Prediction of Intrinsic Ferroelectricity and Large Piezoelectricity in Monolayer Arsenic Chalcogenides

Prediction of intrinsic ferroelectricity and large piezoelectricity in monolayer arsenic chalcogenides

Weiwei Gao∗,† and James R. Chelikowsky∗,†,‡,¶

†Center for Computational Materials, Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712 ‡Department of Physics, The University of Texas at Austin, Austin, TX 78712 ¶McKetta Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712

E-mail: weiwei@ices.utexas.edu; [email protected]

Abstract Keywords

Two-dimensional materials that exhibit spon- ferroelectricity; piezoelectricity; two-dimension taneous electric polarization are of notable in- material; first-principles calculations; polymor- terest for functional materials. However, de- phism spite many two-dimensional polar materials are predicted in theory, the number of experimentally confirmed two-dimensional ferro- Introduction electrics are still far less than bulk ferroelectrics. Materials lacking inversion symmetry may dis- We provide strong evidence that the Pmn2 1 play useful properties such as piezoelectric- phase of arsenic chalcogenides As X (X=S, Se, 2 3 ity and ferroelectricity, which have wide ap- and Te), which include the recently isolated plications in modern industries. In particular, monolayer orpiment, are intrinsic ferroelectrics ferroelectric materials are not only prototypi- and demonstrate strong in-plane piezoelectric- cal systems for studying spontaneous symme- ity. We found the calculated energy barriers for try breaking and structural phase transitions, collectively reversing the electric polarization but also key components for non-volatile mem- or moving a 180◦ domain wall are reasonable ory devices, piezoelectric sensors, photocatal- compared to previously reported ferroelectrics. ysis, and many other technologically impor- We propose a high-symmetry structure (with tant applications.1–3 Driven by the need for fur- Pmmn space group) transforms into the fer- ther miniaturization of electronic devices, re- roelectric Pmn21 phase by a soft B2u phonon arXiv:2007.10575v2 [cond-mat.mtrl-sci] 30 Oct 2020 searchers have devoted significant efforts to re- mode. By studying other soft modes of the duce the thickness of thin-films ferroelectrics.4–6 high-symmetry Pmmn structure, we identify Despite the depolarization field,7–9 which usu- several undiscovered metastable polymorphs, ally inhibits the electric polarization of thin-film including a polar phase (with a P2 space 1 ferroelectrics, a few groups have demonstrated group) with sizable piezoelectricity. ferroelectricity sustains in bulk ferroelectrics with thickness down to ∼ 1 nm.10,11 The recent discovery of ferroelectricity in monolayer or few- layer Van der Waals (vdW) materials offer new

1 opportunities for shrinking the size of ferroelec- Monolayer Monolayer a orpiment b anorpiment tric devices to the atomically thin regime.12–16 Compared to conventional bulk ferroelectrics, a key advantage of two-dimension vdW materials is free of dangling bonds on the surface. As S First-principles calculations also show that a P large number of two-dimensional (2D) materials are piezoelectric.17–19 Remarkably, some of them17,20 even demonstrate giant piezoelectric P effects, which can be more than two-orders-of- y magnitude stronger than bulk piezoelectric ma- z x terials. Currently, our understanding of the funda- P mental physical properties of 2D piezoelectric z y and ferroelectric systems is in an early stage, x and the lack of a robust and economical fabri- z cation process for high-quality 2D ferroelectric P samples hinders mass production and applica- y x tions.21 Among 2D ferroelectric materials pre- Figure 1: The crystal structures of (a) mono- dicted with first-principles theories,20,22–27 only layer orpiment and (b) monolayer anorpiment, a few, such as monolayer SnS,28 SnSe,29 SnTe14 plot with VESTA.31 The red arrows show the and In Se ,15 have so far been synthesized and 2 3 direction of polarization. confirmed to be ferroelectric. First-principles prediction of piezoelectricity or switchable electric polarization in readily fabricated 2D mate- Results and Discussion rials is important for enriching the toolbox of 2D non-centrosymmetric materials with tech- Under ambient conditions, bulk As2S3 can be nological interests. either amorphous or crystalline. Bulk orpi- Through first-principles calculations, we show ment and anorpiment, which were found in nat- ample evidence that three monolayer arsenic ural minerals,32 are two common crystalline chalcogenides (As2X3) with the Pmn21 space As2S3 phases with noncentrosymmetric lay- group will exhibit spontaneous and reversible ered structures bounded by vdW interactions. in-plane polarization. Among these three ma- To date, two-dimensional anorpiment has not terials, the Pmn21 As2S3, i.e., monolayer orpi- been synthesized, while monolayer and few- ment, has recently been isolated through me- layer orpiment have been successfully exfoli- chanical exfoliation.30 Moreover, we predict ated and demonstrates better chemical stabil- the existence of several novel metastable poly- ity than phosphorene under low light condi- 30 morphs of As2S3. Both ferroelectric monolayer tions. Our calculated total energy of mono- orpiment and these new polymorphs can be re- layer orpiment is lower than that of monolayer lated to the soft zone-center modes of a hy- anorpiment by 73 meV/formula unit (f.u.), sug- pothetical high-symmetry phase. Remarkably, gesting better stability of monolayer orpiment our calculations show some of these polymorphs compared to monolayer anorpiment. A finite have large piezoelectric coefficients comparable bandgap is required for sustaining the ferro- to those of group IV-VI compounds.20 electricity of 2D materials with in-plane polarization. Monolayer orpiment and anorpiment have indirect bandgaps around 2.2 eV calculated with the Perdew-Burke-Ernzerhof (PBE) functional.33 The band structures are presented in the Supporting Information.

