Orbital Edelstein effect from the gradient of a scalar potential

In the present work the authors propose a way to induce an Orbital Edelstein Effect (OEE) in a three-dimensional electron gas with a potential $U(z)$ that breaks mirror symmetry $z\to -z$. The potential is proposed to be the source of a finite in-plane orbital magnetization $M_y\propto zp_x$. The wave function is a scalar and has no spin nor internal orbital structure. Upon expansion of the potential in powers of $z$, in a way that an electric field appears, $U(z)=U_0+z\frac{\partial U}{\partial z}+\ldots$, a finite in-plane magnetization results to be proportional to the electric field $\langle M_y\rangle \propto \partial U/\partial z$.

I find the derivation not convincing, tricky and full of loopholes. First of all, it is not clear whether the potential $U(z)$ is a confining potential, that confines the motion in the $xy$ plane or not.

In case it is not a confining potential and it only comprises an electric field $\partial U/\partial z$ (that is the only characteristics of the potential that appears in the entire derivation), then one should consider an electric field pointing in a general direction in the $xz$ plane, with $E_x$ and $E_z\propto \partial U/\partial z$ components. In this case there is only electron drift along the direction of the applied electric field.

In case $U(z)$ is a confining potential, then the origin of the system is arbitrary along the $z$ direction. For example, by introducing a fictitious magnetic field $B_y$ described by a vector potential $A_x=-B_y(z-z0)$, the resulting "orbital magnetic moment operator" would acquire an additional term proportional to $z_0$, and the latter could be chosen to be $z_0=\int dz z n(z)$, so to nullify the expectation value of $\hat{M}_y$.

Also, the introduction of this "orbital magnetic moment operator" is not convincing. What is well defined is the orbital magnetization as $-\partial F/\partial B_y$ (with $F$ the equilibrium free energy, I leave aside a discussion out of equilibrium, but analogous results should follow) in presence of an external magnetic field. The latter is associated to orbital currents and is typically calculated in linear response as a current-current response function, with paramagnetic and diamagnetic terms, and it goes to zero for zero field.

The operator $xp_z-zp_x$ is the angular momentum operator $L_y$ and, upon a rotation in the $xz$ plane such that the momentum is along the drift direction given by the combination of $E_x$ and $\partial U/\partial z$, an ill-defined expressions would appear, as already pointed out by the authors.

For these reasons I find the work not suitable for publication.

Reject

1- The submitted paper addresses a topic of much current interest, the orbital Edelstein effect.
2- A new finding is that the orbital effect scales as the cube of the momentum relaxation time.

1- although a part of the theory is on a high level, there can be technical questions around certain model assumptions.
2- the derived expression does not explicitly depend on the materials' specific electronic structure.

The manuscript of Ado and collaborators addresses the origin of the orbital Edelstein effect. In the manuscript, an induced orbital momentum density is derived for a scalar potential gradient. Two derivations are in fact made, an elementary one and a microscopic derivation. These give both a scaling of the effect as tau^3.

1- In ref. 8 it was shown that in order to have the OEE in a crystal the local point group of the atom should not have inversion symmetry. It would be appropriate to specify in how far the here proposed gradient of the potential is different from the mechanism of ref. 8. When there is no inversion symmetry in z direction, there is a difference of the potential in +z and -z direction. The here considered gradient seems to be a special and less general form of this condition.

2- In section 2.2 there is a subtle change performed from operators in the first equations macroscopic quantities such as the drift velocity. It might be correct, but the expectational value of zv_x is then the product of two quantities, which is dangerous. The authors should specify why their rewriting of <zv_x> is possible.

3- The final equation (12) or (13) is written to depend on tau^3. This might be so when one rewrites the mean free path l, but then a further E also appears. In the end, the equation has then M proportional to E^3, but this is a higher order correction to the OEE. Eq. (25) seems to be different, however.

4- The form of how the electric field is included might also warrant further explanation. It is done through the mentioned substitution of the drift velocity, whereas the applied electric field doesn't appear in the Hamiltonian. There is instead the magnetic field that is derived from the vector potential in the Hamiltonian. An explanation is appropriate here, since E would be the time derivative of the vector potential, but this is zero for the chosen form, i.e., no E field.

The authors should consider points 1 to 4 in the above section and address these.

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