Strong spintronic magnetoelectric effect in layered magnetic metamaterial
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Strong spintronic magnetoelectric effect in layered magnetic metamaterial
P.V.Pyshkin and A.V.Yanovsky
B.Verkin Institute for Low Temperature Physics & Engineering,National Academy of Sciences of Ukraine, 47 Lenin Ave, 61103, Kharkov, Ukraine
It is shown that external magnetic field or magnetization induces electric polarization of micro-scopic isolated magnetic/non-magnetic hybrid structures due to the spin-dependent electron redis-tribution and mutual capacity. The magnetoelectric effect can be very strong, for instance thechange of electric polarization reaches 100 µC/m at a constant magnetic field of only 1 . ÷
10 nanometer structures which tightly fill the space, composing a metamaterial.The effect does not require using exotic compounds. Based on the obtained results we suggest anumber of recommendations for future experimental design.
Spin electronics (spintronics) has been already used indata storage and other devices. Advantages and phys-ical principles of utilizing the spin of the charge carri-ers in electronics are widely discussed, see e.g. recentreviews and the Nobel lectures of the 2007 . In this ar-ticle we show that novel spintronic materials could man-ifest unique properties due to proximity between smallmagnetic/non-magnetic hybrid structures.The magnetic/non-magnetic hybrid structures are im-portant subject of study in spintronics. Here the word“non-magnetic” refers to non-magnetic metal or semicon-ductor, such as Cu or GaAs. And the term “magnetic”means magnetic conductors, e.g. the ferromagnetic met-als (Fe, Co), or some paramagnetic metals which are closeto ferromagnetic instability (Pd), or the diluted magneticsemiconductors (Ga − x Mn x As, Zn − x Be x Se) etc. It iswell known that magnetization induces the giant Zee-man spin splitting of the conduction band in such mag-netic conductors due to the strong exchange interactiondepending on the electron spin direction versus the mag-netization axis. As is the convention, the appropriatespin bands are denoted hereinafter by the indices ↑ forspin-up or ↓ for spin-down.The giant Zeeman splitting significantly changes thespin density of states (DOS) of the conduction electrons.It affects all the spin components of the physical param-eters related to the DOS resulting giant magnetic resis-tance effect , spin-aligning etc. Lots of magneticallyinhomogeneous spintronic devices are proposed as partsof an electrical circuit: the spin-valve , Datta and Dasstransistor , and other spin transistors .In contrast, we discuss the equilibrium states of iso-lated conducting magnetic/non-magnetic structures afterspin relaxation (i) with and (ii) without external mag-netic field and show that heterogeneity of spin ↑ , ↓ bandscould be used to design artificial materials that displaystrong magnetoelectric (ME) effect.We discuss direct ME effect: the induction of an elec-tric polarization by a magnetic field, see e.g. books ,reviews . The inverse effect will be considered else-where. At present, ME effects have attracted great inter-est for lots of applications: sensors, information storage,anti-radar coating and others . The conventional MEeffect arises from interaction of electric and magnetic sub- systems in time-asymmetric ionic crystals or piezoelectricand magnetic subsystems in composite multiferroics .From applied point of view, linear ME effect has specialimportance . But also it requires specific efforts fromfundamental point of view because the magnetic field isa pseudo vector, and electric polarization is a vector. Sothe linear effect in conventional multiferroics is only pos-sible for a special asymmetry, e.g in multi-sublattice in-sulators with rare-earth elements. As it is shown below,the spintronics provides linear ME effect with high accu-racy due to transfer of conduction electrons, which areabsent in dielectric multiferroics.In most experiments (see e.g. Ref. ), the value ofthe ME effect is 10-100 µC/m at a magnetic field of 3 T or more, giant ME effect is 3600 µC/m at 7 T withthe same 10-100 µC/m at 2 T .In this paper, the strong ME effect of quite a differentnature is predicted. It