Odd-triplet superconductivity in single-level quantum dots

We study the interplay of spin and charge coherence in a single-level quantum dot. A tunnel coupling to a superconducting lead induces superconducting correlations in the dot. With full spin symmetry retained, only even-singlet superconducting correlations are generated. An applied magnetic field or attached ferromagnetic leads partially or fully reduce the spin symmetry, and odd-triplet superconducting correlations are generated as well. For single-level quantum dots, no other superconducting correlations are possible. We analyze, with the help of a diagrammatic real-time technique, the interplay of spin symmetry and superconductivity and its signatures in electronic transport, in particular current and shot noise.

Figure 1

A single-level quantum dot is tunnel coupled to two ferromagnetic leads with coupling strength ${\mathrm{\Gamma }}_{\alpha }$, $\alpha =L/R$, magnetized in directions ${\mathbf{n}}_{\alpha }$ and a conventional superconductor with gap $\mathrm{\Delta }$. The magnetization of each ferromagnet encloses an angle ${\varphi }_{\alpha }$ with the $\mathbf{x}$ axis. Tunneling between the superconductor and quantum dot with coupling strength ${\mathrm{\Gamma }}_S$ gives rise to a finite pairing amplitude on the quantum dot. The underlying coordinate system is shown as well.

Figure 2

Andreev current as a function of gate voltage $\delta$ and bias voltage $\mu$. The coupling to the SC is ${\mathrm{\Gamma }}_S=U/2$, the coupling to the paramagnetic leads is ${\mathrm{\Gamma }}_N=U/1000$, and temperature is $k_{B}T=U/100$.

Figure 3

Andreev shot noise for (a) the same lead (left or right) and (b) cross correlations between the left and right lead. Other parameters are as in Fig. 2.

Figure 4

(a) Modulus and (b) phase of the induced even-singlet order parameter ${\mathrm{\Delta }}_e^S$ as a function of gate and bias voltages. Other parameters are as in Fig. 2.

Figure 5

Andreev current for scenario B1, i.e., the proximitized QD is tunnel coupled to ferromagnetic leads. Polarizations are chosen symmetrically $p_{\alpha }=0.8,{\varphi }_{\alpha }=0$. Other parameters are as in Fig. 2.

Figure 6

Andreev shot noise of scenario B1 for (a) the same lead (left or right) and (b) cross correlations between the left and right lead. Other parameters are as in Fig. 5.

Figure 7

(a) Modulus and (b) phase of the induced even-singlet order in scenario B1 for the same parameters as in Fig. 5.

Figure 8

Odd-triplet order parameter in scenario B1. The FM leads yield a finite spin accumulation on the QD, which results in finite ${\mathrm{\Delta }}_o^T$ in the direction of the induced exchange magnetic field. Other parameters are as in Fig. 5.

Figure 9

Andreev current for scenario B2. The leads are assumed to be paramagnetic metals $p_{\alpha }=0$. A local magnetic field $B_x=U/5$ lifts the spin degeneracy on the QD. Other parameters are as in Fig. 2.

Figure 10

Andreev shot noise for scenario B2. (a) The shot noise and (b) the cross correlators. Parameters are as in Fig. 9.

Figure 11

(a) Modulus and (b) phase of the induced even-singlet order parameter scenario B2 for parameters as in Fig. 9.

Figure 12

Odd-triplet order parameter scenario B2. Parameters are as in Fig. 9.

Figure 13

Andreev current for scenario C, here realized with FM leads, finite polarization $p=0.8$, ${\varphi }_L-{\varphi }_R=\pi /2$, and ${\mu }_L=-{\mu }_R$. Other parameters are as in Fig. 2.

Figure 14

(a) Shot noise and (b) cross correlations for scenario C for parameters as in Fig. 13.

Figure 15

(a) Modulus and (b) phase of the even-singlet order parameter for scenario C for parameters as in Fig. 13.

Figure 16

(left column panels (a), (c), and (e)) give the Modulus and the (right column panels (b), (d), and (f)) the phase of the odd-triplet order parameter vector $({\mathrm{\Delta }}_o^{T_x},{\mathrm{\Delta }}_o^{T_y},{\mathrm{\Delta }}_o^{T_z})$.