Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects

Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects

Nanoelectromagnetomechanical systems (NEMMS) open up a new path for the development of high speed autonomous nanoresonators and signal generators that could be used as actuators, for information processing, as elements of quantum computers etc. Those NEMMS that include ferromagnetic layers could b...

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Datum:2012
Hauptverfasser: Gomonay, H.V., Kondovych, S.V., Loktev, V.M.
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Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2012
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Магнетизм
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spelling nasplib_isofts_kiev_ua-123456789-1172652025-02-09T21:38:21Z Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects Gomonay, H.V. Kondovych, S.V. Loktev, V.M. Магнетизм Nanoelectromagnetomechanical systems (NEMMS) open up a new path for the development of high speed autonomous nanoresonators and signal generators that could be used as actuators, for information processing, as elements of quantum computers etc. Those NEMMS that include ferromagnetic layers could be controlled by the electric current due to effects related with spin transfer. In the present paper we discuss another situation when the current-controlled behavior of nanorod that includes an antiferro- (instead of one of ferro-) magnetic layer. We argue that in this case ac spin-polarized current can also induce resonant coupled magnetomechanical oscillations and produce an oscillating magnetization of antiferromagnetic (AFM) layer. These effects are caused by i) spin-transfer torque exerted to AFM at the interface with nonmagnetic spacer and by ii) the effective magnetic field produced by the spin-polarized free electrons due to sd-exchange. The described nanorod with an AFM layer can find an application in magnetometry and as a current-controlled high-frequency mechanical oscillator. The authors acknowledge partial financial support from the Special Program for Fundamental Research of the Department of Physics and Astronomy of National Academy of Sciences of Ukraine. The work of H.G. and S.K. was partially supported by the grant from the Ministry of Education and Science of Ukraine. 2012 Article Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects / H.V. Gomonay, S.V. Kondovych, V.M. Loktev // Физика низких температур. — 2012. — Т. 38, № 7. — С. 801-807 . — Бібліогр.: 41 назв. — англ. 0132-6414 PACS: 85.75.–d, 75.50.Ee, 75.47.–m, 75.47.De https://nasplib.isofts.kiev.ua/handle/123456789/117265 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Магнетизм
Магнетизм
spellingShingle Магнетизм
Магнетизм
Gomonay, H.V.
Kondovych, S.V.
Loktev, V.M.
Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
Физика низких температур
description Nanoelectromagnetomechanical systems (NEMMS) open up a new path for the development of high speed autonomous nanoresonators and signal generators that could be used as actuators, for information processing, as elements of quantum computers etc. Those NEMMS that include ferromagnetic layers could be controlled by the electric current due to effects related with spin transfer. In the present paper we discuss another situation when the current-controlled behavior of nanorod that includes an antiferro- (instead of one of ferro-) magnetic layer. We argue that in this case ac spin-polarized current can also induce resonant coupled magnetomechanical oscillations and produce an oscillating magnetization of antiferromagnetic (AFM) layer. These effects are caused by i) spin-transfer torque exerted to AFM at the interface with nonmagnetic spacer and by ii) the effective magnetic field produced by the spin-polarized free electrons due to sd-exchange. The described nanorod with an AFM layer can find an application in magnetometry and as a current-controlled high-frequency mechanical oscillator.
format Article
author Gomonay, H.V.
Kondovych, S.V.
Loktev, V.M.
author_facet Gomonay, H.V.
Kondovych, S.V.
Loktev, V.M.
author_sort Gomonay, H.V.
title Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
title_short Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
title_full Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
title_fullStr Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
title_full_unstemmed Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
title_sort magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2012
topic_facet Магнетизм
url https://nasplib.isofts.kiev.ua/handle/123456789/117265
citation_txt Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects / H.V. Gomonay, S.V. Kondovych, V.M. Loktev // Физика низких температур. — 2012. — Т. 38, № 7. — С. 801-807 . — Бібліогр.: 41 назв. — англ.
