Vibrational Spectra of MO (M=Sn/Pb) in Their Bulk and Single Layer Forms: Role of Avoided Crossing in their Thermodynamic Properties

Raju K Biswas; Swapan K Pati

arxiv: 1907.08836 · v1 · pith:ABJ3Z3MOnew · submitted 2019-07-20 · ❄️ cond-mat.mtrl-sci

Vibrational Spectra of MO (M=Sn/Pb) in Their Bulk and Single Layer Forms: Role of Avoided Crossing in their Thermodynamic Properties

Raju K Biswas , Swapan K Pati This is my paper

Pith reviewed 2026-05-24 18:42 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci

keywords phonon dispersionavoided crossingspecific heatvibrational entropySnOPbOmonolayerbulk

0 comments

The pith

Avoided crossing between degenerate phonon bands near the X point lowers specific heat and vibrational entropy up to 150 K in bulk SnO, bulk PbO, and monolayer SnO.

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

The paper performs ab initio phonon calculations for bulk and single-layer SnO and PbO. It identifies an avoided crossing between longitudinal acoustic and low-energy transverse optical modes near the X point that arises when two bands become degenerate. This feature alters the phonon density of states and thereby reduces the calculated specific heat and vibrational entropy in the three cases listed above, but only below about 150 K. Monolayer PbO lacks the crossing because of its larger mass contrast and missing interlayer interactions, so its thermodynamic quantities remain higher at low temperature. The calculations also show that long-range Coulomb effects raise Born charges and force constants in the bulk forms relative to the monolayers.

Core claim

Due to avoided crossing between two degenerate bands, the phonon dispersion near the high-symmetry X point lowers specific heat and vibrational entropy in bulk SnO, bulk PbO, and monolayer SnO up to 150 K. Monolayer PbO shows no such crossing and therefore exhibits higher specific heat and vibrational entropy at low temperatures; the difference traces to the large Pb–O mass ratio and the absence of interlayer van der Waals forces that otherwise permit the degeneracy.

What carries the argument

Avoided crossing (Landau degeneracy) between LA and low-energy TO phonon branches near the X point, produced when the modes share the same frequency and the long-range Coulomb interaction lifts the degeneracy.

If this is right

Bulk SnO and PbO exhibit lowered specific heat and entropy from the avoided crossing while monolayer PbO does not.
Monolayer SnO retains the crossing and the associated thermodynamic lowering, behaving like the bulk forms.
Higher Born effective charges and stiffer M–O force constants raise all vibrational frequencies in monolayer PbO relative to its bulk counterpart.
The A1g Raman mode lies above the Eg mode at the zone center in all four systems, matching reported Raman spectra.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Layer thickness or cation mass could be used to switch the presence of the crossing and thereby tune low-temperature heat capacity in related layered oxides.
The same mechanism may appear in other materials whose acoustic and optical branches approach each other at zone-boundary points.
Heat-capacity measurements performed on isotopically substituted samples could test whether the mass ratio alone controls the crossing.

Load-bearing premise

The DFT phonon calculations correctly locate the avoided crossing and integrate the resulting density of states without errors from functional choice or k-point sampling that would change the thermodynamic results.

What would settle it

A phonon dispersion measurement on monolayer PbO that reveals an avoided crossing near the X point, or a specific-heat measurement on bulk SnO that shows no reduction relative to the no-crossing case below 150 K.

read the original abstract

We report ab-initio calculations of the phonon dispersion relation on the bulk and single layer of SnO and PbO. We identify Raman active modes and infrared active modes at the zone center {\Gamma} point. In agreement with experimental observations of Raman spectroscopy measurement, we find that A1g mode is higher in frequency than that of Eg mode. Moreover, the reason behind the shift of A2u mode to higher frequency for monolayer of both SnO and PbO is revealed from our calculations. We also find that long-range Coulomb interaction enhances the dielectric constant and Born effective charges in bulk SnO and bulk PbO, compared to their monolayer. Here, we observe avoided crossing or Landau degeneracy between longitudinal acoustics (LA) and low energetic transverse optical (TO) modes in bulk form of both SnO and PbO. Additionally, monolayer SnO also shows low energetic Raman modes (Eg and A1g) of same frequency as bulk. As a result, we notice avoided crossing between LA and TO modes in monolayer SnO. Interestingly, higher Born effective charge and low dielectric constant enhances self-force constants and the interatomic force constants (IFCs) between the M-O bonds. The enhanced force constants give rise to higher vibrational frequency of phonon modes for monolayer PbO. Our studies reveal that due to avoided crossing between two degenerate bands, the phonon dispersion near high symmetry X point lowers specific heat and vibrational entropy in bulk SnO, bulk PbO and only in monolayer SnO upto temperature 150 K. Moreover, the large mass difference between Pb and Oxygen atoms and absence of interlayer van der Waal interactions give rise to high phonon vibration which reduces the occurrence of band crossing between two degenerate energy levels. The absence of avoided crossing leads higher specific heat and vibrational entropy in monolayer PbO at low temperatures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

