4.3 Model
4.4.6 Spatial dependence
In order to achieve a spatial resolution of the local density of states (LDOS) we consider a symmetric model of three normal islands connected to two superconductors at Ï = 0. Due to symmetry the Green’s functions in the left and right normal nodes are equal. Both nodes are thus called N1 in the following, the central node is calledN2. We consider two different setups. The first is a structure consisting of three normal cavities in series with ballistic coupling between each other. The coupling towards the superconductors is ballistic as well. The second setup consists of three normal nodes in series, which are coupled via tunnel contacts. The first and the last node are coupled to the superconductors via ballistic contacts.
Three chaotic cavities in series
In this system the nodes N1 are connected via a ballistic conductance G1 to the superconductor and via G2 to N2. In each normal node, electron-hole-decoherence is described through Thouless energies ETh1,2, respectively. The system setup is sketched in Fig.4.8. Matrix current conservation in one of the nodesN1 and in the node N2 determines the Green’s function and the LDOS
Figure 4.8: Sketch of a system with three normal nodes connected to two superconductors atÏ= 0via ballistic contacts. The normal nodes are coupled via ballistic contacts as well. Such a geometry can model, for example, a series of three cavities connected by point contacts of different widths. The contacts have the conductancesG1 and G2 and at each normal node a leakage current described byETh1,2 is taken into account.
in the particular node [Naz99; Naz94]:
Iˆ1S+ ˆI12+i(G1 +G2)(E/ETh1 )[ˆ·3,ˆg1(E)] = 0, (4.9)
≠2ˆI12+i(2G2)(E/ETh2 )[ˆ·3,ˆg2(E)] = 0. (4.10) The matrix currentsIˆ12 andIˆ1S are defined according to the definition (4.2). In general, the system is described by three parametersETh1 ,ETh2 , andG1/G2. We fix one parameter by considering only systems with equal mean level spacings in all three normal nodes ”s1 =”2s =”s. Furthermore, we fix
G1G2/((G1+ 3G2)∆)”s =GQ/2. (4.11) In this case for G1/G2 π1the LDOS in N1 and N2 are equal and correspond to the result of Figure 3.1, and for G1/G2 ∫ 1 we have the BCS-DOS inN1
(compare with Fig. 2.1) and again the result of Figure 3.1 in N2. Figure 4.9 shows the LDOS in the two nodes for G1/G2 = 1/500 [Fig. 4.9 (a)] and for G1/G2 = 500 [Fig. 4.9(b)]. Taking eq. (4.11) into account, ETh1 and ETh2 can be expressed in terms of G1/G2. We find
ETh1 = (G1+G2)/GQ”s =∆(G2+G1)(3G2+G1)/(2G1G2), (4.12) ETh2 = 2G2/GQ”s=∆(3G2+G1)/G1. (4.13) In Fig. 4.10, we show the numerical results for the LDOS in both normal
(a) (b)
Figure 4.9: Spatial dependence of the DOS. (a) The DOS forG1/G2 = 1/500 is constant in the normal region. (b) For G1/G2 = 500 the outer nodes are strongly coupled to the superconductors, whereas the inner node shows the standard result for a ballistic cavity (a).
4.4 Results 57
(a) (b)
Figure 4.10: LDOS in N1 (a) andN2 (b) for intermediate values ofG1/G2 betweene≠6 and e6. The white regions in (a) are regions where N(E)/N0 >5.
Whenever a gap appears in the central nodeN2 there appears a gap in the outer nodesN1 as well and vice versa.
nodes in dependence of G1/G2 and energy in the secondary gap region below the superconducting gap edge ∆. Figure 4.10 (a) shows the result for the outer nodes N1, and Fig. 4.10(b) shows the result for the inner node N2. The white region in Fig.4.10 (a) denotes N(E)/N0 >5. The main finding of our calculations is that the LDOS in the two nodes differs only whereN(E)/N0 >0 in both nodes. Whenever it is zero in the central node it is also zero in the outer nodes and vice versa. We thus find a behavior of the secondary gap similar to what is already known from the usual minigap [Pil00]. The width of this gap is not position dependent, only the LDOS above/below the particular gap edge varies with position.
