Exchange Mechanisms. Erik Koch Institute for Advanced Simulation, Forschungszentrum Jülich. lecture notes:

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1 Exchange Mechanisms Erik Koch Institute for Advanced Simulation, Forschungszentrum Jülich lecture notes:

2 Magnetism is Quantum Mechanical QUANTUM MECHANICS THE KEY TO UNDERSTANDING MAGNETISM Nobel Lecture, 8 December, 977 J.H. VAN VLECK Harvard University, Cambridge, Massachusetts, USA Bohr van Leeuwen theorem in a classical system in thermal equilibrium a magnetic field will not induce a magnetic moment Lorentz force perpendicular to velocity does not change kinetic energy Boltzmann statistics occupies states according to energy

3 magnetic moments j( r )= complex wave function: current density e~ im e ( r )r ( r ) ( r )r ( r ) µ = Z orbital magnetic moment r jd 3 = e~ m e hl i = µ B hl i electron spin µ S = g e µ B hs i, g e atomic moments of the order of µb

4 magnetic interaction dipole-dipole interaction E = µ µ 3( ˆR µ )( ˆR µ ) 4 0 c R 3 µ R µ interaction energy of two dipoles µb two Bohr radii a0 apart: E = µ B 4 0 c (a 0 ) 3 = / 37 8 Hartree 0.09 mev expect magnetic ordering below temperatures of about K what about magnetite (Fe3O4) with Tc 840 K?

5 exchange mechanisms coupling of magnetic moments results from the interplay of the Pauli principle with Coulomb repulsion and electron hopping not a fundamental but an effective interaction: model/mechanism

6 Models and Mechanisms The art of model-building is the exclusion of real but irrelevant parts of the problem, and entails hazards for the builder and the reader. The builder may leave out something genuinely relevant; the reader, armed with too sophisticated an experimental probe or too accurate a computation, may take literally a schematized model whose main aim is to be a demonstration of possibility. P.W. Anderson Local Moments and Localized States Nobel Lecture 977

7 Coulomb Exchange Coulomb repulsion between electrons H U = X i<j r i r j consider two electrons in orthogonal orbitals φa and φb Slater determinant of spin-orbitals: a, ; b 0 ( r,s ; r,s )= p a ( r ) (s ) a ( r ) (s ) b ( r ) 0 (s ) b ( r ) 0 (s ) = p a ( r ) a ( r ) (s ) 0 (s ) b ( r ) a ( r ) 0 (s ) (s )

8 Coulomb exchange: same spin when electrons have same spin: σ = σ a, ; b = p a ( r ) b ( r ) b ( r ) a ( r ) (s ) (s ) a, ; b, r r Coulomb matrix-element a, ; b, = (U ab J ab J ba + U ba )=U ab J ab Coulomb integral U ab = exchange integral J ab = Z Z d 3 r Z d 3 r Z d 3 r a( r ) b( r ) r r d 3 r a( r ) b ( r ) b( r ) a ( r ) r r

9 Coulomb exchange: opposite spin when electrons have opposite spin: σ = -σ a,";b#( r,s ; r,s )= p a ( r ) b ( r ) "(s )#(s ) b ( r ) a ( r ) #(s )"(s ) a,#;b"( r,s ; r,s )= p a ( r ) b ( r ) #(s )"(s ) b ( r ) a ( r ) "(s )#(s ) diagonal matrix-elements off-diagonal matrix-elements a, ; b, a"; b# r r r r a, ; b, = U ab a#; b" = J ab Coulomb matrix Uab J ab J ab U ab

10 Coulomb exchange H U = 0 U ab J ab U ab J ab 0 0 J ab U ab U ab J ab C A "" "# #" ## eigenstates triplet: Δεtriplet = Uab Jab "" = a ( r ) b ( r ) b ( r ) a ( r ) "# + #" = a ( r ) b ( r ) b ( r ) a ( r ) + ## = a ( r ) b ( r ) b ( r ) a ( r ) "# #" singlet: Δεsinglet = Uab + Jab = a ( r ) b ( r )+ b ( r ) a ( r )

11 Coulomb exchange orthogonal orbitals φa and φb: Jab > 0 singlet Jab triplet first of Hund s rules: ground-state has maximum spin S L J d-shell more electrons more complicated Coulomb matrix number of electrons Multiplets in Transition Metal Ions

