Hysteresis Characteristics of Generalized Spin-S Magnetic Binary Alloys

Gülşen<EMAIL_ADDRESS>
The Graduate School of Natural and Applied Sciences, Dokuz Eylül University,
Tr-35160 İzmir, Turkey

Ümit<EMAIL_ADDRESS>
Department of Physics, Dokuz Eylül University, TR-35160 Izmir, Turkey

##  1 Abstract

In this study, hysteresis characteristics of the generalized spin-S binary
alloy represented by the formula $A_{c}B_{1-c}$ have been investigated within
the framework of effective field approximation. The binary system consists of
type A (spin-S) and type B (spin-S) atoms which are randomly distributed on a
honeycomb lattice. Both integer and half-integer spin models of two magnetic
atom types are examined. By detailed investigation on hysteresis loops,
multiple hysteresis behaviors are obtained for a given set of Hamiltonian
parameters. Besides, the quantities of hysteresis characteristics as the
hysteresis loop area, remanent magnetization, and coercive field have been
investigated as functions of concentration.

Keywords: Hysteresis characteristics, binary alloy, effective field theory

##  2 Introduction

The development and characterization of the transition metal alloys and rare
earth alloys are high interest due to the emergence of successful applications
in this rapidly evolving field [1]. Multicritical points of rare earth alloys
such as $Tb-Er$, $Dy-Er$, $Tb-Tm$, $Dy-Tm$ and $Gd-Er$ have been investigated
in the presence of crystal field effects [2]. Phase diagrams of magnetic and
non-magnetic transition metal alloys have been studied both experimentally and
theoretically based on Ising-type phenomenological models [3].

Disordered binary alloys represented by $A_{c}B_{1-c}$ can be modeled
theoretically using well-known Ising-like models. Binary alloy systems
consisting of different spin values, such as half integer - integer spin
valued models are investigated by means of the effective field theory (EFT)
[4, 5, 6], mean field theory (MFT) [7, 8, 9] and Monte Carlo (MC) simulations
[10]. Models with half integer - half integer spins are also examined by means
of MFT [11] and also within the two frameworks EFT and MFT [12, 13]. Besides,
several spin systems are modeled such as $S_{A}=1/2$ and another component is
generalized spin-$S$ by use of MFT [14] and within EFT and MFT [12].
Generalized of both spin variables of binary ferromagnetic alloy has been
investigated with competing anisotropies by means of MFT [15]. Besides,
$Fe_{p}Al_{q}$ alloys [16] and $NiBi$ alloys [17] have been constructed on the
Ising model within the framework of EFT. Site-diluted Ising spin model for
$Fe_{1-q}Al_{q}$ alloys have been examined by use of EFT [18, 19] and pair
approximation [20]. The bond disordered Blume-Capel model for ternary
$(Fe_{0.65}Ni_{0.35})_{1-x}Mn_{x}$ and $Fe_{p}Al_{q}Mn_{x}$ alloys have been
studied with mean-field renormalization group (MFRG) method [21]. The Potts-
like model has been utilized to describe $Gd_{1-x}C_{x}$ alloy on basis of MC
method [22].

It is crucial to emphasize that magnetic materials used in important
technological applications are represented by higher spin systems.
$AB_{p}C_{1-p}$ ternary alloy system corresponding to the magnetic Prussian
blue analogs of
$(Ni_{p}^{II}Mn_{1-p}^{II})_{1.5}[Cr^{III}(CN)_{6}]\cdot zH_{2}O$ type
consisting of $S_{A}=3/2$, $S_{B}=1$, $S_{C}=5/2$ have been investigated by
use of MFT [23] and MC [24]. The
$(Fe_{p}^{II}Mn_{1-p}^{II})_{1.5}[Cr^{III}(CN)_{6}]\cdot nH_{2}O$ type analog
consisting of $S_{A}=3/2$, $S_{B}=2$, $S_{C}=5/2$ has been investigated by use
of MFT [25]. Besides, different spin variables have been studied such as
$S_{A}=1/2$, $S_{B}=1$, $S_{C}=3/2$ [26] and $S_{A}=1$, $S_{B}=3/2$,
$S_{C}=1/2$ [27] within the framework of EFT.

On the other hand, hysteresis behavior of magnetically ordered organic and
molecule-based materials has been inspected extensively [28, 29]. A molecular-
based magnetic material $AFe^{II}Fe^{III}(C_{2}O_{4})_{3}$ which corresponds
to ferrimagnetic mixed spin-$2$ and spin-$5/2$ Ising model on a honeycomb
lattice exhibits $2S+1$ magnetic plateaus in the presence of single-ion
anisotropies and at low temperatures. Triple and double hysteresis behaviors
have been found by means of MC [30] and EFT [31] for the given system.

In Ref. [32], Akıncı concluded that crystal field diluted $S-1$ Blume-Capel
model has double and triple hysteresis behavior in the ground state, at
negative large values of the crystal field within the framework of EFT. Then,
generalization of these results to a spin-$S$ $(S>1)$ Blume Capel [33] and
Heisenberg models [34] have been realized. These models display $2S$-windowed
hysteresis loop in this region. These works also focused on the difference
between integer and half integer spin models. While half integer spin model
displays central loop (which is symmetric about the origin), integer spin
model does not exhibit central loop.

Similar results for other systems have been reported in the literature. For
instance, the binary alloy system consisting of spin-$1/2$ and spin-$1$ could
exhibit double hysteresis behavior in a concentration $0<c<0.557$ range within
the framework of EFT [5]. The effects of the symmetric double Gaussian random
field distribution on this system has also been investigated by use of EFT,
and double hysteresis character has been obtained depending on Hamiltonian
parameters [35]. Hysteresis behavior of quenched disordered binary alloy
cylindrical nanowire consisting of the same spin set as the previous model has
been examined by MC [36]. The effects of the concentration and temperature on
disordered $Fe_{x}Al_{1-x}$ alloys have been studied by using first-principle
theory and MC simulation [37]. There are also studies in the literature
involving the hysteresis features of higher spin models. For example, mixed
spin-$1/2$ and spin-$3/2$ Ising ferromagnetic and ferrimagnetic bilayer system
on a honeycomb lattice has been studied within EFT [38]. It is noteworthy
that, multiple hysteresis behaviors have been obtained on the mixed spin-$7/2$
and spin-$3/2$ Ising ferrimagnetic model by use of MC simulation [39].
Furthermore, the dynamical hysteresis properties of mixed spin-$3/2$ and
spin-$5/2$ ferrimagnetic Ising model have been obtained by means of EFT based
on Glauber-type stochastic dynamics [40]. Single and triple dynamical
hysteresis loops have been observed in a spin-$1/2$ Ising bilayer system using
the same method [41].

There are comprehensive experimental studies about magnetic alloy systems
presented. Some of them are, amorphous $Fe_{100-x}B_{x}$ alloys [42],
$Fe$-doped $Au_{x}Pd_{1-x}$ alloys [43], $Mn_{x}Zn_{1-x}F_{2}$ alloys in the
presence of random field [44], $Tb_{x}Y_{1-x}$ and $Tb_{x}Gd_{1-x}$ alloys in
the presence of single-ion anisotropy [45], $Fe_{100-x}Ni_{x}$ and
$Mn_{100-x}Ni_{x}$ alloys [46], $Fe_{100-q}Al_{q}$ [47] and
$Fe_{x}Al_{1-x}Mn_{x}$ alloys [48].

In our recent work, we have investigated the hysteresis behavior of the binary
magnetic alloy that consist of spin-1 and spin-$1/2$ atoms [5]. The main aim
of this work is to generalize the results obtained in our recent work. We want
to determine the hysteresis characteristics of the magnetic binary alloys by
considering both spins which are generalized by spin-$S$ of type-$A$ and
type-$B$ atoms consisting of different spin values. For this aim, the outline
of this paper as follows: In Sec. 3 we briefly present the model and
formulation. The results and discussions are presented in Sec. 4, and finally
Sec. 5 contains our conclusions.

##  3 Model and Formulation

The system consists of randomly distributed $A$ type atoms that have
spin-$S_{A}$ and $B$ type atoms that have spin-$S_{B}$ within the Ising model.
The concentrations of the $A$ type atoms are denoted as $c$, and the $B$ type
atoms are denoted as $1-c$. Therefore the chemical formula is given by
$A_{c}B_{1-c}$. Note that, the lattice has no vacancies. The Hamiltonian of
the binary Ising model is given by

$\mathcal{H}=-J{{\underset{<i,j>}{\overset{}{\displaystyle\sum}}}\left(\xi_{i}\xi_{j}\sigma_{i}\sigma_{j}+\xi_{i}\delta_{j}\sigma_{i}s_{j}+\delta_{i}\xi_{j}s_{i}\sigma_{j}+\delta_{i}\delta_{j}s_{i}s_{j}\right)}-D{{\underset{i}{\overset{}{\displaystyle\sum}}}\left(\xi_{i}\sigma_{i}^{2}+\delta_{i}s_{i}^{2}\right)}-H{{\underset{i}{\overset{}{\displaystyle\sum}}}\left(\xi_{i}\sigma_{i}+\delta_{i}s_{i}\right)},$
(1)

where $\sigma_{i},s_{i}$ are the $z$ components of the spin-$S_{A}$ and
spin-$S_{B}$ operators and they take the values
$\sigma_{i}=-S_{A},-S_{A}+1,\ldots,S_{A}-1,S_{A}$ and
$s_{i}=-S_{B},-S_{B}+1,\ldots,S_{B}-1,S_{B}$, respectively. $J>0$ is the
ferromagnetic exchange interactions between the nearest neighbor spin pairs,
$D$ is the crystal field (single ion anisotropy) and $H$ is the longitudinal
magnetic field. Here, $\xi_{i}=1$ means that lattice site $i$ is occupied by
type-A type atoms and $\delta_{i}=1$ means that lattice site $i$ is occupied
by type-B type atoms. The site occupation number holds the relation
$\xi_{i}+\delta_{i}=1$. The first summation in Eq. (1) is over the nearest-
neighbor pairs of spins and the other summations are over all the lattice
sites.

