Dissipative Transport and Phonon Scattering Suppression via Valley Engineering in Single-Layer Antimonene and Arsenene Field-Effect Transistors
Jiang Cao, Yu Wu, Hao Zhang, Demetrio Logoteta, Shengli Zhang, Marco Pala
DDissipative Transport and Phonon Scattering Suppression via ValleyEngineering in Single-Layer Antimonene and Arsenene Field-EffectTransistors
Jiang Cao, Yu Wu, Hao Zhang, Demetrio Logoteta, Shengli Zhang, and Marco Pala School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094,China a)2)
Key Laboratory of Micro and Nano Photonic Structures (MOE) and Key Laboratory for Information Science ofElectromagnetic Waves (MOE) and Department of Optical Science and Engineering, Fudan University, Shanghai 200433,China b)3)
Dipartimento di Ingegneria dell’Informazione, Università di Pisa, Via G. Caruso 16, 56126 Pisa,Italy MIIT Key Laboratory of Advanced Display Materials and Devices, College of Material Science and Engineering,Nanjing University of Science and Technology, Nanjing 210094, P. R. China c)5)
Université Paris-Saclay, Centre National de la Recherche Scientifique, Centre de Nanosciences et de Nanotechnologies,91120 Palaiseau, France (Dated: 22 January 2021)
Two-dimensional (2D) semiconductors are promising channel materials for next-generation field-effect transistors(FETs) thanks to their unique mechanical properties and enhanced electrostatic control. However, the performanceof these devices can be strongly limited by the scattering processes between carriers and phonons, usually occurring athigh rates in 2D materials. Here, we use quantum transport simulations calibrated on first-principle computations toreport on dissipative transport in antimonene and arsenene n -type FETs at the scaling limit. We show that the widely-used approximations of either ballistic transport or simple acoustic deformation potential scattering result in largeoverestimation of the ON current, due to neglecting the dominant intervalley and optical phonon scattering processes.We also propose a strategy to improve the device performance by removing the valley degeneracy and suppressingmost of the intervalley scattering channels via an uniaxial strain along the zigzag direction. The method is applicableto other similar 2D semiconductors characterized by multivalley transport. I. INTRODUCTION
Recently, two-dimensional (2D) materials have shown awealth of interesting properties and possible technologicalapplications . Their mechanical properties, ultimate thin-ness and absence of surface dangling bonds are particularlysuitable for conventional and flexible electronics . The2D materials with a finite energy bandgap have the potential toreplace Si as the channel material in ultra-scaled metal-oxide-semiconductor field-effect transistors (MOSFETs) for futurenanoelectronics . However, two main issues hinder theadoption of 2D materials for MOSFET applications, namelythe high resistance between metallic contacts and semicon-ducting 2D materials and their relatively low mobility.Experimental measurements of room temperature electronmobilities in 2D materials have so far reported values lowerthan 400 cm V − s − , much smaller than the value forbulk Si (about 1400 cm V − s − ) . When employed inelectron devices in place of 3D semiconductors such as Si, Geand III-V compounds, 2D-material-based MOSFETs do notsuffer from surface roughness scattering , but have a room-temperature mobility limited by defects and electron-phononcoupling (EPC). The former, just as the contact resistance, canin principle be reduced by developing dedicated engineering a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] processes . On the contrary, the electron-phonon scatteringrepresents an intrinsic mechanism and cannot be easily sup-pressed at room temperature.Unfortunately, 2D semiconductors usually have largerelectron-phonon scattering rates with respect to their 3D coun-terparts, due to a higher density of electronic and phononicstates at energies close to the band extrema . The lowphonon-limited mobility in monolayers has been also con-firmed by several first-principle calculations based on densityfunctional theory (DFT) and on the Boltzmann transport equa-tion .In light of these considerations, it appears compulsory totake into account the electron-phonon interactions in the simu-lation of 2D material-based electron devices, in order to obtainreliable quantitative predictions of the device performance atroom temperature. Nevertheless, only a few recent simula-tion studies have fully included the effect of EPC . Inother works, the mobility has been evaluated through the Tak-agi formula , which takes into account only the Bardeen-Shockley acoustic-deformation-potential (ADP) scattering and neglects all the optical and intervalley phonon scatter-ings . However, many 2D semiconductors have multipledegenerate conduction band valleys, which results in a largephase space for intervalley scatterings and in large scatteringrates . The inelastic nature of these scattering processeshas been already shown to deeply affect the transport in elec-tronic devices .More often, only the ballistic regime is considered in sim-ulations. This approximation is usually justified by invoking a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n FIG. 1. (a) Top and side views of the single-layer antimonene andarsenene atomic structure. The unit cell and the direction of the uni-axial strain is also represented. (b) Energy contour plots of the lowestconduction band (CB) in single-layer antimonene. Six-fold degener-ate CB