Superconductor / topological insulator junctions

We have now established the construction and importance of Majorana particles. The question that arises is, what kind of system supports their existence? From Eq. (1.26) it follows that we are looking for a superconductor where pairing between electrons with the same spin happens. This type of pairing is known as triplet pairing.4 _{Moreover, the structures of f andf}† _{are (∼γ}

1±iγ2) are crucial here as well. The most obvious candidate to fit these two criteria is a chiral p-wave superconductor. In the superconductors we have considered so far, the ∆ parameter was a constant (see Section 1.2.3). In a chiralp-wave case, it is momentum dependent and has the structure

∆(~p) = ∆(px±ipy) = ∆e±iφ, (1.29)

where px and py are the momentum components in the xand y direction, respectively, and φ=

tan(py/px). This structure implies that the pair potential ∆ is rotating as a function of momentum.

The direction of the rotation can be either ±, which is called thechirality.

The most well known chiral p-wave superconductor is Sr2RuO4. However, it is very difficult to realise this material experimentally. Besides that, the actual pair potential remains a point of discussion and it is still not proven that Sr2RuO4 has indeed the pairing symmetry described by Eq. (1.29). [37]

a There are, however, other ways to induce triplet pairing in materials. A well known way to do so is using nanowires, which we will discuss in Section 4.1.1. Another possibility, which is the topic of interest here, is bringing a standards-wave superconductor (with a constant ∆ and Cooper pairs with opposite spin) in combination with a topological insulator.

When we first introduced the Majorana particle at the beginning of this section, we considered it as a particle that is half electron/half hole and that is located at zero energy. As discussed in Sec- tion 1.2.5, two superconductors with another material in between them can host an Andreev bound state (see Fig. 1.12a). If this bound state is located at zero energy, does that turn the Andreev bound state into a Majorana bound state? Almost.

a In the conventional case, the material in between the two superconductors is a normal metal, which has the parabolic dispersion relation that was shown in Fig. 1.8a. If the interfaces are abso- lutely perfect, an electron (without hole component) reflects as a hole (without electron component) and there is no interaction between the two bands. Therefore, there are states at zero energy, so in principle, this should work. In reality, however, the interfaces are not perfect. The two bands 4_{The name “triplet pairing” comes from the fact that there are three ways to pair electrons that are symmetric}

under exchange: | ↑↑i,| ↓↓iand| ↑↓i+| ↓↑i. This is opposed to singlet pairing, where we only had| ↑↓i − | ↓↑i, as discussed in Section 1.2.2.

24 CHAPTER 1. INTRODUCING PHYSICAL CONCEPTS

interact and a gap in the dispersion opens, comparable to the dispersion shown in Fig. 1.8b. The zero energy level lies inside this gap and therefore, it is impossible to have a Majorana bound state in this system. [29, 38]

a If we replace the normal metal in the middle by a topological insulator, this problem is solved. Recall that a topological insulator has spin-momentum locking (see Section 1.3.2) and therefore, backscattering is not possible. Hence, a particle inside the topological cannot go back; it can only go into the superconductor. Therefore, the transmission is equal to 1. We will see in Section 4.4.2 that this is a crucial property to host a Majorana particle.

a It can be shown mathematically that the spin-momentum locking of Eq. (1.21) with induced superconductivity results in the same energy spectrum as thepx+ipy superconductor. This idea

was first postulated in the famous work of Fu and Kane. [39] In Chapter 2, we will calculate the current through a superconductor/topological insulator/superconductor junction and it will turn out that we find indeed thep-wave bound state energy spectrum.

2

Modelling multiple Andreev

reflections

There are several ways to detect a Majorana particle. The most common ones are the use of nanowires and the 4π periodic current-phase relation. These two methods will be explained in Chapter 4, when we look at the experimental aspects of detecting a Majorana particle. In this chapter, we focus on yet another method, which considers multiple Andreev reflections (MAR).

a The concept of MAR was explained in Section 1.2.5. MAR occur in junctions with two super- conductors with a different material in between them. A superconductor/normal metal/superconductor (S/N/S) junction is called aconventional Josephson junction. If we replace the normal metal with a topological insulator, we have a superconductor/topological insulator/superconductor (S/TI/S) junction, which is referred to as atopological Josephson junction.

In Section 2.1, we will first discuss the rather simple resistively shunted junction model that has been used to model S/N/S junction. This model does not take MAR into account. The current through a one dimentional S/N/S junction as a result of MAR was first modelled by Averin- Bardas in 1995. Their model is discussed in Section 2.2. As a next step, we are interested in a junction with a topological insulator (TI) in the middle. When a TI is brought into contact with a superconductor, its surface becomes superconducting as well via the proximity effect. This combination of spin-momentum locking and superconductivity allows symmetry protected surface states to host Majorana particles; which has been a hot topic for the past decades. The one dimensional S/TI/S junction and its relation to Majorana particles were considered by Badiane, Houzet and Meyer in 2011. A brief review of their work is given in Section 2.3.

a The S/TI/S model showed a phenomena that is fundamentally different from the S/N/S junction. This is very interesting, however, it is not directly applicable to experiments. The reason is that experimental junctions cannot be considered one dimensional. The current can also flow through the junction under an angle, which makes the system two dimensional. The experimentally obtained result then corresponds to the current averaged over all possible angles. We expanded the existing one dimensional model to two dimensions to make an actual experimental prediction. By going to two dimensions, we are able to take the length of the TI, as well as the chemical potentials of both materials into account. Our expanded model is shown in Section 2.4.

We have obtained some very remarkable results. The angle that we introduced is a measure for the Fermi surface mismatch between the topological insulator and the superconductor. This mismatch shifts the full spectrum. After angle averaging, however, most of the MAR structure was lost. We also showed that in the absence of an applied voltage, the S/TI/S junction hosts a bound state that

26 CHAPTER 2. MODELLING MULTIPLE ANDREEV REFLECTIONS

is similar to the bound state in chiralp-wave superconductors.

2.1

The resistively shunted junction model

A frequently used tool to model S/N/S junctions is the resistively shunted junction (RSJ) model. In the RSJ model, a resistor (R) is put parallel to the S/N/S junction (J J), such that the current can be modelled as the sum of the supercurrentIS (the current in the absence of an applied voltage)

and the normal state currentIN (Ohmic behaviour). This is shown in Fig 2.1.

JJ R

IN

IS

I

Figure 2.1: Resistively shunted junction model.

The supercurrent has amplitudeIc (the critical current) and oscillates as a function of the phase

difference between the two superconductorsφ. In this simple model, the voltageV is given by the magnetic flux quantum (_~/2e) times the phaseφ. Summing the two contributions, we obtain

IRSJ=IS+IN =Icsinφ+ ~

φ

2eR. (2.1)

Because of its simplicity, this model is popular among experimentalists. However, it can only explain the experimental observations to some extent. The RSJ model does not take many physical aspects of the S/N/S junction into account. Especially the quantum properties are left out. For example, whereas the RSJ model gives a smoothI/V curve, experiments show an oscillating sub- harmonic gap structure (i.e. oscillations foreV <∆, where V is the applied voltage and ∆ is the superconducting gap).

a The main phenomenon that is responsible for the current in S/N/S junctions is the effect of multiple Andreev reflections (MAR), which will be the main topic of this chapter.