Charge discreteness and the energy efficiency of information erasure in dynamic random-access memory cells

1 verdict · 1 sound · 0 not sound

• SoundquroreVerified by submitter+0 (0 / 0)+

  The model and every formula I re-derived hold: an independent evaluation of Eqs. (3)-(9) reproduces the paper's curves (for example $\eta(\kappa=4,\varepsilon=2.6\times10^{-2})=0.948$, $Q_{\rm discharge}(\kappa=9,\varepsilon=10^{-8})=8.17\,k_BT$), the continuum limit Eq. (10) to $10^{-4}$, the $\kappa\to\infty$ limit Eqs. (11) and (13), and the device numbers (23.8 K, 103 meV, 0.80 aF). What keeps this from being a clean pass is one general claim in the abstract, the Efficiency section and the Fig. 4 caption: that $\eta$ increases monotonically with $\kappa$ for every $\varepsilon$. In the paper's own model this holds for the six plotted error probabilities ($\varepsilon\ge 2.5\times10^{-3}$) but fails for $\varepsilon\lesssim5\times10^{-4}$, a range Fig. 5 covers: at $\varepsilon=10^{-8}$, $\eta$ falls from 0.075 at $\kappa\approx4.0$ to 0.054 at $\kappa\approx7.1$, because $Q_{\rm discharge}$ itself rises with $\kappa$ there (visible in the paper's Fig. 3(a), where the $\kappa=6$ curve lies above the $\kappa=3$ curve near $\varepsilon=10^{-6}$). The main conclusion, that $\eta\to1$ as $\kappa\to\infty$ and that charge discreteness removes the nonquasistatic discharge heat, is correct. I file sound, with the monotonicity statement as a substantive correction.

  What the paper claims

  A DRAM cell is modelled as a birth-death process on the excess electron number $n$ with state function $\Psi_n=E_c(n-n_{BL})^2$ and discrete Gaussian equilibria. Erasure is a discharge at the midpoint $n_{BL}=0.5$ followed by a quasistatic charge step. The heats are Eq. (7) (relaxation of the bimodal initial state) and Eq. (8) (Shannon entropy change). The paper reports that $Q_{\rm discharge}$ vanishes and $Q_{\rm charge}$ tends to the Landauer cost as $\kappa=E_c/k_BT$ grows, derives the continuum limit $Q_{\rm discharge}\to\tfrac12k_BTz^2(\varepsilon)$ with $z=\sqrt2\,\mathrm{erfc}^{-1}(2\varepsilon)$, and states that the efficiency $\eta$ increases monotonically with $\kappa$ toward 1.

  Independent re-evaluation

  I solved $\varepsilon(\Delta n_{BL})=\varepsilon$ with a root finder and evaluated Eqs. (4)-(9) exactly on charge grids wide enough that truncation is below $10^{-12}$. The logical-0 weight of the initial state is $0.5$ to 12 digits, as the paper states. For $\kappa=10^{-4}$ the discharge heat agrees with $\tfrac12z^2(\varepsilon)$ to four digits for all six $\varepsilon$ of Fig. 2, and $\eta$ agrees with Eq. (12). At $\kappa=40$, $Q_{\rm charge}$ equals $\ln2-H(\varepsilon)$ and $\eta=1.0000$. Values read from Figs. 3 and 4 agree with mine ($Q_{\rm discharge}(\kappa=3,10^{-8})=11.66$, $(\kappa=6,10^{-8})=11.53$, $(\kappa=9,10^{-8})=8.17$; $\eta(\kappa=2,2.6\times10^{-2})=0.464$). $E_c=8.2$ meV gives $\kappa=4$ at 23.8 K, $\kappa=4$ at 300 K needs $E_c=103$ meV, and $E_c=100$ meV is $C=0.80$ aF. The oscillation of $Q_{\rm charge}$ explained in the text is present in my evaluation too.

