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The Meyer-Neldel (MN) rule, also known as entropy-enthalpy compensation, is a peculiar phenomenon that has been reported across diverse fields of the natural sciences, in semiconductors, chemical reactions, biology, etc [1]. In magnetism, it was very recently observed in the decay of the skyrmion lattice [2]. Consider a family of thermally activated processes whose rates k obey an Arrhenius-type law, i.e., k = f0 exp(-βΔE), in which ΔE is the internal energy barrier, β = (kBT)-1 is Boltzmann&#39;s factor, and f0 is the Arrhenius prefactor — the so-called &quot;attempt frequency&quot;. If the Meyer-Neldel rule applies, the prefactor scales like an exponential of the energy barrier. In this work, we compute the mean waiting time between thermally activated magnetization reversals in a nanodisk, with parameters that resemble that of a free CoFeB layer, as used in magnetic random access memories [3]. We use two independent approaches, namely Langer&#39;s theory [4] and forward flux sampling [5]. By varying the perpendicular anisotropy, and the interfacial Dzyaloshinkii-Moriya interaction (DMI), we find that the Arrhenius prefactor can take extreme, seemingly non-physical values up to 1021 Hz, which is orders of magnitude beyond the typically assumed value of 109 Hz, and vary drastically as a function of material parameters. We show that it behaves like an exponential of the energy barrier, ΔE, which stems from a linear variation of the activation entropy, ΔS, with the energy barrier, thereby demonstrating a new case of the Meyer-Neldel rule (Fig. 1). Our results show that the Arrhenius prefactor is not an attempt frequency, because it also contains an important entropic contribution which cannot be neglected in complex systems. This suggests that modelling information retention times with a barrier-independent prefactor in magnetic storage elements is not justified [6].References: [1] A. Yelon, B. Movaghar, and R. Crandall, Reports on Progress in Physics, Vol. 69, p.1145 (2006) [2] J. Wild, T. N. Meier, S. Pollath et al., Science Advances, Vol. 3, p. e1701704 (2017) [3] J. Sampaio, A. Khvalkovskiy, M. Kuteifan et al., Applied Physics Letters, Vol. 108, p.112403 (2016) [4] J. S. Langer, Annals of Physics, Vol. 54, p. 258 (1969) [5] R. J. Allen, P. B. Warren, and P. R. ten Wolde, Physical Review Letters, Vol. 94, p. 018104 (2005) [6] L. Desplat and J.-V. Kim, submitted (2020)

MMM 2020

Entropy-Reduced Retention Times in Magnetic Memory Elements: A Case of the Meyer-Neldel Compensation Rule

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INTERMAG 2021

The Meyer-Neldel (MN) rule, also known as entropy-enthalpy compensation, is a peculiar phenomenon that has been reported across diverse fields of the natural sciences, in semiconductors, chemical reactions, biology, etc [1]. In magnetism, it was very recently observed in the decay of the skyrmion lattice [2]. Consider a family of thermally activated processes whose rates k obey an Arrhenius-type law, i.e., k = f0 exp(-βΔE), in which ΔE is the internal energy barrier, β = (kBT)-1 is Boltzmann's factor, and f0 is the Arrhenius prefactor — the so-called "attempt frequency". If the Meyer-Neldel rule applies, the prefactor scales like an exponential of the energy barrier. In this work, we compute the mean waiting time between thermally activated magnetization reversals in a nanodisk, with parameters that resemble that of a free CoFeB layer, as used in magnetic random access memories [3]. We use two independent approaches, namely Langer's theory [4] and forward flux sampling [5]. By varying the perpendicular anisotropy, and the interfacial Dzyaloshinkii-Moriya interaction (DMI), we find that the Arrhenius prefactor can take extreme, seemingly non-physical values up to 1021 Hz, which is orders of magnitude beyond the typically assumed value of 109 Hz, and vary drastically as a function of material parameters. We show that it behaves like an exponential of the energy barrier, ΔE, which stems from a linear variation of the activation entropy, ΔS, with the energy barrier, thereby demonstrating a new case of the Meyer-Neldel rule (Fig. 1). Our results show that the Arrhenius prefactor is not an attempt frequency, because it also contains an important entropic contribution which cannot be neglected in complex systems. This suggests that modelling information retention times with a barrier-independent prefactor in magnetic storage elements is not justified [6].References: [1] A. Yelon, B. Movaghar, and R. Crandall, Reports on Progress in Physics, Vol. 69, p.1145 (2006) [2] J. Wild, T. N. Meier, S. Pollath et al., Science Advances, Vol. 3, p. e1701704 (2017) [3] J. Sampaio, A. Khvalkovskiy, M. Kuteifan et al., Applied Physics Letters, Vol. 108, p.112403 (2016) [4] J. S. Langer, Annals of Physics, Vol. 54, p. 258 (1969) [5] R. J. Allen, P. B. Warren, and P. R. ten Wolde, Physical Review Letters, Vol. 94, p. 018104 (2005) [6] L. Desplat and J.-V. Kim, submitted (2020)

