Mechanically-controllable strong 2D ferroelectricity and anisotropic optical properties of flexible BiN monolayer
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Mechanically-controllable strong 2D ferroelectricity and anisotropic optical propertiesof flexible BiN monolayer
Peng Chen,
1, 2
Xue-Jing Zhang,
1, 2 and Bang-Gui Liu
1, 2, ∗ Beijing National Center for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China. (Dated: November 28, 2017)Structural, electronic, ferroelectric, and optical properties of two-dimensional (2D) BiN monolayermaterial with phosphorene-like structure are studied in terms of the density functional theory andmodern Berry phase ferroelectric method. Both phonon spectra and molecular dynamics simulationsindicate that the BiN monolayer is a room-temperature stable 2D ferroelectric with polarization aslarge as 580 pC/m. Further studies show that the polarization in the BiN monolayer can be easilyswitched from [100] to [010] direction over the bridging saddle phase by applying a tensile [010]stress of 2.54 N/m or compressive [100] stress of -1.18 N/m. This phase transitions makes its latticeconstants vary in a large range compared to other non-ferroelectric 2D materials. Moreover, throughapplying uniaxial tensile stress parallel to the polarization, one can fix the polarization and changethe semiconductor energy gap from direct to indirect one. The optical properties feature a verystrong anisotropy in reflectivity below the photon energy of 4 eV. All these significant ferroelectric,electronic, and optical properties make us believe that the 2D BiN monolayer can be used to makestretchable electronic devices and optical applications.
I. INTRODUCTION
The two-dimensional material has attracted more andmore attentions in both condensed matter physics andmaterial science due to its wealth of physical phenom-ena because the charges and spins are confined to atwo-dimensional plane[1–7]. These special structural andelectronic properties can produce many unique optical,mechanical, and electronic functions[7] and electronicdevices[8–10]. In most of the applications of ferroelectricmaterials, thin films are used, as usual[11–13], becausethis allows an achievable moderate voltage to switch thepolarization[14]. However, when using thin films, toguarantee the devices work reliably, the quality of thethin film samples and the interfaces are highly demanded.Moreover, there exists a critical thickness[15] for the fer-roelectricity in the traditional perovskite ferroelectric ul-trathin films, because of the imperfect screening of de-polarizing field at the ferroelectric-metal interfaces[16].Therefore, a naturally Van der Waals layered ferroelec-tric materials may be advantageous.The idea of two-dimensional ferroelectricity is veryattractive because 2D layered materials, such as Blackphosphorus, graphene, and MoS , usually have greatmechanical flexibility and can sustain large strains ( ∼ T c is often farbelow room temperature (for two-dimensional isotropicHeisenberg spin models, one has T c = 0 accordingto Mermin-Wagner theorem[21]), two-dimensional ferro-electricity may survive at a relatively high temperature. ∗ [email protected] It is also of interest that two-dimensional ferroelectricityis compatible with two-dimensional semiconductors[16].Recently, ferroelectric and even multiferroic phenomenonhave been proved in several types of two-dimensionalmaterials[22–25]. Because the ridged structure ofblack phosphorus has a unique atomic arrangementorder[26], ferroelectricity is found in phosphorene-likestructures[22–24].In this paper, we use the Bi 6s lone pair to induce astrong ferroelectric polarization in BiN monolayer witha phosphorene-like structure. Our first-principles inves-tigation proves that this two-dimensional BiN is ther-mally and dynamically stable. Strain engineering cal-culations indicate that the ferroelectric polarization andthe semiconductor band gap can be easily manipulatedby applying uniaxial strains. The mechanical propertiesprove that the BiN monolayer is mechanically flexible.Its anisotropic features have a significant effect on its op-tical properties. These features make us believe that thisBiN monolayer with Phosphorene-like structure may beapplicable for stretchable electronic devices and opticalapplications. More detailed results will be presented inthe following.
