#
Influence of the hBN Dielectric Layers on the Quantum Transport Properties of MoS_{2} Transistors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Construction of the MoS${}_{2}$-hBN Heterostructure

#### 2.2. Ab Initio Calculations

#### 2.3. Quantum Transport Simulations

^{†}indicates the Hermitian transposition. All matrices in Equations (1) and (2) have a size ${N}_{O}\times {N}_{O}$, where ${N}_{O}$ is the total number of orbitals in the considered device. The ${\mathsf{\Sigma}}^{\gtrless}$ self-energy contain a boundary ${\mathsf{\Sigma}}^{\gtrless ,B}$ and scattering ${\mathsf{\Sigma}}^{\gtrless ,S}$ term. The former connects the simulation domain with semi-infinite leads and is computed through contour integral techniques [63]. The latter is limited here to electron–phonon interactions and approximated in two different ways, as discussed later. For simplicity, the retarded component of the scattering self-energy is extracted from the lesser/greater ones as [64].

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Roy, K.; Jung, B.; Than, A.R. Integrated Systems in the More-than-Moore Era: Designing Low-Cost Energy-Efficient Systems Using Heterogeneous Components. In Proceedings of the 2010 23rd International Conference on VLSI Design, Bangalore, India, 3–7 January 2010; pp. 464–469. [Google Scholar] [CrossRef]
- Illarionov, Y.Y.; Knobloch, T.; Jech, M.; Lanza, M.; Akinwande, D.; Vexler, M.I.; Mueller, T.; Lemme, M.C.; Fiori, G.; Schwierz, F.; et al. Insulators for 2D nanoelectronics: The gap to bridge. Nat. Commun.
**2020**, 11, 1–15. [Google Scholar] [CrossRef] [PubMed] - Lee, G.H.; Yu, Y.J.; Cui, X.; Petrone, N.; Lee, C.H.; Choi, M.S.; Lee, D.Y.; Lee, C.; Yoo, W.J.; Watanabe, K.; et al. Flexible and transparent MoS
_{2}field-effect transistors on hexagonal boron nitride-graphene heterostructures. ACS Nano**2013**, 7, 7931–7936. [Google Scholar] [CrossRef] [PubMed] - Fiori, G.; Bonaccorso, F.; Iannaccone, G.; Palacios, T.; Neumaier, D.; Seabaugh, A.; Banerjee, S.K.; Colombo, L. Electronics based on two-dimensional materials. Nat. Nanotechnol.
**2014**, 9, 768–779. [Google Scholar] [CrossRef] [PubMed] - Park, W.; Park, J.; Jang, J.; Lee, H.; Jeong, H.; Cho, K.; Hong, S.; Lee, T. Oxygen environmental and passivation effects on molybdenum disulfide field effect transistors. Nanotechnology
**2013**, 24, 095202. [Google Scholar] [CrossRef] [PubMed] - Late, D.J.; Liu, B.; Matte, H.R.; Dravid, V.P.; Rao, C. Hysteresis in single-layer MoS
_{2}field effect transistors. ACS Nano**2012**, 6, 5635–5641. [Google Scholar] [CrossRef] - Cho, K.; Park, W.; Park, J.; Jeong, H.; Jang, J.; Kim, T.Y.; Hong, W.K.; Hong, S.; Lee, T. Electric stress-induced threshold voltage instability of multilayer MoS
_{2}field effect transistors. ACS Nano**2013**, 7, 7751–7758. [Google Scholar] [CrossRef] - Kc, S.; Longo, R.C.; Wallace, R.M.; Cho, K. Computational study of MoS
_{2}/HfO_{2}defective interfaces for nanometer-scale electronics. ACS Omega**2017**, 2, 2827–2834. [Google Scholar] [CrossRef][Green Version] - Salvatore, G.A.; Münzenrieder, N.; Barraud, C.; Petti, L.; Zysset, C.; Büthe, L.; Ensslin, K.; Tröster, G. Fabrication and transfer of flexible few-layers MoS
