#  Multifractality and statistical localization in highly heterogeneous random
networks

Diego Tapias<EMAIL_ADDRESS>Institute for
Theoretical Physics, Georg-August-Universität Göttingen, 37077 Göttingen,
Germany  Peter Sollich<EMAIL_ADDRESS>Institute for
Theoretical Physics, Georg-August-Universität Göttingen, 37077 Göttingen,
Germany  Department of Mathematics, King’s College London, London WC2R 2LS, UK

###### Abstract

We consider highly heterogeneous random networks with symmetric interactions
in the limit of high connectivity. A key feature of this system is that the
spectral density of the corresponding ensemble exhibits a divergence within
the bulk. We study the structure of the eigenvectors associated with this
divergence and find that they are multifractal with the statistics of
eigenvector elements matching those of the resolvent entries. The
corresponding localization mechanism relies on the statistical properties of
the nodes rather than on any spatial structure around a localization centre.
This “statistical localization” mechanism is potentially relevant for
explaining localization in different models that display singularities in the
bulk of the spectrum of eigenvalues.

##  I Introduction

After three decades of an extensive analysis of its main properties, the
configuration model has stood its ground as the simplest but realistic
theoretical model for complex networks bollobas1998random ; newman2018networks
. A particularly appealing property of this model is that the degree
distribution can be freely specified, making it possible to capture a large
range of random networks varying from highly homogeneous (all nodes with same
degree) to very heterogeneous metz2020spectral .

In addition to the topological structure of the network, the edges are often
weighted, physically representing the strength of an interaction between nodes
newman2004analysis ; barrat2004architecture . This is encoded in a matrix that
contains the weights of all edges in the network. For the theoretical
approach, rather than focusing on individual matrices, one reduces them to
their defining properties and represents these in a statistical ensemble of
weight matrices albert2002statistical . The interplay between interaction and
structure can be studied by analyzing, for instance, the spectral properties
of the corresponding ensemble of random matrices (see for instance ref.
kuhn2008spectra  for a classical work in the case of symmetric sparse matrices
or ref. metz2019spectral  for a more recent perspective on non–Hermitian
random matrices).

Analytical methods for computing spectra of configuration model networks can
be found in references dorogovtsev2003spectra ; rogers2008cavity ;
newman2019spectra ; susca2021cavity . Most of the theory is based on
message–passing algorithms and the corresponding cavity (or resolvent)
equations mezard2009information ; susca2021cavity . Recently, in references
metz2020spectral ; silva2022analytic  an analysis of these equations for
highly heterogeneous networks, i.e. those with non-vanishing relative variance
of the degree distribution (in the high connectivity limit), has been
performed. The main result is that in the thermodynamic limit, classical
results of Random Matrix Theory do not apply. This includes in particular
Wigner’s semicircle law to describe the spectrum of eigenvalues. Deviations
from this are accompanied also by other striking properties, such as the
appearance of singularities in the bulk of the spectral density.

In this paper, we study highly heterogeneous random networks using the
negative binomial distribution as the degree distribution for the
configuration model, with interactions sampled from a Gaussian with zero mean
and a variance that scales inversely with the mean connectivity. As has been
shown in reference silva2022analytic , this setting leads to a family of
networks controlled by a single parameter that allows one to cover the range
from highly heterogeneous all the way to essentially homogeneous networks. In
reference silva2022analytic , the resolvent equations were derived and solved
for this system. Their solution revealed that in the case of highly
heterogeneous graphs, there is a divergence in the spectral density at
eigenvalue zero. This can be traced back to the distribution of the local
Density of States, which acquires a power–law tail leading to an infinite
first moment.

Here we investigate the structure of the eigenvectors associated with the
divergence of the spectral density. Our main finding is that they are
localized and exhibit “strong multifractal” behavior. We rationalize this by
finding an anticorrelation between the degree of the node and the
corresponding amplitude of the eigenvector. It turns out that localization
occurs in nodes with low degree (relative to the mean), which typically are
far apart from each other. We denote the corresponding localization mechanism
“statistical localization” as it is driven by the statistical properties of
the node(s). This contrasts with the standard case of Anderson localization on
random networks kravtsov2018non ; tikhonov2021anderson ; garcia2022critical ,
where eigenvectors are localized on nodes around a single localization centre.
We conjecture that the new mechanism of statistical localization is relevant
also for other types of models that exhibit singularities within the bulk of
the spectral density such as the Poisson random graph with Gaussian couplings
kuhn2008spectra  or the sparse Barrat–Mézard trap model tapias2020entropic ;
tapias2022localization .

##  II Heterogeneous weighted random networks

We provide in this section the essential pieces of information that define the
ensemble of random networks that we study. For more details, we refer to the
reader to the original paper silva2022analytic . Let us consider a simple and
undirected network with $N$ nodes. We consider the configuration model with a
negative binomial distribution of degrees $k$,

$\displaystyle
p_{k}=\frac{\Gamma(\gamma+k)}{\Gamma(\gamma)}\frac{1}{k!}\left(\frac{c}{\gamma}\right)^{k}\frac{1}{\left(1+{c}/{\gamma}\right)^{\gamma+k}}$
(1)

where $c$ is the mean connectivity, $\Gamma(\cdot)$ is the Gamma function and
$0<\gamma<\infty$ controls the heterogeneity of the distribution 111In ref.
silva2022analytic , the parameter $\gamma$ is denoted by $\alpha$. We use a
different convention here because we will need $\alpha$ later in the notation
for the spectrum of fractal dimensions.. Indeed, the relative variance of the
degree distribution is given by

$\displaystyle\frac{\sigma^{2}}{c^{2}}=\frac{1}{c}+\frac{1}{\gamma}$ (2)

