Model

Throughout this section, I study a standard Cooper pair splitter geometry. The setup is shown in Fig. 5.17a. It consists of two quantum dots, each of them tunnel-coupled to its own normal (N_L and N_R) lead, and a common superconducting lead (SC). First lets recall the Hamiltonians of the system, which I have already introduced in the second chapter of the thesis (see Secs. 2.2, 2.4.1 and 2.5.3). The Hamiltonian of the system is:

H =HQD+HSC+HT,SC+HIT+HN+HT,N. (5.24) We assume that the level spacings of the dots are large, i.e. each quantum dot has a single spinful orbital, which can be occupied by 0, 1 or 2 electrons, hence the QDs are described

5.4. Transport signatures of an Andreev molecule in a quantum dot – superconductor – quantum dot setup

(0,0)

(↑↓,0)

S(1,1)

(0,↑↓)

(↑↓,↑↓) Γ_LAR,R

Γ_CAR Γ_CAR

γ_EC

(σ,0)

(0,σ)

(σ,↑↓)

(↑↓,σ) Γ_LAR,R

Γ_LAR,L γ_EC γ_EC

Γ_CAR Γ_CAR

t_LR t_LR

tLR

t_LR ε_L, U_L

T₀(1,1) (σ,σ) t_LR

t_SL t_SR

t_NL t_NR

μ_SC=0

μ_NL

SC

_ε μ_NR

R, U_R

N

_R

N

_L

QD

_L

QD

_R

a) b)

Triplet - T Doublet - D Singlet - S

Δ ε_R, U_R ε_L, U_L

Local Andreev reflection - LAR Crossed Andreev reflection - CAR

Elastic cotunneling - EC Interdot tunneling - IT

c)

eff ](0,↑↓)-(0,0)

eff ](↑,↓)-(0,0)

eff ](↑,0)-(0,↑) [H_IT](↑,0)-(0,↑)

[H^EC

[H^LAR [H^CAR

γ_EC Γ_LAR,L

Γ_LAR,L Γ_LAR,R

Figure 5.17: Coherent hybridization between a superconductor and two quantum dots in a Cooper pair splitter. a) Schematics of the Cooper pair splitter setup. b) Invariant subspaces of the effective Hamiltonian H_eff describing the QD–SC–QD setup, also showing the couplings between the basis states. The highlighted processes are illustrated in panel c. c) Examples of the coupling processes encoded in the effective Hamiltonian H_eff. LAR couples, e.g., state|0,0i to state|0,↑↓iby transferring a Cooper pair from SC to QD_R. CAR splits a Cooper pair by filling both QDs with one electron with opposite spins. EC transfers an electron from one QD to the other via a virtual intermediate quasiparticle state in SC. IT transfers an electron from one QD to the other without any interaction with SC.

by two copies of the single impurity Anderson model, H_QD = X

α=L,R

X

σ=↑,↓

ε_αd^†_ασd_ασ +U_αn_α↑n_α↓

!

. (5.25)

Note that the interdot Coulomb repulsion in neglected here, since the superconducting lead between the quantum dots screens this interaction. In the following, I assume identical Coulomb repulsion energies in the two dots, and use this energy scale U = U_L = U_R as the unit of energy.

The SC lead is described by the standard mean-field Bardeen-Cooper-Schrieffer (BCS) Hamiltonian

H_SC =X

kσ

E_kγ_kσ^† γ_kσ. (5.26)

5.4. Transport signatures of an Andreev molecule in a quantum dot – superconductor – quantum dot setup The Hamiltonian of the normal leads is

H_N=X

αkσ

ε_αkc^†_αkσc_αkσ. (5.27)

Tunneling between the leads and the two dots is described by the following terms:

H_T,SC = X

αkσ

t_Sαc^†_Skσd_ασ +h.c.

, H_T,N = X

αkσ

t_{N α}c^†_αkσd_ασ +h.c.

. (5.28)

Finally,

HIT=tLR

X

σ

d^†_LσdRσ+h.c.

(5.29) describes interdot tunneling (IT), i.e., direct tunneling between the quantum dots, with an amplitude t_LR.

The complete Hamiltonian H specified above is infinite-dimensional. However, if the temperature T is low and the superconducting gap ∆ is large, then one may simplify the Hamiltonian by eliminating the superconducting quasiparticles from the description.

Technically, this is done by integrating out the quasiparticles using second-order perturba-tion theory in the SC–QD tunneling termH_T,SC. This procedure yields a 16-dimensional low-energy effective Hamiltonian for the double dot, which describes the superconducting proximity effect on the double quantum dot. Here, I describe this effective Hamiltonian and the procedure to obtain it.

