Nonequilibrium spintronic transport through Kondo impurities

The mean current $J(V)$ and the corresponding differential conductance $G(V)$ through the system calculated at the particle-hole symmetry point ($\varepsilon_{d}=-U/2$) for different values of the lead spin polarization $p$ are presented in Fig. 2. For this we always evaluate the symmetrized current as discussed in App. A.2 [cf. Eq. (14)]. For $p=0$, we observe a zero-bias conductance peak, characteristic of the Kondo effect [6, 7]. However when $p$ is finite, the Kondo temperature is found to decrease with increasing the lead spin polarization. This was predicted to affect the Kondo temperature of the system at equilibrium by using the poor man’s scaling method as [8]

The decrease of the Kondo energy scale with spin polarization can be understood by realizing that by construction with Eq. (5), increasing polarization reduces the hybridization of the suppressed spin orientation. As such, this decreases the rate of spin-flip cotunneling processes responsible for the Kondo effect.

To quantitatively elucidate the influence of $p$ on the Kondo effect, we define the Kondo energy scale $V_{\mathrm{K}}$ in the applied bias voltage as the half maxima point of the conductance curve, i.e., $G(V_{\mathrm{K}})/G(0)=\frac{1}{2}$ at $T=B=0$. In Fig. 3 we present the dependence of $V_{\mathrm{K}}$ obtained from our NRG-tDMRG numerical calculations along with the Kondo temperature $T_{\mathrm{K},p}$ estimated from Eq. (10) by using the poor man’s scaling, and $T_{\mathrm{K}}(p)$ calculated using the equilibrium NRG [32] from the temperature dependence of the linear conductance based on the definition $G(T_{\mathrm{K}})/G(0)=\frac{1}{2}$. Our nonequilibrium data corroborates the general tendency to decrease the Kondo energy scale with increasing the spin polarization $p$. However, Fig. 3 also demonstrates some deviations: $V_{\mathrm{K}}$ is slightly larger than the equilibrium $T_{\mathrm{K}}$, but smaller than the Kondo temperature predicted by the analytical formula (10), after normalizing the Kondo energy scales with respect to their respective values at $p=0$.

We now discuss the behavior of the differential conductance in the case when the energy level is away from the p-h symmetry point ($\varepsilon_{d}=-U/3$), but still in the local moment regime where the strong electron correlations play a vital role. The solid lines in Figure 4(a) show the bias dependence of the conductance with increase in the lead spin polarization $p$, computed at zero external magnetic field. One observes a finite zero-bias peak that gets suppressed when $p$ grows. This effect can be attributed to the emergence of exchange field in the system, cf. Eq (8). The exchange field introduces a spin splitting of the orbital level, which suppresses the Kondo resonance, once $|\Delta\varepsilon_{\rm exch}|\gtrsim T_{\mathrm{K}},V_{\mathrm{K}}$. The color-coded arrows in Fig. 4(a) indicate the magnitude of the exchange field for the corresponding spin polarizations obtained from Eq. (8) with $T=0$. When the exchange field energy approaches the Kondo energy scale of the system, $|\Delta\varepsilon_{\rm exch}|\approx T_{\mathrm{K}}$, the zero-bias conductance becomes suppressed. When increasing the spin polarization further, the differential conductance starts to develop a peak around $V\equiv V_{\mathrm{peak}}\approx|\Delta\varepsilon_{\rm exch}|$, which is a reminiscent of the splitting of the local density of states (lDOS) vs. frequency in the presence of a sufficiently strong local magnetic field. To be specific, the peak in the differential conductance presented in Fig. 4(a) emerges for $p\gtrsim 0.1$. For this value of spin polarization, one can find that $T_{\mathrm{K},p}=3.66\cdot 10^{-5},|\Delta\varepsilon_{\rm exch}^{p}|\simeq 4.4\cdot 10^{-5}=1.2\,T_{\mathrm{K},p}$. Increasing the polarization further, the peak at $V\approx|\Delta\varepsilon_{\rm exch}|$ persists while at the same time, the conductance overall also diminishes.

