# Validating Mean Field Theory in a New Complex, Disordered High-Entropy
Spinel Oxide

Neha Sharma  Department of Physics and Material Science, Thapar Institute of
Engineering and Technology, Patiala 147004, India  Nikita Sharma  Department
of Physics and Material Science, Thapar Institute of Engineering and
Technology, Patiala 147004, India  Jyoti Sharma  UGC-DAE Consortium for
Scientific Research, Khandwa Road, Indore 452001, Madhya Pradesh, India  S. D.
Kaushik  UGC-DAE Consortium for Scientific Research, BARC Campus, Trombay
Mumbai - 400085, India  Sanjoy Kr. Mahatha  UGC-DAE Consortium for Scientific
Research, Khandwa Road, Indore 452001, Madhya Pradesh, India  Tirthankar
Chakraborty<EMAIL_ADDRESS>Department of Physics and Material Science,
Thapar Institute of Engineering and Technology, Patiala 147004, India  Sourav
Marik<EMAIL_ADDRESS>Department of Physics and Material Science, Thapar
Institute of Engineering and Technology, Patiala 147004, India

###### Abstract

The advent of novel high-entropy oxides has sparked substantial research
interest due to their exceptional functional properties, which often surpass
the mere sum of their constituent elements’ characteristics. This study
introduces a complex high-entropy spinel oxide with composition
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. We performed comprehensive
structural (X-ray and Neutron diffraction), microstructural, magnetic, and
local electronic structure investigations on this material. Despite the
material’s high degree of disorder, detailed magnetization measurements and
low temperature neutron powder diffraction studies reveal long-range
ferrimagnetic ordering beginning at 293 K. The sample exhibits a high
saturation magnetization of 766 emu-cm3 (at 50 K), a low coercivity (HC) of
100 Oe (50 K), a high transition temperature (TC) around room temperature, and
high resistivity value of 4000 Ohm-cm at room temperature, indicating its
potential for high density memory devices. The magnetic structure is
determined using a collinear-type ferrimagnetic model with a propagation
vector k = 0,0,0. Various analytical techniques, including modified Arrott
plots, Kouvel-Fischer analysis, and critical isotherm analysis, are employed
to investigate the phase transitions and magnetic properties of this complex
system. Our results indicate a second-order phase transition. Remarkably,
despite the complex structure and significant disorder, the critical exponents
obtained are consistent with the mean field model. The high entropy leads to a
remarkably homogeneous distribution of multiple cations, validating the
approximation of average local magnetic environments and supporting the mean
field theory.

Keywords

##  I Introduction

High entropy oxides (HEOs) have recently emerged as a promising and exciting
field of research due to their extraordinary properties and potential for
revolutionizing of material design and properties Oses  _et al._ (2020);
Aamlid  _et al._ (2023); Miracle and Senkov (2017); Ye  _et al._ (2016). These
compounds are characterized by incorporating multiple elements, typically five
or more in equimolar ratios, into a single crystallographic site. This
incorporation results in enhanced configurational entropy ($\Delta S_{conf}$)
which stabilizes the structure of these compounds, creating what is known as
an entropy-stabilized phase. The first high entropy oxide,
(Mg0.2Co0.2Ni0.2Cu0.2Zn0.2)O was stabilized and investigated by Rost et al.
Rost  _et al._ (2015) in 2015. This pioneering work paved the way for
extensive research into high entropy oxides (HEOs), driven by their easy
synthesis, remarkable stability, exceptional functional properties, and the
unique design flexibility that includes multiple cations in these materials.

HEOs exhibit a range of exceptional properties that make them promising for
various technological applications, including catalysts, thermoelectrics, and
magnetic materials Yan and Zhang (2020); Sarkar  _et al._ (2019); Anandkumar
and Trofimov (2023); Jana and Maiti (2023); Chen  _et al._ (2024); Katzbaer
_et al._ (2023); Cocconcelli  _et al._ (2024); Jin  _et al._ (2023a); Mazza
_et al._ (2022). Among HEOs, the spinel family stands out as an intriguing
material system for studying the interaction between disorder-driven
complexity and the relationship between structural and functional properties.
In the spinel structure AB2O4, A ions are located at tetrahedral sites, while
B ions occupy octahedral sites. The B ions arrange themselves into a
pyrochlore network of interconnected tetrahedra, which leads to substantial
magnetic frustration. Concurrently, the A cations form a diamond-like network
that exhibits antiferromagnetic collinear order, adding to the magnetic
frustration Buttgen  _et al._ (2004); Fritsch  _et al._ (2004). The unique
crystal structure of the spinel system allows for entropy modification at both
A and B sites, providing a rich platform for exploring tunable properties in
spintronics and magnetism. Prior research on high entropy spinel oxides has
revealed intriguing physical and magnetic characteristics, including tunable
magnetism, robust ferrimagnetism, switchable magnetic anisotropy, enhanced
exchange bias effect, and high magnetic frustration Johnstone  _et al._
(2022); Jin  _et al._ (2023b); Sharma  _et al._ (2023, 2024); Sarkar  _et al._
(2023). The high entropy stabilization facilitates precise atomic-scale tuning
of materials, enabling the development of materials with tailored properties
for specific applications. However, synthesizing HEOs with precise
compositions and desired properties remains a significant challenge. This
creates the need for detailed exploration and investigation of new strongly
correlated high entropy oxides with novel structures and intriguing
properties. To date, most studies have focused on the structure and
physicochemical properties of new high entropy oxides, while comprehensive
studies on the homogeneity of the magnetic structure (despite high disorder)
and the nature of magnetic phase transitions through critical analysis are
still missing.

Critical analysis plays a crucial role in understanding the phase transitions
and magnetic properties of complex systems. The investigation of critical
behavior at the ferromagnetic to paramagnetic phase transition involves
studying critical exponents using various analytical techniques such as
modified Arrott plots, Kouvel-Fischer, and critical isotherm analysis. These
analyses reveal the nature of the magnetic phase transition elucidating the
conventional universality classes. In this paper, we studied the critical
behavior of a new high entropy spinel oxide system with the chemical
composition (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.666Fe0.666Cr0.666)O4. Notably,
magnetic cations are incorporated in both A and B sites. The nuclear and
magnetic structure of the material is characterized using X-ray and neutron
powder diffraction measurements. Through various approaches, we performed an
in-depth analysis of the critical exponents around phase transition
temperature to investigate and elucidate the nature of the magnetic phase
transition. Remarkably, despite the high degree of disorder, the derived
critical exponents closely match those predicted by the mean field model of
conventional universality classes. We propose that the high configurational
entropy in this complex oxide may contribute to its conformity to such a
simplified model.

Figure 1:  (a) Crystal structure of the the material (AB2O4 type spinel). (b)
Room temperature FESEM image and corresponding EDS mapping of
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4.

##  II Experimental Details

Synthesis. The polycrystalline material having chemical composition
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.666Fe0.666Cr0.666)O4 are prepared via the
conventional solid-state reaction method.The compound was produced using
stoichiometric amounts of high purity powders, specifically Co3O4 (99.9%), CuO
(99.9%), ZnO (99.9%), NiO (99.9%) and Cr2O3 (99.9%),, MgO(99.99%) and MnO2
(99.9%) and Fe2O3 (99.99%). Using a mortar and pestle, the components were
first combined. After the mixture was homogenized, it underwent multiple heat
treatments before being air quenched and then final sintering of material was
performed at 1373 K for 36 hours.