_{2}thin film transistors to any arbitrary substrate. ACS Nano**2013**, 7, 8809–8815. [Google Scholar] [CrossRef] - Wang, H.; Yu, L.; Lee, Y.H.; Shi, Y.; Hsu, A.; Chin, M.L.; Li, L.J.; Dubey, M.; Kong, J.; Palacios, T. Integrated circuits based on bilayer MoS
_{2}transistors. Nano Lett.**2012**, 12, 4674–4680. [Google Scholar] [CrossRef][Green Version] - Qian, Q.; Li, B.; Hua, M.; Zhang, Z.; Lan, F.; Xu, Y.; Yan, R.; Chen, K.J. Improved gate dielectric deposition and enhanced electrical stability for single-layer MoS
_{2}MOSFET with an AlN interfacial layer. Sci. Rep.**2016**, 6, 1–9. [Google Scholar] [CrossRef] - Park, Y.; Baac, H.W.; Heo, J.; Yoo, G. Thermally activated trap charges responsible for hysteresis in multilayer MoS
_{2}field-effect transistors. Appl. Phys. Lett.**2016**, 108, 083102. [Google Scholar] [CrossRef] - Jiménez-Molinos, F.; Gámiz, F.; Donetti, L. Coulomb scattering in high-κ gate stack silicon-on-insulator metal-oxide-semiconductor field effect transistors. J. Appl. Phys.
**2008**, 104, 063704. [Google Scholar] [CrossRef] - VenkataRatnam, D.; ChowdaryGutta, R.; Kumar, M.R.; Rao, L.E.; Reddy, M.S.S.; Pavan, P.S.; Santhosh, C. Effect of High-K Dielectric materials on Mobility of Electrons. Int. J. Emerg. Trends Eng. Res.
**2020**, 8, 314–316. [Google Scholar] - Ma, N.; Jena, D. Charge scattering and mobility in atomically thin semiconductors. Phys. Rev. X
**2014**, 4, 011043. [Google Scholar] [CrossRef][Green Version] - Han, T.; Liu, H.; Chen, S.; Chen, Y.; Wang, S.; Li, Z. Fabrication and Characterization of MoS
_{2}/h-BN and WS_{2}/h-BN Heterostructures. Micromachines**2020**, 11, 1114. [Google Scholar] [CrossRef] - Sun, J.; Lu, C.; Song, Y.; Ji, Q.; Song, X.; Li, Q.; Zhang, Y.; Zhang, L.; Kong, J.; Liu, Z. Recent progress in the tailored growth of two-dimensional hexagonal boron nitride via chemical vapour deposition. Chem. Soc. Rev.
**2018**, 47, 4242–4257. [Google Scholar] [CrossRef] - Han, X.; Lin, J.; Liu, J.; Wang, N.; Pan, D. Effects of Hexagonal Boron Nitride Encapsulation on the Electronic Structure of Few-Layer MoS
_{2}. J. Phys. Chem. C**2019**, 123, 14797–14802. [Google Scholar] [CrossRef][Green Version] - Uchiyama, Y.; Kutana, A.; Watanabe, K.; Taniguchi, T.; Kojima, K.; Endo, T.; Miyata, Y.; Shinohara, H.; Kitaura, R. Momentum-forbidden dark excitons in hBN-encapsulated monolayer MoS
_{2}. NPJ 2D Mater. Appl.**2019**, 3, 26. [Google Scholar] [CrossRef][Green Version] - Ahmed, T.; Bellare, P.; Debnath, R.; Roy, A.; Ravishankar, N.; Ghosh, A. Thermal History-Dependent Current Relaxation in hBN/MoS
_{2}van der Waals Dimers. ACS Nano**2020**, 14, 5909–5916. [Google Scholar] [CrossRef] - Srivastava, A.; Fahad, M.S. Vertical MoS
_{2}/hBN/MoS_{2}interlayer tunneling field effect transistor. Solid-State Electron.**2016**, 126, 96–103. [Google Scholar] [CrossRef] - Dean, C.R.; Young, A.F.; Meric, I.; Lee, C.; Wang, L.; Sorgenfrei, S.; Watanabe, K.; Taniguchi, T.; Kim, P.; Shepard, K.L.; et al. Boron nitride substrates for high-quality graphene electronics. Nat. Nanotechnol.
