- #1
SchroedingersLion
- 215
- 57
Greetings,
assume we have a 2-dimensional system in the x-y-plane. An electric field is applied in x-direction, a magnetic field is applied in z-direction. As is well-known, the charge carriers get pushed in the y-direction due to the Lorentz-force until the Hall field is strong enough to counteract this motion. In steady state, there will thus be no current in y-direction.
The magneto-conductivity terms are given by
$$J_x = \sigma_{xx} * E_x + \sigma_{xy} *E_y $$
and
$$J_y= \sigma_{yx} * E_x + \sigma_{yy} *E_y
$$
The magneto-resistivities (given by inversion of the ##\sigma## matrix) are given by
$$E_x = \rho_{xx} * J_x + \rho_{xy} *J_y $$
and
$$E_y= \rho_{yx} * J_x + \rho_{yy} *J_y
$$
It holds that ##\sigma_{xx}=\sigma_{yy}## and ##\sigma_{xy}=-\sigma_{yx}##. Same for ##\rho_{ij}##.
A plot of these quantities w.r.t strength of the magnetic field is attached.
I am trying to understand them qualitatively.
It makes sense that ##\sigma_{ii}## decrease with increasing B, as more and more charge carriers are bound to create the Hall field ##E_y##, meaning ##E_x## needs to be stronger and stronger to keep the current flowing.
On the same line, one can argue that ##\sigma_{xy}## has to decrease from 0 into the negative numbers, as the Hall-field removes charge carriers from their motion in x-direction.
First question: Why does this effect saturate? Why is there an extremum in ##\sigma_{ij}##?
Now to the resistivities:
At B=0 (and in an isotropic system), the resistivity ##\rho## is simply the inverse of the conductivity ##\sigma##. Here, however, ##\rho_{xx}## stays constant even though ##\sigma_{xx}## goes to zero.
I was trying to explain it like this: The conductivity describes the strength of the current that gets created by a field. The resistivity gives the resistance against a current that is already flowing. In other words, ##\sigma_{xx}=0## means that no current can flow, whereas ##\rho_{xx}>0## means that a current WOULD experience a resistance if it could flow.
Second question: Does this make sense?
Third question: Why does ##\rho_{yx}## (or, as in the figure, ##-\rho_{xy}##) increase with B?
Is it because the growing B-field increases ##E_y## which, again, draws away charges from their motion in x-direction, effectively increasing the resistance in x-direction?
It's funny, I did all the maths to find expressions for the different matrix components, but it is harder to me to understand it intuitively.
SL
assume we have a 2-dimensional system in the x-y-plane. An electric field is applied in x-direction, a magnetic field is applied in z-direction. As is well-known, the charge carriers get pushed in the y-direction due to the Lorentz-force until the Hall field is strong enough to counteract this motion. In steady state, there will thus be no current in y-direction.
The magneto-conductivity terms are given by
$$J_x = \sigma_{xx} * E_x + \sigma_{xy} *E_y $$
and
$$J_y= \sigma_{yx} * E_x + \sigma_{yy} *E_y
$$
The magneto-resistivities (given by inversion of the ##\sigma## matrix) are given by
$$E_x = \rho_{xx} * J_x + \rho_{xy} *J_y $$
and
$$E_y= \rho_{yx} * J_x + \rho_{yy} *J_y
$$
It holds that ##\sigma_{xx}=\sigma_{yy}## and ##\sigma_{xy}=-\sigma_{yx}##. Same for ##\rho_{ij}##.
A plot of these quantities w.r.t strength of the magnetic field is attached.
I am trying to understand them qualitatively.
It makes sense that ##\sigma_{ii}## decrease with increasing B, as more and more charge carriers are bound to create the Hall field ##E_y##, meaning ##E_x## needs to be stronger and stronger to keep the current flowing.
On the same line, one can argue that ##\sigma_{xy}## has to decrease from 0 into the negative numbers, as the Hall-field removes charge carriers from their motion in x-direction.
First question: Why does this effect saturate? Why is there an extremum in ##\sigma_{ij}##?
Now to the resistivities:
At B=0 (and in an isotropic system), the resistivity ##\rho## is simply the inverse of the conductivity ##\sigma##. Here, however, ##\rho_{xx}## stays constant even though ##\sigma_{xx}## goes to zero.
I was trying to explain it like this: The conductivity describes the strength of the current that gets created by a field. The resistivity gives the resistance against a current that is already flowing. In other words, ##\sigma_{xx}=0## means that no current can flow, whereas ##\rho_{xx}>0## means that a current WOULD experience a resistance if it could flow.
Second question: Does this make sense?
Third question: Why does ##\rho_{yx}## (or, as in the figure, ##-\rho_{xy}##) increase with B?
Is it because the growing B-field increases ##E_y## which, again, draws away charges from their motion in x-direction, effectively increasing the resistance in x-direction?
It's funny, I did all the maths to find expressions for the different matrix components, but it is harder to me to understand it intuitively.
SL