The driving engine for the exponential growth of digital information processing systems is scaling down the transistor dimensions. For decades, this has enhanced the device performance and density. However, the International Technology Roadmap for Semiconductors (ITRS) states the end of Moore’s law in the next decade due to the scaling challenges of silicon-based CMOS electronics, e.g. extremely high power density. The forward-looking solutions are the utilization of emerging materials and devices for integrated circuits, e.g. carbon-based materials. The presentation of my Ph.D. work focuses on graphene, one atomic layer of carbon sheet, experimentally discovered in 2004. Since fabrication technology of emerging materials is still in early stages, transistor modeling has been playing an important role for evaluating futuristic graphene-based devices and circuits. The device has been simulated by solving a quantum transport model based on non-equilibrium Green’s function (NEGF) approach, which fully treats short channel-length electrostatic effects and the quantum tunneling effects, leading to the technology exploration of graphene nanoribbon field effect transistors (GNR FETs) for the future. This research presents a comprehensive study of the width-dependence performance of the GNR FETs and the scaling of its channel length down to 2.5 nanometer, investigating its potential use beyond-CMOS emerging technology.

1. Final Examination
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy in Electrical Engineering
by
Division of Electrical and Computer Engineering
School of Electrical Engineering and Computer Science
Louisianan State University
Mar. 17, 2016
Yaser M. Banadaki
Advisor: Dr. Srivastava

2. Moore’s law (1965) > 2x transistors on a chip ~ 18 months
Scaling transistors > 40% performance, 2X density and 4X memory capacity
ITRS: End of road map > short channel effects in sub-10 nm > increase power density >
heat makes amorphous silicon!
Three intervals of transistor scaling:
 Era of Simple Scaling
Improving lithography
 Transition Region
Complex device geometries
(double gate and tri-gate transistors)
+ High performance materials
(Beyond Silicon)
 Quantum Effects Dominate
(& Atomic dimensions region)
Fundamental change in transistor
operation (beyond FET)

3. 50 years ago: Germanium  Silicon (SiO2 )
one more block up: Silicon  Carbon (allotropes )
ITRS:
Potential solution using
carbon-based FET,
Development and
pre-production on track

4. Graphene : the First truly 2D material with atomic thickness
Discovered by Novoselov & Geim in 2004, Nobel Prize in Physics
(2010) for identification and characterization of graphene
Graphene Superlatives:
 Excellent electronic properties > High carrier velocity + High carrier
concentration > Better switching
 Atomically thin structure (maximum surface/bulk ratio)
> Better gate control over channel
 Compatible with current CMOS fabrication processes
> Potential wafer scale production
 Excellent thermal conductivity (strong carbon-carbon bonding -
Thermal conductivity ~5000 W/mK – 3X diamond + Maximum surface/bulk ratio)
> Better Heat removal
 Excellent Mechanical strength (200×steel, Young’s modulus~1,100 Gpa)
> Flexible and stretchable devices
 Optically Transparent : (97.7 % of white light can transmit)
> Transparent IC.
 Bandgap engineering in all-graphene architecture
 Same graphene used for Channel and interconnect (reduce Rc)
> Low-power IC

5. Graphene (+Challenges) :
1) Mechanical Exfoliation (Scotch-tape method) from
highly oriented pyrolytic graphite (HOPG) [Novoselov and
Geim, 2004].
Challenge: mass production and selective placement
of graphene  not suitable for integrated circuits
2) Epitaxial growth from silicon carbide (SiC) substrates
 (thermal desorption of silicon at high temperatures (>1250 °C))  not suitable for integrated circuits.
etc. …..
Graphene Nanoribbon (GNR) (+Challenges):
No lithography with atomic precision  edge roughness  shorten the mean free path
(edge scattering)  Most of current research are to fabricate smooth-edge GNRs
 unzipping the oxidized MWCNT through mechanical sonication
 Or scalable bottom-up approach methods
[1] Geim, A. K. "Graphene prehistory." Physica Scripta 2012.T146 (2012): 014003.
[2] Lu, Ganhua, et al. "Semiconducting graphene: converting graphene from semimetal to
semiconductor." Nanoscale 5.4 (2013): 1353-1368.