12 Q. Zhang: Calculations of Atomic Multiplets across the Periodic Table MSc thesis, RWTH Aachen 04 atomic multiplets

13 kinetic exchange Coulomb exchange: Coulomb matrix for anti-symmetric wave functions kinetic exchange: only diagonal U, interplay of Pauli principle and hopping toy model two sites with a single orbital hopping between orbitals: t two electrons in same orbital: U φ -t φ one electron Hamiltonian (tight-binding) H = 0 t t 0,, ± = eigenstates ± ± = t

14 direct exchange: same spin two electrons of same spin: basis states,,, Hamiltonian: no hopping, no Coulomb matrix element (Pauli principle) H = ,, εtriplet = 0

15 direct exchange: opposite spin two electrons of opposite spin: basis states,,, (covalent states),,, (ionic states) H = 0 Hamiltonian 0 0 t t 0 0 +t +t t +t U 0 t +t 0 U C A,,,, hopping -t: keep track of Fermi sign!, t,, ( t),

16 direct exchange: opposite spin ± = U ± U + 6 t eigenstates ±,, t, +,, ± = q + ±/(t ) cov =0, cov =, +, ion = U, ion =,, (εtriplet) 0 8 limit U (or t 0):! / t ε+ ionic covalent ε U / t! U +4t /U +! 4t /U

17 downfolding partition Hilbert space H = H 00 T 0 T 0 H resolvent G( ) = ( H) = H 00 T 0 T 0 H inverse of block-matrix G 00 ( ) = H00 + T 0 ( H ) T 0 downfolded Hamiltonian H e H 00 + T 0 ( 0 H ) T 0 good approximation: narrow energy range and/or small coupling

18 inversion by partitioning M = a c b d matrix M = ad bc d b c a M = invert block- matrix A B C D M = Ã C B D A B C D Ã solve C B D = 0 0 AÃ + B C = = (A BD C)Ã CÃ + D C = 0 C = D CÃ Ã = A BD C

19 direct exchange: effective Hamiltonian H e ( )= systematic treatment of limit U (or t 0): downfolding H = t t 0 0 +t +t t +t U 0 t +t 0 U downfolding eliminates ionic states (actually change of basis) t t +t +t U 0 0 U t +t t +t C A t U diagonalize Heff t = 0 t =, +, s = 4t U s =,, triplet singlet

20 direct exchange: effective spin-coupling J direct = triplet singlet =4t /U J > 0 AF coupling triplet Jdirect H e = t U = + t U singlet effective spin-hamiltonian,, SS z z + S + S + S S+ = 4t U S S 4 Heisenberg J

21 keeping track of all these signs towards second quantization Slater determinant (x,x )= p (' (x )' (x ) ' (x )' (x )) corresponding Dirac state, i = p ( i i i i) use operators, i = c c 0i position of operators encodes signs c c 0i =, i =, i = c c 0i product of operators changes sign when commuted: anti-commutation anti-commutator {A, B} := AB+ BA

22 second quantization: motivation specify N-electron states using operators N=0: 0i (vacuum state) normalization: h0 0i = N=: i = c 0i (creation operator adds one electron) normalization: overlap: h i = h0 c c 0i h i = h0 c c 0i N=:, i = c c 0i adjoint of creation operator removes one electron: annihilation operator c 0i =0andc c = ±c c + h i antisymmetry: c c = c c

23 second quantization: formalism vacuum state 0 and set of operators cα related to single-electron states φα(x) defined by: c 0i =0 c,c =0= c,c h0 0i = c,c = h i creators/annihilators operate in Fock space transform like orbitals! field operators Slater determinant ˆ(x) = X ' n (x) c n n p 0 ˆ(x ) ˆ(x )... ˆ(x N ) c N! N...c c 0

24 second quantization: examples H = two-site model with one electron t c c + c c + c c + c c = t X i,j, c j c i H = t two-site model with two electrons c c + c c + c c + c c + U n n + n n = t X i,j, c j c i + U X i n i n i also works for single electron and for more sites easy to handle Slater determinants easy to write down many-body Hamiltonian (become independent of particle number)

