Quantum Critical Metals:
Dynamical Planckian Scaling and Loss of Quasiparticles

Haoyu Hu1,2, Lei Chen1, Qimiao Si1

1Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice
University, Houston, TX 77005, USA

2 Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018
Donostia-San Sebastian, Spain

Strange metals develop near quantum critical points (QCPs) in a variety of
strongly correlated systems. Some of the important issues include how the
quantum critical state loses quasiparticles, how it drives superconductivity,
and to what extent the strange-metal physics in different classes of
correlated systems is connected with each other. In this article, we confront
some of these issues from the vantage point of heavy fermion metals. We will
describe the notion of Kondo destruction and how it leads to a transformation
of the Fermi surface from “large” to “small” across the QCP, a loss of
quasiparticles everywhere on the Fermi surface at the QCP, and a dynamical
Planckian scaling in various properties including charge responses. We close
with a brief discussion about the connections between the strange-metal
physics in heavy fermion metals and its counterparts in the cuprates and other
correlated materials.

##  I Introduction

Large classes of quantum materials contain strongly correlated electrons
Kei17.1 ; Pas21.1 . Many of them feature unconventional superconductivity. One
connection among the strongly correlated systems is illustrated in Fig. 1(a).
The superconducting transition temperature ($T_{c}$) is several percent of the
effective Fermi temperature ($T_{0}$) across several material classes of
strongly correlated superconductors, with each quantity spanning about three
decades. A $T_{c}/T_{0}$ ratio of a few percent qualifies these systems as
“high-$T_{c}$” superconductors, recognizing that this ratio is about two
orders of magnitude smaller in conventional (Bardeen-Cooper-
SchriefferBardeen57 ) superconductors. Another connection lies in their normal
states at $T>T_{c}$, which are often strange metals with a non-Fermi liquid
form of ($T-$linear) electrical resistivity and a slew of other exotic
properties.

a

b c

Figure 1:  (a) Superconducting transition temperature ($T_{c}$) vs. effective
Fermi temperature ($T_{0}$, extracted from entropy and other means) for
various superconductors from different material families (adapted from Ref.
Curro05 , courtesy J. D. Thompson). (b) Illustration of entropy evolution near
a QCP, based on measurements of the uniaxial Grüneisen ratios in CeCu6-xAux at
$x_{c}=0.1$ (from Ref. Grube17 ). Panel b shows the parameter space of the
uniaxial stress along the $a,b,c$ axis. The red arrow marks the direction with
the steepest slope of entropy ${\bm{\nabla}}S$. Away from the QCP, one can
tune the system in directions that are

(cont’d) perpendicular to the ${\bm{\nabla}}S$ direction (the red plane),
along which the Grüneisen ratio is not singular. (c) Illustration of the
entropy maximization near the QCP, in the parameter space of the shear stress
$\sigma_{x,y}$.

The link between the strange-metal normal state and unconventional
superconductivity in heavy fermion systems is particularly striking. Indeed,
heavy fermion metals represent a prototype setting in which quantum critical
metallicity has been elucidated KirchnerRMP , in part because $T_{c}$ being
relatively small in absolute magnitude opens up a large window of temperature
over which the strange-metal properties can be explored. These systems often
have antiferromagnetic (AF) correlations. Heavy fermion superconductors per se
is a venerable topic, and have now grown into a material family of about $50$
members. By contrast, the strange-metal behavior and its association with
quantum criticality have only been the focus during the relatively recent
past.

It is natural for quantum criticality to nucleate unusual properties Mathur98
. Indeed, as a system is tuned towards the quantum critical regime at a given
low (but nonzero) temperature, the entropy is expected to be maximized
ZhuGarst.03 ; Wu2011 . This has been demonstrated in CeCu6-xAux across
multiple tuning parameters, as illustrated in Figs. 1(b,c) Grube17 .
Accordingly, quantum critical systems are particularly soft and are prone to
the formation of unusual excitations and novel phases.

That strange metals develop via quantum criticality is strikingly demonstrated
in heavy fermion metals. We illustrate the point in YbRh2Si2 and CeRhIn5, via
their respective phase diagrams [Fig. 2(a,b)]. Each exhibits an AF order at
ambient conditions. In YbRh2Si2, a magnetic field applied perpendicular to the
tetragonal plane of about 0.7 T (or one applied within the plane of about 66
mT) tunes the system to its QCP Geg02.1 , where a $T$-linear resistivity
occurs over more than three decades in temperature Ngu21.1 . In CeRhIn5, a
quantum critical “fan” develops near a pressure of $2.3$ GPa park-nature06 ;
Kne08.1  with a nearly-$T$-linear resistivity Par08.1 .

a b

Figure 2:  Temperature-control parameter phase diagrams of (a) YbRh2Si2 under
a magnetic field Geg02.1 ; Custers2003  and (b) CeRhIn5 under pressure Par08.1
. The colors represent $\gamma$, determined by a logarithmic derivative of
$\Delta\rho(T)\equiv\rho(T)-\rho(T=0)$ with respect to $\log T$, signifying a
$T^{\gamma}$ dependence.

Theories of metallic QCPs have two general types. One class of theory is based
on the fluctuations of Landau’s order parameter Hertz ; Millis , which
characterizes a spin-density-wave (SDW) order in the present context. In this
standard description, the nonzero ordering wavevector ${\bf Q}$ links narrow
“hot” regions of the Fermi surface to each other; the bulk of the Fermi
surface corresponds to the “cold” region (Fig. 3). The order parameter
fluctuation connects one point on the cold region of the Fermi surface to
another point in the Brillouin zone whose energy is substantially away from
the Fermi energy. Correspondingly, for the electronic states in the cold
region of the Fermi surface, the quantum critical fluctuations have a minimal
effect and the quasiparticles retain their integrity Hlubina95 ; Rosch99 ;
Borges22.1x . The electrical transport will not show the strange-metal
behavior given that the quasiparticles, being long-lived, will short-circuit
the electrical transport. To realize the strange-metal behavior, it is
necessary to detroy the quasiparticles on the entire Fermi surface. This takes
place in the second type of thoery for metallic quantum criticality, which
goes beyond the Landau framework Si-Nature ; Colemanetal ; senthil2004a .

Figure 3:  Schematic illustration of the Fermi surface. Only a small portion
of the Fermi surface, nested by the AF wave vector $Q$, is “hot” (red stripes
with a width of $\sim\sqrt{T}$ for three dimensions). The majority of the
Fermi surface remains “cold”, where Landau quasiparticles are left intact.

In this article, we survey the beyond-Landau quantum criticality. We start by
considering how quasiparticles can be (critically) destroyed. The central
theme here is that, for bad metals such as heavy fermion systems, the
quasiparticles are fragile to begin with and their formation takes place
through a process that is non-perturbative and yet well-understood. This
understanding sets the stage for describing how the quasiparticles can be
destroyed. For heavy fermion metals, the Kondo effect underlies the formation
of heavy quasiparticles and the Kondo destruction leads to their loss. We
suggest that these understandings are relevant to the loss of quasiparticles
in a variety of strongly correlated systems, including the doped cuprates, as
well as the iron-chalcogenide and certain organic superconductors. While our
focus is theoretical, some of the salient experimental developments, e.g.
Refs. paschen2004 ; Friedemann.10 ; shishido2005 ; Schroder ; Aro95.1 ;
Prochaska2020 , will be briefly described.

##  II Quantum critical metals – how to destroy quasiparticles

To see how the quasiparticles can be lost everywhere on the Fermi surface, we
start from their formation away from the QCP.

###  II.1 Quasiparticles: the robust and the fragile

In quantum many-body systems, the low energy physics is analyzed in terms of
building blocks and their symmetry-allowed interactions Pas21.1 .
Traditionally, one takes bare electrons as the building blocks and treat the
interactions as a perturbation Nozieres18 . The notion of quasiparticles
survives up to infinite orders of the order-by-order perturbation series. In
that sense, it is rather robust, and for a long time the validity of Fermi
liquid theory was largely unquestioned for systems in dimensions higher than
one; indeed, Fermi liquid theory was considered to be the only fixed point of
the renormalization-group (RG) flow in such dimensions Shankar1994 . The
quasiparticles correspond to a sharp peak in the spectral function of the bare
electron, and has the physical meaning of a dressed electron with exactly the
quantum numbers of a bare electron or hole in a non-interacting system: charge
$\pm e$ and spin $\frac{\hbar}{2}$. Their Fermi statistics dictates a decay
rate that goes as $(k_{\rm B}T)^{2}$ or $E^{2}$ as temperature ($T$) or energy
($E$, measured from the Fermi energy) goes to zero. The self-energy
$\Sigma({{\bf k}},\omega)$ retains the Fermi liquid form up to infinite orders
of the perturbative expansion Nozieres18 . The quasiparticle weight, $Z_{{\bf
k}}$, is given by $[1-\partial\Sigma({{\bf k}},E)/\partial E]^{-1}$ at the
quasiparticle energy $E$.

