Local polarization dynamics in ferroelectric materials

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Rep. Prog. Phys. 73 (2010) 056502 S V Kalinin et al Length l (nm) 0 2 4 6 8 10 12 14 (a) Radius r (nm) (b) Radius r (nm) (c) Radius r (nm) (d) Radius r (nm) 1.25 0.6 UU cr Saddle point U (V) Free energy Φ (e) 15 10 50 100 5 25 10 5 2 0 0 1 2 3 4 5 15 10 5 11.167 20 40 5 2 0 0 1 2 3 4 5 r (d units) 0.25 15 10 5 16 32 8.05 4 0 0 1 2 3 4 5 U s 4 0 are metastable and the ones with (r,U) 0 arises, corresponding to a metastable domain with r ms and l ms . Finally, for U U cr , the absolute minimum min < 0is achieved for r eq and l eq , corresponding to a thermodynamically stable domain. The value U cr determines the point where the homogeneous polarization distribution becomes absolutely unstable. The minimum point (either metastable {r ms ,l ms } or stable {r eq ,l eq }) and the coordinate origin are separated by the saddle point {r S ,l S }. The corresponding energy (r S ,l S ) = E a is an activation barrier for domain nucleation, while domain parameters {r S ,l S } represent the critical nucleus size. Such 31
Rep. Prog. Phys. 73 (2010) 056502 ‘threshold’ domain nucleation is similar to the well-known where first order phase transition (compare figure 31(e) with the well-known dependence of the free energy profile on the order parameter for first order ferroelectrics). In the thermally induced nucleation limit, the domain nucleation process is analyzed as a thermally activated motion in the phase space of the system along the minimum energy path connecting the origin and one of the local minima. The relaxation time necessary for the stable domain formation at U cr is maximal and the critical slowing down appears in accordance with the general theory of phase transitions. Within the framework of the activation rate theory, the domain nucleation takes place at a higher activation voltage U a determined from the condition (U a ) = E a , corresponding to the activation time τ = τ 0 exp(E a /k B T). For instance, the activation energy E a = 20k B T corresponds to a relatively fast nucleation time τ ∼ 10 −3 s for phonon relaxation time τ 0 = 10 −12 s, while the condition E a 2k B T corresponds to ‘instant’ or thermal nucleation. × The difference between the voltages corresponding to the formation of a saddle point and a stable domain, U S − U cr , determines the width of the thermodynamic hysteresis loop (see figures 31(e)–(g)). More realistic models of piezoresponse hysteresis loop formation consider domain wall pinning effects. In the weak pinning limit, the domain growth in the forward direction is assumed to follow the thermodynamic energy minimum, while with decreasing bias, the domain remains stationary due to domain wall pinning by lattice and atomic defects. For stronger pinning, the domain size is limited by the wall mobility in the tip field and ferroelectric (as, e.g., studied for domain wall dynamic by Molotskii in [311]), and the full analysis of this problem remains the matter of future research. 3.3.2. Analytical treatment for Landauer geometry. The integral expressions for the free energy components in equation (3.1b) are extremely complex, and can be evaluated analytically only for simple domain configurations. Here we summarize the Pade approximations for the individual terms in the free energy (r, l, U) = S (r, l)+ p (r,l,U)+ D (r, l) of the semi-ellipsoidal domain for ferroelectric semiconductors allowing for Debye screening and uncompensated surface charges. The domain wall energy has the form ( r S (r, l) = πψ S lr l + arcsin √ ) 1 − r 2 /l √ 2 1 − r2 /l 2 ( ) ≈ π 2 ψ S lr 1+ 2 (r/l)2 , (3.2) 2 4+π (r/l) where r is the semi-ellipsoid domain radius, l is the domain length. The Pade approximation for the depolarization energy of a semi-ellipsoidal domain including the effects of Debye screening in the material is DL (r, l) = 4πP2 S r2 R d n D (r, l) ε 0 κ 4n D (r, l) +3R d (γ/l) , (3.3a) × 32 n D (r, l) = ⎛ (√ ) (rγ/l)2 ⎝ arcth 1 − (rγ/l) 2 1 − (rγ/l) 2 √ 1 − (rγ/l) 2 S V Kalinin et al ⎞ − 1⎠ (3.3b) is the depolarization factor, κ = √ ε a ε c is the effective dielectric constant of the medium, and, γ = √ ε c /ε a is the anisotropy factor. For an infinite Debye length (i.e. a rigid dielectric), equation (3.3a) becomes the well-known Landauer depolarization energy. The energy of the depolarization field created by the surface charges (σ S − P S ) located on the domain face has the form 4πr 3 R d DS (r, l) ≈ ε 0 (16κr +3πR d (κ + ε e )) ( (σ S − P S ) 2 + 2P S (σ S − P S ) 1+(l/rγ) ) . (3.4) The surface charge density σ S is −P S σ S P S , while σ S =−P S without screening charges. The driving force for the switching process is the interaction energy determined by the electrostatic field structure produced by the probe. The Pade approximation for the interaction energy between a spherical tip and the surface based on image charge series is p (r, l, U) ≈ 4πε 0 ε e UR 0 ∞ ∑ m=0 q m (3.5) × R d ((σ S − P S ) F W (r, 0,d − r m ) +2P S F W (r, l, d − r m )) ε 0 ( (κ + ε e ) R d +2κ √ (d − r m ) 2 + r 2 ) . The image charges q m are located at distances r m from the spherical tip center, where ( ) κ − m εe sinh (θ) q 0 = 1, q m = κ + ε e sinh ((m +1) θ) , (3.6a) r 0 = 0, r m = R 0 sinh (mθ) sinh ((m +1) θ) , cosh (θ) = d R 0 . (3.6b) Here R 0 is the tip radius of curvature, R is the distance between the tip apex and the sample surface, so d = R 0 + R. The function F W (r, l, z) ≈ r 2 √ r2 + z 2 + z + (l/γ) (3.7) is the Pade approximation of a cumbersome exact expression obtained originally by Molotskii [309]. Under the typical condition R ≪ R 0 , i.e. when the tip is in contact with the surface, equation (3.5) can be approximated as R d UC t /ε 0 p (r, l) ≈ (κ + ε e ) R d +2κ √ d 2 + r ( 2 (σ S − P S ) r 2 √ r2 + d 2 + d + 2P S r 2 √ r2 + d 2 + d + (l/γ) ) , (3.8)
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