1 Magnetic frustration and multiferroics Virginie Simonet Institut Néel, CNRS/Université Grenoble Alpes, Grenoble, France HSC18, Grenoble, 2015 21/09/15 2 Outline of the lecture Reminder: Conventional Magnetism Magnetic frustration Introduction Competition of interactions Geometrical frustration Quantum case Role of the anisotropy Multiferroism Introduction About symmetries and chirality Examples Sorry, I will only speak about neutron scattering! HSC18, Grenoble, 2015 21/09/15 3 Reminder: conventional magnetism Temperature << interactions Temperature ≈ interactions Temperature >> interactions Ordered state Order onset Fluctuating disordered small fluctuations Heisenberg Hamiltonian = S~ J˜ S~ H − i ij j i,j J < 0 AFMX Order parameter Order parameter 2 T T=0 TN ≈ JS HSC18, Grenoble, 2015 21/09/15 4 Reminder: conventional magnetism Temperature << interactions Temperature >> interactions Ordered state Disordered state Magnetic frustration : generalities ⟨S (0)⋅S (r)⟩ magnons r 2 • Spin pair correlation function <Si(0)Sj(r) > <Si(0)Sj(r) >= S δij Magnetic frustration : generalities Real space ⟨S (0)⋅S (r)⟩ magnons order Reciprocal space S2 Reciprocal space Intensité w dispersion parameter S(Q) S(Q) BraggBragg peaks relation(s) Paramagnetic r diffuse scattering w(Q) -S2 order ∝ JS parameter Intensité w dispersion Bragg peaks relation(s)Q Q Q w(Q) HSC18, Grenoble, 2015 ∝ JS 21/09/15 magnetic order Q Q k b T 2 T =0 1 magnetic order J S k b T 2 T =0 1 J S 5 Reminder: conventional magnetism Temperature << interactions Temperature >> interactions Ordered state Disordered state Magnetic frustration : generalities ⟨S (0)⋅S (r)⟩ magnons 2 • Spin pair correlation functionr <Si(0)Sj(r) > <Si(0)Sj(r) >= S δij order • Excitations: dispersion relation ω(Q) parameter Intensitéspin waves (magnons) w dispersion Bragg peaks relation(s) w(Q) JS ∝ JS / Q Q HSC18, Grenoble, 2015 magnetic order 21/09/15 k b T 2 T =0 1 J S 6 Magnetic frustration: introduction Is it possible to impede the onset of magnetic ordering? è Yes, through dimensional effects, introduction of disorder and frustration Consequences on spin pair correlations, excitations, neutron signatures? ✓CW Frustration index f = TN HSC18, Grenoble, 2015 21/09/15 7 MagneticMagnetic frustration: introduction frustration : generalities Is it possible to impede the onset of magnetic⟨S (0) ⋅ordering?S (r)⟩ −r /ξ spinons ∝e α 1 r è Yes, through dimensional effects, introduction of disorder and frustration∝ / Consequences on spin pair correlations, excitations, neutron signatures? r MagneticReal space frustration : generalities ⟨S (0)⋅S (r)⟩ −r /ξ spinons ∝e α order∝ 1/ r Reciprocal space parameter S(Q)Intensité w 2-spinons Bragg peaks continuum r diffuse scattering order parameter Intensité w 2-spinons Bragg peaks continuum Q Q diffuse scatteringHSC18, Grenoble, 2015 21/09/15 quantum spin classical spin Q liquids/ice liquids/iceQ k b T quantum spin classical spin 1 2 liquids/ice liquids/ice T =0 1/S J S k b T low dimensionality,2 geometrical frustration, exchange disorder, etc. T =0 1/S 1 J S low dimensionality, geometrical frustration, exchange disorder, etc. 8 MagneticMagnetic frustration:frustration introduction : generalities atomic states ⟨S (0)⋅S (r)⟩ −r /ξ spinons ∝e α ∝1/ r cristallized solid liquid gas r order Magnetic order Spin liquid paramagnet 2-spinons parameter IntensitéS(Q) w Bragg peaks continuum Magnetic states diffuse scattering Q Q HSC18, Grenoble, 2015 21/09/15 quantum spin classical spin liquids/ice liquids/ice k b T 2 T =0 1/S 1 J S low dimensionality, geometrical frustration, exchange disorder, etc. 9 Magnetic frustrationMagnetic : main frustration:parameters introduction to play with Spin anisotropy One or several competingTopology constrains can of not the be satisfied lattice simultaneously Important ingredients: • Topology of the lattice ... Ising XY Heisenberg Norm of the spin Interaction tensors ~ S = 1/2, 1, 3/2 , …, ∞ H =−∑ S i J ij S j ij Competing interactions/degrees of freedom : exotic ground states J<0 ground state entanglement S = 1/2 Ising , S = ∞ degeneracy (quantum) (classical) ? 1 HSC18, (|Grenoble,↑↓ ⟩ − |↓↑ 2015⟩ ) 2 √ 21/09/15 ⟨S (0)⋅S (r)⟩ = ? 10 Magnetic frustrationMagnetic : main frustration:parameters introduction to play with Spin anisotropy One or several competingTopology constrains can of not the be satisfied lattice simultaneously Important ingredients: • Topology of the lattice ... Ising XY Heisenberg • Competing interactions: superexchange (nearest/further neighbors) dipolar, DM Norm of the spin Interaction tensors = S~ J˜ S~ Dij.(Si Sj) H − i ij j ⇥ i,j 3 Si Sj 3(Si rij)(Sj rij) = J S S + Dr · · · H X− i · j nn r 3 − ~r 5 S = 1/2, 1, 3/2 , …,i>j ∞ Hi>j ij S J ijS X X= |−|∑ i |ij | j ij Competing interactions/degrees of freedom : exotic ground states J<0 ground state entanglement S = 1/2 Ising , S = ∞ degeneracy (quantum) (classical) ? 