# Engineering ultra-strong electron-phonon coupling and nonclassical electron
transport in crystalline gold with nanoscale interfaces

Shreya Kumbhakar1,⋆,† Tuhin Kumar Maji1,⋆,† Binita Tongbram1 Shinjan Mandal1
Shri Hari Soundararaj1,3 Banashree Debnath1 T. Phanindra Sai1 Manish Jain1 H.
R. Krishnamurthy1,4 Anshu Pandey2 Arindam Ghosh1,† 1Department of Physics,
Indian Institute of Science, Bangalore 560012, India  2Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012,
India  3Mechanical Engineering Deptartement, University of California,
Riverside  4International Centre for Theoretical Sciences, Tata Institute of
Fundamental Research, Bangalore 560012, India

††preprint: APS/123-QED

Electrical resistivity in good metals, particularly noble metals such as gold
(Au), silver (Ag), or copper, increases linearly with temperature ($T$) for
$T>\Theta_{\mathrm{D}}$, where $\Theta_{\mathrm{{D}}}$ is the Debye
temperature [1, 2]. This is because the coupling ($\lambda$) between the
electrons and the lattice vibrations, or phonons, in these metals is rather
weak with $\lambda\sim 0.1-0.2$ [1], and a perturbative analysis suffices to
explain the $T$-linear electron-phonon scattering rate. In this work, we
outline a new nanostructuring strategy of crystalline Au where this
foundational concept of metallic transport breaks down. We show that by
embedding a distributed network of ultra-small Ag nanoparticles (AgNPs) of
radius $\sim 1-2$ nm inside a crystalline Au shell, an unprecedented
enhancement in the electron-phonon interaction, with $\lambda$ as high as
$\approx 20$, can be achieved. This is over hundred times that of bare Au or
Ag, and ten times larger than any known metal. With increasing AgNP density,
the electrical resistivity deviates from $T$-linearity, and approaches a
saturation to the Mott-Ioffe-Regel scale [3] $\rho_{\mathrm{MIR}}\sim
ha/e^{2}$ for both disorder ($T\rightarrow 0$) and phonon
($T\gg\Theta_{\mathrm{D}}$)-dependent components of resistivity (here, $a=0.3$
nm, is the lattice constant of Au). This giant electron-phonon interaction,
which we suggest arises from the coulomb interaction-induced coupling of
conduction electrons to the localized phonon modes at the buried Au-Ag hetero-
interfaces, allows experimental access to a regime of nonclassical metallic
transport that has never been probed before.

The phenomenon of resistivity saturation in disordered metals, especially
transition metals and compounds that are usually good superconductors at low
temperatures, has remained an open problem in solid-state physics for over 50
years [4, 5, 6, 7, 8, 9, 10, 11, 12]. It is commonly agreed that saturation
occurs when the scattering of electrons by either defects or thermal
excitations (phonons) shrinks the mean free path to the order of inter-atomic
spacing – the so-called Mott-Ioffe-Regel (MIR) limit [3, 7]. The quasi-
particles become incoherent, resulting in a class of ‘bad metals’. Although a
comprehensive theoretical understanding of the saturation remains elusive [9,
10, 11, 12], the electron-phonon coupling (EPC) seems to play a key role, and
resistivity saturation has been linked to, for example, the breakdown of Born-
Oppenheimer approximation and delocalization of electronic states [13],
intermediate coupling of phonons to local electronic levels or hopping
integrals [9, 10] or phonon-driven parallel channels of electrical conduction
[11, 12]. In fact, the limit of extremely strong EPC remains poorly understood
in the context of metallic transport in general, and it is not clear though
whether the scattering rate of electrons would be limited by, for example,
possible universal (Planckian) bound [14, 15], polaronic deformation [12], or
indeed, the stability of the metallic state itself against polaronic self-
trapping [16]. The lack of understanding is partly caused by the fact that in
most naturally occurring or synthesized metallic solids/alloys so far, the EPC
parameter ($\lambda$) has not been found to exceed $\sim 2$ [1], let alone a
systematic tunability of $\lambda$ over a broad range in the same system.
Controlled incorporation of disorder has been shown to assist saturation of
resistivity at high temperature, leading to the so called ‘Mooij correlation’
[17], but the EPC remains largely intrinsic and $\lesssim 1$, and the fate of
metallic transport in the limit $\lambda\gg 1$ remains experimentally unknown.

Figure 1: Structure and electrical transport:(a)-(e) Ag@Au nanohybrids at
different length scales. (a) High-Resolution Transmission Electron Microscopy
(HRTEM) image of single AgNP of high crystallinity is embedded inside a
crystalline matrix of Au. The AgNPs are typically spherical and the interface
between AgNPs and the Au host is sharp even at the atomic scale. (b) The
interface sharpness remains robust even after forming a dense AgNP network
within Au. (c) A snapshot of the intermediate stage of fusing (or, cross-
linking) of the Ag@Au nanohybrids that eventually form a continuous and
compact network upon multiple loading via dropcast. (d) Scanning Electron
Microscope (SEM) image of a Ag@Au film after multiple (about ten) iterations
of dropcast and cross-linking. (e) Optical image of typical Ag@Au film
fabricated on pre-patterned Van der Pauw leads. $I+,I-$ and $V+,V-$ represent
the current and volatge contacts for four-probe resistivity ($\rho$)
measurements. (f) Variation of $\rho$ with temperature ($T$) for films with
different Ag volume fraction $F$ (equivalent to AgNP density), showing
metallic transport down to the lowest temperature. (g) Residual resistivity
($\rho_{0}$), defined as the value of $\rho$ observed at $T\sim 6$ K as a
function of $F$. The red and green points were evaluated from films in Van der
Pauw and Hall bar geometries, respectively. (See Extended Data Fig. 1) (h)
Variation of the $T$-dependent component of $\rho$, obtained by subtracting
$\rho_{0}$ from $\rho$, reveals the emergence of sub-linear behavior in $\rho$
at high temperatures with increasing $F$. The pure AuNP film is represented by
$F=0$. Dashed lines represent fit to the data using the two-component parallel
channel model given by Eq. [3].

Well known methods to engineer EPC in solids often depend on confinement or
localization of both electrons and phonons, enabled by defects, topological
disorder or interfaces that cause acoustic impedance mismatch, lifting of
structural symmetry, etc [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In
semiconductor quantum dots and wells, Fröhlich interaction between the charge
and the electric field from the optical phonons is naturally enhanced when the
chemical bonds are polar in nature [28], but placing an interface, for
example, an antiphase boundary, was shown to increase the Huang-Rhys factor (a
measure of the EPC in optically excited semiconductors) by orders of magnitude
even in nominally weakly polar III-V crystals [18]. In metals, however,
attempts to increase EPC by confining electrons [19], confining phonons [20],
interfacial charge transfer [21], enhanced electron surface scattering [22,
29], optical driving [23], or application of stress [24, 25, 26, 27] resulted
only in moderate increase in $\lambda$ within a factor of $\sim$ two. An
alternate strategy involves core@shell (e.g., Au@Ag or Au@Ag@Pt)
nanostructures, where the effective EPC can be continuously tuned with
core/shell mass fraction owing to sound velocity mismatch at the hetero-
interface [30]. Charge scattering at such heterointerfaces seems to determine
the residual resistivity at low temperatures ($T$) in Ag@Au core@shell
nanostructures [31], but their effect on the EPC has not been investigated.

In this work, we have investigated EPC in noble metal hybrids consisting of a
network of nanometer-sized Ag cores embedded in a crystalline Au matrix. The
small intrinsic EPC and near-identical lattice constants of Ag and Au that
preserve a global translation symmetry, provide a simple platform for
analyzing the metallic state resistivity. Using electrical transport and point
contact spectroscopy, we find that both static disorder and the EPC increase
dramatically with increasing density of Ag nanoparticles (AgNP, core), i.e.,
the proliferation of buried Ag-Au interfaces. At intermediate volume fractions
of Ag, $\lambda$ as high as $\sim 20$ could be observed, over ten times that
of any known metal. This regime is also associated with a strong saturation in
electrical resistivity that could be monitored by varying the EPC over nearly
two orders of magnitude for the first time.

