Sequential mechanism of electron transport in the resonant tunneling diode with thick barriers
ABSTRACT A frequency-dependent impedance analysis (0.1–50 GHz) of an InGaAs/InAlAs-based resonant tunneling diode with a 5-nm-wide
well and 5-nm-thick barriers showed that the transport mechanism in such a diode is mostly sequential, rather than coherent,
which is consistent with estimates. The possibility of determining the coherent and sequential mechanism fractions in the
electron transport through the resonant tunneling diode by its frequency dependence on the impedance is discussed.
Article: Tunneling in a finite superlattice[show abstract] [hide abstract]
ABSTRACT: We have computed the transport properties of a finite superlattice from the tunneling point of view. The computed I‐V characteristic describes the experimental cases of a limited number of spatial periods or a relatively short electron mean free path.Applied Physics Letters 07/1973; · 3.79 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Capacitances in a double-barrier tunneling structure are calculated for the specific sequential electron tunneling regime. Starting from Luryi's (1988) definition of quantum capacitance, the authors model the charge accumulation in the well during the tunneling process using the Fermi-Dirac distribution. Analytical formulas for the total capacitance and conductance are derived. A complete small-signal model is proposed that demonstrates the external capacitance and conductance of the structure and its frequency behavior. The authors show both theoretically and experimentally that the capacitance in a tunneling structure is both bias- and frequency-dependentIEEE Transactions on Electron Devices 10/1991; · 2.06 Impact Factor
ISSN 1063-7826, Semiconductors, 2007, Vol. 41, No. 2, pp. 227–231. © Pleiades Publishing, Ltd., 2007.
Original Russian Text © N.V. Alkeev, S.V. Averin, A.A. Dorofeev, P. Velling, E. Khorenko, W. Prost, F.J. Tegude, 2007, published in Fizika i Tekhnika Poluprovodnikov, 2007,
Vol. 41, No. 2, pp. 233–237.
PHYSICS OF SEMICONDUCTOR
Recently, interest in semiconductor mesoscopic
structures has significantly increased . First of all,
this circumstance is caused by the development of
semiconductor technology allowing the fabrication of
structures with sizes on the order of a few and tens of
nanometers. In such structures, the electron de Broglie
wavelength exceeds the structure size, and the transport
of electrons is mainly controlled by their wave proper-
ties, which results in a large variety of new effects .
These effects disappear if the dephasing time of the
electron wave function is much shorter than the elec-
tron transit time through a structure. The study of
mechanisms of carrier transport in mesoscopic struc-
tures is an important fundamental problem.
One mesoscopic structure is the resonant tunneling
diode (RTD) suggested for the first time by Esaki and
Tsu ; this is one of the first nanoelectronic devices
. It consists of a narrow-gap semiconductor layer,
i.e., quantum well (QW), arranged between two semi-
conductor layers (barriers) with a wider band gap.
These layers are in turn arranged between layers (spac-
ers) of a lightly doped narrow-gap semiconductor fol-
lowed by heavily doped emitter and collector layers.
One or several size-quantization levels arise in the QW.
As a bias voltage is applied, the current through the
RTD flows only when the emitter contains electrons
which can resonantly (i.e., with conservation of energy
and transverse momentum) be tunneled to the QW level
and further to the collector. The RTD features a very
fast response; e.g., it is known that its nonlinear proper-
ties are retained up to ~10 THz . The RTD has other
unique properties: in particular, it is the only nanoelec-
tronic device operating at room temperature, and its
) characteristic contains negative
differential conductance (NDC) portions.
