PHYSICAL REVIEW APPLIED 11, 064063 (2019)

Lattice Dynamic and Instability in Pentasilicene: A Light Single-ElementFerroelectric Material With High Curie Temperature

Yaguang Guo,1 Cunzhi Zhang,1 Jian Zhou,2 Qian Wang,1,3,* and Puru Jena3

1Center for Applied Physics and Technology, Department of Materials Science and Engineering, HEDPS,

BKL-MEMD, College of Engineering, Peking University, Beijing 100871, China2Center for Advancing Materials Performance from the Nanoscale, State Key Laboratory for Mechanical

Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China3Department of Physics, Virginia Commonwealth University, Richmond, Virginia 23284, USA

(Received 20 April 2019; revised manuscript received 6 June 2019; published 26 June 2019)

The compatibility of Si semiconductor technology with current electronic devices has led to a continu-ing search for new Si-based materials with novel properties. The recent synthesis of penta-Si nanoribbons[Phys. Rev. Lett. 117, 276102 (2016); Nano Lett. 18, 2937 (2018); Nat. Commun. 7, 13076 (2016)] isa new addition to this family. However, pentasilicene, a two-dimensional (2D) sheet composed of onlySi pentagons, was found to be unstable in pentagraphenelike configuration and could only be stabilizeddynamically by surface decoration. Using first-principles calculations and a thorough analysis of its imag-inary frequencies, we show that pentasilicene can indeed be made dynamically stable by tilting the Sidimers to reduce the Coulomb repulsion between them. The consequence of this tilting leads to a muchmore interesting discovery: the stabilized pentasilicene breaks the structural inversion symmetries, result-ing in spontaneous electrical polarization and ferroelectricity with a high Curie temperature of 1190 K.This is the first report of a 2D ferroelectric system composed of a light single element, which can havepotential applications in nonvolatile random access memory.

DOI: 10.1103/PhysRevApplied.11.064063

I. INTRODUCTION

Over the past few decades, silicon has been theleading material in the microelectronic industry for high-performance information storage devices. Recently, con-siderable effort has been made to develop nonvolatilephase change materials, such as ferroelectric or ferro-magnetic RAMs, for ultrafast information storage devicesdown to the sub-10-nm size. In the ferroelectric phasechange materials, different ferroelectric phases are sepa-rated by a high-energy barrier owing to their ionic motion.Hence, the ferroelectric phase can be well protected belowthe Curie temperature and the stored information can lastfor over 10 years. In addition, ferroelectric phase transi-tions are displacive and martensitic, which guarantee itsfaster kinetics compared to the currently used reconstruc-tive Ge-Sb-Te alloys [1]. Two-dimensional (2D) materialshave the potential to work as future nonvolatile phasechange materials [2] so that electronic devices can beminiaturized into nanoscale with high information den-sity. Thus far, Si is used as a substrate to grow ferro-electric materials (e.g., HfO2) [3], but 2D ferroelectricphase change materials based exclusively on Si are still

*qianwang2@pku.edu.cn

unknown. The reason is that once a 2D material is basedon a single element (e.g., Si), positive and negative chargestates are no longer separated as they are in conventionalferroelectric materials (e.g., perovskites) [4,5]. This greatlyhinders the potential applications for Si electronics.

In order to separate positive and negative charge cen-ters in a pure 2D Si material, one possible scheme is toinclude both sp2 and sp3 hybridization. Note that con-ventional Si compounds are composed exclusively of sp3

hybridized Si atoms. Recently, 2D silicene (a single atomiclayer Si sheet) composed of only sp2 Si has been dis-covered [6]. Unfortunately, due to its zero band gap [7],2D silicene is not suitable for information storage. Moti-vated by the prediction of pentagraphene [8], a 2D five-membered ring-based carbon allotrope composed of sp2

and sp3 hybridized carbon atoms, attempts have been madeto synthesize a pentasilicene analog. However, theoret-ical studies showed such a structure to be dynamicallyunstable and it can only be stabilized by surface decora-tion [9,10]. Note that penta-Si nanoribbon [11–13], whichexhibits exotic phenomena, such as topologically protectedphases or increased spin-orbit effects [11], has alreadybeen synthesized and can be used as a building blockto produce functional architectures via covalent assem-bly [12]. Hence, the discovery of an intrinsically stable

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YAGUANG GUO et al. PHYS. REV. APPLIED 11, 064063 (2019)

pentasilicenelike structure will be a welcome addition tothe Si family of materials.

