Shot noise suppression in double barrier resonant tunnelling diodes with a Fano factor well below the value of 0.5 is theoretically predicted. This giant
suppression is found to be a signature of coherent transport regime and can occur at zero temperature as a consequence of the Pauli principle or at sufficiently high temperatures above 77 K as a consequence of a long-range Coulomb interaction. These predictions are in agreement with experimental data

Aleshkin, V. Ya

Alkeev, N. V.

Figueiredo, J. M. L.

Ironside, C. N.

Lyubchenko, V. E.

Reggiani, L.

Stanley, C. R.

Semiconductor Science and Technology

Universidade do Algarve

INSTITUTE OF PHYSICS PUBLISHING SEMICONDUCTOR SCIENCE AND TECHNOLOGY
Semicond. Sci. Technol. 18 (2003) L35–L38 PII: S0268-1242(03)60270-1
LETTER TO THE EDITOR
Giant suppression of shot noise in double
barrier resonant diode: a signature of
coherent transport
V Ya Aleshkin1, L Reggiani2, N V Alkeev3, V E Lyubchenko3,
C N Ironside4, J M L Figueiredo4,5 and C R Stanley4
1 Institute for Physics of Microstructures, Nizhny Novgorod GSP-105 603600, Russia
2 National Nanostructure Laboratory of INFM, Dipartimento di Ingegneria dell’ Innovazione,
Universita` di Lecce, Via Arnesano s/n, 73100 Lecce, Italy
3 Institute of Radioengineering & Electronics, Russian Academy of Sciences, Russia
4 Department of Electronic and Electrical Engineering, University of Glasgow, UK
5 Department de Fı´sica, Faculdade de Cieˆncias e Technologia, Universidade do Algarve,
8000-117 Faro, Portugal
Received 3 March 2003, in final form 4 April 2003
Published 25 April 2003
Online at stacks.iop.org/SST/18/L35
Abstract
Shot noise suppression in double barrier resonant tunnelling diodes with a
Fano factor well below the value of 0.5 is theoretically predicted. This giant
suppression is found to be a signature of coherent transport regime and can
occur at zero temperature as a consequence of the Pauli principle or at
sufficiently high temperatures above 77 K as a consequence of a long-range
Coulomb interaction. These predictions are in agreement with experimental
data.
Since its realization [1], the double barrier resonant diode
(DBRD) proved to be an electron device of broad physical
interest because of its peculiar non-Ohmic current voltage
(I–V) characteristic. Indeed, after a strong superOhmic
increase of current, it exhibits a negative differential
conductance and eventually hysteresis effects [2]. Even
the shot noise characteristics are of relevant interest, since
suppressed as well as enhanced shot noise with respect to
its full Poissonian value has been observed (see [3] for a
review on the subject). These electrical and noise features
are controlled by the mechanism of carrier tunnelling through
the double potential barriers. The microscopic interpretation
of these features is found to admit a coherent [4] or a sequential
[5] tunnelling approach. The intriguing feature of these two
approaches is that from the literature, it emerges that both of
them explain the I–V experimental data as well as most of the
shot noise characteristics. Therefore, to our knowledge, there
is no way to distinguish between these two transport regimes
and the natural question whether the tunnelling transport is
coherent or sequential remains an unsolved one.
The coherent approach to shot noise in DBRD has
received wide attention since the first experimental evidence
by Li et al [6] of shot noise suppression with a minimum
value of the Fano factor γ = SI /(2qI) = 0.5, where SI
is the current spectral density and q the absolute value of
the unit charge responsible of current. Remarkably, most of
the coherent approaches developed so far predict a maximum
suppression γ = 0.5 even if there is clear experimental
evidence of suppression below this value (here referred as
giant suppression) [7–9] down to values of γ = 0.25 [8, 9]. To
this purpose, some authors obtained theoretical values of the
Fano factor just below the value of 0.5, 0.45 [10] and 0.38
[11], respectively. However, the physical interpretation of
these results remains mostly qualitative and quoting [3] this
direction of research looks promising but certainly requires
more efforts.
In this letter, we announce that a giant suppression of
shot noise occurring before the peak value of the current is a
signature of coherent transport in DBRDs. To this purpose, we
present a theoretical model which predicts this phenomenon
and is validated by experiments.
The typical structure here investigated is the standard
symmetric double well reported in figure 1. We denote by
0268-1242/03/060035+04$30.00 © 2003 IOP Publishing Ltd Printed in the UK L35
Letter to the Editor
Figure 1. Band diagram of the double barrier structure considered
here under an applied voltage V . The bottom of the conduction band
in the emitter in the well and in the collector coincides at V = 0.
 = L +R the resonant states width and by εr the energy of
the resonant level as measured from the centre of the potential
well. Here L,R are the partial widths due to the tunnelling
through left and right barrier, respectively. For simplicity, we
consider the case of coherent tunnelling when there is only
one resonant state with L = R = /2 and we take unit
square contacts. The kinetic model is developed by assuming
that the electron distribution functions in the emitter and in
the collector of the DBRT are equilibrium-like, with different
electro-chemical potentials Fi :
fi(ε, Fi) = 11 + exp ( ε−Fi
kBT
) (1)
here i = L stands for the emitter, i = R for the collector; kB
is the Boltzmann constant, T the temperature and ε the kinetic
carrier energy. The double barrier transparency D(εz) can be
written in the following form:
D(εz) =
2
4
(εz − εr + qu)2 + 24
(2)
where εz,⊥ are the energies for electron motion perpendicular
and along barriers, and u is the voltage drop between the
emitter and the centre of the potential well (see figure 1). For
the current flowing from the emitter to the collector we found
I = − qm
2π2h¯3
∫ ∞
0
dεz dε⊥D(εz)[fL(ε) − fR(ε)] (3)
with m the carrier effective mass and h¯ the reduced Planck
constant.
The relation between u and the total applied voltage V is
given by
CRV = (CR + CL)u − QQW (4)
where CL,R are the capacitances corresponding to the
emitter and the collector junctions, and QQW = q
(
N+DQW −
NQW
)
, the charge in the quantum well with N+DQW, NQW
the concentrations of charged donors and free electrons,
respectively. The expression for NQW writes
NQW = m
π2h¯2
∫ ∞
0
dεzD(εz)
∫ ∞
0
dε⊥[fL(ε) + fR(ε)]
+
2m
π2h¯2
∫ 0
−qV
dεzD(εz)
∫ ∞
0
dε⊥fR(ε). (5)
In the right-hand side of equation (5), the first term is the
contribution of electrons inside the energy interval in which
there are electron fluxes from the emitter and the collector to
the quantum well; the second term corresponds to the energy
interval where there is only one electron flux from the collector
to the quantum well.
To calculate noise, we shall use the wave packet approach
[12] here implemented to account for the self-consistent
potential following the theory developed for the vacuum
diode [13]. Accordingly, for the spectral density of current
fluctuations, we obtain
SI = q
2m
π2h¯3
∫ ∞
0
dε⊥ dεz
{
D
[(
q+eff
)2
fL(1 − fR)
+
(
q−eff
)2
fR(1 − fL)
] − D2(fL − fR)2} (6)
with q±eff the dimensionless effective charge accounting for the
self-consistent potential [14]
q±eff(ε) = 1 ±
2h¯

