Formal Integration of Electron Scattering Processes via Separation of Dynamical and Geometric Contributions

Abstract

By decoupling the geometric from the dynamical contributions in the scattering processes, we develop a method to compute the scattering matrix of electrons in a one-dimensional coherent conductor connected to two electrodes. In particular, we demonstrate that, in the high-energy regime, the transmission matrix converges to the Berry operator of the system. We showcase the method through several examples featuring different in-plane magnetic field profiles. Notably, our results reveal the possibility of achieving near-perfect spin-flip transmission, highlighting potential applications in spintronics.

Figures (5)

• Figure 1: Schematic of the model: electrons propagates on a 1D wire parametrized by the longitudinal coordinate $y$, under the action of an external, static magnetic field. The wire is composed by three distinct regions: the left lead ($y \leq y_{\rm L}$), the right lead ($y_{\rm R} \leq y$) where the magnetic field is uniform, assuming constant values $\bm{B}_{\rm L}$ and $\bm{B}_{\rm R}$, respectively, and the scattering region ($y_{\rm L}<y<y_{\rm R}$) where instead the magnetic field vector $\bm{B}(y)$ (represented by purple vectors) can vary. Panel a): The plot shows the $y$-varying magnetic field $\bm{B}(y)$ of the scheme I of Sec. \ref{['sec:expI']}, with components given in \ref{['eq:bzI']} setting $q_1=0$ and $q_2=0$. In this specific case, the magnetic field has only a positive $z$ component in the left lead and a negative $z$ components in the right lead. Panel b): The plot shows the $y$-varying magnetic field $\bm{B}(y)$ of the scheme II of Sec. \ref{['sec:expII']}, with components given by Eq. \ref{['eq:bzII']} setting $q_1=0$ and $q_2=0$. In this case, the magnetic field has only a positive $z$ component in the left lead and a negative $x$ components in the right lead. To enhance clarity, we have aligned the directions $\bm{n}_1$ and $\bm{n}_3$, which define the magnetic field orientation along the wire as described in Eq. (\ref{['defBsca']}), with the $x$- and $z$-axis, respectively. Dispersion curves for the leads are shown on the sides of the plots.
• Figure 2: Transmission probabilities as a function of injection energy for Scheme I (Sec. \ref{['sec:expI']}), with $L=3L_Z$. Each panel corresponds to different values of the parameters $q_1$ and $q_2$, which are specified at the top of each panel. The value of $L=3L_Z$ was selected because it enables nearly perfect spin flip ($P_E^{\uparrow \mapsto \downarrow} \approx 1$) over the energy range from 0 to $E_{\rm Z}$ when both parameters are set to zero ($q_1=0$, $q_2=0$), as illustrated in panel a). The insets in all plots display an extended energy range, allowing access to the high-energy limit where $E / E_{\rm Z} \rightarrow \infty$. In this limit, for all cases, the red and green curves ($P_E^{\uparrow \mapsto \downarrow}$ and $P_E^{\downarrow \mapsto \uparrow}$) approach zero, while the orange curve ($P_E^{\uparrow \mapsto \uparrow} = P_E^{\downarrow \mapsto \downarrow}$) approaches 1. This behavior confirms the prediction given by Eq. \ref{['eq:tsolfin1EINF']}, where the transmission matrix at large energies coincides with the Berry operator. Notably, this result is independent of $q_1$ and $q_2$, as the Berry operator depends solely on the magnetic field components in the leads [see Eq. \ref{['eq:holo']}].
