Ballistic transport in nanodevices based on single-crystalline Cu thin films

Domain: chemistry
Published: 2026-03-07
Editorial outcome: major_revision

Abstract
In ballistic transport, the movement of charged carriers remains unimpeded by scattering
events. In this limit, microscopic parameters such as crystal momentum, spin and quantum
phase are well conserved, allowing electrons to maintain their quantum coherence over
longer distances. Nanoscale materials, such as carbon nanotubes, graphene, and nanowires,
can exhibit ballistic transport. However, their scalability in devices is significantly limited.
While deposited metal films offer scalability for nanodevices, their short electronic mean free
paths hinder ballistic transport. In this study, we investigated the electronic transport in
cross-geometry devices fabricated with 90-nm-thick Cu films without grain boundaries. We
demonstrated ballistic transport in devices with channel widths of 150 nm at temperatures
below 85 K via negative bend resistance measurements. Our findings establish a scalable
platform for exploring the intrinsic quantum mechanical properties of Cu, advancing both
the fundamental understanding of quantum transport in metals and its practical
applications in next-generation electronic quantum technologies.

Introduction
When the electronic mean free path exceeds the device size, electrons undergo ballistic transport
within the device, experiencing minimal scattering. Ballistic transport can reveal intrinsic
quantum properties of solid-state materials, as electrons' quantum information—such as crystal
momentum, spin, and quantum phase—is well conserved1-4. Research on quantum properties of
normal metals, including electronic band structure, Fermi surface topology, conductivity, and
magnetoresistance has accelerated in recent years5-12. Cu is used in electronic circuits owing to
its conductivity, reliability, and versatility. It remains crucial in the electronics industry, used for
high-speed data-processing cables and interconnects in advanced semiconductor devices.
Electronic band structure, Fermi surface topology and Fermi velocity of bulk Cu are well
understood through various measurements13-19. However, growing atomically flat and uniform
Cu thin films without grain boundaries has been technically challenging, and in-plane ballistic
transport in Cu thin film nanostructures has not been achieved, limiting the exploitation of their
intrinsic quantum properties. Although polycrystalline Cu thin films prepared by sputtering and
post-annealing have shown low room-temperature resistivities20 (2.0–2.2×10-8 Ω·m), ballistic
transport was not explicitly examined.

In this study, we demonstrated ballistic transport in nanoscale devices composed of atomically
flat single-crystalline Cu(111) thin films (SCCFs) (Supplementary Fig. 1) grown using the atomic
sputtering epitaxy (ASE) technique21. This method addresses challenges in conventional metal-
film deposition, such as surface oxidation and electrical performance degradation due to defects
such as GBs and impurities22. We observed negative bend resistance in cross-geometry devices,
indicating ballistic transport23. The electronic mean free path of the 80-nm-thick SCCF was 150
nm at a temperature below 85 K (Supplementary Fig. 2). Our SCCF enables studies on the quantum
nature of Cu thin films, including topological properties24-26 and phase-coherent quantum
interference27,28. It also suggests potential applications in quantum circuits, spintronic devices, and
Cu interconnect technology29-31.

Results
Ballistic transport in SCCFs

Using standard electron beam lithography and argon-ion etching, Hall bar-shaped devices with
nominal channel widths (W) of 10 μm, 1 μm, 250 nm, and 150 nm were fabricated with a thickness
(t) of ~90 nm (Figs. 1a,b and Supplementary Fig. 3). Here, W denotes the nominal channel width,
which may differ from the actual one. In all devices measured, the actual channel width deviated
by no more than ~10% from the designed width. Cross-sectional transmission electron microscopy
(TEM) images revealed perfectly aligned Cu atoms, with 2.07 Å spacings, in the [111] direction
of the thin film (Fig. 1b), indicating that the thin film retained its high crystalline quality after
fabrication (Supplementary Fig. 4). The bend resistance (R = V /I) was measured in the cross
B B
configuration (Fig. 1c), with current (I) injected from terminal 1 to terminal 5, and bend voltage
(V ) was measured between terminals 3 and 4. In a diffusive transport regime, where the electronic
B
mean free path (l ) is shorter than the device length, the van der Pauw formula R = (ρln2)/(tπ)
mfp B
for symmetric cross geometry can be applied. In this regime, R is expected to be positive32 as the
B
resistivity (ρ) is inherently positive. The temperature (T) dependence of ρ in diffusive metals is
predominantly influenced by electron–acoustic phonon scattering, which follows the Bloch–
Grüneisen formula (equation (1)) derived from the Boltzmann transport theory33,34.

𝑇 5 𝛩𝑅/𝑇 𝑥5
𝜌(𝑇) = 𝜌 +𝛼 ( ) ∫ 𝑑𝑥 (1)
0 𝑒𝑙−𝑝ℎ 𝛩 (𝑒𝑥 −1)(1−𝑒−𝑥)
𝑅 0

Here, ρ represents the residual resistivity due to temperature-independent scattering by defects
0
such as GBs and impurities, α represents the electron–phonon coupling strength, and Θ is
el-ph R
the Debye temperature. As shown in Fig. 1d, the temperature dependence of R for devices with
B
W = 10 and 1 μm can be well fitted35 with equation (1) using fitting parameters α = 8.8 × 10-8
el-ph
Ω·m and Θ = 270 K, indicative of the occurrence of diffusive transport over the entire
R
temperature range. The fitting parameters α and Θ were determined from the fitting analysis
el-ph R
(Supplementary Fig. 5). We also note that Hall measurements of a device with W = 10 μm
(Supplementary Fig. 6) indicate the presence of multiple carrier types at 1.7 K, but an electron-
like band provides the dominant contribution to the conductance (Supplementary Fig. 7 and
Supplementary Table 1). These results support the use of equation (1) as an effective single-
carrier Bloch–Grüneisen description of R (T). The sample used in this study is nearly single-
B
crystalline and would typically be expected to exhibit a Debye temperature close to or even
higher than that of polycrystalline Cu. However, the observed Debye temperature of 270 K is
slightly lower than the commonly reported bulk value (~330 K). We attribute this deviation to
the thin film geometry and the fabrication process. In particular, device fabrication steps such as
Ar ion etching and chemical treatments may introduce surface roughening, partial oxidation, or
other structural imperfections that alter the phonon spectrum and reduce the effective Debye
temperature. Similar reductions in Debye temperature due to dimensional confinement and
surface degradation have also been reported in sputtered Cu thin films36. Indeed, R
B
(T) of the
device with W = 10 μm remains positive and is well described by the Bloch–Grüneisen formula,
consistent with fully diffusive transport in a device W >> l . However, R (T) of devices with W
mfp B
≤ 250 nm, which is comparable to or shorter than l
mfp
, deviates from the value expected from the
Bloch–Grüneisen formula. For the device with W = 150 nm, R even shows negative values
B
below 85 K. Although decreasing W increases the contribution of edge scattering and thus
reduces l (Supplementary Fig. 8), l nonetheless exceeds W = 150 nm and ballistic transport
mfp mfp
persists. We also estimated the mean free path l , ≈ 42 nm using Drude model equation (2):
mfp Drude

𝑙 = 𝑚∗𝑣𝐹
(2)
mfp,Drude 𝑛𝑒2𝜌

Here, we used the measured resistivity 𝜌 = 1.03 × 10-8 Ω·m in the device of W = 150 nm and
the carrier density n = 1.683 × 1029 m-3 extracted from the Hall analysis at 1.7 K (Supplementary
Fig. 9). A Fermi velocity37 v = 1.57 × 106 m/s and an effective mass38 m* = 1.1843 × 10-30 kg
F
from the bulk Cu were used. l , is slightly different from the experimentally inferred l (~
mfp Drude mfp
150 nm at 1.7 K) from the observation of negative R in the device of W = 150 nm. The
B
difference may be attributed to the fact that the electric parameters of the 90-nm-thick, 150-nm-
wide Cu device can differ from those of bulk Cu.

Counter-intuitive negative bend resistance has been observed in high-mobility two-dimensional
(2D) electronic systems such as GaAs/AlGaAs heterostructures and graphene3,4,39 owing to
ballistic transport40, but not in deposited metallic films. In the ballistic regime, electrons from
terminal 5 can propagate to terminal 3 without scattering, as shown by the red arrow in Fig. 1c. As
electrons accumulate near terminal 3, a negative electrical potential builds up, decreasing R ,
B
which becomes negative23. These findings confirm that ballistic transport, previously hidden by
electron scattering at GBs, emerges in an SCCF when GBs are eliminated41. To verify that the
observed negative R is not due to a specific choice of current–voltage configuration or artifact,
B
we performed additional measurements across all contact configurations (Supplementary Fig. 10
and Supplementary Table 2). At high temperatures, the R of the device with W = 150 nm appears
B
slightly lower than that of the other devices with larger W. This difference may arise from
geometric imperfections introduced during fabrication, as supported by simulations in
Supplementary Fig. 11.

Structural defects in the SCCF and a polycrystalline Cu thin film (PCCF)

We analyzed the crystallographic microstructure of the SCCF, as high crystalline quality is
essential for ballistic transport. During thin-film growth, GBs and twin boundaries (TBs) form
owing to lattice mismatch with the substrate. While TBs are unavoidable, the formation of GBs
can be suppressed using a growth technique that considers extended atomic distance mismatch
(EADM)22. The distribution of GBs and TBs was analyzed for a polycrystalline Cu thin film
(PCCF; Figs. 2a–c), an SCCF with minimal GBs (Figs. 2d–f), and an SCCF without GBs (Figs.
2g–i) using misorientation line mapping, electron backscatter diffraction (EBSD), and SEM43,44.
GBs and TBs are denoted by blue and red lines, respectively, in Fig. 2. PCCF usually contains
both GBs and TBs (enlarged misorientation maps of the boxed areas in a, d, and g left panels are
shown in the right upper panels and those of the boxed areas in the upper panels are shown in the
right lower panels). As shown in Fig. 2a, the PCCF has many grains (a conventional PCCF is
expected to have trillions of grains on the two-inch wafer42 and each grain has a different
orientation, as illustrated by the numbered indicators in the right lower panel of Fig. 2a). The Cu
thin films grown by ASE show perfect alignment along the (111) plane (Fig. 2e) despite the
presence of some GBs (Fig. 2d). The GBs observed in the misorientation line map in Fig. 2d are
close to the TBs; thus, they do not appear as different grains in the EBSD map, as they deviate
slightly from the ideal TB condition of 60° (deviation of 1°–2°)45. These GBs can influence
electronic transport, and thus, we used only the SCCF without GBs in this study. A well-grown
SCCF (Fig. 2g) is devoid of GBs, contains only TBs, and exhibits two orientations I and II (Fig.
2i right lower panel). The crystallographic orientation of the SCCF follows either ABC… or ACB...
stacking, satisfying symmetry operation by 60° rotation (Supplementary Fig. 12). Although the
TB density fluctuates across areas, it remains within the same order of magnitude. Unlike GBs,
TBs minimally affect electrical resistivity because: 1) charged defects do not form around TBs,
resulting in negligible potential variation near TB46,47, and 2) well-matched Fermi surfaces of
adjacent grains on either side of a TB reduce electron wave scattering, as the Fermi surface of Cu
remains unchanged with 60° rotation about the (111) axis33,45,47-49.

GB dependence of electrical transport

Figure 3 shows the electronic transport behavior for different GB lengths. L , defined as the
GB
total length of line segments separating the grains in the misorientation line maps, was measured
over an area of 7.7 μm × 22.7 μm using the EBSD technique (Supplementary Fig. 12)43,44. More
GBs generally increase resistivity by acting as scattering sites for electrons. Thus, as the grain size
(d) decreases, i.e., as L increases, the resistivity contribution of the GBs increases. According to
GB
the Mayadas–Shatzkes (MS) model (equation (3)), which quantitatively describes electron
scattering at GBs50-52, the resistivity decreases as the grain size (d) increases, whereas it increases
with the probability of electron scattering at the GBs (S):

𝜌𝑔 = {3[ 1 − 1 𝛼+𝛼2 −𝛼3lnP(1+ 1 )]} −1 (3a)
𝜌𝑖 3 2 𝛼

𝛼 = 𝑙 mfpN𝑆
(3b)
𝑑 1−𝑆

Here, ρ represents the resistivity arising from electron scattering at the GBs, while ρ denotes the
g i
intrinsic resistivity from all other sources excluding GBs. As shown in Fig. 3a, L
GB
dependence of
the resistivity of the device with W = 1 μm at 4.3 K, which is in the diffusive regime, is well fitted
by the MS relationship 𝑑 ∝ (T𝐿 )−1 for a 2D system. Fig. 3b shows L dependence of R at 4.3
GB GB B
K normalized with that at 285 K for devices with W = 150 nm. As shown in Supplementary Fig.
13, for devices with W = 1 μm, the normalized R shows similar L dependence of resistivity as
B GB
the film is in a diffusive regime for all L values. By contrast, for devices with W = 250 nm
GB
(Supplementary Fig. 13) and 150 nm, the MS model cannot explain the normalized R for small
B
values of L , where l exceeds W, and R becomes negative.
GB mfp B

Geometrical effect on the transport measurement for SCCF

To further investigate the ballistic transport behavior under a magnetic field, we investigated
the magnetic field dependence of the bend resistance R (Fig. 4). We symmetrized R to suppress
B B
intermixed signals from slight asymmetries and misalignments between contacts. A magnetic field
B was applied normal to the SCCF, which was parallel to the (111) direction. As the magnitude of
|B| increased, the symmetrized bend resistance 𝑅s gradually increased, because electron
B
deflection due to the Lorentz force caused fewer electrons to reach the opposite terminal 3 (Fig.
4a)40. This ballistic transport behavior can be qualitatively explained using the Landauer–Büttiker
approach23. 𝑅s for a single channel can be expressed in terms of the forward mode-averaged
B
transmission probability P , and the mode-averaged transmission probabilities for turning left P
FW L
and turning right P , as shown in equation (4):
R

ℎ 𝑃 𝑃 −𝑃2
𝑅s = L R FW (4a)
B 𝑒2 𝐷

𝐷 = (𝑃 +𝑃 )[2𝑃 (𝑃 +𝑃 +𝑃 )+𝑃2 +𝑃2] (4b)
L R FW FW L R L R

For B = 0, P exceeds both P and P , and thus, (P P - P 2) becomes negative, leading to 𝑅s <
FW L R L R FW B
0. When a magnetic field is introduced, the ballistic electrons are deflected by the Lorentz force,
causing a reduction in P and an increase in either P or P , which in turn increases 𝑅s. As
FW L L R B
shown in Fig. 4b, the increase in 𝑅
s
with B is more pronounced in the ballistic regime (W = 150
B
nm) than in the diffusive regime (W = 1 μm) because in the ballistic regime, the electron trajectories
are more well-defined. The characteristic magnetic field B
0
at which the cyclotron radius r
c
=
ℏk /eB matches the channel width is estimated to be ~36 T for W = 250 nm. Here, k is the Fermi
F F
wavevector of bulk Cu53, k ≈ 1.36 × 1010 m−1. Consistent with the expectation for ballistic
F
transport54, we observe a monotonic increase of 𝑅s up to our maximum accessible magnetic field
B
of ~10 T that is smaller than B .
0

To characterize the charge carrier types for devices, we performed Hall measurements and plotted
antisymmetrized Hall resistivity 𝜌as for different W at 1.7 K as shown in Fig. 4c (see
xy
Supplementary Fig. 14 for 𝜌as measured at 120 K). The observed nonlinear Hall effect at devices
xy
with W = 10 and 1 µm agrees with the results of a recent study45, which revealed the co-existence
of holes and electrons in <205-nm-thick SCCFs without GBs. However, when the width is reduced
to ≤250 nm, the Hall effect becomes linear, which suggests that electrons become the major charge
carriers. To explain this phenomenon in 1D, we calculated and compared effective band structures
of a 2D Cu(111) thin film (Fig. 4d) along specific k-points with those of a 1D Cu(111) rod (Fig.
4e). In the 2D thin-film band structure, both electron and hole orbits are distinctly visible. In
contrast, the 1D structure's band structure appears more complex due to confined geometry in both
out-of-plane and in-plane directions.

The Fermi surface of the 2D thin film shows splitting of degenerate bands due to confined
geometry along the [111] direction (Fig. 4d, left panel). In the periodic zone scheme representation
(Fig. 4d, right panel), the spectral weight of hole bands is prominent45. The 1D structure's Fermi
surface is featureless, offering negligible insights. Thus, we calculated the effective Fermi surface
via band unfolding to 2D (Fig. 4e, left panel). In the unfolded periodic zone scheme (Fig. 4e, right
panel), the spectral density of hole bands is reduced, disconnecting the hole orbits and allowing
electron orbits to dominate. Although the calculation system size is smaller than the actual device,
it shows the trend of reduced hole orbits as geometry approaches 1D. This explains the
experimental observation that the two-carrier model-induced nonlinear Hall effect becomes linear
with decreasing W. For reference, Supplementary Fig. 15 compares the 3D, 2D, and 1D Fermi
surfaces at a given 𝑘 .
𝑧

In this study, we observed ballistic electronic transport in nanodevices based on Cu(111) thin
films and investigated its dependence on film quality. Hall-bar shaped devices were fabricated
with atomically flat 90-nm-thick SCCFs. We obtained negative bend resistance as direct evidence
of ballistic transport. In a perpendicular magnetic field, ballistic electrons are disturbed, increasing
bend resistance. We determined the distributions of GBs and TBs through EBSD measurement
and showed that the Cu thin-film device’s resistivity primarily depended on GBs, remaining
insensitive to TBs. Our experimental observations and theoretical calculations reveal that as the
Cu(111) thin film transitions from 2D to the 1D limit, hole orbits observed in 2D disappear because
of quantum confinement in both out-of-plane and in-plane directions, suggesting transition from
nonlinear Hall effect, driven by two carriers, to a linear Hall effect.

This paper presents a platform for investigating the intrinsic quantum mechanical properties of
Cu, with the potential to advance high-performance electronic55 and spintronic devices through
preservation of quantum information, including momentum, quantum phase, and spin. This
approach may help address critical reliability challenges in semiconductor technology, such as
Joule heating30 and electromigration31 in Cu interconnects29. Recent discoveries reveal that the
Fermi surface of Cu exhibits a topologically nontrivial genus24, opening the door to various
topology-related experiments in ballistic metals24-26.

Methods
Preparation of thin SCCFs using the ASE technique

The ASE method involves stacking atoms individually without forming clusters, which can
cause arbitrary deposition. To achieve this, we modified the conventional sputtering system. The
network of conducting wires, including cables, in the conventional sputtering system was upgraded
with single-crystalline Cu wires, fabricated by cutting single crystal Cu wafers in a spiral fashion
using wire electrical discharge machining (EDM). The wafers were sliced from a single-crystal
ingot grown by the Czochralski method. To minimize vibration from ambient noise, we
implemented a mechanical noise reduction system. While such vibrations may not significantly
affect conventional thin film growth, especially for PCCFs, they can cause irreversible stacking
faults that disrupt the initial nucleation and lateral growth processes, particularly the coherent
coplanar merging of nuclei. The thin film growth system creates a stable environment for single-
atom deposition, aiming to achieve atomically flat surfaces through precise stacking of single
atoms.

The optimized sputtering conditions were as follows:
• Substrate: A double-sided polished (001) Al 2 O 3 wafer with a thickness of 430 μm.
• Deposition temperature: Approximately 170 °C.
• RF power: 13.56 MHz at 30 W.
• Target-to-substrate distance: 95 mm.
• Base pressure: Maintained at less than 2 × 10−7 Torr.
• Working pressure: 5.4 × 10−3 Torr with an Ar gas (99.9999% (6N)) flow of 50 sccm.
The relationship between the deposition time and thickness of the thin film (or the average growth
rate) was determined from the average deposition time of a 200-nm-thick film grown under
optimal conditions.

