# 3D photonics for ultra-low energy, high bandwidth-density chip data links

Stuart Daudlin1, Anthony Rizzo1,2, Sunwoo Lee3, Devesh Khilwani3, Christine
Ou3, Songli Wang1, Asher Novick1, Vignesh Gopal1, Michael Cullen1, Robert
Parsons1, Alyosha Molnar3, and Keren Bergman1,* 1Department of Electrical
Engineering, Columbia University, New York, NY 10027
2Air Force Research Laboratory Information Directorate, Rome, NY 13441
3 Department of Electrical and Computer Engineering, Cornell University,
Ithaca, NY 14853

††preprint: APS/123-QED

Artificial intelligence (AI) hardware is positioned to unlock revolutionary
computational abilities across diverse fields ranging from fundamental science
carrasquilla2017machine  to medicine he2019practical  and environmental
science kaack2022aligning  by leveraging advanced semiconductor chips
interconnected in vast distributed networks. However, AI chip development has
far outpaced that of the networks that connect them, as chip computation
speeds have accelerated a thousandfold faster than communication bandwidth
over the last two decades dally2021evolution ; mirabbasi2023through . This gap
is the largest barrier for scaling AI performance wang2023zero++ ;
pati2023computation  and results from the disproportionately high energy
expended to transmit data lee2022beyond , which is two orders of magnitude
more intensive than computing miller2017attojoule . Here, we show a leveling
of this long-standing discrepancy and achieve the lowest energy optical data
link to date through dense 3D integration of photonic and electronic chips. At
120 fJ of consumed energy per communicated bit and 5.3 Tb/s bandwidth per
square millimeter of chip area, our platform simultaneously achieves a twofold
improvement in both energy consumption and bandwidth density relative to prior
demonstrations rakowski2018hybrid ; samanta2023direct . These improvements are
realized through employing massively parallel 80 channel microresonator-based
transmitter and receiver arrays operating at 10 Gb/s per channel, occupying a
combined chip footprint of only 0.32 mm2. Furthermore, commercial
complementary metal-oxide-semiconductor (CMOS) foundries fabricate both the
electronic and photonic chips on 300 mm wafers, providing a clear avenue to
volume scaling. Through these demonstrated ultra-energy efficient, high
bandwidth data communication links, this work eliminates the bandwidth
bottleneck between spatially distanced compute nodes and will enable a
fundamentally new scale of future AI computing hardware without constraints on
data locality.

Light, as a medium for communication, has the unique ability to transmit
volumes of data with minimal energy loss. This capability not only sparked the
revolution of internet-based communication over fibre optic networks, but also
holds the potential to significantly expand computing power beyond current
capabilities. Specifically, artificial intelligence (AI) is poised to
dramatically transform the computational landscape if provided with more
efficient data communication between nodes in computer networks wang2023zero++
; wu2023peta . A critical bottleneck to the full implementation of light-based
communication is the conversion of electrical data from inside a computer chip
to optical data. At present, data is stored densely in these semiconductor
chips in compute nodes, but is sent out of the chip through centimeter-long
electrical wires before finally interfacing with optical transmitters in the
form of pluggable optical transceivers. The design of these electrical
channels results in slow data rates that are not scalable without accounting
for a substantial amount of energy consumption lee2022beyond . To overcome
this bottleneck, electrical channels must be condensed and converted into
optical signals within a compact area.

Figure 1: 3D integrated photonic-electronic transceiver. a, Illustration of
the 3D integrated photonic-electronic system combining arrays of electronic
transmitter and receiver cells with arrays of photonic devices. b, Cross-
sectional diagram of the electronic and photonic chips and their associated
material stacks. Both chips consist of a crystalline silicon (c-Si) substrate,
doped silicon devices, and metal interconnection layers. c, A scanning
electron microscope (SEM) image of the cross section of the flip-chip bonded
electronic and photonic chips. d, Image of the wire-bonded transceiver die-
bonded to a printed circuit board and optically coupled to a fibre array with
a U.S. dime for scale. e, Microscope images of the standalone and 3D
integrated chips with the larger photonic chip beneath the flipped electronic
chip. The active photonic circuits occupy the area outlined in blue, while the
rest of the photonic chip area is used to fan out the optical/electrical lanes
for fibre coupling and wire bonding.

Previously, intensive efforts have produced chip-scale transmitters and
receivers (transceivers) towards this goal but have been marked by a lack of
efficiency or scale. These works build on the field of integrated photonics, a
technology that aggregates a multitude of optical components onto a single
integrated chip. In particular, silicon is highly appealing as a material
platform for integrated photonics since it can leverage the tremendous
investment in the complementary metal-oxide-semiconductor (CMOS)
infrastructure used to fabricate microelectronics chips hochberg2010towards .
The silicon photonics technology platform includes devices such as micro-
resonator-based modulators xu2005micrometre ; timurdogan2014ultralow ,
filters, and germanium photodiodes michel2010high  that are compact, efficient
in their electrical-to-optical and optical-to-electrical conversions, and
scalable to many wavelength channels rizzo2023massively ; rizzo2022petabit .
To date, the largest of these systems is comprised of 64 channels of photonics
and electronics on a single chip and achieves 240 femtojoules per communicated
bit (fJ/bit) by the transmitter wade2018bandwidth ; sun2020teraphy ;
wade2021error . However, this system has receiver energy consumption above
1000 fJ/bit and has a limited density from the lateral arrangement of
photonics and electronics on the same two-dimensional chip. While this
monolithic integration of CMOS transistors alongside photonic devices on the
same chip may appear highly appealing sun2015single ; atabaki2018integrating ,
this configuration ‘freezes’ transistors at a given node size and thus cannot
benefit from the further energy efficiency, size, and speed gains of moving to
more advanced CMOS nodes. Alternatively, three-dimensional (3D) integration
combines a more efficient, leading edge CMOS node electronic chip and a
separate photonic chip to improve on these limitations. Ongoing 3D efforts
have demonstrated sub-200 fJ/bit powers from transmitters zheng2011ultralow ;
rakowski2018hybrid ; samanta2023direct ; ban2023highly  and receivers
rakowski2018hybrid ; samanta2023direct ; saeedi201625 , but the chip-to-chip
bond spacings are either significantly larger than the devices themselves
zheng2011ultralow ; rakowski2018hybrid ; ban2023highly ; saeedi201625  or rely
on emerging hybrid bonding technology samanta2023direct . Furthermore, 3D
integrated transceivers have not scaled to more than eight channels
zheng2011ultralow  and have yet to achieve both transmitter and receiver
powers below 100 fJ/bit.

Here, we present the most energy-efficient conversion between electrical data
and optical data and the highest density of data transmission from an
integrated chip-scale system. Our novel approach to photonic transceivers
simultaneously addresses energy efficiency and bandwidth density scaling for
future computing systems. This transceiver demonstration is scaled up to 80
channels while having a scaled-down energy consumption through low-capacitance
connections between low-capacitance photonics and co-designed CMOS electronic
circuits. The data signaling rate per channel is relatively low at 10 gigabits
per second (Gb/s) per channel, which permits the receiver electronics to
operate in an ideal regime for minimized energy consumption (Methods). The
large array of channels compensates for the low per channel data rates and
delivers a high aggregate data rate of 800 Gb/s in a compact area of only 0.32
mm2. From the perspective of interfacing to a processor, a large array of low
data rate channels relaxes signal processing and time multiplexing of the low
data rate streams native to the processor miller2017attojoule ;
georgas2011addressing ; zhu2020photonic . Furthermore, wavelength-division
multiplexing (WDM) sources for these numerous data streams are becoming
readily available with the advent of chip-scale micro-combs gaeta2019photonic
; rizzo2023massively . Our demonstration unlocks the tremendous potential of
light as a high bandwidth and energy-efficient inter-chip communication
medium, offering an immediate solution to the pressing challenge of AI
scaling.

## Results

Figure 2: Transmitter characterization and performance. a, Illustration of the
transmitter experimental test setup showing a single wavelength laser channel
modulated by the transmitter and measured on an oscilloscope. b, Transmitter
cell circuit schematic. c, Optical spectrum of the four channel transmitter
bus. d, Resonance shift of a representative micro-disk as a function of
reverse-bias voltage applied to the vertical p-n junction. e, Transmitted
signal extinction ratio (ER, power of ‘1’ bit divided by power of ‘0’ bit),
insertion loss (IL, power of ‘1’ bit divided by power before the modulator),
and normalized optical modulation amplitude (OMA, normalized power of ‘1’ bit
minus power of ‘0’ bit) with a 1.5 V driver voltage. f, ER and IL at maximum
OMA for a range of driver voltages. g, Measured energy consumption of the
transmitter array for a range of driver voltages and a $\frac{1}{4}$CV2 fit,
where C is the capacitance charged by the driver voltage V. h, Bit error ratio
(BER) measurement of the modulated signal input to a commercial receiver at 1
V, 1.25 V, and 1.5 V driver voltages; received power is the average signal
power at the commercial receiver. i, Eye diagrams for all 80 modulators on the
photonic chip at 10 Gb/s/modulator and 1 dBm input laser power.

We implement this high-density transceiver using compact photonic devices and
dense, 28-nm node co-designed electronic circuits; however, the total density
ultimately depends on the 3D bond spacings. To address this, we employ a high-
density bonding process using copper pillar bumps. An electroplating process
is used to form bumps on the photonic chip with copper pedestals capped with a
layer of tin. The copper-tin bumps are then bonded to a nickel-plated
electronic chip under a thermal and compressive force bringing the chips
together. Figure 1b illustrates the layers on the electronic chip, photonic
chip, and the bonding metals. We push the limits of this bonding technology by
using a 15 $\mu$m spacing and 10 $\mu$m bump diameters (25 $\mu$m pitch) in an
array of 2,304 bonds. This process balances two potential failure modes for
such close spacing: excessive tin causing flow and electrical shorting to
adjacent bonds during bonding, and insufficient tin leading to brittle bonds
li2021scaling . We test our bonding process using cross-sectional scanning
electron micrographs of the bonds, shown in Figure 1c, and by measuring the
force needed to separate the bonded chips. The cross-section analysis reveals
that tin does not flow to adjacent bonds, while the shear test demonstrates a
robust 2.1 kg (114.9 MPa) force required to separate the bonded dies. Modeling
and measurements show each pair of bonds (for a signal and ground) has a 10 fF
capacitance (Methods). Figure 1d shows the assembled transceiver wire-bonded
to a printed circuit board and optically coupled to a fibre array (Methods),
while Figure 1e shows microscope images of the face-up electronic and photonic
chips, and the face-down electronic chip flip-chip bonded on top of the larger
photonic chip. This bonding technique provides an ideal platform to achieve
the required density for chip-to-chip data communication links.

The 3D integrated chip contains an array of 80 transmitter cells and 80
receiver cells; these cells are organized into 20 waveguide buses with four
wavelength channels per bus. Each transmitter cell has a local memory in the
electronic chip that stores a pseudo-random bit sequence. A periodic clock
signal triggers the electronics, and the transmitter cell electronics send out
the programmed bit sequences as voltage pulses incident on the photonic
modulator electrodes. These voltage pulses blue-shift the micro-disk resonance
from a blocking to a non-blocking state and thus modulate an on-resonance
laser line. Figure 2a illustrates the transmitter experiment and Figure 2b
shows a schematic of the transmitter cell, while Figure 2c shows the spectrum
of the modulator bus with four micro-disk resonances. After the transmitter
characterization, we test the receivers, which function similarly: in each
receiver cell, wavelength channels carry signals on the photonic chip and
micro-rings selectively drop wavelengths onto each respective photodiode. The
electronic chip then amplifies the photocurrent generated by the photodiode
and writes the data into the local memory of each receiver cell, as
illustrated in Figure 3a. Figure 3b shows a schematic of the receiver cell.
For performance characterization, an on-chip circuit compares the receiver
memory to expected data and keeps an error count that is periodically read out
of the chip. This architecture of transmitters and receivers fills the array
of channels in the dense area provided by the bonding process.

Figure 3: Receiver characterization and performance. a, Illustration of the
receiver test setup showing a laser line modulated by a commercial transmitter
and received by the 3D integrated photonic-electronic receiver. b, Receiver
cell circuit schematic. c, Responsivity (light to electrical current
conversion efficiency) measurement of the photodiode. d, Optical spectrum of
receiver bus. e, Bit error ratio (BER) test of a receiver cell using a
commercial transmitter signal; received power is the average signal power at
the photodiode.

