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# Effect of excitation power on voltage induced local magnetization dynamics in an ultrathin CoFeB film

## Abstract

Voltage or electric field induced magnetization dynamics promises low power spintronics devices. For successful operation of some spintronics devices such as magnetic oscillators and magnetization switching devices a clear understanding of nonlinear magnetization dynamics is required. Here, we report a detailed experimental and micromagnetic simulation study about the effect of excitation power on voltage induced local magnetization dynamics in an ultrathin CoFeB film. Experimental results show that the resonance line-width and frequency remains constant, whereas cone angle of the magnetization precession increases linearly with square-root of excitation power below threshold value, known as linear excitation regime. Above threshold power, the dynamics enters into nonlinear regime where resonance line-width monotonically increases and resonance frequency monotonically decreases with increasing excitation power. Simulation results reveal that a strong nonlinear and incoherent magnetization dynamics are observed in our experiment above the threshold power which reduces dynamic magnetic signal by suppressing large cone angle of magnetization precession. Moreover, a significant transfer of spin angular momentum from uniform FMR mode to its degenerate spin waves outside of excitation area further restrict the cone angle of precession within only few degrees in our device. Our results will be very useful to develop all-voltage-controlled spintronics devices.

## Introduction

The recent discovery of voltage-controlled magnetic anisotropy (VCMA) has a huge potential for the development of low power spintronics devices fully operated by voltage1, 2. Several physical phenomena like control of coercive field3 and domain wall motion4, 5, magnetization switching1, 6,7,8,9 and coherent ferromagnetic resonance induced by voltage10,11,12 have already been studied by several groups. Although various kinds of material systems have been used for these studies, the multilayer systems formed by 3d-ferromagnetic metals and insulating MgO are found to be one of the most promising candidates among them. In particular, magnetic tunnel junctions (MTJ) formed by CoFeB/MgO/CoFeB are the basic structure of modern spintronics devices due to their large tunneling magnetoresistance (TMR) value at room temperature13, 14, which can be used in non-volatile MRAM devices.

Precessional magnetization dynamics are called in the linear regime when cone angle of magnetization precession remains only within few degrees. When the excitation power crosses threshold value, the trajectory of the magnetization precession is strongly modified and dominated by incoherent magnetization precession, known as nonlinear magnetization dynamics15, 16. The nonlinear magnetization dynamics are essential for the operation of spintronics devices such as magnetic switching devices6, 7, 9, magnetic oscillators17. Until now, most of the voltage induced ferromagnetic resonances are reported in the linear regime10,11,12, 18. In spite of many reports on the magnetic field16, 19, 20 or spin torque15 induced nonlinear magnetization dynamic, only a single report on voltage induced nonlinear dynamics of a spatially confined CoFeB film with perpendicular magnetic anisotropy (PMA) are found21. Here, we study VCMA induced nonlinear magnetization dynamics in an ultrathin CoFeB film. We locally excite magnetization dynamics and systematically increase excitation power to investigate the variation of resonance frequency, line-width, line-shape and cone angle of precession with excitation power. We also performed micromagnetic simulations to visualize the spatial maps of dynamic magnetization at different excitation power and to understand the underlying mechanism of observed dynamic behavior as a function of excitation power. We believe that our study will be very useful to understand the nonlinear magnetization dynamics in an ultrathin ferromagnetic film under local excitation scheme.

## Results

### Schematic of device and working principle

Figure 1 shows the schematic diagram of the device and experimental set up. The device with MTJ structure consists of 3 nm top CoFeB layer (reference layer) and 1.4 nm bottom CoFeB layer (free layer) separated by a 2 nm thick insulating MgO layer (see methods for details). The reference layer was patterned into rectangular shape (2 × 4 µm2) in the middle of a much wider (50 × 100 µm2) free layer (Fig. 1). Both CoFeB layers have PMA at the interface of CoFeB/MgO due to the hybridization of Fe-3d and O-2p orbitals22, 23 and the magnitude of PMA strongly depends upon the thickness of ferromagnetic layer (inversely proportional to thickness)24, 25. The PMA field value of reference layer is less than the demagnetizing field. Therefore, reference layer has in-plane magnetic easy axis24. On the other hand, the PMA of free layer overcomes the demagnetizing field to have a net effective PMA. Therefore, free layer has out-of-plane magnetic easy axis. When dc or rf voltage is applied across the MTJ, the electric field at the interface of CoFeB layers modulates charge or spin density in the Fe-3d orbitals26,27,28, which causes a change in PMA. This change in PMA can be thought as equivalent to a magnetic field along out-of-plane direction which is proportional to out-of-plane component of saturation magnetization (M z)10. For our measurement, rf voltage (V rf) with varying frequency (f) is applied across the MTJ through the capacitor port of bias tee (Fig. 1). When f matches with the ferromagnetic resonance frequency (f FMR) of the free layer under the application of bias magnetic field (H), its magnetization oscillates around effective field direction. The rectified voltage (V rec) is measured through the dc port of bias tee as a function of f varying from 2 to 9 GHz (see methods for details).

### Variation of PMA field with dc voltage

To evaluate the variation of PMA field (μ 0 H k) with voltage, we measured TMR of MTJ as a function of external magnetic field magnitude (μ 0 H) from −320 mT to +320 mT along the y-axis (Fig. 1) under various dc bias voltages (V b) ranging from −0.62 V to +0.62 V. Note that positive voltage means the top electrode has positive potential with respect to the bottom electrode of MTJ. The resistance and area product (RA) for our device is about 100 kΩ.µm2. The TMR of a MTJ depends upon the relative angle between the magnetizations of two CoFeB layers29, i.e. $${\rm{TMR}}\propto \,\sin \,{\varphi }_{F}$$, where ϕ F is the angle between the magnetizations of two CoFeB layers (Fig. 2(a)). Therefore the ratio of in-plane component of free layer magnetization (M y) and saturation magnetization (M s) can be evaluated from normalized TMR curve30 as $${M}_{y}\propto {M}_{s}\,\cos \,{\varphi }_{F}$$. In Fig. 2(b), we plot experimentally measured normalized TMR as a function of +μ 0 H for different values of V b. At μ 0 H = 0 mT, all the curves show their maximum values which indicates that the magnetizations of both layers are aligned almost perpendicular to each other along their easy axes as schematically shown in the same figure. As μ 0 H increases, the magnetization of free layer starts to incline along positive y-axis. Therefore, TMRs decrease monotonically with the increase of +μ 0 H for all bias voltages until they reach to the minimum values at around +190 mT, indicating that the magnetizations of the both CoFeB layers are aligned along the +y-axis for μ 0 H > + 190 mT. The difference in slopes among the curves originates from the modulation of μ 0 H k by V b through VCMA effect31. We evaluate μ 0 H k for each value of V b by using the method mentioned in refs 18, 30 and 31. The effective value of μ 0 H k at V b = 0 V is about 185 mT. In Fig. 2(c), we plot μ 0ΔH k, defined by the difference between μ 0 H k at certain bias voltage and zero bias voltage, as a function of V b. The magnitudes of the gradients of $$|{\mu }_{0}{\rm{\Delta }}{H}_{k}|$$ are 45 and 23 mT/V for negative and positive V b, respectively. The difference between the slopes of μ 0ΔH k for negative and positive V b originates probably due to the two different interfaces (CoFeB/MgO & CoFeB/Ta) of free CoFeB layer in the MTJ22, 31.

### Resonance line-shape and line-width at lower excitation power

We measured V rec as a function of f for different magnitudes of +μ 0 H. Figure 2(d) shows a typical plot of V rec as a function of f measured at μ 0 H = 100 mT and rf power (P rf) = 5 µW. The rf power is low enough to excite FMR in the linear regime. The plot shows two resonance peaks. The resonance peak at 4.72 GHz with large peak-to-peak height (V pp) corresponds to FMR of free layer, whereas the resonance peak at 6.75 GHz with very small value of V pp corresponds to FMR of reference layer. This shows that the reference layer is not efficiently excited by voltage due to near in-plane orientation of magnetization. Note that very low tunneling current density (~106 A.m−2) can not excite reference layer by spin transfer torque (STT) or field like torque (FLT). The FMR spectrum of free CoFeB layer (Fig. 2(d)) can be fitted by linear combination of symmetric and anti-symmetric Lorentzian functions given by refs 18, 32, 33:

$${V}_{{\rm{rec}}}=\frac{{V}_{{\rm{s}}}}{1+{(f-{f}_{{\rm{FMR}}})}^{2}/{\sigma }^{2}}+\frac{{V}_{{\rm{as}}}(f-{f}_{{\rm{FMR}}})/\sigma }{1+{(f-{f}_{{\rm{FMR}}})}^{2}/{\sigma }^{2}},$$
(1)

where, V s and V as are the amplitudes of symmetric and anti-symmetric terms, σ is the half width at half maximum (HWHM) of the spectrum and f FMR is the FMR frequency. For homodyne-detected signal, V s originates from STT induced FMR, whereas V as originates from the FMR induced by VCMA torque and/or FLT18, 33, 34. Peak-to-peak height (V pp) of V rec curves can be obtained by summing up V s and V as. Equation 1 fits well with the experimental resonance spectrum of free layer as shown by solid line in Fig. 2(d). According to the fitting result, V as is around ten times larger than V s. Therefore, the contribution of STT is negligible or much smaller than the combined effect of VCMA torque and FLT. For further confirmation, we plot peak-to-peak value (V pp) of V rec as a function of angle (ϕ F) made by free layer with film plane (see supplementary for details). The VCMA torque induced homodyne signal amplitude should be proportional to sin2 ϕ F cos ϕ F (refs 10 and 21), where V pp should be maximum at ϕ F ~ 55°. The maximum value of V pp in our sample is obtained for ϕ F ~ 59°, which implies voltage excitation of magnetization dynamics. The small deviation from theoretical value may occur due to the stray magnetic field from reference layer10.

