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$\Delta T$-noise in Multiterminal Hybrid Systems

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Pith's one-line read In multiterminal hybrid superconductor-normal devices, temperature-bias noise at a terminal encodes the heat conductances to all other terminals, not just the electrical conductance.

desk verdict Solid scattering-theory results for delta-T noise in multiterminal N/S systems, with the heat-conductance identification valid only for energy-independent scattering but presented as general. read the letter →

arxiv 2411.12572 v1 pith:SPAO2QUR submitted 2024-11-19 cond-mat.mes-hall

Leonardo Pierattelli , Fabio Taddei , Alessandro Braggio This is my paper · ORCID
classification cond-mat.mes-hall PACS 72.70.+m74.45.+c73.23.-b
keywords Delta-Tnoisehybridnormal-superconductingsystemsLandauer-BüttikerscatteringtheorybackgroundexcessheatconductanceAndreevreflectionthermalshot
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 charge-current noise in multiterminal normal-superconductor hybrid devices when the contacts are held at different temperatures, a regime known as $\Delta T$-noise. Using the Landauer–Büttiker scattering approach, the authors split the noise into a background part and an excess (partition) part. Their central claim is that in the presence of temperature biases, with superconducting terminals grounded, the background noise in the self-correlation of a terminal's current contains terms strictly related to the heat conductances to all other terminals, not only the electrical conductance that enters the equilibrium Johnson-Nyquist formula. This would make a charge-noise measurement a potential probe of heat transport. The paper also shows that thermally induced excess noise acquires Andreev-specific interference terms, and it illustrates both effects in a three-terminal NSN junction and in a spin-resolved quantum Hall device.

What carries the argument

The central object is the scattering matrix $s(E)$ of a Bogoliubov–de Gennes description, from which the paper defines transmission functions $\ell^\pm_{ik}(E) = N_i \delta_{ik} - \mathrm{Tr}[s^{ee\dagger}_{ik}s^{ee}_{ik}] \pm \mathrm{Tr}[s^{he\dagger}_{ik}s^{he}_{ik}]$, where $\ell^+$ governs charge transport and $\ell^-$ enters the heat conductance. The argument separates the zero-frequency noise into background noise (Eq. (12), built from Fermi-function factors $F_i^e = 2 f_i^e(1-f_i^e)$) and excess noise (Eq. (13), built from squared Fermi-difference integrals $D^{k\gamma l\delta} = [f^\gamma_k - f^\delta_l]^2$). The load-bearing step is the energy-independent reduction of the background self-correlator to Eq. (15), which uses the unitarity identity $\sum_k \ell^-_{ik} = 0$ to expose the $\ell^-$ (heat-conductance) terms. For the excess noise, the asymptotic slope $\eta = 2\ln 2 - 1$ of the squared-difference integrals defines the thermal shot-noise regime, and the NSN example expresses all results in terms of the normal and Andreev probabilities $R, T, R_A, T_A$.

What would settle it

Calculate the difference between the background self-noise integrand of Eq. (12) and the heat-conductance combination of Eq. (10) for a scatterer whose transmission functions vary with energy on the scale of $k_B T$ (e.g., a resonant level or an Andreev bound state near the Fermi energy); a nonvanishing difference at finite temperature bias would show that the heat-conductance interpretation fails outside the energy-independent limit. Experimentally, one could heat one terminal $k$ of a multiterminal NSN device and compare the rise in the self-noise of a distant terminal $i$ with the independently measured heat conductance $L^{HH}_{ik}$; agreement would confirm the claim only within the energy-independent regime.

