# Structural, magnetic and x-ray absorption spectroscopy studies of new Cr-
based low, medium and high-entropy spinel oxides

Sushanta Mandal  Department of Physics and Materials 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  Tirthankar Chakraborty  Department of Physics and Materials
Science, Thapar Institute of Engineering and Technology, Patiala 147004, India
Sanjoy Kr. Mahatha  UGC-DAE Consortium for Scientific Research, Khandwa Road,
Indore 452001, Madhya Pradesh, India  Sourav Marik<EMAIL_ADDRESS>Department of Physics and Materials Science, Thapar Institute of Engineering
and Technology, Patiala 147004, India

###### Abstract

The emergence of high-entropy oxides has spurred significant research interest
in recent times. These compounds exhibit exotic functional properties that
often transcend simple linear combinations of their constituent elements.
Herein, we present a new series of Cr-based low, medium, and high entropy
spinel oxides with composition NiCr2O4, [Ni0.5Mn0.5]Cr2O4,
[Ni0.33Mn0.33Co0.33]Cr2O4, [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4,
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. We
conducted detailed structural (X-ray and Neutron diffraction),
microstructural, Raman spectroscopy, magnetic, and X-ray absorption
spectroscopy measurements on these materials. Our study reveals that the
incorporation of multiple cations at the A-site of the structure (AB2O4)
significantly modulates the magnetic properties. These compounds exhibit
transitions from complex ferrimagnetic ([Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4) to
antiferromagnetic ([Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4) states, with remarkable
coercivity variations, demonstrating the ability to tailor magnetic responses
through compositional design.

##  I Introduction

Recently, high entropy oxides (HEOs) have emerged as an exciting and promising
field of research, fundamentally reshaping material design and properties 1 ;
2 . The configurational entropy ($\Delta$Sconf.) is greatly enhanced by
randomly incorporating multiple elements (five or more in an equimolar ratio)
into a single crystallographic site. This highly enhanced configurational
entropy can stabilize the structure of a compound, leading to what is termed
an entropy-stabilized phase.

Rost et al. in 2015, stabilized and investigated the first high entropy oxide
(HEO) having composition (Mg0.2Co0.2Ni0.2Cu0.2Zn0.2)O 3 . Subsequently, high
entropy oxides (HEOs) have attracted considerable research interest due to
their facile synthesis, outstanding stability, extraordinary functional
properties, and the augmented design versatility offered by the possibility of
the presence of multiple cations in the structure 1 ; 2 ; 3 ; 4 . Illustrative
examples include tunable magnetism 5 , high exchange bias effect 6 , and
extremely high magnetic frustration 7  in high entropy spinel oxides, spin
reorientation in high entropy perovskites 8 , robust ferrimagnetism in
(CrMnFeCoNi)3O4 film 9 , extremely high value of colossal magnetoresistance in
strongly correlated high-entropy manganites 10 , enhanced catalytic
performance in Al-based high entropy spinels 11 , enhanced corrosion
resistance in high entropy pyrochlores 12 , promising thermoelectric
properties with ultralow (record low) thermal conductivity in high entropy
oxides with a tungsten bronze structure 13  and enhanced thermal radiation 14
in high entropy type spinel oxides. The increased disorder within entropy-
stabilized systems brings significant enhancements in their functional
properties. High entropy stabilization offers a promising framework for finely
tuning materials at the atomic scale, enabling the fabrication of materials
with tailored properties to address specific needs. However, synthesizing high
entropy oxides (HEOs) with precise compositions and desired properties poses a
significant challenge. A precise selection of constituent elements and their
stoichiometric ratios makes it feasible to engineer materials with the desired
characteristics suitable for targeted applications. Thus, a compelling
motivation exists to explore and investigate new strongly correlated high
entropy oxides having novel structures and intriguing properties. Herein, we
present a new series of Cr-based low, medium, and high entropy spinel oxides.
We have highlighted the detailed and systematic structural, magnetic, Raman
spectroscopy, and X-ray absorption spectroscopy studies on these new
compounds.

Spinel oxide system stands as a fascinating category of material system, not
only for their broad utility across diverse applications but also for their
plethora of novel and intriguing physicochemical properties. These include
phenomena such as frustrated magnetism, multiferroic behavior, orbital glass
states, and the emergence of spin-orbital liquids 15 ; 16 ; 17 ; 18 .
Structurally, they adhere to a distinct framework characterized by the general
formula AB2O4, with A and B representing metal ions. Within this structural
framework, metal cations are both in octahedral and tetrahedral coordination
environments, surrounded by oxygen atoms, leading to the formation of two
distinct sets of magnetic sublattices. Conventionally, B cations occupy
octahedral sites, configuring a lattice akin to pyrochlore and thereby
inducing highly frustrated magnetic interactions 19 . The geometrical
frustration within the pyrochlore lattice yields a plethora of intriguing
physical effects, such as multiferroics, Spin-Peierls-like phase transitions,
and quantum magnetic frustration 15 ; 16 ; 17 ; 18 ; 20 . These effects emerge
from the interplay between the lattice’s structure and the orientation of its
spins, highlighting the correlation between structural and spin degrees of
freedom. Conversely, A ions occupy tetrahedral sites, forming an eightfold
coordination reminiscent of a diamond lattice structure 21 . This bipartite
lattice arrangement can be envisaged as comprising two face-centered
interpenetrating cubic (fcc) sublattices, each diagonally shifted by one-
quarter of the lattice parameter. Variations in the composition of magnetic
and non-magnetic cations across both tetrahedral A and octahedral B sites can
give rise to intricate magnetic interactions, thus fostering the emergence of
exotic properties and states such as spiral spin liquid phases 22 ,
distinctive glassy magnetic behaviors 23 , and spin-orbital liquids 24  in
spinel materials. The configuration of the pyrochlore lattice presents a rich
and fertile landscape for both theoretical exploration and experimental
investigation.

Figure 1: Crystal structure of the spinel AB2O4 oxides. B cations (here Cr)
occupy the octahedral sites, A cations (Ni, Mn, Mg, Co, Cu, Zn) occupy the
tetrahedral sites

In the present paper, we have synthesized a new series of Cr-based low,
medium, and high entropy spinel oxides having compositions NiCr2O4,
[Ni0.5Mn0.5]Cr2O4, [Ni0.33Mn0.33Co0.33]Cr2O4, [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4,
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 by
systematically increasing the configuration entropy (and chemical complexity)
in the A site of the Cr-based spinel structure. We have presented detailed and
systematic structural (using X-ray powder diffraction, neutron powder
diffraction, and electron microscopy), magnetic (magnetization), Raman
spectroscopy, and X-ray absorption spectroscopy studies on these new
compounds. A remarkable change in the magnetic properties (from ferrimagnetic
to antiferromagnetic, unusually large coercivity) is observed with a small
change in the A site of the structure. Our study shows the feasibility of
accommodating numerous elements within a single sublattice of a complex spinel
system and highlights prospects for tailoring and optimizing functional
properties in high-entropy stabilized correlated electron systems.

Figure 2:  (a) Enlarged part of the room temperature X-ray diffraction (RT -
XRD) patterns for NiCr2O4, [Ni0.5Mn0.5]Cr2O4, [Ni0.33Mn0.33Co0.33]Cr2O4,
[Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4, [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. (b) - (g) Rietveld refinement plots of the
RT - XRD patterns, variation of the (h) lattice parameters, and (i) Cr - Cr
and Cr-A distances with the radius of the A site for all the samples. The data
for the cubic NiCr2O4 is taken from Ref. 26 .

##  II Experimental Details

Synthesis. The standard solid-state reaction route is used to prepare all the
samples. Stoichiometric amounts of high-purity Co3O4 (99.9%) MnO2 (99.9%), CuO
(99.9%), ZnO (99.9%), NiO (99.9%), MgO (99.9%) and Cr2O3 (99.99%) were
employed. The precursor materials were thoroughly mixed using a mortar and
pestle, followed by multiple heat treatments and air quenching. The final
sintering was conducted at 1373 K for 36 hours.

X-ray Diffraction and Neutron Diffraction. X-ray diffraction (XRD) analysis
was performed on the powdered samples using a PANalytical diffractometer with
(Cu-Kα, $\lambda$ = 1.54056 Å) at ambient conditions. Neutron diffraction
(NPD) data for the [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 and
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 were collected at ambient temperature at
BARC, India ($\lambda$ = 1.48 Å). We have performed the Rietveld refinements
of the diffraction patterns (XRD and NPD) using the fullProf suite software.

Electron Microscopy. Energy Dispersive X-ray Spectroscopy (EDS) mappings were
obtained using a BRUKER XFlash 6160 system, while microstructure analysis was
performed with a ZEISS GEMINI field emission scanning electron microscope (FE-
SEM). High-angle annular dark-field scanning transmission electron microscopy
(HAADF-STEM) was conducted using a JEOL ARM-200F microscope with cold FEG
probe image aberration correction at 200 kV to investigate the crystal
structure. All measurements were conducted at room temperature.

Figure 3:  Representative FE-SEM images and the EDS elemental mapping results for (a) [Ni0.5Mn0.5]Cr2O4, (b) [Ni0.33Mn0.33Co0.33]Cr2O4, (c) [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4, (d) [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and (e) [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. All the materials show excellent homogeneity at the microscopic scale. The elemental ratios obtained from EDS elemental mapping are highlighted in Table I. Table 1:  Compositions of all the materials based on EDS mapping analysis. Sample Compositions (stoichiometric) | Sample Compositions (EDS mapping)
---|---
[Ni0.5Mn0.5]Cr2O4 |  [Ni0.46Mn0.55]Cr1.99O4
[Ni0.33Mn0.33Co0.33]Cr2O4 |  [Ni0.29Mn0.27Co0.45]Cr1.99O4
[Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4 |  [Ni0.23Mn0.24Co0.27Cu0.25]Cr2O4
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 |  [Ni0.18Mn0.23Co0.17Cu0.19Zn0.17]Cr2.04O4
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 |  [Ni0.25Mg0.09Co0.23Cu0.21Zn0.21]Cr2.03O4

Raman Spectra. Raman spectra were collected from powdered samples under
ambient conditions using a LabRAM HR Raman spectrometer (Horiba, France) with
a 532 nm laser source.

X-Ray Absorption Spectroscopy. Soft X-ray absorption spectroscopy (SXAS)
measurements were conducted at the Raja Ramanna Centre for Advanced Technology
(RRCAT) in Indore, India. The SXAS beamline (BL-01) of the Indus-2 synchrotron
source provided an energy resolution of approximately 0.5 eV for the
measurements, all of which were performed at room temperature.

Magnetism. Magnetization measurements, including temperature and magnetic
field dependence, were performed using a Quantum Design MPMS 3 superconducting
quantum interference device (SQUID). Both field-cooled (FC) and zero-field-
cooled (ZFC) modes were utilized. The measurements were conducted over a
temperature range of 4 K to 300 K.