A sufficiently strong electron correlation can lead to other forms of building
blocks for the low-energy physics. For example, heavy fermion systems involve
$f$-electron-based local moments and $spd$-electron-based conduction electron
bands as the starting point for the description of their low-energy properties
Pas21.1 ; StewartRMP ; Coleman-Nature ; Si_Science10 ; KirchnerRMP . In that
case, quasiparticles are fragile, with a weight that is exponentially small.

Consider the Kondo lattice Hamiltonian:

$\displaystyle H_{\rm KL}=\sum_{{\bf k}}\varepsilon_{{\bf k}}c_{{\bf
k}}^{\dagger}c_{{\bf k}}+\sum_{ij}I_{ij}{\bf S}_{i}\cdot{\bf
S}_{j}+\sum_{i}J_{K}{\bf S}_{i}\cdot
c^{\dagger}_{i}\frac{\vec{\sigma}}{2}c^{\phantom{\dagger}}_{i}.$ (1)

The involved building blocks are the $f$-electrons in the form of local
moments, $\bf{S}_{i}$, and a band of $spd$ conduction electrons, $c_{{\bf
k}\sigma}$ with an energy dispersion $\varepsilon^{\phantom{\dagger}}_{{\bf
k}}$. At each site $i$, an AF Kondo interaction $J_{K}$ couples the spin of
the local moment and that of the conduction electrons, ${\bf
s}_{c,i}=(1/2)c_{i}^{\dagger}\vec{\sigma}c_{i}$, where $\vec{\sigma}$ denote
the three Pauli matrices. Across the sites, the local moments are coupled to
each other via an RKKY interaction $I_{ij}$.

We start from the parameter regime when the Kondo interaction succeeds in
driving the formation of a Kondo singlet, which can be pictured as a bound
state between a local moment and the triplet particle-hole combination of the
conduction electrons. Breaking the bound state leads to not only bare
conduction electrons, but also a composite fermion formed between the local
moment and a conduction electron. The composite fermions have the quantum
numbers of a bare electron, and they hybridize with the conduction electrons
to form heavy quasiparticles that have a large effective mass and a small
quasiparticle weight: $Z$ is exponentially small and, in practice, of the
order $10^{-3}$.

When the quasiparticles are this fragile, competing interactions can readily
destroy it.

###  II.2 Quantum criticality from Kondo destruction

The notion of Kondo destruction was introduced in a type of beyond-Landau
quantum criticality. The corresponding QCP is illustrated in Fig. 4(a) Si-
Nature ; KirchnerRMP , which shows a zero-temperature tuning parameter,
$\delta=T_{K}^{0}/I$, describing the ratio of the Kondo coupling to the RKKY
interaction.

a

b c

d e

f g

Figure 4:  (a) Kondo destruction quantum criticality of a Kondo lattice, in
which a Kondo destruction energy scale $E_{\rm loc}^{*}$ vanishes at the QCP.
Here, $T_{\rm N}$, $T_{\rm FL}$ and $T_{0}$ denote

(cont’d) the temperatures for an AF ordering, crossover into a Fermi liquid
state and the initial onset of Kondo correlations, respectively. On the two
sides of the QCP are (b) an AF order without the formation of a Kondo singlet
and (c) a paramagnetic phase with a Kondo-singlet ground state. In the AF
phase, the quasiparticles only involve (d) the conduction electrons, in
contrast to the paramagnetic phase, in which (e) the Kondo singlets in the
ground state yield composite fermions in the excitation spectrum. The Fermi
surface for the AF phase is small (f) in that it only involves the conduction
electrons, while that for the paramagnetic phase is large (g) in that it also
counts the number of local moments. Panel (a) and (b-g) are adapted from Refs.
Si-Nature ; KirchnerRMP  and Ref. Pas21.1 , respectively.

When $\delta$ is sufficiently large, a Kondo singlet is formed in the ground
state, as illustrated in Fig. 4(c). As the RKKY interaction $I$ is increased,
i.e. when the parameter $\delta$ is tuned downward, the RKKY interaction
promotes correlations of a spin singlet between the local moments. This
process is detrimental to the formation of the Kondo singlet. When it
suppresses the Kondo singlet in the ground state, the heavy quasiparticles are
critically lost.

Thus, both the formation and loss of quasiparticles can be considered by
analyzing the fate of the Kondo singlet or, more specifically, the amplitude
of the Kondo singlet in the ground state. Our strategy is to start from the
Kondo side, and see whether and how the dynamical competition of the RKKY
interaction brings about the suppression of this Kondo-singlet amplitude. One
can in principle also start from the opposite end, by analyzing the Kondo
lattice in terms of a quantum nonlinear sigma model representation; the
results of such analyses Si.06 ; Yamamoto07 ; Ong_Jones09 ; Yamamoto-JLTP2010
; Goswami-prl11 ; Goswami-prb14 ; Goswami-prb17 ; CCLiu-prb17  are consistent
with the conclusions we present here.

###  II.3 Kondo lattice – EDMFT analysis

We now turn to the calculations that have provided the basis for the notion of
Kondo destruction. The approach that has played a central role in addressing
not only the formation of the heavy quasiparticles but also their destruction
is an extended dynamical mean field theory (EDMFT) Si.96 ; SmithSi-edmft ;
Chitra2001 . This approach treats the dynamical interplay between the Kondo
and RKKY interactions of a Kondo lattice (Fig. 5). A recent review of the
EDMFT approach can be found in Ref. Hu-EDMFT-review2022 . The EDMFT approach
corresponds to a non-perturbative summation of an infinite series of skeleton
diagrams. They are generated by an effective action functional and are
systematic and conserving.

Figure 5:  Schematic illustration of the Kondo lattice model. Adapted from
Ref. Si-Nature .

In this approach, the fate of the Kondo-singlet amplitude is determined by the
nature of local correlation functions. The latter are determined from a Bose-
Fermi Kondo/Anderson model:

$\displaystyle H_{BFK}$ $\displaystyle=$ $\displaystyle\sum_{{\bf k}}E_{{\bf
k}}c^{\dagger}_{{\bf k}\sigma}c^{\phantom{\dagger}}_{{\bf k}\sigma}+\sum_{{\bf
p}}\omega_{{\bf p}}\bm{\phi}_{-{\bf p}}^{{\dagger}}\bm{\phi}_{{\bf p}}$ (2)
$\displaystyle+J_{K}\bm{S}\cdot\frac{c_{0}^{\dagger}\bm{\sigma}c_{0}}{2}+g:\bm{S}:\cdot\sum_{{\bf
p}}(\bm{\phi}_{{\bf p}}^{{\dagger}}+\bm{\phi}_{-{\bf
p}}^{{\dagger}})+h_{loc}S^{z}\,.$

Here the dispersion $E_{{\bf k}}$ and $\omega_{\bf p}$ characterize a
fermionic and a bosonic bath. Their couplings to the local moment have the
strength of $J_{K}$ and $g$, respectively. Self-consistency equations are
expressed in terms of the local correlators of the Bose-Fermi Kondo model.

More specifically, the self-consistency equations capture the translational
invariance:

$\displaystyle\chi_{\text{loc}}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{\bf q}\chi({\bf q},\omega),$ $\displaystyle
G_{\text{loc}}(\omega)$ $\displaystyle=$ $\displaystyle\sum_{\bf k}G({\bf
k},\omega)\;,$

where the lattice susceptibility and Green’s function are

$\displaystyle\chi({\bf q},\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{M(\omega)+I_{{\bf q}}}\,,$ $\displaystyle G({\bf
k},\epsilon)$ $\displaystyle=$
$\displaystyle\frac{1}{\epsilon+\mu-\epsilon_{\bf k}-\Sigma(\epsilon)}\,.$ (5)

Here $M(\omega)$ and $\Sigma(\epsilon)$ are self-energies for the spin and
single particles, respectively Si.96 ; SmithSi-edmft ; Chitra2001 . The
effective magnetic field $h_{loc}$ is determined self-consistently by
$h_{loc}=-I_{\textbf{Q}}\langle S^{z}\rangle$.

