[PPT] - Transport and optical response of molecular junctions UCSD July PowerPoint Presentation

SLIDE 1

Transport and optical response

f molecular junctions

UCSD July 20-21, 2009

Michael Galperin University of California at San Diego

UCSD July 20-21, 2009 – p.1

SLIDE 2

Introduction

UCSD July 20-21, 2009 – p.2

SLIDE 3

Introduction

Timescale → BO Energy scale

E

L R

M = L+ R

Weak el-ph coupling M ≪

∆E2 + (Γ/2)2

Moderately strong el-ph coupling M ≥

∆E2 + (Γ/2)2

UCSD July 20-21, 2009 – p.2

SLIDE 4

Introduction

J. Phys.: Condens. Matter 19, 103201 (2007)

Science 319, 1056 (2008).

IETS RIETS e-e interaction Noise Non-linear conductance Heating Light-matter interaction

UCSD July 20-21, 2009 – p.2

SLIDE 5

Introduction

L R |1> |2>

Phys. Rev. Lett. 95, 206802 (2005)
J. Chem. Phys. 124, 234709 (2006)

Nano Lett. 9, 758 (2009)

IETS RIETS e-e interaction Noise Non-linear conductance Heating Light-matter interaction

UCSD July 20-21, 2009 – p.2

SLIDE 6

Experiments

Metal enhanced fluorescence (Cy5 on Ag)

J.Zhang et al. Nano Lett. 7, 2101 (2007)

UCSD July 20-21, 2009 – p.3

SLIDE 7

Experiments

Intramolecular photon emission in STM

S.W.Wu et al. Phys. Rev. B 77, 205430 (2008)

UCSD July 20-21, 2009 – p.3

SLIDE 8

Experiments

SERS of molecules on nanoparticles

S.Nie and S.R.Emory. Science 275, 1102 (1997)

UCSD July 20-21, 2009 – p.3

SLIDE 9

Experiments

Simultaneous Raman and conduction

D.R.Ward et al. Nano Lett. 8, 919 (2008)

UCSD July 20-21, 2009 – p.3

SLIDE 10

Experiments

Heating detected by Raman

Z.Ioffe et al. Nature Nanotechnology 3, 727 (2008)

UCSD July 20-21, 2009 – p.3

SLIDE 11

HOMO-LUMO model

Absorption line shape of molecule in biased junction Light induced current in molecular junction Fluorescence from current carrying molecular bridge Current from electronic excitations in the leads Raman spectroscopy of biased junctions

Phys. Rev. Lett. 95, 206802 (2005); 96, 166803 (2006)
J. Chem. Phys. 124, 234709 (2006); 128, 124705 (2008)

Nano Lett. 9, 758 (2009); J. Chem. Phys. 130, 144109 (2009)

UCSD July 20-21, 2009 – p.4

SLIDE 12

HOMO-LUMO model

L R

L R |1> |2>

Fluxes considered electronic current through the molecule energy flow between the molecule and electron-hole excitations in the leads incident

r

emitted photon flux

UCSD July 20-21, 2009 – p.4

SLIDE 13

HOMO-LUMO model

L R

|2> |1> e e e

Fluxes considered electronic current through the molecule energy flow between the molecule and electron-hole excitations in the leads incident

r

emitted photon flux

UCSD July 20-21, 2009 – p.4

SLIDE 14

HOMO-LUMO model

EF |2> |1> e e

Fluxes considered electronic current through the molecule energy flow between the molecule and electron-hole excitations in the leads incident

r

emitted photon flux

UCSD July 20-21, 2009 – p.4

SLIDE 15

HOMO-LUMO model

|2> |1> e ph |2> |1> e ph

Fluxes considered electronic current through the molecule energy flow between the molecule and electron-hole excitations in the leads incident

r

emitted photon flux

UCSD July 20-21, 2009 – p.4

SLIDE 16

HOMO-LUMO model

ˆ H = ˆ H0 + ˆ V ˆ H0 =

m=1,2

εmˆ c†

mˆ

cm +

k∈{L,R}

εkˆ c†

kˆ

ck +

α

ωαˆ a†

αˆ

aα ˆ V = ˆ VM + ˆ VN + ˆ VP ˆ VM =

K=L,R
m=1,2;k∈K
V (MK)

km

ˆ c†

kˆ

cm + H.c.

