# Control of ternary alloy composition during remote epitaxy on graphene

Zachary LaDuca  Materials Science and Engineering, University of Wisconsin-
Madison, Madison, WI 53706  Katherine Su  Materials Science and Engineering,
University of Wisconsin-Madison, Madison, WI 53706  Sebastian Manzo  Materials
Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706
Michael S. Arnold  Materials Science and Engineering, University of Wisconsin-
Madison, Madison, WI 53706  Jason K. Kawasaki  Materials Science and
Engineering, University of Wisconsin-Madison, Madison, WI 53706
<EMAIL_ADDRESS>
###### Abstract

Understanding the sticking coefficient $\sigma$, i.e., the probability of an
adatom sticking to a surface, is essential for controlling the stoichiometry
during epitaxial film growth. However, $\sigma$ on monolayer graphene-covered
surfaces and its impact on remote epitaxy are not understood. Here, using
molecular-beam epitaxial (MBE) growth of the magnetic shape memory alloy
Ni2MnGa, we show that the sticking coefficients for metals on graphene-covered
MgO (001) are less than one and are temperature and element dependent, as
revealed by ion backscattering spectrometry (IBS) and energy dispersive x-ray
spectroscopy (EDS). This lies in stark contrast with most transition metals
sticking on semiconductor and oxide substrates, for which $\sigma$ is near
unity at typical growth temperatures ($T<800\degree$C). By initiating growth
below $400\degree$ C, where the sticking coefficients are closer to unity and
wetting on the graphene surface is improved, we demonstrate epitaxy of Ni2MnGa
films with controlled stoichiometry that can be exfoliated to produce
freestanding membranes. Straining these membranes tunes the magnetic coercive
field. Our results provide a route to synthesize membranes with complex
stoichiometries whose properties can be manipulated via strain.

Remote [1, 2] and van der Waals [3, 4, 5, 6] epitaxy on monolayer graphene-
covered substrates are promising strategies for synthesizing single
crystalline films that are mechanically decoupled from the substrate. In
remote epitaxy, films are thought to grow on graphene-covered substrates with
epitaxial registry to the substrate, due to the “remote” lattice potential of
the substrate potential that permeates through graphene [1, 7]. Applications
include lattice mismatched epitaxy with reduced dislocation densities [8, 9],
etch-free exfoliation of membranes for flexible electronics and re-use of
substrates [1], and discovery of new properties induced by extreme strain and
strain gradients in membranes [10, 11, 12].

A fundamental challenge, however, is controlling the film stoichiometry during
growth on graphene. Due to the weak van der Waals interactions, the sticking
coefficients $\sigma$ for metals on multilayer graphite are typically
$\sigma<0.1$ at room temperature as measured by desorption spectroscopy [13],
x-ray photoemission spectroscopy (XPS) [14, 15], and scanning tunneling
microscopy (STM) [14, 15]. This lies in stark contrast with the typical
$\sigma\sim 1$ for metals on semiconductor, oxide, and metal surfaces [16],
which enables a simple one-to-one correspondence between film stoichiometry
and incident flux ratios. Although $\sigma$ on monolayer graphene-covered
surfaces is anticipated to be closer to unity [16], due to the “remote”
substrate interactions that permeate through graphene [1, 17, 7], sticking on
graphene is less understood and unlikely to be exactly 1. Moreover, the
“remote” argument suggests that $\sigma$ on graphene-covered substrates should
depend on the identity of the substrate. The impact of element-dependent, non
unity sticking coefficients during growth on monolayer graphene is generally
overlooked, in part, because remote epitaxy has focused on compound
semiconductors like GaAs for which the stoichiometry is self-limited by growth
within an adsorption-controlled window [18, 19]. But for more complex
materials like ternary transition metal oxides or intermetallic Heusler
compounds adsorption-controlled growth windows are only accessible in select
cases [20, 2, 21, 22, 23]. Controlling the stoichiometry of these materials
during remote epitaxy or van der Waals epitaxy on graphene [12, 2, 24, 25, 26,
10, 5] in the ultrathin limit will require understanding the sticking
coefficients on graphene.

Here, using MBE growth of the magnetic shape memory alloy Ni2MnGa, we show
that the sticking coefficients for transition metals on graphene-covered
substrates are non unity and both element and temperature dependent. Our
measurements of the stoichiometry by ion backscattering spectrometry (IBS) and
energy dispersive spectroscopy (EDS) for films with thickness 20-80 nm provide
upper bounds for the sticking coefficients of Ni and Mn on graphene/MgO, which
are less than 0.6 at $600\degree$C. Controlling the stoichiometry requires
compensating for the nonunity sticking coefficient on graphene, or initiating
growth at low temperatures where $\sigma$ on graphene is near unity. We
demonstrate epitaxial Ni2MnGa films that can be mechanically exfoliated, and
show how externally applied strain in Ni2MnGa membranes tunes the magnetic
coercive field.

