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 the coherent transport regime and
can occur near zero temperature as a consequence of the Pauli principle or
above about 77 K as a consequence of long range Coulomb interaction. These predictions are validated by 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. 19 (2004) S161–S163 PII: S0268-1242(04)72144-6
Giant suppression of shot noise as
signature of coherent transport in double
barrier resonant diodes
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 & Electronics, Russian Academy of Sciences, Vernadsky
Prospect, Moscow 117454, Russia
4 Department of Electronic and Electrical Engineering, University of Glasgow, Ranking
Building, Dakfield Avenue, Glasgow G12 8LT, UK
Received 28 July 2003
Published 3 March 2004
Online at stacks.iop.org/SST/19/S161 (DOI: 10.1088/0268-1242/19/4/056)
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 the coherent transport regime and
can occur near zero temperature as a consequence of the Pauli principle or
above about 77 K as a consequence of long range Coulomb interaction.
These predictions are validated by experimental data.
1. Introduction
Since its realization, the double barrier resonant diode (DBRD)
has proved to be an electron device of broad physical interest
because of its peculiar non-Ohmic current voltage (I–V)
characteristic [1, 2]. Even the shot noise characteristics are
of relevant interest due to the fact that suppressed as well as
enhanced shot noise with respect to its full Poissonian value
has been observed [3]. The microscopic interpretation of
these features is found to admit a coherent [1] or a sequential
tunnelling [2] approach. The intriguing feature of these two
approaches is that from the existing literature it emerges that
both of them are capable of explaining the I–V experiments as
well as most of the shot noise characteristics [3, 4]. 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.
Here we answer this question by announcing that a giant
suppression of shot noise occurring before the peak value of
the current with a Fano factor (noise power to current ratio)
below 0.5 is a signature of coherent transport in DBRDs. To
this purpose, we have formulated a coherent approach for
transport in DBRDs that includes both the Pauli principle and
the long range Coulomb interaction.
2. Model and results
The typical structure investigated is reported in figure 1. We
denote by  = L + R the resonant state width due to the
tunnelling through left and right barriers, respectively, and by
εr the energy of the resonant level as measured from the centre
of the potential well. 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 assumes that the electron distribution functions in the
emitter and in the collector of the DBRT are Fermi Dirac
equilibrium-like, with electro-chemical potentials shifted by
the value of the applied voltage. For convenience calculations
are carried out using the CGS system. The double barrier
transparency D(εz) is given by [3]
D(εz) = 
2/4
(εz − εr + qu)2 + 2/4 (1)
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
[5]:
I = − qm
2π2h¯3
∫ ∞
−gL
∫ ∞
0
dεz dε⊥ D(εz)[fL(ε) − fR(ε)] (2)
0268-1242/04/040161+03$30.00 © 2004 IOP Publishing Ltd Printed in the UK S161
V Ya Aleshkin et al
quL
qu quR
qV
gL
dQW 
dL
dR
εr
L
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 coincide at V = 0.
with q the electron charge, m its effective mass and h¯ the
reduced Planck constant. In our model, we suppose that in
the emitter there are no electron states with energy below −gL
and that electron states with energy higher than this value are
three dimensional. Furthermore, the relation between u and
the total applied voltage V is calculated self-consistently by
using the condition of electroneutrality of the device [5]. To
calculate noise we implemented the wave packet approach [6]
to account for the self-consistent potential. Accordingly, for
the spectral density of current fluctuations we obtain: [5]
SI = q
2m
π2h¯3
∫ ∞
0
dε⊥
∫ ∞
−gL
dεz
{
D
[(
q+eff
)2
fL(1 − fR)
+ (q−eff)
2fR(1 − fL)
] − D2(fL − fR)2} (3)
with
∣∣q±eff
∣∣ < ∞ a dimensionless effective charge accounting
for the self-consistent potential. When q±eff = 1 the self-
consistent potential is neglected, and equation (3) coincides
with that derived by [6]. From equations (2) and (3) the Fano
factor γ = SI /(2qI) is directly obtained.
We compare theory with two sets of experiments
performed on DBRDs with barriers sufficiently narrow to
expect that coherent tunnelling is of importance. In both cases
the comparison is limited to the voltage region up to the peak
current since theory neglects energy levels higher than the first
one in the quantum well.