##  III Results and Discussion

Figure 2(a) highlights the enlarged part of the RT-XRD data for all the
samples. The RT-XRD patterns confirm the phase purity of all the samples. As
expected NiCr2O4 is found to crystallize in a tetragonal structure (space
group I41/amd, structural transition temperature 310 K 25 ; 26 ) with lattice
parameters of a = b = 5.8494 (1) Å and c = 8.3773 (1) Å. The Jahn-Teller
active Ni2+ in the tetrahedral site leads to the stabilization of a tetragonal
structure. However, dilution in the Ni2+ content in the tetrahedral site
stabilizes the cubic spinel structure. Figure 2(b) - (g) illustrates the
Rietveld refinement patterns of the RT - XRD patterns for all the samples. The
refinement parameters (lattice parameters and atomic parameters) are provided
in the supporting information file 27 . All the samples, except NiCr2O4
crystallize in a cubic spinel structure (space group $Fd\overline{3}m$). The
Chromium cations (Cr3), located at the 16b (0.5, 0.5, 0.5) site do not show
any deviation from the full occupancy. The refined lattice parameters, Cr - Cr
and Cr - A bond distances (Figure 2 (h) and (i)) are found to increase with
increasing the average radius of the A site. Figure 3 (a) - (e) displays the
FE-SEM images along with the results of the elemental mapping for all the
samples. The EDS mapping showcases the stoichiometric chemical composition,
indicating the homogeneous distribution of all elements across the sample at a
micrometer scale. The chemical compositions as obtained from the EDS analysis
are highlighted in Table I.

Furthermore, to investigate the detailed crystal structure, we have collected
the neutron powder diffraction (NPD) patterns at room temperature (RT) for two
high entropy spinel oxides [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. We will use the results of the NPD Rietveld
refinements to describe the crystal structure of these two samples. Figure 4
shows the final plot of the RT-NPD Rietveld refinements for these two samples.
The structural parameters obtained from the RT-NPD refinement for the
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 samples
are summarized in Table II. Cr cations occupy the 16d (0.5, 0.5, 0.5)
positions and form the octahedrally coordinated pyrochlore lattice of the
spinel structure, while A site cations (Ni, Mn, Mg, Co, Cu, Zn) occupy the
eightfold tetrahedral A sites (8a (0.125, 0.125, 0.125)). Oxygen atoms occupy
the 32e (x, x, x) Wyckoff positions. Refinement of the oxygen and Cr occupancy
indicates no deviation from full occupancy. In agreement with the EDS
measurements, occupancy refinements of the A site elements for
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 shows the Mg deficiency in the A site. For
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 no significant deviations from the
stoichiometric composition are observed in the refinement. Fig. 5 presents the
STEM-HAADF image of the [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 along the [101]
direction. The observed STEM image corresponds well with the spinel material’s
anticipated crystal structure, highlighting the excellent structural quality
at the nanoscale. The lattice parameter measured from the STEM images (8.4 Å)
is consistent with the values obtained from RT - NPD (Table II) and RT-XRD
refinements.

Figure 4:  Rietveld refinement plots of the room temperature neutron powder diffraction (NPD) patterns for (a) [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 and (b) [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. Table 2:  Atomic and structural parameters and selected bond lengths obtained from the RT - NPD refinements for [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 and [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. |  [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 |  [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4
---|---|---
Space Group | $Fd\overline{3}m$ | $Fd\overline{3}m$
a = b = c (Å) | 8.3377 (1) Å | 8.3235 (1) Å
Ni (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.21 (1)
Biso | 0.95 (4) | 0.82 (3)
Mn/Mg (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.15 (1)
Biso | 0.95 (4) | 0.82 (3)
Co (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.22 (1)
Biso | 0.95 (4) | 0.82 (3)
Cu (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.21 (1)
Biso | 0.95 (4) | 0.82 (3)
Zn (0.125, 0.125, 0.125) |  |
Occupancy | 0.2 | 0.21 (1)
Biso | 0.95 (4) | 0.82 (3)
Cr (0.5, 0.5, 0.5) |  |
Occupancy | 1.0 | 1.0
Biso | 0.52 (3) | 0.21 (2)
O (x,x,x) |  |
x = 0.26224 (3) | 0.26205 (3) |
Occupancy | 1.0 | 1.0
Biso | 0.36 (3) | 0.26 (1)
Cr - Cr | 2.9478 (1) Å | 2.9428 (1) Å
Cr - A | 3.4566 (1) Å | 3.4508 (1) Å
$\chi^{2}$ | 8.04 | 7.46
RP | 5.84 | 6.98
RWP | 5.37 | 6.5
Figure 5:  STEM - HAADF image along the [101] zone axis for
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4, collected at room temperature. Cr and A site
cations are shown in the figure. Figure 6:  (a) Raman spectra for all the
materials collected at room temperature. (b) Shift of the F2g (1) mode with
the variation of the radius of the A site for all the samples. Figure 7:
Temperature dependent FC and ZFC magnetic susceptibilities for (a)
[Ni0.5Mn0.5]Cr2O4, (b) [Ni0.33Mn0.33Co0.33]Cr2O4, (c)
[Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4. The Curie-Weiss fitting for all the materials
is highlighted in the graph. The inset in Figure (a) shows an anomaly due to
the phase transition driven by Jahn-Teller distortions of the Ni2 in the
inverse susceptibility plot. The inclusion of Mn in the A-site of the
structure dilutes the Jahn-Teller activity of the Ni cations. Therefore, the
structural transition occurs at a lower temperature (transition temperature =
310 K for NiCr2O4). Figure 8:  Temperature dependent (left axis) FC and ZFC
magnetic susceptibilities and (right axis) inverse FC susceptibilities for (a)
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and (b) [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4.
Figure 9:  Magnetic field variation of the magnetizations (M - H) recorded at
4 K for all the materials.

The Raman scattering studies were conducted at room temperature on all samples
to understand their structure and the response of the increased disorder (and
entropy) in the A site of these spinel materials. The room temperature Raman
spectra for six different medium, and high entropy oxides (HEOs) with cubic
structures are shown in Figure 6. The spinel unit cell, comprising 56 atoms,
reduces to 14 atoms in the primitive cell. Based on group theory analysis, for
Cubic spinels (space group Fd - 3m), five first-order Raman active phonons,
represented as $\Gamma$Raman = A1g \+ Eg \+ 3F2g are expected 28 ; 29 . A1g
mode (600-720 cm-1) is due to vibrations (move away) along the direction
connecting an oxygen atom to the tetrahedral A2+ cations. The Eg mode (250-360
cm-1) corresponds to the symmetric bending of the oxygen anion relative to the
octahedral B3+ cations. The F2g (1) mode, appearing between 160-220 cm-1, is
related to a complete translation of the tetrahedral unit, while the higher
frequency F2g (2) and F2g (3) modes (440-590 cm-1) arise from asymmetric
stretching and bending of oxygen, respectively. All five first-order Raman
modes are apparent in all the samples. However, in comparison to NiCr2O4, we
observe a broadening of the A1g peak in the medium and high entropy spinels.
The electronic and high entropy stabilized disorder in the A site of the
spinel structure could contribute to the broadening of the A1g peak. Figure 6
(b) highlights the evolution of the F2g (1) mode with the average radius of
the A site (AB2O4 type) in the structure. It shows that the F2g (1) mode
shifts toward the higher wave number with an increase in the average radius of
the A site of the structure. Previously, it was suggested that F2g (1) mode
corresponds to a translational motion of the entire A-site tetrahedral unit
within the spinel structure 28 . Therefore, our data support that this
vibration primarily involves the tetrahedral cation, as this mode exhibits a
significant and systematic change in the wavenumber among all Raman-active
modes.

Figure 10:  Room temperature XAS data of Cr L - edge for all the materials. Table 3:  Sample compositions, magnetic transition temperatures (TC for ferrimagnetic (FIM) materials and TN for antiferromagnetic (AFM) materials), Coercivity (HC), experimentally obtained and theoretically calculated effective magnetic moments ($\mu_{eff}$), Curie-Weiss temperature ($\theta_{CW}$) and frustration parameters (f = |$\frac{\theta_{CW}}{T_{C}}$|) for all the materials. Sample Compositions |  TC (K) |  HC (4 K) |  $\mu_{eff}$ ($\mu_{B}$) |  $\mu_{eff}$ (theo.) | $\theta_{CW}$ | f | Ref.
---|---|---|---|---|---|---|---
NiCr2O4 | 74 (FIM) | 9kOe | 6.53 |  6.16$\mu_{B}$ | -487 | 6.6 | 25
[Ni0.5Mn0.5]Cr2O4 | 45 (FIM) | 15kOe | 7.12 |  7.18$\mu_{B}$ | -351 | 7.8 | this work
[Ni0.33Mn0.33Co0.33]Cr2O4 | 63 (FIM) | 10.3kOe | 7.4 |  7.01$\mu_{B}$ | -375 | 5.9 | this work
[Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4 | 78 (FIM) | 0.7kOe | 6.86 |  6.72$\mu_{B}$ | -472 | 6 | this work
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 | 40 (FIM) | 0.17kOe | 6.74 |  6.49$\mu_{B}$ | -445 | 11.1 | this work
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 | 35 (AFM) | 0.08kOe | 6.13 |  5.93$\mu_{B}$ | -467 | 13.4 |  this work, 30

Magnetic susceptibility as a function of temperature under field cooled (FC)
and zero field cooled (ZFC) protocol are shown in Figure 7 and 8 for all the
compositions. For medium entropy samples [Ni0.5Mn0.5]Cr2O4,
[Ni0.33Mn0.33Co0.33]Cr2O4, [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4, a sharp upturn is
observed in the FC susceptibility data and is marked as the onset of the
magnetic ordering. The corresponding transition temperatures for different
samples are tabulated in Table 3. Similar to the parent NiCr2O4 these three
samples show ferrimagnetic (FIM) - type ordering. An anomaly in the
susceptibility plot for [Ni0.5Mn0.5]Cr2O4 is observed at 280 K and could be an
indication of tetragonal to cubic structural transition. In NiCr2O4 similar
structural transition occurs at 310 K 25 . At 310 K, a phase transition driven
by cooperative Jahn-Teller distortions occurs and lifts the orbital degeneracy
and reduces the structural symmetry from cubic (Fd-3m) to tetragonal
(I41/amd). The inclusion of Mn in the A-site of the structure (in
[Ni0.5Mn0.5]Cr2O4) dilutes the Jahn-Teller distortion of the Ni cations.
Therefore, the structural transition is occurring at a lower temperature (at
280 K). Further dilution in the Jahn-Teller activity in other medium and high
entropy samples removes the tetragonal distortion in the sample. High
irreversibility between the field-cooled and zero-field-cooled measurements is
observed for these three samples and can be attributed to spin freezing. This
spin freezing likely results from the high degree of structural disorder in
the sample. The magnetic field-dependent magnetization measurements for all
the samples recorded at 4 K are shown in Figure 9. The typical Brillouin shape
of the curves for (Ni0.5Mn0.5)Cr2O4, and (Ni0.33Mn0.33Co0.33)Cr2O4 is a
signature of the typical ferrimagnetic ordering of the samples. Also, an
extremely high value of the coercivity is observed ($\simeq$ 1.5 T, table 3)
for these two samples. However, for [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4 a linear
complex antiferromagnetic type M - H loop is observed. Spinel oxides based on
Cr often exhibit intricate magnetic behaviors. For instance, CoCr2O4
transitions to a ferrimagnetic state at 95 K, followed by long-range non-
collinear magnetic ordering below 31 K 16 ; 31 . NiCr2O4 shows ferrimagnetic
ordering at 74 K followed by a complex antiferromagnetic ordering at 31 K 25 .
The linear M-H characteristic in [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4 could be due
to predominant antiferromagnetic interactions overshadowing ferrimagnetic
interactions at lower temperatures. The FC and ZFC magnetic susceptibilities
as a function of temperature for two high entropy samples
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 are
shown in Figure 8. Similar to other samples both the FC and ZFC susceptibility
for [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 show a ferrimagnetic type sharp upturn at
40 K, then on further cooling irreversibility between FC and ZFC
susceptibility data is observed below 25 K. On the other hand
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 shows a long-range antiferromagnetic ordering
at 35 K 30 . To further investigate the magnetic properties of these samples,
the temperature variation of the susceptibility plots for all the samples are
fitted with the Curie-Weiss expression

