Discovery of Real‐Space Topological Ferroelectricity in Metallic Transition Metal Phosphides

Ferroelectric metals—with coexisting ferroelectricity and structural asymmetry—challenge traditional perceptions because free electrons screen electrostatic forces between ions, the driving force of breaking the spatial inversion symmetry. Despite ferroelectric metals having been unveiled one after another, topologically switchable polar objects with metallicity have never been identified so far. Here, the discovery of real‐space topological ferroelectricity in metallic and non‐centrosymmetric Ni2P is reported. Protected by the rotation–inversion symmetry operation, it is found that the balanced polarity of alternately stacked polyhedra couples intimately with elemental valence states, which are verified using quantitative electron energy‐loss spectroscopy. First‐principles calculations reveal that an applied in‐plane compressive strain creates a tunable bilinear double‐well potential and reverses the polyhedral polarity on a unit‐cell scale. The dual roles of nickel cations, including polar displacement inside polyhedral cages and a 3D bonding network, facilitate the coexistence of topological polarity with metallicity. In addition, the switchable in‐plane polyhedral polarity gives rise to a spin–orbit‐coupling‐induced spin texture with large momentum‐dependent spin splitting. These findings point out a new direction for exploring valence–polarity–spin correlative interactions via topological ferroelectricity in metallic systems with structural asymmetry.

symmetry-confined polarity analysis, electron energy-loss spectroscopy (EELS) measurement and first principles calculations reveal that asymmetric Ni 2 P is a topological ferroelectric metal. Its switchable polyhedral polarity and coupling to degeneracylifting spin polarization present fascinating interactions in such asymmetric metallic materials.
Hexagonal Ni 2 P, with lattice parameters of a ≈ 0.5859 nm and c ≈ 0.3382 nm, is comprised of Ni1-centered tetrahedra (Te) and Ni2-centered square-pyramids (Py) along the c axis (Figure 1a,b). In the presence of 6 rotation-inversion symmetry, only in-plane (IP) polarity is allowed to develop along the <100> directions, as evidenced by polar distortions of the polyhedra (Table S1, Supporting Information). For the sake of clarity, we re-shape the primitive unit cells (PUCs-dashed lines) into equivalently transformed unit cells (ETUCs-thick solid lines) with hexasymmetry ( Figure 1b). The polyhedral polarity is then seen to form head-to-head and tail-to-tail charged configurations at P2-centered axes and ETUC boundaries within the unit cell, respectively. Studies on ordinary ferroelectrics have shown that charged domains with unequal polarization could destabilize the unit-cell-scale wall structures. [19] This indicates that a balance of tetrahedral-pyramidal polarity is a prerequisite for stabilization of the Ni 2 P phase structure.
By taking into account the atomic sites, polyhedral coordination and elemental valence states (set as variables), we establish a series of symmetry equations to balance the polyhedral polarity of Ni 2 P along the IP and out-of-plane (OP) directions. The bond dipole moment (D), [20] which is sensitive to bond length, elemental electronegativity difference, and coordination atom number, was first calculated to describe the polarity of the polyhedra in Ni 2 P.
where D Te and D Py are the dipole moments of the polyhedra, x 1 (= y 1 ) and x 2 (= y 2 ) are the fractional coordinates of the Ni1 and Ni2 atoms, P atoms are located at fixed positions (see Table  S1, Supporting Information) and the variable δ i (i = Ni or P) is the valence state of Ni or P, respectively. Along the z axis, we noticed that m j = …, -1, 0, 1, 2, 3, … for the pyramids and n j = …, -1, 1, 3, 5, …for the tetrahedra with respect to locations of the polyhedra ( Figure S1a, Supporting Information). In the basal plane, a balance of polyhedral polarity yields the expression For a tetrahedron that is sandwiched by a pair of pyramids along the z axis, we have for a pyramid sandwiched by a pair of tetrahedra. We notice that the relation m j-1 + m j+1 = n j always holds. Together with the electroneutrality of the unit cell, we obtain analytic solutions to the equations. It is found that the valence states of Ni (δ Ni ) and P (δ P ) and separation of positive-negative charge centers for the polyhedra are linearly dependent on the fractional coordinates of Ni atoms in the ab plane (see Experimental Section). For the experimentally determined DataSet-1 (Ni1: x 1 = y 1 = 0.7425; Ni2: x 2 = y 2 = 0.4043), [16] we obtain δ Ni1 = −0.427δ P1 , δ Ni2 = −0.547 δ P1 , and δ P2 = 0.923 δ P1 and separation of charge centers