Computational Materials Science 108 (2015) 301–308

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

The 90� domain splitting and electromechanical behaviorsin ferroelectric thin films with triangle anti-dot array

http://dx.doi.org/10.1016/j.commatsci.2015.03.0050927-0256/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Laboratory of Solid State Microstructures andInnovation Center of Advanced Microstructures, Nanjing University, Nanjing210093, China.

E-mail address: liujm@nju.edu.cn (J.-M. Liu).

Y. Zhang a, Y.L. Wang a, P. Chu a, Y.L. Xie a, Z.B. Yan a, J.-M. Liu a,b,⇑a Laboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, Chinab Institute for Quantum Materials, Hubei Polytechnic University, Huangshi 435000, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 December 2014Received in revised form 2 March 2015Accepted 5 March 2015Available online 25 March 2015

Keywords:Ferroelectric thin filmsAnti-dot arrayPiezoelectricity

The 90�-domain structure in ferroelectrics of tetragonal symmetry is one of the core ingredients deter-mining the ferroelectric and electromechanical performances. We perform Monte Carlo simulations onthe splitting of the 90� twin-like domains in ferroelectric thin films with differently oriented triangleanti-dot arrays. It is revealed that the domain splitting is substantially dependent of the array orientation,and thus the piezoelectric and dielectric responses are the orientation-dependent too. The mechanism forthe domain structure evolution is the competition between the dipole–dipole interaction and electrome-chanical energy, similar to the cases of square anti-dot arrays. The present work suggests the role ofgeometry and symmetry of the anti-dot array in modulating the electromechanical behaviors of 90�-domained ferroelectric thin films.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction

Ferroelectric (FE) materials have piezoelectricity and thus elec-tromechanical coupling is one of the core ingredients of physics forFE domain structure [1]. A full understanding of the physics hasbeen of fundamental interest and useful for guiding domain struc-ture control for diverse applications [2–5]. We consider the mostconcerned tetragonal ferroelectrics like BaTiO3 and PbTiO3, anddiscuss the 90�-domain structure which is the most interestedcharacteristic of tetragonal ferroelectrics [6,7], so that the elec-tromechanical and dielectric performances can be improved [8,9].In the phenomenological Landau–Devonshire framework ontetragonal ferroelectrics, the 90� twin-like domain structure isbasically the consequence of competition among the multifoldinteractions, including the dipole–dipole interaction fdi, gradientenergy fgr, elastic energy fel, and electrostrictive energy fes, etc.[10]. The last two terms can be called the electromechanical energyfem = fel + fes. This competition results in a delicate balance amongthese energy terms in the 90�-domain structure [11]. The 90�twin-like domain structure is thus the optimized self-organizedpattern, while the competition between the long-range dipole–dipole interaction and electromechanical energy is dominant indetermining the domain structure [12,13].

One understands that the 90� twin-like structure favors an effi-cient energy release [11], in particular the release of electrome-chanical energy fem (elastic energy). However, this release is farfrom sufficient and there remains substantial energy stored insidethe domains [14,15]. It is known that the piezoelectricity is deter-mined by the dynamic response of the domains against electric fieldor vice versa via the electromechanical coupling. Unfortunately, theelastic energy stored inside domains still imposes high resistanceagainst electric field driven domain wall motion. This case is an ana-log with the pre-stressing mechanism usually employed in concreteconstruction engineering. Therefore, additional efficient energyrelease scheme is still favored. To our knowledge, a promisingapproach to overcome this challenge is to downscale the materialsinto nanostructures such as nano/micro-scale pillars, bubbles, anddots [16–18]. These structures have free boundaries allowing suffi-cient energy release [19]. However, these small-scale structureshave insufficient volume to support mechanical load even thoughthe piezoelectric coefficient itself is large.

A possible and alternative scheme may be to embed mesoscopicanti-dot array into FE lattice, where the anti-dot size or/and arrayperiodicity is on the same order of magnitude as the twin-likedomain width. This scheme has recently been addressed in ourphase-field simulations (Monte Carlo simulations) on FE thinfilms with embedded square anti-dot array [20]. By optimizingthe anti-dot size and array periodicity, it is predicted thatthe electromechanical energy can be efficiently released by the90�-domain splitting process and consequently the piezoelectriccoefficient can be significantly enhanced.

