The perpendicular shape anisotropy-spin transfer torque-magnetic random access memories (PSA-STT-MRAMs) take advantage of the nanopillar free-layer geometry for securing a good thermal stability factor from the shape anisotropy of the nanomagnet. Such a concept is particularly well-suited for small junctions down to a few nanometers. At such a volume size, the nanopillar can be effectively modeled as a Stoner–Wohlfarth particle, and the shape anisotropy scales with the spontaneous magnetization by Ms2. For almost all ferromagnets, Ms is a strong function of temperature; therefore, the temperature-dependent shape anisotropy is an important factor to be considered in any modeling of the temperature-dependent performance of PSA-STT-MRAMs. In this work, we summarize and discuss various possible temperature-dependent contributions to the thermal stability factor and coercivity of the PSA-STT-MRAMs by modeling and comparing different temperature scaling and parameters. We reveal nontrivial corrections to the thermal stability factor by considering both temperature-dependent shape and interfacial anisotropies. The coercivity, blocking temperature, and electrical switching characteristics that resulted from incorporating such a temperature dependence are also discussed, in conjugation with the nanomagnet dimension and coherence volume.

Magnetic tunnel junctions (MTJs) have been intensively investigated fundamentally and developed technologically in the past few decades due to their high potential in future non-volatile magnetic memories. While the end of Moore’s law may be approaching, the scaling of the MTJs has also been one of the main challenges in the development of nonvolatile spin-transfer torque magnetoresistive random access memory (STT-MRAM). The key technology of today’s MTJ is the perpendicular MTJ (p-MTJ) using interfacial anisotropy at a ferromagnet/oxide (e.g., FeCoB/MgO) interface, and high volume manufacturing based on p-MTJs with only tens of nm has been initiated. However, as MTJ units become smaller to meet the ultrahigh-density integration, their thermal stability factor becomes a critical issue, as the interfacial anisotropy inevitably reaches a physical limit in securing a sufficient energy barrier between the two possible states.

As a result, a new concept known as the perpendicular shape anisotropy-spin transfer torque-magnetic random access memories (PSA-STT-MRAMs) has been recently proposed and demonstrated,1–5 which synergizes the shape anisotropy of the free-layer nanomagnet with the interfacial anisotropy to achieve high perpendicular anisotropy by properly engineering the free-layer’s aspect-ratio. This concept is well suited for future scaling of MTJs in the context of using nanomagnets for device building blocks, as the free-layer magnet approaching a Stoner–Wohlfarth (SW) particle with dimensions of a few tens of nm3 or even smaller,5 as illustrated in Fig. 1(a).

FIG. 1.

(a) Schematic illustration of the PSA-STT-MRAM, consisting (from the bottom): buffer layer, reference layer, MgO, layer (for inducing Ki), free-layer, MgO and capping layer. Compared with the conventional MTJ, the free-layer magnet in this PSA-MTJ is made into a nanopillar with tD so that the easy axis is along the thickness direction. (b) Temperature scaling of m(τ): the Bloch model for the temperature dependence (dashed line) and the re-plotted fitting curves (symbols) adopted from previous experimental reports using the Kuz’min temperature scaling with fitting parameters (s,p), for example, materials, Fe (0.35, 4), Co (0.11, 2.5), Ni (0.15, 2.5),18,19Nd5Fe17 (EP: 1.49, 2,5. HA: 0.72, 2.5),20 GdFeB (0.4, 2.5),21 FeCoB (0.65, 2.5),22 YFeB (0.7, 2.5),21Fe16N2 (0.42, 3.8),23CrI3 (bulk: 0, 1.7, Kuz’min scaling: 0.25. ML: 0, 1.7, Kuz’min scaling: 0.22),24CrBr3 (1, 2.5, Kuz’min scaling: 0.31), and EuS (0.8, 2.5, Kuz’min scaling: 0.369).25 

FIG. 1.

(a) Schematic illustration of the PSA-STT-MRAM, consisting (from the bottom): buffer layer, reference layer, MgO, layer (for inducing Ki), free-layer, MgO and capping layer. Compared with the conventional MTJ, the free-layer magnet in this PSA-MTJ is made into a nanopillar with tD so that the easy axis is along the thickness direction. (b) Temperature scaling of m(τ): the Bloch model for the temperature dependence (dashed line) and the re-plotted fitting curves (symbols) adopted from previous experimental reports using the Kuz’min temperature scaling with fitting parameters (s,p), for example, materials, Fe (0.35, 4), Co (0.11, 2.5), Ni (0.15, 2.5),18,19Nd5Fe17 (EP: 1.49, 2,5. HA: 0.72, 2.5),20 GdFeB (0.4, 2.5),21 FeCoB (0.65, 2.5),22 YFeB (0.7, 2.5),21Fe16N2 (0.42, 3.8),23CrI3 (bulk: 0, 1.7, Kuz’min scaling: 0.25. ML: 0, 1.7, Kuz’min scaling: 0.22),24CrBr3 (1, 2.5, Kuz’min scaling: 0.31), and EuS (0.8, 2.5, Kuz’min scaling: 0.369).25 

Close modal

In PSA-STT-MRAMs, assuming the single-domain magnetization reversal of the free-layer, the thermal stability factor, Δ, is expressed as1 

(1)

where μ0 is the permeability in free space, E0 the energy barrier, kB the Boltzmann constant, Ms0 the saturation magnetization of the bulk ferromagnet, and T the absolute temperature. Energy-wise, Kb and Ki are the bulk (magneto-crystalline) and interfacial-anisotropy energy densities, respectively. In conventional MTJs, the primary perpendicular anisotropy contribution comes from the interfacial term, Ki. Dimension-wise, t and D are the thickness and diameter of the ferromagnetic layer, and δN the difference in the dimensionless demagnetization coefficient, a.k.a., the shape anisotropy coefficient, between the perpendicular and in-plane directions. In conventional MTJs, such coefficient δN is close to 1 (when Dt); therefore, the shape anisotropy term in Eq. (1) is usually considered a negative contribution to the PMA, which is mainly sourced by the interfacial anisotropy. However, in PSA-STT-MRAMs, by properly engineering the nanomagnet aspect-ratio t/D, δN can become negative so that the shape anisotropy term, δNMs022μ0t, provides a positive contribution to the thermal stability in synergy with the interfacial term.

