High-Symmetry Polarization Domains in Low-Symmetry Ferroelectrics

Nov 24, 2014 - We present experimental evidence for polygonal domain faceting in the ferroelectric polymer poly(vinylidene fluoride-trifluoroethylene)...
3 downloads 7 Views 2MB Size
Download PDF
Letter pubs.acs.org/NanoLett

High-Symmetry Polarization Domains in Low-Symmetry Ferroelectrics I. Lukyanchuk,*,†,‡ P. Sharma,§ T. Nakajima,∥ S. Okamura,∥ J. F. Scott,⊥ and A. Gruverman*,§ †

Laboratory of Condensed Matter Physics, University of Picardie, Amiens, 80000, France Landau Institute for Theoretical Physics, Moscow, 119334, Russia § Department of Physics and Astronomy, Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, United States ∥ Department of Applied Physics, Tokyo University of Science, Tokyo, 125-8585, Japan ⊥ Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Ave, Cambridge CB3 0HE, United Kingdom ‡

S Supporting Information *

ABSTRACT: We present experimental evidence for polygonal domain faceting in the ferroelectric polymer poly(vinylidene fluoride-trifluoroethylene) (PVDF-TrFE) films with the lower orthorhombic crystallographic symmetry. It is proposed that this effect can arise from purely electrostatic depolarizing forces. We show that, in contrast to magnetic bubble shape domains, where such type of deformation instability has a predominantly elliptical character, the emergence of more symmetrical circular harmonics is favored in ferroelectrics with high dielectric constants.

KEYWORDS: Ferroelectric polymers, domains, faceting, piezoresponse force microscopy

I

symmetry axis in quartz irradiated simultaneously by e-beams and blue laser light.7,8 Brown and Hollingsworth reported domain structures in thiourea compounds with 12-fold symmetry in spite of the fact that the crystals had only 2-fold lattice symmetry.9 A peculiar kind of hexagonal crystallization has been observed in sodium dodecyl sulfate surfactant,10 but there is only a rather general understanding of this phenomenon. Rules, known as Plateau’s laws, determine the formation of hexagonal facets in foams.10 In the isolated films, domain wall orientation becomes a criterion instead of macroscopic space-filling. It is also important to note that typically hexagonal faceting is observed, while pentagons or heptagons are rarely seen. In this work, we present new data on polygonal domain formation in the ferroelectric copolymer of poly(vinylidene fluoride-trifluoroethylene) (PVDF-TrFE) that lacks any high lattice symmetry. We interpret our data in terms of the electrostatic depolarization instability theory developed by Thiele11 to depict the elliptical deformation of small circular domain patterns in ferromagnetic bubble-memory devices.12 A similar elliptic instability of polarization domains was observed

n nature, low-symmetry molecules often self-assemble to produce high-symmetry mesoscopic arrays or bundles of colloids, nanoparticles, proteins, and viruses.1 A similar kind of mesoscopic self-assembly occurs for domain pattern formation in ferroelectric crystals. However, in most cases, the symmetry of such domains is either the same or lower than the crystal symmetry and is fixed in space by the crystallographic axes. On the other hand, there are reports where domain faceting has a greater symmetry than that of the lattice or even incompatible with it. Triangular and hexagonal domains have been observed in lead germanate Pb5Ge3O112 and lithium niobate LiNbO33 crystals with 3-fold polar axes. Relaxation of initially circular domains into pentagonal and hexagonal ones was also reported in [111]-oriented lead zirconate-titanate Pb(Zr,Ti)O3 films.4 In some cases these effects occur under external perturbation, such as laser or e-beam irradiation. A 6fold pattern in laser-irradiated ferroelectric of PbMg1/3Nb2/3O3 with a purely cubic symmetry was observed by Scott et al.5,6 However, it has not been completely clear whether this is driven by beam heating and the thermal conductivity anisotropy of the target or by charging and depolarization fields. In this context, it is very important to compare faceting under TEM with faceting observed in piezoresponse force microscopy (PFM), since the latter does not involve the same degree of sample heating. In earlier studies, Schwarz and Hora reported the 6-fold diffraction patterns appearing along a 2-fold © XXXX American Chemical Society