is related to the spin-dependent re-distribution of free electrons in inhomogeneous magneticconductive structures embedded into dielectric matrix,forming a metamaterial, i.e. nanostructured compositematerial, cf. for example Ref. . In contrast to conven-tional metamaterials the purpose of this one is to reachhigh mutual capacity of meta-atoms (“artificial atoms”)from their closeness.In the beginning let us consider one meta-atom of themetamaterial. Then we will show how ME effect occursand why it increases while these meta-atoms decreasing.The said meta-atom is a standard object in spintron-ics – a micro-sized conductor consisting of two parts: anon-magnetic ( N ) and magnetic ( M ). At achievable sta-tionary magnetic fields, the energy ∆ of giant Zeemansplitting of magnetized M -part (electron volts in ferro-magnetic metals) is much larger than one of N -part Zee-man splitting. Therefore, we will neglect the latter. Theequilibrium of isolated meta-atom is determined by theconstancy of the charge carriers’ total number and equal-ity of the electrochemical potentials of the meta-atom’smagnetic and non-magnetic parts.In equilibrium, the chemical potential level µ M in M -part is the same for electrons with different spins due tothe spin relaxation. Since the electron DOS N ( ε ) de-pends on the energy, the chemical potential µ M differsfrom the chemical potential µ of the system without amagnetic field. Let us analyze µ M dependence on themagnetization and magnetic field.For simplicity, we assume that the temperature is muchlower than Fermi energy and Zeeman splitting energy in M -part such that the Fermi distribution is replaced bya step function. Change of µ M in a magnetic field isconsidered in and experimentally-confirmed in : dµ M d ( gµ B H ) = N − ↓ − N − ↑ (cid:16) N − ↓ + N − ↑ − I (cid:17) , (1)where g is the electronic g-factor, µ B is the Bohr magne-ton, H is the external magnetic field; N ↑ = N ( µ + ∆ / N ↓ = N ( µ − ∆ /
2) are the electron DOS of thecorresponding spin subbands at the Fermi level in therigid band approximation, N ( ε ) is the electron DOS (perone spin) in the absence of the Zeeman splitting, I refersto Stoner parameter characterizing the magnitude of theelectron exchange interaction in M -part. Here the Zee-man splitting is: ∆ = 2( IM + µ B H ) , (2)where M is the magnetization. From here, the variationsof µ M in paramagnetic and ferromagnetic materials dif-fer significantly because paramagnetic magnetization isproportional to H , while ferromagnetic material has itsown magnetization.More specifically, in the case of paramagnetic material:∆ para ≈ − α µ B H, (3)where α = 2 IN < M -part magnetic suscep-tibility for the paramagnetic material. When α ≥ M -part is ferromagnetic (Stoner criterion ), the expression(3) is not applicable, and∆ ferro ≈ IM (4)does not depend on the external magnetic field in themain approximation by H .Using (1)-(4) we obtain the following expression for thechemical potential variation in the case of the ferromag-netic M -part ( α > δµ ferro ( H ) = gµ B H Xα − , (5)where X = ( N ↑ − N ↓ )( N ↑ + N ↓ ) − refers to spin polariza-tion of DOS at the Fermi level. And for the case of para-magnetic M -part with exchange enhancement ( α < δµ para ( H ) = − gµ B N ′ ( ε F )4(1 − α ) N ( ε F ) H , (6)where N ′ ( ε F ) is the first derivative of the DOS with re-spect to the energy at Fermi level. Thus, the chemical FIG. 1. (a) Metamaterial is represented by M - N - I chain,(b) electrical potential for δµ <
0, (c) equivalent circuit potential change for paramagnetic M is ∝ H , while it islinear for ferromagnetic M , both far from the ferromag-netic transition with H . Also it is nonlinear and huge( ∼ IM ) at crossing the transition with H .Under equilibrium condition, the electrochemical po-tentials of M and N parts of the meta-atom are equal,as has been said, for times longer than electron spin relax-ation time. Because of non zero δµ M , the electrochemicalpotentials equilibrate by the transfer of electrons betweenthe N and M parts (transfer direction is determined bythe sign of δµ M ). Now let’s see what happens in a meta-material which consists of these meta-atoms.For simplicity, we consider a chain of identically