2 (a) (b)

Space grp:Pmmn x Pmn21, 0 meV (c) (d)

doubling unit cell

due to soft phonon at X

y y x x P2 meV/f.u. P21/m, 297 meV/f.u. P2/c, 306 meV/f.u. P21212 (unstable) 1, 7

Figure 2: (a) (left) The high-symmetry structure with space group Pmmn and (right) the change of total energy under collective atomic displacements of the Pmmn structure with frozen-in soft phonon mode Au,B1g,B3g, and B2u. The displacement vector ∆Rp is proportional to the polarization vector of phonon p: ∆Rp = Q·up. (b) Schematic plot of the transform from Pmmn structure to monolayer orpiment through the B2u soft mode. The red arrows (not to scale) show the moving direction of corresponding atoms. (c) Unit cell of the metastable P21/m phase and P2/c phase. (d) Schematic plot of the stabilization of the P21212 phase by doubling the unit cell due to soft modes at X point.

The crystal structures of monolayer ar- try) are dynamically stable, while those with senic chalcogenides do not resemble those of the anorpiment-like structure (Pc space group) other well-known 2D materials. As shown demonstrate dynamical instability with imagi- in Fig.1(a), monolayer orpiment is highly nary phonon modes. anisotropic and consists of rings connected Using first-principles methods based on mod- 35,36 by six corner-sharing AsS3 units, which have ern polarization theory, we calculate that a pyramidal shape. Monolayer orpiment has monolayer orpiment has a spontaneous electric Pmn21 symmetry, which includes a mirror- polarization of 71 pC/m. According to the ex- reflection to the xz-plane, but no symmetry perimental structure of bulk orpiment,32,37 the with the yz-plane, as illustrated in Fig.1(a). electric polarization in neighboring layers aligns Such symmetry properties allow a spontaneous in an antiferroelectric order. Therefore, bulk or- electric polarization along the x-axis. In com- piment shows no macroscopic polarization and parison, monolayer anorpiment has a more the net polarization of a few-layer orpiment irregular structure and electric polarization shows an odd-even effect. Only samples with pointing in the y-direction, as illustrated in odd numbers of layers show net electric polar- Fig.1(b). ization. Bulk As2Se3 can be found in mineral As a classical example of displacive transi- laphamite with a similar structure as orpi- tions, the ferroelectric phase transition of per- 34 ment, while bulk As2Te3 with the orpiment- ovskite oxide like PbTiO3 is explained by a like structure is yet to be found. Our calcula- zone-center vibrational mode which vanishes at tions show monolayer As2Se3 and As2Te3 with the phase transition. Similarly, we propose the the orpiment-like structure (Pmn21 symme- ferroelectricity of monolayer orpiment is also

3 (a) (b) p p

As4

As3 As2

As1

Figure 3: (a). (top) Selected intermediate states on the transition path of inverting the polarization direction of monolayer orpiment. Red arrows show the main movement of As ions between consec- utive intermediate structures. The dashed curves are used to represent the atomic displacements that cross the unit-cell boundary. (bottom) The evolution of the total energies of intermediate structures along the transition path. (b). A comparison between the theoretical energy barriers Ebarrier of the polarization-reversing process of arsenic chalcogenide and other ferroelectrics. The values from previous work are all calculated with density functional theory. driven by a soft mode of a high-symmetry struc- the deepest double-well potential curve among ture with space group Pmmn. The unit cell all zone-center soft modes. Therefore the B2u of the Pmmn structure is shown schematically mode is the dominant phonon mode driving the in Fig.2(a). Different from monolayer orpi- structural transition from the unstable Pmmn ment which only has mirror symmetry to xz- structure to the Pmn21 phase of As2S3. As plane, the Pmmn structure of As2S3 has ad- shown in Fig.2(b), the main effect of B 2u op- ditional mirror symmetry with yz-plane. This tical mode is to shift the As atoms along the high-symmetry Pmmn structure is dynamically x-axis and break the reflection symmetry to yz- unstable with five soft optical phonon modes plane. The relative shifts between different S at the Γ point. To quantify contributions of atoms are smaller comparing to the displace- a zone-center soft phonon mode to the struc- ment of As atoms in the B2u mode. tural transition from the high-symmetry Pmmn We also examine the roles of other four soft structure to Pmn21 phase, we calculate the zone-center modes, namely B3g,B1g,B3u, and projection of the atomic displacement vector Au modes, by collectively displacing atomic co-