series Физика низких температур
work_keys_str_mv AT gomonayhv magnetoelasticcouplingandpossibilityofspintronicelectromagnetomechanicaleffects
AT kondovychsv magnetoelasticcouplingandpossibilityofspintronicelectromagnetomechanicaleffects
AT loktevvm magnetoelasticcouplingandpossibilityofspintronicelectromagnetomechanicaleffects
first_indexed 2025-12-01T01:52:49Z
last_indexed 2025-12-01T01:52:49Z
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fulltext © Helen V. Gomonay, Svitlana V. Kondovych, and Vadim M. Loktev, 2012 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 7, pp. 801–807 Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects Helen V. Gomonay1,2, Svitlana V. Kondovych1, and Vadim M. Loktev1,2 1National Technical University of Ukraine “KPI”, 37 Peremogy Ave., Kyiv 03056, Ukraine E-mail: malyshen@ukrpack.net vloktev@bitp.kiev.ua 2Bogolyubov Institute for Theoretical Physics National Academy of Sciences of Ukraine 14-b Metrologichna Str., Kyiv 03680, Ukraine Received February 21, 2012 Nanoelectromagnetomechanical systems (NEMMS) open up a new path for the development of high speed autonomous nanoresonators and signal generators that could be used as actuators, for information processing, as elements of quantum computers etc. Those NEMMS that include ferromagnetic layers could be controlled by the electric current due to effects related with spin transfer. In the present paper we discuss another situation when the current-controlled behavior of nanorod that includes an antiferro- (instead of one of ferro-) magnetic layer. We argue that in this case ac spin-polarized current can also induce resonant coupled magnetomechanical oscil- lations and produce an oscillating magnetization of antiferromagnetic (AFM) layer. These effects are caused by i) spin-transfer torque exerted to AFM at the interface with nonmagnetic spacer and by ii) the effective magnetic field produced by the spin-polarized free electrons due to sd-exchange. The described nanorod with an AFM layer can find an application in magnetometry and as a current-controlled high-frequency mechanical oscillator. PACS: 85.75.–d Magnetoelectronics; spintronics: devices exploiting spin polarized transport or integrated magnetic fields; 75.50.Ee Antiferromagnetics; 75.47.–m Magnetotransport phenomena; materials for magnetotransport; 75.47.De Giant magnetoresistance. Keywords: antiferromagnetic layer, electromagnetomechanical effects, spin-polarized free electrons. 1. Introduction Nanoelectromagnetomechanical systems (NEMMS) that convert electromagnetic energy into mechanical motion and vice versa are now of great interest for several reasons. First of all, NEMMS themselves give yet another manife- station of the coupling between magnetic and mechanical degrees of freedom. Up to now magnetomechanical inte- ractions were the most completely studied for the systems with no electric current (we are talking about the orienta- tional phase transitions, see, e.g. [1], the coupled magnon- phonon modes [2], formation of a magnetoelastic gap [3] etc.). In these cases one can speak about thermodynamic equilibrium and describe the system with the time-in- dependent equations. At the same time in recent years in- vestigations in physics of magnetic phenomena have moved to a new field of spintronics, where not just the current, but the spin-polarized electrical current is a critical component that forms the magnetic properties of — mainly metallic — systems. On the other hand, recently increased attention to NEMMS is also related with their potential applications. In particular, because of small geometrical size, the funda- mental mechanical modes of NEMMS fall into GHz range and corresponding devices could be used as high-fre- quency actuators and transducers of mechanical motion [4] (see also [5] and references therein). Besides, at low tem- peratures (much smaller than the energy of fundamental mode) NEMMS show quantized mechanical behavior and thus could be used for