at d c-Te = −1.45 pm and d c−Py = 1.14 pm. For a mean nickel valence of <δ Ni > = 0.5, the Ni and P valences are δ Ni1 = 0.439, δ Ni2 = 0.561, δ P1 = −1.026, and δ P2 = −0.947. The polyhedral polarity per unit volume (V 0 is volume of the unit cell), with P S = 6D/V 0 = −0.61 μC cm −2 , forms a center-convergent geometry within the ETUC ( Figure 1c). As the Ni coordinates change to x 1 = y 1 = 0.7432 for Ni1 and x 2 = y 2 = 0.4054 for Ni2 (DataSet-2), [21] with the Ni1 and Ni2 atoms individually shifted by ≈0.42 pm and ≈0.66 pm, the polyhedral polarity is reversed and the charge centers are separated by d c-Te = 3.75 pm and d c-Py = -3.00 pm, respectively. Correspondingly, the polyhedral polarity with P S = 1.59 μC cm −2 at <δ Ni > = 0.5 forms a center-divergent geometry. With increasing elemental valence state, the polyhedral polarity increases monotonically ( Figure 1c). According to the theory of topological defects, [15] the IP center-convergent and center-divergent electric dipoles therefore form topological configurations in the order-parameter space of polarity. Given the homotopic relation of dipole counter maps per unit cell along the OP direction, this gives rise to a winding order  ∫ π θ = n d 1 2 = 1 (where dθ is the angular change between neighboring polar vectors) for both cases and a globally null polarization in the crystal ( Figure S1b-g, Supporting Information). It is intriguing to find out the intimate polarity-valence correlation in non-centrosymmetric Ni 2 P. In order to further affirm validity of the symmetry equations in this system, we measured the elemental valence states experimentally using EELS, which records information about the unoccupied local density of states (DOS) of excited atoms. Compositional analysis (Ni:P ≈ 2:1) by energy-dispersive X-ray spectroscopy (EDX) and atomicresolution high-angle annular dark-field scanning transmission electron microscopy confirmed the identity of the structural phase (Figure 1d,e; Figure S2, Supporting Information). With respect to the elemental Ni standard, a positive chemical shift by 1.3 eV is observed for the Ni L 2,3 core edges (corresponding to a 2p → 3d dipole transition), [22] indicating the oxidation of Ni in the phosphide (Figure 2a). The empirical white-line intensity method, which is weakly dependent on local coordination geometry and bonding type, [23,24] was employed to probe the 3d electron occupancy (n 3d ) of Ni (see Experimental Section).
After subtraction of a stepped background and an oxidized surface signal (marked by asterisks in Figure 2a,b), our measurement provides a value of n 3d = 8.49 for Ni in Ni 2 P. Considering the hybridization of the d-band with the conduction band [25] and all possible 3d 10-q 4s q configurations, we obtain a range of 0.31 ≤ δ Ni = 10−q−n 3d ≤ 1.51 for the Ni valence in the phosphide (inset in Figure 2a; Figure S3a,b, Supporting Information). Referring to the Ni 0 standard (3d 9 4s 1 ), the valence of Ni is determined to be δ Ni = 0.51 ± 0.09. Atomic-resolution EELS experiments reveal that the Ni valences are δ Ni1 = 0.41 ± 0.10 and δ Ni2 = 0.61 ± 0.10 ( Figure 2b; Figure S4, Supporting Information). These values, reflecting the coordination discrepancy of Ni1 and Ni2 in the compound, [26] are in good agreement with solutions to the polarity symmetry equations.
Analogously, abrupt P K edges (peak A: 1s → 3p, peak B: 1s → P-3p/Ni-4p dipole transition [27] ) were collected in our experiments for valence determination (Figure 2c; Figure S3c,d, Supporting Information). Defining the intersection of the profile slope with the energy axis as the onset energy, we see that the P K edge in the phosphide shifts negatively by 1.2 eV compared with the P 0 standard ( Figure 2c). This is in sharp contrast to the phosphate (P 5+ ) standard, which shifts positively by 8.1 eV. The proximity of the P K edge onset to the P 0 standard suggests that P is weakly charged in the phosphide. In order to quantify the P valence state, we analyzed X-ray photoelectron spectroscopy (XPS) data for a wide range of P-containing compounds, [26,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] including phosphates, phosphites, and phosphides (see Table S2, Supporting Information). We find that the P-2p 3/2 binding energy (E B ) and the P valence (δ P ) form a bowknot-shaped chart with the node locating at the position of the P 0 standards (Figure 2d). By deducting the constant energy differences, for example, the 1s → 2p 3/2 energy level gap (see Experimental Section), the XPS relation chart becomes applicable to the EELS K edges and the P valence in Ni 2 P is determined to be in the range −3.0 ≤ δ P ≤ −0.62, which is in good agreement with the values derived from the polarity symmetry equations.
