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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NNANO.2013.170

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Supplementary Information

A current-driven single-atom memory

C. Schirm, M. Matt, F. Pauly, J. C. Cuevas, P. Nielaba, and E. Scheer

Content

1. Device fabrication 2. Low-temperature transport measurements 3. Determination of transmission channel probabilities 4. Theoretical model 5. Electromigration in nanometre scale structures 6. Comparison of experimental and theoretical transmission channel probabilities 7. Statistical behaviour of bistable switches 8. Atomic configurations of bistable switches 9. Example of a long-lived bistable switch References

1. Device fabrication

On the polished bronze wafer (200 µm in thickness), a layer of polyimide (2 µm thick) is spin-coated. This layer serves as an electrical insulator and a sacrificial layer in the subsequent etching process. On top of the polyimide, a double layer of electron-beam resists, MMA-MAA/PMMA, is deposited by spin-coating (maximum 5000 rpm). Then electron beam lithography is performed on the prepared wafer. After developing the resist, aluminium (Al), about 110 nm thick, is deposited using electron beam evaporation at a pressure of about 10-8 mbar. Finally, dry etching with O2 plasma in a reactive ion etcher is performed in order to form a free-standing bridgeS1. The scanning electron microscope image of a finalized sample is presented in Fig. 1a of the main text.

2. Low-temperature transport measurements

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Charge transport measurements were performed at low temperature (base temperature 250 mK, sample temperature 250 mK < T < 1.5 K) in a 3He insert with a custom-made mechanically controllable break junction (MCBJ) system, see Fig. 1b. The devices are mounted on the breaking mechanism inside the vacuum chamber of the insert. After cooling down the cryostat, the breaking process of the sample is carried out by the breaking mechanics controlled by a dc motor and a differential screw. While controlling the nanogap distance and thereby breaking and forming Al-Al contacts, the dc conductance as well as the current-voltage characteristics (IVs) are measured by a home-made electronics setup described in Ref. S2. All data were collected by home-made software using the Python platform through an ADWin data acquisition system.

The electromigration process with zigzag current ramps is performed with the low-resistive wiring. During this procedure a magnetic field of 18 mT is applied to suppress superconductivity. When the voltage drop across the sample is below 1 mV, the magnetic field is turned off, the wiring is switched to the high-resolution scheme and IVs are measured, as exemplified in Fig. 3b of the main text.

3. Determination of transmission channel probabilities

Current-voltage curves are recorded in the voltage range from -1 mV to 1 mV. That voltage interval is divided into 1000 equally spaced data points, i.e., 500 points for each polarity. The supercurrent branch at V = 0 is used to correct for offsets of the amplifiers.

To extract transmission eigenchannel probabilities, we use the theory of multiple Andreev reflections (MARs)S3,S4. With the help of this theory we calculate a set of basis functions I(V,τ), i.e., IVs for a single channel with arbitrary transmission values between 0 ≤ τ ≤ 1 with a spacing in τ of 0.001. We fit both polarities of the experimentally recorded IVs separately. The fitting is performed by a home-made python routine, applying a least-squares fitting to the linear combination of the single channel basis functions. Due to the thermal smearing and the finite energy resolution of the setup, the total number of channels that can be disentangled is finite. With the parameters of our experiment the maximum channel number is limited to around 8, depending somewhat on the distribution of the τi. Accordingly, we concentrate here on contacts with a conductance below 3G0 that typically accommodate no more than 8 channels.

We first perform a rough determination of the number of channels. The code allows for up to 9 channels. When contacts with a smaller number of open channels are fitted, the additional channels are found to have a transmission of τi < 0.02. We then repeat the fitting with a reduced set of channels to verify the precision of the numerical values of the individual τi.


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The value of the superconducting gap parameter Δ, used in these initial fits, is estimated from IVs recorded in the tunnelling regime G < 0.01G0 with a single channel, where the IVs reveal a sharp step at Δ. It varies slightly from sample to sample between 181 µeV and 183 µeV.

The independent fitting of the IVs of both polarities allows for additional control. If the best-fit parameters for the τi differ by more than 0.03, we discard the contact. This happened in 10 out of more than 2000 contacts.

