PbSe/PbTe Monolayer Heterostructure

• The PbSe/PbTe monolayer heterostructure is a 2D vertical system with a corrugated Se–Pb–Pb–Te sheet and weak, polar covalent bonds that induce strong anharmonicity.
• First-principles, Boltzmann transport theory, and machine learning reveal ultralow lattice thermal conductivity (≈0.3–0.37 W/mK) dominated by dispersive optical phonons.
• Anisotropic thermoelectric performance is achieved with a maximum ZT≈5.3 along the y-direction, highlighting the impact of directional bonding and charge transport.

Searching arXiv for the specified paper to ground the article in the cited source. The PbSe/PbTe monolayer heterostructure is a two-dimensional out-of-plane (vertical) heterostructure composed of a Se–Pb–Pb–Te corrugated sheet in which two Pb layers are sandwiched between alternately arranged Se and Te layers. In reported calculations, it combines a honeycomb-like corrugated and asymmetric configuration with relatively weak interatomic interactions, anti-bonding states, strong anharmonicity, ultralow lattice thermal conductivity, and highly anisotropic thermoelectric response. The system was investigated by combining first-principles calculations, Boltzmann transport theory, and machine learning methods, with a reported maximum dimensionless figure of merit of approximately $ZT \approx 5.3$ along the $y$ direction at 800 K (Tan et al., 29 Jul 2025).

1. Crystal structure and bonding topology

The monolayer is described as a vertical heterostructure with space group P3m1 (No. 156). Its lattice parameters are $a=b=4.300\ \text{\AA}$, $\alpha=\beta=90^\circ$, and $\gamma=120^\circ$, and the effective thickness is $H_{\rm eff}=6.038\ \text{\AA}$. The unit cell contains eight atoms. The bonding environment is characterized by $d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$ and $d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$, with representative bond angles $\theta_1=97.74^\circ$ and $\theta_2=91.32^\circ$. A standard rectangular unit cell defines the $y$0 and $y$1 transport directions used for transport analysis (Tan et al., 29 Jul 2025).

The structural asymmetry is attributed to the size mismatch between Te, with atomic radius approximately $y$2, and Se, with atomic radius $y$3. In the reported interpretation, this mismatch induces a wrinkled morphology and local bond-angle asymmetry. The result is a corrugated, honeycomb-like sheet rather than a planar binary monolayer.

Electronic bonding indicators support a weakly bonded and electronically delocalized picture. The Electron Localization Function indicates weakly localized electrons on Pb–Se and Pb–Te bonds, and the bonding is described as polar covalent with weak ionic character rather than strongly covalent. Bader charges were reported as approximately Te $y$4, Se $y$5, and Pb $y$6 with average $y$7, substantially smaller in magnitude than nominal ionic charges of $y$8. COHP analysis further shows significantly negative $y$9 for Pb–Se and Pb–Te bonds in the energy window around $a=b=4.300\ \text{\AA}$0 to $a=b=4.300\ \text{\AA}$1 below $a=b=4.300\ \text{\AA}$2, indicating dominant antibonding states.

These bonding features are central to the reported transport behavior. Weak, antibonding character reduces bond stiffness and enhances anharmonicity, providing the electronic-structure basis for the large Grüneisen parameters, strong phonon–phonon scattering, and low lattice thermal conductivity discussed below.

2. Mechanical, dynamical, and thermodynamic stability

Mechanical stability was evaluated from elastic constants satisfying the Born–Huang criteria. The reported values are $a=b=4.300\ \text{\AA}$3, $a=b=4.300\ \text{\AA}$4, $a=b=4.300\ \text{\AA}$5, and $a=b=4.300\ \text{\AA}$6, which satisfy the conditions $a=b=4.300\ \text{\AA}$7, $a=b=4.300\ \text{\AA}$8, and $a=b=4.300\ \text{\AA}$9. The two-dimensional Young’s modulus is approximately $\alpha=\beta=90^\circ$0 and the Poisson ratio is $\alpha=\beta=90^\circ$1 (Tan et al., 29 Jul 2025).

