Theory Of Electromigration Research Articles

Electrical current-assisted reactive crucible melting technique: Case study of the Fe-Sn system

Modern materials science relies heavily on the advancement of combinatorial high-throughput experimental and computational methods to discover new advanced materials. Among these methods, the reactive crucible melting (RCM) technique stands out as particularly promising. However, a significant challenge in RCM combinatorial analysis is the occurrence of "missing" phases. In this study, we successfully addressed this limitation through the application of electrical current stressing. Furthermore, we investigate the influence of electric current stressing on diffusion rates and the phase formation processes within the reactive crucible. We employed Comsol Multiphysics modeling to analyze the experimental results in the framework of electromigration theory, focusing on the distribution of electrical field lines, temperature gradients, and current density values. Additionally, through the integration of finite element calculations with analysis of Electron Backscatter Diffraction (EBSD) and Magneto-Optical Kerr Effect (MOKE) data, we estimated the gradient of internal stresses along the boundaries of the crucible's reaction volume induced by the diffusion of electrically-induced vacancies and adatoms. Due to modest values of the current density in our experiment and the disparity between the rates of crucible body dissolution and vacancy diffusion, no compressive or tensile stresses were observed. Nonetheless, we propose that the developed electrical current-assisted reactive crucible melting technique holds promise for synthesizing metastable phases existing under high positive or negative pressures.

Electromigration enhanced growth kinetics of intermetallics at the Cu/Al interface

The growth kinetic behaviors of the Cu/Al system under current were investigated. The current was shown to significantly promote the growth of the intermetallic (IMC) layer. The growth rate constant of the Al4Cu9 layer under current was approximately 3.26 times higher than that processed without current. Moreover, the current direction leads to a significant difference in the growth rate of the IMC layer, which was attributed to the electromigration effect. Based on the traditional thermal-activated theory and electromigration theory, an electromigration and thermal dual-activated growth kinetic model incorporating the contributions of Joule heat and the nonthermal effect was constructed. The growth kinetic behaviors of the Al4Cu9 layer were investigated by applying this model. The calculated growth activation energies under different current densities were almost constant and close to 130 kJ/mol. The experimental data were applied to verify the model, and the results showed that the model was in good agreement with the experimental results.

Accelerated test device and method for critical chloride concentration initiating steel depassivation in cement-based materials

Abstract In order to overcome the disadvantages of traditional test devices and methods, new accelerated test device and efficient test method for critical chloride concentration (Ccrit) initiating steel depassivation in cement-based materials were developed. Based on the theory of electromigration, a new accelerated test device for Ccrit was invented first by combining with the chloride electromigration system and electrochemical test system. Then the influences of applied voltage and continuous power-on time on electrochemical parameters including open-circuit potential, polarization resistance, and corrosion current density during the steel depassivation process were investigated. Moreover, an efficient test method for Ccrit was proposed based on the quantitative criterion in terms of corrosion current density. Analysis results show that Ccrit can be measured quickly and accurately based on the accelerated test device and method by selecting appropriate applied voltage (e.g., 3–6 V) and continuous power-on time (e.g., 48–72 h).

Synchrotron X-ray study of electromigration, whisker growth, and residual strain evolution in a Sn Blech structure

Synchrotron X-ray analysis of the Sn electromigration behavior and Sn whisker growth in a Blech structure using nano-X-ray fluorescence microscopy and white beam Laue nanodiffraction was conducted. Sn depletion at the cathode and whisker/extrusion formation at the anode were characterized in-situ, and the results obeyed the electromigration kinetics. This electromigration scenario gradually decayed because of the counterbalance between electron wind force and back stress. White beam Laue nanodiffraction analysis showed that a noticeable compressive deviatoric stress in the direction of electron flow built up at the anode of Sn strips, particularly in the roots of Sn whiskers, confirming that electromigration-induced atomic accumulation occurred downstream in a strip and that Sn whiskering was closely related to internal stress resulting from atomic accumulation in confined segments. Finally, a theoretical model based on fundamental electromigration theory revealed that Sn diffused predominately through lattice and grain boundary paths at Sn homologous temperature of 0.6.