$\chi(T)=\frac{C}{(T-\theta_{CW})}$ (1)

where $\theta_{CW}$ is the Curie-Weiss temperature, C is the Curie-Weiss
constant, and the fitted parameters are listed in table III. The calculated
effective magnetic moments (Peff) using the fitted Curie-Weiss constant C show
good agreement with the expected theoretical values (considering the spin-only
contribution) for all the samples. However, all the materials show very high
values of $\theta_{CW}$. All obtained Curie-Weiss temperatures ($\theta_{CW}$)
are negative. This indicates that the overall exchange interactions among the
magnetic ions are predominantly antiferromagnetic. In comparison with the low
entropy and medium entropy samples two high entropy materials show a
significant increase in the frustration parameter f =
|$\frac{\theta_{CW}}{T_{C}}$|. The high atomic-level disorder in these two
high entropy samples likely results in considerable magnetic frustration. The
complex crystal structure, featuring diamond and pyrochlore lattices, along
with the random distribution of magnetic and nonmagnetic cations within
distinct sublattices, is anticipated to foster complex magnetic correlations
and substantial frustration.

In general, the intricate magnetic properties observed in these samples
reflect the presence of multiple competing magnetic interactions. This is
expected in systems where multiple magnetic cations occupy different crystal
field environments, resulting in varied magnetic moments that interact with
diverse exchange interaction strengths and signs. The magnetic transition
temperature is observed to vary with changes in the entropy of the structure,
yet this variation does not follow a consistent increasing or decreasing
trend. Since the transition temperature is dependent on the strength of
exchange interactions, this suggests a complex interplay among the magnetic
ions. This complexity precludes straightforward comparisons across systems
with differing types and proportions of magnetic ions, preventing the
establishment of a systematic trend.

To explore the electronic states and local geometry of the material, XAS
experiments were performed on the high entropy samples. Figure 10 compares the
XAS spectra of the Cr L3,2-edges for all the compounds. In Cr-based spinel
materials, Cr cations occupy the octahedral B site. Figure 10 clearly shows
that all the Cr L3,2 spectra are similar to those of the parent NiCr2O4,
confirming the octahedral environment of the Cr3+ cations in all the samples.
Consequently, all the A-site cations in these samples are expected to have a
charge state of 2+.

##  IV Conclusion

In this study, we successfully synthesized and characterized a new series of
Cr-based low, medium and high-entropy spinel oxides with composition NiCr2O4,
[Ni0.5Mn0.5]Cr2O4, [Ni0.33Mn0.33Co0.33]Cr2O4, [Ni0.25Mn0.25Co0.25Cu0.25]Cr2O4,
[Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. RT-XRD
patterns and RT-EDS elemental mappings are used to determine the crystal
structure and micro-scale homogeneity for all the samples. The STEM-HAADF
image of the [Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4 highlight the excellent
crystalline quality and homogeneity in nanoscale. The lattice parameters and
Cr-Cr distances for all the samples are found to follow the average radius of
the A site. Furthermore, we have used the ambient temperature neutron
diffraction patterns to determine the detailed crystal structure for two high
entropy spinel oxides [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4, and
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. Magnetic measurements reveal that
introducing multiple cations into the A-site of the spinel structure
profoundly affects the magnetic properties, inducing a range of behaviors from
ferrimagnetism to antiferromagnetism, and significantly enhancing coercivity.
High entropy sample [Ni0.2Mn0.2Co0.2Cu0.2Zn0.2]Cr2O4 shows a complex
ferrimagnetic behaviour at 40 K. On the other hand, a small change in the
structure induces antiferromagnetic behavior in
[Ni0.2Mg0.2Co0.2Cu0.2Zn0.2]Cr2O4. Extremely high coercivity is observed for
[Ni0.5Mn0.5]Cr2O4, [Ni0.33Mn0.33Co0.33]Cr2O4. Room temperature XAS confirms
the octahedral environment of the Cr3+ cations in all the samples. The
variation in magnetic transition temperatures, independent of a clear trend
with entropy changes, indicates the sensitivity of magnetic properties to
specific elemental compositions and their distributions within the lattice.
These findings demonstrate the versatility and potential of high-entropy
spinel oxides in developing materials with tailored magnetic properties for
various technological applications. From a broader vantage point, our study
highlights the potential to incorporate a spectrum of elemental components
within a structurally complex oxide framework. This highlights excellent
opportunities for tailoring and refining properties in strongly correlated
materials.

##  V Acknowledgments

S. M. acknowledges the SERB, Government of India, for the (SRG/2021/001993)
Start-up Research Grant. We acknowledge the UGC-DAE Consortium for Scientific
Research, Indore for the XAS and magnetization measurements. UGC-DAE Mumbai is
also acknowledged for the Neutron diffraction experiments. We would like to
thank Dr. Rajeev Rawat (magnetization), Dr. S. D. Kaushik (Neutron
diffraction), Mr. Kranti Kumar Sharma (magnetization) and Rakesh Shah (SXAS)
for their support during the measurements.

## References

  * (1) D. B. Miracle, O. N. Senkov, Acta Mater., 122, 448-511 (2017).
  * (2) Y. F. Ye, Q. Wang, J. Lu, C. T. Liu, Y. Yang, Materials Today, 19, 349-362 (2016).
  * (3) C. M. Rost, E. Sachet, T. Borman, A. Moballegh, E. C. Dickey, D Hou, J. L. Jones, S. Curtarolo, J.-P. Maria, Nat. Commun., 6, 8485 (2015).
  * (4) A. R. Mazza, E. Skoropata, Y. Sharma, J. Lapano, T. W. Heitmann, B. L. Musico, V. Keppens, Z. Gai, J. W. Freeland, T. R. Charlton, M. Brahlek, A. Moreo, E. Dagotto and T. Z. Ward, Adv. Sci., 9, 2200391 (2022).
  * (5) G. H. J. Johnstone, M. U. Gonzalez-Rivas, K. M. Taddei, R. G. A. Sawatzky, R. J. Green, M. Oudah, and A. M. Hallas, J. Am. Chem. Soc., 144, 20590-20600 (2022).
  * (6) N. Sharma, S. Jangid, S. Choudhury, S. Kr Mahatha, R. P. Singh, S. Marik, Applied Physics Letters 123, 161901 (2023).
  * (7) S Marik, D Singh, B Gonano, F Veillon, D Pelloquin, Y Breard, Scripta Mater., 186, 366-369 (2022).
  * (8) M. Cocconcelli, D. Miertschin, B. Regmi, D. Crater, F. Stramaglia, L. Yao, R. Bertacco, C. Piamonteze, S. van Dijken, and A. Farhan, Phys. Rev. B, 109, 134422 (2024).
  * (9) F. Jin, Y. Zhu, L. Li, Z. Pan, D. Pan, M. Gu, Q. Li, L. Chen, H. Wang, Advanced Functional Materials, 33, 2214273 (2023).
  * (10) A. Sarkar, D. Wang, M. V. Kante, L. Eiselt, V. Trouillet, G. Iankevich, Z. Zhao, S. S. Bhattacharya, H. Hahn, R. Kruk, Advanced Materials, 35, 2207436 (2022).
  * (11) R. R. Katzbaer, F. M. Vieira, I. Dabo, Z. Mao and R.E. Schaak, J. Am. Chem. Soc. 145, 6753-6761 (2023).
  * (12) F. Cheng, Z. Meng, C. Cheng, J. Hou, Y. Liu, B. Ren, H. Hu, F. Gao, Y. Miao and X. Wang, Corrosion Science, 218, 111199 (2023).
  * (13) S. S. Jana and T. Maiti, Mater. Horiz., 10, 1848-1855 (2023).
  * (14) Na-Li Chen, Ge-Ting Sun, Cheng-Yu He, Bao-Hua Liu, Hui-Xia Feng, Gang Liu and Xiang-Hu Gao, Materials Today Physics, 42, 101363 (2024).
  * (15) V. Fritsch, J. Hemberger, N. Buttgen, E.-W. Scheidt, H.-A. Krug von Nidda, A. Loidl, and V. Tsurkan, Phys. Rev. Lett., 92, 116401 (2004).
  * (16) V. Kocsis, S. BordÃ¡cs, D. Varjas, K. Penc, A. Abouelsayed, C. A. Kuntscher, K. Ohgushi, Y. Tokura, and I. Kezsmarki, Phys. Rev. B, 87, 064416 (2013).
  * (17) R. Fichtl, V. Tsurkan, P. Lunkenheimer, J. Hemberger, V. Fritsch, H.-A. Krug von Nidda, E.-W. Scheidt, and A. Loidl, Phys. Rev. Lett. 94, 027601 (2005).
  * (18) R. Nirmala, Kwang-Hyun Jang, H. Sim, H. Cho, J. Lee, Nam-Geun Yang, S. Lee, R. M. Ibberson, K. Kakurai and M. Matsuda, Journal of Physics: Condensed Matter, 29, 13LT01 (2017).
  * (19) J. H. Chung, M. Matsuda, S.-H. Lee, K. Kakurai, H. Ueda, T. J. Sato, H. Takagi, K.-P. Hong, and S. Park, Phys. Rev. Lett. 95, 247204 (2005).
  * (20) L. D. C. Jaubert, Y. Iqbal and Harald O. Jeschke, arXiv:2405.13908v1 (2024).
  * (21) J. R. Chamorro, L. Ge, J. Flynn, M. A. Subramanian, M. Mourigal, and T. M. McQueen, Phys. Rev. Materials 2, 034404 (2018).
  * (22) D. Bergman, J. Alicea, E. Gull, S. Trebst, and L. Balents, Nature Phys. 3, 487 (2007).
  * (23) F. L. Buessen, M. Hering, J. Reuther, and S. Trebst, Phys. Rev. Lett. 120, 057201 (2018).
  * (24) M. K. Wallace, Jun Li, P. G. Labarre, S. Svadlenak, D. Haskel, J. Kim, G. E. Sterbinsky, F. Rodolakis, H. Park, A. P. Ramirez, and M. A. Subramanian, Phys. Rev. Materials 5, 094410 (2021).
  * (25) M. R. Suchomel, D. P. Shoemaker, L. Ribaud, M. C. Kemei and R. Seshadri, Phys. Rev. B, 86, 054406 (2012).
  * (26) O. Crottaz, F. Kubel and H. Schmid, J. Mater. Chem., 7, 143-146 (1997).
  * (27) Supporting
  * (28) V. D’Ippolito, G. B. Andreozzi, D. Bersani, P. P. Lottici, Journal of Raman Spectroscopy, 46, 1255–1264 (2015).
  * (29) Z. Wang, S.K Saxena, P. Lazor and H.S.C O’Neill, Journal of Physics and Chemistry of Solids, 64, 425-431 (2003).
  * (30) S Marik, D Singh, B Gonano, F Veillon, D Pelloquin and Y Breard, Scripta Materialia, 183, 107-110 (2020).
  * (31) D. Zakutna, A. Alemayehu, J. Vlcek, K. Nemkovski, C. P. Grams, D. Niznansky, D. Honecker and S. Disch, Phys. Rev. B 100, 184427 (2019).


