Colloquium: Physical properties of group-IV monochalcogenide monolayers
Salvador Barraza-Lopez, Benjamin M. Fregoso, John W. Villanova, Stuart S. P. Parkin, Kai Chang
CColloquium: Physical properties of group-IV monochalcogenide monolayers
Salvador Barraza-Lopez ∗ Department of Physics,University of Arkansas,Fayetteville, AR 72701,USA
Benjamin M. Fregoso
Department of Physics,Kent State University,Kent, OH 44242,USA
John W. Villanova
Department of Physics,University of Arkansas,Fayetteville, AR 72701,USA
Stuart S. P. Parkin
Max Planck Institute of Microstructure Physics,Weinberg 2, Halle 06120,Germany
Kai Chang † Beijing Academy of Quantum Information Sciences, Beijing 100193,China (Dated: September 10, 2020)
We survey the state-of-the-art knowledge of ferroelectric and ferroelastic group-IVmonochalcogenide monolayers. These semiconductors feature remarkable structuraland mechanical properties, such as a switchable in-plane spontaneous polarization, softelastic constants, structural degeneracies, and thermally-driven two-dimensional struc-tural transformations. Additionally, these 2D materials also display selective valleyexcitations, valley Hall effects, and persistent spin helix behavior. After a descrip-tion of their Raman spectra, a discussion of optical properties arising from their lackof centrosymmetry—such as an unusually strong second-harmonic intensity, large bulkphotovoltaic effects, photostriction, and tunable exciton binding energies—is provided aswell. The physical properties observed in these materials originate from (correlate with)their intrinsic and switchable electric polarization, and the physical behavior herebyreviewed could be of use in non-volatile memory, valleytronic, spintronic, and optoelec-tronic devices: these 2D multiferroics enrich and diversify the 2D materials toolbox.
CONTENTS
I. Introduction: The diversity of ultrathin ferroelectrics 2II. Atomistic structure and chemical bonding of O − MX sfrom the bulk to MLs 3III. Experimentally available O − MX MLs 7A. SnS MLs 7B. SnSe and SnTe MLs 7IV. Switching the direction of P on O − MX MLs:demonstrating ferroelectric behavior 8 ∗ [email protected] † [email protected] A. Polarization switching and ultrathin memories basedon in-plane ferroelectric tunnel junctions 9V. Linear elastic properties, auxetic behavior, andpiezoelectricity of O − MX MLs 9VI. Structural degeneracies and anharmonic elastic energy ofO − MX MLs 10VII. Structural phase transition and pyroelectric behavior ofO − MX MLs 11VIII. Electronic, valley and spin properties of O − MX MLs 13A. Electronic band structure 13B. Valleytronics 14C. Persistent spin helix behavior 15IX. Optical properties of O − MX MLs 16 a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p A. Optical absorption 16B. Raman spectra 16C. Second harmonic generation 16D. Bulk photovoltaic effects: injection and shiftcurrents 18E. Photostriction 18F. Excitons 18X. Summary and outlook 19Acknowledgments 19References 19
I. INTRODUCTION: THE DIVERSITY OF ULTRATHINFERROELECTRICS
Ferroelectric materials have a spontaneous, intrinsicpolarization P that can be switched by external electricfields. The first ferroelectric material—Rochelle salt—was discovered about a century ago (Valasek, 1921). De-spite a long history of applications of ferroelectrics inelectric and electronic devices, the modern theory of fer-roelectricity based on Berry phase—which made accu-rate comparisons between theory and experimental mea-surements possible—was not established until the 1990s(King-Smith and Vanderbilt, 1993; Resta, 1994); seeRef. (Rabe et al. , 2007) for more details. From thatpoint on, deep connections of this field with the geome-try and topology of quantum mechanical wave functionshave been pointed out (Bernevig and Hughes, 2013; Van-derbilt, 2018). Ferroelectric behavior is relevant fromboth a fundamental physical perspective as well as ap-plications, and this Colloquium was written to highlightthe physical properties of 2D ferroelectric and ferroelasticmaterials within the group-IV monochalcogenide family(Littlewood, 1980).Researchers have always wondered whether there is acritical thickness for ferroelectric behavior below whichpolarization switching becomes suppressed (Rabe et al. ,2007). Considering non-layered ferroelectric films withan out-of-plane intrinsic polarization P , it was initiallythought that the depolarization field arising from an in-complete cancellation of the space charge and an out-of-plane polarization charge at an electrode-ferroelectric in-terface [see, e.g., Refs. (Black et al. , 1997; Janovec, 1959;Mehta et al. , 1973; Merz, 1956; Triebwasser, 1960)] wouldraise the total energy of the system and eventually sup-press the polarized state. Nevertheless, and as growthtechniques for thin films developed, the experimentallyextracted critical thickness of ferroelectric thin films de-creased from over 100 nm (Feuersanger et al. , 1964) totens of nanometers (Slack and Burfoot, 1971; Tomash-polski, 1974; Tomashpolski et al. , 1974), and eventuallyto only a few unit cells (u.c.s) (Bune et al. , 1998; Ty-bell et al. , 1999). The behavior of ultrathin ferroelectricfilms has been predicted to high precision by first princi-ples calculations, suggesting critical thicknesses of several u.c.s for certain materials (Gerra et al. , 2006; Junqueraand Ghosez, 2003; Meyer and Vanderbilt, 2001; Sai et al. ,2005; Wu et al. , 2004; Zembilgotov et al. , 2002), or single-u.c. thickness for others (Almahmoud et al. , 2010, 2004;Sai et al. , 2009; Zhang et al. , 2014). Concurrently, so-phisticated experiments on select compounds [PbTiO (3u.c.s) (Fong et al. , 2006, 2004), BaTiO (4 u.c.s) (Tenne et al. , 2006, 2009), PbZr . Ti . O (1.5 u.c.s) (Gao et al. ,2017), YMnO (2 u.c.s) (Nordlander et al. , 2019), andBiFeO (1 u.c.) (Wang et al. , 2018a)] continue to pushthe critical thickness toward the single u.c. limit.Meanwhile, a series of ultra-thin layered ferroelectricmaterials—especially attractive for the design and fabri-cation of functional (van der Waals) heterostructures—have been experimentally discovered, including In Se (Cui et al. , 2018a; Ding et al. , 2017; Poh et al. , 2018;Wan et al. , 2018; Xiao et al. , 2018; Xue et al. , 2018a,b;Zheng et al. , 2018; Zhou et al. , 2017), CuInP S (Deng et al. , 2019; Liu et al. , 2016), BA PbCl (Liao et al. ,2015; You et al. , 2018), d1T-MoTe (Yuan et al. , 2019),1T-WTe (Fei et al. , 2018) [which is also a quantumspin Hall material (Asaba et al. , 2018; Fei et al. , 2017;Qian et al. , 2014; Song et al. , 2018; Tang et al. , 2017;Wu et al. , 2018)], and of course, monolayers (MLs) ofgroup-IV monochalcogenides like SnS, SnSe and SnTe.A brief and experimentally-driven summary of layeredferroelectrics is provided in Table I.The discovery of 2D and layered ferroelectrics facili-tates the design of future non-volatile devices that arefully made of 2D material heterostructures. The ex-perimentally verified 2D ferroelectric materials exhibitboth out-of-plane and in-plane switchable spontaneouspolarizations in few-layer films and at room temperature.Some prototype devices have also been demonstrated.For example, a ferroelectric diode in a graphene/ α -In Se heterostructure has a relatively low coercive field of 200kV/cm, and an electric current on/off ratio of ∼ (Wan et al. , 2018). A d1T-MoTe ferroelectric tunnel-ing junction yielded an electric current on/off ratio of1,000 (Yuan et al. , 2019).Among all ultrathin ferroelectrics, a family of ferro-electric semiconductors with moderate band gaps knownas group-IV monochalcogenide MLs—and referred toas M X
MLs henceforth—exhibit outstanding propertiesthat are promising for many applications. By far, theyare the only family of 2D ferroelectrics experimentallyshown to display a robust and switchable in-plane spon-taneous polarization at the limit of a single van der WaalsML at room temperature. Furthermore, many intriguingphysical behaviors in
M X have been theoretically pre-dicted in
M X
MLs such as selective valley excitations,valley Hall effects, persistent spin helix behavior, etc.
Nevertheless, and despite of these attractive theoreti-cal predictions, the experimental growth and characteri-zation remain difficult, partly because of reduced sampledimensions. Therefore, reviewing the current achieve-
TABLE I Experimentally reported layered ferroelectrics, including the space group of the ferroelectric phase, its intrinsicand switchable polarization [in-plane (IP) and out-of-plane (OOP)], preparation methods employed, the critical temperatureabove which a paraelectric phase ensues, the coercive field E c (thicknesses for which T c and E c were determined are addedin parenthesis), and other related properties. If the space group was not specified in the source literature, a prefix “d-” (for distorted ) was added in front of the space group of the corresponding undistorted high-symmetry structure. ML stands formonolayer. P stands for spontaneous polarization.Material Space group P direction Preparation a T c (K) E c (kV/cm) Other properties α -In Se R m IP + OOP ME 700 (4 ML) 200 (5 nm) d = 0 .
