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Modeling of dual material surrounding split gate junctionless transistor as biosensor
Superlattices and Microstructures 135 (2019) 106290
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Modeling of dual material surrounding split gate junctionless transistor as biosensor Muktasha Maji a, *, Gaurav Saini b a b
School of VLSI Design and Embedded Systems, National Institute of Technology, Kurukshetra, India Department of Electronics and Communication Engineering, National Institute of Technology, Kurukshetra, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Junctionless transistor Field effect transistor Biosensor Split gate Dielectric modulation Modeling Sensitivity Double gate Dual material
In this paper an analytical model of dual material surrounding split gate junctionless transistor for application as a biosensor is introduced. The simulation results show fair compatibility with the analytical models developed for the channel center potential and threshold voltage at channel length of 45 nm and 20 nm. For the detection of the bio-molecules, concept of dielectric modu lation has been used. The model shows a satisfactory value of sensitivity in the range of 0–0.27 and 0–0.8 for relative permittivity of biomolecules varying from 1 to 9 at channel length of 45 nm and 20 nm respectively, thus making it an attractive option for use as biosensor. This model has been demonstrated with the aid of TCAD device simulations.
1. Introduction WITH the evolution of CMOS technology into nanoscale club, the challenges to the semiconductor industry is on the rise. This is due to the heavy doping in the ultra-shallow junctions which makes the fabrication process difficult. Further at such scale the effect of short channel effects (SCEs) also increases. To address all this, Colinge et al. [1] proposed and later fabricated a new device called junc tionless transistor (JLT). Conventionally it does not have any junctions and also have no doping gradient. The main advantage of these transistors depends on improved and easier fabrication process. Also, it is less susceptible to SCEs. The basic principle behind working of JLT is a narrow channel which allows full depletion of carriers in the OFF-state and heavy doping for a satisfactory level of current flow in the ON-state. It basically acts as a variable resistor which is controlled by a gate electrode. The electrical characteristics of JLT are similar to that of conventional MOSFETs but with a different physics behind it. The biosensors have huge application in biomedical field mainly because it can help in the early detection of various kinds of diseases like Alzheimer’s, ovarian cancer and few viral diseases. Different types of biosensors have been made such as a nano-mechanic devices [2], optical [3], piezoelectric [4], electrochemical [5] etc. But these devices need expensive equipment and have lofty manufacturing cost. As a result, it led to the exposure of semiconductor devices based biosensors. Among these devices, field effect transistor (FET) biosensors have been extensively used to detect biomolecules [6,7]. FET based biosensors have a simple detection system and does not need a transducer. Further, the use of JLTs as biosensors along with the immense advancement in the fabrication industry offers cheap mass production of these biosensors. The detection is done by considering dry environment. In the literature, split gate structures have been proposed for JLT based biosensor [8]. In this, the center of the gate oxide is carved out to create a cavity with
* Corresponding author. E-mail addresses: [email protected] (M. Maji), [email protected] (G. Saini). https://doi.org/10.1016/j.spmi.2019.106290 Received 1 May 2019; Received in revised form 30 September 2019; Accepted 1 October 2019 Available online 5 October 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 135 (2019) 106290
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Fig. 1. Structure of dual material surrounding split gate JLT. Table 1 Device specifications. Symbol
Quantity
Specification
LCH L1 LCAV L2 tSi tSiO2 tHfO2
Channel Length Length of Metal M1 Cavity Length Length of Metal M2 Silicon Thickness Silicon Dioxide Thickness Hafnium Dioxide Thickness
45 nm 15 nm 15 nm 15 nm 10 nm 1 nm 4 nm
tbio WF1 WF2 kSiO2 kHfO2 ND
Cavity Thickness Work function of metal M1 Work function of metal M2 Dielectric constant of SiO2 Dielectric constant of HfO2 Doping concentration
19 nm 5.5eV 5eV 3.9 22 1019 cm
3
oxides on its either sides. This cavity acts as the biomolecules sensing area. When the biomolecules are present in this area the properties of the transistor change as compared to that in its absence. During the absence of the biomolecules, the cavity is packed with air which has a unit dielectric constant. Further when the biomolecules are there in the cavity, it is assumed to be filled with materials possessing distinct dielectric constant greater than unity. The variation in the permittivity of the biomolecule leads to a change in the threshold voltage of the device. This is used as a procedure to calculate the sensitivity of the device. Different types of JLTs based biosensors have been presented in the literature. Buitrago et al. [9] showed the biosensing capability of JLTs. Further, Narang et al. [10] demonstrated dielectric modulation on a double-gate JLT based biosensor. Ahangari et al. [11] and Parihar et al. [12] presented dual material JLT based biosensors. Barik et al. [13] demonstrated a dual-gated JLT based on Carbon Nano Tube (CNT) for detection of acetylcholine. Chakraborty et al. [14] performed sensitivity analysis of dielectric-modulated gate stack JLT for application as 2
