theoremTheorem[section]

# Continuity of Thresholded Mode-Switched ODEs and Digital Circuit Delay
Models

Arman Ferdowsi  TU WienViennaAustria<EMAIL_ADDRESS>,  Matthias
Függer  CNRS & LMF, ENS Paris-Saclay, Université Paris-Saclay & InriaGif-sur-
YvetteFrance<EMAIL_ADDRESS>,  Thomas Nowak  Université Paris-Saclay, CNRS,
ENS Paris-SaclayInstitut Universitaire de FranceGif-sur-YvetteFrance
<EMAIL_ADDRESS>and  Ulrich Schmid  TU WienViennaAustria
<EMAIL_ADDRESS>
(2023)

###### Abstract.

Thresholded mode-switched ODEs are restricted dynamical systems that switch
ODEs depending on digital input signals only, and produce a digital output
signal by thresholding some internal signal. Such systems arise in recent
digital circuit delay models, where the analog signals within a gate are
governed by ODEs that change depending on the digital inputs.

We prove the continuity of the mapping from digital input signals to digital
output signals for a large class of thresholded mode-switched ODEs. This
continuity property is known to be instrumental for ensuring the faithfulness
of the model w.r.t. propagating short pulses. We apply our result to several
instances of such digital delay models, thereby proving them to be faithful.

mode-switched ordinary differential equations; thresholding operator;
continuity; circuit delay models; faithfulness

This research was supported by the Austrian Science Fund (FWF) project DMAC
(grant no. P32431) and the ANR project DREAMY (ANR-21-CE48-0003).

††journalyear: 2023††copyright: rightsretained††conference: Proceedings of the
26th ACM International Conference on Hybrid Systems: Computation and Control;
May 9–12, 2023; San Antonio, TX, USA††booktitle: Proceedings of the 26th ACM
International Conference on Hybrid Systems: Computation and Control (HSCC
’23), May 9–12, 2023, San Antonio, TX, USA††doi:
10.1145/3575870.3587125††isbn: 979-8-4007-0033-0/23/05

##  1\. Introduction

A natural class of hybrid systems can be described by the dynamics of a
continuous process, which is controlled by externally supplied digital mode
switch signals, and provides a digital output based on whether some internal
signal crosses a threshold, see Fig. 1 for an illustration. Examples are
digitally controlled thermodynamic processes, hydrodynamic systems, and, in
particular, digital integrated circuits. The continuous dynamics of these
systems are described by _Ordinary Differential Equations_ (ODEs) for the
temperature, the pipe’s pressures and fill-levels, or the gate’s currents and
voltages over time. Digital mode switches are used to switch between ODE
systems, e.g., by turning on a heater, closing a valve, or applying an input
transition to a gate’s input. The environment of the hybrid system is only
notified if the temperature or fill-level crosses a threshold, or, in the case
of a digital gate, is said to produce an output transition when some internal
voltage crosses a threshold.

Figure 1. Thresholded mode-switched ODE with a single mode input $i$, the
delayed input $i_{d}$, two continuous states $x,y$, and two thresholded
outputs $\Theta_{\alpha}(x)$ and $\Theta_{\beta}(y)$.

In this work, we consider the composition of such hybrid systems in a circuit,
where digital threshold signals of one component drive mode switch signals of
a downstream component. We give conditions that ensure the continuity of the
outputs of such circuits with respect to their inputs and provide two
application examples in the context of circuit delay models. The proven
continuity property shows that small variations of the inputs lead to small
variations of the output signal, a property that is necessary for digital
circuit models to be consistent with physical analog ODE models.

Digital circuits, continuity, and faithful delay models. Analog simulations of
digital circuits are time-consuming and are thus replaced by digital
simulations whenever possible. Typical application domains that require
simulation of precise circuit transition times are particularly timing-
critical, asynchronous parts of a circuit, e.g., inter-neuron links using
time-based encoding in hardware-implemented spiking neural networks
(BVMRRVB19), where the worst-case delay estimates provided by static timing
analysis techniques are not sufficient for ensuring correct operation.

A mandatory prerequisite for dynamic timing analysis are digital delay models,
which allow to accurately determine the input-to-output delay of every
constituent gate in a circuit. Suitable models must also account for the fact
that the delay of an individual signal transition usually depends on the
previous transition(s), in particular, when they were close. The simplest
class of such models are _single-history delay models_ (FNS16:ToC; BJV06;
FNNS19:TCAD), where the input-to-output delay $\delta(T)$ of a gate depends on
the previous-output-to-input delay $T$.

It has been proved by Függer et al. (FNNS19:TCAD) that a certain continuity
property of single-history models is mandatory for the digital abstraction to
faithfully model the analog reality. In particular, the predicted output
transitions must not be substantially affected by arbitrarily short input
glitches. For example, the constant-low input signal and an arbitrarily short
low-high-low pulse must produce arbitrary close gate output signals. So far,
the only delay model known to ensure this continuity property is the
_involution delay model_ (IDM) (FNNS19:TCAD), which consists of zero-time
Boolean gates interconnected by single-input single-output involution delay
channels. An IDM channel is characterized by a delay function $\delta$, which
is a negative involution, i.e., $-\delta(-\delta(T))=T$. In its generalized
version, different delay functions $\delta_{\uparrow}$ resp.
$\delta_{\downarrow}$ are assumed for rising resp. falling transitions,
requiring $-\delta_{\uparrow}(-\delta_{\downarrow}(T))=T$. Unlike all other
existing delay models, the IDM has been proved to faithfully model glitch
propagation for the so-called short-pulse filtration problem (FNNS19:TCAD),
and is hence the only candidate for a faithful delay model known so far
(FNS16:ToC).

It has also been shown (FNNS19:TCAD) that involution delay functions arise
naturally in a 2-state thresholded hybrid channel model, which consists of a
pure delay component, a slew-rate limiter with a rising and falling switching
waveform, and an ideal comparator (Fig. 2): The binary-valued input $i_{a}$ is
delayed by $\delta_{\mathrm{min}}>0$, which assures causality of channels,
i.e., $\delta_{\uparrow/\downarrow}(0)>0$. For every transition on $i_{d}$,
the generalized slew-rate limiter switches to the corresponding waveform
($f_{\downarrow}/f_{\uparrow}$ for a falling/rising transition). The essential
property here is that the analog output voltage $o_{a}$ is a _continuous_ (but
not necessarily smooth) function of time. Finally, the comparator generates
the output $o_{d}$ by digitizing $o_{a}$ w.r.t. the discretization threshold
voltage $V_{th}$.

Figure 2. Hybrid involution delay channel model (upper part) with a sample
execution (bottom part). Adapted from (FNNS19:TCAD).

Whereas the accuracy of IDM predictions for single-input, single-output
circuits like inverter chains or clock trees turned out to be very good, this
is less so for circuits involving multi-input gates (OMFS20:INTEGRATION). It
has been revealed by Ferdowsi et al. (FMOS22:DATE) that this is primarily due
to the IDM’s inherent lack of properly covering output delay variations caused
by _multiple input switching_ (MIS) in close temporal proximity (CGB01:DAC),
also known as the _Charlie effect_ : compared to the _single input switching_
case, output transitions are sped up/slowed down with decreasing transition
separation time on different inputs. Single-input, single-output delay
channels like IDM cannot exhibit such a behavior.

To capture MIS effects in a 2-input NOR gate, Ferdowsi et al. (FMOS22:DATE)
hence proposed an alternative digital delay model based on a 4-state hybrid
gate model. It has been obtained by replacing the 4 transistors in the RC-
model of a CMOS NOR gate by ideal zero-time switches, which results in one
mode per possible digital state of the inputs
$(A,B)\in\\{(0,0),(0,1),(1,0),(1,1)\\}$. In each mode, the voltage of the the
output signal and an internal node are governed by constant-coefficient first
order ODEs. When an input signal changes its state, the system switches to the
new mode and its corresponding ODEs.

Albeit digitizing this hybrid gate model, using a comparator with a suitable
threshold voltage $V_{th}$ as in Fig. 2, leads to a quite accurate digital
delay model, it turned out to still fail to capture the MIS delay for a rising
output transition. In a follow-up paper (ferdowsi2022accurate), Ferdowsi et.
al. hence introduced a refined gate delay model, where the switching-on of the
pMOS transistors is not instantaneous, but rather governed by a simple time
evolution function $\sim 1/t$, inspired by the Shichman-Hodges transistor
model (ShichmanHodges). The resulting 4-state hybrid model consists of a
single not-constant-coefficient first-order ODE per mode, and has been shown
to accurately model MIS effects.

Whereas the experimental evaluation of the modeling accuracy of the hybrid
models discussed above shows that they outperform the simple IDM model
(OMFS20:INTEGRATION), it is not clear whether they are also _faithful_ digital
delay models. What would be needed here is a proof that the digital delay
models obtained by digitizing hybrid models satisfy the continuity property
required for faithfulness.

Contributions. Our paper answers this question in the affirmative. More
generally, we prove that any thresholded hybrid model like the one shown in
Fig. 1 that satisfies some mild conditions on their ODEs results in a
continuous digitized hybrid model. We then show that the above hybrid gate
models fall into this category, and that the proven continuity implies
faithful short-pulse propagation of any such model. Since the square of a
signal is (proportional to) its power, this also implies a continuity property
from the input signal power to the output signal power. Consequently, these
delay models are indeed promising candidates for the correct and timing+power-
accurate simulation of digital circuits. In more detail:

  1. (1)

We show that any hybrid model, where mode $m$ is governed by a system of
first-order ODEs $\frac{dx}{dt}=f_{m}(t,x)$, leads to a continuous digital
delay model, provided all the $f_{m}$ are continuous in $t$ and Lipschitz
continuous in $x$, with a common Lipschitz constant for every $t>0$ and $m$.

  2. (2)

We prove that the parallel composition of finitely many digitized hybrid gates
in a circuit result in a unique and Zeno-free execution, under some mild
conditions regarding causality. In conjunction with our continuity result, we
prove that the resulting model is faithful w.r.t. solving the canonical short-
pulse filtration problem.

  3. (3)

We demonstrate that the hybrid gate models proposed in (FMOS22:DATE;
ferdowsi2022accurate) satisfy these properties, and are hence continuous and
thus faithful.

Paper organization. In Section 2, we instantiate our general continuity result
(5). Section 3 presents our main continuity result for hybrid gate models (6
and 7), and Section 4 deals with circuit composition. In LABEL:sec:examples,
we provide examples for the hybrid models considered in this work: a simple
heater from literature (henzinger2000theory), the simple hybrid gate model
(FMOS22:DATE), and the advanced gate model (ferdowsi2022accurate). Some
conclusions and directions of future research are provided in
LABEL:sec:conclusions.

##  2\. Thresholded Mode-Switched ODEs

In this section, we provide a generic proof that every hybrid model that
adheres to some mild conditions on its ODEs leads to a continuous digital
delay model. We start with proving continuity in the analog domain and then
establish continuity of the digitized signal obtained by feeding a continuous
real-valued signal into a threshold voltage comparator. Combining those
results will allow us to assert the continuity of digital delay channels like
the one shown in Fig. 2.

