Tip-induced domain protrusion in ferroelectric films with in-plane polarization
TTip-induced domain protrusionin ferroelectric films with in-plane polarization
S. Kondovych,
1, 2, a) A. Gruverman, and I. Luk’yanchuk
1, 4 Laboratory of Condensed Matter Physics, University of Picardie, 33 rue St. Leu, Amiens 80039,France Life Chemicals Inc., Murmanska st. 5, Kyiv, 02660, Ukraine Department of Physics and Astronomy, University of Nebraska, Lincoln, NE 68588-0299,USA Faculty of Physics, Southern Federal University, 5 Zorge Str., 344090, Rostov-on-Don,Russia
Charge manipulation and fabrication of stable domain patterns in ferroelectric materials by scanning probe microscopyopen up broad avenues for the development of tunable electronics. Harnessing the polarization energy and electrostaticforces with specific geometry of the system enables producing the nanoscale domains by-design. Along with that,domain engineering requires mastery of underlying physical mechanisms that govern domain formation. Here, wepresent a theoretical description of the domain formation by a scanning probe microscopy tip in a ferroelectric filmwith strong in-plane anisotropy of polarization. We demonstrate that local charge injection produces wedge-shapeddomains that propagate along the anisotropy axis, whereas the tip-written lines of charge generate a comb-like domainstructure. The results of our calculations agree with earlier experimental observations and allow for the optimization ofthe targeted domain structures.
I. INTRODUCTION
Anisotropy of ferroelectrics and their high sensitivity to ex-ternal electric fields allow for effective control of their func-tional properties for successful implementation of novel fer-roelectric technologies , such as data storage , non-binaryand neuromorphic computing systems , terahertz emittersand detectors , low-dissipate computing circuits with neg-ative capacitance . A great variety of ferroelectric po-larization topological structures , including domains anddomain walls (DW) , vortices , skyrmions , andhopfions makes textured ferroelectric materials attractivenot only for fundamental studies but also for industrial ap-plications in nanoelectronics . Properties of the ferroelec-tric films critically depend on the emerging domain patterns,which ultimately determine the resulting operation modes offerroelectric-based electrically-controllable devices. The de-cisive role here belongs to the unique tunability of the ferro-electric film functionalities by the external parameters suchas strain-induced anisotropy, temperature, sample geometry,mechanical stress, and applied voltage. This makes the fab-rication of stable domain textures with tailored geometry anessential goal of device design and engineering .Whereas the out-of-plane domain structures and corre-sponding switching phenomena in thin ferroelectric crystals,films, and superlattices have been extensively investigatedboth theoretically and experimentally , the staticsand, particularly, dynamics of in-plane ferroelectricity havebeen studied to a much lesser extent. This is unfortunate as theinvestigation of in-plane polarization reversal provides an in-valuable insight into the process of the forward (along the po-lar axis) domain growth, which is difficult to attain in conven-tional ferroelectric switching studies . Recent experimen- a) Electronic mail: [email protected] tal studies performed by visualization of the domain struc-ture on the non-polar surface allowed for comparative analysisof the forward and lateral growth kinetics, direct assessmentof the role of structural defects in domain motion, and effectof screening on the kinetics and stability of the charged do-mains walls . Most of these studies have been performedby means of piezoresponse force microscopy (PFM), wheredomain manipulation on the non-polar surface was realizeddue to the lateral components of the electric field generatedby a PFM probe. Further expansion of this approach to thein-plane polarization control requires a better understandingof the fundamental mechanisms driving the domain growthin the conditions of a highly non-uniform electric field andstrongly anisotropic medium.In this paper, we present a theoretical consideration on theearlier experimental results on the PFM-induced formation ofthe in-plane wedge-shaped domains in uniaxial ferroelectriccrystal diisopropylammonium bromide (DIPA-B) . Model-ing the tip-induced voltage in the experimental setup as apoint-like and linear charge injection, we explain the originof a single wedge-shaped and comb-like domain structuresthat were observed (See Fig. 1a and Fig. 1b, respectively) andcalculate their parameters. Qualitative and quantitative agree-ment of the obtained results with the experiment enables opti-mization and by-design engineering of the domain patterns inferroelectric films with in-plane anisotropy. II. MODELA. Geometry and formation of the domain structure