Connecting the phase-field-crystal model of electromigration with electronic and continuum theories

Electromigration (EM) is the motion of lattice atoms under high-density electric current. It is an important mechanism for structural changes in nanoelectronic devices and a major contributor to the electroplastic effect in structural metals. Recently, a phase-field-crystal (PFC) model for studying EM in metals was developed and shown to successfully capture many important EM-driven structural evolutions at experimentally relevant timescales. In this work, connections between the previous PFC EM model and existing EM theories are established. It is shown that the PFC model can be linked to both the electron-density-based quantum EM theory and the drift-diffusion-based continuum EM theory, therefore filling an important gap in the theoretical and computational study of EM-related phenomena. The numerical method for implementing the PFC EM model is discussed in detail. The well-established Blech effect is quantitatively reproduced using the model.

Electromigration-assisted manipulation of microstructure and properties of metals via cyclic direct current stressing treatment

Electromigration is inevitable in electronic products and detrimental to the reliability. However, we present in this study an innovative material treatment using cyclic direct current stressing for the manipulation of the microstructure and material properties of metals by introducing the basic electromigration theory. The experiment using a pure Al metal was designed to prevent possible interference from complex precipitation occurrence upon the treatment. The present work evidenced the electro-hardening of metals accompanying electrical resistance enhancement upon the cyclic direct current stressing treatment. The cyclic treatment with repeated direct current stressing and liquid nitrogen quench is capable of controlling the grain size and thus the material properties through the competition between the recrystallization (athermal effect dominated) and grain growth (thermal effect dominated) mechanisms. A combined tD (duration time) over n (cycle) parameter describing the energy input per unit cycle is proposed as an efficient index for the optimal properties achievement. In comparison with the conventional electropulsing treatment, the novel treatment as proposed can be applied for testing subjects without the need for a macroscopic deformation pre-treatment.

Contemporary chiral simulators for capillary zone electrophoresis.

For separation of enantiomers in presence of a chiral selector, data obtained with the 1D dynamic simulators SIMUL5complex and GENTRANS are compared to data predicted by PeakMaster 6, a recently released generalized model of the linear theory of electromigration. Four electrophoretic systems with stereoisomers of weak bases were investigated. They deal with the estimation of input data for complexation together with the elucidation of the origin of observed system peaks, the interference of analyte and system peak migration, the change of enantiomer migration order as function of the selector concentration and the inversion of analyte migration direction in presence of a multiply negatively charged selector. For all systems, data predicted with PeakMaster 6 are in agreement with those of the dynamic simulators and simulation data compare well with experimental data that were monitored with setups featuring conductivity and/or UV absorbance detection along the capillary. SIMUL5complex and GENTRANS provide the full dynamics of any buffer and sample arrangement and require very long execution time intervals. PeakMaster 6 is restricted to conventional CZE, is based on an approximate solution of the transport equations, provides data for realistic experimental conditions within seconds and represents a practical tool for an experimentalist.

Generalized model of the linear theory of electromigration and its application to electrokinetic chromatography: Capillary zone electrophoretic systems with complex-forming equilibria

We discuss several possible phenomena in electrophoretic systems with complexing agents present in the background electrolyte. In our previous work, we extended the linear theory of electromigration with the first-order nonlinear term, which originally applied to acid-base equilibria only, by generalizing it to any fast chemical equilibria. This extension provides us with a fresh insight into the well-established technique of elecktrokinetic chromatography (EKC). We combine mathematical analysis of the generalized model with its solution by means of the new version of our software PeakMaster 6, and experimental data. We re-examine the fundamental equations by Wren and Rowe and Tiselius in the frame of the generalized linear theory of electromigration. Besides, we show that selector concentration can increase inside the interacting-analyte zone due to its complexation with the analyte, which contradicts the generally accepted idea of a consumption of a portion of the selector inside the zone. Next, we focus our discussion on interacting buffers (i.e., buffer constituents that form a complex with the selector). We demonstrate how such side-interaction of the selector with another buffer constituent can influence measuring analyte-selector interactions. Finally, we describe occurrence and mobilities of system peaks in these EKC systems.We investigate systems with fully charged analytes and neutral cyclodextrins as selectors. Although the theory is not limited in terms of the charge and/or the degree of (de)protonation of any constituent, this setup allows us to find analytical solutions to generalized model under approximate, yet realistic, conditions and to demonstrate all important phenomena that may occur in EKC systems. An occurrence of system peaks in a system with fully charged selector is also investigated.

