Threshold voltage model for hetero-gate-dielectric tunneling field effect transistors

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 2, April 2020, pp. 1764~1771
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i2.pp1764-1771  1764
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
Threshold voltage model for hetero-gate-dielectric tunneling
field effect transistors
Ajay Kumar Singh, Tan Chun Fui and Tan Wee Xin Wilson
Communication System and Integrated Circuit Design Research Center,
Faculty of Engineering and Technology, Multimedia University, Malaysia
Article Info ABSTRACT
Article history:
Received May 2, 2019
Revised Oct 22, 2019
Accepted Oct 31, 2019
In this paper, a two dimensional analytical model of the threshold voltage for
HGD TFET structure has been proposed. We have also presented the analytical
models for the tunneling width and the channel potential. The potential model
is used to develop the physics based model of threshold voltage by exploring
the transition between linear to exponential dependence of drain current on
the gate bias. The proposed model depends on the drain voltage, gate dielectric
near the source and drain, silicon film thickness, work function of gate metal
and oxide thickness. The accuracy of the proposed model is verified by
simulation results of 2-D ATLAS simulator. Due to the reduction of
the equivalent oxide thickness, the coupling between the gate and the channel
junction enhances which results in lower threshold voltage. Tunneling width
becomes narrower at a given gate voltage for the optimum channel
concentration of 1016
/cm3
. The higher concentration in the source (Ns) causes
a steep bending in the conduction and valence bands compared to the lower
concentration which results in smaller tunneling width at the source-channel
interface.
Keywords:
Hetero-gate dielectric
Poisson equation
TFET
Threshold voltage
Tunneling width
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Ajay Kumar Singh,
Communication System and Integrated Circuit Design Research Center,
Faculty of Engineering and Technology,
Multimedia University,
Jalan Ayer Keroh Lama, 75450-Melaka-Malaysia.
Email: ajay.singh@mmu.edu.my
1. INTRODUCTION
As the conventional MOSFETs are scale to the sub-nanometer region, the close proximity between
the source and drain reduces the gate control over channel which leads to several problems, such as high
subthreshold swing (SS), high leakage current, and short channel effects (SCEs) [1-4]. Tunneling field-effect-
transistors (TFETs) are considered one of the attractive devices to replace the conventional MOSFETs in
the sub-nanometer region [5, 6] due to its lower SS value (< 60 mV/decade) at room temperature.
This advantage translates into low-voltage operation and results in low stand-by power dissipation which
makes TFET more energy-efficient compared to the conventional MOSFETs [7-9]. Despite the better SS, there
are two main drawbacks with TFETs; one is the low ON- current which degrades the performance and
the other is the large ambipolar current [10].
Thus, to increase the ON-current and reduce the ambipolar current, researchers have introduced
hetero-gate-dielectric (HGD) TFETs [11-15]. This device gives lower SS without scarifying the chip density.
The HGD TFET has different dielectric material at the drain and the source side. The characteristics of HGD
TFET device in terms of band bending channel potential, electric field, ON-current and SCEs experimentally
and analytically have been studied extensively in the literature [16-19]. The threshold voltage of the TFET
device is one of the most important electrical parameter. Researchers have extracted the value of threshold

Int J Elec & Comp Eng ISSN: 2088-8708 
Threshold voltage model for hetero-gate-dielectric tunneling field effect transistors (Ajay Kumar Singh)
1765
voltage either using constant current method (at gate voltage for which ID=10-7
A) or transconductance change
method [20, 21]. Since, the constant current method uses an arbitrary value; hence, it has no practical meaning.
In the transconductance change method, the threshold voltage is defined as the gate voltage corresponding to
maxima of the transconductance derivative. The main drawback of this method is that the plot has much
numerical derivative noise which makes its impractical. Analytical models are helpful to design, simulate and
provide a further insight on the working electrical characteristics of the device. Therefore, to understand
the overall performance of a device compact analytical threshold voltage model of HGD TFET structure is
needed but no analytical model is available in the literature for HGD TFET in best of our knowledge except
few analytical models available for TFET [22-25].