X-ray Diffraction and Neutron Diffraction. We have performed X-ray diffraction
(XRD) experiments using the powder of the materials. The powder XRD patterns
were collected at room temperature (RT) using a Rigaku X-ray diffractometer
(Cu-Kα, $\lambda$ = 1.54056 Å). Further, Neutron diffraction (NPD)
measurements were conducted on
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.666Fe0.666Cr0.666)O4 at 10 K and 300 K using a
multi-Position Sensitive Detector (PSD) Focusing Crystal Diffractometer (FCD)
provided by the UGC-DAE Consortium for Scientific Research at the Mumbai
center, located at the National Facility for Neutron Beam Research (NFNBR)
within the Dhruva reactor in Mumbai, India. These measurements were performed
at a wavelength of 1.48 Å. We used a vanadium sample holder, which was
directly exposed to the neutron beam. The diffracted neutrons were recorded
using He3 filled linear position-sensitive detectors, employing the charge
division method.

Field Emission Scanning Electron Microscopy (FESEM), and Energy Dispersive
Spectroscopy (EDS). We utilized field emission scanning electron microscopy
(FE-SEM) and electron diffraction to investigate the microstructure and
chemical homogeneity of the high entropy materials. The chemical composition
was determined using Energy Dispersive X-ray Spectroscopy (EDS) with a BRUKER
XFlash 6160 detector, while the microstructure analysis was conducted with a
ZEISS GEMINI FE-SEM.

Magnetic Measurements. Magnetization measurements were conducted using a
Quantum Design MPMS 3 superconducting quantum interference device (SQUID).
Both field-cooled (FC) and zero-field-cooled (ZFC) modes were employed across
a temperature range from 4 K to 375 K. In addition, multiple M-H loops were
measured at temperatures ranging from 4 K to 310 K, utilizing magnetic fields
varying between 100 Oe and 2 T.

X-Ray Absorption Spectroscopy. Soft X-ray absorption spectroscopy (SXAS)
measurements on the L-edges of the constituent transition metal atoms were
performed at the SXAS beamline (BL-01) of the Indus2 synchrotron source,
located at the Raja Ramanna Centre for Advanced Technology (RRCAT) in Indore,
India. All the measurements were conducted at room temperature with an energy
resolution of approximately 0.5 eV. The CTM4XAS (Charge transfer multiplet
program for X-ray absorption spectroscopy) software was used to simulate the
XAS spectra of various transition metal elements in the sample.

##  III Scaling analysis

The scaling hypothesis and analysis are essential in understanding critical
phenomena, where physical properties remain scale-invariant near the critical
point. The hypothesis states that certain physical quantities, which undergo
changes during the phase transition, can be described as functions of the
external variables driving the transition. Scaling analysis involves
extracting critical exponents from data collapses, facilitating the
classification of transitions based on universal critical behavior that
transcends the microscopic details of individual systems. According to the
scaling hypothesis, the spontaneous magnetization $M_{S}(T)$ below the phase
transition temperature ($T_{c}$), the inverse magnetic susceptibility
$\chi_{0}^{-1}(T)$ above $T_{c}$ and the magnetization at $T_{c}$ should
exhibit the following relationships for a second-order magnetic phase
transition Stanley (1971):

$M_{S}(T)=M_{0}(-\epsilon)^{\beta},\hskip 28.45274pt\epsilon<0$ (1)
$\chi_{0}^{-1}(T)=\Gamma(\epsilon)^{\gamma},\hskip 28.45274pt\epsilon>0$ (2)
$M=XH^{\frac{1}{\delta}},\hskip 28.45274pt\epsilon=0$ (3)

where, $M_{0},\Gamma$ and $X$ are the critical amplitudes and $\beta,\gamma$
and $\delta$ are the critical exponents. $\epsilon$ is the reduced temperature
defined as $\epsilon=\left(\frac{T-T_{c}}{T_{c}}\right)$.

Scaling theory provides a method to relate the critical exponents $\beta$ and
$\gamma$ Cardy (1996); Pathria and Beale (2011), resulting in the formulation
of the magnetic equation of state, which can be expressed as follows Kaul
(1985):

$M(H,\epsilon)=\epsilon^{\beta}f_{\pm}\left(\frac{H}{\epsilon^{\beta+\gamma}}\right)$
(4)

where, $f_{+}(T>T_{c})$ and $f_{-}(T<T_{c})$ are the regular analytical
functions. By introducing renormalized magnetization
$m\equiv\epsilon^{-\beta}M(H,\epsilon)$ and renormalized field
$h\equiv\epsilon^{-(\beta+\gamma)}H$, Equation 4 can be simplified as follows:

$m=f_{\pm}(h)$ (5)

According to Equation 5, scaling relations implies that when plotting $m$
against $h$, the data should form distinctive patterns known as universal
curves which consist of two separate branches: one for temperatures above
$T_{c}$ where $\epsilon$ is positive, and the other for temperatures below
$T_{c}$ where $\epsilon$ is negative Arrott (1957). The emergence of these
universal curves confirms the validity of the scaling relations. Furthermore,
they are essential for accurately determining the critical exponents $\beta$,
$\gamma$, and $\delta$ within the critical regime.

Figure 2:  Rietveld refinement patterns of the (a) 300 K and (b) 10 K neutron
powder diffraction (NPD) patterns for
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. Lower ticks in figure (b)
shows the magnetic peak positions with k = 0, 0, 0. The obtained magnetic
structure is shown as an inset in figure (b), green spheres highlight the B -
site cations and blue spheres highlight the A - site cations.

##  IV Results and Discussions

###  IV.1 Structure and Magnetic Structure

Phase purity and compositional analysis of the sample are verified with the
Rietveld refinement of the room temperature powder X-ray diffraction pattern
(RT-XRD) and Energy-dispersive X-ray spectroscopy (EDS) mapping. The RT-XRD
pattern of (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4 reveals that the
sample crystallizes in a cubic structure with the space group
$Fd\overline{3}m$. Figure 1 (b) presents the ambient temperature field
emission scanning electron microscopy (FESEM) image and EDS elemental mapping
results. Ambient temperature EDS mapping confirms the stoichiometric chemical
composition, indicating a microscale homogeneous distribution of all elements
across the sample. To further investigate the detailed nuclear and magnetic
structures, neutron powder diffraction (NPD) patterns were collected at 10 K
and 300 K. The results from NPD Rietveld refinements are utilized to elucidate
the crystal structure and magnetic structure of
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. Figure 2 presents the
neutron powder diffraction (NPD) Rietveld refinement plots (10 K and 300 K) of
the sample. The structural parameters derived from the 10 K and 300 K NPD
refinements for (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4 are
summarized in Table 1. The crystal structure is depicted in Figure 1 (a).
Here, metal cations occupy both octahedral and tetrahedral coordination
environments, surrounded by oxygen atoms, forming two distinct sets of
magnetic sublattices. Typically, B cations (Cr, Mn, and Fe for the present
sample) occupy octahedral sites, configuring a lattice similar to pyrochlore,
which induces highly frustrated magnetic interactions. Conversely, A cations
(Ni, Mg, Co, Cu, and Zn for the present sample) occupy tetrahedral sites,
creating an eightfold coordination reminiscent of a diamond lattice structure.

