Origin of the Giant Spin-Detection Efficiency in Tunnel-Barrier-Based Electrical Spin Detectors

Efficient conversion of a spin signal into an electric voltage in mainstream semiconductors is one of the grand challenges of spintronics. This process is commonly achieved via a ferromagnetic tunnel barrier, where nonlinear electric transport occurs. In this work, we demonstrate that nonlinearity may lead to a spin-to-charge conversion efficiency larger than 10 times the spin polarization of the tunnel barrier when the latter is under a bias of a few millivolts. We identify the underlying mechanisms responsible for this remarkably efficient spin detection as the tunnel-barrier deformation and the conduction-band shift resulting from a change of applied voltage. In addition, we derive an approximate analytical expression for the detector spin sensitivity $P_{\text{det}}(V)$. Calculations performed for different barrier shapes show that this enhancement is present in oxide barriers as well as in Schottky-tunnel barriers, even if the dominant mechanisms differ with the barrier type. Moreover, although the spin signal is reduced at high temperatures, it remains superior to the value predicted by the linear model. Our findings shed light onto the interpretation and understanding of electrical spin-detection experiments and open paths to optimizing the performance of spin-transport devices.

Figure 1

The nonlinear spin-detection efficiency under bias. (a) A schematic energy-band diagram of the FM/B/NM tunnel contact under a voltage bias, where ${\mu }_v=-e{V}_{\text{app}}$ is the applied electrochemical potential and $w$ and ${\mathrm{\Phi }}_B$ are the width and the height of the barrier. A drawing of the entire spintronic device is also provided for clarity. (b) The computed spin-detection efficiency with tunnel bias for two different spin accumulations ${\mu }_s$ ($w=4$ nm; ${\mathrm{\Phi }}_B=1$ eV; $P_G=50\mathrm{\%}$; $T=1$ K). The dashed line is obtained by canceling the barrier deformation under bias, corresponding to the linear model. (c) The spin-detection efficiency as a function of the applied voltage for different barrier dimensions. The dashed lines correspond to results reported in Ref. [29].

Figure 2

(a) A schematic representation of the barrier deformation for two different values of the applied voltage associated with ${\mu }_v=-e{V}_{\text{app}}$ (light color) and ${\mu }_v^{\mathrm{\prime }}=-e{V}_{\text{app}}-e{V}_{\text{comp}}$ (dark color). (b) Sketches of the corresponding transmission functions under these conditions: the hatched areas correspond to the two contributions to the current variation due to the bias change. $S_1$ is integrated from $-\mathrm{\infty }$ to ${\mu }_v+{\mu }_s/2$ and represents the change of current due to the reduction of the barrier permeability compensated by the increased number of available carriers with a longitudinal energy lower than ${\mu }_v+{\mu }_s/2$. In the linear regime of spin detection, $S_1=0$. $S_2$ is integrated from ${\mu }_v+{\mu }_s/2$ to ${\mu }_v^{\mathrm{\prime }}$ and corresponds to the gain of current resulting from the enhancement of $V_{\text{spin}}$ in comparison with the linear model. (c) The variation of $\sqrt{|S_1-S_2|}$ with the spin voltage and the applied bias. Along the dashed line, the two areas are equal and $V_{\text{comp}}=V_{\text{spin}}$.

Figure 3

The competition between the gain and loss of tunnel current in a spin-detection experiment bearing nonlinear transport. The evolution of integration areas $S_i$ (index $i$ refers to $1$ or $2$) with (a) the applied bias (semilogarithmic scale) and (b) the compensation voltage (linear scale). (c) The relative shift of $S_2$ (blue) in comparison to $S_1$ (orange) for different spin voltages (semilogarithmic scale).

Figure 4

The effect of the variation of the rectangular-barrier dimensions on the spin-detection efficiency. The results are obtained for $P_G=50\mathrm{\%}$ and ${\mu }_s=10$ meV.

Figure 5

The effect of the SC band gap on the spin-detection efficiency. (a) A schematic representation of a FM/B/SC structure for a degenerated SC. The Fermi level is located at an energy value of $ϵ$ above the bottom of the conduction band $E_c$. The barrier and the $E_c$ edge change as the bias is set to ${\mu }_v^{\mathrm{\prime }}$ (without spin accumulation ${\mu }_s=0$—dark color) instead of ${\mu }_v$ (with spin accumulation—light color). (b) A qualitative sketch of the corresponding transmission function. The $S_3$ integral is associated with the current loss due to the energy shift of the band gap. (c) A quantitative analysis of the variation of the areas $S_1$, $S_2$, and $S_3$ with the applied junction voltage for $V_{\text{comp}}=-20$ mV, ${\mu }_s=10$ meV and for two different values of $ϵ$. The areas (for each value of $ϵ$) are normalized by $S_1(0)$ and are offset.

Figure 6

The effect of the variation of the rectangular-barrier parameters—(a) $P_G$, (b) $ϵ$, (c) $w$, and (d) ${\mathrm{\Phi }}_B$—on the spin-detection efficiency when the band gap affects the tunnel transport. The fixed parameters are $P_G=50\mathrm{\%}$, $ϵ=10$ meV, $w=3$ nm, and ${\mathrm{\Phi }}_B=1$ eV. The bottom panels compare results of simulations with the analytical solution from Eq. (14) for $V_{\text{app}}=-0.8$ V.

Figure 7

(a) A schematic representation of the different shapes of the interface barrier, from the background to the front: rectangular, triangular, parabolic, and exponential evolution of the barrier height with the distance from the SC. (b) The energy dependence of the transmission probability through the barrier. (c),(d) The nonlinearity of the spin-detection efficiency with the applied voltage for tunnel barriers with various shapes. The results depicted in (c) show the effect of the barrier shape for a FM/B/NM configuration, while those shown in (d) focus on the FM/B/SC case. The calculations are performed using $w=3$ nm (except for the rectangular barrier, where $w=2$ nm), ${\mathrm{\Phi }}_B=1$ eV, $P_G=100\mathrm{\%}$, $T=1$ K, and $ϵ=50$ meV.

Figure 8

The consequences of the variation of the spin-detection efficiency with the spin accumulation. (a) The computed value of $P_{\text{det}}$ for different spin accumulations in the vicinity of a tunnel rectangular barrier ($w=3$ nm, ${\mathrm{\Phi }}_B=1$ eV, $P_G=30\mathrm{\%}$, $V_{\text{app}}=-800$ mV, $ϵ=50$ meV). (b),(c) numerical simulation of a 3T spin-precession experiment (b) and a 4T spin-diffusion experiment (c). In (b), the spin voltage is plotted (based on the linear model and considering the nonlinear effect) with the corresponding spin lifetime, assuming that both curves are Lorentzian distributions. In (c), the spin-diffusion length is extracted by fitting with the analytical solution of the one-dimensional (1D) spin-diffusion model.

Figure 9

The effect of temperature on the spin-detection efficiency. (a) The variation of $P_{\text{det}}$ with the temperature and the junction bias. (b) A comparison of the temperature effect on different barrier shapes. The calculations are performed using barrier properties as presented in Fig. 7. (a) $ϵ=50$ meV, $P_G=50\mathrm{\%}$, ${\mathrm{\Phi }}_B=1$ eV, and $w=3$ nm. (b) The same, except for the rectangular case: $w=2$ nm.