Electric displacement as the fundamental variable in electronic-structure calculations

Electric displacement as the fundamental variable in electronic-structure calculations CECAM - Centre Européen de Calcul Atomique et Moléculaire EPF Lausanne, Switzerland Conference UC Davis, 6/23/2009

Acknowledgements Nicola Spaldin UC Santa Barbara David Vanderbilt Rutgers University FUNDING: Department of Energy SciDac programme on Quantum simulation of materials and nanostructures, grant number DE-FC02-06ER25794

Ferroelectrics: bulk Methods Macroscopic description Spontaneous P, switchable with E Microscopic mechanisms (e.g. BaTiO 3 ) Unstable polar phonon mode Subtle balance of long-range and short-range interactions

Ferroelectrics: thin films Thin films desirable for applications Ferroelectric memories Gate dielectrics However: Strong size effects Polarization is reduced Y. S. Kim et al., APL 86, 102907 (2005). Polar state relaxes over time D. J. Kim et al., PRL 95, 237602 (2005).

What is the origin of the problem? Strong dependence on electrical boundary conditions No screening: uniform P is suppressed by depolarizing field Imperfect screening: P might be allowed if thick enough Popular explanation: Intrinsic interface effect, Thomas-Fermi model Black and Welser, IEEE Trans. 46, 776 (1999). Imperfect screening P C i No screening P Cfilm C i V(x) V(x) This talk: Interfacial capacitance C I depends critically on chemical bonding We can make C I negative!

What is the origin of the problem? Strong dependence on electrical boundary conditions No screening: uniform P is suppressed by depolarizing field Imperfect screening: P might be allowed if thick enough Popular explanation: Intrinsic interface effect, Thomas-Fermi model Black and Welser, IEEE Trans. 46, 776 (1999). This talk: Interfacial capacitance C I depends critically on chemical bonding We can make C I negative! ρ Imperfect screening P C i No screening METAL P Free carriers Cfilm C i Polarization charge V(x) FERROELECTRIC x V(x)

What theory to use? Want a microscopic description of realistic metal-ferroelectric interfaces Density-functional theory E(ρ) = E Kinetic (ρ) E Hartree (ρ) E XC (ρ) Want to calculate the electrical properties (e.g. polarization, capacitance,...), modern theory of polarization Very recent developments... V P M. Stengel and N. A. Spaldin, PRB 75, 205121 (2007). M. Stengel, N. Spaldin and D. Vanderbilt, Nature Physics 5, 304 (2009).

Outline Methods 1 Methods 2 Interface ferroelectricity via chemical bonding

Finite fields in periodic solids Want to apply a uniform external electric field E Idea: F(ρ, E) = E(ρ) ρ(r)v (r)d 3 r, V (r) = E r The scalar potential is not periodic Finite potential drop V Solution: Rewrite the coupling term ρ(r)v (r)d 3 r ΩE P

Finite fields in periodic solids Want to apply a uniform external electric field E Idea: F(ρ, E) = E(ρ) ρ(r)v (r)d 3 r, V (r) = E r P defined as a Berry phase The scalar potential is not periodic Finite potential drop V Solution: Rewrite the coupling term ρ(r)v (r)d 3 r ΩE P King-Smith and Vanderbilt, PRB 47, 1651 (1993); Resta, Ferroelectrics 136, 51 (1992).

Finite fields in periodic solids The scalar potential is not periodic Finite potential drop V Solution: Electric enthalpy: F(E) = E KS ΩE P I. Souza, J. Iniguez and D. Vanderbilt, PRL 89, 117602 (2002). There is no electronic ground state Zener tunneling instability Solution: Finite k-point sampling makes the electrons nearsighted

Outline Methods 1 Methods 2 Interface ferroelectricity via chemical bonding

Metal-insulator heterostructures Insulator Metal Unit cell (x direction) E gap Valence band Conduction band Conducting in plane, insulating along x I should be able to apply a finite bias potential...

Capacitors at finite E Methods Electric field Conduction band E gap V Valence band Metal Unit cell (x direction) Insulator P and coupling to E defined with Wannier functions The metastable polarized state is well defined as long as V < E gap (no Schottky tunneling) M. Stengel and N. A. Spaldin, PRB 75, 205121 (2007).