Received: August 9, 2014 Revised: October 25, 2014

A

dx.doi.org/10.1021/nl503070f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

also in lipid monolayers.13 However, unlike magnetic bubbles, ferroelectric PVDF-TrFE favors the polygonal faceting instability. Our approach is similar to the recent analysis of the shape of the pinned domains in the lead zirconate titanate (PZT) films,14 based on the Landauer energy approach.15 However, to study the observed faceting instability, we additionally take into account the energy of domain edge fringing field, which was first proposed by Thiele.11 PVDF-TrFE is the best-known ferroelectric polymer widely used in device applications due to its relatively high polarization and piezoelectric coefficients.16 Discovery of ferroelectricity in ultrathin PVDF-TrFE films17 and fabrication of highly ordered nanomesa arrays for nonvolatile memories18 draw additional attention to a possibility of integration of these materials into all-organic electronic systems. It remains a model material for investigation of the mechanism of polarization reversal in organic ferroelectrics. Note that, because of the low symmetry of the order parameter, the domain walls present the simple Ising-like profile. In principle, a more complicated structure, typical for the high-symmetry ferroelectrics,19 can affect the domain wall energy, but it will not substantially change our consideration. The PVDF-TrFE (75/25) copolymer films have been fabricated by spin-coating the VDF/TrFE solution dissolved in diethyl carbonate (3 wt %) on the (111) Pt-sputtered (150 nm) on Ti(5 nm)/SiO2(200 nm)/Si substrate followed by annealing at 120 °C.20 The thickness of the PVDF-TrFE film is 50 nm. X-ray θ−2θ scans showed a clear peak at 19.7°, which corresponded to (110/200) d-spacing of the all-trans ferroelectric β-phase.21 The (110/200) peak points out to the highly textured PVDF-TrFE molecular chains lying parallel to the substrate. A commercial atomic force microscope (Asylum MFP-3D) has been used to generate and visualize the domain structure in the piezoresponse force microscopy (PFM) mode. Rectangular Si cantilevers with a spring constant of 0.3 N/m and the Ti−Pt coated tips with an apex curvature radius of about 20 nm have been used in these studies. Domain visualization has been performed by applying a 1.5 V modulating voltage in the 200− 400 kHz frequency range. The P−E hysteresis loops have been recorded by a ferroelectric test system (Precision LC, Radiant Technologies) from the Au top electrodes deposited by thermal evaporation. Ferroelectric polarization in PVDF-TrFE arises from alignment of molecular dipoles formed by electropositive hydrogen and electronegative fluorine atoms. The all-trans (β-phase) molecular conformation results in the dipoles being oriented nearly perpendicular to the chain axis, and polarization reversal is associated with rotation of these dipoles about the molecular chains. In the crystalline β-phase of highly textured PVDFTrFE, the molecules tend to pack parallel to each other forming a structure with the orthorhombic symmetry (Cm2m (C14 2v ) space group) with the 2-fold polar axis nearly perpendicular to the film plane. In Figure 1a, it can be seen that the molecular chains are aligned in parallel rows, forming a quasi-hexagonal close-packing structure.22 Preliminary PFM testing showed that the as-grown PVDFTrFE films were uniformly polarized downward. Polarization loops were shifted horizontally toward the negative voltage (Figure 1b), suggesting a presence of an internal built-in electric field oriented toward the substrate. This field could be estimated from the polarization loop asymmetry as E = (|V+c | − |V−c |)/h ≈ 105 V/cm, where V−c = −5.2 V and V+c = 4.1 V are

Figure 1. (a) An atomically resolved STM image of the surface of PVDF-TrFE film at room temperature (reprinted with permission from ref 22). (b) Polarization hysteresis loop of PVDF-TrFE film. (c) PFM image of the domain written by a −55 V with the aspect ratio r/h ≈ 4.3, 1 s pulse applied by a PFM tip. (d) PFM image of the same domain 15 min after pulse application. (e) Pentagonal and (f) rectangular domains developing during relaxation of smaller written domains with r/h ≈ 3.6 and r/h ≈ 2.7, correspondingly. The regular dashed-line polygons are sketched to guide the eye.