###  2.1. Continuity of ODE mode switching

For a vector $x\in\mathds{R}^{n}$, denote by $\lVert x\rVert$ its Euclidean
norm. For a piecewise continuous function $f:[a,b]\to\mathds{R}^{n}$, we write
$\lVert f\rVert_{1}=\int_{a}^{b}\lVert f(t)\rVert\,dt$ for its $1$-norm and
$\lVert f\rVert_{\infty}=\sup_{t\in[a,b]}\lVert f(t)\rVert$ for its supremum
norm. The projection function of a vector in $\mathds{R}^{n}$ onto its
$k^{\text{th}}$ component, for $1\leq k\leq n$, is denoted by
$\pi_{k}:\mathds{R}^{n}\to\mathds{R}$.

In this section, we will consider non-autonomous first-order ODEs of the form
$\frac{d}{dt}\,x(t)=f(t,x(t))$, where the non-negative $t\in\mathds{R}_{+}$
represents the time parameter, $x(t)\in U$ for some arbitrary open set
$U\subseteq\mathds{R}^{n}$, $x_{0}\in U$ is some initial value, and
$f:\mathds{R}_{+}\times U\to\mathds{R}^{n}$ is chosen from a set $F$ of
bounded functions that are continuous for $(t,x)\in[0,T]\times U$, where
$0<T<\infty$, and Lipschitz continuous in $U$ with a common Lipschitz constant
for all $t\in[0,T]$ and all choices of $f\in F$. It is well-known that every
such ODE has a unique solution $x(t)$ with $x(0)=x_{0}$ that satisfies
$x(t)\in U$ for $t\in[0,T]$, is continuous in $[0,T]$, and differentiable in
$(0,T)$.

The following lemma shows the continuous dependence of the solutions of such
ODEs on their initial values. To be more explicit, the exponential dependence
of the Lipschitz constant on the time parameter allows temporal composition of
the bound. The proof can be found in standard textbooks on ODEs
(teschl2012ordinary, Theorem 2.8).

###### Lemma 1.

Let $U\subseteq\mathds{R}^{n}$ be an open set and let $f:\mathds{R}\times
U\to\mathds{R}^{n}$ be Lipschitz continuous with Lipschitz constant $K$ for
$t\in[0,T]$ with $T>0$, and let $x,y:[0,T]\to U$ be continuous functions that
are differentiable on $(0,T)$ such that $\frac{d}{dt}\,x(t)=f(t,x(t))$ and
$\frac{d}{dt}\,y(t)=f(t,y(t))$ for all $t\in(0,T)$. Then, $\lVert
x(t)-y(t)\rVert\leq e^{tK}\lVert x(0)-y(0)\rVert$ for all $t\in[0,T]$.

A _step function_ $s:\mathds{R}_{+}\to\\{0,1\\}$ is a right-continuous
function with left limits, i.e., $\lim_{t\to t_{0}^{+}}s(t)=s(t_{0})$ and
$\lim_{t\to t_{0}^{-}}s(t)$ exists for all $t_{0}\in\mathds{R}_{+}$. A _binary
signal_ $s$ is a step function $s:[0,T]\to\\{0,1\\}$, a _mode-switch signal_
$a$ is a step function $a:[0,T]\to F$, $t\mapsto a_{t}$.

Given a mode-switch signal $a$, a _matching output signal_ for $a$ is a
function $x_{a}:[0,T]\to U$ that satisfies

  1. (i)

$x_{a}(0)=x_{0}$,

  2. (ii)

the function $x_{a}$ is continuous,

  3. (iii)

for all $t\in(0,T)$, if $a$ is continuous at $t$, then $x_{a}$ is
differentiable at $t$ and $\frac{d}{dt}\,x_{a}(t)=a_{t}(t,x_{a}(t))$.

For (iii), recall that the domain of $a$ is $F$.

###### Lemma 2.1 (Existence and uniqueness of matching output signal).

Given a mode-switch signal $a$, the matching output signal $x_{a}$ for $a$
exists and is unique.

###### Proof.

$x_{a}$ can be constructed inductively, by pasting together the solutions
$x_{t_{j}}$ of $\frac{d}{dt}\,x_{t_{j}}(t)=a_{t_{j}}(t,x_{t_{j}}(t))$, where
$t_{0}=0$ and $t_{1}<t_{2}<\dots$ are $a$’s switching times in $S_{a}$: For
the induction basis $j=0$, we define $x_{a}(t):=x_{t_{0}}(t)$ with initial
value $x_{t_{0}}=x_{t_{0}}(t_{0}):=x_{0}$ for $t\in[0,t_{1}]$. Obviously, (i)
holds by construction, and the continuity and differentiability of
$x_{t_{0}}(t)$ at other times ensures (ii) and (iii).

For the induction step $j\to j+1$, we assume that we have constructed
$x_{a}(t)$ already for $0\leq t\leq t_{j}$. For $t\in[t_{j},t_{j+1}]$, we
define $x_{a}(t):=x_{t_{j+1}}(t)$ with initial value
$x_{t_{j+1}}=x_{t_{j+1}}:=x_{a}(t_{j})=x_{t_{j}}(t_{j})$. Continuity of
$x_{a}(t)$ at $t=t_{j}$ follows by construction, and the continuity and
differentiability of $x_{t_{j+1}}(t)$ again ensures (ii) and (iii). ∎

Given two mode-switch signals $a$, $b$, we define their distance as

(1) $d_{T}(a,b)=\lambda\big{(}\\{t\in[0,T]\mid a_{t}\neq b_{t}\\}\big{)}$

where $\lambda$ is the Lebesgue measure on $\mathds{R}$. The distance function
$d_{T}$ is a metric on the set of mode-switch signals.

The following 2 shows that the mapping $a\mapsto x_{a}$ is continuous.

###### Theorem 2.

Let $K\geq 1$ be a common Lipschitz constant for all functions in $F$ and let
$M$ be a real number such that $\lVert f(t,x(t))\rVert\leq M$ for all $f\in
F$, all $x\in U$, and all $t\in[0,T]$. Then, for all mode-switch signals $a$
and $b$, if $x_{a}$ is the output signal for $a$ and $x_{b}$ is the output
signal for $b$, then $\lVert x_{a}-x_{b}\rVert_{\infty}\leq
2Me^{TK}d_{T}(a,b)$. Consequently, the mapping $a\mapsto x_{a}$ is continuous.

###### Proof.

Let $S=\\{t\in(0,T)\mid\text{$a$ or $b$ is discontinuous at
$t$}\\}\cup\\{0,T\\}$ be the set of switching times of $a$ and $b$. The set
$S$ must be finite, since both $a$ and $b$ are right-continuous on a compact
interval. Let $0=s_{0}<s_{1}<s_{2}<\cdots<s_{m}=T$ be the increasing
enumeration of $S$.

We show by induction on $k$ that

(2) $\forall t\in[0,s_{k}]\colon\quad\lVert x_{a}(t)-x_{b}(t)\rVert\leq
2Me^{tK}d_{t}(a,b)$

for all $k\in\\{0,1,2,\dots,m\\}$. The base case $k=0$ is trivial. For the
induction step $k\mapsto k+1$, we distinguish the two cases
$a_{s_{k}}=b_{s_{k}}$ and $a_{s_{k}}\neq b_{s_{k}}$.

If $a_{s_{k}}=b_{s_{k}}$, then we have $a_{t}=b_{t}$ for all
$t\in[s_{k},s_{k+1})$ and hence $d_{t}(a,b)=d_{s_{k}}(a,b)$ for all
$t\in[s_{k},s_{k+1}]$. Moreover, we can apply Lemma 1 and obtain

(3) $\forall t\in[s_{k},s_{k+1}]\colon\quad\lVert x_{a}(t)-x_{b}(t)\rVert\leq
e^{(t-s_{k})K}\lVert x_{a}(s_{k})-x_{b}(s_{k})\rVert\enspace.$

Plugging in (2) for $t=s_{k}$ reveals that (2) holds for all
$t\in[s_{k},s_{k+1}]$ as well.

If $a_{s_{k}}\neq b_{s_{k}}$, then $x_{a}$ and $x_{b}$ follow different
differential equations in the interval $t\in[s_{k},s_{k+1}]$. We can, however,
use the mean-value theorem for vector-valued functions (rudin1976principles,
Theorem 5.19) to obtain

(4) $\forall t\in[s_{k},s_{k+1}]\colon\quad\lVert
x_{a}(t)-x_{a}(s_{k})\rVert\leq M(t-s_{k})\enspace\text{and}$ (5) $\forall
t\in[s_{k},s_{k+1}]\colon\quad\lVert x_{b}(t)-x_{b}(s_{k})\rVert\leq
M(t-s_{k}).$

This, combined with the induction hypothesis, the equality
$d_{t}(a,b)=d_{s_{k}}(a,b)+(t-s_{k})$, and the inequalities $1\leq e^{tK}$ and
$e^{s_{k}K}\leq e^{tK}$, implies

$\begin{split}\lVert x_{a}(t)-x_{b}(t)\rVert&\leq\lVert
x_{a}(t)-x_{a}(s_{k})\rVert\\\ &\quad+\lVert
x_{a}(s_{k})-x_{b}(s_{k})\rVert+\lVert x_{b}(s_{k})-x_{b}(t)\rVert\\\ &\leq
2M(t-s_{k})+2Me^{s_{k}K}d_{s_{k}}(a,b)\\\ &\leq
2Me^{tK}(t-s_{k})+2Me^{tK}d_{s_{k}}(a,b)\\\
&=2Me^{tK}\big{(}d_{t}(a,b)-d_{s_{k}}(a,b)\big{)}+2Me^{tK}d_{s_{k}}(a,b)\\\
&=2Me^{tK}d_{t}(a,b)\end{split}$

for all $t\in[s_{k},s_{k+1}]$. This concludes the proof. ∎

We conclude this section with the remark that the (proof of the) continuity
property of 2 is very different from the standard (proof of the) continuity
property of controlled variables in closed thresholded hybrid systems. Mode
switches in such systems are caused by the time evolution of the system
itself, e.g., when some controlled variable exceeds some value. Consequently,
such systems can be described by means of a _single_ ODE system with
discontinuous righthand side (Fil88).

By contrast, in our hybrid systems, the mode switches are solely caused by
changes of digital inputs that are _externally_ controlled: For every possible
pattern of the digital inputs, there is a dedicated ODE system that controls
the analog output. Consequently, the time evolution of the output now also
depends on the time evolution of the inputs. Proving the continuity of the
(discretized) output w.r.t. different (but close, w.r.t. some metric) digital
input signals requires relating the output of _different_ ODE systems.

###  2.2. Continuity of thresholding

For a real number $\xi\in\mathds{R}$ and a function $x:[a,b]\to\mathds{R}$,
denote by $\Theta_{\xi}(x)$ the thresholded version of $x$ defined by

(6)
$\Theta_{\xi}(x):[a,b]\to\\{0,1\\},\quad\Theta_{\xi}(x)(t)=\begin{cases}0&\text{if
}x(t)\leq\xi,\\\ 1&\text{if }x(t)>\xi.\end{cases}$

###### Lemma 3.

Let $\xi\in\mathds{R}$ and let $x:[a,b]\to\mathds{R}$ be a continuous strictly
monotonic function with $x(b)=\xi$. Then, for every $\varepsilon>0$, there
exists a $\delta>0$ such that, for every continuous function
$y:[a,b]\to\mathds{R}$, the condition $\lVert x-y\rVert_{\infty}<\delta$
implies $\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{1}<\varepsilon$.

