Pressure dependence of electronic structure and superconductivity of the MnX (X = N, P, As, Sb)

A recently experimental discovered (Cheng et al., Phys. Rev. Lett. 114, 117001 (2015)) of superconductivity on the border of long-range magnetic order in the itinerant-electron helimagnet MnP via the application of high pressure makes MnP the first Mn-based superconductor. In this paper, we carry out first-principles calculations on MnX (X = N, P, As, Sb) and find superconducting critical temperature TC of MnP sharply increases near the critical pressure PC ≈ 8 GPa, which is in good agreement with the experiments. Electron-phonon coupling constant λ and electronic density of states at the Fermi level N (EF) are found to increase with pressure for MnP, which lead to the increase of TC of MnP. Moreover, we also find that the TC of MnAs and MnSb are higher than MnP, implying that the MnAs and MnSb may be the more potential Mn-based superconducting materials.

Scientific RepoRts | 6:21821 | DOI: 10.1038/srep21821 be seen that the MnP at different pressure up to 10 GPa all have metallic characteristics because the DOS at Fermi level are not zero, which might favor the superconducting behavior. Mn atoms contribute more to DOS than the P atoms at Fermi level and the majority of the density of states near the Fermi level for MnP is attributed to the Mn-3d states. The P-3p bands are overlapped with the Mn-3d bands in the -10-8 eV energy range, representing a hybridization of the P-3p and Mn-3d states to form the covalent bonding. The difference of the spin-up band and spin-down band of Mn-3d orbitals show that they carry very large spin moment in MnP at different pressure. P-3p and P-3s orbitals also have small contribution to the magnetic property of MnP. Literature 6 reveal that MnP undergoes two successive magnetic transitions upon cooling in the absence of a magnetic field. One is a transition from the paramagnetic (PM) to ferromagnetic (FM) state at T C = 291 K, and then a second transition to a double helical state at T s ≈ 50 K, In the FM state, the ordered moment of the Mn spins is about 1.3μ B / Mn. Moreover, as shown in Fig. 3, the pressure up to 10 GPa have less effect on the density of states of MnP. Furthermore, the total density of states at Fermi surface (N (E F )) of MnP are summarized in Table S1 and increase with the pressure increasing.
Vibrational analysis. The calculated phonon dispersions and projected phonon densities of states (PHDOS) of MnP under different pressure are shown in Fig. 4. Absence of any imaginary frequency in the Brillouin zone confrms the dynamical stability of MnP. The modes at the high frequency region are associated with the vibrations of P atoms beating against Mn atoms. The PHDOS of this structure shows that the heavier Mn atoms dominate the low-frequency vibrations, and the lighter P atoms contribute significantly to the high-frequency modes. The phonon calculation results for other Mn-based compounds at different pressure can be seen in Figure S1.
where Θ D is the Debye temperature, λ is the electron-phonon coupling strength, μ * is the Coulomb pseudopotential. MaMillan's strong coupling theory defines an electron-phonon coupling constant (EPC) λ by 8,10,11 λ η ω ω where M is the atomic mass, I 2 is the square of the electron-ion matrix element, ω 2 1 2 is the average squared phonon frequency, N(E F ) is the total density of states at Fermi surface which can be found in Table S1. Furthermore, μ * can be obtained from the empirical relation in the following equation: In this paper, Θ D is calculated using the following expression 12 : where h and k are the Planck and Boltzmann constants, respectively. N A is Avogadro's number, n is the number of atoms in the molecule, M is the molecular weight, and ρ is the density of the crystal. v m is the mean sound velocity, which can be calculated by 13 where B and G are the bulk modulus and shear modulus, respectively. In order to calculate the B and G, firstly, the elastic constants of orthorhombic crystal (C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 and C 23 ) are calculated by applying stress tensors with various small strains onto the equilibrium structures. After obtaining elastic constants, the polycrystalline bulk modulus B and shear modulus G are calculated from the Voigt-Reuss-Hill (VRH) approximations 15 . The calculated density and mechanical modulus are tabulated in Table S2. The evaluated Debye temperature Θ D , electron-phonon coupling strength λ, average phonon frequency < ω 2 > 1/2 and superconducting critical temperature T C are exhibited in Fig. 5. The variation trend of Debye temperature Θ D and average phonon frequency < ω 2 > 1/2 are similar and the Θ D and < ω 2 > 1/2 values of MnP and MnN are larger than MnAs and MnSb. Moreover, the electron-phonon coupling strength λ and superconducting critical temperature T C has the similar variation trend and the λ and T C values of MnP and MnN are smaller than MnAs and MnSb, suggesting that the MnAs and MnSb may be the more potential Mn-based superconducting materials than MnP and MnN. The EPC parameter λ of the compounds is below 0.5, which indicate the electron-phonon interaction is fairly weak. Although the Θ D of MnAs and MnSb are lower than MnP and MnN, the larger EPC parameter λ can mainly directly contribute to higher T C of MnAs and MnSb. In consideration of the N(E f ) values in Table S1, we can infer that the weak electron phonon coupling λ and small N(E f ) are the main factors, which lead to the low T C of MnP and MnN 16 . As has been reported, the application of high pressure reduces continuously the magnetic transition temperatures and eventually suppresses the magnetic order around P C ≈ 8 GPa 7 . With the pressure increasing, the decrease of < ω 2 > 1/2 play an important role to the upward trend of the EPC parameter λ of MnP when the pressure is near 8 GPa. Meanwhile N( F) of MnP also increases under the studied pressure range. The tendency of the two parameters makes λ become higher, which lead to the increase of T C of MnP with increasing pressure. Our computational results are in accordance with the experimental observation in the framework of BCS superconductivity and the deep reason need to be further investigated.

