2DEG - Magnetotransport, quantum Hall effect
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Magnetotransport in 2DEG
Magnetotransport in 2DEG
Contents
• Classical and quantum mechanics of two-dimensional electron gas
• Density of states in magnetic field • Capacitance spectroscopy • (Integer) quantum Hall effect • Shubnikov-de-Haas-oscillations (briefly)
Lorentz force:
Newtonian equation of motion:
Perpendicular to the velocity!
Magnetotransport in 2DEG
Classical and quantum mechanics of 2DEG
Classical motion:
Cyclotron orbit
Cyclotron frequency,
Cyclotron radius,
In classical mechanics, any size of the orbit is allowed.
Magnetotransport in 2DEG
Conductance becomes a tensor:
Relaxation time
Magnetotransport in 2DEG
Conductance and resistance are tensors:
For classical transport,
What happens according to quantum
mechanics?
Equipotential lines
Magnetotransport in 2DEG
Bohr-Sommerfeld quantization rule:
the number of wavelength along the trajectory must be integer.
Only discrete values of the trajectory radius are allowed
Energy spectrum: Landau levels
Wave functions are smeared around classical orbits with
lB is called the magnetic length
Magnetotransport in 2DEG
Classical picture Quantum picture
Magnetotransport in 2DEG
The levels are degenerate since the energy of 2DEG depends only on one variable, n.
Number of states per unit area per level is
Realistic picture
Finite width of the levels is due to disorder
Magnetotransport in 2DEG
Landau quantization (reminder from QM)
Magnetic field is described by the vector-potential,
We will use the so-called Landau gauge,
In magnetic field,
Magnetotransport in 2DEG
Ansatz:
Cyclotron frequency Displacement
Similar to harmonic oscillator
Magnetotransport in 2DEG
Since kx is quantized, , the shift
is also quantized, , so
The values of ky are also quantized,
By direct counting of states we arrive at the same expression for the density of states.
Magnetotransport in 2DEG
Usually the so-called filling factor is introduced as
For electrons, the spin degeneracy
Magnetic field splits energy levels for different spins, the splitting being described by the effective g-factor
For bulk GaAs,
- Bohr magneton
Magnetotransport in 2DEG
An even filling factor, , means that j Landau levels are fully occupied.
An odd integer number of the filling factor means that one spin direction of Landau level is full, while the other is empty.
How one can control chemical potential of 2DEG in magnetic field?
By changing either electron density (by gates), or magnetic field.
We illustrate that in the next slide assuming
Hence, the integrated density of states per Landau level is
Magnetotransport in 2DEG
Metal
Insulator
A series of metal-to-insulator
transitions
A way to measure – magneto-capacitance spectroscopy
Magnetotransport in 2DEG
Insulating spacer
δ-doping
The current at a phase difference π/2 to ac signal is measured by lock-in amplifier
Charge injection changes the 2DEG Fermi level
Magnetotransport in 2DEG
“Chemical” capacitance
The energy, E, is fixed by Vdc
Magnetotransport in 2DEG
The measured capacitance shows the filling of the 2DEG at Vg = 0.77 V, as well as the modulated density of states in perpendicular magnetic fields.
Magnetotransport in 2DEG
The quantum Hall effect
Ordinary Hall effect
Magnetotransport in 2DEG
Klaus von Klitzing, 1980
Si-MOSFET
What is the origin of this fantastic phenomenon?
The following discussion will be
oversimplified
Magnetotransport in 2DEG
Conductance and resistance are tensors:
Therefore small corresponds to small . How comes?
Magnetotransport in 2DEG
Equipotential lines
E E
Magnetotransport in 2DEG
Solution in the absence of scattering
drift velocity of the guiding center
cyclotron radius
Drift of a guiding center + relative circular motion
(Over)simplified explanation: Classical picture
From that (after averaging over fast cyclotron motion):
Magnetotransport in 2DEG
Role of edges and disorder
Cyclotron motion in confined geometry
Classical skipping orbits Quantum edge states
Magnetotransport in 2DEG
Calculated energy versus center coordinate for a 200-nm-wide wire and a magnetic field intensity of 5 T. The shaded regions correspond to skipping orbits associated with edge-state behavior.
Schematic illustration showing the suppression of backscattering for a skipping orbit in a conductor at high magnetic fields. While the impurity may momentarily disrupt the forward propagation of the electron, it is ultimately restored as a consequence of the strong Lorentz force.
Only possible scattering is in forward direction – chiral motion
Magnetotransport in 2DEG
Disorder makes the states in the tails localized!
Sketch of the potential profile at different energies
Lakes and mountains do not allow to come through, except very close to the LL centers
Magnetotransport in 2DEG
Localized states in the tails cannot carry current.
Consequently, only extended states below the Fermi level contribute to the transport. Thus is why Hall conductance is frozen and does not depend on the filling factor!
Localized states in the tails serve only as reservoirs determining the Fermi level
In the region close to E2 electrons can percolate, and this is why transverse conductance is finite.
The above explanation is oversimplified.
And we have not explained yet why the Hall resistance is quantized in .
We will come back to this issue after consideration of one-dimensional conductors.
Magnetotransport in 2DEG
Quantum Hall effect: Application to Metrology
Since 1 January 1990, the quantum Hall effect has been used by most National Metrology Institutes as the primary resistance standard.
For this purpose, the International Committee for Weights and Measures (CIPM) set the imperfectly known constant RK (=quantized Hall resistance on plateau 1) to the then best-known value of RK-90 = 25812.807 Ω.
The relative uncertainty of this constant within the SI is 1x 10-7, and is therefore about two orders of magnitude worse than the reproducibility on the basis of the quantum Hall effect.
The uncertainty within the SI is only relevant where electrical and mechanical units are combined.
Magnetotransport in 2DEG
Using a high-precision resistance bridge, traditional resistance standards are compared to the quantized Hall resistance, allowing them to be calibrated absolutely. These resistance standards serve in their turn as transfer standards for the calibration of customer standards. The measurement system at METAS (Federal office of Metrology, Switzerland) allows a 100 Ω resistance standard to be compared to the quantized Hall resistance with a relative accuracy of 1x10-9. This measurement uncertainty was confirmed in November 1994 by comparison with a transfer quantum Hall standard of the International Bureau of Weights and Measures (BIPM).
Magnetotransport in 2DEG
Shubnikov-de-Haas oscillations
In relatively weak magnetic fields quantum Hall effect is not pronounced.
However, density of states oscillates in magnetic field, and consequently, conductance also oscillates.
Mapping these – Shubnikov-de-Haas- oscillations to existing theory allows to determine effective mass, as well as scattering time.
This is a very efficient way to find parameters of 2DEG
Thus, magneto-transport studies are very popular
Magnetotransport in 2DEG
What has been skipped?
Detailed explanation of the Integer Quantum Hall Effect
Theory of the Shubnikov-de Haas effect
Fractional Quantum Hall Effect (requires account of the electron-electron interaction)
Magneto-transport is a very important tool for investigation of properties of low-dimensional systems
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