Cluster Sliding Ferroelectricity in Trilayer Quasi-Hexagonal C60

doi:10.21203/rs.3.rs-4825496/v1

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Cluster Sliding Ferroelectricity in Trilayer Quasi-Hexagonal C60

https://doi.org/10.21203/rs.3.rs-4825496/v1

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Electric polarization typically originates from non-centrosymmetric charge distributions. Since chemical bonds between atoms of the same elements favor centrosymmetric crystal structures and symmetrically distributed electron charges, elemental ferroelectrics are extremely rare. In comparison to atoms, elemental clusters are less symmetric and typically have various preferred orientations in crystals. Consequently, the assembly of clusters with different orientations tends to break the inversion symmetry. Based on this concept, we show that sliding ferroelectricity naturally emerges in trilayer quasi-hexagonal phase (qHP) C₆₀, a cluster-assembled carbon allotrope recently synthesized. Trilayer qHP C₆₀’s have several ferroelectric structures, which are distinguishable in second-harmonic generation (SHG) responses. Compared to previously found elemental ferroelectrics, trilayer qHP C₆₀’s have sizable band gaps and some of them have both switchable out-of-plane and in-plane polarizations. The out-of-plane and in-plane polarizations of these trilayer systems can be switched independently and enable an easy-to-implement construction of Van der Waals homostructures with ferroelectrically switchable chirality.

Physical sciences/Materials science/Condensed-matter physics/Ferroelectrics and multiferroics

Physical sciences/Materials science/Nanoscale materials/Two-dimensional materials

Physical sciences/Nanoscience and technology/Nanoscale materials/Carbon nanotubes and fullerenes

Non-centrosymmetric two-dimensional (2D) systems can host a plethora of effects with technological implications, including the Rashba effects [1], bulk photovoltaic effects [2, 3], and piezoelectricity [4–6]. Particularly, ferroelectric 2D materials allow for non-volatile switching of electric polarizations in monolayer and few-layer forms. Among them, ferroelectrics with robust out-of-plane (OP) electric polarizations, which are more easily detectable in experiments and implemented in devices, have been pursued by researchers [7–9]. The absence of dangling bonds on the surfaces of 2D ferroelectrics renders them ideal components for building Van der Waals heterostructures with clean and well-defined interface structures, holding promises for applications such as next-generation memory devices [10–14].

Intrinsic 2D ferroelectrics with OP polarizations are rare and mostly compounds. Typically, the chemical bonds between atoms of the same elements favor centrosymmetric structures and charge distributions, which hinder electric polarizations and make elemental 2D ferroelectrics scarce. The discovery of new elemental ferroelectricity can help uncover atypical mechanisms of symmetry breaking. Previous experiments have identified very few elemental ferroelectrics, such as the black phosphorous-like Bi monolayer [15, 16]. However, black-phosphorus-like bismuth [15] and some of the theoretically proposed 2D elemental materials [16–19] only have in-plane electric polarization. In comparison, stable elemental ferroelectrics with both OP polarization and a sizable band gap can be more appealing for experimental verification and practical applications [9, 13, 20, 21]. Recently, sliding ferroelectrics offer new opportunities for designing OP ferroelectricity and have been identified in a series of multilayer Van der Waals materials [7, 8, 22, 23], including the recently identified multi-layer graphene (with four or more layers) [24–26]. Since graphene is a semimetal, it is challenging to apply the standard piezoresponse force microscopy to experimentally confirm the ferroelectricity [27].

