Plasmonic Dispersions Under Electron and Hole Doping

present the corresponding data from full ab initio calculations [319]. Although, the resulting dispersions are on an eV range (for which our model is actually not set-up), we find a remarkable agreement with differences on the order of 100meV to 200meV («10% to20%).

Altogether, we find that the inclusion of static screening effects due to bands at high energies (i.e. background screening) is essential to derive material-realistic plasmonic dispersions upon doping. In contrast to simplified k¨pmodels [321, 322], which utilize bare (or constantly screened) Coulomb matrix elements at this stage, our interaction matrix elements are strongly reduced due to q-dependent screening effects from the electronic bands which are neglected in the k¨p models. As a result of the 2D layer geometry, these dielectric properties cannot be modeled by a simple dielectricconstant but have to be described as a q-dependent dielectric function as discussed in section 3.3.

Figure 4.20.:

Real and imaginary parts of the polarization functions (dxy{dxy channel) and EELS spectra for hole doped MoS2 without (left column) and with (right column) spin orbit coupling. The insets in (a) and (d) illustrate the Fermi surface pockets around K and K¹ in the valence band for a single spin component.

In comparison to the corresponding EELS data in Fig. 4.20 (c), we see that for higher momentum transfers (away from Γ) the EELS maxima closely follow the band-like characteristics of the polarization function. For small momenta around Γ we find a clearly separated band in the EELS spectra, which can not be seen in the real part of the polarization. This separated band arises from the well known ?q-dispersive intra-valley plasmon mode in 2D [321]. Additionally, we find a linear-dispersive mode around K stemming from an inter-valley plasmon [305]. These activation laws are consistent with the generalized expression for the plasmon dispersion relation defined by Eqs. (4.31) and (4.36) and approximated following Ref. [274]

ωpqq “~vFq d

1` rN0Wpqqs²

p1{4q `N0Wpqq, (4.37) where vF is the Fermi velocity, N0 the density of states at the Fermi level and Wpqq the macroscopic background screened Coulomb interaction of the undoped system. In the long-wavelength limit (q Ñ0) the Coulomb potential remains unscreened, i.e. in leading orderW91{q, resulting in a square-root renormalization of the otherwise linear dispersion. However, in the opposite short-range limit, i.e. at the zone boundary, the screened Coulomb potential approaches a constant, and therefore the resulting dispersion of the dielectric function is linear in q, which holds for the polarization function itself as well. Thus, for momenta away from Γ it is sufficient to study the polarization function to understand how the resulting plasmon dispersion will behave.

Of special interest are damping effects which are known to attenuate plasmon modes which merge with the particle-hole continuum. Here, the square-root mode around Γ behaves in a distinctly different manner compared to the linear modes originating at K. At sufficiently small momentum transfers q ă qc the square-root modes are con-siderably separated from the nearby particle-hole continua [Fig. 4.20 (c) and (f)] and therefore better protected from decomposition via hybridization and Landau damping [expressed as non-vanishing imaginary parts of the polarization as shown in Fig. 4.20 (b) and (e)] compared to the linear modes originating at finite momenta. In contrast, the linear plasmon modes are much closer to their neighbouring continua [Fig. 4.20 (c)], which leads to attenuation effects, reflected in reduced oscillator strength and broadening of the peaks. There is a significant difference in the oscillator strengths of these modes which can be several orders of magnitude apart as can be seen in Fig.

4.20 (c) and (f). Hence, in order to clearly detect these linear plasmon modes in ex-periments, it may prove practical to use a logarithmic scale to shield the dominant square-root mode around q“Γ, as it is done in Fig. 4.20 (c) and (f).

When we account for spin-orbit coupling the relative depth of theK andK¹ pockets shifts. In this case momentum transfer of q “ K no longer connects points on the Fermi surface belonging to different hole pockets, which results in two clearly visible characteristics in the polarization of Fig. 4.20 (d): First, at q “ K the scattering process is possible only for a finite energy difference, which opens a finite energy gap

of «250meV. Second, the Fermi surfaces at K and K¹ are now of different sizes but can still be connected with slightly smaller and larger q, resulting in gap-less linear modes originating slightly shifted from K as seen in Fig. 4.20 (d).

We conclude that the plasmonic features in hole doped MoS2 are mainly character-ized by a square-root mode in the vicinity of the Γ point and additional low contri-butions at the Brillouin zone edge which disperse linearly when SOC is disregarded.

As long as SOC is not taken into account and the K valley is occupied solely this is qualitatively very similar to the plasmonic properties of doped graphene [305]. Upon inclusion of SOC the linear plasmon mode around K is shifted leading to a gapped excitation spectra at this point.

Electron Doping

The lowest conduction band is characterized by two prominent minima aroundK and Σ. Without SOC these minima are separated by less than 100meV which is further reduced by considering SOC. Hence, in contrast to the hole doped case, small variations in the electron doping can change the Fermi-surface topology, as shown in the insets of Fig. 4.21 and Fig. 4.22. In order to study these changes, we will neglect the SOC for the beginning and choose two doping levels, resulting in Fermi surfaces comparable to the hole doped case (i.e. K valley occupation only) and a surface with additional pockets at Σ, labeled by low- and high-doping respectively (see Fig. 4.18). Since the K valley is mainly described by dz² orbitals and the Σ valley predominately by dxy

and dx²´y² states, we focus on corresponding diagonal orbital channels in Π_αβ in the following. Off-diagonal elements betweend_z² anddxy{d_x²_´y² orbitals are negligible here and off-diagonal terms between dxy{dx²´y² states are very similar to diagonal dxy{dxy

and dx²´y²{dx²´y² combinations. The corresponding polarization functions are shown along the pathΓ´Σ´K´M ´Γ through the whole Brillouin zone in Fig. 4.21 and Fig. 4.22 for dz² and dxy states, respectively.

