Quantum Pumping
~
Volker J. Sorger
~
University of Texas
22
nd
of March 2005
Course: ‘Physics in Nanostuctures’
Prof. Dr. Q. Niu
Introduction
We usually associate electric current with a dissipative motion of electrons [1]: The
energy provided by an external source eventually becomes heat. However, examples of a
current flowing without dissipation are also well known. For instance, a static magnetic
field magnetizes a metal specimen by causing edge electric currents (Landau
diamagnetism) [1] Moreover, a static magnetic flux threading a metallic ring induces a
persistent current [2]. The electrons remain in equilibrium, and energy does not dissipate.
Pumping is also a way of transferring electric charges, but it is qualitatively different
from both mentioned mechanisms.
Before we go into details, let us review a classical example of pumping (Fig. 1).
Figure 1 Adiabatic Thouless pumping of electrons (A) and water pumping by Archimedean screw
(B). Shift of the potential by one period in A is similar to one turn of the screw handle in B (black
arrows).
The water pump transports particles from a lover to a higher level (applied V
sd
in the
electron pump case). By turning the screw one time the every potential minimum gets
shifted by the period a. There would be no charge transfer if each minimum returned to
its original position: A standing wave does not generate a DC current. However a
superposition of standing waves would be able to do it.
In pumping, periodic AC perturbations of the system yield a DC current. This current,
though not an equilibrium response to an external perturbation, may be entirely adiabatic:
The system always remains in the ground state. In contrast to the more familiar
dissipative rectification of AC current, the charge transferred in each cycle of adiabatic
pumping is independent of the period T. Therefore, at large T, pumping dominates.
Chapter 1 ‘’ Theoretical Basics””
The idea of a quantized particle transport was first introduced by Thouless [1]. The
fundamental question that was posed was: “If the potential is changed slowly in such a
way that it returns to its starting value in time T, is the integrated current of electrons
across the boundary quantized”? For an infinite periodic system with electrons in filled
bands the adiabatic particle transport is quantized.
For simplicity, consider spinless electrons in a one-dimensional channel, subject to a
potential U(x) periodic along the channel: U(x + a) = U(x). Provided the number of
electrons na per period a equals an integer N, the lowest N bands of the energy spectrum
are full while the higher bands are empty. Now let the potential move with some small
velocity, U(x,t) = U(x ? vt). At each point x the potential U(x,t) varies periodically with
time, because U(x) is translationally symmetric. If electrons follow adiabatically the
variation of the potential, then a particle transfer is given by
( )
∫∫
∑ ?
?
?
?
=
T
kt
i
L
kk
dkdtfC
00
2
2π
λλ
ψψ
λ
λπ
, (1)
where T is the time period, ψ
λ
the normalized bloch-wave function in the λ
th
band of the
instantaneous Hamiltonian and k the wave number. Hence a current I = nev is induced,
where v=a/t is the operating frequency of the potential. Over the period T=a/v, a
quantized charge Q =IT = Ne is transferred through any cross section of the channel.
Again the assumption for this result were a one-dimensional noninteracting electron
system at zero temperature in an adiabatic time variation of the potential and an integer
number of filled (perfect) Bloch bands of the instantaneous Hamiltonian [2]
),()(
2
22
2
txUtH
x
m
+?=
?
?h
. (2)
Also note, that the Fermi level has to lie in an energy gap between the filled states and the
empty states, so that the spectrum of the instantaneous Hamiltonian takes such a form
that a energy gap between the first N states and the rest appears. However, it has been
shown, that the quantization of the pumped charge is quite robust and survives, for
example, in the presence of an additional smooth static potential, disorder and many-body
interactions [2].
In Ref [3] Niu studied three different cases of potential systems in one-dimesion: (1) A
linear superposition of two potential, whereas one potential is fixed in space and the other
is translating. (2) An amplitude modulated atomic potential array, with modulating wave
propagation in space and (3) a propagating sound wave. The results for the first case
show that electrons in full bands are locked in those parts of the potential whose Fourier
components correspond to points where the Fermi gap is large. This analytic result turned
out to be exact for sinusoidal potentials (see also Ref [6]). For the second case of
periodically modulated potential the transported electrons are quantized and shows
similarities to the quantum Hall effect in a two-dimensional periodic potential, where the
integer of the Hall conductance is found to be described by a similar equation [7][8][9].
