Thru-Hole Epitaxy: Is Remote Epitaxy Really Remote?

The remote epitaxy was originally proposed to grow a film, which is not in contact but crystallographically aligned with a substrate and easily detachable due to a van der Waals material as a space layer. Here we show that the claimed remote epitaxy is more likely to be nonremote `thru-hole' epitaxy. On a substrate with thick and symmetrically incompatible van der Waals space layer or even with a three-dimensional amorphous oxide film in-between, we demonstratively grew GaN domains through thru-holes via connectedness-initiated epitaxial lateral overgrowth, not only readily detachable but also crystallographically aligned with a substrate. Our proposed nonremote thru-hole epitaxy, which is embarrassingly straightforward and undemanding, can provide wider applicability of the benefits known to be only available by the claimed remote epitaxy.


INTRODUCTION
It is a golden rule that epitaxial growth of a crystalline film is allowed only by direct bonding to a crystalline substrate. Extremely surprising studies reported that a crystalline film was epitaxially grown remotely without direct bonding to an underlying substrate despite an ultrathin defect-free 2D overlayer placed in-between, which was named as remote epitaxy. [1][2][3] An earlier study, for instance, claimed that GaN can be remotely grown through only up to two layers of graphene but not even a single layer of h-BN on underlying GaN. 2 This result was explained by the teleported influence of the GaN substrate that was not completely screened. 2 Once the thickness of an inserted 2D material becomes above a critical value, the grown film is no longer crystallographically aligned with the underlying substrate. 2,3 The remote epitaxy strictly requires not only the defect-free growth of 2D material but also the state-of-the-art transfer with precise layer-number control. Nevertheless, great attention has been paid to remote epitaxy since it has been believed to provide the great benefit of easy separation of the film, which is crystallographically aligned with a substrate. [1][2][3] Several other studies have also demonstrated the facile detachment of a film taken as evidence of remote epitaxy. [4][5][6][7][8] Strictly speaking, however, the easy detachability may not necessarily originate from the 'remoteness' of the remote epitaxy, but simply indicates that the binding between the film and space layer/substrate is weaker than the adhesion of the film to a detacher such as a thermal release tape. In fact, the entire remoteness of the remote epitaxy across the interface has never been rigorously verified yet. The estimated potential fluctuation across the surface of 2D material/substrate given as theoretical evidence of remote epitaxy was only one-tenth of thermal energy at growth temperature, and more importantly, the corresponding potential profile is essentially uncorrelated to that of the bare substrate. 2 Moreover, it turned out that the overlooked connectedness directly to the substrate through the space layer (Extended Data Fig. 7 of Ref. [1]) should not have been neglected. Thus, it is physically more reasonable to presume that the claimed remote epitaxy is not likely to be remote.
In order to support our argument above, we elaborate on the logic related to growth types categorization. Let us consider a single domain grown on a 2D layer transferred onto a single crystalline substrate. The growth type [GT] for this domain can be categorized into one of the following five as shown in Fig. 1  growth with misaligned orientation. These GT's were categorized on basis of alignment (A or A ), easy detachability (B), and the existence of at least one thru-hole in the 2D insertion layer beneath the grown domain (C). The presence of at least one thru-hole in the i-th region of the 2D insertion layer beneath the growth domain is labeled as (C i ). Thus, (C) is equivalent to C 1 ∨ C 2 ∨ · · · ∨ C N where N is the number required for the union of 1-st, 2-nd, · · · , N -th region to fully cover the entire region of 2D insertion layer beneath the growth domain. On the other hand, the logical negation of (C i ) or NOT (C i ), denoted as (C i ), is the complete absence of thru-hole in the i-th region of 2D insertion layer beneath the growth domain. Likewise, (C), indicating NOT (C), is the complete absence of thru-hole in the entire region of 2D insertion layer beneath the growth domain. Of course, these growth types are not necessarily exclusive to one another, and under certain circumstances, mixed growth types may be observed among various domains or within a merged film. However, for the simplicity of discussion, we assume that one growth type is associated with one single domain.
The necessary and sufficient condition for [GT1] remote epitaxy is the combination of (A) and (C). The verification of (C) requires a full series of high-resolution cross-sectional transmission electron microscopy (HR-TEM) images all across the interface, which has never been given yet. This kind of extensive HR-TEM measurements is, however, impractical even for micro-sized domains, so that the verification of (C 1 ), which was typically shown by a single HR-TEM image in the previous papers on remote epitaxy, was provided as evidence for remote epitaxy instead of (C). On the other hand, we named any growth type, which satisfies (A), (B), and (C), as [GT2] growth type X, which is definitely not [GT1] remote epitaxy. As can be readily inferred from Fig. 1 Thus, either remote epitaxy or thru-hole epitaxy cannot be specified on basis of all the combined experimental evidences, claimed by remote epitaxy, which are just necessary conditions of remote epitaxy. Only if (C 1 ) ∧ (C 2 ) ∧ · · · ∧ (C N ) equivalent to (C) were verified, the existence of [GT1] remote epitaxy would be verified. The flaw of logic in the previous papers on remote epitaxy is that the existence of [GT1] remote epitaxy was claimed to be confirmed only by (C 1 ) instead of (C) although other evidences were given. Moreover, our computational simulation revealed that the theoretical evidence of [GT1] remote epitaxy is questionable as described below. With appropriate justification ((i) logic flow chart, (ii) our experimental results, and (iii) our computational simulation), we claim that the existence of remote epitaxy has not been verified yet and is questionable.
It appears that the original proposers of remote epitaxy recently realized the existence of thru-holes and recognized that the previous experimental evidences for remote epitaxy may not be enough to specify either [GT1] remote epitaxy or [GT2] thru-hole epitaxy. 9 In order to find out the role of thru-holes and specify either remote epitaxy or thru-hole epitaxy, they designed and carried out experiments, but they reached a hasty conclusion in which [GT2] was excluded even in the presence of thru-holes. 9 (See Supplementary Note2 for the more detailed explanation.)

