Mic

Abstrfor sbandatomsemiand

G

of itcontcarriprocgrapapplapplbeyounderespdepemoddesctimeillusclarielectMOScarritechn

grapmulttunnpropa πHaman asolvhexaand depeenersimu

IWT

TimeDharmend

croelectronics

ract—We pressimulation of dd quasi-ballist

mistic-tight-bini-implicit numabsorbing bou

Graphene, quan

Graphene is bts novel proptinues to chaier velocities,cess compatibphene a promlications [1,2]lications [3]. ond quasi-staterstand their onse. Towardendent non-edeling dynamicribe the essene evolution sctrate them viaity. With trostatically SFETs with iers will add nical one. The challeng

phene includeti-band tran

neling is a critposed graphenπ-orbital-basedmiltonian mod

lternating-diree the time-agonal lattice,

achieve coendent sourcergy terms arulation region

I. SOLUTIO

Within the tiThis work is supp

e Depedar Reddy, Ps Research Ce

sent an efficiendynamic throutic quantum tnding Hamiltonmerical time evundary conditio

ntum transport

I. INT

being consideperties. Altho

allenge CMOS, the ultimatebility with simising candid] and perhapFor such de

tic to truly timintrinsic frequds this goal,

equilibrium Gic quantum trntial elementscheme and opa simple MAT

these elemself-consistena thermal donly to the

es of simulatie fast (1 nm/sport―the ltical physical ne devices. d nearest-ne

del of graphenection semi-im-dependent S, to maintain omputational e terms and re added to .

ONS OF TIME-Dight-binding

ported by NRI, SW

endentPriyamvada Jenter, The Uni

nt time-dependugh steady-stattransport in gnian, novel altvolution schemons.

t, time depende

TRODUCTION ered for manyough the lackS-like logic e ultrathin boilicon techno

date for radiops for novel evices, howevme-dependentuency limitati, we are deGreen's functransport in grs of the methpen boundary

TLAB-based [ments givennt simulatiodistribution ocomputationa

ing time-depe/fs), quasi-nolatter becauconsiderationTo address ighbor atom

ne is used. Nmplicit schemSchrödinger

stability, conefficiency.

stationary buallow transp

DEPENDENT SC

formalism, thWAN and DARPA

t QuanJadaun, Amitiversity of Tex

dharma

dent NEGF mete intra- and igraphene usinternating diremes, and inje

ence, NEGF

y purposes beck of a band applications, ody and pote

ologies still mo frequency “beyond CM

ver, one must analysis to ions and dynveloping a ttion (NEGF)raphene. Herhod―Hamiltoy conditions―[4] simulationn, future, on of grapof source-injeal burden, no

endent transpoon-dispersive, use band-to-n for essentiall

these challenmistic-tight-binNovel variationme are employ

equation onnserve probab

Potentially ut non-local port through

CHRODINGER Ehe time-depenA CERA program

ntum Tthraj Valsaraxas at Austin, areddy84@gm

ethod inter-

ng an ection ecting

cause gap

high ential make (RF)

MOS” st go fully

namic time-) for re we nian, ―and ns for

e.g., phene ected t the

ort in and

-band ly all nges, nding ns of ed to

the bility,

time self-

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EQ. ndent

Schr

wheHamon-sdimnearon-spote

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i

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AGaukD ispac

ms.

Fipobilinalascoblblda

Transpaj, Leonard F10100 Burnet

mail.com

rödinger equa

ere ψ and Hmiltonian squasite and nearensional (2D)rest-nearest nesite matrix eential, e.g.―anTo numericalle via the wellhod [5]. This ersible evolutime step of Δt f

t tt

ψ ψ

noting n ψ ψ

be rewritten f

i t I H

As an illustrussian wave-pain wave-vectoce, as per Fig.

igure 1. Illustratiooints where the coinding coupling ones (solid or dashternating directi

ssociated four-atooupling between lack solid lines, lack dashed linesashed gray lines.

port inF. Register, at Road, Bldg.

ation is of the

Hψψ

dtdi

H are the wavare matrix, rerest-neighbor ) hexagonal laeighbor matrixelements Hi,i―nd all other mly solve (1), w-proven semiapproach prov

ion and servesfor time integr

t t

ψH

n tψ and Hfor each time s

1/2 1n n H ψ

ration of timacket centeredor space and 2(a),

on of the graphenolors distinguish of the tight-bindinhed, black or graon implicit sche

om unit cell is shatoms within thecoupling betwee

s and coupling b

n Grapand Sanjay K

160, Austin, T

form,

ψ ,

ve-function coespectively. π-bonding o

attice of graphx elements Hi―defined by

matrix elementwe first consid

-implicit Cravides accurates as a referencration, the CN

2

tt

ψ

1/2n n H Hstep as,

1/2ni t I H

me evolutiond around one oinitially well

ne crystal lattice, the sublattice, anng Hamiltonian, ay). For one impeme, ADI1 (seehown in the rectae same unit cell aen unit cells in

between unit cells

pheneK. Banerjee

TX 78758, U.