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In ferrimagnetic films spin-orbit torque (SOT) has been used for current-induced magnetization switching1 but also domain wall2 and skyrmion motion3. Ferrimagnets with interfacial Dzyaloshinskii-Moriya interaction (DMI) are attractive materials due to their low stray field and fast spin dynamics which can lead to ultrasmall2 and fast skyrmions. Yet, so far, the reported skyrmion velocities3 are much lower than of domain walls2. The current-induced dynamics of DMI skyrmions is intrinsically related to the charge-spin conversion efficiency. Here we investigate the DMI and SOT in CoGd ferrimagnetic thin films, in which we previously showed that we can tune the DMI in a range that allows sub-100 nm skyrmion formation4. DMI and SOT both arise from interfacial effects. It is thus important to understand the interplay between these two effects to achieve efficient and fast skyrmion motion. For this purpose, we studied Pt/CoxGd1-x/(Ir, W or Ta) films as a function of the cap layer to systematically vary the DMI and spin-Hall efficiency (SHE). We performed SOT magnetization switching experiments that we correlate to the measurements of DMI energy and SHE. Figure 1 shows an example of SOT switching phase diagram. The switching current density for different field values and cap layer is summarized in Fig. 2(a). W and Ta capping, which induce the largest net DMI, provide the lowest switching current density, thus the most efficient switching. However, Pt/CoGd/Ir requires a smaller in-plane field to achieve full magnetization switching. In the low field region, the films capped by W and Ta exhibit a metastable state [Fig. 2(b)] composed of skyrmion bubbles as suggested by micromagnetic simulations. Our results provide insight into achieving energy-efficient and fast skyrmion devices.References: 1S.-G. Je et al., APL 112, 062401 (2018) 2L. Caretta et al., Nature Nanotechnology 13, 1154 – 1160 (2018) 3S. Woo et al., Nature Communications 9, 959 (2018). 4Y. Quessab et al., Scientific Reports 10, 7447 (2020)

Interplay between spin-orbit torque and Dzyaloshinskii-Moriya interactions in thin amorphous ferrimagnetic films

Antiskyrmions and other unconventional topological excitations can potentially be supported by low-symmetry ferromagnet--heavy metal (FM-HM) thin films by the Dresselhaus-type anisotropic Dzyaloshinskii-Moriya interaction (aDMI) arising at the FM-HM interfaces. This work presents the potential of C2V Ag/Pt/Co-based films to exhibit aDMI and stabilize new chiral magnetic textures. X-ray diffraction 2θ-ω scans of Ag/Pt/Co/Pt films grown epitaxially on Si(110) single crystal substrates show that Co grows predominantly with (10.0) hexagonal closed-packed (hcp) orientation, deviating from the (110)-oriented face-centered cubic (fcc) structure of Ag and Pt. These findings were corroborated by selected area electron diffraction patterns and further supported by a strong in-plane uniaxial anisotropy measured along the hcp c-axis by alternating gradient field magnetometry. Crosstie domain walls in these films were directly observed by Lorentz transmission electron microscopy, which are also predicted from micromagnetic simulations. The materials system introduced above creates the previously unexplored situation where an in-plane easy axis is coupled with a strong (potentially anisotropic) DMI, which may lead to interesting spin configurations. However, to break the z-mirror plane [v1] and create a non-zero DMI, we fabricate thinner Ag/Pt/[Co/Ni]x/Pt films. X-ray diffraction analysis suggest that the hcp structure is preserved throughout the Ni layers. Structural and magnetic observations of this lower symmetry film stack are compared to micromagnetic simulations and discussed in the context of C2V films potentially exhibiting Dresselhaus-type aDMI and hosting antiskyrmions.References: 1. U. Güngördü et al., Physical Review B., 93, 064428 (2016) 2. M. Hoffman et al., Nature Communications., 8, 308 (2017)

