Universitat de Barcelona1990-2010: quantum magnetism Fall 2011 Javier Tejada, Dept. Física Fonamental
ContenidosContent Introduction to magnetism Single Domain Particles Quantum relaxation: 1990-96 Resonant spin tunneling: 1996-2010 Quantum magnetic deflagration Superradiance
Introduction to magnetism• Electrostatic interaction + Quantum Mechanics 2 e Overlapping of wave r12 functions 2 e Is different for S 0 and S 1 r12 Heisenberg Term si s In the Hamiltonian j hamiltonian
Introduction to magnetismThe magnetic moment of an atom has e two contributions: p μorbital1. The movement of the electrons around the nucleus. The electric charges generate magnetic fields while moving2. Electron, like the other fundamental particles, has an intrinsic propierty named spin, which generates a magnetic moment even outside the atom: e e μspin S=1/2 S=-1/2 Hence, the magnetic moment of the atom is the sum of both contributions e μtotal = μorbital + μspin p
Título Introduction to magnetism S 0 S 1 Atoms can be found with two or more interacting electrons. Considering two of them in an atom, the energy of the spin interaction can be calculed: e The system always tends to be at e p the lowest energy state:: J ~ TC ˆ ˆ ˆ J s1 s 2 eff The overlapping of the wave Summation over functions decays exponentially. nearest neighbours
Título Introduction to magnetism Existence of metastable states Magnetic Time dependent hysteresis phenomena Slow relaxation towards the free energy minimum. Global Non-linear thermodynamic effects. equilibrium.
Título Single domain particles• Permanent magnets divide themselves in magnetic domains to minimize their magnetic energy.• There are domain walls between these domains: E ex Exchange energy E ex a (nm ) E an Anisotropy energy E an a Lattice constant
Título Single domain particles E ex 3 5 The exchange energy is so high that Tipically 10 10 E an it is difficult to do any non-uniform rotation of the magnetization. If the particle has R then no domain walls can be formed. This is a SDP:The probabilty of the flip E ex exp( ) 0 and E ex Tcof an individual spin is: T Hence, at low T, the magnetic moment is a T Tc S ct vector of constant modulus:
Single domain particlesThe rotation of M as a whole needs certain energy called magnetic anisotropy.• Relativistic origin: p v – Order of magnitude , with p even. c• Classic description: – Energetic barrier of height: U U (H ) U kV e T Anisotropy Volume constant
Single domain particlesQuantum description: Because the spin is a quantum characteristic, it can pass the barrier by tunnel effect. The tunnel effect, that reveals the Easy axis Hard axis quantum reality of the magnetism, allows the chance of finding the magnetic moment of the particle in two different states simultaneously. U + The action of the observer on the particle will determine its final state!!!
Single domain particlesImportant aspects of SDPs:• Volume distribution: f R f V f U• And orientations:• Their magnetic moments tend to align with the applied magnetic field.
Single domain particles• The particles relax toward the equilibrium state: t M M0 1 S ln 0 Magnetic viscosity• Thermal behaviour ( S T ) – At high temperatures it is easier to “jump” the barrier.• Quantum behaviour (independent of T) – Relaxation due to tunnel effect.
Magnets: memory and relaxation When removing the applied Magnetic solids (ferromagnets) show field, these materials keephysteresis when an external magnetic field is certain magnetization that applied: slowly decreases with time. M HH MR ~ Memory MR ~ ln t Hc H Magnetic solids have memory, and they lose it with time!!! H H t ~ 109 years: Paleomagnets Hc Magnet ~ 5000 Oe t ~ 10 years: credit cards Hc Transformer ~ 1 Oe
Título Quantum relaxation: 1990-96 Magnetic viscosity Magnetic viscosity dependance on T, for low variation with respect T, of a TbFe3 thin film to the magnetic field.