Sample nanofabrication and transport measurement

To fabricate Hall bar devices using the prepared SCCFs, we partially removed sections of single-
crystalline Cu via electron beam lithography and Ar-ion milling, forming Hall bars with varying
linewidths (Supplementary Fig. 3). After the Ar-ion milling, Cu redeposition was observed along
the lithography resist sidewalls (Supplementary Fig. 16), but such regions may have only limited
contribution to conduction because they likely exhibit reduced crystallinity. The Cu atomic spacing
was 2.07 Å along the (111) direction, confirming that the pristine crystalline quality of the SCCF
was maintained even after patterning (Fig. 1b). Temperature dependence of the bend resistance
and magnetic field dependence of the Hall resistivity of the SCCF Hall bar devices were measured
in Oxford Instrument Teslatron PT with a base temperature of 1.7 K connected with low-pass RC
filters. GB dependence of bend resistance, magnetic field dependence of bend resistance and
longitudinal resistance of SCCF Hall bar devices were measured in Oxford Instrument Heliox with
a base temperature of 4.3 K connected with low-pass RC filters. Bend and longitudinal resistance
were measured by the DC IV-sweep method by measuring the voltage drop with a Keithley 2000
voltmeter, and the current was applied to 200 nA through Yokogawa GS610 source measure unit
with a load resistance of 100 kΩ. Hall resistance was measured with Delta measurement method
by measuring the voltage drop with low-noise Keithley 2182 nanovoltmeter, and current with 1
mA was applied through low-noise Keithley 6221 current source.

Structural information of Hall bar pattern

To investigate the cross-sectional structure of ballistic transport in the SCCF, the annular dark-
field (ADF) imaging mode of aberration-corrected scanning transmission electron microscopy
(STEM; JEM-ARM200CF, JEOL) at 300 kV was employed. The angle range of the ADF detector
was set to 45–175 mrad, and the semiconvergence angle of the condenser lens was ≈24 mrad. In
combination with STEM imaging, elemental mapping of the Cu films was performed in the same
STEM image mode using electron energy loss spectroscopy (EELS; Quantum ER965, Gatan).
Cross-sectional TEM sampling was conducted using the Ga-ion milling and slicing method in a
focused-ion-beam scanning electron microscope (SEM; Helios NanoLab 450, Thermo Fisher
Scientific). Low-energy Ar-ion beam milling at 700 V for 10 min was sequentially performed as a
post-surface treatment to remove the damaged surface layer that usually forms during heavy Ga-
ion beam milling.

