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Ferroelectrics from first principles Tips, tricks and pitfalls Nick Bristowe School of Physical Sciences, University of Kent, Canterbury UK CCP9 meeting, April 10th 2017 Ferroelectricity vs ferromagnetism Botharetypesof“ferroics” REVIEWSFerroelectricity – spontaneouspolarizationP (switchablewithanelectricfieldE) Ferromagnetism – spontaneousmagnetizationM (switchablewithamagneticfieldH) REVIEWS NATURE|Vol 442 NATURE|Vol 442|17 Classedbysymmetry: Figure 2 | Time-reversal and spatial-inversion symmetrysymmetry in ferroics. may be represented by by a positive thatlies liesasym asy Figure 2 | Time-reversal and spatial-inversion in ferroics. may be represented a positivepoint point charge charge that Eerenstein, & Scott Nature 442, 759 (2006) a, Ferromagnets. The local magnetic may be represented within a crystallographic unit cellthat thathas has no net There a, Ferromagnets. The local magnetic moment mmoment mayMathur bemrepresented within a crystallographic unit cell netcharge. charge. Th classically by adynamically charge that dynamically traces as an indicated orbit, as indicated dependence, spatial inversionreverses reverses p. classically by a charge that traces an orbit, by the by the dependence, but but spatial inversion p. c,c,Multiferroics Multiferro 1.1 Oxides, interfaces and polarity Ferroelectrics (a) 1.1 Oxides, interfaces and polarity P up Ferroelectrics Pdown (a) Pup e.g. BaTiO3 PbTiO3 (b) Figure 1.2: f“Hysteresis”: Pdown Pup (b) (c) P Figure 1.2: Pdown f P Pdown Pup Pup P (c) P εE Two(ormore)stable polarstates, switchablewith externalfield Pup ε Pdown Pdown (a) Schematic of the o↵-centring for up and down ferroelectric polarisation states. (b) The typical double well shape of the free energy as a of polarisation.for (c) up Theand hysteretic of the polarisation (a) Schematic offunction the o↵-centring downbehaviour ferroelectric with Ferroelectrics are also… Dielectrics Piezoelectric Pyroelectrics Ferroelectrics Ferroelectrics are also… Dielectrics Piezoelectric Pyroelectrics Ferroelectrics Pinduced underE-field Uses:insulators& microwavedielectrics Ferroelectrics are also… Dielectrics Piezoelectric Pyroelectrics Ferroelectrics Pinduced underE-field Uses:insulators& microwavedielectrics Pinduced understrain Uses:Sparkplugs, pressuresensors Ferroelectrics are also… Dielectrics Piezoelectric Pyroelectrics Ferroelectrics Pinduced underE-field Uses:insulators& microwavedielectrics Pinduced understrain Uses:Sparkplugs, pressuresensors SpontaneousP(T) Uses:Tsensors Ferroelectrics are also… Dielectrics Piezoelectric Pyroelectrics Ferroelectrics Pinduced underE-field Uses:insulators& microwavedielectrics Pinduced understrain Uses:Sparkplugs, pressuresensors SpontaneousP(T) Uses:Tsensors SpontaneousandswitchableP Uses:memorydevices,capacitors,high-kdielectrics, optoelectronics,catalysis,“photoferroics”,“multiferroics” Research Activity Scopusresultforpaperspublishedwith“ferroelectric”mentionedinanyfield Talk Outline Highlightseveralbasicissuesforfirstprinciples calculationsofferroelectrics P P 1.1 Oxides, interfaces and polarity (a) up 1.1 Oxides, interfaces and polarity down (a) Pup 1)CalculatingthespontaneousPolarisation (b) (c) f P Pup (b) 2)Calculatingthehysteresisloop Pup Pdown (c) f ε P Pdown Pup P Pup ε P Pdown Pdown Figure 1.2: Figure 1.2: Pdown (a) Schematic of the o↵-centring for up and down ferroelectric polarisation states.ferroelectric (b) The typical (a) Schematic of the o↵-centring for up and down polarisation states. (b) The typical double well shape of the free energy as a 3)Determiningthetypeofferroelectric double well shape of the free energy as a function of polarisation. (c) The hysteretic behaviour of the polarisation with applied electric field, E. function of polarisation. (c) The hysteretic behaviour of the polarisation with applied electric field, E. short-range ionic Coulombic forces, the system would spontaneously polarise. The dp hybridisation is intimately linked with the anomalously large Born e↵ective charges, Z ⇤ , in ferroelectrics. The Born e↵ective charge is dynamical, and should short-range ionic Coulombic forces, the system would spontaneously polarise. The be distinguished from static charges which are usually based on partitioning the Multifunctional magnetoelectrics dp hybridisation is intimately linked with the anomalously large Born e↵ective 4)Finitesizeeffects full electron density in to contributions from di↵erent ions. Instead Z ⇤ relates the change in polarisation upon displacement of an atomic sub-lattice. charges, Z ⇤ , in ferroelectrics. The Born e↵ective charge is dynamical, and should Z ⇤ is large in (Generalized) Magnetoelectric: cross coupled ferroelectrics due to the charge transfers associated with bond formation upon B be distinguished from staticand charges