6. Graphene: Honeycomb-like hexagonal lattice or two interpenetrating
triangular lattices  two carbon atoms in unit cell
 Dispersion Relation: Two sets of three cone-like points K and K’ on the
edge of the Brillouin zone << Dirac points >>
>> CB and VB meet each other  No bandgap !
 Linear dispersion (not quadratic)  second derivative = 0 (massless particle)

7. But, bandgap is required to turn off a FET device.
Figure (Graphene FET): three Fermi levels in correspondence
with three gate voltages:
At VGS1  EF1 inside CB > electrons mostly contribute to IDS
At VGS3  EF3 inside VB > holes mostly contribute to IDS (Ambipolar transport)
At VGS2  EF2 at Dirac Point > n=p > ~Charge neutrality point (CNP)
High carrier concentration at OFF-state due to EG = 0 (Fermi-Dirac distribution)
 High leakage current (small ION/IOFF) -> Not good for logic application
Solution: Patterning large-area graphene into nanoribbon strips
 Quantum-mechanical confinement of carriers in one-dimension
 split 2D energy dispersion into multiple 1D modes and can induced a finite energy gap
Figure (Graphene Nanoribbon FET):
At VGS2  EF2 at middle of wide EG
 small number of carrier contribute in IOFF
 Small IOFF (large ION/IOFF )
 GNR FET: good for logic application

8. GNR = unfolded CNT similar CNT index for GNR
For unit vectors, 𝑎1 and 𝑎2 , CNT Chiral (Circumference) Vector:
Figure (If n =m): Armchair Circumference CNT(n,n) = Zigzag edge GNR(n,n)
GNR form Graphene:
(Cutting from red : Zigzag & from green: Armchair)
Zigzag GNR  (30 degrees rotation)  Armchair GNR
Optical and AFM images of the graphene sheet:
C. Almeida, V. Carozo, R. Prioli, and C. Achete, “Identification of graphene
crystallographic orientation by atomic force microscopy,” Journal of Applied
Physics, vol. 110, no. 8, p. 086101, 2011.

9. Zigzag-edge GNR:
Width confinement  Discretized kT lines in Γ-M path
 all lines pass Dirac point in 2D Brillion zone  no bandgap in
1D energy dispersion  All zigzag GNRs are Metallic.
Armchair-edge GNR:
 Corresponds to Γ-K path in Brillion zone.
 Some lines don’t pass Dirac Point  Semiconducting
 Size of induced EG depends on
1) GNR width (inverse relation) or strength of confinement
2) Type of edge boundary
3) # of dimer atoms, Na, in confined
transverse direction [Slide 28]
 Three family of GNRs
( 1/3 of armchair GNRs has very small bandgap, Na = 3p+2)

10. > Quantum Transport (Schrodinger Equation):
For emerging material and devices like Graphene-based FET:
1) Short Channel length
(mean free path > channel length
weak electron-phonon interaction
 Ballistic Transport)
2) Very important
Tunneling effects
Three carrier transport equations:
> Classical Transport (Newton’s law),
e.g. Drift-diffusion equations
> Semi-classical Transport (Boltzmann Equations)
 Both focuses on scattering effects
(mean free path < channel length)

11. Self-consistent calculation between:
Quantum Transport : Non-equilibrium
Green’s function (NEGF) formalism and
Electrostatic Problem: Poisson Equation
NEGF formalism >> Solving Schrodinger equation under non-
equilibrium condition (external field)
> Atomistic description of channel material
 by constructing Hamiltonian Matrix, [H]
> Effects of contacts on carriers transport in the channel
 by defining Self-energies of contacts and
> Tunneling effects