the quantum measurements and quantum information processing [6–9]. At last, due to high sensitivity to the external fields, including electric, magnet- ic and surface stresses, the NEMMS could be used as the effective tools for biological imaging [5], magnetometry [10,11], for the measurement of magnetoelastic properties and magnetic anisotropy of the materials [12] etc. Helen V. Gomonay, Svitlana V. Kondovych, and Vadim M. Loktev 802 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 7 An effective way to induce nanomechanical oscillations is based on the spin-related phenomena, in particular, on spin transferred torque (STT) predicted by Berger [13] and Slonczewski [14,15]. Flip of the free electron spin at the interfaces between the layers with different magnetic prop- erties is related with the change of the angular momentum and for nanosize objects (like NEMMS) can result in the noticeable rotation, torsion or bending of the sample. Up to dates, combination of nanomechanics and spin- tronics is implemented in the devices that include ferro- mangetic (FM) and nonmagnetic (NM) metallic layers. In a nanowire with an only FM/NM interface the FM layer serves as a polarizer for an electric current, and spin flip processes at the FM/NM interface produce a mechanical torque in the sample [16–19]. Another modification of NEMMS (see [20–22]) is analogous to spin-valves and includes at least two FM layers — one is a polarizer and the magnetization of the other is rotated by STT. Oscilla- tions of magnetization, in turn, induce the mechanical movement, due to the presence of spin-lattice coupling. In the present paper we propose the NEMMS which in- cludes at least one antiferromagnetic (AFM) layer (see Fig. 1) that could be set into motion by spin-polarized cur- rent. Our idea is based on the following facts: i) theoretical predictions [23–25] and experimental evidence [26–30] of STT effects in AFMs; ii) strong (compared to FM) spin- lattice coupling in AFM that reveals itself, e.g., in the pro- nounced magnetoelastic effects like an energy gap for AFMR frequency [3] and shape-induced magnetic aniso- tropy [31,32]. In the framework of hydrodynamic-like ap- proach we analyze the coupled magnetomechanical dy- namics of nanorod consisting of FM, NM and AFM layers and calculate eigen frequencies and current-induced me- chanical and magnetic responses of the system. We show that dissipative and nondissipative components of spin- polarized ac current contribute differently to magnetome- chanical motion and thus could be separated experimental- ly. The proposed device can be also used as a current- driven nanoresonator that produces no magnetic field. The paper is devoted to the 80-th anniversary of the prominent Ukrainian experimentalist Prof. V.V. Eremenko whose contribution into the field of magnetoelasticity is remarkable and is world-wide recognized. 2. Model Let us consider the NEMMS that demonstrates the tor- sional mechanical oscillations, e.g., doubly clamped nano- rod (Fig. 1,a). In general case, torsional dynamics can be viewed as inhomogeneous (space-dependent) rotation of the crystal lattice with respect to some reference state. On the other hand, the magnetics with the strong enough ex- change coupling between the magnetic sublattices have another rotational degrees of freedom, namely, those re- lated with the solid-like rotation of the magnetic sublattices [33]. Lattice and magnetic rotations could be coupled due to, e.g., magnetic anisotropy, magnetoelastic or/and shape effects. Thus, any spin torque transferred to the magnetic layer will induce twisting of the crystal lattice and vise versa, any mechanical torque will induce rotations/oscil- lations of the magnetic subsystem. In what follows we consider a heterostructure that in- cludes a thin (thickness )AFMd metallic AFM layer in- serted just in the middle between two metallic NM rods (each of the length ).AFML d Spin-polarized electric current J flowing through