Superparamagnetic dwell times and tuning of switching rates in perpendicular CoFeB/MgO/CoFeB tunnel junctions

G. Reiss; J. Ludwig; K. Rott

REVIEW 3 major objections 9 minor 28 references

Superparamagnetic dwell times and tuning of switching rates in perpendicular CoFeB/MgO/CoFeB tunnel junctions

T0 review · 3 major / 9 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that thermally driven switching in perpendicular CoFeB/MgO/CoFeB junctions is governed by a ~17 nm activation volume, not the full 70 nm electrode, and that entropic pathway counting resolves the fast-dwell-time puzzle.

desk verdict Entropic explanation of fast MTJ switching is plausible but rests on an untested N! pathway-count ansatz; the tuning-curve demonstration is the solid part. read the letter →

arxiv 1908.02139 v3 pith:34UMDPRQ submitted 2019-08-06 cond-mat.mes-hall cond-mat.mtrl-sciphysics.data-an

G. Reiss , J. Ludwig , K. Rott This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sciphysics.data-an

keywords superparamagnetictunneljunctionsmagneticperpendicularanisotropyArrheniusdwelltimesactivationvolumespin-transfertorquevoltage-controlledstochasticneuralnetworks

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies superparamagnetic magnetic tunnel junctions whose free CoFeB electrode switches thermally on millisecond timescales at temperatures near 50 °C. The measured dwell times follow an Arrhenius law but are orders of magnitude shorter than a single-domain model would allow. The authors argue that including entropic effects – a large number w of possible switching pathways – removes this discrepancy, reducing the effective activation volume to a region of about 17 nm radius. They then use MgO barrier thickness as a control to separate spin-transfer torque from voltage-induced anisotropy changes, and show that switching-rate tuning curves follow Gaussians whose peak shifts with the other input. If correct, this makes such junctions usable as tunable stochastic units for true random number generation and population-coding neural networks.

What carries the argument

The central object is the entropic multiplicity $w$ of switching pathways in the free-energy barrier. The paper writes the mean dwell time as $\bar{\tau} = (\tau_0/w) \exp(\Delta E/k_B T)$, so that a large $w$ shortens the effective prefactor without changing the barrier. To estimate $w$, it assumes switching can nucleate in any of $N$ interchangeable sub-volumes, giving $w \approx N!$; with $\ln w \approx 35$ this yields $N \approx 17$ and an activation radius $r_A = r_E/\sqrt{K/K^*} \approx 17\,\text{nm}$. This entropic prefactor is what reconciles the measured Arrhenius slopes with the fast observed switching.

What would settle it

Measure dwell times on electrodes with different diameters (e.g., 100 nm, 140 nm, 200 nm) at fixed thickness and temperature; if the entropic model is right, the inferred activation radius stays near 17 nm regardless of electrode size, whereas if nucleation is tied to grain size or the full electrode, the radius would scale with electrode size.

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Extended reading notes

Core claim

The central claim is that the inconsistency between measured superparamagnetic dwell times and the single-domain thermal-activation barrier disappears once the entropy $S = k_B \ln w$ of switching pathways is added to the free energy. With the free energy $F = E - T S$, the mean dwell time becomes $\bar{\tau}_{P/AP} = (\tau_0/w) \exp(\Delta E_{P/AP}/k_B T)$. Fitting the field and temperature dependence gives an activation energy $\Delta E$ and an apparent anisotropy $K^* = \Delta E/V_E$ that is 13 to 24 times smaller than the measured anisotropy $K$. Interpreting the ratio as a reduced magnetic activation volume yields $r_A = r_E/\sqrt{K/K^*} = (17\pm4)\,\text{nm}$, and the same volume makes the extracted saturation magnetization fall in the reported 500 kA/m to 1 MA/m range. The paper also reports that the same 17 nm radius follows from counting $w \approx N! \approx 10^{15}$ nucleation pathways, with $N \approx 17$. For the bias-voltage response, the spin-torque contribution scales exponentially with MgO thickness while the anisotropy contribution scales as $1/t_{\text{MgO}}$, allowing the two to be separated; the resulting switching-rate tuning curves are Gaussian in both magnetic and electric fields.

