Neutrons for Quantum Magnetism Bella Lake Helmholtz Zentrum Berlin, - PowerPoint PPT Presentation

Neutrons for Quantum Magnetism Bella Lake Helmholtz Zentrum Berlin, Germany Berlin Technical University, Germany B. Lake; Tartu, Sept 2017 Oliver Pieper, Kolloquium, 28. Mai 2010

Outline Conventional Magnets Long-range magnetic order and spin-wave excitations Origins of Quantum magnetism low spin values, antiferromagnetic, low-dimensional, frustration, spin liquids Neutron scattering for quantum magnets Triple Axis spectrometer, time-of-flight spectrometer Examples of frustrated magnets 0-dimensional magnets e.g. dimer magnets 1.Dimensional magnets e.g. the spin-1/2 chain 2-Dimensional magnets e.g. Square, triangular, kagome, lattice 3-Dimensional magnets e.g. pyrochlore, spin ice and water ice B. Lake; Tartu, Sept 2017

Conventional Magnetism – Magnetic Moments • Electrons possess spin and orbital angular momenta ( s and l ). s 1 • S and L for an ion can be determined by s 2 summing the electronic s and l of the s 4 unpaired electrons s 3 The ionic magnetic moment is m =g s  B S . • S=s 1 + s 2 + s 3 + s 4 +... The Mn 2+ ion, S =5/2 |S| 2 = S(S+1)=35/4 S z =+5/2 • S is the quantum number associated S z =+3/2 with the angular momentum S . S z =+1/2 • S is restricted to take on discrete values either integer or half integer. S z =-1/2 S z =-3/2 S z =-5/2 B. Lake; Tartu, Sept 2017

Conventional Magnetism - Exchange Interactions   S S H J . Heisenberg interactions n m , n m n m , J < 0 ferromagnetic J > 0 antiferromagnetic 1D magnet 3D magnet |J 1 |=|J 2 |, J 3 =J 4 =0 |J 1 |=|J 2 |=|J 3 |=|J 4 | J 1 J 2 J 1 J 2 e.g. KCuF 3 e.g. RbMnF 3 J 3 J 3 J 3 J 3 J 3 J 1 J 2 J 1 J 2 1D alternating magnet 2D magnet |J 1 |  |J 2 |, J 3 =J 4 =0 |J 1 |=|J 2 |=|J 3 |, J 4 =0 J 3 J 3 J 3 J 3 J 3 e.g. CuGeO 3 and e.g. La 2 CuO 4 J 1 J 2 J 1 J 2 CuWO 4 and CFTD           S S x x S S y y S S z z H J   Anisotropic interactions n m , n m n m n m n m , B. Lake; Tartu, Sept 2017

Conventional Magnetism - Ordered Ground State Exchange interactions between magnetic ions often lead to long-range order in the ground state. spin glass ferromagnet antiferromagnet spiral magnet helical magnet B. Lake; Tartu, Sept 2017

Conventional Antiferromagnets T < T N Real Space T > T N • Long-range magnetic order on cooling as thermal fluctuations weaken J 1 J 1 J 1 J 1 J 0 J 0 Reciprocal Space (0,2,1.5) (0,2,0.5) b  (0,2,0) • Magnetic Bragg peaks 40 T N = 50 K c  Intensity (~<S> 2 ) 30 appear below the (0,1,0.5) (0,1,1.5) (0,1,0) 20 transition temperatures Nuclear Bragg 10 peaks and grow as a function of 0 Magnetic Bragg (0,0,0.5) (0,0,1.5) peaks 0 20 40 60 80 (0,0,0) temperature (0,0,1) (0,0,2) Temperature (K) B. Lake; Tartu, Sept 2017

Magnetic Excitations Real Space • Spin-waves are the collective motion of spins, about an ordered ground state (similar to phonons) Reciprocal Space • Observed as a well defined dispersion in energy and wavevector B. Lake; Tartu, Sept 2017

The Origins of Quantum Magnetism • Quantum fluctuations suppress long-range magnetic order, spin- wave theory fails • Quantum effects are most visible in magnets with • low spin values • antiferromagnetic exchange interactions • low-dimensional interactions • frustrated interactions • Quantum effects give rise to exotic states and excitations   H H , n m  n , n m       x x y y z z S S H J J S S S S S S , , , n m n m n m n m n m n m n m          z z H J S S J S S S S n m , n m , n m n m n m B. Lake; Tartu, Sept 2017