this system exerts spin torque to AFM layer due to spin-flip processes at the NM/AFM interface. Thus, the magnetic subsystem serves as a source of the magnetic and, as a result, the mechanical torque for the whole system. The optimal geometry of the magnetic (FM, or polariz- er, and AFM, or “rotator”) layers can be predicted from general principles. Curren-induced STT is parallel to the FM magnetization, FMM , so, FMM should be parallel to the axis of nanorod. On the other hand, the most effective energy transfer between the magnetic and crystal lattices occurs for the modes with the same symmetry. So, an op- timal orientation of the magnetic vectors should allow transversal (with respect to nanorod axis) oscillations with the minimal possible frequency. It should be noted that spin-polarized current acts on AFM layer in three ways. First, STT that is proportional to the spin flux transferred to the magnetic layer and is re- lated with dissipative processes. Second, spin current pro- duces the effective magnetic field sd FMJ∝H M parallel to the spin polarization. Corresponding torque that acts on AFM vector is nondissipative (adiabatic). Third, the cur- rent itself generates an Oersted field which direction and Fig. 1. (Color online) Nanotorsional oscillator. Nanorod made of NM metal with thin AFM section is mechanically clamped be- tween the FM and NM leads (a). The current J that flows from FM to NM lead is polarized in FM ZM direction and gives rise to the torques twisting the AFM vector l in the middle sec- tion (b). Due to magnetic anisotropy, rotation of the magnetic moments through the angle magθ induces rotation of the crystal lattice through the angle latθ . Axes x, y denote the reference frame, while X, Y show the instantaneous orientation of the ro- tated crystal axes. J FM NM AFM MFM MFM –L L –dAFM /2 dAFM /2 θmag θlat Xx I Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 7 803 value within an AFM layer depends upon the geometry of the system. The last contribution is supposed to produce a negligible effect on AFM dynamics and will be disre- garded in the following consideration*. The value of the effective field sdH depends upon the exchange coupling between free and localized spins (so called sd-exchange) and thus can be noticeable, especially in the case of ac cur- rent, as will be shown below. Coupled rotational dynamics of the magnetic and crys- tal lattices can be described phenomenologically in the framework of continuius approach in terms of the Gibbs' vectors = tan ( / 2)α α αϕ θ e that parametrize solid-like rotation of the crystal lattice ( latα⇒ ) and magnetic sub- system ( magα ⇒ ) around an instantaneous rotation axis αe through the angle αθ . Vectors ( , )tαϕ r are the field variables that define the state of the crystal and magnetic lattices at a moment t in a point r . In the simplest case under consideration (thin nanorod) the rotation axis coin- sides with the rod axis, so lat mag .Ze e Time, αθ , and space, ' zα αθ ≡ ∇ θ , derivatives of thus in- troduced generalized coordinates latθ and magθ generate the rotation frequencies and vorticities, correspondingly**. According to Ref. 33, the rotating magnetic frame pro- duces the dynamic contribution into macroscopic magneti- zation, AFMM , of AFM. Thus, with account of the effec- tive magnetic field sd ZH the magnetization of AFM layer is parallel to the nanorod axis Z and its value is ex- pressed as mag mag ad= ( ) = ( ) ,AFM sd AFM AFMM H S j Sχ χ θ + γ θ + γβ γ γ (1) where AFMS is the nanorod crossection area within AFM layer, χ is magnetic susceptibility, γ is gyromagnetic ratio. The last expression in (1) includes the material adia- batic (see below) constant adβ that defines the relation between the effective field ad=sdH jβ and the the current density = / AFMj J S ***. As follows from definition of the effective field ,sdH adβ is proportional to the constant of sd-exchange and to the fraction of free electrons that did not flip their spins at NM/AFM interface. Thus, this con- stant describes the action of nondissipative (adiabatic) component of spin-polarized current, as will be discussed below. The Lagrange function of the system written from the general symmetry considerations takes a form: 2 2 lat lat 1= ( ) ( ) 2 L L dz I z − ⎡ ⎤′θ − κ θ +⎣ ⎦∫L /2 2 mag ad mag lat2 /2 ( ) ( ) . 