Load-bearing premise

The 17 nm activation radius rests on the assumption that switching can nucleate in any of N sub-volumes and that the number of distinct pathways is N!, so a different relationship between pathway count and sub-volume count would change the inferred radius and could erase the agreement with the energy-based estimate.

Editorial extensions

If this is right

Dwell-time statistics of superparamagnetic MTJs can be described by the Arrhenius law once the entropic prefactor is included, with an activation volume of roughly 17 nm radius rather than the full electrode.
The bias-voltage effect separates by MgO thickness: spin-transfer torque dominates thin barriers and voltage-induced anisotropy change becomes relatively stronger for thick barriers.
Switching rate as a function of magnetic field or voltage is a Gaussian whose peak position can be shifted by the other input, satisfying the tuning-curve requirements of population-coding neural networks.
Quasistatic coercivity measurements become time-scale dependent: if the measurement time exceeds the mean dwell time, the apparent coercive field drops to zero even though the intrinsic anisotropy remains.
Using the reduced activation volume instead of the full electrode volume restores physically plausible saturation magnetization values for these CoFeB films.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Going beyond the paper, if the $N!$ pathway count is taken literally, the inferred 17 nm region is comparable to the domain-wall width in similar CoFeB films, suggesting reversal may nucleate and expand from a small reversed domain; time-resolved magnetic imaging of individual switching events could test this directly.
A further implication is that the pathway-counting assumption is the softest link; an independent measurement of the attempt prefactor $\tau_0/w$ from ferromagnetic resonance or from the temperature dependence of the dwell-time distribution would constrain $w$ without relying on combinatorial arguments.
Another testable extension is to vary the electrode diameter at fixed thickness: the entropic model predicts the activation radius should remain near 17 nm, while a model that ties sub-volumes to grain size would predict a different scaling.
Finally, the demonstrated Gaussian, shiftable tuning curves suggest that populations of such junctions could implement basis-function coding; a concrete next step is checking whether the peak-shift behaviour survives when many junctions are averaged in parallel.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The paper tries to explain why dwell times in 140-nm perpendicular CoFeB/MgO/CoFeB tunnel junctions are orders of magnitude shorter than single-domain Arrhenius predicts. The suggestion is entropic: many switching pathways lower the free energy barrier. If right, this is a useful resolution of a known puzzle in superparamagnetic MTJs, and their way of separating spin-torque from voltage-induced anisotropy effects by comparing three MgO thicknesses is a genuinely nice idea. The tuning-curve demonstration—shifting the Gaussian switching-rate peak with the other control parameter—is clean and directly relevant for stochastic-computing hardware.

The central quantitative claim, though, is shakier than the tone suggests. The entropy factor ln(w)≈35 is extracted from the intercept of the same Arrhenius plot that gives the energy barrier, and the conversion to a 17-nm activation radius assumes w≈N! for N sub-volumes. That is an explicit ansatz with no independent support. A telegraph signal between two resistance levels, as in Fig. 1a, is naturally described by one nucleation event followed by fast wall propagation; the number of equivalent paths should then scale as N, not N!. If w≈N, ln(w)≈35 would require ~10^15 sub-volumes, which is incompatible with a 140-nm electrode, and the entropic explanation collapses. The agreement between the entropy-derived radius and the energy-derived one is therefore not yet confirmation—it may be coincidence, because both derive from the same linear fit and the same ΔE.

The paper is also sloppy about error bars. Table I lists no uncertainties, and the voltage-separation fit gives one parameter with ±6/nm on an exponent, so the STT/anisotropy split is indicative, not precise. The claim that the tuning curves 'match exactly the requirements' for neural-like computing is overstated; they match the shape.

What is solid: the data are real, the two-state statistics are clean, the field/voltage dependence follows the expected exponential form, and the barrier-thickness method is worth pursuing. The proposed neurons-in-population-code application is plausible, not proven.

I would send this to a serious referee, but I would tell the referee to focus on the N! assumption and to ask for either a direct test—e.g., diameter scaling of the dwell-time statistics—or a much more cautious presentation of the entropy-derived radius. As is, the paper is a good concept paper with one load-bearing link missing.