Figure 2 reports the experiments of [7] at 77 K and the
results of present calculations at 77 and 4.2 K. Numerical
results make use of the following values for the parameters
entering the model: εr = 104 meV, R = L = 0.48 meV,
carrier concentration at the contacts n = 2 × 1016 cm−3,
same width of the left barrier, right barrier and quantum
well dL = dR = dQW = 5 nm, width of the spacer in the
collector L = 50 nm, m = 0.067m0, with m0 the free electron
mass and κ = 12.9. The only two fitting parameters are εr
and  = 0.5L,R. Their values control the location and the
amplitude of the current peak, respectively, and are chosen
by optimizing the agreement between the experimental and
the calculated I–V characteristics. All other parameters are
provided by the experimental conditions, and we want to stress
that the Fano factor is obtained from the same set of parameter
values. The function gL(uL), uL being the voltage drop in
0
2
4
6
8
(a)
experiment 77 K
theory 77 K
theory 4.2 K
I (m
A)
γ
0.0 0.2 0.4 0.6
0.1
1
(b)
 Voltage (V)
Figure 2. Experimental and calculated dependences of current
(a) and the Fano factor (b) on applied voltage for the structure in [7].
Experiments refer to T = 77 K and theory to T = 4.2 and 77 K,
respectively.
0
10
20
30
40
50
60 (a)
experiment
theory
I(m
A)
0.00 0.25 0.50 0.75
0.1
1
(b)
 Voltage (V)
γ
Figure 3. Experimental and calculated dependences of current
(a) and the Fano factor (b) on applied voltage for the structure in [5]
at T = 300 K.
front of the left barrier edge, is calculated by solving the
Schro¨dinger equation. For the values used, the dependence
gL(uL) is found to be almost linear and well approximated
S162
Giant suppression of shot noise as signature of coherent transport in double barrier resonant diodes
by gL(uL) ≈ 0.44(quL − 1.5kBT ) − 0.07kBT for 77 K
and gL(uL) ≈ 0.28(quL − 110kBT ) − 14kBT for 4.2 K.
The agreement between theory and experiments reported in
figure 2 is considered to be satisfactory. We remark that
numerical results at 4.2 K do not differ significantly from
those at 77 K in agreement with what is claimed in [7]. Thus,
the giant suppression of shot noise predicted by present theory
extends the previous findings of Wei et al [8] and is confirmed
by these experiments.
A recent set of experiments [5] carried out at 300 K on
a DBRD with barriers thinner than those of [7], thus more
adequate to check coherent tunnelling at high temperature,
is reported in figure 3 together with theoretical calculations.
The structure consisted of two 2 nm AlAs layers separated by
6 nm InGaAs quantum well [9]. Measurements were carried
out at 300 K using a noise figure meter (XK5-49), that allows
us to measure simultaneously noise figure and power gain
of two-port networks in the 50  feed circuit. Numerical
results make use of the following values for the parameters
entering the model: εr = 93 meV, R = L = 0.9 meV,
n = 7.5 × 1016 cm−3, dL = dR = 2 nm, dQW = 6 nm
L = 50 nm, gL(uL) = 0.43(quL − 1.5kBT ) + 0.44kBT . Also
here the only two fitting parameters are εr and  = 0.5L,R,
all other parameters being provided by the experimental
conditions. Figure 3(a) reports the I–V characteristics which
show a region of positive differential conductance (PDC) up
to about 0.7 V followed by an instability region. In the same
PDC region, the Fano factor is found to exhibit a suppression
with a minimum value of about 0.4 at around 0.6 V (see
figure 3(b)). The agreement between theory and experiments
is within experimental uncertainty and thus is considered to be
satisfactory even in this case. In particular, here the Coulomb
interaction is found to be determinant in leading to the Fano
factor below the value of 0.5. 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 transport occurs under such a condition.