In this paper, we show that the desired unusual proper-ties of Si can be realized by stabilizing Si in a quasi-2Dpentagonal structure where Si dimers within a confinedlattice are tilted. We label this system as T pentasil-icene. Because the existence of soft modes of pentasiliceneis due to the strong Coulomb interactions between thecharged Si atoms, tilting the Si dimers can reduce Coulombrepulsion and stabilize the system. Using first-principlescalculations combined with the Berry phase method andLandau-Ginzburg theory, we have not only found this tiltedconfiguration to be dynamically stable, but it also leadsto spontaneous electrical polarization, making T pentasil-icene exhibit in-plane ferroelectricity with a very highCurie temperature (TC) of 1190 K. Thus, this ferroelec-tric phase can be well protected under an ambient workingtemperature.

II. COMPUTATIONAL METHODS

Calculations are carried out using first-principlesdensity-functional theory (DFT) with electronic exchange-correlation interaction treated by the Perdew-Burke-Ernzerh (PBE) functional within the generalized gradi-ent approximation (GGA) [14]. The projector augmentedwave (PAW) method [15,16] with plane-wave basis setembedded in the Vienna Ab initio Simulation Package(VASP) [17] is used to optimize the geometrical structuresand study their electronic properties. The energy cutoffis set at 500 eV. The convergence criteria for energyand force are set to 0.0001 eV and 0.01 eV/Å, respec-tively. The Brillouin zone is represented with a 19 × 19 × 1Monkhorst-Pack special k-point mesh [18]. A vacuumspace of 20 Å in the direction perpendicular to the sheetis used to avoid interactions between the two neighbor-ing layers. To ensure an accurate determination of elec-tronic properties, calculations are repeated by using thehybrid Heyd-Scuseria-Ernzerhof functional (HSE06) [19].Phonon calculations are performed by using both the finite-displacement method [20] and density-functional perturba-tion theory (DFPT) method with a 5 × 5 × 1 supercell, asimplemented in the PHONOPY program [21]. The criteriafor the energy and force convergence are 1 × 10−8 eV and1 × 10−6 eV/Å, respectively. The intrinsic electrical polar-ization is calculated by using the Berry phase method [22]as implemented in VASP.

III. RESULTS AND DISCUSSION

A. Geometry and stability analysis

Figure 1(a) shows the structure of pentasilicene, whichshares the same geometric feature as pentagraphene [8].We find that the sp3 (four-fold coordinated) and sp2 (three-fold coordinated) Si atoms are charged, and denote them as

(a) (b)

FIG. 1. (a) Top view of the pentasilicene sheet. The red-dashedsquare is the (l, m)th lattice point of the sheet. Indices a, b, c,d, e, and f represent different atoms in the unit cell. The blueand red numbers denote charge states. (b) Phonon spectrum ofpentasilicene. The inset image shows the soft vibrational modes.

Si4 and Si3, respectively. Through Bader’s charge analysis,the charges on Si3 and Si4 are found to be q0 = 0.61 and−2q0 =−1.22, respectively. The phonon spectrum is plot-ted in Fig. 1(b). One can see that two imaginary branchesappear throughout the entire Brillouin zone, which is con-sistent with previous results [9,10]. In order to understandthe origin of the dynamic instability of this structure, westudy the phonon modes by applying a classical displace-ment model. The total energy is estimated to contain twoparts, namely, conventional elastic energy and electrostaticinteractions between the charged atoms. For simplicity, weadopt a rigid ion model of atoms with Coulomb inter-action and neglect the short-range Born-Mayer potential.Following the general notations in lattice dynamics, theposition of atom κ in the lth lattice is r

(lκ

)and rij =

r(

lj liκj κi

)= r

(ljκj

)− r

( liκi

). Thus, the total energy can be

written as

E = Eelastic + Eelectrostatic

= 12

⎛

⎝12β

∑

〈i,j 〉r2

ij +∑

i,j ,j �=i

ZκiZκj

rij

⎞

⎠ . (1)

Here, i (j ) is the atom index. Z and β refer, respectively,to its charge and the normalized stiffness of the spring(force constant) between the nearest neighbor i and j. Thesecond term accounts for the Coulomb interaction. A fac-tor of 1/2 is used to avoid double counting. In the unit cellof pentasilicene, there are six Si atoms [labeled as a, b, c,d, e, and f in Fig. 1(a)]. Hence, the dynamic matrix of theequation of motion is an 18 × 18 matrix.

Because of the large size of the dynamic matrix, thedimension of such a scalar determinant needs to bereduced. By analyzing the soft vibrational modes, wefind that Si3 atoms prefer to move mainly along the

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LATTICE DYNAMIC AND INSTABILITY IN PENTASILICENE. . . PHYS. REV. APPLIED 11, 064063 (2019)

out-of-plane direction [see Fig. 1(b)]. Hence, we onlyconsider the displacements along the z direction. In thereciprocal space, the dynamic matrix is

M = D + ZCZ, (2)

where D is a 6 × 6 dynamic matrix resulting from thespringlike force without Coulomb interaction. Z is thecharge matrix, Z = diag [−2q0, −2q0, q0, q0, q0, q0,], andmatrix C is independent of the charge and mass of ions