Regards.

Referee Report

3 major / 9 minor

Summary. This manuscript reports time-resolved measurements of thermally driven switching in perpendicular CoFeB/MgO/CoFeB magnetic tunnel junctions with 140-nm-diameter pillars, a 1.1-nm free CoFeB electrode and MgO barriers of 1.2, 1.4 and 1.6 nm. The devices show a clean two-level telegraph with exponentially distributed dwell times. From the temperature and magnetic-field dependence of the mean dwell times the authors extract, for each barrier thickness, an activation energy ΔE and a Zeeman-energy product V·M_S. Because ΔE/V_E corresponds to an apparent anisotropy 12.7 to 24.3 times smaller than the quasistatically measured effective anisotropy K ≈ 330 kJ/m³, they infer a magnetic activation volume V_A = ΔE/K with radius r_A = r_E/√(K/K*) = (17±4) nm; recomputing M_S with V_A brings the saturation magnetization into the literature range. The Arrhenius intercept yields ln w ≈ 35 (w ≈ 10^15), and assuming w ≈ N! pathways among N switching sub-volumes gives N ≈ 17 and an equivalent radius of about 17 nm, which the authors call a remarkable agreement between the energetic and thermodynamic routes. Bias-voltage sweeps at the three barrier thicknesses are decomposed into a spin-torque contribution that decays exponentially with barrier thickness and a voltage-induced anisotropy contribution, and the switching-rate tuning curves obtained by sweeping field at fixed bias and bias at fixed field are Gaussian, matching a requirement for neural-like population coding.

Significance. If the central claims hold, the paper is a useful contribution to the sp-MTJ literature: it provides quantitative evidence that the thermally active volume in these nominally single-domain 140-nm electrodes is far smaller than the electrode (about 17 nm radius), that the independently measured quasistatic anisotropy combined with that volume reconciles the barrier heights, and that the resulting saturation magnetization agrees with literature values. The clean telegraph data, the verified exponential dwell-time statistics, the use of an independent anisotropy measurement as a cross-check, and the demonstration of Gaussian tuning curves shifted by bias voltage are genuine strengths; the paper also candidly admits that the anisotropy/spin-torque separation is not reliable. The specifically entropic result is, however, conditional: the values w ≈ 10^15, N ≈ 17 and the claimed corroboration of the activation volume rest on an unverified w ≈ N! pathway-counting model. As it stands, the paper establishes the reduced activation volume through the energetic analysis but does not establish the entropic mechanism announced in the abstract.

major comments (3)

[Entropic estimate of activation volume (p. 3)] This step is load-bearing for the abstract's claim that including entropic effects leads to a magnetic activation volume much smaller than the electrode. The entropy-derived radius r_A ≈ 17 nm is obtained exclusively from the assumption w ≈ N!, i.e., that the magnetization can switch through any ordering of N sub-volumes; no microscopic justification is given, and the paper flags the step only with 'If we assume'. The assumption is in tension with the data in Fig. 1a, which show switching between exactly two resistance states: a process of a single nucleation event followed by rapid wall propagation would naturally have w ∝ N (the choice of nucleation site), and with w ∝ N the value ln w = 35 would force N ≈ 10^15 sub-volumes and a sub-atomic activation radius, destroying the entropic explanation. Because N is fixed by the same Arrhenius intercept that the model is meant to explain, the 'remarkable' agreement of the two radii tests only the assumed functional form. I ask the authors to justify N! microscopically, test it by measuring ln w versus pillar diameter (for w ≈ N! a doubling of diameter changes ln w by roughly a factor of four plus a large additive term, whereas for w ∝ N it changes by only ln 4), or reframe the entropic estimate as a hypothesis rather than as confirmation.
[Table I and Arrhenius/field fits (pp. 2-3)] The primary extracted quantities — ΔE, V·M_S, K*, M_S and r_A — are reported without error bars; the (17±4) nm uncertainty is only the spread across three barrier thicknesses, not a propagated fit error, and no fit ranges, numbers of points or weighting are given. The value ln w ≈ 35 is the intercept of the same Arrhenius regression and depends fully on the assumed attempt time τ0 = 10^-11 s (Ref. [16]): a factor-of-ten change in τ0 shifts ln w by 2.3 and N by about two to three sub-volumes, and a temperature dependence of K or τ0 over the measured range would bias the slope and intercept simultaneously. Since the paper's central message is an agreement of 'almost the same radii', the absence of error propagation makes that agreement impossible to evaluate; please provide uncertainties derived from the fits and a sensitivity statement for τ0.
[Spin-torque/anisotropy separation (Eq. (3), Fig. 3)] The decomposition of the bias-voltage effect rests on only three barrier thicknesses, and the paper itself states that the anisotropy contribution is 'not very reliable'. The quoted parameters confirm the concern: the spin-torque decay length is 13±6/nm (about 46% relative error) and the anisotropy coefficient is β = (30±15) fJ/(V·m) (about 50% relative error). In addition, the extracted activation energies are non-monotonic in barrier thickness (2.5, 1.3 and 2.4 eV for 1.2, 1.4 and 1.6 nm), which is unexplained and weakens confidence in the thickness trends used for the separation. The abstract's claim that the data allow one to separate the impacts of Zeeman energy, spin-transfer torque and voltage-induced anisotropy is therefore stronger than the evidence supports; the voltage-anisotropy value should be presented as tentative.