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Extended reading notes

Core claim

The paper's central discovery is the exact structure of the background noise for multiterminal hybrid systems. At thermal equilibrium the background noise of terminal $i$ reduces to the Johnson-Nyquist term governed by $\ell^+$ (electrical conductance); but when terminals are held at different temperatures, the self-correlator $\bar{S}_{ii}$ gains a sum over other terminals $k$ of terms proportional to $(T_k - T_i)$ times the transmission functions $\ell^-_{ik}$ that also enter the heat conductance $L^{HH}_{ik}$. In the energy-independent limit the background noise takes the closed form $\bar{S}_{ii}/(2 k_B G_0) = 2 T_i \ell^+_{ii} + \sum_k (T_k - T_i)[\mathrm{Tr}(s^{ee\dagger}_{ik}s^{ee}_{ik}) + \mathrm{Tr}(s^{he\dagger}_{ik}s^{he}_{ik})]$, so the thermal part measures excitation transport irrespective of quasiparticle charge. The paper further shows that the excess noise under temperature bias differs from the voltage-bias case: self-correlators obey partition-type forms built from $\ell^-$ only, and cross-correlators contain an interference term proportional to $\sqrt{R R_A T T_A}$ that is nonzero only when normal and Andreev processes coexist. These features are demonstrated for an NSN junction and for a four-terminal integer quantum Hall bar with a superconducting finger.

Load-bearing premise

The load-bearing premise is that the scattering matrix is energy independent, $s(E) = s$; only then do the background-noise integral and the heat-conductance integral share the same energy kernel, making the claim 'background noise measures heat conductance' exact rather than approximate.

Editorial extensions

If this is right

  • In a grounded-superconductor multiterminal device, the background charge noise of a terminal becomes a combined probe of electrical and heat conductances, so a measurement at a single terminal can give access to heat-transport information that normally requires separate heat-current measurements.
  • The thermal shot-noise self-correlators take the partition form $x(1-x)$ with $x$ the sum of normal and Andreev transmission (or reflection) probabilities, so their noise is set by the total quasiparticle flux regardless of charge sign.
  • The cross-correlator thermal shot noise contains an interference term $\propto \sqrt{R R_A T T_A}$ that exists only when normal and Andreev processes coexist, making the ratio of electrical to thermal shot noise a marker of superconducting proximity.
  • In the chiral quantum Hall device, the sign of the voltage floor in the excess noise reveals which process, spin mixing or Andreev conversion, dominates current partitioning between the edge states.

Reading between the lines

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

  • If the background-noise/heat-conductance link survives contact with experiment, it provides a route to heat metrology in nanoscale hybrid devices where direct heat-current measurement is impractical; the paper identifies this possibility but does not develop a measurement protocol.
  • The energy-independent restriction suggests a testable extension: the mismatch between the noise and heat-conductance kernels at finite energy dependence could itself be used to spectroscopically probe the energy structure of the scattering matrix.
  • The interference term in the thermal shot noise of cross-correlators points toward phase-sensitive detection of Andreev processes, e.g., an experiment that varies the relative phase between normal and Andreev scattering amplitudes (here encoded in φ−ψ) and tracks the resulting noise modulation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a Landauer-Büttiker scattering theory for zero-frequency charge-current noise in multiterminal normal-superconducting systems under both voltage and temperature biases. After recalling the current and Onsager formulas, the authors split the noise into a background part (Eq. 12) and an excess (partition) part (Eq. 13). Their central claim is that, in the presence of temperature differences with grounded superconducting terminals, the background self-correlator contains a contribution involving the heat-conductance-related functions ℓ^-, in contrast to the equilibrium Johnson-Nyquist form; this is made explicit in Eqs. (14)-(15) for an energy-independent scattering matrix. The excess noise is then analyzed in the electrical and thermal shot-noise limits, and the formalism is applied to a three-terminal NSN junction and to a chiral integer quantum Hall device with a superconducting finger, where the authors identify signatures of spin-mixing versus Andreev processes and an interference term in the thermal shot-noise cross-correlator.

