Dynamics of Field Induced Polarization Reversal in Strained Perovskite Ferroelectric Films with c-oriented Polarization
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Dynamics of Field Induced Polarization Reversal in Strained Perovskite FerroelectricFilms with c-oriented Polarization
Laurent Baudry, ∗ Igor A. Luk’yanchuk,
2, 3 and Anna Razumnaya Institute of Electronics, Microelectronics and Nanotechnology (IEMN)-DHS D´epartment,UMR CNRS 8520, Universit´e des Sciences et Technologies de Lille, 59652 Villeneuve d’Ascq Cedex, France Laboratory of Condensed Matter Physics, University of Picardie Jules Verne, Amiens, 80039, France L. D. Landau Institute for Theoretical Physics, Moscow, Russia Physics Department, Southern Federal University, Rostov on Don, 344090 Russia (Dated: September 1, 2018)The field-induced polarization reversal in c -oriented ferroelectric phase of strained perovskite filmhas been studied. We show that in additional to the conventional longitudinal switching mechanism,when c-oriented polarization vector changes its modulus, the longitudinal-transversal and transversalmechanisms when the perpendicular component of polarization is dynamically admixed are possible.The later process can occurs either via the straight-abrupt or initially-continues polarization turnoverscenario. We specified the obtained results for the case of PbTiO and BaTiO ferroelectrics andpropose the experimental methods for their investigation. Dynamical switching properties of ferroelectrics are es-sential for their application in the memory-storage de-vices [1]. The underlying mechanism of polarization re-versal is of special interest for the mostly used pseudo-cubic perovskite crystals that, depending on the orienta-tion of polarization P = ( P , P , P ) can exhibit tetrago-nal, orthorhombic or rhombohedral structural phases inthe ferroelectric state of the bulk material [2]. The situ-ation is more diverse in case of substrate-deposited per-ovskite ferroelectric films in which the substrate-provideddeformation makes the lattice constant c in z -direction(perpendicular to the film surface) different from thein-plane lattice constants a = b already in the high-temperature paraelectric phase with P = 0. In particu-larly, Pertsev, Zembilgotov and Tagantsev [3, 4] studiedthe effect of substrate clamping on PbTiO and BaTiO films and proposed that at least four structural phasescan exist in strain-temperature, u m - T phase diagram(Fig. 1). The so-called c -phase with P = (0 , , P ) oc-curs at high compressive strains whereas the aa -phasewith P = ( P , P ,
0) is realized at high tensile strains.Either ac -phase with P = ( P , , P ) or r -phase with P = ( P , P , P ) can occur at low strains. These phasesare thermodynamically stable and separated by contin-uous (thin) or discontinuous (bold) transition lines inFig. 1.In the present letter we study the uniform polariza-tion switching in PbTiO and BaTiO oxides induced bythe applied electric field and demonstrate that the situ-ation is even more rich. Additional phases can dynami-cally appear during the polarization reversal. We restrictourselves to the c -phase region of u m - T phase diagramand consider the switching process when the initially up-oriented polarization P = (0 , , P ) decreases and thensuddenly drops down under the oppositely applied field E = (0 , , E ) with E < and BaTiO materials we usethe renormalized Landau-Devonshire functional given in FIG. 1. Regions of the longitudinal ( l ), longitudinal-transversal ( lt ) and transversal ( t ) switching regimes and cor-responding separating lines L and L on the phase diagramsof strained films of PbTiO (a) and BaTiO (b), adopted fromRef. [3]. Stability line L determines the type of transversalswitching. Close location of L and L for BaTiO impliesthat it occurs according the initially-continuous turnover ofpolarization ( t c ), whereas the absence of this line for PbTiO means that transversal switching is straight-abrupt ( t a ) [3], for which the account of the six-order terms is knownto be important [5–7]:˜ G ( P , E, T, u m ) = a ∗ (cid:0) P + P (cid:1) + a ∗ P + a ∗ (cid:0) P + P (cid:1) + a ∗ P + a ∗ (cid:0) P + P (cid:1) P + a ∗ P P + a P P P + a (cid:2) P (cid:0) P + P (cid:1) + P (cid:0) P + P (cid:1) + P (cid:0) P + P (cid:1)(cid:3) + a (cid:0) P + P + P (cid:1) + u m s + s − EP . (1)The last term in Eq. (1) presents the field-driving in-teraction with electric polarization. The renormalizedcoefficients a ∗ , a ∗ , a ∗ , a ∗ , a ∗ and a ∗ depend on themisfit strain u m and temperature T whereas other coeffi-cients s , s , a , a and a correspond to its bulkhomologous, as was explicitly specified in Ref. [3].Note that several alternative approaches were proposedto establish the u m - T phase diagram of BaTiO [8–10].Their results are competitive with [3, 4] mostly in relativelocation of r - and ac - phases. This minor difference is notessential for our consideration and can be easily takeninto account for each particular case. In what follows, weconsider the competition between the switching-induced ac and r phases. By substitution of the correspondingorder parameters P = ( P , , P ) and P = ( P , P , P ) in(1) we obtain the following effective functional:˜ G = b P + b P + b P + b P + b P P (2)+ b P P + b P P + b P + b P − EP , where b = 2 a ∗ , b = 2 a ∗ , b = 4 a ∗ , b = 2 a ∗ , b =4 a ∗ , b = 6 a , b = 2 a , b = 2 a , b =6 a for ac -phase case and b = 4 a ∗ , b = 2 a ∗ , b =8 a ∗ + 2 a ∗ , b = 4 a ∗ , b = 4 a ∗ , b = 12 a +12 a , b = 2 a + 4 a , b = 4 a , b = 6 a for r -phase case.Our approach is inspired by that given by Iwata andIshibashi [11] for the case of cubic (unstrained) latticein paraelectric phase. It was demonstrated that de-pending on the strength of the polarization-lattice cou-pling, two reversal mechanisms are possible. For strongcubic anisotropy the switching occurs like in uniaxialone-component ferroelectrics by dynamical change ofthe modulus of the longitudinal polarization component P . For weak anisotropy the transversal component P virtually admixes to P during the process. Such polarization-rotation scenario can, for instance, occursin PbZr x Ti − x O compounds when the anisotropic cou-pling is soften just as the composition parameter x ap-proaches the morphotropic point x ≃ .
44 from above.The distinguishing feature of the substrate-depositedfilms from the bulk cubic case is the strain-induced uni-axial anisotropy that is reflected both by the splitting ofthe critical temperatures in the second order P and P terms and by accounting for the 6th-order cross-coupling terms. To understand the dynamical mechanism of polar-ization reversal we should catch the critical field at whichthe switching instability occurs. Application of an oppo-site electric field leads to the decrease of c -oriented polar-ization which stays yet positive until the critical field isreached. At this stage the field-driven polarization evo-lution, P ( E ) is given by the one-component variationalequation: ∂ ˜ G∂P ! P , =0 = b P + b P + b P − E = 0 . (3)The value of the critical field at which polarizationswitching starts can be obtained from the loss of the posi-tive definiteness of the Hessian matrix H ij = ∂ ˜ G∂P i ∂P j , pre-sented in the extremal point of initial equilibrium P = 0, P = P ( E ) as: H = b + 3 b P + 5 b P , (4) H = b + b P + b P , (5) H = H = 0 . (6)where the dependence P ( E ) is given by Eq.(3). Uponfield increase the competition occurs between the lon-gitudinal and transversal critical fields E ( l ) and E ( t ) ,determined by the conditions H (cid:0) P (cid:0) E ( l ) (cid:1)(cid:1) = 0 and H (cid:0) P (cid:0) E ( t ) (cid:1)(cid:1) = 0. Importantly, the switching occursat the instability field E ( l ) or E ( t ) which is attained firstand the further scenario of polarization vector evolutionis determined by the occurring type of instability.(i) For (cid:12)(cid:12) E ( l ) (cid:12)(cid:12) < (cid:12)(cid:12) E ( t ) (cid:12)(cid:12) the longitudinal ( l ) switchinginstability is realized first and the polarization vector re-verses its direction by change of the amplitude of P frompositive to negative, passing through P = 0.(ii) For (cid:12)(cid:12) E ( t ) (cid:12)(cid:12) < (cid:12)(cid:12) E ( l ) (cid:12)(cid:12) the transversal ( t ) switchinginstability is realized first and the component P is ad-mixed to P after the beginning of the reversal process,just above E ( t ) . Polarization switching has therefore therotational constituent, like in the Iwata and Ishibashimodel.