Next, we discuss about the line-width (HWHM) of FMR spectra. The HWHM (σ) of the measured FMR spectra can be expressed as35

$$\sigma ={\sigma }_{ex}+{\sigma }_{in}={\sigma }_{ex}+\alpha {f}_{FMR}.$$
(2)

Here, σ in is the intrinsic line-width, which originates from intrinsic Gilbert damping, σ ex is the extrinsic line-width, which originates from various extrinsic contributions and α is the intrinsic Gilbert damping constant. The intrinsic line-width is proportional to the f FMR. According to the literature24, the intrinsic damping constant of 1.4 nm CoFeB is about 0.02, which gives the intrinsic resonance line-width (σ in) of about 100 MHz in absence of any extrinsic contribution. We extracted the value of σ as 425 MHz from the fitting of resonance spectrum with Eq. 1. This broad resonance line-width in our device indicates the presence of other extrinsic contributions, which may originate from inhomogeneous distribution of PMA and/or two-magnon scattering. In our case, the device was post-annealed at 300 °C in vacuum for 1 hour. Therefore, the PMA is expected to be uniform throughout the film. Two-magnon scattering which originates from the defects at the interfaces36, 37, may be present only if M s is aligned at an angle (ϕ F) less than 45° with respect to film plane37. In our case, two-magnon scattering has negligible contribution as ϕ F = 58° (>45°) (see supplementary informations). In this study, the magnetization dynamics are locally excited in the centre of a wider thin film, where spin waves are also excited outside the excitation area in addition to uniform FMR underneath the excitation area (see supplementary informations). The eigen-frequencies of the uniform FMR and spin wave modes are almost degenerated due to the small thickness of CoFeB film. This opens up a new channel for relaxation of spins correspond to uniform FMR via spin waves. Therefore, the resonance peak of uniform FMR mode is broadened. Probably, this broad peak of uniform FMR mode and other spin wave modes are not well resolved in our experiment. Therefore, we observed a single and broad resonance peak in our device instead of many isolated peaks.

### Resonance spectra as a function of excitation power

We measured resonance spectra for different values of P rf ranging from 5 µW to 1.41 mW at μ 0 H = 100 mT. In static condition, the free layer magnetization is oriented at an angle of 32° with respect to the film normal at this magnetic field value. We plot V rec, rescaled by a factor of 1/P rf, as a function of f for different values of P rf in Fig. 3(a). With the increase of P rf, the resonance line-width is broadened maintaining anti-symmetric Lorentzian shape and the resonance frequency (f FMR) shifts towards the lower frequency. These features may be the fingerprints of nonlinear magnetization dynamics. However, we didn’t observe hysteresis between the resonance spectra for upward and downward sweeps of f (see supplementary information) in our MTJ unlike the previous reports15, 20. The reason behind this is probably thermal fluctuation of magnetizations due to Joule heating15, 21.

We fitted the resonance spectra with Eq. 1 to find out the parameters V s, V as, f FMR and σ. Figure 3(b) shows the plot of peak-to-peak height (V pp) of V rec curves as a function of P rf. The inset graph shows that the V pp linearly increases with P rf up to about P rf = 30 μW. However, V pp deviates from linear increment at higher values of P rf. In Fig. 3(c), we represent variation of f FMR as a function of V rf, where V rf = (PrfZ0)1/2 is the input rf voltage and Z 0 (=50 Ω) is the characteristic impedance of waveguide. The graph shows that f FMR remains almost constant (4.72 GHz) up to V rf ≈ 0.04 V, (which correspond to 32 μW of rf power) and then monotonically (linearly) decreases with the increase of V rf. We also plot HWHM as a function of V rf in Fig. 3(d). The HWHM remains almost constant (~425 MHz) up to V rf ≈ 0.04 V. After that, HWHM increases almost linearly with the increase of V rf.

### Cone angle of precession

We also roughly estimated cone angle of precession in experiment from peak-to-peak (V pp) value of resonance spectra by using the following expression10, 21:

$$\sin \,{\theta }_{c}=\frac{2}{{r}_{TMR}\,\sin \,{\theta }_{F}}\frac{{R}_{AP}(R+{Z}_{0})}{{R}^{2}}\frac{{V}_{pp}}{{V}_{rf}}.$$
(3)

Here, θ c is the cone angle of precession, r TMR = (R APR P)/R P, R P and R AP are resistances for parallel and anti-parallel orientations of magnetization respectively, R is the resistance at μ 0 H = 100 mT and Z 0 is the characteristics impedance of waveguide. Figure 4(a) shows the plot of the θ c as a function of V rf. Graph shows that θ c linearly increases with V rf up to V rf = 0.04 V, which corresponds to P rf ≈ 32 µW. Above 32 µW, θ c deviates from linear increment with V rf and becomes almost saturated above V rf ≈ 0.27 V (P rf ≈ 1.41 mW), where we get the maximum value of θ c of about 4.35°. All these observations suggest that P rf ≈ 32 µW (V rf ≈ 0.04 V) is the threshold value of rf power above which the dynamics enters into nonlinear regime.

### Micromagnetic simulations

To understand the underlying mechanism of the observed dynamic behaviours as a function of P rf, we performed micromagnetic simulations38 based on Landau-Lifshitz-Gilbert (LLG) equation (see methods). In the simulations, magnetization dynamics were excited by applying sinusoidal rf magnetic field (h rf sin 2πft) along z-axis with frequency f only in the central 2 × 4 µm2 area of CoFeB layer highlighted by green colour in Fig. 5(a). The magnitudes (h rf) of rf magnetic field were taken to be equivalent to the PMA field modulated by applied V rf as calculated from experimental result presented in Fig. 2(c) and Eq. 4 given below21.

$${h}_{rf}={V}_{rf}(1+{\rm{\Gamma }})\frac{\partial {H}_{k}}{\partial {V}_{b}}\,\sin \,{\varphi }_{F}\,\cos \,{\varphi }_{F}.$$
(4)

Here, $$\frac{\partial {H}_{k}}{\partial {V}_{b}}=34.2\mathrm{mT}/V$$ is the average gradient of μ 0ΔH k for positive and negative V b, Γ = (R − Z 0)/(R + Z 0) is the reflection coefficient of V rf from the MTJ. The experimentally measured rectified voltage (V rec) can be expressed as the time-averaged value of the product of oscillating TMR due to precession of magnetization and rf tunnel current as given below18, 33

$${V}_{{\rm{rec}}}=\langle R(t){I}_{rf}(t)\rangle =\langle \frac{R}{1+{p}^{2}\,\cos \,{\varphi }_{F}(t)}\frac{{V}_{{\rm{rf}}}}{R}\,\sin (2\pi ft)\rangle ,$$
(5)

where, R is the resistance, p is a spin polarization (we used 0.5), and ϕ F(t) is time-dependent relative angle between the magnetizations in the free and reference CoFeB layers. As the orientation of reference layer magnetization is nearly in-plane of the film, we simply adopt the angle (ϕ F) made by free layer with the film plane as equals to the angle made by free layer with the reference layer. R(t) = R/(1 + p 2 cos θ F(t)) represents the time-dependent resistance due to oscillation of ϕ F and I rf(t) = V rfsin(2πft)/R represents the time-dependent tunnel current. Therefore, experimentally measured V rec can be reproduced by micromagnetic simulations by using the time variation of ϕ F when applying V rf. Note that ϕ F is the angle made by average magnetization of the green area (Fig. 5(a)) with the film plane. We used α = 0.02 in the simulation.

Figure 5(b) shows the plot of simulated rectified voltage (V rec), rescaled by a factor of 1/P rf, as a function of excitation frequency (f) for different values of P rf. The simulated resonance spectra also look like anti-symmetric Lorentzian in the linear or lower power (up to 36 μW) regime. With the increase of P rf the resonance spectra are broadened gradually like experimental results. The spectra also show decrease of f FMR and peak-to-peak height of V rec/P rf curves with P rf similar to the experimental results. To get into the more details of the underlying mechanism, we show snapshots of the spatial distribution of dynamic magnetization for different values of P rf in Fig. 5(c). The snapshots show that the dynamics is uniform and coherent up to 36 μW of rf power. Above 36 μW, the dynamics starts to become non-uniform and incoherent. The degree of incoherence gradually increases with further increase of P rf. The snapshots clearly show a smooth transition from coherent or linear magnetization dynamics to incoherent or nonlinear magnetization dynamics with a threshold power of about 36 μW. Therefore the dynamic dephasing due to the incoherent nonlinear dynamics reduces peak-to-peak height (V pp) of rectified voltage (V rec). Hence we observe a deviation of V pp from linear increment with P rf and deviation of θ c from linear increment with V rf above threshold value of P rf in experiment as well as in simulation results. Note that this dynamic dephasing also broadens resonance line-width.

In Fig. 4(b) we also plot simulated value of θ c as a function of V rf. The graph shows that θ c linearly increases with V rf up to V rf = 0.04 V (P rf ≈ 32 µW). Above the threshold power of 32 µW, θ c deviates from linear increment with V rf and becomes almost saturated above V rf ≈ 0.25 V (P rf ≈ 1.25 mW), where we get the maximum value of θ c of about 12°. It shows that simulation result qualitatively reproduces the experimental result about the variation of θ c with V rf (Fig. 4(a)). However, simulation result shows much larger cone angle of precession as opposed to the experiment. This can be explained in the following way. We observed broad resonance line-width in experiment due to the spin wave generation outside the excitation area. In this way spin angular momentum corresponds to uniform FMR mode is transferred to its degenerate spin waves. Therefore in our experiment, the cone angle of precession corresponds to uniform FMR can not go beyond 4.35° even at very high excitation power. For exact reproduction of experimental result all the external parameters like extrinsic contributions of damping, temperature, stray field from reference layer and other boundary conditions have to be taken into account, which is not possible in OOMMF simulation. Therefore, broad resonance line-width due to extrinsic contributions like spin wave generation can not be reproduced in micromagnetic simulations. That is why the cone angle of precession in simulation can reach up to 12° unlike experiment.

