Crystalline and Magnetic Anisotropy of the 3d Transition-Metal Monoxides

Transcription

1 Crystalline and of the 3d Transition-Metal Monoxides Institut für Festkörpertheorie und -optik Friedrich-Schiller-Universität Max-Wien-Platz Jena

2 Introduction Crystalline Anisotropy MnO, FeO, CoO, and NiO

3 Introduction MnO, FeO, CoO, and NiO (distorted) rock-salt (rs) structure

4 Introduction MnO, FeO, CoO, and NiO (distorted) rock-salt (rs) structure antiferromagnetic ordering AF2 interesting benchmark materials

5 Introduction Mn 2+ e g MnO, FeO, CoO, and NiO (distorted) rock-salt (rs) structure antiferromagnetic ordering AF2 interesting benchmark materials series of materials successive filling of minority-spin t 2g states study physical trends 3d free atom t 2g d z 2 d xy d x 2 -y 2 d xz d yz octahedral crystal field

6 Crystalline Anisotropy Open Problems What happens at the phase transition? paramagnetic phase ideal rock-salt structure antiferromagnetic phase rhombohedral / monoclinic perturbations

7 Crystalline Anisotropy Open Problems What happens at the phase transition? paramagnetic phase ideal rock-salt structure antiferromagnetic phase rhombohedral / monoclinic perturbations magnetic ordering: antiparallel orientation obvious but: How are they oriented with respect to the crystal? [001] [100]

8 Method Spin Polarized Density Functional Theory (DFT) collinear approximation: E tot (n, n, {R s }) n(x) = n (x) + n (x) m(x) = n (x) n (x) electron density (spin) magnetization density equilibrium: minimum of E tot with respect to n, n, and {R s } Self Consistent Field (SCF) loop n(x) m(x) relaxation of lattice geometry {R s } Hellmann-Feynman forces: Rs E tot = 0 (typically <5 mev/å) geometry with accuracy < 0.01 Å

9 Method Relativistic Corrections account for corrections up to ( ) v 2 c scalar-relativistic corrections (Darwin, mass correction) always included spin-orbit interaction requires non-collinear treatment transversal electron interaction (Breit correction) not included in self-consistent calculation

10 Challenges Spin-Polarized Exchange and Correlation Functional standard approach: local spin density approximation with generalized gradient corrections (GGA) sum rule for XC hole fulfilled interpolation for spin-polarization problem: electrons in TM 3d states strongly localized strong density variations Are local approximations still valid? solution: add a correction functional in real space acting only on TM 3d states Cohesive Energy (ev ) HF HSE rs-af wz-af U (ev ) U Tr(ˆn 2 3d ˆn 3d )

11 Challenges Direcion Dependence of the Magnetization non-collinear approximation: full vector of magnetization density m(x) 2 additional degrees of freedom for rotation besides spin-magnetization also orbital magnetization

12 Crystalline Anisotropy

13 Monoclinic Strain Tensor r e t MnO FeO CoO NiO

14 Monoclinic Strain Tensor r e t MnO FeO CoO NiO

15 Monoclinic Strain Tensor r e t MnO FeO CoO NiO

16 Lattice Distortions Crystalline Anisotropy MnO: rhombohedral distortion angle α GGA+U 0.72 Exp

17 Lattice Distortions Crystalline Anisotropy MnO: rhombohedral distortion angle α GGA+U 0.72 Exp CoO: monoclinic distortion angle β GGA+U 0.80 Exp. 0.30

18 Lattice Distortions Crystalline Anisotropy MnO: rhombohedral distortion angle α GGA+U 0.72 Exp CoO: monoclinic distortion angle β GGA+U 0.80 Exp differences due to orbital occupancy of t 2g sub-shell

19

20 Treatment 2 step procedure step 1 spin-orbit coupling as implemented in VASP step 2 transversal electron-electron interaction due to Breit term approximated by magnetic dipole interaction

21 Spin-Orbit Interaction Crystalline Anisotropy anisotropy energy small (mev) calculated E(θ, φ) = E 0 + E(θ, φ) for various directions θ and φ fit to E = sin 2 θ [K 1 + K 1 cos(2φ)] with anisotropy constants K 1 and K 1 K 1 (mev) K 1 (mev) MnO FeO CoO NiO contributions only for partially filled t 2g states

22 Magnetic Dipole Interaction magnetic dipole interaction energy: E d = 1 µ 0 µ 2 ) 2 4πr1 3 σ l (3 ( ( ˆm l ˆr l ) 2 r 1 1 r l l ) 3

23 Magnetic Dipole Interaction magnetic dipole interaction energy: E d = 1 µ 0 µ 2 ) 2 4πr1 3 σ l (3 ( ( ˆm l ˆr l ) 2 r 1 1 r l l magnetic anisotropy energy in uniaxial symmetry along [111]: E = K1 d sin2 θ ) 3

24 Magnetic Dipole Interaction magnetic dipole interaction energy: E d = 1 µ 0 µ 2 ) 2 4πr1 3 σ l (3 ( ( ˆm l ˆr l ) 2 r 1 1 r l l magnetic anisotropy energy in uniaxial symmetry along [111]: E = K1 d sin2 θ K1 d depends only on material parameters sign(k1 d ) depends only on crystal structure + AF2 ordering ) 3

25 Magnetic Dipole Interaction θ easy axis K d 1 K d 1 > 0: easy axis [111] < 0: easy plane [111] easy plane θ

26 Magnetic Dipole Interaction θ easy axis K d 1 K d 1 > 0: easy axis [111] < 0: easy plane [111] easy plane θ MnO FeO CoO NiO µ (µ B ) a 0 (Å 3 ) K (mev)

27 Magnetic Dipole Interaction θ easy axis K d 1 K d 1 > 0: easy axis [111] < 0: easy plane [111] easy plane θ MnO FeO CoO NiO µ (µ B ) a 0 (Å 3 ) K (mev) contributions small but essential for MnO and NiO

28 Summary good agreement of crystalline anisotropy with experiments for all TMOs FeO and CoO: magnetic anisotropy dominated by spin-orbit coupling MnO and NiO: transversal electron-electron interaction is dominating in total small magnetic anisotropy crystalline and magnetic anisotropy depend on occupancy of t 2g shell