Kondo destruction can be seen from the RG flow of the Bose-Fermi Kondo model.
The RG beta-functions have been determined at one loop from an
$\epsilon$-expansion, which was first carried out in the model with Ising
anisotropy Si.96  and subsequently extended to the model with SU(2) spin
symmetry SmithSi_EPL1999 ; Sengupta . For both the SU(2) and $xy$-spin
symmetry case, the RG analysis has also been carried out to two and higher
loops ZhuSi ; ZarandDemler . Here, $\epsilon$ is specified by the power-law
spectrum of the bosonic bath:

$\displaystyle\rho_{b}(\omega)\equiv\sum_{{\bf p}}\delta(\omega-w_{\bf
p})\propto|\omega|^{1-\epsilon}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}~{}|\omega|<\Lambda\,.$
(6)

Fig. 6(a) illustrates the flow that is associated with the RG beta-functions
in the $\epsilon$-expansion, for a positive $\epsilon$. In the absence of the
bosonic Kondo coupling (i.e. when $g=0$), any nonzero $J_{K}$ flows away from
the decoupled fixed point and towards the Kondo fixed point Hewson . The
bosonic coupling $g$ leads to two new fixed points and a separatrix in the
$J_{K}$-$g$ plane. The critical (red) fixed point controls the physics on the
separatrix, corresponding to a critical destruction of the Kondo phase. On the
right of the separatrix, the system flows to a Kondo-destroyed (green) fixed
point where $J_{K}$ vanishes altogether.

a b

Figure 6:  RG flow of the BFKM based on an $\epsilon$-expansion ZhuSi  (a) and
from studies based on CT-QMC Cai-SU2_19.1  and a large-$S$ expansion Hu22.fix
at $1>s>s^{*}$.

The nature of the critical (red) Kondo destruction fixed point is to be
contrasted with that of the Kondo fixed point. There are three key
distinctions. First, the Kondo fixed piont describes a phase with a nonzero
amplitude of the Kondo singlet in the ground state, whereas, in the phase
associated with the Kondo destruction fixed point, the Kondo-singlet amplitude
vanishes in the ground state.

Second, and concomitantly, the distinction appears in the low-lying
excitations. For the Kondo destruction (red) fixed point, the vanishing of the
Kondo-singlet amplitude in the ground state implies that the weight of any
Landau quasiparticle goes to zero. This can also be explicitly seen from the
finite-size spectrum of the many-body excitations as determined by the
numerical-renormalization-group (NRG) approach: the spectrum can no longer be
fit in terms of a combination of any quasiaprticles Kandala2022 .

Third, the Kondo destruction fixed point is critical and interacting (as
opposed to being Gaussian). Thus, $k_{\rm B}T$ is the only energy scale.
Accordingly, singular responses such as the local spin and charge
susceptibilities have a dynamical Planckian ($\hbar\omega/k_{\rm B}T$)
scaling. This has been seen both in the Kondo destruction fixed point of the
Bose-Fermi Kondo/Anderson model as determined by a dynamical-large-$N$ (where
the index $N$ appears in the spin channel) approach Cai-charge20.1 ; Zhu.04
and in the SU(2) case Cai-charge20.1 ; Kandala2022 .

We close this subsection by noting the extra fixed points that exist beyond
the $\epsilon$-expansion. It turns out that, for small $\epsilon$, the SU(2)
Bose-Fermi Kondo/Anderson model has more fixed points beyond those that are
accessed by the $\epsilon$-expansion method. Fig. 6(b) shows the RG flow
diagram when $\epsilon$ is sufficiently small (or $s=1-\epsilon$ is
sufficiently large) Cai-SU2_19.1 . As $\epsilon$ further increases, the fixed
points are pair-wise annihilated, and this has recently been understood
analytically based on a $1/S$ expansion (where $S$ is the spin size) Hu22.fix
.

###  II.4 Global phase diagram

The Kondo destruction from the EDMFT analysis has inspired considerations of
new quantum phases in the AF Kondo-lattice systems that are not only
distinguished by the Landau order parameters but also by the existence or
absence of the Kondo singlet in the ground state. This has led to a global
phase diagram (Fig. 7) Si.06 ; Si_PSSB10 ; Coleman_Nev  in the two-parameter
space of $J_{K}$, the Kondo coupling, and $G$, which specifies the extent of
the quantum fluctuations in the local-moment magnetism. The $G$ axis captures
the tuning of dimensionality Custers-2012  or geometrical frustration Zhao-
frustration19 ; Mun12 ; Kim-2013 ; Fritsch-2014 .

Figure 7: Global phase diagram of the heavy fermion systems. Adapted from
Refs. Si.06 ; Si_PSSB10 ; Pixley-2014 ; Pas21.1 .

The stability of the ${\rm AF_{S}}$ phase was analyzed in terms of a quantum
nonlinear sigma model representation of the Kondo lattice Si.06 ; Yamamoto07 ;
Ong_Jones09 ; Yamamoto-JLTP2010 . Using the $AF_{S}$ phase as the starting
point, there are three routes for quantum phase transitions to the
paramagnetic heavy fermion ($P_{L}$) phase. Trajectory I describes a direct
transition, with a Kondo destruction QCP at the border of the AF order.
Trajectory II passes through an intermediate ${\rm AF_{L}}$ phase, which
corresponds to the SDW order from the heavy quasiparticles of the ${\rm
P_{L}}$ phase. A Kondo destruction transition takes place inside the AF order,
while the QCP from the AF order to the paramagnetic phase is of the SDW type.
Trajectory III involves an intermediate ${\rm P_{S}}$ phase, which could
involve non-magnetic order such as a valence-bond solid. Generically, the
Luttinger theorem of the Kondo lattice is obeyed, as can be seen from how the
local-moment part and conduction electrons senthil2004a ; Pivovarov.04
respond to the adiabatic insertion of an external flux Oshikawa . The
stability of the paramagnetic heavy fermion ($P_{L}$) phase itself, as
described earlier, is not in doubt.

The three trajectories of quantum phase transitions provide a variety of ways
for the Landau quasiparticles of the paramagnetic heavy fermion phase to be
destroyed. There have been considerable effort in exploring this phase
diagram, both experimentally Custers-2012 ; Zhao-frustration19 ; Mun12 ;
Kim-2013 ; Fritsch-2014  and theoretically Pixley-2014 ; Goswami-prl11 ;
Goswami-prb14 ; Goswami-prb17 ; CCLiu-prb17 ; Toshihiro2018 . In the remainder
of this article, we aim to keep our discussion focused by primarily addressing
the Kondo destruction transition along trajectory I.

##  III  Dynamical Planckian scaling

Analysis of the EDMFT equations has been carried out both analytically and
numerically Si-Nature ; lcp-prb ; GrempelSi ; ZhuGrempelSi ; SunKotliar ;
Glossop07 ; Zhu07 ; Hu20.1x . We focus on the solution of the aforementioned
trajectory I, i.e. a transition of the Kondo destruction type between the
paramagnetic and AF phases.

At the QCP, the dynamical spin susceptibility at the AF wavevector ${\bf Q}$
is found Si-Nature ; lcp-prb  to have the following dynamical Planckian
scaling form:

$\displaystyle\chi({\bf
Q},\omega)=\frac{1}{(-i\hbar\omega)^{\alpha}}W^{-1}\left(\frac{\hbar\omega}{k_{\rm
B}T}\right)\,.$ (7)

Here, $W=A\,\mathcal{M}(\omega/T)$ , with $A$ being a constant prefactor and

$\displaystyle{\cal
M}(\omega/T)=\left(\frac{T}{-i\omega}\right)^{\alpha}\exp\,[\,\alpha\,\psi(1/2-i\omega/2\pi
T)\,]\,,$ (8)

where $\psi$ is the digamma function.

