Luigi Paolasini [email protected] LECTURE 7: “Magnetic excitations”
- Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations.
L. Paolasini - LECTURES ON MAGNETISM- LECT.7 External parameter , as for example the temperature, can drive a symmetry- breaking transition, in which a symmetry element of the system is lost.
Broken rotational and translational symmetry Broken the full rotational symmetry The liquid phase is invariant under any arbitrary The paramagnetic state possesses complete translation and rotation. rotational symmetry The ordered phase have only a reduced set of The ferromagnetic state have a reduced rotational symmetry operations symmetry about the magnetization axis
The ordered phase possesses a lower symmetry with respect to the disordered phase!
L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Symmetry group : Collection of elements and a set of operations that combines them: closure, associativity, existence of an inverse and the identity
A physical system possesses a particular symmetry if the Hamiltonian is invariant with respect to the transformations associated with the element of symmetry group.
Discrete symmetry groups : symmetry group with countable elements (Ex.: point groups, lattice groups, space groups)
Continuous symmetry groups : uncountable continuum of elements (Ex.: rotational symmetry group of a sphere O(3), Lie groups)
Global symmetry : the system is invariant under the symmetry elements of the group applied globally to the entire system.
Local symmetry : Hamiltonian is unchanged when the symmetry operations are applied differently to different point in the space.
L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - Phase transitions occur when an internal symmetry is broken by an external force. - Consequences of broken symmetry are:
PHASE TRANSITION sharp change in behaviour at the critical temperature and the region near the PT is called critical region
RIGIDITY or STIFFNESS resistance of the system against any attempt to change its state
DEFECTS Symmetry broken differently in two adiacent part of the system (grain boundary or magnetic domain wall)
EXCITATIONS At T≠0 the dynamic excitations of the order parameter tend to remove or to weakens the ordered state
L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - Magnetic phases : Phase transitions occur between different magnetic phases as a function of the thermodinamical parameters: Pressure, Magnetic field, Temperature
- Stability of a magnetic structure :
kBT << Jij (exchange constant)
- Phase Transitions : from a long-range magnetic ordered state (low T) to a paramagnetic state (high T) Free energy: - Critical temperature : F= E - TS TC = Curie temperature (ferromagnets) Energy TN = Néel temperature (antiferromagnets) Entropy (Low T) (High T) - Critical fluctuations: the magnetic moments fluctuate in space and time
L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - The order parameter η saturate at low temperature and vanishes above the critical temperature TC - The order parameter can vanishes continuosly (second order transition) or discontinuously (first order transition)
The free energy F(T, η ) of a 2nd order magnetic phase transition Lev Davidovitch Landau can be expanded near Tc as:
where the order parameter η=0 for T>Tc
The thermal equilibrium condition at any T L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Order parameter: Magnetization η = M Notice that odd power terms are null because: Condition for thermal stability: Solutions: Mean-field Magnetization and T>Tc Valid only if T L. Paolasini - LECTURES ON MAGNETISM- LECT.7 In the presence of a magnetic field H Magnetic susceptibility Condition for stability: Mean-field Susceptibility L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Mean field theories fail in describing the systems near Tc because ignore correlations and fluctuations Indipendently of the precise nature of the interactions considered, near Tc the phase transitions shows an Universal behaviour. Reduced temperature L. Paolasini - LECTURES ON MAGNETISM- LECT.7 The microscopic models of magnetic interactions are classified in term of: - dimensionality of the order parameter D - space dimensionality d D (=1, 2, 3) depends on the number of components of Sx Sy and Sz of the spin operator S. Heisenberg model: D=3 Spins are 3D vectors The lattice dimensionality d=1,2,3… Ising model: D=1 Spins are 1D vectors (S=Sz) The lattice dimensionality can be d=1,2,3 … L. Paolasini - LECTURES ON MAGNETISM- LECT.7 The universality classes are defined in term of : - Order parameter dimensionality D - System dimensionality d - short or long range interaction (ex.:covalent or metallic bonding) Within a given universality class the value of the critical esponents are the same and do not depend on the detailed nature of the system Specific heat Magnetization Susceptibility Correlation length L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - 1D lattice of N+1 atoms - All spin ferromagnetically aligned Energy cost to flip the chain: Ground state energy: +J/2 because is already in favourable state +J/2 because cost energy in the new state Entropy gain: S= kB ln N because we can put the defect in any position For a long chain: N!