negative and positive coercive voltages, respectively, and h ≈ 50 nm is the film thickness. The dielectric permittivity of the film was taken as ε ≈ 15.16 Figure 1c,d illustrates the domain faceting effect observed in these films. Application of a voltage pulse in the range from −15 V to −60 V to the PFM tip results in a single circular domain with radius r of about 50−250 nm (Figure 1c) with the polarization oriented against the imprint. After the field is off, the domain starts to slowly relax back, acquiring a hexagonal shape during the process (Figure 1d). Importantly though is that the written domains of smaller sizes relax into the pentagonal (with r ≈ 180 nm) or even rectangular (with r ≈ 130 nm) shapes (Figure 1e and f). Contours of these polygonal domains are shown in the Supporting Information (Figure S2) for a more clear illustration of the domain shapes. We show that both the domain contraction and tendency to expand its perimeter via faceting instability are due to the purely electrostatic force balance. It is known that the depolarizing effect of surface charges due to discontinuity of polarization at the interface plays a critical role in the formation and dynamics of ferroelectric domains. However, the foreseeable periodic Landau−Kittel domain structure23,24 with expected half-period d ≈ (hξ0)1/2 ≈ 10 nm (where ξ0 ≈ 2 nm is the coherence length) was not observed in these PVDFTrFE films. Most probably, this is due to the fact that the depolarization field is initially screened by accumulation of the charged species at the top surface and by the charge carriers of the conductive electrode at the bottom interface of the film. During application of the switching pulse by the PFM tip, the surface screening charges in the vicinity of the tip are dispersed, activating the depolarization forces in the area of the newly generated domain. Subjected to electrostatic tension, which could also stem from the internal built-in field, the switched domain starts to relax back to its equilibrium state until it is B

dx.doi.org/10.1021/nl503070f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

stabilized by the redistributed surface screening charges that eliminate the driving electric field. To account for the forces, acting on the generated domain just after its creation and to reveal the faceting instability, we, following ref 11, distinguish three contributions to the energy Wr of cylindrical domain: Wr = Wdw + WPE + Wd

Wdε = −

(2)

which is determined by DW surface tension σdw. (ii) WPE is the energy of polarization interaction with the electrical field E, which in our case of the relaxing domain is the internal built-in imprint field:

∞

R (θ ) = r +

8r e

∑ (Δrk)cos kθ k=2

h

(6)

Transformation 5 is valid for a single domain in a freestanding ferroelectric film. To account for the influence of conducting substrate, one should apply the mirror boundary condition, provided by one more transformation Wd(h) → (1/ 2)Wd(2h) that we shall use at the end. The equilibrium domain radius is given by the minimum of the energy (eq 1). However, with DW surface tension σdw = ε−1(4πP2) ξ0 ≈ 2 erg/cm2 and ξ0 ≈ 2 nm,28−30 the Wdw is at least of ξ0/h ≈ 25 times smaller than two other contributions WPE and Wdε (unlike in magnetic domains where ξ0 ∼ 50 nm) and can be neglected in electrostatic calculations. A closer look at eqs 3 and 6 reveals that this minimum lies well below the actual radius, at r ≪ h, and is comparable to the Kittel domain width d ≈ 10 nm. This explains the tendency of the newly created domain to relax via viscous contraction. To take into consideration the possibility of shape-breaking instability during the relaxation process, we parametrize the domain boundary in polar coordinates as R = R(θ) and, following ref 11, account its deviation from the cylinder with R = r by emergence of circular harmonics:

(3)

