Neuromorphic computation is based on memristors, which function equivalently to neurons in brain structures. These memristors can be made more efficient and tailored to neuromorphic devices by using ferroelastic domain boundaries as fast diffusion paths for ionic conduction, such as of oxygen, sodium, or lithium. In this paper, we show that the local memristor generates a second, unexpected feature, namely, weak magnetic fields that emerge from moving ferroelastic needle domains and vortices. The vortices appear near ferroelastic “junctions” that are common when the external stimulus is a combination of electric fields and structural phase transitions. Many ferroelastic materials show such phase transitions near room temperatures so that device applications display a “multiferroic” scenario where the memristor is driven electrically and read magnetically. Our computer simulation study of an elastic spring model suggests magnetic fields in the order of 10−7 T, which opens the way for a fundamentally new way of running neuromorphic devices. The magnetism in such devices emerges entirely from intrinsic displacement currents and not from any intrinsic magnetism of the material.

Neuromorphic devices are based on memristive ionic conductivity,1 where the electronic analog of a biological synapse is a memristor, proposed in 19712 and first demonstrated in 2008.3 A memristor resistance can be continuously tuned with strong history dependence. In this configuration, high conductivity indicates a strong synaptic connection, whereas in a low conduction state, the connection becomes weak. The learning and processing abilities of our brain are linked to the strength of the synaptic connection that varies as the information flow between neurons,4–7 and several inorganic materials have been proposed as memristors for neuromorphic computation.8–15 

Ferroelastic materials are particularly adapted for such devices because they contain two useful domain patterns, which increase the efficiency of the memristor switching. The first are needle domains that progress and retract under external electric fields and stresses.16–18 The second nanostructure that dominates ferroelastic materials are domain wall junctions,19 namely, intersections between polar domain walls. Changing temperature or fields will change these nanostructures, which is the topic of this paper. As an example, perovskite materials, such as WO3,20 are particularly promising for such device applications because the ionic mobility of dopants, such as sodium or oxygen, is very high and conductivity change can, for some temperature intervals,21 be as large as a factor of 30 during the metal–insulator phase transition.22 A structural phase transition between the β and γ phase of WO3 occurs near room temperature.23 During this transition, the nano-structure of WO3 changes dramatically, which is a secondary effect that we explore in this paper. These nano-structures are important because they channel ionic diffusion.24–26 The switching of the transport path mainly depends on the regions between domains, namely, domain walls.27 These domain walls are typically polar and have similar crystal structures to ferroelectric materials.28–34 The domains in ferroelastic materials remain non-polar and do not contribute to the sample polarization. Wall-related switches occur typically over extremely short time scales35 and involve reorientations of local polarity. Such changes necessarily induce displacement currents.36–38 These magnetic fields extend to some μT.39 In this scenario, ultrafast wall switching is a feature of fast memristive switching and is expected to occur in neuromorphic devices. The characteristic structural element during the switches, which generates magnetism, is the polarization vortex. If this happens, weak but measurable magnetic signals during neuromorphic operation are expected. In addition to WO3, similar effects were seen in Pb3(PO4)240–43 and cryogenic SrTiO3.44–50 

To quantify the ferroelastic domain structures and the emitted magnetism, we simulate the structural changes using a two-dimensional Landau-type potential with different interatomic interactions, as schematically shown in Fig. 1. The model consists of two sublattices carrying positive (green atoms) and negative (violet atoms) charges. The interactions are divided into short-range interatomic interactions and long-range Coulombic interactions. The pairwise short-range interactions for anion sublattices contain four terms: the harmonic first nearest atomic interactions Ur=20r12 (black springs), the anharmonic second-nearest Landau-type interaction Ur=10r22+8000r22 (gray sticks) along the diagonals in the lattice unit, the fourth-order third-nearest interaction Ur=8r24, and the anharmonic fourth-nearest Landau-type interaction Ur=10r52+5100r52 along diagonals in the lattice unit, where r is the distance between atoms. The interactions for cation atoms are purely harmonic with first-nearest interaction Ur=20r12 (green springs) and second-nearest interaction Ur=1.5r22 along the diagonals. The harmonic springs are related to the elastic interactions and constitute the elastic background in ferroelastic materials. The second-nearest Landau-type interactions are constructed to maintain an equilibrium shear angle of 2°, which was inspired by the typical ferroelastic phase transition of SrTiO3 at 105 K.51 An additional Landau-type spring is added to obtain a reasonable domain wall thickness and stability.52,53 The interaction between the anion and cation is a harmonic spring Ur=1.5r2/22 (red springs) to avoid the collapse of the cation sublattice onto the anion sublattice. The interactions were chosen to be purely harmonic to exclude any polar instabilities inside the bulk. The symmetry properties of the model were discussed in detail in Ref..54. The domain walls are polar due to the flexoelectric effect55–57 and a biquadratic coupling between deformation parameters58,59 near the domain walls. Experimentally, wall dipoles are related to displacements of 6 pm as observed in CaTiO3.28 All the simulations are performed by using the LAMMPS code.60 The domain structures are visualized by using OVITO software.61 

FIG. 1.