12. Step I: calculate TB for a slab with zero potential
> TB: Nearest neighbor orthogonal pz orbitals as basis functions (s, px, and py far from EF)
>> Hamiltonian between αth atom within
nth slab and βth atom within mth slab
δnα,mβ: Kronecker delta, Unα :electrostatic potential energy at the (n,α) atom site,
t: nearest neighbor hopping energy
If (n,α) and (m,β) neighbor 
Edge bond relaxation of atoms along the edges: tedge = 1.12 t
 E-K Diagram (Bandgap comparison)
1D quantum confinement of carriers:
 Open the bandgap, but reduce band linearity near the Dirac point
Correct it using non-parabolic effective mass model (NPEM):
Figure: Green: Constant effective mass model
Blue: TB bands
Red: NPEM
WGNR > NPEM: more important
 of the lowest subbands for a given N atoms
0
, , ,n m n m n m n
H H U       
Step I
Reference for comparison: Sako R, Hosokawa H, Tsuchiya H. Computational study of edge configuration and quantum confinement effects on
graphene nanoribbon transport. Electron Device Letters, IEEE. 2011;32:6-8
2 2
*
( )1
( )
2 2 2
b
g b
b b
g b
E E k k
E k
E m
  
      
  

13. Step III. Construct Retarded Green’s function
E : energy, I : identity matrix,
: Hamiltonian matrix  similar to TB case with 1D
discretization step equal to the slab width (size )
 For Non-parabolic band correction, Hamiltonian depends on energy
through position-energy dependent effective mass model :
: mid-gap energy
1
( ) [ ]b b
b b S DG E EI H 
     
bH
3 ccX a 
*
*
( )
1 ( )
( )
( , )
( )
1 ( )
( )
b
bc
b ib
g
b
b
bv
b ib
g
E E x
m if E E x
E x
m x E
E x E
m if E E x
E x
  
   
   
  
  
   
   ( )b
iE x
N N
and are Self-energies matrices: Null matrix,
except elements and , obtained
by piecewise equations:
(1,1)b
S ( , )b
D N N
2
2
/ 0
2
( 1) 2 0
( ) / ( 1) 2 0 2
( 1) 2 2
b
S D
x x x x
E t x i x x x
x x x x
       


      

     
0
0
( (1)) / 2 (1)
:
( (1) ) / 2 (1)
b b
c ib
S b b
V i
x E E t E E
x E E t E E
   
 
  Ref. Datta, Supriyo. Quantum transport: atom to transistor. Cambridge University Press, 2005.
Step II. For initial potential distribution:
Repeat TB for every slab of the ribbon along transport direction only at k = 0
 Obtain minimum energies of subbands,
+ wavefunctions,
[as a function of longitudinal (transport) direction]
in
nU 
( )b
cE x ( )b
vE x
( )b
n x

14. Step IV. Calculate inflow of carriers from S/D contacts into
channel:
and outflow of carriers from channel into contacts:
where,
𝛤S and 𝛤D : level broadening quantities :
 Continuous energy in contacts broaden the discrete energy levels in channel
: Fermi function of S/D contacts
( ) ( )[ ( ) ( )] ( )b b
b b S D bG E G E E E G E   
   
/ / /( ) ( )b b
S D S D S DE i f E
  
/ / /( ) [1 ( )]b b
S D S D S DE i f E
   
( ) ( )[ ( ) ( )] ( )b b
b b S D bG E G E E E G E   
   
2
( )
1
2
2 [ ( , ; ) ]
b
i
b
n n b
b E x
n i G n n E dE 




   
( )
2 1
2
2 [ ( , ; ) ]
b
iE x
b
n n b
b
p i G n n E dE 

 

  
Then, calculate the inflow and
outflow correlation functions:
Finally, calculate Electron and
hole numbers at (n,α) atom site,
/ / /( )b b b
S D S D S Di 
    
SF FE E DF F DSE E qV 
EF : reference Fermi level of GNR
/ 1
/ ( ) [1 exp( )]S DF
S D
B
E E
f E
K T