The calculated exponent $\alpha$ is fractional, and is close to being 0.75 for
the Ising anisotropic case (between 0.72 and 0.78 when different methods are
used for the calculation) GrempelSi ; Zhu.04 ; Glossop07 ; Zhu07  and about
$0.71$ in the CT-QMC calculation for the SU(2) case Hu20.1x . At a general
wavevector ${\bf q}$, the dynamical spin susceptibility takes the following
form:

$\displaystyle\chi({\bf q},\omega)=\frac{1}{\theta({\bf
q})+A\,(-i\omega)^{\alpha}\mathcal{M}(\omega/T)}\quad.$ (9)

where $\theta({\bf q})=I_{{\bf Q}}-I_{{\bf q}}$.

These theoretical results are important, because they provide the
understanding of the inelastic neutron scattering data measured in CeCu6-xAux
at its quantum critical concentration $x_{c}=0.1$ Schroder  (see also Ref.
Aro95.1 ). The experiments show not only the scaling form but also a
fractional exponent $\alpha\approx 0.75$.

The Kondo destruction QCP also predicted the temperature dependence of the NMR
relaxation rate. When the hyperfine form factor does not have a strong
dependence on the wavevector, the NMR relaxation rate $1/T_{1}$ is determined
by the local spin susceptibility, leading to Si-Nature ; lcp-prb :

${1\over T_{1}}\sim{\rm constant}\,.$ (10)

By contrast, if the hyperfine coupling has a strong ${\bf q}$-dependence
leading to a cancellation of the contributions from the dynamical spin
susceptibility near the AF wavevector (as in the well-known case of the
oxygen-site NMR relaxation rate of the cuprates), the NMR relaxation rate has
the following temperature dependence Si03.1 :

${1\over T_{1}}\sim T^{\alpha}\,.$ (11)

The results from the Si-site NMR experiments in YbRh2Si2 found the NMR
relaxation rate to be strongly dependent on the applied magnetic field Ishida-
prb03 ; Ishida-prl02 . When combined with the $\mu$SR results, they have
allowed the extraction of the relaxation rate $1/T_{1}$ at the quantum
critical magnetic field Si03.1 ; Ishida-prb03 , and the result is consistent
with the prediction of Eq. (10). Whereas the measured Cu-site NMR relaxation
rate in CeCu6-xAux, at $x_{c}=0.1$, is compatible with the expectation given
in Eq. (11) Si03.1 .

Figure 8:  Singular charge response of the hybridization $B$-field in the
Kondo-lattice model at the KD QCP calculated by the EDMFT Cai-charge20.1 .

Importantly, charge response has also been found to be critical Prochaska2020
. This would have been unusual for an SDW QCP, where the singular fluctuations
are in the magnetic sector. Theoretically, at the Kondo destruction QCP, the
engagement of the Kondo process in the quantum criticality suggests the
relevance of the single-particle and charge sectors to quantum criticality. A
charge response that is of particular interest Cai-charge20.1  is

$\displaystyle G_{B}^{\prime}(\tau)=\frac{G(\tau)}{\sqrt{\chi(\tau)}}\,$ (12)

where $G(\tau)$ is the single-electron Green’s function of the correlated
specifies in the Anderson lattice model and $\chi(\tau)$ is the local spin
correlation function, both as a function of the imaginary time $\tau$. For a
vanishing Kondo-singlet amplitude in the ground state, this quantity is the
correlator of a charge-carrying hybridization $B$-field Cai-charge20.1 . The
result of an EDMFT analysis at the Kondo destruction QCP of an SU(2) Anderson
lattice model is shown in Fig. 8. This charge correlator collapses as a
function of $\pi T/\sin(\pi\tau T)$, in a way that is singular and obeys
dynamical Planckian scaling.

##  IV  Transformation of large-to-small Fermi surface and loss of
quasiparticles

An important characteristic of the Kondo destruction quantum criticality is a
transformation of a “large” to “small” Fermi surface across the QCP. This
turns out to be intimately connected to a loss of quasiparticles everywhere on
the Fermi surface at the QCP.

###  IV.1 Large to small Fermi surface transformation across the QCP and loss
of quasiparticles at the QCP

In the paramagnetic phase, the ground state has a nonzero Kondo-singlet
amplitude [Fig. 4(c)]. Correspondingly, composite fermions develop in the low-
energy single-electron excitation spectrum [Fig. 4(e)], and the Fermi surface
is large in the sense that it incorporates both the conduction electrons and
the Kondo-destroyed composite fermions. This is characterized by a nonzero
amplitude, $b^{*}$, for a pole Auerbach-prl86 ; Millis87  of the conduction-
electron self-energy in energy space:

$\displaystyle\Sigma({\bf
k},\omega)=\frac{(b^{*})^{2}}{\omega-\varepsilon_{f}^{*}}\quad.$ (13)

Here the self energy is specified via the Dyson equation: $G_{c}({\bf
k},\omega)=\left[\omega-\varepsilon_{\bf k}-\Sigma({\bf
k},\omega)\right]^{-1}$. Correspondingly, the conduction-electron Green’s
function contains two poles, respectively at energies

$\displaystyle E^{\pm}_{{\bf k}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\left[\,\varepsilon_{\bf
k}+\varepsilon_{f}^{*}\pm\sqrt{(\varepsilon_{\bf
k}-\varepsilon_{f}^{*})^{2}+4(b^{*})^{2}}\,\right]\quad.$ (14)

They correspond to the heavy fermion bands. The nonzero $b^{*}$ specifies a
Kondo resonance and leads to a large Fermi surface. The quasiparticle weight
is $Z_{L}\propto(b^{*})^{2}$ (Fig. 9). The damping rate has the Fermi liquid
$T^{2}$ and $E^{2}$ form.

Figure 9:  The quasiparticle spectral weight $Z_{S}$ and $Z_{L}$ for the small
and large Fermi surfaces respectively as a tuning of $\delta$, showing a
large-to-small Fermi surface transforamtion across the Kondo destruction QCP,
and the loss of quasiparticles everywhere on the Fermi surface at the QCP
KirchnerRMP .

As the system approaches the Kondo destruction QCP, the Kondo-singlet
amplitude goes to zero. The residue of the pole in the conduction-electron
self-energy $\Sigma({\bf k},\omega)$ of Eq. (13) vanishes. Correspondingly,
the quasiparticle weight on the large Fermi-surface, $Z_{\rm L}$ vanishes
(Fig. 9). The absence of the Kondo-singlet amplitude on the other side of the
QCP [Fig. 4(b)], implies the absence of the composite fermion as a well-
defined excitation [Fig. 4(d)]. Correspondingly, the Fermi surface is small
[Fig. 4(f) and Fig. 9] in the sense that it incorporates conduction electrons
only. As the dictated by continuity, the weight of the quasiparticles on the
small Fermi surface, $Z_{\rm S}$, vanishes as the QCP is approached (Fig. 9).

The jump of the large-to-small Fermi surface across the QCP is experimentally
testable Si-Nature ; lcp-prb ; Colemanetal ; senthil2004a ; Pepin-prl07 .
Probing this jump has been the subject of intriguing experiments. Across the
field-induced QCP in YbRh2Si2, a remarkable sequence of measurements
paschen2004 ; Friedemann.10  have identified a rapid isothermal crossover in
the normal Hall coefficient. The crossover width extrapolates to zero in the
$T=0$ limit. This jump of the Hall coefficient provides evidence for a jump in
the Fermi surface across the QCP. Moreover, the location of the crossover maps
out a new temperature scale [Fig. 2(a)] paschen2004 ; Friedemann.10 ;
Gegenwart2007 . Separately, in CeRhIn5, measurements of the de Haas-van Alphen
effect have provided evidence of a sharp jump of the Fermi surface across the
pressure-induced QCP shishido2005 .

How to probe the destruction of quasiparticles at the QCP? In heavy fermion
metals, angle-resolved photoemission spectroscopy (ARPES) measurements have
yet to reach adequate resolution to address the issue. Recent experiments have
reported the results of a new probe of strongly correlated metals, the current
shot noise, which provides fairly direct evidence for the loss of
quasiparticles in the quantum critical regime of YbRh2Si2 Liyang-Chen2022.x .

###  IV.2 Implications of the singular charge response: dynamical Kondo effect
and high-$T_{\rm c}$ superconductivity

That charge responses are singular and obeys dynamical Planckian scaling at a
magnetic QCP carries a special significance. It implicates a charge-spin
entanglement at the Kondo destruction QCP, in spite of a vanishing amplitude
of the Kondo singlet in the ground state. In fact, it has been shown that a
dynamical Kondo correlation persists in this regime: a nonzero Kondo coupling
in the Hamiltonian dictates that the cross local moment-conduction electron
spin correlations operate at any nonzero frequencies Hu20.1x .