Analogous to the hole doped case, we observe in all situations (high and low electron doping) around q “ Γ the expected plasmonic resonances arising from intra-valley scattering (either within the K or the Σ valleys). The structure of the polarization for larger q can be understood by inspecting the Fermi surface shown in Fig. 4.18:

Inter-valley scattering between similar valleys is possible for momentum transfers of q “ K (K Ø K¹ and Σ Ø Σ¹ scattering) and q “ Σ and M (Σ Ø Σ¹ scattering).

Therefore, we expect additionalinter-valley plasmon branches close to these momenta.

In principle K Ø Σ scattering can be found as well (for instance for q “ Σ or q “ M), but with strongly decreased amplitudes due to vanishing overlap matrix elements Md_z2dxy.

Indeed, we find at q«K in all situations without SOC possible excitations at zero energy. In Fig. 4.21 (a) and (b) we see the corresponding K Ø K¹ and in Fig. 4.22 (a) the Σ Ø Σ¹ modes. As expected, at momentum transfers of q “ M and Σ we find gap-less linear modes only within the dxy channel for high doping concentrations

(a) low electron doping (b) high electron doping Figure 4.21.:

Real parts of the polarization functions ford_z²{d_z² scattering at low and high electron doping concentration without SOC. The insets depict the resulting electron pockets within the lowest conduction band.

as shown in Fig. 4.22 corresponding to Σ Ø Σ¹ excitations. In contrast, within the low doping case [Fig. 4.21 (a)] we observe weak and gapped («0.1eV) excitations for these momenta originating from K ØΣ scattering.

While the SOC has a negligible effect on thed_z² valley atK, it splits the dxy{d_x²_´y² valleys at Σ resulting in minima at comparable energies. The corresponding Fermi surface for a single spin component is indicated in the inset of Fig. 4.22 (b). The six Σ points decompose into two distinct sets, Σ and Σ¹. Fermi pockets within each of these subsets are mutually connected by 2π{3 rotations and remain equivalent after inclusion of SOC, while the degeneracy ofΣandΣ¹ is lifted by SOC. As a consequence, the phase space for ΣØ Σ¹ is lost and the gap-less excitations at q« Σ and q «K must vanish, butΣØΣscattering processes are still possible. Consequently, we see in the corresponding polarization for the dxy channel with SOC in Fig. 4.22 (b) gap-less modes only atΓ and M. Since the Fermi surface around K is not changed drastically upon SOC, the corresponding polarization for the dz² channel is very similar to the one obtained without SOC [see Fig. 4.21 (b)].

Substrate Effects to the Plasmonic Dispersion

After having discussed the details of all relevant scattering channels on the basis of the orbital resolved polarization function we now turn our focus to EELS data in order to study the arising acoustic intra-valley plasmonic branch in the vicinity of the Γ point in more detail. Therefore, we investigate how the square-root mode behaves under the influence of different electron-doping levels and by considering varying dielectric

(a) without SOC (b) with SOC Figure 4.22.:

Real parts of the polarization functions for d_xy{dxy scattering at elevated electron doping concentration (K and Σ valleys are partially occupied) without and with SOC. The insets depict the resulting electron pockets within the lowest conduction band for a single spin component.

environments⁹. In Fig. 4.23 we present the resulting spectra for increasing doping levels Ef “ ´1.20eV to Ef “ ´1.05eV (top to bottom) and increasing dielectric constants of the environment (above and below the layer) fromε “1toε“50(left to right). The first three rows belong to doping situations in which the conduction band’s K{K¹ and only one of theΣvalleys are occupied, while the last row corresponds to an occupation of all K{K¹ and Σ{Σ¹ valleys.

In all situations we can clearly identify the electron-hole continua as grey-shaded areas and the plasmonic branch as sharp black lines (in most situations) separated from the continua forq ă1{2ΓΣ. For the free standing layer (i.e.Ď ε“1) we find strong variations of the plasmonic branch in dependence of the doping-level. For the lowest doping level we find a “shoulder” at about 300meV which rises in energy and flattens with increasing doping. As soon as the second Σ¹ valley is occupied, we even find a maximum within the former square-root branch close toΓ. Next to these considerable changes of the acoustic plasmonic branch, we see a broadening of theK- and Σ-valley continua (the former belongs to the “branch” with maxima aroundq “1{2ΓΣĎ and the latter maxima in the near of q “ Σ) which naturally arise due to increasing Fermi surfaces.

The most prominent effect of the dielectric environment is a considerable reduction of the plasmonic energies for small q ă 1{2ΓΣ. Due to its macroscopic nature theĎ environmental screening affects solely these long-wavelength ranges which results in

9The dependence of the dielectric environment is introduced on the level of the underlying Coulomb model which is now describing both, the background screening effects from the neglected inter-band transitionsand those stemming from the dielectric environment for the undoped system as explained in section 4.2.2.

Figure 4.23.:

EELS spectra for an increasing doping level (from top to bottom) and an increasing dielectric screening due to the environment (from left to right). On the very right we depict the corresponding Fermi surfaces / electron pockets within the lowest conduction band for a single spin component.

nearly linear (but still separated) modes for small q and low electron doping levels.

For high doping levels the square-root like shape of the acoustic branch is recovered.

In all screening scenarios shown here, we find a strong increase of the plasmonic en-ergies at small momenta as soon as the second Σ¹ valley becomes occupied resulting in a significant reshaping of the acoustic plasmonic mode. This finding might be an important observation in order to interpret and understand experimental data since the occupation of distinct electron pockets can now be traced back to corresponding changes in the acoustic plasmonic modes.