Spin resonance without spin splitting

We predict that a single-level quantum dot without discernible splitting of its spin states develops a spin-precession resonance in charge transport when embedded into a spin valve. The resonance occurs in the generic situation of Coulomb blockaded transport with ferromagnetic leads whose polarizations deviate from perfect antiparallel alignment. The resonance appears when electrically tuning the interaction-induced exchange field perpendicular to one of the polarizations—a simple condition relying on vectors in contrast to usual resonance conditions associated with energy splittings. The spin resonance can be detected by stationary $dI/dV$ spectroscopy and by oscillations in the time-averaged current using a gate-pulsing scheme. The generic noncollinearity of the ferromagnets and junction asymmetry allow for an all-electric determination of the spin-injection asymmetry, the anisotropy of spin relaxation, and the magnitude of the exchange field. We also investigate the impact of a nearby superconductor on the resonance position. Our simplistic model turns out to be generic for a broad class of coherent few-level quantum systems.

Figure 1

Schematic setup of a QD spin valve, indicating the spin-precession resonance mechanism.

Figure 2

Differential conductance $dI/d{V}_b$ for the setup shown in Fig. 1 for the current from the source into the QD for ${\Gamma }_s=2{\Gamma }_d=0.01U, T=0.05U, W=50U, n_s=n_d=0.99, \alpha =0.01\pi$. The white dashed curve follows from the resonance condition (15). Signatures in the conductance can already be found for $n_s, n_d\gtrsim 0.6$, and $\alpha <0.4\pi$ as discussed in Sec. 3b; here we use larger polarizations and smaller $\alpha$ for illustration purposes.

Figure 3

Main panel (a) and sketched zoom-in (b): Exchange field component along ${\mathbf{n}}_s$ from the source electrode ($B_s$, green), the drain electrode ($B_{d,\parallel }$, blue), and their sum ($B_s+B_{d,\parallel }$, red) as a function of $V_b$ for $V_g=0.375U$, with other parameters as in Fig. 2. (c) Illustration of the spin precession (gray) for different directions of the exchange field (red), taken for different $V_b$ as indicated in (b). The opening angle is maximal for (ii) at $V_b=V_b^{*}$ where Eq. (15) holds.

Figure 4

Differential conductance $dI/d{V}_b$ as a function of bias voltage $V_b$ for gate voltage $V_g=7.5T$, varying (a) the polarization magnitude $n_s=n_d=n$ as indicated for fixed noncollinearity angle $\alpha =0.01\pi$ and (b) varying the angle $\alpha$ as indicated for fixed $n_s=n_d=0.99$. All other parameters are as in Fig. 2.

Figure 5

Numerically computed differential conductance $dI/d{V}_b$ as a function of bias voltage $V_b$ when modifying several parameters but keeping the spin-injection asymmetry (20) fixed. (a) The tunnel couplings are varied as ${\Gamma }_d={\Gamma }_s/[2cos\left(\alpha \right)]={10}_{}^{-3}T\cdots {10}^{-1}T$ in four equidistant steps, keeping $n_s=n_d=0.99$ and $\alpha =0.01\pi$ fixed. The curves are vertically offset by ${10}^{-2}$ with respect to each other. (b) The polarization magnitudes are varied as $n_d=n_s/cos\left(\alpha \right)=0.6\cdots 0.99$ in four equidistant steps, keeping ${\Gamma }_d={\Gamma }_s/2=0.1T$ and $\alpha =0.01\pi$ constant. (c) The noncollinearity angle is varied as $\alpha =0.85\pi \cdots 0.97\pi$ in four equidistant steps, adjusting ${\Gamma }_d={\Gamma }_s/\left[2\sqrt{cos\left(\alpha \right)}\right]=0.1T$ and $n_d=n_s/\sqrt{cos\left(\alpha \right)}=0.99$. The parameters in (a)–(c) are chosen such that $a=2$, the other parameters are $U=20T, V_g=7T$, and $W=1000T$. (d) The interaction energy is varied as $U=40T\cdots 100T$ in four equidistant steps for $n_s=n_d=0.99, {\Gamma }_d={\Gamma }_s/2=0.1T, \alpha =0.01\pi , V_g=0.45U$, and $W=1000T$.

Figure 6

(a) Modification of the quantum-dot spin valve depicted in Fig. 1 including a superconducting terminal. (b) Differential conductance $dI/d{V}_b$ for setup (a) for the current from the source into the QD for ${\Gamma }_s=2{\Gamma }_d=0.01U, {\Gamma }_{sup}=0.75U, T=0.025U, W=50U, n_s=n_d=0.99, \alpha =0.01\pi$. The white (black) dashed curve follows from the resonance condition (15) including (excluding) the effect of the Andreev bound states. We excluded cotunneling from the calculations for (b). We comment on this in Appendix pp2-s4.

Figure 7

Differential conductance $dI/d{V}_b$ as a function of gate voltage $V_g$ and bias voltage $V_b$. In (a), the spin-injection ratio is $a=10$ with ${\Gamma }_s=0.1/\sqrt{cos\left(\alpha \right)}$ and ${\Gamma }_d=0.01/\sqrt{cos\left(\alpha \right)}$ and in (b) the spin-injection ratio is $a=1$ with ${\Gamma }_s={\Gamma }_d=0.01\sqrt{cos\left(\alpha \right)}T$. All other parameters are as in Fig. 2.