In Pursuit of the Anisotropic Dzyaloshinskii-Moriya Interaction: Structural and Magnetic Characterization of Uniaxial Pt/Co-based C2V Thin Films

We investigate the effect of the distribution of pinning sites on the magnetization behavior in systems with Dzyaloshinskii–Moriya interaction (DMI). We consider the standard classical spin Hamiltonian with Heisenberg exchange, DMI, uniaxial anisotropy statistically varying across the lattice sites, and the Zeeman energy. The statistical variation of the anisotropy constants follows from a spatial Gaussian distribution with non-zero mean and standard deviation σ, and serves as a way to model the structural disorder in the sample. First, we discuss the reduction of the model to a lattice resolved mean-field theory and the development of energy minimisation algorithm for solving it. This approach allows to compute systematically and efficiently the magnetisation versus field, M(H), dependence for variable σ and temperature, and can be used for computing qualitative thermodynamic phase diagrams to explore material behaviour and to guide the computationally costly Monte-Carlo simulations. We show that as the σ increases, relative to the strength of exchange interaction and DMI, the nature of the reversal modes observed along a typical M(H) hysteresis loop changes in a certain temperature window. Namely, in ‘clean’ systems with small σ, the reversal proceeds first through the appearance of skyrmion lattices at low external fields, while in ‘dirty’ systems with large σ the reversal is through the nucleation of individual or small groups of skyrmions. We systematically quantify this effect and discuss its broader implications for applications.

The effects of disorder on hysteresis loops in chiral magnets

Threshold logic [1] efficiently implements complex logical functions within a single gate, but its widespread adoption has been impeded by insufficient nonlinearity in conventional transistors. The recent experimental demonstration of cascaded logic gates with current-driven magnetic domain walls [2] operates based on the nonlinearity intrinsic to anisotropic systems, thus naturally enabling threshold logic gates. However, these logic gates are described in [2] as reconfigurable NAND and NOR gates, rather than as three-input minority threshold logic gates. We therefore propose a novel domain wall threshold logic (DWTL) paradigm that enables circuits that are simpler and more efficient than can be achieved with the reconfigurable gates of [2]. Similar to [2], magnetic domains magnetized in the +z (-z) direction represent logic ‘1’ (‘0’), and each in-plane region causes magnetization reversal due to chiral coupling resulting from the interfacial DMI. As demonstrated in [2], DWTL gates produce an output domain that is the inversion of the majority of the inputs. That is, this logic gate performs the minority threshold function, which is significantly more logically expressive than NAND or NOR. By leveraging this minority gate behavior, the one-bit full adder of Fig. 1 can be realized with three logic gates, 80% fewer than the 15 required in [2]. We also propose DWTL functions with logical expressivity increased further by using more than three inputs, and with inputs that have non-uniform impacts on the output magnetization. As shown in Fig. 2a, a four-input DWTL gate can be realized in which one input has twice the width of the other three; therefore, this wider input will have roughly twice the impact on the output magnetization. DWTL gates with greater logical expressivity further reduce the number of devices required to perform logical operations, such as in the three-bit DWTL full adder of Fig. 2b that requires only two gates per bit (87% reduction).References: [1] D. Hampel and R. O. Winder, IEEE Spectrum, Vol. 8, pp. 32-39 (1971) [2] Z. Luo, A. Hrabec, T. P. Dao, et al., Nature, Vol 579, pp. 214-218 (2020)

