# Noise Characterization of STT-based Nanoscale Magnetic Tunnel Junction Magnetic Sensors

Kosuke Komuro, Hugo Nicolas, Bernard Dieny, Daiki Oshima, Takeshi. Kato,

Ricardo C. Sousa

► To cite this version:

Kosuke Komuro, Hugo Nicolas, Bernard Dieny, Daiki Oshima, Takeshi. Kato, et al.. Noise Characterization of STT-based Nanoscale Magnetic Tunnel Junction Magnetic Sensors. Journal of Magnetism and Magnetic Materials, 2025, 629, pp.173215. 10.1016/j.jmmm.2025.173215. hal-05217858

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Submitted on 21 Aug 2025

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# Noise Characterization of STT-based Nanoscale Magnetic Tunnel Junction Magnetic Sensors

Kosuke Komuro<fnref n="1" />, Hugo Nicolas<fnref n="2" />, Bernard Dieny<fnref n="3" />, Daiki Oshima<fnref n="1" />, Takeshi. Kato<fnref n="1" />,<fnref n="4" />, Ricardo C. Sousa<fnref n="3" />

Corresponding author: Kosuke Komuro

E-mail: komuro.kosuke.h2@s.mail.nagoya-u.ac.jp

## Abstract

Recently, the trend towards electrification has led to the widespread use of magnetic sensors in various applications. Magnetic tunnel junction (MTJ)-based magnetic sensors exhibiting very high sensitivity, compactness, and small dimensions could offer significant advantages over existing technologies such as conventional Hall effect sensors. New sensor concepts have been explored such as spin transfer torque (STT)-based MTJ magnetic sensors which have the potential to enable magnetic field detection with fine spatial resolutions of 50 to 100 nm and micro-Tesla resolution. This paper analyzes and describes the noise characteristics of STT-based MTJ sensors. Calculations and experimental results demonstrate that the signal-to-noise ratio (SNR) of the STT-based MTJ sensor is proportional to the square of the thermal stability factor, which is the ratio of the energy barrier of magnetization reversal and the thermal energy. This finding opens the way for future advances through predictive modelling of the sensing element and a deeper understanding of the noise sources in the proposed device.

Keywords: magnetic field sensor, magnetic tunnel junction, spin transfer torque

## 1. Introduction

Magnetic tunnel junction (MTJ) devices are widely utilized in various types of applications, not only for hard drive disk's read heads [1]~[3], but also for magnetic random-access memory (MRAM) cells [4]~[8], position or angle sensors [9]~[11], bio-magnetic sensors [12], [13], or cutting-edge computing devices [14], [15]. Since MTJ-type magnetic sensors have many advantages such as high sensitivity and good compatibility for monolithic integration with CMOS (complementary metal oxide semiconductor) chips, they are expected to take a bigger part in the market dominated by semiconductor-type Hall sensors which are commonly used at present.

Recently, our group has introduced spin transfer torque (STT)-based MTJ sensors, where the sensing element is identical to the MTJ used as memory cells in MRAMs, with perpendicular magnetic anisotropy (PMA) [16]. The sensor measures magnetic fields by detecting variations of reversal voltages of perpendicular anisotropy magnetic tunnel junctions with diameters of less than 100 nm (Section 2.1). Due to the small size of the sensing element, the sensor can measure magnetic fields with high spatial resolution. Such a sensor is attractive for different applications, such as failure analysis of large-scale integration (LSI) circuits, generally requiring a scanning electron microscope to probe the circuit [17]. One major concern of the STT-based MTJ sensor reported by Nicolas et al. is the large noise level of typically 21.8 µT/Hz<fnref n="1" />/<fnref n="2" /> [16] compared to other MTJ-based sensors such as those described in recent reports [18]. In previous publications [16], it was concluded that the noise originates from the stochastic behavior of the magnetic tunnel junction free layer reversal induced by the STT, however, a detailed discussion has not been investigated yet. Hence, this work investigates the noise characteristics of the STT-based MTJ sensors by conducting theoretical calculations and measurements of the STT loops of MTJs with various thermal stability. Moreover, output noise measured in the sensor circuit was compared with the theory to confirm its validity.

