Li diffusion in LixCoO2 probed by muon-spin spectroscopy

Path ahead: Tackling the Challenge of Computationally Estimating Lithium Diffusion in Cathode Materials

In the roadmap toward designing new and improved materials for Lithium ion batteries, the ability to estimate the diffusion coefficient of Li atoms in electrodes, and eventually solid-state electrolytes, is key. Nevertheless, as of today, accurate prediction through computational tools remains challenging. Its experimental measurement does not appear to be much easier. In this work, we devise a computational protocol for the determination of the Li-migration energy barrier and diffusion coefficient, focusing on a common cathode material such as LiNiO2, which represents a prototype of the widely adopted NMC (LiNi1-x-y Mn x Co y O2) class of materials. Different methodologies are exploited, combining ab initio metadynamics, path sampling, and density functional theory. Furthermore, we propose a novel, fast, and simple 1D approximation for the estimation of the effective frequency. The outlined computational protocol aims to be generally applicable to Lithium diffusion in other materials and components for batteries, including anodes and solid electrolytes.

First-Principles Study on the Interplay of Strain and State-of-Charge with Li-Ion Diffusion in the Battery Cathode Material LiCoO2

Cathode degradation of Li-ion batteries (Li+) continues to be a crucial issue for higher energy density. A main cause of this degradation is strain due to stress induced by structural changes according to the state-of-charge (SOC). Moreover, in solid-state batteries, a mismatch between incompatible cathode/electrolyte interfaces also generates a strain effect. In this respect, understanding the effects of the mechanical/elastic phenomena associated with SOC on the cathode performance, such as voltage and Li+ diffusion, is essential. In this work, we focused on LiCoO2 (LCO), a representative LIB cathode material, and investigated the effects of biaxial strain and hydrostatic pressure on its layered structure and Li+ transport properties through first-principles calculations. With the nudged elastic band technique and molecular dynamics, we demonstrated that in Li-deficient LCO, compressive biaxial strain increases the Li+ diffusivity, whereas tensile biaxial strain and hydrostatic pressure tend to suppress it. Structural parameter analysis revealed the key correlation of "Co layer distances" with Li+ diffusion instead of "Li layer distances", as ordinarily expected. Structural analysis further revealed the interplay between the Li-Li Coulomb interaction, SOC, and Li+ diffusion in LCO. The activation volume of LCO under hydrostatic pressure was reported for the first time. Moreover, vacancy formation energy calculations showed that the Li intercalation potential could be decreased under compressive biaxial strain due to the weakening of the Li-O bond interaction. The present findings may serve to improve the control of the energy density performance of layered cathode materials.

Elucidating local diffusion dynamics in nickel-rich layered oxide cathodes

Elucidating Li-ion transport properties is essential for designing suitable methodologies to optimise electrochemical performance in Ni-rich cathodes for high energy density Li-ion batteries. Here, we report the local-scale Li-diffusion characteristics of a series of nickel-rich layered oxide cathodes, prepared via microwave methods, using muon spin relaxation methods. Our results detail the effects of cation dopants, selected for structure stability, on transport properties in candidate nickel-rich chemistries. We find that the local diffusion properties improve with increasing nickel content. Our results demonstrate that these observations are dependant on substitutional effects.

Direct Observation of Dynamic Lithium Diffusion Behavior in Nickel-Rich, LiNi0.8Mn0.1Co0.1O2 (NMC811) Cathodes Using Operando Muon Spectroscopy

Ni-rich layered oxide cathode materials such as LiNi_0.8Mn_0.1Co_0.1O₂ (NMC811) are widely tipped as the next-generation cathodes for lithium-ion batteries. The NMC class offers high capacities but suffers an irreversible first cycle capacity loss, a result of slow Li⁺ diffusion kinetics at a low state of charge. Understanding the origin of these kinetic hindrances to Li⁺ mobility inside the cathode is vital to negate the first cycle capacity loss in future materials design. Here, we report on the development of operando muon spectroscopy (μSR) to probe the Å-length scale Li⁺ ion diffusion in NMC811 during its first cycle and how this can be compared to electrochemical impedance spectroscopy (EIS) and the galvanostatic intermittent titration technique (GITT). Volume-averaged muon implantation enables measurements that are largely unaffected by interface/surface effects, thus providing a specific characterization of the fundamental bulk properties to complement surface-dominated electrochemical methods. First cycle measurements show that the bulk Li⁺ mobility is less affected than the surface Li⁺ mobility at full depth of discharge, indicating that sluggish surface diffusion is the likely cause of first cycle irreversible capacity loss. Additionally, we demonstrate that trends in the nuclear field distribution width of the implanted muons during cycling correlate with those observed in differential capacity, suggesting the sensitivity of this μSR parameter to structural changes during cycling.