Threshold Logic with Current-Driven Magnetic Domain Walls

The understanding of magnetization reversal in STTMRAM cells is of considerable interest. We use time-resolved conductance measurement, micromagnetic simulations and modeling to study it. The samples are fabricated from proprietary supersoft free layers of high volume and low moment. The experiments reveal the oscillatory nature of the conductivity signature of the switching (see figure). Modeling indicates that the conductance oscillations arise from the domain wall motion that happens in the so-called precessional regime above the Walker breakdown, when the domain wall propagation involves back-and-forth motions on top of a global drift. We first extend a collective coordinate model, taking into account the position of the wall and the tilt of the in-plane component of the magnetization within the wall. The mean wall velocity and its oscillations can be understood from the continuous exchange of energy between these two degrees of freedom, and the work supplied by the STT. In discs of diameters 70-100nm, experiments identify several distinct switching paths that occur stochastically and lead to distinct switching durations, while in disks of diameter 40 nm much less variability is found. Micromagnetics is used to understand these switching paths. For disks with 40 nm of diameter, the wall sweeps through the disk in an almost monotonous manner regardless of the wall state (Bloch, Néel or more complex). Conversely at larger diameters, the manner in which the wall crosses the disk depends on the wall state, and sets the variability of the transition time during the switching. The wall oscillation and its advance are essentially stopped if and only if a Bloch line is nucleated from the edge of the wall. This leads to switching times that are usually long compared to the standard deviation; this is one of the main mechanisms determining the write error rate in the ultrafast switching regime. Our model is finally used to design sample specifications with shall lead to rather reproducible propagation times, of direct interest for magnetic memories.

Walker-like domain wall motion in STTMRAM cells: evidence and consequences for the write error rate

Magnetoresistive random access memories (MRAM) based on spin-transfer torque are a scalable low-power nonvolatile memory solution being actively developed by industry, where a perpendicularly magnetized thin disk is reversed by a spin polarized current. During the switching process domain walls (DWs) can form [1-4] and understanding their dynamics in a disk geometry becomes of great interest. Here we show that DW surface tension always leads to oscillatory DW motion and instabilities that affect the switching dynamics. In a disk, a DW generally takes a circular form to minimize the sum of the surface and volume energies of the reversed domain, with the domain boundary perpendicular to the element boundary [1]. In a collective coordinate model the domain wall has two degrees of freedom, its position relative to the center of the disk q and the angle of the spins relative to the normal to the DW Φ. Here, Φ=0 is a Neel and Φ=90 deg. is a Bloch DW. We analyze the dynamics with micromagnetic and the collective coordinate model. Fig. 1a shows micromagnetic simulations of the evolution of the average perpendicular magnetization (m_z) with fixed initial DW position q, such that m_z<0, and different initial angles Φ, with no field or current applied. Whereas both Neel and Bloch DWs states relax towards a reversed state (m_z=-1), an intermediate initial angle DWs (Φ=45 deg) switches. Fig. 1b shows the (q,Φ) phase diagram indicating that the dynamics of domains close to the disk center is sensitive to small changes in the initial phase. Fig. 2 shows the same phase diagram as in Fig. 1b, now driven by a fixed spin-polarized J/Jc= 0.1. The results from both relaxation and switching simulations indicate that the DW phase Φ plays a critical role in the dynamics. A collective coordinate model will be presented which captures aspects of these instabilities and illustrates their physical origin.References: [1] G. Chaves-O'Flynn et al., Phys. Rev. Appl. 4, 024010 (2015) [2] T. Devolder et al., Phys. Rev. B 93, 024420 (2016) [3] P. Bouquin et al., Appl. Phys. Lett. 113, 222408 (2018) [4] J. B. Mohammadi et al., arXiv:2003.13875 (2020)

Micromagnetic instabilities associated with domain wall dynamics in perpendicular magnetized magnetic tunnel junction nanopillars