Theoretical calculations

The total energy and electronic band structure calculations were performed using the first-
principles density functional theory56 as implemented by Kresse and Joubert57 using the projector
augmented-wave method58. The exchange‒correlation functional was modeled using the
generalized gradient approximation in the Perdew–Burke–Ernzerhof form59. All the calculations
were spin-polarized, and the positions of the atoms, as well as the size and shape of the unit cell,
were fully relaxed to obtain the optimized lattice structure. All the atoms of bulk Cu were fully
relaxed until the force on the atom was less than 0.001 eV/Å, and the change in total energy was
less than 10-6 eV. The electron wavefunctions were expanded using a plane-wave basis set with a
cut-off energy of 400 eV for the bulk, slab, and rod calculations. A 12-ML (ML: monolayer)
Cu(111) slab structure was used to simulate the 2D-limit thin film, and a 6-ML×6-ML rod
structure was used for the 1D-limit nanorod. The choice of a 12-ML Cu(111) slab is supported by
thickness-convergence tests on 6-, 9-, and 12-ML slabs (Supplementary Fig. 17). We maintained
a 20-Å-thick vacuum layer to prevent interactions between the periodic images.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "• What are the noteworthy results?\nThe authors report the growth of high-quality Copper thin films (characterized via TEM and detailed electron backscatter\ndiffraction (EBSD) maps) on which electron transport measurements, especially bend resistance and Hall resistivity, support\nthe presence of ballistic transport over length scales of the order of 100 nm.\n• Will the work be of significance to the field and related fields? How does it compare to the established literature? If the work\nis not original, please provide relevant references.\nIf the authors can convincingly make the case that their transport data is conclusive and firmly confirm ballistic electron\ntransport in their Cu thin films, it would be quite a significant achievement. Ballistic copper thin films can have applications in\nexisting integrated circuit technologies and can find future application in quantum circuits.\nThe growth methods and related characterization methods of very similar Cu films were previously published by some of the\nsame authors (see Ref. 18, Kim. S. J., Nature 603, 434 (2022)). What is new here is that these films and their detailed\ncharacterization is combined with detailed charge transport measurements. Some of the authors have extensive experience\nwith such measurements (e.g. H. -W. Jeong et al., Nano Lett. 2024, 24, 15950 (2024)). To the best of my knowledge there is\nno previous work with this detailed connection between Cu thin film morphology and charge transport measurements (at\nvarious temperatures) showing evidence for ballistic transport over such a long length scale (> 100 nm).\n• Does the work support the conclusions and claims, or is additional evidence needed?\nTo the best of my assessment, the work partially supports the claims, and I suggest additional data and/or clarifications\nwhich would strengthen the claims. In the next sections I also suggest some clarifications regarding the data analysis.\n(1) I would respectfully request that you please state clearly in the main manuscript the number of devices which were\nstudied for each set of measurements. Please be specific, how many W=150 nm devices were fabricated? How many of\nthese were characterized using EBSD? And using charge transport? (do the same for all W’s reported). Most importantly:\nplease clearly state if the EBSD data reported in Figs. 2 and 3 are from the same device as transport data in Figs. 3 and 4?\nThe strength of the argument is modified significantly if the data are from different devices or the same devices. If the data\nare from the same devices, please identify clearly which data points in Fig. 3 and 4 correspond to the Grain Boundary (GB)\nmaps in Figs. 2 and 3.\n(2) Please give more information, and show more raw data in the supplemental information, about the charge transport\nmeasurements. Specifically, please show how the bend resistance, RB, is modified when you use various set of contact\nterminals for the current injection and VB measurements. Show the results for all contact configurations in the supplemental\ninformation, for each device showing ballistic transport.\n(3) While I found the discussion of Figure 4 interesting, the data reported only marginally supports the discussion. Indeed,\nthe expected non-linearity (bump) in RB is not visible in Fig. 4b, and its presence in rho_xy in Fig. 4e is speculative at best.\nMoreover, there is no discussion of the magnetic field value Bo at which you expect the cyclotron orbit radius to match the\nwidth of the Hall bar (expected B-field location of the non-linearity). The experimental data in this section does not support\nwell the discussion. The discussion and/or data (how many samples have been measured? Do we have the EBSD data for\nthese?) should be modified to better support each other, or the Figure should be redesigned. I also mention that the cartoon\nof the Hall bar used in Fig. 4 does not match well the actual aspect ratio of the W=150 and 250 nm devices which are at the\ncenter of the discussion. Personally, I find that the data in Fig 4b is more interesting and convincing than the one in Fig. 4e,\nso I would suggest to perhaps recenter the discussion on this.\n• Are there any flaws in the data analysis, interpretation and conclusions? Do these prohibit publication or require revision?\nThe bend resistance and Hall resistivity measurements are very likely to be sensitive to the choice of contacts in the\nnarrowest Hall bars because of edge confinement and the asymmetric shape of the devices. Given the very rough electron-\nbeam lithography for the narrowest Hall bars (poor homogeneity of the thickness and width), it is essential to convince the\nreader that the measurements all agree and point to a single conclusion.\nAdditionally, the data analysis of transport data is based on the van der Pauw method. This method works in the limit of\n“point-like” contacts on the outer edge of the sample. Here the contacts are sometimes even larger than the device itself.\nIt should be clarified what are the error bars (systematic) stemming from using the van der Pauw method in these devices (the\nerror bars should be larger in the narrower devices from several reasons including the non-uniform thickness).\nIn my opinion, the main evidence for the ballistic transport data claim is in Fig. 3b and supplementary Fig. 6a. However, the\nsignal to noise ratio for the claim is close 1, meaning that the scatter of the data point in these two Figures is comparable to\nthe magnitude of the negative bend resistance. If this could be strengthened by addressing the above (showing if the RB\ndata are from multiple contact configurations and from devices whose EBSD maps are reported), then I think that the\nconclusion could be stronger.\nI do command the authors for this work. This is an incredibly complex set of sample preparation and challenging transport\nmeasurements (given the very low resistivity of copper – which gives little signal to noise in transport). This is unique work. I\ncould be convinced of the validity of their claim, but I would need more evidence that the negative bend resistance is\nreproducible and quantitative accurate – and in multiple devices.\n• Is the methodology sound? Does the work meet the expected standards in your field?\nYes, the authors are accomplished and barring the above comments, the work is of very high quality.\n• Is there enough detail provided in the methods for the work to be reproduced?\nYes, except for the points mentioned above.\nReferences\nI found that unfortunately the bibliography is inadequate. Here are a few things which could improve it in my view.\n(i) In the introduction, please add several recent references showing the ongoing development in the field. For example, on\nline 55 the authors say “in recent years 5-9” but Refs. 5-9 are not recent at all.\n(ii) In multiple places no references are given for items which clearly demand one. For example, on line 71 when listing key\nideas to be explored in ballistic copper films, topics such as “the quantum Hall effect”, “quantum confinement”, and “phase-\ncoherent interference” are all mentioned without any reference. I also would ask that you please delete the mention of\n“hydrodynamic electron transport22”. This is a strange point the make because this hydrodynamic regime is reported in\ngraphene which has vastly different electron-electron and electron-phonon scattering lengths than copper, there is no reason\n(except if you discuss it) to expect any hydrodynamic regime in your devices.\n(iii) Several more (and recent) references related to the transport measurements would be appreciated. There are other ways\nto assess if a device is ballistic (e.g. quantum interferences, or conductance quantization), but none of this is mentioned.", "reviewer_2": "The paper discusses ballistic electron transport in 90-nm-thick films made of high-quality Cu. The copper films are grown\nusing the atomic sputtering epitaxy technique, which allows obtaining a very large grain structure with atomically smooth\ninterfaces. An extensive analysis has been performed to confirm grain boundary-free regions on the Cu films over micron-\nsized areas. Transport measurements in the magnetic field of a cross geometry of different sizes support the claim of the\nballistic propagation of electrons at low temperatures.\nThe reported results raise several issues, which raise concerns about the validity of the results and analysis. These issues\nare mainly relevant to transport measurements, but not limited to them. Below, I give a list of the problems that I have found.\n1. The paper does not seem to discuss the available literature on the current state of the problem. At least one paper claims\nthat polycrystalline Cu at room temperature shows better resistivity than that reported in the manuscript. [1]\n2. SFig. 4 shows a 250 nm wide sample. However, I do not understand how the sizes were calculated, as I tried to\ndemonstrate in the image below. It is also unclear what the origin of the Cu regions, highlighted in yellow ellipses in (c), is. If\nit were a standard lithography and lift-off process, I would say these are left over after a lift-off process. The Methods have not\nprovided details on the fabrication method besides lines 397-399. The SEM image in SFig. 3 is blurred, as is evident from a\npoorly focused image, as seen by a substantial difference (presence and absence of small features) on SFig. 3a and SFig.\n3b. I could say that the size determined by SEM is incorrect, as HAADF-STEM achieved subatomic resolution and should be\nmore accurate. Anyway, the image also shows that the profile of the device is not a simple rectangle; moreover, it has\nparasitic parallel (presumably conducting) channels due to Cu left-overs.\n3. Fig.1d shows bend resistance as a function of temperature. I think the experimental results are too close to zero\ntemperature, even for the minimum temperature of T=1.7 K. The behaviour of the 250 nm sample looks very good due to the\nagreement between high-temperature resistances. This can be viewed as perfect uniformity of the grown samples; however,\nthe lower resistance of the narrowest sample is very surprising at high temperatures. This difference is not discussed in the\ntext. The line that separates diffusing and ballistic transport regimes is misleading, as some electrons can propagate\nballistically even in a diffusive regime. When a significant part of the electrons are ballistic, the bend resistance should\ndeviate from the expected behaviour obtained from the theory for the wide samples. the 250 nm wide sample demonstrates a\nclear ballistic transport as it differs from the theory.\n4. αel-ph should have units. The Debye temperature obtained from the fit, 270K, differs from that found in the literature,\nѲR=330K. This can be related to the thickness or grain boundaries or something else. There is no discussion on the topic.\nThe Bloch–Gruneisen formula for (line 103) has a missing power of 5 in the T/ѲR prefactor. Maybe this is why I could not\nreproduce the fitting parameters reported. My graph below is given for the αel-ph=7e-6 Ωcm and ѲR=270K. The Python\ncode is provided.\n5. The authors should provide the parameters they used for calculating the mean free path in formula (2), including the\neffective mass of the electrons in copper, the Fermi velocity, and how they calculated the concentration.\n6. Fig. 4b has different bend resistances at zero magnetic field compared with Fig. 1d. Does the system become more\nballistic between the measurements? It would be good if there were more than one sample using a standard van der Pauw\n(with 90 degrees rotation and inversion of the magnetic field), as the authors used DC, which is sensitive to any DC bias due\nto the Seebeck effect. The ballistic transport complicated it due to the high sensitivity of scattering from the edges. Thus, it is\nhelpful to show that the negative bend resistance is observed in all realisations of the van der Pauw method [2].\n7. The quantum size effect arises when the film's thickness is comparable to the de Broglie wavelength of the electrons in\ncopper. Thus, this effect is highly unlikely to influence the Hall effect [3]. The thickness of 90 nm corresponds to >400 unit\ncells. Independent of the quantization, the high current used in the measurements, 1mA, suggests testing the heating effects\nat low temperatures. Also, the profile of the copper conductor is not rectangular, as can be seen in SFig. 4c, and obviously,\nthis asymmetry is more substantial for narrower samples. There is no such problem in 2D samples, but for 3D wires, the Hall\nvoltage is not the same on the side walls and can generate a classical magnetoresistance in a perpendicular magnetic field.\n8. The main problem of the reported results is massive degradation of mobility (by a factor of 50) in copper compared to\npolycrystal bulk copper, as demonstrated in SFig. 2. Thus, it is unclear why this result could be reported as a novelty. I do\nnot see any simple correlation between the absence of GB and “high” quality of such samples. Probably it is related to scattering\non the edges. In the paper [1], the resistivity of thin films is better at room temperature, which is most relevant to applications.\n9. If the primary source of scattering is related to scattering from the edges, then the narrow samples should have a lower\nmean free path. This is possible if we compare the mean free path of 170 nm with the reported widths of the devices.\nPlease note that the samples of 250 nm and 150 nm wide show ballistic transport; however, it is hard to compare where the\nballistic transport is larger. More negative signals do not mean more ballistic transport. More resistive samples with the same\nmobility give more negative bend resistance. You can see it directly from the formula (4). Assume r0 = h/4e, PL=PR=PLR << PFW, that means most of the electrons are going forward. Then RB≈-r0/PLR. If PLR is larger (for wider samples, assuming\nscattering on the edges), then the negative bend resistance is smaller. In the case PLR=PFW, RB≈0. This means you have\na few scattering events, which means the mean free path is comparable to the size of the sample.\n10. Please check the formula (3), as the reference gives a different interpretation and power of -1. There is also a recent\napplication to gold [4].\nReferences\n[1] A. Mallikarjunan, S. Sharma and S. P. Murarka “Resistivity of Copper Films at Thicknesses Near the Mean Free Path of\nElectrons in Copper Minimization of the Diffuse Scattering in Copper” Electrochem. Solid-State Lett. 3, 437 (2000)\nhttps://iopscience.iop.org/article/10.1149/1.1391172/pdf\n[2] van der Pauw, L.J. \"A method of measuring specific resistivity and Hall effect of discs of arbitrary shape\" (PDF). Philips\nResearch Reports. 13, 1–9 (1958).\nhttps://pearl-hifi.com/06_Lit_Archive/02_PEARL_Arch/Vol_16/Sec_53/Philips_Rsrch_Reports_1946_thru_1977/Philips\nResearch Reports-13-1958.pdf\n[3] G. Govindaraj and V. Devanathan Quantum size effect in thin metal films Phys. Rev. B 34, 5904 (1986).\nhttps://doi.org/10.1103/PhysRevB.34.5904\n[4] Weigang Ma, Xing Zhang and Koji Takahashi \"Electrical properties and reduced Debye temperature of polycrystalline\nthin gold films\" J. Phys. D: Appl. Phys. 43 (2010).\nhttps://iopscience.iop.org/article/10.1088/0022-3727/43/46/465301", "reviewer_3": "Referee Report\nI have reviewed the manuscript with interest. The authors report on ballistic transport in high-quality copper films and present\na systematic study of the impact of grain boundaries on ballistic conduction. These are timely and relevant findings that\ncould, in principle, merit publication. However, the presentation and writing currently do not meet the standards expected for\nNature Communications. Below I outline my specific comments.\nMajor Comments\n1. Debye Temperature\nThe extracted Debye temperature is significantly lower than the commonly reported value for copper (~343 K). The authors\nshould explain this discrepancy. Is this due to material quality, finite-size effects, or the fitting method used? Clarifying this\npoint is important to substantiate the validity of the extracted phonon parameters.\n2. Estimation of Mean Free Path\nCan the authors extract the Hall mobility from their data (e.g., via a multi-band model fit)? If so, they could estimate the\nelectronic mean free path independently. Such a value would provide a useful consistency check for their claims of ballistic\ntransport.\n3. Reversibility of Bend Resistance Measurements\nHave the authors verified the reversibility of their bend resistance experiments—for instance, by reversing current direction\nor interchanging current and voltage probes? Without such tests, it is difficult to rule out geometric effects or other\nmeasurement artifacts as the origin of the observed negative bend resistance.\n4. Figure Quality\nThe quality of several figures needs improvement. In particular, font sizes are too small in Fig. 4a and in the tables of Fig. 2c,\nf, and i, making them difficult to read. Figures must be legible when printed and should meet the journal’s formatting\nstandards.\n5. References and Justification\nSome references appear unrelated to the claims in the main text. For example:\no Line 73: None of the cited papers mention ballistic transport. If these are meant to motivate the use of ballistic copper\ninterconnects, the rationale must be explicitly stated and supported by appropriate references.\n________________________________________\nMinor Comments\nMain Text:\n• L29: We believe that collisions with the sample’s edge cannot be neglected and are most likely still dominating. We\nrecommend rephrasing the statement.\n• L71: The absence of defect-induced scattering alone does not justify the emergence of hydrodynamic transport. In metals,\nthe high carrier density screens the Coulomb interactions needed for this regime. If the authors believe hydrodynamics might\nappear in these SCCF Cu films, they must provide strong motivation and suitable references.\n• L79 (Fig. 1a): Please explain what the red dashed line represents.\n• L107: A reference should be provided to support the reported electron-phonon coupling values, ideally comparing with\neither ab initio or experimental data from the literature.\n• L113: Define all symbols clearly. Specify the values used for Fermi velocity and charge carrier density.\n• L121: A reference should support the statement made here.\n• L194: Clarify whether \"ρ_I\" refers to the total resistivity, including the grain boundary contribution. If so, consider\nreformulating the sentence for precision.\nSupplementary Information:\n• L33: The term \"actual\" is vague—please clarify what is meant.\n• L43: There appears to be a mismatch in line color between the figure and its legend—this should be corrected."}, "author_response": "We deeply appreciate all the reviewers for reviewing our manuscript and providing valuable\nsuggestions and advice to strengthen the manuscript. We have revised our manuscript\naccording to the reviewer’s comments and addressed the issues raised by the reviewers\npoint-by-point in this response letter. Reviewers’ comments are written in black, authors’\nresponses are written in blue, and the changes made in the manuscript are written in red.\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\n• What are the noteworthy results?\nReviewer: The authors report the growth of high-quality Copper thin films (characterized\nvia TEM and detailed electron backscatter diffraction (EBSD) maps) on which electron\ntransport measurements, especially bend resistance and Hall resistivity, support the\npresence of ballistic transport over length scales of the order of 100 nm.\n• Will the work be of significance to the field and related fields? How does it compare to\nthe established literature? If the work is not original, please provide relevant references.\nReviewer: If the authors can convincingly make the case that their transport data is\nconclusive and firmly confirm ballistic electron transport in their Cu thin films, it would be\nquite a significant achievement. Ballistic copper thin films can have applications in existing\nintegrated circuit technologies and can find future application in quantum circuits.\nThe growth methods and related characterization methods of very similar Cu films were\npreviously published by some of the same authors (see Ref. 18, Kim. S. J., Nature 603, 434\n(2022)). What is new here is that these films and their detailed characterization is combined\nwith detailed charge transport measurements. Some of the authors have extensive\nexperience with such measurements (e.g. H. -W. Jeong et al., Nano Lett. 2024, 24, 15950\n(2024)). To the best of my knowledge there is no previous work with this detailed\nconnection between Cu thin film morphology and charge transport measurements (at\nvarious temperatures) showing evidence for ballistic transport over such a long length\nscale (> 100 nm).\nResponse: We would like to express our sincere gratitude for the reviewer’s insightful\nevaluation, valuable comments, and encouraging remarks on this manuscript.\n• Does the work support the conclusions and claims, or is additional evidence needed?\nReviewer: To the best of my assessment, the work partially supports the claims, and I\nsuggest additional data and/or clarifications which would strengthen the claims. In the\nnext sections I also suggest some clarifications regarding the data analysis.\nResponse: We sincerely appreciate the reviewer’s positive evaluation and the suggestions\nfor potential improvements. We will carefully address all the comments and points\nmentioned below to ensure they are fully incorporated.\nReviewer: (1) I would respectfully request that you please state clearly in the main\nmanuscript the number of devices which were studied for each set of measurements.\nPlease be specific, how many W=150 nm devices were fabricated? How many of these\nwere characterized using EBSD? And using charge transport? (do the same for all W’s\nreported). Most importantly: please clearly state if the EBSD data reported in Figs. 2 and 3\nare from the same device as transport data in Figs. 3 and 4? The strength of the argument\nis modified significantly if the data are from different devices or the same devices. If the\ndata are from the same devices, please identify clearly which data points in Fig. 3 and 4\ncorrespond to the Grain Boundary (GB) maps in Figs. 2 and 3.\nResponse: Here, W denotes the nominal (designed) channel width, which may differ from\nthe actual one. In all devices measured, the actual channel width deviated by no more\nthan ~10 % from the designed width. A total of 25 single-crystalline Cu(111) films were\nfabricated into devices for this study. In total, 76 Hall bar devices were prepared: 1 device\nwith W = 10 μm (for comparison with the 1 μm device in Fig. 1d), 25 devices with W = 1\nμm, 25 devices with W = 250 nm, and 25 devices with W = 150 nm. All samples from the\n25 films were measured for their electrical transport properties in Fig. 3b, while a few films\nout of 25 were measured for their electrical transport properties in Figs. 1 and 4. In our\nexperimental procedure, the samples were first grown and then characterized by EBSD.\nDevices were subsequently fabricated from these EBSD-characterized samples, meaning\nthat 100% of the samples used for device fabrication were measured by EBSD in advance.\nRegarding the reviewer’s question, “Most importantly: please clearly state if the EBSD data\nreported in Figs. 2 and 3 are from the same device as transport data in Figs. 3 and 4?”,\neach point in Fig. 3b corresponds to a single film, and each device was fabricated either\non a different film or on a different location of the same film. Therefore, the misorientation\nline map from EBSD shown in Fig. 3b represents only one example among the samples in\neach category, and not all samples in the same category have the same misorientation\nmap. Nevertheless, the samples classified in the same category have comparable GB\ndensities and exhibit similar transport behaviour, as described.