Persistent current (PC) is induced by changing the magnetic flux threading the ring, which can be expressed as I = −∂E∂φgs, where φ is magnetic flux threading the ring. PC is an observable which serves as a measure for the Kondo coupling as its magnitude is propotional to the strength of the Kondo coupling.
4.4.1 K=0
Figure 4.9: Persistent current versus magnetic flux with K = 0. (Reproduced from Ref.[13]) Red dashed line is L ≈ 200. Black solid line is L ≈ 100. (a)L=4n (b) L=4n+1 (c)L=4n+2 (d)L=4n+3 Other parameters of data: ²d = -0.8, tR = tL = 0.4, in units of t.
Before we present our new results on the PC, it proves useful to summarize the previous results[12][13] for K = 0, corresponding to the system of a QD embedded in a mesoscopic
ring as shown in Fig.(4.9). First, since the system is in the Kondo regieme, PC increases with increasing system size L. PC as a function of the magnetic flux is a weak sinusoidal wave when L ∼ ξk0; while it behaves like a metallic ring with high transmission when L ¿ ξk0. Second, there exists a relation between PC for L and L + 2: P CN(φ) = P CN +2(φ + π).
Third, PC for L = odd is much smaller than that for L = even. This can be understood as for L = odd the electrons are fully occupied at the Fermi level, which suppress the Kondo resonance; whileas for L = even there is an unpair electron on the Fermi level, leading to the Kondo resonance.
4.4.2 K 6= 0
A.
Varies K with fixed L:PC versus magnetic flux with different K at a finite size L ∼ 200 is shown in Fig.(4.10).
The general trend in all four cases is that the amplitude of PC gets smaller as K is increased and finally vanishes for K > Kc2, in agreement with the expectation on the suppression of the Kondo effect by the AF coupling. There are detail differences among the four cases. In Fig.(4.10(a) and (c)), with increasing K the persistent current P CL=240near 0.5π < φ < 1.5π decreases more slowly than that for 0π < φ < 0.5π and 1.5π < φ < 2π;
similarily for P CL=242 except for the range of φ is being interchanged. This suggests that the Kondo effect is more robust in 0.5π < φ < 1.5π for L = 4n and in 0π < φ < 0.5π and 1.5π < φ < 2π for L = 4n + 2. Note that we find the similar discontinuous jumps at K = Kc1 mentioned previously in DOS to also appear in PC for L = 4n + 1 and L = 4n + 3; while PC is continuous at at K = Kc1 for the other two cases. Despite the above detail differences, we find two common features in Fig. (4.10) that remain the same as in the case of K = 0: (i). P CN(φ) = P CN +2(φ + π) and (ii). PC for L = even is larger than that for L = odd.
-4
Figure 4.10: PC versus φ with various K. K = 0, 0.015, 0.03 and fixed L at about 240.
(a)L=4n (b) L=4n+1 (c)L=4n+2 (d)L=4n+3 Other parameters of data: ²d = -0.8, tR = tL = 0.4, in units of t.
B.
Varies L with fixed K:Fig.(4.13) and Fig.(4.12) show PC versus magnetic flux at different system sizes. In the following the discussion is separated into two cases: (i). K < KC where PC increases with increasing size L, and (ii). K ≤ KC where PC decreases to 0 as L reaches the thermodynamic limit.
K ≤ KC
-4
Figure 4.11: Persistent current v.s. φ with various L. K=0.008:(a)L=4n+1 (b) L=4n+3;
K=0.015:(c)4n+1 (d)4n+3. Other parameters of data: ²d = -0.8, tR = tL = 0.4, in units
Figure 4.12: Persistent current v.s. φ with various L. K=0.015:(a)L=4n (b) L=4n+2.
Other parameters of data: ²d = -0.8, tR = tL = 0.4, in units of t.
Fig.(4.12) shows PC versus magnetic flux with different size L for K = 0.008 and K = 0.015. For K = 0.008 (Fig.(4.12(a)(b))), as L is increased we find the PC exhibits two different behaviors: near φ = π for L = 4n+1 and φ = 0, 2π for L = 4n+3 PC changes from the behavior in crossover region to the Kondo state at K = 0 (see Fig. (4.9(b)(d));
while out of these ranges of φ its behavior remians the same as in the crossover region but with an increasing amplitude. A discionuous jump is seen to seperate these two regions.
