Peer review of: Evidence for spin-dependent energy transport in a superconductor. (2020)

Reviewers' Comments:  
Reviewer #1: 
Remarks to the Author: 

This manuscript probes spin-energy mode(s) in a superconductor via non-local transport of spin 
polarised quasiparticles within an un-thermalised region. The results are supported by theory 
involving the calculation of the distribution functions for spin up/down quasiparticles. The paper is 
exciting, robust and interesting to the broad readership of scientists working on superconductivity 
and out-of-equilibrium physics in condensed matter systems. 
 
I am happy to support publication in Nature Communications, but request that the authors 
consider the following points: 
 
1. There is a strong assumption about the quasiparticle distribution between the superconducting 
bar and the detector. This is potentially a weakness in the paper given the different injector-
detector distances and the interfacial qualities. Could the authors comment on this in the paper. 
 
2. Fig. 2b shows a plateau around zero energy in the absence of an injected current (no 
quasiparticles). What might be the origin of this plateau? 
 
3. For Fig.5, please interpret why the charge imbalance first increases and then decreases as the 
field is lowered. 
 
3. Some recent (experimental) literature on spin-transport in superconductors is missing from the 
references. Please check your reference list carefully. 
 
 
 
Reviewer #2: 
Remarks to the Author: 
The present manuscript reports on a very interesting experiment which reveals the existence of 
the spin-energy mode excited in a superconductor. 
Previous experiments on the non-equilibrium modes have indeed detected the other three modes 
(charge, spin, and energy). In this regard, the experiment complete the full picture and in a way 
demonstrate the validity of the theories that have been developed for the description of the non-
equilibrium modes in superconductors in Zeeman fields. 
Therefore I would recommend this manuscript to be published. 
 
I have nevertheless some comments and recommendations about the the way the authors present 
the results. The interpretation of the experimental results strongly relies in theory and 
technicalities. Indeed, the authors made an effort to avoid these technicalities in the main text, 
however in certain parts of the main text important explanations are missing and a the reader is 
referred simply to the Supp. Info (SI, see below list of comments). The latter is rather long, so it 
would be good to know exactly where to look at. I recommend the authors to specify either 
section, or equations, or figures when refereeing to the SI. 
 
The spectroscopy technique used in the present experiment to detect the spin-energy mode is in 
my view very interesting. And as I said above, the interpretation of the measurements strongly 
relies on the theoretical model which is the generalization of the Keldsyh kinetic equation for 
superconductors in the presence of a spin-splitting field. This theory was presented in ref. [31]. As 
far as I know it was in that work when for the first time it was realized that due to the spin-
splitting the four different modes are coupled. Indeed, the fact that there are two contributions to 
the charge accumulation (f_T and f_L3) when spin-up and spin-down DOS are different, was 
shown in Eq. (1) of Ref.[31]. Strangely Ref. [31] is only marginal cited in the present manuscript, 
when in my opinion has a central role in the interpretation of the data. The SI of the present 
manuscript contains exactly the same formalism presented for the first time in the Supp. Info of 
Ref.[31]. It is also strange when the authors in the in the Supp. Info write: 
“This chapter provides a detailed overview of the theory presented in [8] and [9], while restricting 
the discussion to the aspects relevant to the experiment. “ 
Indeed, Ref. 8 in the Supp. Info is a review that contains among many other stuff, a detail 
description of the formalism introduced in ref.[31] On the other hand it is obvious that the 
formalism of Ref. 9 in the SI haven’t be used in the present manuscript. Even though Ref. 9 
introduced the concept of spin-energy mode, it did not deal with spin-splitting case and hence all 
kinetic coefficients were spin-independent. The spin-dependence of the kinetic coefficients (Eqs. 7-
8 of the SI in the present manuscript) is crucial for understanding this and other experiments 
when detecting non-equilibrium modes. In my view it would be correct to emphasize the 
importance of the results of Ref.[31] for the present experiment. 
In addition, the importance of the main finding of the present manuscript (the detection of the 
spin-energy mode), can be boosted by adding a brief historical account of the observation of the 
different modes in superconductors and the fact that the spin-energy mode still remained 
unrevealed. 
 