The precision with which the individual τi can be determined depends on the number of open channels and the precise distribution of the τi, because the current contributions are a nonlinear function of the transmission eigenchannel probabilities. The value of the leading channel (denoted below as τ1) can typically be determined with a precision of 0.001, if it is larger than 0.9. If the ensemble of channel transmissions involves several small channels with τi < 0.1, their precision is often limited to roughly 0.05. Consequently, throughout this work, we consider channels to be “open”, when they show a transmission τi > 0.05. Our fitting procedure provides the set of τi with the minimum number of open channels.

For particular combinations of eigenchannels, e.g., when there are channels with a transmission τ2,τ3 < 0.025 in addition to a dominating one with τ1 > 0.8, the fitting with a single low-transmissive channel with τ2* = τ2 + τ3 is normally of similar quality. As a result, in the experiments we find situations that can be described with two channels, while a three-channel fit would also be possible (see e.g. Fig. S5b). In the calculations two channel contacts are untypical. We attribute this discrepancy to the finite precision of the fitting procedure that by itself is mainly limited by the experimental rounding of the IVs.

When we determine the channel distribution of the bistable switches, we typically analyze the first four switching cycles, as visible in Fig. 3. The individual transmission coefficients usually agree to within ±0.02 for all channels in the low-conductance and high-conductance states, respectively.

4. Theoretical model

As mentioned in the main text, we combined classical molecular dynamics (MD) simulations of the formation of atomic-sized contacts with a tight-binding description of the electronic structure to calculate the conduction properties with the help of Green's function techniques. Our methodology proceeds along the lines of Refs. S5-S7. In the following we explain the details of the employed techniques.


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Molecular dynamics simulations. For the MD simulations we used the open source program package LAMMPSS8,S9. Within LAMMPS, we employ the embedded atom method with the semi-empirical potentials from Sheng et al.S10 to model the interactions between the atoms. To generate the geometrical configurations, we start with an ideal fcc lattice, where the crystal direction 〈111〉 lies parallel to the z axis, coinciding with the transport and elongation direction. We did compare our results to a few simulations for wires oriented along the 〈100〉 and 〈110〉 directions. Since no significant differences were found, we concentrate here on the 〈111〉 orientation.

For the MD calculations, the geometry is divided into three parts: Two electrodes and a central wire, attached to them. This is shown in Fig. S1. The electrodes consist of 352 atoms each which are kept fixed during the simulations. The wire is made up of 301 atoms which follow the Newtonian equations of motion. We assume a canonical ensemble (also called NVT ensemble with fixed particle number, volume, and temperature) and use the velocity Verlet integration scheme. The wire possesses an initial length of 3.78 nm. The starting velocities of the atoms in the wire were chosen randomly with a Gaussian distribution to yield an average temperature of T = 250 mK. Because of this randomness, every elongation calculation evolves differently, while a Nosé-Hoover thermostat keeps the temperature fixed. To relax the system, the wire gets equilibrated for 0.1 ns. Finally, the elongation process is simulated by separating one electrode from the other at a constant velocity of 0.7 m/s. During this process, every 10 ps the geometry is recorded. A stretching process needs a total simulation time of about 4 ns, until the contact breaks.

Conductance calculations. To calculate the transport properties of the geometries obtained from the MD simulations, the Landauer-Büttiker formalism is used. We express it in terms of Green’s functions, as explained in Refs. S5-S7. The electronic structure information needed for the evaluation of the transmission is obtained from a Slater-Koster tight-binding parametrizationS11.

In this way, we arrive at a material-specific, atomistic description of the atomic contacts.

Analogously to the MD simulations, the system is divided into three regions for the transport calculations, i.e. the upper and lower electrodes and the central wire (see Fig. S1). As the local environment of the atoms in the central part is very different from the ones in the bulk, we enforce the charge neutrality for all the atoms of the wireS6. Such a neutrality condition is typically a good approximation for metallic systems. The electrodes are considered to be semi-infinite perfect crystals. Their surface Green's functions are computed with the help of a decimation techniqueS7,S12,S13, and we use the same tight-binding parameterization as for the central part to determine their electronic structure.


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Within the Landauer formalismS6,S13, we calculate both the transmission and its decomposition into individual transmission eigenchannels τi. According to the Landauer formula, the transmission probabilities τi of the eigenchannels are related to the conductance via G = G0Σiτi. In the expression, G0=2e2/h is the quantum of conductance.

Figure S1: (a) Ideal fcc starting structure of the atomic contacts. (b) Sample atomic contact at an elongation of 1.74

nm. In both panels we have indicated the partitioning of the contact into the upper and lower electrodes and the

central wire, as used for the MD and transport calculations.