# Self-supervised Contrastive Learning for 6G UM-MIMO THz Communications:
Improving Robustness Under Imperfect CSI

Rafid Umayer Murshed1, Md Saheed Ullah2, Mohammad Saquib1, and Moe Z. Win3

Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>1Department of Electrical and Computer Engineering, The
University of Texas at Dallas, Texas, USA  2Department of Electrical and
Computer Engineering, University of Delaware, Delaware, USA  3Laboratory for
Information and Decision Systems, Massachusetts Institute of Technology,
Massachusetts, USA

###### Abstract

This paper investigates the potential of contrastive learning in 6G ultra-
massive multiple-input multiple-output (UM-MIMO) communication systems,
specifically focusing on hybrid beamforming under imperfect channel state
information (CSI) conditions at THz. UM-MIMO systems are promising for future
6G wireless communication networks due to their high spectral efficiency and
capacity. The accuracy of CSI significantly influences the performance of UM-
MIMO systems. However, acquiring perfect CSI is challenging due to various
practical constraints such as channel estimation errors, feedback delays, and
hardware imperfections. To address this issue, we propose a novel self-
supervised contrastive learning-based approach for hybrid beamforming, which
is robust against imperfect CSI. We demonstrate the power of contrastive
learning to tackle the challenges posed by imperfect CSI and show that our
proposed method results in improved system performance in terms of achievable
rate compared to traditional methods.

###### Index Terms:

Contrastive Learning, Hybrid Beamforming, Imperfect CSI, NYUSIM, Rain Fade,
THz, UM-MIMO, 6G.

##  I INTRODUCTION

Ultra-massive multiple-input multiple-output (UM-MIMO) systems are a promising
solution to meet the increasing demand for high data rates and reliable
communication in next-generation 6G wireless networks. By utilizing a large
number of antennas at both the transmitter and receiver, UM-MIMO systems can
achieve significant improvements in spectral efficiency (SE) and capacity[1].
However, these benefits come with the challenge of obtaining accurate channel
state information (CSI) to design effective beamforming strategies.

Hybrid beamforming, which combines digital and analog beamforming techniques,
is a widely used approach in UM-MIMO systems to reduce hardware complexity and
power consumption[2, 3]. By capitalizing on the strengths of digital and
analog elements, hybrid beamforming adeptly manages the trade-off between
performance and resource efficiency[4]. This strategic alignment not only
enhances the overall SE of communication systems but also addresses the
challenges arising from hardware limitations in emerging technologies.
However, the performance of hybrid beamforming is highly dependent on accurate
CSI[5]. In practice, acquiring perfect CSI is challenging due to various
factors such as channel estimation errors[6], feedback delays[7], and
quantization errors[8]. Therefore, there is a need for robust beamforming
techniques that can perform well even with imperfect CSI. As the demand for
high-performance wireless communication systems continues to grow, the
refinement and better estimation of CSI data in conjunction with robust hybrid
beamforming strategies become pivotal in enhancing the reliability and
efficiency of communication networks.

Harnessing the power of the NYUSIM channel model, an advanced simulation tool
from New York University, we can accurately emulate the complexities of real-
world wireless environments. NYUSIM allows for simulations with RF bandwidths
up to 1 GHz and carrier frequencies reaching 150 GHz, exceeding the scope of
5G technologies [9]. It offers a versatile platform for testing diverse
scenarios like urban microcell (UMi) and macrocell (UMa), as well as indoor
hotspots (InH), through adjustable parameters such as spatial consistency and
human blockage [10]. These features enable a detailed replication of the
multifaceted nature of THz wireless communication environments, making NYUSIM
an invaluable asset for researchers and engineers examining the intricacies of
wireless systems [11].

Building on this foundation, our paper introduces a pioneering self-supervised
contrastive learning-based approach for hybrid beamforming in UM-MIMO THz
systems, particularly under the challenge of imperfect CSI. This method
harnesses contrastive learning, a technique that has demonstrated remarkable
success in diverse domains such as computer vision, natural language
processing, and even seismology[12], to discern and exploit the subtle nuances
in data. By utilizing this approach, we aim to develop a robust beamforming
strategy that can deliver high performance even in the face of CSI
inaccuracies, thereby meeting the demands for more resilient and efficient
next-generation wireless communication systems.

##  II System Model and Problem Formulation

This section presents the system model for UM-MIMO communications and
formulates the hybrid beamforming problem under imperfect CSI conditions. We
discuss the challenges associated with conventional hybrid beamforming
techniques and motivate the use of contrastive learning to address these
challenges.

###  II-A System Model

Our single-user UM-MIMO network features a base station (BS) and user
equipment (UE), each with a uniform linear array (ULA) of $N_{t}$ and $N_{r}$
antennas, respectively, and $N^{t}_{\text{RF}}$ and $N^{r}_{\text{RF}}$ RF
chains for transmission and reception. The network supports $N_{s}$ data
streams, constrained by $N_{s}\leq\min(N^{t}_{\text{RF}},N^{r}_{\text{RF}})$.
The limited RF chains necessitate hybrid digital-analog beamforming, with the
transmitted signal
$\textbf{x}=\textbf{F}\textsubscript{RF}\textbf{F}\textsubscript{BB}\textbf{s},$,
where FBB and FRF are the digital and analog precoders, and s is the
transmitted symbol vector with
${\mathbb{E}}\left\\{{\textbf{s}\textbf{s}^{H}}\right\\}=\frac{1}{N_{s}}\textbf{I}{N_{s}}$.
The received signal is
$\textbf{r}=\sqrt{\rho}\textbf{W}\textsubscript{RF}^{H}\textbf{W\textsubscript{BB}}^{H}\textbf{H}\textbf{x}+\textbf{W\textsubscript{RF}}^{H}\textbf{W\textsubscript{BB}}^{H}\textbf{n},$,
with WRF and WBB as the receiver’s beamformers, $\rho$ as transmit power, and
n as AWGN.

###  II-B Problem Formulation

The beamforming challenge in UM-MIMO systems is twofold, encompassing precoder
and decoder design. In this work, our focus is on demonstrating the
application and potential of self-supervised contrastive learning to enhance
robustness in UM-MIMO communications. Specifically, we concentrate on the
design of the digital part of the beamforming process under the assumption of
a perfectly designed analog beamformer. This approach allows us to explore the
intricate dynamics of the digital domain more thoroughly, leveraging self-
supervised learning techniques for optimization.

Given this context, the precoder design is represented as:

$\begin{gathered}\min_{\textbf{F}_{\text{RF}},\textbf{F}_{\text{BB}}}\left\|\textbf{F}_{\text{opt}}-\textbf{F}_{\text{RF}}\textbf{F}_{\text{BB}}\right\|_{F}\\\
\text{s.t.}\;|(\textbf{F}_{\text{RF}})_{ij}|=1,\;\left\|\textbf{F}_{\text{RF}}\textbf{F}_{\text{BB}}\right\|_{F}^{2}=N_{s}.\end{gathered}$
(1)

Maximizing SE is analogous to the above formulation [13]. The optimal hybrid
precoders closely resemble fully digital precoders in an unconstrained
setting. Utilizing a MIMO channel’s decomposition into parallel independent
channels [14], the data rate increases proportionally to the number of
independent streams, $N_{s}$. The optimal fully digital precoder matrix
$\textbf{F}_{\text{opt}}\in\mathbb{R}^{N_{t}\times N_{s}}$ is derived from the
Singular Value Decomposition (SVD) of the channel matrix H as
$\textbf{H}=\textbf{U}\Sigma\textbf{V}^{H}$ [15]. In this context, the
challenge lies in the absence of direct access to the noise-free optimal fully
digital precoder matrix, $\textbf{F}_{\text{opt}}$. Our formulation must
contend with $\tilde{\textbf{F}}_{\text{opt}}$, a noise-corrupted
approximation of $\textbf{F}_{\text{opt}}$, resulting from the inevitable
imperfections in CSI. This reality necessitates a robust design strategy that
can effectively approximate $\textbf{F}_{\text{opt}}$ and sustain performance
despite the presence of CSI inaccuracies, underlining the critical need for
resilient optimization in the digital beamforming domain. Upon establishing
the digital beamforming matrix, the corresponding analog beamformer can be
determined through a supervised approach, as elucidated in [16].

The SE is calculated using the formula:

$R=\log_{2}\left|\textbf{I}_{M}+\frac{\rho}{N_{s}}\textbf{W}_{t}(\textbf{W}_{t}^{H}\textbf{W}_{t})^{-1}\textbf{W}_{t}^{H}\textbf{H}\textbf{V}_{t}\textbf{V}_{t}^{H}\textbf{H}^{H}\right|,$
(2)

where $\textbf{W}_{t}=\textbf{W}_{\text{RF}}\times\textbf{W}_{\text{BB}}$,
$\textbf{V}_{t}=\textbf{F}_{\text{RF}}\times\textbf{F}_{\text{BB}}$, and
$\rho$ denotes the transmit power.

##  III Proposed Contrastive Learning-Based Approach

In this section, we describe our proposed contrastive learning-based approach
for hybrid beamforming in UM-MIMO systems under imperfect CSI. We first
introduce the concept of contrastive learning and explain how it can be
utilized in the context of hybrid beamforming. We then present the
architecture of our proposed deep learning model and describe the training
procedure and optimization techniques used to learn the optimal beamforming
weights.

###  III-A Contrastive Learning - Background

Contrastive learning is a self-supervised learning paradigm that learns to
encode data by contrasting positive pairs against negative pairs. Fundamental
to this is the contrastive loss function, often formulated as a function of
temperature-scaled dot products between representations. Specifically, for a
pair of representations $u$ and $v$, the contrastive loss can be described as:

$\mathcal{L}_{contrastive}(u,v)=-\log\frac{\exp(u\cdot v/\tau)}{\sum_{w\in
W}\exp(u\cdot w/\tau)}$ (3)

where $W$ contains one positive sample and several negative samples, and
$\tau$ represents the temperature parameter, adjusting the sensitivity of the
loss function to the similarity between samples. This loss encourages the
network to learn an embedding space where positive samples are pulled closer
and negative samples are pushed apart.

Figure 1: Schematic diagram of the proposed self-supervised contrastive DNN
illustrating its architecture and workflow. Figure 2: PCA plot of
$\textbf{F}_{\text{opt}}$ embeddings (a) before and (b) after Contrastive pre-
training, illustrating the model’s ability to cluster augmented samples from
the same channel state.