valleys are indicated. Three types of intervalley scatteringchannels are represented by red arrows. (c) Schematic view of thedouble-gate field-effect transistor investigated in this work. The gatelength L G varies between 4 and 9 nm. The source and drain exten-sions are doped with a donor concentration N D = 3.23 × cm − .The doping region is marked by blue color and stops exactly at thechannel-source and channel-drain interfaces. Two t OX =2 nm thickdielectric layers (HfO ) sandwich the 2D material channel and sep-arate it from the gate electrodes. Transport occurs along the x -axis, z is vertical confinement direction, and the device is assumed to beperiodic along y . (d) local k -space reference frame ( k (cid:107) , k ⊥ ) of a CBvalley and the global one ( k x , k y ), where k x and k y are the wavevectorcomponents along the transport and the transverse (periodic) direc-tion, respectively. the shortness of the considered device, though no experimen-tal evidence of ballistic transport at room temperature for 2Dmaterials other than graphene has been provided yet. There-fore, it is still unclear to which extent the electron-phononscattering affects the room-temperature transport even in de-vices with characteristic lengths of less than ten nanometers.In this paper, we report the results of room-temperature nu-merical simulations of arsenene- and antimonene-based short-channel MOSFETs with double-gate architecture. Arseneneand antimonene are among the most promising candidates aschannel materials in emerging MOSFET devices, due to theirsmall effective masses and correspondingly high theoreticalmobility . However, previous application-oriented stud-ies have been based either on purely ballistic simulations orhave considered only acoustic phonon scattering. A full treat-ment of EPC in these materials is therefore still missing.To this purpose, we use a non-equilibrium Green’s func-tion (NEGF) transport solver including acoustic and optical intra- and intervalley EPCs with parameters fully determinedfrom first-principles calculations. The NEGF transport solveris coupled with a Poisson solver for self-consistently comput-ing the electrostatic potential in the whole device domain.The main findings of this work are not limited to the spe-cific case of arsenene and antimonene, but have more generalimplications and suggest that a more careful treatment of theelectron-phonon coupling is necessary in simulating MOS-FETs based on 2D materials in order to avoid severe overes-timations of performance. In view of our results, we proposean approach to mitigate the detrimental intervalley scatteringthat is applicable not only to the case of antimonene and ar-senene, but also to many similar 2D materials characterizedby multivalley transport. II. RESULTS AND DISCUSSIONA. Conduction band valleys
For the purpose of electronic transport simulation, the con-duction band of arsenene and antimonene [a single-layer ofarsenic/antimony atoms forming a buckled hexagonal honey-comb structure, see Fig. 1(a)] can be described as six ener-getically degenerate valleys, located at the midpoint of the Γ -M line. We note that the next low-energy conduction bandvalley is 0.23 eV (0.27 eV) above the conduction band mini-mum for antimonene (arsenene), thus playing negligible rolesin the electronic transport at 300 K. In our simulations, werestrict ourselves to consider the six low-lying valleys, mod-eled within an anisotropic effective-mass approximation, asdescribed in Sec. IV A. We have verified that this is enough toinclude the relevant states for transport at the doping level andtemperature (300 K) considered in this work. In this regard,we compare in Fig. 2 the transfer characteristics computed inthe ballistic regime within our model for an antimonene-basedMOSFET with the corresponding data obtained by using anatomistic full-band model in Ref. . The good agreement be-tween our results (dashed line) and those obtained within theatomistic full-band model (dots) confirms the validity and ac-curacy of our approach. B. Electron-phonon couplings from first-principles
We compute the relevant electron-phonon couplings fromfirst-principles by using the density functional perturbationtheory (DFPT). The values of the deformation potentials andphonon mode frequencies of intravalley and intervalley EPCsare listed in Table I. The details about the DFPT calcula-tions can be found in Sec. IV C. As shown in Fig. 1(b), threetypes of intervalley EPCs connecting two electronic states lo-cated in different valleys (V1 ↔ V2, V1 ↔ V3, and V1 ↔ V4)are considered. In the transport calculations, all the intraval-ley EPCs are included, while only the intervalley EPCs with asignificant deformation potential ( > × eV/cm) are takeninto account. Ignoring the smaller EPCs has very little ef-fect on the results, since the electron-phonon self-energies de- TABLE I. Conduction band effective masses ( m (cid:107) and m ⊥ ), phonon frequencies ( ω ), and electron-phonon deformation potentials ( Ξ ) calculatedfrom DFT and DFPT for all the phonon modes in antimonene and arsenene. The values of C and Ξ Γ AC are taken from Ref . antimonene arsenene m (cid:107) m ⊥ C [eV/Å ] 1.948 † † Intravalley EPC Ξ Γ LO [eV/cm] 8.07 × × Ξ Γ TO [eV/cm] 4.90 × × ω Γ LO [meV] 20 27 ω Γ TO [meV] 16 23 Ξ Γ AC [eV] 3.265 † † intervalley EPC (V1 → V2) (V1 → V3) (V1 → V4) (V1 → V2) (V1 → V3) (V1 → V4) Ξ ZA [ eV/cm] 3.55 × × × × × × ω ZA [meV] 3.50 8.89 7.26 Ξ TA [ eV/cm] 8.51 × × × × × × ω TA [meV] 4.76 8.97 8.53 7.97 8.97 8.53 Ξ LA [ eV/cm] 2.841 × × × × × × ω LA [meV] 6.58 13.90 10.07 Ξ TO1 [ eV/cm] 4.201 × × × × × × ω TO1 [meV] 17.0 Ξ TO2 [ eV/cm] × × × × × × ω TO2 [meV] Ξ LO [ eV/cm] × × × × × × ω LO [meV] pend quadratically on the deformation potential (see Eq. (5) inSec. IV B). Recently, it has been shown that in antimonene theelectron-transverse acoustic (TA) phonon couplings are for-bidden by symmetry for both intra- and intervalley scatter-ing , in good agreement with our numerical calculation here.From Table I, we can notice that some intervalley EPCshave deformation potential values almost an order of magni-tude larger than those of the intravalley EPCs. Due to the rel-atively low phonon frequencies in antimonene and arsenene,those phonon modes can be thermally populated at roomtemperature, which leads to increased scattering of electronsthrough both emission and absorption of phonons. Therefore,the intervalley EPCs are expected to play a non-negligible rolein the electronic transport and sensibly affect the MOSFETperformance. This is confirmed by our quantum transport sim-ulations, presented in the next section. C. Quantum transport simulations with dissipation
Our aim is to provide quantitative and precise predictionsof the optimum performance of these devices, when all theextrinsic sources of scattering are eliminated. Under these hy-potheses, the electron-phonon interaction represents the dom-inant scattering mechanism at room temperature.Based on the electronic model and the EPC models pre-sented in the previous sections, we have built a dissipativequantum transport solver including all the elastic and inelas-tic electron-phonon scattering mechanisms, while still keep-ing the computational burden at the minimum level. More de-tails on the quantum transport model are given in Sec. IV B. We focus on n -type MOSFET devices. A sketch of thesimulated device structure is drawn in Fig. 1(c). We con-sider a double-gate MOSFET architecture with source anddrain extensions chemically doped at a donor concentration N D = . × cm − and gate lengths L G ranging from 4 to9 nm. The channel is undoped and has the same length of thegates. The effective oxide thickness is chosen to be 0.42 nm(corresponding to a 2 nm thick HfO high- κ dielectric layer),and the supply voltage is V DD =0.55 V, according to the ITRSspecifications .We show in Fig. 2 the transfer characteristics of armchair-antimonene-based MOSFET with L G = I DS = I OFF ≡ . , at V GS =
0. Ac-cordingly, the ON state is at V GS = V DD =0.55 V, and the cor-responding current defines the ON current ( I ON ). The sub-threshold swing (SS) is defined as the inverse slope of the I DS - V GS curve in semi-logarithmic scale in the subthresholdregime ( I DS < [ ∂ log ( I DS ) / ∂ V GS ] − .As shown in Fig. 2 for an antimonene-based MOSFET with L G = FIG. 2. I DS - V GS transfer characteristic in linear (a) and logarith-mic (b) scale for an antimonene-based double-gate transistor with L G =5 nm. The considered electron-phonon scattering processes are:intravalley scattering mediated by acoustic phonons (blue line), in-travalley scattering mediated by acoustic and optical phonons (blackline), and both intervalley and intravalley scattering mediated byacoustic and optical phonons (red line). The results obtained in theballistic transport regime (dashed lines) are compared to the values(gray dots) reported in Ref. . the associated deformation potentials. In quantitative terms,when all the significant scattering processes are included, theON current drops by a factor larger then 4 with respect to theballistic or ADP scattering approximation. Similar results areobtained in the case of the MOSFET based on arsenene andcan be found in the Supplementary Information. More in gen-eral, these data strongly suggest that modeling the transportin 2D-material-based transistors as elastic, either in the ballis-tic or in the ADP scattering approximation, is not sufficient toreliably predict the device performance. On the contrary, in-elastic intravalley and intervalley scattering mechanisms dom-inate and significantly affect the transport, even at gate lengthsas short as 5 nm.To elucidate the effects of EPC on the charge transport allalong the device, we plot in Fig. 3 the current spectrum in theOFF and in the ON states for L G = L G =9 nm).We now focus on the ON state. The current spectrum inFig. 3 (b) and (d), shows that, contrarily to the behavior in theOFF state, electrons are now mainly injected at higher ener-gies with respect to the top of the channel barrier. The cur-rent spectrum in Fig. 3 (b), obtained by including the inelas-tic scattering processes, demonstrates that electrons graduallylose energy by emitting optical phonons. The strong emissionprocesses entail a significant backscattering and explains theconsiderable decrease of the current with respect to the casein which only elastic electron-phonon interactions are consid-ered.Figure 4 summarizes the simulated transfer characteristicsof the antimonene- and arsenene-based MOSFET for L G vary-ing from 4 to 9 nm, and the transport axis along the armchairand zigzag directions. All the relevant electron-phonon scat-tering processes are included. In all the considered cases theantimonene- and arsenene-based devices exhibit very similartransfer characteristics. The extracted SS and the ON currentfor the antimonene-based MOSFET are reported in Fig. 5.The very similar data obtained for the arsenene-based tran-sistor can be found in the Supplementary Information. Weobserve that the armchair and zigzag transport directions arenot equivalent, since the rotation by π /4 needed to switch be-tween them is not a symmetry of the crystal. Particularly,while the density of states and the scattering rates associatedto the six degenerate valleys are invariant under arbitrary rota-tions, the anisotropy of the valleys makes the average effectivemass seen in the armchair and zigzag directions different. Thehigher ON current in the zigzag direction arises from a smalleraverage transport effective mass (cid:104) m ∗ t (cid:105) , and thus a higher groupvelocity. On the other hand, the smaller (cid:104) m ∗ t (cid:105) entails a largersource-to-drain tunneling (STDT), which results in higher SSvalues. D. Reducing phonon scattering by valley-engineering