The transmitter cell within the 80-cell array consumes 50 fJ/bit when driving
the micro-disks with a 1 V swing. This power is dynamic, equal to
$\frac{1}{4}\textrm{CV}^{2}$, where C is the capacitance being charged or
discharged during a bit transition, and V is the charged-to-discharged voltage
miller2012energy . The vertical p-n junction micro-disk enables a low voltage
drive by featuring a higher overlap of the p-n depletion region and the
optical whispering gallery mode of the disk compared to lateral junctions
timurdogan2014ultralow , and results in an electrical-to-optical response of
75 pm resonance shift per applied volt (Fig. 2d). We further characterize this
response with the dynamic insertion loss (IL, the power of a ‘1’ bit divided
by the power before the modulator) and extinction ratio (ER, the power of a
‘1’ bit divided by the power of a ‘0’ bit). Figure 2e shows these metrics,
captured from the modulated signal output of a transmitter cell driven at 1.5
V. In this measurement, the laser wavelength moves into the shifting resonance
and the optical modulation amplitude (OMA, the power of ‘1’ bit minus the
power of ‘0’ bit) of the output signal increases, reaching a maximum at 2.5 dB
IL and 4 dB ER. Figure 2f shows ER and IL at maximum OMA for driver voltages
between 1 and 1.5V. These high ERs and low ILs per volt enable a reduced V in
$\frac{1}{4}\textrm{CV}^{2}$. Capacitance sources include the micro-disk p-n
junction (128 fF), bond pads (10 fF), and capacitances within the driver
circuit (61 fF), combining for a total expected capacitance of 199 fF
(Methods). These capacitances exhibit low values through micro-disk
compactness, miniaturized bonds, and careful design in the 28 nm electronic
chip technology. Figure 2g shows the transmitter energy consumption as all 80
modulators are transmitting data with drive voltages ranging between 1 V to
1.5 V. The $\frac{1}{4}\textrm{CV}^{2}$ model fits 198 fF total capacitance
per cell using this data, aligning closely with the expected 199 fF from the
independently measured and modeled devices. Next, we record eye diagrams for
each of the 80 transmitters on the chip with a drive at 10 Gb/s/transmitter
and 1 dBm laser power before the modulator (Fig. 2i). As every transmitter
modulates, we measure current and voltage on the transmitter power supply for
the previous energy consumption (Fig. 2g). With no optical amplification, the
oscilloscope receiver is the limiting factor of the eye qualities with an 8 uW
input-referred root mean square noise (denoted as input-referred noise here
on). All 80 eye diagrams in the array are open and uniform, which confirms the
high-yield of the bonding process and validates our many-channel approach. As
a further confirmation of transmitter signal quality, a bit error ratio (BER)
test with a reference receiver demonstrates error-free performance (BER $<$
10-12) down to a receiver noise-determined power for 1 V, 1.25 V, and 1.5 V
modulator drives (Fig. 2h). In the following section, our on-chip receiver
shows a dramatically reduced input-referred noise of 480 nW. The 80 channel
photonic transmitter array outputs an aggregate data rate of 800 Gb/s and
occupies an area of 0.15 mm2, demonstrating an unprecedented bandwidth density
of 5.3 Tb/s/mm2.

The receiver cell consumes 70 fJ/bit when receiving a 10 Gb/s signal at -24.85
dBm average power with a $4\times 10^{-10}$ BER. The receiver spends energy as
a static biasing of the electronic amplifier. The photodiode is a vertical
p-silicon, i-germanium, n-germanium diode that efficiently converts optical
signals to electrical current with an efficiency of 1 A/W (Fig. 3c). The
capacitance of this photodiode is crucial since receiver noise is proportional
to the amplifier input capacitance and is compensated by static biasing power
(Methods). Minimizing this noise is critical for reducing laser power sourced
into the link and improving energy efficiency. With a measured photodiode
capacitance of 17 fF and pad capacitances of 10 fF, the simulated input-
referred noise is 300 nW. A BER test is used to evaluate this performance; a
signal from an ideal modulator is used to send a 10 Gb/s data stream into the
chip from which on-chip circuits measure errors in the received bits. On the
photonic chip, a ring resonator on a four-channel bus filters the modulated
signal to a photodiode. Figure 3d shows the four-channel bus spectrum. We then
gradually reduce the signal power while counting errors on the electronic
chip, obtaining the BER curve in Figure 3e. This test reveals that the
receiver has a sensitivity of -24.85 dBm for a $4\times 10^{-10}$ BER,
resulting in a measured input-referred noise of 480 nW using the 19 dB ER
signal. We record the power of the receiver from its power supply at -24.85
dBm input optical power with 70 fJ/bit consumption. This result, along with
the transmitter cell performance, demonstrates that both the receiver and
transmitter consume less than 100 fJ/bit while supporting a massive bandwidth
transfer of 800 Gb/s from the dense array.

Figure 4: Transmitter to receiver data communication link demonstration. a,
Illustration of the transmitter to receiver link showing a 3D integrated
photonic-electronic transmitter modulating four laser lines and a separate
photonic-electronic receiver converting the four data channels back to the
electrical domain. b, Spectrum of the link laser source. c, Eye diagrams of
the four channels after the transmitter. d, Bit error ratios (BERs) of the
data channels after the receiver.

The transmitter and receiver cells independently demonstrate a combined 120
fJ/bit; we next connect them and validate their combined performance. Optical
fibre connects two separate transceivers as a complete data communication
link, with one transceiver functioning as a transmitter and the other as a
receiver (Fig. 4a). A shared clock synchronizes the two electronic chips, and
programmable clock delays in each receiver cell align the transmitted data
with the receiver sampling point. A laser diode array provides four wavelength
channels at -5 dBm power per channel, which feed into a bus on the transmitter
chip. Figure 4b shows the spectrum of these laser channels. Individual
transmitter cells simultaneously modulate each wavelength at 8 Gb/s at a 1.5 V
drive, resulting in open eye diagrams for each channel (Fig. 4c). The signal
powers are too low for detection by the diagnostic oscilloscope receiver, so
we amplify the signal before the oscilloscope and normalize the eye diagrams.
However, the signals do not require amplification for the electronic-photonic
receiver due to its high sensitivity. The average power per channel at each
receiver photodiode is -19.5 dBm. On chip error counters record errors in each
channel in one-minute intervals, revealing a maximum recorded BER of $6\times
10^{-8}$ and a minimum count of no errors in the interval, denoted as 10-12,
in Figure 4d. This result shows that the transmitters and receivers within the
3D integration can form a complete low-power, high-bandwidth link needed for
next-generation computing systems.

## Discussion

Integrated photonic chips present a promising low-power platform to address
the data transfer demands of AI computing. Here, we realized this promise by
demonstrating a scaled-up array of 80 channels on a single electronic-
photonic, densely 3D integrated transceiver. This multi-chip module consumes
minimal energy by virtue of the large number of channels, cutting-edge low-
capacitance bonding technology, co-designed electronic-photonic circuits, and
advanced devices used. While our demonstrated system achieves record
performance in terms of energy efficiency and bandwidth density, the
performance can be further improved in future implementations. Although the
micro-disks used in this demonstration exhibit high performance, resonant
modulators can be developed with lower capacitance timurdogan2014ultralow ;
sun2018128  and a higher electro-optical response timurdogan2014ultralow ;
gevorgyan2022miniature , both of which would decrease the dynamic power of the
transmitter. Similarly, on the receiver side, lower capacitance photodiodes
lischke2015high ; chen20161  could reduce the power and noise of the receiver
architecture. However, miniaturizing photodiodes requires considering a loss
of responsivity chrostowski2015silicon , presenting complex link-level
tradeoffs. Additionally, the energy consumption of the electronic circuits can
be further reduced by moving to a more advanced CMOS node. While our
demonstrated bonding technology is approaching the limit of how closely spaced
tin bonds can be made, further density scaling could be realized through the
development of hybrid bonding samanta2023direct . However, after achieving the
low capacitance value of the bonds demonstrated in this work, pursuing a
further reduction in bond capacitance would yield diminishing returns in terms
of energy efficiency.

Reduced chip-to-fibre optical losses can improve the loss budget of our
demonstrated link, and laser powers may be as low as 47 fJ/bit with a
distributed feedback laser and 30 fJ/bit for a scalable, high channel count
comb laser (Supplementary Note 1). Furthermore, silicon resonators are
sensitive to temperature and fabrication variation, requiring thermal control
circuits sun201645 ; hattink2022streamlined  and, for minimized power, reduced
heat leakage into the environment using methods such as a silicon substrate
removal around resonators ban2023highly ; masood2013comparison ;
rizzo2023ultra . Detailed wafer-scale resonance variation data and approximate
thermal energy contributions across a range of scenarios provide a best case
thermal energy consumption of 71 fJ/bit (Supplementary Note 2). Additionally,
the photonic circuits are highly polarization-sensitive and require
polarization-maintaining fibre or the addition of polarization management
circuits chrostowski2015silicon ; ma2015symmetrical ; park2019ring . Finally,
while we demonstrate high bandwidth density, a higher per-fibre bandwidth and
photonic chip-edge bandwidth density can be achieved with wavelength scaling
by cascading more arrayed channels onto fewer waveguide buses
rizzo2023massively ; rizzo2022petabit . This architecture can combine with
chip-scale frequency combs to generate hundreds of wavelength channels
rizzo2023massively .

While the potential impact of this technology is evident for the advancement
of energy-efficient AI computing, its use may extend to far reaching
applications. These low-power, massively parallel optical links could enable
pervasive device connectivity and transform computing by streamlining resource
allocation through optically-linked, disaggregated, and reconfigurable
computing and memory resources zhu2020photonic ; jouppi2023tpu ;
gonzalez2022optically ; michelogiannakis2022case , revolutionizing the
computing landscape over the next decade and beyond.

## Methods

Transceiver assembly. Separate CMOS foundries are used fabricate the
electronic and photonic chips. The photonic chips were fabricated through the
American Institute for Manufacturing Integrated Photonics (AIM Photonics) on a
custom 300 mm silicon-on-insulator wafer. The AIM Photonics process design kit
(PDK) includes the micro-disks, ring filters, and photodiodes
fahrenkopf2019aim . The electronic chips were fabricated through Taiwan
Semiconductor Manufacturing Company Limited (TSMC) on a shared multi-project
wafer in a 28 nm CMOS process node. Both chips feature a square array of 2,304
aluminium pads at a 25 $\mu$m pitch on their top surfaces. These pads connect
to lower metal layers within each chip and the devices on the silicon layer.

The photonic and electronic chips are then processed post-fabrication before
bonding their pad arrays together. In this step, we core the 300 mm photonic
wafer to a 200 mm wafer and a wafer-level process is used to bump its pads
with electroplated layers of copper and tin. The electronic chips, received as
individual 1.6 mm2 units from a shared wafer, are unsuitable for wafer-level
photolithography-based processes. Alternatively, we adopt a chip-level process
of electroless nickel plating, followed by an additional layer of immersion
gold plating to prevent nickel oxidation. After dicing the bumped photonic
wafer into 6.5 mm by 3 mm chips, a thermo-compression bond is used to connect
the bumped photonic chips to the plated electronic chips.

To power and operate the transceiver, we create electrical connections to the
electronic chip through the bonds to the photonic chip. Metal layers on the
photonic chip wire these connections to large electrical pads on an exposed
edge of the photonic chip. Wire-bonds connect these pads to a printed circuit
board (PCB), which connects to: (i) a micro-controller that programs the
electronic chip, (ii) power sources that supply the electronic chip voltage
rails, and (iii) a radio-frequency (RF) clock generator for the 5 GHz clock of
the electronic chip. This clock line is a coplanar RF waveguide on the PCB
with a matched impedance to 50 $\Omega$ RF cables. Optical fibres couple light
to waveguide buses through silicon nitride edge couplers; these couplers are
on the side of the photonic chip that is opposite the wire-bond pads. A micro-
positioner is used to align a standard single mode fibre v-groove array with
the edge-couplers, which are spaced at a 127 $\mu$m pitch. The assembly
procedures of photonic wafer bumping and bonding, electronic chip plating, and
wire-bonding are conducted at Micross AIT, CVI, and Cornell University,
respectively.

