(bot4) edge [->] node[left] $\ominus_4$  (bot5);

A sketch of region $R_4=(sup,sig)$ of Lemma <ref>, where $i=3$ and $j=1$ and $\ell=4$.
In particular, $v_{3_1}'=v_{1_1}'=v_{4_0}'=v_2'$.
Moreover, $sig(v_2')=\inp$ and $sig(e)=\swap$ if $e\in \{v_1, w_1,w_3, w_4\}$ and $sig(e)=\nop$ otherwise.

[new set = import nodes]
[nodes=set=import nodes]
(bot1) at (0,0);
ıin 0,...,4 (ı) at (ı*1.8cm+1.8cm,0);

(bot1) at (bot1) $\bot_0$;
ıin 0,...,4 (ı) at (ı) $t_{0,\i}$;

(import nodes);

[yshift=-1cm, nodes=set=import nodes]
(bot2) at (0,0);
ıin 0,...,4 (ı) at (ı*1.8cm+1.8cm,0);
ıin 2,3 [red!20] (ı) circle (0.3cm);
(bot2) at (bot2) $\bot_1$;
ıin 0,...,4 (ı) at (ı) $t_{1,\i}$;

(import nodes);

[yshift=-2cm, nodes=set=import nodes]
(bot3) at (0,0);
ıin 0,...,4 (ı) at (ı*1.8cm+1.8cm,0);

(bot3) at (bot3) $\bot_2$;
ıin 0,...,4 (ı) at (ı) $t_{2,\i}$;

(import nodes);

[yshift=-3cm, nodes=set=import nodes]
(bot4) at (0,0);
ıin 0,...,4 (ı) at (ı*1.8cm+1.8cm,0);
ıin 2,3 [red!20] (ı) circle (0.3cm);
(bot4) at (bot4) $\bot_3$;
ıin 0,...,4 (ı) at (ı) $t_{3,\i}$;

(import nodes);

[yshift=-4cm,nodes=set=import nodes]
(bot5) at (0,0);
(bot5) at (0,0) $\bot_4$;
ıin 0,...,4  (ı) at (1.8cm+ ı*1.8cm,0) ;
ıin 0,3,4 [red!20] (ı) circle (0.3cm);
ıin 0,...,4  (ı) at (ı) $t_{4,\i}$;

(import nodes);

(bot1) edge [->] node[left] $\ominus_1$  (bot2);
(bot2) edge [->] node[left] $\ominus_2$  (bot3);
(bot3) edge [->] node[left] $\ominus_3$  (bot4);
(bot4) edge [->] node[left] $\ominus_4$  (bot5);

A sketch of $R_5=(sup,sig)$ of Lemma <ref>, where $i=3$ and $j=1$ and $\ell=4$.
In particular, $v_{i_1}'=v_{j_1}'=v_{\ell_0}'=v_2'$ and $v_{i_1}=v_{j_1}=v_{\ell_0}=v_2$.
Moreover, $sig(v_2')=\inp$, and $sig(e)=\swap$ if $e\in \{v_2,w_4\}$ (notice the sequence $v_2'v_3v_2v_3$  in $T_4'$) and $sig(e)=\nop$ otherwise.

§ THE PROOF OF LEMMA <REF>: CONTINUED

We argue that $\overline{B}_G$ has the $\tau$-ESSP.
Without loss of generality, we assume $\used\in \tau$.
(Otherwise $\free\in \tau$ and this case is similar.)
Recall that $G$ is $3$-bounded, that is, every node-event or its splitting occur at most three times in $\overline{B}_G$.
For a start, it is easy to see that if $e\in \bot\cup W$, then $(e,s)$ is $\tau$-solvable for all relevant $s\in S(\overline{B}_G)$.

Let $i\in \{0,\dots, m-1\}$ and $e\in E(T_i)$ be arbitrary but fixed, and let $j\not=\ell\in \{0,\dots, m-1\}\setminus\{i\}$ such that $e\in E(T_j') \cap E(T_\ell')$ (if they exist).
The following region $R=(sup, sig)$ solves $(e,s)$ for all $s\in S(\overline{B}_G)\setminus (S(T_i') \cup S(T_j')\cup S(T_\ell'))$:
for all $e'\in E'$, if $e'=e$, then $sig(e)=\used$;
if $e'\in \{w_i,w_j,w_\ell\}$, then $sig(e')=\swap$,
otherwise, $sig(e')=\nop$.

Thus, it remains to argue that $(e,s)$ is also solvable is $e$ and $s$ occur in the same $T_i'$.
To do so, we let $i\in \{0,\dots, m-1\}$ and start with he case where $v_{i_0}\in M$ and $v_{i_1}\not\in M$, that is,
$T_i'=t_{i,0}\fBedge{v_{i_0}'} t_{i,1}\fBedge{v_{i_1}} t_{i,2} \fBedge{v_{i_0}} t_{i,3} \fBedge{v_{i_1}} t_{i,4}$, and show $(e,s)$ is $\tau$-solvable for all $e\in \{v_{i_0}, v_{i_0}',v_{i_1}\}$ and all $s\in S(T_i')$.
Recall that $v_{i_0}$ and $v_{i_1}$ unambiguously define an edge of the graph $G$, that is, they never occur bot in two distinct $T_i'\not= T_j'$.
Moreover, by definition of $\overline{B}_G$, a prime-event $v'$ does never occur between an un-split event $u\in V$, that is, $uv'u$ is not possible.

Let $j\not=\ell\in \{0,\dots, m-1\}\setminus\{i\}$, such that $v_{i_0}'\in E(T_j')\cap E(T_\ell') $.
Notice that this implies that $v_{i_1}\not\in E(T_j')\cup E(T_\ell')$.

First of all, we want to define a $\tau$-region $R=(sup, sig)$ that solves $(v_{i_0'}, s)$ for all $s\in \{t_{i,2}, t_{i,3}\}$ such that ${T_i'}^{R}=1\edge{\used}1\edge{\swap}0\edge{\nop}0\edge{\swap}1$.
The following definition is sound and satisfies this requirement:
for all $e\in E'$, if $e=v_{i_0}$, then $sig(e)=\used$;
if $e\in \{w_i, w_j, w_\ell, v_{i_1}\}$, then $sig(e)=\swap$;
otherwise, $sig(e)=\nop$.

To solve the remaining atom $(v_{i_0}', t_{i,4})$, we want to define a $\tau$-region $R=(sup, sig)$ such that ${T_i'}^{R}=1\edge{\used}1\edge{\nop}1\edge{\swap}0\edge{\nop}0$.
Up to isomorphism or renaming, two situations can occur for the shape of $T_j'$ and $T_\ell'$:

$T_j'=t_{j,0}\fBedge{v_{i_0}'} t_{j,1}\fBedge{v_{j_1}} t_{j,2} \fBedge{v_{i_0}} t_{j,3} \fBedge{v_{j_1}} t_{j,4}$,

$T_\ell'=t_{\ell,0}\fBedge{v_{\ell_0}} t_{\ell,1}\fBedge{v_{i_0}} t_{\ell,2} \fBedge{v_{\ell_0}} t_{\ell,3} \fBedge{v_{i_0}'} t_{\ell,4}$.

To define $R$ appropriately, we want to ensure that
${T_j'}^{R}=1\edge{\used}1\edge{\nop}1\edge{\swap}0\edge{\nop}0$ and ${T_\ell'}^{R}=0\edge{\nop}0\edge{\swap}1\edge{\nop}1\edge{\used}1$.
The following definition is sound and satisfies this requirement:
for all $e\in E'$, if $e=v_{i_0}$, then $sig(e)=\used$;
if $e\in \{w_i, w_j, v_{i_0}\}$, then $sig(e)=\swap$;
otherwise, $sig(e)=\nop$.

The following region $R=(sup, sig)$ yields ${T_i'}^{R}=0\edge{\nop}0\edge{\swap}1\edge{\used}1\edge{\swap}0$ and thus solves $(v_{i_0}, s)$ for all $s\in \{t_{i,0}, t_{i,1}, t_{i,4}\}$:
for all $e\in E'$, if $e=v_{i_0}$, then $sig(e)=\used$;
if $e\in \{w_j,w_\ell,v_{i_1}\}$, then $sig(e)=\swap$;
otherwise $sig (e)=\nop$.

The following region $R=(sup, sig)$ yields ${T_i'}^{R}=0\edge{\swap}1\edge{\used}1\edge{\nop}1\edge{\used}1$ and thus solves $(v_{i_1}, t_{i,0})$:
for all $e\in E'$, if $e=v_{i_1}$, then $sig(e)=\used$;
if $e\in \{w_i,v_{i_1}\}$, then $sig(e)=\swap$;
otherwise $sig (e)=\nop$.

Altogether, by the arbitrariness if $i$, we have shown that if $v_{i_0}\in M$ and $v_{i_1}\not\in M$, then all well-defined ESSP atoms $(e,s)$ such that $e\in E(T_i')$ and $s\in S(T_i')$ are $\tau$-solvable for all $i\in \{0,\dots, m-1\}$.
One finds out that the cases where $v_{i_0}\not\in M$ and $v_{i_1}\in M$ as well as  $v_{i_0}, v_{i_1}\in M$ can be solved similar.
This finally proves the $\tau$-ESSP for $\overline{B}_G$ and thus completes the proof of Lemma <ref>.

§ EXTRA?

