Physics: Quantum Classical Expretation Of IQHE)

Table of content

Introduction Two dimensional electron system Hall geometry Classical interpretation of IQHE Quantum mechanical interpretation of IQHE Significance of IQHE References

The integral quantum hall effect (IQHE) was first discovered by Klaus von Klitzing and coworkers in 1980. It was demonstrated by two dimensional electron system in a device called metal oxide semiconductor field effect transistor (MOSFET), under low temperature and large magnetic field. They discovered that the transverse conductance dropped to zero at some values of magnetic field, and concurrently the longitudinal resistance didn’t change its value and hence showing plateaus on figure 1 [1980]. It was a definitely remarkable discovery featured by a wonderful equation R_H=h/ie^2 saying that the hall resistance is quantized where i is an integer. This equation relates physical quantity to fundamental physical constants, and thus can be served as a standard of physical quantities like resistance [].
This essay will explain how the hall resistance is quantized and its contributions to physics world.

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So, before looking at IQHE in details, we can gain some insights from the simple Hall effect. Let’s look at the simple theory of Hall effect. Consider a rectangular metal with height h, width w and length l. Suppose the current I_x=Aj_x is flowing inside along x direction and we apply magnetic field B in z-direction. The magnetic force pushes the electrons to the negative y-direction and a transverse electric field is built up. In equilibrium, the electric force and magnetic force on electrons balance each other or mathematically E_y=vB, where v is the drift velocity of electrons. Then the Hall voltage is measured in y-direction, V_H=wE_y. In term of volume density of electron in the metal, the current density can be written as j_x=nev. The Hall resistance is

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