II. METHOD
The calculations are performed in terms of the den-sity functional theory (DFT)[27, 28] and projector-augmented wave potentials[29], as implemented inVASP[30]. We use the exchange-correlation functionalPBEsol[31] because it is best for solid materials. Weadopt usual atomic pseudopotentials of Bi:6s and N:2s . Our computational model is a supercellconsisting of the BiN monolayer and a vacuum layer of20 ˚A. The plane wave energy cutoff is set to 500 eV.The 11 × × III. RESULTSA. Stable structure and ferroelectricity
FIG. 1. (Color online) Top views, side views, and phononspectra of reference phase (a.1, a.2, a.3), saddle phase (b.1,b.2, b.3), and ferroelectric phase (c.1, c.2, c.3) of BiN mono-layer. The large green ball represent Bi and the black ball N.The red circle in reference means inversion center, and the redarrows in the structures indicate the ferroelectric distortions.
It is well known that the strong ferroelectricity inBiFeO originates from the lone pair of Bi atom. Webelieve that Bi can cause two-dimensional ferroelectric-ity in good structures. Considering that the radius ofBi ion is almost the same as that of Gd ion, weconstruct the BiN monolayer in terms of the structureof GdN compound. On the other hand, this BiN mono-layer can be considered to be made by substituting theP atoms in the famous phosphorene structure by Bi and N atoms. It has center inversion symmetry, as shown inFig. 1 (a.1) and (a.2). It can be seen that the centersof anions and cations coincide, which indicates no ferro-electric polarizations. We will call this phosphorene-likestructure as reference phase in the following.We calculate its zero K phonon spectrum, and findthat the largest soft mode frequency can be found at Γ point, as shown in Fig. 1 (a.3). Further analysisshows that this imaginary frequency is two-fold degen-erate and they imply possible ferroelectric displacementsalong [100] and/or [010] direction. Then, we distortthe reference structure according to the two soft phononmodes. When applying the two modes together, we canget a ferroelectric phase described in Fig. 1 (b.1) and(b.2), and its polarization is along [110] direction. Fur-ther calculated phonon spectra show that the structurewith [110] polarization still has soft modes, as shown inFig. 1(b.3), which indicates this phase is a saddle pointon the energy surface (saddle phase). When adoptingonly one of the modes, we obtain the ferroelectric phasein Fig. 1 (c.1) and (c.2) and the polarization is alongeither [100] or [010] direction. The space group of thesupercell is Pmn2 ( a = 3 . b = 3 . c = 15˚A.The internal coordinates of Bi is (0.5000,0.0000,0.4132)and those of N (0.3866,0.0000,0.5562). There are fouratom planes, in the series of Bi-N-N-Bi, and the Bi-Biplane distance is 2.606˚A. The Bi-N bond lengthes are2.184˚A and 2.278˚A. Further phonon spectra show thatthere is no imaginary frequency in this case, as shownin Fig. 1(c.3), and therefore the structure with [100] or[010] polarization is dynamically stable.Furthermore, we perform ab-initial molecular dynamicsimulations of the BiN monolayer at 300 K, 400 K, 500K, 700 K, and 1100 K. The structures of 300 K and 500Kat 7.5 ps are shown in Fig. 2. We can see that the two-dimensional BiN monolayer can preserve its ferroelectricproperties at least up to 500 K; and when the temper-ature is as high as 700 K, the structure is still stable,but the polarization is substantially reduced; and whenthe temperature is higher than 1100 K, the structure isbroken by the thermal fluctuation. Therefore, we believethat the ferroelectric two-dimensional BiN monolayer isthermodynamically stable beyond 500 K.It is noted that the ferroelectric [100] and [010] phasesare equivalent to each other, with their polarization ori-entating in the two directions, but we still distinguishthem by their different polarization directions becausethese equivalent phases can have different behavior un-der different strains. This is crucial to the discussion ofthe ferroelectric switching in the following. B. Mechanical flexibility