DFT phonon calculations on SnO and PbO link avoided crossings to reduced low-T specific heat in bulk and monolayer SnO but not monolayer PbO, with the numerics plausible yet unverified due to missing convergence data.

read the letter

The paper's central result is that avoided crossings between LA and low-energy TO modes near X reduce specific heat and vibrational entropy up to 150 K in bulk SnO, bulk PbO, and monolayer SnO, while monolayer PbO lacks the crossing and shows higher values. This is attributed to differences in Born charges, dielectric screening, mass contrast between Pb and O, and the absence of interlayer vdW forces in the monolayer PbO case. The calculations also match the experimental Raman ordering with A1g above Eg and explain the A2u upshift in monolayers via enhanced self-force constants from long-range Coulomb terms. These are legitimate new numerical findings for these specific compounds. The work does a clean job separating bulk versus monolayer trends and SnO versus PbO behavior using standard ab-initio phonon methods. The mechanistic story around force constants and degeneracy is internally consistent with the reported dispersions. The main limitation is the absence of any convergence information. There are no tests on k-point density for the DFPT step, supercell size for the force constants, q-grid for the thermodynamic integrals, or sensitivity to the exchange-correlation functional. Small frequency shifts near X would change both the crossing identification and the low-T heat-capacity integrals, so the stress-test concern lands. The paper also omits the actual phonon DOS plots or the integration details that would let a reader judge the size of the effect. No direct comparison to measured specific heat appears either. This is a materials-specific computational study aimed at researchers already working on tin and lead oxide monolayers or on phonon thermodynamics in layered systems. It is not a broad methodological paper. The numerics are new enough and the argument coherent enough that it deserves peer review, mainly so referees can request the missing convergence checks and any available experimental thermodynamic data.

Referee Report

3 major / 1 minor

Summary. The manuscript reports ab initio DFT phonon dispersion calculations for bulk and monolayer SnO and PbO. It identifies Raman/IR active modes at Gamma with A1g > Eg matching Raman data, attributes A2u shifts and enhanced Born charges/dielectric constants in bulk to long-range Coulomb interactions, identifies avoided crossings between LA and low-energy TO modes near X (present in bulk SnO/PbO and monolayer SnO, absent in monolayer PbO) due to force constant enhancements, and concludes that these crossings lower specific heat and vibrational entropy up to 150 K in the former cases while monolayer PbO shows higher values due to mass differences and absent crossings.

Significance. If the avoided-crossing identification and its quantitative effect on low-T thermodynamics hold after verification, the work would demonstrate how dimensionality modulates long-range electrostatic contributions to phonon dispersions and resulting thermodynamic integrals in these oxide systems, offering insight into tailoring vibrational properties in 2D materials.

major comments (3)

[Results and discussion of thermodynamic properties] The central claim that avoided crossing near X lowers Cv and Sv up to 150 K rests on numerical integration of the phonon DOS, but no Cv(T), Sv(T) curves, integration method, q-grid density, or convergence tests for the thermodynamic quantities are shown or described.
[Computational details and phonon dispersion results] Identification of the avoided crossing (and its attribution to long-range Coulomb-enhanced IFCs and Born charges) requires accurate DFPT or finite-displacement calculations, yet no k-point sampling density, plane-wave cutoff, supercell size for IFCs, or sensitivity tests to functional choice (LDA/PBE + vdW) are reported.
[Phonon dispersion near X and thermodynamic implications] The assertion of a measurable reduction in low-frequency DOS due to the crossing lacks any error bars, frequency-shift sensitivity analysis, or explicit comparison to measured specific heat data that would confirm the effect persists to 150 K.

minor comments (1)

[Abstract] The abstract references agreement with Raman spectroscopy but provides no citation to the specific experimental data or reference used for comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments. We address each of the major comments below and indicate the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Results and discussion of thermodynamic properties] The central claim that avoided crossing near X lowers Cv and Sv up to 150 K rests on numerical integration of the phonon DOS, but no Cv(T), Sv(T) curves, integration method, q-grid density, or convergence tests for the thermodynamic quantities are shown or described.

