Current with unequilibrated vibration

As a next step, we discuss the current for the case of unequilibrated vibration for antiparallel ferromagnetic leads. We found clear signatures of the nonequilibrium phonon occupation in terms of a suppression (enhancement) of the current when the phonon occupation of the oscillator decreases (increases) compared to thermal vibration.

For the regime of unequilibrated vibration, we use the full phonon Green’s functions (4.24) and (4.25) to calculate the electron self-energies (4.38) and (4.39) and, hence, the current Eq. (4.50) in the limit γtot (ω, Γl, Γr,T,eV). The results are similar to the previous case for the equilibrated vibration. For instance, the inelastic correction I_in is similar to (4.54) in which we have to replace the thermal phonon occupation nB(ω) with the nonequilibrium occupation ¯nas given by Eq. (4.32). For oscillators with very high quality factor, we have that ¯nis essentiallyn, as given by Eq. (4.32). A similar approach was used in other nanomechanical systems [176]. We consider such an approach reasonable for weak spin-vibration coupling and low current through the dot.At the same time, it is also useful to discuss qualitatively the behavior of the system to understand the possible features appearing in the current-voltage characteristic associated to a strongly cooled or heated oscillator. A more refined self-consistent approach, as discussed in Ref. [108], is beyond the aim of this work.

We start with the discussion of fully polarized leads in the antiparallel configuration pr=−p_l= 1. Notice that, in this case, the elastic contributions to the current vanish:

I₀ = 0 [Eq. (4.51)] and I_ec = 0 [Eq. (4.52)], since the electrons have to change their spin when tunneling from one lead to another. This can happen only through inelastic phonon-assisted spin-flip processes. Therefore, the total current Eq. (4.50) reduces to the inelastic current I_in given by Eq. (4.54) withn_B(ω) replaced by ¯n. At resonance ε_z = ω, such inelastic processes can cool the oscillator, ¯n n_B(ω), for positive voltage eV > 0 [see Fig. 4.10(e)], whereas they can heat the oscillator, ¯n nB(ω), for negative voltageeV <0 [see Fig.4.10(f)].

In Fig. 4.15(a) and (b), we compare the current at resonance for equilibrated [γ0 γ, ¯n'nB(ω)] and unequilibrated vibration [γ0 γ, ¯n'n] for fully polarized antiparallel ferromagnets at finite temperature T = 10ω. Essentially, the nonequi-librium phonon occupation corresponding to the current of Fig. 4.15(b) is shown in

4.7 Current

Figure 4.15: Inelastic current for fully polarized antiparallel ferromagnets (pr =

−p_l = 1) at resonance ε_z =ω, T = 10ω, and Γ = 0.2ω. (a) Equilibrated vibration with ¯n = nB(ω). (b) Unequilibrated vibration with a coupling constant λ = 0.01ω and an intrinsic damping of γ0 = 10⁻⁵ω. The nonequilibrium phonon occupation

¯

n corresponding to the inelastic current in (b) is shown in Fig. 4.11. For eV > 0, the oscillator is strongly cooled ¯nn_B(ω), leading to a suppression of the inelastic current in (b) compared to the case of equilibrated vibration in (a).

Fig. 4.11. For eV > 0, the current in Fig. 4.15(b) is strongly suppressed compared to the case of equilibrated vibration. In this case, the oscillator is cooled close to its quantum ground state (¯n 1) so that electrons can tunnel only through phonon-emission characterized by the rateγ_lr⁻[see Eq. (4.54)]. Since in the cooling regime the relationγ_lr⁻ γ⁺_lr holds, the current flowing through the dot results extremely low.

In other words, increasing the current implies cooling the oscillator more efficiently which turns out in a reduction of the current itself. By contrast, for eV < 0, the current strongly decreases with the voltage for unequilibrated vibration before the regime of instability is reached at some threshold voltage.

In Figs. 4.16 (a) and (b), we show the inelastic current for equilibrated and un-equilibrated vibration and different energy separation εz. The parameters are the same as in Fig. 4.15 but the current is shown as a function of voltage at ε₀ = 0. In Fig.4.16(a), the vibration-assisted spin-flip rates give the largest contribution to the current at resonance εz =ω. In Fig. 4.16 (b), we find that the current for eV > 0 is strongly suppressed compared to the current in Fig. 4.16(a). For eV < 0, the current sharply decreases since the oscillator approaches the mechanical instability.

Out of resonance, the current decreases at larger negative voltages compared to the resonant case. The differences between equilibrated and unequilibrated vibration are also visible in the differential conductancedI_in/dV shown in Figs.4.16 (c) and (d).

As last point, we analyzed the effects of a finite polarization in the current-voltage characteristic. We calculated the full current as given by Eq. (4.50) including the leading elastic term I₀ [Eq. (4.51)] and the elastic correction I_ec [Eq. (4.52)]. In Figs. 4.17 (a) and (b), we compare the current for equilibrated and unequilibrated vibrations at p_r = −p_l = 0.5. Here, we set λ = 0.2ω (ω = 2π·100 MHz) for large spin-orbit coupling estimated by recently reported measurement for the spin-orbit

Figure 4.16: Current for equilibrated (a) and unequilibrated vibration (b) for fully antiparallel polarized ferromagnets pr = −p_l = 1, T = 10ω, ε0 = 0, Γ = 0.2ω, λ= 0.01ω, andγ0 = 10⁻⁵ω. ForeV >0, the current in (b) is suppressed compared to the current in (a). At negative voltages in (b), the oscillator approaches the mechanical instability and sharply decreases. In (c) and (d), we show the differential conductance dIin/dV corresponding to (a) and (b), respectively.

coupling ∆SO in carbon nanotubes [179]. We can still observe the strong suppression of the current at eV >0 compared to equilibrated vibration in Fig.4.17 (a) as well as the sharp decrease of the current when the oscillator approaches the mechanical instability. At positive voltages in Fig.4.17(b), the current is dominated by the elastic current I0, since the oscillator is strongly cooled. The sharp decrease at negative voltages occurs due to the corrections to the current. Note the different scales of the current in Fig.4.17and 4.16. The differential conductances corresponding to the current in Fig. 4.17(a) and (b), are shown in Fig. 4.17(c) and (d), respectively.

Summarizing this section we note that the current follows the nonequilibrium phonon occupation in some regime. Since the current strongly depends on the po-larization and alignment of the ferromagnets, transport measurements with tunable nano-ferromagnetic contacts can provide a feasible way to detect the spin-vibration interaction in suspended CNTQDs.