MAGNETIC ANISOTROPY IN TIGHT-BINDING
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1 MAGNETIC ANISOTROPY IN TIGHT-BINDING Cyrille Barreteau Magnetic Tight-Binding Workshop, London10-11 Sept SEPTEMBRE 2012 CEA 10 AVRIL 2012 PAGE 1
2 SOMMAIRE TB Model TB 0 : Mehl & Papaconstantopoulos P.04 TB LCN : Local charge neutrality P.06 TB mag : Stoner model P.07 Beyond Stoner model: Coulomb Interaction U P.09 TB SO : Spin Orbit Coupling P.13 Determination of parameters TB 0, TB LCN, TB mag, TB SO P.15 Applications Fe from bulk to wire P.20 FePt from bulk to clusters P.27 9 SEPTEMBRE 2012 CEA Sept PAGE 2
3 TB MODEL CEA 10 AVRIL 2012 PAGE 3 9 SEPTEMBRE 2012
4 TB 0 : MEHL AND PAPACONSTANOPOULOS Non magnetic Hamiltonian H0 i i H j j i, j: atoms ij, : orbitals s px py pz dxy dxz d 2 2 xz d d x y 3z 2 r 2 Hopping integrals overlap integrals On-site elements S i j i i H i i, j i H j Two-center SK formulation i, j ( R), S( R) R 9 SEPTEMBRE 2012 CEA SEPT PAGE 4 i f ( ) i
5 Analytical expressions ( R), S( R) ( e fr )exp( gr) F ( R) a b c d exp( R ) F ( R ) 2/3 4/3 2 i i i i Approximately parameters per element Total energy: the MP trick Etot f C i ij C ij ji The usual pair-potential «repulsive term» is hidden in the environment dependence of the on-site elements PRB 54, 4519 (1996) 9 SEPTEMBRE 2012 CEA SEPT PAGE 5 PRB 58, 9721 (1998)
6 TB LCN : LOCAL CHARGE NEUTRALITY Inhomogeneous systems H U ( n n ) i, i LCN i i i,0 1 H U ( n n ) U ( n n ) 2 S i, j LCN i i i,0 j j j,0 i, j Avoid charge transfers between unequivalent atoms LCN = penalization on the local charge 1 tot i j ij 2 occ i, j i E c c H U ( n n ) 2 i tot 0 i i i i,0 Min E / c 1 H H H Double counting E f U n n tot 2 i ( i i,0) i 9 SEPTEMBRE 2012 CEA SEPT PAGE 6 2 LCN
7 TB MAG : STONER MODEL I m 1 i,, i,,0 2 i, i, I I I 1 s p 10 d Exchange splitting 1 H mag 2Ii, mi. 1 i, Stoner criterion Onset of magnetism when: D ( E) Im D ( E) D ( E IM ) ID ( ) 1 0 EF D ( E) E E 1 2 F 1 tot 4 i i, ( ) 4 i, i, E I m ED E de Im occ Double counting i 2 i, 9 SEPTEMBRE 2012 CEA SEPT PAGE 7
8 Fixed spin moment calculation E F F E ED ( E) de ED ( E) de Im tot E D ( E) D ( E IM ) Em ( ) m 0 m 1 m E EF F E ( M ) ED ( E) de ED ( E) de Im tot EF EF 0( ) 0( ) 1 F F 2 EF EF 1 0( ) 0( ) F F 2 M ED E de ED E de N ED E de ED E de 2 E E Im E E Im E c c H E tot i j ij occ i, j Penalization of local magnetization pen E ( m m ) pen i i i,0 i E (cos cos ) pen i i i,0 i 2 2 z i mi 2 i Min E / c 1 H H H tot 0 pen i 9 SEPTEMBRE 2012 CEA SEPT PAGE 8
9 BEYOND STONER MODEL= TB+U H H H 0 int H U c c c c int i i1 i 2 ' i 4 ' i 3 2 i n df ( )orbitals i : atomic site Hartree Fock decoupling (mean field) 1 H U c c c c U c c c c HF int i i 4 i 3 i 2 ' i1 ' i i 4 i 3 ' i 2 ' i1 2 i ' e Um 1m ' ( ) 2m3m 4 d r d r im r 1 im2 '( r ') im ( r) 3 im4 '( r ') r r' e U d r d r ' i ( r) 1 i 2 '( r ') i ( r) 3 i 4 '( r ') r r' U PRB 76, (2007) linear combination( A, B, C) Racah parameters: ( F0, F2, F4) Slater intergrals 9 SEPTEMBRE 2012 CEA SEPT PAGE 9
