Magnetotransport in 2DEG

Classical and quantum mechanics of 2DEG Classical motion: Cyclotron orbit Cyclotron frequency, Cyclotron radius,... Only discrete values of the trajectory radius are allowed Energy spect

Magnetotransport in 2DEG

Lorentz force:

Newtonian equation of motion:

Perpendicular to the velocity!

Classical and quantum mechanics of 2DEG Classical motion:

Cyclotron orbit Cyclotron frequency, Cyclotron radius,

Conductance becomes a tensor:

Relaxation time

Conductance and resistance are tensors :

For classical transport,

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

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

“Chemical” capacitance The energy, E, is fixed by Vdc

The quantum Hall effect

Ordinary Hall effect

Klaus von Klitzing,

1980

Si-MOSFET

The following discussion will be

Conductance and resistance are tensors :

Therefore small corresponds to small How comes?

Equipotential lines

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 nm-wide wire and a magnetic field intensity of 5 T

200-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

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

We will come back to this issue after consideration of dimensional conductors

one-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

Using a high-precision resistance bridge, traditional

resistance standards are compared to the quantized

Hall resistance, allowing them to be calibrated

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

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