Current Understanding of Water Properties inside Carbon Nanotubes

Confined water inside carbon nanotubes (CNTs) has attracted a lot of attention in recent years, amassing as a result a very large number of dedicated studies, both theoretical and experimental. This exceptional scientific interest can be understood in terms of the exotic properties of nanoconfined water, as well as the vast array of possible applications of CNTs in a wide range of fields stretching from geology to medicine and biology. This review presents an overreaching narrative of the properties of water in CNTs, based mostly on results from systematic nuclear magnetic resonance (NMR) and molecular dynamics (MD) studies, which together allow the untangling and explanation of many seemingly contradictory results present in the literature. Further, we identify still-debatable issues and open problems, as well as avenues for future studies, both theoretical and experimental.

Na Diffusion in Hard Carbon Studied with Positive Muon Spin Rotation and Relaxation

The diffusive nature of Na⁺ in Na-inserted hard carbon (C _x Na), which is the most common anode material for a Na-ion battery, was studied with a positive muon spin rotation and relaxation (μ⁺SR) technique in transverse, zero, and longitudinal magnetic fields (TF, ZF, and LF) at temperatures between 50 and 375 K, where TF (LF) denotes the applied magnetic field perpendicular (parallel) to the initial muon spin polarization. At temperatures above 150 K, TF-μ⁺SR measurements showed a distinct motional narrowing behavior, implying that Na⁺ begins to diffuse above 150 K. The presence of two different muon sites in C _x Na was confirmed with ZF- and LF-μ⁺SR measurements; one is in the Na-inserted graphene layer, and the other is in the Na-vacant graphene layer adjacent to the Na-inserted graphene layer. A systematic increase in the field fluctuation rate (ν) with increasing temperature also evidenced a thermally activated Na diffusion, particularly above 150 K. Assuming the two-dimensional diffusion of Na⁺ in the graphene layers, the self-diffusion coefficient of Na⁺ (D _Na ^J) at 300 K was estimated to be 2.5 × 10^-11 cm²/s with a thermal activation energy of 39(7) meV.

Na-ion mobility in P2-type Na0.5MgxNi0.17-xMn0.83O2 (0 ≤ x ≤ 0.07) from electrochemical and muon spin relaxation studies

Sodium transition metal oxides with a layered structure are one of the most widely studied cathode materials for Na⁺-ion batteries. Since the mobility of Na⁺ in such cathode materials is a key factor that governs the performance of material, electrochemical and muon spin rotation and relaxation techniques are here used to reveal the Na⁺-ion mobility in a P2-type Na_0.5Mg_xNi_0.17-xMn_0.83O₂ (x = 0, 0.02, 0.05 and 0.07) cathode material. Combining electrochemical techniques such as galvanostatic cycling, cyclic voltammetry, and the galvanostatic intermittent titration technique with μ⁺SR, we have successfully extracted both self-diffusion and chemical-diffusion under a potential gradient, which are essential to understand the electrode material from an atomic-scale viewpoint. The results indicate that a small amount of Mg substitution has strong effects on the cycling performance and the Na⁺ mobility. Amongst the tested cathode systems, it was found that the composition with a Mg content of x = 0.02 resulted in the best cycling stability and highest Na⁺ mobility based on electrochemical and μ⁺SR results. The current study clearly shows that for developing a new generation of sustainable energy-storage devices, it is crucial to study and understand both the structure as well as dynamics of ions in the material on an atomic level.

