Stability of Polar Vortex Lattice in Ferroelectric Superlattices

Transcription

1 Supporting Information: Stability of Polar Vortex Lattice in Ferroelectric Superlattices Zijian Hong 1, *, Anoop R. Damodaran 2, Fei Xue 1, Shang-Lin Hsu 2, 3, 4, Jason Britson 1, Ajay K. Yadav 2, 3, Christopher T. Nelson 2,3, Jian-Jun Wang 1, James F. Scott 5, Lane W. Martin 2, 3, Ramamoorthy Ramesh 2, 3, 4,, Long-Qing Chen 1,# 1 Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA 2 Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA 3 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 4 Department of Physics, University of California, Berkeley, CA 94720, USA 5 Schools of Chemistry and Physics, University of St Andrews, St Andrews KY16 9ST, UK Corresponding authors: * Mr. Z. Hong, zxh121@psu.edu Dr. R. Ramesh, rramesh@berkeley.edu # Dr. L. -Q. Chen, lqc3@psu.edu 1. Phase-field Methodology For this work, the improper ferroelectric behavior with coupling of multiple order parameters is ignored, polarization vectors P =(Px, Py, Pz) are the only order parameters 1

2 for the sake of simplicity. The system is governed by the time dependent Landau- Ginsburg-Devonshire (LGD) equation: P t = L δf δp (1) Where t, F, L are the time, total free energy and kinetic coefficient, respectively. The total free energy has contributions from Landau, elastic, electric and gradient energies [1]. F = (f Landau + f elastic + f electric + f gradient )dv (2) The Landau free energy density can be expressed by the Landau polynomial, with odd terms being canceled by the symmetry of cubic reference phase. f Landau = α ij P i P j + β ijkl P i P j P k P l + γ ijklmn P i P j P k P l P m P n + (3) The elastic energy density can be calculated from the elastic strain by the following expression: f elastic = 1 2 C ijkle ij e kl = 1 2 C ijkl(ε ij ε ij 0 )(ε kl ε kl 0 ) (4) Where C ijkl, e ij, ε ij, ε ij 0 are the elastic stiffness tensor, elastic strain, total strain and eigen strain, respectively. The eigen strain is defined as the phase transformation strain given by the electromechanical coupling and the lattice strain:ε ij 0 = Q ijkl P k P l + ε lattice. Qijkl is the electrostrictive coefficient tensor. The lattice strain is due to the difference between the psuedocubic lattice constants of PTO (or STO) and the reference state: ε lattice (PTO) = a PTO a ref a ref (5) 2

3 ε lattice (STO) = a STO a ref a ref (6) To solve the elastic equations, the total strain ε ij is separated into homogenous strain ε and heterogeneous strain δε ij. The homogenous strain is defined as the lattice mismatch between reference state a ref and substrate a sub : ε 11 = ε 22 = a sub a ref a ref. Where the effective inplane coherency strain, defined as the differences between homogenous strain and lattice strain: ε effective (PTO) = ε 11 ε lattice (PTO) = a sub a PTO a sub (7) ε effective (STO) = ε 11 ε lattice (STO) = a sub a STO a sub (8) is independent of the reference state [2], hereafter the average pseudo-cubic lattice constant of PTO and STO is chosen as the reference state. The volume integration of heterogeneous strain δε ij is zero, i.e., δε ij V dv = 0. The elastic equilibrium condition is satisfied: σ ij,j = 0 (9) Where σ ij is the elastic stress: σ ij = C ijkl e kl =C ijkl (ε kl ε kl 0 ). In order to incorporate the anisotropic elastic properties of PTO and STO, the elastic stiffness tensor is separated into homogenous part C ijkland perturbation part δc ijkl : C ijkl = C ijkl + δc ijkl (10) σ ij =C ijkl (ε kl ε kl 0 ) = (C ijkl + δc ijkl )(ε kl ε kl 0 ) (11) 3

4 Combine equations 9-11: σ ij x j = x j [(C ijkl + δc ijkl )(ε kl ε kl 0 )] = 0 x j [(C ijkl + δc ijkl )(ε effective + δε ij Q ijkl P k P l )] = 0 (12) The inhomogeneous strain δε ij can be further calculated by the differentiation of local displacement, δε ij = 1 2 (u i,j + u j,i ), where u i,j = u i x j, substitute into equation 12, C ijkl 2 u k = x j x l x [(C ijkl + δc ijkl )(ε effective Q ijkl P k P l )] u (δc k j x ijkl ) (13) j x l The thin film boundary condition is applied, i.e. stress free at the film top: σ i3 filmtop = 0, while the displacement at the film bottom is fixed to zero: u i filmbottom = 0. Equation 13 can be solved via the iteration method [3]. To ensure both high calculation speed and accuracy, the cutoff for the iteration is set as within 0.1% deviation in elastic energy. The electric energy density can be written as: f electric = 1 2 K ijε 0 E i E j E i P i. (14) Whereas 0 is the dielectric permittivity of free space, ij is the background dielectric constants of the superlattice film, E i is the local electric field defined as the gradient of the electric potential E i = i φ. The electrostatic equilibrium must be satisfied, i.e., D i,i = 0 (15) 4