[1 − fL(ε) − fR(ε)][
CL + CR + q
∂NQW
∂u
] ∂I
∂u
. (7)
When q±eff = 1, the self-consistent potential is neglected and
equation (6) coincides with that derived by [12]. At zero
applied voltage, equation (6) satisfies the Nyquist theorem.
Furthermore, the same equation gives SI = ∞ on the border
of the instability region where also dI/dV = ∞. Both tests,
by recovering expected features of the spectral density, prove
the physical reliability of the present approach.
We now consider the case of zero temperature where
Pauli principle is expected to play a dominant role. In
this case, only such states for which fL = 1 and fR = 0
take part to the current transmission. Therefore, the second
term in the right-hand side of equation (6) is zero and from
equation (7), we see that q+eff = 1. These conditions are valid
everywhere with exception of the border stability region where
dI/dV = ∞ = q+eff . In other words, as expected, long-range
Coulomb interactions are suppressed at zero temperature.
Then, we analyse the case of high applied voltages, when
qV > FR , because it is that more close to experiments,
and for convenience, we express the applied voltage and the
electrochemical potentials by the dimensionless parameters:
ξ = 2(qu − εr)/, f = 2FL/. Figure 2 reports the current
(figure 2(a)) and the Fano factor γ (ξ) (figure 2(b)) as the
function of ξ for different values of f . Curves labelled 1, 2
and 3 correspond to f = 1, 15 and ∞, respectively. The
current exhibits the well-known peaked behaviour reducing to
zero at the highest voltages because we neglect resonant states
above the first one. Of interest is the behaviour of the Fano
factor in figure 2(b). The common features of all curves is
the presence of a minimum of γ near to the current peak and
the evidence of full shot noise in regions where the current is
relatively small. Remarkably, the minimum value of the Fano
factor is found to be always less than 0.5, taking the value
0.391 at ξ = −0.801 for f = ∞ and being systematically
lower than this value at decreasing value of f . More precisely,
for f = ∞ the current increase towards the peak is widely
broadened in the positive differential conductance region and,
in close analogy with the one-dimensional system considered
[15], the Fano factor takes the universal expression:
γ (ξ) = 1 − 1
2
{
1 − ξ
(1 + ξ 2)
[
π
2 − arctan(ξ)
]
}
. (8)
L36
Letter to the Editor
(a)
(b)
Figure 2. Dependence of current (a) and Fano factor (b) on applied
voltage in a typical symmetric DBRD at zero temperature. For
convenience, dimensionless current and voltages are used with
I0 = qmLR/(2π 2h¯3) and ξ = 2/(qu − εr). Curves labelled as
1, 2, 3 correspond to values of the dimensionless electrochemical
potentials: f = 1, 15,∞ (f = 2FL/), respectively.
We remark that experiments at temperatures around 4 K [6],
by exhibiting γ = 0.5 fall inside the case of f → ∞ and
well agree with the present model. In particular, the minimum
value of suppression γ = 0.391 is in agreement with the
value of 0.35 found in experiments [7]. From figure 2(b), it is
clear that by decreasing the value of f also the minimum
of the Fano factor decreases systematically up to a zero
value. To understand the physical reason of this behaviour,
we discuss the limiting condition when f  1. In this case,
the width of the electron energy distribution is lower than that
of the resonant state. As a consequence, the transmission
probability becomes the same for all the electrons and we
obtain the standard partition expression for the Fano factor
[12] γ (ξ) = 1 − D(0). When D(0) is close to unity γ tends
to zero as found by Lesovik [16] for the analogous case of
a quantum point contact. We conclude, that the main reason
for shot noise suppression in DBRDs at zero temperature is
connected with Pauli principle and, because of the coherent
tunnelling regime, the Fano factor can take values significantly
lower than 0.5 near to the current peak.
For relatively high temperatures (i.e., 77 K), the
Coulomb interaction is expected to play the main role in
suppressing shot noise. The present coherent approach in
0
20
40
60
(a)
T=300 K
I(m
A)
0,0 0,2 0,4 0,6
0,4
0,6
0,8
1,0
1,2
exp. 60 MHz
exp. 200 MHz
theory
(b)
Voltage (V)
 
γ
0 1 2
0
40
80
Figure 3. Dependence of current (a) and Fano factor (b) on applied
voltage in a symmetric DBRD at room temperature. Continuos
curves refer to theoretical calculations and symbols to experiments.
The insert in figure 3(a) reports the complete set of I–V experiments.
the limit of Boltzmann statistics yields for the Fano factor:
γ =
[
1 +
2h¯

