# Lyapunov Exponents and Phase Transition of Hayward AdS Black Hole

Naba Jyoti Gogoi1<EMAIL_ADDRESS>Saumen Acharjee1
<EMAIL_ADDRESS>Prabwal Phukon1,2<EMAIL_ADDRESS>$1.$
Department of Physics, Dibrugarh University, Dibrugarh 786004, Assam, India
$2.$Theoretical Physics Division, Centre for atmospheric studies, Dibrugarh
University

###### Abstract

In this paper, we study the relationship between the phase transition and
Lyapunov exponents for 4D Hayward anti-de Sitter (AdS) black hole. We consider
the motion of massless and massive particles around an unstable circular orbit
of the Hayward AdS black hole in the equatorial plane and calculate the
corresponding Lyapunov exponents. The phase transition is found to be well
described by the multivaled Lyapunov exponents. It is also found that
different phases of Hayward AdS black hole coincide with different branches of
the Lyapunov exponents. We also study the discontinuous change in the Lyapunov
exponents and find that it can serve as an order parameter near the critical
point. The critical exponent of change in Lyapunov exponent near the critical
point is found to be $1/2$.

Black Hole Thermodynamics,Lyapunov exponent, Phase transition, Hayward black
holes

###### pacs:

04.30.Tv, 04.50.Kd

##  I Introduction

Black hole thermodynamics [1, 2, 3, 4, 5] has its roots in the similarities
between black hole mechanics and thermodynamics. Over the last few decades, it
has emerged as a vibrant and dynamic field of study, drawing considerable
attention from researchers. Over time, numerous remarkable investigations have
unveiled a plethora of fascinating and thought-provoking findings [6, 7, 8, 9,
10, 11, 12, 13, 14]. The introduction of the AdS/CFT correspondence [15, 16]
has motivated researchers for an extensive exploration into the thermodynamics
and phase structure for a number of AdS black holes [17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30]. Different approaches such as Ruppeiner
geometry [34, 35, 36, 37, 38, 39, 40], thermodynamic topology [41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64, 65, 66, 67] etc. have been use to study phase transition from different
perspectives.

Of particular interest, endeavors have been made to link the phase transitions
of black holes with observable phenomena. These efforts have explored
potential connections between black hole phase transitions and various
observational signatures, including characteristics such as quasinormal modes
[68, 69, 70, 71, 72], the circular orbit radius of test particles [73, 74,
75], and the radius of the black hole shadow [76, 77].

One fascinating phenomenon worth mentioning is the motion of particles around
black holes because they can provide some important information regarding the
background spacetimes. For instance, it has been found that the unstable
circular null geodesics can impact the optical appearance of a star
experiencing gravitational collapse, potentially elucidating the exponential
fade-out in luminosity observed during the collapse process [78]. Also, the
null geodesics are found to be useful in explaining the quasinormal modes
(QNMs) of a black hole [79, 80, 81]. Such motion of particles may cause a very
important phenomenon in non-linearly dynamic systems known as chaos and to
study a chaotic system, Lyapunov exponents can be used [82]. It provides a
straightforward means of characterizing the dynamics of a chaotic system by
examining its effective degrees of freedom. Extensive research has been
conducted on the chaotic motion of particles in black hole spacetime [83, 84,
85, 86, 87, 88, 89, 90, 91, 92, 93]. A universal upper bound for the Lyapunov
exponents in thermal quantum systems is found in [94]. Nevertheless, it is
found to be violated in some cases studied in [95, 96].

A recent conjecture proposed in [97] suggests a relationship between Lyapunov
exponents and the phase transition of black holes. The work shows that the
Lyapunov exponents become multivalued during phase transition and become
single valued when there is no phase transition. Furthermore, it is found that
the discontinuous change in the Lyapunov exponents can be treated as an order
parameter, yielding a critical exponent of $1/2$ near the critical point. This
conjecture has been further verified for different black holes [98, 99, 100,
101].

In this work, we extend the study of Lyapunov exponents to a regular black
hole named Hayward AdS black hole which was first proposed by Sean A. Hayward
[102]. This black hole differs notably from conventional black holes like the
Schwarzschild and Reissner-Nordström black holes, which typically exhibit
singularities at their centres. The thermodynamics and phase transition of
Hayward black hole in AdS spacetime have been extensively studied in a number
of remarkable works [103, 104, 105]. We study the Lyapunov exponent of
massless and massive particles in an unstable circular orbit in the equatorial
plane around the Hayward AdS black hole and study its relationship with the
phase transition. We also investigate the behavior of the Lyapunov exponents
near the critical point and calculate the critical exponent.

This paper is organized as follows: In section II, we review the
thermodynamics and phase structure of the Hayward AdS black hole. In section
III, the Lyapunov exponents for massless and massive particles are discussed
in their respective subsections. In this section, we also study the
relationship between Lyapunov exponents and the phase structure of Hayward AdS
black hole. Finally, we conclude our results in section IV.

##  II Thermodynamics and phase structure of Hayward AdS black hole

The static and spherically symmetric Hayward AdS black hole is represented by
the following line element [105]:

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin\theta^{2}d\phi^{2})\quad\text{with}\quad
A=Q_{m}\cos\theta d\phi,$ (1)

and

$f(r)=1+\frac{r^{2}}{l^{2}}-\frac{2Mr^{2}}{r^{3}+q^{3}},$ (2)

where $M$ is the mass of the black hole, $l$ is the AdS length and $q$ is an
integration constant which is related to the total magnetic charge $Q_{m}$ of
the black hole by

$Q_{m}=\frac{q^{2}}{\sqrt{2\alpha}},$ (3)

in which, $\alpha$ is a parameter associated with the non-linear
electromagnetic field. The mass of the black hole $M$ can be calculated by the
condition $f(r_{+})=0$, which yields

$M=\frac{\left(l^{2}+r_{+}^{2}\right)\left(q^{3}+r_{+}^{3}\right)}{2l^{2}r_{+}^{2}}.$
(4)

Here, $r_{+}$ is the horizon radius of the black hole. The Hawking temperature
$T$ and entropy $S$ are given by

$T=\frac{l^{2}\left(r_{+}^{3}-2q^{3}\right)+3r_{+}^{5}}{4\pi
l^{2}r_{+}\left(q^{3}+r_{+}^{3}\right)},$ (5)

and

$S=2\pi\left(\frac{r_{+}^{2}}{2}-\frac{q^{3}}{r_{+}}\right).$ (6)

The first law of thermodynamics has the following form

$dM=TdS+\Psi dQ_{m}+VdP+\Pi d\alpha,$ (7)

where $\Psi$ and $\Pi$ are conjugate parameters corresponding to $Q_{m}$ and
$\alpha$. $P$ is the pressure with its conjugate volume $V$. The Gibbs free
energy can be calculated from the definition as

$F=M-TS=\frac{l^{2}\left(8q^{3}r_{+}^{3}-2q^{6}+r_{+}^{6}\right)+10q^{3}r_{+}^{5}+2q^{6}r_{+}^{2}-r_{+}^{8}}{4l^{2}r_{+}^{2}\left(q^{3}+r_{+}^{3}\right)}.$
(8)

Now, we use dimensional analysis and scale the following quantities as

$\tilde{r}_{+}=r_{+}/l,\quad\tilde{q}=q/l,\quad\tilde{F}=F/l,\quad\tilde{T}=Tl,\quad\text{and}\quad\tilde{M}=M/l.$
(9)

The tilde symbol is used to denote dimensionless quantities.

