arXiv subjects

A. Chainani

Publications and source records attributed to A. Chainani.

At least 19 recordsLinked to original sources

arXiv2026

Kondo scaling of $4f$-electron states and the Kondo singlet breakdown in heavy fermions

The low-energy spin- and charge-sensitive thermodynamic properties of a broad range of strongly correlated 4f-electron systems follow Kondo scaling, with a characteristic Kondo temperature, $T_K$. While the theory is known for thermodynamic properties and high-energy spectroscopies of Kondo materials, the surface sensitivity of electron spectroscopy limits the extent to which Kondo scaling can be quantitatively verified. In this study, bulk-sensitive photon-in photon-out temperature-dependent resonant inelastic X-ray scattering (RIXS), in combination with single-impurity Anderson model (SIAM) calculations, is used to provide quantitative evidence of low- and high-energy Kondo scaling in CeSi$_2$. RIXS Ce M$_5$-edge spectra show a clear decrease in the occupancy of the $f^0$ state as temperature increases accompanied by an increase of the spectral weight of the $f^1\underline L^1$ state, in good agreement with the SIAM calculations. The results demonstrate the breakdown of the Kondo singlet state, coupled with thermal occupation of the low-lying first-excited magnetic states. The RIXS data reveal a temperature evolution of the $f^n$ spectral weights, which is in stark contrast to that extracted from photoemission and inverse photoemission spectroscopies. This study provides an accurate spectroscopic method to determine the Kondo energy $k_B$$T_K$ that is consistent with thermodynamic measurements, and highlights soft X-ray RIXS as a quantitative bulk probe of low- and high-energy-scale hybridization effects in strongly correlated materials.

cond-mat.str-el↗

arXiv2026

Anomalous pressure dependence of the bulk modulus and Yb valence in cubic YbPd

We investigate the Yb valence instabilities in the strongly correlated YbPd compound using resonant X-ray emission spectroscopy as a function of pressure across the charge-order (CO) transition. At a low temperature (T = 30 K) in the CO phase, the Yb $4f$ valence remains nearly constant up to a pressure P$_L$ = 1.5 $\pm {0.2}$ GPa, and then increases gradually at higher pressures. In contrast, at room temperature in the normal phase, an anomalous decrease of the Yb $4f$ valence is observed, without any accompanying structural phase transition. This behavior is corroborated by a systematic pressure-dependent decrease of the unit-cell volume. Based on a Birch-Murnaghan analysis, the compressibility indicates hardening of the lattice with applied pressure up to a distinct kink seen at P$_K$ = 1.6 $\pm {0.2}$ GPa. In contrast, for P $>$ P$_K$, the Yb $4f$ valence saturates and the compressibility reveals a counterintuitive pressure-induced softening. The results show a minimum in the compressibility of YbPd (with $f^{0}$-$f^{1}$ hole-type mixed-valence), reminiscent of the maximum in compressibility seen in the $\gamma$-$\alpha$ first-order isostructural phase transition in cerium (with $f^{0}$-$f^{1}$ electron-type mixed-valence).

cond-mat.str-el↗

arXiv2025

The nexus between negative charge-transfer and reduced on-site Coulomb energy in a correlated topological metal CoTe$_2$

The layered $3d$ transition metal dichalcogenide (TMD) CoTe$_2$ is a topological Dirac Type-II metal. However, the Co $3d$-bands in CoTe$_2$ do not exhibit the expected correlation-induced band narrowing seen in CoO. We address this conundrum by studying the electronic structure of CoTe$_2$ using hard x-ray photoemission spectroscopy (HAXPES), x-ray absorption spectroscopy (XAS) and Resonant-PES. We quantify the on-site Coulomb energy $U_{dd}$ via single-particle partial density of states and the two-hole correlation satellite using valence band Resonant-PES), and obtain $U_{dd}$ = 3.0 eV for CoTe$_2$. Charge-transfer (CT) cluster model simulations of the measured core-level Co $2p$ PES and $L$-edge XAS spectra of CoTe\textsubscript{2} and CoO validate their contrasting electronic parameters:$U_{dd}$ and CT energy $\Delta$ are (3.0 eV, -2.0 eV) for CoTe\textsubscript{2}, and (5.0 eV, 4.0 eV) for CoO, respectively. The $d$-$p$ hybridization strength $T_{eg}$ for CoTe$_2$$<$CoO, and indicates that the reduced $U_{dd}$ in CoTe\textsubscript{2} is not due to $T_{eg}$. The increase in $d^n$-count$\sim$1 by CT from ligand to Co site in CoTe$_2$ is due to a negative-$\Delta$ and reduced $U_{dd}$. Yet, only because $U_{dd}$$>$$\big|\Delta\big|$, CoTe$_{2}$ becomes a topological metal with $p$$\rightarrow$$p$ type lowest energy excitations. The study reveals the nexus between negative-$\Delta$ and reduced $U_{dd}$ required for setting up the electronic structure framework for achieving topological behavior via band inversion in the correlated metal CoTe$_2$.