ˆ

VN =

K=L,R
k=k′∈K
V (NK)

kk′

ˆ c†

kˆ

ck′ˆ c†

2ˆ

c1 + H.c.

ˆ

VP =

α
V (P)

α

ˆ aαˆ c†

2ˆ

c1 + H.c.

UCSD July 20-21, 2009 – p.4

SLIDE 17

HOMO-LUMO model

SE due to electron tunneling ΣMK,mm′(τ1, τ2) =

k∈K

V (MK)

mk

gk(τ1, τ2)V (MK)

km′

’

2 1

projections (WBL and no mixing) Σr

MK,mm′ = −iδmm′ΓMK,m/2

Σ<

MK,mm′(E) = iδmm′fK(E)ΓMK,m

Σ>

MK,mm′(E) = −iδmm′[1 − fK(E)]ΓMK,m

UCSD July 20-21, 2009 – p.4

SLIDE 18

HOMO-LUMO model

SE due to e-h excitations in the contacts ΣNK(τ1, τ2) =

k=k′∈K

V (NK)

kk′

2

gk(τ2, τ1)gk′(τ1, τ2) ×

G22(τ1, τ2)

G11(τ1, τ2)

’

2 1

2 1 2 k2 k1

UCSD July 20-21, 2009 – p.4

SLIDE 19

HOMO-LUMO model

Projections Σ<

NK,mm(E) =

dω 2π BNK(ω, µK) G<

¯ m ¯ m(E + ω)

Σ>

NK,mm(E) =

dω 2π BNK(ω, µK) G>

¯ m ¯ m(E − ω)

with BNK(ω, µK) = 2π

dE
k=k′∈K
V (NK)

kk′

2

×δ(E − εk)δ(E + ω − εk′)fK(E)[1 − fK(E + ω)] ≡ 2π

V (NK)

2 ρe−h

K (ω)

UCSD July 20-21, 2009 – p.4

SLIDE 20

HOMO-LUMO model

Simplified version when ε21 ≫ Γ1,2 Σ<

NK = iBNK

n2 0
Σ>

NK = −iBNK

0 1 − n1
where BNK = BNK(ε21)

UCSD July 20-21, 2009 – p.4

SLIDE 21

HOMO-LUMO model

SE due to coupling to photon field ΣP(τ1, τ2) = i

α

V (P)

α

2

×

Fα(τ2, τ1)G22(τ1, τ2)

Fα(τ1, τ2)G11(τ1, τ2)

’

2 1

2 1 2

UCSD July 20-21, 2009 – p.4

SLIDE 22

HOMO-LUMO model

Σ<

P(E) = α

V (P)

α

2

×

(1 + Nα)G<

22(E + ωα)

NαG<

11(E − ωα)

Σ>

P(E) = α

V (P)

α

2

×

NαG>

22(E + ωα)

(1 + Nα)G>

11(E − ωα)

N0 = 1 for pumping mode (absorption flux)

Nα = 0 for absorbing modes (fluorescence)

UCSD July 20-21, 2009 – p.4

SLIDE 23

HOMO-LUMO model

Simplified version for emission flux when ε21 ≫ Γ1,2 Σ<

P

= iγP(ε21)

n2 0
Σ>

P

= −iγP(ε21)

0 1 − n1
where γP(ω) = 2π

α

V (P)

α

2

δ(ω − ωα)