On the other hand, since the shape anisotropy strongly depends on the saturation magnetization (Ms02), the temperature dependence of such an additional PMA source that is due to the temperature-dependent Ms should not be neglected. So far, however, due to its dominating room temperature applications, discussions regarding the temperature-dependent behavior of PSA-STT-MRAMs has not been comprehensive, with only a few pioneering reports addressing this issue.6,7

In this work, we discuss the temperature dependence of PSA-STT-MRAMs by incorporating and comparing several temperature scaling models. First, we note that in Eq. (1), if a temperature-dependent Ms0(T) and the Curie temperature (Tc) are considered, the shape anisotropy term becomes δNMs02m02(τ)2μ0t, where m0(τ)=Ms0(T)Ms0(0) is the reduced magnetization and τ=TTc is the reduced temperature. In addition, to be more complete, we also include the temperature dependence of the interfacial anisotropy, which usually follows the power scaling law, Ki(T)=Ki(0)[Ms(T)Ms(0)]n according to Callen and Callen,8,9 with Ms being the spontaneous magnetization and n=i(2i+1) corresponding to the ith-order anisotropy. For the 1st-order anisotropy, a scaling power n=3 should be expected. However, different values of n have also been reported which are related to, for example, local magnetic properties and nanofabrication related processes.7,10–15 Here, we also separate the interfacial magnetization, Ms1, with the earlier bulk counterpart, Ms0, and define the reduced interfacial magnetization, m1(τ)=Ms1(T)Ms1(0), and include the temperature dependence as Kim1n(τ).16 Therefore, the modified function for the thermal stability Δ can be expressed as

(2)

Next, it is important to properly treat the temperature-dependent reduced magnetization functions, m0(τ) and m1(τ). To explicitly apply Eq. (2), the reduced magnetization should be determined empirically with appropriate fitting parameters derived from experimental measurements. However, as an evaluation of the temperature model, it is also possible to adopt the temperature scaling equation for m0(τ) and m1(τ) involving material dependent parameters. In treating the temperature-dependent interfacial anisotropy, the Bloch scaling law17 is usually adopted: mbl(τ)=(1T/Tc)v, where v is the scaling parameter whose value is expected to be 3/2. However, many reports in studying the temperature-dependent magnetization found that using such a single parameter cannot well reproduce the experimental results, even for v values that deviate quite much from 3/2, see Fig. 1(b). Alternatively, a general equation for m(τ) with appropriate material-dependent parameters has been proposed by Kuz’min et al., which is expressed as18,19

(3)

In the above equation, the value for p is 5/2 in most of the ferromagnetic materials according to the analysis from the series expansion of low-lying magnetic excitations, and s is a more material dependent parameter, with 0<s<5/2, describing the functional profile of m(τ) as it varies with the reduced temperature. The theoretical derivation of s depends on the intensity of the exchange interaction and the stiffness of the magnetization excitation as reveled by the Heisenberg model. In Fig. 1(b), we re-plot the experimental fitting curves using the Kuz’min temperature model from the previous literature for a collection of materials of interest including metals, metal–alloys, Fe–(B,N) alloys, and also more recent layered van der Waals magnets.20–25 Almost all curves deviate quite much from the Bloch scaling law in which v is the only fitting parameter. In addition, some materials such as Nd5Fe17 exhibit quite different temperature scaling by measurements along the easy-plane (EP) and hard-axis (HA).

These material-dependent features have been shown to be able to be properly accounted for by the parameter s in the Kuz’min model. For example, the nontrivial discrepancy of the temperature-dependent magnetization in the rare-earth(RE)-FeB compounds, i.e., (RE)2Fe14B, RE = Nd, Gd, and Y, etc., has been attributed to the differences in the exchange interactions between RE-Fe as well as the stiffness around localized spin-wave branches, which are both accounted for by the parameter s. Last but not least, it is also found that a small modification of the Kuz’min scaling coefficient away from 1/3 can further improve the fitting to the temperature scaling of some materials.24,25

In order to demonstrate the temperature-dependent effects, we adopt the Kuz’min scaling law and use the parameter for FeCoB (s=0.65,p=5/2), and realistic values of Tc=480 K, Ms0=1.52 T, Kb=1.1×105Jm3, and Ki=2.2×103Jm2, as adopted from the previous literature. The Tc value we use is on the lower end, to also take into account the finite size effect26 which is relevant to the nanopillars with dimensions tens of nm2 or even smaller. It is also noted that these values should be only viewed as one set of example values, as the parameters, including Tc, can be largely different upon materials and devices engineering. We present in Fig. 2(a) the temperature-dependent thermal stability factor calculated by using the different temperature scaling laws and without considering the temperature dependence of Ki(T). As an example, the dimensions of the nanopillar free-layer are set to t=30 nm and D=10 nm (aspect-ratio: t/D=3). More details regarding the nanomagnet’s dimension dependence will be discussed later.

FIG. 2.

(a) Temperature-dependent thermal stability factor Δ for FeCoB calculated by using the Kuz’min reduced magnetization m(τ) (×) and using the Bloch scaling mbl(τ) (dashed line), with different v=1.5,1.75,2.0. Here, the temperature dependence of Ki(T) is not considered. (b) Temperature-dependent thermal stability factor Δ by using the Kuz’min scaling [with parameters for FeCoB (0.65,2.5)] for both m0(τ) and m1(τ), at different interfacial anisotropy coefficient n=0,2,3,4,5.

FIG. 2.

(a) Temperature-dependent thermal stability factor Δ for FeCoB calculated by using the Kuz’min reduced magnetization m(τ) (×) and using the Bloch scaling mbl(τ) (dashed line), with different v=1.5,1.75,2.0. Here, the temperature dependence of Ki(T) is not considered. (b) Temperature-dependent thermal stability factor Δ by using the Kuz’min scaling [with parameters for FeCoB (0.65,2.5)] for both m0(τ) and m1(τ), at different interfacial anisotropy coefficient n=0,2,3,4,5.