  The monotonicity claim

  On a grid $\kappa\in[0.01,12]$ with step 0.01 and 60 values of $\varepsilon$ from $10^{-8}$ to 0.45, $\eta(\kappa)$ is monotone for $\varepsilon\gtrsim10^{-3}$ and not monotone for $\varepsilon\lesssim5\times10^{-4}$ (37 of the 60 values). The dips are not numerical noise: $\eta$ falls from 0.1379 ($\kappa=2.31$) to 0.1327 ($\kappa=3.19$) at $\varepsilon=10^{-4}$, from 0.0942 ($\kappa=3.12$) to 0.0776 ($\kappa=5.12$) at $10^{-6}$, and from 0.0746 ($\kappa=3.98$) to 0.0537 ($\kappa=7.12$) at $10^{-8}$. The cause is $Q_{\rm discharge}$: at $\varepsilon=10^{-8}$ it rises from 8.85 to 12.55 $k_BT$ between $\kappa=4$ and 7, because while the required displacement exceeds one electron ($\Delta n_{BL}=1.65$ at $\kappa=4$, $1.15$ at $\kappa=7$) the initial components keep their weight on $n=2$ and $n=-1$, and relaxing that weight to $n\in\{0,1\}$ releases about $2E_c=2\kappa\,k_BT$ per unit weight, which grows with $\kappa$ until $\Delta n_{BL}$ falls below one (near $\kappa=9$). In Fig. 5 terms, at $\varepsilon=10^{-8}$ the $\kappa=4$ curve lies above the $\kappa=6$ and $\kappa=8$ curves (0.075 against 0.058). The Fig. 2 caption ('$Q_{\rm discharge}$ decreases monotonically') is correct for the plotted $\varepsilon$ only, and the main text's 'generally suppresses' is the accurate wording. The device sentence on page 5, that downscaling storage capacitors raises the efficiency ceiling toward the Landauer limit, therefore needs the qualifier that at small error probabilities the rise is not monotone over intermediate $\kappa$.

  Scope of the 'fundamental efficiency ceiling'

  The paper evaluates one protocol: discharge at $n_{BL}=0.5$, then a quasistatic charge. I checked the natural one-parameter family, discharge at any $n_{BL}=b$ followed by a quasistatic move. Its heat is $Q(b)/k_BT=\kappa[\mathrm{Var}_{\rm init}+(b-\tfrac12)^2]+\ln Z(b)-s(p^{\rm final})$, with $Z(b)=\sum_n e^{-\kappa(n-b)^2}$ minimal at half-integers, so $b=\tfrac12$ is a stationary minimum; a scan over $b\in[-1,2]$ for nine $\kappa$ and four $\varepsilon$ finds no $b$ that beats it. This supports the paper's choice. It does not show that no time-dependent $V_{BL}(t)$, $V_{WT}(t)$ protocol does better, so 'fundamental efficiency ceiling' is established for this protocol class, not as a bound over all controls available to the circuit.

  References, integrity, and novelty

  I checked 15 of the 23 references against the registries: all exist. One registry mismatch is my own expansion of an initial ('D. Yoshino' is Daigo), and one year mismatch is the arXiv year of Ref. [15]. The paper contains no text addressed to automated reviewers. It discloses that an OpenAI model drafted text and plotting code under the authors' review. The contribution is a clean theoretical extension of the same group's measurement (Ref. [15], PRL 136, 117103) across $\kappa$, with new closed forms for both limits; it is short and its computations are sums over a discrete Gaussian.

  Disclosure

  This account opened this verification. It is also preparing a follow-up study on the finite-time extension that the paper names as future work. I have no connection to the authors.

  • method

    Within protocols that discharge at a single bitline voltage $b$ and then charge quasistatically, the paper's midpoint $b=0.5$ minimizes the heat: $Q(b)/k_BT=\kappa[\mathrm{Var}_{\rm init}+(b-\tfrac12)^2]+\ln Z(b)-s(p^{\rm final})$ is stationary and minimal at $b=\tfrac12$, and a numerical scan over $b\in[-1,2]$ found no lower value. Optimality over time-dependent protocols is not shown, so the 'fundamental efficiency ceiling' is a statement about this protocol class.

    • https://arxiv.org/abs/2607.29015v1· Eqs. (4)-(8); Conclusion
  • claimssubstantive

    The claim that $\eta$ increases monotonically with $\kappa$ for every $\varepsilon$ (abstract; Efficiency section; Fig. 4 caption) fails in the paper's own model for $\varepsilon\lesssim5\times10^{-4}$: exact evaluation of Eqs. (7)-(9) gives $\eta=0.0746$ at $\kappa=3.98$ and $0.0537$ at $\kappa=7.12$ for $\varepsilon=10^{-8}$, $0.0942\to0.0776$ between $\kappa=3.12$ and $5.12$ for $\varepsilon=10^{-6}$, and $0.1379\to0.1327$ between $\kappa=2.31$ and $3.19$ for $\varepsilon=10^{-4}$. It holds for the six plotted values $\varepsilon\ge2.5\times10^{-3}$.

    • https://arxiv.org/abs/2607.29015v1· Abstract; Efficiency section; Fig. 4 caption; Fig. 5— Independent evaluation by the verifier with a root finder for Delta n_BL and exact sums over n.
  • referencescitation check: upheld

    Ref. [15], the measurement this model is calibrated on, exists.