###### Proof.

We show the lemma for the case that $x$ is strictly increasing. The proof for
strictly decreasing $x$ is analogous.

Set $\chi=x(a)$. Since $x$ is bijective onto the interval $[\chi,\xi]$, it has
an inverse function $x^{-1}:[\chi,\xi]\to[a,b]$. The inverse function $x^{-1}$
is continuous because the domain $[a,b]$ is compact (rudin1976principles,
Theorem 4.17).

The relation $t\leq x^{-1}(\xi-\delta)$ implies $x(t)+\delta\leq\xi$. Hence,
if $\lVert x-y\rVert_{\infty}<\delta$, then $y(t)\leq x(t)+\delta\leq\xi$ for
all $t\leq x^{-1}(\xi-\delta)$. This means that $\Theta_{\xi}(y)(t)=0$ for all
$t\leq x^{-1}(\xi-\delta)$, so $t>x^{-1}(\xi-\delta)$ for every $t\in[a,b]$
where $\Theta_{\xi}(y)(t)=1$.

By assumption, we have $\Theta_{\xi}(x)(t)=0$ for all $t\in[a,b]$. Thus,

(7)
$\begin{split}\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{1}&=\lambda\big{(}\\{t\in[0,T]\mid\Theta_{\xi}(y)=1\\}\big{)}\\\
&=\lambda\big{(}\\{t\in[0,T]\mid y(t)>\xi\\}\big{)}\\\ &\leq
b-x^{-1}(\xi-\delta).\end{split}$

Note that continuity of $y$ is sufficient to ensure that the set in 7 is
measurable. Since $x^{-1}$ is continuous, we have $x^{-1}(\xi-\delta)\to
x^{-1}(\xi)=b$ as $\delta\to 0$. In particular, for every $\varepsilon>0$,
there exists a $\delta>0$ such that $b-x^{-1}(\xi-\delta)<\varepsilon$. This
concludes the proof. ∎

The following 4 shows that we can drop the assumption $x(b)=\xi$ in 3:

###### Lemma 4.

Let $\xi\in\mathds{R}$ and let $x,y:[a,b]\to\mathds{R}$ be two continuous
functions where $x$ is strictly monotonic. Then, for every $\varepsilon>0$,
there exists a $\delta>0$ such that $\lVert x-y\rVert_{\infty}<\delta$ implies
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{1}<\varepsilon$.

###### Proof.

We again show the lemma for the case that $x$ is strictly increasing. The
proof for strictly decreasing $x$ is analogous.

Let $\varepsilon>0$. We distinguish three cases:

(i) If $x(b)<\xi$, then we have $\Theta_{\xi}(x)(t)=0$ for all $t\in[a,b]$.
Choosing $\delta=\xi-x(b)$, we deduce $y(t)<x(t)+\delta\leq x(b)+\xi-x(b)=\xi$
for all $t\in[a,b]$ whenever $\lVert x-y\rVert_{\infty}<\delta$. Hence, we get
$\Theta_{\xi}(y)(t)=0$ for all $t\in[a,b]$ and thus
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{1}=0<\varepsilon$.

(ii) If $x(a)>\xi$, then we can choose $\delta=x(a)-\xi$ and get
$\Theta_{\xi}(y)(t)=\Theta_{\xi}(x)(t)=1$ for all $t\in[a,b]$ whenever $\lVert
x-y\rVert_{\infty}<\delta$. In particular,
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{1}=0<\varepsilon$.

(iii) If $x(a)\leq\xi\leq x(b)$, then there exists a unique $c\in[a,b]$ with
$x(c)=\xi$. Applying Lemma 3 on the restriction of $x$ on the interval
$[a,c]$, we get the existence of a $\delta_{1}>0$ such that $\lVert
x-y\rVert_{[a,c],\infty}<\delta_{1}$ implies
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[a,c],1}<\varepsilon/2$; herein,
$\lVert\cdot\rVert_{[a,c],\infty}$ and $\lVert\cdot\rVert_{[a,c],1}$ denote
the supremum-norm and the $1$-norm on the interval $[a,c]$, respectively.
Applying Lemma 3 on the restriction of $x$ on the interval $[c,b]$ after the
coordinate transformation $t\mapsto-t$ yields the existence of a
$\delta_{2}>0$ such that $\lVert x-y\rVert_{[c,b],\infty}<\delta_{2}$ implies
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[c,b],1}<\varepsilon/2$. Setting
$\delta=\min\\{\delta_{1},\delta_{2}\\}$, we thus get
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[a,b],1}=\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[a,c],1}+\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[c,b],1}<\varepsilon/2+\varepsilon/2=\varepsilon$
whenever $\lVert x-y\rVert_{[a,b],\infty}<\delta$. ∎

The following 5 shows that the mapping $x\mapsto\Theta_{\xi}(x)$ is continuous
for a given function $x$, provided that $x$ has only finitely many local
optima, i.e., points where $x^{\prime}(t)=0$:

###### Theorem 5.

Let $\xi\in\mathds{R}$ and let $x,y:[0,T]\to\mathds{R}$ be two differentiable
functions. Assume that $x$ has only finitely many local optima. Then, for
every $\varepsilon>0$, there exists a $\delta>0$ such that $\lVert
x-y\rVert_{\infty}<\delta$ implies
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{1}<\varepsilon$. Consequently,
the mapping $x\mapsto\Theta_{\xi}(x)$ is continuous.

###### Proof.

Let $\mathcal{N}=\left\\{t\in[0,T]\mid\text{$x$ has a local optimum at
$t$}\right\\}\cup\\{0,T\\}$, which is finite by assumption, and
$t_{0}<t_{1}<t_{2}<\cdots<t_{m}$ be the increasing enumeration of
$\mathcal{N}$. By the mean-value theorem, the function $x$ is strictly
monotonic in every interval $[t_{k},t_{k+1}]$ for $k\in\\{0,1,2,\dots,m-1\\}$.

Let $\varepsilon>0$. Applying Lemma 4 to the restriction of $x$ on each of the
intervals $[t_{k},t_{k+1}]$, we get the existence of $\delta_{k}>0$ such that
$\lVert x-y\rVert_{[t_{k},t_{k+1}],\infty}<\delta_{k}$ implies
$\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[t_{k},t_{k+1}],1}<\varepsilon/m$
for each $k\in\\{0,1,2,\dots,m-1\\}$. Setting
$\delta=\min\\{\delta_{0},\delta_{1},\delta_{2},\dots,\delta_{m-1}\\}$, we
thus obtain

(8)
$\begin{split}\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[0,T],1}&=\sum_{k=0}^{m-1}\lVert\Theta_{\xi}(x)-\Theta_{\xi}(y)\rVert_{[t_{k},t_{k+1}],1}\\\
&<\sum_{k=0}^{m-1}\varepsilon/m=\varepsilon\end{split}$

whenever $\lVert x-y\rVert_{[0,T],\infty}<\delta$. ∎

##  3\. Continuity of Digitized Hybrid Gate

To prepare for our general result about the continuity of hybrid gate models,
we will first (re)prove the continuity of IDM channels as shown in Fig. 2,
which has been established by a quite tedious direct proof in (FNNS19:TCAD).
In our notation, an IDM channel consists of:

  * •

A nonnegative minimum delay $\delta_{\mathrm{min}}\geq 0$ and a delay function
$\Delta_{\delta_{\mathrm{min}}}(s)$ that maps the binary input signal $i_{a}$,
augmented with the left-sided limit $i_{a}(0-)$ as the _initial value_ 111In
(FNNS19:TCAD), this initial value of a signal was encoded by extending the
time domain to the whole $\mathds{R}$ and using $i_{a}(-\infty)$. that can be
different from $i_{a}(0)$, to the binary signal
$i_{d}=\Delta_{\delta_{\mathrm{min}}}(i_{a})$, defined by

(9) $\Delta_{\delta_{\mathrm{min}}}(i_{a})(t)=\begin{cases}i_{a}(0-)&\text{if
}t<\delta_{\mathrm{min}}\\\ i_{a}(t-\delta_{\mathrm{min}})&\text{if
}t\geq\delta_{\mathrm{min}}\enspace.\end{cases}$

  * •

An open set $U\subseteq\mathds{R}^{n}$, where $\pi_{1}[U]$ represents the
analog output signal and $\pi_{k}[U]$, $k=\\{2,3,\ldots,n\\}$, specifies the
internal state variables of the model. In this fashion,222In real circuits,
the interval $(0,1)$ typically needs to be replaced by $(0,V_{DD})$. we
presume that $\pi_{1}[U]=(0,1)$, i.e., the range of output signals is
contained in the interval $(0,1)$.

  * •

Two bounded functions $f_{\uparrow},f_{\downarrow}:\mathds{R}\times
U\to\mathds{R}^{n}$ with the following properties:

    * –

$f_{\uparrow},f_{\downarrow}$ are continuous for $(t,x)\in[0,T]\times U$, for
any $0<T<\infty$, and Lipschitz continuous in $U$, which entails that every
trajectory $x$ of the ODEs $\frac{d}{dt}\,x(t)=f_{\uparrow}(t,x(t))$ and
$\frac{d}{dt}\,x(t)=f_{\downarrow}(t,x(t))$ with any initial value $x(0)\in U$
satisfies $x(t)\in U$ for all $t\in[0,T]$, recall Section 2.1.

    * –

for no trajectory $x$ of the ODEs $\frac{d}{dt}\,x(t)=f_{\uparrow}(t,x(t))$
and $\frac{d}{dt}\,x(t)=f_{\downarrow}(t,x(t))$ with initial value $x(0)\in U$
does $\pi_{1}\circ x$ have infinitely many local optima, i.e., critical points
with $(\pi_{1}\circ x)^{\prime}(t)=0$.

  * •

An initial value $x_{0}\in U$, with $x_{0}=f_{\uparrow}$ if $i_{a}(0-)=1$ and
$x_{0}=f_{\downarrow}$ if $i_{a}(0-)=0$.

  * •

A mode-switch signal $a:[0,T]\to\\{f_{\uparrow},f_{\downarrow}\\}$ defined by
setting $a(t)=f_{\uparrow}$ if $i_{d}(t)=1$ and $a(t)=f_{\downarrow}$ if
$i_{d}(t)=0$.

  * •

The analog output signal $o_{a}=x_{a}$, i.e., the output signal for $a$ and
initial value $x_{0}$.

  * •

A threshold voltage $\xi=V_{th}\in(0,1)$ for the comparator that finally
produces the binary output signal $o_{d}=\Theta_{\xi}(o_{a})$.

By combining the results from Section 2.1 and 2.2, we obtain:

###### Theorem 6.

The channel function of an IDM channel, which maps from the input signal
$i_{a}$ to the output signal $o_{d}$, is continuous with respect to the
$1$-norm on the interval $[0,T]$.

###### Proof.

The mapping from $i_{a}$ to $o_{d}$ is continuous as the concatenation of
continuous mappings:

  * •

The mapping from $i_{a}\mapsto i_{d}$ is continuous since
$\Delta_{\delta_{\mathrm{min}}}$ is trivially continuous for input and output
binary signals with the $1$-norm.