Figure 10: Illustration of the quantum critical charge fluctuation from the
Kondo entanglement near a Kondo destruction QCP (from Ref. Prochaska2020 ).

Qualitatively, the Kondo destruction QCP features quantum fluctuations between
a phase with a Kondo singlet in the ground state, well-defined composite
fermions and thus a large Fermi surface on the one hand on the one hand, and a
phase without a Kondo singlet in the ground state, absence of composite
fermions and thus a small Fermi surface on the other hand (Fig. 9 and Fig.
10). Since the composite fermions carry both charge and spin, the fact that
they are critically destroyed at the Kondo destruction QCP means that the
charge sector is an inherent component of the quantum criticality (Fig. 10).

This dynamical Kondo effect has important implications for how unconventional
superconductivity develops out of the strange-metal normal state. The Kondo
destruction quantum criticality is robust in that it encodes a large amount of
entropy in the quantum fluctuation spectrum – it amounts to a significant
portion of R$\ln 2$ per $f-$site. The primarily degrees of freedom that is
involved in this amplified quantum fluctuations are spin in nature. Indeed, a
recent calculation using the cluster version of the EDMFT approach has shown a
strong intersite spin-singlet correlations in this quantum critical fluid
Hu2021.sc-x . Through the dynamical Kondo effect, such amplified quantum
fluctuations strongly influence the charge sector, so much so that it leads to
pronounced pairing correlations in the spin-singlet sector. The calculations
show that this process drives unconventional superconductivity with
high-$T_{\rm c}$: the transition temperature reaches a few percent of the
effective Fermi temperature Hu2021.sc-x .

###  IV.3 Dynamical Kondo effect and quantum entanglement

The charge-spin entanglement enabled by the dynamical Kondo effect at the
Kondo destruction QCP provides a strong motive to examine the quantum
entanglement properties across the QCP. Entanglement entropy has been analyzed
near a Kondo destruction QCP of the Bose-Fermi Kondo model Pix-
entanglement15.2 . Recently, through the EDMFT approach, both the entanglement
entropy and mutual information have been calculated across the Kondo
destruction QCP in the Kondo lattice model Hu20.1x . The results enable a
quantum-informational means to characterize the dynamical Kondo effect.

a b

c d

Figure 11:  (a) Doping dependence of the Hall number in the hole-doped
cuprates Bad16.1 . (b) Effective mass enhancement near optimally doped YBCO
under a high magnetic field Ram15.1 . (c) Phase diagram of FeTe1-xSex, with
$T_{N}$, $T_{s}$ and $T_{c}$ respectively representing the AF, structural and
superconducting transition. The orbital-selective Mott transition temperature
($T_{\rm OSMT}$) is defined as the temperature at which the photoemission
spectral weight of the $d_{xy}$ orbital vanishes. (d) Schematic illustration
of a large-to-small Fermi surface transofrmation as $x$ is increased in
FeTe1-xSex, induced by the de-hybridization of the $d_{xy}$ orbital associated
with the OSMT Huang2022 .

##  V Implications and broader contexts

###  V.1 Delocalization-localization transition in other correlated systems

We have emphasized how Kondo destruction corresponds to a delocalization-
localization transition of the $f$-electrons across the QCP. Localization-
delocalization transition of this kind in a metallic environment is emerging
as a unifying theme across the correlated material classes. In the hole doped
cuprates, strange-metal behavior is well established Legros19.1 ; Giraldo-
Gallo2018 ; Berben22.x ; Tak92.1 ; Martin-Tlinear-prb90 ; Phillips-science22 .
Hall effect measurements in YBa2Cu3Oy (YBCO), when combined with the results
in underdoped La2-xSrxCuO4 (LSCO) and overdoped Tl2Ba2CuO6+δ and overdoped
Tl-2201, have implicated a transition between phases with carrier
concentrations $p$ and $1-p$ near the optimal doping [Fig. 11(a)] Bad16.1 .
This is accompanied by the observation of mass enhancement in YBCO near
optimal doping [Fig. 11(b)] Ram15.1 . While this remains an issue of active
discussions Putzke-natphys21 ; Bal03.1 , the notion that Fermi surface
undergoes a small-to-large transformation as a function of hope doing has also
been reported based on recent angle-dependent magnetoresistance (ADMR)
measurements in La1.6-xNd0.4SrxCuO4 Fang2022 . Moreover, recent inelastic
measurements have implicated a QCP near the optimal doping in LSCO Zhu-
LSCO22.1x . All these provide evidence for the relevance of a QCP involving an
electronic localization-delocalization to the physics of optimally hole-doped
cuprates.

A localization-delocalization transition has also been evidenced in other
correlated electron material classes. An orbital-selective Mott transition,
with a large-to-small Fermi surface transformation, has been evidenced in
FeTe1-xSex by ARPES measurements [Fig. 11(c,d)] Huang2022 . In a doped Mott
insulator of organic charge-transfer salts, some preliminary evidence for a
rapid Fermi surface change in a doped Mott insulator has also emerged from
Hall Oik15.1  and thermoelectric Wakamatsu22.1x  measurements, Finally, in
moiré systems, related properties are also being uncovered Cao18.1 ; Cao-
StrangeM20.1 ; Cao-StrangeM20.1 ; Jao-StrangeM22.1 . Intriguingly,
$T_{c}/T_{0}$ of the moiré systems is also on the order of a few percent
Cao18.1 .

###  V.2 $t-J$ models with random exchange interactions

In treatment of a Hubbard-Heisenberg $t-J-U$ model with random $J$
interactions, the model can be studied by dynamical equations that are similar
to the EDMFT equations Joshi2020 ; Dumitrescu2022 ; ChowdhuryRMP . The local
dynamics are also studied by a Bose-Fermi Anderson model, and properties
similar to what we have summarized in this article for the Kondo lattice
systems have recently been discussed Joshi2020 ; Dumitrescu2022 ; ChowdhuryRMP
. In that sense, the physics of the Kondo lattice provides a perspective to
view the doped Mott insulator problem as appearing in the cuprates. On the
other hand, there is a recognition that the cuprate strange metals need to be
addressed starting from the correlations effects in a disorder-free
environment Else2021 . The issue of treating the cuprate strange metal physics
primarily via disordered or clean models of strong electron correlations
remains a subject of investigation.

###  V.3 Loss of quasiparticles under symmetry constraints

We close by pointing to an exemplary new setting for a loss of quasiparticles.
Recent developments have highlighted how strong correlations may drive
correlated metallic topology Lai2018 ; Dzsaber2017 ; Grefe-prb20.1 . This line
of development has pushed the frontier about the interplay between strong
correlations and topology Dzs-giant21.1 ; Chen-Natphys22 . Recently, Ref.
hu2021gapless  shows that, for general interacting electron systems, the set
of all eigenvectors of the single-electron Green’s function form a
representation space of the space group. As such, the space group symmetry
constrains the single-particle excitations even when there are no Landau
quasiparticles. This allows for the realization of non-Fermi liquid topology,
where the quasiparticles are destroyed; combining the considerations of both
symmetry and strong correlations led to several proposed heavy fermion
materials to realize these phases hu2021gapless . Conversely, it points to the
emergence of topological non-Fermi liquid. For heavy fermion systems, this
raises the prospect that lattice symmetry may further enrich their global
phase diagram.

##  VI Summary and outlook

We have highlighted the theme that quasiparticles are fragile to begin with in
strongly correlated metals such as heavy fermion systems and that, in the
Kondo destruction quantum criticality, the quasiparticles are lost at the
delocalization-localization transition of the $f$-electrons. This theme
unveils a hidden Mott transition in an unlikely setting, namely between two
metallic phases. As such, the unusualness of the properties here rivals what
happens in the case of the standard Mott transition. By certain measure, it is
even more striking because, with both sides of the transition being metallic,
the Coulomb interactions are screened and it is more natural to have the
quantum phase transition to be continuous. The loss of quasiparticles at the
Kondo destruction QCP is accompanied not only by such spectacular feature as
dynamical Planckian scaling in the spin and charge dynamics but also in a
sudden transformation between large and small Fermi surfaces across the QCP.