<footnote n="1">¹Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan</footnote>

<footnote n="2">²School of Life Sciences, University of Applied Sciences, Muttenz, Switzerland</footnote>

<footnote n="3">³SPINTEC, Grenoble Alpes Univ., CEA, CNRS, Grenoble-INP, Grenoble 38000, France</footnote>

<footnote n="4">⁴Institute of Materials and System for Sustainability, Nagoya University, Nagoya 464-8603, Japan</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Field detection model by STT magnetization switching in a perpendicular MTJ. (a) If a small $H_{ext}$ is applied, a large switching current is required to reverse the magnetization in the free layer, which indicates that a large bias voltage is necessary and (b) vice versa. $H_{ext}$ can be estimated by detecting the change in the voltage required for the magnetization reversal.</figcaption></figure>

## 2. Methods

### 2.1 Sensor working principle

The concept of magnetic field measurements through the STT magnetization reversal in a MTJ with perpendicular magnetization is illustrated in Fig. 1. The magnetization reversal voltage is directly dependent on the anisotropy field $H_k$ and the external field $H_{ext}$ applied perpendicularly to the MTJ plane. Upon application of an external field $H_{ext}$, the threshold voltage for STT reversal of the free layer magnetization is modified, establishing the basic principle for magnetic field detection in STT-based MTJ sensors. The circuit schematic of the STT-based MTJ sensor based on such principle is shown in Fig. 2. The MTJ device is placed in a typical inverting operational amplifier circuit, and a sinusoidal ac voltage is applied as the input voltage $V_{in}$ in the circuit. The waveform obtained at the output $V_{out}$ of the circuit in Fig. 2 exhibits sinusoidal shape with two discontinuities over one wave cycle. The discontinuous changes of $V_{out}$ are due to magnetization reversals of the free layer in MTJ either from antiparallel to parallel (APP) state or from parallel to antiparallel (PAP) state, since the MTJ resistance depends on the direction of the free layer magnetization relative to the reference layer, leading to changes in the gain of the circuit. When $H_{ext}$ is applied to the MTJ, the required voltage to switch the free layer magnetization of the MTJ is modified. Therefore, $H_{ext}$ can be estimated by measuring the time duration between the $V_{out}$ zero crossing and the discontinuous voltage change due to either APP or PAP magnetization reversal. This is shown in Fig. 2, as indicated by red and blue arrows, respectively. Several detection methods such as using time-to-digital converters or pulse-width-modulation circuits were proposed for this detection [16]. However, the noise level of the final output signal in these methods are similar, and the fundamental origin of the output noise is the stochastic nature of the STT magnetization reversal.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Circuit diagram of a STT-based MTJ sensor. The waveform of the input signal $V_{in}$, the resistance changes of the MTJ, and the output signal $V_{out}$ are shown. $H_{ext}$ can be estimated by detecting the time duration between 0 V crossing and the next signal discontinuity of $V_{out}$ originating from either APP or PAP magnetization reversal, as shown by the red and blue arrows.</figcaption></figure>

## 2.2 Theory of STT switching at room temperature

For an STT-based MTJ sensor operated at room temperature with an input frequency of typically 100 kHz, STT switching probability can be described by a Néel-Arrhenius equation (1) for the thermally activated reversal regime [19]:

$$P = 1 - \exp\left(-\frac{\tau}{\tau_0} \exp\left(-\Delta\left(1 - \frac{V}{V_c(H)}\right)\right)\right) \quad (1)$$

where $\Delta$, $\tau$, $\tau_0$, $V_c$ are respectively the thermal stability factor, pulse width of the switching voltage, reversal attempt time and the critical switching voltage at the temperature of 0 K. $\Delta$ is calculated as $E/k_B T$ where E is the energy barrier for the magnetization reversal, with $k_B$ the Boltzmann constant, and T the temperature of the tunnel junction, typically 300 K for room temperature measurements. Finally, $V_c$ can be calculated as follows [20]:

$$V_c = \frac{2|e|}{\hbar\eta} \alpha M \cdot RA \cdot H_{eff} \cdot t_F \quad (2)$$

where $e$, $\hbar$, $\eta$, $\alpha$, RA and $t_F$ are respectively the elementary charge, Dirac's constant, the STT efficiency, damping constant, the resistance area product of the MTJ and the thickness of the free layer. $H_{eff}$ represents the resulting effective field including $H_k$ and $H_{ext}$ as shown in Fig. 1. $V_c$ is not only determined by the uniaxial effective anisotropy but also by the damping constant $\alpha$ of the free layer in the MTJ. Here, the STT switching probability density (SPD) is defined by the derivative of P with respect to V as given by (3):

$$\frac{\partial P}{\partial V} = \frac{\tau}{\tau_0} \cdot \frac{\Delta}{V_c(H)} \left(1 - \frac{V}{V_c(H)}\right) \cdot \exp\left(-\frac{\tau}{\tau_0} \exp(-\Delta) - \Delta \left(1 - \frac{V}{V_c(H)}\right)\right) \quad (3)$$