Each node $i$ is assigned a degree $k_{i}$ drawn randomly from $p_{k}$, and
nodes are then randomly connected following the standard configuration model
prescription newman2018networks . If there is a link between nodes $i$ and $j$
we set the random variable $c_{ij}=1$, otherwise it is set to zero.
Additionally we consider interactions $J_{ij}$ between nodes as random
variables sampled independently on each link (with $J_{ij}=J_{ji}$) from a
distribution $p_{J}$ with mean zero and standard deviation $J/\sqrt{c}$.
Overall, the weight matrix ${\bm{A}}$ has elements

$\displaystyle A_{ij}=c_{ij}J_{ij}$ (3)

In the high–connectivity limit $c\to\infty$, it makes sense to consider the
distribution of the rescaled degrees $\kappa=k/c$. This distribution
$\nu(\kappa)$ is formally defined as follows

$\displaystyle\nu(\kappa)=\lim_{c\to\infty}\sum_{k=0}^{\infty}p_{k}\,\delta\\!\left(\kappa-\frac{k}{c}\right)$
(4)

For $p_{k}$ as given in eq. (1), one finds that $\nu(\kappa)$ is a Gamma
distribution with shape parameter $\gamma$ and scale parameter $1/\gamma$,
i.e.

$\displaystyle\nu(\kappa)=\frac{\gamma^{\gamma}\kappa^{\gamma-1}{\rm{e}^{-\gamma\kappa}}}{\Gamma(\gamma)}$
(5)

The regime $\gamma<1$ characterizes random networks with strongly
heterogeneous degrees as the relative variance of the degree distribution is
greater than 1 (cf. eq. (2)). In the limit $N\to\infty$ (taken before
$c\to\infty$), the distribution of eigenvalues,
$\rho(\lambda)=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\delta(\lambda-\lambda_{i})$
, of the weight matrix (eq. (3)) exhibits a power law divergence

$\displaystyle\rho(\lambda)\sim|\lambda|^{\gamma-1}\,,\quad|\lambda|\to 0$ (6)

with $0<\gamma<1$. This behavior can be rationalized via the connection
between the resolvent and the spectral density potters2020first , namely

$\displaystyle\rho(\lambda)=\lim_{\epsilon\to 0}\lim_{N\to\infty}\frac{1}{\pi
N}\sum_{i=1}^{N}{\rm{Im}}\,G_{ii}(\lambda-i\epsilon)$ (7)

where the $G_{ii}$ are the diagonal elements of the resolvent matrix
${\bm{G}}=((\lambda-i\epsilon){\bm{I}}-{\bm{A}})^{-1}$, with $\bm{I}$ the
identity matrix. As a matter of fact, networks generated by the configuration
model exhibit locally a tree–like structure newman2018networks , this implies
that we can use the cavity method rogers2008cavity ; susca2021cavity  to
estimate the diagonal elements of the resolvent matrix, also called _local
Density of States_ (lDOS). Ref. silva2022analytic  shows that at $\lambda=0$
the distribution of $y_{i}={\rm{Im}}\,G_{ii}$ in the limit $\epsilon\to 0$ is
given by

$\displaystyle
P_{0}(y)=\frac{\gamma^{\gamma}}{\Gamma(\gamma)J^{\gamma}}\frac{\exp\left(-\frac{\gamma}{Jy}\right)}{y^{\gamma+1}}$
(8)

By virtue of the resolvent identity (eq. (7)), the mean value of the lDOS
distribution determines the spectral density. From the power-law tail of the
distribution (8) one can then clearly see the origin of the divergence in
equation (6).

A final result that we want to introduce here comes from reference
dembo2021empirical . This states that in the limit $c\to\infty$, the spectral
density of the weight matrix $\bm{A}$ is given by the free multiplicative
convolution speicher2009free ; akemann2011oxford  of $\nu(\kappa)$ with the
Wigner semicircle law $\rho_{\rm{W}}(\lambda)$. As an implication, one has
that the weight matrix can be decomposed as the product of two different
matrices, $\bm{X}$ and $\bm{D}$, where the latter is the degree matrix with
elements $D_{ij}=\kappa_{i}\delta_{ij}$ and the former is a random (symmetric)
matrix with zeros on the diagonal and off–diagonal elements drawn from a
Gaussian distribution with mean zero and variance $J^{2}/N$. After
symmetrization, one can write

$\displaystyle{\bm{A}}={\bm{D}}^{1/2}{\bm{X}}{\bm{D}}^{1/2}$ (9)

This decomposition is used in reference silva2022analytic  to compare the
predictions (in particular, for the spectral density) from theory to numerical
results using exact diagonalization, without explicitly constructing the
configuration model. We will use the same approach to obtain the eigenvectors
associated with the modes around $\lambda=0$ numerically.

##  III Wavefunction statistics

We exploit the decomposition in equation (9) to generate an interaction matrix
$\bm{A}$. Then we diagonalize it with the Lanczos algorithm to extract the
eigenvectors with associated eigenvalues closest to zero 222For this purpose
we use an implementation of the _Arpack_ package for the _Julia_ programming
language arpack .. In Figure 1((a)) we show the distribution of the (scaled)
squared eigenvector entries $x_{i}=N|\psi_{i}|^{2}$, which in quantum
mechanical language are squared wavefunction amplitudes. The data are obtained
from independent instances of $\bm{A}$ for three different values of $\gamma$.
The figure suggests that the distributions have power-law tails for large $x$,
with $\gamma$-dependent exponents 333The figure also suggests the existence of
a left power-law tail. This would contribute to the scaling of the exponents
$\tau(q)$ for negative $q$ (see eq. (16)) and equivalently to the multifractal
spectrum for $\gamma\alpha>1$ (cf. eq. (15)). As this piece is not relevant
for the understanding of the localization properties, we do not consider it
further in our analysis.. In fact, we found that the exponents agree with
those for the resolvent, i.e. $P(x)\sim P_{0}(x)\sim x^{-(1+\gamma)}$ for
sufficiently large $x$, as shown in Figure 1((b)). This agreement of the
statistics of the resolvent and eigenvector entries is consistent with
previous results for eigenvectors with multifractal properties, see for
instance monthus2017statistical .