  * •

The mapping $i_{d}\mapsto a$ is a continuous mapping from the set of signals
equipped with the $1$-norm to the set of mode-switch signals equipped with the
metric $d_{T}$, since the points of discontinuity of $a$ are the points where
$i_{d}$ is discontinuous.

  * •

By Theorem 2, the mapping $a\mapsto x_{a}$ is a continuous mapping from the
set of mode-switch signals equipped with the metric $d_{T}$ to the set of
piecewise differentiable functions $[0,T]\to U$ equipped with the supremum-
norm.

  * •

The mapping $x_{a}\mapsto\pi_{1}\circ x_{a}$ is a continuous mapping from the
set of piecewise differentiable functions $[0,T]\to U$ equipped with the
supremum-norm to the set of piecewise differentiable functions $[0,T]\to(0,1)$
equipped with the supremum-norm. Since
$\lVert(x_{1},\dots,x_{n})\rVert_{1}=\lVert x_{1}\rVert_{1}+\dots+\lVert
x_{n}\rVert_{1}$ for every $x\in U$, this follows from
$\lVert\pi_{1}(x)\rVert_{1}\leq\lVert x\rVert_{1}$.

  * •

By Theorem 5, the mapping $\pi_{1}\circ x_{a}\mapsto\Theta_{\xi}(\pi_{1}\circ
x_{a})$ is a continuous mapping from the set of piecewise differentiable
functions $[0,T]\to(0,1)$ equipped with the supremum-norm to the set of binary
signals equipped with the $1$-norm.∎

General digitized hybrid gates have $c\geq 1$ binary input signals
$i_{a}=(i_{a}^{1},\dots,i_{a}^{c})$, augmented with _initial values_
$(i_{a}^{1}(0-),\dots,i_{a}^{c}(0-))$, and a single binary output signal
$o_{d}$, and are specified as follows:

###### Definition 3.1 (Digitized hybrid gate).

A digitized hybrid gate with $c$ inputs consists of:

  * •

$c$ delay functions $\Delta_{\delta_{j}}(s)$ with $\delta_{j}\geq 0$, $1\leq
j\leq c$, that map the binary input signal $i_{a}^{j}$ with initial value
$i_{a}^{j}(0-)$ to the binary signal
$i_{d}^{j}=\Delta_{\delta_{j}}(i_{a}^{j})$, defined by

(10) $\Delta_{\delta_{j}}(i_{a}^{j})(t)=\begin{cases}i_{a}^{j}(0-)&\text{if
}t<\delta_{j}\\\ i_{a}^{j}(t-\delta_{j})&\text{if
}t\geq\delta_{j}\enspace.\end{cases}$

  * •

An open set $U\subseteq\mathds{R}^{n}$, where $\pi_{1}[U]$ represents the
analog output signal and $\pi_{k}[U]$, $k=\\{2,3,\ldots,n\\}$, specifies the
internal state variables of the model.

  * •

A set $F$ of bounded functions $f^{\ell}:\mathds{R}\times U\to\mathds{R}^{n}$,
with the following properties:

    * –

$f^{\ell}$ is continuous for $(t,x)\in[0,T]\times U$, for any $0<T<\infty$,
and Lipschitz continuous in $U$, with a common Lipschitz constant, which
entails that every trajectory $x$ of the ODE
$\frac{d}{dt}\,x(t)=f^{\ell}(t,x(t))$ with any initial value $x(0)\in U$
satisfies $x(t)\in U$ for all $t\in[0,T]$.

    * –

for no trajectory $x$ of the ODEs $\frac{d}{dt}\,x(t)=f^{\ell}(t,x(t))$ with
initial value $x(0)\in U$ does $\pi_{1}\circ x$ have infinitely many local
optima, i.e., critical points with $(\pi_{1}\circ x)^{\prime}(t)=0$.

  * •

A mode-switch signal $a:[0,T]\to F$, which obtained by a continuous choice
function $a_{c}$ acting on $i_{d}^{1}(t),\dots,i_{d}^{c}(t)$, i.e.,
$a(t)=a_{c}(i_{d}^{1}(t),\dots,i_{d}^{c}(t))$.

  * •

An initial value $x_{0}\in U$, which must correspond to the mode selected by
$a_{c}(i_{a}^{1}(0-),\dots,i_{a}^{c}(0-))$.

  * •

The analog output signal $o_{a}=x_{a}$, i.e., the output signal for $a$ and
initial value $x_{0}$.

  * •

A threshold voltage $\xi=V_{th}\in(0,1)$ for the comparator that finally
produces the binary output signal $o_{d}=\Theta_{\xi}(o_{a})$.

By essentially the same proof as for 6, we obtain:

###### Theorem 7.

The gate function of a digitized hybrid gate with $c$ inputs, which maps from
the vector of input signals $i_{a}=(i_{a}^{1},\dots,i_{a}^{c})$ to the output
signal $o_{d}$, is continuous with respect to the $1$-norm on the interval
$[0,T]$.

##  4\. Composing Gates in Circuits

In this section, we will first compose digital circuits from digitized hybrid
gates and reason about their executions. More specifically, it will turn out
that, under certain conditions ensuring the causality of every composed gate,
the resulting circuit will exhibit a unique execution, for every given
execution of its inputs. This uniqueness is mandatory for building digital
dynamic timing simulation tools.

Moreover, we adapt the proof that no circuit with IDM channels can solve the
bounded SPF problem utilized in (FNNS19:TCAD) to our setting: Using the
continuity result of 7, we will prove that no circuit with digitized hybrid
gates can solve bounded SPF. Since unbounded SPF can be solved with IDM
channels, which are simple instances of digitized hybrid gate models,
faithfulness follows.

###  4.1. Executions of circuits

Circuits. Circuits are obtained by interconnecting a set of input ports and a
set of output ports, forming the external interface of a circuit, and a finite
set of digitized hybrid gates. We constrain the way components are
interconnected in a natural way, by requiring that any gate input, channel
input and output port is attached to only one input port, gate output or
channel output, respectively. Formally, a circuit is described by a directed
graph where:

  1. C1)

A vertex $\Gamma$ can be either a circuit input port, a circuit output port,
or a digitized hybrid gate.

  2. C2)

The _edge_ $(\Gamma,I,\Gamma^{\prime})$ represents a $0$-delay connection from
the output of $\Gamma$ to a fixed input $I$ of $\Gamma^{\prime}$.

  3. C3)

Circuit input ports have no incoming edges.

  4. C4)

Circuit output ports have exactly one incoming edge and no outgoing one.

  5. C5)

A $c$-ary gate $G$ has a single output and $c$ inputs $I_{1},\dots,I_{c}$, in
a fixed order, fed by incoming edges from exactly one gate output or input
port.

Executions. An execution of a circuit ${\mathcal{C}}$ is a collection of
binary signals $s_{\Gamma}$ defined on $[0,\infty)$ for all vertices $\Gamma$
of ${\mathcal{C}}$ that respects all the gate functions and input port
signals. Formally, the following properties must hold:

  1. E1)

If $i$ is a circuit input port, there are no restrictions on $s_{i}$.

  2. E2)

If $o$ is a circuit output port, then $s_{o}=s_{G}$, where $G$ is the unique
gate output connected to $o$.

  3. E3)

If vertex $G$ is a gate with $c$ inputs $I_{1},\dots,I_{c}$, ordered according
to the fixed order condition C5), and gate function $f_{G}$, then
$s_{G}=f_{G}(s_{\Gamma_{1}},\dots,s_{\Gamma_{c}})$, where
$\Gamma_{1},\dots,\Gamma_{c}$ are the vertices the inputs $I_{1},\dots,I_{c}$
of $C$ are connected to via edges
$(\Gamma_{1},I_{1},G),\dotsm,(\Gamma_{d},I_{c},G)$.

The above definition of an execution of a circuit is “existential”, in the
sense that it only allows checking for a given collection of signals whether
it is an execution or not: For every hybrid gate in the circuit, it specifies
the gate output signal, given a fixed vector of input signals, all defined on
the time domain $t\in[0,\infty)$. A priori, this does not give an algorithm to
construct executions of circuits, in particular, when they contain feedback
loops. Indeed, the parallel composition of general hybrid automata may lead to
non-unique executions and bizarre timing behaviors known as _Zeno_ , where an
infinite number of transitions may occur in finite time (LSV03).

To avoid such behaviors in our setting, we require all discretized hybrid
gates in a circuit to be _strictly causal_ :

###### Definition 4.1 (Strict causality).

A digitized hybrid gate $G$ with $c$ inputs is strictly causal, if the pure
delays $\delta_{j}$ for every $1\leq j\leq c$ are positive. Let
$\delta_{\mathrm{min}}^{C}>0$ be the minimal pure delay of any input of any
gate in circuit $C$.

We proceed with defining input-output causality for gates, which is based on
signal transitions. Every binary signal can equivalently be described by a
sequence of transitions: A falling transition at time $t$ is the pair $(t,0)$,
a rising transition at time $t$ is the pair $(t,1)$.

###### Definition 4.2 (Input-output causality).

The output transition $(t,.)\in s_{G}$ of a gate G _is caused_ by the
transition $(t^{\prime},.)\in s_{G}^{j}$ on input $I_{j}$ of $G$, if $(t,.)$
occurs in the mode $a_{c}(i_{d}^{1}(t^{+}),\dots,i_{d}^{c}(t^{+}))$, where
$i_{d}^{j}(t^{+})$ is the pure-delay shifted input signal at input $I_{j}$ at
the last mode switching time $t^{+}\leq t$ (see 10) and $(t^{\prime},.)$ is
the last transition in $s_{G}^{j}$ before or at time $t^{+}-\delta_{j}$, i.e.,
$\not\exists(t^{\prime\prime},.)\in s_{G}^{j}$ for
$t^{\prime}<t^{\prime\prime}\leq t^{+}-\delta_{j}$.

We also assume that the output transition $(t,.)\in s_{G}$ _causally depends_
on every transition in $s_{G}^{j}$ before or at time $t^{+}-\delta_{j}$.

Strictly causal gates satisfy the following obvious property:

###### Lemma 4.3.

If some output transition $(t,.)\in s_{G}$ of a strictly causal digitized
hybrid gate $G$ in a circuit $C$ causally depends on its input transition
$(t^{\prime},.)\in s_{G}^{j}$, then $t-t^{\prime}\geq\delta_{j}$.

The following Theorem 4.4 shows that every circuit made up of strictly causal
gates has a unique execution, defined for $t\in[0,\infty)$.

###### Theorem 4.4 (Unique execution).

Every circuit $C$ made up of finitely many strictly causal digitized hybrid
gates has a unique execution, which either consists of finitely many
transitions only or else requires $[0,\infty)$ as its time domain.

###### Proof.

We will inductively construct this unique execution by a sequence of
iterations $\ell\geq 1$ of a simple deterministic simulation algorithm, which
determines the prefix of the sought execution up to time $t_{\ell}$. Iteration
$\ell$ deals with transitions occurring at time $t_{\ell}$, starting with
$t_{1}=0$. To every transition $e$ generated throughout its iterations, we
also assign a _causal depth_ $d(e)$ that gives the maximum causal distance to
an input port: $d(e)=0$ if $e$ is a transition at some input port, and $d(e)$
is the maximum of $1+d(e^{j})$, $1\leq j\leq c$, for every transition added at
the output of a $c$-ary gate caused by transitions $e^{j}$ at its inputs.