In a typical EFT approximation, we consider specific site (denoted by $0$) and
nearest neighbors of it. All interactions of this site can be represented by
$\mathcal{H}_{0}^{A}/\mathcal{H}_{0}^{B}$ if the site $0$ occupied by type-A/B
atoms, respectively. These terms can be treated as local fields acting on a
site $0$,

$\mathcal{H}_{0}^{A}=-\xi_{0}\sigma_{0}\left[J{{\underset{j=1}{\overset{z}{\displaystyle\sum}}}\left(\xi_{j}\sigma_{j}+\delta_{j}s_{j}\right)}+H\right]-\xi_{0}\left(\sigma_{0}\right)^{2}D,$
(2)

$\mathcal{H}_{0}^{B}=-\delta_{0}s_{0}\left[J{{\underset{\delta=1}{\overset{z}{\displaystyle\sum}}}\left(\xi_{j}\sigma_{j}+\delta_{j}s_{j}\right)}+H\right]-\delta_{0}\left(s_{0}\right)^{2}D.$
(3)

By defining spin-spin interaction part of these local fields as,

$E_{0}^{A}=J{{\underset{j=1}{\overset{z}{\displaystyle\sum}}}\left(\xi_{j}\sigma_{j}+\delta_{j}s_{j}\right)},\quad
E_{0}^{B}=J{{\underset{\delta=1}{\overset{z}{\displaystyle\sum}}}\left(\xi_{j}\sigma_{j}+\delta_{j}s_{j}\right)}$
(4)

we can write Eqs. (2) and (3) more compact form as,

$\mathcal{H}_{0}^{A}=-\xi_{0}\sigma_{0}\left[E_{0}^{A}+H\right]-\xi_{0}\left(\sigma_{0}\right)^{2}D,$
(5)

$\mathcal{H}_{0}^{B}=-\delta_{0}s_{0}\left[E_{0}^{B}+H\right]-\delta_{0}\left(s_{0}\right)^{2}D.$
(6)

For obtaining magnetizations ($m_{A},m_{B}$) and quadrupolar moments
($q_{A},q_{B}$) for the system, we can use the exact identities [49] which are
given by

$m_{A}=\frac{\left\langle\left\langle\xi_{0}\sigma_{0}\right\rangle\right\rangle_{r}}{\left\langle\xi_{0}\right\rangle_{r}}=\frac{1}{\left\langle\xi_{0}\right\rangle_{r}}\left\langle\left\langle\frac{Tr_{0}\xi_{0}\sigma_{0}\exp{\left(-\beta\mathcal{H}_{0}^{A}\right)}}{Tr_{0}\exp{\left(-\beta\mathcal{H}_{0}^{A}\right)}}\right\rangle\right\rangle_{r},$

$q_{A}=\frac{\left\langle\left\langle\xi_{0}\sigma_{0}^{2}\right\rangle\right\rangle_{r}}{\left\langle\xi_{0}\right\rangle_{r}}=\frac{1}{\left\langle\xi_{0}\right\rangle_{r}}\left\langle\left\langle\frac{Tr_{0}\xi_{0}\sigma_{0}^{2}\exp{\left(-\beta\mathcal{H}_{0}^{A}\right)}}{Tr_{0}\exp{\left(-\beta\mathcal{H}_{0}^{A}\right)}}\right\rangle\right\rangle_{r},$

$m_{B}=\frac{\left\langle\left\langle\delta_{0}s_{0}\right\rangle\right\rangle_{r}}{\left\langle\delta_{0}\right\rangle_{r}}=\frac{1}{\left\langle\delta_{0}\right\rangle_{r}}\left\langle\left\langle\frac{Tr_{0}\delta_{0}s_{0}\exp{\left(-\beta\mathcal{H}_{0}^{B}\right)}}{Tr_{0}\exp{\left(-\beta\mathcal{H}_{0}^{B}\right)}}\right\rangle\right\rangle_{r},$
(7)

$q_{B}=\frac{\left\langle\left\langle\delta_{0}s_{0}^{2}\right\rangle\right\rangle_{r}}{\left\langle\delta_{0}\right\rangle_{r}}=\frac{1}{\left\langle\delta_{0}\right\rangle_{r}}\left\langle\left\langle\frac{Tr_{0}\delta_{0}s_{0}^{2}\exp{\left(-\beta\mathcal{H}_{0}^{B}\right)}}{Tr_{0}\exp{\left(-\beta\mathcal{H}_{0}^{B}\right)}}\right\rangle\right\rangle_{r}.$

where $Tr_{0}$ is the partial trace over the site $0$,
$\beta=1/\left(k_{B}T\right)$, $k_{B}$ is Boltzmann constant and $T$ is the
temperature. We have two averages here, thermal averages (inner bracket) and
random configurational averages (bracket with subscript $r$). This last
average should be taken into account to include the effect of the random
distribution of the atoms in the system.

Since all relations in Eq. (7) are in the same form, it is enough to derive
one of them for obtaining the final form of the equation. Let us choose the
equation related to $m_{A}$ from Eq. (7).

By writing Eq. (5) in Eq. (7) and performing partial trace operations by using
identity $e^{\xi x}=\xi e^{x}+1-\xi,$ (where $x$ is any real number and
$\xi=0,1$) we can obtain expression in a closed form as

$\frac{\left\langle\left\langle\xi_{0}\sigma_{0}\right\rangle\right\rangle_{r}}{\left\langle\xi_{0}\right\rangle_{r}}=\left\langle\left\langle
f_{m}^{A}\left(E_{0}^{A}\right)\right\rangle\right\rangle_{r},$ (8)

where the function is given by [50]. All definitions of these functions will
be given at the end of this section. By using differential operator technique
[51], Eq. (8) can be written as

$\frac{\left\langle\left\langle\xi_{0}\sigma_{0}\right\rangle\right\rangle_{r}}{\left\langle\xi_{0}\right\rangle_{r}}=\left\langle\left\langle
e^{E_{0}^{A}\nabla}\right\rangle\right\rangle_{r}f_{m}^{A}(x)|_{x=0},$ (9)

where $\nabla$ represents the differential with respect to $x$. The effect of
the differential operator $\nabla$ on an arbitrary function $F$ is given by

$\exp{\left(a\nabla\right)}F\left(x\right)=F\left(x+a\right),$ (10)

with arbitrary constant $a$.

At this stage of the derivation, we have to convert the exponential operator
in average braces to a polynomial form. For this aim using using approximated
van der Waerden identities [52] is typical. This identity is

$\exp{\left(aS\right)}=\cosh{\left(a\eta\right)}+\frac{S}{\eta}\sinh{\left(a\eta\right)},$
(11)

where $\eta^{2}=\left\langle S^{2}\right\rangle$ and $S$ is the spin
eigenvalue. By using $E_{0}^{A}$ of Eq. (4) in Eq. (9) we can obtain
polynomial form of the operator. Then, by using Eq. (10) with the identity
$e^{\xi x}=\xi e^{x}+1-\xi,$ we can obtain equation for $m_{A}$

$m_{A}={{\underset{p=0}{\overset{z}{\displaystyle\sum}}}}{{\underset{q=0}{\overset{z-p}{\displaystyle\sum}}}}{{\underset{r=0}{\overset{p}{\displaystyle\sum}}}}{{\underset{s=0}{\overset{z-q-r}{\displaystyle\sum}}}}{{\underset{t=0}{\overset{q+r}{\displaystyle\sum}}}}C_{pqrst}(-1)^{t}c^{z-p}\left(1-c\right)^{p}\left(\frac{m_{A}}{\eta_{A}}\right)^{q}\left(\frac{m_{B}}{\eta_{B}}\right)^{r}f_{m}^{A}\left(\left[z-2s-2t\right]J,\right)$
(12)

where $\eta_{A}^{2}=q_{A}=\left\langle\sigma^{2}\right\rangle$, and

$C_{pqrst}=\frac{1}{2^{z}}\left(\begin{array}[]{c}z\\\
p\end{array}\right)\left(\begin{array}[]{c}z-p\\\
q\end{array}\right)\left(\begin{array}[]{c}p\\\
r\end{array}\right)\left(\begin{array}[]{c}z-q-r\\\
s\end{array}\right)\left(\begin{array}[]{c}q+r\\\ t\end{array}\right).$ (13)

By using the same steps between Eqs. (8)-(12) to other relations in Eq. (7),
we can obtain equations for other quantities as,

$q_{A}={{\underset{p=0}{\overset{z}{\displaystyle\sum}}}}{{\underset{q=0}{\overset{z-p}{\displaystyle\sum}}}}{{\underset{r=0}{\overset{p}{\displaystyle\sum}}}}{{\underset{s=0}{\overset{z-q-r}{\displaystyle\sum}}}}{{\underset{t=0}{\overset{q+r}{\displaystyle\sum}}}}C_{pqrst}(-1)^{t}c^{z-p}\left(1-c\right)^{p}\left(\frac{m_{A}}{\eta_{A}}\right)^{q}\left(\frac{m_{B}}{\eta_{B}}\right)^{r}f_{q}^{A}\left(\left[z-2s-2t\right]J,\right)$
(14)

$m_{B}={{\underset{p=0}{\overset{z}{\displaystyle\sum}}}}{{\underset{q=0}{\overset{z-p}{\displaystyle\sum}}}}{{\underset{r=0}{\overset{p}{\displaystyle\sum}}}}{{\underset{s=0}{\overset{z-q-r}{\displaystyle\sum}}}}{{\underset{t=0}{\overset{q+r}{\displaystyle\sum}}}}C_{pqrst}(-1)^{t}c^{z-p}\left(1-c\right)^{p}\left(\frac{m_{A}}{\eta_{A}}\right)^{q}\left(\frac{m_{B}}{\eta_{B}}\right)^{r}f_{m}^{B}\left(\left[z-2s-2t\right]J,\right)$
(15)

