First-principles theory of field-effect doping in transition-metal dichalcogenides: Structural properties, electronic structure, Hall coefficient, and electrical conductivity

We investigate how field-effect doping affects the structural properties, the electronic structure, and the Hall coefficient of few-layers transition-metal dichalcogenides by using density-functional theory. We consider monolayers, bilayers, and trilayers of the H polytype of ${\mathrm{MoS}}_2$, ${\mathrm{MoSe}}_2$, ${\mathrm{MoTe}}_2$, ${\mathrm{WS}}_2$, and ${\mathrm{WSe}}_2$ and provide a full database of electronic structures and Hall coefficients for hole and electron doping. We find that, for both electron and hole doping, the electronic structure depends on the number of layers and cannot be described by a rigid band shift. Furthermore, it is important to relax the structure under the asymmetric electric field. Interestingly, while the width of the conducting channel depends on the doping, the number of occupied bands at each given $\mathbf{k}$ point is almost uncorrelated with the thickness of the doping-charge distribution. Finally, we calculate within the constant-relaxation-time approximation the electrical conductivity and the inverse Hall coefficient. We demonstrate that in some cases the charge determined by Hall-effect measurements can deviate from the real charge by up to 50%. For hole-doped ${\mathrm{MoTe}}_2$ the Hall charge has even the wrong polarity at low temperature. We provide the mapping between the doping charge and the Hall coefficient. We present more than 250 band structures for all doping levels of the transition-metal dichalcogenides considered within this work.

Figure 1

Band structure and density of states projected onto atomic orbitals for monolayer ${\mathrm{MoSe}}_2$ without (top) and with (bottom) including spin-orbit coupling. The energy is given relative to the valence-band maximum $E_v$. As apparent in the case without SOC, mainly in-plane states contribute to the valence-band maximum near $K$ (Mo $d_{x^2-y^2}$ and $d_{xy}$, Se $p_{x/y}$), while the maximum near $\Gamma$ is formed by out-of-plane states (Mo $d_{z^2}$, Se $p_z$). This also holds in the SOC case where the valence-band states with mainly in-plane character can be found near $K$ ($j=5/2,m_j=\pm 5/2,\pm 3/2$ and $j=3/2,m_j=\pm 3/2$) while those with more out-of-plane character can be found near $\Gamma$ ($m_j=\pm 1/2$ for both $j=5/2$ and $j=3/2$, see Ref. [51] for the states in terms of spherical harmonics). On the other hand, the conduction-band minimum near $K$ has mainly contributions from out-of-plane Mo and in-plane Se states (without SOC: Mo $d_{z^2}$ and Se $p_{x/y}$, with SOC: Mo $m_j=\pm 1/2$ for both $j=5/2$ and $j=3/2$).

Figure 2

Band structure for bilayer and trilayer ${\mathrm{MoSe}}_2$. The energy is given relative to the valence-band maximum $E_v$. Molybdenum diselenide changes (as all TMDs investigated in this paper) from a direct-band-gap semiconductor to an indirect one when the number of layers is increased.

Figure 3

(a) Schematic illustration of an FET setup in which the 2D metallic system is separated from the gate electrode by a dielectric with dielectric constant ${\varepsilon }_{ox}$ of thickness $d_{ox}$. (b) Equivalent circuit for the overall capacitance seen at the gate electrode.

Figure 4

(Left) Structure of three different structural phases of monolayer ${\mathrm{MoS}}_2$. The H polytype is the one found in the bulk compound where the coordination of the molybdenum (gray) is trigonal prismatic. The T polytype with octahedral coordination can change to the ${\text{T}}^{\prime }$ polytype in which the molybdenum atoms form zigzag chains. The sulfur atoms are shown in yellow. (Right) With increasing electron doping the difference between the total energy of 1T/1T′ structure ($E_{{\text{x-MoS}}_2}^{\mathrm{tot}}$) and the total energy of the 1H polytype ($E_{{\text{1H-MoS}}_2}^{\mathrm{tot}}$) decreases. For a doping larger than $\mathrm{n}=-0.44e/\mathrm{unit}$ cell ($n\approx -4.96\times {10}^{14}{\mathrm{cm}}^{-2}$) $1{\text{T}}^{\prime }\text{−}{\mathrm{MoS}}_2$ is the lowest-energy structure.

Figure 5

Band structure for different FET-induced doping of monolayer ${\mathrm{MoS}}_2$. The figures in the left column are for electron doping while the right column shows the hole-doping case. For monolayer ${\mathrm{MoS}}_2$ mainly the valleys at $K$ are filled and only for a high doping of $n\approx \pm 1.69\times {10}^{14}{\mathrm{cm}}^{-2}$ ($\mathrm{n}=\pm 0.15e/\mathrm{unit}$ cell) a small amount of charge is in the maximum at $\Gamma$ (hole doping) or in the minimum at $Q$ (electron doping). The band structures for more doping levels and all the other TMDs can be found in the Supplemental Material [25] (Figs. S10–S45).

Figure 6

Band structure for different FET-induced doping of trilayer ${\mathrm{MoS}}_2$. The figures in the left column are for electron doping while the right column shows the hole-doping case. In contrast to the monolayer case, the doping at $\Gamma /Q$ (hole/electron doping) is more important in trilayer ${\mathrm{MoS}}_2$. The band structures for more doping levels and all the other TMDs can be found in the Supplemental Material [25] (Figs. S10–S45).