**2010**, 5, 722–726. [Google Scholar] [CrossRef] [PubMed] - Watanabe, K.; Taniguchi, T.; Kanda, H. Direct-bandgap properties and evidence for ultraviolet lasing of hexagonal boron nitride single crystal. Nat. Mater.
**2004**, 3, 404–409. [Google Scholar] [CrossRef] [PubMed] - Shi, Y.; Hamsen, C.; Jia, X.; Kim, K.K.; Reina, A.; Hofmann, M.; Hsu, A.L.; Zhang, K.; Li, H.; Juang, Z.Y.; et al. Synthesis of few-layer hexagonal boron nitride thin film by chemical vapor deposition. Nano Lett.
**2010**, 10, 4134–4139. [Google Scholar] [CrossRef] - Zhang, K.; Feng, Y.; Wang, F.; Yang, Z.; Wang, J. Two dimensional hexagonal boron nitride (2D-hBN): Synthesis, properties and applications. J. Mater. Chem. C
**2017**, 5, 11992–12022. [Google Scholar] [CrossRef] - Lee, C.; Rathi, S.; Khan, M.A.; Lim, D.; Kim, Y.; Yun, S.J.; Youn, D.H.; Watanabe, K.; Taniguchi, T.; Kim, G.H. Comparison of trapped charges and hysteresis behavior in hBN encapsulated single MoS
_{2}flake based field effect transistors on SiO_{2}and hBN substrates. Nanotechnology**2018**, 29, 335202. [Google Scholar] [CrossRef] [PubMed] - Liu, Y.; Ong, Z.Y.; Wu, J.; Zhao, Y.; Watanabe, K.; Taniguchi, T.; Chi, D.; Zhang, G.; Thong, J.T.; Qiu, C.W.; et al. Thermal conductance of the 2D MoS
_{2}/h-BN and graphene/h-BN interfaces. Sci. Rep.**2017**, 7, 43886. [Google Scholar] [CrossRef][Green Version] - Baugher, B.W.; Churchill, H.O.; Yang, Y.; Jarillo-Herrero, P. Intrinsic electronic transport properties of high-quality monolayer and bilayer MoS
_{2}. Nano Lett.**2013**, 13, 4212–4216. [Google Scholar] [CrossRef] - Radisavljevic, B.; Kis, A. Mobility engineering and a metal–insulator transition in monolayer MoS
_{2}. Nat. Mater.**2013**, 12, 815–820. [Google Scholar] [CrossRef] - Cui, X.; Lee, G.H.; Kim, Y.D.; Arefe, G.; Huang, P.Y.; Lee, C.H.; Chenet, D.A.; Zhang, X.; Wang, L.; Ye, F.; et al. Multi-terminal transport measurements of MoS
_{2}using a van der Waals heterostructure device platform. Nat. Nanotechnol.**2015**, 10, 534–540. [Google Scholar] [CrossRef] - Rhodes, D.; Chae, S.H.; Ribeiro-Palau, R.; Hone, J. Disorder in van der Waals heterostructures of 2D materials. Nat. Mater.
**2019**, 18, 541. [Google Scholar] [CrossRef] - Suryavanshi, S.V.; Gabourie, A.J.; Farimani, A.B.; Yalon, E.; Pop, E. Thermal boundary conductance of the MOS
_{2}-SiO_{2}interface. In Proceedings of the 2017 IEEE 17th International Conference on Nanotechnology (IEEE-NANO), Pittsburgh, PA, USA, 25–28 July 2017; pp. 26–29. [Google Scholar] [CrossRef] - Knobloch, T.; Illarionov, Y.Y.; Ducry, F.; Schleich, C.; Wachter, S.; Müller, T.; Waltl, M.; Lanza, M.; Vexler, M.I.; Luisier, M.; et al. On the suitability of hBN as an insulator for 2D material-based ultrascaled CMOS devices. arXiv
**2020**, arXiv:2008.04144. [Google Scholar] - Britnell, L.; Gorbachev, R.V.; Jalil, R.; Belle, B.D.; Schedin, F.; Katsnelson, M.I.; Eaves, L.; Morozov, S.V.; Mayorov, A.S.; Peres, N.M.; et al. Electron tunneling through ultrathin boron nitride crystalline barriers. Nano Lett.
**2012**, 12, 1707–1710. [Google Scholar] [CrossRef][Green Version] - Wachter, S.; Polyushkin, D.K.; Bethge, O.; Mueller, T. A microprocessor based on a two-dimensional semiconductor. Nat. Commun.
**2017**, 8, 1–6. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] [PubMed] - Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci.
**1996**, 6, 15–50. [Google Scholar] [CrossRef] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett.
**1996**, 77, 3865. [Google Scholar] [CrossRef][Green Version] - Grimme, S. Semiempirical GGA-type density functional constructed with a long-range dispersion correction. J. Comput. Chem.