∆R = RP mmn −RP mn21 on soft phonon modes: ordinates of the high-symmetry Pmmn structure by ∆Rp = Q · up, where up is the nor- ∆R malized polarization vector of phonon mode p. η[p] = · up |∆R| As shown in Fig.2(a), one can easily identify structures that correspond to the local min- where u is the normalized polarization vector p ima (shown as small spheres) on the double-well of a zone-center phonon p. We ﬁnd η[B ] = 2u curve of total energy versus general coordinate 86%, and other four soft zone-center modes con- Q. Further relaxing local minimal structures tribute less than 1 percent to ∆R. This is ex- may lead to new metastable phases of As S . pected since the B mode is the only one that 2 3 2u Since the structural relaxation moves the local breaks the inversion symmetry and also has minima that correspond to the B3u mode back

4 to monolayer orpiment, we will not discuss it modes of the high-symmetry Pmmn structure further. of As2Se3 and As2Te3. Like As2S3, they both Interestingly, B3g and B1g modes trans- have a B2u mode driving the displacive transi- form the high-symmetry Pmmn structure into tion to the corresponding Pmn21 phase. As2Se3 metastable P2/c and P21/m phases, respec- also has a metastable phase with P21 space tively. As shown in Fig.2(c), both P2/c and group. In Table1, we list the space groups P21/m phase show unusual one-dimensional and the electric polarization of all stable poly- chain structures consist of interconnected AsS3 morphs studied in this work. We find P of pyramidal units. These two phases show zero Pmn21 phases decreases as the chalcogen el- macroscopic polarization since the dipole mo- ement changes from sulfur to tellurium. A ments of neighboring AsS3 pyramidal units similar trend also appears in IV-VI monolay- point in opposite directions and thus cancel ers.20 We explain this qualitatively with two with each other. arguments. First, The electrical polarization A more complicated case is the soft Au P is positively correlated to the difference be- mode. Relaxing the local minima structure cor- tween the electron negativity of As and the responding to the Au mode leads to a dynam- chalcogen elements. As the chalcogen element ically unstable P21212 structure without a net changes from S to Te, the reduced electroneg- electric polarization. Such an unstable struc- ativity results in a diminished polarization. ture has doubly-degenerate soft phonon modes Second, since the distortion amplitude |Qmin| at the X point which can stabilize the structure at the minima of the double-well potential of by doubling the unit cell along x-axis. The fi- B2u mode decreases as the chalcogen element nal stable structure we find has the P21 space changes from S to Te, the dipole moment, which group symmetry and a rectangle unit cell with is proportional to |Qmin|, also decreases. 20 atoms, as shown in Fig.2(d). The P2 1 phase has a noncentrosymmetric structure with spon- Table 1: Summary of the space group and elec- taneous polarization of 20 pC/m pointing in the tric polarization P of different arsenic chalco- y-direction. We confirmed the stability of P2/c, genides phases. P21/m, and P21 phases from their phonon spec- P (pC/m) tra calculated with density functional perturba- Formula Space group tion theory38 and finite-temperature molecular (The direction of P) 39 dynamics trajectories. These results are pre- Pmn21 sented in Supporting Information. (Monolayer 71 (x) The total energy of the P21 phase As2S3 is 65 orpiment) meV/f.u. lower than that of monolayer anor- As2S3 Pc piment, and only 7 meV/f.u. higher than that (Monolayer 47 (y) of monolayer orpiment. Even though the P2/c anorpiment) and P21/m phases are shown to be metastable, P2/c 0 they have total energies which are about 300 P21/m 0 meV/f.u. higher than that of monolayer or- P21 20 (y) piment, because they are composed with one- Pmn21 54 (x) dimensional chain-like structures bounded by As2Se3 weak vdW forces. We mention only the zone- P21 18 (y) center soft modes of Pmmn structures are stud- As2Te3 Pmn21 45 (x) ied in this work. It is also interesting to study the finite-momentum soft modes, which also The reversibility of electric polarization is a appear in the phonon spectrum of the Pmmn necessary condition for ferroelectrics and also structure and may lead to other interesting important for the application in data storage. polymorphs. Using the nudged-elastic-band method,40 we We perform similar analyses on the soft find a minimal-energy-barrier transition path

