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Current through a nano-device Theory of quantum transport Results Conclusion

Quantum transport through nano-devices A scattering-states numerical renormalization group approach to open quantum systems

Frithjof B. Anders

Institut f¨ ur Theoretische Physik · Universit¨ at Bremen

Hvar, 29. September 2008 Collaborators:

• A. Schiller
• D. Logan, M. Galpin, G. Finkelstein

Current through a nano-device Theory of quantum transport Results Conclusion

Sponsors

KITP Santa Barbara

Current through a nano-device Theory of quantum transport Results Conclusion

Outline

1

Current through a nano-device Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots

2

Theory of quantum transport Open quantum systems The scattering-states NRG

3

Results Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance

4

Conclusion Questions Summary

Current through a nano-device Theory of quantum transport Results Conclusion

Contents

1

Current through a nano-device Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots

2

Theory of quantum transport Open quantum systems The scattering-states NRG

3

Results Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance

4

Conclusion Questions Summary

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

• D. Goldhaber-Gordon, Nature 98

weak coupling

M.Kastner RMP 1992

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

V(g) single particle picture ☞ random conductance peaks

weak coupling

M.Kastner RMP 1992

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

V(g)

no current

single particle picture ☞ random conductance peaks

weak coupling

M.Kastner RMP 1992

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

V(g) single particle picture ☞ random conductance peaks

weak coupling

M.Kastner RMP 1992

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

V(g)

U=e/C

2

physics driven by charging energy Ec = e2/C

weak coupling

M.Kastner RMP 1992

E = e2

2C

• ˆ

N − Ng 2

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

• D. Goldhaber-Gordon, Nature 98

strong coupling

van der Wiel et al. Science 289 (2000)

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

• D. Goldhaber-Gordon, Nature 98

strong coupling

van der Wiel et al. Science 289 (2000)

Current through a nano-device Theory of quantum transport Results Conclusion Kondo effect in a single-electron transistor (SET)

Kondo effect in a single-electron transistor (SET)

single-electron transistor

• D. Goldhaber-Gordon, Nature 98

strong coupling

Goldhaber-Gorden, Nature, 391, 157 (1998)

Current through a nano-device Theory of quantum transport Results Conclusion Zero-bias conductance in carbon nanotube quantum dots

Zero-bias conductance in carbon nanotube quantum dots

Samples

doped Si SiO2 nanotube Vgate Vsource-drain A Single-wall CNT ~2 nm in diameter Measurement: differential conductance = dI/dV (Vgate)

• A. Makarovski et al. PRL 99, 066801 (2007)

see Gleb Finkelstein’s talk

• rbits

semiconducting carbon-nanotubes ☞ two effective orbitals ☞ two conduction channels

Current through a nano-device Theory of quantum transport Results Conclusion Zero-bias conductance in carbon nanotube quantum dots

Zero-bias conductance in carbon nanotube quantum dots

Experiment: A. Makarovski et al. PRL 99, 066801 (2007) Theory: FBA, Logan, Galpin, Finkelstein, PRL 100, 086809 (2008)

• rbits

filled in groups of ☞ two orbits peaks equidistant ☞ two orbits, U ≈ 10meV , J ≈ 0 4 groups with increasing coupling to the leads

Current through a nano-device Theory of quantum transport Results Conclusion Zero-bias conductance in carbon nanotube quantum dots

Zero-bias conductance in carbon nanotube quantum dots

Experiment: A. Makarovski et al. PRL 99, 066801 (2007) Theory: FBA, Logan, Galpin, Finkelstein, PRL 100, 086809 (2008)

• rbits

filled in groups of ☞ two orbits peaks equidistant ☞ two orbits, U ≈ 10meV , J ≈ 0 4 groups with increasing coupling to the leads

Current through a nano-device Theory of quantum transport Results Conclusion Zero-bias conductance in carbon nanotube quantum dots

Zero-bias conductance in carbon nanotube quantum dots

Experiment: A. Makarovski et al. PRL 99, 066801 (2007) Theory: FBA, Logan, Galpin, Finkelstein, PRL 100, 086809 (2008)

• rbits

filled in groups of ☞ two orbits peaks equidistant ☞ two orbits, U ≈ 10meV , J ≈ 0 4 groups with increasing coupling to the leads

Current through a nano-device Theory of quantum transport Results Conclusion

Contents

1

Current through a nano-device Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots

2

Theory of quantum transport Open quantum systems The scattering-states NRG

3

Results Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance

4

Conclusion Questions Summary

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Theoretical approaches to non-equilibrium transport

rate (master) equations Keldysh based approaches: perturbation theory, RG methods: Meir, Wingreen, Ueda, Oguri, W¨