The elastic stiffness constants C ij can be deter-mined by making six finite distortions of the latticeand deriving the elastic constants from the strain-stress FIG. 2. (Color online) Top view snapshots, side view snap-shots, and velocity distributions of ferroelectric BiN mono-layer at 300 K (a.1, a.2, a.3) and 500 K (b.1, b.2, b.3), fromthe ab-initio molecular-dynamics simulations for 7.5 ps. relationship[36]. Since the system is treated as a bulk(in the VASP package) which is a combination of themonolayer and vacuum, the C ij are in GPa. As a result,the 2D elastic stiffness constants should be recovered by C Dij = C ij × c , in which c is the lattice constant contain-ing the vacuum, accordingly the C Dij are in J/m . Then,we can derive Young’s modulus ( Y D ), shear modulus( G D ), and Poisson’s ratios ( ν D ) for the 2D system[37]: Y D k = C D C D − C D C D C D ,Y D ⊥ = C D C D − C D C D C D ,G D = C D ,ν D k = C D C D , ν D ⊥ = C D C D (1)Our calculated Young’s modulus and Poisson’s ratiosare listed in Table I. Due to the polarization, the Young’smodulus (Poisson’s ratio) are anisotropic in the plane ofthe monolayer, being different when the direction is par-allel ( k ) and perpendicular ( ⊥ ) to the ferroelectric ori-entation. The Young’s modulus in the ⊥ direction istwo times larger than that in the k direction. As shownin Table I, Graphene has the largest Young’s and shearmodulus, and MoS and phosphorene have smaller val-ues. It can be seen that the moduli of the BiN mono-layer are larger than those of phosphorene and the in-plane anisotropy is less than that of phosphorene. Al-though the moduli are the second smallest among the TABLE I. The two-dimensional Young’s modulus (Y D k andY D ⊥ in J/ m ), shear modulus (G in J/ m ), and Poisson’sratios ( ν D k and ν D ⊥ ) for the Phosphorene, Graphene, MoS ,and monolayer BiN. The k and ⊥ indicate the ferroelectricdirection and the direction perpendicular to the ferroelectricorientation respectively.Phosphorene[20] Graphene[38] MoS [39] BiN Y D k Y D ⊥ G D ν D k ν D ⊥ four monolayer systems, the shear modulus of the BiNmonolayer is the smallest, nearly half that of phospho-rene. For the Poisson’s ratio, we notice that the BiNmonolayer has the smallest ν D k , which means that whenwe squeeze (stretch) the BiN monolayer along the ferro-electric direction, it expands (shrinks) the smallest in theperpendicular direction. We believe that the BiN mono-layer can preserve its superior mechanical flexibility asphosphorene does. C. Ferroelectric polarization and mechanicalswitching
We calculate the ferroelectric polarization with themodern berry phase ferroelectric theory. The polariza-tion of the BiN monolayer reaches to 580 pC/m whichis larger than that of the predicted GeSn. If we stackthe BiN monolayers to form a 3D layered structure, thepolarization can be as large as 88.8 µ C/cm which iscomparable with the famous room temperature multifer-roic BiFeO [41–43]. The ferroelectric polarization of theBiN monolayer can take one of the [100] and [010] di-rections, and then we think that the BiN monolayer cantake one of the two phases defined to have polarizationalong the [100] and [010] directions, respectively. Theuniaxial stresses parallel and perpendicular to the polar-ization will produce different effects on the BiN mono-layer. When a uniaxial stress is applied, there will be acorresponding strain parallel to the stress and an oppo-site strain perpendicular to the stress. The perpendicularstrain is determined by optimizing the total energy alongthis direction. In Fig. 3 (a), we present the total energiesof the two ferroelectric phases as functions of the parallelstrains. The two vertical black lines indicate the equiv-alent lattice constants along [100] direction (lattice con-stant a ) for the [100] and [010] phases. If focusing on thered line (ferroelectric [010]) in Fig. 3 (a), the strains at theleft lattice constant means the compressive strains per-pendicular to the polarization direction and the strainsat the right lattice constant indicate the tensile strainperpendicular to the polarization direction. The sameconfigurations are applied to the blue line (ferroelectric[100]) in Fig. 3 (a), the only difference is that the strainsare parallel to the polarization direction. We can see thatthere is an energy crossing between the ferroelectric [100](blue line in Fig. 3) and the ferroelectric [010] (red line inFig. 3) at a = 3 . FIG. 3. (Color online) Total energies (a) and ferroelec-tric polarizations (b) of ferroelectric [100] and [010] phases asfunctions of the lattice constant. The two black dashed ver-tical lines indicate the equivalent lattice constants in the twodirections. (c) Energy values along the switching path fromferroelectric [010] phase to ferroelectric [100] phase throughthe saddle phase whose polarization is along [110] direction.