The critical points can be calculated by using the condition

$\frac{\partial\tilde{T}}{\partial\tilde{r}_{+}}=\frac{\partial^{2}\tilde{T}}{\partial^{2}\tilde{r}_{+}}=0,$
(10)

where, we have used the Hawking temperature (5) along with the scaling (9).
There exists a single critical point for each corresponding quantity and their
numerical values are given by

$\tilde{r}_{+c}=0.435773,\quad\tilde{q}_{c}=0.142336,\quad\text{and}\quad\tilde{T}_{c}=0.264695$
(11)

Using the Hawking temperature expression (5) and the scaling (9), we plot
temperature as a function of horizon radius for different values of
$\tilde{q}$ as shown in Figure 1. From this figure, we find different black
hole solutions with different horizon radii $\tilde{r}_{+}$ below the critical
point $\tilde{q}_{c}$. Above $\tilde{q}_{c}$, there exists a single black hole
solution for a range of horizon radius and temperature.

Figure 1: Hawking temperature as a function of horizon radius for different
values of $\tilde{q}$ above (green and brown) and below (yellow and blue) the
critical point $\tilde{q}_{c}=0.142336$ (red).

Now, to study the phase transition, we use Gibbs free energy given by (8). We
express the horizon radius $\tilde{r}_{+}$ as a function of Hawking
temperature $\tilde{T}$ using (5) and find that $\tilde{r}_{+}(\tilde{T})$ is
multivalued. Then, we put $\tilde{r}_{+}(\tilde{T})$ in (8) and finally obtain
the rescaled free energy $\tilde{F}$ as a function of $\tilde{T}$ and
$\tilde{q}$. The Gibbs free energy thus obtained are shown in Figure 2 with
fixed $\tilde{q}$. When $\tilde{q}$ is smaller than the $\tilde{q}_{c}$ we
have three black hole solutions namely, small BH, intermediate BH and large
BH. These three black hole solutions can coexist for
$\tilde{T}_{b}<\tilde{T}<\tilde{T}_{a}$, where $\tilde{T}_{b}$ and
$\tilde{T}_{a}$ are the temperatures at the point $b$ and $a$ respectively.
The temperature at the point $p$ represents the phase transition point
$(\tilde{T}_{p}=0.282789)$. When $\tilde{q}$ is greater than $\tilde{q}_{c}$,
there will be no phase transition as we have only a single black hole
solution.

(a) For $\tilde{q}=0.1<\tilde{q}_{c}$

(b) For $\tilde{q}=0.25>\tilde{q}_{c}$

Figure 2: Gibbs free energy as a function of $\tilde{T}$ for fixed values of
$\tilde{q}$.

##  III Lyapunov exponent and phase transition

The concept of Lyapunov exponent is widely used in the field of dynamical
systems and chaos theory. It quantifies how quickly nearby trajectories in a
system either move apart (diverge) or come together (converge) over time. In
this section, we intend to study the Lyapunov exponents of massless and
massive particles in an unstable circular orbit on the equatorial plane around
the Hayward AdS black hole. While the computation of the Lyapunov exponent is
well known, we will provide a brief overview for the convenience of the
readers. To begin, we commence with the Lagrangian, focusing on the case where
$\theta=\pi/2$. This can be expressed as:

$\mathcal{L}=\frac{1}{2}\Bigl{\\{}-f(r)\Big{(}\frac{dt}{d\tau}\Big{)}^{2}+\frac{1}{f(r)}\Big{(}\frac{dr}{d\tau}\Big{)}^{2}+r^{2}\Big{(}\frac{d\phi}{d\tau}\Big{)}^{2}\Bigr{\\}},$
(12)

where $\tau$ is the proper time and $f(r)$ is given by (2). From the
Lagrangian the canonical momenta of the particle can be easily worked out as:

$p_{t}=\frac{\partial\mathcal{L}}{\partial\dot{t}}=-f(r)\dot{t}=-E,\quad
p_{r}=\frac{\partial\mathcal{L}}{\partial\dot{r}}=\frac{1}{f(r)}\dot{r},\quad
p_{\phi}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\dot{\phi}=L,$
(13)

where $E$ and $L$ are the energy and angular momentum of the particle. Also,
the dots represent the derivatives with respect to proper time $\tau$. From
(13) we can find that

$\dot{t}=\frac{E}{f(r)}\quad\text{and}\quad\dot{\phi}=\frac{L}{r^{2}}.$ (14)

Then, we calculate the Hamiltonian as

$\displaystyle 2\mathcal{H}$
$\displaystyle=2(p_{t}\dot{t}+p_{r}\dot{r}+p_{\phi}\dot{\phi}-\mathcal{L})$
(15)
$\displaystyle=-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}\dot{\phi}^{2}$
$\displaystyle=-\frac{E^{2}}{f(r)}+\frac{\dot{r}^{2}}{f(r)}+\frac{L^{2}}{r^{2}}=-\delta_{1},$

where we have used (14). For timelike geodesic $\delta_{1}=1$ and for null
geodesic $\delta_{1}=0$. Using the definition of effective potential for
radial motion, $V_{r}=-\dot{r}^{2}$ in (15), we find that

$V_{r}=f(r)\Big{[}\delta_{1}-\frac{E^{2}}{f(r)}+\frac{L^{2}}{r^{2}}\Bigr{]}$
(16)

Now, if we write the angular momentum $L$ in terms of the effective potential
$V_{r}$ (setting $E=0$) and plug it into (15) then, the Hamiltonian can be
expressed as

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{V_{r}-E^{2}}{2f(r)}+\frac{\dot{r}^{2}}{f(r)}-\frac{\delta_{1}}{2}$
(17)
$\displaystyle=\frac{V_{r}-E^{2}}{2f(r)}+\frac{f(r)p_{r}^{2}}{2}-\frac{\delta_{1}}{2}$

