where Jk,k′ represents the anti-ferromagnetic coupling which is given by:
Jkk′ = VkσVk∗′σ[ 1
ǫd− ǫk+ U + 1
ǫ′k− ǫd], (1.3)
where ǫk is energy level of conduction electrons, ǫd is energy level of quantum dot, skk′ is the spin of conduction electrons, and Sdis an impurity spin. Because the Kondo resonance lies exactly at the Fermi energy, the contribution of exchange process is only from states around Fermi energy(ǫF). So we set that ǫd ≡ ǫd−ǫF, ǫk ≡ ǫk−ǫF and ǫk ∼ 0. Since ǫd< 0 and the Coulomb potential U is larger than ǫd, Jkk′(∼ JkFkF) is the anti-ferromagnetic coupling(Jkk′ > 0). For U >> |ǫd|, the expression reduces to:
JkFkF ≡ J ∼ −|Vkσ2 |
ǫd = − |Vkσ2 | ǫd− ǫkF
> 0. (1.4)
The second term of Eq.(1.2) is precisely the s-d interaction term written as Hex = J(r) · S. It describes the spin exchange between an impurity and the surrounding conduction electrons. The distance r is measured from the impurity site to conduction electrons. This simple model explains the Kondo problem that resistivity of metals in magnetic impurity bulk system at low temperature will increase logarithmically. We will discuss Kondo effect in QD system in Section 1.1.2 and two-channel Kondo in Section 1.1.4.
1.2 Kondo Effect In Quantum Dot System
Due to the progress of science and technology, scientist can fabricate semiconductor struc-ture under nano-mater scale. The advance of micro-fabrication and cooling technology make a chance that we can research Kondo effect in nano-size system. Kondo physics can be realized in a tunable quantum dot (TQD). TQD is made by the single electron tunneling transistor (SET) with two dimensional electron gas (2DEG) heterostructure, as shown in Fig.(1.3) [1, 18, 21]. SET device has GaAs/AlGaAs layer and multiple elec-trodes, where three gate electrodes on left and the other one on right in the picture.
Then GaAs layer confines 2DEG repelled by electrodes, and induces two tunneling junc-tions under and above it. A metallic island is confined between two tunneling juncjunc-tions called ”quantum dot (QD)”. One of early experiments of Kondo effect in QD system was
1.2. KONDO EFFECT IN QUANTUM DOT SYSTEM
Figure 1.3: (a) Scanning electron micrograph of SET device. The top and down electrodes on the left side and the electrode on the right side are used to operate the barrier of quantum dot. The middle electrode on the left side are used to tune the energy level of QD relative to 2DEG [13, 14]. (b) Schematic SET device [21].
made by D. Gordhrber-Golen et.al in 1998 [13, 14, 17]. The conductance in QD systems is quantized by Coulomb blockade (CB) oscillations, leading to the difference of Kondo effect between QD systems and bulk systems.
1.2.1 Coulomb Blockade Oscillations With Kondo Effect
The original Kondo physics (in bulk system) was introduced in last section, where Kondo effect induce resistivity to be enhanced. This section, we will discuss the Kondo effect in QD system, which is very different from bulk system. In the single electron tunneling transistor (SET) device, the Coulomb blockade (CB) oscillations affects the conductivity [21, 22] as shown in Fig.(1.4). In Fig.(1.4.a), it exhibits conductance increasing as odd number in quantum dot (blue line), and there are peaks at VSD = 0 in Fig.(1.4.b). The peaks at VSD = 0 are temperature dependent, as temperature is lower, peak is higher.
These peaks are due to Kondo effect, so-call Kondo peaks. We will introduce CB and and briefly discuss the results in theory and experiment, then discuss Kondo effect in SET.
1.2. KONDO EFFECT IN QUANTUM DOT SYSTEM
Figure 1.4: (a) Conductance has two different behaviors at even and odd electron number.
(b) The zero bias differential conductance anomaly at VSD∼ 0, where VSD is the voltage difference between source and drain. [1, 13, 14].