The monolayer is correspondingly soft and compliant. The reported $\alpha=\beta=90^\circ$2 is much smaller than that of graphene, approximately $\alpha=\beta=90^\circ$3, and h-BN, approximately $\alpha=\beta=90^\circ$4. The Debye temperature is $\alpha=\beta=90^\circ$5, much lower than BP, approximately $\alpha=\beta=90^\circ$6, and MoS$\alpha=\beta=90^\circ$7, approximately $\alpha=\beta=90^\circ$8. In the source analysis, this low Debye temperature is consonant with low lattice thermal conductivity.

Dynamical stability was established from phonon dispersions with no imaginary modes along $\alpha=\beta=90^\circ$9–M–K–$\gamma=120^\circ$0. Thermodynamic stability was examined by ab initio molecular dynamics on a $\gamma=120^\circ$1 supercell in the NVT ensemble for 10 ps with a 1 fs timestep at 300 K and 700 K. The reported trajectories show stable total energy and intact crystal integrity, without reconstruction or disintegration.

Taken together, these criteria indicate that the corrugated vertical heterostructure is stable within the computational framework used for the transport study. At the same time, its low stiffness and low characteristic phonon energy scales are consistent with the strong anharmonicity that later emerges in the phonon calculations.

3. Computational framework and transport formalism

The reported first-principles calculations used VASP with PAW pseudopotentials. Geometry optimization and forces were treated within GGA-PBE, while HSE06 was used for the accurate band structure. The plane-wave cutoff was 500 eV, the $\gamma=120^\circ$2-mesh was $\gamma=120^\circ$3, the vacuum spacing was 20 Å, and the energy and force convergence thresholds were $\gamma=120^\circ$4 and $\gamma=120^\circ$5, respectively. van der Waals corrections were included with IVDW = 11, and COHP analysis was performed through LOBSTER (Tan et al., 29 Jul 2025).

High-order interatomic force constants were obtained with a machine-learning interatomic potential. The Moment Tensor Potential was trained on AIMD trajectories from a $\gamma=120^\circ$6 supercell at 50, 300, 500, and 700 K, each for 1 ps with 1 fs timestep, minimizing a loss over energies, forces, and stresses with weights $\gamma=120^\circ$7, $\gamma=120^\circ$8, and $\gamma=120^\circ$9. The MTP cutoff was $H_{\rm eff}=6.038\ \text{\AA}$0. Reported errors were approximately energy MAE $H_{\rm eff}=6.038\ \text{\AA}$1 and force MAE $H_{\rm eff}=6.038\ \text{\AA}$2, and phonon dispersions from the MLIP were reported to match finite-displacement DFT well. Second-order IFCs were generated by PHONOPY, third-order IFCs including up to 8th-neighbor interactions on a $H_{\rm eff}=6.038\ \text{\AA}$3 supercell via THIRDORDER.PY with MTP, and fourth-order IFCs by interfacing MTP with FOURTHORDER.PY.

Phonon transport was treated within the Boltzmann transport equation using ShengBTE for three-phonon processes and Fourphonon for four-phonon processes. The $H_{\rm eff}=6.038\ \text{\AA}$4-meshes were $H_{\rm eff}=6.038\ \text{\AA}$5 for three-phonon and $H_{\rm eff}=6.038\ \text{\AA}$6 for four-phonon calculations, with a convergence threshold of $H_{\rm eff}=6.038\ \text{\AA}$7. Boundary or isotope scattering was not added; the reported results correspond to intrinsic three- and four-phonon scattering. The phonon quasiparticle picture was checked by comparing total scattering rates to $H_{\rm eff}=6.038\ \text{\AA}$8 between 400 and 800 K, and the majority of modes satisfy $H_{\rm eff}=6.038\ \text{\AA}$9, supporting use of the BTE.

