Topological interface states -- a possible path towards a Landau-level laser in the THz regime

1. Draws interesting and promising connections between distinct fields of research
2. Good balance of review and new results
3. Provides intuitive understanding of the electronic states under examination

1. It is essential that more detailed calculations of optical processes are included.
2. A stronger motivation of the actual advantages of this proposal in view of devices is needed.
3. Some points of the presentation should be clarified.

The manuscript by Goerbig capitalizes on previous works (including the author's ref.16) on the electronic states at the interface between a topological and trivial 3D material to propose an interesting application of topological interfaces for optical devices. As such, the manuscript contains an interesting and commendable pioneering attempt to connect the normally very distinct worlds of abstract topological condensed matter theory and real-word optoelectronic applications. In a sense, this proposal generalizes what is often done in current semiconductor technology to engineer electronic states of, e.g., quantum wells (the extreme example is the quantum cascade laser) to a novel class of materials with richer possibilities stemming from new alloys with topological band structures. On this basis, I find that the manuscript has the potential to become a very good SciPost paper.

However, before I can recommend it for publication, the author should carefully consider the following list of remarks:

1/ I am very confused when the author speaks of Voigt vs. Faraday geometry and characterizes the optical process in terms of the direction of propagation. As far as I understand, the wavevector of light involved in the optical transition is very small and thus negligible, while what really matters in the selection rules for optical transitions is the direction of polarization of the electric field. The author should correspondingly correct the manuscript to make this point fully clear for the readers.

2/ After eq.(5) (and at a few other places in the ms.) the author says that an in-plane magnetic field may modify the surface-state-width parameter. I do not understand how this can happen, so I recommend that the author adds some explaination to make the manuscript self-contained.

2bis/ A few lines later, the author says 'Within this analogy... "fake" magnetic field oriented in the interface... in the z-direction'. Also this sentence is obscure. I suspect it is somehow connected to my point 2/: if so, the author should clarify.

3/ If I understand correctly, the proposal of Sec.III requires that the upper state of the population-inverted transition is only slightly below the optically pumped state. As such, I don't understand what would be the advantage of using this device as a source of light as compared to directly using the pump laser.

3bis/ Some justification to the parameter dependences shown in formula (31) would also be appreciated. It could be useful to mention this formula also around eq.(7).

4/ To make the manuscript complete, the author could include more levels in fig.2 so to clearly show that no other optical transition to higher levels is indeed possible. As further modification to the figure, he could explicitly indicate the Fermi level (right above 0, I guess) and use a logical color code for the arrows indicating pump and emitted light.

5/ In fig.3, he should also indicate the Fermi energy which is now just below 0. He should add more levels to show that no higher level is populated.

5bis/ As a most important point, the author should add some explicit discussion and/or numerical rate equation calculation that explicitly demonstrates that population inversion is indeed possible and no nasty reabsorption process can take place between the several levels involved in the optical transitions. The arguments on the RHS column of pag.5 (including the mentioned but not calculated depletion of the m=0,n=-1 level) are in fact too qualitative to be fully convincing for optics-oriented readers.

6/ In general, readers would appreciate knowing more on the potential advantage that the LL laser device would have in comparison to other existing or proposed sources of coherent light in the IR range.

7/ I do not generally like very long paragraphs in a research article. I suggest the author to add more \newline's to facilitate the reader in following the different steps of the logical flow.

1. Reinforce the discussion of the optical processes at play and, if possible, make it quantitative.

2. Clarify the role of light propagation direction vs. polarization in the analysis of the Faraday vs. Voigt geometry

3. Update the figures

4. Take all other remarks in due consideration

Original, well written, well argued, at the crossroads of fundamental and applied physics.

In this manuscript, the author proposes a new mechanism to engineer lasing in the THz regime. For that purpose, the author takes advantage of the Landau levels (LLs) made out of Volkov-Pankratov (VP) states in a magnetic field. Those VP states are massive states that appear at smooth interfaces between topologically distinct materials, in addition to the topologically protected massless surface state that survives for abrupt interfaces. Interestingly, the author notices that a population inversion of electrons, induced by resonant optical pumping, can be controlled by a combination of the magnetic field (controlling the LLs spacing) and the interface width (controlling the dispersion of the VP states). The comparison between the magnetic length and the surface-state-width parameter, that somehow plays the role of a fictitious magnetic length, may give rise to two different scenarios of photons emissions, called Faraday and Voigt configurations, in which the electrons relaxation or excitation mechanism implies different transitions between LLs of VP states and LLs of the topological chiral interface state. Besides, those two configurations interestingly correspond to photons emissions in either parallel or perpendicular direction to the external magnetic field. The author also investigates different schemes implying three and four LLs, and argues a possible implementation in a PbSnSe crystal whose concentration in Pb and Sn is known to control a gap inversion between a topologically trivial and a crystalline topological insulator.

Beyond finding this original proposal very appealing, I also find the paper very well written and argued. Many orders of magnitude are provided, an example of a concrete material is suggested, transition rates are discussed. I am confident that this convincing idea will stimulate further promising studies. I am thus very inclined to recommend this paper for publication.

I nevertheless would like to ask one (maybe naive) question to the author, about the sample geometry. All the physics discussed here takes place in the vicinity of a bulk interface in a 3D material (see e.g. insets in Fig. 1 and 2). When the photons are emitted, they are assumed here to propagate freely through the bulk and then escape out the material. In particular, when pumping in the Voigt geometry, a photon is emitted between LLs of the chiral surface state (Faraday emission). This exact same transition exists at the actual surface of the topological material (where no VP exist), and toward which the emitted photon is sent, so that this photon could be reabsorbed at the surface. Could this mechanism, not mentionned in the manuscript, spoil the required lasing effect, at least in that geometry? For instance, we could imagine that the photon could be re-emitted, but in another direction.

I list below a few (possible) typos.

Below equation 2, after [16], is that the n=0 surface state or the m=0 surface state?
Before Part II - one may therefore expect(ed)
Before Eq. 7. I guess \ell should be \ell_S
Below Eq. 7, after \tau, there is a missing reference ‘’[]’’
Beginning of the last paragraph of section III, ‘’goemetry ‘’ -> ‘’geometry’’
Last paragraph, first column, page 5, a reference should be added regarding after ‘’Such Auger processes have been shown…’’
Page 6, first paragraph, there is a latex problem with parenthesis Pb_{(1-x)}.