# Optimizing effective anisotropy in magnetic tunnel junctions for operation at cryogenic temperatures

Sergi Martín Rio, Kintali Subham Senapati, Louis Farcis, Lucile Soumah, Pedro Veiga, Kevin Garello, Ricardo C. Sousa

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Sergi Martín Rio, Kintali Subham Senapati, Louis Farcis, Lucile Soumah, Pedro Veiga, et al. Optimizing effective anisotropy in magnetic tunnel junctions for operation at cryogenic temperatures. Physical Review Applied, 2025, 24 (2), pp.024016. 10.1103/62tn-vnzb. hal-05217689

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Submitted on 21 Aug 2025

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<figure><img src="image_1.png" /></figure>
Optimizing Effective Anisotropy in Magnetic Tunnel Junctions
for Operation at Cryogenic Temperatures

Sergi Martín Rio,*, Kintali Subham Senapati, Louis Farcis, Lucile Soumah, Pedro B. Veiga, Kevin Garello, and Ricardo C. Sousa,

Université Grenoble Alpes,
CEA, CNRS, Grenoble-INP,
SPINTEC, Grenoble 38000, France

(Dated: August 21, 2025)
Abstract

The development of working cryogenic memories is very important for the scalability and fea-
sibility of quantum computing. In this context, FeCoB/MgO-based spin transfer torque magnetic
random access memories emerge as promising candidates to address this challenge. At low tem-
peratures, the reduction in thermal noise allows for a lower energy barrier while retaining sufficient
memory performance. Since the effective anisotropy of the system is directly proportional to the
energy barrier, understanding how it varies with material parameters and temperature is cru-
cial. In this work, we present an experimental method for extrapolating the thin film magnetic
properties measured at room temperature to the electrical properties of nanofabricated devices at
various targeted temperatures. We employ magneto-optical Kerr effect (MOKE) magnetometry to
calculate the effective anisotropy energy density across the entire 100 mm wafer surface at room
temperature. This method is entirely adapted to probe wafers of various sizes, and it is found
to be faster and equivalent to conventional methods such as VSM and SQUID. The temperature
evolution of magnetic anisotropy is further investigated using SQUID magnetometry from 300K
to 10K. Finally, the critical switching voltage from 300K to 40K is measured for various nanofab-
ricated devices with different FeCoB storage layer thicknesses. A model correlating the switching
voltage of these devices with the effective anisotropy at the thin film level is used to explain their
temperature and thickness dependencies. These results confirm the validity of the MOKE method
presented in this work and its potential in selecting devices across the wafer to function at a given
operational target temperature.

# I. INTRODUCTION

The increasing global energy demand calls for the development of more efficient and low-
power electronic devices. This need is further justified by the rapid growth of applications
such as artificial intelligence (AI) and the Internet of Things (IoT) driving an increase in
data center demand. Currently, significant energy is lost as heat dissipation from aggressive
device scaling in current CMOS technology [1–4]. In this context, novel approaches such
as quantum computing offer a promising alternative, with the potential to outperform clas-
sical computers in both speed and energy efficiency. In current laboratory-scale quantum

<footnote>* Contact author: smartinrio@gmail.com</footnote>
computers, quantum circuits are maintained at a few mK, while long cables connect these circuits to processing and memory units operating at room temperature [5]. Although this design is effective for studying the properties and capabilities of a small number of qubits, it is not scalable to systems with hundreds or thousands of them, which is a necessary condition for the future development of the field. Therefore, integrating both processing and memory units with the low-temperature qubit system is critical [5–7].

Perpendicular magnetized FeCoB/MgO-based spin transfer torque magnetic random access memories (pSTT-MRAM) emerge as promising candidates to address this challenge. This technology has been shown to work on a wide range of temperatures, from below 10K [8–13] up to 400K [14, 15]. Their main properties include non-volatility, high speed with nanosecond switching times, low energy consumption, endurance of more than $10^{15}$ cycles, and adjustable retention time ranging from more than 10 years [16] to nanoseconds, enabling them to be used as memory storage units or processing units relying on probabilistic computing [17]. In pSTT-MRAM, a ferromagnetic reference layer and a storage layer, separated by a thin insulating oxide film, enable data storage through switching of the storage layer magnetization via a spin-polarized current. The relative magnetization orientation of the two layers (parallel or antiparallel) results in different resistance states, separated by an energy barrier proportional to the anisotropy of the device. Optimizing the height of this barrier is critical to minimize energy consumption while achieving the desired information retention time. Additionally, a high tunneling magnetoresistance (TMR) ratio is essential for efficient readout and enhanced switching speed. At room temperature and above, a high energy barrier is required to overcome thermal noise and ensure stability, while at cryogenic temperatures, the reduction in thermal noise allows for a lower energy barrier, retaining sufficient memory performance. Thus, optimizing the anisotropy of the magnetic junctions is essential for their performance across a broad temperature spectrum.

When wafer-sized samples are used, the post-nanofabrication process yields thousands of working devices. Hence, automated probers are routinely used for efficient electrical characterization. However, automated probers operating at cryogenic temperatures are rare and costly, making systematic characterisation of devices optimized for low temperature hard to achieve. Therefore, in this study, we present an alternative approach for extrapolating the thin-film magnetic properties measured at room temperature to the electrical properties of nanofabricated devices at various target temperatures. First, magneto-optical Kerr effect
(MOKE) magnetometry is used to measure the effective anisotropy energy density across the unprocessed entire wafer at room temperature. Then, superconducting quantum interference device (SQUID) magnetometry is employed to examine the temperature dependence of saturation magnetization, interface anisotropy, and effective anisotropy. These temperature dependencies are found to follow Bloch and Callen-Callen models. Moreover, three nanofabricated devices, each with different FeCoB storage layer thicknesses, are selected for direct comparison with the thin-film properties. The effective anisotropy of each device is determined by electrical means across a broad temperature range by fitting the switching probability versus magnetic field curves using a thermal activation model [18]. A strong correlation is observed between the temperature dependence of effective anisotropy in both nanofabricated devices and thin films, underscoring the validity of this approach. Finally, the STT switching characteristics of the devices are studied by measuring the resistance versus pulse voltage hysteresis loops over the whole range of temperatures. The critical switching voltage is obtained from the voltage switching probability curves using the same thermal activation model used to extract the anisotropy field at the device level. The critical switching voltage of each device increases monotonically with decreasing temperature. Notably, similar to the trend observed for the effective anisotropy, the switching voltage depends closely on the storage layer thickness and decreases for thicker FeCoB. A thermally activated model relating the switching voltage and the effective anisotropy is used to fit the temperature dependence curves [19]. For low switching voltage a rather good agreement with the model is observed, however for larger voltage values this does not hold. This deviation from the model is attributed to temperature induced variations of the magnetic properties due to Joule heating of the device.