$q_{B}={{\underset{p=0}{\overset{z}{\displaystyle\sum}}}}{{\underset{q=0}{\overset{z-p}{\displaystyle\sum}}}}{{\underset{r=0}{\overset{p}{\displaystyle\sum}}}}{{\underset{s=0}{\overset{z-q-r}{\displaystyle\sum}}}}{{\underset{t=0}{\overset{q+r}{\displaystyle\sum}}}}C_{pqrst}(-1)^{t}c^{z-p}\left(1-c\right)^{p}\left(\frac{m_{A}}{\eta_{A}}\right)^{q}\left(\frac{m_{B}}{\eta_{B}}\right)^{r}f_{q}^{B}\left(\left[z-2s-2t\right]J.\right)$
(16)

Here the functions are defined as [50],

$f_{m}^{A}\left(x,H,D\right)=\frac{{{\underset{k=-S_{A}}{\overset{S_{A}}{\displaystyle\sum}}}}k\exp{\left(\beta
Dk^{2}\right)\sinh{\left[\beta
k\left(x+H\right)\right]}}}{{{\underset{k=-S_{A}}{\overset{S_{A}}{\displaystyle\sum}}}}\exp{\left(\beta
Dk^{2}\right)\cosh{\left[\beta k\left(x+H\right)\right]}}},$ (17)

$f_{q}^{A}\left(x,H,D\right)=\frac{{{\underset{k=-S_{A}}{\overset{S_{A}}{\displaystyle\sum}}}}k^{2}\exp{\left(\beta
Dk^{2}\right)\cosh{\left[\beta
k\left(x+H\right)\right]}}}{{{\underset{k=-S_{A}}{\overset{S_{A}}{\displaystyle\sum}}}}\exp{\left(\beta
Dk^{2}\right)\cosh{\left[\beta k\left(x+H\right)\right]}}}.$ (18)

$f_{m}^{B}\left(x,H,D\right)=\frac{{{\underset{k=-S_{B}}{\overset{S_{B}}{\displaystyle\sum}}}}k\exp{\left(\beta
Dk^{2}\right)\sinh{\left[\beta
k\left(x+H\right)\right]}}}{{{\underset{k=-S_{B}}{\overset{S_{B}}{\displaystyle\sum}}}}\exp{\left(\beta
Dk^{2}\right)\cosh{\left[\beta k\left(x+H\right)\right]}}},$ (19)

$f_{q}^{B}\left(x,H,D\right)=\frac{{{\underset{k=-S_{B}}{\overset{S_{B}}{\displaystyle\sum}}}}k^{2}\exp{\left(\beta
Dk^{2}\right)\cosh{\left[\beta
k\left(x+H\right)\right]}}}{{{\underset{k=-S_{B}}{\overset{S_{B}}{\displaystyle\sum}}}}\exp{\left(\beta
Dk^{2}\right)\cosh{\left[\beta k\left(x+H\right)\right]}}}.$ (20)

Eqs. (12) and (14)-(16) constitute a system of coupled nonlinear equations.
The coefficients in this system are given by Eq. (13). By using functions from
Eqs. (17)-(19) we can solve this system by numerical procedures. After getting
the values of $m_{A},m_{B},q_{A},q_{B}$ from this solution we can calculate
the total magnetization ($m$) and quadrupolar moment ($q$) of the system via

$m=cm_{A}+\left(1-c\right)m_{B},\quad q=cq_{A}+\left(1-c\right)q_{B}.$ (21)

The hysteresis curves can be obtained by sweeping the magnetic field from $-H$
to $H$ and calculating magnetization in each step. The reverse sweep will
yield other branch of the curve, if present.

##  4 Results and Discussion

We consider the following scaled (dimensionless) quantities in this work

$d=\frac{D}{J},t=\frac{k_{B}T}{J},h=\frac{H}{J}.$ (22)

Results have been investigated on a honeycomb lattice (i.e. $z=3$).

In this study, we will discuss the effects of the crystal field and the
concentration on the hysteresis properties of generalized spin-S binary alloy
system. Note that for different selected spin values of these atoms we can
choose both types of atoms as integer and half integer. We have limited this
work by examining the case of $S_{A}<S_{B}$. Note also that, the concentration
of $c$ in the case of $S_{A}<S_{B}$ corresponds to the concentration of $1-c$
in the $S_{A}>S_{B}$ case.

The hysteresis loop can be obtained by calculating the magnetization by
sweeping the magnetic field from $-h$ to $h$ direction and vice versa. The
system prefers the non-magnetic $s=0$ state which is the disordered phase in
the ground state, at large negative large crystal fields, for integer spin
atoms. Besides, the system is exposed to the magnetic $s=\pm 1/2$ ground
states representing by the ordered phase at negative large crystal fields, for
half integer spin atoms. When the magnitude of the magnetic field begins to
align the spins in parallel to the field direction, the system exhibits
transition from mostly occupied ground state $s$ to the next occupied $s+1$
ground state, at low temperatures. If the magnetic field further increases,
$s+1\rightarrow s+2\rightarrow...\rightarrow S$ transitions occur, for both of
integer and half integer spin values. This is the well known plateau like
ground state structure. If we examine these transitions for both positive and
negative directions of the external magnetic field, the magnetization response
of the system could display interesting hysteresis characteristics. These
characteristics are the main investigation area of this work.

Firstly, we choose the spin values of two different types of atoms as half
integer spins. The hysteresis loops are depicted in Fig. 1 with selected
values of the concentrations $c=0.1$, $c=0.3$ and $c=0.8$. Spin variables of
the binary alloy system are $S_{A}=1/2$, $S_{B}=3/2$ in Fig 1 (a) and
$S_{A}=5/2$, $S_{B}=7/2$ in Fig 1 (b). One should notice that $c=0$ and $c=1$
cases correspond to a situation where all lattice sites are occupied by
spin-$3/2$ and spin-$1/2$ atoms in Fig 1 (a), spin-$7/2$ and spin-$5/2$ atoms
in Fig 1 (b), respectively. When the majority of the binary alloy system
consists of $spin-3/2$ atoms, (such as $c=0.1$ in Fig. 1 (a)), the system
displays triple hysteresis (TH) behavior for selected values of the
Hamiltonian parameters $t=0.45$ and $d=-2$ (see Fig 1 (a)). The central loop
imply the ordered phase for this system. When we start adding more type-A
atoms to the system (i.e. rising $c$), the outer windows that appears large
negative and positive values of magnetic field of hysteresis curve, disappear.
The central loop continues with the same structure, i.e. single hysteresis
(SH) is observed. When the majority of the binary alloy composed of type-A
atoms (i.e. $c=0.8$), the ordered phase has been protected in central loop in
the same way. Due to the fact that, the system prefers magnetic $s=\pm 1/2$
ground states of binary alloy which consists of half integer spins. When the
system exposed to a large magnetic field, all spins align parallel to the
field. Magnetization is also greater in the system with high spin variable in
large longitudinal field. For instance, the magnetization related to $c=0.1$
is greater than the case of $c=0.8$ system for $h=2$ (see Figs. 1 (a) and
(b)). As seen in Fig. 1 (b), for the binary alloy system which consists of
higher spin values, the number of windows of the hysteresis loops increase for
a low temperature $t=0.5$ and for crystal field parameter $d=-2$. While
$c=0.1$ case exhibits seven ($2S_{B}$-windowed) windows of hysteresis (7H)
loops, $c=0.3$ and $c=0.8$ cases exhibit five ($2S_{A}$-windowed) windows of
hysteresis (5H) loops . Inset of graph in Fig. 1 (b) shows the central loop
which represent an ordered phase that exists for all concentration values. As
an important result, we can emphasize that, while the concentration decreases,
two additional outermost symmetric windows of hysteresis loops appear. The
magnetization of the system also increases for large longitudinal magnetic
field, as expected. These results are compatible with the results presented in
Ref. [33].

In Fig. 2, we consider a system where both of the spin variables of binary
alloy are integer -values. Hysteresis properties of the system are given for
selected values of concentrations $c=0.1$, $c=0.3$ and $c=0.8$. Spin values
are chosen as $S_{A}=1$ and $S_{B}=2$ in Fig. 2 (a) and $S_{A}=1$ and
$S_{B}=3$ in Fig. 2 (b) for $t=0.5$ and $d=-2$. In the case of $c=0.1$ the
system displays four-windowed hysteresis $(4H)$ loops which have no central
loop (see Fig. 2 (a)). This means, the system has disordered phase which
corresponds to the system with non-magnetic $s=0$ ground state. The physical
mechanism is as follows: with rising field in positive direction, $s=0$ state
begins to evolve into $s=\pm 1$ states, and then these states evolves into
$s=\pm 2$, with rising magnetic field. For the concentration values of $c=0.3$
and $c=0.8$, the outer symmetric two windows of the hysteresis curve disappear
and the system exhibits double hysteresis (DH) character. Magnetization
decreases with increasing concentrations at large applied magnetic field. As
seen in Fig 2 (b), if we fix the spin value of A atoms as $S_{A}=1$, and
increase the spin value of B atoms such that $S_{A}=3$, then the binary alloy
system with concentration $c=0.1$ exhibits $6$-windowed hysteresis $(6H)$
loops, without central loop. A considerable noteworthy result is that the
system displays DH behavior for $c=0.3$ and $c=0.8$, due to the higher spin
value of B atoms. The system eliminates the outermost four symmetric loops to
adapt from the $6$-windowed hysteresis loop behavior to the DH loop behavior.
Regardless of the difference between the spin values of the binary alloys, the
number of outer windows that disappeared will be twice that difference for
concentration that close to the small spin value. To clarify, difference
between the spin values of type-A and type-B atoms are $1$ (2), the number of
outer windows that disappear will be two (four) with increasing components of
A atoms in Fig 2 (a) (Fig. 2 (b)), respectively.