Authors: We agree with this observation. The revised manuscript will include explicit plots of the temperature-dependent specific heat Cv(T) and vibrational entropy Sv(T) for bulk and monolayer SnO and PbO. We will describe the numerical integration method used to compute these quantities from the phonon density of states, specify the q-grid density employed (a dense mesh such as 20x20x20), and report convergence tests with respect to the q-grid to ensure the results are reliable up to 150 K. revision: yes
Referee: [Computational details and phonon dispersion results] Identification of the avoided crossing (and its attribution to long-range Coulomb-enhanced IFCs and Born charges) requires accurate DFPT or finite-displacement calculations, yet no k-point sampling density, plane-wave cutoff, supercell size for IFCs, or sensitivity tests to functional choice (LDA/PBE + vdW) are reported.

Authors: We will expand the computational details section in the revised manuscript to include all relevant parameters: the k-point sampling density for the electronic and phonon calculations, the plane-wave cutoff energy, the supercell sizes used for computing the interatomic force constants, and any sensitivity tests performed with different functionals (LDA, PBE, and with van der Waals corrections). This will substantiate the accuracy of the DFPT calculations and the observed avoided crossings. revision: yes
Referee: [Phonon dispersion near X and thermodynamic implications] The assertion of a measurable reduction in low-frequency DOS due to the crossing lacks any error bars, frequency-shift sensitivity analysis, or explicit comparison to measured specific heat data that would confirm the effect persists to 150 K.

Authors: To address this, we will perform and report a sensitivity analysis by introducing small perturbations to the frequencies near the avoided crossing and recomputing the thermodynamic integrals to show that the reduction in Cv and Sv remains significant. We will also include error estimates associated with the numerical integration. Regarding comparison to experimental data, low-temperature specific heat measurements for these particular systems are not available in the literature to the best of our knowledge; our work provides theoretical predictions that can guide future experiments. revision: partial

Circularity Check

0 steps flagged

No circularity: thermodynamic integrals follow directly from computed phonon dispersions via standard formulas

full rationale

The paper computes phonon dispersions via DFT/DFPT, identifies avoided crossings by inspection of the resulting bands near X, and evaluates Cv and Sv from the phonon density of states using the usual Bose-Einstein integrals. No parameter is fitted to the target thermodynamic quantities, no self-citation supplies a uniqueness theorem or ansatz that forces the result, and the avoided-crossing effect on low-frequency DOS is an output of the dispersion calculation rather than a definitional input. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard DFT approximations for phonons (Born-Oppenheimer, harmonic approximation, supercell finite-displacement or DFPT method) plus the numerical identification of avoided crossing from the computed dispersion; no new entities are postulated.

free parameters (2)

DFT exchange-correlation functional
Choice of functional (likely PBE or similar) controls force constants and dielectric screening; value not stated in abstract.
k-point sampling and plane-wave cutoff
Convergence parameters that affect phonon frequencies and the precise location of avoided crossings.

axioms (2)

domain assumption Harmonic approximation for lattice vibrations is sufficient to compute thermodynamic quantities up to 150 K
Abstract computes specific heat and entropy from phonon dispersion without anharmonic corrections.
domain assumption Long-range Coulomb interactions are correctly captured by the chosen supercell or DFPT implementation
Used to explain enhancement of dielectric constant and Born charges in bulk versus monolayer.

pith-pipeline@v0.9.0 · 5884 in / 1545 out tokens · 34997 ms · 2026-05-24T18:42:35.997120+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A. K. Geim and K. S. Novoselov, Nat Mater, 2007, 6, 183. [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Science, 2004, 306, 666

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Cahen, R

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Kumar, J

P. Kumar, J. Liu, P. Ranjan, Y. Hu, S. S. R. K. C. Yamijala, S. K. Pati, J. Irudayaraj, Cheng, G. J. Small, 2018, 14, 1703346

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T. Daeneke, P. Atkin, R. Orrell-Trigg, A. Zavabeti, T. Ahmed, S. Walia, M. Liu , Y. Tachibana, M. Javaid, A. D. Greentree, S. P. Russo, ACS nano., 2017, 11, 10974-83