## II. SAMPLE PREPARATION AND CHARACTERIZATION BY MOKE

Magnetic stacks are deposited by magnetron sputtering on polished 100 mm diameter Si wafers and annealed at 300 °C for 10 minutes under high vacuum. Their basic structure is as follows: Si wafer/buffer/SAF/W (0.2)/FeCoB (1)/MgO ($t_{MgO}$)/FeCoB ($t_{FeCoB}$)/W (2)/Pt (2). The numbers represent the nominal thickness in nanometers and SAF stands for synthetic antiferromagnet, which is composed of two antiferromagnetically coupled Co/Pt multilayers separated by a thin Ru film. Additionally, to fully take advantage of the wafer
dimensions, both the MgO tunnel barrier and the FeCoB storage layer were deposited in an off-axis configuration, creating two orthogonal thickness wedges, represented here as $t_{\text{MgO}} = 1.1 - 1.7$ nm and $t_{\text{FeCoB}} = 0.9 - 1.9$ nm. The formation of the MgO tunnel barrier is a three-step process: (i) deposition of 0.75 nm of a metallic Mg layer; (ii) natural oxidation of the Mg layer under controlled time and pressure conditions; and (iii) off-axis wedge deposition of additional metallic Mg. As a result, the final thickness of the MgO layer is directly related to the level of oxidation of the tunnel barrier, with variations in $t_{\text{MgO}}$ reflecting this property.

After deposition, the magnetic properties of the stack at the thin-film level are characterized by Magneto-Optical Kerr Effect (MOKE) magnetometry. MOKE magnetometry offers different advantages over other techniques, such as vibrating sample magnetometry (VSM) or superconducting quantum interference device magnetometry (SQUID), as it is both local and non-destructive [20]. Furthermore, MOKE can be adapted to probe wafers of varying dimensions, making it particularly versatile for large-area samples. In MOKE, the polarization state of a monochromatic linearly polarized light beam is rotated at an angle $\Delta\phi$ upon reflection on a magnetic sample. This angular shift is proportional to the magnetization of the sample and will hereafter be referred to as the MOKE amplitude (see Fig. 1(a)). By applying a sweeping static magnetic field, MOKE amplitude versus magnetic field hysteresis loops can be obtained, enabling the characterization of the magnetic state of the film (see Fig. 1(b)). In addition, in our experimental setup, the 100 mm diameter wafer is placed on a moving stage, allowing for a full-wafer mapping of magnetic properties at the thin film level. This approach ensures micrometer-sized spatial resolution of the magnetic properties across the entire wafer surface.

The MOKE amplitude, remanence, and coercive field maps of the stack are shown in Fig. 1(c) with the magnetic field applied perpendicular to the film surface in all cases. By analyzing the maxima and minima across these maps, four distinct magnetic regions can be identified across the wafer, as highlighted in the coercive field map in Fig. 1(c). In the first region, the stack exhibits superparamagnetic (SPM) behavior. In the second region, the film is ferromagnetic with an out-of-plane easy axis (OOP FM). The third region is the OOP-IP spin reorientation transition (SRT) region, leading to a ferromagnet with an in-plane easy axis (IP FM) in the fourth region. These magnetic behaviors can be qualitatively explained by variations in the effective anisotropy energy density $K_{\text{eff}}$ across the wafer. For thin films,
<figure><img src="image_2.png" /><figcaption>FIG. 1. Schematic diagram of the MOKE experimental setup. A linearly polarized monochromatic laser beam reflects off the magnetic stack, which is magnetized out of plane. The reflected beam tilts the polarization axis by an angle $\Delta\phi$, directly proportional to the magnetization of the sample and recorded as a MOKE signal amplitude. The 100 mm wafer is placed on a moving stage that can shift in the X and Y directions, allowing for full wafer mapping of the film magnetic properties. (b) Example of a MOKE amplitude vs OOP magnetic field hysteresis loop at room temperature in a specific region of the wafer. (c) Three different MOKE maps are shown as a function of both MgO tunnel barrier and FeCoB storage layer thicknesses obtained from a single measurement: MOKE amplitude, remanence and coercive field maps (top to bottom). The coercive field map highlights four distinct magnetic easy axis regions in the wafer: superparamagnetic (SPM), out-of-plane ferromagnetic (OOP FM), spin reorientation transition (SRT), and in-plane ferromagnetic (IP FM).</figcaption></figure>

$K_{\text{eff}}$ can be expressed as:

$$K_{\text{eff}} = -\frac{1}{2}\mu_0 M_s^2 + \frac{K_i}{t}, \qquad (1)$$

namely $M_s$ the saturation magnetization per unit volume, $K_i$ the interface anisotropy
energy per unit area, $\mu_0$ the vacuum permeability, and $t$ the thickness of the ferromagnetic storage layer. The first term on the right-hand side is the demagnetizing energy density, while the second term is the interfacial anisotropy one. It is worth noting that the volume anisotropy contribution is negligible for thin films and not considered in Eq. 1.