Refinement of oxygen occupancy shows no deviation from full occupancy. The
cations Co, Mg, Cu, and Zn exclusively occupy the tetrahedral A sites (8a
(0.125, 0.125, 0.125)), while Fe, Cr, and Mn are located in the octahedral B
sites (16d (0.5, 0.5, 0.5)). Ni and Mn cations are found in both
crystallographic sites, with 40% of the Ni cations residing in the octahedral
sites. Despite the high disorder and chemical complexity, the low temperature
(10 K) neutron powder diffraction (NPD) data reveal long-range magnetic
ordering. All peaks with magnetic origins align with the allowed nuclear
reflections, and the magnetic structure is described using the propagation
vector k = (0,0,0). The NPD magnetic reflections for long-range magnetic order
fit best with a collinear ferrimagnetic ordering model, similar to those
proposed for CoFe2O4 and CoFeRhO4 Teillet  _et al._ (1993); Mohanty  _et al._
(2023). The magnetic structure is shown in the inset of figure 2(b). The
absence of the (200) magnetic peak suggests that a non-collinear spin
arrangement is unlikely. According to the Yafet-Kittel model, the presence of
the (200) magnetic Bragg peak would indicate long-range spin canting,
involving the transverse component ordering of the magnetic spins. However,
this peak is not observed at 20.35 $\deg$, while the (400) peak is clearly
seen at 2$\theta$ = 41.39 $\deg$. The magnetic moments in the tetrahedral A
sites (M(Td)) and octahedral B sites (M(Oct)) are detailed in Table 1. We have
observed a discrepancy between the experimentally measured and theoretically
calculated magnetic moments for these sites. The observed reduction in
magnetic moments can be attributed to the high disorder and local canting of
spins. The extremely complex cationic distribution and the presence of
nonmagnetic cations (Mg and Zn) within the structure could result in competing
magnetic interactions, leading to nonuniform local spin canting. This
reduction in magnetic moments due to local spin canting has also been observed
in CoFe2O4 and Ti-doped CoFe2O4 Teillet  _et al._ (1993); Dormann  _et al._
(1987). Nevertheless, both materials are suggested to exhibit collinear
ferrimagnetic long-range magnetic ordering.

Table 1:  Atomic and structural parameters as obtained from the 10 K and 300 K NPD refinements for (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. | 300 K | 10 K
---|---|---
Space Group | $Fd\overline{3}m$ | $Fd\overline{3}m$
a = b = c (Å) | 8.3804 (1) Å | 8.3760 (1) Å
Ni (0.125, 0.125, 0.125) |  |
Occupancy | 0.08 (2) | 0.08 (2)
Biso | 0.60 (2) | 0.41 (2)
Mn (0.125, 0.125, 0.125) |  |
Occupancy | 0.12 (2) | 0.12 (2)
Biso | 0.60 (2) | 0.41 (2)
Mg (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.2
Biso | 0.60 (2) | 0.41 (2)
Co (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.2
Biso | 0.60 (2) | 0.41 (2)
Cu (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.2
Biso | 0.60 (2) | 0.41 (2)
Zn (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.2
Biso | 0.60 (2) | 0.41 (2)
Fe (0.5, 0.5, 0.5) |  |
Occupancy | 0.66 | 0.66
Biso | 0.39 (2) | 0.10 (1)
Cr (0.5, 0.5, 0.5) |  |
Occupancy | 0.66 | 0.66
Biso | 0.39 | 0.10 (1)
Mn (0.5, 0.5, 0.5) |  |
Occupancy | 0.52 (2) | 0.52 (2)
Biso | 0.39 (2) | 0.10 (1)
O (x,x,x) |  |
x = 0.2608 (1) | 0.2608 (1) |
Occupancy | 1.0 | 1.0
Biso | 0.50 (1) | 0.35 (1)
B - B distance | 2.9629 (1) Å | 2.9614 (1) Å
A - B | 3.4743 (1) Å | 3.4725 (1) Å
$\chi^{2}$ | 9.5 | 3.73
RP | 8.25 | 3.65
RWP | 7.60 | 4.92
RMag | - | 2.92
M (Td) | - |  2.2 (1) $\mu$B
M (Oct.) | - |  3.3 (1) $\mu$B
Figure 3:  RT - XAS data and simulated Mn L-edge of the
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. Figure 4:  The simulated
(red) and experimental (black) XAS spectra of (a) Ni L2,3, (b) Fe L2,3 and (c)
Cr L2,3 edges of (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4.. Figure 5:
(a) Temperature dependence of magnetic susceptibility measured under a field-
cooled (FC) protocol with an external field of 1 T. The derivative of the
susceptibility is also shown, indicating the transition occurring around 293 K
(marked by the dashed vertical line). (b) Magnetic field dependence of
magnetization measured at 4 K and 50 K.

###  IV.2 X-ray absorption spectroscopy

X-ray Absorption Spectroscopy (XAS) is the measurement of transition of core
level electron to the states near or above the Fermi level in order to
investigate the electronic structure and local atomic environment of the
absorbing atom. The energy position and shape of the XAS spectra provide vivid
information about the oxidation state, coordination number, and interatomic
distances in a manner that is element-sensitive. Figure 3 displays the
experimental XAS spectra of the Mn L-edge along with the simulated spectra of
Mn3+ in octahedral geometry and Mn2+ in tetrahedral geometry. In order to get
an estimate of relative contributions of octahedral-Mn3+ and tetrahedral-Mn2+
in the studied sample, the simulated spectra of Mn3+ and Mn2+ are mixed in
different proportions and the resultant spectra are compared with the
experimental spectra. We found the resultant simulated spectra with 15% Mn2+
and 85% Mn3+ matches well with the experimental one. The simulated spectra
were generated using the CTM4XAS software package. The software comprises
three theoretical components that corresponds to the atomic multiplet theory,
crystal field theory, and charge transfer theory respectively. It allows to
model and interpret the electronic structure and local atomic environment of
materials by providing detailed simulations of spectral features. For Mn3+
(and 2+), the parameters were set as follows: 10Dq = 1.2 (1.3), Fdd = 1.5
(0.8), Fpd = 1.5 (0.8), Gpd = 1.5 (0.8), Lorentzian broadening = 0.2, and
Gaussian broadening = 0.4. Similarly, the L-edge XAS spectra of Ni, Fe, and Cr
have been simulated and shown in comparison with the experimental spectra in
Figure 4 (a-c). The Ni XAS spectra (Figure 4(a)) indicate the presence of Ni2+
oxidation state. However, the asymmetry in the peak shape suggests that some
Ni may occupy octahedral positions, possibly exchanging places with Mn2+ in
the tetrahedral sites. This trend is also observed in our NPD refinements
(table 1). The Cr XAS spectra confirm a Cr3+ oxidation state. Similarly, the
Fe L2,3-edge spectra (Figure 4(c)) are consistent with Fe3+ oxidation state.
Based on our NPD and RT-XAS analysis, we found that the chemical formula of
the sample can be written as
(Ni0.1Mn0.1Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.56Ni0.1Fe0.66Cr0.66)O4. Nevertheless, we
will refer to our sample as (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4
in the rest of the manuscript.

###  IV.3 Magnetism and Arrott plot

Figure 5 (a) shows the temperature dependence of magnetic susceptibility
($\chi$) measured under a field-cooled (FC) protocol with an external field of
1 T, within temperature range of 2 K to 370 K. As the temperature decreases,
$\chi$ begins to increase sharply below 300 K, indicating the onset of long-
range magnetic ferrimagnetic (FIM) ordering. The transition onset temperature
is determined from the derivative of the curve ($d\chi/dT$), which exhibits a
dip around 293 K, identifying this as the Curie temperature ($T_{c}$). The
magnetization measurements as a function of temperature (Figure 5 (b)) reveal
a sharp magnetization saturation with a very low coercivity of 100 Oe at 50 K.
It highlights a remarkably high saturation magnetization (MS) of 766 emu cm3
at 50 K. Resistivity measurements demonstrate an impressive resistivity of
4000 ohm-cm at room temperature (Figure 1 in the supporting information file
SI (File)). Additionally, the sample reaches magnetic saturation under a very
weak field of 1500 Oe, highlighting its potential as a soft magnetic insulator
material.