10 real orbitals ( ) 4, ( ) 4, 98 U U A B C F F F J U B C F F spherical harmonics (Anisimov) New set of parameters ( A, B, C ) ( U, J, B) 1 7 U A Umm' A C F 25 5 mm' 0 1 U J ( U J ) A A mm' mm' 20 mm' mm' ( 2 4 ) J A B C F F U J A U 7 J 2J 5 9 SEPTEMBRE 2012 A CEA SEPT PAGE A U J U A 5 J 2J 7 A
11 From HF to Stoner ni,, ' ni,, ' n n 1 i, 5 i,, H H ( U n I m ) c c 1 int eff i 2 dd i i i i U (9U 2 J ) / 5 I ( U 6 J ) / 5 eff dd Idd Controls the spin-moment U J, B Controls the orbital-moment and anisotropy 9 SEPTEMBRE 2012 CEA SEPT PAGE 11
12 1 What about TB+V H V c c c c int ij i j ' j ' i 2 ij,, ' i j 1 0 Vij i ( r) j ( r ') i ( r) j ( r ') Vij V0 r- r' R ij Hartree Fock decoupling (mean field) R On-site renormalization V n i i ij j j Im Spin dependent hopping integrals V c c i, j i, j ij j i W W 9 SEPTEMBRE 2012 CEA SEPT PAGE 12 PRB 69, (2004) E
13 SPIN-ORBIT COUPLING HSOC i ( r) Li. S i i z We keep d orbitals only R r r dr 2 2 i, d i, d () x Magnetic anisotropy Band structure anisotropy Orbital moment ML 0.1 B (bulk) M (wire) L B J. Phys.: Cond. Matter 18, 6785 (2006) 9 SEPTEMBRE 2012 CEA SEPT PAGE 13
14 TB PARAMETERS CEA 10 AVRIL 2012 PAGE 14 9 SEPTEMBRE 2012
15 TB 0 Fit on ab-initio band structures and total energy of non-magnetic bulk systems Total energy Band structure 9 SEPTEMBRE 2012 CEA SEPT PAGE 15
16 TB LCN Influence of local charge neutrality on surface band structure No LCN With LCN Rh(111) surface Surf. Sci. 346, 300 (1996) 9 SEPTEMBRE 2012 CEA SEPT PAGE 16 PRB 58, 9721 (1998)
17 TB MAG IFe Determination of the Stoner parameter On-set of magnetism with lattice parameter expansion 0.88,0.95 ev I 0.60eV Pt What about U,J,B parameters I ( U 6 J ) / 5 U / J B / J 0.14 B 0.1eV 9 SEPTEMBRE 2012 CEA SEPT PAGE 17 J. Phys.: Cond. Matter accepted (2012)
18 TB SOC Determination of the Spin-Orbit Coupling constant Non-magnetic band structure Fe Pt 0.06eV [0, 45, 0.55] ev Absolutely not «structure» dependent (same SOC parameter for fcc, bcc, or wire) 9 SEPTEMBRE 2012 CEA SEPT PAGE 18
19 APPLICATIONS CEA 10 AVRIL 2012 PAGE 19 9 SEPTEMBRE 2012
20 IRON FROM BULK TO WIRE BULK IRON Phase stability I Fe 0.95eV Fe bcc stabilized by magnetism Complex magnetic structure of Fe fcc For I 0.88 ev hcp is the ground state.. Fe LS and HS, AF, spin spiral J. Phys.: Cond. Matter 18, 6785 (2006) 9 SEPTEMBRE 2012 CEA SEPT PAGE 20 J. Phys.: Cond. Matter 22, (2010)
21 BULK IRON Complex magnetic structure of Fe fcc M( d) EM ( ) Spin spiral Eq ( ) M( q) 9 SEPTEMBRE 2012 CEA SEPT PAGE 21
22 (110) Band structure IRON SURFACE El () 9 SEPTEMBRE 2012 CEA SEPT PAGE 22 J. Phys.: Cond. Matter 18, 6785 (2006)
