Scattering matrix description

In order to calculate energies of Andreev bound states we apply the scattering matrix formulation of transport through our system. Andreev bound states are formed by multiple Andreev reflections at the superconductors, which transform electrons into holes and vice versa. Their energy is determined by Beenakker’s determinant equation:

deth

e^2i‰≠e^iψSˆh(E)e^≠iψSˆe(E)i

= 0. (6.1)

Sˆe(E)is the electronic scattering matrix of the normal part of the system at energy E. The scattering matrix for holes Sˆh(E) is related to the scattering matrix of electrons via Sˆh(E) = ˆS_e^ú(≠E). In the short junction limit the energy-dependence of Sˆe and Sˆh is weak and can be neglected. The matrix ψ appearing in the Andreev reflection term is given by

6.3 The 4-T ring 95

Figure 6.2: Sketch of the 4-T ring in a scattering matrix description. Each of the nodes is connected by N transport channels to the superconductors. The number of modes in the internal connectorsM_ij can in general be different, but are in this work always equal toM. Scattering at each node is described by a random scattering matrixsˆ⁽ⁱ⁾, which is part of the circular orthogonal ensemble (COE) of RMT [Bro95]. Figure from T. Yokoyama.

ˆ

where the diagonal blocks contain the phases of the four superconducting terminals. ‰= arccos(E/∆) is an energy dependent phase, entering through the Andreev reflection at the superconductors. The energies of Andreev levels are thus determined by the eigenvalue problem (6.1). In order to solve it, the full scattering matrix Sˆe must be expressed through the scattering matrices ˆ

s⁽ⁱ⁾e at each chaotic cavity. The sˆ⁽ⁱ⁾e are described by the circular orthogonal ensemble (COE) of RMT [Bro95]. At each cavity the scattering matrix relates incoming to outgoing modes. For the(0)-node we have

0

Herea0 andb0 are vectors of lengthN describing incoming and outgoing modes towards terminal(0). c_ij are vectors describing modes in the internal connectors going from node j to node i. They have length Mij. Similar equations are valid at the other three nodes. The splitting of the scattering matricessˆ⁽ⁱ⁾e into subblocks is necessary in order to be able to eliminate the internal modes and find an effective scattering matrix that relates incoming external modes directly to outgoing external modes. To achieve this we introduce vectors

˛a =

The first four components of˛cdescribe clockwise - , the second four components counterclockwise propagation of modes inside the ring. This order is not necessary and can be chosen freely, since these are anyway the modes we want to eliminate. Combining Eq. (6.3) for all four nodes and reordering rows and columns appropriately we find permutation matrices describing the correct reordering of subblocks in sˆ_e. Eliminating ˛c from Eq. (6.5) we find for the effective scattering matrix of electrons, which relates incoming modes from the terminals to outgoing modes to the terminals

Sˆ_e = ˆX+ ˆZ[1≠Wˆ]^≠1Y .ˆ (6.6) The dimension of Sˆe is8N◊8N. Our main interest concerns the regime where N is much larger than the internal number of modes M (for simplicity we assume the number of modes in all internal connectors to be equal to M). In this limit at each nodeN≠2M modes coming from the superconductors are just reflected and lead to transmission eigenvaluesT = 0at the particular node. We

6.3 The 4-T ring 97 thus want to reduce the dimension of the scattering matrix to16M ◊16M by decoupling the reflected modes, which only contribute trivial solutions E =∆ to the Andreev spectrum. This can be achieved by using that the2M non-zero transmission eigenvalues at each node are distributed according to

fl(T) = N + 2M spectrum. There are no transmission eigenvalues in the interval 0< T < Tc. However, there are correlations between different transmission eigenvalues, which are beyond this average distribution. Typically these eigenvalues repel each other, similar to charged particles in a Coulomb gas. The distribution of their spacings∆T is usually very well described by a Wigner-Dyson distribution

fl(∆T) = fi 2

∆T

”_T² e^{≠fi4(∆T /”T)2}, (6.8) where”_T is the average spacing of neighboring T eigenvalues. This is the only parameter entering this distribution. In order to generate a random list of eigenvalues, which have the average distribution Eq. (6.7), but at the same time repel each other according to the Wigner-Dyson distribution (6.8), a first eigenvalueT1 is generated randomly according to the average distribution at the lower edge of the spectrum. For the second eigenvalue we estimate the average spacing as the inverse of the average distribution at this point: ”T1 = 1/fl(T1).

Then a random number∆T1 of Wigner-Dyson-distribution with ”T1 as input parameter is generated. T₂ is given by T₂ =T₁+∆T₁. All other values follow accordingly. As soon as we arrive atT = 1 the procedure is stopped. If the list ofT values has the correct length, we use it, if not we discard it and generate a new list. After having generated T-lists for all four nodes, the scattering matrices at these nodes are constructed as follows:

ˆ Tˆ⁽ⁱ⁾are diagonal matrices with the randomly generated transmission eigenvalues on the diagonals and U⁽ⁱ⁾ and U^(i)Õ are random unitary 2M ◊2M matrices from the circular unitary ensemble (CUE). The length of vectors ai andbi is

reduced to 2M. Since Andreev reflected holes follow time reversed trajectories of electrons which compensates for any unitary mixing, the unitary matrices towards the superconductors U^(i)Õ can without loss of generality be chosen to U^(i)Õ =12M. We define Áˆ⁽ⁱ⁾ =1≠Tˆ⁽ⁱ⁾ and introduce 8M ◊8M unitary block

Oˆ₁ andOˆ₂ are8M◊8M permutation matrices which exchange the components of vector˛cappropriately. In the following only the product of both is relevant:

Oˆ ©Oˆ₂Oˆ₁ =

Oˆ is also a permutation matrix and has the property Oˆ² = 1. The effective scattering matrix for the full normal part becomes