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biosensor. Ajay et al. [15] modelled gate underlap JLT as bio-sensor. Ouarghi et al. [16] analysed the impact of triple-material gate on the sensitivity of DNA biosensors. In this paper, gate oxide stack has been used to enhance the device performance. Dual material gate has been used to further boost the performance of the biosensor. A metal of high work function is kept at the source end to reduce leakage current whereas a metal of lower work function is kept near the drain to maintain the ON-state current. Typical sensitivity values reported in literature are in the range of 0–0.2 [8,10,14,15] for relative permittivity varying from 1 to 9. The proposed device has an enhanced sensitivity in the range of 0–0.27 for the same range of relative permittivity. The work discussed in this paper is arranged as follows. Section II includes the simulation setup and device structure. Section III has the developed analytical model of the channel center potential and threshold voltage. Section IV has the results and discussions. In Section V, the conclusion is presented. 2. Simulation setup and device structure Fig. 1 shows the two-dimensional device structure of the proposed biosensor. The standard parameters considered for the analysis of this sensor are listed in Table 1. For 20 nm technology, the channel length has been divided into three equal parts of 6.667 nm each for simulation. We have used metal M1 work function as 5.5eV. The use of materials such as platinum and p þ polycrystalline silicon with high work function has been reported by Chanda et al. [17], Rewari et al. [18], Ghosh et al. [19], Colinge et al. [20] and Lee et al. [21,22]. The utilization of a high work function metal of 5.5 eV may result in a challenge for large scale manufacturing of the proposed biosensor because it is difficult to obtain good interfacing of the gate electrode material with that of the dielectric. Further thermal stabilization of the device becomes difficult. However, it does not have much effect on the general course of the findings. The default cavity length considered is 15 nm. However, it has been varied to study the effect of cavity length on the biosensor sensitivity. It is worth mentioning that the model has been verified for neutral biomolecules only such as streptavidin (K ¼ 2.1) [23], protein (K ¼ 2.50) [24], biotin (K ¼ 2.6) [24], and APTES (K ¼ 3.57) [25]. The existence of a biomolecule in the cavity of the biosensor is simulated by substituting air with unity dielectric constant by a higher dielectric material. Without the loss of generalizability, we have considered K varying from 1 to 9 to study the effects of presence of biomolecules. TCAD [26] has been used for the simulations. It is widely used for research and production purposes in the semiconductor industry. It simulates device behavior based on different fundamental physical models. The models used by us to study the parameters of proposed biosensors are Drift Diffusion model for carrier transport, Field dependent and Constant mobility model, Shock ley–Read–Hall recombination model, Fermi Dirac statistics model, doping dependent mobility degradation model and Auger recombination model. 3. Analytical model 3.1. Channel center potential The channel center potential of the proposed biosensor has been modelled by the help of 2-D Poisson’s equation. We have considered cylindrical coordinate system for the analysis. The 2-D Poisson’s equation [14] can be written as � � 1 ∂ ∂ ψ ðr; zÞ ∂2 ψ ðr; zÞ qN D ¼ (1) r þ r ∂r ∂r ∂z2 εSi Here r is the radius with r ¼ 0 corresponding to the center of the channel and r ¼ R ¼ t2si corresponding to the surface of the channel, z represents the direction along the length of the channel with z ¼ 0 referring to the source end and z ¼ LCH referring to the drain end, ψ(r,z) is the 2-D potential distribution along the channel, q represents the charge of electron and ND is the constant doping concentration. The general solution of Equation (1) can be approximated according to parabolic potential profile as [27]. (2)
ψ ðr; zÞ ¼ AðzÞ þ BðzÞr þ CðzÞr2 The constants A (z), B (z) and C (z) are to be evaluated from the following boundary condition: 1:ψ ðr; zÞjr¼0 ¼ ψ C ðzÞ
(3)
2:ψ ðr; zÞjr¼R ¼ ψ S ðzÞ
(4)
� ∂ψ ðr; zÞ�� 3: ¼0 ∂r � r¼0
(5)
�
∂ψ ðr; zÞ�� Ci fVGS r¼R ¼ ∂r � εsi
VFBi
(6)
ψ S ðzÞg
Here, ΨC(z) represents the channel center potential of the device, ΨS(z) represents the device surface potential, Ci is the effective capacitance in the region i (i ¼ 1,2,3), εsi is the permittivity of Silicon, VGS is the gate voltage applied to the device and VFBi refers to the flat band voltage of the region i. 3
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Capacitance Ci can be written as C1 ¼ C3 ¼
C2 ¼
CSiO2 CHfO2 CSiO2 þ CHfO2
(7)
CSiO2 Cbio CSiO2 þ Cbio
(8)
Let Cx represent the capacitances CSiO2 ; CHfO2 andCbio [14]. Cx ¼
2ε � � x tsi ln 1 þ 2ttsix
(9)
Here εx is the permittivity of the material ‘x’, tx is the thickness of material ‘x’ and tsi is the thickness of silicon. Substituting Equations (3)–(6) we get, � � � � RCi RCi A z ¼ ψ S ðzÞ 1 þ ðVGS VFBi Þ 2εsi 2εsi
(10) (11)
BðZÞ ¼ 0 � � Ci C z ¼ fVGS 2Rεsi
(12)
ψ S ðzÞg
VFBi
Further substituting Equation (10)–(12) in Equation (2) we get equation of potential in region i (i ¼ 1,2,3) as, � � RC RCi Ci r 2 ψ i ðr; zÞ ¼ ψ Si ðzÞ 1 þ i ðVGS VFBi Þ þ fVGS VFBi ψ Si ðzÞg 2εsi 2εsi 2Rεsi
(13)
From Equation (3) we get (14)
ψ C ðzÞ ¼ AðzÞ
Substituting Equation (14) in equation (10), we get relation between surface potential ψ Si ðzÞ and channel potential ψ Ci ðzÞ as
ψ Si ðzÞ ¼
RCi ðVGS VFBi Þ 2εsi þ ψ ðzÞ 2εsi þ RCi 2εsi þ RCi Ci
(15)
Putting Equation (13) in equation (1) a 2nd order differential equation of surface potential is obtained
∂2 ψ Si ðzÞ ∂z2
2Ci 2C ðV VFBi Þ ψ ðzÞ þ i GS ¼ Rεsi Rεsi Si
qN D
(16)
εsi
This equation can thus be written in terms of the channel potential ψ Ci ðzÞ by substituting Equation (15) in (16) as
∂2 ψ Ci ðzÞ ∂z2
1
αi 2
(17)
ψ Ci ðzÞ ¼ βi
The coefficients αi and βi have been mentioned in the Appendix-A. The general solution of differential equation (17) can be written in a general form as z
ψ Ci ðzÞ ¼ mi eαi þ pi e
z
(18)
αi 2 βi
αi
The channel region has been split into three distinct areas as shown in Fig. 1. The channel center potential in each region can be written as. Region 1: z1 � z � z2 z
ψ C1 ðzÞ ¼ m1 eα1 þ p1 e
α1 2 β 1
(19)
α2
z
α2 2 β 2
(20)
z
α3 2 β 3
(21)
z
α1
Region 2: z2 � z � z3 z
ψ C2 ðzÞ ¼ m2 eα2 þ p2 e Region 3: z3 � z � z4 z
ψ C3 ðzÞ ¼ m3 eα3 þ p3 e
α3
The values of the coefficients mi and pi (i ¼ 1, 2 and 3) in Equation (19)–(21) are calculated in terms of channel center potentials V1, V2, V3 and V4 at z1, z2, z3 and z4 respectively.