Since the six-fold degenerate conduction band valleys areinherited from the crystal rotational symmetry, applying anexternal symmetry-breaking uniaxial strain along the zigzagdirection (see Fig. 1) can split the six valleys into four higher-energy and two lower-energy valleys on the armchair axis. Onthe contrary, if the uniaxial strain is along the armchair direc-tion, only two valleys will be pushed up to higher energy . FIG. 3. Current density spectrum in an antimonene-based MOSFET with L G = I DS - V GS transfer characteristics in linear and logarithmic scale of antimonene (Sb) and arsenene (As) based MOSFETs along armchairand zigzag directions, and for L G =4, 5, 7 and 9 nm. In the following, we will consider the uniaxial strain alongthe zigzag direction. A sizable splitting of ≈ ≈ . Theeffect on EPC and effective masses by such small strain hasbeen shown to be negligible . Instead, since the strain shiftsthe energy of the valleys, it effectively turns off the intervalleyscattering channels between the lower-energy and the higher-energy valleys. In addition, the transport becomes anisotropic,since the two lower-lying valleys both have the large effectivemass in the armchair direction and the small one in the zigzagdirection.Figure 6 compares the transfer characteristics of thestrained-arsenene-based MOSFET along the armchair and zigzag directions, for L G =4 and 9 nm. The transfer charac-teristics of the unstrained device are also plotted for reference.The simulations clearly shows an improvement of the ON cur-rent for all the considered configurations. Such an improve-ment is in excess of a factor 2.5 in the zigzag direction for L G =9 nm and amounts to approximately a factor 2 in the arm-chair direction for L G =4 nm. Similar improvements (a fac-tor of 2.4 in the zigzag direction for L G =9 nm and 2 in thearmchair direction for L G =4 nm) are obtained for MOSFETsbased on strained antimonene. The corresponding data are re-ported in the Supplementary Information.The SS and I ON for both arsenene- and antimonene-basedMOSFETs are quantified in Fig. 7 for L G =4, 5, 7, and 9 nm.The plots shows that, while the transistors fabricated by using FIG. 5. ON current and SS as a function of the gate length L G ex-tracted from the transfer characteristics in Fig. 4 of the antimonene-based MOSFET. the two monolayers exhibit almost identical SSs as a func-tion of L G , that based on arsenene can achieve higher valuesof I ON . In both cases, at short channel lengths ( L G < 5 nm)the best performance are obtained when the transport occursalong the armchair direction. Indeed, due to the higher ef-fective mass along this direction, the I ON is less degraded bythe STDT, which results in a better scaling behavior. On thecontrary, at longer gate lengths, the transport along the zigzagdirection is more advantageous, due to the higher injection ve-locity associated to the small effective mass. We remark thatalso the low m ⊥ / m (cid:107) ratio plays a significant role in enhanc-ing the carrier velocity, since at each given energy it entails alarger kinetic component in the transport direction. III. CONCLUSION
We have investigated the dissipative transport in single-layer arsenene and antimonene MOSFETs by performingfirst-principle-based NEGF quantum simulations including allthe relevant intra- and intervalley electron-phonon scatteringmechanisms. We have shown that even for a gate length of5 nm the ON current can be strongly decreased by the opti-cal intravalley and intervalley phonon scatterings, and that, asa consequence, the simulations in the ballistic regime and/oronly including the acoustic phonon scattering can largelyoverestimate the performance of real devices. Due to thecombination of the six-fold degeneracy of the conductionband valleys, the large magnitude of the electron-phonon cou-plings, and the low optical phonon frequencies, the intervalleyelectron-phonon scattering is the dominant scattering mech-anism in the arsenene- and antimonene-based MOSFETs atroom temperature.Moreover, we have investigated a viable approach to mit-igate the impact of intervalley phonon scattering by remov-ing the valley degeneracy via a small uniaxial strain alongthe zigzag direction. Our calculations indicate that for gatelengths larger than 5 nm, the ON current in the zigzag direc- tion can be increased by a factor 2.5. For gate lengths shorterthan 5 nm, selecting the armchair direction as the transportdirection minimizes the source-to-drain tunneling effect andresults in a two times ON-current improvement with respectto the case with unstrained material.Overall, our work can provide useful guidelines to the sim-ulation of transistors based on 2D materials, and suggests thatthere are room and opportunities to overcome the obstacles onthe way towards development of a future 2D-based electron-ics.
