Fabrication of atomic junctions with experimental parameters optimized using ground-state searches of Ising spin computing

Feedback-controlled electromigration (FCE) is employed to control metal nanowires with quantized conductance and create nanogaps and atomic junctions. In the FCE method, the experimental parameters are commonly selected based on experience. However, optimization of the parameters by way of tuning is intractable because of the impossibility of attempting all different combinations systematically. Therefore, we propose the use of the Ising spin model to optimize the FCE parameters, because this approach can search for a global optimum in a multidimensional solution space within a short calculation time. The FCE parameters were determined by using the energy convergence properties of the Ising spin model. We tested these parameters in actual FCE experiments, and we demonstrated that the Ising spin model could improve the controllability of the quantized conductance in atomic junctions. This result implies that the proposed method is an effective tool for the optimization of the FCE process in which an intelligent machine can conduct the research instead of humans.

learning beyond the baseline model that finds the ground state. Our work differs from that presented in previous reports in that we apply the optimization results obtained with the Ising spin computing to actual experiments in the research domain. The ability to optimize FCE parameters using Ising spin computing is equivalent to being able to address this problem with quantum annealing machines/quantum annealers based on the Ising spin model. Compared with the method of simulated annealing 31 , the performance of quantum annealers is superior and advantageous for solving problems with energy landscapes that have complicated tunneling barriers 32 . Therefore, in the future, complex optimization problems concerned with physical phenomena are also foreseen to be solved efficiently using quantum annealers. Our approach opens new avenues for bridging the gap between key problems of parameter optimization in fabricating nanoscale devices and their implementation to Ising spin computing. Figure 1 schematically illustrates the proposed system. We first outline our approach for constructing a database of the FCE (Fig. 1a) and then show how it is "informed" from our database (Fig. 1b). In this study, we focused on the feedback (FB) voltage V FB , which is one of the FB parameters, and collected the data of V FB . V FB determines the amount of voltage reduction, which suppresses the heat of the channel. Therefore, this parameter plays an important role for the control of quantized conductance. In the mapping step (Fig. 1c), the problem of optimizing V FB is mapped into the Ising spin model. The mapping step consists of two parts. One represents the evaluation of fce experimental parameters using cost function. We set the cost function, which evaluates the values of V FB , to map the V FB schedule onto the Ising spin model. Here, the cost function is assumed to become large when the quantized conductance smoothly plateaus, and the step-wise conductance subsequently and clearly drops by G 0 (G 0 = 2e 2 /h = 77.6 μS, where e is the electron charge, and h is Planck's constant). The time-dependent conductance curve corresponds to the change in the structure at the constricted section of the Au nanowires 33 . Therefore, we set five variables to represent the features of the conductance curve. Figure 2a shows a schematic of the conductance trace and applied voltage V and represents the variables in the cost function. P 1 is the number of data points counted from G ref in the quantized conductance that lie within a conductance window G ref ± 0.5 G 0 in each conductance trace, L is the number of data points from G ref to the end of the FB cycle G E , D is the decrease in conductance in the region of EM occurrence, F is the decrease in conductance during one EM cycle between the start of the FB cycle G S and G E , and P 2 is the change in conductance between G min and G max during the application of ramp voltage. Furthermore, N is set as the number of voltage FB cycles during the FCE process. In an ideal control scheme, D and F are close to 1 G 0 , P 1 /L, which means the stability of the conductance trace, is set to 1, and P 2 is set to 0 G 0 . Therefore, the cost function has the form Here, we defined the output using Eq. (1) as the Score of V FB values (X%) at nth times of FB. When the conductance trace is close to these ideal values, the s n (X) increases until infinity. Here, Eq. (1) is applied below 40 G 0 . In this region, the electron transport is in the ballistic regime and the EM is activated with the one-by-one removal of atoms, which is driven by the kinetic energy transfer from single conduction electrons to single Au atoms 34,35 . Hence, using Eq. (1), we can evaluate the V FB values from the point of view of the conductance behavior under the voltage FB process. Figure 2b presents the typical result of the conductance G of the Au nanowires and the voltage V as a function of the process time t when a random V FB schedule is applied to an FCE. The vertical axis of Fig. 2b is the conductance G of the sample over G 0 . Furthermore, it should be noted that the decrease in the conductance of "1" G 0 corresponds to a one-by-one removal of Au atoms 34,35 . Three FB points are indicated by black arrows, and Fig. 2b also presents the results of s n (X) obtained from Eq. (1). From Fig. 2b, the s 11 (40), in which V FB = 40% at FB point 2, indicating smooth plateaus caused by quantized conductance, is much higher than s 10 (20) (at FB point 1) and s 12 (50) (at FB point 3). Therefore, the controllability of the quantized conductance can be evaluated with Eq. (1).
