Patent Application: US-15108805-A

Abstract:
a ferroelectric nanostructure formed as a low dimensional nanoscale ferroelectric material having at least one vortex ring of polarization generating an ordered toroid moment switchable between multi - stable states . such a nanostructure is capable of achieving ultrahigh recording density in non - volatile ferroelectric random access memory and may have applications in piezoelectric sensors , efficient actuators , nano - scale dielectric capacitors for energy storage , and nano - scale ultrasounds for medical use .

Description:
we investigate , from first principles , the ferroelectric properties of batio 3 colloidal nanoparticles — and , in particular , answer whether there is ferroelectricity in fe nanoparticles and how these particles respond to applied electric fields . these properties are found to be unusual and differ from what is commonly believed . here , we further develop and use a first - principles - derived effective - hamiltonian approach [ 18 , 19 ] coupled with monte carlo simulations . ( ideally , one would like to use direct first - principles density - functional theory , but this is currently computationally impracticable .) the effective hamiltonians of refs . [ 18 , 19 ], which are derived from first principles and possess a comparable accuracy , have been successfully applied to many fe materials , including simple batio 3 [ 20 ], pbtio 3 [ 21 ], and knbo 3 [ 22 ] systems , and complex pb ( zr , ti ) o 3 [ 19 ] and pb ( sc , nb ) o 3 [ 23 ] solid solutions . in this approach , local modes { u i } ( i is the cell index ) describe the ferroelectric instability in individual 5 - atom cells ; ui are associated with local electrical dipoles p i via p i = z * u i ( where z * is the effective charge of the local mode ). compared to the original method detailed in ref . [ 18 ], two new developments are made here in order to be able to study fe nanoparticles : ( i ) no supercell periodic boundary conditions are imposed , and the lr dipole - dipole interaction is performed in real - space ( inside the nanoparticles ) rather than in reciprocal space . our simulations with open - boundary condition precisely mimic the experimental situations [ 3 , 5 , 6 ] in which polarizations in fe wires and films are probed by noncontact electrostatic forces without metallic electrodes . by contrast , the calculations of batio 3 thin films in ref . [ 17 ] assume a short - circuit boundary condition with metallic electrodes surrounding the films . also note that , in our real - space implementation without artificial periodicity for finite systems , the potential field generated by every dipole in the nanoparticles — including the depolarization field produced by the charges ( i . e ., uncompensated dipoles ) at nanoparticle surfaces — is precisely computed and properly accounted for . ( ii ) existence of the vacuum surrounding nanoparticles will cause surface - induced atomic relaxations and cell - shape changes ( thus affecting both local modes and local inhomogeneous strains ) near the nanoparticle surfaces . to account for the effect of atomic relaxations on local modes , an interaction between local modes at surfaces and the vacuum ( denoted as mode - vacuum interaction ) is added in the hamiltonian . similarly , an interaction between the inhomogeneous strains and the vacuum ( denoted as local strain - vacuum interaction ) is added to account for the effect of cell - shape changes on local strains . the parameters of these two sr interactions — whose contributions to total energy are analytically similar to the species - dependent intersite coupling terms of ref . [ 19 ]— are determined from first - principles local - density - approximation ( lda ) calculations on batio 3 surfaces . our effective hamiltonian for fe dots thus includes the dominating effects caused by the vacuum on charge redistribution , atomic relaxations , and cell - shape changes near the surfaces . we assume that the surfaces of nanoparticles are bao terminated , since they have lower energies than tio - terminated surfaces [ 24 ]. other interaction parameters used here ( to describe the fe material per se ) are those of ref . [ 20 ] for bulk batio 3 , since ref . [ 17 ] demonstrates that these parameters do not change significantly when going from bulk to nanostructures . local modes { u i } at low temperatures are obtained via temperature - annealing monte carlo simulations . a pressure of − 4 . 