Figure 1: (a) Ion Beam Scattering (IBS) for nominally 20 nm thick Ni2MnGa
films grown on graphene/MgO and MgO at $600\degree$ C, showing reduced
sticking for Ni and Mn on graphene. (b) Energy dispersive X-ray spectroscopy
(EDS) measurements for Ni2MnGa films with nominal thickness 80 nm on gr/MgO
and on MgO. Both IBS and EDS sample was were capped with a protective layer of
Au. (c) EDS intensity ratios $I_{graphene}/I_{MgO}$, tracking temperature and
element dependent changes in the cumulative sticking coefficient for Ni2MnGa
on graphene-covered MgO. Error bars are standard deviations on multiple
regions of a given sample. (d,e) SEM images of the nominally 80 nm thick films
grown at room remperature on graphene/MgO and MgO and capped with Au. (f,g)
SEM images of the nominally 80 nm thick films grown at $625\degree$C on
graphene/MgO and MgO and capped with Au.

Ni2MnGa films were grown by molecular beam epitaxy (MBE) on graphene-covered
MgO (001) substrates. The graphene was grown by chemical vapor deposition on
polycrystalline Cu foils and wet transferred to the MgO (001) substrate using
a poly(methyl methacrylate) (PMMA) handle, Cu etch, and scoop from deionized
water, as described in Refs. [10, 5]. Films with nominal composition Ni2MnGa
and nominal thickness 20-80 nm were grown by MBE using elemental effusion cell
sources, with typically fluxes of $2.2\times 10^{13}$ atoms /
(cm${}^{2}\cdot$s) for Ni and $1.1\times 10^{13}$ atoms / (cm${}^{2}\cdot$s)
each for Mn and Ga. We use the term “nominal” to indicate the composition and
thickness if all of the incident Ni, Mn, and Ga atomic fluxes stuck to the
surface ($\sigma=1$). All samples were capped with $\sim 20$ nm of Au at room
temperature before removal from the MBE system, to avoid oxidation. Fluxes
were measured in situ using a quartz crystal microbalance and calibrated to
absolute fluxes via ex situ Ion Backscattering Spectrometry (IBS) measurements
on calibration samples, grown at room temperature on Si. Energy dispersive
x-ray spectroscopy (EDS, beam energy 10-20 keV, interaction depth of a few
microns) was used to measure relative differences in Ni2MnGa film composition.
The increased depth sampling of IBS and EDS, compared to more surface
sensitive XPS, allows us to sum over all species that stick, including those
that may intercalate [27, 28, 29] or diffuse [30, 31] beneath graphene, rather
than primarily detecting species that reside at the surface.

Fig. 1(a) compares IBS measurements (He+, 4.9 MeV, $\theta=8\degree$) of a
nominally 20 nm thick Ni2MnGa film grown on graphene/MgO with a film grown
directly on MgO. Growth was performed at $600\degree$ C on an MgO substrate
that is half covered with graphene, such that both sides of the sample are
exposed to the same incident atomic fluxes of Ni, Mn, and Ga. We find that the
areal density of Ga on graphene/MgO and on MgO are nearly equal. In contrast,
the areal densities of Ni and Mn on graphene/MgO are only 50 to 60% of the Ni
and Mn on the MgO surface. Similar results are found for EDS measurements of
thicker films. Fig. 1(b) compares EDS measurements for nominally 80 nm thick
Ni2MnGa films grown on graphene/MgO and MgO at $625\degree$C, where again we
observe similar sticking for Ga on the graphene/MgO and MgO sides, and a $\sim
50\%$ reduction of the sticking for Ni and Mn on graphene/MgO compared to MgO.

The large stoichiometry differences in films with nominal thickness 20-80 nm
are at first surprising, since differences in sticking coefficient are
expected to be limited to within the first few atomic layers of growth. For
planar film growth, after a layer of Ni2MnGa covers the graphene or MgO
surface, subsequent Ni2MnGa film growth in both cases should be Ni2MnGa on
Ni2MnGa, and thus the stoichiometries of thick films should converge. We
attribute the large observed stoichiometry differences to a combination island
morphology and reduced sticking coefficients on graphene. Scanning electron
micrographs (SEM) reveal that the nominally 80 nm thick film grown on
graphene/MgO at $625\degree$C has a disconnected island morphology (Fig.
1(f)), which we attribute to poor wetting on low surface energy graphene.
Similar poor wetting has been observed for other films on monolayer graphene-
covered surfaces [30, 32]. In contrast, films grown directly on MgO at the
same temperature have a smoother and more connected morphology (Fig.1(g)).
This morphology on graphene suggests that even after tens of nanometers of
nominal growth, some exposed regions of the graphene remain. Thus our IBS and
EDS measurements result from combined sticking on exposed graphene regions
(where $\sigma<1$) and on Ni2MnGa islands (where $\sigma\sim 1$). This
cumulative sticking coefficient $\sigma^{\prime}$ is therefor an upper bound
for the true sticking coefficient $\sigma$ on graphene in the atomic layer
limit.