Induction basis $\ell=1$: At the beginning of iteration 1, which deals with
all transitions occurring at time $t_{1}=0$, all gates are in their initial
mode, which is determined by the initial values of their inputs. They are
either connected to input ports, in which case $s_{i}(0-)$ is used, or to the
output port of some gate $G$, in which case $s_{G}(0)$ (determined by the
initial mode of $G$) is used. Depending on whether $s_{i}(0-)=s_{i}(0)$ or
not, there is also an input transition $(0,s_{i}(0))\in s_{i}$ or not. All
transitions in the so generated execution prefix $[0,t_{1}]=[0,0]$ have a
causal depth of 0.

Still, the transitions that have happened by time $t_{1}$ may cause additional
_potential future transitions_. They are called future transitions, because
they occur only after $t_{1}$, and potential because they need not occur in
the final execution. In particular, if there is an input transition
$(0,s_{i}(0))\in s_{i}$, it may cause a mode switch of every gate $G$ that is
connected to the input port $i$. Due to Lemma 4.3, however, such a mode
switch, and hence each of the output transitions $e$ that may occur during
that new mode (which all are assigned a causal depth $d(e)=1$), of $G$ can
only happen at or after time $t_{1}+\delta_{\mathrm{min}}^{C}$. In addition,
the initial mode of any gate $G$ that is not mode switched may also cause
output transitions $e$ at arbitrary times $t>0$, which are assigned a causal
depth $d(e)=0$. Since at most finitely many critical points may exist for
every mode’s trajectory, it follows that at most _finitely_ many such future
potential transitions could be generated in each of the finitely many gates in
the circuit. Let $t_{2}>t_{1}$ denote the time of the closest transition among
all input port transitions and all the potential future transitions just
introduced.

Induction step $\ell\to\ell+1$: Assume that the execution prefix for
$[0,t_{\ell}]$ has already been constructed in iterations $1,\dots,\ell$, with
at most finitely many potential future transitions occurring after $t_{\ell}$.
If the latter set is empty, then the execution of the circuit has already been
determined completely. Otherwise, let $t_{\ell+1}>t_{\ell}$ be the closest
future transition time.

During iteration $\ell+1$, all transitions occurring at time $t_{\ell+1}$ are
dealt with, exactly as in the base case: Any transition $e$, with causal depth
$d(e)$, happening at $t_{\ell+1}$ at a gate output or at some input port may
cause a mode switch of every gate $G$ that is connected to it. Due to Lemma
4.3, such a mode switch, and hence each of the at most finitely many output
transitions $e^{\prime}$ occurring during that new mode (which all are
assigned a causal depth $d(e^{\prime})=d(e)+1$), of $G$ can only happen at or
after time $t_{\ell+1}+\delta_{\mathrm{min}}^{C}$. In addition, the at most
finitely many potential future transitions w.r.t. $t_{\ell}$ of all gates that
have not been mode-switched and actually occur at times greater than
$t_{\ell+1}$ are retained, along with their assigned causal depth, as
potential future transitions w.r.t. $t_{\ell+1}$. Overall, we again end up
with at most finitely many potential future transitions, which completes the
induction step.

To complete our proof, we only need to argue that
$\lim_{\ell\to\infty}t_{\ell}=\infty$ for the case where the iterations do not
stop at some finite $\ell$. This follows immediately from the fact that, for
every $k\geq 1$, there must be some iteration $\ell\geq 1$ such that
$t_{\ell}\geq k\delta_{\mathrm{min}}^{C}$. If this was not the case, there
must be some iteration after which no further mode switch of any gate takes
place. This would cause the set of potential future transitions to shrink in
every subsequent iteration, however, and thus the simulation algorithm to
stop, which provides the required contradiction. ∎

From the execution construction, we also immediately get:

###### Lemma 4.5.

For all $\ell\geq 1$, (a) the simulation algorithm never assigns a causal
depth larger than $\ell$ to a transition generated in iteration $\ell$, and
(b) at the end of iteration $\ell$ the sequence of causal depths of
transitions in $s_{\Gamma}$ for $t\in[0,t_{\ell}]$ is nondecreasing for all
components $\Gamma$.

###  4.2. Impossibility of short-pulse filtration

The results of the previous subsection allow us to adapt the impossibility
proof of (FNNS19:TCAD) to our setting. We start with the the definition of the
SPF problem:

Short-Pulse Filtration. A signal contains a pulse of length $\Delta$ at time
$T_{0}$, if it contains a rising transition at time $T_{0}$, a falling
transition at time $T_{0}+\Delta$, and no transition in between. The _zero
signal_ has the initial value 0 and does not contain any transition. A circuit
solves Short-Pulse Filtration (SPF), if it fulfills all of:

  1. F1)

The circuit has exactly one input port and exactly one output port. (Well-
formedness)

  2. F2)

If the input signal is the zero signal, then so is the output signal. (No
generation)

  3. F3)

There exists an input pulse such that the output signal is not the zero
signal. (Nontriviality)

  4. F4)

There exists an $\varepsilon>0$ such that for every input pulse the output
signal never contains a pulse of length less than or equal to $\varepsilon$.
(No short pulses)

We allow the circuit to behave arbitrarily if the input signal is not a single
pulse or the zero signal.

A circuit solves bounded SPF if additionally:

  1. F5)

There exists a $K>0$ such that for every input pulse the last output
transition is before time $T_{0}+\Delta+K$, where $T_{0}$ is the time of the
first input transition. (Bounded stabilization time)

A circuit is called a forward circuit if its graph is acyclic. Forward
circuits are exactly those circuits that do not contain feedback loops.
Equipped with the continuity of digitized hybrid gates and the fact that the
composition of continuous functions is continuous, it is not too difficult to
prove that the inherently discontinuous SPF problem cannot be solved with
forward circuits.

###### Theorem 4.6.

No forward circuit solves bounded SPF.

###### Proof.

Suppose that there exists a forward circuit that solves bounded SPF with
stabilization time bound $K$. Denote by $s_{\Delta}$ its output signal when
feeding it a $\Delta$-pulse at time $0$ as the input. Because $s_{\Delta}$ in
forward circuits is a finite composition of continuous functions by Theorem 7,
$\lVert s_{\Delta}\rVert_{[0,T],1}$ depends continuously on $\Delta$, for any
$T$.

By the nontriviality condition (F3) of the SPF problem, there exists some
$\Delta_{0}$ such that $s_{\Delta_{0}}$ is not the zero signal. Set
$T=2\Delta_{0}+K$.

Let $\varepsilon>0$ be smaller than both $\Delta_{0}$ and $\lVert
s_{\Delta_{0}}\rVert_{[0,T],1}$. We show a contradiction by finding some
$\Delta$ such that $s_{\Delta}$ either contains a pulse of length less than
$\varepsilon$ (contradiction to the no short pulses condition (F4)) or
contains a transition after time $\Delta+K$ (contradicting the bounded
stabilization time condition (F5)).

Since $\lVert s_{\Delta}\rVert_{[0,T],1}\to 0$ as $\Delta\to 0$ by the no
generation condition (F2) of SPF, there exists a $\Delta_{1}<\Delta_{0}$ such
that $\lVert s_{\Delta_{1}}\rVert_{[0,T],1}=\varepsilon$ by the intermediate
value property of continuity. By the bounded stabilization time condition
(F5), there are no transitions in $s_{\Delta_{1}}$ after time $\Delta_{1}+K$.
Hence, $s_{\Delta_{1}}$ is $0$ after this time because otherwise it is $1$ for
the remaining duration $T-(\Delta_{1}+K)>\Delta_{0}>\varepsilon$, which would
mean that $\lVert s_{\Delta_{1}}\rVert_{[0,T],1}>\varepsilon$. Consequently,
there exists a pulse in $s_{\Delta_{1}}$ before time $\Delta_{1}+K$. But any
such pulse is of length at most $\varepsilon$ because $\lVert
s_{\Delta_{1}}\rVert_{[0,\Delta_{1}+K],1}\leq\lVert
s_{\Delta_{1}}\rVert_{[0,T],1}=\varepsilon$. This is a contradiction to the no
short pulses condition (F4). ∎

We next show how to simulate (part of) an execution of an arbitrary circuit
${\mathcal{C}}$ by a forward circuit ${\mathcal{C}}^{\prime}$ generated from
${\mathcal{C}}$ by the unrolling of feedback loops. Intuitively, the deeper
the unrolling, the longer the time ${\mathcal{C}}^{\prime}$ behaves as
${\mathcal{C}}$.

###### Definition 4.7.

Let ${\mathcal{C}}$ be a circuit, $V$ a vertex of ${\mathcal{C}}$, and $k\geq
0$. We define the $k$-unrolling of ${\mathcal{C}}$ from $V$, denoted by
${\mathcal{C}}_{k}(V)$, to be a directed acyclic graph with a single sink,
constructed as follows:

The unrolling ${\mathcal{C}}_{k}(I)$ from input port $I$ is just a copy of
that input port. The unrolling ${\mathcal{C}}_{k}(O)$ from output port $O$
with incoming channel $C$ and predecessor $V$ comprises a copy of the output
port $O^{(k)}$ and the unrolled circuit ${\mathcal{C}}_{k}(V)$ with its sink
connected to $O^{(k)}$ by an edge.

The 0-unrolling ${\mathcal{C}}_{0}(B)$ from hybrid gate $B$ is a trivial
Boolean gate $X_{v}$ without inputs and the constant output value $v$ equal to
$B$’s initial digitized output value. For $k>0$, the $k$-unrolling
${\mathcal{C}}_{k}(B)$ from gate $B$ comprises an exact copy of that gate
$B^{(k)}$. Additionally, for every incoming edge of $B$ from $V$ in
${\mathcal{C}}$, it contains the circuit ${\mathcal{C}}_{k-1}(V)$ with its
sink connected to $B^{(k)}$. Note that all copies of the same input port are
considered to be the same.

To each component $\Gamma$ in ${\mathcal{C}}_{k}(V)$, we assign a value
$z(\Gamma)\in\mathds{N}_{0}\cup\\{\infty\\}$ as follows: $z(\Gamma)=\infty$ if
$\Gamma$ has no predecessor (in particular, is an input port) and
$\Gamma\not\in\\{X_{0},X_{1}\\}$. Moreover, $z(X_{0})=z(X_{1})=0$, $z(V)=z(U)$
if $V$ is an output port connected by an edge to $U$, and $z(B)=\min_{c\in
E^{B}}\\{1+z(c)\\}$ if $B$ is a gate with its inputs connected to the
components in the set $E^{B}$. LABEL:fig:unrolling shows an example of a
circuit and an unrolled circuit with its $z$ values.


