Since the first observation of the Kondo effect in semiconductor quantum dots 123, various aspects of Kondo physics have been revealed, owing to the artificial tunability and flexibility of the systems, e.g., an enhanced Kondo effect with an even number of electrons at the spin-singlet-triplet degeneracy 4, the SU(4) Kondo effect with S = 1/2 and orbital degeneracy 5, and the bonding and antibonding states between the Kondo resonant levels in coupled quantum dots 67. One of the major issues which still remain unsolved in the Kondo physics is the observation of the Kondo singlet state, so-called Kondo screening cloud. The size of the screening cloud is evaluated as L_K = ħv_F/T_K, where v_F is the Fermi velocity and T_K is the Kondo temperature. There have been several theoretical works on L_K, e.g., ring-size dependence of the persistent current in an isolated ring with an embedded quantum dot 8, Friedel oscillation around a magnetic impurity in metal 9, and spin-spin correlation function 1011.

We focus on the Kondo effect in a quantum dot embedded in an Aharonov-Bohm (AB) ring. In this system, the conductance shows an asymmetric resonance as a function of energy level in the quantum dot, so-called Fano-Kondo effect. This is due to the coexistence of one-body interference effect and many-body Kondo effect, which was studied by the equation-of-motion method with the Green function 12, the numerical renormalization group method 13, the Bethe ansatz 14, the density-matrix renormalization group method 15, etc. This Fano-Kondo resonance was observed experimentally 16. The interference effect on the value of T_K, however, has not been fully understood 1718.

In our previous work 19, we examined this problem in the small limit of AB ring using the scaling method 20. Our theoretical method is as follows. First, we create an equivalent model in which a quantum dot is coupled to a single lead. The AB interference effect is involved in the flux-dependent density of states in the lead. Second, the two-stage scaling method is applied to the reduced model, to renormalize the energy level in the quantum dot by taking into account the charge fluctuation and evaluate T_K by taking spin fluctuation 21. This method yields T_K in an analytical form.

The purpose of this article is to derive an analytical expression of T_K for the finite size of the AB ring, using our theoretical method. We find two characteristic lengths. One is the screening length of the charge fluctuation, Lc = ℏvF∕|ε̃0| with ε̃0 being the renormalized energy level in the quantum dot, which appears in the first stage of the scaling. The other is the size of Kondo screening cloud, L_K, which is naturally obtained in the second stage. In consequence, the analytical expression of T_K is different for situations (i) L_c ≪ L_K ≪ L, (ii) L_c ≪ L ≪ L_K, and (iii) L ≪ L_c ≪ L_K, where L is the size of the ring. We show that T_K strongly depends on the magnetic flux Φ penetrating the AB ring in cases (ii) and (iii), whereas it hardly depends on Φ in case (i).

Our model is shown in Figure 1a. A quantum dot with an energy level ε₀ is connected to two external leads by tunnel couplings, V_L and V_R. Another arm of the AB ring (reference arm) and external leads are represented by a one-dimensional tight-binding model with transfer integral -t and lattice constant a. The size of the ring is given by L = (2l + 1)a. The reference arm includes a tunnel barrier with transmission probability of T_b = 4x/(1 + x)² with x = (W/t)². The AB phase is denoted by ϕ = 2πΦ/Φ₀, with flux quantum Φ₀ = h/e. The Hamiltonian is

H ( 0 ) = H dot + H leads + ring + H T ,

H dot = ∑ σ ε 0 d σ † d σ + U n ^ ↑ n ^ ↓ ,

H leads + ring = ∑ i ≠ 0 ∑ σ ( - t a i + 1 , σ † a i , σ + h . c . ) + ∑ σ ( W e i ϕ a 1 , σ † a 0 , σ + h . c . ) ,

H T = ∑ σ ( V L d σ † a - l , σ + V R d σ † a l + 1 , σ + h . c . ) ,

where d σ † and d_σ are creation and annihilation operators, respectively, of an electron in the quantum dot with spin σ. a i , σ † and a_i,σ are those at site i with spin σ in the leads and the reference arm of the ring. n ^ σ = d σ † d σ is the number operator in the dot with spin σ. U is the charging energy in the dot.

We consider the Coulomb blockade regime with one electron in the dot, -ε₀, ε₀ + U ≫ Γ, where Γ = Γ_L + Γ is the level broadening. Γ α = π ν 0 V α 2 , with ν₀ being the local density of states at the end of semi-infinite leads. We analyze the vicinity of the electron-hole symmetry of -ε₀ ≈ ε₀ + U.