34 pm/V b (1 ML) β (cid:48) -In Se d- R m IP ME >
473 (100 nm)CuInP S Cc IP + OOP c ME >
320 (4 nm)BA PbCl Cmc IP ME >
300 (2 ML) 13 (bulk) P ∼ µ C/cm d1T-MoTe d- P m OOP ME, MBE 330 (1 ML)1T (cid:48) -WTe P nm OOP ME 350 (2 − P (cid:54) = 0 only when ≥ P mn IP MBE, PVD >
300 (1 −
15 ML) 10.7/25 (1/9 ML)SnSe ML P mn IP MBE 380 ∼
400 (1 ML) 140 (1 ML)SnTe ML P mn IP MBE 270 (1 ML) a MBE, molecular beam epitaxy; ME, mechanical exfoliation; PVD, physical vapor deposition b Piezoelectric coefficient c Vanishing in-plane polarization below a critical thickness of 90 −
100 nm. ments and spurring a broader interest in this field pro-vided the motivation to write this Colloquium. Despitethe existence of several reviews focusing on the compu-tational (Cui et al. , 2018b; Wu and Jena, 2018), exper-imental/computational (Guan et al. , 2019), and exper-imental/theoretical (Titova et al. , 2020) aspects of 2Dferroelectrics, an all-encompassing review dedicated tothe physical behavior of
M X
MLs is still missing.The structure of this Colloquium is as follows. Theatomistic structure of O − M X s in the bulk and MLsis discussed in Sec. II. Atomistic coordination, the na-ture of their chemical bond, group symmetries, orderparameters, as well as unexpected atomistic configura-tions experimentally obtained are covered in this Section.Sec. III introduces the three members of this family (SnS,SnSe, and SnTe) that have been grown at the ML limit.Experimental characterization, including the verificationof polarization at exposed edges in ML nanoplates, canbe found there as well. The experimental ferroelectricswitching of SnS, SnSe, and SnTe MLs is discussed inSec. IV; novel memory concepts based on an in-plane fer-roelectric switching are also introduced in that Section.Linear elastic properties, structural degeneracies, andfinite temperature thermal behavior (including phasetransitions) are covered in Secs. V, VI, and VII, respec-tively. In a nutshell, O − M X
MLs are much softer thangraphene, hexagonal boron nitride MLs, and transitionmetal dichalcogenide (TMDC) MLs. Their linear elasticproperties, unusually large piezoelectric coefficients, andauxetic behavior are described in Sec. V. The elastic en-ergy landscape is introduced in Sec. VI, which permitsunderstanding the structural degeneracies of these 2Dferroelectrics, and the structural phase transitions thatare discussed in Sec. VII.Electronic and optical properties of O − M X
MLs are the subjects of Secs. VIII and IX. The electronic prop-erties are discussed in a gradual manner that includesband structures and valley properties without spin-orbitcoupling, and a subsequent exposition of (spin-enabled)persistent spin helix behavior. Optical properties includethe anisotropic absorption spectra, Raman spectra, SHG,injection and shift currents, photostriction, and excitoniceffects. A summary and outlook is presented in Sec. X.A unified and consistent notation has been deployedto streamline the discussion. In particular, the choice ofcrystallographic axes is such that orthogonal lattice vec-tors a , , a , , and a , correspond to crystallographicvectors a , b , and c and point along the x − , y − , and z − direction, respectively [ a ( a ) is the so-called arm-chair (zigzag) direction]. These choices will lead to amodification of space group labeling, a redefinition ofhigh-symmetry points in the electronic band structure,and to the relabeling of tensors from some of the sourceliterature. The benefit from this effort is a self-containeddiscussion that is not interrupted from a lack of a stan-dard notation. In addition, given that the structureof these materials evolves as a function of mechanicalstrain, temperature, electric field, and optical illumina-tion, structural variables with a zero subindex representtheir value on a ground state configuration at zero tem-perature and without external perturbations. II. ATOMISTIC STRUCTURE AND CHEMICALBONDING OF O − MX S FROM THE BULK TO MLS
Group-IV monochalcogenides are binary compoundswith a chemical formula
M X , where M is a group IVAelement and X belongs to group VIA in the PeriodicTable. Even though carbon, silicon, lead, oxygen, andeven polonium belong to these groups, M X compoundscontaining these elements will not be discussed here forthe following reasons: SiS MLs possess a ground statestructure with
P ma et al. , 2016) whichlacks a net intrinsic electric polarization; see Ref. (Ka-mal et al. , 2016) concerning a lower-energy structure for2D SiSe, too. Pb X compounds lack a net P regardlessof the number of layers (more on this later). Similarly,materials such as 2D SiO, GeO, and SnO display a non-ferroelectric litharge structure (Kamal et al. , 2016; Lefeb-vre et al. , 1998). This way, M will either be germanium(Ge) or tin (Sn) while X represents sulphur (S), selenium(Se), or tellurium (Te) in what follows.GeTe and SnTe are rhombohedral (R − phase) and GeS,GeSe, SnS, and SnSe turn orthorhombic (O − phase) inthe bulk (Littlewood, 1980). As illustrated in Fig. 1(a),O − M X compounds have a layered structure. One MLrefers to a van der Waals layer [or two atomic layers (2ALs)] or half of an O − M X u. c.Strictly speaking, the intrinsic switchable polarization P should not be showcased as a vector on periodic struc-tures. Therefore, in certain theoretical discussions, wewill utilize an order parameter p (parallel to P ) thatremains well-defined on periodic structures. The letter p stands for projection , and this order parameter is definedin the next paragraph.Consider the vector r XM starting at X atom 1 andending at the nearest M atom (atom 2) in the lower MLseen in Fig. 1(a). The positions of the remaining twoatoms within the lower ML (3 and 4) are obtained by ascrew operation about the x − axis, or by a diagonal ( n )glide operation about the z − axis applied to r XM . Call-ing r (cid:48) XM the vector starting at ( X ) atom 3 and endingat ( M ) atom 4, we define p = r XM + r (cid:48) XM . The mirrorsymmetry along the x − z plane makes p · ˆ y = 0, whilethe screw operation renders p · ˆ z = 0, so that p is par-allel to the longer lattice vector a , . These symmetriesalso render a zero intrinsic polarization along the y − and z − directions. Ferroelectric O − M X
MLs belong to spacegroup 31 (Kamal et al. , 2016; Rodin et al. , 2016) [usuallywritten as
P nm but labeled P mn for lattice vectorsas drawn in Fig. 1(c)], and a top view of their anisotropicu. c. is provided in Fig. 1(c) within solid lines. Phonondispersion calculations demonstrate the structural stabil-ity of these MLs (Singh and Hennig, 2014). A zero valueof the order parameter θ in a ML with dissimilar latticeconstants ( a , (cid:54) = a , ) leads onto a paraelectric structurewith p = belonging to symmetry group 59 ( P mmn ).The alternating direction of p (or antipolar coupling)within each ML arises from the inversion center shownin the side view along the x − z plane in Fig. 1(a) (theatoms related by inversion are joined by dash-dot lines).Bulk O − M X s belong to space group 62 [
P nma (Gomesand Carvalho, 2015), or
P cmn with the lattice vectorsemployed here]. The side view of the x − z plane inFig. 1(a) also contains the u. c. boundaries in solid line,interatomic distances d , < d , < d , , a tilt angle θ , FIG. 1 (a) Structure of bulk O − MX s, with the M atomshown in gray (big) and the X atom in yellow (small) cir-cles. Left: 3D view of two MLs with antiparallel polarization.Right: side view along the x − z plane showing a , and a , lattice vectors ( a , points into the page). The inversion cen-ter swaps the direction of p at consecutive MLs (red arrows),and p = | p | ∝ θ . Interatomic distances d , , d , , and d , as well as ∆ h are shown, too. (b) Electronic density showinga lone pair ζ , and ∆ h . Bold (shaded) bold circles stand fortin (sulfur or selenium). Adapted from Ref. (Lefebvre et al. ,1998) with permission. Copyright, 1998, American PhysicalSociety. (c) Top view of a O − MX ML. ∆ α measures the de-viation from 90 ◦ of the rhombus highlighted by dashed lines;∆ α = 0 when a , = a , . (d) The a , /a , ratio (propor-tional to ∆ α ) is tunable by the compound’s average atomicnumber ¯ Z . Adapted from Ref. (Mehboudi et al. , 2016a) withpermission. Copyright, 2016, American Chemical Society. and the height ∆ h of an X atom relative to its nearest M atom (Kamal et al. , 2016). A net switchable P ensues inbinary compounds lacking inversion symmetry, which isthe case for individual MLs of O − M X s (Fei et al. , 2015;Gomes and Carvalho, 2015; Gomes et al. , 2015; Singh andHennig, 2014; Tritsaris et al. , 2013; Zhu et al. , 2015). Theatomistic structure and the in-plane p of O − M X
MLs,can be understood on the basis of the chemistry of blackphosphorus (BP) MLs as follows.Carbon belongs to group IVA and graphite has four va-lence electrons and an sp hybridization. Phosphorus (P)belongs to group VA, and black phosphorus has five va-lence electrons and displays an sp hybridization (Kamal et al. , 2016). In (three-fold coordinated) graphite, threeatoms form strong ( σ ) in-plane bonds and the fourth ( π )electron protrudes out of plane. BP is three-fold co-ordinated as well, having its closest neighboring atomat a distance d and two additional atoms located at aslightly larger distance d . Given that a phosphorus atomcontains five valence electrons, such three-fold coordina-tion requires the existence of two additional non-bondedelectrons [known as a lone pair ( ζ )] per atom. Unlike TABLE II Interatomic distances in bulk SnS and SnSe. N isthe number of neighbors at any given distance. Taken fromRef. (Lefebvre et al. , 1998).Material atoms d (˚A) N Material atoms d (˚A) N SnS Sn-S 2.63 ( d , ) 1 SnSe Sn-Se 2.74 ( d , ) 12.66 ( d , ) 2 2.79 ( d , ) 23.29 ( d , ) 2 3.34 ( d , ) 23.39 1 3.47 1Sn-Sn 3.49 2 Sn-Sn 3.55 2S-S 3.71 4 Se-Se 3.89 43.90 2 3.94 2 graphene, which maintains a planar configuration withtwo atoms in its u. c., lone pairs confer BP MLs witha puckered structure and a rectangular u. c. containingfour atoms.Similar to hexagonal boron nitride—which is made outof a group IIIA element (B) and a group VA element (N)and is isostructural to graphite—Fig. 1(a) indicates thatO − M X s are isostructural to BP. Table II shows thatbulk SnS has similar distances d and d for a three-fold atomistic coordination, and the same can be saidof interatomic distances in bulk SnSe, also listed in theTable. The equivalent to a lone pair ζ is assigned to themore negatively charged X atom in Fig. 1(b) (Lefebvre et al. , 1998). The reader may notice that ∆ h is positivein Fig. 1(a) and negative in Fig. 1(b): its sign determinescertain elastic properties that will be discussed in Sec. V.The rhombic distortion angle ∆ α (Chang et al. , 2016)shown in Fig. 1(c) indicates the anisotropy of the u.c. andis related to the ratio of lattice constants a , /a , asfollows (Barraza-Lopez et al. , 2018): a , a , = 1 + sin ∆ α cos ∆ α , (1)or ∆ α (cid:39) a , a , − α is expressedin radians.Letting Z M ( Z X ) be the atomic number of atom M ( X ) and defining the average atomic number ¯ Z = ( Z M + Z X ) /
2, Fig. 1(d) illustrates a decaying exponential de-pendence of a , a , − Z (Mehboudi et al. , 2016a). a , a , − reversible strain (Wu andZeng, 2016) or tetragonality ratio , and it correlates with P in bulk ferroelectrics (Lichtensteiger et al. , 2005).Fig. 1(d) indicates that—at zero temperature—latticevectors turn more equal (unequal) on heavier (lighter) M X
MLs. Having equal lattice vectors, Fig. 1(d) showsthat Pb-based
M X
MLs are paraelectric [ P = 0; a be-havior experimentally confirmed on PbTe MLs; see Sup-porting Information in Ref. (Chang et al. , 2016)] and arenot discussed here for that reason. Ferroelectric O − M X
