Probing Spin Accumulation induced Magnetocapacitance in a Single Electron Transistor

The interplay between spin and charge in solids is currently among the most discussed topics in condensed matter physics. Such interplay gives rise to magneto-electric coupling, which in the case of solids was named magneto-electric effect, as predicted by Curie on the basis of symmetry considerations. This effect enables the manipulation of magnetization using electrical field or, conversely, the manipulation of electrical polarization by magnetic field. The latter is known as the magnetocapacitance effect. Here, we show that non-equilibrium spin accumulation can induce tunnel magnetocapacitance through the formation of a tiny charge dipole. This dipole can effectively give rise to an additional serial capacitance, which represents an extra charging energy that the tunneling electrons would encounter. In the sequential tunneling regime, this extra energy can be understood as the energy required for a single spin to flip. A ferromagnetic single-electron-transistor with tunable magnetic configuration is utilized to demonstrate the proposed mechanism. It is found that the extra threshold energy is experienced only by electrons entering the islands, bringing about asymmetry in the measured Coulomb diamond. This asymmetry is an unambiguous evidence of spin accumulation induced tunnel magnetocapacitance, and the measured magnetocapacitance value is as high as 40%.

] is exchange-induced by the much thicker Py as a result of proximity effect 1,2 . Therefore, the initial value for P(x,t) is non-zero, spatial-independent, and takes up a value close to the one found in Py, i.e., P 0 ≈ P Py . At x=x a , there is a barrier interface Al 2 O 3 /Al where electrons tunnel into Al from Co, giving rise to spin-dependent current density, which can be estimated based on Jullière model: where P Co is spin polarization of Co, and J=-I/A In All-P configuration, we can assume P Co = P 0 . The spin-dependent but spatialindependent current density is J ± = J 1± P 0 ( ) 2 , and thus there is no spin-accumulation.
On the other hand, there is charge screening, which in All-P configuration is calculated below. First, the boundary condition of the spin-dependent E-field (gradient of V ± x,t is determined such that ∂V + ∂x = ∂V − ∂x = − ΔV d is a constant at x=x a , while vanishes at x=x b ( Figure S1a). The latter is set under the assumption that the electric field vanishes at the Al/Py interface. We then consider the spin-dependent driven-drift current caused by the boundary E-field, i.e., J ext ,± x,t where σ Al is the intrinsic conductivity of Al. Plugging this in the continuity equation we obtain spin-dependent depletion causing by external E-field after an infinitesimal time dt: ( ) is the level spacing inside Al. As a result, a steady back- Using again the continuity equation, we obtain the refill caused by internal back-diffusion for each spin channel: , where D ± x,t  Figure S1b. This deviation from the initial uniform charge density n ±,0 , i.e., e n ± x,t ( ) − n ±,0 ⎡ ⎣ ⎤ ⎦ , modifies the chemical potential profile V ± x,t ( ) , as described by the spin-dependent Poisson's equation: , where ε 0 is free space electric permittivity. At this point V ± x,t ( ) can be solved and plugged back in equation (2) to form a complete iteration loop. We note that, in All-P configuration, even though spin-up and spin-down charge densities are different, but due to nonzero P(x,t), the chemical potential for spin-up, and spin-down are the same at all In addition, the average chemical potential The total charge density perturbation is then given by Where n 0 = n +,0 + n −,0 . Accordingly, the interfacial capacitance C Al = eΔn x can be calculated. profiles. The dark area shows initial uniform free electron density inside Al and the green line shows initial Note that both V x ( ) and Δn x ( ) follow an exponential decay with the screening length x, as depicted in Figure S1c, but the calculated C Al is constant everywhere inside Al. This screening length x turns out to be the same as that for the case of non-magnetic junctions 3 , in which P(x,t)=0, and n + x serves as a baseline for the calculations of extra C S in the AP-configurations.

S2. Spin-dependent diffusion lengths
In AP-configuration, we assume P Co = −P 0 , and equation (1) is reduced to Figure 3a in Main Text). Consequently, the spin dependent current density obeys diffusion equation, which in steady state is given as: After each dt, we have J ± x,t + dt ( ) = J ± x,t ( )+ dJ ± x,t ( ), and the accumulated/depleted spin density is 4-6 : This spin-accumulation causes a difference in the chemical potential for spin-up and spindown electrons. This difference in turn triggers spin flips and that produces a "refill" of spin density: , where n ± x,t ( )δ ± x,t ( )− n ∓ x,t ( )δ ∓ x,t ( ) is the chemical potential difference. In the mean time, spin accumulation also cause a difference in diffusion coefficient for each spin, ( ) , which depends on their respective chemical potential. As a result, each spin has a different diffusion length, since λ ± ∝ D ± . Following Eq. (2) and (3), the total spin density at t=t+dt is the sum of all contributions, i.e., This is then plugged back in Eq. (4) to complete the new iteration loop. Finally, upon convergence, the system reaches a steady state so that the sums ( ) and n ± x ( ) distributions. By updating Eq. (5), we deduced C S in APconfiguration (in the Main Text).

S3. Formation of Charge dipole by Spin Accumulation
Our device is composed of tunnel junctions that comprise of structure as follow:  Figure S2b). As electron density determine diffusion coefficient, thus the spin component that is accumulating has a longer diffusion length as compared to the one that is depleting. And the total accumulated spin across Al must equate the total depleted spin, due to the conservation of charge.

S4. Numeric calculation of Δ TMC value
Within our model, the Δ TMC value increases with the spin diffusion length λ. For λ→0, there is no spin accumulation and the All-P case is recovered so that Δ TMC →0.
Contrarily, in the case of λ→∞, it is not possible to flip spins, and the energy required to flip a single spin (e 2 /2C S in Fig. 3c) becomes infinite so that Δ TMC →100%. In the numeric calculations, due to some technique issues (such as limited memory and CPU speed), we could only get a Δ TMC value of up to 80%. The calculated Δ TMC as a function of spin diffusion length λ using the experimental device parameters and the reported polarization values 7 for Co (35%) and Py (40%) is shown in Fig. S3. It is found that, for a given λ, the