whichfields are usually based on partitioning the response to electric magnetic cation displacements. full electron density in to contributions from di↵erent ions. Instead Z ⇤ relates the Macroscopically, ferroelectric phase transitions can be described phenomenochange in polarisation upon displacement of an atomiclogically sub-lattice. Z ⇤ is large in within thermodynamic theory. + 5)Multiferroics cation-displacements. E H This is the Landau-Ginzburg-Devonshire ferroelectrics due to the charge transfers associated with bond formation upon B Polarization, P Magnetization, M P M Macroscopically, ferroelectric phase transitions can be described phenomeno- 7 1. Calculating P 1. Calculating P 1. Calculating P 1. Calculating P 1. Calculating P 1. Calculating P 1. Calculating P 1. Calculating P Modern theory of polarisation Defined in terms of Wannier centres r r 2 " = $ w n ( r # R) i occ n,i - + + - + - + z ! P P: Dipole moment per unit volume of point charges DEFINED UP TO POLARISATION QUANTA 1. Calculating P Modern theory of polarisation Defined in terms of Wannier centres r r 2 " = $ w n ( r # R) i occ n,i - + + - + - + z ! P (jumped by a polarisation quantum) P: Dipole moment per unit volume of point charges DEFINED UP TO POLARISATION QUANTA 1. Calculating P For centrosymmetric bulk systems (or more generally for this, any symmetry transforming z into -z), Pz should be invariant: Pz = -Pz Infinitesystem– notoneP,butasetofallowedPvalues should be invariant: which is the case for the two values Centrosymmetric materialscanhave2setsbysymmetry: {….. -2P0, -P0, 0, P0, 2P0 ……} P0:quantum {….. -3P0/2, -P0/2, P0/2, 3P0/2 ……} Originof“polar”surfaces Non-Centrosymmetric materials: {……. -2P0+Ps , -P0+Ps , Ps , P0+Ps , 2P0+Ps ……} Ferroelectric materials: {……. -2P0-Ps , -P0-Ps , -Ps , P0-Ps , 2P0-Ps ……} 1. Calculating P For centrosymmetric bulk systems (or more generally for this, any symmetry transforming z into -z), Pz should be invariant: Pz = -Pz Infinitesystem– notoneP,butasetofallowedP values should be invariant: which is the case for the two values Centrosymmetric materialscanhave2setsbysymmetry: {….. -2P0, -P0, 0, P0, 2P0 ……} P0:quantum {….. -3P0/2, -P0/2, P0/2, 3P0/2 ……} Originof“polar”surfaces Non-Centrosymmetric materials: {……. -2P0+Ps , -P0+Ps , Ps , P0+Ps , 2P0+Ps ……} Ferroelectric materials: {……. -2P0-Ps , -P0-Ps , -Ps , P0-Ps , 2P0-Ps ……} Whicharethe relevantpairs? 2. Determining hysteresis (b) a)WhatusPup- Pdown? (c) P f Pdown Figure 1.2: Pup Pup ε P Pdown (a) Schematic of the o↵-centring for up and down ferroelectric P : {……. -2P +P , -P +P , P , P +P , 2P +P ……} 0 s 0 s 0 polarisation states. (b) The uptypical double well shape ofs thes free0 energy as as function of polarisation. (c) The hysteretic behaviour of the polarisation with Pdown :{……. -2P 0-Ps , -P0-Ps , -Ps , P0-Ps , 2P0-Ps ……} applied electric field, E. short-range ionic Coulombic forces, the system would spontaneously polarise. The 2. Determining hysteresis 2P0+Ps P 2P0-Ps P0+Ps P0-Ps Ps Modeamplitude -Ps -P0+Ps 0-Ps 1.1-POxides, interfa -2P0+Ps 1.1 Oxides, interfaces and polarity (a) 1.1 Oxides, interfaces and polarity -2P0-Ps (a) Pup Pdown Pup Pdown 2. Determining hysteresis 2P0+Ps P 2P0-Ps P0+Ps P0-Ps Ps Modeamplitude -Ps -P0+Ps 0-Ps 1.1-POxides, interfa -2P0+Ps 1.1 Oxides, interfaces and polarity 1.1 Oxides, interfaces and polarity -2P0-Ps Connectthedots! (a) (a) Pup Pdown Pup Pdown 2. Determining hysteresis 2P0+Ps P 2P0-Ps P0+Ps P0-Ps Ps Modeamplitude -Ps -P0+Ps 0-Ps 1.1-POxides, interfa -2P0+Ps 1.1 Oxides, interfaces and polarity 1.1 Oxides, interfaces and polarity -2P0-Ps Connectthedots! (a) (a) Pup Pdown Pup Pdown 2. Determining hysteresis F P 1.1 Oxides, interfa 1.1 Oxides, interfaces and polarity (a) 1.1 Oxides, interfaces and polarity (a) Pup Pdown Pup Pdown 2. Determining hysteresis F= aP2 + F bP4 E= P 1.1 Oxides, interfa 1.1 Oxides, interfaces and polarity (a) 1.1 Oxides, interfaces and polarity (a) Pup Pdown Pup Pdown 2. Determining hysteresis P P Ε (a) (b) 2. Determining hysteresis P P Veryapproximate estimateofhysteresis Inreality,dominatedby domainwallmotion Ε E (a) (b) 2. Determining hysteresis P P Veryapproximate estimateofhysteresis Inreality,dominatedby domainwallmotion Ε E (a) (b) 2. Determining hysteresis P P Veryapproximate estimateofhysteresis Inreality,dominatedby domainwallmotion Ε E (a) Requiresmesoscopicmodelling Beyondscopefortoday (b) polarities and, similarly, can be computed the mode th ary conditions, which is particularly useful in studying ferroelectric systems. from Furthermore, 2.reformulation Determining hysteresis eigenvector and the of theand magnetizations citors suggests an appealing in terms ofknowledge free charges potentials,inwhi duced byan displacing individual atment of stresses and strains. Using PbTiO example, we showatoms. that our technique enabl 