We quantify the correlation between the nearest-neighbor V FB values through the following formula: n n n n trans , 1 1 where the combination of s n (X) and s n+1 (Y) represent s n, n+1 (X, Y) occurring at V FB of X% and Y% during an FCE process. In Fig. 2c, we show the mutual information between V FB values. For example, s 1, 2 (10,20) represents the evaluation for V FB transition from 10% to 20% at the first to the second voltage FB cycles. In the FCE method, generally, Au nanowires should gradually narrow with the progress of FB cycles/steps, which is due to the EM. In other words, the conductance of Au nanowires should decrease with the evolution of FB processes. Here we hypothesize that a correlation exists between subsequent feedback steps because the EM proceeds by removing one atom at a time. Therefore, the correlation between nearest-neighbor V FB values is assumed to be equivalent to considering the junction states between FB cycles. The pixels in Fig. 2c illustrate the fact that the transitions of the V FB values strongly influence the evaluated values of an FCE process. As a result, we obtain the suitable combination of V FB values for the controllability of quantized conductance during the FCE.
www.nature.com/scientificreports www.nature.com/scientificreports/ Mapping of fce schedules onto ising spin model. We map the V FB schedule onto the Ising spin model using the cost function described above. The formulation of H Ising , the system Hamiltonian (total energy of the system), for the Ising spin model can be described by  www.nature.com/scientificreports www.nature.com/scientificreports/ Typically, a real-world problem formulation requires conversion to the Ising Hamiltonian. In this research, the V FB schedule is mapped into fully connected two-dimensional Ising spin cells consisting of N (row) × 9 (column) spins. As a result, the minimum value of the system energy and the corresponding optimum provide the solution to the original problem. In this model, each row corresponds to the order of V FB values (i, j ∈ 1, …, N) and each column corresponds to the V FB values (x, y ∈ 1,…, 9). Each spin (σ i, x or σ j, y ) takes a binary value of "+1" or "0, " which represents an up spin or a down spin. Furthermore, a mapped V FB schedule is finally represented by a trace of spin value "+1. " The strength of the connection between spin (i, x) and spin (j, y) is denoted by J ix, jy . Each spin is also subject to the external magnetic field of h i, x . The formulation of the Ising Hamiltonian for the optimization of the FCE process is explained below. First, to select the highly rated schedules, the term is set into the system Hamiltonian, where W x, y is based on Eq. (2). For example, W 1, 2 is decided based on the evaluation of the transition of V FB from 10% to 20%. In addition, δ i, j is equal to 1 (i = j) or 0 (otherwise). Penalty terms are added such that spin states that do not correspond to valid solutions are penalized. Then, to ensure no two V FB overlap, we introduce a penalty term, Finally, to ensure that the total number of spins on "+1" is exactly N, is added to the system Hamiltonian. Hence, the cost function for V FB scheduling, which is represented by H FCE , is given as where A, B, C, and D are positive constants. A valid solution will have H overlap = H total = 0.