Fig. 1a-d show the high-resolution transmission electron microscopy (HRTEM,
Fig. 1a-b; TEM Fig. 1c) and scanning electron microscopy (SEM, Fig. 1d) image
of the hybrid at increasing length scales. The building block consists of
solution-processed $\sim 20–30$ nm Au shells each of which encloses multiple
AgNPs of $\sim 2$ to $5$ nm diameter (Fig. 1a,b). The shells are subsequently
fused, or ‘cross-linked’, and compacted to form macroscopic films on a glass
substrate with pre-patterned electrical leads (Fig. 1e, Extended data Fig. 1).
The Methods and Supplementary Information section I-III describe the chemical
synthesis, characterization, and the film making processes in detail. Fig. 1a
(and also Extended data Fig. 2) emphasizes the sharp interface between AgNPs
and the Au shell, which was found to be the case irrespective of the density
$(\approx F/r_{\mathrm{Ag}}^{3})$ of the AgNPs (here, $r_{\mathrm{Ag}}$ and
$F=V_{\mathrm{Ag}}/(V_{\mathrm{Ag}}+V_{\mathrm{Au}})$ are the radius of the
AgNP and the net relative volume fraction of Ag in the hybrid, respectively).
Typically, we synthesize AgNPs of $r_{\mathrm{Ag}}\approx 1-2$ nm and vary $F$
to tune the concentration of AgNPs, and thereby the inter-AgNP distance,
$d_{\mathrm{Ag}}=2r_{\mathrm{Ag}}/F^{1/3}$, and overall interface density,
$F/r_{\mathrm{Ag}}$.

Figure 2: Point contact spectroscopy of Ag@Au hybrid films: (a) Schematic and
processes: The sensitivity of point contact spectroscopy to the phonon
spectrum arises from the inelastic scattering of high energy electrons from an
emitter (metallic tip) by the phonons in the sample. The tip is biased with a
voltage ($V$) while the sample is grounded thereby driving current ($I$)
across the tip-sample nanocontact of dimension $d_{\mathrm{pc}}$ and
resistance $R_{\mathrm{pc}}$. An electron with initial energy $\epsilon_{1}$
gains an additional energy $eV$ while passing from the tip to the sample for a
ballistic contact where $l\gg d_{\mathrm{pc}}$, $l$ being the mean free path
of the electrons. Straight lines represent electrons, while curly lines
indicate the emission of phonons responsible for the dissipation of the excess
energy. In the case of a diffusive contact, since $l\ll d_{\mathrm{pc}}$ the
energy relaxation at the orifice occurs via emitting multiple phonons, whose
contributions accumulate towards a finite
$\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ even at energies where the phonon
density-of-states vanish (i.e. $eV\gg k_{\mathrm{B}}\Theta_{\mathrm{D}}$). The
resulting weakly varying background at large $V$ is directly proportional to
the electron-phonon coupling $\lambda$. (b) Top panel shows the bias
dependence of measured $\mathrm{d}V/\mathrm{d}I$ (i.e., $R_{\mathrm{pc}}$) and
$\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ for a typical film with Ag volume
fraction $F=0.3$. The bottom panel shows the bias dependence of normalized
point contact spectrum i.e.
$(\rho/R_{\mathrm{pc}})\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ for different
point contact resistances which converge to a geometry-independent background
(black dotted line). (c) Bias dependence of
$(\rho/R_{\mathrm{pc}})\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ for films with
different $F$ measured at $T\approx 5$ K demonstrating an increasing
background value with increasing $F$.

Fig. 1f shows the $T$-dependence of the electrical resistivity ($\rho$) for
Ag@Au nanohybrid films of different $F$ between $6$ K and $300$ K. All the
films showed metallic behavior where $\rho$ decreases monotonically with
decreasing $T$ with very little or no evidence of upturn even at $T\sim 0.3$ K
(Extended data Fig. 3). Thorough compaction and crosslinking result in
geometric uniformity and electrical homogeneity better than $\sim 20\%$
(Extended data Fig. 1) and low background resistivity $\rho\sim
0.2~{}\mu\Omega.\mathrm{m}$ obtained in identically prepared films of bare Au
nanoparticle (i.e., $F=0$). The incorporation of AgNPs causes $\rho$ to
increase rapidly, which decreases again when the composite becomes Ag-rich
($F\rightarrow 1$). This is seen in the variation in the residual resistivity
$\rho_{0}$ (defined as $\rho$ at $T\approx 6$ K) with $F$, shown in Fig. 1g.
Remarkably, in the intermediate range of $F\sim 0.4–0.8$, $\rho_{0}$ is nearly
constant and pinned to the magnitude of $\sim 30-40~{}\mu\Omega.$m, which is
scale of the Mott-Ioffe-Regel limit [3], $\rho_{\mathrm{MIR}}=3\pi^{2}\hbar
a/e^{2}\approx 10~{}\mu\Omega.m$ of metallic resistance for Au ($a=0.3$ nm, is
the lattice constant). At low $F$ ($\leq 0.4$), $\rho_{0}$ increases linearly
with the overall Ag-Au interface per unit volume, suggesting that the
scattering of the electrons occurs dominantly at the buried Ag-Au interfaces
(Extended data Fig. 4) [31].

Figure 3: Electron-phonon coupling in Ag@Au hybrids: (a) Electron-phonon
coupling constant ($\lambda$) estimated from point contact measurements and
electrical transport ($\rho-T$) data as function of Ag filling $F$. Inset
shows exponential decay of $\lambda$ with the distance between two AgNP-Au
interfaces $\sim(d_{\mathrm{Ag}}-2r_{\mathrm{Ag}})$, where $d_{\mathrm{Ag}}$
and $r_{\mathrm{Ag}}$ are the average center-to-center distance and radius,
respectively, of the AgNPs. The dashed line represents decay length scale
$\sim 3.2$ nm.(b) Room temperature resistivity ($\rho_{300~{}\mathrm{K}})$ for
different materials is normalized by the respective Mott-Ioffe-Regel
resistivity ($\rho_{\mathrm{MIR}}$) and plotted as a function of electron-
phonon coupling constant $\lambda$. Red, blue, green, and yellow open circles
represent non-superconducting metals, metals/alloys that superconduct at low
$T$, intermetallic compounds, and high-$T_{c}$ cuprates, respectively (see
Supplementary Information VI for detail). Filled red and purple circles
represent films of pure Au nanoparticle and Ag@Au hybrids for different $F$
values, respectively. (c) (Top Panel): A schematic of the electrochemical
potential of electrons at the Ag and Au sites across the Ag@Au nanohybrid,
$\epsilon_{0}$ being the potential difference between them. Electrons transfer
from a higher onsite potential at Ag to a lower potential in Au. (Bottom
Panel): Theoretical computation of the excess electron occupancy $\delta n$ in
a square lattice toy model (See Methods and Supplementary Information Section
VII), where Ag is embedded inside Au. (d) Variation of the theoretically
computed EPC, $\lambda^{\mathrm{(calc)}}$ with the average excess electron
occupancy on Au, $\langle{\delta n_{\mathrm{Au}}}\rangle$. The dashed line
shows a quadratic dependence of $\lambda^{\mathrm{(calc)}}$ with
$\langle{\delta n_{\mathrm{Au}}}\rangle$. Inset shows the extrapolation of
this quadratic dependence of $\lambda^{\mathrm{(calc)}}$ to higher values of
$\langle{\delta n_{\mathrm{Au}}}\rangle\sim 0.5$. The shaded region of the
inset depicts the regime of charge transfer found experimentally in the Ag@Au
nanohybrids.

The $T$-dependent component of electrical resistivity, i.e., $\rho-\rho_{0}$,
separately shown in Fig. 1h, contains two important features. First, the
increase in the overall magnitude of $\rho-\rho_{0}$ with increasing $F$ at
any given $T$, implies increasing contribution from phonons to resistivity,
and thus enhancement in the ‘effective’ EPC. Secondly, incorporation of the
AgNPs also makes the $T$-dependence of $\rho$ increasingly sublinear at high
temperatures for $T>\Theta_{\mathrm{D}}$, where $\Theta_{\mathrm{D}}\sim 150$
K is the Debye temperature of Au. The coexistence of sub-linear $T$-dependence
of $\rho$ in disordered metals with large EPC, such as the A15 compounds, has
been known for many years [5, 7], but it is not expected in crystalline noble
metals such as Au or Ag (or their alloys). The sublinearity makes the Bloch-
Grüneisen formula ($\rho_{\mathrm{BG}}(T)$) for metallic resistivity, which
derives $\rho\propto T$ for $T>\Theta_{D}$ by treating the EPC perturbatively,
evidently inadequate except for $F=0$ and $1$ (Extended data Fig. 5). This
necessitates an alternative experimental tool to quantify the EPC parameter
$\lambda$ in this case.