In this paper, compact analytical models for the channel potential, tunneling width and the threshold
voltage of the HDG TFET have been derived using parabolic potential approximation method to solve
the 2-D Poisson’s equation. The accuracy of the proposed models are validated by comparing the model results
with the results available in the literature as well as 2-D ATLAS TCAD simulator results with a close
agreement. It is observed that as the length of high-k region near the source reduces, the conduction band
becomes shallow which makes band-to-band tunneling difficult and increases the threshold voltage.
The structure of this paper is given as follows: Section 2 describes the analytical models for the channel
potential, tunneling width and threshold voltage. Section 3 describes the simulation results whereas section 4
concludes the paper.
2. PROPOSED ANALYTICAL MODELS
Figure 1 shows the schematic of the proposed n-channel HGD TFET and its coordinate system.
The whole channel region is divided into two: High-k region near source to control the tunnelling current
(relative permittivity of kr1 and oxide thickness, tox1=3 nm), Low-k region near drain (relative permittivity of
kr2 and oxide thickness, tox2=3 nm) to minimize the ambipolar current. The doping concentration of
p+ source region is 1020
/cm3
, channel region is 1016
/cm3
and drain region n+ is 1018
/cm3
. L1 is the length of
high-k region and (L-L1) is length of low-k region respectively where L is total gate length. The quantum
confinement effect in the analysis is ignored because the film thickness (𝑡 𝑠𝑖) is greater than 3 nm. Neglecting
the fixed carrier oxide change, the 2-D Poisson equation for n-channel HGD TFET in High-k and Low-k (SiO2)
regions are given as [18].
Figure 1. Schematic of the n-channel HGD TFET device
𝑑2
𝜑𝑗(𝑥,𝑦)
𝑑𝑥2
+
𝑑2
𝜑𝑗(𝑥,𝑦)
𝑑𝑦2
=
−𝑞𝑁𝑐𝑗
𝜖 𝑠𝑖
(1)
