Conclusion

4.3. Electrical tuning of Rashba spin-orbit interaction in multigated InAs nanowires Nevertheless it does not influence the finding that l_SO can be tuned by the gate voltages even if the conductance and the average electron density is kept constant.

4.3. Electrical tuning of Rashba spin-orbit interaction in multigated InAs nanowires

a) d)

e)

f) b)

c)

-8 0 8

80 120 160 200

l SO (nm)

V_SG1 (V) Measurement Simulation

V_BG = 15 V l_SO,Bi = 280 nm

0 20 40 60

80 120 160

V_SG1 = V_SG2 = 0 V l_SO,Bi = 160 nm l SO (nm)

V_BG (V)

Measurement Simulation -2

0 2 4

1 2 3 4 5

y

zzz z

y

y y

V(r) (V)

V(r) (V) ρ(εF) (eV -1cm -3)

ρ(εF) (eV -1cm -3) 40 nm

Figure 4.13:Results of the numerical simulation. a),b) The electric potential and the density of states (DOS) at the Fermi energy, respectively, in the vicinity of the nanowire (NW), forVSG1 = 11 V,VSG2=−7.7 V,VBG= 15 V. The black lines mark the NW/SiOx/vacuum boundaries. The low DOS values at the surface of the NW is the result of the relatively coarse resolution of the grid (5 nm). c) The spin relaxation length in the asymmetric side gate configuration, the measurement data is the same as in Fig. 4.12a (black squares), and the result of the numerical calculation (red curve). d),e) Same as a) and b), for the BG-tuned measurement withV_SG1 =V_SG2 = 0 V, VBG= 58 V. f) Same as c), for tuning with BG.

4.3. Electrical tuning of Rashba spin-orbit interaction in multigated InAs nanowires

5

Proximitized quantum dots

A future step in quantum information processing is to construct fault-tolerant com-putational architectures. Topologically protected exotic states arising in superconducting hybrid structures provide a platform to implement partially or fully fault-tolerant ar-chitectures. The fault-tolerance means that the computational basis states are always degenerate, hence the system is insensitive to external noises. The protected operations are preformed by moving around the these particles, called braiding.

A roadmap to implement a fully fault-tolerant architecture follows the utilization of the following particles: Majorana fermions [11–17,20–28], parafermions [29–31], Fibonacci anyons [190]. In this series more and more logical gates are performable in a protected way. The current step is to realize the Majorana fermion-based platform. These particles were already reported in different systems, like in semiconducting nanowire-based setups [171, 176] or in chains of magnetic adatoms on superconducting surfaces [80, 81].

Besides the usual proposals, where the Kiteav model is realized by a microscopic Hamiltonian, there are ideas to construct artificial topological platforms. Such a system was investigated theoretically by Sau et al. in Ref. [63], where the Kiteav model is realized by alternating chain of quantum dots and superconducting islands (see Fig. 5.1). In the Majorana chain the neighboring sites – dots – are coupled in two different ways, imple-menting the two different coupling mechanism of the original proposal of Kitaev. First is the direct tunnel coupling between the dots and second the crossed Andreev reflection, which creates (annihilates) electrons on two neighboring sites.

Figure 5.1: The proposed Majorana chain of Sau et al.. The Kitev chain is realized by a series of quantum dots (green) coupled by superconducting islands (blue). The tunability is provided by local gate electrodes (red). The figure is adapted from Ref. [63].

In this chapter I will investigate different building blocks of the Majorana chain to examine the feasibility of the model.

In a strongly coupled superconductor – quantum dot hybrid structures subgap Shiba states arise by the hybridization of the states of the dot and the superconductor (see Sec. 2.5.3). These states extend into the superconductor to a finite distance. The key to implement the couplings between the neighboring quantum dots of the Majorana chain lies in the overlap of the Shiba states hosted by the dots. With the state-of-the-art fabrication techniques the achievable closest spacing of the dots is 80−100 nm. However, both the theory of Shiba systems and the STM experiment preformed on Shiba states indicates that the extension is limited to few nanometers [69,71,73,76] (also see Sec. 2.5.2), making the implementation of the Majorana chain proposal elusive.