C(

qκ κ ′

)= −2 exp

{iq · r

[l l′κ κ ′

]}

×∑

l′,lr−3

(l l′κ κ ′

). (3)

The square root of the eigenvalues of dynamic matrix Mis the phonon frequency. An exact analytical solution ofthis dynamic matrix is very challenging. To get insightsinto the phonon behavior, we apply the nearest-neighborinteraction approximation (i.e., only consider the leadingterm of the Coulomb force) and write Newton’s equationof motion as follows:

Mal,m = β1[(cl,m − al,m) + (dl,m − al,m) + (el,m − al,m)

+ (fl,m − al,m)] + C3−4A [(cl,m − al,m)

+ (dl,m − al,m) + (el,m − al,m) + (fl,m − al,m)],

Mbl,m = β1[(cl,m − bl,m) + (dl+1,m − bl,m)

+ (el+1,m+1 − bl,m) + (fl,m+1 − bl,m)]

+ C3−4A [(cl,m − bl,m) + (dl+1,m − bl,m)

+ (el+1,m+1 − bl,m) + (fl,m+1 − bl,m)],

Mcl,m = β1[(al,m − cl,m) + (bl,m − cl,m)]

+ β2(el,m+1 − cl,m) + C3−3R (el,m+1 − cl,m)

+ C3−4A [(al,m − cl,m) + (bl,m − cl,m)],

Mdl,m = β1[(al,m − dl,m) + (bl−1,m − dl,m)]

+ β2[fl−1,m − dl,m] + C3−3R (fl−1,m − dl,m)

+ C3−4A [(al,m − dl,m) + (bl−1,m − dl,m)],

Mel,m = β1[(al,m − el,m) + (bl−1,m−1 − el,m)]

+ β2(cl,m−1 − el,m) + C3−3R (cl,m−1 − el,m)

+ C3−4A [(al,m − el,m) + (bl−1,m−1 − el,m)],

Mfl,m = β1[(al,m − fl,m) + (bl,m−1 − fl,m)]

+ β2(dl+1,m − fl,m) + C3−3R (dl+1,m − fl,m)

+ C3−4A [(al,m − fl,m) + (bl,m−1 − fl,m)], (4)

where M is the mass of a Si atom, β1 and β2 are the springstiffnesses of Si3-Si4 and Si3-Si3, and the C3−3

R and C3−4A

are, respectively, the constants of Coulomb repulsive andattractive forces, which have the following expressions:

C3−3R = − 2|ZSi3

i ||ZSi3j |

|�rSi3i − �rSi3

j |3, C3−4

A = 2|ZSi3i ||ZSi4

j ||�rSi3

i − �rSi4j |3

. (5)

By solving these equations (see Text S1 in the Supplemen-tal Material for details [23]), we find that the degenerateimaginary frequencies in the pentasilicene sheet [see Fig.1(b)] are mainly determined by the sum of β1, β2, C3−4

A ,and C3−3

R . It is confirmed that the first three terms are pos-itive, while C3−3

R is negative (details can be found in TextS2 of the Supplemental Material [23]). Thus, the Coulombrepulsions from Si3-Si3 contribute a negative sign to M,which destabilizes the system.

This suggests that an effective way to stabilize pentasil-icene would be to reduce the Coulomb repulsive force.Similar to Pandey’s chain buckling of a Si(111) (2 × 1)reconstructed surface [24–26], we find that a small tilt ofSi3 dimers can lower the energy and stabilize the phonondispersions, leading to the formation of a stable structure,T pentasilicene, as shown in Fig. 2(a). Different from theflat Si3 dimer in pentasilicene [Fig. 1(b)], one Si3 atommoves away from the sheet (denoted as Si3,o, toward thesp3 configuration) and the other Si3 atom moves towardthe sheet (denoted as Si3,i, toward the sp2 configuration).Note that the movement of Si3 atoms follows the eigenvec-tors of the soft modes [see red arrows in Fig. 1(b)]. Aftergeometry optimization, the bond length of Si3,o-Si3,i is2.26 Å, and they carry charges q3,o = 0.70 and q3,i = 0.52,respectively. The space group of T pentasilicene is reducedfrom P421m to P21. Compared with pentasilicene [Fig.1(a)], the numerator

(ZSi3

i ZSi3j

)of Eq. (5) decreases, while

the denominator(∣∣�rSi3

i − �rSi3j

∣∣3) increases, thereby reduc-ing the value of C3−3

R . Consequently, the total energy ofthe system is lowered by 81 meV/cell due to the reduced

(a) (b)

FIG. 2. (a) Atomic geometry and (b) phonon spectrum alongthe high symmetric k-path of T pentasilicene.