cond-mat.str-el↗

arXiv2025

Role of on-site Coulomb energy and negative-charge transfer in a Dirac semi-metal NiTe$_2$

Angle-resolved photoemission spectroscopy (ARPES) combined with band structure calculations have shown that the layered transition metal dichalcogenide(TMD) NiTe$_2$ is a type-II Dirac semimetal. However, conflicting conclusions were reported regarding the role of electron correlations in NiTe$_2$. We study core-levels and valence band electronic structure of single crystal NiTe$_2$ using soft and hard x-ray photoemission spectroscopy(SXPES, HAXPES), X-ray absorption spectroscopy(XAS) and Ni $2p-3d$ Resonant-PES to quantify electronic parameters in NiTe$_2$. The Ni $3d$ on-site Coulomb energy ($U_{dd}$) is quantified from measurements of the Ni $3d$ single particle density of states(DOS) and the two-hole correlation satellite. The Ni $2p$ core level and $L$-edge XAS spectra are analyzed by charge-transfer (CT) cluster model calculations using the experimental $U_{dd}$, and it shows that NiTe$_2$ exhibits a negative CT energy $\Delta$. A comparative analysis of NiO $L$-edge XAS confirms its well-known strongly correlated CT insulator character, with a larger $U_{dd}$ and positive $\Delta$. The $d$-$p$ hybridization strength $T_{eg}$ for NiTe$_2$$<$NiO, and shows that $T_{eg}$ is not responsible for reducing $U_{dd}$ in NiTe\textsubscript{2} compared to NiO. The negative-$\Delta$ and a reduced $U_{dd}$ leads to the increase in $d^n$ count on the Ni site in NiTe$_{2}$ by nearly one electron. However, importantly, since $U_{dd}$$>$$|\Delta|$, a finite repulsive $U_{dd}$ results in pushing $d$-states away from Fermi level and this is required to make NiTe$_{2}$ a moderately correlated Dirac semi-metal with band inversion in the $p$-$p$ type lowest energy excitations.

cond-mat.str-el↗

arXiv2023

Unusual band evolution and persistence of topological surface states in high-T_C magnetic topological insulator

Understanding the mechanism of ferromagnetism in ferromagnetic topological insulators (TIs) is a key to realize exotic time-reversal-symmetry-broken quantum phases. However, electronic states relevant to the ferromagnetism are highly controversial. Here we report angle-resolved photoemission spectroscopy on (CrxSb1-x)2Te3 thin films, high-Curie-temperature (T_C) ferromagnetic TIs, spanning the non-doped (T_C=0 K) to highly-doped (T_C=192 K) region. We found that, upon Cr doping to Sb2Te3, the bulk valence-band valley exhibits filling-in behavior while retaining band inversion, leading to the formation of a nearly-flat band in high-T_C regime and evolution from a six-petal flower to a Star-of-David Fermi surface. Despite the weakening of spin-orbit coupling with Cr doping, the Dirac-cone state persists up to the highest-T_C sample, and shows a clear magnetic-gap opening below TC accompanied with an unexpected band shift, signifying its strong coupling with spontaneous ferromagnetism. The present result lays the foundation for understanding the interplay between band topology and ferromagnetism in TIs.

cond-mat.mes-hall↗

arXiv2023

Oxygen on-site Coulomb energy in Pr$_{1.3-x}$La$_{0.7}$Ce$_x$CuO$_{4}$ and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and its relation with Heisenberg exchange