The strain to induce the phase transition of the the[100] phase is only 2%. This strain can be produced byapplying a parallel compressive stress of -1.18 N/m, ora perpendicular tensile stress of 2.54 N/m, which can beeasily achieved in the two-dimensional materials. Theenergy barrier between these two phases has been cal-culated with the NEB method, as shown in Fig. 3 (c).We find out that the energy barrier is equivalent to81 meV/f.u. if the ferroelectric [100] phase is switchedthrough the reference phase (Fig. 1 (a.1) and (a.2)), butthe energy barrier reduces to 10.5 meV/f.u. if the ferro-electric [100] phase is switched through the saddle phase(Fig. 1 (b.1) and (b.2)) that has polarization along the[110] direction. Because of the stress-induced phase tran-sition, the lattice constant of the BiN monolayer can varyin a very large range, which makes it very flexible. Fromthe blue line of the tensile strain part (ferroelectric [100])in Fig. 3 (a), we can also see that the energy increases very slowly with the strain, which indicates the BiNmonolayer is very mechanically flexible. On the otherhand, we can also fix the polarization by applying tensilestress parallel to the polarization, as shown in Fig. 3(a).All these properties can make the BiN monolayer a verypromising candidate for stretchable electronic devices.
D. Mechanical manipulation of optical properties
The band structures of the BiN monolayer under a se-ries of uniaxial stresses are calculated. We have foundthat the energy band gaps can be tuned by the uniax-ial stress, changing from direct gap to indirect gap. InFig. 4, the black dashed lines in (a) and (b) indicate theband structures of the ferroelectric [010] and [100] phases,respectively. In the Fig. 4 (a), we can see that it is a di-rect gap of 1.5 eV for the ferroelectric [010] phase, andit becomes an indirect band gap when compressive uni-axial stress is applied, reaching to an indirect gap of 1.0eV at the parallel strain of − FIG. 4. (Color online) Energy bands of the the ferroelectric[010] (a) and [100] (b) phases, with the [100] lattice constantchanging from 3.30 to 3.80 ˚A.
We calculate frequency-dependent dielectric constantsand present the dielectric functions parallel and perpen-dicular to the ferroelectric directions in Fig. 5. As shownin the anisotropic Young’s modulus, the anisotropic fea-ture is also reflected in the optical properties of the BiN (a) (b)
FIG. 5. (Color online) The imaginary part of dielectric func-tion (a) and reflectivity (b) of the BiN monolayer as functionsof photon energy parallel (red) and perpendicular (blue) tothe ferroelectric directions. The inset describes the partialenergy band structure with arrows indicating the transitions. monolayer. The peak ∆⋆ along the ferroelectric direc-tion can only be found around 1.5 eV in the Fig. 5(a).This unique peak comes from the direct band gap alongthe Γ → X ( ∆ ) direction. Since the compressive stressapplied to the ⊥ direction can tune the direct gap intoan indirect gap, the ∆⋆ feature can be changed by apply-ing such stress. The reflectivity below 4 eV is stronglyanisotropic, as shown in Fig. 5(b), which implies that forthe photon energy between 0 and 3.5 eV, the perpendic-ular reflectivity is at least twice the parallel reflectivity.Such strong anisotropy should be useful for designing newapplications. IV. DISCUSSION AND CONCLUSION