The principal parameters of the system are illustrated inFig. 1c. We consider a ferroelectric film of thickness h withthe in-plane spontaneous polarization directed along the x-axis, P = ( P s , , ) , deposited on the dielectric substrate withthe dielectric constant ε d . The uniaxial dielectric tensor of a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b FIG. 1. Wedge-shape domain formation in the ferroelectric film withuniaxial in-plane anisotropy. (a) and (b) Experimental domains generated by PFM tip in a planar organic DIPA-B ferroelectric slabunder the different applied voltages for point-localized and linear dis-tribution of the tip-injected charges respectively (reprinted with per-mission from Adv. Mater. 27, 7832 (2015). Copyright 2015 JohnWiley and Sons). The yellow and brown colors correspond to thedomains with the opposite orientation of polarization vectors (bluearrows). (c) Cross-section view of the domain in the film. The tip-generated voltage pulse V injects localized positive charge Q intothe ferroelectric film of thickness h with in-plane spontaneous po-larization P s and dielectric tensor ˆ ε . The film is deposited on thedielectric substrate with dielectric constant ε d . The x -axis of the co-ordinate system is directed along P s , and the z -axis is perpendicularto the film surface. (d) View from the top. The injected charge Q attracts the negative depolarization charge − q and repulses the pos-itive one + q , resulting in the appearance of an elongated domainof reversed polarization, − P s , of the length l and width d . (e) Thelinear charge injected by the moving along the y -axis tip extrudes amultiple-domain comb-like structure with the period D . the film has the diagonal components ˆ ε = ( ε , ε , ε ) =( , η , η ) ε , where η > Q , positive fordefiniteness, into the sample and generates an elongatedwedge-like domain as shown in Fig. 1d (view from the top).The domain grows along the x -axis, acquiring the width d and length l . The depolarization charges due to polariza-tion reversal at the tip-application point and at the domainending are equal respectively to ∓ q (cid:39) ∓ ( hd ) P s . The dy-namics of the domain evolution after charge Q injection isgoverned by the ponderomotive force acting on the termi-nal charge + q from the charge Q − q at the domain origin,composed from the injected charge Q and the depolarizationcharge − q . In case Q > q the interaction has the repulsive character that pushes the domain termination point away fromthe tip-induced charge Q , resulting in protrusion of the do-main in the plane of the film until the increasing energy ofthe DW compensates the electrostatic force. This defines theequilibrium domain length. We take the domain width pro-portional to the effective charge at the domain origin and present this dependence as κ hP s d = Q − q , where κ isthe material-dependent proportionality coefficient to be foundfrom the experimental data.Another experimental result is related to the in-plane do-main formation by moving the electrically-biased PFM tipover the non-polar surface of the DIPA-B sample in the direc-tion perpendicular to the polar axis injecting a linear chargewith density λ Q . This stimulates the partial polarization re-orientation at one semi-space with the formation of the com-pensating depolarization charge due to the head-to-head (ortail-to-tail) polarization junction at the line. Such reorienta-tion causes the protrusion of a number of charges into thematerial and generates a comb-like domain pattern with theperiod D as shown in Fig. 1e. The effective charge density atthe line can be estimated as λ Q − λ q , where λ q = α P s h is thelinear depolarization charge due to polarization re-orientationin domains, and parameter α (cid:39) d / D < , is proportional to λ Q − λ q .We assume that κ hP s d = ( λ Q − λ q ) D with experimentally-defined coefficient κ .Although the mechanism of domain formation is similar forboth cases, the quantitative parameters of domains depend onthe dimensionality of the injected charge regions, as will beshown further. B. Total energy of the system
The optimal geometric parameters of the domain structureare obtained by minimization of the total energy of the systemthat includes the electrostatic energy and the DW energy: W tot = W el + W dw . (1)The DW energy, W dw , of the single domain shown in Fig. 1dis written as 2 σ dw hl , where the surface tension of the DW ofthickness ξ (cid:39) σ dw (cid:39) ξ P / ε ε . Here, ε is the vacuum permittivity, and factor 2 accounts for twoDWs forming the domain. In the case of the comb-like domainstructure, the DW energy per unit of length along the chargedline is expressed as 2 σ dw hl / D .The electrostatic energy W el describes the interaction of thecharges + q , located at the terminal point of the domains withthe tip-induced charges at the domain origin, Q − q for the caseof the localized charge and λ Q − λ q for the charged line. Wefurther derive this energy for the cases of the point-localizedand linear charge injection to find the equilibrium domain pa-rameters. C. Electrostatic energy of the single domain
