The Quantum Hall Effect (QHE)

Abstract

The observations of the quantum Hall effect (QHE) , and The Fractional Quantum Hall Effect (FQHE), which is mentioned in Sect. 14.6, were made possible by advances in the preparation of high mobility materials with electrons and/or holes serving as physical realizations of a 2D electron gas . The MOSFET devices (see Sect. 12.2) and the modulation-doped heterostructures (see Sect. 12.3) that give rise to the formation of a 2D electron gas in a narrow interface region of typical samples. In this Chapter we present a simple view of the physics of the quantum Hall effect and the two-dimensional electron gas used to study the physics of the quantum Hall Effect.

Problems

14.1

• (a) Assume that the conductivity tensor ${\mathop {\sigma }\limits ^{\leftrightarrow }}$ is an off–diagonal matrix,

  $$ {\mathop {\sigma }\limits ^{\leftrightarrow }} = \begin{bmatrix} 0&-ie^2/h\\ ie^2/h&0\\ \end{bmatrix} $$

  Show that the measured Hall resistances $R_H$ and longitudinal resistances $R_x$ are independent of sample geometry.

• (b) In order to observe the Quantum Hall Effect , why is it necessary for the electron gas be two dimensional?

14.2

In the integral quantum Hall effect , the electron density n of a 2D free electron gas is given by

$$\begin{aligned} n = i\frac{eB}{hc} \end{aligned}$$

(14.42)

where i is an integer denoting the number of occupied magnetic energy levels, and the electron density n depends only on the magnetic field B.

• (a) How do you reconcile this quantization of n with the 2D electron density measured in zero magnetic field?

• (b) Suppose now that we have a 2D semiconductor with 4-fold symmetry having elliptical constant energy surfaces centered at the Brillouin zone boundary, where the E($\mathbf {k}$) relation for pocket #1 is written as

  $$\begin{aligned} E(\mathbf {k}) = \frac{\hbar ^2k_x^2}{2m_{xx}} + \frac{\hbar ^2k_y^2}{2m_{yy}}, \end{aligned}$$

  (14.43)

  in which $m_{xx} = m_0$ and $m_{yy} = 0.1m_0$. Starting with $n_0$ electrons/cm$^2$ at zero magnetic field, find the magnetic field B$_c$ (for $\mathbf {B}\parallel z$-axis) at which all the carriers in pocket #1 are transferred to carrier pocket #2.

• (c) What is $\rho _{xx}$ and $\rho _{xy}$ for this value of magnetic field B$_c$?

• (d) How do you reconcile the results in part (b) with the results in part (a)?

14.3

Suppose that you have a modulation doped (n–type) quantum well structure composed of layers of GaAs/$\mathrm{Ga}_{1-x}\mathrm{Al}_x\mathrm{As}$ such that the bulk carrier density in the GaAs is $10^{16}$/cm$^3$ and the width of the quantum well is 80$\mathrm \AA$. (Use $m_e^* = 0.07 m_0$, $E_{g1} = 1.42$ eV for GaAs, and $E_{g2} =1.70\,eV$ for $\mathrm{Ga}_{1-x}\mathrm{Al}_x\mathrm{As}$ and $\varDelta E_c = 3 \varDelta E_v$ for the band offsets). For simplicity in calculating the energy levels, use the energy eigenvalues of the infinite well.

• (a) What is the quantum well widths range so that two bound states are contained in the quantum well at zero magnetic field. How many Landau levels are occupied at a field of 10 Tesla applied normal to the two dimensional electron gas? What is the fractional occupation of the last Landau level? The fractional occupation refers to the number of occupied states in the Landau level compared to the total number of states in the Landau level obtained from the degeneracy factor.

• (b) Give design parameters for a quantum well that has only 1 bound state level, and this level has a filling factor or fractional occupation of 1/3.