To estimate $\lambda$ independently, we have performed point contact
spectroscopy on the Ag@Au nanohybrid films. Fig. 2a and 2b schematically
explain the experimental arrangement and the underlying physical processes,
respectively (see Methods and Extended data Fig. 6 for more detail). $\lambda$
is estimated from the energy-derivative of the resistance $R_{\mathrm{pc}}$ of
a nanoscale contact between the film and a metallic Pt/Rh tip as (see
derivation in Methods) [32],

$\lambda\approx\frac{3.4\pi
ne\hbar}{16m}d_{\mathrm{pc}}\left[\frac{\mathrm{d}R_{\mathrm{pc}}}{\mathrm{d}V}\right]_{V\rightarrow\infty}.$
(1)

where $V$, $n$ and $m$ are the tip-sample bias, electron density in Au and
electronic mass, respectively, and $d_{\mathrm{pc}}=\rho/R_{\mathrm{pc}}$ is
the contact diameter (Maxwell regime). The upper panel of Fig. 2c shows
typical $V$-dependence of $R_{{\mathrm{pc}}}$ and
$\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$. At large $V$, i.e.
$e|V|\gg\epsilon_{t}$, where $\epsilon_{\mathrm{t}}\sim
k_{\mathrm{B}}\Theta_{\mathrm{D}}\sim 10-20$ meV is the energy scale beyond
which the Migdal-Eliashberg function (Eq. 2)
$\alpha^{2}\mathcal{F}(\omega)\rightarrow 0$,
$d_{\mathrm{pc}}\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ becomes independent of
the geometric details of the contact, as seen from the convergence of the
traces at different $R_{\mathrm{pc}}$ (lower panel of Fig. 2c). Fig. 2d shows
the $V$-dependence of $d_{\mathrm{pc}}\mathrm{d}R_{{\mathrm{pc}}}/{d}V$ for
different values of $F$. The rapid increase in its large-$V$ magnitude with
increasing $F$ confirms the enhancement in the EPC with the incorporation of
AgNPs.

Fig. 3a shows the magnitude of $\lambda$, obtained from Eq. 14 using
$d_{\mathrm{pc}}\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ values at large $V$, as
function of $F$. Remarkably, we find $\lambda$ can be as high as $\approx 20$
for $F\sim 0.5–0.7$, before dropping to $\sim 1$, expected for a film of small
AgNPs [29]. Both the magnitude and the $F$-dependence of $\lambda$ from point
contact spectroscopy are consistent within $20\%$ with the estimates obtained
by fitting the Bloch-Grüneisen formula at low $T$ ($\leq 100$ K) (open symbols
in Fig. 3a, see Methods and Extended data Fig. 5 for details). Such large
$\lambda$ is unprecedented in metallic solids, and exceeds those with strong
EPC, for example the A15 compounds, by at least a factor of $\sim$ ten (Fig.
3b, also see Supplementary Information section VI for details). Intriguingly,
the extended correlation between $\lambda$ and the normalized resistivity in
Fig. 3b suggests Ag@Au hybrids to be an ‘extreme’ case of a metal where the
electron-phonon scattering drives $\rho\rightarrow\rho_{\mathrm{MIR}}$ even at
room temperature.

Surface scattering can increase EPC in nanostructured metal films compared to
bulk [22], but such enhancements are within a factor of $\sim$ two, much
smaller than the enhancement observed here. The near-exponential decay of
$\lambda$ with the separation, $d_{\mathrm{Ag}}-2r_{\mathrm{Ag}}$, between
neighbouring Ag-Au interfaces (inset of Fig. 3a), however, suggests the effect
could arise from a (screened) Coulomb interaction between the interfaces and
the conduction electrons. Indeed, the formation of the solution processed
Ag@Au core-shell nanoparticle hybrids, and their stability against galvanic
replacement, are critically dependent on the interfacial charge transfer that
results in to electric dipoles across the hetero-interface [33, 34, 35, 36].
The X-ray photoelectron spectroscopy (XPS) in these nanohybrids suggests
average electron doping of the Au atoms by as much as $\sim 0.3-0.4$ per atom
for $F=0.5$ (Extended Data Fig. 7 and Supplementary Information II.B). To
verify if interfacial charge transfer can indeed cause substantial enhancement
in $\lambda$, we computed the EPC using a two-dimensional lattice with a
periodic square array of Ag atoms embedded within a matrix of Au atoms where
the on-site charge excess charge ($\delta n$) depends on the difference
($\epsilon_{0}$) between the Ag and Au site energies and the long-range
Coulomb interaction between electrons (Fig. 3d (schematic), Methods and
Supplementary Information Section VII). The radial dipoles across the
interface, formed when the Ag|δn|+ and Au|δn|- sites (Fig. 3d, lower panel)
assume opposite oxidation states, couple strongly to the lattice phonons via
long-range Coulomb interactions and generate additional contributions to the
Migdal-Eliashberg function:

$\alpha^{2}\mathcal{F}(\omega)=A\sum_{k,q}|g(k,q)|^{2}\delta(\varepsilon_{k})\delta(\varepsilon_{k+q})\delta(\omega-\omega_{q})$
(2)

where $k$ is the electron wave vector, $q$ and $\omega_{q}$ are the phonon
wave vector and frequency, respectively; $A$ is a normalization constant, and
$g(k,q)$ are the the electron-phonon matrix elements. The nanostructuring and
the resulting charge redistribution generates additional contribution to
$g\propto(\delta n)V_{0}$, where $V_{0}$ embodies the strength of the inter-
site Coulomb interaction and $\delta n$ is the on-site excess electron
occupancy (see Methods and Supplementary Information Section VII for the
analytical and computational details). Fig. 3e shows the calculated EPC,
$\lambda^{\mathrm{(calc)}}$, normalized to its value for $\delta n\rightarrow
0$, as a function of the average electron transfer $\langle\delta
n_{Au}\rangle$ to the Au atoms. We note that $\lambda^{\mathrm{(calc)}}$
increases nearly by a factor of $\sim$ two within the calculated range
$\langle\delta n_{Au}\rangle\leq 0.07$ (limited by the finite lattice size),
and second, $\lambda^{\mathrm{(calc)}}\propto\langle\delta n_{Au}\rangle^{2}$
at larger $\langle\delta n_{Au}\rangle$. The latter, when extrapolated to the
experimentally estimated $\langle\delta n_{Au}\rangle\sim 0.4$, suggests an
enhancement of $\lambda^{\mathrm{(calc)}}$ (shaded region in the inset of Fig.
3e), which is within a factor of $\sim 2-3$ of that observed in experiments.

We now focus on the resistivity saturation at $T\gg\Theta_{\mathrm{D}}$, which
probably is the ‘smoking gun’ signature of the strong emergent EPC in Ag@Au
hybrids. In fact, our ability to vary $\lambda$ by over a factor $\sim 200$
(from bare gold film to Ag@Au nanohybrid at $F\approx 0.7$), allows access to
the phonon contribution to resistivity dynamically from weak to ultra-strong
coupling regime on a single material platform for the first time. In Fig. 4,
we plotted the high-temperature segment ($300$ K $\geq T\geq 150$ K, i.e.,
$T\geq\Theta_{\mathrm{D}}$) of ($\rho-\rho_{0}$) shown in Fig. 1h for all $F$,
where the temperature axis is scaled by the corresponding $\lambda$, obtained
from the point contact measurements. Two key observations can be summarized as
follows: First, the collapse of the resistivity traces for different $F$ onto
a single one suggests $\lambda T$ would continue to be the ‘scaling variable’
that determines the resistivity even at very large $\lambda$, although the
perturbative limit with linear scattering rate $\approx 2\pi
k_{\mathrm{B}}\lambda T/\hbar$, is expectedly recovered only when
$\lambda\rightarrow 0$ (dashed line). Second, the sublinearity in
$(\rho-\rho_{0})$ at large $\lambda T$, representing ‘resistivity saturation’,
can be modeled with a parallel resistor channel as,