Close modal

First, we note that if the temperature-dependent Ms is not considered [m(τ)=1], a significant overestimation of Δ can be expected compared to that using the Kuz’min scaling. At around room temperature (300 K), the difference can be as large as δΔ50. Such a discrepancy further increases as temperature increases, which can be close to δΔ100 near the Curie temperature of FeCoB. This result shows the importance of considering the temperature-dependent magnetization in analyzing PSA-STT-MRAMs, especially for higher temperature related applications (150°C). On the other hand, the Bloch scaling seems to always underestimate Δ, regardless of the chosen v (example curves are shown for v=1.5,1.75, and 2). The comparison between the Bloch scaling and the Kuz’min model indicates the important role of the material-dependent parameter s in determining the energy barrier between the two states.

Figure 2(b) compares the different temperature-dependent Δ caused by the interfacial anisotropy Ki(T). Here, the Kuz’min model for the reduced magnetization is used for all the plotted curves (same shape anisotropy), however, different values of the interfacial anisotropy scaling coefficient (n) are compared, in which n=0 (dashed line) indicates a temperature-independent Ki. Other example curves for n=2,3,4,5 are shown, where n=3 is expected from theory for the 1st-order anisotropy. Experimental studies have shown that the m3(τ) scaling is generally successful in predicting the temperature dependence of interface-driven p-MTJs such as FeCoB/MgO and Fe/MgO. The temperature-dependent Ki(T) only becomes significant to Δ at around room temperatures and up to Tc. In general, an overestimation on Δ (in the range of δΔ1020) can be expected if one does not consider any temperature-dependent Ki (n=0), see Fig. 2(b). In addition, the different choices of n only account for a small modification to Δ, around δΔ< 10. However, since interfaces are generally rather sensitive to extrinsic thin-film parameters, the actual temperature dependence of interfacial MTJs is usually underestimated in a theoretical modeling. Furthermore, abrupt temperature-driven property changes may also arise,4 which may be attributed to multiple magnetic phases and/or exchange bias at the interface. On the other hand, the temperature-dependent shape anisotropy often reflects a more intrinsic dependence with the material properties, i.e., saturation magnetization, which makes the temperature-dependent properties of PSA-MTJs more reliable in performance and predicable with theoretical modeling.4,6,7

Another critical issue that would affect the temperature scaling related to Ki contribution is whether a separated interfacial magnetization, Ms1 should be considered, which is different from the bulk Ms0, as illustrated in Fig. 3(a). This treatment makes general sense, since many layer-resolved studies have indicated the existence of an attenuated interface/surface magnetization value (Ms1) or even a dead-layer, and the anisotropy is of interfacial origin that should then naturally scale only with the Ms1. As an example, such a consideration has recently found a great relevance in the temperature scaling of Fe/MgO interfacial p-MTJs.27 In PSA-MTJs, the separation of Ms0 and Ms1 seems more critical as both the shape (m02) and interfacial (m1n) anisotropies contribute to the thermal stability factor.

FIG. 3.

(a) Schematic illustration of the PSA-STT-MRAMs after considering an interfacial layer with an attenuated saturation magnetization Ms1=r×Ms0, for the temperature scaling of the interfacial anisotropy. (b) Temperature-dependent thermal stability factor Δ at different attenuation coefficient r, where r=1 corresponds to no inter-layer magnetization. (c) Temperature-dependent thermal stability factor Δ at different thermal coefficients b=0.25,0.5, and 0.75, for three r values (r=0.4,0.6,0.9).

FIG. 3.

(a) Schematic illustration of the PSA-STT-MRAMs after considering an interfacial layer with an attenuated saturation magnetization Ms1=r×Ms0, for the temperature scaling of the interfacial anisotropy. (b) Temperature-dependent thermal stability factor Δ at different attenuation coefficient r, where r=1 corresponds to no inter-layer magnetization. (c) Temperature-dependent thermal stability factor Δ at different thermal coefficients b=0.25,0.5, and 0.75, for three r values (r=0.4,0.6,0.9).

Close modal

Here, we take into account an interfacial magnetization that is attenuated by a factor of r with respect to the bulk value, i.e., r=Ms1Ms0, to properly account for the temperature scaling of Ki. For simplicity, we adopt the Kuz’min scaling for both m0(τ) and m1(τ), however, it is noted that the most explicit function should be determined empirically with fitting parameters derived from experimental measurements, as the Kuz’min scaling parameter may also be different for the bulk and interfacial components.

Figure 3(b) shows the effect of such an attenuating factor r on the temperature scaling of Δ, where r=1 indicates Ms0=Ms1, and r=0 represents an interfacial dead-layer. The separation of interfacial and bulk magnetization seems quite influential. Not considering the interfacial magnetization effect leads to an overall underestimation of Δ. In general, this effect becomes rather significant at lower temperatures. Nevertheless, at around room temperature, a non-trivial modification of Δ about 50–100 is still predicted (for a practical range of r, 0.60.7).

The inclusion of the surface magnetization reduction also brings up a necessary discussion of the thermal expansion effect to the temperature-dependent scaling. Due to the sputtered hetero-junction formed between the thin FeCoB layer and the MgO barrier, the interfacial PMA can become quite sensitive to lattice strain.29 In fact, strain engineering has been a key topic among other practical solutions for bench-marking MTJ performance.30,31 Such an interfacial strain effect would become more significant in thinner MTJ devices. Resultantly, the temperature-dependent thermal expansion effect should non-trivially factor in to the temperature-dependent interfacial anisotropy, especially when considering a magnetization reduction at the interface (Ms0Ms1). Usually, the temperature-dependent thermal expansion effect follows a scaling behavior of (1bTTc), where b is the thermal expansion coefficient.6,32,33 Such a correction factor should be superimposed to the existing Callen–Callen law for interfacial anisotropy, therefore, the temperature-dependent interfacial component becomes Ki(T)=Ki(1bTTc)m1n(τ).

