Molecular electronics: a new challenge for O(N) methods

1

Molecular electronics: a new challenge for O(N) methods

Roi Baer and Daniel Neuhauser (UCLA)

Institute of Chemistry and Lise Meitner Center for Quantum ChemistryThe Hebrew University of Jerusalem, Israel

IPAM, April 2, 2002

2

Collaboration

Derek Walter, PhD. Student (UCLA) Prof. Eran Rabani, Tel Aviv University Oded Hod, PhD. student (Tel Aviv U) Acknowledgments:

Israel Science Foundation Fritz Haber center for reaction dynamics

3

Overview

Molecular electronics is interesting Formalism O(N3) algorithm: non-interacting electrons Possible O(N) algorithm Electron correlation: O(N2) algorithm

4

Introduction

Why are coherent molecular wires interesting?

5

Conductance of C60

(a)

I

QDV

R1,C1

R2,C2

-1.0 -0.5 0.0 0.5 1.0

-0.8

-0.4

0.0

0.4

I (nA)

Tip Voltage (V)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(c)

(b)

(a)

dI/dV (a

.u.)

0.0

0.2

0.4

0.6

0.8

1.0

T = 4.2 K

T = 4.2 KdI/d

V (a.u.)

0.0 0.3 0.6 0.9 1.20.0

0.3

0.6

0.9

dI/dV

Tip Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0T = 300 K

dI/dV (a

.u.)

Tip Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

-0.8

-0.4

0.0

0.4

I (nA)

Tip Voltage (V)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(c)

(b)

(a)

dI/dV (a

.u.)

0.0

0.2

0.4

0.6

0.8

1.0

T = 4.2 K

T = 4.2 K

dI/dV (a

.u.)

0.0 0.3 0.6 0.9 1.20.0

0.3

0.6

0.9

dI/dV

Tip Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0T = 300 K

dI/dV (a

.u.)Tip Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

-0.8

-0.4

0.0

0.4

I (nA)

Tip Voltage (V)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

(c)

(b)

(a)

dI/dV (a

.u.)

0.0

0.2

0.4

0.6

0.8

1.0

T = 4.2 K

T = 4.2 K

dI/dV (a

.u.)

0.0 0.3 0.6 0.9 1.20.0

0.3

0.6

0.9

dI/dV

Tip Voltage (V)

-1.0 -0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0T = 300 K

dI/dV (a

.u.)

Tip Voltage (V)

Voltage [V]

dI/dV [a.u]

1.0

0.5

0.0

-1.0 0.0 1.0

T=4.2 K

STM

tip

Tunnel

Junction 1

Tunnel

Junction 2

(b)

D. Porath and O. Millo, J. Appl. Phys. 81, 2241 (1997).

6

Conductance of a nanotube

S. Frank and W. A. de Heer et al, Science 280, 1744 (1998).

7

Conductance of C6H4S2

Reed et al,

Science 278,252 (1997)

Chen et al,

Science 286,1550 (1998)

8

Coherent electronics

Size: ~ 1013 logic gates/cm2 (108) Response times: 10-15 sec (10-9)

Quantum effects: Interference Uncertainty Entanglement Inclonability

9

Interference effects

de-Broglie: electrons are waves Interference Nonlocal particle nature

Electrons are not photons! Fermions: cannot scatter into “any

energetically open state” Correlated: inelastic collisions, Coulomb

blockade… Tunneling: reducing/killing interference

effects, sensitive

10

A simple wire

W: Huckel parameters S D M: chain of 20 “gold” atoms, G G MW coupling = b Expect: current should grow with b

Units: eV

30 Carbons long

ML MR

V

bb

11

Sometimes more is less

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

b=1b=1.5b=2b=2.5b=3

I (e

2 /h V

olt)

V (Volt)

Inversion

12

Current from transmittance

R LI I I

Landauer current formula

R LI n E T E dE

30 Carbons long

ML MR

V

2eV 2eV

1

1 LL E

n Ee

13

Just because of the coupling…

0

0.2

0.4

0.6

0.8

1

-9 -8 -7 -6 -5 -4

b=1b=2

T(E

)

E (eV)

14

A switch based on interference

Simplest model of interference effects

2 4

6 8

10

15

Current-Voltage

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2

0246810

V (Volt)

I (e

2 /h V

olt)

Destructive

Constructive

2 4

6 8

10

16

Fermi Wave length

=L

a=CC

Band bottom

Totally bonding

=2a Band top:

Totally non bonding

F=4a Band middle

Half filling

17

XOR gate based on interference

V1

V2

Current I

V1V2I

000

110

011

101

18

SensitivityDFT electronic structure. Molecule connected to gold wire, acting as a lead