3 as The magnetic response to an electric field is readily variables within the density functionalFiniteFieldMethods formalism. odern theory of polarization1 P has in the theory of the ferroelectric t could previously be inferred only now be computed with quantum rst-principles density functional s focused on bulk ferroelectric e balance between covalency and ferroelectricity. Over time, these the effects of external parameters lds2,3 . Of particular note is the od for performing calculations at P4 . The ability to compute crystal as a function of P provides an nshire and related semiempirical (a) obtained from InthelinearP(E)regime,wecanuse these expressions. Indeed, the equilibrium value of un for applied E and zero magnetic field is “frozenphonon”approach Sand I C A Lthe R E Berry-pha VIEW LE 117201 (2008) energy PEHKSY P v through PRL the 101, Kohn–Sham X 1 d ¼ to the pmoment polarization The P lattice (ref.contributions 1).un(For thepolarization we fix the latti ni E i ; and magnetithus(5) gainin C i zationwill are be discussedn shortly.) theof compu vectors; strains The minimum U require sim N fixed from D is which given the by induced magnetization 1 X is obtained as considered P ¼ pd u (3) IR latt;i Ε !0 ni n sponse, wh n¼1 NIR X X 1 @E 1 @U @EKS @P Indeed, su m ¼ " e.g.Iniguez ¼ ! E þ p u M jand = ⌦ (D · = 0nj n playing a elec;ij 4⇡P) i PRL101117201(2008) ! @H ! N @v D 0 @v @v j 0 n¼1 i 1 X polar crys IR M latt;j ¼ !0X Nn¼1 IR pm nj un ; (4) X X 1 1 m Comparing respectively. with the fixed-E approach 2,3 ¼GobeyondlinearP(E)regimebydefiningelectric !elec;ij E ithe þ dielectric pnj of ofrefs pdninth E i :IRin pdn is polarity the !0 from Cn iBorn effecelectric enthalpy given by mode, F which can be obtained the atomic i is n¼1 enthalpyfunctionalinDFTmethods Then, linear we see that tive charges and the mode eigenvector [10]. The pm n cothe mode-decomposed lattice-mediated part efficients are the magnetic analogue of the dielectric Fand, (E,v) Ecan ⌦ E ·P(v) KS (v) polarities similarly, be from the mode ME response is=given bycomputed eigenvector and the knowledge of the magnetizations inNIR individual atoms. X NIR d m ducedSouza,Iniguez,VanderbiltPRL89117602(2002) by displacing X pni pnj 1 Umari &Pasquarello PRL89,157602(2002) The magnetic to an !latt;ij ¼ response !latt;nij ¼electric field is readily : (b) Ε coupling t included tin which (6)will there derivation constant m ofusing thethe I ( tural varia dynamical Results (7) m proposed 3. Types of ferroelectrics DFThasplayedanimportantroleinunderstanding“origin”ofP CohenNature358136(1992) 3. Types of ferroelectrics HybridizationleadstoanomalousBorneffectivecharges PH. GHOSEZ, X. GONZE, PH. LAMBIN, AND 6766 J.-P. MICHENAUD TABLE I. Born effective charges for cubic BaTi03. Shell model Ions 1.63 7.51 2.9 6.7 —2.4 —4.8 Ba Zo —2.71 —3.72 'Reference 4. "Reference 9. Zhong et al. " a„)(=4.00 A a„ii=3.94 A a„))=3.67 A 2.75 7. 16 —2. 11 —5.69 2.74 7.29 —2.13 —5.75 2.77 7.25 —2.15 —5.71 2.95 7.23 —2.28 —5.61 Ghosez etalPRB516765(1995) of Zo ( —5.71) and Zo i ( —2. 15) refer, respectively, to a displacement of the oxygen and +7.25. For 0, the values II ion along the Ti-0 direction or perpendicular to it. Our results are in good agreement with those obtained by Zhong, King-Smith, and Vanderbilt (also at the theoretical volume) from finite difference of polarization and globally reproduce the values deduced by Axe. The accordance with the shell model is only qualitative and illustrates the limited precision obtained within such an empirical description. We note however that this shell model was not designed to reproduce the dielectric properties of BaTi03. The large value of Z~; (7.51) orbitals is well known and was already pointed out from and linear combinatight-binding models, experiments, tion of atomic orbitals calculations. It was highlighted more recently by Cohen from first principles as an essential feature of ABO3 ferroelectric compounds. In this context, it seemed realistic, following Harrison, to focus on 0 2p-Ti 3d hybridization changes to explain intuitively large anomalous contributions. went bePosternak, Resta, and Baldereschi yond this credible assumption. They showed for KNb03 that the anomalous contributions disappear when the interaction between the 2p and Nb 4d orbitals is artificially suppressed. 