25 Hartree-Fock (, ) = ansatz: Slater determinant sin( ) c + cos( ) c sin( ) c + cos( ) c E 0 energy expectation value E( ", #) = t (sin " sin # + cos " cos # ) (cos " sin # + sin " cos # ) +U sin " sin # + cos " cos # minimize wrt θ and θ HF orbitals respect symmetry of model: restricted Hartree-Fock (RHF) here: θ = θ = π/4 HF allowed to break symmetry: unrestricted Hartree-Fock (UHF) here: θ = π/ θ

26 Hartree-Fock energy expectation value for θ = π/ θ E HF ( ) / t U=t 0 /4 /

27 Hartree-Fock exact RHF UHF E / t E RHF = t + U/ E UHF = U / t ERHF for U apple t t /U for U t E exact = U ( p + 6t /U ) 4t /U

28 direct kinetic exchange singlet triplet -t direct exchange U virtual hopping -t /U

29 superexchange TMOs: negligible direct hopping between d-orbitals instead hopping via oxygen t pd U d symmetry: only one oxygen-p involved in hopping H = X d X i n i + p n p t pd X i ε d ε p c i c p + c p c i + U d X i n i n i

30 superexchange: same spin H = X d X i n i + p n p t pd X i c i c p + c p c i + U d X i n i n i oxygen-p full, two d-electrons of same spin H = 0 t pd t pd t pd U d + pd 0 t pd 0 U d + pd A c c p c p c 0 c c p c c 0 c c c p c 0 H e =(t pd,t pd ) (Ud + pd ) 0 0 (U d + pd ) tpd t pd t pd U d + pd

31 superexchange: opposite spin t pd +t pd t pd +t pd t pd 0 U d + pd t pd 0 t pd +t pd 0 0 U d + pd t pd t pd 0 +t pd 0 0 U d + pd 0 +t pd 0 +t pd 0 +t pd U d + pd 0 +t pd +t pd 0 0 t B pd 0 +t pd 0 U d t pd 0 +t pd 0 U d 0 A 0 0 t pd t pd +t pd +t pd 0 0 (U d + pd ) c c p c p c 0 c c p c p c 0 c c p c c 0 c c c p c 0 c c p c c 0 c c c p c 0 c p c p c c 0 c c c p c p 0 c c c c 0

32 H e = H 00 + T 0 superexchange: opposite spin H + T ( H 00 T 0 H T 0 T 0 H T H T H T 0 = t pd U d + pd 0 0 t 4 pd (U d + pd ) H ) T T0 expand in /Ud U d + U d + pd singlet-triplet splitting: J = 4t 4 pd (U d + pd ) U d + U d + pd

33 ferromagnetic superexchange t pd t pd t px d Jxy p x ε d ε p U d p y t py d 80 o superexchange 90 o superexchange hopping only via oxygen-p pointing in direction connecting d-orbitals no hopping connecting d-orbitals but Coulomb exchange on oxygen double exchange

34 ferro superexchange: same spin 0 0 t pd t pd 0 t pd U d + pd 0 t pd t pd 0 U d + pd t pd 0 t pd t pd (U d + pd ) J xy C A c c x c x c y c y c 0 c c c x c y c y c 0 c c x c x c y c c 0 c c c x c y c c 0 t px d J xy p x U d p y H e = tpd 4tpd 4 U d + pd (U d + pd ) (U d + pd ) J xy t py d

35 ferro superexchange: opposite spin t pd 0 t pd t pd 0 t pd 0 0 t pd 0 U d + pd t pd 0 0 t pd 0 U d + pd t pd t pd U d + pd 0 t pd 0 B 0 t pd U d + pd 0 t pd 0 0 t pd 0 t pd 0 (U d + pd ) J A xy t pd 0 t pd J xy (U d + pd ) H e = = + t pd U d + pd 0 0 t pd U d + pd + 4tpd 4 (U d + pd ) 4(U d + pd ) Jxy 4tpd 4 (U d + pd ) 4tpd 4 J xy (U d + pd ) 4(U d + pd ) Jxy (U d + pd ) J xy! c c x c x c y c y c 0 c c x c x c y c y c 0 c c c x c y c y c 0 c c c x c y c y c 0 c c x c x c y c c 0 c c x c x c y c c 0 c c c x c y c c 0 c c c x c y c c 0 (Ud + pd ) +J xy +J xy (U d + pd ) (as for same spin) singlet-triplet splitting J = 4tpd 4 J xy (U d + pd ) 4(U d + pd ) Jxy