$\frac{1}{\rho-\rho_{0}}=\frac{1}{\rho_{\mathrm{BG}}}+\frac{1}{\rho_{||}}$ (3)

where $\rho_{\mathrm{BG}}(T\geq\Theta_{\mathrm{D}})=2\pi
mk_{\mathrm{B}}\lambda T/ne^{2}\hbar$ [1, 2] (See Methods for details), and
$\rho_{||}$ is the resistivity of a parallel non-classical channel whose
universality, temperature dependence, or even existence, have been questioned
many times, but without a satisfactory answer so far [5, 7, 11]. The solid
line fit in Fig. 4 corresponds to $\rho_{||}\approx 20~{}\mu\Omega.$m implying
that the phonon contribution to resistivity can be described by an ‘ideal’
Bloch-Grüneisen behavior in parallel to a $T$-independent non-classical
channel of resistivity close to the MIR-limit. In fact, the variation in
$\rho(T)$ over the entire experimental temperature range can be satisfactorily
captured by using the full form of $\rho_{\mathrm{BG}}(T)$ and Eq. 3 (dashed
lines in Fig. 1h). See Methods and Supplementary Information Section V for
further discussions on the parallel resistor formula and other fit protocols.

Figure 4: Scaling of resisitivity and electron-phonon coupling strength:
Universality in the $T$-dependent component of $\rho$ for
$T>\Theta_{\mathrm{D}}\sim 150$ K (left panel) for all $F$ with $T$ scaled
with the corresponding EPC parameter $\lambda$ from point contact spectroscopy
(right panel). The dashed line in the right panel indicates the ‘Planckian’
resistivity $(=2m\pi k_{\mathrm{B}}\lambda T/(\hbar ne^{2}))$. The solid line
is a fit to $(\rho-\rho_{0})$ according to the two-component parallel channel
model given by Eq. [3] (see text).

Our experiment confirms the long-suspected inevitability of resistivity
saturation in metals [6], irrespective of the strength of the EPC. However,
there are deeper consequences. The first concerns the question, is there a
universal bound to the EPC for a metal to exist? The persistence of metallic
transport in Ag@Au hybrids with $\lambda\gg 1$, is a priori at odds with a
stability bound observed in Monte Carlo calculations on the Holstein model
[16], or other fundamental ‘Planckian limits’ to dissipation or thermalization
in metals [14, 15]. An important consideration would, however, be the
heterogeneous nature of our system, where the electrons couple to engineered
vibration modes of a foreign species, rather than those of the host lattice
itself. While in-homogenous polaronic instabilities could lead to a thermally
activated parallel channel in transport [11, 12] (See Supplementary
Information Section IV), the apparent suppression of the quantum correction to
conductivity, and indeed that of Anderson localization itself, even for very
strong disorder at $F\sim 0.5–0.8$ at low $T$ (down to $\sim 0.3$ K) are
unexpected. These suggest a significant modification of the phase coherent
effects in our nanohybrids, whose origin is not clear at present but inelastic
scattering at the buried interfaces could have a role to play. Third, there is
also a discrepancy with the models of resistivity saturation built on the
breakdown of Born-Oppenheimer approximation and Matthiessen’s rule, for
example, polaronic deformation of disorder [37] or phonon-assisted
delocalization [13, 11, 12]. These mechanisms often involve crossover to
negative temperature coefficient of resistivity at
$\rho\sim\rho_{\mathrm{MIR}}$, i.e., ‘Mooij correlation’ [17, 37], which was
not observed in any of our samples. Finally, we doubt that electron-electron
interaction, which can impact transport in multiple ways including assisting
delocalization of carriers [38], introducing hydrodynamic viscosity [39] or
suppression of resistivity saturation itself [7, 5, 8], is of importance in
our case. This is because the charge carrier density in the hybrids, measured
by the low-field Hall effect (Extended data Fig. 8), is found to be the same
as that of crystalline Au in which electronic interaction is known to be weak.

In conclusion, we have reported the realization of a metallic hybrid, composed
of ultra-small silver cores dispersed in a crystalline gold matrix, in which
the electrical resistivity shows saturation as the silver core density and
temperature are increased. Both electrical transport and point contact
spectroscopy reveal that the electron-phonon coupling $\lambda$ in these
engineered metallic hybrids can be as large as $\sim 20$, more than ten times
than any known metallic solid. Our experiments outline a novel strategy to
modify some of the fundamental properties of metallic solids utilizing buried
interfaces at the nanoscale.

⋆Equal contribution
†Corresponding authors:
<EMAIL_ADDRESS>
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## Methods

### Chemical synthesis

Ag@Au nanohybrids (Ag@Au NHs) were synthesized using a colloidal approach [31,
40]. The synthesis process involved two sequential stages: the reduction of
AgNO3 with ice-cold NaBH4 to form Ag nanoparticles (AgNPs) in an aqueous
solution (in ultrapure water, purity $\sim 18.2~{}\mathrm{M}\Omega$.cm)
containing NaOH, NH4Br, KI, and CTAB as a capping agent, followed by the
introduction of HAuCl4 at $40\hskip 1.42271pt^{\circ}$C with continuous
stirring. In situ, UV-Vis absorption spectroscopy was employed to monitor the
synthesis process, and the reaction was terminated by adding isopropyl alcohol
(IPA), resulting in nanohybrid agglomeration. The solution was then
centrifuged at $10,000$ rpm for $15$ minutes to remove excess CTAB and isolate
the nanohybrids.

### Characterization

UV-Vis Spectroscopy: UV-vis absorption spectroscopy was used to assess the
formation of AgNPs and Ag@Ag nanohybrids. The addition of NaBH4 in the
reaction mixture containing AgNO3, at a specific ’wait time’ ($t$) resulted in
a prominent peak around $\sim 393$ nm, corresponding to the localized surface
plasmon resonance (LSPR) absorption band of ultra-small AgNPs. Upon
introducing HAuCl4, the LSPR peak underwent a redshift to $\sim 524$ nm over
approximately $1000$ seconds, indicating the formation of a thicker shell due
to spontaneous interdiffusion. To ensure a well-defined AgNP surface, we
employed a strategic approach. Monitoring the SPR band shift with UV-Vis
spectroscopy, we terminated the reaction by adding isopropyl alcohol (IPA)
approximately $30$ seconds after HAuCl4 addition.

X-Ray Photo-electron Spectroscopy: An Axis Ultra K$\alpha$ X-ray photoelectron
spectrometer with a monochromatized photon energy of $\sim 1486.6$ eV was used
for all XPS measurements. To minimize moisture absorption, the sample was
quickly inserted into the load-lock of the instrument, pumped in the entry
chamber until the pressure around $10^{–8}$ mbar was reached, and subsequently
transferred to the analysis chamber. The individual core-level spectra were
corrected for charging using C $1s$ at $284.5$ eV as standard. The peak
fitting of the individual core-levels was done using Casa XPS software.

Transmission Electron Microscopy: Transmission Electron Microscopy (TEM) was
employed to examine the structural characteristics of an Ag@Au nanohybrid. The
FEI TITAN Themis TEM operating at $300$ kV, which offers a point resolution of
$\sim 0.2$ nm and an energy spread of $0.136$ nm, was utilized to capture all
TEM images. High-Resolution Transmission Electron Microscopy (HRTEM) images
were obtained along the zone axis to assess the Ag-Au interface and
investigate any defects that may have arisen within the bimetallic entities.
To prepare the sample for TEM imaging, the nanoparticle underwent multiple
cleaning cycles in a CHCl3 : methanol mixture ($1:3$ ratio), followed by
centrifugation at $15000$ rpm for $5$ minutes. The resulting precipitated
sample was then dispersed in chloroform and deposited onto a carbon-coated TEM
grid, which was subsequently dried under vacuum overnight.