0 antipolar X5 mode contribution, in complete contrast to the eigendisplacements (η) of the FE unstable mode for t case of perovskite oxides [12]. of NaBF3 compounds, again comparing with BaTi To understand the differences that we have found between In all fluoride cases, we find a strong A-site cont oxide and fluoride perovskites, we next calculate the Born with a substantial contribution also from F∥ but a n effective charges (Z ∗ ) of selected ABF3 cubic perovskites, and contribution from the B-site ion. The dominance of th HybridizationleadstoanomalousBorneffectivecharges we compare them in Table II with the values for BaTiO3 . As displacement can also be seen in Fig. 3, where w expected for ionic compounds, the Born effective charges of with red arrows the eigendisplacements for the FE the fluoroperovskites are close to their nominal ionic charges, NaMnF3 . The trend in A-site displacement magnitud -P. PH. AND MICHENAUD PH. with X. LAMBIN, J. GHOSEZ, GONZE, the most anomalous value—a deviation of −0.71e for the A- and B-site relative masses: the A-site cont F∥ —resulting from a small but nonzero charge transfer from F to the FE mode eigendisplacements for A = Li, Na TABLE Born effective for cubic I. charges BaTi03. to B as the atoms move closer to each other. This is in striking and Cs in ANiF3 are 0.347, 0.186, 0.138, 0.081, an contrast to the oxides, where the Born effective charges are respectively, while the contributions from Ni across Shell model et al. " Ions 00 A 67 A a„)(=4. a„))=3. a„ii=3. strongly anomalous.Zhong For example, the corresponding oxygen94 A series are 0.003, −0.019, −0.053, −0.073, and −0.0 in63BaTiO3 is −3.86e, indicating2.74 a much stronger To explore 2.75 2.77 2.95 the origin of the unstable mode, we 2.9 anomaly 1. Ba charge transfer and change of7.29 hybridization acthe 7.25 Table III 7. 23 on-site interatomic force constants (I 7.51 7. 16 6.7 dynamical Since key ingredient —2.4 companying —2.71the Ti-O displacement. —2. 11 —2.a 13 —2.in15 the cubic —2.NaBF 28 3 series. The on-site IFC gives the ferroelectricity —5.61 atom when it is displaced in the cryst —4.8 explaining —3.72 —5.in69perovskite oxides —5.75 is the presence —5.71 an individual Zo all the other atoms are kept fixed [27]. Even in high-s phases with structural instabilities, the on-site IFCs ar 'Reference 4. TABLE II. Born effective charges, Z ∗ (e), and eigendisplacements Ghosez etalPRB516765(1995) positive (indicating a restoring force) as a periodic c "Reference 9. of the FE unstable mode, η, in NaBF3 and BaTiO3 . X⊥ and X∥ indicate 3. Types of ferroelectrics 6766 the Born effective charge of the anion (F or O) when it is displaced and +7.25. For ( —2. 15) refer, is well known and was already pointed out from parallel perpendicular the B-X orbitals bond. values of 5.71) and toZo 0, theAsopposedto“Geometricferroelectrics”withnominalcharges Zo or( — TABLE III. On-site IFCs (first four columns) and larg i experiments, and linear combinatight-binding models, II respectively, to a displacement of the oxygen Z∗ ion along the Ti-0 direction or perpendicular to it. Our rethose obtained A B by X X∥ sults are in good agreement with System Zhong, ⊥ King-Smith, and Vanderbilt (also at the theoretical volume) Nominal 1 2 −1 −1 from finite difference of polarization and globally reproduce NaMnF3 1.17 2.21 −0.83 −1.72 with the shell the values deduced by Axe. The accordance NaVF3 1.18 1.99 −0.71 −1.75 model is only qualitative and illustrates the limited precision NaZnF3 1.15 2.22 −0.84 −1.69 obtained within such an empirical description. We note howNaNiF3 1.15 2.05 −0.74 −1.73 the to reproduce ever that this shell model was not BaTiO designed 2.75 7.37 −2.14 −5.86 3 dielectric properties of BaTi03. The large value of Z~; (7.51) atomic IFCs (last three columns) of ABF3 (eV/Å2 ). X⊥ a η tion of atomic orbitals calculations. It was highlighted defined as in Table II. more recently by Cohen from first principles as an essential A B X X∥ of ABO3⊥ ferroelectric feature it compounds. FEmodeappearsduetoionicsizemismatch System A In this B context, X⊥ X∥ B − X∥ B − B∥′ seemed realistic, following Harrison, to focus on 2p-Ti 3d 0.174 −0.005 −0.020 0.17 large 13.14anomalous 0.76 13.35 −4.62 −2.77 to explainNaMnF hybridization intuitively changes−0.088 3 0.181 −0.018 −0.019 −0.077Resta, 0.23 19.00 went 0.88 be17.29 −6.78 −1.96 NaVF contributions. and3 Baldereschi Posternak, 0.177 −0.025assumption. −0.084 0.43 for 12.63 1.22 that 13.81 −4.39 −3.21 NaZnF this credible They 3showed KNb03 yond −0.005 0.186 −0.002 −0.025 −0.068 disappear 0.60 17.07 1.63 16.13 −5.69 −2.62 NaNiF3 when the anomalous contributions the interaction Garcia-CastroetalPRB89104107(2014) 0.001 8.27 13.09 6.74 sup11.70 2.71 −13.83 BaTiO 3 between0.098 the −0.071 and Nb 4d orbitals is artificially 2p −0.155 pressed. 