Significance. The analytic framework is clean, and the central algebraic step leading to Eq. (15) follows correctly from Eq. (12) and unitarity with no fitted parameters. If the heat-conductance interpretation of background charge noise is valid, the paper offers a concrete, experimentally relevant route to access heat-transport information from charge-noise measurements. The NSN and chiral-device examples produce specific, falsifiable predictions (partition forms x(1−x), an AR/SM interference term, and sign changes controlled by the relative strengths of Andreev and spin-mixing processes). The paper is also careful to state its key limit in the derivation section. The main reservation, detailed in the major comments, is that the abstract and conclusions present the heat-conductance dependence without the energy-independence qualification that the derivation actually requires.

major comments (3)
  1. [Abstract; Sec. II.C.1, Eqs. (10), (12), (14), (15)] The abstract and conclusions present the dependence of the background self-correlator noise on heat conductance as a general property of multiterminal hybrid systems, but the derivation of Eq. (15) in Sec. II.C.1 explicitly assumes an energy-independent scattering matrix, s(E)=s. For energy-dependent scattering, the background self-correlator in Eq. (12) contains an integral of F_k(E) ℓ^-_{ik}(E), whereas the linear-response heat conductance in Eq. (10) involves E^2(−∂f/∂E) ℓ^-_{ik}(E); these two kernels coincide only when ℓ^-_{ik}(E) is constant on the scale of k_B T. The paper should either add the energy-independence qualification to the abstract and conclusions, or quantify the error for weakly energy-dependent ℓ^-_{ik}(E).
  2. [Sec. III.B.1, Eqs. (35)-(36)] Equations (35) and (36) as printed contain stray symbols 'r' and 'z' and appear to be intended as square roots, namely √{−ℓ^-_{i1}(1+ℓ^-_{i1})} and √{−ℓ^+_{31}ℓ^+_{41}}. If that is the intended reading, the normal limit is problematic: for a single normal channel of transmission τ, ℓ^-_{31}=−τ, so Eq. (35) would give s̃S_{33}/(2G_0 η k_B T_1)=√{τ(1−τ)}, whereas the known thermal shot noise for a normal channel is τ(1−τ) (see Ref. [37]). The authors should correct the typography and verify the physical content of these expressions.
  3. [Sec. III.B, Eq. (30)] The 4×4 matrix t(E) in Eq. (30) has duplicated entries in its third and fourth columns: seh_{31}, seh_{41}, shh_{31}, and shh_{41} appear twice, while the corresponding elements for contact 2 (seh_{32}, seh_{42}, shh_{32}, shh_{42}) are missing. Since this matrix is used to construct the ℓ^± matrix in Eq. (31) and the subsequent noise expressions, the authors should correct the typo and verify any consequences for Eqs. (31)-(36).
minor comments (4)
  1. [Sec. III.A, Fig. 2] The device scheme appears twice in the text with the same caption; one copy should be removed.
  2. [Eq. (17)] The notation '6kBT' and '18kBT' should use explicit k_B (e.g., 6 k_B T) for clarity and consistency with the rest of the text.
  3. [Sec. IV] The phrase 'Andreev's processes' should be 'Andreev processes'.
  4. [Eq. (16)] The sign function sgn(α) is defined in Sec. II.A, but it would be helpful to restate its definition at its first use in Eq. (16), since the notation appears again in several later equations.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the noise–heat-conductance relation is derived from the scattering matrix, not fitted or defined into existence; the energy-independence caveat is a limitation, not a circular step.

full rationale

The derivation is self-contained. The background noise formula, Eq. (12), follows from the standard scattering-theory current correlator, while the heat conductance enters through the Onsager matrix Eq. (10); both are expressed in terms of the same transmission functions ℓ^± defined in Eq. (5), but neither quantity is defined in terms of the other. In the energy-independent limit, the noise integral becomes 2k_B T_k ℓ^-_ik, whereas the heat conductance becomes (π²/3)(k_B²/e²)G0 T ℓ^-_ik, so Eq. (15) is an algebraic consequence, not an assumption disguised as a result. The thermal shot-noise results are explicit evaluations of Eq. (16) for model scattering matrices, with no fitted parameters. Self-citations in the reference list are contextual and not load-bearing for the central derivation. The main caveat, that the heat-conductance interpretation is exact only for energy-independent s(E), is stated in Sec. II C 1 even though the abstract omits it; this is a correctness/generality concern, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claims rely on standard scattering theory and the stated idealizations (grounded superconductors, energy independence for closed forms, no interactions). The example-specific scattering probabilities are model parameters, not fitted to data, and do not enter the main result. There is no fitting or invented physical entity.

