Ferromagnetism vs. Antiferromagnetism in Double Quantum Dots: Results from the Bethe Ansatz Robert Konik, Brookhaven National Laboratory Hubert Saleur, Saclay/University of Southern California Andreas Ludwig, University of California, Santa Barbara Galileo Galilei Institute for Theoretical Physics Arcetri, Firenze October 1, 2008 Title indicates a conflict between what the physics actually is in the double dots and what one might expect it to be from elementary arguments

Overview Themes: 1. Show classic Tsvelik-Wiegmann exact solution of single level Anderson model can be extended to more generalized dot systems 2: Show that from these exact solutions, interesting (Kondo) physics arise that is non-perturbative in nature 3. Discuss prospects of extending exact solvability to non-equilibrium problems

Outline Brief introduction to quantum dots Kondo physics Bethe ansatz solution of double quantum dots Exact linear response conductance at T=0 Friedel sum rule - RKKY mediated Kondo effect Beyond T=0 linear response - Finite temperature - Non-equilibrium

Quantum Dots: What are they? Small groups (N~102-103) of electrons confined to a small enough region such that - Electronic levels are discrete - Dot Coulombic effects are important Dots interact with large electron reservoirs or “leads” Dot-lead coupling is origin of all the physics I will discuss

Built from gated heterostructures Semiconductor Dots quantum dot Built from gated heterostructures

Fabrication of Semiconductor Quantum Dots Gates on top of GaAs/AlGaAs heterostructure segregate a portion of the two dimensional electron gas two dimensional electron gas (2DEG) Gates can arranged so that a double dot structure is obtained (here, in parallel) in parallel: Chang et al. PRL 92, 176801 (2004) in series: Petta et al. PRL 93, 186802 (2004)

Double Nanotube Quantum Dots Gates allow chemical potentials of dots to be tuned independently N. Mason, M. Biercuk, C. Marcus, Science 303 (2004) 655

Regime of Interest: Dots with Single Active Level At low temperatures and/or large level spacing, only level nearest Fermi energy is relevant EF δE lead L lead R V V dot lead L lead R

Energy Scales of a Dot with a Single Level εd+U εd Δ~V2 μR μL tunnel barriers between dot-lead Vg energy of second occupation gate voltage making dot tunable energy of singly occupied dot

How Does Kondo Physics Arise in a Single Level Dot At symmetric point of dot (U + 2εd = 0), one electron sits on the dot level. U/2 EF EF U/2 Isolated electron acts like magnetic impurity: Kondo physics

What is Kondo Physics? Two regimes separated by TK, the Kondo temperature: T << TK: low temperature behaviour lead L lead R Dot electron forms a singlet with electrons in leads T >> TK: high temperature behaviour Dot electron interacts only weakly with electrons in leads

Signature of Kondo Physics, T<<TK impurity density of states, ρ(ω) Dynamically generated spectral weight at the Fermi level (Abrikosov-Suhl resonance) ω -TK TK

Kondo Physics in Double Quantum Dots: Possibilities Suppose we two electrons on the two dots and effective channel of electrons coupling to the dots: L R Ferromagnetic RKKY interaction binds electrons into triplet – underscreened spin-1 Kondo effect (standard picture) i) S=1 L R Direct singlet formation between dot electrons – no Kondo effect ii) L R Singlet formation as mediated by electrons in leads – RKKY Kondo effect iii)

Double Quantum Dots in Parallel Two dots are not capacitively or tunnel coupled

Anderson-like Hamiltonian of Double Dots lead electrons dot-lead tunneling chemical potential of dots Coulomb repulsion Model is exactly solvable via Bethe ansatz -RMK (PRL 99, 076602 (2007), New J. Phys. 9, 257 (2007)) Must explicitly check, however, that the multi-particle wavefunctions are of Bethe ansatz form

Constraints on Parameter Space i) Model is integrable if only one channel couples: for consistency need VL1/VR1 = VL2/VR2 ii) Energy of double occupancy is the same on the two dots U1 + 2εd1 = U2 + 2εd2 εd1 and εd2, cannot be adjusted independently iii) U1(V1L2+V1R2) = U2 (V2L2+V2R2) Small perturbations on these constraints do not affect the physics

Landauer-Büttiker Transport Theory left lead right lead RMK, H. Saleur, A. Ludwig PRL 87 (2001) 236801 PRB 66 (2002) 125304 - With μL>μR charge flows from left to right L-B requires knowledge of transmission probabilities, T(ε), for electrons with energies, ε, in the range μL>ε>μR; current J is then Integrability allows one to both determine the correct scattering states to use as well as their exact scattering matrices (i.e. T(ε))