0 0 temperature, T, 3. Types of ferroelectrics isotropy) in the absence of strain and an external electric field, the free f (T, P ) = a1 P 2 + a11 P 4 + a111 P 6 , ProperFerroelectrics y of a ferroelectric can be expressed as a function of polarisation, P , and erature, T , Landaudescription where the coefficients are temperature dependent. Perhaps the m 2 6 f (T, P ) = a11,Ptakes + a11 P 4form, + a111aP coefficient, the = 1 , ↵(T T0 ), where ↵(1.2) is a positiv T0 is the phase transition temperature. Additionally the coeffici the coefficients are temperature dependent. Perhaps the most important mines the type of transition; first or second order. For a second or cient, a1 , takes the form, a1 = ↵(T T0 ), where ↵ is a positive constant and a11 is positive. When T > T0 , a1 is positive, and the free energy the phase transition temperature. Additionally the coefficient, a11 , deterwith minimum at P = 0, and the material therefore in the parae Curie-Weisslaw the type of transition; first or second order. For a second order transition, instead T < T0 , the free energy is a double well (see figure 1.2) positive. When T > T0 , a1 is positive, and p the free energy is a single well at P = ±PS , where PS = a1 /(2a11 ) is the spontaneous pola minimum at P = 0, and the material therefore in the paraelectric state. If #$$ = "R #$$ = Local tions associated with the R point of the cubic first Brillouin zone, respectively. We also compute the following quantities: 3. Types of ferroelectrics !NZ!"2 2 &'u$( − 'u$(2), V % ok BT 0.02 <u >=<u >=<u > 1 2 3 0.01 0.00 0.20 TriggeredFerroelectrics (b) !2" N2 2 2 &'"R, $( − '"R,$( ), VkBT R3c Pm3m I4/mcm 0.15 AFD motions (radians) where “' (” denotes statistical averages and where u$ and Landaudescription "R,$ are the $ component of u and "R, respectively. N is the number of sites in the supercell, while V is its volume. Z! is 0.10 the Born effective charge associated with the local mode, kB < ! > =< ! > 1 R 2 R 2 4 2 2 is the Boltzmann constant, and %o is the permittivity of 0.05 vacuum. #$$ are the diagonal elements of the dielectric sus"R 23,24 <! > ceptibility tensor, while #$$ is the staggered 3 R 25 susceptibility associated with "Kisanonpolarmode,ais+ve . andc–ve R,$ 0.00 Figure 1!a" shows the 2predicted x-, y-, and z-Cartesian 350 If|cK |>a,PbecomesdoublewellwhenKiscondensedin coordinates !'u1(, 'u2(, and 'u3(" of 'u( in BFO, while Fig. (c) Neithermodehastobe“primary”modeifcouplingcislarge 1!b" displays '"(R !with '"1(R, '"2(R, and '"3(R denoting the 300 x-, y-, and z-Cartesian coordinates of '"(R", and Fig. 1!c" 250 reports the susceptibilities. Figures 1!a" and 1!b" indicate that, as consistent with experiments !see Ref. 9 and refer200 ences therein", our numerical scheme predicts a R3c ground <" >=<" > 11 22 state that exhibits a DoesnotfollowCurie-Weisslaw: polarization pointing along the &111) 150 <" > direction !since 'u1( = 'u2( = 'u3( ! 0 at the lowest tempera33 100 tures" and a tilting of the oxygen octahedra about the &111) axis !since '"1(R = '"2(R = '"3(R ! 0 at the lowest tempera50 tures". Figure 1 also indicates a first-order FE-to-paraelectric transition from a rhombohedral R3c phase to a tetragonal 0 0 500 1000 1500 2000 I4 / mcm state that solely exhibits AFD motions about the Temperature (K) &001) axis, about TC * 1075 K as indicated by the nonzero value of '"3(R above *1075 K in Fig. 1!b" and the vanishFIG. 1. Supercell average 'u( of the local mode vectors &panel Holakovský, Phys.1075 Status ing of 'u( via the jump seen in Fig. 1!a" around K. Solidi B 56, 615 1973 !a"), AFD-related '"( quantity &panel !b"), and #$$ dielectric sus'"2(R 79 100105 2009 R Figure 1!a" thus reveals that 'u1(, 'u2(,&'uBellaiche, Kornev 3(, '"1(R, and PRB ceptibilities &panel !c") as a function of temperature in BFO and as are the OPs of this FE transition. The I4 / mcm phase exists F = aP + bP + cP K 2000 AFD susceptibility Dielectric susceptibility 1500 <" <" ! 11 ! 33 >=<" ! > 22 > 1000 500 0 900 950 1000 1050 1100 1150 1200 3. Types of ferroelectrics ImproperFerroelectrics Landaudescription 1 1 2 E = A0 P + B0 P 4 + C12φ12φ22 2 4 1 1 + A1φ12 + B1φ14 + C01φ12 P 2 2 4 1 1 + A2φ22 + B2φ24 + C02φ22 P 2 2 4 +λφ1 φ2 P Otheranharmonic couplingspossible (Podd) 3. Types of ferroelectrics ImproperFerroelectrics Landaudescription f1,2 µ (T-Tc)1/2 ® P µ f1.f2 µ (T-Tc) ® c independent of T Bousquet et al Nature 452 732 (2008) A.P. Levanyuk and D.G. Sannikov, Sov. Phys. Usp. 17, 199 (1974) 4. Finite size effects Ultrathin-films miniaturizationoftransistors,developmentoflayer-by-layergrowthtechniques 4. Finite size effects Ultrathin-films miniaturizationoftransistors,developmentoflayer-by-layergrowthtechniques Epitaxialstrain 4. Finite size effects Ultrathin-films “Strainengineering” 2 + csP2 F = aP2 + bP4 + KsPHYSICAL REVIEW B 72, 144101 !2005" Diéguez,Rabe&VanderbiltPRB 72 144101 (2005) 4. Finite size effects Ultrathin-films “Strainengineering” 2 + csP2 F = aP2 + bP4 + KsPHYSICAL REVIEW B 72, 144101 !2005" Diéguez,Rabe&VanderbiltPRB 72 144101 (2005) 4. Finite size effects Opencircuitboundaryconditionsi.e.filmsinair σ bound ! ! ρtotal ∇⋅E = ε0 ! ! = P ⋅ n = Pz ε 0 ΔEz = σ free − ΔPz FEinopen-circuitcreatesa“depolarizingfield” F = aP2 + bP4 - E.P E.P termdominatescurvatureofPtobecomepositive! nd redox nce of a ernative creening 4. Finite size effects PHYSICAL REVIEW