2 dAFM AFM dAFM S dz j U − ⎡ ⎤χ + θ + γβ − θ −θ⎢ ⎥ γ⎢ ⎥⎣ ⎦ ∫ (2) Here κ is a torsion modulus (rigidity) that can be ex- pressed through the elastic modula and the dimensions of the sample once the geometry is known, mag lat( )U θ −θ is the energy of the magnetic anisotropy which depends upon the relative orientation of the magnetic moments with re- spect to crystal lattice (see Fig. 1,b). A specific (per unit length) moment of inertia of nanorod, ( )I z ≡ 2 2 rod ( )x y dxdy≡ ∫ρ + , is supposed to be different in NM, ( ) ,NMI z I≡ / 2 | |AFMd z L≤ ≤ and in AFM, ( ) ,AFMI z I≡ | | / 2AFMz d≤ regions, here rodρ is the nanorod density. In Eq. (2) we have neglected inhomogeneous exchange interactions (terms with mag )′θ that are vanishingly small for a thin (below the characteristic domain wall thickness) AFM layer. We also assume that κ is constant along the rod, generalization for a more complicated case is straight- forward. Dissipative phenomena within an AFM layer that arise from the STT and internal damping are described with the help of generalized potential (or Rayleigh dissipation func- tion) [36] as follows: /2 2 dis mag mag2 /2 = , dAFM AFM AFM AFM dAFM j S dz − ⎛ ⎞βγ χ θ − θ⎜ ⎟⎜ ⎟γγ⎝ ⎠ ∫R (3) where AFMγ is a half-width of AFMR that characterizes the damping. We have also taken into account that the cur- rent polarization is parallel to the rod axis, .FM ZM The above introduced material constant disβ that de- scribes dissipative component of spin-polarized current needs some special explanation. The value dis jβ is equal to spin-flux that is transferred to the unit volume of AFM layer due to spin-flip scattering of the conduction electrons at NM/AFM interface. Thus, two constants, adβ and dis ,β though having different physical dimensions, are in a certain sense complementary: the greater is one, the smaller is other. * According to Refs. 34 and 35 typical value of current-induced Oersted field is 1 kOe. For FM materials with characteristic fields of reorientation 0.1–1 kOe the effect of Oersted field can be significant. However, in AFMs with strong exchange coupling and high Néel temperature (FeMn, IrMn, NiO) the typical value of spin-flop field is higher and falls into 1–10 kOe range. Thus, the ef- fect of the Oersted field can be neglected, at least in the first approximation. ** In general case, frequency is a vector and vorticity is a second rank tensor. *** Stricktly speaking, current density j is defined by the effective (Sharvin) crossection which in the case of inhomogeneous rod can differ from .AFMS Helen V. Gomonay, Svitlana V. Kondovych, and Vadim M. Loktev 804 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 7 Damping of the mechanical oscillations are accounted by the corresponding Rayleigh function with the damping constant latγ : 2 lat lat lat 1= ( ) . 2 L L dz I z − γ θ∫R (4) Functions (2), (3) and (4) together with the boundary conditions lat ( ) = 0Lθ ± (doubly clamped rod) generate the system of dynamic equations for the angles latθ , magθ that unambiguously describes the nanorod state. Oscillato- ry behavior of a system implies small deflections of lat mag,θ θ from equilibrium zero values. To this end, mag- netic anisotropy can be approximated as mag lat( )U θ −θ ≈ 2 2 2 mag lat( ) /(2 ),AFMR≈ χΩ θ −θ γ where AFMRΩ is AFMR frequency of the mode that corresponds to homogeneous* (within AFM layer) rotation of the magnetic moments around Z-axis. It should be stressed that the constant of magnetic anisotropy, 2 2/AFM AFMRK ≡ χΩ γ , is defined by spin-orbit or dipole interactions and thus includes contribu- tion of magnetoelastic nature. 