We substitute this modified Ki(T) contribution to our analysis in Eq. (2). Figure 3(c) shows the effect of different thermal expansion coefficients b at three different Ms1 attenuation ratios (r=0.9, 0.6, and 0.4). First, we note that the thermal expansion effect to the calculated Δ is most dramatic at intermediate to high (approaching Tc) temperature ranges, e.g., T=200400 K. At lower temperatures, the shape anisotropy become more dominant (Ms02), rendering the surface magnetization effects nearly negligible. Second, the effective modification to Δ increases as the attenuation ratio decreases. When r is large, the thermal expansion effect is nearly negligible, however, for smaller r values such as r=0.4, a discrepancy of δΔ>50 can be obtained via changing b from 0.25 to 0.75, at 300 K.

Last but not least, we note that one other contribution which have not been accounted for is the temperature-dependent magnetocrystalline anisotropy. For soft ferromagnets with significant δN, like in the case of PSA-STT-MRAMs, the magnetocrystalline anisotropy usually plays a less critical role and therefore is generally neglected. However, for materials with large magnetocrystalline anisotropy, the temperature dependence should further account for the effect of Kb(T), and a nontrivial extra correction may be then expected besides the shape and interfacial anisotropy effect considered herein.

Next, we shift our focus to the temperature-dependent coercivity. Unlike the thermal stability factor, the coercivity is a more extrinsic parameter that depends not only the magnetic anisotropy, but also the dimension and shape of the nanomagnet, as well as the magnetization reversal mechanisms, such as coherent rotation or domain wall nucleation and propagation. In the context of SW particles, the volume size for a coherent rotation satisfies V<Vcoh(Lcoh)3, in which Lcoh is the coherence length. For Ni, Co, and Fe, the coherence lengths are 25, 15, and 11 nm, respectively.34–36 As a result, to properly model the coercivity, the volume size of the free-layer relative to the coherence volume needs to be taken into account. To this purpose, we define a reduced volume, Vred=V/Vcoh.

First, for a given volume size, V=πD24t, the nanomagnet dimension (t,D) directly impacts the shape coefficient δN. For a nanopillar free-layer, the demagnetization coefficients Nx=Ny and Nx+Ny+Nz=1, therefore δN=NzNx, where z is along the film normal direction. In Fig. 4(a), we plot the shape coefficient δN following the theoretical equations for two cases: the oblate spheroid (t<D) and prolate spheroid (tD), which have been used in describing the PSA-STT-MRAMs design window in many earlier reports.1,2,4,5

FIG. 4.

(a) Calculated shape anisotropy coefficient δN=NzNx as a function of the nanomagnet dimension (t,D). For t/D<1, the oblate spheroid case is used and for tD, the prolate spheroid case is used. (b) Mapping of the thermal stability factor Δ as a function of the nanomagnet dimension (t,D) at T=300 K. Scale-bar: [200,200]. The calculated t vs D corresponding to the coherence volume (dotted) and the maximum macrospin thickness (dashed) of Ni, Co, and Fe are also plotted.

FIG. 4.

(a) Calculated shape anisotropy coefficient δN=NzNx as a function of the nanomagnet dimension (t,D). For t/D<1, the oblate spheroid case is used and for tD, the prolate spheroid case is used. (b) Mapping of the thermal stability factor Δ as a function of the nanomagnet dimension (t,D) at T=300 K. Scale-bar: [200,200]. The calculated t vs D corresponding to the coherence volume (dotted) and the maximum macrospin thickness (dashed) of Ni, Co, and Fe are also plotted.

Close modal

Figure 4(b) shows the thermal stability factor Δ as a function of t and D at T=300 K, using Tc=480 K, Ms0=1.52 T, Kb=1.1×105Jm3, and Ki=5.0×103Jm2. It features two high-Δ regions: (i) one at the bottom-right corner that corresponds to the conventional interfacial anisotropy MTJ. Here, high enough Δ (usually Δ>80) may not be obtained for smaller D values such as D<20 nm. (ii) A much larger one near the top that corresponds to the PSA-MTJ. Here, high enough Δ can be realized even for D values 10 nm or less thanks to the shape anisotropy contribution. Separating the above two high-Δ regions is a dark region, which represents the case for in-plane shape anisotropy.

In Fig. 4(b), we also overlay curves representing the coherence volume, Vcoh, using the values for Ni, Co, and Fe, respectively. Under a certain Vcoh curve, the magnetization reversal satisfies the coherent rotation criteria, and above it, domain wall processes or a mixed reversal mechanism may be relevant. From our calculation, it is seen that the PSA-MTJ region from thermal stability point of view has an appreciable overlap with the coherent rotation region for VcohNi, however, only a small overlap for VcohFe and VcohCo. This suggests that the domain-wall-driven magnetization reversal may still be largely relevant in (Fe,Co)B PSA-STT-MRAMs. However, such a scenario could easily change via proper engineering of material parameters including the anisotropy energies and the coherence length. Besides, in order to maintain a certain volume size, one needs to increase t significantly if D has to decrease due to, for example, ultrahigh density considerations.

In addition, another constraint comes directly from the aspect ratio (t/D) affecting the thermal stability factor due to the limitation of the macrospin approach, i.e., there is an upper limit of thermal stability when we increase t/D, since the magnetization, beyond such a threshold t/D, would favor a reversal via domain wall nucleation and propagation.3 Therefore, we estimate the maximum macrospin thickness, tmax, as a function of the MTJ diameter using the method presented in Perrissin et al.3 and plot three curves for Ni, Fe, and Co in Fig. 4(b), respectively. We can see from Fig. 4(b) that we need to consider both the coherence volume and the maximum macrospin thickness effects whenever we attempt to utilize MTJs with large PSA (Δ>80), especially for Co- and Fe-based MTJs. Last but not the least, to mitigate the maximum macrospin thickness issue, we may also adopt the method of inserting a thin non-magnetic layer in the free layer.5 

To model the temperature-dependent coercivity, Hc(T), we write the temperature-dependent anisotropy, K(τ)=Δ×kBTcτ/V, and therefore, Hc(T)=μ0gK(τ)Ms0m0(τ)×{1[In[t/t0]Δ(τ)]1/α}, where In[t/t0] is a factor related to the time necessary for jumping over the energy barrier and is usually estimated as about 25, g and α are fitting parameters related to the anisotropy axis and the size of the nanomagnet. We use g=2,α=2 to approach the magnetization reversal in the context of SW particle with a coherent rotation mode.