MLs with a , (cid:54) = a , have similar structures and hencedisplay similar physical behavior; this observation willpermit drawing meaningful comparisons between differ- TABLE III Net charge transfer ∆ Q (in e ) from atom M toatom X and change in electronegativity ∆ ξ = ξ X − ξ M (ineV) for O − MX MLs. Taken from Ref. (Kamal et al. , 2016).Material ¯ Z ∆ Q ∆ ξ Material ¯ Z ∆ Q ∆ ξ GeS ML 24 0.815 0.57 SnS ML 33 0.980 0.62GeSe ML 33 0.649 0.54 SnSe ML 42 0.855 0.59GeTe ML 42 0.372 0.09 SnTe ML 51 0.596 0.14 ent experimental and theoretically studied compoundswithin this material family.Continuing the discussion of chemistry, one observes inTable III a correlation between the charge transfer ∆ Q [or ionicity (Littlewood, 1980)] from the group IVA ele-ment onto the one belonging to group VIA, and Pauling’sdifference in electronegativity ∆ ξ . Although an interplayamong covalent, ionic, and resonant bonding has beenargued to describe M X s, a new type of bonding (called metavalent , and thought of as a combination of ‘metallic’and ‘covalent’) has been proposed to classify these ma-terials (Kooi and Wuttig, 2020; Raty et al. , 2019; Ron-neberger et al. , 2020). Variables employed to identifythe appropriate type of bonding include the coordina-tion number, the electronic conductivity, the dielectricconstant (cid:15) ∞ , the bond polarizability, and the lattice an-harmonicity. Setting up a two-dimensional map wherethe horizontal axis is the charge transfer ∆ Q and thevertical axis (named electron sharing ) is a measure ofelectronic exchange and correlation (Raty et al. , 2019),metavalent compounds sit in between covalently-bondedand metallic materials. In the bulk, materials such asGeS, GeSe, SnS, and SnSe are assigned a covalent bond-ing, while GeTe can display either covalent or metavalentbonding depending on its phase [R and cubic (C) phasesbeing metavalent and the O phase being covalent]; bulkSnTe, PbS, PbSe, and PbTe are assigned a metavalentcharacter (Kooi and Wuttig, 2020; Raty et al. , 2019).The in-plane u. c. area ( i.e. , | a , × a , | ) of O − M X sis a function of the number of MLs (Dewandre et al. ,2019; Hu et al. , 2015; Poudel et al. , 2019; Ronneberger et al. , 2020; Yang et al. , 2018), a feature observed inBP as well (Shulenburger et al. , 2015) that is relatedto the thickness-dependent spatial distribution of lonepairs. Such a dependence of | a , × a , | on thicknessis not observed in more traditional 2D materials such asgraphene and TMDCs.Leaving a detailed discussion of ultrathin film creationand characterization to Sec. III, Figs. 2(a) and 2(b) dis-play few-ML SnS and SnTe films and provide strikingexamples of unexpected structure: indeed, while bulkSnS displays the P cmn group symmetry, SnS grown onmica can take on a ferroelectrically-coupled (sometimeslabeled AA) stacking sequence for up to fifteen MLs(Higashitarumizu et al. , 2020) [Fig. 2(a)], with a
P cmn group symmetry acquired by subsequent MLs on thicker
FIG. 2 (a) Cross-sectional STEM images of few-ML SnSgrown on mica exhibit ferroelectric coupling for up to 15MLs. Reproduced from Ref. (Higashitarumizu et al. , 2020)with permission. Copyright, 2020, Springer Nature. (b) Ul-trathin SnTe grown on epitaxial graphene develops an anti-ferroelectric coupling, as demonstrated by band bending atthe exposed edges of few-layer nanoplates (subplots i throughvi). Lowermost diagrams indicate electrostatic interactions atexposed edges upon antiferroelectric coupling. Reproducedfrom (Chang et al. , 2019a) with permission. Copyright, 2019,John Wiley and Sons. (c and d) Demonstration of layer-ing in ferroelectrically-coupled ultrathin SnTe by the unevendistance d i among layers. Reproduced from (Yang et al. ,2018) and (Liu et al. , 2018) with permission. Copyright, 2017,American Institute of Physics and 2018, American PhysicalSociety. (e) A SnTe bilayer with antiferroelectric (AFE) cou-pling has a lower total energy when compared to a ferroelec-tric (FE) coupled one. Reproduced from (Kaloni et al. , 2019)with permission. Copyright, 2019, American Physical Society. films. Non-polar, thick SnS can be switched into a fer-roelectric phase by an external electric field (Bao et al. ,2019). Additional experimental M X morphologies in-clude GeS nanowires created along an axial screw dislo-cation (Sutter et al. , 2019) and the antiferroelectrically-coupled ultrathin SnTe grown on (metallic) epitaxialgraphene (Chang et al. , 2019a) that is discussed next.Bulk SnTe displays a metavalent, R − phase in the bulk.Grown on a metallic substrate, ultrathin SnTe flakes witha 3-ML, bilayer, or ML thicknesses were characterizedwith a scanning tunneling microscope (STM), which per-mits elucidating their in-plane polarization P switchingfrom the band bending of the conduction band edge ob-served in Fig. 2(b). These STM spectra were capturedalong the dashed straight lines at subplots (i) through(vi) in Fig. 2(b) cutting through the nanoplates’ edges(Chang et al. , 2019a, 2020, 2016). (Additional details on the determination of P will be provided in Sec. III.) Bandbending is almost non-existent in SnTe bilayers, whichimplies an antipolar coupling among MLs, and showsthat the bonding of SnTe transitions from metavalentin the bulk to covalent in ultrathin films (Ronneberger et al. , 2020).Three theoretical works (Liu et al. , 2018; Ronneberger et al. , 2020; Yang et al. , 2018) explain the layered na-ture of ultrathin SnTe. They were performed using ei-ther the local density approximation [LDA (Perdew andZunger, 1981)] or the generalized gradient approxima-tion as implemented by Perdew, Burke, and Ernzer-hof [PBE (Perdew et al. , 1996)] for exchange-correlation(XC) within density-functional theory (Martin, 2004)and assume a bulk-like ( i.e., ferroelectric ) stacking of suc-cessive MLs in freestanding SnTe configurations as theone depicted in Fig. 2(c).Bulk SnTe features a Peirels distortion—a result of thecompetition among electron delocalization and localiza-tion (Ronneberger et al. , 2020)—that creates a net polar-ization along its diagonal and distorts a cubic lattice intoa rhombohedral one. As a result: (i) a bulk u. c. has bothin-plane and an out of plane polarization and, consider-ing two atomic layers as a ML, (ii) consecutive MLs arecoupled ferroelectrically. This is different to O − M X s,compounds with no net out-of-plane polarization and anantipolar coupling among successive MLs; see Fig. 1(a).Nevertheless, the depolarization field quenches the out-of-plane polarization of SnTe films (Liu et al. , 2018), cre-ating an in-plane lattice expansion (Yang et al. , 2018)and a separation between MLs resulting in the layeredstructure seen in Fig. 2(d). Freestanding SnTe films withferroelectric coupling have an intrinsically higher T c thantheir bulk counterpart due to an interplay among hy-bridization interactions and Pauli repulsion. Addition-ally, electron sharing (Raty et al. , 2019) increases withdecreasing thickness, imparting chemical bonds with amore covalent character (Ronneberger et al. , 2020).Most computational works on M X s that employdensity-functional theory make use of the PBE approxi-mation (Perdew et al. , 1996) to XC. Yet, and as seen inFig. 2(e), the experimentally observed antipolar couplingand the magnitude of ∆ α on bilayer SnTe films is recov-ered when using self-consistent van der Waals [vdW-DF-cx (Berland and Hyldgaard, 2014)] interactions (Kaloni et al. , 2019). In any case, Figs. 2(a) and 2(b) indicatethat the details of the initial surface are crucial for thetype of atomistic structure formed by ultrathin M X films(Kooi and Wuttig, 2020).
FIG. 3 (a) Atomic force microscopy topographic image ofa SnS ML on mica. Reproduced from Ref. (Higashitarumizu et al. , 2020) with permission. Copyright, 2020, Springer Na-ture. (b) STM topography of a SnTe ML nanoplate on epi-taxial graphene, with head-to-tail 90 ◦ domains having an in-trinsic polarization P indicated by arrows. Adapted fromRef. (Chang et al. , 2016). Copyright, 2016, American Associ-ation for the Advancement of Science. (c) Low temperature dI/dV spectra across the dashed arrow in the inset for a SnSeML nanoplate on graphene. (c and d) Room temperatureSTM topography and dI/dV : higher dI/dV implies a largerelectronic charge. Taken from Ref. (Chang et al. , 2020) withpermission. Copyright, 2020, American Chemical Society. III. EXPERIMENTALLY AVAILABLE O − MX MLSA. SnS MLs
Room-temperature in-plane ferroelectricity wasdemonstrated in few-ML SnS by a combination ofpiezo-force microscopy (PFM), second harmonic gener-ation (SHG), and electric transport experiments (Bao et al. , 2019). In order to overcome PFM’s weaknessin detecting the in-plane polarization of O − M X s, SnSfilms were grown by molecular beam epitaxy (MBE)on corrugated graphite substrates so that P was notperfectly perpendicular to the PFM tip and a finitepolarization signal could be picked up. Ferroelectricdomains and PFM hysteresis loops were resolved on 10nm thick SnS films grown on mica, and a SHG signal was collected, too. The modification of film morphologyillustrates the difficulties of traditional techniques suchas PFM to characterize ultrathin ferroelectric films withan in-plane P switching, and the need to develop newtechniques to characterize these ferroelectrics withoutchanging morphology. Two-terminal devices were fabri-cated on a 15 nm thick SnS film grown on mica whichwas subsequently transferred onto a doped Si substratecovered by 300 nm thick SiO . Hysteresis was found inthe I − V curves with a coercive field of 10.7 kV/cmand a maximum I on /I off ratio of ∼ et al. , 2019).The creation and characterization of SnS MLs has beenreported subsequently (Higashitarumizu et al. , 2020).SHG signals—a signature of lack of inversion symmetryand ferroelectricity that will be discussed from a com-bined theory/experiment perspective in Sec. IX.C—weredetected in SnS ML flakes grown via physical vapor depo-sition (PVD) on (insulating) mica. Two-terminal devicespatterned onto these as grown flakes display hysteresisin I − V loops—yet another signature of ferroelectricitythat will be discussed in Sec. IV. Consistent with the MLarrangement depicted in Fig. 2(a), ferroelectricity is de-tected in SnS films composed of up to fifteen MLs, includ-ing those composed of an even number of MLs . Ferroelec-tricity is unexpected in even-ML O − M X films becausethey are assumed to be centrosymmetric, lacking a netpolarization according to the
P cmn group symmetry. Acoercive field of 25 kV/cm was found for 9-ML thick SnSby electric transport measurements. An apparent rem-nant polarization as large as P r ∼ µ C/m was exper-imentally determined, much larger than the theoreticalvalue of 240 −
265 pC/m listed in Table IV, and probablyan artifact due to the relatively high conductance of SnS.
B. SnSe and SnTe MLs
The first experimentally discovered 2D ferroelectricin the O-
M X family is the SnTe ML grown by MBEon (metallic) graphene (Chang et al. , 2016; Chang andParkin, 2019) and characterized by STM (Chang et al. ,2019a, 2016, 2019b) in Fig. 3(b). As seen in Figs. 3(c) and3(d), SnSe MLs have been grown by MBE on graphene,too (Chang et al. , 2020). Due to the metallic substratein which these are grown, the techniques employed tocharacterize ultrathin SnSe and SnTe films are differentand complementary to those employed for SnS. AlthoughSTM is an unconventional tool to study ferroelectrics,its extreme surface sensitivity and access to the materi-als’ local electronic structure are advantageous for study-ing ultrathin ferroelectric flakes with an in-plane intrin-sic polarization, where PFM lacks sensitivity and mayeven damage ultrathin samples. STM measurements arehelped by the fact that these ultrathin films are not insu-
TABLE IV Spontaneous in-plane polarization P (in pC/m)as determined by DFT with the PBE XC functional.Material ¯ Z P a P b P c Material ¯ Z P a P b P c GeS ML 24 484 480 486 SnS ML 33 260 240 265GeSe ML 33 357 340 353 SnSe ML 42 181 170 190GeTe ML 42 − −
308 SnTe ML 51 − − a Ref. (Wang and Qian, 2017b). b Ref. (Rangel et al. , 2017). c Our calculations. lators but semiconductors, such that a tunneling currentcan be established into the metallic substrate (Chang andParkin, 2020).Being a vector, P has a magnitude P , an orientation,and sense of direction. SHG can only tell orientation,while two-terminal electric measurements and STM [seeFigs. 2(b-e)] can determine orientation and sense of di-rection. These three techniques require additional cali-bration to uncover the magnitude ( P ), whose calculatedvalues are listed in Table IV.Similar to subplots (v) and (vi) in Fig. 2(b) showingband bending of SnTe MLs on STM topography images,SnSe ML nanoplates display the band bending seen inFig. 3(c) as a result of bound charges accumulated atthe nanoplates’ edges, reflecting the in-plane polariza-tion P of these 2D ferroelectrics. The direction of P forSnSe and SnTe MLs—shown by arrows with a P label inFigs. 2(b) and 3(b-e)—is identified by band bending atnanoplate edges, and by the difference of lattice param-eters a and a as extracted from atomically resolvedSTM images. Stripe-shaped ∼ ◦ “head-to-tail” do-mains are observed in SnTe monolayer plates in Fig. 3(b)(Chang et al. , 2016), while 180 ◦ domains are formed inSnSe MLs (Chang et al. , 2020). The different type ofdomains formed in SnSe and SnTe MLs has to do with alattice commensuration of SnSe MLs on graphene (Chang et al. , 2020). A decrease of Sn vacancy concentration by2 ∼ et al. , 2016). Electronicband gaps of SnSe and SnTe MLs (obtained by the deter-mination of the valence and conduction band edges via dI/dV measurements) are listed in Table VII.Band bending disappears in SnTe MLs at 270 K(Chang et al. , 2016), but it can still be observed at 300 Kin SnSe MLs, implying a robust in-plane ferroelectricityat room temperature in 2D SnSe. According to variabletemperature dI/dV mapping experiments, T c reaches380 −
400 K for SnSe MLs, a promising magnitude forroom-temperature applications. A theoretical descrip-tion of thermally-driven structural transformations canbe found in Section VII.