**2006**, 27, 1787–1799. [Google Scholar] [CrossRef] - Yu, L.; Ranjan, V.; Lu, W.; Bernholc, J.; Nardelli, M.B. Equivalence of dipole correction and Coulomb cutoff techniques in supercell calculations. Phys. Rev. B
**2008**, 77, 245102. [Google Scholar] [CrossRef][Green Version] - Jiang, S.; Greengard, L.; Bao, W. Fast and Accurate Evaluation of Nonlocal Coulomb and Dipole-Dipole Interactions via the Nonuniform FFT. arXiv
**2013**, arXiv:1311.4120. [Google Scholar] [CrossRef][Green Version] - Popescu, V.; Zunger, A. Extracting E versus $\overrightarrow{k}$
effective band structure from supercell calculations on alloys and impurities. Phys. Rev. B
**2012**, 85, 085201. [Google Scholar] [CrossRef][Green Version] - Mostofi, A.A.; Yates, J.R.; Lee, Y.S.; Souza, I.; Vanderbilt, D.; Marzari, N. wannier90: A tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun.
**2008**, 178, 685–699. [Google Scholar] [CrossRef][Green Version] - Szabó, Á.; Rhyner, R.; Luisier, M. Ab initio simulation of single-and few-layer MoS
_{2}transistors: Effect of electron-phonon scattering. Phys. Rev. B**2015**, 92, 035435. [Google Scholar] [CrossRef] - Klinkert, C.; Szabó, A.; Stieger, C.; Campi, D.; Marzari, N.; Luisier, M. 2-D Materials for Ultrascaled Field-Effect Transistors: One Hundred Candidates under the Ab Initio Microscope. ACS Nano
**2020**, 14, 8605–8615. [Google Scholar] [CrossRef] [PubMed] - Kühne, T.D.; Iannuzzi, M.; Del Ben, M.; Rybkin, V.V.; Seewald, P.; Stein, F.; Laino, T.; Khaliullin, R.Z.; Schütt, O.; Schiffmann, F.; et al. CP2K: An electronic structure and molecular dynamics software package-Quickstep: Efficient and accurate electronic structure calculations. J. Chem. Phys.
**2020**, 152, 194103. [Google Scholar] [CrossRef] [PubMed] - VandeVondele, J.; Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys.
**2007**, 127, 114105. [Google Scholar] [CrossRef][Green Version] - Goedecker, S.; Teter, M.; Hutter, J. Separable dual-space Gaussian pseudopotentials. Phys. Rev. B
**1996**, 54, 1703–1710. [Google Scholar] [CrossRef][Green Version] - Togo, A.; Tanaka, I. First principles phonon calculations in materials science. Scr. Mater.
**2015**, 108, 1–5. [Google Scholar] [CrossRef][Green Version] - McClellan, C.J.; Yalon, E.; Smithe, K.K.; Suryavanshi, S.V.; Pop, E. Effective n-type doping of monolayer MoS
_{2}by AlO x. In Proceedings of the 2017 75th Annual Device Research Conference (DRC), South Bend, IN, USA, 25–28 June 2017; pp. 1–2. [Google Scholar] - Suh, J.; Park, T.E.; Lin, D.Y.; Fu, D.; Park, J.; Jung, H.J.; Chen, Y.; Ko, C.; Jang, C.; Sun, Y.; et al. Doping against the native propensity of MoS
_{2}: Degenerate hole doping by cation substitution. Nano Lett.**2014**, 14, 6976–6982. [Google Scholar] [CrossRef] - McDonnell, S.; Addou, R.; Buie, C.; Wallace, R.M.; Hinkle, C.L. Defect-dominated doping and contact resistance in MoS
_{2}. ACS Nano**2014**, 8, 2880–2888. [Google Scholar] [CrossRef] - Houssa, M.; Iordanidou, K.; Dabral, A.; Lu, A.; Pourtois, G.; Afanasiev, V.; Stesmans, A. Contact resistance at MoS
_{2}-based 2D metal/semiconductor lateral heterojunctions. ACS Appl. Nano Mater.**2019**, 2, 760–766. [Google Scholar] [CrossRef] - Iannaccone, G.; Bonaccorso, F.; Colombo, L.; Fiori, G. Quantum engineering of transistors based on 2D materials heterostructures. Nat. Nanotechnol.