minor comments (9)

[Eq. (4)] The Gaussian tuning curve as printed, ν = ν0·exp(½((C−C0)/σ)²), is inverted (it has a minimum at C0 and maxima at the wings); even in a letter, the missing minus sign will confuse readers, since Fig. 4 shows peaked curves. It should read ν = ν0·exp(−½((C−C0)/σ)²).
[Abstract] The abstract uses the spelling 'Zeman energy' twice; the standard spelling is 'Zeeman energy'.
[Title consistency] The in-paper title 'Tuning superparamagnetism in perpendicular magnetic tunnel junctions' (p. 1) differs from the submission title 'Superparamagnetic dwell times and tuning of switching rates in perpendicular CoFeB/MgO/CoFeB tunnel junctions'; the two should be made consistent.
[Table I] Table I uses decimal commas (e.g., 9,28; 2,5) inconsistently with the main text, and the column header 'V_E/A M_S, Anm²' does not state the power-of-ten prefactor of the units.
[Fig. 1 caption] The caption of Fig. 1b contains 'the mean well times ¯τP/AP'; 'well' should be 'dwell'.
[Measurement details (p. 2)] The manuscript does not state how many junctions were measured for each barrier thickness, how many switching events were collected per dwell-time value, or the temperature range and number of points used in the Arrhenius fits; these details are needed to judge the fit quality.
[Supplement] The grain-size result from supplement appendix C is the only microstructural evidence supporting granularity and is used to compare with the 17 nm radius; a one-sentence summary of the measured grain size should appear in the main text.
[Bias-voltage discussion (p. 3)] The statement that 'for thick barriers the change of the anisotropy can have larger impact' is not borne out by the data in Fig. 3b, where the spin-torque term (about 0.045 eV/V) still exceeds the anisotropy term (about 0.021 eV/V) at t_MgO = 1.6 nm; please clarify the intended claim.
[Ref. [8]] Reference [8] contains a typo ('EUropean patent submission') and should be formatted as a standard patent reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: activation-volume estimates use independent slope/intercept parameters and external anchors.

full rationale

The paper's two activation-volume estimates are not equivalent by construction. The energy-based radius rA = rE/sqrt(K/K*) uses the Arrhenius slope (Delta E), the independently measured anisotropy K from quasistatic film data, and the electrode radius; the entropy-based radius uses the Arrhenius intercept ln w approx 35 plus an explicit N! pathway-count ansatz. These are different fit parameters (slope versus intercept) of the same data, so agreement between the resulting radii is a nontrivial consistency check rather than a mathematical identity. The MS values in Table I are anchored to literature values, and the spin-torque and anisotropy-change separations are compared with external results (Endo et al.). The N! ansatz is an unverified model assumption, but the paper explicitly labels it as an assumption ('If we assume...'), and an unverified model assumption is a correctness risk, not circularity. No load-bearing self-citation: the skyrmion entropy reference [15] is from an independent group, and the authors' own patent [8] plays no role in the derivation. Therefore no step reduces to its own input by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The key ingredients are fitted energy barriers, an inferred entropy factor, and an ad hoc combinatorial model for the number of switching pathways. The attempt time and anisotropy field are taken from prior literature and quasistatic measurements, respectively.