###  III-B Contrastive Learning for Robust UM-MIMO

Leveraging self-supervised contrastive learning, our approach aims to enhance
hybrid beamforming performance under imperfect CSI. We begin with a batch of
optimal beamforming matrices $\textbf{F}_{\text{opt}}$ as a base for
augmentation. By adding AWGN noise 111Our proposed methodology uniformly
applies to other noise models. , we generate two augmented samples,
$\tilde{\textbf{F}}_{\text{opt}}$, for each $\textbf{F}_{\text{opt}}$,
creating positive pairs, while the rest within the batch serve as negative
samples. This generates a rich dataset that captures the variance in channel
conditions due to noise. Our contrastive loss function then works to cluster
the $D_{E}$ dimensional embeddings (as depicted in Fig. 1) of augmented
samples from the same channel state, driving the model to learn robust
representations despite noisy inputs.

Fig. 2 presents principal component analysis (PCA) plots of the embeddings
before and after contrastive training. For visualization, we select five
samples and generate two additional augmented samples for each by adding
noise. Initially, the PCA plot reveals significant dispersion among the
original and their corresponding augmented samples, reflecting the impact of
the noise. Post-contrastive training, the PCA plot indicates that the
embeddings of the original and augmented samples have converged, demonstrating
the efficacy of our training method in creating tightly clustered
representations resilient to CSI noise variations.

These representations are pivotal for our hybrid beamforming, where the
digital beamformer is designed using embeddings from the contrastive pre-
training stage. By doing so, the model is trained to discern the subtle
differences between optimal and suboptimal beamforming strategies, even when
the CSI is less than perfect. The final stage training employs a bidirectional
LSTM combined with separation networks, culminating in a robust hybrid
beamforming solution suited for real-world UM-MIMO deployments.

Figure 3: Detailed view of the contrastive pre-training network showcasing the
projection head for embedding generation. Figure 4: Architecture of the final-
stage training network with bidirectional LSTM blocks for generating the
beamforming matrix.

###  III-C Architecture of the Proposed Contrastive Network

Our contrastive pre-training network employs a multi-branch architecture to
independently process phase and magnitude inputs of beamforming matrices
before fusion as depicted in Fig. 1. Each branch consists of a series of 1D
convolutional (Conv) blocks, with time-distributed (T.D) operations to handle
the temporal aspects of our data. The network accepts
$\textbf{F}_{\text{opt}}$ phase and magnitude inputs of dimension
($N_{t}$,$N_{s}$).

The first Conv block in each branch includes separable convolutional layers
with 16 filters of kernel size 256, succeeded by batch normalization and ReLU
activation. Subsequent blocks increment the filter count and decrement the
kernel size, intensifying feature specificity. After three Conv blocks, depth-
wise separable convolutional layers aim to efficiently learn spatial
hierarchies. Each Conv block is succeeded by a max-pooling layer to reduce
dimensionality and emphasize salient features.

Following feature extraction, branches merge, forming a fusion network that
advances through a projection head. This projection head, discardable after
training, is composed of a shallow multilayer perceptron (MLP) with a flatten
layer, ensuring a succinct $D_{P}$ dimensional representation of the $D_{E}$
embeddings for the contrastive loss function. The architecture is meticulously
designed to ensure the embeddings from the phase and magnitude of
$\textbf{F}_{\text{opt}}$ converge, facilitating effective contrastive
learning across diverse channel conditions. The detailed architecture of the
contrastive pre-training network is shown in Fig. 3.

###  III-D Beamforming Matrix Generation

The final stage of training of our network employs a novel architecture
specifically designed for generating beamforming matrices. The network
architecture is composed of a bidirectional Long Short-Term Memory (LSTM)
block with multiple LSTM layers that process the embeddings generated from the
contrastive stage as shown in the prediction head of Fig. 1.

The bidirectional LSTM block, with 256, 128, and 128 units respectively in
each subsequent layer, ensures the capture of temporal dependencies within the
data. This is essential for learning the dynamics within the phase and
magnitude representations of $\textbf{F}_{\text{BB}}$. The output from the
LSTM is concatenated, resulting in a feature vector of size 200, which
encapsulates temporal features from both directions.

Concurrently, the network’s lower branch processes the magnitude and phase
separately. Each path features a stack of dense layers with 100 neurons,
followed by a reduction to 50. These dense layers, selected for efficiency,
are accompanied by swish activations to emphasize salient features. The
outputs are then flattened and input into subsequent dense layers. The
detailed architecture of the contrastive pre-training network is displayed in
Fig. 4.

The LSTM and lower branch outputs work by combining temporal and spatial
features. This composite feature set is then refined through additional dense
layers, culminating in the final magnitude and phase outputs of
$\textbf{F}_{\text{BB}}$, each with dimensions $(N_{\text{RF}},N_{s})$. This
architecture facilitates a sophisticated learning mechanism for predicting
beamforming matrices adept at navigating the complexities of real-world UM-
MIMO systems.

In this architecture, the self-supervised characteristic emerges from the fact
that the input $\textbf{F}_{\text{opt}}$ is also used to calculate the loss
function which is formulated analogously to (1). Here $\textbf{F}_{\text{RF}}$
is obtained via the product of the input $\textbf{F}_{\text{opt}}$ and pseudo-
inverse of the generated output $\textbf{F}_{\text{BB}}$. This innovative
approach underpins the training process, ensuring that the network’s output,
$\textbf{F}_{\text{BB}}$, is consistently aligned with the optimal beamforming
matrices. By integrating the input as part of the loss computation, the
network is intrinsically guided toward solutions that inherently respect the
underlying physical model of UM-MIMO systems.

Figure 5: Plot of SE with SNR for perfect CSI scenarios. Figure 6: Performance
comparison in perfect CSI scenarios with that at a CSI SNR of 36 dB. Figure 7:
Plot of SE with SNR for different CSI imperfection scenarios for (a) Proposed
DNN and (b) PE-ALT-MIN.

##  IV Simulation Results and Performance Analysis

We evaluate the performance of our proposed approach through extensive
simulations under various imperfect CSI conditions. We compare our proposed
method’s achievable rate with existing hybrid beamforming techniques and show
that our contrastive learning-based approach outperforms traditional methods,
particularly under severe CSI imperfections.

###  IV-A Environment

Our framework has been developed utilizing Python, Tensorflow 2.12, and Keras
2.12. Numpy is employed for computations, while MATLAB 2023b is utilized for
generating certain visual representations. The model is trained on Google
Colab Pro+ using a system with 83.5 GB of RAM, 166.8 GB of disk space, and an
NVIDIA A100-SXM4-40GB GPU to accelerate the training process. After training,
the model’s size is compact ($\thickapprox$ 1.3 MB), enabling it to be
deployed on regular PC hardware and edge computing platforms.

###  IV-B Data Generation

The NYUSIM channel simulator (version 3.1) is utilized for generating channel
data with a carrier frequency of 100 GHz and an RF bandwidth of 800 MHz[17].
We contemplate a uniform rectangular array, square in shape, comprising 1024
elements for the transmitter, and a receiver array with 16 elements, both
featuring a standard element spacing of 0.5 wavelength. We took into account a
rain rate of 0.1 mm/hr to incorporate rain fade and emulate real-life
scenarios[18]. The parameters for dataset generation are outlined in Table I.

To train and validate our proposed DNN model, we generate 50,000 channel
realizations using the previously specified configuration. We additionally
produce an additional 10000 channel realizations (test-set) to evaluate the
efficacy of the proposed model.

TABLE I: NYUSIM Simulation Parameters Parameters | Values
---|---
Scenario | UMa
Frequency | 100 GHz
RF Bandwidth | 800 MHz
TX and RX Antenna Type | URA
Number of TX Antennas | 1024
Number of RX Antennas | 16
Environment | LOS
T-R Separation Distance |  $10-30$ m
TX Power | 50 dBm
Base Station Height | 35 m
User Terminal Height | 1.5 m
Barometric Pressure | 1013.25 mbar
Humidity | 50 %
Temperature | 20 ∘C
Rain Rate | 0.1 mm/hr

###  IV-C Performance under Perfect CSI

Before discussing our model’s efficacy under imperfect CSI, it is illustrative
to consider its performance with perfect CSI knowledge, serving as a benchmark
for the universality of our approach. In the assessment of our model’s
performance under perfect CSI conditions, the results exhibit a promising
alignment with the theoretical optimum. The graph shown in Fig. 5 plots SE
against signal-to-noise ratio (SNR), where our proposed DNN approach and the
baseline Alternating Minimization (PE-Alt-MIN) method are compared against the
optimal beamforming performance. Here, the optimal digital beamformer is
obtained directly via the SVD of the channel matrix. As SNR increases, both
the proposed DNN and PE-Alt-MIN closely track the optimal performance, merging
with the optimal curve at most SNR values. This indicates that our approach
achieves an equivalent SE to Alt-MIN under perfect CSI, as shown by the
overlapping lines on the graph. The simulation reinforces the DNN’s capacity
to approximate optimal beamforming techniques, validating the efficacy of our
model in ideal conditions, where we presume an optimal analog beamformer is in
place.

###  IV-D Performance under Imperfect CSI

To examination of our model’s resilience under imperfect CSI, we first compare
its performance at a CSI SNR of 36 dB (close to perfect CSI) and under perfect
CSI conditions. The results are presented in Fig. 6, where the plots also show
the performance of the benchmark PE-Alt-MIN and the optimal digital beamformer
under both perfect CSI and noisy CSI (36dB). The SE of our DNN approach
maintains a near-constant level, demonstrating robustness even as the CSI SNR
varies. This is contrasted with the Alt-MIN algorithm, which exhibits a
significant drop in performance with decreasing CSI SNR. Here, the SE achieved
by the DNN in both perfect CSI conditions and 36dB CSI SNR remains exactly the
same and hence only the latter is shown in the plot. The numerical results
reveal a gain of more than 10% in SE compared to other methods, affirming the
DNN’s effectiveness.

For a more comprehensive performance analysis, we vary the extent of CSI
imperfection and examine the SE achieved by the algorithms in such scenarios.
As further illustrated in Fig. 7(a), the SE of our DNN remains stable and
performs comparably to the optimal beamforming scenario, signifying the
model’s insensitivity to CSI imperfections. In comparison, the SE of Alt-MIN,
as depicted in Fig. 7(b), reduces drastically with the increment of noise in
CSI, emphasizing the DNN’s superior noise resilience. Particularly, the SE of
Alt-MIN decreases by almost 17 times at a CSI SNR of 20 dB, indicating a steep
performance degradation under increased noise levels. This stark disparity
highlights our DNN’s superior adaptability, substantiating its viability for
deployment in UM-MIMO systems where perfect CSI is a rarity. The DNN’s
enduring high SE across the spectrum of CSI SNR values not only attests to its
adaptability but also signals a stride toward dependable, high-caliber
wireless communication systems, capable of weathering the CSI uncertainty
inherent in real-world scenarios.