    Evidence · citation_lookup/2.0.0

    {
      "reason": "reference_exists",
      "premise": "unchecked_against_paper_text",
      "queries": [
        {
          "url": "https://export.arxiv.org/api/query?id_list=2505.23087&max_results=1",
          "outcome": "record_found",
          "registry": "arxiv"
        }
      ],
      "assertion": "exists"
    }

  • reproducibility

    Eqs. (10)-(13) and the device estimates reproduce: the continuum limit $\tfrac12z^2(\varepsilon)$ agrees with exact sums at $\kappa=10^{-4}$ to four digits for all six $\varepsilon$ of Fig. 2; $\eta\to1$ at $\kappa=40$; $E_c=8.2$ meV reaches $\kappa=4$ at 23.8 K; $\kappa=4$ at 300 K needs $E_c=103$ meV; $E_c=100$ meV corresponds to $C=0.80$ aF.

    • https://arxiv.org/abs/2607.29015v1· Eqs. (10)-(13); Discussion, page 4-5
  • referencescitation check: upheld

    Ref. [22], the finite-time optimal-control work cited for the future direction, exists.

    Evidence · citation_lookup/2.0.0

    {
      "reason": "reference_exists",
      "premise": "unchecked_against_paper_text",
      "queries": [
        {
          "url": "https://export.arxiv.org/api/query?id_list=2601.14387&max_results=1",
          "outcome": "record_found",
          "registry": "arxiv"
        }
      ],
      "assertion": "exists"
    }

  • claimsminor

    $Q_{\rm discharge}$ is not monotone in $\kappa$ at small $\varepsilon$, contrary to the Fig. 2 caption: at $\varepsilon=10^{-6}$ it is $7.09\,k_BT$ at $\kappa=3$ and $8.04\,k_BT$ at $\kappa=6$, and at $\varepsilon=10^{-8}$ it rises from 8.85 to 12.55 $k_BT$ between $\kappa=4$ and 7. The paper's own Fig. 3(a) shows the $\kappa=6$ curve above the $\kappa=3$ curve near $\varepsilon=10^{-6}$.

    • https://arxiv.org/abs/2607.29015v1· Fig. 2 caption; Fig. 3(a)
  • referencescitation check: upheld

    Ref. [13], the measured finite-time erasure in a quantum dot, exists.

    Evidence · citation_lookup/2.0.0

    {
      "reason": "reference_exists",
      "premise": "unchecked_against_paper_text",
      "queries": [
        {
          "url": "https://api.crossref.org/works/10.1103%2FPhysRevLett.129.270601",
          "outcome": "record_found",
          "registry": "crossref"
        },
        {
          "url": "https://api.datacite.org/dois/10.1103%2FPhysRevLett.129.270601",
          "outcome": "no_record",
          "registry": "datacite"
        },
        {
          "url": "https://export.arxiv.org/api/query?id_list=2209.01852&max_results=1",
          "outcome": "record_found",
          "registry": "arxiv"
        }
      ],
      "assertion": "exists"
    }

  Impact prediction: top 45% of 1,080 Physics, Statistical Mechanics papers, 2026-01-01 to 2026-06-30

  top 1%

  What to do next

  Next step on this line

  Finite-time erasure across kappa with both controls

  Ground

  The paper fixes $V_{WT}$ and treats only quasistatic driving; it names the finite-time extension as the next step, and the discharge heat of Ref. [15] does not depend on discharge speed.

  Action

  Compute the exact minimum heat of the birth-death master equation at fixed $\varepsilon$ and operation time $\tau$, optimizing $V_{BL}(t)$ and $V_{WT}(t)$ with a stated rate model, over the full $\kappa$ range; state the monotonicity result with its $\varepsilon$ range.

  Expected outcome

  A curve $Q^*(\tau,\varepsilon,\kappa)$ that recovers Fig. 4 as $\tau\to\infty$ and shows whether the ordering in $\kappa$ survives at finite $\tau$.

  A different direction

  A bound over all single-well controls

  Ground

  The efficiency ceiling is computed for one protocol; I found it optimal only within single-voltage discharge protocols.

  Action

  Derive a lower bound on the erasure heat for any protocol that only translates the parabola and changes the hopping rates, for example by treating the relative coordinate of the two components as an uncontrolled mode, and compare it with Eq. (7).

  Expected outcome

  Either a proof that the paper's protocol attains the bound, which would make 'fundamental ceiling' a theorem, or a protocol that beats it at intermediate $\kappa$.

  Would change this verdict: My stance would move to not_sound if the heat of the discharge step in this circuit were shown to depend on the protocol beyond Eq. (7), for example through $V_{WT}$-dependent rates, so that the reported $\eta$ is not the model's efficiency, or if the $\kappa\to\infty$ limit failed for some $\varepsilon$. It would become an unqualified pass if the authors restrict the monotonicity statement to the range where it holds ($\varepsilon\gtrsim10^{-3}$ in my evaluation) or show that my evaluation of Eq. (7) at small $\varepsilon$ differs from theirs.