5 for reversing the electric polarization of an infi- ers two orders of magnitudes from 0.6 meV to nite large monolayer orpiment. We show im- 116 meV. The Ebarrier of As2X3 are notably portant intermediate structures and the cor- smaller than those of two room-temperature 16 responding energies on the transition path of ferroelectrics, namely Td-WTe2 and mono- 42 As2S3 in Fig.3 (a). In the polarization- layer d1T-MoTe2, and a few predicted fer- 45 23 reversing process, three important intermediate roelectrics such as GeS and Sc2CO2, but 0 49 25 structures labeled as A, B, and A are found. A much larger than those of CuInP2S6, In2Se3, and A0 are related by a 180◦ rotation around SnS45 and so on. Such comparisons suggest the z-axis. B is structurally akin to the un- that the energy barriers of switching the po- stable P21212 structure. The initial and final larization direction of Pmn21 As2X3 are within structures on the transition path correspond to a proper range. monolayer orpiment with electric polarization pointing in opposite directions. The process of initial intermediate final reversing electric polarization goes in the se- quence {initial → A → B → A0 → final}, which consists of four major steps. Each step mainly involves shifting a single As atom along the x-axis. For example, in the first step {initial → A}, the major structural change is the displacement of As1 (i.e. the arsenic atom at the bottom of the unit cell shown in Fig.3 (a)) along the negative x-direction. Similarly, in the second step {A → B}, we observe the movement of As2 along the negative x-direction to pass the boundary of the unit cell. We em- Figure 4: (top panel) Selected structures on phasize that the switching process we presented the process of moving the 180◦ domain wall in here may not correspond to the global mini- As2S3. The shaded regions highlight the do- mum barrier, since the changes of lattice vec- main boundaries. (bottom panel) Total ener- tors are not considered in our nudged-elastic- gies on the process of moving the domain walls band calculations and the electric dipoles can of As2S3, As2Se3, and As2Te3. never be switched simultaneously in real situations. Previous work also shows that including In practical situations, domain-wall shifting the variation of lattice vectors can further lower and domain growing mediate the process of the energy barrier.23,41 Nevertheless, our theo- reversing electric polarization of ferroelectrics. retical energy barrier Ebarrier provides an up- Formation energies of domain walls and energy per bound for the activation energy of the real barriers for moving domain walls indicate how polarization-reversing process. With the simi- difficult it is to form and grow a domain, respec- lar approach, the energy profiles for reversing tively. We studied the atomistic structure of a ◦ the electric polarization of As2Se3 and As2Te3 few 180 domain walls parallel to the x-axis in ◦ are calculated and shown in the Supporting In- As2X3. For example, the structure of the 180 formation. domain wall in As2S3 is shown schematically in In Fig.3 (b), we compare calculated en- the top panel of Fig.4. Our calculations in- ergy barriers Ebarrier of As2X3 with those of dicate the energy costs of forming and moving other ferroelectrics, which are either studied the 180◦ domain wall along x-axis are reason- experimentally15,16,25,28,42–49 or solely predicted able compared to other ferroelectrics. In de- 26,27,45,50–52 in theory. Except FeTiO3, LiNbO3, tail, the calculated domain-wall formation ener- dw AgBiP2Se6, and PbTiO3, the Ebarrier in Fig.3 gies Eform are 89, 105, and 124 meV/f.u. (i.e., are calculated with the nudged-elastic-band 43, 50, and 58 meV/A)˚ for monolayer As2S3, method. The range of calculated E cov- barrier As2Se3, and As2Te3, respectively. Previous cal-

6 dw culations show the Eform of group IV-VI ma- structural flexibility motivates us to investigate terials range from 8 meV/A˚ to 116 meV/A,˚ 45 the piezoelectricity of arsenic chalcogenides. which covers those of 2D As2X3 ferroelectrics. We summarize the calculated elasticity ten- dw 25 The Eform of In2Se3 is 220 meV/f.u., compa- sor Cij and piezoelectric tensor elements eij rable to those of As2X3. Assuming the thickness and dij in Table2. More details of calcu- of monolayer As2X3 is 6.0 A,˚ we convert the lating these tensor elements are presented in dw Eform of As2S3, As2Se3, and As2Te3 to be 115, Supporting Information. Obviously, the piezo- 2 133, and 155 mJ/m , which are in the same or- electric strain coefficients dij of Pmn21 and Pc der as those of some ferroelectric oxides, such as phases are one-order-of-magnitude larger than 2 ◦ PbTiO3 (132 mJ/m for 180 domain wall and those of common two-dimensional polar materi- 2 ◦ 53 19 35.2 mJ/m for 90 domain wall) and BiFeO3 als such as 2H-MoSe2 (d11 = 3.73 pm/V), 2H- 2 54 19 (205 to 1811 mJ/m ), but much higher than WSe2 (d11 = 2.79 pm/V), hexagonal group 2 53 18 that of BaTiO3 (7.5 mJ/m ). III-V materials (0.02 < d11 < 5.50 pm/V), Using nudged-elastic-band method, we cal- and multilayer janus transition metal chalco- dw 17 culate the energy barriers Ebarrier for mov- genide MoSTe (5.7 < d33 < 13.5 pm/V). ing the 180◦ domain walls are 233 meV/f.u., On the other hand, the piezoelectric stress con- 128 meV/f.u., and 35 meV/f.u. (i.e., 113 stants eij of arsenic chalcogenides are compara- meV/A,˚ 54 meV/A,˚ and 16 meV/A)˚ for mono- ble with those of 2H-MoSe2, 2H-WSe2, and so 17–19 layer As2S3, As2Se3, and As2Te3, respectively, on. This indicates the large dij coefficients as shown in the bottom panel of Fig.4. This of arsenic chalcogenides is originated from their ◦ suggests the 180 domain wall of As2X3 be- superior flexibility, i.e., small elasticity tensor comes easier to shift as the chalcogen element components. Compared to group IV-VI mono- X changes from sulfur to tellurium. Com- layers with giant piezoelectricity, the dij coeffi- dw pared to bulk ferroelectrics, the Ebarrier of cients of Pmn21 and Pc phases are on the same monolayer As2Te3 and As2Se3 are of the same order as that of GeS, but two to four-times order-of-magnitude as those of bulk corundum smaller than those of SnS, SnSe, and GeSe.20 derivatives ranging from 14 meV/f.u. to 197 The piezoelectric coefficients of P21 are much 55 meV/f.u. Compared to other two-dimensional smaller than other phases. Interestingly, P21 dw ferroelectrics, Ebarrier of monolayer As2S3 is As2Se3 shows weak negative piezoelectric effect more than an order-of-magnitude higher than along y-direction. those of group IV-VI two-dimensional ferro- In summary, we employ ab initio methods to electrics (less than 1.6 meV/A),˚ 45 but in simi- predict the intrinsic ferroelectricity and strong lar order with that of monolayer In2Se3, which piezoelectricity in arsenic chalcogenides, which ranges from 280 meV/f.u. to 400 meV/f.u.25 include the recently isolated monolayer orpi- These comparisons suggest that the energy ment. By analyzing the soft optical modes of costs for forming and moving the 180◦ domain the high-symmetry Pmmn structures of arsenic wall of As2X3 are reasonable. chalcogenides, we find these soft modes can Similar to monolayer group IV-VI compounds lead to several undiscovered metastable poly- and black phosphorene, monolayer arsenic morphs. The Pmn21 ferroelectric phases can be chalcogenides studied in this work are super related to the soft B2u phonon mode of a high- flexible. We calculate Young’s modulus of symmetry Pmmn structure. We investigate the monolayer orpiment to be 8.6 N/m along the feasibility of switching the electrical polariza- x-axis and 33.6 N/m along the y-axis, which tion in the Pmn21 phase. The energy barrier of is more than one-order-of-magnitude smaller coherently flip all electrical dipoles and that of than those of graphene (345 N/m)56,57 and also moving a 180◦-domain wall in two-dimensional significantly smaller than that of black phos- Pmn21 As2X3 are in a proper range compared phorene (21 ∼ 56 N/m).58,59 To our knowl- with other ferroelectrics. Moreover, superior edge, monolayer orpiment is among the softest structural flexibility results in large piezoelec- 2D material ever fabricated. Such remarkable tric responses in a few polymorphs. Such a

