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The Bloch Point in Uniaxial Ferromagnets As a Quantum Mechanical Object Andriy Borisovich Shevchenko1 and Maksym Yurjevich Barabash2*

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The Bloch Point in Uniaxial Ferromagnets As a Quantum Mechanical Object Andriy Borisovich Shevchenko1 and Maksym Yurjevich Barabash2* Shevchenko and Barabash Nanoscale Research Letters 2014, 9:132 http://www.nanoscalereslett.com/content/9/1/132 NANO EXPRESS Open Access The Bloch point in uniaxial ferromagnets as a quantum mechanical object Andriy Borisovich Shevchenko1 and Maksym Yurjevich Barabash2* Abstract Quantum effects such as tunneling through pinning barrier of the Bloch Point and over-barrier 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 materials Background magnetic fluctuations of such type in DWs of various Mesoscopic magnetic systems in ferromagnets with a ferro- and antiferromagnetic materials were considered in uniaxial magnetic anisotropy are nowadays the subject of [7-11]. The quantum tunneling between states with differ- considerable attention both theoretically and experimen- ent topological charges of BLs in an ultrathin magnetic tally. Among these systems are distinguished, especially film has been investigated in [12]. domain walls (DWs) and elements of its internal structure - Note that in the subhelium temperature range, the DWs vertical Bloch lines (BLs; boundaries between domain wall and BLs are mechanically quantum tunneling through the areas with an antiparallel orientation of magnetization) and pining barriers formed by defects. Such a problem for the Bloch points (BPs; intersection point of two BL parts) [1]. case of DW and BL in a uniaxial magnetic film with The vertical Bloch lines and BPs are stable nanoformation strong magnetic anisotropy has been investigated in with characteristic size of approximately 102 nm and [13] and [14], respectively. Quantum depinning of the considered as an elemental base for magnetoelectronic DW in a weak ferromagnet was investigated in article and solid-state data-storage devices on the magnetic base [15]. At the same time, the BPs related to the nucleation with high performance (mechanical stability, radiation [16-18] definitely indicates the presence of quantum prop- resistance, non-volatility) [2]. The magnetic structures erties in this element of the DW internal structure, too. similar to BLs and BPs are also present in nanostripes The investigation of the abovementioned problem for the and cylindrical nanowires [3-6], which are perspective BP in the DW of ferromagnets with material quality factor materials for spintronics. (the ratio between the magnetic anisotropy energy and It is necessary to note that mathematically, the DW and magnetostatic one) Q > > 1 is the aim of the present work. its structural elements are described as solitons, which have We shall study quantum tunneling of the BP through topological features. One of such features is a topological defect and over-barrier reflection of the BP from the charge→ which characterized a direction of magnetization defect potential. The conditions for realization of these vector M reversal in the center of magnetic structure. Due effects will be established, too. to its origin, the topological charges of the DW, BL, and BP are degenerated. Meanwhile, in the low temperature → Methods range (T <1 K), M vector reversal direction degeneration Quantum tunneling of the Bloch point can be lifted by a subbarier quantum tunneling. Quantum Let us consider a domain wall containing vertical BL and BP, separating the BL into two parts with different signs of * Correspondence: [email protected] the topological charge. Introducing a Cartesian coordinate 2Technical Centre, National Academy of Science of Ukraine, 13 Pokrovskya St, Kyiv 04070, Ukraine system with the origin at the center of BP, the axis OZ is Full list of author information is available at the end of the article directed along the anisotropy axis, OY is normal to the © 2014 Shevchenko and Barabash; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Shevchenko and Barabash Nanoscale Research Letters 2014, 9:132 Page 2 of 6 http://www.nanoscalereslett.com/content/9/1/132 plane of the DW. According to the Slonczevski equations coercive force of a defect on the characteristic size of [1], one can show that in the region ofp theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi domain wall the DW, vertical BL, or BP. Δ < r ≤ Λ,whereΔ is the DW width, r ¼ x2 þ z2 , Λ ¼ Δ Substituting Equation 4 into Equation 3, and taking into pffiffiffiffi ‘ ’ Q is the characteristic size of BL, the Bloch point deforms account that in the point z0 =0,the potential energy W a magnetic structure of BL, as is described by the following has a local metastable minimum (see Figure 1), we obtain ‘vortex solutions’ [19]. the following expression: π2 −1=2 3 tgϕ ¼ z=x ð1Þ Q MSHc z WzðÞ¼ − 0 þ dz2 ð5Þ 0 2 3 0 ϕ where→ = arctg My/Mx are the components of the vector pffiffiffiffiffi M. In this case, a distribution of the magnetization along where d ¼ Λ 2ε, ε =1− H/Hc < < 1 (we are considering −1 the axis OY has the Bloch form: sinθ = ch (y/Δ), where the magnetic field values H close Hc, that decreases sig- θ is the polar angle in the chosen coordinate system. nificantly the height of the potential barrier). In addition, It is noted that it is the area which mainly contributes potential W(z0) satisfies the normalization condition to m = Δ/γ2 (γ is the gyromagnetic ratio) - the effective ÀÁ BP ; ¼ mass of BP [19]. It is natural to assume that the above- Wz0;1 z0;2 0 mentioned region of the DW is an actual area of BP. pffiffiffiffiffi where z =0 and z ; ¼ 3Λ 2ε are the barrier Taking into account Equation 1 and assuming that the 0,1 0 2 coordinates. motion of BP along the DW is an automodel form ϕ = ϕ It should be mentioned that Equation 5 corresponds to (z − z , x), z is the coordinate of the BP's center), we can 0 0 the model potential proposed in articles [13-15] for the in- write after a series of transformations the energy of vestigation of a tunneling of DW and vertical BL through interaction of the Bloch point WH with the external → → the defect. magnetic field H ¼ −He as follows: y y Following further the general concepts of the Wentzel- Z Kramers-Brilloin (WKB) method, we define the tunneling ∂ϕ 2 amplitude P of the Bloch point by the formula W H ¼ −MSπΔz0H dxdz cosϕ j ¼ ≈−MSπ ΔΛz0H; ∂z z0 0 Δ< ≤Λ r Pe expðÞ−B ð2Þ Zz0;2 2 _ where B ¼ ℏ jjzmBP dz and ℏ is the Planck constant. where MS is the saturation magnetization. To describe the BP dynamics caused by magnetic field H z0;1 and effective field of defect H ,wewillusetheLagrangian d After variation of the Lagrangian function L and inte- formalism. In this case, using Equation 2 and the ‘potential gration of the obtained differential equation with the ’ ¼ z_ 2 − ðÞ energy in the Lagrangian function L mBP Wz0 , _ 2 boundary condition in the point z0 =0, z0→0; and t → we can write it in such form −∞, which corresponds to the pinning of the BP on a de- Zz fect in the absence of the external magnetic field, we will 0 ÀÁÀÁ ′ ′ find the momentum of the BP and, hence, the tunneling WzðÞ¼−M2π2ΛΔ dz H−H z ð3Þ 0 S d exponent 0 Zz0;2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Expanding H (z ) in series in the vicinity of the defect 2 d 0 B ¼ 2m WzðÞdz ð6Þ position, its field can be presented in the following form: ℏ BP z0;1 ÀÁ ðÞ¼ −ðÞ− 2= 2 ð Þ Hd z0 Hc 1 z0 d 2D 4 Taking into account Equation 5, the expression (6) can be rewritten in the following form: where Hc is the coercive force of a defect, d is the coordin- 3 1=2 5=4 − ∂2 8Δ Qh ε 2 ¼ 1 Hd j ¼ c ðÞπ 2 ð Þ ate of its center, D ∂ 2 ¼ , D is the barrier width. B 4 MS 7 Hc z z0 d ℏω It is reasonable to assume0 that the typical change of M defect field is determined by a dimensional factor of where hc = Hc/8MS, ωM =4πγMS. ∂2 =∂ given inhomogeneity. It is clear that in our case, Hd Temperature Tc at which the quantum regime of the BP 2e =Λ2 Λ z0 Hc and hence D ~ . Note also that the abovemen- motion takes place can be derived from relations (5) and ¼ = tioned point of view about defect field correlates with the (7), taking into account the relation T c W max kBB , results of work [20], which indicate the dependence of where Wmax is the maximal value of the potential barrier, Shevchenko and Barabash Nanoscale Research Letters 2014, 9:132 Page 3 of 6 http://www.nanoscalereslett.com/content/9/1/132 Figure 1 Potential W(z0) caused by magnetic field H and effective field of defects Hd. pffiffiffiffiffi kB is the Boltzmann constant. Thus, in accordance with point z0;2 ¼ 3Λ 2ε at τ = 0 and back to the point z0,1 at the above arguments, we obtain τ = ∞ pffiffiffi pffiffiffiffiffi 2 1=2 1=4 2 = z ¼ 3Λ 2ε=ch ðÞω τ ; ω ¼ ω h ðÞ2ε =2 ð9Þ W ¼ 2 QðÞ4πM 2Δ3h ε3 2 in in in M c max 3 S c Further, in defining the instanton frequency, we shall and consider the validity of use of WKB formalism for the pffiffiffi description of the BP quantum tunneling.
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