In Fig. 5(a), we plot the temperature-dependent thermal stability factor for selective nanopillar dimensions (D=3,5,8,15,30 nm) with a fixed aspect-ratio (t/D=3). A curve representing the Co coherence volume is also shown (dashed line) as a reference. Figure 5(b) shows the corresponding temperature-dependent coercivity referenced to a low temperature value Hc(5K). The temperature-dependent Δ changes rapidly with the increase of the volume size and is basically irrelevant to the critical coherence volume. For coercivity, the temperature profile Hc(T) also exhibits a strong dependence on the nanomagnets’ dimension in the coherent rotation region (V<Vcoh), especially for the blocking temperature (TB), at which the Hc(T) crosses zero. However, the Hc(T) profile, as well as the TB value, does not change much with further increasing the dimension, once the volume size exceeds the coherence volume. In addition, the Bloch function of m(τ) (solid line) also cannot well reproduce the Hc(T) profile, and it tends to underestimate the blocking temperature, especially at regions beyond the coherence volume.

FIG. 5.

(a) Temperature-dependent thermal stability factor, Δ calculated for selective nanopillar dimensions (t,D), D=3,5,8,15,30 nm, with a fixed aspect-ratio (t/D=3). (b) The corresponding calculated temperature-dependent coercivity, Hc(T). The dashed lines in (a) and (b) represent the results obtained using the coherence volume of Co. For D=30 nm, the calculated Hc(T) for m(τ)=1 and for m(τ) using the Bloch function are also plotted as references.

FIG. 5.

(a) Temperature-dependent thermal stability factor, Δ calculated for selective nanopillar dimensions (t,D), D=3,5,8,15,30 nm, with a fixed aspect-ratio (t/D=3). (b) The corresponding calculated temperature-dependent coercivity, Hc(T). The dashed lines in (a) and (b) represent the results obtained using the coherence volume of Co. For D=30 nm, the calculated Hc(T) for m(τ)=1 and for m(τ) using the Bloch function are also plotted as references.

Close modal

Recently, it has been found interesting and technological relevant to study MTJs that operate in the superparamagnetic regime, a.k.a. stochastic MTJs, near and above the MTJs’ blocking temperature. The fluctuation of magnetization in stochastic MTJs can be used for high-dimensional optimization or sampling problems in probabilistic computing.16,37 Usually, the blocking temperature for SW particles with small volume size V and small shape coefficient δN is proportional to the potential barrier that only weakly depends on the temperature effect of magnetization. However, for larger particles with larger shape anisotropy, as in the present case, the correction factor attributed to the temperature variation becomes significant. This effect can be already observed in Fig. 5(b) for the different nanomagnet dimensions. This is because the correction on Hc(T) attributed to the temperature-dependent shape anisotropy is realized by the reduced magnetization m(τ), which directly results in the correction factor m(τB), where τB is the reduced blocking temperature. At lower temperatures, m(τ)1, a constant energy barrier is a good approximation to estimate the blocking temperature. On the other hand, if m(τ) varies significantly from m(0)=1, such as near Tc, then the determination of TB cannot neglect the correction effect arising from the m(τ).

To further evaluate the dimension effect on the blocking temperature, we calculated the Hc(T) for a range of D values, as shown in Fig. 6(a), while keeping the aspect-ratio the same (t/D=3) to ensure a same δN. A curve representing the coherence volume is also plotted as a reference (solid-line) to indicate the coherent rotation region. The blocking temperature, as derived from the coercivity function, is described as TB=E(τB)/[kBIn(t/t0)]. The dependence of TB on the reduced volume, Vred, can be then solved numerically. The result is plotted in Fig. 6(b). In the limit of small volume size, e.g., Vred0.10.3, TB is small and varies almost linearly with Vred, since m(τB) approaches 1 as τB approaches 0. As the nanopillar volume grows beyond Vred0.3, τB begins to deviate from the linear behavior with respect to Vred, since m(τB) decreases nonlinearly toward 0 and the correction to m(τB) becomes more significant. Such a nonlinear behavior also depends on the nanomagnet dimension (for a given V), and is more pronounced for higher aspect-ratio nanomagnet.

FIG. 6.

(a) (Dashed line) Temperature-dependent coercivity Hc(T) at a range of nanopillar dimension D increasing from 1 to 60 nm with a step size of 1 nm. The aspect-ratio is kept as t/D=3. (Solid line) Hc(T) reference calculated for a nanopillar dimension corresponding to the coherence volume of Co. (b) The numerically-solved blocking temperature, TB, from (a), vs the reduced volume Vred=V/Vcoh, at an fixed aspect-ratio t/D=3.

FIG. 6.

(a) (Dashed line) Temperature-dependent coercivity Hc(T) at a range of nanopillar dimension D increasing from 1 to 60 nm with a step size of 1 nm. The aspect-ratio is kept as t/D=3. (Solid line) Hc(T) reference calculated for a nanopillar dimension corresponding to the coherence volume of Co. (b) The numerically-solved blocking temperature, TB, from (a), vs the reduced volume Vred=V/Vcoh, at an fixed aspect-ratio t/D=3.

Close modal

Considering the potential applications of PSA-STT-MRAMs at both lower and higher temperatures, we further discuss and compare the thermal stability factor map Δ(t,D) at selective temperatures, T=50,200,380 K, in Fig. 7. At 50 K and below (down to the cryogenic temperature), the shape- and interfacial-induced stability merge together below D20 nm. In addition, it seems practical to secure a good thermal stability even for a nanopillar dimension below 10 nm. The overlap of the two regions indicates a synergistic effect of the shape and interfacial anisotropies. As temperature increases, both regions shrink at about the same rate. However, it is still possible to identify a reasonably large design window for different (t,D) combinations at a wide range of temperature from 100 to 300 K. At higher temperatures, the acceptable range for D becomes narrower (centered around 20 nm) and higher aspect-ratio of the nanomagnet is also desirable.

FIG. 7.