In the previous examinations, the binary alloy system had only one of the
ordered (half integer spin valued binary alloys) or disordered (integer spin
valued binary alloys) phases at low temperatures and large negative crystal
field values. To find out which phase does the system exhibit in binary alloys
consisting of spin values integer and half integer, we choose spin value of A
atoms as integer and spin value of B atoms as half integer in Fig 3.
Hysteresis behaviors of binary alloy consisting of (a) $S_{A}=1$, $S_{B}=3/2$,
(b) $S_{A}=1$, $S_{B}=7/2$ and (c) $S_{A}=2$, $S_{B}=5/2$ spin values have
quite striking results. In figure 3 (a), hysteresis curves are shown for the
values of the concentration $c=0.1$, $c=0.3$, $c=0.5$ and $c=0.8$ for $t=0.46$
and $d=-1.8$. For the case of $c=0.1$, namely majority of the lattice sites
consist of type B atoms, the system has TH behavior with central loop (see the
curves related to the $c=0.1$ in Figs. 3 (a), (b) and (c)). For the
concentration value $c=0.3$, the central loop disappears and the outer
symmetric two loops become narrower. Since the central loop is lost, the
system should exhibit a phase transition in this concentration ranges. For the
value of $c=0.3$, the system does not have a magnetic ground state at low
magnetic field values. As the applied magnetic field increases, smaller
magnetic field induces the transition between the ground states of the system.
Contrary to the common belief, the difference between the new ground states
dominated by the B atoms are less than one. The transition from TH to DH can
be arranged systematically as follows: The number of windows that disappeared
will be two times of difference between two spin values of type A and type B
atoms, i.e. $2(S_{B}-S_{A})$ windows disappear. The difference between
$S_{A}=1$ and $S_{B}=3/2$ spins are $1/2$, so one loop disappears, which is
the central loop. For concentration value of $c=0.5$, the system exhibits
paramagnetic hysteresis (PH) which prefers $s=0$ state (see the curves for
$c=0.5$ in Fig. 3 (a), (b) and (c)). When the concentration value increases to
$c=0.8$, due to the spin value of A atoms, $2S_{A}$-windowed (namely DH)
behavior appears. Depending on the concentration of the binary alloy, DH for
$c=0.3$ and $c=0.8$ are different from each other due to the difference
between their ground states. In this case, $TH-DH-PH-DH$ hysteresis behaviors
occur. The first one (TH) corresponds to the ordered phase and the others (DH,
PH and other DH) are disordered phase. Phase transition between two phases
causes $2(S_{B}-S_{A})=1$ windows to disappear. In Fig 3 (b) Hamiltonian
parameters are set as $t=0.46$ and $d=-2$ for various concentrations. For
$c=0.1$ system exhibits $7$-windowed hysteresis loops with central loop which
is demonstrated in the upper inset. For the value of $c=0.3$, the system
exhibits DH behavior which is demonstrated in lower inset. The observed DH
loops are also narrower than the concentric loops. So, transition from ordered
to a disordered phase occurs between these concentration ranges. As the
concentration of type-A atoms increases in the system, we observe that some
loops disappear according to the new ground states, which are dominated by the
B atoms. From the general result mentioned above, $2(S_{B}-S_{A})=5$ windows
should disappear. Four of them are the outermost symmetric windows and the
other is the central loop as can be seen in Fig 3 (b). For the value of
$c=0.5$, the system exhibits disordered phase and then for $c=0.8$ the
innermost double symmetric loops appear, which are also demonstrated in upper
inset of Fig. 3 (b). For alloys consisting mostly of atoms type-A, the ground
state will be dominated by atoms type-A, so it has different center from the
other DH loops. In brief, $7H-DH-PH-DH$ hysteresis behaviors are observed as
the concentration increases. In Fig 3 (c), the system exhibits $5H-4H-PH-4H$
hysteresis behaviors (while the concentration increases) for $t=0.47$ and
$d=-2$, respectively. Transition from the ordered phase ($(5H)$ windowed loop)
to the disordered phase ($(4H)$ windowed loop) causes $2(S_{B}-S_{A})=1$
window disappears, which is the central loop. The other $4$-windowed loops get
narrower. For the value of $c=0.5$ system has paramagnetic hysteresis (PH)
loop. It is worth to emphasize here that, for $c=0.8$, the innermost
$4$-windowed hysteresis loops (which have different centers from the other
$4$-windowed loops) appear.

When we generalized integer-half integer results, for low temperatures and
negative large crystal fields, $2S_{B}-2S_{A}-PH-2S_{A}$-windowed hysteresis
loops are observed while the concentration increases for $S_{A}<S_{B}$. The
first one is the ordered phase and the other three are in disordered phase.
The transition from $2S_{B}$ to $2S_{A}$-windowed loop causes $2(S_{B}-S_{A})$
disappearing windows. The last $2S_{A}$-windowed hysteresis appears as
innermost loops due to the case of $S_{A}<S_{B}$.

In Fig 4, we have investigated binary alloy model consisting of half integer -
integer spins such as (a) $S_{A}=1/2$, $S_{B}=1$, (b) $S_{A}=3/2$, $S_{B}=2$
and (c) $S_{A}=3/2$, $S_{B}=3$. In Fig 4 (a), the system exhibits $DH-PH-PH-
SH$ hysteresis behaviors for the values of $t=0.46$ and $d=-1.6$. In Fig 4
(b), $4H-2H-PH-3H$ hysteresis behaviors are obtained for $t=0.5$ and $d=-1.8$.
In Fig 4 (c), $6H-2H-PH-3H$ hysteresis behaviors are observed for $t=0.5$ and
$d=-1.8$. Firstly, $2S_{B}$-windowed hysteresis loops are observed for lower
concentration values, as expected. If the concentration of the system is
increase from $c=0.1$ to $c=0.3$, the innermost DH disappears due to the new
ground states dominated by atoms of type-B (see Fig. 4 (b)). The system
exhibits paramagnetic hysteresis behavior for $c=0.5$. If the concentration of
the system is increases from $c=0.5$ to $c=0.8$, the innermost
$2S_{A}$-windowed hysteresis appears with a central loop. The first three
hysteresis behaviors ($4H-2H-PH$) correspond to disordered paramagnetic phase
and the last one is ($3H$) the ordered phase. The phase transition occurs
between these two phases.

The effects of crystal field parameter are examined for the binary alloys
consisting of integer-half integer spin model as, can be seen in Fig 5. Spin
values of the system are selected as $S_{A}=1$, $S_{B}=7/2$ and for the
parameters $c=0.1$ and $t=0.5$ in Fig. 5 (a), $c=0.8$ and $t=0.46$ in Fig. 5
(b). As an evolution of the hysteresis loops with changing crystal field,
$SH-6H-6H-4H-2H$ is observed (see Fig. 5 (a)). We concluded by comparison of
Fig. 3 (b) and Fig. 5 (a) from which we see that, by increasing the
temperature, the central loop disappears for the value of $d=-2$. The system
exhibits SH behavior, which is represented by ordered phase for $d=-1$, and
disordered phase for lower values of the crystal field. As the crystal field
increases to a negative large values, the outermost symmetric loops gradually
disappear and the windows start to be separated from each other. As seen in
Fig. 5 (b) $SH-SH-DH-PH-PH$ hysteresis behaviors are observed for given values
of the crystal field. Phase transition occurs when the crystal field parameter
changes from $d=-1$ to $d=-2$. When the system consists of mostly type-A
atoms, multiple hysteresis behavior is replaced by SH and DH behaviors. In
Fig. 6, we have investigated the effects of crystal field on binary alloy
consisting of half integer-integer spins at the temperature $t=0.46$. For the
spin values of $S_{A}=3/2$, $S_{B}=2$, the system displays $SH-4H-4H-2H$
hysteresis behaviors for $c=0.1$ and for selected crystal field parameters
(see Fig. 6 (a)). The system exhibits a phase transition with increasing
crystal field values, in the region between $d=-0.5$ and $d=-2$. Hysteresis
behaviors which are observed as $SH-3H-DH-PH$ for $c=0.9$, respectively can be
seen in Fig. 6 (b). The central loop disappears when passing from $3H$ to $DH$
behavior, and phase transition occurs in this range.

In order to investigate the evolution of the hysteresis properties, we
describe the quantities hysteresis loop area (HLA), remanent magnetization
(RM) and coercive field (CF). HLA is defined as the area covered by hysteresis
loop in $(m,h)$ plane, and it describes the loss of energy due to the
hysteresis. The RM is residual magnetization in the system after an applied
magnetic field is removed. CF is the intensity of the magnetic field needed to
change the sign of the magnetization.

In Figs. 7, 8 and 9 we demonstrate the variation of the quantities (HLA, RM
and CF, respectively) with the temperature, for one or both of two spin
variables chosen as integer or half integer spin. Chosen spin values at
crystal field $d=0$ are (a) $S_{A}=1/2$, $S_{B}=3/2$, (b) $S_{A}=1$,
$S_{B}=2$, (c) $S_{A}=1$, $S_{B}=3/2$ and (d) $S_{A}=3/2$, $S_{B}=2$. Besides,
the selected values of the concentrations are $c=0.0$, $c=0.5$ and $c=1.0$ in
these figures. In Fig. 7 (a) HLA drops to zero as the temperature increases.
Note that, the HLA of $c=0.0$ is greater than the HLA for $c=0.5$. In $c=1.0$
case there is no HLA, which means a paramagnetic hysteresis loop. Rising
temperature drives the system to a disordered phase, due to the thermal
agitations. In Fig. 7 (b) for integer-integer spin model, HLA exists for $c=1$
concentration at low temperatures and the value of HLA is greater than Fig. 7
(a). Therefore, more energy dissipation occurs. The temperature values that
HLA vanishes, increases. For $c=0$ the value of HLA has remained constant at
low temperatures as in Fig 7 (b). If we fix spin value of A atoms as
$S_{A}=1$, and increase the spin value of B atoms such that $S_{B}=3/2$, HLA
curves of $c=0.0$ and $c=0.5$ decrease (compare Fig 7 (b) and (c)). If we
fixed the spin value of B atoms as $S_{B}=2$, then increase the spin value of
A atoms such that $S_{A}=3/2$, HLA curves of $c=0.5$ and $c=1.0$ get higher
(compare Fig. 7 (b) and (d)). As seen in Figs. 8 for RM, and Fig. 9 for CF,
decreasing behavior occurs with the increasing concentration. According to the
different spin variables, results of RM and CF are consistent with HLA’s
behavior that is obtained above.

