Electronic structures of 1D systemsFrom atom wire and molecular bridges to scanning probe microscopy M.Tsukada Waseda Univ

The 1st A3 Foresight Summer SchoolJune 19-22, Seoul National Univ.

outline

1 Introduction2 Theoretical methods (RTM,NEGF)3 Quantum transport in atomic wire eigen channels, quantization 4 Transport through molecules resonant tunneling, Coulomb blockade

5 Large loop current in molecules6 Some topics of CNT cap, junction, helical shape, persistent current

7 Effect of molecular vibration8 Magnetism of atomic wire9 Theory of SPM what and how SPM sees the nano-world?

10 Summary and outlook

View of the systems treated in this lecture

Atomic wires

Molecular bridges

STM,AFM,KFM

Contact problem

quantizationCoherent vs Dissipation

Loop current

Resonant tunneling

Atomistic observation

Atom manipulation

Explore functions

Local gatesControl of level postions

Trends of nano-technology

Molecular quantum bit

source

gate

drain

Tunnel junction

Invention of transistorRoad of IC, LSI

Down sizingby top down

Economical and physical limit

Electronics using single molecule

Characterization by SPM

Novel quantum function

Bottom upProvided by Prof.Y.Wada

Atom and Molecular bridge structures

1 Bottom up formation of nano-devices

2 Novel functionality utilizing enhanced Quantum nature

Nonlocality

Multiplicity

InstantaneousState transitionReduction of wavepacket

Quantum entangled state

Nature of electron transport?Single electron processCoherent transportDissipation processesLight emissionFET, Switching, Sensing …

Theory can play very essential role for exploring these new area

Coherent Transportvs　　 Dissipative Hopping Transport

Strong bottle neck　　　　 even for strong coupling

€

Δn<<Δθ

Dissipative Hopping TransportSingle electron process

Weak bottle neck　 Effective interaction between the states around EF 　

€

Δn>>Δθ

Coherent quantum transport

Competition or Coexistence of the two regimes?

Uncertainty relation

Phase of the state

€

Δθ

€

ΔnMolecule

Electrode

Electrode

θ,n[ ] = iElectron number in molecule

Quantum Conductance of Au Point Contact Lars Olesen

W tip wetted with Au

Au surface

Retraction of the tip from the surface

2e2

h=

112.9kΩ

Quantization value of conductance

Concept for the First-principles Recursion Transfer Matrix Method

ΨL

ΨR

Left electrode wave

Right electrode wave

EFL

EFReV

With an appropriate boundary condition, are calculatedFrom the right and left electrode wave-functions DFT potential determined.This is equivalent to Non-equlibrium Green’s function approach.

ΨL and Ψ R

RTM　method details

ΨE,iL/Rr()=expikPrP()expiGPjrP()ψE,iL/RGPj,z()j∑a zp , E( )U L /R zp+1,E( )−b zp,E( )U L /R zp,E( ) + c zp,E( )U L /R zp−1,E( ) =0

U L / R z, E( )⎡⎣ ⎤⎦i, j

=ΨE,iL /R GP

j( )

a zp , E( ) =I −16

h2V zp+1( )

b zp , E( ) =2I +53

h2V zp( )

c zp , E( ) =I −16

h2V zp−1( )

Vij z( ) =

12

kP+GPi 2

−E⎛⎝⎜

⎞⎠⎟δ ij +

1S

Veff r( )∫∫ exp i GPj −GP

i( )rP( )drP

T L / R zp−1,E( ) = b zp,E( )−a zp,E( )T L /R zp,E( )⎡⎣ ⎤⎦−1

c zp,E( )

T L / R zp ,E( ) =U L /R zp+1,E( )U L /R zp,E( )−1

z0 , z1,.. zk , zk+1,...., zN −1,zN

First-Principles

Laue representation of the wave

Transfer matrix

Matrix difference equation

Ratio matrix

Recursion relation

Coefficient matrix

RTM　method details　２

U L zp( ) =KL

+( )pΛ + KL

−( )pR p≤0( )

KR+( )

pT p> N( )

⎧⎨⎪

⎩⎪

KL / R± =

gL /R± GP

1( ) 0 . . 0

0 gL /R± GP

2( ) . . 0

. . . . .

. . . . .

