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2. physics

Quantum Wires and Quantum Point
Contacts
Quantum conductance
Classification of quasi-1D systems
Quasi-1D systems with width w comparable to the
de Broglie wavelength
Quantum wires and point contacts
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Quantum Hall effect
Klaus von Klitzing
The accuracy
δρxy/ρxy is of the
order of 3 × 10−10.
Quantisation in units RK=h/e2
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
A quasi-1D system
6.45 k
Superconductivity is suppressed
when the room temperature
resistance is greater than the
superconducting resistance
quantum, Rq=h/4e2
Quantum wires and point contacts
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Focused Ion Beam quantum wires fabrication
YBa2Cu3O7−x
Ga - ions
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Wave functions and eigenvalues
A strip of conducting material about 10 nanometers or less in width
and thickness that displays quantum-mechanical effects.
Split gate
Interface
Simple, but realistic model:
Effective potential
Electron density
Parabolic potential well
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Parabolic confinement model
Potential:
Schrödinger equation:
Solution:
Harmonic oscillator!
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Energy spectrum in parabolic confinement
set of 1D bands
Density of states:
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Energy spectrum and density of states for parabolic confinement
F
Quantum wires and point contacts
F
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Diffusive quantum wires
Top view of a diffusive
quantum wire (L = 40 μm,
geometric width wg = 150 nm),
patterned into a Ga[AI]AsHEMT by local oxidation.
The bright lines define the
walls (a close-up is shown in
the inset).
TH
The electrostatic wire width w can be tuned by applying
voltages to the two in-plane gates IPG 1 and IPG2.
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Magnetic field effect in parabolic confinement
Parabolic confinement:
Then we have the equation very similar to that leading to
Landau quantization, however with replacements:
The resulting equation is:
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Magneto-electric modes
Magnetic field increases the confinement strength
and can lead to depopulation of magneto-electric
modes.
(at Fermi surface)
As a result, density of states oscillates as a function
of magnetic field. However, the oscillations are not
periodic in 1/B.
In very strong magnetic fields,
come back to the case of 2DEG.
Quantum wires and point contacts
, we
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Quantum wire: comparison with experiment
The magnetoresistivity of the
wire shown earlier.
The most prominent feature is
the oscillation as a function of B,
which detects the magnetic
depopulation of the wire modes.
Due to quantum
interference
The positions of the oscillation minima are plotted vs. 1/ B as full
circles in the inset. Here, the line is a least squares fit to theory,
which gives an electronic wire width of 158 nm.
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Boundary scattering
Boundary scattering in wires is important. If the
roughness scale is less or of the order of the Fermi
wave length, then the scattering is diffusive.
For smooth edges the scattering is mirror, it does
not influence conductance very much.
Magneto-transport is specifically sensitive to the
boundary scattering. Depending on the ratio
between the wire width and cyclotron radius
electron spend different time near the edges.
It influences the conductance, see next slide.
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Boundary scattering and ‘wire peak’
Diffusive scattering at the
wire boundaries causes a
magnetoresistivity peak at
w = O.55rc rc = (ħ/eB)1/2
The main figure shows these
peaks for wires of different
widths, studied at QW made
by ion beam implantation,
which generates particularly
rough, diffusive walls.
Wire peaks in QWRs defined by top gates of similar widths (shown in
the inset) are much weaker (the reflection is more specular). The
length of the wires was L = 12 μm.
For QWRs defined by top gates, by cleaved edge overgrowth, or by
local oxidation, one can safely neglect boundary scattering.
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Ballistic quantum wires
First system that have been studied – ballistic point contact
GaAlAs-HEMT
split gate
Two traces
correspond to
different
concentrations tuned
by illumination.
T = 60 mK
Gate voltage
Conductance is quantized in units of
in the absence of magnetic field
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Quantum Hall effect
Klaus von Klitzing
The accuracy
δρxy/ρxy is of the
order of 3 × 10−10.
Quantisation in units RK=h/e2
Magnetotransport in 2DEG
Quantization in quantum point contacts
The quantization seems to be similar to the quantum Hall effect.
However:
• No Landau quantization, no impurities in the contact region
• The typical accuracy is
• Conductance quantization usually disappears at L > 2 μm while
quantum Hall effect can be observed in samples of millimetre
sizes.