Importantly, we find the cumulative sticking on graphene/MgO is highly
temperature and element dependent. Fig. 1(c) plots the EDS intensity ratio on
the graphene/MgO versus on MgO ($I_{graphene}/I_{MgO}$), for a series of
nominally 80 nm thick Ni2MnGa films as a function of growth temperature. We
normalize to the intensity on the MgO side, since the sticking coefficients
for metals directly on MgO are nominally 1. Thus the ratio
$I_{graphene}/I_{MgO}$ is approximately equal to the cumulative sticking
coefficient on graphene/MgO. Cumulative sticking for Ni and Mn on graphene/MgO
are lowest at high substrate temperature and approach 1 for all three elements
below $400\degree$ C. We attribute this temperature dependence on graphene to
the combined decreased desorption rate and smoother morphology with less
exposed graphene at lower temperature (Fig. 1(d)). We attribute the reduced
sticking on graphene, compared to MgO, to relatively weak van der Waals
interactions between metal adsorbates and graphene. Indeed, density functional
theory calculations suggest that adsorption energies for metals on graphene
are of order $E_{a}\sim 0.5$ eV, compared to $E_{a}\sim 3$ eV for Au on the Au
(111) surface [16]. Surprisingly, the sticking coefficient for Ga on graphene
is near unity and independent of temperature, despite the fact that Ga has a
higher vapor pressure than Ni or Mn. We speculate the increased sticking of Ga
on graphene may arise from reactions or from intercalation beneath graphene.
Ga, In, and Sn are known to intercalate at graphene/SiC interfaces [27], and
Au is known to intercalate between sheets of graphite [28].

Our findings suggest that control of the Ni2MnGa film stoichiometry on
graphene requires compensating for the non-unity sticking coefficients at high
growth temperature or initiating the growth at lower temperatures where the
cumulative sticking coefficients are closer to 1. Lower temperature growth is
also beneficial for promoting a smoother morphology on graphene (Fig. 1(d)).
Once the interface has formed and the graphene layer is buried, growth can
resume under more normal temperatures and fluxes.

Figure 2: Out of plane X-ray diffraction scans (Cu $K\alpha$) of Ni2MnGa films
grown on MgO and on graphene/MgO at $600\degree$C, $400\degree$C, and
$370\degree$C, compared to a film grown directly on MgO. Asterisks * denote
MgO substrate reflections and “x” denotes secondary phase reflections.

For simplicity adopt the strategy of growth at a fixed lower temperature. Fig.
2 compares x-ray diffraction patterns for Ni2MnGa films grown at $370\degree$
C, $400\degree$ C, and $600\degree$ C on graphene/MgO, with a film grown by
directly on MgO. We find that the sample grown at $600\degree$ C on
graphene/MgO displays several impurity reflections, marked by “x,” consistent
with large deviations from stoichiometry observed by IBS and EDS (Fig. 1).
Films grown at $400\degree$C and below on graphene display only the Heusler
Ni2MnGa reflections and no impurity reflections. Interestingly, the films on
graphene/MgO display both $00L$ and $HH0$ reflections, indicating both (001)
and (110) oriented growth, whereas epitaxy directly on MgO produces only (001)
oriented growth (black curve).

Figure 3: (a) Azimuthal $\phi$ scans for a Ni2MnGa film grown on graphene/MgO
(001). The off axis $010$ reflections track the in-plane orientation of
Ni2MnGa domains with (110) out of plane orientation. The $101$ reflections
track the in-plane orientation of the (001) domain. (b) Domain orientations of
Ni2MnGa (blue) with respect to MgO (001) (black) determined from (a).

Azimuthal $\phi$ scans reveal that both (110) and (001) oriented Ni2MnGa
domains on graphene/MgO (001) have well defined in-plane orientations with
respect to the underlying MgO substrate, despite the presence of the
polycrystalline graphene interlayer. In Fig. 3(a), the four-fold pattern of
Ni2MnGa 101 reflections is rotated by 45 degrees with respect to the MgO 101.
This indicates that the (001) Ni2MnGa domain has a 45 degree rotated cube on
cube epitaxial relationship to MgO, i.e. Ni2MnGa (001) [110] $\parallel$ MgO
(001) [100], and a $2\%$ tensile lattice mismatch (Fig. 3(b)). For the (110)
domain, we observe a four-fold pattern of 010 Ni2MnGa reflections aligned with
the MgO 101. This indicates two rectangular domains, labelled A and B, with
orientations Ni2MnGa (110) [001] $\parallel$ MgO (001) [010] and Ni2MnGa (110)
[001] $\parallel$ MgO (001) [100] (Fig. 3(b)). For these (110)-orientated
domains, the mismatch between Ni2MnGa $d_{110}$ and the MgO $a$ lattice
spacings is $2\%$, while the mismatch in the orthogonal in-plane direction
($a_{Ni2MnGa}$ vs $a_{MgO}$) is much larger and would require a larger
supercell to produce a commensurate structure. We speculate that the presence
of the graphene interlayer may relax the constraints of direct epitaxy in
which there are direct bonds formed between film and substrate, and allow for
alternative film orientations that lower the total energy. Similar new
epitaxial structures have been observed in the form of rotated superstructures
for GdPtSb films on graphene/sapphire [5]. Further studies are required to
understand why the (110) domain appears on graphene/MgO and not directly on
MgO.