In FeCoB/MgO-based magnetic systems, the hybridization of Fe 3d orbitals with the O 2p orbitals at the FeCoB/MgO interface produces a large perpendicular magnetic anisotropy contribution [21, 22], while the demagnetizing term favors in-plane magnetization. Hence, the easy magnetization axis is determined by the sign of $K_{\text{eff}}$, which is negative for IP FM and positive for OOP FM, as described by Eq. 1. Fig. 1(c) and Fig. S1 show that the MOKE amplitude increases linearly with $t_{\text{FeCoB}}$, indicating that the saturation magnetization per unit volume of FeCoB is constant throughout the wafer. This implies that the demagnetizing term in Eq. 1 is uniform across the wafer. Additionally, after the annealing process, the interfacial properties between the FeCoB and MgO layers are established, suggesting that the interfacial anisotropy energy $K_i$ primarily varies with the MgO thickness, which reflects the level of oxidation of the tunnel barrier. Consequently, the specific spin reorientation transition (SRT) thickness of the FeCoB, which governs the transitions between the OOP and IP magnetic regions, depends on the MgO thickness. This feature is observed in the MOKE maps from Fig. 1(c), where the SRT region shifts to lower FeCoB values as the MgO thickness increases. Furthermore, the superparamagnetic behavior (SPM) observed in regions with thinner FeCoB is attributed to the formation of a magnetically dead layer (DL), caused by the diffusion of W atoms into the film during the annealing process [23, 24]. Based on the MOKE amplitude map, the thickness of this dead layer is estimated to be 0.71 nm, corresponding to the storage layer thickness where the MOKE amplitude approaches zero (see Appendix S1).

### III. DETERMINATION OF THE EFFECTIVE ANISOTROPY USING MOKE

The energy barrier height between the two stable states is driven by the effective anisotropy values. Hence, accurately determining the anisotropy at the thin-film level is critical for predicting the STT switching characteristics of nanofabricated devices. In our sample, the cross-wedge thicknesses of both MgO tunnel barrier and the FeCoB storage layer set a range of effective anisotropy, from positive values (OOP FM region) to negative
values (IP FM region). For specific applications and operational temperature ranges, precise anisotropy values are required, making the determination of effective anisotropy across the entire wafer a matter of considerable importance. Experimentally, $K_{\text{eff}}$ can be extracted using several techniques, including VSM [25, 26], SQUID [27], ferromagnetic resonance (FMR) [28], Brillouin light scattering (BLS) [29–31] and electrical transport measurements [32]. However, these techniques frequently require the sample size to be reduced to a few millimeters to fit the physical limitations of the experimental setup. Furthermore, a comprehensive measurement of both $K_{\text{eff}}$ and $K_i$ across the wafer as a function of MgO and FeCoB thicknesses would require significant effort to cut the wafer into multiple small pieces. To address these limitations, we propose a non-destructive and expedited method based on MOKE magnetometry. This approach enables the extraction of both $K_{\text{eff}}$ throughout the wafer and $K_i$ as a function of MgO thickness, offering a faster alternative.

Magnetic anisotropy is generally defined as the energy required to reorient the magnetization from the easy axis to the hard axis of magnetization. In a hysteresis loop, this energy is represented by the area enclosed between the loops corresponding to the easy and hard axes [25]. In isotropic systems, this area difference vanishes. In anisotropic systems, when the magnetic field is applied along the OOP or IP direction and the easy axis aligns correspondingly (OOP or IP), a square-shaped hysteresis loop is obtained, resulting in an area close to zero. Conversely, when the easy axis is IP (OOP) and the field is applied OOP (IP), the hysteresis loop becomes linear until saturation, leading to a non-zero area and indicating $K_{\text{eff}} < 0$ ($K_{\text{eff}} > 0$). In the MOKE maps from Fig. 1(d) an OOP magnetic field is applied. In this case, the hysteresis loops from the OOP FM and SRT regions are square-shaped, resulting in $K_{\text{eff}} \approx 0$. However, as the FeCoB thickness increases, the easy axis gradually transitions from SRT to IP FM, requiring a larger magnetic field to saturate the film. This results in increasingly linear hysteresis loops and nonzero areas ($K_{\text{eff}} < 0$). Fig. 2(a) shows the calculated effective anisotropy energy density map as a function of MgO and FeCoB thicknesses derived from the MOKE amplitude loops in Fig. 1(d). Fig. 2(b) depicts representative MOKE loops for increasing FeCoB thicknesses at a fixed MgO thickness of approximately 1.4 nm, illustrating the variation in loop shape.

It is important to mention that the maximum magnetic field in our experimental setup is limited to 2300 Oe, which is insufficient to fully saturate the hard-axis hysteresis loops far from the spin-reorientation transition (SRT) region. Additionally, even a slight OOP
<figure><img src="image_3.png" /></figure>

FIG. 2. (a) Effective anisotropy energy density $K_{\text{eff}}$ map as a function of MgO and effective FeCoB storage layer thicknesses. The map highlights the regions corresponding to: out-of-plane ferromagnetic (OOP FM), reorientation (SRT), and in-plane ferromagnetic (IP FM). (b) MOKE amplitude vs OOP magnetic field hysteresis loops for various FeCoB effective thicknesses. These loops represent points along the blue arrow in (a), all taken at a fixed MgO thickness of 1.4 nm. (c) $K_{\text{eff}}t_{\text{eff}}$ vs $t_{\text{eff}}$ curves measured using VSM (green squares), SQUID (red star), and MOKE (orange dots), all at a fixed MgO thickness of 1.4 nm. The dashed black line corresponds to the fit using Eq. 1, with the slope set to match the FeCoB saturation magnetization value (1700 emu/cm$^3$). (d) Interface anisotropy $K_i$ dependence on MgO thickness, showing a decreasing trend with increasing MgO thickness.

component in an applied IP magnetic field can cause the film to align in the OOP direction due to the large perpendicular magnetic anisotropy, affecting the shape of the IP hysteresis loops and the calculation of $K_{\text{eff}}$. Due to these constraints, our method exclusively uses an OOP magnetic field to extract the anisotropy. With an alternative setup capable of applying stronger magnetic fields, both OOP and IP field orientations could be used for a more comprehensive anisotropy analysis.