Computation of Scattering Amplitudes Exact solvability means one can construct exact many-body eigenfunctions: The momenta are quantized according to the Bethe ansatz equations: gives scattering amplitude N. Andrei, Phys. Lett. 87A, 299 (1982) These quantization conditions can be recast as T(p) = sin2((δimp(p))

T=0 Linear Response Conductance in Double Quantum Dots one electron on dots forming a singlet with conduction electrons; standard Kondo effect L R Linear response conductance Rapid variation due to interference between dots L R εd1 εd2 decreasing dots’ chemical potential with εd1-εd2 kept fixed two electrons on dots forming singlet; system is p-h symmetric L R RKKY Kondo effect

Evidence for Kondo Physics at the Particle-Hole Symmetric Point Impurity Density of States Spin-Spin Correlation Function Slave boson mft RMK + M. Kulkarni Ferromagnetic like correlations exist between dots although over all ground state is a singlet An Abrikosov-Suhl resonance exists = ¼ for ferro = -¾ for antiferro

The Friedel Sum Rule in Double Dots Friedel sum rule states: Scattering phase of electrons at Fermi energy is proportional to the number of electrons displaced by impurity, Ndisplaced δ = πNdisplaced G = 2e2/h sin2(δ) Ndisplaced has two contributions Electrons on dots deviations in electron density in leads Langreth, Phys. Rev. 150, 516 (1966)

Two difficulties in doing so: Beyond Linear Response at T=0: Finite Temperature and Out-of-Equilibrium Challenge: Compute transmission probability, T(ε), at finite energies Two difficulties in doing so: 1) Non-unique construction of electron 2) Does this construction of the electron behave well under map from original (two leads 1,2) to integrable basis (even, odd) Yes, maybe. charge excitations one parameter family used; reproduces Kondo physics spin excitations

Finite Temperature Linear Response Conductance in Single Dots: Universal Scaling Curve no parameter fit Kaminski et al. numerical renormalization group - Costi et al.

Finite Temperature Linear Response Conductance in Double Quantum Dots Dot levels well separated Dot levels close together εd1 εd2 L R

Out-of-Equilibrium Conductance in a Single Dot In zero magnetic field, ansatz fails: At large voltages, the computed conductance does not behave as G ~ 1/log2(V/TK) At small voltages, produces incorrect coefficient, α, of second order term in voltage Likely culprit: Wrong choice of one parameter family of scattering states Why? Ansatz does work at qualitative level in large magnetic fields where ambiguity of choice is lifted

Out-of-Equilibrium Magnetoconductance Observations Bethe Ansatz Computations T=0 peak value occurs at Δμ<H; agrees with experiments on carbon nanotube dots Kogan et al. Phys. Rev. Lett. 93, 166602 (2004) (μR-μL)/H We can make similar computations for the noise at large fields and bias (Gogolin, RMK, Ludwig, and Saleur, Ann. Phys. (Leipzig) 16, 678 (2007))

Other Strategies for Out-of-Equilibrium Transport: Open Bethe Ansatz Our approach: Equilibrium scattering data out of equilibrium results Open Bethe ansatz: attempt to construct wavefunctions of Bethe type with non-equilibrium boundary conditions Mehta and Andrei, PRL 96, 216802 (2006) μL μR left lead right lead Wavefunctions encode the different number of electrons in the two leads

Nature of Wavefunctions in Open Bethe Ansatz In the interacting resonant level model, to construct such wavefunctions, single particle wavefunctions with removable singularities must be used: |Ψ(x)|2 |Ψ(0)|2 |Ψ(0)|2 ≠ |Ψ(0+)|2≠ |Ψ(0-)|2 |Ψ(x)|2 Probabilitistic interpretation of wavefunction? Can physics be changed by changing wavefunction on a set of measure zero?

Conclusions Exactly solvable variants of the Anderson model applicable to multi-dot systems exist - double dots in series (RMK, PRL 99, 076602 (2007)) - dots exhibiting Fano resonances (RMK, J. Stat. Mech L11001 (2004)) These solutions can exhibit unexpected physics: - singlet formation in a double dot where a triplet might be expected Out-of-equilibrium no generic approach employing exact solvability seems to be available

Finite Temperature Conductance Through a Single Dot Conductance, G TK For the dot, it is the conductance (not the resistance) that increases as T is lowered Kondo singlet free spin