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Le Bellac, J

D. Le Bellac, J. M. Kiat, P. Garnier, Journal of Solid State Chemistry, 1995, 114, 459-68. [10] Y. Ogo, H. Hiramatsu, K. Nomura, H. Yanagi, T. Kamiya, M. Hirano and H. Hosono, Appl. Phys. Lett., 2008, 93, 032113. [11] Y. W. Li, Y. Li, T. Cui, L. J. Zhang, Y. M. Ma, and G. T. Zou, J. Phys. Condens. Matter, 2007, 19, 425230

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Govaerts, R

K. Govaerts, R. Saniz, B. Partoens, and D. Lameon, Phys. Rev. B, 2013, 87, 235210

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[8] [9]

Seixas, A

L. Seixas, A. Rodin, A. Carvalho, A. C. Neto, Phys. Rev. Lett., 2016, 116, 206803

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W. Zhou, N. Umezawa, Phys. Chem. Chem. Phys., 2015, 17,1781617820

work page 2015

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Gonze, J

X. Gonze, J. C. Charlier, D. C. Allan, M. P. Teter, Physical Review B, 1994, 50, 13035

work page 1994

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Geurts, S

J. Geurts, S. Rau, W. Richter, F. J. Schmitte, Thin Solid Films, 1984, 121, 217

work page 1984

[13] [14]

Meyer, G

M. Meyer, G. Onida, A. Ponchel, and L. Reining, Comput. Mater. Sci., 1998, 10, 319

work page 1998

[14] [15]

Y. W. Li, Y. Li, T. Cui, L. J. Zhang, Y. M. Ma, and G. T. Zou, J. Phys. Condens. Matter, 2007, 19, 425230

work page 2007

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Walsh and G

A. Walsh and G. W. Watson, Phys. Rev. B, 2004, 70, 235114

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Boroni, P

S. Boroni, P. Giannozzi,and A. Tesla, phys. Rev. Lett., 1987, 58,1861

work page 1987

[18] [19]

Giannozzi, S

P. Giannozzi, S. De Gironcoli, P. Pavone, S. Baroni, Physical Review B., 1991, 43, 7231

work page 1991

[19] [20]

A. Togo, I. Tanaka, Scripta Materialia, 2015, 108, 1-5

work page 2015

[20] [21]

Cochran and R

W. Cochran and R. A. Cowley, J. Chem. Phys. Solids, 1962, 23, 447

work page 1962

[21] [22]

Molina-Sanchez, L

A. Molina-Sanchez, L. Wirtz, Physical Review B, 2011, 84, 155413

work page 2011

[22] [23]

Gonze, C

X. Gonze, C. Lee, Physical Review B., 1997, 55, 10355

work page 1997

[23] [24]

X. Wang, F. X. Zhang, I. Loa, K. Syassen, M. Hanfland, and YL. Mathis, physica status solidi(b), 2004, 241, 3168-3178

work page 2004

[24] [25]

Jr. W. J. Moore, L. Pauling, J. Am. Chem. Soc., 1941, 63, 13921394

work page 1941

[25] [26]

Born and K

M. Born and K. Huang, (Oxford University Press, Oxford, 1954)

work page 1954

[26] [27]

Kokalj, Journal of Molecular Graphics and Modelling, 1999, 17, 176-9

A. Kokalj, Journal of Molecular Graphics and Modelling, 1999, 17, 176-9

work page 1999

[27] [28]

Devaquet, Pure Appl

A. Devaquet, Pure Appl. Chem., 1975, 41, 455

work page 1975

[28] [29]

Walsh, G

A. Walsh, G. W. Watson, Journal of Solid State Chemistry. 2005, 178, 1422-8

work page 2005

[29] [30]

Semeniuk, A

O. Semeniuk, A. Csik, S. Ko¨ke´nyesi and A. Reznik, J. Mater. Sci., 2017, 52, 7937–7946

work page 2017

[30] [31]

Baleva and V

M. Baleva and V. Tuncheva, J. Solid State Chem., 1994, 110, 36–42

work page 1994

[31] [32]

Koval, R

S. Koval, R. Burriel, M. G. Stachiotti, M. Castro, R. L. Migoni, M. S. Moreno, A. Varela, C. O. Rodriguez, Physical Review B. 1999, 60, 14496. Figure 1: Crystal structure of MO (M=Sn/Pb). (a) monolayer (b) viewed along z-axis, (c) bulk. The arrows in (b) depict lattice parameters a and b along the x and y directions, respectively. The interlayer distance ...

work page 1999