We study the electronic structure of electron-doped Pr$_{1.3-x}$La$_{0.7}$Ce$_{x}$CuO$_{4}$ (PLCCO ; $T_{c}$ = 27 K, x = 0.1) and hole-doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ (Bi2212 ; $T_{c}$ = 90 K) cuprate superconductors using x-ray absorption spectroscopy (XAS) and resonant photoemission spectroscopy (Res-PES). From Res-PES across the O K-edge and Cu L-edge, we identify the O 2p and Cu 3d partial density of states (PDOS) and their correlation satellites which originate in two-hole Auger final states. Using the Cini-Sawatzky method, analysis of the experimental O 2p PDOS shows an oxygen on-site Coulomb energy for PLCCO to be $U_{p}$ = 3.3$\pm$0.5 eV and for Bi2212, $U_{p}$ = 5.6$\pm$0.5 eV, while the copper on-site Coulomb correlation energy, $U_{d}$ = 6.5$\pm$0.5 eV for Bi2212. The expression for the Heisenberg exchange interaction $J$ in terms of the electronic parameters $U_{d}$, $U_{p}$, charge-transfer energy $\Delta$ and Cu-O hopping $t_{pd}$ obtained from a simple Cu$_2$O cluster model is used to carry out an optimization analysis consistent with $J$ known from scattering experiments. The analysis also provides the effective one band on-site Coulomb correlation energy $\tilde{U}$ and the effective hopping $\tilde{t}$. PLCCO and Bi2212 are shown to exhibit very similar values of $\tilde{U}$/$\tilde{t}$ $\sim$9-10, confirming the strongly correlated nature of the singlet ground state in the effective one-band model for both the materials.

cond-mat.str-el↗

arXiv2022

Connecting the one-band and three-band Hubbard models of cuprates via spectroscopy and scattering experiments

The one-band and three-band Hubbard models which describe the electronic structure of cuprates indicate very different values of effective electronic parameters, such as the on-site Coulomb energy and the hybridization strength. In contrast, a comparison of electronic parameters of several cuprates with corresponding values from spectroscopy and scattering experiments indicates similar values in the three-band model and cluster model calculations used to simulate experimental results. The Heisenberg exchange coupling $J$ obtained by a downfolding method in terms of the three band parameters is used to carry out an optimization analysis consistent with $J$ from neutron scattering experiments for a series of cuprates. In addition, the effective one-band parameters $\tilde{U}$ and $\tilde{t}$ are described using the three band parameters, thus revealing the hidden equivalence of the one-band and three-band models. The ground-state singlet weights obtained from an exact diagonalization elucidates the role of Zhang-Rice singlets in the equivalence. The results provide a consistent method to connect electronic parameters obtained from spectroscopy and the three-band model with values of $J$ obtained from scattering experiments, band dispersion measurements and the effective one-band Hubbard model.

cond-mat.str-el↗

arXiv2021

Unconventional Superfluidity in a model of Fermi-Bose Mixtures

A finite-temperature ($T>0$) study of a model of a mixture of spin-zero hardcore bosons and spinless fermions, with filling fractions $\rho_B$ and $\rho_F$, respectively, on a two-dimensional square lattice with composite hopping $t$ is presented. The composite hopping swaps the locations of a fermion and a boson that occupy nearest-neighbor sites of the lattice. The superfluid order parameter $\psi$, the femion hopping amplitude $\phi$, the chemical potential $\mu$, the free energy minimum $\tilde{F}$ and entropy $S$ are calculated in the limit $\rho_B+\rho_F=1$ within a mean-field approximation, and lead to a phase diagram in the $\rho_F - T$ plane. This phase diagram consists of a metallic superfluid phase under a dome-shaped $T(\rho_F)$, and insulating normal liquid and insulating normal gas phases outside the dome. These phases are separated by coupled discontinuous transitions as indicated by jumps in $\psi$ and $\phi$. The maximum critical transition temperature $T_c$ is observed very close to $\rho_F = 1/2$. While $\tilde{F} (T)$ is continuous with a derivative discontinuity at $T=T_c (\rho_F)$ for $0 <\rho_F \le 1/2$ (first-order transition), it becomes {\em discontinuous} for $\rho_F>1/2$ (zeroth-order transition), where the entropy becomes negative for a range of temperatures below $T_c$. The ratio of $T_c$ to Fermi band width agrees remarkably with the ratio of $T_c$/$T_F$ (where $T_F$ is the Fermi temperature) of unconventional superfluids and superconductors like Fermi-Bose mixtures, the high-$T_c$ cuprates, iron-based and hydride superconductors, that exhibit experimental values of $T_c$ spread over nine orders of magnitude from $\sim 200$nK to $\sim 260$K.

cond-mat.mtrl-sci↗

arXiv2020

Evolution of valence-specific spin states and local distortions in La$_{2-x}$Sr$_x$CoO$_4$