To investigate the nature of the phase transition in detail, we conducted a
critical analysis as described below. In conventional practices, critical
exponents and critical temperature are initially derived from Arrott plots
assuming mean field interactions within the system which is characterized by
$\beta=0.5$ and $\gamma=1$. Thus, an Arrott plot consists of plotting
isotherms as $M^{2}$ versus $H/M$ which, in ideal situation, yields a series
of parallel straight lines near $T_{c}$, with the line at $T=T_{c}$ passing
through the origin. From this plot, $\chi_{0}^{-1}(T)$ and $M_{S}(T)$ can be
determined from the intercepts of the curves on positive $H/M$ and $M^{2}$
axes respectively.

Figure 6 displays the Arrott plot around $T_{c}$ for this system. Except the
one with mean field model (Figure 6 (a)), the nonlinear nature of the curves,
even at high fields, indicates that these values of the critical exponents are
not appropriate. Additionally, the order of the magnetic phase transition can
be determined from the slope of the curves using the Banerjee criterion
Banerjee (1964). The positive slope of the curves confirms that the phase
transition is of second order.

Figure 6:  $M^{1/\beta}$ vs. $(H/M)^{1/\gamma}$ showing (a) Arrott plots
considering mean field model ($\beta=0.5,\gamma=1$) and modified Arrott plots
considering (b) 3D-Ising model ($\beta=0.325,\gamma=1.24$), (c) tricritical
mean field model ($\beta=0.25,\gamma=1$), (d) 3D-Heisenberg model
($\beta=0.365,\gamma=1.336$).

Even though the mean field model produces reasonably parallel straight lines
in the higher field region, more accurate values of the exponents are
rigorously investigated beyond these approximations to gain deeper insights
into the microscopic nature of second-order phase transitions. A modified
Arrott plot, known as the Arrott-Noakes equation, is employed as follows:

$\left(\frac{H}{M}\right)^{\gamma}=a\frac{(T-T_{c})}{T}+bM^{\frac{1}{\beta}}$
(6)

where, $a$ and $b$ are constants. This approach represents the most general
form of the equation of state near the critical point, adhering to the scaling
laws. Within this magnetic equation of state, different theoretical models
correspond to distinct sets of critical exponents, such as the 3D-Ising model
($\beta=0.325$, $\gamma=1.24$), the tricritical mean field model
($\beta=0.25$, $\gamma=1$), and the 3D-Heisenberg model ($\beta=0.365$,
$\gamma=1.336$) Pathria and Beale (2011); Le Guillou and Zinn-Justin (1977);
Pelissetto and Vicari (2002). Modified Arrott plots according to these models
are shown in Figure 6(b), (c), and (d) respectively which reveals curved lines
instead of straight lines. Notably, none of the plots exhibit a line passing
through the origin near the $T_{C}$ of 293 K. Consequently, these observations
implies that these conventional models are not appropriate to explain the
behavior of our systems definitively. This necessities further modification of
the critical exponents in order to get an optimal equation of state that
explains the critical behavior.

Determining a suitable set of exponents that would yield parallel straight
lines, as described by Equation 6, is challenging since it depends on two free
parameters, $\beta$ and $\gamma$. To avoid fitting inaccuracies and systematic
errors, we adopted a rigorous iterative approach to determine the values of
$\beta$ and $\gamma$. This involves utilizing a modified Arrott plot (Equation
6) with initial estimates of $\beta$ and $\gamma$, derived from intercepts of
linear extrapolations of isotherms in the high-field region on the
$M^{1/\beta}$ and $(H/M)^{1/\gamma}$ axes, respectively. These estimates of
$M_{S}(T)$ and $\chi_{0}^{-1}(T)$ are then employed to fit Equations 1 and 2.
From Equation 1, the slope of the straight-line fit of $\log[M_{S}(T)]$ versus
$\log(\epsilon)$ provides a refined value for $\beta$. Similarly, Equation 2
allows us to obtain a new value for $\gamma$ from the straight-line fitting of
$\log[\chi_{0}^{-1}(T)]$ versus $\log(\epsilon)$. During the fitting process,
$T_{c}$ remains a variable parameter, adjusted to achieve the best fit. These
new values of $\beta$, $\gamma$, and $T_{c}$ are used in Equation 6 and the
iterative process is continued until convergence is achieved. Following this
procedure, optimal values $\beta=0.48$ and $\gamma=1.10$ are obtained for the
present system. The resulting plots using these exponents show reasonably good
parallel straight lines, as illustrated in Figure 7, and therefore, these
exponents values are consistent with the critical behavior of the present
system.

Figure 7:  Modified Arrott plot: $M^{1/\beta}$ vs. $(H/M)^{1/\gamma}$
isotherms with $\beta=0.48$ and $\gamma=1.10$. The dashed line highlights the
isotherm at $T_{C}=293$ K and passes through the origin.

At low field strengths, the lines exhibit curvature. This behavior is typical
in many other complex oxide systems, as the magnetization represents a
macroscopic average of many misaligned domains Aharoni  _et al._ (2000);
Pramanik and Banerjee (2009); Triki  _et al._ (2013); Jia  _et al._ (2019);
Chakraborty and Elizabeth (2023). At high field strengths, the isotherms align
as nearly parallel straight lines. The extrapolated straight line from the
linear region of the isotherm at 293 K passes through the origin, indicating
that the corresponding temperature is consistent with previously determined
$T_{c}=293$ K. To verify the reliability of the critical exponents and $T_{c}$
associated with the modified Arrott plot, $M_{S}(T)$ and $\chi_{0}^{-1}(T)$
are extracted from Figure 7 following the process mentioned above and plotted
against temperature in Figure 8. By fitting the $M_{S}(T)$ using Equation 1
and $\chi_{0}^{-1}(T)$ with Equation 2 yield $\beta=0.53\pm 0.02$ with
$T_{c}=293.0\pm 0.5$ K and $\gamma=1.13\pm 0.04$ with $T_{c}=293.0\pm 0.80$
respectively. These critical exponents and $T_{c}$ values closely match those
derived from the modified Arrott plot in Figure 7. This consistency signifies
successful convergence in the iterative approach, validating the reliability
and accuracy of the obtained critical exponents.

Figure 8:  The temperature variation of the spontaneous magnetization
$M_{S}(T)$ and inverse susceptibility $\chi_{0}^{-1}$ obtained from the
extrapolation of modified Arrott plot in Figure 7. The values of $\beta$,
$\gamma$ and $T_{c}$ are obtained from the straight line fittings of Equation
1 and 2.

###  IV.4 Kouvel-Fischer Plot

In order to determine the critical exponents and $T_{c}$ more accurately, the
data of $M_{S}(T)$ and $\chi_{0}^{-1}(T)$ should be analyzed using the Kouvel-
Fischer method Kouvel and Fisher (1964):

$M_{S}(T)\left[\frac{dM_{S}(T)}{dT}\right]^{-1}=\frac{(T-T_{c})}{\beta}$ (7)
$\chi_{0}^{-1}(T)\left[\frac{d\chi_{0}^{-1}(T)}{dT}\right]^{-1}=\frac{(T-T_{c})}{\gamma}$
(8)

According to these equations, plotting $M_{S}(T)[dM_{S}(T)/dT]^{-1}$ and
$\chi_{0}^{-1}(T)[d\chi_{0}^{-1}(T)/dT]^{-1}$ against temperature $T$ should
result in linear relationships. The slopes of these lines are inversely
proportional to the critical exponents $\beta$ and $\gamma$, respectively. A
significant advantage of the Kouvel-Fischer plot is that it enables the
determination of $T_{c}$ without prior knowledge of its value. This is
achieved by extrapolating the linear fits to their points of intersection with
the temperature axis, as demonstrated in the Kouvel-Fischer plot for the
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4 system in Figure 9.

