PhD work on Graphene Transistor

Final Examinationin Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy in Electrical Engineeringby

Division of Electrical and Computer EngineeringSchool of Electrical Engineering and Computer Science

Louisianan State University

Mar. 17, 2016

Yaser M. Banadaki

Advisor: Dr. Srivastava

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 transistoroperation (beyond FET)

50 years ago: Germanium Silicon (SiO2 )

one more block up: Silicon Carbon (allotropes )

ITRS: Potential solution usingcarbon-based FET,Development and pre-production on track

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

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.

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)

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

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.

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 on1) 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)

> 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)

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

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 nH 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 bb b

g b

E E k kE k

E m

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 ( )

( )

bbc

b ib

g

bb

bvb ib

g

E E xm if E E x

E xm x E

E x Em 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

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

22 [ ( , ; ) ]

bi

b

n n b

b E x

n i G n n E dE

( )2 1

22 [ ( , ; ) ]

biE 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 EDF F DSE E qV

EF : reference Fermi level of GNR

/ 1

/ ( ) [1 exp( )]S DF

S D

B

E Ef E

K T

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 rr

,n nn p

( )rr

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

hI T E f E f E dE

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

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

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 VGSstrong 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

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 ratioLCH 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

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.

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:

Complementary operationFigure: 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 )

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


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


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

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]

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)

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)

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

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

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


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)


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

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 material5) 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

Thank you

Any Question?

Green’s functions

> a mathematical construction

to solve differential equationssubject to specific initial orboundary 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.

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)


Appendix 3: Density of States for 3D, 2D and 1D systems


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.

0

( ) ( )b bn f E D E dE

[ ( ( )) ( ( )))]2

n b bCHGNR b FS C b FD C

b

q LQ 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

qI 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

qI 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 EqI k T T

h E E k T qV E

* (1/2) 3/22

3/2

( 2 )exp( )

9 2

b b bBTBT

m ET

q F

h

b

C b chE E

FS F SE E qV

FD F DE E qV

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

Appendix 7: ITRS Curves (2013 - 2028)

Appendix 8: NSF grants on 2D materials

National science foundation (NSF) addresses two-dimensional (2D) materialsresearch under Atomic-layer Research and Applications (2-DARE) topic, urgingproposals 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

PhD work on Graphene Transistor

Engineering

Transcript of PhD work on Graphene Transistor