Theory
What is the Quantum Hall Effect?
The Hall effect occurs when a sample with a current running through it is placed in a magnetic field (Figure 1). The magnetic field deflects charge carriers to the edge of the sample, so an electric field builds up across the sample width. The electric field causes a voltage drop across the sample. This voltage is called the Hall voltage. In the presence of strong magnetic fields, the Hall voltage becomes quantized. This is called the Quantum Hall Effect. The Hall voltage is often expressed as a resistance by dividing by the current through the sample.
The applied magnetic field causes the charge carriers to rotate (classically), and the direction of the magnetic field and the type of charge carrier determine direction of rotation. In the center, the charge carriers are able to complete full orbits. However, at the edge of the sample, the charge carriers bounce off of the edge. This means they are not able to complete full rotations. Since the magnetic field imposes a direction of rotation, the charge carriers cannot move backwards after they bounce of off the edge! This would require the charge carriers to change their direction of rotation. By the same logic, the edge states are dissipation-less. Dissipation of current occurs when charge carriers scatter off of an irregularity in the structure or material of the sample. On the edge of the sample, the charge carriers can still scatter from impurities, but they are forced to continue moving in their original direction in semicircular paths by the magnetic field. This means that scattering does not diminish the current running through the edge states. They can thus carry current with no resistance. Figure 3 shows the semicircular edge paths, and the full rotations of the charge carriers in the bulk of the sample.
Dissipation-less Chiral Edge States
The wavefunctions associated with each Landau level are localized in the y direction [1]. (Across the width of the sample. see Figure 1 for a definition of the axes). This means that the charge carrier associated with each wave function was likely to be found within a narrow width in the y direction. A top down view of the sample is shown in Figure 2. The lines show the positions of the localized wave functions. Because the wavefunctions are localized, there is a set of states that live on the edge of the sample. These states are called edge states. These edge states can carry current, so they have a conductance. These edge states are chiral, meaning that they can only carry current is one direction along the sample's edge. This chirality is imposed by the magnetic field.
Landau Quantization
When placed in a magnetic fields, charge carriers rotate in circles. Thus, the magnetic field imposes boundary conditions on the their movement, causing the charge carrier energy levels to become quantized. These quantized charge carrier energy levels are called Landau levels.
Hall Conductance
The Hall conductance is defined as the current through the sample divided by the Hall voltage.
The Hall conductance is thus quantized when the Hall voltage is quantized. The Hall voltage is the same as the total edge state conductance. This can be shown by calculating the total edge current on both sides of the sample . This is not done here, and the forms of the Hall conductance in both bilayer and monolayer graphene are shown below [2].
e is the electron charge, h is Planck's constant, and N is called the Landau number. It is an integer that is determined by the highest energy Landau level occupied on the edge. Both equations are examples of the Integer Quantum Hall effect, since N is an integer. Changing either the applied magnetic field or the charge carrier density of the sample changes N, so sweeping over either of these reveals a discrete pattern of plateaus in the Hall conductance. This is shown in Figure 4. These are linear equations in N. Because the denominator of the slope is Planck's constant, fitting measured Hall conductances to these equations is a way to determine Planck's constant.
Shubnikov-De Haas Oscillations
The longitudinal resistance (In the direction of the current in Figure 1) is also interesting. It is briefly discussed, although the theory is not explained in depth. Sweeping over either the charge carrier density or applied magnetic field also affects the charge carriers in the bulk of the sample. The bulk resistance undergoes Shubnikov-De Haas (SDH) oscillations. The resistance is zero, except at the transitions between plateaus in Hall resistance/conductance [1]. Sweeping over the charge carrier density thus produces a patter of peaks and troughs, as shown in Figure 4.
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References
[1] Leggett, Anthony J. “Lecture 7: The Integral Quantum Hall Effect.” 2010.
[2] Geim, A. K., and K. S. Novoselov. “The Rise of Graphene.” Nature Materials, vol. 6, no. 3, 2007, pp. 183–191. doi:10.1038/nmat1849.