(iii) There can exist also the mixed longitudinal-transversal (lt) regime when the polarization reversalstarts according to longitudinal scenario at E = E ( l ) butthe transversal component P virtually appears at thelater stage of the process.The polarization evolution in l , lt and t switchingregimes is sketched in Fig. 2. We presume that they areseparated by crossover lines L and L in u m - T phasediagram and find the condition of their existence. The t -type switching can have either initially-continous ( t c )or straight-abrupt ( t a ) character as will be specified later.According to the given above consideration thetransversal component P can dynamically admix tothe component P during polarization reversal if thepolarization-dependent Hessian matrix element H be-comes negative in course of the switching. This occur FIG. 2. Reversal of the polarization vector as function ofincreasing with time switching electric field during longitu-dinal ( l ), longitudinal-transversal ( lt ), transversal straight-abrupt ( t a ) and transversal initialy-continuous ( t c ) switching.Solid lines present the thermodynamically stable field-inducedstates whereas the dashed lines present the dynamically-virtual states appearing during the abrupt switching process. e.g., when coefficient b is negative. Then, when thedropping-down polarization goes through the state withvanishing P , the element H , according Eq. (5) ac-quires the negative sign in the vicinity of P = 0. Thepolarization vector will experience the instability towardsthe transversal deviation and the lt regime will be real-ized. Therefore the crossover line L between l and lt regimes is given by the condition:L : b ( u m , T ) = 0 . (7)Noteworthy that the line L can be found in u m - T phasediagram as the prolongation of the paraelectric aa phasetransition line located in u m > u m < lt and t switchingregimes can be found by equating the critical fields E ( l ) and E ( t ) or, what is equivalent and easier, by equatingthe corresponding longitudinal and transversal criticalpolarizations P ( l )3 = P (cid:0) E ( l ) (cid:1) , P ( t )3 = P (cid:0) E ( t ) (cid:1) calcu-lated at these fields. The latter can be found from Eqs. H (cid:16) P ( l )3 (cid:17) = 0 and H (cid:16) P ( t )3 (cid:17) = 0 as: P ( l ) 23 = (cid:0) b − b b (cid:1) / − b b , (8) P ( t ) 23 = (cid:0) b − b b (cid:1) / − b b . (9)Condition P ( l )3 = P ( t )3 determines the crossover line L between lt and t regimes:L : b b − b b b b − b b = 5 b b − b b b b − b b . (10)To be more specific we delimit the location of l , lt and t switching regimes and corresponding crossover lines L and L on phase diagram of strained PbTiO and BaTiO FIG. 3. Time dependence of the longitudinal, I l = d P d t andtransversal, I t = d P d t polarization currents for (a) transversal( t ), (b) longitudinal-transversal ( lt ) and (c) longitudinal ( l )switching regimes for PbTiO . Panels (a), (b) and (c) corre-spond to the points A, B and C in Fig. 1 (a). The cross andcircle markers indicate the beginning of the longitudinal andtransversal polarization reversal process correspondingly. films using the taken from [3] strain and temperature de-pendencies of coefficients of functional (2) and examiningseparately the cases of transitions through the ac and r phases.In the case of PbTiO all these regimes are clearly vis-ible and are located inside the region of thermodynam-ically stable c -phase as shown in Fig. 1 (a). To studythe transient polarization dynamics we select the repre-sentative points for each transition region (points A, Band C in Fig. 1 (a)) and numerically solve the Landau-Khalatnikov kinetic equations. L i d P i d t = − δ ˜ GδP i , (11)for each polarization component P i = P i ( t ). Here L i arethe corresponding damping coefficients. The results arepresented in Fig. 3 in form of experimentally measurablelongitudinal and transversal polarization currents I l = d P d t and I t = d P d t .Point A is selected for the l -switching region at u m = − . T = 500 ◦ C. As it follows from Fig. 3 (a)the polarization current has only the longitudinal compo-nent that is characteristic for the longitudinal switchingregime. Point B corresponds to the lt -switching