Transcript Conductance of a spin-1 QD: signatures of the two

Conductance of a spin-1 QD:
two-stage Kondo effect
Anna Posazhennikova
Institut für Theoretische Festkörperphysik,
Uni Karlsruhe, Germany
Collaborators: Babak Bayani (TFP, University Karlsruhe, Germany)
Piers Coleman (Rutgers University, NJ, USA)
Les Houches, June 19, 2006
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
H  Ed  d d  Un n

Possible states
History of Kondo effect
Q2: why does the formation of LM lead to ρmin ?
• 1964 – Jun Kondo, Hamiltonian


H Kondo  H 0  J AFM s 0 S
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
l0
History of Kondo effect
Q4: what happens in more realistic situation with
l0
•1980 Blandin, Nozieres
•1984 Andrei, Tsvelik, Wiegman
Perfectly Screened KE
FL
Underscreened KE
Overscreened KE
NFL
NFL
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
Kondo Hamiltonian
diagonalization
J1, J2 – two coupling constants =>
two screening channels
Conductance of a spin-1 QD:
one-channel => two channel crossover
Pustilnik,
Glazman, PRL’01
Reminder:
Hubbard-Stratonovich transformation
Kondo Model
Anderson Model
Schrieffer-Wolff transformation
2 channel Anderson H-an (infinite U)
Auxiliary particle representation of the Anderson
Hamiltonian
Introduce auxiliary particles
Schwinger bosons
+ constraint
auxiliary fermions (holons)
Novel large-N approximation
T-matrix for one channel
for different temperatures
and at finite voltage
Conductance of a spin-1 QD: expectations (reminder)
One-channel
Log correction due to USK!
Two-channel
Current through the dot
Follow 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 expressed
In terms of t-matrices of single channels
Simplifications: linear conductance – elastic scattering => results
TK2/TK1
3
2
1
1 – 10
2 – 100
3 – 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