Figure 8

Differential conductance $dI/d{V}_b$ as a function of bias voltage $V_b$ for ${\Gamma }_s={\Gamma }_d=0.5T, n_s=n_d=0.99, \alpha =0.01\pi , U=40T$, and $W=1000T$. For the gate voltage that restores the particle-hole symmetry, $\tilde{\delta }=0$, the spin resonance is absent and the broad zero-bias anomaly of Refs. [24, 28] is visible. For $\tilde{\delta }=0.025$, when the particle-hole symmetry is absent, the conductance profile for small bias is by contrast completely dominated by the spin resonance. We chose here rather larger tunnel couplings to compare with the above-cited references. We note that the conductance is shown there for strictly antiparallel polarizations, $\alpha =0$, which has negligible impact on the zero-bias anomaly as compared to the case $\alpha =0.01\pi$ considered here. Note that the zero-bias anomaly does not appear in Fig. 7 because the tunnel couplings are smaller there, yet our spin resonance persists for these parameter values.

Figure 9

Stationary current as a function of bias voltage $V_b$ up to $O\left(\Gamma \right)$ and $O\left({\Gamma }^2\right)$ as indicated. (a) Strongly underdamped case ($b\gg 1$) in the large-$U$ limit. The current is computed first numerically from the extension of Eq. (13) to the finite-$U$ case for $U=1000T/3$ (red, denoted by $I$) and approximated by formula (31) in the limit of $U\to \infty$ (blue, denoted by $I^{appr}$). We also show the current (32) for zero spin accumulation (dashed black line, denoted by $I_0$). The other parameters are $V_g=75T/3, {\Gamma }_s=2{\Gamma }_d=0.2T/3, n_s=n_d=0.99, \alpha =0.01\pi$, and $W=1000T/3$. This choice of parameters implies $b\gtrsim 5$ [given by Eq. (37)] for both $O\left(\Gamma \right)$ and $O\left({\Gamma }^2\right)$ at the resonance. The approximated current and the numerically computed current match well but not perfectly. The main reason for the deviation is that the resonance does not appear here under strong Coulomb blockade conditions, as required for Eq. (31) to be strictly valid. These conditions are met in Fig. 10 below, where approximation and numerical solution match perfectly. However, if we go deeper into the Coulomb blockade regime here, the resonance disappears in $O\left(\Gamma \right)$, cf. Ref. [26]. Therefore, to make a comparison between the $O\left(\Gamma \right)$ and $O\left({\Gamma }^2\right)$ current, we considered the resonance closer to the single-electron tunneling regime. (b) Critically damped case ($b\approx 1$) in the finite-$U$ case: Stationary current up to $O\left(\Gamma \right)$ and $O\left({\Gamma }^2\right)$ as a function of bias voltage $V_b$ numerically calculated from Eq. (13) for $V_g=5T$ with all other parameters as in Fig. 2, implying $b\approx 0.5$ for $O\left(\Gamma \right), b\approx 0.2$ for $O\left({\Gamma }^2\right)$. Note that our approximation formula (31) cannot be applied for the finite-$U$ case employed here.

Figure 10

(a) Schematics of the pulsing scheme. (b) Stationary current as function of $V_g$, obtained by solving Eq. (13) exactly ($I_{st}$, green), by neglecting the spin accumulation, i.e., forcing $\mathbf{S}=0$ ($I_0$, dashed black), and by taking the approximation (31) near resonance ($I_{st}^{appr}$, red), see Sec. 3f. (c)–(e) Average current $\overline{I}={\int }_0^t(d{t}^{\prime }/t)I_s\left(t^{\prime }\right) (t\gg \tau ,{\tau }^0)$ (green curves) as a function of $\tau$ for three different $V_g$ as indicated and for fixed ${\tau }^0=2\times {10}^3/T=0.46{\tau }_P$, and $V_g^0=30T$. The times ${\tau }^0$ and $\tau$ are given in units of the precession period at resonance, ${\tau }_P\approx 4.7\times {10}^3/T$. The current is offset by ${\overline{I}}_{st}$, the current that would flow if the QD were in the stationary state at each instant of time. Also plotted is the spin component along the drain polarization $\mathbf{S}·{\widehat{\mathbf{n}}}_d$ (blue curves) computed from Eq. (13) for initial condition $\mathbf{S}={\widehat{\mathbf{n}}}_s/2$ and $p_1=1-p_0=1$. Throughout we used $n_s=n_d=0.99$ (see caption of Fig. 2), $\alpha =0.005\pi , V_b=50T, W=500T, {\Gamma }_s=0.15T, {\Gamma }_d=0.1T$. The plots are obtained by numerically solving the analytically derived kinetic equations (13) in the limit $U\to \infty$ using the scheme discussed in Appendix pp2-s3. To make use of analytical results, we need a tiny angle $\alpha$ here. For finite $U$, this restriction is unnecessary.

Figure 11

(a) Ratio $\overline{I}/{\overline{I}}_{st}$ as a function of the duration ${\tau }^0$ in units of the electron dwell time ${\tau }_T^0=1/I_{st}^0\approx 4.7\times {10}^4/T$. Here $\overline{I}$ is the average current (39) for the pulsing scheme and ${\overline{I}}_{st}$ is the stationary current obtained if the QD was in the stationary state all the time. The gate voltage $V_g=V_g^{*}=59.8T$ is tuned to the resonance and $\tau =2500/T\approx 0.53{\tau }_P$ so that a nearly maximal enhancement of the current occurs after the precession. (b) Average current $\overline{I}$ and time-dependent current $I_s\left(t\right)$ as function of the pulse time $\tau$ with ${\tau }_P\approx 4.7\times {10}^3/T$ and ${\tau }^0=10/T=2.1\times {10}^{-3}{\tau }_P\ll \tau$. We subtract the stationary current $I_{st}$ flowing at gate voltage $V_g=V_g^{*}$. The results shown in (a) and (b) are computed for a single pulse, $N=1$ [$t={\tau }^0+\tau$ in Eq. (39)], and all other parameters are the same as in Fig. 10.