[new set = import nodes, scale =0.7]

[nodes=set=import nodes]
ıin 0,...,4  (ı) at (ı*1.2cm, 0) ;

ıin 2,4  (ı) at (ı) $1$;
ıin 0,1,3  (ı) at (ı) $0$;
(import nodes);
0 ->["$\nop$"]1->["$\swap$"]2->["$\swap$"]3->["$\swap$"]4;

[new set = import nodes, scale =0.7]

[xshift=6cm, nodes=set=import nodes]
ıin 0,...,4  (ı) at (ı*1.2cm, 0) ;

ıin 3,4  (ı) at (ı) $1$;
ıin 0,1,2  (ı) at (ı) $0$;
(import nodes);
0 ->["$\nop$"]1->["$\nop$"]2->["$\swap$"]3->["$\nop$"]4;

[xshift=12cm, nodes=set=import nodes]
ıin 0,...,4  (ı) at (ı*1.2cm, 0) ;

ıin 1,2,3,4  (ı) at (ı) $1$;
ıin 0  (ı) at (ı) $0$;
(import nodes);
0 ->["$\swap$"]1->["$\nop$"]2->["$\nop$"]3->["$\nop$"]4;

[new set = import nodes]

[nodes=set=import nodes]
ıin 0,...,10  (ı) at (ı*1.14cm, 0) ;

ıin 0,...,4  (sı) at (ı) $t_{0,\i}$;
ıin 5  (sı) at (ı) $s_1$;
ıin 6,...,10  int(ı-6) (sı) at (ı) $t_{1,\pgfmathresult}$;

(x2) at (10*1.14, -1) $x_2$;
(import nodes);
s0 ->["$v_0$"]s1->["$v_1$"]s2->["$v_0$"]s3->["$v_1$"]s4->[swap, "$y_1$"]s5->[swap, "$z_1$"]s6 ->["$v_0$"]s7->["$v_2$"]s8->["$v_0$"]s9->["$v_2$"]s10;

[yshift=-2cm,nodes=set=import nodes]
ıin 0,...,10  (ı) at (10*1.14cm-ı*1.14cm, 0) ;

ıin 0,...,4  (tı) at (ı) $t_{2,\i}$;
ıin 5  (tı) at (ı) $s_3$;
ıin 6,...,10  int(ı-6) (tı) at (ı) $t_{3,\pgfmathresult}$;

(x4) at (0, -1) $x_4$;
(import nodes);
t0 ->[swap, "$v_0$"]t1->[swap, "$v_4$"]t2->[swap, "$v_0$"]t3->[swap, "$v_4$"]t4->["$y_3$"]t5->["$z_3$"]t6 ->[swap, "$v_1$"]t7->[swap, "$v_2$"]t8->[swap, "$v_1$"]t9->[swap, "$v_2$"]t10;
t10->[swap, "$y_4$"]x4;

[yshift=-4cm, nodes=set=import nodes]
ıin 0,...,10  (ı) at (ı*1.14cm, 0) ;

ıin 0,...,4  (uı) at (ı) $t_{3,\i}$;
ıin 5  (uı) at (ı) $s_5$;
ıin 6,...,10  int(ı-6) (uı) at (ı) $t_{4,\pgfmathresult}$;
(import nodes);
u0 ->["$v_2$"]u1->["$v_3$"]u2->["$v_2$"]u3->["$v_3$"]u4->[swap, "$y_5$"]u5->[swap, "$z_5$"]u6 ->["$v_3$"]u7->["$v_4$"]u8->["$v_3$"]u9->["$v_4$"]u10;
x4->[swap, "$z_4$"]u0;

The TS $A_G$ as introduced in Section <ref> that originates from Example <ref>.

[new set = import nodes]

[nodes=set=import nodes]
ıin 0,...,10  (ı) at (ı*1.14cm, 0) ;

ıin 0,...,4  (sı) at (ı) $t_{0,\i}$;
ıin 5  (sı) at (ı) $s_1$;
ıin 6,...,10  int(ı-6) (sı) at (ı) $t_{1,\pgfmathresult}$;

(x2) at (10*1.14, -1) $x_2$;
(import nodes);
s0 ->["$v_0$"]s1->["$v_1$"]s2->["$v_0$"]s3->["$v_1$"]s4->[swap, "$y_1$"]s5->[swap, "$z_1$"]s6 ->["$v_0$"]s7->["$v_2$"]s8->["$v_0$"]s9->["$v_2$"]s10;

[yshift=-2cm,nodes=set=import nodes]
ıin 0,...,10  (ı) at (10*1.14cm-ı*1.14cm, 0) ;

ıin 0,...,4  (tı) at (ı) $t_{2,\i}$;
ıin 5  (tı) at (ı) $s_3$;
ıin 6,...,10  int(ı-6) (tı) at (ı) $t_{3,\pgfmathresult}$;

(x4) at (0, -1) $x_4$;
(import nodes);
t0 ->[swap, "$v_0$"]t1->[swap, "$v_4$"]t2->[swap, "$v_0$"]t3->[swap, "$v_4$"]t4->["$y_3$"]t5->["$z_3$"]t6 ->[swap, "$v_1$"]t7->[swap, "$v_2$"]t8->[swap, "$v_1$"]t9->[swap, "$v_2$"]t10;
t10->[swap, "$y_4$"]x4;

[yshift=-4cm, nodes=set=import nodes]
ıin 0,...,10  (ı) at (ı*1.14cm, 0) ;

ıin 0,...,4  (uı) at (ı) $t_{3,\i}$;
ıin 5  (uı) at (ı) $s_5$;
ıin 6,...,10  int(ı-6) (uı) at (ı) $t_{4,\pgfmathresult}$;
(import nodes);
u0 ->["$v_2$"]u1->["$v_3$"]u2->["$v_2$"]u3->["$v_3$"]u4->[swap, "$y_5$"]u5->[swap, "$z_5$"]u6 ->["$v_3$"]u7->["$v_4$"]u8->["$v_3$"]u9->["$v_4$"]u10;
x4->[swap, "$z_4$"]u0;

TS $B_G$ built from $A_G$ and the splitting () as defined for Lemma <ref>.

[new set = import nodes, scale =0.8]

[nodes=set=import nodes]
(A) at (-1,0) $A_G=$;
ıin 0,...,5  (ı) at (ı*2cm, 0) ;
(6) at (12.25,0);

(0) at (0) $T_0$;
(1) at (1) $c_0$;
(2) at (2) $T_1$;
(3) at (3) $c_2$;
(4) at (4) $\dots$;
(5) at (5) $c_{m-1}$;
(6) at (6) $T_{m-1}$;

(import nodes);
0 ->["$y_1$"]1->["$z_1$"]2->["$y_2$"]3->["$z_2$"]4->["$y_{m-1}$"]5->["$z_{m-1}$"]6;

Synthesis is the task to find for a given specification an implementing system if it exists.
Obviously, there are at least variable parts of this problem, namely which kind of specification, which kind of system is sought and which kind of implementation-relation between them is required.
In the case of flip-flop net synthesis, the specification is a directed labeled graph $A$, also called transition system (TS), and the searched system is a flip-flop Petri net $N$ whose reachability satisfies the requirements of the implementation-relation.
Three different more or less strict requirements are considered:
First, the strictest one requires that $N$ implements $A$ up to isomorphism, that is, he reachability graph of $N$ and $A$ are isomoprhic;
second, a weaker one requires that $N$ simulates $A$ up to language-equivalence, that is, there is a surjective simulation map between $N$'s reachability graph and $A$;
finally, at third one requires that $N$ embeds $A$, that is, there is a injective simulation map between the reachability graph of $N$ and $A$.
Every just introduced synthesis instance defines a corresponding decision problem that asks if, for a given $A$, a flip-flop net $N$ that fulfills the implementation requirements exists.
Unfortunately, for all kinds of implementations, $N$ does not always exists.
To convert a non-implementable $A$ to an implementable one $A'$, label-splitting is an option, that is, relabelling the edges of $A$ such that differently labelled edges are different.
Using the label-splitting technique, generally, $A$ can always be converted into $A'$ such that there is always a searched net $N$ for the latter.
However, it is desired to split as few labels as possible, which leads to consider label-splitting as an optimization problem, respectively decision problem.
In this paper, we show that the label-splitting problem for flip-flop nets is NP-complete.

Altogether, this finally leads to the following decision problems that are the main subject of this paper:

label-splitting for $\tau$-State Separation Property (LS$\tau$-SSP)

a TS $A=(S,E,\delta, \iota)$, a natural number $\kappa$.

Decide:   whether there exists a $E'$-label-splitting $B$ of $A $ such that  $\vert E' \vert \leq \kappa$ that has the $\tau$-SSP.

label-splitting for $\tau$-Event State Separation Property (LS$\tau$-ESSP)

a TS $A=(S,E,\delta, \iota)$, a natural number $\kappa$.

Decide:   whether there exists a $E'$-label-splitting $B$ of $A $ such that $\vert E' \vert \leq \kappa$ that has the $\tau$-ESSP.

label-splitting for $\tau$-Synthesis (LS$\tau$-Synthesis)

a TS $A=(S,E,\delta, \iota)$, a natural number $\kappa$.

Decide:   whether there exists a $E'$-label-splitting $B$ of $A $ such that $\vert E' \vert \leq \kappa$ that has both the $\tau$-SSP and the $\tau$-ESSP.


# High-field 1/f noise in hBN-encapsulated graphene transistors

A. Schmitt<EMAIL_ADDRESS>Laboratoire de Physique de l’Ecole
normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université
de Paris, 24 rue Lhomond, 75005 Paris, France  D. Mele  Laboratoire de
Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne
Université, Université de Paris, 24 rue Lhomond, 75005 Paris, France  Univ.
Lille, CNRS, Centrale Lille, Univ. Polytechnique Hauts-de-France, Junia-ISEN,
UMR 8520-IEMN, F-59000 Lille, France.  M. Rosticher  Laboratoire de Physique
de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 24 rue Lhomond, 75005 Paris, France  T. Taniguchi
Advanced Materials Laboratory, National Institute for Materials Science,
Tsukuba, Ibaraki 305-0047, Japan  K. Watanabe  Advanced Materials Laboratory,
National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan  C.
Maestre  Laboratoire des Multimatériaux et Interfaces, UMR CNRS 5615, Univ
Lyon, Université Claude Bernard Lyon 1, F-69622 Villeurbanne, France  C.
Journet  Laboratoire des Multimatériaux et Interfaces, UMR CNRS 5615, Univ
Lyon, Université Claude Bernard Lyon 1, F-69622 Villeurbanne, France  V.
Garnier  Université de Lyon, MATEIS, UMR CNRS 5510, INSA-Lyon, F-69621
Villeurbanne cedex, France  G. Fève  Laboratoire de Physique de l’Ecole
normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université
de Paris, 24 rue Lhomond, 75005 Paris, France  J.M. Berroir  Laboratoire de
Physique de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne
Université, Université de Paris, 24 rue Lhomond, 75005 Paris, France  C.
Voisin  Laboratoire de Physique de l’Ecole normale supérieure, ENS, Université
PSL, CNRS, Sorbonne Université, Université de Paris, 24 rue Lhomond, 75005
Paris, France  B. Plaçais<EMAIL_ADDRESS>Laboratoire de Physique
de l’Ecole normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 24 rue Lhomond, 75005 Paris, France  E. Baudin
<EMAIL_ADDRESS>Laboratoire de Physique de l’Ecole normale
supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de
Paris, 24 rue Lhomond, 75005 Paris, France

###### Abstract

Low-frequency 1/f noise in electronics is a conductance fluctuation, that has
been expressed in terms of a mobility ”$\alpha$-noise” by Hooge and
Kleinpenning. Understanding this noise in graphene is a key towards high-
performance electronics. Early investigations in diffusive graphene have
pointed out a deviation from the standard Hooge formula, with a modified
expression where the free-carrier density is substituted by a constant density
$n_{\Delta}\sim 10^{12}\;\mathrm{cm^{-2}}$. We investigate hBN-encapsulated
graphene transistors where high mobility gives rise to the non-linear
velocity-saturation regime. In this regime, the $\alpha$-noise is accounted
for by substituting conductance by differential conductance $G$, resulting in
a bell-shape dependence of flicker noise with bias voltage $V$. The same
analysis holds at larger bias in the Zener regime, with two main differences:
the first one is a strong enhancement of the Hooge parameter reflecting the
hundred-times larger coupling of interband excitations to the hyperbolic
phonon-polariton (HPhP) modes of the mid-infrared Reststrahlen (RS) bands of
hBN. The second is an exponential suppression of this coupling at large
fields, which we attribute to decoherence effects. We also show that the HPhP
bands control the amplitude of flicker noise according to the graphene-hBN
thermal coupling estimated with microwave noise thermometry. The phenomenology
of $\alpha$-noise in graphene supports a quantum-coherent bremsstrahlung
interpretation of flicker noise.