3. Coupled magnetomechanical dynamics Let us consider small oscillations induced by ac current 0= cosj j tω . Corresponding equations for the space de- pendent functions lat ( )zθ and mag ( )zθ in neglection of damping could be reduced to a form: 2 2 2lat lat2 2 2 2 ( ) ( ) ( ) AFM AFMR AFMR d S z I z dz ⎡ ⎤θ Θ χΩ κ +ω + θ =⎢ ⎥ γ Ω −ω⎢ ⎥⎣ ⎦ 2 dis ad 02 2 ( ) = ( ) , ( ) AFMR AFM AFMR i S z j β − χβ ω Ω − Θ γ Ω −ω 2 lat dis ad mag 02 2 2 2 ( ) = ( ), ( ) AFMR AFMR AFMR i j z ⎡ ⎤Ω θ γ β − χβ ω θ + Θ⎢ ⎥ Ω −ω χ Ω −ω⎢ ⎥⎣ ⎦ (5) where form-function ( ) = 1zΘ inside the AFM layer (| | / 2)AFMz d≤ and vanishes outside it ( | | / 2AFMz d≥ ). Analysis of Eqs. (5) shows that the spin-polarized cur- rent produces a mechanical torque (r.h.s. of the first equa- tion) and thus is a motive force for torsional oscillations. The value of the torque is proportional to the magnetic anisotropy constant 2 AFM AFMRK ∝ Ω and the thickness of AFM layer (factor ( )zΘ ) and can increase greatly in the vicinity of AFMR ( AFMRω→Ω ). Physical interpretation of this fact is quite obvious: mechanical torque occurs due to spin-lattice coupling within AFM layer and should be proportional to its thickness and coupling constant, the current acts directly on the magnetic subsystem and indi- rectly on the mechanical one, thus the largest effect should be observed at AFMR frequency. * As it was already mentioned above, we consider only long-wave motions of AFM subsystem, so-called macrospin approximation. Fig. 2. (Color online) Torsional modes and spectrum of AFM-based nanorod. Low-frequency torsional modes, ph= nkω v , = 0,1,2n induced by STT. Relative amplitude of torsional angle, lat ( )zθ , is frequency dependent. Low panel schematically shows the position of AFM layer (the thickness = 0.02AFMt L is slightly exaggerated) (a). Spectrum of eigen modes (schematically). In the absence of coupling (upper panel) the mechanical modes though smeared (half-width latγ ) are well separated due to the rather high value of quality factor latQ . The magnetic modes ( = AFMRω Ω ) are degenerated and have a pronounced width ( AFMγ ). Magnetomechanical coupling (lower panel) results in the (1) (“red” online) shift of the mechanical modes and small (2) (“blue” online) shift of the magnetic modes (shown by solid vertical lines). While the shifted mechanical modes are still well distinguishable, the spectrum of the shifted magnetic modes falls completely into the line width (b). a b 1.00 0.75 0.50 0.25 0 –0.25 –0.50 –1.0 –0.5 0 0.5 1.0 n = 0 n = 1 n = 2 z/L ph 0kv ph 1kv ph 2kv θ I , a rb . u ni ts γlat γAFM ΩAFMR ω ω 1 1 1 1 1 1 2 2 AFM Magnetoelastic coupling and possibility of spintronic electromagnetomechanical effects Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 7 805 3.1. Oscillation modes and spectrum The rod under consideration has two types of the tor- sion eigen modes, symmetric ( lat lat( ) = ( )z zθ θ − ) and anti- symmetric ( lat lat( ) = ( )z zθ −θ − ) with respect to space in- version. From Eqs. (5) it follows that in the present geometry the spin-polarized current can excite only sym- metric modes that show maximum deflection latθ within an AFM layer ( 0z ≈ ). In the first approximation (taking into account that / 1AFMd L ) the symmetric modes (see Fig. 2,a) could be represented as ( )( ) ( ) lat lat( , ) = (0)e cos , nn n i t nz t k zωθ θ (6) 2 ( )( )( ) mag lat2 2 ( ) ( , ) = (0)e , nnn i tAFMR AFMR z z t ωΩ Θ θ θ Ω −ω were the allowed wave vector = (2 1) / (2 )nk n Lπ + is cal- culated from the boundary conditions. Corresponding ei- gen frequencies ( )nω calculated from Eqs. (5) are the fol- lowing: {( ) 2 2 2 ph 1= (1 ) 2 n AFMR n nk±ω Ω + +λ ±v 1/21/22 2 2 2 2 2 2 ph ph( (1 ) ) 4 ,AFMR n n n n AFMRk k ⎫⎡ ⎤± Ω − +λ + λ Ω ⎬⎣ ⎦ ⎭ v v (7) where 1/2 ph = ( / )Iκv is the phonon velocity and I ≡ (1/ 2 ) ( )L LL I z dz−≡ ∫ is the averaged moment of inertia. Following the notions of Ref. 20, we have introduced in Eq. (7) the coupling coefficient 2 2 ph2 AFM AFM n n K