How Li diffusion in spinel Li[Ni1/2Mn3/2]O4 is seen with μ ±SR

The diffusive behavior in a spinel-type Li+ ion battery material, Li[Ni1/2Mn3/2]O4, has been studied with positive and negative muon spin rotation and relaxation (μ ±SR) measurements in the temperature range between 200 and 400 K using a powder sample. The implanted μ + locates at an interstitial site near O2− ion so as to form a O–H like bond, while the implanted μ − is mainly captured by an oxygen nucleus, resulting in the formation of muonic oxygen. This means that local magnetic environments in Li[Ni1/2Mn3/2]O4 were investigated from the two different sites in the lattice, i.e., one is an interstitial site for μ +SR and the other is an oxygen site for μ −SR. Since both μ +SR and μ −SR detected an increase in the fluctuation rate of a nuclear magnetic field for temperatures above 200 K, the origin of this increase is clearly confirmed as Li diffusion. Assuming a random walk process with the hopping of thermally activated Li+ between a regular Li site and the nearest neighboring vacant octahedral sites, a self-diffusion coefficient of Li+ was found to range above 10−11 cm2/s at temperatures above 250 K with an activation energy of about 0.06 eV.

Operando optical tracking of single-particle ion dynamics in batteries

The key to advancing lithium-ion battery technology-in particular, fast charging-is the ability to follow and understand the dynamic processes occurring in functioning materials under realistic conditions, in real time and on the nano- to mesoscale. Imaging of lithium-ion dynamics during battery operation (operando imaging) at present requires sophisticated synchrotron X-ray^1-7 or electron microscopy^8,9 techniques, which do not lend themselves to high-throughput material screening. This limits rapid and rational materials improvements. Here we introduce a simple laboratory-based, optical interferometric scattering microscope^10-13 to resolve nanoscopic lithium-ion dynamics in battery materials, and apply it to follow cycling of individual particles of the archetypal cathode material^14,15, Li_xCoO₂, within an electrode matrix. We visualize the insulator-to-metal, solid solution and lithium ordering phase transitions directly and determine rates of lithium diffusion at the single-particle level, identifying different mechanisms on charge and discharge. Finally, we capture the dynamic formation of domain boundaries between different crystal orientations associated with the monoclinic lattice distortion at the Li_0.5CoO₂ composition¹⁶. The high-throughput nature of our methodology allows many particles to be sampled across the entire electrode and in future will enable exploration of the role of dislocations, morphologies and cycling rate on battery degradation. The generality of our imaging concept means that it can be applied to study any battery electrode, and more broadly, systems where the transport of ions is associated with electronic or structural changes. Such systems include nanoionic films, ionic conducting polymers, photocatalytic materials and memristors.

Machine learning approach to muon spectroscopy analysis

In recent years, artificial intelligence techniques have proved to be very successful when applied to problems in physical sciences. Here we apply an unsupervised machine learning (ML) algorithm called principal component analysis (PCA) as a tool to analyse the data from muon spectroscopy experiments. Specifically, we apply the ML technique to detect phase transitions in various materials. The measured quantity in muon spectroscopy is an asymmetry function, which may hold information about the distribution of the intrinsic magnetic field in combination with the dynamics of the sample. Sharp changes of shape of asymmetry functions-measured at different temperatures-might indicate a phase transition. Existing methods of processing the muon spectroscopy data are based on regression analysis, but choosing the right fitting function requires knowledge about the underlying physics of the probed material. Conversely, PCA focuses on small differences in the asymmetry curves and works without any prior assumptions about the studied samples. We discovered that the PCA method works well in detecting phase transitions in muon spectroscopy experiments and can serve as an alternative to current analysis, especially if the physics of the studied material are not entirely known. Additionally, we found out that our ML technique seems to work best with large numbers of measurements, regardless of whether the algorithm takes data only for a single material or whether the analysis is performed simultaneously for many materials with different physical properties.

In Situ Diffusion Measurements of a NASICON-Structured All-Solid-State Battery Using Muon Spin Relaxation

In situ muon spin relaxation is demonstrated as an emerging technique that can provide a volume-averaged local probe of the ionic diffusion processes occurring within electrochemical energy storage devices as a function of state of charge. Herein, we present work on the conceptually interesting NASICON-type all-solid-state battery LiM2(PO4)3, using M = Ti in the cathode, M = Zr in the electrolyte, and a Li metal anode. The pristine materials are studied individually and found to possess low ionic hopping activation energies of ∼50-60 meV and competitive Li+ self-diffusion coefficients of ∼10-10-10-9 cm2 s-1 at 336 K. Lattice matching of the cathode and electrolyte crystal structures is employed for the all-solid-state battery to enhance Li+ diffusion between the components in an attempt to minimize interfacial resistance. The cell is examined by in situ muon spin relaxation, providing the first example of such ionic diffusion measurements. This technique presents an opportunity to the materials community to observe intrinsic ionic dynamics and electrochemical behavior simultaneously in a nondestructive manner.