Generalized model of the linear theory of electromigration and its application to electrokinetic chromatography: Theory and software PeakMaster 6-Next Generation.

The linear theory of electromigration, including the first-order nonlinear approximation, is generalized to systems with any equilibria fast enough to be considered instantaneous in comparison with the timescale of peak movement. For example, this theory is practically applied in the electrokinetic chromatography (EKC) mode of the CZE. The model enables the calculation of positions and shapes of analyte and system peaks without restricting the number of selectors, the complexation stoichiometry, or simultaneous acid-base equilibria. The latest version of our PeakMaster software, PeakMaster 6-Next Generation, implements the theory in a user-friendly way. It is a free and open-source software that performs all calculations and shows the properties of the background electrolyte and the expected electropherogram within a few seconds. In this paper, we mathematically derive the model, discuss its applicability to EKC systems, and introduce the PeakMaster 6 software.

The dynamics of band (peak) shape development in capillary zone electrophoresis in light of the linear theory of electromigration.

The continuity equations that describe the movement of ions in liquid solutions under the influence of an external stationary electric field, as it is utilized in electrophoresis, were introduced a long time ago starting with Kohlrausch in 1897. From that time on, there have been many attempts to solve the equations and to discuss the results. In electrophoresis, special attention has always been devoted to the peak shapes obtained by the detector since the shapes have a tight connection with the phenomena taking place during electromigration and influence the efficiency and selectivity of the separation. Among these phenomena, the most important is electromigration dispersion. In this commented review paper, we compare various models of electromigration, try to find points that connect them, and discuss the range of their validity in light of the linear and nonlinear theory of electromigration.

Electromigration in 3D-IC scale Cu/Sn/Cu solder joints

The electromigration effect on the three-dimensional integrated circuits (3D-IC) scale solder joints with a Cu/Sn(25–50 μm)/Cu configuration was investigated using a field-emission scanning electron microscope (FE–SEM) combined with electron backscatter diffraction (EBSD) analysis system. Electron current stressing for a few days caused the pronounced accumulation of Cu6Sn5 in specific Sn grain boundaries (GBs). The EBSD analysis indicated that both the β-Sn crystallographic orientation and GB orientation play dominant roles in this accumulation. The dependencies of the Cu6Sn5 accumulation on the two above factors (i.e., Sn grain orientation and GB orientation) can be well rationalized via a proposed mathematic model based on the Huntington and Grone's electromigration theory with the Cu anisotropic diffusion data in a β-Sn lattice.

Real-time study of electromigration in Sn Blech structure

Electromigration has become a critical issue for reliability in Sn-based conducting materials that are used in advanced microelectronic packages with micro-joint integration. In this study, a Blech structure was used to characterize the Sn electromigration behavior of various strip lengths (L0=10–100μm). We established a mathematical model based on the fundamental electromigration theory to describe the correlation between the residual strip length (L) and the current stressing time (t). A three-stage mechanism was proposed to rationalize the Sn electromigration in the Blech structure. A good agreement between the mathematical model and experimental data was obtained, advancing our understanding of Sn electromigration.

Real-time X-ray microscopy study of electromigration in microelectronic solder joints

Electromigration (EM) in microelectronic solder joints was investigated via transmission X-ray microscopy (TXM) in conjunction with three-dimensional computed microtomography (3D μ-CT). This nondestructive analysis provided visualization of the EM dynamics in the encapsulated solder joints and showed the linear dependence of the Cu pad depletion on the current stressing time. Quantitative analysis of the interstitial Cu diffusion in β-Sn based on Huntington's EM theory offered rationalization of this EM-induced degradation mechanism.

Generalized model of electromigration with 1:1 (analyte:selector) complexation stoichiometry: Part II. Application to dual systems and experimental verification