The thermal stability factor (Δ) as a function of the nanomagnet dimension (t,D) at selective temperatures: (a) 50 K, (b) 200 K, and (c) 380 K. Scale bar: [200,200].

FIG. 7.

The thermal stability factor (Δ) as a function of the nanomagnet dimension (t,D) at selective temperatures: (a) 50 K, (b) 200 K, and (c) 380 K. Scale bar: [200,200].

Close modal

We also evaluate the critical current density (Jc0) as a function of the MTJs’ dimension and the temperature in a current switching scenario via the spin-transfer torque (STT). The critical current density, Jc0=8αeγΔkBT/(μBgSTT), where γ is the gyromagnetic ratio, e is the electron charge, α is the damping constant, μB is the Bohr magneton, and gSTT is the STT efficiency.38gSTT is given by p/2/(1+p2cosθ) in the single reference layer MTJ and p/(1p4cos2θ) in the dual reference layer MTJ, where p is the spin polarization factor and θ is the initial angle between the magnetizations of the free layer and reference layer.39 We use a dual reference layer structure to perform our calculations. To obtain the STT switching curves, we normalize the resistance using the parallel resistance of the PSA-STT-MRAM structure, and therefore, only need to consider the voltage-dependent tunnel magnetoresistance ratio (TMR). The TMR is given by TMR(0)/[1+(Vsw/Vh)2], where Vsw is the applied (bias) voltage across the MTJ and Vh is the bias voltage where TMR(Vh) is half of TMR(0). As an example, we show the MTJ-diameter-dependent STT switching loops for different temperatures in Fig. 8.

FIG. 8.

The STT switching loops as a function of the MTJ diameter (D) at selective temperatures: (a) 50 K, (b) 200 K, and (c) 380 K. The MTJ thickness is set to 30 nm.

FIG. 8.

The STT switching loops as a function of the MTJ diameter (D) at selective temperatures: (a) 50 K, (b) 200 K, and (c) 380 K. The MTJ thickness is set to 30 nm.

Close modal

The STT switching simulations capture the essential physics since the simulated curves are qualitatively similar to the experimental observations.1 We further plot the critical current density map Δ(t,D) at selective temperatures, T=50,200,380 K, in Fig. 9. The four different sizes presented in Fig. 8 are also indicated on the map. At a fixed size, or a fixed aspect-ratio, the switching current density of the MTJ nanomagnet strongly depends on the temperature. For example, for a nanomagnet dimension of (30, 8), the switching current can be a few times different as one goes from cryogenic temperature to above room temperature.

FIG. 9.

The critical current density as a function of the nanomagnet dimension (t,D) at selective temperatures. The MTJ thickness is 30 nm. The calculated t vs D corresponding to the coherence volume of Ni, Co, and Fe are also plotted. The four dots correspond to the four MTJ dimensions discussed in Fig. 8.

FIG. 9.

The critical current density as a function of the nanomagnet dimension (t,D) at selective temperatures. The MTJ thickness is 30 nm. The calculated t vs D corresponding to the coherence volume of Ni, Co, and Fe are also plotted. The four dots correspond to the four MTJ dimensions discussed in Fig. 8.

Close modal

Similar to our analysis in the thermal stability factor map, we again overlay the coherence volume curves, using the values for Ni, Co, and Fe, respectively. Under the curve, the magnetization reversal satisfies the coherent rotation criteria without necessarily triggering the domain wall processes. From our calculation, it is seen that the coherent current-switching model is still limited to the smaller nanomagnet dimensions in the PSA-STT-MRAM window, which suggests that the domain-wall-driven magnetization reversal may be largely relevant in the current-driven switching scenario. Furthermore, we notice from a recent report that the switching could be driven by domain nucleation and domain wall motion even when the magnet size is small, but as long as t/D is larger than one.28 The corresponding temperature dependent behaviors with respect to these domain wall processes, though beyond the scope of the current work, may deserve more future investigations.

In summary, we provided a model to analyze the temperature dependence of shape-anisotropy magnetic tunnel junctions. The thermal stability factor, described in Eq. (1) is modified to include the temperature-dependent shape and interfacial anisotropy, Eq. (2), which is particular important for nanopillar-shaped free-layer of ferromagnetic materials. The corrections on field-switching coercivity, Hc(T), blocking temperature, TB, and current-driven switching using STTs are subsequently derived and presented, which attributes to the effect of temperature-dependent spontaneous magnetization m(τ).

Finally, we note that another important effect that has not been accounted for in this work is a quantitative analysis regarding nanomagnet’s finite size effect. The finite size effect of nanostructures is mainly related to the geometrical confinement to the correlation length, causing a reduction in the ordering temperature Tc. In addition, this effect will be further modified by the free surface effect as the nanomagnet size approaches the ultrafine limit, smaller than the effective range of spin–spin interaction.26 Last but not least, we point out that in many electric-current-induced switching experiments, additional temperature-dependent parameters need to be further enclosed, such as the spin-transfer torque efficiency, magnetic damping, resistance-area product, and so on. However, these factors as well as their temperature behaviors may be worthy of a separate study.

