Resonant current-in-plane spin-torque diode effect in magnet$-$normal metal bilayers

I enjoyed reading the manuscript “Resonant current-in-plane spin-torque diode effect in magnet–normal metal bilayers”. The authors discuss the spin-torque diode effect in a magnet-normal metal bilayer, where the current is applied in plane. In contrast to earlier works, the authors include in their description the longitudinal spin transport from magnons and electrons (besides the coherent transverse spin transport that is crucial for the resonance). For this, they use a magneto-electric circuit description.

This subject deserves consideration. The paper is well written, very educational, and mostly clear. In particular, up to Eq.(28), I was able to easily follow the train of thought. However, once the authors discuss the quadratic response, the results seem a little rushed in comparison. I would recommend filling in some details, see below. Some revisions regarding my questions/remarks are in order before I can recommend publication.

1) The authors mention in the introduction that this spin torque diode in N|F bilayers with in-plane currents is the counterpart to the spin torque diode in F|N|F trilayers. What would be a potential advantage of the former?

2) I am confused about some details in the magneto-electric circuit diagram of Fig.2:

a) Why are spin currents (solid lines) and charge currents (dashed lines) connected at a vertex (black dot)? This seems to falsely suggest that spin actually can be transformed into charge, and we can apply Kirchhoff’s junction rule somehow. Furthermore, the spin-Hall elements look like variable resistors. Isn’t it clearer to represent the spin-Hall element by a “two-port device” with four terminals: spin current in/out and charge current in/out? 


b) The arrows of the solid lines are all pointing up, suggesting that spin is flowing from the ground of F to the ground of N. This seems a little odd.

c) It seems that you write the symbol for the charge current $\delta \bar{i}$ next to the solid line of spin current. Why?

3) The discussion about the quadratic response [in particular from Eq. (32) to Eq. (37)] seems to brief. For example, why is Eq. (37) a relevant impedance? The setup seems to suggest that we have to add $Z_N$ in series and not in parallel with the impedances from the magnet. Is this why there is a prime at $Z'_\parallel$? More details on the exact calculation would help.

4) While the resonance peak positions are given by Eq. (26), I am wondering if there is a simple expression for the width of these resonances. Also, is the only effect of including the longitudinal spin transport an overall increase of the response coefficients $r_\Omega$? If so, this should be stressed more. Again, the insight regarding the origin of the impedance $Z’_\parallel$ would help.

5) The discussion around Eq. (27) is a little confusing. It says that the current is expanded to linear order in $\boldsymbol{\mu}_s$ and $\dot{\bf{m}}$ although it is already linear in these quantities. I suppose the authors perform an expansion of the spin current in the “generalized forces” $\boldsymbol{\mu}_s$ and $m_\perp$ to obtain $j^z_{sm\perp}(t)=-\frac{g_{\uparrow\downarrow}}{4\pi}\left(\mu_{s\perp}+ i \dot{m}_\perp\right)$. The same procedure then gives rise to Eq. (29) in second order, where $j^z_{sm\perp}(t)$ is again the first-order result, correct? If so, please state this more clearly.

6) The authors state in the introduction, “that not only resonances at the uniform ferromagnetic resonance frequency, but also at higher magnon frequencies are included in our calculation.” Is this in reference to Eq. (41)-(44) describing longitudinal spin transport via magnons in the magnet? Furthermore, looking at Ref. 27, it seems Eq. (44) uses the limit of large temperatures, $k_B T\gg \hbar \omega_0$, in order to carry out the integration. For consistency, Eq. (41) and Eq. (43) should be carried out under the same approximation.

7) In Eq.(32), is it supposed to be $j^{z,(2)}_{sm\parallel,\Omega}$ or are other second-order contributions considered?

Small cosmetic things I noticed while reading:

i) Different usage of hyphens in “spin-torque diode effect” in title vs “spin-torque-diode effect” in introduction.

ii) Below Eq. (29), it should say eliminating both $m_{\perp,\omega}$ and $m_{\perp,\omega}^*$.

iii) In Eq. (16) and also in the text, $\mu_{m,\omega}$ is used instead of $\mu_{sm\parallel,\omega}$.

iv) In Eq. (17), $m_{\perp}(\omega)$ is used instead of $m_{\perp,\omega}$ .

v) In Eq. (41) and Eq. (43), factors $\hbar$ are missing.

vi) At various points the $z$ index of the spin current is dropped.

vii) The convention F|N vs N|F is switched multiple times (title, abstract, text, caption of table and figures).

Ask for minor revision

The authors present a theory of the resident in plane spin-diode effect, including longitudinal spin transport by conduction electrons and incoherent magnons.
The authors address a new problem of including the longitudinal spin current and show how this affects the second order in electric field response of the normal metal ferromagnet bilayer.

In my opinion, the work is new, relevant and well written and hence meets the acceptance criteria of the journal.
Therefore, I recommend this article for publication after a few minor revisions.

Below you find my comments and questions about the article.

Although the authors discussed numerical estimates for gold iron and platinum YIG bilayers, it will be nice to read a comment on how to measure this unit directional transport experimentally. I believe it will be insightful if the article includes a short sentence that states one can measure this effect experimentally via the electrical resistivity—measuring the current while applying a constant voltage.

Equation (11) contains spin currents of which the meaning can be inferred but are not explicitly introduced. I would prefer the different currents to be explicitly introduced.

Between equation (14) and (15) the authors state: “solving the equation of motion for the conducting electrons, incoherent magnons and coherent magnons.” It's not clear to me which equations the authors refer to.

Typically, the magnetization direction is controlled by an external magnetic field. Could you comment on how the presence of an external magnetic field fits in with your results? This discussion would probably fit around equation (25).

I would prefer a bit more discussion after equation (33) and (34) especially relating to the spin-diode effect. Although the diode effect is captured by these equations, a short discussion of this effect would be appreciated.

As a final question, why can dipole-dipole interactions be ignored or how they would change the results?

Ask for minor revision