The lattice thermal conductivity was written in the scalar form

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$0

and in tensor form

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$1

The group velocity is

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$2

the Grüneisen parameter is

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$3

and the relaxation time was expressed through Matthiessen’s rule,

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$4

Electronic transport was evaluated with BoltzTraP for $d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$5, $d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$6, and $d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$7 within the rigid band approximation, with chemical potential scanned to emulate doping. Carrier relaxation times were estimated from two-dimensional deformation potential theory under acoustic phonon-limited scattering. The conductivity and Seebeck coefficient were given as

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$8

$d({\rm Pb\!-\!Se})=2.854\ \text{\AA}$9

with

$d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$0

The 2D mobility and relaxation time were written as

$d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$1

$d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$2

and the electronic thermal conductivity followed the Wiedemann–Franz relation

$d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$3

4. Phonon spectrum, anharmonicity, and ultralow lattice thermal conductivity

The phonon structure comprises three acoustic branches, ZA, TA, and LA, and nine optical branches. The highest phonon frequency is approximately $d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$4, which is low compared with ZnSe at approximately $d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$5 and MoS$d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$6 at approximately $d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$7. The acoustic branches lie approximately in the range $d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$8–$d({\rm Pb\!-\!Te})=3.006\ \text{\AA}$9, while optical branches span approximately $\theta_1=97.74^\circ$0–$\theta_1=97.74^\circ$1, with a clear acoustic–optical gap. The ZA branch is particularly flat, indicating very low group velocity and suppressed acoustic heat transport. A pronounced PhDOS peak near approximately $\theta_1=97.74^\circ$2 signals strong acoustic–optical coupling. Pb dominates the low-frequency acoustic region from $\theta_1=97.74^\circ$3 to $\theta_1=97.74^\circ$4 and also contributes non-negligibly in the $\theta_1=97.74^\circ$5–$\theta_1=97.74^\circ$6 optical range (Tan et al., 29 Jul 2025).

With both three- and four-phonon scattering included, the reported lattice thermal conductivity at 300 K is $\theta_1=97.74^\circ$7 along the $\theta_1=97.74^\circ$8 direction and $\theta_1=97.74^\circ$9 along the $\theta_2=91.32^\circ$0 direction. These values compare with $\theta_2=91.32^\circ$1 and $\theta_2=91.32^\circ$2, respectively, when only three-phonon scattering is included. The temperature dependence follows a power law $\theta_2=91.32^\circ$3. For three-phonon scattering alone, the exponents are $\theta_2=91.32^\circ$4 and $\theta_2=91.32^\circ$5, close to $\theta_2=91.32^\circ$6 behavior. With four-phonon scattering included, the exponents increase to $\theta_2=91.32^\circ$7 and $\theta_2=91.32^\circ$8, indicating stronger temperature dependence due to higher-order anharmonicity. Using these fitted exponents, the reported 800 K estimates are approximately $\theta_2=91.32^\circ$9 along $y$00 and $y$01 along $y$02.

A notable result is that optical phonons carry approximately 59% of the lattice thermal conductivity. At 300 K under four-phonon scattering, the optical contribution is approximately 58% along $y$03, and it exceeds 50% in both directions and in both scattering models. This is explicitly contrasted with the conventional view that long-mean-free-path acoustic phonons dominate heat transport. In the reported interpretation, optical modes dominate because they attain relatively large group velocities, exceeding $y$04 in parts of the spectrum, with a maximum group velocity of approximately $y$05; corrugation and asymmetry mix acoustic and optical character while flattening acoustic branches; and despite strong scattering, the large mode density and sizeable optical-group velocities yield major net contributions to $y$06.

Anharmonicity is correspondingly strong. The magnitudes of the Grüneisen parameters are large across both acoustic and optical branches, reaching $y$07 up to approximately 20 at 300 K. Frequency-resolved scattering rates $y$08 and $y$09 span approximately $y$10 to $y$11 and exhibit peaks around approximately $y$12 and $y$13. The low-frequency peak is associated with Pb-dominated vibrations and coupling between the lowest optical and acoustic branches. Three-phonon emission rates increase with frequency, while absorption rates decrease with frequency. Large absorption phase space occurs at low frequencies from $y$14 to $y$15, whereas large emission phase space appears at high frequencies around $y$16–$y$17. Optical branches with Pb and Se provide abundant high-frequency channels.