IV. METHODSA. The anisotropic effective mass model
The conduction band of Sb and As monolayers has six-fold degenerate valleys located at the midpoint of the Γ -Mpaths. These valleys have a larger effective mass along thehigh-symmetry Γ -M direction, denoted by m (cid:107) . The effectivemass along the direction perpendicular to Γ -M is denoted by m ⊥ . By using local k -coordinates for each valley, we can writethe conduction band energy in the anisotropic effective massmodel as ¯ h m (cid:32) k ⊥ m ⊥ + k (cid:107) m (cid:107) (cid:33) = E , (1)where ¯ h is the reduced Planck constant, m the electron mass,and k ⊥ , k (cid:107) the wavevector components along the Γ -M and theorthogonal direction, respectively. For transport calculations,we need to express k ⊥ and k (cid:107) in terms of k x and k y , namelythe wave vector components in the transport and transversedirections, respectively (see Fig. 1 (d)): (cid:40) k (cid:107) = cos θ k x + sin θ k y k ⊥ = − sin θ k x + cos θ k y (2)where θ denotes the angle between the k (cid:107) -axis and the trans-port direction x .The kinetic energy of electrons in terms of k x and k y reads:¯ h m (cid:20) k x (cid:18) cos θ m (cid:107) + sin θ m ⊥ (cid:19) + k y (cid:18) cos θ m ⊥ + sin θ m (cid:107) (cid:19) + k x k y sin θ cos θ (cid:18) m (cid:107) − m ⊥ (cid:19)(cid:21) = E . The Hamiltonian in the form employed in the simulationsis obtained by replacing k x with the differential operator − i ∂ / ∂ x . We enforce periodic boundary conditions in thetransverse direction, thus the Hamiltonian maintains a para-metric dependence on k y . The value of θ to be considered foreach valley depends on the chosen transport direction (arm-chair or zigzag). FIG. 6. I DS - V GS transfer characteristics in linear and logarithmic scale for a MOSFET based on arsenene with an uniaxial strain of 2% alongthe zigzag direction. Two different gate lengths ( L G = L G extracted from the transfer characteristics in Fig. 6 of the MOSFETsbased on strained antimonene (Sb) and arsenene (As). B. Quantum transport
To simulate the electron transport, we adopt the Keldysh-Green’s function formalism , which allows us to include theelectron-phonon coupling by means of local self-energies. Weself-consistently solve the following equations for every val-ley (cid:40) ( EI − H − Σ SD − Σ ph ]) G = IG ≶ = G ( Σ ≶ SD + Σ ≶ e − ph ) G † (3)where E is the electron energy, I is the identity matrix, H isthe anisotropic effective mass Hamiltonian matrix, Σ SD and Σ e − ph are the retarded self-energies corresponding to sourceand drain contacts and to the electron-phonon coupling, G isthe retarded Green’s function, and Σ ≶ SD , Σ ≶ e − ph and G ≶ are thecorresponding lesser-than and greater-than quantities. Equa-tions (3) are spatially discretized over a uniform mesh withstepsize equal to 0.5 Å.For the intravalley acoustic phonons, the local self-energiesat the i -th discrete space site along the transport direction at room temperature can be expressed in the quasi-elastic ap-proximation as Σ ≶ ac , ν ( i , i ; k y , E ) = Ξ k B TC ∑ q y G ≶ ν ( i , i ; k y − q y , E ) , (4)where C is the elastic modulus, Ξ ac the acoustic deformationpotential, k B the Boltzmann constant, T the temperature, q y the transverse phonon wavevector and ν is the valley index.For the intravalley optical phonons, the self-energies relatedto a phonon with branch λ can be expressed as Σ ≶ λ Γ , ν ( i , i ; k y , E ) = ¯ h Ξ λ Γ ρω Γ λ ∑ q y { G ≶ ν ( i , i ; k y − q y , E ∓ ¯ h ω Γ λ ) N ( T , ω Γ λ )+ G ≶ ν ( i , i ; k y − q y , E ± ¯ h ω Γ λ ) (cid:2) N ( T , ω Γ λ ) + (cid:3) } , (5)where ρ is the mass density, N ( T , ω Γ λ ) is the Bose-Einsteindistribution at temperature T of a phonon with frequency ω Γ λ at Γ point, and the upper/lower sign of the term ¯ h ω Γ λ corre-sponds to lesser/greater-than self-energies.The self-energies associated to the intervalley scattering areanalogously obtained as: Σ ≶ λ q , ν ( i , i ; k y , E ) = ¯ h Ξ λ q ρω q λ ∑ ν (cid:48) (cid:54) = ν ∑ q y { G ≶ ν (cid:48) ( i , i ; k y − q y , E ∓ ¯ h ω q λ ) N ( T , ω q λ )+ G ≶ ν (cid:48) ( i , i ; k y − q y , E ± ¯ h ω q λ ) (cid:2) N ( T , ω q λ ) + (cid:3) } , (6)where q is the phonon wavevectors connecting two degeneratevalley minima, and the three scattering processes shown inFig. 