302 Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308

Nevertheless, it should be mentioned that a hidden issue for the90�-domain structure is the lattice symmetry. For a three-dimen-sional (3D) tetragonal FE lattice, there are six FE domain variantsstemming from the cubic symmetry of the paraelectric parent lat-tice [21]. For a FE thin film with in-plane polarization as an exam-ple, the parent lattice has square symmetry. We thus consider the2D in-plane FE domains which have four equivalent variants. Inthis case, the square anti-dot array can induce domain splittingover the whole lattice [20]. It will be of interest to see the domainsplitting behavior if an anti-dot array different from the squaresymmetry, such as triangle anti-dot array, is embedded into thelattice. This is the major motivation of the present work.

In this paper, we will investigate the 90�-domain structures inFE thin films with embedded triangle anti-dot arrays. As a repre-sentative case, we choose isosceles right triangle anti-dot arraywith different orientations. We first perform Monte Carlo sim-ulations on the 2D square FE lattice with periodic boundary condi-tions as an approximation to infinite tetragonal FE thin films. Then,we compute the piezoelectric and dielectric responses to checkhow the embedded triangular anti-dot array modulates theseresponses. It should be mentioned that extension to the 3D latticeis direct.

2. Model and simulation methodology

We start from a 2D L � L square lattice, and the energy on eachsite i (or energy density) can be represented by two order parame-ters: electric dipole vector P(r) = (Px,Py) and elastic displacementvector u(r) = (ux,uy), where x and y are the coordinate axes.Following earlier work, we consider the same energy terms asthe Landau–Devonshire phenomenological theory, including theLandau energy fld, gradient energy fgr, dipole–dipole interactionfdi, elastic energy fel, electrostrictive energy fes, and electrostaticenergy fse [20,22,23]. For such a lattice, the total energy densitycan be written as:

f total ¼ f ld þ f gr þ f di þ f em þ f se; ð1Þ

where electromechanical energy fem = fel + fes. For the Landauenergy, we consider a six-order polynomial:

f ld ¼ A1ðP2x þ P2

yÞ þ A11ðP4x þ P4

yÞ þ A12ðP2x P2

yÞ þ A111ðP6x þ P6

yÞ

þ A112ðP4x P2

y þ P4yP2

x Þ; ð2Þ

where A1, A11, A12, A111, and A112 are the Landau coefficients. A1 > 0and A11 < 0 are required for the first-order transitions. A1 is sup-posed to be temperature (T) dependent and can be written asA1 = A10(T � T0), where T0 is the critical temperature for the FE tran-sition, and A10 = (2e0C)�1 with vacuum permittivity e0 and Curieconstant C. The gradient energy which expresses the domain wallenergy is:

f gr ¼12

G11ðP2x;x þ P2

y;yÞ þ G12Px;xPy;y þ G44ðPx;y � Py;xÞ2 þ G044ðPx;y þ Py;xÞ2h i

;

ð3Þ

with positive gradient coefficient G11, G12, G44, G044, and Pi,j = @Pi/@rj.The dipole–dipole interaction can be written as:

f di ¼1

4peX

hjiPi � Pj

jri � rjj3� 3Pi � ðri � rjÞ½Pj � ðri � rjÞ�

jri � rjj5

" #; ð4Þ

In the strict case, the calculation of this term should takeaccount of whole lattice for the long-range feature. Precise numeri-cal methods like the Ewald summation scheme are used but verytedious [24]. Here, one can choose a preset truncating distance R0

from the central site to make the computation simple. For a 2D lat-tice, this scheme is sufficiently reliable (R0 = 8 in our simulation)

and a pre-checking with a comparison with the Ewald summationscheme was done [15,26]. It should be mentioned that another effi-cient scheme to deal with the computation of fdi is the spectraliterative perturbation (SIP) method [25].

