4. Molybdenum Disulfide Monolayers 107
4.5. Intra- and Inter-Valley Plasmonic Excitations
4.6.2. Heterostructures Induced by Heterogeneous Dielectric Environ-
Now we turn to structured dielectric environments as depicted in Fig. 4.24. Here, the broken translational symmetry makes GW calculations numerically extremely de-manding. As an alternative, we switch to a model system that mimics the essential gap-opening mechanisms and interaction effects present in semiconducting TMDCs.
At the same time it allows us to study the influences of a structured dielectric envi-ronment on the local density of states (LDOS) and the resulting spatial variation of the band gap.
Model Description in the Hartree-Fock Approximation
In order to model the lowest conduction and the highest valence band of monolayer MoS2 we consider a two-band model in momentum space described by a single-particle Hamiltonian of the form
HSPpkq “
˜ǫm||l“0pkq tK tK ǫm||l“˘2pkq
¸
. (4.42)
Here,ǫm||l“0pkqis meant to describe molybdenumdz2 states andǫm||l“˘2pkqthe combined dxy and dx2´y2 contributions. To simplify the model as much as possible, we approxi-mate ǫm||l“0pkq “ ǫ||pkq and ǫm||l“˘2pkq “ ´ǫ||pkq with the tight-binding dispersions of 2D hexagonal lattices
ǫ||pkq “t ÿ6
j“1
exppiδj¨kq, (4.43)
where t describes “in-plane” hopping between Mo nearest neighbours which are con-nected by δj. The gap arising in this model is a hybridization gap opened due to tK. In Fig. 4.26 we show the resulting band-structures [obtained by diagonalization of the
M K
Figure 4.26.:
Sketch of the band-gap opening mechanism within the applied model. The dashed lines represent a zoom of the band structure as resulting without any hybridization. The full lines visualize the full band structure, i.e. with hybridization due to tK. The region within these dispersions significantly differ is δk.
Hamiltonian given in Eq. (4.42)] for tK “ 0 (red/blue) and non-zero tK (black). For tK “0we find two non-hybridized bands with opposite slopes and a crossings between Γ and K (in the vicinity of Σpoint) and between Γ and M. Upon increasing tK ą0 the two bands are shifted by˘tK (atΓ) and hybridize at the band crossings. Thereby, a global band gap is opened which is the same mechanism as phenomenologically de-scribed in section 4.1.2 for semiconducting TMDC materials10. For the band-structures shown in Fig. 4.26 the hopping parameters are chosen to reproduce the band width (W} «1.0eV) and the DFT band gap (∆«2.0eV) of MoS2. In fact, Eq. (4.42) is the most simple description of a 2D semiconductor and thus more generic.
In real-space we realize such a system with the help of two hexagonal lattices with lattice constantsa0which arevertically separated byc. These two planes thus represent the ml “ 0 and ml “ ˘2 subsystems. The in-plane lattice constants are chosen corresponding to MoS2 a0 “ 3.18Å and the vertical distance is set to c “a{4 in the following.
As we show in section A.7.2, the inclusion of the Coulomb interaction gives rise to a real-space self-energy in the Hartree-Fock approximation according to
Σij “δij
ÿ
l
2Uilδnl
loooooomoooooon
Hartree
´Uijxc:jciy looooomooooon
Fock
, (4.44)
10Furthermore, we find comparable variations of the orbital characters throughout the Brillouin zone within the hybridized bands. The direct band gap at K is not reproduced which traces back to the neglect of the second ml“ ˘2band. The latter can be seen, for instance, in the three-orbital nearest-neighbour TB model by Liuet al.from Ref. [234].
-8 -4 0 4 8 -3
-2 -1 0 1 2
-8 -4 0 4 8 -8 -4 0 4 8
unit cell
-8 -4 0 4 8 -8 -4 0 4 8
2 1 2 1
2 1= 1 2 1= 1 = 2 = 2 2 1= 3
4 2 2 2 4 2 3 2
LDOS(a.u.)
EEf(eV)
Figure 4.27.:
Local density of states for unit cells along lines perpendicular to the dielectric interfaces.
Negative unit-cell numbers correspond to areas with ε1 and positive numbers to ε2. In all panels ε2 is fixed to 15while ε1 is decreased from15 to 5from left to right.
whereδnl “ xnˆly ´n¯ is the deviation from the average occupationn,¯ Uij “Upri,rjqis the interaction energy between electrons or ions at sites ri and rj and c:i (ci) are the corresponding electronic creation (annihilation) operators. Spin indices are suppressed as Σij is spin diagonal. The structured dielectric environments, as depicted in Fig.
4.24, enter the model via the interaction matrix elementsUij which are correspondingly screened by the dielectric function εij resulting in
Uij “ ż
ε´1pri,rlqvprl,rjqd3rl. (4.45) For heterogeneous environments the corresponding background-screened Coulomb in-teraction can be obtained from a solution of the Poisson equation. In general, numerical schemes need to be employed for this purpose, although analytical results exist for sim-plified situations. For instance, the potential for a setup as shown in Fig. 4.24 (c) with zero layer height can be analytically derived, as we sketch out in section A.7.3. For such a scenario we are now able to (self-consistently) calculate Σij and the resulting LDOS for arbitrary ε1 and ε2.
In Fig. 4.27 we present the LDOS along lines perpendicular to the dielectric inter-faces for situations with varying ε2{ε1 ratios and fixed ε2 “ 15 (thus decreasing ε1).