UCSD July 20-21, 2009 – p.4

SLIDE 24

Flux expression

IB = +∞

−∞

dE 2π Tr [ΣB

<(E) G>(E) − ΣB >(E) G<(E)]

with B0 = . . . P0, 22 or minus P0, 11 for absorption flux ML or minus MR for current through the junction P, 11 or minus P, 22 for fluorescence

UCSD July 20-21, 2009 – p.5

SLIDE 25

Absorption line shape

General expression Iabs(ω0) = +∞

−∞

dE 2π

Σ<

P0,22(E) G> 22(E) − Σ> P0,22(E) G< 22(E)

Simplified version (Lorentzian)

ε1 ≪ µL,R ≪ ε2 (low bias) coupling to the photon field is weak Γ1,2 ≪ ε21, |ε1,2 − EF| Iabs(ω0) =

V (P)
2
Γ

(ε2 − ω0 − ε1)2 + (Γ/2)2 × ΓM,1ΓM,2 Γ1Γ2

UCSD July 20-21, 2009 – p.6

SLIDE 26

Absorption line shape

1 2 3 4

0 (eV)

4 8 12 ( 10

+9)

Iabs (photons/s) =1.5 V =1.2 V =1.1 V =1.0 V =0.5 V

ε21 = 2 eV γP = 10−6 eV T = 300 K BNL = BNR = 0.1 eV ΓML/R,1 = 0.01 eV ΓML/R,2 = 0.2 eV

partial population of LUMO (HOMO) distortes the Lorentzian shape

UCSD July 20-21, 2009 – p.6

SLIDE 27

Light induced current

Radiation field in resonance with the molecular optical transition Molecules with strong charge-transfer transitions

DMEANS (4-Dimethylamino-4’-nitrostilbene) 7D (ground) → 31D (first excited singlet) all-trans Retinal in Poly-methyl methacrylate films 6.6D → 19.8D (1Bu electronic state) 40Å CdSe nanocrystals 0D → 32D (first excited state)

If optical charge transfer is parallel to the wire axis

ptical pumping → charge flow between the two leads

UCSD July 20-21, 2009 – p.7

SLIDE 28

Light induced current

General expression Isd = +∞

−∞

dE 2πTr [Σ<

ML(E) G>(E) − Σ> ML(E) G<(E)]

Simplified version (ω0 ∼ ε21, Φ = 0, Γ1,2 ≪ ε21) Isd = |V (P) |2

Γ

(ε2 − ω0 − ε1)2 + (Γ/2)2 ΓML,1ΓMR,2 − ΓML,2ΓMR,1 Γ1Γ2 The yield of the effect Yc = Isd Iabs

Φ=0

= ΓML,1ΓMR,2 − ΓML,2ΓMR,1 ΓM,1ΓM,2

UCSD July 20-21, 2009 – p.7

SLIDE 29

Light induced current

1 2 3 4

0 (eV)

1 3 5 ( 10

10)

Isd (A)

Φ = 0 ε21 = 2 eV γP = 10−6 eV T = 300 K BNL = BNR = 0.1 eV ΓML/R,1 = 0.2 eV ΓML,2 = 0.02 eV ΓMR,2 = 0.3 eV V (P) = 10−3 eV

peak at the HOM0-LUMO gap frequency

UCSD July 20-21, 2009 – p.7

SLIDE 30

Light induced current

0.0 0.5 1.0

(V)

5 10 15 20 ( 10-7)

Isd (A)

0=1.5 eV 0=1.7 eV 0=1.9 eV

Φ = 0 ε21 = 2 eV γP = 10−6 eV T = 300 K BNL = BNR = 0.1 eV ΓML,1/MR,2 = 0.2 eV ΓML,2/MR,1 = 0.02 eV V (P) = 0.02 eV

If the level position is pinned to the contact to which it is coupled stronger → NDR

UCSD July 20-21, 2009 – p.7

SLIDE 31

Fluorescence

Light emission from STM junctions e excites surface plasmon which later emits time-dependent potential of a tunneling e → electronic excitation of the molecule → fluorescence current carrying situation with excited state formed with a finite probability → photon emission. . .

