This document summarizes the results of simulations performed to analyze the performance of graphene nanoribbon tunnel field-effect transistors (TFETs). Key findings include: (1) Subthreshold swing and on/off current ratio vary inversely with channel width and positively with channel length; (2) Contact doping positively correlates with subthreshold swing and on/off ratio up to 0.28eV doping; and (3) Drain bias weakly positively correlates with subthreshold swing and negatively correlates with on/off ratio. The best performance was obtained for a device with length of 40nm, width of 5nm, drain bias of 0.1V, and contact doping of 0.24eV, achieving a subthreshold swing

1. Mark Cheung
Department of Electrical and Computer Engineering,
University of Virginia, Charlottesville, VA 22904, USA
1

2. This lecture will cover:
Field-effect transistor (FET) review
Motivation for TFET
Device design and simulation
Literature review
Simulation results
2

3. Field-effect transistor (FET)
review
 Switch
 On: ID is high
 Off: ID is low
Landauer Formula:
𝐼𝐷 =
𝑞
ℎ −∞
∞
𝑑𝐸𝐷(𝐸 − 𝑈)(
𝛾1𝛾2
𝛾1 + 𝛾2
)[𝑓1 𝐸 − 𝑓2 𝐸 ]
3

4. Motivation
"Intel," 2011. Available: http://www.carthrottle.com/why-chemistry-dictates-an-electric-vehicle-
future/
4

5. Current-voltage (IV) curve
 Subthreshold Swing SS (mV/dec):
𝜕𝑉𝑔𝑠
𝜕log(𝐼𝑑)
 Power P=(1/2)C𝑉𝑑
2
f+VdIloff
Ioff
Ion
~60 mV/dec
MOSFET IV Curve
5
𝜕𝑉𝐺𝑆
𝜕log(𝐼𝐷)
= 𝑙𝑛 10 ∗ (
𝜕𝐼𝐷
𝜕𝑉𝐺𝑆
∗
1
𝐼𝐷,𝑜𝑛
)−1
≈ 60 mV/dec

6. 6
Tunnel Field Effect Transistor (TFET)

7. Tunnel Field Effect Transistor (TFET)
𝐼𝑑 =
2𝑒
ℎ
𝑊
𝐸𝑐
𝑠
𝐸𝑣
𝑐ℎ
𝑇 𝐸 − 𝑈 𝑓𝑠 𝐸 − 𝑓𝑑 𝐸 𝑑𝐸
7
Off
On
𝐸𝑐
𝐸𝑣
q∆𝑉𝐺
λ
Channel
Source Drain
𝑓𝑠 𝐸

8. Device design and simulation
µ1
µ2
[𝛴]1
Source Drain
Gate
𝑉𝐷𝑆
𝐼𝐷𝑆
[𝛴]2
[H]
Green Function: 𝐺 = (𝐸𝐼 − 𝐻 − Σ 1 − Σ 2) −1
8

9. Graphene Nanoribbon (GNR)
Subbands Transmission
9

10. Relevant Functions (analytical)
SS=
𝜕𝑉𝑔𝑠
𝜕log(𝐼𝑑)
= 𝐥𝐧 𝟏𝟎 ∗ (
𝝏𝑰𝒅
𝝏𝑽𝒈𝒔
∗
𝟏
𝑰𝒅,𝒐𝒏
)−𝟏

𝝏𝑰𝒅
𝝏𝑽𝒈𝒔
= 𝒆
𝝏𝑰𝒅
𝝏𝑬𝒗
𝒄𝒉 =
𝟐𝒆𝟐
𝒉
(
𝝏𝑻𝑾𝑲𝑩
𝑬𝒗
𝒄𝒉 𝑭 𝑬𝒗
𝒄𝒉 + 𝑻𝑾𝑲𝑩
𝝏𝑭(𝑬𝒗
𝒄𝒉)
𝝏𝑬𝒗
𝒄𝒉 )
 𝑻𝑾𝑲𝑩 = 𝒆
−
𝟒𝜦 𝟐𝒎∗𝑬𝒈
𝟑
𝟐
𝟑𝒉 ∆𝝓+𝑬𝒈
 F 𝑬𝒗
𝒄𝒉 = 𝐸𝑐
𝑠
𝐸𝑣
𝑐ℎ
𝑓𝑠 𝐸 − 𝑓𝑑 𝐸 𝑑𝐸
10
J. Knoch, S. Mantl and J. Appenzeller, "Impact of dimensionality on the performance of tunneling
FETs: Bulk versus one-dimensional devices," ScienceDirect, vol. 51, pp. 572-78, 2007.