4
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 2. Calibration of TCAD simulation models with experimental data [28].
mi ¼
Vi
α2i βi
� 1
pi ¼
Viþ1
α2i βi 1
�
e
�
2ðZiþ1
αi
e
�
2ðZiþ1
αi
αi
Zi
(22)
Zi Þ
α2i βi e
Vi
Ziþ1
α2i βi e
Viþ1
Ziþ1
αi
Zi
(23)
Zi Þ
The potentials at source end and drain end are known and are given as [14]. V1 ¼ 0
(24)
V4 ¼ VDS
(25)
Here VDS is the drain to source applied potential. The other potentials V2 and V3 can be obtained from the electric field continuity equation at z ¼ z2 and z ¼ z3. � � d ψ C1 ðzÞ �� d ψ C2 ðzÞ�� ¼ z¼z z¼z dz � 2 dz � 2 � d ψ C2 ðzÞ �� d ψ C3 ðzÞ z¼z ¼ dz � 3 dz
� � � z¼z � 3
(26) (27)
Solving this using Crammer’s rule we get � � � F1 S12 � � � � F2 S22 � � V2 ¼ �� � � S11 S12 � � S21 S22 �
(28)
� � � S11 F1 � � � � S21 F2 � � V2 ¼ �� � � S11 S12 � � S21 S22 �
(29)
The coefficients F1, F2, S11, S12, S21 and S22 have been mentioned in the Appendix-A.
5
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 3. Plot of channel center potential versus position along z-direction for different biomolecules for (a) 45 nm channel length (b) 20 nm channel length.
3.2. Threshold voltage In JLTs the threshold voltage corresponds to that gate voltage when the minimum channel center potential is zero. The position of the least channel-center potential (Zvth Þ depends on the metal gate work function and is calculated to be the minimum of the threshold values in each region. It can be calculated as d ψ C1 ðzÞ ¼0 dz Zvth
(30) 6
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 4. Plot of threshold voltage versus relative permittivity of different biomolecules for (a) 45 nm channel length (b) 20 nm channel length.
Applying this in Equation (18) we get � � αi pi Zvth ¼ ln 2 mi
(31)
Substituting Equation (31) in Equation (18) we have minimum channel center potential as pffiffiffiffiffiffiffiffi ψ Cimin ðzÞ ¼ 2 mi pi αi 2 βi Putting value of mi and pi from Equation (22) and (23) we get threshold voltage in region i as
7
(32)
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 5. Plot of variation of sensitivity of the device with relative permittivity of different biomolecules for (a) 45 nm channel length (b) 20 nm channel length.
VTHi ¼
w2i þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w22i 4w1i w3i 2w1i
(33)
The overall threshold voltage of the device can be calculated as minimum (VTHi). The coefficients w1i, w2i and w3i [15] are given in Appendix B. 4. Results and discussion Calibration to validate the TCAD simulation models with experimental data [28] has been done in Fig. 2. The proposed model has 8
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 6. Plot of variation of sensitivity of the device with LCAV for 45 nm technology.
Fig. 7. Plot of variation of sensitivity of the device with tbio for 45 nm technology.
been compared with TCAD simulation. The channel center potential and the threshold voltage have been plotted. We find decent similarity between the proposed model and simulation. Further the sensitivity has also been plotted. Since sensitivity is an important parameter for biosensors, variations of sensitivity with respect to various parameters have also been shown. Fig. 3 (a) shows the changes in the surface potential along the length of the channel when different biomolecules are placed in the cavity for 45 nm technology and Fig. 3. (b) shows the same for 20 nm technology. When the dielectric constant of the biomolecules increases the minimum surface potential decreases in both the cases. This is because when dielectric constants of biomolecules present in the cavity increases, the effective gate capacitance also increases along with vertical electric field and channel center potential resulting in a change in the flat band voltage in the cavity region. Variation in the threshold voltage is a key consideration in case of biosensors as it helps to determine the sensitivity of the device. We can see in Fig. 4. (a) and (b) that with increasing permittivity of biomolecules the threshold voltage increases. This is because with decrease in channel center potential, more gate voltage is needed to completely deplete the channel, thereby increasing the threshold voltage. We also observe that with reduction in channel length the threshold voltage reduces for each biomolecules thereby affecting its sensitivity. The biosensor sensitivity is determined as the ratio of the difference in threshold voltage of the device when the biomolecule is present and absent in the cavity to the threshold voltage of the device when the biomolecule is absent in the cavity [15]. 9
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 8. Plot of electron concentration versus position along z-direction for different biomolecules for 45 nm technology.
Fig. 9. Plot of drain current versus gate voltage for different biomolecules for 45 nm technology.
S¼
VTHðK>1Þ
(34)
VTHðK¼1Þ
VTHðK¼1Þ
From Fig. 5. (a) and (b) we can see that the sensitivity of biosensor is increasing with increase in the relative permittivity of the biomolecule existing in the cavity. This can be explained from the increase in threshold voltage as shown in Fig. 4 (a) and (b) respectively. We can clearly see that this device has good sensitivity and can thus be used in practical applications for sensing of the biomolecules. Since the biosensor sensitivity is important in determining its usability, variation of sensitivity with respect to length of cavity (LCAV) and thickness of cavity (tbio) have also been studied. The proposed model shows good agreement with that of simulation. Fig. 6 plots the variation in the sensitivity of biosensor with the length of cavity (Lcav) for a particular biomolecule present in the cavity for 45 nm technology. Here we have considered K ¼ 9 for simulation. It is clear that with increasing Lcav the sensitivity S of the device increases. This is because with increasing length of biomolecule cavity, the effective gate capacitance increases, lowering the channel center potential and hence increasing the threshold voltage as well as sensitivity. Fig. 7 plots the variation in the sensitivity of biosensor with the thickness of the cavity (tbio) for a particular biomolecule present in the cavity for 45 nm technology. Here we have considered K ¼ 9 for simulation. It is seen that with increasing tbio the sensitivity S of the 10