5 The electric displacement D i = K ij ε 0 E j + P i, substitute into Equation 15: ε 0 K ij 2 φ x i x j = P i,i (16) Close-circuit boundary condition is employed, with potential fixed as zero at the SLs top and the film bottom. With the pseudo-cubic assumption, the gradient energy density can be given by: f gradient = 1 2 G ijklp i,j P k,l (17) Where G ijkl is the gradient coefficient tensor, P i,j stands for the spatial differential of the polarization vector: P i,j = P i x j. More detailed description of film-based phase-field method as well as iteration method for solving anisotropic elastic equations can be found elsewhere [3-5]. 2. Initial setup and parameters The simulation system is consisted of three parts: 30 unit cell of DSO substrate, 198 unit cell of SLs layer which has alternating layer of 10 unit cells of PTO, 1 unit cell of interfacial layer and 10 unit cells of STO. Periodic boundary condition is applied for the lateral (x and y) dimension while a superposition method is used for the z dimension. The simulation system size is , corresponding to 80 nm 80 nm 100 nm in real space. The normalized time step for simulation is t = 0.02,where the normalizer is t 0 = 1 a 0 L, a 0 = a 1PTO T=300K. Here L is the kinetic coefficient, which is related to the mobility of the domain wall. 5

6 The simulation parameters are mainly taken from previous reports, including Landau coefficients, elastic stiffness, electrostrictive coefficients for both PTO [6] and STO [7]. The lattice parameter for DSO substrate to calculate the misfit strain is taken from Chen et al. [8]. For the sake of simplicity, the normalized gradient coefficients are set as homogenous across the whole superlattice: G 11 = 0.6 G 12 = 0 G 44 = 0.3 G 44 = 0.3, where G = G G 110, G 110 = C -2 m 4 N [4]. The background dielectric constant is set as 40 for the two layers [9-11]. For the interfacial layer, the potentials as well as the properties are set as the average of PTO and STO. Random noise with magnitude less than C/m 2 is added as the initial nuclei. 3. Energy density plot of vortex and flux-closure structure: the contribution from elastic, electric, Landau and gradient energies. We first analyze the spatial distributions of different energy contributions within a single vortex. Figure S1(a) clearly shows the vortex core as the region of highest energy density. By separating out individual contributions from the elastic, electric, chemical, and polarization gradient energy (Figure S1b-S1e, respectively), we learn that the elastic, chemical and gradient energies dominate and taper out almost radially away from the vortex core. This can be understood as a consequence of the large lattice distortions (higher elastic energy), highly diminished and near zero polarization (higher Landau chemical energy as indicated by the double well energy curve), and large gradient energies from rapid change in both, the magnitude and direction of polarization that exists at the vortex core. This high energy density core allows a polarization 6

7 distribution with largely in-plane component of polarization at the interface between PTO and STO, and that with a largely out-of-plane polarization in the region between a vortex and a neighboring antivortex. Such a distribution serves to minimize depolarization-energy related effects (Figure S1c) at the ferroelectric/paraelectric interface that would otherwise destabilize ferroelectricity, while at the same time adopting an elastically favorable out-of-plane polarized configuration in the region. Moreover, the magnitude of ferroelectric polarization away from the vortex core rapidly approach values where contributions from the chemical landau energy are the most negative. In other words, the introduction of a high energy density vortex core, serves to minimize the volume averaged total energy density in the PTO layers under the given strain and electrostatic boundary conditions. It is also noteworthy that under similar electrostatic and strain boundary conditions, thicker period superlattices form traditional flux-closure structures [12] with uniformly polarized c (tetragonal, out-ofplane polarized)- and a (tetragonal, in-plane polarized)- domains that are separated by well-defined domain walls and characterized by a total energy density distribution (Figure S1f, computed for n=50) that is maximal at the 180 o domain wall (separating c+ and c- domains) followed by 90 o domain walls (separating c and a domains). In summary, these studies reveal length-scale related criterion, in addition to favorable elastic and electrical boundary conditions, towards the stabilization of vortex states in these superlattices. 7

