JOURNAL OF APPLIED PHYSICS 101, 073709 共2007兲
Oscillation of gate leakage current in double-gate
metal-oxide-semiconductor field-effect transistors
V. Nam Doa兲 and P. Dollfus
Institut d’Electronique Fondamentale, Bâtiment 220—UMR8622, CNRS, Université Paris Sud,
91405 Orsay, France
共Received 3 November 2006; accepted 2 February 2007; published online 13 April 2007兲
Using the nonequilibrium Green’s function method, gate current characteristics are investigated for
nanometer-scaled double-gate metal-oxide-semiconductor field-effect transistor. The mode-space
approximation is, at the first stage of the calculation, used to obtain self-consistently the potential
profile and the charge distribution in the structure. This solution is then used to solve the
two-dimensional transport equation to extract the desired quantities. In addition to the dependence
of the gate-leakage current on the gate bias and on the oxide thickness, our calculation shows the
oscillation behavior of the leakage current versus the drain voltage. It is explained as the result of
the strong quantization of electronic states inside the device, giving a resonant-like character to the
tunneling of charges from source and drain contacts to the gates. This effect is strongly dependent
on the gate length. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2716874兴
I. INTRODUCTION
Scaling down the size of field-effect transistors 共FETs兲,
one has to face with arising of quantum effects that may
strongly affect their operation and performance. Since
charges can move from source to drain by tunneling through
the channel, the off-current value and the subthreshold swing
may be dramatically degraded.1 Additionally, because of the
quantum mechanical confinement of the electron states in the
channel, the carrier density is low at the Si/ SiO2 interface
and reaches its maximum inside the Si film, contrary to the
classical picture,2,3 which results in a reduction of gate capacitance. As a consequence of ultra-thin gate oxide requirements, electrons can tunnel to the gate, creating the so-called
gate-leakage current that is considered as one of the most
severe challenge to take up for next complementary metaloxide-semiconductor 共CMOS兲 generations, especially regarding the power consumption. To date, a lot of
experimental4,5 and theoretical works have focused on this
problem. On the theoretical point of view, both
semiclassical6 and quantum mechanical approaches have
been developed 共see Refs. 2, 7,8, and 9, and references
therein兲. Except for the semiclassical approach of Ref. 6
coupled with two-dimensional 共2D兲 Monte Carlo simulation
of transistor, most of them are based on the one-dimensional
description of tunneling from the quasibounded states in the
inversion layer of MOS capacitors, which is accurate but not
enough to understand the gate current behavior in the full
range of transistor operating conditions.
In this article we present a new interesting feature of the
gate leakage current in ultra-thin double-gate metal-oxidesemiconductor field-effect transistors 共DG MOSFETs, depicted in Fig. 1兲: the oscillation versus the drain bias. As
explained later, this phenomenon is a consequence of the fact
that electrons tunnel out-of/into quantized electronic states
inside the device through the gate oxide layer. Besides, we
systematically investigate the dependence of this current on
the gate voltage, the oxide thickness, and the gate length.
This study is based on the nonequilibrium Green’s function
共NEGF兲 formalism,10,11 which is considered as one of the
most powerful tools to treat the carrier transport in nanoscaled FETs since quantum mechanical effects can be rigorously included. In small devices, where the transport may be
considered as essentially ballistic, the NEGF method simply
leads to a current formula similar to that of the Landauer
formalism.11,12 However, it still costs expensive computational time and memory for the exact treatment of 2D problems. Therefore, efficient approximations have been developed as the so-called mode-space approach.13,14 As proved
elsewhere, e.g., in Refs. 13 and 15, this approximation works
well in DG-MOSFET with body thickness less than 5 nm.