For four-phonon processes, both Normal and Umklapp channels were included, with Umklapp processes dominant in thermal resistance. The channel taxonomy comprises combination $y$18, redistribution $y$19, and splitting $y$20, and redistribution $y$21 is identified as the most influential channel in reducing $y$22. The acoustic–optical gap at approximately $y$23–$y$24 suppresses three-phonon phase space, especially absorption, because of energy conservation constraints. Four-phonon processes bridge this gap and activate additional redistribution and splitting channels where three-phonon scattering is limited, making four-phonon contributions comparable near the gap.

The cumulative conductivity analysis shows extremely short characteristic phonon mean free paths. The mean free path corresponding to 50% of $y$25 at 300 K is approximately $y$26 along $y$27 and $y$28 along $y$29. The volumetric heat capacity saturates at high temperature, with a relatively small maximum of approximately $y$30. The reported interpretation is that further suppression of $y$31 by nanostructuring is limited because the dominant heat carriers already have very short mean free paths.

5. Electronic structure and charge transport

The monolayer is semiconducting with an indirect band gap. Reported band gaps are $y$32 within PBE and $y$33 within HSE06, while inclusion of SOC yields $y$34 in PBE and $y$35 in HSE06. Because the SOC effect on the gap and dispersion is described as modest, SOC was neglected in the transport calculations (Tan et al., 29 Jul 2025).

The band-edge character is asymmetric. The valence-band maximum is relatively flat, corresponding to a large hole effective mass and large Seebeck coefficient, whereas the conduction-band minimum is dispersive, corresponding to a small electron effective mass and high mobility. The projected density of states indicates that the CBM is Pb-dominated and the VBM is mainly Se-derived.

At 300 K, the reported electron effective mass is approximately $y$36 in both $y$37 and $y$38. For holes, the effective masses are approximately $y$39 along $y$40 and $y$41 along $y$42. For $y$43-type transport, the deformation potentials are approximately $y$44 along $y$45 and $y$46 along $y$47, and the elastic moduli are approximately $y$48 and $y$49, respectively. The corresponding mobilities are approximately $y$50 and $y$51, and the relaxation times are approximately $y$52 and $y$53. For $y$54-type transport, the deformation potentials are approximately $y$55 along $y$56 and $y$57 along $y$58, the mobilities are approximately $y$59 and $y$60, and the relaxation times are approximately $y$61 and $y$62. The source analysis notes that electron mobilities and relaxation times exceed hole values by approximately $y$63–$y$64, consistent with the more dispersive CBM.

The Seebeck coefficients at 300 K are large for both carrier types. The reported maxima are approximately $y$65 along $y$66 and $y$67 along $y$68 for $y$69-type transport, and approximately $y$70 along $y$71 and $y$72 along $y$73 for $y$74-type transport. Electrical conductivity generally increases with temperature and carrier concentration, but large $y$75 yields moderate $y$76 at optimal conditions, typically around $y$77. The power factor $y$78 is larger for $y$79-type than for $y$80-type transport and is higher along $y$81 than along $y$82. The maximum reported power factor, for $y$83-type transport along $y$84, is approximately $y$85. The electronic thermal conductivity remains below approximately $y$86 across the explored conditions, and $y$87-type $y$88 exceeds $y$89-type $y$90.

These electronic trends set up the thermoelectric trade-off that governs the final $y$91. The conduction band supports higher mobility and higher power factor, while the valence band favors lower $y$92 and larger Seebeck response.

6. Thermoelectric performance, anisotropy, and interpretive limits

The figure of merit was evaluated using

$y$93

The reported results show pronounced anisotropy and doping dependence. Although the power factor is higher for $y$94-type transport, $y$95 is higher for $y$96-type transport because $y$97 is lower for holes (Tan et al., 29 Jul 2025).

