Prediction of Coupled Electronic and Phononic Ferroelectricity in Strained 2D h-NbN: First-Principles Theoretical Analysis

Using first-principles density functional theoretical analysis, we predict coexisting ferroelectric and semimetallic states in a two-dimensional monolayer of h-NbN subjected to an electric field and in-plane strain ($\epsilon$). At strains close to $\epsilon =4.85\%$, where its out-of-plane spontaneous polarization changes sign without inverting the structure, we demonstrate a hysteretic response of its structure and polarization to an electric field, and uncover a three-state ($P=\pm P_o$, 0) switching during which h-NbN passes through Dirac semimetallic states. With first-principles evidence for a combination of electronic and phononic ferroelectricity, we present a simple model that captures the energetics of coupled electronic and structural polarization, and show that electronic ferroelectricity arises in a material which is highly polarizable (small band gap) and exhibits a large electron-phonon coupling leading to anomalous dynamical charges. These insights will guide the search for electronic ferroelectrics, and our results on 2D h-NbN will stimulate development of piezofield effect transistors and devices based on the multilevel logic.

Figure 1

(a) Energies of planar and buckled structures of h-NbN as a function of $\epsilon$, which cross at $\epsilon =5.26\%$. Inset shows the dependence of frequencies of $\mathrm{\Gamma }$ optical and $M$ acoustic modes as a function of $\epsilon$, showing a structural instability emerging at $\epsilon =5.31\%$. (b) Electric polarization and buckling ($\mathrm{\Delta }=d_{\mathrm{Nb}}^z-d_{\mathrm{N}}^z$) as a function of $\epsilon$. Inset shows the structure of monolayered h-NbN. The sign of electric polarization reverses at $\epsilon =4.85\%$, while the sign of buckling $\mathrm{\Delta }$ is unchanged. $\mathrm{\Delta }$ vanishes for $\epsilon \ge 6.5\%$, and $P$ vanishes too.

Figure 2

(a) Electronic structure, (b) phonon dispersion, and (c) energetics of buckling ($\mathrm{\Delta }$) that is relevant to polarization switching in h-NbN at $\epsilon =5\%$. In (c), energy of the planar structure of h-NbN (configuration 3) is the reference. Ferroelectric states correspond to configurations 1 and 5, and the states 2 and 4 are the transition states giving energy barriers to be crossed during switching. (d) Polarization as a function of change in buckling of h-NbN [configuration 5 in (c)] at $\epsilon =5\%$ and $\epsilon =4.7\%$, revealing that the sign of $P$ reverses without inversion of the structure, i.e., change in sgn($\mathrm{\Delta }$). Inset shows that polarization changes sign with $\mathrm{\Delta }$, and vanishes for the flat structure.

Figure 3

Response of h-NbN ($\epsilon =5\%$) to an electric field, (a) hysteretic switching of buckling ($\mathrm{\Delta }=d_{\mathrm{Nb}}^z-d_{\mathrm{N}}^z$) as a function of electric field (high values), (b) corresponding polarization exhibiting hysteretic behavior. Here, $D$, ${\mathrm{\Delta }}^{+}$ ($D$, ${\mathrm{\Delta }}^{-}$) label a Dirac semimetallic state with a positive (negative) $\mathrm{\Delta }$, and GSM indicates a gapped semimetallic state. $D$ labels Dirac semimetallic state and $N$ is the equilibrium semiconductor. A weak hysteresis in buckling response to a low electric field (c), and polarization as a function of buckling obtained using our model Hamiltonian [inset of Fig. 3]. The arrows in (a) indicate the directions of changes in electric field along two paths, and the same symbols are used in other panels.