Figure 4: (a) X-ray diffraction before and after membrane exfoliation. (b)
SQUID magnetometry of a relaxed Ni2MnGa film on graphene/MgO (dark blue), and
on the same sample after exfoliation and rippling to create a strained Ni2MnGa
membrane (light blue). The measurement was performed at 100 K with field
oriented within the film plane.

Finally, we show that applying external strains to exfoliated membranes tunes
magnetic properties. We exfoliate membranes by adhering the film to a glass
slide using crystal bond, then peeling the film from the graphene/MgO. After
exfoliation we observe only the 110 and 001-type film reflections and no
substrate reflections, as shown in Fig. 4(a). We then apply ripples to the
membrane to induce strain. The rippling was performed by adhering a tensile
strained polyurethane film to the exfoliated Ni2MnGa membrane, heating to
approximately $150\degree$C to release the Ni2MnGa/polyurethane bilayer from
the crystalbond, and relaxing to impart ripples upon contraction of the
polyurethane. Further details of the rippling procedure are described in Ref.
[10].

Fig. 4(b) shows SQUID magnetometry measurements for a Ni2MnGa film on
graphene/MgO, and the same film after exfoliation and subjected to strain in
the form of rippling, measured at 100 K with field oriented in plane. We find
that strain and/or strain gradients enhance the coercive field, from 400 to
650 Oe. The membrane has thickness 80 nm, ripple period of 8 microns, and peak
to peak height of 3 microns. Assuming a sinusoidal shape we we estimate the
peak magnitudes of strain to be $|\epsilon|<3.6\%$, if no plastic deformation
[10]. The strain tunable coercive field may be useful for strain-assisted
reading and writing of magnetic memory.

In summary, we showed that sticking coefficients for metals on graphene-
covered substrates are non unity and highly dependent on element and
temperature. IBS and EDS measurements of films with tens of nanometers
thickness provide upper bounds for the sticking coefficients on graphene/MgO:
$\sigma<0.6$ for Ni and Mn and $\sigma<1$ for Ga at $600\degree$C. Surface
sensitive measurements in the monolayer limit are required to fully quantify
the atomic sticking coefficients on graphene, and understand the effects of
changing the underlying substrate and effects of defects and contaminants at
the graphene/substrate interface. In particular, the lattice potential
permeation argument of remote epitaxy[6, 7] suggests that the sticking
coefficients should also depend on the identity of the substrate. We show that
synthesis at lower temperature $\leq 400\degree$C enables phase pure epitaxy
of Ni2MnGa films on graphene/MgO. Similar strategies may apply to remote and
van der Waals epitaxy of other materials with complex stoichiometries, for
which adsorption-controlled growth windows are not accessible.

## Acknowledgment

We thank Greg Haugsted for IBS/RBS measurements. This work was primarily
supported by the Air Force Office of Scientific Research grant FA9550-21-0127
(Z.L. and J.K.K.). Preliminary Heusler synthesis was supported by the National
Science Foundation DMR-1752797 (Z.L., S.M., and J.K.K.). Graphene synthesis
via CVD supported by the U.S. Department of Energy, Office of Science, Basic
Energy Sciences, Grant DE-SC0016007 (K.S. and M.S.A.).

We gratefully acknowledge the use of x-ray diffraction facilities supported by
the NSF through the University of Wisconsin Materials Research Science and
Engineering Center under Grant No. DMR-1720415.