The dotted lines in Fig. 4(a) correspond to the case in which the system has no exchange field (i.e., $p=0$), but there is an external magnetic field applied, whose magnitude equals the exchange field calculated from Eq. (8) according to the spin polarizations mentioned in Fig. 4(a). This comparison shows two major differences between the exchange field and the magnetic field. Firstly, a strong enough exchange field suppresses the split-Kondo peak in the differential conductance significantly more strongly and only leaves a residual conductance derived from the hybridization side peaks energies $V\simeq\varepsilon_{d}$ [note the log-scale in Fig. 4 (a)]. This is mainly attributed to the fact that the Kondo scale gets reduced with increasing the spin polarization [cf. Fig. 3], such that the ratio $|\Delta\varepsilon_{\rm exch}|/T_{\mathrm{K}}$ is enhanced for the presence of an exchange field when compared to a local magnetic field. Secondly, the location of the split Kondo peak for finite $p$ occurs at slightly higher voltages than for the case of a local magnetic field. The latter effect may be attributed to $B\approx\Delta\varepsilon_{\rm exch}$ representing a lowest-order estimate. The explicit dependence of $V_{\mathrm{peak}}$ on the spin polarization $p$ in the two above-discussed cases is shown in Fig. 4(b). For comparison, we also present the $p$-dependence of $\Delta\varepsilon_{\rm exch}$ and $T_{\mathrm{K},p}$ estimated from the respective analytical formulas. One can see that indeed the split Kondo peak emerges when $|\Delta\varepsilon_{\rm exch}|\gtrsim T_{\mathrm{K},p}$. Moreover, by comparing $\Delta\varepsilon_{\rm exch}$ and $T_{\mathrm{K},p}$, one can find that these two energy scales become equal for $p=0.0834$. Keeping in mind that this is an approximate estimate, our numerical results corroborate this tendency very well. The split-Kondo peak shows a slightly nonlinear behavior around low spin polarizations. We fit the $V_{\mathrm{peak}}^{2}$ data against $p^{2}$ to unveil any behavior of the form $V_{\mathrm{peak}}{\color[rgb]{.0,.5,.2}{\sim}}\sqrt{p^{2}-p_{0}^{2}}$. Both the fits for the exchange field and the corresponding magnetic field give essentially the same value of $p_{0}{\color[rgb]{.0,.5,.2}{\simeq 0.071}}$ indicated by the grey vertical line on panel Fig. 4(b). The prefactor of the fit is exactly one ($1.001$) within numerical accuracy in the presence of polarization, having $V_{\rm peak}\simeq\sqrt{(\Delta\varepsilon_{\rm exch})^{2}-(\gamma p_{0})^{2}}$ [cf. Eq. (9)]. This is also clearly seen in Fig. 4(b) in that the fit exactly coincides with $\Delta\varepsilon_{\rm exch}$ for larger $p$. In the case of a substitute local magnetic field $B=\Delta\varepsilon_{\rm exch}^{p}$ but unpolarized leads, the fit reads $V_{\rm peak}\simeq 0.860\sqrt{(\Delta\varepsilon_{\rm exch})^{2}-(\gamma p_{0})^{2}}$. This systematically offsets the peaks with the dashed data in Fig. 4(a) by a constant factor $0.860$ towards slightly smaller values of the bias voltage, yet leads to a disappearance of the split-peak at around the same polarization $p_{0}$. On the semilog scale in Fig. 4(b) this change in the prefactor simply shifts the fits vertically relative to each other as also reflected in the data for the full polarization range.

The symbols on the left vertical axis in Fig. 4(a) correspond to the linear response data obtained by NRG, which is equivalent to the differential conductance for $V\to 0$. As also seen in later figures, while we have good overall consistency [e.g., see inset of Fig. 6(b)], there are minor quantitative differences in the NRG-tDMRG results while comparing with the linear-response NRG results. These are attributed to the different parametrization and discretization schemes. Specifically, linear conductance within linear response in NRG can be obtained strictly at $V=0^{+}$ [33]. In constrast, the NRG-tDMRG approach always must assume a small but finite voltage in the presence of a finite level spacing with the objective to numerically compute a steady-state current via a real-time simulation.

In Fig. 5 we study the influence of external magnetic field on the split Kondo peak exhibited by the system detuned out of the p-h symmetry point assuming the lead spin polarization $p=0.2$. We observe a full restoration of the zero-bias Kondo resonance by an applied magnetic field with magnitude that can counterbalance the spin splitting induced by the exchange field, see the curve for $B=1.1\,\Delta\varepsilon_{\rm exch}^{p}$ in Fig. 5. However, a further increase in magnetic field is shown to suppress the zero-bias peak again. This behavior qualitatively matches the experimental results discussed in the Fig. 2 of the Ref. [14]. As seen from the color-coded arrows in Fig. 5, the position of the split Kondo resonance corresponds to $V\approx|B_{\mathrm{tot}}^{\mathrm{eff}}|$ as defined in Eq. (7). The revival of the Kondo resonance can be distinctly observed from the inset of Fig. 5 where $G(V\rightarrow 0)$ exhibits a maximum around $B_{\rm max}\simeq\Delta\varepsilon_{\rm exch}^{p}$ such that $B_{\mathrm{tot}}^{\mathrm{eff}}\simeq 0$ [Eq. (7)]. More precisely, from in the inset of Fig. 5, $B_{\rm max}=1.12\,|\Delta\varepsilon_{\rm exch}^{p=0.2}|$, with the small difference primarily attributed to the perturbative nature of the analytic formula Eq. (8). The prefactor approximately coincides with a similar scale factor already encountered with Fig. 4(b) where $B=\Delta\varepsilon_{\rm exch}^{p}$ also underestimated the peak position by an approximate factor $1/0.860=1.16$.

