Spin Waves and Magnetic Anisotropy in Pulse
Electro-deposited Nanocrystalline Ni
Pavan Venu Prakash Madduri and S. N. Kaul
School of Physics, University of Hyderabad, Central University P.O., Hyderabad-500046
Andhra Pradesh, India
Corresponding author’s e-mail: kaul.sn@gmail.com, Tel.: +91-40-23134322; Fax: +91-40-23010227
Abstract
Spin-wave stiffness at 0 K, D(0), and thermal renormalization of D (magnetocrystalline anisotropy constant, K1(T))
in nanocrystalline Ni samples with average crystallite sizes d
= 12(1) nm and 22(1) nm have been determined from thermomagnetic, M(T), data (M-H isotherms) taken at magnetic
fields in the range 25 Oe ≤ H ≤ 90 kOe. D(0) increases from
D(0) = 285(7) meVÅ2 for d = 12 nm to D(0) =298(7) meVÅ2
for d = 22 nm. At 2 K, K1 is smaller by one (two) order (s) of
magnitude in d = 22 nm (12 nm) than in bulk Ni.
Keywords: Magnetic excitations; Spin waves; Nanocrystalline
ferromagnets; Approach-to-saturation; Magnetocrystalline
anisotropy.
Introduction
Despite intense efforts to ascertain the
crystallite size (d) dependence of the magnetic
properties of nanocrystalline (nc-) ferromagnets, Fe, Co,
Ni, many aspects of the magnetic behavior still remain
obscure. One such feature pertains to the nature of lowlying magnetic excitations. In sharp contrast with the
strong departures from the spin-wave T3/2 variation of
magnetization for T ≤ 350 K reported [1] in nc-Ni with
d = 10 nm, small angle neutron scattering experiments
[2] reveal the existence of well-defined spin waves in
the nc-Ni with d = 49 nm. It is not clear if the reduction
in d changes the nature of low-lying magnetic excitations or if below a certain value of d, it is harder to
excite spin waves at low and intermediate temperatures.
The present work addresses this basic issue.
Results and discussion
The Williamson-Hall plots, constructed from
the x-ray diffraction patterns taken on two samples of
pulse electrodeposited (PED) Ni at room temperature,
yielded d = 12(1) nm and 22(1) nm. Figure 1 demonstrates that the conventional spin-wave (SW) relation
for Heisenberg ferromagnets M(T, H) = M(0, H) – g μB
 {n-3/2 exp(- n tH)} × [ kBT / 4 D(T)] 3 / 2, with D(T) =
D(0) (1 - D2 T2 ), D(0) = 285(7) meVÅ2 [298(7)
meVÅ2] and D2 = 2.06(3) × 10-6 K-2 [2.04(7) × 10-6 K-2]
describes (continuous curves) the observed M(T)
(symbols) in the sample with d = 12 nm [d = 22 nm]
quite well. In the above SW expression, the reduced
Fig 1: Spin-wave fits (continuous curves) to the M(T)
data (symbols).
temperature tH = gμBH / kBT allows for the fieldinduced energy gap in the SW spectrum. D(0)  (2 / 3)
D(5K), reported [2] for d = 49 nm and bulk Ni.
M-H isotherms in the temperature range 2 K 
T  300 K, are analyzed in terms of the ‘approach-tosaturation’ relation M(T,H) = MS(T){1 - [a(T) / H] [b(T) / H2]} + hf H. Magnetocrystalline anisotropy
coefficient K1(T) determined from the relation [3] b(T)
= 0.0762 [K1(T) / MS(T)]2, turns out to be one order of
magnitude smaller than in bulk Ni. The reasons for the
reduced D(0) and K1(T) in nc-Ni have been discussed.
Acknowledgment
This work was supported by the DST under
Grant no: SP/S2/JCB-18/2010. P. V. P. Madduri thanks
CSIR, India, for the Senior Research Fellowship.
References
[1] H. E. Schaefer et al., Nanostruct. Mater. 1,
523 (1992).
[2] J. Weissmüller et al., Phys. Rev. B 63, 214414
(2001).
[3] S. Chikazumi, Physics of Magnetism (John Wiley,
New York, 1964) p. 277.