3 Collective Magnetic Behaviour
71
3.2 Case Studies: Superspin Glasses
Spin glasses are formed by atomic systems where there is structural disorder and
competing ferro- and antiferromagnetic interaction giving rise to frustrated spins.
The properties of model spin glasses such as Cu(Mn) and Fe 0.5 Mn 0.5 TiO 3 have
been extensively studied and exhibit universal properties as to the existence of a
second-order phase transition (revealed from static and dynamic scaling analyses)
and infinite relaxation times and non-equilibrium dynamics manifested by ageing,
memory and rejuvenation phenomena at temperature below T g [18]. Measurable
physical manifestations of superspin glass states are found in systems with strong
enough dipolar interaction. Dipolar interaction can in randomly packed systems
gives rise to dynamic frustration, i.e. depending on how neighbouring particles flip
their magnetization direction, the dipolar moment on a specific particle changes sign
with time. Strong enough means that the dipolar interaction strength causes a glass
temperature that exceeds the blocking temperature (at the observation time (about
10 s) of magnetization versus temperature measurements) by a factor of two or
more. For instance, if the dipolar interaction E dd /k B = 100 K, the anisotropy of the
particles E ap /k B < log(τ /τ 0) × T B ~ 1250 K (considering T B = 50 K, on experimental
observation time (10 s) and τ 0 = 10
−11 s). Translated to observation times, this
implies that the relaxation time of the particles at T g , τ p ~ 10
−5 s. When T g exceeds
T B by a factor of five, the relaxation time of the particles at T g becomes τ p ~ 10
−9
s. To measure critical slowing down on the time scales of standard ac-susceptibility
experiments (1–10,000 Hz, t obs = 1/ω ~ 0.16–1.6 × 10
−5 s) the relaxation time of
the slowest particles should always be much shorter than the observation time of the
probe. In systems with weak interparticle interaction, this criterion is not fulfilled.
On the other hand, as mentioned above (see Sect. 3.1.4), other manifestations of
collective phenomena, such as ageing and memory, are readily observed in systems
with wide distributions of particle sizes and comparably weak dipolar interparticle
interaction.
3.2.1 Frozen Ferrofluids
Ferrofluids allow continuous tuning of the particle density and thus the strength
of the interparticle dipolar interaction. Dense randomly packed frozen ferrofluids
show magnetic ageing behaviour due to collective non-equilibrium dynamics introduced by dipolar dynamic frustration [14]. Interparticle interaction broadens the
relaxation function at low temperatures and shifts the maximum in the low-field
ZFC magnetization versus temperature curve to higher temperatures. In cases of
strong dipolar interaction and narrow particle size distributions, critical slowing
down indicating a spin glass like transition is observed on the time scales of ACsusceptibility experiments. An example of this is shown in Fig. 3.5 citing results
from AC-susceptibility measurements on a system of FeC nanoparticles at different
71
3.2 Case Studies: Superspin Glasses
Spin glasses are formed by atomic systems where there is structural disorder and
competing ferro- and antiferromagnetic interaction giving rise to frustrated spins.
The properties of model spin glasses such as Cu(Mn) and Fe 0.5 Mn 0.5 TiO 3 have
been extensively studied and exhibit universal properties as to the existence of a
second-order phase transition (revealed from static and dynamic scaling analyses)
and infinite relaxation times and non-equilibrium dynamics manifested by ageing,
memory and rejuvenation phenomena at temperature below T g [18]. Measurable
physical manifestations of superspin glass states are found in systems with strong
enough dipolar interaction. Dipolar interaction can in randomly packed systems
gives rise to dynamic frustration, i.e. depending on how neighbouring particles flip
their magnetization direction, the dipolar moment on a specific particle changes sign
with time. Strong enough means that the dipolar interaction strength causes a glass
temperature that exceeds the blocking temperature (at the observation time (about
10 s) of magnetization versus temperature measurements) by a factor of two or
more. For instance, if the dipolar interaction E dd /k B = 100 K, the anisotropy of the
particles E ap /k B < log(τ /τ 0) × T B ~ 1250 K (considering T B = 50 K, on experimental
observation time (10 s) and τ 0 = 10
−11 s). Translated to observation times, this
implies that the relaxation time of the particles at T g , τ p ~ 10
−5 s. When T g exceeds
T B by a factor of five, the relaxation time of the particles at T g becomes τ p ~ 10
−9
s. To measure critical slowing down on the time scales of standard ac-susceptibility
experiments (1–10,000 Hz, t obs = 1/ω ~ 0.16–1.6 × 10
−5 s) the relaxation time of
the slowest particles should always be much shorter than the observation time of the
probe. In systems with weak interparticle interaction, this criterion is not fulfilled.
On the other hand, as mentioned above (see Sect. 3.1.4), other manifestations of
collective phenomena, such as ageing and memory, are readily observed in systems
with wide distributions of particle sizes and comparably weak dipolar interparticle
interaction.
3.2.1 Frozen Ferrofluids
Ferrofluids allow continuous tuning of the particle density and thus the strength
of the interparticle dipolar interaction. Dense randomly packed frozen ferrofluids
show magnetic ageing behaviour due to collective non-equilibrium dynamics introduced by dipolar dynamic frustration [14]. Interparticle interaction broadens the
relaxation function at low temperatures and shifts the maximum in the low-field
ZFC magnetization versus temperature curve to higher temperatures. In cases of
strong dipolar interaction and narrow particle size distributions, critical slowing
down indicating a spin glass like transition is observed on the time scales of ACsusceptibility experiments. An example of this is shown in Fig. 3.5 citing results
from AC-susceptibility measurements on a system of FeC nanoparticles at different