Where, the subscript j takes value 1 and 2 for High-k and Low-k regions respectively, 𝜑𝑗(𝑥,𝑦) is the 2-D
electrostatic potential, 𝜖 𝑠𝑖 is the dielectric constant of the silicon, 𝑁𝑐𝑗 = 𝑁𝑐 is the channel doping concentration.
Assuming, a parabolic potential profile along the film thickness (i.e. along y-direction), the 2-D electrostatic
potential in the channel can be expressed as [18]:
𝜑𝑗(𝑥,𝑦) = 𝑎0 + 𝑎𝑗1 𝑦 + 𝑎𝑗2 𝑦2
(2)
Where, 𝑎0𝑗 , 𝑎𝑗1, aj2 are constants and function of x-only. These constants can be determined by using
the following Boundary Conditions (BCs):

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 10, No. 2, April 2020 : 1764 - 1771
1766
𝑑2 𝜑 𝑗(𝑥,𝑦)
𝑑𝑦2
|
𝑦=0
= 0, 𝜑𝑗(𝑥,𝑦)|
𝑦=
𝑡 𝑠𝑖
2
= 𝜙 𝑠𝑗(𝑥),
𝑑2 𝜑 𝑗(𝑥,𝑦)
𝑑𝑦2
|
𝑦=
𝑡 𝑠𝑖
2
= −
𝐶 𝑜𝑥1
𝜖 𝑠𝑖
[𝑉′
𝐺𝑆𝑓 𝑗
+ 𝜙 𝑆𝑗(𝑥)]
Where 𝜙 𝑆𝑗(𝑥) is surface potential, 𝑉′
𝐺𝑆𝑓 𝑗
= 𝑉𝐺𝑆 − 𝑉𝑓𝑏. 𝑉𝑓𝑏 is the flat-band voltage which is given as
𝑉𝑓𝑏 = 𝜙 𝑚 − [𝑥 +
𝐸𝑔
2
+
𝑘𝑇
𝑞
ln(
𝑁𝑐
𝑛𝑖
)] (3)
𝜙 𝑚 is the metal work function, 𝑥 is the silicon electron affinity and 𝐸𝑔 is forbidden gap of Si. Using,
these Boundary conditions, we get
𝑎0𝑗 = 𝜙 𝑠𝑗(𝑥) − 𝑎𝑗2
𝑡 𝑠𝑖
4
2
, and 𝑎𝑗2 = −
𝐶 𝑜𝑥𝑗
𝜖 𝑠𝑖 𝑡 𝑠𝑖
(𝑉′
𝐺𝑆𝑓 − 𝜙 𝑆𝑗(𝑥)) (4)
Substituting (2) into (1), the surface potential 𝜙 𝑆𝑗(𝑥), in the respective region, must satisfy the following 2-D
scaling equation
𝑑2
𝜙𝑠𝑗(𝑥)
𝑑𝑥2
−
1
𝜆2
𝑗
𝜙 𝑆𝑗(𝑥) +
𝜎𝑗
𝜆2
𝑗
= 0 (5)
Where 𝜆𝑗 = √
2𝐶 𝑜𝑥𝑗
is known as characterise length or scaling length, Coxj=(εoxj/toxj), 𝜎𝑗 is long channel surface
potential and given as
𝜎𝑗 = −
𝑞𝑁𝑐
𝜖 𝑠𝑖
𝜆𝑗
2
− 𝑉′ 𝐺𝑆𝑓
The general solution of the differential equation (5) is expressed as:
𝜙 𝑆𝑗(𝑥) = 𝐴𝑗 𝑒
𝑥
𝜆 𝑗 + 𝐵𝑗 𝑒
−
𝑥
𝜆 𝑗 + 𝜎𝑗
(6)
Where 𝐴𝑗 and 𝐵𝑗are constant and determined using the BCs:
𝜙 𝑆𝑗(𝑥=0)|
𝑠𝑜𝑢𝑟𝑐𝑒 𝑒𝑛𝑑
= 𝑉𝑏𝑖1 ,𝜙 𝑆𝑗(𝑥=𝐿)|
𝑑𝑟𝑎𝑖𝑛
= 𝑉𝑏𝑖2 + 𝑉𝑑𝑠
𝑑𝜑 𝑠1
𝑑 𝑥
|
𝑥=𝐿1
=
𝑑𝜑 𝑠2
𝑑 𝑥
|
𝑥=𝐿1
, and 𝜙 𝑆1(𝑥=𝐿1) = 𝜙 𝑆2(𝑥=𝐿1) (7)
Where, 𝑉𝑏𝑖2 =
𝑘𝑇
𝑞
ln(
𝑁 𝑑
𝑛 𝑖
), 𝑉𝑏𝑖1 = −
𝑘𝑇
𝑞
ln(
𝑁𝑠
𝑛 𝑖
).