Flicker (1/$f$) noise is a type of noise that dominates the spectral density
at low frequencies. Ubiquitous in condensed matter and quantum conductors, it
has also been measured in a large variety of systems ranging from biology to
economics. In electronics, flicker noise is a conductance fluctuation revealed
as an excess low-frequency current noise, of spectral density
$S_{I}=CI^{2}/f$, superimposed on the Johnson-Nyquist thermal noise
$S_{th}=4Gk_{B}T$ at finite bias current $I$. Despite almost a century of
research on $1/f$ noise since Johnson measured it in vacuum tubes
[Johnson1925pr, ], its origin remains controversial. In semiconductors, $1/f$
noise is usually well described using the McWorther model relying on carrier
number fluctuations due to defect traps [McWorther, ; lowfreq, ]. Alternative
mechanisms based on carrier-number-fluctuation or involving fluctuations of
mobility have been proposed. As pointed out by Hooge and Kleinpenning
[Hooge1969pla, ], the noise amplitude $C$ is a bulk effect that can be
conveniently expressed in terms of a mobility noise, called $\alpha$-noise,
with an intensive Hooge parameter $\alpha_{H}=NC$, where $N$ is the number of
carriers participating to the conductance. Measurements in both metals
[Hooge1969physica, ] and semiconductors [Hooge1981rep, ] have yielded a quasi-
universal Hooge parameter $\alpha_{H}\sim 2-5\times 10^{-3}$. A leading
interpretation of the $\alpha$-noise relies on a distribution of two-level
fluctuators that locally and elastically modulate electronic transmission
[Hooge1981rep, ,Kogan, ]. An alternative interpretation, based on a quantum
theory proposed by Handel [Handel1975, ; Handel1996pss, ], points to an
infrared (IR) bremsstrahlung origin which leads to a $1/f$ noise spectrum with
a universal $\alpha_{H}=2\alpha_{0}/\pi=4.6\times 10^{-3}$ in the quantum-
coherent regime, where $\alpha_{0}=1/137$ is the vacuum fine-structure
constant that controls light-matter coupling. This interpretation puts
emphasis on an inelastic origin of conductance noise; a critical discussion
can be found in Refs.[Nieuwenhuizen1987, ; Weissman1988, ]. These contrasted
interpretations take root in the ambivalent nature of conductivity, which is
both a momentum relaxation coefficient $\sigma_{1}(t)=(J/E)(t)$, and an energy
relaxation parameter in the Joule power density $\sigma_{2}(t)=(P/E^{2})(t)$.
Whereas $\sigma_{1}=\sigma_{2}$ in average, they may differ in their time
dependencies due to different scattering mechanisms and times, fingerprinted
in a low-frequency noise $\delta\sigma(t)$.

Graphene is an attractive platform for flicker noise investigation because of
the tunability of charge carrier density and polarity. This versatile material
also presents strong potential for electronics [Balandin2013, ] for which
flicker noise is an important performance limit [Mavredakis2013acsaem, ]. Most
experiments so far have been carried out at low to moderate bias with a $1/f$
corner frequency, separating the low-frequency $1/f$ noise from the high-
frequency thermal white noise, in the sub-MHz range [Lin2008, ;
Rumyantsev2010, ], and mostly on SiO2-supported devices with a low mobility
$\mu\lesssim 0.1\;\mathrm{m^{2}/Vs}$. Flicker noise in these diffusive
graphene transistors was characterized by a doping-independent flicker
amplitude $A=fS_{I}/I^{2}LW\in[10^{-7},10^{-6}]\;\mathrm{\mu m^{2}}$
[Balandin2013, ] (or $[0.1,1]\;\mathrm{nm^{2}}$), where $L$ and $W$ the length
and width of the graphene channel. This doping-independent value for graphene
violates the Hooge empirical formula where $A=\alpha_{H}/n$ is assumed to be
inversely proportional to carrier density $n$, based on the assumption that
electrons behave independently. It can be re-conciliated with a collective
picture on substituting $n$ by a constant density $n_{\Delta}$, which origin
remains to be clarified. Some reports also point to a flicker noise dip at
charge neutrality [Rumyantsev2010, ; Takeshita2016, ], but charge neutrality
corresponds to a non-local mesoscopic regime that deviates from standard
local-transport $\alpha$-noise conditions. The same occurs under intense
magnetic fields, with a field-induced noise reduction [Rehman2022nanoscale, ],
leading to a full suppression enforced by conductance quantization
[Kalmbach2016, ]. Flicker noise suppression was also recently reported in the
different system of a break-junction, where it is found to accompany shot-
noise suppression at full transmission [Shein2022, ]. A deviation from the
above $A=Cte$ diffusive graphene flicker noise was reported in moderate-
mobility suspended samples, where a noise reduction was observed
[Kumar2015apl, ].

The high-mobility hBN-encapsulated graphene transistors investigated in the
present work constitute a testbed for $\alpha$-noise as they allow studying
flicker noise in an extended electric-field range where different scattering
and relaxation mechanisms succeed one another in increasing bias. In the
perspective of testing the bremsstrahlung interpretation, the use of hBN-
encapsulates is an additional asset, as hBN not only provides a superior
mobility [Dean2010nnano, ], but also a very characteristic mid-infrared (MIR)
near-field electromagnetic environment [Dai2015nnano, ; Kumar2015nl, ], with
its two Reststrahlen (RS) bands, $\hbar\Omega_{I}=95$-$100\;\mathrm{meV}$ and
$\hbar\Omega_{II}=170$-$200\;\mathrm{meV}$. As a matter of fact, recent
experiments have shown that the hyperbolic light of these RS bands strongly
couples to graphene electronic transport [Yang2018nnano, ; Baudin2020adfm, ].
The three electronic transport regimes observed in increasing electric field
are: (i) the mobility-limited Drude regime $J=\sigma E=ne\mu~{}E$ at low
field, (ii) the velocity saturation regime for $E\gtrsim E_{sat}=v_{sat}/\mu$
where $v_{sat}$ is the phonon-limited saturation velocity, and finally (iii)
the interband Zener regime characterized by a doping and bias-independent
differential conductivity $\sigma_{Z}$. Based on combined transport, flicker
and thermal noise characterization in a series of high-quality samples, we
show that : i) the standard analysis of flicker noise in graphene can be
consistently extended to the non-linear saturation regime and, ii) the
interband Zener contribution of gapless graphene can be accounted for by
introducing a semi-empirical formula in the spirit of the bremsstrahlung
interpretation, accounting for the enhanced electromagnetic coupling of
interband excitations and decoherence effects.

We use a series of ten hBN-encapsulated graphene transistors, of a large size
$(L,W)=4-25\;\mathrm{\mu m}$ and low edge-contact resistance $R_{c}$,
exhibiting varied but large mobility $\mu=2$-$15\;\mathrm{m^{2}/Vs}$,
saturation velocity $v_{sat}=0.2$-$0.8v_{F}$, and Zener conductivity
$\sigma_{Z}=0.1$-$1\;\mathrm{mS}$, as described in Supplementary Table-SI-1.
They map a broad range of hBN dielectric thickness
$t_{hBN}=50-160\;\mathrm{nm}$ and a variety of gating: (IR reflecting) Au
bottom gates, (IR absorbing) graphite bottom gates or SiO2-insulated Si back
gates. We also use two different hBN grades: the high-pressure high-
temperature (HPHT) from NIMS [Taniguchi2007, ], and the polymer-derived
ceramic (PDC) from Lyon [Maestre2022, ,Pierret2022MatRes, ] which behave
differently. All devices are embedded in coplanar waveguides for room-
temperature probe-station measurement of DC transport, sub-MHz flicker noise,
and microwave thermal noise (see Fig. 1-a). Thanks to their high mobility,
transistors sustain large saturation currents ($\sim 1\;\mathrm{mA/\mu m}$),
rejecting the $1/f$ noise corner frequency ($f_{c}=CI^{2}/4Gk_{B}T\propto I$)
in the low GHz range. As a matter of fact, the flicker noise tails are visible
in the GHz noise spectra and allow for a sanity check of the sub-MHz
measurements.

Current noise is analyzed with a modified Hooge formula $S_{I}=C(GV)^{2}/f$,
obtained by substituting the total DC current $I$ by the differential current
$GV$ to account for non-linear transport [Mavredakis2020acsaem, ]. Here $V$ is
the drain-source voltage obtained after subtraction of the contact-voltage
drop ($R_{c}I$), $G=(ne\mu(E)+\sigma_{Z})W/L$ is the differential conductance
with $\mu(E)=\mu(0)/(1+E/E_{sat})^{2}$ the optical phonon-limited mobility
with $E_{sat}$ and $v_{sat}=\mu(0)E_{sat}$ the saturation field and velocity.
Following Ref.[Yang2018nnano, ], we correct for drain-gating effect by biasing
transistors along constant density lines on applying a bias-dependent gate
voltage $V_{g}(V)=V_{g}(0)+\beta V$. This biasing procedure allows rejecting
drain pinch-off effect, such as investigated in Ref.[Schmitt2022, ], while
securing quasi-homogeneous doping and electric fields, up to small gradients
that become negligible at large doping. Low-frequency noise characterization
is enriched by Johnson-Nyquist thermal noise measurements, performed in the
microwave ($1$-$10\;\mathrm{GHz}$) band where flicker noise contribution is
negligible [Yang2018nnano, ]. This allows estimating the electronic
temperature $T_{N}$ and extract the thermal conductivity $dP/dT_{N}$ to the
hBN substrate which turns on in the Zener regime [Baudin2020adfm, ], where
$P=VI/LW$ is the areal Joule power.

