Pith. sign in

REVIEW 3 major objections 7 minor 87 references

Accessing Few-Layer CrI$_3$ Magnetoelasticity Through Bulk Single Crystals

T0 review · 3 major / 7 minor · reviewed 2026-07-31 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Magnetostriction in bulk CrI3 is so surface-sensitive that few-layer uniaxial strain physics can be quantified without exfoliation.

desk verdict Solid uniaxial magnetostriction data that cleanly separate surface AFM from bulk FM responses in CrI3 and give usable pressure derivatives; the few-layer-without-exfoliation claim is useful but rests on T*∼J⊥. read the letter →

arxiv 2607.24222 v1 pith:6F62MZI7 submitted 2026-07-27 cond-mat.str-el cond-mat.mtrl-sci

J. Arneth , M. Möller , D. A. S. Kaib , P. P. Stavropoulos , A. Razpopov , M. Jonak , S. Spachmann , M. Abdel-Hafiez
show 4 more authors
S. Biswas K. Riedl R. Valentí R. Klingeler
This is my paper
classification cond-mat.str-elcond-mat.mtrl-sci
keywords CrI3magnetostrictionvanderWaalsmagnetsuniaxialstrainsurfaceantiferromagnetismmagnetoelasticcouplingfew-layermagnetismspin-fliptransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that high-resolution magnetostriction on bulk CrI3 single crystals is dominated by the thin antiferromagnetic surface layers rather than the ferromagnetic bulk. That surface signal lets the authors separate in-plane from out-of-plane uniaxial strain effects on both the bulk ferromagnetic phase and the surface antiferromagnetic phase. They extract large pressure derivatives for the surface spin-flip field and the surface interlayer coupling—about thirty times stronger than the corresponding bulk response. Because those surface layers magnetically match few-layer CrI3, the claim is that the magnetoelastic parameters needed for 2D devices can be read out from ordinary bulk crystals. A sympathetic reader cares because real devices inevitably impose anisotropic strain, and this route supplies directional tuning numbers that hydrostatic-pressure work cannot isolate.

What carries the argument

The surface magnetostriction discontinuity at the spin-flip field B*, converted by the Clausius-Clapeyron relation into uniaxial pressure derivatives of B* and of the surface interlayer coupling J_SAFM_⊥ (under the identification T* ∼ J_SAFM_⊥).

What would settle it

Apply controlled uniaxial c-axis pressure to the same bulk crystals and remeasure B* and T*; if ∂B*/∂pc and ∂T*/∂pc deviate strongly from 0.5 T/GPa and 44 K/GPa, or if the magnetostriction jump at B* vanishes when surfaces are passivated or removed, the central claim fails.

Watch

Extended reading notes

Core claim

Magnetostriction in bulk CrI3 is unexpectedly sensitive to the surface antiferromagnetic phase below T* ≃ 50 K. From the length jump at the surface spin-flip field B* and Clausius-Clapeyron analysis, the authors obtain ∂B*/∂pc = 0.50(6) T/GPa and ∂ln(J_SAFM_⊥)/∂pc ≃ 90 %/GPa, values that exceed bulk ferromagnetic strain effects by a factor of ~30. With supporting ab-initio magnetoelastic couplings and cluster mean-field modeling, this shows that few-layer CrI3 magnetoelasticity can be studied on bulk single crystals without exfoliation.

Load-bearing premise

The surface ordering temperature is set essentially only by the interlayer coupling, so its pressure derivative equals that of J_SAFM_⊥; if other energies dominate T*, the quoted 90 %/GPa does not apply to the coupling.

Editorial extensions

If this is right

  • Uniaxial c-axis compression strongly stabilizes antiferromagnetic interlayer order in surface and few-layer CrI3.
  • In-plane and out-of-plane uniaxial strain tune bulk and surface magnetism in opposite directions and can be applied selectively.
  • Few-layer magnetoelastic parameters can be extracted from bulk crystals instead of only from exfoliated flakes.
  • Hydrostatic-pressure results on CrI3 are largely c-axis-dominated and must be read with the large surface contribution in mind.
  • The factor-of-30 surface enhancement makes bulk dilatometry a practical proxy for 2D spin-lattice coupling studies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Other layered magnets that keep a monoclinic surface stacking may show the same surface-dominated magnetostriction, giving a general bulk route to few-layer magnetoelasticity.
  • Intentional surface passivation, capping, or stacking control should switch the large magnetostrictive jump on or off, directly testing the surface assignment.
  • The large calculated magnetoelastic single-ion and Γ′ terms imply that moment reorientation, not only exchange, is a primary strain handle for anisotropy in CrI3 devices.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The manuscript reports high-resolution capacitance-dilatometry magnetostriction and magnetization measurements on bulk CrI3 single crystals, complemented by DFT-based magnetoelastic calculations and self-consistent cluster mean-field (SCCMFT) simulations. The central experimental observation is that the c-axis magnetostriction shows a pronounced discontinuous jump at the surface-antiferromagnetic (SAFM) spin-flip field B* = 2.1 T — accounting for ~40% of the total magnetostriction although the associated magnetization jump is only ~5% of M_sat — demonstrating that bulk magnetostriction is unexpectedly sensitive to the ~10% surface-layer fraction. Via Maxwell and Clausius-Clapeyron relations the authors extract ∂B*/∂p_c = 0.50(6) T/GPa (Eq. 2), the spin-flip entropy ΔS* = 8.4(12) mJ/mol K (Eq. 3), ∂T*/∂p_c = 44(7) K/GPa (Eq. 4), and — identifying T* with the interlayer coupling scale — ∂ln J_SAFM_⊥/∂p_c ≃ 90%/GPa (Eq. 5), which they contrast with ∂ln T_C/∂p_c ≃ 3%/GPa for the bulk ferromagnet (a factor of ~30). In-plane data separately resolve the bulk anisotropy field and a surface anisotropy field B_a^ab/SAFM = 5.8(7) T, consistent with Raman results on few-layer samples. The theory side reproduces the shapes of dL/L(B) for both field directions and decomposes the response into bulk, surface intralayer, and interlayer contributions. The authors conclude that few-layer CrI3 magnetoelasticity can be accessed through bulk crystals without exfoliation.