0 . . . gL /R± GP

n( )

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

gL / R± GP

i( ) =

6 + 5βL /Ri

6 −βL /Ri ±i 1−

6 + 5βL /Ri

6 −βL /Ri

⎛

⎝⎜⎞

⎠⎟

2

6 + 5βL /Ri

6 −βL /Ri m

6 + 5βL /Ri

6 −βL /Ri

⎛

⎝⎜⎞

⎠⎟

2

−1

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

βL / Ri < 0

βL / R

i = h2 VL / R +1

2kP + GP

i 2− E

⎛⎝⎜

⎞⎠⎟

T zN +1( ) =KR+

at the boundary in the right electrode

U z0( ) =T z−1( ) KL+T z−1( )−I( ) KL

+ −KL−( )

−1Λ

ΨL

ΨR

Left electrode wave

Right electrode wave

Left jelliumelectrode

Right jellium electrode

z0 , z1,.. zk , zk+1,...., zN −1,zN

βL / Ri > 0

→ T zN( ) → T zN −1( ) → T zN −2( ) → ........... → T z0( ) → T z−1( ) From the recursion relation

→ U z1( ) → U z2( ) → U z3( ) → .......... → U zN( ) By multiplingAll the transefer matrix is solved

T zp( )

Exact solvable case

Open-Nonequilibrium System

Tunnel Current Density and Barrierd=12au Vs=2V

Current is confinedIn a narrow channel !!

Barrier above Fermi level

Calculation withFirst principles RTM

N.Kobayashi, K.Hirose and M.Tsukada,J.J.Appl. Phys. 35(1996)3710

what happens at the atomic contact？

Vs=2V

Barrier height and current density

Conductance at the atomic contact

Conductance at the contact is close to the value of the quantization unit

Atom extraxtion by the tipNa tip

Na surface

0V

5V

8V

Surface bias

Electron cloudis pushed outfrom the nagative tip

When a bridge structureof the electron cloud is formed, the atom just ahead of the tip feels a strong puliing force

The change of electron density by bias voltage

Surface positive Surface negativeTotal charge

Reduction of the Potential Barrier by the bias

Tip height 8au=0.42nmTip negative

Tip positive

Potential barrier is remarkably reduce by the applied bias voltage

Eigen Channel decomposition

Conductance Landauer formula

Quantization unit=

€

112.9kΩ

J = ev E( )

ρ E( )2πEF

EF +eV

∫ dE =ev EF( )×2

2πhv EF( )×eV =

2e2

hV

U L zp( ) =KL

+( )pΛ + KL

−( )pR p≤0( )

KR+( )

pT p> N( )

⎧⎨⎪

⎩⎪

Left electrode wave

ΛR

ΨL Tinitial

Reflection coeff.matrix

Transmission coeff.matrixDiagonalize T +T

Unitary transformation of the original channnel

Eigen-channels

G =2e2

htr T +T( ) G =

2e2

hti

i∑

In the eigen channel rep.

Transmission Probability of the channel i

If ti =1

Conductance of Jellium Cylinder

Quantized conductance in semiconductor mesoscopic system

Increase of negative gate voltageNarrowing of the channel width at CNumer of the channnels decreases one by one

Gate VoltageGate voltage

Conductanc

e2e2

h

⎛

⎝⎜⎞

⎠⎟

Quantum conductance of Al,Na atomic wire

0.99

0.47X2

Channel DOS and Channel Transmission of Al Atom Wire

Channel DOS and Channel Transmission of Na Atom Wire

Loop current seen in the bent Al atomic wire

RjloopC∫ dr =0

Impossible in the flame work of classical electromagnetic theory

Remarkable Quantum Effect

C

Inclusion of Non-Local Term in RTM

€

−h2

2m∇2 + Vloc (r)

⎡

⎣ ⎢

⎤

⎦ ⎥ϕ E ,i

αlm r( ) +δVlα r( )φlm

α = Eϕ E ,iαlm r( )

A particular solution with one nonlocal term (αlm) present.

€

ϕE ,iαlm (r) = GE ,i(r,r ')δVl

α ( ′ r )φlmα ( ′ r )∫ dr '

Transform into integral formlm

atom α

(decomposition)

( ) ( )rr αα φδ lmlV|

is present at small regimes.

Very fast calculation !