5. Electromigration in nanometre scale structures

In macroscopic theories, the current density is considered to be the cause of electromigration, as confirmed by many experimental studies (see for instance Ref. S14 and references therein). In simple terms, the model considers two main force contributions: the direct force and the so-called wind force. The direct force is given by the action of the electrical field onto the ions, while the wind force arises since the travelling electrons transfer momentum to the atoms proportional to their velocity. When the cross section of metal wires becomes smaller than a few nanometres, transport changes from diffusive to ballistic, and the response of the metal contact to high current bias starts to deviate from the one of macroscopic wiresS15-S18. It has been understood that


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electromigration in microstructures and nanowires down to a few tens of nanometres in width is thermally driven and very high local temperatures have been estimated. When performing the process at low temperature, the local temperature gradients increase, making the process in general less controlledS19. For very small wire thicknesses the behaviour changes markedly, indicating that the quantum transport regime is being approached where the macroscopic theories break downS19-S24. For thin contacts with cross sections in the order of a few nanometres and conductances in the range of up to 10G0 self-healing processes as well as a non-monotonous evolution of the conductance as a function of time or bias have been reportedS21,S23,S24. The crossover from the macroscopic to the quantum regime has not been treated theoretically yet. Furthermore, in the quantum regime the mechanisms are not completely understood. It is not possible anymore to distinguish the classical contributions direct force and wind force, because the local electric fields are strongly interlinked with the local current flowS16. More importantly, it is unclear at present which quantity controls the electromigration process in this regime: the current, the current density, the voltage, the local electric field, or the dissipated power.

So far, only a few examples of ab initio treatments of current-induced forces have been reported. Brandbyge et al. calculated the forces exerted by transport currents onto the atoms in an atomic scale gold wire to be in the order of 0.05 to 1 nN for voltages in the range of 100 mV to 2 VS16. We use this observation to estimate the order of magnitude of the current-induced forces from the switching voltages observed in our experiment. To the best of our knowledge no quantitative treatment of electromigration forces in Al atomic-scale wires has been reported. However, we performed analogous experiments with Au contacts at comparable bias currentsS25. This suggests that the order of magnitude of the forces is comparable for atomic size contacts of both metals.

Below we provide a more detailed discussion of electromigration mechanisms in atomic-size contacts, including experimental data. Besides that, we estimate energy barriers and forces that are needed for the switching process based on our simulations of the mechanical stretching. Although our theory does not treat the electromigration or bistability after training directly, we show that the computed forces are in the same range as those expected due to currents, indicating that our calculations can be used to understand the involved atomic rearrangements.

5.1 Experimental results

Irreversible atomic rearrangements. In our experiments the electromigration is performed at low temperature at variance with most studies that investigate the process at room temperatureS19-S24. In addition we start the electromigration only in the regime of rather low conductance and small


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cross sections.

For elucidating the electromigration mechanism we discuss Fig. S2 where the conductance of an atomic Al contact as a function of applied bias current is shown (measured at T ≈ 300 mK). The contact has been prepared by mechanical deformation at low temperature. Initially the contact has a conductance of Gi = 0.72G0. We start by sweeping the current in the positive direction (P1). Several abrupt steps tending to higher conductance values are observed. At I = 26 µA the contact exhibits a conductance of approximately Gf = 1.1G0. The current is then lowered, swept in the negative direction to –26 µA (P2) and finally up again to +26 µA (P3). During the down and up sweep and further subsequent ones (P4 and P5), no conductance jumps are observed (apart from a very small step around -24 µA). The slight but smooth variation of the conductance with bias is typical for atomic contactsS26, and other examples exhibit a slight decrease rather than an increase (see also Fig 1e of the main text). This indicates that the behaviour is not caused by local heating but by quantum interferenceS27. We find that such self-stabilizations as in Fig. S2 are typical for small contacts with a starting conductance below a few G0. Furthermore, we stress the following important finding indicative of the quantum regime: We do not observe an evolution of the conductance with time even though a constant high-bias current might be applied, at variance with reports for thicker wires. This shows that the electromigration is not thermally driven.