Resonant spin tunneling onmollecular magnets• Identical to single domain particles• Quantum objectsObjetos cuánticos[M i , M j ] 2i M B ijk k |M| ~ μB Quantum[M i , M j ] M iM j M jM i B M k |M| » μB Classic Empirically, the magnetic moment is considered in a quantum way if |M| ≤ 1000μB 2 2H A DS z ES x M(H,T) univocally determined by D and E
ResonantTítulo spin tunneling onmollecular magnets • Application of an external field: Zeeman term H S - Longitudinal component of the field (H || easy axis) Moves the levels. - Transverse component of the field (H easy axis) Allows tunnel effect. • The tunnel effect is possible for certain values of the field; resonant fields.
Resonant spin tunneling onmollecular magnetsThe spin energy levels are moved by an applied magnetic fieldFor multples of the resonant field (HR, 2HR, 3HR, …) theenergy of two levels is the same, producing quantumsuperposition, allowing the tunneling. This is known as Szmagnetic resonance Sz -Sz -Sz
Resonant spin tunneling onmollecular magnets -3-2 12 -4 3 -5 4 -6 5 -7 6 -8 7 -9 8 -10 9 B = B0 10 Magnetic field
Resonant spin tunneling onmollecular magnets -3-2-10 1 -4 2 -5 3 -6 4 -7 5 -8 6 -9 7 -10 8 9 B = 2B0 Magnetic field 10
ResonantTítulo spin tunneling onmollecular magnets• After a certain time, the relaxation becomes exponential: M t M eq t 1 exp H t• Peaks on the relaxation rate Γ(H) at the resonances:
A.C. measurements• TB depends on measuring frequency K V0 TB ln 1 / 0
Quantum magnetic deflagration Avalanche ignition produced by SAW:Surface Acustic Waves (SAW) are low frequency acoustic phonons(below 1 GHz)Coaxial cable connected to an Agilent microwave signal generatorChange in magnetic moment registered in a rf-SQUID magnetometer Hz Coaxial cable LiNbO3 IDT Mn12 crystal substrate c-axis Conducting stripes
Quantum magnetic deflagration κ U(H) v exp τ0 2k B T f This velocity is well fitted: κ = 0.8·10-5 m2/s• The speed of the avalanche Tf (H = 4600 Oe) = 6.8 Kincreases with the applied Tf (H = 9200 Oe) = 10.9 Kmagnetic field• At resonant fields the • The ignition time shows peaks atvelocity of the flame front the magnetic fields at which spinpresents peaks. levels become resonant.
Superradiancie – All spins decay to the fundamental level coherently, with the emission of photons. -1 -3-2 0 1 2 -4 -5 3 -6 4 -7 5 -8 6 -9 7 -10 8 9 B = 2B0 10
SuperradianciaThis kind of emission (SR) has carachteristical propierties that make it different from other more common phenomena like luminiscence I Luminescence τ1 t I L L~λ Superradiancie τSR λ t
Rotational Doppler Effect 2 B FMR 0 H n I n Hn 0 I 2 H Hn 1 Hn I 2 B I measured H ~ 2 . 5 Oe produced by r ~ 1nm particles
Rotational Doppler Effect Occupied states L L 1 En n n 1 B H 2I 2 En n ~ B H 1/ 2 k BT E n ~ k BT n~ B H T ~ 2K n 100 B H ~ 0 . 17 mK
Rotational Doppler Effect• Change in frequency observed due to rotation:• RDE in GPS systems (resonance of an LC circuit) – Resonant frequency insensitive to magnetic fields Resonance• RDE in Magnetic Resonance systems – Resonant frequency sensitive to magnetic fields Resonance
Rotational Doppler Effect• Article: S. Lendínez, E. M. Chudnovsy, and J. Tejada Phys. Rev. B 82, 174418 (2010)• Expression for ω’Res are found for ESR, NMR and FMR. Resonance • Exact expression depends on type of resonance (ESR, NMR or FMR) • Depends on anisotropy
Rotational Doppler Effect• Ω ≈ 100 kHz Ω << ωRes << Δω • ωRes ≈ GHz BUT• ESR and FMR: • Δω ≈ MHz Position of maximum can be determined with accuracy of 100 kHz ≈ Ω • ωRes ≈ MHz• NMR: • Δω ≈ kHz Ω ≈ Δω ESR and Gyromagnetic FMR: tensor (shape,...) anisotropy• κ ≠ 1 needed Hyperfine NMR: interactions
Magnetic VorticesMagnetic vortices are bi-dimensional magnetic systems whose magnetic equilibriumconfiguration is essentially non-uniform (the vortex state): the spin field splits into twowell-differentiated structures, 1) the vortex core consisting of a uniform out-of-planespin component whose spatial extension is ∼ 10nm and 2) the curling magnetizationfield (in-plane spin component), characterized by a non-zero vorticity value.We study disk-shaped magnetic vortices.The application of an in-plane magnetic field yieldsthe displacement of the vortex core perpendicularlyto the field direction.The vortex core entirely governs the lowfrequency spin dynamics: applying a superpositionof a static magnetic field (∼ 100Oe) and an ACmagnetic field (∼ 10Oe), the vortex shows aspecial vibrational mode (called ’slowtranslational/gyrotropic mode’), consistingof the displacement of the vortex core as a whole, following a precessional/gyrotropic movement around the vortex centre. Its characteristic frequency belongsto the subGHz range.