(

olumn matrixHere we con

only on the hene (Fig. 1),

Hi,j = τo = −3.03y an electrots set to zero. er discretizati

ank-Nicolson e, unitary andce here. Assu

N is

2t t

ψ.

/ 2t t , Eq

nψ .

n, we considof the Dirac pl-localized in

represented by thnd the nearest tigh

represented by thplementation of

e Section III), thangle. In this casare indicated by ththe x direction bs in y direction b

.S.A

(1)

x and nsider

two- with 3 eV, static

on of (CN)

d time uming

(2)

q. (2)

(3)

der a points

real-

he ht-he an he se, he by by

SISPAD 2012, September 5-7, 2012, Denver, CO, USA

SISPAD 2012 - http://www.sispad.org

51 ISBN 978-0-615-71756-2

The matrcoefinitiawaveand predgrapsprealittle2(b) withconfcontchar

Astep.speethe othanpracour h1.0 fmoti0.1 leadiexpe

Wstep simuelemcomalter

Figat (

FiguGra

(a)

t )0,(r

two-element rix describing fficients of theally here, in ge-packet contavalence band

dominately linphene within ads from the

e change in that t = 14 fs. T

h the given firmed by antrasts markedracteristic of p

Accurate simul. For an expli

ed of 1 nm/fs aorder of 0.1 n

n or equal to atice, we have half-implicit sfs time step rion of the wafs time step. ing edge of thected.

With the basic increases su

ulation regionments of the Hmputational e

rnating-directi

gure 2. Snapshots(a) t = 0 and (b) t=

(a)

ure 3. Snapshot aphene at 3 fs for

exp2

12

r

column matrixthe relation b

e two sub-lattgeneral it can ains equal cond, and extendnear dispersithe relevant center in a a

he ring thickneThe principle d

initial pseudnalytic resultsdly to the articles in a silation, howevicit time-evoland nearest-ne

nm would sugabout 0.1 fs salso found thscheme(s). Aesults in slow

ave-packet by Note that in

he wave-pack

CN method, tuper-linearly n due to thHamiltonian rffort, we inion implicit (A

s of time evolutio=14 fs.

of a time evolvea time step of (a)

io 2 22rr

x is the so-calbetween the ctices of graphbe a function ntributions frods well into ion/constant energy range

asymmetric riness with timedirection of mdo-spin phases for this simbroadening-G

ingle band of wver, requires alution schemeeighbor inter-

ggest the use oimply to track

his estimate to As shown in Fwed and qualit

comparison n the latter caket travels abo

the computatiwith size o

he far-from-drequired in (3ntroduce tw

ADI) methods

on of initially Ga

(b)

ed initially Gauss) 0.1 fs and (b) 1

(b)

i

iD

1k .

lled “pseudo-scomplex amplhene. While

of r as well. om the conduboth. Due tocarrier speed, the wave-pang-like shape , as shown in

motion is assoce. This behample system

Gaussian-evoluwell-defined ma quite small e, the fixed caatomic spacinof a time stepk a wave fronbe appropriat

Fig. 3, the usetatively inaccuto the result

ase (Fig. 3(a)out 3 nm in 3

ional cost per of the considdiagonal non3). To reduceo split ope.

aussian wave-pac

sian wave-packetfs.

(4)

spin” litude fixed This

uction o the d in acket with

n Fig. ciated avior,

[6], ution mass. time

arrier ng on p less nt. In te for e of a urate for a ) the fs as

time dered nzero e the erator

Ioperand

Hophopphalf Howrectaby Couis cCoulinescouplinesAnyand tridithe t

AfollolattiEachby iin Fof thonsistep

The consfrom3H −

Fshowan iADIthe

cket

t on Figthe

In the first arator H into qthe time step

1/

1/

2

2

nx

ny

ti

ti

I H

I H

pping in the ping in nomin

f of the time-wever, there isangular splitticonsidering t

upling betweenconsidered in upling betweens) is considerpling betweens) is consider

y on-site contHy. This op

iagonal matrixtime step, respAnother approows the natuce. We subdivh of the three gnoring all boig1, as illustrahe Hi, their site coupling is is now split i

