# Dimensionality of the Superconductivity in the Transition Metal Pnictide WP

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{8}

^{9}

^{*}

## Abstract

**:**

## 1. Introduction

_{6}octahedra [8], shown in Figure 1. Four of the six bonds are inequivalent due to the space group anisotropy [8]. In this compound, the spatial extension of the W-5d orbitals induces a large overlap and a strong coupling with the neighboring p-orbitals, resulting in a distortion of the crystal structure more pronounced compared to that of CrAs and MnP [8,9]. Moreover, the spin-orbit coupling of W-5d electrons is stronger than that of 3d electrons of CrAs and MnP [9]. In particular, the 3d materials display strong electron correlations, narrow bandwidths, and robust magnetism. On the other hand, the 5d materials exhibits increased hybridization, more diffuse orbitals, and a strong spin-orbit coupling competing with magnetic, crystal-field, many-body Coulomb, and other interactions leading to novel and exotic behaviors [10,11]. Moreover, the relativistic shifts in orbital energies, combined with spin-orbit and bandwidth effects, drive band inversions leading to topological phases and enhanced Rashba splittings.

## 2. Sample Preparation Method and Experimental Details

^{3}. Figure 2a shows the powder X-ray diffraction (XRD) data on WP at room temperature. The data show that WP crystallizes in a MnP-type orthorhombic structure (space group Pnma, No. 62) with lattice parameters $a=0.57222\left(6\right)\mathrm{nm}$, $b=0.32434\left(9\right)\mathrm{nm}$, and $c=0.62110\left(6\right)\mathrm{nm}$. The b-axis direction is parallel to the longest direction of the sample. The energy-dispersive X-ray spectroscopy (EDX) was performed to check the chemical composition of the grown single crystals. Figure 2b shows the typical EDX spectrum of an individual crystal. Only two elements, W and P, are detected. The average ratio of the elements at different locations in the crystals is 50.9:49.1, which is close to the 1:1 stoichiometry of the compound. Further details on the fabrication procedure and the structural, compositional, and transport characterizations are reported elsewhere [8]. The electrical resistance measurements below 2 K were performed by the standard four-probe technique in a top-loading Helium-3 refrigerator with a superconducting magnet with fields up to 15 T.

## 3. Theoretical and Experimental Results

#### 3.1. Theoretical Calculations

#### 3.2. Experimental Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The orthorhombic crystal structure of tungsten phosphide WP with space group Pnma. Orange and blue spheres indicate W and P ions, respectively, with nonequivalent lattice positions of the W ions labeled as ${\mathrm{W}}_{A}$, ${\mathrm{W}}_{B}$, ${\mathrm{W}}_{C}$, ${\mathrm{W}}_{D}$. Face-sharing WP${}_{6}$ octahedra are shaded in gray.

**Figure 2.**(

**a**) Room-temperature X-ray diffraction patterns and the Rietveld refinement of WP. Open circles, solid line, and lower solid line represent experimental, calculated, and difference XRD patterns, respectively. The inset shows the SEM image for WP single crystal. (

**b**) EDX result of WP single crystal. The average ratio of the elements is close to the 1:1 stoichiometry of the compound. The inset shows the SEM image for WP single crystal (×430 magnification).

**Figure 3.**Fit of the DFT bands (red lines) using the tight-binding model (blue lines) along the high-symmetry path of the orthorhombic Brillouin zone. The Fermi level is at zero energy.

**Figure 4.**Partial density of states relative to the W-5d states (continuous line) and P-3p states (dotted line). The d states are predominant close to the Fermi level, while the p states are far from the Fermi level, which is set at zero energy.

**Figure 5.**Fermi surface of WP in the first Brillouin zone with spin-orbit coupling. In panels (

**a**–

**d**) we show the contributions of the four different bands that cut the Fermi level. The color code denotes the Fermi velocity.

**Figure 6.**Angular dependence of the superconducting upper critical field ${H}_{c2}$ at 0.3 K. Black circles and green squares represent the measured transition fields (defined by the 90% criterion) with positive and negative field polarity, respectively. The lines are the theoretical fits to the experimental data for the angular dependence of the critical field. Red lines represent the theoretical dependencies according to the Ginzburg-Landau model for a superconductor with 3D character and anisotropic effective mass, blue lines to the Tinkham model for a superconductor with 2D character.

**Figure 7.**The normalized excess conductivity plotted as a function of the temperature in a magnetic field ${\mu}_{0}H$ ranging from 1.5 to 10.5 mT.

**Figure 8.**The critical temperature as a function of the applied magnetic field ${\mu}_{0}H$. Error bars correspond to the transition width $\mathsf{\Delta}{T}_{c}$ evaluated from the 10–90% criterion. The inset refers to the transition width $\mathsf{\Delta}{T}_{c}$ versus ${\mu}_{0}H$. The solid red line is the linear best fit to the experimental data.

**Figure 9.**Scaling plots of $(\mathsf{\Delta}\sigma /{\sigma}_{n})\left({H}^{1/3}{T}^{-2/3}\right)$ as a function of $(T-{T}_{c}\left(H\right))/{\left(TH\right)}^{2/3}$ for the transition curves in Figure 7 at magnetic fields $\ge 4\mathrm{mT}$, for the 3D Ullah-Dorsey model of the paraconductivity described by Equation (7).

Parameters | Values |
---|---|

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{100}$ | $-0.284$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{200}$ | $0.009$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{300}$ | $-0.004$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{010}$ | $-0.195$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{020}$ | $0.081$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{030}$ | $-0.053$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{001}$ | $0.049$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{002}$ | $-0.032$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{A}}^{003}$ | $0.004$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{B}}^{100}$ | $0.082$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{C}}^{001}$ | $0.001$ |

${t}_{{\mathrm{W}}_{A}{\mathrm{W}}_{C}}^{001}$ | $0.319$ |

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**MDPI and ACS Style**

Nigro, A.; Cuono, G.; Marra, P.; Leo, A.; Grimaldi, G.; Liu, Z.; Mi, Z.; Wu, W.; Liu, G.; Autieri, C.;
et al. Dimensionality of the Superconductivity in the Transition Metal Pnictide WP. *Materials* **2022**, *15*, 1027.
https://doi.org/10.3390/ma15031027

**AMA Style**

Nigro A, Cuono G, Marra P, Leo A, Grimaldi G, Liu Z, Mi Z, Wu W, Liu G, Autieri C,
et al. Dimensionality of the Superconductivity in the Transition Metal Pnictide WP. *Materials*. 2022; 15(3):1027.
https://doi.org/10.3390/ma15031027

**Chicago/Turabian Style**

Nigro, Angela, Giuseppe Cuono, Pasquale Marra, Antonio Leo, Gaia Grimaldi, Ziyi Liu, Zhenyu Mi, Wei Wu, Guangtong Liu, Carmine Autieri,
and et al. 2022. "Dimensionality of the Superconductivity in the Transition Metal Pnictide WP" *Materials* 15, no. 3: 1027.
https://doi.org/10.3390/ma15031027