Capacitance models and measurements. We identify several sources of
capacitance that affect energy efficiency: (i) chip pads, (ii) bump
parasitics, (iii) the micro-disk junction, (iv) the photodiode junction, and
(v) electronic driver capacitance. Extended Data Figure 1a depicts these
capacitance sources. These capacitances are determined through electrostatic
simulations, circuit model simulations, and empirical measurements. Focusing
initially on the bumps and photonic chip pads, an electrostatic solver (Ansys
Maxwell) is used to simulate the photonic chip pad-to-substrate capacitance (4
fF) and bump-to-bump parasitic capacitance ($<1$ fF). Similarly, we simulate
an electronic-chip pad model with extracted parasitic capacitances (Cadence
Virtuoso), yielding an electronic chip pad capacitance of 6 fF. This extracted
circuit model simulation also results in an effective 61 fF capacitance inside
the electronic driver.

Experimental methods are used determine the capacitances of the micro-disk and
photodiode junctions, along with a validation of the simulated pad
capacitance. A vector-network analyzer (VNA, Keysight P5007A) is used to
record an electrical RF reflection from probed devices. We fit the magnitude
and phase of the reflected wave across RF frequencies to a reflection from a
lumped complex impedance. Extended Data Figure 1b shows the imaginary
impedances of the devices and their associated fitted capacitor impedances.
During these measurements, an RF bias tee provides a DC reverse bias voltage
to the device. An electronic calibration module (Keysight N4693D) is used to
calibrate out the VNA response of the RF cable up to the output of the bias
tee. The bias tee connects to a 25 $\mu$m pitch RF probe (FormFactor
InfinityXT) and this probe response is calibrated out of the results by de-
embedding its unlanded response as an electrical open. After this calibration,
measurements of the photonic pad and bumped photonic pad show capacitances of
3 fF and 4 fF, respectively. Subsequent measurements are used to first de-
embed these pad responses and then measure the photodiode and micro-disk
capacitances. The measured photodiode capacitance is 17 fF.

As anticipated from a p-n junction, the micro-disk capacitance decreases with
an increasing reverse bias. Extended Data Figure 1c shows the measured
capacitance functions of the four micro-disks. The energy spent per bit
transition is the integral of this capacitance function weighted by the
difference of supply voltage and output voltage miller2012energy . A midpoint
Riemann sum of the micro-disk capacitances, weighted by reverse bias voltage
subtracted from 1 V, between 0 and 1 V bias voltage, yields an effective disk
junction capacitance for dynamic energy consumption of 128 fF (averaged across
the four micro-disks). A summation of the pad, driver, and junction
capacitances results in a 199 fF transmitter capacitance. This result is in
excellent agreement with the capacitance directly measured from the energy
consumption of the transmitter (198 fF), validating the capacitance models and
measurements.

Transmitter characterization. Each transmitter result is experimentally
measured from the 3D integrated photonic-electronic chip. An exception is the
DC electro-optic response measurement, for which a separate photonic chip with
the same modulator design is used. We apply a voltage to a probe to reverse
bias the modulator at varying DC voltages, and use an optical spectrum
analyzer (Keysight 8164B) to record each response. The remainder of the
transmitter, receiver, and link characterizations employ an optical switch
(Polatis 1000n 24x10). This switch optically connects equipment and devices-
under-test and minimizes fibre mating cycles. This approach streamlines the
measurement process and eliminates potential power discrepancies that might
stem from fibre mating inconsistencies. An optical spectrum analyzer (Aragon
BOSA 400) is used to measure the transmitter bus spectrum. For dynamic data
transmission, a micro-controller is used to write a different PRBS6 pattern
into the 64-bit registers in each of the 80 transmitter cells. Next, all
modulators transmit this data simultaneously as the electronic chip is
clocked, and all data registers are driven out of the chip to their respective
micro-disks. The 64-bit pattern transmitted by each modulator repeats
indefinitely as the chip clock is running. In this state, we record the eye
diagrams of each modulator, dynamic characteristics of the modulators, and
transmitter energy consumption from the electronic driver array voltage rail.
A narrow linewidth tunable laser (Santec TSL-210) is used as the light source
in these measurements. Laser light travels through a fibre polarization
controller and then into the chip. An oscilloscope (Tektronix DSA8300 with an
80C01 Optical Sampling Module) is used to receive modulated light for dynamic
characterization and eye diagrams. In the bit error ratio test, modulated
light initially passes through a variable optical attenuator (VOA) before
reaching a commercial receiver (Thorlabs RXM40AF). The commercial receiver
converts the optical signals into electrical signals that are read by a bit
error ratio tester (BERT, Anritsu MU195040A). We sweep the received optical
power with the VOA and record errors from the BERT to construct the
transmitter BER curves.

Receiver characterization. An ideal modulation source and an on-chip bit-error
checker circuit are used to characterize the receiver cell performance.
Separately, a tunable laser (Keysight 8164B) and a DC electrical probe landed
on photodiode pads are used to measure the photodiode responsivity; the probe
applies a reverse bias voltage and senses photocurrent from a known input
laser power. Next, for the dynamic characterization, we use an ideal
modulation source (Thorlabs MX35E) consisting of an internal laser and a
lithium niobate Mach-Zehnder modulator. A pulsed pattern generator (PPG,
Anritsu MU195020A) is used to drive the modulator with a repeating 64-bit
PRBS6 pattern. The signal travels through fibre and a polarization controller
before coupling into the photonic chip. Voltage is applied to a doped-silicon
resistor adjacent to the ring filter to generate heat and tune the ring filter
resonance to the desired wavelength channel. The ring resonator drops the
signal to a photodiode, which converts it from light into photocurrent for the
electronic chip to then amplify. Timing circuits continuously write the
received bits into a 64-bit long memory in a cycle. For timing, a programmable
timing offset circuit in the electronic chip and a timing offset of the PPG
align the incoming data to the receiver sampling point. A split clock source
synchronizes the receiver chip and PPG clock frequencies. As the final step,
readouts from the serial programming port display the saved received bits and
confirm data reception. However, the serial port cannot update fast enough to
give a bit error ratio in a short time frame. Instead, an on-chip error-
counter circuit in each cell compares the received memory with pre-programmed
expected bits and, if there is a discrepancy, it adds an error to an on-going
count. Readouts from the serial port display this count and we obtain a BER
curve as we sweep signal power using a VOA inside the ideal modulation source.

Link demonstration. The link demonstration combines the previously described
experimental setups of the transmitter and receiver. A microcontroller sets
the sent bits in a transmitter chip and a second microcontroller reads the
received bits in a separate receiver chip. Four channels of data transmit
through the link at 8 Gb/s/channel and serial port readouts from the receiver
record this data, along with an on-going error count for each channel. A
shared clock signal synchronizes the two transceivers and a programmable delay
block in each receiver cell delays the receiver sampling point with respect to
the transmitter clock. A distributed feedback laser array (Thorlabs PRO8) is
used as the four channel optical source for the link. An arrayed waveguide
grating multiplexes these different wavelengths of light from the laser array
onto a single fibre. We place polarization controllers before the transmitter
chip and before the receiver chip to ensure optimal coupling into the
fundamental transverse electric (TE) mode of each waveguide. The optical
switch is used to direct light from the transmitter to an erbium-doped fibre
amplifier (EDFA) and an oscilloscope for each eye diagram; the switch then
directs light back to the 3D integrated receiver for BER measurements without
amplification. Optical losses in this link amount to 14.5 dB. These losses are
from several sources: three chip-to-fibre interfaces at 3 dB each account for
9 dB, the modulation insertion loss is 2.5 dB, a modulation penalty accounts
for 1.5 dB (this is the difference between the optical power of a ‘1’ bit and
the average power), and an extra 1.5 dB of power is lost through the optical
switch and fibre connectors throughout the link.

Electronic circuit architectures. Transmitted data starts as bits in the
memory of each transmitter cell. Timing circuits running on a 5 GHz input
clock (half transmitted data rate) generate memory read addresses and two-to-
one multiplexer select signals. Circuits in the data path operate at a voltage
supply of 1 V, except for the driver, which operates between 1 and 1.5 V.
Extended Data Figure 3a shows the driver circuit. Inside of the design, high
speed 1 V transistors in a cascode configuration prevents transistor junction
breakdown from supply voltages exceeding 1 V. The main driver branch (M5-M8)
has wide transistors to reduce the switching delay on modulator capacitance
(Cmicrodisk = 128 fF). A capacitor (Ccoupling = 183 fF) ensures a high
switching speed while the auxiliary branch (M1-M4) holds the DC voltage level.

The receiver circuit senses a modulated photodiode current, amplifies it to
digital levels at the supply rail voltage, and de-serializes the signal before
writing it into internal memory. Extended Data Figure 3b shows the amplifier
circuit, which uses an inverter-based trans-impedance amplifier (TIA) as an
initial gain stage followed by an equalizer and inverters. A programmable
current digital-to-analog converter (DAC) at the amplifier input supplies a
current (IDAC) that cancels the DC offset of the photodiode current. The TIA
stage has a high feedback resistance for a high gain (Rf = 18.6 k$\Omega$).
This resistor equates to a lower input resistance (Rin = 2.1 k$\Omega$) from
the Miller Effect, however, it combines with the input capacitance
(Cphotodiode = 17 fF, Cpad = 10 fF) for a low frequency pole. As a remedy, an
active inductor circuit in the subsequent equalizer stage cancels out the TIA
pole (Req = 3.1 k$\Omega$, Ceq = 33.6 fF). After the equalizer, ensuing
inverters ensure the output swings between 0 and 1 V. An isolated, 1 V power
supply of the receiver amplifiers mitigates supply noise.

The TIA dominates the receiver amplifier energy consumption and its energy per
bit is the static biasing power divided by the data rate. However, the TIA
design introduces trade-offs between noise, bandwidth, and power. Equation 1
shows how the receiver signal-to-noise ratio, SNR, relates to receiver energy
per bit (E/bitRX), input signal (I), input capacitance (C), and channel
bandwidth (BW). Supplementary Note 3 provides a derivation of Equation 1. This
relationship sets a boundary on channel data rate scaling. With constant SNR
and C, the design can expand BW with an increase in the input signal. In this
context, the energy per bit remains constant, with the growing BW balancing
out added laser power. This could imply an indefinite data rate scaling.
However, a rise in BW necessitates wider TIA transistors, which subsequently
contribute significantly to the input capacitance, C. This sequence results in
a degradation of SNR at the receiver for high BWs, establishing a cap on the
energy-efficient per-channel data rate. To achieve higher data rates without
compromising energy, the focus should be on parallel data communication across
multiple channels. Similar conclusions have been made in other studies, which
advocate for parallel channels operating at moderate data rates
miller2017attojoule ; georgas2011addressing .

$SNR\sim\left(\frac{I}{BW}\right)^{2}\frac{E/bit_{RX}}{C^{2}}.$ (1)

## Acknowledgements

This work was supported by the U.S. Defense Advanced Research Projects Agency
(DARPA) under PIPES Grant HR00111920014 and by the U.S. Advanced Research
Projects Agency-Energy (ARPA-E) under ENLITENED Grant DE-AR000843. S.D
acknowledges support by the National Science Foundation (NSF) GRFP under Grant
DGE-1644869. We thank G. Keeler for leading the PIPES program, N. Abrams and
M. Hattink for helpful discussions, and the engineering teams at AIM/SUNY Poly
Photonics, Micross AIT, and CVI for their roles in the transceiver fabrication
and assembly.

## Author Contributions

S.D. designed the photonic chip and developed the 3D bonding process. S.D. led
transceiver testing and photonic device analysis with assistance from A.R.,
S.W., A.N., and V.G. A.R. compiled chip designs for the custom photonic wafer
run. S.W., D.K., C.O., and A.M. designed the electronic chip and conducted
bring-up tests of the electronic chip. S.D. and M.C. designed the printed
circuit boards. R.P. gathered wafer-level micro-disk fabrication variation
data. A.M. and K.B. supervised the project.

## Competing Interests

The authors declare no competing interests.