UCSD July 20-21, 2009 – p.8

SLIDE 32

Fluorescence

Frequency resolved spectrum I′

em(ω)

= ρP(ω) +∞

−∞

dE 2π

Σ<

P,11(E, ω) G> 11(E) − Σ> P,11(E, ω) G< 11(E)

Overall emission intensity

Itot

em

= ∞ dω I′

em(ω)

= +∞

−∞

dE 2π

Σ<

P,11(E) G> 11(E) − Σ> P,11(E) G< 11(E)

UCSD July 20-21, 2009 – p.8

SLIDE 33

Fluorescence

When coupling to radiation field is weak and Γ1,2 ≪ ε21 I′

em(ω)

= γP(ω)

+∞

−∞

dE 2π fL(E + ω)ΓML,2 + fR(E + ω)ΓMR,2 (E + ω − ε2)2 + (Γ2/2)2 × [1 − fL(E)]ΓML,1 + [1 − fR(E)]ΓMR,1 (E − ε1)2 + (Γ1/2)2

Itot

em

= γP(ε21)

n2 [1 − n1]

UCSD July 20-21, 2009 – p.8

SLIDE 34

Fluorescence

When in addition µL ≫ ε2 > ε1 ≫ µR Itot

em = γP

ΓML,2ΓMR,1

Γ1Γ2 Also in this case Isd = 1

m=1,2

ΓML,mΓMR,m Γm + BN + γP

ΓML,2ΓMR,1

Γ1Γ2 So that the yield Yem = Itot

em

Isd = γP

ΓMR,2 ΓMR,1Γ1 + ΓML,1 ΓML,2Γ2 + BN + γP

UCSD July 20-21, 2009 – p.8

SLIDE 35

Fluorescence

Conditions for the higher yield here ΓMR,2 < ΓMR,1 ΓML,1 < ΓML,2 are opposite to the light induced case ΓML,1ΓMR,2 > ΓML,2ΓMR,1 Light induced current

L R

1 2 3

Fluorescence

L R

1 2 3

UCSD July 20-21, 2009 – p.8

SLIDE 36

Fluorescence

5 10 15 ( 10

7)

Iem

tot (s

1)

1 2 ( 10

5)

Isd (A)

BNL=BNR=0.1 eV 5 10 15 ( 10

7)

Iem

tot (s

1)

1 2 ( 10

5)

Isd (A)

BNL=BNR=1 eV 1 2 3 4 5 6

(V)

2 6 10 ( 10

7)

Iem

tot/Isd BNL=BNR=1.0 eV BNL=BNR=0.1 eV

T = 300 K ε21 = 2 eV ΓMK,m = 0.1 eV γP = 10−6 eV BNL = BNR = 0.1 eV

emission and e-h excitations compete for the same LUMO → HOMO transition

UCSD July 20-21, 2009 – p.8

SLIDE 37

Fluorescence

Fermi population features in the lineshape

1 2 3 4

(eV)

1 3 5 7 ( 10

9)

Iem’( )

=5 V =3 V =2 V =1 V

UCSD July 20-21, 2009 – p.8

SLIDE 38

Fluorescence

1 2 3 4 5 6

(V)

1 2 ( 108)

dIem

tot/d (s

1/V)

MK 3

BNK 10

1 2 3 4

(eV)

1 3 5 7 ( 109)

Iem’( )

=3 V

Linewidth is more sensitive to ΓMK than BN since ΓN,1 = BNn2 ΓN,2 = BN[1 − n1] while [1 − n1], n2 ≪ 1 for low bias

UCSD July 20-21, 2009 – p.8

SLIDE 39

Fluorescence

1 2 3 4 5 6

(V)

4 8 12 16 ( 107)

Iem

tot (s

1)

=0.99 =0.66 =0.50

η = ΦL/Φ = ΓMR,m/Γm = BNR/BN η → 1 no emission (either LUMO is empty

r HOMO is full)