Coulomb Blockade
Coulomb blockade oscillation appears due to strong Coulomb potential system, that’s why it called as Coulomb blockade. In the SET device, Coulomb potential affect the electron tunneling between leads and dot, illustrated in Fig.(1.5). The first order tunneling is blocked by the Coulomb blockade, where U is Coulomb potential between the electron of quantum dot and lead. The CB Hamiltonian related to constant interaction model was constructed as [21, 22]:
HDCB =X
µ
ξµc†µcµ+ E(N), E(N) = ECN2− eVgN (1.5)
, where E(N) is the interaction term of the system, including gate voltage term as eVgN, where gate voltage leads to a electric field that increases the energy of dot electrons. In general, we have to simultaneously think about source, drain, and gate voltage. Here we lump all terms in a gate voltage term. The energy Ec = 2Ce2 , where C is the capacitance of a single electron. The bias voltage leads to µL≥ µdot ≥ µR, and the bias voltage difference between the left lead (source) and the right lead (drain) was defined as VSD. Because of the inequality of chemical potential, electrons can flow, therefore an added electron from source excites dot energy from Edot(N − 1) to Edot(N), then an electron hops from dot to
1.2. KONDO EFFECT IN QUANTUM DOT SYSTEM
Figure 1.5: The illustration of Coulomb blockade. The first order tunneling is blocked by Coulomb potential U, the cotunneling (2nd) solution is solvable. The additional energy for N to N + 1 state is E2/C + ∆E. The voltage spacing between source and drain is defined as VSD which is eV here.
drain, leading dot energy from Edot(N) to Edot(N − 1). The tunneling occurs when
αeVg(N) = E(N + 1) − E(N), (1.6)
where α = Cg/Cis the ratio of gate capacitance to total capacitance, called ”gate cou-pling”. The αeVg(N) is similar to chemical potential of quantum dot, which is
eVg(N) ≡ µdot(N) = Edot(N) − Edot(N − 1) = (n − 1 2)e2
C − eVg+ EN, (1.7) the remain terms defined as EN. The additional energy is given by ∆µdot, where
∆µdot ≡ µdot(N) − µdot(N − 1) = e2
C + EN − EN −1 = e2
C − ∆E. (1.8)
1.2. KONDO EFFECT IN QUANTUM DOT SYSTEM
The irregular spacing of the single electron levels is defined as ∆E. When charging energy
e2
C is much larger than ∆E, CB oscillations is dependent on it. The peak spacing of CB as a function of gate voltage is given by
∆Vg = ∆µ(N)/eα = (e/c2+ ∆E)/eα, (1.9) while condition Eq.(1.6) gives the gate voltage of N-th Coulomb peak. We take differential of E(N) in Eq.(1.5) with respect N, we obtain the optimum number of particles,
Nopt= eVg/2Ec (1.10)
When optimum number is
• Integer: There is an energy gap for adding electrons.
• Half-integer: There are two degenerate charge states, then electrons can transit.
As shown in Fig.(1.4.a), it exhibits difference results indicated different number of electron on quantum dot, and Fig.(1.4.b) exhibits non-equilibrium differential conductance with anomaly behavior at VSD ∼ 0 [1]. The electron number can be changed by gate tuning. The CB oscillation affect the conductance by electron number, then we discuss how Kondo effect can overcome CB in QD.
Kondo Effect In Quantum Dot System
The tunable QD is sometimes similar to individual artificial magnetic impurity, which leads to Kondo screening for (T < Tk). When optimum number is odd, the QD with a single electron which is occupying the top-most quantum state, which is similar to a magnetic impurity. In other words, Kondo screening occurs with a single spin-degenerate energy state ǫd, no Kondo effect when optimum number is even. Kondo effect appears when (T < Tk), where the Kondo temperature Tk found to be:
TK = [UΓ]12e−π(µ−ǫd)2Γ , (1.11) where (ǫd) is dot level, (Γ) is a coupling between leads and dot, and chemical po-tential µ [5]. These parameters all influence Kondo temperature. The Kondo effect is