On the basis of the experimental validation, first-principles calculations were performed to explore the switching mechanism of polyhedral polarization. In a fully relaxed condition, that is, a = a 0 , the center-convergent topology has the most stable state and the polyhedral polarity is P S = 8.7 μC cm −2 at a fixed value of <δ Ni > = 0.5 (see Experimental Section). In the presence of isotropic IP compressive strain, we find that the Ni1 and Ni2 atomic coordinates in Ni 2 P linearly couple with each other, both of which increase monotonically with an increase of the compressive strain (Figure 3a; Figure S5a, Supporting Information). As the shifts of Ni1 and Ni2 atoms individually exceed 2.0 and 1.6 pm, the polyhedral polarization is reversed after exceeding a critical value of a = 0.98a 0 (Figure 3b). Correspondingly, the convergent-type topological configuration changes to divergent-type when a < 0.98a 0 , for example, P S = 14.8 μC cm −2 at a = 0.95a 0 . By monitoring local charges in the atomic spheres and considering the Ni valences determined from the experiments, our first principles calculation also reproduces the reversal of polyhedral polarity under the straining conditions ( Figure S5b, Supporting Information). Meanwhile, we found that the shift of Wannier functions as a function of a/a 0 is consistent with the atomic shifts.
www.advmat.de www.advancedsciencenews.com By varying the Ni1 atomic site step by step and relaxing the Ni2 atoms, or vice versa, the system energy is observed to change in a parabolic manner. As a function of the Ni1 and Ni2 fractional coordinates, we see that the applied compressive strain creates a bilinear double-well potential together with the fully relaxed state (a = a 0 ) (Figure 3c,d). Specifically, as the applied strain exceeds the critical value of ε = (a-a 0 )/a 0 = −2%, the tunable bilinear double-well potential corresponds to the reversible polarization states of the polyhedra, resembling that in ordinary ferroelectrics such as Hf 0.5 Zr 0.5 O 2 . [13] In the presence of compressive strain, our calculation also evidences a dramatic change of the 3D Fermi surfaces resulting from volume contraction and associated polarity reversal of the polyhedra (Figure 3e,f). The calculated DOS indicates that the metallic conductivity is dominated by NiNi bonding from the neighboring polyhedra, which applies to the compressively strained situation as well ( Figures S6 and S7, Supporting Information).
In ordinary ferroelectric semiconductors, it is known that inversion-symmetry breaking is intimately correlated with Rashba physics. [45] To further corroborate the topological polarization presented above, spin-orbit interaction was considered in our calculations. As a result of the absence of structural inversion symmetry in Ni 2 P, a complicated momentum-dependent lifting of spin degeneracy of the electronic states is established. Specifically, some electronic states with finite k z and k || , sensing the switchable local IP polarization at specific straining conditions, are presented for comparison (Figure 4a-c). On the one hand, we find that the spin-orbit coupling induces spin splitting of bands in the range of 20 to 100 meV, reaching the order of magnitude of Rashba-type spin splitting of the Au (111) surface state. [46] On the other hand, from the z component of spin integrated over all atoms, we clearly see changes of the spin polarization with reversal of the polyhedral polarity, for example, the states at −0.8 eV near the zone boundary. Even when the local polarization is compensated, that is, at a = 0.98a 0 , the spin splitting remains finite. This behavior is analogous to the "hidden spin polarization" observed in an inversion-symmetric crystal, which allows for these effects despite complete compensation of the lattice polarization. [47] Our results show that apart from semiconductors, the Rashba effect with substantial spin splitting is also observable in metals such as Ni 2 P. This modifies the Fermi surface states (Figure 4e,f) and may consequently couple to the transport properties. directions. c) P K edges in Ni 2 P and in P 0 and LiFePO 4 standards, with onsets defined by the intersections of slopes (gray solid lines) with the energy axis (gray dashed lines). d) Bowknot-shaped binding energy (E B ) plotted as a function of P valence (δ P ) from XPS P-2p 3/2 data of P-containing compounds (see Table S2, Supporting Information). Replication of the E B -δ P relation to the P K edges referring to the P 0 standards quantifies the δ P range (solid green line) in Ni 2 P. The green circle marks δ P = −1.02.