Figure 7

Position of the different band minima/maxima $E_i$ (left) with respect to the Fermi level and relative amount of doping charge per valley $n_{\alpha }$ (right) as a function of doping for monolayer, bilayer, and trilayer ${\mathrm{MoS}}_2$. “$Q$” labels the conduction-band minimum halfway between $K$ and $\Gamma$. In each graph, the scale and the units for the lower $x$ axis are given in the lowest graph, while those of the upper $x$ axis are given in the uppermost graph. Two different line styles for the two spin-orbit-split conduction-band minima at $K$ were used in order to enhance the readability. Lines are guides for the eye.

Figure 8

Position of the different band minima/maxima $E_i$ (left) with respect to the Fermi level and relative amount of doping charge per valley $n_{\alpha }$ (right) as a function of doping for monolayer, bilayer, and trilayer ${\mathrm{MoSe}}_2$. “$Q$” labels the conduction-band minimum halfway between $K$ and $\Gamma$. In each graph, the scale and the units for the lower $x$ axis are given in the lowest graph, while those of the upper $x$ axis are given in the uppermost graph. Lines are guides for the eye.

Figure 9

Position of the different band minima/maxima $E_i$ (left) with respect to the Fermi level and relative amount of doping charge per valley $n_{\alpha }$ (right) as a function of doping for monolayer, bilayer, and trilayer ${\mathrm{MoTe}}_2$. “$Q$” labels the conduction-band minimum halfway between $K$ and $\Gamma$. In each graph, the scale and the units for the lower $x$ axis are given in the lowest graph, while those of the upper $x$ axis are given in the uppermost graph. Lines are guides for the eye.

Figure 10

Position of the different band minima/maxima $E_i$ (left) with respect to the Fermi level and relative amount of doping charge per valley $n_{\alpha }$ (right) as a function of doping for monolayer, bilayer, and trilayer ${\mathrm{WS}}_2$. “$Q$” labels the conduction-band minimum halfway between $K$ and $\Gamma$. In each graph, the scale and the units for the lower $x$ axis are given in the lowest graph, while those of the upper $x$ axis are given in the uppermost graph. Lines are guides for the eye.

Figure 11

Position of the different band minima/maxima $E_i$ (left) with respect to the Fermi level and relative amount of doping charge per valley $n_{\alpha }$ (right) as a function of doping for monolayer, bilayer, and trilayer ${\mathrm{WSe}}_2$. “$Q$” labels the conduction-band minimum halfway between $K$ and $\Gamma$. In each graph, the scale and the units for the lower $x$ axis are given in the lowest graph, while those of the upper $x$ axis are given in the uppermost graph. Lines are guides for the eye.

Figure 12

Band structure of trilayer ${\mathrm{WSe}}_2$ for a high hole doping of $n\approx +1.03\times {10}^{15}{\mathrm{cm}}^{-2}$ ($\mathrm{n}=+1e/\mathrm{unit}$ cell).

Figure 13

Planar-averaged doping-charge distribution along $z$ for trilayer ${\mathrm{MoS}}_2$ for a doping of $n\approx \pm 2.25\times {10}^{13}{\mathrm{cm}}^{-2}$ ($\mathrm{n}=\pm 0.02e/\mathrm{unit}$ cell, upper panel) and $n\approx \pm 1.69\times {10}^{14}{\mathrm{cm}}^{-2}$ ($\mathrm{n}=\pm 0.15e/\mathrm{unit}$ cell, lower panel). The dashed line within the graph indicates the end of the barrier potential, while the sketch above shows the position of the atoms (gray: Mo, yellow: S).

Figure 14

Relative doping charge per layer for bilayer and trilayer of the different dichalcogenides with increasing FET doping.

Figure 15

Total spin-valley degeneracy $\nu$ as a function of doping for the monolayer, bilayer, and trilayer system of all investigated TMDs. The total spin-valley degeneracy has been calculated by summing the spin and valley degeneracies of all doped valleys. Lines are guides for the eye.

Figure 16

Ratio of the inverse Hall coefficient $R_{xyz}^{-1}$ to the doping charge $\mathrm{n}$ as a function of doping for the monolayer (black, solid), bilayer (red, dashed), and trilayer (green, dash-dotted) of all investigated TMDs for a temperature of $T=300\text{K}$. Note also the different range of the ordinate in the case of ${\mathrm{MoTe}}_2$.

Figure 17

Hall coefficient $R_{xyz}$ as a function of doping for monolayer ${\mathrm{MoTe}}_2$ and temperatures $T=300\text{K}$, $T=100\text{K}$, and $T=50\text{K}$. The inset shows the band structure for a critical doping of $n\approx 1.48\times {10}^{14}{\mathrm{cm}}^{-2}$ ($\mathrm{n}=+0.16e/\mathrm{unit}$ cell). The specific band structure of ${\mathrm{MoTe}}_2$ with the nearly linear dispersion along $K\to \Gamma$ and the changing sign of the effective mass when increasing the distance to $K$ leads to the large difference between $R_{xyz}^{-1}$ and $\mathrm{n}$.

Figure 18

DOS at the Fermi energy $E_F$ (left, $T=0\text{K}$) and in-plane conductivity ${\sigma }_{xx}/\tau$ (right, $T=300\text{K}$) as a function of doping for the monolayer (black, solid), bilayer (red, dashed), and trilayer (green, dash-dotted) of all investigated TMDs. Note also the different range of the ordinate for $\mathrm{DOS}(E_F$) in the case of ${\mathrm{WS}}_2$ and ${\mathrm{WSe}}_2$. Lines are guides for the eye.