Figure 1: (a) Distribution of the (logarithm) of squared wavefunction
amplitudes, $x_{i}=N|\psi_{i}|^{2}$, for three different eigenvectors of
interaction matrices of size $N=2^{14}$ generated for different heterogeneity
parameters $\gamma$. (b) Estimation of the power law tail of the distribution
$P(x)\sim x^{-g(\gamma)}$, fitted from data for $x>x_{*}$ with
$x_{*}=20\,x^{\rm{typ}}$ well above the mode ($\sim x^{\rm{typ}}$) of the
distribution. Error bars show statistics over 128 instances of $\bm{A}$, using
the 50 eigenvectors with eigenvalues closest to $\lambda=0$ for each instance.

The power-law tail motivates the following _ansatz_ for the whole distribution
above a scale $x_{\min}(N)$, close to the mode of the distribution

$\displaystyle P(x)=b(N)x^{-(\gamma+1)}$ (10)

with $b(N)$ and $x_{\min}(N)$ functions to be determined. Their scaling can be
estimated by using the normalization of $P(x)$ and its first moment
monthus2017statistical ; the latter follows from the normalization of the
eigenvectors, $1=\sum_{i}|\psi_{i}|^{2}=N^{-1}\sum_{i}x_{i}$. The first
condition, namely, $\int P(x)\,dx=1$ gives

$\displaystyle 1\approx\int_{x_{\min}}^{N}P(x)\,dx\approx\gamma
bx_{\min}^{-\gamma}$ (11)

whereas the second one, namely $\int xP(x)\,dx=1$, yields

$\displaystyle 1\approx\int_{x_{\min}}^{N}xP(x)\,dx\approx\gamma
bN^{-\gamma+1}$ (12)

These two relations imply $b\sim N^{\gamma-1}$ and $x_{\min}\sim
N^{1-1/\gamma}$. As $x_{\min}$ sets the scale from which the power–law becomes
valid, we identify this as the typical value of the distribution. This scaling
is confirmed using results from Exact Diagonalization as can be appreciated in
Figure 2.

Figure 2: Scaling of typical value defined as
$x^{\rm{typ}}={\rm{e}}^{\langle\ln x\rangle}$ with system size for
$\gamma\in\\{1/3,2/3\\}$. The prediction $x^{\rm{typ}}\sim N^{1-1/\gamma}$ is
shown by the dashed lines.

Substituting the scaling for $b(N)$ in our ansatz (eq. (10)), we write the
distribution $P(x)$ in the generic way

$\displaystyle P(x)=\frac{A}{x}N^{\gamma-1}x^{-\gamma}$ (13)

with $A=O(N^{0})$ a normalization constant. The spectrum of fractal dimensions
$f(\alpha)$, which is defined as de2014anderson ; kravtsov2015random

$\displaystyle f(\alpha)=\lim_{N\to\infty}\ln\left(xNP(x)\right)/\ln N$ (14)

where on the r.h.s. $x=N^{1-\alpha}$ or conversely $\alpha=1-\ln x/\ln N$.
Substitution of eq. (13) into eq. (14) gives the expression

$\displaystyle f(\alpha)=\gamma\alpha\,,\quad{\rm{for}}\quad\gamma\alpha\leq
1$ (15)

where the upper cutoff $\alpha_{\max}=\frac{1}{\gamma}$ corresponds to the
lower cutoff $x_{\min}$. Finally, the Legendre transform of equation (14)
gives the set of exponents characterizing the $q$–th moment of the
distribution $I_{q}(N)=\sum_{i}|\psi_{i}|^{2q}\propto N^{-\tau(q)}$ The
Legendre transform relation reads explicitly monthus2017statistical

$\displaystyle-\tau(q)=\max_{\alpha}\,[f(\alpha)-q\alpha]$ (16)

Substitution of $f(\alpha)$ (eq. (15)) into equation (16) yields

$\displaystyle\tau(q)=\begin{cases}\frac{q}{\gamma}-1\qquad&q\leq\gamma\\\
0\qquad&q>\gamma\end{cases}$ (17)

Figure 3 compares this result with Exact Diagonalization.

Figure 3: (a) Scaling of the wavefunction moments as encoded in the exponents
$\tau(q)$ for $N=2^{14}$. The dashed lines correspond to the linear piece of
the theoretical prediction (17). See numerical details in the appendix.

##  IV Discussion

The scaling of the moments $I_{q}$ as given by equation (17) has been found
across different models and the corresponding phase has been described
variously as “quasilocalized” kravtsov2015random , or localized with
multifractal properties monthus2016localization , or localized with “strong
multifractal” behavior garcia2022critical . The corresponding models are the
Gaussian Rosenzweig–Porter, random Levy matrices, and the Anderson model on
small–world networks, respectively. References kravtsov2015random ;
monthus2016localization  point to a power–law (instead of exponential)
localization as the origin of this kind of behavior. In contrast, reference
garcia2022critical  (see also garcia2020two ) focusses on the existence of a
length scale that characterizes the decay of wave functions along the typical
branches as responsible for the scaling above. These mechanisms, however, do
not seem to be operative for our system.

