- #1
Pi-Bond
- 302
- 0
Homework Statement
Electrons in a ferromagnet whose spins are oriented in the direction of, or opposite to, the internal magnetisation carry independent currents I+ and I−. This leads to the material behaving as though it has different conductivity σ+ and σ− for each of the two current components. These currents may be thought of as flowing through parallel resistances. Two ferromagnetic layers with opposite magnetisation are placed next to each other as shown in the figure.
Each layer has a thickness t and area A. When electrons pass from one layer to the other
their spin-direction remains unchanged. A voltage U is placed across the layers in series
with an external resistor R.
1. Show that the total resistance of the circuit is
[itex]R_0 = \frac{t}{2A} (\frac{1}{\sigma_+} + \frac{1}{\sigma_-})+R[/itex]
2. If an external magnetic field above a certain strength is applied to the system, the two ferromagnetic layers will be magnetised in the same direction. Show that the total resistance is now
[itex]R_H = \frac{2t}{A(\sigma_+ + \sigma_-)} +R[/itex]
Homework Equations
Resistivity
[itex]\rho=\frac{1}{\sigma}[/itex]
Resistance
[itex]R=\rho\frac{l}{A}[/itex]
The Attempt at a Solution
At first I though that the equivalent circuit for the "component" in the middle would be a parallel one with resistances due to σ+ and σ−. In this case the equivalent resistance would be
[itex]R_{eq} = \frac{t}{A}(\frac{1}{\sigma_+ + \sigma_-})[/itex]
But this is evidently wrong. Then I thought that each of the two layers has the parallel configuration, and these two layers are in series. In this case the equivalent resistance would be twice the above, which is still wrong.
So it is clear I have the geometry wrong. Can anyone explain the correct geometry?
Last edited by a moderator: