Full proximity treatment of topological superconductors in Josephson-junction architectures

Experiments on planar Josephson-junction architectures have recently been shown to provide an alternative way of creating topological superconductors hosting accessible Majorana modes. These zero-energy modes can be found at the ends of a one-dimensional channel in the junction of a two-dimensional electron gas (2DEG) proximitized by two spatially separated superconductors. The channel, which is below the break between the superconductors, is not in direct contact with the superconducting leads, so that proximity coupling is expected to be weaker and less well controlled than in the simple nanowire configuration widely discussed in the literature. This provides a strong incentive for this paper which investigates the nature of proximitization in these Josephson-junction architectures. At a microscopic level we demonstrate how and when it can lead to topological phases. We do so by going beyond simple tunneling models through solving self-consistently the Bogoliubov-de Gennes equations of a heterostructure multicomponent system involving two spatially separated $s$-wave superconductors in contact with a normal Rashba spin-orbit-coupled 2DEG. Importantly, within our self-consistent theory we present ways of maximizing the proximity-induced superconducting gap by studying the effect of the Rashba spin-orbit coupling, chemical potential mismatch between the superconductor and 2DEG, and sample geometry on the gap. Finally, we note (as in experiment) a Fulde-Ferrell-Larkin-Ovchinnikov phase is also found to appear in the 2DEG channel, albeit under circumstances which are not ideal for the topological superconducting phase.

Figure 1

(a) Schematic diagram of a 2DEG in proximity to two spatially separated superconducting leads which form a Josephson junction. By tuning the strength of either the applied in-plane magnetic field $B$ or the phase difference $\varphi$ between the two superconductors, the system can be tuned into the topological superconducting phase which hosts Majorana zero modes ($\gamma$) at the end of the junction. (b) Schematic diagram of a nanowire proximitized by a superconductor. The system becomes a topological superconductor, which hosts Majorana zero modes ($\gamma$) at the end of the nanowire, when the strength of the magnetic field $B$ is above a certain critical value.

Figure 2

Profile of pair amplitudes (top panel) and energy spectra (bottom panel) of the planar Josephson junction for the case: (Left panel) Zeeman is only in the junction ($E_{Z,J}=0.17$ and $E_{Z,L}=0$) and (right panel) Zeeman is uniform across the 2DEG ($E_{Z,J}=E_{Z,L}=0.17$). Note that the presence of the Zeeman field in the 2DEG below the superconductor ($E_{Z,L}$) reduces the induced pair amplitude and proximity gap in the 2DEG [panels (b) and (d)]. The black dashed lines in the top panel denote the boundaries between the superconductors and the 2DEG. The parameters used are ${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1,\alpha =0.05,{\mathrm{\Delta }}_0=0.3$ $[\xi =v_F/\left(\pi {\mathrm{\Delta }}_0\right)=2.12/k_F]$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,W=6/k_F,D_{\mathrm{SC}}=10/k_F$, and $D_{2\mathrm{DEG}}=4/k_F$.

Figure 3

Understanding effects of chemical potential mismatch on proximitization. Energy spectra of the normal part of the Hamiltonian of the superconductor (SC) and 2DEG for the case where (a) ${\mu }_{\mathrm{S}}={\mu }_{2\mathrm{DEG}}$ and (b) ${\mu }_{\mathrm{S}}\gg {\mu }_{2\mathrm{DEG}}+{\alpha }^2$. For the case where (a) ${\mu }_{\mathrm{S}}={\mu }_{2\mathrm{DEG}}$, the mismatch between the Fermi momenta of the SC and 2DEG gets larger for increasing SOC strength $\alpha$ while for the case where (b) ${\mu }_{\mathrm{S}}\gg {\mu }_{2\mathrm{DEG}}+{\alpha }^2$, the mismatch between the Fermi momenta of the SC and 2DEG is weakly dependent on the SOC strength $\alpha$. In summary, the dependence of the proximity gap ${\mathrm{\Delta }}^{\mathrm{prox}}$ on $\alpha$ is weaker for the case where the SC chemical potential is much larger than the 2DEG chemical potential.

Figure 4

Effects of SOC on the spectral gap for the case where there is no magnetic field. Top panel: For small chemical potential mismatch, e.g., $\delta \mu ={\mu }_{\mathrm{S}}-{\mu }_{2\mathrm{DEG}}=0$, the gap depends strongly on the 2DEG SOC. The gap decreases with increasing SOC strength as shown in panels (a)–(c) because there is a larger mismatch between the Fermi momentum of the superconductor and Rashba spin-orbit-coupled 2DEG as the SOC strength increases. Bottom panel: For large chemical potential mismatch, e.g., $\delta \mu ={\mu }_{\mathrm{S}}-{\mu }_{2\mathrm{DEG}}=9$, the gap depends weakly on the 2DEG SOC [see panels (d)–(f)] as there are more occupied subbands in superconductors with large ${\mu }_S$. This implies that for an incident electron coming from one of the bands of the 2DEG, there is a band in the superconductor with a momentum close to the incident momentum. The parameters used are ${\mu }_{2\mathrm{DEG}}=1,E_{Z,J}=E_{Z,L}=0,{\mathrm{\Delta }}_0=0.3$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,W=6/k_F,D_{\mathrm{SC}}=10/k_F$, and $D_{2\mathrm{DEG}}=4/k_F$.

Figure 5

Energy spectra of planar Josephson junctions for different junction widths: (a) $W=6/k_F$, (b) $W=30/k_F$, and (c) $W=80/k_F$. The spectral gap decreases with increasing junction width $W$. The parameters used are: ${\mu }_{2\mathrm{DEG}}=1,{\mu }_{\mathrm{S}}=10,E_{Z,J}=E_{Z,L}=0,{\mathrm{\Delta }}_0=0.3$ $[\xi =v_F/\left(\pi {\mathrm{\Delta }}_0\right)=2.12/k_F]$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,D_{\mathrm{SC}}=10/k_F$, and $D_{2\mathrm{DEG}}=4/k_F$.

Figure 6

Thickness effects. Profile of pair amplitude (top panel) and energy spectra (bottom panel) of planar Josephson junctions for zero Zeeman field and different thickness of 2DEG: $D_{2\mathrm{DEG}}=4/k_F$ (left panel), $D_{2\mathrm{DEG}}=8/k_F$ (middle panel) and $D_{2\mathrm{DEG}}=11/k_F$ (right panel). Note that the thicker the 2DEG is, the smaller is the induced superconducting gap in the 2DEG. The black dashed lines in the top panel denote the boundaries between the superconductors and the 2DEG. The parameters used are ${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1,\alpha =0.05,E_{Z,J}=E_{Z,L}=0,{\mathrm{\Delta }}_0=0.3$ $[\xi =v_F/\left(\pi {\mathrm{\Delta }}_0\right)=2.12/k_F]$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,W=6/k_F,\mathrm{and}{D}_{\mathrm{SC}}=10/k_F$.

Figure 7

Summary figure showing how ${\mathrm{\Delta }}^{\mathrm{prox}}$ depends on 2DEG thickness, $\delta \mu$, and SOC. (a) Induced gap ${\mathrm{\Delta }}^{\mathrm{prox}}/{\mathrm{\Delta }}_0$ as a function of the 2DEG thickness $D_{2\mathrm{DEG}}$ for SOC strength $\alpha =0.05$ and ${\mu }_{\mathrm{S}}={\mu }_{2\mathrm{DEG}}=1$. The induced gap decreases with increasing 2DEG thickness. (b) Induced gap ${\mathrm{\Delta }}^{\mathrm{prox}}/{\mathrm{\Delta }}_0$ as a function of chemical potential difference ($\delta \mu ={\mu }_{\mathrm{S}}-{\mu }_{2\mathrm{DEG}}$) calculated for ${\mu }_{2\mathrm{DEG}}=1,D_{2\mathrm{DEG}}=4/k_F$ and several values of SOC strength $\alpha$. For small $\alpha$, the induced gap decreases with increasing $\delta \mu$. For large $\alpha$, the induced gap has nonmonotonic dependences on $\delta \mu$ where it first increases with increasing $\delta \mu$, rises to a maximum, and after reaching the maximum it decreases with increasing $\delta \mu$. (c) Induced gap ${\mathrm{\Delta }}^{\mathrm{prox}}/{\mathrm{\Delta }}_0$ as a function of SOC strength $\alpha$ for $D_{2\mathrm{DEG}}=4/k_F$ and different values of $\delta \mu$. For small $\delta \mu$, the induced gap depends strongly on $\alpha$ where it decreases with increasing $\alpha$. For the case where ${\mu }_{\mathrm{S}}$ is much bigger than ${\mu }_{2\mathrm{DEG}}$, the induced gap depends weakly on $\alpha$. The parameters used for the above plots are $W_{\mathrm{SC}}=20/k_{\mathrm{F}},W=6/k_{\mathrm{F}},D_{\mathrm{SC}}=10/k_{\mathrm{F}},E_{Z,J}=E_{Z,L}=0,{\mathrm{\Delta }}_0=0.3$, and ${\mu }_{2\mathrm{DEG}}=1$.