The equations of motion in proper time configuration can be derived from the
Hamiltonian as:

$\displaystyle\dot{r}=\frac{\partial\mathcal{H}}{\partial p_{r}}=f(r)p_{r},$
(18) $\displaystyle\dot{p_{r}}=-\frac{\partial\mathcal{H}}{\partial
r}=\frac{V_{r}-E^{2}}{2f(r)^{2}}f^{\prime}(r)-\frac{1}{2}p^{2}f^{\prime}(r)-\frac{V^{\prime}_{r}}{2f(r)},$

where the primes denote the derivatives with respect to $r$. Now, we can
linearize these equations of motion about the circular orbit $r_{0}$ and
calculate the linear stability matrix $K$ in terms of the coordinate time $t$
as:

$\begin{pmatrix}0&\frac{f(r_{0})}{\dot{t}}\\\
-\frac{V_{r}^{\prime\prime}(r_{0})}{2f(r_{0})\dot{t}}&0\end{pmatrix}$ (19)

The eigenvalue of (19) gives the Lyapunov exponent

$\lambda=\sqrt{-\frac{V^{\prime\prime}_{r}(r_{0})}{2\dot{t}^{2}}},$ (20)

where we have dropped $\pm$ for simplicity. Also, $\lambda$ is real when
$V^{\prime\prime}_{r}(r_{0})<0$.

###  III.1 Massless particle (null geodesic)

To calculate the Lyapunov exponent for massless particle we set
$\delta_{1}=0$. The condition for unstable geodesic is
$V_{r}(r_{0})=V^{\prime}_{r}(r_{0})=0$ and $V^{\prime\prime}_{r}(r_{0})<0$.
From (16) and using these conditions we can find that

$\frac{E}{L}=\frac{\sqrt{f(r_{0})}}{r_{0}}$ (21)

Plugging this in (14) we get

$\dot{t}=\frac{L}{r_{0}\sqrt{f(r_{0})}}$ (22)

Now, we find the radius of the unstable circular orbit by the condition
$V^{\prime}_{r}(r_{0})=0$ and $V^{\prime\prime}_{r}(r_{0})<0$. To work this
out, we calculate the first derivative of $V_{r}(r_{0})$ and equate it to zero
with $\delta_{1}=0$. i.e.,

$V^{\prime}_{r}(r_{0})=-\frac{2L^{2}\Bigl{\\{}r_{0}^{5}\left(r_{0}-3\tilde{M}\right)+2r_{0}^{3}\tilde{q}^{3}+\tilde{q}^{6}\Bigr{\\}}}{r_{0}^{3}\left(\tilde{q}^{3}+r_{0}^{3}\right){}^{2}}=0$
(23)

Then, we solve the obtained equation for $r_{0}$ and check for
$V^{\prime\prime}_{r}(r_{0})<0$. Finally, we use the expression for mass (4)
with scaling (9) to find $r_{0}$ as a function of $\tilde{q}$ and
$\tilde{r}_{+}$. We observe that $r_{0}$ is independent of the angular
momentum $L$. The explicit expression is not shown intentionally for
simplicity. Also, the second derivative of the effective potential is given as

$V^{\prime\prime}_{r}(r_{0})=\frac{6L^{2}\Bigl{\\{}r_{0}^{5}\tilde{q}^{3}\left(2\tilde{M}+3r_{0}\right)+r_{0}^{8}\left(r_{0}-4\tilde{M}\right)+3r_{0}^{3}\tilde{q}^{6}+\tilde{q}^{9}\Bigr{\\}}}{r_{0}^{4}\left(\tilde{q}^{3}+r_{0}^{3}\right){}^{3}}$
(24)

The Lyapunov exponent $\lambda$ is then calculated for $r_{0}$ using (22),
(24) in (20). We observe that $\lambda$ depends on $\tilde{q}$ and
$\tilde{r}_{+}$ and it is independent of angular momentum $L$. The plot of
Lyapunov exponent $\lambda$ for null geodesic as a function of $\tilde{q}$ and
$\tilde{r}_{+}$ is shown in Figure 3. In the figure, no black hole solution
exists in the black region. This can be understood if we observe the Hawking
temperature $\tilde{T}$ which is negative for
$\sqrt[3]{2}\tilde{q}<\sqrt[3]{3\tilde{r}^{5}+\tilde{r}^{3}}$.

Figure 3: Lyapunov exponent $\lambda$ as a function of $\tilde{q}$ and
$\tilde{r}_{+}$ for unstable circular null geodesics. Black region represents
non-physical region.

For a better understanding of the Lyapunov exponent for unstable circular null
geodesics, we plot $\lambda$ in a 2D plane for fixed values of $\tilde{q}$ in
Figure 4. The gray area in the figure represents a non physical region because
of the Hawking temperature being negative $(\tilde{T}<0)$. On the black curve,
the temperature is zero $(\tilde{T}=0)$. The figure also shows that for
smaller value of $\tilde{r}$, the Lyapunov exponent $\lambda$ increases as we
decrease the values of $\tilde{q}$. As we gradually increase $\tilde{r}_{+}$,
the Lyapunov exponent curves for different values of $\tilde{q}$, start to
coalesce. In fact, as $\tilde{r}_{+}$ or $\tilde{q}$ tends towards infinity,
$\lambda$ approaches to $1$.

Figure 4: 2D view of Lyapunov exponent curves for unstable circular null
geodesics with different values of $\tilde{q}$. Gray area represents the non-
physical region.

We can write the Lyapunov exponent $\lambda$ in terms of temperature
$\tilde{T}$ by expressing $\tilde{r}_{+}$ as a function of $\tilde{T}$ from
the Hawking temperature expression. As discussed earlier
$\tilde{r}_{+}(\tilde{T})$ is multivalued and hence we obtain multiple
functions of Lyapunov exponent which defines different phases of the Hayward
AdS black hole. We have shown $\lambda$ as a function of $\tilde{T}$ for fixed
values of $\tilde{q}$ in Figure 5. The left figure (5(a)) shows $\lambda$ for
$\tilde{q}<\tilde{q}_{c}=0.142336$. Here, we have three different regions of
Hayward AdS black hole which are small black hole (blue), intermediate black
hole (red) and large black hole (green). The point $\tilde{T}_{p}=0.282789$ is
the phase transition temperature. For $\tilde{T}_{b}<\tilde{T}<\tilde{T}_{a}$
the Lyapunov exponent $\lambda$ has three branches and all the three black
hole solutions (small BH, intermediate BH and large BH) coexist in this
region. As temperature is raised from $\tilde{T}_{b}$ to $\tilde{T}_{a}$,
$\lambda$ for small black hole and large black hole decreases slightly and
$\lambda$ for intermediate black hole increases from their respective
positions. Also, $\lambda\to 1$ when $\tilde{T}\to\infty$. For a $\tilde{q}$
value greater than $\tilde{q}_{c}$, say $\tilde{q}=0.25$, the Lyapunov
exponent $\lambda$ is single valued for any value of temperature $\tilde{T}$
as shown in 5(b). In this case, there exists a single black hole solution and
phase transition is not possible. From 5(b), we observe that the trend of
$\lambda$ initially exhibits a slight increase, followed by a decrease as we
gradually raise $\tilde{T}$. As $\tilde{T}$ approaches infinity, $\lambda$
tends toward $1$.