7 −10 Table 2: Elasticity tensor elements (N/m) and piezoelectric coeﬃcients (10 C/m for eij and pm/V for dij).

Space group (Point group) Formula C11 C22 C12 e11 e12 d11 d12

As2S3 11.07 43.38 10.40 4.36 -1.75 55.7 -17.4 Pmn21 (C2v) As2Se3 13.86 41.76 10.03 6.71 -1.49 61.7 -18.4 As2Te3 18.09 34.65 9.72 9.09 -1.48 61.9 -21.6

C11 C22 C12 e21 e22 d21 d22

As2S3 18.63 16.29 3.42 -0.85 0.22 -5.0 2.4 P21 (C2) As2Se3 21.51 23.76 2.65 -0.91 -0.33 -4.1 -0.9

Pc (Cs) As2S3 21.63 9.25 8.00 -2.77 2.69 -34.7 59.1 unique combination of unusual structures, pli- molecular dynamics simulation, transition path ability, strong piezoelectricity, and predicted for reversing electric polarization, and struc- ferroelectricity make monolayer arsenic chalco- ture parameters of arsenic chalcogenides poly- genides new platforms of studying polar mate- morphs. This material is available free of charge rials. They are also convincing candidates for via the internet at http://pubs.acs.org. small-sized, flexible electronic devices. Computation Details: Our first-principles calculations are based on pseudopotential den- References sity functional theory implemented in Quantum (1) Li, W.; Ji, L.-J. Perovskite ferroelectrics Espresso60,61 and PARSEC.62,63 More techni- go metal free. Science 2018, 361, 132–132. cal details are presented in Supporting Infor- 37–39,64–68 mation, which cites these references. (2) Scott, J. F.; Paz de Araujo, C. A. Fer- roelectric Memories. Science 1989, 246, Acknowledgement 1400–1405. (3) Fang, L.; You, L.; Liu, J.-M. Ferroelectric W.G. and J.R.C. acknowledge support from a Materials for Energy Applications; John subaward from the Center for Computational Wiley & Sons, Ltd, 2018; Chapter 9, pp Study of Excited-State Phenomena in Energy 265–309. Materials at the Lawrence Berkeley National Laboratory, which is funded by the U.S. De- (4) Setter, N.; Damjanovic, D.; Eng, L.; partment of Energy, Office of Science, Basic Fox, G.; Gevorgian, S.; Hong, S.; Energy Sciences, Materials Sciences and Engi- Kingon, A.; Kohlstedt, H.; Park, N. Y.; neering Division under Contract No. DEAC02- Stephenson, G. B.; Stolitchnov, I.; 05CH11231, as part of the Computational Ma- Taganstev, A. K.; Taylor, D. V.; Ya- terials Sciences Program. Computational re- mada, T.; Streiffer, S. Ferroelectric thin sources are provided by the Texas Advanced films: Review of materials, properties, and Computing Center (TACC). applications. Journal of Applied Physics 2006, 100, 051606. Supporting Information Avail- (5) Park, M. H.; Lee, Y. H.; Kim, H. J.; Kim, Y. J.; Moon, T.; Kim, K. D.; able Müller, J.; Kersch, A.; Schroeder, U.; The supporting information presents more de- Mikolajick, T.; Hwang, C. S. Ferroelec- tails on piezoelectric tensors, phonon spectra, tricity and Antiferroelectricity of Doped