Initially , it was assumed that the electron trans-
port through the RTD is coherent, i.e., electrons pass
through RTD barriers in the same way as the light wave
passes through the Fabry–Perot cavity: the barrier
transparency resonantly increases when the energy of
electrons flying from the emitter to the barrier
approaches the energy of one of the size quantization
levels in the RTD QW. Another mechanism of the elec-
tron transport in the RTD was proposed in . This
mechanism is also resonant, but incoherent; it is often
referred to as the “sequential” one. According to this
mechanism, the electron tunnels through the RTD in
two stages: initially from the emitter through the first
barrier to the QW level, and then from this level
through the second barrier to the collector. It is assumed
in this case that both tunneling events are independent,
i.e., the information on the initial phase of the electron
wave function is completely lost due to inelastic colli-
sions for the time of electron residence in the QW. In
, it was noted that the classification of resonant tun-
neling into coherent and incoherent has no physical
meaning, since a certain degree of coherence is
required to discretize the electron spectrum in the well,
i.e., the dephasing time of the wave function should
exceed the double electron time of flight through the
well. Otherwise, counterpropagating electron waves in
the cavity are incoherent, and the discrete electron
Sequential Mechanism of Electron Transport in the Resonant
Tunneling Diode with Thick Barriers
N. V. Alkeev ^, S. V. Averin, A. A. Dorofeev
, W. Prost
Institute of Radio Engineering and Electronics, Russian Academy of Sciences (Fryazino Branch),
pl. Vvedenskogo 1, Fryazino, Moscow oblast, 141190 Russia
Pulsar Research Institute, Moscow, 105187 Russia
Solid State Electronics Department, Gerhard-Mercator-University, 47057 Duisburg, Germany
Submitted June 6, 2006; accepted for publication June 15, 2006
, P. Velling
, and F. J. Tegude
tunneling diode with a 5-nm-wide well and 5-nm-thick barriers showed that the transport mechanism in such a
diode is mostly sequential, rather than coherent, which is consistent with estimates. The possibility of determin-
ing the coherent and sequential mechanism fractions in the electron transport through the resonant tunneling
diode by its frequency dependence on the impedance is discussed.
PACS numbers: 73.40.Gk, 85.30.Mn
—A frequency-dependent impedance analysis (0.1–50 GHz) of an InGaAs/InAlAs-based resonant
ALKEEV et al.
energy spectrum becomes continuous. It was also noted
that the tunneling “coherence” is significantly affected
by the nonidentity and inhomogeneity of the entrance
and exit barriers over the RTD area. In the case of non-
identical barriers, different amplitudes of interfering
electron waves cause a decrease in the wave function
amplitude in the QW and hence, a decrease in the trans-
mittance. We note that the nonidentity of RTD barriers
will have a much stronger effect in thin barriers, rather
than in thick ones whose reflectance is close to unity.
In one recent paper  devoted to an analysis of
electron transport mechanisms in the RTD, it was pro-
posed to distinguish three mechanisms: “coherent,”
“sequential,” and “completely incoherent.” In the
coherent mechanism, the major fraction of electrons
tunnel through the RTD, retaining their phases. This
mode is best described by the model proposed in . In
the sequential mechanism, almost all electrons lose
their phase memory and are thermalized in the QW. In
this case, the resonant levels in the QW are strictly
determinate. To describe this transport mechanism, the
model proposed in  is most adequate. Finally, in the
case of the completely incoherent transport mecha-
nism, the dephasing time of the electron wave function
tends to zero, and the QW resonant levels disappear,
while the density of states in the QW tends to the bulk
one. Moreover, the NDC portions in the
istic of the RTD also disappear .
The calculations of  showed that both coherent
and sequential mechanisms of the electron transport in
the RTD result in the same
i.e., it is impossible to determine the electron transport
mechanism implemented in an individual RTD by ana-
characteristics. An interesting method
was proposed in , in which the total current through
the RTD contains a certain coherent current fraction. It
was known that the spectral density of the shot noise
during sequential electron tunneling through two barri-
ers cannot be lower than half the Poisson value
However, the calculations of  showed that the spec-
tral density of shot noise can be much lower than
ing the coherent electron transport through the RTD.
The experimental data  on shot noises measured in
RTDs with thin (2 nm) barriers showed that there is a
coherent component in the total RTD current.
In this study, we propose a method, based on fre-
quency-dependent impedance analysis, for detecting
the current fraction caused by the sequential transport
mechanism in the total RTD current. The essence of the
method is as follows. As noted above, in the sequential
transport mechanism, the electron tunnels through barri-
ers independently; therefore, while an electron tunnels
through one barrier, a bias current should flow through
another barrier, which provides the total-current continu-
ity . Measurement of this bias current allows one to
judge the contribution of the sequential transport mecha-
nism to the total current through the RTD.