( ) ( )rr αα φδ lmlV|

Å@)',(, rriEG constructed from )(, rLiEΦ )(, rR

iEΦ and ;Green’s function for the local pseudo-potential

systemGE,I(r,r’)

( ) ∑+Φ=Ψlm

lmiE

RLlm

RLiE

RLiE C

α

αα ϕ )()( ,

//,

/, rrr

Dependence of the Conductance on the Length of atomic-wires

Si

Wire

Dot

Density of States

N=1

N=2

N=4

N=6 Mixed atoms at the contact determine the magnitude of conductances.

1D wire

Conductance through Al atomic-wires with various atoms mixed at contactsK.Hirose,N.Kobayashi, M.Tsukada, Phys.Rev.B69 (2004) 2454121 First principles RTM calculation with non-local pseudo-potential

Na

Cl

Al

Where does the bias drop in the wire ?

Bias = 5V

Local polarization (s-orbital)

Spread-out (p-orbital)

Potential difference

Charge difference ( )

without wire

Bias drop is determined by the local polarization.One impurity gives a significant influence!

Ez~

)0,()5,( VV rr ρρ −

Resonant tunneling and quantum coherencemolecular bridges

Kondo state

Large loop currentPersistent current

Quantum entanglement

Resonant tunnelingCoherent coupling

MoleculeElectrode

Electrode

Free electrons Localized spinDegenrate states

Internal deg. of freedom

Free electrons

Resonant Tunneling,Bottle neck of the coherent transport, and Coulomb blockade

Double tunnel junctions and bridge systems

€

T (E) =T1T2

1− 2 R1R2 cosθ(E ) + R1R2

€

T (E) =T1T2

{ T1 + T2( ) / 2}2 + 2(1− cosθ(E ))

≅Γ1Γ2

E − E0( )2

+ Γ1 + Γ2( )2

/ 4

€

Γi =dEdθ

Ti

€

T(E )−∞

∞

∫ dE =Γ1Γ2

Γ1 + Γ2

resonant tunneling

€

T1

€

T2

€

R1

€

R2

€

E0

€

EFL

€

EFR

100% transmission, if the energy is

exactly tuned at E0

Contribution to the conductance from the Whole energy range

In proportion to the width of the resonance!

Nano-structures sandwiched between the planer electrodes

First Principles Recursion Transfer Matrix Method N.Kobayashi and M.Tsukada, Jpn. J.Appl. Phys., 38 (1999) 3805

Removing vacuum gap・・・1D character is appeared!

Dimension crossover by the bottleneck

Resonant tunneling０ -dimensional

Band conduction１ -dimensional

Channel transmission

spectra

１ Dim Channel

Atomic wire with good contact

Atom or molecular bridge with poor contact 　　And resonant tunneling systems

E

E

E

Transmission Prob.

Tunneling through Kondo Resonant State

T (E) =T1T2

{ T1 +T2( ) / 2} 2 + 2(1−cosθ(E))

=cos2 ηs(E)−ηa(E)( )

Phase shift of Symmetric, Antisymmetric Scattering waves

E → 0　　

ηS 0( ) − η a 0( ) =π

2

When the energy crosses with the resonant level, Phase shift abruptly increses by　　　　　　　 Resonant tunneling

Kondo resonant peak always sticks to the Fermi level, thus transmission probability is almost unity over wide range of the gate voltage.

A.Kawabata, in Transport phenomena in Mesoscopic Systems, (Springer, ‘91)

π

How electron flows through electrode-

molecule-electrode?

Coherent in the hole system

Dissipated hopping

1)Coherent throughout whole systems resonant tunneling process　　

2)Incoherent with the electrodes、 but coherent within the molecule？

3)Incoherent both between the molecule and electrodes, as well as within the molecule　　

　　　 Intra -molecular hopping

the same as small molecular aggregates

Single molecular bridge

Organic molecular thin film(EL)

Transmission assisted by molecular states

Coulomb blockade, single electron processes

Partially coherent within a molecule

Coulomb blockade and single electron process

€

φ t( ) =eh

V (t)dt−∞

t

∫ =e

hCQ(t)dt

−∞

t

∫

€

˙ φ t( ) =e

hCQ t( )

€

φ,Q[ ] = ie

Phase and electron number(charge) are conjugate quantities with each other

€

EC =CV − e( )

2

2C−

CV( )2

2C=

e2

2C− eV

Energy change just after the tunneling event of an electron

EC < 0

Γ V ,T( ) =1

e2rf (E){1− f (E − Ec )}dE

−∞

∞

∫

=V − e / 2C

er[1− exp{−V − e / 2C

kBT}]