8 In order to give a more qualitative view point, the Line plot along X and Z through the vortex center is also plotted in Figure S1, which reveals the dominant role of gradient energy near the vortex core. Generally, in the X direction through vortex core (Figure S1g), the Landau energy and elastic energy is lower while along Z (Figure S1h), the Landau energy and electric energy is lower along X. The surrounding area of the vortex core always has the highest energy and decay in a circular manner. (g) (h) Figure S1 Energetics of one vortex in (PTO) 10/(STO) 10 superlattices grown on a DSO substrate calculated from phase-field simulations. Spatial distributions of (a) the total energy density, (b) the elastic energy density, (c) the electric energy density, (d) the chemical energy density, and (e) the 8

9 gradient energy density, (f) total energy density of one flux-closure in (PTO) 50/(STO) 50 superlattices grown on a DSO substrate. Line plot of energy densities across the vortex core from the vortex center (g) along X direction, and (h) along Z direction. In an attempt to better distinguish between traditional flux-closure structure and this vortex state, the structure as well as energy densities of flux-closure structure are also computed. Both simulation results and the TEM polarization vector plot for PTO 50/STO 50 shows a classical fluxclosure structure composed of four 90 a/c domain wall and one 180 c+/c- domain wall (Figure S2a- S2b). The formation of flux-closure can be understood through the further separation of each individual energy (elastic, electric, Landau, gradient) contribution. (a) (b) Figure S2 Vector plot of flux-closure structure in PTO 50/STO 50. (a) From phase-field predication. (b) Experimental HRTEM mapping. As shown in Figure S3(a), the elastic energy is much higher near the 180 domain wall region and relatively lower in the c domain region. The 180 wall is rod-like in shape stretched along Z direction. For electric energy density, however, the 90 domain walls has the highest energy. While the a domain region has generally lower electric energy density (Figure S3b). The high Landau energy region can be found near both 9

10 the 90 and 180 domain walls (Figure S3c). While the Landau energy is lower inside the c and a domains, suggesting that the magnitude of polarization is similar in the c and a domains. It is also worthwhile mentioning that the Landau energy in the small triangular a domain near the interface is higher due to decreased magnitude of polarization through the interfacial effect. The gradient energy is largest near the 180 wall due to the increased domain wall width as compared to 90 wall (Figure S3d). The two 90 wall separating c domain and large a domain is also higher gradient energy wall as compared to the 90 wall that s separating c domain and smaller a domain since the polarization in the smaller a domain is suppressed. The line plot of the energy densities across the 180 wall center for flux-closure is also provided (Figures S3e and S3f), which shows that the domain wall is elongated along Z direction which can minimize the domain wall area as compared to the circular shape of the vortex. Note that the Landau energy density is lower than the vortex structure due to a larger magnitude of polarization in higher periodicity. (a) (b) 10

11 (c) (d) (e) (f) Figure S3 The energetics of flux-closure. Spatial resolution of (a) elastic energy density, (b) electric energy density, (c) Landau energy density and (d) gradient energy density. Line plot of energy densities across the vortex core from the 180 domain wall center along (e) X direction, and (f) Z direction. The differences in the vortex and flux-closure structure can be addressed: the polarization is continuous rotating in the vortex structure while only sudden changes near the distinct domain wall area in the flux-closure domain. From the energetic perspective, the total energy is changing in a circular way from the vortex center for vortex structure while flux-closure domain generally has higher energy in the domain wall regions. 11

12 The computed energy density of various phases with respect to periodicity can reveal the phase transition sequence with increasing periodicity. The complete phase diagram and total energy density is given in the main article, while the contribution from each energies is provided herein. As can be seen in Figure S4(a), the elastic and gradient energy has a sudden change at n=10 due to the a1/a2 to vortex transition. The decrease in elastic energy can fully compensate the gradient energy increase, which leads to a slightly lower energy density of vortex state. From Figure S4(b), the Landau energy change is trivial at the transition. Meanwhile, the differences between the two phases are larger at higher periodicity, leading to more stable vortex state at higher periodicity. The electric energy has a small jump from the a1/a2 state to vortex state, primary owing to the increased out-of-plane polarization. The total energy density difference between vortex and a-twin structure is increased with increasing periodicity from n=10 to n=16. Figure S4(c) shows the continuous energy change from vortex structure at low periodicity to flux-closure at high periodicity. The Landau and gradient energy generally decrease with increasing periodicity as has been discussed in the main text. The elastic energy increases due to larger strain in the STO layer with smaller polar magnitude. The smooth change in the individual energy densities (elastic, electric, gradient, Landau) leads to smooth change in the total energy density, which indicate that the vortex to flux-closure transition is more or less second order transition in nature. 12