Unfortunately, it does not allow to investigate the oxideleakage effects because of difficulty of treating the boundary
condition for the wave function. To overcome this obstacle
we propose an efficient procedure to extract the essential
information on the gate-leakage current as follows: First, in
the mode-space approximation, a self-consistent procedure is
used to find out the potential profile and the carrier distribution in the device under a given transport condition. Second,
the resulting self-consistent potential is used to solve exactly
the quantum transport equation, i.e., to determine every 2D
Green’s function, from which desired quantities are extracted. Practically, this procedure is simple but it guarantees
FIG. 1. 共Color online兲 Schematic of the DG MOSFETs.
a兲
Electronic mail: van-nam.do@ief.u-psud.fr
0021-8979/2007/101共7兲/073709/6/$23.00
101, 073709-1
© 2007 American Institute of Physics
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073709-2
J. Appl. Phys. 101, 073709 共2007兲
V. N. Do and P. Dollfus
a potential profile very close to the exact one and it strongly
reduces computational requirements compared to a full 2D
treatment.
The paper is organized as follows: Sec. II is devoted to
the description of the model and its treatment using the
NEGF approach. The gate-leakage current is reformulated in
an appropriate form for the aim of numerical calculation.
Section III presents the numerical results and their discussions. The last section is the conclusion.
II. FORMALISM
As usually done, we consider a model describing the
transport of electrons in a DG MOSFET 共see Fig. 1兲 in the
ballistic regime using this standard effective mass
Hamiltonian13,14
H=兺
␯
−
再冋 冉 冊 冉 冊
冉 冊冎
−
ប2 ⳵ 1 ⳵
ប2 ⳵ 1 ⳵
−
+ Ec共x, y兲
2 ⳵ x mx␯ ⳵ x
2 ⳵ y m␯y ⳵ y
ប2 ⳵ 1 ⳵
2 ⳵ z mz␯ ⳵ z
,
册
共1兲
where ␯ 共running from 1 to 6兲 is the valley index of the
␯
silicon electronic structure; mx/y/z
are the effective masses
along the three direction in the valley ␯; Ec共x , y兲 is the bottom of conduction band, which includes the discontinuity of
the conduction band between the oxide and the silicon, and
the potential U共x , y兲 created by all charges in the device. For
simplicity, we now ignore the band index ␯.
The three-dimensional Hamiltonian before is actually
decoupled into two independent parts, the two first terms and
the last one, wherein the eigenstates of the latter are simply
the plane wave and the associated eigenvalues
具z兩k典 =
1
冑L z
eikz ; ⑀k =
ប 2k 2
.
2mz
共2兲
The spirit of the mode-space approximation is to completely
decouple the Hamiltonian by choosing an appropriate representation basis for the two first terms, for example 兵兩i典
丢 兩␣i典其, where i denotes the position xi along the OX direction in the real space 共real-space representation兲 and 具y 兩 ␣i典
= ␾␣i 共y兲 is the eigenvector of the second term in Eq. 共1兲 with
x = xi, associated with the eigenvalue E␣i ⬅ E␣共xi兲,
冋
−
冉 冊
册
ប2 ⳵ 1 ⳵
+ Ec共xi, y兲 ␾␣i 共y兲 = E␣i ␾␣i 共y兲.
2 ⳵ y my ⳵ y
共3兲
So that, if we neglect the i dependence of ␾␣i 共y兲, in the basis
兵兩i典 丢 兩␣i典 丢 兩k典其 the Hamiltonian Eq. 共1兲 is completely decoupled and the transport of carriers can be described by a onedimensional 共1D兲 effective Hamiltonian
Heff = −
冉 冊
ប2 ⳵ 1 ⳵
+ 关⑀k + E␣共x兲兴.
2 ⳵ x mx ⳵ x
共4兲
The quantum transport equation is, in principle, written
in terms of three independent Green functions, for example
the retarded 共Gr兲, advanced 共Ga兲, and lesser 共G⬍兲 functions.
By denoting T as the matrix form of the first term of Heff, the
retarded Green function matrix is given by
关G␣r 共E, k兲兴 = 关共E − ⑀k − E␣兲I − 关T兴 − 关⌺rS共E, k兲兴
r
− 关⌺D
共E, k兲兴兴−1 ,
共5兲
r
where 关⌺S共D兲
兴 is the retarded self-energy matrix describing
the coupling between the source 共drain兲 and the reservoir.