Honeycomb layered oxides: structure, energy storage, transport, topology and relevant insights

The advent of nanotechnology has hurtled the discovery and development of nanostructured materials with stellar chemical and physical functionalities in a bid to address issues in energy, environment, telecommunications and healthcare. In this quest, a class of two-dimensional layered materials consisting of alkali or coinage metal atoms sandwiched between slabs exclusively made of transition metal and chalcogen (or pnictogen) atoms arranged in a honeycomb fashion have emerged as materials exhibiting fascinatingly rich crystal chemistry, high-voltage electrochemistry, fast cation diffusion besides playing host to varied exotic electromagnetic and topological phenomena. Currently, with a niche application in energy storage as high-voltage materials, this class of honeycomb layered oxides serves as ideal pedagogical exemplars of the innumerable capabilities of nanomaterials drawing immense interest in multiple fields ranging from materials science, solid-state chemistry, electrochemistry and condensed matter physics. In this review, we delineate the relevant chemistry and physics of honeycomb layered oxides, and discuss their functionalities for tunable electrochemistry, superfast ionic conduction, electromagnetism and topology. Moreover, we elucidate the unexplored albeit vastly promising crystal chemistry space whilst outlining effective ways to identify regions within this compositional space, particularly where interesting electromagnetic and topological properties could be lurking within the aforementioned alkali and coinage-metal honeycomb layered oxide structures. We conclude by pointing towards possible future research directions, particularly the prospective realisation of Kitaev-Heisenberg-Dzyaloshinskii-Moriya interactions with single crystals and Floquet theory in closely-related honeycomb layered oxide materials.

Tracer diffusion coefficients of Li+ ions in c-axis oriented LixCoO2 thin films measured by secondary ion mass spectrometry

Lithium diffusion is a key factor in determining the charge/discharge rate of Li-ion batteries. Herein, we study the tracer diffusion coefficient (D*) of lithium ions in the c-axis oriented LiCoO2 thin film using secondary ion mass spectrometry (SIMS). We applied a step-isotope-exchange method to determine D* in the Li-extracted LixCoO2. The observed values of D* ranged from 2 × 10-12 to 3 × 10-17 cm2 s-1 depending on the compositions in the range of 0.4 < x < 1.0. Approaching the stoichiometric composition (x = 1.0), D* decreases steeply to the minimum, which can be explained by the vacancy diffusion mechanism. Electrochemically determined diffusion coefficients corrected by thermodynamic factors are found to be in good agreement with D* determined by our method, over a wide range of compositions. The c-axis diffusion was explained by the migration of Li+ ions from one layer to another through additional diffusion channels, such as antiphase boundaries and a pair of Li antisite and oxygen vacancies in cobalt oxide layers.

Magnetism and ion diffusion in honeycomb layered oxide [Formula: see text]

In the quest for developing novel and efficient batteries, a great interest has been raised for sustainable K-based honeycomb layer oxide materials, both for their application in energy devices as well as for their fundamental material properties. A key issue in the realization of efficient batteries based on such compounds, is to understand the K-ion diffusion mechanism. However, investigation of potassium-ion (K[Formula: see text]) dynamics in materials using e.g. NMR and related techniques has so far been very challenging, due to its inherently weak nuclear magnetic moment, in contrast to other alkali ions such as lithium and sodium. Spin-polarised muons, having a high gyromagnetic ratio, make the muon spin rotation and relaxation ([Formula: see text]SR) technique ideal for probing ions dynamics in these types of energy materials. Here we present a study of the low-temperature magnetic properties as well as K[Formula: see text] dynamics in honeycomb layered oxide material [Formula: see text] using mainly the [Formula: see text]SR technique. Our low-temperature [Formula: see text]SR results together with complementary magnetic susceptibility measurements find an antiferromagnetic transition at [Formula: see text] K. Further [Formula: see text]SR studies performed at higher temperatures reveal that potassium ions (K[Formula: see text]) become mobile above 200 K and the activation energy for the diffusion process is obtained as [Formula: see text] meV. This is the first time that K[Formula: see text] dynamics in potassium-based battery materials has been measured using [Formula: see text]SR. Assisted by high-resolution neutron diffraction, the temperature dependence of the K-ion self diffusion constant is also extracted. Finally our results also reveal that K-ion diffusion occurs predominantly at the surface of the powder particles. This opens future possibilities for potentially improving ion diffusion as well as K-ion battery device performance using nano-structuring and surface coatings of the particles.

