1/$f$ noise of a tiny tunnel magnetoresistance sensor originated from a wide distribution of bath correlation time

In most experiments, the voltage noise of a TMR sensor is measured under a constant direct current, $I$. Assuming that the measured voltage, $V$, is proportional to the resistance, $R$, the power spectrum of voltag, $S_{VV}(f)$, is proportional to the power spectrum of resistance, $S_{RR}(f)$, as

Introducing the angular frequency, $\omega=2\pi f$, the power spectrum of resistance is defined as

where $\langle\ \rangle$ represents the statistical average. Substituting Eq. (6) into Eq. (8), $S_{RR}(\omega)$ is expressed as

where $S_{m_{y^{\prime}}m_{y^{\prime}}}(\omega)$ is the power spectrum of $m_{y^{\prime}}$ defined as

We define the Fourier transform of a function $f(t)$ as $f(\omega)=\int_{-\infty}^{\infty}f(t)\exp\left(-i\omega t\right)dt$. Substituting the inverse Fourier transform of $m_{y^{\prime}}$ into Eq. (10) and performing some algebra, we obtain

The Fourier transform of $m_{y^{\prime}}$ can be obtained by solving the equations of motion in the Fourier space.

The equations of motion of $\bm{m}$ is given by the following generalized Langevin equation [30, 31, 32],

where $\dot{\bm{m}}(t)$ is the time derivative of $\bm{m}(t)$, $\gamma$ is the gyromagnetic ratio, and $\alpha$ is the Gilbert damping constant. The effective magnetic field acting on $\bm{m}$ is given by

The memory function is defined as

where $\tau_{c}$ is the bath correlation time. The thermal agitation field, $\bm{r}$, is a random field satisfying $\langle r_{j}\rangle=0$ and

where subscripts $j$ and $k$ denotes $x$, $y$, $z$, $y^{\prime}$, or $z^{\prime}$. The constant $\mu$ is defined as

where $k_{B}$ is the Boltzmann constant, $T$ is temperature, and $\Omega$ is the volume of the FL. From Eqs. (15) and (16) we see that the magnitude of the thermal agitation field is of the order of $\sqrt{\alpha}$ because $\mu$ is of the order of $\alpha$. The stochastic LLG equation with the Markovian damping derived by Brown [33] is reproduced in the limit of $\tau_{c}\to 0$ because $\lim_{\tau_{c}\to 0}\nu(t-t^{\prime})=2\delta(t-t^{\prime})$, where $\delta(t-t^{\prime})$ is Dirac’s delta function. It should be noted that 1/$f$ noise cannot be derived from the LLG equation with the Markovian damping because many physical processes with different time scale is required to generate 1/$f$ noise.

Since the FL of a typical TMR sensor is made of a ferromagnetic material with $\alpha\ll 1$, we focus on terms up to the first order of $\alpha$ in the equations of motion. We also assume that $m_{x}$, $m_{y^{\prime}}$, $r_{x}$, $r_{y^{\prime}}$, and $r_{z^{\prime}}$ are small enough to linearize the equations of motion in terms of these small variables. Equation (III.2) can be approximated as

where

Following Ref. [32], we approximate the non-Markovian damping term in Eqs. (III.2) and (III.2) up to the first order of $\alpha$. Successive application of the integration by parts gives the following linearized equations of motion up to the order of $\alpha$,

where

Details of the derivation of the above equations will be provided in Appendix A. In the Fourier space, the equations of motion are expressed as

The solutions are obtained as

where

From Eq. (31), the correlation of $m_{y^{\prime}}(\omega)$ and $m_{y^{\prime}}(\omega^{\prime})$ is expressed as

where we use the fact that $r_{x}$ and $r_{y^{\prime}}$ do not correlate with each other. The correlation $\langle m_{y^{\prime}}(-\omega)m_{y^{\prime}}(\omega^{\prime})\rangle$ is obtained by replacing $\omega$ with $-\omega$ in Eq. (III.3)

Following Ref. [34], the correlation of thermal agitation fields in the Fourier space is obtained as

The correlation $\langle r_{j}(-\omega)r_{k}(\omega^{\prime})\rangle$ is obtained by replacing $\omega$ with $-\omega$ in Eq. (34).

Substituting Eqs. (III.3) and (34) into Eq. (III.1), the power spectrum of $m_{y^{\prime}}$ is expressed as

where

In the low frequency regime satisfying $\omega\ll\omega_{0}$ and $\omega\ll\omega_{1}$, Eq. (35) can be approximated by the Lorentzian function as

Since $\omega_{0}$ and $\omega_{1}$ are of the order of 0.1 GHz $\sim$ 10 GHz for conventional TMR sensors [7], the low frequency condition is clearly satisfied for the frequency range of the bio-magnetic signal, i.e. less than a few hundred Hz.

Bath correlation time, $\tau_{c}$, is the decay time of the correlation of thermal agitation field as shown in Eq. (15). Thermal agitation field is produced by many kinds of sources or baths such as dipolar coupling with magnons in the reference layer and spin orbit coupling with phonons. Since $\tau_{c}$ depends on the relaxation mechanism of the bath, different relaxation modes in different baths have their own $\tau_{c}$. Instead of discussing $\tau_{c}$ for some specific types of baths, we just assume a distribution of $\tau_{c}$ and analyze the effect of the distribution of $\tau_{c}$ on the low frequency power spectrum of voltage. Assuming a wide distribution of $\tau_{c}$, we derive an analytical expression of the power spectrum of the magnetic 1/$f$ noise.