 

15. Step V. Find a new potential energy:
Solve 3D Poisson:
: dielectric permittivity
: net charge density distribution
= S/D doping profile &
> Gate electrodes U=VG : Drichlet B.C.
> Insulator boundaries: Neumann B.C.
(assumed perpendicular E to boundary = zero)
Step VI: Check the convergence condition
If No, replace by and go to step II.
If YES, go to the next step.
Step VII: Determine transmission function,
Calculate current :
(Landauer formalism)
( )nU r
r
.[ ( ) ( )] ( )nr U r qQ r  
r r r
( )Q r
r
,n nn p 
( )r
r
old
n nU U   
old
nU  nU 
( ) ( )b
b
T E T E 
( ) [ ]b b
b S b D bT E Trace G G
  
2
( )[ ( ) ( )]DS S D
q
h
I T E f E f E dE


 

16. Figure: Total charge density and 1D potential profile:
CG = Series combination of CQ and CINS :
Quantum capacitance:
Insulator capacitance:
 Typical silicon MOSFET: small CINS  dominant in CG
 smaller tINS by scaling  same gate electrostatic control on the short
channel
GNR: Atomically thin in vertical direction and quantum
mechanically confined in transverse direction (1D system)
 Small density of state  small CQ + high-k dielectrics and
high-geometry gate (large CINS) CQ  dominant in CG
1 1 1
G Q insC C C
 
Q SC dQ dV
ins OXC dQ dV
n nQ n p  

17.  GNR FET performance by scaling LCH for two GNRs when vertical scaling, tINS,
become less important by approaching QCL.
GNR FET Structure:
 Double gate MOSFET-like structure
 Insulator layers: Aluminum nitride (AlN) [High-K : k=9 and tins = 1nm]
(AlN: good reproducibility and uniformity + small phonon scattering in epitaxial graphene)
 GNR extensions on both sides of the intrinsic channel  doped with
concentration: 0.01 n-type dopants per carbon atom  Ohmic contact

18. Transfer characteristics:
a) Imin at a charge neutrality point (CNP) for a given VDS
b) By VDS  BTBT + DIBL  Imin
c) SS ~ 60 mV/decade
IDS versus VDS for different VGS strong saturation region
(good MOSFET behavior) even at LG = 5nm
Transfer characteristics for different LG of
1) GNR(7,0), EG = 1.53 eV, 2) GNR(13,0), EG = 0.86 eV

19. Scaling Effects on Static Metric of GNR FET:
1) OFF-current: LCH [15 nm  2.5nm]  IOFF
LCH  Decrease channel potential barrier’s
height  increase thermionic carriers emission over barrier
& width  increase direct carrier tunneling through barrier
GNR(13,0): smaller EG and lighter m*  BTBT
(BTBT between channel hole states and drain electron states)
GNR(7,0): smaller IOFF  robustness to short channel effects
 narrower GNR: promising (below both ILP and IHP)
2) ION/IOFF ratio
LCH  ION/IOFF ratio since IOFF , but ION
 ION/IOFF ratio: 1/6 LCH > 2WGNR  BTBT
 GNR(13,0): just good for HP design
100 /nA m
10 /nA m

20. 3) Subthreshold swing (SS) (limit of 60 mV/decade at RT)
a) SSGNR(7,0) < SSGNR(13,0)
b) Compare with 90 mV/decade for 10nm-scaled Si MOSFET [1]
and 125 mV/decade for FinFET [1]
4) Drain-induced barrier lowering (DIBL)
 Barrier lowering at the beginning of channel due to the increase in VD
a) DIBLGNR(7,0) < DIBLGNR(13,0)
Higher # of subbands  lower gate ability to control thermionic carriers by VD
b) LDOS(x,E): DOS as a function of position
(Blue: lowest DOS, Red: Highest DOS)
 LG = 15 nm  No change in current spectrum by increasing VD
 LG = 2.5 nm  Barrier + Direct tunneling through barrier
[1] S. Hasan, J. Wang, and M. Lundstrom, “Device design and manufacturing issues for 10 nm-scale MOSFETs: a computational study,” Solid-State Electronics, vol.
48, no. 6, pp. 867-875, 2004.