13 (a) (b) (c) Figure S4 Energy densities of a 1/a 2 twin structure, vortex and flux-closure structure. (a) Elastic, Gradient (b) electric and Landau energy density comparisons for a 1/a 2 and vortex (c) The energy densities at larger periodicities for vortex and flux-closure structures. 4. Analytical expression for the periodicity dependence. In a 2-D model, the total free energy of a thin film can be explicitly rewritten as: 13

14 F = { a 1 P x 2 + a 3 P z 2 + a 11 P x 4 + a 33 P z 4 + a 12 P x 2 P z 2 + a 111 (P x 6 + P z 6 ) +a 112 (P x 4 P z 2 + P x 2 P z 4 ) + g 0 2 [( P x x )2 + ( P x z )2 + ( P z x )2 + ( P z z )2 ] 1 2 ε 0ε 11 E x ε 0ε 33 E z 2 E x P x E z P z (18) } dxdz where the a parameters are the Landau potentials under thin film boundary conditions [13], g0 is the isotropic gradient energy coefficient, Ex and Ez are the x, z components of the total electric field, respectively. ε 11 and ε 33 are the background dielectric constants. The dielectric displacements Dx and Dz can be written as: D x = ε 0 ε 11 E x + P x (19) D z = ε 0 ε 33 E z + P z (20) Minimizing equation18 with respect to Px and Pz, we have: 2a 1 P x + 4a 11 P x 3 + 2a 12 P x P z 2 + 6a 111 P x 5 +a 112 (4P x 3 P z 2 + 2P x P z 4 ) g 0 ( 2 P x x 2 2a 3 P z + 4a 33 P z 3 + 2a 12 P z P x 2 + 6a 111 P z 5 +a 112 (4P z 3 P x 2 + 2P z P x 4 ) g 0 ( 2 P z x 2 The electrostatic equilibrium needs to be satisfied, i.e., + 2 P x z 2 ) = E x (21) + 2 P z z 2 ) = E z (22) D x x + D z z = 0 (23) Whereas the electric field satisfies the following relationship: E x z = E z x (24) The continuity of the out-of-plane dielectric displacement give rise to the following boundary condition: 14

15 D z (x, z = 0) = 0 (25) D z (x, z = a f ) = 0 (26) where z=0 and z=a f denote the ferroelectric/dielectric interfaces, x denote any lateral position. It is difficult to solve equations analytically under the boundary condition of equations 25 and 26. Let s first consider the simplest case, in the vicinity of the vortex core, where the polarization is very small, the high order Landau terms of equation 18 can be omitted. If we were only to expand to the second order in the Landau energy, Equations 21 and 22 can be reduced to: 2a 1 P x g 0 ( 2 P x x 2 2a 3 P z g 0 ( 2 P z x 2 We shall look for the periodical solution along x for Pz: + 2 P x z 2 ) = E x (27) + 2 P z z 2 ) = E z (28) P z = Sin(k 1 x)f(z) (29) Perform separation of variables for Px, and assume: P x = q(x)h(z) (30) Plug equations 29, 30 into equations 27, 28, 23, 24 and separate the variables again, we can get: 2a 1 dh(z) dz q(x) g 0 ( d2 q(x) dx 2 dh(z) + d3 h(z) dz dz 3 q(x)) = (2a 3 k 1 f(z) + g 0 k 1 3 f(z) k 1 d 2 f(z) dz 2 )Cos(k 1x) (31) 15

16 (2a 1 ε 0 k ) dq(x) dx h(z) g 0 ( d3 q(x) dx 3 h(z) + d2 h(z) dq(x) dz 2 dx ) + Sin(k 1 x)[ (2a 3 ε 0 k ) df(z) dz g 0 ( d3 f(z) dz 3 2 df(z) k 1 )] = 0 (32) dz Solving (31) and (32) and separating the variables, we can get: dq(x) dx ~ Sin(k 1x) (33) q(x) = τ Cos(k 1 x) (34) Similarly, we can look for the periodic solutions along z direction, i.e., f(z) = P 0 Sin(k 2 z) (35) Plug in the solutions back to equations (31) and (32), we can get: dh(z) dz ~ Sin(k 2z) (36) h(z) Cos(k 2 z) (37) Combine the solutions together, we can get: P z = P 0 Sin(k 1 x)sin(k 2 z) (38) P x = P 0 Cos(k 1 x)cos(k 2 z) (39) Applying the boundary condition of (25)-(26), we can get: k 2 = π a f (40) 16