Figure S2: Conductance of an Al atomic contact as a function of the bias current. The measurement was performed

at a bath temperature of T = 300 mK with a small magnetic field applied for suppressing the superconductivity. The

black solid curve (P1) shows the behaviour when increasing the current for the first time, starting at zero current up

to + 26 µA. The red solid curve (P2) shows the first sweep from +26 µA to -26 µA. The remaining curves of

different line styles and colours (P3 to P5) depict the behaviour of all subsequent current sweeps.


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We interpret the findings displayed in Fig. S2 as follows. Initially, due to the cold working process the atoms in the constriction forming the atomic contact are not in their equilibrium positions, but are trapped in local energy minima. When applying a sufficient voltage, the resulting local electrical field and current density (both quantities are proportional to the voltage in classical physics) will move those atoms in places with the lowest energy barriers to energetically more favourable neighbour positions. When reducing the voltage, the atoms remain in their new positions, even if the bias is reversed. Only higher voltages of either polarity are able to cause further rearrangements.

Switching histograms. In the following, we try to estimate the required energies and forces to perform the switching. In Fig. S3a and S3b we show histograms of the switching currents and switching voltages of roughly 140 experimentally investigated bistable switches operating in the conductance range of 0.5G0 < G < 2.5G0. Every contact contributes with two values in each plot, namely the current or voltage to switch to the low or high conductance state. The switching current shows a maximum at 10 µA with the peak ranging from 5µA to 20 µA, and the switching voltage is maximal at a value of 100 mV, while the main contribution to the peak is located between 50 and 200 meV. In Fig. S3c and S3d we show density plots of conductance vs. switching current or switching voltage, respectively. Again, two pairs of data points (I,G) or (V,G) enter in the plots for each switch, namely those where the switching occurs to the GL or GH state. The density plots reveal that the switching voltages stay at similar values throughout the considered conductance range. Their scatter is broader at the lowest conductances considered and is reduced at higher ones, explaining the rather narrow distribution of switching voltages in the histogram of Fig. S3b. For the currents we see that higher values are needed for rearrangements at higher conductances. The scatter at high G explains the rather broad tail in the histogram in Fig. S3a with currents of up to 50 µA. The narrow distribution of the switching voltages and its relative independence of the conductance indicate that the switching is voltage-controlled rather than current-controlled.

Using this information, we correlate now the switching voltage with energy barriers and forces. When assuming that the conduction electrons transfer their complete energy to the ions, they can provide an energy as high as the applied potential, i.e. around 100 meV. Following Brandbyge et al.S16, a voltage of 100 mV translates into a force in the order of 0.05 nN for Au contacts. We assume that the relationship in Al is similar, perhaps even with a tendency to larger forces, because the transport channels in Al are not completely openS6,S28-S30. In reality, the whole voltage does not drop at those atoms that are rearranging. The thin metallic leads will for instance provide a series resistance, reducing the local potential drop somewhat.


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In conclusion, our experimental data show that the switching of atomic size contacts is controlled by the voltage. For the switching processes realized here, we estimate that the energy barriers and forces involved are in the range of 50 meV to 200 meV and approximately 0.02 to 0.2 nN, respectively.

Figure S3: Histograms of (a) the switching current and (b) the switching voltage, constructed from 138 bistable

switches with conductance values ranging from 0.5G0 to 2.5G0. Density plots of conductance vs. (c) switching

current and (d) switching voltage are shown for a slightly larger conductance range. Bin sizes are (a) 5 µA, (b) 50

mV, (c) 2.5 µA, and (d) 25 mV. The bin size of the conductance scale in panels (c) and (d) is 0.146G0.

5.2 Theoretical results

Microscopic simulations of electromigration phenomena have been performed as yet only for systems with a small number of dynamic atomsS15-S17. It is beyond the scope of this work to extend them to large contacts with hundreds of atoms, as investigated here. In order to identify the driving mechanism for the atomic rearrangements in the switches, we estimate the energy barriers and tensile force changes between adjacent configurations in a stretching process.

