Andreev reflection and cyclotron motion at superconductor—normal-metal interfaces

We investigate Andreev reflection at the interface between a superconductor and a two-dimensional electron system (2DES) in an external magnetic field such that cyclotron motion is important in the latter. A finite Zeeman splitting in the 2DES and the presence of diamagnetic screening currents in the superconductor are incorporated into a microscopic theory of Andreev edge states, which is based on the Bogoliubov-de Gennes formalism. The Andreev-reflection contribution to the interface conductance is calculated. The effect of Zeeman splitting is most visible as a double-step feature in the conductance through clean interfaces. Due to a screening current, conductance steps are shifted to larger filling factors and the formation of Andreev edge states is suppressed below a critical filling factor.

Figure 1

Semiclassical picture of Andreev edge states. Electrons and holes in skipping orbits along a clean $S\text{−}N$ interface are successively Andreev reflected into each other. In the $S$ region, quasiparticle excitations exist only on the (extremely short) length scale of the superconducting coherence length. The presence of a uniform supercurrent parallel to the interface results in a shift of electron and hole orbits in the normal region. Andreev reflection couples electron and hole orbits with guiding-center coordinate $\pm X-\lambda$, respectively. $\lambda$ denotes the magnetic penetration depth.

Figure 2

(Color) Effect of Zeeman splitting on the Andreev-reflection contribution to the $S\text{−}N$ interface conductance $G_{\mathrm{AR}}$. A double-step structure is clearly visible for the dot-dashed curve which has been calculated for an ideal interface $(s=1,w=0)$ and $g=-20$, ${\Delta }_0=0.3\mathrm{meV}$, ${ϵ}_{F,s}={ϵ}_{F,n}=10\mathrm{meV}$, and $m_s=m_n=0.035m_e$. This feature turns out to be obscured at larger values of the superconducting gap (${\Delta }_0=3\mathrm{meV}$ for all solid curves), in particular when an interface barrier is present (nonzero values of $w$ as indicated for each curve).

Figure 3

(Color) Effect of Zeeman splitting for a realistic $S$-2DEG junction realized, e.g., by a typical $\mathrm{Nb}\mathrm{N}∕\mathrm{In}\mathrm{Ga}\mathrm{As}$ hybrid system: ${ϵ}_{F,s}=7\mathrm{eV}$, ${ϵ}_{F,n}=10\mathrm{meV}$, $m_s=m_e$, $m_n=0.035m_e$, ${\Delta }_0=3\mathrm{meV}$, $w=0$, and for various values of the gyromagnetic factor $g$ as indicated. For these parameters, the condition $\nu =2$ is realized at a magnetic field of approximately $3\mathrm{T}$, which is well below the upper critical field of typical NbN electrodes (Ref. 29) where $H_{c2}>14\mathrm{T}$ at temperatures $⩽1.4\mathrm{K}$.

Figure 4

(Color) Diamagnetic screening currents lead to a shift of conductance steps. Data shown are calculated for an ideal interface $(s=1,w=0)$, $g=0$, ${\Delta }_0=0.3\mathrm{meV}$, ${ϵ}_{F,s}={ϵ}_{F,n}=10\mathrm{meV}$, and $m_s=m_n=0.035m_e$ for various values of the magnetic penetration depth $\lambda$.

Figure 5

(Color) Andreev-reflection transport at high magnetic fields through a nonideal $S\text{−}N$ interface with diamagnetic screening current present. Results shown are for $\lambda =5\mathrm{nm}$ and various values of $w$ as indicated, with all other parameters as given in Fig. 4.

Figure 6

(Color) Effect of the diamagnetic screening current at a realistic $S$-2DEG interface such as in a $\mathrm{Nb}\mathrm{N}∕\mathrm{In}\mathrm{Ga}\mathrm{As}$ hybrid system, but for $g=0$. Sample parameters are the same as for Fig. 3.