## Data Availability

The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

## Correspondence

Correspondence and requests for materials should be addressed to K.B. (email:
bergman@ee.columbia.edu).

## References

  * (1) Carrasquilla, J. & Melko, R. G.  Machine learning phases of matter.  _Nature Physics_ 13, 431–434 (2017).
  * (2) He, J. _et al._ The practical implementation of artificial intelligence technologies in medicine.  _Nature medicine_ 25, 30–36 (2019).
  * (3) Kaack, L. H. _et al._ Aligning artificial intelligence with climate change mitigation.  _Nature Climate Change_ 12, 518–527 (2022).
  * (4) Dally, W. J., Keckler, S. W. & Kirk, D. B.  Evolution of the graphics processing unit (gpu).  _IEEE Micro_ 41, 42–51 (2021).
  * (5) Mirabbasi, S., Fujino, L. C. & Smith, K. C.  Through the looking glass—the 2023 edition: Trends in solid-state circuits from isscc.  _IEEE Solid-State Circuits Magazine_ 15, 45–62 (2023).
  * (6) Wang, G. _et al._ Zero++: Extremely efficient collective communication for giant model training.  _arXiv preprint arXiv:2306.10209_ (2023).
  * (7) Pati, S., Aga, S., Islam, M., Jayasena, N. & Sinclair, M. D.  Computation vs. communication scaling for future transformers on future hardware.  _arXiv preprint arXiv:2302.02825_ (2023).
  * (8) Lee, B. G., Nedovic, N., Greer, T. H. & Gray, C. T.  Beyond cpo: A motivation and approach for bringing optics onto the silicon interposer.  _Journal of Lightwave Technology_ 41, 1152–1162 (2022).
  * (9) Miller, D. A.  Attojoule optoelectronics for low-energy information processing and communications.  _Journal of Lightwave Technology_ 35, 346–396 (2017).
  * (10) Rakowski, M. _et al._ Hybrid 14nm finfet-silicon photonics technology for low-power tb/s/mm 2 optical i/o.  In _2018 IEEE Symposium on VLSI Technology_ , 221–222 (IEEE, 2018).
  * (11) Samanta, A. _et al._ A direct bond interconnect 3d co-integrated silicon-photonic transceiver in 12nm finfet with-20.3 dbm oma sensitivity and 691fj/bit.  In _2023 Optical Fiber Communications Conference and Exhibition (OFC)_ , 1–3 (IEEE, 2023).
  * (12) Wu, Z. _et al._ Peta-scale embedded photonics architecture for distributed deep learning applications.  _Journal of Lightwave Technology_ (2023).
  * (13) Hochberg, M. & Baehr-Jones, T.  Towards fabless silicon photonics.  _Nature photonics_ 4, 492–494 (2010).
  * (14) Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M.  Micrometre-scale silicon electro-optic modulator.  _nature_ 435, 325–327 (2005).
  * (15) Timurdogan, E. _et al._ An ultralow power athermal silicon modulator.  _Nature communications_ 5, 1–11 (2014).
  * (16) Michel, J., Liu, J. & Kimerling, L. C.  High-performance ge-on-si photodetectors.  _Nature photonics_ 4, 527–534 (2010).
  * (17) Rizzo, A. _et al._ Massively scalable kerr comb-driven silicon photonic link.  _Nature Photonics_ 1–10 (2023).
  * (18) Rizzo, A. _et al._ Petabit-scale silicon photonic interconnects with integrated kerr frequency combs.  _IEEE Journal of Selected Topics in Quantum Electronics_ 29, 1–20 (2022).
  * (19) Wade, M. _et al._ A bandwidth-dense, low power electronic-photonic platform and architecture for multi-tbps optical i/o.  In _2018 European Conference on Optical Communication (ECOC)_ , 1–3 (IEEE, 2018).
  * (20) Sun, C. _et al._ Teraphy: An o-band wdm electro-optic platform for low power, terabit/s optical i/o.  In _2020 IEEE Symposium on VLSI Technology_ , 1–2 (IEEE, 2020).
  * (21) Wade, M. _et al._ An error-free 1 tbps wdm optical i/o chiplet and multi-wavelength multi-port laser.  In _Optical Fiber Communication Conference_ , F3C–6 (Optica Publishing Group, 2021).
  * (22) Sun, C. _et al._ Single-chip microprocessor that communicates directly using light.  _Nature_ 528, 534–538 (2015).
  * (23) Atabaki, A. H. _et al._ Integrating photonics with silicon nanoelectronics for the next generation of systems on a chip.  _Nature_ 556, 349–354 (2018).
  * (24) Zheng, X. _et al._ Ultralow power 80 gb/s arrayed cmos silicon photonic transceivers for wdm optical links.  _Journal of Lightwave Technology_ 30, 641–650 (2011).
  * (25) Ban, Y. _et al._ Highly optimized o-band si ring modulators for low-power hybrid cmos-sipho transceivers.  In _2023 Optical Fiber Communications Conference and Exhibition (OFC)_ , 1–3 (IEEE, 2023).
  * (26) Saeedi, S., Menezo, S., Pares, G. & Emami, A.  A 25 gb/s 3d-integrated cmos/silicon-photonic receiver for low-power high-sensitivity optical communication.  _Journal of Lightwave Technology_ 34, 2924–2933 (2016).
  * (27) Georgas, M., Leu, J., Moss, B., Sun, C. & Stojanović, V.  Addressing link-level design tradeoffs for integrated photonic interconnects.  In _2011 IEEE Custom Integrated Circuits Conference (CICC)_ , 1–8 (IEEE, 2011).
  * (28) Zhu, Z. _et al._ Photonic switched optically connected memory: An approach to address memory challenges in deep learning.  _Journal of Lightwave Technology_ 38, 2815–2825 (2020).
  * (29) Gaeta, A. L., Lipson, M. & Kippenberg, T. J.  Photonic-chip-based frequency combs.  _nature photonics_ 13, 158–169 (2019).
  * (30) Li, Z. _et al._ Scaling solder micro-bump interconnect down to 10 um pitch for advanced 3d ic packages.  In _2021 IEEE 71st Electronic Components and Technology Conference (ECTC)_ , 451–456 (IEEE, 2021).
  * (31) Miller, D. A.  Energy consumption in optical modulators for interconnects.  _Optics express_ 20, A293–A308 (2012).
  * (32) Sun, J. _et al._ A 128 gb/s pam4 silicon microring modulator with integrated thermo-optic resonance tuning.  _Journal of Lightwave Technology_ 37, 110–115 (2018).
  * (33) Gevorgyan, H., Khilo, A., Wade, M. T., Stojanović, V. M. & Popović, M. A.  Miniature, highly sensitive moscap ring modulators in co-optimized electronic-photonic cmos.  _Photonics Research_ 10, A1–A7 (2022).
  * (34) Lischke, S. _et al._ High bandwidth, high responsivity waveguide-coupled germanium pin photodiode.  _Optics express_ 23, 27213–27220 (2015).
  * (35) Chen, H. _et al._ \- 1 v bias 67 ghz bandwidth si-contacted germanium waveguide pin photodetector for optical links at 56 gbps and beyond.  _Optics Express_ 24, 4622–4631 (2016).
  * (36) Chrostowski, L. & Hochberg, M.  _Silicon photonics design: from devices to systems_ (Cambridge University Press, 2015).
  * (37) Sun, C. _et al._ A 45 nm cmos-soi monolithic photonics platform with bit-statistics-based resonant microring thermal tuning.  _IEEE Journal of Solid-State Circuits_ 51, 893–907 (2016).
  * (38) Hattink, M., Dai, L. Y., Zhu, Z. & Bergman, K.  Streamlined architecture for thermal control and stabilization of cascaded dwdm micro-ring filters bus.  In _Optical Fiber Communication Conference_ , W2A–2 (Optica Publishing Group, 2022).
  * (39) Masood, A. _et al._ Comparison of heater architectures for thermal control of silicon photonic circuits.  In _10th International Conference on Group IV Photonics_ , 83–84 (IEEE, 2013).
  * (40) Rizzo, A. _et al._ Ultra-efficient foundry-fabricated resonant modulators with thermal undercut.  In _CLEO: Science and Innovations_ , SF2K–6 (Optica Publishing Group, 2023).
  * (41) Ma, Y. _et al._ Symmetrical polarization splitter/rotator design and application in a polarization insensitive wdm receiver.  _Optics express_ 23, 16052–16062 (2015).
  * (42) Park, A. H., Shoman, H., Ma, M., Shekhar, S. & Chrostowski, L.  Ring resonator based polarization diversity wdm receiver.  _Optics express_ 27, 6147–6157 (2019).
  * (43) Jouppi, N. _et al._ Tpu v4: An optically reconfigurable supercomputer for machine learning with hardware support for embeddings.  In _Proceedings of the 50th Annual International Symposium on Computer Architecture_ , 1–14 (2023).
  * (44) Gonzalez, J. _et al._ Optically connected memory for disaggregated data centers.  _Journal of Parallel and Distributed Computing_ 163, 300–312 (2022).
  * (45) Michelogiannakis, G. _et al._ A case for intra-rack resource disaggregation in hpc.  _ACM Transactions on Architecture and Code Optimization (TACO)_ 19, 1–26 (2022).
  * (46) Fahrenkopf, N. M. _et al._ The aim photonics mpw: A highly accessible cutting edge technology for rapid prototyping of photonic integrated circuits.  _IEEE Journal of Selected Topics in Quantum Electronics_ 25, 1–6 (2019).

Figure Extended Data 1: Device capacitances. a, Schematic of the transmitter
and receiver capacitance sources; 1: electronic driver circuit, 2: electronic
chip pads, 3: bump parasitics, 4: micro-disk PN junction, 5: photodiode PIN
junction, 6: photonic chip pads. b, Measured imaginary impedances of devices
and their fitted capacitor impedances. c, Measured micro-disk junction
capacitance as a function of reverse bias voltage. Figure Extended Data 2:
Transmitter and receiver electronic circuit schematics. a, Transmitter driver
circuit schematic and driver highlighted in the transmitter illustration; data
as Vin drives the micro-disk as Vout. b, Receiver circuit schematic and the
receiver highlighted in the receiver illustration; amplification converts Iin
to Vout. In the schematics, a multiplier labels each transistor width as a
multiple of a ‘1x’ transistor. The ‘1x’ transistor has a width of 500 nm and
length of 30 nm in the transmitter and 300 nm by 30 nm in the receiver.


# Magnetic properties of perovskites
Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3: Monte Carlo simulations and
experiments

Yethreb<EMAIL_ADDRESS>Hung T<EMAIL_ADDRESS>corresponding author, Mohamed<EMAIL_ADDRESS>and E. K.
Hlil<EMAIL_ADDRESS>1Sfax University, Faculty of Sciences of
Sfax, LM2EM, B.P. 1171, 3000, Sfax, Tunisia.
2 Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris
Université, CNRS, UMR 8089
2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France.
3 Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000,
Grenoble, France.

###### Abstract

This work presents the remarkable experimental magnetocaloric properties of
the perovskites Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3, including the
magnetic entropy change $|\Delta S_{m}|$ and the Relative Cooling Power (RCP).
To understand these striking properties, we elaborate in this paper a model
and use Monte Carlo (MC) simulations to study it for comparison. For the
model, we take into account nearest-neighbor (NN) interactions between
magnetic ions Mn3+($S=2$) and Mn4+($S=3/2$) and the interactions between these
Mn ions with the magnetic Pr ions. The crystal is a body-centered tetragonal
lattice where the corner sites are occupied by Mn ions and the center sites by
Pr and Sr ions in their respective concentrations given in the compound
formula. We use an Ising-like spin model describing a strong anisotropy on the
$z$ axis. We show that pairwise interactions between ions cannot reproduce the
large plateau of the magnetization experimentally observed below the phase-
transition temperature. By introducing for the first time a many-spin
interaction between Mn ions, we obtain an excellent agreement with
experiments. Fitting the experimental Curie temperature $T_{C}$ with the MC
transition temperature, we estimate the value of the effective exchange
interaction in the system. From this value, we estimate various exchange
interactions between ions: the dominant one is that between Mn3+ and Mn4+
which is at the origin of the ferromagnetic ordering below $T_{C}$. We also
studied the applied-field effect on the magnetization in the region below and
above $T_{C}$. The obtained MC results for $|\Delta S_{m}|$ are in agreement
with experiments performed for applied fields from 1 to 5 Tesla. MC results of
RCP are also shown and compared to experimental ones. Various other physical
quantities obtained from MC simulations including internal energy and specific
heat, versus temperature are also shown and discussed.