Figure 1: DC transport and flicker noise in the high-mobility graphene
transistor GRS5. a) Experimental setup for low-frequency and GHz noise
measurements, with the graphene transistor embedded in a coplanar waveguide.
b): current-voltage curves for doping $n=[0-1.3]10^{12}\;\mathrm{cm^{-2}}$
corresponding to a Fermi energy $\epsilon_{F}=[0-130]\;\mathrm{meV}$. c):
Noise spectra in increasing bias at $n=0.6~{}10^{12}~{}\mathrm{cm^{-2}}$.
Flicker noise appears as plateaus of $fS_{I}(f)$. Dashed lines are linear fits
of $fS_{I}(f)$ of the data with slopes and extrapolates $(fS_{I})(0)$
measuring respectively the white noise and flicker noise. The fall down of
noise below $0.2\;\mathrm{MHz}$ is due to the cut-off frequency of the bias-
tee. d): bell-shape dependence of $fS_{I}(0)$ versus $V$ for
$n=[0-0.8]10^{12}\;\mathrm{cm^{-2}}$; inset shows a semi-log plot of
$A=fS_{I}/(GV)^{2}LW$ for same doping values indicating an exponential decay
of $A$ with bias.

Figure 1 summarizes the experimental technique (panel a) and procedure (panels
b-d). Figure 1-b shows typical current-voltage curves measured in the typical
transistor “GrS5”. They show the successive low-bias linear ohmic behavior,
the velocity-saturation regime for $V>V_{sat}\simeq 0.25\;\mathrm{V}$, and
finally the interband Zener regime. The saturation of intraband current can be
consistently attributed to the scattering by the optical phonons of the lower
Reststrahlen (RS) band of hBN ($\hbar\Omega_{I}=95-100\;\mathrm{meV}$)
[Yang2018nnano, ]. The Zener regime is accompanied by the onset of energy
relaxation by optical phonons of the upper RS band of hBN
($\hbar\Omega_{II}=170-200\;\mathrm{meV}$) [Yang2018nnano, ]. Its main
signature is electron cooling, that is observed in the noise temperature
(inset of Fig.3-c) by a decrease of the temperature-field slope. This change
of slope is typical of single-layer graphene; in bilayer graphene this
radiative cooling is even more drastic and eventually takes the form of a
temperature drop with bias [Baudin2020adfm, ].

The differential conductivity (not shown) is accurately fitted using the sum
of saturation ($\sigma_{sat}$) and Zener ($\sigma_{Z}$) contributions
[Meric2008, ; Yang2018nnano, ] :

$\sigma(E)=ne\frac{\mu(0)}{(1+E/E_{sat})^{2}}+\sigma_{Z}\qquad,$ (1)

where $\mu(0)$ is the extrapolated low-bias electronic mobility, $n$ is the
carrier density, and $E=V/L$. $\mu(0)$ agrees with standard mobility $\mu$
deduced from the low-bias $G(V_{g})$ dependence. $G=\sigma W/L$, and its
constituents $G_{sat}$ (or $\sigma_{sat}$) and $G_{Z}$ (or $\sigma_{Z}$), are
the associated differential conductance (conductivity). Eq.(1) is used to
extract DC transport parameters $\mu(0)$, $E_{sat}$ or $v_{sat}=\mu(0)E_{sat}$
that are listed in Table-SI1.

For each bias and doping, we measure the current noise in the
$f=[0.1-1]\;\mathrm{MHz}$ frequency band, as shown in Fig.1-c. Flicker $1/f$
noise corresponds to plateaus $fS_{I}(f)=a+bf$, where $a$ is the amplitude of
flicker noise and $b$ measures the instrumental background noise. The
$\alpha$-noise amplitude $a$ (denoted $fS_{I}(0)$ in Fig.1-d and thereafter)
has an asymmetric bell-shape bias dependence, and a strong doping dependence
$fS_{I}\propto n^{2}$. The noise amplitude $A=fS_{I}/(GV)^{2}(WL)$ is plotted
in the inset of Fig.1-d. Its low-bias extrapolate $A\sim 10^{-7}\;\mathrm{\mu
m^{2}}$ is consistent with diffusive graphene values [Balandin2013, ]. At high
bias we observe an amplitude suppression, with $A$ dropping below
$10^{-8}\;\mathrm{\mu m^{2}}$, following an exponential dependence $A\propto
e^{-E/E_{\Lambda}}$ (inset of Fig.1-d), with a characteristic electric field
$E_{\Lambda}\sim 0.1\;\mathrm{V/\mu m}$. The doping dependence $fS_{I}\propto
n^{2}$ is quite generic; it suggests a doping-independent velocity flicker
noise, $fS_{v}LW=Av^{2}$ (with $v=\mu(E)E$), as illustrated in Supplementary
Information Fig.SI-2.

Figure 2: DC transport and flicker noise in the high-mobility graphene
transistor “Lyon1”. a): current-voltage curves, for doping
$n=[0.6-1.4]10^{12}\;\mathrm{cm^{-2}}$, and noise thermometry (inset) show no
fingerprint of Zener tunneling and cooling. b): Quasi-scaling of the velocity
flicker noise $fS_{v}LW(E)$ as a function of electric field $E=V/L$. The
representative $n=0.7\;10^{12}\;\mathrm{cm^{-2}}$ (resp.
$n=1.1\;10^{12}\;\mathrm{cm^{-2}}$) data are well fitted by Eq.(2) with
$A=6.4\;10^{-7}\;\mathrm{\mu m^{2}}$ and $v_{sat}=0.82~{}v_{F}$ (resp.
$A=4.6\;10^{-7}\;\mathrm{\mu m^{2}}$ and $v_{sat}=0.65~{}v_{F}$).

We start our analysis by focusing on the velocity saturation regime, which is
exemplified in sample “Lyon1” where the Zener contribution to transport is
minimized to $\sigma_{Z}\simeq 0.1\;\mathrm{mS}$ (Table SI-1), as deduced from
the fitting of the differential conductance with Eq.(1) in Fig.2-a data.
Consistently, noise thermometry (inset of Fig.2-a) shows little fingerprint of
Zener cooling, as opposed to other samples where Zener cooling shows up as a
prominent breakout of the $T_{N}(E)$ dependence (see Supplementary
Informations Fig.SI-3). The origin of Zener transport suppression in sample
“Lyon1” is discussed below. Importantly, the flicker noise in Fig.2-b reduces
to its velocity-saturation contribution $fS_{sat}LW=A[\mu(E)E]^{2}$, which is
entirely determined by the differential mobility $\mu(E)$, according to

$fS_{sat}LW=A\times
v_{sat}^{2}\left[\frac{E/E_{sat}}{(1+E/E_{sat})^{2}}\right]^{2}\quad.$ (2)

Velocity-flicker in Eq.(2) has a quadratic low-bias onset
$fS_{sat}LW=A(\mu(0)E)^{2}$, consistent with standard Hooge law
$fS_{I}LW=A(I)^{2}$, a peak $fS_{sat}LW=Av_{sat}^{2}/16$ at $E=E_{sat}$, and a
bias tail $fS_{sat}LW=Av_{sat}^{2}(E_{sat}/E)^{2}$. Orange and brown dashed
lines in Fig.2-b are theoretical fits with Eq.(2) of the $n=0.7$ and
$n=1.1\;10^{12}\;\mathrm{cm^{-2}}$ data. The quasi-scaling observed here (and
in Figs.SI-2 for the other samples) relies on the weak doping-dependence of
$v_{sat}$ (and $E_{sat}=v_{sat}/\mu(0)$ as $\mu(0)$ is doping independent).
From the amplitude of the $n=7.10^{11}\;\mathrm{cm^{-2}}$ data we deduce
$A\simeq 6.4\;10^{-7}\;\mathrm{\mu m^{2}}$ which is indeed consistent with
diffusive-graphene literature data [Balandin2013, ]. Setting
$A=2\alpha_{0}/\pi n_{\Delta}$ according to the modified Hooge-Handel formula,
we infer $n_{\Delta}\simeq 0.7\;10^{12}\;\mathrm{cm^{-2}}$.

Figure 3: Velocity flicker noise $fS_{v}LW$ for $6$ typical devices at a
representative doping $n=6\;10^{11}\;\mathrm{cm^{-2}}$ (dots), and their fits
with Eqs.(2) and (3) for the two-fluid model. Green and red lines correspond
to the saturation and Zener contributions respectively, and the orange lines
correspond to their sum. Successive panels correspond to devices of increasing
mobility $\mu=4\rightarrow 15~{}\mathrm{m}^{2}/\mathrm{Vs}$ (see Table SI-1).
Values of the fitted parameters $A$ and $E_{\Lambda}$ are summarized in Table
SI-1. Insets show for each device the noise temperature $T_{N}$ as function of
bias and doping measured in the $f=[1-10]\;\mathrm{GHz}$ range, as detailed in
the Supplementary Information section (Fig.SI-3)

We now turn to the more general case where a significant Zener contribution is
observed at high field, in both the DC transport (Fig.SI-1) and in the noise
thermometry (Fig.SI-3 and insets of Figs.3), and quantified in Table-SI1.
Figs.3 compare velocity flicker noise at the representative doping
$n=6.10^{11}\;\mathrm{cm^{-2}}$ for $6$ devices of the series. As seen in the
figure, the bias dependencies generally deviate from the velocity-saturation
bell-shape of Fig.2-b (reproduced in Fig.3-a) and described by Eq.(2). To
explain this phenomenology we include in the velocity flicker a semi-empirical
Zener correction to the current flicker formula in the form

$fS_{Zener}LW=Be^{-E/E_{\Lambda}}\times
v_{Z}^{2}\left[\frac{E}{E_{\Lambda}}\right]^{2}\quad,$ (3)

where $B=\alpha_{g}/n_{\Delta}\sim 100\times A$, with
$\alpha_{g}=e^{2}/4\pi\epsilon_{0}\epsilon_{hBN}\hbar v_{F}\simeq 0.70$ the
fine-structure constant for hBN-encapsulated graphene ($\epsilon_{hBN}=3.4$
[Pierret2022MatRes, ]), $E_{\Lambda}$ the decoherence field discussed in
Fig.1-d (inset), and $v_{Z}=\frac{\sigma_{Z}}{en_{\Delta}}E_{\Lambda}\sim
0.01v_{F}$ a characteristic Zener velocity deduced from Zener conductivity
$\sigma_{Z}$, the characteristic doping $n_{\Delta}$, and $E_{\Lambda}$. The
Zener contribution also has a bell-shape bias dependence, which is however
different with a $E^{2}$ low bias dependence, a peak
$fS_{Zener}LW=4Bv_{Z}^{2}/e^{2}$ at $E=2E_{\Lambda}$ and an exponential tail.
Inspection of Eqs.(2) and (3) shows that the two contributions have similar
peak amplitudes for $v_{sat}/v_{Z}\sim
8e^{-1}\sqrt{\frac{\pi\alpha_{g}}{2\alpha_{0}}}\simeq 36$. Their relative
amplitudes depend on $n_{\Delta}$ and $E_{\Lambda}$, which enter in $v_{Z}$
($A$ and $B$ are simply proportional), and constitute the $2$ fitting
parameters used below. Their peak positions are shifted by a factor $2$ for
$E_{sat}\sim E_{\Lambda}$, in such a way that $S_{Zener}$ prominently affects
the high-field tail of the flicker noise, when $S_{sat}$ accounts for the
flicker noise onset.