Figure 9:  Kouvel-Fischer plot of spontaneous magnetization $M_{S}$ and
inverse susceptibility $\chi_{0}^{-1}$. Solid straight lines highlight the
linear fit of the data.

The analysis yields estimates for the critical exponents and transition
temperature, namely, $\beta=0.502\pm 0.004$ and $T_{c}=292.03\pm 0.09$ K,
based on the first fitting, and $\gamma=1.13\pm 0.04$ and $T_{c}=293.08\pm
0.80$ K, based on the second fitting. These values are consistent with
previously obtained ones.

###  IV.5 Critical isotherm exponent

Figure 10:  The log-log plot of isothermal magnetization (M) with applied
magnetic field (H) for this high entropy oxide collected at their respective
TC = 293 K. The solid lines are the linear fits of critical isotherm relation
described in Equation (3) to extract the $\delta$ value.

The critical isotherm exponent, $\delta$, characterizes the power-law behavior
of the isothermal magnetization near the critical point, providing comparisons
among various systems with similar critical behavior. According to Equation 3,
$\ln(M)$ versus $\ln(H)$ plot at the transition temperature should be a
straight line with $\delta$ being its slope. For the present system, the high-
field region yields $\delta=3.43\pm 0.02$ as shown in Figure 10. Additionally,
the Widom theorem relates Widom (1964) $\delta$ to $\beta$ and $\gamma$
through Equation

$\delta=1+\frac{\gamma}{\beta}$ (9)

Using the values of $\beta$ and $\gamma$ from the modified Arrott and Kouvel-
Fischer plots, we obtain $\delta=3.29\pm 0.006$ and $3.25\pm 0.009$,
respectively. The consistency of these values, obtained through different
methods, confirms that the system obeys Widom scaling and demonstrates the
accuracy of the estimated exponents ($\beta$ and $\gamma$) within experimental
precision.

###  IV.6 Conformity of scaling law

The critical exponents obtained through various methods are summarized in
Table 3 alongside their theoretical values predicted for different
universality classes.

Table 2: Values of critical exponents $\beta$ , $\gamma$ and $\delta$ as obtained from the modified Arrott plot, Kouvel–Fisher plot, and the critical isotherm are given for (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. Critical exponents corresponding to various universality classes are also given for comparison.  System or Models | Technique | $\beta$ | $\gamma$ | $\delta$ | References
---|---|---|---|---|---
HEO | Modified Arrott plot | 0.48 $\pm$ 0.01 | 1.10 $\pm$0.01 | 3.29$\pm$0.02 | This work
HEO | Kouvel–Fisher | 0.50$\pm$ 0.02 | 1.13 $\pm$0.01 | 3.26$\pm$0.03 | This work
Mean field model Theory | Theory | 0.5 | 1.0 | 3.0 | Kaul (1985)
3D-Ising model Theory | Theory | 0.325 | 1.241 | 4.82 | Kaul (1985)
3D-Heisenberg model Theory | Theory | 0.365 | 1.386 | 4.80 | Kaul (1985)
Tricritical mean field model | Theory | 0.25 | 1 | 5 | Kaul (1985)

The comparison shows that the critical exponents obtained through various
methods match closely with the mean field model amongst various conventional
universality classes. It is crucial to further validate the consistency of the
critical analysis and the obtained critical exponents by examining the
universal scaling behavior, as described by Equation 5.

Figure 11:  Scaling plots (renormalized magnetization
$m\equiv\epsilon^{-\beta}M(H,\epsilon)$ vs. renormalized field
$h\equiv\epsilon^{-(\beta+\gamma)}H$) below and above $T_{c}$. Values of
$\beta$ and $\gamma$ that were determined from the modified Arrott plot are
used.

The plot of the renormalized magnetization ($m$) against the renormalized
field ($h$), as obtained using the values of $\beta$ and $\gamma$ obtained
from the modified Arrott plots, is shown in Figure 11. Notably, all data
points converge into two distinct curves, one below $T_{c}$ and the other
above $T_{c}$. This data collapse evidences the presence of a renormalized
interaction within the critical regime. Renormalization interaction refers to
the collective impact of fluctuations and interactions among the order
parameters within the system, which becomes increasingly significant as the
critical point is approached. This enhanced interaction modifies the behavior
of the system, leading to the observed data collapse and demonstrating the
existence of universal scaling laws near the critical point. Moreover, the
confirmation of data collapse validates the estimated critical exponents and
$T_{C}$.

###  IV.7 Interpretation of the critical exponents from high entropy aspect

It is concluded from various analysis of this study that $\beta=0.48$,
$\gamma=1.1$, $\delta=3.3$ and $T_{C}=293$ K is a consistent set of critical
exponents. This closely matches with the exponents of the conventional mean
field model. This is a noteworthy result of the system. It is worth mentioning
here that critical behavior of many complex oxide materials (non-high entropy)
with novel magnetism and functional properties, such as various perovskite,
double perovskite, spinel and their derivatives have been extensively
investigated Zhou  _et al._ (2008); Borah and Ravi (2020); Xu  _et al._
(2016); Singh  _et al._ (2020); Yadav  _et al._ (2021). Very often, it is a
common observation that their critical exponents are unique and do not belong
to any of the conventional universality classes. There are many factors which
causes the phase transition unconventional in these systems. Few well accepted
facts are- inevitable oxygen vacancies, cluster formation and chemical
homogeneity etc lading to mixed valence of the transition metals and resulting
in distribution of exchange interaction strengths Chakraborty and Elizabeth
(2023). From this perspective, it is remarkable to note that despite being a
high entropy oxide—a complex system with multiple transition metal elements in
both A and B sites—our present system adheres to the mean field model. This
model simplifies the understanding of complex many-body systems by averaging
the effects of all other particles on any given particle. In magnetism, it
does not treat all spins individually. It Substitutes the individual spin -
spin interaction between electrons by the interaction of the spins with an
average or effective interaction which is basically the internal magnetic
field proposed in Weiss molecular field theory of ferromagntism. Though this
is an oversimplified assumption, it is interesting to observe that this
picture holds consistently in such a complex system with numerous magnetic
atoms with diverse spin moments in A and B sites. This is however, not very
uncommon in case of cubic spinel systems. Experimental observations in many
spinel oxides yield critical exponents close to the mean field predictions,
suggesting the validity of mean field theory in describing their magnetic
behavior. A list of such spinel systems consistent with mean field model is
given in Table III.

Table 3: Spinel systems where critical exponents are consistent with the mean-field model Sr No | Spinel Oxide | $\beta$ | $\gamma$ | $\delta$ | Theory | References
---|---|---|---|---|---|---
1 | CoGa1.2Fe0.8 O4 | 0.49 | 1.00 | 3.01 | Close to MFT | Hu  _et al._ (2022)
| Ferromagnetic (TC=236K) |  |  |  |  |
2 | Co1+xCrxFe2-x O4 | 0.44 | 0.99 | 3.21 | Close to MFT | Islam and Hossain (2023)
| Ferrimagnetic (TC=636K) |  |  |  |  |
3 | Ni0.6Cd0.2Cu0.2 Fe2O4 | 0.44 | 1.03 | 3.31 | Close to MFT | Kouki  _et al._ (2019)
| Ferromagnetic (TC=655K) |  |  |  |  |
4 | Mn0.5Zn0.5Fe2O4 | 0.48 | 1.07 | 3.34 | Close to MFT | Anwar and Koo (2022)
| Ferromagnetic (TC=345K) |  |  |  |  |

Possible reasons for these systems to follow mean field model might be their
high symmetric cubic crystal structure, the nature of superexchange
interactions, and the uniform distribution of cations which make it possible
that the assumptions of mean field theory are well-satisfied in these
materials. Our high entropy spinel oxide system
(Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4 is also based on spinel with
cubic structure. While a similar rationale might reasonably apply to our
system, the distinct high-entropy nature of our system suggests that entropy
could also play a crucial role in our case. It is possible that the high
entropy leads to a homogeneous distribution of multiple cations, as it is
energetically favourable. This ensures that averaging of the local magnetic
environments (also observed in 10 K NPD refinement) is a reasonably good
approximation of the system and make it favourable to satisfy the mean field
theory.