Outline Methods 1 Methods 2 Interface ferroelectricity via chemical bonding

Is E a good choice? Methods Ferroelectrics (e.g. PbTiO 3 ) have a polar instability in the centrosymmetric configuration Double-well potential Start from P = 0 Apply a small E field Large response I cannot access the unstable region! This region is necessary for modeling, so the answer is NO Energy [mev] 40 20 0-20 -40-1 0 1 Polarization [C/m 2 ]

Is E a good choice? Methods Ferroelectrics (e.g. PbTiO 3 ) have a polar instability in the centrosymmetric configuration Double-well potential Start from P = 0 Apply a small E field Large response I cannot access the unstable region! This region is necessary for modeling, so the answer is NO Idea: Try constraining D = E 4πP instead... Energy [mev] 40 20 0-20 -40-1 0 1 Polarization [C/m 2 ]

Solution: Fixed-D method Linear coupling Fixed-E } F(E) = min v {E KS (v)ωe P(v) Closed-circuit capacitor at given V V E Quadratic coupling Fixed-D U(D) = min v { E KS (v) Ω 8π [D 4πP(v)]2 } Open-circuit capacitor at given Q Full control over the electrical boundary conditions In open circuit the polar instability is removed M. Stengel, N. Spaldin and D. Vanderbilt, Nature Physics 5, 304 (2009).

Solution: Fixed-D method Linear coupling Fixed-E } F(E) = min v {E KS (v)ωe P(v) Closed-circuit capacitor at given V Quadratic coupling Fixed-D U(D) = min v { E KS (v) Ω 8π [D 4πP(v)]2 } Open-circuit capacitor at given Q V E Q D -Q Full control over the electrical boundary conditions In open circuit the polar instability is removed M. Stengel, N. Spaldin and D. Vanderbilt, Nature Physics 5, 304 (2009).

Further advantages of using D: locality principle D is constant throughout a layered heterostructure... E P D A B C...if we know A, B and C we can predict ABC! U ABC (D) = U A (D) U B (D) U C (D) Quantum theory old-school electrostatics

Piezoelectricity: treatment of strains V E V E V or Q remain constant under an applied strain, not E or D Proper treatment of piezoelectric effects Relationship to the fields: V = E a 3, Q = D (a 1 a 2 ) 4π Legendre transform: F(V ) = min Q [U(Q) QV ]

Piezoelectricity: treatment of strains Q Q Q D D V or Q remain constant under an applied strain, not E or D Proper treatment of piezoelectric effects Relationship to the fields: V = E a 3, Q = D (a 1 a 2 ) 4π Legendre transform: F(V ) = min Q [U(Q) QV ] Q

Outline Methods Interface ferroelectricity via chemical bonding 1 Methods 2 Interface ferroelectricity via chemical bonding

Interface ferroelectricity via chemical bonding Computational models: four capacitors Pt SrRuO 3 Pt Pt 2:AO ATiO 3 AO:Pt 2 Pt N=8.5 BaTiO 3 PbTiO 3 SrO:TiO 2 TiO 2:SrO SrRuO 3 ATiO 3 SrRuO 3 Pt SrRuO 3 N=6.5 Ru Ti O Pt A=Ba,Pb (not shown)

Stability of the paraelectric state Interface ferroelectricity via chemical bonding Inverse capacitance ( ) 2 C 1 = 4π d 2 U D=0 S dd 2 Stability against polar distortion C 1 > 0 stable C 1 < 0 unstable U [mev] 10 0-10 U ~ C -1 D 2-20 -10 0 10 20 D [µc cm -2 ] Results for C 1 S [m 2 /F] Impact of bulk vs. interface effects? SrRuO 3 Pt PbTiO 3 0.44-1.43 BaTiO 3-1.55-7.92

Interface ferroelectricity via chemical bonding Local dielectric response: ferroelectric regime Inverse permittivity profile: ɛ 1 (z) = de(z) dd D=0 Strong intrinsic effect, localized at the interface Series capacitor model C 1 = 2C 1 I NC 1 bulk Interface term C 1 I : SrRuO 3 Pt PbTiO 3 1.73 1.26 BaTiO 3 2.28 0.04 Units of inv. capacitance density, m 2 /F Inverse permittivity 0.01 0-0.01-0.02-0.03 SRO RuO 2 SrO TiO 2 AO TiO 2 STO PTO BTO Pt Pt AO TiO 2 AO TiO 2 PTO BTO

Critical thickness for ferroelectricity Interface ferroelectricity via chemical bonding Series capacitor model: C 1 (N) = 2C 1 I NC 1 bulk Critical thickness for FE: C 1 = 0 N crit = 2C1 I C 1 bulk SRO Pt PbTiO 3 8 6 BaTiO 3 5 0 Thickness in number of unit cells Inv. capacitance C -1 [m 2 F -1 ] 4 2 0-2 -4 PTO/Pt BTO/Pt BTO/SRO PTO/SRO STO/SRO 2 4 6 8 10 Film thickness N [unit cells] Paraelectric Ferroelectric N crit = 0 in BaTiO 3 /Pt?!

