Electronic Conduction of Mesoscopic Systems and Resonant States Naomichi Hatano Institute of Industrial Science, Unviersity of Tokyo Collaborators: Akinori

Electronic Conduction of Mesoscopic Systems and Reson

ant States

Electronic Conduction of Mesoscopic Systems and Reson

ant StatesNaomichi Hatano

Institute of Industrial Science, Unviersity of TokyoCollaborators: Akinori Nishino (IIS, U. Tokyo)　　　　　　 Takashi Imamura (IIS, U. Tokyo)　　　　　　 Keita Sasada (Dept. Phys., U. Tokyo)　　　　　　 Hiroaki Nakamura (NIFS) 　　　　　　 Tomio Petrosky (U. Texas at Austin)　　　　　　 Sterling Garmon (U. Texas at Austin)

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ContentsContents

1.Conductance and the Landauer Fo

rmula

2.Definition of Resonant States

3. Interference of Resonant States an

d the Fano Peak

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What are mesoscipic systems?What are mesoscipic systems?

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T. Machida (IIS, U. Tokyo)

S. Katsumoto (ISSP, U. Tokyo)

T. Machida (IIS, U. Tokyo)

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Theoretical modelingTheoretical modelinglead

Scatterer(Quantum Dot, …)

Cross section of a lead

lead

? kT

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k

ε(k) 2πL

Landauer formulaLandauer formulaS. Datta “Electronic Transport in Mesoscopic Systems”

Perfect Conductor

L

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I = envv>0

<ε (k)<

∑ =eL

dε(k)d(hk)

=v>0

<ε (k)<

∑ ehL

Lπ

dε

∫ =e

h −

e=e

hV

k

ε(k)

2πL

G =e

hConductance of a Perfect ConductorSpin

Density

n =1/L

Voltage differenc

e

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So, what was the conductance?So, what was the conductance?

I =GV =e

hV

V =RI G =R−

G− =he

=.9 kΩ[ ]

Conductance is the inverse of the resistance.

Be aware! R =ρLS does not hold!

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Perfect Conductor


Contact resistance G− =he

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L

ScattererProbability T

I = T(k)envv>0

<ε (k)<

∑ =eL

T(k)dε(k)d(hk)

=v>0

<ε (k)<

∑ ehL

Lπ

T(ε)dε

∫ ;e

hT(F )

−e

=e

hT(F )V

G =e

hT Conductance in general

Linear response

Calculates at the Fermi energy

Gate voltage

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L

G− =he

T=

he+

he−T

T

Contact resistance “Raw” resistance of a scatterer

ScattererProbability T

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V VTransmission probability: T(E)

Scatterer

Conductance (Inverse Resistance)

G =e

hT(EF )

Note R =ρ L S does not hold.

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Example: 3-state quantum dotExample: 3-state quantum dotKeita Sasada: Ph. D. Thesis (2008)

ε0 t = 0, ε1 t = −0.3, ε 2 t = 0.5

v01 t = 0.8, v12 t = 0.4, v20 t = 0.5

tα t = tβ t = 1

⎧

⎨⎪

⎩⎪

Resonance Peak(AsymmetricFano Peak)

Trans. Prob. T

Fermi Energy

Con

duct

ance

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ContentsContents


rmula



d the Fano Peak

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Definition of resonance: 1Definition of resonance: 1

Aeikx

CeikxBe−ikx

S(E)≡r(E) ′t (E)t(E) ′r (E)

⎛

⎝⎜

⎞

⎠⎟≡

B(E)A(E)

′C (E)′A (E)

C(E)A(E)

′B (E)′A (E)

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′A e−ikx

′B eikx′C e−ikx

Pole → or , whereA(E)=0 ′A (E) = 0 E ∈£

where E =E(k)

Resonance: Pole of Trans. Prob. (S-Matrix)

G(E)=e

ht(E) =

e

hC(E)A(E)

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Siegert condition (1939)Resonance: Eigenstate with outgoing waves only.

a−ax

Be−iKx CeiKx

−

h2

2m

d 2

dx2+ V (x)

⎛

⎝⎜⎞

⎠⎟ψ (x) = Eψ (x)

V(x)

−V

Fei ′K x +Ge−i ′K x

E =

hK

m=

h ′K

m−V

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a−ax

Be−iKx CeiKx

Fei ′K x +Ge−i ′K x

V(x)

−V

Even solutions: B C, F G

E =

hK

m=

h ′K

m−V

F cos ′K a=CeiKa

−F ′K sin ′K a=iCKeiKa

Odd solutions: B C, F G

− ′K tan ′K a = iK

′K 2 tan2 ′K a = −K 2 =

2mV

h2− ′K 2

α =± %V cosα

α =± %V sinα

α ≡ ′K a, %V ≡

2mVa2

h2

⎛

⎝⎜⎞

⎠⎟

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Im Kn < 0Re Kn

><0 ⇔ ImEn

<>0

Eigen-wave-number EigenenergyBound state

a = %V =

a = %V =

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Non-Hermiticity of open systemNon-Hermiticity of open system