We create an equivalent model to the Hamiltonian (1), following Ref. 19. First, we diagonalize the Hamiltonian H_leads+ring for the outer region of the quantum dot. There are two eigenstates for a given wavenumber k; |ψk,→〉 represents an incident wave from the left and partly reflected to the left and partly transmitted to the right, whereas |ψ_k,←〉 represents an incident wave from the right and partly reflected to the right and partly transmitted to the left. Next, we perform a unitary transformation for these eigenstates

( ∣ ψ ° k 〉   ∣ ψ k 〉 ) = ( ∣ ψ k , → 〉   ∣ ψ k , ← 〉 )   ( A k B k * B k − A k * ) ,

where A_k and B_k are determined so that 〈 d | H T | ψ ̄ k 〉 = 0 with dot state |d〉. In consequence, mode |ψ_k〉 is coupled to the dot via H_T, whereas ∣ ψ ° k 〉 is completely decoupled.

Neglecting the decoupled mode, we obtain the equivalent model in which a quantum dot is coupled to a single lead. In a wide-band limit, the Hamiltonian is written as

H = ∑ σ ε 0 d σ † d σ + U n ^ ↑ n ^ ↓ + ∑ k , σ ε k a k , σ † a k , σ + ∑ k , σ V ( d σ † a k , σ + h . c . ) ,

with V = V L 2 + V R 2 and density of states in the lead

ν ( ε k ) = ν 0 1 - R b cos ε k + D 0 ε T - P ( ϕ ) sin ε k + D 0 ε T .

Here, D₀ is the half of the band width, k_F is the Fermi wavenumber, R_b = 1 - T_b, and

P ( ϕ ) = α T b ( 1 - T b ) cos ϕ ,

where α = 4Γ_LΓ_R/(Γ_L + Γ_R)² is the asymmetric factor for the tunnel couplings of quantum dot.

The AB interference effect is involved in the flux-dependent density of states in the lead, υ(ε_k) in Eq. (6). As schematically shown in Figure 1(b), υ(ε_k) oscillates with the period of ε_T, where ε_T = ħv_F/L is the Thouless energy for the ballistic systems. We assume that ε_T ≪ D₀.

We apply the two-stage scaling method to the reduced model. In the first stage, we consider the charge fluctuation at energies of D ≫ |ε₀|. In this region, the number of electrons in the quantum dot is 0, 1, or 2. We reduce the energy scale from bandwidth D₀ to D₁ where the charge fluctuation is quenched. By integrating out the excitations in the energy range of D₁ < D < D₀, we renormalize the energy level in the quantum dot ε₀. In the second stage of scaling, we consider the spin fluctuation at low energies of D < D₁. We make the Kondo Hamiltonian and evaluate the Kondo temperature.

In the first stage, the charge fluctuation is taken into account. We denote E₀, E₁, and E₂ for the energies of the empty state, singly occupied state, and doubly occupied state in the quantum dot, respectively. Then the energy levels in the quantum dot are given by ε₀ = E₁ - E₀ for the first electron and ε₁ = E₂ - E₁ for the second electron. When the bandwidth is reduced from D to D - |dD|, E₀, E₁, and E₂ are renormalized to E₀ + dE₀, E₁ + dE₁, and E₂ + dE₂, where

d E 0 = - 2 V 2 ν ( - D ) D + E 1 - E 0 | d D | , d E 1 = - V 2 ν ( D ) D + E 0 - E 1 + V 2 ν ( - D ) D + E 2 - E 1 | d D | , d E 2 = - 2 V 2 ν ( D ) D + E 1 - E 2 | d D | ,

within the second-order perturbation with respect to tunnel coupling V. For D ≫ |E₁ - E₀|, |E₂ - E₁|, they yield the scaling equations for the energy levels

d ε i d ln D = 2 ν 0 V 2 f ( k F L , ϕ ) sin D ε T ,

where i = 0, 1 and

f ( k F L , ϕ ) = R b sin k F L - P ( ϕ ) cos k F L .

By the integration of the scaling equation from D₀ to D 1 ≃ | ε ̃ 0 | , we renormalize the energy levels in the quantum dot ε_i to ε ̃ 1 :

ε ̃ i - ε i ≃ 2 ν 0 V 2 f ( k F L , ϕ ) Si | ε 0 | ε T - Si D 0 ε T ,

where

Si ( x ) = ∫ 0 x sin ξ ξ d ξ .

Si(x) goes to 0 as x → 0 and π/2 as x → ∞.

From Equation 10, we conclude that

ε ̃ i ≃ ε i - π ν 0 V 2 f ( k F L , ϕ ) ,

when ε T ≫ | ε ̃ 0 | , and ε ̃ i = ε i when ε T ≪ | ε ̃ 0 | . These results can be rewritten in terms of length scale. We introduce Lc = ℏvF∕|ε̃0| , which corresponds to the screening length of charge fluctuation. When L ≪ L_c, the renormalized level ε ̃ i is given by Equation 11. When L ≫ L_c, the energy level is hardly renormalized and is independent of ϕ.

