Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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�Heff = � ks JD−δD = JD + νJ 2 D Low-Temperature Conduction of a Quantum Dot 121 ξ k ψ † 1ks ψ 1ks + JD−δD(s1 · S) , |ξk|
122 M. Pustilnik and L.I. Glazman dot and the leads. The maximal possible conductance in this case is 2e 2 /h. To obtain a more precise bound, we discuss in this section the conductance in the strong coupling regime T ≪ TK. We start with the zero-temperature limit T =0.Asdiscussedabove,the ground state of the Kondo model (62) is a singlet [56], and, accordingly, is not degenerate. Therefore, the t-matrix of the conduction electrons interacting with the localized spin is completely characterized by the scattering phase shifts δs for electrons with spin s at the Fermi level. The t-matrix is then given by the standard scattering theory expression [63] − πν Ts(0) = 1 2i (Ss − 1) , Ss =e 2iδs , (82) where Ss is the scattering matrix for electrons with spin s, which for a single channel case reduces to its eignevalue. Substitution of (82) into(71) yields 1 G(0) = G0 2 � s sin 2 δs (83) for the conductance, see (71). The phase shifts in (82), (83) are obviously defined only mod π (that is, δs and δs + π are equivalent). This ambiguity can be removed if we set to zero the values of the phase shifts at J =0in(62). In order to find the two phase shifts δs, we need two independent relations. The first one follows from the invariance of the Kondo Hamiltonian (62) under the particle-hole transformation ψks → sψ † −k,−s (here s = ±1 for spinup/down electrons). The particle-hole symmetry implies the relation for the t-matrix Ts(ω) =−T ∗ −s(−ω) , (84) valid at all ω and T .Inviewof(82), it translates into the relation for the phase shifts at the Fermi level (ω =0)[66], δ↑ + δ↓ =0. (85) The second relation follows from the requirement that the ground state of the Hamiltonian (62) is a singlet [66]. In the absence of exchange (J =0) and at T = 0, an infinitesimally weak (B → +0) magnetic field acting on the dot’s spin, HB = BS z , (86) would polarize it; here B is the Zeeman energy. Since free electron gas has zero spin in the ground state, the total spin in a very large but finite region of space V surrounding the dot coincides with the spin of the dot, 〈S z 〉J=0 = −1/2. If the exchange with electron gas is now turned on, J>0, a very weak field will not prevent the formation of a singlet ground state. In this state, the total spin within V is zero. Such change of the spin is possible if the numbers of spin-up and spin-down electrons within V have changed to compensate for the
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Lecture Notes in Physics Editorial
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W. Dieter Heiss (Ed.) Quantum Dots:
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Preface Nanoscale physics, nowadays
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Contents A Guide for the Reader ...
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A Guide for the Reader Quantum dots
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The Renormalization Group Approach
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RG for Interacting Fermions 5 where
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where � is a shorthand: � ≡ 3
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RG for Interacting Fermions 9 These
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RG for Interacting Fermions 11 is i
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RG for Interacting Fermions 13 the
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RG for Interacting Fermions 15 this
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RG for Interacting Fermions 17 equa
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� p � dθp = g 2π RG for Inter
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RG for Interacting Fermions 21 the
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References RG for Interacting Fermi
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Semiconductor Few-Electron Quantum
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1.2 Implementations Semiconductor F
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GaAs n-AlGaAs AlGaAs GaAs Semicondu
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ε1 ε0 e Semiconductor Few-Electro
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250 Ω twisted pair 20 MΩ powder
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10 5 0 -5 Semiconductor Few-Electro
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G ( e / h) 2 2 1 0 Semiconductor Fe
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V R (V) V R (V) Semiconductor Few-E
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V L (V) -1.084 -1.082 -1.080 -1.078
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B // Semiconductor Few-Electron Qua
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a RESERVOIR 200 nm Semiconductor Fe
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172 C.W.J. Beenakker (τ−,τ+), w
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Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

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