1(
CL + CR + q
∂NQW
∂u
) ∂I
∂u
]2
(9)
where the suppression is found to be controlled by the
differential conductance dI/du as expected [13]. For suitable
parameters of the DBRD, the above equation predicts the
possibility of a giant shot noise suppression to values of
the Fano factor well below the value of 0.5, in contrast
to what found for the sequential tunnelling model where at
most suppression is of 0.5. The above possibility has been
verified by room temperature experiments in a DBRD with
barriers sufficiently thin to expect coherent tunnelling. The
structure consists of two 2 nm AlAs layers separated by 6 nm
InGaAs quantum well [17]. Measurements are carried out
using a noise figure meter (XK5-49), that allows to measure
simultaneously noise figure and power gain of two-port
networks in the 50  feed circuit. Simultaneously with the
noise, the I–V curve was measured. Noise measurements
at frequencies 60 and 200 MHz showed the same results
within an experimental uncertainty at worst of 20%, thus
indicating that 1/f noise contribution is negligible. Figure 3
reports the I–V characteristics (see figure 3(a)) and the
Fano factor (see figure 3(b)). Here the I–V characteristic
[see figure 3(a)] shows a region of positive differential
L37
Letter to the Editor
conductance (pdc) up to about 0.7 V followed by an instability
region terminated around 1.3 V as reported in the insert of
figure 3(a). In the same pdc region, the Fano factor is
found to exhibit a suppression with a minimum value of
about 0.4 around 0.6 V [see figure 3(b)]. Present theory
is compared with experiments in the positive differential
resistance region of the I–V characteristic. Theoretical
calculations performed on the basis of equations (3) and (6)
make use of the following values for the parameters entering
the model: m = 0.045m0, with m0 the free electron mass,
 = 1.3 meV, εr = 85 meV, CL = 2.04pF,CR = 0.274pF ,
electron concentration on the border of the left barrier n =
2.4 × 1017 cm−3. The values of CL and CR are optimized
for the fitting, while other parameters are typical of the
materials used for DBRDs. The agreement between theory
and experiments catch the relevant features of the experiments
and is considered to be satisfactory for the purpose of the
paper. Thus, the comparison between theory and experiments
supports the physical interpretation of the giant suppression of
shot noise in DBRDs and in particular confirms that coherent
tunnelling occurs under such a condition.
The reason why shot noise suppression is more effective
for coherent than for sequential tunnelling can be physically
explained as follows. Starting from the fact that the two
mechanisms responsible for shot noise suppression are Pauli
principle an Coulomb interaction, we note that both of them
act simultaneously for coherent and sequential tunnelling. Let
us consider the first of them, which is the most relevant at zero
temperature, in the case when the Fermi energy is so small that
all the electrons exhibit the same transparency. For coherent
transport, it is possible to have the transparency equal to unity,
which implies γ = 1 − D according to Lesovik findings [16].
For sequential transport, the total transparency is always less
than unity, since the flux of electrons reflected by the barrier
always exists. As a consequence, under coherent transport for
the case D = 1, there is no noise; by contrast, under sequential
transport the presence of scattering always introduces noise.
This example illustrates why Pauli principle is more efficient
in suppressing shot noise under coherent than sequential
transport. By considering Coulomb interaction, which is
more relevant at higher temperatures, we recall that in the
absence of collisions it provides giant shot noise suppression
as in the vacuum tubes [13] because electron reflection in this
case is due only to Coulomb interaction. It is clear that the
presence of scattering provides additional random mechanisms
for electrons returning to the emitter and, therefore, provides
additional source of noise. Even this example shows that
Coulomb interaction is more efficient in suppressing shot noise
under coherent than sequential transport.
In conclusion, we have investigated coherent tunnelling
by implementing the wave packet approach for transport in
DBRDs that includes both Pauli principle and long-range
Coulomb interaction. In agreement with existing results,
we have found that around zero temperature shot noise is
suppressed because of Pauli principle alone. Moreover, under
normal conditions (i.e., f → ∞) the suppression exhibits
a Fano factor of 0.5 in a wide region of applied voltages,
with a minimum of 0.391 near the current peak in agreement
with experiments. Interestingly, we have found that shot
noise can be suppressed well below the value of 0.5 also
because of Coulomb interaction. This giant suppression is
here confirmed by experimental measurements performed at
room temperature. Therefore, shot noise suppression below
one-half of the full Poissonian value is proven to be a signature
of coherent tunnelling against sequential tunnelling in double
barrier resonant diodes. We finally want to stress that the main
reason of the difference between these approaches stems from
the fact that the sequential tunnelling is based on a master
equation [18, 19] for treating fluctuations of carrier numbers
inside the quantum well while coherent tunnelling uses the
quantum partition noise within the wave-packet approach
[12]. The master equation describes implicitly a sequential
mechanism for a carrier entering/exiting from the well and,
as a consequence, its intrinsic limit coincides with that of two
independent resistors (or vacuum diodes) connected in series
and each of them exhibiting full shot noise. This system
yields a maximum suppression of shot noise down to the value
of 0.5. By contrast, partition noise, inherent to the wave-
packet formalism, can be fully suppressed down to zero in
the presence of a fully transparent barrier and/or of Coulomb
interaction like in vacuum diodes.
Acknowledgments
Prof V Volkov of Moscow Institute of Radioengineering and
Electronics is thanked for having suggested the problem of this
research. Partial support from the Italian Ministry of Foreign
Affairs through the Volta Landau Centre (the fellowship of
VYaA), and the SPOT-NOSED project IST-2001-38899 of the
EC is gratefully acknowledged.
References
[1] Tsu R and Esaki L 1973 Appl. Phys. Lett. 22 562
[2] Goldman V, Tsui D and Cunningham J 1987 Phys. Rev. Lett.
58 1256
[3] Blanter Y M and Bu¨ttiker M 2000 Phys. Rep. 336 1
[4] Chang L L, Esaki L and Tsui R 1974 Appl. Phys. Lett. 24 593
[5] Luryi S 1985 Appl. Phys. Lett. 47 490
[6] Li Y P, Zaslavsky A, Tsui D C, Santos M and Shayegan M
1990 Phys. Rev. B 41 8388
[7] Brown E R 1992 IEEE Trans Electron Devices 39 2686
[8] Przadka A, Webb K J, Janes D B, Liu H C and Wasilewski Z R
1997 Appl. Phys. Lett. 71 530
[9] Kuznetsov V, Mendez E, Bruno J and Pham J 1998 Phys. Rev.
B 58 R10159
[10] Egues J C, Hershfield S and Wilkins J V 1994 Phys. Rev. B
49 13517
[11] Jahan M M and Anwar A F M 1995 Solid-State Electron.
38 429
[12] Martin T and Landauer R 1992 Phys. Rev. B 45 1742
[13] Van der Ziel A 1954 Noise (New York: Prentice Hall)
p 89
[14] Aleshkin V Ya, Reggiani L, Alkeev N V, Lyubchenko V E,
Ironside C N, Figueiredo J M L and Stanley C R 2003
Preprint cond-matter/0304077v1
[15] Averin D V 1993 J. Appl. Phys. 73 2593
[16] Lesovik G B 1989 JETP Lett. 49
[17] Alkeev N V, Lyubchenko V E, Ironside C N, Figuiredo J M L
and Stanley C R 2000 J. Commun. Technol. Electron.
45 911
[18] Davies J H, Egues J C and Wilkins J W 1995 Phys. Rev. B
55 11259
[19] Iannaccone G, Macucci M and Pellegrini B 1997 Phys. Rev. B
55 4539
L38


Giant suppression of shot noise in double barrier resonant diode: a signature of coherent transport

Shot noise suppression below 1/2 of the full Poisson value in double barrier resonant diodes is confirmed to be a signature of coherent rather than sequential tunnelling transport. We reply to the arguments of the previous comment which dispute the above claim. We anticipate the development of a rigorous theory that improves previous approaches without contradicting the essential findings we recently reported (Aleshkin et al 2003 Semicond.
Sci. Technol. 18 L35)

