Calculate the activation energy for surface diffusion in silver

The projector augmented-wave PAW [2] method was used to represent the ion-core electron interactions. For Cu, the cutoff radius for the above psuedopotential is 2. For Ag, the cutoff radius for the above pseudopotential is 2. The structural convergence criteria were 0. A Cu FCC experimental bulk lattice constant of 3. A five layered 3 x 3 unit slab of Cu was used for surface calculations where the bottom three layers were frozen during the optimization and the top two layers were unconstrained Figure 1a.

A plane wave energy cutoff of eV and Monkhorst-Pack [4] k-point mesh of 5 x 5 x 1 was used for all surface slab calculations after proper convergence tests described in the next section. Transition state was located using the climbing image nudged elastic band CI-NEB method [5] 5 images by relaxing the reaction tangent force below 0. Transition state was verified to contain a single imaginary frequency along the reaction coordinate.

It determines the Hessian matrix matrix of the second derivatives of the energy with respect to the atomic positions using density functional perturbation theory []. Convergence Tests. Convergence test for K-points mesh sampling the brilliouin zone was done at a plane wave energy cut off of eV by varying the k-points against the single point energy of Cu surface. The relative energies are plotted against the number of irreducible k-points as shown in Figure 1b. The energy varied by an amount 0.

However, the energy fluctuated when the K-points were raised further. To reduce the computation cost, lower K-points were chosen due to limited availability of computational resources which should be a reasonable starting point because we are mostly concerned with a difference in energies between different systems and not the absolute numbers themselves.

The energy changed only by 0. Finally, the k-points mesh of 5x5x1 with 13 irreducible K-points and a plane wave energy cut off of eV was considered suitable for all subsequent surface calculations of adsorption energies. A Cu surface has two types of three-fold binding sites available for adsorption by a silver atom — hollow-fcc and hollow-hcp sites shown in Figure 2a. The figure shows adjacent hollow-fcc and hollow-hcp sites. A hollow fcc site is a three fold site with no atom directly below but with an atom in the layer next to it.

A hollow hcp site is a three fold site with an atom directly below and with no atom in the layer next to it. The adsorption energies for the two adjacent three-fold sites are labelled in Figure 2b. Both sites show the same adsorption energy of 2. These two states obtained were then studied for diffusion between the two adjacent sites and the activation energy was probed for the same. Figure 2: a Cu surface showing two adjacent hollow-fcc and hollow-hcp sites of adsorption, b Ag atom adsorbed on a hollow-fcc site on Cu and c Ag atom adsorbed on a hollow-hcp site on Cu Adsorption energies for the two states of adsorption are also shown for comparison.

Bottom and top Au layers act as contact pads, Ag NWs act as effective bottom and top electrodes for the CF formation, and silk acts as insulating switching layer. A metallic CF can be formed in response to the application of a large voltage to the electrodes. In non-volatile switching, the CF can remain in the connected state for a relatively long time until another voltage pulse induces its disconnection.

On the other hand, in the case of volatile switching, the CF retracts back to the electrodes immediately after set transition. When a positive sweep voltage is applied, the current increases sharply, indicating the formation of the metallic CF.

The compliance current I C across the device controls the size of the formed CF. After the positive voltage sweeps back to zero, the device still remains in the low resistance state. As a negative voltage is applied, the device switches to the high resistance state indicating CF disruption. When a positive sweep voltage is applied, the current increases sharply indicating CF formation.

However, the device quickly switches back to its original high resistance state after the forcing voltage is swept to zero. Bidirectional volatile switching is observed, as the application of a negative voltage also generates a metastable CF. Given the random nature of the Ag NW network in the host material, the distance between the two electrodes in the switching region might be affected by variability.

Each device might in fact have a different distance between the active NWs, thus resulting in a variation of threshold voltage from device to device. Even within the same device, the weakest point where the filament is formed might change location or distance at each cycle, thus resulting in a cycle-to-cycle variation of threshold voltage. The threshold voltage distribution of our devices were shown to have a relatively good uniformity both from cycle to cycle, and from device to device 20 , thus indicating a relatively small variation of the distance between Ag NWs.

