The effect of ferromagnetism on the critical supercurrent in topological insulator Josephson junction

Abstract

To guide the potential applications of topological insulator, we provide a theoretical investigation for a finite temperature and an arbitrary length scale Josephson junction in the Furusaki–Tsukada formula. We have shown theoretically that a large degree of control over the generated critical supercurrent can be obtained by changing directions of the magnetizations in ferromagnet. The special importance for experimental measurements is the asymmetric character, oscillatory feature arising in the Fermi energy window, and energy shift phenomenon unveiled by a top gate voltage, which are independent of magnetization amplitude.

Appendix

In this section, we use a standard formula of retarded Green’s function to obtain the DC Josephson current of the topological insulator Josephson junction. The retarded Green’s function can be constructed by the following four types of quasiparticle injection processes: type 1 (electron-like quasiparticle incident from the left), type 2 (hole incident from the left), type 3 (electron-like quasiparticle incident from the right), and type 4 (hole incident from the right).

For an electron-like quasiparticle incident from the left superconductor lead, the wave functions in the type 1 scenario can be written as below:

$$\varPsi_{1} (x) = \left\{ {\begin{array}{*{20}l} {\varPsi_{S}^{e + } + r^{1} \varPsi_{S}^{e - } + r_{A}^{1} \varPsi_{S}^{h - } ,} \hfill & {x \le 0} \hfill \\ {t^{1} \varPsi_{S}^{e + } + t_{A}^{1} \varPsi_{S}^{h + } ,} \hfill & {x \ge l} \hfill \\ \end{array} } \right.$$

(8)

For a hole-like quasiparticle incident from the left superconductor lead, the wave functions in the type 2 scenario can be written as the following forms:

$$\varPsi_{2} (x) = \left\{ {\begin{array}{*{20}l} {\varPsi_{S}^{h + } + r^{2} \varPsi_{S}^{h - } + r_{A}^{2} \varPsi_{S}^{e - } ,} \hfill & {x \le 0} \hfill \\ {t_{A}^{2} \varPsi_{S}^{e + } + t^{2} \varPsi_{S}^{h + } ,} \hfill & {x \ge l} \hfill \\ \end{array} } \right.$$

(9)

For an electron-like quasiparticle incident from the right superconductor lead, the wave functions in the type 3 scenario can be written as below:

$$\varPsi_{3} (x) = \left\{ {\begin{array}{*{20}l} {t^{3} \varPsi_{S}^{e - } + t_{A}^{3} \varPsi_{S}^{h - } ,} \hfill & {x \le 0} \hfill \\ {\varPsi_{S}^{e - } + r^{3} \varPsi_{S}^{e + } + r_{A}^{3} \varPsi_{S}^{h + } ,} \hfill & {x \ge l} \hfill \\ \end{array} } \right.$$

(10)

For a hole-like quasiparticle incident from the right superconductor lead, the wave functions in the type 4 scenario can be written as the following forms:

$$\varPsi_{4} (x) = \left\{ {\begin{array}{*{20}l} {t_{A}^{4} \varPsi_{S}^{e - } + t^{4} \varPsi_{S}^{h - } ,} \hfill & {x \le 0} \hfill \\ {\varPsi_{S}^{h - } + r^{4} \varPsi_{S}^{h + } + r_{A}^{4} \varPsi_{S}^{e + } ,} \hfill & {x \ge l} \hfill \\ \end{array} } \right.$$

(11)

Because of the translational invariance along y-axis, the wave functions $\varPsi_{i} \left( {x, y} \right) = \varPsi_{i} \left( x \right)e^{iqy}$ in the superconducting leads. Thus, we can write the retarded Green’s function as $G(x,\;x^{\prime},\;y,\;y^{\prime}) = \sum\nolimits_{q} {G(x,x^{\prime})e^{{iq(y - y^{\prime})}} }$. Based on Eqs. 8–11, the retarded Green’s function G(x, x ^′) can be written as

$$ G(x,x^{\prime}) = \left\{ \begin{array}{l} \alpha_{1} \varPsi_{3} (x)\varPsi_{1}^{T} (x^{\prime}) + \alpha_{2} \varPsi_{3} (x)\varPsi_{2}^{T} (x^{\prime}) \\ \; + \alpha_{3} \varPsi_{4} (x)\varPsi_{1}^{T} (x^{\prime}) + \alpha_{4} \varPsi_{4} (x)\varPsi_{2}^{T} (x^{\prime}), \quad {x \le x^{\prime}} \\ \beta_{1} \varPsi_{1} (x)\varPsi_{3}^{T} (x^{\prime}) + \beta_{2} \varPsi_{1} (x)\varPsi_{4}^{T} (x^{\prime}) \\ \; + \beta_{3} \varPsi_{2} (x)\varPsi_{3}^{T} (x^{\prime}) + \beta_{4} \varPsi_{2} (x)\varPsi_{4}^{T} (x^{\prime}),\quad x \ge x^{\prime} \\ \end{array} \right. $$

(12)

where $\varPsi_{i}^{T} (x)$ with i = 1 − 4 are the wave functions corresponding to the conjugate processes of Ψ _i (x). The parameters α _i and β _i with i = 1 − 4 can be determined by the boundary conditions of Green’s function

$$G(x + 0,x) - G(x - 0,x) = \hbar^{ - 1} v_{F}^{ - 1} (i\tau_{z} \sigma_{y} )$$

(13)

where τ _z is the Pauli matrix actions in the electron–hole space.

The DC Josephson current for topological insulator Josephson junction is determined by electric charge conservation rule

$$\partial_{t} P + \partial_{x} J_{x} + S = 0$$

(14)

where $P = e(\varPsi_{ \uparrow }^{\dag } \varPsi_{ \uparrow } + \varPsi_{ \downarrow }^{\dag } \varPsi_{ \downarrow } )$, $J_{x} = iev_{F} (\varPsi_{ \uparrow }^{\dag } \varPsi_{ \downarrow } - \varPsi_{ \downarrow }^{\dag } \varPsi_{ \uparrow } )$, and $S = 2e\text{Im} [\Delta^{ * } \varPsi_{ \downarrow } \varPsi_{ \uparrow } - \Delta^{ * } \varPsi_{ \uparrow } \varPsi_{ \downarrow } ]$ are electric charge density, electric current, and source term, respectively.

Therefore, after straightforward but tedious derivation following Ref. [27], we find that the total Josephson current is given by

$$I = \frac{e\Delta }{2\hbar }\sum\limits_{\sigma q} {k_{B} T\sum\limits_{{\omega_{n} }} {\frac{1}{{2\varOmega_{n} }}} } \left( {k_{n}^{e} + k_{n}^{h} } \right)\left( {\frac{{r_{n}^{1} }}{{k_{n}^{e} }} - \frac{{r_{n}^{2} }}{{k_{n}^{h} }}} \right)$$

(15)

where $k_{n}^{e}$, $k_{n}^{h}$, $r_{n}^{1}$, and $r_{n}^{2}$ are obtained from $k_{S}^{e}$, $k_{S}^{e}$, $r_{A}^{1}$, and $r_{A}^{2}$ by the analytic continuation E → iω _n , the Matsubara frequencies are $\omega_{n} = \pi k_{B} T\left( {2n + 1} \right)$ with n = 0, ±1, ±2,…, and $\varOmega_{n} = \sqrt {\omega_{n}^{2} + \Delta^{2} }$.