According to electrostatics, the energy of the interactionbetween charges + q and Q − q is written as W el = q ϕ ( x ) ,where ϕ ( x ) is the potential of the charge Q − q at the loca-tion x of the charge q . This potential is found as a solution ofthe Poisson’s equation with appropriate boundary conditions(see Appendix A for details). At distances larger than the filmthickness, x (cid:29) h , the dependence ϕ ( x ) is written as: ϕ ( x ) (cid:39) ϕ ( ) Φ (cid:16) x Λ (cid:17) . (2)Here, Φ ( ζ ) = H ( ζ ) − N ( ζ ) denotes the difference be-tween the zero-order Struve and Neumann functions , andthe Λ and ϕ ( ) are the characteristic length and characteristicpotential, respectively: Λ (cid:39) √ ε ε + ε d ε d η h , ϕ ( ) = Q − q ε ε η h , (3)where the expression for Λ is given for ε , ε , ε d (cid:29)
1, for ar-bitrary dielectric constants see Eq.(A6) in Appendix A. Thischaracteristic length separates the regions with different typesof the electrostatic interaction . Namely, for x (cid:28) Λ , theinteraction manifests the 2D logarithmic behavior, ϕ ( x ) ∼− ln ( x / Λ ) , whereas for x (cid:29) Λ the 3D Coulomb decay of thepotential is obtained, ϕ ( x ) ∼ Λ / x , see Fig. 2a. The charac-teristic potential ϕ ( ) estimates the value of the potential at x ∼ Λ . FIG. 2. Electrostatic potential of the tip-generated field in the ferro-electric film. (a) The purple line corresponds to the space variationof the potential, ϕ ( x ) , created by the point-localized tip-injectedcharge. (b) The potential, ϕ ( x ) , induced by the tip-drawn linearcharge (red line). The black line represents the auxiliary function g ( x ) , which forms the expression (4) for ϕ ( x ) . The distance x inboth plots is scaled in units of the characteristic length Λ and poten-tials are scaled in units of characteristic potentials, ϕ ( ) in (a) and ϕ ( ) in (b). The dashed lines depict the limit cases of the functionalbehavior at x (cid:29) Λ and at x (cid:28) Λ , as described in the text. D. Electrostatic energy of the comb-like domain structure
The electrostatic energy (per unit of length along thecharged line) of the interaction of the depolarization charges located at the termination of the comb-like domains with thetip-induced linear charge is written as W el = λ q ϕ , where theelectrostatic potential ϕ is due to the field created by the DW.This potential at x (cid:29) h is expressed as: ϕ ( x ) (cid:39) ϕ ( ) (cid:104) ln (cid:16) c x Λ (cid:17) + g (cid:16) x Λ (cid:17)(cid:105) , ϕ ( ) = λ q − λ Q πε ε d , (4)where ln c (cid:39) .
577 is the Euler’s constant, the function g ( ζ ) =( π / − Si ζ ) sin ζ − Ci ζ cos ζ is the auxiliary trigonometricintegral function , the characteristic length Λ is given byEq. (3), and ϕ ( ) is the characteristic potential of the systemat x ∼ Λ . At small distances, x (cid:28) Λ , the g -function behavesas g ( ζ ) (cid:39) − ln ( c ζ ) + πζ /
2, and the resulting potential has thelinear behaviour. At large distances, x (cid:29) Λ , the vanishing be-haviour of g ( ζ ) (cid:39) − / ζ results in the logarithmic decay ofthe potential (Fig. 2b). III. RESULTSA. Single domain
The equilibrium length, l , of the single domain is calculatedby the minimization of the total energy W tot (Eq. (1)) with re-spect to x . The final expression, relating l and the domainwidth d , which depends on the injected charge, is written as Φ − (cid:18) l Λ (cid:19) = − Λ ξ κ η d , (5)where Φ − ( ζ ) ≡ d Φ / d ζ . The dependence l ( d ) in limitcases, which correspond to different regimes of domaingrowth, is found explicitly from Eq.(5) using the asymp-totic expressions for Φ ( ζ ) . The equilibrium domain lengthis estimated as l (cid:39) ( κ η / πξ ) d for intermediate lengths h (cid:28) l (cid:28) Λ . For the longer domains with l (cid:29) Λ , it is givenby l (cid:39) ( κ η Λ / πξ ) / d . B. Comb-like domain structure
Minimization of W tot in the case of multiple-domain comb-like structure gives implicitly the dependence l ( d ) , f (cid:18) l Λ (cid:19) = πε d ξ Λ D κ ε hd , (6)where f ( ζ ) is related to g ( ζ ) as f ( ζ ) = g (cid:48) ( ζ ) + / ζ . Thisfunction decreases monotonously and for the large values ofargument is simplified as f ( ζ ) (cid:39) / ζ . We obtain that for large l (cid:29) Λ , the equilibrium domain length depends on the domainwidth d as: l (cid:39) ( κ ε h / πε d ξ D ) d . IV. DISCUSSION