These salient properties allow us to connect the strange metallicity of heavy
fermion metals with that of a variety of strongly correlated systems. The
strange metallicity in the cuprates and organic systems naturally develop in
the backdrop of the parent Mott insulator phase. In the iron-based
superconductors, recent experiments have provided evidence for the proximate
orbital-selective Mott phase. Finally, in moiré systems, where strange metal
behavior has also been observed, correlated insulating phases may well be
considered as the result of electron localization. It appears to be no
coincidence that the strange metal behavior develops in all these strongly
correlated material classes and that superconductivity emerges with a high-Tc.
By extension, it seems likely that, in most if not all of these systems,
strange metallicity is underlined by the loss of quasiparticles on the entire
Fermi surface. Exploring the issues in the new settings and from the new
perspectives described earlier, promise to deepen the understanding about
quantum critical metals and uncover new connections that the strange metal
physics of heavy fermion metals may have with that of the other distinct and
diverse material classes.

###### Acknowledgements.

We would like to thank the late D. R. Grempel, A. Cai, L. Y. Chen, L. Deng, M.
T. Glossop, P. Goswami, K. Grube, K. Ingersent, A. Kandala, S. Kirchner, J.
Kono, X.-W. Li, C.-C. Liu, D. Natelson, E. M. Nica, J. H. Pixley, L.
Prochaska, C. Setty, J. L. Smith, S. Sur, J. D. Thompson, H. von Löhneysen, Y.
Wang, S. Wirth, J.-W. Wu, R. Yu, J.-X. Zhu, L. J. Zhu and, particularly S.
Paschen and F. Steglich, for collaborations on work pertaining to the subject
of this article. We are grateful to J. D. Thompson for supplying Fig. 1(a).
This work has been supported in part by the NSF Grant No. DMR-2220603 (Q.S.)
the AFOSR under Grant No. FA9550-21-1-0356 (H.H.), the Robert A. Welch
Foundation Grant No. C-1411 (L.C.), and the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (Grant
Agreement No. 101020833) (H.H).

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# Transformation of Analog to Digital Resistive Switching in Cu Implanted
ITO/NiO/Ag Device for Neuromorphic Applications

Sourav Bhakta  School of Physical Sciences, National Institute of Science
Education and Research (NISER) Bhubaneswar, an OCC of HBNI, Jatni-752050,
Odisha, India  Pratap K. Sahoo<EMAIL_ADDRESS>School of Physical
Sciences, National Institute of Science Education and Research (NISER)
Bhubaneswar, an OCC of HBNI, Jatni-752050, Odisha, India  Center for
Interdisciplinary Sciences (CIS), NISER Bhubaneswar, HBNI, Jatni-752050,
Odisha, India

###### Abstract

Both analog and digital resistive switching are essential components in the
neuromorphic computing system. This work reports the influence of Cu ions for
the transformation of analog to digital resistive switching in Indium–Tin-
Oxide(ITO)/NiO/Ag memristor devices. The undoped and low-concentration Cu
doping illustrates the analog switching, whereas higher doping demonstrates
the digital characteristics. At higher bias voltage, the Schottky barrier
($\phi_{B}$) is developed at both ITO/NiO and NiO/Ag interfaces. The
increasing and decreasing of current conduction with the escalating number of
cycles for both the polarity in undoped and low doped is elucidated by the
electrode-dominated mechanism in terms of reduction and enhancement of
Schottky barrier height at the interface, respectively. The digital switching
characteristic due to the formation and rupturing of the vacancy filament at
higher doped sample is induced due to the boosting of vacancies above the
critical amount using ion implantation. The synergic effect of current
conduction due to local Cu migration and oxygen vacancies can be utilized as a
learning and forgetting process for neuromorphic applications.

††preprint: AIP/123-QED

##  I INTRODUCTION

The next generation technological advancement and complex computation
capabilities demand high data processing in miniaturized storage devices.
Hence, intense research is in progress for an alternative way to the
conventional CMOS-based technology to overcome the shortcomings of Si-based
flash memories with the physical and technological limitations of building an
advanced memory device over the past few decades. Resistive random access
memory (RRAM) exhibits the potential interest to replace these limitations due
to its simple structure, rapid operation, high-density integration, clear
switching events, good reversibility, neuromorphic computing, and 3D staking
compatibility Kim, Jeong, and Hwang (2011); Waser and Aono (2007); Baek  _et
al._ (2004); Sun  _et al._ (2015); Marković  _et al._ (2020). Among various
binary metal oxides such as TiO2 Cao  _et al._ (2009), ZnO Chen  _et al._
(2012), HfO2 Privitera  _et al._ (2013), CeO2 Younis  _et al._ (2013), CuxO
Dong  _et al._ (2007), NiO is one of the suitable candidates due to having
simple cubic rack salt structure and intrinsic wide gap (3.70 eV) for RRAM
applications to fulfill the above criteria Seo  _et al._ (2004). The digital
RRAM stored information by switching resistance abruptly in the oxide layer
from a high resistance state (HRS) to a low resistance state (LRS) and vice
versa. On the other hand, analog switching shows a gradual change in
conductance with the bias, which has drawn enormous interest among researchers
due to its intrinsic analogy to biological synapses for implementing brain-
inspired neuromorphic computing applications. Such analog memory analogous to
artificial synapses can be utilized to understand the highly efficient bio-
mimicking computing system Del Valle  _et al._ (2018). Various approaches,
such as doping, interface engineering, implantation, etc., have been adopted
for the betterment of the memristor parameter to realize the high-performance
device Shi  _et al._ (2021); Deng, Pang, and Wu (2014). Mitra et al. Mitra,
Chakraborty, and Menon (2014) showed the anticlockwise bipolar resistive
switching in chemically synthesized ITO/NiO/Ag heterojunction. The report
explains the switching mechanism in terms of rupture and recovery of
conducting filament in the oxide layer incorporated with the migration of
oxygen ions and interfacial effects. The effect of Schottky contact of ITO/NiO
for the variation of resistive switching is reported by No et al. No  _et al._
(2013). Electrode-dependent digital and analog switching in NiO-based
memristors is investigated by Swathi et al. Swathi and Angappane (2022). Saleh
et al. Saleh  _et al._ (2010) explained the impact of Ni (260 and 75 KeV) and
O (80 and 25 KeV) ion implantation in the variation of resistive switching of
Pt/NiO/Pt devices. Elliman showed the fabrication of Ta2O5/TaOx
heterostructure by O ion implantation beyond the conventional film deposition
techniques to observe the application of implantation for improving the non-
volatile RRAM device. Ning et al. Deng, Pang, and Wu (2014) investigated the
change in resistive switching characteristics in HfO2 by implanting 30 KeV Al,
Cu, and N ions at a dose of $1\times 10^{14}$ ions/cm2. The report
investigated a remarkable improvement in device-to-device uniformity by only
Cu ion implantation. Yen et al. Yen, Chin, and Gritsenko (2021) observed high
stability and excellent retention in 10 KeV As+ ion implanted SiNx memristor
compared to the nonimplanted device. Hardly reports were found showing both
analog and digital switching in NiO. However, the analog to digital resistive
switching using Cu-ion implantation in ITO/NiO/Ag devices has not been
explored.

In this report, we investigated the impact of 100 KeV Cu ion implantation in
ITO/NiO/Ag memristor to observe the transformation from analog to digital
switching. The introduction of vacancy defects in NiO induces digital
switching in high-dose Cu doping, which is found to be bulk-dependent; in
contrast, analog switching is electrode-dominated (Pristine and low Cu dose
sample). The decrease and increase of the Schottky barrier have been observed
for pristine and low-dose samples. This enhances and reduces the current
conduction with the number of cycles in pristine and low-dose samples,
respectively. The current conduction behavior can be combined together for
neuromorphic applications. The digital switching is activated in high-dose
samples due to the boosting of vacancies above the critical amount in the NiO
matrix by ion implantation.