RESULTS AND DISCUSSION
Based on the argument made above, we hypothesized that the claimed remote epitaxy may simply be 'thru-hole' epitaxy composed of two processes through nanoscale holes sparsely distributed in the space layer: nucleation on the exposed substrate and lateral growth 10 over the space layer. To verify our proposed hypothesis, we demonstratively grew epitaxial GaN domains under various et mediocri conditions, which are far from those required by the state-of-the-art growth and transfer indispensable in the claimed remote epitaxy. By doing so, we reveal that our proposed nonremote thru-hole epitaxy can equally well provide in an unprecedentedly straightforward manner the benefits, which are supposedly e The maximum potential differences ∆V observed between the Ga and N sites of a Ga-N dimer considered as an initial seed depicted in (a-c). The infinity indicates the 2D overlayer without the r-sapphire substrate. The topmost atomic layer overlaid on each color-coded potential profile is a guide for the eyes. The color bars indicate the potential variation relative to the average potential value set to be zero. Note that the potential variations on the overlayers were color-coded within a much narrower range. Each 2D space layer/r-sapphire is composed of a 2 × 3 rectangular cell of the r-sapphire and a rectangular cell modified from a ( √ 13 × √ 13)R13.9 • cell of a 2D overlayer, which somewhat mimics a naturally-occurring incommensurate stacking.
accompanied only by the claimed remote epitaxy, such as facile detachability and crystallographic alignment with a substrate. In the following, we first provide computational results showing the unfeasibility of remote epitaxy and then experimental results verifying thru-hole epitaxy.
Computational evidence questioning remote epitaxy

6
The remote epitaxy proposed by earlier studies 1-8 was validated mainly by the surface potential distribution calculated over a space layer complying with that of a substrate. We discussed some questions raised from the previous calculations in Supplementary Note3. To verify whether such validation is indeed valid, we evaluated the electrostatic potential distributions on various surfaces including not only bare substrates such as sapphire, Si, GaAs, GaN, and LiF but also those substrates with a 2D graphene or h-BN overlayer, which will later be denoted as 'overlayer/substrate', using the first-principles density functional theory (DFT) calculations. As an exemplary demonstration, we display the surface structures of the bare r-sapphire, graphene/r-sapphire, and h-BN/r-sapphire in the top view and their calculated surface potential profiles in Figs. 2(a-c), respectively. Here we emphasize that a computationally-convenient small commensurate stacking configuration introduced to resolve the lattice mismatch between the substrate and the 2D overlayer instigates an artificially periodic potential fluctuation, which would lead to a misinterpretation, as discussed later. To mimic a naturally-occurring incommensurate stacking, we thus constructed relatively a large supercell structure composed of 2D overlayer/r-sapphire.
As shown in the lower part of Fig. 2a, the bare r-sapphire without a space layer generates a huge potential variation, which is attributed to both a strong ionic characteristic of Al-O bonds and an uneven surface configuration of the substrate. When a monolayer (n = 1) of the 2D space overlayer, either graphene or h-BN, covers the r-sapphire, not only is the surface potential variation drastically reduced due to the screening of the space layer, but it does not reflect the shape of the potential profile on the r-sapphire substrate, as shown in the lower parts of Figs. 2b and c. With one more layer (n = 2) of the overlayer material, not to mention the complete dissimilarity between its surface potential distribution and that on the bare substrate, the surface potential variation becomes much smaller, being almost close to that over the overlayer itself without the substrate, since the second layer is essentially flat as shown in Fig. 2d. Furthermore, the potential variations for n≥3 cases were found to be almost the same as that of either isolated graphene or h-BN, as shown in Supplementary   Fig. 1.
To examine whether the remote epitaxy would indeed have been eventuated on defect-free 2D space materials, we estimated the potential difference ∆V undergone by a Ga-N dimer regarded as a primordial growth seed that should anchor on the 2D overlayer with a similar orientation to on the bare substrate to guarantee the remote epitaxy. Figure. 2e shows ∆V as a function of the layer numbers n of 2D graphene and h-BN overlayers. Even with n = 1, ∆V was calculated to be only 1 ∼ 2 % of that on the bare sapphire, and rapidly converged to the value on the 2D overlayer without the substrate when n≥2. For comparison, over other substrates, such as Si, GaAs, GaN, and LiF, we evaluated their potential variations at d = 3.0Å, near which the 2D space layer would locate, above their surfaces, to know over which substrate the potential variations emerge through a 2D space layer. As shown in Supplementary Fig. 2, the magnitude of the evaluated potential variation only over the GaN substrate is similar to that over the r-sapphire, but those over the other substrates are much smaller. Thus, the teleportation of substrate potential variations through a 2D space layer is not likely to occur over any kind of substrates as seen in Fig. 2 for r-sapphire.
We further investigated the potential variations on the c-and m-sapphire substrates while increasing the layer number n of the h-BN space material from n = 0 to n = 3, whose trends are essentially identical to that on the r-sapphire case, as shown in Supplementary Fig. 3.
To verify the effect of an artificially-generated periodicity caused by the strain applied to forcibly match the lattice mismatch between the substrate and the space layer as mentioned above, we evaluated the surface potential profiles over various stacking configurations of h-BN/c-sapphire, the combination of which clearly evinces such stacking effect. As displayed in Supplementary Fig. 4, the calculated potential variations are very sensitive to a choice of stacking configurations. Especially, in small supercell configurations, local potential variations look as if that of the substrate would be reflected over the 2D space layer, but this is a misleading artifact caused by such a forcibly matched stacking to make it commensurate.
Therefore, the potential fluctuation of 2D material/substrate does not truly reflect the orientation and periodicity of the underlying substrate if the artifact originating from artificial stacking configurations is simply excluded. For a detailed explanation, see Supplementary Note4.
Based on our calculations, the remoteness of the claimed remote epitaxy 1-8 is conceptually and strongly questionable, and thus it should be replaced by a genuine growth mechanism, nonremote thru-hole epitaxy. The proposed thru-hole epitaxy enabled us to grow epitaxial GaN domains, which still exhibit all the benefits from the seemingly remote epitaxy, in the growth regime even prohibited by the claimed remote epitaxy.
Thru-hole epitaxy enabled by connectedness To verify our proposed growth mechanism, we grew GaN domains on a thick and poly- It is the first direct evidence of our proposed thru-hole epitaxy, which is applicable to any systems and also even to the claimed remote epitaxy. It is farfetched that the claimed remote epitaxy was attributed to remoteness, but it is rather reasonable to connectedness secured by thru-hole epitaxy, as verified by our computational results described above. In a previous study, as a matter of fact, this same kind of connectedness was also observed on small areas but unfortunately disregarded as a minor effect. 1 Being above the critical layer thickness or losing the connectedness is the reason why the claimed remote epitaxy fails. Thus, we claim that the connectedness serving as a control parameter is utilized as a true criterion not only for the claimed remote epitaxy but also for the thru-hole epitaxy.
We showed how to control the extent of the connectedness by intentionally adjusting the thickness of and deliberately not improving the quality of a 2D space layer. ( Supplementary   Fig. 10) We emphasize that the better the quality of a 2D space layer the lower the density of thru-holes. As a result, the weaker the connectedness the smaller the critical value of the thickness, above which the claimed remote epitaxy or the thru-hole epitaxy is forbidden. 2,4 Therefore, the thru-hole epitaxy of crystallographically aligned GaN on h-BN/sapphire does not require the state-of-the-art perfection of h-BN at all.
The readily facile detachability of a grown film crystallographically aligned with an underlying substrate is the exclusive benefit of the claimed remote epitaxy. To check whether the grown film by thru-hole epitaxy is readily detachable as well, we carried out the detachability experiment. Supplementary Fig. 11 shows that GaN domains were readily detached simply by using a thermal release tape although they were connected to the substrate by thru-holes. (Supplementary Fig. 12) We emphasize that there is a critical connectedness below which all the benefits of the claimed remote epitaxy can be obtainable. Unlike the stringent requirement imposed by the claimed remote epitaxy, the connectedness below such critical connectedness is readily achievable. (Supplementary Fig. 13)