These self-energy matrices can be calculated exactly
r
关⌺S共D兲
共E, k兲兴i, i⬘ = t2x gS共D兲共E, k兲␦k, k⬘␦i, i⬘␦i, 1共N兲 ,
共6兲
tx = ប2 / 2mx␯a2
where
is the hopping parameter between the
device and the contact; gS共D兲共E , k兲 is the reservoir-surface
Green function, which satisfies this normal algebra equation
2
− ␭S共D兲共E, k兲gS共D兲 + 1 = 0,
t2x gS共D兲
共7兲
with the convenient sign for the root to guarantee the Green
function to be finite. In this equation ␭S共D兲共E , k兲 = E − ⑀共k兲
+ 2tx − E␣共x1共N兲兲.16
The advanced Green function and the advanced selfenergies are Hermitian adjoints of the corresponding retarded
matrices: Ga = 关Gr兴+ and ⌺a = 关⌺r兴+. The lesser Green function is thus expressed as
G␣⬍共E, k兲 =
=
兺
a
G␣r 共E, k兲⌺⬍
c 共E, k兲G␣共E, k兲
共8兲
兺
⌫c␣共E, k兲兩G␣r 共E, k兲兩 2 f c共E兲,
共9兲
c=S, D
c=S, D
␣
r
a
= i关⌺S共D兲
− ⌺S共D兲
兴 and f c is the Fermi function aswhere ⌫S共D兲
sociated with the contact c. To determine the conduction
band profile Ec共x , y兲, one needs to calculate the electron
density which is given by this formula
关ne兴i, j =
冑
冕
1
ab
⫻
2mzkBT
兺
␲2ប2 c=S, D
dE兩␾␣i共j兲兩2关Ac␣共E兲兴iF− 1
2
冉 冊
␮c − E
,
k BT
共10兲
E = E − ⑀ k;
关Ac␣共E兲兴i
where
we
have
introduced
␣
r
2
= 关⌫c 共E兲兴ic兩关G␣共E兲兴i,ic兩 ; a and b are the grid spacings along
the OX and OY directions. We then put the electron density
into the Poisson equation to take out the potential profile
U共x , y兲.
Once the the conduction band is self-consistently calculated, we use it to solve Eq. 共1兲 in the full real space, i.e.,
considering every Green function as a four-index matrix as
for instance
关G␣r 共E, k兲兴i, i⬘ → 关Gr共E, k兲兴i, j; i⬘, j⬘ ,
共11兲
and Eq. 共5兲 is replaced by
r
共E, k兲
关Gr共E, k兲兴 = 关共E − ⑀k兲I − H − ⌺rS共E, k兲 − ⌺D
− ⌺Tr共E, k兲 − ⌺Br共E, k兲兴−1 ,
共12兲
r
where 关⌺T共B兲
共E , k兲兴 is the self-energy matrix describing the
coupling between the top 共bottom兲 gate and the device.
These coupling self-energy matrices are generally determined through the surface Green functions, which, in Eq.
共6兲, becomes a two-index matrix satisfying this matrix equation
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073709-3
J. Appl. Phys. 101, 073709 共2007兲
V. N. Do and P. Dollfus
关gT−1共E, k兲兴i; i⬘ = 共E − ⑀k − 2ty − UG
i 兲␦i; i⬘
− 关Tx兴i; i⬘ − t2y 关gT共E, k兲兴i; i⬘ ,
共13兲
where ty = ប2 / 2m␯y b2 is the hopping parameter between the
top gate and the device; and 关Tx兴 is the matrix describing the
kinetic energy on the top-gate surface. We distinguish the
two cases of metal and polysilicon gate by setting UG = 0 for
the former case and UG
i = Ui,1共Ny兲 for the latter one.
To extract the gate-leakage current, we use the Landauer
formula by noting that the current is due to the contribution
of all carriers from the source and the drain tunneling
through the oxide layer to the gate. Accordingly, IG is characterized by the transmission coefficients TS/G and TD/G and
it can be expressed as IG = IS/G + ID/S, where
IS共D兲/G = e
冑
2mzkBT
␲ 3ប 4
− F− 1
2
冉
冕
␮G − E
k BT
冋 冉
dETS共D兲/G共E兲 F− 1
冊册
2
,
␮S共D兲 − E
k BT
冊
共14兲
and
Ny
iG
2
G
r
2
⌫S共D兲
兺
1, j ⌫i, 1兩G1, j; i, 1兩 .
j=1 i=i
TS共D兲/G = 兺
FIG. 2. 共Color online兲 Gate leakage current density in the 关6/2/0.5兴 device at
various gate voltages, from 0.1 to 0.6 V 共upward兲. Temperature: 300 K.
