T. K. T. Nguyen, M. N. Kiselev, and V. E. Kravtsov The Abdus Salam ICTP, Trieste, Italy Effect of magnetic field on thermoelectric coefficients of a single electron transistor at almost perfect transmission Regional ICTP’s school Hanoi, 24 December 2009 Cond-mat/ arXiv:

Outline: Thermo-electric transport Beenakker – Staring theory of thermopower for quantum dot Matveev – Andreev theory for open quantum dot Our work: thermopower of an open quantum dot in a magnetic field Conclusion

Transport Coefficients Ielectric current density Jparticle current density J Q heat flux, heat current density µchemical potential Ttemperature Vvoltage, electrostatic potential Quelle: R.D. Barnard Thermoelectricity in Metals and Alloys (1972)‏ Themo-electric effect: the conversion of temperature difference to electric voltage and vice versa.

Kelvin-Onsager relation, and Cutler-Mott formula Transport relations Thermopower General relations for the thermo-electric coefficients

Beenakker - Staring rate equation approach [PRB 46 (1992)] Model Classical regime TP oscillations at different temperatures Effects of spin degeneracy

Matveev-Turek theory: Effects of cotunneling [PRB 65 (2002)] Mechanisms of transport Sequential tunneling Cotunneling Coulomb peak position Cotunneling dominates in valeys Sequentional tunneling dominates at peaks

Matveev-Andreev theory: exact solution at zero and infinite magnetic field [PRB 66 (2002), PRL 86 (2001)] Model Prediction: non-Fermi-Liquid behaviour of thermo-electric coefficients at low T scaling in maximum Goal: describe thermo-transport in strong coupling regime corresponding to 2-channel Kondo

Just to bear in mind: 2CK is unstable fixed point 2CK Line where Matveev-Andreev theory is valid 1CK Motivation Q: What happens if we deviate from unstable separatrix? In order to do it we introduce magnetic field which is a relevant perturbation.

Puzzles of the Matveev-Andreev (MA) theory: two limiting cases Spinless fermions QPC is fully spin-polarized Spinful fermions QPC is non-polarized Q: How does one regime crossover to another one? FL behavior NFL behavior

MA energy scales 0

Generalization of MA theory for finite magnetic field Setup Two-fold effects of the magnetic field What is more important: asymmetry of the Fermi velocities or asymmetry of the reflection coefficients? A: asymmetry of reflection coefficients lead to much more pronounced effects. Approximation: we ignore effects of curvature and take into account the effects of reflection amplitudes asymmetry only!!! Justification: B< B c - Field which makes one fermionic component fully reflecting

Model, assumptions and approximations: Model Asymmetry of reflection amplitudes due to magnetic field is taken into account Asymmetry of Fermi-velocities due to spectrum curvature is ignored Discreteness of levels in the dot is ignored Assumptions:

Model: method of solution Step 1: Mapping onto Anderson-like model Step 2: Diagonalization of the model and calculation of the Green’s functions Step 3: Calculation of the conductance and thermo-conductance

Main results of the theory: New energy scale Resolution of the second puzzle: See also Le Hur, PRB 64 (2001), PRB 65 (2002)

Fermi-liquid properties are restored at finite magnetic field !!! For Effects of magnetic field on thermo-electric coefficients Resolution of the first puzzle:

Effects of magnetic field on thermo-electric coefficients Summary of results and predictions:

Energy scales in magnetic field Non Fermi liquid – Fermi liquid crossover 0 1CK B B 2CK

Conclusions There is a new energy scale which enters the thermo-electric coefficients at finite magnetic field. This energy scale characterizes the asymmetry of reflection amplitudes (asymmetry of backward scattering) from the QPC. At critical field B c the QCP completely reflects one fermionic component providing a crossover to fully spin polarized QPC description. Effects of spectra curvature (finite mass) result in sub-leading corrections to the thermo-electric coefficients. At any finite magnetic field 2-channel Kondo effect description of the thermoelectric coefficient fails. Magnetic field restores the Fermi-Liquid behaviour of the thermo-transport properties. The universality class of the problem in the presence of magnetic field corresponds to 1-channel Kondo effect. Another possibility of the 2CK suppression is associated with a finite source-drain voltage or noise. Strong dependence of thermo-electric coefficients on magnetic field is predicted and to be verified experimentally.