4.4.2. Frequency Dependence

¯

We next consider the dependence of the spin current on the frequency of precession Ω = 2*π*/T . Here we change the period of precession T = *Nδt* by varying the time interval *δt* while the number of divisions remains fixed to *N* = 20. All other parameters and initial conditions are set as before.

In Figure 6, we plot the dependence of the averaged spin current *<sup>I</sup>*spin against the normalized frequency Ω ¯ ≡ Ω/*<sup>ω</sup>*u.

The frequency dependence of *<sup>I</sup>*spin features two characteristic regimes: a low-frequency regime, where *<sup>I</sup>*spin depends linearly on Ω (Ω ¯ 0.0025) and a high-frequency regime, where *<sup>I</sup>*spin exhibits oscillations with respect to Ω. These characteristics are explained by comparing the time interval *δ*¯*t* and the relaxation time *τ*¯*r* of the population of dot electrons (*τ*¯*r* ≈ 5 in the present case; see Figure 5A). For lower frequencies, for which *δ*¯*t <sup>τ</sup>*¯*r*, the numerator of the time integral of *J*spin(*t*) in Equation (36) becomes constant because the instantaneous spin current has already vanished at a certain ¯*t δ*¯*t* (see Figure 5A), which results in the linear dependence of *<sup>I</sup>*spin on Ω ¯ . As Ω ¯ becomes larger and the time interval satisfies *δ*¯*t <sup>τ</sup>*¯*r*, the angle *φ* changes during the relaxation process. In this situation, the electron dynamics exhibits two extreme features; when *δ*¯*t* is an integer multiple of the period of the Larmor precession *h*/2*M*, we have resonance enhancement of the transition between half-filled states |0, 1 and |1, 0 by the sudden change of *φ* to exhibit a maximum of *<sup>I</sup>*spin, whereas it is anti-resonantly suppressed to exhibit a minimum when *δ*¯*t* is a half-integer multiple of the period [58].

**Figure 6.** Frequency dependence of the temporal average of spin current *<sup>I</sup>*spin. The division number of the step-like precession is now set to *N* = 20. With fixed *δφ* = *π*/10, the frequency is changed by changing *δt*. The frequency dependence exhibits two characteristic features: the spin current depends linearly on Ω ¯ for Ω ¯ 0.0025, whereas it exhibits oscillation with respect to Ω ¯ for Ω ¯ - 0.0025. The other parameters are the same as in Figure 5.

Calculating the spin current for different values of *θ*, we find that the spin polarization of the spin current exhibits a dependence on *θ* in that for 0 < *θ* < *π*/2 the spin polarization is antiparallel to the *z*-axis, whereas for *π*/2 < *θ* < *π* the spin polarization is parallel to the *z*-axis. For *θ* = 0, *π*/2, *π*, the spin current vanishes because the spin flip in the quantum dot does not occur for *θ* = 0, *π* or the two half-filled states in the dot |1, 0 and |0, 1 degenerate for *θ* = *π*/4 (see Equation (29)).

Finally, we note that the averaged spin current *<sup>I</sup>*spin diverges with respect to Ω¯ . The divergence is caused by the accumulation of a nonzero impetus of current *J*spin(*t*) just after the sudden change of *φ* (see Figure 5). In Reference [60], we showed that the nonzero impetus of *J*spin(*t*) is an unphysical effect caused by the Born-Markovian approximation, and the divergence is eliminated by taking into account the non-Markovian effect by keeping the upper bound of the time integration in (10) finite.