**2018**, 13, 183–191. [Google Scholar] [CrossRef] [PubMed] - Jena, D.; Banerjee, K.; Xing, G.H. Intimate contacts. Nat. Mater.
**2014**, 13, 1076–1078. [Google Scholar] [CrossRef] [PubMed] - Yang, L.; Majumdar, K.; Du, Y.; Liu, H.; Wu, H.; Hatzistergos, M.; Hung, P.; Tieckelmann, R.; Tsai, W.; Hobbs, C.; et al. High-performance MoS2 field-effect transistors enabled by chloride doping: Record low contact resistance (0.5 kΩ· μm) and record high drain current (460 μA/μm). In Proceedings of the 2014 Symposium on VLSI Technology (VLSI-Technology): Digest of Technical Papers, Honolulu, HI, USA, 9–12 June 2014; pp. 1–2. [Google Scholar]
- Das, S.; Chen, H.Y.; Penumatcha, A.V.; Appenzeller, J. High performance multilayer MoS
_{2}transistors with scandium contacts. Nano Lett.**2013**, 13, 100–105. [Google Scholar] [CrossRef] - Chhowalla, M.; Jena, D.; Zhang, H. Two-dimensional semiconductors for transistors. Nat. Rev. Mater.
**2016**, 1, 1–15. [Google Scholar] [CrossRef] - Shen, P.C.; Su, C.; Lin, Y.; Chou, A.S.; Cheng, C.C.; Park, J.H.; Chiu, M.H.; Lu, A.Y.; Tang, H.L.; Tavakoli, M.M.; et al. Ultralow contact resistance between semimetal and monolayer semiconductors. Nature
**2021**, 593, 211–217. [Google Scholar] [CrossRef] - Sohier, T.; Calandra, M.; Mauri, F. Two-dimensional Fröhlich interaction in transition-metal dichalcogenide monolayers: Theoretical modeling and first-principles calculations. Phys. Rev. B
**2016**, 94, 085415. [Google Scholar] [CrossRef][Green Version] - Ziogas, A.N.; Ben-Nun, T.; Fernández, G.I.; Schneider, T.; Luisier, M.; Hoefler, T. A data-centric approach to extreme-scale ab initio dissipative quantum transport simulations. In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, Denver, CO, USA, 17–19 November 2019; pp. 1–13. [Google Scholar]
- Datta, S. The non-equilibrium Green’s function (NEGF) formalism: An elementary introduction. In Proceedings of the Digest International Electron Devices Meeting, San Francisco, CA, USA, 8–11 December 2002; pp. 703–706. [Google Scholar] [CrossRef]
- Brück, S.; Calderara, M.; Bani-Hashemian, M.H.; VandeVondele, J.; Luisier, M. Efficient algorithms for large-scale quantum transport calculations. J. Chem. Phys.
**2017**, 147, 074116. [Google Scholar] [CrossRef] - Lake, R.; Klimeck, G.; Bowen, R.C.; Jovanovic, D. Single and multiband modeling of quantum electron transport through layered semiconductor devices. J. Appl. Phys.
**1997**, 81, 7845–7869. [Google Scholar] [CrossRef][Green Version] - Chang, J.; Register, L.F.; Banerjee, S.K. Ballistic performance comparison of monolayer transition metal dichalcogenide MX
_{2}(M = Mo, W; X = S, Se, Te) metal-oxide-semiconductor field effect transistors. J. Appl. Phys.**2014**, 115, 084506. [Google Scholar] [CrossRef][Green Version] - Lü, J.; Wang, J.S. Coupled electron and phonon transport in one-dimensional atomic junctions. Phys. Rev. B
**2007**, 76, 165418. [Google Scholar] [CrossRef][Green Version] - Afzalian, A. Ab initio perspective of ultra-scaled CMOS from 2D-material fundamentals to dynamically doped transistors. Npj 2D Mater. Appl.
**2021**, 5, 1–13. [Google Scholar] [CrossRef]