We assume that $\tau_{c}$ is uniformly distributed in the range of $\tau_{c,\mathrm{min}}\leq\tau_{c}\leq\tau_{c,\mathrm{max}}$ and has the probability distribution defined as $\rho(\tau_{c})=1/(\tau_{c,\mathrm{max}}-\tau_{c,\mathrm{min}})$. The superimposition of $S_{m_{y^{\prime}}m_{y^{\prime}}}(\omega)$ for all $\tau_{c}$ is given by

As a function of $\tau_{c}$, $\hat{\gamma}_{1}$ is almost unity except around the peak at $\tau_{c}=1/\sqrt{\omega_{0}\omega_{1}}$, which is of the order of ns. Since the peak value of $\hat{\gamma}_{1}$ is as small as $1+(\alpha/2)\sqrt{\omega_{1}/\omega_{0}}$, and $\omega$ is assumed to be much smaller than $\omega_{0}$ and $\omega_{1}$, Eq. (38) can be approximated as

where $\tau_{c,\mathrm{diff}}=\tau_{c,\mathrm{max}}-\tau_{c,\mathrm{min}}$. Since $\arctan(x)/x$ is a monotonically decreasing function of $x$ for $x>0$ and $\lim_{x\to 0}\arctan(x)/x=1$, $S_{m_{y^{\prime}}m_{y^{\prime}}}(\omega)$ is a monotonically decreasing function of $\omega$ and take a maximum value of $2\gamma^{2}\mu/\omega_{0}^{2}$ in the limit of $\omega\to 0$.

When $\omega\tau_{c,\mathrm{min}}\ll 1$, the second term in the square bracket of Eq. (III.4) can be neglected and $S_{m_{y^{\prime}}m_{y^{\prime}}}(\omega)$ is approximated as

Figure 2 shows $S_{m_{y^{\prime}}m_{y^{\prime}}}(\omega)$ given by Eq. (40) normalized by $2\gamma^{2}\mu/\omega_{0}^{2}$ for $\tau_{c,\mathrm{max}}$=10 s (black solid), 1 s (red dotted), and 0.1 s (blue dashed). The values at $\omega$ = $1/\tau_{c,\mathrm{max}}$ are indicated by the green circles. All curves are almost flat for $\omega\ll 1/\tau_{c,\mathrm{max}}$ and inversely proportional to $\omega$ for $\omega\gg 1/\tau_{c,\mathrm{max}}$.

Assuming a wide distribution of $\tau_{c}$ satisfying $\omega\tau_{c,\mathrm{min}}\ll 1$ and $\omega\tau_{c,\mathrm{max}}\gg 1$, we have $\arctan\left(\omega\tau_{c,\mathrm{min}}\right)=0$ and $\arctan\left(\omega\tau_{c,\mathrm{max}}\right)=\pi/2$. Then the power spectrum can be approximated as

which is inversely proportional to the angular frequency, $\omega$ (=$2\pi f$). From Eqs. (7), (9) and (41) the voltage power spectrum is given by

This is the main result of this paper. The obvious difference from other models of low frequency magnetic noise [35, 21, 36, 37, 23, 38] is that Eq. (42) has the term $1/\tau_{c,\mathrm{max}}$ as information of the distribution of the bath correlation time. It should be noted that the 1/f noise of a tiny TMR sensor we derived is response to the thermal agitation fields that exhibit 1/f power spectrum as the superimposition of the Lorentzian power spectrum.

We compare the derived 1/$f$ noise of the macrospin model with the experimental results of a conventional TMR sensor with large junction area reported in Ref. [7]. The Hooge parameter, $\alpha_{H}$, is a convenient measure to compare the 1/f noise between different MTJs, which is defined as

where $S_{VV}^{\rm wh}$ is the power spectral density of the white noise, $V_{b}$ is the bias voltage, and $A$ is the area of the MTJ. The typical value of the Hooge parameter of conventional TMR sensors is about $10^{-6}\sim 10^{-11}$ $\mu$m² [26, 27, 7, 28]. From Eq. (6), the bias voltage is given by

From Eqs. (16), (42), (43), and (44), the Hooge parameter of a tiny TMR sensor is obtained as

where $d$ is the thickness of the FL.

To compare Eq. (III.5) with the experimental results of a conventional TMR sensor, we determine the junction parameters by fitting the bias field dependence of the resistance shown in Figs. 2(a) of Ref. [7]. The parameters are determined as $M_{s}$=0.93 MA/m, $\mu_{0}H_{k}$=1.0 mT, $\mu_{0}H_{p}$= 2.15 mT, $R_{0}$=10.7 $\Omega$, $\bar{R}$=10.3 $\Omega$, $P^{2}$=0.74. Figure 3(a) shows the bias field, $\mu_{0}H_{b}$, dependence of the sensitivity defined as

where $R_{\rm max}$ is the maximum value of the resistance. The experimental results indicated by the yellow curves are well reproduced by the theoretical results represented by the black curves.

Figure 3(b) shows the bias field dependence of the Hooge parameter, $\alpha_{H}$. The experimental results are indicated by the yellow circles. The black solid curve represents the theoretical results multiplied by $10^{6}$. For calculation of the Hooge parameter, the following parameters are assumed: $\alpha$ = 0.05, $d$ = 2 nm, $\tau_{c,\mathrm{max}}$ = 1 s, and $T$ = 300 K. The field dependences of the Hooge parameter and the sensitivity look very similar because the Hooge parameter is proportional to the square of the sensitivity along the $y^{\prime}$ axis as indicated by Eqs. (6), (42), and (46).

The results show that the Hooge parameter of a tiny TMR sensor is much smaller than that of the conventional TMR sensor with the same sensitivity by a factor of $10^{-6}$. If a number of tiny MTJs are connected in parallel to reproduce the same resistance as a conventional TMR sensor, the power spectrum of the magnetic 1/$f$ noise can be reduced by a factor of $10^{-6}$ without reducing sensitivity.