8 Thin HfO2 -Based Films. Advanced Ma- (15) Zhou, Y.; Wu, D.; Zhu, Y.; Cho, Y.; terials 2015, 27, 1811–1831. He, Q.; Yang, X.; Herrera, K.; Chu, Z.; Han, Y.; Downer, M. C.; Peng, H.; Lai, K. (6) Böscke, T. S.; Müller,J.; Bräuhaus,D.; Out-of-Plane Piezoelectricity and Ferro- Schröder, U.; Böttger, U. Ferroelectric- electricity in Layered α-In Se Nanoflakes. ity in hafnium oxide thin films. Applied 2 3 Nano Letters 2017, 17, 5508–5513. Physics Letters 2011, 99 . (16) Sharma, P.; Xiang, F.-X.; Shao, D.- (7) Junquera, J.; Ghosez, P. Critical thickness F.; Zhang, D.; Tsymbal, E. Y.; Hamil- for ferroelectricity in perovskite ultrathin ton, A. R.; Seidel, J. A room-temperature films. Nature 2003, 422, 506–509. ferroelectric semimetal. Science Advances (8) Mehta, R. R.; Silverman, B. D.; Ja- 2019, 5 . cobs, J. T. Depolarization fields in thin ferroelectric films. Journal of Applied (17) Dong, L.; Lou, J.; Shenoy, V. B. Large Physics 1973, 44, 3379–3385. In-Plane and Vertical Piezoelectricity in Janus Transition Metal Dichalchogenides. (9) Wurfel, P.; Batra, I. P.; Jacobs, J. T. Po- ACS Nano 2017, 11, 8242–8248. larization Instability in Thin Ferroelectric Films. Phys. Rev. Lett. 1973, 30, 1218– (18) Blonsky, M. N.; Zhuang, H. L.; 1221. Singh, A. K.; Hennig, R. G. Ab Ini- tio Prediction of Piezoelectricity in (10) Fong, D. D.; Stephenson, G. B.; Streif- Two-Dimensional Materials. ACS Nano fer, S. K.; Eastman, J. A.; Auciello, O.; 2015, 9, 9885–9891. Fuoss, P. H.; Thompson, C. Ferroelectric- ity in Ultrathin Perovskite Films. Science (19) Duerloo, K.-A. N.; Ong, M. T.; 2004, 304, 1650–1653. Reed, E. J. Intrinsic Piezoelectricity in Two-Dimensional Materials. The Jour- (11) Lee, S. R.; Baasandorj, L.; Chang, J. W.; nal of Physical Chemistry Letters 2012, Hwang, I. W.; Kim, J. R.; Kim, J.-G.; 3, 2871–2876. Ko, K.-T.; Shim, S. B.; Choi, M. W.; You, M.; Yang, C.-H.; Kim, J.; Song, J. (20) Fei, R.; Kang, W.; Yang, L. Ferroelectric- First Observation of Ferroelectricity in ity and Phase Transitions in Monolayer 1 nm Ultrathin Semiconducting BaTiO3 Group-IV Monochalcogenides. Phys. Rev. Films. Nano Letters 2019, 19, 2243–2250. Lett. 2016, 117, 097601.

(12) Fei, Z.; Zhao, W.; Palomaki, T. A.; (21) Cui, C.; Xue, F.; Hu, W. J.; Li, L. J. Two- Sun, B.; Miller, M. K.; Zhao, Z.; Yan, J.; dimensional materials with piezoelectric Xu, X.; Cobden, D. H. Ferroelectric and ferroelectric functionalities. npj 2D switching of a two-dimensional metal. Na- Materials and Applications 2018, 2 . ture 2018, 560, 336–339. (22) Wu, M.; Zeng, X. C. Intrinsic Ferroe- (13) Liu, F. et al. Room-temperature ferroelec- lasticity and/or Multiferroicity in Two- tricity in CuInP 2 S 6 ultrathin ﬂakes. Na- Dimensional Phosphorene and Phospho- ture Communications 2016, 7 . rene Analogues. Nano Letters 2016, 16, (14) Chang, K.; Liu, J.; Lin, H.; Wang, N.; 3236–3241, PMID: 27096689. Zhao, K.; Zhang, A.; Jin, F.; Zhong, Y.; (23) Mehboudi, M.; Dorio, A. M.; Zhu, W.; Hu, X.; Duan, W.; Zhang, Q.; Fu, L.; van der Zande, A.; Churchill, H. O. H.; Xue, Q. K.; Chen, X.; Ji, S. H. Dis- Pacheco-Sanjuan, A. A.; Harriss, E. O.; covery of robust in-plane ferroelectricity in atomic-thick SnTe. Science 2016, 353, 274–278.