Conclusion
In summary, the electronic structure, lattice dynamics, elastic properties and superconductivity of MnX (X = N, P, As, Sb) are investigated by means of the first-principles within the LSDA+ U method. The majority of the density of states near the Fermi level for MnP is attributed to the Mn-3d states and the total density of states at Fermi surface (N (E F )) of MnP increase with the pressure increasing. The increasing EPC parameter λ makes the superconducting critical temperature T C of itinerant helimagnet MnP become higher than 1 K when its long-range magnetic order is completely suppressed by the application of high pressure around P C ≈ 8 GPa, which is in consistent with the experimental observation and provide theoretical identification for the experimental finding that breaks the general wisdom about the Mn's antagonism to superconductivity. In addition, the T C of MnAs and MnSb are found to be higher than MnP, which indicates that the MnAs and MnSb may be the more potential Mn-based superconducting materials. This work would provide guidelines for future experimental investigations and hope that such an investigation might contribute some further understanding to the superconductivity of MnP under high pressure.

Methods
In this paper, the electronic structure calculations with high accuracy for the stable MnX (X = N, P, As, Sb) are performed using the on-the-fly generated (OTFG) pseudopotentials 17 implemented in Cambridge Serial Total Energy Package (CASTEP) code based on the density functional theory (DFT). The exchange-correlation energy is calculated using local spin-polarized density approximation (LSDA). For strong correlated systems, these functionals are unable to give the correct ground state. we have selected the LSDA+ U (U is the Hubbard energy) method in this calculation. The U values are tested and selected by experiment and theory from the references. The U value is chosen as 6 eV in this wok. The dispersion interactions correction proposed by Grimme is considered in terms of DFT+ D2 scheme in this work 18 . For different atomic species, the valence orbitals and electrons for pseudo-atoms are Mn 3d 5 4s 2 , N 2s 2 2p 3 , P 3s 2 3p 3 , As 4s 2 4p 3 and Sb 5s 2 5p 3 . The electronic wave functions are expanded in a plane-wave basis set with a cutoff energy of 800 eV and appropriate Monkhorst-Pack mesh of 4 × 6 × 8 is chosen for all compounds to ensure that enthalpy calculations are well converged to better than 1 meV/atom. In the geometrical optimization, all forces on atoms are converged to less than 0.005 eV/Å. The phonon calculations and electron-phonon coupling (EPC) calculations are carried out using the linear response theory through the Quantum ESPRESSO package 19 . The kinetic energy cutoff is set 90 Ry. And the q-point mesh of the electron-phonon interaction matrix element adopted 4 × 4 × 4.