17 Assuming the periodicity along x is d, we can get: k 1 = π d (41) The analytical solutions from equations (38)-(41), as well as the vortex structure simulated by the phase-field simulation are plotted in Figure S5(a) and Figure S5(b), respectively. It is shown that the analytical solution can well reproduce the characteristics of the vortex structure even from the simplest model. Fig.S5 Vortex structure by (a) analytical calculation (b) phase-field simulation in (PTO) 10/(STO) 10. The analytical solution refers to Equations 38-41, with P 0 = P 0 = 0.5 C/m 2 and a f = d = 10 unit cells. Plug in the solutions back to (31) and (32), one can get: ( 2a 1 + g 0 (k k 2 2 ))k 2 P 0 = ( 2a 3 + g 0 (k k 2 2 ))k 1 P 0 (42) ε 0 k 11 k 1 P 0 (2a 1 + g 0 (k k 2 2 )) + k 1 P 0 = ε 0 k 33 k 2 P 0 (2a 3 + g 0 (k k 2 2 )) + k 2 P 0 (43) 17

18 Since a 1 < a 3 < 0 and k 1, k 3, P 0, P 0 > 0, one can find: 2a 3 g 0 < (k k 2 2 ) < 2a 1 g 0 (44) In the case where the substrate strain in the ferroelectric layer is small (less than 0.2% between PTO and DSO using the parameters given in the main article), one can estimate that: k k 2 2 a 1+a 3 g 0 (45) a 1 + a 3 2a 10 (T T c ) (46) Now, let s expand the Landau terms to fourth order, the minimization with respect to Pz and Px will yield: 2a 1 P x + 4a 11 P x 3 + 2a 12 P x P z 2 g 0 ( 2 P x x 2 2a 3 P z + 4a 33 P z 3 + 2a 12 P z P x 2 g 0 ( 2 P z x 2 Separating variables again, rewrite Px and Pz such that: + 2 P x z 2 ) = E x (47) + 2 P z z 2 ) = E z (48) P x = h(x)t(z) (49) P z = f(x)u(z) (50) Combine (49), (50) and (21), (22), one can write: 2a 1 h(x) dt(z) dz + 12a 11h(x) 3 t(z) 2 dt(z) + 2a dz 12h(x) dt(z) dz f(x)2 u(z) 2 + 4a 12 h(x)t(z)f(x) 2 u(z) du(z) dz g 0 ( d2 h(x) dx 2 dt(z) dz + d3 t(z) h(x)) dz 3 = 2a 3 u(z) df(x) dx 4a 12 u(z)f(x)t(z) 2 h(x) dh(x) dx + 12a 11u(z) 3 f(x) 2 df(x) + 2a dx 12u(z) df(x) dx h(x)2 t(z) 2 + g 0 ( d2 u(z) dz 2 18 df(x) dx + d3 f(x) dx3 u(z)) (51)

19 (2a 1 ε 0 k )t(z) dh(x) dx 4ε 0 k 11 a 12 h(x)t(z)f(x)u(z) 2 df(x) dx 1)f(x) du(z) dz + 12ε 0k 11 a 11 h(x) 2 t(z) 3 dh(x) + 2ε dx 0k 11 a 12 t(z) dh(x) dx f(x)2 u(z) 2 + g 0ε 0 k 11 ( d3 h(x) dx 3 t(z) + d2 t(z) dh(x) ) + (2a dz 2 dx 3ε 0 k ε 0 k 33 a 33 f(x) 3 u(z) 2 du(z) + 2ε dz 0k 33 a 12 f(x) du(z) dz h(x)2 t(z) 2 + 4ε 0 k 33 a 12 u(z)t(z)f(x)h(x) 2 dt(z) dz g 0ε 0 k 33 ( d3 u(z) dz 3 f(x) + d2 f(x) du(z) ) = 0 (52) dx 2 dz Again it is difficult to solve the equations analytically, we can use iterative perturbation method to estimate. Let s consider the zeroth approximation as: f(x) = Sin(k 1 x) (53) After some algebra, we can write: (2a 12 dt(z) dz (2a 1 dt(z) dz g 0 d 3 t(z) dz 3 u(z)2 +4a 12 t(z)u(z) du(z) dz )h(x) + 12a 11t(z) 2 dt(z) dz h(x)3 + )Sin2 (k 1 x)h(x) g 0 dt(z) dz d 2 h(x) dx 2 d = (2a 3 k 1 u(z) g 2 u(z) 0 k dz g 0 u(z)k a 11 k 1 u(z) 3 ) Cos(k 1 x) 12a 11 k 1 u(z) 3 Cos 3 (k 1 x) + 2a 12 u(z)t(z) 2 Cos(k 1 x)h(x) 2 + 4a 12 u(z)t(z) 2 Sin(k 1 x)h(x) dh(x) dx (54) [(2a 1 ε 0 k )t(z) g 0 ε 0 k 11 d 2 t(z) dz 2 ] dh(x) dx + 12ε 0k 11 a 11 t(z) 3 2 dh(x) h(x) + dx 2ε 0 k 11 a 12 t(z)u(z) 2 Sin 2 (k 1 x) dh(x) + 4ε dx 0k 1 k 11 a 12 t(z)u(z) 2 Sin(k 1 x)cos(k 1 x)h(x) g 0 ε 0 k 11 t(z) d3 h(x) dx 3 + [(2a 3 ε 0 k ) du(z) d g dz 0 ε 0 k 3 u(z) 33 + dz 3 19