Interactions among analyte forms that undergo simultaneous dissociation/protonation and complexation with multiple selectors take the shape of a highly interconnected multi-equilibrium scheme. This makes it difficult to express the effective mobility of the analyte in these systems, which are often encountered in electrophoretical separations, unless a generalized model is introduced. In the first part of this series, we presented the theory of electromigration of a multivalent weakly acidic/basic/amphoteric analyte undergoing complexation with a mixture of an arbitrary number of selectors. In this work we demonstrate the validity of this concept experimentally. The theory leads to three useful perspectives, each of which is closely related to the one originally formulated for simpler systems. If pH, IS and the selector mixture composition are all kept constant, the system is treated as if only a single analyte form interacted with a single selector. If the pH changes at constant IS and mixture composition, the already well-established models of a weakly acidic/basic analyte interacting with a single selector can be employed. Varying the mixture composition at constant IS and pH leads to a situation where virtually a single analyte form interacts with a mixture of selectors. We show how to switch between the three perspectives in practice and confirm that they can be employed interchangeably according to the specific needs by measurements performed in single- and dual-selector systems at a pH where the analyte is fully dissociated, partly dissociated or fully protonated. Weak monoprotic analyte (R-flurbiprofen) and two selectors (native β-cyclodextrin and monovalent positively charged 6-monodeoxy-6-monoamino-β-cyclodextrin) serve as a model system.

Electromigration of bivalent functional groups on graphene

Chemical functionalization of graphene holds promise for various applications ranging from nanoelectronics to catalysis, drug delivery, and nanoassembly. In many of these applications, it is critical to assess the rates of electromigration-directed motion of adsorbates along the surface of current-carrying graphene due to the electron wind force. In this paper, we develop an accurate analytical theory of electromigration of bivalent functional groups (epoxide, amine) on graphene. Specifically, we carefully analyze various factors contributing to the electron wind force, such as lattice effects and strong scattering beyond Born approximation, and derive a simple analytical expression for this force. Further, we perform accurate electronic-structure-theory calculations to parametrize the obtained analytical expression. The obtained results can be generalized to different functional groups and adsorbates, e.g., alkali atoms on graphene.

Electromigration in thin-film solder joints

Intermetallic compound (IMC) formation and growth are critical reliability issues with interconnected microelectronics, especially for small-scale solder joints in three-dimensional integrated circuits (3D IC) packaging. In this study, the growth behavior of IMCs in ultrathin 3D-IC solder joints under electromigration was investigated using a field-emission scanning electron microscope (FE-SEM) equipped with an electron backscatter diffraction (EBSD) analysis system. The joints consisted of a Cu/Sn/Cu configuration with a Sn thickness of ~30μm. We determined a strong dependence of the IMC growth and Cu pad depletion on the β-Sn orientation upon electron current stressing. When the c-axis of the Sn grain was oriented in the direction of the electron flow, aggressive CuSn IMC growth and Cu pad depletion occurred at the anode and cathode, respectively. A mathematical analysis based on crystallography and electromigration theory was used to rationalize the strong dependences that were observed.

Graphene Nanoelectrodes: Fabrication and Size-Dependent Electrochemistry

The fabrication and electrochemistry of a new class of graphene electrodes are presented. Through high-temperature annealing of hydrazine-reduced graphene oxides followed by high-speed centrifugation and size-selected ultrafiltration, flakes of reduced graphene oxides (r-GOs) of nanometer and submicrometer dimensions, respectively, are obtained and separated from the larger ones. Using n-dodecanethiol-modified Au ultramicroelectrodes of appropriately small sizes, quick dipping in dilute suspensions of these small r-GOs allows attachment of only a single flake on the thiol monolayer. The electrodes thus fabricated are used to study the heterogeneous electron transfer (ET) kinetics at r-GOs and the nanoscopic charge transport dynamics at electrochemical interfaces. The r-GOs are found to exhibit similarly high activity for electrochemical ET reactions to metal electrodes. Voltammetric analysis for the relatively slow ET reaction of Fe(CN)6(3-) reduction produces slightly higher ET rate constants at r-GOs of nanometer sizes than at large ones. These ET kinetic features are in accordance with the defect-dominant nature of the r-GOs and the increased defect density in the nanometer-sized flakes as revealed by Raman spectroscopic measurements. The voltammetric enhancement and inhibition for the reduction of Ru(NH3)6(3+) and Fe(CN)6(3-), respectively, at r-GO flakes of submicrometer and nanometer dimensions upon removal of supporting electrolyte are found to significantly deviate in magnitude from those predicted by the electroneutrality-based electromigration theory, which may evidence the increased penetration of the diffuse double layer into the mass transport layer at nanoscopic electrochemical interfaces.