1/2

n=1

ε−1 ε+1

⎡1 c ⎤ 8r Wdε ≈ −(2πr )h2⎢ 2 ln 1/2 + ε ⎥(2P)2 ⎣ε ε⎦ e h

This contribution is proportional to the domain volume πr2h, and factor 2 is introduced because of the domain repolarization. (iii) Wd is the depolarization energy, which is determined by the surface charge of density σ = ±P due to polarization termination at the film surface. To outline how this contribution drives the domain shape instability, we note that Wd is nothing else but the energy excess of the cylindrical capacitor with the surface charge Q = ±(2P)πr2, which is oppositely polarized with respect to the original ferroelectric slab. The capacitance of such a finite-size capacitor with ε = 1 was first calculated by Kirchhoff in 187725 in the limiting case of r ≫ h and two years later generalized by Lorentz26 in terms of the elliptic function for the arbitrary aspect ratio h/r. In late of 1960s, Thiele performed similar calculations for the energy of the cylindrical magnetic domain,11 but again with ε = 1.27 Note that the edge-fringing energy for ferroelectric domains was also calculated in ref 27 for a particular case when ε inside and outside the film were comparable. We summarize their results for the case of r ≫ h as Wd1 ≈ −(2πr )h2(2P)2 ln

β=

(5)

where (i) Wdw is the domain wall (DW) energy

WPE = 2(πr 2h)PE

∞

∑ (−β)n W1d(nh)

where Wd1(h) = 2q2[r−1 − (h2 + ρ2)−1/2] is the energy of dipolar interaction in vacuum at ε = 1 (see SI for details). Since each domain can be considered as an ensemble of interacting parallel dipoles, eq 5, according to principle of superposition, can also be used to find the depolarization energy Wdε of such a domain for an arbitrary value of ε if its energy Wd1 is known for ε = 1 (see also SI). 2 n 2 3 Using summations ∑∞ n=1n (−β) = −((ε − 1)/4ε ) and cε = n 2 (4ε2/(ε2 − 1))∑∞ (−β) n ln n ≈ 0.84 (ε ≫ 1), we transform n=1 eq 4 to

(1)

Wdw ≈ (2πrh)σdw

4ε 2 ε2 − 1

(4)

(7)

with small amplitudes |Δrk| ≪ r. To address the question of domain stability under perturbation 7 we, following ref 11, expand the variation of the energy as

where e ≅ 2.71 and the index “one” means ε = 1.27 Expression 4 is similar to the well-known energy of the edge fringing field of the finite-size capacitor.25,26 Importantly, the negative sign of Wd1 is explained by the repulsion of the edgeforming dipoles. Caused by the long-range electrostatic forces, this term is not simply proportional to the DW surface, but also contains the nonlocal logarithmic correction, dependent on the integral domain shape. Acting against WPE, which minimizes the domain volume, it determines the tendency of DW to increase its surface by either faceting or roughening deformations. The size and shape of domain is, therefore, the result of a balance between contributions of different terms in eq 1. The comprehensive analysis of possible domain instabilities has been carried out by Thiele for ferromagnetic materials, which are characterized by the almost uniform permittivity.11 To apply these results to ferroelectric films, one has to find out how Thiele’s approach can be generalized for the case of the films with arbitrary dielectric constant ε > 1. Let us consider two dipoles with charges ±q, located across the slab with permittivity ε and thickness h and separated by a distance ρ. Their interaction energy can be calculated as

W = Wr +

1 2

∞

″ (k) + W PE ″ + Wd″(k)](Δrk)2 ∑ [Wdw k=2

(8)

π hσdwk 2 r

(9)

where ″ (k ) = Wdw

is the energy variation due DW surface energy, ″ = W PE

2 1 ∂ WPE = 2πhPE 2 ∂r 2

(10)

is the harmonics interaction with imprint field and Wd″(k) is the depolarization field contribution (see also SI for details). For ε = 1 and in the limit r ≫ h, this term was calculated in ref 11 as ″ (k) ≈ −π (2P)2 Wd1 C

h2 2 8er k ln r kh

(11)

dx.doi.org/10.1021/nl503070f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