INSTITUTE OF PHYSICS PUBLISHING SEMICONDUCTOR SCIENCE AND TECHNOLOGY
Semicond. Sci. Technol. 19 (2004) 665–666 PII: S0268-1242(04)69928-7
REPLY
Reply to Comment on ‘Giant suppression
of shot-noise in double barrier resonant
diode: a signature of coherent transport’
V Ya Aleshkin1, L Reggiani2, N V Alkeev3, V E Lyubchenko3,
C N Ironside4, J M L Figueiredo4 and C R Stanley4
1 Institute for Physics of Microstructures, Nizhny Novgorod GSP-105 603600, Russia
2 National Nanostructure Laboratory of INFM, Dipartimento di Ingegneria dell’ Innovazione,
Universita` di Lecce, Via Arnesano s/n, 73100 Lecce, Italy
3 Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow,
Russia
4 Department of Electronic and Electrical Engineering, University of Glasgow, Glasgow, UK
Received 27 October 2003
Published 23 March 2004
Online at stacks.iop.org/SST/19/665 (DOI: 10.1088/0268-1242/19/5/N04)
Abstract
Shot noise suppression below 1/2 of the full Poisson value in double barrier
resonant diodes is confirmed to be a signature of coherent rather than
sequential tunnelling transport. We reply to the arguments of the previous
comment which dispute the above claim. We anticipate the development of
a rigorous theory that improves previous approaches without contradicting
the essential findings we recently reported (Aleshkin et al 2003 Semicond.
Sci. Technol. 18 L35).
In a recent letter [1] we showed that suppression of shot-noise
in resonant diodes below 1/2 of full shot-noise value 2qI ,
with q the unit charge and I the steady current, is a signature
of coherent against sequential tunnelling transport regime.
The proof stems on the fact that the theoretical approach we
developed for the coherent tunnelling regime predicts under
suitable bias and temperature conditions a Fano factor below
1/2 while the standard sequential tunnelling regime [2, 3]
never allows for a value of the Fano factor below 1/2. These
findings were in agreement with experiments.
In their comment, Blanter and Bu¨ttiker (BB) contest our
theoretical discussion on the basis of the results they obtained
on the same subject [4, 5] and which never allow for a Fano
factor drop below 1/2. To this purpose the comment reports
several arguments on which we reply in order of appearance.
In our opinion, the sequential tunnelling regime cannot be
interpreted as the semiclassical limit of the coherent tunnelling
regime when considering current noise. The microscopic
noise source in each regime are of a different nature, even if
both correctly recover the Nyquist relation under equilibrium,
because the sequential and coherent tunnelling are different
mechanism for electron transfer in the resonant tunnelling
diode. In the coherent transport there is no scattering during
tunnelling while scattering in the quantum well is essential for
sequential tunnelling. Moreover, the increase in the number of
channels does not introduce scattering. Consider the simple
case when the Coulomb interaction is neglected. Then, in
the sequential tunnelling regime the fluctuations are described
by the differential rates controlling the relaxation of carrier
number fluctuations inside the two terminal device through
the contacts [3]. By contrast, in the coherent tunnelling
regime the fluctuations are described by the transparency
of the whole device through the partition noise mechanism.
These two descriptions are inherently different and provide in
general different results. Therefore, we do not agree on the
expected equivalence of phase coherent and sequential results
and on the estimated quantum corrections reported in the
comment.
Concerning the details of calculations which lead to our
equation (8) we note the following. We consider high voltage
regime, v > EF . However, to our knowledge neither explicitly
nor implicitly we assumed E0  EF , and thus we believe that
equation (8) is rigorously obtained. We add that a similar
expression for the Fano factor was obtained in [6, 7] and
0268-1242/04/050665+02$30.00 © 2004 IOP Publishing Ltd Printed in the UK 665
Reply
that also in that case a Fano factor below 1/2 was predicted
under suitable conditions and for the one-dimensional case.
Moreover, their results [6, 7] can be easy reproduced from our
equation (6) when ε⊥ is excluded from integration. Now the
natural question arises: What is the reason for the discrepancy
of our results with those of BB [4]? In our opinion such
a discrepancy arises because of the infinite small resonance
width approximation which was used in BB [4], when before
their equation (28) they replaced at an intermediate stage of
calculations the Breit–Wigner expression with a delta function.
Moreover, when the Coulomb interaction is neglected, under
this delta approximation the Fano factor found by BB [4]
follows from equation (8) of our work [1] (ξ  −1). In
other words, because of this delta approximation BB simply
neglected the voltage interval where the Fano factor is below
1/2.
Concerning the reasonability of the assumptions made, we
confirm the use of energy independent tunnelling rates. We
also confirm that current and noise are never exactly zero in the
presence of an applied voltage. This is clear by carrying out
a numerical calculation of the transparency for the structure
drawn in the Comment. There is always nonzero transparency
for electrons with energy above the band edge even when the
resonant level is below it. Thus, the conclusion concerning the
absence of the current and noise made by BB and used in their
equation (14) [4] when the resonant level is below the band
edge of the emitter is again the result of an approximation
and does not correspond to a rigorous quantum mechanical
description of tunnelling. Therefore, we disagree that by
including an energy dependent tunnelling rate there are regions
of applied voltages with exactly zero current and zero noise,
unless one is forcing this result. We also disagree that finding a
Fano factor below 1/2 is strictly related to this approximation.
As it is clear from figure 2 of our paper [1], there is wide region
up to ξ = −10 for f = ∞ and f = 15 where Fano factor is
below 1/2. In this region the resonant level is situated higher
than the band edge of the emitter and the energy difference
between them is greater than , thus the influence of the edge
on L is neglibible in calculating the transparency. It means
that the Fano factor can drop below 1/2 in the case of an energy
dependent L,R if in any point of the mentioned region of ξ
values L = R. As stated in the letter, we believe that in the
coherent regime it is the presence of the (1-D) factor (D being
the transparency of the device) which allows for a suppression
below 1/2, in analogy with the original findings of Lesovik
[8] for the case of a point contact.
We agree that the simplicity of the model can be a source
of discrepancy between experiments and theory, and that more
investigations are needed to shed new light on the subject. We
are presently engaged in this effort.
Concerning the inclusion of Coulomb effects, we
presented as equation (6) in [1], we never found negative
values of the current spectral density. On the other hand,
we agree that the role of Coulomb interaction in general, and
at zero temperature in particular, should be reconsidered more
accurately than reported in equation (6). To this purpose we
are working out a theoretical model following a full quantum
mechanical approach [9]. In any case, at present we have not
found results which contradict the essential findings reported
in the letter.
Acknowledgments
Partial support from the Italian Ministry of Foreign Affairs
(MAE) through the Volta Landau Center (the fellowship of
VYaA) and from the cofin-03 project ‘Modelli e misure di
rumore in nanostrutture’ financed by the Italian Ministry
for Education University and Research (MIUR) is gratefully
acknowledged.
References
[1] Aleshkin V Ya et al 2003 Semicond. Sci Technol. 18 L35
[2] Iannaccone G, Macucci M and Pellegrini B 1997 Phys. Rev. B
55 4539
[3] Aleshkin V Ya and Reggiani L 2001 Phys. Rev. B 64 245333
[4] Blanter Y M and Bu¨ttiker M 1999 Phys. Rev. B 59 10217
[5] Blanter Y M and Bu¨ttiker M 2000 Phys. Rep. B 336 1
[6] Averin D V 1993 J. Appl. Phys. 73 2593
[7] Wei Y, Wang B, Wang J and Guo H 1999 Phys. Rev. B 60
16900
[8] Lesovik G B 1989 JETP Lett. 49
[9] Aleshkin V Ya et al 2003 Preprint cond-mat/0304077
666


