Electromigration: Theory and Simulation

[This was auto-generated. There may be mispellings.]

Alright, good afternoon everyone.

We will start our webinar on "Electromigration: Theory and Simulation." We are going to cover theory and simulation results, but mostly we will cover the theory so that we can understand what type of equations are being sold behind the scenes when we are using packages like ANSYS.

So we are Ozen Engineering. We are the ANSYS Channel Partner of the Year back in 2015, 2018, and 2021. And we are a channel partner for ANSYS. We are involved with the ANSYS software tools. We do training, mentoring, and also consulting.

Quickly jumping into electromigration, we'll talk about the introduction and the background. So what is electromigration? Electromigration is a physical phenomenon of mass transport in interconnects when the metallization is subject to stresses induced by high electrical current density.

High electrical current density. And then, you know, what this results in is the atomic flux divergence. This atomic flux divergence, either positive or negative, it's either going to create void nucleation or helix formation.

Electromigration has been a problem in the interconnect reliability for some period of time. So electromigration has been a problem in the interconnect reliability for some period of time. And so I'm going to try to explain it now.

Electromigration causes voids if there are more atoms flowing out of the domain than flowing in. That's what we call mass depletion. And it results in the creation of helix when there are more atoms flowing in than flowing out. And that results in mass accumulation.

You can see some pictures that describe that. Please note the scale here is at the micron 10 micron level: a hillock formation in the lower picture and maybe in the upper picture, the formation of a void.

So, electron migration is primarily the study of atomic flux distribution, but there are more components to it. It's most significant in applications where current densities are high. And as to why current densities are high, as you know, in electronics, everything is becoming smaller.

As the physical size of the components is becoming smaller, what's happening is the current in the smaller size is still the same. And thus, the current density is going to get really high.

The term electron migration was coined by Professor Huntington because he didn't like the German use of the word electro transport. And so that was back in the 1950s. So the term electron migration stuck in the industry.

There are two main forces that affect the ionized atoms in a conductor: the first one is the electric field-induced direct electrostatic force, F sub e, and the force due to momentum exchange with other charge carriers, F sub p.

For metallic conductors, F sub p is caused by the electron wind and acts in a direction toward the flow of charged carriers. And F sub p is in a direction opposite to that of the electric field. So, the net force on an ion in the electric field is F sub e minus F sub p.

Over time, this electron wind force moves a significant number of atoms far enough from their original positions that a gap can develop in the metal, thereby preventing the flow of electricity.

The atomic flux divergence is defined by the number of atoms flowing into a region per unit time not equal to the number flowing out of that region per unit time.

So, if the number of atoms flowing into a region per unit time is not equal to the number of atoms flowing out of that region per unit time, then whatever it is, we call the flux divergence.

When there is a deeply negative flux, or going into a ritual movement state, then the fluctuation of flux happens. It's no longer kehr as in the original equation. The concentration of high current and high temperature causes the formation of these voids.

The void propagates across the entire solder interface and leads to a connection failure.

There are four different physics that we need to capture: first, what we call electron migration; the second one is thermal migration because electron migration is going to cause temperature rise; the third one is stress migration when the temperatures are high; then that's going to cause stresses in the structure; and the fourth one is the elevation of the insulation layer.

These are the four things that we need to account for and also the atomic concentration gradient. The traditional electron migration models are based on the solution of diffusion equations induced by driving forces: temperature gradient induced driving force and the atomic concentration gradient.

So, we will address both of these. First, we will start with the traditional electromigration model, which is based on the convection-diffusion equation, which takes this form here. The traditional electromigration equation looks like this.

And in this equation, C is the atomic concentration, D is diffusivity, and alpha is the reduced electromigration force.

And alpha, we are going to get more into this because, in the calculation of alpha, it's just going to be an important calculation where we have to take into account the different physics.

But staying with the traditional equation, alpha is force divided by K, which is K is this Boltzmann constant, and T is the temperature.

And the force F in the numerator can be rewritten as Z star, which is the effective charge, multiplied by the electronic charge, multiplied by resistivity, multiplied by the current density. So, as you can see from here, alpha is directly related to current density.

And as current density goes up, alpha goes up. So, in the traditional electromigration equation, again, listing it here, the question is, can this equation be solved in ANSYS? And when I say ANSYS, I'm talking about ANSYS mechanical at this point in time.

Can we solve this equation in ANSYS mechanical? And the answer is yes. And here, I'm going to explain how you can make use of ANSYS to solve this traditional electromigration equation. In ANSYS, the first law of thermodynamics states that the thermal energy is conserved.

And here, in finite element formulation, this is a transient temperature fields are calculated in finite element formulation. So, if you apply that to a differential control volume, then the equation is opened up and it looks like this, the bottom equation.

And in this equation, as you can see here, it's a three-dimensional equation. That means we can solve the problem in 3D. So, we can actually solve the traditional electromigration equation in three dimensions, which, you know, ANSYS is a 3D code.