21. GNR(18,0): WGNR  EG
at VD & VG  band bending
LDOS: Quantum interference pattern due to the incident and reflected waves in the
generated quantum well in the valence band of channel.
 Charge in EG due to BTBT tunneling
IDS – VDS : no saturation region  not suitable for logic operation
Thus, after on-set of BTBT tunneling (~ WGNR & bias)
BTBT  device performance (not due to short channel)
BTBT
Scaling Effects on Static Metric of GNR FET:

22. Complementary operation
Figure: Voltage Transfer Characteristics of GNRFET-based Inverter:
 Blue: 5nm GNR(7,0): AINV = 4.6 & NM = 33% VDD
 Pink: 5nm GNR(13,0) : AINV = 4.1 & NM = 29% VDD (BTBT )
 Black: 2.5nm GNR(13,0): AINV = 3.7 & NM = 24% VDD (Direct tunneling through barrier )
 Green: 5nm GNR(18,0): AINV = 1.6 and diminish NM (BTBT )

23. Scaling Effects on Switching Attributes of GNR FET:
Gate-channel Capacitance (C-V curve):
a) LG  CG while same behavior Vs. VG (same DOS for a GNR)
b) By approaching CNP  QGNR  CQ  CG [CG ~ CQ (due to small DOS)]
c) CG maximized after VTH (higher subbands get populated and saturated)
1) Intrinsic cut-off frequency
 Comparison of intrinsic upper limit of GNR FET performance
a) fGNR(13,0) > fGNR(7,0) ( EG & m* )
b) fT Vs. LCH at VG = 0.4V and 0.7V
 LCH < 7.5 nm : no drop of fintrinsic
for GNR(13,0) by scaling VG
/ (2 )T m G
f g C

24. Scaling Effects on Switching Attributes of GNR FET:
2) Intrinsic gate-delay time
Figure: vs. ION/IOFF ratio for scaling LCH [10nm  2.5nm] (CS) and scaling VG [0.9V  0.7V] (VS)
a) GNR(13,0) : EG & m*  drive currents  & IOFF  ION/IOFF
b) Objective: small slope of Vs. ION/IOFF (ION/IOFF & ) while scaling VDD (smaller switching power)
c) When GNR FET operates at saturation region: Same slopes for all CS and VS
G GS DSC V I 


 
d) Both GNR(7,0) and GNR(13,0): outperform LP and HP
projection of ITRS,
e.g. GNR(13,0): 50X smaller than scaled 5nm MOS FET
in year 2028


25. Scaling Effects on Switching Attributes of GNR FET:
3) Intrinsic power-delay product (PDP) [average energy consumed per switching event]
Fig. : PDP of GNR(13,0) and GNR(7,0) by scaling LCH and VG
a) LG  PDP (~switching power) but leakage current (~static power)
b) For scaled VG and LCH  PDP of GNR(13,0)
c) ITRS: PDP reduce from [2013] to [2025] for &
 GNR(7,0):
 GNR(13,0): (promising for switching transistors)
7.5GL nm 0.7DDV V~ 0.8( / )fJ m ~ 0.37( / )fJ m
~ 0.45( / )fJ m
~ 0.18( / )fJ m