In this section we analyze the effect of finite temperature on the split Kondo resonance. Figure 6 shows the bias voltage dependence of the differential conductance for various temperatures calculated for $\varepsilon_{d}=-U/3$ and $p=0.2$. One can see that increasing $T$ results in the suppression of the split Kondo peak, which completely disappears once the thermal energy exceeds the induced exchange splitting. Increasing temperature still further overall suppresses the differential conductance. The suppression of the split-Kondo peak is accompanied with a weak increase of the conductance at zero bias for temperatures corresponding to the splitting of the lDOS due to the exchange field, as seen in the inset of Fig. 6(b). This can be used to estimate the temperature where the splitting in the differential conductance disappears. The split-Kondo peak can survive up to a maximum temperature $T_{\mathrm{max}}$ defined as the temperature in which $G(V\rightarrow 0,T)=G(V=V_{\mathrm{peak}},T)$. For the spin polarization $p=0.2$, we estimate $T_{\mathrm{max}}=2.06\cdot 10^{-2}\,\Gamma=T_{\mathrm{K},0.2}$.

The $V\rightarrow 0$ differential conductance is equivalent to linear-response in thermal equilibrium. The latter is readily obtained by NRG, with a direct comparison shown in the inset of Fig. 6(b). Overall, we observe good quantitative agreement. The points corresponding to the temperatures plotted in the main panels are marked by the same color-matched symbols (squares). Since linear response can be efficiently obtained by NRG, this permits a more dense set of data points in the inset.

The weak increase in the linear-response conductance for a finite temperature can be explained by examining the energy dependence of the equilibrium local density of states, i.e., the impurity spectral function, assuming that this lDOS changes only weakly at low temperatures $T\lesssim\mathrm{max}(T_{\mathrm{K}},|\Delta\varepsilon_{\rm exch}|)$. The linear-response conductance $G=\Sigma_{\sigma}G_{\sigma}$ is obtained from the spectral function using $G_{\sigma}(T)=\frac{\pi e^{2}}{h}\int_{-D}^{D}d\omega\,\Gamma_{\sigma}A_{\sigma}(\omega)[-f^{\prime}(\omega)]$ [33], where $A_{\sigma}(\omega)=-\frac{1}{\pi}\mathrm{Im}\,G_{\sigma}(\omega)$ is the spin-resolved spectral function based on the retarded impurity Green’s function $G_{\sigma}(\omega)$, and $f^{\prime}(\omega)$ is the derivative of the Fermi function at temperature $T$. Now if the exchange field due to polarization is sufficiently strong, $|\Delta\varepsilon_{\rm exch}|\gtrsim T_{\mathrm{K}}$, this will already split the spin-averaged lDOS at equilibrium, as shown for $p=0.2$ in Fig. 7. When temperature is increased, the transport window widens, and thus encompasses more weight from the split peaks. Assuming that the lDOS only changes weakly by turning on a small temperature $T\lesssim\mathrm{max}(T_{\mathrm{K}},|\Delta\varepsilon_{\rm exch}|)$, the contributions from the peak in the spectral function around $\omega\approx|\Delta\varepsilon_{\rm exch}|$ will therefore increase the linear-response conductance up to $T\lesssim|\Delta\varepsilon_{\rm exch}|$, where it reaches a maximum before it starts to decrease.

An explicit temperature dependence of the split-Kondo peak conductance for a few selected values of spin polarization is shown in Fig. 8. This figure is determined at finite bias voltage $V(p)\approx|\Delta\varepsilon_{\rm exch}^{p}|$, i.e. at the voltage corresponding to the location of the split Kondo peak $V_{\mathrm{peak}}$ shown in Fig. 4. As seen by the vertical markers in Fig. 8, $V$ agrees well with $|\Delta\varepsilon_{\rm exch}|$ for large polarization $p$, but clearly starts to differ for smaller $p$, given that there is no peak at finite $V$ for $p\lesssim 0.0834$. By starting from the peak conductance, one can now clearly see in Fig. 8 the decrease of the remaining side Kondo resonance as the temperature increases. The logarithmic decrease in the split-Kondo peak conductance at higher temperatures has been experimentally observed in the Fig. 3a of Ref.[9]. In the case of $p=0.1$, the split Kondo peak just emerged, having $\Delta\varepsilon_{\rm exch}^{p=0.1}\approx T_{\mathrm{K}}$, as can be observed from Fig. 4 and the vertical blue lines in Fig. 8. Hence, we can see a slight non-monotonic behavior arising from the interplay between the Kondo effect and the exchange field. More generally, one can infer from Figs. 6 and 8, for the split-Kondo regime, i.e., sufficiently strong polarization $p$ with $|\Delta\varepsilon_{\rm exch}|>T_{\mathrm{K}}$, that $G_{V}$ vs. $T$ [$G_{T}$ vs. $V$] will exhibit a non-monotonic behavior if $T<|\Delta\varepsilon_{\rm exch}|$ [$V<|\Delta\varepsilon_{\rm exch}|$], yet a monotonic decay if $T\gtrsim|\Delta\varepsilon_{\rm exch}|$ [or $V\gtrsim|\Delta\varepsilon_{\rm exch}|$], respectively. We also note that the temperature dependence of the non-equilibrium differential conductance at $V\approx|\Delta\varepsilon_{\rm exch}|$ does not show a universal dependence. This can be understood by realizing that the system is then out of the Kondo regime.