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With our obtained knowledge, we can now identify the superconducting gap matrix ∆(k) from equation (5.40) as an appropriate order parameter for the superconducting phase tran- sition. The gap matrix represents the Cooper pairing between electrons, so in the normal, non-superconducting state it must be zero since there are no Cooper pairs present. Below the transition point it evolves continuously and when it is non-zero, the symmetry of the system is changed discontinuously since it breaks the U(1) gauge symmetry, as we have seen in chapter

2. We have also seen that the gap matrix can be written in terms of simpler functions. In the case of spin-singlet pairing all information is contained in a scalar ψ(k) that is even ink, while for spin-triplet pairing all information is contained in a vectord(k) that is odd ink.

In this section we will identify the possible superconducting states of a system with the irreducible representations of the symmetry group of the Hamiltonian. This group-theoretical approach to classify the possible superconducting states was started in the 1980’s by various physicists [56,57,58,59] and is extensively reviewed in [45].

6.3.1 Symmetry Properties of the Superconducting Order Parameter

Let us first have a closer look at the symmetry properties of the gap matrix. The Hamiltonian H from equation (5.34) has a certain symmetry group G that leaves H invariant. This group Gcontains in general multiple different symmetry groups:

• Gc, the symmetry group of the crystal lattice;

• SU(2), the spin rotation symmetry group;

• T, the time-reversal symmetry group.

TheU(1) gauge symmetry has been discussed before and simply means we can multiply every- thing by a phase factoreiφ. The symmetry groupGcrepresents the symmetry of the crystal and could for example beD3dfor CuxBi2Se3, as has been discussed in sections4.1.3and4.2.4. The

latter two require some additional explanation. A full treatment of spin rotations can be found in chapter 3 of [12] and is beyond the scope of this thesis. For us it is sufficient to know that the groupSU(2) describes rotations of the spins of particles and that the Hamiltonian remains in- variant under such rotations. The time-reversel symmetry groupT contains just two elements: the identity and the time-reversal operatorT. Simply said, T reverses the time-ordering of the system19. This means for example that particle trajectories, and thereby their momentum and acceleration, are reversed while their position remains invariant. It turns out that time-reversal also leads to complex conjugation. For a full treatment of time-reversal symmetry, see chapter 13 of [46] or chapter 4 of [12].

Of course the elements of G can also act on ∆(k),ψ(k), and d(k). Let us examine this in more detail. In truth, the elements act on the constituents of ∆(k), which are the operators cks, so we need to use equation (5.39) and (5.40) to determine its effect on ∆(k).

• An elementg of U(1) acts oncks by multiplying it by a phase factoreiφ so we have

g(cks) =eiφcks (6.12) and thus g(∆(k)) =ei2φ∆(k). (6.13) It follows that g(ψ(k)) =ei2φψ(k) g(d(k)) =ei2φd(k) (6.14)

• The elements g of the crystal symmetry group Gc act only on position and momentum vectors, so we have

g(cks) =cg(k)s. (6.15)

We thus simply have

g(∆(k)) = ∆(g(k)) g(ψ(k)) =ψ(g(k)) g(d(k)) =d(g(k)).

(6.16)

• The elements from the spin-rotation group SU(2) mix the different spin-states, but to understand this we need to dig deeper in spin-space, so we will just state the result

g(∆(k)) =D₍T_S)(g)∆(k)D(S) g(ψ(k)) =ψ(k)

g(d(k)) =D(R)(g)d(k),

(6.17)

19

Actually, time-reversion is an ill-chosen name. It would be more appropriate to call it motion-reversion.

whereD(R)(g) is a 3×3 matrix representation ofgthat denotes the equivalent rotation in

regular space with the addition that inversion does not change the sign ofd(k). In other words, inversion of spin does not change the sign of d(k). More detail can be found in [60].

• Similarly we won’t give the details for the time-reversal operator but just give the result T(∆(k)) =σy∆∗(−k)σy

T(ψ(k)) =ψ∗(−k) T(d(k)) =−d∗(−k).

(6.18)

6.3.2 The Effect of Spin-Orbit Coupling

In section3.2we have seen that spin-orbit coupling gives rise to interesting effects. In particular we have seen that spin cannot be taken independently from the momentum anymore. This means that transformation by elements of the crystal symmetry groupGcmentioned above have to accompanied by the appropriate spin-rotation and vice versa. Thankfully the correspondence between both is clear, if we transform the momentumkin a certain, we have to transform the spin in the same way. We have seen that ψ(k) remains invariant under spin-rotations so in the case of spin-singlet pairing nothing changes. In the case of spin-triplet pairing we have for an elementg of Gc

g(d(k)) =D₍T_R)(g)d(g(k)) (6.19) whereD(R)(g) is the matrix representation of the element ofSU(2) corresponding tog. Also, the

spin-up and spin-down labels are not suited anymore and we have to use the pseudo-spin labels + and−introduced in section 3.2instead. But, taking into account that spin and momentum have to be transformed simultaneously, this makes no formal difference and we can keep on labelling them with up and down, keeping in mind that these labels refer to the pseudo-spin states instead of the real spin states.