**Figure 1.**Hexagonal cell built from two MoS${}_{2}$ and hBN monolayers viewed under different angles. The yellow, gray, turquoise, and magenta spheres represent sulfur, molybdenum, boron, and nitrogen atoms, respectively. The y-axis refers to the stacking direction, while x and z are the in-plane axis. (

**a**) Top view of the cell. (

**b**) Bottom view of the cell. (

**c**) Front view of the cell aligned with the x-axis. (

**d**) Same as (

**c**), but rotated of 45 degrees.

**Figure 2.**(

**a**) Electronic dispersion of an isolated MoS${}_{2}$ (red dots) and the MoS${}_{2}$ layer extracted from the MoS${}_{2}$-hBN heterostructure (blue dots). Both dispersions have been computed with VASP along the same path corresponding to the primitive unit cell of MoS${}_{2}$ via band unfolding, as described in Ref. [42]. (

**b**) Phonon dispersion of MoS${}_{2}$-hBN as obtained with CP2K. (

**c**) Corresponding vibrational density-of-states (VDOS) projected onto the molybdenum and sulfur atoms (orange area) and onto the boron and nitrogen atoms (blue area).

**Figure 3.**Schematic view of a single-gate monolayer MoS${}_{2}$ field effect transistor. The source, drain, and gate regions measure ${L}_{Source}=12.5$ nm, ${L}_{Drain}=12.5$ nm, and ${L}_{Channel}=15$ nm, respectively. The top and bottom dielectric are hBN and SiO${}_{2}$. The red (cyan) box refers to the DFT domain 1 (2), which contains a MoS${}_{2}$ monolayer (with a hBN single-layer on top of it). The shaded hBN atoms and the SiO${}_{2}$ substrate only enter Poisson’s equation and do not participate in the transport calculation.

**Figure 4.**Transfer characteristics ${I}_{d}-{V}_{gs}$ at ${V}_{ds}$ = 0.7 V of the MoS${}_{2}$ FET in Figure 3. The solid line with circles (dashed line with crosses) represents the current for the MoS${}_{2}$ (MoS${}_{2}$-hBN) device when the pseudo-scattering model of Equation (4) is turned on. The triangle (star) indicates the ON-state current of the MoS${}_{2}$ (MoS${}_{2}$-hBN) FET when both the electron and phonon populations are driven out of equilibrium. They are shown on the linear scale corresponding to the right y-axis.

**Figure 5.**(

**a**) Comparison of the electron density in the DFT domain 1 (pure MoS${}_{2}$ device, solid red line) and DFT domain 2 (MoS${}_{2}$-hBN device, dashed blue line with dots) at a gate-to-source voltage ${V}_{gs}=$0.4 V. (

**b**) Electron density in the hBN layer included in the DFT domain 2 at the same gate voltage as in (

**a**).

**Figure 6.**(

**a**) Energy- and position-resolved electron current in the DFT domain 1 at ${V}_{gs}$ = 0.7 V and ${V}_{ds}$ = 0.7 V in the presence of electron–phonon scattering according to Equations (7) and (8). Red indicates high current concentrations and green indicates no current. The black curve represents the conduction band edge. The dashed blue line close to the source (drain) indicates the Fermi level in the left (right) contact. (