Fluorescence in STM Φ should fall at the molecule-substrate interface

Spacers reduce energy losses into substrate (BN) But enable light emission at the molecule

UCSD July 20-21, 2009 – p.8

SLIDE 40

Current from e-h excitations

L R

L R |1> |2>

MG, A.Nitzan, M.A.Ratner, PRL 96, 166803 (2006)

UCSD July 20-21, 2009 – p.9

SLIDE 41

Current from e-h excitations

L R

|2> |1> e e e

Fluxes considered electronic current through the molecule energy flow between the molecule and electron-hole excitations in the leads

UCSD July 20-21, 2009 – p.9

SLIDE 42

Current from e-h excitations

EF |2> |1> e e

Fluxes considered electronic current through the molecule energy flow between the molecule and electron-hole excitations in the leads

UCSD July 20-21, 2009 – p.9

SLIDE 43

Current from e-h excitations

ˆ H = ˆ H0 + ˆ V ˆ H0 =

m=1,2

εmˆ c†

mˆ

cm +

k∈{L,R}

εkˆ c†

kˆ

ck ˆ V = ˆ VM + ˆ VN ˆ VM =

K=L,R
m=1,2;k∈K
V (MK)

km

ˆ c†

kˆ

cm + H.c.

ˆ

VN =

K=L,R
k=k′∈K
V (NK)

kk′

ˆ c†

kˆ

ck′ˆ c†

2ˆ

c1 + H.c.

UCSD July 20-21, 2009 – p.9

SLIDE 44

Current from e-h excitations

Isd = IL

sd + Ie−h sd

IL

sd

= e

+∞

−∞

dE 2π

m=1,2

Γ(ML)

m

Gr

mm(E)Γ(MR) m

Ga

mm(E)

× [fL(E) − fR(E)] Ie−h

sd

= e B

n(ML)

2

Γ(MR)

1

Γ1 − n(MR)

1

−n(MR)

2

Γ(ML)

1

Γ1 − n(ML)

1

Γ(MK)

m

≪ ε21 is assumed in the last

UCSD July 20-21, 2009 – p.9

SLIDE 45

Current from e-h excitations

For strong bias (e.g. µR ≪ ε1 < ε2 ≪ µL) IL

sd = e

m=1,2

Γ(ML)

m

Γ(MR)

m

Γm sgn(µL − µR) Ie−h

sd

= e B×

Γ(ML)

2

Γ(MR)

1

Γ1Γ2 θ(µL − µR) − Γ(ML)

1

Γ(MR)

2

Γ1Γ2 θ(µR − µL)

UCSD July 20-21, 2009 – p.9

SLIDE 46

Current from e-h excitations

L R

1

UCSD July 20-21, 2009 – p.9

SLIDE 47

Current from e-h excitations

L R

2

UCSD July 20-21, 2009 – p.9

SLIDE 48

Current from e-h excitations

L R

3

UCSD July 20-21, 2009 – p.9

SLIDE 49

Current from e-h excitations

Molecules with strong charge-transfer transitions

DMEANS (4-Dimethylamino-4’-nitrostilbene) 7D (ground) → 31D (first excited singlet) all-trans Retinal in Poly-methyl methacrylate films 6.6D → 19.8D (1Bu electronic state) 40Å CdSe nanocrystals 0D → 32D (first excited state)

If charge transfer is parallel to the wire axis e-h excitation → charge flow

UCSD July 20-21, 2009 – p.9

SLIDE 50

Current from e-h excitations

2
1

1 ( 10

5)

Isd (A)

e-h Landauer Isd

1,2 (ML/R)=0.1 eV

6
4
2

2 4 6 e (eV)

1

1 ( 10

5)

Isd (A)

2 (MR)=0.01 eV 2 (ML)=0.19 eV 1 (ML/R)=0.1 eV

T = 300 K ε1 = 0 eV ε2 = 2 eV Γ(M)

