**In many modern technologies, changes in entropy can irreversibly alter the microscopic structures of advanced circuits. Until now, however, theoretical equations to describe this process have been incomplete. In his research, Prof King-Ning Tu at the University of California, Los Angeles revisits these concepts to derive an updated equation. His work provides the first accurate model for predicting when microcircuits will eventually fail and could provide crucial guidance for how the latest electrical devices can keep up with the latest technological demands.**

In the coming years, the electrical devices we use in our everyday lives look set to rely increasingly on advanced applications including 5G and artificial intelligence. As this happens, demands are rapidly growing for devices with smaller sizes, lower power requirements, and lower costs – all while achieving higher functionalities. Recently, however, these required advances in hardware appear to have hit a wall. From the 1970s until just a few years ago, increases in computing power reliably followed ‘Moore’s law’, stating that the number of transistors which can fit onto a single chip tended to double every two years. Unfortunately, even as modern silicon-based technologies become increasingly sophisticated, it appears that they are no longer keeping up with this trend.

One critical aspect of this slowdown relates to the reliability of the microscale circuits which are integrated into these devices. Unless they are fully optimised, these circuits can exhibit weak points at the soldered joints between their components, which fail quickly compared to the rest of the circuit when large electrical currents flow through them. At these points, device operation can eventually lead to unwanted alterations in the circuit’s microscopic structure, leaving it unable to carry the quantities of electrical current required by the device. Until now, however, researchers have not fully accounted for these changes, leading to large uncertainties regarding the reliability of modern devices.

### Interacting flows

Within the latest designs of microcircuit, failures are rooted in the co-existence of flows of electrons, atoms, and heat. As they interact with each other, these flows can trigger irreversible changes in circuit microstructures, which can be described through the principles of ‘non-equilibrium thermodynamics’. In particular, ‘electromigration’ is caused by the interaction between flows of electrons and atoms, while ‘thermomigration’ arises from the combined flows of atoms and heat. In addition, ‘stress migration’ is caused by the motions of atoms driven by mechanical stresses in the circuit, opening up empty ‘voids’ which can impede its performance.

To account for changes in circuit microstructure, researchers and engineers must rely on theoretical predictions which describe their influences mathematically.

To account for these effects, researchers and engineers must rely on theoretical predictions which describe their influences mathematically. The quantity most commonly used to measure the reliability of electronic devices is their ‘mean time to failure’ (MTTF) following the beginning of their operational lifetimes. In addition, their designers must consider how the future demands of applications like 5G and artificial intelligence may require these devices to carry electrical currents at higher densities. If these values are accurately determined, engineers can define the maximum current density that can be applied to a product, without any risk of failure within its required lifetime. However, making these predictions is no simple task.

### Irreversible changes

In order to accurately calculate the MTTF of an electrical device, researchers must consider how these interactions result in changes in the entropy within their microstructures. Entropy is an important concept in the wider field of thermodynamics. It describes the degree of randomness and disorder in molecular structures. Here, its importance can be imagined through a simple thought experiment involving a closed box – half filled with liquid water, and half with vapour. If conditions inside the box are kept at a constant temperature and pressure, both fluids will exchange molecules at a constant rate – meaning entropy does not change, and the whole system stays in equilibrium.

The situation will change entirely if a hole is punched in the top of the box, allowing vapour to escape while heat is applied to the water, so that its rate of evaporation increases. Here, equilibrium can still be reached if the amount of vapour produced by the water matches the amount escaping through the hole. However, since vapour is passing into the surrounding air, the overall system is now ‘open’, and entropy increases in the system comprising the box and its surrounding environment. Furthermore, there is no way to draw the escaping vapour back into the box, making the process irreversible.

*Figure 2. (a) Schematic diagram of flip chip technology, in which two solder joints are connected by an Al interconnect on the chip side. (b) The simulation of current density distribution in the pair of solder joints. The Al interconnect is hotter than the solder joints.*

In the same way, interactions between flows of electrons, atoms, and heat in an open microelectronic system can raise its overall entropy, triggering the irreversible changes which eventually lead to device failures. In 1969, physicist James Black used this concept to derive an equation relating the MTTF to the electrical current density, and the energy required to activate microstructural changes through electromigration. Yet, as Prof King-Ning Tu at the University of California, Los Angeles, points out, this description was still far from complete.