\nMore specifically, as the reviewer requested, the three misorientation line maps presented\nin the lower panel of Fig. 3b correspond to the representative films located around -9\nlog(L /m), -5 log(L /m), and -3 log(L /m) in the graph, respectively.\nGB GB GB\nWe have added the words 'nominal' in the captions of Fig. 1a, Fig. 3a and Fig. 4a.\n“Fig. 1a, Scanning electron microscope (SEM) image of a Hall bar device with a nominal\nchannel width (W) of 150 nm.'\nWe have added the word \"nominal\" at page 5, line 91 in the main text.\n\"Using standard electron beam lithography and argon-ion etching, Hall bar-shaped devices\nwith nominal channel widths (W) of 10 μm, 1 μm, 250 nm, and 150 nm were fabricated\nwith a thickness (t) of ~90 nm (Figs. 1a,b and Supplementary Fig. 3).\"\nWe have added the following sentence at page 5, line 92-94 in the main text to explain\nnominal (designed) channel width W.\n“Here, W denotes the nominal channel width, which may differ from the actual one. In all\ndevices measured, the actual channel width deviated by no more than ~10 % from the\ndesigned width.”\nWe have added the words 'nominal (designed)' in the caption of Supplementary Fig. 3a.\n“SEM image of an SCCF Hall bar device with a nominal channel width W = 1 μm.”\nWe have added the words 'nominal (designed)' in the caption of Supplementary Fig. 4a.\n“SEM image of an SCCF Hall bar device with a nominal channel width (W) of 250 nm.”\nReviewer: (2) Please give more information, and show more raw data in the supplemental\ninformation, about the charge transport measurements. Specifically, please show how the\nbend resistance, R, is modified when you use various set of contact terminals for the\nB\ncurrent injection and V measurements. Show the results for all contact configurations in\nB\nthe supplemental information, for each device showing ballistic transport.\nResponse: We thank the reviewer for the constructive comment. To directly address how\nthe measured bend resistance (R) depends on the choice of current and voltage terminals,\nB\nwe performed systematic measurements using both DC and Delta-mode techniques. For\neach representative Cu cross device (nominal width of 150 nm) exhibiting signatures of\nballistic transport at 4.3 K, we carried out DC current-voltage (IV) sweeps across eight\ndistinct contact configurations, covering both forward and reverse arrangements of current\nand voltage terminals (Table R1-1).\nConfiguration Current leads Voltage leads\n# (I⁺ → I⁻) (V⁺ → V⁻)\nTable R1-1 | Newly added Supplementary Table 1. Contact terminal combinations for\nbend resistance measurements. Current (I⁺ → I⁻) and voltage (V⁺ – V⁻) terminal\nconfigurations used in the eight measurement setups shown in Figs. R1-1c-e. Terminals\nare labelled 1 to 4 in counterclockwise order starting from the left.\nFig. R1-1 | Newly added Supplementary Fig. 7. a, The device marked with a red arrow\nin Fig. 3b was used for the measurements shown in b–e. b, Scanning electron microscope\n(SEM) image of a representative device from a. c, Measured bend resistance (R) at 285 K\nB\nvia Delta-mode measurement across 8 different configurations in Table R1-1. d, Measured\nR at 4.3 K via DC measurement across 8 different configurations, including forward and\nB\nreverse arrangements of current and voltage terminals. e, Measured R at 4.3 K via Delta-\nB\nmode measurement across 8 different configurations.\nWe present here an additional result among the dataset shown in Fig. 3b (Fig. R1-1a),\nwhich was not presented in the main manuscript. Cu film thickness of this device is 80 nm,\nand its SEM image is shown in Fig. R1-1b.\nEach R value was extracted from the slope of the IV curve measured between -300 μA\nB\nand +300 μA. As shown in Fig. R1-1d, all eight configurations yielded consistently negative\nR values, though the magnitude varied. This indicates that the observed negative R is\nB B\nnot tied to a particular terminal arrangement but is a robust feature of transport in these\nnanoscale devices.\nTo further confirm the reliability of these measurements, we also performed Delta-mode\nmeasurements on eight configurations at 4.3 K (Fig. R1-1e). This method alternates the\nbias current between +300 μA and -300 μA with a delay time of 300 ms, reducing low-\nfrequency noise and improving precision for low-resistance signals. The results closely\nmatch the DC data in both sign and magnitude, reinforcing that the negative R\nB\nobserved is intrinsic and not a measurement artifact.\nWe then extended the Delta-mode measurements to room temperature (285 K) for the\nsame configurations (Fig. R1-1c). Although the extracted R values showed increased\nB\nvariation up to ±20 mΩ, the overall magnitude remained low, and no sign inversion was\nobserved. This variation is consistent with realistic lithography-induced asymmetries of\ngeometry, such as terminal misalignment depicted in Fig. R1-2a. To support this, we\nconducted finite element method simulations using commercial package (COMSOL\nMultiphysics®) with introducing vertical misalignments of up to 30 nm between opposing\nterminals.\nFig. R1-2 | Newly added Supplementary Fig. 8. a, Schematic of the Cu cross geometry\nused in the simulations (width 150 nm, thickness 90 nm), where the vertical misalignment\nbetween opposing terminals was systematically varied. b, Simulated R as a function of\nB\nvertical misalignment (-30 nm to +30 nm) for device widths of 150 nm and 250 nm.\nAs shown in Fig. R1-2b, even a small misalignment of 10 nm can induce R changes of\nB\n~10 mΩ, which is quantitatively consistent with the experimental spread.\nIn summary, the negative R observed at low temperature is reproducible across all\nB\nconfigurations and two measurement techniques. The configuration-dependent variation\nin R is well explained by plausible fabrication-induced asymmetries, and does not\nB\nundermine the intrinsic ballistic nature of electron transport in our Cu thin films.\nWe have added the following sentence at page 6, line 138-144 in the main text to explain\nthe additional experiments showing that negative R is consistently observed across\nB\ndifferent configuration.\n“To verify that the observed negative R is not due to a specific choice of current–voltage\nB\nconfiguration or artifact, we performed additional measurements across all contact\nconfigurations (Supplementary Fig. 7 and Supplementary Table 1).”\nWe have added these Figs. R1-1, R1-2 and Table R1-1 to the Supplementary Fig. 7,8,\nSupplementary Table 1, and corresponding Supplementary note in revised Supplementary\ninformation (page 8-11).\nReviewer: (3) While I found the discussion of Figure 4 interesting, the data reported only\nmarginally supports the discussion. Indeed, the expected non-linearity (bump) in R is not\nB\nvisible in Fig. 4b, and its presence in rho_xy in Fig. 4e is speculative at best.\nResponse: We thank the reviewer for raising this point. As the reviewer pointed out the\nnonlinear feature (“bump”) is not visible in Fig. 4b. This is expected, as Fig. 4b presents the\nbend resistance (R) of a device, which predominantly reflects the longitudinal resistivity\nB\n(ρ) in diffusive regime (device with W=1 μm) due to its geometric configuration. In such\nxx\ncases, R shows a parabolic dependence on magnetic field arising from cyclotron motion\nB\nof charge carriers under the Lorentz force [5]. In contrast, in the ballistic regime (device\nwith W=150 nm), the increase in R (equivalently, suppression of negative R) arises\nB B\nbecause more ballistic electrons present at B = 0 are redirected to terminal 4 as the\nmagnetic field increases, consistent with the Landauer–Büttiker formalism [44]. Unlike the\nHall resistivity (ρ) shown in Fig. 4c (Fig. 4e in original manuscript), R in this regime is not\nxy B\nsensitive to multiband effects such as electron–hole coexistence, which underlie the\nnonlinearity discussed in our previous studies.\nIn contrast, Fig. 4c presents Hall resistivity data in 80 nm-thick Cu films, where quasi-2D\ntransport characteristics begin to emerge. Nonlinear behavior in ρ, attributed to the\nxy\ncoexistence of electrons and holes, has been consistently observed in dozens of devices\nwith thicknesses below ~80 nm, as reported in our previous work [Ref. 49: Ok, J. M. et al.,\nAdv. Mater. 36, 2403783 (2024)]. As shown in Fig. R1-3, this nonlinear Hall response is\nmost pronounced in 40 nm-thick Cu(111) single-crystal films, but begins to appear near\n200 nm thickness. In strong magnetic fields, the 200 nm and 40 nm samples clearly exhibit\nelectron- and hole-dominated behaviour, respectively, while the 80 nm sample lies at the\ncrossover between 3D and 2D regimes, exhibiting a field-dependent inflection point. This\nbehaviour reflects a balance between electron and hole contributions that evolves with\nfield strength and film thickness.\nWhile the nonlinear feature in ρ appears subtle in Fig. 4c, it remains qualitatively\nxy\nconsistent with prior systematic studies. R and ρ reflect different physical responses: R\nB xy B\nis a nonlocal resistance sensitive to geometric current path and ballisticity, whereas ρ\nxy\nmore directly probes the effective carrier type. Therefore, nonlinearity in ρ due to\nxy\nmultiband effects may not manifest in R in Fig. 4b.\nB\nWe have clarified these points in the revised manuscript to better distinguish the regimes\nand measurement sensitivities relevant to Fig. 4.\nFig. R1-3 | (Fig. 2 of Ref. 34) Fermi surfaces and Hall effects in 3D and 2D Cu. a,\nCalculated Fermi surface contour of Cu in 3D limit with open necks only along three\ndirections of {111}. b, All curves measured for the single-crystal samples thicker than 1 μm\nare linear and show almost no temperature dependence meaning for carriers definitely to\nbe electrons in 3D regime. c, Hall effect of polycrystalline Cu (PCCF) with a thickness of\n82nm. d–h) measured nonlinear Hall effects of SCCFs with thicknesses of d, 205 nm,\ne, 80\nnm, f, 40 nm, g, 12 nm, and h, 10 nm.\nThe key point we intended to convey in the discussion of Fig. 4c is that the hole carriers\nobserved in the 2D regime are transformed back into electrons when entering the 1D\nballistic regime. The nonlinear feature (“bump”) is observed in devices (of thickness 80 nm)\nwith channel widths (W) of 1 μm and 10 μm, consistent with the behaviour previously\ndemonstrated in 80 nm-thick films (see Fig. R1-3e), where sufficient two-dimensional\nconfinement gives rise to multiband transport involving both electron and hole carriers.\nHowever, as the channel width is reduced to 250 nm and 150 nm, this nonlinearity gradually\nvanishes. This trend is understood in terms of the Fermi surface evolution shown in Fig. 4e\n(Fig. 4g in original manuscript), which indicates that hole-like carriers present in the 2D\nregime are converted into electron-like carriers as the system approaches the 1D limit.\nTherefore, the absence of a nonlinear bump in the 250 nm (violet) and 150 nm (red) curves\nin Fig. 4b is expected and consistent with this dimensional crossover.\nReferences\n[5] Pippard, A. B. Magnetoresistance in metals. vol. 2. Cambridge University Press. (1989).\n[44] Beenakker, C. et al. Billiard model of a ballistic multiprobe conductor. Phys. Rev. Lett.\n63, 1857 (1989).\nReviewer: Moreover, there is no discussion of the magnetic field value Bo at which you\nexpect the cyclotron orbit radius to match the width of the Hall bar (expected B-field\nlocation of the non-linearity).\nResponse: We thank the reviewer for pointing out the importance of discussing the\ncharacteristic magnetic field scale B at which the cyclotron radius matches the channel\n0\nwidth. In our ballistic devices, the bend resistance R is negative at zero magnetic field,\nB\nindicating that electrons can travel directly from the current source to the opposite voltage\nterminal without scattering, which is a signature of ballistic transport. As the magnetic field\nincreases, the Lorentz force gradually curves these trajectories, reducing the transmission\nto the opposite terminal and thereby suppressing the negative R.\nB\nUsing the relation for cyclotron radius r=ℏk/eB with the Fermi wavevector of bulk Cu [R1]\nc F\nk≈1.36×1010 m−1, the magnetic field required to give r similar to the channel width\nF c\nW=250 nm is approximately B≈36 T. We have observed the monotonic increase of R up\nto our maximum magnetic field of ~10 T that is smaller than B as expected in ballistic\n0\ntransport [R2].\nWe have added the following sentence to the revised manuscript as follows (lines 258–262\nof page 12):\n“The characteristic magnetic field B at which the cyclotron radius r=ℏk/eB matches the\n0 c F\nchannel width is estimated to be ~36 T for W = 250 nm. Here, k the Fermi wavevector of\nF\nbulk Cu57 k≈1.36×1010 m−1. Consistent with the expectation for ballistic transport58, we\nF\nobserve a monotonic increase of R up to our maximum accessible magnetic field of ~10\nB\nT that is smaller than B.”\n[R1, Newly added Ref. 57] Balcerzak, T. Calculation of the Fermi wave vector for thin films,\nquantum wires and quantum dots. Mater. Sci.-Pol. 24 (2006).\n[R2, Newly added Ref. 58] Banszerus, L. et al. Ballistic transport exceeding 28 μm in CVD\ngrown graphene. Nano Lett. 16, 1387 (2016).\nReviewer: The experimental data in this section does not support well the discussion.\nThe discussion and/or data (how many samples have been measured? Do we have the\nEBSD data for these?) should be modified to better support each other, or the Figure\nshould be redesigned.\nResponse: We performed EBSD measurements on all films prior to device fabrication,\nafter\nwhich the devices were made. After fabricating the device structures using these films,\nthe\nsurfaces become roughened due to sidewalls and other processing steps, making EBSD\nmeasurements impossible on the device structures themselves. Instead, the crystallinity of\nthe device structures was verified using SEM and STEM analyses. While STEM is a\ndestructive technique and therefore cannot be applied to many devices, the internal\nstructures of the devices we examined were confirmed to maintain their crystallinity as\nshown in Fig. 1b.\nReviewer: I also mention that the cartoon of the Hall bar used in Fig. 4 does not match\nwell the actual aspect ratio of the W=150 and 250 nm devices which are at the center of\nthe discussion. Personally, I find that the data in Fig 4b is more interesting and convincing\nthan the one in Fig. 4e, so I would suggest to perhaps recenter the discussion on this.\nResponse: We revised device schematics in Fig. 4a and in the Supplementary Fig. 11 (Fig.\n4c in the original manuscript) to represent actual ratio of thickness to width (=90 nm :\n150\nnm) as shown in Fig. R1-4. In line with the reviewer’s suggestion to focus more on Fig. 4b,\nwe moved the Hall-measurement schematic and the Hall-resistivity (ρ) at 120 K dataset\nxy\n(which were previously Figs. 4c and d, respectively) to the Supplementary Fig. 11. While\nwe agree that Fig. 4b presents convincing results for ballistic transport, we believe that Fig.\n4c also provide essential complementary information on charge carrier types in a device\nwith very narrow width. Therefore, we prefer to maintain the current balance of discussion\namong the figures, as each highlights a different and important aspect of our findings.\nFig. R1-4 | Revised Fig. 4\nNewly added Supplementary Fig. 11\nWe have revised the following sentence at page 12, line 263-267 in the revised manuscript.\n“To characterize the charge carrier types for devices, we performed Hall measurement and\nplotted antisymmetrized Hall resistivity 𝜌as for different W at 1.7 K as shown in Fig. 4c\nxy\n(see Supplementary Fig. 11 for 𝜌as measured at 120 K). The observed nonlinear Hall effect\nxy\nat devices with W = 10 and 1 µm agrees with the results of a recent study49, in which\nreveals the co-existence of holes and electrons in <205-nm-thick SCCFs without GBs. ”\n• Are there any flaws in the data analysis, interpretation and conclusions? Do these prohibit\npublication or require revision?\nReviewer: The bend resistance and Hall resistivity measurements are very likely to be\nsensitive to the choice of contacts in the narrowest Hall bars because of edge confinement\nand the asymmetric shape of the devices. Given the very rough electron-beam lithography\nfor the narrowest Hall bars (poor homogeneity of the thickness and width), it is essential\nto convince the reader that the measurements all agree and point to a single conclusion.\nResponse: We thank the Reviewer for highlighting this important concern regarding the\npotential influence of fabrication inhomogeneity on our transport measurements. We fully\nacknowledge that, due to the resolution limit of electron-beam lithography, achieving\nuniform width and thickness becomes particularly challenging in our narrowest Hall bar\ndevices with W=150 nm. In this regime, small variations in line edge roughness or\nalignments can indeed introduce geometric asymmetries that may affect both the bend\nresistance (R) and Hall resistivity.\nB\nTo directly assess the impact of contact configuration and geometric asymmetry, we\nperformed additional measurements on a representative device exhibiting negative R as\nB\ndiscussed above in Figs. R1-1 d-e. In this device, R was measured across all eight\nB\npermutations of current and voltage terminals. Although the magnitude of R varied across\nB\nconfigurations presumably due to lithographic imperfections, the sign of R remained\nB\nconsistently negative. This strongly indicates that the observation of negative R is not an\nB\nartifact arising from a particular choice of contacts, but rather reflects an intrinsic feature\nof the device's transport behaviour.\nTo further quantify the effect of geometric inhomogeneity, we conducted finite-element\nsimulations using COMSOL Multiphysics® at room temperature (Fig. R1-2). These\nsimulations focused on vertical misalignment between opposing terminals as a\nrepresentative form of asymmetry. We find that a misalignment as small as 10 nm can\ninduce changes in R on the order of 10 mΩ, which is comparable to the experimental\nB\nvariations observed. This provides a plausible explanation for the spread in measured R\nB\nvalues and supports the interpretation that random lithographic deviations are responsible\nfor the configuration-dependent scatter.\nIn summary, while geometric variations due to fabrication limits are unavoidable at the\n150 nm scale, our combined experimental and simulation results confirm that the\nemergence of negative R is robust and reproducible, and not contingent on a particular\nB\ncontact configuration. These findings reinforce the validity of our conclusion that negative\nR reflects ballistic transport in these nanoscale Cu devices.\nB\nReviewer: Additionally, the data analysis of transport data is based on the van der Pauw\nmethod. This method works in the limit of “point-like” contacts on the outer edge of the\nsample. Here the contacts are sometimes even larger than the device itself. It should be\nclarified what are the error bars (systematic) stemming from using the van der Pauw\nmethod in these devices (the error bars should be larger in the narrower devices from\nseveral reasons including the non-uniform thickness).\nResponse: We appreciate the Reviewer’s comment regarding the applicability of the van\nder Pauw method in our devices, which do not feature strictly “point-like” contacts. The\ncurrent and voltage leads are defined with the narrow central Cu cross (~150 nm ×\n~150 nm), where all transport phenomena are confined. The arms of the cross maintain a\nwidth of 150 nm and gradually widen toward the wire-bonding regions.\nFrom a theoretical standpoint, the so-called “Greek cross” geometry (Fig. R1-5a) we employ\nis one of the most robust configurations for minimizing systematic errors from finite\ncontact size. As described in the references [R3, R4], the error E can be quantitatively\nestimated using the following empirical relation (Eq. R1)\nE=1-ρ/ρ=(0.59±0.006)exp[-(6.23±0.002)a/w] (Eq. R1),\nm\nwhere ρ and ρ denote the actual and measured resistivities, respectively. For a cross with\nm\narm width w and contact length a, the relative error in extracted resistivity becomes\nnegligible when the ratio w/(w+2a) is small. In our case, the terminals have a tapered shape,\nmaintaining a width of 150 nm over a length of 200 nm before gradually widening toward\nthe contact pad. Given that the contact length is at least 200 nm as shown in Fig. R1-5b,\nthe corresponding resistivity error is on the order of 10-4. In summary, although our\ncontacts are not point-like in the strictest sense, both theoretical considerations and\ngeometric parameters indicate that the finite contact size introduces negligible systematic\nerror in our transport analysis.\nFig. R1-5 | a, Schematic of Greek cross geometry. b, SEM image of the real device with\nW=150 nm.\nReferences\n[R3] David, J. M. et al. A numerical analysis of various cross sheet resistor test\nstructures. Solid-State Electronics 20, 539-543 (1977)\n[R4] Lindemuth, et al. Hall effect measurement handbook: a fundamental tool for\nsemiconductor material characterization. Lake Shore Cryotronics, Inc. (2020).\nReviewer: In my opinion, the main evidence for the ballistic transport data claim is in Fig.\n3b and supplementary Fig. 6a. However, the signal to noise ratio for the claim is close 1,\nmeaning that the scatter of the data point in these two Figures is comparable to the\nmagnitude of the negative bend resistance. If this could be strengthened by addressing\nthe above (showing if the RB data are from multiple contact configurations and from\ndevices whose EBSD maps are reported), then I think that the conclusion could be stronger.\nResponse: We acknowledge that the data exhibit noticeable scatter in Fig. 3b and\nSupplementary Fig. 6a, and we provide the following clarifications to strengthen the\ninterpretation.\nFirst, as discussed above, the fabrication of narrow Hall bars (W=150 nm~250 nm) is\ninherently susceptible to lithographic asymmetry due to the resolution limit of the electron-\nbeam lithography. To evaluate the impact of such geometric imperfections, we performed\nfinite-element simulations (Fig. R1-2), where we systematically introduced vertical\nmisalignments of up to ±30 nm between opposing terminals. These simulations show that\neven a misalignment of 10 nm can cause an R variation of ~10 mΩ, consistent with the\nB\nlevel of scatter observed in our experimental data. While we cannot directly identify the\nmisalignment present in each individual device, this result confirms that the experimentally\nobserved spread in R is well within the range expected from realistic fabrication tolerances.\nB\nSecond, we would like to clarify the relationship between the R data and EBSD analysis.\nB\nEBSD measurements were performed for all films used in Fig. 3b and Supplementary Fig. 6a\nand each data point corresponds to a distinct Cu film, and thus has its own unique EBSD\nmap. The specific EBSD image shown in Fig. R3b is a representative example from this set.\nMore specifically, among the total of 25 devices fabricated and measured, negative R was\nB\npredominantly observed in those with negligible grain boundary density, while those with\nhigher boundary densities tended to show suppressed or non-negative values. These\nresults provide strong, statistically meaningful evidence linking the structural quality of the\nCu film (as determined by EBSD) to the emergence of ballistic transport behaviour.\nTaken together, the negative R signal is reproducible, physically interpretable, and\nB\nconsistently correlates with film crystallinity. These combined findings reinforce the\nconclusion that the emergence of negative R marks the transition to the ballistic regime,\nB\nand that this conclusion is not undermined by the observed data variation.\nReviewer: I do command the authors for this work. This is an incredibly complex set of\nsample preparation and challenging transport measurements (given the very low resistivity\nof copper – which gives little signal to noise in transport). This is unique work. I\ncould be convinced of the validity of their claim, but I would need more evidence that the negative bend resistance is\nreproducible and quantitative accurate – and in multiple devices.\nResponse: We sincerely thank the Reviewer for the encouraging and thoughtful comments.\nAs rightly noted, one of the major challenges in our study arises from the exceptionally\nlow resistivity of the single-crystalline Cu thin films, even lower than that of bulk copper\nand the contact electrodes. This also results in very small voltage signals, requiring careful\nnoise suppression. To ensure reliable measurements, we employed low-noise electrical\nfilters and utilized two complementary techniques, DC and Delta-mode. The consistency\nof the results across both methods confirms that the observed signals are robust and not\ndependent on the measurement scheme.