1 HSC18, (|Grenoble,↑↓ ⟩ − |↓↑ 2015⟩ ) 2 √ 21/09/15 ⟨S (0)⋅S (r)⟩ = ? 11 Magnetic frustrationMagnetic : main frustration:parameters introduction to play with Spin anisotropy One or several competingTopology constrains can of not the be satisfied lattice simultaneously Important ingredients: • Topology of the lattice ... Ising XY Heisenberg • Competing interactions: superexchange (nearest/further neighbors) dipolar, DM Norm of the spin Interaction tensors = S~ J˜ S~ Dij.(Si Sj) H − i ij j ⇥ i,j 3 Si Sj 3(Si rij)(Sj rij) = J S S + Dr · · · H X− i · j nn r 3 − ~r 5 S = 1/2, 1,Magnetic 3/2 , …, i>j frustration∞ :H i>jmainij parametersS J ijS to play with X X= |−|∑ i |ij | j ij • Anisotropy,Spin anisotropy spin dimension and norm (classicalTopology versus of thequantum) lattice Competing interactions/degrees of freedom : exotic ground states S=1/2, 1, 3/2, …, ∞ ... J<0 groundIsing state XY Heisenberg entanglement Ising , S = ∞ degeneracy S = 1/2 • Disorder,Norm thermal of the spinand quantum fluctuationsInteraction (quantum) tensors (classical) ? 1 HSC18, (|Grenoble,↑↓ ⟩ − |↓↑ 2015⟩ ) 2 √ ~ 21/09/15 S = 1/2, 1, 3/2 , …, ∞ H =−∑ S i J ij S j ⟨S (0)⋅S (r)⟩ = ? ij Competing interactions/degrees of freedom : exotic ground states J<0 ground state entanglement S = 1/2 Ising , S = ∞ degeneracy (quantum) (classical) ? 1 (|↑↓ ⟩ − |↓↑ ⟩ ) √2 ⟨S (0)⋅S (r)⟩ = ? 12 Magnetic frustration: competition of interactions One or several competing constrains can not be satisfied simultaneously Through competing interactions: Square lattice with Ising spins Spin chain with Ising spins AFM J1 HSC18, Grenoble, 2015 21/09/15 13 Magnetic frustration: competition of interactions One or several competing constrains can not be satisfied simultaneously Through competing interactions: Square lattice with Ising spins Spin chain with Ising spins AFM J1 ≈AFM J2 ? ? HSC18, Grenoble, 2015 21/09/15 14 Magnetic frustration: competition of interactions One or several competing constrains can not be satisfied simultaneously Through competing interactions: Square lattice with Ising spins Spin chain from Ising to Heisenberg (XY) spins AFM J1 ≈AFM J2 ? ? ✓ Helical order for J1≤4J2 J with ✓ = arccos(− 1 ) 4J2 HSC18, Grenoble, 2015 21/09/15 15 Magnetic frustration: competition of interactions One or several competing constrains can not be satisfied simultaneously Through competing interactions: Square lattice with Ising spins Spin chain from Ising to Heisenberg (XY) spins AFM J1 ≈AFM J2 ? ? ✓ HSC18, Grenoble, 2015 21/09/15 16 Magnetic frustration: competition of interactions One or several competing constrains can not be satisfied simultaneously Through competing interactions: Square lattice with Ising spins Spin chain from Ising to Heisenberg (XY) spins AFM J1 ≈AFM J2 ? ? ✓ è Prevents magnetic order or find a compromise with exotic magnetic orders (non collinear) HSC18, Grenoble, 2015 21/09/15 17 Geometrical frustration One or several competing constrains can not be satisfied simultaneously Through the geometry of the lattice: Frustration magnétique PolygonsFrustration with odd magnétique number of edges Antiferromagnetic Ex. triangles, interactions Des contraintes en compétitionPentagons, tetrahedron ne peuvent pas être satisfaites simultanément Des contraintes en compétition ne peuvent pas être satisfaites simultanément ≠ Interactions AFM ≠ Interactions AFM Ising Géométrie : ? Géométrie : Polygones à nombre Polygones à nombre impair de côtés ? impair de côtés ? ? et interactions ? et interactions ? antiferromagnétiques antiferromagnétiques ? Ex. triangles,Ex. triangles, ? pentagones pentagones J2 J2 J J1 1 ChiralitéChiralité Degeneracy (measured by finite entropy) 6" 6" 18 Geometrical frustration One or several competing constrains can not be satisfied simultaneously Through the geometry of the lattice: Frustration magnétique PolygonsFrustration with odd magnétique number of edges Ex. triangles, Des contraintes en compétitionPentagons, tetrahedron ne peuvent pas être satisfaites simultanément Des contraintes en compétition ne peuvent pas être satisfaites simultanément ≠ Interactions AFM ≠ Interactions AFM Ising Géométrie : ? Géométrie : Polygones à nombre Polygones à nombre impair de côtés ? impair de côtés ? ? et interactions ? et interactions ? antiferromagnétiques XY or Heisenberg antiferromagnétiques ? Ex. triangles,Ex. triangles, ? pentagones pentagones J2 J2 J J1 1 ChiralitéChiralité = J(S1.S2 + S2.S3 + S3.S1) H − J = (S + S + S )2 + cst − 2 1 2 3 HSC18, Grenoble, 2015 21/09/15 6" 6" 19 Geometrical frustration One or several competing constrains can not be satisfied simultaneously Through the geometry of the lattice: Frustration magnétique