Tunnel coupled normal nodes with ballistic coupling to the superconductors In this section we calculate the position resolved density of states for a system where the normal nodes are connected via tunnel contacts, and only the coupling to the superconductors is ballistic. Such a system simulates a diffusive normal metal, which can in circuit theory be described by a series of tunnel contacts [Naz09]. A sketch of the system we have in mind is shown in Fig. 4.11. Again, we consider equal level spacings of the normal nodes ”s1 =”s2 =”s and use the second condition ”sG1G2/((G1+ 3G2)∆) =GQ/2, which gives equal LDOS in both nodes for G1/G2 π1, corresponding to the result of Fig. 3.1(a). These two conditions define the Thouless energies in N1 and N2 in the same way as
Figure 4.11: A system consisting of three tunnel-coupled normal nodes which are connected to two superconductors at Ï= 0 via ballistic contacts. Such a geometry can model a diffusive normal metal, which is known to be describable in circuit theory as a series of tunnel contacts [Naz09]. As before, at each normal node a leakage current described byETh1,2 must be taken into account.
before via Eq. (4.12) and Eq. (4.13). For G1/G2 ∫1we find the BCS DOS in N1. Due to the tunnel coupling between the normal nodes, the result in N2 is now different from Fig. 4.9 (b). It corresponds to a single node of ETh =∆, which is coupled to the superconductors via tunnel contacts. There appears no secondary gap. These results for both nodes N1 and N2 in the limits of G1/G2 ∫1and G1/G2 π1 are shown in Figure4.12.
In the limit G1/G2 π1 (here r = 1/500) we find equal results for the two different contact types in the normal part. This shows that for strongly coupled normal nodes the contact type between the normal nodes is not important.
This is because in this limit, the Green’s function is constant inside the normal part and thus the matrix current vanishes for any contact type (commutator in Eq. 4.2 vanishes).
For G1/G2 ∫ 1 (here G1/G2 = 500) the situation is different depending on whether there are tunnel contacts or ballistic contacts between the normal nodes. The explanation is the same as before: For G1/G2 ∫ 1 the Thouless energy of the central node approachesETh2 =∆. Furthermore, the outer normal nodes are strongly coupled to the superconductors, whereas the central normal node is only weakly coupled to the outer normal nodes. We thus find the DOS in the outer nodes approaching the BCS DOS (shown in Fig. 2.1), whereas the DOS in the central node approaches the result of a single node with ETh =∆
4.4 Results 59
(a) (b)
Figure 4.12: Spatial dependence of the DOS in the two limiting cases of large and small ratioG1/G2. We focus on the secondary gap region belowE =∆. (a) The DOS forG1/G2 = 1/500is constant in the normal region and corresponds to the DOS for a system of a single node ofETh =∆, coupled ballistically to the superconductors. This result is almost the same like Fig. 4.9 (a). (b) For G1/G2= 500 the outer nodes are strongly coupled to the superconductors and have a DOS close to the BCS DOS, whereas the inner node shows the result for a cavity coupled via tunnel contacts to the superconductors atETh=∆without a secondary gap.
connected to the superconductors via tunnel contacts. For such a system, no secondary gap exists, and we see that the DOS remains finite in the outer nodes as well. The DOS in the outer nodes approaches the BCS DOS forG1/G2 æ Œ without completely vanishing for any finite value of G1/G2. This supports our claim that a gap in the DOS of the normal region either has to exist in every point or in none. Figure4.13 shows the DOS in both normal nodesN1 and N2 in the secondary gap region for intermediate values of the parameter G1/G2.
From the results for the two system types considered in this chapter we conclude that the secondary gap not only appears in the LDOS of a singular point, but as well in the integrated DOS of a finite region. Depending on the parameters of the normal region, not for every system a secondary gap appears.
However, if it appears in one point, it exists also in every other point of the normal part.
(a) (b)
Figure 4.13: LDOS in N1 (a) andN2 (b) as G1/G2 is increased from e≠6 to e6. The white regions correspond toN(E)/N0 >5in (a) andN(E)/N0 >2.5in (b). Whenever a gap appears in the central nodeN2 there appears a gap in the
outer nodes N1 as well. ForG1/G2∫1 no gap exists.
4.5 1D scattering model
The secondary gap we found for diffusive Josephson systems was calculated using Green’s function techniques in the quasiclassical approximation. Whereas this method is very powerful in calculating expectation values of physical observables, it does not provide a simple intuitive explanation for the absence of Andreev levels in the secondary gap region and the dependence of these levels on the phase difference Ï between the superconductors. In this section, we investigate a simple 1D scattering model which is able to explain the secondary
“Smile”-gap qualitatively. However, since we deal with diffusive or chaotic scattering systems with large conductance, we should not expect to reproduce the details of 3D solutions.