Current density calculation in anisotropic ferromagnetic film

[Mooorshum]

Homework Statement

I have implemented a finite element geometry of a thin ferromagnetic film with conductive points on it's surface to calculate the current density distribution in the film's volume.

Relevant Equations

∇ ⋅ ( - σ ∇ V) = 0

Hi everyone! First time posting here, so I apologize in advance for any bad formatting. I am working on a mathematical model of an anisotropic magnetoresistive sensor. The central part of such a device is a Wheatstone bridge that consists of four current-carrying ferromagnetic (FM) films, the electrical resistance of which changes when an external magnetic field is introduced. An important thing to mention is that typically the current carrying FM film has highly conductive strips added on top of it at a 45 degree angle relative to the long axis of the film. There are multiple reasons for this, but I won't go into any details, as it is not really relevant to the problem at hand. The conductivity of these strips is 1-2 orders higher than that of the FM material, so they act as a shunt, changing the inclination of the current flowing through the film, as can be seen here:

The change in resistivity of the FM film is dependent on the angle

between the current j and magnetization vector M at a certain point of the FM volume, and is governed by the Voigt-Thompson law:

, (1)
where

is the resistivity when there is no applied magnetic field,

is the change of resistivity (which is proportional to the magnetoresistive ratio of the FM material).
​

Therefore, to find the resistivity change of a single FM film, we need to know the magnetization and current density distribution in the whole sample.
I have implemented a finite element micromagnetic model in matlab, for which I solve the Landau-Lifshitz-Gilbert equation numerically in the constituent cells of the FM film. This allows me to obtain the magnetization configuration for a given external field:

Now I need to calculate the current density distribution in the film, and this is where I am running into some problems.​

I assume that we have two contact surfaces on opposite parts of the film, between which there is a voltage V0. The voltage is applied to two horizontal strips on top of the film, located at opposite ends of it (these contacts are of the same highly conductive material as the inclined strips). For simplicity, we can assume that one of these strips is grounded:

The electric field can be expressed via the electrostatic potential:

E = - ∇ V (2)​

The current density can be expressed as:

j = σ E, (3)​

We also have the conservation of charge for the electrostatic case:

∇ ⋅ j = 0, (4)​

From Eq. 2, 3, 4 we can derive:

∇ ⋅ ( - σ ∇ V) = 0 (5).​

Eq. 5 let's me calculate the electrostatic potential, which lets me find the electric field from Eq. 2, after which I can find the current density distribution via Eq. 3.

Now, Eq. 5 is in the continuous form. To apply it to my finite element geometry, I need to discretize it. If the points in space are denoted by their indices {i, j, k}, then by expanding Eq. 5, and using the finite difference approximations for the first and second order partial derivatives, I can express the electrostatic potential at a given point as:

V{i, j, k} = 1/6 * 1/σ{i, j, k} * ( σ{i+1, j, k} - σ{i-1, j, k} )*( V{i+1, j, k} - V{i-1, j, k} ) / 4h² + σ{i, j, k} * (V{i+1, j, k} + V{i-1, j, k})/h² + ... (6)
(repeated twice, shifting the indices every time) ).​

I also have to keep in mind the boundary conditions for V and j:

• Current cannot flow out any external surface of the sample, except for the contact points, so the normal component of the current density at the boundary faces must be equal to zero.
• The electrostatic potential at the contact surfaces is fixed (V0 and 0 respectively).

An important note is that the FM film's conductivity σ is dependent on the local current density. Therefore, I am solving Eq. 6 for a fixed conductivity distribution, then updating the values of j at each point, after which I calculate the new values of σ, and solve Eq. 6 again. This continues until j and σ converge.

When I implemented this method, I expected to get a current density distribution similar to what is shown in the first picture, however, that is not what I got. Instead of following a zigzag pattern, the current flow seems to be unidirectional, with exceptions near the contact surfaces:

Why could this be happening? I am certain that there is no error in Eq. 6. I have also set the conductivity of the strips at a constant value, so σ only changes in the ferromagnetic portion of the sample. Could I have missed something? Perhaps, the derivation of Eq. 6 is insufficient to describe the shunting effect?