For the last case of a traveling sound wave a actual experiment has been made [10, see
also chapter ‘experimental results’]. The idea for such an approach is to send a coherent
surface acoustic sound wave (SAW) in a one-dimensional system and then to measure the
induced current. Some conditions for this are that the wavelength has to be as short as the
Fermi wavelength of the electrons and the amplitude should be large enough to generate a
Fermi gap. Also, the size of the induced gap should be large compared with the frequency
in order for the adiabatic approximation to be valid.
Recently the theory of adiabatic quantum pumping (AQP) has been extended to pumping
in interacting wires, luttinger-liquid regimes and finite temperatures [5]. Assuming
interactions of the wire with its surroundings, the current behaves as a power law of the
temperature with an exponent depending on the interactions. Fig (2) shows the low-
frequency behavior for different temperatures T.
Chapter 2 “Towards a quantum pump”
An easy way of understanding the quantum adiabatic particle transport (QAPT) is to
assume a rigidly propagating potential U(x-vt), with τ=a/v, where a and τ are the spartial
and temporal periods respectively. The Galilean principle and the adiabatic theorem now
ensure that the electron number densities shifts all together with the potential [4] and the
integral of the number density over the spartial period is an integer n for n filled bands,
which is also the number of transported electrons per period τ.
If we are consider that only the lowest band is filled, the corresponding many-body state
is determinant by Bloch-waves, but can also be written as a determinant of single-particle
Wannier states [11]. Therefore the particle density can be written as
() ( )
∑
∞<<∞?
?=
j
tjaxWtx
2
,,ρ (3)
where W(x,t) is the Wannier function for the filled band of the instantaneous
Hamiltonian. Let’s define W(x,t) as an eigenstate of the position operator projected onto
the subspace of the Bloch band,
() () ( ) () ( )txWtltxWtPxtP ,, =
)
)
)
(4)
where ()tP
)
is the projection operator. Thus, the eigenposition l(t) is the center of mass of
the density profile ()
2
,taW [12]. A intrinsic ability of the Wannier-functions is that they
are localized in space. Focusing on this feature and taking into account that the other
eigenstates of () ()tPxtP
)
)
)
are of the form ( )tjaxW ,? with their eigenpositions at
, it can be shown that in the case of an remaining open Fermi-gap l(t)
vary continuously with U(x,t). Hence all we have to do is follow the positions of the
Wannier-functions. When the potential returns to its initial form the sets {lj(t)} of
eigenpositions must return to their former form as well, so each Wannier state must have
shifted its center of mass by
() () jatltl
j
+=
()( ) natltl =?+τ , yielding in a particle transport is equal to
n and therefore quantized.
As described in [3] the current of mutually non connected potentials can produce a global
induced particle transport. A given total-potential of the form
() ()( 3/,, jaxutAtxU
j
j∑
∞<<∞?
= ) (5)
where u(x) are the local potentials, can mime a phase-coherent amplitude oscillation with
a traveling envelope with a being the spatial period. Additionally, U(x,t) is chosen such to
contain 3 local potentials. Fig (3) shows the time development of these potential One can
easily see, that after a period of τ/3 the potential has propagated to the right by a/3,
yielding in each Wannier-function in a given Bloch band to have shifted a distance of
ma+a/3, where m is an integer, leading to a non-zero particle transport of 3m+1.
Figure 2, The first time evolution of the potential U(x,t) in Eq. 5 during the first third temporal
period
How can such a local potential be created? - Due to modern Nano-Device-Fabrication-
Techniques it is relatively easy and straight forward to design a structure like presented in
Fig(4) of the spatial dimensions in the order L<1μm with normal lithography methods
like Electron-Beam-Lithography (EBL) or Focused-Ion-Beam (FIB) on common Si/SiO
2
doped wavers with thin-film-evaporation techniques and Photo-lift-off processes.