• lfe, Kroha, Flensberg,

Paaske, Schoeller, K¨

• nig,...

flow equation: Kehrein finite-size methodes: time-dependent DMRG, Schmitteckert scattering-states approaches: Hershfield, Metha, Andrei, Han, Oguri

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Theoretical approaches to non-equilibrium transport

rate (master) equations Keldysh based approaches: perturbation theory, RG methods: Meir, Wingreen, Ueda, Oguri, W¨

• lfe, Kroha, Flensberg,

Paaske, Schoeller, K¨

• nig,...

flow equation: Kehrein finite-size methodes: time-dependent DMRG, Schmitteckert scattering-states approaches: Hershfield, Metha, Andrei, Han, Oguri

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Theoretical approaches to non-equilibrium transport

rate (master) equations Keldysh based approaches: perturbation theory, RG methods: Meir, Wingreen, Ueda, Oguri, W¨

• lfe, Kroha, Flensberg,

Paaske, Schoeller, K¨

• nig,...

flow equation: Kehrein finite-size methodes: time-dependent DMRG, Schmitteckert scattering-states approaches: Hershfield, Metha, Andrei, Han, Oguri

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Theoretical approaches to non-equilibrium transport

rate (master) equations Keldysh based approaches: perturbation theory, RG methods: Meir, Wingreen, Ueda, Oguri, W¨

• lfe, Kroha, Flensberg,

Paaske, Schoeller, K¨

• nig,...

flow equation: Kehrein finite-size methodes: time-dependent DMRG, Schmitteckert scattering-states approaches: Hershfield, Metha, Andrei, Han, Oguri

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Theoretical approaches to non-equilibrium transport

rate (master) equations Keldysh based approaches: perturbation theory, RG methods: Meir, Wingreen, Ueda, Oguri, W¨

• lfe, Kroha, Flensberg,

Paaske, Schoeller, K¨

• nig,...

flow equation: Kehrein finite-size methodes: time-dependent DMRG, Schmitteckert scattering-states approaches: Hershfield, Metha, Andrei, Han, Oguri

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Open quantum systems: finite size representations

Goldhaber-Gordon, Nature 1998

• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 J [ e/h ] µSD = 0.8 µSD = 0.6 µSD = 0.4 Time T [ h/t ] µ = 0

TD-DMRG calculation: Schneider, Schmitteckert, arXiv:0601389

Problems: Coulomb repulsion U, charging energy Ec: non–perturbativ time-dependent DMRG

DMRG boundary condition: closed quantum system Kondo problem: transient time ∝ 1/TK

How do we simmulate an open system with a finite chain?

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Boundary condition

lead device lead

• pen quantum system

L

Problem: steady-state limit transient currents: lim

L→∞ lim t→∞ J(t) = 0

steady-state currents: J∞ = limt→∞ limL→∞ J(t) > 0

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Boundary condition

lead device lead

• pen quantum system

L

Problem: steady-state limit transient currents: lim

L→∞ lim t→∞ J(t) = 0

steady-state currents: J∞ = limt→∞ limL→∞ J(t) > 0

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

the challenge

description of steady-state currents for arbitrary V , T, U, H using a finite size system

the problem

Coulomb interaction: Kondo effect, non-perturbative already in equilibrium in finite-size system representation, boundary condition?

a solution

FBA, PRL 101, 066804 (2008) and J. Phys.: Condens. Matter 20, 195216 (2008)

1 open quantum system: scattering states (Hershfield, Han,

Andrei, Oguri ..)

2 density operator at finite bias but U = 0! (Hershfield 1993) 3 TD-NRG: evolving the system into the interacting limit

(FBA, A. Schiller, 2005, 2006)

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

the challenge

description of steady-state currents for arbitrary V , T, U, H using a finite size system

the problem

Coulomb interaction: Kondo effect, non-perturbative already in equilibrium in finite-size system representation, boundary condition?

a solution

FBA, PRL 101, 066804 (2008) and J. Phys.: Condens. Matter 20, 195216 (2008)

1 open quantum system: scattering states (Hershfield, Han,

Andrei, Oguri ..)

2 density operator at finite bias but U = 0! (Hershfield 1993) 3 TD-NRG: evolving the system into the interacting limit

(FBA, A. Schiller, 2005, 2006)

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

the challenge

description of steady-state currents for arbitrary V , T, U, H using a finite size system

the problem

Coulomb interaction: Kondo effect, non-perturbative already in equilibrium in finite-size system representation, boundary condition?

a solution

FBA, PRL 101, 066804 (2008) and J. Phys.: Condens. Matter 20, 195216 (2008)

1 open quantum system: scattering states (Hershfield, Han,

Andrei, Oguri ..)