共15兲
G1
Here, for simplicity, we just keep the diagonal elements of
the tunneling rates and iG1, iG2 denote the coordinates of the
gate ends.
III. RESULTS AND DISCUSSION
Using the earlier simulation procedure we analyze the
gate leakage current in DG-MOSFETs. For short, the simulated structures are denoted 关Lg / TSi / Tox兴, where Lg, TSi, and
Tox, given in nanometers, are the gate length, the silicon
body thickness, and the gate oxide thickness, respectively.
Source/channel and channel/drain junctions are assumed to
be abrupt with source 共S兲 and drain 共D兲 regions uniformly
doped to 1020 cm−3 and not overlapped by the gate. We have
considered the leakage current in case of poly-silicon gate
and we have realized that the gate current always oscillates
as a function of drain bias in the regime of weak inversion.
However, when increasing the gate voltage the oscillation is
suppressed and, as a general behavior, the current tends to
decrease when increasing the drain bias. Although the gate
has to work as a conductor rather than a semiconductor, in
order to highlight the phenomenon we essentially present
below the results for the case of a work function corresponding to a weakly doped gate, wherein the phenomenon can be
clearly described. However, at the end of the article we make
a brief comparison of the gate current in the two cases of
weakly and heavily doped gates for the same operating state
of the device.
In Fig. 2 the gate leakage current in 关6/2/0.5兴 is plotted
versus the drain bias for various gate voltages. Obviously,
pronounced oscillations occur at small VGS and suppress at
high VGS. Before going to explain such features of the gate
leakage current we plot in Fig. 3 the distribution of this current along the gate length. Note that, for instance when
VDS = 0 V, in the two cases of low and high gate voltage, the
shape of this current distribution is different. This is because
the gate surface electronic states contribute differently to the
leakage current. When VGS is low, obviously only the lowest
level of such states is important, thus the distribution of IG
takes the simple form shown in Fig. 3共a兲. However, when
VGS is high, IG get the multihump form as shown in Fig. 3共b兲
since several states contribute to the current. When increasing the drain bias, the area limited by the curve strongly
varies on a large portion of the gate surface. It is obviously
due to the drain-induced drop of potential in the channel.
Particularly, one can realize the oscillation of such area in
Fig. 3共a兲 while it decreases monotonically in Fig. 3共b兲. This
behavior is in agreement with the results shown in Fig. 1. We
can define a particular value of energy, denoted as Eoff, such
that electrons with E ⬍ Eoff hardly tunnel to the gate. In the
case of weakly doped gate, i.e., the gate Fermi level E f is
below the bottom of conduction band Ec in the gate, Eoff can
FIG. 3. 共Color online兲 Distribution of leakage-current along the gate at
various drain biases, VDS = 0.0 V 共circles兲, 0.08 V 共squares兲, 0.14 V 共diamonds兲, 0.2 V 共up-triangles兲, and 0.3 V 共down-triangles兲. 共a兲 Weak inversion
regime 共VGS = 0.1 V兲 and 共b兲 strong inversion regime 共VGS = 0.6 V兲.
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073709-4
V. N. Do and P. Dollfus
FIG. 4. 共Color online兲 Leakage-current spectral density in the 关6/2/0.5兴 device at two typical values of gate voltage, 共a兲 0.1 and 共b兲 0.6 V. Temperature:
300 K.
be well identified with Ec, but in the case of heavily doped
gate Eoff may be roughly defined in the inferior vicinity of
E f 共Eoff ⬍ E f 兲, so that the number of unoccupied states 1
− f G at Eoff is small. Additionally, one should notice the fact
that Eoff does not depend on VDS but on VGS only.