8 gpa is used in simulations to correct the lda error in lattice constant . results presented here are obtained for 50 k . for simplicity , we assume batio 3 dots to be rectangular , since adopting a spherical or a square cross section leads only to a minor difference . the rectangular dots are denoted as n x × n y × n z , where n x , n y , n z are the numbers of five - atom cells contained in the dots along the pseudocubic [ 100 ], [ 010 ], and [ 001 ] directions , respectively . the average size of a nanoparticle is thus given by l =( l x l y l z ) 1 / 3 , where l = n i a ( i = x , y , z ) are the lengths along the three cartesian axes ( a = 4 . 0 å is the lattice constant of batio 3 ). for the clarity of demonstration , our prediction for local modes { u i } will be presented on a certain perovskite b - site plane , which is specified by its normal direction and its order index among the equivalent planes ( e . g ., the y = 6 plane is the 6 th plane having a normal direction along the y axis ). local modes { u i } on the y = 6 plane of a 12 × 12 × 12 dot are depicted in fig1 a . significant ferroelectric off - center displacements can be clearly seen in this small ( l = 4 . 8 nm ) dot . the local - mode magnitude & lt ; lul & gt ;, averaged over all 5 - atom cells , is 0 . 043a in this dot . this is remarkably large and comparable with the value of 0 . 039a found in bulk batio 3 . these large off - center displacements , indicating the existence of significant dipoles in each cell , have an important implication — that is , a large amount of macroscopic polarization can be generated by aligning these local dipoles with electric fields . note that this effect cannot be achieved in nonferroelectric nanoparticles without off - center displacements , unless a huge electric field is applied . in fact , other calculations we performed ( not shown here ) predict that large off - center displacements yielding a magnitude & lt ; lul & gt ;]= 0 . 052a occur even in the tiny 4 × 4 × 4 dot ( l = 1 . 6 nm ). conceivably , such a small size is likely the limit achievable in experiments . these results thus suggest that there is virtually no critical size for ferroelectric instability in nanoparticles . for the dipoles on the dot surfaces , analysis of fig1 a shows that the parallel - to - surface components of these dipoles prefer to point along opposite directions on two opposite surfaces , while the ( small ) normal - to - surface components tend to bulge out due to the surface - induced atomic relaxations . fig1 a further reveals that the local dipoles in small dots prefer to rotate from cell to cell , forming an unusual and complex “ vortex - like ” pattern ( similar to the ones found in some magnetic compounds ). note that this vortex pattern in fe dots is different from the ferroelectric pattern in bulk ( where all dipoles are aligned along the same direction , as predicted by similar - size supercell effective - hamiltonian simulations or assumed in five - atom first - principles calculations ). interestingly , as a result of the dipole pattern in fig1 a the total macroscopic polarization is found to be zero ( i . e ., [ u ]= 0 ). furthermore , we found that the vortex pattern of displacements does not alter appreciably when turning off and on the mode - vacuum sr interaction . this suggests another important conclusion , namely , that ferroelectric instability in dots is not much affected by the surface local environments or , equivalently , that the capping organic matrix materials used in experiments should not affect the ferroelectric properties of the dots . unlike in ref . [ 17 ] where charges in metallic electrodes are able to move freely and will thus cause strong screening , the organic capping materials used in colloidal fe nanoparticles [ 1 , 2 , 7 ] are insulators of large gap (˜ 10 ev ) and the resulting screening is very small . as the dot increases in size to 24 × 24 × 24 , the local fe displacements tend to order between each other via the formation of eight rather uniform ferroelectric domains as shown in fig1 d . the polarization of each domain in fig1 d is found to point along one of the eight pseudocubic [± 1 ± 1 ± 1 ] directions . the macroscopic polarization of the entire dot remains as zero . the displacement pattern in fig1 a results from a new balance ( with respect to the bulk case ) between the lr and sr interactions in dots , and can be simply explained as follows . first , let us consider the lr dipole - dipole interaction alone for two isolated dipoles ( with four specific orientations illustrated in fig1 e ); the lowest - energy configuration is that these two dipoles both point at the same direction along the dipole axis ( case ( i ) in fig1 e ). indeed , it can be seen in fig1 a that the dipoles belonging to a same row have their parallel - to - dipole - axis ( i . e ., the z axis ) components aligned along the same direction . ( note that dipoles on the dot surfaces are exceptions , see below .) next , let us select a given row in fig1 a , and note that the dipoles located near the nanoparticle surfaces in this row tend to have large parallel - to - surface (“ in - plane ”) components with their normal components suppressed by the vacuum ; these in - plane components ( being perpendicular to the dipole axis ) shall flip their directions ( since case ( ii ) has a lower energy than case ( iii ) in fig1 e ). however , this flip of the component perpendicular to the z axis does not occur within the nearest cell ( see fig1 a ), since we found that it will otherwise drastically increase the short - range energy . instead , the flip occurs across the entire dot , forming the unusual pattern in fig1 a . now we turn our attention to electric - field effects in nanoparticles . more precisely , we are particularly interested in revealing the size dependences of these effects . we decide to elongate the nanoparticles only along the z axis ( that is , the applied - field direction ) to mimic quantum wires , partly because increasing the dot size along all three dimensions is computationally prohibitive . fig2 a shows the resulting net z axis mode average ( u z ) and clearly indicates that the same electric field induces a larger polarization per five - atom cell in a long wire than in a short wire . field effects in fe dots thus depend substantially on sizes . this is the first time that the size dependence of field - induced polarization is firmly established ( to our knowledge ). fig2 a further reveals that a small electric field in long wires drives a rapidly increasing polarization and thus easily turns a ( macroscopically ) paraelectric fe nanostructure into a ( macroscopically ) ferroelectric phase . here we define the poling field e pf as the field that is needed to drive a net average displacement ( u z ) equal to the bulk value of 0 . 02a . the poling fields for different - size wires are given in fig2 a , and drastically decrease when increasing size . more precisely , fitting our theoretical data in fig2 b as a function of size gives e pf = 38 . 1966 / l z n with n = 0 . 7821 , where e pf is in unit of 10 8 vwm and l z in unit of nm . our predictions provide explanations and / or suggest reinterpretations of many experimental results . for instance , no detectable polarizations were probed by electrostatic force microscopy along the perpendicular directions of both as - deposited batio 3 nanowires [ 3 ] and as - grown pzt films [ 5 ]. this can be simply explained by the vanishing net polarizations we found in dots and in wires ( see e . g ., fig1 a ). our findings further suggest that the nondetectable polarization in pzt films may not be due to the electrostatic passivation of additional charges at sample surfaces as it was speculated [ 5 ], but rather results from the “ intrinsic ” arrangement of local dipoles . moreover , it was found experimentally [ 3 ] that a field of 3 . 6 × 10 8 v / m is needed to “ write ” a polarization along the perpendicular direction of an 18 nm diameter batio 3 wire , which is in excellent agreement with our predicted value of 3 . 98 × 10 8 v / m obtained from the formula given above . we now provide a microscopic understanding of the field - induced responses in fe dots . we find that the responses in fig2 a can be separated into three stages according to their ( u z )- vs - field behaviors , and here we use the 12 × 12 × 12 dot to illustrate these stages . ( i ) at stage i ( that occurs at field e & lt ; 10 9 v / m ), the dipoles pointing opposite to the field direction are sequentially flipped ( fig1 b ). the flip process occurs first near the domain boundary , while dipoles on the dot surface are found to be more resistant to the applied field ( see fig1 b ). stage i generates a polarization that varies linearly with the field strength ( fig2 a ). the slope ( i . e ., dielectric susceptibility ) is given in fig2 b for different wires and is found to increase linearly with the wire length . ( 2 ) at stage ii ( 10 9 & lt ; e & lt ; 4 × 10 9 v / m ), the dipoles start to rotate towards the field direction as the dipole flips of stage i have been completed ( fig1 c ). stage ii differs from the polarization rotation in bulk [ 25 , 26 ] in that , prior to the rotation , local dipoles in dots are not parallely aligned as in bulk . interestingly , this second stage yields a strong nonlinear field dependence of polarization ( fig2 a ). ( 3 ) at stage iii ( e & gt ; 4 × 10 9 v / m ), with the dipoles having all been previously rotated along the field direction , the magnitude of each dipole then starts to be enlarged by the field , resulting in a nearly linear field - dependent polarization again . finally , we examine the electromechanical response in fe wires . the field - induced η 3 strains are depicted in fig2 c . at stage i , the strain response is found to be surprisingly small though there is a rapid increase in polarization , which suggests that polarization does not couple with strain during the dipole flipping process . at stage ii , the strain increases evidently with the field strength ( fig2 c ); the strain - vs - field slope is the piezoelectric coefficient d 33 . the d 33 values are given in fig2 d for different wires , and exhibit a monotonous increase with size . interestingly , the d 33 coefficients in nanoparticles (˜ 10 pc / n ) are found to be much smaller than in bulk (˜ 77 pc / n ), suggesting that the electromechanical response can be drastically modified by varying sizes . in summary , we find : ( i ) large ferroelectric off - center displacements exist in very small (˜ 5 nm ) dots . this result solves a long - standing puzzling question in experiments , namely , whether there exists ferroelectricity in colloidal fe dots under zero field . this discovery also opens a possibility of tremendous increase in fe - memory density . ( ii ) fe displacements in dots exhibit an unusual and hitherto - unknown vortex pattern . this pattern is found to cause rather peculiar field - induced polarization responses . ( iii ) the ferroelectric instability in dots is found to be robust against the organic capping materials . ( iv ) the poling field is predicted to decrease drastically with increasing size , which is important for practical controls of fe nanostructures by use of external electric fields . ( v ) the polarization responses of fe dots at stage ii are found to be strongly nonlinear , while the electromechanical responses in dots are found to be remarkably smaller than those in the bulk . we perform further ab initio studies of ferroelectric nanoscale disks and rods of technologically - important pb ( zr , ti ) o 3 solid solutions , and demonstrate the existence of previously unknown phase transitions in zero - dimensional ferroelectric nanoparticles . the minimum diameter of the disks that display low - temperature structural bistabilitv is determined to be 3 . 2 nm , enabling an ultimate nferam density of 60 × 10 12 bits per square inch — that is , five orders of magnitude larger than those currently available [ 30 ]. our results suggest an innovative use of ferroelectric nanostructures for data storage , and are of fundamental value for the theory of phase transition in systems of low - dimensionalitv . one main difference between ferroelectric ( fe ) nanostructures and ( infinite ) bulk materials is the existence in the former of depolarizing fields , because of the uncompensated charges at the surface [ 43 ]. the depolarizing field ( the magnitude of which can - be as high as 10 4 kv cm − 1 ) is able to quench spontaneous polarization ; this is consistent with the recent experimental and theoretical findings that the ground states of finite free - standing fe samples remain paraelectric at very low temperature ( 50k ) [ 3 , 32 ]. to induce sizable polarizations , external electric fields [ 3 , 5 ]— or equivalently , as pointed out in ref . [ 32 ], short - circuit boundary conditions [ 17 , 21 ]— are needed to screen the depolarizing field . as the external fields vanish , the induced polarizations are expected to relax to the ground state of non - polarity , with the relaxation time decreasing exponentially with reducing size , which hampers the miniaturization of nanoscale nferams . further , a large external electric field is needed to reverse the polarization and to swap charges in electrodes in order to rewrite data bits . whether phase transitions still occur in low - dimensional structures has been a subject of long - standing fundamental interest for understanding and revealing collective interactions [ 27 , 28 , 41 , 42 ]. for instance , ref . [ 27 ] predicts that the thermal fluctuation of acoustic phonons and the entropy due to domain formation disfavor long - range ordering of local dipoles in one - dimensional systems , without any transition to phases with spontaneous polarization . similarly , using a spin - lattice model , mermin and wagner showed that no spontaneous magnetic ordering exists in one - dimensional systems [ 28 ]. here we report state - of - the - art ab initio simulations , which lead to the discovery that ( 1 ) phase transitions do in fact exist in zero - dimensional fe nanostructures ; ( 2 ) these phase transitions differ profoundly from those occurring in bulk material , in the sense that they lead to the formation of spontaneous toroid moment [ 44 ] rather than spontaneous polarization , below a critical temperature ; ( 3 ) the unusual characteristics of the resulting low - temperature phases promise the generation of new nferam devices with remarkable capabilities . our simulations also reveal the precise role of finite - size effects on toroid moments . we study free - standing nanoparticles of perovskite pb ( zr 0 . 