For the energy-related aspects, we use the evolution of the total energy E during the elongation


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processes to analyse its jump heights. As an example, E is shown in Fig. S4a as a function of the displacement for the contact discussed in Fig. 4 of the main manuscript. The curve can be divided into elastic stages, showing a monotonous increase of E when bonds are stretched, and plastic stages with sudden, sharp decreases due to rearrangements of atoms when bonds break. Since we are interested in the current-induced rearrangements in small atomic contacts, we restrict our analysis to those parts of the energy evolution with a total conductance between 0.5G0 and 2.5G0, similar to the range used in the experimental plots in Fig. S3. It is visible in Fig. S4a that the energy jump heights in that later part of the curve are clearly smaller than those of larger contacts in the beginning of the stretching process. In the simulations, the continuous, microscopic energy evolution is perturbed by random forces due to the thermostat. In order to get rid of these energy fluctuations, we apply an appropriate smoothing by a repeated averaging over neighbouring data points. (We take averages over 32 neighbouring data points.) The barrier height ΔE is now defined as the difference between a local maximum and the following minimum in the stretching evolution. While the energy evolution in the main panel of Fig. S4a is not smoothed, the inset shows the smoothed curve in the relevant conductance range 0.5G0 < G < 2.5G0. The red-coloured part of the curve in the inset finally illustrates the region between the high and low conductance states of Fig. 4, compatible with the experimental situation in Fig. 3. A histogram of the determined energy barriers is shown in Fig. S4b. For the red histogram the data from all 100 curves of our stretching studies are used with 0.5G0 < G < 2.5G0. The green (blue) histograms select only the relevant conductance range for those 5 (7) opening traces that are compatible with the switches shown in Fig. 3 (Fig. S7a). The energy jump height (“energy barrier”) histogram shown in Fig. S4b exhibits a broad distribution, while most points are in the range of 50 meV to 300 meV.

In the same manner we analyse the tensile force F and its jump height ΔF (“force barrier”). (We compute the force via F = A Σi pzz,i. Here, pzz,i is the z-component of the stress tensor of the atom i and A is the area of the supercell perpendicular to the elongation direction, which coincides with the z axis.) Fig. S4c and S4d show a sample force evolution and the force histograms for the same data as used to create Fig. S4a and S4b, respectively. The evolution of F displays the same characteristics as those of the energy: a continuous increase of forces when bonds are stretched and sudden, sharp decreases when bonds break. The comparison of Fig. S4a and S4c shows that each change in E is accompanied by a corresponding change in F, as expected. Only in the final stage of the stretching process, when single atoms are involved in the rearrangements, ΔF is well below 1 nNS6. The distribution of the force barriers in Fig. S4d features a maximum close to 0.2 nN. Most values are in the interval between 0.05 and 0.5 nN.


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Figure S4: Evolution of (a) the total energy E as a function of the electrode displacement for the stretching process

shown in Fig. 4 of the main text. The energy scale has been set to zero at the lowest value in the opening trace. The

inset depicts those part of the curve corresponding to 0.5G0 < G < 2.5G0, and has been smoothed. The red part of the

curve highlights the energy evolution between the configurations displayed in Fig. 4b and 4c, i.e. those compatible

with the switching process in Fig. 3. (b) Histogram of the energy barriers ΔE. “All configurations” (red) corresponds

to the compilation of all the data obtained from 100 stretching events with 0.5G0 < G < 2.5G0. In contrast,

“Configurations Fig. 3” and “Configurations Fig. S7a” consider only selected stretching processes that contain two

states compatible with the switches shown in Figs. 3 and S7a, respectively. The energy changes for the relevant

stretching processes are then taken into account only between those two configurations that correspond to the low

and high conductance states. (c,d) The same as in (a,b) but for the force F.


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5.3 Discussion and conclusions

While we have not attempted to simulate the experimental electromigration processes, we find that our simulations of the mechanical stretching of contacts yield data consistent with the electromigration mechanism. Thus, the theoretically determined typical energy barriers of 50 meV to 300 meV (Fig. S4b) are in reasonable agreement with the experimental estimate of around 50 to 200 meV (Fig. S3a). Furthermore, we find that forces in the range of 0.2 nN lead to rearrangements of single atoms in the constriction. Based on Ref. S16 and the expectation of slightly higher current-induced forces in Al than in Au, the theoretical data centred around 0.2 nN is again compatible with the experimentally determined switching voltages. The close connection between mechanical and electrical manipulations is further illustrated in Ref. S31 which reports an experiment in which the reversible switching between atomic configurations has been induced by stretching and pushing rather than by electromigration.