PACS numbers: 5.10.Ln;64.30.-t;75.50.Cc

††preprint: APS/123-QED

##  I INTRODUCTION

Changes observed in Earth’s climate since the industrial revolution was mostly
driven by increase dramatically of global energy consumption for cooling. As
well as, using conventional gas compression refrigeration systems have shown
dangerous effects on the earth because it is consumed 15% of electricity
generated globally and account for 10% of global greenhouse gas (GHG)
emissions into the atmosphere resulting in a global warming that we have
observed over the last century period and which has become an important
environmental issue worldwide. In 2016, for example, scientists concluded that
the global electricity usage for cooling was 2000 TW h, or 18.5 % of annual
electricity consumption [1] and thanks largely demand for cooling,
particularly in the world’s warmer regions, global energy consumption for
cooling could explode tenfold by 2050 [2], which has also motivated the
exploration of Photovoltaics (PV) production for cooling [3] to conserve
energy and electricity, and alternative coolers to reduce the negative impacts
of global warming. Already today, debating global warming and energy
efficiency scenarios and input them into globally validated models for cooling
demand, photovoltaic (PV) electricity generation, thermal storage and looking
for alternative coolers are the most critical blind spots in today’s energy
debate. Nowadays, among the most prominent worldwide in the new generation of
refrigerants, Magnetic Refrigeration (MR), based on the magnetocaloric effect
(MCE), becomes a promising technology and alternative refrigeration systems
that can replace the conventional gas refrigeration technique [4]. In contrast
to a compression cycle the MR can be an eco-friendly refrigeration technology
[5], due to the use of magnetic materials that acts as a refrigerant, which
produces no ozone-depleting chemicals (CFCs), hazardous chemicals (NH3), or
greenhouse gases (HCFCs and HFCs) [6]. MR has presented another advantage like
large refrigeration efficiency and the potential to reduce energy use by 30%
and requires no refrigerant. The MCE is an intrinsic property of magnetic
materials, is defined as the changes in the magnetic entropy ($|\Delta
S_{m}|$) and hence the temperature of the material when placed in a magnetic
field under adiabatic conditions [7-9]. MCE was first discovered by Warburg
(1881) with iron [10], it paved the way for further developments. The reader
is referred to Ref. [11] for a very good review of historical developments,
both experimentallly and theoretically, during the following 50 years. Though
Gd-based alloys are the most studied materials which exhibits excellent
magnetocaloric properties, but its high cost which could be as high as 4,000
kg–1 limits their use as magnetic refrigerant [12]. In fact, a huge part of
current researchers is orienting its attention on some new MCE materials
having the best magnetic properties at a low production cost. The most
promising are the rare-earth transition-metal manganites of general formula
Ln1-xAxMnO3 (Ln = trivalent rare-earth, A = divalent alkaline earth) because
of their exciting properties such as colossal magnetoresistance (CMR) and
colossal magnetocaloric effect (MCE) [13-19]. These rare earth manganese
oxides also show several other interesting physical properties like
magnetodielectric behavior [20], multiferroicity [21], large magneto-optic
responses [22]and orbital ordering [23]. We note that new researches [24,25]
have been recently carried out to show the high potential of nanopowder
perovskites for magnetic refrigeration application, which is urgently crucial
for green and environmentally friendly technology. To get a clear idea about
the performance of materials used in magnetic refrigeration devices, the main
requirements for a magnetic material are the large magnetic entropy change
$|\Delta S_{m}|$, the large spontaneous magnetization as well as the sharp
drop in the magnetization associated with the ferromagnetic to paramagnetic
transition at the Curie temperature $T_{C}$ [26, 27]. As known, large magnetic
entropy changes were reported in other materials such as Pr0.6Sr0.4MnO3 [28].
It is distinguished by a maximum change of magnetic entropy $|\Delta
S_{m}^{max}|$ corresponding to 3.58 J/kg K under a magnetic field of 5T and a
Relative Cooling Power corresponding to 159.37 J/kg which make these materials
good candidates for magnetic refrigeration applications. Furthermore, many
experimental works were made for the perovskite Pr1-xSrxMnO3 material.
However, these experimental works allow only a partial understanding of the
compound. A better understanding of the roles of each microscopic interaction
in of this magnetic system requires more theoretical and numerical studies.
The aim of our work is to study the magnetic and magnetocaloric properties of
the perovskite Pr0.9Sr0.1MnO3 material by performing experiments and by Monte
Carlo simulations to compare to experimantal data. For the modeling, we use an
Ising-like spin model with various interactions based on experiment
observations. To fit with experiments, in addition to pairwise interactions
between magnetic ions, we introduce a new many-spin interaction term between
Mn ions. As seen in this paper, this model is justified by a comparation with
the experimental measurements performed on this material.

Before introducing our perovskte compound, let us emphasize that rich and
various perovskite compounds due to doping are currently attracting a
considerable attention [29]. They have many remarkable properties because of
the complex interplay among spins which induces a rich phase diagram [30-33].
There are many applications in particular in spintronics [33-35]. The diagram
in the phase space defined by concentration $x$, temperature $T$, magnetic
field $H$, superexchange (SE) $J$ is not quite clear yet for different
compounds. Jonker and Van Santen [36] have studied ferromagnetic compounds of
manganese with perovskite structure. Their properties can be understood as the
result of a strong ferromagnetic (positive) exchange interaction between
nearest neighboring Mn ions via intercalated oxygen: Mn3+-O-Mn4+.
Note that the double exchange (DE) mechanism developed by Zener [37-39]
explains the existence of ferromagnetism and the metallic behavior at low
temperatures. There is now a consensus to recognize that the interesting
properties observed in perovskites are fundamentally originated from the DE
mechanism along the link Mn3+-O-Mn4+. This characteristics is at the origin of
a new interesting observed phase transition in doped manganites [40,41] from a
magnetically-ordered phase to the disordered phase. Recent refinement of
experimental techniques and the improvement of the sample quality have made
possible to discuss critical phenomena of this transition [42].

In this paper we experimentally investigate magnetic properties of perovskite
manganite Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3 using by Powder X-ray
diffraction (XRD) technique and the SQUID magnetometer developed in Louis Neel
Laboratory at Grenoble, France. In order to fit with the experimental results,
we propose a new model including a many-spin interaction in addition to the
pairwise exchange interactions between magnetic ions. We then study the model
with Monte Carlo (MC) simulations. Note that the compound Pr0.55Sr0.45MnO3 has
been experimentally studied by Fan et al [43] and by Saw et al [44]. Other
concentrations of Pr, namely Pr0.5Sr0.5MnO3 and Pr0.6Sr0.4MnO3, have been
studied in Refs. [45,46,47].

We use a discrete spin model to express the strong Ising-like anisotropy along
the $z$ axis and we take into account various types of interactions between
spins of Mn4+, Mn3+ and Pr3+. This model is justified by an excellent
agreement with experimental magnetization measurements performed on this
material.

The paper is organized as follows: in section II, we describe our experiment.
Our theoretical model and the MC method are shown in section III. Results are
shown and compared to experimental data in section IV. Our concluding remarks
are given in section V.

##  II Experiment

Polycrystalline sample Pr0.9Sr0.1MnO3 were prepared using Sol-Gel method from
initial pure oxides; Pr6O11, SrCO3 and Mn2O3, with nominal purities higher
than 99.9%. Identification of the crystalline phase and structural analysis
were carried out by Powder X-ray diffraction (XRD) with a diffractometer
(SIEMENS D500) using $\lambda_{CuKa}$ ($\lambda=1.54056\AA$) radiations at
room temperature with a scan step of 0.02∘ in the range $20^{\circ}\leq
2\theta\leq 80^{\circ}$. Crystal structure parameters were refined using
FullProf program based on the Rietveld method [48,49]. Magnetization
measurements versus temperature (T) and magnetic applied field used for MCE
studies, were performed using the SQUID magnetometer developed in Louis Neel
Laboratory at Grenoble, France.

Let us detail the sample preparation: polycrystalline
Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3 oxides were prepared by Sol–Gel
method, and single perovskite phase was obtained by appropriate heat treatment
in accordance with the following reaction: 0.15 Pr6O11 \+ 0.1 SrCO3 \+ 0.5
Mn2O3 $\rightarrow$ Pr0.9Sr0.1MnO3 +$\delta$ CO2.

The appropriate amounts of high purity (99.9%) Pr6O11, SrCO3 and Mn2O3 powder
precursors were dissolved in nitric acid and distilled water. The solution was
then heated at 90°C on a hot plate under constant stirring to eliminate the
excess water and to obtain a homogeneous solution. Subsequently, citric acid
and ethylene glycol were added under thermal agitation to obtain a viscous
gel. The gel was then dried at 300°C and calcinated at 600°C for 5 hours
resulting in fine powder. Finally, the powders were ground and mixed in a
mortar for 10 min and then placed in platinum crucibles and compressed into
pellets 12 mm in diameter and 2 mm thick. The pellets were finally sintered at
1100°C for 3 hours.

Experimental results will be shown in section IV together with the MC results
for comparison.

##  III Model and Method

###  III.1 Model

We consider the body-centered tetragonal (bct) lattice with the following
pairwise-interactions (see Fig. 1a):

${\cal
H}_{p}=-\sum_{<i,j>}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}-\mu_{0}H\sum_{<i>}S_{i}$
(1)

where $\mathbf{S}_{i}$ is the spin at the lattice site $i$, $\sum_{<i,j>}$ is
made over spin pairs coupled through the exchange interaction $J_{ij}$. $H$ is
a magnetic field applied along the $z$ axis.

Before defining explicitly the interactions, let us discuss about the spin
model we shall use. We suppose that the spins of Mn ions lie on the $z$ axis
with a strong uniaxial anisotropy. We have first tried to calculate using the
Heisenberg model with a strong anisotropy but we did not get an agreement with
experimental data. On the other hand, using a discrete Ising-like spin model,
we obtain an excellent agreement with the experimental magnetization in the
whole temperature range as shown in the next section. So, we shall use the
Ising spins with different spin amplitudes depending on the ion kind. We shall
write $S_{i}$ instead of $\mathbf{S}_{i}$ etc in the following. Note that
$S_{i}$ represents the Ising spin of amplitude $S$.

Let us recall that there are two kinds of Mn ions which occupy the corner
sites of the bct lattice in Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3:
Mn4+ with spin amplitude $S=3/2$ and Mn3+ with $S=2$. Due to the doping, the
positions of Mn3+ and Mn4+ are at random. The Pr and Sr ions occupy the
centered sites of the bct lattice with their respective concentrations. It is
experimentally found that interaction between neighboring Mn3+ and Mn4+ as
well as that between Mn3+ and Mn3+ are strongly ferromagnetic while that
between Mn4+ and Mn4+ is weakly antiferromagnetic. Hereafter we shall take the
following interactions between nearest-neighbors (NN) between different kinds
of magnetic Mn ions:

$J_{1}$: Interaction coupling of a Mn3+ ion with a NN Mn3+ ion,
$J_{2}$: Interaction coupling of a Mn3+ ion with a NN Mn4+ ion,
$J_{3}$: Interaction coupling of a Mn4+ ion with a NN Mn4+ ion,
$J_{4}$: Interaction coupling of a Pr ion with a Mn3+ ion,
$J_{5}$: Interaction coupling of a Pr ion with a Mn4+ ion
$J_{6}$: Interaction coupling between two Pr ions on the adjacent bct units.

Note that the two outer electrons of the Pr ion occupy the orbitals 4f2, so
its spin is 1 (Hund’s rule), and $L=3+2=5$. So, for less than half filled
shell , $J=|L-S|=4$. However, as we are interested in spin-spin interactions
of the compound which are responsible for the magnetic transition (not $J-J$
interaction), the orbitals $L$ are not taken into account.