The green parts (V
1
and V
2
) are the local potential producing local gates, whereas V
3
is
an extra Top-gate and the quantum wire (blue) . The basic idea of the device is to create a
traveling potential along the wire, generated via capacitance coupling to the gates, on
which phase shifted voltages are applied. The Top-gate V
3
can modulate the conductance
of each segment of the wire and can tune the conductance. If all Top-gates are operated at
the same voltage at the same time then they act like a global back-gate we typical
transconductance behavior is shown [13].
Appropriate programming of the voltages on the gates should then produce a traveling
potential wave. If the potential is placed just above the lowest energy level right into the
Fermi-gap and each level is occupied by 2 electrons (Pauli Principle) then with each
cycle we can pump 2 electrons. However, if we go into the Coulomb-Blockade(CB)-
regime (sufficient small sizes and low temperatures) [14] then we can place the potential
offset right into this coulomb-gap yielding in a single-electron-pump.
Figure 3, Schematic structure of the proposed charge pump. In blue the underneath nanowire
Chapter 3 “Accuracy & Application”
The basic formula concerning the accuracy of the pumped current is I=nev. The
frequency can be measured very accurate using the AC Josephson Effect. On the other
hand effects that contribute to the integer n are: thermal activation (~ exp{-?/k
B
T} and
non-adiabatic-excitations (~exp{-?/hv}) should be relatively easy controlled, where k
B
is
the Boltzmann constant and T the temperature. Effects from disorder and many-body
interactions have also been seen not to contribute as long as the Fermi gap remains open.
Furthermore the correction to the quantization due to the finiteness of the system (the
Fermi-gap closes at the end of the system) is of the order of exp-{L/l} where L is the
system length and l the electron-electron-correlation length.
The accuracy also depend on the leaking current through the device. In Ref. [37] this
probability is given to be
?
?
?
?
?
?
?∝Γ
max
0
2
2
exp mV
h
b
h
E π
where b is the external potential thickness, V
max
the height, E
0
the lowest energy level.
For a GaAs-AlGaAs quantum wire (width = 100nm) heterostructure formed in a 2DEG
with typical values of
nmb
meVEEE
EV
meVE
mm
e
200
687.1
10
562.0
067.0
01
0max
0
=
=?=?
=
=
=
The leaking probability for one cycle is less than 10
-11
for a pumping frequency >
200MHz. If instead of the assumed quantum wire e.g. a single walled nanotube (SWNT)
with typical diameters of 1.5nm were used the this number can be further decreased. In
[37] it is also shown that the deviation of the necessary equilibrium condition for
adiabatic pumping due to a finite rate of potential change decreases exponentially with
the frequency:
?
?
?
?
?
?
?
?
?
?∝Γ
max
2
3
mVb
hv
E
epx
mequilibriu
π
h
and one finds for e.g. ν=230MHz a correction of 10
-8
.
Drawing the attention to the different used energies we can summarize the conditions for
QAPT-pumping:
Energy Chargingspacing levelBox in Particle Blockade) (Qulomb Noise thermallimit Adiabatic
in wordsor
arg
<<<<
echthermo
EEEhν
:parameters learchieveab realistic lfor typica (NT)) Nanotubs and (QD)Dot (Quantum
devices size-nanofor valuessome shows 1 Table [15]. plots staircase'''' V-I
normalby measuredly convenient becan energy spacing-level thesmaller,actually isEth if however,
regime, Blockade Qulomb in the operate toEan greater th becan Energy Themal that theNote
sd
?