### Film preparation

The drop-cast technique was employed to fabricate the Ag@Au NH film onto
prepatterned Cr$/$Au contacts (with thickness of $\approx$ 10 nm$/$60 nm)
arranged in various lead configurations on a glass substrate. Prior to drop-
casting onto the pre-patterned leads, the sample was dissolved in CHCl3.
Subsequently, the sample was dried at 70 ∘C for 30 seconds and washed with
deionized (DI) water followed by KOH solution and IPA to eliminate any excess
CTAB and achieve a chemically sintered cross-linked nanostructure. This
process was repeated ten times for each film, resulting in an average film
thickness of $t_{\mathrm{f}}\approx 3\pm 0.5~{}\mu$m and a diameter of
$\approx 4$ mm, typically covering the leads (Fig. 1e of Main Manuscript,
Extended Data Fig. 1).

### Electrical Measurement

Four-probe resistivity of the sample was measured down to temperature ($T$)
$\sim 6$ K in a home-built cryostat by passing a DC current of $\sim
100~{}\mu$A with Keithley 6221 and measuring the voltage with Keithley 2182A.
Voltage across multiple contacts were recorded using Keithley 3700 Multiplexer
card. The voltage was measured in delta mode to cancel any thermo-emf across
the contacts. Resistivity from $T\sim 10$ K down to $T\sim 0.3$ K was measured
in a He3 cryostat.

### Fitting of $\rho-T$ data

The resistivity ($\rho$) of metal with electron-phonon interaction playing the
dominant role of scattering can be expressed in terms of the Bloch Grüneisen
[1, 2] form as:

$\rho(T)=\rho_{0}+\rho_{\mathrm{BG}}(T)$ (4)

where $\rho_{0}$ is the residual resistivity, and

$\rho_{\mathrm{BG}}=\frac{2\pi\lambda
k_{\mathrm{B}}/\Theta_{\mathrm{D}}}{(n/m)e^{2}}\left(\frac{T}{\Theta_{\mathrm{D}}}\right)^{5}\int_{0}^{\Theta_{\mathrm{D}}/T}\frac{x^{5}}{(e^{x}-1)(1-e^{-x})}dx$
(5)

is the Bloch Grüneisen form of resistivity arising from electron-phonon
scattering. $\Theta_{\mathrm{D}}$, the Debye temperature, and $\lambda$, the
electron-phonon coupling constant can be estimated by fitting the $\rho-T$
data with Eq. [4]. As shown in Fig. 1(h) and Extended Data Figure 5(a), we
have fitted the $\rho-T$ data of AuNP and AgNP films using Eq. [4].
$\Theta_{\mathrm{D}}\sim 170$ K, and $\lambda\sim 0.45$ for Au, and
$\Theta_{\mathrm{D}}\sim 190$ K and $\lambda\sim 1$ for Ag are estimated as
fit parameters. The increased value of $\lambda$ as compared to the bulk value
of $\sim 0.2$ for Au and Ag could be attributed to the nanostructuring in the
film and increased electron scattering from the surfaces [22]. For Ag@Au
films, Eq. [4] cannot describe $\rho-T$ for the entire range of $T$. Extended
data Fig. 5(b) shows that the transport data for $F=0.5$ deviates from the
low-temperature BG fit ($T\leq 100~{}\mathrm{K})$ to the data. For fitting the
$\rho-T$ data for Ag@Au hybrid films, Eq. [2] in main manuscript is used where
$\rho_{\parallel}$, $\Theta_{\mathrm{D}}$ and $\lambda$ are the parameters of
fit. $\Theta_{\mathrm{D}}$ obtained from fitting $\rho-T$ of Ag@Au films with
Eq. [4] in the low $T$ range, and with Eq. [2] with a parallel conduction
channel, are consistent and lies within the range of $150-170$ K for all films
(Extended Data Figure 5(c)). $\Theta_{D}$ being close to the Debye temperature
of Au ($\Theta_{\mathrm{D,Au}}\sim 170$ K) in all cases indicates the
electrical conduction occurs primarily within the host lattice of Au.
$\lambda$ derived from the low-$T$ Bloch Grüneisen fit is shown in Fig. 3(a).
However, $\lambda$ estimated from parallel channel fit is slightly
overestimated ($\sim 20\%$) and probably less accurate since the scattering
mechanism with this model even at low $T$ is not purely electron-phonon
mediated. $\rho_{\parallel}$ estimated from the parallel channel fit, given by
Eq. [2] of main manuscript, is plotted in Extended Data Figure 5(d) as a
function of $F$ after normalizing with $\rho_{\mathrm{MIR}}$.
$\rho_{\parallel}\approx 8-25~{}\mu\Omega.$m is $T$-independent, and lies
within a factor of two of $\rho_{\mathrm{MIR}}\sim 10~{}\mu\Omega$.m, which
drives the saturation of $\rho(T)$.

### Point contact Measurements

Experimental setup: A sharp Pt/Rh metallic tip is brought in contact with the
film in a controlled manner with the help of nanopositioners (attocubes and
piezo tubes) as indicated in the schematic of the experimental set-up in Fig.
2(a) of main manuscript and Extended Data Figure. 6. The tip-sample chamber is
loaded inside a home-built cryostat that could be cooled down to $T\sim 5$ K.

Theoretical framework for analysis: Point contact measurements are done with
the technique of modulation spectroscopy [32, 41] which measures the higher-
order derivatives of a signal through its AC components at harmonics of a
definite frequency. The circuit for measurement of the point contact voltage
is shown in Extended data Fig. 6. A mixed AC+DC current
$I+i_{\mathrm{m}}\mathrm{sin}(\omega t)$ is passed through the sample using a
constant current circuit. AC voltage from a lock-in amplifier and DC voltage
from Keithley 2400 are added with an op-amp adder. A series resistance
$R_{\mathrm{s}}\gg R_{\mathrm{pc}}$ is used to achieve the constant current in
the circuit. The voltage across the tip-sample contact can be represented as a
Taylor series expansion of the $I-V$ curve as follows:

$\displaystyle V={}$ $\displaystyle f(I+i_{\mathrm{m}}\mathrm{sin}(\omega
t))=\sum_{k=1}^{\infty}\frac{i^{k}_{\mathrm{m}}}{k!}\frac{\mathrm{d}V^{k}}{\mathrm{d}^{k}I}\mathrm{sin}^{k}(\omega
t)$ $\displaystyle={}$ $\displaystyle V+i_{\mathrm{m}}\mathrm{sin}(\omega
t)\frac{\mathrm{d}V}{\mathrm{d}I}+\frac{1}{2!}\frac{\mathrm{d}^{2}V}{\mathrm{d}I^{2}}i_{\mathrm{m}}^{2}\mathrm{sin}^{2}(\omega
t)$
$\displaystyle+\frac{1}{3!}\frac{\mathrm{d}^{3}V}{\mathrm{d}I^{3}}i_{\mathrm{m}}^{3}\mathrm{sin}^{3}(\omega
t)+...$ $\displaystyle={}$ $\displaystyle
1\left(V+\frac{i_{\mathrm{m}}^{2}}{4}\frac{\mathrm{d}^{2}V}{\mathrm{d}I^{2}}+...\right)+$
$\displaystyle\mathrm{sin}(\omega
t)\left(i_{\mathrm{m}}\frac{\mathrm{d}V}{\mathrm{d}I}+\frac{i_{\mathrm{m}}^{3}}{8}\frac{\mathrm{d}^{3}V}{\mathrm{d}I^{3}}+...\right)-$
$\displaystyle\mathrm{cos}(2\omega
t)\left(\frac{i_{\mathrm{m}}^{2}}{4}\frac{\mathrm{d}^{2}V}{\mathrm{d}I^{2}}+\frac{i_{\mathrm{m}}^{4}}{12}\frac{\mathrm{d}^{4}V}{\mathrm{d}I^{4}}+...\right)+..$
(6)

Grouping terms of the same amplitude of the modulation frequency $\omega$
gives

$V=V_{0}+\sum_{i=n}^{\infty}(a_{2n-1}\mathrm{sin}((2n-1)\omega
t)+a_{2n}\mathrm{cos}(2n\omega t))$ (7)

For a small AC current $i_{\mathrm{m}}\ll I$, the higher-order terms in the
series expansion, which vary as $i^{n}$ can be neglected in each group and the
amplitude of the voltage at frequency $n\omega$ becomes proportional to the
$n$-th order derivative .i.e
$a_{n}\propto{\mathrm{d}^{n}V}/{\mathrm{d}I^{n}}$. Also, it is to be noted
that the odd harmonics are in-phase (sine component) with the source AC signal
$i_{\mathrm{m}}\mathrm{sin}(\omega t)$, and even harmonics are out of phase,
at $90^{\circ}$ (cosine component) with the signal.
Hence, the first order derivative can be estimated from the amplitude of the
$\omega$ component at $0^{\circ}$ phase as in Eq. [6].

