Polarization, Magnetic Ordering and Structural Instabilities in Biferroic Materials: A First-principles Study

1. Polarization, Magnetic Ordering and Structural Instabilities in Biferroic Materials: A First-principles Study Umesh V. Waghmare Theoretical Sciences Unit J Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560 064 INDIA http://www.jncasr.ac.in/waghmare Funded by Department of Science and Technology, Govt of India

2. Collaborator • Nirat Ray (JNCASR) • Motivated from earlier work(s) in collaboration with experimental groups of C N R Rao (JNCASR), A K Sood and S B Krupanidhi (IISc)

3. Outline • Introduction: Ferroics and biferroics • Polarization Puzzle in biferroics • Our new scheme to compute polarization • A biferroic family of chromites: Interplay between magnetic ordering and structural instabilities (phonons)

4. Ferroics and Multiferroics • Smart materials: Sense external stimulus (sensor) Actuate a controlled response (actuator) • Memory effects Switchability between different states

5. INTRODUCTION Eg. Piezoelectrics Mechanical response Mechanical stimulus Electrical signal Electrical response

6. Ferroelastic Material Strain T Eg. Lead Ortho Phosphate

7. Important Fields and Couplings in : smart materials Stress strain piezoelectric piezomagnetic magnetization polarization magnetoelectric Magnetic Field Electric Field

8. History M = X E

9. INTRODUCTION (out of 122)

10. M Spontaneous Magnetization P Spontaneous Polarization Classification of some Functional Materials FERRO-MAGNETIC FERRO-ELECTRIC FERRO-ELASTIC Biferroic : Ferroelectric + Ferro/Antiferro-magnetic

12. Mechanisms in multiferroic ((A)FM+FE) • Lone pair of A atom: FE and d electrons of B atom: FM Eg. BiMnO3, BiFeO3 • (Ref. Wang et al, Science 299, 1719 (2003)) • 2. Geometric (Structural) ferroelectricity in a magnetic compound, Eg. YMnO3, InMnO3 (hexagonal) • (Ref. Van Aken et al, Nature Mat. 3, 164 (2004)). • 3. Magnetoelastic structural modulation to give FE in a magnetic compound. Eg. TbMnO3, DyMnO3 • (Ref. Kimura et al, Nature 426, 55 (2003)). • 4. Proposed Mechanism: Superposition of two different charge-ordered states to give a dipole moment. • Eg. La1-xCaxMnO3, x<0.5 • (Ref. Efremov at al, Nature Mat 3, 853 (2004)).

13. Polarization Puzzle in Biferroics: A scheme to determine P Nirat Ray and U V Waghmare, a preprint (2007).

14. Polarization in biferroics Values calculated from first-principles do not always agree with experiment. Values estimated with a simple ionic model often do not agree with experiment. Example: bulk BiFeO3: Bi ions are quite off-centered with respect to their centro-symmetric position, giving a large dipole! But the measured value of P is rather small (by almost a factor of 10).

15. Summary: BiFeO3 Bulk Epitaxial

16. Mystery of Polarization: BiFeO3 FE PE With Neaton, Spaldin, Rabe et al.PRB 71, 014113 (05).

17. : does not work for an infinite crystal Polarization: Berry phase (Ref. King-smith and Vanderbilt, Phys. Rev. B47, R1651 (1993); R. Resta, RMP 66, 899 (94)). P has to be defined through a changeΔP, arising from an adiabatic flow of charge when a system is changed from one state to another: (R: direct space lattice vector) P forms a lattice

18. P of a centrosymmetric phase can be nonzero! P = eR/V/2: half integer quantum P A real material example: Biferroic InMnO3 (isostructural to YMnO3) P=19 µC/cm2 (as obtained using Berry phase) But for the ref. PE structure: P = half integer quantum = 27 µC/cm2 ΔP = 8 µC/cm2 should be a measured P Polarization Issue: example InMnO3 Serrao, Waghmare, Rao, et al, J. Appl. Phys 100, 076104 (2006). A similar consideration holds true for YMnO3! Van Aken, Spaldin et al, Nat. Mat. 3, 164 (2004).

19. PTO Four Unit Cells PTO Single Unit Cell Pz calculated using Modern Theory of P depends on the choice of unit cell! Unit cell is doubled in directions PERPENDICULAR to the direction along which we calculate polarization Because the P quantum in the Modern Theory of Polarization depends on the choice of unit cell. Note that changes in P(if smaller than quantum P) are independent of the unit cell!

20. Issues with P • Interpretation of absolute polarization estimated in MTP with observed P (order parameter of ferroelectrics) is tricky, and should be done with care. • Can the non-abelian geometric phases (Г matrices) help? Bhattacharjee and Waghmare, Phys Rev B 71, 045106 (05)). Band-by-band decomposition of P. Eg. Superlattices (Vanderbilt et al, PRL 06). • How to use Г matrices in obtaining P that is easier to connect with experimental P? (Nirat Ray and U. V. Waghmare, a preprint).