We present X-ray spectroscopic evidence for the evolution of valence-specific spin states and tetragonal distortions in single-layer cobaltates. Measurements of Co $L_3$-edge resonant inelastic X-ray scattering reveal the $t_{2g}$ electronic structure of Co for hole-doped La$_{2-x}$Sr$_x$CoO$_4$ ($x$ = 0.5, 0.7 and 0.8). As the Sr-doping $x$ increases, the tetragonal splitting of the $t_{2g}$ states of high-spin Co$^{2+}$ decreases, whereas that of low-spin Co$^{3+}$ increases and the fraction of high-spin Co$^{3+}$ increases. The results enable us to clarify the origin of the change of magnetic anisotropy and in-plane resistivity in a mixed-valence cobaltate caused by the interplay of spin-orbit coupling and tetragonal distortion.

cond-mat.str-el↗

arXiv2019

Coupled First-Order Transitions In A Fermi-Bose Mixture

A model of a mixture of spinless fermions and spin-zero hardcore bosons, with filling fractions $\rho_F$ and $\rho_B$, respectively, on a two-dimensional square lattice with {\em composite} hopping $t$ is presented. In this model, hopping swaps the locations of a fermion and a boson at nearest-neighbor sites. When $\rho_F+\rho_B=1$, the fermion hopping amplitude $\phi$ and boson superfluid amplitude $\psi$ are calculated in the ground state within a mean-field approximation. The Fermi sector is insulating ($\phi=0$) and the Bose sector is normal ($\psi=0$) for $0 \le \rho_F < \rho_c$. The model has {\em coupled first-order} transitions at $\rho_F = \rho_c \simeq 0.3$ where both $\phi$ and $\psi$ are discontinuous. The Fermi sector is metallic ($\phi>0$) and the Bose sector is superfluid ($\psi>0$) for $\rho_c < \rho_F < 1$. At $\rho_F=1/2$, fermion density of states $\rho$ has a van Hove singularity, the bulk modulus $\kappa$ displays a cusp-like singularity, the system has a density wave (DW) order, and $\phi$ and $\psi$ are maximum. At $\rho_F=\rho_{\kappa} \simeq 0.81$, $\kappa$ vanishes, becoming {\em negative} for $\rho_{\kappa}<\rho_F<1$. The role of composite hopping in the evolution of Fermi band dispersions and Fermi surfaces as a function of $\rho_F$ is highlighted. The estimate for BEC critical temperature is in the subkelvin range for ultracold atom systems and several hundred kelvins for possible solid-state examples of the model.

cond-mat.str-el↗

arXiv2019

Electronic structure investigation of GdNi using X-ray absorption, magnetic circular dichroism and hard x-ray photoemission spectroscopy

GdNi is a ferrimagnetic material with a Curie temperature Tc = 69 K which exhibits a large magnetocaloric effect, making it useful for magnetic refrigerator applications. We investigate the electronic structure of GdNi by carrying out x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) at T = 25 K in the ferrimagnetic phase. We analyze the Gd M$_{4,5}$-edge ($3d$ - $4f$) and Ni L$_{2,3}$-edge ($2p$ - $3d$) spectra using atomic multiplet and cluster model calculations, respectively. The atomic multiplet calculation for Gd M$_{4,5}$-edge XAS indicates that Gd is trivalent in GdNi, consistent with localized $4f$ states. On the other hand, a model cluster calculation for Ni L$_{2,3}$-edge XAS shows that Ni is effectively divalent in GdNi and strongly hybridized with nearest neighbour Gd states, resulting in a $d$-electron count of 8.57. The Gd M$_{4,5}$-edge XMCD spectrum is consistent with a ground state configuration of S = 7/2 and L=0. The Ni L$_{2,3}$-edge XMCD results indicate that the antiferromagnetically aligned Ni moments exhibit a small but finite magnetic moment ( $m_{tot}$ $\sim$ 0.12 $\mu_B$ ) with the ratio $m_{o}/m_{s}$ $\sim$ 0.11. Valence band hard x-ray photoemission spectroscopy shows Ni $3d$ features at the Fermi level, confirming a partially filled $3d$ band, while the Gd $4f$ states are at high binding energies away from the Fermi level. The results indicate that the Ni $3d$ band is not fully occupied and contradicts the charge-transfer model for rare-earth based alloys. The obtained electronic parameters indicate that GdNi is a strongly correlated charge transfer metal with the Ni on-site Coulomb energy being much larger than the effective charge-transfer energy between the Ni $3d$ and Gd $4f$ states.