B 85, 024106 (2012) Screeningmechanism1:Ionicadatoms Metal FE Air N. C. BRISTOWE et al. FIG. 1. Schematic illustration of the conventional (left) and red (center) mechanisms for ferroelectric screening in the absence o top electrode. The presence of a biased tip can promote an alterna Bristoweetal redox mechanism that provides an external circuit for the screen electrons PRB (right).85 024106 (2012) enough to show bulklike features in the center, and 3 u cells of BTO was experimentally shown to be thick enou for ferroelectricity.1 We use a dipole correction to simul open-circuit boundary conditions, enforcing zero macrosco 4. Finite size effects Screeningmechanism2:Zenertunneling Aguado-Puente,Bristoweetal PRB92 03548(2015) Yinetal PRB92 115406(2015) 4. Finite size effects Screeningmechanism3:Domains w d Fluxclosure: Aguado-Puente&Junquera PRB85 1841052012 in0 monodomain Curves are calcu parameters obta 0 plot % = 1 V, re -100 G (MJ/m ) 3 G (MJ/m ) 3 ◦ domain phasesstructure, (monodomain andG polydomain), we need tothe compare width [53,54] γ w/d, propor180the domain in which straight ofwhere the material, elec = stripes 4. Finite size effects evolution of direction the energy ofbea thin film intothe constant can calculated to all tionality withthe thethickness same width inbethe perpendicular thetwo scenarios. For the polydomain phase, we assume a domaindifferent wall, have an out-of-plane polarization of the same 2 ◦ 8.416P 180 domain structure, in orientations. which straightFor stripes ofidealized the material, Screeningmechanism3:Domains magnitude but with alternating such (27) γ = all with the same width in the direction perpendicular to the 3 1/2 π ε0 [1 + (εx εfound ] in tetragonal version of the domain structures typically z) domain wall, have an out-of-plane polarization of the same ferroelectric ◦ thin films, the energy per wunit of volume of the for 180 stripebut domains [55]. The width ofFor domain walls in a magnitude with alternating orientations. such idealized polydomain phase can be expressed as d -100 where % case version of the domain structures typically found in tetragonal typical ferroelectric is vanishingly small and remnant electric -200 by surface red ! ferroelectric thin films, the energy per unit of volume of the fields in polydomain configurations decay exponentially away G + G = U + , (26) poly elec nm. Interesting polydomain phase can be expressed as w from the surface and the domain wall, thus, except in the 0 2t -200 predicts that ! of a domain where the energy per unit of area wall and limit!ofis thicknesses ofpoly a few G + Gelecwe = U unit + cells, , can approximate (26) polarization d w is the domain width. The electrostatic energy G due to w |P | ∼ PS throughout the film and U = U0 (PS elec ) as a constant. 0 2 crossover betw stray fields in the polydomain configuration is proportional to FIG. 8. (Color onlin where isStrayfield: theand energy per unit of area a domain wall and Using this ! result differentiating Eq. of (26) with respect to w thickness the thetodomain width [53,54] G = γ w/d, where the proporin monodomain (black w is the domain width. The electrostatic energy G due to elec elec find the equilibrium domain width for a given thickness of formation of a tionalitystray constant bepolydomain calculated configuration to be are8.calculated fo fields can in the is proportional to CurvesFIG. (Color onli the film, one obtains the well-known Kittelwhere law [53] overobtained the break the domain width [53,54] G = γ w/d, the propor- parameters fr in monodomain (blac elec2 Kittellaw: 8.416P tionality constant be calculated !d to be plotCurves % parameters, = are 1 V,calculated realistic th (27) γ = can 2 3 1/2 (28) parameters obtained f π ε0 [1w+ = (εx εz ) . 2 ] 4.6 nm, well w γ 8.416P plot % = 1 V, realistic (27) = The for 180◦ stripe domains γ[55]. width of domain walls in a grown%in=ferr 3 ε [1 + (ε ε )1/2 ] π case where C/ 0 x z Substituting this expression for the equilibrium domain typical ferroelectric is vanishingly small and remnant electric at which ◦ by surface redoxthe prot for 180 stripe domains [55]. The width of domain walls in a width into Eq. (26) leads to the following expression for the fields in polydomain configurations decay exponentially away case where % = C/ the atoms move from an sp2 environment towards sp3 bonding, resulting in polarization in the z direction [9]. In Table I, we report the lowest TO and LO phonon Ferroelectrictypesthatareresistanttodepolarizingeffects frequencies and dielectric constants, as well as band gaps, pP ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! 