QD [meV] NT [meV] Current [nA]
?E
1,7 @ d=100nm
~1 for 2μm NT
3pA @ 20MHz
E
th
=k
B
T 1.0 @ 10K 0.4 @ 4K
0.16 nΑ @ 1GHz
E=hν
0.00008 @ 20MHz 0.004 @ 1GHz
3 nΑ @ 20GHz
Table 1. Overview of level-splitting-energy Thermo-Energy and Pumping induced Energy for
Quantum-Dots with typical scaling and Nanotubes (NT) in meV. The pumped current for different
Frequencies assuming weak pumping regime is shown as well
Applications
The device is a high-precision DC-current source. Also it could be used as a precise
pump for electric charges. The DC-output (pumped charges) can be well controlled by
the number of operating cycles and can be therefore used to measure capacitances with
high precision [4]. By pumping a certain amount of electrons and measuring the overall
pumped charge, it should be possible to measure the fundamental electron charge e.
Upon succeeding of realization of a current standard, the triangle voltage, resistance,
current would be completed (Fig 5) and Ohm’s law could be satisfied:
2
2
2
2
h
e
h
eIRU ×=== (6)
where voltage is measured by the Josephson-Effect and R by the Quantum-Hall-Effect.
Figure 4, Conceptual relationship among the three quantum effects. Each device is characterized by
a quantum number, e, e/h, or h/e^2. The output of a device is equal to its input times its quantum
number and an integer factor.
Chapter 4 “Continuative Theory ”
In recent years, electron pumps consisting of small semiconductor quantum dots have
received considerable experimental and theoretical attention [21,22]. A quantum dot is a
small metal or semiconductor island, confined by gates, and connected to the outside
world via point contacts. Several different mechanisms have been proposed to pump
charge through such systems, ranging from a low-frequency modulation of gate voltages
in combination with the Coulomb blockade[21,22] to photon-assisted transport at or near
a resonance frequency of the dot [23,24]. Their applicability depends on the characteristic
size of the system and the operation frequency. Those systems can be described by a
classical ‘peristalic’ electron pump [27], Fig [6].
Most experimental realizations of electron pumps in semiconductor quantum dots make
use of the principle of Coulomb blockade. If the dot is coupled to the outside world via
tunneling point contacts and the E
th
is smaller than E
charge
, the charge on the dot is
quantized and apart from degeneracy points, transport is inhibited as a result of the high
energy cost of adding an extra electron to the dot. Pothier et al. constructed an electron
pump that operates at arbitrarily low frequency and with a reversible pumping direction
[21,22]. The pump consists of two weakly coupled quantum dots in the Coulomb
blockade regime and
operates via a mechanism that closely resembles a peristaltic pump: Charge is pumped
through the double dot array from the left to the right and electron-by-electron as the
voltage ()tU ωsin
1
∝ of the left dot reaches its minima and maxima before the voltage
()φω ?∝ tU sin
2
of the right one [22]. The pumping direction can be reversed by
reversing the phase difference f of the two gate voltages.
A similar mechanism was proposed by Spivak, Zhou, and Beal Monod for an electron
pump consisting of single quantum dot only [25]. In this case a DC-current is generated
by adiabatic variation of two different gate voltages that determine the shape of the
nanostructure, or any other pair of parameters X
1
and X
2
, like magnetic field or Fermi
energy, that modify the quantum mechanical properties of the system, see Fig. 7. The
magnitude of the current is proportional to the frequency v with which X
1
and X
2
are
varied and for small variations to the product of the amplitudes δ X
1
and δ X
2
. The
direction of the current depends on microscopic quantum properties of the system, and
need not be known a priori from its macroscopic properties. As in the
case of the double-dot Coulomb blockade electron pump of Ref. [22], the direction of the
current in the single-dot parametric pump of Spivak et al.[25] can be reversed by
reversing the phases of the parameters X
1
and X
2
. An important difference between the
two mechanisms is that a parametric electron pump like the one in Ref. [25] does not
require that the quantum dot is in the regime of Coulomb blockade: it operates if the
dot is “open”, meaning well coupled to the leads by means of ballistic point contacts.
Experimentally, an electron pump in an open, chaotic (non-ballistic) quantum dot has
been realized [20].