Magnetostatically coupled monodomain patterns have been studied for use as cellular automata and logic gates [1, 2]. Here the external magnetic field clocked switching is deterministic, and the magneto-optic Kerr (MOKE) effect microscopy or magnetic force microscopy (MFM) characterization only detects slow dynamics. We investigate similar systems, but with the use of spintronics for fast, all-electronic control and measurement. This is relevant both for low power magnetic logic, and for detection of fast dynamics in coupled superparamagnets that could be used in reservoir computing. We have demonstrated the ability to deterministically switch the state of a magnetic tunnel junction (MTJ) using spin orbit torque (SOT) [3], and this is used to switch the control dot at the beginning of the coupled patterns (Fig.1a). We have also detected magnetization changes in patterned nanodot MTJs using tunnel magnetoresistance (TMR) and a conductive atomic force microscopy (C-AFM) probe tip [4], and this is used to detect the dot magnetization at different locations. The dots were patterned by E-beam lithography from an MTJ thin film stack with a 3 nm thick CoFeB free layer. Fig.1b shows the schematic for a control dot and a neighboring secondary dot. Mumax3 simulations were conducted to match the magnetic field generated by the control dot with the coercive field of the secondary dot to facilitate the signal processing in magnetically coupled dots [5]. Fig.2 shows the magnetic field profile along +x direction. Inside the secondary dot, the magnetic field ranged from 2 to 15 mT, comparable to the 8 mT coercive field of the secondary dot, as shown in the Fig.2 inset. This ensures us that we could propagate a signal inside magnetically coupled dots. C-AFM measurements test the predicted switching and propagation along the nanodot chain of Fig.1.References: 1. R. P. Cowburn and M. E. Welland, Science 287 1466 (2000). 2. A. Imre, et al., Science 311 205 (2006). 3. M. Bapna, et al., Phys. Rev. Applied 10, 024013 (2018). 4. B.Parks, et al., Phys. Rev. Applied 13, 014063 (2020). 5. A. Vansteenkiste et al., AIP Adv. 4, 107133 (2014).

Spintronic control of magnetostatically coupled dot assemblies

In the racetrack or similar technologies, information (bit) stored in the direction of the magnetic moment of domains placed next to each other in a wire is shifted when electrical current passes along or parallel to the wire. While using electrons as ‘carrier’, these technologies sometimes lack the deterministic motion of the bits causing data integrity issues [1]. In this work, we are proposing a more deterministic, ‘carrier’ free transmission of data in an insulating ferromagnetic wire where the data is encoded in the oscillation frequencies of magnetic moments. Fig. 1 shows the schematic view of the device, including a thin transmission strip made of soft magnetic material. An Antiferromagnet (AF)/Ferromagnet (F) coupled perpendicular spin torque oscillator (STO) [2] is used to generate the full 360o in-plane magnetization rotation in a small confined region of the transmission strip that is also the free layer of the STO. This precession at STO end generates a traveling rotational magnetization wave along the wire, as seen in Fig. 1. While the spatial wavelength is increasing with distance, the rotational frequency along the wire remains constant, which is a key for our data transmission method. By changing the current amplitude according to the transmission bits, STO frequency can be tuned. Even at 800 nm away from the STO end (Fig. 2), the variation of the rotational frequency with time matches the variation of input STO frequency (or the input current pulse), affirming reliable data transmission. For this micromagnetic simulation, archetypal values for F such as α = 0.01 and Ms = 800 emu/cc are considered. 1 μm wire with a width of 10 nm and thickness of 2 nm is discretized using a grid size of 4nm x 2nm x 2nm.References: [1] Foerster M. et al., Domain Wall Memory Device (2016). In: Xu Y., Awschalom D., Nitta J. (eds) Handbook of Spintronics. Springer, Dordrecht [2] J.-G. Zhu, A. Shadman, IEEE Trans. Magn., vol. 55, no. 3, 2019.