3

3

3

dti

dti

dti

I

I

I

parameter η iserving half-i

m the left-− η(H1 + H2 +Figs. 5(a)-(c) iwing essentialinitially GausI, ADI1, and Acomputationa

gure 4. Illustratioe ADI2 method.

approach, ADquasi-rectanguin half such th

/2 1/2

/2 1

n

n

ψ

ψ

nominal y dnal x directionstep, and vices more than oning of the Hamthe rectanguln atoms within

both Hx andn unit cells inred only in Hn unit cells inred only in Htributions to Hperator splittinx equations inpectively. oach we use tural rotationavide the Hamcomponents H

onds in the coated in Fig. 4. strength is has split equallyinto three sub-

1/31

2/32

13

n

n

n

H ψ

H ψ

H ψ

is set to 3/2 soimplicit. (I.e.-hand/implicit+ H3) from theillustrate the ally identical sssian wave-paADI2 method

al effort for th

on of the H = H

DI1, we splitular coordinatehat (2) becom

1/2

1/2

2

2

ny

nx

ti

ti

I H

I H

direction is fn is fully implie-versa duringne way to permiltonian. Wear unit cell n the unit celld Hy but at n the x directHx but at fuln the y direc

Hy but again H are split eng leads to pn the first and

to split the Hal symmetry

miltonian as HHamiltonians orresponding Since each bo

alved therein, y among the th-steps as,

3

3

3

dti

dti

dti

I H H

I H H

I H H

o that the meth, such that ηt side(s) oe right-hand/eaccuracy of thsnapshots obtacket using thds, respectivelhe three diffe

H1 + H2 + H3

t the Hamiltoes with Hx an

mes,

1/2

n

n

ψ

ψ

.

fully expliciticit during theg the second

rform such a qe obtain Hx anshown in Fi

l (solid black lhalf-strength

tion (dashed bl strength to.

ction (dashed at full strengvenly betweepentadiagonald second halv

Hamiltonian, Aof the grap

H = H1 + H2 +Hi are construdirection δi shond appears into ½to again.hree Hi. The

1

1/32

2/33

n

n

n

H ψ

H ψ

H ψ

.

hod remains nη(H1 + H2 +of (6) eexplicit sides(shese ADI methained at 3 fs he reference ly. Fig. 5(d) serent methods

split Hamiltonian

onian d Hy,

(5)

t and e first

half. quasi-nd Hy ig. 1. lines)

½to. black And gray

gth to. en Hx l and ves of

ADI2, phene + H3. ucted hown n two . The time

(6)

norm-+ H3) quals s).) hods, from non-

hows s as a

n of

52

funcan ~ADIwithgrowdoesmatrrequanalyfor aconv

IIF

(C) dynaleft (regio

CThe

L ψ

wavrLψ

equa

HerecentrelemThe

FigpseAD(blafun

(a

(c

ction of the nu~20 nm wide I methods exhh simulation rwth with the s not divergerix solver useuires slightly ysis finds thea given time venience may

I. ABSORBIN

For the sake osimulation re

amic probabil(L) and right (on is ψC, ψL, aConsider injectotal wave-fu

in rL L ψ ψ w

e-function whis any reflect

ation is transfo

didt

ψ

ψ

ψ

e, HC, HL, anral, left and r

ments represenprobability so

gure 5. Snapshoeudospin angle ofDI2 methods, andack), ADI1 (das

nction of simulatio

a)

c)

umber of atomgraphene rib

hibit only linregion size, non-ADI met

e earlier is a ed in MATLAless effort p

e ADI2 methostep. The difbe the best de

NG AND INJEC

of the currentegion throughlity/charge cu(R) “leads,” wand ψR, respecction first, frounction in the where n

Liψ is

hose time evolted wave. Theormed into the

0

rL L

C CL

R

ψ H HH H

ψ H

nd HR are thright regions, nt the couplingource term Sψ

ot of an initiaf 60o at 3fs, obta

d (d) computationshed red), and Aon region size.

(

ms in the simbbon of varyinear growth inin contrast tothod. (That th

testament toAB). While t

per time step,od to be slighfferences are eterminant of

CTING BOUNDA

t discussion, h which we

urrent flow, cwhere the wavctively. om the left leleft lead can

s the assumelution in the le time-depende inhomogene

0LC

C CR

RC R

H ψ

H H ψ

H H ψ

he Hamiltoniarespectively. g between theS on right-han

ally Gaussian wained using (a) nonal time per time ADI2 (dash-dot

(b)

(d)

mulation regionng lengths. n simulation o the super-lhe effort requo the UMFPAthe ADI1 me, a more dethtly more accusmall enoughwhich to use.