We further plot simulated values of f FMR as a function of V rf in Fig. 3(c). The simulated results also show that f FMR remains almost constant up to about V rf = 0.04 V (threshold power). Above threshold power, f FMR decreases almost linearly with V rf like the experimental result due to the non-linear dynamics. However, we observed a steeper variation of f FMR with V rf in the simulation than that of the experiment. In the linear regime, the precessional axis of the magnetization makes an angle of 32° with respect to out-of-plane direction which is essentially the direction of the effective magnetic field (H eff) determined by bias magnetic field, PMA field and demagnetizing field. With the increase of P rf, the cone angle of precession increases which in fact decreases M z (ref. 15). Therefore, H eff and hence f FMR decreases with the increase of P rf. In our simulation, the steeper decrease of f FMR with V rf was observed mainly due to the larger cone angle of precession as opposed to the experiment. A complete quantitative reproduction of experimental results is not possible as the effect of Joule heating and extrinsic contributions to the resonance line-width can not be taken into account. These Joule heating may also change the PMA of CoFeB film. Therefore the simulation result may differ from experimental results. Nevertheless, the micromagnetic simulations qualitatively reproduce all the important features of dynamics behaviour observed in the experiment.

## Discussions

A complete knowledge about the voltage induced nonlinear magnetization dynamics is essential for the application in future spintronics devices. However, the behaviour of nonlinear magnetization dynamics is quiet different than that of the linear dynamics. The magnetization dynamics in the nonlinear regime are governed by second order Suhl process16, 39 or four magnon scattering. In this process two k = 0 magnons are annihilated to create two k ≠ 0 (+k and −k) magnons19, 40. In other words a spontaneous transfer of angular momentum from coherent uniform precession of magnetizations to incoherent non-uniform precession of magnetizations occurs. This agrees with the observed behaviour in micromagnetic simulations. The dynamic dephasing among these incoherent k ≠ 0 magnons decreases the amplitude (V pp) or cone angle of precession (θ c) of dynamic magnetization than that of the expected values. The relaxation of magnetization dynamics in the nonlinear regime is also governed by intrinsic four magnon scattering40. The spin relaxation rate increases with the increase of four magnon scattering39. Again, the line-width of the resonance spectra is linearly proportional to the relaxation rate. As the four magnon scattering process increases with the increase of P rf, we observed a monotonic increment of HWHM as a function of V rf or P rf.

In conclusion, we have shown a systematic transition from linear magnetization dynamics to nonlinear magnetization dynamics for local excitation of magnetization by experiment and micromagnetic simulation study. Simulations results demonstrate that the coherent and linear magnetization dynamics becomes incoherent and nonlinear above a certain threshold value of rf power. This incoherent nonlinear dynamics significantly reduces dynamic signal i.e. cone angle of magnetization precession and increases resonance line-width. Voltage induced local magnetization dynamics is essential for exciting spin waves in future spintronics devices. We show that locally excited uniform magnetization dynamics also excites spin waves, almost degenerate to uniform FMR mode, outside the excitation area in ultrathin magnetic films. The resonance line-width correspond to uniform FMR mode is enhanced due to the channelling of angular momentum from uniform FMR inside excitation area to spin wave mode outside the excitation area. We have further shown that this broad resonance line-width does not allow cone angle of magnetization precession to exceed only above few degrees in our device. We think our study will be very useful for the development of all-voltage-controlled spin waves based logic devices.

## Methods

### Sample fabrication

The devices were fabricated by a multistep fabrication method. At first, film-stacking structure used in this study were prepared on a thermally oxidized Si(001) substrate by rf sputtering at room temperature at a base pressure of 10−9 Torr. The structure consists of the following layers (nominal thicknesses in nanometres are stated in parentheses): Si/Ta(5)/Ru(20)/Ta(5)/Co20Fe60B20(1.4)/MgO(2)/Co20Fe60B20(3)/Ta(5)/Ru(5). In second step, the free layers with dimension 50 × 100 μm2 were defined by maskless photolithography followed by Ar+ ion milling down to Si-substrate. In the third step, reference layers with dimension 2 × 4 μm2 were defined in the middle of free layers (Fig. 1(a)) by maskless photolithography followed by Ar+ ion milling down to MgO layer and subsequent deposition of 40 nm thick Al2O3. The dimension of reference layer defines the dimension of MTJ. In the fourth step, 100 nm thick Al2O3 was deposited everywhere except on top of the reference layer (for top electrode) and in the vicinity of the edge of free layer (for bottom electrode). Finally, the contacts were designed by photolithography followed by deposition of Ti(5)/Au(100) by electron beam evaporation. Fabricated devices were post-annealed at 300 °C in vacuum under a perpendicular magnetic field of 600 mT for one hour. High temperature annealing over 300 °C decreased µ0 H k due to intermixing of the interface between CoFeB and Ta41. For a good thermal stability of the surface magnetic anisotropy, one can use W buffer layer instead of Ta42.

### Experimental Measurement

The ferromagnetic resonance in CoFeB layers were excited by applying rf voltage across the MTJ. The rf voltage (V rf) produces rf electric field (E rf) at the interfaces of the CoFeB/MgO junction. The E rf modulates interfacial PMA of CoFeB layer. When frequency of V rf matches the FMR frequency (f FMR) of free layer, magnetization dynamics is excited. Magnetization dynamics produces an oscillatory TMR (depends upon the relative angle between the magnetizations of two layers) at the same frequency than that of the V rf. The mixing of oscillatory TMR and small rf tunnel current generates finite dc voltage (called as rectified voltage), which is measured by a multimeter connected to the dc port of the bias tee. To remove the background not coming from FMR, we measured reference spectra at μ 0 H = 300 mT correspond to each rf power (P rf) and subtracted from spectrum corresponds to each bias magnetic field and P rf. At μ 0 H = 300 mT, the magnetizations of free layer and reference layer are expected to be parallel to each other. Therefore precession of free layer magnetization does not produce any time-dependent TMR.

### Micromagnetic Simulations

The micromagnetic simulations were performed by using Object Oriented Micromagnetic Framework (OOMMF) software based on Landau-Lifshitz-Gilbert equation of motion38 given by:

$$\frac{{\rm{d}}{\bf{M}}}{{\rm{d}}t}=-\,\gamma {\bf{M}}\times {{\bf{H}}}_{{\rm{eff}}}+\alpha {\bf{M}}\times \frac{{\rm{d}}{\bf{M}}}{{\rm{d}}t}\cdot$$

The STT term is neglected because of its small contribution in our MTJ. In this equation, M is saturation magnetization, γ is the gyromagnetic ratio, α is the Gilbert damping constant and H eff is the effective magnetic field composed of bias magnetic field, PMA field and demagnetizing field. For the simulations, a model sample with dimension 10 µm (x) × 10 µm (y) × 1.4 nm (z) was considered (Fig. 5(a)). The mess size was taken as 20 nm × 20 nm × 1.4 nm. We adopted the anisotropy energy density of 1.012 × 105 J/m3 in out-of-plane direction, saturation magnetization µ 0 M s of 1.5 T, gyromagnetic ratio of 28 GHz.T−1 and exchange stiffness constant of 20 pJ.m−1. The exchange stiffness constant was found from ref. 43, whereas other parameters were extracted from experimental results (see supplementary information). At first, the ground state of magnetization was prepared by applying a bias magnetic field along y-direction. The magnetization dynamics was then excited by applying a sinusoidal rf magnetic field (h z = h rf sin 2πft) (Eq. 4) with frequency f. As V rf modulates PMA of free CoFeB layer only in the area underneath the reference CoFeB layer, h z was applied only in the central 2 × 4 µm2 area of CoFeB layer marked by green colour in Fig. 5(a). The magnitudes of h z i.e. h rf were chosen equivalent to the PMA field modulated by applied V rf as calculated from experimental result shown in Fig. 2(b) and Eq. 4. The time varying average magnetization components were also extracted from the green area to mimic the experimental condition.

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## Acknowledgements

We would like to thank Dr. Olivier Rousseau and Dr. Susumu Ogawa for fruitful discussions.

## Author information

Authors

### Contributions

Y.O. and H.T. supervised the study. B.R. and K.M. fabricated devices. B.R. performed measurements and numerical simulations. B.R. analyzed data and wrote the manuscript with the help of Y.F. All the authors discussed the results and commented to improve the quality of manuscript.

### Corresponding author

Correspondence to YoshiChika Otani.

