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