## References

  * Kim et al. [2017] Y. Kim, S. S. Cruz, K. Lee, B. O. Alawode, C. Choi, Y. Song, J. M. Johnson, C. Heidelberger, W. Kong, S. Choi, et al., Nature 544, 340 (2017).
  * Yoon et al. [2022] H. Yoon, T. K. Truttmann, F. Liu, B. E. Matthews, S. Choo, Q. Su, V. Saraswat, S. Manzo, M. S. Arnold, M. E. Bowden, et al., Science Advances 8, eadd5328 (2022).
  * Koma [1992] A. Koma, Thin Solid Films 216, 72 (1992).
  * Ren et al. [2021] F. Ren, B. Liu, Z. Chen, Y. Yin, J. Sun, S. Zhang, B. Jiang, B. Liu, Z. Liu, J. Wang, et al., Science Advances 7, eabf5011 (2021).
  * Du et al. [2022] D. Du, T. Jung, S. Manzo, Z. LaDuca, X. Zheng, K. Su, V. Saraswat, J. McChesney, M. S. Arnold, and J. K. Kawasaki, Nano Letters 22, 8647 (2022).
  * Kim et al. [2014] J. Kim, C. Bayram, H. Park, C.-W. Cheng, C. Dimitrakopoulos, J. A. Ott, K. B. Reuter, S. W. Bedell, and D. K. Sadana, Nature communications 5, 1 (2014).
  * Kong et al. [2018] W. Kong, H. Li, K. Qiao, Y. Kim, K. Lee, Y. Nie, D. Lee, T. Osadchy, R. J. Molnar, D. K. Gaskill, et al., Nature materials 17, 999 (2018).
  * Bae et al. [2020] S.-H. Bae, K. Lu, Y. Han, S. Kim, K. Qiao, C. Choi, Y. Nie, H. Kim, H. S. Kum, P. Chen, et al., Nature nanotechnology 15, 272 (2020).
  * Liu et al. [2022] B. Liu, Q. Chen, Z. Chen, S. Yang, J. Shan, Z. Liu, Y. Yin, F. Ren, S. Zhang, R. Wang, et al., Nano Letters 22, 3364 (2022).
  * Du et al. [2021] D. Du, S. Manzo, C. Zhang, V. Saraswat, K. T. Genser, K. M. Rabe, P. M. Voyles, M. S. Arnold, and J. K. Kawasaki, Nature communications 12, 1 (2021).
  * Du et al. [2023] D. Du, J. Hu, and J. K. Kawasaki, Applied Physics Letters 122, 170501 (2023).
  * Kum et al. [2020] H. S. Kum, H. Lee, S. Kim, S. Lindemann, W. Kong, K. Qiao, P. Chen, J. Irwin, J. H. Lee, S. Xie, et al., Nature 578, 75 (2020).
  * Arthur and Cho [1973] J. Arthur and A. Cho, Surface Science 36, 641 (1973).
  * Lopez-Salido et al. [2005] I. Lopez-Salido, D. C. Lim, and Y. D. Kim, Surface science 588, 6 (2005).
  * Howells et al. [2002] A. R. Howells, L. Hung, G. S. Chottiner, and D. A. Scherson, Solid State Ionics 150, 53 (2002).
  * Appy et al. [2014] D. Appy, H. Lei, C.-Z. Wang, M. C. Tringides, D.-J. Liu, J. W. Evans, and P. A. Thiel, Progress in surface science 89, 219 (2014).
  * Rafiee et al. [2012] J. Rafiee, X. Mi, H. Gullapalli, A. V. Thomas, F. Yavari, Y. Shi, P. M. Ajayan, and N. A. Koratkar, Nature materials 11, 217 (2012).
  * Arthur Jr [1968] J. Arthur Jr, Journal of Applied Physics 39, 4032 (1968).
  * Cho and Arthur [1975] A. Cho and J. Arthur, Progress in Solid State Chemistry 10, 157 (1975), ISSN 0079-6786.
  * Jalan et al. [2009] B. Jalan, P. Moetakef, and S. Stemmer, Applied Physics Letters 95, 032906 (2009).
  * Ihlefeld et al. [2009] J. F. Ihlefeld, W. Tian, Z.-K. Liu, W. A. Doolittle, M. Bernhagen, P. Reiche, R. Uecker, R. Ramesh, and D. G. Schlom, IEEE transactions on ultrasonics, ferroelectrics, and frequency control 56, 1528 (2009).
  * Theis et al. [1998] C. D. Theis, J. Yeh, D. G. Schlom, M. Hawley, and G. Brown, Thin Solid Films 325, 107 (1998).
  * Shourov et al. [2020] E. H. Shourov, R. Jacobs, W. A. Behn, Z. J. Krebs, C. Zhang, P. J. Strohbeen, D. Du, P. M. Voyles, V. W. Brar, D. D. Morgan, et al., Physical Review Materials 4, 073401 (2020).
  * Ma et al. [2021] C.-H. Ma, L.-S. Lu, H. Song, J.-W. Chen, P.-C. Wu, C.-L. Wu, R. Huang, W.-H. Chang, and Y.-H. Chu, APL Materials 9, 051115 (2021).
  * Jia et al. [2021] R. Jia, H. S. Kum, X. Sun, Y. Guo, B. Wang, P. Fang, J. Jiang, D. Gall, T.-M. Lu, M. Washington, et al., Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films 39, 040405 (2021).
  * Dai et al. [2022] L. Dai, J. Zhao, J. Li, B. Chen, S. Zhai, Z. Xue, Z. Di, B. Feng, Y. Sun, Y. Luo, et al., Nature Communications 13, 2990 (2022).
  * Briggs et al. [2020] N. Briggs, B. Bersch, Y. Wang, J. Jiang, R. J. Koch, N. Nayir, K. Wang, M. Kolmer, W. Ko, A. De La Fuente Duran, et al., Nature materials 19, 637 (2020).
  * Gierz et al. [2010] I. Gierz, T. Suzuki, R. T. Weitz, D. S. Lee, B. Krauss, C. Riedl, U. Starke, H. Höchst, J. H. Smet, C. R. Ast, et al., Physical Review B 81, 235408 (2010).
  * Ebert [1976] L. B. Ebert, Annual Review of Materials Science 6, 181 (1976).
  * Strohbeen et al. [2021] P. J. Strohbeen, S. Manzo, V. Saraswat, K. Su, M. S. Arnold, and J. K. Kawasaki, ACS Applied Materials & Interfaces 13, 42146 (2021).
  * Morrow et al. [2016] W. K. Morrow, S. J. Pearton, and F. Ren, Small 12, 120 (2016).
  * Alaskar et al. [2014] Y. Alaskar, S. Arafin, D. Wickramaratne, M. A. Zurbuchen, L. He, J. McKay, Q. Lin, M. S. Goorsky, R. K. Lake, and K. L. Wang, Advanced Functional Materials 24, 6629 (2014).