Extended demonstration of thru-hole epitaxy
Our experimental results showed that the connectedness by thru-holes is crucial to achieving the crystallographic alignment of a film with a 2D-layer-covered substrate. As long as the connectedness is maintained, the thru-hole epitaxy was found to be valid regardless of either the type or the thickness of a space layer. Our proposed mechanism might, however, be challenged by the speculation that the growth was partly initiated over the h-BN regions the thickness of which was less than a critical value of the claimed remote epitaxy. To unambiguously exclude the speculation, we introduced an additional 50-nm-thick SiO 2 space layer between h-BN and r-sapphire to form SiO 2 /r-sapphire with and without h-BN overlayer, which should completely suppress the remoteness. Subsequently, nanoscale openings were produced in SiO 2 to achieve connectedness. This experimental configuration fully guarantees the thru-hole epitaxy but completely eliminates the possibility of the remote epitaxy.
As expected from the thru-hole epitaxy, GaN domains grown in these experimental configurations were crystallographically aligned with r-sapphire as shown in Figs. 5(a-c). This  result indicates that we successfully achieved the connectedness by intentionally creating nanoscale openings, (See Methods section) verified by TEM analysis as shown in Fig. 5d and Supplementary Fig. 14. The thru-hole epitaxy is robustly manifested even with a SiO 2 space layer.
Moreover, we also successfully detached those [1120]-oriented GaN domains from h-BN/SiO 2 /r-sapphire simply by using a thermal release tape as shown in Supplementary   Fig. 15. Here we would like to emphasize that the crystallographic alignment is associated with the connectedness, whereas h-BN or any 2D van der Waals space layer plays a role in not transferring the crystallographic information through but only in allowing the film grown above to be readily detachable.
We have shown that ostensibly 'remote' epitaxy based on remoteness is evinced by the nonremote thru-hole epitaxy originating from connectedness. Our proposed thru-hole epitaxy mechanism maintains every advantage (e.g., undemanding detachability, crystallographical alignment with an underlying substrate, independence of space layer symmetry) of the remote epitaxy in an embarrassingly straightforward and undemanding manner. It was also demonstrated that our method can be readily extended to hBN/substrate even with a thick SiO 2 film in-between without compromising the advantages described above. This Interlayer interactions were incorporated with Grimme-D2 van der Waals correction. 20 To reduce long-range interactions from neighboring cells located along the out-of-plane or growth direction, we included a sufficiently large vacuum region of 20Å. The Brillouin zone (BZ) of each structure was sampled using a separation of 0.04Å −1 k-point mesh according to the Monkhost-Pack scheme. 21 The structures were relaxed until all forces became smaller than 0.02 eV/Å. The exchange-correlation potential was excluded in potential fluctuation maps because exchange-correlation potential on a long distance from the surface is incorrect within standard DFT. 2,22 Various 2D overlayer/sapphire supercell structures were constructed to avoid artificially generated periodic potentials.