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
device increases. This is because with increasing thickness of biomolecule cavity, the effective gate capacitance increases, lowering the channel center potential and hence increasing the threshold voltage as well as sensitivity. Fig. 8 plots the electron concentration along the channel of the biosensor along the position along z-direction for different bio molecules for 45 nm technology. The trend in electron concentration can be explained on the fact that with increasing dielectric constant of the biomolecule present in the cavity the channel center potential is lowered, consequently lowering the electron con centration in the channel as well. Fig. 9 shows the drain current versus gate voltage for different biomolecules in the cavity for 45 nm technology. The greater current ratio with the increasing dielectric constant is due to the reduction in the channel center potential. Due to this the sensitivity of this device is also improved. 5. Conclusion In this paper, the models for channel surface potential and threshold voltage of dual material surrounding split gate junctionless transistor have been derived. Sensitivity of the device has been defined in terms of change in threshold voltage of the device. The sensitivity analysis of the device has been done with respect to parameters such as length of the cavity and thickness of the cavity. The simulation results show fair compatibility with the analytical models developed for the channel center potential, threshold voltage and sensitivity at channel length of 45 nm and 20 nm. The decent value of sensitivity of the device along with easy fabrication process of junctionless transistor makes the proposed device an attractive option in the biosensor industry. Acknowledgement We would like to acknowledge the Ministry of Electronics and Information Technology(MeitY) sponsored SMDP-C2SD program for providing the necessary tools. Appendix A sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4tsi εsi þ Ci t2si αi ¼ 16Ci βi ¼
�
1
VGS
α2i
VFBi þ
0 L1 α1
B C 2 C þ α2 β 1 B 1 B L1 A @eα1 e
0
2L1 2L1
e α1 S12 ¼
S21 ¼
S22 ¼
e L1 α1
B C 2 2 2 B C L2 A þ α2 β2 B Lcav @ α2 α3 e e e
B F2 ¼ VDS B @ Lα2 e3
þ
2Lcav α2
þ1
2Lcav α2
1
e
e
1
0
1 2L1 α1 2L1 α1
e
e
2Lcav α2
e
e
Lcav α2
2 Lcav α2
e
2Lcav α2
þ 1
2Lcav α2
1
e e
e
Lcav α2
2L2
þ
e α3 þ 1 2L2
e α3
e Lcav α2
B þ 1C 2 B C C þ α23 β3 B L2 @ A α 1 e3 e
2 Lcav
e α2
2Lcav α2 2Lcav α2
e
þ 1C C C 1A
0
1 2Lcav α2
Lcav α2
1
B þ 1C 2 B C C þ α22 β2 B Lcav @e α2 A e 1
0
1
e α1 þ 1
�
0
1
B 2 F1 ¼ Vbi B @ Lα1 e1 e
S11 ¼
qND tsi qND t2si þ 4Ci 16εsi
1
11
1
L2 α3
e
2L2 α3
e
2L2 α3
þ 1C C C 1A
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Appendix B 4τ1i τ3i
w1i ¼ 1 w2i ¼ 2θi
4 τ2i τ3i þ τ1i τ4i þ ϕf
w3i ¼ θi 2
4 τ2i τ4i þ ϕf θi qN D
θi ¼
1 e
τ2i ¼
Ziþ1
Hi
Ziþ1
αi
e
Zi
e
Zi
αi
Ziþ1
αi
�
Zi
Ziþ1
e
Ziþ1
τ3i ¼
αi
θi Þe
ðVi
ϕf 2
VFBi
εSi αi 2 βi
τ1i ¼
�
αi
Zi
Zi
θi Þ
ðJi
e
Ziþ1
αi
Zi
τ1i
1
τ4i ¼ Vi
θi
τ2i
Hi ¼ E1i
F1i
G1i
Ji ¼ E2i
F2i
G2i
0
1
Ziþ1 Zi 2ðZiþ1 1B e αi B 2 e αi E1i ¼ þ B 2ðZiþ1 αi @ 2ðZiþ1αi Zi Þ e 1 e αi
0
F2i ¼
11 2ðZiþ1
Be αi B þ θi B 2ðZ @ iþ1αi e
Zi Þ Zi Þ
C þ 1C CC CC AA 1
1 Zi
αi @
þ
e e
2ðZiþ1
Zi Þ
2ðZiþ1
Zi Þ
αi αi
þ 1C C C A 1
θi F1i 0
2ðZiþ1 1B Be αi G1i ¼ B 2ðZiþ1 αi @ e αi
G2i ¼
þ 1C C C A 1
0
Zi
0 1B B Ziþ1αi Be
Zi Þ
1
0 B C Ziþ1 2@Vi θ i A e αi 1B B E2i ¼ B 2ðZiþ1 Zi Þ αi @ e αi 1
F1i ¼
Zi Þ
1 Zi Þ Zi Þ
þ 1C C C A 1
θi G1i
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.spmi.2019.106290.
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Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
References [1] J.P. Colinge, et al., Nanowire transistors without junctions, Nat. Nanotechnol. 5 (3) (February 2010) 225–229. [2] J. Fritz, et al., Translating biomolecular recognition into nano-mechanics, Science 288 (5464) (April 2000) 316–318. [3] S.W. Oh, et al., Calixarene derivative as a tool for highly sensitive detection and oriented immobilization of proteins in a microarray format through noncovalent molecular interaction, FASEB J. 19 (10) (June 2005) 1335–1337. [4] Y.T. Yang, et al., Zeptogram-scale nanomechanical mass sensing, Nano Lett. 6 (4) (March 2006) 583–586. [5] T.G. Drummond, M.G. Hill, J.K. Barton, Electrochemical DNA sensors, Nat. Biotechnol. 21 (10) (October 2003) 1192–1199. [6] Y. Cui, Q. Wei, H. Park, C.M. Lieber, Nanowire nanosensors for highly sensitive and selective detection of biological and chemical species, Science 293 (5533) (August 2001) 1289–1292. [7] K.I. Chen, B.-R. Li, Y.T. Chen, Silicon nanowire field-effect transistor-based biosensors for biomedical diagnosis and cellular recording investigation, Nano Today 6 (2) (April 2011) 131–154. [8] S. Singh, B. Raja, S.K. Vishvakarma, Analytical modeling of split-gate junction-less transistor for a biosensor application, Sensing and Bio-Sensing Research 18 (April 2018) 31–36. [9] E. Buitrago, G. Fagas, M.F.B. Badia, Y.M. Georgiev, M. Berthom� e, A.M. Ionescu, Junctionless silicon nanowire transistors for the tunable operation of a highly sensitive, low power sensor, Sens. Actuators B Chem. 183 (July 2013) 1–10. [10] R. Ajay, M. Narang, M. Saxena, Gupta, Investigation of dielectric modulated (DM) double gate (DG) junctionless MOSFETs for application as a biosensors, Superlattice Microstruct. 85 (September 2015) 557–572. [11] Z. Ahangari, Performance assessment of dual material gate dielectric modulated nanowire junctionless MOSFET for ultrasensitive detection of biomolecules, RSC Adv. 6 (92) (September 2016) 89185–89191. [12] M.S. Parihar, A. Kranti, Enhanced sensitivity of double gate junctionless transistor architecture for biosensing applications, Nanotechnology 26 (14) (March 2015) 145201. [13] M.A. Barik, R. Deka, J.C. Dutta, Carbon nanotube-based dual gated junctionless field-effect transistor for acetyl choline detection, IEEE Sens. J. 16 (2) (January 2016) 280–286. [14] A. Chakraborty, A. Sarkar, Analytical modeling and sensitivity analysis of dielectric-modulated junctionless gate stack surrounding gate MOSFET (JLGSSRG) for application as biosensor, J. Comput. Electron. 16 (3) (September 2017) 556–567. [15] R. Ajay, M. Narang, M. Saxena, Gupta, Modeling of gate underlap junctionless double gate MOSFET as bio-sensor, Mater. Sci. Semicond. Process. 