Solution searches of fce experimental parameters using ising spin computing. Here, we first provide and set the optimal solution for our simulation to measure the performances of our Ising spin computing. According to the database, which consists of 50 samples, the average number of FB steps in the FCE process in a sample was approximately ten. Therefore, in the calculation experiments, we set 90 spins (N = 10, N × 9 = 90) with 10,000 iterative cycles. A single iterative cycle describes the flips of all spins chosen at random. Furthermore, the update of spins depends on the temperature T through the simulated annealing procedure 31 with the aim of escaping from a local minimum. In each step of this algorithm, a spin is given a small random displacement and the resulting change, ΔH, in the energy of the system is computed. The probability that the configuration is accepted is P(ΔH) = exp(−ΔH/T). Random numbers uniformly distributed in the interval (0, l) are selected and compared to P(ΔH). If they are less than P(ΔH), the new configuration is retained; if not, the original configuration is used to start the iteration.
In the test problem we set, at the global minimum with a system energy H FCE of −4,175, the final spin status appears and traces the following: V FB (%) = 60 → 70 → 80 → 90 → 10 → 20 → 30 → 40 → 50 → 60. Figure 3a shows the system energy H FCE and temperature T as a function of the number of iterative cycles. In this case, the computation is complete within seconds because we used conventional computing resources. Over time the energy decreases and is characterized by upward and downward movements. After 10,000 iterative cycles, the system energy reaches −4,175. Therefore, the system energy is considered to have reached the global minimum we set through the ground-state searches of the Ising spin computing. Figure 3c also presents the final spin status of the system at the global minimum energy. Each spin is expressed by a square. The spin state in orange in Fig. 3c represents the optimal solution set of V FB scheduling with H FCE = −4,175. Based on these results, Ising spin computing can determine the optimal solution of experimental parameters for the V FB schedule.
To demonstrate the utility of our Ising spin computing, we apply it to a database containing conductance measurements of 50 Au nanowires. Figure 3b plots the process of the ground-state search based on the experimental data stored in the database. As seen in Fig. 3b, the energy profiles exhibit similar characteristics to those in Fig. 3a. Furthermore, the final ground-state spin configuration in Fig. 3d represents a suitable V FB schedule that alternates between V FB of 20% and 60% with time. This result traces the spins with highly evaluated values of V FB obtained from the database consisting of 50 samples. Thus, Ising spin computing progressed from the entirely random data of the V FB schedule towards an indication of the experimental parameters for the V FB schedule. Clearly, an ordered V FB schedule can be generated from the disordered data stored in the database by using the proposed Ising spin computing model. Score of V fB schedules obtained with solution searches. Next, we investigated the ability of our Ising spin computing to search for the ground-state solution. We did this by comparing the score of the V FB scheduling obtained from the Ising spin computing and random selection. Here, we defined the Score of V FB schedule as Notice that s n,n+1 (X, Y) is s N,1 (X, Y) when n = N. In other words, Eq. (8) evaluates the entire V FB scheduling. Because N is set to 10 in this study, a V FB schedule consisting of 10 FBs is evaluated by Eq. (8), and S 10 is calculated. The detailed calculation procedure of S 10 is as follows. First, a V FB schedule consisting of 10 FBs is generated with Ising spin computing or random selection. Next, s 1,2 , s 2,3 , …, and s 10,1 are scored by the database shown in Fig. 2c. Finally, S 10 is calculated by adding s 1,2 , s 2,3 , …, and s 10, 1 . Figure 4a presents a histogram of the Score process obtained from the Ising spin computing (orange histogram) and random selection (blue histogram) when both are performed 1,000 times. All initial spin values were random in all experiments described in this paper to explore the various conditions. The blue histogram in Fig. 4a exhibits uniform Gaussian-like score distributions, www.nature.com/scientificreports www.nature.com/scientificreports/ which represent the validity of the database. In addition, the distribution of the histogram shifting to the right indicates that the highly rated V FB schedule is selected. From Fig. 4a, the orange histogram of the Ising spin computing is significantly shifted to the right as compared with the histogram obtained from random selection. This result indicates that Ising spin computing can automatically select the highly evaluated V FB schedules through the ground-state searches of the solutions. Figure 4b-d show the top three final spin statuses corresponding to each Score process . The highest scoring solution (Fig. 4b) is considered the optimum schedule of V FB . Further, the results of the scoring solutions for the second and third place in Fig. 4c,d indicate that the heuristic solutions almost agree with the optimal schedule. Therefore, the local minimum solutions obtained by Ising spin computing are also considered suitable for use in the FCE method. These results show that Ising spin computing could discover a remarkable amount of information about the FCE by starting from disorderly generated data; that is, these are the results of applying machine learning to disordered data to obtain order in exploring the experimental parameters of the FCE method.
