Abstract
Quantum effects such as tunneling through pinning barrier of the Bloch Point and overbarrier reflection from the defect potential of one have been investigated in ferromagnets with uniaxial strong magnetic anisotropy. It is found that these phenomena can be appeared only in subhelium temperature range.
Keywords:
Quantum tunneling; The bloch point; Domain walls; Vertical bloch lines; uniaxial magnetic film; Quantum depinning; Magnetic field; Potential barrier; Ferromagnetic materialsBackground
Mesoscopic magnetic systems in ferromagnets with a uniaxial magnetic anisotropy are nowadays the subject of considerable attention both theoretically and experimentally. Among these systems are distinguished, especially domain walls (DWs) and elements of its internal structure  vertical Bloch lines (BLs; boundaries between domain wall areas with an antiparallel orientation of magnetization) and Bloch points (BPs; intersection point of two BL parts) [1]. The vertical Bloch lines and BPs are stable nanoformation with characteristic size of approximately 10^{2} nm and considered as an elemental base for magnetoelectronic and solidstate datastorage devices on the magnetic base with high performance (mechanical stability, radiation resistance, nonvolatility) [2]. The magnetic structures similar to BLs and BPs are also present in nanostripes and cylindrical nanowires [36], which are perspective materials for spintronics.
It is necessary to note that mathematically, the DW and its structural elements are
described as solitons, which have topological features. One of such features is a
topological charge which characterized a direction of magnetization vector
Note that in the subhelium temperature range, the DWs and BLs are mechanically quantum tunneling through the pining barriers formed by defects. Such a problem for the case of DW and BL in a uniaxial magnetic film with strong magnetic anisotropy has been investigated in [13] and [14], respectively. Quantum depinning of the DW in a weak ferromagnet was investigated in article [15]. At the same time, the BPs related to the nucleation [1618] definitely indicates the presence of quantum properties in this element of the DW internal structure, too. The investigation of the abovementioned problem for the BP in the DW of ferromagnets with material quality factor (the ratio between the magnetic anisotropy energy and magnetostatic one) Q > > 1 is the aim of the present work. We shall study quantum tunneling of the BP through defect and overbarrier reflection of the BP from the defect potential. The conditions for realization of these effects will be established, too.
Methods
Quantum tunneling of the Bloch point
Let us consider a domain wall containing vertical BL and BP, separating the BL into
two parts with different signs of the topological charge. Introducing a Cartesian
coordinate system with the origin at the center of BP, the axis OZ is directed along
the anisotropy axis, OY is normal to the plane of the DW. According to the Slonczevski
equations [1], one can show that in the region of the domain wall Δ < r ≤ Λ, where Δ is the DW width,
where ϕ = arctg M_{y}/M_{x} are the components of the vector
It is noted that it is the area which mainly contributes to m_{BP} = Δ/γ^{2} (γ is the gyromagnetic ratio)  the effective mass of BP [19]. It is natural to assume that the abovementioned region of the DW is an actual area of BP.
Taking into account Equation 1 and assuming that the motion of BP along the DW is
an automodel form ϕ = ϕ(z − z_{0}, x), z_{0} is the coordinate of the BP's center), we can write after a series of transformations
the energy of interaction of the Bloch point W_{H} with the external magnetic field
where M_{S} is the saturation magnetization.
To describe the BP dynamics caused by magnetic field H and effective field of defect H_{d}, we will use the Lagrangian formalism. In this case, using Equation 2 and the ‘potential
energy’ in the Lagrangian function
Expanding H_{d}(z_{0}) in series in the vicinity of the defect position, its field can be presented in the following form:
where H_{c} is the coercive force of a defect, d is the coordinate of its center,
It is reasonable to assume that the typical change of defect field is determined by
a dimensional factor of given inhomogeneity. It is clear that in our case,
Substituting Equation 4 into Equation 3, and taking into account that in the point z_{0} = 0, the ‘potential energy’ W has a local metastable minimum (see Figure 1), we obtain the following expression:
where
where z_{0,1} = 0 and
Figure 1. Potential
It should be mentioned that Equation 5 corresponds to the model potential proposed in articles [1315] for the investigation of a tunneling of DW and vertical BL through the defect.
Following further the general concepts of the WentzelKramersBrilloin (WKB) method, we define the tunneling amplitude P of the Bloch point by the formula
where
After variation of the Lagrangian function
Taking into account Equation 5, the expression (6) can be rewritten in the following form:
where h_{c} = H_{c}/8M_{S}, ω_{M} = 4πγM_{S}.
Temperature T_{c} at which the quantum regime of the BP motion takes place can be derived from relations
(5) and (7), taking into account the relation
and
Substituting into the expressions (7) and (8), the numerical parameters corresponding to uniaxial ferromagnets: Q ~ 5–10, Δ ~ 10^{−6} cm, 4πM_{S} ~ (10^{2} − 10^{3}) Gs, H_{c} ~ (10 − 10^{2}) Oe [19] (see also articles [20,21], in which the dynamic properties of BP in yttriumiron garnet were investigated), γ ~ 10^{7} Oe^{−1} s^{−1}, for ϵ ~ 10^{−4} − 10^{−2}, we obtain B ≈ 1–30 and T_{c} ~ (10^{−3} − 10^{−2}) К.
The value obtained by our estimate B ≤ 30 agrees with corresponding values of the tunneling exponent for magnetic nanostructures [22], which indicate the possibility of realization of this quantum effect. In this case, as can be seen from the determination of the BP effective mass, in contrast to the tunneling of the DW and vertical BL through a defect, the process of the BP tunneling is performed via the ‘transfer’ of its total effective mass through the potential barrier.