##  V Conclusion

In this paper, we have proposed a novel contrastive learning-based approach
for hybrid beamforming in UM-MIMO systems under imperfect CSI conditions. The
efficacy of the proposed method is substantiated through numerical results,
which reveal a marked improvement in system performance—specifically in
achievable rate—over conventional techniques. Notably, at a CSI SNR of 20 dB,
our approach maintains stable SE, in stark contrast to traditional beamforming
algorithms which exhibit a seventeenfold decrease in SE. Additionally, our
unique approach showcases potential in handling non-instantaneous CSI
challenges, enhancing adaptability for user mobility and dynamic environments,
thereby paving the way for addressing key 6G UM-MIMO communication issues,
including energy-efficient resource allocation.

## References

  * [1] I. F. Akyildiz and J. M. Jornet, “Realizing ultra-massive mimo (1024$\times$ 1024) communication in the (0.06–10) terahertz band,” _Nano Communication Networks_ , vol. 8, pp. 46–54, 2016.
  * [2] M. S. Ullah, S. C. Sarker, Z. B. Ashraf, and M. F. Uddin, “Spectral efficiency of multiuser massive mimo-ofdm thz wireless systems with hybrid beamforming under inter-carrier interference,” in _2022 12th International Conference on Electrical and Computer Engineering (ICECE)_. IEEE, 2022, pp. 228–231.
  * [3] A. F. Molisch, V. V. Ratnam, S. Han, Z. Li, S. L. H. Nguyen, L. Li, and K. Haneda, “Hybrid beamforming for massive mimo: A survey,” _IEEE Communications magazine_ , vol. 55, no. 9, pp. 134–141, 2017.
  * [4] M. S. Ullah, R. U. Murshed, and M. F. Uddin, “Beyond traditional beamforming: Singular vector projection techniques for mu-mimo interference management,” _arXiv preprint arXiv:2311.03741_ , 2023.
  * [5] A. Liu and V. K. Lau, “Impact of csi knowledge on the codebook-based hybrid beamforming in massive mimo,” _IEEE Transactions on Signal Processing_ , vol. 64, no. 24, pp. 6545–6556, 2016.
  * [6] Z. Gao, Y. Mei, and L. Qiao, “Spatially common sparsity based adaptive channel estimation and feedback for fdd massive mimo,” in _Sparse Signal Processing for Massive MIMO Communications_. Springer, 2023, pp. 51–81.
  * [7] E. M. Ghourab, L. Bariah, S. Muhaidat, P. C. Sofotasios, M. Al-Qutayri, and E. Damiani, “Secure relay selection with outdated csi in cooperative wireless vehicular networks: A dqn approach,” _IEEE Access_ , 2023.
  * [8] H. Chang, X. Liang, H. Li, J. Shen, X. Gu, and L. Zhang, “Deep learning-based bitstream error correction for csi feedback,” _IEEE Wireless Communications Letters_ , vol. 10, no. 12, pp. 2828–2832, 2021.
  * [9] H. Poddar, S. Ju, D. Shakya, and T. S. Rappaport, “A tutorial on nyusim: Sub-terahertz and millimeter-wave channel simulator for 5g, 6g and beyond,” _IEEE Communications Surveys & Tutorials_, 2023.
  * [10] S. Ju and T. S. Rappaport, “Millimeter-wave extended nyusim channel model for spatial consistency,” in _2018 IEEE Global Communications Conference (GLOBECOM)_. IEEE, 2018, pp. 1–6.
  * [11] S. Sun, G. R. MacCartney, and T. S. Rappaport, “A novel millimeter-wave channel simulator and applications for 5g wireless communications,” in _2017 IEEE international conference on communications (ICC)_. IEEE, 2017, pp. 1–7.
  * [12] R. U. Murshed, K. Noshin, M. A. Zakaria, M. F. Uddin, A. Amin, and M. E. Ali, “Real-time seismic intensity prediction using self-supervised contrastive gnn for earthquake early warning,” _arXiv preprint arXiv:2306.14336_ , 2023\.
  * [13] H. Kasai, “Fast optimization algorithm on complex oblique manifold for hybrid precoding in millimeter wave MIMO systems,” in _2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP)_ , 2018, pp. 1266–1270.
  * [14] A. Goldsmith, _Wireless Communications_. Cambridge University Press, 2005.
  * [15] T. Peken, S. Adiga, R. Tandon, and T. Bose, “Deep learning for SVD and hybrid beamforming,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 10, pp. 6621–6642, 2020.
  * [16] R. U. Murshed, Z. B. Ashraf, A. H. Hridhon, K. Munasinghe, A. Jamalipour, and M. F. Hossain, “A cnn-lstm-based fusion separation deep neural network for 6g ultra-massive mimo hybrid beamforming,” _IEEE Access_ , vol. 11, pp. 38 614–38 630, 2023.
  * [17] A. B. ZEKRI and R. AJGOU, “Study of mmwave channels for different scenarios,” in _2019 6th International Conference on Image and Signal Processing and their Applications (ISPA)_. IEEE, 2019, pp. 1–6.
  * [18] A. A. Budalal, I. Shayea, M. R. Islam, M. H. Azmi, H. Mohamad, S. A. Saad, and Y. I. Daradkeh, “Millimetre-wave propagation channel based on nyusim channel model with consideration of rain fade in tropical climates,” _IEEE Access_ , vol. 10, pp. 1990–2005, 2021.


# Autonomous Ratcheting by Stochastic Resetting

Pulak K<EMAIL_ADDRESS>(corresponding author), Shubhadip
Nayak1, Jianli Liu2, Yunyun Li2, and Fabio
Marchesoni2,3222fabio.marchesoni@pg.infn.it(corresponding author) 1 Department
of Chemistry, Presidency University, Kolkata 700073, India  2 Center for
Phononics and Thermal Energy Science, Shanghai Key Laboratory of Special
Artificial Microstructure Materials and Technology, School of Physics Science
and Engineering, Tongji University, Shanghai 200092, China  3 Dipartimento di
Fisica, Università di Camerino, I-62032 Camerino, Italy

###### Abstract

We propose a generalization of the stochastic resetting mechanism for a
Brownian particle diffusing in a one-dimensional periodic potential: randomly
in time, the particle gets reset at the bottom of the potential well it was
in. Numerical simulations show that in mirror asymmetric potentials,
stochastic resetting rectifies the particle’s dynamics, with maximum drift
speed for an optimal average resetting time. Accordingly, an unbiased Brownian
tracer diffusing on an asymmetric substrate can rectify its motion by adopting
an adaptive stop-and-go strategy. Our proposed ratchet mechanism can model
directed autonomous motion of molecular motors and micro-organisms

The notion of stochastic resetting (SR) is attracting growing attention (see
Ref. SR_rev  for a recent review). This term refers to the sudden interruption
of a stochastic process after random time intervals, followed by its starting
anew, possibly after a further latency time, with same initial conditions.
Diffusion under SR is a non-equilibrium stationary process, which found
applications in search contexts Maj_PRL1 , optimization of randomized computer
algorithms Zecchina , and in many biophysical problems Reuveni1 ; Reuveni2 .
Surprisingly, under SR the otherwise infinite mean first passage time of a
freely diffusing Brownian particle Redner  from an injection point to an
assigned target point becomes finite, and, most notably, can be minimized for
an optimal choice of the resetting time, $\tau$ Maj_PRL2 ; Reuveni3 . Many
analytical methods earlier developed in the theory of homogeneous stochastic
processes Redner ; Gardiner  can be generalized to study diffusion under SR,
for instance, to calculate the mean-first-exit time (MFET) of a reset particle
out of a one-dimensional (1D) domain Pal  or potential well Reuveni4 . In
general, SR speeds up (slows down) diffusive processes characterized by random
escape times with standard deviation larger (smaller) than the respective
averages Reuveni2 .

Figure 1: Autonomous ratcheting by stochastic resetting: (a) schematics; (b)
$\langle v\rangle$ vs $\tau$ for different $D_{0}$; (c) MSD vs $t$ for
$D_{0}=1.5$ and different $\tau$. The asymptotic dependence is linear in $t$
(dashed line), while the horizontal plateaus for very low $\tau$ come close to
the square half-width of the relevant probability density peak,
$\langle(x-x_{0})^{2}\rangle\simeq 2D_{0}\tau$, given in the text (see, e.g.,
the solid line for $\tau=10^{-3}$). Values of the diffusion constant, $D$,
obtained by fitting Eq. (3) are plotted in Fig. 2(c). Numerical simulations
for the ratchet potential of Eq. (2) with $L=1$ and $\tau_{0}=0$.  Figure 2:
Simulation data analysis: (a) $p(x;\tau,D_{0})$ for different values of
$D_{0}$ and $\tau$; (b) $\langle v\rangle$ vs. $\tau$ for different $D_{0}$.
The dashed curves are our predictions respectively for small $\tau$, $\langle
v\rangle=L/\langle T(\tau)\rangle$, and in the strong noise regime, $\langle
v\rangle=\delta x/\tau$ with $\delta x=(L_{L}-L_{R})/2$; (c) fitting
parameter, $D$, of the diffusion law of Eq. (3), also for different $D_{0}$.
Our analytical estimates for small and large $\tau$ (see text) are represented
respectively by dashed and solid curves. Numerical simulations for the ratchet
potential of Eq. (2) with $L=1$ and $\tau_{0}=0$.

In this Letter we propose an SR mechanism with degenerate resetting point. Let
us consider an overdamped Brownian particle of coordinate $x$, diffusing in a
1D periodic potential, $V(x)$, of period $L$. We assume for simplicity that
the potential unit cells have one minimum each at $x_{n}=x_{0}+nL$, with
$n=0,\pm 1,\dots$. Upon resetting, the particle stops diffusing and falls
instantaneously at the bottom of the potential well it was in; it will resume
diffusing after a latency time $\tau_{0}\geq 0$, see Fig. 1(a). By this
mechanism of autonomous SR, we intend to model the dynamics of small motile
tracers (like bacteria or micro-robots Wang ) capable of switching their
internal engine on and off. In the case of undirected motility, the tracer
would perform an unbiased Brownian motion. Let us further assume that the
barriers separating two adjacent potential minima are asymmetric under mirror
reflection, i.e., $V(x-x_{0})\neq V(-x+x_{0})$ (ratchet potential RMP ).
Extensive numerical simulations show that (i) SR rectifies diffusion in a
ratchet potential. The particle’s net drift speed, $\langle v\rangle$, reaches
a maximum for an optimal value of the resetting time, $\tau$, which strongly
depends on the potential profile, see Fig. 1(b); (ii) SR suppresses spatial
diffusion. For large observation times, the particle’s mean-square
displacement (MSD) turns proportional to time (normal diffusion); the relevant
diffusion constant increases sharply with the resetting time in correspondence
with the maximum of the drift speed, see Fig. 1(c).

While this variant of the SR mechanism may be reminiscent of a flashing
ratchet with pulsated temperature pla , here the diffusing tracer exploits the
substrate spatial asymmetry to autonomously rectify its random motion in the
absence of external time-dependent fields of force or gradients RMP ; JCP1 ;
JCP2 , simply by time-operating its internal engine to adjust to the substrate
itself.