26.  Energy of the first four subbands at CNP vs. WGNR :
a) Repeating pattern of upper subbands
b) Larger EG of GNR(3p+1,0) than neighbors (Black)
c) Energy of 2nd subband GNR(3p+1,0) is near 1st subband,
(Blue)
 m* of first 4 subbands at CNP for members of two GNR
families (3p+1,0), (3p,0).
a) m2* of GNR(3p+1,0) is small too,
 can significantly contribute to carrier transport 
Single band approximation is inaccurate for GNR(3p+1,0)
, but used in [1], [2] !
b) 1st subband from [2] by arrow (good agreement)
c) m1* crosses m2* & m3* crosses m4*
 need accurate m* extraction using TB calculation
[1] R. Sako, H. Hosokawa, and H. Tsuchiya, “Computational study of edge configuration and quantum confinement effects on graphene nanoribbon transport,”
Electron Device Letters, IEEE, vol. 32, no. 1, pp. 6-8, 2011.
[2] H. Raza, and E. C. Kan, “Armchair graphene nanoribbons: electronic structure and electric-field modulation,” Physical Review B, vol. 77, no. 24, pp. 245434, 2008.
 Removing or adding one edge atom with a period-three
modulation significantly change EG of GNRs
EG : GNR(3p+1,0) > GNR(3p,0) > GNR(3p+2,0)
[Semiconducting] [metallic]

27. GNR FET Structure:
 Double gate MOSFET-like structure: Maximize gate electrostatic control over GNR channel
 LG = 7.5 nm, LS = LD = LG : doping concentration of 0.01 n-type dopants per carbon atom
 Equivalent oxide thickness (EOT) = 0.5nm  Double Insulator layers: High-k dielectric: HfO2
[𝛆r = 24, tox = 1.2nm] + Buffer layer: h-BN [𝛆r = 4, interlayer spacing 3A].
h-BN buffer layers:
a) similar lattice constant as graphene (~ %1.7 mismatch)
b) free of dangling bonds and charge traps
Promote growth of uniform and charge trapping free
high-k dielectric & protect GNR against environmental influence.
c) Difference in interaction energy between
carbon–nitrogen and carbon–boron 
Open bandgap in graphene.
(No experimental observation yet due to the
lack of crystallographic alignment )
50E meV 
Objective: Width-dependent of GNRFET for two semiconducting
families of armchair GNRs(N=3p,0) and GNRs(N=3p+1,0)

28.  IDS - VDS for two GNR groups of (3p+1,0) and (3p,0):
a) strong saturation region in LG = 7.5 nm
b) Drive current : GNRs (3p,0) ~ 2X GNRs (3p+1,0) [more
subbands contribute in carrier transport for a same bias (EG )]
 Transfer characteristics for two GNR groups of (3p+1,0) and (3p,0):
a) WGNR  curve shift to VG (smaller VTH)
b) WGNR  Both ION and IOFF
 VTH Vs. WGNR:
a) WGNR  EG  VTH
[For small EG, small VG can induced enough carriers in subbands]
e.g. At VG=0.4V
GNR(24,0), WGNR=3.07nm  VTH=0.18V & IDS=18µA
GNR(25,0), WGNR=3.19nm  VTH=0.32V & IDS=5.6µA
(only one carbon atom)

29. Scaling Effects on Static Metric of GNR FET
1) IOFF
a) WGNR  EG  ITH and IBTBT  IOFF
b) IOFF(GNRs (3p,0)) > IOFF(GNRs (3p+1,0))
c) Green: EG (GNR(12,0)) = EG (GNR(19,0)) = 0.6eV  Different IOFF
 GNR(19,0): m1* = 0.075m0 and m2* = 0.085m0  IOFF = 4.9 µA/µm
but GNR(12,0): m1* = 0.055m0  IOFF = 11 µA/µm
d) Blue: constant parabolic effective mass model  underestimate IDS
WGNR  band-linearity  Importance of non-parabolic effective
mass model .
e) ITRS (2025): IOFF (LP)= 30 pA/μm, IOFF (HP)= 100 nA/μm.
Satisfied by  First 3 members of GNRs(3p+1,0) + first of GNRs(3p,0)
f) Red: Possible induced bandgap of h-BN layer
 reduce IOFF , more important for wider GNRs
50E meV 