Following the integration of the motion equation of the BP obtained via the Lagrangian
function variation, we find the its instanton trajectory z_{in} and the instanton frequency of the Bloch point ω_{in} (see review [23]), which characterize its motion within the space with an ‘imaginary’ time τ = it: from the point z_{0,1} = 0 at τ = −∞ to the point
Further, in defining the instanton frequency, we shall consider the validity of use of WKB formalism for the description of the BP quantum tunneling. As known [24], the condition of applicability of the WKB method is the fulfillment of the following inequality:
where p is momentum, m is the quasiparticle mass, and F is the force acting on it.
In our case
Setting the abovementioned parameters of the ferromagnets and defect into Equation 11, it is easy to verify that this relationship is satisfied, that in turn indicates the appropriateness of use of the WKB approximation in the problem under consideration.
Let us estimate the effect of dissipation on the tunneling process of the BP. To do
this, we compare the force F, acting on the quasiparticle, with the braking force
The analysis of this expression shows that
Note also that the mechanism of breaking force has been investigated in the work [25] and is associated with the inclusion of relaxation terms of exchange origin in the LandauLifshiz equation for magnetization of a ferromagnet [26].
Results and discussion
The overbarrier reflection of the Bloch point
In the above, it was mentioned that tunneling of DW and vertical BL is carried out via subbarrier transition of small parts of the area of DW or the length in case BL. In this case, both DW and vertical BL are located in front of a potential barrier at a metastable minimum that makes possible the process of their tunneling. At the same time, the BP depinning occurs via ‘transmission’ through the potential barrier instantly of entire effective mass of the quasiparticle. This result indicates that the presence of a metastable minimum in the interaction potential of BP with a defect (in contrast to DW or BL) is not necessary. Moreover, it means that there exists a possibility of realization for BP of such quantum effect as overbarrier reflection of a quasiparticle from the defect potential. In this case, the velocity at which BP ‘falls’ on the barrier may be determined by a pulse of magnetic field applied to the BP. And, as we shall see bellow, the potential of interaction between the BP and a defect has a rather simple form. Obviously, the effect is more noticeable in the case when the BP energy is not much greater than the height of the potential barrier U_{0}.
Using the formula (2), we represent the dynamics equation for the BP in a pulsed magnetic field H_{y}(t) = H_{0}χ(1 − t/T) in the form
where v = ∂z_{0}/∂t is the BP velocity, χ(1 − t/T) is the Heaviside function, H_{0} is the amplitude, and T is the pulse duration.
By integrating the Equation 12 for
Note that the study, performed for time
We assume that defect is located at z_{0} = 0. Then, by expanding the potential of interaction of BP with the defect, U_{d}(z_{0}), in a series near this point and taking Equation 2 into account, we can write down
where in accordance with the formula (2), the height of the potential barrier is U_{0} = π^{2}Λ^{2}ΔM_{S}H_{c}.
Note that phenomenological expression for defecteffective field H_{d} (see formula (4)) follows from the series expansion of the potential U_{d}(z_{0}) near the inflection point. It was at this point that there is maximum field of defect. It is natural to assume that if BP has overcome the barrier in this point, then the tunneling process is probable in general.
Using the WKB approximation, and following the formalism described in [27,28], we determine the coefficient of overbarrier reflection of the Bloch Point R by the formula
where
Taking into account the expression for the potential (14), from Equation 15, we find
where the parameter ϵ′ = (E_{BP} − U_{0})/E_{BP} < < 1 (recall that we consider the case when the energy E_{BP} close to U_{0}).
Using the formula (13), Equation 16 can be rewritten as
Substituting into the expressions (15) and (17), the ferromagnet and defect parameters, at ϵ′ ≥ 5 × 10^{−5} we obtain R ≤ 0.1, which is in accordance with criterion of applicability of Equation 15 (see [28]).
Note that from Equations 15 and 16, it follows that R → 0 at U_{0} → 0, i.e., we obtain a physically consistent conclusion about the disappearance of the effect of overbarrier reflection in the absence of a potential barrier.
Based on the obvious relation,
Let us consider the question about validity of applicability of the WKB approximation to the problem under consideration. Since in the given case E_{BP} ≈ U_{0}, then the conditions of ‘quasiclassical’ behavior of the Bloch point and the potential barrier actually coincide and, in accordance with [24], are reduced to the fulfillment of the inequality
where
Using the explicit form of U_{0}, Equation 18 can be rewritten as
An analysis of this inequality shows its fulfillment for the values ϵ′ ≥ 10^{−4}, that in fact is a ‘lower estimate’ for this parameter. In a critical temperature
An estimate of the expression (19) shows that
Conclusions
It is shown that in the subhelium temperature range, the Bloch point manifest themselves as a quantum mechanical object. Thus, the BP may tunnel through the pining barrier formed by the defect and overbarrier reflection from the defect potential. In this case, since the overbarrier reflection of the BP and subbarrier tunneling of the BP occur in pulse and permanent magnetic fields, respectively, the practical possibility to study these quantum phenomena separately exists. Moreover, the experimental realization of the mentioned phenomena can be the basis for the creation of new methods of diagnostic of ferromagnetic materials and sensitive methods for studying an internal structure of their DWs.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
ABS and MYB read and approved the final manuscript.
References

