ElEleecctrontron--vvibibrarationtion ssccaatteritteringng ... · Current (Meir-Wingreen) TThhee...
Transcript of ElEleecctrontron--vvibibrarationtion ssccaatteritteringng ... · Current (Meir-Wingreen) TThhee...
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ElectronElectron--vibration scatteringvibration scatteringin molecular transportin molecular transport
Alessandro Pecchia
CNR/CNISM - University of Roma “Tor Vergata”
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CollaborationsCollaborations
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http://flourish.org/upsidedownmap/
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ElectronElectron--vibron scattering using NEGFvibron scattering using NEGF
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ElectronElectron--vibron scattering using NEGFvibron scattering using NEGF
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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
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( ) ( ) ( ) ( )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)
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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:
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k
[Porezag, et al Phys. Rev. B 51 (1995) 12947]
H [ ]V n n
onsite atomic energy levels
two-centre density superposition
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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:
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[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
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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
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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:
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IETSIETS –– molecular fingerprintsmolecular fingerprints
I
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Elastic
InelasticV
h/e V
h/e V
h/e V
dI/d
Vd
2I/d
V2
h/e
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Alkanethiols on AuAlkanethiols on Au
Reed’s experiment Calculations match and enhanceexperimental assignment
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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
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Geometry dependenciesGeometry dependencies
higher energy
lower energy
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Kusmerick et al. Nano Lett. (2004)
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Selection/propensity rulesSelection/propensity rules
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Selection/propensity rulesSelection/propensity rules
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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
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Lost operators: axes C2x and C2y, inversion, Plane xy
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. 125 (2006) 184702
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BDT Buttiker channelsBDT Buttiker channels†)( MRML GGg Diagonalize
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Black: total A1 + A2 + B1 + B2 red: highest gii blue: second highest gii
2 1
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Junction orbitalsJunction orbitals
i j k l
Mil
Rkl
Mjk
Lij
MRML
GG *
†
)()Tr(
)(
g
GGg
diagonalize L,R => Junction-Conductance Orbitals
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2
,
Tr( ) " | " | "L M Rii ij jj
i j
G gLeft-junctioninteraction
Right-junctioninteraction
Through-moleculeconnectivity
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Junction ChannelsJunction Channels
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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
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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
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1 11 1A AA A
Propensity rule:
L,R are mainly A1
A1 (ag) modes are favouredB1
B1
B1
A1
A1
A1
A1
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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
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,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
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Simmetry and MOsSimmetry and MOs
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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
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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
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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)
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Inelastic Interference patternsInelastic Interference patterns
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Power DissipationPower Dissipation
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Power DissipationPower Dissipation
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Molecular heating&coolingMolecular heating&cooling
Thermal effects at the molecular scale represent an increasingly ‘hot’ topic
Theoretical and experimental challenges to measure nanoscale temperatures
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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
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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
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Set up a steady-state solution for the vibronic populations
( ( )) 0q
q q q q A
dNR J N n T
dt
Rate equation:
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Emission ratesEmission rates
2
[ ( ) ( ) ( ) ( )]q q qW E Tr E G E E G E dEh
q
q
q
WR
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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
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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
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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
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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
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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
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Dissipation ratesDissipation rates
The vibron lifetimes areproportional to the linewidths
The C60 is relaxed on the substrate
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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
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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)
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e-h pair formation
( )q q q q molq q
U N n T
Definition of molecular temperature:
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Model vs ExperimentsModel vs Experiments
Theory predictions Experiments
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Tip-induced cooling
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Power diss. in the moleculePower diss. in the molecule
Pmol P=IV
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Pmol<<IV
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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
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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 ?
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Temperature vs biasTemperature vs bias
C60 on Si(100)C60 on Cu(110)
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SummarySummary
Incoherent electron-vibron interactions:
- We can understand propensity rules in terms of coupling with L/Rprojected LDOS
- We can predict and analyze IETS
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Disadvantages:
- Approximate method (requires testing)
- Repulsive potentials are needed for relaxations
- We can model and understand molecular heating
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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
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Thank you
Carlo Caringi, Student
Prof. Jeff Reimers Univ. of Sidney
Gemma Solomon (PhD Sidney- Post Doc Chicago)
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Molecular ChannelsMolecular Channels
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Diagonalize GM Molecular-Conductance Orbitals
EPSRC Symposium Warwick, August 24-28, 2009 36
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Interference betweenmolecular-conductance
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Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush Nano Lett. 6 (2006)
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Interference effectsInterference effects
EPSRC Symposium Warwick, August 24-28, 2009 37
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