COMPETING INTERESTS
The authors declare no competing interests.

DATA AVAILABILITY
All data are available in the main text or the supplementary information.

Crystallographic alignment of a grown film with an underlying substrate:
The crystallographic alignment is a consequence of epitaxial growth involving direct bonding between a film and an underlying substrate. So, this evidence can be readily understood in terms of [GT2] thru-hole epitaxy as well and is only a necessary condition for [GT1] remote epitaxy.

Crystallographic alignment limited by layer number of 2D insertion material:
The crystallographic alignment claimed in claimed remote epitaxy was observed only on monolayer or bilayer graphene. Such alignment in this limited condition was regarded as the main feature of claimed remote epitaxy. This can be also easily and self-consistently explained by thru-hole epitaxy as follows. Even a state-of-art 2D layer transferred onto a target substrate has some unavoidable holes, which can serve as nucleation spots for thru-2 hole epitaxy, although the size of holes may vary from monovacancy to a-few-micrometers.
The stacking of 2D layers containing holes would decrease the number density of thru-holes, so that the number density of potential nucleation spots would decrease. One important factor here is that how fast the number density of thru-holes decreases with an increasing number of stacking. If the quality of the 2D layer is excellent, only a few stacking would immediately block all the holes whereas thru-holes still survive even after stacking several times if it is mediocre. In any case, if the number density and size of thru-holes get smaller than critical values, [GT2] thru-hole epitaxy becomes less dominant over [GT5] growth with misaligned orientation.

Ionicity dependence of crystallographic alignment:
In claimed remote epitaxy, ionicity was regarded as a key factor to obtain crystallographic alignment. Such aligned domains of ionic material can be readily explained by thru-hole epitaxy as long as the size and number density of thru-holes are larger than their critical values for crystallographic alignment. These critical values vary with materials properties such as ionicity. For example, if two different materials with and without ionicity are separately grown on the same 2D insertion layer on the substrate made of their respective materials, adatoms of the nonionic material would be less attracted toward thru-holes by the exposed substrate than those of the ionic material. That is because the range of attractive interaction from the exposed area of the nonionic material is much shorter than that of the ionic material. Thus, the formation of misaligned domains on the 2D insertion layer would be more probable with the nonionic material than the ionic material because the actual size and number density of thru-holes are smaller than critical values for the nonionic material but larger than those for the ionic material. In other words, nonionic material would be more likely to be grown as [GT5] growth with misaligned orientation than ionic material on the same 2D insertion layer. It should be noted that this kind of ionicity-dependence of interaction range is not typically observed in conventional ELOG because the size of an individual hole or opening area is so sufficiently large that stochastic diffusion of adatoms toward opening areas is more dominant. 3