20 g 0 ε 0 k 33 k 1 2 du(z) dz ]Sin(k 1x) + 12ε 0 k 33 a 33 u(z) 2 du(z) dz Sin3 (k 1 x) + (2ε 0 k 33 a 12 du(z) dz t(z)2 + 4ε 0 k 33 a 12 u(z)t(z) dt(z) dz )Sin(k 1x)h(x) 2 = 0 (55) We can find that h(x) has the following form: h(x) = Cos(k 1 x) + γ 1 Cos(3k 1 x) + H. O. T (56) where H.O.T represent the high order terms. Plug in (56) to (54) and (55) and neglect the high order terms, we can get the first order estimation: f(x) = Sin(k 1 x) + γ 2 Sin(3k 1 x) + H. O. T (57) Similarly, the z-components can be estimated: u(z) = P 0 (Sin(k 2 z) + γ 3 Sin(3k 2 z) + H. O. T) (58) t(z) = P 0 (Cos(k 2 z) + γ 4 Cos(3k 2 z) + H. O. T) (59) Plug the solutions back to (54), and subtract the first order terms, one can find: ( 2a a 11P a 12P g 0 (k k 2 2 ))k 2 P 0 = ( 2a a 33P a 12P g 0 (k k 2 2 ))k 1 P 0 (60) Now, let s go back and expand the Landau terms to six order, solving equations (21)-(24) together, one can get self-consistent solutions: h(x) = Cos(k 1 x) + γ 1 Cos(3k 1 x) + γ 1 Cos(5k 1 x) + H. O. T f(x) = Sin(k 1 x) + γ 2 Sin(3k 1 x) + γ 2 Sin(5k 1 x) + H. O. T u(z) = P 0 (Sin(k 2 z) + γ 3 Sin(3k 2 z) + γ 3 Sin(5k 2 z) + H. O. T) t(z) = P 0 (Cos(k 2 z) + γ 4 Cos(3k 2 z) + γ 4 Cos(5k 2 z) + H. O. T) (61) Plug in to equations (21)-(24), subtract only the first order terms: 20

21 [2a a 11P a 12P a 111P a 112(P P 0 2 P 0 2 ) + g 0 (k k 2 2 )]k 2 P 0 = [ 2a a 33P a 12P a 111P a 112(P P 0 2 P 0 2 ) + g 0 (k k 2 2 )]k 1 P 0 (62) Similarly, in the case where the substrate strain is small, one can find: g 0 (k k 2 2 ) ( a 1 + a (a 11P a 33 P 0 2 ) a 12(P P 0 2 ) a 111(P P 0 4 ) a 112(P P P 0 2 P 0 2 )) (63) In the vortex phase, P 0 P 0, If we were to define: a 1 = 1 2 (a 1 + a 3 ) + ( 9 16 a a a 12)P ( a a 112)P 0 4 (64) Equation 63 can be rewritten as: g 0 (k k 2 2 ) 2a 1 (65) Applying conditions (40), (41), Equation (65) can be expressed as: If we were to write: 2a 1 = g 0 ( π2 d 2 + π2 a f 2) (66) a 0 2 = g 0 2a 1 π2 Where a 0 is the characteristic length related to the gradient energy coefficients and the modified Landau coefficients. While the parameter g 0 a 1 is related to δ, δ is the width of 180 o domain wall [14]. At room temperature, for short period PTO/STO superlattices, if the spontaneous polarization is estimated to be ~0.5 C/m 2, using the parameters given by Li et al. [4], one can estimate the characteristic length to be ~ (67)