The general physics of a quantum pump has been the subject of several theoretical
analyses [26,27]. Zhou et al. demonstrated that at low temperatures both the magnitude
and the sign of the pumped charges are sample specific quantities, and the typical value
in disordered chaotic systems [like in 20] turns out to be determined by quantum
interference effects, rather than classical trajectories. Another general expression for the
average transmitted charge current was derived by Brouwer [27] under the adiabatic
condition and based on the time dependent S-matrix method [28], which appears to be
quite successful for adiabatic weak pumping. In his work he considered a parametric
electron pump through an open system in a scattering approach. The main result is a
formula for the pumped current in terms of the scattering matrix S(X
1
, X
2
). Such a
formula is the analogue of the Landauer formula, which relates the conductance G =
dI/dV of a mesoscopic system with two contacts to a sum over the matrix elements S
αβ
.
The formular for the transported current is
∑∑
?
?
?
?
?
=
∈ β
αβαβ
α
π
φω
δ
21
1
21
Im
2
sin
X
S
X
S
XXe
I (7)
Like the Landauer formula, equation (7) equation is valid for a phase coherent system at
zero temperature and to linear response in the amplitudes δX
1
and δX
2
. It captures both a
classical contribution to the current and the quantum interference corrections.
Quantum corrections can be important in the mesoscopic regime, especially if there is no
‘classical’ mechanism that dominates the pumping process [20, 25].
Results of Eqn (7): First, for a phase coherent quantum system, the out-of-phase variation
of any pair of independent parameters will give rise to a DC-current to order ω. Second, I
is not quantized, unlike in the case of the electron pumps that operate in the regime of
Coulomb blockade [21, 22].
Figure 7. (a) A open quantum with two parameters X1 and X2 that describe a deformation of the
shape of the dot. As X
1
and X
2
are varied periodically, a DC-current I is generated. (b) In one period,
the parameters X
1
(t) and X
2
(t) follow a closed path in the parameter space. The pumped current
depends on the enclosed area A in the (X
1
, X
2
) parameter space.
The electron systems discussed in [1-4] are all ‘closed’ systems. The theory has been
expanded to open systems and non-exact eigenfunctions as one fine in real experiments
[6]. The electron energy levels are broadened due to inelastic processes at T > 0 and, in
the case of an open system, are further broadened due to finite dwell time. It’s been
demonstrated that at low T both the magnitude and the sign of Q are sample specific. The
typical value of Q in disordered (chaotic) systems turns out to be determined by quantum
interference effects and one finds that it is much larger than the one in ballistic systems.
This enhancement manifests of the well-known fact that at low temperatures, all
electronic characteristics of mesoscopic samples are extremely sensitive to changes in the
scattering potential [5–8].
Let us now consider the sample sketched in Fig. 8 with two gates (labeled by α={1, 2}),
biased with AC voltages of the same frequency and with a phase shift
21
δδδ ?=
() ( )
αα
δ+?= tVtV sin
0
(8)
Let us assume that the potential induced in the metal by the voltages V
α
is screened with a
screening length r
0
much less than L
x
and
() ( )
() () ()
?
?
?
?
?
?
?
?
???=
+?=
2
2
0,
4
sin
00
zZ
W
xrrV
ttg
WL
rCV
g
y
αα
αα
θθ
δ
(9)
where C is the capacitance of the gate, W >> r0 is the width of the gate along the z
direction, Z
1,2
are the z coordinates of the center of the gate 1,2 and θ(z) is the
step function: θ (z>0)=1 while θ (z<0)=0 [6]. With the volume of the sample v = L
x
L
y
L
z
and the total number of electrons inside the sample N, the average of the transported
charge can be written as
() N
rCV
efQ
F
z
2
2
00
0
sin
?
?
?
?
?
?
?
?
?=
νμ
δ (10)
where f
0
is a geometry-dependent factor ~ 1. Note also that the pumped current has a
sinusoidal phase-shift dependence (for weak pumping), given by I
max
for
221
π
δδ =? and
π2
min
?= nI , where n is an integer including 0. This sinusoidal dependence has been
measured by Switkes et al. [20].
Figure 8. Geometry of the considered sample. Shaded bars represent gates 1 and 2.
crosses represent random scatterers.