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### Competing Interests

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Rana, B., Fukuma, Y., Miura, K. et al. Effect of excitation power on voltage induced local magnetization dynamics in an ultrathin CoFeB film. Sci Rep 7, 2318 (2017). https://doi.org/10.1038/s41598-017-02427-3

• Accepted:

• Published:

• ### Dynamic electromagnonic crystal based on artificial multiferroic heterostructure

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• , Andrei V. Drozdovskii
• , Andrey A. Nikitin
• , Alexander A. Semenov
• , Dmytro A. Bozhko
• , Alexander A. Serga
• , Burkard Hillebrands
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• ### Towards magnonic devices based on voltage-controlled magnetic anisotropy

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• ### Experimental Demonstration of a Spin Logic Device with Deterministic and Stochastic Mode of Operation

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Equivariant Analytic Localization of Group Representations

Laura Smithies Kent State University, Kent, OH
Available Formats:
Electronic ISBN: 978-1-4704-0321-8
Product Code: MEMO/153/728.E
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AMS Member Price: $32.40 Click above image for expanded view Equivariant Analytic Localization of Group Representations Laura Smithies Kent State University, Kent, OH Available Formats:  Electronic ISBN: 978-1-4704-0321-8 Product Code: MEMO/153/728.E  List Price:$54.00 MAA Member Price: $48.60 AMS Member Price:$32.40
• Book Details

Memoirs of the American Mathematical Society
Volume: 1532001; 90 pp
MSC: Primary 22; 18;

The problem of producing geometric constructions of the linear representations of a real connected semisimple Lie group with finite center, $G_0$, has been of great interest to representation theorists for many years now. A classical construction of this type is the Borel-Weil theorem, which exhibits each finite dimensional irreducible representation of $G_0$ as the space of global sections of a certain line bundle on the flag variety $X$ of the complexified Lie algebra $\mathfrak g$ of $G_0$.

In 1990, Henryk Hecht and Joseph Taylor introduced a technique called analytic localization which vastly generalized the Borel-Weil theorem. Their method is similar in spirit to Beilinson and Bernstein's algebraic localization method, but it applies to $G_0$ representations themselves, instead of to their underlying Harish-Chandra modules. For technical reasons, the equivalence of categories implied by the analytic localization method is not as strong as it could be.

In this paper, a refinement of the Hecht-Taylor method, called equivariant analytic localization, is developed. The technical advantages that equivariant analytic localization has over (non-equivariant) analytic localization are discussed and applications are indicated.

Graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

• Chapters
• Introduction
• 1. Preliminaries
• 2. The category $\mathcal {T}$
• 3. Two equivalences of categories
• 4. The category $D^b_{G_0}(\mathcal {D}_X)$
• 5. Descended structures
• 6. The category $D^b_{G_0}(\mathcal {U}_0(\mathfrak {g}))$
• 7. Localization
• 8. Our main equivalence of categories
• 9. Equivalence for any regular weight $\lambda$
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Volume: 1532001; 90 pp
MSC: Primary 22; 18;

The problem of producing geometric constructions of the linear representations of a real connected semisimple Lie group with finite center, $G_0$, has been of great interest to representation theorists for many years now. A classical construction of this type is the Borel-Weil theorem, which exhibits each finite dimensional irreducible representation of $G_0$ as the space of global sections of a certain line bundle on the flag variety $X$ of the complexified Lie algebra $\mathfrak g$ of $G_0$.

In 1990, Henryk Hecht and Joseph Taylor introduced a technique called analytic localization which vastly generalized the Borel-Weil theorem. Their method is similar in spirit to Beilinson and Bernstein's algebraic localization method, but it applies to $G_0$ representations themselves, instead of to their underlying Harish-Chandra modules. For technical reasons, the equivalence of categories implied by the analytic localization method is not as strong as it could be.

In this paper, a refinement of the Hecht-Taylor method, called equivariant analytic localization, is developed. The technical advantages that equivariant analytic localization has over (non-equivariant) analytic localization are discussed and applications are indicated.

Graduate students and research mathematicians interested in topological groups, Lie groups, category theory, and homological algebra.

• Chapters
• Introduction
• 1. Preliminaries
• 2. The category $\mathcal {T}$
• 3. Two equivalences of categories
• 4. The category $D^b_{G_0}(\mathcal {D}_X)$
• 5. Descended structures
• 6. The category $D^b_{G_0}(\mathcal {U}_0(\mathfrak {g}))$
• 7. Localization
• 8. Our main equivalence of categories
• 9. Equivalence for any regular weight $\lambda$
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## Friday, March 17, 2017

### Kryptos - The Cipher (Part 2)

This is Part 2 about Kryptos and the first post can be found here. In this post i will focus more on
speculations, brainstorming and solutions attempts.

One noticeable fact is, that the letters KRYPTOS somehow are involved in the decryption process of all previous ciphers. In K1 and K2 they were directly used as one of the keywords in the Vigenère-Variant Quagmire3. For K3 there are several ways to transpose the ciphertext in order to reveal the plaintext, but one of them has to do with ordering/reordering the letters of KRYPTOS alphabetically. So, it seems plausible, that also in K4 these letters play a role in one or the other way. A further hint towards this is, that the letters of KRYPTOS all appear on the right side or in direct neighborhood on the left side, as marked below:

$\small{\texttt{25| E C D M R I P F E I M E H N L S S T T R T V D O H W ? }}$$\small{\texttt{ O B }}$$\small{\texttt{ K R }}$
$\small{\texttt{26| U O X O G H U L B S O L I F B B W F L R V Q Q P R N G K S }}$$\small{\texttt{ S O }} \small{\texttt{27| }}$$\small{\texttt{ T }}$$\small{\texttt{ W T Q S J Q S S E K Z Z W A T J K L U D I A W I N F B N }}$$\small{\texttt{ Y P }}$
$\small{\texttt{28| V T T M Z F P K W G D K Z X T J C D I G K U H U A U E K C A R }}$

The position of the seven letters for KRYPTOS are all touching each other. This seems too strange to be by chance and i am not the first who mentioned this [1].

Whats is the correct representation to start with? The ciphertext has 97 characters, this is almost $7\times 14$ and, as just explained, $7$ is an important number, since it is also the number of characters in KRYPTOS. So if we arrange the ciphertext in a rectangle of this size $7\times 14$, we get:

$\small{\texttt{01| O B K R U O}}$
$\small{\texttt{02|X O G H U L B}}$
$\small{\texttt{03|S O L I F }}$$\small{\texttt{ B B }} \small{\texttt{04|W F L R V }}$$\small{\texttt{ Q Q }}$
$\small{\texttt{05|P R N G K }}$$\small{\texttt{ S S }} \small{\texttt{06|O T W T Q S J}} \small{\texttt{07|Q }}$$\small{\texttt{ S S }}$$\small{\texttt{ E K }}$$\small{\texttt{ Z Z }}$
$\small{\texttt{08|W A T J K L U}}$
$\small{\texttt{09|D I A W I N F}}$
$\small{\texttt{10|B N Y P V }}$$\small{\texttt{ T T }} \small{\texttt{11|M Z F P K W G}} \small{\texttt{12|D K Z X T J C}} \small{\texttt{13|D I G K U H U}} \small{\texttt{14|A U E K C A R }} Since the ciphertext has one letter less, we skipped the first position to create a tiny breach in the upper left-hand corner 😉. In this case i marked all double letters that occur in this representation. All but one are in the last two columns. Again, i am not the first to mention this [1]. So, what does it mean? The probability that this happens by chance is low (i think somewhere between ( if 0.01% and 0.1%) ). Furthermore, his two intentional errors from K1 and K2 have something to do with double letters: I L L U S I O N → I Q L U S I O N and vice versa U N D E R G R O U N D → U N D E R G R U U N D Ed Scheidt said [2] that he used some kind of masking technique to hide the properties of the english language and to increase the difficulty of cryptanalysis. Has the masking something to do with this double letters? The goal of masking is to alter the frequency distribution of the characters to hide the language based properties. But the Vigenère-Method itself can be seen already as a kind of masking method, since the character distribution changes after encryption in contrast to a simple substitution or transposition cipher. Jim Gillogly, the first solver of the ciphers K1 - K3 speculates the following about K4 [3]: "I suspect it's running key, or combined polyalphabetic sub and transposition, or perhaps autokey. The only likely periodicity appears to be at period 25, but that may well just be chance" Jim Gillogly A Running Key Cipher, especially with a strong key is almost perfectly secure. If the key is obtained from a really random source, then this gives a One-Time-Pad, a provable secure cipher. The methods Sanborn used for K1-K3 were extremely straight forward. Nothing special, no new variations. Solvable in a matter of seconds with modern computers and software. One easy way to prevent the rest of the world to attack the cipher K4 directly is to permute the ciphertext, such that even the correct decryption process and key yields garbage. However, this contradicts the clues that the consecutive characters 'N Y P V T T M Z F P K' translate to 'B E R L I N C L O C K'. ┄┄┄┄ Or is this argument wrong? Does Sanborn really say that these 11 characters decrypt one-to-one to Berlin Clock or does the clue only refer to the positions, i.e., 64,65,66,67,68,69? There are even two remarks that can be made about the ciphertext characters, that make up the clue, i.e., NYPVTTMZFPK. They contain: 1. the two characters 'Y', 'P' from the block of characters that contain all KRYPTOS letters 2. the only two the letters that only appear once in the ciphertext: 'M' and 'Y' ### ⬛ Hill Cipher Approach The Hill Cipher is a type of matrix encryption, that uses a secret n \times n matrix to decrypt n characters of the plaintext. So equal n-grams that are n places apart, give the same ciphertext. But for larger n and a tiny ciphertext, as in case of K4, information about the single characters are masked properly. Since the line 15 of the Vigenère-Table contains an extra character 'L' (which is wrong at this place), that, together with the two rows above and the row below, yields 'H I L L', it is possible, that Sanborn hints towards the usage of the Hill Cipher. To construct the secret matrix$$\textbf{S} = \begin{pmatrix} s_{0,0} & s_{0,1} & s_{0,2}\\ s_{1,0} & s_{1,1} & s_{1,2}\\ s_{2,0} & s_{2,1} & s_{2,2}\\ \end{pmatrix} $$maybe Sanborn used the intentional errors from from K1 to K3 as also mentioned in [4]. Lets take a closer look: If you e.g. pick n=3 then you encrypt via \begin{pmatrix} s_{0,0} & s_{0,1} & s_{0,2}\\ s_{1,0} & s_{1,1} & s_{1,2}\\ s_{2,0} & s_{2,1} & s_{2,2}\\ \end{pmatrix} \begin{pmatrix} p_{3i+0} \\ p_{3i+1} \\ p_{3i+2} \\ \end{pmatrix} = \begin{pmatrix} c_{3i+0} \\ c_{3i+1} \\ c_{3i+2} \\ \end{pmatrix} \pmod{26} whereof p_{3i+j} is the plaintext character at position 3i+j and c_{3i+j} is the ciphertext character at position 3i+j. In short this is$$\textbf{S}\cdot \textbf{p} = \textbf{c} \pmod{26} $$Decryption than simply is: \textbf{p} = \textbf{S}^{-1}\cdot\textbf{c} \pmod{26} This shows that it is important that \textbf{S} is invertible modulo 26, i.e.,\text{Det}(\textbf{S}) \not\equiv 0\pmod{26}. However, there is no reason not to encrypt n \times n characters at once via: \begin{pmatrix} s_{0,0} & s_{0,1} & s_{0,2}\\ s_{1,0} & s_{1,1} & s_{1,2}\\ s_{2,0} & s_{2,1} & s_{2,2}\\ \end{pmatrix} \begin{pmatrix} p_{3i+0} & p_{3i+3} & p_{3i+6}\\ p_{3i+1} & p_{3i+4} & p_{3i+7}\\ p_{3i+2} & p_{3i+5} & p_{3i+8}\\ \end{pmatrix} = \begin{pmatrix} c_{3i+0} & c_{3i+3} & c_{3i+6}\\ c_{3i+1} & c_{3i+4} & c_{3i+7}\\ c_{3i+2} & c_{3i+5} & c_{3i+8}\\ \end{pmatrix} \pmod{26} In short this is$$\textbf{S}\cdot \textbf{P} = \textbf{C} \pmod{26} $$Since the determinant is multiplicative, it must hold$$ \text{Det}(\textbf{S})\text{Det}(\textbf{P}) = \text{Det}(\textbf{C}) \pmod{26} $$Lets make use of our clues BERLIN and CLOCK. Accidentally (?), this 11 characters block starts at a position that is one after a multiple of 9 (64 = 9\cdot 7+1). Hence, if a 3\times 3 Hill Cipher was used, we can start here with a new 3\times 3 block. If \alpha is the function that assigns each character a unique integer then we have \begin{pmatrix} s_{0,0} & s_{0,1} & s_{0,2}\\ s_{1,0} & s_{1,1} & s_{1,2}\\ s_{2,0} & s_{2,1} & s_{2,2}\\ \end{pmatrix} \begin{pmatrix} \alpha(\text{B}) & \alpha(\text{L}) & \alpha(\text{C})\\ \alpha(\text{E}) & \alpha(\text{I}) & \alpha(\text{L})\\ \alpha(\text{R}) & \alpha(\text{N}) & \alpha(\text{O})\\ \end{pmatrix} = \begin{pmatrix} \alpha(\text{N}) & \alpha(\text{V}) & \alpha(\text{M})\\ \alpha(\text{Y}) & \alpha(\text{T}) & \alpha(\text{Z})\\ \alpha(\text{P}) & \alpha(\text{T}) & \alpha(\text{F})\\ \end{pmatrix}\pmod{26} If this is solvable, depends on the \alpha-function. If one picks the canonical choice$$\alpha(A) = 0, \alpha(B) = 1, ..., \alpha(Z) = 25$$then it must hold for the known plaintext revealed by Sanborn's clue: \text{Det}(\textbf{S}) \text{Det} \begin{pmatrix} 1 & 11 &2 \\ 4 & 8 & 11 \\ 17 &13 &14 \end{pmatrix} = \text{Det} \begin{pmatrix} 13 & 21& 12 \\ 24 & 19& 25 \\ 15&19& 5 \end{pmatrix}\pmod{26} but it is$$ \text{Det}(\textbf{S})\cdot 20 = 23 \pmod{26} $$which is not solvable, since \text{gcd}(20,26) \nmid 23. One could change the \alpha function, e.g., to$$\alpha(A) = 1, \alpha(B) = 2, ..., \alpha(Z) = 26$$and than has to check if \text{Det}(\textbf{S}) \text{Det} \begin{pmatrix} 2 & 12 & 3 \\ 5 & 9 & 12 \\ 18 &14 &15 \end{pmatrix} = \text{Det} \begin{pmatrix} 14& 22& 13\\ 25& 20& 0\\ 16&20& 6 \end{pmatrix}\pmod{26} This time it is$$ \text{Det}(\textbf{S})\cdot 24 = 18\pmod{26} $$which is solvable for \text{Det}(\textbf{S}) \pmod{26} \in \{4,17\}. We compute the inverse of the secret matrix S modulo 13:$$ \textbf{P}^{-1} = \begin{pmatrix} 2 & 12 & 3 \\ 5 & 9 & 12 \\ 18 &14 &15 \end{pmatrix}^{-1}= \begin{pmatrix} 10 & 4 & 0 \\ 1 & 12 & 11 \\ 7 &10 &8 \end{pmatrix} \pmod{13} $$hence$$ \textbf{S} = \begin{pmatrix} 14& 22& 13\\ 25& 20& 0\\ 16&20& 6 \end{pmatrix} \begin{pmatrix} 10 & 4 & 0 \\ 1 & 12 & 11 \\ 7 &10 &8 \end{pmatrix} = \begin{pmatrix} 6 & 8 & 8 \\ 10 & 2 & 12 \\ 1 &0 &8 \end{pmatrix} \pmod{13} $$However, the extension of the matrix S to the integer ring modulo 26 shows that there is no solution to the equation (by testing all 2^9 possibilities to add 13 to the entries). Maybe the approach is correct, but the alphabet, i.e., the alpha function is wrong and there are way to much possible alpha functions to test. In [4] they also tested the alphabet:$$ \texttt{KRYPTOSABCDEFGHIJLMNQUVWXZ} $$with no success. I tested several further alphabets, e.g.,$$ \texttt{KOMITEABCDFGHJLNPQRSUVWXYZ, SANBORCDEFGHIJKLMPQTUVWXYZ}$$,$$ \texttt{JIMSANBORCDEFGHKLPQTUVWXYZ, CIPHERABDFGJKLMNOQSTUVWXYZ}$$and indeed some gave me a secret matrix \textbf{S}, however, none of them contains only integers from the intentional errors nor did they reveal any further english plaintext words when applied to the further ciphertext. However, one does not need to bruteforce the whole secret matrix \textbf{S} with possible 26^9 = 5.429.503.678.976 combinations, but rather can only (and has to do anyway) bruteforce the alphabet's scrambling keyword and test some usual choices for the alpha-function. And if a solutions is possible, the secret matrix \textbf{S} can be computed efficiently. ### ⬛ Running Key Approach As written in the in the beginning, Jim Gillogly suggests, that it could be a Running Key Cipher. Also, if you e.g., use the free tool Cryptocrack and let it analyze the ciphertext in order to determine the used cipher method with the most likely score, it outputs on position one the Running Key Cipher with a fairly large gap to the second candidate (Quagmire 2). It is reasonable, that Ed Scheidt suggested to use the Running Key Cipher for K4 in order to make it the hardest crypto puzzle of the four ciphers. If this is true, then in order to solve K4 you need: 1. The Running Key, i.e., a text which has the same length as K4, which Sanborn used as the key to encrypt the plaintext 2. The actual encryption method. The Running Key cipher can be executed with different methods, that define how to encrypt a plaintext letter at position i with the running key letter at position i. You can use a Vigenère-Table, a Beaufort approach and nowadays the bitwise XOR method is common. To find the key, you can for instance search the following sources: Running Key source: Book(s) If you assume, that Sanborn uses the Vigenère-Cipher with the standard TABULA RECTA for the actual encryption, then you get that the key for the revealed plaintext part 'BERLINCLOCK' is 'MUYKLGKOR'.This does not seem to be a part of an english book or text. When you change the Vigenère-Tableau to, e.g., the one on the statue, the key becomes 'SYMLWRHZN'. Again, no english words or fragments. Only if you find a Vigenère-Tableau or a different method, where the key is an english word or parts of english words, it is probably worthwhile to start searching for an appropriate book. Running Key source: K1-K3 It is totally reasonable, that Sanborn used the previous ciphers K1-K3 as the source for the Running Key. However, if this would be the case, it think K4 would have been cracked already years ago, since this is too straightforward. Maybe he used some arithmetic progressions or non-linear functions on the texts, to permute the key among the characters. Remark. This reminds me a little bit of the solution attempt from Scott, who revealed BERLIN at the correct position 64-69 by using as his first step, the first 11 characters of K1 as the Quagmire3 key. ▚ Running Key source: PRNG - Wichmann-Hill. A perfect source for a Running Key Cipher is a pseudo random number generator (PRNG). It creates a sequence of random looking integers, that could be reconstructed deterministically if you know the initial seeds of the generator. The one PRNG that seemed interesting to me, is the Wichmann-Hill pseudo number generator, which uses 3 integers as seeds. There are a lot of PRNGs, but i decided to pick this one first, because: 1. It was developed 1982, hence was known to the time KRYPTOS was created 2. It contains the word "Hill" in its name, which may be related to the 'HILL' in the last column (so roughly the same argument as to test the Hill Cipher) 3. Uses three integers as the input seed, which may be related to the previous errors of K1-K3 or maybe the gps coordinates from K2 According to the original definition, the three seeds are integers between up to 30.000 and the generator outputs integers between 0 and 1 rounded to 6 decimals places. The length of the sequence (i.e. its cycle length) is 6.953.607.871.644. Since i can not test all 30.000^3 = 27.000.000.000.000 possible seeds up to the named cycle length, i tested only the tiny, but perhaps reasonable range:$$ 0 \leq s1,s2,s3 \leq 50$$and i assumed that the$(64\pm 1)$th up to the$(69 \pm 1)\$th output was used to encrypt BERLINCLOCK. For the method of encryption is used again Vigenère with different keywords and alphabets. And, has you probably guess, none of the random characters obtained from the PRNG could be used to decrypt the clue correctly. However, it think it is another approach, which is perhaps worth to check a little bit more.