2 density operator at finite bias but U = 0! (Hershfield 1993) 3 TD-NRG: evolving the system into the interacting limit

(FBA, A. Schiller, 2005, 2006)

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

the challenge

description of steady-state currents for arbitrary V , T, U, H using a finite size system

the problem

Coulomb interaction: Kondo effect, non-perturbative already in equilibrium in finite-size system representation, boundary condition?

a solution

FBA, PRL 101, 066804 (2008) and J. Phys.: Condens. Matter 20, 195216 (2008)

1 open quantum system: scattering states (Hershfield, Han,

Andrei, Oguri ..)

2 density operator at finite bias but U = 0! (Hershfield 1993) 3 TD-NRG: evolving the system into the interacting limit

(FBA, A. Schiller, 2005, 2006)

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

the challenge

description of steady-state currents for arbitrary V , T, U, H using a finite size system

the problem

Coulomb interaction: Kondo effect, non-perturbative already in equilibrium in finite-size system representation, boundary condition?

a solution

FBA, PRL 101, 066804 (2008) and J. Phys.: Condens. Matter 20, 195216 (2008)

1 open quantum system: scattering states (Hershfield, Han,

Andrei, Oguri ..)

2 density operator at finite bias but U = 0! (Hershfield 1993) 3 TD-NRG: evolving the system into the interacting limit

(FBA, A. Schiller, 2005, 2006)

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

Modelling of the single-electron transistor (SET)

• D. Goldhaber-Gordon, Nature 98

H =

• α=L,R,kσ

εkσαc†

kσαckσα +

• σ

E d

σ d† σdσ + Und ↑nd ↓

+

• kασ

tα

• d†

σckσα + c† kσαd

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

1: Open quantum system: boundary condition

γεασ

device left-moving and right-moving scattering states γ†

εασ

γεασ: itinerant states correct boundary condition complex: current carring states

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

1: Open quantum system: boundary condition

γεασ

device left-moving and right-moving scattering states γ†

εασ

γεασ: itinerant states correct boundary condition complex: current carring states

Current through a nano-device Theory of quantum transport Results Conclusion Open quantum systems

1: Open quantum system: boundary condition

γεασ

device left-moving and right-moving scattering states γ†

εασ

γεασ: itinerant states correct boundary condition complex: current carring states

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

1: Open quantum system: scattering states

U = 0 steady-state density operator: exact solution steady-state density operator (Hershfield 1993): ˆ ρ0(U = 0) = 1 Z0 e−β(H0− ˆ

Y0)

ˆ Y0 =

• α=L,R
• σ

µα

• dεγ†

εασγεασ

γ†

εασ scattering- states operator for itinerant α = L/R states

(Hershfield PRL 1993, Oguri PRB 2007)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

1: Open quantum system: scattering states

U = 0 steady-state density operator: exact solution steady-state density operator (Hershfield 1993): ˆ ρ0(U = 0) = 1 Z0 e−β(H0− ˆ

Y0)

ˆ Y0 =

• α=L,R
• σ

µα

• dεγ†

εασγεασ

γ†

εασ scattering- states operator for itinerant α = L/R states

(Hershfield PRL 1993, Oguri PRB 2007)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

2: Open quantum system: interacting nano-device U = 0

left mover µ right mover µr

l

fictious junction

Wilson chain for right and left-movers (FBA, PRL (2008)) local d-orbital ( ¯ V =

• V 2

L + V 2 R, rα = Vα/ ¯

V ) dσ = 1 ¯ V

• α

Vαdσα dσα = ¯ V

• dε
• ρ(ε)|G d

0 (ε + iδ)|eiΦ0(ε)γεασ

novel NRG: discretisation of the scattering states

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

2: Open quantum system: interacting nano-device U = 0

left mover µ right mover µr

l

fictious junction

Wilson chain for right and left-movers (FBA, PRL (2008)) local d-orbital ( ¯ V =

• V 2

L + V 2 R, rα = Vα/ ¯

V ) dσ = 1 ¯ V

• α

Vαdσα dσα = ¯ V

• dε
• ρ(ε)|G d

0 (ε + iδ)|eiΦ0(ε)γεασ

novel NRG: discretisation of the scattering states

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

2: Open quantum system: interacting nano-device U = 0

left mover µ right mover µr

l

fictious junction

price: current operator energy depended I(µL, µR) = G0 e

• σ

dε π [fL(ε) − fR(ε)] ℑmG d

σ (ε − iδ)