FIG. 4 (a) Ferroelectric resistive switching of a SnS ML onmica via source/drain bias. Taken from Ref. (Higashitaru-mizu et al. , 2020) with permission. Copyright, 2020, SpringerNature. (b) Ferroelectric switching on a SnSe ML is achievedby bias voltage pulses V P applied on the STM tip at a point onthe graphene substrate close the SnSe ML plate. (c) Consec-utive dI/dV images along a ferroelectric switching sequenceof a SnSe ML at room temperature. The pulses were ap-plied at the point indicated by the cross in the first panel,and white dashed lines indicate a 180 ◦ domain wall. Adaptedfrom Ref. (Chang et al. , 2020) with permission. Copyright,2020, American Chemical Society. (d) Ferroelectric switchingof a SnTe ML by an STM tip. Taken from Ref. (Chang et al. ,2016) with permission. Copyright, 2016, American Associa-tion for the Advancement of Science. IV. SWITCHING THE DIRECTION OF P ON O − MX MLS: DEMONSTRATING FERROELECTRIC BEHAVIOR O − M X
MLs can only be considered ferroelectrics ifthey can be controllably switched by an external electricfield. For this purpose, a two-terminal device was builtby adding silver contacts to the SnS ML grown on mica,and the current I D was measured as the drain bias V D was swept from − − ◦ domainsin a SnSe ML nanoplate. Demonstrating ferroelectriccontrol, the polarization of the whole plate can be re-versed by this approach. Statistical studies suggest acritical in-plane electric field of domain wall movement of E (cid:107) ,c = 1 . × V/cm. Polarization switching was alsodemonstrated by applying bias voltage pulses to the STMtip at the surface of SnTe MLs, which locally switches fer-roelectric domains by the domain wall motion highlightedwithin red circles in Fig. 4(d).
FIG. 5 (a) Schematics of upward (top) and downward (bot-tom) band bending near a ferroelectric’s edge, for a chemicalpotential near the valence band edge. The Fermi energy ofthe electrode was set to be E = 0. (b)-(d) Schematic of thedevice (b) and band diagrams for the “on” (c) and “off” (d)sates. Adapted from Ref. (Shen et al. , 2019a). Copyright,2019, American Physical Society. A. Polarization switching and ultrathin memories based onin-plane ferroelectric tunnel junctions
Ferroelectrics find applications in nonvolatile memo-ries due to their switchable bistable ground states (Scottand Paz de Araujo, 1989). First-generation ferroelec-tric memories use the surface charge in a ferroelectriccapacitor to represent data (Evans and Womack, 1988).As a result, discharging the capacitor to measure thecharge destroys the stored data, so the capacitor mustbe recharged after reading. A second generation of thesememories probes the ferroelectric polarization using atunneling-electroresistance effect (Tsymbal and Kohlst-edt, 2006) within a metal-ferroelectric-metal junction inwhich an out-of-plane P exists within the ferroelectricthin film. The tunneling potential barrier is determinedby the out-of-plane polarization in the ferroelectric layer.It may be possible to create in-plane ferroelectric mem-ories by adding an insulator and a top gate to the two-terminal device shown in Fig. 4(a). Indeed, if P pointsin-plane, such as in M X
MLs, the band bending at theferroelectric materials’ edge can be read out with metalcontacts (Shen et al. , 2019a). As depicted by the depen-dency of I D on V D in Fig. 4(a), the upward or downwardband bending drawn in Fig. 5(a) could represent the “on”or “off” state respectively, so that the information storedis read nondestructively.Fig. 5(b) shows a ferroelectric thin film sandwichedbetween a metallic substrate and a wide-band-gap insu-lator. The writing and reading electrodes are depositedat opposite edges of the top insulator. If P lies in-planealong the + x direction, it will induce opposite net chargesat the ferroelectric boundaries along that direction. De-pending on the polarization direction (either + x or − x ),the band bending near the reading electrode could be upward or downward, leading to “on” and “off” states[Figs. 5(c) and Figs. 5(d), respectively]. Reading is non-destructive because the electric field generated by thereading voltage is perpendicular to the ferroelectric’s po-larization. Using Landauer’s conductance formalism andsuitably chosen parameters, currents of the order of µA and I on /I off ratios of the order of 10 have been pre-dicted (Shen et al. , 2019a). See Refs. (Shen et al. , 2019b)and (Kwon et al. , 2020) for additional memory devicesbased on O- M X s. V. LINEAR ELASTIC PROPERTIES, AUXETICBEHAVIOR, AND PIEZOELECTRICITY OF O − MX MLS
The physical properties of 2D materials can be tunedby strain (Amorim et al. , 2016; Naumis et al. , 2017).In linear elasticity theory, the strain tensor is definedas (cid:15) ij (cid:39) (cid:16) ∂u i ∂x j + ∂u j ∂x i (cid:17) , where u = ( u x , u y , u z ) is thedisplacement field u = r − r away from a structuralconfiguration that minimizes the structural energy.The constitutive relation establishes a linear depen-dence among the stress tensor σ ij and (cid:15) ij : σ ij = C ijkl (cid:15) kl , where C ijkl is the elasticity tensor. Symme-try restrictions on O − M X
MLs imply that only C xxxx , C xxyy , C yyyy , and C xyxy are non-zero (Fei et al. , 2015;Gomes et al. , 2015); subindices xx and yy label com-pressive/tensile (normal) strain, and xy is a shear strain.Using Voigt notation ( xx → yy →
2, and xy → C , C , C , and C and their magnitudes arelisted in Table V. C , C , and C are obtained by fitting against theelastic energy landscape shown in Fig. 6(a), in which (cid:15) = ∆ a a , = a − a , a , and (cid:15) = ∆ a a , . Consistent with thechange in area in going from to bulk to a ML, elastic con-stants tend to be slightly softer in MLs than in the bulk.Ref. (Gomes et al. , 2015) provides the Young’s modulusfor GeS, GeSe, SnS and SnSe as well, which is an orderof magnitude smaller that its magnitude of 340 N/m forgraphene (Lee et al. , 2008). Additionally, C and C are smaller when compared to their values for MoS andGaSe MLs, listed in Table V as well. The shear elasticcoefficient C in Table V is as small as C : shear strainchanges the magnitude of ∆ α in Fig. 1(c), implying thatdistortions by such an angle are as soft as a compressionor elongation along the a direction. In light of Table V, M X
MLs are soft
2D materials with anisotropic elasticproperties.The Poisson’s ratio ν determines the rate of contrac-tion in transverse directions under longitudinal uniaxialload. Most materials have a positive Poisson’s ratio but,as discussed in Refs. (Gomes et al. , 2015; Jiang and Park,2014; Kong et al. , 2018; Liu et al. , 2019a) and summa-rized in Table VI, the buckled structure of O − M X
MLsdepicted in Fig. 1(c) confers them with negative ratios0
TABLE V Relaxed-ion components of the elastic tensor C ij for MX MLs in N/m. Adapted from Refs. (Gomes et al. ,2015) and (Fei et al. , 2015).
Material ¯ Z C C C C GeS ML 24 15.24 − − − − − − − − − − − − ML a
130 130 32 –GaSe ML b
83 83 18 – a Ref. (Duerloo et al. , 2012). b Ref. (Li and Li, 2015).TABLE VI Sign of ∆ h and linear Poisson’s ratio ν ij of a BPML and of group-IV monochalcogenide MLs. Negative valuesof ν ij indicate auxetic behavior. Adapted from Refs. (Jiangand Park, 2014; Kong et al. , 2018; Liu et al. , 2019a).Material ¯ Z ∆ h ν yx ν xy ν zx ν zy BP 15 0 0.400 0.930 0.046 − − .
