Inverse Tunnel Magnetocapacitance in Fe/Al-oxide/Fe3O4

Magnetocapacitance (MC) effect, observed in a wide range of materials and devices, such as multiferroic materials and spintronic devices, has received considerable attention due to its interesting physical properties and practical applications. A normal MC effect exhibits a higher capacitance when spins in the electrodes are parallel to each other and a lower capacitance when spins are antiparallel. Here we report an inverse tunnel magnetocapacitance (TMC) effect for the first time in Fe/AlOx/Fe3O4 magnetic tunnel junctions (MTJs). The inverse TMC reaches up to 11.4% at room temperature and the robustness of spin polarization is revealed in the bias dependence of the inverse TMC. Excellent agreement between theory and experiment is achieved for the entire applied frequency range and the wide bipolar bias regions using Debye-Fröhlich model (combined with the Zhang formula and parabolic barrier approximation) and spin-dependent drift-diffusion model. Furthermore, our theoretical calculations predict that the inverse TMC effect could potentially reach 150% in MTJs with a positive and negative spin polarization of 65% and −42%, respectively. These theoretical and experimental findings provide a new insight into both static and dynamic spin-dependent transports. They will open up broader opportunities for device applications, such as magnetic logic circuits and multi-valued memory devices.


In this Supplementary Information section, we present a detailed derivation of the inverse TMC and provide the parameters used in the calculation of iTMC-V curves shown in the main text.
Derivation of the inverse TMC. The calculation of the inverse TMC is performed using DF model (combined with Zhang formula and parabolic barrier approximation) and SDD model. The DF model is a useful tool for the calculation of dynamic dielectric constant in a variety of insulating solid and liquid systems 1−4 . This model can also be applied to the inverse TMC in MTJs, because it has been successful in accounting for the normal TMC in dielectric-based spintronic devices, such as MgO-based MTJs and FeCo-MgF nanogranular films 5,6 . Based on the model 7 , the complex dielectric constant ε * can be generally represented by where ε ∞ and ε 0 are the high-frequency and static dielectric constants, ω is the angular frequency and τ is the relaxation time. The exponent β, indicating the distribution of relaxation time, is between 0 and 1. Since the real part of ε * is proportional to the capacitance, the capacitance C P(AP) DF ( f ) as a function of frequency f for the P(AP) configuration in MTJs can be expressed by where C ∞, P(AP) and C 0, P(AP) are the high-frequency and static capacitances, τ P(AP) is the relaxation time and β P(AP) is the exponent showing the distribution of relaxation time, respectively, for the P(AP) configuration. After a straightforward calculation of equation (2), we can obtain According to Julliere formula 8 , the relation between τ P and τ AP in FM 1 /insulator/FM 2 (P 1 > 0 and P 2 < 0) is given by 3 Therefore, under no bias voltage, as the inverse TMC ratio is defined by we can find the frequency characteristics of the iTMC ratio by substituting C ∞, P(AP) , C 0, P(AP) , β P(AP) , τ P and P 1(2) in equations (3)−(5).
In addition to this procedure, the following three models are taken into account to describe the inverse TMC at a finite bias voltage; i) Zhang formula, ii) parabolic barrier approximation and iii) SDD model. According to Zhang's theory 9 where τ P(AP),0 is the relaxation time at zero bias voltage in the P(AP) configuration; equation (4) is modified to τ AP,0 = τ P,0 (1 − |P 1 ||P 2 |)/(1+|P 1 ||P 2 |). Therefore, τ P(AP) in equaiton (3) should be replaced to τ P(AP),V , described by equation (6), under the application of the bias voltage.
Next, we consider the bias voltage dependence of the effective barrier thickness, which contributes to the measured capacitance. The potential profile in the barrier is assumed to be a parabolic function, which is known as a good approximation to describe tunneling process, such as ac tunneling transport explained by DF model 5,6 and fluctuation-induced tunneling (FIT) proposed by Sheng 10 . In this parabolic barrier approximation, the potential function ϕ(u) under the bias voltage V can be expressed by eVu u u u x is the distance from the surface of the one side electrode, d is the barrier thickness, ϕ 0 is the barrier height in the absence of the bias voltage and e is the electron charge.
Since the solution of ϕ(u)=eV is u 1 = eV/4ϕ 0 and u 2 = 1 (for u 1 < u 2 ), the effective barrier thickness d eff can be represented by Therefore, from equations (3), (6) and (7), the capacitance C P(AP),V DF-ZP ( f ) at a finite applied voltage in the P(AP) configuration, based on Zhang model and parabolic barrier approximation, can be written by Finally, we incorporate spin capacitance caused by the difference between spin-up and spin-down diffusion length, described by SDD model 11 . According to this model, accumulation of minority spins and depletion of majority spins, taking place at the interface between the ferromagnetic layer and insulator, form a tiny screening charge dipole. This dipole gives rise to an addition serial capacitance, which corresponds to spin capacitance. The screening charge density in P(AP) configuration is given by en P(AP) ( ) = en 0,P(AP) exp( − x/ξ), where x is the distance from the interface between one-side electrode and insulator, ξ is a characteristic screening length and is the screening charge for P(AP) configuration and ΔV is the electrical potential difference applied in the charging space, it is given by C P(AP),V SDD = eSn P(AP) (x)dx/dV(x), where S is a junction area and V(x) is an electrical potential as a function of x in the charging space, i.e., where γ is an adjustable positive parameter of much smaller than 1.0. From these equations, we can find a simple formula: Since this screening charge acts as a serial capacitance, V in equations (6)−(8) should be modified into (1 − γ)V. Therefore, these equations are replaced to 0 , P(AP) P(AP) , P(AP) From these procedures, the capacitance C P(AP),V ( f ) at a finite bias voltage V in P(AP) configuration is given by The behavior of charge accumulation, contributing to C P(AP),V DF-ZP ( f ) and C P(AP),V SDD , is illustrated in Fig. 2 in the main text. The equivalent circuit of the MTJ is also shown. Consequently, as the iTMC ratio in the presence of the applied voltage is defined by we can obtain the frequency characteristics and bias dependence of the iTMC ratio using equations (9)−(14).  Tables 1 and 2, respectively. Table 1 Parameters used in the calculation of iTMC-V curves shown in Fig. 4 in the main text.

Positive bias
Sample