The $K_{\text{eff}}$ map shown in Fig. 2(a) was obtained by fitting each MOKE amplitude vs magnetic field hysteresis loop with a sigmoidal function of the type:
$$m(H_{\text{OOP}}) = a \arctan \left( \frac{H_{\text{OOP}}}{b} \right), \qquad (2)$$

where a and b are fitting parameters, $H_{\text{OOP}}$ is the OOP applied magnetic field, and $m \in [-1, 1]$ is the normalized MOKE amplitude. The corresponding fits are represented by a black dashed line in Fig. 2(b). An analytical expression for $K_{\text{eff}}$ can be obtained (see Appendix S2) by expressing $H_{\text{OOP}}$ as a function of $m$ in Eq. 2 and performing the integral over $m$ from 0 to 1, i.e. $K_{\text{eff}} = (\mu_0 M_s / 2) \int_0^1 H_{\text{OOP}}(m) dm$. This expression reads

$$K_{\text{eff}} = \frac{\mu_0 M_s}{2} \left\{ H_{\text{sat}} \left[ 1 - a \arctan \left( \frac{H_{\text{sat}}}{b} \right) \right] + \frac{1}{2} ab \ln \left( \frac{H_{\text{sat}}^2}{b^2} + 1 \right) \right\} \quad (3)$$

namely $M_s$ the FeCoB saturation magnetization and $H_{\text{sat}}$ the saturation magnetic field. The three fitting parameters ($a$, $b$, and $H_{\text{sat}}$) are obtained from experimental MOKE loops, and the FeCoB $M_s$ is independently determined using VSM and SQUID, corresponding to a room temperature value of approximately 1700 emu/cm$^3$. The area representing $K_{\text{eff}}$ is highlighted in orange over the loops in Fig. 2(b).

At a fixed MgO thickness, the interface anisotropy $K_i$ can be estimated by plotting $K_{\text{eff}}t_{\text{eff}}$ vs $t_{\text{eff}}$, where $t_{\text{eff}}$ is the FeCoB effective thickness, i.e. $t_{\text{FeCoB}} - t_{\text{DL}}$, with $t_{\text{DL}} = 0.71$ nm representing the dead-layer thickness. According to Eq. 1, the slope of this curve corresponds to the demagnetizing term, while the y-intercept corresponds to $K_i$. Fig. 2(c) shows this relationship for $t_{\text{MgO}} = 1.4$ nm, corresponding to the horizontal dashed line in Fig. 2(a), alongside values obtained via VSM and SQUID. A clear overlap is observed between the results from VSM, SQUID, and the MOKE method presented here. The slope of the curve is fixed according to the saturation magnetization value (1700 emu/cm$^3$), and the resulting interface anisotropy is approximately 1.6 mJ/m$^2$, consistent with values reported for FeCoB/MgO systems [33–35]. This analysis was extended to all available MgO thicknesses in the wafer, and the variation of the interface anisotropy as a function of MgO thickness is shown in Fig. 2 (d). As expected, $K_i$ decreases with increasing MgO thickness due to a reduction in the oxygen content within the barrier, which reduces the number of Fe-O bonds at the interface contributing to perpendicular magnetic anisotropy [16]. The next step towards the optimization of devices working at cryogenic temperature is to understand the temperature dependence of $K_{\text{eff}}$, $M_s$, and $K_i$. This is described in the next section.
IV. TEMPERATURE DEPENDENCE OF MAGNETIC ANISOTROPY

The temperature dependence of magnetic anisotropy was investigated using SQUID magnetometry. The stack structure employed was identical to the one described in Section III, with two main differences: the SAF multilayer was omitted, and the FeCoB reference layer was reduced to 0.4 nm to make it non-magnetic. These changes were implemented in order to focus solely the magnetic features of the free layer, omitting any possible contribution from both SAF structure and reference layer during SQUID measurements. The film was deposited on a polished Si substrate of 5 × 5 mm² using the same conditions and annealing process as in Section II and its structure is as follows: Si wafer/W (0.2)/FeCoB (0.4)/MgO (1.4)/FeCoB (1.45)/W (2)/ Pt (2). The numbers represent the nominal thickness in nanometers. In this case, both the MgO tunnel barrier and effective FeCoB layer have a constant thickness of 1.4 nm and 0.95 nm, respectively, leading to an out-of-plane easy axis.

The effective magnetic anisotropy was measured by sweeping an in-plane applied magnetic field from 0 to 20 kOe to align the magnetization in the plane of the film. The area over the magnetization versus field (M-H) curve represents the effective anisotropy of the film, which increases as the temperature decreases, as shown in Fig. 3(a). The calculation of this area followed the procedure described in the previous Section, in which the hysteresis curve was fitted using Eq. 2, and the area was subsequently calculated using Eq. 3. The saturation magnetization $M_s$ at each temperature was determined as the average magnetization value between 10 and 20 kOe. Thus, both $K_{eff}$ and $M_s$ were directly extracted from the M-H curves and $K_i$ was deduced using Eq. 1. Only one-quarter of the total hysteresis loop was measured at each temperature to reduce the measurement time. The temperature dependence of $M_s$ is shown in Fig. 3(b). It increases from approximately 1700 emu/cm³ at room temperature to 2200 emu/cm³ at 5K. The evolution of $M_s$ was fitted using a standard Bloch model:

$$M_s(T) = M_s(0) [1 - \gamma T^{\alpha}], \quad (4)$$

where $M_s(0)$ is the value of $M_s$ at zero temperature, and $\gamma$ and $\alpha$ are fitting parameters. The dashed black line in Fig. 3(b) shows the fit to the experimental data using this model. Fig. 3(c) shows the relationship between the interface anisotropy $K_i$ and the normalized saturation magnetization $M_s(T)/M_s(0)$. This relationship is described by the Callen-Callen law [36] (dashed line):
<figure><img src="image_4.png" /><figcaption>FIG. 3. (a) Magnetization vs IP magnetic field (M-H) at 300K, 200K, 100K, and 5K. As the temperature decreases, both the saturation magnetization and effective anisotropy increase. The effective anisotropy is represented by the area above the M-H curve. (b) Temperature dependence of the saturation magnetization $M_s$ extracted from the M-H loops in (a). The magnetization at the field plateau with the field is plotted against temperature. The dashed line represents the fit to the data using Eq. (5), with fitting parameters $\alpha = 1.76$ and $\gamma = 1.2 \times 10^{-5} K^{-\alpha}$. (c) Interface anisotropy $K_i$ plotted against normalized magnetization $M_s(T)/M_s(0)$ across the temperature range. The dashed line represents the expected dependence according to the Callen-Callen law (Eq. 6), with a fitted exponent $\beta = 2.07$. (d) Effective anisotropy energy density $K_{eff}$ temperature dependence, obtained experimentally from the M-H loops (black circles). The fit incorporates the temperature dependence of both saturation magnetization and interface anisotropy. The different dashed lines show calculated $K_{eff}$ for various FeCoB effective thicknesses.</figcaption></figure>

$$K_i(T) = K_i(0) \left[ \frac{M_s(T)}{M_s(0)} \right]^\beta, \qquad (5)$$

where $K_i(0)$ is the interface anisotropy at zero temperature and $\beta$ is a fitting exponent close to 3 for uniaxial anisotropy [36]. However, in this experiment and in accordance with
previous reports in similar systems [37–39], the best fit for $\beta$ was close to 2. This deviation
is attributed to extrinsic factors such as the influence of the magnetic dead layer thickness or
surface inhomogeneities, as predicted theoretically [40]. Finally, the temperature dependence
of $K_{\text{eff}}$ is shown in Fig. 3(d). The experimental points are represented by black circles,
while the dashed line shows the calculated $K_{\text{eff}}$ using Eq. 1, incorporating the temperature
dependencies of $M_s$ and $K_i$ from Eqs. 4 and 5. The different curves in the figure represent
the calculated values of $K_{\text{eff}}$ for various FeCoB storage layer thicknesses. These curves
suggest that in wafer regions where the easy axis is IP FM or OOP FM, $K_{\text{eff}}$ becomes more
negative in the former and more positive in the latter. As a result, none of these regions
will be suitable for cryogenic memory applications. At the same time, the narrow spin
reorientation region $K_{\text{eff}}$ is vanishingly small at room temperature, resulting in the desired
moderate to small $K_{\text{eff}}$ values at cryogenic temperatures. This result suggest that devices
placed along the SRT region will be the best suited for cryogenic memory applications. The
correlation between the magnetic properties observed at the thin film level and the electric
performance of nanofabricated devices is the topic of next section.

V. FROM THIN FILMS TO NANOFABRICATED DEVICES

Magnetic tunnel junctions (MTJs) with a nominal diameter of 50 nm were fabricated from
the original stack using multiple lithography and etch steps. To investigate the correlation
between the magnetic properties observed at the thin film level and fabricated MTJ electrical
performance, three devices with varying FeCoB thicknesses were selected. Device 1 features
a storage layer with 0.9 nm nominal effective thickness, well within the OOP FM easy
axis region. Device 2 has a slightly thicker FeCoB, around 0.95 nm, exhibiting also an
OOP easy axis and corresponds to the same thickness as the stack measured by SQUID
for direct comparison. Finally, device 3 has a storage layer thickness of 1.00 nm, near the
spin-reorientation region, showing very low coercive field at room temperature.

The magnetic properties at the device level were obtained by performing hysteresis loops
of electrical resistance against OOP magnetic field (R-H loops) over a wide range of temper-
ature, from 300K to 40K. The tunneling magnetoresistance ratio (TMR) of all three devices
increases while decreasing the temperature (see Appendix S3), as observed in previous re-
ports for similar systems [41, 42]. Furthermore, the field switching probability distribution
curves obtained from the R-H loops measured, were used to extract the effective anisotropy field by fitting to a thermal activation model [18]:

$$P_{\text{sw}}(H_{\text{OOP}}) = 1 - \exp\left[\frac{-H_k f_0 \sqrt{\pi}}{2R\sqrt{\Delta}} \text{erfc}\left[\sqrt{\Delta}\left(1 - \frac{H_{\text{OOP}}}{H_k}\right)\right]\right]. \quad (6)$$

Here, $H_k = 2K_{\text{eff}}/(\mu_0 M_s)$ is the anisotropy field, $\Delta$ is the thermal stability factor, $f_0 \approx 1$ GHz is the attempt frequency, and $R$ is the experimental field sweep rate. A total of 100 R-H loops were measured at each temperature for statistical accuracy. To illustrate this, Fig. 4(a) shows the R-H loops of Device 3 ($t_{\text{eff}} = 1.00$ nm) and their corresponding $P_{\text{sw}}(H_{\text{OOP}})$ at 50K. The dashed line represents the fitting of the curve according to Eq. 6. Moreover, Fig. 4(b) depicts the temperature evolution of the coercive field $H_c$ for each device. As expected, the coercive field increases with decreasing temperature, and its evolution can be described using a thermally activated model:

$$H_c(T) = H_k(0) \left(1 - \frac{T}{T_B}\right)^{1/\lambda}, \quad (7)$$

$H_k(0)$ being the effective anisotropy field at 0K, $T_B$ the blocking temperature, and $\lambda \in [1, 2]$ a fitting exponent. The blocking temperature is the temperature at which $H_c$ vanishes, and in our case it ranges from 273K to 418K. The fitting exponent $\lambda$ was approximately 1.2 for all three devices. Fig. 4(c) shows the evolution of $H_k$ with temperature for each device and compares it with the values obtained by SQUID at the thin film level, represented by black dots in the figure. The thin film values were calculated from the definition of effective anisotropy, i.e. $H_k = 2K_{\text{eff}}/(\mu_0 M_s)$, while those of the devices were extracted by fitting the switching probability versus OOP magnetic field curves to Eq. 6. In particular, the clear overlap between the thin film and Device 2 values (both having a 0.95 nm storage layer effective thickness) indicates that the anisotropy of the stack remains unchanged after the nanofabrication process. In addition, the effective anisotropy value, as predicted, is tuned by varying the FeCoB thickness. All the dashed lines in Fig. 4(c) represent the calculated temperature dependence of $H_k$ using Eq. 1 and incorporating the temperature dependence of both $M_s$ and $K_i$ obtained from SQUID (see Eqs. 4 and 5), being $t_{\text{eff}}$ the only varying quantity. This result demonstrates that, regardless of the thickness, the temperature evolution of the effective anisotropy is well described using the same fitting parameters and exponents obtained by SQUID at the thin-film level. Additionally, due to this property, it
allows us to estimate for a given MgO thickness the maximum storage layer thickness with OOP FM easy axis (i.e. $K_{eff} > 0$) at any given temperature. This critical thickness can be derived from Eq. 1 and expressed as