Subsequently, the elastic energy is written as:

f el ¼12

C11ðu2x;x þ u2

y;yÞ þ C12ux;xuy;y þ12

C44u2x;y; ð5Þ

with the elastic coefficients C11, C12, C44, and ui,i = @ui/@ri, ui,j = @ui/@rj + @uj/@ri. Then, the electrostrictive energy reads:

f es ¼ �ux;xgxx � uy;ygyy � ux;ygxy; ð6Þ

where gxx ¼ Q11P2x þ Q12P2

y , gyy ¼ Q11P2y þ Q12P2

x , gxy = Q44PxPy, andQ11, Q12, Q44 are the electrostrictive coefficients. Finally, the electro-static energy is

f se ¼ �E � Pi; ð7Þ

with applied electric field E = (Ex,Ey).Given an anti-dot array, we assume P(r) � 0 and ui,j(r) � 0

(i, j = x,y) for the sites within anti-dots, which means the anti-dotsare empty (full of air). Such kind of model could be dealt with moreeasily. This scheme corresponds to realistic structure which may beexperimentally realized by micro-fabrication [27]. We considerisosceles right triangle anti-dots with four different orientations,where the hypotenuses are parallel to [11] or [1 �1] directions andthe two legs align along the x-axis and y-axis, respectively.Mathematically, mechanical balance conditions on the boundariesbetween anti-dots and matrix can be handled by solving the time-dependent Ginzburg–Landau (TDGL) equation [21,28,29] or usingthe SIP method [30]. However, following earlier work [20], we stillemploy Monte Carlo simulation to track the domain patterns forthe sake of convenience and contrast [10]. A reasonable assump-tion here is that the response speed of elastic strain is much fasterthan that for dipole relaxation, such that the mechanical equilib-rium is established just after an electric dipole flip event occurs[13]. In our simulation, we choose BaTiO3 thin films as objects ofsimulation and relevant materials parameters are taken fromliterature [10]. We choose lattice constant a0, Landau coefficienta0 with a0 = a10(T1 � T0) and a10 > 0, and polarization P0 at T = T1

far below T0 as reference values to normalize other parameters.Term |a0|P2

0, which has an energy unit, is used to normalize thoseenergy terms [31], and thus the dimensionless parameters arelisted in Table 1 [12,32].

3. Results and discussion

We start from a FE lattice free of any anti-dot. The lattice isgradually annealed from a high T� T0 to a sufficiently lowT� T0. The details of simulation were described in earlier work[25] and no description is given here. Fig. 1(a) shows the simulatedFE domain pattern with lattice size L = 64. One can observe thewell-aligned stripe-like 90�-domains, while domain walls alignalong the [1 �1] direction. The results are in accord with those basedon phase-field simulations using the TDGL equation [13,15,33].Subsequently, an anti-dot array is embedded into the lattice, andan adequately long-time relaxation of domain pattern is allowed.We carefully check the total energy by choosing various annealingpaths to make sure that the domain structure reaches the energy-lowest state (ground state) eventually.

As a reference, we first look at the domain structure in a latticewith a 4 � 4 square anti-dot array, as shown in Fig. 1(b), where theanti-dot size is Ls = 6. It is seen that the stripe-like domains nearthe anti-dots split regularly [20]. Then we embed isosceles righttriangle anti-dot arrays with different orientations into the latticerespectively and perform extensive simulations upon different

Table 1Parameters chosen for the simulation (1/4pe = 1, Boltzmann constant kB = 1).

Parameter (unit) Value Parameter (unit) Value Parameter (unit) Value

A10 (a10) 0.10 A11 (a11P20/|a0|) �0.24 A12 (a12P2

0/|a0|) 4.50

A111 (a111P40/|a0|) 0.49 A112 (a112P4

0/|a0|) 1.20 G11 (g11/a20|a0|) 1.60

G12 (g12/a20|a0|) 0.00 G44 (g44/a2

0|a0|) 0.80 G044 (g044/a20|a0|) 0.80

C11 (c11/|a0|P20) 2.75 C12 (c12/|a0|P2

0) 1.79 C44 (c44/|a0|P20) 0.54

Q11 (q11/|a0|) 0.142 Q12 (q12/|a0|) �0.0074 Q44 (q44/|a0|) 0.0157L (a0) 64 T0 (Tc/(Tc � T1)) 5.00

Fig. 1. Simulated in-plane domain patterns for lattice free of anti-dot (a), lattice with square anti-dot array (Ls = 6) (b), lattice with the type-(�1 1) anti-dot array (c), lattice withthe type-(�1 �1) anti-dot array (d), lattice with the type-(1 �1) anti-dot array (e), and lattice with the type-(11) anti-dot array (f), respectively. The red arrows indicate thedirection of polarization and the anti-dot array periodicity is 16. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version ofthis article.)

Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308 303

anti-dot size Ls and arrays, where Ls is the hypotenuse length. Forconvenience, we choose the anti-dot hypotenuse orientation assignature of the array, and we consider four representative typesof arrays: type-(�11), type-(�1 �1), type-(1 �1), and type-(11). The

simulated domain structures for the four cases are shown inFig. 1(c)–(f), given the 4 � 4 arrays and Ls = 7. It is clear that theanti-dot array orientation has notable impact on the domain pat-tern. An immediate claim is that local domain splitting will occur

Fig. 2. Simulated fdi-contours for the lattices (a) free of anti-dot, (b) with the type-(�1 1) anti-dot array, (c) with the type-(�1 �1) anti-dot array, (d) with the type-(1 �1) anti-dotarray, (e) with the type-(11) anti-dot array, respectively. The anti-dot array periodicity is 16.

304 Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308

if domains near the anti-dots have polarizations perpendicular tothe legs, so that the polarizations near the anti-dots flip to be par-allel to the edges. If the polarization is not perpendicular to oneedge, such a splitting will not necessarily occur. It is seen thehigher the probability of domain splitting, the closer to ±p/2 theangle hP between the polarization P and one nearby triangle edge.The underlying mechanism is the efficient release of the depolar-ization field on the anti-dot edges via the domain splitting. Thisdepolarization field is attributed to the dipole–dipole interactionand electromechanical energy.

We take the lattice with the type-(�11) anti-dot array as anexample, as shown in Fig. 1(c). It is seen that the original dipolesnext to anti-dots are perpendicular to the triangle legs, and thus

the depolarization field has to be suppressed by the domain split-ting, while the original dipoles near the triangle hypotenuses havehP ±p/4. In contrast, for the type-(1 �1) array shown in Fig. 1(e),the domain structure remains roughly unchanged since all thedipoles near the triangle legs are parallel to the edges. For thetype-(�1 �1) and type-(11) arrays, shown in Fig. 1(d) and (f), similaranalysis can be made and one sees local domain splitting eventson those locations at which the original domains are perpendicularto the triangle edges.

It seems that the size and orientation of the anti-dots plus thearray periodicity all have influences on the domain structure split-ting. The anti-dot array embedding has two impacts. First, an anti-dot itself allows the release of local elastic energy in surrounding

Fig. 3. Simulated fem-contours for the lattices (a) free of anti-dot, (b) with the type-(�11) anti-dot array, (c) with the type-(�1 �1) anti-dot array, (d) with the type-(1 �1) anti-dotarray, (e) with the type-(11) anti-dot array, respectively. The anti-dot array periodicity is 16.

Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308 305

region. Second, the depolarization field near the anti-dot edges hasto be suppressed by the domain splitting. The twofold effect can beunderstood by looking into the energy landscape. We calculate thespatial contours of terms fdi and fem. Fig. 2(b)–(e) shows the con-tours of term fdi surrounding one anti-dot with Ls = 7 for the fourdifferent orientations, while the contour for the lattice free ofanti-dot (Ls = 0) is shown in Fig. 2(a) as a reference. It is clearlyrevealed that for Ls = 0, the lattice has higher fdi on the domainwalls than that inside domains. With the embedded anti-dotarrays, the regions near the two triangle legs have the lowest fdi,while the regions near the triangle hypotenuses have the highest

fdi with the angle hP ±p/4 for almost all local dipoles. Obviously,the anti-dot array embedding results in higher fdi averaged overthe lattice.