I will be extremely grateful for any tips or suggestions, as I cannot figure out what is wrong.

 

[Tom.G]

Just a WAG (Wild As* Guess) here.

The current will take the path of least resistance between the conductive strips.
The strips are all parallel with each other.
Assuming homogenous substrate, the path of least resistance is a straight line to the adjacent strip.
QED

Cheers,
Tom

 

[Mooorshum]

Just a WAG (Wild As* Guess) here.

The current will take the path of least resistance between the conductive strips.
The strips are all parallel with each other.
Assuming homogenous substrate, the path of least resistance is a straight line to the adjacent strip.
QED

Cheers,
Tom

Hi, Tom!

First of all, thank you for answering.
You are absolutely correct, the whole point of these high-conductivity regions is that they are supposed to shunt the current, so that it "jumps" to the next strip. So the current should indeed flow at a 45 degree angle (relative to the X axis) in the ferromagnetic medium between adjacent strips (with a slight deviation due to the spatially variant resistivity of the magnetic substrate).

Perhaps, my solution of Eq. 5 might be yielding different results due to the boundary conditions not being fully defined? I've only defined them at the edges of the whole geometry, but not at the interfaces between the ferromagnetic and high-conductivity regions. If I'm not mistaken, the tangential component of the electric field should remain the same across the boundary. However, I'm not sure whether this is accounted for in Eq. 5 by default. Should I define the aforementioned bc separately?

What are your thoughts on this?

 

[fluidistic]

Hi, Tom!

First of all, thank you for answering.
You are absolutely correct, the whole point of these high-conductivity regions is that they are supposed to shunt the current, so that it "jumps" to the next strip. So current should indeed flow at a 45 degree angle (relative to the X axis) in the ferromagnetic medium between adjacent strips (with a slight deviation due to the spatially variant resistivity of the magnetic substrate).

Perhaps, my solution of Eq. 5 might be yielding different results due to the boundary conditions not being fully defined? I've only defined them at the edges of the whole geometry, but not at the interfaces between the ferromagnetic and high-conductivity regions. If I'm not mistaken, the tangential component of the electric field should remain the same across the boundary. However, I'm not sure whether this is accounted for in Eq. 5 by default. Should I define the aforementioned bc separately?

What are your thoughts on this?

Did you test your code on a simpler problem (single material)?
Normally you solve eq. 5 with appropriate boundary conditions, you shouldn't impose conditions on E nor derived quantities directly, I think.
For these kinds of problems, I would use fenicsx, with mixed elements (since M affects V and vice versa). Also, unless I missed it, we don't see which eq. you're solving for M.

 

[Mooorshum]

Did you test your code on a simpler problem (single material)?
Normally you solve eq. 5 with appropriate boundary conditions, you shouldn't impose conditions on E nor derived quantities directly, I think.
For these kinds of problems, I would use fenicsx, with mixed elements (since M affects V and vice versa). Also, unless I missed it, we don't see which eq. you're solving for M.

I did test it for a ferromagnetic film with no strips added, and the current seemed to behave the way it should. Do you know what bc I should include? I solve Eq. 6 in every cell except for those that are subject to bc, and then I impose them. If I can't do that, then how can I account for the bc? Generally curious, as I don't have any experience of solving such problems.
As for M, I solved the LLG equation for the same finite element sample. M does not depend on V; we consider it to be a purely magnetic property. So apart from the change in resistivity, there is no direct coupling between M and V

 

[fluidistic]

I did test it for a ferromagnetic film with no strips added, and the current seemed to behave the way it should. Do you know what bc I should include? I solve Eq. 6 in every cell except for those that are subject to bc, and then I impose them. If I can't do that, then how can I account for the bc? Generally curious, as I don't have any experience of solving such problems.
As for M, I solved the LLG equation for the same finite element sample. M does not depend on V; we consider it to be a purely magnetic property. So apart from the change in resistivity, there is no direct coupling between M and V

I see, then no need to use mixed elements.
Could you please define one of the electrode just half of what it is (say, keep only the top half). So that we are 100% sure the current density shouldn't be homogeneous around it.
Those current density vectors you plotted are plotted at each points of your grid? They are scaled by magnitude, or something else?