(a) For $\tilde{q}=0.1<\tilde{q}_{c}$

(b) For $\tilde{q}=0.25>\tilde{q}_{c}$

Figure 5: Lyapunov exponent as a function of temperature $\tilde{T}$ for
unstable circular null geodesics.

Now, we study the difference of Lyapunov exponents in the phase transition
point of Hayward AdS black hole. At the small- large phase transition point
$p$, the Lyapunov exponent for small and large black hole is respectively
denoted as $\lambda_{s}$ and $\lambda_{l}$. With different values of
$\tilde{q}$, the phase transition temperature $\tilde{T}_{p}$ changes and for
these values of $\tilde{T}_{p}$ we calculate the difference of Lyapunov
exponents $\Delta\lambda=\lambda_{s}-\lambda_{l}$. The phase transition
vanishes at the critical point $\tilde{q}=\tilde{q}_{c}$ as the two extreme
points of $\tilde{T}$ vs $\tilde{r}_{+}$ curve coincides. At this point
$\tilde{T}_{p}=\tilde{T}_{c}$ and $\lambda_{s}=\lambda_{l}=\lambda_{c}$ which
results $\Delta\lambda=0$. The critical value of Lyapunov exponent
$\lambda_{c}$ can be calculated by inserting the critical values given in (11)
and it is found as $\lambda_{c}=1.2118$. We represent the
$\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ curve in Figure
6.

Figure 6: $\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ curve
for null geodesics.

In this figure, we see that the change in Lyapunov exponent $\Delta\lambda$ is
non-zero at the phase transition. As the phase transition temperature
$\tilde{T}_{p}$ slowly moves towards the critical temperature $\tilde{T}_{c}$,
the difference in Lyapunov exponent $\Delta\lambda$ non-linearly decreases. At
the critical point $\tilde{T}_{p}/\tilde{T}_{c}=1$ and
$\Delta\lambda/\lambda_{c}=0$. Such behaviors of the parameter $\Delta\lambda$
indicate that $\Delta\lambda$ acts as an order parameter.

To study the critical behavior of $\Delta\lambda$ we calculate the critical
exponent, a numerical value that characterizes the behavior of a physical
system near its critical point. The relation of critical exponent $\delta$ and
$\Delta\lambda$ is defined as [98]:

$\Delta\lambda\equiv\lambda_{s}-\lambda_{l}\sim|\tilde{T}-\tilde{T}_{c}|^{\delta}.$
(25)

To calculate $\delta$ we follow the method provided in [106]. We rewrite the
horizon radius at phase transition point and the Hawking temperature as

$\tilde{r}_{p}=\tilde{r}_{c}(1+\Delta),$ (26)

and

$\tilde{T}(\tilde{r}_{+})=\tilde{T}_{c}(1+\epsilon),$ (27)

where $|\Delta|\ll 1$ and $|\epsilon|\ll 1$. The Lyapunov exponents can be
expanded using Taylor series about the critical point $\tilde{r}_{c}$ as

$\lambda=\lambda_{c}+\left[\frac{\partial\lambda}{\partial\tilde{r}_{+}}\right]_{c}d\tilde{r}_{+}+\mathcal{O}(\tilde{r}_{+}),$
(28)

where the subscript $``c"$ is used to represent values at the critical point.
Using (26) and (28) we can find

$\Delta\tilde{\lambda}=\frac{\Delta\lambda}{\lambda_{c}}=\frac{\lambda_{s}-\lambda_{l}}{\lambda_{c}}=\frac{\tilde{r}_{c}}{\lambda_{c}}\left[\frac{\partial\lambda}{\partial\tilde{r}_{+}}\right]_{c}(\Delta_{s}-\Delta_{l}),$
(29)

where the subscript $``s"$ and $``l"$ represents small and large black hole
branch. Here, we have also used
$\lambda_{s}(\tilde{r}_{c})-\lambda_{l}(\tilde{r}_{c})=0$. Similarly, we can
Taylor expand Hawking temperature about the critical point $\tilde{r}_{c}$ and
find

$\tilde{T}=\tilde{T}_{c}+\frac{\tilde{r}_{c}^{2}}{2}\left[\frac{\partial^{2}\tilde{T}}{\partial\tilde{r}_{+}^{2}}\right]_{c}\Delta^{2},$
(30)

where we have omitted the higher order terms. Finally, using (29) and (30) we
can obtain

$\Delta\tilde{\lambda}=k\sqrt{t-1},$ (31)

where $t=\frac{\tilde{T}}{\tilde{T}_{c}}$ and

$k=\frac{\sqrt{\tilde{T}_{c}}}{\lambda_{c}}\left[\frac{1}{2}\frac{\partial^{2}\tilde{T}}{\partial\tilde{r}_{+}^{2}}\right]^{-1/2}_{c}\left[\frac{\partial\Delta\lambda}{\partial\tilde{r}_{+}}\right]_{c}.$
(32)

Therefore, the critical exponent $\delta$ of $\Delta\lambda$ near the critical
point is $1/2$ which is same as that of the order parameter in VdW fluid. In
Figure 7 we are focusing on the parameter $\Delta\lambda$ near the critical
point $\tilde{T}_{c}$. The black dot represents the parameter $\Delta\lambda$
(scaled with $\lambda_{c}$) for massless particles. We have calculated the
value of $k$ numerically and subsequently, using (31) find that

$\Delta\tilde{\lambda}=\frac{\Delta\lambda}{\lambda_{c}}=1.77499\sqrt{t-1}=1.77499\sqrt{\tilde{T}_{p}/\tilde{T}_{c}-1},$
(33)

which is represented by the magenta curve in Figure 7 and this serves a good
fit for $\Delta\lambda/\lambda_{c}$ (black dots in the figure). This further
confirms that the critical exponent for $\Delta\lambda$ is $1/2$.