Data Transmission in an Insulator utilizing Rotational Magnetization Wave

It has been shown both theoretically [1] and experimentally [2] that a spectrum analyzer based on an externally driven spin torque nano-oscillator (STNOs) can perform high-speed, wide-band spectrum analysis in a microwave frequency band. In such a spectrum analyzer, an STNO generates a signal whose frequency is tuned in a linear manner over a scanning bandwidth Δf. Theoretical estimations show that when employing an STNO in spectrum analysis, the following can be achieved: a Δf of 10 GHz, a linear scanning rate ρ=Δf/T (where T is the scanning period) of up to 2 GHz/ns, all in a frequency range of operation between 200 MHz and 70 GHz [3]. The fast speed of frequency analysis in a STNO-based spectrum analyzer can be achieved mainly due to the nano-scale size of the STNO, and the consequent ns characteristic scanning times. At a sufficiently large scanning rate ρ, the accuracy of the frequency determination ΔF (defined as the minimum separation required to distinguish two neighboring frequencies) of an STNO-based spectrum analyzer is fundamentally limited by the bandwidth theorem. In contrast, at low scanning rates, this accuracy is limited by the STNO generation linewidth [4], which is determined by the thermal (or random-walk) noise that is causing the jitter of the STNO generation frequency. Fig. 1 shows the frequency accuracy ΔF as a function of the scanning rate ρ for the STNO-based spectrum analyzer [2], where the gray solid line is the result of numerical modeling, dash-dotted line is a result of the experimental measurement [2], and dashed line shows the limit due to the “bandwidth” theorem. It is evident that at sufficiently large scanning rate, ΔF approaches the fundamental limit ΔF =1/T, whereas at smaller ρ the ΔF is limited by the magnitude of the STNO linewidth.References: [1] S. Louis, et al., Appl. Phys. Lett., 113, p. 112401, (2018). [2] A. Litvinenko, et al., “Experimental demonstration of a rapid sweep-tuned spectrum analyzer with temporal resolution based on a spin-torque nano-oscillator,” arXiv preprint arXiv:2004.03508, (2020). [3] S. Louis, “Development of a spectrum analyzer using a spin torque nano-oscillator,” Ph.D. dissertation, Department of Electrical and Computer Engineering, Oakland University, (2020). [4] Joo-Von Kim, et al., Phys. Rev. Lett. 100, 017207 (2008).

The influence of a thermal noise on the accuracy of a spectrum analyzer based on a spin torque nano-oscillator

Development of complementary metal–oxide–semiconductor (CMOS) logic is expected to approach its fundamental limit as the scaling down of the CMOS technology is reaching an end. As a route to extend the technology roadmap beyond traditional CMOS logic, novel spin-based logic architectures are being developed to provide nonvolatile data retention, near-zero leakage, and scalability. In particular, architectures based on magnetic domain walls (DWs) take the advantage of the fast motion, high density, non-volatility and flexible design of DWs to process and store information [1,2]. Such schemes have so far relied on DW manipulation and clocking using an external magnetic field, which hinders their implementation in dense, large-scale chips. Here we demonstrate a method for performing all-electric logic operations and cascading using DW racetracks [3]. Our concept for the magnetic DW logic is based on efficient DW motion induced by spin-orbit torques (SOTs) [4] and recently developed chiral coupling between adjacent magnets where the magnetic anisotropy competes with the interfacial Dzyaloshinskii–Moriya interaction (DMI) [5,6]. First we created a DW inverter, the essential basic building block for all implementations of Boolean logic, by embedding a narrow in-plane magnetized region in a racetrack with out-of-plane magnetization and demonstrate the operation of the DW inversion on application of an electric current (Fig. 1a). Building on the principles used to construct the DW inverter, we then fabricated reconfigurable NAND and NOR logic gates, making our concept for current-driven DW logic functionally complete. Finally, we cascaded several NAND gates to build XOR and full adder gates, demonstrating electrical control of magnetic data and device interconnection in logic circuits (Fig. 1b). Our work provides a viable platform for scalable all-electric magnetic logic, paving the way for memory-in-logic applications.References: [1] Allwood, D. A. et al. Science 309, 1688–1692 (2005). [2] Parkin, S. & Yang, S.-H. Nature Nanotechnol. 10, 195–198 (2015). [3] Luo, Z. et al. Nature 579, 214-218 (2020). [4] Manchon, A. et al. Rev. Mod. Phys. 91, 035004 (2019). [5] Luo, Z. et al. Science 363, 1435 (2019). [6] Dao, P. D. et al. Nano Lett. 19, 5930-5937 (2019).

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