On the one hand, if $x\mapsto\ell(w, x)$ is $\xi$-Lipschitz uniformly on $\W$, then

$$|\G|\leq \frac{\xi}{m}\sum_{i=1}^m\E_{\Pp_W, X_i^\star}[\Wass(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W, X_i^\star})+\Wass(\Pp_{\bar X_i|X_i^\star}, \Pp_{\bar X_i|W, X_i^\star})]\,.$$

On the other hand, assume <ref>. if $x\mapsto\nabla_w\ell(w, x)$ is $\xi$-Lipschitz uniformly on $\W$, then

$$|\G|\leq \frac{\xi}{m}\sum_{i=1}^m\sum_{k=1}^\infty \ee_{k-1}\E_{\Pp_W, X_i^\star}[\Wass(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W_k, X_i^\star})+\Wass(\Pp_{\bar X_i|X_i^\star}, \Pp_{\bar X_i|W_k, X_i^\star})]\,.$$

Fix $p>1$ and let $r=p/(p-1)$. On the one hand, if $|\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{2\xi}{m}\sum_{i=1}^m\,(I^{(p)}(W; X_i|X_i^\star)+1)^{1/p}\,.$$

On the other hand, assume <ref>. If $|\nabla_w\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{2\xi}{m}\sum_{i=1}^m\sum_{k=1}^\infty\ee_{k-1}(I^{(p)}(W_k;X_i|X_i^\star)+1)^{1/p}\,.$$

On the one hand, if $|\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{\xi}{m}\sum_{i=1}^m \,\E_{\Pp_{W, X_i^\star}}\left[\sqrt{\chi^2(\Pp_{X_i|W, X_i^\star}\|\Pp_{X_i|X_i^\star})}+\sqrt{\chi^2(\Pp_{\bar X_i|W, X_i^\star}\|\Pp_{\bar X_i|X_i^\star})}\right]\,.$$

On the other hand, assume <ref>. If $|\nabla_w\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{\xi}{m}\sum_{i=1}^m\sum_{k=1}^\infty\ee_{k-1}\E_{\Pp_{W, X_i^\star}}\left[\sqrt{\chi^2(\Pp_{X_i|W_k, X_i^\star}\|\Pp_{X_i|X_i^\star})}+\sqrt{\chi^2(\Pp_{\bar X_i|W_k, X_i^\star}\|\Pp_{\bar X_i|X_i^\star})}\right]\,.$$

On the one hand, if $|\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{2\xi}{m}\sum_{i=1}^m\,\E_{\Pp_{W, X_i^\star}}\left[\TV(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W, X_i^\star})+\TV(\Pp_{\bar X_i|X_i^\star}, \Pp_{\bar X_i|W, X_i^\star})\right]\,.$$

On the other hand, assume <ref>. If $|\nabla_w\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{2\xi}{m}\sum_{i=1}^m\sum_{k=1}^\infty\ee_{k-1}\left[\TV(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W_k, X_i^\star})+\TV(\Pp_{\bar X_i|X_i^\star}, \Pp_{\bar X_i|W_k, X_i^\star})\right]\,.$$

On the one hand, if $|\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{2\xi}{m}\sum_{i=1}^m\sqrt{2\LI(W;X_i|X_i^\star)}\,.$$

On the other hand, assume <ref>. If $|\nabla_w\ell(w, x)|\leq\xi$, uniformly on $\W$ and $\X$, then

$$|\G|\leq \frac{2\xi}{m}\sum_{i=1}^m\sum_{k=1}^\infty\ee_{k-1}\sqrt{2\LI(W_k;X_i|X_i^\star)}\,.$$

§.§ Summary table

Several explicit bounds that can be derived within our general framework of Section <ref> are reported in Table <ref>. The first column states the regularity condition required on the loss. However, we refer to the corresponding propositions for the detailed assumptions of each bound. All bounds are stated for $\xi=1$. The last columns give the literature references for each bound, to the best of our knowledge. However, this bibliography should be taken as a mere guideline, as there might possibly be missing references. Those bounds that we could not find in the literature are marked as “New”.