Figs.3 also show fits of the velocity flicker data (orange lines) with a two-
fluid model where $S_{v}=S_{sat}+S_{Zener}$ from Eqs.(2) and (3), using
$n_{\Delta}$ and $E_{\Lambda}$ as the only free parameters. The velocity-
saturation (green lines) and Zener (red lines) components are displayed in
each panel. A good agreement with the two-fluid model is observed with,
however, some deviations at extremely high bias $E\gtrsim 0.5\;\mathrm{V/\mu
m}$ in few devices (see e.g. ”InOut1” in Fig.3-b), with an high-bias noise
tail at a non-zero value, possibly due to an additional spurious mechanism
that remains to be clarified. The Zener contribution is tiny in the ”Lyon1”
device (Fig.3-a), justifying the analysis of Fig.2-b in terms of the only
saturation contribution. A similar trend is observed in ”Lyon2” (Fig.3-d)
which is made with the same PDC-grade [Maestre2022, ] hBN material. Other
devices, fabricated with high-pressure high-temperature HPHT-grade hBN from
NIMS [Taniguchi2007, ], exhibit more prominent HPhP cooling in the Zener
regime (see Figs.3 insets and Figs.SI-3). In these devices, single-fluid fits
are unable to map the observed field dependencies, hence the justification of
our ’two-fluid’ (intraband saturation and interband Zener) analysis. The
saturation contribution (green lines in Figs.3) is mostly responsible for the
onset of flicker noise at low bias, with a slope increasing with mobility
$\mu(0)$. The Zener contribution (red lines in Figs.3) becomes significant at
larger bias, with a characteristic peak field at $2E_{\Lambda}>E_{sat}$,
explaining the slower decrease of flicker noise at high bias. The amplitudes
of the two contributions are comparable with, as a general trend, an intraband
saturation term becoming more prominent in high-mobility devices.

We now discuss the sample-dependent parameters $A$ (or $n_{\Delta}$) and
$E_{\Lambda}$, which are extracted from above fits and listed in Table-SI1. We
obtain $A\in[2-7]10^{-7}\;\mathrm{\mu m^{2}}$, corresponding to
$n_{\Delta}\in[0.7-2.3]10^{12}\;\mathrm{cm^{-2}}$. These values, obtained in a
broad $\mu(0)\in[2-15]\;\mathrm{m^{2}/Vs}$ mobility range, confirm the
literature data $A\in[10^{-7}-10^{-6}]\;\mathrm{\mu m^{2}}$ reported for
diffusive SiO2-supported graphene (dark-blue rectangle in Fig. 4-a),
suggesting that $A$ is essentially impurity-scattering independent. This
observation is further illustrated in the comparison between devices “Goyave2”
(Table SI-1) and “AuS3” (Fig.3-f), which yield the same $A$ in spite of a
factor $7$ in mobility. The similarity of flicker noise amplitude $A$ in the
impurity-dominated (linear low-mobility) and OP-phonon-dominated (non-linear
high-mobility) scattering regimes, reflects one more time its universality.
The characteristic field for the Zener correcting term, $E_{\Lambda}=75\pm
25~{}\mathrm{mV/\mu m}$, is quite similar for the $6$ representative devices.
It can be translated in terms of a Zener junction length $\Lambda=\sqrt{\hbar
v_{F}/eE_{\Lambda}}=100\pm 15~{}\mathrm{nm}$, which shows a reduced dispersion
when compared with the quite spread values of the Zener conductivity
$\sigma_{Z}\in[0.1-1]\;\mathrm{mS}$.

As seen in Fig.4-a all values of $A$ lie in the light-blue domain
$A\in[2-6]10^{-7}\;\mathrm{\mu m^{2}}$, where the two bounds correspond to
$n_{\Delta}\in[0.7-2.3]10^{12}\;\mathrm{cm^{-2}}$. We can convert this doping
range in terms of an energy range $\tilde{\Delta}=\hbar v_{F}\sqrt{\pi
n_{\Delta}}$ relying on the massless dispersion of graphene. Doing so we find
that $\tilde{\Delta}\in[\Delta_{1},\Delta_{2}]=[0.09-0.17]\;\mathrm{eV}$ is
actually bounded by the lower and upper Reststrahlen bands of hBN. These two
energies have been associated in Ref.[Yang2018nnano, ] to momentum relaxation
(out-of-plane optical phonons of lower RS band) and energy relaxation (in-
plane optical phonons of upper RS band) respectively. This finding suggests
that flicker noise in high-mobility graphene does map the ambivalent nature of
conductance as an admixture of momentum and energy relaxations parameters.

The correlation of flicker noise to energy relaxation is even more obvious in
the Zener contribution characterized by a large $B\approx 100\;A$ and the
characteristic velocity $v_{Z}=\sigma_{Z}E_{\Lambda}/en_{\Delta}$ in Eq.(3).
The semi-log plot $v_{Z}^{2}(dT_{N}/dP)$ in Fig.4-b indeed shows a strong
decrease of the Zener flicker amplitude with the thermal resistance to the hBN
substrate. This is exemplified by transistor ”Lyon1”, with a large thermal
resistance to the hBN substrate due to weak Zener cooling, and a tiny Zener
flicker amplitude. We attribute this reduced Zener cooling to the larger
structural disorder in the two ”Lyon”-grade hBN-based devices, where hBN is
fabricated through the PDC route. As a matter of fact, hBN structural quality
is essential to ensure free propagation of hyperbolic phonon polaritons in the
hBN bulk; their strong backscattering/damping in ”Lyon”-grade hBN devices is
likely to suppress the MIR electromagnetic coupling, implying lower Zener
conductivity and radiative cooling, resulting in a smaller Zener flicker
noise.

Figure 4: Flicker-noise correlations to mobility and Zener cooling. Panel a):
Noise amplitude $A$ as function of mobility for 10 devices of the series
(dots). The light-blue rectangle corresponds to our hBN-encapsulated graphene
devices. Panel b): Zener-flicker amplitude $v_{Z}^{2}$ as function of the
thermal resistance $dT_{N}/dP$ in the Zener interband regime extracted from
GHz noise thermometry measurements (insets of Fig.3) for the device series.

We now interpret the interband Zener term in Eq.(3), which is introduced as a
correction to the intraband saturation term Eq.(2). It is characterized by a
large coupling constant $\alpha_{g}\sim 100\times\alpha_{0}$, that is affected
by a field reduction factor $e^{-E/E_{\Lambda}}$ with a nearly sample-
independent characteristic field $E_{\Lambda}\simeq 75\;\mathrm{mV/\mu m}$.
This characteristic field can be associated to the energy scale
$\varepsilon_{\Lambda}=\sqrt{e\hbar v_{F}E_{\Lambda}}=7\;\mathrm{meV}$, and
length $\Lambda=\hbar v_{F}/\varepsilon_{\Lambda}=100\;\mathrm{nm}$ of Zener
tunneling. In this interpretation, $\varepsilon_{\Lambda}$ and $\Lambda$ are
the characteristic energy and length of coherent tunneling, which are limited
by inelastic HPhP scattering. They control the Hooge parameter
$\alpha_{g}e^{-E/E_{\Lambda}}$ in its quantum-coherent version by Handel. Note
in passing that the incoherent variant of quantum theory of flicker noise
involves a large reduction factor $v_{F}^{2}/c^{2}\ll 1$, where $c$ is the
speed of light, with respect to the coherent one $\alpha_{H}=2\alpha_{0}/\pi$
[Handel1996pss, ]. Further insight into this coherent mechanism requires a
comprehensive theoretical approach, including geometrical antenna effects in
free-light and HPhP emissions, which is still elusive and beyond the scope of
the present experimental report.

The conclusion of our experimental report of $\alpha$-noise in hBN-
encapsulated graphene is twofold. In a first part we recover and generalize
the Hooge formula for non-linear intraband transport by substituting the total
current $I$ with the differential current $GV$. We give an interpretation of
the constant flicker noise prefactor, previously revealed in diffusive
graphene, in terms of a scattering density $n_{\Delta}$, defined by an
effective energy $\tilde{\Delta}$ interpolating between the elastic-scattering
energy $\Delta_{1}$ and the inelastic-scattering energy $\Delta_{2}$. In a
second part, we extend the Hooge formula to include the interband transport
contribution, which is prominent in high-mobility graphene at high field. We
propose a compact formula, in the spirit of the Hooge-Handel interpretation of
flicker noise, with a strongly enhanced interband Hooge parameter reflecting
the large coupling of graphene electron-hole pairs to near-field hyperbolic
phonon polaritons of hBN [Baudin2020adfm, ]. The graphene case corresponds to
the collective strongly-inelastic variant, where IR bremsstrahlung constitutes
the main energy-relaxation mechanism in the Zener regime where $\alpha_{H}\sim
1$. The superposition of the two flicker terms in the two-fluid $\alpha$-noise
analysis accounts for the observed field dependencies in a series of graphene
transistors with varied IR environments. Experiment indicates that the
interband coupling constant is affected by an exponential suppression factor
accounting for decoherence effects in the HPhP coupling. Our observations that
i) intraband Hooge parameter approaches the Handel limit
$\alpha_{H,intra}=2\alpha_{0}/\pi$ , ii) interband parameter
$\alpha_{H,inter}\sim\alpha_{g}$ is strongly enhanced by a factor $100$, and
iii) that it is sensitive to decoherence effects via the
$\exp(-E/E_{\Lambda})$ factor, iv) and the correlation of flicker noise with
thermal conductance to the hBN substrate, constitute four qualitative
indications supporting the quantum-coherent bremsstrahlung interpretation of
flicker noise. In addition, this electromagnetic coupling interpretation
provides a natural explanation for the ubiquitous scatter in flicker data by
ascribing it to the variability in the IR environment. In a broader
perspective, our work promotes flicker noise, in complement to DC transport
and noise thermometry, as a powerful semi-quantitative tool to decipher the
physical mechanisms at stake in graphene transistors under extreme bias.