ori-ented M - N meta-atoms, Fig.1(a), separated by the in-sulator I . The results for such a quasi-one-dimensionalsystem is extended to three-dimensional systems.As noted above, the appearance of δµ M induces the re-distribution of charge carriers and subsequently the for-mation of an electric potential difference between M and N pieces . This effect can be regarded as “magneto-contact potential difference”, i.e. a part of the contactpotential difference that depends on magnetic field. Sothe electric charges appear at the interfaces M / N , M / I , N / I . Thus the chain can be represented as an equiva-lent electrical circuit shown in Fig.1(c). In this figure, the M - N - I chain is a series of the two alternating capacities: C MN is the capacity of M / N interface and C N IM is thecapacity of N / I / M interface. The difference in electronchemical potential from change of a magnetic field is rep-resented as power sources with the voltage of U = e − δµ connected to each of M - N capacitances.It can be seen from the Fig.1(c) that the extra chargesare concentrated at the M / N and N / I / M interfaces.They are proportional to the capacity of the interfaces: q MN = U C MN , q N IM = U C
N IM because of the poten-tial equality for the interconnected plates of C MN and C N IM . The electric potential as function of coordinate isshown at Fig.1(b). Using the formula for a flat capacitor,we get: C N IM = εS/ (4 πw ), C MN = S/ (4 πa ), where ε is the permittivity of I , and S is a cross-section, w isthe distance between M - N meta-atoms, a is a distanceof charge separation on M / N interface (of the order ofscreening length). The process of spatial separation of acharge results in appearance of the electric polarization : P = 1 V cell Z + ∞−∞ dt Z V cell dV j , (7)where V cell is the volume of elementary cell and j is adensity of current in charge redistribution and insulatorpolarization processes. Thus, using (7) we obtain thatthe external magnetic field induces an electric polariza-tion: P ( H ) = δµ ( H ) γ ( ε − πe ( w + l ) . (8)So this is the ME effect. Here 0 < γ < M / N structures in it, but not on the directionof the applied magnetic field. Also, as can be seen from(8) polarization will increase when dielectric permittivityof I increases, the size of M / N structures decreases, andwhen they are located more compactly. Additionally, theexpression (8) does not include parameters of material N .This is due to the fact that the calculation does not takeinto account changes of the Fermi level by filling the elec-tron quantum states. This correction is necessary onlyfor extremely small meta-atoms (of the order of severalinteratomic distances). Note that it is the permittivity ε = 1 that makes nonzero electric polarization. Other-wise only net quadrupole will be created. So, insulator ε = 1 breaks the symmetry in charge redistribution, alsoit distinctly contributes to the total dipole moment.The linear ME effect calculated by (8) and (5) is shownon Fig.3 for different thicknesses of N / M / I layers andthe following parameters: M -layer is Fe, N -layer is Cu,insulator – barium titanate (BaTiO ) with ε = 1400;we set γ = 0 .
5, single multilayer thickness l + w = 10 − cm for dashed line, 3 × − cm for solid line, 10 − cmfor dotted line. As the graphs shows, the magnitude of P for submicron structures comes up to tens of µC/m that is comparable with results Ref. , and is obtainedin quite a low magnetic field. The nanostructures (about10 nanometers) are even better and the correspondingpolarization is about hundreds of µC/m at the samemagnetic field. It is obvious from (8) that the P can beincreased by using the dielectric with a greater ε , suchas the copolymers, ceramics, etc., or nonstationary non-intrinsic colossal ε , see e.g. and references in it. To FIG. 2. The metamaterial consisting of a two-layer platesmagnet/non-magnet in a dielectric matrix.FIG. 3. The metamaterial Cu/Fe/BaTiO electric polariza-tion dependence on the external magnetic field. Dashed line: l + w = 10 − cm ; solid line: l + w = 3 × − cm ; dottedline: l + w = 10 − cm achieve the record even for ε &