i
In the local picture of the gate leakage current IG
, where
i denotes the position along the gate, its oscillation versus
VDS is straightforward. The maxima correspond to the cases
of alignment between Eoff and one of the quantized levels in
the channel and the minima occur when Eoff is in between
two such levels. This picture resembles that in a double barrier structure 共DBS兲 when increasing the bias.16 Comparing
these two pictures, Eoff apparently plays a role similar to the
emitter conduction band in the DBS, i.e., the tunneling cutoff
energy. The total gate leakage current can be separated as
IG = Iground + Iosci, where Iground and Iosci are due to the contribution of local currents that are insensitive and sensitive to
VDS, respectively. When the gate voltage is high enough, the
channel potential barrier is low so that at the source-end and
on a large part of the channel the quantized energy levels are
quite flat and weakly sensitive to VDS. The contributions to
Iosci only come from the drain-end of the channel, which
results in an oscillatory part Iosci much smaller than Iground. In
contrast, when the gate voltage is low, the channel potential
barrier is high but short channel effects make it sensitive to
VDS on the whole channel length. As a consequence, almost
the full gate area provides contributions to Iosci. The magnitudes of Iground and Iosci become comparable and the gate
leakage current significantly oscillates as a function of VDS.
To strengthen the mechanism analyzed earlier we plot in
Fig. 4 the spectral density of this current in the two typical
cases of weak 共VGS = 0.1 V兲 and strong 共VGS = 0.6 V兲 inversion regime at various drain voltages. Figure 4共a兲 shows significant peaks and their harmonious changing versus the
drain bias. The evolution of IG共E兲 vs VDS concretely seems to
be: the peaks for VDS = 0.14 V, denoted as 共0.14兲␣ for short,
move to 共0.20兲␣, and 共0.20兲␣+1 become 共0.30兲␣; the value of
Eoff in this case is about 0.26 eV, i.e., close to the main peak
J. Appl. Phys. 101, 073709 共2007兲
FIG. 5. 共Color online兲 Gate length effect on the leakage current plotted in
the strong inversion regime 共VDS = 0.1 V兲, Lg = 3 nm 共lozenges兲, Lg = 6 nm
共squares兲, and Lg = 12 nm 共circles兲. Temperature: 300 K.
of IG共E兲, which explains why IG regularly oscillates. Moreover, this global picture can be deeper understood if we note
that the position of the peak 共0.14兲1 is mostly the same as
that of 共0.30兲1 共these two VDS values correspond to the current maxima while the minimum is for VDS = 0.20 V, see Fig.
1兲. The main contribution to IG virtually comes from the gate
portion close to the source where it is weakly sensitive to the
drain bias. The other part of the channel is sensitive to VDS
and gives the oscillatory contribution to IG. In contrast, when
VGS = 0.6 V, there exist very thin peaks localized at high energy relative to Eoff ⯝ −0.23 eV 关see Fig. 3共b兲兴. Obviously,
such peaks of the leakage current spectral density simply
contribute to the magnitude of the total current but do not
cause significant oscillations.
From Fig. 3, the tendency of decreasing leakage current
as a function of VDS is straightforward to understand. The
fact is that the gate leakage current is due to the contribution
of two particle fluxes from the source and from the drain.
When VDS = 0 V, these two flux components are equal in
their magnitude but they are opposite. Thus there is, of
course, no drain current but IG is maximal. When increasing
VDS, since the potential in the channel is dropped in the
drain-end of the channel, the flux from the drain decreases,
and may even cause a negative contribution to IG. Meanwhile the flux contributing to IG from the source mostly does
not change 共since there is no potential drop in the source-end
of the channel兲 in spite of increasing total particle flux. Consequently, it results in the decreasing of the leakage current
as VDS increases.
The effect of gate length on the current oscillation for
VGS = 0.1 V is shown in Fig. 5. We consider the gate length of
3, 6, and 12 nm, all other things being equal 共Tox = 0.5 nm,
TSi = 2 nm兲. The result shows that the amplitude of the oscillations are strongly affected by the gate length as it influences the potential barrier in the channel. For a gate length of
12 nm, the short channel effects can be neglected and the
channel potential is influenced by VDS only at the drain-end
of the channel that becomes the only position in the channel
giving rise to oscillations of the current. As a consequence,
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073709-5
J. Appl. Phys. 101, 073709 共2007兲
V. N. Do and P. Dollfus
FIG. 6. 共Color online兲 共a兲 Gate leakage current density vs gate voltages in
the 关6 / 2 / Tox兴 devices 关Tox = 0.5, . . . , 1.0共nm兲兴 with drain voltage VDS
= 0.0 V. Temperature: 300 K. 共b兲 Dependence of gate leakage current on the
oxide thickness for VDS = 0 V and VGS = 0.6 V.
