Unconventional superconducting phases in multilayer films with layer-dependent Rashba spin-orbit interactions

We present resistivity measurements of bilayer and trilayer films with layer-dependent Rashba spin-orbit interactions. A sharp upturn is observed in the temperature dependence of the parallel upper critical field. We show that it corresponds to a transition from a complex-stripe phase to a helical phase. Moreover, we propose the phase diagram of a multilayer system, which is obtained from experimental observations with the support of numerical calculations. These results pave the way for the clarification of non-trivial superconducting states in multilayer systems composed of two-dimensional Rashba superconductors.

Figure 1

Superconducting phases in multiple monolayer systems. (a) Theoretical model for a bilayer system in a parallel magnetic field $H^{\parallel }$. The row of red circles denotes the 2D superconductor with ${\alpha }_m$, where ${\alpha }_m$ is the Rashba spin-orbit coupling of the $m\mathrm{th}$ layer. When each layer is decoupled from the other layer ($t_{\perp }=0$), the superconducting order parameter of the upper (lower) layer is given by ${\mathrm{\Delta }}_1\left(\mathit{r}\right)=\mathrm{\Delta }{\mathrm{e}}^{\mathrm{i}\mathit{q}·\mathit{r}}$ [${\mathrm{\Delta }}_2\left(\mathit{r}\right)=\mathrm{\Delta }{\mathrm{e}}^{-\mathrm{i}\mathit{q}·\mathit{r}}$] as the helical phase with a finite center-of-mass momentum $\mathit{q}(-\mathit{q})$. In contrast, when each layer is coupled to the other layer ($t_{\perp }\ne 0$), using interlayer Josephson coupling $\delta , {\mathrm{\Delta }}_1$ (${\mathrm{\Delta }}_2$) is given by ${\mathrm{\Delta }}_1\left(\mathit{r}\right)=\mathrm{\Delta }({\mathrm{e}}^{\mathrm{i}\mathit{q}·\mathit{r}}+\delta {\mathrm{e}}^{-\mathrm{i}\mathit{q}·\mathbf{r}})\simeq \mathrm{\Delta }{\mathrm{e}}^{\mathrm{i}\mathit{q}·\mathit{r}}{[1+{\delta }^2+2\delta cos(2\mathit{q}·\mathit{r})]}^{1/2}$ {${\mathrm{\Delta }}_2\left(\mathit{r}\right)=\mathrm{\Delta }(\delta {\mathrm{e}}^{\mathrm{i}\mathit{q}·\mathit{r}}+{\mathrm{e}}^{-\mathrm{i}\mathit{q}·\mathit{r}})\simeq \mathrm{\Delta }{\mathrm{e}}^{-\mathrm{i}\mathit{q}·\mathit{r}}{[1+{\delta }^2+2\delta cos(2\mathit{q}·\mathit{r})]}^{1/2}$} as the complex-stripe (CS phase, which is characterized by both phase and magnitude modulations. (b) Theoretical model for the CS phase of a trilayer system in $H^{\parallel }$. In the second (intermediate) layer, local space inversion symmetry is preserved; therefore, ${\mathrm{\Delta }}_2$ for the second layer is given by ${\mathrm{\Delta }}_2\left(\mathit{r}\right)\sim {\mathrm{e}}^{\mathrm{i}\mathit{q}·\mathit{r}}+{\mathrm{e}}^{-\mathrm{i}\mathit{q}·\mathit{r}}\sim cos(\mathit{q}·\mathit{r})$ as the Larkin-Ovchinnikov (LO) state.

Figure 2

Superconducting transition in multiple monolayer films. (a), (b) Sheet resistance $R_{\mathrm{sq}}$ as a function of temperature $T$ for (a) bilayer film with $d_{\mathrm{Sb}}=3.75$ nm and (b) trilayer film with $d_{\mathrm{Sb}}=3.0$ nm in parallel magnetic fields $H^{\parallel }$, varying in 1 T steps from 0 to 11 T. The insets of panels (a) and (b) show the schematic drawing of the bilayer and trilayer films, respectively.

Figure 3

Parallel and perpendicular upper critical magnetic fields of multiple monolayer films. (a)–(f) Temperature dependence of $H_{\mathrm{c}2}$ for the bilayer and trilayer films for different thickness $d_{\mathrm{Sb}}$ of an Sb spacer layer. The closed (open) circles denote the data of $H_{\mathrm{c}2}^{\parallel }$ ($H_{\mathrm{c}2}^{\perp }$), which are determined from the midpoint of the resistive transition. The dashed lines are linear fits to $H_{\mathrm{c}2}^{\perp }$, from which ${\xi }_{\mathrm{GL}}\left(0\right)$ is obtained. The solid blue (red) lines are obtained by substituting ${\xi }_{\mathrm{GL}}\left(0\right)$ and $d_{\mathrm{bilayer}}=d_{\mathrm{Sb}}+2d_{\mathrm{Pb}}$ ($d_{\mathrm{trilayer}}=2d_{\mathrm{Sb}}+3d_{\mathrm{Pb}}$) into Eq. (1) [Eq. (2)] for each bilayer (trilayer) film (see text). The open diamonds denote the data of $H_{\mathrm{c}2}^{\parallel }$, which are determined from $T_{\mathrm{c}}$ at which zero resistance is observed.

Figure 4

Theoretical calculation for superconducting states of trilayer systems. (a) Numerical calculations of ${\mu }_{\mathrm{B}}{H}_{\mathrm{c}2}^{\parallel }/k_{\mathrm{B}}{T}_{\mathrm{c}0}$ as a function of $T/T_{\mathrm{c}0}$ with different Rashba spin-orbit coupling $({\alpha }_1,{\alpha }_2,{\alpha }_3)=(125,0,-125)$ (solid red line), $({\alpha }_1,{\alpha }_2,{\alpha }_3)=(75,0,-75)$ (dashed red line) and $({\alpha }_1,{\alpha }_2,{\alpha }_3)=(0,0,0)$ (dashed black line), where the unit of Rashba spin-orbit coupling ${\alpha }_m$ is in meV. (b),(c) Center-of-mass momentum $q$ and the interlayer Josephson coupling $\delta$ as a function of $T/T_{\mathrm{c}0}$ at $H^{\parallel }=H_{\mathrm{c}2}^{\parallel }$. (d) Phase diagram for the trilayer system with $({\alpha }_1,{\alpha }_2,{\alpha }_3)=(125,0,-125)$.