Figure 8

Topological phase diagrams for a proximitized Josephson junction with vanishing chemical potential mismatch $\delta \mu =0$. (a) Class BDI and (b) Class D phase diagram as functions of $E_Z$ and $\varphi$. Each region is labeled by different $\mathbb{Z}$ topological invariants in the BDI phase diagram. The ${\mathbb{Z}}_2$ invariant gives the parity of the $\mathbb{Z}$ index. The topological invariant $Q_{{\mathbb{Z}}_2}=-1$ and $Q_{{\mathbb{Z}}_2}=1$ corresponds to the odd and even $\mathbb{Z}$ indices which in turn indicates the topological and trivial phases of class D. The parameters used are ${\mu }_{\mathrm{S}}={\mu }_{2\mathrm{DEG}}=1,\alpha =0.05,{\mathrm{\Delta }}_0=0.3$, $W_{\mathrm{SC}}=20/k_F,W=6/k_F,D_{\mathrm{SC}}=10/k_F$, and $D_{2\mathrm{DEG}}=4/k_F$.

Figure 9

Comparison of topological phase diagrams (a) without (${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1$) and (b) with chemical potential mismatch (${\mu }_{\mathrm{S}}=20,{\mu }_{2\mathrm{DEG}}=1$). Class D phase diagrams for two different values of chemical potential differences between the superconductors and the 2DEG: (a) ${\mu }_{\mathrm{S}}=1$ and ${\mu }_{2\mathrm{DEG}}=1$ and (b) ${\mu }_{\mathrm{S}}=20$ and ${\mu }_{2\mathrm{DEG}}=1$. The ${\mathbb{Z}}_2$ invariant $Q_{{\mathbb{Z}}_2}=-1$ and $Q_{{\mathbb{Z}}_2}=1$ indicate the topological and trivial phases of class D. The chemical potential of the 2DEG is renormalized by the chemical potential of the superconductors resulting in a difference between the effective chemical potential of the 2DEG below the superconductor and that of the 2DEG in the junction. This difference increases as the mismatch between the superconductor and 2DEG chemical potential becomes larger which in turn increases the amplitude of normal reflections in the 2DEG. As a result, for a larger chemical potential mismatch, the phase diagram becomes more stripelike (less dependent on $\varphi$) and the critical Zeeman field for $\varphi =\pi$ shifts to a larger value. The parameters used are ${\mu }_{2\mathrm{DEG}}=1,\alpha =0.05,{\mathrm{\Delta }}_0=0.3$, $W_{\mathrm{SC}}=20/k_F,W=6/k_F,D_{\mathrm{SC}}=10/k_F$, and $D_{2\mathrm{DEG}}=4/k_F$.

Figure 10

Evolution of the energy spectrum of a planar Josephson junction across the topological phase transition for $\varphi =0$ (upper panel) and $\varphi =\pi$ (lower panel). The topological transition is characterized by a gap closing at $k_x=0$ where the critical field at which the topological transition occurs is the smallest at $\varphi =\pi$. The critical fields for $\varphi =0$ and $\varphi =\pi$ are $E_Z=0.11$ [panel (b)] and $E_Z=0.0053$ [panel (e)], respectively. Energy spectra shown correspond to the phase diagram of Fig. 8. The gap closes and reopens at $k_x=0$ as the Zeeman field $E_Z$ is, respectively, tuned towards and away from the critical field. (a),(d) The system is in the trivial phase; (b),(e) the system undergoes a topological phase transition with a gap closing at $k_x=0$; (c),(f) the system is in the topological phase. Shown here are only a few low-energy states close to zero energy where the energy levels closest to zero energy are shown by red lines. Here, we take the Zeeman field to be uniform ($E_{Z,J}=E_{Z,L}=E_Z$) in the 2DEG. The parameters used are ${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1,{\mathrm{\Delta }}_0=0.3$, $\alpha =0.05,W_{\mathrm{SC}}=20/k_F,W=6/k_F,D_{\mathrm{SC}}=10/k_F$, and $D_{2\mathrm{DEG}}=4/k_F$.