1 are considered.The total lesser-than and greater-than self-energies Σ ≶ e − ph are obtained by summing the self-energies associated to allthe different scattering processes. The retarded self-energy isthen computed by Σ e − ph ( i , i ; k y , E ) = [ Σ > e − ph ( i , i ; k y , E ) − Σ < e − ph ( i , i ; k y , E )] , (7)The real part of the retarded self-energy, which only con-tributes to a shift of the energy levels, is neglected.Equations (3-7) are iteratively solved within the so-calledself-consistent Born approximation (SCBA) until the con-vergence is achieved. The SCBA convergence ensures thatthe electronic current is conserved along the device struc-ture. Once the SCBA loop achieves the convergence, the elec-tron and transport current densities are determined from theGreen’s functions of all the valleys n ( i ) = − i π ∑ ν , k y (cid:90) E max E min d E G < ν ( i , i ; k y , E ) , (8)and J x ( i ) = eh ∑ ν , k y (cid:90) E max E min d E [ H i , i + G < ν ( i + , i ; k y , E ) − H i + , i G < ν ( i , i + k y , E )] , (9)where the integration range from E min to E max includes allthe energy states contributing to transport. We discretize theenergy into a uniform grid with a stepsize ∆ E = ∆ E = ∆ E is further discretized by defining 3 quadra-ture nodes. The charge and current densities are computed byfirst performing an integration over each interval by using theGauss-Legendre quadrature rule and then adding the contri-butions of all the intervals. The 2D charge density is used toset up the Poisson equation, which is solved in the device x - z cross-section. The simulation runs over a loop between theGreen’s function equations and the Poisson problem until anoverall self-consistent solution is reached. C. First-principles calculation details
The DFT simulations were performed in a plane wave basisby means of the Quantum Espresso suite . The antimoneneand arsenene monolayers were simulated by using scalar rel-ativistic norm-conserving pseudopotentials based on the lo-cal density approximation exchange-correlation functional. Inboth cases, a 16 × × × × q -point grid and a16 × × k -point grid.The electron-phonon matrix elements between two elec-tronic states | n k (cid:105) and | n k + q (cid:105) connected by a phonon mode λ of wavevector q were computed within the DFTP framework as g kq , mn λ = (cid:113) ¯ h / ( M ω λ q ) (cid:104) m k + q | ∆ V KS q λ | n k (cid:105) , (10)where M is the unit cell mass, m and n are band indices, and ∆ V KS q λ is the lattice periodic part of the perturbed Kohn-Shampotential expanded to first order in the atomic displacement.The g kq , mn λ were computed on a 16 × × k -point grid anda 8 × × q -point grid, and then interpolated on a denser gridby projection on a basis of maximally localized Wannier func-tions by using the wannier90 code and the EPW code .Six Wannier orbitals were employed to reproduce accuratelythe conduction and valence bands close to the bandgap andachieve good spatial localization of the Wannier orbitals. Theintravalley and intervalley electron-phonon coupling were de-scribed by the matrix element at q = q = k (cid:48) − k , respec-tively, where k (cid:48) and k correspond to two degenerate valleyminima. The deformation potentials were finally computed as Ξ λ q = | g kq , mn λ | × (cid:113) M ω λ q / ¯ h . (11) Supporting InformationAcknowledgements
This work is supported by Natural Science Foundationof Jiangsu Province under Grants No. BK20180456 andBK20180071, and National Natural Science Foundation ofChina under Grant No. 11374063. M.P. acknowledges fi-nancial support from the Agence Nationale de la Recherchethrough the "2D-ON-DEMAND" project. S.Z. acknowledgesfinancial support from the Training Program of the Major Re-search Plan of the National Natural Science Foundation ofChina (Grant No. 91964103). H.Z. acknowledges financialsupport from the Shanghai Municipal Natural Science Foun-dation under Grant No. 19ZR1402900.
