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ElectronElectron--vibration scatteringvibration scatteringin molecular transportin molecular transport

Alessandro Pecchia

CNR/CNISM - University of Roma “Tor Vergata”

CollaborationsCollaborations

EPSRC Symposium Warwick, August 24-28, 2009 2

http://flourish.org/upsidedownmap/

ElectronElectron--vibron scattering using NEGFvibron scattering using NEGF


ElectronElectron--vibron scattering using NEGFvibron scattering using NEGF

Quantum TransportQuantum Transport

1

0( ) [ ]r r r rL RG E E H

Electron propagator

eμL

μR

Device regionLeft contact

ee

Right contact

Electron in-scattering functions


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

2[ ]

L

R

eI Tr G G dE

h

L R

Kinetic equation for the electronic density

Current (Meir-Wingreen)

The DFTB approachThe DFTB approach

DFTB = DFT based Tight-Binding method

0

[ ] [ ] repk k k XC

k

E n n H n E n r E

0k

kH E S cKohn-Sham equation:

Density Functional:


k

[Porezag, et al Phys. Rev. B 51 (1995) 12947]

H [ ]V n n

onsite atomic energy levels

two-centre density superposition

SelfSelf--consitent extensionconsitent extension

0 ( )SccH H H nSelf consistent H:

(2)

,

( )1( ) ( ') '

' ( ')xcv r

E n r n r drdr q qr r r

Second order energy term:


[Frauenheim, et al. J. Phys.: Cond. Matt. 14 (2002) 3015]

1

2SccH S q

[Elstner, et al. Phys. Rev. B 58 (1998) 7260]

Correcting term in H obtained by variational theorem

electronelectron--phonon couplingsphonon couplings

= † †

,

ˆ ˆ ( )qq q

q

H H c c c c a a

The electron-phonon coupling Hamiltonian is derived by expanding to first orderthe TB-Hamiltonian with respect to the atomic positions.

H


02

,, ,

ˆq q

II q I I

Q

He

m Q

, , ,( ) ' ( ') ( ')

2q q

ph qq

iE dE G E E D E

Electron phonon self-energy, Born approximation:

IETSIETS –– molecular fingerprintsmolecular fingerprints

I


Elastic

InelasticV

h/e V

h/e V

h/e V

dI/d

Vd

2I/d

V2

h/e

Alkanethiols on AuAlkanethiols on Au

Reed’s experiment Calculations match and enhanceexperimental assignment


W. Wang, T. Lee, I. Kretzschmar & M.Reed Nano Lett. (2004) 4(4) 643-646

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124, 094704

Geometry dependenciesGeometry dependencies

higher energy

lower energy


Kusmerick et al. Nano Lett. (2004)

Selection/propensity rulesSelection/propensity rules


Selection/propensity rulesSelection/propensity rules

Conductance point groupConductance point group

Lost operators: axes C2x and C2y, inversion, Plane xy

• GM commutes with all molecular point-group symmetryoperators

• L and also R only commute with operators that do notinterchange the electrodes

• eg. Benzenedithiol (BDT): Full point group = D2h

Conductance point group = C2v


Lost operators: axes C2x and C2y, inversion, Plane xy

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. 125 (2006) 184702

BDT Buttiker channelsBDT Buttiker channels†)( MRML GGg Diagonalize


Black: total A1 + A2 + B1 + B2 red: highest gii blue: second highest gii

2 1

Junction orbitalsJunction orbitals

i j k l

Mil

Rkl

Mjk

Lij

MRML

GG *

†

)()Tr(

)(

g

GGg

diagonalize L,R => Junction-Conductance Orbitals


2

,

Tr( ) " | " | "L M Rii ij jj

i j

G gLeft-junctioninteraction

Right-junctioninteraction

Through-moleculeconnectivity

Junction ChannelsJunction Channels


Eigenvalue A1 A2 B1 B2

1 0.19 0.001 0.035 0.008

2 0.010 0.005 0.0004

3 0.0015 810-8 410-10

4 710-5 910-14

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush Nano Lett. 6 (2006)

Energy

Inelastic channelsInelastic channels

2( ) Tr

q

r r a ainc q L q R q

q eV

eI eV G G G G dE

h

2

,

| |L q Rq ii ij jj

i j

I A1


1 11 1A AA A

Propensity rule:

L,R are mainly A1

A1 (ag) modes are favouredB1

B1

B1

A1

A1

A1

A1

Diagonalize spectral matricesDiagonalize spectral matrices

2( ) Tr

q

a r r ainc q L q R q

q eV

eI eV G G G G dE

h

* 2( ) [( ) ] | |L R L q Rq q q ii ij jj

ij

strength Tr A A A A

Block-Diagonalize according to transportsymmetries


,L RA Diagonalize the A matrices and transformthe electron-phonon coupling matrices

Injectionchannel

Extractionchannel

Through-moleculeinelastic scattering

1 32 Totally symmetric =

symmetries

2( ) | |L q Rq ii ij jj

ij

strength A A

Simmetry and MOsSimmetry and MOs


Simmetry and MOsSimmetry and MOs

Eigenvalue A1 A2 B1 B2

1 3.44 0.011 27.4 2.26

2 0.144 0.071

3 0.128


Weighted Junction Cond. OrbitalsWeighted Junction Cond. Orbitals

Few dominant channels with rep.