In order to find the mechanism that governs the behavior of our model, we
construct finite instances of the networks generated with the configuration
model. Then we investigate the correlation between the squared eigenvector
entry (or “mass”) and the degree of the node. Our finding is that low–degree
nodes, typically leaves (i.e. nodes with degree one), concentrate the mass of
the eigenvectors. For mildly heterogenous networks, i.e. those with
$1<\gamma<2$, the anticorrelation between degree and eigenvector mass is
easily visible in a scatter plot (c.f. Figure 4). If we extrapolate this
finding to $c\to\infty$, we expect that for a given instance nodes with small
relative degree will concentrate the mass of the eigenvector, and what is more
relevant, that those nodes may be far apart from each other on the network.
Indeed, we find that the dominant nodes from figure 4 lie at distances of the
order of the graph diameter from each other. Thus what matters for
localization is the identity of each node and its statistical properties,
rather than its spatial location in the network. In this sense, we say that
the system exhibits “statistical localization” and that this leads to the
multifractal behavior encoded in the exponents (17).

To illustrate further the mechanism described above, consider the Bouchad trap
model on a sparse random graph margiotta2018spectral . The ground state of
this system is the Boltzmann distribution, and the set of exponents
characterizing the scaling of the moments is of the form (17), with
temperature instead of $\gamma$ as the control parameter (see equation 4.15 in
reference riccardothesis ). Thus, for finite instances, the nodes that would
carry most of the mass of the vector are the ones with the largest energies,
corresponding to the deepest traps. Those nodes are spatially uncorrelated and
are instead identified by local statistical properties.

At this point, it is worth comparing our results with earlier observations of
localization in heterogenous networks. As a representative example of this
class we consider the Laplacian on Poisson (Erdös–Rényi) random graphs
biroli1999single ; kuhn2008spectra  (see also monasson1999diffusion ). In both
cases, for small $|\lambda|$, localization is driven by low degree nodes.
However, for our model, this happens in the bulk of the spectrum and is
accompanied by a divergence in the spectral density; whereas for the Poisson
random graph, localization happens in the (Lifshitz khorunzhiy2006lifshitz )
tail of the spectrum, which is separated from the bulk by a mobility edge.
Additionally, while it has been observed that localized states in this tail
are centred on geometric defects with abnormally low connectity
biroli1999single  and that the density of states is dominated by low degree
nodes kuhn2008spectra , it remains an open question whether those states
exhibit multifractal behavior and the existence of multiple localization
centres. This would be interesting to study in future work.

Figure 4: Correlation among the scaled eigenvector mass,
$x_{i}=N|\psi_{i}|^{2}$ and the node degree $k_{i}$ for the eigenvector with
eigenvalue closest to $0$ for an instance constructed via equation (3), for a
finite configuration model network with mean connectivity $c=400$ and
$\gamma=1.2$ and of size $N=8187$. (This network is the giant connected
component of the original configuration model graph with $2^{13}=8192$ nodes.)

In conclusion, we have analysed the wavefunction statistics for highly
heterogeneous random graphs from the configuration model defined by a negative
binomial degree distribution and in the limit of high connectivity. We have
found that the distribution of the eigenvector mass for the modes that
contribute to the divergence of the spectral density is a power-law, with the
same exponent as the one for the local Density of States. Those eigenvectors
exhibit strong multifractality, and for finite graphs are essentially
localized in the lowest-degree nodes. On this basis, we introduce the concept
of “statistical localization” that is to be contrasted with the standard one
of “spatial localization”.

We leave for future work an extensive analysis of the eigenstate correlations
of these modes as has been done for the Anderson model tikhonov2021eigenstate
or the Gaussian Rosenzweig-Porter model kravtsov2015random , in order to
investigate specific footprints of the statistical localization mechanism. It
will also be interesting to carry out an analysis similar to the one in this
paper for the sparse Barrat-Mézard trap model tapias2020entropic ;
tapias2022localization , which like the model considered here exhibits
divergences in the bulk of the spectral density. Work in this direction is in
progress. Finally, we point out that the phase described in earlier studies as
a “frozen phase” has similar phenomenology to the one observed here
chamon1996localization ; carpentier2001glass ; chou2014chalker , and thus may
potentially also be viewed as an instance of the notion of statistical
localization that we have introduced here. A quantitative analysis along these
lines is left as an interesting task for future work.

*

##  Appendix A Numerical details

We generate $2^{21}/N$ instances of weight matrices of size
$N\in\\{2^{13},2^{14}\\}$, constructed according to equation (9). For each
instance, we extract the 50 eigenvectors closest to $\lambda=0$. Then, for
each eigenvector $\psi^{(\alpha)}$ with entries $\psi^{(\alpha)}_{i}$ we
compute the $q$–moment (of the squared entries) as
$I_{q}=\sum_{i=1}^{N}|\psi^{(\alpha)}_{i}|^{2q}$. Finally, we obtain the
typical value over all the vectors as $I_{q}^{\rm{typ}}={\rm{e}}^{\langle\ln
I_{q}\rangle}\sim N^{-\tau(q)}$ and estimate $\tau(q)$ as:

$\displaystyle\tau(q)=-\frac{\ln I_{q}^{\rm{typ}}(N)-\ln
I_{q}^{\rm{typ}}(N-\delta N)}{\ln(\delta N)}$ (18)

with $N=2^{14}$ and $\delta N=2^{13}$.