$$t_{\text{eff}}^{C}(T) < \frac{2K_{i}(0)}{\mu_{0}M_{s}^{2}(0)} (1 - \gamma T^{\alpha})^{\beta-2}, \qquad (8)$$

namely $K_i(0)$ and $M_s(0)$ the interface anisotropy energy and saturation magnetization at zero temperature, respectively, and $\alpha, \beta$ and $\gamma$ the fitting parameters defined in Eqs. 4 and 5 and obtained by SQUID that best describe the temperature dependence of $K_i$ and $M_s$. In our particular case, $\beta \approx 2$, resulting in an almost constant critical thickness $t_{\text{eff}}^C$ throughout the temperature range. This indicates that, in our wafer, only the devices placed along the spin reorientation region are the best candidates for working at moderate switching voltage at cryogenic temperatures. Lastly, to assess the validity of the temperature dependence determined by SQUID, Fig. 4(d) shows the calculated interface anisotropy for all three devices and the thin film. In particular, all curves overlap in a single master curve, since $K_i$ is independent of $t_{\text{FeCoB}}$. The thin film values were extracted from the SQUID measurements, while $K_i$ values were calculated for the devices using $K_i = t_{\text{eff}}[(\mu_0/2)M_s(H_k + M_s)]$. The dashed line corresponds to Eq. 5 using the fitting parameters and exponents obtained by SQUID.

Figure 4 linked the magnetic properties observed at thin film level to those measured in nanofabricated devices. The last step in our analysis is to evaluate the STT switching performance of these devices and their dependence on anisotropy. In this way, in the absence of field, 100 ns voltage pulses with sweeping amplitude were used to switch the magnetization of the storage layer and obtain resistance versus voltage hysteresis loops (R-V loops). Similarly to the extraction of $H_k$ from the field switching probability distribution, the critical switching voltage $V_{c0}$ was deduced from the voltage switching probability distribution (see Appendix S4). Figure 5(a) shows the temperature dependence of $V_{c0}$ for all devices studied. It is observed that the critical switching voltage increases monotonically with decreasing temperature regardless of the storage layer thickness. In particular, as expected, the magnitude of $V_{c0}$ depends closely on the thickness, being larger for thinner storage layers. The same phenomenon is observed for $K_{eff}$ (see Fig. 4(c)). A thermally activation model relating the switching voltage with the effective anisotropy can be used to understand the temperature and thickness dependencies [19]:
$$V_{c0}(T) = V_{c0}(0) \left[ 1 - \frac{k_B T}{K_{\text{eff}}(T) A t} \ln(\tau/\tau_0) \right], \quad (9)$$

where $V_{c0}(0)$ is the critical switching voltage at zero temperature, $k_B$ is Boltzmann constant, $\tau_0 \approx 1$ ns is the attempt time, and $\tau = 100$ ns is the pulse duration. The volume of the MTJ is represented by $A t$, being $A = \pi(d/2)^2$ the area with MTJ nominal diameter $d = 50$ nm, and $t$ is the storage layer effective thickness. The dashed lines in the figure represent the fit of the experimental points to Eq. 9 considering the same effective anisotropy values as obtained by SQUID, leaving $V_{c0}(0)$ as the only fitting parameter. The fit is relatively good for Device 3, with effective thickness of 1.00 nm and belonging to the spin reorientation region. In contrast, an important deviation from the experimental points is observed for devices 1 and 2, which belong to the OOP FM region and have a storage layer thicknesses of 0.90 and 0.95 nm. Two sources of uncertainty may explain the observed deviations. On the one hand, the diameter of the MTJ is fixed to its nominal value of 50 nm. However, after the nanofabrication process, the impact of ion beam etching on side walls or some defects generated by this physical etching may modify the actual diameter of the MTJ, ultimately altering the temperature dependence in Eq. 9. Secondly, the effective anisotropy values used for the fit correspond those obtained from SQUID measurements, which were also in accordance with the values of effective field obtained from the R-H loops (see Fig. 4(c)). However, these values were obtained in experiments in which the heat dissipated through the MTJ was negligible, since the reading voltage was very small (10-30 mV). In this case, the pulse switching voltages are higher and the 100 ns pulse is enough time to reach an equilibrium temperature higher than the low bias measurements, resulting in a lower effective anisotropy value during the reversal. Consequently, $K_{\text{eff}}$ in Eq. 9 might be an overestimation of the real anisotropy value. The rise in temperature due to Joule heating during pulse application has already been observed and studied in magnetic tunnel junctions [11, 43]. The dissipated power density can be calculated $P = V^2/(R \times A)$, where $V$ is the applied voltage amplitude and $R \times A$ the resistance-area product of the junction. The temperature rise can be estimated by multiplying this power dissipation by a proportionality factor $a_T$. This proportionality factor is stack- and geometry-dependent and for similar systems values $\approx 4$ K/(mWµm²) have been reported [11, 43, 44]. Assuming $R \times A = 15\Omega\mu m^2$ and $V = 0.5V$, the estimated temperature rise would be close to 70 K, similar to reports on similar systems [11]. Given that the dissipated power scales as $V^2$ and that the experimental points from
device 3 fit the model, the data and model appear consistent, thereby supporting the local heating hypothesis.