36 double exchange double exchange involves both, full Coulomb matrix and hopping mixed-valence compound: non-integer filling of d-orbital d-electrons can hop even when U is large simple model: two sites with two orbitals each -tbb b a Jab b a -taa

37 double exchange t bb Sz=3/ H = b a Jab t bb t bb J ab ± = J ab ± t bb ± =,, ±,, =, b ±, b, a b-electron hops against background of half-filled a-orbitals

38 double exchange Sz=/ J ab H = 0 J ab t bb t bb 0 J ab J ab 0 t bb t bb 0 J ab J ab 0 t bb t bb J ab C A ground state 0 = J ab t bb,, +,, +,, +,, +,, +,, 6 =, b +, b, a +, a +, b +, b, a hopping electron aligns a-electrons ferromagnetically (teleports local triplet into triplet of a-electrons)

39 double exchange Sz=3/ Sz=/ Sz= / Sz= 3/ t bb J ab

40 double exchange alternative model: assume passive orbitals with many electrons (large Hund s rule spin) example: eg electrons hopping against tg background consider these spins fixed with quantization axis tilted by ϑ relative to each other rotation of quantization axis d b" = cos( /) c b" sin( /) c b# d b# = sin( /) c b" + cos( /) c b# hopping mixes spins t bb c b c b = t bb + cos( /) d b + sin( /) d b t bb c b c b = t bb sin( /) d b + cos( /) d b c b c b

41 double exchange assume a-spins cannot be flipped no J terms 4 independent 4 4 Hamiltonians for tbb << Jab tilt merely reduces width of b-band ± = J ab ± t bb cos( /) again, hopping of b-electron prefers ferro-aligned a-electrons

42 orbital ordering same model, but now one electron per orbital H = t bb t aa 0 0 +t aa +t bb t bb +t aa U ab J ab 0 t aa +t bb 0 U ab J ab C A H e U ab J ab t aa + t bb t aa t bb t aa t bb taa + tbb = (t aa t bb ) t aa t bb U ab J ab U ab J ab effective interaction between orbitals: orbital singlet/triplet

43 orbital ordering: opposite spins H = t bb t aa t aa +t bb t bb t aa t aa +t bb t bb +t aa 0 0 U ab 0 J ab 0 t aa +t bb U ab 0 J ab 0 0 t bb +t aa J ab 0 U ab t aa +t bb 0 J ab 0 U ab C A

44 orbital-ordering: opposite spin H e U ab J ab 0 (t aa + t bb )U ab t aa t bb U ab (t aa + t bb )J ab t aa t bb J ab t aa t bb U ab (t aa + t bb )U ab t aa t bb J ab (t aa + t bb )J ab (t aa + t bb )J ab t aa t bb J ab (t aa + t bb )U ab t aa t bb U ab C A = = U ab U ab J ab J ab Uab apple t aa t bb J ab (taa + tbb )J ab t aa t bb U ab (taa + tbb )J ab J ab t aa + tbb t aa t bb J ab U ab t aa t bb taa + tbb apple U ab + J ab J ab (t aa t bb ) +t aa t bb spin-exchange orbital-exchange simultaneous coupling of spins and orbital occupations spin- and orbital-exchange tend to have opposite sign 3 4

45 summary exchange mechanisms dominant magnetic interaction in materials not a fundamental but an effective interaction: model/mechanism Coulomb exchange: off-diagonal Coulomb matrix-elements; ferromagnetic coupling (Hund s rule) kinetic exchange: only diagonal Coulomb matrix-elements & hopping direct exchange: anti-ferromagnetic spins: virtual hopping -4t /U superexchange: hopping via O-p orbitals tends to be anti-ferromagnetic (80 o superexchange) but 90 o superexchange is ferromagnetic double exchange: hopping electrons align spins ferromagnetically orbital ordering: exchange interaction between orbital occupations

46 summary H U = 0 U ab J ab U ab J ab 0 0 J ab U ab U ab J ab C A direct exchange Coulomb exchange: ferro (Hund s rule) kinetic exchange: anti-ferro superexchange

47 summary double exchange: often ferro t px d J xy p x t bb U d p y t py d J ab orbital-ordering