When considering that the noise of the STT-based MTJ sensor is generated by the stochastic behavior of the STT switching, we assumed the noise power N to be the variance of the switching probability $\sigma^2$. Moreover, we defined the average bias voltage of the magnetization reversal to be the value where the switching probability P is 50 %. Hereafter, we refer to this voltage as $V_{50}$. Although the STT-based sensor in [16] detects $H_{ext}$ by measuring the timing of the signal discontinuities in $V_{out}$, it is equivalent to treat $V_{50}$ as the sensor's signal. Here, $V_{50}$ is written as:
$$V_{50} = V_c(H) \left( 1 + \frac{1}{\Delta} \ln \left( -\frac{\tau}{\tau_0} \ln(1-P) \right) \right) \quad (4)$$

The SPD is maximum at $V_{50}$. Therefore, the signal-to-noise ratio (SNR) of the STT switching can be obtained by calculating $V_{50}^2/N$.

## 2.3 Experimental procedure

The MTJs used in this work are fabricated on 100 mm Si wafers, with a perpendicular anisotropy magnetic stack based on a conventional FeCoB/MgO/FeCoB layer configuration. Physical etching methods are then used to create the nanopillar to nominal diameters between 50 to 100 nm. Further details about the MTJs used are described in [16].

The junctions were then measured in a probing system to record the voltage and magnetic field hysteresis loops. For each MTJ, voltage hysteresis loops were measured under different external field values, allowing for the characterization of the switching probabilities (SP) of each MTJ pillar. Fig. 3 shows a typical STT switching probability curve of an MTJ pillar of 50 nm nominal diameter. Dashed lines in Fig. 3 show data fitting to equation (1), to obtain $V_{50}$ and SPD curves using equation (3), and used in the following discussion of the SNR in STT-based MTJ sensors.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Example of STT switching probability (SP) curves measured in a typical STT-MTJ pillar. Blue and red circles represent the experimental results of PAP and APP reversals, respectively. These data are fitted to the (1) to extract the values of $\Delta$, $\tau$, $\tau_0$, $V_c$. The switching probability density can be calculated by differentiating the fitted curves with respect to the bias voltage.</figcaption></figure>

# 3. Results and discussions

## 3.1 Numerical estimation

Using (3) and (4), $N$ and $V_{50}$ were numerically calculated and used to investigate the behavior of the SNR. $\tau_0$ was kept constant as 1 ns [19], [21]. For the calculations, $\tau$ was fixed at 1 µs, corresponding the experimental pulse width measurements, while $\Delta$, $V_c$ values ranged respectively from 30 to 70 and from 0.3 to 0.5 V. Fig. 4 shows the numerical calculation of (a) thermal stability $\Delta$ dependence of SNR in dB scale calculated by $10 \log_{10}(V_{50}^2/N)$, (b) voltage dependence of SPD where $\Delta = 30$ or 50 and $V_c = 0.4$ V, and (c) voltage dependence of SPD where $V_c = 0.3$ V or 0.5 V and $\Delta = 50$. The calculation was conducted under the assumption that $\Delta$ and $V_c$ are independent. From Fig. 4 (a), the SNR was found to depend strongly on $\Delta$, while $V_c$ has no particular impact on SNR. These phenomena can be also qualitatively understood from Figs. 4 (b) and (c). When $\Delta$ was increased, $V_{50}$ also increased, and the $\sigma$ decreased. This indicates that large reversal voltages and small noises are obtained at the same time. On the other hand, when $V_c$ increases, $V_{50}$ and $\sigma$ increase simultaneously as shown in Fig. 4 (c), resulting in no influence on the sensor's SNR. Therefore, the SNR of the STT-based MTJ sensor is expected to depend only on the thermal stability factor $\Delta$.

