The g DFTB method applied to transport in Si nanowires and carbon nanotubes 1 Dip. di Ingegneria...

The gDFTB method applied to transport in Si nanowires and

carbon nanotubes

1 Dip. di Ingegneria Elettronica, Universita` di Roma Tor Vergata2 Computational Material Science, Universitat Bremen

Alessandro Pecchia1 L. Latessa1, Th. Frauenheim2, A. Di Carlo1

ACS - San Francisco 2006

RRD

DRDDR

LDL

H

H

H

0

0

H

,r aE i S H G I

Retarded (r) and advanced (a) Green functions are defined as follow

Let us write H and G in a block form

RRDRL

DRDDL

LRLDL

GGG

GGG

GGG

G

NEGF + DFTB = gDFTB

1 RLDD HEG

LDLDLL g

RDRDRR g Self-energies

Device region

Lead Lead

HDHL HR

LD RD

- +


Non equilibrium density

In order to compute V(r) we need the local (r)

1

( )2

dE G Ei

Density Matrix

( ) ( ) ( ) ( ) ( )L RL RG E i E f E i E f E

We can build the n.e. density matrix

L R

( ) ( ) ( ) ( )r aG E G E E G E


Self-consistent loop0μνH

<G

Density matrix

ρ

Multigrid Poisson solver

Self-consistent

solutions iq eR RG Σ

Evaluation of<G

Green’s function

1μνH

External potential

Hartree term

Exchange-correlation (LDA)+ +


Implementation details

<G Is performed by countour integration and has been parallelized (MPI)

1 RLDD HEG

• All matrices stored in dense format

• Green’s functions computed by direct inversions

• Sparse storage

• Implementation of a block-iterative construction

Old gDFTB (2003-2005)New gDFTB (2006-)


Block-iterative algorithm

The device G.F. are computed with an iterative algorithm

1, , , , 1 1, 1 1,[ ]L L L L L L L L L L L Lg ES H T g T

1) Computation of partial Green’s

2) Computation of equilibrium Green’s

, , , , 1 1, 1 1, ,L L L L L L L L L L L L L LG g g T G T g

3) Additional blocks needed for non-equilibrium

, , , 1 1,L n L L L L L LG g T G

HPLHPLHPLHPLHPL

†

†

†

†

L

L

L

L

H

H

H

H

H


Profiling

SWCNT(20,0): 2880 atoms, 36 Principal Layers

2,5 nm

14,7 nm

60,2

5,40

20

40

60

80

Tempo (s)

OldgDFTB

NewgDFTB

1743

1860

500

1000

1500

2000

Peak Memory(MB)

OldgDFTB

NewgDFTB

Single node (P4 - 3.2 GHz), Single energy point


Self-consistent potential

eV


Poisson Equation

The Poisson’s equation is solved with a 3D Multi-grid algorithm.

( ) ( )n r q n r

2 4V n

Discretize in real space

This allows to solve complex boundary conditions (bias, gate)

2-terminals gated

coaxially-gated

4-terminals


projection back in AO

3 ( ) ( )V d rV r n r

0 1( )

2H H S V V

Now we need to project the solution into the local basis set

Can be viewed as an approximation of the rigorous matrix elements of V(r).

This is consistent with standard DFTB


Summary gDFTB implementation

Construction of H and S directly in sparse format

Solution of Green’s functions via block-iterative methods

Current Bottle-neck: Poisson equation

- Very efficient for 1D type systems- Memory scales linearly (depends on PL size)- Can be used for O(N) calculations even in equilibrium- Considerable speed-up and memory save

- Dense matrices never allocated

- Multigrid with uniform grid, dense storage!- Need to implement more efficient methods (finite elements with adaptive grids)


Applications of gDFTB

symmetric moleculesymmetric molecule

Molecular Electronics.

Incoherent Transport andInelastic Tunneling Spectroscopy

A. Pecchia et al., Nano Lett. 4, 2109 (2004)

G. Solomon et al., J. Chem Phys 124, 094704 (2006)

A. Pecchia, A. Di Carlo, Introducing molecular electronics, Springer Series, (2005)

A. Pecchia, A. Di Carlo, Molecular Electronics: Analysis design and simulations, Elsevier (2006)


Applications to CNT and SiNW


Coaxially gated CNT

VD VS

=0

VG

Semiconducting CNT

Insulator (εr=3.9)

10 nm

1.5 nm

x

yz

CNT contact

(INFINEON - Düsberg)


Atomic Forces VGATE = 5 V

Ang.

An

g.