Probability of the electron tunneling

φ,n[ ] = i

€

e2C

€

I

Based on a naïve assumption that electron with the energy E sees the Fermi level of the counter electrode shifted with the energy Ec

Coulomb blockade and single electron process2

I

More accurate treatment with including the coupling with external electromagnetic environment

H env =

e2n2

2C+

e2nλ2

2Cλ

+he

⎛⎝⎜

⎞⎠⎟

2 12Lλ

φ−φλ( )2⎧

⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪λ∑

Hamiltonian of the whole system

€

H = Ek + eV( )ck+ck

k∑ + Eqcq

+cq + Tqkcq+cke

− iφ

qk∑

q∑ + h .c. + H env

€

n = cq+cq

q∑

Zero bias conductance anormaly

φ,n[ ] = i Equivalent to the many-harmonic- oscillators system

Coulomb blockade and single electron process3

Tunneling probability between electrodes

€

rΓ V( ) =

1e2R

dEq−∞

∞

∫ dEk f Ek( ) 1− f Eq + eV( ){ }P(Ek − Eq )

Dissipation spectral fumction

€

€

P E( ) =1

2πhexp J t( ) +

ih

Et ⎧ ⎨ ⎩

⎫ ⎬ ⎭dt

−∞

∞

∫

Correlation function of phase

€

J t( ) = φ t( ) −φ 0( )[ ]φ 0( )Tunneling current

€

I = er Γ V( ) −

s Γ V( ){ }

€

I V( ) =1eR

1− exp −eVkBT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪E

1− exp −E

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

P eV − E( )dE−∞

∞

∫

≈1eR

eV − E( )P E( )dE−∞

∞

∫

=V −

e

2CR

Zero bias conductance anormaly

Coulomb blockade and single electron process4 Multi-junctions and Coulomb diamond

() )2(22 22221 eQVCCeCQCeQCVeCE ++=−++=Δ− () )2(22 12212 eQVCCeCQCeQCVeCE +−=−−++=Δ+ () )2(22 12212 eQVCCeCQCeQCVeCE ++−=−++−=Δ−

If above 4 energies are all positive

electron tunneling is prohibited

e

2C1

V

e

2C2

e

2−

e

2

−e

2C1

−e

2C2

Q

Q

V

() )2(22 22221 eQVCCeCQCeQCVeCE +−−=−−+−=Δ+

ΔE1+

ΔE1−

ΔE2−

ΔE2+

No tunneling

Vgate

(Vgate )

Single electron tunneling process

Wilkins et alPhys. Rev.Lett. 63(1989)801

Many other examples;Ag cluster/GaAs surfaceDye molecules embedded in oxides thin film

Current

Voltage (V)-0.2 0.0 0.2

T=4.2K

STM tip

Al oxide

Al substrate

In fine particle

Coulomb blockade and resonant tunneling

€

C1 << C2,R1 << R2

€

V(n) =e

C2

n +12

⎛ ⎝

⎞ ⎠

€

n = 0,±1,±2,....Energy of n-th ionized state

Density Functional, First-Principles RTM and Non-Equilibrium Green’s Function calculations for the molecular bridges

Conductance of Benzene di-thiolate

SSSemi-infinite jellium electrode

- 4 - 2 0 2 4Voltage V

- 400

- 200

0

200

400

I-V characteristic

Differential conductance

1

0

2

4

6

8

0

ecnatcudno

C2e

2h

- 4 - 2 0 2 4Voltage V

HOMO-LUMO

First-Principles RTM method by Hirose (NEC)

- 8 - 6 - 4 - 2 0 2 40

2.5

5

7.5

10

12.5

15

Energy [eV]

DO

S [

eV-1]

π *πσ

Exp. Reed et al

I-V Characteristics with various contacts

d=2 a.u.

- 4 - 2 0 2 4Voltage V

0

2

4

6

8

10

ecnatcudnoC

2e2h

- 4 - 2 0 2 4Voltage V

- 400

- 200

0

200

400

tnerruC

ｵA

I-V characteristic Differential conductance

HOMO-LUMO state

- 4 - 2 0 2 4Voltage V

- 10

- 5

0

5

10

tnerruC

ｵA

I-V characteristic

tunneling

- 15 - 10 - 5 0 5 10 15Distance a.u.

- 15

- 10

- 5

0

5

laitnetoPVe

average local effective potential

tunneling regime

molecule

d=8 a.u.