Interface ferroelectricity via chemical bonding Chemical bonding at the AO-Pt interface Unsupported AO-terminated interfaces Both O and A in the atop site Frustrated bonding environment, two competing effects: Pt-O attraction (bonding) Pt-A repulsion (non-bonding) δ Pt 2 1 d(0) δ AO P=0 d (Å) δ Pt (Å) δ AO (Å) A=Ba 2.26 0.46 0.12 A=Pb 2.11 0.34 0.19 Pt-O 2.0 Å in bulk oxide phases

Interface bonding and ferroelectricity Interface ferroelectricity via chemical bonding Idea: Loose interface bonds favor P Electrode P Force constants at the interface: PbTiO 3 BaTiO 3 KPtO L 3.9-0.1 (K ij = 2 U z i z j, units of 10 3 a.u.) Computational experiment: Artificially soften Pt-O bond in PbTiO 3 /Pt

Interface bonding and ferroelectricity Interface ferroelectricity via chemical bonding Idea: Loose interface bonds favor P 0.01 Pt Electrode Force constants at the interface: PbTiO 3 BaTiO 3 KPtO L 3.9-0.1 (K ij = 2 U z i z j, units of 10 3 a.u.) Computational experiment: Artificially soften Pt-O bond in PbTiO 3 /Pt P Inverse permittivity 0-0.01-0.02-0.03 Pt AO TiO 2 AO TiO 2 PTO BTO

Interface bonding and ferroelectricity Interface ferroelectricity via chemical bonding Idea: Loose interface bonds favor P 0.01 Pt "Softened" Pt-O bond Electrode Force constants at the interface: PbTiO 3 BaTiO 3 KPtO L 3.9-0.1 (K ij = 2 U z i z j, units of 10 3 a.u.) Computational experiment: Artificially soften Pt-O bond in PbTiO 3 /Pt P Inverse permittivity 0-0.01-0.02-0.03 Interface ferroelectricity driven by Pt-O bond! Pt AO TiO 2 AO TiO 2 PTO BTO

What about the polar ground state? Interface ferroelectricity via chemical bonding So far: paraelectric reference structure Out of curiosity: look at P in the ferroelectric state (BaTiO 3 /Pt) N =1, 2, 3, 5, 7 Standard short-circuit Polarization and distortions increase the thinner the film Chemical bonding mechanism even stronger in the FE state! Layer rumpling [Å] Polarization [C/m 2 ] 0.5 0.4 0.3 0.2 0.1 0 0.55 0.5 0.45 0.4 N=1 goes polar 2 4 6 8 10 12 14 Layer number Bulk 0.35 0 2 4 6 8 Film thickness

Negative dead layer regime Interface ferroelectricity via chemical bonding Potential SRO contact BaTiO3 ferroelectric E P SRO contact Layer rumpling [Å] 0.5 0.4 0.3 0.2 0.1 N=1 goes polar Dead layer Dead layer 0 2 4 6 8 10 12 14 Layer number 0.55 Pt contact Undead layer P E Undead layer Pt contact Polarization [C/m 2 ] 0.5 0.45 0.4 Bulk x coordinate 0.35 0 2 4 6 8 Film thickness

Summary Methods Layer-by-layer device design via constrained-d density-functional theory M. Stengel, N. A. Spaldin and D. Vanderbilt, Nature Physics 5, 304 (2009). E P D A B C Enhancement of ferroelectricity driven by electrode-film bonding M. Stengel, D. Vanderbilt and N. A. Spaldin, 1 d(0) 2 δ Pt δ AO P=0 Nature Materials 8, 392 (2009). d() P d()

Microscopic analysis of C 1 Decomposition into LO polar modes ( C 1 = (C ) 1 i,x (K L ) 1 4π S ) 2 ij Z L Three basic ingredients: 1. Electronic capacitance C (frozen ionic positions) ij Zj,x L 2. LO dynamical charges Z L (lattice-electric field coupling) 3. Interatomic force constants K L ij (bond stiffness )

Purely electronic effects (C and Z L ) C (frozen ions) depends on center of induced ρ PTO/Pt better than BTO/Pt! ρ [10-2 e bohr -3 ] 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Pt Pt 2 AO TiO 2 AO TiO 2 PTO BTO Dynamical charges are smaller near the interface PTO/Pt similar to BTO/Pt! L Callen charge Z 1.0 0.5 Pt 0.0-0.5 Pt / PTO Ti Pb O2 O1 Pt Pt / BTO Ti Ba O2 O1