208 0.411GeSe ML 33 − − − − − when the out-of-plane ( z − direction) is considered. Thesubindices of ν ij in Table VI indicate the (linear) Pois-son’s ratio along the i direction due to a load along the j direction as defined in Fig. 1. Negative values of ν ij areindicative of auxetic behavior, i.e., an elongation (com-pression) occurs along the i direction when these 2D ma-terials are elongated (compressed) along the j direction.According to Table VI, there is a direct correlation be-tween a positive ∆ h in Fig. 1(a) and a negative ν zx .The third-order piezoelectric tensor d ijk links ∆ P = P − P with the applied strain (cid:15) jk . Using Voigt notationfor the last two entries of the piezoelectric tensor and forthe applied strain, Fig. 6(b) displays a ten times largermagnitude of d for GeS, GeSe, SnS, and SnSe when con-trasted with the piezoelectric coefficients of quartz andother polar materials (Fei et al. , 2015). VI. STRUCTURAL DEGENERACIES ANDANHARMONIC ELASTIC ENERGY OF O − MX MLS
As it turns out, the elastic energy landscape fromwhich elastic properties were discussed in Sec. V is non-linear. Its non-linearity underpins the highly anharmonicvibrational properties and a propensity of O − M X
MLsfor sudden changes in ferroelectric, structural, electronic,spin, and optical properties with temperature.Turning the ∆ a /a , = ( a − a , ) /a , axis inFig. 6(a) into a and ∆ a /a , into a , and increas-ing the range for both a and a from which thestructural energy E ( a , a ) is computed, the elastic en-ergy landscape ∆ E ( (cid:15) , (cid:15) ) = ∆ E ( a , a ) = E ( a , a ) − FIG. 6 (a) Elastic energy landscape of a SnSe ML (in units ofK/u.c.) as a function of uniaxial strain along the a and a di-rections. Adapted from Ref. (Gomes et al. , 2015) with permis-sion. Copyright, 2015, American Physical Society. (b) Piezo-electric coefficient d of GeS, GeSe, SnS and SnSe MLs andother known piezoelectric materials. Adapted from Ref. (Fei et al. , 2015) with permission. Copyright, 2015, American In-stitute of Physics. (c) Elastic energy landscape over a largerrange of values for a and a ; point A corresponds to theground state unit cell shown in Fig. 1(c). (d) Two-dimensionallowest-energy path joining degenerate ground states A and B ; the shaded rectangle corresponds to ± E , ∆ α , θ , and P in going through the pathshown in (c); the dashed vertical line cuts through ∆ E = 0and thus shows ∆ α , θ and P . Subplots (c-e) are adaptedfrom Ref. (Barraza-Lopez et al. , 2018) with permission. Copy-right, 2018, American Physical Society. E ( a , , a , ) shown in Fig. 6(c) ensues. Given that E ( a , , a , ) = E ( a , , a , ) on 2D materials with arectangular u.c. ( a , > a , ), the elastic energy land-scape has two degenerate structures labeled A and B in Fig. 6(c) (Mehboudi et al. , 2016a; Wang and Qian,2017b). O − M X s have eight degenerate u.c.s, occurringupon a mirror reflection with respect to the x − z or x − y planes, or by an exchange of x − and y − coordinates(Mehboudi et al. , 2016a). Nevertheless, an inversion withrespect to the x − y plane does not change the orientationnor the sense of direction of P and is usually disregardedwhen describing degeneracies for that reason; the four re-maining degenerate ground state u.c.s are displayed as aninset in Fig. 6(d). They have projections p → , ↑ , ← ,and ↓ , that are reminiscent of discrete clock models—wellknown tools to discuss order-by-disorder 2D transforma-tions in Statistical Mechanics (Potts, 1952) that provide1important insight into the finite temperature behavior ofO − M X
MLs (Mehboudi et al. , 2016a).The saddle point c in Fig. 6 indicates the minimumelastic energy necessary to switch in between ferroelec-tric states A and B . It is situated at ( a c , a c ), with a c = (cid:16) − √ (cid:17) a , + a , √ (Poudel et al. , 2019). The five-fold coordinated u.c. at point c is paraelectric (Mehboudi et al. , 2016a) and it belongs to symmetry group 129( P /nmm , or P mm /n with our choice of axes) (Vil-lanova et al. , 2020). J s is the energy difference in betweenthe five-fold coordinated paraelectric u.c. at point c andany of the degenerate ferroelectric ground states ( i.e., the one at point A with polarization along the positive x − direction): J s = E c − E A, → = E ( a c , a c ) − E ( a , , a , ). J s indicates the ease of a ferroelastic transformationamong a pair of degenerate structures shown in Fig. 6(d).As it will be discussed in Sec. VII, it is a qualitative esti-mator of the critical temperature T c at which a ferroelec-tric to paraelectric transition takes place in these 2D ma-terials. [ a , = a , = a c for Pb X MLs in Fig. 1(d), whichhence have a single non-degenerate structural groundstate and J s = 0.]The white path r ( a , a ) in Fig. 6(c) provides thelowest-energy distortion that is necessary to turn degen-erate structure A into B elastically, and it is projectedinto a 2D plot in Fig. 6(d). For a SnSe ML, points A and B are located at distances r = 0 .
134 ˚A and r = − .
134 ˚Aalong this path; point c is located at r = 0 ˚A. The rangeof values utilized to extract linear elastic properties inSec. V can be seen as a yellow rectangle in Fig. 6(d).The anharmonicity of the elastic energy landscape isestablished by the double-well potential ∆ E ( r ) seen assubplot (i) in Fig. 6(e). The magnitude of J s for a SnSeML [as computed with the vdW-DF-CX (Berland andHyldgaard, 2014) XC functional] can also be seen in thatplot. The dependency of ∆ E on the path coordinate r is bistable ( i.e., fundamentally non-harmonic). The evolu-tion of ∆ α , θ , and the polarization P along r —includingthe four possible orientations of p ( P )—is displayed assubplots (ii), (iii), and (iv) in Fig. 6(e). The area inlight yellow in Fig. 6(e) corresponds to the ± .
5% strainwithin which ∆ E can be fitted to a parabola, and where∆ α , θ , and P are linear on r (Fei et al. , 2015).The vertical dashed line, crossing through ∆ E = 0shows ∆ α , θ , and P ; i.e. , the magnitudes of thesevariables in a u. c. like the one seen in Fig. 1(c). Theangle ∆ α is positive for r > a > a ), zero at r = 0( a = a ), and negative for r < a < a ). The an-gle θ , in turn, points along the (positive or negative) x − direction when r >
0, it is zero at r = 0, and itpoints along ± y when r <
0. Importantly, θ and P arelinearly proportional ( P ∝ θ ) and P = 0 when θ = 0and ∆ α = 0 in an elastic transformation in which lat-tice parameters can vary. The possibility of switching P gives rise to a combined ferroelectricity and ferroelastic- FIG. 7 (a) Exponential dependency of J s on ¯ Z . This plotalso shows the dependence of J s (and therefore of a , , a , ,∆ α , P , and θ ) on the XC functional employed. Taken fromRef. (Poudel et al. , 2019) with permission. Copyright, 2019,American Physical Society. (b) Experimental thermal depen-dency of ∆ α for few-layer SnTe on epitaxial graphene. A para-electric phase (P=0) ensues at temperatures above T c = 270K in SnTe MLs. Adapted from Ref. (Chang et al. , 2016) withpermission. Copyright, 2016, American Association for theAdvancement of Science. ity; i.e. , to multiferroic behavior in O − M X
MLs (Wangand Qian, 2017b; Wu and Zeng, 2016).
VII. STRUCTURAL PHASE TRANSITION ANDPYROELECTRIC BEHAVIOR OF O − MX MLS
Structural degeneracies underpin strong anharmonicelastic properties, soft phonon modes, and structuralphase transitions. Taking J s —the relevant energy scalein the system—as an ad-hoc exchange parameter, a clockmodel with r = 4 degenerate states yields the follow-ing relation among T c and J s : T c = 1 . J s (Potts,1952). The Potts model also has a prescription in casethat only a subset of two degenerate states is available( e.g., → and ← ), which could occur in a constrainedscenario in which a and a keep zero-temperature mag-nitudes (Fei et al. , 2016): calling J r the energy barrierunder such constrained configuration, Potts dictates that T (cid:48) c = 2 . J r (Potts, 1952). Numerical calculations indi-cate that J r ≥ . J s , so that T (cid:48) c ≥ . T c . The message isthat structural constraints lead to an increased T c .The ferroelectric-to-paraelectric transition tempera-ture T c of O − M X
MLs calculated at the DFT level (Mar-tin, 2004) has a strong dependency on the choice of XCfunctional, and it is unclear that the PBE XC functionalought to provide the most accurate description of thethermal behavior of O − M X s. The strong dependencyof T c on XC functional can be already foreseen in themagnitude of J s displayed in Fig. 7(a), which containspredictions with LDA, PBE, multiple non-empirical vander Waals implementations (Berland et al. , 2015), andeven the recent SCAN+rVV10 (Peng et al. , 2016) XCfunctional (Poudel et al. , 2019). Ab initio molecular dynamics (AIMD) calculations2 (a) M od i f y e l ec t r on i c d i s t r i bu t i on ( i n t e r a t o m i c f o r ces ) ( m od i f y c r i t i ca l t e m p e r a t u r e ) < | n | > NPTNVT C oun t s ( ) T=130 KT=170 K
Lifetime (ps)
Increase constraints(increase critical temperature) (iv) T c =180 K ( J s =50 K/u.c.) (vi) T c ’’=394 K ( J r =110 K/u.c.) (i) T c =212 K ( J s =149 K/u.c.) (ii) T c ’=440 K ( J r =350 K/u.c.) AIMD, NPT ensemble AIMD, NVT ensemble UOV model v d W - D F - C X X C PBE X C (v) T c ’=320 K ( J r =110 K/u.c.) (iii) T c ’’=730 K ( J r =350 K/u.c.)
03 0 400 800 0 400 8000 400 800 0 400 8000 400 800 0 400 80002 0 200 4000 200 400 0 200 400 0 200 4000 200 400 0 200 400 030202 036024 02403 036036024 ∆ α ( deg r ee ) ∆ α ( deg r ee ) θ ( deg r ee ) θ ( deg r ee ) ∆ α ( deg r ee ) ∆ α ( deg r ee ) θ ( deg r ee ) θ ( deg r ee ) Temperature (K)Temperature (K) Temperature (K)Temperature (K) Temperature (K)Temperature (K) ∆ α ( deg r ee ) ∆ α ( deg r ee ) θ ( deg r ee ) θ ( deg r ee ) (b) Topological charges:
Below T c Above T c Temperature (K) ᴽ θ )(P ᴽ θ )(P ᴽ θ )(P ᴽ θ )(P ᴽ θ )(P ᴽ θ )(P P>0 P=0P>0 P=0 P>0 P=0P>0 P = P>0 P=0P>0 P=0
FIG. 8 (a) Temperature dependence of ∆ α and θ for a SnSe ML in AIMD calculations employing two XC functionals, andobtained within the NPT [subplots(i) and (iv)] and NVT ensembles [subplots(ii) and (v)]. Results from a more constrainedunidirectional optical vibration (UOV) model are presented in subplots (iii) and (vi). The ferroelectric phase occurs when θ >
0. Note that ∆ α = 0 implies θ = 0 but that θ = 0 does not necessarily imply ∆ α = 0. (b) The structural transformationsdescribed by AIMD can be understood in terms of a fluctuating connectivity ( i.e. , topology) of these 2D ferroelectrics. Pairsof topological charges have a temperature-dependent lifetime and their number saturates at temperatures within T c . Adaptedfrom Ref. (Villanova et al. , 2020) with permission. Copyright, 2020, American Physical Society. performed on freestanding SnSe MLs using the NPT en-semble (in which containing walls are allowed to move toaccommodate for thermal expansion) provide the follow-ing information: (i) T c is larger in GeSe MLs and bilayersthan it is in SnSe MLs and bilayers, owing to the smaller¯ Z and hence larger barrier J s [Fig. 7(a)] for GeSe; (ii)for a given O − M X , T c increases with increasing numberof MLs (Chang et al. , 2016; Mehboudi et al. , 2016b). Aslight dependency of T c in the size of the simulation su-percell has been documented, too (Barraza-Lopez et al. ,2018; Mehboudi et al. , 2016b). In agreement with J s ’s in-verse dependency on ¯ Z , experiments indicate a T c largerthan 400 K for SnSe ( ¯ Z = 42) on graphene (Chang et al. ,2020), and T c =270 K for SnTe MLs ( ¯ Z = 51) on thesame substrate [Fig. 7(b) (Chang et al. , 2016)].The upper row in Fig. 8(a) shows a progression of T c estimates for a freestanding SnSe ML that were ob-tained using the vdW-DF-CX XC functional (Berlandand Hyldgaard, 2014). From left to right, the figure dis-plays the thermal behavior of ∆ α and θ when (i) usingthe NPT ensemble (in which containing walls move sothat the material remains at atmospheric pressure), (ii)the NVT ensemble (in which the supercell volume V isfixed and containing walls do not move), and (iii) a uni-directional optical vibration (UOV) model in which onlyone unidirectional vibrational mode—out of twelve—isemployed and the containing walls do not move either(Fei et al. , 2016). The point is that (as already foreseenby the Potts model a few lines above) T c increases withadded constraints. Energy barriers J s and J r are listedin that Figure, too.Briefly said, AIMD calculations carried out with theNPT ensemble yield the smallest magnitude of T c (212 K). Although the compounds are not the same (a free-standing SnSe ML in calculations and a SnTe ML ongraphene in experiment), the decay of ∆ α in the cal-culations seen in subplot (i) of Fig. 8(a) indicates aphenomenology consistent with experiment in Fig. 7(b)(Barraza-Lopez et al. , 2018; Chang et al. , 2016). Whenconstraining the SnSe ML by not permitting its area toincrease at finite temperature, the structural transitionwithin the NVT ensemble necessarily requires additional(thermal) energy to take place, raising T (cid:48) c up to 440 K anddisplaying ∆ α > T c . Despite the existence of nearlydegenerate vibrational modes oscillating along both x − and y − directions, the highly constrained UOV modelonly permits an optical vibration along the x − direction(an oscillatory mode valid only at the Γ − point) and thusyields the largest T (cid:48)(cid:48) c = 730 K, still showing ∆ α > T c as a and a are kept fixed [Eqn. (1)]. ∆ α [Fig. 7(b)] isnot a relevant order parameter for subplots (ii) and (iii) inFig. 8(a) because a and a retain their zero-temperaturevalues in these models.The increasing sequence of critical temperatures ob-served with increasing mechanical constraints is inde-pendent of the XC approximation, as the lower row inFig. 8(a) shows a similar phenomenology when the PBEXC functional is employed. T c , T (cid:48) c , and T (cid:48)(cid:48) c have smallervalues than those obtained with the vdW-DF-CX XCfunctional (Villanova et al. , 2020). As illustrated inFig. 8(b), the structural transition is underpinned bychanges in the connectivity of the 2D lattice as the twoatoms defining the angle θ in Fig. 1(a) rotate about theout-of-plane z − axis; this change in connectivity confersa topological character to the structural transformation(Kosterlitz, 2016; Villanova et al. , 2020; Xu et al. , 2020).3Pyroelectricity is the creation of electricity by a tem-perature gradient. Given the direct proportionality be-tween θ and P , the temperature derivative of θ inFig. 8(a) gives a direct insight into the pyroelectric prop-erties of O − M X s (Mehboudi et al. , 2016b).The effects of substrates such as Ni, Pd, Pt, Si, Ge,CaO, and MgO on the morphology and properties ofSnTe MLs (including charge transfer and atomistic dis-tortions) have been studied (Fu et al. , 2019). SinceO − M X
MLs are presently grown on substrates, the ef-fect of the substrate-
M X interaction on the transitiontemperature is an important avenue for further theory.Along these lines, the elastic energy barriers J s of GeSe,GeTe, SnS, SnSe, and SnTe have been shown to vanishunder a modest hole doping of 0.2 | e | /u.c., where e is theelectron’s charge (Du et al. , 2020; Zhu et al. , 2020).We indicated in Sec. II that O − M X s are isostructuralto BP. This makes BP MLs doubly-degenerate upon ex-change of x − and y − coordinates, and suggests that BPMLs may also undergo a phase transition at finite tem-perature. Nevertheless, considering J s as an approximatemeasure of T c , one observes J s > ,
000 K/u.c. for BPin Fig. 7(a) regardless of XC functional. Such magnitudeis so large that a BP ML melts rather than undergoinga ferroelastic to paraelastic transition (Mehboudi et al. ,2016a), thus explaining the lack of thermally-driven et al. , 2014b).] Thepropensity to undergo thermally-driven 2D transitions isa crucial aspect that sets 2D ferroelectrics apart fromother 2D materials such as graphene, TMDCs with a 2Hsymmetry, and BP MLs.