ψ p2 ψΩ

= −h2 ψ (x)∗ ′′ψ (x)dx− L

L

∫ = −h2 ψ (x)∗ ′ψ (x)⎡⎣ ⎤⎦x=− L

L+ h2 ′ψ (x)∗

− L

L

∫ ′ψ (x)dx

ψ p2 ψΩ

∗= −h2 ψ (x) ′′ψ (x)∗dx

− L

L

∫ = −h2 ψ (x) ′ψ (x)∗⎡⎣ ⎤⎦x=− L

L+ h2 ′ψ (x)

− L

L

∫ ′ψ (x)∗dx

2i Im ψ Htotal ψ Ω

=−ihm

Re ψ (x)∗pψ (x)x=L+ψ (x)∗(−p)ψ (x)

x=−L( )

Im ψ Htotal ψ Ω

=−hm

Reψ pn ψ ∂Ω

N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187

H total =p

m+V(x)

2i Im ψ p ψ

Ω=−h ψ (x)∗ ′ψ (x)−ψ (x) ′ψ (x)∗⎡⎣ ⎤⎦x=−L

L=−

hm

Re ψ (x)∗pψ (x)⎡⎣ ⎤⎦x=−L

L

where ImV (x)≡0

Im En<>0 ⇔ ReKn

><0

Ω=[−L, L]

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Non-Hermiticity of open systemNon-Hermiticity of open system

∂∂t

Ψ(t) Ψ(t)Ω

=2

hIm Ψ(t) H total Ψ(t)

Ω= −

1

mRe Ψ(t) pn Ψ(t)

∂Ω

N. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187

Im En > 0 ⇔ ReKn < 0

ih∂∂tΨ(t) =Htotal Ψ(t)

∂∂t

Ψ(t) Ψ(t)Ω

= −i

hΨ(t) H total Ψ(t)

Ω− Ψ(t) H total

† Ψ(t)Ω( ) =

2

hIm Ψ(t) H total Ψ(t)

Ω

Im En < 0 ⇔ ReKn > 0Ω

“Anti-resonant state as an eigenstate“Resonant state” as an eigenstate

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Im Kn < 0Re Kn

><0 ⇔ ImEn

<>0

Eigen-wave-number EigenenergyBound state

Resonant stateAnti-resonant state

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Eigenfunction of resonant stateEigenfunction of resonant stateN. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187

Im En<>0 ⇔ ReKn

><0 ⇒ ImKn < 0

En =

h

mKn

⇒ ImEn =h

mReKn ImKn

CeiKn xBe−iKnx

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Particle-number conservationParticle-number conservationN. Hatano, K. Sasada, H. Nakamura and T. Petrosky, Prog. Theor. Phys. 119 (2008) 187

En =

h

mKn

⇒ ImEn =h

mReKn ImKn

⎛

⎝⎜⎞

⎠⎟

vn =

hm

ReKn

x

NΩ = Ψ(t) Ψ(t) Ω dx−L(t)

L(t)

∫ =e−t ImEn /h ψ ψΩ

dx−L(t)

L(t)

∫ ≈e−t ImEn /h e x ImKndx−L(t)

L(t)

∫: e− ImEnt+L(t) ImKn =exp −

th

ImEn + thm

ReKn ImKn⎛⎝⎜

⎞⎠⎟=constant

L−L L(t)−L(t)

L(t)≡vnt=t

hm

ReKn

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Bound, resonant, anti-resonant statesBound, resonant, anti-resonant states

K

Bound state

Resonantstate

Anti-resonantstate

Continuum

E

Bound state

Resonant state

Anti-resonantstate

Continuum

E =h

mK

Branch point

Branch cut

ψ n (x) ~e−iKn x

eiKn x

⎧⎨⎩

(Far left)

Im Kn < 0 Re Kn><0 ⇔ ImEn

<>0

(Far right)

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Tight-binding modelTight-binding model−t ψ l +1 +ψ l −1( ) + Vlψ l = Eψ l

Vl =0ψ l = eiklis an eigenstate for

E =−tcoskDispersion relation:

kππ0

Continuum limit Impurity bound state

energyband

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t t

Bound, resonant, anti-resonant statesBound, resonant, anti-resonant states

K

Bound state

Resonantstate

Anti-resonantstate

Continuum

E

Bound state

Resonant state

Anti-resonantstate

Continuum

E =−tcosK

Branch point

Branch cut

ψ n (x) ~e−iKn x

eiKn x

⎧⎨⎩

(Far left)

Im Kn < 0 Re Kn><0 ⇔ ImEn

<>0

(Far right)

π π

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H leadH lead

Fisher-Lee relationFisher-Lee relation

T (E)∝ xL

E−H total + iη

xR

= xL

E−Heff (E)

xR

H eff (E)=Hscatterer + Veff (E)