Significance. If the interpretation holds, this is a useful and timely contribution: uniaxial-strain response of 2D vdW magnets is a recognized experimental bottleneck (most work is hydrostatic or computational), and the demonstration that a bulk thermodynamic probe can cleanly separate surface (few-layer-like) and bulk magnetic subsystems is genuinely enabling. Strengths that deserve explicit credit: (i) the thermodynamic analysis is parameter-free on the experimental side — Eqs. (2)–(4) use only measured discontinuities obtained by area-conserving constructions, the molar volume, and phase-boundary slopes, with stated uncertainties; I verified the arithmetic of Eqs. (2)–(4) is internally consistent; (ii) the BFM/SAFM assignment is cross-checked against the disappearance of anomalies above T*, the hysteresis at B*, and the agreement of B_a^ab/SAFM with independent Raman data; (iii) the work yields falsifiable numbers (∂B*/∂p_c, ∂T*/∂p_c) directly testable by uniaxial-pressure experiments; (iv) the SCCMFT decomposition (Fig. A7) gives a transparent microscopic rationalization of why the surface channel dominates dL_c/L_c. The theory is honestly presented as fitted rather than predictive in its关键

major comments (3)
  1. [§III.A, Eq. (5)] Eq. (5) and the abstract: the step from the measured ∂ln T*/∂p_c = 90%/GPa to ∂ln J_SAFM_⊥/∂p_c = 90%/GPa rests on the assumption that T* is controlled 'essentially only' by J_SAFM_⊥ (T* ~ J_SAFM_⊥). For an A-type antiferromagnet with strong easy-axis single-ion anisotropy (Table I: A_c = -0.25 meV, the largest single coupling after J) and strong intralayer FM exchange, T* in general depends on J_⊥, the intralayer couplings, and A_c; the authors themselves invoke a distribution of layer-dependent J_⊥ (§III.A, discussion of the smeared B_a^ab/SAFM). This identification carries the paper's headline number ('quantify the uniaxial strain dependence of J_SAFM_⊥' in the abstract) and part of the factor-of-30 claim. The authors should either (a) justify it quantitatively — e.g., a mean-field or SCCMFT estimate of how T* scales with J_⊥ versus with A_c and J, given that ˜A_c = -1.56 meV is by fa
  2. [§III.C (Discussion), Eqs. (10)-(11)] The comparison between theory and experiment for the key quantity is described as 'good agreement' when the numbers are ∂ln J_SAFM_⊥/∂p_c ≃ 50%/GPa (theory) versus 90%/GPa (experiment) — nearly a factor of two apart, and the theory value itself uses ˜J_SAFM_⊥ = -2.0 meV obtained by fitting the SCCMFT output to the experimental magnetostriction (§III.B), so the comparison is not independent. Relatedly, the predicted in-plane magnetostriction overshoots the measured (dL_ab/L_ab)_max by an order of magnitude (Eq. 10 vs. Fig. 1(a)), which the text acknowledges but does not fold into an assessment of the quantitative reliability of the framework. Given that J_SAFM_⊥ (Eq. 8), ˜J_SAFM_⊥, A_c (adjusted to -0.125 meV, Table I footnote), κ_ab/κ_c (order-of-magnitude proxies from hydrostatic CrBr3 data), and C ≈ 0.06 are all fitted or estimated, the paper should present the theory-experiment compar
  3. [§III.C, paragraph on κ_ab, κ_c, C] The compressibility inputs that convert the SCCMFT magnetostriction coefficients to absolute dL/L values — κ_ab ≈ 0.01 GPa^-1 and κ_c ≈ 0.02 GPa^-1 — are extrapolated from hydrostatic linear compressibilities of CrBr3 (Ref. [65]) via two qualitative corrections (softer compound; uniaxial softer than hydrostatic). Since these enter Eqs. (10)-(11) and the 50%/GPa theory estimate, and since the surface layers may have different elastic properties than the bulk (as the authors note), the absolute-magnitude discussion rests on an uncontrolled input. The authors should either cite or measure CrI3 elastic constants (or bound κ_i from their own thermal-expansion/pressure data, cf. Ref. [38]), or explicitly propagate the uncertainty in κ_i into the quoted theory numbers. This is load-bearing only for the quantitative theory claims, not for the experimental Eqs. (2)-(4).
minor comments (7)
  1. [§III.A / Eq. (9)] §III.A: the magnetostriction coefficient is defined in the text as λ_i = ∂L_i/∂B (dimensions of length/field), while Eq. (9) defines it as (1/L_i)∂L_i/∂B_j (dimensionless per tesla). Please make the notation consistent throughout.
  2. [Eqs. (2)-(3)] Eq. (2): the dimensional conversion from ΔM* in μB/f.u. to SI units (needed to obtain T/GPa) is not shown; a brief footnote or explicit conversion factor would help readers verify the Clausius-Clapeyron applications. Similarly, state how ∂B*/∂T entering Eq. (3) was extracted given the weak temperature dependence of B* at low T (fit range, use of hysteresis midpoints).
  3. [§II (Methods)] Methods: the surface-layer fraction x = 0.9 is estimated from the low-field M/M_s ratio; please quantify the uncertainty on x and note its (minor) impact on the SCCMFT total observables, since x enters Eq. (E1).
  4. [Various] Typos: 'since the the ferromagnetic bulk moments' (§III.A); Table I note reads 'SCCMFT alculations'; also check 'RESUL TS'/'SUMMAR Y' headings and spacing artifacts (may be PDF-extraction issues, but worth a proofread).
  5. [Appendix A, Fig. A2] Fig. A2(b): the determination of B_a^ab/SAFM via the 'onset of virtually constant magnetostriction' is inherently subjective for a feature smeared over ~2 T; the quoted uncertainty 5.8(7) T should reflect this (e.g., by showing the construction for more than one temperature). The asterisk-marked experimental artifact should be described in the caption.
  6. [§III.A] §III.A, QCEP paragraph: the inference from a ~0.1 T hysteresis at 2 K to a quantum-critical-endpoint scenario is plausible but the argument would benefit from stating explicitly what additional measurement (e.g., the T-dependence of ΔB* or of ΔS*) would discriminate QCEP from a simple first-order line.
  7. [Abstract] Abstract: the 'factor of ~30' compares ∂ln T*/∂p_c with ∂ln T_C/∂p_c (ordering temperatures), while the sentence presents it as the ratio of 'uniaxial strain dependence of J_SAFM_⊥ and B*' to bulk strain effects — please align the wording with what is actually compared.