2. ANALYSIS OF EXPERIMENTAL RESULTS 
In the ultrahigh frequency (microwave) range, it is
common practice to characterize semiconductor
devices by small-signal equivalent circuits (ECs). Such
an approach allows better understanding of carrier
transport mechanisms in various semiconductor
devices, as well as optimum design of various micro-
wave devices. Based on small-signal ECs of the RTD,
nonlinear ECs are developed.
A large number of RTD ECs have been proposed
[13–18]; however, all these ECs are based on the
sequential electron transport mechanism. We have not
found in the available publications an RTD EC which
would describe its high-frequency properties for the
coherent transport mechanism; however, we believe
that the high-frequency properties of the RTD can be
best represented by the EC in this case (Fig. 1). Indeed,
in the coherent-transport mechanism, electrons tunnel
through RTD barriers as if through a single whole;
therefore, the differential conductance of a structure
can be characterized by the quantity
the geometric capacitance between the heavily doped
emitter and collector of the RTD, and
contact resistance. We note that the EC shown in Fig. 1
is the EC of the conventional tunneling diode and the
Schottky barrier diode.
It is also worth noting that a number of ECs describ-
ing the high-frequency properties of the RTD for the
sequential mechanism of the electron transport involve
an inductance [13–15]. For example, the inductance
was introduced in  to explain specific features in the
frequency dependence of the RTD impedance. In
[14, 15], the inductance simulated delay of electrons in
the RTD QW with current flowing through it. In ,
an RTD EC consisting of two parallel
nected in series was suggested, where the impedance
can depend on frequency. Each
one of the RTD barriers. This EC was further developed
in [17, 18].
Figure 2 shows the EC from , which is almost
identical to that of . The conductances
this EC are tunneling conductances of the first and sec-
ond barriers. The capacitances
ble to take into account the displacement currents
. In Fig. 1,
is the ohmic
make it possi-
coherent transport mechanism.
High-frequency EC of the RTD in the case of the
SEQUENTIAL MECHANISM OF ELECTRON TRANSPORT 229
through the first and second barriers, while electrons
tunnel through the second and first barriers, respec-
tively. According to , the capacitance
the charge accumulation in the RTD QW. Finally, in
, it was proposed to describe the RTD impedance
using a functional dependence, which cannot be associ-
ated with any EC.
Due to the large number of proposed RTD ECs, the
problem arises to select an EC best describing the mea-
sured frequency-dependent impedance of an individual
RTD. In , to select an EC, we proposed to use the
nonlinear regression method. The essence of this
method is as follows. At all points
range, at which the RTD impedance
sured, the impedance
) of the assumed RTD EC
was calculated. Then the goal function
of the frequency
) was mea-
By varying the parameters of EC components, the
of function (1) was determined and the
values of the components at which
to be the values of RTD EC components. The degree of
agreement between the measured impedance and the
impedance of the assumed RTD EC can be judged from
, as well as visually by the degree of agreement
between the curves of the measured RTD impedance
and the impedance of the assumed EC in the Smith dia-
gram . To more reasonably judge which theoretical
dependence better fits the experimental curve, we can
use the methods of mathematical statistics. In cases
similar to that under consideration, the Pearson (
terion is most frequently used .
In , at various bias voltages, we measured the
RTD impedance at 401 points uniformly distributed in
the frequency range of 0.1–50 GHz. The RTD under
study contained InAlAs barriers 5 nm thick, its InGaAs
QW width was also 5 nm. The experiment is described
in more detail in . Figure 3 shows a part of the
Smith diagram with three frequency dependences of the
is the experimental frequency
dependence of the RTD impedance at a bias of 0.3 V
is the frequency dependence of the imped-
ance for the EC of Fig. 1 at optimum values of its com-
ponents (i.e., at
characteristic difference between these two curves, i.e.,
the coherent mechanism does not quite adequately
describe the electron transport in the RTD under study.
Note that we analyzed the influence of the barrier inho-
mogeneity over the RTD area on the shape of the fre-
quency dependence of the RTD impedance in the case
of the coherent electron transport. The analysis per-
formed showed that the changes in the curve shape are
insignificant and lie within the experimental error.