Figure 1: Hysteresis evolutions of the binary alloy system which consist of
half integer-half integer spin model such as (a) $S_{A}=1/2$, $S_{B}=3/2$ with
temperature $t=0.45$ and (b) $S_{A}=5/2$, $S_{B}=7/2$ spin variables with
$t=0.5$ for selected concentration values $c=0.1$, $c=0.3$ and $c=0.8$ at
$d=-2$ crystal field parameter.

Figure 2: Hysteresis evolutions of the binary alloy system which consist of
integer-integer spin model such as (a) $S_{A}=1$, $S_{B}=2$ and (b) $S_{A}=1$,
$S_{B}=3$ spin variables for selected concentration values $c=0.1$, $c=0.3$
and $c=0.8$, with the temperature $t=0.5$ and $d=-2$ crystal field parameter.

Figure 3: Hysteresis evolutions of the binary alloy system which consists of
integer- half integer spin model such as (a) $S_{A}=1$, $S_{B}=3/2$, (b)
$S_{A}=1$, $S_{B}=7/2$ and (c) $S_{A}=2$, $S_{B}=5/2$ spin variables for a
given set of Hamiltonian parameters.

Figure 4: Hysteresis evolutions of the binary alloy system which consists of
half integer-integer spin model such as (a) $S_{A}=1/2$, $S_{B}=1$, (b)
$S_{A}=3/2$, $S_{B}=2$ and (c) $S_{A}=3/2$, $S_{B}=3$ spin variables for a
given set of Hamiltonian parameters.

Figure 5: Hysteresis evolutions of the binary alloy system which consists of
integer- half integer spin model such as $S_{A}=1$, $S_{B}=7/2$ for the
concentrations (a) $c=0.1$ and (b) $c=0.8$ for a given set of Hamiltonian
parameters.

Figure 6: Hysteresis evolutions of the binary alloy system which consists of
half integer-integer spin model such as $S_{A}=3/2$, $S_{B}=2$ for the
concentrations (a) $c=0.1$ and (b) $c=0.9$ for a given set of Hamiltonian
parameters.

Figure 7: Variation of HLA with the temperature of the binary alloy system
which consists of (a) $S_{A}=1/2$, $S_{B}=3/2$, (b) $S_{A}=1$, $S_{B}=2$, (c)
$S_{A}=1$, $S_{B}=3/2$ and (d) $S_{A}=3/2$, $S_{B}=2$ spin variables for
selected values of concentrations $c=0.0$, $c=0.5$ and $c=1.0$ at $d=0$
crystal field parameter.

Figure 8: Variation of RM with the temperature of the binary alloy system
which consists of (a) $S_{A}=1/2$, $S_{B}=3/2$, (b) $S_{A}=1$, $S_{B}=2$, (c)
$S_{A}=1$, $S_{B}=3/2$ and (d) $S_{A}=3/2$, $S_{B}=2$ spin variables for
selected values of concentrations $c=0.0$, $c=0.5$ and $c=1.0$ at $d=0$
crystal field parameter.

Figure 9: Variation of CF with the temperature of the binary alloy system
which consists of (a) $S_{A}=1/2$, $S_{B}=3/2$, (b) $S_{A}=1$, $S_{B}=2$, (c)
$S_{A}=1$, $S_{B}=3/2$ and (d) $S_{A}=3/2$, $S_{B}=2$ spin variables for
selected values of concentrations $c=0.0$, $c=0.5$ and $c=1.0$ at $d=0$
crystal field parameter.

##  5 Conclusion

In conclusion, hysteresis characteristics of the generalized spin-S magnetic
binary alloy system represented by $A_{c}B_{1-c}$ have been investigated
within the framework of effective field theory. The system consists of
type-$A$ and type-$B$ atoms with the concentrations $c$ and $1-c$,
respectively. Results of the generalized spin-$S$ binary alloy model are
discussed as one or both of the spins of $A$ and $B$ atoms are selected as
integer or half integer spin values.

The effects of the concentration and crystal field parameters of the magnetic
binary alloy model strongly depend on whether the spins of the atoms are
integer or half-integer. As consistently by the related literature, the
special cases ($c=0$, $c=1$) of binary alloy system exhibit $2S$-windowed
hysteresis character. The evolution of the multiple hysteresis loops are
observed for higher spin valued alloy system for the large negative value of
crystal field at low temperatures. Integer (half-integer) spin valued binary
alloy system exhibits disordered (ordered) phase in this region. It has been
found that the number of outer windows that disappeared will be twice that
difference between spin values when majority of binary alloy composed of type
A atoms in the case of $S_{A}<S_{B}$.

The significantly remarkable point of our results is that the effect of
concentration is one of the most important parameters affecting the hysteresis
behavior in the system. Since the majority of the lattice sites consist of
half-integer spin values, the center loop lost and the system exhibits phase
transition as the concentration increases for integer-half integer system.
Besides, the magnetic ground state of the system disappears in the low
magnetic field. As the applied magnetic field increases, the transition
between the other ground states of the system occurs with a very small
magnetic field value. We have decided that the difference of the magnetization
values between the new ground states dominated by the B atom is less than $1$.
For alloys with concentration consisting mostly integer spin valued atoms, the
ground state will be dominated by type-A atoms. It is remarkable generalized
result that $2S_{B}-2S_{A}-PH-2S_{A}$-windowed hysteresis loops are observed
with increasing concentrations for integer-half integer system. The transition
from $2S_{B}$ to $2S_{A}$-windowed loop causes $2(S_{B}-S_{A})$ vanishing
windows. When the majority of binary alloy is composed of integer spin values,
inner (low magnetic field) ground states of the system disappear at low
magnetic field as we started to add more half-integer spin to the half
integer-integer system. As concentration rises, the magnetic ground states
dominated by half-integer spin appear and the system is in an ordered phase
anymore.

It has been demonstrated that the outermost symmetric loops disappear
gradually and windows are separated from each other, as the crystal field
parameter is increased in the negative direction. Besides, the quantities of
hysteresis loops have been investigated with the variation of the temperature.
Rising temperature drags the system into a disordered phase due to the thermal
agitations. As the concentration increases from $c=0$ to $1$, HLA, RM and CF
decreases for all binary alloy system which is one or both of two spin
variables chosen as integer or half integer spin model. These quantities
increase as the spin value gets higher.

We hope that the results obtained in this work may be beneficial form both
theoretical and experimental points of view.

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# Photocathode characterisation for robust PICOSEC Micromegas precise-timing
detectors

M. Lisowska<EMAIL_ADDRESS>R. Aleksan  Y. Angelis  S. Aune  J.
Bortfeldt  F. Brunbauer  M. Brunoldi  E. Chatzianagnostou  J. Datta  K.
Dehmelt  G. Fanourakis  S. Ferry  D. Fiorina111Now at Gran Sasso Science
Institute, Viale F. Crispi, 7 67100 L’Aquila, Italy. K. J. Floethner  M.
Gallinaro  F. Garcia  I. Giomataris  K. Gnanvo  F.J. Iguaz222Now at SOLEIL
Synchrotron, L’Orme des Merisiers, Départementale 128, 91190 Saint-Aubin,
France. D. Janssens  A. Kallitsopoulou  M. Kovacic  B. Kross  C.C. Lai  P.
Legou  J. Liu  M. Lupberger  I. Maniatis333Now at Department of Particle
Physics and Astronomy, Weizmann Institute of Science, Hrzl st. 234, Rehovot,
7610001, Israel. J. McKisson  Y. Meng  H. Muller  R. De Oliveira  E. Oliveri
G. Orlandini  A. Pandey  T. Papaevangelou  M. Pomorski  M. Robert  L.
Ropelewski  D. Sampsonidis  L. Scharenberg  T. Schneider  E. Scorsone  L.
Sohl444Now at TÜV NORD EnSys GmbH & Co. KG. M. van Stenis  Y. Tsipolitis  S.
Tzamarias  A. Utrobicic  I. Vai  R. Veenhof  L. Viezzi  P. Vitulo  C. Volpato
X. Wang  S. White  W. Xi  Z. Zhang  Y. Zhou

###### Abstract

The PICOSEC Micromegas detector is a precise-timing gaseous detector based on
a Cherenkov radiator coupled with a semi-transparent photocathode and a
Micromegas amplifying structure, targeting a time resolution of tens of
picoseconds for minimum ionising particles. Initial single-pad prototypes have
demonstrated a time resolution below $\sigma$ = 25 ps, prompting ongoing
developments to adapt the concept for applications. The achieved performance
is being transferred to robust multi-channel detector modules suitable for
large-area detection systems requiring excellent timing precision. To enhance
the robustness and stability of the PICOSEC Micromegas detector, research on
robust carbon-based photocathodes, including Diamond-Like Carbon (DLC) and
Boron Carbide (B4C), is pursued. Results from prototypes equipped with DLC and
B4C photocathodes exhibited a time resolution of $\sigma$ $\approx$ 32 ps and
$\sigma$ $\approx$ 34.5 ps, respectively. Efforts dedicated to improve
detector robustness and stability enhance the feasibility of the PICOSEC
Micromegas concept for large experiments, ensuring sustained performance while
maintaining excellent timing precision.

