Local polarization dynamics in ferroelectric materials

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Rep. Prog. Phys. 73 (2010) 056502 Initial and boundary conditions for polarization in equation (4.8) are ( P 3 (r,t 0) = P 0 (x) , P 3 − λ ∂P )∣ 3 ∣∣∣z=0 = 0. ∂z (4.9) Polarization distribution P 0 (x) satisfies equation (4.8) at zero external bias, V e = 0. Reported extrapolation length λ values are 0.5–50 nm [357]. Hereafter, second order ferroelectrics with large extrapolation length, λ ≫ √ ξ, are considered. Infinite extrapolation length λ →∞corresponds to the situation of perfect atomic surface structure without defects or damaged layer. 4.3.2.2. Probe-induced domain formation and ferroelectric breakdown through the film. The analysis of the domain switching produced by the probe field requires the analytical description of the depolarization field produced by the counter domain wall. Since for most ferroelectrics the tip size is larger than the correlation length, L ⊥ ≪ d, this approximation is used hereinafter. Applying direct variational method to the linearized solution of equation (4.8), we obtained that in the actual region z ≫ L z , the gradient effects lead to the unessential renormalization of expansion coefficient α as α → α R = α(1+(1+γ −2 )L 2 ⊥ /d2 ). The approximation is rather rigorous outside the domain wall region. Hence, coupled equations (4.7) and (4.8) can be rewritten as ⎧ ⎪⎨ γP 2 (P 3) ∂2 ϕ ∂z + ∂2 ϕ 2 ∂x + ∂2 ϕ 2 ∂y = 0, 2 α R P 3 + βP3 ⎪⎩ 3 + δP 3 5 =−∂ϕ ∂z , ϕ (x,y,z = 0) = V e (x,y) , P 3 (r ≫ d,z < 0) →−P S . (4.10) Polarization-dependent anisotropy factor γ P [P 3 ] is introduced as √ ε33 b 1 γ P (P 3 ) = + ( ε 11 ε 11 ε 0 αR +3βP3 2 +5δP ). 3 4 (4.11) Note that the dielectric susceptibility χ ∼ (α R +3βP 2 3 + 5δP 4 3 )−1 is positive for thermodynamically stable states. Then the spatial distribution of the polarization can be found as the solution of the nonlinear algebraic equation α R P 3 (ρ,z) + βP 3 3 (ρ,z) + δP 5 3 (ρ,z) = E 3 (ρ,z) . (4.12) We emphasize that the effective field E 3 is the sum of the probe field E e (ρ, z) and depolarization field E W (ρ, z). The left-hand side of equation (4.12) describes the conventional ferroelectric hysteresis. Thus, under the absence of the pinning field, a thermodynamically stable domain wall boundary ρ(z) can be determined from equation (4.12) as the coercive point, i.e. under the condition α R +3βP3 2(ρ, z) +5δP 3 4 (ρ, z) = 0 valid at coercive field: E 3 (ρ, z) = E c . 47 S V Kalinin et al The intrinsic coercive field E c is well known [358]as ⎧ √ 2 3 √ − α3 R 3 β , for the second order ferroelectrics, ⎪⎨ 2 ( E c = 2β + √ ) 9β 5 2 − 20α R δ ( ) 3/2 2α R ⎪⎩ × −3β − √ , for the first order. 9β 2 − 20α R δ (4.13) Note that this analysis essentially reproduces early arguments of Kolosov [123], stating that the domain size in a PFM experiment corresponds to the region in which tip-induced field exceeds coercive field. Here, we obtain a similar result; however, the field is now intrinsic (rather than macroscopic) coercive field renormalized by the depolarization field of the nascent domain. The bias dependences of the domain length l(V ) and radius r(V) calculated from the equation E 3 (ρ, z) = E c are shown in figure 45 for typical ferroelectric materials including LiTaO 3 , PbTiO 3 and PbZr 40 Ti 60 O 3 in three limiting cases: (i) Perfect screening of domain wall depolarization field by free charges. For this case there are no resulting charge at the wall and no depolarization field (see dashed curves in figure 45). (ii) No motion of the charged domain wall by depolarization field and no screening charges. This case has unclear physical interpretation and shown by dotted curves in figure 45 for comparison. (iii) The motion of the charged domain wall by the maximal depolarization field is considered. The situation is typical in the absence of screening or very slow screening (see solid curves in figure 45). The remarkable aspect of the above analysis is that the domain radius r calculated from equations (4.12) isalways finite at finite intrinsic domain wall width L ⊥ ≠ 0. This reflects the fact that spontaneous polarization re-orientation takes place inside the localized spatial region, where the resulting electric field absolute value is more that thermodynamic coercive field, i.e. |E 3 | > E c , while the hysteresis phenomenon appeared in the range |E 3 | < E c as anticipated within the LGD approach considering nonlinear correlation effects. The domain breakdown through the sample depth appears for infinitely thin domain walls (L ⊥ → 0), i.e. under the absence of domain wall correlation energy (ξ,η → 0). The microscopic origin of the domain tip elongation in the region where the probe electric field is much smaller than the intrinsic coercive field is the positive depolarization field appearing in front of the moving charged domain wall (see figure 44(c)). Note, that the activationless hysteresis phenomenon calculated within the LGD approach corresponds to the metastable state, in contrast to activation mechanism of the stable domain formation calculated within the energetic LM approach. Within the rigid LM approach the domain walls are regarded infinitely thin and polarization absolute value is constant: −P S outside and +P S inside the domain (if any). Semi-ellipsoidal domain radius r and length l are calculated