Chapter 5 “Experimental Results”
( comment of the editor: due to the long paper, the results are shortly presented, most of
them are already mentioned anyway )
Ref [21]
A Quantized current in an lateral Qunatum dot, defined by metal gates in a two-
dimensional electron Gas (2DEG) of GaAsAl/AlGaAs heterostructure was observed. By
modulating the tunnel barriers in the 2DEG with two phase-shifted rf-signals and
operating in the Coulomb-blockade regime, quantized current plateaus in the current-
voltage characteristics at integer multiples of ef where f is the rf-frequency,
demonstrating an integer number of electrons pass through the quantum dot in each rf
cycle were observed.
Ref [20]
An open quantum dot- based parametric electron pump has been fabricated. Two gates
with oscillating voltages control the deformation of the shape of the dot (see Fig. 9 lower,
middle). The pumped DC voltage V
dot
is measured to vary with the phase difference φ=π
between the two gate voltages and is antisymmetric about φ = π. At low pumping
amplitude, the experimental data gave V
dot
~sin(φ). The amplitude of the pumped signal is
found to increase nonlinearly with the driving force and it decays with temperature T as a
power law
9.0
1
T
∝ [29].
Figure 10. Pumped dc voltage V
dot
as a function of the phase difference between two shape-
distorting ac voltages and magnetic field B. Note the sinusoidal dependence on and the symmetry
about B = 0 (dashed white line). (B) Plot of V
dot
( ) for several different magnetic fields (solid
symbols) along with fits of the form V
dot
= A
0
sin + B
0
(dashed curves). (C) Schematic of the
measurement set-up and micrograph of device 1. The bias current is set to 0 for pumping
measurements. (A) Standard deviation of the pumping amplitude, (A
0
), as a function of ac pumping
frequency. The slope is ~40 nV/MHz for both device 2 (solid symbols) and 3 (open symbols). Circular
symbols represent a second set of data taken for device 3.
Figure 12. Standard deviation of the pumping amplitude, (A
0
), as a function of the ac driving
amplitude A
ac
, along with fits to (A
0
) A
ac
2
below 80 mV (dashed line), (A
0
) A
ac
(solid line), and
(A
0
) A
ac
1/2
(dotted line) above 80 mV. The lower inset shows that the sinusoidal dependence of
V
dot
( ) at small and intermediate values of A
ac
(solid curve, A
ac
= 100 mV) becomes nonsinusoidal for
strong pumping (dotted curve, A
ac
= 260 mV), but maintains V
dot
( ) = 0, as required by time-reversal
symmetry. The upper inset is a schematic of the loop swept out by the pumping parameters X
1
and
X
2
. The charged pumped per cycle can be written in terms of an integral over the surface enclosed
by the loop
Also Ref. [30] has supporting results for Switkes et al. work. The current transported by
weak and strong tunneling fit very well to the measured results in [20] Fig. 13.
Figure 13. left: the pumped currents versus the phase difference f for w=3.0 for different gate
voltages V
1
and V
3
for the strong pumping regime. One can see a shift of the pumped current and the
latter one is not equal 0 any more for f/p=1.right: pumped current for the weak pumping regime.
Ref [10]
A detailed experimental study of the quantized acoustoelectric current induced by a
surface acoustic wave (SAW) in a one-dimensional channel defined in a GaAs-
AlxGa12xAs heterostructure by a split gate is reported. The current measured as a
function of the gate voltage demonstrates quantized plateaus in units of I=e f where e
is the electron charge and f is the surface acoustic wave frequency. The quantization is
due to trapping of electrons in the moving potential wells induced by the surface acoustic
wave, with the number of electrons in each well controlled by electron-electron repulsion.
Therefore it can be described classically. The experimental results demonstrate that
acoustic charge transport in a one-dimensional channel may be a viable means of
producing a standard of electrical current.