high-k electrostatic instability, can favor selection of those surfaces that are most close to the observed polygonal faceting. The dynamic buckling effects during domain contraction can also favor the particular (hexagonal) shape-breaking as was shown, for example, for the case of the relaxing drops in dipolar ferrofluids.32 To conclude, we have discovered polygonal faceting of domains in ferroelectric polymer PVDF-TrFE films, which have the lower orthorhombic symmetry. We interpret this effect in terms of the electrostatic depolarization instability originally used by Thiele for magnetic bubble domains. Unlike magnetic bubbles with the k = 2 elliptical instability, ferroelectrics favor the instability with higher-k modes. The described approach can be applied to a wider range of instabilities in quite unrelated materials, including 6-fold faceting in surfactants10 in thiourea inclusion compounds with k = 129 and in multiple domain state in LiNbO3.33

A cylindrical domain is unstable with respect to axial symmetry-breaking modes if the corresponding coefficients in eq 8 are negative: ″ (k) + W PE ″ + Wd″(k) < 0 Wdw

(12)

The instability comes from the negative logarithmic term in eq 11. However, to explore the instability condition (eq 12), we should generalize the expression 11 for arbitrary ε > 1 using the transformation 5. Summing the log-terms in lowest in h/r order, we obtain Wd″ε(k) = −

c ⎤ 8er πh2 ⎡ 1 + ε ⎥(2P)2 k 2 ⎢⎣ 2 ln r ε kh ε⎦ −1

(13)

Comparison of eq 9 (with σdw = ε (4πP )ξ0) and eq 13 shows that the DW surface energy gives only a small correction πξ0/h ≈ 0.1 to cε and, therefore, can be again neglected. Now, using the transformation W″dε(k,h) → (1/2)W″dε(k,2h), we include the effect of the bottom electrode and arrive at the instability condition: ⎡1 ⎤ εE r 8er < 2k 2⎢ ln + cε ⎥ ⎣ ε k(2h) ⎦ P (2h)

2

■

ASSOCIATED CONTENT

S Supporting Information *

Technical derivation of dipolar interaction energy and correspondence of our results to Thiele’s calculations for magnetic domains. This material is available free of charge via the Internet at http://pubs.acs.org.

(14)

For ε ≫ 1, we can neglect the log term and reduce condition 14 to simpler form: (εE/P)(r/h) < 3.4k2. Graphic analysis shown in Figure 2 reveals that the shapebreaking instability develops first for the high k-modes, which

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS I.L. thanks Prof. Zapolsky for fruitful discussion of faceting problems and acknowledges support by the FP7-ITN-NOTEDEV project.

■

REFERENCES

(1) Glotzer, S. C. Science 2012, 337, 453. (2) Shur, V. Ya.; Gruverman, A.; Letuchev, V. V.; et al. Ferroelectrics 1989, 98, 29. (3) Scrymgeour, D. A.; Gopalan, V.; Itagi, A.; Saxena, A.; Swart, P. J. Phys. Rev. B 2005, 71, 184110. (4) Ganpule, C. S.; Roytburd, A. L.; Nagarajan, V.; Hill, B. K.; Ogale, S. B.; Williams, E. D.; Ramesh, R.; Scott, J. F. Phys. Rev. B 2001, 65, 014101. (5) Scott, J. F.; O’Sullivan, R. A. Nature 1996, 382, 305. (6) Scott, J. F.; O’Sullivan, R. A.; Reich, M. H. Physica A 1996, 233, 655. (7) Schwarz, H.; Hora, H. Appl. Phys. Lett. 1969, 15, 349. (8) Schwarz, H. Nature 1970, 225, 1173. (9) Brown, M. E.; Hollingsworth, M. D. Nature 1995, 376, 323. (10) Flesselles, J.-M.; Magnasco, M. O.; Libchaber, A. Phys. Rev. Lett. 1991, 67, 2489. (11) Thiele, A. A. Bell System Techn. J. 1969, 48, 3287; ibid. 1971, 50, 711. (12) Malozemoff, A. P.; Slonczewski, J. C. Magnetic domain walls in bubble materials; Academic Press: London, 1979. (13) Keller, D. J.; Korb, J. P.; McConnell, H. M. J. Phys. Chem. 1987, 91, 6417. (14) Ivry, Y.; Durkan, C.; Chu, D.; Scott, J. F. Adv. Funct. Mater. 2014, 24, 2267. (15) Landauer, R. J. Appl. Phys. 1957, 28, 227. (16) Lovinger, A. Science 1983, 220, 1115.