###### keywords:

Gaseous detectors , Micromegas , Photocathodes , Timing resolution

††journal: NIM A

[inst1]organization=European Organization for Nuclear Research (CERN), 1211
Geneve 23, Switzerland

[inst2]organization=Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

[inst3]organization=IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-
Yvette, France

[inst4]organization=Department of Physics, Aristotle University of
Thessaloniki, University Campus, GR-54124, Thessaloniki, Greece

[inst5]organization=Center for Interdisciplinary Research and Innovation
(CIRI-AUTH), Thessaloniki 57001, Greece

[inst6]organization=Department for Medical Physics, Ludwig Maximilian
University of Munich, Am Coulombwall 1, 85748 Garching, Germany

[inst7]organization=Dipartimento di Fisica, Università di Pavia, Via Bassi 6,
27100 Pavia, Italy

[inst8]organization=INFN Sezione di Pavia, Via Bassi 6, 27100 Pavia, Italy

[inst9]organization=Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794-3800, USA

[inst10]organization=Jefferson Lab, 12000 Jefferson Avenue, Newport News, VA
23606, USA

[inst11]organization=Institute of Nuclear and Particle Physics, NCSR
Demokritos, GR-15341 Agia Paraskevi, Attiki, Greece

[inst12]organization=Helmholtz-Institut für Strahlen- und Kernphysik,
University of Bonn, Nußallee 14–16, 53115 Bonn, Germany

[inst13]organization=Laboratório de Instrumentacão e Física Experimental de
Partículas, Lisbon, Portugal

[inst14]organization=Helsinki Institute of Physics, University of Helsinki,
FI-00014 Helsinki, Finland

[inst16]organization=University of Zagreb, Faculty of Electrical Engineering
and Computing, 10000 Zagreb, Croatia

[inst17]organization=European Spallation Source (ESS), Partikelgatan 2, 224 84
Lund, Sweden

[inst18]organization=State Key Laboratory of Particle Detection and
Electronics, University of Science and Technology of China, Hefei 230026,
China

[inst19]organization=Physikalisches Institut, University of Bonn, Nußallee 12,
53115 Bonn, Germany

[inst20]organization=Friedrich-Alexander-Universität Erlangen-Nürnberg,
Schloßplatz 4, 91054 Erlangen, Germany

[inst21]organization=CEA-LIST, Diamond Sensors Laboratory, CEA Saclay, F-91191
Gif-sur-Yvette, France

[inst22]organization=Queen’s University, Kingston, Ontario, Canada

[inst23]organization=National Technical University of Athens, Athens, Greece

[inst24]organization=Ruđer Bošković Institute, Bijenička cesta 54., 10 000
Zagreb, Croatia

[inst25]organization=Department of Physics and Astronomy, University of
Florence, Via Giovanni Sansone 1, 50019 Sesto Fiorentino, Italy

[inst26]organization=University of Virginia, USA

Figure 1: (a) PICOSEC detection concept: A charged particle passing through a
Cherenkov radiator generates UV photons, which are converted into electrons on
a photocathode. The electrons are multiplied in the gas volume in two stages
and induce a signal on the anode. (b) Typical PICOSEC waveform: The signal
displays a fast electron peak and a slow ion tail. Note that the figures are
not drawn to scale [11].

##  1 Introduction

The intense interest in advancing technologies for precise-timing detectors
has been driven by demanding environments anticipated in forthcoming High
Energy Physics experiments. Meeting the criteria of achieving an excellent
time resolution, ensuring stable long-term operation and providing large-area
coverage is crucial for adapting the concept for physics applications. The
PICOSEC Micromegas (hereafter PICOSEC) project [1] undertakes the efforts to
develop a robust multi-channel gaseous detector, targeting a time resolution
of $\mathcal{O}(10)$ ps for Minimum Ionising Particles (MIPs). Initial single-
pad prototypes equipped with a Cesium Iodide (CsI) photocathode have
demonstrated a time resolution below $\sigma$ = 25 ps [1], prompting further
developments [2] \- [13]. Although preliminary measurements of alternative
approaches have been conducted previously [2, 5, 11, 13], a comprehensive
characterisation of different carbon-based photocathodes has not been
reported. This work aims to advance the development of robust photocathodes
for PICOSEC detectors while maintaining an excellent time resolution.

Within the scope of this paper, measurements of three different photocathode
materials, including CsI, Diamond-Like Carbon (DLC) and Boron Carbide (B4C),
are reported. The samples were manufactured by various research institutes,
including the European Organization for Nuclear Research (CERN), the
University of Science and Technology of China (USTC), the French Atomic Energy
Commission (CEA) and the European Spallation Source (ESS).

##  2 PICOSEC Micromegas detection concept

The PICOSEC Micromegas detection concept is depicted in Fig. 1a [1]. The
detector is designed to minimize time jitter from primary ionization in the
gas by using a Cherenkov radiator. A charged particle passing through the
crystal radiator generates a cone of ultraviolet (UV) photons which are
partially converted into primary electrons on a photocathode coated on the
window. Typically, the radiator in a PICOSEC detector is a 3 mm thick
Magnesium Fluoride (MgF2) crystal which transmits light above 120 nm. The
window is coated with a 3 nm thick chromium (Cr) conductive interfacial layer
that serves as a contact for the high voltage (HV). The UV photon-to-electron
converter is an 18 nm thick semi-transparent CsI photocathode known for its
high Quantum Efficiency (QE) for photons below 200 nm. All primary electrons
are created at the same surface, eliminating uncertainty about the ionisation
location and thereby minimising time jitter. The gas volume is filled with a
mixture of 80$\%$ Ne, 10$\%$ CF4 and 10$\%$ C2H6 at ambient pressure. The
typical electric fields across the pre-amplification and amplification gaps
are approximately 40 kV/cm and 20 kV/cm, correspondingly. In the presence of a
strong electric field, the extracted electrons successively ionise gas
molecules, causing electron multiplication, which first occurs in the pre-
amplification gap and, after passing through the Micromegas mesh, continues in
the amplification gap. The total gain achieved is on the level of
$\mathcal{O}(10^{5}-10^{6})$. The amplified electrons move towards the anode,
while the ions travel to the mesh. Their movement induce a signal on the
anode, which is then amplified and read out by a digitiser. A typical PICOSEC
waveform displaying a fast electron peak and a slow ion tail is illustrated in
Fig. 1b. The leading edge of the electron peak determines a Signal Arrival
Time (SAT).

Several optimisation studies were conducted on a 10 mm diameter active area
single-pad PICOSEC detector to enhance high voltage stability, reduce noise,
improve signal integrity and achieve uniform timing response, using a
simplified assembly process [12]. The design includes improvements to the
detector board, vessel, mechanical parts and electrical connections for both
high voltage and signal. A new single-channel detector board is shown in Fig.
2.

To enhance the robustness and stability of the PICOSEC detector, research on
robust photocathodes is underway. The initial prototype involved a CsI
photocathode due to its high QE and UV sensitivity, nonetheless, the material
is vulnerable to damage from ion blackflow, discharges and humidity. The
robustness of the photocathode is crucial for maintaining detector efficiency
and timing resolution during long-term operation. Potential alternatives
include using protective layers or other materials, with carbon-based
photocathodes like DLC and B4C being the most promising candidates.

Figure 2: Single-channel detector board with a 10 mm diameter active area.

##  3 Experimental methodology

R&D activities within the PICOSEC project have covered aspects ranging from
design and production, through assembly, to measurements in both laboratory
conditions as well as with particle beams. Prototypes were assembled in a
clean-room environment and tested in the laboratory. The ASSET (A Small Sample
Evaporation Test) setup was developed to characterise photocathodes,
facilitating transparency and QE measurements, as well as ageing studies [14,
15]. The time resolution of the detectors was evaluated using 150 GeV/c muon
beams at the CERN SPS H4 beamline.

###  3.1 ASSET photocathode characterisation device

The ASSET setup was developed to characterise photocathodes for gaseous
radiation detectors, as in the PICOSEC project [5]. The primary aims are to
quantify QE and transparency of the photocathodes as well as their degradation
due to ion backflow. Fig. 3 provides an overview of the ASSET setup.

Figure 3: Overview of the ASSET photocathode characterisation device: The
setup uses UV light from a deuterium lamp and a VUV monochromator, features a
collimating mirror chamber, a beam splitter and two calibrated CsI PMTs to
measure light intensity and stability within a wavelength range of 140-200 nm,
peaking at 160-180 nm. It allows for measurements in reflective and
transmission mode, as well as studies of the ageing process due to ion
backflow [5].

Measurements utilise UV light from a system consisting of a deuterium lamp
(McPherson, Model 632) and a Vacuum UltraViolet (VUV) monochromator
(McPherson, Monarch). The system is flushed with high-purity nitrogen to
prevent absorption of the short-wavelength portion of the UV spectrum. The
monochromator is attached to the measurement chamber through an extension
containing a collimating mirror chamber that focuses the light. A beam
splitter divides the light into two beams. Two calibrated CsI photomultiplier
tubes (PMTs) register the light intensity: one measures the amount of light at
the sample position, while the other monitors light stability during
measurements. The wavelength range for measurements is limited to about 140 nm
on the low-wavelength side by the transparency of the MgF2 lens and to about
200 nm on the high-wavelength side by the deuterium lamp’s light intensity.
The highest light intensity is observed at 160-180 nm.

The ASSET setup allows for measurements in reflective and transmission modes.
In both modes, the extraction grids record the current generated by electrons
extracted from the photocathodes. The reflective mode measures the QE of
photocathodes, while the transmission mode evaluates both QE and transparency.
The measurements are conducted in a vacuum of 10-6 mbar, achieved using pre-
pumps and turbomolecular pumps. Additionally, the device enables the the
collection of determined amounts of ion current on samples to study potential
charge-induced ageing effects. Further details about the ASSET photocathode
characterisation device can be found in [5].

The primary function of the ASSET setup is the measurement of photocurrent
from photocathode samples. The QE is calculated from current measurements
using the formula

$QE=\frac{N_{e}}{N_{ph}}=\frac{\frac{I_{s}}{e}}{\frac{I_{PMT}}{e\cdot
C_{PMT}}}=\frac{I_{s}\cdot C_{PMT}}{I_{PMT}},$ (1)

where $N_{e}$ is the number of electrons extracted from the sample (measured
on the sample), $N_{ph}$ is the number of photons that arrive at the sample
(measured with the PMT), $I_{s}$ is the current measured on the sample (offset
subtracted), $I_{PMT}$ is the current measured on the PMT (offset subtracted),
$C_{PMT}$ is the calibration factor of PMT (depends on wavelength) and $e$ is
the elementary charge.

The current measured by the PMT is used to calculate the transparency (T) of
the samples, given by

$T=\frac{{I}_{PMT,~{}s.~{}in}}{{I}_{PMT,~{}s.~{}out}},$ (2)

where ${I}_{PMT,~{}s.~{}in}$ is the PMT current measured with the sample in
the measurement position and ${I}_{PMT,~{}s.~{}out}$ is the PMT current
measured with the sample out of measurement position. Transparency
measurements in ASSET was performed in the VUV range between 140 nm and 200
nm. Transparency can also be assessed across the visible light spectrum, from
200 nm to 800 nm, using a spectrophotometer (PerkinElmer, Lambda 650 UV/VIS
Spectrometer).

