Two-channel model for spin-relaxation noise

We develop a two-channel resistor model for simulating spin transport with general applicability. Using this model, for the case of graphene as a prototypical material, we calculate the spin signal consistent with experimental values. Using the same model we also simulate the charge and spin-dependent $1/f$ noise, both in the local and nonlocal four-probe measurement schemes, and identify the noise from the spin-relaxation resistances as the major source of spin-dependent $1/f$ noise.

Figure 1

Schematic diagram of a nonmagnetic channel (gray) with spin-polarized contacts (purple) for a four probe (a) charge transport and (b) spin transport measurement scheme. A region of length $l\approx {\lambda }_{\text{s}}$ is modeled as $n=3$ basic units connected in series (c), each formed by an equivalent circuit of a spin-up resistance $R_{\text{ch}}^{\uparrow }$ (red) and a spin-down resistance $R_{\text{ch}}^{\downarrow }$ (green), connected via a spin-relaxation resistance $R_{\uparrow \downarrow }$ (blue). (d) A two-channel model for spin transport is constructed by replacing the transport channel with a series connection of basic units from (c), and by modeling the spin-polarized contacts with two resistors $R_{\text{C}}^{\uparrow }$ and $R_{\text{C}}^{\downarrow }$.

Figure 2

Spin signal as a function of injector-detector separation $L$. For the circuit simulations, we use the experimentally obtained $P\sim 5\%, {\lambda }_{\text{s}}\sim 1.5 \mu \mathrm{m}, R_{\text{sq}}\sim 400\mathrm{\Omega }$, and $w=1.7\mu \mathrm{m}$.

Figure 3

Circuit model for $1/f$ noise (at 1 Hz) for (a) a single-channel region of resistance $R$, under an applied current $I$, with a current noise source $i=\sqrt{\gamma }I$, has total noise $V$. (b) Same region as in (a), represented as a series of three resistors, each with a scaled $i_n=\sqrt{3\gamma }I$, which keeps the total noise consistent $V$ [20], using Eq. (5). (c) Transition to a two-channel model in the limit of fast spin relaxation, where the channel resistors $R_{\text{ch}}^{\uparrow (\downarrow )}=2R/3$ each have an equivalent noise current source $i_n=\sqrt{6\gamma }\times I/2$. (d) Introduction of the spin-relaxation resistance $R_{\uparrow \downarrow }$. For an unpolarized current $I$, there is no current present at these resistors, so they do not contribute to $1/f$ noise. To keep a consistent $V$ for cases (a)–(d), we must consider $i_n$ in the spin channel resistors as independent noise sources. (e) Full two-channel model, as in Fig. 1, including also noise sources for the contact resistances and for the spin-relaxation resistances. The latter contribute to the total noise, as in a spin injection geometry there is now a spin current present in the channel. A noise voltage $v_n$ appears between C3–C4 due to noise current $i_n$ in the circuit.

Figure 4

Charge and spin-dependent $1/f$ noise measurements for the experimental device of Ref. [12]. The green spectrum is the charge $1/f$ noise measured in the local four-probe geometry. The black spectrum is the nonlocal thermal noise background, for $I=0$ in Fig. 3. The red spectrum is the measured total nonlocal $1/f$ noise, which is the sum of the thermal noise, the background charge noise, and the spin-dependent noise (magnitude denoted by the red arrow). The blue spectrum is the spin-independent background (thermal and charge backgrounds), measured when the spin accumulation is suppressed by an applied out-of-plane magnetic field. The horizontal lines indicate the noise levels measured at 1 Hz, and the dots the corresponding results from the simulations.