Subsequently, we look at the fem-contours, as shown inFig. 3(b)–(e) for the four cases, while the contour at Ls = 0 is shownin Fig. 3(a) for reference. It is seen that the lattice free of anti-dotaccommodates high fem over the whole lattice. In contrast, the lat-tice with the type-(�11) array, which experiences domain splitting,has the lowest averaged fem although the fem-contour is inhomoge-neous, as shown in Fig. 3(b). The lattice with the type-(1 �1) array,which has no domain splitting and thus wide domains, shows high

Fig. 4. (a) Simulated effective longitudinal piezoelectric coefficient deff33 as a function of Ls for the lattices with four types of triangle anti-dot arrays. The deff

33 contours for thefour types of anti-dot arrays are shown in (b)–(e), respectively. The anti-dot array periodicity is 16.

306 Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308

fem-contour too, as shown in Fig. 3(d). As mentioned above, the lat-tices in Fig. 3(c) and (e) accommodate local domain splitting andthus have lower fem.

The above results clearly demonstrate that the electromechani-cal energy fem can be efficiently released by the domain splitting,although the local density of dipole–dipole interaction may beenhanced as compensation. It is noted that energy terms fld, andfgr may change too upon the domain splitting, but the variationsare small and can be neglected. In the overall sense, the depolariza-tion field near the anti-dot triangle edges is the key driving forcefor the splitting initiation, while the electromechanical energydrives the splitting across the whole region between two neighboranti-dots. As the reduction in fem is more than the gain in fdi, thedomain splitting is thermodynamically spontaneous.

One of the consequences of the domain splitting is theenhanced electromechanical responses in sufficiently split domain

structure, i.e. enhanced piezoelectric and dielectric responses. Theanti-dot embedding itself also functions the energy release. For thetriangle anti-dot arrays, these responses are dependent of the arrayorientation. We evaluate the effective longitudinal piezoelectric

coefficient deff33 = dg[10]/dE[10], where g[10] = hQ11P2

x þ Q 12P2yi, E[10]

denotes the electric field along the x-axis, and h i represents thespatial averaging over the whole lattice including anti-dots [33].We choose �0.2 < E[10] < 0.2, within which a good linear depen-dence of term g[10] on field E[10] is shown and the slope is defined

as deff33 . For possible comparison with experiments, we have a rough

estimation of the realistic d33, following the procedure given in

Refs. [12,32]. It is noted that deff33 has the unit of Q11P2

x=E, with Q11

has the unit of q11/C11. The parameters q11, C11, Px, and E are nor-malized by |a0|, |a0|P2

0, P0, and |a0|P0, respectively. One can obtainthe unit of d33 to be 1/|a0|P0, where |a0| = 1011 N m2/C2, P0 = 1

(a)

(b)

Fig. 5. Simulated ac dielectric permittivity real part (a) and imaginary part (b) forlattices with Ls = 0 (free of anti-dot) and lattices with the four types of triangle anti-dot arrays, respectively. The anti-dot array periodicity is 16.

Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308 307

C/m2. By this way, we have the calculated deff33 in the unit of pC/N.

Fig. 4(a) plots the evaluated deff33 as a function of Ls for the four types

of 4 � 4 arrays. The error bars for the data are ±10%, upon averag-

ing over ten simulation cycles. It is seen that the deff33 for all the four

cases increases with Ls and reaches the peak at Ls = 7 for the type-(�11) and type-(1 �1) arrays, and at Ls = 9 for the type-(�1 �1) and type-(11) arrays. It is evidenced that, first, the lattices embedded withthe triangle anti-dot arrays do have significantly enhanced piezo-

electric performance. The peaked deff33 can be one order of magni-

tude larger than that of the lattice free of anti-dot. Second, thepiezoelectric performance is indeed orientation-dependent. Thelattice with the type-(1 �1) array, which shows nearly no domain

splitting, has the lowest deff33 , suggesting the better electromechani-

cal performance of the domain structure with narrower stripewidth. In the other words, the domain splitting may benefit tothe performance. However, it is not the domain structure with

the type-(�11) anti-dot array that exhibits the highest deff33 .