In order to expand the family of low-dimension ferroelectrics, we propose to examine cluster-assembled materials. Compared to atomic crystals, systems that consist of inter-linked clusters are more likely to break the inversion symmetry. For example, despite C₆₀ being known as one of the most stable and symmetric clusters, it is still less symmetric than real atoms, which have perfect spherical symmetry. In few-layer forms, two-dimensional materials consisting of C₆₀ with different orientations are more likely to break the inversion symmetry compared to atomic crystals [28, 29]. A recently synthesized polymeric fullerene is one of the ideal candidates for constructing cluster sliding 2D ferroelectrics [30–32]. By exploiting this observation, in this Letter, we demonstrate that trilayer quasi-hexagonal phase C₆₀’s have a variety of polar stacking configurations, which are stable elemental ferroelectrics with both sizable band gaps and out-of-plane electric polarizations. These phases can be conveniently distinguished through second-harmonic-generation (SHG) measurements[33, 34]. More importantly, some trilayer qHP C₆₀ have decoupled in-plane and out-of-plane polarizations, which can leads to an easy-to-implement scheme for building chiral vdW homostructures with ferroelectrically switchable chirality.

In experiments, 2D quasi-hexagonal phase (qHP) and quasi-tetragonal phase (qTP) C₆₀ have been successfully synthesized [30–32]. The qHP C₆₀ demonstrates better stability [29, 35] than qTP C₆₀ and can exist in both monolayer and few-layer forms. In qHP C₆₀, fullerenes adopt two possible orientations[29–31], which can be distinguished from the highlighted pentagons on the top of a C₆₀, as shown in Fig. 1 (a).

The monolayer qHP C₆₀ has a centrosymmetric structure. In stable few-layer forms, a fullerene tends to align with the center of the triangle consisting of three nearest-neighboring fullerenes from the adjacent layer, as observed in experiments [30, 31]. There are two distinct bilayer configurations (labeled as AB and AB’) of qHP C₆₀, which are both centrosymmetric, as shown schematically in Fig. 1 (b) and (c). In both cases, the bottom layer is shifted relative to the top layer by approximately \(\:b/3\), where \(\:b\) is the lattice constant along the \(\:b\)-direction.

Next, to examine whether trialyer qHP C₆₀ can be noncentrosymmetric, we enumerate all possible trilayer qHP C₆₀’s and identify six different stacking structures. The naming convention and schematic plots of all these trilayer structures are presented in Supplemental Fig. S1 and S2. Among them, two stacking configurations (AB’A’, AB’C’) possess both out-of-plane and in-plane polarizations, two stacking configurations have only in-plane polarizations, and the other two are centrosymmetric. The stacking configurations, point group, and electric-polarization directions are summarized in Table 1. In the following discussion, we focus on the trilayer phases with out-of-plane polarizations.

Figure 1. (a) (left) The structural model of monolayer qHP C₆₀. The blue and yellow pentagon rings are used to highlight C₆₀ clusters in two different orientations. (right) A simplified illustration of the qHP C₆₀ monolayer, where the blue and yellow spheres represent two different orientations of the C₆₀. (b) Bilayer qHP C₆₀ in the AB and (c) AB’ stacking configuration, with the inversion center labeled by the red star. (d) Trilayer C₆₀ with AB’A’ stacking configuration; (e) The isosurface plot (with isosurface value \(\:\pm\:6.5\times\:{10}^{-5}\:e/\text{B}\text{o}\text{h}{\text{r}}^{3}\)) of the differential charge density of the AB’A’ structure, where yellow and green isosurfaces indicate electron accumulation and depletion due to non-centrosymmetric layer stacking; (f) The relative energies of 6 stacking configurations and the out-of-plane polarization of the polar structure.

Table 1

The symmetry properties and polarization directions of 6 different trilayer stacking configurations.
Stacking configurations	Point Group	Out-of-plane (OP) polarization	In-plane (IP) polarization
AB’A’, AB’C’	m	✓	✓
ABA, AB’A	mm2	×	✓
ABC, AB’C	2/m	×	×

Figure 1 (d) schematically depicts the non-centrosymmetric AB’A’ stacking configuration with both IP and OP polarization. To better illustrate the origin of its polarization, we calculated the charge density differences \(\:{\Delta\:}\rho\:\left(\varvec{r}\right)\) between qHP C₆₀ layers in the AB’A’ stacking configuration and free-standing C₆₀ monolayers. As shown in Fig. 1 (e), due to the weak interlayer interaction, the redistributed charge density mostly occurs at the boundary of the Van der Waals gap between adjacent layers and evidently reveals the asymmetric shape in the AB’A’ structure. In addition, the planar-averaged distribution \(\:{\Delta\:}\rho\:\left(z\right)=\int\:{\Delta\:}\rho\:\left(\varvec{r}\right)dxdy\) along the z-axis quantitatively demonstrates symmetry broken, giving rise to the out-of-plane polarization (see Supplemental Fig. S3 for more details).