Interatomic potential for our generic ferroelastic model. The non-linear elastic interactions are listed in the text.

FIG. 1.

Interatomic potential for our generic ferroelastic model. The non-linear elastic interactions are listed in the text.

Close modal

The dominant microstructure in ferroelastic materials is the comb pattern, which consists of needle arrays with various distances between the needles. The simulated stress-driven propagation of comb patterns consists of 5 vertical thin needles with 3 atomic layers for each [Fig. 2(a)]. By controlling the external stress, vertical needles were stabilized at the same distance from the surface with equal needle distances of 0.7 nm. A shear deformation drives the propagation of the comb pattern with a strain rate 2.7 × 108 s−1. All the vertical needles were activated at the same time with a gradual increase in speed before reaching a steady state at a velocity of 0.24 km/s [Fig. 2(d)]. They finally slow down when moving close to the lower surface or interface. Two atomic configurations with a time interval Δt = 3.0 ps during the steady state motion were chosen to calculate the local current density and the corresponding magnetic moment. Relative displacements between cations and anions [Fig. 2(b)] and the inducing displacement current vortex array [Fig. 2(c)] are observed between the vertical needles. The total magnetic moment is 0.000 73 µB, where μB = 9.274 × 10−24 JT−1 is the magnetic moment of an electron. The magnetic field density for this current vortex array is 6.2 × 10−7 T.

FIG. 2.

Displacement currents and the corresponding magnetic fields produced by the propagation of the comb pattern. (a) The atomic images of the movement of needle domains by the external stress. The colors are coded according to the atomic-level shear strain. (b) The relative displacement of cations and anions. Ionic displacements are amplified by a factor of 30 for clarity. (c) The displacement currents near the moving needle tips generate magnetic fields in the out-of-twin-plane direction. The current densities in the vector maps are amplified by a factor of 1 × 1018 for clarity. (d) The averaged position of the needle tips with five needles as a function of time during needle propagation (after Lu et al.36).

FIG. 2.

Displacement currents and the corresponding magnetic fields produced by the propagation of the comb pattern. (a) The atomic images of the movement of needle domains by the external stress. The colors are coded according to the atomic-level shear strain. (b) The relative displacement of cations and anions. Ionic displacements are amplified by a factor of 30 for clarity. (c) The displacement currents near the moving needle tips generate magnetic fields in the out-of-twin-plane direction. The current densities in the vector maps are amplified by a factor of 1 × 1018 for clarity. (d) The averaged position of the needle tips with five needles as a function of time during needle propagation (after Lu et al.36).

Close modal

We now add temperature as a second variable. Many potential memristors possess structural phase transitions near room temperature. We simulate the effect of a para-elastic to ferroelastic transition on the domain structure. Starting from the single domain state (symmetry breaking strain in our simulation is ɛxy = 0.035), we first increase the temperature with low heating rate through the phase transition temperature Ttr. Then, we repeatedly heat and cool the structure with sequences of domain nucleation and migration and annihilation of the polar domain walls. The evolution of the potential energy and temperature as function of time is shown in Fig. 3(a). Starting from the high temperature paraelastic phase [“A” in Fig. 3(b)], the structure is cooled down below Ttr with the formation of typical stripe ferroelastic domains [“B” in Fig. 3(b)]. As the temperature decreases further, ferroelastic domains evolve with the migrations of the domain walls [“C” in Fig. 3(b)]. Under the second cooling and heating cycle, more complex domains with vertical and horizontal domain walls and domain wall junctions occur at low temperature [“F” in Fig. 3(b)].

FIG. 3.

Cooling and heating cycles and evolutions of ferroelastic domain structures. (a) Time evolution of the temperature and potential energy. (b) Domain structures evolution during the cooling and heating cycles. The colors are coded by the atomic-level polar displacement directions. The temperatures A–F are shown in Fig. 3(a).

FIG. 3.

Cooling and heating cycles and evolutions of ferroelastic domain structures. (a) Time evolution of the temperature and potential energy. (b) Domain structures evolution during the cooling and heating cycles. The colors are coded by the atomic-level polar displacement directions. The temperatures A–F are shown in Fig. 3(a).

Close modal

These complex domain structures are expected to form close to the injection points of the memristor. The domain walls act as fast diffusion paths, which show that the low temperature phase is the most effective memristor. The needle domains will generate magnetic signals during their propagation. In addition, when needle domains of orthogonal orientations intersect and form wall junctions, they generate displacement current vortices [Fig. 4(b)].

FIG. 4.

(a) Transient displacement current vector map of the ferroelastic domain structures during the cooling and heating cycles shown in Fig. 3. The displacement currents are of the order of 10−18 A. Panel (c) shows the local magnetic field profile corresponding to the vortex in panel (b). The local magnetic field density in panel (c) is 2.0 × 10−6 T when averaged over the area in panel (b).