Reply to comment on 'Giant suppression of shot-noise in double barrier resonant diode: a signature of coherent transport' 

Sapientia

INSTITUTE OF PHYSICS PUBLISHING SEMICONDUCTOR SCIENCE AND TECHNOLOGYSemicond. Sci. Technol. 18 (2003) L35–L38 PII: S0268-1242(03)60270-1LETTER TO THE EDITORGiant suppression of shot noise in doublebarrier resonant diode: a signature ofcoherent transportV Ya Aleshkin1, L Reggiani2, N V Alkeev3, V E Lyubchenko3,C N Ironside4, J M L Figueiredo4,5 and C R Stanley41 Institute for Physics of Microstructures, Nizhny Novgorod GSP-105 603600, Russia2 National Nanostructure Laboratory of INFM, Dipartimento di Ingegneria dell’ Innovazione,Universita` di Lecce, Via Arnesano s/n, 73100 Lecce, Italy3 Institute of Radioengineering & Electronics, Russian Academy of Sciences, Russia4 Department of Electronic and Electrical Engineering, University of Glasgow, UK5 Department de Fı´sica, Faculdade de Cieˆncias e Technologia, Universidade do Algarve,8000-117 Faro, PortugalReceived 3 March 2003, in final form 4 April 2003Published 25 April 2003Online at stacks.iop.org/SST/18/L35AbstractShot noise suppression in double barrier resonant tunnelling diodes with aFano factor well below the value of 0.5 is theoretically predicted. This giantsuppression is found to be a signature of coherent transport regime and canoccur at zero temperature as a consequence of the Pauli principle or atsufficiently high temperatures above 77 K as a consequence of a long-rangeCoulomb interaction. These predictions are in agreement with experimentaldata.Since its realization [1], the double barrier resonant diode(DBRD) proved to be an electron device of broad physicalinterest because of its peculiar non-Ohmic current voltage(I–V) characteristic. Indeed, after a strong superOhmicincrease of current, it exhibits a negative differentialconductance and eventually hysteresis effects [2]. Eventhe shot noise characteristics are of relevant interest, sincesuppressed as well as enhanced shot noise with respect toits full Poissonian value has been observed (see [3] for areview on the subject). These electrical and noise featuresare controlled by the mechanism of carrier tunnelling throughthe double potential barriers. The microscopic interpretationof these features is found to admit a coherent [4] or a sequential[5] tunnelling approach. The intriguing feature of these twoapproaches is that from the literature, it emerges that both ofthem explain the I–V experimental data as well as most of theshot noise characteristics. Therefore, to our knowledge, thereis no way to distinguish between these two transport regimesand the natural question whether the tunnelling transport iscoherent or sequential remains an unsolved one.The coherent approach to shot noise in DBRD hasreceived wide attention since the first experimental evidenceby Li et al [6] of shot noise suppression with a minimumvalue of the Fano factor γ = SI /(2qI) = 0.5, where SIis the current spectral density and q the absolute value ofthe unit charge responsible of current. Remarkably, most ofthe coherent approaches developed so far predict a maximumsuppression γ = 0.5 even if there is clear experimentalevidence of suppression below this value (here referred asgiant suppression) [7–9] down to values of γ = 0.25 [8, 9]. Tothis purpose, some authors obtained theoretical values of theFano factor just below the value of 0.5, 0.45 [10] and 0.38[11], respectively. However, the physical interpretation ofthese results remains mostly qualitative and quoting [3] thisdirection of research looks promising but certainly requiresmore efforts.In this letter, we announce that a giant suppression ofshot noise occurring before the peak value of the current is asignature of coherent transport in DBRDs. To this purpose, wepresent a theoretical model which predicts this phenomenonand is validated by experiments.The typical structure here investigated is the standardsymmetric double well reported in figure 1. We denote by0268-1242/03/060035+04$30.00 © 2003 IOP Publishing Ltd Printed in the UK L35Letter to the EditorFigure 1. Band diagram of the double barrier structure consideredhere under an applied voltage V . The bottom of the conduction bandin the emitter in the well and in the collector coincides at V = 0. = L +R the resonant states width and by εr the energy ofthe resonant level as measured from the centre of the potentialwell. Here L,R are the partial widths due to the tunnellingthrough left and right barrier, respectively. For simplicity, weconsider the case of coherent tunnelling when there is onlyone resonant state with L = R = /2 and we take unitsquare contacts. The kinetic model is developed by assumingthat the electron distribution functions in the emitter and inthe collector of the DBRT are equilibrium-like, with differentelectro-chemical potentials Fi :fi(ε, Fi) = 11 + exp ( ε−FikBT) (1)here i = L stands for the emitter, i = R for the collector; kBis the Boltzmann constant, T the temperature and ε the kineticcarrier energy. The double barrier transparency D(εz) can bewritten in the following form:D(εz) =24(εz − εr + qu)2 + 24(2)where εz,⊥ are the energies for electron motion perpendicularand along barriers, and u is the voltage drop between theemitter and the centre of the potential well (see figure 1). Forthe current flowing from the emitter to the collector we foundI = − qm2π2h¯3∫ ∞0dεz dε⊥D(εz)[fL(ε) − fR(ε)] (3)with m the carrier effective mass and h¯ the reduced Planckconstant.The relation between u and the total applied voltage V isgiven byCRV = (CR + CL)u − QQW (4)where CL,R are the capacitances corresponding to theemitter and the collector junctions, and QQW = q(N+DQW −NQW), the charge in the quantum well with N+DQW, NQWthe concentrations of charged donors and free electrons,respectively. The expression for NQW writesNQW = mπ2h¯2∫ ∞0dεzD(εz)∫ ∞0dε⊥[fL(ε) + fR(ε)]+2mπ2h¯2∫ 0−qVdεzD(εz)∫ ∞0dε⊥fR(ε). (5)In the right-hand side of equation (5), the first term is thecontribution of electrons inside the energy interval in whichthere are electron fluxes from the emitter and the collector tothe quantum well; the second term corresponds to the energyinterval where there is only one electron flux from the collectorto the quantum