References G. Fiori, F. Bonaccorso, G. Iannaccone, T. Palacios, D. Neumaier,A. Seabaugh, S. K. Banerjee, and L. Colombo, “Electronics based on two-dimensional materials,” Nat. Nanotechnol. , 768–779 (2014). K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. Castro Neto, “2Dmaterials and van der Waals heterostructures,” Science (2016). J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross, K. L. Seyler, W. Yao,and X. Xu, “Valleytronics in 2D materials,” Nat. Rev. Mater. , 16055(2016). S. Yang, C. Jiang, and S.-h. Wei, “Gas sensing in 2D materials,” Appl.Phys. Rev. , 021304 (2017). C. Jin, E. Y. Ma, O. Karni, E. C. Regan, F. Wang, and T. F. Heinz, “Ultrafastdynamics in van der Waals heterostructures,” Nat. Nanotechnol. , 994–1003 (2018). X. Liu and M. C. Hersam, “2D materials for quantum information science,”Nat. Rev. Mater. , 669–684 (2019). S. Das, D. Pandey, J. Thomas, and T. Roy, “The Role of Grapheneand Other 2D Materials in Solar Photovoltaics,” Adv. Mater. , 1802722(2019). S. Zhang, S. Guo, Z. Chen, Y. Wang, H. Gao, J. Gómez-Herrero, P. Ares,F. Zamora, Z. Zhu, and H. Zeng, “Recent progress in 2D group-VA semi-conductors: from theory to experiment,” Chem. Soc. Rev. , 982–1021(2018). B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis,“Single-layer MoS2 transistors,” Nat. Nanotechnol. , 147–150 (2011). H. Wang, L. Yu, Y.-H. Lee, Y. Shi, A. Hsu, M. L. Chin, L.-J. Li, M. Dubey,J. Kong, and T. Palacios, “Integrated Circuits Based on Bilayer MoS2 Tran-sistors,” Nano Lett. , 4674–4680 (2012). M. Chhowalla, D. Jena, and H. Zhang, “Two-dimensional semiconductorsfor transistors,” Nat. Rev. Mater. , 16052 (2016). J. Cao, D. Logoteta, S. Özkaya, B. Biel, A. Cresti, M. G. Pala, and D. Es-seni, “Operation and Design of van der Waals Tunnel Transistors: A 3-DQuantum Transport Study,” IEEE Trans. Electron Devices , 4388–4394(2016). G. Pizzi, M. Gibertini, E. Dib, N. Marzari, G. Iannaccone, and G. Fiori,“Performance of arsenene and antimonene double-gate MOSFETs fromfirst principles,” Nat. Commun. , 12585 (2016). D. K. Polyushkin, S. Wachter, L. Mennel, M. Paur, M. Paliy, G. Iannac-cone, G. Fiori, D. Neumaier, B. Canto, and T. Mueller, “Analogue two-dimensional semiconductor electronics,” Nat. Electron. , 486–491 (2020). A. Allain, J. Kang, K. Banerjee, and A. Kis, “Electrical contacts to two-dimensional semiconductors,” Nat. mater. , 1195–1205 (2015). A. Allain and A. Kis, “Electron and Hole Mobilities in Single-LayerWSe2,” ACS Nano , 7180–7185 (2014). L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, andY. Zhang, “Black phosphorus field-effect transistors,” Nat. Nanotechnol. ,372–377 (2014). Y. Liu, X. Duan, Y. Huang, and X. Duan, “Two-dimensional transistorsbeyond graphene and tmdcs,” Chem. Soc. Rev. , 6388–6409 (2018). T. Sohier, D. Campi, N. Marzari, and M. Gibertini, “Mobility of two-dimensional materials from first principles in an accurate and automatedframework,” Phys. Rev. Materials , 114010 (2018). L. Cheng, C. Zhang, and Y. Liu, “Why Two-Dimensional Semiconduc-tors Generally Have Low Electron Mobility,” Phys. Rev. Lett. , 177701(2020). S. Poncé, E. R. Margine, and F. Giustino, “Towards predictive many-bodycalculations of phonon-limited carrier mobilities in semiconductors,” Phys.Rev. B , 121201 (2018). S. Poli, M. G. Pala, T. Poiroux, S. Deleonibus, and G. Baccarani, “SizeDependence of Surface-Roughness-Limited Mobility in Silicon-NanowireFETs,” IEEE Trans. Electron Devices , 2968–2976 (2008). C. Grillet, D. Logoteta, A. Cresti, and M. G. Pala, “Assessment of theElectrical Performance of Short Channel InAs and Strained Si NanowireFETs,” IEEE Trans. Electron Devices , 2425–2431 (2017). W. Liao, S. Zhao, F. Li, C. Wang, Y. Ge, H. Wang, S. Wang, and H. Zhang,“Interface engineering of two-dimensional transition metal dichalcogenidestowards next-generation electronic devices: recent advances and chal-lenges,” Nanoscale Horiz. , 787–807 (2020). K. Kaasbjerg, K. S. Thygesen, and K. W. Jacobsen, “Phonon-limited mo-bility in n -type single-layer MoS from first principles,” Phys. Rev. B ,115317 (2012). M. V. Fischetti and W. G. Vandenberghe, “Mermin-Wagner theorem, flexu-ral modes, and degraded carrier mobility in