V LI k λ ≡ v (8) which is proportional to the magnetic anisotropy of the whole AFM layer (with the volume AFM AFM AFMV d S≡ ). Expression (7) for eigen frequencies is analogous to one obtained in Ref. 20 for a nanorod with the FM layer. The expression (7) confirms quite obvious conclusion that the spectrum of nanorod consists of two branches — high-frequency quasimagnetic, ( )n +ω , and low-frequency quasimechanical (torsional), ( )n −ω . In the limit 0nλ → the quasimagnetic frequency ( )n AFMR+ω →Ω and quasime- chanical one ( ) ph n nk−ω → v . Further analysis of current-induced dynamics can be simplified due to specification of “small” and “large” quantities. The frequency of the torsional fundamental, “zero”, mode for a nanosized rod ( 30–100L ∝ nm, 3 ph 5 10∝ ⋅v m/s) is ph 0 10 100k ∝ ÷v GHz. Characteristic AFMR frequency for a bulk sample of a typical AFM with high Néel temperature (FeMn, IrMn, NiO) is noticeably greater, / 2 150–1000AFMR AFMRν ≡ Ω π ∝ GHz*, depend- ing on the mode type [38–40]. So, in contrast to FM, where the fundamental frequency of the mechanical oscillations is close to the FMR frequency [20], for the nanorods with AFM layer ph 0AFMR kΩ v . However, for higher har- monics (with 10–100)n ∝ the crossing of frequencies ph( n AFMRk ∝Ωv ) is possible. The coupling constants 1 0< < < 1n n−λ λ λ… . For example, for a typical AFM Ir 20 Mn 80 the anisotropy constant 510AFMK ∝ J/m 3 [37], so, for the 50 50 2× × nm AFM layer 2 0 10−λ ∝ . However, it should be stressed that the constant 0λ in AFM is substantially larger than for analogous FM layer (e.g., for Fe the value 3 0 10−λ ∝ [20]), due to the difference in magnetic anisotropy. The quality factor of the mechanical oscillations, lat ph 0 lat= / (2 )Q k γv , strongly depends upon the surface effects but even in the worst case is as large as 310 [10]. The quality factor of the metallic magnetic subsystem, mag = / (2 )AFMR AFMQ Ω γ , is much smaller, e.g., for the metallic FM the quality factor 2 mag 10Q ∝ [20]. Thus, the spectrum of the mechanical and magnetic ex- citations (Eq. (7)) for a typical AFM-based nanorod has the following features (see Fig. 2,b): — in the absence of coupling ( = 0λ ) the spectrum of the mechanical modes consists of thin ( lat 1Q ) well- separated lines. The spectrum of the magnetic modes is degenerated ( = AFMRω Ω ), corresponding line is rather thick; — far from the crossing the coupling-induced shift of the frequencies, ( ) 2 2 2 ph ph= (1 / 2 )n n n n AFMRk k−ω −λ Ωv v , ( ) =n +ω 2 2 2 ph= (1 / 2 ),AFMR n n AFMRkΩ +λ Ωv is vanishingly small. So, “mechanical” modes are still well separated, while the splitting of the “magnetic” modes is below the line width; — in the vicinity of crossing the splitting of the me- chanical and magnetic modes is substantially greater, ( ) = (1 /2).n AFMR n±ω Ω ± λ Damping processes are defined mainly by the magnetic subsystem, so, corresponding qual- ity factor is close to magQ . Thus, the magnetic and me- chanical modes could be resolved providing mag > 1.nQλ 3.2. Current-induced oscillations From the properties of oscillation spectrum it follows that current-induced behavior of nanorod is different in the low-frequency ( AFMRω Ω ) and high-frequency ( )AFMRω∝ Ω ranges. Let us consider them separately. In the low-frequency range the last term in the l.h.s. of the first of Eqs. (5) is small (∝ λ ) and can be neglected. To this end, torsion angle of mechanical oscillations is expressed as * For the small samples AFMRν can be smaller due to the size effects, see, e.g. [37]. Helen V. Gomonay, Svitlana V. Kondovych, and Vadim M. Loktev 806 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 7 0 lat ph 0 ( ; ) = 4 AFMV j z kIL π θ ω × ωγ v dis ad 22 2 lat ( ) sin [( | |) / ] e , ( / ) ( / 4 ) ( / )cos sin ii L z c L c Q L c φβ − χβ ω − ω × ω + π ω (9) where φ is the frequency dependent phase shift with re- spect to j , in the vicinity of resonance / 2φ→ π . It can be easily seen from Eq. (9) that the current- induced torsional oscillations have clearly defined reson- ance character at ( ) ph= n nk−ω ω ≈ v . Space dependence of lat ( )zθ at a given ω (see Fig. 2,a) is close to the mechani- cal eigen modes. The resonant amplitude obtained from Eq. (9) is ( ) lat 0 dis ad ph 0lat 2 2 ph 0 (res) = 2 1 n AFMQ V j i k nIL k β⎛ ⎞θ + χβ =⎜ ⎟+⎝ ⎠γ v v 0 lat 0 dis ad ph 02 2 = . 