Cu

rre

nt (

nA

)

Bias (Volt)

19

Quantum conductance formalism

( )ˆ1 ˆlimTr H N

R RtI Z e e I tb bm- - ×

®¥

é ù= ê úë û

rr

ˆ ˆ ˆ ˆ,l e l e l

iI qN q H Né ù= - = - ê úë û

&h

L R

IR

Wire

hR=1hR=0

( )ˆ ˆiH t iHt

R RI t e I e-

=h h ˆ

Tr H NZ e eb bm- ×é ù= ê úë û

rr

R. Baer and D. Neuhauser, submitted (2002).

20

Weak Bias: Linear Response

( ) ( ) ( )ˆ1 ˆˆlimTrlH N

lR e l RtG qZ e I I t

b mm m b¢ - -é ù -=ë û

®¥

é ù= ê úë û

Conductivity is a current-current correlation formula

R. Kubo, J Phys. Soc. Japan 12, 570 (1957).

21

Non-interacting Electrons

( ) ( )2 e

R l lRl R

qI F E N E dE

h ¹

= å ò

( )( )

1

1 llF

eb e me

-=

+

NlR(E) = cumulative transmission probability (from l to R)

R. Landauer, IBM J. Res. Dev. 1, 223 (1957).

22

Calculating conductance

Non-interacting particle formalism 4 step O(N3) algorithm

23

Step #1: Structure under bias

Use SCF model like DFT/HF etc. Optimize structure and e-density

ss

+

+

+

+

+

+

+

-

-

-

-

-

-

-

Right slabLeft slab

24

Step #2: Add Absorbing boundaries

effL RH H i

D. Neuhauser and M. Baer, J. Chem. Phys 90, 4351 (1989)

ss

+

+

+

+

+

+

+

-

-

-

-

-

-

-

Left slab Right slab

LG RG

25

Step #3: Trace Formula

( ) ( ) ( )†4 L RN E Tr G E G Eé ù= G Gë û

( ) ( ) 1G E E H -= -

T. Seideman and W. H. Miller, J. Chem. Phys. 96, 4412 (1992).

26

Step #4: Current formula

L R LRI F E F E N E dE

1

1 llF

e

l leVm m= +

(Landauer formula)

0

0.2

0.4

0.6

0.8

1

1.2

-1.5 -1 -0.5 0 0.5 1 1.5

DF

0

0+eV/2

0-eV/2

27

Efficient O(N3) Implementation

*

16Re L Rn

nm mnnm n m

e BI W W

h ff=

-å

( ) ( )L Rn

n

F E F EB dE

E f-

=-ò

N(E) is spiky Integrate energy analytically

†L LW U U= G

n n nHU Uf=

28

O(N) Algorithm

N(E) is averaged over E → A sparse part of G needed

The trace can be computed by a Chebyshev series

All energies computed in single sweep: integration is trivial

( ) ( ) ( )†

†

4 L RN E Tr G E G E

Tr T T

é ù= G Gë ûé ù= ë û ( )2 L RT G E= G G

R. Baer, Y. Zeiri, and R. Kosloff, Phys. Rev. B 54 (8), R5287 (1996).

0

0.2

0.4

0.6

0.8

1

1.2

-1.5 -1 -0.5 0 0.5 1 1.5

DF

0

0+eV/2

0-eV/2

29

Including electron correlation

Time Dependent Density Functional Theory

30

Linear response

, ,t

I t L G t t E t dt

r r

, ,I LG E r r

Uniform, weak, time dependent electric field:

----

++++

2

220

t

E t E e

31

Building the model

Small jellium sandwich

Large jellium sandwich

Embed small in large

Frozen Jellium (leads)

Dynamic system (w+contacts)

ImaginaryImaginary

potentialpotential

ImaginaryImaginary

potentialpotential

32

The setup for C3

a) Dynamic density

b) Frozen density

c) Total Density

d) Kohn-Sham potential

33

Conductance of C3

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.25 0.5 0.75 1

z2 = -8 a

0

z2 = -4 a

0

z2 = 0 a

0

z2 = 4 a

0

z2 = 8 a

0

G(z

2,z0;

) [g

o]

[au]

34

Are correlations important?

Conductance is smaller by a factor 10. Possible reason: the same reason that

causes DFT to underestimate HOMO-LUMO gaps

35

Summary

Molecular electronics Theory of conductance Linear scaling calculation of conductance Importance of electrson-electron correlations TDDFT is expensive and at least O(N2)