Instead, the lattices with local domain splitting, as the cases with

the type-(�1 �1) and type-(11) arrays, show higher deff33 than that for

the lattice with the type-(�11) array, implying that the type-(�1 �1)and type-(11) arrays themselves can release the energy more effi-ciently than the type-(�11) and type-(1 �1) arrays. In Fig. 4(b)–(e) areplotted the lattice d33-contours for the four types of anti-dot arraysat Ls = 7, while the d33-contour for the lattice free of anti-dot isroughly 10 and homogeneous over the whole lattice. It is shownthat embedding the anti-dot array does allow the domain walls toexhibit higher d33 for all the four cases. Finally, it should be men-

tioned that the predicted deff33 values of 100 pC/N are definitely

quite high for BaTiO3 thin films in such microwave range.Without the embedded anti-dot array, the real d33 would be 10–20 pC/N, typical values for thin films.

Another consequence of the domain splitting associated withthe anti-dot array embedded lattice is the enhanced dielectric

response in the microwave frequency [15], because the dielectricresponse in this frequency range comes from the domain wallvibration driven by ac electric field E(t). Therefore, more domainwalls and free boundaries contribute higher dielectric permittivity.The ac dielectric permittivity e(x) can then be calculated byP(x) = e(x)E(x), where P(x) and E(x) are the Fourier trans-formations of polarization P(t) and electric field E(t), as describedin details in Ref. [15]. The Monte Carlo step (mcs) is used to scaletime t. Similarly, the dielectric permittivity e(x) = P(x)/E(x),where P(x) and E(x) have the unit of P0�mcs and |a0|P0�mcs,respectively. The e(x) has a unit of 1/|a0|. Fig. 5(a) and (b) showthe simulated real and imaginary parts of the dielectric permittiv-ity spectra for all the cases. Clearly, the lattices with the embeddedanti-dot arrays do show much higher dielectric response than thatfor the lattice free of anti-dot (Ls = 0), consistent with the fact thatthe anti-dot array embedded lattices have larger piezoelectriccoefficient than the lattice free of anti-dot. It is also noted thatthe evaluated dielectric constant is also consistent with measuredvalues in the microwave frequency range (GHz).

The present triangle anti-dot array structure can be seen as analternative to the square anti-dot structure investigated earlier[20], while they both improve the ferroelectric and piezoelectricproperties effectively. However, the triangle anti-dot array is dif-ferent from the square one in symmetry. We consider four differ-ent orientations and the performances among them are diverse.From the analysis of the domain patterns with fdi- and fem-contoursof four orientations, one can clearly understand the roles of thesetwo energy terms on the domain splitting. At the same time, theelectromechanical and dielectric responses are orientation-depen-dent, which may be a reference to design more complex anti-dotstructures.

Finally, it should be mentioned that the present work only con-siders the effect of triangle anti-dot arrays of four different ori-entations on the in-plane stripe-like 90� domain structure in FEthin films. We do not deal with the 3D cases where in-planedomains coexist with out-of-plane domains which deserve forfuture investigation. It is understood that realistic domain struc-tures would have more complicated behaviors than the presentresults, but the underlying physics should be similar. The main taskhere is to modulate the 90�-domain structures effectively byembedding different anti-dot arrays. By analyzing the roles of theanti-dot symmetry in controlling ferroelectric, piezoelectric, anddielectric properties of FE thin films, the present simulations pro-vide an approach in practically engineering domain structure inFE materials for specific electromechanical applications.

4. Conclusion

In summary, we have investigated the in-plane 90� domainstructure and piezoelectric/dielectric responses of tetragonal FEthin films with triangle anti-dot arrays. It has been found thatthe domain splitting is dependent of the orientation and dimensionof triangle anti-dot array. The delicate balance of the dipole–dipoleinteraction and electromechanical energy results in the domainsplitting. Simultaneously, the calculations indicate considerablyenhanced electromechanical performance and dielectric response,which are, however, orientation-dependent. The present workoffers an alternative approach to control FE domain structuresand their features for practical electromechanical applications.

Acknowledgments

This work was supported by the National 973 Projects of China(Grant Nos. 2015CB654602 and 2011CB922101), the NaturalScience Foundation of China (Grant Nos. 51431006 and 11234005).

308 Y. Zhang et al. / Computational Materials Science 108 (2015) 301–308

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