# Integer Quantum Hall effect, conductivity & edge states

I'm confused about the conductivity and the edge states in the IQHE. On the plateaux, we zero the longitudinal conductivity and resistivity, right? So is it really true, that on the plateaux, there is no current flowing in the longitudinal direction, only in transverse? From this image, it looks to me, that the edge states carry the current in longitudinal direction and that there is no current flowing in transverse direction. What did I miss understand?

Greetings

## 1 Answer

The two dimensional conductivity ($$\sigma$$) and resistivity ($$\rho$$)tensors are defined by $$j_a= \sigma_{ab}E_b,\\ E_b= \rho_{ab}j_b$$ respectively. Here $$a,b$$ stand for the $$x,y$$ directions. This means that $$\sigma_{ab}$$ is the inverse matrix to $$\rho_{ab}$$. On an IQHE plateau $$\sigma_{ab}= \frac{ne^2}{h}\left(\matrix{0&1\cr -1&0}\right)_{ab}$$ and the inverse matrix is $$\rho_{ab}=\frac{h}{ne^2}\left(\matrix{0&-1\cr 1&0}\right)_{ab}.$$ We see that both longitudinal conductiites $$\sigma_{xx}$$ and $$\sigma_{yy}$$ are zero as are the longitudinal resitivities $$\rho_{xx}$$ and $$\rho_{yy}$$. There can certainly be non-zero currents and voltages however. It is just that the current and voltage must be perpendicular to each other. You can have a current in the $$x$$ direction and an $${\bf E}$$ field in the $$y$$ direction. In your pictured Hall bar there is net left-to-right current, but only a top-to-bottom potential drop: $$j_x= \frac{ne^2}{h} E_y.$$

• But on a plateaux, we have both longitudinal resistivity and conductivity equal to zero. If there would be a current in x-direction, this would be a superconducting state, right? Isn't it true that zero longitudinal resistivity and conductivity just means that the current is not able to flow in longitudinal direction? – Motionx Apr 29 '20 at 14:49
• No. As I explained, it just means that the voltage drop is pependicular to the current. The situation is similar to a superconductor in that ${\bf j}\cdot {\bf E}=0$ means that there is no dissipation, but it is different from superconductivity in many ways. – mike stone Apr 29 '20 at 15:05
• Thanks for your answer. Sorry for my confusion. But then im still confused about the interpretation. What exactly means zero conductivity in longitudinal direction? So if we have current in x-direction & no dissipation, why is this different than a superconductor? – Motionx Apr 29 '20 at 15:41
• In superconductor we can have a current with no electric field. On a QHE plateau a current always has an electric field at right-angles to it. Zero longitudinal conductivity means that given an ${\bf E}$ field there is no curent parallel to ${\bf E}$. There can be a current caused by ${\bf E}$ and at right angles to ${\bf E}$. The matrix expressions I wrote in my answer make this clearer than words can. – mike stone Apr 29 '20 at 15:53
• ok, but in the pictured Hall bar, we have the source & drain both on different potentials in the longitudinal direction. You wrote, that there is only a top to botton potential drop, why isn't there also a right to left potential drop? I think there is something fundamental which I didn't unterstand. How can one understand, that on a IQHE plateaux both longitudinal resistivity & conductivity is zero at the same time, what is the mechanism behind? – Motionx Apr 29 '20 at 16:04