As shown in Fig. 1 (f), the energy differences between different stacking configurations are on the order of 0.1 meV/atom, which is beyond the chemical accuracy of DFT calculations. Accordingly, these stacking configurations are nearly degenerate ground-state structures. In the Berry phase formulation of macroscopic polarization, the calculated vertical polarizations of polar structures AB’A’ and AB’C’ are 0.22 and 0.25 pC/m, respectively. This is on the same order of magnitude as the out-of-plane polarization of tetralayer graphene [24], InSe [8], and many others [20]. The ABA and AB’A structures exhibit in-plane polarization of 0.17 pC/m and 0.18 pC/m, respectively, but no out-of-plane polarization. DFT calculations reveal trilayer qHP C₆₀ with different stacking configurations have a band gap of around 0.8 eV (as shown in Supplemental Fig S4). Compared to metal or semi-metallic ferroelectrics [24, 27, 36], the finite band gap and OP polarizations of trilayer qHP C₆₀ make it easier to measure the electric polarization in experiments and more suitable for diverse applications.

SHG response is a well-established technique for identifying layer stacking of 2D materials with variations of centrosymmetric structures [33, 37]. Figure 2 (a) shows the calculated frequency-dependent second-order nonlinear susceptibility tensor component \(\:{\chi\:}_{yyy}^{\left(2\right)}\) of AB’A’ and AB’C’ structures. The highest peak of \(\:{\chi\:}_{yyy}^{\left(2\right)}\) for the AB’C’ structure corresponds to a photon energy of 1.213 eV (equivalent to a wavelength of 1022 nm), while the highest peak for the AB’A’ structure is located at 1.329 eV (corresponding to 933 nm). The dependencies of other second-order nonlinear susceptibility tensor components are presented in Supplemental Material Fig. S5. Under an incident light with a wavelength of 1022 nm and an incident angle of \(\:{\theta\:}\) = 0°, the SHG responses of two trilayer structures are plotted as a functions of polarization angle \(\:\varphi\:\) in Fig. 2 (b) and (c) [38, 39]. The maximum parallel components \(\:{\varvec{I}}_{\parallel\:}\left(\varphi\:\right)\) of both AB’A’ and AB’C’ configurations appear at \(\:\varphi\:=90^\circ\:\), but with a substantial difference in their magnitude. Additionally, the vertical component \(\:{\varvec{I}}_{\perp\:}\left(\varphi\:\right)\) of AB’A’ peaks at \(\:\varphi\:=\:0^\circ\:,\) with its maximum value comparable to the parallel component. In contrast, the vertical component intensity of AB’C’ is only one-twentieth of its parallel component. Clearly, SHG responses can serve as a useful tool for distinguishing trilayer qHP C₆₀ stacking configurations.

Similar to other sliding ferroelectrics [30–32], all stacking configurations are related by the relatively translational displacements of C₆₀ layers, i.e., interlayer sliding. For example, the displacement of the middle layer by \(\:(\varvec{a}+\varvec{b})/2\) directly transforms AB’A’ into ABA’. One should note that ABA’ is equivalent to the A’B’A structure and has an out-of-plane polarization opposite to that of AB’A’. Consequently, the polarization reversing process is achieved under the process depicted in Fig. 3 (b). The calculated energy barrier required to achieve this transition is about 7 meV/atom, which is on the same order-of-magnitude as the sliding energy barrier of tetra-layer graphene (about 4 meV/unit cell) [24]. Moreover, the symmetry of the AB’A’ structure suggests that displacement of the middle layer by \(\:(-\varvec{a}+\varvec{b})/2\), \(\:(\varvec{a}-\varvec{b})/2\), \(\:-(\varvec{a}+\varvec{b})/2\) results in the same structure as that of \(\:(\varvec{a}+\varvec{b})/2\). This can be illustrated by the energy landscape as a function of different translational movements of the middle layer (see Supplemental Fig. S6).