It is known, by the theory of critical phenomena, that the nature of a phase
transition depends on a few parameters: the spin nature (Ising, XY,
Heisenberg, Potts,..), the nature of their interaction and the space
dimension. Only when the interaction is short-range ferromagnetic (or non
frustrated antiferromagnetic), the transition obeys the universality class
depending only on the nature of spin and the space dimension: one has 2D Ising
universality class, 3D Ising universality class, 3D XY universality class, 3D
Heisenberg universality class…. However, when a disorder is introduced, or a
frustration due to competing interactions takes place … the nature of the
transition changes . It may have other critical exponents in the presence of a
disorder, or it becomes of the first order when the system is frustrated (see
recent reviews in Ref. [50]). The transition experimentally observed for our
compound is of the second order with a magnetization plateau below $T_{c}$.
Our compound has a smalll disorder but there is no frustration. For the
modeling, we have tried the Heisenberg model, the continuous Ising model, …
with various pairwise interactions, but we did not find the magnetization
plateau and the sharp second-order transition experimentally observed. The
idea to introduce the multispin interaction to reproduce the experimental
observations, as we will see later, was successful. After several trial forms,
the following term is found to well describe the experimental magnetization
curve:

${\cal H}_{m}=-K\sum_{i}S_{i}S_{i1}S_{i2}S_{i3}S_{i4}$ (2)

where $K$ is the interaction strength and the sum runs over all Mn sites and
the spins $S_{i1},\ S_{i2},\ S_{i3}$ and $S_{i4}$ are the NN of the spins
$S_{i}$ on the $xy$ plane (see Fig. 1b).

Figure 1: (a) An example of ion distribution on the bct lattice: black, red
and green circles represent Mn3+, Mn4+ and Pr3+, respectively, (b) Spin of the
Mn ion at the site $i$ interacts simultaneously with four Mn spins at sites
$i1$, $i2$, $i3$ and $i4$ in the multi-spin interaction given by Eq. (2). See
text for comments.

Let us discuss about the multi-spin interaction given in Eq. (2). First, we
take just a plaquette shown in Fig. 1b: we see that there are many degenerate
configurations: all spins are parallel, any two spins among $i1$,$i2$, $i3$
$i4$ are reversed and four spins $i1$,$i2$, $i3$ $i4$ are reversed. Of course,
when the plaquette is connected to surrounding plaquettes and when we take
into account the pairwise interactions, the degeneracy is reduced.
Nevertheless, this deneneracy favors a sharp second-order transition
experimentally observed for the compound under study. At low temperatures, the
multi-spin interaction favors the ferromagnetic state against single-spin
excitations. This explains the magnetization plateau experimentally observed.

At this stage, it is worth to emphasize that the Heisenberg model is the
lowest order involving only two spins because of the initial assumption of the
overlap between the spin-dependent wave functions of only two neighboring
atoms (Hartree-Fock approximation, see Ref. 51, pp. 55-60). In materials,
however the interaction between one spin with its neighbors is simultaneous,
but the demonstration for the multispin interaction as in the case of two-body
Heisenberg model is at present impossible. There was an attempt to make a
fourth-order pertubation expansion for the Heisenberg model. It results in a
term of 3-spin and a term of 4-spin interactions [52]. But for the Ising
model, the demonstration by exact methods has been done. The reader is
referred to Ref. [53] for the references on the demonstration of various
multispin interactions. Note that recently there has been an increasing number
of papers using the multispin interaction for various purposes [53-55].

We have conducted standard MC simulations on samples of dimension $N=L\times
L\times L$, where $L$ is the number of bct cells in each of the $x$, $y$ and
$z$ directions. Periodic boundary conditions are used in all directions.
Simulations have been carried out for different lattice sizes ranging from
$12^{3}$ to $30^{3}$ lattice cells to check finite-size effects. The results
shown below are those of $30^{3}$ lattice size (the finite-size effect is no
more significant from $20^{3}$).

The procedure of our simulation can be split into two steps. The first step
consists in equilibrating the lattice at a given temperature. The second step,
when equilibrium is reached, we determine thermodynamic properties by taking
thermal averages of various physical quantities [56,57]. Starting from a
random spin configuration as the initial condition for the MC simulation, we
have calculated the internal energy per spin $E$, the specific heat $C_{V}$,
the magnetic susceptibility $\chi$, the magnetization of each sublattice and
the total magnetization, as functions of temperature $T$ and magnetic field
$H$. The MC run time for equilibrating is about $10^{5}$ MC steps per spin.
The averaging is taken, after equilibrating, over $10^{5}$ MC steps. A large
number of runs have been carried out to check the reproductivity of results
shown below.

The statistiical averages of the $z$ spin component of Mn3+ and Mn4+ and Pr3+
($\langle S_{1}\rangle$, $\langle S_{2}\rangle$ and $\langle S_{3}\rangle$,
respectively) and the Edwards-Anderson order parameter $Q_{EA}$ for these
three sublattices $\ell=1,2,3$ are defined by

$\displaystyle\langle S_{\ell}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N_{\ell}}\langle\sum_{i\in\ell}S_{i}\rangle$ (3)
$\displaystyle Q_{EA}(\ell)$ $\displaystyle=$
$\displaystyle\frac{1}{N_{\ell}}\sum_{i\in\ell}\langle S_{i}\rangle$ (4)

where $\langle...\rangle$ indicates the statistical time average and the sum
is taken over Mn3+ ($\ell=1$) or Mn4+ ($\ell=2$) or Pr3+ ($\ell=3$), with
$N_{\ell}$ being the number of spins of each kind. Note that $Q_{EA}$ is
calculated by first taking the time average of each spin and secondly taking
the spatial average over all spins. This parameter is used to calculate the
freezing degree of the spins when a long-range ordering is absent or the
nature of ordering is unknown such as in spin glasses or in disordered systems
[57-60].

The total magnetization $M$ is defined by

$\langle M\rangle=g\mu_{B}(\langle S_{1}\rangle+\langle S_{2}\rangle+\langle
S_{3}\rangle)$ (5)

where $g$ and $\mu_{B}$ are the effective gyromagnetic factor and the Bohr
magneton. Note that experimental data on the total magnetization (magnetic
moment) is given by $M$. The magnetizations of Mn3+, Mn4+ and Pr3+ were not
measured separately, therefore the gyromagnetic factor $g$ which relates the
total spin to $M$ should be understood as an ”effective $g$”.

The average internal energy $E$ per spin, the specific heat $C_{V}$ per spin
and the susceptibility $\chi$ per spin are defined by

$\displaystyle\langle E\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N}\langle({\cal H}_{p}+{\cal H}_{m})\rangle$ (6)
$\displaystyle C_{V}$ $\displaystyle=$
$\displaystyle\frac{1}{k_{B}T^{2}}[\langle E^{2}\rangle-\langle E\rangle^{2}]$
(7) $\displaystyle\chi$ $\displaystyle=$
$\displaystyle\frac{1}{k_{B}T}[\langle M^{2}\rangle-\langle M\rangle^{2}]$ (8)

##  IV Results - Comparison with Experiment

###  IV.1 How to fit the MC results with experiments

Experiments found that $J_{1}$ and $J_{2}$ dominate and give rise to the
ferromagnetic ordering up to very high temperatures $T_{C}=220$ K for
Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3. For MC simulations, after many
trial sets of parameteers, we found the following set which reproduces the
experimental magnetization:

$\displaystyle J_{1}$ $\displaystyle=$ $\displaystyle+6.0J,\ J_{2}=+6.0J,\
J_{3}=-0.1J,\ J_{4}=-6.0J,\ J_{5}=-3.0J,\ J_{6}=-0.1J$ (9) $\displaystyle C$
$\displaystyle=$ $\displaystyle 0.5$ (10) $\displaystyle K$ $\displaystyle=$
$\displaystyle 1.9$ (11)

where $J$ is equal to 1, namely $J$ is the MC energy unit, $C$ is the
reduction coefficient applied to the interaction between Pr-Pr on the $z$
axis.

Using the above interaction values we obtained the MC transition temperature
$T_{C}(MC)=57.11$. In order to fit the MC transition temperature with the
experimental value $T_{C}(exp)=220$ K, we have to multiply all the interaction
values given above by $J=220/57.11$. Let us show the MC curve and the
experimental magnetization in Fig. 2.

Figure 2: MC result (violet squares) and experimental magnetization (green
void circles) versus temperature $T$ in Kelvin, are shown for comparison. See
text for comments.

We note the following important points:

(i) The experimental magnetization shows a flatness up to the phase transition
temperature. This is unusual in magnetic materials. At first, we have tried to
modify the values of $J_{1},...,J_{6}$ to get the agreement, but we failed. At
best the MC curve coincides with the experimental one up to $T=200$ K and
starts to decrease from that temperature to make the transition at $T=220$K.
The deficit of the MC magnetization between 200 and 220 K with respect to the
experimental one is solved by introducing the multispin interaction between Mn
ions on the $xy$ plane: this finally gives the good MC result in excellent
agreement with the experimental magnetization.

(ii) The fall of the magnetization curve at $T_{C}$ is very sharp. However, as
there is no discontinuity of the magnetization at $T_{C}$, the transition is
thus of second-order. This is confirmed by the internal energy and the
specific heat which will be shown below.

In order to estimate the amplitudes of physical parameters in real units, we
use the mean-field approximation [50]:

$T_{C}=\frac{2}{3k_{B}}ZS_{eff}(S_{eff}+1)J_{eff}$ (12)

where $Z=6$Mn+$8\times 0.9$Pr=13.72 is effective coordination number at a Mn
site and $S_{eff}$ the effective spin length which can be taken as the average
on the Mn4+, Mn3+ and Pr3+ using their concentrations: $S_{eff}=(0.9\times
2+0.1\times 1.5+0.9\times 1)/(0.9+0.1+0.9)=1.5$. Putting $T_{C}=220$ K in Eq.
(12), we obtain $J_{eff}\simeq=0.0005525$ eV=5.52 K.

From the fit of $T_{C}$ above, it is easy to determine each of the exchange
interactions defined earlier, in Kelvin. For example, $J_{1}=6J=6\times
220/57.11\simeq 23$ K. Note that in magnetic materials with Curie temperatures
at room or higher temperatures the exchange interaction is of the order of
several dozens of Kelvin [61,62] which is the same order of magnitude as what
we found here. Other interactions can be calculated in the same manner. Note
that using such a mean-field approximation, we obtain the order of magnitude
of interaction parameters, but not to a good precision.

Now, what is the real value of the energy in the real unit ? To answer this
question, there are two ways to do:

1\. We use the classical ground-state energy with the effective parameters
calculated above:

$\displaystyle E_{0}(MF)$ $\displaystyle=$ $\displaystyle-0.5\times Z\times
S_{eff}^{2}\times J_{eff}$ (13) $\displaystyle=$ $\displaystyle-0.5\times
13.72\times 1.5^{2}\times 55.25\times 10^{-5}\ \mbox{eV}$ (14)
$\displaystyle=$ $\displaystyle-852.73\times 10^{-5}\mbox{ eV}$ (15)
$\displaystyle\simeq$ $\displaystyle-8.53\ \mbox{meV}$ (16)

2\. We use the MC result of the GS energy using the interaction values given
in Eqs. (9)-(11) which is $E_{0}/J\simeq 68$ when $T\rightarrow 0$. This value
has been obtained with $k_{B}=1$ to simplify the MC simulation. To recover the
real value of $E_{0}$, we write

$\displaystyle E_{0}(MC)$ $\displaystyle=$ $\displaystyle-68\times
3.8582\times k_{B}$ (17) $\displaystyle=$ $\displaystyle-262.3576\times
1.380649\times 10^{-23}\ \mbox{Joules}$ (18) $\displaystyle=$
$\displaystyle-362.21\times 10^{-23}\times 6.242\times 10^{18}\ \mbox{eV}$
(19) $\displaystyle=$ $\displaystyle-2260.92\times 10^{-5}\ \mbox{ eV}$ (20)
$\displaystyle\simeq$ $\displaystyle-22.61\ \mbox{meV}$ (21)

This value is almost three times higher than that of the mean-field
approximation in Eq. (16). It is interesting to calculate the effective
exchange interaction from this value: one has

$\displaystyle J_{eff}$ $\displaystyle=$ $\displaystyle E_{0}(MC)/[-0.5\times
Z_{eff}\times S_{eff}^{2}]$ (22) $\displaystyle=$ $\displaystyle
22.61/[-0.5\times 13.72\times 1.5^{2}]\ \mbox{meV}$ (23) $\displaystyle=$
$\displaystyle 1.4648\ \mbox{meV}$ (24) $\displaystyle\simeq$ $\displaystyle
14.46\ \mbox{K}$ (25)

This value is of course more reliable than the value 5.52 K deduced from the
mean-field approximation. It is in the range of exchange interactions found in
magnetic materials [60] and in another family of perovskite compound [61].