So, you can do it in 3D or you can do it in 2D or you can also do it in 1D. So, in this equation, rho is density, C is specific heat, T is temperature, t is time. So, electromigration is definitely a transient phenomena. You'd create voids over time or you create hillocks over time.

So, that's why we have to do a transient problem. So, if we apply Fourier's law to thermal gradients, then we are going to get this additional equation in there. And if we add the first law of thermodynamics to Fourier's law, then we are going to end up with this equation at the top.

Now, if we expand the terms, then, as you'll see here, you will get the three dimensional representation over here. And if you'd like to reduce it to, let's say, 30 degrees, then if you put this in the green, you will get this kind of equation and you can still get this, this kind of equation.

To one dimension, then you can set nu y nu z ky kz to zero. Then you're going to get the equation in just one dimension. In one dimension.

So if we rearrange the terms there from time transient thermal simulation in ANSYS and compare it to the traditional electromigration equation, as you can see here, those equations look very much alike.

So then you can say, okay, D in the traditional electromigration equation is k sub x, which is thermal conductivity in the x direction divided by rho c, density times specific heat. So, now we can also define a non-dimensional Peclet number. And the Peclet number is defined by this equation here.

Again, it's non-dimensional. And in that equation, nu is the magnitude of the velocity vector. And L is the velocity vector. And L is the velocity vector. And L, capital L, is the element length along the velocity vector.

So if we are solving in 1D and interested in just one dimensions, how the current flows from, say, left to right, excuse me, capital L is the element length. And we can also modify this for orthotropic materials. But at this point, we'll just let it go. We'll just look into one-dimensional solution.

So as you can see from this formulation, alpha, the electron migration force, can now be represented as a functional Peclet number. And as you can see here, alpha becomes two times the Peclet number divided by the current conducting length. So they are very much interrelated.

And in ANSYS mechanics, the time transition is very difficult for the time transition solution to be valid. The Peclet number needs to be less than one. Otherwise, it creates unstable solutions.

And just as an example, if we set Peclet number to 0.1, then L times alpha becomes 0. 2. Now, why are we doing this? Because we'd like to come up with a normalized time and solve this equation by a finite element formulation in ANSYS. So we have a one-day electron migration class.

And one of the first workshops we go through is really going through this equation. And we are trying to prove that ANSYS can be used to solve this electron migration equation. And this electron migration equation can be solved by the solution of this equation.

And this electron migration equation was sold back in the 1990s in reference one, where they produce the results that you see here, where the x-axis is basically normalized time. And the y-axis is concentration, the atomic concentration. So the challenge here is, can we reproduce these curves?

Using ANSYS? So the answer is yes, it can be done. And these are the boundary conditions you can use in ANSYS. You can create a one-dimensional model or two-dimensional model. And using these boundary conditions, you can reproduce the same results that were given in a paper back in 1990s.

And as you see here, the results are exactly the same. For the solution, we can use the same equation. So we can use the same equation for the solution of traditional electron migration equation. On the left-hand side, we are looking at results obtained from running a transient simulation in ANSYS.

And on the right-hand side are the results that were published in a paper. Published in a paper and the results were presented in that paper in 1990s. So, and we also explain how this can be done. This is the first step.

So you can use the input file that's used in ANSYS mechanical to solve the same exact problem. And from here, you can reproduce the same results again. So in our class, basically, we go through this. We explain every single thing in the ANSYS input file.

So that way we covered the traditional electron migration equation. And so and you can see here, you can see that the results are the same. So and you can use ANSYS to solve the traditional electron migration equation. Now, how about the advanced electron migration modeling?

So please note that this convection-diffusion equation is the traditional equation. However, it is not the full result. It doesn't give you the full result for what we call the advanced electron migration simulation. Because we need to take into account these additional driving forces.

These additional driving forces are just are these four forces. Electrical field force, thermal gradient force, stress gradient force, and atomic concentration gradient. And to be able to account for these, what we have to do is look at this electron migration equation. And the Q vector here.

Which is the normalized atomic flux. We need to divide it up and take into account Q sub E, which is the forces due to electrical field. Q sub T, which is the thermal gradient forces. Q sub S, which is the stress gradient forces. And Q sub C, which is the atomic concentration gradient.

So if we take a look at that again. Q sub E plus Q sub T plus Q sub S plus Q sub C. And open these up. So for example, Q sub E, that's the additional driving force due to electrical field. It can be expressed in this equation here. The thermal gradient is expressed in this equation here.

Stress gradient in this equation here. And the atomic concentration gradient expressed here. So let's see. So we need to add all of these to each other. And we note that all these equations have some terms that are common. And we note that, you know, it's basically this capital D.

