4. Molybdenum Disulfide Monolayers 107
4.4. Optical Properties Under Electron-Hole Excitations
4.4.1. Absorption Spectra
The linear absorption spectrum of MoS2 is calculated by solving the semiconductor Bloch equation (SBE)
d
dtψhepk, tq “ ´irε˜hpkq `ε˜epkq ´iγsψhepk, tq (4.21)
`iΩhepk, tq r1´fhpk, tq ´fepk, tqs in time for the microscopic inter-band polarizations
ψhepk, tq “ xahpkqaepkqy (4.22) and the conventional population functions
fhpk, tq “A
a:hpkqahpkqE
and fepk, tq “@
a:epkqaepkqD
, (4.23)
where we set ~ “ 1. Here, ae{hpkq (a:e{h) annihilates (creates) an electron or hole with momentum k in the conduction or valence band, respectively. Since non-linear optical properties will not be considered, we neglect the equations of motion for the population functions and assume Fermi distributions of the electrons and holes. ε˜e{hpkq describe electron or hole energies which might berenormalized due to additional charge carriers in the valence or conduction bands resulting from electron or hole doping or electron-hole excitations (called plasma in the following). Ωhepk, tq is the generalized Rabi energy given by
Ωhepk, tq “dhepkq ¨Eptq ` 1 A
ÿ
k1h1e1
Vkkeh11kkhe11ψh1e1pk, tq (4.24) containing the plasma-screened Coulomb potential V, dipole transition matrix ele-ments dhepkq and the electric field Eptq. Furthermore, we assumed an artificial de-phasing of γ “ 10meV. The macroscopic polarization of the system as a response to the electric field is calculated as
Pptq “ ÿ
k,h,e
pψkhedhek `c.c.q. (4.25) From this, the linear absorption spectrum is obtained using classical electrodynamics by considering an electric field propagation vertical to the single-layer plane of δ-extension which yields reflection (R) and transmission (T) coefficients and thus the absorption by α “1´R´T. More details on each ingredient and the numerics can be found section A.5.1.
Σ Σ Σ
K K
K ´0.6%strain
`0.6%strain unstrained
2 2.2 2.4 2.6
0.2 0.2 0.2
0.4 0.4 0.4
0.6 0.6 0.6
energy (eV)
absorption
A B
A1 C
Figure 4.15.:
Ground-state absorption spectra for free-standing MoS2 using a0 “ 3.16, 3.18 and 3.20Å from top to bottom, the middle panel is representing the unstrained monolayer.
A phenomenological homogeneous broaden-ing of 10meV (hwhm) has been used. The spectra are terminated on the high-energy side by the onset of the unbound continuum states. Strain-induced band shifts cause the conduction-band minimum to change from K to Σ, thereby creating an indirect band gap. Relevant segments of the band-structure, including SOC, are shown as in-sets.
Ground-State Spectral Properties
For the zero-density case (i.e. no additional electrons or holes), the ground-state absorption spectrum, which corresponds to fλpkq “ 0, V “ W and ε˜ “ ε (hence no additional renormalizations, neither to ε˜nor to V), is shown in Fig. 4.15 for the unstrained case (middle) and˘0.6%biaxial strain of the monolayer5 (top and bottom).
We find in all three cases a series of peaks comparable to the spectra discussed in section 4.1.4. In more detail, our calculations for the unstrained case yield two peaks around 2.1eV and 2.2eV corresponding to the excitonic A and B transitions separated by the valence-band splitting of 130meV. While the splitting is in very good agreement with experimental findings (see section 4.1.4), the absolute positions of these peaks are slightly blue-shifted. This is most likely an artifact arising due to not fully converged GW calculations. As discussed in section 4.1.2, it is a tough task to gain highly reliable band gaps within aG0W0 calculation concerning thek-mesh convergence. On the basis of the data presented in Ref. [197] we estimate that a fully converged calculation would yield a band-gap reduction of„100meV. Since the binding energies of570and580meV for A and B, respectively, are not changed we expect a red-shift of all spectra by a comparable amount. Thereby the A peak would be in the range of1.9eV as observed in experiments.
Regarding strain, we find a redshift of the spectrum with increasing lattice constant (trend in Fig. 4.15 from top to bottom). No present work is known to us where
bi-5 These strains correspond to lattice constants ofa0 “ 3.16Å anda0 “ 3.20Å. We redid all GW band-structure calculations and Wannier interpolations and obtained band gaps of 2.80eV and 2.66eV atK{K1, respectively. Hence, strain decreases the direct band gap. In the former case (a0 “3.16Å), the direct band gap is lost due to a lowering of the conduction-band minimum at Σ, shown in the insets of Fig. 4.15. We do not change the Coulomb-interaction model since lattice constant variations of this order are known to have a negligible impact [204].
Figure 4.16.:
Normalized excitonic wave functions for the ground-state A peak, the first excited-state A1 peak and the C peak (from left to right) for the unstrained lattice constant of a0 “3.18Å.
Shown is the extent over the first Brillouin zone with the K (lowermost corner) and K1 (topmost corner) valleys in the six corners and theΓ point in the middle. While the ground-state wave function is positive over the whole Brillouin zone, the excited-ground-state wave function crosses the zero plane along one closed line (see white/red circles around the maxima at K in the middle panel).
axial tensile strain is systematically studied in freestanding monolayer MoS2. However, in comparison to mono-axial strain of MoS2 on a substrate our results of 110meV{% exceeds the literature values by about a factor of two [248, 293].