Circularity Check

1 steps flagged · score 2.0 of 10

Core experimental strain derivatives are independent Clausius–Clapeyron/Maxwell results; only a mild theory-side calibration of J̃_SAFM_⊥ to the same magnetostriction is then re-compared as ‘agreement’.

  1. fitted input called prediction [Sec. III.B (after Eq. 8) and Sec. III.C Discussion (theory–experiment J_SAFM_⊥ pressure comparison)]
    "By comparing our cluster-mean field results, discussed further below, to experiments, we further estimate its magnetoelastic coupling as J̃_SAFM_⊥ = −2.0 meV. ... the values used in our simulations, together with the above estimates, yield ∂ln(J_SAFM_⊥)/∂pc ≃ 50 %/GPa, which is in good agreement with the value estimated from our thermodynamic analysis in Eq. 5."

    J̃_SAFM_⊥ is tuned so the simulated surface magnetostriction jump matches experiment. The experimental ∂ln(J_SAFM_⊥)/∂pc in Eq. 5 is extracted from that same magnetostriction discontinuity (via Clausius–Clapeyron and T*∼J). Converting the fitted J̃ with estimated κ_c and reporting ‘agreement’ with Eq. 5 is therefore partly by construction, not an independent ab-initio prediction of the surface pressure derivative. The experimental Eq. 5 result itself remains independently measured.

full rationale

The paper’s load-bearing claims—∂B*/∂pc = 0.50(6) T/GPa, ∂T*/∂pc = 44(7) K/GPa, and the inferred ∂ln(J_SAFM_⊥)/∂pc ≃ 90%/GPa—are obtained from measured jumps Δ(dLc/Lc)*, ΔM*, and the B*(T) slope via standard thermodynamic identities (Eqs. 1–5). Those identities do not presuppose the numerical answers. Setting the bare surface interlayer scale by the classical spin-flip condition J_SAFM_⊥ = (1/3)gμ_B B* (Eq. 8) is ordinary parameter fixing, not a claimed first-principles prediction of B*. The only mild circularity is on the theory side: J̃_SAFM_⊥ is adjusted so SCCMFT magnetostriction matches experiment, then converted with estimated κ_c into ∂ln(J_SAFM_⊥)/∂pc ≃ 50%/GPa and called ‘good agreement’ with the thermodynamic 90%/GPa that itself comes from the same length-change discontinuity. That agreement is partly forced by the fit and is not an independent validation; it is not, however, what carries the experimental factor-of-~30 claim. Bulk ab-initio J̃ couplings and the prior thermal-expansion comparison [38] are separate and non-circular. No self-definitional loop or uniqueness-import chain is present. Score 2 reflects one non-load-bearing fitted-input comparison only.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

Experimental claims rest on standard Maxwell and Clausius-Clapeyron thermodynamics plus the identification of magnetostriction anomalies with bulk vs surface spin configurations established in prior CrI3 literature. The large quoted surface pressure sensitivity further assumes T* is set by J_SAFM_⊥. Theory adds fitted interlayer and magnetoelastic surface parameters, an adjusted single-ion anisotropy, a fixed surface volume fraction, and elastic proxies from isostructural CrBr3.

free parameters (7)
  • J_SAFM_⊥ = 0.08 meV
    Effective surface interlayer Heisenberg coupling fixed so the classical spin-flip field equals experimental B*≈2.1 T.
  • ˜J_SAFM_⊥ = -2.0 meV
    Magnetoelastic derivative of surface interlayer coupling chosen by matching SCCMFT magnetostriction to experiment.
  • A_c (SCCMFT) = -0.125 meV
    Single-ion anisotropy reduced from ab-initio value for improved quantitative critical fields (Table I note).
  • surface layer fraction x = 0.9
    Weight of bulk vs surface in combined observables estimated from low-field M/M_s.
  • κ_ab, κ_c uniaxial compressibilities = ≈0.01 and 0.02 GPa^-1
    Order-of-magnitude uniaxial compressibilities adopted as slightly larger than hydrostatic k_i of CrBr3 to convert theory to absolute dL/L.
  • C (ab vs c magnetoelastic scaling) = ≈0.06
    Constant relating ∂J/∂L_c to ∂J/∂L_ab from ab-initio Poisson-like response under constrained c strain.
  • DFT+U / Kanamori parameters = U=5 eV, J_H=1 eV, λ_p=0.5 eV
    Interaction and ligand SOC parameters in exchange extraction (U=5 eV, J_H=1 eV, λ_p=500 meV; structural U_eff=2 eV).
assumptions (6)
  • standard math Maxwell relation ∂L_i/∂B_j|_p = -∂M_j/∂p_i|_B and Clausius-Clapeyron for first-order B*(p,T) hold for the measured discontinuities.
    Used to extract ∂B*/∂pc (Eq. 2), ΔS* (Eq. 3), and ∂T*/∂pc (Eq. 4) from jumps in length and magnetization.
  • domain assumption T* is dominated by the surface interlayer coupling, T* ∼ J_SAFM_⊥, so logarithmic pressure derivatives are interchangeable (Eq. 5).
    Converts the thermodynamic ∂T*/∂pc into the headline ∂ln(J_SAFM_⊥)/∂pc ≃ 90%/GPa and the factor-of-~30 vs bulk.
  • domain assumption Magnetostriction anomalies at B_c_sat, B*, B_a^ab/BFM, and B_a^ab/SAFM map onto bulk FM polarization, surface spin-flip, and surface in-plane anisotropy fields as in prior bulk/few-layer CrI3 studies.
    Assignment underpins separating BFM vs SAFM uniaxial pressure responses throughout Results.
  • domain assumption Surface layers of bulk crystals (C2/m, A-type AFM) microscopically represent few-layer CrI3 magnetoelasticity.
    Stated motivation and summary claim that bulk experiments access few-layer physics without exfoliation.
  • ad hoc to paper Magnetostriction coefficient factorizes as sum over (∂J/∂L)(∂M/∂J) with ∂J/∂L_c = -C ∂J/∂L_ab in the linear elastic regime (Eq. 9).
    Approximation taken from prior α-RuCl3 work and used to simulate dL_i/L_i from SCCMFT susceptibilities.
  • domain assumption Interlayer coupling may be treated as a single effective Heisenberg J_⊥ in classical mean-field between quantum intralayer clusters (SCCMFT).
    Methods/Appendix D; absorbs further-neighbor and anisotropic interlayer terms into one constant.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Accessing Few-Layer CrI$_3$ Magnetoelasticity Through Bulk Single Crystals." pith.science (2026). https://pith.science/paper/6F62MZI7

