Abstract
The control of the magnetism of ultrathin ferromagnetic layers using an electric field, rather than a current, has many potential technologically important applications. It is usually insisted that such control occurs via an electric field induced surface charge doping that modifies the magnetic anisotropy. However, it remains the case that a number of key experiments cannot be understood within such a scenario. Much studied is the spinsplitting of the conduction electrons of nonmagnetic metals or semiconductors due to the Rashba spinorbit coupling. This reflects a large surface electric field. For a magnet, this same splitting is modified by the exchange field resulting in a large magnetic anisotropy energy via the DzyaloshinskiiMoriya mechanism. This different, yet traditional, path to an electrically induced anisotropy energy can explain the electric field, thickness and material dependence reported in many experiments.
Introduction
The possibility of controlling the magnetic anisotropy of thin ferromagnetic films using a static electric field E is of great interest since it can potentially lead to magnetic random access memory (MRAM) devices which require less energy than spintorquetransfer random access memory STTMRAM^{1,2,3,4,5,6,7}. Thin magnetic films with a perpendicular magnetic anisotropy (PMA) are important for applications^{8,9}. That an interfacial internal electric field might be used to engineer such a PMA is also of great interest. Experiment^{10,11} has indeed shown that such a PMA might, in turn, be modified by an externally applied electric field, however the data is usually interpreted in terms of changes to the electronic contribution to magnetic anisotropy due to the surface doping induced by the applied electric field^{3,11,12}.
The theory of the fieldinduced changes of the magnetic anisotropy reflecting surface doping is invariably developed in terms of band theory^{13,14,15,16,17}. The results for both the bulk and thin films can be adequately understood in terms of second order perturbation theory^{18} in which the relevant matrix elements of the spinorbit interaction are between full and empty states. Large contributions come from regions where different dbands (almost) cross. That such crossings should be close to the Fermi surface leads to a strong doping dependence in such theories. Nakamura et al.^{19} pointed out that the strong negative applied field dependence of the PMA for an isolated monolayer (ML) of Fe(001) arises directly from band splitting rather than from doping. In this case, an E perpendicular to the film breaks reflection symmetry causing a large spinorbit splitting of dlevels near the Fermi surface. As will be explain below, despite these important theoretical developments, a clear explanation of a number of key experiments is still lacking.
Here we develop a simple analytic theory for the existence and electrical control of the PMA based upon the Rashba spinorbit interaction^{20,21,22} and the single band Stoner model of magnetism. We exhibit the somewhat delicate, but very interesting, competition between the Rashba spinorbit fields and the exchange interaction, reflecting electron correlations. This theory can potentially lead to a very large magnetic anisotropy arising from the internal electric fields E_{int} which exist at, e.g., ferromagnetic/metal and ferromagnetic/oxide insulator interfaces but modified by the addition of an applied electric field E_{ext}. There is a Rashba splitting of the band structure leading to a quadratic, (E_{int} + E_{ext})^{2}, contribution to the magnetic anisotropy, contrasting with a linear in E_{ext} doping effect.
Results
Model
This comprises a band Stoner model with the Rashba interaction added^{23}:
where p is the electron momentum operator, S the order parameter, σ the Pauli matrices and α_{R} = eη_{so}E the Rashba parameter proportional to η_{so}, which characterises the spinorbit coupling. The electric field is taken to be perpendicular to the plane of the system and is perpendicular to and makes an angle θ to the direction, as in Fig. 1(a).
Illustrative nonmagnetic example
Consider the Rashba effect in nonmagnetic two dimensional electron gases or surface states on noble metals, e.g., a surface state of Au. As shown in the methods, the single particle energy
as the spin quantum number σ = ±1. The momentum shift and
identified in the inset of Fig. 1(b), reflects the single particle energy gain relative to zero electric field, i.e., E = 0 and hence α_{R} = 0. For the three dimensional problem and there is no equivalent momentum shift in k_{z}. For the surface state of Au, E_{R} ≈ 3.5 meV^{24,25} exemplifying the energy scale.
The study of such surface states and differences in chemical potentials, also helps set the scale for E. Between a metal and the vacuum, or dielectric, the E ~ 10 V/nm near the Cu (100) surface^{26}, reflects the electron image potential. For the Ag/Cu(111) system^{27} an E of similar magnitude occurs at the metal interface. The difference in the (111) chemical potentials^{28} of Cu(4.96 eV) and Ag(4.74 eV) is reflected by a potential increase from Cu to Ag. The chemical potential of Au(5.31 eV) implies a similar potential change and E between Cu and Au, but of the opposite sign. It is the selectrons which penetrate the core and determine the spinorbit parameter η_{so}. The example Cu/Ag/Au therefore illustrates the higher/lower potential of these selectrons in the second/third transition series relative to the similar electrons in the 3d elements.