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# Innovating at Speed and at Scale: A Next Generation Infrastructure for
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Richard A. Gottscho  Lam Research Corporation, Fremont, CA 94538, USA  Edlyn
V. Levine  America’s Frontier Fund, Arlington, VA 22202, USA  Department of
Physics, Harvard University, Cambridge, MA 02138, USA  Tsu-Jae King Liu
Department of Electrical Engineering and Computer Sciences, University of
California Berkeley, Berkeley, CA 94720, USA  Paul C. McIntyre  Department of
Materials Science and Engineering, Stanford University, Stanford, CA 94305,
USA  Stanford Synchrotron Radiation Lightsource, Menlo Park, CA 94025, USA
Subhasish Mitra  Department of Electrical Engineering, Stanford University,
Stanford, CA 94305, USA  Department of Computer Science, Stanford University,
Stanford, CA 94305, USA  Boris Murmann  Department of Electrical Engineering,
Stanford University, Stanford, CA 94305, USA  Jan M. Rabaey  Department of
Electrical Engineering and Computer Sciences, University of California
Berkeley, Berkeley, CA 94720, USA  Sayeef Salahuddin  Department of Electrical
Engineering and Computer Sciences, University of California Berkeley,
Berkeley, CA 94720, USA  Willy C. Shih  Harvard Business School, Boston, MA
02163, USA  H.-S. Philip Wong  Department of Electrical Engineering, Stanford
University, Stanford, CA 94305, USA

###### Abstract

Semiconductor innovation drives improvements to technologies that are critical
to modern society. The country that successfully accelerates semiconductor
innovation is positioned to lead future semiconductor-driven industries and
benefit from the resulting economic growth. It is our view that a next
generation infrastructure is necessary to accelerate and enhance semiconductor
innovation in the U.S. In this paper, we propose such an advanced
infrastructure composed of a national network of facilities with enhancements
in technology and business models. These enhancements enable application-
driven and challenge-based research and development, and ensure that
facilities are accessible and sustainable. The main tenets are: a challenge-
driven operational model, a next-generation infrastructure to serve that
operational model, technology innovations needed for advanced facilities to
speed up learning cycles, and innovative cost-effective business models for
sustainability. Ultimately, the expected outcomes of such a participatory,
scalable, and sustainable nation-level advanced infrastructure will have
tremendous impact on government, industry, and academia alike.

##  I Introduction

Semiconductor technologies are foundational to all modern digital economies.
As the U.S. House-passed America COMPETES act heads for reconciliation with
the Senate, proposals for supporting the rejuvenation of domestic
semiconductor manufacturing are arriving from multiple quarters. All of the
published proposals we have seen so far advocate spending more on research,
training the workforce for the future, and leveraging the infrastructure that
the writers either already have in place or hope to build. But a question we
should ask is what kind of next-generation research and development (R$\&$D)
infrastructure will be needed to meet future challenges in the face of the
technological and economic obstacles that lie ahead? Equally important, what
business model will ensure sustainability of these investments over the many
years it will take to regain that leadership? That is what we hope to address
here.

In recent years, the cost and time associated with the introduction and
development of new semiconductor technologies have drastically increased, and
in many cases are prohibitive nsfFoundry ; nsfCMOSX . Growth in complexity of
semiconductor devices and systems are a driving factor to these expense and
delay increases, and has led to challenges in design robustness nsfEDA . For
the U.S. to remain a technology leader, it is imperative to accelerate the
pace of innovation and expand the number of innovators that have access to the
relevant device technology, design, and prototyping capabilities to innovate
at speed and at scale.

Meaningful innovation must be proven relevant to and integrated with
commercial state-of-the-art devices before emerging semiconductor technologies
can transition to market applications. To be both relevant and sustainable,
novel approaches are needed for both the infrastructure used to conduct
research and commercialization activities, and for the business models to
operate and sustain that infrastructure.

##  II Mission-Driven Infrastructure

Semiconductor R$\&$D infrastructure has diverse requirements that depend
heavily upon the specific challenges to be addressed and the part of the
semiconductor ecosystem that is engaged in solving these challenges. Figure 1
depicts the various axes that define the relevant ecosystem.

The first axis follows the traditional stack model of design, with increasing
abstraction from materials and devices, to architectures, systems, and
applications. The second axis follows five components, which together form the
building blocks of any information system: analog, digital, memory,
communication, and security src . A final axis concerns the application spaces
and the associated systems, namely cloud, edge, and extreme edge
bryant2020nanotechnology . Indeed, the future requirements of the cloud
(compute and data centers) are vastly different from the edge (mobiles,
internet routers, edge computers, 5+G), and even further removed from the
extreme edge (IoT, industrial internet, internet of actions, robots and
autonomous systems).

Figure 1: The semiconductor and information technology innovation space. A
network of facilities organized to achieve a moonshot covers sufficiently
large sub-spaces to provide meaningful R$\&$D environments.

The axes in Figure 1 represent the enormous space of diverse information
technology challenges. This space is too large to be addressed by a single
R$\&$D facility that covers only a subdomain. Research centers built to
support the development of singular technologies such as advanced CMOS, 3D
integration, storage, or analog/mixed-signal are examples of infrastructure
limited to subdomains. Focused, disconnected infrastructure is the de-facto
situation today for facilities at universities, company R$\&$D facilities, and
national labs. This approach to infrastructure is not scalable, participatory,
or sustainable within the evolving research enterprise required by the
advancing complexity of information systems.

Instead, we propose that the next generation of semiconductor R$\&$D
infrastructure be designed and built around a set of moonshot goals OSTP .
Moonshots should be chosen to cover a substantial fraction of the space of
Figure 1, providing context to the advances in each individual element. Such
infrastructure should contain sufficient flexibility and adaptability to allow
for exploration of a variety of new approaches both within and outside a
moonshot program. This approach embraces co-design or co-optimization for
which innovative solutions are created by considering multiple axes or aspects
of a problem simultaneously.