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The presence of switchable local electric polarization and the similarity to a double-well potential (Figure 3c,d) indicate that the asymmetric Ni 2 P shows an analogy to the ordinary ferroelectric semiconductors. [13] Nevertheless, the coexistence of topological polarity with metallicity distinguishes the nonpolar Ni 2 P from the ordinary ferroelectrics, which are insulators and undergo phase transitions. Its unique structural, electrical, and spin-splitting properties therefore lead us to define a new material category-topological ferroelectric metal. It is noteworthy that the rotation-inversion symmetry plays an important role in enabling the coexistence of structural asymmetry with metallicity. [48] On the one hand, the unique stacking of acentric polyhedra, allowing IP polar displacements of Ni cations, constructs the unit-cell-scale short-range topological polar order in real space, whose long-range correlation is implemented by lattice translational symmetry. On the other hand, the Ni cations in the polyhedral cages form a 3D bonding network through the NiNi bonds, which are responsible for the bulk conductivity (see Figures S6 and S7, Supporting Information). As a result of the dual roles of the Ni cations, the conflicting constraints to electron behavior in insulators are lifted in such systems.

Experimental Section
Sample Synthesis and Specimen Preparation: The Ni 2 P nanoparticles on carbon paper and nanosheets on Ni foam were prepared by gastransport phosphorization procedures under inert gas atmospheres. More synthetic details, X-ray diffraction and relevant characterization were presented elsewhere. [53][54][55] Scanning electron microscopy (JEOL, JSM-7400F) and STEM were used for morphological characterization. For TEM observations, a piece of Ni 2 P/carbon paper was crushed using an agate mortar in ultrapure water. The suspension solution was dropped onto a Cu grid, dried on a plate oven (≈60 °C) and then cooled to room temperature. The nanosheets on Ni foam were first scraped using a blade. Then the above procedure was applied. The elemental Ni standard was prepared using a focused ion beam system (FEI Helios Nanolab 400s).

Structural, Electrical Characterization, and EELS, EDX Experiments:
In these experiments, the phase structures of the nickel phosphides were usually examined using EDX and selected area electron diffraction before performing EELS experiments. The Dr. Probe software package was used  = a 0 ), respectively. The boundary corresponds to reversal of the polyhedral polarization at a = 0.98 a 0 . c) Ground state energy change obtained by fixing the Ni1 atomic sites and relaxing the Ni2 sites at a = a 0 and a = 0.95a 0 , respectively. d) Ground state energy change obtained by fixing the Ni2 atomic sites and relaxing the Ni1 sites at a = a 0 and a = 0.95a 0 , respectively. The insets illustrate the change in topological polarity (pyramids-blue; tetrahedra-green) corresponding to the tunable double-well potential (yellow dotted lines). e,f) 3D plots of the Fermi surface (without spin-orbit coupling) for a = a 0 and a = 0.95a 0 , respectively. The three sheets are annotated by green/magenta, cyan/red, and blue/yellow in terms of increasing order of the band index.
www.advmat.de www.advancedsciencenews.com for image simulation, [56] and CrystalMaker and VESTA software packages were used for drawing the crystal structures. Electrical transport measurements showed that the Ni 2 P exhibits metallic conductivity with a resistivity of ≈30 μΩ cm at room temperature. [57] An FEI Titan 80-200 ChemiSTEM microscope, which was equipped with a Super-X energy-dispersive X-ray spectrometer, STEM annular detectors and a Gatan Enfinium ER (model 977) spectrometer with DUAL EELS option (allowing calibration of core-loss spectra by low-loss spectra), was used to collect EDX and EELS results. The EEL spectra were collected using a convergence semi-angle of 24.7 mrad and a collection semi-angle of 29 mrad at a dispersion of 0.25 eV per pixel. The energy resolution was 1.25 eV based on the full width at half maximum of the zero-loss peak. As a result of surface oxidation, shoulder peaks beside the Ni L 2,3 edges were recorded from the phosphide. [58] Valence State Quantification: The white line intensity method is used to determine the 3d electron occupation and oxidation state of Ni. Apart from power law background subtraction, the Ni L 2,3 spectra are deconvoluted using the Fourier-ratio method [59] to remove multiple scattering components when necessary, that is, when the specimen thickness t/λ p > 0.35 (where the mean free path λ p is ≈100 nm). After the removal of a continuum background modeled by the Hartree-Slater step function with a step height ratio of 3.3:1 at the peaks, [60] the white line intensities of the L 3 and L 2 edges are integrated and normalized by a 50 eV continuum energy window (starting 50 eV above the L 3 edge onset), defined as I 3d . As a calibration, the quality of the empirical formula 1.06(1 0.094 ) 3d 3d for obtaining the 3d electron occupancy is examined for the Ni 0 standard, for which the 3d 9 4s 1 configuration has the lowest energy. This measurement reveals that n 3d = 8.77 for the Ni 0 standard, which is in an excellent agreement with a previous (3d 8.8 4s 1.2 ) report. [24,25] On this basis, the n 3d in the phosphides is systematically calculated together with removal of the oxidized surface signal.