An important capability of the ASSET measurements is conducting ageing
studies. The procedure starts with an X-ray beam entering the irradiation
chamber filled with a gas mixture of Ar/CO2 (70/30 %) at ambient pressure that
ionises the gas and creates primary electrons. The primary electrons are
attracted to multiplication wires, where an avalanche occurs in the high
electric field region near the wires, producing additional electrons and ions.
The electrons from he avalanche are collected on the wires, while the ions
move towards the grounded sample. This process accumulates charge on the
sample, allowing for the quantification of the drop in QE after exposure.

###  3.2 Characterisation with particle beams

Particle beam campaigns are conducted to measure the time resolution of
prototypes assembled in various configurations. The measurements are conducted
using 150 GeV/c muon beams at the CERN SPS H4 beamline. The test setup
includes a beam telescope facilitating triggering, timing and tracking
capabilities. An example telescope configuration is shown in Fig. 4. Precise
particle tracking is achieved utilising three triple Gas Electron Multiplier
(GEM) detectors with a spatial resolution below 80 µm. These devices use APV25
front-end ASICs to shape the signals from all electrode channels, which are
subsequently digitised using the Scalable Readout System (SRS) [16]. The GEMs
are operated in a gas mixture of Ar/CO2 (70/30 %) at ambient pressure. A
micro-channel plate photomultiplier tube (MCP-PMT, Hamamatsu, R3809U-50)
serves as the timing reference and data acquisition (DAQ) trigger. The
telescope can be utilised for testing several PICOSEC prototypes
simultaneously.

Different components individually contribute to the overall time resolution in
a detector system, with the total time resolution being the sum of variances
from each of these contributions. It can be represented by a simplified model

${\sigma_{tot}}^{2}={\left({\sigma_{SPE}}\over{\sqrt{N_{PE}}}\right)}^{2}+{\sigma_{t_{0}}}^{2}+{\sigma_{n}}^{2}+...$
(3)

The first term is the time resolution of the PICOSEC prototype measured with
MIPs, where $\sigma_{SPE}$ is the time resolution achieved with SPE events and
$N_{PE}$ is the number of photoelectrons extracted from the photocathode [2].
The second term is the contribution of the reference device. The third term is
the time jitter caused by noise in the detector and the connected external
electrical circuit. The time resolutions discussed in this paper include
contributions from all these components.

The photocathode performance is assessed by comparing $N_{PE}$ they generate
during the process. The number is estimated as the ratio of the signal
amplitude induced by a single photoelectron (SPE) to the signal amplitude
induced by a MIP. SPE measurements are performed using a light-emitting diode
(LED), while the multiple photoelectron (MPE) measurements are conducted with
muon beams. The signals are amplified by a custom-developed charge-sensitive
ARTH amplifier [2] and read out by an oscilloscope (LeCroy WR8104, 10 GS/s
sampling rate, 1 GHz bandwidth).

The procedure for calculating $N_{PE}$ involves several steps. Firstly, the
maximum amplitude for each waveform is determined. Secondly, a histogram of
all maximum amplitudes is plotted. The noise component is fitted using a
Gaussian distribution, while the signal component is fitted using an equation
based on a Pólya distribution of the form

$P_{n}=\frac{(\theta+1)^{\theta+1}}{\bar{n}\Gamma(\theta+1)}\left(\frac{n}{\bar{n}}\right)^{\theta}e^{-(\theta+1)n/\bar{n}},$
(4)

where: $n$ is the number of charges produced, $\bar{n}$ the mean avalanche
size, and $\theta$ the shape parameter. From this Pólya distribution, the mean
value is calculated. Finally, the mean amplitude of the MPE is divided by the
mean amplitude of the SPE to obtain the NPE for a given photocathode. The
$N_{PE}$ is given by the formula

$N_{PE}=\frac{{V}_{MIP}}{{V}_{SPE}},$ (5)

where ${V}_{MIP}$ is a MIP mean amplitude, ${V}_{SPE}$ is a SPE mean
amplitude.

Quantifying the time resolution of the PICOSEC detector requires a reference
device with significantly superior timing precision. To fulfill this
condition, the MCP-PMT with a time resolution below 6 ps in the central region
[1, 2] serves as the reference for the PICOSEC prototypes. The devices are
aligned to each other, ensuring that particles passed through the active area
of both. To amplify and read out the signal, a custom-developed RF pulse
amplifier cards optimized for PICOSEC (with built-in discharge protection up
to 350 V, 650 MHz bandwidth, 38 dB gain, 75 mW power consumption [10, 17]) are
used in conjunction with an oscilloscope.

In the analysis, the constant fraction discrimination (CFD) method is employed
to determine the timestamp at which the signal reaches a specific fraction of
its peak amplitude, thereby addressing the issue of time walk [1]. The leading
edge of the electron peak is fitted with a sigmoid function to identify the
signal position in time at 20$\%$ of the its amplitude. The SAT is calculated
as the difference between the timestamps of the PICOSEC detector and the
reference device. The SAT distribution is analysed using a double Gaussian fit
and time resolution of the detector system is computed as the standard
deviation of this distribution.

Figure 4: The beam telescope equipped with three triple-GEMs for particle
tracking, an MCP-PMT for timing reference as well as DAQ trigger, and PICOSEC
prototypes for testing.

The results presented in this paper are derived from calculations conducted
after applying several selection criteria (cuts) to the triggered events.
Initially, a time window cut is implemented, selecting events within 300 ps of
the median time difference of all recorded signals to exclude off-time events
and noise fluctuations. Additional cuts are applied to set signal amplitude
limits. Events with amplitudes below 1% of the dynamic range, categorised as
empty events, and those above 99%, classified as saturated waveforms, are
rejected. Furthermore, a geometrical cut is essential due to the Cherenkov
radiator’s emission of a UV photon cone at approximately 45-degree angle. To
ensure accurate time resolution measurements, only fully contained events
within the detector’s active area should be included. Specifically, a cut of a
4 mm diameter circle around the pad center for a 10 mm diameter detector
equipped with a 3 mm thick radiator (or a larger diameter, if otherwise
specified) was implemented. Signals from tracks that pass outside this central
region show reduced amplitude because of the partial loss of NPE beyond the
area of the channel.

##  4 Photocathode characterisation

Three different photocathode materials including Cesium Iodide, Diamond-Like
Carbon and Boron Carbide have been examined. The fabrication, characterisation
and performance evaluation of these photocathode materials have been detailed,
highlighting their strengths and challenges in achieving precise timing in
advanced detection systems. To ensure consistency of the results, all
measurements presented in this paper were performed using identical single-pad
non-resistive detectors featuring a 10 mm diameter active area (or 15 mm if
otherwise specified), with pre-amplification and amplification gaps of
approximately 127 µm. The detectors were operated in a sealed mode,
maintaining a gas pressure of 990 $\pm$ 5 mbar for the purpose of comparison.

###  4.1 Cesium Iodide

Cesium Iodide is the most widely used material for photocathodes in gaseous
radiation detectors due to its high QE and its sensitivity to UV radiation.
The baseline photocathode used in the PICOSEC detector consists of an 18 nm
thick CsI layer deposited onto a 3 mm thick MgF2 substrate with a 3 nm thick
Cr conductive interfacial layer, exceeding 12 photoelectrons per MIP [1],
Previous studies on transparency in the VUV range have shown that the pure
MgF2 substrate has approximately 80% transparency, which decreases to 45% with
the addition of the Cr layer, and further to 10% with the CsI layer [5].

Particle beam measurements were performed using a PICOSEC prototype equipped
with a CsI photocathode, operated in a sealed mode at a gas pressure of 985
mbar. At voltage settings of V${}_{\text{C}}$ = -430 V on the cathode and
V${}_{\text{A}}$ = 265 V on the anode, the detector exhibited a time
resolution of $\sigma$ = 15.8 $\pm$ 2.2 ps, as illustrated in Fig. 5.

Figure 5: SAT distribution of a single-pad prototype, featuring a 10 mm
diameter active area, equipped with an 18 nm thick CsI photocathode with a 3
nm thick Cr conductive layer, operated in a sealed mode at a gas pressure of
985 mbar. The voltage settings were V${}_{\text{C}}$ = -430 V on the cathode
and V${}_{\text{A}}$ = 265 V on the anode. The histogram consists of the data
after implementing a geometrical cut of a 4 mm diameter circle around the pad
center to include only fully contained events. The results of a double
Gaussian fit yield a time resolution of $\sigma$ = 15.8 $\pm$ 2.2 ps.

Despite exhibiting excellent time resolution, CsI is vulnerable to damage from
ion backflow, discharges and humidity. Fig. 6 illustrates the decline in QE
observed in a CsI sample before and after multiple irradiation steps,
providing evidence of ion bombardment affecting CsI. Studies demonstrated that
after accumulating a charge of 6 mC/cm2, the QE for CsI decreased by 80%,
whereas for B4C, it decreased by 45%, indicating a more rapid degradation of
CsI [5]. One possible solution to reduce the impact of ion backflow on CsI and
mitigate degradation involves introducing protective layers, such as MgF2 or
Lithium Fluoride (LiF) [18]. Nonetheless, tests conducted within the PICOSEC
project revealed that these layers inhibited electron extraction, leading to
reduced QE [5]. Another approach to address the issue of non-robustness is to
explore alternative materials, with carbon-based photocathodes like DLC and
B4C being the most promising candidates.

###  4.2 Diamond-Like Carbon

Diamond-Like Carbon represents an alternative photocathode material that
offers greater resistance to environmental influences compared to CsI. DLC
belongs to a class of amorphous carbon materials, exhibiting excellent
properties including chemical and thermal stability, hardness and robustness.
Initial measurements of DLC photocathodes have been conducted by collaborators
from USTC [13], including tests of various layer thicknesses, measurements of
QE, NPE and time resolution as well as aging studies. DLC samples consistently
demonstrated good performance and robustness, making them a strong candidate
for PICOSEC photocathodes.

