Many-body resonances in double quantum-dot systems
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1 Many-body resonances in double quantum-dot systems Akinori Nishino Kanagawa University
2 Collaborators Naomichi Hatano IIS, University of Tokyo Takashi Imamura RCAST, University of Tokyo Gonzalo Ordonez Butler University
3 3 Summary We study double quantum-dot (DQD) systems with an interdot Coulomb interaction Exact many-electron scattering eigenstates Appearance of many-body bound states Binding strength Resonant poles of scattering Many-body resonant states Lead 1 (source) Quantum dot 1 Lead 2 (drain) Quantum dot 2
4 Part I: Electron transport in open quantum-dot systems
![Experiments: Quantum dot (QD) 5 2D elec. systems are realized on heterostructures of semiconductors. Kondo effect in QD [D. Goldhaber-Gordon et al., Nature391(1998)156] [W. G. van der Wiel et al.](/docs-images/94/121857834/images/5-1.jpg ", Science289(2000)2105] source drain Conductance e2/h Quantum dot on GaAs/AlGaAs T=15 800mK Source-drain voltage mv Mesoscopic systems Finite bias voltage (The system is far from equilibrium.")
5 Experiments: Quantum dot (QD) 5 2D elec. systems are realized on heterostructures of semiconductors. Kondo effect in QD [D. Goldhaber-Gordon et al., Nature391(1998)156] [W. G. van der Wiel et al., Science289(2000)2105] source drain Conductance e2/h Quantum dot on GaAs/AlGaAs T=15 800mK Source-drain voltage mv Mesoscopic systems Finite bias voltage (The system is far from equilibrium.) Coulomb interactions for the electrons localized on QD
6 Open quantum systems Closed systems Open systems equilibrium equilibrium Bound states Equilibrium states in the TD limit? Scattering states Nonequilibrium steady states between equilibrium states?
7 Landauer formula 7 Coherent electron transport in Transmission noninteracting prob. cases reservoir Open systems reservoir Average Electrical electric current conductance Interacting cases Many-electron scattering eigenstates
8 Open quantum-dot (QD) system 8 Interacting resonant-level model (IRLM) ( a minimal model for QD systems with interactions) Lead 1 Lead 2 Spinless electrons Lead-dot coupling Single energy level Coulomb interaction
9 Calculation of current in IRLM Bethe ansatz approach [P. Mehta & N. Andrei, PRL96(2006)216802] Numerical renormalization group (NRG) [L. Borda, K. Vladar & A. Zawadowski, PRB75(2007) ] Nonequilibrium Greens function (NEGF) approach [B. Doyon, PRL99(2007)076806] [A. Golub, PRB76(2007)193307] Time-dependent DMRG (TDDMRG) [E. Boulat, H. Saleur & P. Schmitteckert, PRL101(2008)140601] Extension of Landauer formula with exact scattering eigenstates [A.N., T. Imamura & N. Hatano, PRL102(2009)146803] Functional RG (FRG) & Real time RG in frequency space [C. Karrasch et al., EPL90(2010)30003]
10 Universal electric current 10 [A.N., T. Imamura & N. Hatano,PRB86(2011)035306] For the IRLM with linearized dispersion relations, Negative differential conductance appears for U > 0! Formation of many-body bound states
11 Calculation of current in IRLM Bethe ansatz approach [P. Mehta & N. Andrei, PRL96(2006)216802] Numerical renormalization group (NRG) [L. Borda, K. Vladar & A. Zawadowski, PRB75(2007) ] Nonequilibrium Greens function (NEGF) approach [B. Doyon, PRL99(2007)076806] [A. Golub, PRB76(2007)193307] Negative differential conductance for U > 0 Time-dependent DMRG (TDDMRG) [E. Boulat, H. Saleur & P. Schmitteckert, PRL101(2008)140601] Extension of Landauer formula with exact scattering eigenstates [A.N., T. Imamura & N. Hatano, PRL102(2009)146803] Functional RG (FRG) & Real time RG in frequency space [C. Karrasch et al., EPL90(2010)30003]
12 Summary of Part I and comments We need to consider the effect of the Coulomb interaction for the open QD systems. Many-electron scattering states play important roles to study the electron transport in the systems. The effect of the Coulomb interaction appear in physical quantities through many-body bound states. We apply the approach to other QD systems. Many-body bound states Many-body resonances
13 Part II: Many-body resonances in double quantum-dot systems
14 Double quantum-dot (DQD) systems Lead 1 Lead 2 Spinless electrons Linear dispersion relation Lead-dot coupling Dot-dot coupling Single energy level Interdot Coulomb interaction
![[Tanaka & Kawakami, PRB 72 (2005) 085304] T-shaped Parallel:](/docs-images/94/121857834/images/15-1.jpg "Serial: [W. G. van der Wiel et al., Science289, 2015 (2000) ] [H.")
15 Relation to other DQD systems Various DQD systems are reproduced at special values of parameters. cf. [Tanaka & Kawakami, PRB 72 (2005) ] T-shaped Parallel: Serial: [W. G. van der Wiel et al., Science289, 2015 (2000) ] [H. Jeong et al., Science293, 2221(2001)]
16 One-electron scattering eigenstates Eigenstates with plane-wave incident states Reflected wave Transmitted wave Incident wave : level-width matrices
17 Resonances in one-electron scattering Resonant poles Poles of in the k-complex plane Poles of in the k-complex plane
18 Scattering states & resonant states Scattering eigenstates: plane-wave incident states Resonant states: incident elec. is on DQD Divergent in x No incoming electron! cf. Properties of resonant states in QD systems [Hatano, Sasada, Nakamura & Petrosky, PTP119(2008)187]
19 Roles of resonant poles Resonant poles Position of peaks Width of peaks i) Resonant peaks of conductance at U=0 (known) (Landauer formula)
20 Conductance at U =0 T-shaped DQD Parallel DQD Serial DQD Asymmetric peak Fano resonance cf.[k. Kobayashi et al., PRB70(2004)035319] [N. Hatano, K. Sasada & G. Ordonez, JPSJ80(2011) ]
21 Roles of resonant poles Resonant poles Position of peaks Width of peaks i) Resonant peaks of conductance at U=0 (known) (Landauer formula) ii) Binding strength of two-body bound states in interacting cases (U > 0)
22 Exact two-electron scattering eigenstates Eigenstates with two-electron plane-wave incident states Transmitted wave 22 2-body bound states
23 Two-body bound states & resonant poles Two-electron transmitted wave Plane wave Plane wave Plane wave Plane wave 2-body bound state Resonant poles Resonant states appear as two-body bound states!
24 Two-body resonant states One-body resonant states an incident elec. on DQD Two-body resonant states two incident elec. on DQD No incoming electron! The 2-body resonant state is the resonant state of 2-body bound states!
25 25 Summary We studied the DQD systems with arbitrary lead-dot and dot-dot couplings and an interdot Coulomb interaction. Exact many-electron scattering eigenstates Relation between many-body bound states and many-body resonant poles One-body and two-body resonant states Future problems Calculation of electric current for the DQD systems How can we observe many-body resonant states?
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