Easy detachability:
It should be noted that easy detachability does not necessarily suggest a complete absence of direct bonding between a film and an underlying substrate. Instead, it simply indicates that the adhesive force between the grown film and a thermal release tape is large enough to break the binding force between the grown film and a space layer/substrate. It can be easily inferred that increasing the layer number of 2D insertion material would decrease the number density of thru-holes, so that detachability would be improved as well. Moreover, the detachability would be enhanced by reducing the size of thru-holes. In both situations, Nevertheless, it is very challenging to achieve such vdW epitaxy or to grow 3D singlecrystalline films on dangling-bond-free 2D materials, 2 because of small potential fluctuation and the absence of nucleation centers. To overcome this limitation in most cases, various defects playing roles as nucleation centers were introduced on a 2D surface using surface modification techniques. 3,4 Moreover, vdW epitaxy has been observed relatively more often on h-BN than on graphene because the potential fluctuation on the latter surface is even smaller than that on the former surface. Another important point is whether the potential profile reflects the symmetry of underlying substrate or 2D material for remote or vdW epitaxy, respectively, or not. As clearly shown in our manuscript, the teleported potential fluctuation does not reflect the symmetry of the underlying substrate, whereas the surface potential profile of h-BN or graphene does show its complete symmetry. We also agree with the reviewer that the formation of a large nucleate with a radius larger than the critical radius is an important key issue for epitaxy, and our estimated potential fluctuation across the surface of 2D material/substrate is even larger than that of vdW epitaxy. However, we would like to point out two points: (i) the structural configurations of large nucleates formed during the growth process would be quasi-random and thus their symmetry would not be necessarily consistent with the symmetry of substrate; and (ii) even if there is a large nucleate that has the same symmetry of substrate by accident, it is unlikely that such nucleate settles on 2D material with crystallographic alignment to the underlying substrate, because the potential profile of 2D material/substrate does not reflect the symmetry of the underlying substrate as mentioned above and in our manuscript.
Some of the previous papers on remote epitaxy showed that their potential/charge profiles reflect the characteristics of the underlying substrates. It turned out that such results had been obtained only under certain constrained conditions as described in the following. In the paper by Kong et al. 5 , which is one of the original papers explaining the concept of remote epitaxy, they did not show the total potential distribution U tot of 2D material/substrate, which is the true potential governing growth processes. Instead, they intentionally calculated a potential distribution defined as U = U tot − U 2D , to reflect mainly substrate contribution by subtracting the 2D material contribution, U 2D . We would like to emphasize again that the true potential governing growth processes is not this potential difference, but the total potential. Moreover, they used small supercell sizes to enforce commensurability between 2D material and substrate by introducing large strain, resulting in the creation of an artificial symmetry reflecting that of the underlying substrate. Such structural constraint is, of course, unavoidable since a periodic system is required in calculations. That is why we considered various stacking configurations (relatively different orientations and supercell sizes) of 2D material/substrate to exclude constraint-induced artifacts in our calculation. As shown in Fig. 2 in our main manuscript and Supplementary Fig. 4, U tot depends strongly on the stacking configuration (relative orientation and supercell size) and thus does not follow the symmetry of the underlying substrate. Another work 6 showed the charge density with a small supercell instead of potential distribution. In that paper, they computed charge density using ρ = ρ tot − ρ 2D to extract the substrate contribution, rather than the total charge density ρ tot of 2D material/substrate, which is a true charge density responsible for the growth process. Therefore, the existence of vdW epitaxy cannot be direct evidence for remote epitaxy, the concept of which is still not regarded to be validated by DFT calculations, but rather strongly questionable.
There is a (101) Bragg peak of GaN near 2θ = 37 • the intensity of which is several orders smaller than that of the c-GaN (002) Bragg peak.  indicating that they were directly grown on SiO 2 /r-sapphire without a h-BN space layer.