\nWe fully agree that demonstrating reproducibility and quantitative accuracy of the\nobserved negative bend resistance (R) is essential to support our central claim. To directly\nB\naddress this, as kindly suggested by reviewer, we performed systematic transport\nmeasurements on an independent device, distinct from the one presented in the main\nmanuscript. This additional device also exhibited clear signatures of ballistic transport. As\nshown in Fig. R1-1, we carried out both DC and Delta-mode measurements at 4.3 K across\neight distinct contact configurations, encompassing all permutations of current and voltage\nterminals. Despite expected variation in magnitude due to geometric asymmetries, the sign\nof R remained consistently negative across all configurations and in both measurement\nB\nmodes. Finite-element simulations further show that realistic levels of misalignment can\nquantitatively reproduce the scatter observed in R among different devices. These results\nB\nprovide strong evidence that the negative R is not a measurement artifact or a\nB\nconsequence of specific terminal arrangement, but a robust and reproducible feature of\nballistic transport in these nanoscale Cu devices. We believe this additional dataset directly\nresponds to the Reviewer’s request and further strengthens the validity of our conclusions.\n• Is the methodology sound? Does the work meet the expected standards in your field?\nReviewer: Yes, the authors are accomplished and barring the above comments, the work\nis of very high quality.\n• Is there enough detail provided in the methods for the work to be reproduced?\nReviewer: Yes, except for the points mentioned above.\nReferences\nReviewer: I found that unfortunately the bibliography is inadequate. Here are a few things\nwhich could improve it in my view.\n(i) In the introduction, please add several recent references showing the ongoing\ndevelopment in the field. For example, on line 55 the authors say “in recent years 5-9” but\nRefs. 5-9 are not recent at all.\nResponse: We appreciate the reviewer’s comment regarding the inadequacy of recent\nreferences in the introduction. In response, we have revised the sentence on line 55 to\nbetter reflect ongoing developments in the field of quantum transport in normal metals.\nSpecifically, we have added the following recent references:\n• R5 (2019) reported how Fermi surface topology can give rise to pronounced\nmagnetoresistance effects in metals.\n• R6 (2013) demonstrated extremely large magnetoresistance in the nonmagnetic\nmetal PdCoO₂ despite its simple Fermi surface.\n• R7 (2024) proposed a theoretical framework where linear magnetoresistance in 3D\nDirac metals can arise from guiding center motion related to quantum geometric\neffects.\nThese references (now cited as Refs. 10–12) have been incorporated into the revised\nintroduction to better represent the recent progress in the field.\nReferences\n[R5, Newly added Ref. 10] Zhang, ShengNan, et al. Magnetoresistance from Fermi surface\ntopology. Phys. Rev. B 99, 035142 (2019).\n[R6, Newly added Ref. 11] Takatsu, Hiroshi, et al. Extremely large magnetoresistance in the\nnonmagnetic metal PdCoO 2. Phys. Rev. Lett. 111, 056601 (2013).\n[R7, Newly added Ref. 12] Kaplan, Daniel et al. Unification of nonlinear anomalous Hall\neffect and nonreciprocal magnetoresistance in metals by the quantum geometry. Phys. Rev.\nLett. 132, 026301 (2024).\n(ii) In multiple places no references are given for items which clearly demand one. For\nexample, on line 71 when listing key ideas to be explored in ballistic copper films, topics\nsuch as “the quantum Hall effect”, “quantum confinement”, and “phase-coherent\ninterference” are all mentioned without any reference. I also would ask that you please\ndelete the mention of “hydrodynamic electron transport22”. This is a strange point the\nmake because this hydrodynamic regime is reported in graphene which has vastly different\nelectron-electron and electron-phonon scattering lengths than copper, there is no reason\n(except if you discuss it) to expect any hydrodynamic regime in your devices.\nResponse: We appreciate the reviewer’s comment regarding the lack of appropriate\nreferences and the inapplicability of the hydrodynamic electron transport regime to our\ncopper thin-film devices. In the revised manuscript, the mention of hydrodynamic electron\ntransport has been removed, as this regime is not expected for copper. We have also\nreplaced the term quantum confinement effect with the more appropriate confined\ngeometry effect, which is widely used in the thin metal film literature. Furthermore, we\nhave added authoritative references for each key concept mentioned in this section:\nQuantum Hall effect [R8], quantum size effect in thin metal films [R9], and phase-coherent\ninterference [R10]. These changes ensure that each concept is appropriately supported by\na reference and that the terminology aligns with established usage in the field. In the main\ntext, these references are cited as [27], [28], and [29].\nWe have revised the following sentence to caption of Fig. 4d in the revised manuscript as\nfollows (lines 232–233 of page 11):\n“The inner surfaces of 2D limit Fermi surface are splitted bands due to confined geometry\nalong the out-of-plane direction (left panel) Calculated effective Fermi surfaces of 2D Cu\nin periodic zone scheme (right panel).”\nWe have revised the following sentence to revised manuscript as follows (lines 271–275 of\npage 12-13):\n“In the 2D thin-film band structure, both electron and hole orbits are distinctly visible. In\ncontrast, the 1D structure's band structure appears more complex due to confined\ngeometry in both out-of-plane and in-plane directions.\nThe Fermi surface of the 2D thin film shows splitting of degenerate bands due to confined\ngeometry along the [111] direction (left panel of Fig. 4d in revised manuscript).”\nWe have revised the following sentence to revised Supplementary information as follows\n(lines 161–162 of page 15):\n“This result suggests that the electronic band structure of the 90-nm-thick SCCF, modified\nby confined geometry effect perpendicular to the film plane, has semimetal-like properties.”\nWe have revised the following sentence to revised Supplementary information as follows\n(lines 167–168 of page 15):\nIn addition, the semimetal-like behaviour disappears possibly because of additional\nconfined geometry effects along the channel width direction.\nWe have added the following references R8-R10 (27-29) to the revised manuscript as\nfollows (lines 71–72 of page 3):\n“Our SCCF enables studies on the quantum nature of Cu thin films, including topological\nproperties24-26, the quantum Hall effect27, confined geometry effect28, phase-coherent\nquantum interference29,30 and quantized conductance31,32 applicable to other metallic thin\nfilms.”\nReferences\n[R8, Newly added Ref. 27] Yoshioka, D. The quantum Hall effect. Vol. 133. Springer Science\n& Business Media, r\n[R9, Newly added Ref. 28] Zhigal'skii, G. P. et al. The physical properties of thin metal films.\nVol. 13. CRC Press, 71~75 (2003)\n[R10, Newly added Ref. 29] Van Haesendonck. C. et al. The Aharonov-Bohm Effect in\nNormal Metals. Europhys. News 19, 89-93 (1988)\n(iii) Several more (and recent) references related to the transport measurements would be\nappreciated. There are other ways to assess if a device is ballistic (e.g. quantum\ninterferences, or conductance quantization), but none of this is mentioned.\nResponse: We thank the reviewer for suggesting the inclusion of additional and recent\nreferences related to transport measurements and alternative signatures of ballistic\ntransport. In the revised manuscript, we have added references covering two important\napproaches not previously mentioned: quantum interference and conductance\nquantization. For quantum interference, we now cite the Aharonov–Bohm effect in both\nsuperconducting normal metals [R10] and Nb systems [R11]. For conductance quantization\nin the ballistic regime, we have included both a theoretical treatment for metallic narrow\nchannels [R12] and experimental observations in GaAs 2DEG quantum point contacts [R13].\nThese additions broaden the contextual basis for assessing ballistic transport beyond our\nprimary measurement approach.\nWe have added the following references R10-R13 (29-32) to the revised manuscript as\nfollows (lines 71–72 of page 3):\n“Our SCCF enables studies on the quantum nature of Cu thin films, including topological\nproperties24-26, the quantum Hall effect27, confined geometry effect28, phase-coherent\nquantum interference29,30, and quantized conductance31,32 applicable to other metallic thin\nfilms.”\nReferences\n[R10, Newly added Ref. 29] Van Haesendonck C. et al. The Aharonov–Bohm Effect in\nNormal Metals. Europhys. News 19, 89–93 (1988).\n[R11, Newly added Ref. 30] Imry, Y. et al. Quantum interference and the Aharonov–Bohm\neffect. Sci. Am. 260, 56–65 (1989).\n[R12, Newly added Ref. 31] Isawa, Y. Quantized conductance of metallic narrow channels\nin ballistic regime. J. Phys. Soc. Jpn. 57, 3457–3462 (1988).\n[R13, Newly added Ref. 32] Liu, R. C. et al. Quantum interference in electron collision.\nNature 391, 263–265 (1998).\nReviewer #2 (Remarks to the Author):\nReviewer: The paper discusses ballistic electron transport in 90-nm-thick films made of\nhigh-quality Cu. The copper films are grown using the atomic sputtering epitaxy technique,\nwhich allows obtaining a very large grain structure with atomically smooth interfaces. An\nextensive analysis has been performed to confirm grain boundary-free regions on the Cu\nfilms over micron-sized areas. Transport measurements in the magnetic field of a cross\ngeometry of different sizes support the claim of the ballistic propagation of electrons at\nlow temperatures.\nThe reported results raise several issues, which raise concerns about the validity of the\nresults and analysis. These issues are mainly relevant to transport measurements, but not\nlimited to them. Below, I give a list of the problems that I have found.\nResponse: We appreciate the reviewer for the valuable indication and comments. We have\nexamined reviewer’s comments below in details and carefully revised our manuscript\naccording to the reviewer’s comments and addressed the issues raised by the reviewer’s\npoint-by-point in this response letter.\nReviewer: 1. The paper does not seem to discuss the available literature on the current\nstate of the problem. At least one paper claims that polycrystalline Cu at room temperature\nshows better resistivity than that reported in the manuscript. [1]\nResponse: We thank the Reviewer for pointing out the importance of discussing prior\nliterature relevant to the current state of the field. The study by Mallikarjunan et al. [R14]\nindeed reports exceptionally low resistivity values of 2.0~2.2 μΩ·cm for polycrystalline Cu\nfilms with thicknesses of 30~40 nm, fabricated by sputtering followed by annealing. These\nvalues are remarkably low and almost comparable to those obtained from our single-\ncrystalline Cu films, as shown in Fig. R2-1 (reproduced from Fig. 6e of Ref. 46 in revised\nmanuscript, Ha T. et al., Nat. Commun. 14, 685 (2023)).\nIn the study by Ha T. et al., the thickness-dependent resistivity was evaluated using three\nindependent methods including DC electrical transport, Fourier-transform infrared\nspectroscopy (FTIR), and time-domain terahertz spectroscopy, all of which showed excellent\nagreement with each other and with theoretical predictions based on the Fuchs–\nSondheimer (FS), Namba, and effective medium approximation (EMA) models.\nThe extremely low resistivity reported by Mallikarjunan et al. implies that their sputtered\nfilms, although polycrystalline, likely possessed exceptionally high crystalline quality and\nminimal scattering from grain boundaries. We agree that this work is relevant and now we\ninclude it in the revised manuscript (line 62-64, page 2) along with a discussion comparing\nit to our results as follows:\n“Although polycrystalline Cu thin films prepared by sputtering and post-annealing have\nshown low room-temperature resistivities (2.0–2.2 μΩ·cm)20, ballistic transport was not\nexplicitly examined.”\nFig. R2-1 | (Fig. 6e of Ref. 46) Ha, T. et al. Coherent consolidation of trillions of nucleation\nfor mono-atom step-level flat surfaces. Nat. Commun. 14, 685 (2023).\nReferences\n[R14, Newly added Ref. 20] Mallikarjunan, A. et al. Resistivity of copper films at thicknesses\nnear the mean free path of electrons in copper minimization of the diffuse scattering in\ncopper. Electrochem. Solid-State Lett. 3, 437–438 (2000).\n[46 in revised manuscript] Ha, T. et al. Coherent consolidation of trillions of nucleation for\nmono-atom step-level flat surfaces. Nat. Commun. 14, 685 (2023).\nReviewer: 2. SFig. 4 shows a 250 nm wide sample. However, I do not understand how the\nsizes were calculated, as I tried to demonstrate in the image below. It is also unclear what\nthe origin of the Cu regions, highlighted in yellow ellipses in (c), is. If it were a standard\nlithography and lift-off process, I would say these are left over after a lift-off process. The\nMethods have not provided details on the fabrication method besides lines 397-399. The\nSEM image in SFig. 3 is blurred, as is evident from a poorly focused image, as seen by a\nsubstantial difference (presence and absence of small features) on SFig. 3a and SFig.\n3b. I could say that the size determined by SEM is incorrect, as HAADF-STEM achieved\nsubatomic resolution and should be more accurate. Anyway, the image also shows that\nthe profile of the device is not a simple rectangle; moreover, it has parasitic parallel\n(presumably conducting) channels due to Cu left-overs.\nResponse: We thank the Reviewer for the detailed evaluation of Supplementary Figure 4\nand the comments regarding device geometry, width determination, and the presence of\nresidual Cu regions. In response, we have revised the Supplementary Methods and added\nfurther clarification and analysis as follows. Here, W denotes the nominal (designed)\nchannel width, which may differ from the actual one. In all devices measured, the actual\nchannel width deviated by no more than ~10 % from the designed width.\nFirst, regarding the device width: the nominal width of 250 nm refers to the designed value\ndefined by electron-beam lithography. However, we acknowledge that the actual width of\nthe specific device shown in Supplementary Fig. 4b is approximately 280 nm (Fig. R2-2b),\nas determined from high-resolution electron microscopy. To prevent confusion, we have\nupdated Supplementary Fig. 4 and its caption to reflect the actual width of the measured\ndevice.\nWe have added the following sentence to the revised manuscript as follows (lines 92–94\nof page 5):\n“Here, W denotes the nominal channel width, which may differ from the actual one. In all\ndevices measured, the actual channel width deviated by no more than ~10 % from the\ndesigned width.”\nFig. R2-2 | Revised Supplementary Fig. 4. Cross-sectional TEM image of ballistic\ntransport in SCCF. a, SEM image of an SCCF Hall bar device with a nominal channel width\n(W) of 250 nm. b, Low-magnification ADF-STEM image displaying the cross-sectional view\nof the SCCF Hall bar device, with a red arrow indicating the actual width of the measured\ndevice. c, EELS-based elemental map of Cu. d–f, Enlarged atomic structure images for the\nregions marked by white boxes in c.\nSecond, concerning “the Cu regions highlighted in yellow ellipses in panel (c)”, while the\nannotated image was not included in the Reviewer’s comments we received from the\nJournal Nature Communications, we assume that the Reviewer is referring to the sidewall\nstructures visible in the cross-sectional STEM image that we highlighted with yellow ellipses\nin Fig. R2-3a. To investigate the composition of these regions, we performed EELS\n(electron energy-loss spectroscopy) measurements at three positions labeled #1–#3 in Fig. R2-2a.\n• #1 corresponds to the Al 2 O 3 sapphire substrate,\n• #2 to the center of the Cu film, and\n• #3 to the residual sidewall structure.\nThe presence of Cu signals at positions #2 through #3 confirms that these sidewall\nstructures contain copper (Fig. R2-2c). Importantly, the EELS spectrum at position #3 also\nshows a distinct oxygen K-edge at around 535~550 eV (Fig. R2-3b), suggesting that this\nregion likely contains oxidized copper (CuOx). We expect that the observed sidewall\nstructure originates from the re-deposition of etched Cu atoms onto the sidewalls of the\nPMMA resist during the Ar ion etching process used to define the Hall bar geometry (Fig.\nR2-3d). As a result, this region likely exhibits reduced crystallinity and significantly higher\nresistivity compared to the pristine single-crystalline Cu film. Therefore, its contribution to\nthe overall conduction is expected to be insignificant. Furthermore, in our van der Pauw\ngeometry, where current injection and voltage sensing are confined to the central Cu cross,\nthe presence of such sidewall structures does not affect the interpretation of the observed\nballistic transport behaviour.\nFinally, regarding the image quality in Supplementary Fig. 3: the apparent difference in\nresolution between panels 3a and 3b-c arises from the different magnifications used.\nPanel\n3a provides a lower-magnification overview of the entire structure, while panels 3b and 3c\nwere acquired at higher magnification to resolve finer features such as edge definition and\nsurface uniformity.\nWe hope these clarifications address the Reviewer’s concerns and reinforce the robustness\nof our device characterization.\nWe have added the following sentence to the revised manuscript as follows (lines 331–333\nof page 14):\n“After the Ar-ion milling, Cu redeposition was observed along the lithography resist\nsidewalls (Supplementary Fig. 8), but such regions may have only limited contribution to\nconduction because they likely exhibit reduced crystallinity.”\nWe have added Fig. R2-3 to the Supplementary Fig. 13 and corresponding Supplementary\nnote in revised manuscript as follows (page 17-18).\nFig. R2-3 | Newly added Supplementary Fig. 13. Cross-sectional transmission electron\nmicroscopy (TEM) and Electron energy loss spectroscopy (EELS) analysis of a 250 nm-\nwide Cu Hall bar. a, Cross-sectional TEM image showing a thin oxidized layer near the\nfilm edge. b, EELS spectra of O measured at positions #1~#3 marked in a. c, EELS spectra\nof Cu measured at positions #1~#3 marked in a. d, Schematic of Ar⁺ ion etching showing\nCu re-deposition on the PMMA sidewall and formation of CuO.\nx\nReviewer: 3. Fig.1d shows bend resistance as a function of temperature. I think the\nexperimental results are too close to zero temperature, even for the minimum temperature\nof T=1.7 K. The behaviour of the 250 nm sample looks very good due to the agreement\nbetween high-temperature resistances. This can be viewed as perfect uniformity of the\ngrown samples; however, the lower resistance of the narrowest sample is very surprising\nat high temperatures. This difference is not discussed in the text. The line that separates\ndiffusing and ballistic transport regimes is misleading, as some electrons can propagate\nballistically even in a diffusive regime. When a significant part of the electrons are ballistic,\nthe bend resistance should deviate from the expected behaviour obtained from the theory\nfor the wide samples. the 250 nm wide sample demonstrates a clear ballistic transport as\nit differs from the theory.\nResponse:\n(1) Low-temperature range in Fig. 1d\nWe thank the Reviewer for the thoughtful feedback on Fig. 1d. First, Fig. R2-4 presents a\nzoomed-in view of Fig. 1d, where the lowest temperature measured was 1.7 K. The data\nmay appear to extend to zero temperature due to the wide x-axis range (0–300 K), making\nthe low-temperature region visually compressed.\nFig. R2-4 | Zoomed-in view of Fig. 1d showing the bend resistance data in the temperature\nrange 0–5 K.\n(2) The discrepancy of R between 150 nm and 250 nm devices at high-temperature\nB\nSecond, we agree that the difference in high-temperature R values between the 150 nm\nB\nand 250 nm devices is noticeable. We believe this discrepancy is not due to a fundamental\nchange in transport mechanism but is more likely a result of geometric imperfections\nintroduced during fabrication especially when fabricating very narrow devices. In particular,\nthe 150 nm device width approaches the resolution limit of our SEM-based electron-beam\nlithography, making it more susceptible to electrode misalignment and patterning\nasymmetries. Even small deviations in geometry can lead to measurable changes in R.\nB\n(3) Robustness of negative R across terminal configurations and measurement techniques\nB\nWe present here an additional result among the dataset shown in Fig. 3b (Fig. R2-5a),\nwhich was not presented in the main manuscript. Cu film thickness of this device is 80 nm,\nand its SEM image is shown in Fig. R2-5b.\nTo directly address how the measured bend resistance (R) depends on the choice of\nB\ncurrent and voltage terminals, we performed systematic measurements using both DC and\nDelta-mode techniques.\nFor each representative Cu cross device (150 nm width) exhibiting signatures of ballistic\ntransport at 4.3 K, we carried out DC current-voltage (IV) sweeps across eight distinct\ncontact configurations, covering both forward and reverse arrangements of current and\nvoltage terminals (Table R2-1). Each R value was extracted from the slope of the IV curve\nB\nmeasured between -300 μA and +300 μA. As shown in Fig. R2-5d, all eight configurations\nyielded consistently negative R values, though the magnitude varied. This indicates that\nB\nthe observed negative R is not tied to a particular terminal arrangement but is a robust\nB\nfeature of transport in these nanoscale devices.\nConfiguration Current leads Voltage leads\n# (I⁺ → I⁻) (V⁺ → V⁻)\nTable R2-1 | Newly added Supplementary Table 1. Contact terminal combinations for\nbend resistance measurements. Current (I⁺ → I⁻) and voltage (V⁺ – V⁻) terminal\nconfigurations used in the eight measurement setups shown in Figs. R2-5c-e. Terminals\nare labelled 1 to 4 in counterclockwise order starting from the left.\nFig. R2-5 Newly added Supplementary Fig. 7. a, The device marked with a red arrow in\nFig. 3b was used for the measurements shown in b–e. b, Scanning electron microscope\n(SEM) image of a representative device from a. c, Measured bend resistance (R) at 285 K\nB\nvia Delta-mode measurement across 8 different configurations in Table R2-1. d, Measured\nR at 4.3 K via DC measurement across 8 different configurations, including forward and\nB\nreverse arrangements of current and voltage terminals. e, Measured R at 4.3 K via Delta-\nB\nmode measurement across 8 different configurations.\nWe measured a different device, selected from the dataset shown in Fig. 3b but not used\nin the main manuscript as shown in Fig. R2-5a. Cu film thickness of this device is 80 nm,\nand its SEM image is shown in Fig. R2-5b. To further confirm the reliability of these\nmeasurements, we also performed Delta-mode measurements on eight configurations at\n4.3 K (Fig. R2-5e). This method alternates the bias current between +300 μA and -300 μA\nwith a delay time of 300 ms, reducing low-frequency noise and improving precision for\nlow-resistance signals. The results closely match the DC data in both sign and magnitude,\nreinforcing that the negative R observed is intrinsic and not a measurement artifact.\nB\nWe then extended the Delta-mode measurements to room temperature (285 K) for the\nsame configurations (Fig. R2-5c). Although the extracted R values showed increased\nB\nvariation up to ±20 mΩ, the overall magnitude remained low, and no sign inversion was\nobserved. This variation is consistent with realistic lithography-induced asymmetries of\ngeometry, such as terminal misalignment depicted in Fig. R2-6a. To support this, we\nconducted finite element method simulations using commercial package (COMSOL\nMultiphysics®) with introducing vertical misalignments of up to 30 nm between opposing\nterminals. As shown in Fig. R2-6b, even a small misalignment of 10 nm can induce R\nB\nchanges of ~10 mΩ, which is quantitatively consistent with the experimental spread.\nFig. R2-6 | Newly added Supplementary Fig. 8. a, Schematic of the Cu cross geometry\nused in the simulations (width 150 nm, thickness 90 nm), where the vertical misalignment\nbetween opposing terminals was systematically varied. b, Simulated R as a function of\nB\nvertical misalignment (-30 nm to +30 nm) for device widths of 150 nm and 250 nm.\n(4) Removal of schematic boundary line in Fig. 1d\nIn light of the reviewer’s point, we also agree that the sharp boundary line between ballistic\nand diffusive transport in Fig. 1d may be overly schematic and may not capture the gradual\ncrossover regime where partial ballistic behaviour occurs. In the revised version, we have\nremoved this boundary line to avoid a misleading binary interpretation and now describe\nas shown in Fig. R2-7.\nFig. R2-7 | Revised Fig.1d\nWe have added Figs. R2-5, R2-6 and Table R2-1 to the Supplementary Figs. 7, 8,\nSupplementary Table 1, and corresponding Supplementary note in revised manuscript\n(page 8-11):\nWe have added the following sentence at page 6, line 138-141 in the main text to explain\nthe additional experiments showing that negative R is consistently observed across\nB\ndifferent configuration\n“To verify that the observed negative R is not due to a specific choice of current–voltage\nB\nconfiguration or artifact, we performed additional measurements across all contact\nconfigurations (Supplementary Fig. 7 and Supplementary Table 1).”