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YAGUANG GUO et al. PHYS. REV. APPLIED 11, 064063 (2019)

Coulomb repulsion. The enhanced stability is verified bythe phonon spectrum as shown in Fig. 2(b), where no imag-inary frequency exists throughout the entire Brillion zone,confirming the dynamic stability of T pentasilicene. Notethat the T pentasilicene sheet should be confined within aconstrained lattice during the geometry optimization, oth-erwise it gradually undergoes a transformation into thepredicted siliconeet phase [27]. This implies that a properconfining material (e.g., a metal substrate) is needed to pro-duce T pentasilicene in experiments (see Text S3 of theSupplemental Material for more details [23]).

Next, we analyze the Si3 dimers from the electronicstructure point of view. The evolution of the tilted dimerbond formation is plotted in Fig. 3(a), where the dark solidcircles represent the Si3 atoms. Each Si3 in the dimer is ina distorted sp2 configuration, which leaves one pz orbitalas a dangling bond. These two pz orbitals hybridize andform a π -π* pair. However, because of the larger atomicsize of a Si atom as compared to that of a C atom [28],the π bonds are not strong enough to stabilize the pen-tasilicene structure. Therefore, a separation in π -π* levelsoccurs to further reduce the energy. This is similar to theJahn-Teller distortion and electrons on Si3,i site transfer tothe Si3,o atom [see arrows in Fig. 2(a)], forming a tilteddimer.

In Fig. 3, we also compare the bonding configurationof T pentasilicene with other previously discovered pen-tasheets [8,9,29–32]. For pentagraphene [8] [see Fig. 3(b)],

the π -π* interaction is strong with a large strain energy.Hence, the flat three-fold coordinated carbon dimers canbe maintained and the symmetric configuration is dynam-ically stable. This is similar to the spontaneous bucklingof Pandey’s sp2 dimer chain in a Si(111) (2 × 1) recon-structed surface, while no buckling occurs on diamondC(111) owing to the strong strain energy [24–26]. Forpenta-B2C [29] [see Fig. 3(c)], since B has one valenceelectron less than Si, there is no extra electron on the π

orbital. Thus, no extra reduction in energy can be achievedby dimer tilting. Hence, the flat configuration remains.For penta-CN2, SiN2, or SiP2 [30,31] [see Fig. 3(d)], thedimers consist of group V element atoms, which have oneelectron more than Si. Since π and π* orbitals are fullin these cases, energy cannot be reduced by dimer tilt-ing. As for the hydrogenated pentasilicene [9] [see Fig.3(e)], H and Si3 atoms form a sigma bond, which satu-rates the dangling bond. In this case, all of the electronsare paired, hence reducing the energy and keeping the flatconfiguration of the Si dimers.

B. Electronic structure

Next, we examine the electronic properties of T pen-tasilicene. In Fig. 4(a), one can see that the band edgesare mainly contributed by the Si3 atoms, similar to thosein pentagraphene. Since the dimer tilting further enlargesthe energy separation between the π -π* levels, the T

(a)

(b) (c) (d) (e)

FIG. 3. (a) Understanding the dimer tilting in T pentasilicene. Bonding diagram of (b) pentagraphene, (c) penta-B2C, (d) penta-CN2,SiN2, SiP2, and (e) hydrogenated pentasilicene.

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LATTICE DYNAMIC AND INSTABILITY IN PENTASILICENE. . . PHYS. REV. APPLIED 11, 064063 (2019)

(a) (b)

FIG. 4. (a) Band structure of T pentasilicene at the HSE06 levelprojected on Si4 atoms (red) and Si3 atoms (blue). (b) Chargedensity difference in Si3 dimer.

pentasilicene has a larger band gap of 0.65 eV (at theHSE06 level) than that of the original configuration(0.45 eV, see Fig. S4 in the Supplemental Material [23]).The valence band maximum (VBM) and conduction bandminimum (CBM) are located at different positions in theBrillouin zone, showing the signature of an indirect bandgap. We note that there exists a sub-VBM close to theposition of the CBM on the �-M path, thus the T pen-tasilicene can be considered as a quasidirect band-gapsemiconductor. The value of this gap is close to that of thethree-dimensional (3D) h-Si6 allotrope [33], which maylead to a wider range of applications for T pentasilicenein electronics as compared with the zero-gap silicene. Tobetter visualize the charge redistribution in the tilted Si3dimer, we calculate the charge density difference as shownin Fig. 4(b). The red and blue regions represent electronaccumulation and depletion, respectively. It is clear thatelectrons on the Si3,i site transfer to the Si3,o site, whichvalidates the bonding diagram in Fig. 3(a).