well.To calculate noise, we shall use the wave packet approach[12] here implemented to account for the self-consistentpotential following the theory developed for the vacuumdiode [13]. Accordingly, for the spectral density of currentfluctuations, we obtainSI = q2mπ2h¯3∫ ∞0dε⊥ dεz{D[(q+eff)2fL(1 − fR)+(q−eff)2fR(1 − fL)] − D2(fL − fR)2} (6)with q±eff the dimensionless effective charge accounting for theself-consistent potential [14]q±eff(ε) = 1 ±2h¯[1 − fL(ε) − fR(ε)][CL + CR + q∂NQW∂u] ∂I∂u. (7)When q±eff = 1, the self-consistent potential is neglected andequation (6) coincides with that derived by [12]. At zeroapplied voltage, equation (6) satisfies the Nyquist theorem.Furthermore, the same equation gives SI = ∞ on the borderof the instability region where also dI/dV = ∞. Both tests,by recovering expected features of the spectral density, provethe physical reliability of the present approach.We now consider the case of zero temperature wherePauli principle is expected to play a dominant role. Inthis case, only such states for which fL = 1 and fR = 0take part to the current transmission. Therefore, the secondterm in the right-hand side of equation (6) is zero and fromequation (7), we see that q+eff = 1. These conditions are valideverywhere with exception of the border stability region wheredI/dV = ∞ = q+eff . In other words, as expected, long-rangeCoulomb interactions are suppressed at zero temperature.Then, we analyse the case of high applied voltages, whenqV > FR , because it is that more close to experiments,and for convenience, we express the applied voltage and theelectrochemical potentials by the dimensionless parameters:ξ = 2(qu − εr)/, f = 2FL/. Figure 2 reports the current(figure 2(a)) and the Fano factor γ (ξ) (figure 2(b)) as thefunction of ξ for different values of f . Curves labelled 1, 2and 3 correspond to f = 1, 15 and ∞, respectively. Thecurrent exhibits the well-known peaked behaviour reducing tozero at the highest voltages because we neglect resonant statesabove the first one. Of interest is the behaviour of the Fanofactor in figure 2(b). The common features of all curves isthe presence of a minimum of γ near to the current peak andthe evidence of full shot noise in regions where the current isrelatively small. Remarkably, the minimum value of the Fanofactor is found to be always less than 0.5, taking the value0.391 at ξ = −0.801 for f = ∞ and being systematicallylower than this value at decreasing value of f . More precisely,for f = ∞ the current increase towards the peak is widelybroadened in the positive differential conductance region and,in close analogy with the one-dimensional system considered[15], the Fano factor takes the universal expression:γ (ξ) = 1 − 12{1 − ξ(1 + ξ 2)[π2 − arctan(ξ)]}. (8)L36Letter to the Editor(a)(b)Figure 2. Dependence of current (a) and Fano factor (b) on appliedvoltage in a typical symmetric DBRD at zero temperature. Forconvenience, dimensionless current and voltages are used withI0 = qmLR/(2π 2h¯3) and ξ = 2/(qu − εr). Curves labelled as1, 2, 3 correspond to values of the dimensionless electrochemicalpotentials: f = 1, 15,∞ (f = 2FL/), respectively.We remark that experiments at temperatures around 4 K [6],by exhibiting γ = 0.5 fall inside the case of f → ∞ andwell agree with the present model. In particular, the minimumvalue of suppression γ = 0.391 is in agreement with thevalue of 0.35 found in experiments [7]. From figure 2(b), it isclear that by decreasing the value of f also the minimumof the Fano factor decreases systematically up to a zerovalue. To understand the physical reason of this behaviour,we discuss the limiting condition when f  1. In this case,the width of the electron energy distribution is lower than thatof the resonant state. As a consequence, the transmissionprobability becomes the same for all the electrons and weobtain the standard partition expression for the Fano factor[12] γ (ξ) = 1 − D(0). When D(0) is close to unity γ tendsto zero as found by Lesovik [16] for the analogous case ofa quantum point contact. We conclude, that the main reasonfor shot noise suppression in DBRDs at zero temperature isconnected with Pauli principle and, because of the coherenttunnelling regime, the Fano factor can take values significantlylower than 0.5 near to the current peak.For relatively high temperatures (i.e., 77 K), theCoulomb interaction is expected to play the main role insuppressing shot noise. The present coherent approach in0204060(a)T=300 KI(mA)0,0 0,2 0,4 0,60,40,60,81,01,2exp. 60 MHzexp. 200 MHztheory(b)Voltage (V) γ0 1 204080Figure 3. Dependence of current (a) and Fano factor (b) on appliedvoltage in a symmetric DBRD at room temperature. Continuoscurves refer to theoretical calculations and symbols to experiments.The insert in figure 3(a) reports the complete set of I–V experiments.the limit of Boltzmann statistics yields for the Fano factor:γ =[1 +2h¯1(CL + CR + q∂NQW∂u) ∂I∂u]2(9)where the suppression is found to be controlled by thedifferential conductance dI/du as expected [13]. For suitableparameters of the DBRD, the above equation predicts thepossibility of a giant shot noise suppression to values ofthe Fano factor well below the value of 0.5, in contrastto what found for the sequential tunnelling model where atmost suppression is of 0.5. The above possibility has beenverified by room temperature experiments in a DBRD withbarriers sufficiently thin to expect coherent tunnelling. Thestructure consists of two 2 nm AlAs layers separated by 6 nmInGaAs quantum well [17]. Measurements are carried outusing a noise figure meter (XK5-49), that allows to measuresimultaneously noise figure and power gain of two-portnetworks in the 50  feed circuit. Simultaneously with thenoise, the I–V curve was measured. Noise measurementsat frequencies 60 and 200 MHz showed the same resultswithin an experimental uncertainty at worst of 20%, thusindicating that 1/f noise contribution is negligible. Figure 3reports the I–V characteristics (see figure 3(a)) and theFano factor (see figure 3(b)). Here the I–V characteristic[see figure 3(a)] shows a region of positive differentialL37Letter to the Editorconductance (pdc) up to about 0.7 V followed by an instabilityregion terminated around 1.3 V as reported in