[**Comment on “Domain Structure in a Superconducting Ferromagnet”**]{}\
According to Faurè and Buzdin [@FB] in a superconducting ferromagnet a domain structure with a period small compared with the London penetration depth $\lambda$ can arise. They claim that this contradicts the conclusion of Ref. that ferromagnetic domain structure in the Meissner state of a superconducting ferromagnet is absent at equilibrium. Actually, there is no contradiction: The results of Ref. have only been misunderstood.

First of all it is necessary to properly define what is a ferromagnetic domain structure. A distinctive feature of a ferromagnetic state is a nonzero average spontaneous magnetization $\vec M$ in a [*macroscopic*]{} volume. This takes place even in a ferromagnet with domains, since in ferromagnets the domain size $l$ is macroscopic. It depends on the size and shape of the sample and on the orientation of $\vec
M$ with respect to the sample surface. For example, in a ferromagnetic slab of thickness $L$, but infinite in other directions, there are no domains if $\vec M$ is parallel to the slab surface. But if $\vec M$ is normal to the surface the stripe domains of the macroscopic size $l \propto
\sqrt{L}$ appear at equilibrium [@LL].

On the other hand, from the very beginning of studying the coexistence of the ferromagnetism and superconductivity it was known that competition between ferromagnetism and superconductivity may lead to structures with periodic variation of the $\vec M$ direction in space. The period of these structures is determined by the intrinsic parameters of the material, is normally smaller than $\lambda$, and does not depend on the size and shape of the sample. Appearance of this structure means that ferromagnetism has lost competition with superconductivity and the “superconducting ferromagnet” is not a ferromagnet in a strict sense: this is an antiferromagnetic structure with a large but finite period. Various types of such structures were known: cryptoferromagnet alignment of Anderson and Suhl [@AS], spiral structure of Blount and Varma [@BV], or domain structure of Krey [@Krey]. One can find these and other references in the review [@BB] cited in Ref. . The second paragraph in Ref. clearly emphasized the difference between the ferromagnetic macroscopic domains and these structures (let us call them intrinsic domain structures) and specifically warned that the paper addressed the case when the material is stable with respect to formation of intrinsic domains.

Faurè and Buzdin [@FB] considered the intrinsic domain structure, which was analyzed by Krey [@Krey] more than 30 years ago. They rederived the structure parameters obtained by him. The domain size given by Faurè and Buzdin in Eq. (7), $l
\sim \tilde w^{1/3} \lambda^{2/3}$, coincides with that given by Krey in his Eq. (30) (apart from notations). Here $\tilde w \sim (K/2\pi M^2) \delta$, $K$ is the energy of the easy-axis anisotropy, and $\delta$ is the domain-wall thickness. The condition for formation of this structure obtained by Krey also coincides with that of Faurè and Buzdin: $\lambda > \tilde w$. Thus in the limit $L\to \infty$ they obtained the intrinsic domain structure in the state which is globally antiferromagntic. The structure can appear in any sample whatever its demagnetization factors are, in particular, in the slab of thickness $L$ independently of whether $\vec M$ is normal or parallel to the slab plane. Certainly the results of Ref. cannot be relevant for this state as clearly warned there. Faurè and Buzdin claimed that their results for thin slabs (small $L$) disagree with Ref. , though Ref. did not consider finite-$L$ corrections at all addressing (like Refs. ) only the macroscopic limit, when $L$ exceeds any intrinsic scales (including $\lambda$) or any combination of them. Only then the difference between intrinsic domains and [*macroscopic*]{} domains has a clear meaning.

Though time and again Faurè and Buzdin stressed contradiction to Ref. , in reality they confirmed its conclusion: If the superconducting ferromagnet is stable with respect to formation of intrinsic domains, macroscopic domains also do not appear. They claim that the area of stability, for which the analysis of Ref. is relevant, corresponds to “the nonrealistic limit of vanishing $\lambda$”. In reality Krey’s stability condition $ \lambda <
\tilde w \sim (K/2\pi M^2)\delta$ (not $ \lambda \ll \tilde w $ !) is not so severe and allows the values of $\lambda$ essentially larger than the domain-wall width $\delta$. Indeed, the ratio $K/2\pi M^2$, which is called the quality factor of the magnetic material, can be rather high. This is required for various applications of magnetic materials [@MS]. The quality factor is especially high for weak ferromagnetism, which is the most probable case for the coexistence of ferromagnetism and superconductivity.

E.B. Sonin\
Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel\
PACS numbers: 75.60.Ch, 74.25.Ha, 74.90.+n

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