FIG. 4.

(a) Transient displacement current vector map of the ferroelastic domain structures during the cooling and heating cycles shown in Fig. 3. The displacement currents are of the order of 10−18 A. Panel (c) shows the local magnetic field profile corresponding to the vortex in panel (b). The local magnetic field density in panel (c) is 2.0 × 10−6 T when averaged over the area in panel (b).

Close modal
Based on Maxwell's theory and the Biot–Savart law, we determine the magnetic fields,
Bri=μ04πk=1N/2rirk×jkrirk3,
(1)
where Bri is the magnetic field at position ri, μ0 is the vacuum permeability (μ0 = 4π × 10−7 T m A−1), and N is the number of atoms. Figure 5(a) shows the dynamic change in the magnetic field as functions of time. The magnetic fields are emitted from two specific events: domain nucleation, migration, and annihilation when cooled and heated near Ttr. Non-zero averaged magnetic fields with magnitudes of several 10−7 T emerge to the breaking of the structural centro-symmetry caused by the presence of complex twins and the displacement current vortices. When the system is paraelastic or frozen, the magnetic field is inactive (Bri0). The key for the magnetism emission lies in the thermally induced ferroelastic domain evolutions and the displacement current vortices. The domain structures during the cooling and heating trajectory, “A,” “B,” and “C” in Fig. 5(a), are shown in Fig. 5(b). The out-of-plane magnetic field profiles are shown in Fig. 5(c), which shows the magnitude contrast for the paraelastic phase, dynamic ferroelastic domains, and the frozen domains.
FIG. 5.

Evolution of the magnetic field as a function of time. (a) Time dependence of temperature and the induced magnetic field. (b) Domain structures indicated by “A,” “B,” and “C” in panel (a). (c) Magnetic field profiles corresponding to the moments indicated by “A”–“C.” The domain structures in panel (b) are coded by the atomic-level shear strain (ɛxy). The maps in panel (c) are coded by the local magnetic field values. Blue planes in panel (c) indicate the zero magnetic field density. The magnetic field of the state B does not average to zero over short time intervals [e.g., 50 ps, red shadows in panel (a)] so that magnetic signals are predicted.

FIG. 5.

Evolution of the magnetic field as a function of time. (a) Time dependence of temperature and the induced magnetic field. (b) Domain structures indicated by “A,” “B,” and “C” in panel (a). (c) Magnetic field profiles corresponding to the moments indicated by “A”–“C.” The domain structures in panel (b) are coded by the atomic-level shear strain (ɛxy). The maps in panel (c) are coded by the local magnetic field values. Blue planes in panel (c) indicate the zero magnetic field density. The magnetic field of the state B does not average to zero over short time intervals [e.g., 50 ps, red shadows in panel (a)] so that magnetic signals are predicted.

Close modal

In summary, we have shown that macroscopic, transient magnetism can be emitted from neuromorphic devices for two reasons. The nucleation and progression of the typical needle domains generate displacement currents on either side of a micro-structural needle domain (Fig. 2). The second mechanism is the nucleation of domain junctions62 where each junction is decorated by a vortex (Fig. 4). These vortices carry displacement currents, which, in turn, generate magnetism of some 10−7 T. Given these results, it is now possible that the readout from neuromorphic devices can be extended to magnetic signals in addition to the electric signals used previously. This technology has potentially tremendous advantages as it allows a second channel for the information transfer between artificial neurons. Preliminary squid observations has indeed shown magnetic signals in low temperature SrTiO3, but much more research is needed to develop neuromorphic devices based on this idea.

Conversely, in complex biological systems, magnetic brain signals are well known to exist during brain activities .63 While most can be related to known biochemical processes, others seem to be unknown for its physical origin. In particular, transcranial magnetic stimulation (TMS) has evolved from an almost magical curiosity and a simple tool for neurophysiologists to a mainstay of non-invasive neuromodulation, gaining acceptance as a candidate treatment in psychiatry, neurology, and, perhaps, other clinical specialties.64 Our simple simulations can, at this stage, not clarify any of these biological issues, but it remains a tantalizing possibility that weak magnetism is possibly related to artificial neurons, as in neuromorphic computation or even in biological systems.

We thank Beena Kaliski and Shai Rabkin (Bar Illan University, Israel) and Gustau Catalan (ICN2 Oxide Nanophysics Group, Catalonia) for their help in the first attempts to develop device applications. Guangming Lu acknowledges the financial support by the National Natural Science Foundation of China (Grant No. 12304130) and the Doctoral Starting Fund of Yantai University (Grant No. 1115-2222006). EKHS acknowledges the EPSRC (Grant No. EP/P024904/1) and EU Marie Sklodowska-Curie 2020 program (Grant No. 861153).

The authors have no conflicts to disclose.

Guangming Lu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal). Ekhard K. H. Salje: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

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