Electronic structures of 1D systems From atom wire and molecular bridges to scanning probe...
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Transcript of Electronic structures of 1D systems From atom wire and molecular bridges to scanning probe...
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