C. Ferroelectricity

The charge redistribution leads us to explore the electri-cal polarization of the T pentasilicene sheet. We find thattilting the Si3 dimer not only lowers the total energy ofthe system, but also induces a polarized bond between thetwo nonequivalent Si3 atoms (see Text S4 of the Supple-mental Material for more details [23]). By applying themodern Berry phase theory [22,34], we calculate the elec-tric polarization vector P0 of the T pentasilicene sheet, andfind that the in-plane polarization is 18.7 pC/m (along the xdirection), while the out-of-plane component is zero. Thisconfirms that T pentasilicene is a pure Si-based 2D fer-roelectric material, adding another member to the single-element ferroelectric family of As, Sb, and Bi monolayers[35]. Note that the phosphorene sheet is not electricallypolarized [35]. T pentasilicene extends this family to thethird row elements in the periodic table. Ferroelectricity inT pentasilicene raises another question: is an antiferroelec-tric (AFE) state energetically more stable? To study this,

we construct a 2 × 2 supercell with a checkerboard pat-tern, where the polarization is opposite in the neighboringunit cells (see Fig. S6 in the Supplemental Material [23]).After geometry optimization, we find that the AFE con-figuration transforms into the ferroelectric (FE) structure,which implies that the AFE state does not exist.

One of the most important properties in a ferroelectricmaterial is its phase transition. Because the Berry phasecalculations based on DFT are performed at zero temper-ature, it is necessary to study the temperature range overwhich the ferroelectric order can survive. In general, softmodes theory is a powerful tool to study the phase tran-sition in bulk ferroelectrics [32]. A way to estimate TC isby simulating the behavior of phonons at a finite tempera-ture. Unfortunately, this technique is not fully developed.Alternatively, we adopt the Landau-Ginzburg theory [36]to study the ferroelectric phase transition. Since thermo-dynamics governs the phase diagram, the key problemis to represent the free energy (E) as a function of theorder parameter (P). To this end, we define two geomet-ric parameters h1 and h2 as the bridge to connect E withP [see Fig. 5(a)]. When h1 = h2 = 0 (Phase A), the systemconverts back to the original pentasilicene phase, wherethe ferroelectric effect vanishes due to the inversion sym-metry. When h1, h2 > 0 or h1, h2 < 0, the structure becomespolarized. Especially for h1 = h2, there are two stable ener-getically degenerate phases (B and B′), which correspondto the T pentasilicene structures. We note that phases B andB′ can be transformed into each other by spatial inversion.Thus, if there is a polarization (PB) in the B phase, theremust be an inverse polarization (−PB) in the B′ phase.

Within the Landau-Ginzburg theory, we use the φ4

model [37] to write E as

E =∑

i

[A2

(P2i ) + B

4(P4

i )

]+ C

2

∑

〈i,j 〉(Pi − Pj )

2, (6)

where i and j denote the lattice points and symbol 〈〉 inthe summation represents the nearest neighbor lattice site.Parameters A, B, and C are determined by fitting to theDFT data. The first two terms correspond to the energycontributed by the polarization in each unit cell, and shoulddescribe the anharmonic double-well shape of the poten-tial profile. Note that because of spatial inversion, thereonly exist even powers. We neglect the higher-order terms,which are found to be less important. To obtain the valuesof parameters A and B, we relate E and P by mapping aseries of h1 and h2; a direct 2D mapping is computationallycostly. According to the structural symmetry of pentasil-icene, we note that the variations of h1 and h2 are equalto each other. Thus, we only need to consider the one-dimensional (1D) parameter space, that is, h1 = h2 = h, tosimplify our simulation. In practice, we calculate the totalenergy and polarization for each value of h. In Fig. 5(b), weplot the variation of energy (E) with respect to polarization

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YAGUANG GUO et al. PHYS. REV. APPLIED 11, 064063 (2019)

(a) (b)

(c) (d)

FIG. 5. (a) Schematic side views of thetwo degenerate ferroelectric phases (B andB′) and the nonpolar phase (A). (b) Double-well potential versus polarization for T pen-tasilicene. Eb is the potential barrier. A, B,and B′ correspond to the phases in (a). (c)The dipole-dipole interaction energy in Tpentasilicene calculated by using the mean-field method. (d) Variation of polarizationwith temperature obtained from MC simu-lations for T pentasilicene.

(P). One can see that the energy profile has a double-well shape (purple balls), which is an important feature offerroelectric materials. We observe that when h is largerthan 0.57 Å, the monotonic relationship between P andh is destroyed (see Text S5 in the Supplemental Material[23]). As a result, in Fig. 5(b), we only show the DFTresults with h < 0.57 Å, rather than the complete shapeof the double-well potential. Even so, because the transi-tion is between phases A and B (B′), the data presented inFig. 5(b) is enough to estimate the Curie temperature. Byfitting the power function, we obtain the values of param-eters A (−0.962) and B (2.89 × 10−3). The last term ofEq. (6) represents the dipole-dipole couplings between theneighboring cells, which are calculated using the super-cell approach and the mean-field approximation. The DFTresults with a parabola shape are plotted in Fig. 5(c), whichvalidates the use of the second-order approximation. Byfitting the points, we find parameter C to be 0.316.