21. Non-abelian Geometric phases and polarization in crystals

22. Connecting Bloch functions at k and k+Δk Overlap matrix S : Overlap matrix, S, in terms of R and Γk as, Parallel transportedwave functions satisfy “parallel transport” gauge: Discretized Parallel Transport and Г Bloch functions of the form : Determination of R and Γ matrices  Singular Value Decomposition of overlap matrix, S=U∑V† Rotate wave functions at k+Δk byM = (UV† )* → S’ = R’ Eigenvalues of Г matrix, τi = eigenvalues ofPrP * (2π/a)

23. Ionic polarization • Ionic positions (di) are remapped between [-0.5,0.5) • Electronic polarization Centre of ionic charge defined as: Each eigenvalue of Γ matrix folded between [-0.5,0.5) is shifted by -1 or +1 so that it is closer to the centre of an ionic charge å Z d i i = i d å center Z i i Electronic Polarization Our scheme to determine P Justification: Implicitly, use paths expressed in the space of ionic displacements

24. Real-space decomposition of P:Distribution of atomic and electronic centres ionic ionic electronic electronic ionic Note that this distribution is independent of choice of unit cell Our scheme ties computation of both ionic and electronic P together! electronic

25. YCrO3 with A-type AFM ordering Polarization as a function of parameter “s” Discontinuity in the Polarization as calculated in MTP, when the |polarization| becomes greater than (half the) quantum of polarization.

26. Rare- Earth Chromites: A new family of biferroics Discovered at JNCASR

27. Experimental Background: Temperature variation of magnetization of YCrO3 Dielectric Hysteresis at different temperatures for YCrO3 Variation of the TE and TN with radius of rare earth • Magnetic measurements show magnetic hysteresis • P-E measurements show hysteresis. • Heavy rare-earth chromites HoCrO3, ErCrO3,YbCrO3, LuCrO3 and YCrO3 belong to a new family of multiferroics canted antiferromagnetism and ferroelectricity. C. R. Serrao, A. K. Kundu, S. B. Krupanidhi, U. V. Waghmare, and C. N. R. Rao, Phys Rev B 72, 220101(R) (2005) J. R. Sahu, C. R. Serrao, N. Ray, U. V. Waghmare and C. N. R. Rao, J. Mater. Chem. (comm) 17, 42–44 (2007)

28. Coupling between magnetic ordering and structural instabilities in rare-earth chromites N. Ray and U.V. Waghmare, preprint Motivation • Learn from Chromites and design of new biferroic materials • Try to explore the origin of very small value of polarization (< 3 μC/cm2) in these materials  Arising from competing structural instailities? • Look into the proposal of locally broken centrosymmetry and globally preserved centrosymmetry (CNR Rao et al, J Phys C (2007)).

29. Structural Parameters, INPUT Methodology BEC u0+ujβ OUTPUT P DFT code PWSCF Force, Fiα Energy, E FROZENPHONON CALCULATIONS DIAGONALISATION ujβ: Displacement of jth atom in β-direction Fiα:Force on ith atom in α-direction Eigenvalues of D  ω2

30. High density of Cr 3d states at Fermilevel in PM phase • Instability in structure • Stabilisation: Spin polarisation and/or Structural distortion AO projected Density of states (DOS) PM FM

31. Cr Cr Cr Cr Cr Oxygen p-orbital Cr eg t2g eg t2g Cr Cr - d orbital Cr - d orbital Rombhohedral Symmetry Cubic structure with AFM ordering G- type AFM : All bonds are antiferromagnetic

32. G-AFM

33. Orbital Projected DOS for AFM ordering • LUMO has both Lu and Cr d-orbital character

34. Born Effective Charges (BEC’s) for AFM structure, along the Z-axis (xx component) • Anomalous value of Z* on the A-cation • Ferroelectricity driven by A-cation? La Lu Size of A cation is important: La, being too large, does not undergo off-centering.

35. Phonon Frequencies LaCrO3 Paraelectric at all temperatures • No Г-point instabilities in the PM form but, exhibits weak ferroelectric instability upon FM ordering. • Substantial softening of the Г25 mode in the FM phase as compared to the PM phase LuCrO3 Ferroelectric • Strong FE instabilities in both PM and FM orderings. • Г25 mode becomes strongly unstable in the FM phase

36. Unstable phonon modes for AFM ordering • R-point modes are the dominant instabilities • R25 mode  Oxygen rotations • Most unstable! • Competition between R25andГ15instabilities R25 mode Z Y X

37. Г25 Mode Г 15 Mode A-cations moving opposite to B cations andOxygens (A-off centre mode) Oxygen displacements (Non-polar mode) R25 Mode R15 Mode Displacement of A-cations (and small Oxygen displacements) Rotation of Oxygen octahedra

38. Change in Cr-O-Cr Bond angle Change in Cr-O Bond length • Modes involving change in Cr-O-Cr bond angle are more stable in the AFM (Eg. The Г25 mode and the stable Г15 mode). • Modes involving change in the Cr-O bond length ( R25’ and R2’) become softer in PM ordering.

39. Cr-O-Cr  153° Stabilization of FM ordering ? By freezing in the Г25 mode along the <100> and <111> directions Involves motion of only oxygen ions (Non-polar mode) Г25 mode Stabillizes FM phase (though large distortions)

40. Change in B-cation: Fe instead of Cr • Fe : d5 configuration stronger superexchange interaction between eg orbitals (as compared to t2g orbitals of Cr) • Opposite effect on the modes involving changes in the Fe-O-Fe bond angle

41. Effect of Correlations on phonons

42. Summary Non-abelian geometric phases • Help with estimation ofpolarization that is more readily interpreted Rare Earth Chromites • G-type antiferromagnetic ordering most stable in accordance with superexchange arguments (Insulating in AFM phase) • Most dominant instabilities associated with rotation of oxygen octahedra • Certain (Eg. Г25) instabilities depend sensitively on magnetic ordering • Ferroelectric structural distortions driven by the size of A-cations