Using, these four Boundary Conditions, we get;
𝐴2 =
𝑞11−𝐵2
𝛾
, 𝐵2 =
𝑞12
𝛾𝑒
−
𝐿
𝜆2−𝛿𝑒
𝐿
𝜆2
, 𝐴1 = 𝛼𝐴2 + 𝛽𝐵2 + 𝑅11 𝑒−
𝐿
𝜆1 and 𝐵1 = 𝑉𝑏𝑖1 − 𝜎1 − 𝐴1
Where, 𝑅11 = √
𝜎1−𝜎2
2
, 𝛼 =
1
2
𝜆1 [
1
𝜆1
−
1
𝜆2
] 𝑒
𝐿1(
1
𝜆2
−
1
𝜆1
)
, 𝛽 =
1
2
𝜆1 [
1
𝜆1
+
1
𝜆2
] 𝑒
−𝐿1(
1
𝜆1
+
1
𝜆2
)
, 𝑞11 = 𝑅11 𝑒
−
2𝐿1
𝜆1 + 𝜎1 (𝑒
𝐿1
𝜆1 −
1
2
) +
𝜎2
2
− 𝑉𝑏𝑖1 𝑒
−
𝐿1
𝜆1 , 𝑞12 = (𝑉𝑏𝑖2 + 𝑉𝑑 𝑠 − 𝜎2)𝛾 − 𝑞11 𝑒
𝐿
𝜆2 𝛾 = 2𝛼𝑆𝑖𝑛ℎ (
𝐿1
𝜆1
) − 𝑒
𝐿1
𝜆1 , 𝛿 = 2𝛽𝑆𝑖𝑛ℎ (
𝐿1
𝜆1
) − 𝑒
−
𝐿1
𝜆1
Therefore, electrostatic potential in the respective region is given as;
𝜑𝑗(𝑥,𝑦) = 𝜙 𝑠𝑗 [1 +
𝐶 𝑜𝑥𝑗
(
𝑡 𝑠𝑖
2
4
− 𝑦2
)] +
𝐶 𝑜𝑥𝑗
𝑉𝑔𝑠𝑓 (
𝑡 𝑠𝑖
2
4
− 𝑦2
) (8)
At the interface, 𝑦 =
𝑡 𝑠𝑖
2
(8) reduces to 𝜑𝑗 (𝑥,
𝑡 𝑠𝑖
2
) = 𝜙 𝑠𝑗(𝑥) . The centre potential is obtained for y=0 in (8).
From analysis it is observed that surface potential is smaller than the centre potential particularly
at the source and drain ends respectively. Tunnelling barrier width (𝑤 𝑏) is an important parameter to determine
the threshold voltage of the TFET. Tunnelling width exhibits a transition from strong dependence to weak
dependence on the gate voltage at threshold voltage. 𝑤 𝑏 is the lateral distance between the source channel

1767
interface (𝑥 = 0) and the point in the channel when surface potential changes by (
𝐸 𝑔
𝑞
). Neglecting source/drain
depletion region due to heavy doping, the tunnelling width is obtained after solving eqn. (8) and given as;
𝑤 𝑏
𝜆1
= ln [
ᵹ1
2
{1 + √1 −
4ᵹ
ᵹ1
2 }] (9)
Where, ᵹ = (
𝐵1
𝐴1
) and ᵹ1 =
(𝐴1+𝐵1−
𝐸 𝑞
𝑞
)
𝐴1
The physical definition of threshold voltage Vth for TFETs is the gate voltage which indicates
the transition between the strong control and weak control of the tunneling energy barrier width at the source
tunnel junction [20]. Tunneling barrier width exhibits a transition from strong dependence to weak dependence
on gate voltage at gate threshold voltage. At this inflection point, x = 𝑤 𝑏 and the surface potential reaches to
value of 𝑉𝑑𝑠 +
𝑘𝑇
𝑞
ln
𝑁 𝑑
𝑁 𝑐
[23]. Substituting 𝑥 =̃ 𝑤 𝑏 in eqn. (8) and equating it to value of
𝑘𝑇
𝑞
ln
𝑁 𝑑
𝑁 𝑐
, the analytical
expression for the threshold voltage is;
Vth =
h11 − √h11
2
− L ∗ h12
2
(10)
Where, ℎ11 =
(
𝐸 𝑞
𝑞
)+ᵹ11ᵹ14+ᵹ12ᵹ13−(𝛼11ᵹ12+ᵹ11+ᵹ11ᵹ14−ᵹ13)
ᵹ12+ᵹ14−ᵹ12ᵹ14
, ℎ12 =
𝛼11ᵹ11+𝛼11ᵹ13−𝛼11(
𝐸 𝑞
𝑞
)−ᵹ11ᵹ13
ᵹ12+ᵹ14−ᵹ12ᵹ14
𝛼11 = 𝑉𝑑𝑠 +
𝑘𝑇
𝑞
ln
𝑁 𝑑
𝑁𝑐
−
𝑞𝑁𝑐
𝜖 𝑠𝑖
𝜆1
2
− 𝑉𝑓𝑏 , ᵹ11 = 𝛼𝜉13 + 𝛽𝜉11 + 𝑅11 𝑒
−
𝐿1
𝜆1 , ᵹ12 = 𝛽𝜉12 − 𝛼𝜉14, ᵹ13 = 𝑉𝑏𝑖1 −
𝑞𝑁𝑐
𝜖 𝑠𝑖
𝜆1
2
− 𝑉𝑓𝑏 −
ᵹ11, ᵹ14 = (1 − ᵹ12), 𝜉11 = (
𝜂12
𝜂14
) , 𝜉12 = (
𝜂13
𝜂14
), 𝜉11 =
𝜂11−𝜉11 𝛿
𝛾
, 𝜉11 = (
𝑒
−
𝐿1
𝜆1+𝜉12 𝛿
𝛾
), 𝜂11 = (𝑅11 𝑒
−
2𝐿1
𝜆1 − 𝑉𝑏𝑖1 𝑒
−
𝐿1
𝜆1) +
(
𝑞𝑁𝑐
𝜖 𝑠𝑖
𝜆1
2
− 𝑉𝑓𝑏)(𝑒
−
𝐿1
𝜆1 −
1
2
) +
1
2
(
𝑞𝑁𝑐
𝜖 𝑠𝑖
𝜆1
2
+ 𝑉𝑓𝑏), 𝜂12 = (𝑉𝑏𝑖2 + 𝑉𝑑𝑠)𝛾 − (
𝑞𝑁𝑐
𝜖 𝑠𝑖
𝜆2
2
+ 𝑉𝑓𝑏) 𝛾 − 𝜂11, 𝜂13 = 𝛾 − 𝑒
−
𝐿1
𝜆1, 𝜂14 =
𝛾𝑒
−
𝐿