In your case, since you apply a known voltage V0, I would define Dirichlet boundary conditions on V, which is what you did, I believe. I am more familiar with FEniCSx (finite elements), but some parts are abstracted out of the user, such as exactly how the boundary conditions are applied. We specify on which cells the b.c. are applied, but not sure exactly what is going on internally.
What is the resolution of your mesh? Increasing by a factor of 2 doesn't change anything?

 

[Mooorshum]

I see, then no need to use mixed elements.
Could you please define one of the electrode just half of what it is (say, keep only the top half). So that we are 100% sure the current density shouldn't be homogeneous around it.
Those current density vectors you plotted are plotted at each points of your grid? They are scaled by magnitude, or something else?

In your case, since you apply a known voltage V0, I would define Dirichlet boundary conditions on V, which is what you did, I believe. I am more familiar with FEniCSx (finite elements), but some parts are abstracted out of the user, such as exactly how the boundary conditions are applied. We specify on which cells the b.c. are applied, but not sure exactly what is going on internally.
What is the resolution of your mesh? Increasing by a factor of 2 doesn't change anything?

I'm not exactly sure what you mean by only half of the electrode, but here is a basic description of how I calculate j (notice that the blue and grey cells have a spatially variant conductivity, so j cannot be homogenous in that region):

1. First of all, I define an initial approximation for V(i,j,k) and j(i,j,k) across the whole geometry. The initial V simply decreases from V0 to 0 across the x axis. The current between strips is initially shunted and of equal magnitude everywhere. Here is how the shunted current is supposed to flow between strips (not all cells are shown, but I hope you get the idea):

2. For the inner cells (the opaque grey ones) I solve Eq. 6 and then impose the nearest grey cell's potential on the boundary cells (which are shown as transparent). Since current can't flow out of the sample (except for from the contact cells), I make the normal component of current in the boundary cells equal to zero.
3. Now that I have V in each cell, I calculate E, and then j.
4. I update the conductivity of the magnetic cells, and solve Eq. 6 again.

This repeats until convergence for j and σ is reached.

The sample size is 200 by 50 by 6 cubic cells, each of which has a side length of 5nm. I haven't tried increasing it, as the magnetic calculations start taking too long.

Now that I think about it, when I am solving for V, the high conductivity areas are not included, since they do not have an upper neighbor, and are only one cell thick (I can't define the derivatives for them, so Eq. 6 breaks down). Since I then impose the voltage in these cells, their conductivity does not influence the value of V in the other cells around them (I think?), so only the magnitude of j changes, which is what I am currently getting. This might be what is causing the issue. Sadly, I don't have the code on me, so I'll only be able to check whether this is the case on Monday. I'll make them 2 cells thick, so that they are partially present when solving for V. Perhaps, this will help.

Thank you for taking your time with me! I think I might actually be on the right path to making this whole thing work :)

btw, do you know if I need to also account for the boundary interface between materials (the continuity of the tangential value of E), or should Eq. 6 be sufficient?

 

[Mooorshum]

I see, then no need to use mixed elements.
Could you please define one of the electrode just half of what it is (say, keep only the top half). So that we are 100% sure the current density shouldn't be homogeneous around it.
Those current density vectors you plotted are plotted at each points of your grid? They are scaled by magnitude, or something else?

In your case, since you apply a known voltage V0, I would define Dirichlet boundary conditions on V, which is what you did, I believe. I am more familiar with FEniCSx (finite elements), but some parts are abstracted out of the user, such as exactly how the boundary conditions are applied. We specify on which cells the b.c. are applied, but not sure exactly what is going on internally.
What is the resolution of your mesh? Increasing by a factor of 2 doesn't change anything?

I'm back. I tried changing the strip thickness, so they cover the entirety of the film (in the z direction), and I also added boundary conditions between the FM and conductive areas (both contacts and strips). This resulted in the current flowing properly through the strips, but in the magnetic medium it's still mostly oriented along the x axis, while in reality it should be perpendicular to the strip edge. At the boundaries it is perpindicular, due to the fact that E_tangential_1 = E_tangential_2 and j1_normal = j2_normal (so that current doesn't accumulate at the interface)
Any idea what else could work?