A1 (a1g , b1u) and B1 (b3u ,b2g) :

2| |L q RI A A

B1

B1

B1

A1

A1

A1

A1

A1


1 1 1 1A BA B

11 11B BA A

2| |L q Rq ii ij jj

ij

I A AB1

Gagliardi, Solomon, Pecchia, Frauenheim, Di Carlo, Reimers, Hush, PRB 75, 174306 (2007)

Inelastic Interference patternsInelastic Interference patterns


Power DissipationPower Dissipation


Power DissipationPower Dissipation

Molecular heating&coolingMolecular heating&cooling

Thermal effects at the molecular scale represent an increasingly ‘hot’ topic

Theoretical and experimental challenges to measure nanoscale temperatures


Z. Huang et al. Nano Lett. 6, 1240 (2006)

N. Néel et al. PRL 98, 065502 (2007)

Z. Ioffe et al. Nature Nanotech., on-line doi:10.1038/nnano.2008.304

electronelectron--vibron scatteringvibron scattering

Elastic

V

μL

μ

R

Left contact

Device regionRight contact

Inelastich/e

1( ) [ ]r DFT r r rL R scattG E ES H

, , , , ,2 Im( )rL R L R L R L R L Rif if

( ) [ ]r aL R scattG E G G


Set up a steady-state solution for the vibronic populations

( ( )) 0q

q q q q A

dNR J N n T

dt

Rate equation:

Emission ratesEmission rates

2

[ ( ) ( ) ( ) ( )]q q qW E Tr E G E E G E dEh

q

q

q

WR


Delicate cancellation of terms which relays on current conservation

( 1)q q q q qR N E N A

0( )q q q

q

q q q

n T J EN

J A E

A. Pecchia et al., Phys. Rev. B 75, 035401 (2007)

( ( ))q q q q AR J N n T

SC phonon populationSC phonon population

- The calculation of Rq relays on a delicate cancellation of terms

- Current conservation is a necessary condition

- We use a predictor-corrector strategy and SCBA

SCBA LOOP

( ) ( ) ( 1) ( )q q q q q q q q qE N G E N G E

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


2

[ ( ) ( )]q q q qE Tr G E G E dEh

2

[ ( ) ( )]q q q qA Tr G E G E dEh

0( )q q q

q

q q q

n T J EN

J A E

( ) ( ) ( ) ( )G E G E E G E

SC POPULATION LOOP

CC6060 burning experimentburning experiment

Tip approaches at fixed V until C60 cracks.Molecules can be selectively burned

10-5

10-4

IC

rack

[A

]

-4,0 -3,5 -3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5

10

100

1000

I[n

A]

Z [Å]

A

B

C

D

E


0,5 1,0 1,5 2,0 2,5 3,0

10-5

Bias [V]

G. Shulze et al., Phys. Rev. Lett. 100, 136801 (2008)

P = Icrack ∙Vcrack is not a constantbut shows features and plateaux

Dissipation ratesDissipation rates

The vibron lifetimes areproportional to the linewidths

The C60 is relaxed on the substrate


Vibron density of states.Shaded area Cu phonon DOS.

The vibron lifetime (Jq) are extractedadapting the Green’s functionsformalism to the molecule/substratecoupled dynamical system.

2

,

1Im ( )q q

q q

q

J e e

Heating and cooling processesHeating and cooling processes

μ

L

μ

R

μ

L

μ

R

μ

L

μ

R

μ

L

μ

R

Phonon emission (Eq) Phonon absorption 1 (Aq)

Phonon absorption 2 (Aq) Phonon dissipation (Jq)


e-h pair formation

( )q q q q molq q

U N n T

Definition of molecular temperature:

Model vs ExperimentsModel vs Experiments

Theory predictions Experiments


Tip-induced cooling

Power diss. in the moleculePower diss. in the molecule

Pmol P=IV


Pmol<<IV

Cooling effectsCooling effects

When the resonance approaches the bias windowseveral regimes take place subsequently:a) Resonant e-h absorption that cools the moleculeb) Intermediate regime with competing effects

a

b

c a b c


b) Intermediate regime with competing effectsc) Emission take over absorption

In the regimes a) and b) absorption wins over emission and themolecular temperature can decrease within a bias interval.

Temperature decreases <> Power dissipated increases

“Negative Differential Thermal Resistence” regime ?

Temperature vs biasTemperature vs bias

C60 on Si(100)C60 on Cu(110)


SummarySummary

Incoherent electron-vibron interactions:

- We can understand propensity rules in terms of coupling with L/Rprojected LDOS

- We can predict and analyze IETS


Disadvantages:

- Approximate method (requires testing)

- Repulsive potentials are needed for relaxations

- We can model and understand molecular heating

AcknowledgementsAcknowledgements

Prof. Aldo Di Carlo

Matthias Auf Der Maur, Post Doc

Fabio Sacconi, Post Doc

Alessio Gagliardi, Post Doc

Michail Povolotsky, Post Doc

Giuseppe Romano, PhD

Gabirele Penazzi, PhD

Marco Pacini, Student

Luca Salvucci, Student

Carlo Caringi, Student


Thank you

Carlo Caringi, Student

Prof. Jeff Reimers Univ. of Sidney

Gemma Solomon (PhD Sidney- Post Doc Chicago)

Molecular ChannelsMolecular Channels

i j k l

Mil

Rkl

Mjk

Lij

MRML

GG *

†

)()Tr(

)(

g

GGg

Diagonalize GM Molecular-Conductance Orbitals


i ij

Mjj

Rji

Mii

Lij

i

Mii

Rii

Lii GGG *2 )'('''|'|'')Tr(g

Current through eachmolecular-conductance

orbital

Interference betweenmolecular-conductance

orbitals

Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush Nano Lett. 6 (2006)

Interference effectsInterference effects


i ij

Mjj

Rji

Mii

Lij

i

Mii

Rii

Lii GGGT *2 )'('''|'|'')Tr(g

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