1. science
2. physics

Magnetotransport in 2DEG
Content
• 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
Magnetotransport in 2DEG
Classical and quantum mechanics of 2DEG
Classical motion:
Lorentz force:
Perpendicular to the velocity!
Newtonian equation of motion:
m*v2/rc = evB; v=eBrc/m*; =v/2rc; c=2=eB/m*
Cyclotron orbit
Cyclotron frequency,
Cyclotron radius,
In classical mechanics, any size of the orbit is allowed.
Magnetotransport in 2DEG
Diffusive transport
Between scattering events electrons move like free
particles with a given effective mass.
In 1D case the relation between the final velocity and the
effective free path, l, is then
Assuming
where
is the drift velocity
while
is the typical velocity and introducing the
collision time as
we obtain in the linear
approximation:
Mobility
Update of solid state physics
4
Diffusion motion of electron in magnetic field
ωcτ ≤ 1
In magnetic field
“friction”
Lorentz force
Update of solid state physics
5
Conductivity tensor
Magnetic field is applied in the z-direction, B = (0, 0, B) Important quantity is the
product of the cyclotron
frequency,
by the relaxation time,
S is a geometry factor
Here vi are the components of the drift velocity vector. Solving this system
of equations for j gives j = ^
σE with conductivity as a tensor,
Resistivity tensor,
Update of solid state physics
6
Classical Hall effect
S is a geometry factor
For classical transport,
xy = -c/0 =
-(eB/m)/(ne2/m) = -B/en
Equipotential lines
Hall coefficient
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:
ωcτ ≥ 1
Landau levels
Wave functions are smeared around
classical orbits with
rn = lB (n+1)1/2;
lB= (ħ/cm)1/2
lB is called the magnetic length. It is radius
of classical electron orbit for n = 0.
  v/r; r v/; mv2/2= ħc(n+1/2); vn0 = (ħc/m)1/2; lB= rn0 = (ħ/cm)1/2
Magnetotransport in 2DEG
Wave functions in rotationally-invariant gauge
Classical picture
Magnetotransport in 2DEG
Quantum picture
Density of states
Two dimensional system
, periodic boundary conditions
Momentum is quantized in units of
A quadratic lattice in k-space, each of them is g-fold
degenerate (spin, valleys).
Assume that
, the limit of continuous spectrum.
Number of states between k and k+dk:
Update of solid state physics
10
Density of states in 2D
Number of states per volume per the region
k,k+dk
Density of states -Number of states per volume per the
region E,E+dE. Since
3
Update of solid state physics
11
Density of states in different dimensions
Electron density of states in the effective mass
approximation as a function of energy, in one, two, and
three dimensions
Update of solid state physics
12
Modification of density of states
The levels are degenerate since the energy of 2DEG depends only on one variable, n.
Number of states per unit area per level is

m*
Realistic picture
ωcτ ≥ 1
Finite width of the levels is due to disorder
Magnetotransport in 2DEG
Landau quantization
Magnetic field is described by the vector-potential,
We will use the so-called Landau gauge,
curl F =
In magnetic field,
Magnetotransport in 2DEG
Harmonic oscillator in Landau quantization
Displacement
Cyclotron frequency
Similar to harmonic oscillator
Magnetotransport in 2DEG
2DEG quantization conditions in Landau gauge
Since kx is quantized,
kx,n = 2n/Lx , the shift
is also quantized and oscillators are centered at
positions
, so
2
2
The values of ky are also quantized,
2
By direct counting of states we arrive at the same
expression for the density of states.
Magnetotransport in 2DEG
2DEG quantization in Landau gauge
The eigenfunctions of the x–y Hamiltonian are thus plane waves in the
xdirection, multiplied with Hermite polynomials in the y-direction. The
positions of the harmonic oscillator potentials yn in the y-direction are
given by the wave numbers kx that satisfy the boundary condition.
Magnetotransport in 2DEG
Filling factor in Landau quantization
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
Meaning of filling factor
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
Varying occupation of Landau levels
Metal
Insulator
A series of
metal-to-insulator
transitions
A way to measure – magneto-capacitance spectroscopy
Magnetotransport in 2DEG
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
Magnetotransport in 2DEG
“Chemical” capacitance
“Chemical” capacitance
The energy, E, is fixed by Vdc
Magnetotransport in 2DEG
Magneto-capacitance spectroscopy
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 classical Hall effect
Ordinary Hall effect
Magnetotransport in 2DEG
Quantum Hall effect
Klaus von Klitzing,
1980
Si-MOSFET
The accuracy
δρxy/ρxy is of the
order of 3 × 10−10.
What is the origin of this fantastic phenomenon?
Magnetotransport in 2DEG
Resistivity in Quantum Hall effect
Conductivity and resistivity are tensors:
Therefore small
corresponds to small
. How comes?
Magnetotransport in 2DEG
The observation σxx = 0 implies
that no current flows in the xdirection when a voltage is
applied in the x-direction. On the
other hand, ρxx = 0 means that
applying a current in the xdirection causes no voltage drop
in the x-direction.