Model. The simulated particle dynamics was formulated in terms of the Langevin
equation (LE),

$\dot{x}=-V^{\prime}(x)+\xi(t),$ (1)

where $\xi(t)$ denotes a stationary zero-mean valued Gaussian noise with
autocorrelation $\langle\xi(t)\xi(0)\rangle=2D_{0}\delta(t)$ (white noise) and
$V(x)$ is the standard ratchet potential RMP ,

$V(x)=\sin(2\pi x/L)+(1/4)\sin(4\pi x/L),$ (2)

with asymmetric barriers of height $\Delta V=(3/2)(1+2/\sqrt{3})^{1/2}\simeq
2.20$. The potential unit cell $[0,L]$ has a maximum (barrier) at
$x_{b}=(L/2\pi)\arccos[(\sqrt{3}-1)/2]\simeq 0.19L$ and a minimum (well
bottom) at $x_{0}=L-x_{b}\simeq 0.81L$, with curvatures
$\omega_{0}^{2}=V^{\prime\prime}(x_{0})=-V^{\prime\prime}(x_{b})=(2\pi/L)^{2}(3\sqrt{3}/2)^{1/2}\simeq
63.6/L^{2}$, see Fig. 1(a). The asymmetric potential wells have right/left
slopes of different lengths, $L_{R,L}$, with $L_{L}=x_{0}-x_{b}=L-L_{R}\simeq
0.62L$. In addition to the thermal fluctuations and the ratchet potential, the
particle is subjected to resetting to the attracting local substrate minimum
after a random time that is taken from exponential distribution with mean
$\tau=1/r$. Where, $r$ is the resetting rate. Partly motivated by technical
issues in experiments, earlier worksMajumdar-A ; Majumdar-B ; Majumdar-C  had
considered the case when the particle was reset to a fully randomly chosen
position. Along with the restart protocol, the Eq.(1) was numerically
integrated by means of a standard Milstein scheme Kloeden , to compute the
drift speed $\langle v\rangle=\lim_{t\to\infty}[\langle x(t)\rangle-x(0)]/t$,
and the asymptotic MSD,

$\langle\Delta x^{2}(t)\rangle=\langle x^{2}(t)\rangle-\langle
v\rangle^{2}t^{2}\equiv 2Dt,$ (3)

of a particle under stationary conditions (with or without SR), (Figs. 1 and
2), and the MFET’s, $\langle T_{R,L}(\tau)\rangle$, for a reset particle
injected at the bottom of the well, $x_{0}$, to first exit it through the left
(right) barrier, $x_{b}$ ($x_{b}+L$) (Fig. 3).

Rectification under SR. The key features of the resulting SR ratchet are
illustrated in the bottom panels of Fig. 1: the particle motion gets rectified
with net speed $\langle v(\tau)\rangle$ [Fig. 1(b)] and asymptotic diffusion
constant, $D(\tau)$, a monotonically increasing function of the SR time [Fig.
1(c)]. Rectification is maximum in an optimal $\tau$ range, as $D$ approaches
an stationary value (the same as in the absence of SR).

To explain ratcheting under SR we anticipate two properties of the statistics
of particle escape out of a potential well, summarized in Fig. 3. In the
absence of resetting, i.e., for asymptotically large $\tau$, the probability
current density of the process is zero, which rules out rectification RMP ,
$\langle v(\infty)\rangle=0$. Things change upon decreasing the SR time, as
proven by the $\tau$-dependence of the splitting probabilities,
$\pi_{R,L}(\tau)$, for the particle to exit a potential well through the
right/left barrier. Upon lowering $\tau$, the asymmetry ratio
$\pi_{R}/\pi_{L}$ in Fig. 3(b) grows monotonically, the effect being more
apparent at low noise, $D_{0}\ll\Delta V$, so that $\langle v(\tau)\rangle>0$.
We qualitatively explain this property with the increased asymmetry of the
probability density Redner  of the reset particle around the potential minima
[Fig. 2(a)]. On the other hand, the data of Fig. 3(b) clearly show that in the
limit $\tau\to 0$, $\langle T_{R}(\tau)\rangle$ diverges exponentially, so
that we anticipate $\langle v(\tau\to 0)\rangle=0+$. The combination of these
two opposite effects determines the typical resonant profile of the $\langle
v(\tau)\rangle$ curves.

Slow SR. More in detail, the data of Fig. 2(b) suggest that $\langle
v(\tau)\rangle$ decays asymptotically like $\tau^{-1}$. This behavior can be
easily explained in the strong noise regime with $D_{0}\gg\Delta V$ and
$\langle T_{R,L}(\tau)\rangle\ll\tau$. Under this condition, the particle
executes many barrier crossings before being reset at the bottom of a $V(x)$
well. At resetting, it is caught in average to the left of the well bottom;
hence, at each resetting the particle jumps to the right an average distance,
$\delta x=\bar{x}-x_{0}>0$, $\bar{x}$ being the center of mass of the
(periodic) particle’s stationary probability density function,
$p(x;\tau,D_{0})$, in the potential well with bottom at $x=x_{0}$.
Accordingly, the particle gets rectified with positive net drift speed
$\langle v(\tau)\rangle=\delta x/\tau$. In the strong noise regime,
$p(x;\tau,D_{0})$, approaches a uniform distribution; hence, $\delta
x=(L_{L}-L_{R})/2$, in good agreement with the numerical data of Fig. 2(a).

Figure 3: Exit time statistics: (a) MFET’s out of a potential well through the
right/left barrier, $\langle T_{R,L}\rangle$, or through either barrier,
$\langle T\rangle$; (b) splitting probabilities, $\pi_{R,L}$, for right/left
exits (solid/empty symbols), vs $\tau$ for different $D_{0}$ (see legends).
Note that $\langle T\rangle=\pi_{R}\langle T_{R}\rangle+\pi_{L}\langle
T_{L}\rangle$ Redner . The dashed curve in (a) represents the analytical
estimate of $\langle T(\tau)\rangle\simeq\langle T_{R}(\tau)\rangle$ for
$\tau\to 0$, Eq. (4), with $\langle T(\infty)\rangle=T_{K}$ (see text). The
dashed lines in (b) are the large $D_{0}$ limits of
$\pi_{R,L}(\infty)=L_{L,R}$ (with the corresponding ratio displayed in the
inset). The horizontal lines in the inset of (a) are the expected $\langle
T_{R}\rangle/\langle T_{L}\rangle$ ratios for $D_{0}\to 0$ (upper) and
$\infty$ (lower, see text). (c) unconstrained right/left MFPT’s, $\langle
T^{\rm(u)}_{R,L}(\tau)\rangle$, $x_{0}\to x_{0}\pm L$, (empty/solid symbols)
for different values of $D_{0}$; inset: small-$\tau$ dependence of the MFPT’s
for multiple cell transitions, $x_{0}\to x_{0}+nL$, for $D_{0}=1$. Simulation
data are compared with the relevant estimates of Eqs. (5) (large $\tau$) and
(6) (small $\tau$). Numerical simulations for the ratchet potential of Eq. (2)
with $L=1$ and $\tau_{0}=0$.

Upon decreasing the noise strength, $\delta x$ diminishes for two reasons, as
illustrated in Fig. 2(a). Firstly, in the absence of SR, i.e., for
$\tau\to\infty$, the probability density, $p(x;\tau,D_{0})$, approaches its
thermal equilibrium form, $p(x;\infty,D_{0})={\cal N}\exp(-V(x)/D_{0})$, with
${\cal N}$ an appropriate normalization constant. For $D_{0}\ll\Delta V$,
$p(x;\infty,D_{0})$ shrinks around $x_{0}$, that is, $\delta x$ diminishes.
Secondly, by lowering $D_{0}$ in the presence of SR, i.e., for finite $\tau$,
$\langle T(\tau)\rangle$ grows comparable with $\tau$. Accordingly, barrier
escape and resetting events grow correlated, which invalidates the above
estimate of the particle’s drift speed. However, numerical data confirm that
$\langle v(\tau)\rangle$, though strongly suppressed, keeps decaying
asymptotically like $1/\tau$, even at low noise.

Fast SR. The plots of $p(x;\tau,D_{0})$ for the lowest $\tau$ values in Fig.
2(a) consist of a central peak tapering off with asymmetric slow-decaying
tails on both sides. In the limit $\tau\to 0$, (i) the peak gets sharper and
more symmetric, while remaining centered at the resetting point, $x_{0}$. Its
square half-width can be easily calculated for $D_{0}\ll\Delta V$, by
approximating $V(x)\simeq\omega_{0}^{2}(x-x_{0})^{2}/2$ and averaging over the
SR time, that is, $\langle(x-x_{0})^{2}\rangle\simeq
2D_{0}\tau/(1+2\omega_{0}^{2}\tau)$; (ii) the tails get thinner but more
asymmetric. This behavior is consistent with the $\tau$-dependence of the
escape asymmetry ratio, $\pi_{R}/\pi_{L}$, displayed in Fig. 3(b) Redner .

The sharp decay of $\langle v(\tau)\rangle$ for $\tau\to 0$ proves that fast
SR eventually suppresses the interwell particle diffusion. In such limit, as
shown in Fig. 3, the particle tends to jump to the right, with
$\pi_{R}(\tau)\gg\pi_{L}(\tau)$ and, therefore, $\langle
T(\tau)\rangle\simeq\langle T_{R}(\tau)\rangle$ with $\langle
T(\tau)\rangle\gg\tau$. Under these conditions, the resulting drift speed can
be easily estimated under renewal theory approximation Cox , that is $\langle
v(\tau\rangle=L/\langle T_{R}(\tau)\rangle$.

To calculate $\langle T_{R}(\tau)\rangle$ we had recourse to the analytical
results of Ref. Reuveni4  for Brownian diffusion under SR in the presence of a
constant bias. We made contact with Eq. (6) there, by replacing the constant
bias with the effective (right-to-left) restoring force of our ratchet
potential, $\Delta V/L_{R}$. In the limit $\tau\to 0$, the MFET for the
transition to the adjacent well on the right, $x_{0}\to x_{0}+L$, is twice the
MFET for the transition $x_{0}\to x_{b}+L$, that is

$\langle T(\tau)\rangle\simeq\langle T_{R}(\tau)\rangle\simeq
2\tau\exp\left({\Delta V}/{2D_{0}}+{L_{R}}/{\sqrt{D_{0}\tau}}\right).$ (4)

Of course, this approximation holds good only for $\pi_{R}(\tau)\simeq 1$
($\pi_{L}(\tau)\simeq 0$), and its agreement with the numerical data improves
upon decreasing the noise strength, i.e., for $D_{0}\lesssim\Delta V$, as
shown in Fig. 3(a). On making use of this estimate for $\langle
T_{R}(\tau)\rangle$, we closely reproduced also the raising branches of the
$\langle v(\tau)\rangle$ curves in Fig. 2(b).

Diffusion under SR. Regarding the intrawell diffusion, we remind that in the
absence of SR, the MFET from $x_{0}$ to $x_{0}\pm L$ amounts to the standard
Kramers’ time Gardiner  $T_{K}=(2\pi/\omega_{0}^{2})\exp(-\Delta V/D_{0})$. By
the same token, one concludes that for $\tau\to\infty$, $\langle
T_{R}\rangle\simeq\langle T_{L}\rangle$, with both MFET’s tending to $T_{K}$
for $D_{0}/\Delta V\to 0$, and their ratio, $\langle T_{R}\rangle/\langle
T_{L}\rangle$ approaching $(1+L/L_{L})/(1+L/L_{R})\simeq 0.72$ in the opposite
limit, $D_{0}/\Delta V\to\infty$. On the other hand, for large $\tau$ the
splitting probabilities can be easily computed assuming no SR (see Sec. 5.2.7
of Ref. Gardiner ); their limits for $D_{0}/\Delta V\to 0$ (and $\to\infty$)
are respectively $\pi_{R,L}(\infty)=1/2$ (and $L_{L,R}/L$), as shown in Fig.
3(b).