1,2 = 0.2 eV

in asymmetric case e-h is significant

UCSD July 20-21, 2009 – p.9

SLIDE 51

Current from e-h excitations

Distance dependence Γ(MK)

m

= A(MK)

m

exp

−α(MK)

m

R

B(K) = β(K)/R3

UCSD July 20-21, 2009 – p.9

SLIDE 52

Current from e-h excitations

2 4 6 8 10 12 14 R (Angstrom) 10

11

10

10

10

9

10

8

10-7 10-6 10

5

Isd (A)

e-h Landauer Isd =3 V

A(ML/R)

1

= 0.27 eV A(ML)

2

= 0.52 eV A(MR)

2

= 0.027 eV α(MK)

m

= 1 Å−1 β(K) = 0.01 eV

UCSD July 20-21, 2009 – p.9

SLIDE 53

Raman Spectroscopy

D.R.Ward et al. Nano Lett. 8, 919 (2008)

UCSD July 20-21, 2009 – p.10

SLIDE 54

Model

ˆ H = ˆ H0 + ˆ V (e−v) + ˆ V (et) + ˆ V (v−b) + ˆ V (e−h) + ˆ V (e−p) ˆ V (e−v) electron-vibration interaction ˆ V (et) electron transfer ˆ V (v−b) thermalization of vibration ˆ V (e−h) energy transfer ˆ V (e−p) coupling to radiation field

Nano Lett. 9, 758 (2009); J. Chem. Phys. 130, 144109 (2009)

UCSD July 20-21, 2009 – p.11

SLIDE 55

Model

ˆ H0 =

m=1,2

εm ˆ d†

m ˆ

dm + ωvˆ b†

vˆ

bv +

k∈L,R

εkˆ c†

kˆ

ck +

β

ωβˆ b†

βˆ

bβ +

α

ναˆ a†

αˆ

aα ˆ V (e−v) =

m=1,2

V (e−v)

m

ˆ Qv ˆ d†

m ˆ

dm ˆ V (et) =

K=L,R
k∈K;m
V (et)

km ˆ

c†

k ˆ

dm + V (et)

mk ˆ

d†

mˆ

ck

ˆ

V (v−b) =

β

U (v−b)

β

ˆ Qv ˆ Qβ ˆ V (e−h) =

k1=k2
V (e−h)

k1k2

ˆ d†

1 ˆ

d2ˆ c†

k1ˆ

ck2 + H.c.

UCSD July 20-21, 2009 – p.11

SLIDE 56

Model

Coupling to the laser field ˆ V (e−p) =

α
U (e−p)

α

ˆ d†

1 ˆ

d2ˆ aα +

∗

U (e−p)

α

ˆ a†

α ˆ

d†

2 ˆ

d1

+
α
k∈{L,R}
m=1,2
V α

kmˆ

c†

k ˆ

dm + V α

mk ˆ

d†

mˆ

ck

× (ˆ

aα + ˆ a†

α)

UCSD July 20-21, 2009 – p.11

SLIDE 57

Model

Introducing excitation operators ˆ D ≡ ˆ d†

1 ˆ

d2 ˆ Dmk ≡ ˆ d†

mˆ

ck m = 1, 2 k ∈ {L, R} and after small polaron transformation ˆ V (e−p) =

α
ˆ

a†

α ˆ

Oα + ˆ O†

αˆ

aα

ˆ

Oα =

∗

U (e−p)

α

ˆ D ˆ X +

k∈{L,R}
V α

1k ˆ

D1k ˆ X†

1 + V α k2 ˆ

Dk2 ˆ X2 + V α

k1 ˆ

Dk1 ˆ X1 + V α

2k ˆ

D2k ˆ X†

2

≡ ˆ

O(M)

α

+ ˆ O(1)

α + ˆ

O(2)

α + ˆ

O(3)

α + ˆ

O(4)

