Conductance of a spin-1 QD:two-stage Kondo effect

Anna PosazhennikovaInstitut für Theoretische Festkörperphysik,

Uni Karlsruhe, Germany

Les Houches, June 19, 2006

Collaborators: Babak Bayani (TFP, University Karlsruhe, Germany)

Piers Coleman (Rutgers University, NJ, USA)

Outline

• Introduction: Kondo effect in bulk and mesoscopic systems

• Spin-1 QD: theoretical expectations

• Model and T-matrix analysis

• Conductance calculations

• Conclusions and outlook

Introduction: Kondo Effect

History of Kondo effect

System: metallic host + magnetic impurity• 1930 – ρmin in some alloys

• 1950 – Χcurie – measurements showed that a LM forms in those alloys, which exhibit ρmin

Q1: why does LM form?• 1961 - P. W. Anderson: U is large enough compared to interlevel

spacing

Atomic limit of Anderson model

nUnddEH d

Possible states

History of Kondo effect

Q2: why does the formation of LM lead to ρmin ?• 1964 – Jun Kondo, Hamiltonian

SsJHH AFMKondo 00

Perturbation theory breaks down at TK

TK is the only scale in the problem

Q3: why does ρ saturates at low temperatures?• 1970 – conjecture of Anderson and Yuval – GS is a paramagnetic spin singlet confirmed by• 1971 - K.Wilson – NRG

Up to here: orbital momentum 0l

History of Kondo effect

Q4: what happens in more realistic situation with 0l•1980 Blandin, Nozieres•1984 Andrei, Tsvelik, Wiegman

Perfectly Screened KE Underscreened KE Overscreened KE

FL NFLNFL

USK, OSK – inaccessible in bulk materials Mesoscopics?

Introduction: Kondo Effect in Quantum Dots

Introduction: Kondo Effect in QD

Realization of spin-1 QD: singlet-triplet transition in zero magnetic field

Neven Nodd

Spin-1 QD: two-channel Kondo effect

Set up: spin, coupled to left (L) and right (R) leads

L R

diagonalization

J1, J2 – two coupling constants =>

two screening channels

Kondo Hamiltonian

Conductance of a spin-1 QD: one-channel => two channel crossover

Pustilnik,Glazman, PRL’01

Reminder:

Kondo Model Anderson Model

Hubbard-Stratonovich transformation

Schrieffer-Wolff transformation

2 channel Anderson H-an (infinite U)

Auxiliary particle representation of the Anderson Hamiltonian

Introduce auxiliary particles

+ constraint

Schwinger bosons

auxiliary fermions (holons)

Novel large-N approximation

T-matrix for one channel

and at finite voltage

for different temperatures

Conductance of a spin-1 QD: expectations (reminder)

One-channel Two-channel

Log correction due to USK!

Current through the dotFollow method proposed by Meir, Wingreen, PRL 1992

-Keldysh Green‘s functions

Current throught the dot

Single-channel contributions + interference term

Goal: calculation of the dot‘s Green‘s functions D

One-channel current

Note: the dot GF is proportional to the t-matrix of conduction electrons

Results: conductance – one channel

Linear conductance

Voltage-dependence of conductance

Results: conductance two-channel QD

Everything is messed up by the interchannel GFs => the current is not expressedIn terms of t-matrices of single channels

Simplifications: linear conductance – elastic scattering => results

1

2

3TK2/TK1

1 – 102 – 1003 – 1000

Comparison with NRG

Hofstetter, Schoeller, PRL 2002

NRG gives qualitatively same results

Conclusions and future work• We calculated analytically transport in a spin-1 QD in

case of one and two-channels• In case of one –channel singular conductance is obtained

– signature of US Kondo effect• In case of two-channels interference effects are observed

– conductance is suppressed at low T

Future projects

•Improve the results for two-channel conductance, taking into account the inelastic scattering terms

•Inclusion of magnetic field•Inclusion of spin – relaxation effects