22 nm. Equation 66 can be simplified as: 1 d a f 2 = 1 a 0 2 (68) Substitute d/af with the aspect ratio r, and multiply equation 68 by af 2 at both sides, Equation 4 in the main article can be deduced: = a f r 2 a2 (69) 0 5. Synthesis and Characterization of superlattice thin films a. Synthesis: Laser-MBE Superlattices of (SrTiO3)n/(PbTiO3)n (shorthand: n/n) were synthesized on DyScO3 (110)o substrates via RHEED-assisted pulsed laser deposition (PLD) method, where n refers to the thickness in unit cells of respective materials oriented in the [001]pc (psuedocubic) direction. Superlattice films of 100nm thick were grown at a set temperature of 630 C and chamber pressure of 100 mtorr. While the growth of PbTiO3 was achieved using a lead rich polycrystalline target with composition Pb1.2TiO3, growth of SrTiO3 targets was done using a single crystalline target. For the growth of superlattice films, a buffer layer of SrRuO3 was used which was grown at set temperature of 700 C and growth pressure of 50 mtorr. A laser fluence of 1.5 J/cm 2 was used on all three targets. The growth of PbTiO3 and SrTiO3 was monitored using RHEED (Reflection High Energy Electron Diffraction) in-situ characterization technique, with Frank-van der Merwe growth mode present throughout the growth of 100nm thick superlattice film. During the growth, the RHEED oscillations were monitored along the [1 1 0] direction of the DyScO3 (110)o substrate for the specular 22

23 spot on the RHEED pattern. After growth, films were cooled down at an oxygen pressure of 50 Torr to ensure full oxidation of the samples. b. Structural characterization i. X-ray Diffraction (XRD) All superlattice films were characterized by X-ray diffraction using a PANalytical X Pert Pro diffraction machine which uses Cu Kα radiation. Symmetric XRD scans were first performed to confirm the periodicity of superlattice structures as characterized by in-situ RHEED during growth. Reciprocal space maps studies were also performed to ensure that superlattice films are strained to the substrate. ii. Transmission Electron Microscopy (TEM) Atomic resolution High-Angle Annular Dark-field (HAADF) STEM images of superlattice films were obtained using TEAM 0.5 electron microscope located at National Center for Electron Microscopy, Lawrence Berkeley National Laboratory. TEAM 0.5 is an aberration-corrected electron microscope, and it uses commercial TITAN electron microscope manufactured by FEI. Electron transparent TEM samples were prepared using mechanical polishing with a 0.5 wedge for cross-sectional TEM. TEAM 0.5 microscope operated at a voltage of 300kV was used to perform scanning TEM, and High-angle annular detector was employed to obtain atomic-resolution Z- contrast images. The polar displacement mapping on these atomic resolution STEM images was obtained using algorithm detained elsewhere [15, 16]. 23

24 c. TEM verification of the simulated phase diagram Figure S6 HRTEM polar mapping of the PTO/STO superlattices with different periodicities. (a) 4/4 with inplane polarization; (b) 10/10 showing the vortex configuration; (c) 16/16 transition from vortex to flux-closure, elongation of the vortex core and decreasing of the aspect ratio; (d) 50/50 showing the flux-closure domain. This study confirms the phase diagram in figure 3 of main article. In order to further verify the simulated phase diagram, HRTEM displacement vector plot is shown in Figure S6. It is revealed that at lower periodicity (i.e., 4/4, meaning superlattice consisting of 4 layers of PTO and 4 layers of STO), the inplane polarization is observed, confirmed the ferroelastic domain configuration by planar TEM shown in the figure 3 of main text (Figure S6a). Further increasing of the periodicity, i.e., 10/10, shows the continuous rotation of polarization (polar vortex), as depicted in both this paper and Ajay et al. [17], which has been thoroughly studied in the main text of this paper. With a higher periodicity, for instance 16/16, elongation of the vortex core to a rode-like configuration with decreasing aspect ratio is clearly shown 24

25 (Figure S6c). This can be viewed as a transition point from vortex to flux-closure state. And eventually, at large periodicity (50/50), the flux-closure domain with distinct domain wall is given in Figure S6d. The above observation eloquently confirmed the phase diagram (Figure 3 of main article) predicted by phase-field simulations. d. Calculated phases for the (PTO)10/(STO)m (m=2-100) superlattices (a) (c) (e) (g) (b) (d) (f) (h) Figure S7 Phase diagram for (PTO) 10/(STO) m. m=2, planar view (a) and cross-section view (b), a/c domain; m=3, planar view (c) and cross-section view (d), mixed a/c and vortex; m=20, planar view (e) and cross-section view (f), mixed a and vortex; m=100, planar view (g) and cross-section view (h), a 1/a 2; The (PTO)10/(STO)m (m=2-100) phase diagram is calculated by the phase-field simulation. As shown in Figure S7(a) and (b), when STO layer is thin (i.e., only two unit cells), it is highly polarized by the PTO layers. The polarization in the STO layer can be as large as 0.3 C/m 2, which is comparable to BaTiO3, the large polarization in the STO layer is also observed by the recent experimental studies in ultrathin STO film 25