Including four-phonon scattering increases $y$98 relative to three-phonon-only calculations. At 800 K for $y$99-type transport, the reported maxima are approximately $a=b=4.300\ \text{\AA}$00 along $a=b=4.300\ \text{\AA}$01 and $a=b=4.300\ \text{\AA}$02 along $a=b=4.300\ \text{\AA}$03, compared with approximately $a=b=4.300\ \text{\AA}$04 and $a=b=4.300\ \text{\AA}$05, respectively, when only three-phonon scattering is included. The optimal value along $a=b=4.300\ \text{\AA}$06 is attributed to ultralow $a=b=4.300\ \text{\AA}$07, approximately $a=b=4.300\ \text{\AA}$08 using the fitted exponent $a=b=4.300\ \text{\AA}$09, together with moderate $a=b=4.300\ \text{\AA}$10, large $a=b=4.300\ \text{\AA}$11 in the range approximately $a=b=4.300\ \text{\AA}$12–$a=b=4.300\ \text{\AA}$13, and respectable $a=b=4.300\ \text{\AA}$14 around $a=b=4.300\ \text{\AA}$15. The optimal carrier concentrations are reported to fall in the experimentally accessible range of approximately $a=b=4.300\ \text{\AA}$16–$a=b=4.300\ \text{\AA}$17.

The anisotropy of $a=b=4.300\ \text{\AA}$18 is linked to structural asymmetry and wrinkled corrugation, which produce directional differences in bond angles, local stiffness, and velocity and scattering distributions. The $a=b=4.300\ \text{\AA}$19 direction exhibits both lower $a=b=4.300\ \text{\AA}$20 and higher power factor, reflecting directional differences in effective masses, deformation potential constants, and elastic constants. The reported discussion relates this overall anisotropy to the P3m1 symmetry, the finite $a=b=4.300\ \text{\AA}$21 angle of $a=b=4.300\ \text{\AA}$22, and the heterogeneous bonding environment across the Pb–Se and Pb–Te sides.

The same analysis also outlines design implications. Carrier concentration optimization, by electrostatic gating or chemical doping, is presented as a route to maximize the power factor without driving $a=b=4.300\ \text{\AA}$23 too high. The source further suggests that aligning devices along the $a=b=4.300\ \text{\AA}$24 direction could exploit the superior power factor and lower $a=b=4.300\ \text{\AA}$25, and that maintaining the wrinkled geometry is important because substrate-induced modifications of corrugation and phonon spectra could alter the optical-mode dominance and low thermal conductivity. Because the dominant mean free paths are only approximately $a=b=4.300\ \text{\AA}$26–$a=b=4.300\ \text{\AA}$27, grain-boundary engineering is described as having limited additional effect relative to intrinsic anharmonicity; interfacial phonon engineering and orientation control are presented as more plausible routes at the device scale.

Several assumptions delimit the quantitative interpretation. The rigid band approximation neglects band-structure renormalization at high doping, so results at extreme doping are treated as trends rather than absolutes. Relaxation times derived from acoustic deformation-potential theory omit impurity, polar optical phonon, and electron–electron scattering, which may moderately overestimate $a=b=4.300\ \text{\AA}$28, and therefore $a=b=4.300\ \text{\AA}$29 and $a=b=4.300\ \text{\AA}$30. The Wiedemann–Franz treatment uses a nominal Lorenz number, so deviations from the degenerate limit could shift $a=b=4.300\ \text{\AA}$31. On the phonon side, the BTE treatment includes intrinsic three- and four-phonon scattering only, omitting extrinsic boundary, isotope, and defect scattering. The four-phonon $a=b=4.300\ \text{\AA}$32-mesh is reported as converged to $a=b=4.300\ \text{\AA}$33, but phase-space sampling in strongly anharmonic systems remains demanding. The MLIP was trained on short AIMD windows and finite supercells; although validated against DFT dispersions, residual errors can still influence higher-order IFCs and scattering rates quantitatively.

A central conceptual point concerns a common expectation in thermal transport: that long-mean-free-path acoustic phonons necessarily dominate heat conduction. In this heterostructure, the reported calculations indicate the opposite. Weak Pb–Se and Pb–Te bonding with antibonding occupation reduces stiffness and enhances anharmonicity, while honeycomb-like corrugation and asymmetry flatten acoustic branches, especially ZA, yet preserve dispersive optical modes with high group velocity. This combination yields both ultralow $a=b=4.300\ \text{\AA}$34 and a dominant optical-phonon contribution. Within the framework of the study, the PbSe/PbTe monolayer heterostructure is therefore presented as a thermoelectric system in which weak interactions, corrugation-induced phonon dispersion reshaping, and higher-order phonon scattering are inseparable from the high and anisotropic $a=b=4.300\ \text{\AA}$35 response.