Figure 7: $\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ curve
for null geodesics near critical point. The black dots represents
$\Delta\lambda$ (scaled with $\lambda_{c}$) for different values of
$\tilde{q}$. The magenta curve is for
$\Delta\lambda/\lambda_{c}=1.77499\sqrt{\tilde{T}_{p}/\tilde{T}_{c}-1}$

###  III.2 Massive particle (timelike geodesic)

For timelike geodesics we chose $\delta_{1}=1$. The condition
$V(r_{0})=V^{\prime}(r_{0})=0$, provides us the following relations for energy
and angular momentum as

$E^{2}=\frac{2f\left(r_{0}\right){}^{2}}{2f\left(r_{0}\right)-r_{0}f^{\prime}\left(r_{0}\right)},$
(34)

and

$L^{2}=\frac{r_{0}^{3}f^{\prime}\left(r_{0}\right)}{2f\left(r_{0}\right)-r_{0}f^{\prime}\left(r_{0}\right)}.$
(35)

Therefore, from (14) we obtain

$\dot{t}=\frac{1}{\sqrt{f\left(r_{0}\right)-\frac{1}{2}r_{0}f^{\prime}\left(r_{0}\right)}}$
(36)

The radius $r_{0}$ of the unstable circular geodesic is calculated from
$V^{\prime}(r_{0})=0$ with $\delta_{1}=1$ and checked for
$V^{\prime\prime}(r_{0})<0$. Unlike the null geodesics case, here $r_{0}$
depends on the angular momentum $L$. We have not shown the explicit form of
$r_{0}$ for simplicity. Also, the second derivative of potential is

$V^{\prime\prime}_{r}(r_{0})=\frac{\begin{aligned}
6\tilde{L}^{2}\Bigl{\\{}\tilde{q}^{3}\tilde{r}^{5}\left(2\tilde{M}+3\tilde{r}\right)+\tilde{r}^{8}\left(\tilde{r}-4\tilde{M}\right)+3\tilde{q}^{6}\tilde{r}^{3}+\tilde{q}^{9}\Bigr{\\}}\\\
+2\tilde{r}^{4}\Bigl{\\{}\left(\tilde{q}^{3}+\tilde{r}^{3}\right)^{3}-2\tilde{M}\left(-7\tilde{q}^{3}\tilde{r}^{3}+\tilde{q}^{6}+\tilde{r}^{6}\right)\Bigr{\\}}\end{aligned}}{\tilde{r}^{4}\left(\tilde{q}^{3}+\tilde{r}^{3}\right)^{3}}.$
(37)

The effective potential $V_{r}$ for unstable timelike null geodesics can be
written as a function of $\tilde{r}$, $\tilde{r}_{+}$ and $\tilde{q}$ using
the expression of $f(r)$ and $\tilde{M}$ in (16). The $V_{r}-\tilde{r}$
relation is shown in Figure 8 for different values of $\tilde{r}_{+}$ with
$\tilde{q}=0.1$. Here, we have set $L=20l$ and $E=0$. In the figure, the black
dots represent the maximum of the effective potential for which
$V^{\prime\prime}_{r}<0$ corresponding to unstable equilibria. The minimum of
$V_{r}$ for which $V^{\prime\prime}_{r}>0$ corresponds to stable equilibria.
The figure also shows that the maximum of $V_{r}$ decreases with the increase
of $\tilde{r}_{+}$ and for $\tilde{r}_{+}=0.5$ there is no maximum. This
implies that the unstable timelike geodesic will disappear for large value of
$\tilde{r}_{+}$.

Figure 8: Effective potential $V_{r}$ as a function $\tilde{r}$ for
$\tilde{q}=0.1$. Red, green and blue curves are respectively for
$\tilde{r}_{+}=0.122$, $\tilde{r}_{+}=0.24$ and $\tilde{r}_{+}=0.5$.

Using (36) and (37) in (20) we can calculate the Lyapunov exponent $\lambda$
for unstable circular timelike geodesics. In this case, $\lambda$ depends on
$L$, $\tilde{q}$ and $\tilde{r}_{+}$. For simplicity we have avoided writing
the explicit form of $\lambda$. The three dimensional representation of the
Lyapunov exponent $\lambda$ for unstable circular timelike geodesic is shown
in Figure 9 where we have chosen $L=20l$. In the figure, the black area
represents non-physical region with negative temperature. The unstable region
exist for smaller values of $\tilde{r}_{+}$ and $\lambda$ in this region is
represented by the blue surface. In the white region, there is no unstable
timelike circular orbits and therefore, $\lambda$ vanishes in this region.

Figure 9: Lyapunov exponent $\lambda$ as a function of $\tilde{q}$ and
$\tilde{r}_{+}$ for unstable circular timelike geodesics. Black region
represents non-physical region.

The two dimensional view of the Lyapunov exponent for timelike geodesic is
shown in the Figure 10. Here, the non-physical region with negative
temperature is shaded as gray.

Figure 10: 2D view of Lyapunov exponent curves for unstable circular geodesics
with different values of $\tilde{q}$. Gray area represents the non-physical
region.

To study the relationship between phase transition and Lyapunov exponent, we
write $\lambda$ in terms of $\tilde{T}$ using the Hawking temperature. The
Lyapunov exponent $\lambda$ for $L=20l$ is shown in Figure 11. The left figure
(11(a)) shows $\lambda$ for $\tilde{q}=0.1$ which is below the critical value
$\tilde{q}_{c}=0.142336$. For this value of $\tilde{q}$, the Lyapunov exponent
$\lambda$ is multivalued and it has three branches. These branches corresponds
to three different phases or three black hole solutions of Hayward AdS black
hole which can coexist for $\tilde{T}_{b}<\tilde{T}<\tilde{T}_{t}$ . The phase
transition from small black hole to large black hole occurs at the temperature
$\tilde{T}=\tilde{T}_{p}=0.2871785$. For $\tilde{q}=0.25>\tilde{q}_{c}$ there
is no phase transition and the Lyapunov exponent $\lambda$ is found to be
single valued. Unlike the massless particles case, $\lambda$ for massive
particles vanishes and become zero at the temperature point
$\tilde{T}=\tilde{T}_{t}$.

(a) For $\tilde{q}=0.1<\tilde{q}_{c}$

(b) For $\tilde{q}=0.25>\tilde{q}_{c}$

Figure 11: Lyapunov exponent as a function of temperature $\tilde{T}$ for
unstable circular timelike geodesics.