##  V Conclusions

In conclusion, We have stabilized and studied a new high entropy spinel oxide
with composition (Ni0.2Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.66Fe0.66Cr0.66)O4. The
comprehensive structural (using NPD and X-ray diffrration), microstructural,
magnetic, and X-ray absorption spectroscopy analyses reveal that despite the
high degree of disorder, this oxide exhibits long-range ferrimagnetic ordering
starting at 293 K. The material’s impressive high saturation magnetization of
766 emu cm-3, low coercivity of 100 Oe, high transition temperature around
room temperature, and outstanding resistivity of 4000 ohm-cm at room
temperature underscore its potential for high-density memory devices. The NPD
analysis and RT-XAS measurements highlight the actual chemical composition of
the materials as (Ni0.1Mn0.1Mg0.2Co0.2Cu0.2Zn0.2)(Mn0.56Ni0.1Fe0.66Cr0.66)O4.
Our investigation on the magnetic structure, using a collinear-type
ferrimagnetic model with a propagation vector k = 0,0,0, along with various
analytical techniques such as modified Arrott plots, Kouvel-Fischer analysis,
and critical isotherm analysis, confirms a second-order phase transition. The
critical exponents obtained are consistent with the mean field model,
demonstrating that high entropy in this material leads to a homogeneous
distribution of multiple cations, thereby validating the approximation of
averaging local magnetic environments and supporting the mean field theory. We
would like to emphasize that this is the only work on the critical behavior
around the phase transition temperature on a complex high entropy oxide. This
study not only highlights the potential of incorporating a diverse range of
elemental components on both the crystallographic sites of a complex spinel
oxide framework but also highlights the excellent opportunities for tailoring
and enhancing the properties of strongly correlated materials.

##  VI Acknowledgments

S. M. acknowledges the SERB, Government of India, for the (SRG/2021/001993)
Start-up Research Grant and UGC DAE CSR for the CRS project (project No. -
CRS/2023-24/01/1001). We acknowledge the UGC-DAE Consortium for Scientific
Research (CSR), Indore for the magnetization and XAS measurements. We would
like to thank Dr. Rajeev Rawat, Dr. Mukul Gupta, Mr. Rakesh Sah, Mr. Avinash
Wadikar, Mr. Kranti Sharma and Ms. Soumya Shephalika for their support during
the magnetic and XAS measurements. UGC-DAE CSR, Mumbai is also acknowledged
for the Neutron diffraction experiments.

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# Towards Chip-in-the-loop Spiking Neural Network Training via Metropolis-
Hastings Sampling

Ali Safa1,2, Vikrant Jaltare3, Samira Sebt3, Kameron Gano3, Johannes
Leugering3,
Georges Gielen1,2, Gert Cauwenberghs3 1imec, Leuven, Belgium, 2ESAT, KU
Leuven, Belgium, 3University of California at San Diego, La Jolla, USA
<EMAIL_ADDRESS>
###### Abstract

This paper studies the use of Metropolis-Hastings sampling for training
Spiking Neural Network (SNN) hardware subject to strong unknown non-
idealities, and compares the proposed approach to the common use of the
backpropagation of error (backprop) algorithm and surrogate gradients, widely
used to train SNNs in literature. Simulations are conducted within a chip-in-
the-loop training context, where an SNN subject to unknown distortion must be
trained to detect cancer from measurements, within a biomedical application
context. Our results show that the proposed approach strongly outperforms the
use of backprop by up to $27\%$ higher accuracy when subject to strong
hardware non-idealities. Furthermore, our results also show that the proposed
approach outperforms backprop in terms of SNN generalization, needing
$>10\times$ less training data for achieving effective accuracy. These
findings make the proposed training approach well-suited for SNN
implementations in analog subthreshold circuits and other emerging
technologies where unknown hardware non-idealities can jeopardize backprop.

###### Index Terms:

Spiking Neural Networks, Metropolis-Hastings sampling, chip-in-the-loop
training.

##  I Introduction

In recent years, Spiking Neural Networks (SNNs) [1] have attracted much
attention for extreme-edge AI applications such as healthcare monitoring,
wearables and other battery-power IoT applications where machine learning
tasks must be run with the tightest of energy, latency and area budgets [2, 3,
4, 5, 6]. Compared to conventional Deep Neural Networks (DNNs) [7], SNNs make
use of event-driven binary activations (or spikes), closely replicating how
biological neurons generate spikes in the brain [8]. This spiking, event-
driven nature, makes SNNs highly appropriated for ultra-low-power applications
since SNNs only consume dynamic energy when a spike is emitted [9]. This
significantly reduces power consumption when implemented in neuromorphic
hardware, compared to frame-based DNNs processing continuous values [10, 11,
12]. In addition, their binary activation nature drops the need for costly
weight multiplication hardware since inner products $\bar{W}^{T}\bar{s}$
between weight vectors $\bar{W}$ and binary spike vectors $\bar{s}$ can be
reduced to a simple addition of the weight elements corresponding to a spiking
entry in $\bar{s}$, further reducing energy and area overheads in neuromorphic
hardware [14, 15].

In addition to the hardware-efficient working properties of SNNs, many teams
are currently exploring the implementation of SNNs in analog sub-threshold
hardware instead of less power-efficient digital solutions [16, 17, 18, 19].
But an additional challenge that comes with analog sub-threshold SNN designs
is the sensitivity of the underlying hardware to Process, Voltage, and
Temperature (PVT) variations, as well as distortions, hysteresis, and non-
linearities arising from the biasing of transistors in the subthreshold regime
[20, 21]. Due to the dependency of these non-idealities to PVT variations, it
is often hard to build accurate software models of the obtained SNN chip since
its properties can vary both with time and across different produced chips
[22].

Still, current SNN training approaches such as the widely-used backpropagation
of error (backprop) algorithm coupled with the use of surrogate gradients [23]
(for alleviating the ill-defined nature of the Dirac spike derivative) do
require the availability of a software model describing the behavior of each
spiking neuron in the hardware [24, 25]. This jeopardizes the direct
deployment of an SNN model trained offline in analog SNN implementations,
where an accurate model of each chip is not always available. Indeed, since
the underlying hardware does not precisely match the SNN software model, a
direct deployment of the trained model would result in significant losses in
accuracy [26].

Figure 1: Chip-in-the-loop training of SNN hardware. This work studies
Metropolis-Hastings sampling, where the SNN weights $W$ are learned (inferred)
using Bayes’ rule with $P(W|D),P(W)$ the posterior and prior densities, and
$L(D|W)$ the likelihood density given the data $D$ (see Section II-A). The
proposed method is compared to conventional back-propagation of error, where
the weights $W$ are learned via gradient descent for minimizing the error $E$.

To alleviate this issue, chip-in-the-loop training approaches [27, 28] (see
Fig. 1) have been proposed where the actual SNN hardware is used for the
inference (or forward propagation) pass of the training procedure, and the
inference result is then used to evaluate the error function ($E$ in Fig. 1)
and perform backprop accordingly. But still, these approaches rely on the
assumption that the surrogate gradients used during chip-in-the-loop training
are still valid gradient approximations even in the presence of unknown SNN
non-idealities, which might not always be the case in current and future
analog implementations, as the limits of energy and area efficiency are pushed
further [1, 12, 13].