ARY CONDITIO

consider a cewish to con

coupled to “ove-function in

ead for specifithen be writteed-given inclead is knowndent Schrödineous/NEGF fo

rL

C S

R

ψ

ψ ψ

ψ

.

ans of the iso The off-diag

e adjacent regnd-side is give

wave-packet withon-ADI (b) ADI1step for the nonAblue) methods a

n for Both time inear uired ACK ethod tailed urate

h that

ONS entral nsider open”

each

ficity. en as ident

n, and ger’s

orm,

(7)

olated gonal gions. en by

Ccoultraninderequof thtimebounreasstatiextrcentassuperhmultthat the latte

FempabsofiniteffecomTo averlarggrapleadHowposiregirefle

Tthe tGauwidecons6(a)potecomat th15 mlong(NOleft showfuncThe regisimu

h a 1 (c) ADI as a

ψ

Consider nextld derive bosmitting bou

ependent NEGuired for the these requires ke step, the pndary, or at onable approxionary but at apolation of ttral region tumptions abohaps aided byti-band transpabrupt variatlead boundar

er approach toFor these reaploy stationaryorbing complete length, succtive, howeve

mbination of cavoid reflect

rage complex e to complet

phene―wave-d and reflect wever, the ratition, particulon and the ection in and oTo first illustrtime evolution

ussian wave-pe by ~19 nmsider three sce: an ~85 nm

ential (NOCAmplex absorbinhe boundary tomeV at its rigg lead regioOCAP-L). (Wi

lead is essenw the probabiction of time

reflection fron in the NOCulation region

0

LC

L

S

i

H

H

ψ

t absorption boundary selfundary condiGF. Howevtime-dependenkeeping trackpast history

least a sigximation. It mleast local-in

the time-depeto (just) acrout its approy a transversport and quastions in the wry may rema QTBCs challsons, for simy but spatiallex potentials―ch as used iner, there are soomplex potention from thepotential with

tely absorb a-function befoall the way

te of change arly near thelead, cannot of itself. rate the absorpn of an initial

packet out ofm long section

enarios for thelong lead reg

AP); the samng potential (Co the central r

ght-side hard-won with no ith the predomntially of no ility density wfor the non-Arom the lead CAP case is cn is effectivel

0

inLC

inLdt

d

ψ

ψ.

by the leads.f-energies bations (QTBC

ver, non-stationt system. In

k of, and integof the wav

gnificant perimay be possibn-time approxendent wave-fross the bouximate form

se mode expai-non-dispersi

wave-functionain at the boulenging.

mplicity, and ly non-local p―self-energien electromagome basic req

ntial and lead e end of thehin the lead many injected―ore it can reaback to the of the comp

e boundary bebe so fast

ption boundarlly predominaf an approximn of graphenee right lead, a

gion with no cme lead regioCAP) linearly region up to awall boundary

complex abminately right-

consequence)within the simuADI, ADI1 an

end back inclear. In the Nly a perfect in

In principleased on quaCs) as for onary QTBCsprinciple, crerating over at

ve-function atod thereof f

ble to provide ximate QTBCfunction withiundary, based

(such as inansion). Howive transport,

n induced far undary, make

for flexibilityposition-depens―within lea

gnetics [8]. Tquirements thalength must m

e finite leadsmust be suffici―and very faach the end osimulation re

plex potential etween the ceas to cause

ry conditions ately right-dirmately (~) 21e is simulatedas illustrated incomplex absoon with an a

ramped froma purely imagy; and an ~17bsorbing pote-directed wave). Figures 6(bulation region

nd ADI2, methnto the simul

NOCAP-L casenfinite lead w

(8)

, one antum time-s are

eation t each t the for a non-

Cs via in the d on n [7]

wever, such from

e this

y, we ndent

ads of To be at the meet. s, the iently ast in of the egion.

with entral back

only, rected 1 nm

d. We n Fig rbing

added m zero ginary 0 nm ential e, the b)-(d) n as a hods. lation e, the

within

53

the cthe wbackassobetwthe althoand miniFinaindecomregio

Tcons~21 nm lenerthe mwava 20priorwithrectaimmneceachisimubackobsethe elengnecetraveof Fprop

Figleafunwitbla

(c)