##  I Data availability

Data are available at the DOI : https://doi.org/10.5281/zenodo.7632159

##  II Author declaration

###  II.1 Conflict of interest

The authors have no conflict of interest to disclose.

###  II.2 Author contributions

AS, EB and BP conceived the experiment. AS conducted device fabrication and
measurements, under the guidance of DM and MR in the early developments. TT,
KW, CM, CJ and VG have provided the hBN crystals. AS, EB and BP developed the
models and theoretical interpretations. AS, DM, CV, JMB, GF, CV, EB and BP
participated to the data analysis. AS wrote the manuscript with assistance of
EB and BP, and contributions from the coauthors.

###  II.3 Acknowledgments

Authors thank Gerbold Ménard for his critical reading of the manuscript. The
research leading to these results has received partial funding from the
European Union Horizon 2020 research and innovation program under grant
agreement No.881603 ”Graphene Core 3”, and from the French ANR-21-CE24-0025-01
”ELuSeM”.

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# Insights into star formation and dispersal from the synchronisation of
stellar clocks

Núria Miret-Roig1 João Alves1 David Barrado2 Andreas Burkert3,4,5 Sebastian
Ratzenböck1,6 & Ralf Konietzka4,7,8

###### Abstract

Age is one of the most fundamental parameters of stars, yet it is one of the
hardest to determine as it requires modelling various aspects of stellar
formation and evolution. When we compare the ages derived from isochronal and
dynamical traceback methods for six young stellar associations, we find a
systematic discrepancy. Specifically, dynamical traceback ages are
consistently younger by an average of $\langle\Delta_{\rm Age}\rangle=5.5\pm
1.1$ Myr. We rule out measurement errors as the cause of the age mismatch and
propose that $\Delta_{\rm Age}$ indicates the time a young star remains bound
to its parental cloud before moving away from its siblings. In this framework,
the dynamical traceback “clock” starts when a stellar cluster or association
begins to expand after expelling most of the gas, while the isochronal “clock”
starts earlier when most stars form. The difference between these two age-
dating techniques is a powerful tool to constraint evolutionary models, as
isochronal ages cannot be younger than dynamical traceback ages. Measuring the
$\Delta_{\rm Age}$ accurately and understanding its variations across
different environments will provide further information on the impact of local
conditions and stellar feedback on the formation and dispersal of stellar
clusters.

University of Vienna, Department of Astrophysics, Türkenschanzstraße 17, 1180
Wien, Austria; e-mail<EMAIL_ADDRESS>
Centro de Astrobiología (CSIC-INTA), Depto. de Astrofísica, ESAC Campus,
Camino Bajo del Castillo s/n, 28692, Villanueva de la Cañada, Madrid, Spain

Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Scheinerstr.
1, 81679 Munich, Germany

Max-Planck Institute for Extraterrestrial Physics, Giessenbacherstr. 1, 85748
Garching, Germany

Excellence Cluster ORIGINS, Boltzmannstrasse 2, 85748 Garching, Germany

University of Vienna, Research Network Data Science at Uni Vienna, Kolingasse
14-16, 1090 Wien, Austria

Ludwig-Maximilians-Universität München, Geschwister-Scholl Platz 1, 80539
Munich, Germany

Harvard University Department of Astronomy, Center for Astrophysics | Harvard & Smithsonian, 60 Garden St. Cambridge, MA 02138, USA

Ages are crucial to understanding most astrophysical processes. In particular,
the ages of young stars are fundamental for studying the formation and early
evolution of stars and planets [1, 2], establishing timescales for
protoplanetary disk dissipation [3], determining stellar masses from O-stars
to substellar objects (brown dwarf, bound planets and free-floating planets
[4]), and studying the recent star formation history of the solar
neighbourhood [5]. However, estimating precise stellar ages for individual
stars (and even clusters) is complicated since different methodologies work
best for different ages and masses, and the results are often strongly model-
dependent [6, 7, 8, 9]. Stellar clusters and associations containing co-eval
stars constitute excellent benchmarks for age-dating methods because they
represent snapshots in evolution for stellar ensembles across a large range of
masses (at a given metallicity). Absolute ages for young stellar associations
can be obtained from evolutionary models (e.g., isochrone fitting, lithium
depletion boundary, and asteroseismology) or the analysis of their kinematics
(e.g., dynamical tracebacks or assuming linear expansion).

Isochrone fitting is one of the most common techniques to determine stellar
ages. The excellent photometry and distances from Gaia [10] have provided
systematic isochronal ages for hundreds of previously known clusters and
associations and thousands of Galactic stellar ensembles that have been
recently identified [11, 12, 13, 14]. However, isochrones are based on stellar
evolutionary models, strongly dependent on the complex physical processes
included, and are particularly uncertain for young, pre-main sequence stars
and low-mass stars [15, 16, 17], where different families of models still have
discrepancies of about a factor of two. In addition, young stars are
photometrically variable (due to activity) and are affected by interstellar
extinction, all of which make isochronal age dating problematic [18, 19, 20].
An alternative, potentially model-independent technique known as the lithium
depletion boundary (LDB, for which there is a sharp gap in the luminosity
between fully convective Li-rich and Li-poor stars) can yield precise ages and
represent excellent benchmarks to test evolutionary models [21, 22, 23, 24,
25, 26, 27]. However, detecting LDBs requires many hours of spectroscopic
observing time on large telescopes and this technique is only effective for
clusters with ages $>20$Myr.

Dynamical traceback ages constitute another model-independent method that can
be used to complement isochronal ages. This technique assumes that an
expanding group of unbound stars formed when they occupied a minimum volume of
space [28, 29]. These ages denote the traceback time required for the system
to reach its minimum volume configuration. Precise 3D velocities for
individual stars are critical to tracing the trajectories of stars back in
time, and this has been a limiting factor for many years [30, 31, 32, 33, 34].
Thanks to Gaia astrometry and complementary spectroscopic surveys such as
APOGEE [35], we can now measure 3D velocities with precisions below 1 km/s.
This allows us to calculate stellar orbits a few tens of Myr back in time with
precisions of a few parsecs. The availability of high-precision kinematic and
positional measurements has recently revived interest in traceback analyses
[29, 36, 37, 38, 39, 40, 41, 42].

Historically, there have been significant age discrepancies, of more than 50%,
in the ages determined in young stellar clusters and associations [7, 8, 9].
This was due to a combination of contaminated samples and significant
uncertainties in the observables and the models. Since the beginning of the
Gaia era, we have censuses of co-eval stars with an unprecedented degree of
completeness and low contamination. Additionally, Gaia and complementary
spectroscopic surveys have significantly reduced the uncertainties in the 6D
phase space parameters (positions and velocities) of many stars producing
dynamical traceback ages more accurately than in the past. These recent
dynamical traceback ages have narrowed the gap between ages obtained from
evolutionary models and dynamical tracebacks [29, 40], enabling a detailed
comparison of these two independent methods for the first time.

In Figure 1, we compare ages from evolutionary models (isochrone fitting and
lithium) and dynamical traceback ages for six benchmark associations where
homogeneous and accurate dynamical traceback ages are available (see Methods
section). We find a mean systematic offset, $\langle\Delta_{\rm Age}\rangle$ =
$\langle$evolutionary models – dynamical traceback age$\rangle$, of $5.5~{}\pm
1.1$ Myr and of $5.7~{}\pm 1.1$ Myr when excluding $\rho$ Oph, the youngest
group for which we did not find signs of expansion yet. This offset is evident
despite the still large uncertainties on the age determinations with different
techniques and samples. The correction of errors and biases on the current age
determinations could reduce the $\Delta_{\rm Age}$ we measure, but we argue
that they can not completely account for it.

Despite the improvements in observations and models, significant uncertainties
still affect age determinations. Since the methods to determine stellar ages
are complicated functions of the observables, it is challenging to propagate
them to the final age uncertainty. Estimating and including the systematic
uncertainties of evolutionary models in the final age determination is
difficult, if not impossible. We included studies using different techniques
(isochrone fitting and lithium) to illustrate the variability of ages within a
single region, denoting the uncertainties of ages from evolutionary models. We
estimated the uncertainties in the dynamical traceback ages by simulating a
mock association and convolving it with realistic observational errors (see
the Methods section). Uncertainties introduced by methodological choices, such
as the Galactic potential and the associations’ size estimator, are smaller
than the uncertainties in the 3D velocities and the sample selection [29, 41].
The errors on the dynamical traceback ages are $\lesssim 1-2$ Myr for
associations younger than 40 Myr. The $\Delta_{\textup{Age}}$ we observe in
Figure 1 (and report in Table 1) is larger than the scatter of ages from
different evolutionary models and larger than the errors from dynamical
traceback ages, indicating that uncertainties alone cannot explain this
discrepancy. This is especially true for the youngest groups ($\rho$ Oph,
$\nu$ Sco, $\beta$ Sco and $\delta$ Sco).

To investigate the origin of the $\Delta_{\textup{Age}}$, it is necessary to
precisely define the birth time of the cluster (the time the clock starts) for
evolutionary models and dynamical traceback ages. The initial time for
evolutionary models is difficult to establish since the early stages of core
formation and collapse are very quick ($\lesssim 10^{5}$ yr) and hard to
observe (due to large amounts of extinction). Some authors have suggested that
stars are born when most of the material in the envelope has collapsed onto
the disk and the central protostar becomes observable at infrared wavelengths
[43, 44], while others place the time zero a bit earlier, at the moment when
the core becomes optically thick [45]. In any case, the differences between
these two stages should be short, given the rapidness of these initial stages.

