by calculating the Wilson loops in half of the mirror planes. With the Wilson loop, the mirror Chern number is defined as ${C}_{M}={X}_{+i}-{X}_{-i}$, where ${X}_{\pm i}$ are the differences in numbers of Wilson bands with positive and negative slopes crossing a horizontal reference line in half of the mirror plane in the subspace with eigenvalues $\pm i$ [8].

_{z}= 0, π, and k

_{y}= 0 planes are sufficient to derive the mirror Chern numbers of all mirror symmetric planes. From the analysis above, we can directly read mirror Chern numbers at 0.3 GPa from the Wilson loops colored by eigenvalues of mirror operators in Figure 6. ${C}_{M}$ are 2 for both k

_{z}= 0 and k

_{y}= 0 planes, and it is zero for k

_{z}= π plane. The mirror Chern numbers change when we increase pressures. We list the mirror Chern numbers at different pressures in Table 1, and the corresponding Wilson loops can be found in Figure S2 of the Supplementary Materials. Table 1 shows that ZrRuAs is always a topological crystalline insulator with nontrivial mirror Chern numbers at different pressures. Therefore, we conclude that ZrRuAs remain topological in all pressures studied in our experiment.

## 4. Conclusions

_{c}increases from 7.44 K at ambient pressure and reaches a maximum value of 7.93 K at 2.1 GPa, then decreases with a dome-like behavior. The slight change of T

_{c}s of ZrRuAs indicates that the superconducting state is robust to external pressure. Through ab initio band structure calculations, we find that the application of pressure does not qualitatively change the electronic and topological nature of ZrRuAs until the high-pressure regime. Based on the observation that both topological properties and superconducting state are robust in ZrRuAs under high pressure, our research could not only intrigue studies of the interplay between topological electronic states and superconductivity but also inspire the experimental searching for possible topological superconductivity.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Crystal structure of TT’X (take ZrRuAs as an example). T and T

^{’}are represented by red and blue spheres, respectively, and the two types of X environment are indicated by the yellow and green spheres. The unit cell is shown by the parallelepiped. (

**b**) EDS and optical photograph of ZrRuAs. (

**c**) SEM images of ZrRuAs. (

**d**) Single crystal diffraction pattern of Fe

_{2}P-type structure on (010) plane.

**Figure 2.**Electrical resistivity of ZrRuAs without (

**a**) and with (

**c**) field. The inset of (

**a**) is the electrical resistivity of ZrRuAs lower than 14 K. (

**b**) Temperature dependence of the magnetic susceptibility of ZrRuAs, measured in an applied field of 10 Oe using both the ZFC and FC protocols. (

**d**) The upper critical field H

_{c}

_{2}versus the transition temperature T

_{c}for ZrRuAs. The solid red line represents a fit to the G-L formula. (

**e**) The field-dependent magnetization was recorded at various temperatures of ZrRuAs. For each temperature, H

_{c}

_{1}was determined as the value where M(H) starts deviating from linearity (see red dashed line). (

**f**) The lower critical field H

_{c1}versus the transition temperature T

_{c}for HfRuP and ZrRuAs. The dashed line represents a linear fitting.

**Figure 3.**Electrical resistivity of ZrRuAs (

**a**) and HfRuP (

**b**) as a function of temperature at various pressures. Temperature-dependent resistivity of ZrRuAs (

**c**) and HfRuP (

**d**) in the vicinity of the superconducting transition.

**Figure 5.**Electronic band structure and of ZrRuAs at 0.3 GPa without (

**a**) and with (

**b**) SOC. Green and red lines are valances and conduction bands. The inset in (

**a**) shows a crossing along M-K in the absence of SOC. The shadow area in (

**b**) indicates a continuous gap between valance and conduction bands when the SOC is turned on. (

**c**) The nodal lines are presented on the k

_{z}= 0 plane when SOC is turned off. Red lines are nodal lines. Dashed lines indicate the first Brillouin Zone.

**Figure 6.**Mirror symmetric planes and Wilson loops at 0.3 GPa. (

**a**) Orange, green, and yellow planes are mirror symmetric planes at k

_{z}= 0, π, and k

_{y}= 0, respectively. Wilson loops on the k

_{z}= 0, π, and k

_{y}= 0 are presented in (

**b**–

**d**), respectively. The horizontal dashed lines in each plot are the reference lines. Blue lines and green lines denote the flows of Wannier charge centers for states in subspaces of mirror +i and −i eigenvalues, respectively.

**Table 1.**Mirror Chern number under different pressures. The first row denotes pressures in the unit of GPa. The second and third rows are mirror Chern numbers for k

_{z}= 0 and k

_{y}= 0 planes, respectively. Mirror Chern numbers for k

_{z}= π are always 0 under pressures in the table.

Pressure (GPa) | 0.3 | 0.9 | 2.1 | 6.0 | 13.5 | 17.1 | 24.0 | 29.8 |
---|---|---|---|---|---|---|---|---|

C_{M} (k_{z} = 0) | 2 | −2 | 2 | −2 | 2 | 2 | 2 | 2 |

C_{M} (k_{y} = 0) | 2 | −2 | −2 | 2 | 2 | −2 | −2 | −2 |

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**MDPI and ACS Style**

Li, C.; Su, Y.; Zhang, C.; Pei, C.; Cao, W.; Wang, Q.; Zhao, Y.; Gao, L.; Zhu, S.; Zhang, M.;
et al. Pressure-Tuning Superconductivity in Noncentrosymmetric Topological Materials ZrRuAs. *Materials* **2022**, *15*, 7694.
https://doi.org/10.3390/ma15217694

**AMA Style**

Li C, Su Y, Zhang C, Pei C, Cao W, Wang Q, Zhao Y, Gao L, Zhu S, Zhang M,
et al. Pressure-Tuning Superconductivity in Noncentrosymmetric Topological Materials ZrRuAs. *Materials*. 2022; 15(21):7694.
https://doi.org/10.3390/ma15217694

**Chicago/Turabian Style**

Li, Changhua, Yunlong Su, Cuiwei Zhang, Cuiying Pei, Weizheng Cao, Qi Wang, Yi Zhao, Lingling Gao, Shihao Zhu, Mingxin Zhang,
and et al. 2022. "Pressure-Tuning Superconductivity in Noncentrosymmetric Topological Materials ZrRuAs" *Materials* 15, no. 21: 7694.
https://doi.org/10.3390/ma15217694