These remarks are useful to interpret the MSD data sets of Fig. 1(c).
Numerical simulation indicates that diffusion at large times, $t\gg\langle
T(\tau)\rangle$, is normal, as anticipated by the fitting law of Eq. (3). At
small $\tau$, a transient plateau for $t\lesssim\langle T(\tau)\rangle$,
$\langle\Delta x^{2}\rangle\simeq 2D_{0}\tau$, marks the particle relaxation
inside a single potential well [with $\langle\Delta x^{2}\rangle$ of the order
of the square half-width of the $p(x;\tau,D_{0})$ peak estimated above]. The
$\tau$-dependence of the asymptotic diffusion constants, $D$, is reported in
Fig. 2(c). For large $\tau$, the $D(\tau)$ curves approach the horizonal
asymptotes Gardiner , $D=L^{2}/2T_{K}$, as to be expected in the absence of
SR. Vice versa for very short SR times, the diffusion constant is well
approximated by $D(\tau)=L^{2}/2\langle T_{R}(\tau)\rangle$, as predicted by
the renewal theory for a process with average escape time constant $\langle
T_{R}(\tau)\rangle$ Cox . In both $\tau$ limits, our phenomenological
arguments are supported by numerical simulation.

Figure 4: Rectification speed, $\langle v(\tau)\rangle$, of the SR ratchet of
Fig. 1 with $\tau_{0}=0.1$ (filled symbols) and $\tau_{0}=0$ (empty symbols),
for different $D_{0}$. The zero-latency data have been rescaled according to
Eq. (7). Inset: $\langle v(\tau)\rangle$ of a flashing ratchet with dichotomic
noise strength, $D_{0}(t)$, switching between 0 (fixed waiting time
$\tau_{0}=0.1$) and $D_{0}$ (random waiting times exponentially distributed
with average $\tau$), compared with the flashing ratchet in the main panel
(circles).

Comparison with standard diffusion under SR. Numerical data in Fig. 3(a) show
that by decreasing the SR time, $\langle T_{R}(\tau)\rangle$ keeps being
larger than $\langle T_{L}(\tau)\rangle$. Moreover, $\langle T(\tau)\rangle$
grows monotonically with $\tau$, i.e., the MFET out of the potential well is
not optimized by resetting. Of course, the predicted SR optimization of the
average passage times Maj_PRL2  is still detectable, but only for the
unconstrained transitions $x_{0}\to x_{b}$ with $x\geq x_{b}$, and $x_{0}\to
x_{b}+L$ with $x\leq x_{b}+L$. In panel (c) of Fig. 3 we investigated the same
transitions as in panel (a), except for the reflecting barriers, which were
shifted at $\mp\infty$. The corresponding right/left unconstrained MFPT
curves, $\langle T^{\rm(u)}_{R,L}(\tau)\rangle$, overlap throughout the entire
$\tau$ range. Furthermore, all MFPT curves diverge for $\tau\to\infty$, as to
be expected due to the lack of a reflecting barrier Maj_PRL2 . In the absence
of SR (i.e., for $\tau\to\infty$), the particle still diffuses over the
substrate like a free particle, but with the reduced effective diffusion
constant, $D=L^{2}/2T_{K}$, of Eq. (3) [see fits in Fig. 2(c)]. This suggests
rewriting Eq. (7) of Ref. Maj_PRL2  as

$\langle T^{\rm(u)}_{R,L}(\tau)\rangle/\tau=\exp{(L/\sqrt{D\tau})}-1,$ (5)

a formula that well reproduces the large-$\tau$ branches of the curves in Fig
3(c) with no additional fitting parameters. In the inset of the same figure,
we analyze the small-$\tau$ dependence of the MFPT’s for the right transitions
$x_{0}\to x_{0}+nL$ with $n=1,2,\dots$ and reflecting barriers at $-\infty$.
By applying the heuristic argument invoked to derive Eq. (4), we obtain the
working approximate estimate,

$\langle T^{\rm(u)}_{R,L}(\tau)\rangle/\tau\simeq\exp({\Delta
V}/{2D_{0}}+{nL}/{\sqrt{D_{0}\tau}}),$ (6)

which holds for $n$-cell transitions to the right/left at vanishingly small
$\tau$. Note that here, contrary to Eq. (6), we make use of the free diffusion
constant, $D_{0}$.

Concluding remarks. The SR ratcheting mechanism introduced above can be
readily generalized to more realistic cases when resetting takes a finite time
SR_rev , $\tau_{0}$, called here latency time. The relevant net ratchet speed
turns out to be a function of both $\tau$ and $\tau_{0}$, $\langle
v(\tau,\tau_{0})\rangle$, which can be related with the zero-latency speed,
$\langle v(\tau,0)\rangle$, through a simple time rescaling, namely

$\langle v(\tau,\tau_{0})\rangle=\langle v(\tau,0)\rangle/(1+\tau_{0}/\tau)),$
(7)

as illustrated in Fig. 4(a).

This instance of SR ratchet lends itself to a simple laboratory demonstration.
We start again from the LE (1) with the potential of Eq. (2) but, instead of
implementing the SR protocol with latency time $\tau_{0}$, we now assume a
dichotomic noise strength, $D_{0}(t)$, with $D_{0}=0$ for fixed time
intervals, $\tau_{0}$, and $D_{0}(t)=D_{0}$ for random time intervals
exponentially distributed with average $\tau$. The resulting LE describes a
rectifier, which could be classified as a special case of flashing ratchet pla
. In one regard the two rectification mechanisms are apparently similar: in
both cases the particle rests at the bottom of a potential well for the time
interval, $\tau_{0}$, before resuming Brownian diffusion, because either reset
that way (SR ratchet) or given enough time to relax there (flashing ratchet
with $\omega_{0}^{2}\tau_{0}\gg 1$). As shown in the inset of Fig. 4, for the
same choice of the tunable parameters, $D_{0},\tau$ and $\tau_{0}$, the
rectification power of the two ratchets is almost identical. Therefore, one
can utilize a ratchet with dichotomic noise strength to experimentally
demonstrate the rectification properties of the proposed SR ratchet. However,
an important difference between these two ratchets is also noteworthy. The
flashing ratchet is fueled by an external source capable of “heating and
cooling” the particle or its substrate Libchaber ; Sano . SR ratcheting with
finite latency time, instead, can be controlled by the particle itself, by
autonomously regulating its own internal motility mechanism for maximum
efficiency.

In summary, we have proposed a new protocol of stochastic resetting, whereby a
particle diffusing on a one-dimensional substrate, gets reset not at a fixed
point, but rather at one of the degenerate minima of the substrate. We
investigated, both numerically and analytically, the diffusion properties of
the reset particle and showed that for spatially asymmetric substrates the
particle gets rectified with direction determined by the substrate profile,
and optimal speed depending on the resetting time. We argue that, thanks to
such a mechanism, a motile system (biological and synthetic, alike) can
exploit the substrate asymmetry to autonomously direct its motion, for
instance, by randomly switching on and off its propulsion engine at an
appropriate rate.

## Acknowledgements

Y.L. is supported by the NSF China under grants No. 11875201 and No.
11935010\. P.K.G. is supported by SERB Core Research Grant No.
CRG/2021/007394.

## Data Availability

The data that support the findings of this study are available within the
article.

## Conflict of interest

The authors have no conflicts to disclose.

## References

  * (1) M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, J. Phys. A: Math. Theor. 53, 193001 (2020).
  * (2) L. Kusmierz, S. N. Majumdar, S. Sabhapandit, and G. Schehr, First order transition for the optimal search time of Lévy flights with resetting, Phys. Rev. Lett. 113, 220602 (2014).
  * (3) A. Montanari and R. Zecchina, Optimizing searches via rare events, Phys. Rev. Lett. 88, 178701 (2002).
  * (4) S. Reuveni, M. Urbakh, and J. Klafter, Role of substrate unbinding in Michaelis–Menten enzymatic reactions Proc. Natl. Acad. Sci. USA 111, 4391 (2014).
  * (5) S. Reuveni, Optimal stochastic restart renders fluctuations in first passage times universal, Phys. Rev. Lett. 116, 170601 (2016).
  * (6) S. Redner, A Guide to First-Passage Processes (Cambridge University Press, UK, 2001).
  * (7) C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985).
  * (8) M. R. Evans and S. N. Majumdar, Diffusion with stochastic resetting, Phys. Rev. Lett. 106, 160601 (2011).
  * (9) A. Pal and S. Reuveni, First passage under restart, Phys. Rev. Lett. 118, 030603 (2017).
  * (10) A. Pal and V. V. Prasad, First passage under stochastic resetting in an interval, Phys. Rev. E 99, 032123 (2019).
  * (11) S. Ray, D. Mondal, and S. Reuveni, Péclet number governs transition to acceleratory restart in drift-diffusion, J. Phys A: Math. and Theor. 52 255002 (2019).
  * (12) J. Wang, Nanomachines: Fundamentals and Applications (Wiley-VCH, Weinheim, 2013).
  * (13) P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009).
  * (14) P. Reimann, Brownian motors: Noisy transport far from equilibrium, Phys. Rep. 361, 57 (2002).
  * (15) P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1992).
  * (16) D. R. Cox, Renewal Theory (Methuen, London, 1970)).
  * (17) L. P. Faucheux, L. S. Bourdieu, P. D. Kaplan, and A. J. Libchaber, Optical thermal ratchets, Phys. Rev. Lett. 74, 1504 (1995).
  * (18) F. D. Ribetto, S. E. Deghi, H. L. Calvo, and R. A. Bustos-Marún, A dynamical model for Brownian molecular motors driven by inelastic electron tunneling, J. Chem. Phys. 157, 164102 (2022).
  * (19) J. Valdiviezo, P. Zhang, D. N. Beratan, Electron ratcheting in self-assembled soft matter, J. Chem. Phys. 155, 055102 (2021).
  * (20) B. Besga, A. Bovon, A. Petrosyan,S. N. Majumdar, and S. Ciliberto, Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results. Phys. Rev. Res., 2, 032029 (2020).
  * (21) B. Besga, F. Faisant, A. Petrosyan, S. Ciliberto,and S. N. Majumdar, Dynamical phase transition in the first-passage probability of a Brownian motion Phys. Rev. E, 104, L012102 (2021).
  * (22) G. Tucci, A. Gambassi, S. N. Majumdar, and G. Schehr, First-passage time of run-and-tumble particles with noninstantaneous resetting, Phys. Rev. E, 106, 044127 (2022).
  * (23) H. R. Jiang, N. Yoshinaga, and M. Sano, Active motion of a Janus particle by self-thermophoresis in a defocused laser beam, Phys. Rev. Lett. 105, 268302 (2010).