In Sec. 5.1 I will investigate the spatial extension of a Shiba state formed in the InAs nanowire-based superconductor-quantum dot hybrid device by non-local transport measurements. The tunnel electrode the probes the Shiba state is placed at a distance of 100−200 nm from the quantum dot. The signatures of the Shiba states in the current implies that it extends to a much larger distance in the presented system compared to previous reports on STM setups. I will show that the key to the much larger extension lies in the geometrical properties of the setup, which makes it effectively one-dimensional, and removes the fast decay implied by the higher spatial dimensions.

Regarding the complexity the next building block to analyze is the QD–SC–QD sub-system. In the remaining of the my thesis I will focus on that. This block was widely investigated previously by Cooper pair splitter measurement, where the dots were only weakly coupled to the superconductor to filter the flow of the electrons [55–62]. To provide a significant coupling between the dots of the Majorana chain, they have to be strongly coupled to the superconductor. In Sec. 5.2 I will investigate how the presence of the Shiba state modifies the operation of a Cooper pair splitter. We will see that the Shiba state gives similar signatures as the Cooper pair splitting signal, but this is present even when the splitting is prohibited by the blockage of the current in one arm. The results tell us that one has to be careful during the analyzes the splitting effect, since it is mimiced by the Shiba state.

In Sec. 5.3 I will give a proposal to experimentally prove the spin singlet character of the individual split Cooper pairs in the QD–SC–QD system. The proposed experiments is based on the state-of-the-art spin qubit toolkit. After introducing the experimental scheme, the material related non-idealities will be discussed, how they limits the proposed experiment, and they can by overcame.

Finally in Sec. 5.4 I will analyze the the different coupling mechanisms arising the QD–SC–QD system starting from a microscopic Hamiltonian. I will show that besides the usually used local and crossed Andreev processes there a third, superconductivity related coupling of the dot states, the elastic cotunneling. Subsequently, I will investigate in detail the ground state properties and the excitation spectrum of the QD–SC–QD system with different coupling strengths and I will show that the detailed transport characterization allows for the identification of the dominant coupling mechanism.

These works are important steps on the way to realize the Majorana chain and to understand the building blocks of it.

5.1. Magnetic field enhancement of non-local Shiba signal

5.1 Magnetic field enhancement of non-local Shiba signal

As I discussed in the theory background chapter there are two different platforms which can host Shiba states formed around magnetic impurities in superconductors (see Sec. 2.5). First, when magnetic atoms are deposited on superconducting substrates and second, when quantum dots are attached to a superconducting electrode. The first setup is widely studied by STM groups and it was found that the Shiba state decays on a very short length scale in the superconductor: it extends up to 1 nm on 3-dimensional [69, 73]

and up to 10 nm on 2D superconductors [71, 76]. In quantum dot-based geometries the spatial extension of the Shiba states has not been addressed yet.

In this section I will show that the Shiba state formed around a quantum dot can extend into the superconductor to a distance one order of magnitude larger than in STM geometries. I have studied the extension of the Shiba by measuring thenon-local current in a tunnel probe attached to the superconductor about 100 nm away from the dot.

Surprisingly, I have found a resonance-like increase in the differential conductance of the tunnel probe when the Shiba state was tuned on resonance, indicating that the Shiba wavefunction has a finite weight at the tunnel probe. Furthermore, I will show that in finite magnetic field the extension of the Shiba state is increased significantly further, indicated by the strong increase of the conductance enhancement. Finally, I will compare my experimental findings with numerical renormalization group (NRG) calculations.¹

The results presented here opens the way to the realization of artificial topological devices, like the Majorana chain [63], which realizes the famous Kiteav model in a chain-like structure formed of the alternating series of quantum dots and superconductors. With the state-of-the-art fabrication techniques the currently available closest spacing of the dots is about 100 nm, comparable to the extension of the Shiba state. This suggests that coupling the neighboring dots via the superconductor is indeed feasible.