Phonon-limited transport in two-dimensional materials: A unified approach for ab initio mobility and current calculations

In this paper, we present an ab initio methodology to account for electron-phonon interactions in two-dimensional (2D) materials, focusing on transition-metal dichalcogenides (TMDCs). It combines density functional theory and maximally localized Wannier functions to acquire material data and relies on the linearized Boltzmann transport equation (LBTE) and the nonequilibrium Green’s functions (NEGF) method to determine the transport properties of materials and devices, respectively. It is shown that both the LBTE and NEGF methods return very close mobility values, without the need to adjust any parameters. The excellent agreement between the two approaches results from the inclusion of nondiagonal entries in the electron-phonon scattering self-energies. The NEGF solver is then used to shed light on the “current versus voltage” characteristics of a monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ transistor, highlighting how the interactions with phonons impact both the current magnitude and its distribution. The mobility of other TMDCs is considered as well, demonstrating the capabilities of the proposed technique to assess the potential of 2D channel materials in next-generation logic applications.

Figure 1

Top view of a $\mathrm{Mo}\mathrm{S}$${}_2$ monolayer constructed by replicating either the primitive cell of $\mathrm{Mo}\mathrm{S}$${}_2$ (green-shaded area on the left) or the orthorhombic unit cell of width $R_z$ (blue-shaded area on the right). Electrons are confined along the $y$ axis (out of plane). The primitive hexagonal unit cell of $\mathrm{Mo}\mathrm{S}$${}_2$ is used to calculate the electronic structure of the material, generate MLWFs, and solve the LBTE system, while the orthorhombic unit cell is utilized to construct the full device Hamiltonian that enters the NEGF equations.

Figure 2

(a) Schematic of a 2D single-gate transistor structure featuring a $\mathrm{Mo}\mathrm{S}$${}_2$ monolayer as the channel region. The lengths of the source, drain, and gate are denoted as $L_s$, $L_d$, and $L_g$, respectively. (b) Detailed view of the channel region, which is divided into orthorhombic unit cells. The corresponding Hamiltonian blocks are indicated; block $b$ is connected with itself (${\mathbf{\text{H}}}_{bb}$) as well as its next- (${\mathbf{\text{H}}}_{bb+1}$) and previous- (${\mathbf{\text{H}}}_{bb-1}$) neighbor blocks.

Figure 3

Top view of a $\mathrm{Mo}\mathrm{S}$${}_2$ monolayer, illustrating the interaction range considered when constructing the scattering self-energies ${\mathbf{\Sigma }}_{mn}^{e\text{-ph}}(E,k_z)$. Interactions between atoms $n$ and $m$ (highlighted in red) are included if their distance is smaller than a predefined cutoff radius $r_{\mathrm{cut}}$—i.e., if $|r_n-r_m|<r_{\mathrm{cut}}$. For distances exceeding this cutoff radius, the self-energy entries ${\mathbf{\Sigma }}_{mn}^{e\text{-ph}}(E,k_z)$ are set to zero.

Figure 4

Electronic (a)–(d) and phonon (e)–(h) band structure of monolayer $\mathrm{Mo}\mathrm{S}$${}_2$, $\mathrm{Mo}\mathrm{Se}$${}_2$, $\mathrm{WS}$${}_2$, and $\mathrm{WSe}$${}_2$, respectively. In (a)–(d), the plane-wave DFT (solid lines) and MLWF (circles) band structures are compared to each other. The position of the second conduction-band minimum at $Q$ is indicated along the high-symmetry path. The zero energy level is set to the conduction-band minimum of each material.

Figure 5

(a) Electronic and (b) phonon band structure of silicon. The same plotting conventions as in Fig. 4 are used.

Figure 6

Phonon-limited mobility of the considered TMDCs—(a) $\mathrm{Mo}\mathrm{S}$${}_2$, (b) $\mathrm{Mo}\mathrm{Se}$${}_2$, (c) $\mathrm{WS}$${}_2$, and (d) $\mathrm{WSe}$${}_2$—as a function of the carrier density for a temperature $T=100\mathrm{K}$ (dashed line with triangles), $T=200\mathrm{K}$ (dashed line with diamonds), and $T=300\mathrm{K}$ (dashed line with squares). All these results have been obtained with LBTE. The stars in subplots (a), (c), and (d) refer to NEGF calculations.

Figure 7

The channel resistance of monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ as a function of the sample length, at temperatures of 200 K and 300 K and for electron concentrations $n=1\times {10}^{12}{\mathrm{cm}}^{-2}$ and $n=5\times {10}^{13}{\mathrm{cm}}^{-2}$. The symbols represent simulation results, while the lines are used as fits and serve as inputs to the $dR/dL$ method [73].

Figure 8

Transfer characteristics $I_d$-$V_{gs}$ at $V_{ds}=0.6\mathrm{V}$ of a single-gate monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ field-effect transistor in the quasiballistic limit of transport ($I_{QB}$, blue curve with circles). The on-state current in the pure ballistic limit ($I_{\mathrm{ON},B}$, green triangle) and in the presence of electron-phonon interactions according to Eq. (35) ($I_{\mathrm{ON},e\text{-ph}}$, red star) are indicated. Both a linear and a log scale are provided for $I_{QB}$.

Figure 9

Spectral on-state current ($I_d$ as a function of the energy $E$ and position $x$) for a single-gate monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ field-effect transistor in (a) the ballistic limit of transport and (b) when the proposed electron-phonon scattering model is turned on. Red indicates high current concentration, while no current flows through the green regions. The conduction-band edge is marked by a black dashed line. The Fermi levels of the left ($E_{\text{FL}}$) and right ($E_{\text{FR}}$) contacts are highlighted in blue. The location at which the injection velocity is extracted is marked with $v_{inj}$.

Figure 10

Phonon-limited mobility of monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ as a function of the $\mathbf{\text{k}}$/$\mathbf{\text{q}}$-point density. At a $301\times 301\times 1\mathbf{\text{k}}$/$\mathbf{\text{q}}$-point grid, the phonon-limited mobility shows only a 0.3% deviation from the value obtained with a denser $501\times 501\times 1\mathbf{\text{k}}$/$\mathbf{\text{q}}$-point grid.

Figure 11

Phonon-limited mobility of monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ as calculated with LBTE (green line with squares), the maximum value of the Hamiltonian derivative $dH/dQ$ (blue line with triangles), and the mean value of $dH/dQ$ (red line with diamonds) are reported as a function of the number of atoms interacting with an arbitrary point of reference.

Figure 12

Electrical current flowing through 10-nm-long monolayer $\mathrm{Mo}\mathrm{S}$${}_2$ and $\mathrm{WSe}$${}_2$ samples under a flat-band potential and an applied voltage of 1 mV as a function of the cutoff radius $r_{\mathrm{cut}}$ applied to Eq. (35). The relative changes of the current between $r_{\mathrm{cut}}=5$ and 12 Å (18%), $r_{\mathrm{cut}}=11$ and 12 Å (6%) for $\mathrm{Mo}\mathrm{S}$${}_2$, and $r_{\mathrm{cut}}=5$ and 13.2 Å (2.5%) for $\mathrm{WSe}$${}_2$ are indicated.