Further comments/questions: 
 
1) Why is it important that the detector is a superconductor and not a normal metal? 
2) What would be the result in Fig. 4 if the S-detector is spin-split,? 
3) In page 11 it is written “As expected, we do not observe f_L3 at Jdet2 and Jedt3 …” and a 
reference to the SI. is given. Maybe it is better to say with words in the main text why this is 
expected, or refer to the exact section of the SI where this is discussed. 
4) Fig. 3c-d: the authors claim that “the black lines fall at the location of a step-like feature in the 
colour map, as expected”. I would say that this true in the theory panel d. However, in the 
experimental panel c there are certain regions in the (Vdet, Iinj) diagram in which this does not 
hold. Is there a simple explanation why the experimental data is not as expected in these regions? 
5) Minor typos: In page 10 Delta_D and Delta_det are used for the same quantity. The same with 
E_Z and mu_B H 
6) Also in page 10 it is written : “The difference between them, which is the charge imbalance, is 
maximal at Iinj ≈ 8nA, corresponding to maximal spin polarisation of the injection current, i.e. 
when the injection voltage is just below the coherence peak of the second spin species. “ What 
does “second spin species” refer here to? 
 
 
 
Reviewer #3: 
Remarks to the Author: 
The manuscript addresses an interesting question: how to experimentally resolve the spin-
dependent quasiparticle populations from the different branches of the excitation spectrum of a 
superconductor. In previous literature, QP distributions have often been probed via 
superconducting tunnel spectroscopy. The present manuscript examines how a variant of this 
technique fares when excitations with different spin are not degenerate in energy due to a Zeeman 
splitting, and leverages this to observe a mixed electron-hole-spin component of the electron 
distribution. This tests an experimentally less explored part of the understanding of the 
quasiparticle aspect of superconducting spintronics. These are the novel contributions of the 
manuscript. 
 
The experimental approach taken is to perform nonlocal conductance measurements in a lateral 
geometry, using S-probe junctions to be able to use conductance maps to resolve energy 
structure. To recognize the spin-dependent imbalance part, the energy and magnetic field 
dependency is measured, and compared to the expectations from quantum-kinetic simulations, 
also done by the authors here. This is a valid approach to take, and the analysis performed 
appears reasonable. The semi-quantitative agreement between experiment and theoretical results 
is good, given the fairly simplified model. 
 
However, I think the discussion of the two expected contributions to the charge imbalance needs 
to be clarified in the main text. From the discussion of Figs. 4 and 5, one gets the impression that 
only f_L3 is important at large H, but this is not unambiguously stated. To clarify the situation, the 
authors could explicitly show both contributions (f_T and f_L3) to charge imbalance separately in 
the theory curves from their simulations, not only the total CI. The authors could perhaps also 
comment on the relation to the Tinkham-like excitation picture. 
 
The manuscript is written with sufficient technical information, and the supplement clarifies details 
of interest to experts. 
 
I believe the achievement is interesting enough for publication in Nat. Comms., and that a suitably 
revised manuscript could be considered for publication. 
 
 
Technical comments: 
 
- In Fig. 1(a-b), what quantity are the shaded curves more precisely? Excess quasiparticles? The 
x-axis indicates N_up/down, but they probably are not N=rho*f from panels (a) and (c) as in Eq. 
(1). It's also somewhat unclear what the "multiplied five times" in the caption means. I think the 
illustration needs to be revised. 
 
- The assumption is that the detector electrodes are not spin-split by external field, due to the Pt 
layer. How sensitive are the results vs. this assumption, e.g., based on allowing for small spin 
splitting in their theory model? 
 
- In Eq. (3), f(E) should probably be f_noneq(E) as in the supplement? Also check the sign of V in 
(3). 
 
- For accessibility to non-expert readers, the qualitative description of the spectroscopy method 
should explain also the V_det < 0 side, so that the discussion of the symmetry of the figures can 
be understood in comparison to that in Eq. (1). 
 
- On p. 11 "odd component of the data" --- odd in which quantity? "... which comes from f_L3" --- 
at this point in the discussion, this point has not yet been established. 
 
- Fig. 5a) caption has reference to "top axis", but there is no scale or labels on the top. I'd 
recommend the authors to carefully check the main text in entirety. 
 
- In Ref. [41], "such measurements of excess QPs have been made in extended junctions" --- 
which references?