Figure 6: Ageing studies of a CsI photocathode performed in the ASSET setup:
Decrease in QE after multiple irradiation steps, indicating the impact of ion
bombardment on CsI [5]. Figure 7: Picture of DLC photocathodes with various
layer thicknesses deposited at the CERN MPT workshop using a magnetron
sputtering technique.

The DLC photocathodes presented in this paper were fabricated using a
magnetron sputtering technique at the CERN Micro-Pattern Technologies (MPT)
workshop. DLC photocathodes with thicknesses ranging from 1 nm to 100 nm were
deposited on glass and MgF2 substrates, with some samples including a Cr
interfacial layer. Fig. 7 depicts the DLC photocathodes with the various
thicknesses produced during one of the two deposition campaigns. The process
proved to be reproducible across both campaigns.

Profilometer measurements were performed to investigate the thicknesses of the
50 nm and 100 nm layers. For layers in the range of a few nanometers,
specifically from 1.5 nm to 4.5 nm, thicknesses were estimated by scaling the
coating time. Fig. 8 presents the results of transparency measurements
conducted in the wavelength range from 200 nm to 800 nm using a
spectrophotometer. The data demonstrate a correlation between the estimated
thicknesses and the measured transparency. The transparency of samples with a
thickness of a few nanometers in the VUV range is approximately 60%.

Figure 8: Transparency measurements of the DLC layers with thicknesses ranging
from 1.5 nm to 100 nm deposited on MgF2 and glass substrates, conducted in the
wavelength range from 200 nm to 800 nm using a spectrophotometer.

Surface resistivity measurements were conducted, exhibiting a correlation
between the thickness of the DLC layer and its resistivity, as shown in Fig.
9. The higher resistivity of the DLC photocathodes on the MgF2 radiator
compared to the glass substrate suggests that the layer deposited on the MgF2
is thinner, potentially due to the lower adhesion of the crystal.

Figure 9: Surface resistivity measurements of the DLC photocathodes,
presenting a correlation between estimated thicknesses and resistivity. The
higher resistivity of the DLC photocathodes on the MgF2 radiator compared to
the glass substrate suggests a thinner deposited layer on the MgF2, possibly
due to lower crystal adhesion.

The DLC photocathodes, with thicknesses ranging from 1.5 nm to 3.5 nm, were
characterised during particle beam measurements. The measurements were
performed at a gas pressure of 990 mbar. The NPE generated by the DLC
photocathodes varied between 2.5 and 3. The 10 mm diameter active area
detector featuring a 1.5 nm thick DLC photocathode deposited directly on the
radiator and operated at voltages of V${}_{\text{C}}$ = -500 V on the cathode
and V${}_{\text{A}}$ = 275 V on the anode demonstrated the best performance,
achieving a time resolution of $\sigma$ = 31.9 $\pm$ 1.3 ps, as illustrated in
Fig. 10. Thicker samples exhibited comparable time resolution values, with a
deviation of approximately 4 ps.

Figure 10: SAT distribution of a prototype equipped with a 1.5 nm DLC
photocathode. The voltage settings were V${}_{\text{C}}$ = -500 V on the
cathode and V${}_{\text{A}}$ = 275 V on the anode. The results show a time
resolution of $\sigma$ = 31.9 $\pm$ 1.3 ps.

The DLC photocathodes deposited directly on the radiator without a Cr
conductive interfacial layer are sufficient for studying photocathode
performance. Nonetheless, in high-rate environments, a Cr layer is essential
to mitigate charging-up effects and voltage drops, particularly for samples
with larger surface areas. Consequently, samples with Cr were evaluated,
showing a decrease in transparency of approximately 30% and 2 ps worse time
resolution.

To enhance UV photon production, a thicker radiator was introduced. At the
same time, the Cherenkov cone diameter was widened. A prototype with a 15 mm
diameter active area, equipped with a 2.5 nm thick DLC photocathode deposited
on a 5 mm thick MgF2 radiator, was tested. The analysis involved applying a
geometric cut, consisting of a 5 mm diameter circle around the pad center, to
include only fully contained events. The detector operated at of
V${}_{\text{C}}$ = -490 V on the cathode and V${}_{\text{A}}$ = 275 V on the
anode yielded a time resolution of $\sigma$ = 28.0 $\pm$ 1.4 ps.

###  4.3 Boron Carbide

Boron Carbide is a chemical compound from the carbide group, known for its
crystalline structure and exceptional hardness, often used as a substitute for
diamond. The initial B4C photocathodes were developed at CEA, where a wide
range of thicknesses was tested, yielding promising results [2, 11]. More
detailed studies discussed in this paper were conducted with B4C samples
fabricated using a sputtering technique at ESS. During the coating campaigns,
B4C photocathodes with thicknesses ranging from 7 nm to 15 nm were deposited
on MgF2 radiators with a Cr conductive interfacial layers.

Profilometer measurements were conducted to determine the thickness of the
photocathodes. The 7 nm layer was estimated based on the scaling of the
coating time, considering the machine’s resolution limits. Transparency
measurements of the B4C layers, conducted in the VUV wavelength range,
demonstrated a correlation between estimated thicknesses and transparency,
with values ranging from 40% for the thinnest layer to 20% for the thickest.

The B4C photocathodes were evaluated through particle beam measurements. The
prototypes were operated at a gas pressure of 990 mbar. The NPE created by the
B4C photocathodes ranged between 2 and 4, with higher PE production observed
in thinner layers. The detector equipped with a 9 nm thick B4C photocathode
and operated at V${}_{\text{C}}$ = -490 V on the cathode and V${}_{\text{A}}$
= 275 V on the anode, exhibited the best performance, achieving a time
resolution of $\sigma$ = 34.5 $\pm$ 1.5 ps, as shown in Fig. 11. The time
resolution of the remaining B4C samples decrease, with the thickest sample
showing a degradation of approximately 10 ps.

Figure 11: SAT distribution of a prototype equipped with a 9 nm B4C
photocathode. The voltages were at V${}_{\text{C}}$ = -490 V on the cathode
and V${}_{\text{A}}$ = 275 V on the anode. The results show a time resolution
of $\sigma$ = 34.5 $\pm$ 1.5 ps.

##  5 Discussion

In the comparative analysis of photocathode materials, distinct performance
characteristics were observed among CsI, DLC, and B4C. CsI exhibited the
highest QE, exceeding 12 photoelectrons per MIP, whereas DLC yielded QE values
ranging between 2.5 and 3, while B4C between 2 and 4. Despite B4C being more
transparent than DLC, both materials exhibit similar efficiency, generating
comparable NPE. NPE correlates with the achievable time resolution. CsI
showcased the best performance with $\sigma$ $\approx$ 15.8 ps, followed by
DLC with $\sigma$ $\approx$ 32 ps and B4C with $\sigma$ $\approx$ 34.5 ps.
Although demonstrating excellent timing results, CsI also exhibited a faster
degradation in QE compared to carbon-based materials. Furthermore, carbon-
based materials displayed notably greater resistance to humidity. Considering
the demand for robust detectors with timing specifications that allow a margin
on operating condition, DLC and B4C emerge as promising candidates.
Consequently, a detector configuration comprising a single-pad prototype with
a 15 mm diameter active area resistive Micromegas of 20 M$\Omega$/$\Box$,
utilising a 1.5 nm DLC photocathode and an amplifier integrated on the outer
PCB, was tested. Operating the device at V${}_{\text{C}}$ = -530 V on the
cathode and V${}_{\text{A}}$ = 275 V on the anode, a time resolution of
$\sigma$ = 31.4 $\pm$ 0.6 ps was achieved within a 9 mm diameter circle around
the pad center, including only fully contained events. A uniform time response
across this region, with an RMS = 38.8 $\pm$ 0.3 ps, was observed, as depicted
in Fig. 12.

##  6 Conclusions

The work described in this paper is focused on enhancing the robustness of
PICOSEC detectors. The research delves into the characterisation of robust
carbon-based photocathodes, such as DLC and B4C. Results obtained from
prototypes equipped with DLC and B4C photocathodes revealed time resolutions
of approximately $\sigma$ $\approx$ 32 ps and $\sigma$ $\approx$ 34.5 ps,
correspondingly. Efforts dedicated to detector developments increase the
feasibility of the PICOSEC concept for experiments requiring sustained
performance while maintaining excellent timing precision. Current developments
include research on alternative materials, such as titanium to replace Cr, and
carbon-based nanostructures for use as photocathodes. Simultaneously, a 10×10
cm2 robust photocathode, incorporating a conductive interlayer to prevent a
voltage drop, will be tested with a 100-channel prototype [10] and a SAMPIC
digitiser [19]. In view of improving the stability of the detector, the
production of a high-rate 10×10 cm2 Micromegas with double-layer DLC for
vertical charge evacuation and evaluation of rate capability is ongoing.
Additionally, efforts to enhance spatial resolution involve testing high-
granularity layouts. Scaling up the PICOSEC detector by tiling 10×10 cm2
modules or developing larger prototypes are the next steps.

Figure 12: Time resolution map of a detector configuration consisting of a 15
mm diameter active area resistive Micromegas of 20 M$\Omega$/$\Box$, a 1.5 nm
DLC photocathode and an integrated amplifier. The black circle indicates the
active area of the detector, while the red circle highlights fully contained
events.

## Acknowledgements

We acknowledge the support of the CERN EP R&D Strategic Programme on
Technologies for Future Experiments; the RD51 Collaboration, in the framework
of RD51 common projects; the DRD1 Collaboration; the PHENIICS Doctoral School
Program of Université Paris-Saclay, France; the Cross-Disciplinary Program on
Instrumentation and Detection of CEA, the French Alternative Energies and
Atomic Energy Commission; the French National Research Agency (ANR), project
id ANR-21-CE31-0027; the Program of National Natural Science Foundation of
China, grants number 11935014 and 12125505; the COFUND-FP-CERN-2014 program,
grant number 665779; the Fundação para a Ciência e a Tecnologia (FCT),
Portugal; the Enhanced Eurotalents program, PCOFUND-GA-2013-600382; the US CMS
program under DOE contract No. DE-AC02-07CH11359; this material is based upon
work supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics under contracts DE-AC05-06OR23177.

## References

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