@misc{pith2026260724222,
  author       = {Pith},
  title        = {Pith review of: Accessing Few-Layer CrI$_3$ Magnetoelasticity Through Bulk Single Crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6F62MZI7}},
  note         = {Machine review of arXiv:2607.24222}
}
abstract

The persistence of ferromagnetic long-range order in monolayers of the van der Waals semiconductor CrI$_3$ opens new routes for spintronic applications based on two-dimensional quantum magnets. In the fabrication of such devices, the constituent materials inevitably experience anisotropic strain, which modifies their intrinsic electronic properties. At the same time, strain can serve as a powerful tuning parameter, driving the material to desired regimes. While several theoretical studies have investigated the effect of biaxial in-plane strain on CrI$_3$ numerically, experiments are widely limited to the application of hydrostatic pressure. Here, we perform high-resolution magnetostriction experiments on bulk CrI$_3$ samples, and \textit{ab-initio}-based magnetoelastic calculations, to elucidate the role of uniaxial lattice strain on the magnetic properties. Our data show that magnetostriction in CrI$_3$ is unexpectedly sensitive to surface effects, which enables us to investigate the influence of in-plane and out-of-plane strain separately, in both the bulk ferromagnetic (BFM) phase emerging at $T_{\rm C}=61\,\mathrm{K}$ and the surface antiferromagnetic (SAFM) phase below $T^* \simeq 50\,\mathrm{K}$. In particular, we quantify the uniaxial strain dependence of the surface interlayer coupling $J^{\rm SAFM}_{\perp}$ and the surface spin-flip field $B^*$, which drastically exceed the strain effects in the BFM phase by a factor of $\sim 30$. The large magnetostrictive response allows us to study the magnetoelastic coupling in few-layer CrI$_3$ through experiments on bulk single crystals, without requiring exfoliation.

Figures

Figures reproduced from arXiv: 2607.24222 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Isothermal magnetostriction and (b) magnetization for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Isothermal magnetostriction (blue) and differential magnetic susceptibility (orange) for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Magnetic phase diagrams obtained from dilatometry (black data points) and magnetization (blue data points) mea [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: In this description, J⊥ represents an effective interlayer coupling constant absorbing other, possibly further￾2 Since the interlayer coupling J⊥ is incorporated classically in the cluster mean-field method, the distinction between the R3 and C2/m crystal structures ef…
Figure 4
Figure 4. Figure 4: FIG. 4. Low-temperature Cr site honeycomb network in CrI [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Magnetization curves obtained from the self-consistent ED calculation for applied field [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Focusing on the in-plane magnetic field B||ab (panel (b)), the simulation successfully reproduces both the sign and overall shape of the experimental data. Additionally, it captures the finite dLab/Lab as observed in the field range of approximately 6.3−8 T. The result…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

87 extracted references

  1. [65]

    Riedl, D

    K. Riedl, D. Amoroso, S. Backes, A. Razpopov, T. P. T. Nguyen, K. Yamauchi, P. Barone, S. M. Winter, S. Picozzi, and R. Valent ´ ı, Microscopic origin of magnetism in monolayer 3d transition metal dihalides, Physical Review B106, 035156 (2022)

    2022

  2. [38]

    Bushick, S

    K. Bushick, S. Chae, Z. Deng, J. T. Heron, and E. Kioupakis, Boron arsenide heterostructures: lattice-matched heteroin- terfaces and strain effects on band alignments and mobility, npj Comput. Mater.6, 3 (2020)

    2020

  3. [1]

    Magnetization The simulated magnetization curves for magnetic fields applied out-of-planeB∥c(a) and in-planeB∥ab(b) are displayed in Fig. 5. Each plot displays the total magnetization in black, and its constituent contributions from bulk and surface magnetization in green and yellow, respectively. ComparingM tot with the lowest temperatureT= 2 K experimen...

  4. [2]

    Magnetostriction Beyond magnetization, we simulated the magnetostriction i.e., the field-induced relative length changes, as measured in the experiment, following the same approximation for the magnetostriction coefficient as in Ref. [64]: 1 Li ∂Li ∂Bj pi = 1 V ∂Mj ∂pi Bj ≈ κi Li V X J ∈{J, K,Γ′,...} ∂J ∂Li ∂M ∂J Bj ,(9) whereVis volume,κ i =− 1 Li ∂Li ∂p...

  5. [3]

    A1 forB∥c(a,c) andB∥ab(b,d)

    Magnetostriction The isothermal magnetization and corresponding (differential) magnetic susceptibility of CrI 3 at various temper- atures is shown in Fig. A1 forB∥c(a,c) andB∥ab(b,d). As forB∥c, the spin-flip transition in the SAFM phase is suppressed and occurs at smaller fieldsB ∗ as the temperature increases. Furthermore, the hysteresis atB ∗ is dimini...