We offer two examples to illustrate the potential scope, intent and scale of a
moonshot vision:

Moonshot 1: A Virtual Model of the World Many companies are investing in
developing a virtual model of the physical world. Examples include Meta’s
Metaverse and NVIDIA’s Omniverse. The realization of such virtual models could
have huge impacts in domains such as health care, agriculture, climate
control, mobility, etc. Current technology falls far short of the required
energy efficiency needed to create a functional digital copy of the world,
making breakthrough advances in the basic information processing units (here
called xPUs) essential src . Historically, the energy efficiency of graphical
processing units (GPUs) has doubled every two years. We envision that a
90,000X improvement in energy efficiency of xPUs over the next 25 years will
be necessary to enable a scalable virtual model of the world, requiring a 2.5X
increase in energy efficiency every two years. Achieving this moonshot will
require breakthroughs on many fronts, including nanometer-scale components,
innovative logic and memory integration, advances in packaging and
interconnects, mixtures of photonics and electronics, and entirely new chip
architectures to implement the next generation of efficient artificial
intelligence algorithms.

Moonshot 2: A Physical Portal to the Virtual World Benefiting from the
tremendous amount of information and knowledge enabled by the virtual world
envisioned by Moonshot 1 requires far richer human-computer interfaces with
capabilities that extend beyond what is possible today. Augmented VR/AR
headsets are an example of such a portal. Today’s VR/AR headsets weigh 500
grams ( 1 pound), have 2-3 hours of battery life, and are priced at $\$$300\.
Within 25 years, we envision VR/AR headsets with far richer human-computer
interfaces and capabilities that weigh 10 grams or less, have 16 hours of
battery life, provide data protection and security, and are priced in the same
range as today’s mobile phones. Achieving this moonshot will require
innovation spanning multiple disciplines including small form factor smart
sensors and actuators, mixed-signal data in-sensor data processing, neuro-
inspired information extraction and interpretation, high density low-power and
low-leakage data storage, efficient communication, embedded intelligence,
intrinsic security, advanced small form-factor packaging, and software stacks
that are sufficiently small so as not to load down the system, yet are
scalable.

Other moonshots can be envisioned, such as the realization of human brain-like
functionality in a form factor the size of the human skull at the same average
power budget of 20 W, or enabling radiation-hardened cloud computing in space.
The examples given serve as models and are indicative of the very different
requirements imposed on the next-generation infrastructure depending upon the
moonshot and the application space. One thing these moonshots have in common
is the need for an accelerated pace of hardware development.

##  III General Requirements for Next-Generation Infrastructure

To increase the pace of hardware innovation, there are three common challenges
that a next-generation infrastructure must resolve. These challenges are
encountered when integrating new materials, processes, and devices to realize
systems that meet the metrics dictated by a moonshot:

  1. 1.

The first challenge is combining established materials and processes in an
innovative way to reach desired goals. Foundries are not amenable to changing
their standard process flow or parameters - e.g. dopant concentration, film
thickness, etc. - to implement a new device. Since most foundry facilities are
operated at full capacity to maintain profitability, they cannot be used to
run the necessary experiments to develop new technology.

  2. 2.

The second challenge is even more complex and involves the introduction of new
materials and processes. It is almost impossible to contract a foundry to
perform the steps needed for new material introduction due to concerns about
contamination and tolerance of the tools.

  3. 3.

The third challenge is ensuring co-innovation and co-optimization across the
device-to-system stack. System complexity has grown across all layers of the
stack and the cost of incremental improvements within a layer have become
extremely high. Continuation of siloed research domains in materials, devices,
architectures, and systems is consequently unsustainable.

Next-generation infrastructure will need to be designed and built to overcome
these challenges and meet the specific research and development tasks posed by
a given moonshot vision. Unconventional thinking is needed to establish the
necessary infrastructure requirements to ensure success. We propose the
following requirements be met by future facilities:

Enablement of Frontier Research in CMOS and Emerging Devices: Advances in both
CMOS circuits and emerging devices must be enabled by new facilities built in
the U.S. The co-innovation and co-optimization of CMOS devices with emerging
devices is critical to unleash the full potential of new technologies nsfCMOSX
. U.S. government research funding has mostly been targeted towards emerging
devices nsfBeyond . Although such funding should continue, it is not
surprising that research activity on advanced transistors is virtually non-
existent in the U.S. This puts U.S. researchers at a global disadvantage
because CMOS is the common enabler of all other technologies and the
technologies developed for CMOS are broadly applicable for the manufacture of
new devices and systems. Revenue coming from high-performance transistor
technology dwarfs that from any other technology. Therefore, it is important
to ensure that prototyping facilities allow integrated research on both CMOS
and other emerging devices.

A Hierarchy of Experimental and Prototyping Facilities: Next-generation
infrastructure should incorporate multiple facilities to address innovation
required for moonshots that span a significant subdomain of Figure 1.
Laboratories will be needed to support new materials research and chemical
synthesis, new equipment development and process integration, and experimental
pilot lines for device and package prototyping. The requirements, tool sets,
and expertise needed for each of these facilities differs greatly. Prototyping
facilities will additionally need to be varied to enable experimentation on a
hierarchy of wafer sizes from small substrates (coupons), to 150 mm, 200 mm
wafers, and finally 300 mm wafer pilot lines. Enabling experimentation on
small substrates in coupon facilities is critical since it allows for lower-
cost, flexibility, and broad access to rapid experimentation for the research
community. This is in contrast to 300-mm wafer pilot lines which are
necessarily more rigid and not economically feasible for new materials or
process development. Different process flows enabling memory, logic, and
analog device prototyping must also be included and connected with advanced
packaging capabilities.