There are many experiments^{24,25,29,30,31} which put in evidence the Rashba splitting in two dimensional electron gases, surface states of noble metals, bulk layered systems and e.g., of a surface state of ferromagnetic Yb.
Magnetic case  origin of the magnetic anisotropy energy
In the methods it is shown the θ dependent single particle energy, i.e., the equivalent of Eq. (2) for the magnetic case is:
The direction of the now θ dependent momentum shift changes sign as the spin index σ = ±1. These shifts also change sign with for a given σ. This “magnetic Rashba splitting” with is observed for the surface state of Yb^{31}.
Also in the methods the DzyaloshinskiiMoriya (DM) and pseudodipolar (PD) contributions to the magnetic anisotropy are highlighted by contrasting the perpendicular and parallel orientations of order parameter to the plane.
Assuming (J_{0}S)^{2} > (α_{R}k_{x})^{2} and retaining the θdependent terms up to the order of E^{2} in (4), we obtained our principal result:
for the magnetic anisotropy energy, with
where 〈 〉 denotes an average over the Fermi sea (see methods). The Rashba spinorbit interaction produces a uniaxial anisotropy energy which, as in the DzyaloshinskiiMoriya theory^{32,33,34,35}, comprises a direct second order in E easy plane pseudodipolar interaction and an indirect contribution proportional to E^{2}/J_{0}S reflecting the competition between the first order in E, RashbaDzyaloshinskiiMoriya and exchange fields.
Competition between the DzyaloshinskiiMoriya and pseudodipolar contributions
Clearly Eq. (5) implies an E^{2} dependent PMA results if T > J_{0}S/2, i.e., when the DM is larger than the PD term. Taken literally, the Stoner model Eq. (1), with its quadratic dispersion, predicts the ratio of the DM and PD contributions to the PMA. The result, (see methods), depends upon the spatial dimension. In two dimensions the PD and DM terms cancel although higher order terms (O(α_{R}^{4})) lead to a PMA while in three dimensions the DM term is −(4/5) E_{R} cos^{2} θ and an inplane magnetisation is favoured. Lastly, a two dimensional system with a highly anisotropic conductivity might be modelled as a series of parallel one dimensional chains. For chains the DM contribution −(4/3) E_{R} cos^{2} θ which dominates the PD energy E_{R} cos^{2} θ, appropriate when is inplane and perpendicular to the chains. Corresponding to the hardest axis, when inplane but rather parallel to the chains, there is neither a DM or PD contribution to the magnetic anisotropy energy. There is thereby a predicted electric field dependence of the inplane anisotropy as seen in early experiments^{3}, given the large compressive strain that arises in these experiments.
However, for the real problem of 3d magnets, a quadratic dispersion is not at all realistic and the crystal potential V(r) must be accounted for, see methods. For 3d elements the wavefunction ψ is well localised within the atomic sphere and the averages, e.g., and hence T, are very much increased as compared to the above naïve estimates. In reality, the DM contribution will invariably lead to a PMA.
Discussion
The resulting anisotropy energy can be very large. The work already cited^{24,25,26,27} on conducting but nonmagnetic materials helps set the scales. The value of the scaling prefactor E_{R} in Eq. (5) for the surface state of Au is ~ 3.5 meV or about 35 T in magnetic field units and very much larger than the typical ~ 1 T demagnetising field. If a Au film is polarised by contact with an ultrathin ferromagnet, the second factor, 2T/J_{0}S, in Eq. (5) for the field inside a Au surface layer can be quite large ~ 5 leading to a PMA and indeed ultrathin Fe on Au does have such a PMA^{36,37}. Ultrathin ferromagnetic films in contact with, e.g., Ru, Pd, Pt and Ta, etc., also are found to have a PMA^{6,12,38,39}.
Schematically shown in Fig. 2(a) is the potential seen by electrons in a free standing ultrathin ferromagnetic film. At the surface, the potential reflects an electron's image charge but reaches the vacuum level within a few atomic spacing. As already discussed, this results in a finite large electric field E ~ 10 V/nm at each surface but in opposite senses. Assuming an appreciable spinorbit coupling in the interface region, this results, in turn, in a Rashba field B_{R} which also changes sign between the two surfaces for a given momentum. Thus, for a perfectly symmetric film, the ferromagnetically polarised electrons see no average field B_{R}. This symmetry can be broken by the application of an external electric field as shown in Fig. 2(b). The electric field is increased at one surface and decreased at the other doubling the net effect. In contrast, for this same symmetric situation, the surface charges are opposite and doping effects must cancel. Experimentally applied fields of 1 V/nm are relatively easy to achieve implying a ~ 10% change in the surface anisotropy. Experiments^{40} with a 1.5 nm Fe_{80}B_{20} sandwiched between two MgO layers are perhaps closest to this situation although the thickness 1.5 nm and 2.5 nm of these layers are not equal. Roughly consistent with our estimate [see Fig. 2(c), case(i)], there is^{40} an ~ 15% symmetric contribution to the magnetic anisotropy for an applied voltage of 2 V.