G d

0σ(z)

= r2

R ≪ dσR|d† σR ≫ (z) + r2 L ≪ dσL|d† σL ≫ (z)

Meir, Wingreen 1993

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

3: Open quantum system: interacting nano-device U > 0

U

left mover µ right mover µr

l

fictious junction

scattering-states NRG: switch on HU = Und

↑nd ↓: backscattering

d†

σdσ

= r2

Rnd Rσ + r2 Lnd Lσ

• density part

+ rRrL

• d†

LσdRσ + d† RσdLσ

• backscattering

absorb density part H0

U = αα′ r2 αr2 α′nα↑nα′↓ in ρ0 → ˜

ρ0 ˜ ρ0 exact in the tunneling regime!

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

3: Open quantum system: interacting nano-device U > 0

U

left mover µ right mover µr

l

fictious junction

scattering-states NRG: switch on HU = Und

↑nd ↓: backscattering

d†

σdσ

= r2

Rnd Rσ + r2 Lnd Lσ

• density part

+ rRrL

• d†

LσdRσ + d† RσdLσ

• backscattering

absorb density part H0

U = αα′ r2 αr2 α′nα↑nα′↓ in ρ0 → ˜

ρ0 ˜ ρ0 exact in the tunneling regime!

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

3: Open quantum system: interacting nano-device U > 0

U

left mover µ right mover µr

l

fictious junction

Scattering-states NRG: using the TD-NRG (FBA, Schiller 2005, 2006) ˆ ρ∞ = lim

T→∞

1 T T dτe−iHf τ ˜ ρ0eiHf τ calculate finite U non-equilibrium G d

σ (ω)

novel NEQ-GF TD-NRG algorithm

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

3: Open quantum system: interacting nano-device U > 0

U

left mover µ right mover µr

l

fictious junction

Scattering-states NRG: using the TD-NRG (FBA, Schiller 2005, 2006) ˆ ρ∞ = lim

T→∞

1 T T dτe−iHf τ ˜ ρ0eiHf τ calculate finite U non-equilibrium G d

σ (ω)

novel NEQ-GF TD-NRG algorithm

FBA, J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

Summary of scattering-states NRG

scattering-state basis continuum: boundary condition of an

• pen quantum system

discretizing in left and right movers expand local orbital in a left and a right moving part: dσL,R TD-NRG ρ∞ = lim

T→∞

1 T T e−iHf tρ0eiHf t calculate NEQ spectral functions FBA, J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

Summary of scattering-states NRG

scattering-state basis continuum: boundary condition of an

• pen quantum system

discretizing in left and right movers expand local orbital in a left and a right moving part: dσL,R TD-NRG ρ∞ = lim

T→∞

1 T T e−iHf tρ0eiHf t calculate NEQ spectral functions FBA, J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

Summary of scattering-states NRG

scattering-state basis continuum: boundary condition of an

• pen quantum system

discretizing in left and right movers expand local orbital in a left and a right moving part: dσL,R TD-NRG ρ∞ = lim

T→∞

1 T T e−iHf tρ0eiHf t calculate NEQ spectral functions FBA, J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

Summary of scattering-states NRG

scattering-state basis continuum: boundary condition of an

• pen quantum system

discretizing in left and right movers expand local orbital in a left and a right moving part: dσL,R TD-NRG ρ∞ = lim

T→∞

1 T T e−iHf tρ0eiHf t calculate NEQ spectral functions FBA, J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion The scattering-states NRG

Summary of scattering-states NRG

scattering-state basis continuum: boundary condition of an

• pen quantum system

discretizing in left and right movers expand local orbital in a left and a right moving part: dσL,R TD-NRG ρ∞ = lim

T→∞

1 T T e−iHf tρ0eiHf t calculate NEQ spectral functions FBA, J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion

Contents

1

Current through a nano-device Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots

2

Theory of quantum transport Open quantum systems The scattering-states NRG

3

Results Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance

4

Conclusion Questions Summary

Current through a nano-device Theory of quantum transport Results Conclusion Benchmarking of the NEQ-GF TD-NRG algorithm

Equilibrium spectra: evolution from U = 0 to finite U

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10

ω/Γ

0.1 0.2 0.3 0.4

ρ(ω)