**b**) Same as (

**a**), but for the MoS${}_{2}$-hBN device (DFT domain 2). (

**c**) Energy-resolved electron current at the position $x=2.4\phantom{\rule{3.33333pt}{0ex}}$nm. The solid red line refers to DFT domain 1 in (

**a**), the dashed blue line refers to DFT domain 2 in (

**b**). The vertical dotted line indicates the position of the Fermi level in the source contact. (

**d**) Same as (

**c**), but for $x=14.6\phantom{\rule{3.33333pt}{0ex}}$nm. (

**e**) Same as (

**c**), but for $x=24.4\phantom{\rule{3.33333pt}{0ex}}$nm.

**Figure 7.**(

**a**) Energy- and position-resolved phonon energy current under the same conditions and for the same device as in Figure 6a. Red (blue) indicates positive (negative) currents and white indicates no current. The location of the highest phonon generation rate is marked by a vertical dashed line. (

**b**) Same as (

**a**), but for the MoS${}_{2}$-hBN device. (

**c**) Energy-resolved phonon population generated in the MoS${}_{2}$ FET (solid red line) and in the MoS${}_{2}$-hBN device (dashed blue curve with dots) close to their source contact in the phonon energy range $0<\hslash \omega <60$ meV. For the MoS${}_{2}$-hBN structure, the phonons generated in MoS${}_{2}$ (orange curve with squares) are separated from those created in hBN (green curve with stars). The sum of both lines gives the dashed blue line with dots. Inset of (

**c**), same as (

**c**) but on a logarithmic scale and over the entire MoS${}_{2}$-hBN phonon energy spectrum, i.e., from 0 to 187 meV.

**Figure 8.**Phonon dispersion of MoS${}_{2}$-hBN (blue dots) and pure MoS${}_{2}$ (red dots) as obtained with CP2K in the energy range $0<\hslash \omega <60\phantom{\rule{3.33333pt}{0ex}}$meV. The breathing mode at $\hslash \omega =8\phantom{\rule{3.33333pt}{0ex}}$meV is highlighted by a black circle.

**Figure 9.**Sketch of the first six phonon modes of MoS${}_{2}$-hBN. The yellow, violet, blue, and magenta spheres represent sulphur, molybdenum, boron, and nitrogen atoms, respectively. The red (blue) dashed arrow represents the direction of displacement of MoS${}_{2}$(hBN). (

**a**) LA mode: the two atomic planes move in phase along the x-axis. (

**b**) TA mode: same as (

**b**), but along the z-axis. (

**c**) ZA mode: same as (

**a**), but along the y-axis. (

**d**) First shearing mode: the two atomic planes move with opposite phase along the x-axis. (

**e**) Second shearing mode: same as (

**d**), but along the z-axis. (

**f**) Breathing mode: same as (

**d**), but along the y-axis.

**Figure 10.**Effective lattice temperature ${T}_{eff}$ as a function of the position along the transport direction (x-axis) in DFT domain 1 (solid red curve) and DFT domain 2 (dashed blue line with dots). The contributions coming from the different layers of the heterostructure are shown as a green line with stars for hBN and as an orange line with squares for MoS${}_{2}$.

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**MDPI and ACS Style**

Fiore, S.; Klinkert, C.; Ducry, F.; Backman, J.; Luisier, M. Influence of the hBN Dielectric Layers on the Quantum Transport Properties of MoS_{2} Transistors. *Materials* **2022**, *15*, 1062.
https://doi.org/10.3390/ma15031062

**AMA Style**

Fiore S, Klinkert C, Ducry F, Backman J, Luisier M. Influence of the hBN Dielectric Layers on the Quantum Transport Properties of MoS_{2} Transistors. *Materials*. 2022; 15(3):1062.
https://doi.org/10.3390/ma15031062

**Chicago/Turabian Style**

Fiore, Sara, Cedric Klinkert, Fabian Ducry, Jonathan Backman, and Mathieu Luisier. 2022. "Influence of the hBN Dielectric Layers on the Quantum Transport Properties of MoS_{2} Transistors" *Materials* 15, no. 3: 1062.
https://doi.org/10.3390/ma15031062