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Electromigration
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== Step Bunching due to Electromigration == Step bunching on DC-heated sublimating vicinal crystal surfaces of Si(111) was observed by A. Latyshev et al. in 1989.<ref name="Latyshev1989">A. V. Latyshev, A. L. Aseev, A. B. Krasilnikov, and S. I. Stenin, "Transformations on clean Si(111) stepped surface during sublimation," Surface Science '''213''', 157 (1989).</ref> Soon after, Stoyan Stoyanov advanced a model in which as the reason for step bunching is identified the biased diffusion of the adatoms.<ref name="Stoyanov1991">S. Stoyanov, "Electromigration induced step bunching on Si surfaces—how does it depend on the temperature and heating current direction?", Japanese Journal of Applied Physics '''30''', 1 (1991).</ref> In 1998, Stoyanov and Tonchev extended Stoyanov’s model by incorporating step-step repulsions<ref name="Stoyanov1998">S. Stoyanov and V. Tonchev, "Properties and dynamic interaction of step density waves at a crystal surface during electromigration-affected sublimation," Physical Review B '''58''', 1590 (1998). DOI: [10.1103/PhysRevB.58.1590](https://doi.org/10.1103/PhysRevB.58.1590)</ref> and derived a scaling relation for the minimal step-step distance in a bunch under diffusion-limited sublimation, non-transparent steps, and step-down current conditions: <math> l_{\min} \sim N^{-2/3} </math> where <math> N </math> is the number of steps in the bunch, and the proportionality coefficient has the dimension of length. This scaling law has been confirmed by numerous experimental studies.<ref name="Fujita1999">K. Fujita, M. Ichikawa, and S. S. Stoyanov, "Size-scaling exponents of current-induced step bunching on silicon surfaces," *Physical Review B* '''60''', 16006 (1999). DOI: [10.1103/PhysRevB.60.16006](https://doi.org/10.1103/PhysRevB.60.16006)</ref><ref name="Homma2000">Y. Homma and N. Aizawa, "Electric-current-induced step bunching on Si(111)," Physical Review B '''62''', 8323 (2000). DOI: [10.1103/PhysRevB.62.8323](https://doi.org/10.1103/PhysRevB.62.8323)</ref><ref name="Gibbons2006">B. J. Gibbons, S. Schaepe, and J. P. Pelz, "Evidence for diffusion-limited kinetics during electromigration-induced step bunching on Si(111)," Surface Science '''600''', 2417 (2006). DOI: [10.1016/j.susc.2006.02.025](https://doi.org/10.1016/j.susc.2006.02.025)</ref><ref name="Usov2011">V. Usov, C. O. Coileain, and I. V. Shvets, "Experimental quantitative study into the effects of electromigration field moderation on step bunching instability development on Si(111)," Physical Review B '''83''' (2011). DOI: [10.1103/PhysRevB.83.245429](https://doi.org/10.1103/PhysRevB.83.245429)</ref> In 2018, Toktarbaiuly et al. reported electromigration-induced step bunching on vicinal W(110) surfaces. Their study revealed that step bunching occurred for both step-up and step-down current directions at the same temperature, T = 1500°C, with distinct size-scaling exponents depending on the current direction.<ref name="Toktarbaiuly2018">O. Toktarbaiuly et al., "Electromigration-induced step bunching on tungsten (110) surfaces," Physical Review B '''97''', 035436 (2018). DOI: [10.1103/PhysRevB.97.035436](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.035436)</ref> More recently, Usov et al. (2020) demonstrated that electromigration-induced step bunching is not limited to silicon surfaces but can also occur on dielectric surfaces, such as sapphire (Al₂O₃(0001)).<ref name="Usov2020">V. Usov et al., "Revealing electromigration on dielectrics and metals through the step-bunching instability," Physical Review B '''102''', 035407 (2020). DOI: [10.1103/PhysRevB.102.035407](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.035407)</ref> This study suggests that the fundamental mechanism of step bunching on W(110), Al₂O₃(0001), and Si(111) follows similar principles. Moreover, annealing W(110) offcut in the [001] direction with an up-step current produced a morphology where the bunch edges formed zigzag segments meeting at right angles.
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