Magnetic VorticesWe have studied an array of These hysteresis loops correspondpermalloy (Fe81 Ni19) disks with to the single domaindiameter 2R = 1.5 μm and thickness (SD)⇐⇒Vortex transitions. For theL = 95 nm under the application of range of temperaturesan in-plane magnetic field up to explored, the vortex linear regime1000 Oe in the range of in the ascending branch shouldtemperatures 2 − 300 K. extend from 300 Oe to 500 Oe at least.
Magnetic Vortices a) Temperature dependence of both MZFC(H) and MFC(H). b) Isothermal magnetic measurements along the descending branch of the hysteresis cycle, Mdes(H), from the SD state (H = 1KOe)
Magnetic Vortices The FC curve is the magnetic equilibria of the system. a) Normalized magnetization (M(t) − Meq)/ (M(0) − Meq) vs. ln t curves measured for two different applied fields (H = 0 and 300 Oe) at T = 2 K. b) Thermal dependence of the magnetic viscosity S(T) for H = 0 and 300 Oe.
Magnetic Vortices Conclusions 1) The existence of structural defects in the disks could be a feasable origin of the energy barriers responsible for the magnetic dynamics of the system. We consider that these defects are capable of pinning the vortex core,when the applied magnetic is swept, in an non-equilibrium position. 2) Thermal activation of energy barriers dies out in the limit T → 0. Our observation that magnetic viscosity S(T) tends to a finite value different from zero as T → 0 indicates that relaxations are non-thermal in this regime (underbarrier quantum tunneling).
Magnetic Vortices Theoretical modeling Rigid model of the shifted vortex ⇒ The vortex core is described as a zero-dimensional object whose dynamics is ruled by Thiele’s equation. The Langrangian is given by L = Gy·x − W(r), where r = (x, y) are the coordinates of the vortex core in the XY plane, G is the modulus of its gyrovector and W(r) is the total magnetic energy of the system. We consider the vortex core as a flexible line that goes predominantly along the z direction, so that r = r(z, t) is a field depending on the vertical coordinate of the vortex core, z. The whole magnetic energy (including the elastic and the pinning potential) is described via a biparametric quartic potential given by where μ and h are the magnetic moment of the dot, respectively the modulus of external magnetic field (applied in the y direction), λ is the elastic coefficient and κ and β are the parameters of the potential energy.
Magnetic VorticesIn absence of applied magnetic field (h = 0), the obtained expressions for thecrossover temperature Tc and the depinning exponent Seff are ,respectively, where c is a numerical factor of order unity. Experimentally wehaveand for a measurable tunneling rate Seff cannot exceed 25−30. From all thesewe deduce the estimates andFinally, from these values of the parameters of the pinning potential we canestimate the width of the energy barrier, which is given by the expressionand the order of magnitude of the heigth of the barrier, which is