We can see in Fig. 3 that the measurement accuracy
allows us to select a better EC for the RTD under study
than the EC shown in Fig. 1; therefore, we searched for
for all known (to us) RTD EC to find the best EC
version. For ECs containing an inductance [13–15], it
was found that the inductance
ECs with inductances were reduced to the EC shown in
Fig. 1 and featured the same
conclusion that these ECs describe the high-frequency
properties of the RTD under study with inadequate
accuracy. Satisfactory agreement with the experimental
curve at all bias voltages at the RTD was obtained for
ECs given in [16–18].
in Fig. 3 is the frequency dependence of the
impedance for the EC of  at
fits experimental curve
(see Fig. 2) is more than ten times smaller than
the EC in Fig. 1.
The table lists the values of the EC components
shown in Fig. 2 at
which allows the conclusion that circuits
describe the emitter and collector barriers, respectively.
Indeed, an electron-enriched layer is formed near the
emitter, and the emitter–QW capacitance increases. In
). We can see that there is a
. This fact allows the
for the EC of 
much better than
. We can see
. We can see that
RTD EC from .
0.1 GHz0.1 GHz0.1 GHz
3 3 3
2 2 2
1 1 1
50 GHz50 GHz50 GHz
range of 0.1–50 GHz: (
dependence of RTD impedance at a bias of 0.3 V; (
dependence of the impedance of the optimized RTD EC
shown in Fig. 1; and (
) the dependence of the impedance
of the RTD EC shown in Fig. 2.
Frequency dependences of the impedance in the
) the experimental frequency
Vol. 41 No. 2 2007
ALKEEV et al.
contrast, there is an electron-depleted layer near the
collector; therefore, the collector–QW capacitance
decreases. The root-mean-square deviations of the
parameters of EC components can be calculated in the
same way as in .
The closely related values of Gmin for ECs of
[16, 18] can be explained by the fact that these ECs dif-
fer by the components connecting points a and b in
Fig. 2 (in , it is a capacitance; in , points a and
b are connected). If the potentials of points a and b are
close to each other (this situation occurs when both
RTD barriers are identical), the ECs of [16, 18] are
almost identical and the accuracy of these experiments
does not allow detection of difference in Gmin of these
Taking into account the fact that the electron trans-
port in the RTD can simultaneously proceed by coher-
ent and sequential mechanisms, we also minimized the
goal function for the EC consisting of two parallel
branches that describe the coherent (circuit of Fig. 1)
and sequential (circuit of Fig. 2) transport. An analysis
showed that Gmin in this case almost does not differ
from Gmin for the EC shown in Fig. 2. To detect an
appreciable difference between the minimums of goal
functions in the first and second cases, hence, to deter-
mine the fractions of coherent and sequential transport
mechanisms in the current flowing through the RTD,
the impedance measurement accuracy should be
improved, and the frequency measurement range
should be extended, which is quite achievable for mod-
ern measuring instruments.
Thus, the EC corresponding to the sequential trans-
port mechanism  describes the frequency depen-
dence of the RTD impedance much better than the EC
corresponding to the coherent mechanism of electron
transport. On this basis, it can be concluded that the
major fraction of electrons tunnel in the studied RTD by
the sequential mechanism.
Let us consider some estimates confirming that the
sequential mechanism of the electron transport is dom-
inant in the RTD under study. As noted in , the
sequential mechanism of electron transport in the RTD
takes place if the condition
is satisfied. Here, τ0 is the doubled electron time of
flight through the well; τp is the dephasing time of the
electron wave function, which is identified with the
time of electron free path in ; and τn is the so-called
“radiative” lifetime, or the electron lifetime with
respect to electron tunneling from the well. The first
inequality expresses the condition for the formation of
resonant levels in the QW. At the well width a, τ0 =
2a/ν1, where ν1 = (2ε1/m*)1/2 is the velocity of trans-
verse electron motion at the first resonant level in a well
with the energy ε1. For the studied RTD, ε1 = 70 meV
and ν1 = 6.1 × 105 m/s, from which it follows that τ0 ≈
1.7 × 10–14 s. According to estimations, τp ≈ 10–12 s in
InGaAs at room temperature. We note that the hetero-
interface roughness was disregarded in this case. First
of all, this roughness is caused by the imperfection of
the technology of RTD layer growth and results in a
decrease in the dephasing time. Therefore, τp is an
upper-bound estimate in the case under consideration.