Principal questions:
• How finite conductance can exist without scattering?
• Where the heat is released?
• What is the reason for quantization?
Why is the conductance
quantized in units 2e2/h?
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Conductance quantization in QPC: a simple model
A barrier with
tunneling probability T,
connecting 1D wires
Let us calculate the current in a symmetric system:
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Conductance quantization in QPC
For a 1D system the density of states is inversely
proportional to the velocity!
f
E
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Conductance quantization in QPC
At T=1,
Here we have assumed that the distributions in the
leads are described by the Fermi functions, i. e., the
leads are in the equilibrium.
Such an equilibrium is achieved only at the distances
larger than the electron mean free path. Thus, the
dissipation takes place in the leads close to the
contacts.
What we have calculated is the total resistance of the
device, which can be understood as an intrinsic contact
resistance.
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Multimode conductance in QPC
What happens if the wire is quasi-1D, i. e., has several
modes?
Landauer formula,
- transmission amplitudes
If there are no inter-mode scatterings, then the
transport is called adiabatic.
This is the case for the wires with smooth modulation
of the width.
In case of pronounced backscattering the shape of the
conductance steps is modified.
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Conductance in QPC – influence of shape
Smooth
Sharp
A model for smooth point contacts
leads to the following criterion for
the transition regions to be narrow:
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Conductance in QPC – saddle-like shape
Conductance of a QPC with
a saddle point potential,
characterized by ωx and
ωy, calculated for zero
temperature. The steps
vanish as soon as ωy
becomes smaller than ωx.
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Conductance in QPC – influence of temperature
The conductance steps
vanish at a characteristic
temperature given by the
energy spacing of the
modes
.
Conductance
steps tends to saturate at
low Θ.
The steps are thermally smeared as soon as the full width at halfmaximum of ∂ f /∂E equals Δ. This is the case for Δ = 2kBΘln(3+ 2√2)
≈ 3.52kBT.
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Conductance in QPC – influence of magnetic field
The modes of a QPC get depleted
by perpendicular magnetic fields
since the magnetic field increases
the confinement and the energy
separation between the modes
becomes larger. Simultaneously,
quantum Hall states are formed in
the 2DEG, which influences the
resistance measurements.
Depopulation of QPC modes by a magnetic
field. As B is increased, the width of the
conductance plateaus increases. At B = 1.8 T
and above, the spin degeneracy gets lifted
and additional plateaus evolve at odd
integers of e2/h.
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Two-mode QPC – influence of magnetic field
The conductance
quantization
in
units of 2e2/h is
expected. If two
degenerate
modes (belonging
to
different
ladders)
cross
the
chemical
potential,
the
conductance will
change by 4e2/h.
The evolution of steps in two-mode QPC in perpendicular magnetic field. The
channels in the y-direction are labelled by l and in the z-direction by m.
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Transconductance measurements of two-mode QPC
Transconductance dG/dUsg as a
function of magnetic field. Dark
regions
correspond
to
low
transconductance: here, the chemical
potential lies in between two adjacent
modes. The bright lines (high
transconductance) reflect the mode
change. As a function of By, both
ladders show a positive dispersion.
For magnetic fields in the x-direction,
the dispersion of the m = 1 states is
reversed
Transconductance dG/dUsg as a function of
(a) By and (b) Bx. The bright lines represent
the QPC modes. (c, d) The measurements
are compared to the energy spectrum of an
elliptical parabolic confinement.
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Two-mode QPC –theoretical description
Perpendicular field (By)
Parallel field
The energy in the QPC is
proportional to Usg. The lever
arm α = dE/dUsg can be
estimated by temperaturedependent
measurements,
which give the energy spacing
between adjacent modes. A
value of α ≈ 0.02 eV/V is found.
The calculated spectra agree very well with the experimental result for ωy = 2 meV
and ωz = 5 meV.
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The “0.7 structure”
The Landauer formula explaining steps is applicable only if nothing happens
inside the wire – no interactions, inelastic processes, etc. It feils to address
some problems.
Example: puzzle
of 0.7 plateaus
Conductance of a
QPC as a function of
the spit-gate voltage.
(a) The "0.7 feature" becomes sharper with temperature increase.