Now, we calculate the change in the difference of Lyapunov exponent
$\Delta\lambda$ for different values of $\tilde{q}$. The plot of
$\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ is shown in
12(a). The figure shows that the parameter $\Delta\lambda$ is non-zero at the
phase transition and it non-linearly decrease as the phase transition
temperature $\tilde{T}_{p}$ slowly approaches to the critical temperature
$\tilde{T}_{c}$. At the critical point $\tilde{T}_{p}/\tilde{T}_{c}=1$ and
$\Delta\lambda/\lambda_{c}=0$. The critical behavior of $\Delta\lambda$ is
shown in 12(b). Near the critical point, $\Delta\lambda/\lambda_{c}$ (black
dots in the figure) is well represented by

$\frac{\Delta\lambda}{\lambda_{c}}=1.89539\sqrt{\tilde{T}_{p}/\tilde{T}_{c}-1},$
(38)

which confirms that the critical exponent of $\Delta\lambda$ is $1/2$.

(a) Large range view of $\Delta\lambda/\lambda_{c}$ vs
$\tilde{T}_{p}/\tilde{T}_{c}$ curve.

(b) $\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ curve near
the critical point.

Figure 12: $\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ curve
for timelike geodesics. The black dots represents $\Delta\lambda$ (scaled with
$\lambda_{c}$) for different values of $\tilde{q}$. The magenta curve is for
$\Delta\lambda/\lambda_{c}=1.89539\sqrt{\tilde{T}_{p}/\tilde{T}_{c}-1}$.

##  IV Conclusion

In this paper, we have studied the link between Lyapunov exponents and the
phase structure of 4D Hayward AdS black hole. We have calculated the Lyapunov
exponents for massless and massive particles in an unstable circular orbit of
the black hole in the equatorial plane and study its behavior.

For the massless particles, below the critical value of $\tilde{q}_{c}$, the
Lyapunov exponent $\lambda$ has three different branches each of them
corresponding to three different phases (SBH, IBH and LBH) of the Hayward AdS
black hole. Above the critical value of $\tilde{q}_{c}$, the Lyapunov exponent
$\lambda$ has a single branch. In this case, there is no phase transition.
Which implies that $\lambda$ is multivalued when there is a phase transition.
Also, $\lambda$ tends to $1$ when temperature $\tilde{T}$ tends to infinity
implying that there is no terminating temperature for $\lambda$ in case of
unstable circular null geodesics in Hayward AdS black hole.

The motion of massive particles around the Hayward AdS black hole in unstable
circular orbit is defined by the timelike geodesics. We observe that below the
critical value of $\tilde{q}_{c}$, the Lyapunov exponent $\lambda$ is
multivalued and its three different branches corresponds to three different
phases of Hayward AdS black hole. Above the critical value of $\tilde{q}_{c}$
there is no phase transition and the Lyapunov exponent is found to be single
valued. In the massive particle case, there is terminating temperature
$\tilde{T}_{t}$ for Lyapunov exponent $\lambda$ at which it tends to zero.

In both the massless and massive particle cases we have studied the
discontinuous change in the Lyapunov exponent $\lambda$. We have plotted the
$\Delta\lambda/\lambda_{c}$ vs $\tilde{T}_{p}/\tilde{T}_{c}$ curve and observe
that when the Hayward AdS black hole undergoes small-large black hole phase
transition, $\lambda$ moves from $\lambda_{s}$ to $\lambda_{l}$ with a non-
zero change in the Lyapunov exponent $\Delta\lambda$. At the critical point
$\Delta\lambda$ vanishes. The parameter $\Delta\lambda$ for Hayward AdS black
hole, acts as an order parameter and the critical exponent near the critical
point is $1/2$.

It will be interesting to study how this conjecture holds in different
ensembles of black holes in different spacetime. We plan to do so in our
future work.

*

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#

CMOS Monolithic Pixel Sensors based on the Column-Drain Architecture for the
HL-LHC Upgrade

K. Moustakas<EMAIL_ADDRESS>M.Barbero  I. Berdalovic  C. Bespin
P.Breugnon  I. Caicedo  R. Cardella  Y. Degerli  N. Egidos Plaja  S. Godiot
F.Guilloux  T. Hemperek  T. Hirono  H. Krüger  T. Kugathasan  C. A. Marin
Tobon  P. Pangaud  H. Pernegger  P. Riedler  P. Rymaszewski  E. J. Schioppa
W. Snoeys  M. Vandenbroucke  T. Wang  N. Wermes  Physikalisches Institut,
Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany  CERN
Experimental Physics Department, CH-121 Geneve 23, Switzerland  Centre de
physique des particules de Marseille, 163 Avenue de Luminy, Marseille, France
IRFU, CEA-Saclay, Gif-sur-Yvette Cedex, 91191 France

###### Abstract

Depleted Monolithic Active Pixel Sensors (DMAPS) constitute a promising low
cost alternative for the outer layers of the ATLAS experiment Inner Tracker
(ITk). Realizations in modern, high resistivity CMOS technologies enhance
their radiation tolerance by achieving substantial depletion of the sensing
volume. Two DMAPS prototypes that use the same ”column-drain” readout
architecture and are based on different sensor implementation concepts named
LF-Monopix and TJ-Monopix have been developed for the High Luminosity upgrade
of the Large Hardon Collider (HL-LHC) .
LF-Monopix was fabricated in the LFoundry 150 nm technology and features pixel
size of $50x250~{}\mu m^{2}$ and large collection electrode opted for high
radiation tolerance. Detection efficiency up to 99% has been measured after
irradiation to $1\cdot 10^{15}~{}n_{eq}/cm^{2}$. TJ-Monopix is a large scale
$(1x2~{}cm^{2})$ prototype featuring pixels of $36x40~{}\mu m^{2}$ size. It
was fabricated in a novel TowerJazz 180 nm modified process that enables full
depletion of the sensitive layer, while employing a small collection electrode
that is less sensitive to crosstalk. The resulting small sensor capacitance
($<=3~{}fF$) is exploited by a compact, low power front end optimized to meet
the 25ns timing requirement. Measurement results demonstrate the sensor
performance in terms of Equivalent Noise Charge (ENC) $\approx 11e^{-}$,
threshold $\approx 300~{}e^{-}$, threshold dispersion $\approx 30~{}e^{-}$ and
total power consumption lower than $120~{}mW/cm^{2}$.

