5.3: Quantum Hall Effect
- Page ID
- 18967
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Around a hundred years after the discovery of the Hall effect, the quantum Hall effect was discovered. Klaus von Klitzing discovered the integer quantum Hall effect in 1980 and won the physics Nobel prize for it in 1985 [63]. In 1998, Robert Laughlin, Horst Störmer, and Daniel Tsui won the physics Nobel prize for the discovery of the fractional quantum Hall effect [64]. The integer quantum Hall effect is observed in two dimensional electron gases which can occur, for example, in an inversion layer at the interface between the semiconductor and insulator in a MOSFET [59]. As in the Hall effect, a current is applied in one direction, and the Hall voltage is measured in the perpendicular direction. Following Fig. 5.1.1, assume that a current is applied along the \(\hat{a}_x\) direction in the presence of an external magnetic field in the \( \hat{a}_z\) direction. The voltage \(V_{AB}\) is measured, and Hall resistance \(R_H\) is calculated. The quantum Hall effect is observed at low temperatures and in the presence of strong applied magnetic fields. In such situations, the Hall resistance has the form
\[R_H = \frac{h}{q^2 \cdot \mathfrak{n}} \nonumber \]
where \(h = 6.626 \cdot 10^{-34} J \cdot s\) is the Planck constant and \(\mathfrak{n}\) is an integer [59]. This effect is called the quantum Hall effect because \(R_H\) can take only discrete values corresponding to integer values. Values of the Hall resistance can be measured extremely accurately, to 2.3 parts in \(10^{10}\) [59]. The fractional quantum Hall effect is observed in highly ordered two dimensional electron gases in the presence of very strong magnetic fields, and it involves quantum mechanical electron-electron interactions [65].
The formal definition of the ohm relies on definitions of the meter, kilogram, and second. The kilogram is defined with respect to the weight of a physical object made of platinum and iridium housed in the International Bureau of Weights and Measures in France [59]. Multiple national labs, including the National Institute of Standards and Technology in the United States, have come up with an experimental means of defining the ohm involving the quantum Hall effect. This standardized definition of the ohm is accurate to one part in \(10^9\) which is more accurate than previous definitions involving the kilogram, meter, and second [59]. Because of the high accuracy with which the integer quantum Hall effect can be measured, scientists have proposed using experiments involving it to standardize the measurement of the Planck constant and the definition of the kilogram instead of relying on a definition involving a physical object. These new standards have not been adopted yet, but they may be implemented as early as 2019 [66].