(b) In strong parallel magnetic fields, the spin degeneracy is removed
and additional plateaus are visible at odd integers of e2/h. As B is
reduced, the spin-split plateau at G = e2/h evolves into the 0.7
feature
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Four-terminal resistance measurements
How one can avoid measuring contact resistance?
Use four-probe measurements!
Four-terminal resistance
measurements on a ballistic
quantum wire.
A quantum well is grown in
[100] direction, and three
tungsten gate stripes are
evaporated on top. The wafer
is then cleaved, and a
modulation-doped layer of
Al0.3Ga0.7As is grown along the
[110] direction. The wire
extends along the cleave.
Cleaved Edge Overgrowth
4-terminal experiments
beautifully confirm the
theory of 1D ballistic
transport
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Four-terminal resistance measurements: result
The two-terminal resistance of
the wire (A) along gate 2 and its
four-terminal resistance (B) are
compared.
A
B
In four-terminal measurements
resistance vanishes!
The inset shows the ratio between the four-terminal resistance and the two-terminal resistance as
a function of the probability T for electrons to be transmitted between the wire and the probe.
The lower trace in the main figure has been performed at Rwp = 250 kΩ, i.e. T ≈ 0.05 in the inset.
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Current distribution in QPC
An instrument to measure current distribution in a QPC is
Scanning Gate Microscopy (SGM)
The tip introduces a movable depletion region which scatters
electron waves flowing from the quantum point contact (QPC).
An image of electron flow is obtained by measuring the effect
the tip has on QPC conductance as a function of tip position.
G in total current
Spatial distribution
Explanation: the branching of current flux is due
to focusing of the electron paths by ripples in the
background potential.
Topinka et al., Nature 2001
More sophisticated SGM technique
allows analyzing e-e interaction.
The experimental current distribution
can be explained by account of nonequilibrium effects in QPC.
Jura et al., preprint 2010
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The Landauer–Büttiker formalism
Adiabatic transport:
Landauer formula:
Transition probabilities
Transition amplitudes
S
Wire
D
To get accurate quantization of conductance T needs to be very close to 1 for
occupied modes and 0 for empty modes
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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
Edge states – 1D states in 2DEG
What happens near edges of
2DEG in magnetic field?
Skipping orbits, or edge states
Quantum mechanics:
squeezing of wave
function near edges
leads to increase in
the Landau level
energy
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Current flow in 2DEG
• Flow is strictly one-dimensional
• All electrons near one edge move in the same
direction while at the opposite edge move in the
opposite direction
Current flow diagram of a
Hall bar for two occupied
Landau levels:
To be scattered back an
electron has to traverse the
whole sample, T=1.
Hall bar edge behaves as an ideal ballistic quantum wire!
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Landauer–Büttiker formalism
•The transmission probability of contact p into contact q is
Tq←p = Tqp. It is possible to have Tqp > 1, since more than one
mode may connect the two contacts. Tqp does not have to be an
integer, Tpp is a backscattering probability.
•The total current emitted by contact p is Ip, while μp is the
electrochemical potential of contact p.
•An “ideal” contact absorbs all incoming electrons and
distributes the emitted electrons equally among all outgoing
modes, such that they are filled up to μp, assuming zero
temperature.
The Landauer formula generalizes to the Büttiker formula is:
Quantum wires and point contacts
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Landauer–Büttiker matrix
For the problem above the Landauer - Büttiker formula gives:
Here G = j(2e2/h). By
choosing μd = 0 as a
reference
potential,
and after eliminating
the drain current:
The solution gives:
Vs = V3 = V4
V 1 = V2 = 0
Is = GVs
The accuracy of the quantization is so much more accurate than in
a QPC because backscattering is greatly suppressed.
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Current flow in 2DEG
If the edges are in an equilibrium, or there is no backscattering from
contacts, then the potential difference between the edges is just the
same as between the longitudinal contacts.
Then each channel contributes to the “Hall” current as an ideal
ballistic quantum wire.
This picture can be checked by
inserting
a
gate
causing
backscattering in some channels.
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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.
Magnetotransport in 2DEG
Gate controlled current flow in 2DEG
The resistances depends on filling factor N in the ungated region and M in the gated region:
Left: at suitable gate voltages, the inner one of the two edge states gets reflected. Right:
For a 2DEG in the regime of filling vector 2, with spin-split edge states, a plateau at Rxx =
h/2e2 is observed as a function of the gate voltage. The dip around a gate voltage of
−0.2V can be explained within a trajectory network formed below the gate.