71 (November 2017) 240–251. [16] M. Ouarghi, Z. Dibi, N. Hedjazi, Impact of triple-material gate and highly doped source/drain extensions on sensitivity of DNA biosensor, J. Comput. Electron. 17 (4) (December 2018) 1797–1806. [17] M. Chanda, P. Dey, S. De, C.K. Sarkar, Novel charge plasma based dielectric modulated impact ionization MOSFET as a biosensor for label-free detection, Superlattice Microstruct. 86 (October 2015) 446–455. [18] S. Rewari, V. Nath, S. Haldar, S.S. Deswal, R.S. Gupta, Improved analog and AC performance with increased noise immunity using nanotube junctionless field effect transistor (NJLFET), Appl. Phys. A 122 (12) (December 2016), 1049. [19] B. Ghosh, M.W. Akram, Junctionless tunnel field effect transistor, IEEE Electron. Device Lett. 34 (5) (May 2013) 584–586. [20] A. Kranti, et al., Junctionless nanowire transistor (JNT): properties and design guidelines, in: 2010 Proceedings of the European Solid State Device Research Conference, Sevilla, 2010, pp. 357–360. [21] C.W. Lee, et al., Performance estimation of junctionless multigate transistors, Solid State Electron. 54 (2) (2010) 97–103. [22] C.W. Lee, et al., Junctionless multigate field-effect transistor, Appl. Phys. Lett. 94 (5) (2009), 053511. [23] K.W. Lee, et al., “An underlap field-effect transistor for electrical detection of influenza, Appl. Phys. Lett. 96 (3) (January 2010), 033703. [24] S. Busse, V. Scheumann, B. Menges, S. Mittler, Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods, Biosens. Bioelectron. 17 (8) (August 2002) 704–710. [25] A. Densmore, et al., Spiral-path high-sensitivity silicon photonic wire molecular sensor with temperature-independent response, Opt. Lett. 33 (6) (2008) 596–598. [26] TCAD Sentaurus Device User’s Manual, Synopsys, Mountain View, CA, 2014. [27] K.K. Young, “Short-channel effects in fully depleted SOI MOSFET’s, IEEE Trans. Electron Devices 36 (2) (February 1989) 399–402. [28] Y.K. Choi, et al., Ultrathin-body SOI MOSFET for deep-sub-tenth micron era, IEEE Electron. Device Lett. 21 (5) (May 2000).
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Modeling of dual material surrounding split gate junctionless transistor as biosensor Muktasha Maji a, *, Gaurav Saini b a b
School of VLSI Design and Embedded Systems, National Institute of Technology, Kurukshetra, India Department of Electronics and Communication Engineering, National Institute of Technology, Kurukshetra, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Junctionless transistor Field effect transistor Biosensor Split gate Dielectric modulation Modeling Sensitivity Double gate Dual material
In this paper an analytical model of dual material surrounding split gate junctionless transistor for application as a biosensor is introduced. The simulation results show fair compatibility with the analytical models developed for the channel center potential and threshold voltage at channel length of 45 nm and 20 nm. For the detection of the bio-molecules, concept of dielectric modu lation has been used. The model shows a satisfactory value of sensitivity in the range of 0–0.27 and 0–0.8 for relative permittivity of biomolecules varying from 1 to 9 at channel length of 45 nm and 20 nm respectively, thus making it an attractive option for use as biosensor. This model has been demonstrated with the aid of TCAD device simulations.
1. Introduction WITH the evolution of CMOS technology into nanoscale club, the challenges to the semiconductor industry is on the rise. This is due to the heavy doping in the ultra-shallow junctions which makes the fabrication process difficult. Further at such scale the effect of short channel effects (SCEs) also increases. To address all this, Colinge et al. [1] proposed and later fabricated a new device called junc tionless transistor (JLT). Conventionally it does not have any junctions and also have no doping gradient. The main advantage of these transistors depends on improved and easier fabrication process. Also, it is less susceptible to SCEs. The basic principle behind working of JLT is a narrow channel which allows full depletion of carriers in the OFF-state and heavy doping for a satisfactory level of current flow in the ON-state. It basically acts as a variable resistor which is controlled by a gate electrode. The electrical characteristics of JLT are similar to that of conventional MOSFETs but with a different physics behind it. The biosensors have huge application in biomedical field mainly because it can help in the early detection of various kinds of diseases like Alzheimer’s, ovarian cancer and few viral diseases. Different types of biosensors have been made such as a nano-mechanic devices [2], optical [3], piezoelectric [4], electrochemical [5] etc. But these devices need expensive equipment and have lofty manufacturing cost. As a result, it led to the exposure of semiconductor devices based biosensors. Among these devices, field effect transistor (FET) biosensors have been extensively used to detect biomolecules [6,7]. FET based biosensors have a simple detection system and does not need a transducer. Further, the use of JLTs as biosensors along with the immense advancement in the fabrication industry offers cheap mass production of these biosensors. The detection is done by considering dry environment. In the literature, split gate structures have been proposed for JLT based biosensor [8]. In this, the center of the gate oxide is carved out to create a cavity with
* Corresponding author. E-mail addresses: [email protected] (M. Maji), [email protected] (G. Saini). https://doi.org/10.1016/j.spmi.2019.106290 Received 1 May 2019; Received in revised form 30 September 2019; Accepted 1 October 2019 Available online 5 October 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 1. Structure of dual material surrounding split gate JLT. Table 1 Device specifications. Symbol
Quantity
Specification
LCH L1 LCAV L2 tSi tSiO2 tHfO2
Channel Length Length of Metal M1 Cavity Length Length of Metal M2 Silicon Thickness Silicon Dioxide Thickness Hafnium Dioxide Thickness
45 nm 15 nm 15 nm 15 nm 10 nm 1 nm 4 nm
tbio WF1 WF2 kSiO2 kHfO2 ND
Cavity Thickness Work function of metal M1 Work function of metal M2 Dielectric constant of SiO2 Dielectric constant of HfO2 Doping concentration
19 nm 5.5eV 5eV 3.9 22 1019 cm
3
