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 . 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 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

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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. As known ε1=4 1=2ℏω 2 hc M [24], the condition of applicability of the WKB method T c ¼ ð8Þ 12kB is the fulfillment of the following inequality:

Substituting into the expressions (7) and (8), the nu- mℏjjF =p3 << 1 ð10Þ merical parameters corresponding to uniaxial ferro- −6 2 3 where p is momentum, m is the quasiparticle mass, and F magnets: Q ~5–10, Δ ~10 cm, 4πMS ~(10 − 10 ) 2 is the force acting on it. Gs, Hc ~(10− 10 ) Oe [19] (see also articles [20,21], in pffiffiffiffiffi ¼ ω2 ξ ω ξ ξeΛ ε which the dynamic properties of BP in yttrium-iron garnet In our case F mBP in , p = mBP in , 2 . Then, − − − − were investigated), γ ~107 Oe 1 s 1,forε ~10 4 − 10 2,we taking into account Equation 9, we will rewrite Equation 10 −3 −2 obtain B ≈ 1–30 and Tc ~(10 − 10 ) К. in the following way: The value obtained by our estimate B ≤ 30 agrees with ℏγ2ω−1 −1=2ðÞε −5=4Q−1=Δ3 << ð Þ corresponding values of the tunneling exponent for mag- M hc 2 1 11 netic nanostructures [22], which indicate the possibility of realization of this quantum effect. In this case, as can be Setting the abovementioned parameters of the ferro- seen from the determination of the BP effective mass, in magnets and defect into Equation 11, it is easy to verify contrast to the tunneling of the DW and vertical BL that this relationship is satisfied, that in turn indicates through a defect, the process of the BP tunneling is per- the appropriateness of use of the WKB approximation in formed via the ‘transfer’ of its total effective mass through the problem under consideration. the potential barrier. Let us estimate the effect of dissipation on the tunnel- Following the integration of the motion equation of ing process of the BP. To do this, we compare the force ~ the BP obtained via the Lagrangian function variation, F, acting on the quasiparticle, with the brakingp forceffiffiffiffiffi F , we find the its instanton trajectory zin and the instanton which in our case is approximately αωMωinΛ 2εmBP , −3 −2 frequency of the Bloch point ωin (see review [23]), which where α ~10 − 10 is the magnetization decay param- characterize its motion within the space with an ‘imagin- eter. Taking into account the explicit form of F,we ary’ time τ = it: from the point z0,1 =0 at τ = −∞ to the obtain Shevchenko and Barabash Nanoscale Research Letters 2014, 9:132 Page 4 of 6 http://www.nanoscalereslett.com/content/9/1/132

