Slide 1 by Oi4S6F

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									 Coulomb Blockade and
Non-Fermi-Liquid Behavior
 in a Double-Dot Device

          Avraham Schiller
       Racah Institute of Physics

  Eran Lebanon         (Rutgers University)
  Frithjof B. Anders   (Bremen University)


        Special Thanks to Yuval Oreg
Outline:
  The single-channel Kondo effect

 The Kondo effect in ultra-small quantum dots

 The two-channel Kondo effect

 The Coulomb blockade in quantum boxes

  Two-channel Kondo physics in a lead-dot-box device

      From spin to charge two-channel Kondo effects
      Entanglement of spin and charge
      Discontinuity in the conductance

  Conclusions
Resistivity minimum: The Kondo effect
    Fe in Cu




       T(K)                         T(K)

Franck et al., 1961    De Haas & ven den Berg, 1936



Tmin  c       1/5
               imp
                      Enhanced scattering at low T
The Kondo Effect: Impurity moment in a metal




   A nonperturbative energy scale emerges       TK  exp  1 / J 
   Below TK impurity spin is progressively screened

   All initial AFM couplings flow to a single strong-coupling fixed point

   A sharp resonance is formed at the Fermi energy for T<TK
Local-moment formation: The Anderson model




                V                         ed + U
|ed|




H imp  e d  n  Un  n  hybridization with
           
                             conduction electrons
    The Anderson model: spectral properties

Kondo resonance




              ed           EF           ed+U

    A sharp resonance of width TK develops at EF for T<TK
    Unitary scattering for T=0 and <n>=1
Ultra-small quantum dots as artificial atoms


        VL                        VR
                 U

                            e
                            d




   Lead              Q.D.           Lead
   Electrostatically-defined semiconductor quantum dots

                Goldhaber-Gordon et al., Nature 1998




                             Plunger
                              gate




                                               Quantum dot
Temperature
 depedence       Field dependence
                       Two-channel Kondo effect
                                                             
          Η 
                 
                   e   
                   1, 2 k ,
                                
                                k
                                    
                                    c
                                    
                                    k 
                                           ck
                                                   JSimp  s (0)
                                                    1, 2
Extra channel
    index




                                    T 0          r 

                Impurity spin is overscreened by two identical channels

                A non-Fermi-liquid fixed point is approached for T<<TK
         One- versus two-channel Kondo effect


Property     One channel    Two channel      Non-Fermi-liquid

                              1
S (T  0)        0              log( 2)      Residual entropy
                              2

  C /T          1 / TK     log( TK / T )   Diverging coefficient g



 (T  0)      1 / TK      log( TK / T )   Diverging susceptibility
       Requirements for the realization
       of the two-channel Kondo effect

        Two independent conduction bands


        Equal coupling strength to the two bands


        No scattering of electrons between the bands


        No applied magnetic field acting on the impurity spin


Is realization of the two-channel Kondo effect at all possible?
   The Coulomb blockade in quantum box


Quantum box: Small metallic grain or large semiconductor
                quantum dot with sizeable Charging energy
                EC but dense single-particle levels D




Charging energy:
                   2                                 2
              Q                                 e
      E (Q)       VBQ                   EC 
              2C0                              2C0
                          Energy for charging box with one electron
Charging of a quantum box
Two-channel Kondo effect in the charge sector
                                                                       (Matveev ‘91)



                                                      Focus on EC>>kBT and on
                                                      vicinity of a degeneracy point

                                                      Introduce the charge isospin
                                                     2 z  N  1 N 1  N N
                                                               N 1 N



 H          e k ck ck 
                     
                                            
                                     t ckL cqB   cqB ckL   z eV
                                                      
                                                                          
       L , B k ,
                                 k , q ,


              Channel index                      Lowering and raising isospin operators
       Two-channel Kondo dictionary for the
           Charging of a quantum box

Two-channel Kondo               Charging of a quantum box


Spin index                     Isospin index               

Channel index                  Physical spin               

Exchange interaction       J   Tunneling matrix element    2t

Magnetic Field             H    Deviation from deg. point   eV

Bandwidth                  D    Charging energy             EC

Diverging susceptibility       Diverging capacitance       C
Spin two-channel Kondo effect in a lead-dot-box device
                                    (Oreg and Goldhaber-Gordon ‘03)


  In an ordinary two-lead device:
  Inter-lead spin-exchange spoils
   the two-channel Kondo effect
Spin two-channel Kondo effect in a lead-dot-box device
                                    (Oreg and Goldhaber-Gordon ‘03)