α

UCSD July 20-21, 2009 – p.11

SLIDE 58

Raman flux

Photon flux from mode α into the system Jα(t) ≡ − d dt < ˆ a†

α(t)ˆ

aα(t) > = − t

−∞

dt′ [F <

α (t, t′) G> α (t′, t) + G> α (t, t′) F < α (t′, t)

−F >

α (t, t′) G< α (t′, t) − G< α (t, t′) F > α (t′, t)]

Fα(τ, τ ′) = − i < Tc ˆ aα(τ) ˆ a†

α(τ ′) >

Gα(τ, τ ′) = − i < Tc ˆ Oα(τ) ˆ O†

α(τ ′) >

UCSD July 20-21, 2009 – p.12

SLIDE 59

Raman flux

Scattering-theory on the Keldysh contour One pumping mode i Empty final modes {f} Steady-state photon flux to a final mode f Jf = +∞

−∞

d(t − t′) F >

f (t′ − t) G< f (t − t′)

UCSD July 20-21, 2009 – p.12

SLIDE 60

Raman flux

2nd order perturbation for G<

f (t − t′)

in coupling to the initial mode i Ji→f = +∞

−∞

d(t − t′)

c

dτ1

c

dτ2 F >

f (t′ − t) Fi(τ1, τ2)

× < Tc ˆ O†

f(t′) ˆ

Of(t) ˆ O†

i(τ1) ˆ

Oi(τ2) > we have 54 = 625 channels (different ˆ

O)

we have 3 × 3 = 9 diagrams (positions of t1 and t2)

UCSD July 20-21, 2009 – p.12

SLIDE 61

Raman flux

Choice of diagrams on the Keldysh contour i is pumping mode populated by one photon F <

i (t1 − t2) = −ie−iνi(t1−t2)

f are accepting modes not populated F >

f (t′ − t) = −ie−iνf(t′−t)

nly rates are of interest

UCSD July 20-21, 2009 – p.12

SLIDE 62

Raman flux

J(nR)

i→f =

+∞

−∞

d(t − t′) t

−∞

dt1 t′

−∞

dt2 e−iνi(t1−t2) eiνf(t−t′) < ˆ Oi(t2) ˆ O†

f(t′) ˆ

Of(t) ˆ O†

i(t1) >

t1 t2 t’ t

UCSD July 20-21, 2009 – p.12

SLIDE 63

Raman flux

J(iR)

i→f =

+∞

−∞

d(t − t′) +∞

t

dt1 +∞

t′

dt2 e−iνi(t1−t2) eiνf(t−t′) < ˆ O†

f(t′) ˆ

Oi(t2) ˆ O†

i(t1) ˆ

Of(t) >

t t’ t2 t1

UCSD July 20-21, 2009 – p.12

SLIDE 64

Raman flux

J(intR)

i→f

= +∞

−∞

d(t − t′) t

−∞

dt1 +∞

t′

dt2 2Re

e−iνi(t1−t2) eiνf(t−t′)

< ˆ O†

f(t′) ˆ

Oi(t2) ˆ Of(t) ˆ O†

i(t1) >

t’

t1 t t2 t2 t t1 t’

UCSD July 20-21, 2009 – p.12

SLIDE 65

Molecular Raman

1 2 3

V (V)

1 2 3 10

4

I (a.u.)

8 10 12 10

13

i

{ f} (a.u.)

¯ J

ǫ2 − ǫ1 = 2 eV normal Raman ∼ n1(1 − n2) ⇒ 1 → 1

4

inverse Raman ∼ n2(1 − n1) ⇒ 0 → 1

4

total Raman 1 → 1

2

UCSD July 20-21, 2009 – p.13

SLIDE 66

Molecular Raman

1 2 3

V (V)

1 2 3 10-14

i i v (a.u.)

anti-Stokes Stokes

¯ J

ǫ2 − ǫ1 = 2 eV junction heating → increase in anti-Stokes

UCSD July 20-21, 2009 – p.13

SLIDE 67

Molecular Raman

1

1

( f- i)/

v

6 12 10-14

i f (a.u.)