oxides on its either sides. This cavity acts as the biomolecules sensing area. When the biomolecules are present in this area the properties of the transistor change as compared to that in its absence. During the absence of the biomolecules, the cavity is packed with air which has a unit dielectric constant. Further when the biomolecules are there in the cavity, it is assumed to be filled with materials possessing distinct dielectric constant greater than unity. The variation in the permittivity of the biomolecule leads to a change in the threshold voltage of the device. This is used as a procedure to calculate the sensitivity of the device. Different types of JLTs based biosensors have been presented in the literature. Buitrago et al. [9] showed the biosensing capability of JLTs. Further, Narang et al. [10] demonstrated dielectric modulation on a double-gate JLT based biosensor. Ahangari et al. [11] and Parihar et al. [12] presented dual material JLT based biosensors. Barik et al. [13] demonstrated a dual-gated JLT based on Carbon Nano Tube (CNT) for detection of acetylcholine. Chakraborty et al. [14] performed sensitivity analysis of dielectric-modulated gate stack JLT for application as 2
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
biosensor. Ajay et al. [15] modelled gate underlap JLT as bio-sensor. Ouarghi et al. [16] analysed the impact of triple-material gate on the sensitivity of DNA biosensors. In this paper, gate oxide stack has been used to enhance the device performance. Dual material gate has been used to further boost the performance of the biosensor. A metal of high work function is kept at the source end to reduce leakage current whereas a metal of lower work function is kept near the drain to maintain the ON-state current. Typical sensitivity values reported in literature are in the range of 0–0.2 [8,10,14,15] for relative permittivity varying from 1 to 9. The proposed device has an enhanced sensitivity in the range of 0–0.27 for the same range of relative permittivity. The work discussed in this paper is arranged as follows. Section II includes the simulation setup and device structure. Section III has the developed analytical model of the channel center potential and threshold voltage. Section IV has the results and discussions. In Section V, the conclusion is presented. 2. Simulation setup and device structure Fig. 1 shows the two-dimensional device structure of the proposed biosensor. The standard parameters considered for the analysis of this sensor are listed in Table 1. For 20 nm technology, the channel length has been divided into three equal parts of 6.667 nm each for simulation. We have used metal M1 work function as 5.5eV. The use of materials such as platinum and p þ polycrystalline silicon with high work function has been reported by Chanda et al. [17], Rewari et al. [18], Ghosh et al. [19], Colinge et al. [20] and Lee et al. [21,22]. The utilization of a high work function metal of 5.5 eV may result in a challenge for large scale manufacturing of the proposed biosensor because it is difficult to obtain good interfacing of the gate electrode material with that of the dielectric. Further thermal stabilization of the device becomes difficult. However, it does not have much effect on the general course of the findings. The default cavity length considered is 15 nm. However, it has been varied to study the effect of cavity length on the biosensor sensitivity. It is worth mentioning that the model has been verified for neutral biomolecules only such as streptavidin (K ¼ 2.1) [23], protein (K ¼ 2.50) [24], biotin (K ¼ 2.6) [24], and APTES (K ¼ 3.57) [25]. The existence of a biomolecule in the cavity of the biosensor is simulated by substituting air with unity dielectric constant by a higher dielectric material. Without the loss of generalizability, we have considered K varying from 1 to 9 to study the effects of presence of biomolecules. TCAD [26] has been used for the simulations. It is widely used for research and production purposes in the semiconductor industry. It simulates device behavior based on different fundamental physical models. The models used by us to study the parameters of proposed biosensors are Drift Diffusion model for carrier transport, Field dependent and Constant mobility model, Shock ley–Read–Hall recombination model, Fermi Dirac statistics model, doping dependent mobility degradation model and Auger recombination model. 3. Analytical model 3.1. Channel center potential The channel center potential of the proposed biosensor has been modelled by the help of 2-D Poisson’s equation. We have considered cylindrical coordinate system for the analysis. The 2-D Poisson’s equation [14] can be written as � � 1 ∂ ∂ ψ ðr; zÞ ∂2 ψ ðr; zÞ qN D ¼ (1) r þ r ∂r ∂r ∂z2 εSi Here r is the radius with r ¼ 0 corresponding to the center of the channel and r ¼ R ¼ t2si corresponding to the surface of the channel, z represents the direction along the length of the channel with z ¼ 0 referring to the source end and z ¼ LCH referring to the drain end, ψ(r,z) is the 2-D potential distribution along the channel, q represents the charge of electron and ND is the constant doping concentration. The general solution of Equation (1) can be approximated according to parabolic potential profile as [27]. (2)
ψ ðr; zÞ ¼ AðzÞ þ BðzÞr þ CðzÞr2 The constants A (z), B (z) and C (z) are to be evaluated from the following boundary condition: 1:ψ ðr; zÞjr¼0 ¼ ψ C ðzÞ
(3)
2:ψ ðr; zÞjr¼R ¼ ψ S ðzÞ
(4)
� ∂ψ ðr; zÞ�� 3: ¼0 ∂r � r¼0
(5)
�
∂ψ ðr; zÞ�� Ci fVGS r¼R ¼ ∂r � εsi
VFBi
(6)
ψ S ðzÞg
Here, ΨC(z) represents the channel center potential of the device, ΨS(z) represents the device surface potential, Ci is the effective capacitance in the region i (i ¼ 1,2,3), εsi is the permittivity of Silicon, VGS is the gate voltage applied to the device and VFBi refers to the flat band voltage of the region i. 3
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Capacitance Ci can be written as C1 ¼ C3 ¼
C2 ¼
CSiO2 CHfO2 CSiO2 þ CHfO2
(7)
CSiO2 Cbio CSiO2 þ Cbio
(8)
Let Cx represent the capacitances CSiO2 ; CHfO2 andCbio [14]. Cx ¼
2ε � � x tsi ln 1 þ 2ttsix
(9)
Here εx is the permittivity of the material ‘x’, tx is the thickness of material ‘x’ and tsi is the thickness of silicon. Substituting Equations (3)–(6) we get, � � � � RCi RCi A z ¼ ψ S ðzÞ 1 þ ðVGS VFBi Þ 2εsi 2εsi
(10) (11)