2 ΔZ ¼ ðZ zz zz Þm =N , and polarizations for a variety of m Hyperferroelectrics ABC ferroelectrics; those with imaginary LO frequencies are, by definition, hyperferroelectrics. The relatively small LOisunstable(inadditiontoTO) ProperferroelectricwhichlowP,andlarge ΔZ!zz ≈ 3 and large ðϵ∞ Þzz ≈ 10–20 both contribute to the Weakdepolarizingfields weak depolarization fields in these materials [for reference, cubic perovskites typically have ΔZ!zz ≈ 5 and ðϵ∞ Þzz ≈ 6]. Garrity,Rabe &VanderbiltPRL 112 127601 (2014) Both the small effective charges and large dielectric constants of ABC ferroelectrics are consequences of the Improperferroelectrics covalent bonding and resulting small band gaps of these semiconductors. In Fig. 1(c), we show the phonon Pdrivenbynon-polareffectivefield dispersion for the hyperferroelectric LiBeSb, which is a BalanceofenergyscalesdetermineswhetherPcanremaininopencircuit previously synthesized material [23,24]. We see that the lowest frequency phonon mode for q → 0 is unstable regardless of the direction from which Γ is approached. In order to investigate the electric equation of state of hyperferroelectrics, we use a simple first-principles-based Stengel,Fennie &Ghosez 094112 (2012) model.PRB We86first define a dimensionless polar internal degree 4. Finite size effects 22 mirror plane onfor thethe central metal a complexof structure, exponentially the 2 layer. The amplitude this field,and E d , decays modesymmetry , y, determined sameTiO tetragonal cellrelaxation geometry as in has depends on (1) the in screening length of metallic the region beyond thepolarization first RuO2 of was continued until thethe maximum force on case the atoms was smaller correlayer. It spreads over a both monitor the supercell. Only monodomain was considered, metal and the the thin film, which than 40 meVÅ21. The symmetry constraint prevents the appearance characteristic distance that is related to the screening length of the sponding to a situation that can be achieved experimentally9. In of ferroelectricity, and the resulting structures correspond to the electrode. Each dipole is responsible for electrostatic potential drops Fig. 2, the symbols show the evolution of the energy (with respect to optimized paraelectric states. Starting from the previous relaxed in the same direction, so that the only way to preserve short-circuit the referencetaken paraelectric phase) when freezing-in displaceboundary conditions along the structure will be the appearance of a configurations as reference, we then searched for theatomic existence y of increasing amplitudes. Calculations have been donedepolarizing for electric field inside the ferroelectric film, illustrated in ofments a ferroelectric instability, moving BaTiO atoms continuously 3 differentthe thicknesses of the BaTiO layer. 2 points Fig.a 3b for different values of the induced polarization. The following displacement pattern of 3the bulkFigure tetragonal soft out 22 change behaviourfor at athe critical m ¼ 6as(26 For of this field, E d , depends on (1) the screening length of mode , y,ofdetermined same thickness tetragonalaround cell geometry in Å).amplitude the supercell. Only the monodomain case was considered, corre- the metal and the polarization of the thin film, which both monitor sponding to a situation that can be achieved experimentally9. In Fig. 2, the symbols show the evolution of the energy (with respect to the reference paraelectric phase) when freezing-in atomic displacements y of increasing amplitudes. Calculations have been done for different thicknesses of the BaTiO3 layer. Figure 2 points out a effect of the depolarizing field for a realistic system and dem change of behaviour at a critical thickness around m ¼ 6 (26 Å). For 4. Finite size effects Capacitors Closedcircuitwithrealelectrodes– imperfectscreening letters to nature its crucial role. First, it is responsible for the suppre ferroelectricity at a finite size. Second, even well above th thickness, the depolarizing field reduces the amplitud spontaneous polarization, which only slowly approaches value for films of increasing size (Fig. 2 inset). The results reported here have been obtained for a de insulating Figure 2 Evolution of the energy as a function ofperfectly the soft-mode distortion y. First- BaTiO3 crystal, whereas typical ferr principles results (symbols) and electrostatic model results (lines) are shown for capacitors are well-known to exhibit high leakage currents different thicknesses of the ferroelectric thin film: m ¼ 2 (circles, full line), m ¼ 4 monitored Poole–Frenkel mechanism in the case (upward-pointing triangles, dotted), m ¼ 6 (diamonds, short-dashed), by m ¼ the 8 (squares, 29 dot-dashed), and m ¼ 10 (downward-pointing triangles, long-dashed). The magnitude oxide electrodes ). Itof might therefore be questioned whethe Figure 2 Evolution of the energy as a function soft-mode distortion y. First- of BaTiO3 atoms y is reported as a percentage of of thethebulk soft-mode displacements Figure 1 Structure of a typical ferroelectric capacitor. a, Schematic