2deg - magnetotransport, quantum hall effect

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!


Classical and quantum mechanics of 2DEG

Classical motion:

Cyclotron orbit

Cyclotron frequency,

Cyclotron radius,

In classical mechanics, any size of the orbit is allowed.


Conductance becomes a tensor:

Relaxation time


Conductance and resistance are tensors:

For classical transport,

What happens according to quantum

mechanics?

Equipotential lines


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


Classical picture Quantum picture


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


Landau quantization (reminder from QM)

Magnetic field is described by the vector-potential,

We will use the so-called Landau gauge,

In magnetic field,


Ansatz:

Cyclotron frequency Displacement

Similar to harmonic oscillator


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.


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


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


Metal

Insulator

A series of metal-to-insulator

transitions

A way to measure – magneto-capacitance spectroscopy


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


“Chemical” capacitance

The energy, E, is fixed by Vdc


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.


The quantum Hall effect

Ordinary Hall effect


Klaus von Klitzing, 1980

Si-MOSFET

What is the origin of this fantastic phenomenon?

The following discussion will be

oversimplified


Conductance and resistance are tensors:

Therefore small corresponds to small . How comes?


Equipotential lines

E E


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):


Role of edges and disorder

Cyclotron motion in confined geometry

Classical skipping orbits Quantum edge states


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


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


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.


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.


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).


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


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

2deg - magnetotransport, quantum hall effect

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