cond-mat.str-el↗

arXiv2019

Dimensionality reduction and band quantization induced by potassium intercalation in 1$T$-HfTe$_2$

We have performed angle-resolved photoemission spectroscopy on transition-metal dichalcogenide 1$T$-HfTe$_2$ to elucidate the evolution of electronic states upon potassium (K) deposition. In pristine HfTe$_2$, an in-plane hole pocket and electron pockets are observed at the Brillouin-zone center and corner, respectively, indicating the semimetallic nature of bulk HfTe$_2$, with dispersion perpendicular to the plane. In contrast, the band structure of heavily K-dosed HfTe$_2$ is obviously different from that of bulk, and resembles the band structure calculated for monolayer HfTe$_2$. It was also observed that lightly K-dosed HfTe$_2$ is characterized by quantized bands originating from bilayer and trilayer HfTe$_2$, indicative of staging. The results suggest that the dimensionality-crossover from 3D (dimensional) to 2D electronic states due to systematic K intercalation takes place via staging in a single sample. The study provides a new strategy for controlling the dimensionality and functionality of novel quantum materials.

cond-mat.mtrl-sci↗

arXiv2018

Band structure and Fermi surfaces of the reentrant ferromagnetic superconductor Eu(Fe0.86Ir0.14)2As2

The electronic structure of the reentrant superconductor Eu(Fe$_{0.86}$Ir$_{0.14}$)$_{2}$As$_{2}$ (T$_c$ = 22 K) with coexisting ferromagnetic order (T$_M$ = 18 K) is investigated using angle-resolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). We study the in-plane and out-of-plane band dispersions and Fermi surface (FS) of Eu(Fe$_{0.86}$Ir$_{0.14}$)$_{2}$As$_{2}$. The near E$_F$ Fe 3d-derived band dispersions near the $\Gamma$ and X high-symmetry points show changes due to Ir substitution, but the FS topology is preserved. From momentum dependent measurements of the superconducting gap measured at T = 5 K, we estimate an essentially isotropic s-wave gap ($\Delta\sim5.25\pm 0.25$ meV), indicative of strong-coupling superconductivity with 2$\Delta$/k$_{B}$T$_{c}\simeq$ 5.8. The gap gets closed at temperatures T $\geq$ 10 K, and this is attributed to the resistive phase which sets in at T$_M$ = 18 K due to the Eu$^{2+}$-derived magnetic order. The modifications of the FS with Ir substitution clearly indicates an effective hole doping with respect to the parent compound.

cond-mat.mtrl-sci↗

arXiv2017

Coulomb correlations intertwined with spin and orbital excitations in LaCoO$_3$

We carried out temperature-dependent (20 - 550 K) measurements of resonant inelastic X-ray scattering on LaCoO$_3$ to investigate the evolution of its electronic structure across the spin-state crossover. In combination with charge-transfer multiplet calculations, we accurately quantized the renormalized crystal-field excitation energies and spin-state populations. We show that the screening of the on-site Coulomb interaction of 3d electrons is orbital selective and coupled to the spin-state crossover in LaCoO$_3$. The results establish that the gradual spin-state crossover is associated with a relative change of Coulomb energy versus bandwidth, leading to a Mott-type insulator-to-metal transition.

cond-mat.str-el↗

arXiv2016

Evidence for weakly correlated oxygen holes in the highest-T$_{c}$ cuprate superconductor HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$

We study the electronic structure of HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$ (Hg1223 ; T$_{c}$ = 134 K) using photoemission spectroscopy (PES) and x-ray absorption spectroscopy (XAS). Resonant valence band PES across the O K-edge and Cu L-edge identify correlation satellites originating in O 2p and Cu 3d two-hole final states, respectively. Analyses using the experimental O 2p and Cu 3d partial density of states show quantitatively different on-site Coulomb energy for the Cu-site (U$_{dd}$ = 6.5$\pm$0.5 eV) and O-site (U$_{pp}$ = 1.0$\pm$0.5 eV). Cu$_{2}$O$_{7}$-cluster calculations with non-local screening explain the Cu 2p core level PES and Cu L-edge XAS spectra, confirm the U$_{dd}$ and U$_{pp}$ values, and provide evidence for the Zhang-Rice singlet state in Hg1223. In contrast to other hole-doped cuprates and 3d-transition metal oxides, the present results indicate weakly correlated oxygen holes in Hg1223.