To summarise, the combination of the experimental and theoretical results leads to two important conclusions. First, the electronic energies and resulting forces caused by electromigration are sufficient to induce individual bond ruptures and thus atomic dislocations similar as for mechanical deformations. It is also important to mention that the experimentally observed switching thresholds are not much higher than the theoretical ones. This supports the interpretation that the switching is performed by a single or a very small number of atoms. Second, the comparison explains naturally why the motion of the atoms stops after a single dislocation, while the driving current is still applied: The forces exerted by the travelling electrons or by the locally acting electrical field increase with the applied voltage until the contact rearranges. When this local strain is relaxed by a dislocation, the local electrical field changes and is insufficient to cause further bond breakings. When the bias is reversed, the atoms remain in this relaxed configuration until the field surmounts a threshold that lets a reconfiguration become more favourable by overcoming the energy barrier. As stated in the manuscript, the bistable situation arises in roughly 15% of the trials. Reaching a bistable situation implies that the inverted bias causes the reverse motion of the same atom and requires an energy landscape in which all other configurations are prohibited by high barriers. These two conditions are not automatically fulfilled, explaining the limited probability to find bistable switches.

Finally, we note that the typical switching energies are higher than the thermal energy at room temperature. This allows for operating the switches at room temperature as well, provided that oxidation and other detrimental processes can be avoided. In this regard, Au might be a more suitable material than Al. Since the Debye temperature of Au is lower than those of Al, slightly lower switching voltages can be expected. For both metals it has been demonstrated that work


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hardening enhances the number of observable maxima in the conductance histogramsS32,S33. The interpretation of this finding is that the work hardened wires allow less variability of atomic-size contacts. In the same sense, it is common knowledge in the field in particular when using STMs for producing the contacts, that “good” Au contacts, i.e. contacts with conductance values corresponding to integer multiples of G0, are more frequently observed when the system has been trained by repeated indentation of the tip into the counter electrode. We believe that the alternating current protocol that we apply here has a similar, but probably more locally acting work hardening effect. In the same spirit, atomic contacts of the refractory metals W, Mo, V, Ti appear as promising candidates for room temperature atomic switches.

6. Comparison of experimental and theoretical transmission channel probabilities

In this section, we present comparisons of differently obtained experimental transport properties, namely those of mechanically prepared and electromigrated contacts. We show that similar transmission channel distributions are obtained in both cases, demonstrating that the bistable switches can be related to mechanically stretched contacts. In addition, these results are compared to 100 simulated stretching curves to assess the accuracy of our modelling. The study of the opening of transmission channels as a function of the conductance as well as of the transmission channel histograms indicates that due to their good agreement, we can use the simulations to identify the number of atoms that is involved in a switching process.

6.1 Transmission channel opening

Figure S5 shows the mean channel transmissions for τ1 to τ4 as a function the conductance. Note that the channels are labelled according to their size: The channel with the highest transmission is τ1, the second highest τ2, etc. We compare the theoretical results (“Theory”), obtained from the computed opening traces, to the data acquired from superconducting measurements on bistable switches (“Bistable”) and on mechanically prepared contacts (“Stretched”). The data set “Stretched” corresponds to the initialization stage of the bistable switches. They are obtained by tuning the bending beam such that a junction of several 10G0 yields a conductance below 3G0. Before starting the electromigration protocol, a single measurement of the IV in the superconducting state is performed to obtain the transmission channel distribution. This data is collected in the set “Stretched”, while “Bistable” contains the τi of the high- and low-conductance state of the first switching cycle of each switch. Within the standard deviation, we observe a good agreement between all three data sets.

For monovalent metals the channels open rather one-by-oneS5,S26,S29, i.e. the second channel


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opens only when the first channel has almost reached its maximum value of τ1 = 1. In contrast, it is visible from Fig. S5 that in Al the second and third channel have a considerable contribution before the first one has saturated. This coincides with the expectations for multivalent metals, where several electronic bands contribute to the density of states at the Fermi energy.

The comparison also shows deviations in the details of the channel distributions. For instance, the theory slightly overestimates the strength of the highest transmission channel τ1 compared to the experimental findings. Consequently, the less transmitting channels have to be somewhat underestimated to keep the total conductance fixed. It is expected that disorder in the electrodes and additional scattering processes, which are not accounted for in the calculations, reduce the experimental transmission valuesS29.

Figure S5: (a-d) Transmission probabilities τi of the most transmitting eigenchannels i = 1,2,3,4 as a function of the

conductance. The three curves represent the experimental (“Bistable”, “Stretched”) and theoretical (“Theory”)

averages, while error bars show the standard deviation of the data points. The conductance bin size is 0.1G0.