fce experiments with results of ground-state searches of experimental parameters. Figure 5a,b present the conductance traces recorded during the FCE with the schedule of the 1st Score process obtained from the Ising spin computing and random V FB scheduling, respectively, which are prepared as a www.nature.com/scientificreports www.nature.com/scientificreports/ reference. In the initial stage denoted as region 1, where the conductance is larger than 40 G 0 , the conductance decreases continuously because of the EM of Au atoms. This result indicates that region 1 shows the bulk regime of electrical conduction. However, as the conductance decreases to less than 40 G 0 , conductance plateaus and steps at and near integer multiples of 1 G 0 are observed, as seen in region 2. This result implies that region 2 shows the ballistic conduction regime and that EM proceeds with the behavior of the one-by-one removal of single Au atoms 34,35 . Here, to compare both methods from the viewpoint of the controllability of conductance quantization, enlarged views of region 2 are shown in Fig. 5c,d. The conductance changes in steps, consisting of plateaus and jumps. Further, Fig. 5e,f present histograms of the conductance values derived from the EM data of Fig. 5c,d, respectively, in the range G ≤ 45 G 0 . As the conductance changes whenever the atomic structure changes at the metal contacts, stable structures can yield preferred peaks in the conductance histograms 33 . Therefore, the sharp peaks indicate that particular conductance values and their corresponding contact configurations are more stable. Conductance plateaus and steps at and near integer multiples of G 0 are observed in Fig. 5f as compared to Fig. 5e. We further applied Eq. (1) to each conductance trace in the range G ≤ 40 G 0 to compare V FB scheduling in terms of Score VFB . Figure 5g,h show the Score VFB and the insets show the highest evaluated conductance traces. As shown in Fig. 5g,h, the results of the optimal V FB scheduling obtained from Ising spin computing are superior to those from random V FB scheduling not only from the conductance histograms but also from the viewpoint of Score VFB . In addition, the conductance trace in the inset of Fig. 5h shows conductance plateaus around 10 G 0 . These results suggest that the ballistic conduction regime in the electrical conduction is precisely controlled by the V FB scheduling obtained by the ground-state solution of the Ising spin computing, despite the complicated phenomenon of mass transport in electromigrated nanowires.
To gain insights into the dynamics of EM in Au nanowires, we accumulated V C and plotted its histogram in Fig. 6a (applied using random V FB schedule) and Fig. 6b (applied using optimized V FB schedule). V C is the maximum value of the junction voltage V J for each FB cycle 34,35 . Each histogram consists of data collected from 22 samples. In addition, each inset in Fig. 6a,b shows a histogram constructed from the V C accumulated for the region of G ≥ 40 G 0 . Umeno et al. 34,35 reported that eV C can be regarded as "the chemical potential for atom migration" and, hence, the obtained histogram can be interpreted as "the spectrum of electromigration". As seen in Fig. 6a,b, major peaks coincide with the activation energies for the surface potential of Au adatoms [36][37][38] , results that are similar to those in ref. 35 . Hence, Fig. 6a,b clearly indicate that the process of EM at the Au nanowires is driven by the microscopic kinetic energy transfer from a single conduction electron to a single metal atom 34,35 . In addition, the difference between Fig. 6a,b appears in the range from 0.26 to 0.4 eV, showing a major peak at 0.3-0.32 eV in the case of Fig. 6b. This value is equal to the activation energy for the self-diffusion on the (100) surfaces, which was predicted by the embedded atom model 38 . This is due to the difference in the V FB scheduling in the ballistic regime (G ≤ 40 G 0 ) between the random V FB schedule (Fig. 6a) and the V FB schedule optimized by Ising spin computing (Fig. 6b). The findings obtained here provide a solid basis for understanding the migration of Au atoms at atomic junctions driven by a controlled EM method.