††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Discovery of superconductivity in technetium-borides at moderate pressures

Xiangru Tao  MOE Key Laboratory for Non-equilibrium Synthesis and Modulation
of Condensed Matter, Shaanxi Province Key Laboratory of Advanced Functional
Materials and Mesoscopic Physics, School of Physics, Xi’an Jiaotong
University, 710049, Xi’an, Shaanxi, P.R.China  Aiqin Yang  MOE Key Laboratory
for Non-equilibrium Synthesis and Modulation of Condensed Matter, Shaanxi
Province Key Laboratory of Advanced Functional Materials and Mesoscopic
Physics, School of Physics, Xi’an Jiaotong University, 710049, Xi’an, Shaanxi,
P.R.China  Yundi Quan  MOE Key Laboratory for Non-equilibrium Synthesis and
Modulation of Condensed Matter, Shaanxi Province Key Laboratory of Advanced
Functional Materials and Mesoscopic Physics, School of Physics, Xi’an Jiaotong
University, 710049, Xi’an, Shaanxi, P.R.China  Biao Wan  Key Laboratory of
Material Physics of Ministry of Education, School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou 450052, Henan, P.R.China
Shuxiang Yang<EMAIL_ADDRESS>
Zhejiang laboratory, Hangzhou, Zhejiang, P.R.China  Peng Zhang
<EMAIL_ADDRESS>
MOE Key Laboratory for Non-equilibrium Synthesis and Modulation of Condensed
Matter, Shaanxi Province Key Laboratory of Advanced Functional Materials and
Mesoscopic Physics, School of Physics, Xi’an Jiaotong University, 710049,
Xi’an, Shaanxi, P.R.China

###### Abstract

Advances in theoretical calculations boosted the searches for high temperature
superconductors, such as sulfur hydrides and rare-earth polyhydrides. However,
the required extremely high pressures for stabilizing these superconductors
handicapped further implementations. Based upon thorough structural searches,
we identified series of unprecedented superconducting technetium-borides at
moderate pressures, including TcB (P63/mmc) with superconducting transition
temperature $T_{\text{c}}$ = 20.2 K at ambient pressure and TcB2 (P6/mmm) with
$T_{\text{c}}$ = 23.1 K at 20 GPa. Superconductivity in these technetium-
borides mainly originates from the coupling between the low frequency
vibrations of technetium-atoms and the dominant technetium-4d electrons at the
Fermi level. Our works therefore present a fresh group in the family of
superconducting borides, whose diversified crystal structures suggest rich
possibilities in discovery of other superconducting transition-metal-borides.

##  I Introduction

The discovery of superconductivity in mercury Onnes (1911) motivated a
centurial race for superconductors of higher temperatures. Owing to the
progress in theoretical calculations, Pickard and Needs (2011); Wang  _et al._
(2012a); Oganov and Glass (2006); Giustino (2017) numerous high-temperature
superconducting hydrides have been discovered in the past decade, including
H3S ($T_{\text{c}}$ $\approx$ 191 - 204 K at 200 GPa) Duan  _et al._ (2014);
Drozdov  _et al._ (2015) and LaH10 ($T_{\text{c}}$ $\approx$ 274 - 286 K at
210 GPa) of record high superconducting transition temperature. Liu  _et al._
(2017); Geballe  _et al._ (2018); Somayazulu  _et al._ (2019); Drozdov  _et
al._ (2019) However, stable presence of these superconducting hydrides
requests very high pressures, which largely limits their potential
implementations.

Among all BCS-superconductors, borides represent a unique category with
superconductivity at relatively low pressures. MgB2 has the highest
superconducting transition temperature, $T_{\text{c}}$=39 K, among all BCS-
type superconductors at ambient pressure. Nagamatsu  _et al._ (2001) Up to
today, discovered bulk superconducting borides of the same stoichiometry as
MgB2 include CaB2 ($T_{\text{c}}$ $\sim$ 50 K Choi  _et al._ (2009) or 9.4 -
28.6 K Yu  _et al._ (2022) at ambient pressure, theory), NbB2 ($T_{\text{c}}$
$\sim$ 9.2 K at ambient pressure, experiment Schirber  _et al._ (1992);
Yamamoto  _et al._ (2002); Takeya  _et al._ (2004)), OsB2 ($T_{\text{c}}$=2.1
K at ambient pressure, experiment Singh  _et al._ (2007)), RuB2
($T_{\text{c}}$=1.6 K at ambient pressure, experiment Singh  _et al._ (2007)),
ScB2 ($T_{\text{c}}$=1.5 K at ambient pressure, experiment Samsonov (1980)),
WB2 (maximum $T_{\text{c}}$=15 K at 100 GPa, experiment Pei  _et al._ (2022)),
ZrB2 ($T_{\text{c}}$=5.5 K at ambient pressure, experiment Leyarovska and
Leyarovski (1979)). Superconducting borides of other stoichiometry include
X7B3 (X=Re and Ru with $T_{\text{c}}$=3.3 and 2.58 K respectively at ambient
pressure, experiment Buzea and Yamashita (2001); Kawano  _et al._ (2003)),
Re3B ($T_{\text{c}}$=4.8 K at ambient pressure, experiment Kawano  _et al._
(2003)), X2B (X=Mo, Re, Ta and W with $T_{\text{c}}$=5.07, 2.8, 3.12 and 3.22
K respectively at ambient pressure, experiment Buzea and Yamashita (2001)), XB
(X=Hf, Nb, Mo, Ta and Zr with $T_{\text{c}}$=3.1, 8.25, 0.5, 4.0 and 2.8-3.4 K
respectively at ambient pressure, experiment Buzea and Yamashita (2001)), FeB4
($T_{\text{c}}$=2.9 K at ambient pressure, theory and experiment Kolmogorov
_et al._ (2010); Gou  _et al._ (2013)), XB5 (X=Na, K, Rb, Ca, Sr, Ba, Sc and Y
with $T_{\text{c}}$=17.5, 14.7, 18.6, 6.6, 6.8, 16.3, 14.2 and 12.3 K
respectively at ambient pressure, theory Xie  _et al._ (2023)), BeB6
($T_{\text{c}}$=24 K at 4 GPa, theory Wu  _et al._ (2016)), CB6
($T_{\text{c}}$=12.5 K at ambient pressure, theory Xia  _et al._ (2017)), MgB6
($T_{\text{c}}$=9.5 K at 32.6 GPa, theory Duan  _et al._ (2019)), XB6 (X=Nb,
La, Th and Y with $T_{\text{c}}$=3.0, 5.7, 0.74 and 7.1 K respectively at
ambient pressure, experiment Buzea and Yamashita (2001)), XB7 (X=Li, Na, K,
Mg, Ca and Sr with $T_{\text{c}}$=21.56, 18.33, 26.20, 29.31, 7.68 and 12.67 K
respectively at ambient pressure, theory Han  _et al._ (2023a)), RbB6 and RbB8
($T_{\text{c}}$=7.3 - 11.6 K and 4.8 - 7.5 K at ambient pressure respectively,
theory Zhang  _et al._ (2023)), YB6 ($T_{\text{c}}$=7.2 K at ambient pressure,
experiment Lortz  _et al._ (2006)), LaB8 ($T_{\text{c}}$=14 K Ma  _et al._
(2021) or 20 K Liang  _et al._ (2021) at ambient pressure, theory), XB12
(X=Nb, La, Th and Y with $T_{\text{c}}$=3.0, 5.7, 0.74 and 7.1 K respectively
at ambient pressure, experiment Buzea and Yamashita (2001); Akopov  _et al._
(2019); Teyssier  _et al._ (2007)), or even ternary borides like SrB3C3
($T_{\text{c}}$=22 K at 23 GPa, theory and experiment Zhu  _et al._ (2023)).
Especially, the recent experimental discovery of superconducting MoB2 with
$T_{\text{c}}$=32 K at 100 GPa ignites further enthusiasm in looking for
superconducting transition-metal-borides at ambient pressure or at least
relatively lower pressures.Pei  _et al._ (2023)

The potential superhardness of metal-borides further merits their values.
Among the superconducting metal-borides listed above, several are superhard
due to the strong covalent bonding between the metal-boron and the boron-boron
atoms in crystals. Superconducting FeB4 has nanoindentation hardness of 62$\pm
5$ GPa. Gou  _et al._ (2013) Other superconducting metal-borides with
superhardness includes OsB2 ($\geq$ 2000 kg/mm2, R. W. Cumberland  _et al._
(2005)) YB6 (Vickers hardness of 37.0 GPa L.-P. Ding  _et al._ (2021)), RbB6
(Vickers hardness of 19.7 GPa Zhang  _et al._ (2023)), RbB8 (Vickers hardness
of 36.9 GPa Zhang  _et al._ (2023)), XB7 (X=Li, Na, K, Mg, Ca, Sr with Vickers
hardness of 12.0, 21.3, 22.5, 5.6, 20.4 and 25.1 GPa Han  _et al._ (2023a))
and ZrB12 (Vickers hardness of 40 GPa Ma  _et al._ (2017)).

Figure 1: (a) Formation enthalpy of predicted structures in technetium-boron
binary system at 0 GPa, 90 GPa and 180 GPa. Thermodynamically stable
structures are marked by red-filled dots on convex hull (black solid line);
thermodynamically meta-stable structures are marked by blue ’x’. Composition
ratio is defined by $N_{B}/(N_{T_{c}}+N_{B}$), where $N_{B}$ and $N_{T_{c}}$
represent the number of atoms in the formula unit. (b) Composition-pressure
phase diagram of thermodynamically stable structures in technetium-boron
binary system.

After the discovery of $MgB_{2}$ with $T_{c}$=39 K, there have been extensive
efforts in searching for similar metal-borides superconductors. Unfortunately,
the outcomes are discouraging. $T_{c}$ of most metal-borides are below 10 K as
presented above, except the recently discovered isostructure $MoB_{2}$ with
$T_{c}$=32 K at 100 GPa. The superconducting mechanism of $MoB_{2}$ is
suggested being very different from $MgB_{2}$. In $MgB_{2}$ the B-p electrons
play dominant role for its superconductivity, An and Pickett (2001); Kortus
_et al._ (2001) while in $MoB_{2}$ its Mo-4d electrons contribute majorly. Pei
_et al._ (2023) This raises two essential questions, (1) can we found other
superconductors of different structures than $MgB_{2}$, with transition
temperatures at least above 10 K and at not very high pressures? (2) whether
the superconducting mechanism of $MoB_{2}$ applies to other superconducting
transition metal-borides?

Technetium-borides have been extensively investigated because of their
outstanding mechanical properties. Trzebiatowski and Rudzinski (1964);
Armstrong (1979); Pallas and Larsson (2006); Wang (2007); Wang  _et al._
(2008); Aydin and Simsek (2009); Li  _et al._ (2010); Chen and Jiang (2010);
jie Zhao and Xu (2012); Deligoz  _et al._ (2012); Zhong  _et al._ (2013);
Zhang  _et al._ (2013); Ying  _et al._ (2014); Kuang  _et al._ (2015); Van Der
Geest and Kolmogorov (2014); Wu and Yang (2014); Zhang  _et al._ (2015); Ying
_et al._ (2016); Miao  _et al._ (2017); Ying  _et al._ (2018); Wu  _et al._
(2023) Three technetium-borides have long been synthesized by experiment at
ambient pressure, Trzebiatowski and Rudzinski (1964) Tc3B (Cmcm) of the
orthorhombic structure, Tc7B3 (P63/mmc) and TcB2 (P63/mmc, Vickers hardness
38.4 GPa Wu  _et al._ (2023) or 39.4 GPa Ying  _et al._ (2018)) of the
hexagonal structure. Later theoretical calculations also proposed three
stoichiometry of TcB, TcB3 and TcB4. Li  _et al._ (2010); Wu and Yang (2014);
Zhang  _et al._ (2015); Van Der Geest and Kolmogorov (2014); Miao  _et al._
(2017); Ying  _et al._ (2016, 2018) First-principle DFT calculations by Li et
al. Li  _et al._ (2010) suggest that hexagonal TcB (P$\bar{6}$m2) could be
energetically stable. Structural searches by Wu et al. Wu and Yang (2014)
found a thermodynamically stable TcB (Cmcm) structure above 8 GPa. Later
structrual searches by Zhang et al. Zhang  _et al._ (2015) argues that TcB
(P$\bar{3}$m1, Vickers hardness 30.3 GPa) could be energetically more stable
than the above two structures. Structural predictions by Van Der Geest et al.
Van Der Geest and Kolmogorov (2014) suggest there are two thermodynamically
stable structures, TcB(Pnma) and TcB4 (P63/mmc), at 30 GPa. First-principle
DFT calculations by Miao et al. Miao  _et al._ (2017) reported
thermodynamically stable TcB3 (P$\bar{6}$m2, Vickers hardness 29 GPa)
structure at above 4 GPa. Structural searches by Ying et al. Ying  _et al._
(2016, 2018) suggested two structures, TcB3 (P$\bar{6}$m2, Vickers hardness
30.7 GPa) and TcB4 (P63/mmc, Vickers hardness 32.4 GPa), are thermodynamically
stable at 0 and 100 GPa respectively.

