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NANO-CURRENTS WITHOUT A VOLTAGE VIA QUANTUM PUMPING:
A WAVE PACKET APPROACH
Matthew Galligan, Kunal K. Das
Department of Physics, Fordham University, Bronx NY 10458
Background
Method
Current in Nano Structures
Goals
In large scale electrical circuits, current is measured as an average of large number of collisions of electrons
in the wires. This measure only works if there are enough electrons to define an average that is unchanging.
In extremely small wires, the electron no longer “bounce” around in the wire but rather moves ballistically.
At this point the classical approach to current no longer is valid and quantum affects become noticeable.
The drive to make computers and other electronic devices smaller and smaller is leading to nanoscale (~10 9meters) circuit elements. It has become become necessary to understand, and better control, the way
current flows in these regimes. Quantum pumps offer a potentially better alternative to generating currents
in nano structures without applying a voltage bias.
Quantum Pumping
Quantum pumping creates a current without a voltage bias. By creating a local potential that varies in time
cyclically (like a pump) it is possible to induce a current in the nano structure without needing to create a
voltage bias. This is useful because it is posible to have more control over the flow of the electrons. Also
quantum pumping can use much less energy than creating a bias. Nano-technology will eventually use
some form of quantum pumps as computers and electronic devices shrink in size.
Split-Step Operator
The split-step operator method is a tool to propagate a quantum mechanical wave-packet by stepping it
through a sequence of extremely small time steps, in this case .002 seconds. This is done many times to
simulate the path of the wave-packet over a longer time period. Time evolution of a wave function over a
time period is through the formula:
H Hamiltonian
ˆ
i

t
T Kinetic term
ˆ 
ˆ  V̂
 t  e   0 where 
V Potential term

 
•Study the transport of single electrons through a quantum pump operatiing at
arbitrary rates or velocities.
•Probe the equivalence of localized states and extended states representation in
the presence of time varying potentials
Reasons for Interest
Current analytical formulas for the propagation of an electron in the Quantum
regime through a potential barrier assumes many electrons and the position of
any one individual electron is unimportant, but rather their behavior as a group is
studied. As a result, description of current is always in terms of extended states
(plane wave representation) of the electrons, but when the current can comprise
of one electron at a time through finite-size structures, a localized picture of
individual electrons seems more appropriate, particularly in the presence of time
varying potentials . In this study we probe the limits of validity of those
analytical formulas by modeling electrons with localized packets of varying
widths instead of extended plane wave state, and look at the differences.
Model
• Construct four Gaussian wave packets that each represents a single electron,
one with x0 and k0, x0 and -k0, -x0 and k0, and –x0 and –k0.
•Model a simple time-varying potential by a moving quantum potential barrier
initially at the origin with v0=0.
•Use the split-step operator method to model the electron propagation and
interaction with the barrier as it moves
•Determine the current using:
2
J   dk k   k 
•Repeat the process with the potential barrier back at the origin and with a
different velocity of the barrier till a maximum desired value
The split step operator breaks the formula into two parts, doing the kinetic energy part solely in momentum
space where the operator is a simple multiplication instead of involving derivatives and the potential energy
part in position space for the same reason. A Fourier Transform is capable of moving between momentum
space and position space, and in this case a Fast Fourier Transform (FFT) was used.
v0
k0
1.
2.
Use a Fast Fourier transform (FFT) to convert the position space representation of the wavepacket into a momentum-space representation
Time-evolve the wave function by the kinetic part of the Hamiltonian for half-time-step
e
 iT t
2
e
- k0
k0
 ik t
4 m
Use the inverse FFT to recover the position space wave packet.
Time-evolve the wave function by the potential energy part of the Hamiltonian for the fulltime-step:
 iV t
5.
6.
Use the forward FFT to get the momentum-space representation.
Time-evolve the wave function by the kinetic part of the Hamiltonian for the remaining halftime-step
Reverse FFT to recover the position-space representation of the wave-packet.
This entire process only propagates the wave packet through one time step and as a result, is
repeated many times.
7.
8.
-k0
2
3.
4.
e
One technique used to expedite the
process was to only propagate the wave
packets that would actually interact with
the potential barrier, which starts at x0=0
and +v0. For example, the wave packet
with –x0 and –k0 would never interact
with the barrier while the packet with
+x0 and –k0 would always interact with
the potential. The wave packet with –x0
and +k would only interact if k>v. The
wave packet with +x and +k would
interact only when k<v. The case where
k=v was done separately.
- x0

x0
Limits
The numerical method used showed very good agreement with the analytical
values for plane waves when the wave packet was very wide. As a result
different widths of wave packet were used to test the limit of this method. For all
of them we choose a Gaussian profile with initial amplitude
 2  


 

1/ 4
Square barriers can be used to model delta functions as when the square barrier is much thinner than the wave
packet and the area under the square barrier is the same as that under the delta function
Conclusions
Confirmed that it is possible to simulate an extended plane wave with a sufficiently broad wave packet.
Now have the ability to model the entire regime including non-adiabatic regime.
Can apply these models to any arbitrarily time varying potential.
Wave functions of different widths were used and then the results
were analyzed next to each other. As is evident from the graphs
below, the wider the wave function, the more accurate the results.
The wave functions below are representative of this fact and by
varying the parameters it is possible to maintain an accurate picture
of very localized wave functions.
 =.0002, .002
0.00012
0.00008
0.00004
150
100
50
50
100
150
1
velocity of the potential barrier
4
3.
6
3.
2
2.
8
2
2.
4
1.
6
0
1.
2
The solid line is the analytical solution using plane waves the
circles are the model with only two wave packets and the crosses
are the model with all four wave packets using a square barrier.
The 4-wavepacket model agrees with the plane wave model
exactly.
0.
8
The straight line is the adiabatic approximation and
the curved line is the actual results for a delta
function
2
0.
4
current
3
0
1.
2.
3.
2
 e   x  x0  