  6. [4]

    A3 for different static magnetic fields applied along thecaxis

    Magnetic susceptibility The real part of the dynamic magnetic susceptibility measured with a small excitation fieldH ac = 5 Oe oscillating atf= 500 Hz is depicted in Fig. A3 for different static magnetic fields applied along thecaxis. AtB= 0 T, upon cooling from high temperatures,χ ′ is characterized by a steep increase atT C = 61 K marking the onset of f...

  7. [5]

    A4, with results from the above settings, as well as an additional relaxation where we also turn on SOC

    Strain relaxation results The structural summary of the relaxations is seen in Fig. A4, with results from the above settings, as well as an additional relaxation where we also turn on SOC. We see that with and without SOC there is no appreciable change. During the relaxation we have kept the conventionalc-axis fixed, seen in panel (a) as a perfectly strai...

  8. [6]

    Extracting magnetic exchanges Given the aforementioned structural results, we are within the framework used in Ref. [51] to calculate the magnetic exchanges, by using the extracted metal to ligand hoppingsd−pas well as the on-site atomic energy difference between the metaldand ligandpmanifolds, as found from wannierization. An effective I spin-orbit coupl...

Show all 87 references
  1. [7]

    Solve layerAwith the bare cluster HamiltonianH A and determine its ground state|Ψ A⟩, H0 |ΨA⟩=E A|ΨA⟩.(D1)

  2. [8]

    Measure the spin expectation value on layerA, ⟨SA j ⟩ ≡ ⟨ΨA|SA j |ΨA⟩.(D2)

  3. [9]

    Use these expectation values as input to construct the mean-field term acting on layerB, H A→B ⊥ ⟨SA j ⟩, J⊥ =J ⊥ X ⟨ij⟩⊥ ⟨SA j ⟩S i,(D3) whereJ ⊥ denotes the effective interlayer coupling entering the surface SCCMFT description. 18

  4. [10]

    Solve layerBwith the corresponding mean-field Hamiltonian and obtain the ground state|Ψ B⟩, H0 +H A→B ⊥ |ΨB⟩=E B|ΨB⟩.(D4)

  5. [11]

    Measure the local spin expectation values on layerB, ⟨SB j ⟩ ≡ ⟨ΨB|SB j |ΨB⟩.(D5) and use them to construct the mean-field term acting back on layerA, H B→A ⊥ ⟨SB j ⟩, J⊥ =J ⊥ X ⟨ij⟩⊥ ⟨SB j ⟩S i,(D6)

  6. [12]

    Solve layerAagain with the updated Hamiltonian H0 +H B→A ⊥ |ΨA⟩=E A|ΨA⟩,(D7)

  7. [13]

    At convergence, the resulting pair of states (|ψ A⟩,|ψ B⟩) define the SCCMFT solution for the two coupled layers

    Repeat steps 2–6 iteratively until the mean-field terms H B→A ⊥ , H A→B ⊥ ,(D8) or equivalently the moments⟨S A j ⟩and⟨S B j ⟩, no longer change within a chosen numerical toleranceε. At convergence, the resulting pair of states (|ψ A⟩,|ψ B⟩) define the SCCMFT solution for the ...

  8. [14]

    Huang, G

    B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nature546...

    2017

  9. [15]

    Marian, D

    D. Marian, D. Soriano, E. Cannav´ o, E. G. Marin, and G. Fiori, Electrically tunable lateral spin-valve transistor based on bilayer CrI3, npj 2D Mater. Appl.7, 42 (2023)

    2023

  10. [16]

    M. Zhu, H. Yao, L. Jiang, and Y. Zheng, Theoretical model of spintronic device based on tunable anomalous Hall conduc- tivity of monolayer CrI 3, Appl. Phys. Lett.116, 022404 (2020)

    2020

  11. [17]

    J. F. Sierra, J. Fabian, R. K. Kawakami, S. Roche, and S. O. Valenzuela, Van der waals heterostructures for spintronics and opto-spintronics, Nature Nanotechnology16, 856 (2021)

    2021

  12. [18]

    G.-Y. Hong, S. Zhao, F. Jin, L. Liu, L. Yang, Q. Zhang, M. Tan, K. Li, C. Hu, H.-B. Sun, and L. Lin, Asymmetric valley polarization of excitons and trions in antiferromagnet-semiconductor MoSe 2/CrI3 heterostructures, ACS Nano19, 34880 (2025)

    2025

  13. [19]

    Nnokwe, C

    C. Nnokwe, C. J. Cunningham, W. Liu, G. Ye, T. Nguyen, K. Wu, C. Hemesath, C. Sadler, P. Lukashev, Z. Zhai, B. Lv, J.-A. Yan, P. M. Shand, A. J. Stollenwerk, and R. He, Tuning the magnetic properties of CrI3 using Ni thin film deposition for applications in spintronic devices,...

    2025

  14. [20]

    S.-D. Guo, J. He, and Y. S. Ang, Achieving fully compensated ferrimagnetism through two-dimensional CrI 3/CrGeTe3 heterojunctions, Appl. Phys. Lett.127, 232401 (2025)

    2025

  15. [21]

    C ¸ aha, A

    I. C ¸ aha, A. U. Ahmad, L. Boddapatti, M. Ba˜ nobre-L´ opez, A. T. Costa, A. N. Enyashin, W. Li, P. Gargiani, M. Valvi- dares, J. Fern´ andez-Rossier, and F. L. Deepak, One-dimensional CrI3 encapsulated within multi-walled carbon nanotubes, Commun. Chem.8, 155 (2025)

    2025

  16. [22]

    N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett.17, 1133 (1966)

    1966

  17. [23]

    J. L. Lado and F. Fernˆ andez-Rossier, On the origin of magnetic anisotropy in two dimensional CrI3, 2D Mater.4, 035002 (2017)

    2017

  18. [24]

    D.-H. Kim, K. Kim, K.-T. Ko, J. Seo, J. S. Kim, T.-H. Jang, Y. Kim, J.-Y. Kim, S.-W. Cheong, and J.-H. Park, Giant Magnetic Anisotropy Induced by LigandLSCoupling in Layered Cr Compounds, Phys. Rev. Lett.122, 207201 (2019)

    2019

  19. [25]