26 [18]. With increasing STO layer thickness, the depolarization strength increases which decrease the polarization in the STO layer as indicated in the main text. This leads to the formation of vortices for m>=3, as shown in Figure S7 (c) and (d). Further increasing of the STO layer thickness leads to the mixed phase of vortices and a domains, as clearly shown in Figure S7(e) and (f), with the increasing of vortex ratio. Eventually, the polarization of the STO layer drops to almost zero, which makes outof-plane polarization highly unfavorable, as a result, pure a1/a2 twin domains form despite the large elastic penalty (Figure S7g and S7h). To further reveal the phase transition with different STO layer thickness, we calculate the energies accompanying the transitions (Figure S8). The total energy of the PTO layer drops with increasing STO layer due to the decreasing of the electric energy in the PTO layer, while a reverse trend is observed in the STO layer since the electric energy increases rapidly with smaller internal field (Figure S8a). An energy plateau is formed where the vortex ratio is large. The elastic energy shows reverse behavior as the electric energy, showing the competition of this two energies towards the formation of vortex state (Figure S8b). The gradient energy has a big jump when the vortex forms at the intermediate periodicity (Figure S8c). The Landau energy decreases for STO with increasing STO thickness since the polar state is unfavorable for bulk STO, while the decreasing of the polar magnitude in PTO could leads to higher Landau energy in this layer (Figure S8d). The energy consideration could not only help the explanation of the phase transition phenomenon, but also helpful in the further design of new materials with novel topological structures. 26

27 (a) (b) (c) (d) Figure S8 Energies for (PTO) 10/(STO) m. (a)average total energy of the two layers as a function of n; (b)-(d) shows the elastic, gradient and Landau energies, respectively. References [1] Chen, L. J. Am. Ceram. Soc. 2008, 91, [2] Xue, F.; Wang, J.; Sheng, G.; Huang, E.; Cao, Y.; Huang, H.; Munroe, P.; Mahjoub, R.; Li, Y.; Nagarajan, V.; Chen, L. Acta Mater.2013, 61(8), [3] Wang, J.; Ma, X.; Li, Q.; Britson, J.; Chen, L. Acta Mater. 2013, 61(20), [4] Li, Y.; Hu, S.; Liu, Z.; Chen, L. Acta Mater. 2002, 50, [5] Li, Y.; Hu, S.; Liu, Z.; Chen, L. Appl. Phys. Lett. 2002, 81, 427. [6] Haun, M.; Furman, E.; Jiang, S.; Mckinstry, H.; Cross, L. J. Appl. Phys. 1987, 62, [7] Sheng, G.; Li, Y.; Zhang, J.; Choudhury, S.; Jia, Q.; Gopalan, V.; Schlom, D.; Liu Z.; Chen, L. Appl. Phys. Lett. 2010, 96, [8] Chen, Z.; Damodaran, A.; Xu, R.; Lee, S.; Martin, L. Appl. Phys. Lett. 2014, 104, [9] Tagantsev, A. Ferroelectrics 2008, 375,

28 [10] Tagantsev, A. Ferroelectrics 1986, 69, [11] Zheng, Y.; Woo, C. Nanotechnology 2009, 20, [12] Tang, Y.; Zhu, Y.; Ma, X.; Borisevich, A.; Morozovska, A.; Eliseev, E.; Wang, W.; Wang, Y.; Xu, Y.; Zhang, Z.; Pennycook, S. Science 2015, 348, [13] Gu, Y.; Rabe, K.; Bousquet, E.; Gopalan V.; Chen, L. Phys. Rev.B 2012, 85, [14] Landau L.; Lifshitz, E. Electrodynamics of continuum media; Pergamon Press: Oxford, [15] Ophus, C.; Nelson, C.; Ciston, J. Ultramicroscopy 2016, 162, 1 9. [16] Nelson, C.; Winchester, B.; Zhang, Y.; Kim, S.; Melville, A.; Adamo, C.; Folkman, C.; Baek, S.; Eom, C.; Schlom, D.; Chen, L.; Pan, X. Nano Lett. 2011,11(2), [17] Yadav, A.; Nelson, C.; Hsu, S.; Hong, Z.; Clarkson, J.; Schlepüetz, C.; Damodaran, A.; Shafer, P.; Arenholz, E.; Dedon, L.; Chen, D.; Vishwanath, A.; Minor, A.; Chen, L.; Scott, J.; Martin L.; Ramesh, R. Nature (London)2016, 530, [18] Lee, D.; Lu, H.; Gu, Y.; Choi, S.; Li, S.; Ryu, S.; Paudel, T.; Song, K.; Mikheev, E.; Lee, S.; Stemmer, S.; Tenne, D.; Oh, S.; Tsymbal, E.; Wu, X.; Chen, L; Gruverman, A.; Eom, C. Science 2015, 349,