Go to Kryptos - The Cipher (Part 3)

[1] Elonka Dunin, Slides - Def Con 12, Jul. 14 - Aug. 01, 2004
[2] http://kryptools.com/hints.htm
[3] http://www.elonka.com/kryptos/
[4] Craig Bauer, Gregory Link & Dante Molle, James Sanborn’s Kryptos and the matrix encryption conjecture, pages: 541-552, Cryptologica, published online: 27 Apr 2016

1. This is very nice, good job. I'm leaning towards a Hill Cipher myself, thinking mod25? removing both ll's from the line in which the additional L is on the Viginere Table.

2. I like the mathematical approach.

*Sanborn has stated that N=B, Y=E, P=R, etc.
*Kryptos was designed to be solved with pen and paper. Additionally, Sanborn has admitted to not being very mathematically inclined. This suggests that the techniques that he used are not complex mathematical operations.

Some observations:
* The ciphertext corresponding to letters that appear twice, C and L differ by the distances from each other.
* BER and CK can be derived by adding (mod26) the three previous cyphertext characters, i.e., NFB+NYP=BER, MZF+PKW+CKC.
Unfortunately, these don't work globally, but might hint at a solution.

1. Greetings. Please provide link where Sanborn said N=B, Y=E and so on as I cannot find. In the article he says those positions equate to BERLINCLOCK. Secondly, the whole point of a cipher in the field, is such that people who are not mathematically inclined can still encrypt a message without understanding the underlying algorithm. Doesn't take a genius to implement an advanced ultra secure cipher on paper. I tend to agree with you that there aren't any earth shattering math operations in the encryption stage, but I'd give Sanborn a little more credit. He's pretty bright.

2. In my third post i also mentioned this, which i got from here: https://kryptosfan.wordpress.com/tag/jim-sanborn/


text:
# Direct electric field control of the skyrmion phase in a magnetoelectric insulator

## Abstract

Magnetic skyrmions are topologically protected spin-whirls currently considered as promising for use in ultra-dense memory devices. Towards achieving this goal, exploration of the skyrmion phase response and under external stimuli is urgently required. Here we show experimentally, and explain theoretically, that in the magnetoelectric insulator Cu2OSeO3 the skyrmion phase can expand and shrink significantly depending on the polarity of a moderate applied electric field (few V/μm). The theory we develop incorporates fluctuations around the mean-field that clarifies precisely how the electric field provides direct control over the free energy difference between the skyrmion and the surrounding conical phase. The quantitative agreement between theory and experiment provides a solid foundation for the development of skyrmionic applications based on magnetoelectric coupling.

## Introduction

To realise skyrmion-based applications (skyrmionics)1,2,3,4,5, research into the creation, control and stabilisation of skyrmions is in an active phase1,6,7,8,9,10,11,12,13. In this context, it could seem problematic that in bulk materials the skyrmion phase is stable only for a narrow interval at finite temperature (T) just below the magnetic ordering temperature TC, and under an applied magnetic field (H)6,14,15,16,17. In Cu2OSeO3 for example, the skyrmion phase spreads down in T by just 3.5% of TC, occupying no more than 1% of the total ordered phase space that is otherwise dominated by topologically trivial helical or conical phases7,17,18. This limited skyrmion phase space is observed also in other known bulk hosts of skyrmions14,15,16,19. On the other hand, the finite extent of the skyrmion phase pocket can be considered to present an interesting advantage, since relatively small perturbations can dramatically influence the skyrmion phase stability. It follows that the ability to enhance or suppress the skyrmion phase space in a sample can provide a flexible platform for the respective creation or destruction of skyrmions, a process that can be technologically useful. Here we describe a simple and reliable mechanism for the stabilisation and destabilisation of the skyrmion lattice (SkL) phase by exploiting electric (E) fields applied to the insulating magnetoelectric material Cu2OSeO3.

To date, several approaches for skyrmion manipulation have been demonstrated that make use of either moderate electric currents, electric fields, or thermal gradients6,8,9,10,11,12,13,20,21,22,23. In addition, progress towards tuning the bulk skyrmion phase stability was also demonstrated using both applied uniaxial24,25 and hydrostatic pressure18. For possible applications however, the use of E fields to manipulate skyrmions and skyrmion phase stability in insulators offers several potential advantages for applications, yet this approach remains still little-explored experimentally6,8,23,26. Moreover, the E field control of the SkL phase stability remains an outstanding theoretical issue.

Here we report a combined experimental and theoretical study of skyrmion phase stability under moderate E-fields (V/μm) in the model insulating skyrmion host Cu2OSeO3. We use the microscopic probe of small-angle neutron scattering (SANS) to show that in Cu2OSeO3, the extent of the skyrmion phase stability either expands or shrinks in both T and H, dependent on the E field polarity. Theoretically, we address the role of the E field using first order perturbation theory to evaluate the free energy of the underlying phases. We show how an applied E field causes a relatively small shift of the SkL free energy that is nonetheless commensurate with the mean-field free energy difference between the competing skyrmion and conical phases, and thus dramatically controls the SkL phase stability. For the quantitative description of the experimental phase diagram, we develop a new approach for treating the fluctuative part of the free energy contributed by quasiparticle modes around TC. The inclusion of these modes proves pivotal for the correct evaluation of the free energy difference between the competing skyrmion and conical phases, both with and without E field, and thus represents an improved approach more generally for the calculation of the skyrmion phase diagram.

## Results

### Controlling skyrmion phase stability using electric fields

From recent bulk susceptibility χ(H) measurements of Cu2OSeO323, it was suggested that skyrmions may be “created” or “annihilated” by applying a dc E-field in suitable parts of the temperature–magnetic field (T,H) phase diagram. In that study23 the skyrmion phase is identified by a small drop in χ(H), which serves as an indirect indication for the existence of the skyrmion phase.

We have used the tool of SANS to observe directly the microscopic skyrmionic magnetism in Cu2OSeO3 and its response to an applied dc E-field. In SANS the SkL phase is typically evidenced by a sixfold symmetric diffraction pattern, consistent with the so-called multi-q (triple-q) magnetic structure ansatz for the SkL described by three propagation q-vectors rotated by 120° with respect to each other (note that both ±q each give a SANS diffraction spot)6,14,27. In our SANS experiments we oriented the sample so that E||H|| [111], since according to previous bulk measurements7, the effect of the E field is expected to be maximal in this geometry. By measuring with the neutron beam also along [111], scattered intensity is only observed due to the SkL phase; for both the conical phase with q||H and the zero field helical phase with q||{100} type directions6, the expected scattering lies well out of the detector plane.

Figure 1 summarizes the direct E-field control of the SkL phase stability in Cu2OSeO3. Representative SANS data collected at constant T = 56.8 K, and at (H,E) coordinates selected to emphasise the E field effect, are shown in Fig. 1a–d. The initial states at T = 56.8 K were always prepared after zero-field cooling. By applying H = 17 mT at E = 0, the system the system is located on the border of the SkL phase and mostly in the conical phase, as evidenced by a weak SkL signal on the SANS detector (Fig. 1a). Without changing T or H, applying a positive E field of +5.0 V/μm leads to the appearance of the characteristic 6-fold diffraction pattern in the SANS image (Fig. 1b), demonstrating the creation of a well-developed SkL. Conversely, starting from the SkL state at H = 0 38 mT and E = 0 (Fig. 1c), application of a negative electric field E = −2.5 V/μm erases the SkL as evidenced by the disappearance of the SANS diffraction peaks (Fig. 1d).

Since in our experimental geometry it is only scattering from the SkL that contributes to the observed SANS intensity, we determined the extent of the SkL phase by evaluating the total intensity within the region of interest on the detector (the annular range within the white rings shown in Fig. 1a–d) as a function of T, H, and E. The resulting stability range of the skyrmion phase is shown in Fig. 1e for zero (green), positive (red) and negative (blue) E fields. We find that a positive E field of +5.0 V/μm expands the skyrmion pocket so that it becomes twice larger (in temperature), while a negative E field of just E = −2.5 V/μm shrinks the pocket by a factor of two. These SANS results provide microscopic experimental evidence for the electric field control of the skyrmion phase. Moreover, the results are quantitatively consistent with those of the indirect measurements reported in ref.23 after rescaling the data to take into account the difference between the ranges of applied electric field explored in the two studies.

### Optimum conditions for stabilising and destabilising the skyrmion phase

Examining our SANS data more closely reveals the systematic manner by which applied E fields modify the SkL stability in the phase diagram. Figure 2a,b, show H-scans of the total scattered SANS intensity from the SkL for various E fields, and at T = 55.8 K and 56.8 K, respectively. At each T, the extent in H over which the SkL intensity is observed at E = 0 becomes enhanced under positive E fields, and suppressed under negative E fields. This demonstrates clearly the importance of the E field polarity on either enhancing or suppressing SkL stability. The effect is very pronounced as seen in Fig. 2a; at T = 55.8 K a negative E field can completely destabilise the SkL that is otherwise stable in the unperturbed state (E = 0).