VIII. ELECTRONIC, VALLEY AND SPIN PROPERTIESOF O − MX MLSA. Electronic band structure
TMDC MLs such as 2H-MoS display an indirect-to-direct band gap crossover at the ML limit. As indicatedin Table VII, the experimental band gap of MoS in-creases from 1.29 eV in the bulk up to 1.90 eV in aML due to quantum confinement (Mak et al. , 2010), andthe valence band maxima (VBM) and conduction bandminima (CBM) are both located at the high-symmetry ± K − points in these MLs. Similarly, quantum confine-ment leads to an increase of the electronic band gapof BP (Table VII), and its electronic bands are highlyanisotropic. The VBM and CBM are both located at theΓ − point in BP MLs (Tran et al. , 2014).Tritsaris, Malone, and Kaxiras (Tritsaris et al. , 2013)studied the electronic properties of SnS down to the MLlimit. As indicated in Table VII, the electronic bandgap increases as these materials are thinned down, too. FIG. 9 Electronic band structure of O − MX MLs. With theexception of the SnTe ML, these calculations were carried outwith the HSE06 XC functional. GeSe MLs have direct bandgaps, and the GeTe ML subplot was modified to account foran indirect band gap smaller by just 0.01 eV than the di-rect band gap. The inset in the SnTe subplot displays thehigh-symmetry points and the direction of p . Adapted fromRef. (Singh and Hennig, 2014) with permission. Copyright,2014, American Institute of Physics. The GeS subplot wastaken from Ref. (Gomes et al. , 2016) with permission. Copy-right, 2016, American Physical Society. The SnTe subplotwas adapted from Ref. (Chang et al. , 2019b) with permission.Copyright, 2019, American Physical Society. Nevertheless (and unlike the case for 2H-TMDs and BPMLs), the VBM and CBM are not located at high-symmetry points in the first Brillouin zone (BZ).With reciprocal lattice vectors b = πa , (1 , ,
0) and b = πa , (0 , , X (lo-cated at b / Y (at b / S (at b / b / − M X
MLs have their VBM away from high-symmetrypoints, at about ± (0 . − . X and their CBM at about ± (0 . − . Y for an indirect band gap. The exceptionis GeSe, having a CBM at ± . X and an direct bandgap (Gomes and Carvalho, 2015; Gomes et al. , 2016; Shiand Kioupakis, 2015; Singh and Hennig, 2014). Elec-tronic band structure calculations within the GW ap-proximation (Deslippe et al. , 2012) were carried out inRefs. (Shi and Kioupakis, 2015; Tuttle et al. , 2015) and(Gomes et al. , 2016); their band gaps are listed in Ta-ble VII. The electronic band structure of O − M X s turnsmore (less) anisotropic for lighter (heavier) compounds,for which a , /a , in Fig. 1(d) takes on larger (smaller)values. Going across chemical elements, the band gapfor O − M X
MLs in Table VII is tunable with ¯ Z : ittakes its largest magnitude for lighter compounds (GeS,¯ Z = 24) and it is smaller for the heaviest compound(SnTe, ¯ Z = 51).From an experimental perspective, hole-doped SnTeMLs acquire a domain structure observed as dark verticallines in Figs. 3(b) and 10(a). As seen in Figs. 10(b) and10(d), the spatially resolved dI/dV spectra [proportionalto the sample’s local density of states (LDOS)] features4 TABLE VII Electronic band gap (in eV) for BP, O − MX sand MoS (ML and bulk) with PBE and HSE XC functionals(Heyd et al. , 2003), from GW calculations (Deslippe et al. ,2012), or experiment. PBE values for GeTe and SnTe MLs,and for bulk SnTe were computed by us. Material ¯ Z PBE HSE GW Exp. PBE HSE Exp.(ML) (ML) (ML) (ML) (bulk) (bulk) (bulk)BP 15 0.90 a a b a a c GeS 24 1.65 a a d a a e GeSe 33 1.18 a a d,f a a g GeTe 42 0.87 0.33 h h i SnS 33 1.38 a a a a e SnSe 42 0.96 a a f j a a k SnTe 51 0.68 1.6 l m MoS − n n o n n o a Ref. (Gomes and Carvalho, 2015). b Ref. (Wang et al. ,2015). c Ref. (Keyes, 1953). d Ref. (Gomes et al. , 2016). e Ref. (Malone and Kaxiras, 2013). f Ref. (Shi andKioupakis, 2015). g Ref. (Vaughn et al. , 2010). h Ref. (Di Sante et al. , 2013). i Ref. (Park et al. , 2009). j Ref. (Chang et al. , 2020). k Ref. (Parenteau and Carlone,1990). l Ref. (Chang et al. , 2016). m Ref. (Dimmock et al. ,1966). n Ref. (Shi et al. , 2017). o Ref. (Mak et al. , 2010). electronic standing wave patterns across domains for en-ergies below the VBM that provide indirect informationinto these materials’ electronic properties.The standing wave patterns observed at 4 K are in-duced by the electronic band mismatch at the two sidesof a 90 ◦ domain walls [see domains with P forming ∼ ◦ angles in Figs. 2(b) and 3(b)]. As Fig. 10(c) shows, theband apexes along the Γ − Y direction are 0.3 eV belowthose seen along the Γ − X direction. Such mismatch ofhole momentum prevents a direct (elastic, unscattered)transmission of holes through domain walls, giving rise toa peculiar reflection resulting in standing waves (Chang et al. , 2019b).In fact, and as depicted in Fig. 10(c), the reflectionoff a domain wall occurs via a momentum transfer q oc-curring within each hole band. This observation impliesthat the standing wave pattern is an indirect measureof the electronic band structure around the VBM. Fromthe Fourier transform of the standing wave pattern inFig. 10(d), a single branch of the energy dispersion withscattering vector q [inset in Fig. 10(c)] is experimentallyresolved in Fig. 10(e) (Chang et al. , 2019b); note that P ( T = 4 K ) (cid:39) P . Although spin-orbit interaction in-duces band splitting at the VBM, the contribution fromthe two spin components to the standing wave patternare exactly the same because of time reversal symmetry. B. Valleytronics
The band curvature (cid:126) /m ∗ at the VBM and CBM—where m ∗ is the effective mass—is used to estimate holeand electron conductivities of semiconductors. When de- FIG. 10 (a) STM topographic image of ∼ ◦ domains ona SnTe ML. (b) dI/dV spectra acquired along the white ar-row in (a) at energies around the VBM. (c) Mismatched holebands at opposite sides of a 90 ◦ domain wall. Constant energycontours with opposite spin components are colored in blue orred, respectively. (d) Electronic standing wave pattern acrossa 540 ˚A wide domain. (e) The Fourier transform of (d) revealsthe energy dispersion of scattering vectors. The dashed whitecurve is the E ( q ) dispersion as obtained from the electronicband structure. Adapted from Ref. (Chang et al. , 2019b) withpermission. Copyright, 2019, American Physical Society.FIG. 11 (a) Valley selection under an external oscillatingelectric field. Different valleys are excited depending on thefield’s linear polarization. (b) Valley separation under a staticelectric field. Depending on the polarization of the field, dif-ferent valleys flow along perpendicular directions. Taken fromRef. (Rodin et al. , 2016) with permission. Copyright, 2016,American Physical Society. TABLE VIII Anisotropic effective masses of holes ( m h /m )and electrons ( m e /m ) at VBM and CBM along the x and y − directions as shown in Fig. 1(c) for O − MX MLs.Material ¯ Z ( m h /m ) x ( m h /m ) y ( m e /m ) x ( m e /m ) y GeS ML 24 − . − .
94 0.24 0.57GeSe ML 33 − . − .
32 0.17 0.34GeTe ML 42 − . − .
16 0.08 0.32SnS ML 33 − . − .
27 0.20 0.22SnSe ML 42 − . − .
14 0.14 0.14SnTe ML 51 − . − .
