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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)
Magnetotransport in 2DEG
Classical and quantum mechanics of 2DEG
Classical motion:
Lorentz force:
Perpendicular to the velocity!
Newtonian equation of 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,
Equipotential lines
What happens
according to quantum
mechanics?
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
Magnetotransport in 2DEG
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
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:
Displacement
Cyclotron frequency
Similar to harmonic oscillator
Magnetotransport in 2DEG
Since kx is quantized,
is also quantized,
, the shift
, 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
- Bohr magneton
For bulk GaAs,
Magnetotransport in 2DEG
An even filling factor,
levels are fully occupied.
, means that j Landau
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
The following
discussion will be
oversimplified
What is the origin of this fantastic phenomenon?
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
(Over)simplified explanation: Classical picture
Solution in the absence of scattering
cyclotron
radius
Drift of a guiding center + relative
circular motion
From that (after averaging over fast cyclotron motion):
Magnetotransport in 2DEG
drift velocity of the
guiding center
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 200nm-wide wire and a magnetic field intensity of 5 T.
The shaded regions correspond to skipping orbits
associated with edge-state behavior.
Only possible scattering is in forward direction –
chiral motion
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.
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 onedimensional 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 electronelectron interaction)
Magneto-transport is a very important tool for investigation
of properties of low-dimensional systems
Magnetotransport in 2DEG