5 ti 0 . 5 ) o 3 ( pzt ) solid solution — the most promising candidate for nano - nferam and nano - mems [ 8 , 33 ]. the investigated nanoparticles all have a cylindrical shape , with diameter d and height h ( both in units of bulk lattice constant a = 4 å ); the cylindrical z - axis is chosen to be along the pseudocubic [ 001 ] direction , with the x and y axes along the [ 100 ] and [ 010 ] directions , respectively . particles with d & gt ; h and d & lt ; h are referred to as disks and rods , respectively , and each particle is named as ( d , h ). a variety of combinations of d and h , ranging from 5 to 30 , are considered here . technically , a first - principles - derived effective hamiltonian [ 18 , 19 ] is used to determine the energetics and local dipoles in each perovskite five - atom cell . ( this hamiltonian has been shown to reproduce well the observed thermodynamic behavior of bulk pzt , including the occurrence of an unusual monoclinic phase for a small range of ti composition [ 19 , 35 ].) nanoparticles surrounded by vacuum are mimicked without periodic boundary conditions ; details of our approach for fe nanoparticles are described elsewhere [ 31 ]. the validity of this approach was demonstrated by the accurate determination of the poling fields in batio 3 dots [ 31 ], as well as by the theoretical study of ultrathin pzt films under compressive strains that yields a 180 ° stripe domain [ 45 ], in agreement with experimental observation [ 46 ]. here we focus on atomically disordered pzt nanostructures , which are consistent with the chemical nature of bulk pzt [ 36 ]. for all simulated nanoparticles , the total net polarization p = n − 1 σp i ( where p i is the local dipole of the cell i located at r i , and n is the number of cells in the simulation ) is found to be null down to 10 k . unlike previous studies that mainly focus on spontaneous polarization as the evidence of phase transition , we , on the other hand , discover a new order parameter — namely , the toroid moment g of polarization — defined as g = ( 2 n ) − 1 σ i r i × p i ( 1 ) fig3 a shows that the z - component of toroid moment , g z , of the ( 19 , 14 ) disk increases sharply below 600k while being zero at higher temperature ( the g x and g y components remain nearly null at all temperatures ). this indicates that an ordering associated with local dipoles occurs in this zero - imensional disk below a certain critical temperature , t c . we further found that the specific heat ( not shown here ) exhibits a hump around 550k , which provides a quantitative measure of t c and further confirms the existence of a phase transition . in order to gain a microscopic insight into this unusual phase transition , fig3 b shows the contribution of each ( 001 ) plane in the ( 19 , 14 ) disk to the total g z at high ( 768k ) and low ( 64 k ) temperature . one can clearly see that the moments of individual planes are all small and random at 768 k . on the other hand , these moments markedly increase in magnitude , and also spontaneously order along the z direction , when the temperature is lowered . these low - temperature local toroid moments are predicted to be nearly identical in each ( 001 ) plane , except near the surface layers . the resulting ordered phase , which we denote as phase a , is characterized by either clockwise or anti - clockwise vortices in each z plane , with a vortex in the central z plane displayed in the inset of fig3 b . we now explore how this peculiar a phase , and its characteristics , evolve as a function of diameter d for a fixed height h ( chosen here to be 14 ). the most notable results are that : ( 1 ) the a phase is found to be stable ( at low temperature ) “ only ” above a critical value of d — denoted by d c , a , and equal to 8 when h = 14 ; ( 2 ) low - temperature toroid moment g z increases in magnitude as the diameter becomes larger , with an inflexion point occurring in the g 2 - d curve when meeting the d = h equality — that is , when nanorods become nanodisks ( see fig4 a ); ( 3 ) the t c of nanorods ( d & lt ; h ) markedly increases as d increases , whereas nanodisks ( d & gt ; h ) exhibit a diameter - insensitive t c ( see fig4 b ). result ( 1 ), indicated above , raises the question of what structurally happens