Rep. Prog. Phys. 73 (2010) 056502 Domain radius (nm) 10 3 10 2 10 1 Critical bias V cr r~V 2/3 r~V 1/3 LGD-approach LM-approach Envelope Agronin data Activation bias V a 10 Coercive 10 2 10 3 bias V c Applied bias V (V) Figure 46. Diagram demonstrating the main features of probe-induced domain formation calculated within the LGD approach (solid curves), LM approach (dashed curves) and their envelope (dotted curve). LNO material parameters are the same as in figure 45, ψ S = 0.35Jm −2 , probe surface separation d = 50 nm. Squares are experimental data reported by Agronin et al [359]. Reproduced from [351]. from the free energy excess consisting of the interaction energy, the domain wall surface energy ψ S and the depolarization field energy. Nonlinear correlation energy contribution is absent within the rigid approximation. Within the LM approach, the depolarization field energy vanishes as 1/l, while the interaction energy is maximal at l →∞, the condition of negligible surface energy leads to the domain breakdown l →∞and the subsequent macroscopic region re-polarization even at infinitely small bias (if only VP S > 0), while the hysteresis phenomena or threshold bias (saddle point) are absent. Under finite domain wall energy, the critical bias V cr and energetic barrier E a of stable domain formation exist. Activation (or nucleation) bias V a is determined from the condition E a (V a ) = nk B T,where the numerical factor n = 1 ...25. Usually V a ≫ V cr for thick films. In figure 46 we compare the main features of the probeinduced domain formation calculated within the intrinsic LGD approach and the energetic LM approach. For consistency between the approaches we used the Zhirnov expression [354] for the domain wall surface energy ( ψ S = √ 1+ 2 (( Q 2 11 + ) )) Q2 12 s11 − 2Q 11 Q 12 s 12 β ( s11 2 − ) η s2 12 × (−2α)3/2 , (4.14) 3β where Q ij are the electrostriction tensor, s ij are the elastic compliances. 4.3.2.3. Probe interaction with domain walls. The analytical model for the interaction of the biased SPM probe and existing domain walls opens the way for experimental studies of microscopic mechanisms of domain wall polarization interaction with electric field that can be studied in strongly inhomogeneous fields of biased force microscope probe. This problem is similar to that of domain wall pinning on a charged S V Kalinin et al impurity, where the SPM probe acts as a ‘charged impurity’ with controlled strength (controlled by tip bias) positioned at a given separation from the domain wall. In this context, the problem of the infinitely thin ferroelectric domain wall interaction with a charged point defect was considered by Sidorkin [360]; however, neither correlation effects (e.g. finite intrinsic width of domain walls) nor rigorous depolarization field influence were taken into account. For the description of domain wall equilibrium position the Laplace tension conception (whose applicability to ferroelectrics has not been studied in detail) was used instead of the conventional LGD theory, thermodynamic Miller–Weinreich approach [86] or their combination with molecular dynamics and Monte-Carlo simulations as proposed by Rappe et al [88]. However, they studied domain wall profile changes in homogeneous external field. Below we consider the interaction of ferroelectric 180 ◦ domain wall polarization with a strongly inhomogeneous electric field of biased force microscope probe within LGD thermodynamic approach using direct variational method (see figure 47). The solution of equation (4.8) for the initial flat domain wall profile positioned at x = x 0 is P 0 (x) = P √ S tan h((x − x 0 )/2L ⊥ ), where the correlation length is L ⊥ = −η/2α, and the spontaneous polarization is P 2 S =−α/β. Polarization distribution at the sample surface was derived as P 3 (x,y,0) ≈ P 0 (x) − √ ε11 ε 0 −2α P V d 2 × √ d2 + x 2 + y ( 2 d 2 + x 2 + y 2 + L ⊥ d ). The variational amplitude P V should be found from Landau– Khalatnikov equation τ d dt P V + w 1 P V + w 2 (x 0 ) PV 2 + w 3PV 3 = V (t) , (4.15) with parameters w 1 = 1, w 2 (x 0 ) =− √ 3βP S x 0 −2αε11 ε 0 √ (L ⊥ + d) 2 + x0 2 4α 2 (L ⊥ + d) , βε 11 ε 0 w 3 = 4α 2 (L ⊥ + d) 2 . In the decoupling approximation and object transfer functions approach (see [182, 183, 186, 352]), Pade approximations for the bias V dependence of effective piezoelectric response P R(V ) = u 3 (x = 0,y = 0)/V were found as PR(V,x 0 ) = d0 eff (x 0 ) − ε 0ε 11 (4.16) γ 4∑ B i (γ ) · ln (e + b i (γ )/C) P V (V,x 0 ) × (b i=1 i (γ ) +ln(e + b i (γ )/C)) (L ⊥ + d ln (e + b i (γ )/C)) , where d0 eff(x 0) is the bias-independent PFM profile of the flat 180 ◦ domain wall located at distance x 0 from the tip apex. Response d0 eff(x 0) was √ calculated in [361]. Dielectric anisotropy factor γ = (ε 33 + ε33 b )/ε 11. Constants e ≈ 2.718 28 ... is the natural logarithm base 48
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