BðZÞ ¼ 0 � � Ci C z ¼ fVGS 2Rεsi
(12)
ψ S ðzÞg
VFBi
Further substituting Equation (10)–(12) in Equation (2) we get equation of potential in region i (i ¼ 1,2,3) as, � � RC RCi Ci r 2 ψ i ðr; zÞ ¼ ψ Si ðzÞ 1 þ i ðVGS VFBi Þ þ fVGS VFBi ψ Si ðzÞg 2εsi 2εsi 2Rεsi
(13)
From Equation (3) we get (14)
ψ C ðzÞ ¼ AðzÞ
Substituting Equation (14) in equation (10), we get relation between surface potential ψ Si ðzÞ and channel potential ψ Ci ðzÞ as
ψ Si ðzÞ ¼
RCi ðVGS VFBi Þ 2εsi þ ψ ðzÞ 2εsi þ RCi 2εsi þ RCi Ci
(15)
Putting Equation (13) in equation (1) a 2nd order differential equation of surface potential is obtained
∂2 ψ Si ðzÞ ∂z2
2Ci 2C ðV VFBi Þ ψ ðzÞ þ i GS ¼ Rεsi Rεsi Si
qN D
(16)
εsi
This equation can thus be written in terms of the channel potential ψ Ci ðzÞ by substituting Equation (15) in (16) as
∂2 ψ Ci ðzÞ ∂z2
1
αi 2
(17)
ψ Ci ðzÞ ¼ βi
The coefficients αi and βi have been mentioned in the Appendix-A. The general solution of differential equation (17) can be written in a general form as z
ψ Ci ðzÞ ¼ mi eαi þ pi e
z
(18)
αi 2 βi
αi
The channel region has been split into three distinct areas as shown in Fig. 1. The channel center potential in each region can be written as. Region 1: z1 � z � z2 z
ψ C1 ðzÞ ¼ m1 eα1 þ p1 e
α1 2 β 1
(19)
α2
z
α2 2 β 2
(20)
z
α3 2 β 3
(21)
z
α1
Region 2: z2 � z � z3 z
ψ C2 ðzÞ ¼ m2 eα2 þ p2 e Region 3: z3 � z � z4 z
ψ C3 ðzÞ ¼ m3 eα3 þ p3 e
α3
The values of the coefficients mi and pi (i ¼ 1, 2 and 3) in Equation (19)–(21) are calculated in terms of channel center potentials V1, V2, V3 and V4 at z1, z2, z3 and z4 respectively.
4
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 2. Calibration of TCAD simulation models with experimental data [28].
mi ¼
Vi
α2i βi
� 1
pi ¼
Viþ1
α2i βi 1
�
e
�
2ðZiþ1
αi
e
�
2ðZiþ1
αi
αi
Zi
(22)
Zi Þ
α2i βi e
Vi
Ziþ1
α2i βi e
Viþ1
Ziþ1
αi
Zi
(23)
Zi Þ
The potentials at source end and drain end are known and are given as [14]. V1 ¼ 0
(24)
V4 ¼ VDS
(25)
Here VDS is the drain to source applied potential. The other potentials V2 and V3 can be obtained from the electric field continuity equation at z ¼ z2 and z ¼ z3. � � d ψ C1 ðzÞ �� d ψ C2 ðzÞ�� ¼ z¼z z¼z dz � 2 dz � 2 � d ψ C2 ðzÞ �� d ψ C3 ðzÞ z¼z ¼ dz � 3 dz
� � � z¼z � 3
(26) (27)
Solving this using Crammer’s rule we get � � � F1 S12 � � � � F2 S22 � � V2 ¼ �� � � S11 S12 � � S21 S22 �
(28)
� � � S11 F1 � � � � S21 F2 � � V2 ¼ �� � � S11 S12 � � S21 S22 �
(29)
The coefficients F1, F2, S11, S12, S21 and S22 have been mentioned in the Appendix-A.
5
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 3. Plot of channel center potential versus position along z-direction for different biomolecules for (a) 45 nm channel length (b) 20 nm channel length.
3.2. Threshold voltage In JLTs the threshold voltage corresponds to that gate voltage when the minimum channel center potential is zero. The position of the least channel-center potential (Zvth Þ depends on the metal gate work function and is calculated to be the minimum of the threshold values in each region. It can be calculated as d ψ C1 ðzÞ ¼0 dz Zvth
(30) 6
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 4. Plot of threshold voltage versus relative permittivity of different biomolecules for (a) 45 nm channel length (b) 20 nm channel length.
Applying this in Equation (18) we get � � αi pi Zvth ¼ ln 2 mi
(31)
Substituting Equation (31) in Equation (18) we have minimum channel center potential as pffiffiffiffiffiffiffiffi ψ Cimin ðzÞ ¼ 2 mi pi αi 2 βi Putting value of mi and pi from Equation (22) and (23) we get threshold voltage in region i as
7
(32)
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 5. Plot of variation of sensitivity of the device with relative permittivity of different biomolecules for (a) 45 nm channel length (b) 20 nm channel length.
VTHi ¼
w2i þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w22i 4w1i w3i 2w1i
(33)
The overall threshold voltage of the device can be calculated as minimum (VTHi). The coefficients w1i, w2i and w3i [15] are given in Appendix B. 4. Results and discussion Calibration to validate the TCAD simulation models with experimental data [28] has been done in Fig. 2. The proposed model has 8
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 6. Plot of variation of sensitivity of the device with LCAV for 45 nm technology.
Fig. 7. Plot of variation of sensitivity of the device with tbio for 45 nm technology.
been compared with TCAD simulation. The channel center potential and the threshold voltage have been plotted. We find decent similarity between the proposed model and simulation. Further the sensitivity has also been plotted. Since sensitivity is an important parameter for biosensors, variations of sensitivity with respect to various parameters have also been shown. Fig. 3 (a) shows the changes in the surface potential along the length of the channel when different biomolecules are placed in the cavity for 45 nm technology and Fig. 3. (b) shows the same for 20 nm technology. When the dielectric constant of the biomolecules increases the minimum surface potential decreases in both the cases. This is because when dielectric constants of biomolecules present in the cavity increases, the effective gate capacitance also increases along with vertical electric field and channel center potential resulting in a change in the flat band voltage in the cavity region. Variation in the threshold voltage is a key consideration in case of biosensors as it helps to determine the sensitivity of the device. We can see in Fig. 4. (a) and (b) that with increasing permittivity of biomolecules the threshold voltage increases. This is because with decrease in channel center potential, more gate voltage is needed to completely deplete the channel, thereby increasing the threshold voltage. We also observe that with reduction in channel length the threshold voltage reduces for each biomolecules thereby affecting its sensitivity. The biosensor sensitivity is determined as the ratio of the difference in threshold voltage of the device when the biomolecule is present and absent in the cavity to the threshold voltage of the device when the biomolecule is absent in the cavity [15]. 9
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Fig. 8. Plot of electron concentration versus position along z-direction for different biomolecules for 45 nm technology.