𝜆2 − 𝛿𝑒
𝐿
𝜆2
3. RESULTS AND ANALYSIS
The proposed analytical models are simulated for the following values; L=50 nm, tox1=tox2=3 nm,
tsi=10 nm, Vds=0.7 V, Ns=1020
/cm3
, Nd=1018
/cm3
, Nc=1016
/cm3
unless and until specified. As seen from
Figure 2, a sharp band bending occurs for tox1=3 nm and tox2=5 nm due to reduction of equivalent oxide
thickness which enhances the coupling between the gate and the channel junction.
Figure 2. Band diagram for different combination of oxide thickness in two regions
The surface potential mainly varies within 10 nm from source-channel interface that is entirely in
high-k region whereas away from tunneling space (mainly in low-k region) it remains almost constant which
reflects that tunneling is only controlled by high-k dielectric. This finding suggests that the optimum tunneling
length is set to be at 10 nm for higher ON current. Figure 3(a) predicts that the surface potential takes larger
value when the channel concentration is 1016
/ cm3
. The effect of Nc on surface potential mainly occurs in high-

 ISSN: 2088-8708
1768
k region for x≤10 nm. This result gives the optimum value of the channel doping in the HGD TFET device.
From Figure 3(b) it is observed that electric field takes larger value when high-k region is occupied by higher
dielectric material and the lateral electric field reduces along the channel and takes minimum value in
the middle of the channel. The stronger electric field near the source-channel interface enhances tunneling
probability whereas lower electric field near drain lowers the ambipolar current. The comparison results of
the tunneling width of our proposed model with the 2-D Silvaco simulator and ref [16] results are shown in
Figure 4. A slight difference between proposed and ref [16] results is due to structural difference whereas
a good agreement with the Silavco 2-D simulation results is observed. For the fair comparison, the simulation
is performed for the same value of the parameters as reported in the ref. [16]. The surface potential close to
the source end becomes more abrupt due to increased doping concentration ratio between the source and drain
regions, which results in improved electrical characteristics and narrow tunneling width which results in lower
threshold voltage irrespective of the channel length Figure 5(a). As the length of high-κ region reduces,
conduction band becomes shallow which makes band-to-band tunneling difficult and increases the threshold
voltage Figure 5(b). Table 1 gives the comparison results of the proposed threshold voltage model with ref [13]
results. The two results show close proximity with slight difference due to dual material gate in ref. [13].