Hence, it is crucial to investigate model-free approaches (such as Bayesian
Inference techniques) for SNN training, which do not require the use of the
surrogate gradient method and the assumptions that it entails. To this end,
this paper proposes a novel approach for training SNNs via Metropolis-Hastings
sampling [29] (which performs the inference of the SNN weights $W$ given the
input data $D$ following Bayes’ rule in Fig. 1), and motivate its use for SNN
hardware subject to strong, unknown non-idealities.

The main contributions of this paper are the followings:

  1. 1.

We propose a novel approach for training SNNs via Metropolis-Hastings sampling
[29] for chip-in-the-loop training of SNN hardware subject to potential
distortion and non-idealities, such as in analog sub-threshold SNN
implementations.

  2. 2.

We show that our proposed Metropolis-Hastings SNN training approach
significantly outperforms the use of conventional surrogate gradient backprop
in terms of accuracy when the SNN hardware is subject to unknown distortion
non-idealities.

  3. 3.

We also show that our proposed Metropolis-Hastings SNN training approach
enables the SNN to generalize significantly better to unknown data during test
inference, remarkably needing more than one order of magnitude less training
data in order to achieve similar or higher accuracy compared to backprop.

This paper is organized as follows. Section II introduces our proposed
Metropolis-Hastings SNN training method for chip-in-the-loop learning in SNN
hardware subject to strong non-linear distortion and defects. Section III
shows our results and comparison against conventional surrogate gradient
backprop training. Finally, conclusions are provided in Section IV.

##  II Methods

###  II-A Chip-in-the-loop SNN training via Metropolis-Hastings

In this paper, we propose to use Metropolis-Hastings sampling [29] as a
Bayesian method for training the SNN weights, without the need for a precise
model detailing the underlying SNN hardware and its non-idealities.
Metropolis-Hastings provides a computationally tractable way for estimating
the posterior distribution of the SNN weights $P(W|D)$ given the data $D$
following Bayes’ rule [29]:

$P(W_{n+1}|D)=\frac{L(D|W_{n})P(W_{n})}{P(D)}$ (1)

where $n$ is the current iteration of the Bayesian inference process, $W_{n}$
denotes the SNN weights at step $n$, $L(D|W_{n})$ is the likelihood
probability distribution, $P(W_{n})$ is the prior distribution, $P(D)$ is the
data distribution and $P(W_{n+1}|D)$ the posterior distribution of $W$ at step
$n+1$.

After the convergence of the iterative process (1), the SNN weights $W^{*}$
are obtained as the a posteriori mean value [30]:

$W^{*}=\int W^{*}P(W^{*}|D)dW^{*}$ (2)

The rest of this section describes the definition of the probability density
functions in (1), followed by a description of how each probability density is
used by the Metropolis-Hastings algorithm for training the SNN in Algorithm 1.

####  II-A1 Likelihood density model

We consider an arbitrary SNN architecture with $n$ layers and with the last
layer working as a classification output layer, with as much output leaky
integrate and fire (LIF) neurons as the number of classes in the task to be
solved. We denote $W$ as the tensor containing all the SNN weights.

To compute the Likelihood density, we first feed the training input data
$D_{i},\forall i=1,...,N_{train}$ to the SNN which produces output spikes
$\bar{s}_{out}[k]$ (where $i$ denotes the index of the data point in the train
set and $k$ denotes the time step).

Since the goal is to steer the spiking rate of the output classification layer
to correspond to the one-hot encoded class labels in the train set, we compute
the output spiking rate as:

$\bar{r}_{out,i}=\frac{1}{T}\sum_{k=1}^{T}\bar{s}_{out}[k]$ (3)

which contains values between $0$ and $1$ (with $T$ the total number of time
steps).

Then, we compute the mean square error (MSE) between the output rate vector
$\bar{r}_{out,i}$ of the SNN and the one-hot encoded label $\bar{y}_{i}$
corresponding to the $i^{th}$ data sample in the train set:

$MSE=\frac{1}{N_{train}}\sum_{i=1}^{N_{train}}||\bar{y}_{i}-\bar{r}_{out,i}||_{2}^{2}$
(4)

Then, the MSE is used to form a Gaussian likelihood density (since maximizing
a Gaussian likelihood is similar to minimizing the MSE error, see [32]):

$L(D|W)=\frac{1}{\sqrt{2\pi}v}e^{\frac{-MSE}{2v^{2}}}$ (5)

where $v^{2}$ is the variance of the chosen Likelihood function.

####  II-A2 Prior density model

As prior distribution for each SNN weight $W_{j}$, we use a Gaussian
distribution with variance $p^{2}$:

$P(W_{j})=\frac{1}{\sqrt{2\pi}p}e^{\frac{-W^{2}_{j}}{2p^{2}}}$ (6)

####  II-A3 Posterior density inference

We use Metropolis-Hastings sampling [29] (Algorithm 1) to estimate the
posterior density over the weights $P(W|D)$ by using (5) as the likelihood
model and (6) as the prior model. Metropolis-Hastings sampling is an implicit
approach for maximizing the posterior without explicitly using Bayes’ rule,
which becomes significantly more complex to compute as the dimension of the
weight tensor $W$ (i.e., the number of SNN weights) becomes important [31].
Algorithm 1 describes the Metropolis-Hastings SNN training procedure.

Algorithm 1 Metropolis-Hastings SNN Training

0: $D$ : training data and labels, $c^{2}$ : weight sampling variance, $v^{2}$
: Likelihood density variance in (5), $p^{2}$ : Prior density variance in (6).

1: $\Lambda\xleftarrow{}c^{2}I$ //diagonal covariance matrix with variance
$c^{2}$

2: $W_{1}\sim N(0,\Lambda)$

3: for $n=1$ to end do

4: $W_{p}\sim N(W_{n},\Lambda)$ // Sample random proposal $W_{p}$

5: Run SNN with weights $W_{p}$ and $W_{n}$ and data $D$ to compute
$L(D|W_{p})$ and $L(D|W_{n})$ (see Section II-A1)

6:
$\alpha(W_{p}|W_{n})=\min\\{1,\frac{L(D|W_{p})P(W_{p})}{L(D|W_{n})P(W_{n})}\\}$

7: With probability $\alpha(W_{p}|W_{n})$ set $W_{n+1}=W_{p}$, else keep
$W_{n+1}=W_{n}$

8: end for

###  II-B LIF neuron with distortion defects

In order to study the effect of potentially strong sub-threshold non-
linearities and possible circuit defects causing additional non-linear
behavior, we use the following modified LIF neuron model with an arbitrary
non-linearity $f_{\sigma}$ applied to the membrane potential accumulation
$V_{k}$:

$\begin{cases}V_{k+1}=\alpha V_{k}+(1-\alpha)I_{syn}\\\
V_{k+1}=f_{\sigma}\\{V_{k+1}\\}\\\ S_{out}=1\hskip 3.0pt\text{{if}}\hskip
3.0ptV_{k+1}\geq\mu\hskip 3.0pt\text{{else}}\hskip 3.0pt0\\\ V_{k+1}=0\hskip
3.0pt\text{{if}}\hskip 3.0ptV_{k+1}\geq\mu\hskip 3.0pt\text{{or }}\hskip
3.0ptV_{k+1}<0\end{cases}$ (7)

where $I_{syn}$ is the input synaptic current, $\alpha$ is the membrane decay
parameter, $\mu$ is the spiking threshold, $S_{out}$ is the spiking output and
the non-linearity $f_{\sigma}$ is defined as:

$f_{\sigma}=x+\frac{\sigma}{2}x^{2}+\frac{\sigma}{6}x^{3}\approx\exp{(\sigma
x)}$ (8)

where $\sigma$ defines the strength of the non-linearity (the higher $\sigma$,
the more non linear).