Since the total scattered SANS intensity is indicative of both the population (relative to the conical phase) and quality of SkLs in the sample, the maximum intensity in the H scans like those shown in Fig. 2a,b is identified to represent the optimal SkL stability at each E and T. We present the T-dependence of this intensity peak at each E field in Fig. 2c, along with T windows identified as optimal for either enhancing or suppressing the SkL stability using the E field. Within the T window of 53–55.5 K, positive E fields enhance the SkL phase stability relative to the case at E = 0, while in the T-window of 55–56.5 K the SkL phase stability is readily suppressed by the negative E field relative to the E = 0 case. Positioning a sample at 55.5 K allows the demonstration of either a significant enhancement or suppression of the skyrmion phase at E = 0 by using E-fields of opposite polarity. In what follows, we develop a theory capable of explaining these observations quantitatively.

### Free energy in electric fields

The underlying mechanism for the E field driven enhancement or suppression of the skyrmion phase stability relies upon the magnetoelectric (ME) coupling in insulating Cu2OSeO3, which microscopically originates from the d-p hybridisation mechanism27,28,29,30,31. The emergent electric dipole moment P = λ(SySz, SzSx, SxSy) is generated by the underlying spin structure S(r) = (Sx, Sy, Sz), with the coupling between the magnetic and electric degrees of freedom described by a ME coupling parameter λ of relativistically small magnitude. Crucially, because the skyrmion phase now carries a non-vanishing electric-dipole moment, the ME coupling results in a P·E shift of the skyrmion free energy in E-field. This perturbation renormalises the elementary helices upon which the skyrmion phase is built, and slightly distorts the SkL6,32.

In this work, we apply the ME perturbation to the free energy described by an effective Ginsburg-Landau functional with Dzyaloshinski-Moriya interaction (DMI), and consider carefully the critical fluctuations that favour the skyrmion phase with respect to the competing conical phase (see Methods). Due to the relativistically small size of λ, the dimensionless E-field itself is rather small so that, $${\rm{\ae }}=\lambda E\mathrm{/4}D{k}_{0}\ll 1$$, and we can build a perturbation theory in $${\rm{\ae }}$$; for the modified free energy, neglecting all the terms of order $${{\rm{\ae }}}^{{\rm{2}}}$$ and higher. Our finding is that perturbations of the fluctuative terms are important only at second order, while the mean-field energy already shifts in the first order due to direct ME and nonlinear contributions (see Methods). The corresponding shift in free energy of the skyrmion phase depends on the direction of E-field (see Fig. 2d–f), thus either enhancing (E > 0) or suppressing (E < 0) the skyrmion phase stability. While at first sight it can be surprising that the perturbation due to only a moderate applied E-field can play such a crucial role here, this is facilitated by the very close competition between the skyrmion and conical phases already in the mean-field.

### Calculation of the phase diagram

To calculate the response of the phase diagram to applied E-field, we use a new approach developed on the basis of effective models presented in refs6,14,33 (see Methods). Compared with these earlier studies, the new approach is self-consistent in the way that it reproduces the phase diagram, provides a deeper understanding of the role of quasiparticle modes near TC, and includes the path-integral approach presented previously for calculating the fluctuative free energy14 as a limiting case. We thus treat the first-order perturbation in E field on top of the mean-field solution, and add the fluctuative contributions that stabilise the SkL in the bulk. The main contribution of the E field here is captured by the shift of the mean-field free energy difference between the SkL and conical phases, while the fluctuative shift under voltage remains quadratically small.

Figure 2d,e each show calculations of the H and E field-dependent free energy difference between the skyrmion and conical phases, at T = 55.8 K and T = 56.8 K, respectively. In general, the calculations show the minima of the free energy difference curves deepen with increasingly positive E field. At T = 55.8 K changing between sufficiently negative and positive E fields can shift the curve so that the conical phase favoured for all H in a negative E field becomes unstable towards the skyrmion phase formation in the positive E field. This result is consistent with the experimental data shown in Fig. 2a. We also identify semi-quantitative agreement between these calculations and our experiments; the H location of the minima in the calculated free-energy difference curves [Fig. 2d,e] correspond well to the values of H where the peaks of SANS intensity are observed in the scans shown in Fig. 2a,b. In addition, we can link the calculated T and E field dependence of the free energy difference minima shown in Fig. 2f to our experimental data. The T s at which the free energy difference between the SkL and conical phases vanish for the theoretical E fields correspond to the T windows identified from the experimental data as optimal for either enhancing or suppressing SkL stability with E [Fig. 2c]. This correspondence between experiment and theory provides firm support for the validity of our theoretical treatment of the E-field effect on the SkL phase stability.

Independent of any applied E field, our theory provides a more general understanding of SkL stability on an intuitive, pictorial level: the critical fluctuations (waves) are superposed on top of the variationally minimised free energies. There are three critical modes $${\omega }_{{\bf{k}}}^{\mathrm{(0,1,2)}}$$ around the mean-field (see Methods), with $${\omega }_{{\bf{k}}}^{\mathrm{(0)}}$$ soft on the sphere $$|{\bf{k}}|={k}_{0}$$, which means that it costs very little energy to add many such fluctuations if they are coherent with the helix k0. Thus $${\omega }_{{{\bf{k}}}_{0}}^{\mathrm{(0)}}$$ is the so-called “dangerous” mode since it results in a Van-Hove-like singularity at TC and eventually breaks down the ordered phases into the disordered (paramagnetic) phase33. Below TC the symmetry-breaking can be observed using SANS by either a six-fold pattern (SkL phase) or a two-fold pattern (helical or conical phases), both circumscribed on a sphere of radius $$|{\bf{k}}|={k}_{0}$$ in reciprocal space. Our calculation shows that the skyrmion phase is favoured because adding fluctuations generates more entropy in the skyrmion phase. This analysis also leads to a qualitative criterion for capturing the magnetic-field-independent breakdown of the ordered phases at TC (see Methods). Asymptotically, the main contribution of the fluctuative free energy is given in the short-scale physics, where Cu2OSeO3 is “almost” a ferromagnet, thus reproducing the result of the path-integral approach14 as a limiting case. The model described here captures the qualitative physics of the system, as exemplified by the theoretical phase diagram shown in Fig. 3. To date, a quantitative theory for the skyrmion phase diagram under electric fields has been missing.

## Discussion

In some respects, the effect of E field observed here resembles that achieved due to applied uniaxial pressure24,25 since, the SkL phase stability can be either enhanced or suppressed by appropriate selection of the uniaxial stress direction relative to the direction of H. However, integrating the pressure effect on skyrmion stability into a technological setting is very challenging. In contrast, the E field is a versatile and reliable external parameter; providing an efficient control of both the skyrmion position6,8,26 and, as we show here the stability of the phase as a whole. Since the E field controlled expansion or contraction of the skyrmion phase can occur in general for an insulating ME skyrmion host with any TC, our findings are very attractive for applications at room T; for a device layer of thickness 100 nm the skyrmion phase in a sample can be entirely destabilised (erased) or restabilised (written) with less than 1 V, a voltage compatible with modern microelectronics.

The present study further lays both theoretical and experimental foundations for fully exploring alternative H and E field configurations, not only in reciprocal-space measurements like SANS, but also by real-space imaging techniques such as cryo-Lorentz transmission electron microscopy (LTEM) on technologically-relevant, nanometrically-thin specimens. Crucial next steps experimentally concerning the skyrmion writing and erasing includes exploring directly the expected E field-driven switching hysteresis between the competing conical and skyrmion states, and the associated volatility of remnant states in zero biasing E field. In addition, learning how E field influences out-of-equilibrium and metastable skyrmion configurations in confined geometries can provide progressive insights for assessing the merits of insulating skyrmions for practical uses.

In conclusion, we have demonstrated both theoretically and experimentally the mechanism by which a moderate electric field can either enhance or suppress skyrmion phase stability in the magnetoelectric chiral magnet Cu2OSeO3. In addition, we have provided the parameters by which our theoretical approach achieves quantitative agreement with experiment, and which can be extended towards describing the effect of E field on both stable and metastable skyrmion states, these being of paramount technological importance.

## Methods

### Small-angle neutron scattering (SANS)

For the SANS experiment, we used a single crystal crystal grown using chemical vapour transport34. The crystal with a TC = 58 K was of mass 6 mg and volume 3.0 × 2.0 × 0.50 mm3 with the thinnest axis ||[111], and $$[\bar{1}\bar{1}2]$$ vertical. The sample was mounted onto a bespoke sample stick designed for applying dc E-fields35, and oriented with the orthogonal $$[\bar{1}\bar{1}2]$$ and $$[1\bar{1}0]$$ directions lying in the SANS scattering plane. For the SANS images shown in Fig. 1, the $$[\bar{1}\bar{1}2]$$ direction is aligned with the vertical axis, and the $$[1\bar{1}0]$$ direction aligned with the horizontal axis. In our experiments we achieved E-fields ranging from +5.0 kV/mm to −2.5 kV/mm. Evidence of electrical breakdown was detected for E-fields outside this range.

The sample was loaded into a horizontal field cryomagnet at the SANS-II beamline, SINQ, PSI. The magnetic field (μ0H) was applied parallel to both the [111] direction of the sample and the incident neutron beam to give the experimental geometry E||μ0H||[111]. In this geometry, the SANS signal is only detected from the skyrmion phase, which typically presents as a hexagonal scattering pattern with propagation vectors q μ0H. In this geometry, we avoid detecting any SANS signal due to either of the neighbouring helical (q ||{100}) or conical phases (q ||μ0H), since the propagation vectors of these phases lie well out of the SANS detector plane.