To gain more insight into the nature of the MoS2 bound states, we use the Fourier transform of the microscopic polarizationsψhepk, tqfor each transition between valence and conduction bands theu,
ψpk, ωq “ ÿ
theu
ż`8
´8
dt ψhepk, tqeiωt, (4.26) to obtain what we refer to as the excitonic wave function,
χpk, ωq “ ψpk, ωq
Epωq . (4.27)
For excitation with circularly polarized light, theA-peak wave function, corresponding to an energy of 2.07eV, is shown in the left panel of Fig. 4.16 for the unstrained lattice. We find strong contributions at the K valley due to direct dipole transitions, whereas the contributions from theK1 valley are mainly caused by weak dipole matrix elements, augmented by Coulomb mixing with the K valley. The Bohr radius of the real-space wave function is1nm, in agreement with Ref. [194]. Higher excited states of theAexciton are found for example at2.35(A1) and2.45eV (A2) and can be identified by nodes of the wave functions (A1 wave function is shown in the middle panel of Fig.
4.16). The series of bound K-valley states ends at the onset of the continuum of
unbound states which corresponds to the quasiparticle band gap (line endings in Fig.
4.15 on the high-energy side). We identify the prominent peak at about2.5eV with the C transition discussed in Ref. [194]. In the ground-state spectra, it is superimposed with an excited state from the K valley. The corresponding wave function is shown in the right panel of Fig. 4.16 and exhibits a strong component around the Γ point.
Finite-density Spectral Properties
The SBE offer the distinct advantage that the influence of excited carriers can be explicitly included, giving access to the density-dependent optical response. We assume the system to be in a thermal quasi-equilibrium state described by Fermi functions fλpkq with given temperature, carrier density, but different chemical potentials for electrons and holes. Experimentally, this situation might be realized by exciting the system optically, such that equal electron and hole densities are generated, and letting relaxation processes bring the system into a quasi-equilibrium state.
The presence of carriers in the system leads to Pauli blocking of occupied states, plasma screening of the Coulomb interaction, as well as band-structure and Rabi-frequency renormalizations. The Coulomb interaction is screened due to the optically excited plasma via
Vkkλλ11kkλ1λ1 “ε´1plasmap|k´k1|qWkkλλ11kkλ1λ1 (4.28) in addition to the dielectric screening of the background charges as discussed in section 4.2.2. The Fermi functions and band-structure renormalizations are calculated self-consistently in advance before we solve the SBE. Thus, the Fermi functions do not experience dynamical changes in time. The Pauli blocking is naturally included due to the occurrence of population factors in the SBE as well as in formula describing the plasma screening [see Eq. (A.14)]. The renormalizations are treated in the Coulomb-Hole Screened-Exchange (COHSEX) approximation as described in section A.5.1.
In Fig. 4.17 room-temperature spectra for carrier densities from 0 to1013cm´2 are shown. We find a sizable band-gap shrinkage of more than 500meV (right panel of Fig. 4.17), causing the higher peaks from the K valley to be successively absorbed into the band edge. The relative exciton absorption strength decreases (which is called bleaching) at moderate carrier densities. The positions ofA(andB) transitions exhibit a redshift from2.07to 2.00eV (2.21 to2.15eV) with increasing carrier density, which is a consequence of the competition between the shrinking band gap and the reduction of the binding energy due to plasma screening and Pauli blocking. The real-space Bohr radius increases from 1.0nm in the unexcited system to 1.9nm at a density of 3¨1012{cm2, where the exciton is almost fully absorbed by the band edge. Unlike the K-valley exciton, the C-peak exciton is stable against increasing carrier densities up to 1013{cm2. Since the C-peak’s wave function exhibits significant weight all over the Brillouin zone, it can not be described within a simplified effective-mass picture at the
2.0 2.2 2.4 2.6 0.4
0.4 0.4 0.4 0.4 0.4
0.2 0.2 0.2 0.2 0.2 0.2
10−1 100 101
−7
−5
−3
10−1 100 101 0
1 2 3
0{cm2 1¨1011{cm2 3¨1011{cm2 1¨1012{cm2 3¨1012{cm2 1¨1013{cm2
density (1012/cm2) density (1012/cm2)
absorption
energy (eV)
gapshift(0.1eV)bindingenergy(0.1eV)
Figure 4.17.:
Left: Monolayer MoS2 optical absorption spectra for carrier densities from0to1013{cm2 and 300K. Calculations are shown for the unstrained structure. Right: Gap shifts and binding energies belonging to theA (solid line) andB (dashed line) exciton transitions for increasing carrier densities.
K and K1 valleys alone. The overall finite density results are strongly supported by recent experimental findings [289].
Finite-density results under strain are provided in section A.5.2 and exhibit the same qualitative behaviour. A difference shows up in the amount of redshift of the K-valley exciton, which is due to the different population of conduction band minima and corresponding Hartree-Fock renormalizations as well as plasma screening contributions at elevated carrier densities. This in turn is a consequence of the changing energetic position of theΣpoint under strain. The results again demonstrate that a description of spectral properties in the excited system requires sampling of the whole first Brillouin zone and not just the K and K1 valleys.