\nWe have added the following sentence at page 6, line 141-144 in the main text to explain\nthe possible effect of fabrication-induced geometric imperfections, as supported by our\nsimulations.\n“At high temperatures, the R of the device with W = 150 nm appears slightly lower than\nB\nthat of the other devices with larger W. This difference may arise from geometric\nimperfections introduced during fabrication, as supported by simulations in Supplementary\nFig. 8.”\nReviewer: 4. αel-ph should have units. The Debye temperature obtained from the fit, 270K,\ndiffers from that found in the literature, Ѳ =330K. This can be related to the thickness or\nR\ngrain boundaries or something else. There is no discussion on the topic. The Bloch–\nGruneisen formula for (line 103) has a missing power of 5 in the T/Ѳ prefactor. Maybe\nR\nthis is why I could not reproduce the fitting parameters reported. My graph below is given\nfor the α =7e-6 Ωcm and Ѳ =270K. The Python code is provided.\nel-ph R\nResponse: We thank the Reviewer for the careful reading and constructive comments\nregarding the Bloch–Grüneisen analysis.\nFirst, we have added the appropriate unit for the electron–phonon coupling coefficient α\nin the line 111 of page 5 of the main text; its unit is Ω·m.\nSecond, we agree that the extracted Debye temperature Θ = 270 K differs from the\nD\ncommonly reported bulk value Θ ≈ 330 K. This discrepancy is not unusual and can arise\nD\nfrom a range of factors that influence phonon spectra and scattering processes in metallic\nsystems. In particular, the Debye temperature is known to be sensitive to:\n1. Purity and defect density (e.g., vacancies, dislocations, or oxidation),\n2. Crystal structure and orientation (single-crystalline vs. polycrystalline),\n3. Surface condition (roughness, oxidation, contamination),\n4. Measurement technique and configuration, and\n5. Dimensionality or finite thickness effects in thin films.\nWhile our samples are nearly single-crystalline and thus might be expected to exhibit a\nDebye temperature close to or even higher than bulk Cu, the thin film geometry and\npossible defects introduced during fabrication steps (e.g., etching, lift-off, surface exposure\nto ambient atmosphere) may have reduced the effective Debye temperature. This trend is\nconsistent with prior reports on thin Cu films, where reduced dimensionality and surface\ndegradation are known to lower the observed Debye temperature [R15].\nWe have added the following sentence to the revised manuscript as follows (lines 113–121\nof page 5-6):\n“The sample used in this study is nearly single-crystalline and would typically be expected\nto exhibit a Debye temperature close to or even higher than that of polycrystalline copper.\nHowever, the observed Debye temperature of 270 K is slightly lower than the commonly\nreported bulk value (~330 K). We attribute this deviation to the thin film geometry and the\nfabrication process. In particular, device fabrication steps such as Ar ion etching and\nchemical treatments may introduce surface roughening, partial oxidation, or other\nstructural imperfections that alter the phonon spectrum and reduce the effective Debye\ntemperature. Similar reductions in Debye temperature due to dimensional confinement\nand surface degradation have also been reported in sputtered copper thin films40.“\nLastly, we sincerely thank the Reviewer for identifying the missing fifth power in the\nprefactor of the Bloch–Grüneisen formula in line 106. We have corrected the equation in\nthe revised manuscript to include the appropriate (T/Θ)5 dependence. Importantly, we\nD\nconfirm that the correct expression was used for our data fitting (Fig. 1d) in the original\nmain text, and that the reported parameters in the original main text remain valid. Here,\nwe attached our previously used original MATLAB code and result (Fig. R2-8) for generating\nB-G theory line in Fig. 1d.\nCode:\nf = figure;\nf.Position(3:4) = [460 420];\nTD=270; % TD,c value: RMSE Fitting result from Fig.1d\nc=8.8e-8;\ntCu = 89.275e-9;\nrho0 = 4.6287e-09;\nT_BG = [linspace(0,300,101)]';\nf=@(x) (x.^5)./((exp(x)-1).*(1-exp(-x)));\nfor t=1:101\nT = T_BG(t);\nrho_BG(t) = c*(T/TD)^5*integral(f,0,TD/T);\nend\nR=(log(2)/(pi*tCu))*(rho_BG+rho0);\nplot(T_BG, R*1e3,'.','markersize',5,'color',[0 0 0]);\nylim([0, 80]);\nset(gca,'fontsize',16);\nxlabel('Temperature (K)');ylabel('{\\it R} (mΩ)');\nFig. R2-8 | Bloch–Grüneisen function with (T/Θ)5 dependence, using Θ = 270 K, α\nD D el–\n=8.8 × 10-8 Ω·m, and ρ=4.6287 × 10-9 Ω·m, generated using MATLAB. The dashed line\nph 0\nindicates the ρ value.\n[R15, Newly added Ref. 40] Timalsina, Yukta P. et al. Effects of nanoscale surface roughness\non the resistivity of ultrathin epitaxial copper films. Nanotechnology 26, 075704 (2015).\nReviewer: 5. The authors should provide the parameters they used for calculating the\nmean free path in formula (2), including the effective mass of the electrons in copper,\nthe Fermi velocity, and how they calculated the concentration.\nResponse: In formula (2), we evaluated the mean free path using the effective mass\nm*=1.1843×10-30) kg as reported in [R16], and the Fermi velocity v=1.57 × 106 m/s [R17].\nF\nElectron density n=8.49×1028 m-3 was obtained from the free-electron model assuming one\nconduction electron per Cu atom [R17].\nWe have added the following sentence (and an effective mass m* = 1.1843 × 10-30 kg) and\nreferences 41(R11), 42(R12) to the revised manuscript as follows (lines 129–131 of page 6):\n“Here, l is calculated using equation (2) with a Fermi velocity41 v = 1.57 × 106 m/s an\nmfp F\nelectron density42 n = 8.49 × 1028 m⁻³, and an effective mass42 m* = 1.1843 × 10-30 kg.”\nReferences\n[R16, Newly added Ref. 41] Shoenberg, D. The de haas–van alphen effect in copper.\nNature 183, 171-171 (1959).\n[R17, Newly added Ref. 42] Kittel, C. et al. Introduction to solid state physics. John Wiley\n& Sons (2018).\nReviewer: 6. Fig. 4b has different bend resistances at zero magnetic field compared with\nFig. 1d. Does the system become more ballistic between the measurements? It would be\ngood if there were more than one sample using a standard van der Pauw (with 90 degrees\nrotation and inversion of the magnetic field), as the authors used DC, which is sensitive to\nany DC bias due to the Seebeck effect. The ballistic transport complicated it due to the\nhigh sensitivity of scattering from the edges. Thus, it is helpful to show that the negative\nbend resistance is observed in all realisations of the van der Pauw method [2].\nResponse:\n(1) Difference between zero-field R in Fig. 1d and Fig. 4b\nB\nWe appreciate the Reviewer’s observation. Figures 1d and 4b were obtained from the same\ndevice, but possibly different measurement configurations. As discussed in our response\nto Comment 3 (Figs. R2-5, R2-6, Table R2-1), R can vary by ~10 % depending on the\nB\nmeasurement configuration. This likely accounts for the ~10% difference in zero-field R\nB\nvalues between Fig. 1d and Fig. 4b.\n(2) Thermoelectric offsets and edge scattering\nWe fully agree with the Reviewer’s concern that in DC measurements, parasitic\nthermoelectric effects such as the Seebeck effect could introduce spurious voltage offsets,\nparticularly in metals like Cu with low resistivity. We also acknowledge that in the ballistic\ntransport regime, where edge scattering becomes highly relevant, verifying the robustness\nof the negative R across various measurement configurations is essential.\nB\nTherefore, we performed a comprehensive set of control measurements on a separate\ndevice, different from the one shown in the main Figure 1, which also exhibited clear\nsignatures of ballistic transport.\n(3) Reproducibility across standard van der Pauw configurations and techniques\nAs detailed in our response to Comment 3 (Figs. R2-5, R2-6, Table R2-1), we have\nalready verified that the negative R is reproducibly observed across all measurement\nB\nconfigurations consistent with the van der Pauw method, including different current–\nvoltage terminal arrangements and measurements with both DC and Delta-mode\ntechniques. For a representative 150 nm wide Cu cross device, all eight terminal\narrangements listed in Table R2-1, including 90° rotations, yielded consistently negative\nR values (Figs. R2-5d–e), confirming that the negative bend resistance is intrinsic and\nB\nnot configuration-specific. The Delta-mode data closely matched the DC results in both\nsign and magnitude, and the variation observed at room temperature (up to ±20 mΩ)\nmay be explained by plausible fabrication-induced asymmetries such as terminal\nmisalignment, as supported by our finite-element simulations shown in Fig. R2-6.\n(4) Magnetic-field dependence across rotated configurations\nWe measured the magnetic-field dependence of R from –9 T to +9 T for two contact\nB\nconfigurations rotated by 90° on the same device, as shown in Fig. R2-9. In both\nconfigurations, R increased with increasing |B|, showing the same trend as in Fig. 4b,\nB\nconfirming that the observed magnetic-field dependence is consistent across the two\nconfigurations.\nIn summary, the negative R observed at low temperature is reproducible across all\nB\nconfigurations and two measurement techniques. The configuration-dependent variation\nin R is well explained by reasonable fabrication-induced asymmetries, and does not\nB\nundermine the intrinsic ballistic nature of electron transport in our Cu thin films. In\naddition, magnetic field dependence of R for two 90°-rotated contact configurations on\nB\nthe same device shows the same increase with |B| (Fig. R2-9), consistent with Fig. 4b and\nfurther corroborating the robustness of this behaviour.\nFig. R2-9 | Magnetic-field dependence of R for two contact configurations rotated by 90°\nB\non the same device.\nReviewer: 7. The quantum size effect arises when the film's thickness is comparable to the\nde Broglie wavelength of the electrons in copper. Thus, this effect is highly unlikely to\ninfluence the Hall effect [3].\nResponse: We appreciate the reviewer’s comment and agree that “quantum size effect”\nwas not the appropriate term in this context. We have replaced “quantum confinement”\nwith “confined geometry” and revised the main text accordingly (page 12-13, lines 272–\n275) as follows:\n“In contrast, the band structure of the 1D structure appears much more complex due to\nconfined geometry in both the out-of-plane and in-plane directions.\nThe Fermi surface of the 2D thin film exhibits splitting of degenerate bands due to confined\ngeometry along the [111] direction (Fig. 4d, left panel).”\nWe have revised the following sentence to revised Supplementary information as follows\n(lines 161–163 of page 15):\n“This result suggests that the electronic band structure of the 90-nm-thick SCCF, modified\nby confined geometry effect perpendicular to the film plane, has semimetal-like properties.”\nWe have revised the following sentence to revised Supplementary information as follows\n(lines 167–169 of page 15):\n“In addition, the semimetal-like behaviour disappears possibly because of additional\nconfined geometry effects along the channel width direction.”\nReviewer: The thickness of 90 nm corresponds to >400 unit cells. Independent of the\nquantization, the high current used in the measurements, 1mA, suggests testing the\nheating effects at low temperatures. Also, the profile of the copper conductor is not\nrectangular, as can be seen in SFig. 4c, and obviously, this asymmetry is more substantial\nfor narrower samples. There is no such problem in 2D samples, but for 3D wires, the Hall\nvoltage is not the same on the side walls and can generate a classical magnetoresistance\nin a perpendicular magnetic field.\nResponse: We thank the reviewer for raising this important point. To directly address the\npotential influence of heating at low temperatures, we performed bend resistance (R)\nB\nmeasurements using delta-mode excitation at different current amplitudes (100, 300, 600,\nand 1000 μA). As shown in Fig. R2-10a, the R values remained essentially unchanged\nB\nacross this range, indicating negligible heating effects. We further confirmed this by\nperforming DC IV measurements across the –1000 μA to +1000 μA range (Fig. R2-10b),\nwhich yielded a linear current–voltage slope throughout, again suggesting that heating-\ninduced nonlinearity is absent in our measurement regime.\nWe acknowledge that, due to the nature of the Ar ion milling process, it is challenging to\nachieve perfectly vertical etching relative to the substrate, which can lead to deviations\nfrom an ideal rectangular cross-sectional profile. This limitation may arise geometrical\nnonuniformity, so we applied anti-symmetrisation with respect to magnetic field polarity\nin all our measurements to eliminate this contribution from nonuniformity.\nThese combined results provide strong evidence that the observed R signals are not\nB\ninfluenced by heating effects. Although some degree of geometric asymmetry is inevitable\ndue to fabrication limits, its effect on magnetoresistance has been effectively removed\nthrough anti-symmetrisation. Therefore, the extracted R values faithfully reflect the\nB\nintrinsic transport behaviour of our Cu thin film devices.\nFig. R2-10 | Verification of negligible heating effects in bend resistance measurements. a,\nBend resistance (R) measured using delta-mode measurement at different current\nB\namplitudes (100, 300, 600, and 1000 μA) for a representative 150 nm Cu device at 4.3 K. b,\nDC current–voltage (IV) characteristics measured over ±1000 μA range for the same device.\nReviewer: 8. The main problem of the reported results is massive degradation of mobility\n(by a factor of 50) in copper compared to polycrystal bulk copper, as demonstrated in SFig.\n2. Thus, it is unclear why this result could be reported as a novelty. I do not see any simple\ncorrelation between the absence of GB and “high” quality of such samples. Probably it is\nrelated to scattering on the edges. In the paper [1], the resistivity of thin films is better at\nroom temperature, which is most relevant to applications.\nResponse: We believe there may have been a misunderstanding of Supplementary Fig. 2.\nThis figure does not present mobility data; rather, it shows the mean free path (MFP) as a\nfunction of temperature. The “bulk” reference value is from a single-crystal bulk sample,\nnot a polycrystalline sample. The values at 205 nm and 80 nm correspond to the MFP in\nsingle-crystalline thin films. The observed ∼50× reduction in MFP with decreasing thickness\nis a natural consequence of increased scattering rates from surfaces and interfaces, as\ndescribed by well-established models. In the Fuchs–Sondheimer (FS) formalism [55 in\nrevised version of manuscript], the resistivity of a thin film is expressed as\nρ(t)=ρ [1+(3/8)(1-p)(l /t) ]\nbulk bulk\n, where ρ is the bulk resistivity and l is the bulk mean free path. Here, l refers to\nbulk bulk bulk\nthe average distance an electron travels between scattering events in an idealized bulk\nsingle crystal without surface or interface scattering, and is determined solely by intrinsic\nphonon and impurity scattering. t is the film thickness, and p is the specularity parameter\n(0 = fully diffuse, 1 = fully specular). As t becomes comparable to or smaller than l ,\nbulk\nsurface scattering becomes dominant, leading to a pronounced reduction in the effective\nmean free path l .\nmfp\nThe reviewer’s comment that there is “no simple correlation” between the absence of grain\nboundaries (GBs) and the high quality of such samples is not consistent with established\nliterature. GBs are well known to act as strong scattering centers that degrade conductivity\nand electron coherence, as demonstrated across metals, semiconductors, and oxides.\nThe\nMayadas–Shatzkes (MS) and FS models are the most widely used frameworks to describe\nresistivity increases due to GB and surface scattering, respectively. In ultrathin, GB-free\nfilms, our experimental data follow the predictions of these models. Numerous prior studies\nhave shown GB scattering to be a dominant contributor to resistivity enhancement [R18–\nR22, 46 in revised manuscript]. The room-temperature resistivity of copper with a purity of\n99.99% (4N) and that of 99.99999% (7N) show almost no difference; rather, their electronic\nproperties depend on the number of grain boundaries (GBs). By eliminating these grain\nboundaries, the resistivity was experimentally reduced by 13–15%. [R22]\nReferences\n[55 in revised manuscript] Fuchs, K. The conductivity of thin metallic films according to the\nelectron theory of metals. Math. Proc. Camb. Philos. Soc. 34, Cambridge University Press,\n(1938).\n[R18] Chawla, J. S. et al. Electron scattering at surfaces and grain boundaries in Cu thin\nfilms and wires. Phys. Rev. B 84, 235423 (2011).\n[R19] Bishara, H. et al. Understanding grain boundary electrical resistivity in Cu: the effect\nof boundary structure. ACS Nano 15, 16607–16615 (2021).\n[R20] Barmak, K. et al. Surface and grain boundary scattering in nanometric Cu thin films:\na quantitative analysis including twin boundaries. J. Vac. Sci. Technol. A 32, 061503 (2014).\n[R21] Coffey, K. R. Electron scattering in metallic thin films. In Metallic Films for Electronic,\nOptical and Magnetic Applications: Structure, Processing and Properties 422–453, Elsevier\n(2014).\n[R22] Cho, Y. C. et al. Copper better than silver: electrical resistivity of the grain-free single\ncrystal copper wire. Cryst. Growth Des. 10, 2780–2785 (2010).\n[46 in revised manuscript] Ha, T. et al. Coherent consolidation of trillions of nucleation for\nmono-atom step-level flat surfaces., Nat. Commun. 14, 685 (2023).\nThe alternative hypothesis that edge scattering is responsible for the observed behaviour\nis also not supported by our measurements. Edge scattering is strongly geometry-\ndependent and would produce distinct trends for devices of different widths (e.g., 1 μm,\n250 nm, and 150 nm). However, the key transport features in our data remain consistent\nacross all widths, indicating that edge scattering is not the dominant mechanism.\nRegarding the resistivity values, Ref. [R18] reports 1.79 μΩ·cm, which is lower than our 2.02\nμΩ·cm at 286 K for a 200-nm-thick single-crystalline film. This is indeed impressive,\nparticularly given their polycrystalline sample. This value of 1.79 μΩ·cm even falls below\nvalues predicted by theoretical models. While we cannot fully evaluate their result without\ndetailed crystallographic information, we acknowledge this as a remarkable achievement\nand respect the authors for it. That said, our work is not aimed at achieving low room-\ntemperature resistivity. Rather, our central finding is that, in the absence of grain\nboundaries, electron transport becomes ballistic. We hope this distinction is clear.\nReviewer: 9. If the primary source of scattering is related to scattering from the edges,\nthen the narrow samples should have a lower mean free path. This is possible if we\ncompare the mean free path of 170 nm with the reported widths of the devices. Please\nnote that the samples of 250 nm and 150 nm wide show ballistic transport; however, it is\nhard to compare where the ballistic transport is larger. More negative signals do not mean\nmore ballistic transport. More resistive samples with the same mobility give more negative\nbend resistance. You can see it directly from the formula (4). Assume r0 = h/4e, PL=PR=PLR << PFW, that means most of the electrons are going forward. Then RB≈-r0/PLR. If PLR is larger (for wider samples, assuming\nscattering on the edges), then the negative bend resistance is smaller. In the case PLR=PFW, RB≈0. This means you have\na few scattering events, which means the mean free path is comparable to the size of the sample.\nResponse: We appreciate the reviewer’s insightful comments regarding the interpretation\nof negative bend resistance (R) and its relationship to ballistic transport. We agree that,\nB\nas shown in Eq. 4, the magnitude of R is not a simple monotonic function of 𝑃, 𝑃 , and\nB 𝐿 𝑅\n𝑃 , and thus a straightforward quantitative measure of ballisticity is not always possible.\n𝐹𝑊\nFor this reason, we do not attempt to extract a quantitative metric for universal ballisticity\nsolely from R magnitude.\nB\nTo examine the possible role of edge scattering, we compared resistivity values measured\nin Hall bars with different channel widths (Fig. R2-11). The observed increase in resistivity\nwith decreasing channel width indicates that edge scattering becomes more significant in\nnarrower devices and influences their transport behaviour, as expected. This also aligns\nwith decreasing MFP with decreasing film thickness.\nFig. R2-11 | Newly added Supplementary Fig. 6. Resistivity (ρ ) vs. actual channel\nxx\nwidth in Hall-bar devices.\nEven in the presence of edge scattering, it is important to note that in our study all Hall\nbar devices with different widths (W = 1 μm, 250 nm, 150 nm) were fabricated\nsimultaneously on the same Cu thin film within each measurement set. This ensures that\nintrinsic material parameters such as resistivity and mobility are identical across devices,\nand that their edge properties including roughness is also comparable. Therefore, any\nvariation in R can be attributed solely to geometric effects rather than differences in\nB\nmaterial quality.\nTo isolate this geometric effect, we analysed the normalized bend resistance, R(4.3 K)/R\nB B\n(285 K), across 25 films, each containing all three device geometries. As shown in Commented [조1]: 27에서 25로 수정\nSupplementary Fig. 6a–c in original manuscript (Supplementary Fig. 10a–c in revised\nmanuscript), negative R values become increasingly prominent in narrower devices,\nB\nparticularly when the mean free path exceeds W. Negative R was observed only in a\nB\nsubset of films, indicating a geometry-induced crossover to the ballistic regime.\nTherefore, while we agree that R magnitude is not a universal indicator of ballisticity,\nB\nunder our controlled experimental design, where devices of different widths are fabricated\nsimultaneously on the same film, it serves as a reliable and interpretable comparative\nmetric. Supplementary Figs. 6a–c should be considered collectively, as each reflects\nmeasurements from devices of different geometries on the same film, enabling meaningful\ncomparison of ballistic transport signatures.\nWe have added the following sentence to the revised manuscript as follows (lines 127–129\nof page 6).\n“As W decreases, edge scattering becomes progressively non-negligible and shortens l\nmfp\n(Supplementary Fig. 6), but l still exceeds W=150 nm and ballistic transport is observed.”\nmfp\nReviewer: 10. Please check the formula (3), as the reference gives a different interpretation\nand power of -1. There is also a recent application to gold [4].\nResponse: We sincerely thank the reviewer for pointing out the inconsistency in the\nexponent used in formula (3). Upon re-examining the original reference and related recent\nliterature, including Ref. [4], we confirm that the correct expression indeed involves a power\nof –1. We have revised the formula accordingly in the manuscript to reflect the proper\ninterpretation. We greatly appreciate the reviewer’s careful reading and precise correction.\nReferences mentioned in Reviewer #2’s comments.\n[1] A. Mallikarjunan, S. Sharma and S. P. Murarka, \"Resistivity of Copper Films at Thicknesses\nNear the Mean Free Path of Electrons in Copper: Minimization of the Diffuse Scattering in\nCopper,\" Electrochem. Solid-State Lett. 3, 437 (2000).\n[2] van der Pauw, L. J. A method of measuring the resistivity and Hall coefficient on lamellae\nof arbitrary shape. Philips Tech. Rev. 20, 220–224 (1958).\n[3] G. Govindaraj and V. Devanathan, \"Quantum size effect in thin metal films\", Phys. Rev.\nB 34, 5904 (1986).\n[4] Weigang Ma, Xing Zhang and Koji Takahashi, \"Electrical properties and reduced Debye\ntemperature of polycrystalline thin gold films\", J. Phys. D: Appl. Phys. 43 (2010).\nReviewer #3 (Remarks to the Author):\nReferee Report\nReviewer: I have reviewed the manuscript with interest. The authors report on ballistic\ntransport in high-quality copper films and present a systematic study of the impact of\ngrain boundaries on ballistic conduction. These are timely and relevant findings that could,\nin principle, merit publication. However, the presentation and writing currently do not meet\nthe standards expected for Nature Communications. Below I outline my specific comments.\nResponse: We appreciate the reviewer for the valuable comments. We have examined\nreviewer’s comments below in details and carefully revised our manuscript according to\nthe reviewer’s comments and addressed the issues raised by the reviewer’s point-by-point\nin this response letter.\nMajor Comments\nReviewer: 1. Debye Temperature\nThe extracted Debye temperature is significantly lower than the commonly reported value\nfor copper (~343 K). The authors should explain this discrepancy. Is this due to material\nquality, finite-size effects, or the fitting method used? Clarifying this point is important to\nsubstantiate the validity of the extracted phonon parameters.