We used incident neutrons with a wavelength of 10.8 Å (Δλ/λ = 10%). The scattered neutrons were detected using a position-sensitive multidetector. The SANS measurements were done by rotating (‘rocking’) the sample and cryomagnet ensemble over angles that brought the various SkL diffraction spots onto the Bragg condition at the detector. Data taken at 70 K in the paramagnetic state were used for background subtraction. Before starting each μ0H-scan, the sample was initially zero field-cooled from 70 K to a target temperature, with the E-field applied when thermal equilibrium was achieved. The E-field was maintained during the μ0H-scan. At each T we define the μ0H extent of the SkL phase as that over which SANS intensity is detected. We use this criterion to extract the parametric extent of the SkL phase for (μ0H,T,E) as shown in Figs 1 and 2. See Supplemental Material for more details.

### Mean-field free energy

The effective mean-field theory is based on the coarse-grained magnetisation approach $$M({\bf{r}})={M}_{s}{\bf{S}}({\bf{r}})$$ as described in14. One starts with the mean-field approach with free energy

$$F[{\bf{M}}]=\langle {{\rm{\Theta }}}_{T}\,{{\bf{M}}}^{2}+J{(\nabla {\bf{M}})}^{2}+D\,{\bf{M}}\cdot (\nabla \times {\bf{M}})+U{{\bf{M}}}^{4}-{\bf{H}}\cdot {\bf{M}}\rangle$$
(1)

with spatial average $$\langle \mathrm{...}\rangle =\int \frac{dV}{V}\mathrm{...}$$, and $${{\rm{\Theta }}}_{T}\propto \alpha (T-{T}_{C})$$ near TC, J is the Heisenberg stiffness and D is DMI, H is the magnetic field, and the higher-order term U grants the formation of the crystalline phase14. In the mean-field, the interplay between Heisenberg and DMI energies determines the helical vector as k0 = D/2J. The long-range-ordered hexagonal skyrmion lattice is approximated as $${\bf{S}}({\bf{r}})\simeq {\bf{m}}+\mu {\sum }_{{{\bf{q}}}_{n}}{{\bf{S}}}_{{{\bf{q}}}_{n}}{e}^{i{{\bf{q}}}_{n}{\bf{r}}+i{\varphi }_{n}}+{\rm{c}}{\rm{.c}}{\rm{.}}$$, where the summation runs over the magnetocrystalline wave vectors q1 + q2 + q3 = 0. In the mean-field, the skyrmion phase is slightly gapped with respect to the conical phase, however the two are closely competing. Further details of the mean-field theory described in ref.14.

### Perturbation theory in electric fields

The magneto-electric coupling in Cu2OSeO3 is relativistically small, so the perturbation parameter is $${\rm{\ae }}=\lambda E\mathrm{/4}D{k}_{0}\ll 1$$. It is sufficient to use the first order perturbation theory on top of the non-perturbed free energy. We go to the rotated frame defined by the magnetic field direction along [111], and re-write the free energy. The first order perturbation theory gives eigenvectors:

$$\begin{array}{l}|{{\bf{S}}}_{{\bf{k}}}^{({\rm{\ae }})}\rangle =|{{\bf{S}}}_{{\bf{k}}}^{\mathrm{(0)}}\rangle +\sum _{n\ne 0}|{{\bf{S}}}_{{\bf{k}}}^{(n)}\rangle \frac{\langle {{\bf{S}}}_{{\bf{k}}}^{(n)}|{\hat{ {\mathcal H} }}^{({\rm{\ae }})}|{{\bf{S}}}_{{\bf{k}}}^{\mathrm{(0)}}\rangle }{{\varepsilon }_{{\bf{k}}}^{\mathrm{(0)}}-{\varepsilon }_{{\bf{k}}}^{(n)}}+{\mathscr{O}}({{\rm{\ae }}}^{2}),\end{array}$$
(2)

which are now the basis for constructing the distorted skyrmion lattice. For other (H, E)-field configurations, we re-do the calculations in the new rotated frames. See Supplemental Materials for further details.

### Fluctuation-induced phase stabilisation

We use a new approach, which captures as a limiting case the fluctuation free energy from14. The essential physics is captured already in Gaussian (noninteracting) fluctuations with free energy density

$$\begin{array}{l}\begin{array}{c}{F}_{{\rm{fluct}}}=\sum _{i}\sum _{{\bf{k}}}^{|{\bf{k}}| < {\rm{\Lambda }}}{\omega }_{{\bf{k}}}^{(i)}{f}_{{\bf{k}}}^{(i)}-T\,{{\rm{S}}}_{{\rm{fluct}}},\end{array}\end{array}$$
(3)

where Λ = 2π/a is the natural cut-off, $${f}_{{\bf{k}}}^{(i)}$$ is the critical modes distribution, and the entropy of Gaussian fluctuations is

$$\begin{array}{l}\begin{array}{c}{{\rm{S}}}_{{\rm{fluct}}}=\sum _{i}\sum _{{\bf{k}}}^{|{\bf{k}}| < {\rm{\Lambda }}}\{(1+{f}_{{\bf{k}}}^{(i)})\mathrm{ln}(1+{f}_{{\bf{k}}}^{(i)})-{f}_{{\bf{k}}}^{(i)}\,\mathrm{ln}\,{f}_{{\bf{k}}}^{(i)}\}\end{array}\end{array}$$
(4)

in the case of bosons. Fluctuations around mean-field are described by the generalised susceptibility $${\chi }_{ij}^{-1}({\bf{r}},{\bf{r}}\text{'})=\frac{1}{T}\frac{{\delta }^{2}F}{\delta {M}_{i}({\bf{r}})\delta {M}_{j}({\bf{r}}\text{'})}$$, giving rise to several collective modes (See Supplemental Material). On the local scale $$(k\gg J/D)$$, the chiral magnet is reminiscent of a ferromagnet, so the modes behave asymptotically ωk k2 for large k, thus asymptotically $${F}_{{\rm{fluct}}}\simeq \,\mathrm{log}\,\beta {\omega }_{{\bf{k}}}\propto \,\mathrm{log}\,{k}^{2}$$, which covers the model of ref.14. The main contribution to (5) is given by the short length-scale (“ferromagnetic”) physics,

$$\begin{array}{l}{\rm{\Delta }}{F}_{{\rm{fluct}}}\simeq \frac{10U}{\pi Da}\langle {{\bf{S}}}_{{\rm{SkL}}}^{2}-{{\bf{S}}}_{{\rm{con}}}^{2}\rangle T,\end{array}$$
(5)

The electric field also slightly affects the fluctuative energy, because it modifies the correlation length near TC and so renormalises Jeff, which is neglected here as a higher-order $$({{\rm{\ae }}}^{2})$$ effect. See [Supplemental Material] for further details.

### Parameters of the effective model

For our numerical calculations we use TC = 58 K, which approximately sets the Heisenberg stiffness as J = 4.85 × 10−23 Jm/A2. From the SANS measurement we establish directly the modulation period of 60 nm, which estimatively differs by a few percents from the mean-field value 2π/k0, because the mean-field ordering vector k0 = D/2J is slightly renormalised by the fluctuations near TC. This sets the “bare” DM interaction entering (1) as D = −9.85 × 10−15 J/A2. The lattice parameter is a = 8.91 × 10−10 m, which gives the natural cutoff Λ = 2π/a ≈ 70k0. The saturation magnetization in Cu2OSeO3 is Ms = 1.11 × 105 A/m and scales with temperature as $${M}_{s}(T)={M}_{s}{(1-{(T/{T}_{C})}^{{\alpha }_{1}})}^{{\alpha }_{2}}$$, with α1 = 1.95 and α2 = 0.39336. We choose the nonlinear coupling responsible for SkL formation U = 6.2 × 10−6 Jm−1 A−2 and Landau parameter $${\alpha }_{T}={\theta }_{T}/J{k}_{0}^{2}(T-{T}_{C}^{{\rm{m}}{\rm{.f}}{\rm{.}}})=3.5\,{{\rm{K}}}^{-1}$$. For the qualitative phase diagram shown in Fig. 3, we use a symmetric-response model $$({\text{ae}}^{1})$$, for which the best fit to SANS data is for $${\text{ae}}^{1}=0.02$$, which corresponds here to E = ±5 × 106 V/m coupled with λ/Dk0 = 9.23 × 10−3μm/V to the underlying spin structure through ME mechanism.

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## Acknowledgements

We thank Achim Rosch, Naoto Nagaosa and Jiadong Zang for fruitful discussions. The work was supported by the Swiss National Science Foundation, its Sinergia networks “NanoSkyrmionics” and “Mott Physics Beyond the Heisenberg Model (MPBH)”, and projects 153451, 166298 and P2ELP2_175278. Neutron scattering experiments were carried out at the Swiss Spallation Neutron Source (SINQ), Paul Scherrer Institut, Switzerland.

## Author information

Authors

### Contributions

A.J.K. performed calculations, J.S.W. performed SANS measurements, H.M.R. planned the project. A.J.K., J.S.W., H.M.R. interpreted results. A.M. provided crystals for SANS, M.B. and I.Z. provided technical support. A.J.K., J.S.W., H.M.R. wrote the manuscript.

### Corresponding author

Correspondence to A. J. Kruchkov.

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### Competing Interests

The authors declare no competing interests.

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Reprints and Permissions

Kruchkov, A.J., White, J.S., Bartkowiak, M. et al. Direct electric field control of the skyrmion phase in a magnetoelectric insulator. Sci Rep 8, 10466 (2018). https://doi.org/10.1038/s41598-018-27882-4

• Accepted:

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