The work at the Oakland University was supported by the U.S. National Science Foundation (NSF) under Grant Nos. ECCS-1933301 and ECCS-1941426. The work at the University of Arizona was supported by the U.S. National Science Foundation under Grant No. DMR-1905783. The work at HKUST was supported by the UROP program and HKUST-Kaisa Joint Research Institute grant (No. OKT21EG08).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
K.
Watanabe
,
B.
Jinnai
,
S.
Fukami
,
H.
Sato
, and
H.
Ohno
, “
Shape anisotropy revisited in single-digit nanometer magnetic tunnel junctions
,”
Nat. Commun.
9
,
L1
(
2018
).
2.
B.
Jinnai
,
K.
Watanabe
,
S.
Fukami
, and
H.
Ohno
, “
Scaling magnetic tunnel junction down to single-digit nanometers—Challenges and prospects
,”
Appl. Phys. Lett.
116
,
160501
(
2020
).
3.
N.
Perrissin
,
S.
Lequeux
,
N.
Strelkov
,
A.
Chavent
,
L.
Vila
,
L. D.
Buda-Prejbeanu
,
S.
Auffret
,
R. C.
Sousa
,
I. L.
Prejbeanu
, and
B.
Dieny
, “
A highly thermally stable sub-20 nm magnetic random-access memory based on perpendicular shape anisotropy
,”
Nanoscale
10
,
12187
(
2018
).
4.
N.
Perrissin
,
G.
Gregoire
,
S.
Lequeux
,
L.
Tillie
,
N.
Strelkov
,
S.
Auffret
,
L. D.
Buda-Prejbeanu
,
R. C.
Sousa
,
L.
Vila
,
B.
Dieny
, and
I. L.
Prejbeanu
, “
Perpendicular shape anisotropy spin transfer torque magnetic random-access memory: Towards sub-10 nm devices
,”
J. Phys. D: Appl. Phys.
52
,
234001
(
2019
).
5.
B.
Jinnai
,
J.
Igarashi
,
K.
Watanabe
,
T.
Funatsu
,
H.
Sato
,
S.
Fukami
, and
H.
Ohno
, “High-performance shape-anisotropy magnetic tunnel junctions down to 2.3 nm,” in 2020 IEEE International Electron Devices Meeting (IEDM), San Francisco, CA (IEEE, 2020), pp. 24.6.1–24.6.4.
6.
S.
Lequeux
,
N.
Perrissin
,
G.
Grégoire
,
L.
Tillie
,
A.
Chavent
,
N.
Strelkov
,
L.
Vila
,
L. D.
Buda-Prejbeanu
,
S.
Auffret
,
R. C.
Sousa
,
I. L.
Prejbeanu
,
E.
Di Russo
,
E.
Gautier
,
A. P.
Conlan
,
D.
Cooper
, and
B.
Dieny
, “
Thermal robustness of magnetic tunnel junctions with perpendicular shape anisotropy
,”
Nanoscale
12
,
6378
(
2020
).
7.
J.
Igarashi
,
B.
Jinnai
,
V.
Desbuis
,
S.
Mangin
,
S.
Fukami
, and
H.
Ohno
, “
Temperature dependence of the energy barrier in X-1X nm shape-anisotropy magnetic tunnel junctions
,”
Appl. Phys. Lett.
118
,
012409
(
2021
).
8.
E. R.
Callen
and
H. B.
Callen
, “
Static magnetoelastic coupling in cubic crystals
,”
Phys. Rev.
129
,
578
(
1963
).
9.
E. R.
Callen
and
H. B.
Callen
, “
The present status of the temperature dependence of magnetocrystalline anisotropy, and the l(l+1)/2 power law
,”
J. Phys. Chem. Sol.
27
,
1271
(
1966
).
10.
Y.
Fu
,
I.
Barsukov
,
J.
Li
,
A. M.
Gonçalves
,
C. C.
Kuo
,
M.
Farle
, and
I. N.
Krivorotov
, “
Temperature dependence of perpendicular magnetic anisotropy in CoFeB thin films
,”
Appl. Phys. Lett.
108
,
142403
(
2016
).
11.
H.
Sato
,
P.
Chureemart
,
F.
Matsukura
,
R. W.
Chantrell
,
H.
Ohno
, and
R. F. L.
Evans
, “
Temperature-dependent properties of CoFeB/MgO thin films: Experiments versus simulations
,”
Phys. Rev. B
98
,
214428
(
2018
).
12.
E. C. I.
Enobio
,
M.
Bersweiler
,
H.
Sato
,
S.
Fukami
, and
H.
Ohno
,
Jpn. J. Appl. Phys. Part 1
57
,
04FN08
(
2018
).
13.
J. M.
Iwata-Harms
,
G.
Jan
,
H.
Liu
,
S.
Serrano-Guisan
,
J.
Zhu
,
L.
Thomas
,
R.-Y.
Tong
,
V.
Sundar
, and
P.-K.
Wang
, “
High-temperature thermal stability driven by magnetization dilution in CoFeB free layers for spin-transfer-torque magnetic random access memory
,”
Sci. Rep.
8
,
14409
(
2018
).
14.
J.
Igarashi
,
J.
Llandro
,
H.
Sato
,
F.
Matsukura
, and
H.
Ohno
, “
Magnetic-field-angle dependence of coercivity in CoFeB/MgO magnetic tunnel junctions with perpendicular easy axis
,”
Appl. Phys. Lett.
111
,
132407
(
2017
).
15.
J. G.
Alzate
,
P.
Khalili Amiri
,
G.
Yu
,
P.
Upadhyaya
,
J. A.
Katine
,
J.
Langer
,
B.
Ocker
,
I. N.
Krivorotov
, and
K. L.
Wang
, “
Temperature dependence of the voltage controlled perpendicular anisotropy in nanoscale MgO/CoFeB/Ta magnetic tunnel junctions
,”
Appl. Phys. Lett.
104
,
112410
(
2014
).
16.
C.
Safranski
,
J.
Kaiser
,
P.
Trouilloud
,
P.
Hashemi
,
G.
Hu
, and
J. Z.
Sun
, “
Demonstration of nanosecond operation in stochastic magnetic tunnel junctions
,”
Nano Lett.
21
,
2040
(
2021
).
17.
N. W.
Ashcroft
and
N. D.
Mermin
,
Solid State Physics
(
Saunders College
,
1976
).
18.
M. D.
Kuz’min
, “
Shape of temperature dependence of spontaneous magnetization of ferromagnets: Quantitative analysis
,”
Phys. Rev. Lett.
94
,
107204
(