Dynamical traceback ages measure the time since a group of stars was most
concentrated. A fundamental assumption of the current traceback methods is
that stars move only affected by the gravitational pull of the Galactic
potential. This is a valid assumption in many cases, given the low densities
of these young associations, which are currently unbound. However, it might
not have always been the case if, at birth time, there was enough gas to bind
the association members together. A second important assumption of this
strategy is that all stars were formed nearly simultaneously ($<1$ Myr). This
assumption is supported by observations of star-forming regions [46, 47, 48]
and the colour-magnitude diagrams of the samples considered in this study.
However, small gradients (of up to 6 Myr) cannot yet be ruled out in star-
forming regions spread out in large areas [49].

Observations of unbound and expanding young clusters and associations [50]
prove that gas removal must happen at early stages. However, this process
might not be instantaneous and it has been proposed as a reason to explain the
offset between isochrones and dynamical traceback ages [37, 36, 39]. Young
stars only start moving in free trajectories in the Galactic potential after
dissipating a significant fraction of the parent gas. In Figure 2, we
illustrate a timeline with our interpretation of the $\Delta_{\textup{Age}}$
measured in this study. Extragalactic observations have recently found similar
constraints for the gas embedded phase (1–5 Myr) on larger scales (of about
100 pc) and with different strategies [51, 52, 53]. In addition, recent star
formation simulations have found that the parent gas of clusters and OB
associations dissipates in 2–7 Myr [54, 55, 56].

We have investigated the dependence of the $\Delta_{\textup{Age}}$ as a
function of the number of members in the association (see Figure 3), which is
a relatively easy metric to obtain. The decreasing $\Delta_{\textup{Age}}$
with increasing association mass suggests that regions with numerous feedback
events dissipate the gas more quickly. This trend is favoured by a Bayesian
linear fit that considers uncertainties in the $\Delta_{\textup{Age}}$ (see
Methods section). This relation is consistent with extragalactic observations
[57] and was predicted by hydrodynamical simulations where the gas is expelled
by photoionisation radiation and stellar winds [58]. Other studies have found
the timescale for gas dissipation to depend on the initial mass, compactness
of the cluster, presence of OB stars and supernovae, core radius, star
formation efficiency, or the velocity dispersion [59, 60, 61, 62].
Investigating the dependence of the $\Delta_{\textup{Age}}$ as a function of
these parameters requires precise parameter estimates, which are currently
unavailable and are left for future work.

The results of this work suggest that the $\Delta_{\textup{Age}}$ constitutes
an observational measurement of the duration of the embedded phase and the
timescale of gas dissipation. Interestingly, the mean $\langle\Delta_{\rm
Age}\rangle=5.5\pm 1.1.$ Myr is similar to the crossing time of the typical
initial conditions of star formation. For instance, when considering an
initial dispersion in positions ($\sigma_{\textup{pos}}$) of 10 pc and an
initial dispersion in velocities ($\sigma_{\textup{vel}}$) of 2 km/s, which
aligns with our observations of young stellar associations studied here, the
crossing time is $t_{C}=\sigma_{\textup{pos}}/\sigma_{\textup{vel}}\sim 5$
Myr. This coincidence might indicate that the gas removal timescale is of the
order of the dynamical crossing time. In this context, we can speculate that
the $\Delta_{\textup{Age}}$ increase for less populated associations might be
due to their lower density, enlarging the dynamical crossing timescale. So,
lower-mass associations might form in less dense molecular environments with
larger crossing timescales.

More precise ages of stellar clusters and associations in different
environments are needed to confirm the connection between the
$\Delta_{\textup{Age}}$ and the timescale for gas dissipation. For this kind
of study, identifying distinct kinematically coherent coeval groups in large
star-forming regions is vital [48, 63, 64]. However, only in the solar
vicinity can we have the best precision in 6D phase space coordinates,
reducing the number of regions where this technique is currently applicable.
Large and precise ($<1$ km/s) 3D velocity catalogues are necessary to identify
robust censuses of stars (with high completeness and low contamination) and
compute precise stellar orbits. These observational requirements determine
which associations are better candidates for traceback ages depending on
parameters such as the distance, the peculiar motion of the region, and the
coverage of extensive spectroscopic surveys. We emphasise the importance of
considering all the assumptions and uncertainties related to age
determination, including the physics enclosed in models, the sample selection,
and the need for homogeneous and precise radial velocity samples. Extensive
spectroscopic surveys like APOGEE, WEAVE [65] and 4MOST [66] are essential to
complement the excellent astrometry of Gaia. Finally, the difference between
isochrone and dynamical ages naturally sets a lower limit for the ages that
evolutionary models provide: they cannot be younger than dynamical ages. This
can be an unexplored powerful instrument to test evolutionary models.

Table 1: Properties of the young stellar associations considered.

Association | $N$ | $d$ | DT age | Isoc. age | $\Delta_{\rm Age~{}All}$ | $\Delta_{\rm Age~{}Isoc}$ | $\Delta_{\rm Age~{}Lit}$
---|---|---|---|---|---|---|---
|  | (pc) | (Myr) | (Myr) | (Myr) | (Myr) |
$\rho$ Oph |  $415\pm 20$ [36] | 139 | 0.0$\pm$0.3 [36] | 3.8–5.7 | $4.4\pm 1.0$ | $4.3\pm 0.6$ |
$\nu$ Sco |  $143\pm 12$ [36] | 139 | 0.3$\pm$0.5 [36] | 5.5–7.2 | $6.1\pm 1.0$ | $5.2\pm 0.9$ |
$\beta$ Sco |  $182\pm 13$ [36] | 153 | 2.4$\pm$1.7 [36] | 7.6–13.2 | $6.7\pm 2.6$ | $5.6\pm 1.6$ |
$\delta$ Sco |  $425\pm 21$ [36] | 142 | 4.6$\pm$0.6 [36] | 7.2–10.2 | $4.5\pm 1.3$ | $2.6\pm 1.1$ |
$\beta$ Pic |  $236\pm 15$ [29] |  40 |  $18.5_{-2.4}^{+2.0}$ [29] | 20.2–24.0 | $4.6\pm 2.0$ | $1.7\pm 3.0$ | $5.3\pm 1.9$
Tuc-Hor |  $94\pm 10$ [40] |  47 |  $38.5_{-8.0}^{+1.6}$ [40] | 41.8–46.3 | $6.7\pm 3.8$ | $3.3\pm 5.6$ | $7.8\pm 6.6$
Mean |  |  |  |  | $5.5\pm 1.1$ | $3.8\pm 1.5$ | $6.6\pm 1.8$
Mean excl. $\rho$ Oph |  |  |  |  | $5.7\pm 1.1$ | $3.7\pm 1.7$ |

Notes. This table displays the name of the association (Col. 1), the number of
candidate members and errors, $\sqrt{N}$, (Col. 2), the distance (Col. 3), the
dynamical traceback age used for this study (Col. 4), the age range from
isochrone fitting (only values with an uncertainty $<5$ Myr are considered)
(Col. 5), the mean $\Delta_{\rm Age}$ and standard deviation using all precise
literature determinations (Col. 6), the $\Delta_{\rm Age}$ from the PARSEC
isochronal ages determined in this work with the same sample of stars used for
the traceback analysis (Col. 7), and the mean $\Delta_{\rm Age}$ from Lithium
ages (Col. 8). The two bottom rows contain the mean $\Delta_{\rm Age}$ and the
mean $\Delta_{\rm Age}$ excluding the youngest group $\rho$ Oph.

Figure 1: Systematic offset between ages from evolutionary models and
dynamical traceback ages. The two panels show the one-to-one relation (top)
and the absolute age difference defined as $\Delta_{\rm Age}$ = evolutionary
models – dynamical traceback age (bottom). The figure shows ages from
isochrone fitting (circles) and lithium (squares). Marker colours represent
different associations indicated in the legend. The position and error of the
data points represent the ages and uncertainties from the studies listed in
Supplementary Table 1. The zero offsets (thick line), the mean offset of
$5.5\pm 1.1$ Myr (thin line), and the standard deviation (grey shaded areas)
are indicated. The blue shaded area represents the mean and standard deviation
of the bias in dynamical traceback ages measured in mock simulations (see
Methods). We included a $2\cdot 10^{5}$ yr offset to all the dynamical ages on
the x-axis for visualisation purposes. Figure 2: Diagram representing the
different phases of the formation of stellar clusters and associations. Ages
from evolutionary models measure the time since protostars have accreted most
of their final mass and become pre-main sequence stars. Dynamical traceback
ages measure the time since the association becomes unbound and starts to
expand. The offset between these two techniques ($\Delta_{\rm Age}$) measures
the timescale of the embedded phase, during which stars are still bound to the
parent gas cloud.  Figure 3: $\Delta_{\textup{Age}}$ as a function of the
number of association members. The data points and errors are listed in Table
1 under the column $\Delta_{\textup{Age All}}$ and $N$. The black line shows
the best fit and the shaded area the $1\sigma$ uncertainties.

Sample selection

While fitting isochrones to Gaia absolute colour-magnitude diagrams is
possible for many stellar groups, dynamical traceback ages require expensive
spectroscopic observations and careful treatment of 3D velocities [29]. We
selected six young stellar associations for which there is a recent dynamical
traceback age based on Gaia astrometry plus precise ($\sim 0.5$ km/s) radial
velocities from ground-based observations, APOGEE and Gaia [29, 36, 40]. These
are benchmark associations, covering ages $<40$ Myr, with homogeneous radial
velocities, and for which the dynamical traceback ages were determined with
the same methodology. To minimise possible biases due to different samples of
stars, we obtained an isochrone fitting age [48] with the same stars used for
the traceback analysis and the PARSEC v1.2S models [67]. The results of these
fits are shown in Supplementary Figure 1. The small scatter in these CMDs can
be explained by various factors including differential extinction (especially
likely in the youngest groups, $\rho$ Oph and $\nu$ Sco), photometric
variability, multiplicity (almost all the stars are between the fitted
isochrone and the equal-mass binary sequence, 0.75 mag brighter than the fit)
or contamination from a nearby young association.

In Table 1, we summarise the properties of the six associations considered in
this study. We note that the number of members for the $\beta$ Pic and Tuc-Hor
associations is more uncertain due to the difficulty of identifying members in
low-density associations with strong projection effects due to their proximity
($\lesssim 50$ pc). In these two cases, we decided to take the number of
candidate members compiled from different studies in the literature, knowing
that there is a degree of contamination from field stars and young members of
other associations but also other members might remain unnoted. We took
$\sqrt{N}$ as the uncertainty on the number of association members. Extensive
surveys of precise radial velocities are crucial to revisit the censuses of
these associations.