∞, => S!∞, F !-∞ Free energy: and as far as T>0, the long range order is never reached because just the presence of one defect F= E - TS break the long range order in d=1 J kB ln N L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - Phonons and magnons are quasi-particle associated to the lattice and spin excitations - They are characterized by a frequency ω and a wavevector q - Dispersion relationship between energy !ω and momentum !q - Magnons and phonons are BOSONS, and they are described by symmetric wavefunction with respect to the exchange of particle positions Lattice waves => Phonons collective acoustic and optic lattice vibrations Spin-waves => Magnons collective magnetic excitations associated to the in-phase precession of the spin moments L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - Semi-classical derivation - Linear chain of aligned spins (S=Sz, Sx=Sy=0) Time dependence of expected value of Small perturbation Sz~S, Sx,Sy< S S y j j+1 x j j+1 y x L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Raising and lowering operators are defined as: Commutation relations L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - Quantum mechanical derivation - Heisenberg Hamiltonian for a linear chain of spins Ground state |Ψ> 2 E0=NS J eigenvalue of the ground state |Ψ> |j> excited state: spin flip at site j ΔS=1 total change of the spin Magnons are Bosons! Notice that |j> is not an eigenstate of the Hamiltonian because: L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Plane waves solution in form of a spin-flip excitation propagating along the 1D-dimensional chain Flipped spin state |j> delocalized along the chain - Total spin of the perturbed state |j> is: NS- ΔS = NS-1 - The total energy solution of H |j> = E(q) |j> is: E0 - Magnon dispersion relation: - If (qa)<<1 => parabolic dispersion: L. Paolasini - LECTURES ON MAGNETISM- LECT.7 - At low temperature the density of states is g(q) dq ~ q2 and then g(ω )dω ~√ω dω - In order to calculate the number of magnons at finite temperature T we need to integrate over the all frequencies the magnon density of states multiplied for the Bose factor: Reduced magnetization at low T due to the magnon excitations: Block law T3/2 L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Electron with momentum k+q and spin down is excited in the state k with spin up Fermi surfaces Δ=exchange splitting - Electron-hole excitations between filled and empty spin-split bands - Broadening of energy => Short timescale fluctuations - Single electron excitation In paramagnetic state the spin wave excitations are overdamped and are called paramagnons. L. Paolasini - LECTURES ON MAGNETISM- LECT.7 The triple axis spectrometer (TAS) (Brockhouse 1952) Q=G222+q φ k’ k q Energy conservation Q Kinematical condition L. Paolasini - LECTURES ON MAGNETISM- LECT.7 L. Paolasini -SoNS F.P. Ricci 2012- 22 Different possibilities to cross the dispersion surface S (Q,ω) in 4D space - In genertal 3 parameters are varied between (θM,θA ,φ Ω) - Different choices for the same scattering point Constant-Q scans: We scan the energy axis when Q is constant - θM (or θA) is keep fixed - θA (or θM) and φ Ω are varied in order to keep Q constant Constant-E scans: We scan one of the Q axis when (Ei-Ef) is keep constant - Both θM and θA are keep fixed - Q is varied along a particular trajectory by varying φ Ω L. Paolasini -SoNS F.P. Ricci 2012- 23 Inelastic structure factor Bose-Einsten Thermal population factor Phonon Polarization factor 24 L. Paolasini -SoNS F.P. Ricci 2012- 24 L. Paolasini -SoNS F.P. Ricci 2012- 25 Magnons (or spin waves) are collective magnetic excitations associated to the precession of the spin moments Magnon stiffness constant related to the exchange interaction L. Paolasini -SoNS F.P. Ricci 2012- 26 L. Paolasini, and G.H. Lander, J. All. Comp. 303-304 (2000) 232 and ref. therein L. Paolasini - LECTURES ON MAGNETISM- LECT.7 Spin wave in metals (Band like model) Broadening of energy => Short timescale fluctuations Stoner excitations => single electron excitation Stoner Δ excitations dispersion Σ Spin Energy wave 0 kmin k Reduced wavevector Exchange splitting Δ and Stoner gap Σ. Δ E energy F Σ k min Spin down Electron Spin up Electron wavevector k L. Paolasini - LECTURES ON MAGNETISM- LECT.7