2 1AFMR Q j i k n λ γ β⎛ ⎞+ χβ⎜ ⎟+⎝ ⎠Ω χ v (10) Here the factor i reflects the phase shift of the torsion an- gle with respect to current. As seen from Eq. (10), rotation of lattice results from two effects induced by spin-polarized current, namely, dissipative STT ( dis∝β ) and adiabatic effective spin- induced field ( ad∝ β ). The first contribution diminishes with the frequency ( n∝ ) growth, while the second one is frequency independent (at least, for AFMRω Ω ). More- over, STT-induced term is phase-shifted with respect to current, while adiabatic term is in phase with current. This opens a way to separate these contributions by measuring current dependence of resonant torsional oscillations. An amplitude of the corresponding magnetic oscilla- tions differs from ( ) lat (res)nθ by the factor 0 lat(1 2 ),i Q+ λ as seen from the following ( ) 0 dis mag 0 lat ad ph 02(res) = (1 2 ) . 2 1 n AFMR j i Q i k n γ β⎛ ⎞θ + λ − χβ⎜ ⎟+⎝ ⎠χΩ v (11) It also depends upon both dissipative and nondissipa- tive current-induced contributions, however, phase shift with respect to current is much more complicated due to the term with 0 latQλ . Time derivative ( ) mag (res) =nθ ( ) ph mag= (res)n ni k θv is proportional to magnetization of AFM layer (see Eq. (1) and thus can be detected experi- mentally. In the high-frequency range the magnetic modes with different n are almost degenerated. So, the current induces mechanical, ( )0 lat dis ad2 15 (res) = , 16 AFM AFM AFMR AFMR Q V j i IL θ β + χβ Ω γ Ω (12) and magnetic, ( )0 mag dis ad2(res) = AFM AFMR AFMR Q j i γ θ − β + χβ Ω × χΩ 2 2 ph 0 02 15 1 8 AFM AFMR k Q ⎛ ⎞ ⎜ ⎟× + λ ⎜ ⎟Ω⎝ ⎠ v (13) oscillations with the frequency AFMRω ≈ Ω . 4. Conclusions In the present paper we considered new aspect of mag- netoelastic interactions and studied magnetomechanical oscillations induced by spin-polarized current for the sim- plest case of twisting nanorod. Our calculations demon- strate that ac spin-polarized current can excite quasime- chanical (torsional) as well as quasimagnetical modes. It is interesting to note that the ac spin-polarized current affects the AFM layer in the case of strong scattering at NM/AFM interface (due to STT effect) and in the case of weak scattering as well (due to the effective sd -exchange field “injected” with free electrons into AFM layer). Ratio between dissipative and nondissipative contribution is pro- portional to the phase shift between mechanical oscilla- tions and current and thus can be measured experimentally in the low frequency range. An amplitude of quasimechanical mode depends upon the geometry of the sample (see Eq. (10)) and can be en- hanced by diminishing the moment of intertia (e.g., by us- ing carbon nanotubes [41]) and by enlarging AFM volume .AFMV However, if the thickness of AFM layer, ,AFMd becomes greater than the free path of spin-polarized elec- trons, contribution of dissipative (STT) part will be reduced. The effectiveness of the described electric-through- magnetic-to-mechanical energy conversion can be increas- ed by using nanorod with periodical FM/NM/AFM struc- ture, however this system needs additional treatment and is out of scope of this paper. In this work we considered torsional oscillations of the effectively one dimensional structure. Analogous results could be obtained for nanobeams that show flexional oscil- lations. The authors acknowledge partial financial support from the Special Program for Fundamental Research of the De- partment of Physics and Astronomy of National Academy of Sciences of Ukraine. 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