regionand is taken at u m = − . T = 250 ◦ C. As shownin Fig. 3 (b) both components of polarization currentare observed but the transversal one is excited after thelongitudinal one and vanishes earlier than the longitu-dinal one. Point C is taken in the t -switching regionat u m = − . T = 0 ◦ C. As shown in Fig. 3 (c)the longitudinal and transversal polarization currents areexcited simultaneously.In the case of BaTiO [Fig. 1 (b)] one can observe onlythe l and t -switching regimes. The lt -switching regimeis difficult to detect because of the very close location ofthe lines L and L .An important issue for the t -type switching is the dy-namical behavior of polarization just upon reaching thetransversal instability field. Under certain conditions theintermediately-stable ac - or r - phase can be induced justabove E ( t ) . Then, the continuous (as function of thefield) turnover of polarization through this phase willprecede the abrupt rotational drop-down. To distin-guish between the shown in Fig. 2 initially-continous ( t c )and straight-abrupt ( t a ) transversal switching process westudy the global stability of functional (2) with respect tosmall deviations ∆ P , ∆ P about the equilibrium point P ( t )1 = 0 and P ( t )3 = P (cid:0) E ( t ) (cid:1) exactly at E = E ( t ) . Thisis a peculiar problem since at E = E ( t ) the coefficient H before (∆ P ) is equal to zero and the higher-orderterms should be taken into account. Following the catas-trophe theory we keep only the most relevant terms andpresent the expansion of (2) as:˜ G ≈ ρ (∆ P ) + µ (∆ P ) (∆ P ) + 12 λ (∆ P ) , (12)where ρ = H (see (5)), µ = ∂ ˜ G∂P ∂P = b P ( t )3 +2 b P ( t )33 and λ = ∂ ˜ G∂P = b . Transformation x = (∆ P ) and z = (∆ P ) maps the problem onto thestudy of quadratic functional λx + µxy + ρy . Thelast one is globally unstable at λρ > µ that providesthe straight-abrupt switching at E & E ( t ) . At λρ < µ this functional is locally stable and at the initial stage ofreversal process the P component develops continuouslyas a function of the field. Using the given above defini-tion of λ , ρ , µ and excluding P ( t )3 according Eq. (9) we,after some algebra, present the line L , separating thesetwo regimes on the u m - T phase diagram by equation:L : P R = Q . (13)with P = b b b − b b + 2 b b − b b , (14) Q = 5 b b b − b b b ,R = 3 b b − b b + 8 b b − b b b . The t c and t a switching regions being located below andabove this line correspondingly.Thoughtful analysis of equation (13) for BaTiO caseshows that the line L is located very close to the lineL (see Fig. 1 b) which means that ” t c -switching” alwaysoccurs through the intermediate field-induced ac -phase.In contrast, the line L does not exist in c -phase regionof the phase diagram of PbTiO [Fig. 1 (a)] which im-plies the ” t a -switching” through the intermediate r -phasetakes place.In this letter we have demonstrated the existence ofdifferent polarization reversal regimes in strained pseudo-cubic ferroelectric PbTiO and BaTiO films. Depend-ing on the temperature and on the misfit strain one candistinguish the polarization reversal governed by the lon-gitudinal, transversal or mixed longitudinal-transversalswitching regimes. All three mechanisms can be ob-served in PbTiO compounds. In BaTiO compoundsonly the longitudinal and transversal mechanisms can bedetected. The later occurs through the intermediate ac -phase with initial continuous turnover of the polarizationvector as function of the field. The dynamic appearanceof the transversal polarization during transition can beobserved by the time-resolved piezo-force microscopy orby the in-field Raman spectroscopy sensitive to the po-larization vector variation. Note, however that situationcan be even more complex if the 180 o ferroelectric do-mains exist in the initial c-phase or/and 90 o transversalferroelastic domains emerge during the switching process.The study of such scenarios can be done on the basis ofpresented above calculations.This work was supported by FP7 ITN-NOTEDEV andIRSES-SIMTECH MC mobility programs. ∗ [email protected][1] J. F. Scott, Ferroelectric Memories , Advanced micro-electronics, Springer, 2000.[2] M. E. Lines and A. M. Glass,
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