Based on the effective Hamiltonian, as given in Eq. (6),we performed MC simulation to estimate TC. To eliminatethe effect of translational symmetry, a 50 × 50 supercell isused. We run more than 2 000 000 MC steps to reach ther-mal equilibrium, followed by at least another 2 000 000MC steps to obtain thermal averages. The results of theMC simulation are plotted in Fig. 5(d). One can see thatthe intrinsic TC of the T pentasilicene monolayer is about1190 K, which is inferred from the abrupt drop of polar-ization around this temperature. It is important to point outthat the magnitude of TC is rather large. This is mainly

caused by the higher configurational energy barrier (Eb)as shown in Fig. 5(b). In the above, we have demonstratedthat the total energy difference between the original andtilted pentasilicene is 81 meV per unit cell. This valueis close to kBTC (102.55 meV), indicating that Eb playsa dominant role in determining TC of T pentasilicene. Inaddition, we note that stable tilted Si dimers have beenobserved on a Si (001) surface at low temperatures [38].When the temperature is elevated to 200 K, the dimeratoms start to flip in a double-well potential separated byan energy barrier of 100 meV [39]. This potential profile isexactly same as that of T pentasilicene in our study. Moreinterestingly, analysis of the reflection high-energy elec-tron diffraction (RHEED) rocking curve shows that thereexists a phase transition of a Si (001) surface around 900 K.This corresponds to the transition of the Si dimer struc-ture from the asymmetric to symmetric configuration [40].This phenomenon accounts for the metallization of the Si(001) surface [41]. As mentioned before, the band gap ofthe T pentasilicene (0.65 eV) is larger than that of theoriginal structure (0.45 eV). Because of these similarities(double-well potential with comparable Eb and reductionof electronic band gap), it is easy to understand that the TCof T pentasilicene (1190 K) is close to the phase transitiontemperature of the Si (001) surface. Thus, we conclude thatthe TC value obtained from the MC simulation is reliable.

The large TC indicates the stability of the ferroelec-tric phase, making T pentasilicene a desirable material fordevice applications at ambient temperature. For instance,

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it can potentially replace ferromagnetic RAM devices,as T pentasilicene combines the features of a semicon-ductor and ferroelectricity, while most ferromagnets aremetals. In addition, compared with the insulating tradi-tional ferroelectric materials (e.g., perovskites [4,5] andpolyvinylidene difluoride [42,43]) or binary oxides (e.g.,HfO2/ZrO2 [3]), T pentasilicene has a narrow band gapwith dispersive band edges, as shown in Fig. 4(a). Thus,it could entail both functions of nonvolatile memory andmanipulation of signals, and better transport properties canbe expected to achieve high device performance. The cal-culated coercive electric field of T pentasilicene is about0.2 V/Å. This may be a major concern for its data stor-age, because such a large value implies a high workingvoltage, which would lead to a high writing energy. How-ever, such an electric field strength can be reduced at afinite temperature. In addition, unlike general ferroelectricmaterials, there is no lattice strain change during the phasetransition of T pentasilicene. This stress-free phase transi-tion does not have the usual stress accumulation that couldinduce material damage after many cycles, thus ensuringits reversibility.

The fundamental properties of T pentasilicene implythat it can work as a potential ferroelectric device once itis fabricated. However, the device performance does notuniquely depend on the intrinsic properties of a material.For practical applications of T pentasilicene, some otherfactors need to be considered, such as interface effects bysubstrate and its behavior under an external field. One alsohas to consider protecting it by nonreactive layers (suchas BN), since sp2 hybridized Si is chemically reactive.Hence, future insightful studies are expected to confirm ourpredictions on T pentasilicene.

IV. CONCLUSIONS

In summary, using a classical displacement model com-bined with DFT calculations, we show that the largeCoulomb repulsion between charged Si3 atoms accountsfor the instability of pentasilicene. We predict an alterna-tive dynamically stable T pentasilicene sheet composed ofonly Si five-membered rings, which exhibits in-plane fer-roelectricity. Our study realizes spontaneous electric polar-ization in a light single-element system. Using Landau-Ginzburg theory, we estimate the Curie temperature ofthis material to be 1190 K. Such a high TC suggests arobust ferroelectric stability of the T pentasilicene sheet,with potential for applications in nanoelectronic devices.Given that the T pentasilicene sheet is solely composed ofSi atoms, it could be integrated into the well-developedSi semiconductor wafer technology. We hope that ourstudy will stimulate more research into this material andour prediction can be experimentally validated in the nearfuture.