11. Literature Review: MOSFET/TFET IV
of different material system
A. M. Ionescu and H. Riel, "Tunnel field-effect transistors as energy-efficient
electronics switches," Nature, vol. 479, pp. 329-337, 2011.
11

12. Literature Review: varying gate
overlap & differential voltage
Gate overlap improves SS
without degrading Ion and Ioff
Differential voltage between top and bottom gate
for a double gate TFET correlates positively with Ion/Ioff
Fiori, G.; Iannaccone, G., "Ultralow-Voltage Bilayer Graphene Tunnel FET," Electron Device Letters,
IEEE , vol.0, no.10, pp.1096,1098, Oct. 2009 doi: 10.1109/LED.2009.2028248
12

13. Literature Review: varying drain-
side gate underlap & drain doping
X. Yang, J. Chauhan, J. Guo, and K. Mohanram “Graphene tunneling FET and its applications in
low-power circuit design,” VLSI, pp. 263-268, 2010
13
Drain-side gate underlap and drain doping reduce the
ambipolar IV characteristics without sacrificing Ion/Ioff and SS

14. Result: varying channel width
14
Channel width varies inversely with
SS and correlates negatively
(exponential) with Ion/Ioff
𝐼𝑑 =
2𝑒
ℎ
𝑊
𝐸𝑐
𝑠
𝐸𝑣
𝑐ℎ
𝑇 𝐸 − 𝑈 𝑓𝑠 𝐸 − 𝑓𝑑 𝐸 𝑑𝐸

15. Result: varying channel width
y = 381.85e-0.554x
R² = 0.9697
y = 3E+08e-2.043x
R² = 0.979
1
10
100
1,000
10,000
100,000
1,000,000
0
50
100
150
200
250
0 1 2 3 4 5 6 7 8
ratio
SS(mV/dec)
width (nm)
SS (mV/dec)
Ion/Ioff
15
Channel width varies inversely with SS and
correlates negatively (exponential) with Ion/Ioff

16. Results: varying channel length
16
Off
On
𝐸𝑐
𝐸𝑣
q∆𝑉𝐺
λ
Channel
Source Drain
𝑓𝑠 𝐸

17. Results varying channel length
17
y = 30782ln(x) - 70513
R² = 0.7531
1
10
100
1,000
10,000
100,000
0
20
40
60
80
100
120
140
160
180
0 20 40 60 80 100 120 140 160 180
Ratio
SS
(mV/dec)
length (nm)
SS (mV/dec)
Ion/Ioff
Channel length varies inversely with SS and
correlates positively (logarithmic) with Ion/Ioff

18. Results: varying doping in contacts
18
Channel doping correlates positively with SS (exponential) and
positively with Ion/Ioff (exponential) up until doping of around 0.28eV
Off
On
𝐸𝑐
𝐸𝑣
q∆𝑉𝐺
λ
Channel
Source Drain
𝑓𝑠 𝐸

19. Results: varying doping in contacts
y = 0.1836e15.587x
R² = 0.8899
y = 20.708e32.662x
R² = 0.9263
y = 3E+07e-33.01x
1
10
100
1,000
10,000
100,000
1,000,000
0
10
20
30
40
50
60
70
0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4
ratio
SS
(mV/dec)
doping (eV)
SS (mV/dec)
Ion/Ioff
19
Channel doping correlates positively with SS (exponential) and
positively with Ion/Ioff (exponential) up until doping of around 0.28eV

20. Results: varying drain bias
20
Drain bias correlates positively with SS (linear & weak)
and negatively with Ion/Ioff (exponential)
Off
On
𝐸𝑐
𝐸𝑣
q∆𝑉𝐺
λ
Channel
Source Drain
𝑓𝑠 𝐸

21. Results: varying drain bias
y = 366373e-26.58x
R² = 0.9464
1
10
100
1,000
10,000
100,000
1,000,000
0
2
4
6
8
10
12
0 0.05 0.1 0.15 0.2 0.25
ratio
SS
(mV/dec)
vd (V)
SS (mV/dec)
Ion/Ioff
21
Drain bias correlates positively with SS (linear & weak)
and negatively with Ion/Ioff (exponential)

22. Conclusion
 SS of 6.4 mV/dec and Ion/Ioff of >25,000 were
obtained for length=40nm, width=5nm, vd=0.1 V, and
doping=0.24eV.
 Further analysis is required to balance the trade-offs
among size, power, and performance.
 In comparison to a MOSFET, high Ion/Ioff ratio and
steep SS over several decades indicate GNR TFET’s
superiority for ultra-low-voltage applications.
22