conclusions
In this study, we established a novel design method for determining FCE parameters using Ising spin computing. Optimization of the parameters of the FCE process using the Ising spin computing method provides an alternative to the conventional method with human intervention that requires expert knowledge of the physical processes and methods involved. The advantages of using the Ising spin model over traditional machine-learning methods for small training datasets have been reported 27,28 . Further, we showed that the FCE process can be optimized with a small database of only 50 samples. Therefore, in areas of research where datasets with a small number of relevant samples may be more common, Ising spin computing may be the algorithm of choice. However, this improvement could be even greater when expanding the database and/or optimizing more FCE parameters such as G TH and V STEP , using Ising spin computing because the database is too small. In fact, as the size of the training dataset increased, the results improved to a certain degree in "a Higgs optimization problem 27 " and "a simplified computational biology problem 28 . " Furthermore, in the case of maximum-cut problems, the accuracy of the solutions remained almost constant for various problem sizes 39 . Hence, the same tendency is expected to be obtained in FCE parameter optimization with Ising spin computing.
In this study, ground-state searches of Ising spin computing were performed by simulated annealing, which is a classical method. However, in the future, the development of quantum annealers may make it possible to optimize the experimental parameters of FCE using quantum annealing considering that various optimization problems have been solved with commercially available quantum annealers [23][24][25][26][27][28][29][30] . Therefore, the quantum annealing technique is expected to provide appropriate solutions or optimize the experimental parameters of FCE. Furthermore, the Ising spin model features intrinsically nonlocal correlations that can lead to substantially more compact representations of many-body quantum states. Therefore, their effectiveness for the optimization of the experimental parameters of the FCE method makes them a powerful tool for applications beyond the formation of nanogaps and/or controlling quantized conductance in nanowires, including understanding complicated quantum-mechanical systems.

Methods
Au nanowire fabrication and design. A Si wafer with a thermally grown SiO 2 layer is used as the substrate; next, a positive tone resist was spin coated onto the substrate. The electrode pattern with a constriction of ~100 nm was fabricated using electron-beam lithography. A standard development procedure was applied to remove the resist by inserting the substrate into a development solution, after which the substrate was transferred into propanol to terminate the development process. The resist layer served as a mask for deposition of the metal layer, after which the sample was immersed in acetone for the lift-off process to remove the redundant part of the metal layer. electrical characterization. A source-measure unit was used for sourcing the voltage (up to 10 V and with 50 μV sensitivity) and measuring the voltage (up to 10 V with 10 μV sensitivity) and the current (up to 10 mA with 10 nA sensitivity), allowing the characterization of current-voltage behavior with a time resolution of 25 ms. All measurements were carried out under a two-terminal arrangement. The FCE experiments for the Au nanowires were performed at room temperature under ambient conditions using a real-time operating system (RTOS)-based controller, which is specifically designed to run FCE applications with precise timing and reliability. The system consisted of a host PC, RTOS, and target samples of Au nanowires. In this system, we first The major peaks agree well with the surface self-diffusion potentials of Au atoms. A remarkable finding is that the large peak at 0.12-0.14 eV coincides with the activation energy for self-diffusion on the (111) surfaces 36 . The small peak visible at 0.26-0.28/0.24-0.26 eV is close to the self-diffusion on the (110)-(1 × 1) surfaces predicted by molecular dynamics simulations 37 . Furthermore, 0.3-0.32 eV is close to the theoretically predicted activation energy for the self-diffusion energy on the (100) surfaces 38 .