2.5 Bound states in superconductors
2.5.4 QD–SC–QD system
2.5. Bound states in superconductors
|σ,0i state couples only to |0, σi. Hence the singlet ground state gains more energy with increasingtS compared to the doublet one.
An important difference of the Shiba limit compared to the ABS limit, that there are two different excitations in the Shiba limit, first one can add/remove an electron to/from the quantum dot, which I will calldot-like excitation in the following. And second, one can add/remove a quasiparticle to/from the superconductor, which I will callquasiparticle-like excitation.
Using a more realistic description of the superconductor – like keeping the BCS Hamiltonian and performing a first order perturbation in the tunnel coupling – one can obtain a similar spatial dependence for the Shiba state in SC–QD systems as for the Shiba states in the case of magnetic impurities (see Sec. 2.5.2).
Intermediate limit
In a more realistic situation, where bothU and ∆ are finite both the double occupation of the quantum dot and the presence of the quasiparticles has to be taken into account.
I will illustrate this case via the ZBA model, with the only difference compared to the Shiba limit that the double occuaption of the dot is allowed, i.e. the model is described by Eqs. (2.37), (2.38) and (2.4). The particle number eigenstates of the dot are |0iQD,
| ↑iQD, | ↓iQD and | ↑↓iQD and of the superconductor are |0iSC, | ↑iSC and | ↓iSC. I.e.
altogether there are 12 states. The tunneling, HTSZBA couples them in the following way, obtained similarly as in the Shiba limit: in the singlet subspace
|0,0i ↔ | ↑,↓i − | ↓,↑i ↔ | ↑↓,0i, (2.39) and in the doublet subspace
|0, σi ↔ |σ,0i ↔ | ↑↓, σi. (2.40) These are illustrated on panel a of Fig. 2.22. Similarly as in the Shiba limit, the triplet combinations of the (1,1) states remain uncoupled.
The energy spectrum in the absence, presence of tunnel coupling and the excitation spectrum is plotted on panel b, c and d of Fig. 2.22, respectively, using U = 3.5∆.
Similarly to the ABS limit, the lowest energy excitations form an eye-shaped crossing, but their energy saturates at ∆. Upon increasing the tunnel coupling the doublet phase shrinks, similarly as in the ABS limit. One expects a similar, dome-shaped phase diagram as in the ABS limit, but the disappearance of the doublet phase requires such a strong coupling, which lies outside of the validity regime of the ZBA. Hence a quantitatively accurate phase diagram requires a more sophisticated framework.
2.5. Bound states in superconductors
-5 -4 -3 -2 -1 0 1 2 -5
-4 -3 -2 -1 0 1 2
-5 -4 -3 -2 -1 0 1 2 -5
-4 -3 -2 -1 0 1
2 ε/Δ
ε/Δ
U=3.5Δ
ε/Δ
E/Δ
E/Δ
tS=0.4Δ
U=3.5Δ ΔE/Δ
tS a)
c) d)
SC QD
tS
tS
doublets tS
tS
singlets b)
S
S D S D S
tS=0
-5 -4 -3 -2 -1 0 1 2 0.0
0.2 0.4 0.6 0.8 1.0 1.2
Figure 2.22: Subgap state of SC–QD systems in the intermediate regime, with U = 3.5∆ a) Tunnel coupling hybridizes the dot states with the quasiparticle states. b) Energy spectrum without tunnel coupling. Blue (red) lines are the states without (with) quasiparticles. The red shading illustrates the quasiparticle continuum, that is neglected in the model. c) Energy spec-trum withtS= 0.4∆. Here the blue line denotes the subgap states, formed by the hybridization.
The gray dashed lines energies with tS = 0 (same as on panel b). d) Excitation spectrum for tS = 0.2∆,0.5∆,0.8∆. Blue lines correspond to the transition between the subgap states on panel c.
2.5. Bound states in superconductors (see Fig. 2.23a). The QDs are described by Eq. (2.7) and the tunneling Hamiltonian is extended for both sites,
HTS = X
α=L,R
X
kσ
tSαh d†ασ
ukγkσ+σvkγ−k¯† σ +
ukγkσ† +σvkγ−k¯σ
dασi
, (2.41) where, for simplicity, I assume that the tunneling to the left and right dot occurs at same position.
Crossed Andreev reflection - CAR Elastic cotunneling - EC
a)
eff ](↑,↓)-(0,0) [HECeff ](↑,0)-(0,↑)
[HCAR
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
b)
in units of U
εL/U εL/U
εR/U εR/U
[HECeff ](0,σ)-(σ,0) [HECeff ](↑↓,σ)-(σ,↑↓)
Figure 2.23:a) Illustration of the CAR and EC processes. b) Two examples of the dependence of the EC matrix elements on the on-site energies of the QDs.
Using the second-order quasi-degenerate perturbation theory, the tunnel coupling to the superconductor lead gives rise to two additional terms in the effective Hamiltonian besides the LAR coupling, the CAR and the EC terms. For the details of the calculation see Appendix A.2. The CAR coupling can be summarized as
HeffCAR = ΓCAR
d†R↑d†L↓+d†L↑d†R↓+h.c.
, (2.42)
where ΓCAR=−πtSLtSRρ0andρ0is the normal-state DOS of the superconductor, assumed to be constant. On the level of the model presented here the effective parameters ΓLAR,α and ΓCAR are related to each other, ΓCAR = p
ΓLAR,L·ΓLAR,R (See, e.g., Ref. [138]).
2.5. Bound states in superconductors
However, taking into account a finite spatial separation of the quantum dots the CAR amplitude becomes suppressed by the distance [139] and henceforth one can consider ΓLAR,α and ΓCAR as independent parameters. The finite separation can be taken into account by localizing the tunnelings to different points, giving rise to an eikδr factor for one of the tunneling amplitudes in momentum space in Eq. (2.41).
The EC term HeffEC describes single-electron tunneling between the quantum dots via the superconductor. This term, in contrast to the LAR and CAR coupling, has a strong dependence on the on-site energiesεL,εRof the quantum dots. For example, the EC matrix element coupling the | ↑,0i and |0,↑i states is well approximated by (see Appendix A.2)
HeffEC
(0,↑)−(↑,0) =γEC(εL+εR), (2.43) whereas the matrix element coupling the | ↑,↑↓i and | ↑↓,↑i states is
HeffEC
(↑↓,↑)−(↑,↑↓) =−γEC(εL+εR+UL+UR), (2.44) where the strength of the EC mechanism is characterized by the dimensionless parameter γEC = ΓCAR/∆. See Eq. (A.16) for the complete list of the matrix elements.
To illustrate the dependence of these matrix elements on the quantum dots’ on-site energies, I plot the two matrix elements shown in Eqs. (2.43) and (2.44) in Fig. 2.23b, usingγEC = 0.02. The two matrix elements vanish atεL=−εRand εL+UL =−εR−UR, respectively. Of course, if such a contribution vanishes, then higher-order terms neglected here may actually be important.
Note that certain works using the EC mechanism neglect the energy dependence of its matrix elements [43, 137, 140–144]. Indeed, introducing a finite distance between the quantum dots one can construct models in which the EC term is constant in leading order.
However, these contributions are formally equivalent with previously introduced interdot tunnel coupling, HDQDT (see Eq. (2.8)), hence they can be incorporated in to the same term.
I will discuss the effect of the CAR and EC coupling in the excitation spectrum in detail in Sec. 5.4.
3
Experimental techniques
In this chapter I will shortly introduce the general steps of the sample fabrication and the common properties of the electrical transport measurements. This chapter serves to explain the main concepts, the details are given in each results sections separately.