U=1 U=2 U=4 U=6 U=8 U=10

ω/Γ

0.1 0.2 0.3 0.4 U=1 U=2 U=4 U=6 U=8 U=10

benchmarks using the single lead SIAM

• F. Anders J. Phys.: Condens. Matter 20, 195216 (2008)

Current through a nano-device Theory of quantum transport Results Conclusion Steady-state spectra

Steady-state spectra: symmetric junction – ΓL/ΓR = 1

• 10 -9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6 7 8 9 10

ω/Γ

0.1 0.2 0.3

ρ(ω)

V=0.5*10

• 4

V=0.01 V=0.1 V=0.5 V=1 V=2 V=5

• 1
• 0.5

0.5 1 ω/Γ 0.1 0.2 0.3 ρ(ω) V=0.5*10

• 4

V=0.01 V=0.1 V=0.5 V=1 V=2 V=5

U/Γ = 8, R = ΓL/ΓR = 1, µL = −V /2, µR = V /2

Current through a nano-device Theory of quantum transport Results Conclusion Steady-state spectra

Steady-state spectra: tunnel regime – ΓL/ΓR = 1000

• 10 -9
• 8
• 7
• 6
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5 6 7 8 9 10

ω/Γ

0.1 0.2 0.3

ρ(ω)

V=0.5*10

• 4

V=0.01 V=0.1 V=0.5 V=1 V=2

• 1
• 0.5

0.5 1 ω/Γ 0.1 0.2 0.3 ρ(ω) V=0.5*10

• 4

V=0.01 V=0.1 V=0.5 V=1 V=2

U/Γ = 8, R = ΓL/ΓR = 1000, µL = −r2

RV , µR = r2 LV

Current through a nano-device Theory of quantum transport Results Conclusion Differential conductance

Differential conductance

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

V/Γ

0.5 1 1.5 2

G/G0

R=1 R=10 R=100 R=1000 ρ(ω,V=0)

U/Γ = 8, R = ΓL/ΓR = 1

Current through a nano-device Theory of quantum transport Results Conclusion Differential conductance

Differential conductance: finite magnetic field

• 3
• 2
• 1

1 2 3

V/Γ

0.5 1 1.5 2

G in [e

2/h] H=0. H=0.1 H=0.2 H=0.4

U/Γ = 8, R = ΓL/ΓR = 1

Current through a nano-device Theory of quantum transport Results Conclusion Differential conductance

Comparision with Han, Heary PRL 99, 2007

1 2 3 4 5

V/Γ

0.2 0.4 0.6 0.8 1

G in [2e

2/h] Han,Heary PRL 2007 S-NRG

U/Γ = 5, εf /Γ = −2.5, R = ΓL/ΓR = 1, T = 0.04

Current through a nano-device Theory of quantum transport Results Conclusion Differential conductance

Differential conductance: finite T evolution

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

V

0.2 0.4 0.6

G in [e

2/h] T=2*10

• 4

T=0.05 T=0.1 T=0.4

U/Γ = 12, εf /Γ = −1.5, R = ΓL/ΓR = 10

Current through a nano-device Theory of quantum transport Results Conclusion

Contents

1

Current through a nano-device Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots

2

Theory of quantum transport Open quantum systems The scattering-states NRG

3

Results Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance

4

Conclusion Questions Summary

Current through a nano-device Theory of quantum transport Results Conclusion Questions

Questions limitation of the NEQ GF algorithm at finite bias? comparision with analytical results, Keldysh for U/Γ < 1? calculate currents directly with a complex scattering states NRG? extension to more complex interacting regions

Current through a nano-device Theory of quantum transport Results Conclusion Questions

Questions limitation of the NEQ GF algorithm at finite bias? comparision with analytical results, Keldysh for U/Γ < 1? calculate currents directly with a complex scattering states NRG? extension to more complex interacting regions

Current through a nano-device Theory of quantum transport Results Conclusion Questions

Questions limitation of the NEQ GF algorithm at finite bias? comparision with analytical results, Keldysh for U/Γ < 1? calculate currents directly with a complex scattering states NRG? extension to more complex interacting regions

Current through a nano-device Theory of quantum transport Results Conclusion Questions

Questions limitation of the NEQ GF algorithm at finite bias? comparision with analytical results, Keldysh for U/Γ < 1? calculate currents directly with a complex scattering states NRG? extension to more complex interacting regions

Current through a nano-device Theory of quantum transport Results Conclusion Summary

Summary The scattering-states NRG novel approach to open quantum systems correct boundary condition includes the U = 0 and tunnel regime exactly use the TD-NRG for evolving ρr(ω, U = 0) into the full ρr(ω) novel algorithm to calculate NEQ Green functions