Ferroelectricity in strained ${\mathrm{Hf}}_2{\mathrm{CF}}_2$ monolayer

Low-dimensional ferroelectrics are highly desired for applications and contain exotic physics. Here a functionalized MXene ${\mathrm{Hf}}_2{\mathrm{CF}}_2$ monolayer is theoretically studied which manifests a nonpolar to polar transition upon moderate biaxial compressive strain. Accompanying this structural transition, a metal-semiconductor transition occurs. The in-plane shift of the unilateral fluorine layer leads to a polarization pointing out-of-plane. Such ferroelectricity is unconventional, similar to the recently proposed interlayer-sliding ferroelectricity but not identical. Due to its specific hexapetalous potential energy profile, the possible ferroelectric switching paths and domain walls are nontrivial, which are mediated via the metallic paraelectric state. In this sense, the metallic walls can be manipulated by reshaping the ferroelectric domains.

Figure 1

Possible structures of MXene ${\mathrm{Hf}}_2{\mathrm{CF}}_2$. (a–c) Side views of different configurations of ${\mathrm{Hf}}_2{\mathrm{CF}}_2$. (d) Top view of model 2. The biaxial compressive strain is indicated by arrows.

Figure 2

(a) Ground state phase diagram of ${\mathrm{Hf}}_2{\mathrm{CF}}_2$ as a function of the lattice constant. Left axis: energy per f.u. The energy of optimized model 2 is taken as the reference. Right axis: band gap. Upper axis: the corresponding biaxial strain, defined as $\varepsilon =(a-a_0)/a_0$, where $a_0$ and $a$ are the in-plane lattice constant of the pristine (model 2) and strained MXene, respectively. (b) Phonon spectra of models 2 and 3, under different biaxial compressive strains. Model 2 is dynamically stable without strain (left) and under $\varepsilon =-3\%$ compressive strain (middle). Model 3 is also stable in this condition (right).

Figure 3

The representative electronic structures of ${\mathrm{Hf}}_2{\mathrm{CF}}_2$, which show a metal-semiconductor transition upon compressive strain. (a, b) Unstrained model 2. (a) DOS shows that the electronic state around the Fermi level is mostly from Hf's $5d$ orbitals. (b) The corresponding band structure. Blue: without spin-orbit coupling (SOC); red: with SOC. (c) Band structure for strained model 3, in which an indirect band gap is opened. The HSE06 result is similar to the GGA one, with a slightly larger band gap.

Figure 4

(a) Comparison between the paraelectric phase and ferroelectric phase. The C layer is aligned as the common reference level. The Hf-C layer distances are symmetric in the paraelectric phase (i.e., $d_1=d_2$), which become asymmetric in the ferroelectric one (i.e., $d_1\ne d_2$). Blue broken line: the middle of Hf bilayer. (b) The 2D polarization and associated dipole moment as a function of compressive strain. (c) The energy barrier of possible ferroelectric switching path for ${\mathrm{Hf}}_2{\mathrm{CF}}_2$ at compressive conditions (solid curve: $\varepsilon =-1.9\%$ and broken curve: $\varepsilon =-3\%$).

Figure 5

(a) For each F layer, there are three equivalent shifting directions (named A, B, and C and indicated by yellow arrows) from the paraelectric phase to the ferroelectric phase. (b, c) The energy landscapes as a function of F displacements, obtained using the NBE method. The $+/-$ indicates the shifting for upper/lower F layer, which are also the sign of polarization (up/down). Here two plots are presented: (b) only involves a single F layer shifting, and (c) contains two F layers shifting. The A/B/C domains are always isolated by high-energy barriers.

Figure 6

(a) Sketch diagram of a ferroelectric domain wall between the ${\mathrm{A}}^{-}$ and ${\mathrm{B}}^{+}$ domains. (b) The relaxed domain wall. The local dipole moments, characterized by $\mathrm{\Delta }d=(d_1-d_2)/2$, are indicated by arrows. The value of $\mathrm{\Delta }d$ tends to be zero around the domain wall. (c) The DOS of a supercell containing the domain walls between ${\mathrm{A}}^{-}$ and $B^{+}$. The metallicity is expected from the domain wall. (d) The corresponding electron density distribution, integrated from $-0.4\mathrm{eV}$ to the Fermi level. It is obvious that these electrons appear only at the domain walls.