9 Kumar, P.; Barraza-Lopez, S. Two- Highly Anisotropic Mechanical and Op- Dimensional Disorder in Black Phos- tical Properties of 2D Layered As2S3 phorus and Monochalcogenide Monolay- Membranes. ACS Nano 2019, 13, 10845– ers. Nano Letters 2016, 16, 1704–1712, 10851. PMID: 26866878. (31) Momma, K.; Izumi, F. VESTA3 for three- (24) Gao, Y.; Wu, M.; Zeng, X. C. Phase tran- dimensional visualization of crystal, volu- sitions and ferroelasticity–multiferroicity metric and morphology data. Journal of in bulk and two-dimensional silver and Applied Crystallography 2011, 44, 1272– copper monohalides. Nanoscale Horiz. 1276. 2019, 4, 1106–1112. (32) Kampf, A.; Downs, R.; Housley, R.; Jenk- (25) Ding, W.; Zhu, J.; Wang, Z.; Gao, Y.; ins, R.; Hyrsl, J. Anorpiment, As2S3, the Xiao, D.; Gu, Y.; Zhang, Z.; Zhu, W. Pre- triclinic dimorph of orpiment. Mineralogi- diction of intrinsic two-dimensional ferro- cal Magazine 2011, 75, 2857–2867. electrics in In 2 Se 3 and other III 2 -VI 3 van der Waals materials. Nature Commu- (33) Perdew, J. P.; Burke, K.; Ernzer- nications 2017, 8 . hof, M. Generalized Gradient Approxima- tion Made Simple. Phys. Rev. Lett. 1996, (26) Chandrasekaran, A.; Mishra, A.; 77, 3865–3868. Singh, A. K. Ferroelectricity, An- tiferroelectricity, and Ultrathin 2D (34) Stergiou, A. C.; Rentzeperis, P. J. Electron/Hole Gas in Multifunctional The crystal structure of arsenic selenide, Monolayer MXene. Nano Letters 2017, As2Se3. Zeitschrift f¨urKristallographie - 17, 3290–3296. Crystalline Materials 1985, 173, 185–191.

(27) Lin, L.-F.; Zhang, Y.; Moreo, A.; (35) Resta, R. Macroscopic polarization in Dagotto, E.; Dong, S. Frustrated Dipole crystalline dielectrics: the geometric Order Induces Noncollinear Proper Ferri- phase approach. Rev. Mod. Phys. 1994, electricity in Two Dimensions. Phys. Rev. 66, 899–915. Lett. 2019, 123, 067601. (36) King-Smith, R. D.; Vanderbilt, D. Theory (28) Higashitarumizu, N.; Kawamoto, H.; of polarization of crystalline solids. Phys. Lee, C.-J.; Lin, B.-H.; Chu, F.-H.; Rev. B 1993, 47, 1651–1654. Yonemori, I.; Nishimura, T.; Wak- (37) Mullen, D.; Nowacki, W. Reﬁnement of abayashi, K.; Chang, W.-H.; Nagashio, K. the crystal structures of realgar, AsS and Purely in-plane ferroelectricity in mono- orpiment, As2S3. Zeitschrift f¨urKristallo- layer SnS at room temperature. Nature graphie - Crystalline Materials 1972, 136, Communications 2020, 11, 2428. 48.

(29) Chang, K.; K¨uster, F.; Miller, B. J.; (38) Baroni, S.; de Gironcoli, S.; Dal Corso, A.; Ji, J.-R.; Zhang, J.-L.; Sessi, P.; Barraza- Giannozzi, P. Phonons and related crystal Lopez, S.; Parkin, S. S. P. Microscopic properties from density-functional pertur- Manipulation of Ferroelectric Domains bation theory. Rev. Mod. Phys. 2001, 73, in SnSe Monolayers at Room Tempera- 515–562. ture. Nano Letters 2020, 20, 6590–6597, PMID: 32809837. (39) Berendsen, H. J. C.; Postma, J. P. M.; van ˇ Gunsteren, W. F.; DiNola, A.; Haak, J. R. (30) Siˇskins, M.; Lee, M.; Alijani, F.; van Molecular dynamics with coupling to an Blankenstein, M. R.; Davidovikj, D.; external bath. The Journal of Chemical van der Zant, H. S. J.; Steeneken, P. G. Physics 1984, 81, 3684–3690.