05 0.13 0.14 termined along orthogonal ( x − and y − ) directions, itprovides information about the anisotropy of the chargecarriers’ conductivity. The effective masses at the VBM( m ∗ = m h ) and CBM ( m ∗ = m e ) for multiple O − M X
MLs (expressed in terms of the electron’s mass m ) arelisted in Table VIII (Gomes et al. , 2016). With the ex-ception of GeS MLs, these effective masses are smallerto those of MoS [which range within − (0.44-0.48) m for holes and 0.34-0.38 m for electrons (Cheiwchancham-nangij and Lambrecht, 2012; Peelaers and Van de Walle,2012)], implying sharper hole/electron pockets at theVBM/CBM on O − M X
MLs than those existing inmore traditional materials for valleytronic applications(Schaibley et al. , 2016). Such sharpness of the valence(conduction) band curvature permits stating that holes(electrons) belong to a given valley .Valleytronics refers to the use of the electron/holepockets at the CBM or VBM as information carriers(Schaibley et al. , 2016), which requires creating valley-specific gradients of charge carriers; i.e. , a valley po-larization . Achieving valley polarization requires liftingthe valley degeneracy; something that has been demon-strated in TMDC MLs (Rycerz et al. , 2007; Xiao et al. ,2012). In these 2D materials, valleys at time-reversedstates + K and − K in the BZ couple to the circular po-larization of light so that a pseudospin (“up” or “down”)quantum number can be associated with each valley. Val-ley polarization occurs because right-hand polarized pho-tons only excite the carriers in + K valley, and left-handpolarized photons only excite those in − K valley (Cao et al. , 2012; Mak et al. , 2012; Sallen et al. , 2012; Zeng et al. , 2012). When an in-plane electric field is appliedacross graphene bilayers or TMDC MLs, carriers with“up” and “down” pseudospins acquire a transverse veloc-ity in opposite directions because of the opposite Berrycurvature in + K and − K valleys, giving rise to a valleyHall effect (Mak et al. , 2014; Shimazaki et al. , 2015; Sui et al. , 2015).Unlike a BP ML, which has a single valley centeredaround the Γ − point, the sharp band curvature of theVBM and CBM of O − M X
MLs listed in Table VIII per-mits considering them two-valley materials, too. Theyfeature a valley along the Γ − X direction (the V x val-ley), and another valley along the Γ − Y direction ( V y valley) (Rodin et al. , 2016), and valley-selective opticalexcitation can be realized in these 2D materials using lin-early polarized light (Hanakata et al. , 2016; Rodin et al. ,2016; Shen et al. , 2017; Xu et al. , 2017).Figure 11(a) shows valleys located along the Γ − X andΓ − Y lines in the BZ. First principles calculations andsymmetry analysis show that x -polarized photons havea much higher probability to excite carriers in the V x valley. Similarly, carriers in the V y valley can be read-ily excited by y -polarized light almost exclusively. Inother words, a specific valley can be selectively excitedby controlling the polarization of the incident light [thismechanism does not distinguish sign: for instance, both − (Γ − X ) and +(Γ − X ) are both “the V x valley”]. Asan alternative mechanism to produce valley polarizationby means of time-reversal symmetry, an in-plane staticelectric makes carriers excited from the V x valley [eitherlocated along the − (Γ − X ) or +(Γ − X ) line] bend inopposite directions, generating a valley Hall effect illus-trated in Fig. 11(b). A similar effect occurs when excitingthe V y valley. Additional transport effects arising fromnon-linear electric fields will be discussed in Sec. IX.D. C. Persistent spin helix behavior
So far, we have considered the electronic propertiesof O − M X
MLs without concern for spin polarization.Spin-orbit coupling can create various types of spin split-ting near the band edges, as well as spin Hall effectsin O − M X
MLs (S(cid:32)lawi´nska et al. , 2019).
Zeeman-like spin splitting is the prominent mechanism in TMDCs(Xiao et al. , 2012). On the other hand, a
Rashba-like spin orbit coupling occurs due to the spin-orbit field −→ Σ SOF ( k ) = α ( ˆ P × k ), where α is the spin-orbit couplingstrength, ˆ P = P /P is the direction of the intrinsicelectric polarization in ferroelectrics, and k is the quasi-particle (electron or hole) crystal momentum [Fig. 12(a)].As indicated in Sec. II, the out-of-plane component of P is quenched in ultrathin SnTe, making P = P ˆ x .Since 2D materials lack crystal momentum along the z − direction, −→ Σ SOF ( k ) ∝ (ˆ x × k ), with k = ( k x , k y , − X ( k y = 0) high symmetry line, that it points along the z − direction for k y (cid:54) = 0, and it reverts direction wheneither k y or P change sign (Lee et al. , 2020). Ro-tating P into the y − direction [see Fig. 6(d)] changesthe orientation of the spin-split bands. The strengthof the spin-orbit coupling increases with atomic number Z , as broadly reported for O − M X
MLs (Chang et al. ,2019b; Liu et al. , 2019b; Rodin et al. , 2016; Shi andKioupakis, 2015) and other 2D ferroelectrics (Ai et al. ,2019; Di Sante et al. , 2015; Kou et al. , 2018; Wang et al. ,2018b). Recalling that Pb-based
M X
MLs lack an in-trinsic polarization [ a , = a , in Fig. 1(d) and hence P = 0], the best immediate candidates for 2D ferro-6 FIG. 12 (a) Conventional and (b) out-of-plane spin Rashbaeffects at the bottom of the conduction band. In subplot(b), the out-of-plane spin Rashba effect is realized in a SnTeML, and both the valence and conduction bands are shown.Adapted from Ref. (Lee et al. , 2020) with permission. Copy-right, 2020. American Institute of Physics. electric Rashba semiconductors within
M X
MLs are tel-lurides GeTe and SnTe.Space group P2 mn has the following symmetries: (i)the identity E ; (ii) ¯ C x : a two-fold rotation around the x − axis ( C x ), followed by a translation τ = ( a + a ) / M xy : a reflection by the xy plane followed by τ ; and (iv) a reflection about the xz plane ( M xz ) (Rodin et al. , 2016). Adding time-reversalsymmetry as customarily defined ˆ T = iσ y K (where K represents complex conjugation) the following effectivespin Hamiltonian applies at the top of the valence bandand at the bottom of the conduction band (Absor andIshii, 2019; Lee et al. , 2020):ˆ H = (cid:126) m ∗ (cid:0) k x + k y (cid:1) + (cid:0) αk y + α (cid:48) k x k y + α (cid:48)(cid:48) k y (cid:1) σ z , (2)with m ∗ , α , α (cid:48) , and α (cid:48)(cid:48) to be fitted from band struc-ture calculations [Fig. 12(b)]. This Hamiltonian does nothave contributions from in-plane spin components up tothird order in momentum, leading to a persistent spinHelix effect with a tunable out-of-plane spin . Eqn. (2)is similar to a Dresselhaus model for a bulk zinc blendecrystal oriented along the [110] direction (Dresselhaus,1955). Estimates of the spin-orbit coupling α in M X sare two to three orders of magnitude larger than those inIII-V semiconductor quantum well structures (Lee et al. ,2020), and their wavelength of the spin polarization λ issmaller than that obtained for other Rashba semiconduc-tors (Absor and Ishii, 2019) which permits smaller lateraldevice dimensions. O − M X
MLs are a 2D platform forpersistent spin helix dynamics (Bernevig et al. , 2006).SnTe ML spin transistors may be designed to have achannel length of λ /4 to be electrically switched in theferroelectric channel or magnetically switched in the fer-romagnetic drain (Lee et al. , 2020). Another proposal isan all-in-one spin transistor based on the spin Hall ef-fect, where the inverse spin Hall effect charge current isdetuned by an out-of-plane electric field which [accordingto Fig. 12(a)] breaks the persistent spin helix state downand induces spin decoherence (Slawinska et al. , 2019). TABLE IX Transformation rules for k x , k y , σ x , σ y , and σ z under the symmetry operations of group P2 mn. The point-group operations are defined as ˆ C x = iσ x , ˆ M xz = iσ y , andˆ M xy = iσ z . Adapted from Ref. (Absor and Ishii, 2019) toreflect the choice of lattice vectors used in this work.Symmetry operation ( k x , k y ) ( σ x , σ y , σ z )ˆ E ( k x , k y ) ( σ x , σ y , σ z )ˆ C x = iσ x ( k x , − k y ) ( σ x , − σ y , − σ z )ˆ M xz = iσ y ( k x , − k y ) ( − σ x , σ y , − σ z )ˆ M xy = iσ z ( k x , k y ) ( − σ x , − σ y , σ z )ˆ T = iσ y K ( − k x , − k y ) ( − σ x , − σ y , − σ z ) IX. OPTICAL PROPERTIES OF O − MX MLSA. Optical absorption
Optical absorption reflects the anisotropy of the elec-tronic band structure: linearly polarized light with po-larization parallel to the x − direction leads to a smallerabsorption energy gap when contrasted with light whosepolarization is parallel to the y − axis (Gomes and Car-valho, 2015) [this effect can be observed in Fig. 16, whereGW corrections and Bethe-Salpeter electron-hole inter-actions have been added]. The symmetry imposed by the P /nmm structural transformation at T ≥ T c should bereflected on a symmetric optical absorbance (Mehboudi et al. , 2016b). According to Shi and Kioupakis, the ab-sorbance of O − M X
MLs is unusually strong in the visi-ble range (Shi and Kioupakis, 2015).