Winding Number: Analogous to the skyrmions, [61,62] by mapping the polyhedral polarity of an ETUC into a circular order-parameter space, [6,15,63] summation of the total angle changes gives rise to the winding number n d  ∫ π θ = 1 2 = 1 for both convergent-type and divergent-type topological configurations of Ni 2 P. This holds true for two  .
where a is the lattice constant, and d c-Te versus d c-Py was irrelevant to the valence states. Taking into account the volume space occupied by the polyhedra, that is, V 0 /6 where V 0 is the volume of the PUC or ETUC (see Figure S1, Supporting Information), the bond dipole moment could be converted to polarization, that is, |P S | = |6D/V 0 |. For clarity, changes in the quantities as a function of Ni1 and Ni2 fractional coordinates are shown in Figure S8, Supporting Information. For DataSet-2, [21] the valence relations were δ Ni1 = −0.444δ P1 , δ Ni2 = −0.555 δ P1 , and δ P2 = 0.997 δ P1 . At <δ Ni > = 0.5, δ Ni1 = 0.444, δ Ni2 = 0.556, δ P1 = −1.001, and δ P2 = −0.998. Referring to the Ni 0 -3d L (L = 8.8) [25] configuration, that is, <δ Ni > = 0.31, the upper boundary solution to the equations yielded <δ Ni1 > = 0.274, <δ Ni2 > = 0.346, <δ P1 > = −0.640, and <δ P2 > = −0.591 for the two datasets. Referring to the 3d 10 configuration, that is, <δ Ni > = 1.50, the lower boundary solution yielded <δ Ni1 > = 1.324, <δ Ni2 > = 1.676, <δ P1 > = −3.041, and <δ P2 > = −2.918. Considering the 3Ni1-2P1 and 3Ni2-1P2 constituent per unit cell, the solutions to the symmetry equations of polarity gave the valence range of −3.0 ≤ δ P ≤ −0.616. First Principles Calculations: The theoretical work was performed based on first principles density functional theory implemented in the Vienna ab initio Simulation Package (VASP) [64] and the FLEUR code. [65] Where indicated, the spin-orbit coupling was considered in a self-consistent manner. The exchange-correlation effect was treated with the Perdew, Burke, and Ernzerhof functional. [66] An energy cutoff of 400 eV, with the energy convergence criterion set at 10 −6 eV, was used in the VASP calculations. A Γ-centered k-mesh of 10 × 10 × 18 was employed to converge the self-consistent field calculations of Ni 2 P. For metallic Ni 2 P, the Z 2 topological invariants [67] cannot be calculated, indicating that there is an essential difference between topological spin and topological polarity in the band structures. From the ab initio calculations, it was difficult to obtain local polarizations, but the trends can be followed by monitoring local charges in atomic spheres and using the polarity equations, the charge neutrality condition, Equation (8), and the experimentallyobserved Ni valence as calibrating conditions. Specifically, the polarity (D) of a bond formed by a pair of atoms (A, B) at positions (x A , x B ) and with local charges (δ Α , δ Β ) was expressed by D = (x A − x B )(δ Α − δ Β ) so that 4 1/2 /9 /9 Assuming that the Ni valence was given by the deviation of charge inside the atomic sphere from a nominal value δ Ni = α(9−q Ni ) and similarly δ P = β(θ − q P ), the constants α, β, and θ could be determined from D Py = −D Te , Equation (8) and δ Ni1 +δ Ni2 = 1.0 (see Figure S5, Supporting Information). It was seen that at the compensation point (a/a 0 = 0.982) the relations between the valences given in the main text were almost perfectly fulfilled.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.