2005
).
19.
M. D.
Kuz’min
,
M.
Richter
, and
A. N.
Yaresko
, “
Factors determining the shape of the temperature dependence of the spontaneous magnetization of a ferromagnet
,”
Phys. Rev. B
73
,
100401(R)
(
2006
).
20.
D. Yu.
Karpenkov
,
K. P.
Skokov
,
M. B.
Lyakhova
,
I. A.
Radulov
,
T.
Faske
,
Y.
Skourski
, and
O.
Gutfleisch
, “
Intrinsic magnetic properties of hydrided and non-hydrided Nd5Fe17 single crystals
,”
J. Alloys Compd.
741
,
1012
(
2018
).
21.
M. D.
Kuz’min
,
D.
Givord
, and
V.
Skumryev
, “
Why the iron magnetization in Gd2Fe14B and the spontaneous magnetization of Y2Fe14B depend on temperature differently
,”
J. Appl. Phys
107
,
113924
(
2010
).
22.
H.
Jian
,
K. P.
Skokov
,
M. D.
Kuz’min
,
I.
Radulov
, and
O.
Gutfleisch
, “
Magnetic properties of (Fe, Co)2B alloys with easy-axis anisotropy
,”
IEEE Trans. Magn.
50
,
2104504
(
2014
).
23.
I.
Dirba
,
C. A.
Schwobel
,
L. V. B.
Diop
,
M.
Duerrschnabel
,
L.
Molina-Luna
,
K.
Hofmann
,
P.
Komissinskiy
,
H.-J.
Kleebe
, and
O.
Gutfleisch
, “
Synthesis, morphology, thermal stability and magnetic properties of α-Fe16N2x nanoparticles obtained by hydrogen reduction of γ-Fe2O3 and subsequent nitrogenation
,”
Acta Mater.
123
,
214
(
2017
).
24.
D. A.
Wahab
et al., “
Quantum rescaling, domain metastability, and hybrid domain-walls in 2D CrI3 magnets
,”
Adv. Mater.
33
,
2004138
(
2021
).
25.
M. D.
Kuz’min
and
A. M.
Tishin
, “
Temperature dependence of the spontaneous magnetisation of ferromagnetic insulators: Does it obey the 3/2–5/2–β law?
,”
Phys. Lett. A
341
,
240
(
2005
).
26.
R.
Zhang
and
R. F.
Willis
, “
Thickness-dependent curie temperatures of ultrathin magnetic films: Effect of the range of spin-spin interactions
,”
Phys. Rev. Lett.
86
,
2665
(
2001
).
27.
F.
Ibrahim
,
A.
Hallal
,
A.
Kalitsov
,
B.
Dieny
, and
M.
Chshiev
, “Unveiling temperature dependence mechanisms of perpendicular magnetic anisotropy at Fe/MgO interfaces,” arXiv:2011.02220 (2020).
28.
N.
Caçoilo
,
S.
Lequeux
,
B. M. S.
Teixeira
,
B.
Dieny
,
R. C.
Sousa
,
N. A.
Sobolev
,
O.
Fruchart
,
I. L.
Prejbeanu
, and
L. D.
Buda-Prejbeanu
, “Magnetization reversal driven by spin-transfer torque in perpendicular shape anisotropy magnetic tunnel junctions,” arXiv:2005.06024 (2021).
29.
X.
Li
,
G.
Yu
,
H.
Wu
,
P. V.
Ong
,
K.
Wong
,
Q.
Hu
,
F.
Ebrahimi
,
P.
Upadhyaya
,
M.
Akyol
,
N.
Kioussis
,
X. F.
Han
,
P. K.
Amiri
, and
K. L.
Wang
, “
Thermally stable voltage-controlled perpendicular magnetic anisotropy in Mo/CoFeB/MgO structures
,”
Appl. Phys. Lett.
107
,
142403
(
2015
).
30.
Z.
Zhao
,
M.
Jamali
,
N.
D’Souza
,
D.
Zhang
,
S.
Bandyopadhyay
,
J.
Atulasimha
, and
J.-P.
Wang
, “
Giant voltage manipulation of MgO-based magnetic tunnel junctions via localized anisotropic strain: A potential pathway to ultra-energy-efficient memory technology
,”
Appl. Phys. Lett.
109
,
092403
(
2016
).
31.
L. M.
Loong
,
X.
Qiu
,
Z. P.
Neo
,
P.
Deorani
,
Y.
Wu
,
C. S.
Bhatia
,
M.
Saeys
, and
H.
Yang
, “
Strain-enhanced tunneling magnetoresistance in MgO magnetic tunnel junctions
,”
NPG Sci. Rep.
4
,
6505
(
2014
).
32.
W. J.
Carr
, Jr., “
Temperature dependence of ferromagnetic anisotropy
,”
Phys. Rev.
109
,
1971
(
1958
).
33.
K. M.
Lee
,
J. W.
Choi
,
J.
Sok
, and
B. C.
Min
, “
Temperature dependence of the interfacial magnetic anisotropy in W/CoFeB/MgO
,”
AIP Adv.
7
,
065107
(
2017
).
34.
D. J.
Sellmyer
,
M.
Zheng
, and
R.
Skomski
, “
Magnetism of Fe, Co and Ni nanowires in self-assembled arrays
,”
J. Phys.: Condens. Matter
13
,
R433
(
2001
).
35.
R.
Skomski
, “
Nanomagnetics
,”
J. Phys.: Condens. Matter
15
,
R841
(
2003
).
36.
L.
He
and
C. P.
Chen
, “
Effect of temperature-dependent shape anisotropy on coercivity for aligned Stoner-Wohlfarth soft ferromagnets
,”
Phys. Rev. B
75
,
184424
(
2007
).
37.
K.
Hayakawa
,
S.
Kanai
,
T.
Funatsu
,
J.
Igarashi
,
B.
Jinnai
,
W. A.
Borders
,
H.
Ohno
, and
S.
Fukami
, “
Nanosecond random telegraph noise in in-plane magnetic tunnel junctions
,”
Phys. Rev. Lett.
126
,
117202
(
2021
).
38.
H.
Wang
,
W.
Kang
,
Y.
Zhang
, and
W.
Zhao
, “
Modeling and evaluation of sub 10 nm shape perpendicular magnetic anisotropy magnetic tunnel junctions
,”
IEEE Trans. Electron. Dev.
65
,
5537
(
2018
).
39.
Z.
Diao
,
A.
Panchula
,
Y.
Ding
,
M.
Pakala
,
S.
Wang
,
Z.
Li
,
D.
Apalkov
,
H.
Nagai
,
A.
Driskill-Smith
,
L.-C.
Wang
,
E.
Chen
, and
Y.
Huai
, “
Spin transfer switching in dual MgO magnetic tunnel junctions
,”
Appl. Phys. Lett.
90
,
132508
(
2007
).