Notably, the energy barrier can be reduced when the polarization reversing process goes through intermediate states with stable stacking configurations. For example, Fig. 3 (b) also schematically illustrates the transition process AB’A’⇢ABA⇢ABA’ (= A’B’A). One may note that the intermediate state ABA is non-polar and equivalent to A’B’A’. This process may be understood by first sliding the top layer C₆₀ by \(\:(\varvec{a}+\varvec{b})/2\), followed by sliding the bottom layer C₆₀ by \(\:(\varvec{a}+\varvec{b})/2\). Similarly, another transition path AB’A’⇢;AB’A⇢ABA’ (= A’B’A) is accomplished by first shifting the bottom layer and then the top layer. The energy barriers for these two transition paths are around 3 meV/atom.

While the AB’A’ structure resembles the Bernal stacking order of graphene, the AB’C’ structure resembles the rhombohedral stacking configuration [25]. Consequently, the polarization reversal process of AB’C’ is distinct from that of AB’A’ and can be achieved by sliding the bottom layer and middle layer by \(\:\varvec{a}/2+\varvec{b}/3\) and \(\:\varvec{b}/3\), as shown in Fig. 3 (d). This transition path leads to the final structure ACB’, which is equivalent to C’B’A. Even though this transition path involves a simultaneous movement of two layers, the energy barrier is only around 2 meV/atom, which is lower than many reported 2D ferroelectrics [8, 40–42]. Similar to the AB’A’ structure, the polarization reversal path of AB’C’ can be achieved by passing through a meta-stable intermediate state, such as ACA. This can be accomplished by shifting the top layer by \(\:-\varvec{a}/2-\varvec{b}/2\), followed by sliding the bottom layer by \(\:\varvec{a}/2+\varvec{b}/2\) to get ACB’ (= C’B’A, which has a lower energy barrier of ~ 1 meV/atom.

In addition to the reversal of polarization, switching between different polarization states can also be effectively achieved by interlayer sliding [43]. We calculated the transition energy barriers between four stacking configurations with different out-of-plane and in-plane polarizations, as shown in Fig. 4. Overall, the energy barriers required for transitioning between different polarization states are close to those required to reverse the direction of polarization. For instance, the energy barrier for transitions AB’C’⇢;AB’A’, AB’A’⇢AC’A’, and AC’A’⇢AC’B’ are 0.8, 1.7, and 0.8 meV/atom, respectively.

Notably, the AB’A’ and AC’A’ configurations are physically equivalent, since they are related by a rotation around c-axis by \(\:{180}^{\circ\:}\). Similarly, AB’C’ and AC’B’ structures are physically equivalent. As a result, AB’A’ and AC’A’ structures (or AB’C’ and AC’B’ structures) have the same OP polarization but opposite IP polarizations. In other words, the out-of-plane and in-plane polarizations can be reversed independently in trilayer qHP C₆₀ systems. This property is different from monolayer In₂Se₃ [9] and many others [20], whose OP polarization direction is coupled with the IP polarization direction. Markedly, the decoupling of IP and OP polarization can be a key feature that allows one to construct Van der Waals structures with ferroelectrically switchable chirality.