3. Conclusions
We have investigated coherent tunnelling by implementing
the wave packet approach for transport in DBRDs that
includes both the Pauli principle and the long range Coulomb
interaction. In agreement with existing results, we have
found that at 4.2 K shot noise is suppressed mostly because
of the Pauli correlation. Moreover, the suppression exhibits
the Fano factor of 0.5 in a wide region of applied voltages,
with a minimum of 0.391 at 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 existing experiments at 77 K and recent
measurements performed at room temperature. Therefore,
shot noise suppression below one-half of the full Poissonian
value is proved to be a signature of coherent tunnelling
against sequential tunnelling in double barrier resonant diodes.
We finally want to stress that the main reason for the
difference between these approaches stems from the fact that
the sequential tunnelling is based on a master equation [4]
for treating fluctuations of carrier numbers inside the quantum
well while coherent tunnelling uses the quantum partition noise
within the wave packet approach [6]. 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 such as in
vacuum diodes.
Acknowledgments
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] Chang L L, Esaki L and Tsui R 1974 Appl. Phys. Lett. 24 593
[2] Luryi S 1985 Appl. Phys. Lett. 47 490
[3] Blanter Y M and Bu¨ttiker M 2000 Phys. Rep. 336 1
[4] Iannaccone G, Macucci M and Pellegrini B 1997 Phys. Rev. B
55 4539
[5] 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
[6] Martin T and Landauer R 1992 Phys. Rev. B 45 1742
[7] Brown E R 1992 IEEE Trans. Electron. Devices 39 2686
[8] Wei Y, Wang B, Wang J and Guo H 1999 Phys. Rev. B 60 16900
[9] Alkeev N V, Lyubchenko V E, Ironside C N, Figueiredo J M L
and Stanley C R 2002 J. Commun. Technol. Electron. 47 228
S163


Giant suppression of shot noise as signature of coherent transport in double barrier resonant diodes

Sapientia

INSTITUTE OF PHYSICS PUBLISHING SEMICONDUCTOR SCIENCE AND TECHNOLOGYSemicond. Sci. Technol. 19 (2004) S161–S163 PII: S0268-1242(04)72144-6Giant suppression of shot noise assignature of coherent transport in doublebarrier resonant diodesV 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 & Electronics, Russian Academy of Sciences, VernadskyProspect, Moscow 117454, Russia4 Department of Electronic and Electrical Engineering, University of Glasgow, RankingBuilding, Dakfield Avenue, Glasgow G12 8LT, UKReceived 28 July 2003Published 3 March 2004Online at stacks.iop.org/SST/19/S161 (DOI: 10.1088/0268-1242/19/4/056)AbstractShot 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 the coherent transport regime andcan occur near zero temperature as a consequence of the Pauli principle orabove about 77 K as a consequence of long range Coulomb interaction.These predictions are validated by experimental data.1. IntroductionSince its realization, the double barrier resonant diode (DBRD)has proved to be an electron device of broad physical interestbecause of its peculiar non-Ohmic current voltage (I–V)characteristic [1, 2]. Even the shot noise characteristics areof relevant interest due to the fact that suppressed as well asenhanced shot noise with respect to its full Poissonian valuehas been observed [3]. The microscopic interpretation ofthese features is found to admit a coherent [1] or a sequentialtunnelling [2] approach. The intriguing feature of these twoapproaches is that from the existing literature it emerges thatboth of them are capable of explaining the I–V experiments aswell as most of the shot noise characteristics [3, 4]. Therefore,to our knowledge there is no way to distinguish betweenthese two transport regimes and the natural question whetherthe tunnelling transport is coherent or sequential remains anunsolved one.Here we answer this question by announcing that a giantsuppression of shot noise occurring before the peak value ofthe current with a Fano factor (noise power to current ratio)below 0.5 is a signature of coherent transport in DBRDs. Tothis purpose, we have formulated a coherent approach fortransport in DBRDs that includes both the Pauli principle andthe long range Coulomb interaction.2. Model and resultsThe typical structure investigated is reported in figure 1. Wedenote by  = L + R the resonant state width due to thetunnelling through left and