We expect at much larger system size the Kondo phase will be restored eventually over the entire range of φ. We have also investigated PC for L = 4n, 4n + 2 (not shown here) and find the same qualitative behaviors except that insted of jumps we find a continuous change in PC at K = Kc1. Also, P C4n+2φ∼0 or P C4nφ∼π remain the same behaviors as that for K = 0 for all the sizes we investigate, suggesting that they stay in the Kondo phase from the start.
On the other hand, for a larger K, K = 0.015, as L increases from L ∼ 60 to L ∼ 240 PC stays in the crossover region with an increasing amplitude. It is expected as for larger value of K one must go to much larger system size to observe the restoring of the Kondo effect in PC.
K ≥ KC
As shown in Fig.(4.13), since K > Kc in this case, PC decreases in amplitude with increasing system size L and finally the system reaches the local spin-singlet state with vanishing PC. Note that P C4nφ∼0and correspondingly P C4n+2φ∼π are always vanishingly small for all the sizes we investigate, suggesting the systems are already in the local spin-singlet states from the start for these ranges of φ, which is consistent with our mean-field phase diagram. However, for the remaining ranges of φ the systems start from the crossover region for smaller sizes and reach finally to the local spin-singlet state at large size. This implies that by applying magnetic flux, we may change the ground state of our system at finite sizes from the local spin-singlet state to the crossover region which is inferenced by the Kondo effect.
0 0.5 1 1.5 2 -2
-1 0 1 2
L=60 L=120 L=180 L=240
0 0.5 1 1.5 2
L=62 L=122 L=182 L=242
I/I
0φ(π)
K=0.036 K=0.036
4n 4n+2
φ(π)
(a) (b)
Figure 4.13: Persistent current with various L. (a)L=4n (b) L=4n+2. Other parameters of data: ²d = -0.8, tR = tL = 0.4, K=0.036, in units of t.
Chapter 5
Conclusions and Further Studies
5.1 Conclusions
We have studied the Kondo effect in a side-coupled-quantum-dot system where one dot is embedded in a mesoscopic ring via large-N slave-boson mean-field approach. The com-petition between Kondo effect and antiferromagnetic (AF) coupling gives rise to the K–T transition in thermodynamic limit and the competition also weaves the phase diagrams which is classified into two case. For case I, L=4n, L=4n+1, and L=4n+3, the crossover regime between Kondo phase and spin-singlet phase occurs for K < Kc; while for case II, L=4n+2, the crossover regime exists for K > Kc. For further studies of how the AF cou-pling suppresses the Kondo effect, three observables, TK, ρQD(ω), and PC, are calculated.
For a fixed K > Kc, the system approaches to spin-singlet state where the magnitudes of the three observables go to zero with increasing system size. For a fixed K < Kc, Tk and PC increase in magnitude and ρQD(ω) develops into a single peak centered at ω = 0 with increasing size. Using the finite size scaling of Tk, we have shown that Tkfollows an univer-sal function of 1/(LT∗) for (1/L) < T∗, where T∗ is unambiguously identified the charac-teristic Kosterlitz-Thouless crossover energy scale:T∗ = cfTkexp (−πfTk/ | K − Kc|). For a fixed size L, we find the ground state stay in Kondo state when K < Kc1, it crossover to the spin-singlet state when Kc1 < K < Kc2, and it finally reaches to spin-singlet state
50
when K > Kc2. For L=4n+1 amd L=4n+3, first order jumps exist in all of the observables between the crossover regime and Kondo state. Whether these first order jumps are due to the artifacts of mean-field approach should be clarified by other approaches. However, unlike the previous studies on the side-coupled doubel quantum dot system embedded in 2DEG where assume the density of state is a constant, the key finding in this thesis is the K–T transition with finite critical point Kc in a side-coupled DQD system embedded in a mesoscopic ring. Whether this finite Kcis due to the artifacts of mean-field theory or due to the more singular density of states of 1D tight-binding ring needs further investigations.