free parameters (5)

Activation energy ΔE = 2.5 eV (t=1.2 nm), 1.3 eV (t=1.4 nm), 2.4 eV (t=1.6 nm)
Extracted from the slope of ln(τ/τ0) versus 1/kBT for each device.
Product V_A·M_S (or V_E/A·M_S) = 9.28, 8.33, 6.29 (table units, t=1.2/1.4/1.6 nm)
Obtained from the derivative of ln(τ) with respect to μ0H at fixed temperature.
Entropy factor ln(w) = ≈35
Taken from the intercept of the Arrhenius plot; used to infer N≈17 via w≈N!.
Voltage-induced anisotropy coefficient β = (30±15) fJ/V·m
From the anisotropy contribution Δ+ to the voltage dependence; large uncertainty and said to be 'not very reliable'.
Spin-torque prefactor parameters = Δ- = -(20±2) pJ/V · exp(-(13±6)/nm · tMgO)
Fit of Δ- to an exponential decay across three MgO thicknesses.

assumptions (6)

domain assumption Arrhenius-Neel-Brown model with exponential dependence of dwell time on barrier height
Central to eq. (1); assumed to hold for the sp-MTJ switching.
domain assumption Attempt time τ0 ≈ 10^-11 s constant from literature
Taken from Ref [16]; used to compute ln(w) from the intercept; no direct measurement on these devices.
domain assumption Entropy of switching paths S = k_B ln(w), with free energy F = E - T S
Imported from skyrmion decay paper [15]; used to modify the Arrhenius prefactor.
ad hoc to paper Number of pathways w ≈ N! and activation volume radius r_A = r_E/√N
Combinatorial ansatz introduced to translate the entropy factor into a spatial size; no independent support.
domain assumption Collinear magnetization and field, so barrier changes linearly with H'
Used in eq. (2) and in extracting V_A·M_S.
domain assumption Quasistatically measured K applies to the activation volume
Used to convert ΔE to V_A; assumes the same anisotropy constant governs the small fluctuating volume.

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Cite this review

Pith. "Pith review of Superparamagnetic dwell times and tuning of switching rates in perpendicular CoFeB/MgO/CoFeB tunnel junctions." pith.science (2026). https://pith.science/paper/34UMDPRQ

@misc{pith2026190802139,
  author       = {Pith},
  title        = {Pith review of: Superparamagnetic dwell times and tuning of switching rates in perpendicular CoFeB/MgO/CoFeB tunnel junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34UMDPRQ}},
  note         = {Machine review of arXiv:1908.02139}
}

read the original abstract

Thin electrodes of magnetic tunnel junctions can show superparamagnetism at surprisingly low temperature. We analysed their thermally induced switching for varying temperature, magnetic and electric field. Although the dwell times follow an Arrhenius law, they are orders of magnitude too small compared to a model of single domain activation. Including entropic effects removes this inconsistency and leads to a magnetic activation volume much smaller than that of the electrode. Comparing data for varying barrier thickness then allows to separate the impact of Zeman energy, spin-transfer-torque and voltage induced anisotropy change on the dwell times. Based on these results, we demonstrate a tuning of the switching rates by combining magnetic and electric fields, which opens a path for their application in noisy neural networks.

Figures

Figures reproduced from arXiv: 1908.02139 by the authors.

**Figure 2.** FIG. 2: Mean dwell times [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3: a) [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: Tuning curves for a 1 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [16]

Beaujour, D

J.-M. Beaujour, D. Ravelosona, I. Tudosa, E.E. Fullerton, and A.D. Kent. Ferromagnetic resonance linewidth in ultrathin ﬁlms with perpendicular magnetic anisotropy. Physical Review B, 80(180415(4)), 2009

work page 2009

[2] [1]

Ikeda, K

S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno. A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junc- tion. Nature materials, 9(9):721–724, 2010

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