Although technetium is rare in nature, the technetium-based compounds are
under investigations by multiple disciplines in the past. Perovskites ATcO3 (A
= Ca, Sr, Ba) attracted extensive interests due to their extremely high
antiferromagnetic Neel temperatures (750–1200 K). Avdeev  _et al._ (2011);
Franchini  _et al._ (2011); Rodriguez  _et al._ (2011); Mravlje  _et al._
(2012) Recently, techentium hydrides were theoretically predicted and then
experimentally synthesized under high pressure. Li  _et al._ (2016); Zhou  _et
al._ (2023)

Previous works on technetium-borides are concentrated on the high hardness and
the high incompressibility of technetium-borides, while the explorations for
superconductivity are absent. Considering the fact that series of transition-
metal-borides have been found superconducting at relatively low pressures,
there is no reason to rule out the possible presence of superconductivity in
technetium-borides. Besides, current results about the thermaldynamically
stable phases of technetium-borides are highly diversified. Therefore, we
choose to search the technetium-boron binary system for new superconductors at
low or even ambient pressures. A comprehensive phase diagram of all
thermodynamically stable technetium-borides up to 180 GPa has been derived. We
also found five new superconducting technetium-borides of metastable states,
including TcB (P63/mmc), TcB2 (P6/mmm), Tc2B (I4/mcm), Tc3B (P4/mmm) and TcB
(Cmcm), which remain dynamically stable at relatively low pressures. The
mechanical properties of these superconducting technetium-borides have been
investigated as well.

##  II Methods

The structure prediction for technetium-boron binary crystals is performed by
CALYPSO packageWang  _et al._ (2012b). The electronic structures and the
phonon properties are calculated using QUANTUM-ESPRESSO (QE) package Giannozzi
_et al._ (2009). The plane-wave kinetic-energy cutoff and the charge density
energy cutoff are 100 Ry and 400 Ry, respectively. Optimized norm-conversing
pseudopotential with valence electron configurations of Tc-4p64d55s2 and
B-2s22p1 and Methfessel-Paxton smearing Methfessel and Paxton (1989) width of
0.02 Ry are used. Since technetium is radioactive without stable isotopes, we
adopted the average mass of technetium and boron at 97.907 and 10.811 atomic
mass units throughout our calculations. The dynamic matrix and the electron-
phonon coupling (EPC) constant $\lambda$ are calculated using the density-
functional perturbation theory Baroni  _et al._ (2001). Superconducting
transition temperature is estimated following the Allen-Dynes modified
McMillan equation Allen and Dynes (1975),

$\displaystyle
T_{\text{c}}=\frac{\omega_{\text{log}}}{1.2}\exp\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{\ast}\left(1+0.62\lambda\right)}\right],$
(1)

in which $\lambda$ is the average EPC parameter, $\omega_{\text{log}}$ is the
logarithmic average frequency, and the Coulomb pseudopotential Morel and
Anderson (1962) $\mu^{*}$ = 0.12. Mechanical properties including Vickers
hardness are estimated following models by Chen et al. and Tian et al. Chen
_et al._ (2011); Tian  _et al._ (2012). Calculation details are referenced to
the supplementary information SI .

##  III Results and discussion

###  III.1 Convex hull and phase diagram

Figure 2: Superconducting transition temperature of TcB2 (P6/mmm), TcB
(P63/mmc), Tc2B (I4/mcm), Tc3B (P4/mmm) and TcB (Cmcm) as a function of
pressure.

We have done variable-composition and fixed-composition structure searches in
the Tc-B system at pressures of 0, 90 and 180 GPa. Thermodynamically stable
structures and the derived composition-pressure phase diagram are presented in
Fig. 1. Three existing technetium-borides at ambient pressure, Tc3B (Cmcm),
Tc7B3 (P63/mmc) and TcB2 (P63/mmc), have been successfully identified. Tc3B
(Cmcm) is thermodynamically stable up to 180 GPa in our study. In contrast,
Tc7B3 (P63/mmc) and TcB2 (P63/mmc) stop being energetically favorable above 60
GPa and 139 GPa respectively, and a new TcB2 (I41/amd) thermodynamically
stable phase shows up above 170 GPa. Two previously predicted structures, TcB3
(P$\bar{6}$m2) Miao  _et al._ (2017); Ying  _et al._ (2016) and TcB4 (P63/mmc)
Van Der Geest and Kolmogorov (2014); Ying  _et al._ (2018), also have been
found in our calculations, which are thermodynamically stable above 2 and 35
GPa, respectively. We found TcB (P21) structure being thermodynamically stable
above 24 GPa, then transfers into Pmn21 structure at 63 GPa, and finally into
the previously predicted Pnma structure Van Der Geest and Kolmogorov (2014) at
160 GPa. We also discovered technetium-borides of two new stoichiometries,
Tc3B4 and Tc2B. Tc3B4 (C2/m) is thermodynamically stable above 151 GPa. Tc2B
(C2/m) structure is thermodynamically stable above 23 GPa, then transfers into
Fddd structure at 60 GPa. The crystal structure information of all
thermodynamically stable phases are presented in Table IV of supplementary
information SI .

Table 1: Total electronic DOS at Fermi level N(EF), EPC parameter $\lambda$, logarithmic average frequency $\omega_{log}$, and superconducting transition temperature $T_{\text{c}}$ of the superconducting technetium-borides at their lowest dynamically stable pressures. Formula | Space | P | N(EF) | $\lambda$ | $\omega_{log}$ | $T_{\text{c}}$
---|---|---|---|---|---|---
| group | (GPa) | (states/eV/f.u.) |  | (cm-1) | (K)
TcB2 | P6/mmm | 20 | 1.41 | 1.85 | 125.1 | 23.1
TcB | P63/mmc | 0 | 1.63 | 1.56 | 126.1 | 20.2
Tc2B | I4/mcm | 0 | 1.75 | 0.85 | 165.1 | 10.9
Tc3B | P4/mmm | 0 | 2.85 | 0.92 | 162.9 | 12.9
TcB | Cmcm | 30 | 1.05 | 0.96 | 135.5 | 11.5
Figure 3: Crystal structures of superconducting technetium-borides. (a) TcB2
(P6/mmm) (b) TcB (P63/mmc) (c) Tc2B (I4/mcm) (d) Tc3B (P4/mmm) and (e) TcB
(Cmcm). The technetium and the boron atoms are represented by spheres of coral
and green colors, respectively.

###  III.2 Superconductivity of thermodynamically metastable technetium-
borides

We also examined potential superconductors in technetium-borides, including
all thermodynamically stable structures and the thermodynamically metastable
structures within range of 300 meV above the convex hull. Totally five
thermodynamically metastable technetium-borides have been found
superconducting at 180 GPa, including TcB2 (P6/mmm, 42 meV/atom above the
hull), TcB (P63/mmc, 255 meV/atom above the hull), Tc2B (I4/mcm, 2 meV/atom
above the hull), Tc3B (P4/mmm, 248 meV/atom above the hull) and TcB (Cmcm, 25
meV/atom above the hull). These five superconducting technetium-borides stay
dynamically stable at decreased pressures. The minimum dynamical stable
pressures of TcB2 (P6/mmm) and TcB (Cmcm) are 20 and 30 GPa respectively,
while TcB (P63/mmc), Tc2B (I4/mcm) and Tc3B (P4/mmm) are dynamically stable
even at ambient pressure.

The superconducting transition temperatures of all five technetium-borides
increase at decreased pressure in Fig. 2. We also summarize the
superconducting transition temperature of the five technetium-borides at their
lowest dynamically stable pressures, together with their total electronic DOS
at the Fermi level N(EF), the EPC parameter $\lambda$ and the logarithmic
average frequency $\omega_{log}$ in Table 1. TcB2 (P6/mmm) has the highest
superconducting transition temperature of 23.1 K at 20 GPa, which comes from
its largest EPC parameter $\lambda$=1.85. In contrast, TcB (Cmcm) has much
lower superconducting transition temperature of 11.5 K at 30 GPa due to its
small EPC parameter $\lambda$=0.96. Superconducting transition temperature of
TcB (P63/mmc), Tc2B (I4/mcm) and Tc3B (P4/mmm) at 0 GPa are 20.2, 10.9, and
12.9 K, respectively. Although the EPC parameters $\lambda$ of these
metastable technetium-borides are not small, their logarithmic average
frequency $\omega_{log}$ are rather low at maximumly 165.1 cm-1, which limited
their superconducting transition temperature. This is in sharp contrast with
MgB2, which has smaller $\lambda$ = 0.87 but much larger $\omega_{log}$ = 504
cm-1 and the highest BCS-type superconducting transition temperature of 39 K
at ambient pressure. Kong  _et al._ (2001)

The thermodynamically metastable nature of the discovered superconducting
technetium-borides doesn’t necessarily exclude their experimental synthesis.
Metastable materials have long been synthesized and implemented, Aykol  _et
al._ (2018) typically like fullerene C60. As to superconductor, DFT
calculations predict NdH9 (P63/mmc) is 35 meV/atom above the convex hull at
150 GPa, yet being successfully synthesized with $T_{\text{c}}$ $\approx$ 4.5
K. Zhou  _et al._ (2020) Especially, several metastable borides have been
predicted superconducting in recent structural searches. Xia et al. discovered
thermodynamically metastable CB6 with superconducting transition temperature
of 12.5 K at ambient pressure. Xia  _et al._ (2017) Zhang et al. also found
thermodynamically metastable RbB6 (Pm-3m) and RbB8 (Immm) with superconducting
transition temperatures of 7.3–11.6 and 4.8–7.5 K at ambient pressure,
respectively. Zhang  _et al._ (2023) These works further validate the
importance and necessity of our discoveries of superconducting technetium-
borides.