Quantifying Diffusion through Interfaces of Lithium-Ion Battery Active Materials

Detailed understanding of charge diffusion processes in a lithium-ion battery is crucial to enable its systematic improvement. Experimental investigation of diffusion at the interface between active particles and the electrolyte is challenging but warrants investigation as it can introduce resistances that, for example, limit the charge and discharge rates. Here, we show an approach to study diffusion at interfaces using muon spin spectroscopy. By performing measurements on LiFePO₄ platelets with different sizes, we determine how diffusion through the LiFePO₄ (010) interface differs from that in the center of the particle (i.e., bulk diffusion). We perform ab initio calculations to aid the understanding of the results and show the relevance of our interfacial diffusion measurement to electrochemical performance through cyclic voltammetry measurements. These results indicate that surface engineering can be used to improve the performance of lithium-ion batteries.

Microscopic photoelectron analysis of single crystalline LiCoO2 particles during the charge-discharge in an all solid-state lithium ion battery

We report synchrotron-based operando soft X-ray microscopic photoelectron spectroscopy under charge-discharge control of single crystalline LiCoO₂ (LCO) particles as an active electrode material for an all solid-state lithium-ion battery (LIB). Photoelectron mapping and the photoelectron spectrum of a selected microscopic region are obtained by a customized operando cell for LIBs. During the charge process, a more effective Li extraction from a side facet of the single crystalline LCO particle than from the central part is observed, which ensures the reliability of the system as an operando microscopic photoelectron analyzer that can track changes in the electronic structure of a selected part of the active particle. Based on these assessments, the no drastic change in the Co 2p XPS spectra during charge-discharge of LCO supports that the charge-polarization may occur at the oxygen side by strong hybridization between Co 3d and O 2p orbitals. The success of tracking the electronic-structure change at each facet of a single crystalline electrode material during charge-discharge is a major step toward the fabrication of innovative active electrode materials for LIBs.

Mechanistic insights of Li+ diffusion within doped LiFePO4 from Muon Spectroscopy

The Li⁺ ion diffusion characteristics of V- and Nb-doped LiFePO₄ were examined with respect to undoped LiFePO₄ using muon spectroscopy (µSR) as a local probe. As little difference in diffusion coefficient between the pure and doped samples was observed, offering D_Li values in the range 1.8–2.3 × 10⁻¹⁰ cm² s⁻¹, this implied the improvement in electrochemical performance observed within doped LiFePO₄ was not a result of increased local Li⁺ diffusion. This unexpected observation was made possible with the µSR technique, which can measure Li⁺ self-diffusion within LiFePO₄, and therefore negated the effect of the LiFePO₄ two-phase delithiation mechanism, which has previously prevented accurate Li⁺ diffusion comparison between the doped and undoped materials. Therefore, the authors suggest that µSR is an excellent technique for analysing materials on a local scale to elucidate the effects of dopants on solid-state diffusion behaviour.

Microscopic muon dynamics in the polymer electrolyte poly(ethylene oxide)

The microscopic dynamics of protons (H^{+}) in poly(ethylene oxide) (PEO) have been investigated through a study of implanted positive muons (Mu^{+}), which can be considered a light proton analog. The exponential decay of the muon spin polarization in zero magnetic field indicated that Mu^{+} hopping is in the fast fluctuation limit between 140 and 310 K and the relaxation rate was found to be sensitive to the glass transition. Mu^{+} dynamics in PEO was monitored via the relaxation of the muon spin polarization in a transverse field of 10 mT. Activated hopping of Mu^{+} was observed above the glass transition temperature with an activation barrier of 122±1 meV. The temperature dependence of the diamagnetic muon polarization in PEO can be explained by diffusion of radiolytic electrons.

