Peer review of: Spin-phonon relaxation from a universal ab initio density-matrix approach (2020)

Reviewers' comments: 
Reviewer #1 (Remarks to the Author): 
 
In this manuscript, Xu and co-workers present a numerical approach to simulate spin dynamics in 
materials in presence of spin orbit coupling and electron-phonon interaction. They apply their method 
to Silicon, Iron and transition metal dichalcogenides materials (TMD), and compare favorably their 
temperature dependent spin lifetime with available experimental data. 
 
Overall, this study is of high quality and indeed brings a very interesting perspective on phonon-
mediated spin transport and spin relaxation phenomena. However, there are some essential points 
that must be clarified and some extra calculations which would be necessary to reach a fully 
consistent and useful contribution to the community. 
 
The approach developed by the authors does not allow one to include static disorder which will break 
translational invariance and play a key role at low temperatures. Additionally, the Mathiessen rule is 
expected to break down for low dimensional materials such as few layers TMD, and static defects such 
as vacancies are likely to also generate inelastic disorder very specific to the breaking of crystal 
symmetries. To which extent this could affect some of the conclusion of the paper is difficult to say, 
but cannot be disregarded. Actually for TMD, the authors should add an important calculation which is 
to show the result of the in-place spin lifetime using the same electron-phonon driven scattering times. 
 
I also have a couple of true concerns about the methodology as well as the results presented. The 
authors state that their method allows the consideration of Elliott-Yafet (EY) and D’yakonov-Perel’ (DP) 
spin relaxation on equal footing, yet all of their results appear to show only EY spin relaxation. I do 
not doubt that their approach might also capture DP spin relaxation behavior, but the main message 
of this paper seems to be the universality and flexibility of their method. Without some results backing 
up their claims of generality, this paper loses some of its impact, in my opinion. For this reason, I 
would recommend they also show results for a system where DP is the dominant spin relaxation 
mechanism. The conventional DP mechanism actually presents a few universalities such as the 1/2 
ratio between out-of-plane and in-plane spin lifetime, or the inverse scaling of the spin lifetime with 
momentum scattering. Since the scaling seems here different, the ratio presents some particular 
interest, and the authors should show what they obtain for TMDs. 
 
Next, I am particularly concerned about the assumption that the spin density decays exponentially, as 
given in Eq. (2). This assumption is fine for EY and DP, but as soon as one leaves the motional 
narrowing regime the DP mechanism breaks down and the spin density is characterized by a 
relaxation plus a coherent precession. This occurs when spin-orbit coupling is strong, i.e., exactly the 
scenario encountered in TMD. Can the authors verify that the time-dependent decay of the spin 
density is consistent with the spin relaxation time extracted with the assumption of purely exponential 
decay? I would feel more comfortable about their method and results if they could do this for at least 
one data point. Looking at the full dynamics can also bring much more information, and may even 
shed light on what is actually being measured, given the coincidence between the simulations and the 
experimental results. I think that in-plane vs. out-of-plane spin dynamics in TMD may also show quite 
different behaviors. 
 
Finally, it would be truly interesting to see what result is obtained by the proposed methodology for 
the clean case of graphene. This should be straightforward for the authors given the simplicity of the 
material, and would allow a direct comparison with analytical results or results which have little 
temperature-dependence below 200K (see A.W. Cummings and S. Roche, Phys. Rev. Lett. 116 (8), 
086602 (2016) and D. V. Tuan et al., Scientific Reports 6, 21046 (2016)). It is indeed an open 
question to know if substrate effects, pure dephasing or (maybe) the electron-phonon coupling 
(discussed in Fratini et al, Phys. Rev. B 88, 115426 (2013)) are the ultimate sources of spin 
decoherence in ultraclean graphene. 
 
Adding some data in that respect would make the manuscript of genuine relevance for the field of spin 
transport. 
 
 
 
 
Reviewer #2 (Remarks to the Author): 
 
This paper presents a comprehensive theoretical approach to calculate spin relaxation times fully from 
first principles. The approach is formally based on density-matrix theory within the Lindblad formalism, 
where the spins relax due to a combination of electron-phonon scattering and spin-orbit coupling. In 
this way, the important Elliot-Yafet and Dyakonov-Perel mechanisms are both covered, without any 
bias towards one or the other. The input to the calculations consists of the electronic band structure 
(calculated with DFT, including spin-orbit coupling) and the electron-phonon scattering elements. All of 
these are obtained using standard electronic structure methods. The final step is then to combine 
these ingredients in a Fermi's-Golden-Rule-like formula for the spin-phonon relaxation time. The 
formalism is carefully derived in the supplemental information. 
 
The approach is then applied to three different materials: silicon, iron, and the transition metal 
dichalcogenides MoS2 and MoSe2. Both electron and hole spin relaxation times are calculated as a 
function of temperature, and compared with available experimental data. The agreement is very good, 
which suggests that the approach presented here is powerful and versatile. Moreover, the results are 
carefully analyzed in terms of underlying mechanisms such as valley relaxation and magnetic-field 
dependence. Several new insights are obtained, which will be of interest to experts in the field. 
 
The paper is scientifically sound, well written and very clear. The research is in an area of topical 
interest (the material science of spintronics). The approach to calculating spin lifetimes appears 
superior to what has been previously done in the literature. This work is likely to have broad impact in 
the field. I recommend publication. 
 
I have only a few relatively minor comments. 
 
1. I would suggest to slightly alter the title of the paper. The phrase "in disparate materials" is both 
too vague and too narrow. "Disparate" is usually understood to mean two utterly incompatible things 
or concepts, to the extent that they cannot even be compared with each other. This, I believe, is not 
what the authors mean here (they just mean very different materials). I think that it would be better 
to drop this phrase completely, and just call the paper "Spin-phonon relaxation from a universal ab 
initio density-matrix approach". 
 
2. On page 2 of the main paper (left column, bottom line), and also in equation (1) in the 
supplemental material (middle formula), the quantity G is defined to contain a square root of a delta 
function. This is formally ill-defined: square roots of delta functions have no formal meaning. I 
understand that in the final formula (when the G's are properly combined) all works out just fine, but 
nevertheless this is problematic. I would suggest to redefine the quantities to avoid square roots of 
delta functions. I checked the paper by Rosati, Dolcini and Rossi, and I could not find a square root of 
a delta function there (they used a Gaussian regularization of the delta function). 

3. In the section "Outlook", first paragraph, line 14, it should be MoS2 and MoSe2.