\nResponse: We thank the Reviewer for the careful reading and constructive comments\nregarding the Debye temperature. We agree that the extracted Debye temperature\nΘ_R=270K differs from the commonly reported bulk value Θ_R≈330 K. This discrepancy is\nnot unusual and can arise from a range of factors that influence phonon spectra and\nscattering processes in metallic systems. In particular, the Debye temperature is known to\nbe sensitive to:\n1. Purity and defect density (e.g., vacancies, dislocations, or oxidation),\n2. Crystal structure and orientation (single-crystalline vs. polycrystalline),\n3. Surface condition (roughness, oxidation, contamination),\n4. Measurement technique and configuration, and\n5. Dimensionality or finite thickness effects in thin films.\nWhile our samples are nearly single-crystalline and thus might be expected to exhibit a\nDebye temperature close to or even higher than bulk Cu, the thin film geometry and\npossible defects introduced during fabrication steps (e.g., etching, lift-off, surface exposure\nto ambient atmosphere) may reduce the effective Debye temperature. This trend is\nconsistent with prior reports on thin Cu films, where reduced dimensionality and surface\ndegradation are known to lower the observed Debye temperature [R15].\nWe have added the following sentence to the revised manuscript as follows (lines 113–121\nof page 5-6):\n“The sample used in this study is nearly single-crystalline and would typically be expected\nto exhibit a Debye temperature close to or even higher than that of polycrystalline copper.\nHowever, the observed Debye temperature of 270 K is slightly lower than the commonly\nreported bulk value (~330 K). We attribute this deviation to the thin film geometry and the\nfabrication process. In particular, device fabrication steps such as Ar ion etching and\nchemical treatments may introduce surface roughening, partial oxidation, or other\nstructural imperfections that alter the phonon spectrum and reduce the effective Debye\ntemperature. Similar reductions in Debye temperature due to dimensional confinement\nand surface degradation have also been reported in sputtered copper thin films40.“\nReferences\n[R15, Newly added Ref. 40] Timalsina, Yukta P. et al. Effects of nanoscale surface roughness\non the resistivity of ultrathin epitaxial copper films. Nanotechnology 26, 075704 (2015).\nReviewer: 2. Estimation of Mean Free Path\nCan the authors extract the Hall mobility from their data (e.g., via a multi-band model fit)?\nIf so, they could estimate the electronic mean free path independently. Such a value would\nprovide a useful consistency check for their claims of ballistic transport.\nResponse: We calculated l using Eq. 2 in the main text and obtained a value of 170\nmfp\nnm, which is consistent with the observation that negative value of R begins to appear\nB\nfor channel width W between 150 nm and 250 nm. In formula (2), we evaluated the\nmean free path using the effective mass m*=1.1843 × 10-30 kg as reported in [R16], and\nthe Fermi velocity v=1.57 × 106 m/s [R17]. Electron density n=8.49×1028 m-3 was\nF\nobtained from the free-electron model assuming one conduction electron per Cu atom\n[R17].\nWe have added the following sentence (and an effective mass m* = 1.1843 × 10-30 kg) and\nreferences 41 (R16), 42 (R17) to the revised manuscript as follows (lines 129–131 of page 6):\n“Here, l is calculated using equation (2) with a Fermi velocity41 v = 1.57 × 106 m/s, an\nmfp F\nelectron density42 n = 8.49 × 1028 m⁻³, and an effective mass42 m* = 1.1843 × 10-30 kg.”\nReferences\n[R16, Newly added Ref. 41] Shoenberg, D. The de Haas–van alphen effect in copper.\nNature 183, 171-171 (1959).\n[R17, Newly added Ref. 42] Kittel, C. et al. Introduction to solid state physics. John Wiley\n& Sons (2018).\nReviewer: 3. Reversibility of Bend Resistance Measurements\nHave the authors verified the reversibility of their bend resistance experiments—for\ninstance, by reversing current direction or interchanging current and voltage probes?\nWithout such tests, it is difficult to rule out geometric effects or other measurement\nartifacts as the origin of the observed negative bend resistance.\nResponse: We thank the reviewer for the constructive comment. We present here an\nadditional result among the dataset shown in Fig. 3b (Fig. R3-1a), which was not presented\nin the main manuscript. Cu film thickness of this device is 80 nm, and its SEM image is\nshown in Fig. R3-1b.\nTo directly address how the measured bend resistance (R) depends on the choice of\nB\ncurrent and voltage terminals, we performed systematic measurements using both DC and\nDelta-mode techniques.\nFor each representative Cu cross device (150 nm width) exhibiting signatures of ballistic\ntransport at 4.3 K, we carried out DC current-voltage (IV) sweeps across eight distinct\ncontact configurations, covering both forward and reverse arrangements of current and\nvoltage terminals (Table R3-1). Each R value was extracted from the slope of the IV curve\nB\nmeasured between -300 μA and +300 μA. As shown in Fig. R3-1d, all eight configurations\nyielded consistently negative R values, though the magnitude varied. This indicates that\nB\nthe observed negative R is not tied to a particular terminal arrangement but is a robust\nB\nfeature of transport in these nanoscale devices.\nConfiguration Current leads Voltage leads\n# (I⁺ → I⁻) (V⁺ → V⁻)\nTable R3-1 | Contact terminal combinations for bend resistance measurements.\nCurrent (I⁺ → I⁻) and voltage (V⁺ – V⁻) terminal configurations used in the eight\nmeasurement setups shown in Figs. R3-1c-e. Terminals are labelled 1 to 4 in\ncounterclockwise order starting from the left.\nFig. R3-1 | a, The device marked with a red arrow in Fig. 3b was used for the measurements\nshown in b–e. b, Scanning electron microscope (SEM) image of a representative device\nfrom a. c, Measured bend resistance (R) at 285 K via Delta-mode measurement across 8\nB\ndifferent configurations in Table R3-1. d, Measured R at 4.3 K via DC measurement across\nB\n8 different configurations, including forward and reverse arrangements of current and\nvoltage terminals. e, Measured R at 4.3 K via Delta-mode measurement across 8 different\nB\nconfigurations.\nWe measured a different device, selected from the dataset shown in Fig. 3b but not used\nin the main manuscript as shown in Fig. R3-1a. Cu film thickness of this device is 80 nm,\nand its SEM image is shown in Fig. R3-1b. To further confirm the reliability of these\nmeasurements, we also performed Delta-mode measurements on eight representative\nconfigurations at 4.3 K (Fig. R3-1e). This method alternates the bias current between\n+300 μA and -300 μA with a delay time of 300 ms, reducing low-frequency noise and\nimproving precision for low-resistance signals. The results closely match the DC data in\nboth sign and magnitude, reinforcing that the negative R observed is intrinsic and not a\nB\nmeasurement artifact.\nWe then extended the Delta-mode measurements to room temperature (285 K) for the\nsame configurations (Fig. R3-1c). Although the extracted R values showed increased\nB\nvariation up to ±20 mΩ, the overall magnitude remained low, and no sign inversion was\nobserved. This variation is consistent with realistic lithography-induced asymmetries of\ngeometry, such as terminal misalignment depicted in Fig. R3-2a. To support this, we\nconducted finite element method simulations using commercial package (COMSOL\nMultiphysics®) with introducing vertical misalignments of up to 30 nm between opposing\nterminals. As shown in Fig. R3-2b, even a small misalignment of 10 nm can induce R\nB\nchanges of ~10 mΩ, which is quantitatively consistent with the experimental spread.\nFig. R3-2 | a, Schematic of the Cu cross geometry used in the simulations (width 150 nm,\nthickness 90 nm), where the vertical misalignment between opposing terminals was\nsystematically varied. b, Simulated R as a function of vertical misalignment (-30 nm to\nB\n+30 nm) for device widths of 150 nm and 250 nm.\nIn summary, the negative R observed at low temperature is reproducible across all\nB\nconfigurations and two measurement techniques. The configuration-dependent variation\nin R is well explained by plausible fabrication-induced asymmetries, and does not\nB\nundermine the intrinsic ballistic nature of electron transport in our Cu thin films.\nWe have added the following sentence at page 6, line 138-141 in revised manuscript to\nexplain the additional experiments showing that negative R is consistently observed across\nB\ndifferent configuration:\n“To verify that the observed negative R is not due to a specific choice of current–voltage\nB\nconfiguration or artifact, we performed additional measurements across all contact\nconfigurations (Supplementary Fig. 7 and Supplementary Table 1).”\nReviewer: 4. Figure Quality\nThe quality of several figures needs improvement. In particular, font sizes are too small in\nFig. 4a and in the tables of Fig. 2c, f, and i, making them difficult to read. Figures must be\nlegible when printed and should meet the journal’s formatting standards.\nResponse: We thank the reviewer for pointing out the formatting issues. In the revised\nversion, we changed Fig. 4 from seven panels (a–g) to five panels (a–e). In response, we\nreduced the size of panels 4d and 4e in revised Fig. 4 to make the fonts in panels 4a–e\nlarger (Fig. R3-3), and we have increased the font sizes in Fig. 4a as well as in the tables\nshown in Fig. 2c, f, and i (Fig. R3-4) to improve legibility and ensure compliance with the\njournal’s formatting standards.\nFig. R3-3 | Revised Fig. 4\nFig. R3-4 | Revised Fig. 2\nReviewer: 5. References and Justification\nSome references appear unrelated to the claims in the main text. For example:\no Line 73: None of the cited papers mention ballistic transport. If these are meant to\nmotivate the use of ballistic copper interconnects, the rationale must be explicitly stated\nand supported by appropriate references.\nResponse: We thank the reviewer for pointing this out. In the text near line 73, we did not\nintend to cite references about ballistic transport at room temperature. Instead, we tried\nto emphasize that, under the same geometry, our single-crystalline Cu films exhibit strongly\nsuppressed grain-boundary scattering compared to polycrystalline Cu, resulting in a longer\neffective mean free path even in the diffusive regime.\nThis suppression of scattering mitigates the size-effect resistivity increase predicted by\ngrain-boundary and surface-scattering models, thereby reducing RC delay (τ=R×C),\nlowering power consumption and suppressing Joule heating (P=I2R). Collectively, these\nfactors enable faster signal transmission, reduced thermal load, and improved\nelectromigration reliability in scaled interconnects. References [33–35 in revised manuscript]\nin the main text directly address the relationship between resistivity scaling, Joule heating,\nand electromigration limits in nanoscale Cu interconnects, providing appropriate support\nfor our statements.\nReferences\n[33 in revised manuscript] Merchant, S. M. et al. Copper interconnects for semiconductor\ndevices. JOM 53, 43–48 (2001).\n[34 in revised manuscript] Chiang, T.-Y. et al. Impact of Joule heating on scaling of deep\nsub-micron Cu/low-k interconnects. 2002 Symp. on VLSI Tech. Dig. Tech. Pap. (Cat. No.\n01CH37303). IEEE (2002).\n[35 in revised manuscript] Zahedmanesh, H. et al. Copper electromigration; prediction of\nscaling limits. IITC. IEEE (2019).\nMinor Comments\nMain Text:\nReviewer: • L29: We believe that collisions with the sample’s edge cannot be neglected\nand are most likely still dominating. We recommend rephrasing the statement.\nResponse: We thank the reviewer for the valuable comment. We agree that edge collisions\ncannot be neglected and may dominate in narrow devices. In line with this, we observe an\nincrease in resistivity (ρ) as W decreases (Fig. R3-5), indicating that edge scattering\nxx\nbecomes progressively more significant, thereby increasing ρ and reducing the mean free\nxx\npath (l ). Nevertheless, even in the narrowest devices (W = 150 nm), we observe negative\nmfp\nbend resistance (R), implying that l still exceeds W. While we kept the general definition\nB mfp\nof ballistic transport at L29 unchanged, we revised the later part of the manuscript\n(lines 126–131 of page 6) to clarify that edge scattering becomes significant in narrower\ndevices, while negative R still still supports this interpretation.\nB\n“For the device with W = 150 nm, R even shows negative values below 85 K. As W\nB\ndecreases, edge scattering becomes progressively non-negligible and shortens l\nmfp\n(Supplementary Fig. 6), but l still exceeds W=150 nm and ballistic transport is observed.\nmfp\nHere, l is calculated using equation (2) with a Fermi velocity41 v = 1.57 × 106 m/s, an\nmfp F\nelectron density42 n = 8.49 × 1028 m⁻³, and an effective mass42 m* = 1.1843 × 10-30 kg,\ntypical for Cu.\nFig. R3-5 | Newly added Supplementary Fig. 6. Resistivity (ρ ) vs. actual channel width\nxx\nin Hall-bar devices.\nReferences\n[R16, Newly added Ref. 41] Shoenberg, D. The de haas–van alphen effect in\ncopper. Nature 183, 171-171 (1959).\n[R17, Newly added Ref. 42] Kittel, C. et al. Introduction to solid state physics. John Wiley\n& Sons (2018).\nReviewer: • L71: The absence of defect-induced scattering alone does not justify the\nemergence of hydrodynamic transport. In metals, the high carrier density screens the\nCoulomb interactions needed for this regime. If the authors believe hydrodynamics might\nappear in these SCCF Cu films, they must provide strong motivation and suitable references.\nResponse: We thank the reviewer for the insightful comment. We agree that the discussion\non hydrodynamic transport was not directly relevant to the main focus of the manuscript.\nAccordingly, we have removed the corresponding sentence from the revised version.\nReviewer: • L79 (Fig. 1a): Please explain what the red dashed line represents.\nResponse: We thank the reviewer for their careful reading. We have updated the caption\nof Fig. 1a to clarify that the red line indicates the position along which the cross-sectional\ntransmission electron microscopy (TEM) sample was prepared using focused ion beam\nmilling (FIB) for the analysis shown in Fig. 1b.\nWe have added the following sentence to the caption of Fig. 1a in the revised\nmanuscript:\n\"The red line indicates the position where the cross-sectional TEM sample was\nprepared using focused ion beam milling.\"\nReviewer: • L107: A reference should be provided to support the reported electron-phonon\ncoupling values, ideally comparing with either ab initio or experimental data from the\nliterature.\nResponse: We thank the reviewer for the suggestion. We performed a root-mean-square\nerror (RMSE) fitting of the R(T) data from the 10 μm-wide device using the Bloch–\nB\nGrüneisen formula [R23] to extract the electron–phonon coupling parameter α .\nSince\nel-ph\nthe device size is much larger than the estimated electronic mean free path (~170 nm at\n1.7 K), the system is in a fully diffusive regime, where the temperature dependence of\nresistance is primarily determined by electron–acoustic phonon scattering.\nTo quantitatively extract the fitting parameters, we varied α and Debye temperature Θ\nel-ph R\nover a physically reasonable range and generated a two-dimensional RMSE map, as shown\nin Fig. R3-6. For each parameter set, a Bloch–Grüneisen curve was calculated and compared\nto the experimental data. The optimal values were α =8.8×10−8 Ω·m and Θ = 270 K,\nel-ph R\nwhich minimize the RMSE. As shown in Fig. 1d, the resulting fitting curves match well with\nthe measured R(T) data for both the 10 μm and 1 μm devices.\nB\nWe have added the following sentence to the revised manuscript as follows (lines 112–113\nof page 5):\n“The fitting parameters α and Θ were determined from the fitting analysis\nel-ph R\n(Supplementary Fig. 5).”\nWe have revised the following reference 39 (R18) to the revised manuscript as follows\n(lines 109–112 of page 5):\n“As shown in Fig. 1d, the temperature dependence of R for devices with W = 10 and 1\nB\nμm can be well fitted39 with equation (1) using fitting parameters α = 8.8 × 10-8 Ω·m\nel-ph\nand Θ = 270 K, indicative of the occurrence of diffusive transport over the entire\nR\ntemperature range.”\nFig. R3-6 | Newly added Supplementary Fig. 5. RMSE fitting map for Bloch–Grüneisen\nanalysis. Root-mean-square error (RMSE) map calculated by varying the electron–phonon\ncoupling constant α and Debye temperature to fit the measured R(T) data of the 1 μm\nel-ph B\ndevice. The optimal fitting parameters were extracted at the point of minimum RMSE. The\nfitting result values are indicated with a dash.\nReferences\n[R23, Newly added Ref. 39] Bid, A. et al. Temperature dependence of the resistance of\nmetallic nanowires of diameter⩾ 15 nm: Applicability of Bloch-Grüneisen theorem. Phys.\nRev. B. 74, 035426 (2006).\nReviewer: • L113: Define all symbols clearly. Specify the values used for Fermi velocity and\ncharge carrier density.\nResponse: We thank the reviewer for pointing this out. In the revised text, we have clarified\nthe definition of all symbols and explicitly stated the values used in the calculation of the\nmean free path. Specifically, we used a Fermi velocity of v= 1.57 × 106 m/s [R16] and an\nF\nelectron density of n = 8.49 × 1028 m⁻³, which are standard values for copper [R17].\nReferences\n[R16, Newly added Ref. 41] Shoenberg, D. The de haas–van alphen effect in copper.\nNature 183, 171-171 (1959).\n[R17, Newly added Ref. 42] Kittel, C. et al. Introduction to solid state physics. John Wiley\n& Sons (2018).\nReviewer: • L121: A reference should support the statement made here.\nResponse: We thank the reviewer for the suggestion. We have repositioned reference [R18]\nto follow the mention of \"negative\" and added a new reference [R19] to support the\ndiscussion on effect of grain boundary scattering on electronic mean free path.\nWe have revised the following references 23 (R24) to the revised manuscript as follows\n(lines 69–70 of page 3):\n“We observed negative bend resistance in cross-geometry devices, indicating ballistic\ntransport.23” Commented [조2]: Ref 추가\nWe have revised the following references 23,45 (R24,R25) to the revised manuscript as\nfollows (lines 135–138 of page 6):\n“As electrons accumulate near terminal 3, a negative electrical potential builds up,\ndecreasing R, which becomes negative. These findings confirm that ballistic transport,\nB\npreviously hidden by electron scattering at GBs, emerges in an SCCF when GBs are\neliminated.”\nReferences\n[R24, 23 in revised manuscript] Büttiker, M. Four-terminal phase-coherent conductance.\nPhys. Rev. Lett. 57, 1761 (1986).\n[R25, 45 in revised manuscript] Wu, Wen, et al. Influence of surface and grain-boundary Commented [조3]: 44에서 45로 수정\nscattering on the resistivity of copper in reduced dimensions. Appl. Phys. Lett. 84, 2838\n(2004).\nReviewer: • L194: Clarify whether \"ρ_I\" refers to the total resistivity, including the grain\nboundary contribution. If so, consider reformulating the sentence for precision.\nResponse: We thank the reviewer for the comment. In the Mayadas–Shatzkes theory [53\nin revised manuscript], ρ refers to the intrinsic resistivity in the absence of grain boundary\ni\nscattering, arising from bulk mechanisms such as phonons, impurities, and other defects.\nThe contribution from grain boundary scattering is accounted for separately in the model\nthrough a multiplicative correction factor involving the reflection coefficient and grain size.\nWe have revised the sentence in line 214-215 of page 10 accordingly to clarify this point.\n“Here, ρ represents the resistivity arising from electron scattering at the GBs, while ρ\ng i\ndenotes the intrinsic resistivity from all other sources excluding GBs.”\nReferences\n[54 in revised manuscript] Mayadas, A. F. et al. Electrical-resistivity model for polycrystalline\nfilms: the case of arbitrary reflection at external surfaces. Phys. Rev. B 1, 1382 (1970).\nSupplementary Information:\nReviewer: • L33: The term \"actual\" is vague—please clarify what is meant.\nResponse: We thank the reviewer for pointing this out. To avoid ambiguity, we have\nremoved the term “actual” from the caption of Supplementary Fig. 5c in the revised version.\nReviewer: • L43: There appears to be a mismatch in line color between the figure and its\nlegend—this should be corrected.\nResponse: We thank the reviewer for pointing out the inconsistency between the figure\nand its legend. We have corrected the legend in Supplementary Fig. 2 by changing the\ncolor for the 80 nm curve from gray to green to match the figure as shown in Fig. R3-7.\nFig. R3-7 | Revised Supplementary Fig. 2.\nDecision on Nature Communications manuscript NCOMMS-25-28257A\nWe deeply appreciate the opportunity to submit a second revision of our manuscript. We are\ngrateful to all three reviewers for their thoughtful assessments and we thank them for their\ndetailed and valuable comments that helped us further strengthen the analysis. In particular, we\nare grateful to Reviewer #2 for the thorough evaluation and insightful comments that helped us\nimprove the manuscript, and to Reviewers #1 and #3 for their positive assessments and for\nrecognizing that our previous revision satisfactorily addressed their concerns. We have revised our\nmanuscript according to the comments from Reviewer #2 and addressed the issues point-by-point\nin this response letter. Reviewer’s comments are written in black, authors’ responses are written\nin blue, and the changes made in the manuscript are written in red.\nReview #2’s comments: The authors significantly improved both the main text and the\nsupplementary material to support their methodology and conclusions. Proper van der Pauw\nmeasurements and electron heating measurements provide strong support for the validity of\nthe experimental data. Here, I am convinced they unambiguously demonstrated ballistic\nconductance in high quality Cu thin films.\nMajor corrections needed\nHowever, I still cannot agree that this paper meets the standards of the Nature Communications\njournal due to multiple inconsistencies in data interpretation and analysis. Thin Cu films were\nstudied in [Adv. Mater. 36, 2403783 (2024)][49]. (It is hard to believe that the band structure of\nCu can be modified in 90-nm-thick films. Nevertheless, ARPES and transport measurements\nconfirm the presence of holes and electrons in Cu films.) They demonstrated the presence of\nholes as a key characteristic of transport in thin Cu films. By measuring the nonlinear Hall effect\noriginating from two types of carriers in continuous films (equivalent to 10 μm wide devices in\nthe reviewed paper), they were able to extract electron and hole concentrations for different\ntemperatures. The authors reproduced Hall effect measurements and found some agreement. I\nguess we can also trust the evaluated concentrations and mobilities from the two types of\ncarrier models, which were presented in the supplementary of [24] in Fig. S10. Surprisingly, the\nvalues can be roughly taken as follows: carrier density for holes, n, –5.2×1028 m-3; carrier density\nh\nfor electrons, n, – 0.5×1028 m-3; hole mobility, μ, – 130 cm2V-1s-1; hole mobility, μ, – 420 cm2V-\ne h e\n1s-1. Hole mobility is smaller than the electron mobility by a factor of 3; however, the\nconcentration of electrons is 10 times smaller than the concentration of holes.\nTherefore, if two types of carriers are taken into account, conductivities are comparable, and the\n“bulk” resistivities are not possible to disentangle by neglecting one of the values as was done\nin equation (2) from the main text. I suggest paying more attention to the parameters used in\nthis simple formula. Not just “typical for Cu” but the values obtained from your experiment or\nfrom real measurements in [49]. If you use the formula in the following form: l =m*vμ/e, then\nmfp F e\nyou will not need concentration and resistivity. I have l =460 nm for electrons with the given\nmfp\nparameter, but I think they should be adjusted.\nAuthors’ Response:\nWe thank the reviewer for raising this important point. First, we believe that the reviewer meant\nto refer to Fig. S10 in Ref. [49], not in Ref. [24], since Ref. [49] was mentioned later in the review.\nFigure S10 in Ref. [49] shows the Hall resistivity of a 40 nm-thick single-crystalline Cu film (SCCF),\nalong with the corresponding carrier density and mobility extracted via a two-carrier model.\nIn\ncontrast, most of our data on ballistic transport were measured from 90-nm-thick SCCFs with\nnarrow widths. As shown in Ref. [49], carrier type, concentration, and mobility exhibit thickness\ndependence. Therefore, it is not straightforward to apply the parameters obtained from a 40-nm-\nthick SCCF to explain the transport properties of devices fabricated from 90-nm-thick SCCFs.\nMoreover, Fig. 4c in this work demonstrates that different channel widths lead to variations in\ncarrier properties due to confined geometry effects.