10 (40) Henkelman, G.; Jónsson, H. Improved (49) Brehm, J. A.; Neumayer, S. M.; Tao, L.; tangent estimate in the nudged elastic O’Hara, A.; Chyasnavichus, M.; band method for finding minimum energy Susner, M. A.; McGuire, M. A.; paths and saddle points. The Journal of Kalinin, S. V.; Jesse, S.; Ganesh, P.; Chemical Physics 2000, 113, 9978–9985. Pantelides, S. T.; Maksymovych, P.; Balke, N. Tunable quadruple-well ferro- (41) Barraza-Lopez, S.; Kaloni, T. P.; electric van der Waals crystals. Nature Poudel, S. P.; Kumar, P. Tuning the Materials 2020, 19, 43–48. ferroelectric-to-paraelectric transition temperature and dipole orientation of (50) Ai, H.; Song, X.; Qi, S.; Li, W.; group-IV monochalcogenide monolayers. Zhao, M. Intrinsic multiferroicity in Phys. Rev. B 2018, 97, 024110. two-dimensional VOCl2 monolayers. Nanoscale 2019, 11, 1103–1110. (42) Yuan, S.; Luo, X.; Chan, H. L.; Xiao, C.; Dai, Y.; Xie, M.; Hao, J. Room- (51) Xu, B.; Xiang, H.; Xia, Y.; Jiang, K.; temperature ferroelectricity in MoTe2 Wan, X.; He, J.; Yin, J.; Liu, Z. Monolayer down to the atomic monolayer limit. Na- AgBiP2Se6: an atomically thin ferroelec- ture Communications 2019, 10, 1775. tric semiconductor with out-plane polarization. Nanoscale 2017, 9, 8427–8434. (43) Huan, T. D.; Sharma, V.; Rossetti, G. A.; Ramprasad, R. Pathways towards ferro- (52) Jia, Y.; Zhao, M.; Gou, G.; Zeng, X. C.; electricity in hafnia. Phys. Rev. B 2014, Li, J. Niobium oxide dihalides NbOX2: 90, 064111. a new family of two-dimensional van der Waals layered materials with intrin- (44) Sang, X.; Grimley, E. D.; Schenk, T.; sic ferroelectricity and antiferroelectricity. Schroeder, U.; LeBeau, J. M. On the Nanoscale Horiz. 2019, 4, 1113–1123. structural origins of ferroelectricity in HfO2 thin films. Applied Physics Letters (53) Meyer, B.; Vanderbilt, D. Ab initio study 2015, 106, 162905. of ferroelectric domain walls in PbTiO3. Phys. Rev. B 2002, 65, 104111. (45) Wang, H.; Qian, X. Two-dimensional multiferroics in monolayer group IV (54) Lubk, A.; Gemming, S.; Spaldin, N. A. monochalcogenides. 2D Materials 2017, First-principles study of ferroelectric do- 4, 015042. main walls in multiferroic bismuth ferrite. Phys. Rev. B 2009, 80, 104110. (46) Cui, C. et al. Intercorrelated In-Plane and Out-of-Plane Ferroelectricity in Ultrathin (55) Ye, M.; Vanderbilt, D. Domain walls and Two-Dimensional Layered Semiconductor ferroelectric reversal in corundum deriva- In2Se3. Nano Letters 2018, 18, 1253– tives. Phys. Rev. B 2017, 95, 014105. 1258. (56) Kudin, K. N.; Scuseria, G. E.; Yakob- (47) Beckman, S. P.; Wang, X.; Rabe, K. M.; son, B. I. C2F, BN, and C nanoshell elas- Vanderbilt, D. Ideal barriers to polar- ticity from ab initio computations. Phys. ization reversal and domain-wall motion Rev. B 2001, 64, 235406. in strained ferroelectric thin films. Phys. Rev. B 2009, 79, 144124. (57) Akinwande, D. et al. A review on mechanics and mechanical properties of 2D ma- (48) Ye, M.; Vanderbilt, D. Ferroelectricity in terials—Graphene and beyond. Extreme corundum derivatives. Phys. Rev. B 2016, Mechanics Letters 2017, 13, 42 – 77. 93, 134303. (58) Jiang, J.-W.; Park, H. S. Mechanical properties of single-layer black phosphorus.

11 Journal of Physics D: Applied Physics (67) Troullier, N.; Martins, J. L. Eﬃcient pseu- 2014, 47, 385304. dopotentials for plane-wave calculations. Phys. Rev. B 1991, 43, 1993–2006. (59) Wei, Q.; Peng, X. Superior mechanical ﬂexibility of phosphorene and few-layer (68) Nye, J. F. Physical properties of crystals: black phosphorus. Applied Physics Letters their representation by tensors and matri- 2014, 104, 251915. ces; Clarendon Press: Oxford, UK, 1985.

(60) Giannozzi, P. et al. Advanced capabilities for materials modelling with QUANTUM ESPRESSO. Journal of Physics: Con- densed Matter 2017, 29, 465901.

(61) Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open- source software project for quantum simulations of materials. Journal of Physics: Condensed Matter 2009, 21, 395502.

(62) Chelikowsky, J. R.; Troullier, N.; Saad, Y. Finite-diﬀerence-pseudopotential method: Electronic structure calculations without a basis. Physical Review Letters 1994, 72, 1240–1243.

(63) Kronik, L.; Makmal, A.; Tiago, M. L.; Alemany, M. M. G.; Jain, M.; Huang, X.; Saad, Y.; Chelikowsky, J. R. PARSEC – the pseudopotential algorithm for real- space electronic structure calculations: recent advances and novel applications to nano-structures. physica status solidi (b) 2006, 243, 1063–1079.

(64) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuse- ria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406.

(65) Corso, A. D. Pseudopotentials periodic table: From H to Pu. Computational Mate- rials Science 2014, 95, 337 – 350.

(66) Vanderbilt, D. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Phys. Rev. B 1990, 41, 7892– 7895.

12 Graphical TOC Entry

Pmmn Pmn21

y x