Origin of low xx in Quantum Hall effect
Equipotential lines
E
E
0 = -ne2/m*
For classical transport,
For quantum transport,
j
0
1
-1
0
Magnetotransport in 2DEG
Simplified explanation of Quantum Hall effect
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
Edges states in Quantum Hall effect
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
Lev Davidovich Landau: The Nobel Prize in Physics 1962
Born: 22 January 1908, Baku, Russian
Empire (now Azerbaijan)
Died: 1 April 1968, Moscow, USSR (now
Russia)
Affiliation at the time of the award:
Academy of Sciences, Moscow, USSR
Prize motivation: "for his pioneering
theories for condensed matter,
especially liquid helium"
Field: condensed matter physics,
superfluidity
Prize share: 1/1
Magnetotransport in 2DEG
Klaus von Klitzing: The Nobel Prize in Physics 1985
Born: 28 June 1943, Schroda,
German-occupied Poland (now Poland)
Affiliation at the time of the award:
Max-Planck-Institut für
Festkörperforschung, Stuttgart,
Federal Republic of Germany
Prize motivation: "for the discovery
of the quantized Hall effect"
Field: condensed matter physics
Prize share: 1/1
Magnetotransport in 2DEG
Origin of extended localized states
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
Influence of disorder
‘The disorder does something truly remarkable: it increases the
insulating regions of the parameter range (the parameter is the
electron density or the magnetic field) from points to extended
intervals in real samples, while the extended metallic regions of
the ideal sample are reduced to very small intervals.’ TH
Magnetotransport in 2DEG
Quantum Hall effect in three dimensions?
Quantum Hall effect vanishes in
three-dimensional free electron
gases because the motion in the
direction
of
field
remains
unaffected. A periodic superlattice
generates bands with bandgaps b in
the meV regime, i.e. comparable to
ħωc for moderate magnetic fields.
The corresponding density of
states thus develops gaps in
sufficiently large magnetic fields,
such that the quantum Hall effect
should be visible as soon as b gets
smaller than ħωc.
Landau quantization in three-dimensional systems. (a) Large magnetic field condenses free electron
gas into Landau levels in the (x, y) plane, while kz remains continuous. (b) The density of states for a
free electron gas (bold line), and of a periodic superlattice in the z-direction (dashed lines).
Magnetotransport in 2DEG
Main features of Quantum Hall effect
•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.
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
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
Since h/e2 and 2e/h are the
only “fundamental constants”
which can be measured with
such a high precision, that
conventional
values
were
introduced for metrological
applications, one may think to
fix these numbers instead of
fixing e and h which are not
directly
accessible
by
experiment with high accuracy.
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Klaus von Klitzing: the new SI
From presentation in The Royal Society, London, 2011
Magnetotransport in 2DEG
Metrology and Quantum Hall effect
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
Shubnikov-de-Haas oscillations
In
Shubnikov–de
Haas
(SdH)
oscillations at small and intermediate
magnetic fields, i.e. for ωcτ < 1,
quantum Hall effect is weak, but ρxx(B)
oscillates and does not vanish. The
magnetic field induces weak modulation
of the density of states. As a
consequence, at the Fermi level the
screening properties of the electron
gas oscillate with magnetic field. For
short-range scattering potentials an
analytic expression for ρxx(B) was
derived by Ando.
SdH oscillations of a 2DEG as a function of B
(Top). Bottom: Temperature dependence of
some oscillations. The temperatures are Θ = 1,
2, 4, 6, 8, 10, 12, 15, and 20 K.
Magnetotransport in 2DEG
Ando formula for Shubnikov-de-Haas oscillations
ρxx(B) was derived by Ando:
Both the effective mass and τq can be extracted from measuring the
temperature dependence of the SdH oscillations.
For a resonance:
Plot of ln(A/Θ) vs.
Θ gives a straight
line with a slope of:
For low field oscillations sinh x ≈ x
Magnetotransport in 2DEG
Examples of Shubnikov-de-Haas experiments
In quasi-2DEG each two-dimensional subband causes Shubnikov– de Haas
oscillations
Longitudinal magneto-resistivity in a GaAs–
lxGa1−xAs HEMT with two occupied
subbands. Light ionizes residual neutral
donors increasing the electron density. The
two SdH frequencies correspond to the
partial electron densities in the two
subbands.
Magnetotransport in 2DEG
A parabolic and positive
magneto-resistivity
around B = 0 is the
consequence
of
two
occupied subbands
too
that can be regarded as
resistors in parallel. The
total conductivity tensor
is obtained by simple
addition of the individual
conductivity
tensors.
Suggesting
different
scattering times τ1 and τ2:
Mapping of wave function probability density
Inserting a δ function U0δ(z − z0) in a potential generates energy shifts ΔEi of the
energy eigenvalues Ei, where ΔEi is proportional to the probability density of the
corresponding eigenstate at z0: ΔEi = U0|ψ(z)|2. With U0 known, |ψ(z)|2 can be
determined by measuring ΔEi.