1, one need M so thatthe H would cross the critical magnetic field point.If the M -layer is paramagnetic with a strong exchangegain (e.g. Pd), the dependence of the electric polariza-tion H , according to (6), (8) is quadratic and can beenhanced using M -s which are close to satisfaction ofthe Stoner criterion of ferromagnetism and have a strongDOS dependence on the energy near Fermi level: dN ( ε ) dε (cid:12)(cid:12)(cid:12)(cid:12) ε = ε F >> N ( ε F ) ε F . In summary, we have substantiated that metamaterialconsisting of oriented isolated inhomogeneous magneticconductors can exhibit strong (regarding known rangeof ME effect) electric polarization response on a mag-netic field. It is interesting that the polarization directionis determined by location and orientation of the N - M structures in the material. This feature allows to sep-arate predicted effect even from other ME effects thatcan be imposed, in particular, if the intrinsic ME cou-pling is present in insulator I . In such a case we proposechanging the sample orientation relative to the externalmagnetic field in the measurements.It has been shown that linear ME effect occurs when M -part of meta-atoms is ferromagnetic. As indicatedabove, the linear ME effect is interesting in that H is apseudo vector, and P is a vector. In our effect P is com-posed of the true scalar (average energy of the electronmagnetic moment in magnetic field) and true vector (thedirection of electron transfer during the equilibration).The accuracy of the linear behavior is very high becauseit’s determined by δµ/ε F ∼ − ÷ − .In the case of the paramagnetic M -parts, ME effect isquadratic in H . In particular, it can be detected by afrequency-doubled response to a variable magnetic field.Here we have applied equilibrium approach. It is validfor quasi-static magnetic field when the characteristic fre- quency is less than the reverse time of the nuclear andelectron spin relaxation (10 ÷ sec − ). The resultsof the study in variable electromagnetic fields of higherfrequency will be published later.Notice one more application of the effect. Since themagnetized state is preserved after ferromagnetic tran-sition, the electric polarization of the metamaterials canalso retain for a long time. Therefore, these materialsmay be used as magnetically driven electrets, i.e. sub-stances with long-lasting electrically polarized state.Finally, we emphasize that the electric dipole momentof standalone meta-atom is extremely small, because thecharge separation is confined only within a very thin layerin the vicinity of the M - N interface because of absenceof neighbors’ mutual capacity. Therefore, only the nanos-tructured metamaterial exhibits strong ME response.This work was supported by the grant of the NAS ofUkraine Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. ,323 (2004); S. D. Bader and S. S. P. Parkin, “Spintronics”, AnnualReviews of Condensed Matter Physics, , 71-88 (2010). A.Fert, Rev. Mod. Phys. , 1517 (2008), Peter A.Gr¨unberg, Rev. Mod. Phys. 80, 1531 (2008). A.V.Scherbakov, A.V.Akimov, D.R.Yakovlev, W.Ossau,L.Hansen, A. Waag and L. W. Molenkamp, Appl. Phys.Lett. , 162104 (2005). N.F.Mott, Proc. R. Soc. London, Ser. A , 699 (1936). R.Fiederling, M.Keim, G.Reuscher, W.Ossau, G.Schmidt,A.Waag and L. W. Molenkamp, Nature 402, 787 (1999). M.N. Baibich , J.M. Broto , A. Fert , F. Nguyen Van Dau, F. Petroff , P. Eitenne, G. Creuzet , A. Friederich & J.Chazelas , Phys. Rev. Lett. , 2472 (1988). Binasch G., P. Grunberg, F. Saurenbach, and W. Zinn,Phys. Rev. B , 4828 (1989). S. Datta and B. Das, Appl. Phys. Lett. , p. 665 (1990). J. Carlos Egues, Guido Burkard, and Daniel Loss, Appl.Phys. Lett. , 2658 (2003). D. J. Monsma, J. C. Lodder, Th. J. A. Popma, B. Dieny,Phys. Rev. Lett. , 5260, (1995). M. Johnson, J. Appl. Phys. , 6714 (1994). M. Johnson, Science , 320 (1993). R.N.Gurzhi, A.N.Kalinenko, A.I.Kopeliovich,A.V.Yanovsky, Appl. Phys. Lett. , 4577 (2003). Landau L.D., Lifshitz E.M., Pitaevskii L.P., Electrody-namics of Continuous Media, Pergamon, 1984; Magneto-electric interaction phenomena in crystals, ed. by A. J.Freeman, H. Schmid, L.- [a.o.], 1975. G. A. Smolenskii, I. E. Chupis, Sov. Phys. Usp. R123 (2005); I. E. Chupis, Low Temp.Phys. , 477 (2010); Pyatakov A P, Zvezdin A K, Sov.Phys. Usp.
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