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6.2 Transmission channel histograms

Figure S6 shows the distribution of transmission channel probabilities of contacts with a conductance in the range of 0.8G0 < G < 1.1G0, i.e., we concentrate here on junctions with a conductance in the range of single-atom contacts. Considering the first 4 channels, we compare the experimental data collected by current-voltage measurements for the electromigrated bistable switches (Fig. S6a) with mechanically stretched contacts (Fig. S6b) as well as with the theoretical results from 100 stretching simulations for wires grown along the 〈111〉 direction (Fig. S6c). These so-called “transmission channel histograms”, taken for a certain conductance range, contain similar information as Fig. S5, but illustrate more clearly the distribution of the transmission values around their mean, which is merely summarized by the standard deviation in Fig. S5.

Figure S6: Comparison of experimental and theoretical transmission channel histograms of single-atom contacts

with a conductance in the range of 0.8G0 < G < 1.1G0. Shown are the histograms for the four channels with the

largest transmission probabilities τi with i = 1,2,3,4 for (a) bistable electromigrated switches, (b) measured

mechanically stretched contacts, and (c) simulated mechanically stretched contacts.

The comparison of the three panels of Fig. S6 reveals clear similarities in the distribution of the individual channels for all data sets. There is a rather broad and shallow maximum arising from 0.5 < τ1 < 1.0 as well as a high probability to find channels i = 2,3 with a transmission of up to


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0.3. The contribution of the forth channel, τ4, does not appear in the simulations, but is also faint in the experiments. The probability to find a very transparent channel τ1 is somewhat overestimated by the simulations, as was already observed in Fig. S5.

Overall, the comparison below suggests that the experimental configurations obtained by electromigration are comparable to those obtained by mechanical stretching, and both are in good agreement with our simulation results. We therefore conclude that the simulations fulfil another important condition, necessary to identify the geometrical changes taking place in the experimentally realized atomic switches.

7. Statistical behaviour of bistable switches

For producing the bistable switches we tried several variations of the electromigration protocol. The alternating-current protocol described in the manuscript resulted in the highest ratio of long-lived bistable switches. Some of the switches lived for more than 10 hours and 500 repetitions (see section 9). We did not explore the maximum lifetime systematically but rather gathered statistical data on the typical behaviour of the switches. We usually recorded 4 switching cycles per switch (see Figs. 3 and S7) and then closed or opened the contact further by bending the substrate to explore another configuration. As mentioned in the main text, we concentrated on those switches that locked into the bistable state after up to 20 irregular steps. This happened in roughly 15% of the trials. Increasing the number of maximally allowed irregular steps to 40, a higher percentage was obtained. Thus the training period can be adjusted to obtain a higher ratio of bistable switches.

Summarizing the typical behaviour deduced from more than 150 of these bistable switches in the conductance range of up to 3G0, there is a preference to switch without a change in the number of open channels or with a difference of 1. Only in rare cases and when ΔG > 0.8G0, their number changes by 2 or more. When the number of channels changes, in the majority of the cases (~90%) the state with the higher number of channels has the higher conductance, but sometimes (~10%) this is opposite (see also Fig. S7). In rare cases also switches operating between three or four discrete conductance states were observed.

Two additional examples of switches are shown in Fig. S7. Fig. S7a represents a switch of the type (i) of the manuscript between 4 and 3 open channels, while the reconfigurations in Fig. S7b lead to 3 and 2 channels. In the latter case, the higher conductance state results from a smaller number of open channels. (Note that in the third repetition for the switch in Fig. S7a, a fitting


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with 5 channels gave a marginally better agreement than the former 4 channel configuration.)

Figure S7: Same as Fig. 3a of the main text but for two different bistable systems. The total conductance changes

between (a) GL = 0.95G0, GH = 2.26G0 and (b) GL = 0.61G0, GH = 0.92G0. The transmission channel probabilities are

(a) τ1 = 0.61, τ2 = 0.20, τ3 = 0.12 (blue), τ1 = 0.89, τ2 = 0.84, τ3 = 0.26, τ4 = 0.24 (red) and (b) τ1 = 0.39, τ2 = 0.12, τ3

= 0.10 (blue) and τ1 = 0.78, τ2 = 0.12 (red). The given values are the best-fit results obtained for the first cycle. The

mean values over all shown repetitions are (a) τ1 = 0.61, τ2 = 0.21, τ3 = 0.09 (blue), τ1 = 0.90, τ2 = 0.83, τ3 = 0.28, τ4

= 0.21 (red) and (b) τ1 = 0.39, τ2 = 0.13, τ3 = 0.09 (blue), τ1 = 0.78, τ2 = 0.12 (red).