right barriers, respectively, and byεr the energy of the resonant level as measured from the centreof the potential well. For simplicity, we consider the case ofcoherent tunnelling when there is only one resonant state withL = R = /2 and we take unit square contacts. The kineticmodel assumes that the electron distribution functions in theemitter and in the collector of the DBRT are Fermi Diracequilibrium-like, with electro-chemical potentials shifted bythe value of the applied voltage. For convenience calculationsare carried out using the CGS system. The double barriertransparency D(εz) is given by [3]D(εz) = 2/4(εz − εr + qu)2 + 2/4 (1)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 found[5]:I = − qm2π2h¯3∫ ∞−gL∫ ∞0dεz dε⊥ D(εz)[fL(ε) − fR(ε)] (2)0268-1242/04/040161+03$30.00 © 2004 IOP Publishing Ltd Printed in the UK S161V Ya Aleshkin et alquLqu quRqVgLdQW dLdRεrLFigure 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 coincide at V = 0.with q the electron charge, m its effective mass and h¯ thereduced Planck constant. In our model, we suppose that inthe emitter there are no electron states with energy below −gLand that electron states with energy higher than this value arethree dimensional. Furthermore, the relation between u andthe total applied voltage V is calculated self-consistently byusing the condition of electroneutrality of the device [5]. Tocalculate noise we implemented the wave packet approach [6]to account for the self-consistent potential. Accordingly, forthe spectral density of current fluctuations we obtain: [5]SI = q2mπ2h¯3∫ ∞0dε⊥∫ ∞−gLdεz{D[(q+eff)2fL(1 − fR)+ (q−eff)2fR(1 − fL)] − D2(fL − fR)2} (3)with∣∣q±eff∣∣ < ∞ a dimensionless effective charge accountingfor the self-consistent potential. When q±eff = 1 the self-consistent potential is neglected, and equation (3) coincideswith that derived by [6]. From equations (2) and (3) the Fanofactor γ = SI /(2qI) is directly obtained.We compare theory with two sets of experimentsperformed on DBRDs with barriers sufficiently narrow toexpect that coherent tunnelling is of importance. In both casesthe comparison is limited to the voltage region up to the peakcurrent since theory neglects energy levels higher than the firstone in the quantum well.Figure 2 reports the experiments of [7] at 77 K and theresults of present calculations at 77 and 4.2 K. Numericalresults make use of the following values for the parametersentering the model: εr = 104 meV, R = L = 0.48 meV,carrier concentration at the contacts n = 2 × 1016 cm−3,same width of the left barrier, right barrier and quantumwell dL = dR = dQW = 5 nm, width of the spacer in thecollector L = 50 nm, m = 0.067m0, with m0 the free electronmass and κ = 12.9. The only two fitting parameters are εrand  = 0.5L,R. Their values control the location and theamplitude of the current peak, respectively, and are chosenby optimizing the agreement between the experimental andthe calculated I–V characteristics. All other parameters areprovided by the experimental conditions, and we want to stressthat the Fano factor is obtained from the same set of parametervalues. The function gL(uL), uL being the voltage drop in02468(a)experiment 77 Ktheory 77 Ktheory 4.2 KI (mA)γ0.0 0.2 0.4 0.60.11(b) Voltage (V)Figure 2. Experimental and calculated dependences of current(a) and the Fano factor (b) on applied voltage for the structure in [7].Experiments refer to T = 77 K and theory to T = 4.2 and 77 K,respectively.0102030405060 (a)experimenttheoryI(mA)0.00 0.25 0.50 0.750.11(b) Voltage (V)γFigure 3. Experimental and calculated dependences of current(a) and the Fano factor (b) on applied voltage for the structure in [5]at T = 300 K.front of the left barrier edge, is calculated by solving theSchro¨dinger equation. For the values used, the dependencegL(uL) is found to be almost linear and well approximatedS162Giant suppression of shot noise as signature of coherent transport in double barrier resonant diodesby gL(uL) ≈ 0.44(quL − 1.5kBT ) − 0.07kBT for 77 Kand gL(uL) ≈ 0.28(quL − 110kBT ) − 14kBT for 4.2 K.The agreement between theory and experiments reported infigure 2 is considered to be satisfactory. We remark thatnumerical results at 4.2 K do not differ significantly fromthose at 77 K in agreement with what is claimed in [7]. Thus,the giant suppression of shot noise predicted by present theoryextends the