the insert offigure 3(a). In the same pdc region, the Fano factor isfound to exhibit a suppression with a minimum value ofabout 0.4 around 0.6 V [see figure 3(b)]. Present theoryis compared with experiments in the positive differentialresistance region of the I–V characteristic. Theoreticalcalculations performed on the basis of equations (3) and (6)make use of the following values for the parameters enteringthe model: m = 0.045m0, with m0 the free electron mass, = 1.3 meV, εr = 85 meV, CL = 2.04pF,CR = 0.274pF ,electron concentration on the border of the left barrier n =2.4 × 1017 cm−3. The values of CL and CR are optimizedfor the fitting, while other parameters are typical of thematerials used for DBRDs. The agreement between theoryand experiments catch the relevant features of the experimentsand is considered to be satisfactory for the purpose of thepaper. Thus, the comparison between theory and experimentssupports the physical interpretation of the giant suppression ofshot noise in DBRDs and in particular confirms that coherenttunnelling occurs under such a condition.The reason why shot noise suppression is more effectivefor coherent than for sequential tunnelling can be physicallyexplained as follows. Starting from the fact that the twomechanisms responsible for shot noise suppression are Pauliprinciple an Coulomb interaction, we note that both of themact simultaneously for coherent and sequential tunnelling. Letus consider the first of them, which is the most relevant at zerotemperature, in the case when the Fermi energy is so small thatall the electrons exhibit the same transparency. For coherenttransport, it is possible to have the transparency equal to unity,which implies γ = 1 − D according to Lesovik findings [16].For sequential transport, the total transparency is always lessthan unity, since the flux of electrons reflected by the barrieralways exists. As a consequence, under coherent transport forthe case D = 1, there is no noise; by contrast, under sequentialtransport the presence of scattering always introduces noise.This example illustrates why Pauli principle is more efficientin suppressing shot noise under coherent than sequentialtransport. By considering Coulomb interaction, which ismore relevant at higher temperatures, we recall that in theabsence of collisions it provides giant shot noise suppressionas in the vacuum tubes [13] because electron reflection in thiscase is due only to Coulomb interaction. It is clear that thepresence of scattering provides additional random mechanismsfor electrons returning to the emitter and, therefore, providesadditional source of noise. Even this example shows thatCoulomb interaction is more efficient in suppressing shot noiseunder coherent than sequential transport.In conclusion, we have investigated coherent tunnellingby implementing the wave packet approach for transport inDBRDs that includes both Pauli principle and long-rangeCoulomb interaction. In agreement with existing results,we have found that around zero temperature shot noise issuppressed because of Pauli principle alone. Moreover, undernormal conditions (i.e., f → ∞) the suppression exhibitsa Fano factor of 0.5 in a wide region of applied voltages,with a minimum of 0.391 near the current peak in agreementwith experiments. Interestingly, we have found that shotnoise can be suppressed well below the value of 0.5 alsobecause of Coulomb interaction. This giant suppression ishere confirmed by experimental measurements performed atroom temperature. Therefore, shot noise suppression belowone-half of the full Poissonian value is proven to be a signatureof coherent tunnelling against sequential tunnelling in doublebarrier resonant diodes. We finally want to stress that the mainreason of the difference between these approaches stems fromthe fact that the sequential tunnelling is based on a masterequation [18, 19] for treating fluctuations of carrier numbersinside the quantum well while coherent tunnelling uses thequantum partition noise within the wave-packet approach[12]. The master equation describes implicitly a sequentialmechanism for a carrier entering/exiting from the well and,as a consequence, its intrinsic limit coincides with that of twoindependent resistors (or vacuum diodes) connected in seriesand each of them exhibiting full shot noise. This systemyields a maximum suppression of shot noise down to the valueof 0.5. By contrast, partition noise, inherent to the wave-packet formalism, can be fully suppressed down to zero inthe presence of a fully transparent barrier and/or of Coulombinteraction like in vacuum diodes.AcknowledgmentsProf V Volkov of Moscow Institute of Radioengineering andElectronics is thanked for having suggested the problem of thisresearch. Partial support from the Italian Ministry of ForeignAffairs through the Volta Landau Centre (the fellowship ofVYaA), and the SPOT-NOSED project IST-2001-38899 of theEC is gratefully acknowledged.References[1] Tsu R and Esaki L 1973 Appl. Phys. Lett. 22 562[2] Goldman V, Tsui D and Cunningham J 1987 Phys. Rev. Lett.58 1256[3] Blanter Y M and Bu¨ttiker M 2000 Phys. Rep. 336 1[4] Chang L L, Esaki L and Tsui R 1974 Appl. Phys. Lett. 24 593[5] Luryi S 1985 Appl. Phys. Lett. 47 490[6] Li Y P, Zaslavsky A, Tsui D C, Santos M and Shayegan M1990 Phys. Rev. B 41 8388[7] Brown E R 1992 IEEE Trans Electron Devices 39 2686[8] Przadka A, Webb K J, Janes D B, Liu H C and Wasilewski Z R1997 Appl. Phys. Lett. 71 530[9] Kuznetsov V, Mendez E, Bruno J and Pham J 1998 Phys. Rev.B 58 R10159[10] Egues J C, Hershfield S and Wilkins J V 1994 Phys. Rev. B49 13517[11] Jahan M M and Anwar A F M 1995 Solid-State Electron.38 429[12] Martin T and Landauer R 1992 Phys. Rev. B 45 1742[13] Van der Ziel A 1954 Noise (New York: Prentice Hall)p 89[14] Aleshkin V Ya, Reggiani L, Alkeev N V, Lyubchenko V E,Ironside C N, Figueiredo J M L and Stanley C R 2003Preprint cond-matter/0304077v1[15] Averin D V 1993 J. Appl. Phys. 73 2593[16] Lesovik G B 1989 JETP Lett. 49[17] Alkeev N V, Lyubchenko V E, Ironside C N, Figuiredo J M Land Stanley C R 2000 J. Commun. Technol. Electron.45 911[18] Davies J H, Egues J C and Wilkins J W 1995 Phys. Rev. B55 11259[19] Iannaccone G, Macucci M and Pellegrini B 1997 Phys. Rev. B55 4539L38