$\displaystyle a_{1}={}$ $\displaystyle
i_{\mathrm{m}}\frac{\mathrm{d}V}{\mathrm{d}I}$
$\displaystyle\frac{\mathrm{d}V}{\mathrm{d}I}={}$
$\displaystyle\frac{a_{1}}{i_{\mathrm{m}}}$ (8)

The second order derivative can be derived from the amplitude of the $2\omega$
component at $90^{\circ}$ phase as in Eq. [6].

$\displaystyle a_{2}={}$
$\displaystyle-\frac{i_{\mathrm{m}}^{2}}{4}\frac{\mathrm{d}^{2}V}{\mathrm{d}I^{2}}$
$\displaystyle\frac{\mathrm{d}^{2}V}{\mathrm{d}I^{2}}={}$
$\displaystyle-\frac{4a_{2}}{i_{\mathrm{m}}^{2}}$ (9)

Both $\omega$ and $2\omega$ components of the voltage difference across the
tip-sample junction are acquired simultaneously with two lock-in amplifiers as
denoted by $V_{\omega}$, and $V_{2\omega}$ in Extended Data Fig. 6
respectively, after amplification with SR 560. The sample is grounded through
current-voltage amplifier SR 570, which allows us to constantly monitor the
current through the sample and hence tune the point contact resistance
($R_{\mathrm{pc}}$).
Since a lock-in amplifier shows the rms value of a signal, the amplitude of
the signal is $\sqrt{2}$ times the measured value i.e.
$a_{n}=V_{n\omega}\sqrt{2}$, where $V_{n\omega}$ is the measured signal at
$n\omega$. If $V_{\mathrm{AC}}$ is the voltage at the sine output of the
lockin amplifier, the AC current through the sample is
$i_{\mathrm{m}}=V_{\mathrm{AC}}\sqrt{2}$. $V_{\omega}$, being the measured
voltage by the lock-in amplifier at $\omega$ component and $0^{\circ}$ phase,
the amplitude of the voltage drop across the tip and sample is expressed using
Eq. [8] as

$\frac{\mathrm{d}V}{\mathrm{d}I}=\frac{a_{1}}{i_{\mathrm{m}}}=\frac{V_{\omega}\sqrt{2}}{V_{\mathrm{AC}}\sqrt{2}/R_{\mathrm{s}}}=\frac{V_{\omega}}{V_{\mathrm{AC}}/R_{\mathrm{s}}}$
(10)

Similarly, $V_{2\omega}$, being the signal measured by the lockin amplifier at
$2\omega$ component and $90^{\circ}$ phase, the second order derivative is
expressed using Eq. [9] as

$\frac{\mathrm{d}^{2}V}{\mathrm{d}I^{2}}=-\frac{4V_{2\omega}\sqrt{2}}{(\sqrt{2}V_{\mathrm{AC}}/R_{\mathrm{s}})^{2}}=-\frac{4V_{2\omega}}{\sqrt{2}(V_{\mathrm{AC}}/R_{\mathrm{s}})^{2}}$
(11)

$\mathrm{d}V/\mathrm{d}I$ corresponds to the point contact resistance
$R_{\mathrm{pc}}$ and $\mathrm{d}^{2}V/\mathrm{d}I^{2}$ is related to the
derivative of $R_{\mathrm{pc}}$ with bias as
$\mathrm{d^{2}}V/\mathrm{d}I^{2}=R_{\mathrm{pc}}\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$.

With appropriate multi-stage vibration isolation, $R_{\mathrm{pc}}$ ranging
from 500 $\Omega$ to $2$ K$\Omega$ could be stabilized typically in the Ag@Au
hybrid films with attocube and piezo controllers by monitoring the current
through the sample. The modulation AC current $i_{\mathrm{m}}$ was typically
fixed at $1-5~{}\mu$A, whereas the DC current $I$ was varied till $\sim
100-200~{}\mu$A in magnitude. $R_{\mathrm{pc}}$ can be expressed as a
combination of ballistic Sharvin resistance ($R_{\mathrm{sh}}={16\rho l}/{3\pi
d_{\mathrm{pc}}^{2}}$) and diffusive Maxwell resistance [32]
($R_{\mathrm{M}}=\rho/d_{\mathrm{pc}}$).

$R_{\mathrm{pc}}=\frac{16\rho l}{3\pi
d_{\mathrm{pc}}^{2}}+\frac{\rho}{d_{\mathrm{pc}}}$ (12)

where $v_{\mathrm{F}}$ is the Fermi velocity, and $d_{\mathrm{pc}}$ is the
diameter of the point contact orifice as shown in the schematic of Fig. 2(b)
of the main manuscript. Using a typical resistivity of $10~{}\mu\Omega$.m and
Drude mean free path $l\approx 0.1$ nm, we get $d_{\mathrm{pc}}\approx 10$ nm
for $R_{\mathrm{pc}}\approx~{}1$ k$\Omega$ from Eq. [12]. This indicates a
diffusive nature of the contact since $l\ll d_{\mathrm{pc}}$. For an ideal
ballistic point contact, the derivative of the point contact resistance
represents the Migdal Eliashberg spectral function
$g(\epsilon)=\alpha^{2}\mathcal{F}(\epsilon)$ [32], which is the probability
of specific phonon modes (with energy $\epsilon$) to decay into an electron-
hole pair and resembles closely with the phonon density of states in most
cases.

$\frac{1}{R_{\mathrm{sh}}}\frac{\mathrm{d}R_{\mathrm{pc}}}{\mathrm{d}V}=\frac{8ed_{\mathrm{pc}}}{3\hbar
v_{\mathrm{F}}}g(\epsilon)|_{\epsilon=eV}$ (13)

The integral of $g(\epsilon)$ is a measure of the electron-phonon coupling
constant.

$\lambda=2\int^{\infty}_{0}\frac{g(\epsilon)}{\epsilon}d\epsilon$ (14)

Hence, from the measurement of the point contact spectrum, we can estimate the
electron-phonon coupling parameter.

In the diffusive regime, the spectrum is modified as [32]

$\displaystyle\frac{1}{R_{\mathrm{sh}}}\frac{\mathrm{d}R_{\mathrm{pc}}}{\mathrm{d}V}=$
$\displaystyle\frac{8ed_{\mathrm{pc}}}{3\hbar
v_{F}}[g(\epsilon)+\gamma\int_{0}^{\infty}\frac{g(\omega)}{\omega+\omega_{0}}d\omega+$
(15a)
$\displaystyle\frac{\gamma}{2}\frac{eV}{eV+\omega_{0}}g(\epsilon)]_{\epsilon=eV}$
(15b)

$\gamma\approx 0.58$ is a geometrical factor arising from the shape of the
orifice. The last two terms represent correction to the ballistic expression
due to a background signal arising from the successive generation of phonons
by the energized electrons due to increased collisions at the orifice
($d_{\mathrm{pc}}\gg l$) and then a further scattering of these phonons from
electrons or defects at the point contact. For strong reabsorption of phonons,
[32], the phonon escape frequency
$\omega_{0}=\omega_{\mathrm{D}}l_{\mathrm{ph}}l_{\mathrm{r}}/d_{\mathrm{pc}}^{2}$
is an order or less than the phonon Debye frequency ($\omega_{\mathrm{D}}$) as
$d_{\mathrm{pc}}\gg l_{\mathrm{ph}},l_{\mathrm{r}}$, with $l_{\mathrm{ph}}$,
and $l_{\mathrm{r}}$ being the inelastic phonon-electron scattering length and
elastic phonon-defect scattering lengths respectively. In this case,
$\omega_{0}\rightarrow 0$. Since, $g(e|V|\gg\hbar\omega_{\mathrm{D}})=0$, the
point contact spectrum can be approximated as follows:

$\displaystyle\frac{1}{R_{\mathrm{sh}}}\frac{\mathrm{d}R_{\mathrm{pc}}}{\mathrm{d}V}$
$\displaystyle\approx\frac{8ed_{\mathrm{pc}}}{3\hbar
v_{\mathrm{F}}}g(\epsilon),$ (16a)
$\displaystyle\epsilon\leq\hbar\omega_{\mathrm{D}},\text{Ballistic regime}$
$\displaystyle\approx\frac{8ed_{\mathrm{pc}}}{3\hbar
v_{\mathrm{F}}}[g(\epsilon)[1+\frac{\gamma}{2}]+\gamma\frac{\lambda}{2}],$
$\displaystyle\epsilon\leq\hbar\omega_{\mathrm{D}},\text{Diffusive regime}$
(16b) $\displaystyle\approx 0,$ (16c)
$\displaystyle\epsilon\geq\hbar\omega_{\mathrm{D}},\text{Ballistic regime}$
$\displaystyle\approx\frac{8ed_{\mathrm{pc}}}{3\hbar
v_{\mathrm{F}}}\frac{\gamma}{2}\lambda,$ (16d)
$\displaystyle\epsilon\geq\hbar\omega_{\mathrm{D}},\text{Diffusive regime}$

The magnitude of the point contact spectrum depends on the contact dimension
$d_{\mathrm{pc}}$. Hence after normalizing the experimentally measured value
of $\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ with $d_{\mathrm{pc}}$, we get a
quantity independent of the contact geometry and reflecting $g(\epsilon)$ or
$\lambda$, the fundamental material properties. Hence, we simplify the above
equations as:

$\displaystyle{d_{\mathrm{pc}}}\frac{\mathrm{d}R_{\mathrm{pc}}}{\mathrm{d}V}$
$\displaystyle\approx\frac{16m}{\pi ne\hbar}g(\epsilon),$
$\displaystyle\hbar\omega_{\mathrm{D}}\leq eV,\text{Ballistic regime}$ (17a)
$\displaystyle\approx\frac{16m}{\pi
ne\hbar}[g(\epsilon)[1+\frac{\gamma}{2}]+\gamma\frac{\lambda}{2}],$
$\displaystyle\hbar\omega_{\mathrm{D}}\leq eV,\text{Diffusive regime}$ (17b)
$\displaystyle=0,$ $\displaystyle\hbar\omega_{\mathrm{D}}\gg
eV,\text{Ballistic regime}$ (17c) $\displaystyle\approx\frac{16m}{\pi
ne\hbar}\frac{\lambda}{3.4},$ $\displaystyle\hbar\omega_{\mathrm{D}}\leq
eV,\text{Diffusive regime}$ (17d)

For a ballistic point contact, it is thus possible to study the phonon
spectrum in $g(\epsilon)$. However, in a diffusive regime for strong
reabsorption of phonons, there is a non-zero background which makes it
difficult to distinguish the phonon peaks in $g(\epsilon)$. Nonetheless,
$\lambda$ can be evaluated from the background value
($e|V|\gg\epsilon_{\mathrm{t}}\sim\hbar\omega_{\mathrm{D}}$) of
$d_{\mathrm{pc}}\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ at
$eV\gg\hbar\omega_{\mathrm{D}}$. In the diffusive regime, the point contact
resistance can be approximated by the Maxwell resistance i.e
$d_{\mathrm{pc}}\approx\rho/R_{\mathrm{pc}}$. Hence, for a given
$R_{\mathrm{pc}}$ and $\rho$, where a small corrective factor $F^{-1/3}$
(usually $\lesssim 20$% for most films), is multiplied to the latter for $F>0$
to account for the difference in the effective ‘surface’ disorder as opposed
to that in the bulk transport, we can estimate $d_{\mathrm{pc}}$ using the
above relation and following Eq. [17d] derive the value of $\lambda$ from the
normalized $\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ as:

$\lambda=3.4\frac{\pi
ne\hbar}{16m}\left[\frac{\rho}{R_{\mathrm{pc}}}\frac{\mathrm{d}R_{\mathrm{pc}}}{\mathrm{d}V}\right]_{V\rightarrow\infty}$
(18)

The diffusive nature of the point-contact is experimentally established by the
scaling of $\mathrm{d}R_{\mathrm{pc}}/\mathrm{d}V$ at different values of
$R_{\mathrm{pc}}$ (and hence $d_{\mathrm{pc}}$) assuming
$d_{\mathrm{pc}}=\rho/R_{\mathrm{pc}}$.

## Data availability

Source data are provided with this paper. Data that support the plots within
this paper, and other findings of this study are available from the
corresponding author upon reasonable request.

## Code availability

The codes that support the findings of this study are available from the
corresponding author upon reasonable request.

## Competing interests

The authors declare no competing interests.

Extended Data Fig. 1: Film Uniformity: (a). Schematic of Van der Pauw leads
[42] used for most electrical measurements. The gold-coloured regions indicate
the deposited Cr/Au electrodes and the grey circle indicates the dropcast
film. (b) Optical image of a typical film on Hall bar leads. The bottom panel
shows the schematic of the Hall bar leads, with red dots indicating the
current contacts. The intermediate leads have been used for resistivity
measurements as indicated by the green points of Fig. 1g of the main
manuscript. All configurations of leads, provide identical results in
resistivity measurements. (c) and (d) show the thickness profiles of typical
films in Van der Pauw and Hall bar geometry measured via optical profilometry
along the blue lines indicated in Fig. 1a and bottom panel of Fig. 1b,
respectively. $x$ represents the spatial extent of the film. The average
thickness is $2-3~{}\mu$m, represented by the dotted lines. (d) and (e) show
the temperature ($T$) dependence of the resistivity ($\rho$) normalized by the
residual resistivity ($\rho_{0}$, defined as the resistivity at $T\sim 6$ K)
along multiple voltage channels for Ag filling
$F=V_{\mathrm{Ag}}/(V_{\mathrm{Ag}}+V_{\mathrm{Au}})=0.5$ and $0.06$,
respectively. $I+,I-$, and $V+,V-$ indicate the current and voltage probes
respectively. Insets show the typical images of the films at respective values
of $F$. Variation of $\rho/\rho_{0}$ is within $10\%$ along different channels
which was observed in all films.

Extended Data Fig. 2: High-Resolution Transmission Electron Microscopy (HRTEM)
Image of Ag@Au Interface: (a) presents a high-resolution image of the Ag@Au
nanohybrid with atomic precision, where the darker region represents the AgNPs
surrounded by the lighter region, the Au matrix. The zoomed-in image (b)
focuses on the interface between the AgNPs and the Au matrix, marked by a red
squre in (a), revealing a well-defined sharp boundary between the AgNPs and
the Au matrix. This indicates that the AgNPs are encapsulated by the Au
matrix, forming a core-shell structure. The absence of visual line defects in
these images indicates the high-quality crystalline nature of the nanohybrid.

Extended Data Fig. 3: Low-temperature transport data: (a), (b), (c), and (d)
represent the temperature ($T$) dependence of the average normalized
resistivity ($\rho/\rho_{0}$) for films with different Ag filling
$F=0.7,0.5,0.25$, and $0.14$, respectively. The resistivity measurements were
performed down to $T\sim 0.3$ K. Irrespective of $F$, no evidence of an upturn
in resistance with decreasing $T$ (negative temperature coefficient of
resistance), e.g., due to activation of electrons across inter-grain tunnel
barrier through tunneling or variable range hopping in case of weakly
interconnected or granular assembly of nanoparticles [43, 44, 45, 46, 47, 48,
49, 50], or any other disorder-mediated quantum coherent processes such as
weak localization [51], was observed down to $T\sim 0.3$ K. The upper limit in
the correction to conductivity ($\sigma$) for $F\sim 0.5$ is $\sim
100~{}\Omega^{-1}.\mathrm{m}^{-1}$, which is at least an order of magnitude
smaller than that for bare Au film.