To discuss these phenomena analytically, calculations mentioned above were approximated as follows. An expression of $\sigma^2$, noise power $N$, cannot be found analytically from the SPD. Thus, we approximated the shape of SPD by a triangle as shown in
Fig. 4 (b) and (c). From this approximation, the standard deviation $\sigma$ of SPD can be expressed as proportional to the inverse of the SPD maximum. Because the total area of SPD is equal to 1, $\sigma$ can be written as follows:

$$ \sigma \propto \frac{1}{\max\left(\frac{\partial P}{\partial V}\right)} \propto \frac{V_c(H)}{\Delta} \qquad (5) $$

In addition, according to (3), $V_{50}$ is proportional to $V_c$, therefore, the SNR can be approximated as:

$$ \mathrm{SNR} = \frac{V_{50}^2}{\sigma^2} \propto \frac{V_c(H)^2}{V_c(H)^2/\Delta^2} \propto \Delta^2 \qquad (6) $$

This suggests that the SNR is mainly determined by the square of the thermal stability $\Delta$ and is independent of $V_c$, which hereafter will be referred to as “$\Delta^2$ relation”.

<figure><img src="image_5.png" /><figcaption>Fig. 4. (a) Numerical calculations of the switching SNR as a function of $\Delta$ for various values of $V_{c0}$. (b) and (c) show the impact of $\Delta$ and $V_{c0}$ on the switching probability density SPD. The triangles drawn over each curve are approximations of the SPD having the same height as the original distribution, and keeping the total area under the curve the same and equal to 1.</figcaption></figure>

## 3.2 MTJ variation dependence

Fig. 5 shows the comparison between the experimental SNR estimated from different MTJs and the analytical expression in the previous section. Fig. 4 (a) shows the SNR in dB scale as a function of $\Delta_0$, which is $\Delta$ at $H_{ext} = 0$ Oe. The experimental plots are averaged values of the SNR estimated from APP and PAP switching, and the dashed black curve shows the $\Delta^2$ fitting result. In the $\Delta^2$ relation, the SNR is expressed as follow:

$$ \mathrm{SNR} = a(\Delta - \Delta_c)^2 \qquad (7) $$

where $\alpha$ and $\Delta_c$ are a proportionality constant and an offset, respectively. The numbers of the SNR here are not fixed in dB scale. In Fig. 5 (a), $\Delta$ was replaced by $\Delta_0$. The calculation and the experiments showed a good agreement, and the SNR increases monotonously with $\Delta$ independently of $V_c$. This data confirms the expected proportionality of SNR with $\Delta^2$. From the fitting, $\alpha$ and $\Delta_c$ were estimated to be 0.59 and 6.8 $k_B T$, respectively. The finite number of $\Delta_c$ appears as a consequence of the approximation used for $\sigma$. Fig. 5 (b) shows the SNR as a function of $V_{c0}$, which is equal to $V_c$ at $H_{ext} = 0$ Oe. The SNR was confirmed to be independent of $V_{c0}$ which is consistent with equation (6).
<figure><img src="image_6.png" /><figcaption>Fig. 5. SNR as a function of (a) thermal stability $\Delta_0$ and (b) critical voltage $V_{c0}$, where $\Delta_0$ and $V_{c0}$ are thermal stability and critical switching voltage at $H_{ext} = 0$ Oe, respectively. Red circles are experimental results obtained by the applied voltage dependence of the STT switching probability of different MTJ cells, and the black dashed line shows the fit based on the $\Delta^2$ relation. This relation explains the experimental $\Delta$ dependence of SNR. Moreover, the experimental SNR was confirmed to be independent of $V_c$ as shown in (b).</figcaption></figure>

### 3.3 Field dependence

As mentioned in Section 3.1, the thermal stability factor is determined by the uniaxial anisotropy of the free layer in the MTJ and external field $H_{ext}$ applied. Therefore, we investigated the field dependence of the SNR in the STT-based MTJ sensor.

First, we measured $V_{50}$ and $\sigma$ by measuring the switching probability under different field values. The red and dark circles in Fig.6 (a) show the results of $V_{50}$ and $\sigma$ in APP switching, and the blue and navy crosses show those of PAP switching. The positive direction of $H_{ext}$ is the same direction as $H_{eff}$ as in Fig. 1. In APP switching, $V_{50}$ increased corresponding to the increase of $H_{ext}$ while $\sigma$ decreased, and vice versa in PAP switching. Fig. 6 (b) is a plot of the SNR versus $H_{ext}$ obtained from Fig.6 (a). The red circles are the results of APP switching, and the blue crosses correspond to PAP switching. In both modes, the SNR increased when $H_{ext}$ was applied to the same direction of the free layer magnetization $M$ before the reversal. The external field $H_{ext}$ modifies the energy barrier $\Delta$, by adding or subtracting from the effective anisotropy as follows [22][23]:

$$ \Delta = \Delta_0 \left( 1 \pm \frac{H_{ext}}{H_k} \right)^v \quad (8) $$

Here, $v$ is the power of the equation which is dependent on the relative angle of $H_k$ and $H_{ext}$, anisotropy potential shape [19]. The sign of the second term is determined by the relative direction of $M$ and $H_{ext}$, e.g., the sign becomes positive if they are initially oriented in the same direction adding to the energy barrier, or reducing it for opposite directions. $\Delta_0$ was obtained by fitting the experimental switching probability using equation (1) with $\Delta$ modified as in (8). The field modified $\Delta$ versus SNR is plotted in Fig. 6 (c), as measured in one representative MTJ device. Both APP and PAP switching exhibited almost the same trend, and the SNR was confirmed to increase with the increase of $\Delta$. Similarly to Fig. 5 (a), the dashed black line shows the fitting based on the $\Delta^2$ relation from equation (6). Then, by fitting the experimental data recorded from the PAP switching, we obtain values of $\alpha = 0.60$ and $\Delta_c = 7.6$ k_BT, very similar to the values discussed in Section 3.2. The fitted curve again confirms the validity of the relation in these new measurements.
<figure><img src="image_7.png" /><figcaption>Fig. 6. (a) $V_{50}$ and $\sigma$ responses to varying $H_{\text{ext}}$. The red circles and the blue crosses show the results of $V_{50}$ in APP and PAP switching, and the dark red circles and the navy crosses stand for the switching noise of APP and PAP modes, respectively. (b) SNR vs $H_{\text{ext}}$ dependence by measuring the switching probability of an MTJ under different $H_{\text{ext}}$. (c) SNR as a function of $\Delta$, where $\Delta$ was calculated using $H_{\text{ext}}$ by (7). The black dashed line indicates the fitting based on $\Delta^2$ relation using (5).</figcaption></figure>

### 3.4 Comparison with fabricated sensor measurements

Finally, we discuss whether the $\Delta^2$ relation can explain the SNR obtained from the experimental sensor circuit as reported in previous work [16]. In earlier sections, the SNR was defined in (6) as $\text{SNR} = V_{50}^2/\sigma^2$. As initially explained, the STT-based MTJ sensor circuit measures the time difference between APP or PAP reversal and the instant when $V_{\text{in}}$ crosses 0 V, as depicted in Fig. 2. Here, unlike Fig. 2, a triangular waveform with a frequency of 30 kHz was applied as $V_{\text{in}}$ instead of using a sinusoidal waveform. The reason for using the triangular waveform is that the timing of signal discontinuities reacts more linearly to $H_{\text{ext}}$. Fig. 7 (a) shows the SNR of the sensor output as a function of $H_{\text{ext}}$ using junctions with different values of $\Delta_0$. The output was obtained using the APP reversals. Green crosses and brown circles show results of junction sensors with $\Delta_0 = 47.3$ (sample A) and $\Delta_0 = 62.7$ (sample B), respectively. Higher SNR was confirmed for larger values of $\Delta_0$ or when $H_{\text{ext}}$ adds to the energy barrier. Fig. 7 (b) shows $\Delta$ vs the SNR of the sensor output obtained from APP and PAP reversals in sample A. The values of $\Delta$ in the horizontal axis were calculated from equation (8). As $\Delta$ increases, the SNR in both APP and PAP reversals also increases as expected. However, one can note that the SNR obtained from APP reversal varies more significantly than that of PAP reversal, which is different compared to the direct STT switching measurements in Section 3.3. In that case, the $\Delta$ dependence of SNR obtained from APP and PAP reversals coincided well each other, see Fig. 6 (b). The dashed lines in Fig. 7 (b) indicate the fitting of the experimental data assuming a $\Delta^2$ dependence. Using equation (7), the fitting parameter $\alpha$ and $\Delta_c$ were estimated to be 52.3 and 41.8 for APP reversal and 92.3 and 32.6 for PAP reversal, respectively. A possible reason for the large change in SNR and the value of $\Delta_c$ is considered to be the different thermal activation situation between the case of STT switching and the sensor output measurements. In the STT switching probability measurements, the pulse voltages were applied as 1 µs pulses, while a triangular waveform was applied continuously for sensor measurements. This continuous ac voltage will lead to an increase in temperature thereby affecting the energy barrier. Note that $\Delta$ used in Fig. 6 were obtained from the STT measurements which did not take account of Joule heating of the MTJ under continuous voltage application. Moreover, the $\Delta^2$ approximation is derived from an initial configuration [23] where the initial state of the junction free layer is
set parallel or anti-parallel to the reference layer and the amplitude of spin-polarized current is constant for every reversal attempt. However, when a sweeping ac voltage is applied as $V_{in}$, the initial state of the free layer might already be modified by the spin-polarized current even prior to the reversal, possibly explaining the observed differences [23]. Additional studies including the dynamic response of the junction in the sensor circuit, accounting for heating and the continuously varying spin current are needed to clarify the energy barrier and SNR dependence in sensor operation.