• Figure 3: Scheme I, numerical results of the Hilbert-Schmidt norms $\|t_E(L)-\mathcal{U}_{y_{\rm L}\to y_{\rm R}}\|_{\rm HS}$ and $\|r_E(L)\|_{\rm HS}$ (blue, orange, and green dotted curves) as a function of injection energy $E$. The dashed red lines represent the exact solution for the magnetic wall configuration in Scheme I (Appx. \ref{['appx:infinitesimalscattering']}), showing $\|t^{I}_E(L)-\mathcal{U}_{y_{\rm L} \to y_{\rm R}}\|_{\rm HS}$ and $\|r^{I}_E(L)\|_{\rm HS}$. In panel a) we vary $q_1$ ($q_1 = 0, 1, 10$) with $q_2 = 0$ fixed, while in panel b) we fixes $q_1 = 0$ and varies $q_2$ ($q_2 = 0, 1, 10$). In both cases, the distance between the dashed red curve and the dotted curves decreases as $q_1$ or $q_2$ increases, confirming that Scheme I (Sec. \ref{['sec:expI']}) approaches the magnetic wall configuration ($\bm{n}_{\rm L}=\bm{n}_3$, $\bm{n}_{\rm R}=-\bm{n}_3$) in the limit $q_1\to \infty$ (with $q_2=0$) or $q_2 \to \infty$ (with $q_1=0$), as discussed in Appx. \ref{['appx:magneticwall']}. For the sake fo completeness, fixing $q_1=q_2=0$ and $\beta_0(y)=1$, Equation \ref{['eq:bzI']} gives $B_1(y) = B_0 \, \sin^2\!(\frac{\pi (y - y_{\rm L})}{y_{\rm R} - y_{\rm L}})$ and $B_3(y) = B_0 \, \cos\!(\frac{\pi (y - y_{\rm L})}{y_{\rm R} - y_{\rm L}})$, which correspond to the magnetic field vector of Fig. \ref{['fig:ex1']}a).
• Figure 4: Transmission probabilities as a function of injection energy for Scheme II (Sec. \ref{['sec:expII']}), with $L=6L_Z$. Each panel corresponds to different values of the parameters $q_1$ and $q_2$, which are specified at the top of each panel. The value of $L=6L_Z$ was selected because it enables nearly perfect spin mixing ($P_E^{\downarrow \mapsto -} \approx 1$) over the energy range from 0 to $E_{\rm Z}$ when both parameters are set to zero ($q_1=0$, $q_2=0$), as illustrated in panel a). The insets in all plots display an extended energy range, allowing access to the high-energy limit where $E / E_{\rm Z} \rightarrow \infty$. In this limit, for all cases, the red, green and orange curves ($P_E^{\uparrow \mapsto +}$, $P_E^{\downarrow \mapsto -}$ and $P_E^{\downarrow \mapsto +} = P_E^{\uparrow \mapsto -}$) approach 1/2. This behavior confirms the prediction given by Eq. \ref{['eq:tsolfin1EINF']}, where the transmission matrix at large energies coincides with the Berry operator. Notably, this result is independent of $q_1$ and $q_2$, as the Berry operator depends solely on the magnetic field components in the leads [see Eq. \ref{['eq:holo']}].
• Figure 5: Scheme II, numerical results of the Hilbert-Schmidt norms $\|t_E(L)-\mathcal{U}_{y_{\rm L}\to y_{\rm R}}\|_{\rm HS}$ and $\|r_E(L)\|_{\rm HS}$ (blue, orange, and green dotted curves) as a function of injection energy $E$. The dashed red lines represent the exact solution for the magnetic wall configuration in Scheme II (Appx. \ref{['appx:magneticwall']}), showing $\|t^{\text{II}}_E(L)-\mathcal{U}_{y_{\rm L} \to y_{\rm R}}\|_{\rm HS}$ and $\|r^{\text{II}}_E(L)\|_{\rm HS}$. In panel a) we vary the integer $q_1$ ($q_1 = 0, 1, 10$) while keeping $q_2 = 0$ constant. In panel b) we fix $q_1 = 0$ and vary $q_2$ ($q_2 = 0, 1, 10$). In both cases, the distance between the dashed red curve and the dotted curves decreases as either $q_1$ or $q_2$ increases. However, the green curve in the inset of panel b) exhibits noticeable jumps, causing it to diverge from the red dashed curve. Although we did not plot the distance $\|r_E(L)\|_{\rm HS}$ for larger values of $q_2$ with $q_1=0$, we verified that this distance approaches the red dashed curve $\|r^{\text{II}}_E(L)\|_{\rm HS}$ as $q_2 \to \infty$. This behavior supports our earlier discussion in Appx. \ref{['appx:magneticwall']}: in the limiting cases where $q_1\to\infty$ (with $q_2=0$) and $q_2 \to \infty$ (with $q_1=0$), Scheme II (Sec. \ref{['sec:expII']}) reduces to the magnetic wall configuration with $\bm{n}_{\rm L}=\bm{n}_3$ and $\bm{n}_{\rm R}=\bm{n}_1$. For the sake of completeness, fixing $q_1=q_2=0$ and $\beta_1(y)=\beta_2(y)=0$, Equation \ref{['eq:bzII']} gives $B_1(y) = B_0 \, \sin^2\!(\frac{\pi (y - y_{\rm L})}{2(y_{\rm R} - y_{\rm L})})$, $B_3(y) = B_0 \, \cos^2\!( \frac{\pi (y - y_{\rm L})}{2(y_{\rm R} - y_{\rm L})})$, which correspond to the magnetic field vector of Fig. \ref{['fig:ex1']}b).