Which is denoted as D sub 0. Exponential function E sub A divided by K sub T. So that's the diffusivity with D0 being the thermally activated diffusion coefficient. And we know what the Stefan-Boltzmann constant is. That's a constant, as the name implies.

But we get the absolute temperature and activation energy. And it's really easy to get the temperature from the finite element model. So again, we look at the force due to the electrical field. And that has a Z star, which is the effective charge.

And the electric charge, that's a constant, as given here. The resistivity can also be expressed in this formulation. And then the current density vector is also given here. And the thermal gradient terms look like this.

And so electron migration induced void formation results in localized increased current density. Resulting in these current, what we call current crowding. And joule heating is proportional to the square of the current. So all of this can be accounted for in these equations.

And these thermal gradients will also result in stress gradients. Where we look into the local hydrostatic stress components.

And if we take into account, finally, the atomic concentration gradient, which we have looked at before, and add them all to each other, then we are going to get this complicated expression, the Q vector, which contains all the stress gradient, the thermal gradient, atomic concentration, and the electron force.

So it looks complicated, but it's all doable. It's just that we need to keep track of all these different parameters here, which are basically ANSYS inputs. And if we get it all together, then we can reduce this down to this equation here. So as you see here, the Z star is the electron force.

Q star is the thermal force. Omega is the stress force. And C is the atomic concentration. So now this is what I call the advanced electron migration equation. And you can still use ANSYS to solve this advanced formulation. And to do this, there are really three steps.

The first step is to do a coupled electric thermal finite element analysis. And the second step is to do a thermal stress finite element analysis. And then the third step is correction for atomic concentration calculation. So in ANSYS mechanical, you can actually use solid 226 element.

Solid 226 element, as you see, it has multiple degrees of freedom, structural, temperature, volts, current density and concentration. So you can use solid 226 elements in 3D. In addition to that, this solid 226 element has a lot of different material properties.

In case you would like to add more complexity, and you can add a lot of this non-linearities to the problem, like material non-linearities. And in terms of physics, the solid 226 element is really very versatile. It covers anything that you see in this.

Anything from thermal electric to structural magnetic to thermal diffusion, electric diffusion. So you can actually just use a subset of these. Or you can use the full formulation like shown here, which is structural thermal electric diffusion.

So this is the most advanced electron migration modeling here. And this solid 226 element has it in there. So. And so which ANSYS tools can be used for electron migration modeling? Just like you have seen here, ANSYS mechanical enterprise can be used.

And ANSYS mechanical enterprise is where you can use the solid 226 element for this. And we have an additional what we call EBU, electronics business unit tool. ANSYS SI wave. You can also use SI wave. So let me briefly talk about SI wave.

So we are going to switch gears, go from ANSYS mechanical to SI wave. In SI wave, there is a what we call Black's formulation. So Black developed a model to estimate the median time to failure of a conductor due to a failure. And this is the current density of a conductor due to electron migration.

He published the model in a paper written in 1969. And this is his formulation primarily. This is mean time to failure in hours. This is a, as you see, electron migration calculation due to current density. This is not taking into account all the temperature and stress gradients.

It's just taking into account the current density. But it's a lot quicker simulation. So if you have a complicated PCB design like the one that you're seeing here, all you have to do is in SI wave, you specify the experimental constants, activation energy.

And SI wave is going to give you nice looking plots, contour plots, where on the left hand side, your SI wave is going to tell you what's the mean time to failure. On your complicated PCB design, you will be able to see, for example, in this case here, 2,590 hours.

After 2,590 hours, you can expect electron migration failure in these red zones here. So you will be able to plot mean time to failure in SI wave along the line. Along with other plots like the one that you see on the right hand side, the current density.

And you can see that current density is really high at the same location where we are seeing mean time to failure of really hours of the order of 2, 000. So in conclusion, ANSYS software tools can be successfully used to model electron migration.

You can either use ANSYS Mechanical or for a quick calculation, you can use ANSYS SI wave. And the engineers can redesign under the guidance of simulation results from ANSYS to avoid any failures due to electron migration.

And so that we can address what reference five said as the electron migration has been the most persistent threat to interconnect reliability. So ANSYS made it easy so that we can do through quick calculations to be able to predict electron migration failure.

So here are the references that we were talking about in this presentation here. And if you would like to know more, please sign up for our class. We have a one day class on electron migration. So there are lots of different ANSYS software tools as well that we look into and we provide support.

And also in offices, we have multiple offices on the west coast as well as the east coast. We also are in Canada. And so here's our contact information. Please, if you'd like to get more information, you can send an email to info at Ozen. dot com.

If you're interested in ANSYS software tools, you can send us an email at sales at Ozen. dot com. If you are already our client and get. Like to get some support, just send it to support at Ozen. dot com. Our website is www.ozoneinc.com. And here are our office locations.

So with that, I'd like to say thank you and thank you for your attention. If you have any questions, please feel free to ask at this point. Thank you.