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Transcript

EMR spectra in nanoparticles. Quantatization approach A magnetic nanoparticle is considered as a giant cluster of N microscopic spins s coupled by strong ferromagnetic exchange interaction. Only the ground multiplet is considered to be populated (assuming large enough separation between the lowest multiplet with the maximum spin S = Ns and other multiplets). The energy spectrum of this giant spin is governed by the standard spin Hamiltonian commonly used in the EPR spectroscopy. In the simplest case of axial crystalline field it reads: Hˆ 2 B S DS n (1) where is the modulus of the electron gyromagnetic ratio; D is the parameter of the axial anisotropy; Sn is the spin projection on the anisotropy axis n making an angle with the external magnetic field B. The anisotropy field Ha = - (2S+1)D/ ; the negative D is related to the easy axis anisotropy. The energy spectrum consists of 2S +1 levels Em,. corresponding to different values of the spin projection S z m , m = -S,…,S. In the high-field approximation (B >> D): Em, 3 cos 2 1 mB Dm 2 2 (2) The electron magnetic resonance (EMR) spectrum is represented as a sum of all allowed transitions between the adjacent levels with the change m = 1 in the magnetic quantum number. For a single crystalline nanoparticle with the orientation , the EMR spectrum can be found as: f ( B ) g B B S m, m S m , Wm , (3) where Bm, is the resonance field of the individual transition (m, m+1), Bm, 1 3 cos 2 1 ; 2m 1D 2 (4) g (B – Bm,) is the corresponding form-factor; m, is the equilibrium population difference between the adjacent energy levels (the Boltzmann distribution at the temperature T is assumed); and Wm, AS S 1 mm 1 (5) is the transition probability (the factor A is proportional to the microwave power). Finally, for a sample containing many randomly oriented nanoparticles, the overall spectrum reads: S 0 S GB sin d f ( B)dm (6) In the last expression the sum over m is replaced by the corresponding integral (at S>>1). Substantial difference between this and classical approach [Yu. L. Raikher and V. I. Stepanov, Sov. Phys.- JETP 75, 764 (1992); Yu. L. Raikher, V. I. Stepanov, Phys. Rev. B 50, 6250 (1994)] is that individual contributions are taking into account from every quantum transition in the multi-level energy spectrum of the giant spin cluster. Unlike the classical model, this approach explains a narrow spectral component and low-field signals observed experimentally in many cases. To fit the experimental data, one should specify the individual line shape g(B-Bm,). Either Lorentzian or Gaussian may be tested, assuming the line-width dependent on m. Spreading and fluctuations of D and n are typical for magnetic nanoparticles. This results in the line broadening which increases with m, see Eqs.3 and 4. The narrow spectral component originates from transitions with low m, which are insensitive to imperfections. The corresponding levels lie high enough, see Eq. (2), and become depopulated at low temperatures. This agrees with the observed disappearance of the narrow feature upon cooling. Note, a shift of the main EMR spectrum towards low fields is frequently observed as the temperature decreases. This shift is caused by the second-order anisotropy terms and can be fitted with introducing a term proportional to m2 into Eq.(4). References N. Noginova, F. Chen, T. Weaver, E. Giannelis, A. Burlinos, and V.A. Atsarkin. “Magnetic resonance in nanoparticles: between ferro- and paramagnetism”, Journal of Physics: Condensed Matter, 19, 246208 (2007) N. Noginova, T. Weaver, E.P. Giannelis, A.B. Bourlinos, V. A. Atsarkin, V. V. Demidov. “Observation of multiple quantum transitions in magnetic nanoparticles”. Phys. Rev. B 77, 014403 (2008))