Figure 11

Evidence for FFLO pairing. Profile of the real part of the pair amplitude of planar Josephson junctions with 2D Rashba SOC of different strength: $\alpha =0$ (left panel), $\alpha =0.3$ (middle panel), and $\alpha =0.5$ (right panel). Upper panel shows the color plots of the real part of the pair amplitude $\mathrm{Re}[F/{\mathrm{\Delta }}_0]$, and the lower panel is the linecuts of the pair amplitude along the $y$ direction at different values of $z$. As shown in the bottom panel, the pair amplitude oscillates along the $y$ direction with a characteristic oscillation length $\lambda =E_Z/\left(2v_F\right)$ that decreases with increasing $\alpha$ (as $v_F$ increases with increasing $\alpha$). The black dashed lines in the upper panel denote the boundary between the superconductors and the 2DEG. The parameters used are ${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1,E_{Z,J}=E_{Z,L}=0.3,{\mathrm{\Delta }}_0=0.3$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,W=80/k_F,D_{\mathrm{SC}}=10/k_F,\mathrm{and}{D}_{2\mathrm{DEG}}=4/k_F$.

Figure 12

The change of Fermi surfaces of a 2DEG due to an in-plane magnetic field $B$ along the junction ($x$ direction) for the case of (a) zero SOC and (b) finite SOC strength. The Fermi surfaces in the absence and presence of $B$ are represented by light and dark colors, respectively. (a) In the absence of a Zeeman field, the Fermi surfaces of a 2DEG without SOC are doubly degenerate. When an in-plane Zeeman field is applied, the Fermi surfaces of the up and down spins enlarge and shrink radially in momentum by $E_Z/v_F$ while keeping the two Fermi surfaces concentric. The superconducting term $\mathrm{\Delta }$ pairs up electrons with opposite spin from different Fermi surfaces. (b) The 2D Rashba SOC causes a clockwise and anticlockwise spin orientation (represented by red and blue arrows, respectively). The applied in-plane Zeeman field along the $x$ direction shifts the inner and outer Fermi surfaces in the opposite direction along $k_y$ by $E_Z/v_F$. The superconductivity term $\mathrm{\Delta }$ pairs up electrons with opposite spin from the same Fermi surface.

Figure 13

Evidence for FFLO. Profile of the real part of the pair amplitude of a planar Josephson junction with a 1D Rashba SOC ($\alpha {\partial }_y{\sigma }_x$) of different strengths: $\alpha =0$ (left panel), $\alpha =0.3$ (middle panel), and $\alpha =0.5$ (right panel). Upper panel shows the color plots of the real part of the pair amplitude $\mathrm{Re}[F_0/{\mathrm{\Delta }}_0]$ and lower panel shows the linecuts of the pair amplitude along the $y$ direction at different values of $z$. As shown in the lower panel, the pair amplitude oscillates along the $y$ direction with a characteristic oscillation length $\lambda =E_Z/\left(2v_F\right)$ that decreases with increasing $\alpha$ (as $v_F$ increases with increasing $\alpha$). The black dashed lines in the upper panel denote the boundaries between the superconductors and the 2DEG. The parameters used are ${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1,E_{Z,J}=E_{Z,L}=0.2,{\mathrm{\Delta }}_0=0.3$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,W=80/k_F,D_{\mathrm{SC}}=10/k_F,\mathrm{and}{D}_{2\mathrm{DEG}}=4/k_F$.

Figure 14

Evidence for FFLO. Profile of the real part of the pair amplitude for the effective model of a planar Josephson junction [Eq. (25) of the main text]. The pair amplitudes are calculated for a 2D Rashba SOC of different strengths: $\alpha =0$ (left panel), $\alpha =0.5$ (middle panel), and $\alpha =0.8$ (right panel). Upper panel shows the color plots of the real part of the pair amplitude $\mathrm{Re}[F_0/{\mathrm{\Delta }}_0]$, and lower panel shows the linecuts of the pair amplitude along the $y$ direction at different values of $z$. As shown in the lower panel, the pair amplitude oscillates along the $y$ direction with a characteristic oscillation length $\lambda =E_Z/\left(2v_F\right)$ that decreases with increasing $\alpha$ (as $v_F$ increases with increasing $\alpha$). The parameters used are ${\mu }_{\mathrm{S}}=1,{\mu }_{2\mathrm{DEG}}=1,E_{Z,J}=E_{Z,L}=0.3,{\mathrm{\Delta }}_0=0.3$, $\varphi =0,W_{\mathrm{SC}}=20/k_F,W=80/k_F,\mathrm{and}{D}_{2\mathrm{DEG}}=4/k_F$.