Fig. 5(b) depicts the variation of the critical switching voltage with the FeCoB thickness at 77K. It is observed that the switching voltage magnitude decreases with increasing FeCoB thickness, and it could be further lowered by using thicker storage layers until the critical thickness, which defines the OOP-IP easy axis transition (see Eq. 8), is reached. In our system, MTJs similar to Device 3 (1.50 nm) are close to this limit, therefore additional optimization in the tunnel barrier oxidation conditions or stack structure should be addressed to target lower operational switching voltages. Nevertheless, this result highlights the potential of the method presented in this work to select, from room temperature measurements at the thin film level, those regions of the wafer potentially suitable for cryogenic memory applications.

## VI. CONCLUSIONS

We presented an experimental method based on MOKE magnetometry for rapidly extracting the effective anisotropy of magnetic systems applicable to full wafers. The accuracy of this extraction was validated by direct comparison with values obtained using conventional methods such as VSM and SQUID. The interface anisotropy and saturation magnetization values of the FeCoB/MgO-based magnetic stack investigated were determined to be 1.6 mJ/m² and 1700 emu/cm³, respectively. Moreover, employing the MOKE method presented in this work, a monotonic decrease in interface anisotropy was obtained with increasing MgO thickness with sub-nanometer resolution. Once the validity of the MOKE method was confirmed, the temperature evolution of anisotropy was studied using SQUID magnetometry on a sample 5 × 5 mm² with a similar magnetic stack. The saturation magnetization was found to follow a typical Bloch law with a scaling exponent α = 1.76, while the interface anisotropy adhered to the Callen-Callen law with an exponent β ≈ 2. The deviation from the conventional β = 3 is attributed to the influence of the dead layer or interface inhomogeneities. Additionally, three magnetic tunnel junctions with a nominal diameter of 50 nm and varying effective thicknesses of the FeCoB storage layer (0.90 nm, 0.95 nm, and 1.00 nm) were nanofabricated. The coercive field ($H_c$) and the effective anisotropy ($H_k$) were determined via electrical resistance measurements against out-of-plane magnetic field
hysteresis loops. These results indicate a close correlation between the magnetic properties obtained at the thin-film level and those at the device level. The temperature dependence of $H_k$ was consistent with that obtained by SQUID magnetometry. Finally, 100 ns current pulses were employed to study the STT switching characteristics of the devices as a function of temperature. Notably, the device with the lowest effective anisotropy (Device 3, with $t_{eff} = 1.00$ nm) exhibited the lowest critical switching voltage across the entire temperature range, in agreement with model predictions. These results validate the MOKE method presented and demonstrate its effectiveness in selecting wafer devices to operate at specific target temperatures. This work proposes a methodology for assessing effective anisotropy, enabling combinatorial layer stack optimization in large-area wafers.
<figure><img src="image_5.png" /><figcaption>FIG. 4. Three devices from the same stack with different FeCoB layer thickness were selected to study STT switching: Device 1 (D1, 0.90 nm, yellow squares), device 2 (D2, 0.95 nm, red triangles), and device 3 (D3, 1.00 nm, green stars). For direct comparison the thin film data is included in (d) (TF, 0.95 nm, black dots), with the corresponding legend in (d). (a) Electrical resistance vs OOP magnetic field loops and their corresponding field switching probability distribution $P_{sw}$ for device 3 at 50K. The fitting according to Eq. 6 is shown by the black dashed line. (b) Coercive field dependence with temperature obtained from R-H loops and their evolution according to Eq. 7. (c) Effective anisotropy field dependence with temperature obtained from the fitting of the field switching probability distribution curves. The $H_k$ of the thin film is calculated from its definition: $H_k = 2K_{eff}/(\mu_0 M_s)$. The dashed lines correspond to $H_k(T) = -M_s(T) + 2K_i(T)/(\mu_0 M_s(T)t_{eff})$, being $M_s(T)$ and $K_i(T)$ described by Eqs. 4 and 5, respectively. The only tunning parameter is $t_{FeCoB}$. (d) Interface anisotropy as a function of temperature for all three devices and thin film. The dashed line corresponds to the fit according to Eq. 5.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Temperature dependence of the critical switching voltage for each device and its fit according to Eq. 9. (b) critical switching voltage dependence on FeCoB storage layer thickness at 77K.</figcaption></figure>
ACKNOWLEDGMENTS

This work was supported by the French National Research Agency in Project CRYMCO
(ANR-20-CE24-0009) and in part through the 'France 2030' program (PEPR PRESQUILE-
ANR-22-PETQ-0002).
# Appendix S1: Determination of FeCoB Dead Layer Thickness from MOKE Amplitude Maps

When remanence approaches 100%, the MOKE amplitude is directly proportional to the magnetization of the film. The dead layer thickness $t_{DL}$ is defined as the thickness of the FeCoB storage layer at which the magnetization becomes negligible. The method used to determine $t_{DL}$ involves identifying the FeCoB thickness where the MOKE amplitude vanishes for remanence values greater than 90%. The extrapolation of these data points, illustrated in Fig. S1 as a red line, yields a $t_{DL}$ value of approximately 0.71 nm for our sample. This value is consistent with the dead layer thickness reported for similar FeCoB-based thin films.

<figure><img src="image_7.png" /></figure>

FIG. S1. MOKE amplitude signal data against FeCoB storage layer thickness for remanence values larger than 90% (blue dots). The extrapolation of these points is represented as a red line in the graph and the X intercept corresponds to the dead layer thickness of the system. The MOKE amplitude and remanence maps are shown to highlight the region employed for the calculation, which corresponds to the points inside the vertical dashed lines connecting both maps.
## Appendix S2: Calculating Effective Anisotropy from the Area in MOKE Amplitude vs Field Loops

