Quantum Capacitance Effects Quantum Capacitance Effects In Carbon Nanotube In Carbon Nanotube Field-Effect DevicesField-Effect Devices

L. Latessa, A. Pecchia, A. Di Carlo

P. Lugli

University of Rome ‘Tor Vergata’

Lehrstuhl fur Nanoelectronic, Munchen

Devices based on field effect (CNTFET)Devices based on field effect (CNTFET)

Planargeometry top/bottom

gate(IBM)

VerticalGeometry

coaxialgate

(Infineon)

• ULSI Devices • Quasi-1D coherent transport over long distances• Si integrable technology

gDFTBgDFTB

rErH kkk

ˆ

Green’s functions Density Functional Tight-Binding(A.Pecchia, L.Latessa, A.Di Carlo)

Atomistic simulations: Quantum treatment

rnErnErnTrnE XCHartree

0

Approximated DFT:

(2)0

,

12

repi i i

i

E n H q q E

NEGF ExtentionNEGF Extention

1

2dE G

i

0q q S

Density Matrix

Mullikan Charges

2 4

'Hn V n

H G nSCCloop

L R

A. Pecchia, A. Di Carlo, Rep Prog Phys 67, 1-65 (2004)

3D Poisson Multigrid3D Poisson Multigrid

r

:0 iii CCMetal Gate rrVrC

r

Insulating dielectricand CNT

:0CCi rrVrC

2

rrVrr

4

The cylindrical gate is added as an

appropiate Dirichlet boundary condition

Coaxially gated CNTFET Coaxially gated CNTFET Working mechanisms:

• Schottky barrier modulation at the contacts

• local modulation of channel conductance

Model:

• semiconducting CNT(10,0)• neglect Schottky barrier • Charge injection from p-doped CNT contacts

CNTFET control capacitiesCNTFET control capacities

CD

CS

CG

CG

Cox

CQ

Vg

VCNT

1 1 1

G Q oxC C C

2D fC e D E

1

lE

EC

Q

1 1 1

Q E DC C C

(S.Luryi, Appl.Phys.Lett. 52, 501 (1988), sistemi 2DES)

Gate capacitance Gate capacitance Gate is a Macroscopic metal

CNT

Oxide

In a well-tempered MOS CG >> CS,CD

In a classic MOS CQ >> Cox => modulation depends on Cox

Vds can influence the channel charge and barrier

In 1D systems CQ is small

2

ln( / )oxG CNT

CR R

Caratteristiche di uscita Caratteristiche di uscita IIDSDS/V/VDSDS

dEffGGTrh

eI AR

1212

2

Source

Drain

Source

Drain

Calculation of CCalculation of CQQ

insC

Bare Potential

Partial ScreeningQC

insCGV

QV

QQ

QC

V

Over-screening in CNT Over-screening in CNT

Negative CQ Ctot > Cins The can CNT better than a

metal!

rs=5.03

rs=6.71

rs=10.06

rs=20.12

rs=6.71

rs=5.03

Negative compressibilityNegative compressibility

J.P.Eisenstein et al., Phys.Rev.Lett. 68, 647 (1992)

L.Calmels, A.Gold, Phys. Rev.B. 53, 10846 (1996)

HFAFQ

CNT

K

K

EeC

L 02

1

Experimental measurements in GaAs quantum wells

Analytic results for quasi-1D electron systems

K0 : free electrons compressibilityKHFA: Hartree-Fock exchange compress.

1 2K nn

1

QC Q

Many-body exchange effectMany-body exchange effect

Charge carrier density nC: TOTAL SCREENING CNT ~ macroscopico

High density:PARTIAL SCREENING

Dominated by DOS

Low density: OVER-SCREENINGExchange interaction preveal

Calcolo DFT

CQ= e2(EF)

2

0

XCQ F

KC e E

K

XC in XC in gDFTBgDFTB

rs=5.03

rs=6.71

rs=10.06

rs=20.12

In DFTB the XC term is treated with a Hubbard on-site energy

Substituting UH with Uee accounts only for Hartree

The capacity becomes positive again

Quantum Capacitance vs DOSQuantum Capacitance vs DOS1/DOS

KK

EeC

L

F

CNT 02

1

Over-screening and modulationOver-screening and modulation

VGS

Unconventional Trans-characteristicsPossible applications ?

VGS

ConclusionsConclusions

Comprehensive calculation of Quantum Capacity in a CNT(10,0) using NEGF

XC effects can give deviations from “classical” CQ

Negative CQ below a crytical carrier density which compares well with analytic results in quasi 1D systems

More complicated behaviour when more subbands are occupied