Chapter 6 “Further Research and Applications”
theoretical investigations of parametric pumping have focused on open and transparent
device structures [23-29] For electron pumps operating in the CB regime, the
energy level spacing ?E=E
i
+1 - E
i
of the device is in general much smaller than the
charging energy, where Ei is the ith single electron level. Therefore, in the CB regime ?E
is irrelevant to the pumping operation. However, the phenomenon of resonance-assisted
electron pumping for which ?E plays the most important role is examined in [29] and
focuses on the resonance tunneling regime [31] for which charging energy is of no
concern although the device is not transparent. The results indicate that electron pumping
is drastically modified by the resonance states such that the pumped current obtains a
very large value at a resonance point. As the Fermi energy is varied (which can be
controlled by e.g. gate voltage), the pumped current can reverse its direction as a result of
competition between two pumping parameters X
1
and X
2
.
It is generally understood that although the electron reservoirs are in thermal equilibrium
during the pumping process, the time dependent pumping potential pumps out electrons
and thus produces the Joule heat along with the dissipation at the same time. Recently,
the physics of such thermal transport has been investigated.[32,33,34,35,36] Avron et
al.[32] have derived the lower bound for the dissipation and defined an optimal pump
which is noiseless. Moskalets and Buttiker [33] derived a formula for the heat flow and
the noise in the weak pumping regime. Wang et al.[35] extended the theory to the strong
pumping and finite frequency regime.
Furthermore there are several papers proposing a adiabatic quantum pump where the
transport happens through a carbon nanotube. Due to their unique features (e.g. stiffness
(less disorders), only 2 channel transport, clear Coulomb-Blockade ability, large ballistic
transport length (~ μm), …) they seem to be a promising for this kind of application [38-
48]. So far only theoretical papers have touched this field proposing SAW-approach
and/or lithographical designed voltage leads for top-gated devices. Local gating on
Nanotubes has been fairly understood as Marcus et al has shown [51][52], however, so
far no paper has been published presenting a nanotube-adiabatic-quantum-pump.
In [49] it is found out that QAPT in NT’s indeed can happen. Furthermore due to the
particular electronic properties of the nanotube, the pumped current is found to show a
remarkable parity effect near the resonant levels, with a rather sensitive dependence on
the control parameters of the device such as deformation strength, the amplitude and the
phase difference of the gate voltage. Their approach using a nonequilibrium Green’s
function Theory shows as the Fermi energy is varied, the pumped current is found to
oscillate in a regular fashion as a result of competition of resonant levels. Because of the
resonant nature of the pumping, the pumped current (or originally in the paper they plot
the injectivity defined as dN
i
/dX
i,
where i=1,2 stands for the two local gates) shows
nonsinusoidal dependence on the phase difference of the pumping parameters, consistent
with experimental findings for Ref. 20.
Recent research concentrates on heat- and spin current in relation of QAPT pumping. Ref
[53] for instance calculated the heat current, pumped current and transmission coefficient
for a NT QAPT-pump using tight-binding models and time-dependent scattering matrix
theory [54,36]. They found that the pumped current (I
p
) depends on the pumping
amplitudes (V
p
). The ideal case would be V
p
=0V [Fig. 15, Inset] and for V
p
~10
-5
V the
heat current is proportional to the square of V
p
showing weak-pumping. For V
p
=10
-3
V the
regime of strong-pumping is entered and the heat current increases linearly Fig. 15,16.
The pumped current and the heat current have only large values near the resonant levels
and are highly correlated to the transmission coefficient. The existence of the two
distinctive peaks in the injectivity-plot can be explained by calculating the total DOS for
such a system [49].
Another general idea is to use QAPT-pumping to generate a spin current. Ref. [53] has a
proposal using NT’s. In the presence of magnetic field, the carbon-nanotube-based
quantum pump can function as a spin pump, a molecular device by which a DC pure spin
current without accompanying charge current is generated at zero bias voltage via a
cyclic deformation of two device parameters. The pure spin current is achieved when the
Fermi energy is near the resonant level of the quantum pump. They find that the pure spin
current is sensitive to system parameters such as pumping amplitude, external magnetic
field, and gate voltage.
Another theoretical paper using NT’s for QAPT is Ref. [50]. They assume a SAW
approach which could produce DE minnibandgaps as large as 10meV equivalent to a
T=116K almost opening the room-temperature-working-point-regime.
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