Extended Data Fig. 4: Variation of residual resistivity with the density of
Ag-Au interface area at low fractional filling ($F\lesssim 0.5$): (a) Residual
resistivity ($\rho_{0}$), defined as the resistivity at $T\sim 6$ K, as a
function of the volume fraction $F$ of Ag at the fixed AgNP radius
($r_{\mathrm{Ag}}\approx 1$ nm). (b) $\rho_{0}$ as a function of
$r_{\mathrm{Ag}}$ at a fixed $F(\approx 0.5)$. (c) $\rho_{0}$ as a function of
$F/r_{\mathrm{Ag}}$, the interface area of the AgNP per unit volume of the
hybrid film. The collapse of all points onto a single linear behavior in
$F/r_{\mathrm{Ag}}$ indicates that the Ag-Au interfaces are the dominant
sources of scattering of electrons, as opposed to random atomic scale defects
or alloying-related disorder. Within a simple Landauer-Büttiker formalism
[31], where
$\rho_{0}\approx(h\lambda_{\mathrm{F}}^{2}/e^{2})\times(F/r_{\mathrm{Ag}})$,
we get the Fermi wavelength $\lambda_{\mathrm{F}}\sim 1.3$ nm (dashed line),
which is within a factor of $\sim 2-3$ of that of crystalline Au.

Extended Data Fig. 5: Fit of $\rho-T$ data: (a) Bloch-Grüneisen fit (using Eq.
[4]) to $\rho-T$ data for Au nanoparticle ($F=0$) and Ag nanoparticle ($F=1$)
films. Electron-phonon coupling constant $\lambda\approx 0.45$, and Debye
temperature $\Theta_{\mathrm{D}}\approx 160~{}\mathrm{K}$ for Au, and
$\lambda\approx 1$ and $\Theta_{\mathrm{D}}\approx 190~{}\mathrm{K}$ for Ag
are derived as fit parameters. (b) Eq. [4] is used to fit the $\rho-T$ data
for $T\leq 100$ K in a film of Ag fraction
$F=V_{\mathrm{Ag}}/(V_{\mathrm{Ag}}+V_{\mathrm{Au}})=0.33$ giving
$\Theta_{\mathrm{D}}\sim 160$ K and $\lambda\sim 18$. However, extrapolation
of the fit to high $T>100$ K shows the deviation of the data from linearity.
The sub-linearity of $\rho–T$ for $T\geq\Theta_{\mathrm{D}}$ naturally
indicates the inadequacy of the Bloch-Grüneisen fit for films with $0<F<1$.
(c) $\Theta_{\mathrm{D}}$ obtained from the Bloch-Grüneisen fit to $\rho-T$
data at $T<100$ K and parallel channel model for entire $T$ range is plotted
as a function of $F$. $\Theta_{\mathrm{D}}$ varies between $150-170$ K in all
cases, which is close to that of Au ($\Theta_{\mathrm{D,Au}}\sim 180$ K). (d)
$T$-independent parallel channel resistivity ($\rho_{||}$), obtained as a fit
parameter in the parallel channel fit (Eq. [2] of main manuscript) to $\rho-T$
data is plotted with $F$ after normalizing with Mott-Ioffe-Regel resistivity
[3] $\rho_{\mathrm{MIR}}\approx 3\pi^{2}\hbar^{2}a/e^{2}\approx
10~{}\mu\Omega$.m.

Extended Data Fig. 6: Schematic of the electrical circuit for point-contact
measurements: The tip is mounted on an attachment comprising of attocubes and
piezo, which can control the movement of the tip in x,y and z direction down
to the precision of a nanometer. The sample prepared on pre-patterned glass
substrate is loaded under the tip. This entire sample-tip chamber is loaded
inside a home-built cryostat which can go down to temperature $\sim 5$ K.
Mixed AC+DC current ($I+i(\omega)$, $\omega$ being the frequency of the AC
component) is passed to the tip by using a voltage source and series
resistance $R_{\mathrm{s}}$. The voltage across the tip-sample contact is then
amplified with a voltage amplifier and measured with two lockin amplifiers
simultaneously at frequencies $\omega$ and $2\omega$. The sample is grounded
through a current-to-voltage amplifier, which converts even a small amount of
current through the sample into a measurable voltage drop. The current through
the sample is measured to monitor the tip-sample resistance.

Extended Data Fig. 7:  X-Ray Photoelectron Spectroscopy (XPS) spectrum of the
Ag@Au nanohybrid: High-resolution peak for (a) Au $4f$, and (b) Ag $3d$. The
$4f$ peaks of Au shift toward low binding energy, indicating n-type doping in
the Au-matrix. At the same time, the $3d$ peaks of Ag shift towards higher
binding energy, which indicates that Ag is losing electrons to Au. All the
peaks are normalized with respect to the C=C peak at $284.5$ eV which is used
as a charge correction reference. Extended Data Fig. 8: Hall Measurements: (a)
Top and bottom panels show the schematics of two diferrent lead geometries (G1
and G2) used for Hall measurements. Yellow lines show the depositd Cr/Au
electrodes and the light grey region is the dropcast film. $I+,I-$ and $V+,V-$
indicate the current and voltage probes respectively. (b) Low field hall
measurements at different temperatures ($T$) varying from $0.3-300$ K
illustrated with a typical film with Ag fraction
$F=V_{\mathrm{Ag}}/(V_{\mathrm{Ag}}+V_{\mathrm{Au}})=0.5$. Hall resistance
($V_{\mathrm{H}}/I$, $V_{\mathrm{H}}$, and $I$ are the measured hall voltage
and current respectively) is plotted as a function of magnetic field ($B$) at
different temperatures ($T$). The dotted line indicates the expected linear
variation of the Hall resistance as a function of $B$, which was the case in
all nanoparticle films measured during the course of this experiment. (c) The
electron density $n~{}(\sim 10^{28}~{}\mathrm{m}^{-3})$ for different $F$
normalized to the magnitude of that ($n_{\mathrm{Au}}$) obtained in bare AuNP
film. This indicates that the electron density in the Ag@Au hybrids is the
same to that of bare Au, irrespective of Ag fraction $F$.

## Computational details

The theoretical calculations for the charge transfer were done on a 2D ‘toy
model’ where a periodic array of $(4\times 4)$ clusters of “Ag” sites are
surrounded by 48 “Au” sites forming a superlattice of $(8\times 8)$ 2D
supercells. The calculations were carried out using the model Hamiltonian
described in Supplementary Information Section VII. The relative ratio of the
number of atoms for Ag:Au has been taken to be 1:3 to resemble a typical
fraction $F=0.25$ of the Ag@Au core@shell nanohybrid. Due to the mismatch of
the local potential seen by the conduction electrons localized in Wannier
orbitals at the Ag and Au sites, as shown in Fig. 3c, and the long-range
Coulomb interactions between electrons occupying these Wannier orbitals, there
is charge transfer at each site, indicated by the excess electron occupancy
$\delta n$. The amount of charge transferred can be tuned by varying the
onsite potential difference between Au and Ag atoms $\epsilon_{0}$. As shown
in Fig. 3c of the text, Au and Ag atoms become electron and hole-doped,
respectively, for most positive values of $\epsilon_{0}$, which is also
consistent with the X-ray photoelectron spectroscopy (XPS) data.
The parameters used in our study are as follows: the nearest neighbor
distance, $d_{1}$, within the square lattice considered is set to $4.10$ Å,
corresponding to the lattice spacing in FCC-Ag/Au. The nearest neighbor
hopping between all the atomic sites is set at $1$ eV, and the hopping decay
factor, $\xi_{0}$, is adjusted to yield a second-nearest neighbor hopping of
$0.5$ eV. The on-site Coulomb interaction energy is fixed at $U_{0}=2$ eV, and
$V_{0}$, which controls the strength of the long range coulomb interactions,
is chosen such that $V_{0}/d_{1}=1$ eV. The difference in work function
between Au and Ag atoms is incorporated into the on-site potentials, with
$(\epsilon_{j}^{\text{Ag}}-\epsilon_{j}^{\text{Au}})$ ranging from $0.1$ eV to
$1.5$ eV in our calculations. The charge density obtained from solving the
self-consistent mean field equations at
$(\epsilon_{j}^{\text{Ag}}-\epsilon_{j}^{\text{Au}})=1.5$ eV is presented in
Fig. 3d of the main manuscript. More detailed figures are provided in the
Supplementary Information Section VII.
Phonons are computed with spring constants of $1.0$ for nearest neighbor atoms
and $0.5$ for next-nearest neighbors, in arbitrary units. Electronic and
phononic band spectra are calculated on $(16\times 16)$ $\mathbf{k,q}$ grids
within the Brillouin zone of the supercell, and are used to determine the
electron-phonon coupling strength $(\lambda^{\mathrm{(calc)}})$ using the
‘double delta’ approximation.