two-dimensional crystals withbroken horizontal mirror symmetry,” Phys. Rev. B , 155413 (2016). T. Sohier, M. Calandra, and F. Mauri, “Two-dimensional Fröhlich interac-tion in transition-metal dichalcogenide monolayers: Theoretical modelingand first-principles calculations,” Phys. Rev. B , 085415 (2016). A. Szabó, R. Rhyner, and M. Luisier, “Ab initio simulation of single- andfew-layer MoS transistors: Effect of electron-phonon scattering,” Phys.Rev. B , 035435 (2015). D. Logoteta, J. Cao, M. Pala, P. Dollfus, Y. Lee, and G. Iannaccone, “Cold-source paradigm for steep-slope transistors based on van der Waals hetero-junctions,” Phys. Rev. Research , 043286 (2020). S. Takagi, A. Toriumi, M. Iwase, and H. Tango, “On the universality of in-version layer mobility in Si MOSFET’s: Part I-effects of substrate impurityconcentration,” IEEE Trans. Electron Devices , 2357–2362 (1994). J. Bardeen and W. Shockley, “Deformation Potentials and Mobilities inNon-Polar Crystals,” Phys. Rev. , 72–80 (1950). Y. Wang, P. Huang, M. Ye, R. Quhe, Y. Pan, H. Zhang, H. Zhong, J. Shi,and J. Lu, “Many-body Effect, Carrier Mobility, and Device Performanceof Hexagonal Arsenene and Antimonene,” Chem. Mater. , 2191–2201(2017). T. Sohier, M. Gibertini, D. Campi, G. Pizzi, and N. Marzari, “Valley-Engineering Mobilities in Two-Dimensional Materials,” Nano Lett. ,3723–3729 (2019). M. G. Pala, C. Grillet, J. Cao, D. Logoteta, A. Cresti, and D. Esseni, “Im-pact of inelastic phonon scattering in the off state of tunnel-field-effect tran-sistors,” J. Comput. Electron. , 1240–1247 (2016). C. Klinkert, Á. Szabó, C. Stieger, D. Campi, N. Marzari, and M. Luisier,“2-D Materials for Ultrascaled Field-Effect Transistors: One Hundred Can-didates under the Ab Initio Microscope,” ACS Nano , 8605–8615 (2020). Y. Wu, B. Hou, C. Ma, J. Cao, Y. Chen, Z. Lu, H. Mei, H. Shao, Y. Xu,H. Zhu, Z. Fang, R. Zhang, and H. Zhang, “Effects of intervalley scat-terings in thermoelectric performance of band-convergent antimonene,”(2020), arXiv:2001.07499 [cond-mat.mtrl-sci]. “See “International Technology Roadmap for Semiconductors” (last ac-cessed November 25, 2020),” . A. Fetter and J. Walecka,
Quantum Theory of Many-Particle Systems (Dover, New York, 2003). K. Rogdakis, S. Poli, E. Bano, K. Zekentes, and M. G. Pala, “Phonon-and surface-roughness-limited mobility of gate-all-around 3C-SiC and Sinanowire FETs,” Nanotechnology , 295202 (2009). P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavaz-zoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. DalCorso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann,C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari,F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbrac-cia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari,and R. M. Wentzcovitch, “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys.:Condensed Matter , 395502 (19pp) (2009). P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Ca-landra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna,I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. D. Jr,A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerstmann,F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçük-benli, M. Lazzeri, M. Marsili, N. Marzari, F. Mauri, N. L. Nguyen, H.-V.Nguyen, A. O. de-la Roza, L. Paulatto, S. Poncé, D. Rocca, R. Sabatini,B. Santra, M. Schlipf, A. P. Seitsonen, A. Smogunov, I. Timrov, T. Thon-hauser, P. Umari, N. Vast, X. Wu, and S. Baroni, “Advanced capabilities formaterials modelling with QUANTUM ESPRESSO,” J. Phys.: CondensedMatter , 465901 (2017). G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Giber-tini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M.Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. No-hara, Y. Nomura, L. Paulatto, S. Poncé, T. Ponweiser, J. Qiao, F. Thöle,S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A.Mostofi, and J. R. Yates, “Wannier90 as a community code: new featuresand applications,” J. Phys.: Condensed Matter , 165902 (2020). S. Poncé, E. Margine, C. Verdi, and F. Giustino, “Epw: Electron–phononcoupling, transport and superconducting properties using maximally local-ized wannier functions,” Comput. Phys. Commun.209