Clearly the intrinsic Rashba field B_{R} is modified when the materials adjacent to a 3d ferromagnet (F) are different. In a number of experiments an insulator I, often MgO, lies to one side and a normal metal (N), e.g., Au, Pt, Pd, Ta, or Ru, “cap” completes a trilayer system. The potential, Figs. 2(d) and (e), will increase in passing from Fe to MgO but in passing from the F to Nlayer the potential will either increase, Fig. 2(d), or decrease, Fig. 2(e). As discussed above, the relevant potential would be expected to increase, Fig. 2(d), in passing from a 3d element, e.g., Co, Fe, Ni, or Cu, to a 4d transition metal such as Ru, Pd, Ag but decrease, Fig. 2(e), for the 5d elements, e.g., Au, Pt, Ta. The latter case is particularly favourable since the intrinsic Rashba fields have the same sense and add. In addition, the 5d elements have a larger spinorbit coupling, resulting in a larger α_{R} and hence are more likely to produce a sizeable PMA. If the electric field decreases at the FI interface, the average Rashba field increases in the first case [Fig. 2(f)] when the effects of the surfaces tend to cancel and, as illustrated in Fig. 2(g), decreases in the second case when the inverse is true [see Fig. 2c, case (ii) and (iii)]. Experiment^{12} indeed shows an opposite field dependence for such systems with Pd(4d) and Pt(5d) Nlayers. That the sign of the electric field contribution to the PMA reflects the Nlayer whereas the field is applied to the opposite surface between F and I supports the current Rashba model. This is in stark contrast with the popular surface doping model^{3,12}, for which the effects of surface doping are limited by the (possibly magnetically modified) FermiThomas screening length. In reality^{12}, the screening length is estimated to be much less than 1 nm and much too short for there to be an appreciable doping effect of the Pt or Pd layers that are typically distant by a few nanometers.
Simulating a large applied electric field E, the required asymmetry might be controlled in NFN trilayers by varying in a systematic manner, at the monolayer level, the thickness of one of the normal metal layers and by using metals with different spinorbit couplings. In reality the effect of the substrate transmitted to and through, the bottom normal metal will imply an asymmetry even for largish Nlayer thickness. Indeed the PMA surface term for Au/Fe(110)/Au(111) structures does show an nonmonotonic dependence on the top Au layer thickness^{36}. Experiments^{37} for Fe layers on vicinal Ag(001) and Au(001) surfaces and which undergo a symmetry breaking (5 × 20) surface reconstruction manifest an inplane surface term reflecting this broken symmetry and which is larger for Au, with its stronger spinorbit coupling, than for Ag.
It is predicted that the surface coercivity field H_{c} is proportional to (E_{int} + E_{ext})^{2} where E_{int} is the internal electric field corresponding to the zerobias Rashba contribution to the anisotropy. Such a nonlinear field dependence is observed, e.g., for the inplane contribution for a (Ge,Mn)As/ZrO_{2} surface^{3}. In other experiments^{10} with CoFeB/MgO/CoFeB structures there is qualitative difference between the E dependence of the anisotropy field H_{c} of the, “top” and “bottom”, CoFeB layers of this three layer structure, even when they have similar thicknesses. The bottom layer has a larger H_{c} and is roughly linear while H_{c} becomes highly nonlinear as H_{c} → 0 as would be expected as E_{ext} → −E_{int}.
The most direct experimental test of the model is the observation of the band splittings for a model Rashba system with a variable contact with an itinerant ferromagnet. This can result in giant magnetic anisotropy (GMA) energies. For example an E_{R} ~ 100 meV (or ~ 1000 T) is reported in angleresolved photoemission spectroscopy (ARPES) measurements^{41} on bulk BiTeI. For a thin film of this, or similar material, in contact with an itinerant ferromagnet such as Fe, a suitable exchange splitting J_{0}S, tuned to the order of E_{R}, might be induced and a GMA will result. ARPES performed as a function of the direction of the magnetisation m might determine both E_{R} and the momentum dependence of the exchange splitting leading to estimates of both the PM and DM contributions and which might be directly compared with magnetisation and magnetic resonance measurements. The electrical control of such a GMA has evident important application for nonvolatile memory applications. There are clearly many more complicated embodiments of such a device.