Some bounds that can be derived with the framework from Section <ref>

Assumption ($\forall w\in\W$)     Bound     Prop     Ref

$\ell(w, X)$ $1$-SG     $\sqrt{2 I(W;S)/m}$     <ref>     [Russo and Zou, 2019]

$\nabla_w\ell(w, X)$ $1$-SG     $\sum_k\varepsilon_{k-1}\sqrt{2 I(W_k;S)/m}$     <ref>     [Asadi et al., 2018]

$\ell(w, \cdot)$ $1$-Lipschitz    $\E_{\Pp_W}[\Wass(\Pp_S, \Pp_{S|W})]/\sqrt m$     <ref>     [Lopez and Jog, 2018]

$\nabla_w\ell(w, \cdot)$ $1$-Lipschitz    $\sum_k\varepsilon_{k-1}\E_{\Pp_W}[\Wass(\Pp_S, \Pp_{S|W_k})]/\sqrt m$     <ref>     New

$\ell(w, X)$ $1$-SG     $e^{1/e}\sqrt{p}(I^{(p)}(W;S)+1)^{1/p}/\sqrt{m(p-1)}$     <ref>     [Aminian et al., 2021]

$\nabla_w\ell(w, X)$ $1$-SG     $e^{1/e}\sqrt{p}\sum_k\varepsilon_{k-1}(I^{(p)}(W_k;S)+1)^{1/p}/\sqrt{m(p-1)}$     <ref>     New

$\V_{\Pp_X}[\ell(w, X)]\leq 1$     $\E_{\Pp_W}[\chi^2(\Pp_{S|W}\|\Pp_S)^{1/2}]/\sqrt m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\|\mathbb C_{\Pp_X}[\nabla_w\ell(w, X)]\|\leq 1$     $\sum_k\varepsilon_{k-1}\E_{\Pp_W}[\chi^2(\Pp_{S|W}\|\Pp_S)^{1/2}]/\sqrt m$     <ref>     New

$\ell(w, X)$ $1$-SG     $\sum_i\sqrt{2 I(W;X_i)}/m$     <ref>     [Bu et al., 2019]

$\nabla_w\ell(w, X)$ $1$-SG     $\sum_i\sum_k\varepsilon_{k-1}\sqrt{2 I(W_k;X_i)}/m$     <ref>     [Zhou et al., 2022]

$\ell(w, \cdot)$ $1$-Lipschitz    $\sum_i\E_{\Pp_W}[\Wass(\Pp_X, \Pp_{X_i|W})]/m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\nabla_w\ell(w, \cdot)$ $1$-Lipschitz    $\sum_i\sum_k\varepsilon_{k-1}\E_{\Pp_W}[\Wass(\Pp_X, \Pp_{X_i|W_k})]/ m$     <ref>     New

$\E_{\Pp_X}[|\ell(w, X)-\bar\ell(w)|^{p/(p-1)}]\leq1$     $\sum_i(I^{(p)}(W; X_i)+1)^{1/p}/m$     <ref>     New

$\E_{\Pp_X}[\|\nabla_w\ell(w, X_i)-\overline{\nabla_w\ell}(w)\|^{p/(p-1)}]\leq1$     $\sum_i\sum_k\varepsilon_{k-1}(I^{(p)}(W_k; X_i)+1)^{1/p}/ m$     <ref>     New

$\V_{\Pp_X}[\ell(w, X)]\leq 1$     $\sum_i\E_{\Pp_W}[\chi^2(\Pp_{X_i|W}\|\Pp_X)^{1/2}]/m$     <ref>     New

$\|\mathbb C_{\Pp_X}[\nabla_w\ell(w, X)]\|\leq1$     $\sum_i\sum_k\varepsilon_{k-1}\E_{\Pp_W}[\chi^2(\Pp_{X_i|W_k}\|\Pp_X)^{1/2}]/ m$     <ref>     New

$|\ell|\leq 1$     $\sum_i\E_{\Pp_W}[\TV(\Pp_X, \Pp_{X_i|W})]/m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\|\nabla_w\ell\|\leq 1$     $\sum_i\sum_k\varepsilon_{k-1}\E_{\Pp_W}[\TV(\Pp_X, \Pp_{X_i|W_k})]/m$     <ref>     New

$|\ell|\leq 1$     $\sum_i\sqrt{2 \LI(W;X_i)}/m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\|\nabla_w\ell\|\leq 1$     $\sum_i\sum_k\varepsilon_{k-1}\sqrt{2 \LI(W_k;X_i)}/m$     <ref>     New