The effective anisotropy values presented in Fig. 2(a) from the main text were extracted by calculating the area over the MOKE amplitude vs. field loops according to Eq. 3. In our setup, the magnetic field was applied normal to the film plane, thus the easy-axis square-shaped MOKE hysteresis loops present in the out-of-plane ferromagnetic (OOP FM) and spin reorientation transition (SRT) regions (see Fig. 1(c) in the main text) resulted in zero-valued areas. As the FeCoB thickness increases, the easy axis transitions from SRT to IP FM, and a larger field is needed to saturate the film, resulting in nonzero areas in those regions. The effective anisotropy, identified as the area over the MOKE loop, is defined by the integral:

$$K_{\text{eff}} \approx -\frac{\mu_0 M_s}{2} \int_0^1 H_{\text{OOP}}(m) \, dm. \qquad (\text{S1})$$

We can use Eq. 2 in the main text to express $H_{\text{OOP}}$ as a function of $m$ to obtain:

$$H_{\text{OOP}}(m) = b \tan \left(\frac{m}{a}\right). \qquad (\text{S2})$$

A representative MOKE amplitude vs. field hysteresis loop is shown in Fig. S2. In this case, the effective anisotropy corresponds to the yellow area, denoted as A1. Substituting Eq. S1 into S2 one obtains:

$$K_{\text{eff}} = A_1 = -\frac{m u_0 M_s}{2} \int_0^1 b \tan \left(\frac{m}{a}\right) dm = \frac{\mu_0 M_s}{2} ab \ln \left|\cos \left(\frac{1}{a}\right)\right|, \qquad (\text{S3})$$

where $a$ and $b$ are extracted by fitting the MOKE loop. This method would calculate directly the area $A_1$ and therefore $K_{\text{eff}}$. However, it is not difficult to realize that it is
numerically unstable since the function arctan(x) used to fit the loops converges to ±π/2 for $x \to \pm\infty$, which means that the parameter a, which defines the amplitude of the loop, must converge to ±2/π so that the total values at saturation are ±1. Inserting a = ±2/π into Eq. S3 results in values close to zero inside the logarithm, resulting in numerical instabilities in many of the considered loops. An alternative, also analytical approach to calculate A₁ is to calculate first the green area in Fig. S2, denoted as A₂, and then subtract the joint area between them. For A₂, we use Eq. 2 and the integral becomes

$$A_2 = -\frac{\mu_0 M_s}{2} \int_0^{H_{\text{sat}}} a \arctan \left( \frac{H_{\text{OOP}}}{b} \right) dH_{\text{OOP}}. \qquad (S4)$$

The prefactor is added to maintain units of energy density. $H_{\text{sat}}$ sets the upper limit of the integral and corresponds to the saturation field. This field is extracted systematically from each loop by selecting the minimum field value at which the MOKE loop is 1. Eq. S5 can be solved analytically, resulting in

$$A_2 = \frac{\mu_0 M_s}{2} \left[ a H_{\text{sat}} \arctan \left( \frac{H_{\text{sat}}}{b} \right) - \frac{b}{2} \ln \left( \frac{H_{\text{sat}}^2}{b^2} + 1 \right) \right] \qquad (S5)$$

Now, from Fig. S2 it is easy to see that $A_1 + A_2 = 1 \times H_{\text{sat}}$. Therefore, adding the prefactor $(\mu_0 M_s)/2$ to the total area to have units of energy density, the effective anisotropy can be expressed as:

$$K_{\text{eff}} = A_1 = \frac{\mu_0 M_s}{2} H_{\text{sat}} - A_2 = \frac{\mu_0 M_s}{2} \left\{ H_{\text{sat}} \left[ 1 - a \arctan \left( \frac{H_{\text{sat}}}{b} \right) \right] + \frac{1}{2} ab \ln \left( \frac{H_{\text{sat}}^2}{b^2} + 1 \right) \right\}, \qquad (S6)$$

This corresponds to Eq. 3 in the main text.
<figure><img src="image_8.png" /><figcaption>FIG. S2. Representative normalized MOKE hysteresis loop showcasing the different areas $A_1$ and $A_2$ employed to calculated effective anisotropy. The saturation field $H_{sat}$ is also depicted.</figcaption></figure>

## Appendix S3: Temperature Evolution of Electrical Resistance and Tunneling Magnetoresistance

Hysteresis loops of electrical resistance against OOP magnetic field (R-H loops) were performed over a wide range of temperatures, from 300K to 40K. An example of a typical R-H loop can be observed in Fig. 4(a) from the main text. The high and low resistance states correspond to the antiparallel and parallel relative magnetization states. Fig. S3(a) shows the dependence on temperature of each resistance state for each individual device. The parallel states is almost constant in the whole range of temperatures, while the antiparallel one increases linearly for lower temperature values, as expected [41]. Moreover, the tunneling magnetoresistance, defined as $TMR = (R_{AP} - R_P)/R_P \times 100$, also follows a linear increase for decreasing temperatures (see Fig. S3(b)). In memory applications, high TMR values are desired in order to achieve faster detection, thus improving overall device performance.
<figure><img src="image_9.png" /></figure>

FIG. S3. Evolution with temperature of (a) parallel (dots) and antiparallel (crosses) resistance states, and (b) tunneling magnetoresistance (TMR) for Device 1 (yellow), Device 2 (red), and Device 3 (green).

## Appendix S4: Calculation of the Critical Switching Voltage from the Resistance versus Applied Voltage Hysteresis Loops

The extraction of the critical switching voltage of each device as a function of temperature has been obtained in an analogous manner to that of the extraction of the effective anisotropy field. Instead of fitting electrical resistance versus OOP magnetic field hysteresis loops, the probability switching distribution curve was obtained from electrical resistance versus applied voltage hysteresis loops (R-V loops). This fitting is exemplified in Fig. S4 for Device 1 (with $t_{eff} = 0.90$ nm) at 295K. The top part of the figure contains the 30 experimental R-V loops, while the bottom part is the corresponding voltage-switching probability distribution. Included in the figure are the extracted fitting parameters. This routine was repeated for each device studied at different temperatures.
<figure><img src="image_10.png" /><figcaption>FIG. S4. Electrical resistance vs applied voltage loops (top) and their corresponding voltage switching probability distribution for device 1 ($t_{eff} = 0.90$ nm) at 295K (bottom). The fitting according to Eq. is represented by the dashed line.</figcaption></figure>
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