Applicability and limitations of affinity capillary electrophoresis and vacancy affinity capillary electrophoresis methods for determination of complexation constants

ACE and vacancy affinity capillary electrophoresis (VACE) are the commonly used methods for determination of complexation constants by CE. The applicability and limitations of these methods were tested experimentally and by means of simulations using our simulation software Simul 5 Complex. It was shown that while the ACE method provides reliable and precise values of complexation parameters, those determined by VACE can be incorrect especially in the case of strong complexation. The effective mobilities of the system peaks in the VACE method, and consequently, the resulting complexation parameters were found to be a function of concentration of the analyte present in the BGE. Development of system peaks in VACE is discussed in the frame of the linear theory of electromigration. Dependence of mobility of system peaks on the composition of the BGE cannot be characterized by a simple analytical expression as in the case of ACE method. Thus, the VACE method fails and the resulting complexation constants might seriously differ from the reality.

Fundamentals

The Electrophoresis series on Fundamentals has run for ten years, so this issue marks an important anniversary. Ten years ago we wondered whether a Fundamentals issue should be established; but it has since become a success story: it is among the most cited issues of Electrophoresis. Among scientists, it is the theoreticians who contribute most to Fundamentals, with the result that this special issue has established itself as a common platform where theoreticians can meet and share their results and knowledge. In almost all the papers in the Fundamentals series, you will find many formulae and mathematical equations. The beauty of mathematics when applied to physical and chemical problems lies in its ability to formulate models from first principles. The problem, of course, is the solution of the mathematical models, either analytically or numerically. If the mathematical model is correct, it explains the observed phenomena, and also predicts as yet unknown phenomena. And if the experimental results are not in accordance with the prediction of the model, either the mathematical model is incorrect (which, of course, is the less favorable case), or it is incomplete and additional principles should be considered. What will you find in this issue? The first paper is a review on high-voltage power supplies to capillary andmicrochip electrophoresis. The subsequent series of three papers in the section “Electromigration in microchip channels” analyzes various aspects of chip electrophoresis, especially the phenomena connected with electroosmosis. In spite of my long engagement with electromigration, I am still fascinated by the fact that it is a thin diffusion layer (only a few nanometers thick) at the boundary between solid and liquid phases that is responsible for such macroscopic phenomena as the electroosmotic flow. Computer simulation can still reveal various aspects of electromigration. The papers from my group describe the approximate nonlinear mathematical model of electromigration and a new version of PeakMaster software, which can now predict the shape of system zones. The new PeakMaster 5.3 remains free and publicly available. The following three papers deal with an important physicochemical interaction often used in electrophoresis: the ability of analytes to form weak complexes with other constituents of the background electrolyte. This is utilized, for example, by affinity electrophoresis or in chiral electrophoretic separations. My group interacted with the Thormann group to formulate the dynamical mathematical model of electromigration coupled with complexforming equilibria. The result was two independent simulation programs, the extended Gentrans and Simul 5 Complex, capable of explaining the phenomena observed in such systems. All five papers in the section “Theory of electromigration” deal in various ways with the most important quantity in electrophoresis – mobility. Mobility indeed deserves such attention, as it is the key parameter responsible for the discrimination of analytes during separation. Of all the papers in this section, the paper by Wernersson is worth a special mention as it utilizes molecular modeling for the prediction of electrophoretic mobility. Also the paper by Phillies is innovative and brings up an original idea of “an inversion on perspective: instead of treating the migrating species as the objects of experimental importance and the support medium as being of importance only because it facilitates separations, one treats the polymer solution as being of experimental importance, and the choice of migrating species as being of interest only because different species may probe different aspects of polymer dynamics. The last section, “New approaches in electromigration”, offers seven methodological papers on various aspects of electrophoresis. I would like to express great thanks to all the authors who contributed to this issue, as well as to the referees for their careful work. The series of Special Issues of Electrophoresis on Fundamentals will continue next year and hopefully will remain a great source of fundamental knowledge in this field.

Electromigration of Tritium in V-4Cr-4Ti

In order to determine effective charge Z* of tritium in V-4Cr-4Ti alloy, an electromigration experiment has been carried out at temperatures of 330, 373 and 423 K in an electric field strength of about 5 V/m. Tritium concentration profile in a steady state was measured with an imaging plate (IP) technique. According to a simple electromigration theory, the effective charge Z* of tritium was determined from the tritium profiles. Z* obtained increases from 0.7 to 0.95 with increasing temperature. Comparison with the value of Z* for pure V is made and influences of alloying elements on tritium electromigration are discussed.