### Accounting for thermomigration

In the past, the weak soldered links in electrical circuits were often made using pure metals, including aluminium and copper. Since the process of thermomigration – where structural changes are triggered by the combined flows of atoms and heat – only occurs in metal alloys, it did not contribute to the MTTF in these cases. Because of this, the quantity was left out of Black’s original equation, which provided a valuable basis for calculating the MTTF in most devices. However, with the rise of increasingly sophisticated microcircuits required for 5G and artificial intelligence applications, this situation is now changing rapidly.

In these more modern systems, designed to carry electrical currents at larger densities, solder joints are composed of alloys which can quickly heat up during device operation. Because of this, heat must be rapidly dissipated in order to avoid system failure, introducing temperature gradients as high as 1,000°C per centimetre in microscale components. In turn, heat can flow far more readily in these systems, triggering irreversible structural changes through thermomigration. So far, MTTF calculations using Black’s equation have neither accounted for this effect, nor that of stress migration – limiting researchers’ understanding of how and when failures will occur.

Prof Tu’s work has produced a unified model for the MTTF, which incorporates the combined influences of electromigration, thermomigration,

and stress migration.

### Modifying Black’s equation

Through his research, Prof Tu has addressed this issue by revisiting certain aspects of the mathematics describing entropy production. In doing this, he has now re-derived Black’s equation, to account for all interactions between flows of electrons, atoms, and heat throughout electrical microcircuits. Through his derivations, he has now shown for the first time that temperature must be factored into the equation to accurately predict a device’s MTTF. In addition, Prof Tu has resolved a long-standing controversy regarding the influence of current density. Contrary to some previous arguments, his derivations show that the relationship between this value and the MTTF was correct in Black’s original equation.

Most importantly, Prof Tu’s work has produced a unified model for the MTTF, which incorporates the combined influences of electromigration, thermomigration, and stress migration. Ultimately, this model is the first mathematical description to fully account for all of these microscopic entropy changes. Therefore, it could soon enable accurate simulations of electrical devices and their failure mechanisms, and may also lead to important practical technological applications in the coming years.

### Improving predictions

Today, a growing number of cutting-edge technologies are built using microscale electronic circuits, fabricated on the surfaces of silicon chips. Such leading efforts are focused on producing ‘2.5-dimensional’ integrated circuits, where blocks of microcircuit-carrying semiconducting materials are combined without stacking them on top of each other. This arrangement sets these devices apart from their 3D counterparts, where the technology required for combining circuits is still new and extremely difficult.

With a reliable basis for calculating the MTTF in these 2.5-dimensional microcircuits, researchers and engineers could better ensure that alloy solder joints can be implemented more effectively, minimising the vulnerability of these systems in their weak points. Through further efforts to implement such practices into large-scale manufacturing, Prof Tu’s work will ensure that the latest devices can keep pace with the ever-evolving demands of applications including 5G and artificial intelligence.

For any irreversible process, its MTTF equation can be obtained based on entropy production, provided that both the flux and the driving force can be defined. However, the parameters in the MTTF equation must be determined from experimental data.

**How could microcircuits in the future be adapted to lengthen their mean time to failure?**

In semiconductor manufacturing, because the product quantity is extremely large, high reliability is critically important, especially the electronic 3D integrated circuit for advanced consumer electronic products which are now widely used for distance teaching and home office. Any design of a new electronic device should include ‘design for reliability’ so that the effect of Joule heating, thermal gradient, and stress center can be reduced for low power applications. It is most important to obtain MTTF equations based on entropy production so that the lifetime of the device can be evaluated and early failure removed.

**References**

- Tu, K.N. & Gusak, A.M. (2019). A unified model of mean-time-to-failure for electromigration, thermomigration, and stress-migration based on entropy production.
*Journal of Applied Physics*, 126(7), 075109. Available at: https://doi.org/10.1063/1.5111159

**DOI**

10.26904/RF-134-158161

### Research Objectives

Prof Tu studies the ‘mean time to failure’ (MTTF) for electromigration.

### Funding

This work is supported by Ministry of Science and Technology, Taiwan (grant No. MOST 109-2634-F-009-028) and the Center for Emergent Functional Matter Science of National Chiao Tung University from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan.

### Collaborators

Prof A. M. Gusak, Dept. of Physics, Cherkasy National University, Ukraine

### Bio

**Contact**

International College of Semiconductor Technology

National Chiao Tung University

1001 University Road, Hsinchu, Taiwan 300, ROC

E: kntu@ucla.edu

T: +886 3 5712121, ext 55309

W: http://www.seas.ucla.edu/eThinFilm/