Recent in situ transmission electron microscope TEM observations 17 , 26 of metallic CFs revealed that Ag and Cu atoms from the CF tend to form clusters rather than out-diffuse toward the host dielectric material. The clustering of the metal atoms leads to the formation of nanoclusters and nano-spheres Figure 2a—c shows the time evolution of the simulated Ag CF between the two electrodes, which might represent two Ag NWs in the structure of Fig.

MD simulation of surface diffusion. The color indicates the free energy per atom with lighter blue representing atoms with higher free energy, hence higher diffusivity. Bulk atoms show constant positions, while the surface atoms migrate toward their closest electrodes. Atoms in the CF bottleneck region also show a high tendency to migration to either the top or bottom electrode.

For simplicity, isotropic surface diffusion and axisymmetric along z CF surface are assumed. According to MD simulations, the metallic CF can spontaneously break as a result of atomic surface diffusion driven by the minimization of the system energy. The atomic surface diffusion originates from the gradient of surface atomic vacancy concentration or the gradient of the surface atomic chemical potential 30 , resulting in a tendency to minimize the surface energy Atoms in the bulk of the CF instead remain fixed in their lattice sites of the crystal Fig.

Based on the Gibbs-Thomson effect, the surface atomic flux J s along an arbitrary surface can be modeled by assuming isotropic surface diffusion, leading to 32 :. Note that the driving force for the surface diffusion in Eq. More details about the simulation method and results are reported in the Supplementary Fig. Surface diffusion leads to a segmentation of the CF and the formation of one or more intermediate clusters, as shown in Fig. A similar ovulation effect 33 has been recently reported by in situ TEM observations 17 , 26 , 34 as the by-product of Ag nano-filament clustering in an insulating material.

Note that the specific morphological evolution of the filament is affected by the hydrophobicity between filament and electrode materials, initial filament shape Supplementary Note 6 , Supplementary Fig. However, the generic isotropic simulation shown in Fig. Morphological changes induced by surface diffusion.

The color in the figures indicates the surface curvature, where surface points with higher curvature are more likely to induce surface diffusion, thus providing a suitable spot for CF rupture. The ovulation effect in b , where nano-clusters result from the disconnection of a thin CF, is consistent with in situ TEM observations of Ag nano-clusters evolving from Ag nanoscale CFs 17 , 26 , Reducing the diameter of the CF by 5 times from Fig.

The dependence of lifetime on the initial CF diameter can explain the transition from volatile switching Fig. To highlight the size-dependent lifetime, Fig. The power law with exponent 4 in our surface-diffusion model arises from Eq.

The steep increase for higher d 0 marks the onset of the CF stability, hence non-volatile switching, which corresponds to the equivalence of the two principle radii of surface curvature Fig. CF size-lifetime scaling law. The stability of this shape originates from the two principle radii of surface curvature having equal modulus and opposite signs. Results in Fig. To further validate our model, we conducted time-resolved experiments to measure the lifetime of Ag CFs as a function of I C.

Figure 5a shows the setup of applied voltage pulse, while the inset shows the measurement configuration, consisting of a RS device connected to a load resistance R L in serial.

The R L aimed at limiting the current during the set transition. The figure also shows the calculated conductance evolution G t from Eq. Note the large variation of lifetime, which might originate from anisotropic surface diffusion along different crystalline directions, or inaccurate control of the CF size by I C Different types of dielectric materials would also impact on the lifetime by affecting parameter B in Eq. Despite these variations, the surface diffusion model accounts for the size dependence of lifetime over a broad range of experimental conditions and samples.

The temperature dependent lifetime also relies on the surface activation energy Q s. For the estimations of the parameter B and the predicted size-lifetime lines at different temperatures in Fig. This is consistent with the data projected from the temperature dependent retention times in Cu and Ag RS devices by fitting with the Arrhenius rule, which gives the value of Q s in the range of 1. Universal explanation for volatile and non-volatile switching.

A voltage pulse 2. As I C increases, the lifetime of the Ag filament increases as a result of the size-dependent surface diffusion. The values of the surface activation energy Q S was estimated as 1. Calculations at various temperatures are also shown. The results account for the transition from volatile switching to non-volatile switching in RS devices Fig.

The lifetime of this type of filament, which controls the vacuum tube lifetime, is generally in the range of hundreds of hours. At nanoscale dimensions, the morphological changes driven by the minimization of surface energy are highly accelerated to the range of observable time scale at room temperature, which is at the origin of a liquid-like pseudoelasticity 31 observed in situ.

