World Journal of Nano Science and Engineering, 2012, 2, 171-175 Published Online December 2012 (
New Model for Drain and Gate Current of Single-Electron
Transistor at High Temperature
Amine Touati, Samir Chatbouri, Nabil Sghaier, Adel Kalboussi
Laboratory of Microelectronics and Instrumentation, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia
Received May 28, 2012; revised June 6, 2012; accepted July 3, 2012
We propose a novel analytical model to describe the drain-source current as well as gate-source of single-electron tran-
sistors (SETs) at high temperature. Our model consists on summing the tunnel current and thermionic contribution. This
model will be compared with another model.
Keywords: Single-Electron Transistor (SET); Master Equation; Orthodox Theory; Tunnel Current; Thermionic Current;
1. Introduction
The phenomenal success of semiconductor electronics
during the past three decades was based on the scaling
down of silicon field effect transistors (MOSFET). The
most authoritative industrial forecast, the International
Technology Roadmap for Semiconductors (ITRS) [1]
predicts that this exponential progress of silicon MOS-
FETs and integrated circuits will continue at least for the
next 15 years (“Moore’s Law”) [2]. However, prospects
to continue the Moore law, a very important device: the
single-electron transistor was first suggested in 1985 and
first implemented two years later. This device attracted
much attention because of their nano feature size and less
power consumption. Moreover SETs are suitable for
several applications such as memories, multiple-valued
logic (MVL)… due to the discrete number of electrons in
a coulomb island.
SETs characteristics are very different from those of
MOSFETs. In both of them, electrostatic effects are
dominant, but, due to the existence of Coulomb blockade;
electrons are not so free to move from source to drain,
due to of tunnel junctions. The Coulomb blockade effect:
that is the electrostatic repulsion experienced by an elec-
tron approaching a small negatively charged region, lim-
its the number of electrons in the island. As a result, for
given values of gate and drain voltages, only a range of
charge is possible for tunneling.
Our day extensive research has been conducted on
fabrication, design, and modeling of SET, that has also
been an active area. Monte Carlo simulation has been
widely used to model SETs. SIMON [3] and MOSES [4]
are two most popular SET simulators for circuit analysis
and systems containing more than a few SETs but vali-
dated in ambient temperature range. Several SET ana-
lytical models, each of them based on the orthodox the-
ory, can notably name the models proposed for metallic
SETs by the following:
Uchida et al. [5] proposed an analytical SET model
for resistively symmetric devices (RS = RD) and valid
for DS
, later Inokawa et al. [6] extended
this model to asymmetric SETs but does not account
for the background charges effect.
Recently a compact analytical model (named MIB) [7]
for SET device, which is applicable for 3
and wide-range of temperature, and valid for single/
multiple gate symmetric/asymmetric device, is taken
that the only one direction flow to minimize the num-
ber of exponential terms. MIB model can be used for
both digital and analog SET circuit design and for
both pure SET and hybrid CMOS-SET circuit simula-
CΣ represents the total capacitance of the SET-island:
  (1)
CG1, CG2, CD and CS represent the capacitances of first
gate, second gate (when exists), tunnel drain and tunnel
source junctions respectively.
Two conditions ensure that the transport of charges
through the metallic island is governed by:
1) Charging the island with an additional charge takes
the time Δt = RTC, which is the RC-time constant of the
quantum dot.
2) The charging energy required to add a single elec-
tron with charge e to the quantum dot is: ΔEC = e²/CΣ.
The system will respect Heisenberg’s uncertainty relation:
opyright © 2012 SciRes. WJNSE
ΔECΔt > h, which leads to: RT > h/e² 26 k. Where e is
electronic charge and h is Planck’s constant. This condi-
tion is needed to make the charge on the island a well-
defined quantity.
3) Another necessary criterion to observe in single-
charge-tunneling effects, the charging energy EC = e²/CΣ,
must be much greater than the thermal fluctuations en-
ergy Eth = kBT 25 meV, to add an electron to the island.
Where kB is Boltzmann’s constant and T is the tempera-
2. Model Description
2.1. Tunnel Current Calculation
In this section, we will only consider a system with a
double-junction that is made of normal metals for which
the free energy will be determined. The energetic con-
siderations are important because, if one knows how to
calculate the change in the system’s free energy ΔF for a
tunneling event, then one can calculate the rate at which
this particular process occurs. When the leads and the
island are normal metals, and once all of the tunneling
rates are known, the tunneling current through the device
can be determined. For the case of electron transport
through the SET, let us consider the tunneling between
two electrodes separated by a barrier. The Fermi energies
of the two electrodes are offset from each other by an
amount VDS = eV.
With some approximations the calculation allows us to
obtain the rate from the source state to the drain state
According to the orthodox theory [8-10], can also write
the tunneling rate in a more general form: the free energy
ΔF = –eV , that is:
 