###### keywords:

Pixel detectors , DMAPS , Front end electronics

###### PACS:

29.40.Wk , 29.40.Gx

##  1 Introduction

Monolithic Active Pixel Sensors (MAPS) constitute an attractive alternative in
high energy physics experiments as the building blocks of vertex detectors in
high precision tracking applications. Their prominent advantage stems from the
fact that the manufactured devices contain both the sensor and the front end
electronics in the same silicon crystal and are ready to be used without the
need for the expensive and labor-intensive process of fine pitch bump-bonding.
Monolithic pixel sensors have already been successfully used in experiments
with low radiation environments [1, 2], but their radiation tolerance was
limited. Advancements in CMOS imaging technologies that enable the use of high
resistivity substrate and high voltage biasing can be exploited to achieve
full depletion of the sensitive volume and generation of high electric fields
to ensure fast charge collection by drift. During the phase-II of the HL-LHC
upgrade, the ATLAS ITk will be improved [3] to cope with the unprecedented
levels of radiation and ten times higher hit rate. A dedicated CMOS
collaboration has been established to develop and characterize fully Depleted
MAPS (DMAPS), for the outer layers of the ATLAS ITk. These devices are
required to tolerate particle fluence up to $1.5\cdot 10^{15}~{}n_{eq}/cm^{2}$
and a Total Ionizing Dose (TID) up to 50 Mrad. Moreover they must be able to
comply with the 25ns timing requirement of the ATLAS experiment. DMAPS
prototypes manufactured in different technologies have been reported with
encouraging results regarding the sensor radiation tolerance [4, 5, 6, 7].
Additionally, multiple nested wells offered by modern CMOS processes allow for
complex readout circuitry to be implemented inside the pixel, enabling the use
of fast readout architectures.

Two large scale DMAPS prototypes have been developed for the ATLAS ITk based
on a common ”column-drain” readout architecture derived from the FE-I3 front
end chip [8]. The simple in-pixel logic does not severely constrain the pixel
dimensions and the hit rate capabilities of this approach are well established
and exceed the requirements of the ITk outer layers. A different sensor
implementation concept was pursued in each chip. The first prototype, called
LF-Monopix, is based on a large collection electrode and is manufactured in
the LFoundry 150 nm HV-CMOS process. The second prototype, called TJ-Monopix,
employs a small collection electrode and is fabricated in the TowerJazz 180 nm
imaging process using a novel modification to enhance radiation tolerance [9].
The radiation hardness of LF-Monopix and the implemented large collection
electrode sensor has been proven by beam test results. Full functionality of
TJ-Monopix after irradiation is demonstrated for the first time and its analog
performance is evaluated by laboratory and radioactive source measurement
results reported in this work.

##  2 Sensor Implementation

A crucial factor in the design of monolithic pixel sensors, that affects
radiation tolerance and performance of the detector is the geometry of the
collection electrode. In Figure 1 the cross section of the LF-Monopix pixel
sensor is depicted. The collection electrode is formed by a large very deep
n-well that includes all the in-pixel electronics. A high resistivity ($>$2
k$\Omega\cdot$cm) p-type substrate constitutes the sensitive volume that can
be biased to high voltages. Its depth can vary from $750~{}\mu m$ to
$100~{}\mu m$ after backside processing. The generated strong and uniform
electric field ensures high radiation tolerance to Non Ionizing Energy Loss
(NIEL) damage and high charge collection efficiency [4, 6]. The disadvantage
of this implementation is the large sensor capacitance ($\approx
300-400~{}fF$) that degrades the analog performance in terms of noise and
signal rise time, and has to be compensated with increased power consumption.
Additionally, significant design efforts are needed to reduce the crosstalk
noise from the in-pixel digital activity that is coupled to the input node.

In contrast, a different concept is pursued in TJ-Monopix. A small n-well is
used as the collection electrode and the in-pixel electronics are implemented
around it. The most important benefit of this geometry is the very small
sensor capacitance ($\approx 3fF$) that allows for very low noise and fast
signal timing while keeping the power consumption minimal. Furthermore,
crosstalk noise coupling is substantially reduced. The PMOS transistor n-wells
are shielded by deep p-well implants. A 25$~{}\mu$m thick, high resistivity
($>$1 k$\Omega\cdot$cm) p-type epitaxial layer that is grown on a low
resistivity substrate is used as the sensitive volume. To fully deplete the
epitaxial layer and enhance radiation tolerance, a process modification has
been developed [9]. As depicted in Figure 1, a planar n- layer that realizes
two p-n junctions is implanted over the full pixel area. As a consequence, the
depletion boundary is extended laterally, to the otherwise undepleted part of
the sensitive layer, even for small reverse bias voltages. Small pixel size is
essential to reduce the distance between collection electrodes and ensure that
the electric field is not degraded in the area between pixels. Measurement
results from a test chip manufactured in the modified process to assess the
sensor performance are very promising [5]. Charge collection efficiency
remains uniform and high ($>$95%) and the signal timing is still fast after
irradiation to $1\cdot 10^{15}~{}n_{eq}/cm^{2}$.

Figure 1: 1 LF-Monopix and 1 TJ-Monopix sensor design cross section

##  3 Chip Design and Architecture

###  3.1 Chip Architecture

Figure 2: Chip architecture of LF-Monopix and TJ-Monopix

Both prototypes follow the same basic architecture scheme that is depicted in
Figure 2. Their differences in terms of readout logic are limited to
implementation details and layout organization. A 40MHz Bunch Crossing ID
(BCID) time stamp, generated by a gray counter is distributed over the whole
matrix. After a hit pulse is produced by a pixel’s analog front end, the
Leading Edge (LE) and Trailing Edge (TE) information are stored in local in-
pixel RAM memories. Arbitration over the common data bus is based on a token
that is propagated with priority through the column from top to bottom and
through different columns from left to right. A busy token flag is sent to the
external Data Acquisition (DAQ) controller that initiates and controls the
readout sequence. A local read signal is produced in the pixel that has been
hit and has the highest priority to transmit the hit data. The data received
from the RAM sense amplifiers at the end of each column are arbitrated by the
end of column logic (EoC), and sent to the DAQ system using a 160 Mbps serial
link. A trigger memory has not been implemented in the current prototypes and
hit data is sent out immediately.

###  3.2 LF-Monopix Chip Design

The LF-Monopix chip [6, 7, 10] is the successor to CCPD_LF and LF-CPIX [4, 6]
development line in 150 nm LFoundry technology and the first to incorporate a
full standalone fast readout. It consists of a 129x36 matrix of pixels with
$50x250~{}\mu m^{2}$ size, while its total size is $10x9.5~{}mm^{2}$. Special
guard ring structures enable high voltage biasing up to 280 V before
breakdown. An AC coupled charge sensitive amplifier (CSA) is used to amplify
the input signal. The discriminator can be tuned by an in-pixel 4-bit DAC to
reduce threshold dispersion. Each pixel consumes $\approx 36~{}mW$ of power
yielding a total analog power consumption equal to $300~{}mW/cm^{2}$. The
matrix is not uniform and consists of nine pixel flavors with different
discriminator and CSA implementations. To reduce crosstalk noise, the peak
transient currents of the readout logic and therefore the injected current to
the sensitive node must be minimized. To this end, techniques that include
current steering logic and column bus readout though a current-limiting source
follower have been used [10].