In the second stage, we consider the spin fluctuation at low energies of D < D₁. For the purpose, we make the Kondo Hamiltonian via the Schrieffer-Wolff transformation,

H Kondo = ∑ k , σ ε k σ a k σ † a k σ + H J + H K ,

H J = J ∑ k ′ , k [ S + a k ′ ↓ † a k ↑ + S − a k ′ ↑ † a k ↓ †   + S z ( a k ′ ↑ † a k ↑ − a k ′ ↓ † a k ↓ ) ] ,

H k = K ∑ k ′ , k ∑ σ σ k ′ σ † a k σ ,

where S + = d ↑ † d ↓ , S - = d ↓ † d ↑ and S z = ( d ↑ † d ↑ - d ↓ † d ↓ ) ∕ 2 are the spin operators in the quantum dot. The density of states in the lead is given by Equation 6 and half of the band width is now D 1 ≃ | ε ̃ 0 | . H_J represents the exchange coupling between spin 1/2 in the dot and spin of conduction electrons, whereas H_K represents the potential scattering of the conduction electrons by the quantum dot. The coupling constants are given by

J = V 2 1 | ε ̃ 0 | + 1 ε ̃ 1 , K = V 2 2 1 | ε ̃ 0 | - 1 ε ̃ 1 .

By changing the bandwidth, we renormalize the coupling constants J and K so as not to change the low-energy physics within the second-order perturbation with respect to H_J and H_K. Then we obtain the scaling equations of

d J d ln D = - 2 ν 0 J 2 1 - f ( k F L + π 2 , φ ) c o s D ε T - 4 ν 0 J K f ( k F L , φ ) s i n D ε T ,

d K d ln D = - 2 ν 0 3 4 J 2 + 4 K 2 f ( k F L , φ ) s i n D ε T ,

The energy scale D where the fixed point (J → ∞) is reached yields the Kondo temperature.

Scaling equations (15) and (16) are analyzed in two extreme cases. In the case of D ≫ ε_T, the oscillating part of the density of states ν(ε_k) is averaged out in the integration 22. Then the scaling equations are effectively rewritten as

d J d ln D ≃ - 2 ν 0 J 2 ,

d K d ln D ≃ 0 .

In the case of D ≪ ε_T, the expansion around the fixed point 23 yields

K J ≃ 3 8 c ,

2 ν ( D ) J = 1 + O ( c 2 ) ln ( 1 + ξ ) ,

where ξ = D/T_K - 1 and

c ≃ 2 f ( k F L , ϕ ) 1 - f ( k F L , ϕ ) D ε T .

Now we evaluate the Kondo temperature in situations (i) L_c ≪ L_K ≪ L, (ii) L_c ≪ L ≪ L_K, and (iii) L ≪ L_c ≪ L_K, where L_K = ν_Fħ/T_K. In situation (i), ε_T ≪ T_K and thus J and K follow Equations 17 and 18 until the scaling ends at D ≃ T_K. Integration of Equation 17 from D₁ to T_K yields

T K ≃ | ε 0 | exp ( - 1 2 ν 0 J ) ≡ T K ( 0 ) ,

where J = V 2 ( | ε 0 | - 1 + ε 1 - 1 ) .

In situation (iii), D₁ ≪ ε_T. Then the scaling equations (19) and (20) are valid in the whole scaling region (T_K < D < D₁). From the equations, we obtain

T K ( ϕ ) ≃ | ε 0 | T K ( 0 ) | ε 0 | f ( ϕ ) ,

where f(ϕ) = [1 - f(k_FL + π/2, ϕ)]^-1.

In situation (ii), T_K ≪ ε_T ≪ D₁. The coupling constants, J and K, are renormalized following Equations 17 and 18 when D is reduced from D₁ to ε_T and following Equations 19 and 20 when D is reduced from ε_T to T_K. We match the solutions of the respective equations around D = ε_T and obtain

T K ( ϕ ) ≃ ε T e γ T K ( 0 ) ε T e γ f ( ϕ ) ,

where γ ≈ 0.57721 is the Euler constant.

The different expressions of T_K(ϕ) in the three situations can be explained intuitively. In situation (i), ε_T ≪ T_K. Then the oscillating part of the density of states ν(ε_k) with period ε_T is averaged out in the scaling procedure (Figure 2a). As a result, the magnetic-flux dependence of T_K disappears. In situation (iii), T_K ≪ ε_T. Then ν(ε_k) is almost constant in the scaling (Figure 2c). The Kondo temperature significantly depends on the magnetic flux through the constant value of ν(0) at the Fermi level.

We have theoretically studied the Kondo effect in a quantum dot embedded in an AB ring. The two-stage scaling method yields an analytical expression of the Kondo temperature T_K as a function of AB phase ϕ of the magnetic flux penetrating the ring. We have obtained different expressions of T_K(ϕ) for (i) L_c ≪ L_K ≪ L, (ii) L_c ≪ L ≪ L_K, and (iii) L ≪ L_c ≪ L_K, where L is the size of the ring, Lc = ℏvF∕|ε̃0| is the screening length of the charge fluctuation, and L_K = ħν_F/T_K is the screening length of the charge fluctuation, i.e., size of Kondo screening cloud. T_K strongly depends on ϕ in cases (ii) and (iii), whereas it hardly depends on ϕ in case (i).

AB: Aharonov-Bohm.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.