Networking and Integration of Facilities: The facilities constituting next
generation infrastructure should be networked and integrated allowing for easy
access to research teams, transfer of wafers, and shared data flows for
developments to transition from device to systems level and vise-versa.
Existing university laboratories, national laboratories, and commercial R$\&$D
facilities should be incorporated within this network. Next generation
research infrastructure must additionally be connected with commercial
manufacturers to enable transition of new technologies to volume production.
Standards for common technology platforms such as transistor elements, memory
units, and parameters for heterogeneous integration, should be established in
partnership with industry whenever possible. These standards will be important
for cases in which novel systems can be built on top of existing silicon
infrastructure. However, next generation infrastructure must also enable the
many innovative nanomaterial, nanofabrication, and nanodevice concepts that,
by nature of their novelty, do not conform to any standards.

Enhanced Virtual Process Infrastructure: Virtual models, often referred to as
digital twins, of the tools and process flows used to prototype, scale, and
manufacture new devices will drastically increase cycles of learning by
enhancing empirical understanding of the introduction of new materials and
process alterations. Digital twins of the process flows should be created to
parallel the new physical infrastructure. These digital twins will be informed
by data collected from across all facilities within the network to ensure a
statistically significant data volume. Such data currently exist only within
company proprietary firewalls. Open access to data generated in networked
facilities will accelerate the development of virtual models. Continuous
infusion of new data from ongoing experiments and prototyping will further
enable digital twins of the manufacturing processes to learn and improve.
Ultimately, the digital twins that are envisioned will enable researchers to
simulate their innovation integrated with CMOS across all process steps to
predict feasibility or improve yield upon implementation. Researcher access to
the prototyping facilities is then essential to physically prove out concepts
that have been modeled in the virtual world and to provide further data,
thereby enabling a virtuous cycle of learning.

Enhanced Design Tools and Flows: Current hardware innovation severely lags the
pace of software development. Long design cycles are necessary for a new
device to evolve from concept to adoption in next-generation systems. To
short-circuit this long dependency chain, we envision the creation of digital
twins of devices and compute systems that enable virtual and physical
prototypes to co-evolve, with some parts existing in the virtual and others in
the physical world. Digital twins of devices and systems allow for the
injection of envisioned devices in virtual form into actual compute systems,
and, vice-versa, the construction of full-scale, in-silico prototypes with
newly developed devices. Achieving these digital twins requires the creation
of design tools that are integrated closely with the physical and digital twin
process infrastructure, and enable the early exploration and meaningful
comparison between design alternatives. These tools should cover the following
grounds: design space exploration and co-design across the stack, beyond
traditional chip design to full system design, complete system design
verification, design for robustness in light of innovative computational
paradigms, integrity and security, and co-creation with software.

Cost Effective, Sustainable, and Accessible: Next-generation infrastructure
must be both cost-effective and accessible to researchers across many
organizations including startups, universities, and mid- to large-scale
companies. The hierarchy of facilities will require various business models to
ensure sustainability. Common to all the facilities is the need for dedicated
teams of cross-functional engineers to support new designs, software,
controls, integration, and data flows of this next generation infrastructure.
Similar to user facilities at the national labs, this permanent staff will be
responsible for creating an accessible, world class user program, technique
development, and research programs in collaboration with user groups. This
permanent staff cannot be assignees from companies as their allegiance lies
with the company where they are employed. As of today, this type of
environment does not exist in the U.S. for semiconductor research.

Next generation infrastructure must enable rapid innovation by increasing
cycles of learning across the development process: design, prototyping,
scaling, and production. This is possible by creating the envisioned national
network of semiconductor materials laboratories, equipment development
facilities, and hierarchy of prototyping lines along with their digital twins.
By leveraging small diameter substrates at the onset, multiple technology
vectors can be explored rapidly to support development of technologies needed
to achieve the moonshots, without the fear of contamination in large wafer
facilities. Digital twins of manufacturing processes and of the computational
system under development further accelerate rates of learning in an ever
upward spiral of innovation.

##  IV Technology Innovations to Enable Next-Generation Infrastructure

Many of the aforementioned requirements are possible to achieve with existing
technology. However, some technical advances are needed to truly realize what
is envisioned.

The equipment for next generation infrastructure will need to be carefully
chosen and may need to be custom built. Existing toolsets, such as 300 mm
wafer processing tools, are designed for high-volume manufacturing (HVM)
rather than high-throughput, iterative research. These tools are optimized for
a very narrow window of operating conditions necessary to achieve the high
number of wafers/hour, process uniformity, and yield needed for HVM. There are
many processing tools in a 300 mm fabrication facility that each support only
a single (baseline) process. Thus, even though a 300 mm facility may be
intended for prototyping, fear of a tool breaking down can lead to fab-like
restrictions on its operation, limiting researchers exploring new materials
and process conditions. Building custom tool sets for research purposes,
including tools for coupon prototyping lines, is the likely solution.
Alternatively, tool manufacturers could specify a wider range of operating
conditions that are allowed, while guaranteeing that the baseline process can
be easily restored.

Rapid prototyping is possible to achieve using coupon facilities; however,
graduating a successful coupon prototype to a wafer prototype line as
envisioned in the hierarchy of facilities is non-trivial. Doing so is
contingent on developing a means to scale results from a coupon to a full
wafter. This scaling problem is extraordinarily complex. Ideally, we would be
able to push a button and the results from rapid coupon prototyping would be
scaled and transformed to a wafer-scale design that can then be fabricated.
This is not possible today and research advances are needed to achieve this
goal.

Fortunately, the same technologies that will underlie the development of a
Virtual Model of the World as articulated in Moonshot 1 and the Physical
Portal into the Virtual World described by Moonshot 2 can be deployed to
address the coupon-to-wafer scaling challenge. Moonshot 1 necessarily includes
the digital twin of the manufacturing process envisioned in the enhanced
virtual process infrastructure. Digital twins of process flows across multiple
wafer sizes that learn through the continuous collection of data, leveraging
of machine learning, and combination with existing physics-based models of
materials properties and reactivities, have the potential to effectively
determine a solution from the high-dimensional parameter space of the coupon-
to-wafer problem.