Malozemoff AP, Slonczewski JC: Magnetic Domain Walls in Bubble Materials. New York: Academic Press; 1979.

Konishi A: A newultradensity solid state memory: Bloch line memory.

Klaui M, Vaz CAF, Bland JAC: Headtohead domainwall phase diagram in mesoscopic ring magnets.
Appl. Phys. Lett. 2004, 85:5637. Publisher Full Text

Laufenberg M, Backes D, Buhrer W: Observation of thermally activated domain wall transformations.
Appl. Phys. Lett. 2006, 88:052507. Publisher Full Text

Nakatani Y, Thiaville A, Miltat J: Headtohead domain walls in soft nanostrips: a refined phase diagram.

Vukadinovic N, Boust F: Threedimensional micromagnetic simulations of multidomain bubblestate excitation spectrum in ferromagnetic cylindrical nanodots.

Takagi S, Tatara G: Macroscopic quantum coherence of chirality of a domain wall in ferromagnets.
Phys. Rev. B 1996, 54:9920. Publisher Full Text

Shibata J, Takagi S: Macroscopic quantum dynamics of a free domain wall in a ferromagnet.
Phys. Rev. B 2000, 62:5719. Publisher Full Text

Galkina EG, Ivanov BA, Savel’ev S: Chirality tunneling and quantum dynamics for domain walls in mesoscopic ferromagnets.

Ivanov BA, Kolezhuk AK: Quantum tunneling of magnetization in a small area – domain wall.

Ivanov BA, Kolezhuk AK, Kireev VE: Chirality tunneling in mesoscopic antiferromagnetic domain walls.

Dobrovitski VV, Zvezdin AK: Macroscopic quantum tunnelling of solitons in ultrathin films.
JMMM 1996, 156:205. Publisher Full Text

Chudnovsky EM, Iglesias O, Stamp PCE: Quantum tunneling of domain walls in ferromagnets.
Phys. Rev. B 1992, 46:5392. Publisher Full Text

Shevchenko AB: Quantum tunneling of a Bloch line in the domain wall of a cylindrical magnetic domain.
Techn. Phys. 2007, 52:1376. Publisher Full Text

Dobrovitski VV, Zvezdin AK: Quantum tunneling of a domain wall in a weak ferromagnet.

Lisovskii VF: Fizika tsilindricheskikh magnitnykh domenov (Physics of Magnetic Bubbles). Moscow: Sov. Radio; 1982.

Thiaville A, Garcia JM, Dittrich R: Micromagnetic study of Blochpointmediated vortex core reversal.

Kufaev YA, Sonin EB: Dynamics of a Bloch point (point soliton) in a ferromagnet.

Zubov VE, Krinchik GS, Kuzmenko SN: Anomalous coercive force of Bloch point in iron single crystals.

Kabanov YP, Dedukh LM, Nikitenko VI: Bloch points in an oscillating Bloch line.

Gornakov VS, Nikitenko VI, Prudnikov IA: Mobility of the Bloch point along the Bloch line.

Chudnovsky EM: Macroscopic quantum tunneling of the magnetic moment.
J. Appl. Phys. 1993, 73:6697. Publisher Full Text

Vaninstein AI, Zakharov VI, Novikov VA, Shifman MA: ABS of instantons.

Landau LD, Lifshitz EM: Kvantovaya mekhanika (Quantum Mechanics). Moscow: Nauka; 1989.

Galkina EG, Ivanov BA, Stephanovich VA: Phenomenological theory of Bloch point relaxation.
JMMM 1993, 118:373. Publisher Full Text

Bar’yakhtar VG: Phenomenological description of relaxation processes in magnetic materials.

Pokrovskii VL, Khalatnikov EM: К voprosu о nadbarjernom otrazhenii chastiz visokih energiy (On supperbarrier reflection of high energy particles).

Elyutin PV, Krivchenkov VD: Kvantovaya mekhanika (Quantum Mechanics). Moscow: Nauka; 1976.