<figure><img src="image_8.png" /><figcaption>Fig. 7. (a) External field $H_{ext}$ dependence of SNR of the sensor output obtained by the time difference between the moment of $V_{in}$ = 0 V and APP reversal. Green crosses and red circles are the data for MTJs with $\Delta_0$ = 47.3 (sample A) and 62.7 (sample B), respectively. (b) SNR as a function of $\Delta$, which is calculated from $\Delta_0$ and $H_{ext}$ using (7). The data were taken from sample A and the SNR was obtained using APP and PAP reversals. The black dashed line indicates the fitting based on $\Delta^2$ relation using (5). Unlike in Fig. 6 (b), the SNRs obtained by APP and PAP reversals are slightly different.</figcaption></figure>

## 3.5 Discussion

The SNR of STT-based junction sensors was confirmed to depend on the energy barrier as $\Delta^2$. We discuss this relation qualitatively as follows: if the energy barrier of the magnetization reversal of the free layer is high, the required voltage to reverse the magnetization increases, thereby resulting in a large signal. Therefore, the stochastic variability of the switching voltage due to the thermal activation of the free layer magnetization is reduced in both APP and PAP reversals. The result is a higher SNR, obtained by increasing the thermal stability $\Delta$ of the free layer.

## 4. Conclusion

In this paper, we investigated the noise properties of STT-MTJ-based magnetic sensors. First, in order to describe the underlying mechanisms impacting the SNR of the STT-based field sensor, we defined the signal to be the average switching voltage $V_{50}$ and the noise power to be the dispersion of switching voltage $\sigma^2$. The SNR of the sensor was numerically calculated based on the framework of thermally activated STT switching. We demonstrated that the SNR has a strong correlation with the thermal stability factor $\Delta$ of the MTJ free layer. The STT switching probabilities of different MTJs were measured under various static applied fields, again confirming the correlation between $\Delta$ and SNR. In addition, by approximating the shape of STT switching probability density as a triangular distribution, it can be shown that the SNR is proportional to the square of thermal stability $\Delta^2$. The $\Delta^2$ dependence explains the SNR as obtained from STT switching measurement results. It can be concluded that the increase of $\Delta$ realizes both a large output signal and low noise, presenting a way to enhance the performance of STT-based MTJ sensors. This result gives us a first method to improve the SNR of STT-based sensors, by targeting the development of high $\Delta$ junctions.

Finally, we compared the $\Delta^2$ relation with the SNR obtained from an initial STT sensor circuit demonstrator. We found that SNR measured in the STT switching measurements and the sensor circuit exhibited similar trends. However, the fitted slope $\alpha$ and offset $\Delta_c$ based on the $\Delta^2$ relation were completely different between the two situations. The reason for this is possibly due to the continuous bias voltage application in sensor circuit, which is expected to cause Joule heating and a magnetization reversal process different from the pulse experiment. Further studies of STT switching under a continuous sweeping voltage are required to refine the SNR dependence on the energy barrier in STT-based magnetic field sensors.
## Funding

This work was partially supported by the European Research Council via grant reference ERC-2022-PoC2 (NANOSENSE No. 101100599) and by the French National Research Agency in part through the "France 2030" program PEPR SPIN under Project ADAGE (ANR-22-EXSP-0006). This work was supported in part by JST A-STEP Grant Number JPMJTR24R7, Japan. This work was also supported in part by the Research Program "Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials" in "Network Joint Research Center for Materials and Devices". A part of this work was performed under the Project of Creation of Life Innovation Materials for Interdisciplinary and International Researcher Development of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. A part of this work was conducted at the Next-generation biomaterials Hub, Nagoya University, JPMXP1224NU5222, supported by Advanced Research Infrastructure for Materials and Nanotechnology in Japan of MEXT, Japan.

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