The obtained results allow not only for the qualitative de-scription of the formation mechanism of the wedge-shapeddomains in the uniaxial ferroelectric films but also for thequantitative comparison with the available experimental data.Fig. 3 demonstrates the dependence of the geometrical param-eters of the string domains created by the local application ofthe voltage pulse with amplitude V , which injects a localizedcharge Q , proportional to V . In this figure, we compare theexperimental data for length and width of the shown in Fig. 1asingle domain in DIPA-B slab of thickness h = l ( d ) described by Eq. (5) with P s = µ C/cm , ε = ε = κ (cid:39)
1, and ξ (cid:39) Λ is estimated from (3) with ε d (cid:39)
12 (for Si substrate)and η = ( ε / ε ) / (cid:39) . Λ (cid:39) µ m. The domain width d at given voltage was taken from the best linear fit of theexperimental data (the blue line). FIG. 3. Length and width of the tip-induced single domains as afunction of the applied voltage. The experimental data for the domainlength and width in the DIPA-B slab are denoted by the diamondand circle markers respectively. The blue line corresponds to the bestlinear fit of the experimental domain width d . The red line refers tothe calculated domain length l . Note that the more precise variational approach would ac-count for the spatial distribution of the depolarization chargeover the entire domain boundary, revealing the wedge form ofdomains, similar to that observed in the experiment (Fig. 1a).On the quantitative level, this results in the dimensionless cor-rection coefficient that does not affect the obtained functionaldependence of the domain parameters on the voltage, given inFig. 3.Our calculations demonstrate a very good match with theexperimental data and can be used as the base for the designand optimization of the tip-generated domains in ferroelectricfilms with in-plane anisotropy. In addition, since the domainlength appears to be larger than the characteristic length Λ , thederived above approximate expression l (cid:39) ( κ η Λ / πξ ) / d can be employed for practical use.Estimation of parameters of the domain comb-like structuregenerated by the moving PFM tip has a more qualitative char- acter. The challenge here is to properly take into account theinteraction between the charges at the terminal points of do-mains. This collective effect may not only affect the quantita-tive estimations given by Eq. (6) but also lead to the instabilityof the alignment of the terminal charges in a line. The repul-sion between these charges will result in the variable lengthof the domains, as seen in Fig. 1b. However, the order-of-magnitude estimation of the characteristic (average) domainlength gives l (cid:39) ( κ ε h / πε d ξ D ) d (cid:39) µ m with d / D (cid:39) . κ (cid:39)
1, and d ∼ V ,which provides a satisfactory agreement with the experiment(Fig. 1b). ACKNOWLEDGMENTS
This work was supported by the H2020-MSCA-RISE ac-tions ENGIMA (Grant No: 778072) and MELON (Grant No:872631). We also acknowledge the assistance of I.F. Lab andMileage Mobility Action.
DATA AVAILABILITY
Data sharing is not applicable to this paper as no new datawere created or analyzed in this study.
Appendix A: Methods
The electrostatic potential created by a point-like charge Q inside the ferroelectric layer (Fig. 1c) is found as a solution ofPoisson’s equation, η∂ x ϕ + ∂ y ϕ + ∂ z ϕ = − Q ε ε δ ( x , y , z ) , (A1)with boundary conditions at the ferroelectric interfaces: z = ϕ = ϕ + , ε ∂ z ϕ = ∂ z ϕ + ; (A2) z = − h : ϕ = ϕ − , ε ∂ z ϕ = ε d ∂ z ϕ − . (A3)Here, δ ( x ) is the Dirac delta-function, ϕ ± denote the elec-trostatic potentials in the regions above and below the film,respectively; ∇ ϕ ± =
0. We search for the solution in form: ϕ ( x , y , z ) ∼ (cid:90) ∞ (cid:90) ∞ e ± z √ k y + η k x cos ( k x x ) cos ( k y y ) dk x dk y . (A4)At the film surface, z =
0, at distances larger than the filmthickness, x (cid:29) h , we obtain the expression for the electrostaticpotential ϕ ( x , y ) : ϕ ( x , y ) (cid:39) Q − q ε ε η h Φ (cid:32) (cid:112) x + η y Λ (cid:33) , (A5)that gives Eq. (2) at y =
0. In general case of arbitrary valuesof dielectric constants, the characteristic length, Λ , is givenby: Λ = ( ε a + √ ε ε ) ( ε b + √ ε ε ) ε ( ε a + ε b ) h , (A6)which in case of high permittivity materials with ε , ε , ε d (cid:29) − h < z < λ Q along y axis, thesolution of Poisson’s equation, η∂ x ϕ + ∂ z ϕ = − λ Q h ε ε δ ( x ) , (A7)with the boundary conditions (A2)-(A3) results in Eq. (4), seeRef. for details. M. Dawber, K. M. Rabe, and J. F. Scott, Rev. Mod. Phys. , 1083 (2005). J. Scott, Science , 954 (2007). J. F. Scott,
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