\nIn the previous version of the manuscript, we used resistivity measured from the W = 10 μm\ndevice and carrier density from bulk Cu in the literature. However, in response to the reviewer’s\ncomment, we have now revised our approach for estimating the mean free path. Specifically, we\nuse the measured resistivity (𝜌 = 1.03 × 10-8 Ω·m) and the carrier density (n =\nxx,W=150 nm fit\n1.683×1029 m-3) extracted from Hall measurements in the W = 150 nm device. This device shows\nnearly linear Hall response at low temperatures, as shown in Fig. R1 (newly added as\nSupplementary Fig. 9), indicating electron-dominant transport.\nUsing these values, we calculated the mean free path 𝑙 =\n𝑚∗𝑣𝐹\nby using a simple Drude\nmfp,Drude 𝑛𝑒2𝜌\nmodel with the Fermi velocity41 v = 1.57 × 106 m/s and an effective mass42 m* = 1.1843 × 10-30\nF\nkg from bulk Cu. This gave 𝑙 ~ 42 nm. This value is slightly different from the\nmfp,Drude\nexperimentally inferred mean free path l (~150 nm at 1.7 K) from the observation of negative\nmfp\nR with W = 150 nm, which may be attributed to deviations in the electronic parameters of the\nB\n90-nm-thick, 150 nm-wide Cu device compared to bulk Cu.\nAdditionally, we would like to note that Eq. (2) in our manuscript, l = m∗v/(ne2ρ), is\nmfp,Drude F\nalgebraically identical to the formula l = m∗vμ/e suggested by the reviewer, where μ=1/(neρ).\nmfp F\nTherefore, we can use either of these formulas for the mean free path calculation.\nFig. R1 (Newly added Supplementary Fig. 9) | Hall response of the 90-nm-thick SCCF device\nwith a nominal channel width (W) of 150 nm at 1.7 K.\nThanks to the reviewer’s suggestion, we now include Fig. R1 as Supplementary Figure 9 and have\nmentioned it in the main text.\nWe have revised the following sentence on page 5, lines 123–138 in the main text.\n“Indeed, R(T) of the device with W = 10 μm remains positive and is well described by the Bloch–\nB\nGrüneisen formula, consistent with fully diffusive transport in a device W >> l . However, R(T)\nmfp B\nof devices with W ≤ 250 nm, which is comparable to or shorter than l , deviates from the value\nmfp\nexpected from the Bloch–Grüneisen formula. For the device with W = 150 nm, R even shows\nB\nnegative values below 85 K. Although decreasing W increases the contribution of edge scattering\nand thus reduces l (Supplementary Fig. 6), l nonetheless exceeds W = 150 nm and ballistic\nmfp mfp\ntransport persists. We also estimated the mean free path l , ≈ 42 nm using Drude model\nmfpDrude\nequation (2):\n𝑙 =\n𝑚∗𝑣𝐹\n(2)\nmfp,Drude 𝑛𝑒2𝜌\nHere, we used the measured resistivity 𝜌 = 1.03 × 10-8 Ω·m and the carrier density n = 1.683 ×\n1029 m-3 extracted from the Hall analysis at 1.7 K in the device of W = 150 nm (Supplementary\nFig. 9). A Fermi velocity41 v = 1.57 × 106 m/s and an effective mass42 m* = 1.1843 × 10-30 kg from\nF\nthe bulk Cu were used. l , is slightly different from the experimentally inferred l (~ 150 nm\nmfpDrude mfp\nat 1.7 K) from the observation of negative R in the device of W = 150 nm. The difference may\nB\nbe attributed to the fact that the electronic parameters of the 90-nm-thick, 150-nm-wide Cu\ndevice can differ from those of bulk Cu.\nReference\n[41] Shoenberg, D. The de Haas–van Alphen effect in copper. Nature 183, 171–171 (1959).\n[42] Kittel, C. et al. Introduction to solid state physics. John Wiley & Sons (2018).\n[49] Ok, J. M. et al. Hole‐Carrier‐Dominant Transport in 2D Single‐Crystal Copper. Adv. Mater. 36,\n2403783 (2024).\nIn this light, the analysis of the bend resistance is also incorrect. Note that this means the mean\nfree path (twice as large as the width of the sample or even much larger than the film’s thickness)\nagrees with the statement about the effect of confinement, as a standing wave can be formed in\na narrow sample. In general, holes cannot be neglected in your analysis.\nAuthors’ Response:\nWe thank the reviewer for the comment. However, However, it seems there was some confusion\nabout what we intended to clarify at the outset. Fig. 4c demonstrates that variations in channel\nwidths lead to differences in carrier properties due to geometric confinement, as further illustrated\nin Figs. 4d and e, and discussed in our response to the reviewer’s earlier question on the\ncalculation.\nSpecifically, devices with W = 1 μm and W = 10 μm exhibit nonlinear Hall resistance, indicating\ncontributions from more than two types of carriers. In contrast, the W = 150 nm devices show\nlinear Hall resistance, implying that only a single type of carrier dominates. Accordingly, in our\nanalysis of the narrow device with W = 150 nm, only electron carriers are considered.\nWe emphasize that the confined geometry effect discussed here is distinct from the quantum\nconfinement effect raised by the reviewer. Quantum confinement leads to the formation of\nstanding waves in a narrow device with the number of modes determined by the ratio of device\nsize to the Fermi wavelength. In our case, however, geometric confinement modifies carrier\nproperties without invoking quantum standing-wave effects, and this distinction has been\naddressed in our analysis.\nAlso, the Bloch–Grüneisen formula should not be used in this simple form. I suggest the following\nDrude model to try (assuming successful extraction of mobilities and concentrations at low\ntemperatures).\nAuthors’ Response:\nWe thank the reviewer for the insightful comment and for suggesting a more detailed calculation\nmethod. In previous work (Ref. [49]), the Hall responses of films thicker than ~200 nm were well\ndescribed by a single-carrier model, while films thinner than ~40 nm were best described by a\ntwo-carrier model. The 80–90 nm films in our study fall between these two regimes and naturally\nexhibit more complex transport behavior. In this intermediate regime, electrons are the dominant\ncarriers, but a subset of electrons and holes with different mobilities could also contribute.\nThe 80-nm-thick films in Ref. [49] and the 90-nm-thick devices with W ≥ 1 μm shown in Fig.\n4c display two inflection points in the R as-B curve, which is very different from the behavior\nxy\nobserved in films with other thicknesses. Neither a single- nor two-carrier model can reproduce\nthese two inflection points, indicating the presence of more than two carrier components. As a\nresult, we employed a three-carrier model to fit the Hall measurements of the 90-nm-thick\ndevices with W ≥ 1 μm in this work.\nDuring the revision process, we measured the Hall response of a 90-nm-thick device with W =\n10 μm from T = 1.7 K to 160 K and observed that the Hall resistance curve becomes nonlinear\nbelow approximately 60 K, as shown in Fig. R2.\nFig. R2 (Newly added Supplementary Fig. 6) | Hall response of the 90-nm-thick SCCF device\nwith W = 10 μm at various temperatures.\nWe performed a three-carrier fit for the Hall measurement at 1.7 K (Fig. R3), and extracted the\ncarrier densities and mobilities, summarized in Table R1. While we agree with the reviewer that\nincorporating all relevant carriers for the Bloch–Grüneisen (BG) analysis would be more rigorous,\nwe note that one electron-type dominates the conductivity (≈80 % contribution at 1.7 K). This\ndominant electron-type also contributes more significantly to the conductance at higher\ntemperatures, as evidenced by the Hall resistance becoming more linear at higher temperature\n(Fig. 4c). Therefore, the minority carriers (~20% at 1.7 K) do not substantially affect the overall\ntemperature dependence of ρ(T).\nIncluding all three carriers in the BG analysis would require introducing temperature-dependent\ndensities and mobilities for each carrier, along with several additional phonon-scattering\nparameters. This would substantially increase the number of fitting variables. Since only a single\nρ(T) curve is available, such a multivariable fit would be difficult to constrain in a stable and\nreliable manner. A fully extended analysis of this nature would require a separate in-depth study,\nwhich is beyond the scope of the present manuscript.\nIn the revised manuscript, we clarify that the simplified BG expression is used to capture the\ndominant electron channel. We also explain that the minority carriers (~20 % contribution at 1.7\nK) are not sufficient to significantly alter the temperature dependence of ρ(T) at low temperatures.\nWe sincerely appreciate the reviewer’s valuable insight and kindly ask for understanding on this\npoint.\nFig. R3 (Newly added Supplementary Fig. 7) | Three-carrier analysis of the magnetotransport\nof the 90-nm-thick SCCF Hall bar device with W = 10 μm at 1.7 K. (a) Antisymmetrized Hall\nresistance 𝑅as(B) (symbols) and the corresponding Three-carrier fit (solid line). (b) Symmetrized\nxy\nlongitudinal magnetoresistivity 𝜌s (B) of the same device (symbols) and the corresponding Three-\nxx\ncarrier fit (solid line). The model includes two electron-like and one hole-like carriers. For each\ncarrier i, the longitudinal and transverse conductivity components are given by\nσ (B)=neμ/[1+(μB)2] and σ (B)=sneμ2B/[1+(μB)2], where n is the carrier density, μ is the\nxx,i i i i xy,i i i i i i i\nmobility and s is the charge sign (− 1 for electrons and +1 for holes). The total conductivities are\ni\nσ =∑σ and σ =∑σ , and the fitted n, μ and conductivities are summarized in Supplementary\nxx i xx,i xy i xy,i i i\nTable 1. The simultaneous Three-carrier fit reproduces both 𝑅as(B) and 𝜌s (B) with coefficients of\nxy xx\ndetermination R2 = 0.994 and 0.991, respectively.\nConductivity\ncontribution\nElectron #1 8.76×1028 m-3 100 cm2/V-1·s-1 1.40×108 S/m 76 %\nHole #1 3.67×1027 m-3 595 cm2/V-1·s-1 3.50×107 S/m 19 %\nElectron #2 7.73×1026 m-3 765 cm2/V-1·s-1 9.47×106 S/m 5 %\nTable R1 (Newly added Supplementary Table 1) | Three-carrier fitting results obtained from the\nmagnetotransport of the 90-nm-thick SCCF Hall-bar device with W = 10 μm at 1.7 K.\nThanks to the reviewer’s suggestion, we now include Fig. R2, R3 and Table R1 as Supplementary\nFigure 6, 7 and Supplementary Table 1 and have mentioned it in the main text.\nWe have added the following sentence on page 5–6, lines 110–114 in the main text.\n“We also note that Hall measurements of a device with W = 10 μm (Supplementary Fig. 6) indicate\nthe presence of multiple carrier types at 1.7 K, but an electron-like band provides the dominant\ncontribution to the conductance (Supplementary Fig. 7 and Supplementary Table 1). These results\nsupport the use of equation (1) as an effective single-carrier Bloch–Grüneisen description of R(T).”\nB\nReference\n[49] Ok, J. M. et al. Hole‐Carrier‐Dominant Transport in 2D Single‐Crystal Copper. Adv. Mater. 36,\n2403783 (2024).\nMinor corrections\n• “When the electronic mean free path is longer than the device size, electrons exhibit\nballistic transport without significant scattering from phonons…” Personally, this phrase might be\nconfusing, as I can choose any size of the device. In this case, it is unclear what can be called\nsignificant, as ballistic transport is a relative quantity, but scattering probability has an absolute\nvalue. If you are referring to an elastic MFP, then it means the average distance an electron travels\nbetween any scattering event. In that case, it is not particularly relevant to your experiment, as\nyou have a transport MFP.\nAuthors’ Response: We appreciate the reviewer’s comment. In the opening sentence, our intent\nwas to provide a general, qualitative definition of ballistic transport. We were not implying that\nthe intrinsic electronic properties of the material change with device size. To clarify this point, we\nhave revised the following sentence on page 2, lines 48–49 in the main text.\n“When the electronic mean free path exceeds the device size, electrons undergo ballistic transport\nwithin the device, experiencing minimal scattering.”\nWe also clarify that scattering rate during transport within devices can depend on device geometry\nsuch as film thickness and channel width. When the film thickness or channel width is reduced,\nsurface and sidewall scattering become more prominent and the scattering probability increases.\nThis dependence on geometry is explicitly described in the Fuchs–Sondheimer model for thin\nmetallic films [R1] and in studies of metallic nanowires [R2]. In this context, “mean free path” refers\nto the transport mean free path extracted from our device-level measurements. The transport\nmean free path describes the electron’s behavior in the specific device, where both elastic\nscattering and inelastic scattering play a role, while the elastic mean free path characterizes only\nelastic events. Since the ‘electron mean free path’ in this work is obtained from transport\nmeasurements, it should be interpreted as a transport mean free path.\n• “Ballistic transport reveals intrinsic quantum properties of solid-state materials, as electrons’\nquantum information—such as crystal momentum, spin, and quantum phase—is well conserved.”\nTransport MFP and spin coherence length are different quantities. Quantum phase is even more\nsubtle in metals and is related to a phase coherence length. I expect it to be significantly larger\nat low temperatures than the smallest 150 nm device you have used [Phys. Rev. B 68, 085413\n(2003)].\nAuthors’ Response: We appreciate the reviewer’s clarification regarding the distinction between\nmean free path and phase-coherence length, spin coherence length. In metallic systems at low\ntemperature, it is generally expected that the phase-coherence length and spin-relaxation length\ncan exceed the transport mean free path, because phase coherence is mainly lost through\ninteractions with the environment via scatterings. For instance, a study by Villamor et al. [R4]\nobserved spin relaxation lengths between 500 nm and 1 μm in Cu nanowires with thicknesses of\n40–100 nm.\nWe have revised the following sentence on page 2, lines 49–51 in the main text.\n“Ballistic transport can reveal intrinsic quantum properties of solid-state materials, as electrons’\nquantum properties—such as crystal momentum, spin, and quantum phase—is well conserved”\nremains appropriate in the context of our work.\n• “However, growing atomically uniform and ultraflat Cu thin films without crystalline defects such\nas GBs has been technically challenging, and ballistic transport in Cu thin film nanostructures has\nnot been achieved, limiting the exploitation of their intrinsic quantum properties.” I am not sure\nif it is correct. There are measurements of ballistic hot electron transport in Cu [Phys. Rev. B 78,\n174426 (2008)][J. Vac. Sci. Technol. B 34, 040801 (2016)].\nAuthors’ Response: We appreciate the reviewer’s valuable comment. The ballistic electron\nemission microscopy (BEEM) studies cited by the reviewer [Phys. Rev. B 78, 174426 (2008), J. Vac.\nSci. Technol. B 34, 040801 (2016)] probe hot electrons injected from an STM tip, typically 1–2 eV\nabove the Fermi level, propagating vertically through multilayer metal structures. In such\nexperiments, an attenuation length λ is extracted from the exponential decay of the BEEM collector\ncurrent I(d) ∝ exp(−d/λ), as a function of the metal layer thickness d. This λ characterizes the\nc\ndecay of nonequilibrium, hot-electron-currents and is strongly influenced by interface transmission\nand Schottky barrier effects, so it should be regarded as an effective hot-electron- decay length\nrather than an intrinsic mean free path for equilibrium electrons inside Cu.\nIn contrast, our study addresses low-bias, equilibrium in-plane transport of electrons near the\nFermi level in single-crystalline Cu nanostructures. The negative bend resistance observed in the\nW = 150 nm device at 1.7 K provides direct evidence that the in-plane mean free path of\nequilibrium electrons exceeds the device width (l ≥150 nm). This length scale is physically\nmfp\ndistinct from the hot-electron attenuation lengths reported in BEEM studies, and the measurement\ngeometry and analysis framework are also different.\nWe have revised the following sentence on page 2, lines 57–60 in the main text.\n“However, growing atomically uniform and ultraflat Cu thin films without grain boundaries has\nbeen technically challenging, and in-plane ballistic transport in Cu thin film nanostructures has\nnot been achieved, limiting the exploitation of their intrinsic quantum properties.”\n• “Our SCCF enables studies on the quantum nature of Cu thin films, including topological\nproperties, the quantum Hall effect, confined geometry effect, phase coherent quantum\ninterference and quantized conductance applicable to other metallic thin films.” This is a highly\nspeculative suggestion. Any metallic film has a high electron concentration, which impedes\nobservation of QHE. The quantized transport is indeed possible; however, the quality of the film\nis irrelevant, as this phenomenon is related to atomic-size quantum point contacts. One needs to\ncreate a discrete energy spectrum in a metallic system. It is achieved only for a few monolayer-\nthick metallic films. Considering the effect of quantum confinement for metallic films thicker than\n10 nm is unreasonable, especially when they have a 2D cross-section (effectively a 1D wire), where\na rectangular quantum well should be discussed with a smaller distance between the energy levels.\nAuthors’ Response: We appreciate the reviewer’s valuable comment. We agree that suggesting\nthe quantum Hall effect, quantized transport, and the effect of quantum confinement in our Cu\nthin films may be premature. We have removed these claims from the manuscript.\nWe have revised the following sentence on page 3, lines 70–72 in the main text.\n“Our SCCF enables studies on the quantum nature of Cu thin films, including topological\nproperties24-26 and phase-coherent quantum interference27,28.”\n• Fig.1. “d”-label is missing.\nAuthors’ Response: We thank the reviewer for pointing this out. The missing “d” label in Fig. 1\nhas been added.\nFig. R4 | Revised Fig. 1.\n• Equation 4a is incorrect. It gives expression for a ballistic channel, not for a sample to\nbe compared with measurements. The number of channels should be included.\nAuthors’ Response: We thank the reviewer for this comment. In our formulation, the equation\n(4a) was not intended to represent a single-channel case. The parameters P , P, P, in the\nFW L R\nmanuscript already account for the number of channels. Therefore, we clarify that these\nparameters represent the mode-averaged transmission probabilities. This point has been clarified\nin the revised text.\nWe have revised the following sentence on page 13, lines 254–256 in the main text.\n“This ballistic transport behavior can be qualitatively explained using the Landauer–Büttiker\napproach23. 𝑅s can be expressed in terms of the forward mode-averaged transmission probability\nB\nP , and the mode-averaged transmission probabilities for turning left P and turning right P, as\nFW L R\nshown in equation (4):”\n• “A 12-ML (ML: monolayer) Cu(111) slab structure was used to simulate the 2D-limit thin film,\nwhere a 6-ML×6-ML rod structure was used for the 1D-limit nanorod. We maintained a 20-Å-\nthick vacuum layer to prevent interactions between the periodic images.” I am not an expert;\nhowever, it is quite unusual to conclude the band structure behavior of a relatively large crystal,\n150 nm and 80 nm, based on a single calculation. I would rather investigate the trend by increasing\nthe number of layers: 4ML, 8ML, 12ML, etc.\nAuthors’ Response:\nWe appreciate the reviewer’s concern regarding the validity of using a 12-ML thick Cu(111) slab\nto model the behavior of a 150 nm film. We agree that rigorous testing of convergence with\nrespect to slab thickness is necessary for reliable calculation.\nWe believe that a 12-ML slab is sufficient to represent the system for the following reasons, which\nare now quantitatively supported by the requested convergence test:\nTo directly address the reviewer’s suggestion, we have investigated the convergence of the\nelectronic structure, specifically the Fermi surface topology, using Cu(111) slabs of varying\nthickness (6-ML, 9-ML and 12-ML).\nAs shown in Figure R5, the calculated Fermi surfaces demonstrate rapid convergence.\n The 6-ML slab already exhibits a qualitative equivalence in the Fermi surface topology\ncompared to the 9-ML and 12-ML slabs.\n The Fermi surface features calculated using the 12-ML slab are effectively converged with\nrespect to the 9-ML slab, confirming that the central layers of the 12-ML slab accurately\nsimulate the bulk electronic states relevant to the thin film.\nThis rapid convergence is also physically expected for Cu(111). Owing to its close-packed nature,\nthe Cu(111) surface is known to be structurally robust and bulk-like, exhibiting minimal surface\nreconstruction and only negligible interlayer relaxation (e.g., a very small contraction of the first\ninterlayer spacing, on the order of ~0.7%) [R5]. Consequently, slab calculations for Cu(111) are\nknown to converge quickly with thickness for electronic and related surface properties, and our\nconvergence results in Figure R5 are consistent with this established behavior.\nBased on both the new quantitative evidence provided in Figure R5 and the established\nunderstanding of the robustness of the Cu(111) surface [R5], we conclude that the 12-ML Cu(111)\nslab thickness provides a converged DFT description for simulating the Cu(111) thin film in our\nstudy.\nThanks to the reviewer’s suggestion, we now include Figure R5 as Supplementary Figure 17 and\nhave mentioned it in the main text.\nWe have added the following sentence on page 13, lines 379–381 in the main text.\n“The choice of a 12 ML Cu(111) slab is supported by thickness-convergence tests on 6-, 9-, and\n12-ML slabs (Supplementary Fig. 17).”\nFig. R5 (Newly added Supplementary Fig. 17) | Convergence of the Two-Dimensional Fermi\nSurface Topology with Slab Thickness. Comparison of the effective Fermi surface (FS)\nintersections with the Cu(111) slice plane, calculated using density functional theory (DFT) for\nslabs of increasing thickness: a, 6-monolayer (ML), b, 9-ML, and c, 12-ML. To allow direct\ncomparison with bulk electronic states, the effective Fermi surfaces are unfolded to the primitive\nunit cell of bulk Cu along the [111] direction at different 𝑘 values. The color scale indicates the\nz\nspectral weight of the effective bands, demonstrating the rapid convergence of the electronic\nstructure (band topology and associated features) as the slab thickness increases.\nReference\n[R1] Fuchs K. The conductivity of thin metallic films. Proc. Cambridge Philos. Soc. 34, 100–108.\n(1938).\n[R2] Chawla R. et al. Electron scattering at surfaces and grain boundaries in Cu thin films and\nwires. Phys. Rev. B 84, 235423. (2011).\n[R3] Waser, R. ed. Nanotechnology, Volume 3: Information Technology. John Wiley & Sons\n(2008).\n[R4] Villamor, E. et al. Contribution of defects to the spin relaxation in copper nanowires. Phys.\nRev. B 87, 094417 (2013).\n[R5] Da, S. et al. Converged properties of clean metal surfaces by all-electron first-principles\ncalculations. Surface science 600, 703-715 (2006).\n[41] Shoenberg, D. The de Haas–van Alphen effect in copper. Nature 183, 171–171 (1959).\n[42] Kittel, C. et al. Introduction to solid state physics. John Wiley & Sons (2018).\n[44] Beenakker, C. et al. Billiard model of a ballistic multiprobe conductor. Phys. Rev. Lett.\n63, 1857 (1989).\n[49] Ok, J. M. et al. Hole‐Carrier‐Dominant Transport in 2D Single‐Crystal Copper. Adv. Mater.\n36, 2403783 (2024)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "I thank the Authors for the very detailed, rigorous, and transparent answers to my questions and the modifications that they\nmade. The additional data, clarifications, and discussion have significantly improved the paper in my view.\nI have no more question and recommend publication.", "reviewer_2": "What are the noteworthy results? ... To the best of my knowledge, it is not the first measurement of ballistic transport in Cu,\nbut rather the first measurements of ballistic transport in lateral devices.\nWill the work be of significance to the field and related fields? How does it compare to the established literature? If the work\nis not original, please provide relevant references. ... Yes, I agree that the ballistic transport in Cu is a significant\nachievement. However, in terms of applications at room temperature, it has shown no advancement.\nDoes the work support the conclusions and claims, or is additional evidence needed? ... The main conclusion is well\nsupported, and no further measurements are required.\nAre there any flaws in the data analysis, interpretation and conclusions? Do these prohibit publication or require revision? ...\nI think the analysis can be significantly improved before publication. It needs another major revision. All the details are in the\nattached file.\nIs the methodology sound? Does the work meet the expected standards in your field? ... Yes, the measurements are done on\nthe required level of trust.\nIs there enough detail provided in the methods for the work to be reproduced? ... I think so.", "reviewer_3": "In their extensive review, the authors have addressed all points."}, "author_response": ""}, {"round": 3, "reviewer_comments": {"reviewer_2": "I see no more objections to publication in Nature Communications. The quality of the experiment and fitting can support the\nclaims in the paper."}, "author_response": ""}]

Decision letter