##  II Experimental Procedure

The ITO substrate was brought from MTI Corporation. The substrate was cleaned
with acetone, isopropyl alcohol, and de-ionized water, respectively, after
sonicating with each of them for 5 minutes. The cleaned ITO substrate was
mounted to a self-designed Radio-Frequency (RF) sputtering system
perpendicular to the NiO target (2-inch diameter, $99.99\%$ purity). The
substrate-to-target distance was kept fixed at 6.5 cm during the deposition.
The sputtering parameters, such as RF power, Ar gas flow rate, base pressure,
and deposition pressure, were 100 W, 15 sccm, 6.77 $\times 10^{-6}$ mbar, and
5 $\times 10^{-3}$ mbar, respectively. The as-grown samples were annealed
using the plasma-induced chemical vapor deposition (PECVD) system in a vacuum
of 5.63 $\times 10^{-2}$ mbar. The annealed NiO sample (pristine) was
implanted using 100 keV Cu ions at room temperature, scanning over 10 $\times$
10 mm area of the sample surface using the electrostatic scanner for uniform
implantation. The sample implanted with ion fluences $5\times 10^{15}$ and
$2\times 10^{16}$ ions/cm2 are denoted as sample B and sample C, respectively.
Low ion flux at $2\times 10^{12}$ ions/cm2 was maintained to avoid beam
heating. The range of 100 keV Cu in NiO is 80 nm, calculated from Stopping and
Range of Ions in Matters (SRIM) simulations Ziegler, Ziegler, and Biersack
(2010). The simulation shows that nuclear energy loss (2286 KeV/$\mu m$)
dominates over the electronic energy loss 203.8 KeV/$\mu m$, more than 11
times. Hence, the incident ion can displace the host atoms from their lattice
site through elastic collision. This yields the creation of vacancy defects in
the system. The phase identification of pristine and implanted NiO samples was
carried out using the Rigaku Smartlab X-Ray diffractometer with Cu
$K_{\alpha}$ source ($\lambda=1.5418\AA$). The thickness and surface
morphology of the samples are investigated using cross-sectional Field
Emission Scanning Electron Microscopy (FESEM) and Atomic Force Microscopy
(AFM) images. The top electrode on the NiO sample was accomplished by
conducting Ag paste. The I-V measurements of the ITO/NiO/Ag memristor devices
were performed with a voltage sweep step of 40 mV using a two-terminal
Keithley 2450 source meter.

##  III Results and Discussions

Figure 1:  The X-ray diffraction spectra of ITO substrate, pristine, sample B,
and sample C.

The X-ray diffraction (XRD) spectra of the ITO substrate, pristine, sample B,
and sample C are shown in Fig. 1. The dashed line in Fig. 1 corresponds to the
peak position of the ITO substrate. The solid line corresponds to the NiO peak
position. The NiO peaks are observed at 37.31, 43.30, 62.73, 75.37, and 79.40∘
for all the samples. These peaks agree well with the JCPDS card number 780429,
which identifies the space group of Fm$\bar{3}$m with the cubic phase of NiO.
The XRD spectra confirm that the grown NiO film is polycrystalline in nature.

Figure 2:  (a) Crossectional FESEM images of the pristine sample. The AFM
images of the (b) pristine, (c) sample B, and (d) sample C.

The thickness of the as-grown film is evaluated using the crossectional FESEM
images. Fig. 2(a) shows that the film thickness of the pristine sample is
found to be 185 nm. The surface morphology of the pristine and implanted
samples (sample B and sample C) are examined by AFM images, as shown in Fig.
2(b-d), respectively. Pristine sample shows the continuous growth of the NiO
thin films. The surface of the implanted samples doesn’t show a drastic
variation compared to the pristine samples, rather than the roughness. The
Root Mean Squared (RMS) surface roughness of the pristine is 8.82 nm, and it
increases to 9.02 and 9.38 nm in samples B and C, respectively, due to Cu ion
implantation.

Figure 3:  (a) Schematic of ITO/NiO/Ag RRAM device for I-V measurements. The
current-voltage curve of the 1st, 25th, 50th, 75th, and 100th cycles of (b)
pristine and (c) sample B. (d) The variation of the current (1st cycle) with
an applied bias voltage of pristine and implanted samples with increased ion
fluences.

The schematic of the NiO-based ITO/NiO/Ag memristor device is shown in Fig.
3(a). The bias voltage is stressed between the top Ag electrode and the bottom
ITO substrate. Fig 3(b) and 3(c) shows the variation of current with applied
voltage of NiO memristor of pristine and sample B of the 1st, 25th, 50th,
75th, and 100th cycles for positive and negative bias sweeping. The I-V curve
of the 1st cycle with the variation of ion fluences is demonstrated in Fig.
3(d). An abrupt change of current at a particular voltage signifies the
digital switching in sample C, which will be discussed later. In contrast,
pristine and sample B show the bipolar analog switching in the -2 to +2
voltage sweep range. The small-scale asymmetry is observed in pristine. This
indicates the formation of the Schottky barrier at the NiO/Ag interface, with
NiO being a p-type semiconductor and Ag an n-type metal. The ITO/NiO junction
also forms the Schottky barrier, as reported by Swathi and No et al. Swathi
and Angappane (2022); No  _et al._ (2013). The Schottky barrier at both
interfaces is crucial for analog switching. The anticlockwise hysteresis is
observed for both the samples in both positive (0 V $\rightarrow$ 2 V
$\rightarrow$ 0 V) and negative voltage sweep regions (0 V $\rightarrow$ -2 V
$\rightarrow$ 0 V). In pristine, the current increases with increasing cycles
at both the positive and negative bias sweeping region. In contrast, the
opposite scenario is observed in sample B. The decrease of the current in
sample B may be attributed to the defect formation in the system. Cu
implantation creates a lot of vacancies, interstitial, and substitutional
defect states in the film, and the damage increases with ion fluences. When
these defect states are stressed by sufficient fields for a long period, the
carrier’s movement may be disrupted. This can yield a change in the Schottky
barrier height and Fermi level, which is responsible for decreasing the
current with the consecutive increase in the number of cycles. The negative
voltage sweep exhibits a bigger hysteresis compared to the positive voltage
sweep. The increase and decrease in conductance in pristine and sample B can
be useful for biological synapses Wang  _et al._ (2012). In biological
systems, presynaptic and postsynaptic neurons communicate themselves through
the synaptic node. The variation in the concentration of the synapses plays a
key role in the brain’s memory and learning process Yang  _et al._ (2013). The
short-term memory (STM) followed by long-term memory (LTM) is one of the basis
of human memory loss, according to Psychological studies Wixted and Ebbesen
(1991). The frequent stimulation of the biological system can transform the
STM into LTM McGaugh (2000). In this work, the increasing or decreasing
current of the NiO-based memristor can be modulated gradually with the number
of cycles by taking the combination of pristine and sample B. Hence, using
this analog resistive switching of NiO, synaptic learning and forgetting
characteristics can be stimulated and employed in neuromorphic systems. The
increasing current with consecutive pulses in pristine can be assigned as a
learning process, and the decreasing current in sample B is analogous to the
forgetting process. Now, to investigate the mechanism of analog switching at
the interface, we fitted the double logarithmic I-V curve of the 1st positive
cycle linearly. Fig. 4(a) and 4(b) show the linear fitting of double
logarithmic I-V curve and ln(I) vs $V^{1/2}$ plot (Schottky plot) of the 1st
positive cycle for pristine, respectively. The Ohmic condition is satisfied
under the applied low bias voltage (0 V $\rightarrow$ 0.9 V), as the curve is
fitted well linearly with a slope of 1.09, as seen in Fig. 4(a). But, under a
high bias voltage, the deviation of slope from 1 indicates other conductive
mechanisms, such as Schottky emission Biju  _et al._ (2011), space charge
limited conduction Won Seo  _et al._ (2009), Fowler Nordheim tunneling Chang
Lee  _et al._ (2013), and Poole Frenkel emission Park and Lee (2016) that can
dominate over Ohmic conduction. The good linear fitting observed in Fig. 4(b)
at a high bias voltage region suggests the formation of Schottky contact at
both interfaces. The good linear fitting of the Schottky plot at the high bias
voltage region and the asymmetry characteristic signifies the domination of
Schottky emission compared to the other conduction mechanism. We used the
following Richardson-Schottky equation to understand the I-V characteristics
Zhong  _et al._ (2014):

$I=AB^{*}T^{2}exp[{-q\phi_{B}+\frac{q}{kT}\sqrt{\frac{qE}{4\pi\epsilon_{i}}}}]$
(1)

Where, A = Conduction area, $B^{*}$ = Effective Richardson constant
($\frac{4\pi qm_{n}^{*}k_{0}^{2}}{h^{3}}$), T = Absolute Temperature, k =
Boltzmann constant, q = electric charge, E = $\frac{V}{d}$, d = NiO film
thickness, $\phi_{B}$ = Schottky barrier, $\epsilon_{i}$ Dielectric constant
of NiO.