H scatterer

H total =Hscatterer + H lead

Veff Veff

Complex effective potential: H. Feshbach, Ann. Phys. 5 (1958) 357

G(E)=e

hxL

E−Heff (E)

xR

dεL

dkdεR

dk

S. Datta “Electronic Transport in Mesoscopic Systems”

Complex potential eikx

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Conductance and resonanceConductance and resonance

G(E)=e

hxL

E−Heff (E)

xR

dεL

dkdεR

dk

Green’s function: Inverse of a finite matrix

↓

Conductance for real energy

Resonance from poles in complex energy plane

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ContentsContents


rmula



d the Fano Peak

29/3429/34

N-state Friedrichs modelN-state Friedrichs model

xα

Hα

Hd,α

Hd

Hα

Hαn

d0

Keita Sasada: Ph. D. Thesis (2008)

• All leads are connected to the site d0

• Time reversal symmetry is not broken (no magnetic field)

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Conductance formula

N-state Friedrichs modelN-state Friedrichs model

ρleads E( ) ≡1

π t 2 4t 2 − E2

Local DOS of discrete eigenstates:

gα→ β E( ) =gα→ β

max

± −

ρeigen E( )ρleads E( )

⎛

⎝⎜⎞

⎠⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

ρeigen E( ) ≡

1

2π

d0 ψ n%ψ n d0

E − Enn=1

2 N

∑

Local DOS of leads:

　　　 Maximum conductnace from lead α to lead β

Sign depends on the inner structure of the dot and

E

gα→ βmax :

t 2 ≡ tα t( )

α∑(where　　　　　　　 )

Bound st., Res. st., Anti-res. st.


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Interference of discrete statesInterference of discrete states

Bound states Resonance pair(Res. and Anti-res.)

: Interference between B and R

ρeigen E( ) ≡

1

2π

d0 ψ n%ψ n d0

E − Enn=1

2 N

∑ = ρb E( )b

∑ + ρ rpair E( )

r∑


Discrete eigenstates

G(E) ~C1 E −Er

res( ) +C0

E−Erres( )

+ Ei

res( ) =

qε + rε +

Asymmetry of a conductance peak

q: Fano parameter

ρeigen E( )( )2

→ρ r

pair E( ) × ρb E( )

ρ rpair E( ) × ρ ′r

pair E( )

⎧⎨⎩⎪ : Interference between R and R

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T-shape quantum dot (N=2) T-shape quantum dot (N=2)ε1 t = 0

v01 t = 1

tα t = tβ t = 1

⎧

⎨⎪

⎩⎪

　 Bound state: 2　 Resonant state: 1

Anti-resonant state: 1

Interference between each bound state and the resonace pair determines the asymmetry of the conductance peak.

Bound state 1 Bound state 2

Anti-resonant state

Resonant state

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3-state quantum dot (N = 3)3-state quantum dot (N = 3)Keita Sasada: Ph. D. Thesis (2008)

ε0 t = 0, ε1 t = −0.3, ε 2 t = 0.5

v01 t = 0.8, v12 t = 0.4, v20 t = 0.5

tα t = tβ t = 1

⎧

⎨⎪

⎩⎪

Bound state 1 Bound state 2

Anti-resonant state 2

Anti-resonant state 1

Resonant state 1

Resonant state 2Interference between the resonance pairs 1 and 2 determines the asymmetry of the conductance peal.

　 Bound state: 2　 Resonant state: 2

Anti-resonant state: 2

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Fano parameterFano parameter

ρb E( ) ~1

E − Eb

ρ ′rpair E( ) ~

1

E − E ′rres( )

2

G(E) ~C1 E −Er

res( ) +C0

E−Erres( )

+ Ei

res( ) =

qε + rε +

Large when close

: Interference between B and Rρeigen E( )( )2

→ρ r

pair E( ) × ρb E( )

ρ rpair E( ) × ρ ′r

pair E( )

⎧⎨⎩⎪ : Interference between R and R

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SummarySummary

- Electronic conduction and resonance scattering

- Definition and physics of resonant states

- Particle-number conservation

- Interference between resonant states

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Discetization of Schrödinger equationDiscetization of Schrödinger equation

−

h2

2m

d 2

dx2+ V (x)

⎛

⎝⎜⎞

⎠⎟ψ (x) = Eψ (x)

−

h2

2m

ψ (x + Δx) + ψ (x − Δx) − 2ψ (x)

Δx2+ V (x)ψ (x) = Eψ (x)

−t ψ l +1 +ψ l −1( ) + Vlψ l = Eψ l

H = −t cl+†cl + cl

†cl+( ) +Vlcl†cl⎡⎣ ⎤⎦

l=−∞

∞

∑

Tight-binding model

Documents

Electronic Conduction of Mesoscopic Systems and Resonant States Naomichi Hatano Institute of Industrial Science, Unviersity of Tokyo Collaborators: Akinori