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Conductance quantization in 2DEG
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Edge states in quantum Hall effect regime
Structure of spinless edge states in the QHE regime.
(a)-(c) One-electron picture of edge states. (a) Top view on the 2DEG plane near the edge. (b) Adiabatic
bending of Landau levels along the increasing potential energy near the edge. (c) Electron density as a
function of the distance to the boundary.
(d)-(t) Self-consistent electrostatic picture. (d) Top view of the 2DEG near the edge. (e) Bending of the
electrostatic potential energy and the Landau levels. (f) Electron density as a function of distance to the
middle of the depletion region.
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Other one-dimensional systems
A mechanically controlled
break junction for observing
conductance quantization in
an Al QPC.
The elastic substrate
is bent by pushing rod with a
piezoelectric element.
The thin Al bridge,
fabricated by electron beam
lithography, can be broken
and reconnected for many
cycles.
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Mechanically controlled break junction
Top: Conductance as a
function of the deformation
time of a gold junction (an
STM setup was used). The
observed steps are usually
not quantized in units of
2e2/h. Bottom: A histogram
of
many
consecutive
sweeps of the upper type,
however, reveals that steps
of the expected height
dominate.
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Conductance in QPC – saddle-like shape
Conductance of a QPC with
a saddle point potential,
characterized by ωx and
ωy, calculated for zero
temperature. The steps
vanish as soon as ωy
becomes smaller than ωx.
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Carbon nanotubes
Discovered by S. Iijima, 1991
FET
Electrostatic potential
Modelling
Melting
Space
Tvisting
Quantum wires and point contacts
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Carbon nanotubes applications
Fiber
Spinning
Nanotubes
Electronics
Pick and place
Nanotechnology
Types
Properties, applications
Making carbon nanotubes
Graphite sheet can be rolled in a tube, either single-wall,
or multi-wall.
The graphene lattice and lattice vectors a1 and a2.
A wrapping vector n1a1+n2a2 = 4a1+2a2 is shown.
The shaded area of graphene will be rolled into a tube so that the
wrapping vector encircles the waist of the NT. The chiral angle
measured between a1 and the wrapping vector.
Carbon nanotubes
φ is
52
Carbon nanotubes classification
The geometry of an unstrained NT is described by a
wrapping vector. The wrapping vector encircles the waist
of a NT so that the tip of the vector meets its own tail.
A NT with wrapping vector 5a1+5a2,
φ = 30°.
Chrial angle φ can vary between 0˚ and 30˚ (any wrapping
vector outside this range can be mapped onto 0˚ <
symmetry transformation).
Carbon nanotubes
φ < 30˚ by a
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Carbon nanotubes properties
The most eye-catching features of these structures are their
electronic, mechanical, optical and chemical characteristics,
which open a way to future applications.
For commercial application, large quantities of purified nanotubes
are needed.
Electrical conductivity. Depending on their chiral vector, carbon nanotubes
with a small diameter are either semi-conducting or metallic.
The differences in conducting properties are caused by the molecular
structure that results in a different band structure and thus a different band
gap
Mechanical strength. Carbon nanotubes have a very large Young modulus in
their axial direction.
The nanotube as a whole is very flexible because of the large length.
Therefore, these compounds are potentially suitable for applications in
composite materials that need anisotropic properties.
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Quantization around a graphene cylinder
In a cylinder such as a NT, the electron wave
number perpendicular to the cylinder’s axial
direction,
, is quantized.
For zero chiral angle
Electron states near EF are defined by the intersection of allowed k with
the dispersion cones at the K points.
Carbon nanotubes
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Momentum quantization in carbon nanotubes
The quantized
are determined by the boundary condition
where j is an integer and D is the NT diameter
The exact alignment between allowed k values and the K points of
graphene is critical in determining the electrical properties of a NT
Consider NTs with wrapping indices of the form (n1,0).