1. close to good contact

2. bad contact

d

Strong non-linear behavior

d

Effect of the local tunnel barrier disapepearence

Non-equlibrium Green’s function approach for Molecular bridges

€

TSD(E)= Tr[ΓSGR(E)ΓDGA(E)]

€

GR(E)=(E −Hc −ΣSR −ΣD

R)−1

ΓS = i[ΣSR −ΣS

A]

ΣSR = ′t 2gS

R

Iij =2eh Im[HijGij

<]

Green’s function

Segment current between i and j

Transmission function

ρ r( ) =1

2π iG< r,r, E( )dE∫

I =2eh

fS − fD( )TSD E( )∫ dE

current

Retarded ….Advanced

GA(E)=(E−Hc−ΣSA−ΣD

A)−1

ΣDR = ′t 2gD

R ΣSA = ′t 2gS

A ΣDA = ′t 2gD

A

Vertex ΓD = i[ΣDR −ΣD

A]

Electron density distributionLesser Green’s function

G< E( ) = fp E( )GRΓpGA

p=S,D∑

Transmission spectrum of phenalenyl molecule

′t =t

2⇒Resonant tunneling

′t = t⇒ metallization of the molecule

Energy E / t

Tra

nsm

ission

fu

nctio

n

G0 =2e2

h⎛

⎝⎜⎞

⎠⎟

Connection of Phenalenyl molecules to the electrodes

K.Tagami, L.Wang and M.Tsukada,NANO LETTERS 4 (‘04) 209

SOMOlevel

-0.5 0.0 0.5 eV

-0.5 0.0 0.5 eV

tran

smissio

ntran

smissio

n

Source/drain coreesponds to nodes of SOMO

Phenalenyl molecular bridge

SOMO orbital

Some more different connections

For non-vanishingtransmission at E=0(SOMO level)Both lead sites should be 　 β

α

β

L.Wang, K.Tagami and M.Tsukada,Jpn.J.Appl.Phys., 43 (2004) 2779

β β

β

α

Metal Porphyrin Polymerdependence on polymerization degree

Transmission Spectra of tape-porphyrin molecules

K.Tagami and M.Tsukada, e-J., of Surf. Sci. and Nanotech., 1 (2003) 45Bias window

n=6

Peak in the bias widow contributes conductance

Connection with the electrode and I-V curveConnection with the electrode and I-V curve

K. Tagami et al, e-J. Surf. Sci. Nanotech. 1 , 45 (2003).

N=6

B

C

D

B

D

C

K. Tagami et al, Curr. Appl. Phys 3 (2003) 439.

Molecular device using loop currentMolecular device using loop current

Localised spin direction can be controlled by the bias polarity

Linear and helical polymerized benzothiopheneK.Tagami,M.Tsukada,Y.Wada,T.Iwasaki and H.Nishide, J.Chem.Phys., 119 (‘03)7491

After doping I2Molecular solenoid

Spin transport of phenoxy radical

With OH

With O radical Spin dependent transmission

Conduction Switching of Alkane Molecule on Au(111) by Conformation Change

M. Suzuki, S. Fujii, S. Wakamatsu, U. Akiba,

and M.Fujihira, Nanotechnology 15, S150 (2004).

BCO cHex C51-pentanethiolate

C5 molecules embedded in BCO-SAM Membrane are observed as bright spots.

They are blinking!

Conduction Switching of Alkane Molecule by Conformation change on Au(111)

Bicyclo[2,2,2]octylmethylthiolate

1-pentanethiolate

Stable structure

Quasi-stable structure

K.Tagami and M.Tsukada,e-J. Surface Sci. and Nanotech., 2 (‘04) 186

Bright stable structureAnd dark quasi-stable structure appears!

1-pentanethiolate

electron increased region

electron decreased region

- 0.10 e / cell

Differential Charge

0.3 V/Å

Electrostatic Potential

sample bias = +5.55Vtip distance = 15.0Å

C6H8/Si(001)in strong field

0.5ML

Effective Screening Medium methodO.Sugino and M.Otani (ISSP, Univ. of Tokyo)Physical Review B 73, 115407 (2006).

Vs = 0.0 V

Vs = +6.5 V

C10H6/Si(001)ina strong field 0.5ML 吸着系

HOMO of Si(001)surface

ナフタレンの芳香族性を保持

Crossing of substrate HOMO-LUMO and Surface NOMO-LUMO