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

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G0 Low-Temperature Conduction of a Quantum Dot 125 1 dI dV ∝ � �−2 max {T,B,eV } ln , (91) characteristic for the weak coupling regime of the Kondo system. Consider now a zero-temperature transport through a quantum dot with S =1/2 inthe presence of a strong field B ≫ TK. In accordance with (91), the differential conductance is small compared to G0 both for eV ≪ B and for eV ≫ B. However, the calculation in the third order of perturbation theory in the exchange constant yields a contribution that diverges logarithmically at eV = B [47]. The divergence arises because of the partial restoration of the coherence associated with the formation of the Kondo singlet: at eV = B the scattered electron has just the right amount of energy to allow for a real transition with a flip of spin. However, the full development of resonance is inhibited by a finite lifetime of the excited spin state of the dot [53, 75]. As a result, the peak in the differential conductance at eV ∼ B is broader and lower [53] then the corresponding peak at zero bias in the absence of the field. Even though for eV ∼ B ≫ TK the system is clearly in the weak coupling regime, a resummation of the perturbation series turns out to be a prohibitively difficult task, and the expression for the shape of the peak is still unknown. This problem remains to be a subject of active research, see e.g. [76] and references therein. One encounters similar difficulties in studies of the effect of a weak ac signal of frequency Ω � TK applied to the gate electrode [77] on transport across the dot. In close analogy with the usual photon-assisted tunneling [78], such perturbation is expected to result in the formation of satellites [79] at eV = n�Ω (here n is an integer) to the zero-bias peak in the differential conductance. Again, the restoration of coherence responsible for the formation of the satellite peaks is limited by the finite lifetime effects [80]. The spin degeneracy is not the only possible source of the Kondo effect in quantum dots. Consider, for example, a large dot connected by a singlemode junction to a conducting lead and tuned to the vicinity of the Coulomb blockade peak [28]. If one neglects the finite level spacing in the dot, then the two almost degenerate charge state of the dot can be labeled by a pseudospin, while real spin plays the part of the channel index [28, 81]. This setup turns out to be a robust realization [28, 81] of the symmetric (i.e. having equal exchange constants) two-channel S =1/2 Kondo model [54]. The model results in a peculiar temperature dependence of the observable quantities, which at low temperatures follow power laws with manifestly non-Fermi-liquid fractional values of the exponents [82]. It should be emphasized that in the usual geometry consisting of two leads attached to a Coulomb-blockaded quantum dot with S = 1/2, only the conventional Fermi-liquid behavior can be observed at low temperatures. Indeed, in this case the two exchange constants in the effective exchange Hamiltonian (52) are vastly different, and their ratio is not tunable by conventional means, see the discussion in Sect. 5.1 above. A way around this TK
126 M. Pustilnik and L.I. Glazman difficulty was proposed recently in [83]. The key idea is to replace one of the leads in the standard configuration by a very large quantum dot, characterized by a level spacing δE ′ and a charging energy E ′ C .AtT ≫ δE′ , particle-hole excitations within this dot are allowed, and electrons in it participate in the screening of the smaller dot’s spin. At the same time, as long as T ≪ E ′ C ,the number of electrons in the large dot is fixed. Therefore, the large dot provides for a separate screening channel which does not mix with that supplied by the remaining lead. In this system, the two exchange constants are controlled by the conductances of the dot-lead and dot-dot junctions. A strategy for tuning the device parameters to the critical point characterized by the two-channel Kondo physics is discussed in [84]. Finally, we should mention that the description based on the universal Hamiltonian (15) is not applicable to large quantum dots subjected to a quantizing magnetic field H⊥ [85, 86]. Such field changes drastically the way the screening occurs in a confined droplet of a two-dimensional electron gas [87]. The droplet is divided into alternating domains containing compressible and incompressible electron liquids. In the metal-like compressible regions, the screening is almost perfect. On the contrary, the incompressible regions behave very much like insulators. In the case of lateral quantum dots, a large compressible domain is formed near the center of the dot. The domain is surrounded by a narrow incompressible region separating it from another compressible ring-shaped domain formed along the edges of the dot [88]. This system can be viewed as two concentric capacitively coupled quantum “dots” – the core dot and the edge dot [85, 88]. When the leads are attached to the edge dot, the measured conductance is sensitive to its spin state: when the number of electrons in the edge dot is odd, the conductance becomes large due to the Kondo effect [85]. Changing the field causes redistribution of electrons between the core and the edge, resulting in a striking checkerboard-like pattern of high- and low-conductance regions [85, 86]. This behavior persists as long as the Zeeman energy remains small compared to the Kondo temperature. Note that compressible regions are also formed around an antidot –apotential hill in a two-dimensional electron gas in the quantum Hall regime [89]. Both Coulomb blockade oscillations and Kondo-like behavior were observed in these systems [90]. 7 Summary Kondo effect arises whenever a coupling to a Fermi gas induces transitions within otherwise degenerate ground state multiplet of an interacting system. Both coupling to a Fermi gas and interactions are naturally present in a nanoscale transport experiment. At the same time, many nanostructures can be easily tuned to the vicinity of a degeneracy point. This is why the Kondo effect in its various forms often influences the low temperature transport in meso- and nanoscale systems.
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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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4.5 QPC Versus SET Semiconductor Fe
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