For fair comparison we have taken L=50 nm, L1=8 nm, kr2=21, kr1=3.9, metal work function 4.0 eV and
4.4 eV as suggested in ref. [13]. Increased relative dielectric constant in high-κ region results in stronger control
of gate over the channel which gives larger carrier tunnel from source to channel and hence lowers the threshold
voltage irrespective of the metal type Figure 5(c). This reduction is more for lower work function gate metal
due to larger band overlap which reduces the tunneling width and increases the tunneling probability at
the source side. Figure 5(d) shows the comparison between proposed model results and 2-D ATLAS simulator
results with an excellent matching for the higher dielectric gate material due to controlled leakage current.
Figure 5(e), shows an excellent matching between two results for Vds=0 V due to absence of DIBL effect
whereas appreciable difference is observed for L≤20 nm and Vds=0.7 V due to the negligence of SCEs in
proposed model.
(a) (b)
Figure 3. (a) Surface potential along channel for different channel concentration,
(b) Variation of electric field along channel for different relative permittivity combination
Figure 4. Comparison of tunneling width for proposed model, Silvaco 2-D simulator and ref. [16] results

1769
(a) (b)
(c) (d)
(e)
Figure 5. (a) Threshold voltage of HGD TFET device versus channel length for different concentrations
combination in the source and drain regions, (b) Threshold voltage variation with tox for different
tunneling length (L1), (c) Threshold voltage variation with relative permittivity in region I for two gate
metals, (d) Comparison of threshold voltage with proposed model and 2-D ATLAS simulator results,
(e) Comparison of threshold voltage with proposed and 2-D ATLAS simulator
Table 1. Threshold Voltage comparison with ref. [13] results
Φm1 (eV) Φm2 (eV) Threshold voltage (V)
Proposed Ref [13
4.0 4.0 0.349 0.34
4.4 4.4 1.10 0.74

 ISSN: 2088-8708
1770
4. CONCLUSION
A two dimensional analytical model of threshold voltage has been proposed using parabolic
approximation. The proposed model shows a good agreement with the 2-D ATLAS simulator results.
As the gate voltage increases, the tunneling width reduces which lowers the threshold voltage.
Threshold voltage of the device also reduces as the dielectric constant of the tunnel space region increases and
the length of high-k region increases.
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BIOGRAPHIES OF AUTHORS
Dr. Ajay Kumar Singh is working as an Associate Professor in Faculty of Engineering and
Technology, Multimedia University-Malaysia since 2014. He has supervised 4 PhDs and 6 Master
theses. His areas of interest are modeling of submicron MOS devices, Low power VLSI circuit
design and Renewable energy source. He has published more than 80 research papers in various
International Journals and conferences. Dr Singh has served in many technical committees of
International conferences and reviewer in many journals. He has authored 3 books and contributed
two chapters in edited book. Dr Singh is senior member of IEEE and Fellow-IET (India).
Mr. Tan Chun Fui is academician in Multimedia University, Melaka. He completed his Bachelor
of Science in Mathematics (BSc Honours), in the area of Applied Mathematics in year 2009 from
Universiti Teknologi Malaysia (UTM).. He obtained a Master of Science in Engineering
Mathematics, in the area of Non-linear Waves and Solitons. At present he is pursuing his PhD from
MMU-Melaka on the problem related to the HGD TFET device.
Tan Wee Xin Wilson his bachelor degree in Electronic Engineering from Multimedia University-
Melaka-Malaysia. He was a Research Assistant in Center of Communication system and ICs Design-
Multimedia University. Mr Tan was also Research Intern in EDA Lab, National Chung Cheng
University, Taiwan. At present he is a Tapeout engineer in GLOBALFOUNDRIES- Singapore.
His interest is in IC Design/process and Analog/Digital circuit design.

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