(a)

considered 25wave-function

k and forth ciated result

ween the CAP accuracy of tough accurate

channel lenimize the latally, we notependent of th

mputational buons that will bo illustrate sider transientnm wide met

long central rergetic 95 meVmetallic subbae-function is r fs time constr to saturatio

hin the “slice”angle along t

mediately adjacessary. As shoeved within ulation periodk from the riervable via a sevanescing of

gth of the rigessary; again, ersing both wa

Fig. 7(b) also posed ADI me

gure 6. (a) Geomead) NOCAP-L snction density in th (b) Non ADI, ack) and CAP (so

50 fs simulation of maximumacross the lethe referencand NOCAP

the absorbinge here, the comgth has not tter and, thu

e that the rehe size of theurden will dbe required forinjection alot through stetallic armchaiegion. n

Liψ no

V electron reland. Howeverramped expontant, so it is non. Also, n

Liψ

”―correspondthe x directiocent to the cen

own in Fig. 7, the central r

d. Moreover, ight lead at astanding wavef the probabilight lead wasthe probabilitays across theevidence the

ethods.

etries used for Csimulations as dthe central region(c) ADI 1 and, (dlid red) and NOC

(

on period as thm velocity 1 ead region, we ideal resulresults, there

g potential apmbination of a

yet been fuus, the compquired length

e central regiodecreases wir device simulong with theeady-state tranr graphene ribminally correative to the Dr, the amplitunentially towaot truly mono was defined

ding to the thicon in Fig. 1―ntral region, wnear-steady-s

region well wthere is no a

any time, as e pattern. (To ity with positis extended bty need only be length of the

continued ef

AP, NOCAP anddescribed in texn as a function od) ADI 2 methodCAP-L (solid blue

(d)

(b)

here is no timnm/fs to trav

which makeslt. The agreefore, demonst

pproach. Howabsorbing poteully optimizeputational burh of the leadon, so the relith larger celation. e absorption,nsport througbbon, with ansponds to a mDirac point wude of the incard saturation -energetic, at

d as nonzero ckness of the ―of the left which is all thtate condition

within the 14apparent reflewould be reshow more clion in the leadbeyond what be absorbed bee lead.) The reffectiveness o

d (effectively infixt. Total probabf time for simula

ds, for NOCAP (se) simulations.

me for verse s the ment trates

wever, ential ed to rden. ds is lative entral

, we gh an n ~23 mono-within

ident with least only gray lead

hat is ns are 40 fs ction adily early

d, the was

efore esults of the

Fillusleft by rcoulneed

WdepesteatranHamnumabsoframExteposs

[1] YY. CScale[2] HBaseno. 9[3] SReddCMO2032[4] MMath[5] JevalucondPhilo[6] G"WaB 78[7] Lof MMultPhys[8] indu

Figwiththe eigestealeadAD

(a)

inite bility ation solid

Finally, we wistrated above and right lead

region and sould be placed ded to simulat

We have presendent NEGFdy-state quassport in gr

miltonian, nomerical time-orbing boundamework were ension to musible.

Y.-M. Lin, C. DChiu, A. Grill, ae Epitaxial GraH. Wang, A. Hed Ambipolar R9, pp. 906–908,S. K. Banerjee, dy, and A. H. OS Application2–2046, 2010. MATLAB versihWorks Inc., 20J. Crank, and uation of solutduction type,” osophical SocieG. M. Maksimave packet dyna8, 235321 (2008L. F. Register, Mesoscopic Stidimensional Ts. Vol. 69 (10), C. Leforestier

uced dissociation

gure 7. (a) Snapshin the central renon-ADI implem

en-mode of the ady-state with a 2d. (b) A subset

DI2 methods with

ish to point oin terms of a

ds, the Hamilturce terms and

anywhere wte a particular

III. CO

sented a framF method for si-ballistic inraphene usinovel alterna-evolution sary conditions

demonstrateulti-graphene-

REFER

Dimitrakopouloand P. Avouris, aphene,” SciencHsu, J. Wu, J. KRF Mixers,” Iee 2010. L. F. Register, MacDonald, “

ns,” Proceeding

ion 7.13.0. Nati011. P. Nicolson, “

tions of partial Mathematical

ety, vol. 43, no.mova, V. Ya. Damics in a mon8 ). U. Ravaioli, an

Systems With Time-Dependenpp. 7153-7158 and R. E. W

n,” J. Chem. Ph

shots of probabiegion and a long rmentation, with a

ribbon, althoug20 fs time constan

of the results oessentially indist

out that, althoua central simutonian is not ad, separately,

within a plansystem of int

ONCLUSION mework for asimulation of

ntra- and intng an atomating directischemes, ands. The essentied via illustr-layer system

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ominally propagatxponentially towource term in the non-ADI, ADI1 lts.

d and n and vided gions ne as

time-rough antum nding

mplicit and

f this tions. so be

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bridge

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Appl.

laser 983.

tion with ting

ward left and

54