K points are
always at 1/3
(or 2/3) position
When n1 is a multiple of 3 (n1 = 3q where q is an integer) there is an
allowed
that coincides with K. Setting j = 2q one gets
= 2j/n1 = 4q/3q = 4/3
Carbon nanotubes
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Metallic and semiconducting carbon nanotubes
n1 = 3q+p
Depending on the direction
of the CN axis and the
circumference of the CN,
the kx of one mode may or
may not hit a K-point of the
graphite sheet’s Brillouin
zone, and hence a metallic or
a semiconducting CN results.
It can be shown that CNs are
metallic if (2n1 + n2) is an
integer multiple of 3.
Carbon nanotubes
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Carbon nanotubes: density of states and energy bands
Energy bands E(k) for the metallic nanotubes
(9,6) and (7,4). The Fermi level is atE=0. In both
cases two electron bands cross the Fermi level
at k≥0, each corresponding to a conducting
channel in the nanotube. [Saito'98]
The calculated density of states of (a)
semiconducting and (b) metallic carbon
nanotubes. The Fermi energy is at E = 0. The
bandgap equals 0.7 eV in the semiconducting CN.
Carbon nanotubes: conductance quantization
In CNT, conductance is quantized in units G0
Transport in nanotubes is ballistic. The nanotubes
were not damaged even in the relatively high applied
voltage such as 6 V that should correspond to a
current density of more than 107 𝐴/𝑐𝑚2. Typically, such
current would heat a 1 μm long nanotube with 20 nm
in diameter to a temperature 𝑇𝑚𝑎𝑥= 20000 𝐾.
Quantum wires and point contacts
[Frank'98]
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Potential applications of CNTs
The way in which nanotubes are formed is not exactly known. The growth
mechanism is still a subject of controversy, and more than one mechanism
might be operative during the formation of CNTs.
• Hydrogen storage
• Composite materials
• Li intercalation (for Li-ion batteries)
• Electrochemical capacitors
• Molecular electronics
-field emitting devices: flat panel displays, gas discharge tubes in
telecom networks, electron guns for electron microscopes, AFM tips and
microwave amplifiers;
- transistors
- nano-probes and sensors (AFM tips, etc)
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Carbon nanotube transistor
Gate capacitance per unit
width of CNTFET is about
double
that
of
pMOSFET.
The
compatibility with high- k
gate dielectrics is definite
advantage for CNTFETs.
About twice higher carrier
velocity of CNTFETs than
MOSFETs comes from
the increased mobility
and the band structure.
CNTFETs, in addition,
have about four times
higher transconductance.
Carbon nanotubes
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Circuitry of ballistic wires and QPCs
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Series connection
Two QPCs in
series is a nonOhmic system:
the series
resistance is much
smaller than the
sum of the
individual QPC
resistances
One needs quantum mechanics to understand such behavior
– after the first QPC the electron beam gets collimated,
and then it is prepared to pass through the second QPC
without backscattering.
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Parallel connection
If two QPCs are connected in parallel, special effect in
magnetoresitance occur.
Shown are the effects of commensurability between
the cyclotron orbits and the QPC spacing
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Open questions
We avoided most interesting questions posed by
electron-electron interaction in combination with
quantum mechanical correlations.
In particular, we did not discuss:
• Fractional quantum Hall effect
• Kondo effect in QPC and quantum dots
• Luttinger liquid effects, important for 1D systems
These effects belong to cutting edge of modern
nanoscience and nanotechnology.
Quantum wires and point contacts
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Home activity for Tue. 10 and Wed. 11 March
a) Read: TH, Chapter 6 ‘Quantum Wires and Quantum Point Contacts’ (pp 196241). Try to understand behaviour of Quantum Wires and Quantum Point
Contacts. Identify issues that you would like to discuss on Practical, Wed. 11
March.
b) Work with questions to Chapter 6:
and other questions.
c) Address exercises in TH on pp. (219-241). Can you discuss them on practical?
d) Choose date for full presentation. Available dates: March 4, 11, 25; April 22, 29;
May 6, 13, 20. Please notify of your choice. Please notify of the date of
examination suitable for you. Otherwise you will be allocated to a date.
Examination will take place on May 26 and 27 (two groups). Students should do
compulsory presentation on one of the dates above to be allowed to take
examination. Please send pdf or ppt files of presentations to
[email protected] to put them on the web.
Introduction
66
MENA5010 presentations
List of full presentations :
Hans Jakob Sivertsen Mollatt - presented
Jon Arthur Borgersen
- presented
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
Introduction
67