ACKNOWLEDGMENTS

This work is partially supported by grants fromthe National Science Foundation China (Grant No.NSFC-21773004), and the National Key Research andDevelopment Program of the Ministry of Science andTechnology of China (Grants No. 2016YFE0127300and No. 2017YFA0205003), and is supported by theHigh-Performance Computing Platform of Peking Uni-versity, China. Y. G. acknowledges the Project fundedby China Postdoctoral Science Foundation (Grant No.2019M650289). J. Z. acknowledges the Young TalentSupport Plan of Xi’an Jiaotong University. P. J. acknowl-edges support by the U.S DOE, Office of Basic EnergySciences, Division of Material Sciences and Engineeringunder Grant No. DE-FG02-96ER45579.

[1] J. Hegedüs and S. Elliott, Microscopic origin of the fastcrystallization ability of Ge-Sb-Te phase-change memorymaterials, Nat. Mater. 7, 399 (2008).

[2] M. Wu and P. Jena, The rise of two-dimensional van derWaals ferroelectrics, Wiley Interdiscip. Rev.: Comput. Mol.Sci. 8, e1365 (2018).

[3] J. Muller, T. S. Böscke, U. Schröder, S. Mueller, D.Bräuhaus, U. Böttger, L. Frey, and T. Mikolajick, Ferro-electricity in simple binary ZrO2 and HfO2, Nano Lett. 12,4318 (2012).

[4] M. Dawber, K. M. Rabe, and J. F. Scott, Physics of thin-filmferroelectric oxides, Rev. Mod. Phys. 77, 1083 (2005).

[5] N. Sai, A. M. Kolpak, and A. M. Rappe, Ferroelectricity inultrathin perovskite films, Phys. Rev. B 72, 020101 (2005).

[6] P. Vogt, P. De Padova, C. Quaresima, J. Avila, E.Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G.Le Lay, Silicene: Compelling Experimental Evidence forGraphenelike Two-Dimensional Silicon, Phys. Rev. Lett.108, 155501 (2012).

[7] S. Cahangirov, M. Topsakal, E. Aktürk, H. Sahin, and S.Ciraci, Two-and One-Dimensional Honeycomb Structuresof Silicon and Germanium, Phys. Rev. Lett. 102, 236804(2009).

[8] S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, and P.Jena, Penta-graphene: A new carbon allotrope, Proc. Natl.Acad. Sci. U.S.A. 112, 2372 (2015).

[9] Y. Ding and Y. Wang, Hydrogen-induced stabilization andtunable electronic structures of penta-silicene: A computa-tional study, J. Mater. Chem. C 3, 11341 (2015).

[10] Y. Aierken, O. Leenaerts, and F. M. Peeters, A first-principles study of stable few-layer penta-silicene, Phys.Chem. Chem. Phys. 18, 18486 (2016).

[11] J. I. Cerdá, J. Sławinska, G. Le Lay, A. C. Marele, J. M.Gómez-Rodríguez, and M. E. Dávila, Unveiling the pentag-onal nature of perfectly aligned single-and double-strand Sinano-ribbons on Ag(110), Nat. Commun. 7, 13076 (2016).

[12] G. Prévot, C. Hogan, T. Leoni, R. Bernard, E. Moyen,and L. Masson, Si Nanoribbons on Ag(110) Studied byGrazing-Incidence x-Ray Diffraction, Scanning TunnelingMicroscopy, and Density-Functional Theory: Evidence of

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a Pentamer Chain Structure, Phys. Rev. Lett. 117, 276102(2016).

[13] S. Sheng, R. Ma, J. B. Wu, W. Li, L. Kong, X. Cong, D.Cao, W. Hu, J. Gou, J. W. Luo, P. Cheng, P. H. Tan, Y. Jiang,L. Chen, K. Wu, The pentagonal nature of self-assembledsilicon chains and magic clusters on Ag(110), Nano Lett.18, 2937 (2018).

[14] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gra-dient Approximation Made Simple, Phys. Rev. Lett. 77,3865 (1996).

[15] G. Kresse and D. Joubert, From ultrasoft pseudopotentialsto the projector augmented-wave method, Phys. Rev. B 59,1758 (1999).

[16] P. E. Blöchl, Projector augmented-wave method, Phys. Rev.B 50, 17953 (1994).

[17] G. Kresse and J. Furthmüller, Efficient iterative schemes forab initio total-energy calculations using a plane-wave basisset, Phys. Rev. B 54, 11169 (1996).

[18] H. J. Monkhorst and J. D. Pack, Special points forBrillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).

[19] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid function-als based on a screened Coulomb potential, J. Chem. Phys.118, 8207 (2003).

[20] K. Parlinski, Z. Li, and Y. Kawazoe, First-Principles Deter-mination of the Soft Mode in Cubic ZrO2, Phys. Rev. Lett.78, 4063 (1997).

[21] A. Togo, F. Oba, and I. Tanaka, First-principles calcula-tions of the ferroelastic transition between rutile-type andCaCl2-type SiO2 at high pressures, Phys. Rev. B 78, 134106(2008).