the oscillation amplitude is smaller and the current peak
spacings are greater than those previously obtained for Lg
= 6 nm. In contrast, for Lg = 3 nm, the enhanced short channel
effects increase the amplitude of oscillations and they seem
to mix some current peaks. Furthermore, the short channel
effects are so strong here that the electron injection in the
channel increases under the effect of VDS, which enhances
the gate current and the amplitude of oscillation.
Our calculation also reemphasizes the conclusions about
the dependence of the leakage current on the gate voltage, as
well as the exponential law of IG as a function of Tox in the
important case of VDS = 0 V, as shown in Fig. 6. Such results
were already derived using semiclassical models6 or quantum mechanical 1D models2,7,9 and importantly they are confirmed by experimental data.5 However, according to our results here one can conclude that using semiclassical and/or
quantum 1D models is not appropriate to fully describe the
elegant properties of the gate leakage current in the ultrathin
and ultrashort MOSFET structures.
We now plot in Fig. 7 the gate leakage current in the two
cases of weakly and heavily doped gate. In the former case,
in order the device to be in the same operating state as that of
the latter one, a higher gate voltage is to be applied and the
gate current magnitude is also greater. Besides, the negative
contribution of the particle fluxes to the gate current is also
revealed in Fig. 7. In the latter case 共high gate doping兲, there
exists a range of high drain voltage where the gate current is
negative. Physically, at high VDS, electrons are accelerated by
the electric field and almost all of them then move to the
drain. Consequently, the difference between local Fermi
level in the channel and that of the gate E f decreases. The
gate leakage current, therefore, decreases as VDS increasing
as previously discussed in the article. A flux of electrons can
even tunnel from the gate through the gate oxide layer to the
channel in the drain-end of the channel. One also notices that
the current peak spacings in the two cases are not the same
since the relative position of Eoff to the quantized energy
levels in the two cases is different. We should mention that
FIG. 7. 共Color online兲 Leakage current in two cases of weakly and heavily
doped polysilicon gate. The devices 共关6/2/0.5兴兲 are in the same operating
state in the weak inversion regime. Temperature: 300 K.
the model used here neglects the accumulation and depletion
effects in the polysilicon gate. However, these effects are
currently on the way of studying and we do not expect them
to qualitatively change the results obtained here.
IV. CONCLUSION
In conclusion, we have presented some important investigations of the gate leakage current in double-gate MOSFET
structures using the NEGF technique. Once again, this technique proves its power to describe quantum mechanically the
properties of an open system, which is extremely important
to understand the operation and performance of nanoscaled
devices. The problem of device-reservoirs coupling is exactly treated in the Green’s function approach in terms of the
self-energies while it is a big obstacle in other methods. Our
quantum mechanical treatment shows for the first time the
oscillation of the gate leakage current versus the drain bias.
This interesting feature is then explained using a picture
which is similar to the resonant tunneling through double
barrier structures. The effects of gate length as well as the
gate doping regimes are also discussed. Additionally, our results reconfirm the available conclusions about the gate bias
and oxide thickness dependences of the gate leakage current,
which were previously deduced using both quantum and
semiclassical 1D models, in qualitative agreement with the
experimental data.
ACKNOWLEDGMENTS
This work has been partially done with the support of the
European Community under Contract No. IST-506844 共NoE
SINANO兲. The authors would like to thank V. L. Nguyen,
from the Institute of Physics and Electrics, Hanoi, Vietnam,
for helpful discussions.
1
J. Wang and M. Lundstrom, Tech. Dig. - Int. Electron Devices Meet. 2002,
707 共2002兲.
2
E. Cassan, P. Dollfus, S. Galdin, and P. Hesto, IEEE Trans. Electron Devices 48, 715 共2001兲.
3
L. Ge, F. Gamiz, O. G. Workman, and S. Veeraraghavan, IEEE Trans.
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073709-6
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