ω 2 − 3 ~= ¼ α= 1=2ðÞε 1=4 decay Mt <<10 10 ), allows us to neglect the effect on F F 2 hc 2 ~ the process of the braking force F eαωMmBPv: ~ The analysis of this expression shows that F =F << 1 We assume that defect is located at z0 = 0. Then, by −2 −1 −4 −2 −3 −2 expanding the potential of interaction of BP with the defect, at 10 ≤ hc ≤ 10 , ε ~10 − 10 and α ~10 − 10 .The obtained result indicates that at the consideration of the Ud(z0), in a series near this point and taking Equation 2 into BP quantum tunneling process, the effect of breaking force account, we can write down can be neglected. ÀÁ ðÞ¼ − 2= Λ2 ð Þ Note also that the mechanism of breaking force has Ud z0 U0 1 z0 2 14 been investigated in the work [25] and is associated with the inclusion of relaxation terms of exchange origin in where in accordance with the formula (2), the height of the Landau-Lifshiz equation for magnetization of a fer- 2 2 the potential barrier is U0 = π Λ ΔMSHc. romagnet [26]. Note that phenomenological expression for defect- effective field H (see formula (4)) follows from the series Results and discussion d expansion of the potential Ud(z0) near the inflection point. The over-barrier reflection of the Bloch point It was at this point that there is maximum field of defect. In the above, it was mentioned that tunneling of DW It is natural to assume that if BP has overcome the barrier and vertical BL is carried out via sub-barrier transition in this point, then the tunneling process is probable in of small parts of the area of DW or the length in case general. BL. In this case, both DW and vertical BL are located in Using the WKB approximation, and following the for- front of a potential barrier at a metastable minimum that malism described in [27,28], we determine the coefficient makes possible the process of their tunneling. At the of over-barrier reflection of the Bloch Point R by the ‘ ’ same time, the BP depinning occurs via transmission formula through the potential barrier instantly of entire effective mass of the quasiparticle. This result indicates that the −β R ¼ e ð15Þ presence of a metastable minimum in the interaction zà potential of BP with a defect (in contrast to DW or BL) Z0;2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ − 2 ðÞ− ðÞ Ã Ã is not necessary. Moreover, it means that there exists a where ℏ Im dz 2mBP EBP Ud z z0;2,andz0;2 possibility of realization for BP of such quantum effect as à z0;1 over-barrier reflection of a quasiparticle from the defect potential. In this case, the velocity at which BP ‘falls’ on are the roots of the equation EBP − Ud(z0)=0. the barrier may be determined by a pulse of magnetic field Taking into account the expression for the potential applied to the BP. And, as we shall see bellow, the poten- (14), from Equation 15, we find tial of interaction between the BP and a defect has a rather pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi β ¼ π Δε′=ℏ ð Þ simple form. Obviously, the effect is more noticeable in 2mBPEBP U0 16 the case when the BP energy is not much greater than the where the parameter ε′ =(E − U )/E < < 1 (recall that height of the potential barrier U . BP 0 BP 0 we consider the case when the energy E close to U ). Usingtheformula(2),werepresentthedynamics BP 0 Using the formula (13), Equation 16 can be rewritten equation for the BP in a pulsed magnetic field H (t)= y as H0χ(1 − t/T)intheform 1=2 ′ −1 3 1=2 ~ 2 β ¼ πðÞ2MSHc ε γ Δ Q =ℏ ð17Þ mBP∂v=∂t þ F ¼ π ΛΔMSHyðÞt ð12Þ Substituting into the expressions (15) and (17), the where v = ∂z /∂t is the BP velocity, χ(1 − t/T)isthe −5 0 ferromagnet and defect parameters, at ε′ ≥ 5×10 we Heaviside function, H is the amplitude, and T is the 0 obtain R ≤ 0.1, which is in accordance with criterion of pulse duration. applicability of Equation 15 (see [28]). ≤ << α−1ω−1 By integrating the Equation 12 for T t M ,we Note that from Equations 15 and 16, it follows that find the velocity of the Bloch point at the end of the R → 0atU → 0, i.e., we obtain a physically consistent π2 ΛΔ 0 magnetic field pulse: v(t)= MS H0T/mBP. Accordingly, conclusion about the disappearance of the effect of over- the energy of the BP in current time range EBP is given by barrier reflection in the absence of a potential barrier. = = 1 2 1 2 − = τeΔ mBP ¼ 4 MS 1 2 ¼ 2= ¼ π2ω 2 2Λ2Δ 2= ð Þ Based on the obvious relation, ω Q EBP mBPv 2 M T H0 32 13 U0 M Hc and the numerical data, given above, we determine τ, the << α−1ω−1 Note that the study, performed for time t M characteristic time of interaction of BP with the defect (or with taking into account the value of the magnetization 0.6 ≤ ωMτ ≤ 2.3. It is easy to see that τ satisfies the relation Shevchenko and Barabash Nanoscale Research Letters 2014, 9:132 Page 5 of 6 http://www.nanoscalereslett.com/content/9/1/132

2 ωMτ < ωMt ~10− 10 , which together with an estimate methods for studying an internal structure of their for R indicates on the possibility of the quantum DWs. phenomenon under study. In this case, the analysis of Competing interests expressions (13) and (14) shows that the amplitude of a The authors declare that they have no competing interests. 1/2 pulsed magnetic field is H0 ~4π(MSHc) /ωMT <8MS, which is consistent with the requirement for values of the Authors' contributions planar magnetic fields applied to DW in ferromagnets [1]. ABS and MYB read and approved the final manuscript. Let us consider the question about validity of applic- Author details ability of the WKB approximation to the problem under 1G.V. Kurdymov Institute of Metal , National Academy of Science of 2 consideration. Since in the given case E ≈ U , then the Ukraine, 36 Vernadskogo pr., Kyiv 142, 03680, Ukraine. Technical Centre, BP 0 National Academy of Science of Ukraine, 13 Pokrovskya St, Kyiv 04070, conditions of ‘quasi-classical’ behavior of the Bloch Ukraine. point and the potential barrier actually coincide and, in accordance with [24], are reduced to the fulfillment of Received: 4 December 2013 Accepted: 28 February 2014 Published: 19 March 2014 the inequality pffiffiffiffiffiffiffiffiffiffiffiffiffiffi References 1. Malozemoff AP, Slonczewski JC: Walls in Bubble Materials. δz0 mBPU0=ℏ >> 1 ð18Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi New York: Academic Press; 1979. δ ¼ Δ ðÞ− = ≈Δ ε′: 2. Konishi A: A new-ultra-density solid state memory: Bloch line memory. where z0 2 EBP U0 U0 2 IEEE Trans. Magn. 1838, 1983:19. Using the explicit form of U0, Equation 18 can be re- 3. Klaui M, Vaz CAF, Bland JAC: Head-to-head domain-wall phase diagram in written as mesoscopic ring magnets. Appl. Phys. Lett. 2004, 85:5637. 4. 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doi:10.1186/1556-276X-9-132 Cite this article as: Shevchenko and Barabash: The Bloch point in uniaxial ferromagnets as a quantum mechanical object. Nanoscale Research Letters 2014 9:132.

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