Figure 4: Total and partial electronic DOS, phonon dispersion relation, phonon
density of states (PHDOS), Eliashberg functional $\alpha^{2}F(\omega)$ and
integrated EPC parameter $\lambda(\omega)$ of superconducting technetium-
borides at their lowest dynamically stable pressures. From top to bottom, (a)
TcB2 (P6/mmm, 20 GPa), (b) TcB (P63/mmc, 0 GPa), (c) Tc2B (I4/mcm, 0 GPa), (d)
Tc3B (P4/mmm, 0 GPa) and (e) TcB (Cmcm, 30 GPa).

###  III.3 Crystal structures

The crystal structures of five superconducting technetium-borides in our study
are presented in Fig.3. TcB2 (P6/mmm) shares exactly the same crystal
structure with MgB2 and MoB2. TcB (P63/mmc) has TiAs-type structure, in which
the rhombus Tc-layers are AB-stacking along the c-axis and the rhombus
B-layers are sandwiched between the neighbouring Tc-layers. Tc3B (P4/mmm) has
square Tc-layers stacking in ABB-pattern along the c-axis, and the square
B-layers locate between the two Tc-layers of BB-pattern. Tc2B (I4/mcm)
consists of square Tc-layers AB-stacking along the c-axis, where the
neighbouring Tc-layers are twisted by 37.2 degrees. The B-layers in Tc2B
(I4/mcm) are sandwiched between the neighbouring Tc-layers as well. In TcB
(Cmcm), the square Tc-layers stack in ABCD-pattern along the b-axis, and the
B-atoms form zig-zag chains along the c-axis between the AB and CD Tc-layers.
The angle of the zig-zag chain of B-atoms is around 108.8 degree. Crystal
structure information of the superconducting technetium-borides are referenced
to Table III in the supplementary information. SI

###  III.4 Electronic structures

The electronic DOS of superconducting technetium-borides at their lowest
stabilizing pressure are presented in the left columns of Fig.4. The
electronic DOS of all technetium-borides share certain features. The total DOS
at the Fermi level are dominated by the states of Tc-4d bands. Although the
B-2p DOS have considerable weight away the Fermi level, its contribution is
minor at the Fermi level, if not being zero. The B-2s DOS almost vanish around
the Fermi level, which makes B-2s bands almost irrelevant for electronic
conduction. Our DOS results of technetium-borides are in close resemblance to
another transition-metal-boride MoB2 Pei  _et al._ (2023), while in obvious
contrast with the alkali-earth-metal boride MgB2 An and Pickett (2001); Kortus
_et al._ (2001) or alkali-metal boride RbB6. Zhang  _et al._ (2023) In either
MgB2 An and Pickett (2001); Kortus  _et al._ (2001) or RbB6 Zhang  _et al._
(2023), the major DOS at the Fermi level are contributed by the B-p bands. But
in either the superconducting technetium-borides of our study or MoB2, Pei
_et al._ (2023) the 4d electronic states play dominant roles around the Fermi
level.

The electronic band structure and the Fermi surface of superconducting
technetium-borides are presented in Fig. S1 of supplementary information SI .
For all five superconducting technetium-borides, either their band structure
or their Fermi surface show obvious electronic dispersion in three dimensions.
Typically for example TcB2, which shares the same cyrstal structure as MgB2
and MoB2, has three dimensional Fermi surface like MoB2 Pei  _et al._ (2023);
Quan  _et al._ (2021) while being distinct from the quasi-two-dimensional
Fermi surface of MgB2. Kortus  _et al._ (2001); Choi  _et al._ (2002)

###  III.5 Dynamical stability and electron-phonon coupling

The phonon spectrum, the PHDOS, the Eliashberg functional
$\alpha^{2}F(\omega)$ and the corresponding integrated EPC constant $\lambda$
of superconducting technetium-borides at their lowest dynamically stable
pressures are presented in the right columns of Fig.4. There is no sign of
imaginary frequency in the phonon spectrum of all five superconducting
technetium-borides, which proves the dynamical stability of these structures
at the corresponding pressures. The distribution of the PHDOS and the
Eliashberg spectral functional $\alpha^{2}F(\omega)$ of superconducting
technetium-borides show clear separation between the low frequency phonon
modes of the heavier Tc-atoms and the high frequency phonon modes of the
lighter B-atoms. This enables us to separate the integrated EPC constant
$\lambda$ into two parts, the EPC from Tc-atoms $\lambda_{T_{c}}$, and the EPC
from B-atoms $\lambda_{B}$. The ratio of EPC from oscillation of Tc-atoms
relative to the total EPC, $\lambda_{T_{c}}$/$\lambda$, are 0.883, 0.910,
0.873, 0.907 and 0.911 for TcB2 (P6/mmm), TcB (P63/mmc), Tc2B (I4/mcm), Tc3B
(P4/mmm) and TcB (Cmcm), respectively. It indicates superconductivity in these
five technetium-borides mainly originates from the coupling between the Tc-4d
electrons and the low frequency phonon modes of Tc-atoms. At least three
isotopes of technetium have reasonably long half lives (Tc-97, Tc-98 and Tc-99
at 4.2$\times 10^{6}$, 6.6$\times 10^{6}$ and 2.13$\times 10^{5}$ years,
respectively). Therefore, we suggest experiments on the isotope effects of
technetium to examine our prediction.

The superconducting mechanism of our predicted technetium-borides is similar
to that in transition-metal-borides MoB2, whose superconductivity mainly
originates from the coupling between the Mo-4d electrons and the low frequency
Mo-phonon modes. Pei  _et al._ (2023) However, the superconducting scenarios
in alkali-earth-metal boride MgB2 An and Pickett (2001); Kortus  _et al._
(2001) and alkali-metal boride RbB6 Zhang  _et al._ (2023) are very different
in that the couplings between B-2p electrons and the high frequency B-phonon
modes play dominant roles.

The presence of BCS superconductivity or not in transition metal borides such
as Mn-B, Mo-B, Tc-B, Re-B and Ru-B are likely being related to the magnitude
of correlations of d-electrons in the transition metal. In MnB2, Mn-3d
electrons are localized due to their strong correlations, which leads to
magnetism and the suppression of BCS-superconductivity. In contrast, the
superconductivity in MoB2, technetium-borides, Ru7B3 and Re7B3 are preserved
where the correlations of d-electrons in Mo, Tc, Ru and Re are relatively
smaller.

Observations in phonon spectrum, PHDOS and EPC of superconducting technetium-
borides are consistent with their relatively smaller logarithmic average
frequency $\omega_{log}$ as listed in Table 1, since Tc-atoms are much heavier
than B-atoms. The enhanced $\lambda$ plus small $\omega_{log}$ characters of
TcB2 have also been seen in iso-structural superconductor TlBi2 of heavy
atomic mass, Yang  _et al._ (2023) with $\lambda$=1.4, $\omega_{log}$=37
$cm^{-1}$ and rather low $T_{\text{c}}$=5.5 K. Recent work suggests
introduction of hydrogen atoms into non-superconducting transition-metal
boride Ti2B2 will result in superconducting Ti2B2H4 ($T_{\text{c}}$=48.6 K at
ambient pressure), through expansion of the frequency range of phonon spectrum
and consequently enlarged electron-phonon coupling Han  _et al._ (2023b).
Similar hydrogenation probably helps in elevating the superconducting
transition temperatures of technetium-borides by enlarging $\omega_{log}$.

Another interesting observation on the discovered superconducting technetium-
borides is that the Fermi levels of TcB2 and TcB fall closely above the very
peak positions of their DOS, as shown in Fig. 4. Since the EPC in technetium-
borides is controlled by the coupling between the Tc-4d electrons and the
oscillation of Tc-atoms, slight hole-doping could lower the Fermi level, thus
enhances the effective number of electrons participating into the
superconducting pairing and therefore enlarges the superconducting transition
temperatures.

Table 2: Vickers hardness of superconducting technetium-borides at their lowest dynamically stable pressures.  Formula | Space Group | P | Hv,Chen Chen  _et al._ (2011) | Hv,Tian Tian  _et al._ (2012) |
---|---|---|---|---|---
|  | (GPa) | (GPa) | (GPa) |
TcB2 | P6/mmm | 20 | 10.6 | 12.0 |
TcB | P63/mmc | 0 | 2.8 | 4.8 |
Tc2B | I4/mmm | 0 | 11.8 | 13.0 |
Tc3B | P4/mmm | 0 | 10.0 | 11.3 |
TcB | Cmcm | 30 | 12.2 | 13.7 |

###  III.6 Hardness

We also calculated the Vickers hardness of discovered superconducting
technetium-borides as presented in Table 2, including . At ambient pressure,
TcB (P63/mmc), Tc2B (I4/mcm) and Tc3B (P4/mmm) have Vickers hardness values of
2.8-4.8, 11.8-13.0 and 10.0-11.3 GPa, respectively. The Vickers hardness of
TcB2 (P6/mmm) and TcB (Cmcm) are 9.8-11.3 GPa and 12.2-13.7 GPa at pressures
of 20 and 30 GPa, respectively. The superconducting technetium-borides in our
study have relatively lower hardness values than previously stated
superconducting borides of superhardness, for example RbB6 (Pm-3m, Vickers
hardness of 19.7 GPa at the ambient pressure) and RbB8 (Immm, Vickers hardness
of 36.9 GPa at the ambient pressure). Zhang  _et al._ (2023) Other mechanical
parameters including elastic constants Cij, bulk modulus B, and shear modulus
G at their lowest dynamically stable pressures are also calculated. Mechanical
stability criteria Mouhat and Coudert (2014) related to the elastic constants
of these superconducting technetium-borides are fulfilled as presented in
supplementary information SI .

##  IV Conclusion

In summary, we have conducted thorough structural searches in the technetium-
boron binary system. An updated composition-pressure phase diagram of
technetium-borides up to 180 GPa have been derived, including two new
stoichiometries as Tc3B4 and Tc2B. More importantly, we also found five
unprecedented superconducting technetium-borides which remain dynamically
stable at moderate or even ambient pressures. Among these thermodynamically
metastable superconducting technetium-borides, TcB2 (P6/mmm) has the highest
superconducting transition temperature of 23.1 K at 20 GPa, and TcB (P63/mmc)
has the highest superconducting transition temperature of 20.2 K at ambient
pressure. The superconductivity in these technetium-borides mainly originate
from the coupling between the dominant presence of Tc-4d electronic states
around the Fermi level and the low frequency vibration modes of the
technetium-atoms, which is closely analogous to another transition-metal-
boride MoB2. Our calculations not only identified superconducting TcB2
(P6/mmm) of the same crystal structure as MgB2 and MoB2, but also discovered
series of superconducting technetium-borides with diversified crystal
structures. This work proves the rich structures and stoichiometries in
superconducting technetium-borides at high pressures, thus sheds lights on the
necessity of extended researches in discovery of new superconducting
transition-metal borides.

##  V Acknowledgement

P. Z., Y.D. Q. and S.X. Y. designed the project; X.R. T. conducted the
structure searches; P. Z. and A.Q. Y. calculated the electronic structure, the
phonon spectra and the superconducting transition temperatures; all authors
prepared the manuscript together. This work is supported by the National
Natural Science Foundation of China No. 11604255 and the Natural Science Basic
Research Program of Shaanxi No. 2021JM-001. Shuxiang Yang is supported by the
Key Research Projects of Zhejiang Lab (Grants No. 2021PB0AC02). The
computations are performed at the TianHe-2 national supercomputing center in
Guangzhou and the HPC platform of Xi’an Jiaotong University.

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