Let us show the MC energy $E$ versus temperature $T$(K) and the specific heat
$C_{V}$ versus $T$ in Fig. 3.

Figure 3: (a) Internal energy per spin (in meV) versus temperature $T$ in
Kelvin, (b) Specific heat (in meV/K) versus $T$. Line is guide to the eye.

We note that

(i) The $E$ curve shows a change of curvature at $T=220$ K which confirms the
value of $T_{C}$ in Fig. 2. There is no sign of discontinuity of $E$ at
$T_{C}$ confirming that the transition is of second order. Note that in a
first-order transition, the order parameter (magnetization) and the internal
energy are discontinuous at $T_{C}$ [ 63,64]. This is not the case here.

(ii) The energy at $T=0$ is -22.61 meV.

(iii) The peak of $C_{V}$ confirms the change of curvature of $E$ at $T_{C}$.
The rather large width of $C_{V}$ at $T_{C}$ and the finite peak height
confirm again the second-order nature of the transition.

We show now in Fig. 4 the $z$ component three sublattice magnetic ions
$\langle S(i)\rangle\ (i=1,2,3)$. We see here that the Mn3+ ($S(1)$) and Mn4+
($S(2)$) have the same sign, namely they order ferromagnetically, while $S(3)$
(Pr ions) is ordered antiferromagnetically with the Mn ions. These data are MC
results, they were not available by experiments.

Figure 4:  Sublattice $z$ spin components $S(1)$, $S(2)$ and $S(3)$ versus
temperature $T$.

We show in Fig. 5 the MC result of the internal energy for magnetic field
$\mu_{0}H$ ranging from 0.03 to 5 Tesla.

Figure 5:  Energy versus $T$ for $\mu_{0}H=0.03,1,2,3,4,5$ Tesla (from left to
right). See text for comment

As seen in Fig. 5, the transition temperature increases with increasing $H$,
and the transition is less and less sharp. This is well known in magnetic
systems under an applied field. Rigorously speaking, there is no phase
transition in a ferromagnet under an applied magnetic field since the global
magnetization never goes to zero at finite $T$.

###  IV.2 Magnetocaloric Effect - Magnetic Entropy Change

We apply the magnetic field on the system at a given $T$. The curves are shown
in Fig. 6a for $T$ below and above $T_{C}$. For clarity, experimental data in
this range of $T$ are separately shown in Fig. 6b for comparison. We obtain a
good qualitative agreement between the two sets of curves. We have some
remarks:

i) The MC magnetization curves far below $T_{C}$ are larger than the
experimental curves at low $H$. We think that this is because experimental
samples are polycrystalline ones which have domains resulting in low $M$ at
low $H$ and low $T$, in contrast to MC samples which are on a lattice,

ii) For $T$ close to $T_{C}$ and above $T_{C}$, the agreement between
experiments and MC results is quite good.

Figure 6:  Effect of magnetic field $\mu_{0}H$ (Tesla) on the magnetization
(a) obtained from MC simulations, from top line : $T=180$ K, 185 K, 190 K, …,
260 K ( bottom line), (b) Experimental data for the same system in the same
temperature range indicated on the figure.

Let us show now the results on the magnetocaloric effect of our compound. This
effect is characterized by the magnetic entropy change, called $|\Delta
S_{m}|$, and the Relative Cooling Power (RCP). Let us note that $|\Delta
S_{m}|$ is not the statistical entropy defined for a given $(T,H)$. $|\Delta
S_{m}|$, the magnetic entropy change, is calculated by applying a field
progressively from 0 to $\mu_{0}H$ at a given temperature. $|\Delta S_{m}|$ is
given by the thermodynamic Maxwell formula

$|\Delta S_{m}(T,H)|=\int_{0}^{H}\left[\frac{\delta M(T,H_{i})}{\delta
T}\right]_{H_{i}}\ \mu_{0}dH_{i}$ (26)

where $\delta M$ is the change of the magnetization at $H$ when $T$ varies
$T\rightarrow T+\delta T$. This formula is discretized and used in experiments
as

$|\Delta
S_{m}(T,H)|=\sum_{i}\left[\frac{M_{i}-M_{i+1}}{T_{i+1}-T_{i}}\right]\mu_{0}\Delta
H_{i}$ (27)

The experimental magnetic entropy change $|\Delta S_{m}|$ using this formula
is shown in Fig. 7: the peak temperature increases very slightly with
increasing $H$ but the peak height is higher with larger $H$. The large peaks
of $|\Delta S_{m}|$ indicate the smootth change from the ferromagnetic phase
to paramagnetic phase. This is a common feature of doped perovskite compounds
which have more or less domains, defects and structural inhomogeneities [65].

.

Figure 7:  Experimental magnetic entropy change $|\Delta S_{m}|$ (J/(kgKelvin)
versus temperature $T$ for $\mu_{0}H=$1, 2, 3, 4, 5 Tesla (from bottom to
top). See text for comment.

To compare with experimental magnetic entropy change, we carry out the
calculation of $|\Delta S_{m}|$ using Eq. (27) as in experiments with the MC
magnetization shown in Fig. 6a. This gives the result shown in Fig. 8. Several
remarks are in order:

i) $|\Delta S_{m}|$ obtained from MC simulations show a peak at each value of
$\mu_{0}H$ from 1 Tesla to 5 Tesla. The value of the peak increases with
increasing $H$, in agreement with experiments shown in Fig. 7. The peak
temperarure slightly increases with increasing $H$ as observed in experiments.

ii) Unlike experiments, the MC $|\Delta S_{m}|$ comes to zero below $T\simeq
180$ K. This is due to the fact that the MC magnetization is higher than the
experimental one at low $T$ and low $H$ as said earlier. Below $T=180$ K the
MC system is highly magnetized so that low $H$ does not make effect. Unlike
experimental samples, the MC system is on a lattice, there are thus no
vacancies, no dislocations … The system is, though disordered, more
homogeneous so that the peak widths are narrower.

.

Figure 8:  MC result for magnetic entropy change $|\Delta S_{m}|$
(J/(kgKelvin) versus temperature $T$ for $\mu_{0}H=$1, 2, 3, 4, 5 Tesla (from
bottom to top). See text for comment.

The formula used to calculate the RCP is

$RCP(H)=|\Delta S_{max}(H)|\times\Delta T$ (28)

where $|\Delta S_{max}(H)|$ is the maximum value of $|\Delta S_{m}(H)|$ and
$\Delta T$ the temperature range at the full width at half maximum.

We show in Table 1 our experimental Relative Cooling Power (RCP) and the MC
RCP. As seen, there is a difference between experiments and MC results. This
difference comes from very large temperature range at the full width at half
maximum in experimental data (Fig. 7) as discussed above. Nevertheless, our
compound studied here, both experimentally and numerically, has very high RCP
which is very important for high efficiency in applications using the
magnetocaloric effect.

Table 1:  Experimental Relative Cooling Power and MC results of
Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3, for $\mu_{0}H=1,2,3,4,5$
Tesla. See text for comments.

##  V Conclusion

We have shown in this paper magnetic properties of the perovskite compound
Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3 observed experimentally. These
include the magnetization as functions of $T$ and $H$ and the magnetic entropy
change at the transition from the ferromagnetic phase to the disordered phase
at $T_{C}\simeq 220$ K. We deduce the RCP and find that it varies with $H$ to
a very high value, very interesting for magnetocaloric applications. To model
this compound to fit with the experimental magnetization which shows a plateau
up to the transition temperature, we introduce a five-spin interaction term
between the Mn ions in the Hamiltonian, in addition to the different pairwise
interactions between magnetic ions Mn3+, Mn4+ and Pr3+. MC simulations have
been carried out. We have found an excellent agreement between the MC
magnetization and the experimental one. We have deduced from the fitting the
values of various exchange interactions. Other physical quantities such as
internal energy and specific heat have been calcultated as functions of $T$.

We have also compared the experimental magnetic entropy change $|\Delta
S_{m}(H)|$ with the results from MC simulations. There is an excellent
agreement on the peak values of $|\Delta S_{m}(H)|$. However, for a given $H$
the experimental result show a broad maximum while the MC one show a sharper
peak. As a consequence, the MC RCP is lower than the experimental one.

The overall agreement found in this paper between experiments and the
theoretical model elaborated for the compound
Pr0.9Sr0.1Mn${}_{0.9}^{3+}$Mn${}_{0.1}^{4+}$O3 is remarkable. The multi-spin
interaction introduced here is new, it allows to reproduce the experimental
magnetization plateau below $T_{C}$ and other experimental data. Work is
underway to study the case of other concentrations of Pr-Sr and of Mn3+-Mn4+.

###### Acknowledgements.

Yethreb Essouda is indebted to the CY Cergy Paris University for hospitality
during her working visits.