6.3.3 Superconducting States and Irreducible Representations

We have already seen in equation (6.10) that the superconducting order parameter must have the symmetry of a linear combination of basis functions belonging to the irreducible represen- tations of the symmetry group of the Hamiltonian. It turns out that the different irreducible representations actually correspond to different lower symmetry states, each having its own transition temperature Tc [45,59]. One intuitive way to look at this, is by noting that each irreducible representation is isomorphic to a normal subgroup of the symmetry groupG of the Hamiltonian, and that hence the irreducible representations cannot be the same phase of the material. We will assume that one of these has the highest transition temperature and that this temperature is far higher than the other transition temperatures, so that we can neglect the other irreducible representations. We thus identify the superconducting state of the material with a singlek-dimensional irreducible representation ρ of G, and we write

∆(k) = k

X

n=1

η(ρ, n)∆(ρ, n;k) (6.20)

where the ∆(ρ, n;k) form basis functions of ρ and the η(ρ, n) are coefficients. Note that the ∆(ρ, n;k) can easily be constructed from scalar or vector basis functions that are often tabulated, for example in table 4.4. Now we are in a position to expand the free energy in terms of the order parameter, and then obtain the gap matrix for the superconducting state by minimization

of the free energy. In our discussion of the Landau theory in section 6.2we have seen that we can also use the set of coefficients η(ρ, n) and that this expansion is given by equation (6.11). But in this case we only consider one irreducible representation so the expansion becomes a bit simpler: F =F0+aρ T −Tc Tc X n |η(ρ, n)|2+fρ(η4), (6.21) where thefρ(η4) again denotes the fourth order terms inηthat are invariant under the symmetry groupG.

Now recall from introductory quantum mechanics that spin and angular momentum are good quantum numbers when they are separately conserved and in that case we can decompose eigenfunctions of the Hamiltonian in their spin and orbital part. In a similar way the U(1) gauge symmetry and the time-reversal symmetry can be seen as independent from the SU(2) spin and crystal symmetry. Hence we can decompose the basis functions and split off the crystal symmetry part. In the remainder of this thesis we will thus only consider this part of the basis functions, as we will be interested in the behaviour of the superconducting states under transformations from the crystal symmetry.

6.3.4 Superconducting States in CuxBi2Se3

We are now in a position to classify the possible superconducting states of CuxBi2Se3 according

to their behaviour under the symmetry transformations of the crystal by listing their ψ(k) or d(k). For each irreducible representation of D3d, we can fill in the appropriate fρ(η4), which can be found in table 6.1, in the free energy expansion (6.21) and minimize this free energy. This will give the appropriate ψ(k) or d(k) for different restrictions on the coefficients Ai in the fourth order terms fρ(η4). For 1-dimensional representations the result is simple: there is only one option, namely that ψ(k) ord(k) must be a basis function of the irreducible representation. Therefore we can use the basis functions listed in table4.4, noting that pi now represents the projection of the k-vector on the i-axis and we write ki instead. For the two- dimensional irreducible representations, it can be a linear combination of the two different basis functions. We will not carry out the minimization ourselves but simply cite the result, noting that the irreducible representations ofD3d correspond to irreducible representations ofD6h, as was mentioned in section 4.2.4. The superconducting states are listed in table 6.2. The last column contains alternative scalar basis functions for the irreducible representations. These are also displayed on the cover of this thesis (top: A1g and A2g; bottom: A1u and A2u).

We also know that the basis functions transform according to their irreducible representation, which is isomorphic to one of the factor groups of D3d, as we have seen in section 4.2. In particular the kernel of the representation consists of the elements that are mapped to the identity matrix, thus these elements leave the basis functions invariant. In other words, the kernel is the symmetry group of the basis functions and thus of the order parameter, and the elements of D3d that are missing from the kernel tell us what symmetries are broken. The trivial representation for example has the full group D3d as kernel, meaning it does not break any symmetries in addition to the U(1) gauge symmetry that is always broken in the superconducting state. Also, recall that the odd representations are the representations that do not contain the inversion operationI in their kernel, thus their basis functions change sign under inversion. But this means that the state must be a spin-triplet state, as we have seen in section 5.3.3. An overview of the irreducible representations and the symmetries they break is given in table 6.3.

Representation fρ(η4)

A1g,A2g,A1u,A2u A1|η|4

Eg,Eu A1(η(1)2+η(2)2)2+A2(η(1)∗η(2)−η(1)η(2)∗)2

Table 6.1– Fourth order invariant terms inη for the irreducible representations ofD3d. Adapted

from [45]. ρ Ai ψ(k) or d(k) Ψ A1g - 1,kx2+k2y,kz2 1 A2g - kxky(k2x−3ky2)(ky2−3k2x) pz(p2x+p2y) cos(3θ) Eg A2 <0 kxkz kykz A2 >0 kz(kx+iky) A1u - kxxˆ+kyy, kˆ zˆz (p2x+p2y) cos(3θ) A2u - kyxˆ−kxyˆ pz(p2x+p2y)|cos(3θ)| Eu A2 <0 kzˆx,kxˆz kzˆy,kyˆz A2 >0 (kx+iky)ˆz,kz(ˆx+iˆy)

Table 6.2– Possible superconducting states inD3dfor each irreducible representationρ, for different

relations between the coefficientsAi. Adapted from [45].

Irreducible Representation Kernel Broken Symmetries

A1g D3d None A2g {E, I, C3a, C3b, S6a, S6b} 3C2 and 3σd Eg {E, I} 2C3, 3C2, 3σd and 2S6 A1u {E, C3a, C3b, C2a, C2b, C2c} I, 3σd and 2S6 A2u {E, C3a, C3b, σda, σdb, σdc} I, 3C2 and 2S6 Eu {E} All

Table 6.3 – Overview of the kernels of the irreducible representations ofD3d and the symmetries

that are broken in states belonging to each representation, in addition to theU(1) gauge symmetry. Broken symmetries are listed with their conjugacy classes.