The crystal structure of the BiN monolayer originatesfrom the famous faced center cubic GdN[44]. It iswell known that GdN can preserve ferroelectric understrains[45]. It is reasonable to believe that the ferro-electric distortion without strain in the BiN monolayercomes from the Bi lone-pair 6s electrons. The BiNmonolayer is mechanically flexible according to ourYoung’s modulus calculation, being comparable with the phosphorene which has been proved to be the superiormechanical flexibility material[20]. The shear modulus ofthe BiN monolayer is even smaller than that of phospho-rene, which explains why the energy barrier is so smallfor switching the ferroelectric polarization from [100]to [010] direction through the saddle phase. From theFig. 3, the ferroelectric phase transition between [100]and [010] can be induced by 2% strain, correspondingto the lattice constant ranging from 3.469 ˚A to 3.642˚A. Considering the superior mechanical flexibilities,the BiN monolayer can be very easily stretched in alarge range of lattice constant. Moreover, due to theferroelectric polarization, the symmetry was brokenbetween the [100] and [010] direction, which induces theanisotropic features of the energy band structure andthe optical properties.In summary, we have studied a two-dimensional BiNmonolayer with phosphorene-like structure which hasbeen proven to tending to preserve ferroelectricity. Ourcalculated phonon spectra and MD calculations haveshow its structural stability. Our DFT and Berry phasebased modern ferroelectric theory studies have shownthat the BiN monolayer has a 2D ferroelectricity with aslarge polarization as 580 pC/m, being comparable withBiFeO in the 3D case. Further mechanical studies showthat the polarization in this BiN monolayer can be easilyswitched from [100] to [010] direction over a saddle phasethrough overcoming a 10.5 meV/f.u. barrier by applyingsmall tensile [010] stress of 2.54 N/m or compressive [100]stress of -1.18 N/m, corresponding to a strain of 2%.This phase transition in this 2D BiN monolayer makesits lattice constant change in a very large range in com-parison with other non-ferroelectric materials. Moreover,through applying uniaxial tensile stress perpendicular tothe polarization, one can fix the ferroelectric polariza-tion and change the semiconductor energy gap, betweendirect and indirect. A very strong anisotropy has beenfound in the optical reflectivity when the photon energyis below 4 eV. All these features make us believe thatthe BiN monolayer as a two-dimensional material can beused to achieve stretchable electronic devices and opticalapplications. ACKNOWLEDGMENTS
This work is supported by the Nature Science Foun-dation of China (No.11574366), by the Strategic Pri-ority Research Program of the Chinese Academy ofSciences (Grant No.XDB07000000), and by the De-partment of Science and Technology of China (GrantNo.2016YFA0300701). The calculations were performedin the Milky Way [1] K. S. Novoselov, A. Mishchenko, A. Carvalho, andA. H. C. Neto, Science , 9439 (2016).[2] A. J. Mannix, B. Kiraly, M. C. Hersam, and N. P.Guisinger, Nature Reviews Chemistry , 0014 (2017).[3] A. Carvalho, M. Wang, X. Zhu, A. S. Rodin, H. Su, andA. H. C. Neto, Nature Reviews Materials , 16061 (2016).[4] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross,K. L. Seyler, W. Yao, and X. Xu, Nature Reviews Ma-terials , 16055 (2016).[5] M. Chhowalla, D. Jena, and H. Zhang, Nature ReviewsMaterials , 16052 (2016).[6] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,and M. S. Strano, Nature Nanotechnology , 699 (2012).