Fig. 9. Plot of drain current versus gate voltage for different biomolecules for 45 nm technology.
S¼
VTHðK>1Þ
(34)
VTHðK¼1Þ
VTHðK¼1Þ
From Fig. 5. (a) and (b) we can see that the sensitivity of biosensor is increasing with increase in the relative permittivity of the biomolecule existing in the cavity. This can be explained from the increase in threshold voltage as shown in Fig. 4 (a) and (b) respectively. We can clearly see that this device has good sensitivity and can thus be used in practical applications for sensing of the biomolecules. Since the biosensor sensitivity is important in determining its usability, variation of sensitivity with respect to length of cavity (LCAV) and thickness of cavity (tbio) have also been studied. The proposed model shows good agreement with that of simulation. Fig. 6 plots the variation in the sensitivity of biosensor with the length of cavity (Lcav) for a particular biomolecule present in the cavity for 45 nm technology. Here we have considered K ¼ 9 for simulation. It is clear that with increasing Lcav the sensitivity S of the device increases. This is because with increasing length of biomolecule cavity, the effective gate capacitance increases, lowering the channel center potential and hence increasing the threshold voltage as well as sensitivity. Fig. 7 plots the variation in the sensitivity of biosensor with the thickness of the cavity (tbio) for a particular biomolecule present in the cavity for 45 nm technology. Here we have considered K ¼ 9 for simulation. It is seen that with increasing tbio the sensitivity S of the 10
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
device increases. This is because with increasing thickness of biomolecule cavity, the effective gate capacitance increases, lowering the channel center potential and hence increasing the threshold voltage as well as sensitivity. Fig. 8 plots the electron concentration along the channel of the biosensor along the position along z-direction for different bio molecules for 45 nm technology. The trend in electron concentration can be explained on the fact that with increasing dielectric constant of the biomolecule present in the cavity the channel center potential is lowered, consequently lowering the electron con centration in the channel as well. Fig. 9 shows the drain current versus gate voltage for different biomolecules in the cavity for 45 nm technology. The greater current ratio with the increasing dielectric constant is due to the reduction in the channel center potential. Due to this the sensitivity of this device is also improved. 5. Conclusion In this paper, the models for channel surface potential and threshold voltage of dual material surrounding split gate junctionless transistor have been derived. Sensitivity of the device has been defined in terms of change in threshold voltage of the device. The sensitivity analysis of the device has been done with respect to parameters such as length of the cavity and thickness of the cavity. The simulation results show fair compatibility with the analytical models developed for the channel center potential, threshold voltage and sensitivity at channel length of 45 nm and 20 nm. The decent value of sensitivity of the device along with easy fabrication process of junctionless transistor makes the proposed device an attractive option in the biosensor industry. Acknowledgement We would like to acknowledge the Ministry of Electronics and Information Technology(MeitY) sponsored SMDP-C2SD program for providing the necessary tools. Appendix A sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4tsi εsi þ Ci t2si αi ¼ 16Ci βi ¼
�
1
VGS
α2i
VFBi þ
0 L1 α1
B C 2 C þ α2 β 1 B 1 B L1 A @eα1 e
0
2L1 2L1
e α1 S12 ¼
S21 ¼
S22 ¼
e L1 α1
B C 2 2 2 B C L2 A þ α2 β2 B Lcav @ α2 α3 e e e
B F2 ¼ VDS B @ Lα2 e3
þ
2Lcav α2
þ1
2Lcav α2
1
e
e
1
0
1 2L1 α1 2L1 α1
e
e
2Lcav α2
e
e
Lcav α2
2 Lcav α2
e
2Lcav α2
þ 1
2Lcav α2
1
e e
e
Lcav α2
2L2
þ
e α3 þ 1 2L2
e α3
e Lcav α2
B þ 1C 2 B C C þ α23 β3 B L2 @ A α 1 e3 e
2 Lcav
e α2
2Lcav α2 2Lcav α2
e
þ 1C C C 1A
0
1 2Lcav α2
Lcav α2
1
B þ 1C 2 B C C þ α22 β2 B Lcav @e α2 A e 1
0
1
e α1 þ 1
�
0
1
B 2 F1 ¼ Vbi B @ Lα1 e1 e
S11 ¼
qND tsi qND t2si þ 4Ci 16εsi
1
11
1
L2 α3
e
2L2 α3
e
2L2 α3
þ 1C C C 1A
Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
Appendix B 4τ1i τ3i
w1i ¼ 1 w2i ¼ 2θi
4 τ2i τ3i þ τ1i τ4i þ ϕf
w3i ¼ θi 2
4 τ2i τ4i þ ϕf θi qN D
θi ¼
1 e
τ2i ¼
Ziþ1
Hi
Ziþ1
αi
e
Zi
e
Zi
αi
Ziþ1
αi
�
Zi
Ziþ1
e
Ziþ1
τ3i ¼
αi
θi Þe
ðVi
ϕf 2
VFBi
εSi αi 2 βi
τ1i ¼
�
αi
Zi
Zi
θi Þ
ðJi
e
Ziþ1
αi
Zi
τ1i
1
τ4i ¼ Vi
θi
τ2i
Hi ¼ E1i
F1i
G1i
Ji ¼ E2i
F2i
G2i
0
1
Ziþ1 Zi 2ðZiþ1 1B e αi B 2 e αi E1i ¼ þ B 2ðZiþ1 αi @ 2ðZiþ1αi Zi Þ e 1 e αi
0
F2i ¼
11 2ðZiþ1
Be αi B þ θi B 2ðZ @ iþ1αi e
Zi Þ Zi Þ
C þ 1C CC CC AA 1
1 Zi
αi @
þ
e e
2ðZiþ1
Zi Þ
2ðZiþ1
Zi Þ
αi αi
þ 1C C C A 1
θi F1i 0
2ðZiþ1 1B Be αi G1i ¼ B 2ðZiþ1 αi @ e αi
G2i ¼
þ 1C C C A 1
0
Zi
0 1B B Ziþ1αi Be
Zi Þ
1
0 B C Ziþ1 2@Vi θ i A e αi 1B B E2i ¼ B 2ðZiþ1 Zi Þ αi @ e αi 1
F1i ¼
Zi Þ
1 Zi Þ Zi Þ
þ 1C C C A 1
θi G1i
Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.spmi.2019.106290.
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Superlattices and Microstructures 135 (2019) 106290
M. Maji and G. Saini
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