 
where β–1 = kBT is the thermal energy.
The tunneling rate in the reverse direction is simply
obtained by reversing the sign of the bias voltage.
The current in the device is due to the sequential tun-
neling of electrons through the source and drain junctions
simultaneously. The cotunneling phenomenon is ignored.
Assuming no charge accumulation on the island at
steady state, one can determine the probability p(n) by
requiring the total probability of tunneling into a state to
be equal to the total probability of tunneling out of it.
The master Equations (8)-(10) to determine p(n) is:
 
1111 11
n,nnn,nnn ,nn ,nn
n,n SD
  
 
 
Γn+1,n is the tunneling rates from the state (n) to the
state (n + 1) and Figure 1 describes the electron transi-
tion between different states of SET and illustrates the
concepts of the tunneling rates.
Here we assume that the electron tunneling rate toward
the positive potential is much higher than the electron
tunneling rate in the opposite direction.
Allows to find the probability p(n), the idea is to con-
sider the location of the translated point inside the stable
zone, set of states [–N, N] are needed to determine the
current depending on the values of VDS, and take advan-
tage of the periodicity of VGS, then we calculate p(n) for n
[–N, N]. The approach simply consists in calculating
the number N and how much the point is translated along
VGS direction:
where C is the floor function. If 3
, the cal-
culated N is then 2 and the current expression is:
 
 
 
 
 
  (5)
To determine p0 (and then calculate all the probabili-
ties) can be solved subject to the normalization condi-
2.2. Thermionic Contribution
Now we will add the contribution of the thermionic cur-
rent [11] and the total current between drain and source
electrodes; where the source is connected to ground; is
given by:
 
 
n + 1n
nn ± 1
P(n + 1)
Figure 1. The inflow and outflow electron through the is-
land with suitable concepts of tunneling rates between left
electrode (G) and right electrode (D).
Copyright © 2012 SciRes. WJNSE
 
