Conductance oscillations of antiferromagnetic layer tunnel junctions

We study conductance oscillations of antiferromagnetic layer tunnel junctions composed of antiferromagnetic topological insulators such as ${\mathrm{MnBi}}_2{\mathrm{Te}}_4$. In the presence of an in-plane magnetic field, we find that the two terminal differential conductances across the junction oscillates as a function of field strength. Notably, the quantum interference at weak fields for the even-layer case is distinctive from the odd-layer case due to the scattering phase shift $\pi$. Consequently, the differential conductance vanishes (maximized) at integer magnetic flux quanta for even-layer (odd-layer) junctions. The conductance oscillations manifest the layer-dependent quantum interference in which symmetries and scattering phases play essential roles. In numerical calculations, we observe that the quantum interference undergoes an evolution from superconducting quantum interference device-like to Fraunhofer-like oscillations as the junction length increases.

Figure 1

Schematic view of the ALTJ. Out-of-plane electric fields are applied to the left ($x<-L_x/2$) and the right ($x>L_x/2$) regions. The junction region is assumed to be free of external electric fields. The purple and green colors depict top and bottom surface states crossing the Fermi level, respectively. An in-plane magnetic field is applied along the $y$-axis.

Figure 2

Band alignment of the ALTJ for the calculation of the transmission probability. In the left region, where $V_g\left(x\right)<0$, only the top surface state (purple) crosses the Fermi energy $E_F$. In the right region, where $V_g\left(x\right)>0$, the bottom surface state (green) crosses the Fermi energy. In the junction region, top and bottom surface states are nearly degenerate such that both cross the Fermi energy.

Figure 3

(a) By combining scattering matrices $S_t, S_b$, and $S_{tb}$, we calculate the transmission probability $T={\left|t\right|}^2 {\mathrm{\Phi }}_t^{+/-}$ (${\mathrm{\Phi }}_b^{+/-}$) are the right/left going modes of top (bottom) surface states. (b)–(d) Possible scattering processes for thick films. (b) Most dominating scattering processes for weak connection between top and bottom layers. Note that two scattering processes in (b) form a quantum interference loop enclosing a magnetic flux (if present). (c) Next relevant scattering process for thick layers. (d) Third-order scattering between top and bottom layers, typically negligible compared to (b) and (c).

Figure 4

Symmetries of ALTJs due to the antiferromagnetic ordering across the layers of ALTJs. The up and down arrows represent the direction of magnetization. In (a), the odd-layer films exhibits space inversion symmetry between top and bottom layers of ALTJs. In (b), ALTJs of even-layer films acquire a combined inversion symmetry with space inversion and additional spin inversion.

Figure 5

Conductance oscillations $G\left(B\right)/G_0$ of short ALTJs of odd-layer films (left panels) and even-layer films (right panels). The conductance oscillations of odd- and even-layer films at $V_g=-20\text{meV}$ are depicted in (a) and (b), respectively. The density plots of $G\left(B\right)/G_0$ are provided in (c) and (d) for odd- and even-layer films, respectively. We use parameters $E_F=36\text{meV}, m_{\text{e}}=m_{\text{o}}=35.02\text{meV}$, and $m_{tb}=0.35\text{meV}$ for 11 and 10 quintuple layers. The length of the junction is $L_x=4\text{nm}$, which is much smaller than $\hslash v_F/E_F=9\text{nm}$.

Figure 6

Conductance oscillations $G/G_0=T\left(B\right)$ of ALTJs as a function of magnetic field $B$ for (a), (c), and (e) odd and (b), (d), and (f) even layers with different junction lengths $L_x=4,18,40\text{nm}$. For better displays of the nodes $G/G_0\sim 0$, we provide $G/G_0$ with log scale as an upper inset of each panel. The numerical calculations are obtained with 11 and 10 layers with $E_F=36$ meV.