8. Atomic configurations of bistable switches

In Fig. S7a a switch is shown that changes from 4 to 3 open channels and back. With our atomistic simulations we were able to identify a typical geometric rearrangement corresponding to this situation. In Fig. S8a we show the conductance versus the electrode displacement for a selected stretching process. At a displacement of 1.75 nm the total conductance and the τi agree with the experimental situation to within 0.2G0 and 0.2 for each channel, respectively. The same tolerances are fulfilled at the displacement of 1.85 nm, when the number of open channels has decreased from 4 to 3. The corresponding configurational changes are shown in Fig. S8b and S8c. We find an equivalent transition (regarding both conductance values and τi) and equivalent configurational changes in 6 of the 100 simulated stretching events. This can be considered as a significant portion, since in the experiments we find 13 realizations changing from 4 to 3 channels in a total of 150 switches. (These 150 switches exhibit channel number combinations between 2 and 7.) Out of these 13, 9 examples have a channel distribution compatible with the one shown in Fig. S7a.

As it is visible in Fig. S8, the atomic rearrangements require the rupture of two bonds, namely


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one at atom 1 and another one at atom 2. The switch of Fig. S7a is thus consistent with a wire that initially exhibits two atoms in its narrowest constriction and changes to a dimer. Since bonds at the two dimer atoms break, we call this a “two-atom switch”.

Figure S8: (a) Calculated opening trace showing the conductance as a function of the electrode displacement

together with the corresponding decomposition into transmission channels. Also the number of open channels is

indicated for the criterion τi > 0.05. (b) The configuration of the wire at the displacement of 1.71 nm with a

conductance of G ≈ 2.17G0 and 4 open channels with transmissions τ1 = 0.97, τ2 = 0.73, τ3 = 0.26, τ4 = 0.18 and (c) at

1.75 nm with G ≈ 0.91G0 and with 3 open channels τ1 = 0.62, τ2 = 0.20, τ3 = 0.09. Both displacements are marked by

vertical dashed lines in (a). The red dashed lines in (b) indicate the bonds that are broken to form the configuration

shown in (c).

The second type of switch that occurs with enhanced abundance is the 3 channel to 3 channel switch described in the main text. Using the error bounds indicated above, we have found 5 stretching events out of 100 with transmission properties that are compatible with the particular experimental switching process, displayed in Fig. 3a. For 1 event, the two conductance states emerge from the elastic deformation of a single-atom contact. The remaining 4 of the 5 events, on the other hand, are consistent with the type of geometrical change indicated in Fig. 4 of the main text, which involves the transition from a single-atom contact to a dimer contact. Counting again the total number of experimental bistable switches with 3 channels in both states, we find 17 in total, out of which 8 are also quantitatively equivalent to the example shown in Fig. 3a. The situations thus occur at a similar frequency both in the experiment and in the simulations.

The dimer is the typical end configuration of a stretched Al wire before rupture, as has been reported before in Refs. S6 and S30. Here we find that this end situation can be achieved via two


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preferred pathways, namely starting from the short one-atom contact (monomer) with three channels and from the contact with two atoms in the cross section yielding 4 channels. Both of these pathways are reflected in an enhanced probability to find these channel combinations in the experimentally investigated switches.

9. Example of a long-lived bistable switch

For some of the switches we tested their long-time stability for measurement periods of up to 10 h and more than 500 switching cycles. Fig. S9 displays an example recorded at T = 250 mK in which the conductance switches between 1.3G0 and 1.8G0 reproducibly back and forth 526 times. At around 9.25 h the switching current changed, resulting in a slightly increased switching frequency. After 9.9 h we stopped the measurement and formed a new contact by bending the substrate. Equivalent tests have been performed for higher conductance values (with GL ≈ 5-7G0) with similar findings regarding the stability.


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Figure S9: Long-time behaviour of a bistable atomic switch with more than 500 cycles recorded during around 10 h

at T = 250 mK. The upper trace in each frame shows the conductance (left axis), while the lower trace represents the

control current (right axis). After the measurement period the data acquisition has been stopped intentionally.


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supplementary information supplementary information...combination of the single channel basis...

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