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Electron phase coherence
Interference is the addition (superposition)
of two or more waves that result in a new
wave pattern.
Simple example: Two slit experiment
φ
d
L
Electron phase coherence
Q1: Why do we use a two-slit set?
Q2: What to we need to observe interference from
two different sources?
A: We can observe interference of two sources, if
they are coherent.
Two signal are coherent if the phase difference,
stable.
Electron phase coherence
, is
According to quantum mechanics, electrons can
behave as waves.
What is the role of phase coherence?
To answer this question let us discuss the effect
which would not exist in the absence of
interference – the Aharonov-Bohm effect.
An important difference between electrons and
electromagnetic waves is that electrons have a
finite charge.
Electron phase coherence
Aharonov-Bohm effect
In classical mechanics the motion of a charged particle is not
affected by the presence of magnetic fields in regions from
which the particle is excluded.
For a quantum charged particle there can be an observable
phase shift in the interference pattern recorded at the detector.
The Aharonov-Bohm effect
demonstrates that the electromagnetic
potentials, rather than the electric and
magnetic fields, are the fundamental
quantities in quantum mechanics.
Electron phase coherence
Let us make a confined tube of
magnetic field
1
2
Will the interference pattern
feel this magnetic field?
For a plane wave, the wave function
The phase gain along some way is then
As we know, in a magnetic field
Additional phase difference
Electron phase coherence
Aharonov-Bohm Effect for Nanowires
Φ
Magnetic flux quantum
Electron phase coherence
Aharonov-Bohm oscillations,
Webb 1985, Au
Fourier analysis shows
that there are also weak
oscillations with half
period
Electron phase coherence
High order interferences
Here the clock-wise and counterclockwise paths are exactly the same:
The waves “strengthen” each other,
constructive interference
The backscattering probability is
enhanced.
Sharvin, 1981
Finite magnetic field destroys the
interference, the period being a
half of the period in the ring.
Electron phase coherence
Altshuler, Aronov, Spivak
(AAS) oscillations
The period is 2Φ0
Experiment by Sharvin,
1981, Mg-coated human
hair (different samples,
1.12 K)
AB oscillations vanish in an ensemble of small ring since
the phases φ0 are random.
In contrast, AAS oscillations survive ensemble averaging.
Electron phase coherence
Test of the ensemble averaging, Umbach 1986
Ag loops, 940x940 nm2, width of the wires 80 nm
Fourier
series
N-dependence
of the AB
oscillations
amplitude
Electron phase coherence
Weak localization
Let us calculate the probability for an electron to
move from point 1 to point 2 during time t. in terms of
the transition amplitude, Ai, along different paths.
Classical probability
Interference
contribution
Vanishes for most paths since phases are almost random
Electron phase coherence
Consider now a close loop with two
identical paths traversed in opposite
directions.
Then the amplitude Aj is just a time
reversal of Ai. Hence
The backscattering probability is enhanced by factor 2!
This is a predecessor of localization.
This effect is called the weak localization since the relative
number of closed loops is small.
However, the effect is very important since it is sensitive
to very weak magnetic fields.
Electron phase coherence
Let us calculate the probability for an
electron to return to the “interference
volume” during the time dt.
One obtains:
Interference volume
Volume of diffusive
trajectory
Thus, the relative quantum correction is
decoherence time
Electron phase coherence
D – diffusion
constant, b –
thickness, d dimensionality
In 2D case, introducing the sheet conductance
we get
This contribution is suppressed by very weak magnetic
fields,
where bending of the trajectories by magnetic field is
still not important. This value corresponds to one
magnetic flux quantum trapped in a typical interfering
trajectory.
Electron phase coherence
Experiment:
Si/SiGe quantum
well
Weak localization is a very important phenomenon – it
allows find the decoherence time, spin-orbit interaction,
etc.
Electron phase coherence
Universal conductance fluctuations
Let us look at the resistance of the
wire we have already seen, in a weak
magnetic field, ωc < 0.45ω0
WL
CF
The patterns are not
random, the are
fingerprints of the
system
Electron phase coherence
How they can be explained?
For a rectangular sample LxW the Drude formula can be
written as
Conductance
per mode
Electron phase coherence
Number
of modes
How they can be explained?
For a rectangular sample LxW the Drude formula can
be written as
Conductance per mode
Number of modes
From the Landauer formula,
Now we are interested in
Electron phase coherence
For the transition amplitude,
Since reflections are dominated by few events, they can
be considered as uncorrelated. Then
Now we have to calculate the variance of the
reflections
Electron phase coherence
Result:
Averaging
and substituting
we get:
Universal conductance
fluctuations (UCF)
Thus
Electron phase coherence
Typical fluctuation is very large, and relative
fluctuation does not decay
as it would
follow from statistical physics.
Quantum low-dimensional systems do not possess the
property of self-averaging.
At relatively large temperatures this properties is
restored due to decoherence.
At
,
Electron phase coherence
What happens in magnetic field?
Usually it is assumed that in a magnetic field the sample properties
change significantly.
Thus one can think that at each magnetic field a sample represents a
realization of an ensemble (so-called ergodic hypothesis) .
Consequently, one can average over magnetic fields rather then over
different samples.
As a result, we have fluctuations as a function of
magnetic field, the theory being rather complicated.
Electron phase coherence
Phase coherence in ballistic systems
Electrostatic Aharonov-Bohm Effect
Split gates
Interference pattern
Electron phase coherence
Tunneling transport
Classical motion
Quantum tunneling
E
V0
1
2
a
Electron phase coherence
3
4 unknown coefficients, and 4 boundary conditions at
: continuity of wave functions and their
derivatives (currents)
we find transition and
reflection amplitudes, t and r, respectively
We show expression for the transition probability,
Tunneling
Quantum effects:
Over-barrier reflection
Electron phase coherence
Resonant tunneling and S-matrices
Electron phase coherence
Properties:
For a symmetric system
they imply
What will happen with
transmission probability if
we place two barriers in
series?
Electron phase coherence
+
Geometric series
Transmission probability:
Electron phase coherence
+…
At some values of
For small transmission,
Since the phase is given by the relation
we have
Electron phase coherence
We observe that at E=Es and
T1=T2 the total
transparency is one though partial transparency
are very small.
This phenomenon is called the resonant tunneling
– it is fully due to quantum interference.
Often people introduce the attempt frequency as
and partial escape rates,
s-resonance
Electron phase coherence
. Then near the
Thus, the double-barrier structure behaves as an optical
interferometer.
Will resonant tunneling survive if the barriers are far
from each other?
Unfortunately, NO, since then the coherence would be
lost.
Decoherence can be modeled by introducing a reservoir
between the barriers which can absorb and re-emit the
electrons whose phase memory gets lost.
Electron phase coherence
Current-voltage curve:
Negative differential conductance!
Electron phase coherence
Transmission through a ballistic quantum ring
Calculation can be done using Smatrix approach
Novel feature – triple junctions,
described by the matrix
Electron phase coherence
Open ring, AB
oscillations
Closed ring
Electron phase coherence
Summary
In this lecture, we have discussed
• The Aharonov-Bohm effect in mesoscopic
conductors
• Weak localization
• Universal conductance fluctuations
• Phase coherence in ballistic 2DEGs
• Resonant tunneling
Electron phase coherence