Quantum Magnetism - Low Spin Value          z z H J S S J S S S S n m , n m , n m n m n m • Fluctuations have the largest effect for low spin values For S=1/2, changing S z by 1 unit reverses the spin direction • S=5/2 |S| 2 = S(S+1)=35/4 S=1/2 |S| 2 = S(S+1)=3/4 S z =+5/2 S z =+3/2 Spin Up (  ) S z =+1/2 S z =+1/2 S z =-1/2 Spin Down (  ) S z =-1/2 S z =-3/2 S z =-5/2 B. Lake; Tartu, Sept 2017

Antiferromagnetic Exchange Interactions • Parallel spin alignment is an eigenstate of the Hamiltonian and the ground state of a ferromagnet. • Antiparallel spin alignment (Néel state) is not an eigenstate of the Hamiltonain and is not the true ground state of an antiferromagnet. J>0 J>0          z z H J S S S S S S antiferromagnetic ferromagnetic 1,2 1 2 1 2 1 2      H J / 4 J J 1,2 1 2 1 2          H J / 4 J / 4 S 1 =1/2 S 2 =1/2 S 1 =1/2 S 2 =1/2 1,2 1 2 1 2 1 2 S z =1/2 S z =1/2 S z =1/2 S z =-1/2 B. Lake; Tartu, Sept 2017

Low-Dimensional Interactions For three-dimensional magnets each magnetic ion has six neighbours For a one-dimensional magnet there are only two neighbours Neighbouring ions stabilize long-range order and reduce fluctuations          z z H J S S J S S S S n m , n m , n m n m n m 1D S=1/2 3D S=1/2 J 1 J 1 J 0 J 0 B. Lake; Tartu, Sept 2017

Frustrated Interactions • In some lattices with antiferromagnetic interactions it is impossible for the spins to satisfy all the bonds simultaneously, this phenomenon is known as a geometrical frustration. • Long-range order is suppressed as the spins fluctuate between the different degenerate configurations. ? J J J J J Anisotropy produces frustration if the anisotropy is incompatible with the spin direction favoured by the interactions B. Lake; Tartu, Sept 2017

Examples of Quantum Magnets • Quantum magnets are characterised by suppression of magnetic order, T N << T CW and < S > < S , in some cases the magnet never orders. • The excitations are broadened and renormalised with respect to spin- wave theory, and can be characterised by different quantum numbers. • New theoretical approaches are required to understand these systems Spin liquids, • no local order, no static magnetism highly entangled, dynamic ground state, topological order, spinon excitations B. Lake; Tartu, Sept 2017

Neutron Scattering for Quantum Magnets Magnetic order is suppressed therefore most information about quantum magnets comes from their excitations. It is important to resolve the excitations as a function of energy - inelastic neutron scattering. Since the excitations are often diffuse – wide detector coverage is useful 14 B. Lake; Tartu, Sept 2017

Conservation Laws and Scattering Triangles   1 1 1        Conservation of energy  2 2  2 2 2 E E mv mv k k i f i f i f 2 2 2 m        Q k k Q k k Conservation of momentum i f i f k f Scattering triangles - elastic ħ  =0; | k i | = | k f | 2  scattered k i neutrons k f Q Wavevector Incident Q transfer neutrons 2  k i k i Scattering triangles - Inelastic ħ   0; | k i |  | k f | Neutron loses energy Neutron gains energy An excitation is created An excitation is destroyed k f Q Q k f 2  2  k i k i B. Lake; Tartu, Sept 2017

The Magnetic Cross-section 2      2 k d m     2 2          f       2 2   p p k s V s E E E k k k    i f   s f f f i i i 2   2 m 2 d dE k i i i f   , s , s i i i f f V - the magnetic interaction between neutron and electrons The electrons in an atom possess spin and orbital angular momentum, both of which give rise to an effective magnetic field. The neutrons interact with this field because they possess a spin moment The interaction between a neutron at point R away from an electron with momentum l and spin s is       ˆ ˆ         s R l R 2 1          j j j j   0 N B . V = .B curl      magnetic n  2 2 4  R R        j     2     V b r r nuclear j j m j B. Lake; Tartu, Sept 2017