For
a
parabolic
potential,
superposition of a constant electric
Al0.3Ga0.7As
field
displaces
the
potential
without changing its shape.
Two subband densities are
measured via SdH oscillations.
Al-Ga-As + (z-z0)AlxGa1-xAs
Measured differences of the probability
density between subbands 1 and 2 as a
function of z for two different electron
densities. The different symbols denote
different spike heights, i.e. an Al
concentration of x = 0.05, 0.1, and 0.15,
respectively.
Magnetotransport in 2DEG
Displacement of the quantum Hall plateaux
With symmetrical peaks in the density of
states Hall plateaus are cantered around
integer filling factors : if we
extrapolate the classical Hall slope into
the quantum Hall regime, it should
intersect the plateaus at their centre.
With asymmetrical peaks, centres of Hall
plateaus shift.
Repulsive scatterers shift a fraction of the states
within a peak of the density of states to higher
energies, which results in a shift of the quantum Hall
plateaus to larger magnetic fields. Likewise,
predominantly attractive scatterers shift the quantum
Hall plateaus to smaller magnetic fields.
Magnetotransport in 2DEG
Effect of parallel magnetic field in 2DEG
Parallel field tunes the spin splitting. It also adds to the electrostatic
confinement, shifting position of the potential well and energies of subbands
to higher values: ‘diamagnetic shift’. In addition it increases the effective
mass of charge carriers in the direction perpendicular to B, but in the plane
of the electron gas and enhances separation of energy levels.
Left: A parabolic quantum well in perpendicular and parallel magnetic fields. Center: Fermi sphere in a
magnetic field applied in the x direction (full line), in comparison to the Fermi sphere for B = 0 (dashed
line). Right: The confining potential in the z-direction for B = 0 (dashed line) shifts and narrows in a
parallel magnetic field (full line). This leads to a diamagnetic shift of the energy levels, and an increase
in the subband separation.
Magnetotransport in 2DEG
2DEG in parallel magnetic field : experiment
Magneto-resistivity at different
temperatures in a parabolic quantum
well with three occupied subbands in
the tilted magnetic fields. The
oscillation in 1 T < B < 2 T is
attributed to the second subband,
while oscillations at higher magnetic
fields stem from the first subband.
There are no oscillations in parallel
field.
(a) Plot of ρxx(B) for a tilt angle of zero (θ = 0).
The minima are due to the depletion of
subbands 3 and 2. As the sample is tilted, a
perpendicular magnetic field component
generates SdH oscillations. (b) Temperature
dependent measurements allow one to
determine the effective electron mass.
Magnetotransport in 2DEG
2DEG in parallel field : results of experiment
Shubnikov–de Haas (SdH) oscillations in a
parabolic quantum well in parallel magnetic
field together with Hall measurements
performed in order to determine the total
electron density allow to calculate properties
of electrons in different subbands. The upper
subband is depleted by parallel field, while
electron density in lower subbband approaches
the total electron density at strong parallel
magnetic fields. There is visible increase of
effective mass in parallel field. The increase
in quantum scattering time τq is an unexpected
feature.
Experiment in tilted magnetic field provides information
about the electron densities in the lowest two subbands
(a), the effective mass (b) and the scattering times (c) as
functions of parallel component of B. Most of the data are
in reasonable agreement with available theoretical models.
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 investigating
properties of low-dimensional systems
Magnetotransport in 2DEG
Home activity for Tue. 24 and Wed. 25 February
a) Read: TH, Chapter 6 ‘Experimental Techniques’ (pp 166-195). Try to understand
behaviour of 2D electron gas in strong magnetic field. Identify issues that you
would like to discuss on Practical, Wed. 25 Febuary.
b) Work with questions to Chapter 6: Questions and other questions.
c)
Refresh Chapters 1 and 2 in TH focusing on Hall effect, pp 24-26, the electronic
density of states, pp. 66-68, and the magneto-resistivity tensor on pp. 65-66.
d) Address exercises in TH on pp. (194-195). Can you discuss them on practical?
e) Prepare short presentations (for those who did not do this). Choose date for full
presentation. Available dates: February 18, 25; March 4, 11, 18, 25; April 22, 29;
May 6, 13. Please notify of your choice. Please send pdf files of presented short
and full talks to [email protected] to put them on web.
Introduction
57
MENA5010 presentations
List of full presentations :
Hans Jakob Sivertsen Mollatt – presented
Jon Arthur Borgersen February 25
Knut Sollien Tyse March 18
Ymir Kalmann Frodason March 18
Simen Nut Hansen Eliassen- March 25
Steinar Kummeneje Grinde - April 22
Bjørn Brevig Aarseth April 29
There was very impressive short presentation by Daniel Wolseop Lee on February 11!
Introduction
58
Nobel Laurete Lecture by Professor Klaus von Klitzing
Hall effect
Magnetotransport in 2DEG