## References

  * (1) [1] The Future of Cooling. IEA, 2018. doi:https:/ / www.iea.org/reports/the-future-of-cooling.
  * (2) [2] E. A. Goldstein, A. P. Raman, and S. Fan, “Sub-ambient non-evaporative fluid cooling with the sky”, Nature Energy, vol. 2, 2017. doi: 10.1038/nenergy.2017.143.
  * (3) [3] H. S. Laine, J. Salpakari, E. E. Looney, H. Savin, I. M. Peters, and T. Buonassisi, “Meet ing global cooling demand with photovoltaics during the 21st century”, Energy Environ. Sci, vol. 12, pp. 2706–2716, 2019. doi: 10.1039 /c9ee00002j.
  * (4) [4] J. Amaral et al., “Magnetocaloric effect in Er- and Eu-substituted ferromagnetic La-Sr manganites”, J. Magnetism and Magnetic Materials, vol. 290, pp. 686–689, 2005. doi: 10.1016/j. jmmm.2004.11.337.
  * (5) [5] V. Franco, J. S. Blazquez, J. J. Ipus, J. Y. Law, L. M. Moreno-Ramrez, and A. Conde, Prog. Mater. Sci, vol. 93, p. 112, 2018.
  * (6) [6] Z. Weng, N. Haque, G. M. Mudd, and S. M. Jowitt, “Assessing the energy requirements and global warming potential of the production of rare earth elements”, J. Cleaner Production, vol. 139, pp. 1282–1297, 2016. doi: https :/ / doi.org/10.1016/j. jclepro.2016.08.132. [Online]. Available: https : / / www.sciencedirect.com/science/article/pii/ S0959652616312963.
  * (7) [7] K. A. G. Jr, V. K. Pecharsky, and A. O. Tsokol, “Recent developments in magnetocaloric mate rials”, Rep. Prog. Phys, vol. 68, pp. 1479–1539, 2005. doi: 10.1088/0034-4885/68/6/R04.
  * (8) [8] A. Pathak, I. Dubenko, H. Karaca, S. Stadler, and N. Ali, Appl. Phys. Lett., vol. 97, p. 062 505, 2010.
  * (9) [9] V. K. Pecharsky and J. K. A. Gschneidner, “Heat capacity near first order phase transitions and the magnetocaloric effect: An analysis of the errors, and a case study of Gd5(i2Ge2) and dy”, J. Appl. Phys., vol. 86, no. 11, p. 6315, 2005. doi:10.1063/1.371734.
  * (10) [10] E. Warburg, “Magnetische untersuchungen”, J. Ann. Phys., vol. 249, pp. 141–164, 1881. doi: https : / / doi . org / 10 . 1002 / andp . 18812490510.
  * (11) [11] Anders Smith, Who discovered the magnetocaloric effect? Warburg, Weiss, and the connection between magnetism and heat, Eur. Phys. J. H volume 38, 507-517 (2013); DOI: 10.1140/epjh/e2013-40001-9.
  * (12) [12] M.-H. Phan and S.-C. Yu, “Review of the magnetocaloric effect in manganite materials”, J. Magn. Magn. Mater., vol. 308, pp. 325–340, 2007. doi: https : / / doi . org / 10 . 1016 / j . jmmm.2006.07.025.
  * (13) [13] C. L. Zhang, D. H. Wang, Q. Q. Cao, Z. D. Han, H. C. Xuan, and Y. W. Du, “Magnetostructural phase transition and magnetocaloric effect in off-stoichiometric Mn1.9-xN ixGe alloys”, J. Appl. Phys. Lett., vol. 93, p. 122 505, 2008. doi: https : / / doi . org / 10 . 1063 / 1 . 2990649.
  * (14) [14] C. Zhang, D. Wang, Q. Cao, S. Ma, H. Xuan, and Y. Du, J. Phys. D: Appl. Phys., vol. 43, p. 205 003, 2010.
  * (15) [15] T. Samanta, I. Dubenko, A. Quetz, S. Tem- ple, S. Stadler, and N. Ali, J. Appl. Phys. Lett., vol. 100, p. 052 404, 2012.
  * (16) [16] N. T. Trung, L. Zhang, L. Caron, K. H. J. Buschow, and E. Brück, “Giant magnetocaloric effects by tailoring the phase transitions”, J. Appl. Phys. Lett., vol. 96, p. 172 504, 2010. doi: https://doi.org/10.1063/1.3399773.
  * (17) [17] N. Trung, V. Biharie, L. Zhang, L. Caron, K. Buschow, and E. Brück, “From single- to double-first-order magnetic phase transition in magnetocaloric Mn1-xCrxCoGe compounds”, J. Appl. Phys. Lett., vol. 96, p. 162 507, 2010. doi: https://doi.org/10.1063/1.3399774.
  * (18) [18] J. Wang et al., J. Appl. Phys. Lett., vol. 89, p. 262 504, 2006.
  * (19) [19] E. Liu et al., “Stable magneto-structural coupling with tunable magnetoresponsive effects in hexagonal ferromagnets”, J. Nat. Commun., vol. 3, p. 873, 2012. doi: 10.1038/ncomms1868.
  * (20) [20] N. S. Rogado, J. Li, A. W. Sleight, and M. A. Subramanian, “Magnetocapacitance and magnetoresistance near room temperature in a ferromagnetic semiconductor: La2NiMnO6”, J. Adv. Mater., vol. 17, pp. 2225–2227, 2005. doi: https://doi.org/10.1002/adma.200500737.
  * (21) [21] M. Azuma, K. Takata, T. Saito, S. Ishiwata, Y. Shimakawa, and M. Takano, “Designed ferromagnetic, ferroelectric Bi2NiMnO6”, J. Am. Chem. Soc., vol. 127, pp. 8889–8892, 2005. doi: https://doi.org/10.1021/ja0512576.
  * (22) [22] T. S.-D. Hena Das Molly De Raychaudhury, “Moderate to large magneto-optical signals in high Tc double perovskites”, J. Appl Phys Lett., vol. 92, p. 201 912, 2008. doi: https://doi. org/10.1063/1.2936304.
  * (23) [23] R. Kusters, J. Singleton, D. Keen, R. Mc- Greevy, and W. Hayes, “Magnetoresistance measurements on the magnetic semiconductor Nd0.5Pb0.5MnO3”, J. Physica B: Condensed Matter, vol. 155, pp. 362–365, 1989. doi: https://doi.org/10.1016/0921-4526(89) 90530-9.
  * (24) [24] Ziyu Wei, N.A. Liedienov, Quanjun Li , A.V. Pashchenko, Wei Xu, V. A. Turchenko, Mengyun Yuan, I.V. Fesych, G.G. Levchenko, Influence of post-annealing, defect chemistry and high pressure on the magnetocaloric effect of non-stoichiometric La0.8-xK0.2Mn1+xO3 compounds, Ceramics International 47 (2021) 24553–24563.
  * (25) [25] Ziyu Wei, A. V. Pashchenko, N. A. Liedienov, I. V. Zatovsky, D. S. Butenko, Quanjun Li, I. V. Fesych, V. A. Turchenko, E. E. Zubov, P. Yu. Polynchuk, V. G. Pogrebnyak, V. M. Poroshini and G. G. Levchenko, Multifunctionality of lanthanum–strontium manganite nanopowder, Phys. Chem. Chem. Phys., 2020,22, 11817-11828, https://doi.org/10.1039/D0CP01426E.
  * (26) [26] J. B. Goodenough, “Theory of the role of covalence in the perovskite-type manganites [LaM(II)]MnO3”, J. Phys. Rev., vol. 100, pp. 564-573, 1955. doi:10.1103/PhysRev . 100.564. [Online]. Available: https://link. aps.org/doi/10.1103/PhysRev.100.564.
  * (27) [27] M.-H. Phan, S.-C. Yu, and N. H. Hur, “Excellent magnetocaloric properties of La0.7Ca0.3-xSrxMnO3 ($0.05\leq x\leq 0.25$) single crystals”, J. Appl. Phys. Lett., vol. 86, p. 072 504, 2005. doi: https://doi.org/10. 1063/1.1867564.
  * (28) [28] A. Nasri, E. Hlil, A.-F. Lehlooh, M. Ellouze, and F. Elhalouani, “Study of magnetic transition and magnetic entropy changes of Pr0.6Sr0.4MnO3 and Pr0.6Sr0.4Mn0.9Fe0.1O3 compounds”, Eur. Phys. J. Plus, vol. 131, p. 110, 2016. doi: 10 . 1140 / epjp / i2016 - 16110-y.
  * (29) [29] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 (2001).
  * (30) [30] T. Hotta, A. Feiguin, and E. Dagotto, Phys. Rev. Lett 86, 4922 (2001).
  * (31) [31] E. Dagotto, Science 309 (2005) 257, N. Nagaosa and Y. Tokura, Science 288, 462 (2000).
  * (32) [32] M. S. Kim, J. B. Yang, Q. Cai, X. D. Zhou, W. J. James, W. B. Yelon, P. E. Parris, D. Buddhikot and S. K. Malik, Phys. Rev. B 71, 014433 (2005).
  * (33) [33] M. B. Salamon, M. Jaime, The physics of manganites: structure and transport, Rev. Mod. Phys. 73, 583 (2001).
  * (34) [34] X. Zhu, U. Yuping, X. Luo, L. Hechang, W. Bosen, S. Wenhai, Y. Zhaorong, D. Jianming, SH. Dongqi, D. Shixue, J. Magn. Magn. Mater. 322, 242 (2010).
  * (35) [35] R. Von Helmolt, J. Wecker, B. Holzapfel, L. Schultz, K. Samwer, Phys. Rev. Lett. 71, 2331 (1993).
  * (36) [36] G. H. Jonker, J. H. Van Santen, Physica 16, 337 (1950).
  * (37) [37] C. Zener, Phys. Rev. 81, 440 (1951).
  * (38) [38] C. Zener, Phys. Rev. 82, 403 (1951).
  * (39) [39] C. Zener, Phys. Rev. 83, 299 (1951).
  * (40) [40] C. S. Hong, N. H. Hur, Y. N. Choi, Solid State Com. 131, 779 (2004).
  * (41) [41] E. Bose, S. Karmakar, B. K. Chaudhuri, S. Pal, Solid State Com. 145, 149 (2008).
  * (42) [42] N. Furukawa, Y. Motome, Appl. Phys. A, 74, 1728 (2002).
  * (43) [43] J. Fan, L. Pi, L. Zhang, W. Tong, L. S. Ling, B. Hong, Y. G. Shi, W. C. Zhang, D. Lu, Y. H. Zhang, Magnetic and magnetocaloric properties of perovskitemanganite Pr0.55Sr0.45MnO3, Physica B 406, 2289 (2011).
  * (44) [44] A. K. Saw, S. Hunagund, R. L. Hadimani, V. Dayal, Magnetic phase transition, magnetocaloric and magnetotransport properties in Pr0.55Sr0.45MnO3 perovskite manganite, Materials Today: Proceedings, Volume 46, Part 14, pp. 6218-6222 (2021).
  * (45) [45] S. Chaffai, W. Boujelben, M. Ellouze, A. Cheikh-rouhou, J.C. Joubert, A comparative study of the physical properties of Pr0.5Sr0.5MnO3 and Pr0.5Sr0.5MnO3 manganites, Physica B vol. 321, pp. 74-78 (2002).
  * (46) [46] Abir. Nasri, E.K. Hlil, A.-F. Lehlooh, M. Ellouze, and F. Elhalouani, Study of magnetic transition and magnetic entropy changes of Pr0.6Sr0.4MnO3 and Pr0.6Sr0.4Mn0.9Fe0.1O3 compounds, Eur. Phys. J. Plus vol. 131, p. 110 (2016).
  * (47) [47] M. Ellouze, W. Boujelben, A. Cheikhrouhou, H. Fuess, R. Madar, Structure, magnetic and electrical properties in the praseodymium deficient Pr(0.8-x) h(x) Sr(0.2) MnO3 manganites oxides, Journal of Alloys and Compounds vol. 352, pp. 41-47 (2003).
  * (48) [48] B. Cullity, Elements of X-ray Diffraction, 2nd edition. Addison-Wesley, London, 1978.
  * (49) [49] J. Rodriguez-Carvajal, Fullprof98. Laboratoire Leon Briouillon, 1998.
  * (50) [50] H. T. Diep (Editor), Frustrated Spin Systems, 3rd Edition, World Scientific (2020).
  * (51) [51] H. T. Diep, Theory of Magnetism: Application to Surface Physics, World Scientific, Singapore (2014).
  * (52) [52] R. Singer, F. Dietermann, and M. Fähnle, Spin interactions in bcc and fcc fe beyond the Heisenberg model, Phys. Rev. Lett., 107:017204, Jun 2011.
  * (53) [53] L. Turban, One-dimensional Ising model with multispin interactions, J. Phys. A: Math. Theor. 49 (2016) 355002 (16pp), doi:10.1088/1751-8113/49/35/355002.
  * (54) [54] Thomas R. Bergamaschi, Tim Menke, William P. Banner, Agustin Di Paolo, Steven J. Weber, Cyrus F. Hirjibehedin, Andrew J. Kerman, and William D. Oliver, Distinguishing multi-spin interactions from lower-order effects, Phys. Rev. Applied 18 (2022) 4, 044018.
  * (55) [55] Francisco C. Alcaraz, Rodrigo A. Pimenta, and Jesko Sirker, Ising analogs of quantum spin chains with multispin interactions, Phys. Rev. B 107, 235136 – Published 20 June 2023.
  * (56) [56] N. Metropolis, A. W. Rosenbluth, M, N, Rosenbluth, and A. H. Teller, J. Chem. Phys. 21, 1087 (1953).
  * (57) [57] V.-Thanh Ngo, D.-Tien Hoang, H. T. Diep and I. A. Campbell, Effect of Disorder in the Frustrated Ising FCC Antiferromagnet: Phase Diagram and Stretched Exponential Relaxation, Modern Phys. Lett. B 28, 1450067 (2014).
  * (58) [58] S. F. Edwards and P. Anderson, J. Phys. F 5, 965 (1975).
  * (59) [59] K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986).
  * (60) [60] M. Mézard, G. Parisi and M. Virasoro, Spin Glass Theory and Beyond, World Scientific, New Jersey (1987).
  * (61) [61] See Y. Magnin and H. T. Diep, Monte Carlo Study of Magnetic Resistivity in Semiconducting MnTe, Phys. Rev. B 85, 184413 (2012) and experimental works cited therein.
  * (62) [62] Samia Yahyaoui, Sami Kallel, H. T. Diep, Magnetic properties of perovskites La0.7Sr0.3Mn0.7(3+)Mn(0.3-x)(4+)Ti(x)O3: Monte Carlo simulation versus experiments, Journal of Magnetism and Magnetic Materials 416, 441–448 (2016).
  * (63) [63] H. T. Diep, Statistical Physics: Fundamentals and Application to Condensed Matter, chapter 13: Interacting Spin Systems: Phase Transitions, World Scientific (2015).
  * (64) [64] H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Reviews of Modern Physics, 71, No. 2 (1999).
  * (65) [65] Nikita A. Liedienov et al., Spin-dependent magnetism and superparamagnetic contribution to the magnetocaloric effect of non-stoichiometric manganite nanoparticles, Applied Materials Today 26, 101340 (2022).
  * (66)