  In an ordinary two-lead device:
  Inter-lead spin-exchange spoils
   the two-channel Kondo effect
Spin two-channel Kondo effect in a lead-dot-box device
                                    (Oreg and Goldhaber-Gordon ‘03)


  In an ordinary two-lead device:
  Inter-lead spin-exchange spoils
   the two-channel Kondo effect




    Inter-lead spin-exchange is
                                                     Quantum
     blocked in a lead-dot-box
                                                       box
       device, for kBT < EC !
       Quantum                            Quantum
         box                                box




       Quantum                            Quantum
         box                                box




       Quantum                            Quantum
         box                                box




Tunneling is blocked by the Coulomb blockade
          A second screening channel is dynamically generated
              for temperatures below the charging energy


             A spin two-channel Kondo effect should develop
                   if JBox and JLead are tuned to be equal




Note: The above scenario assumes the formation of a stable local
      moment on the dot, and quantized charge on the box !



    Our goal:    A detailed quantitative theory of this scenario

                 Extension to regimes with charge fluctuations
 Lead—Quantum dot—Quantum box setting
          (Courtesy of D. Goldhaber-Gordon)




                                              Leads


Quantum
  box


                    Quantum dot
                                    The model


H     e   
                     c 
                           c
                    k  k  k 
                                                   
                                                                    
                                      e d d d  Un d  nd   EC N Box  N B
                                                                     ˆ                2

     
      L , B k ,                          

                                      
                                                 
                                   t ck d  d ck   
                                  L, B




Hybridization widths:     t
                                               2
 Method of solution:   Wilson’s numerical renormalization group

                                         (E. Lebanon, AS, F.B. Anders, PRB 2003)




   Strategy:     Fix L and tune B in search of a two-channel Kondo effect




                                                        ( B g )2
Hallmark of spin two-channel Kondo effect:      (T )            ln(TK / T )
                                                        20k BTK



                                             Definition of TK
Symmetric dot: 2ed + U = 0                   L  EC  0.1D


                                                  NB  0




  Two-channel point is found for any U, including U = 0

  Spin two-channel effect persists in the mixed-valent regime
    TK versus U for a symmetric dot


                                               Analytic estimate
                                              for stable moment




                                               Exponentially
                                                  small




TK is significantly enhanced in the mixed-valent regime
Can become of the order of the charging energy EC
        Dependence of TK on the gate voltage NB for U = 0




                                                                     Perfect
                                                                 Transmission
                                                                 for B  B2CK




Prediction of bosonization treatment near perfect transmission

  Spin 2-channel Kondo effect related to perfect transmission through dot
            Two-channel line and charging curve
              for ``realistically large’’ U/EC = 20

2-channel
             E
   line                                               L  EC  0.1D
              D
                  C B
                            A                               NB  0




                                                      Shift in Coulomb
                                                          staircase
      Origin of shift in Coulomb staircase

   Note: shift in staircase occurs for       B  L , EC

   Diagonalize first the link between the dot and box



  Lead                            Dot                  Box

                             tL         tB

                     1        1           1
        local
                                       
                       d  c1  c1 d        
                                               d d
                     2          2            2

Site immediately coupled to the box is only half occupied
Entanglement of spin and charge within the two-channel Kondo effect



                 ˆ
           e2 d N
 C (T ) 
          2 EC dN B




                                                                           2ed/U
      NB  0




Both magnetic susceptibility and charge capacitance diverge logarithmically,
but with different Kondo scales (i.e., slopes)

  Continuous transition from spin to charge 2-channel Kondo effect
           Zero-temperature conductance


   L  EC  0.1D               NB  0           U D



                V




Discontinuous jump in the conductance across the two-channel point
Scaling of the conductance with distance from critical point

         B  B CK
               2
                                              U  2e d  2 D

                                               L  EC  0.1D




                                                B  B CK
                                                      2
      Conclusions
Quantum-dot devices offer a unique opportunity
to study the two-channel Kondo effect.

Exploiting the Coulomb blockade, one can measure
the two-channel Kondo effect in a double-dot device.

Among the exotic features found:
   A continuous transition from a spin to a charge
   two-channel Kondo effect.
   Entanglement of spin and charge.

   A discontinuity in the T = 0 conductance.
   Enhancement of the Kondo temperature away
   from the local-moment regime.

								
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