Electrical control of spin dynamics in finite one-dimensional systems

We investigate the possibility of the electrical control of spin transfer in monoatomic chains incorporating spin impurities. Our theoretical framework is the mixed quantum-classical (Ehrenfest) description of the spin dynamics, in the spirit of the $s$-$d$ model, where the itinerant electrons are described by a tight-binding model while localized spins are treated classically. Our main focus is on the dynamical exchange interaction between two well-separated spins. This can be quantified by the transfer of excitations in the form of transverse spin oscillations. We systematically study the effect of an electrostatic gate bias $V_g$ on the interconnecting channel and we map out the long-range dynamical spin transfer as a function of $V_g$. We identify regions of $V_g$ giving rise to significant amplification of the spin transmission at low frequencies and relate this to the electronic structure of the channel.

Figure 1

Model system investigated in this work: Two localized spins are electronically connected by a monoatomic wire, where electrons can flow. An electrostatic gate is applied to some of the interconnecting sites.

Figure 2

Spin excitation of one-dimensional wire. (a) Time dependence of the spin polarization on the first site ($\Delta s_1^z$) as obtained from the numerical integration of the EOMs and directly from Eq. (6). (b) Initial spin polarization $s^z\left(t_0\right)$ as a function of site index. (c) Time and space evolution of $\Delta s_i^z$ with the color shade representing the magnitude of absolute value of $\Delta s_i^z\left(t\right)$. Note that $2\tau$ is the wave-packet round-trip time as also seen in the bottom panel.

Figure 3

dFT$\left[\Delta s_i^z\left(t\right)\right](k,\omega )$ for different band filling: (a), (d) ${\rho }_0=0.5$; (b), (e) ${\rho }_0=0.6;$ and (c), (f) ${\rho }_0=0.8$. The top panels show the exact analytical excitation spectrum for a homogeneous TB chain without local spins. The bottom panels show the results of our numerical simulations. The intersite distance $a$ is arbitrary. The color shade in the bottom panel represents the absolute value of dFT$\left[\Delta s_i^z\left(t\right)\right](k,\omega )$ and is logarithmically scaled for better contrast.

Figure 4

Schematic of the gate dependence of the ground-state energy spectrum of ${\widehat{H}}_{\mathrm{el}}$. According to their weights at the three subsections of the chain ${\Omega }_{\mathcal{L}},\mathcal{M},\mathcal{R}={\sum }_{i\in \mathcal{L},\mathcal{M},\mathcal{R}}{\left|c_{\mathrm{in}}\right|}^2$ we distinguish three types of states depending on which of the three partial weights is the greatest. We use black circles for the case of ${\Omega }_{\mathcal{L}}$, red squares for ${\Omega }_{\mathcal{M}}$, and green diamonds for ${\Omega }_{\mathcal{R}}$. Note that the eigenstates corresponding to the two ungated regions (${\Omega }_{\mathcal{L}}$ and ${\Omega }_{\mathcal{R}}$) are degenerate by symmetry, as the gate is applied in the exact center of the wire. The inset shows a magnification of an area of intermediate $V_g$, illustrating the situation of avoided crossings. A short chain with $N=30$ sites is used for simplicity.

Figure 5

dFT$\left[\Delta s_i^z\left(t\right)\right](k,\omega )$ for different values of the gate potential: (a) $V_g=0.2\gamma$, (b) $V_g=2.2\gamma$, and (c) $V_g=4.6\gamma$. The color shade is identical to that in Fig. 3.

Figure 6

Time evolution of the $x$ components of the localized spins, $S_1^x\left(t\right)$ (blue solid line) and $S_2^x\left(t\right)$ (red dashed line) [left] and the corresponding spectra, dFT$\left[S_i^x\left(t\right)\right]\left(\omega \right)$, [right] for three different gate voltages (a), (d) $V_g=0.1\gamma$; (b), (e) $V_g=2.2\gamma$; and (c), (f) $V_g=4.6\gamma$. The frequency domain is represented by the dimensionless quantity $\hslash \omega /\gamma$ and ${\overline{\omega }}_L=\hslash {\omega }_L/\gamma =1.16$ for $\gamma =1$ eV. Note that in the case of $V_g=2.2\gamma$ different scales are used for the magnitudes of $S_1^x\left(t\right)$ and $S_2^x\left(t\right)$ and their spectra [axes corresponding to $S_2^x\left(t\right)$ are marked in red and are shown on the opposite sides of the graph].

Figure 7

(a) Time evolution of the energy transfer, $\Delta E\left(t\right)$, between electronic and localized spins for $V_g=2.2\gamma$. The inset shows the maximum value of the energy transfer, $\Delta E_{\max }$, for a fixed duration of simulation (2 ps) as a function of $V_g$. A red arrow indicates the particular voltage for which $\Delta E\left(t\right)$ is plotted. (b) Evolution of the electron populations of different parts of the chain for $V_g=2.2\gamma$: left (black solid line), middle (red dashed line), and right (green dotted line). The two black dashed horizontal lines show the equilibrium populations of the gate-free parts (upper line) and of the gated middle section (lower line), i.e., the populations corresponding to the case when the gate is applied in the ground state.

Figure 8

(a) dFT$\left[s_i^x\left(t\right)\right](k,\omega )$ for $V_g=0$. (b) The difference between dFT$\left[s_i^x\left(t\right)\right](k,\omega )$ for $V_g=2.2\gamma$ and for $V_g=0$ defined as $\Delta =|\mathrm{dFT}\left[s_i^x\left(t\right)\right]{(k,\omega )}_{V_g=2.2\gamma }-\mathrm{dFT}\left[s_i^x\left(t\right)\right]{(k,\omega )}_{V_g=0}|$. The color shade represents the absolute value of dFT$\left[s_i^x\left(t\right)\right](k,\omega )$ and $\Delta$, respectively, and is logarithmically scaled for better contrast.

Figure 9

Adiabatic variation with the gate voltage of a few eigenvalues around the Fermi level (we have used ${\varepsilon }_0=0$ and $\left|\gamma \right|=1$ eV, hence $E_{\mathrm{F}}=0$ at $V_g=0$). Marked with black lines are the pairs of eigenstates $(i,i+1)$ which give rise to the highest (in absolute value) dynamical amplitude $A_{i,i+1}^N$ at the last site $N$.

Figure 10

(a) The maximum single-mode amplitude $A_{\max }^N$ and an estimate for the total low-frequency amplitude $A_{\mathrm{sum}}^N$ (see text for details) as a function of the applied gate voltage. (b) The frequency of the mode related to the pair of eigenstates ($i=15$) around the Fermi level, adiabatically evolved with the gate voltage and (c) its corresponding amplitude as defined in Eq. (A1).