previous findings of Wei et al [8] and is confirmedby these experiments.A recent set of experiments [5] carried out at 300 K ona DBRD with barriers thinner than those of [7], thus moreadequate to check coherent tunnelling at high temperature,is reported in figure 3 together with theoretical calculations.The structure consisted of two 2 nm AlAs layers separated by6 nm InGaAs quantum well [9]. Measurements were carriedout at 300 K using a noise figure meter (XK5-49), that allowsus to measure simultaneously noise figure and power gainof two-port networks in the 50  feed circuit. Numericalresults make use of the following values for the parametersentering the model: εr = 93 meV, R = L = 0.9 meV,n = 7.5 × 1016 cm−3, dL = dR = 2 nm, dQW = 6 nmL = 50 nm, gL(uL) = 0.43(quL − 1.5kBT ) + 0.44kBT . Alsohere the only two fitting parameters are εr and  = 0.5L,R,all other parameters being provided by the experimentalconditions. Figure 3(a) reports the I–V characteristics whichshow a region of positive differential conductance (PDC) upto about 0.7 V followed by an instability region. In the samePDC region, the Fano factor is found to exhibit a suppressionwith a minimum value of about 0.4 at around 0.6 V (seefigure 3(b)). The agreement between theory and experimentsis within experimental uncertainty and thus is considered to besatisfactory even in this case. In particular, here the Coulombinteraction is found to be determinant in leading to the Fanofactor below the value of 0.5. Thus, the comparison betweentheory and experiments supports the physical interpretation ofthe giant suppression of shot noise in DBRDs and in particularconfirms that coherent transport occurs under such a condition.3. ConclusionsWe have investigated coherent tunnelling by implementingthe wave packet approach for transport in DBRDs thatincludes both the Pauli principle and the long range Coulombinteraction. In agreement with existing results, we havefound that at 4.2 K shot noise is suppressed mostly becauseof the Pauli correlation. Moreover, the suppression exhibitsthe Fano factor of 0.5 in a wide region of applied voltages,with a minimum of 0.391 at 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 existing experiments at 77 K and recentmeasurements performed at room temperature. Therefore,shot noise suppression below one-half of the full Poissonianvalue is proved to be a signature of coherent tunnellingagainst sequential tunnelling in double barrier resonant diodes.We finally want to stress that the main reason for thedifference between these approaches stems from the fact thatthe sequential tunnelling is based on a master equation [4]for treating fluctuations of carrier numbers inside the quantumwell while coherent tunnelling uses the quantum partition noisewithin the wave packet approach [6]. The master equationdescribes implicitly a sequential mechanism for a carrierentering/exiting from the well and, as a consequence, itsintrinsic limit coincides with that of two independent resistors(or vacuum diodes) connected in series and each of themexhibiting full shot noise. This system yields a maximumsuppression of shot noise down to the value of 0.5. By contrast,partition noise, inherent to the wave packet formalism, canbe fully suppressed down to zero in the presence of a fullytransparent barrier and/or of Coulomb interaction such as invacuum diodes.AcknowledgmentsPartial support from the Italian Ministry of Foreign Affairsthrough the Volta Landau Centre (the fellowship of VYaA),and the SPOT-NOSED project IST-2001-38899 of the EC isgratefully acknowledged.References[1] Chang L L, Esaki L and Tsui R 1974 Appl. Phys. Lett. 24 593[2] Luryi S 1985 Appl. Phys. Lett. 47 490[3] Blanter Y M and Bu¨ttiker M 2000 Phys. Rep. 336 1[4] Iannaccone G, Macucci M and Pellegrini B 1997 Phys. Rev. B55 4539[5] 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[6] Martin T and Landauer R 1992 Phys. Rev. B 45 1742[7] Brown E R 1992 IEEE Trans. Electron. Devices 39 2686[8] Wei Y, Wang B, Wang J and Guo H 1999 Phys. Rev. B 60 16900[9] Alkeev N V, Lyubchenko V E, Ironside C N, Figueiredo J M Land Stanley C R 2002 J. Commun. Technol. Electron. 47 228S163

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Giant suppression of shot noise as signature of coherent transport in double barrier resonant diodes

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