Advances in high throughput metrology that provide rapid, high quality data
will further enhance the learning available with virtual models. Rapid
prototyping depends on rapid cycles of learning, which are in turn limited by
the availability of quality metrology data. For example, in learning how to
fabricate a new device, the transmission electron micrograph (TEM) is
considered to be the most accurate available data. Unlike data generated by
self-driving automobiles which employ inexpensive sensors and large fleets of
automobiles to drive around and collect data, a TEM is complex, expensive, and
laborious. Emphasis should be placed on high throughput, non-destructive,
real-time metrologies to accelerate learning cycles, lower costs, and feed the
digital twins.

Connecting facilities with different capabilities will require solving
challenges with coupon and wafer flows that go outside one laboratory and into
another. This requires carefully defining the conditions of the wafer and
controlling the handling and shipping environment from the time the wafer is
processed to the time it is ready for the next process. Depending on the
materials and device sensitivities, different protocols and levels of control
will be required. These protocols and control limits will need to be included
in any new material development. Standard methodologies and metrology
references will be essential. To enable rapid prototype development between
facilities, it will be important to establish logistics that minimize delays,
especially for queue time sensitive samples.

##  V Business Model Innovations to Enable Sustainable Facilities

The infrastructure described here is more than just a collection of
facilities. It will need funding to cover capital, operational, and research
costs, but it also needs to attract a supporting ecosystem of suppliers that
provides engineering and design services, and an active and robust network of
engineers and innovators. The cost of running such an advanced semiconductor
R$\&$D environment will not be cheap. IMEC is a 640M Euro per year operation
with 4,500 employees; the NYCreates Albany facility is a $\$$300M per year
operation. While government grants can provide start-up funding, what’s needed
are business models that enable sustainable operation of facilities for many
decades.

The U.S. government should consider a model that encourages a competition of
ideas. One that was successfully used by the National Science Foundation (NSF)
in the National Strategic Computing Initiative and precursor programs was a
competition to build advanced high-performance computing centers nsfNSCI .
Competitive proposals for new facilities not only had to meet technical
specifications, but also had to be accompanied with business plans and initial
financing contributions from partners that demonstrated long term
sustainability. Additionally, because there will need to be a collection of
facilities that address the broad spectrum of activities described above, the
U.S. government may benefit from having overarching program management to
provide coordination.

The government could award funding to build multiple advanced semiconductor
infrastructure facilities – both on the strength of technical proposals and
the strength of commitments from industrial, academic, and other partners. By
encouraging contributions from teams of organizations with diverse skills and
points of view, we can encourage a portfolio of approaches across centers to
the difficult technological challenges ahead. Contributions might be in the
form of funding or the commitments of resources, and the magnitude and length
of commitments can be a positive factor in the awarding of centers.

The users of the facilities will likewise be granted access on a competitive
basis. A significant advantage of such “bottom up” mechanisms for assigning
time on shared resources is that they facilitate a wide variety of high
risk/high reward research efforts to occur in parallel, although they require
developing and sustaining a cadre of highly professional staff to support
varied efforts that push the state of the art and, often, to develop new
methods. This approach has been central to the success of the U.S. Department
of Energy’s scientific user facilities that have fostered ground-breaking
research advances across multiple science and engineering disciplines for many
decades.

As the government considers awarding subsidies to semiconductor manufacturers,
it should consider designating some portion of those funds to purchase foundry
capacity and foundry resources for exploratory research to support these
proposed infrastructure centers. It could then award this capacity on a
competitive basis to researchers who seek to demonstrate new device or process
technologies, facilitating a critical process for scale-up of promising new
technologies that would otherwise have a great deal of difficulty in accessing
such capabilities.

Having a customer for the next-generation semiconductor infrastructure centers
will ensure commercial market discipline. NASA has used this model
successfully over many decades to fuel advances in aeronautical research. With
its Aircraft Energy Efficiency programs nasaE3  and Commercial Orbital
Transportation Services (COTS) program nasaCOTS , it ensured that there was a
market pull for the technological developments that it funded and underwrote a
lot of risk. This approach also encourages market competitive operations
rather than subsidy-hungry facilities and avoids many of the concerns
regarding IP rights when accepting government funding.

A streamlined approach to rapid proposal development and awarding is needed to
encourage private-sector participation to build new infrastructure.
Incorporation of national and university laboratories into the network of
facilities will provide additional incentive to private sector companies to
participate by providing access to capabilities, such as advanced metrology,
that they do not have on their own.

##  VI Conclusion

Semiconductors are a foundational technology that underpin a majority of
technological advances for the betterment of society: from economic
development and national security to combating climate change and improving
medicine and healthcare. The technical community has a responsibility to
formulate a research agenda for accelerating the pace of innovation that
includes a plan for the R$\&$D infrastructure necessary to fulfill the high
expectations of society. The research agenda advocated here is mission-
oriented and challenge-driven, aiming to cover the relevant subdomains of the
semiconductor innovation space that is spanned by the three axes of systems,
layers of abstraction, and component technologies.

It is essential that the next generation R$\&$D infrastructure built to meet
this research agenda is scalable, participatory, and sustainable so as to
broaden the base of innovation and lower the access barrier for technology
translation to commercial reality. Here, we have proposed technology and
business model enhancements for facilities that enable application-driven and
challenge-based innovation founded on a competition of ideas. While this
infrastructure is anchored in one nation, it should have participation from
technology leaders around the world for the simple reason that only the best
ideas deserve focused attention. By inviting partnership with other nations
that complement each other, this national infrastructure can serve to
accelerate the U.S. as a global leader in R$\&$D and enable international
collaboration to reach for the common good.

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