INSTITUTE OF PHYSICS PUBLISHING SEMICONDUCTOR SCIENCE AND TECHNOLOGYSemicond. Sci. Technol. 19 (2004) 665–666 PII: S0268-1242(04)69928-7REPLYReply to Comment on ‘Giant suppressionof shot-noise in double barrier resonantdiode: a signature of coherent transport’V Ya Aleshkin1, L Reggiani2, N V Alkeev3, V E Lyubchenko3,C N Ironside4, J M L Figueiredo4 and C R Stanley41 Institute for Physics of Microstructures, Nizhny Novgorod GSP-105 603600, Russia2 National Nanostructure Laboratory of INFM, Dipartimento di Ingegneria dell’ Innovazione,Universita` di Lecce, Via Arnesano s/n, 73100 Lecce, Italy3 Institute of Radioengineering and Electronics, Russian Academy of Sciences, Moscow,Russia4 Department of Electronic and Electrical Engineering, University of Glasgow, Glasgow, UKReceived 27 October 2003Published 23 March 2004Online at stacks.iop.org/SST/19/665 (DOI: 10.1088/0268-1242/19/5/N04)AbstractShot noise suppression below 1/2 of the full Poisson value in double barrierresonant diodes is confirmed to be a signature of coherent rather thansequential tunnelling transport. We reply to the arguments of the previouscomment which dispute the above claim. We anticipate the development ofa rigorous theory that improves previous approaches without contradictingthe essential findings we recently reported (Aleshkin et al 2003 Semicond.Sci. Technol. 18 L35).In a recent letter [1] we showed that suppression of shot-noisein resonant diodes below 1/2 of full shot-noise value 2qI ,with q the unit charge and I the steady current, is a signatureof coherent against sequential tunnelling transport regime.The proof stems on the fact that the theoretical approach wedeveloped for the coherent tunnelling regime predicts undersuitable bias and temperature conditions a Fano factor below1/2 while the standard sequential tunnelling regime [2, 3]never allows for a value of the Fano factor below 1/2. Thesefindings were in agreement with experiments.In their comment, Blanter and Bu¨ttiker (BB) contest ourtheoretical discussion on the basis of the results they obtainedon the same subject [4, 5] and which never allow for a Fanofactor drop below 1/2. To this purpose the comment reportsseveral arguments on which we reply in order of appearance.In our opinion, the sequential tunnelling regime cannot beinterpreted as the semiclassical limit of the coherent tunnellingregime when considering current noise. The microscopicnoise source in each regime are of a different nature, even ifboth correctly recover the Nyquist relation under equilibrium,because the sequential and coherent tunnelling are differentmechanism for electron transfer in the resonant tunnellingdiode. In the coherent transport there is no scattering duringtunnelling while scattering in the quantum well is essential forsequential tunnelling. Moreover, the increase in the number ofchannels does not introduce scattering. Consider the simplecase when the Coulomb interaction is neglected. Then, inthe sequential tunnelling regime the fluctuations are describedby the differential rates controlling the relaxation of carriernumber fluctuations inside the two terminal device throughthe contacts [3]. By contrast, in the coherent tunnellingregime the fluctuations are described by the transparencyof the whole device through the partition noise mechanism.These two descriptions are inherently different and provide ingeneral different results. Therefore, we do not agree on theexpected equivalence of phase coherent and sequential resultsand on the estimated quantum corrections reported in thecomment.Concerning the details of calculations which lead to ourequation (8) we note the following. We consider high voltageregime, v > EF . However, to our knowledge neither explicitlynor implicitly we assumed E0  EF , and thus we believe thatequation (8) is rigorously obtained. We add that a similarexpression for the Fano factor was obtained in [6, 7] and0268-1242/04/050665+02$30.00 © 2004 IOP Publishing Ltd Printed in the UK 665Replythat also in that case a Fano factor below 1/2 was predictedunder suitable conditions and for the one-dimensional case.Moreover, their results [6, 7] can be easy reproduced from ourequation (6) when ε⊥ is excluded from integration. Now thenatural question arises: What is the reason for the discrepancyof our results with those of BB [4]? In our opinion sucha discrepancy arises because of the infinite small resonancewidth approximation which was used in BB [4], when beforetheir equation (28) they replaced at an intermediate stage ofcalculations the Breit–Wigner expression with a delta function.Moreover, when the Coulomb interaction is neglected, underthis delta approximation the Fano factor found by BB [4]follows from equation (8) of our work [1] (ξ  −1). Inother words, because of this delta approximation BB simplyneglected the voltage interval where the Fano factor is below1/2.Concerning the reasonability of the assumptions made, weconfirm the use of energy independent tunnelling rates. Wealso confirm that current and noise are never exactly zero in thepresence of an applied voltage. This is clear by carrying outa numerical calculation of the transparency for the structuredrawn in the Comment. There is always nonzero transparencyfor electrons with energy above the band edge even when theresonant level is below it. Thus, the conclusion concerning theabsence of the current and noise made by BB and used in theirequation (14) [4] when the resonant level is below the bandedge of the emitter is again the result of an approximationand does not correspond to a rigorous quantum mechanicaldescription of tunnelling. Therefore, we disagree that byincluding an energy dependent tunnelling rate there are regionsof applied voltages with exactly zero current and zero noise,unless one is forcing this result. We also disagree that finding aFano factor below 1/2 is strictly related to this approximation.As it is clear from figure 2 of our paper [1], there is wide regionup to ξ = −10 for f = ∞ and f = 15 where Fano factor isbelow 1/2. In this region the resonant level is situated higherthan the band edge of the emitter and the energy differencebetween them is greater than , thus the influence of the edgeon L is neglibible in calculating the transparency. It meansthat the Fano factor can drop below 1/2 in the case of an energydependent L,R if in any point of the mentioned region of ξvalues L = R. As stated in the letter, we believe that in thecoherent regime it is the presence of the (1-D) factor (D beingthe transparency of the device) which allows for a suppressionbelow 1/2, in analogy with the original findings of Lesovik[8] for the case of a point contact.We agree that the simplicity of the model can be a sourceof discrepancy between experiments and theory, and that moreinvestigations are needed to shed new light on the subject. Weare presently engaged in this effort.Concerning the inclusion of Coulomb effects, wepresented as equation (6) in [1], we never found negativevalues of the current spectral density. On the other hand,we agree that the role of Coulomb interaction in general, andat zero temperature in particular, should be reconsidered moreaccurately than reported in equation (6). To this purpose weare working out a theoretical model following a full quantummechanical approach [9]. In any case, at present we have notfound results which contradict the essential findings reportedin the letter.AcknowledgmentsPartial support from the Italian Ministry of Foreign Affairs(MAE) through the Volta Landau Center (the fellowship ofVYaA) and from the cofin-03 project ‘Modelli e misure dirumore in nanostrutture’ financed by the Italian Ministryfor Education University and Research (MIUR) is gratefullyacknowledged.References[1] Aleshkin V Ya et al 2003 Semicond. Sci Technol. 18 L35[2] Iannaccone G, Macucci M and Pellegrini B 1997 Phys. Rev. B55 4539[3] Aleshkin V Ya and Reggiani L 2001 Phys. Rev. B 64 245333[4] Blanter Y M and Bu¨ttiker M 1999 Phys. Rev. B 59 10217[5] Blanter Y M and Bu¨ttiker M 2000 Phys. Rep. B 336 1[6] Averin D V 1993 J. Appl. Phys. 73 2593[7] Wei Y, Wang B, Wang J and Guo H 1999 Phys. Rev. B 6016900[8] Lesovik G B 1989 JETP Lett. 49[9] Aleshkin V Ya et al 2003 Preprint cond-mat/0304077666

https://sapientia.ualg.pt/bitstream/10400.1/1193/1/Artigo%202003%20Giant%20suppression%20of%20shot%20noise%20in%20double%20barrier%20resonant%20diode.pdf

Giant suppression of shot noise in double barrier resonant diode: a signature of coherent transport

Abstract

Similar works

Full text

Available Versions

Universidade do Algarve

Universidade do Algarve

Sapientia

Sapientia