    I. Lee, J. Cen, O. Molchanov, S. Feng, W. L. Huey, J. van Tol, J. E. Goldberger, N. Trivedi, H.-Y. Kee, and P. C. Hammel, Spin-orbit coupling controlled two-dimensional magnetism in chromium trihalides, Phys. Rev. B113, 014401 (2026)

    2026

  20. [26]

    De Vita, S

    A. De Vita, S. Stavri´ c, R. Sant, N. B. Brookes, I. Vobornik, G. Panaccione, S. Picozzi, M. Wolf, L. Rettig, R. Ernstorfer, and T. Pincelli, Orbital mixing and strong Hund’s coupling stabilize spin order in the van der Waals ferromagnet CrI 3, Phys. Rev. B113, 205120 (2026)

    2026

  21. [27]

    L. Chen, J. Chung, T. Chen, C. Duan, A. Schneidewind, I. Radelytskyi, D. J. Voneshen, R. A. Ewings, M. B. Stone, A. I. Kolesnikov, B. Winn, S. Chi, R. A. Mole, D. H. Yu, B. Gao, and P. Dai, Magnetic anisotropy in ferromagnetic CrI 3, Phys. Rev. B101, 134418 (2020)

    2020

  22. [28]

    Jonak, E

    M. Jonak, E. Walendy, J. Arneth, M. Abdel-Hafiez, and R. Klingeler, Low-energy magnon excitations and emerging anisotropic nature of short-range order in CrI 3, Phys. Rev. B106, 214412 (2022)

    2022

  23. [29]

    Sivadas, S

    N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and D. Xiao, Stacking-Dependent Magnetism in Bilayer CrI 3, Nano Lett. 18, 7658 (2018)

    2018

  24. [30]

    Jiang, C

    P. Jiang, C. Wang, D. Chen, Z. Zhong, Z. Yuan, Z.-Y. Lu, and W. Ji, Stacking tunable interlayer magnetism in bilayer CrI3, Phys. Rev. B99, 144401 (2019)

    2019

  25. [31]

    Soriano, C

    D. Soriano, C. Cardoso, and J. Fern´ andez-Rossier, Interplay between interlayer exchange and stacking in CrI 3 bilayers, Solid State Commun.299, 113662 (2019)

    2019

  26. [32]

    M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales, Coupling of Crystal Structure and Magnetism in the Layered, Ferromagnetic Insulator CrI 3, Chemistry of Materials27, 612 (2015)

    2015

  27. [33]

    Y. Liu, L. Wu, X. Tong, J. Li, J. Tao, Y. Zhu, and C. Petrovic, Thickness-dependent magnetic order in CrI3 single crystals, Sci. Rep.9, 13599 (2019)

    2019

  28. [34]

    S. Li, Z. Ye, X. Luo, G. Ye, H. H. Kim, B. Yang, S. Tian, C. Li, H. Lei, A. W. Tsen, K. Sun, R. He, and L. Zhao, Magnetic-Field-Induced Quantum Phase Transitions in a van der Waals Magnet, Phys. Rev. X10, 011075 (2020)

    2020

  29. [35]

    B. Niu, T. Su, B. A. Francisco, S. Ghosh, F. Kargar, X. Huang, M. Lohmann, J. Li, Y. Xu, T. Taniguchi, K. Watanabe, D. Wu, A. Balandin, J. Shi, and Y. Cui, Coexistence of Magnetic Orders in Two-Dimensional Magnet CrI 3, NANO Lett. 20, 553 (2020)

    2020

  30. [36]

    Huang, G

    B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, W. Yao, P. Jarillo-Herrero, and X. Xu, Electrical control of 2D magnetism in bilayer CrI 3, Nature Nanotech.13, 544 (2018)

    2018

  31. [37]

    Zhang, J.-Y

    Z. Zhang, J.-Y. You, B. Gu, and G. Su, Emergent magnetic states due to stacking and strain in the van der Waals magnetic trilayer CrI3, Phys. Rev. B104, 174433 (2021)

    2021

  32. [39]

    Yastrubchak, T

    O. Yastrubchak, T. Wosi ski, J. Z. Domaga a, E. Usakowska, T. Figielski, B. P´ ecz, and A. L. T´ oth, Misfit strain anisotropy in partially relaxed lattice-mismatched InGaAs/GaAs heterostructures, J. Phys. Condens. Matter16, S1 (2004)

    2004

  33. [40]

    Zhang, W

    Y. Zhang, W. Si, Y. Jia, P. Yu, R. Yu, and J. Zhu, Controlling strain relaxation by interface design in highly lattice- mismatched heterostructure, Nano Lett.21, 6867 (2021)

    2021

  34. [41]

    K. Liu, F. Jin, X. Zhang, K. Liu, Z. Zhang, E. Hua, J. Zhang, H. Ye, G. Gao, C. Ma, L. Wang, and W. Wu, Anisotropic Strain-mediated Symmetry Engineering and Enhancement of Ferroelectricity in Hf 0.5Zr0.5O2/La0.67Sr0.33MnO3 het- erostructures, Adv. Funct. Mater.33, 2209925 (2023)

    2023

  35. [42]

    Zhang, Q

    W.-B. Zhang, Q. Qu, P. Zhu, and C.-H. Lam, Robust intrinsic ferromagnetism and half semiconductivity in stable two- dimensional single-layer chromium trihalides, J. Mater. Chem. C3, 12457 (2015). 22

    2015

  36. [43]

    Webster and J.-A

    L. Webster and J.-A. Yan, Strain-tunable magnetic anisotropy in monolayer CrCl 3, CrBr 3, and CrI 3, Phys. Rev. B98, 144411 (2018)

    2018

  37. [44]

    J. Liu, P. Mo, M. Shi, D. Gao, and J. Lu, Multi-scale analysis of strain-dependent magnetocrystalline anisotropy and strain-induced villari and Nagaoka-Honda effects in a two-dimensional ferromagnetic chromium tri-iodide monolayer, J. Appl. Phys.124, 044303 (2018)

    2018

  38. [45]

    S. I. Vishkayi, Z. Torbatian, A. Qaiumzadeh, and R. Asgari, Strain and electric-field control of spin-spin interactions in monolayer cri3, Phys. Rev. Mater.4, 094004 (2020)

    2020

  39. [46]

    T. Song, Z. Fei, M. Yankowitz, Z. Lin, Q. Jiang, K. Hwangbo, Q. Zhang, B. Sun, T. Taniguchi, K. Watanabe, M. A. McGuire, D. Graf, T. Cao, J. Chu, D. H. Cobden, C. R. Dean, D. Xiao, and X. Xu, Switching 2D magnetic states via pressure tuning of layer stacking, Nat. Mater.18, 12...