[7] S. Das, J. A. Robinson, M. Dubey, H. Terrones, andM. Terrones, Annual Review of Materials Research , 1(2015).[8] S. C. Abrahams, Ferroelectrics , 307 (1993).[9] K. T. Butler, J. M. Frost, and A. Walsh, Energy &Environmental Science , 838 (2015).[10] R. Ramesh and N. A. Spaldin, Nature Materials , 21(2007).[11] L. W. Martin and A. M. Rappe, Nature Reviews Mate-rials , 16087 (2016).[12] N. Setter, D. Damjanovic, L. Eng, G. Fox, S. Gevor-gian, S. Hong, A. Kingon, H. Kohlstedt, N. Y. Park,G. B. Stephenson, I. Stolitchnov, A. K. Taganstev, D. V.Taylor, T. Yamada, and S. Streiffer, Journal of AppliedPhysics , 051606 (2006).[13] D. G. Schlom, L.-Q. Chen, C.-B. Eom, K. M. Rabe, S. K.Streiffer, and J.-M. Triscone, Annual Review of Materi-als Research , 589 (2007).[14] M. Dawber, K. M. Rabe, and J. F. Scott, Reviews ofModern Physics , 1083 (2005).[15] J. Junquera and P. Ghosez, Nature , 506 (2003).[16] K. M. Rabe, C. H. Ahn, and J.-M. Triscone, Physics offerroelectrics: a modern perspective , Vol. 105 (Springerscience & business media, 2007).[17] K. S. Kim, Y. Zhao, H. Jang, S. Y. Lee, J. M. Kim, K. S.Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, and B. H. Hong,Nature , 706 (2009).[18] C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science ,385 (2008).[19] A. Castellanos-Gomez, M. Poot, G. A. Steele, H. S. J.van der Zant, N. Agrait, and G. Rubio-Bollinger,Nanoscale Research Letters , 233 (2012).[20] Q. Wei and X. Peng, Applied Physics Letters , 251915(2014).[21] N. D. Mermin and H. Wagner, Physical Review Letters , 1133 (1966).[22] M. Mehboudi, A. M. Dorio, W. Zhu, A. van der Zande, H. O. H. Churchill, A. A. Pacheco-Sanjuan, E. O. Harriss,P. Kumar, and S. Barraza-Lopez, Nano Letters , 1704(2016).[23] M. Wu and X. C. Zeng, Nano Letters , 3236 (2016).[24] H. Wang and X. Qian, 2D Materials , 015042 (2017).[25] L. Seixas, A. Rodin, A. Carvalho, and A. C. Neto, Phys-ical Review Letters , 206803 (2016).[26] F. Xia, H. Wang, and Y. Jia, Nature Communications , 4458 (2014).[27] P. Hohenberg and W. Kohn, Physical Review , 864(1964).[28] W. Kohn and L. J. Sham, Physical Review , 1133(1965).[29] G. Kresse and D. Joubert, Physical Review B , 1758(1999).[30] G. Kresse and J. Furthm¨uller, Physical Review B ,11169 (1996).[31] J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,Physical Review Letters , 136406 (2008).[32] H. J. Monkhorst and J. D. Pack, Physical Review B ,5188 (1976).[33] A. Togo and I. Tanaka, Scripta Materialia , 1 (2015).[34] G. Mills, H. J´onsson, and G. K. Schenter, Surface Science , 305 (1995).[35] R. D. King-Smith and D. Vanderbilt, Physical Review B , 1651 (1993).[36] Y. L. Page and P. Saxe, Physical Review B , 104104(2002).[37] M. Elahi, K. Khaliji, S. M. Tabatabaei, M. Pourfath, andR. Asgari, Physical Review B , 115412 (2015).[38] A. Bosak, M. Krisch, M. Mohr, J. Maultzsch, andC. Thomsen, Physical Review B , 153408 (2007).[39] K. Liu, Q. Yan, M. Chen, W. Fan, Y. Sun, J. Suh, D. Fu,S. Lee, J. Zhou, S. Tongay, J. Ji, J. B. Neaton, and J. Wu,Nano Letters , 5097 (2014).[40] Q. Peng and S. De, Physical Chemistry Chemical Physics , 19427 (2013).[41] J. Wang, Science , 1719 (2003).[42] G. Catalan and J. F. Scott, Advanced Materials , 2463(2009).[43] J.-G. Park, M. D. Le, J. Jeong, and S. Lee, Journal ofPhysics: Condensed Matter , 433202 (2014).[44] C.-G. Duan, R. F. Sabiryanov, J. Liu, W. N. Mei, P. A.Dowben, and J. R. Hardy, Physical Review Letters ,237201 (2005).[45] H. M. Liu, C. Y. Ma, C. Zhu, and J.-M. Liu, Journal ofPhysics: Condensed Matter23