156
J. Masell and K. Everschor-Sitte
alters the local magnetization direction, the generation of domain walls is not necessarily pairwise, see Fig. 7.3b. One idea is to exploit that the magnetic profile around
an inhomogeneity is twisted and, therefore, spin-torques can act on this part by both,
further twisting and pulling on the magnetic texture [41]. Increasing the applied current will enhance the twisting until a domain wall structure is built, that at the critical
current density j
c
e rips off and travels along the system. Such a creation mechanism
also works in a minimal model consisting of exchange and anisotropy interaction and
basic STTs [41]. In this setup, domain walls are created periodically with a period
T that depends on the applied current strengths j e , or respectively on the effective
spin velocity v e as
T ∼ (v e − v
c
e )
−1/2
∼ ( j e − j
c
e )
−1/2
,
(7.7)
where the exponent is independent of the microscopic details. This universal behavior
of the shedding period T can be proven by explicitly solving for the magnetic profile
and its shedding period in the one-dimensional model including only exchange and
anisotropy interactions. Furthermore, it is valid for a large class of magnetic systems
independent of the details of the microscopic Hamiltonian, including the applicability
for higher dimensions [42]. The required assumptions are (i) presuming a translationally invariant model away from the inhomogeneity and (ii) neglecting non-adiabatic
spin-torque terms. The argument for the universal exponent in the shedding period
is based on combining three ingredients:
(1) the postulate of a critical current density j
c
e above which there will be no statically
stable solution and the created magnetic texture rips off the inhomogeneity,
(2) the behavior of the magnetic structure in the “just still static limit”, i.e., for
j e j
c
e and
(3) the “just dynamic limit”, i.e. for j e j
c
e .
For the last two, one employs that the magnetic profile at the critical point will not
differ too much in these two limits. The main influence on the magnetic structure
will be a (time-dependent) shift in the position x 0 where the structure is centered
in combination with a mild perturbation on the profile. Solving the LLGS equation
in these two limits, yields for the “just still static” limit the relation j
c
e − j
s
e ∼ x
2
0
and for the “just dynamic limit” ∂ t x 0 = j
d
e − j
s
e , where j
s
e is the current strength
in the just still static limit and j
d
e in the just dynamic limit. These relations are
the simplest, that satisfy the expected behavior: (i) the velocity of the domain wall
depends linearly on the current strength beyond the threshold value and (ii) inverting
the direction of the current should, in principle, create the domain wall structure in
the opposite direction. Eliminating j
s
e allows to calculate the period of the magnetic
texture formation T ∼ ( j e − j
c
e )
−1/2 and thus explains the universal dependence.
Note that this universal behavior holds independent of the dimension, provided
the above mentioned assumptions are satisfied. In dimensions higher than one the
precise shape of the created magnetic texture cannot be calculated analytically. Based
on topology, one can, however, conclude that the winding number during the production process must be conserved, opening up the possibility to shed more complex
topological structures and their anti-particles.
J. Masell and K. Everschor-Sitte
alters the local magnetization direction, the generation of domain walls is not necessarily pairwise, see Fig. 7.3b. One idea is to exploit that the magnetic profile around
an inhomogeneity is twisted and, therefore, spin-torques can act on this part by both,
further twisting and pulling on the magnetic texture [41]. Increasing the applied current will enhance the twisting until a domain wall structure is built, that at the critical
current density j
c
e rips off and travels along the system. Such a creation mechanism
also works in a minimal model consisting of exchange and anisotropy interaction and
basic STTs [41]. In this setup, domain walls are created periodically with a period
T that depends on the applied current strengths j e , or respectively on the effective
spin velocity v e as
T ∼ (v e − v
c
e )
−1/2
∼ ( j e − j
c
e )
−1/2
,
(7.7)
where the exponent is independent of the microscopic details. This universal behavior
of the shedding period T can be proven by explicitly solving for the magnetic profile
and its shedding period in the one-dimensional model including only exchange and
anisotropy interactions. Furthermore, it is valid for a large class of magnetic systems
independent of the details of the microscopic Hamiltonian, including the applicability
for higher dimensions [42]. The required assumptions are (i) presuming a translationally invariant model away from the inhomogeneity and (ii) neglecting non-adiabatic
spin-torque terms. The argument for the universal exponent in the shedding period
is based on combining three ingredients:
(1) the postulate of a critical current density j
c
e above which there will be no statically
stable solution and the created magnetic texture rips off the inhomogeneity,
(2) the behavior of the magnetic structure in the “just still static limit”, i.e., for
j e j
c
e and
(3) the “just dynamic limit”, i.e. for j e j
c
e .
For the last two, one employs that the magnetic profile at the critical point will not
differ too much in these two limits. The main influence on the magnetic structure
will be a (time-dependent) shift in the position x 0 where the structure is centered
in combination with a mild perturbation on the profile. Solving the LLGS equation
in these two limits, yields for the “just still static” limit the relation j
c
e − j
s
e ∼ x
2
0
and for the “just dynamic limit” ∂ t x 0 = j
d
e − j
s
e , where j
s
e is the current strength
in the just still static limit and j
d
e in the just dynamic limit. These relations are
the simplest, that satisfy the expected behavior: (i) the velocity of the domain wall
depends linearly on the current strength beyond the threshold value and (ii) inverting
the direction of the current should, in principle, create the domain wall structure in
the opposite direction. Eliminating j
s
e allows to calculate the period of the magnetic
texture formation T ∼ ( j e − j
c
e )
−1/2 and thus explains the universal dependence.
Note that this universal behavior holds independent of the dimension, provided
the above mentioned assumptions are satisfied. In dimensions higher than one the
precise shape of the created magnetic texture cannot be calculated analytically. Based
on topology, one can, however, conclude that the winding number during the production process must be conserved, opening up the possibility to shed more complex
topological structures and their anti-particles.