Chemistry Reference
In-Depth Information
Numerical solution shows that the dynamical matrix
of the nonsecular DME
has exactly 2
S
1)
2
eigenvalues. One of
the real eigenvalues is zero and corresponds to the thermal equilibrium. Complex
eigenvalues occur in complex conjugate pairs. This behavior is similar to that of
the secular DME. At low temperatures in the regime of thermal activation and
weak tunneling (overdamped resonances), there is one nonzero real eigenvalue
that is much smaller than all other real eigenvalues and real parts of complex
eigenvalues. In the case of underdamped resonances that corresponds to the ground-
state tunneling, there are three eigenvalues, one real and two complex, which
describe the slow dynamics. This slow dynamics is captured analytically by Eq.
(237).
Figure 1 shows the zero-temperature escape rate versus the bias field
H
z
in
the generic MM model with
B
+
1 real eigenvalues out of the total (2
S
+
0 in Eq. (141). The striking feature is the spin
tunneling at resonance fields that leads to the increase of the escape rate by many
orders of magnitude. Most of the points have been obtained from the secular
DME, the points at resonances and between them have been obtained from the
semisecular DME, and the analytical result of Eq. (274) is drawn in the vicinity of
resonances. Near the zero-field resonance, Eq. (244) is shown. The characteristic
“shoulder” described by this equation is well reproduced by the numerical result.
As mentioned above, the secular approximation can yield unphysically high escape
rates at resonances. However, the resonances are narrow and there are no secular
points in this numerical calculation that hit them. Resonances with
k
=
1
,
2
,
and
3 are overdamped and can be approximately described by Eq. (281). Resonances
with
k
=
4 in Fig. 1 are underdamped, so that the peaks reach the value given
by Eq. (276). The latter is of order
≥
10
6
s
−
1
and coincides with the spin-
phonon rate between the adjacent levels that is the highest possible rate achievable
off-resonance at temperatures exceeding the energy barrier.
Figure 2 shows the time dependence of spin polarization
∼
3
×
S
z
at different reso-
nances in Fig. 1. The relaxation is exponential for the overdamped resonances, as
well as off resonance (not shown). In contrast, at underdamped resonances with
k
≥
5 there are damped oscillations described by three different relaxation rates
in Eq. (263). In the case of exponential relaxation, it is sufficient to identify the
escape rate with the smallest real eigenvalue of the dynamical matrix. In the case
of underdamped resonances there are three slow eigenvalues, and obtaining the
correct value of the escape rate requires the use of the integral relaxation time.
Escape rate versus the bias field at different temperatures in the generic model
is shown in Fig. 3. All data were obtained from the numerical solution of the
semisecular DME. The anisotropy value
D
0
.
66 K has been chosen to fit the
barrier height in Mn
12
(see below). As expected, the escape rate increases with
temperature, faster off-resonance than on-resonance. One can see (especially clear
for
T
=
1) that at nonzero temperatures the tunneling peak may
consist of several peaks of different width on top of each other [11]. Broad peaks
=
2 K and
k
=