Surface diffusion effects also control general properties in nanoscale world, such as the lack of sharp tipped metal coated probes for atomic force microscopy 48 compared to the covalent bonded crystallized silicon or diamond probes 49 , and the instability of ultrathin metallic NWs, e. In this work, we provide evidence for surface diffusion being the fundamental mechanism for the filament rupture in RRAM devices.

Other works have previously proposed that out-diffusion, rather than surface diffusion, acts as the leading dissolution process in other materials systems, such as Ni in NiO We note, however, that out-diffusion is not expected to play a major role in the materials systems considered in our work, namely Ag or Cu in silk and oxide compounds. This is independently supported by at least two evidences: first, recent TEM observations reveal the presence of Ag or Cu clusters rather than homogeneously distributed Ag within the doped silicon-oxide layer.

All these results indicate that Ag and Cu have relatively low solubility in silicon oxide, thus resulting in the clustering of Ag or Cu atoms. The low solubility of Ag and Cu in the host materials can be understood as the result of the low reactivity of Ag and Cu elements and the strong chemical stability of Si-O valence bond see Supplementary Note 7 for extended discussion.

Out-diffusion might be non-negligible when the metal has a relatively large solubility in the host materials, such as Ag in Ag 2 S 52 and Cu in CuS The volatile switching in RS devices has previously been interpreted as the consequence of the atomic clustering to minimize the Gibbs-Thomson energy at the interface between the filament and the host material 17 , 18 , 19 , Our interpretation of filament shape evolution induced by surface diffusion also has roots in the Gibbs-Thomson effect, given the dependence on surface curvature radii in Eq.

Another common phenomenon induced by Gibbs-Thomson effect is Ostwald ripening 55 , 56 , 57 , 58 , which has been proposed to control the evolution of the particles obtained by filament fragmentation Ostwald ripening instead may be responsible for the post-lifetime evolution of the filament particles.

In any case, the driving force for the surface diffusion and Ostwald ripening both can be traced back to the Gibbs-Thomson effect see Supplementary Note 10 and Supplementary Fig. In RS devices, CFs of various sizes can be electrically formed, resulting in a large range of electrically measurable lifetime.

According to our surface diffusion model, the ultimate stability of the CF for non-volatile switching arises from a stable capillary bridge between the top and bottom electrodes. Note that this condition might result in a relatively large CF, which conflicts with other requirements of RS devices for practical applications in non-volatile memories.

For instance, RS devices should also be easily erasable at relatively low current, which is necessary to minimize the IR voltage drop across lines in crosspoint arrays.

An excessive voltage drop, in fact, would lead to unwanted disturbs to unselected cells. The tradeoff between small operating currents and CF stability requires careful device and materials optimization, toward the minimization of the free energy at the CF surface. Note that experimental data show no tradeoff between endurance and operation current see Supplementary Note 11 for more discussion.

Conversely, minimization of the lifetime in the range of few ns might enable the operation of RS device as fast and efficient selector device for crosspoint arrays 20 , Materials engineering should guide the selection of CF materials for selector technology to enhance the surface energy and the related surface diffusion effects In summary, we propose a universal surface diffusion mechanism for the spontaneous rupture of metallic CFs in filamentary RS devices.

Results provide a general framework for understanding the stability of nanoscale structures, and designing RS devices for a wide range of applications, e. The aqueous solution of silk fibroin for fabricating the device was prepared according to the reported method Then, 0. For this device, the effective active electrodes are equivalent to two adjacent AgNWs in the composite film or one Au electrode and one Ag NW electrode.

Keithley semiconductor parameter analyzer was employed to measure the DC electrical characteristics. The interactions between Ag atoms were described by the EAM potential The initial filament shape is that of two-truncated cones Fig. Isotropic surface diffusion and the CF surface of revolution by Eq.

The morphological evolutions of the CF geometry are simulated in dimension-less form The constant l unit: m is the scaling ratio between the CF size and the dimensionless one, defining the spatial dimension of the filament. For each step, the curvatures of the filament surface for each segment were calculated, thus the changes of the morphological profile were obtained from Eq.

A finite differential method was used to calculate the differential values of surface curvatures and morphological change rates. The source code is available from the authors upon request. The data that support the findings of this study are available from the corresponding author upon reasonable request.

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