23 IRON SURFACE Magnetic anisotropy energy (MAE) MAE Convergence / k points monolayers Orbital moment Easy axis: surface Enhancement of anisotropy De-quenching of orbital moment Bruno s formula E(, ) L(, ) Works well for saturated systems 4 9 SEPTEMBRE 2012 CEA SEPT PAGE 23
24 IRON WIRE Magnetic anisotropy energy (MAE) Band structure MAE Band structure anisotropy (origin of AMR) Easy axis reversal 9 SEPTEMBRE 2012 CEA SEPT PAGE 24
25 IRON WIRE Beyond Stoner: Orbital Polarization Stoner TB+(U,J,B) Is the Orbital Polarization ansatz verified? H U J B H B L L TB (,, ) TB z z? PRB 76, (2007) 9 SEPTEMBRE 2012 CEA SEPT PAGE 25
26 IRON ATOMIC CONTACTGAMR 2.5 B 1.5 B 2 magnetic solutions hysteresis EPL 83, (2008) 9 SEPTEMBRE 2012 CEA SEPT PAGE 26
27 FePt L10: FROM BULK TO CLUSTERS BULK FePt L10 structure c a exp A 2 cexp 3.72A c a exp exp 1.36 Vexp 27.7 A 3 a Very high magnetic uniaxial anisotropy MAE=1.4meV/fu (exp.) 9 SEPTEMBRE 2012 CEA SEPT PAGE 27
28 TB model H U ( n n ) i i U ( n n ) i i 0 0 LCN LCN i i d i, d i, d i id 0 nid, n 0 Fe d adjusted to reproduce electronic and magnetic properties of FePt L10, 6.6 n Charge neutrality, Pt d U U 20eV d d orbital filling M 3 M 0.35 Fe B Pt B 9 SEPTEMBRE 2012 CEA SEPT PAGE 28 J. Phys.: Cond. Matter accepted (2012)
29 BULK FePt Phase stability Functional effect Structural effect 9 SEPTEMBRE 2012 CEA SEPT PAGE 29
30 BULK FePt MAE (c/a) DFT TB 9 SEPTEMBRE 2012 CEA SEPT PAGE 30
31 Clusters Fe Pt R N Concentric spheres 9 SEPTEMBRE 2012 CEA SEPT PAGE 31
32 Clusters FM vs AFM Larger Stoner parameters favors FM Pt surfaces favors AFM (Fe favors FM) 9 SEPTEMBRE 2012 CEA SEPT PAGE 32
33 Clusters MAE FM favors uniaxial anisotropy AFM favors in-plane anisotropy 9 SEPTEMBRE 2012 CEA SEPT PAGE 33
34 Clusters MAE(N) (per formula unit) MAE of clusters is below the bulk value 9 SEPTEMBRE 2012 CEA SEPT PAGE 34
35 CONCLUSION CEA 10 AVRIL 2012 PAGE 35 9 SEPTEMBRE 2012
36 advantages TB is an efficient and versatile method TB is useful to obtain trends by tuning parameters Magnetism is well described by simple Stoner models SOC is easily described in a TB scheme Local basis are well suited to electronic transport formalism disadvantages Often (very) painful to determine parameters It should be handled with care: always check its transferability MAE is a subtle quantity 9 SEPTEMBRE 2012 CEA SEPT PAGE 36
37 Special Thanks to F. Ducastelle M.C. Desjonquères D. Spanjaard C.C. Fu R. Soulairol FeCr G. Autès P. Habibi Spin polarizedtransport 9 SEPTEMBRE 2012 CEA SEPT PAGE 37
38 THANK YOU FOR YOUR ATTENTION PAGE 38 CEA 10 AVRIL 2012 Commissariat à l énergie atomique et aux énergies alternatives Centre de Saclay Gif-sur-Yvette Cedex T. +33 (0) F. +33 (0) DSM IRAMIS SPCSI 9 SEPTEMBRE 2012 Etablissement public à caractère industriel et commercial RCS Paris B
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