at low temperature in fe nanoparticles with diameter smaller than d ca . we numerically found that below a second critical diameter ( to be referred to as d cb and equal to 6 when h = 14 ), no vortices exist in any plane , yielding a vanishing total toroid moment . on the other hand , substantial off - center displacements still occur , with the spontaneous polarization being still null . the resulting macroscopically “ non - polar ” and “ non - toroidal ” phase can be characterized as spin - glass type [ 38 ] and will be denoted as the sg phase . another feature that we discover is that the a phase does not directly transform into this glass - type sg phase , when decreasing the diameter . in fact , when d ranges between d c , a and d c , b , a new structural phase — to be referred to as the b phase — forms . the arrangement of the local dipoles in this b phase is depicted in fig3 c , for the ( 7 , 28 ) nanorod and at low temperature . like the a phase and unlike the sg phase , phase b exhibits vortices with non - zero local toroid moments . however , unlike the a phase , the toroid moments of phase b have x and y components but no z component . specifically , fig3 c shows that , in the ( 7 , 28 ) nanorod , four vortices appear in the central x ( as well as y ) cross - section — and intriguingly , each vortex is found to have a diameter nearly the size of d , with two neighboring vortices having opposite local toroid moments . the total toroid moment , unlike the local toroid moments , is thus relatively small in the b phase . this b phase can be thought as an intermediate phase between the a phase ( characterized by large local and total toroid moments ) and the sg phase ( for which there is neither local nor total toroid moment ), occurring via the quasi - annihilation of the total , but not local , toroid moment . furthermore , fig3 c shows that the edges of the vortices on the x plane go through the centers of the vortices in the y plane : two sets of vortices in the b phase are therefore interconnected like links in a chain . as a result , the local toroid moments adopt a helix - like ordering with a period λ = 2d , and this b phase is 16 - fold degenerate . fig4 a shows the evolution of the magnitude of total tbroid moment g z =( g x 2 + g y 2 ) 1 / 2 in the b phase , as a function of the rod height at low temperature . we can see that this magnitude is relatively small with respect to the g z of the a phase , and decreases non - monotonically as h increases . moreover , fig4 b shows that the critical temperature at which the b phase forms is considerably lower than the t c at which the a phase appears . the local dipoles in ordered a phase bear a remarkable resemblance to the molecule orientations in the so - called smectic ( respectively , cholesteric ) phase of liquid crystals [ 37 ]— and the ordered dipoles in b phase resemble orientations in the cholesteric phase . the above fe patterns in a or b phases are found to be robust , in the sense that they do not depend significantly on the surface termination of nanoparticles , or on whether the short - range interaction between vacuum and the local modes beyond the first surface layer is included or not in the simulations , since these patterns are predominantly determined by the long - range electrostatic interaction . to understand the stabilities of phases a and b , the total energy e tot and dipole energy e dip were determined , and are shown in fig4 c at a fixed and low ( 64 k ) temperature . we find that the delicate balance between non - dipolar and dipolar interactions ( such balance is believed to play a critical role in affecting the collective behavior of ferroelectrics [ 12 ], and is here described quantitatively by the ratio r = ie n - dip / e dip i , where e n - dip = e tot − e dip ) is , surprisingly , mostly size - independent — being 0 . 92 and 0 . 97 for the a and b phases , respectively . however , the stability energy e tot progressively decreases in magnitude when d is reduced in phase a , which explains the size - dependence of the critical temperature ( for the a phase to appear ) in fig4 b . furthermore , ie tot i of phase b (˜ 4 mev ) is substantially smaller than its counterpart in phase a , hence leading to much lower critical temperatures . the occurrence of phase transitions in zero - dimensional nanoparticles , in contrast with the predictions for one - dimensional systems [ 27 , 28 ], has a rather simple explanation . fe domains prevent phase transitions from occurring in one - dimensional systems because they are able to lower the helmholtz free energy f = e tot − ts by increasing the entropy s . on the other hand , these domains become energetically