Conductance oscillations with magnetic field of a two-dimensional electron gas–superconductor junction

We develop a theory for the current-voltage characteristics of a two-dimensional electron gas–superconductor interface in magnetic field at arbitrary temperatures and in the presence of surface roughness. Our theory predicts that, in the case of disordered interface, the higher harmonics of the conductance oscillations with the filling factor are strongly suppressed as compared with the first one; it should be contrasted with the case of the ideal interface for which amplitudes of all harmonics involved are of the same order. Our findings are in qualitative agreement with recent experimental data.

Figure 1

The device which we investigate consists of a superconductor, 2DEG, and a normal conductor. An electron injected from the normal conductor to 2DEG in the integer quantum Hall regime transfers through an edge state to the superconductor. It reflects into a hole and an electron which returns to the normal contact through the other edge states.

Figure 2

(Color online) The propagation of quasiparticles in the quasiclassical approximation can be described in terms of rays. The state of a ray can be found from the equations of the classical mechanics. (a) and (b) show what happens if an electron ray from an edge state of 2DEG comes to the superconductor. The electron ray reflects at the point $y=y_0$ from the superconductor into electron and hole rays (normal and Andreev reflections). They reflect, in turn, at the position $y_1$ from the superconductor generating two other electron (red lines) and two other hole rays (blue lines), and so on. To find the probabilities, e.g., $R_{he}(E,n_o,n_i)$, it is necessary to know the sum of the amplitudes of the eight holes that appear after the last beam reflection at the point $y_3$. Drawing (a), we assume that Andreev approximation (Ref. 4) is applicable [see Eq. (19)] and the 2DEG-S interface is ideally flat.

Figure 3

(Color online) When the conditions of the Andreev approximation are violated, we cannot use the assumption that an Andreev-reflected hole velocity is exactly opposite to the velocity of the incident quasiparticle at the superconductor electron. Then, the quasiparticle rays propagate along the 2DEG-S interface as sketched in the figure. The orbits are organized as if the scattering occurs not at the 2DEG-S interface but from the interface in the superconductor lying at some distance from the 2DEG-S interface. According to Ref. 13, Andreev reflection couples electron and hole orbits with the guiding center $x$ coordinates $-\delta \pm X$, where $\delta \simeq l_B^{2}m{v}_s∕\hslash$, $l_B=\sqrt{\hslash c∕eB}$, and the value of $v_s$ should be taken at the 2DEG-S interface. For superconductors wider than the London penetration length ${\lambda }_M$, $\delta ={\lambda }_M$. (b) illustrates schematically how Andreev and normal quasiparticle reflections occur at the superconducting interface along which a supercurrent flows. Indices $i$ and $o$ label the incident and reflected quasiparticles, respectively.

Figure 4

(Color online) The oscillations with $\nu$ of the dimensionless zero-bias conductance at $T=0$. The parameters are as follows. Fermi wave vector $k_F^{\left(2\mathrm{DEG}\right)}=2\times {10}^6{\mathrm{cm}}^{-1}$ and $\delta \varphi ∕4=20$ that correspond to NbN film with a width of the order of $100\mathrm{nm}$. The $L=3\mu \mathrm{m}$ ($L∕2R_c\simeq 5$ at $\nu =25$) for the lower (black thin) curve and $L=0.6\mu \mathrm{m}$ ($L∕2R_c\simeq 1$ at $\nu =25$) for the upper (red thick) curve. We neglect the Zeeman splitting, which is typically small. The 2DEG-S interface scattering amplitudes $r_{ab}$ were taken according to the BTK model (Ref. 15), with $Z=0.6$. The curves were produced using Eq. (23).

Figure 5

(Color online) The propagation of quasiparticles in the quasiclassical approximation along the 2DEG-S interface in the presence of disorder [see (a)]. It corresponds to the transitions between the Andreev edge states, as shown in (b), which are induced by relatively weak short-range disorder. Disorder at the edges of the 2DEG-S interface leads to the orbits depicted in (c). Strong disorder destroys Andreev edge states. There are no oscillations in the conductance because quasiparticles reflected from the strongly disordered 2DEG-S surface are incoherent. Their orbits are shown in (d).

Figure 6

(Color online) Disorder averaging for the transmission probabilities. The loop corresponds to the trace in Eq. (31). The × vertices are $\Phi ,{\Phi }^{†}$; the solid lines represent $M,M^{†}$; and the dashed lines are the $⟨\Phi {\Phi }^{†}⟩$ correlators.

Figure 7

(Color online) It is convenient to split the $⟨\Phi {\Phi }^{†}⟩$ line into the sum of two. The first one corresponds to the ${\delta }_{ip}$ term in Eq. (32) and the second one to the $\Lambda$ term [(a)]. The parallel lines show the part of the bubble [Eq. (31)] where the zero order in $\Lambda$ is taken (“diffusion”). It does not carry interference information, (b). (c) Typical contribution to the average of the bubble of the first order in $\Lambda$ is schematically shown. The diamond denotes the place where the $\Lambda$ line is inserted. It is this diamond that provides oscillations of the current with $\nu$. In order to find the contribution to the current of the first order in $\Lambda$, one should sum all $n$ diagrams like those shown in (c) that differ by the position of the diamond.

Figure 8

(Color online) The figure shows the dependence of the conductance on $\nu$ at $T=0$ for the case of the disordered 2DEG-S interface. The parameters used for this graph: length $L$ of the S-2DEG interface and the normal conductance of the interface, are the same as for the black thin curve in Fig. 4. The comparison with the data of Refs. 6, 8 suggests $e^{-\eta }\simeq 0.1$ that we choose for this figure. The harmonics $g_2,g_3,\dots$, do not contribute to the conductance in the case of the disordered interface in contrast to the clean interface case of Fig. 4. Remarkably, features of the conductance oscillations, such as suppression of the harmonics higher than $g_1$, the order of the oscillation amplitude, etc., agree qualitatively with the experimental data.

Figure 9

(Color online) The temperature dependence of the conductance. The visibility of the conductance oscillations is maximal at $T=0$ and is strongly suppressed as $T$ tends to $T_c$. We plot the temperature dependence of the conductance at the maximum using Eq. (1) for the case of strong disorder at the interface; parameters correspond to the conductance maximum of the curve at $\nu \approx 23$ (Fig. 8).