free parameters (2)
  • NSN scattering probabilities R, T, R_A, T_A = unspecified (constrained R+T+R_A+T_A=1)
    Model parameters of the NSN example (Eq. 20), used to produce explicit noise expressions (23)-(28). They do not affect the general background-noise result Eq. (15).
  • Chiral device AR/SM parameters a, b, r, t = unspecified (a+b=1, r+t=1)
    Model parameters for the superconducting finger and spin-mixer in the quantum Hall example (Eqs. 37-38), used in Eqs. (39)-(40).
assumptions (6)
  • standard math Landauer-Buttiker scattering formalism for average currents and noise (Eqs. 4, 12, 13) is valid for coherent elastic transport.
    Assumed from refs. [10,61,62]; not re-derived.
  • domain assumption All superconducting regions share a common electrochemical potential mu_s=0 and are grounded.
    State in Sec. II.A and used in Eq. (7); floating superconductors would change the sum rule.
  • domain assumption Scattering matrix is energy-independent for the closed-form results (Eqs. 14-16, 23-28, 35-36).
    Required to express background noise linearly in T and to identify ℓ^- with heat conductance; the abstract does not mention this.
  • domain assumption No electron-electron interactions between chiral edge channels in the quantum Hall example.
    Stated in Sec. III B: 'We also neglect any possible effect of the electron-electron interaction between the two channels [48]'.
  • standard math Unitarity of the scattering matrix and particle-hole symmetry relations among the Nambu blocks.
    Used to derive Eq. (6), Eq. (7), and the PHS relations for s^eh, s^hh (Sec. II.A, III.A).
  • domain assumption Spin degeneracy and single-channel assumption in the NSN example.
    Sec. III A: 'spin degenerate, with only one open transport channel per lead'.

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Pith. "Pith review of $\Delta T$-noise in Multiterminal Hybrid Systems." pith.science (2026). https://pith.science/paper/SPAO2QUR

@misc{pith2026241112572,
  author       = {Pith},
  title        = {Pith review of: $\Delta T$-noise in Multiterminal Hybrid Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPAO2QUR}},
  note         = {Machine review of arXiv:2411.12572}
}
abstract

The study of charge current fluctuations (noise) can give useful insights into the properties of nanoscale systems. In this work, the peculiar properties of noise in multiterminal hybrid normal-superconducting systems are explored in the thermal out-of-equilibrium regime, i.e., when temperature biases are present ($\Delta T$-noise). Using the Landauer-B\"uttiker approach, we identify two contributions: background noise and excess noise, analyzing them when both electrical and thermal biases are applied. When temperature biases are present, and superconducting terminals are grounded, we find that the first contribution depends not only on the electrical conductance, as the Johnson-Nyquist at equilibrium, but also on a quantity strictly related to the heat conductance. This is our first main result. On the other hand, the second contribution shows, as expected, additional terms originating from the partitioning of currents into different transport channels, including the ones associated with Andreev reflection processes. However, noise induced by the temperature differences unveil also interference terms that cannot be present either in voltage bias or in the absence of any Andreev processes. Finally, we apply the results obtained to two different specific physical situations. The first is a generic three-terminal normal-superconductor-normal system, and the second is a device based on spin-resolved co-propagating chiral channels in the integer quantum Hall regime with a superconducting region. In these example setups, we investigate mainly the shot-noise regimes, when high-voltage or high-temperature biases are applied. We find remarkable differences between the two limits, which ultimately show the different nature of electrically and thermally induced charge current fluctuations.

Figures

Figures reproduced from arXiv: 2411.12572 by the authors.

Figure 1
Figure 1. FIG. 1. Plots of the integrals [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Essentially, in these plots, we represent the typi [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Scheme of the NSN device described in the text, with FIG. 2. Scheme of the NSN device described in the text, with [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: FIG. 3. Scheme of the hybrid superconducting-quantum Hall didibd iStiIII BFiliit FIG. 3. Scheme of the hybrid superconducting-quantum Hall [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4. Excess noise for the cross-correlator [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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Forward citations

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