Crucially, the arbitrary non-linearity in (8) is purposefully taken to
benchmark the proposed MH Sampling approach for SNN training under challenging
unknown distortions and defects. Therefore, using this challenging distortion
model will enable our experimental results and discussions to also cover the
case of less challenging distortion models that might be found in different
circuits.

This type of non-linearity can for example be found in the context of analog
sub-threshold LIF circuits [16, 17], where MOSFETs biased in the sub-threshold
region $V_{GS}<V_{T}$ exhibit a non-linear behavior (9) in terms of their
drain current $I_{D}$ in function of their gate-source voltage $V_{GS}$, which
is dependent on unknown Process Voltage and Temperature (PVT) variations.

$I_{D}\propto\exp{(\frac{qV_{GS}}{nkT})}$ (9)

where $\frac{nkT}{q}$ is the thermal voltage.

Hence, the sensibility of analog sub-threshold LIF neuron designs, to unknown
PVT variations, serves as strong further motivation for the study of novel
model-free approaches to SNN training, such as the Metropolis-Hastings
sampling technique detailed in Section II-A.

##  III Simulation Results

###  III-A SNN setup

We conduct our experiments using the popular Wisconsin Breast Cancer (WBC)
dataset [33] for the binary detection of cancer from bio-signal measurements.
The dataset features $569$ data points as vectors composed of $30$ measurement
features for each data point.

As SNN architecture, we use a 1-hidden-layer fully-connected SNN using LIF
neurons subject to the non-ideality in (7), which becomes equivalent to a
regular LIF neuron when the distortion strength $\sigma=0$. We use $3$ neurons
in the hidden layer and $2$ output neurons corresponding to each class in our
cancer detection problem (class $1$: has cancer, class $2$: does not have
cancer).

Before the SNN learning phase, we normalize the entries in each data vector
$\bar{x}$ to the $[0,1]$ range via
$\bar{x}\xleftarrow{}\frac{\bar{x}}{\max\bar{x}}$ and we perform a random
$80\%/20\%$ train-test split of the complete dataset before a Poisson encoding
[34] of each data point into a vector of spike trains with $T$ time bins,
where the probability of having a spike in each time bin for the $i^{th}$
entry in $\bar{x}$ is a Bernoulli process with probability $x_{i}$. We
arbitrary set $T=10$ to limit the time complexity of the system (the larger
$T$, the longer the SNN latency and hence, the higher the power consumption
when implemented in hardware).

For training the SNN, we use and compare two different learning approaches: 1)
the proposed Metropolis-Hastings method described in Section II-A and 2)
conventional backprop using the surrogate gradient technique as the dominant
approach for training SNNs in literature [23, 24, 25].

####  III-A1 Metropolis-Hastings parameters

We use the Metropolis-Hastings SNN training approach described in Algorithm 1,
Section II-A with parameters $c=0.5$, $v=1$ and $p=10$ in Algorithm 1.

####  III-A2 Backprop parameters

We train the SNN via backprop in a similar manner as in [23, 24, 25], by using
a Gaussian surrogate (10) as the replacement gradient of the LIF neuron non-
linearity.

$S_{out}^{\prime}(V)\approx\frac{1}{\sqrt{2\pi}}e^{-2V^{2}}$ (10)

where $S_{out}$ is the spiking output and $V$ is the membrane potential (see
Eq. 7). For training, we use the Adam optimizer with learning rate
$\eta=0.001$ for a total of $100$ epochs with batch size $32$.

SNN evaluation is carried via a 5-fold train-test procedure where each fold
uses a different random train-test partitioning. The final test accuracy and
its standard deviation are computed over the $5$ folds and reported during the
experiments in Fig. 2 and 3 (as the point marker and the shaded area
respectively).

###  III-B Accuracy in function of distortion strength $\sigma$

In this Section, we compare the test accuracy obtained via the proposed
Metropolis-Hastings approach against the use of conventional surrogate
gradient backprop, as a function of the distortion strength $\sigma$. It can
be clearly seen in Fig. 2 that the test accuracy achieved by the proposed
Metropolis-Hastings sampling is significantly higher than the accuracy
obtained via backprop when $\sigma\geq 2$ (a gain of $+27\%$ for $\sigma=4$
and $+4\%$ for $\sigma=3$). On the other hand, test accuracy is similar for
both methods when $\sigma<2$, since the difference between the distorted LIF
model and the ideal LIF model used by the surrogate gradient backprop training
is reduced.

Figure 2: Test accuracy in function of distortion strength $\sigma$. While backprop accuracy strongly degrades when $\sigma\geq 3$, Metropolis-Hastings accuracy is not affected by the increase in LIF distortion.  |
---|---
(a) No distortion ($\sigma=0$) | (b) Low distortion ($\sigma=1$)
|
(c) Mild distortion ($\sigma=2$) | (d) Strong distortion ($\sigma=2$)
Figure 3: Test accuracy in function of the train set size.

###  III-C SNN generalization performance

In this Section, we explore the generalization ability of the SNN obtained via
each training approach, by studying the test accuracy as a function of the
number of training data points in the train set. Indeed, achieving faster
generalization by using less training data is highly attractive in chip-in-
the-loop training scenarios since it will significantly reduce memory
consumption and training time, thereby reducing power consumption if chip-in-
the-loop training is used on-line to e.g. reconfigure SNN hardware at the
extreme edge.

Fig. 3 respectively show the generalization performance of each training
approach for $\sigma=0,1,2,3$. It can be clearly seen on all plots that the
Metropolis-Hastings training approach leads to significantly better SNN
generalization compared to backprop. When no distortion is used ($\sigma=0$ in
Fig. 3), the Metropolis-Hastings approach already achieves $+10\%$ vs.
backprop in test accuracy when using only $25$ data points in the train set,
and still achieves higher test accuracy when more training data is used. This
is thought to be due to the unique characteristics that come with Baye’s rule
(1), placing emphasis on the prior distribution as a means to alleviate the
effects of uncertainty and noise. In the case of Fig. 3, this uncertainty is
due to a significant lack of training data. Therefore, as expected, the MH
model outperforms the traditional backprop model in terms of generalization
ability.

When distortion is added for $\sigma=1,2,3$, the gain in accuracy obtained via
Metropolis-Hastings compared to backprop becomes even more significant (see
Fig. 3). These experiments clearly demonstrate the superior generalization
ability of the proposed Metropolis-Hastings SNN training approach,
significantly reducing the amount of training data that needs to be collected
and used, reducing the training time, power consumption, and memory
requirements when e.g., using chip-in-the-loop training at the extreme edge to
reconfigure the SNN hardware.

##  IV Conclusion

This paper has provided a study of Metropolis-Hastings sampling as a novel
approach for chip-in-the-loop SNN training, where the underlying SNN hardware
is subject to strong distortion non-idealities. Simulation results using the
Wisconsin Breast Cancer detection dataset have shown that the proposed
approach strongly outperforms the use of backprop by up to $27\%$ higher
accuracy under LIF neuron distortion. In addition, it was shown that the
proposed approach outperforms backprop in terms of SNN generalization, needing
$>10\times$ less training data for achieving effective classification
accuracy. Hence, our findings demonstrate that the proposed training approach
is well-suited for SNN implementations in analog subthreshold circuits and
other emerging technologies where unknown hardware non-idealities such as
distortion and hysteresis can jeopardize the use of backprop.

## Acknowledgment

This research was partially funded by a Long Stay Abroad grant from the
Flemish Fund of Research - Fonds Wetenschappelijk Onderzoek (FWO) - grant
V413023N. This research received funding from the Flemish Government under the
“Onderzoeksprogramma Artificiele Intelligentie (AI) Vlaanderen” programme.

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