Self-excited oscillations of charge-spin accumulation due to single-electron tunneling

We theoretically study electronic transport through a layer of quantum dots connecting two metallic leads. By the inclusion of an inductor in series with the junction, we show that steady electronic transport in such a system may be unstable with respect to temporal oscillations caused by an interplay between the Coulomb blockade of tunneling and spin accumulation in the dots. When this instability occurs, a new stable regime is reached, where the average spin and charge in the dots oscillate periodically in time. The frequency of these oscillations is typically on the order of 1GHz for realistic values of the junction parameters.

Figure 1

Charge and spin transport through a layer of quantum dots sandwiched between a N and a F lead subjected to an external magnetic field $\vec{B}$ is studied. The junction is symmetrically biased by the voltage $2V_b$.

Figure 2

Electronic energy scheme ($\mathbf{P}$ configuration) showing the nonmagnetic band (N) and the bands for $\uparrow$ (minority) and $\downarrow$ (majority) spins in the F. The latter are split by $2I$, where $I$ is the ferromagnetic exchange energy. The applied bias voltage $2V_b$ shifts the electron energy by $\pm e{V}_b$ in N/F lead, respectively, from the chemical potential ${\mu }_0$. Here ${ϵ}_{\downarrow }$ and ${ϵ}_{\uparrow }$ are the spin-dependent energy levels in the dots.

Figure 3

Single-dot IVC with corresponding on-dot charge $\left(P_c\right)$ and spin $\left(P_s\right)$ accumulation probabilities for: (a) an N-type junction with ${\gamma }_{\uparrow }=0.085$ and ${\gamma }_{\downarrow }=0.115$ $\left(\xi =0.1\right)$ and (b) an F-type junction with ${\gamma }_{\uparrow }=8.5$ and ${\gamma }_{\downarrow }=11.5$ $\left(\xi =10\right)$. The current is normalized as $\overline{j_0}=j_0/e{\Gamma }_{\text{N}}$. We take ${ϵ}_{\uparrow }=2{ϵ}_{\downarrow }$, temperature $k_{B}T/\Delta ϵ=0.05$ and introduce the dimensionless voltages $\overline{V}=eV/\Delta ϵ$; ${\overline{V}}_{\sigma }={ϵ}_{\sigma }/\Delta ϵ$. One can see that the IVC is not symmetric with respect to voltage biasing $\overline{V_b}\to -\overline{V_b}$. Also, in NF biasing, the N-type junction produces a NDC around ${\overline{V}}_{\uparrow }=2$ in contrast to the F-type junction. For FN biasing the NDC is absent in both cases.

Figure 4

(a) An IVC section with a NDC around $V_b=V_{\uparrow }$ appears for parameters ${\gamma }_{\uparrow }$ and ${\gamma }_{\downarrow }$ within the range corresponding to the shaded area based on the condition $\vartheta \left({\gamma }_{\uparrow },{\gamma }_{\downarrow }\right)<0$. (b) The same condition expressed in terms of the normalized densities of states ${\overline{g}}_{\sigma }\equiv g_{\text{F}}^{\sigma }/g_{\text{N}}$ for various values of the geometrical asymmetry parameter $\xi$ characterizing N junctions $\left(\xi <1\right)$ and F junctions $\left(\xi >1\right)$. The diagonal line divides the region of parameters for the $\mathbf{P}$ configuration $\left({\overline{g}}_{\downarrow }>{\overline{g}}_{\uparrow }\right)$ and the $\mathbf{A}$ configuration $\left({\overline{g}}_{\downarrow }<{\overline{g}}_{\uparrow }\right)$. Evidently, NDC exists in $\mathbf{P}$ configuration only.

Figure 5

Circuit scheme used for the instability investigation: $C$ is the intrinsic junction capacitance, $\mathcal{L}$ is the inductance of the circuit, and $J$ is the current flowing through the inductor.

Figure 6

The instability condition in $\left({\gamma }_{\uparrow },{\gamma }_{\downarrow }\right)$-parameter space: the light gray area shows the range of parameters in which $R_d<0$ and the instability arises for $\mathcal{L}>{\mathcal{L}}_c$. The dark gray area shows the range of parameters in which $R_d>0$ and the instability arises for ${\mathcal{L}}_c<\mathcal{L}<{\mathcal{L}}_c^{\diamond }$. Here $V_b=V_{\uparrow }$ and $k_{B}T/\Delta ϵ=0.05$.

Figure 7

(a) The limit cycle in $\left(\overline{V},P_c,P_s\right)$ space around the fixed point $\left(0.5,0.905,-0.743\right)$. (b) The limit cycle in the $\left(\overline{V},\overline{J}\right)$ space around the fixed point (0.5,0.102) at the IVC (gray curve). We use scales: $\overline{V}=eV/\Delta ϵ$, $\overline{J}=J/e{\Gamma }_{\text{N}}\mathcal{N}$, $\overline{\mathcal{L}}=\mathcal{L}{e}^2{\Gamma }_{\text{N}}^2\mathcal{N}/\Delta ϵ$, and $\overline{\omega }=\omega /{\Gamma }_{\text{N}}$. Voltage is measured with respect to ${\overline{V}}_0=\left({\overline{V}}_{\uparrow }+{\overline{V}}_{\downarrow }\right)/2$. Here the choice of parameters is ${\overline{V}}_b={\overline{V}}_{\uparrow }$, $k_{B}T/\Delta ϵ=0.05$, ${\gamma }_{\uparrow }=0.085$, ${\gamma }_{\downarrow }=0.115$, $\xi =0.1$, $\overline{\mathcal{L}}=8.0$ with critical values for instability ${\overline{\mathcal{L}}}_c=7.8712$, and ${\overline{\omega }}_c=0.5457$. This choice of parameters ($\mathbf{P}$ configuration) leads to an IVC with NDC and the frequency of stable oscillations $\overline{\omega }=0.5456$.

Figure 8

The limit cycle appearing as the result of the instability in the system with positive differential conductance at the stationary IVC (gray curve) in $\mathbf{A}$ configuration, (a) around the stationary points $\left(0.5,0.981,-0.948\right)$ in $\left(\overline{V},P_c,P_s\right)$ space, and (b) around stationary point (0.5,0.0206) in $\left(\overline{V},\overline{J}\right)$ space. Used scales and voltage origin are the same as in Fig. 7. The choice of parameters is ${\overline{V}}_b={\overline{V}}_{\uparrow }$, $k_{B}T/\Delta ϵ=0.05$, ${\gamma }_{\uparrow }=0.08$, ${\gamma }_{\downarrow }=0.02$, and $\overline{\mathcal{L}}=45.0$. The critical values between which the instability takes place are ${\overline{\mathcal{L}}}_c=41.112$; ${\overline{\omega }}_c=0.5883$ and ${\overline{\mathcal{L}}}_c^{\diamond }=257.301$; ${\overline{\omega }}_c^{\diamond }=0.3272$. The frequency of the stable oscillations $\overline{\omega }=0.5865$.