unfavorable in zero - dimensional nanostructures ( therefore allowing the existence of phase transitions ) because the typical size dpd of three - dimensionally confined zero - dimensional domains ( denoted as particle domains ) is on the order of 100 nm according to experimental measurement [ 47 ] and thus substantially larger than the size of the studied nanoparticles . we shall point out that d pd is much larger than the size of planar domains in bulk materials [ 48 ], because the small surface - to - bulk ratio in the latter case decreases the strain energy at domain interface and thus allows narrower domains . also , d pd is significantly different ( as it should be ) from the small size (˜ 2 nm ) of stripe domains found in compressed ultrathin films [ 45 , 46 ] where the substrate - enhanced out - of - plane polarization causes the attractive electrostatic interaction between different domains to become dominating , and in contrast , the strain energy at the domain interface due to short - range bond distortion plays only a minor role ( which explains why the domain size can be small ). theoretical determination of d pd is hindered by the limits of computing facilities . in either a or b phase , the local dipoles are ordered in a specific way that minimizes the depolarizing field , while simultaneously forming toroid moments , as in some magnetic systems [ 39 ]. a possible alternative way to eliminate the depolarizing field is to form 180 ° domains with polarization in each region pointing along + z or − z direction . this configuration is found , however , to be less stable by our constrained simulation ( in which only the z component of each dipole is non - zero and allowed to relax ): the energy per 5 - atom cell in the ( 19 , 14 ) disk of such 180 ° domains is determined to be − 11 mev , much higher than that of the ground state with toroid moment (− 32 mev ). we now consider if the existence of a macroscopic toroid moment , as in phase a — rather than a spontaneous polarization , as in bulk ferroelectrics — can lead to the design of new or improved technological devices . here it is important to realize that phase a is bistable ( that is , the toroid moment can be equivalently parallel or anti - parallel to the z axis ). unlike the situation in bulk ferroelectncs where states with differently - oriented polarization can be accessed via a static external electric - field , we can switch from one minimum of phase a to the other by applying a time - dependent magnetic field . this magnetic field , generating a curling electric field via ∇× e =−∂ b /∂ t , interacts with the total toroid moment of the pzt particles , as described by the energetic term e int =−( 2n ) − 1 σ i p i [ r i ×(∇× e )] g ∂ b /∂ t . the coercive field ∂ b /∂ t to switch the toroid moment of the ( 19 , 14 ) disk is estimated from the total energy e tot to be 1 . 6 mv / å 2 ( the real coercive field may be different because of nucleation and tunneling ). storing data using switchable macroscopic toroid moment could be superior to using spontaneous polarization , because generating a magnetic field — unlike the generation of electric field — does not require electrode contact which is challenging to make in nanoscale devices . furthermore , a large number of particles have to be arranged into regular arrays for memory nanodevices to be efficient , but they should not interact strongly ( in order to avoid the so - called “ cross - talk ” problem ). phase a does not exhibit any macroscopic polarization , nor produce a strong electric field that has a long - range character . the vortex structure of phase a in a single nanoparticle can therefore be switched without modifying the states of its neighboring particles . consequently , the toroid carriers of information can thus be packed considerably more densely than the conventional carriers of polarization , giving rise to a marked improvement in the density of ferroelectric recording . for instance , the minimum diameter that we found being able to generate hi - stable toroid states is 3 . 2 nm . this produces an ultrahigh storage density of 60 tbit inch − 2 , which is five orders of magnitude larger than current nferam capability [ 30 ] of 0 . 2 gbit inch − 2 . such a promising capability also far surpasses the 1 gbit inch − 2 density of typical magnetic recording . we also examined the electric field generated by the toroid phase in the ( 19 , 14 ) disk , and found that this field is measurable in the proximity of the nanodisk ( being about 5 × 10 6 vm − 1 at 2 nm above the disk ), though it decays away quickly ( thus implying that virtually no cross - 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