###  3.3 TJ-Monopix Chip Design

Promising results of the modified process test chip encouraged the design of
two large scale prototypes based on the same technology and front-end, called
MALTA [11] and TJ-Monopix [7, 12] featuring different readout schemes. The
small sensor capacitance yields a very high input signal equal to $15~{}mV$
for the Most Probable Value (MPV) $\approx 1500~{}e^{-}$. The high input
signal to noise ratio can be exploited by a compact, non-conventional, low
power front-end derived from the ALPIDE chip [13], that has been optimized for
fast timing performance to meet the 25ns requirement. TJ-Monopix chip layout
is shown in Figure 3. It features a 448x224 small size pixel ($36x40~{}\mu
m^{2}$) matrix, while the total chip dimensions are $1x2~{}cm^{2}$. The analog
power consumption is $<65~{}mW/cm^{2}$, and combined with the 6-bit time stamp
distribution the total power consumption is only $120~{}mW/cm^{2}$. No in-
pixel threshold tuning has been implemented since the front-end has been
carefully designed for low threshold dispersion ($<30~{}e^{-}$). The matrix is
split in four flavors. Apart from the standard pixel variation that features a
PMOS input reset scheme, a novel leakage current compensation circuit and the
possibility to bias the sensor with high voltage from the front-side are
explored. A low power column bus readout variation is also implemented.

Figure 3: TJ-Monopix chip layout

##  4 Measurement Results

###  4.1 LF-Monopix

LF-Monopix chips have been irradiated with neutrons up to $1.5\cdot
10^{15}~{}n_{eq}/cm^{2}$ at the Jozef Stefan Institute reactor while the chips
received an additional 1 Mrad background TID. The breakdown voltage has been
measured equal to $280~{}V$ before irradiation [7], while after irradiation it
drops to $\approx 200~{}V$ and is still adequate to fully deplete the sensor.
By applying a sophisticated threshold tuning algorithm based on the pixel
noise occupancy, the threshold was tuned to $1500~{}e^{-}$ with a dispersion
of $\approx 100~{}e^{-}$. The achieved threshold value is low compared to the
MPV, which is higher than 12 keV for an unirradiated sample and 4.5 keV after
irradiation [6]. While the threshold value is not significantly affected after
irradiation, the ENC did increase from $200~{}e^{-}$ to $350~{}e^{-}$ due to
the background TID, with a dispersion equal to $30-70~{}e^{-}$ depending on
the CSA flavor. The input charge to output voltage gain has been measured
equal to $12~{}\mu V/e^{-}$ and is not degraded after irradiation. Figure 4
shows the high detection efficiency (98.9%) of an irradiated LF-Monopix chip
that was cooled down to $-40^{o}~{}C$, measured in an 2.5GeV electron beam
[6]. The noise occupancy during the measurement was $<10^{-6}/25~{}ns$.

Figure 4: LF-Monopix detection efficiency measurement after irradiation to
$1.5\cdot 10^{15}~{}n_{eq}/cm^{2}$ [6]

###  4.2 TJ-Monopix

While TJ-Monopix wafers were received in February 2018, and extensive
characterization is currently in progress, first laboratory results are
available. TJ-Monopix chips were also irradiated up to $1.5\cdot
10^{15}~{}n_{eq}/cm^{2}$ and 1 Mrad backround TID, and remain fully
functional. Since the sensor is fully depleted, the p-well potential inside
the matrix can be separate from the epitaxial layer potential. The minimum
p-well bias voltage is -6 V, while the substrate biasing is limited by the
minimum voltage before punchthrough between the p-well and the epitaxial layer
occurs and is equal to -20 V [9]. The maximum front-side biasing voltage when
using the special AC coupled flavor is +50V. The noise performance and
threshold were measured for the PMOS input reset flavor with p-well biasing=-5
V and substrate biasing=-20 V. ENC was measured equal to $11~{}e^{-}$ with
$0.8~{}e^{-}$ dispersion in agreement with simulation and the gain has been
measured equal to $0.4~{}mV/~{}e^{-}$. The high gain and low noise are a
direct consequence of the small sensor capacitance. As shown in Figure 5, the
threshold was measured equal to $271~{}e^{-}$ with dispersion equal to
$30.7~{}e^{-}$. The noise occupancy that corresponds to these measurements is
$\approx 4\cdot 10^{-8}/25~{}ns$. After irradiation, ENC is increased to
$20~{}e^{-}$ and the threshold is increased to $470~{}e^{-}$ with dispersion
equal to $50~{}e^{-}$.

Figure 6 depicts the spectrum of an ${}^{55}Fe$ source that was measured using
the AC coupled HV flavor pixels and the analog ToT information. The bias
voltages were set to p-well=0 V, substrate=-20 V, HV=+30 V. The measurement
was repeated using an irradiated chip. The K$\alpha~{}(5.9KeV)$ and
K$\beta~{}(6.5KeV)$ peaks produced by the electron capture decay of
${}^{55}Fe$ to ${}^{55}Mn$ are visible for the unirradiated case while for the
irradiated chip, only the K$\alpha$ peak is clearly visible. It must be noted
that the peak amplitude is degraded after irradiation due to the different
front end settings, that were applied to increase the threshold and lower the
noise occupancy.

Figure 5: TJ-Monopix threshold scan: PMOS input reset flavor, p-well=-5V,
substrate=-20V Figure 6: TJ-Monopix ${}^{55}Fe$ spectrum before and after
irradiation to $1.5\cdot 10^{15}~{}n_{eq}/cm^{2}$: HV-AC coupled pixel flavor,
p-well=-0V, substrate=-16V, HV=+30V

##  5 Conclusion

Two large scale prototype chips based on the same fast readout architecture
were implemented to demonstrate the feasibility of depleted monolithic active
pixel sensors for the harsh radiation environment of the ATLAS ITk. The LF-
Monopix radiation tolerance has been proven by means of $\approx 99\%$
efficiency after irradiation to $1.5\cdot 10^{15}~{}n_{eq}/cm^{2}$ while the
TJ-Monopix chip remains functional after the same level of irradiation and
first measurement results demonstrate the high analog performance of the small
collection electrode implementation.

## Acknowledgments

This work is supported by the H2020 project AIDA-2020, GA no. 654168, and by
the H2020 project STREAM, GA no. 675587

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