Upper Scorpius and Ophiuchus. Upper Scorpius and Ophiuchus are among the
youngest ($<10$ Myr) and closest (140 pc) star-forming regions to the Sun,
part of the Scorpius-Centaurus complex. Despite its proximity, there has been
a wide debate on their census and age [68, 69, 70, 71, 72]. Recently, several
works have identified different sub-populations in this region [63, 37, 36,
73, 48, 74], showing a complex star formation picture and providing an
explanation for the previous age discrepancies. We selected four groups with
the most robust membership determination when comparing the results between
different studies. These are the $\rho$ Oph, $\nu$ Sco, $\beta$ Sco and
$\delta$ Sco, which have dynamical traceback ages of $\lesssim 5$ Myr [36] and
for which we measured isochrone ages of 4–8 Myr (see Table 1 and Supplementary
Table 1).

$\beta$ Pictoris moving group. The $\beta$ Pictoris moving group ($\beta$ Pic)
is one of the closest young local associations in the solar neighbourhood,
located at an average distance of only 40 pc from the Sun. It has an age of 20
Myr measured with isochrone fitting [75, 76, 77, 78, 79], dynamics [80, 29,
41], and lithium depletion boundary [81, 25, 76, 82, 27]. For this study, we
took the members from Miret-Roig et al. 2020 [29], which have a dynamical
traceback age of $18.5_{-2.4}^{+2.0}$ Myr and for which we measured an
isochrone age of $20.2\pm 2.2$ Myr.

Tucana-Horologium association. The Tucana-Horologium association (Tuc-Hor) is
located at around 47 pc, and has an age of 30–50 Myr from isochrone fitting
[83, 78], dynamics [38, 40], and lithium depletion boundary [81, 84, 27]. For
this study, we took the members from Galli et al. 2023 [40], which have a
dynamical traceback age of $38.5_{-8.0}^{+1.6}$ Myr and for which we measured
an isochrone age of $41.8\pm 3.4$ Myr.

Estimating the observational bias in dynamical traceback ages

We simulated a mock association of different ages [34] to estimate the
uncertainties on dynamical traceback ages and determine whether observational
errors could account for the $\Delta_{\rm Age}$. We used as a centroid the
present positions and velocities of $\beta$ Pic, and we traced this centroid
back in time using Galpy and the MWPotential14 Galactic potential [85]. We
generated 500 particles around the centroid with an initial dispersion in
positions $\sigma_{\textup{pos}}=10$ pc and an initial dispersion in
velocities $\sigma_{\textup{vel}}=2$ km/s. We integrated all the particles
forwards in time to the present configuration, where we convolved the true
positions and velocities with realistic observational uncertainties from Gaia
plus precise spectroscopic radial velocities, $\sigma_{\textup{err, pos}}=0.1$
pc, $\sigma_{\textup{err, vel}}=0.5$ km/s. Finally, we integrated the orbits
of all the particles back in time twice the age of the association and
determined the dynamical traceback age [34, 29]. We repeated this process 1
000 times and computed the mean and the standard deviation of the dynamical
traceback age. For associations younger than 40 Myr, the difference between
the true age in the simulation and the estimated dynamical traceback age is of
$\lesssim 1$ Myr (see the blue shaded area in Fig. 1). This is similar to the
observational biases in dynamical traceback ages recently measured in $\beta$
Pic [41].

Measuring $\Delta_{\text{Age}}$

The associations considered in this study are well-known, and many previous
works determined ages from evolutionary models using different methodologies,
models (mainly with the PARSEC isochrones, especially the recent works, but
not all the studies considered), and stellar censuses (see Supplementary Table
1). To test the robustness of our results, we took three reference samples to
compute the $\Delta_{\text{Age}}$.

  * •

All, $\Delta_{\rm Age~{}All}$. This sample includes all the previous age
determinations from evolutionary models with an uncertainty of less than 5
Myr. It is a heterogeneous sample regarding methodology, evolutionary models,
and association membership.

  * •

Isochronal ages (this work), $\Delta_{\rm Age~{}Isoc}$. This sample only
includes the ages from isochrone fitting, with the PARSEC v1.2S evolutionary
models determined in this work. These ages have the advantage of being
homogeneous in technique and evolutionary models, and they use the same sample
of stars used for the dynamical traceback ages. However, they do not account
for systematic biases among different evolutionary models. With non-standard
models (i.e., models that incorporate additional surface physics such as
magnetic fields and/or rotation), the differences between evolutionary models
could be larger [86, 87].

  * •

Lithium ages, $\Delta_{\rm Age~{}Lit}$. Using only the lithium ages, we
calculated the mean $\Delta_{\text{Age}}$. In principle, this sample is less
model-dependent, although significant age differences have been found
depending on the evolutionary models considered [27]. This sample is
homogeneous in methodology but heterogeneous in the evolutionary models used
and association membership.

We computed the mean $\Delta_{\rm Age}$ for each association and sample and
then averaged the values from different associations to obtain the final
$\langle\Delta_{\rm Age}\rangle$ for each of the three reference samples (see
Table 1). These three values are compatible within $1\sigma$ uncertainties,
however, some differences are noticeable. The homogenous comparison between
isochronal ages and dynamical traceback ages, $\langle\Delta_{\rm
Age~{}Isoc}\rangle$, gives the smallest value, although not consistent with a
null $\Delta_{\rm Age}$, supporting the conclusions of this study. The
comparison with Lithium ages, $\langle\Delta_{\rm Age~{}Lit}\rangle$, is the
largest, although only available for the two oldest associations in our
sample, $\beta$ Pic and Tuc-Hor. For further discussion in the text, we take
the $\langle\Delta_{\rm Age~{}All}\rangle$ as the reference, as it is more
representative of the biases of different age determinations.

$\rho$ Oph is the youngest association included in this study, for which we
could not determine a dynamical traceback age. The $\Delta_{\text{Age}}$
determined in this group may be a lower limit if the association has not
started to expand yet. Therefore, we also computed the
$\langle\Delta_{\text{Age}}\rangle$ excluding $\rho$ Oph (see Table 1). These
values are decimals of Myr older than the result we obtained including this
group, suggesting that $\rho$ Oph is very close to starting free expansion.
This is also observable in Figure 1, where the $\Delta_{\text{Age}}$ of $\rho$
Oph is within the scatter of the rest of the groups.

Modeling the relation between $\Delta_{\text{Age}}$ and the number of
association members

We employed Bayesian linear regression analysis to investigate the relation
between $\Delta_{\text{Age}}$ and the number of association members (see Fig.
3). This approach offers a robust framework by providing a distribution of
plausible fitting parameters rather than a single best-fitting line. To
quantify and test the correlation between these two quantities, we used a
linear model $y=mx+b$, where $y$ represents $\Delta_{\text{Age}}$ and $x$
denotes the number of members per association. The slope ($m$) and intercept
($b$) of the regression line represent the parameters of interest. We assumed
Gaussian deviations from the linear model with zero mean and a known
$\Delta_{\text{Age}}$ variance ($\sigma$). For a given association number of
members ($x_{i}$), $\Delta_{\text{Age}}$ value ($y_{i}$) and uncertainty
($\sigma_{i}$), listed in Table 1, slope ($m$), and intercept ($b$), the
density of observed age differences $p(y_{i}|x_{i},\sigma_{i},m,b)$ becomes:

$p(y_{i}\mid
x_{i},\sigma_{i},m,b)=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\text{exp}\left(-\frac{(y_{i}-mx_{i}-b)^{2}}{2\sigma_{i}^{2}}\right).$
(1)

We obtained the likelihood by factoring these conditional probabilities for
the $N=6$ data points:

$\ \mathcal{L}=\prod_{i=1}^{N}p(y_{i}\mid x_{i},\sigma_{i},m,b),$ (2)

and the posterior probability density function (PDF) by applying Bayes’
theorem:

$p(m,b\mid\\{y_{i}\\}_{i=1}^{N},I)\propto p(\\{y_{i}\\}_{i=1}^{N}\mid
m,b,I)\,p(m,b\mid I).$ (3)

The PDF $p(m,b,f\mid I)$ describes our prior knowledge of the regression
parameters $(m,b)$, while $I$ summarizes all the prior knowledge of the
$x_{i}$ and $\sigma_{i}$. We employed weakly informative priors with normal
densities for $m$ and $b$, with a standard deviation of $10$ Myr/member and 10
Myr, respectively. We set the mean density for the intercept ($b$) to $5.5$
Myr, which corresponds to the average $\Delta_{\text{Age}}$ across the data
set, and a zero-mean prior density for the slope parameter ($m$), assuming no
correlation between $x$ and $y$.

We used the python code PyMC [88], which implements the Markov Chain Monte
Carlo (MCMC) methodology, to sample the posterior PDF. For each parameter
$(m,b)$, we computed its maximum a posteriori (MAP) position, representing the
best-fitting line plotted in Figure 3. The grey shaded area shows the
$1\sigma$ confidence interval within which approximately $68\%$ lines are
expected to lie. The complete PDF distributions for the $(m,b)$ parameters are
displayed in Supplementary Figure 2.

We computed the marginal posterior PDF of the slope parameter
$p(m\mid\\{y_{i}\\}_{i=1}^{N},I)$ to determine the likelihood of a negative
correlation between the number of association members and
$\Delta_{\text{Age}}$. We find a small probability (approximately $5\%$) of
the slope being positive ($m>0$):

$\int_{0}^{\infty}\,dm\,p(m\mid\\{y_{i}\\}_{i=1}^{N},I)\approx 0.05.$ (4)

The Bayesian linear model substantiates a negative correlation between the
$\Delta_{\text{Age}}$ and the number of members in the association. Future
studies with larger samples are needed to re-evaluate this trend with higher
statistical power.

The data that support the findings of this study is available in the Tables in
the manuscript.

We acknowledge the constructive feedback of three anonymous referees that
helped to improve the clarity and quality of the manuscript. D. Barrado is
supported by Spanish MCIN/AEI/10.13039/501100011033 grant PID2019-107061GB-C61
and No. MDM-2017-0737. S. Ratzenböck acknowledges funding by the Austrian
Research Promotion Agency (FFG, https://www.ffg.at/) under project number
FO999892674.

N. Miret-Roig led the analysis and wrote the manuscript. S. Ratzenböck
computed the isochrone fitting ages reported in this study and did the
Bayesian modelling in Figure 3. All authors participated in the scientific
discussion.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to N.M.R
(nuria.miret.roig@univie.ac.at).

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See pages - of Supplementary_Information.pdf