23. Future direction
 Link experimental results with analytical equations
 Adjust simulation to account for experimental
challenges
 Include scattering (inelastic & elastic)
 Alternative TFET designs
23

24. Appendix: Simulation Design
(continue)
 Tight-binding Hamiltonian model
 TFET setup:
 Channel doping
 Tri-gate
 Non-equilibrium green function (NEGF)
 Assumptions:
 Room temperature
 ballistic transport
 electrodes are infinite electron reservoir
 steady state
24

25.  𝐼𝑑 =
2𝑒
ℎ
𝑊 𝐸𝑐
𝑠
𝐸𝑣
𝑐ℎ
𝑇 𝐸 − 𝑈 𝑓𝑠 𝐸 − 𝑓𝑑 𝐸 𝑑𝐸
 𝐺 = (𝐸𝐼 − 𝐻 − Σ 1 − Σ 2) −1
 E : energy matrices from the electronic band structure
 H : hamiltonian matrix
 Σ 1,2 : self energy matrices from the contacts
 Σ 1=Γ1𝑓1 , Σ 2=Γ2𝑓2
 Γ: broadening matrices due to coupling with contacts
 f: fermi functions describing number of electrons
 𝐺𝑛 = 𝐺 Γ1𝑓1 + Γ2𝑓2 𝐺+
 Electron density per unit energy
Appendix: NEGF
25

26. Appendix: NEGF (continue)
 T(E)=Trace(Γ1𝐺Γ2𝐺+
)
 Average transmission at different energy
 U=𝑈𝐿 + 𝑈𝑁
 Potential energy effecting the DOS , and hence the transmission T
 𝑈𝐿 =
𝐶𝐺
𝐶𝐸
(−𝑞𝑉𝐺)+
𝐶𝐷
𝐶𝐸
(−q𝑉𝐷)
 𝑈𝑁 =
𝑞2
𝐶𝑒
∆N
 𝑓(𝐸) =
1
1+𝑒
𝐸−µ
𝑘𝑇
 Probability that an electron will be at an energy state E given the
fermi level µ, and temperature T
 𝐼𝑑 =
2𝑒
ℎ
𝑊 𝐸𝑐
𝑠
𝐸𝑣
𝑐ℎ
𝑇 𝐸 − 𝑈 𝑓𝑠 𝐸 − 𝑓𝑑 𝐸 𝑑𝐸
26

27. Appendix: Relevant functions
(continue)
SS=
𝜕𝑉𝑔𝑠
𝜕log(𝐼𝑑)
= 𝐥𝐧 𝟏𝟎 ∗ (
𝝏𝑰𝒅
𝝏𝑽𝒈𝒔
∗
𝟏
𝑰𝒅,𝒐𝒏
)−𝟏

𝝏𝑰𝒅
𝝏𝑽𝒈𝒔
= 𝒆
𝝏𝑰𝒅
𝝏𝑬𝒗
𝒄𝒉 =
𝟐𝒆𝟐
𝒉
(
𝝏𝑻𝑾𝑲𝑩
𝑬𝒗
𝒄𝒉 𝑭 𝑬𝒗
𝒄𝒉
+ 𝑻𝑾𝑲𝑩
𝝏𝑭(𝑬𝒗
𝒄𝒉)
𝝏𝑬𝒗
𝒄𝒉 )
 𝑻 𝑬 = 𝒆
−𝟐 𝟎
𝜦 𝟐𝒎∗
𝒉𝟐 𝒒𝝃𝒙 𝒅𝒙
 𝑻 𝑬𝒗
𝒄𝒉
= 𝑻𝑾𝑲𝑩 = 𝒆
−
𝟒𝜦 𝟐𝒎∗𝑬𝒈
𝟑
𝟐
𝟑𝒉 ∆𝝓+𝑬𝒈
 ∆𝝓 = 𝑬𝒗
𝒄𝒉
− 𝑬𝒄
𝒔
 𝑼 = 𝑬𝒗
𝒄𝒉
+ 𝑬𝒈
 𝑼 − 𝑬𝒄
𝒔
= −𝒒𝝃𝒙
 F 𝑬𝒗
𝒄𝒉
= 𝐸𝑐
𝑠
𝐸𝑣
𝑐ℎ
𝑓𝑠 𝐸 − 𝑓𝑑 𝐸 𝑑𝐸
27
J. Knoch, S. Mantl and J. Appenzeller, "Impact of dimensionality on the performance of tunneling
FETs: Bulk versus one-dimensional devices," ScienceDirect, vol. 51, pp. 572-78, 2007.