Figure 2. Graphical solution of instability condition 14 for PVDF film with ε ≈ 15,16 E/P ≈ 1.6 as a function of the aspect ratio r/h. Values of k are the instability harmonic numbers, corresponded to the domain shape, presented by pictograms. Black points correspond to the domains presented in Figure 1d−f. Dotted lines present the approximate solutions for high ε.

have a higher symmetry than the crystal background. Notably, for domains with initial aspect ratios 4.3, 3.6, and 2.7, the hexagonal, pentagonal, and quadratic faceting instabilities (with k = 6, 5, and 4 correspondingly) can occur, whereas the elliptic deformation with k = 2, compatible with 2-fold symmetry of PVDF-TrFE are electrostatically protected. However, to know which of the electrostatically allowed symmetry-breaking modes is unstable during domain relaxation, we should go beyond the linear-stability analysis, since several factors are coming into the play. For instance, the DW pinning in the atomic planes shown in Figure 1a can be critical for the mode selection. Minimization of the anisotropic pinning energy, known as Wulff construction,31 acting along with the D

dx.doi.org/10.1021/nl503070f | Nano Lett. XXXX, XXX, XXX−XXX

Nano Letters

Letter

(17) Ducharme, S.; Fridkin, V. M.; Bune, A. V.; et al. Phys. Rev. Lett. 2000, 84, 175. (18) Hu, Z.; Tian, M.; Nysten, B.; Jonas, A. M. Nat. Mater. 2009, 8, 62. (19) Toledano, P.; Guennou, M.; Kreisel, J. Phys. Rev. B 2014, 89, 134104 and references therein.. (20) Mabuchi, Y.; Nakajima, T.; Chaudhuri, K.; Evans, J. E.; Browning, N. D.; Okamura, S. Jpn. J. Appl. Phys. 2011, 50, 09NA05. (21) Bai, M.; Ducharme, S. Appl. Phys. Lett. 2004, 85, 3528. (22) Cai, L.; Qu, H.; Lu, C.; Ducharme, S.; Dowben, P. A.; Zhang, J. Phys. Rev. B 2004, 70, 155411. (23) Landau, L.; Lifshitz, E. Phys. Z. Sowjet. 1935, 8, 153. (24) Kittel, C. Phys. Rev. 1946, 70, 965. (25) Kirchhoff, G. M. Deutsch. Akad. Wiss. Berlin 1877, 144. See also Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Elsevier: New York, 1985. (26) Lorentz, L. Wied. Ann. 1879, 7, 161. (27) Bratkovsky, A. M.; Levanyuk, A. P. Phys. Rev. Lett. 2000, 85, 4614. (28) Lines, M. E.; Glass, A. M. Principles and Applications of Ferroelectrics and Related Materials; Oxford University Press: New York, 1977. (29) Stephanovich, V. A.; Luk’yanchuk, I. A.; Karkut, M. G. Phys. Rev. Lett. 2005, 94, 047601. (30) De Guerville, F.; Luk’yanchuk, I.; Lahoche, L.; El Marssi, M. Mater. Sci. Eng. B 2005, 120, 16. (31) Wulff, G. Zeit. Krystallogr. Mineral. 1901, 34, 449. (32) Langer, S. A.; Goldstein, R. E.; Jackson, D. P. Phys. Rev. A 1992, 46, 4894. (33) Ievlev, A. V.; Jesse, S.; Morozovska, A. N.; et al. Nat. Phys. 2014, 10, 59.

E

dx.doi.org/10.1021/nl503070f | Nano Lett. XXXX, XXX, XXX−XXX