B. Raman spectra
Raman spectroscopy is employed to determine thethickness of layered materials (Castellanos-Gomez et al. ,2014; Li et al. , 2012). As indicated in Sec. II, the atomicbonds evolve with the number of layers in O − M X
MLs(Poudel et al. , 2019; Ronneberger et al. , 2020), whichshould leave signatures in the Raman spectra. Indeed,Raman modes B u , B g , A g and B g are shown formonolayer and bulk SnS in Fig. 13(a), and shift as afunction of the number of MLs is seen in Fig. 13(b) (Park et al. , 2019). Experimentally-determined Raman signa-tures for ultrathin SnS are displayed in Fig. 13(c) forcomparison (Higashitarumizu et al. , 2020). C. Second harmonic generation
Within a semiclassical picture, the SHG originatesfrom the non-sinusoidal motion of carriers inside crystalslacking inversion symmetry, leading to a quadratic effectin the electric field in O-MX MLs that is forbidden inthe bulk. SHG is widely utilized in applications rangingfrom table-top frequency multipliers, surface symmetry7
FIG. 13 (a) Relevant phonon modes for Raman spectra ofML and bulk SnS. (b) Raman shift of SnS as a functionof MLs. Subplots (a) and (b) are adapted from Ref. (Park et al. , 2019) with permission. Copyright, 2020, Springer Na-ture. (c) Experimentally observed thickness dependence ofRaman spectrum at 3 K. The peaks in the hatch are due tothe substrate. From Ref. (Higashitarumizu et al. , 2020) withpermission. Copyright, 2020, Springer Nature. probes, and photon entanglement in quantum comput-ing protocols, among others (Boyd, 2020).If the incoming electric field is homogeneous andmonochromatic E a = E aω e − iωt + c.c., the second-orderpolarization of the crystal oscillates at twice the drivingfrequency: P a = (cid:88) bc χ abc ( − ω ; ω, ω ) E bω E cω e − i ωt + c.c. , (3)where χ abc ( − ω ; ω, ω ) is the SHG response tensor, a isthe cartesian direction of the created electric field, and b and c are the cartesian directions of the incident electricfields. Far away from the source, the irradiated field isgiven by E ∼ d P dt (Jackson, 1998).SHG has been reported for ultrathin noncentrosym-metric samples of MoS and h-BN with odd layer thick-nesses (Li et al. , 2013). The angular dependence of SHGalso reveals the rotational symmetry of the crystal lattice,and can therefore be used to determine the orientationof crystallographic axes (Attaccalite et al. , 2015; Janisch et al. , 2014; Kim et al. , 2013; Kumar et al. , 2013; Li et al. ,2013; Malard et al. , 2013; Zhou et al. , 2015). This effecthas been theoretically (Panday and Fregoso, 2017; Wangand Qian, 2017a) and experimentally (Higashitarumizu et al. , 2020) studied in O − M X
MLs, too.Following the choice of axes in Fig. 1(c), the O − M X
ML defines the xy -plane and the polar axis (the directionof P ) lies along the positive x -direction. Its point grouponly allows for non-zero xzz , xyy , xxx , yyx , zxz compo-nents of χ abc (plus the components obtained by exchang-ing of the last two indices, χ abc = χ acb ). As exemplified FIG. 14 (a) Absolute and imaginary SHG tensor χ abb ( − ω, ω, ω ) for a SnS ML as a function of the outgoingphonon frequency 2 ω . (b) Comparison between the experi-mental SHG tensor for GaAs(001) from (Bergfeld and Daum,2003) and the computed one for a GeSe ML, which is an orderof magnitude larger. Subplots (a) and (b) are adapted fromRef. (Panday and Fregoso, 2017) with permission. Copyright,2017, Institute of Physics. (c) Experimental polarization de-pendence of SHG at 425 nm and room temperature for aSnS ML under perpendicular polarization. The inset axesshow the x − and y − directions corresponding to these definedin Fig. 1(c). Reproduced from Ref. (Higashitarumizu et al. ,2020) with permission. Copyright, 2020, Springer Nature. for a SnS ML in Fig. 14(a), the SHG spectrum displayspeak values within the visible spectrum that can be anorder of magnitude larger than those reported in GaAs(Bergfeld and Daum, 2003) [Fig. 14(b)] or a MoS ML(Malard et al. , 2013; Wang and Qian, 2017a). The SHGspectrum is anisotropic, and | χ xyy | > | χ xxx | > | χ xzz | holds approximately true for all frequencies (Panday andFregoso, 2017). This is a counterintuitive result, as themaximum response along the polar axis occurs for inci-dent optical fields that are perpendicularly polarized tothe polar axis.The role of the spontaneous polarization P in thelarge SHG response tensor and in other nonlinear re-sponses is an active area of investigation. The largemagnitude of the SHG in O − M X
MLs seems to be acombination of many factors, including their reduced di-mensionality and in-plane polarization. Indirect evidencesuggests that the in-plane P enhances the SHG by es-tablishing mirror symmetries that strongly constrain con-tributions from certain regions within the BZ (Panday et al. , 2019).O − M X
MLs grown on insulating substrates permitperforming optical experiments, and Fig. 14(c) is an ex-perimental demonstration of the anisotropic behavior ofthe SHG of a SnS ML (Wang and Qian, 2017a) at roomtemperature, using an 850-nm laser as the excitationsource (Higashitarumizu et al. , 2020). The largest SHGoccurs along the y − axis. As discussed in Sec. IV, thesense of direction of P can be set by a combination ofSHG and transport measurements.8 D. Bulk photovoltaic effects: injection and shift currents
The bulk photovoltaic effect (BPVE) is the genera-tion of a dc current upon illumination in materials thatlack inversion symmetry. It has been extensively stud-ied in bulk ferroelectrics (Ivchenko and Ganichev, 2016;Sturman and Sturman, 1992), topological insulators (Ho-sur, 2011), 2D ferroelectrics (Kushnir et al. , 2019, 2017;Panday et al. , 2019; Rangel et al. , 2017), Weyl semimet-als (Chan et al. , 2017; de Juan et al. , 2017; Rees et al. ,2019; Shvetsov et al. , 2019), BN nanotubes (Kr´al et al. ,2000), among other materials. Many seemingly unre-lated BPVEs have been shown to have a common origin(Fregoso, 2019; Sipe and Shkrebtii, 2000). The BPVEis much larger in 2D ferroelectrics than in bulk ferro-electrics, potentially overcoming the low solar energy ef-ficiency conversion found in the latter (Rappe et al. , 2017;Spanier et al. , 2016; Tan et al. , 2016).The BPVE differs from other photovoltaic effects inthree important ways: (i) it is proportional to the inten-sity of the optical field; (ii) it produces large open-circuitphotovoltages, i.e. , larger than the energy band gap; and(iii) it depends on the polarization state of light. Thesecharacteristics imply, respectively, (i) that the BPVE isa second order effect in the optical field, (ii) that it isan ultrafast phenomena occurring before thermalizationtakes place at the CBM (VBM), and (iii) that the BPVEresponse tensor has a real and an imaginary component.The real component ( σ ) determines the response to lin-early polarized light; the imaginary component ( η ) isthe response to circularly polarized light. Denoting anhomogeneous optical field by E , the BPVE can then beschematically written as (Sturman and Sturman, 1992): J bpve = η E × E ∗ + σ E . (4)The first term is the so-called ballistic current, injec-tion current, or circular photogalvanic effect , and it van-ishes for linear polarization. The injection current iscreated by an unequal momentum relaxation into timereversal states (Ivchenko and Ganichev, 2016; Sturmanand Sturman, 1992) or by unequal carrier pumping ratesinto time-reversed states ± k (Fregoso, 2019; Sipe andShkrebtii, 2000). In addition, and related to spin ef-fects discussed in Sec. VIII.C, the chirality of circularly-polarized light couples to the spin of charge carriers togenerate a spin current in spin-orbit coupled systems(Chan et al. , 2017; Hosur, 2011; Sturman and Sturman,1992). Fig. 15(a) shows the spectrum of the injection cur-rent tensor for a GeSe ML. The only non-zero componentis η yyx and, as a consequence, injection current can onlyflow perpendicularly to the polar ( x ) axis. The injectioncurrent tensor η reaches peak values of 10 A/V s inthe visible spectrum (1 . − et al. , 2019),which is many orders of magnitude larger than its peakvalue in MoS MLs (Arzate et al. , 2016).
FIG. 15 (a) Injection current and (b) shift current for a GeSeML [with axes defined as in Fig. 1(c)]. The injection currentof CdSe is also plotted for comparison. Subplot (a) is adaptedfrom Ref. (Panday et al. , 2019). Copyright, 2019, AmericanPhysical Society. Subplot (b) is taken from Ref. (Rangel et al. ,2017). Copyright, 2017, American Physical Society.
The second term in Eqn. (4) is the shift current, alsoknown as the linear photogalvanic effect , and its micro-scopic interpretation is still under debate. A popularinterpretation is that the shift current arises from a shiftof the electron in real-space when it absorbs a photon(von Baltz and Kraut, 1981). This is reasonable sincethe Wannier centers of charge are spatially separatedin materials that break inversion symmetry. In a sec-ond interpretation, the quantum coherent motion of apair of dipoles moving in k − space originates shift cur-rents (Fregoso, 2019), which vanish for incident circularlypolarized light. Fig. 15(b) shows the shift-current spectrafor a GeSe ML. There is a broad maximum of the orderof 150 µ A/V (Rangel et al. , 2017) in the visible range(1 − e.g. , σ ∼ . µ A/V in BiFeO (Youngand Rappe, 2012)]. These results demonstrate the uniquepotential of O − M X
MLs for optoelectronic applications.
E. Photostriction
Photostriction is the structural change induced by ascreened electric polarization P resulting from photoex-cited electronic states: optical excitations will lead to aconcurrent compression of lattice vector a and a com-paratively smaller increase of a for an overall reductionin the unit cell area. The structural change documentedfor SnS and SnSe MLs is 10 times larger than that ob-served in bulk ferroelectric BiFeO , making O − M X
MLsan ultimate platform for this effect (Haleoot et al. , 2017).
F. Excitons
An exciton is an electron-hole pair hosted within amaterial, whose description therefore goes beyond thesingle-particle picture employed thus far. Excitons dis-play a strong dependency on dimensionality, being morestrongly bound in low-dimensional systems due to a re-duced Coulomb screening, and hence relevant at roomtemperature (Gomes et al. , 2016). Upon laser irradiation,9
FIG. 16 (a) Absorption spectra of GeS, GeSe, and SnSe MLswith (w e − h ) and without (w/o e − h ) electron-hole interac-tions for light polarized along the x − direction ( i.e. , parallelto P or along the y − direction (perpendicular to P . Twopeaks are identified for GeS MLs: peak 1 arises from a directtransition at the Γ − point, and peak 2 from a direct transi-tion at the V x valley (see Figure 9). For both GeSe and SnSeMLs, the only excitation within the band gap correspondsto a direct transition at the V x valley. (b) Subplots for GeSand GeSe MLs were adapted from Ref. (Gomes et al. , 2016).Copyright, 2016, American Physical Society. The subplotsfor the SnSe ML were adapted from Ref. (Shi and Kioupakis,2015). Copyright, 2015, American Chemical Society. a MoS ML displays isotropic excitons with a binding en-ergy of 0.55 eV on a graphitic substrate (Splendiani et al. ,2010; Ugeda et al. , 2014). In turn, anisotropic excitonsin BP MLs have been shown to have a larger binding en-ergy of 0.8-0.9 eV (Rodin et al. , 2014a; Tran et al. , 2014;Wang et al. , 2015). Possible optoelectronic applicationsof O − M X
MLs also call for a deep understanding of exci-tons (Gomes et al. , 2016). Given that they contribute tothe dielectric environment, substrates in which 2D ma-terials are placed may need to be accounted for whencomparing experiment and theory.The exciton binding energy has been calculated forfreestanding GeS, GeSe, and SnSe MLs. It increases fromless than 0.01 eV in the bulk to 1.00 eV on a GeS ML,0.32-0.40 eV on a GeSe ML, and 0.27-0.30 eV in a SnSeML. As seen in Fig. 16(a), the binding energy is larger inmaterials with small ¯ Z , for which the absorbance is moreanisotropic when shining light with polarization alongthe x − or y − directions (Gomes et al. , 2016; Shi andKioupakis, 2015). Concomitant with a stronger bindingand anisotropic absorbance, lighter O − M X
MLs hostmore localized and anisotropic excitons [see Fig. 16(b)].An analytical, Mott-Wannier model has been employedto account for the effect of the supporting substrate—which in general lowers the exciton binding energy—inRef. (Gomes et al. , 2016).
X. SUMMARY AND OUTLOOK
Two-dimensional and ultrathin ferroelectrics are gain-ing increased attention. They complement two-dimensional semimetal graphene, insulator hexagonalboron nitride, a large number of 2D semiconductors, andtwo-dimensional magnets. This Colloquium describes thestructural, mechanical, electronic, and optical propertiesof group-IV monochalcogenide MLs in a comprehensivemanner, including recent developments such as the ex-perimental realization of SnS and SnSe MLs, and noveltheoretical results such as spin helix behavior, theoreti-cal Raman spectra, and bulk photovoltaic effects, to be-come the most up-to-date reference on these materials.While there are a number of challenges still to be resolvedconcerning chemical stability, exfoliation or growth, andtheir stacking into functional layered materials, these ul-trathin ferroelectric and ferroelastic materials have al-ready diversified and enriched the library of layered andtwo-dimensional functional materials. Their prospectiveuse in memory, valley, and optoelectronic applicationscan provide the motivation and justification to drive fur-ther progress in this area.
ACKNOWLEDGMENTS
We thank P. Kumar, L. Bellaiche, J. E. Moore, L. V.Titova, T. Rangel and L. Fu. S.B.-L. and J.W.V. ac-knowledge funding from the US Department of Energy,Office of Basic Energy Sciences (Early Career AwardDE-SC0016139) and DOE-NERSC contract No. DE-AC02-05CH11231. S.S.P.P. and K.C. were supportedby Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation), Project number PA 1812/2-1.
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