view of the system redistribution associated with the finite conductivity of th principles results and electrostatic modelofresults (lines) are shown for considered here: a BaTiO3 film between SrRuO3 electrodes that are short-circuited. (y ¼ 1(symbols) corresponds to the distortion the bulk tetragonal ferroelectric phase).The energy different thicknesses of the ferroelectric film:asmreference. ¼ 2 (circles, full line), m ¼ could 4of the spontaneous electric further screen the depolarizing field and af of the paraelectric phase isthin taken Inset, evolution The whole structure is grown on top of a SrTiO3 substrate. The epitaxy is such that (upward-pointing triangles, m ¼ 6 (diamonds, short-dashed), mdeduced ¼ 8 (squares, polarization, Ps,dotted), SrTiO3 (001) k SrRuO3(001) k BaTiO3 (001)k SrRuO3(001). b, Atomic representation of with thickness. For eachJunquera thickness, P is from the amplitude of y current densities30 (J < 1025 A cm22, &Ghosez s conclusions. For usual dot-dashed), and m ¼ 10 (downward-pointing triangles, long-dashed). The magnitude of the related supercell that is simulated (for the case m ¼ 2). at the minimum. y is reported as a percentage of the bulk soft-mode displacements of BaTiO Figure 1 Structure of a typical ferroelectric capacitor. a, Schematic view of the system and foratomsfields of the order of those of Fig. 3b), screenin Nature423506(2003) considered here: a BaTiO film between SrRuO electrodes that are short-circuited. (y ¼ 1 corresponds to the distortion of the bulk tetragonal ferroelectric phase).The energy © 2003 Nature Publishing Group NATURE | VOL 422 | 3 APRIL 2003 | www.nature.com/nature 507 uncompensated surface polarization charges ðjpol < 1026 of the paraelectric phase is taken as reference. Inset, evolution of the spontaneous The whole structure is grown on top of a SrTiO substrate. The epitaxy is such that polarization, P , with thickness. For each thickness, P is deduced from SrTiO (001) k SrRuO (001) k BaTiO (001)k SrRuO (001). b, Atomic representation of the amplitude of y would require a timescale of the order of 0.1 s. Compare the related supercell that is simulated (for the case m ¼ 2). at the minimum. switching of the polarization (typically ,1029 s), this is a v © 2003 Nature Publishing Group NATURE | VOL 422 | 3 APRIL 2003 | www.nature.com/nature 507 can therefore conclude that this process cannot process. We Effectivescreeninglength the film switching back to the paraelectric state below th thickness. For m larger than the critical value and after a su long time, the finite conductivity could allow the appea additional screening charges. However, as it happens for coming from the electrodes, the finite spread of this extr 3 3 3 3 3 3 3 3 s s M 5. Multiferroics Multiferroic:materialcombiningtwoormoreferroic parameters Ferromagnetic: ● N N N S S S N N N S S S Ferroelectric: M P Ferroelectric: P + - + - + + - - + + - - ● Hysteresis: Multifunctional magnetoelectrics (Generalized) Magnetoelectric: cross coupled response to electric and magnetic fields Non-volatile data-storage! Magnetoelectric: - + Polarization, P E H Claude Ederer First principles studies of multiferroic materials P M Magnetization, M Possibleapplications: i.e. control of the magnetic M (electric P) phase with an applied electric E (magneticRAM:electricwrite/magneticread H) field - Magnetoelectric - 4-statememory 5. Multiferroics Ferroelectricity as a lattice property FE lattice distortion, Q, has identical symmetry properties as the polarization, i.e. Q∝ P, involve small atomic distortions Chemicalincompatibility (Spaldin,J.Phys.Chem.B104 (2000)) Pb FErequires“d0-ness” Ti O FMrequirespartiallyfilledd cubic-paraelectric Commonroutetoavoidthisrule: Space group Pm3m P + – Off-centering drivenby hybridisation betweenemptyd andoccupied2p Cohen Nature 358, 136 (1992) tetragonal-ferroelectric Space group P4mm - haveFEandFMactiveionsondifferentsiteeg BiFeO3EuTiO3 Cornell University! School of Applied and Engineering Physics ! Problem:fewFEbuildingblockstostartfrom+weakcoupling? fenniegroup.aep.cornell.edu! 42 5. Multiferroics Ferroelectricity as a lattice property FE lattice distortion, Q, has identical symmetry properties as the polarization, i.e. Q∝ P, involve small atomic distortions Chemicalincompatibility (Spaldin,J.Phys.Chem.B104 (2000)) Pb FErequires“d0-ness” Ti O FMrequirespartiallyfilledd cubic-paraelectric Commonroutetoavoidthisrule: Space group Pm3m P + – Off-centering drivenby hybridisation betweenemptyd andoccupied2p Cohen Nature 358, 136 (1992) tetragonal-ferroelectric Space group P4mm - haveFEandFMactiveionsondifferentsiteeg BiFeO3EuTiO3 Cornell University! School of Applied and Engineering Physics ! Problem:fewFEbuildingblockstostartfrom+weakcoupling? fenniegroup.aep.cornell.edu! Tomorrow’stalk:UsingimproperFEtocreatemultiferroics 42