    2019

  40. [47]

    T. Li, S. Jiang, N. Sivadas, Z. Wang, Y. Xu, D. Weber, J. Goldberger, K. Watanabe, T. Taniguchi, C. J. Fennie, K. F. Mak, and J. Shan, Pressure controlled interlayer magnetism in atomically thin CrI 3, Nat. Mater.18, 1303 (2019)

    2019

  41. [48]

    Subhan, I

    F. Subhan, I. Khan, and J. Hong, Pressure-induced ferromagnetism and enhanced perpendicular magnetic anisotropy of bilayer CrI3, J. Phys.: Condens. Matter31, 355001 (2019)

    2019

  42. [49]

    Mondal, M

    S. Mondal, M. Kannan, M. Das, L. Govindaraj, R. Singha, B. Satpati, S. Arumugam, and P. Mandal, Effect of hydrostatic pressure on ferromagnetism in two-dimensional cri 3, Phys. Rev. B99, 180407(R) (2019)

    2019

  43. [50]

    Ghosh, D

    A. Ghosh, D. Singh, T. Aramaki, Q. Mu, V. Borisov, Y. Kvashnin, G. Haider, M. Jonak, D. Chareev, S. A. Medvedev, R. Klingeler, M. Mito, E. H. Abdul-Hafidh, J. Vejpravova, M. Kalb` aˇ c, R. Ahuja, O. Eriksson, and M. Abdel-Hafiez, Exotic magnetic and electronic properties of la...

    2022

  44. [51]

    Arneth, M

    J. Arneth, M. Jonak, S. Spachmann, M. Abdel-Hafiez, Y. O. Kvashnin, and R. Klingeler, Uniaxial pressure effects in the two-dimensional van der waals ferromagnet cri 3, Phys. Rev. B105, L060404 (2022)

    2022

  45. [52]

    Kocsis, D

    V. Kocsis, D. A. S. Kaib, K. Riedl, S. Gass, P. Lampen-Kelley, D. G. Mandrus, S. E. Nagler, N. P´ erez, K. Nielsch, B. B¨ uchner, A. U. B. Wolter, and R. Valent ´ ı, Magnetoelastic coupling anisotropy in the Kitaev materialα−RuCl3, Phys. Rev. B105, 094410 (2022)

    2022

  46. [53]

    Arneth, R

    J. Arneth, R. Kalaivanan, R. Sankar, K.-Y. Choi, and R. Klingeler, Competing interactions and the effects of uniaxial out-of-plane perturbations in the honeycomb antiferromagnet Na 2Co2TeO6, Phys. Rev. Mater.9, 094001 (2025)

    2025

  47. [54]

    S. Gass, P. M. Cˆ onsoli, V. Kocsis, L. T. Corredor, P. Lampen-Kelley, D. G. Mandrus, S. E. Nagler, L. Janssen, M. Vojta, B. B¨ uchner, and A. U. B. Wolter, Field-induced transitions in the kitaev materialα−rucl 3 probed by thermal expansion and magnetostriction, Phys. Rev. B1...

    2020

  48. [55]

    Spachmann, A

    S. Spachmann, A. Elghandour, S. Selter, B. B¨ uchner, S. Aswartham, and R. Klingeler, Strong effects of uniaxial pressure and short-range correlations in Cr 2Ge2Te6, Phys. Rev. Res.4, L022040 (2022)

    2022

  49. [56]

    Spachmann, S

    S. Spachmann, S. Selter, B. B¨ uchner, S. Aswartham, and R. Klingeler, Strong uniaxial pressure dependencies evidencing spin-lattice coupling and spin fluctuations in Cr 2Ge2Te6, Phys. Rev. B107, 184421 (2023)

    2023

  50. [57]

    Arneth, K.-Y

    J. Arneth, K.-Y. Choi, R. Kalaivanan, R. Sankar, and R. Klingeler, Signatures of a quantum critical endpoint in the Kitaev candidate Na2Co2TeO6, Phys. Rev. B110, L140402 (2024)

    2024

  51. [58]

    X. Mi, X. Li, L. Zhang, Y. Gu, A. Wang, Y. Li, Y. Chai, and M. He, Precisely tracking critical spin fluctuations using the ac magnetostriction coefficient: A case study on the Kitaev spin liquid candidate Na 3Co2SbO6, Phys. Rev. B111, 014417 (2025)

    2025

  52. [59]

    P. K. Mukharjee, S. Erdmann, L. Wang, J. Kaiser, A. Jesche, P. Puphal, M. Isobe, M. Hepting, B. Keimer, P. Gegenwart, and A. A. Tsirlin, Anisotropic magnetoelastic coupling in the honeycomb magnet Na 3Co2SbO6, Phys. Rev. B113, 205148 (2026)

    2026

  53. [60]

    K¨ uchler, T

    R. K¨ uchler, T. Bauer, M. Brando, and F. Steglich, A compact and miniaturized capacitance dilatometer for measuring thermal expansion and magnetostriction, Rev. Sci. Instrum.83, 095102 (2012)

    2012

  54. [61]

    Werner, W

    J. Werner, W. Hergett, M. Gertig, J. Park, C. Koo, and R. Klingeler, Anisotropy-governed competition of magnetic phases in the honeycomb quantum magnet Na3Ni2SbO6 studied by dilatometry and high-frequency ESR, Phys. Rev.B95, 214414 (2017)

    2017

  55. [62]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B54, 11169 (1996)

    1996

  56. [63]