GATE

GATE

Forces

[Ang]

Application of VG changesCNT diameter


Screening problem

QinsG CCC

111

insC

QC

GV

CNTVQuantum correction to the induced charge

Vext

Distance

V

G

CNT electron gas

CNT is not able to accumulate the electronic charge to completely screen the gate bias (λ > electron gas extension)

λ CNTgate

rins RR

LC

/log

2 0

CNT completely screens the external field.

Classical electrostatics: charge induced on the CNT is

λ


HOWEVER: In a CNT the DOS is not the only contribution. Many body correction should be considered (XC)

Screening in CNT: DOS limit

Why charge induction is limited?

DO

S

[Latessa et al., Phys. Rev. B 72, 035455 (2005)]Pauli exclusion principle limits the induced electrons to the number allowed by filling the DOS

FCNTQ EDOSeVQC 2/


Many-body corrections

0

22

K

KEDOSe

neC FQ

Compressibilityof an interacting

electron gas

Compressibilityof an non-interact

electron gas

1 2K nn

1

QCQ

Compressibility Capacity


Evidence of Negative K

Eisenstein, Pfeiffer and West, PRL 68, 674 (1992)Eisenstein, Pfeiffer and West, PRB 50, 1760 (1994)

N (1011 cm-2)

0.0

0.1

-0.1

-0.20.0 0.2 0.4 0.6 0.8 1.0 1.2

Nc

Compressibility of 2D electron gas

1/ 220 1 ce d N

N N

Thomas-Fermi screening

In 1D systems things can be more complicated because of D(E)

Including exchange


Negative CQ in CNTs

CNT ox

G Q ox

V C

V C C

ox QG

g Q ox

C CdQC

dV C C

Overscreening, CQ<0

[Latessa et al., Phys. Rev. B 72, 035455 (2005)]


Negative CQ in CNTs

Critical charge density nTS

TOTAL SCREENING

High density limit:PARTIAL SCREENING

CQ approaches e2DOS(EF)

gDFTB calculation

CQ proportional to DOS

Low density limit: OVER-SCREENING

Fit to analytic modelChalmers, PRB 52, 10841 (‘95)

Fogler, PRL 94, 056405 (2005)


XC in DFTB

Hubbard and e-e repulsion integrals

0

( )[ ] [ ] ( ) ( ') '

( ')

xcxc xc i

k k

k

v rv n v n n r n r drdr

n r

T.A. Niehaus, PRA 71, 022508 (2004)


Diameter Dependence

CNT (13,0) CNT (10,0) CNT (7,0)


Output characteristics

dEEEfEEfETh

eI SFDF ,,

2h

VDS < 0

p pi

EF,SEF,D

Drain Source

“Electrostatic saturation”

L. Latessaet al.: IEEE Trans. Nanotechnol., in press


• Small sub-threshold swing

(theoretical limit for silicon MOSFET: 60 mV/dec)• Ion/Ioff ~ 108

• Unipolar behavior

Trans-characteristics

DS

G

Id

dVS

log


Band-to-well tunneling

Generation of confined states in a quantum well


SiNW FET

SiO2 shells has been removed and silicon is terminated with H

D. D. D. Ma. et al., Science, vol. 299, pp. 1874-1877, 2003

L. J. Lauhon, et al., Nature, vol. 420, pp. 57-61, 2002.

Coaxially gated Si nanowire FET


Geometry relaxations

d<10 nm

10<d<20 nm

d>30 nm

1.22 nm

0.87 nm


Device geometry

P doped region P doped regionIntrinsic regionoxide

oxide

1.2 nm (2.4 nm)

7.7 nm

3.6 nm

6 nm

Drain Source


CNT vs SiNW

CNT-FET

FQ EDOSeC 2

SiNW-FET

FQ EDOSeC 2

6 nm6 nm


Differences in S

Coax. gated (7,0) CNTFET SiNW – FET

|VGS| (V)

C

urre

nt,

I DS (

A)

S = 180 mV/decS = 75 mV/dec


Conclusions of part II

Atomistic Density Functional approach can be extended to account for current transport in molecular devicesby using self-consistent non-equilibrium Green function.

We use gDFTB is a good compromise between simplicity and reliability but there is room for improvement.

The use of a Multigrid Poisson solver allows for study very complicated device geometries

CNT and Silicon Nanowire FET has been studied with gDFTB

Quantum capacitance in CNT is governed by XC

Gate control in SiNW FETs is more delicate


Surface Green’s function

The surface G.F. is computed by iteration (decimation technique)

HPLHPLHPLHPLHPL

g

†

†

†

†

L

L

L

L

H

H

H

H

H

12345 -> Converged surface Green’s function

Lopez Sancho et al., J. Phys. F: Met. Phys. 14 1205 (1984); ibid., 15 851 (1985)

The g DFTB method applied to transport in Si nanowires and carbon nanotubes 1 Dip. di Ingegneria...

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