High resolution x-ray absorption and emission spectroscopy of Li x CoO2 single crystals as a function delithiation

The effect of delithiation in Li _x CoO₂ is studied by high resolution Co K-edge x-ray absorption and x-ray emission spectroscopy. Polarization dependence of the x-ray absorption spectra on single crystal samples is exploited to reveal information on the anisotropic electronic structure. We find that the electronic structure of Li _x CoO₂ is significantly affected by delithiation in which the Co ions oxidation state tending to change from 3+  to 4+. The Co intersite (intrasite) 4p-3d hybridization suffers a decrease (increase) by delithiation. The unoccupied 3d t _2g orbitals with a _1g symmetry, containing substantial O 2p character, hybridize isotropically with Co 4p orbitals and likely to have itinerant character unlike anisotropically hybridized 3d e _g orbitals. Such a peculiar electronic structure could have significant effect on the mobility of Li in Li _x CoO₂ cathode and hence the battery characteristics.

Li-ion diffusion in Li intercalated graphite C6Li and C12Li probed by μ+SR

We have demonstrated that a local magnetic probe, μ⁺SR, provides a self diffusion coefficient of Li in Li intercalated graphites.

Enhanced Interfacial Kinetics and High-Voltage/High-Rate Performance of LiCoO2 Cathode by Controlled Sputter-Coating with a Nanoscale Li4Ti5O12 Ionic Conductor

The selection and optimization of coating material/approach for electrode materials have been under intensive pursuit to address the high-voltage induced degradation of lithium ion batteries. Herein, we demonstrate an efficient way to enhance the high-voltage electrochemical performance of LiCoO₂ cathode by postcoating of its composite electrode with Li₄Ti₅O₁₂ (LTO) via magnetron sputtering. With a nanoscale (∼25 nm) LTO coating, the reversible capacity of LiCoO₂ after 60 cycles is significantly increased by 40% (to 170 mAh g^-1) at room temperature and by 118% (to 139 mAh g^-1) at 55 °C. Meanwhile, the electrode's rate capability is also greatly improved, which should be associated with the high Li⁺ diffusivity of the LTO surface layer, while the bulk electronic conductivity of the electrode is unaffected. At 12 C, the capacity of the coated electrode reaches 113 mAh g^-1, being 70% larger than that of the uncoated one. The surface interaction between LTO and LiCoO₂ is supposed to reduce the space-charge layer at the LiCoO₂-electrolyte interface, which makes the Li⁺ diffusion much easier as evidenced by the largely enhanced diffusion coefficient of the coated electrode (an order of magnitude improvement). In addition, the LTO coating layer, which is electrochemically and structurally stable in the applied potential range, plays the role of a passivation layer or an artificial and friendly solid electrolyte interface (SEI) layer on the electrode surface. Such protection is able to impede propagation of the in situ formed irreversible SEI and thus guarantee a high initial columbic efficiency and superior cycling stability at high voltage.

Unveiled magnetic transition in Na battery material: μ+SR study of P2-Na0.5VO2

Muon-spin spectroscopy has clarified that the magnetic transition occurs not at 13 K but at 2 K in P2-Na_0.5VO₂.

Evolution of ferromagnetism from a frustrated state in LixNi(2-x)O2 (0.67 < x < 0.98)

Two-dimensional triangular-lattice antiferromagnetic systems continue to be an interesting area in condensed matter physics and LiNiO2 is one such among them. Here we present a detailed experimental magnetic study of the quasi-stoichiometric LixNi2-xO2 system (0.67 < x < 0.98). It exhibits a variety of magnetic ground states-namely spin glass, cluster glass, re-entrant spin glass and ferromagnetic. This study deals with the magnetic properties of these four distinct ground states. The spin glass state is evidenced by the frequency-dependent peak shift as well as the time-dependent slow dynamics (magnetic relaxation, magnetic memory effect etc). By tuning the Li deficiency in a controlled manner, an increase in the ordering temperature is observed. Most strikingly, with the Li deficiency the nature of the magnetic ground state is changed from spin glass to ferromagnetic, with two intermediate states-namely cluster glass and re-entrant spin glass. The critical behaviour of the re-entrant spin glass is also studied here. The critical exponents (β, γ and δ) are extracted from the modified Arrot plot, Kouvel-Fisher method, and critical isotherm analysis. The critical exponents match with the long-range mean-field model. The values of the critical exponents are confirmed by the Widom scaling law: δ = 1 + γβ(-1). Furthermore, the universality class of the scaling relations is verified, where the scaled m and scaled h collapse into two branches. Finally, based on our observations, a phase diagram is constructed.