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

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Low-Temperature Conduction of a Quantum Dot 117 To get an idea about the physics of the Kondo model (see [55] for recent reviews), let us first replace the fermion field operator s1 in (62) by a singleparticle spin-1/2 operator S1. The ground state of the resulting Hamiltonian of two spins �H = J(S1 · S) is obviously a singlet. The excited state (a triplet) is separated from the ground state by the energy gap J1. This separation can be interpreted as the binding energy of the singlet. Unlike S1 in this simple example, the operator s1 in (62) is merely a spin density of the conduction electrons at the site of the “magnetic impurity”. Because conduction electrons are freely moving in space, it is hard for the impurity to “capture” an electron and form a singlet. Yet, even a weak local exchange interaction suffices to form a singlet ground state [56, 57]. However, the characteristic energy (an analogue of the binding energy) for this singlet is given not by the exchange constant J, but by the so-called Kondo temperature TK ∼ ∆ exp(−1/νJ) . (64) Using ∆ ∼ δE and (56) and (22), one obtains from (64) the estimate � � δE ln ∼ TK 1 νJ ∼ e2 /h EC . (65) GL + GR δE Since Gα ≪ e 2 /h and EC ≫ δE, the r.h.s. of (65) is a product of two large parameters. Therefore, the Kondo temperature TK is small compared to the mean level spacing, TK ≪ δE . (66) It is this separation of the energy scales that justifies the use of the effective low-energy Hamiltonian (51), (52) for the description of the Kondo effect in a quantum dot system. The inequality (66) remains valid even if the conductances of the dot-leads junctions Gα are of the order of 2e 2 /h. However, in this case the estimate (65) is no longer applicable [58]. In our model, see (61)–(63), one of the channels (ψ2) of conduction electrons completely decouples from the dot, while the ψ1-particles are described by the standard single-channel antiferromagnetic Kondo model [5, 55]. Therefore, the thermodynamic properties of a quantum dot in the Kondo regime are identical to those of the conventional Kondo problem for a single magnetic impurity in a bulk metal; thermodynamics of the latter model is fully studied [59]. However, all the experiments addressing the Kondo effect in quantum dots test their transport properties rather then thermodynamics. The electron current operator is not diagonal in the (ψ1,ψ2) representation, and the contributions of these two sub-systems to the conductance are not additive. Below we relate the linear conductance and, in some special case, the non-linear differential conductance as well, to the t-matrix of the conventional Kondo problem.
118 M. Pustilnik and L.I. Glazman 5.2 Linear Response The linear conductance can be calculated from the Kubo formula � ∞ 1 G = lim dt e ω→0 � ω iωt �� Î(t), Î(0) �� , (67) where the current operator Î is given by 0 Î = d e � ˆNR − dt 2 ˆ � NL , Nα ˆ = � ks c † αks c αks (68) Here ˆ Nα is the operator of the total number of electrons in the lead α. Evaluation of the linear conductance proceeds similarly to the calculation of the impurity contribution to the resistivity of dilute magnetic alloys (see, e.g., [60]). In order to take the full advantage of the decomposition (61)–(63), we rewrite Î in terms of the operators ψ1,2. These operators are related to the original operators cR,L representing the electrons in the right and left leads via � � � �� ψ1ks cos θ0 sin θ0 cRks = . (69) ψ2ks − sin θ0 cos θ0 cLks The rotation matrix here is the same one that diagonalizes matrix ˆ J of the exchange amplitudes in (51); the rotation angle θ0 satisfies the equation tan θ0 = |tL0/tR0|. With the help of (69) we obtain ˆNR − ˆ � NL = cos(2θ0) ˆN1 − ˆ � N2 − sin(2θ0) � ks � ψ † 1ks ψ 2ks +H.c. � (70) The current operator Î entering the Kubo formula (67) is to be calculated with the equilibrium Hamiltonian (61)–(63). Since both ˆ N1 and ˆ N2 commute with Heff, the first term in (70) makes no contribution to Î. When the second term in (70) is substituted into (68) and then into the Kubo formula (67), the result, after integration by parts, can be expressed via 2-particle correlation functions such as 〈ψ † 1 (t)ψ2 (t)ψ† 2 (0)ψ1 (0)〉 (see Appendix B of [61] for further details about this calculation). Due to the block-diagonal structure of Heff, see (61), these correlation functions factorize into products of the single-particle correlation functions describing the (free) ψ2-particles and the (interacting) ψ1-particles. The result of the evaluation of the Kubo formula can then be written as � � G = G0 dω − df � 1 � [−πν Im Ts(ω)] . (71) dω 2 Here G0 = 2e2 h sin2 (2θ0) = 2e2 h s 4|t 2 L0 t2 R0 | (|t2 L0 | + |t2 , (72) R0 |)2
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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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