cond-mat.str-el↗

arXiv2015

Jahn-Teller distortion driven magnetic polarons in magnetite

The first known magnetic mineral, magnetite (Fe$_3$O$_4$), has unusual properties which have fascinated mankind for centuries; it undergoes the Verwey transition at $T_{\rm V}$ $\sim$120 K with an abrupt change in structure and electrical conductivity. The mechanism of the Verwey transition however remains contentious. Here we use resonant inelastic X-ray scattering (RIXS) over a wide temperature range across the Verwey transition to identify and separate out the magnetic excitations derived from nominal Fe$^{2+}$ and Fe$^{3+}$ states. Comparison of the RIXS results with crystal-field multiplet calculations shows that the spin-orbital $dd$ excitons of the Fe$^{2+}$ sites arise from a tetragonal Jahn-Teller active polaronic distortion of the Fe$^{2+}$O$_6$ octahedra. These low-energy excitations, which get weakened for temperatures above 350 K but persist at least up to 550 K, are distinct from optical excitations and best explained as magnetic polarons.

cond-mat.str-el↗

arXiv2015

Temperature-Dependence of Magnetically-Active Charge Excitations in Magnetite across the Verwey Transition

We have studied the electronic structure of bulk single crystals and epitaxial films of magnetite Fe$_3$O$_4$. Fe $2p$ core-level spectra show clear differences between hard x-ray (HAX-) and soft x-ray (SX-) photoemission spectroscopy (PES), indicative of surface effects. The bulk-sensitive spectra exhibit temperature ($T$)-dependent charge excitations across the Verwey transition at $T_V$=122 K, which is missing in the surface-sensitive spectra. An extended impurity Anderson model full-multiplet analysis reveals roles of the three distinct Fe-species (A-Fe$^{3+}$, B-Fe$^{2+}$, B-Fe$^{3+}$) below $T_V$ for the Fe $2p$ spectra, and its $T-$dependent evolution. The Fe $2p$ HAXPES spectra show a clear magnetic circular dichroism (MCD) in the metallic phase of magnetized 100-nm-thick films. The model calculations also reproduce the MCD and identify the magnetically distinct sites associated with the charge excitations. Valence band HAXPES shows finite density of states at $E_F$ for the polaronic metal with remnant order above $T_V$, and a clear gap formation below $T_V$. The results indicate that the Verwey transition is driven by changes in the strongly correlated and magnetically active B-Fe$^{2+}$ and B-Fe$^{3+}$ electronic states, consistent with resistivity and bulk-sensitive optical spectra.

cond-mat.str-el↗

arXiv2014

Frustration effects and role of selective exchange coupling for magnetic ordering in the Cairo pentagonal lattice

The Cairo pentagonal lattice, consisting of an irregular pentagonal tiling of magnetic ions on two inequivalent sites (3- and 4-co-ordinated ones), represents a fascinating example for studying geometric frustration effects in two-dimensions. In this work, we investigate the spin $S$ = $1/2$ Cairo pentagonal lattice with respect to selective exchange coupling (which effectively corresponds to a virtual doping of $x$ = $0, 1/6, 1/3$), in a nearest-neighbour antiferromagnetic Ising model. We also develop a simple method to quantify geometric frustration in terms of a frustration index $\phi(\beta,T)$, where $\beta$ = $J/\tilde{J}$, the ratio of the two exchange couplings required by the symmetry of the Cairo lattice. At $T = 0$, the undoped Cairo pentagonal lattice shows antiferromagnetic ordering for $\beta \le \beta_{crit} = 2$, but undergoes a first-order transition to a ferrimagnetic phase for $\beta >$ $\beta_{crit}$. The results show that $\phi(\beta,T = 0)$ tracks the transition in the form of a cusp maximum at $\beta_{crit}$. While both phases show frustration, the obtained magnetic structures reveal that the frustration originates in different bonds for the two phases. The frustration and ferrimagnetic order get quenched by selective exchange coupling, and lead to robust antiferromagnetic ordering for $x$ = 1/6 and 1/3. From mean-field calculations, we determine the temperature-dependent sub-lattice magnetizations for $x$ = $0, 1/6$ and $1/3$. The calculated results are discussed in relation to known experimental results for trivalent Bi$_2$Fe$_4$O$_9$ and mixed valent BiFe$_2$O$_{4.63}$. The study identifies the role of frustration effects, the ratio $\beta$ and selective exchange coupling for stabilizing ferrimagnetic versus anti-ferromagnetic order in the Cairo pentagonal lattice.

cond-mat.mtrl-sci↗