$|\ell|\leq 1$     $2\sqrt{2 I(W;S|S^\star)/m}$     <ref>     [Steinke and Zakynthinou, 2020]

$\|\nabla_w\ell\|\leq 1$     $2\sum_k\varepsilon_{k-1}\sqrt{2 I(W_k;S|S^\star)/m}$     <ref>     New

$\ell(w, \cdot)$ $1$-Lipschitz    $\E_{\Pp_{W,S^\star}}[\Wass(\Pp_{S|S^\star}, \Pp_{S|W, S^\star}) + \dots\tablefootnote{Here and in the following, ``$\,\dots$'' should be read as: ``Take the same expression on the left and replace $\Pp_{S|W, S^\star}$ with $\Pp_{\bar S|W, S^\star}$ (or $\Pp_{X_i|W, X_i^\star}$ with $\Pp_{\bar X_i|W, X_i^\star}$).''.}]/\sqrt m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\nabla_w\ell(w, \cdot)$ $1$-Lipschitz    $\sum_k\varepsilon_{k-1}\E_{\Pp_{W,S^\star}}[\Wass(\Pp_{S|S^\star}, \Pp_{S|W_k, S^\star}) + \dots]/\sqrt m$     <ref>     New

$|\ell|\leq 1$     $2e^{1/e}\sqrt{p}(I^{(p)}(W;S|S^\star)+1)^{1/p}/\sqrt{m(p-1)}$     <ref>     New

$\|\nabla_w\ell\|\leq 1$     $2e^{1/e}\sqrt{p}\sum_k\varepsilon_{k-1}(I^{(p)}(W_k;S|S^\star)+1)^{1/p}/\sqrt{m(p-1)}$     <ref>     New

$|\ell|\leq 1$     $2\E_{\Pp_{W, S^\star}}[\chi^2(\Pp_{S|W, S^\star}\|\Pp_{S|S^\star})^{1/2}+\dots]/\sqrt m$     <ref>     New

$\|\nabla_w\ell\|\leq 1$     $2\sum_k\varepsilon_{k-1}\E_{\Pp_{W,S^\star}}[\chi^2(\Pp_{S|W, S^\star}\|\Pp_{S|S^\star})^{1/2}+\dots]/\sqrt m$     <ref>     New

$|\ell|\leq 1$     $2\sum_i\sqrt{2 I(W;X_i|X_i^\star)}/m$     <ref>     [Haghifam et al., 2020]

$\|\nabla_w\ell\|\leq 1$     $2\sum_i\sum_k\varepsilon_{k-1}\sqrt{2 I(W_k;X_i|X_i^\star)}/m$     <ref>     New

$\ell(w, \cdot)$ $1$-Lipschitz    $\sum_i\E_{\Pp_{W, X_i^\star}}[\Wass(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W, X_i^\star})+\dots]/m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\nabla_w\ell(w, \cdot)$ $1$-Lipschitz    $\sum_i\sum_k\varepsilon_{k-1}\E_{\Pp_{W, X_i^\star}}[\Wass(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W_k, X_i^\star})+\dots]/ m$     <ref>     New

$|\ell|\leq 1$     $2\sum_i(I^{(p)}(W; X_i|X_i^\star)+1)^{1/p}/m$     <ref>     New

$\|\nabla_w\ell\|\leq 1$     $2\sum_i\sum_k\varepsilon_{k-1}(I^{(p)}(W_k; X_i|X_i^\star)+1)^{1/p}/ m$     <ref>     New

$|\ell|\leq 1$     $\sum_i\E_{\Pp_{W, X_i^\star}}[\chi^2(\Pp_{X_i|W, X_i^\star}\|\Pp_{X_i|X_i^\star})^{1/2}+\dots]/m$     <ref>     New

$\|\nabla_w\ell\|\leq 1$     $\sum_i\sum_k\varepsilon_{k-1}\E_{\Pp_{W, X_i^\star}}[\chi^2(\Pp_{X_i|W_k, X_i^\star}\|\Pp_{X_i|X_i^\star})^{1/2}+\dots]/ m$     <ref>     New

$|\ell|\leq 1$     2$\sum_i\E_{\Pp_{W, X_i^\star}}[\TV(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W, X_i^\star})+\dots]/m$     <ref>     [Rodríguez-Gálvez et al., 2021]

$\|\nabla_w\ell\|\leq 1$     2$\sum_i\sum_k\varepsilon_{k-1}\E_{\Pp_{W, X_i^\star}}[\TV(\Pp_{X_i|X_i^\star}, \Pp_{X_i|W_k, X_i^\star})+\dots]/m$     <ref>     New

$|\ell|\leq 1$     $2\sum_i\sqrt{2 \LI(W;X_i|X_i^\star)}/m$     <ref>     New

$\|\nabla_w\ell\|\leq 1$     $2\sum_i\sum_k\varepsilon_{k-1}\sqrt{2 \LI(W_k;X_i|X_i^\star)}/m$     <ref>     New