V=0 V V=2.5 V

¯ J

ǫ2 − ǫ1 = 2 eV νi = ǫ2 − ǫ1

UCSD July 20-21, 2009 – p.13

SLIDE 68

Molecular Raman

1

1

( f- i)/

v

5 10 15 10-14

i f (a.u.) (e-h)=0.004 eV (e-h)=0.010 eV

¯ J

ǫ2 − ǫ1 = 2 eV νi = ǫ2 − ǫ1 e-h excitations compete with Raman

UCSD July 20-21, 2009 – p.13

SLIDE 69

Molecular Raman

1

1 2

( i- )/

v

6 10 10-13

i

{ f} (a.u.)

V=0 V V=2.5 V

¯ J

ǫ2 − ǫ1 = 2 eV heating → anti-Stokes

UCSD July 20-21, 2009 – p.13

SLIDE 70

Molecular Raman

0.0 0.01 0.02

(e-h) (eV)

0.2
0.1

0.0

/

0.0 0.01 0.02

m (eV)

0.3
0.2
0.1

/

0.001

¯ J(intR)

νi→νi−ωv ¯

Jνi→νi−ωv ¯ J(intR)

νi→νi−ωv ¯

Jνi→νi−ωv

νf = νi − ωv 2 slits experiment? Γ(e−h) is responsible for switching

UCSD July 20-21, 2009 – p.13

SLIDE 71

Molecular Raman

1 2 3

V (V)

500 1500

T (K) TS-aS Tv

TS−aS = ωv/ ln ¯ Jνi→νi−ωv ¯ Jνi→νi+ωv at low V anti-Stokes disappears

UCSD July 20-21, 2009 – p.13

SLIDE 72

Molecular Raman

1.0 1.01 1.02

TS-aS/Tv

0.005 0.01 0.015 0.02

m (eV)

0.0 0.01 0.02

(e-h) (eV)

0.2
0.1

0.0

/

0.0 0.01 0.02

m (eV)

0.3
0.2
0.1

/

0.001

¯ J(intR)

νi→νi−ωv ¯

Jνi→νi−ωv ¯ J(intR)

νi→νi−ωv ¯

Jνi→νi−ωv

UCSD July 20-21, 2009 – p.13

SLIDE 73

Metal-to-Molecule

J.R.Lombardi et al. JCP 84, 4174 (1986)

UCSD July 20-21, 2009 – p.14

SLIDE 74

Metal-to-Molecule

Origin of the (metal-to-molecule) peak dE1 2π dE2 2π

K=L,R

S(K)<

2

(E1) G>

2 (E2)

νi + E1 + ωvvin − E2 − ωvv′ + iΓ(e−h)/2 At µL = µR = EF and for T → 0 integral on E1 yields ln

(EF − E2 + ωv(vin − v′) + νi)2 + (Γ(e−h)/2)2

D where D is leads half-bandwidth. This gives a peak at νi = E2 − EF − ωv(vin − v′)

UCSD July 20-21, 2009 – p.14

SLIDE 75

Metal-to-Molecule

1.0 1.1 1.2

i (eV)

10 20 10-20

i i- v (a.u.)

20

¯ J

ε2 − EF = 1 eV νf = νi − ωv increase in Γ2 eliminates the peak

UCSD July 20-21, 2009 – p.14

SLIDE 76

Metal-to-Molecule

0.0 0.1 0.2

m (eV)

4 8 12 10-21

V=0V

i=1eV

molecule met-to-mol 0.01 0.02 0.03

m (eV)

4 8 10-23

V=1V

i=0.5eV

i i- v

¯ J

UCSD July 20-21, 2009 – p.14

SLIDE 77

Thanks!

Prof. Abraham Nitzan

Tel Aviv University

Prof. Mark A. Ratner

Northwestern University

UCSD July 20-21, 2009 – p.15