In conclusion, it is suggested that the Rashba magnetic field due to the internal electric field in the surface region of an ultrathin ferromagnet can make an important contribution to the perpendicular magnetic anisotropy. Such surface fields might be modified by application of an applied electric field. Since the internal fields at two surfaces tend to cancel, an asymmetry between the surfaces is important. Such an asymmetry is caused by different metal and insulator caping layers. These ideas are consistent with a large number of experiments.
Methods
The nonmagnetic case
This corresponds to Eq. (1) with J_{0} = 0. It is solved by taking the axis of quantisation to be perpendicular to the inplane k as in Fig. 1(b). The eigenstates are e^{i}^{k·r}s〉 and , where the Rashba magnetic field in energy units is defined as , with μ_{B} the Bohr magneton and g the gfactor, leaving the spin state s〉 to be determined. There are two concentric Fermi surfaces. The energy splitting , where Δ is the value for , with k_{F}_{↑,↓} the Fermi wave number for the spin up/down (σ = ±1) band. For the surface state of Au, Δ ≈ 110 meV while E_{F} ≈ 420 meV giving the E_{R} ≈ 3.5 meV cited in the text. The magnetic case. The full Eq. (1) is solved by defining axes such that lies in the y–zplane and . The total field, which defines the axis of quantisation, . It is assumed that, for a 3d ferromagnet J_{0}S ~ 0.5–1.0 eV and gμ_{B}B_{R} < J_{0}S, i.e., the Rashba is smaller than the exchange splitting. To second order in gμ_{B}B_{R}, where and where differs in direction from by a small angle δ where tan δ ≈ α_{R}(k_{x}^{2} cos^{2}θ + k_{y}^{2})^{1/2}/J_{0}S. The linear in k_{x} term, α_{R}k_{x} sin θ, causes a shift in Fermi sea to give the the single particle energy Eq. (4).
With perpendicular to the plane, i.e., , the exchange and Rashba fields are orthogonal and hence the net energy for a single electron Eq. (4) is
The axis of quantisation is tilted by away from the zaxis as shown in Fig. 1(c). The σ = ±1 electrons gain/lose an energy that is even in E. This arises from the competition of the Rashba field, perpendicular to , with the exchange field. Such a competition generates a second order in E contribution to the magnetic anisotropy and is identified with the DzyaloshinskiiMoriya (DM) mechanism^{32,33,34,35}.
Now take parallel to the yaxis, i.e., . The ycomponent of B_{R} is parallel to the exchange field and is combined with the kinetic energy. The Fermi sea is shifted along the xaxis and lowered by E_{R} as shown in Fig. 1(d). This energy gain corresponds to a pseudodipolar (PD) contribution to anisotropy energy^{35} which favours an inplane magnetisation. On the other hand, the xcomponent of B_{R}, which is perpendicular to , gives rise to a correction to the effective exchange field. The direction of the moment tilts away from the yaxis in the direction perpendicular to the wave vector by as shown in Fig. 1(d). The single particle energy, Eq. (4), is now,
where the shift k_{0} is the same as in Eq. (2) but only along the xaxis.
The effective exchange field in Eq. (8) is smaller than that in Eq. (7) due to the absence of a k_{x}^{2} term. This indicates that the overall DM contribution favours a perpendicular while the PD term favours an inplane . This exchange field changes sign with σ = ±1.
Evaluation of the DzyaloshinskiiMoriya and pseudodipolar contributions
Needed for T in Eq. (5) are the Fermi sea averages and , determined analytically, for quadratic dispersion, in one, two and three dimensions respectively. For an isotropic system these averages are related to J_{0}S via
which determines the ratio 2T/J_{0}S in the principal result, Eq. (5), given in the text.
Role of the crystal potential
The effects of the crystal potential are exhibited by considering a wave function which is a linear combination of plane waves, where K are the reciprocal lattice vectors and the a_{K} are determined by V (r). While not convenient for 3d electrons, at least in principle, such an expansion in the true, rather than crystal, momentum states is always possible. The PD contribution, E_{R} cos^{2} θ is independent of the momentum k + K. However , where 〈 〉_{BZ} is the average over the first Brillouin zone. For 3d electrons, the average and hence T, are dominated by the a_{K} for largish K. It follows T is significantly increased with the consequences discussed in the text.
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Acknowledgements
This work is partly supported by KAKENHI (No. 24740247) from MEXT, Japan.
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S.E.B. identified the problem. S.E.B., J.I. and S.M. performed the analytical calculations, analysed the data and wrote the manuscript. S.E.B. and J.I. devised and prepared the figures. All authors reviewed the manuscript.
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Barnes, S., Ieda, J. & Maekawa, S. Rashba SpinOrbit Anisotropy and the Electric Field Control of Magnetism. Sci Rep 4, 4105 (2014). https://doi.org/10.1038/srep04105
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DOI: https://doi.org/10.1038/srep04105
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