,tunnel ,thermionicDS DSDS
VI VIV (7)
,thermionic thermionic
A: is the Richardson constant;
kB: is the Boltzmann’s constant;
h: is the Planck’s constant;
S: is the area of the junction;
m0: is the free electron mass;
mox: is the mass of the electron in the oxide;
φ0: is the height of the potential barrier;
l: is the thickness of the oxide;
ε0: is the vacuum permittivity;
εr: is the relative permittivity of the dielectric.
3. Results and Discussion
3.1. Verification with Dubuc et al. Model
In order to validate our model, the I-V, we have taken as
a benchmark, the same conditions and parameters of the
SET realized at the University of Sherbrook [12,13], de-
scribed in Table 1 such metallic devices which are made
with Ti and TiOX tunnel junctions can run at relatively
higher temperature.
From Table 1 we can deduce that CΣ = 0.35 aF, Tmax =
530 K and –1.37 V VDS 1.37 V. For T = 300 K the
charging energy EC = e²/2CΣ 0.45 eV > 10 kBT. The
comparison was established between our model and the
model of Dubuc et al. [14]. Figure 2 shows the evolution
of the IDS vs. VDS of Dubuc model with our model, the
two results are in good agreement with the experiment
data with a shifting.
For VDS 0.6 V the transport of electron is still by
thermionic effect, so the electrons have sufficient energy
Table 1. Electronic parameters description of the SET, ma-
nufactured from titanium and its oxide [12,13].
Description Value
Junction area
Dielectric thickness
Ti/TiOX barrier height
Effective electron mass in TiOX
TiOX dielectric constant
SET drain capacitance, CD
SET source capacitance, CS
SET gate capacitance, CG
SET drain resistance, RD
SET source resistance, RS
10 nm × 2 nm
8 nm
0.35 eV
0.06 aF
0.06 aF
0.23 aF
4.5 × 107 Ohms
1.5 × 107 Ohms
Figure 2. IDS-VDS curve simulated with: (a) The Dubuc model
at 296 K (), 336 K () and 430 K (). VGS = 0 V. The ther-
mionic contribution (dashed line) to the total drain current
model at 433 K (continuous line) is shown in the inset graph
and [14]; (b) IDS-VDS verification of our model.
to blow up the energy barrier was created by the tunnel
junction. On the other hand the Coulomb staircase is
transformed to an, practically, continuous regime in this
field the transfer become by flow and not by packet.
The increase is observed indicating a switch of the
dominant transport mechanism. For VDS 0.6 V the tun-
neling current is predominant. Since this value, the ther-
mionic emission can be assumed as the dominant trans-
port mechanism, and suppresses tunneling effects. The
temperature is one of the de-coherence factors, as it usu-
ally tends to reduce the impact of the quantization of the
energy. Also note that as the temperature increases, so
does the current amplitude.
We have also simulated the VGS vs. IDS curves of single
electron transistor with ours model at 336 K for different
VDS values Figure 3. The result of simulation is shown as
Figure 3. The Coulomb blockade phenomenon persists
at high temperature. The effect of VDS voltage is to modu-
late the depth of quantum well that is describe, in the
Copyright © 2012 SciRes. WJNSE
Figure 3. Coulomb oscillations of SET obtained by our
model with the parameters in Table 1 at T = 336 K for dif-
ferent VDS voltages.
curve by a non-zero current for VGS = 0 V and the gate
starts to lose control over the drain current. Contrariwise,
in the Dubuc model, the gate-current was not established.
4. Experiment Results
Because of the wave nature of electrons, some of the out-
going electrons are reflected when they reach the drain,
which reduces the density of emission current; this may
explain the shifting between the theoretical curves and
the experiment curves. We can compensate this by in-
troducing a new physical term that introduce acceptable
physical effects and associated directly to the structure of
the transistor.
But recent models for thermionic emission assume a
spatial distribution of the barrier height to take the inho-
mogeneities of the charges in the interface into account;
the barrier height will have a temperature dependence
which can be described by an effective potential barrier
φ* [15]. However, the effect of temperature on device-
size must also be taken into account; then area S in Equa-
tion (9) will dened a newly effective area 1 i.e. impact
of the dot size dispersion on the thickness. At high tem-
perature dependence on IDS is hypothesized as the reason
why thermionic emission was observed only for T > 300
Figure 4 reproduces the Coulomb staircase, and shows
results for our two empirical values of α (α = 1 and α =
10) for T = 296 K and 336 K.
Now, is clearly, our model gives an accurate result
when compared the experiments ones. It is clear that the
rates outside the range of validity of model have to be
modified for negative bias. Since the model considers
only that the two most-probable charging states and the
probabilities of taking these states p(n) and p(n + 1) are
already know and one direction flow. The difference is
more clear in the reverse bias region, (IDS,min for the two
Figure 4. IDS-VDS empirical model validation (a) for T = 296
K we have chosen α = 10; and (b) for T = 336 K we have
chosen α = 1.
curves plotted in Figure 4 are the same 10 nA) demon-
strating the excess current that can be attributed to image
force lowering to tunneling currents through the barrier.
5. Conclusion
A physically based analytical SET model within the or-
thodox theory is developed for to describe the phenom-
ena at high temperature. This new model can reproduce
not only the transport property in low and high tempera-
ture but also the effects of structure parameters with good
agreement for wide gate and drain bias. Modeling and
simulation of SET are very important to understand be-
havior, and characteristic before start fabricating the de-
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