For this, I consider the Hamiltonian without the N leads, H_QD+H_SC+H_T,SC+H_IT. (I will take into account the N leads later to describe transport.) Assuming ∆ U and further neglecting the QD–SC tunnelingH_T,SC, the 16-dimensional quasiparticle-free low-energy subspace is energetically well-separated from other states containing a finite num-ber of quasiparticles. The low-energy subspace is spanned by theproduct basis, the prod-ucts of particle-number eigenstates of each quantum dot, namely, (|0i^L,| ↑i^L,| ↓i^L,| ↑↓i^L)⊗ (|0i^R,| ↑i^R,| ↓i^R,| ↑↓i^R), where the arrows denote the spin states of the electrons. In the following I will use the notation|i, ji=|ii^L⊗|ji^R. I perform second-order Schrieffer-Wolff perturbation theory in the tunneling term H_T,SC to obtain the effective Hamiltonian for the 16-dimensional low-energy subspace. See Appendix A.2 for the derivation and the va-lidity conditions. As a result of this procedure, one finds that the QD–SC tunnelingHT,SC

generates three coupling terms in the effective Hamiltonian: (i) a local (single-dot) pairing term, calledlocal Andreev reflection (LAR), (ii) a non-local (interdot) pairing term, called crossed Andreev reflection (CAR), and (iii) an effective interdot tunneling term, called elastic cotunneling (EC):

H_eff =H_QD+H_eff^LAR+H_eff^CAR+H_eff^EC+H_IT, (5.30) In Heff, the second and third terms read as

H_eff^LAR = −X

α

Γ_LAR,α

d^†_α↑d^†_α↓+h.c.

H_eff^CAR = Γ_CAR

d^†_R↑d^†_L↓+d^†_L↑d^†_R↓+h.c.

. (5.31)

5.4. Transport signatures of an Andreev molecule in a quantum dot – superconductor – quantum dot setup

Lets remind ourselves that effective parameters Γ_LAR,α and Γ_CAR are related to each other on the level of the presented model, i.e. neglecting the spatial separation of the quantum dots. (See, e.g., Ref. [138] and Appendix A.2). However in the rest of this work, I will consider Γ_LAR,α and Γ_CAR as independent parameters, since the CAR mechanism is expected to be suppressed, when a finite distance between the QDs is introduced [139].

The fourth term describes of H_eff describes the EC process, for two exemplary matrix elements see Eqs. (2.43) and (2.44). Similarly to HIT, the EC term describes the tunnel-ing event of an electron between the dots, but via an intermediate state, when a stunnel-ingle quasiparticle is present in the superconductor. As I discussed it in Sec. 2.5.4 the matrix elements of H_eff^EC have a strong dependence on the on-site energies of the quantum dots, ε_L and ε_R and hence it has to be distinguished from the IT term.

Importantly, fermion parity and spin are conserved in this effective model. This implies that the 16-dimensional effective Hamiltonian has a block structure; more precisely, there are 6 orthogonal subspaces that are not mixed by the effective Hamiltonian. These invari-ant subspaces are shown in Fig. 5.17b. The first invariant subspace (Singlet - S, top panel of Fig. 5.17b) contains the five spin-singlet states with even number of electrons on the qunatum dots: the empty and the doubly occupied states (|0,0i,|0,↑↓i,| ↑↓,0i,| ↑↓,↑↓i), and the spin-singlet combination of the (1,1) states, |S(1,1)i= ^√1

2(| ↑,↓i − | ↓,↑i). The second and third invariant subspaces (Doublet - D, middle panel of Fig. 5.17b) contain the states with odd number of electrons. Since the magnetic field is not taken into account here, these 8 states are decomposed into two invariant subspaces with different total spin z component [(| ↑,0i,|0,↑i,| ↑,↑↓i,| ↑↓,↑i) and (| ↓,0i,|0,↓i,| ↓,↑↓i,| ↑↓,↓i)]. Each en-ergy eigenvalue in one Doublet subspace has an equal partner in the spectrum of the other Doublet subspace. The three spin-triplet combinations of the (1,1) states, i.e.| ↑,↑i,| ↓,↓i and|T₀(1,1)i= ^√1₂(| ↑,↓i+| ↓,↑i) remain uncoupled from each other and from the other invariant subspaces, and these three states have the same energy eigenvalue.

In Fig. 5.17b, the arrows visualize the tunneling-induced matrix elements coupling the basis states of the effective Hamiltonian. Note that in Fig. 5.17b, I use the singlet-triplet basis instead of the product basis. The tunneling processes giving rise to the coupling matrix elements indicated by red arrows on Fig. 5.17b, are illustrated in Fig. 5.17c. E.g., one of the LAR matrix elements corresponds to transferring a Cooper pair from the su-perconductor to QD_R through a virtual intermediate state, in which one electron occupies QD_R and one quasiparticle is present in the superconductor. The analogous CAR matrix element corresponds, again, to extracting a Cooper pair from the superconductor, but in this case the electrons end up in different quantum dots. Both the EC and the IT matrix elements correspond to the transfer of an electron from one quantum dot to the other.

In the case of EC, there is an intermediate virtual state with one quasiparticle in the superconductor, but in the case of IT, the tunneling is direct. The difference between EC and IT processes results an important difference of their matrix elements: for EC, they depend on the on-site energies [see, e.g., Eqs. (2.43) and (2.43)], while for IT they do not.

[see Eq. (5.29)].

In what follows, I will rely on the numerically obtained eigenvaluesE_χ and eigenstates

|χiof the effective HamiltonianHeffof Eq. (5.30). Furthermore, I will useU =UL=URas the energy unit, Γ_LAR,L = Γ_LAR,R = 0.25U, and will focus on the parameter range Γ_CAR∈ [0,0.1]U, γ_EC ∈ [0,0.15], t_LR ∈ [0,0.1]U. To convert the results here to physical units,

5.4. Transport signatures of an Andreev molecule in a quantum dot – superconductor – quantum dot setup one can use, e.g., U = 1 meV; then the above numbers correspond to an experimentally realistic parameter set.

In document Spin-orbit interaction and superconductivity in InAs nanowire-based quantum dots devices (Pldal 135-139)