T. K. T. Nguyen, M. N. Kiselev, and V. E. KravtsovThe Abdus Salam ICTP, Trieste, Italy

Effect of magnetic field on thermoelectric coefficients of a single electron transistor

at almost perfect transmission

Regional ICTP’s schoolHanoi, 24 December 2009

Cond-mat/ arXiv:0912.4632

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

• I electric current density• J particle current density

• JQ heat flux, heat current density

• µ chemical potential• T temperature• V voltage, electrostatic potential

T

Ve

T

L

T

LT

L

T

L

J

J

Je

I

QQ

22221

21211

211 e

TL

e

T

e

STL

2

12

2222 STTL

2112 LL

Quelle: R.D. Barnard Thermoelectricity in Metals and Alloys (1972)

Themo-electric effect: the conversion of temperature difference to electric voltage and vice versa.

00lim

I

th

T T

VS

qT

E

TG

GS

I

T

0 FE

BB

dE

dG

G

Tk

q

kS

3

2

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 spindegeneracy

Matveev-Turek theory: Effects of cotunneling [PRB 65 (2002)]

Mechanisms of transport

Sequential tunneling

Cotunneling

Coulomb peak position

Cotunneling dominates in valeysSequentional 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 behaviourof thermo-electric coefficients at low T

scaling in maximum

Goal: describe thermo-transport in strong coupling regimecorresponding to 2-channel Kondo

Just to bear in mind: 2CK is unstable fixed point

2CK

Line where Matveev-Andreev theory is valid

1CK

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< Bc - 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 scaleResolution 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 fieldNon Fermi liquid – Fermi liquid crossover

0

1CKB

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 Bc 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 predictedand to be verified experimentally.