30. Scaling Effects on Static Metric of GNR FET
2) ION/IOFF ratio
a) WGNR  EG  ION/IOFF ratio
b) ION/IOFF (GNR(7,0)) = 4.5e9  350X > LP
c) h-BN layer, EG  improve ION/IOFF
3) subthreshold swing (~ standby power dissipation)
a) WGNR  SS (approach to 60mv/decade)
b) SSGNRs(3p+1,0) < SSGNRs (3p,0)
e.g. GNR(7,0) : SS = 67 mV/dec & GNR(6,0) : SS = 90 mV/dec

31. Scaling Effects on Switching Attribute of GNR FET
1) Transconductance, gm
a) gm vs. VG  linear dependence to VG around VTH ,
followed by a maximum plateau region ~ current
saturation
b) For similar WGNR  GNR(3p,0): larger gm & drive current
e.g. At VG = 0.45V GNR(24,0): gm = 82µS, IDS = 22µA
GNR(25,0): gm = 62µS, IDS = 8µA

32. Scaling Effects on Switching Attribute of GNR FET
2) Gate-channel capacitances:
a) At EF = mid-gap  CMIN (CG ~ CQ )
b) When EF shifts from mid-gap toward higher energies:
1st and 2nd sub-GNR(25,0) get populated around 0.2eV
 Max peak followed by Min plateau in CG around VG = 0.2V.
c) By increasing EF toward 0.8eV,
 3rd and 4th subbands get populated
 CG increases again around VG = 0.8 V.
So, Maximum peak followed by a minimum
plateau ~~ Fermi level passes a peak in the
density of state of GNRs
WGNR  subbands populate at higher energy
 Max CG shifts to VG
Comparing peak and min: CGNR(3p,0) < CGNR(3p+1,0)
Figure: DOS in color bar for
different GNRs .
Blue  bandgap energy (Small DOS)
Red  highest DOS (peaks)

33. Scaling Effects on Switching Attribute of GNR FET
3) Intrinsic cut-off frequency, fT
a) WGNR  max fT at VG
b) fT GNR(3p,0) ~ 2× fT GNR(3p+1,0) & at lower VG and higher IDS
e.g. GNR (25,0): fT = 25.5 THz at VG = 0.5V & ID = 11.5µA
GNR (24,0): fT = 55 THz at VG = 0.35V & ID = 14µA
4) Intrinsic gate-delay time,
a) WGNR 
b) GNR(3p,0) : Smaller




34. In sub-10 nm channel  leakage current  power density
Alternative channel material: Graphene , but EG = 0  Nanoribbon (1-3 nm width)
Not available fabrication  optimization and prediction by modeling and simulation.
Quantum tunneling in sub-10 nm channel  quantum-based transport model
NEGF formalism: 1) short gate-length electrostatic effects , 2) Direct source-to-drain tunneling in short channel
3) Band-to-band tunneling at source and drain junctions , 4) atomistic description of channel material
5) Effects of contacts on carriers transport in the channel
This work:  Presents accurate and relatively fast numerical algorithm based on NEGF formalism
 with non-parabolic band correction using energy-position effective mass Hamiltonian.
 Evaluates the scaling effects and the width-dependent of graphene nanoribbon on static
metrics and switching attributes of the double gate GNR FETs with high-k dielectric materials.
Scaling down channel length (from 15 nm to 2.5nm) shows:
 Narrow GNRs have superior static performance than wider GNRs (robustness to short channel effects), e.g.
better IOFF, ION/IOFF, subthreshold swing.
 Wider GNR channel shows higher ON-state performance, e.g. better switching power and speed
 Armchair GNRs (3p+1,0) : better static metric & Armchair GNRs (3p,0) : better switching attribute

35. Thank you
Any Question?

38. Green’s functions
> a mathematical construction
to solve differential equations
subject to specific initial or
boundary conditions.
Appendix. 1: Schrodinger to Green’s function
Green’s function: impulse response
of Schrodinger equation
Reference: Datta, Supriyo. Quantum transport: atom to transistor. Cambridge University Press, 2005.