If the second inequality in (2) is not met, a major frac-
tion of electrons will coherently tunnel through the
RTD barriers. The time τn can be estimated by the for-
mulas given in : τn ≈ 5 × 10–10 s in the case under
consideration. Thus, the estimations show that the
sequential mechanism of electron transport is mostly
implemented in the studied RTD.
(i) A frequency-dependent impedance analysis of
the InGaAs/InAlAs-based RTD with rather thick
(5 nm) barriers showed that the RTD equivalent circuit
corresponding to the coherent mechanism of electron
transport, as well as all known equivalent circuits cor-
responding to the sequential mechanism and containing
an inductance, are not applicable to describing the
high-frequency properties of the studied RTD.
(ii) Good agreement with experiment was attained
for the equivalent circuit corresponding to the sequen-
tial mechanism, in which each RTD barrier is described
by a parallel RC circuit. Thus, the electron transport in
the studied RTD occurs mostly by the sequential mech-
anism. Calculations confirmed the validity of this con-
(iii) It was attempted to determine the fractions of
the coherent and sequential mechanisms in the total
electron current through the RTD; however, the accu-
racy of the performed experiments was not adequate.
This study was supported in part by the Russian
Foundation for Basic Research, project no. 04-02-17177.
1. C. W. J. Berenakker and H. van Houten, Solid State
Phys. 44, 1 (1991).
2. R. Tsu and L. Esaki, Appl. Phys. Lett. 22, 562 (1973).
3. Technology Roadmap for Nanoelectronics, European
Commission IST Programme Future and Emerging
Technologies, 2nd ed. (2000).
4. W. R. Frensley, Appl. Phys. Lett. 51, 448 (1987).
5. S. Luryi, Appl. Phys. Lett. 47, 490 (1985).
0.55 1800.098 1160.34 21
SEMICONDUCTORS Vol. 41 No. 2 2007
SEQUENTIAL MECHANISM OF ELECTRON TRANSPORT 231
6. A. S. Tager, Élektron. Tekh., Ser. 1: Elektron. SVCh,
No. 9 (403), 21 (1987).
7. G. Iannaccone and B. Pellegrini, Phys. Rev. B 52, 17406
8. Y. Hu and S. Stapleton, J. Appl. Phys. 73, 8633 (1993).
9. T. Weil and B. Vinter, Appl. Phys. Lett. 50, 1281 (1987).
10. V. Ya. Aleshkin, L. Reggiani, N. V. Alkeev, et al., Phys.
Rev. B 70, 115321 (2004).
11. Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1
12. N. V. Alkeev, V. E. Lyubchenko, P. Velling, et al.,
Radiotekh. Élektron. (Moscow) 49, 886 (2004) [J. Com-
mun. Technol. Electron. 49, 833 (2004)].
13. J. M. Gering, D. A. Crim, D. G. Morgan, and P. D. Cole-
man, J. Appl. Phys. 61, 271 (1987).
14. E. R. Brown, C. D. Parker, and T. C. L. G. Solner, Appl.
Phys. Lett. 54, 934 (1989).
15. M. N. Feiginov, Appl. Phys. Lett. 78, 3301 (2001).
16. F. W. Sheard and G. Toombs, Solid-State Electron. 32,
17. J. Genoe, S. Stapleton, and O. Berolo, IEEE Trans. Elec-
tron Devices 38, 2006 (1991).
18. J. P. Mattia, A. L. McWhorter, R. J. Aggarwal, et al.,
J. Appl. Phys. 84, 1140 (1998).
19. H. P. Joosten, H. J. M. F. Noteborn, K. Kaski, and
D. Lenstra, J. Appl. Phys. 70, 3141 (1991).
20. V. F. Fusco, Microwave Circuits: Analysis and Com-
puter-Aided Design (Prentice-Hall, Englewood Cliffs,
N.J., 1987; Radio i Svyaz’, Moscow, 1990).
21. S. L. Akhnazarova and V. V. Kafarov, Optimization of
Experiments in Chemistry and Chemical Technology
(Vysshaya Shkola, Moscow, 1978) [in Russian].
Translated by A. Kazantsev
SPELL: 1. dephasing, 2. thermalized