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formation

To realize a QD in the LMR we fix the gate voltage ²d such that

²d ¿ −|Γ| ¿µ¿ |Γ| ¿²d+U, (3.8)

4_{For completeness we mention all possible regimes of a single-level AM here: themixed valenceregime}

(MV) is characterized by large charge fluctuations, realized for²d∼µor (²d+U)∼µ. That regime where

the local level is either empty (²dÀµ) or doubly occupied (²d+U ¿µ) is called theempty orbitalregime

ensuring that the QD is singly occupied,hˆni ≈1. For this particular choice of²dthe empty and the doubly occupied states in the QD can be disregarded, since they are energetically highly unfavorable. Thus, the impurity is either occupied with an ⇑or ⇓electron (for the rest of the first part of this thesis we use the following notation: σ =⇑/⇓ corresponds to a spin σ electron in the impurity andσ =↑/↓ to a spinσ electron in the CB). Thus, the impurity spin can be described by a spin operator ˆS, ˆS≡ 1

2 P

µνd†µσµν dν.

Due to virtual excitations, see Fig. 3.3, there is still a small (but finite) probability for the QD to contain zero or two electrons, even though condition (3.8) is fulfilled. For

|²d| ÀΓ, these virtual excitations are appropriately described by second order perturbation theory in the tunneling between the local level and the CB-electrons. Due to the Pauli- principle, virtual processes are only possible between CB and impurity electrons with anti-parallel spin [34]. As these processes lower the energy of the system by an amount ∆E, the effective Hamiltonian - describing the low-energy properties of the system - should include a term that favors anti-parallel alignment between electron spin in the QD and the CB.

There are two types of virtual processes,hole-like (excitations to an unoccupied local level; lower middle panel in Fig. 3.3) and electron-like (excitations to a double occupied local level; upper middle panel in Fig. 3.3) processes. A hole (electron)-like process lowers5 _the energy of the system [35] by ∆Eh (∆Ee)

∆Eh(²k) = V 2_[1−f(² k)] ²k−²d (3.9) ∆Ee(²k) = V2_[f(² k)] U+²d−²k , (3.10)

with the Fermi function of the leads f(²k) = 1/(1 +e(²k−µ)/kBT).

Fig. 3.3 sketches all relevant virtual processes. The impurity spin might: (i) remain unchanged (either with σ =⇑ orσ =⇓), (ii) flip its spin from ⇑→⇓ (in two possible ways - corresponding to a transition from the left side to the right side in Fig. 3.3) or (iii) flip its spin oppositely from ⇓→⇑(also in two possible ways - corresponding to a transition from the right side to the left side in Fig. 3.3). As the total spin is conserved, the CB electron spin has to flip oppositely to the impurity spin [in the processes (ii) and (iii)] or remain unchanged [in process (i)]. A compact representation of all those processes is realized by rewriting them with spin operators. Consequently, a QD in the LMR shall contain a term ˆ S·ˆs0 = £ Sz_sz 0+12 ¡ S+_s− 0 +S−s+0 ¢¤

with the QD spin operator ˆS, the CB spin operator ˆs0, ˆ s0 ≡ 1₂ P kk0 P µνα †

skµσµναsk0_ν [with α_skσ as defined in Eq. (3.6)] and the usual definition of the ladder operatorsS+/− _ands+/−

0 , respectively. A spin-flip event between the left and the right ground state shown in Fig. 3.3, for example, is described by the operator S−_s+

0. Due to the Fermi functions, only states ²k > µ [²k ≤ µ] contribute in Eq. (3.9) [Eq. (3.10)]. As we restricted ourselves to ²d ¿ µ ¿ ²d +U, see Eq. (3.8), we can approximate the denominators in Eqs. (3.9) and (3.10) by their smallest possible values,

5_{Virtual excitations lower the energy by an amount ∆E≈} V2 Eexc−Eini.

ε

ε

+U

ε

+U

ε

Figure 3.3: Sketch of a QD in the LMR with its two possible ground states: the electron in the QD might either be a spin ⇑ (left side) or a spin ⇓ electron (right side). In the middle of the figure the possible virtual excitations (obeying the Pauli principle), due to hybridization between impurity and lead electrons, are shown. The lower middle panel shows a hole-like excitation, realized by a CB hole that tunnels into the QD. The upper panel in the middle shows an electron-like excitation, where a CB electron tunnels into the QD. The energy gain due to these processes is described in Eq. (3.9) and (3.10). As the left and the right state in the figure are connected by virtual transitions the possibility of a spin-flip inside the QD exists!

i.e. ²k,k>µ−²d≈µ−²d and U +²d−²k,k≤µ≈U +²d−µ. Thus, virtual processes lower the systems energy by an amount

J = V 2 µ−²d + V 2 U +²d−µ . (3.11)

Summing up all possible virtual transitions sketched in Fig. 3.3, we arrive at an effective

Hamiltonian for the LMR.

ˆ

Heff = ˆH`+ 2JSˆ·ˆs0, (3.12)

with the local (Heisenberg) couplingJ between the impurity spin and the conduction elec- tron spins.

This heuristic arguments illustrated the general concept of a Schrieffer-Wolff (SW) transfor- mation. For a detailed, more rigorous discussion, see Ref. [30] or [36]. A SW transformation enables one to project a general Hamiltonian (which is the single-level AM here) into a spe- cial subspace of its full Hilbert space (the LMR). The corresponding effective Hamiltonian, Eq. (3.12), describing the LMR is known as the Kondo Hamiltonian ˆHeff = ˆHK.

In this Section we argued, based on a perturbation expansion in the tunneling, how the impurity can lower its energy by means of virtual transitions. In particular processes where the impurity spin flips will become of great interest below, when the Kondo effect will be discussed.

3.2

The Kondo effect

The non-trivial physics associated with the presence of magnetic impurities in a solid is referred to as the Kondo effect. The experimental discovery of a shallow minimum in the resistivity of metals that contain magnetic impurities (at temperatures T ∼ 10K) triggered big interest in this field. Kondo was able to relate this phenomenon to spin-flip scattering events. Kondo could show that this scattering mechanism becomes more and more dominant when the temperature of the system is lowered successively.