39. Appendix 2: Hamiltonian:
> Quantum mechanics operator corresponding to the total energy of system.
For material: Solution of Hamiltonian matrix (Eigenvalue problem)  E-k diagram
(band structure)
Reference: Datta, Supriyo. Quantum transport: atom to transistor. Cambridge University Press, 2005.

40. Appendix 3: Density of States for 3D, 2D and 1D systems
Reference: Datta, Supriyo. Quantum transport: atom to transistor. Cambridge University Press, 2005.

41. Why GNR FET requires accurate numerical simulation ?
(CQ point of view)
1) CQ increases at ON-state (EF ~ subbands)
 Fully dominant CQ in analytical models is not accurate assumption.
2) CQ changes with GNR width (EF ~ subbands)
3) For short channel: channel potential changes not only by VG, but
also VD and VS
4) Full band calculation is required:
e.g. GNR (6,0) and GNR (10,0) : same Eg = 1.1eV
But, different location of upper subbands + DOS
 Different CQ vs. VG
GNR(10,0): smaller and wider CQ : 2nd subband close to 1st subband +
+ Both subbands have larger m* than GNR(6,0)
Q SC dQ dV
Why early theoretical
modeling is important ?
Prediction of materials
characteristics and devices
performance for accelerating
and directing research field.

42. 0
( ) ( )b bn f E D E dE

  [ ( ( )) ( ( )))]
2
n b bCH
GNR b FS C b FD C
b
q L
Q n E E E n E E E      
, ,
,
( )n
CAP i CH i FB i ch
i G B
Q C V V q

    
ch
0
2
( )[ ( ( )) ( ( ))]b b
T FS C FD C
b
q
I T E f E E E f E E E dE
h

      
2
[ln(1 exp(( ) / )) ln(1 exp(( ) / ))]b b
T B FS C FD C
b
q
I k T E E kT E E kT
h
     
, ,
,
1 exp(( ) / ) max( 2 ,0)2
[ ln( ) ]
1 exp(( ) / ) 2
CH D b F B CH D b
BTBT B BTBT
b b F B CH D b
qV E E k T qV Eq
I k T T
h E E k T qV E
   
 
  

* (1/2) 3/22
3/2
( 2 )
exp( )
9 2
b b b
BTBT
m E
T
q F
 
 
h
b
C b chE E  
FS F SE E qV 
FD F DE E qV 

43. 1) NEGF formalism in mode space approach
 GNR transverse confinement  discrete 1D subbands
 solve transport equations for a few lowest subbands
participate in carrier transport (subbands near EF)
e.g. for GNR(25,0) : Charge density  4 subbands
Drain Current  2 subbnads
2) increasing GNR width
 increase TB computational time (TBCT):
So, total time:
Time advantage of our algorithm:
One TB for slab with zero potential + TB for every GNR slab only
at wave vector k = 0 
This non-parabolic effective mass (NPEM) model : two orders
of magnitude saving in computational time for GNR FET with
0.001 eV energy grid.
( / 3 )G cc SCTBCT L a N 
( / 3 ) /G cc SC ETBCT TBCT L a N N  

44. Appendix 7: ITRS Curves (2013 - 2028)

45. Appendix 8: NSF grants on 2D materials
National science foundation (NSF) addresses two-dimensional (2D) materials
research under Atomic-layer Research and Applications (2-DARE) topic, urging
proposals on theory and modeling in this research area, as follows (quoted):
“Theory and modeling is vital to any new field, particularly in nanotechnology
and for research on nanomaterials. Proposals in this thrust are strongly
encouraged as there is a critical need for better understanding the observed
properties, as well as, prediction of new characteristics. The role of theory and
modeling to investigate structure-property correlations in 2D layered materials,
and the development of modeling tools for the exploration of 2D atomic layers
and devices will be important for accelerating research.”
Reference: http://www.nsf.gov/pubs/2015/nsf15502/nsf15502.htm