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 Current through a nano-device 1 Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots Theory of quantum transport 2 Open quantum systems The scattering-states NRG Results 3 Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance Conclusion 4 Questions Summary
Current through a nano-device Theory of quantum transport Results Conclusion Contents Current through a nano-device 1 Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots Theory of quantum transport 2 Open quantum systems The scattering-states NRG Results 3 Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance Conclusion 4 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 weak coupling M.Kastner RMP 1992 D. Goldhaber-Gordon, Nature 98
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 weak coupling V(g) M.Kastner RMP 1992 single particle picture ☞ random conductance peaks
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 weak coupling no current V(g) M.Kastner RMP 1992 single particle picture ☞ random conductance peaks
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 weak coupling V(g) M.Kastner RMP 1992 single particle picture ☞ random conductance peaks
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 weak coupling 2 U=e/C V(g) M.Kastner RMP 1992 physics driven by charging energy � � 2 E = e 2 ˆ N − N g E c = e 2 / C 2 C
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 strong coupling D. Goldhaber-Gordon, Nature 98 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 strong coupling D. Goldhaber-Gordon, Nature 98 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 strong coupling D. Goldhaber-Gordon, Nature 98 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 V source-drain V gate A orbits nanotube SiO 2 doped Si Measurement: differential conductance semiconducting = dI/dV (Vgate) carbon-nanotubes ☞ two effective orbitals Single-wall CNT ~2 nm in diameter ☞ two conduction channels A. Makarovski et al. PRL 99, 066801 (2007) see Gleb Finkelstein’s talk
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 orbits filled in groups of ☞ two orbits peaks equidistant ☞ two orbits, U ≈ 10 meV , J ≈ 0 4 groups with increasing coupling Experiment: A. Makarovski et al. PRL 99, 066801 (2007) to the leads Theory: FBA, Logan, Galpin, Finkelstein, PRL 100 , 086809 (2008)
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 orbits filled in groups of ☞ two orbits peaks equidistant ☞ two orbits, U ≈ 10 meV , J ≈ 0 4 groups with increasing coupling Experiment: A. Makarovski et al. PRL 99, 066801 (2007) to the leads Theory: FBA, Logan, Galpin, Finkelstein, PRL 100 , 086809 (2008)
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 orbits filled in groups of ☞ two orbits peaks equidistant ☞ two orbits, U ≈ 10 meV , J ≈ 0 4 groups with increasing coupling Experiment: A. Makarovski et al. PRL 99, 066801 (2007) to the leads Theory: FBA, Logan, Galpin, Finkelstein, PRL 100 , 086809 (2008)
Current through a nano-device Theory of quantum transport Results Conclusion Contents Current through a nano-device 1 Kondo effect in a single-electron transistor (SET) Zero-bias conductance in carbon nanotube quantum dots Theory of quantum transport 2 Open quantum systems The scattering-states NRG Results 3 Benchmarking of the NEQ-GF TD-NRG algorithm Steady-state spectra Differential conductance Conclusion 4 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¨ olfe, Kroha, Flensberg, Paaske, Schoeller, K¨ onig,... 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¨ olfe, Kroha, Flensberg, Paaske, Schoeller, K¨ onig,... 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¨ olfe, Kroha, Flensberg, Paaske, Schoeller, K¨ onig,... 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¨ olfe, Kroha, Flensberg, Paaske, Schoeller, K¨ onig,... 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¨ olfe, Kroha, Flensberg, Paaske, Schoeller, K¨ onig,... 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 0.6 µ SD = 0.8 0.4 µ SD = 0.6 0.2 µ SD = 0.4 J [ e/h ] 0.0 -0.2 -0.4 Time T [ h/t ] -0.6 µ = 0 TD-DMRG calculation: Schneider, Schmitteckert, Goldhaber-Gordon, Nature 1998 arXiv:0601389 Problems: Coulomb repulsion U , charging energy E c : non–perturbativ time-dependent DMRG DMRG boundary condition: closed quantum system Kondo problem: transient time ∝ 1 / T K 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 open quantum system lead lead device L Problem: steady-state limit transient currents: lim L →∞ lim t →∞ J ( t ) = 0 steady-state currents: J ∞ = lim t →∞ lim L →∞ J ( t ) > 0
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