It is noteworthy that intrinsic chirality in 2D Van der Waals materials is uncommon and have been pursued [44]. Chirality in low-dimension materials can be artificially introduced by twisting [45, 46], kirigami [47, 48], adsorption of chiral molecules [49, 50], and other methods [51, 52], which are challenging to manipulate and do not allow for nonvolatile switching of chirality through electric fields. To demonstrate the concept of ferroelectrically switchable chirality, we consider two slabs of AB’A’ trilayers that are rotated by \(\:\theta\:={90}^{\circ\:}\) with each other, as shown in Fig. 5. In practice, the rotation angle \(\:\theta\:\) can take any value ranging between \(\:0\) to \(\:{180}^{\circ\:}\). As shown in Fig. 5, the top A’B’A’ trilayer has an OP polarization \(\:{\mathbf{P}}_{OP1}\) pointing along the \(\:z\)-direction and an IP polarization \(\:{\mathbf{P}}_{IP1}\) pointing along the \(\:x\)-direction, while the bottom A’B’A’ trilayer has an OP polarization \(\:{\mathbf{P}}_{OP2}\) pointing along the \(\:z\)-direction and an IP polarization \(\:{\mathbf{P}}_{IP2}\) pointing along the \(\:y\)-direction. Such hexa-layer configuration has a total polarization given by \(\:({\mathbf{P}}_{IP1},{\mathbf{P}}_{IP2},{\mathbf{P}}_{OP1}+{\mathbf{P}}_{OP2})\), that forms a basis with right-handed chirality. The chirality can be readily switched by applying an out-of-plane electric field that reverse the OP polarization while keeping the IP polarization unchanged. This leads to a new configuration with polarization \(\:({\mathbf{P}}_{IP1},{\mathbf{P}}_{IP2},{-\mathbf{P}}_{OP1}-{\mathbf{P}}_{OP2})\), which has left-handed chirality. As we noted, the key for this ferroelectric chirality switching is to decouple the OP and IP polarizations, i.e., switching the OP polarization independently.

In conclusion, based on symmetry analysis and first-principles calculations, a few non-centrosymmetric stacking modes of trilayer qHP C₆₀ with out-of-plane or in-plane electrical polarizations are predicted. The energy barriers for switching the electrical polarizations are comparable with those of other sliding ferroelectrics. Such unconventional elemental ferroelectricity originates from the orientational dependent structure of C₆₀ clusters. The multiple ferroelectric stacking states of trilayer qHP C₆₀ can be clearly detected and distinguished via second harmonic generation measurement. We also propose an easy-to-implement scheme that takes advantage of the decoupled in-plane and out-of-plane polarizations to construct Van der Waals few-layer systems with non-volatile switchable chirality. Moreover, sliding ferroelectricity is not limited to trilayer systems. It can be generalized to few-layer qHP C₆₀ with more than three layers. For instance, we enumerate tetralayer qHP C₆₀ configurations (as shown in Supplemental Table S1). Among them, 11 tetralayer stacking configurations are centrosymmetric. In the future, it would be interesting to investigate the sliding ferroelectricity of tetra- or penta-layer qHP C₆₀ or other cluster-assembled systems. Furthermore, the interstitial space between C₆₀ cages implies the possibility of intercalating transition metal ions, which may in turn, introduce multiferroicity to the system.

The calculations were based on pseudopotential density functional theory (DFT) implemented in the Quantum Espresso (QE) package [53, 54]. Projector Augmented Wave (PAW) pseudopotentials which are taken from the SSSP library[55] and the Perdew–Burke–Ernzerhof (PBE) [56] exchange-correlation functional with DFT-D3 dispersion correction [57] were used. A vacuum layer of 30 Å was adopted to isolate the artificial interaction between periodic images in the -direction. For structural relaxation, the convergence threshold of force and energy were Ryd/Bohr and Ryd, respectively. The electron wave function was represented using the plane-wave basis set, with a kinetic-energy cutoff of 50 Ry. For the Brillouin zone (BZ) sampling, a 4×2×1 Monkhorst-Pack k-point grid was employed [58]. Electronic polarization was calculated using the Berry phase formulation of macroscopic polarization [59, 60]. The energy barriers of ferroelectric switching pathways were obtained using the nudged elastic band (NEB) method [61]. The SHG susceptibility was calculated with the independent particle approximation[38, 39, 62], using a self-developed code interfaced with QE (more details in Supplemental Materials).

Acknowledgement:

We acknowledge the support by the National Natural Science Foundation of China (12104080, 12374253, and 12074053). Computational resources are provided by the Supercomputer Center at Wuzhen.

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Cluster Sliding Ferroelectricity in Trilayer Quasi-Hexagonal C60

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