    Koepernik and H

    K. Koepernik and H. Eschrig, Full-potential nonorthogonal local-orbital minimum-basis band-structure scheme, Phys. Rev. B59, 1743 (1999)

    1999

  57. [64]

    P. P. Stavropoulos, X. Liu, and H.-Y. Kee, Magnetic anisotropy in spin-3/2 with heavy ligand in honeycomb Mott insulators: Application to CrI 3, Phys. Rev. Res.3, 013216 (2021)

    2021

  58. [66]

    Liu and C

    Y. Liu and C. Petrovic, Three-dimensional magnetic critical behavior in CrI 3, Phys. Rev. B97, 014420 (2018)

    2018

  59. [67]

    Garst and A

    M. Garst and A. Rosch, Sign change of the Gr¨ uneisen parameter and magnetocaloric effect near quantum critical points, Phys. Rev. B72, 205129 (2005)

    2005

  60. [68]

    A. J. Millis, A. J. Schofield, G. G. Lonzarich, and S. A. Grigera, Metamagnetic Quantum Criticality in Metals, Phys. Rev. Lett.88, 217204 (2002)

    2002

  61. [69]

    Beneke and M

    C. Beneke and M. Vojta, Divergence of the Gr¨ uneisen ratio at symmetry-enhanced first-order quantum phase transitions, Phys. Rev. B103, 174420 (2021)

    2021

  62. [70]

    Gegenwart, F

    P. Gegenwart, F. Weickert, M. Garst, R. S. Perry, and Y. Maeno, Metamagnetic Quantum Criticality in Sr3Ru2O7 Studied by Thermal Expansion, Phys. Rev. Lett.96, 136402 (2006). 23

    2006

  63. [71]

    Weickert, M

    F. Weickert, M. Brando, F. Steglich, P. Gegenwart, and M. Garst, Universal signatures of the metamagnetic quantum critical endpoint: Application to CeRu 2Si2, Phys. Rev. B81, 134438 (2010)

    2010

  64. [72]

    McCreary, T

    A. McCreary, T. T. Mai, F. G. Utermohlen, J. R. Simpson, K. F. Garrity, X. Feng, D. Shcherbakov, Y. Zhu, J. Hu, D. Weber, K. Watanabe, T. Taniguchi, J. E. Goldberger, Z. Mao, C. N. Lau, Y. Lu, N. Trivedi, R. Vald´ es Aguilar, and A. R. Hight Walker, Distinct magneto-Raman sign...

    2020

  65. [73]

    Ghosh, H

    A. Ghosh, H. J. M. J¨ onsson, D. J. Mukkattukavil, Y. Kvashnin, D. Phuyal, P. Thunstr¨ om, M. Ag ˚ aker, A. Nicolaou, M. Jonak, R. Klingeler, M. V. Kamalakar, T. Sarkar, A. N. Vasiliev, S. M. Butorin, O. Eriksson, and M. Abdel-Hafiez, Magnetic circular dichroism in theddexcita...

    2023

  66. [74]

    Spachmann, A

    S. Spachmann, A. Elghandour, M. Frontzek, W. L¨ oser, and R. Klingeler, Magnetoelastic coupling and phases in the skyrmion lattice magnet Gd 2PdSi3 discovered by high-resolution dilatometry, Phys. Rev. B103, 184424 (2021)

    2021

  67. [75]

    Stockert, N

    U. Stockert, N. Leps, L. Wang, G. Behr, S. Wurmehl, B. B¨ uchner, and R. Klingeler, Pr magnetism and its interplay with the Fe spin-density wave in PrFeAsO1−xFx (x= 0,0.15), Phys. Rev. B86, 144407 (2012)

    2012

  68. [76]

    Cen and H.-Y

    J. Cen and H.-Y. Kee, Determining Kitaev interaction in spin- S honeycomb Mott insulators, Phys. Rev. B107, 014411 (2023)

    2023

  69. [77]

    D. A. S. Kaib, S. Biswas, K. Riedl, S. M. Winter, and R. Valent ´ ı, Magnetoelastic coupling and effects of uniaxial strain in α−RuCl 3 from first principles, Phys. Rev. B103, L140402 (2021)

    2021

  70. [78]

    O. Lis, D. Kozlenko, S. Kichanov, E. Lukin, I. Zel, and B. Savenko, Structural, Magnetic and Vibrational Properties of Van Der Waals Ferromagnet CrBr3 at High Pressure, Materials16, 454 (2023)

    2023

  71. [79]

    A. M. Le´ on, J. W. Gonz´ alez, J. Mej ´ ıa-L´ opez, F. Crasto de Lima, and E. Su´ arez Morell, Strain-induced phase transition in CrI3 bilayers, 2D Mater.7, 035008 (2020)

    2020

  72. [80]

    Xu, Q.-J

    C. Xu, Q.-J. Wang, B. Xu, and J. Hu, Effect of biaxial strain and hydrostatic pressure on the magnetic properties of bilayer CrI3, Front. Phys.16, 53502 (2021)

    2021

  73. [81]

    Ouyang, B.-C

    X.-F. Ouyang, B.-C. Lei, X.-M. Li, W.-Z. Zhou, and F.-P. Ouyang, The mechanism of strain tunable magnetic anisotropy and electronic properties of bilayer CrI 3, Physica B Condens. Matter706, 417015 (2025)

    2025

  74. [82]

    Z. Wang, I. Guti´ errez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamo˘ glu, E. Giannini, and A. F. Morpurgo, Very large tunneling magnetoresistance in layered magnetic semiconductor CrI 3, Nat. Commun.9, 2516 (2018)

    2018

  75. [83]

    Kresse and J

    G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B47, 558 (1993)

    1993

  76. [84]

    J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov, G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke, Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces, Phys. Rev. Lett.100, 136406 (2008)

    2008

  77. [85]

    S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study, Phys. Rev. B57, 1505 (1998)

    1998

  78. [86]

    Opahle, K

    I. Opahle, K. Koepernik, and H. Eschrig, Full-potential band-structure calculation of iron pyrite, Phys. Rev. B60, 14035 (1999)

    1999

  79. [87]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Polat, Y. Feng...

    2020

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.