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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
 mB  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  1D
 
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,  AS S  1  mm  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
GB    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))