[22] R. D. King-Smith and D. Vanderbilt, Theory of polarizationof crystalline solids, Phys. Rev. B 47, 1651 (1993).

[23] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevApplied.11.064063 for lattice dynam-ics of pentasilicene (Text S1); determination of the signof spring stiffness, Coulomb repulsive force constant, andCoulomb attractive force constant (Text S2); T pentasil-icene on metal substrate (Text S3); band structure of pen-tasilicene (Fig. S4); formation of dipoles in T pentasilicene(Text S4); antiferroelectric configuration in T pentasilicene(Fig. S6); and relationship between the polarization P andheight of dimer tilting h in pentasilicene (Text S5).

[24] A. Scholze, W. G. Schmidt, and F. Bechstedt, Structureof the diamond (111) surface: Single-dangling-bond versustriple-dangling-bond face, Phys. Rev. B 53, 13725 (1996).

[25] K. C. Pandey, New π -Bonded Chain Model for Si(111)-(2×1) Surface, Phys. Rev. Lett. 47, 1913 (1981).

[26] F. Bechstedt, A. A. Stekolnikov, J. Furthmuller, and P.Kackell, Origin of the Different Reconstructions of Dia-mond, Si, and Ge(111) surfaces, Phys. Rev. Lett. 87,016103 (2001).

[27] Z. Wang, M. Zhao, X.-F. Zhou, Q. Zhu, X. Zhang, H. Dong,A. R. Oganov, S. He, and P. Grünberg, Prediction of novel

stable 2D-silicon with fivefold coordination, arXiv preprintarXiv:1511.08848 (2015).

[28] P. Borlido, C. Roedl, M. A. L. Marques, and S. Botti,The ground state of two-dimensional silicon, 2D Mater. 5,035010 (2018).

[29] M. Naseri, J. Jalilian, and A. Reshak, Electronic and opticalproperties of pentagonal-B2C monolayer: A first-principlescalculation, Int. J. Mod. Phys. B 31, 1750044 (2017).

[30] S. Zhang, J. Zhou, Q. Wang, and P. Jena, Beyond graphiticcarbon nitride: Nitrogen-rich penta-CN2 sheet, J. Phys.Chem. C 120, 3993 (2016).

[31] H. Liu, G. Qin, Y. Lin, and M. Hu, Disparate straindependent thermal conductivity of two-dimensional penta-structures, Nano Lett. 16, 3831 (2016).

[32] P. A. Fleury, J. F. Scott, and J. M. Worlock, Soft PhononModes and the 110° K Phase Transition in SrTiO3, Phys.Rev. Lett. 21, 16 (1968).

[33] Y. Guo, Q. Wang, Y. Kawazoe, and P. Jena, A new sili-con phase with direct band gap and novel optoelectronicproperties, Sci. Rep. 5, 14342 (2015).

[34] R. Resta, Macroscopic polarization in crystalline dielectrics:The geometric phase approach, Rev. Mod. Phys. 66, 899(1994).

[35] C. Xiao, F. Wang, S. A. Yang, Y. Lu, Y. Feng, and S. Zhang,Elemental ferroelectricity and antiferroelectricity in group-V monolayer, Adv. Funct. Mater. 28, 1707383 (2018).

[36] R. Fei, W. Kang, and L. Yang, Ferroelectricity and PhaseTransitions in Monolayer Group-IV Monochalcogenides,Phys. Rev. Lett. 117, 097601 (2016).

[37] J. C. Wojdel and J. Iniguez, Testing simple predictors forthe temperature of a structural phase transition, Phys. Rev.B 90, 014105 (2014).

[38] R. A. Wolkow, Direct Observation of an Increase in Buck-led Dimers on Si(001) at Low Temperature, Phys. Rev.Lett. 68, 2636 (1992).

[39] J. Dabrowski and M. Scheffler, Self-consistent studyof the electronic and structural properties of the cleanSi(001) (2×1) surface, Appl. Surf. Sci. 56–8, 15 (1992).

[40] Y. Fukaya and Y. Shigeta, Phase Transition From Asym-metric to Symmetric Dimer Structure on the Ge(001) Sur-face at High Temperature, Phys. Rev. Lett. 91, 126103(2003).

[41] L. Gavioli, M. G. Betti, and C. Mariani, Dynamics-InducedSurface Metallization of Si(100), Phys. Rev. Lett. 77, 3869(1996).

[42] M. Poulsen and S. Ducharme, Why ferroelectric polyvinyli-dene fluoride is special, IEEE Trans. Dielectr. Electr. Insul.17, 1028 (2010).

[43] Y. J. Park, S. J. Kang, B. Lotz, M. Brinkmann, A.Thierry, K. J. Kim, and C. Park, Ordered ferroelectricPVDF-TrFE thin films by high throughput epitaxy fornonvolatile polymer memory, Macromolecules 41, 8648(2008).

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