After taking logarithm on both sides, the equation 1 reflects as:

$ln(I)=ln(AB^{*}T^{2})-\frac{q\phi_{B}}{kT}+\frac{q}{kT}\sqrt{\frac{q}{4\pi\epsilon_{i}d}}\sqrt{V}$
(2)

The intercept value at the ln (I) axis after linear fitting of the Schottky
plot in positive and negative bias at higher field region (Fig. 4(b)) (0.9 V
$\rightarrow$ 2 V) provides the Schottky barrier height. In this case, we used
the Richardson constant as $B^{*}$ = 119.8 K/$K^{2}.cm^{2}$ by considering
$m_{n}^{*}$ = 1.0 and $m_{0}$ = $9.1\times 10^{-31}$ Kg. The variation of the
Schottky barrier with the number of cycles for positive and negative voltage
sweep of pristine and sample B is shown in Fig. 5(a) and 5(b), respectively.
The energy band diagram of Schottky contact is illustrated in Fig. 5(c). In
pristine, the Schottky barrier decreases with the increase of the number of
cycles for both positive and negative bias. The obstruction for the carrier
reduces due to the decrease in the Schottky barrier at the interface. Hence,
the reduction of $\phi_{B}$ with increasing cycle increases the conduction in
pristine for both positive and negative voltage sweep. In contrast, the
decrease of the current with the number of cycles in sample B is due to the
increase of the Schottky barrier at the interface for both positive and
negative voltage sweep. The increase in the barrier height may be attributed
to the creation of defects by Cu ion implantation in the NiO thin films. The
carriers need more energy to overcome the barrier height, which decreases the
current conduction in sample B. Hence, the prevailing conduction mechanism in
analog RRAM devices is governed by the interface-dominant Schottky effect
rather than filament formation.

Figure 4:  (a) The double logarithmic I-V plot and (b) ln(I) vs $V^{1/2}$ plot
with liner fitting for first positive voltage scan of pristine.  Figure 5:
The variation of Schottky barrier with the increase of the number of cycles in
(a) pristine, (b) sample B for positive and negative voltage scan. (c) The
schematic of Schottky contact at NiO/Ag interface.  Figure 6:  (a) The
current-voltage curve of the 1st, 25th, 50th, 75th, and 100th cycles of sample
C, (b) The LRS and HRS state with an increasing number of cycles for the
endurance test of sample C. The HRS and LRS was read at 0.3 V.  Figure 7:  The
schematic of the proposed filamentary model for the (a) zero bias, (b) SET,
and (c) RESET process in digital resistive switching of sample C.

Sample C shows the digital bipolar resistive switching in the same voltage
sweeping range (-2 V $\rightarrow$ 0 V $\rightarrow$2 V and 2 V $\rightarrow$
0 V $\rightarrow$ -2 V), as shown in Fig. 6(a). An electroforming process was
observed at the lower voltage region due to the presence of high vacancy
defect concentration in the sample, induced at high Cu ion fluences. The
higher fluence sample is transformed into a digital RRAM device through the
electroforming process, whereas the pristine and lower fluence samples (sample
B) show the signature of analog switching. The current jumps at 0.54 V for the
first cycle and reaches the LRS state, defining it as a SET process. The
device moves from LRS to the HRS at -1.15 V by dropping the current abruptly,
defining the RESET process. The SET and RESET cycles of the memristor (sample
C) are clearly observed in Fig. 6(a). The SET voltage increases with
increasing the number of cycles, while the RESET voltage reduces for the
higher one with respect to the initial cycle. The hysteresis developed
drastically in sample C compared to the pristine and sample B. The mechanism
of such an abrupt change of current at the SET and RESET point is understood
through the vacancy filamentary model. Now, the endurance of the device is
examined by plotting the HRS and LRS with the increasing number of cycles, as
shown in Fig. 6(b). The current was read at 0.3 V. The graph shows that HRS
exhibits a negligible fluctuation with the number of cycles, whereas the LRS
is relatively stable with almost zero slope. The clear ON-OFF is sustained up
to the 100th cycle. The gradual increase in the gap between LRS and HRS (as
HRS exhibits a positive slope) suggests that the device can store information
for a long period of time. Hence, ion implantation with proper ion energy and
species can be an alternate technique to improve the endurance capability of
RRAM devices.

The schematic of the filamentary model through oxygen vacancy for
understanding the current conduction mechanism in sample C is shown in Fig. 7.
Lee et al. Lee, Magyari-Köpe, and Nishi (2010) reported that lower formation
energy is required for oxygen vacancy than Ni vacancy, and the certain
configured oxygen vacancies form clusters help to form the conducting path. In
sample C, high fluence boosts the sufficient number of vacancies in the
matrix, which helps in filament formation. In zero bias condition (Fig. 7(a)),
no vacancy filament formed, although a lot of vacancy defects are already
present in the system. After applying the sufficient positive bias voltage,
the vacancies and oxygen ions are generated. The oxygen ions will migrate
toward the Ag electrode due to high drift velocity compared to vacancies. The
existing large amount of induced and generated vacancies in the matrix
accumulate to form the vacancy filament. The path between the top and bottom
electrode is completed through the formation of vacancy filament at the SET
point (Fig. 7(b)), and the current jumps abruptly to reach the LRS state. In
the RESET process, the oxygen ions migrate back toward the bottom electrode.
During migration, they recombine with the oxygen vacancies. The quick
annihilation of the vacancies will rupture the filament (Fig. 7(c)), and the
current will drop sharply at the RESET voltage. Hence, the formation and
rupture of the vacancy filament explain the SET and RESET process of the
ITO/NiO/Ag device (implanted with a sufficient high fluence of 2$\times$1016
ions/cm2). In neuromorphic systems, synaptic weights play a crucial role in
determining the strength of the connections between neurons. Digital RRAM
cells can be used to store these weights by encoding them in different
resistance states. By adjusting the device to a SET or RESET state, RRAM can
attain various resistance levels, allowing for precise modulation of synaptic
weights. The quick SET and RESET operations of RRAM devices provide rapid
updates to these weights, enabling high-speed learning and processing. The
delayed SET with increasing cycles is compatible with slow learning, whereas
the quick RESET indicates high processing. The analog switching is modulated
by the interface-dominated Schottky effect at the NiO/Ag and ITO/NiO
interface, whereas the bulk-dominated vacancy filamentary model influences the
digital switching in sample C. The NiO film is modified by the creation of a
lot of vacancy defects using Cu ion implantation, which plays the underlying
role of transforming analog to digital switching of ITO/NiO/Ag memristor
device after the threshold fluence.

###  III.1  CONCLUSION:

In conclusion, we investigated the analog to digital transformation of bipolar
resistive switching in 100 KeV Cu-implanted ITO/NiO/Ag memristor devices.
Pristine and sample B show analog switching, but the digital switching
characteristic developed in sample C. The Schottky barrier is formed at the
NiO/Ag and ITO/NiO interfaces. The observed analog switching in pristine and
sample B is interface barrier dominant due to the Schottky effect rather than
the bulk domination. The reduction of $\phi_{B}$ for both the polarity
increases the current conduction in pristine with the increase of the number
of cycles. The opposite scenario, i.e., the lowering in current due to an
increase of $\phi_{B}$ with the number of cycles, is observed in sample B. The
combination of pristine and sample B can be utilized as a learning and
forgetting process for neuromorphic applications. The sudden jump of current
from OFF to ON and ON to OFF state at SET and RESET voltage in sample C is
elucidated by the formation and rupturing of bulk-dominated vacancy
filamentary model. In this case, the high Cu ion fluence boosts the vacancies
in sample C above the critical vacancies (compared to pristine and sample B)
which is sufficient in forming the filament for exhibiting such digital
switching. Slow learning is associated with high SET points with the
increasing number of cycles, whereas quick SET exhibits the possibility of
high processing.

###### Acknowledgements.

The authors thank the National Institute of Science Education and Research
(NISER), DAE, Government of India, for funding this research work through
project number RIN-4001. Sourav Bhakta and Pratap Sahoo also acknowledge the
staff of the Inter-University Accelerator Center, New Delhi, for their
constant support and for providing a stable beam for 100 KeV Cu ion
implantation.

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