The very left panel of Fig. 4.27 corresponds to a homogeneous dielectric environment, while the very right panel shows the LDOS for a strongly heterogeneous situation. In all panels, two main characteristics can be clearly seen: The van-Hove singularities (from the M point, see Fig. 4.26) as maxima in the LDOS, and spatially dependent band gaps Egapprq as energy ranges where the LDOS vanishes between the singularities11.
11Since an artificial broadeningδ“5meV is involved in the evaluation of the LDOS it never vanishes
Egapprqis clearly reduced in theε2 regions (on the right hand side of each panel) com-pared to theε1 area (left hand side) as a result of stronger external screening effects of theε2 substrate and correspondingly reduced Coulomb interaction. For all given ε2{ε1 ratios, we find a nearly vanishing conduction-band offset (CBO) between both regions, while the corresponding band gaps can be tuned precisely. Thus, the ratio between the CBO and the band gaps can be controlled in these kind of heterostructure, allowing e.g.
for optimal solar cell setups [342, 343]. For all heterogeneous situations (ε2{ε1 ą 1), the overall variation of the band gaps along the spatial direction is reminiscent of a heterojunction band diagram of type-II.
This kind of band diagram will arise in all systems shown in Fig. 4.24, although the effect of the structured environment is strongest in the setups corresponding to the panels Fig. 4.24 (a) and Fig. 4.24 (c). In order to obtain strong effects in the other situations the substrate ε in Fig. 4.24 (b) or capping layer ε3 in Fig. 4.24 (d) should have a small polarizability compared to the adsorbed molecules or ε1{2, respectively.
Most importantly for electronic functionalities and particularly regarding electronic transport in these heterojunctions, the band-gap changes within less than 5 unit cells around the interface, which holds for the whole range of ε2{ε1 ratios shown in Fig.
4.27. In lateral heterojunctions made from stitching together different TMDCs, a comparable length-scale has been reported [339]. Thus, we find a similar behaviour with the difference, that here the heterojunction does not arise from different materials, but is induced externally by structuring the dielectric substrate.
There are intrinsic and extrinsic factors limiting the length scale over which band-gap variations in dielectrically induced heterojunctions can be realized. The major extrinsic factor determining the sharpness of the induced band-gap variation, is the length-scale on which the dielectric environment changes, which depends on experimental substrate or adsorbate preparation procedures. There are several experimental ways to realize nearly atomically sharp variations of the dielectric polarizability of the environment of a 2D material. Examples range from the extreme case of substrates containing holes [344, 345, 193], patterned adsorption of polarizable molecules [346, 347, 348, 349], and intercalation or adsorption of atoms [350, 351] to self-organized growth of structured dielectrics by epitaxial means [352, 353, 354, 355].
A lower intrinsic bound δr for the length scale, on which the band-gap variation takes place is defined by the spatial extent of the self-energy which can be deduced qualitatively from the underlying model. According to Eq. (4.44) the range of the self-energy is limited by the real-space decay range of the correlation functions xc:jciy. In reciprocal space this extent translates to the region δk (see right panel of Fig.
4.26) in which we find significant hybridization between the two layers which we can approximate on the basis of the Wannier Hamiltonian from Eq. (4.42). Using this Hamiltonian we find a hybridization gap of the order „ tK around ǫ}pkq “ 0, as illustrated in the right panel of Fig. 4.26. Thus, the single-particle band structure is
completely. Therefore we consider values smaller than0.02as zero.
-5 0 5
-5 0 5
-5 0 5
-5 0 5
AA x 5 BA
AB BB x 5
−0.6 −0.4 −0.2Σ0.0(eV) 0.2 0.4 0.6
R( ˚A)
Figure 4.28.:
Real-space representation of the Hartree-Fock self-energyΣij in the ε1 “5 area of the system shown in Fig. 4.27 (unit cell “´10”). The grey lines mark the hexagonal Wigner-Seitz unit cells. Note, the diagonal elements were enhanced by a factor of 5.
significantly changed due to the hybridization in a region extending aboutδk“tK{vF (see right panel of Fig. 4.26) with vF being the Fermi velocity which is proportional to the band width W 9 t. By the uncertainty principle, the momentum-space extent δk translates into a range δr 9 1{δk « t{tK of the correlation functions xc:jciy in real space. As a consequence, Σij is generically limited to the scale of a few unit cells as long as hybridization (tK) and band width (W 9 t) are similar in size.
This finding is reflected in the numerical data for the non-local real-space self-energy Σij depicted in Fig. 4.28. In analogy to the discussion of the MoS2 GW self-energy in the homogeneous case, we show in Fig. 4.28 the self-energy in the middle of theε1“5 area of the heterostructure (unit cell “´10” in the right most panel of Fig. 4.27). Here, the local dielectric environment is essentially homogeneous and thus comparable to the fully homogeneous case. The off-diagonal self-energy terms shown in Fig. 4.28 are by definition non-local, as they describe modulations of interlayer couplings (separated by Rz “c), but significant contributions are limited to a single unit cell. For orbitals in the same layer (diagonal panels in Fig. 4.28), the self-energy is smaller, and substantial contributions are limited to about two unit cells. Hence, the real-space structure of the model self-energy in this homogeneous-like area of the system is quite similar to the self-energy in MoS2 obtained from full ab initio calculations (see Fig. 4.28).
More specifically, the Hartree contribution of the self-energy ΣH [see Eq. (4.44)] is diagonal and has hardly any effect on the band structure. Especially the non-local Fock terms ΣF, as we show in more detail in section A.7.4, increase the band gap by modifying the hybridization (as seen in ΣAB{ΣBA in Fig. 4.28). This tendency is independent of the dielectric constant and inherent to all semiconducting 2D materials in which band gaps result from hybridization effects. Consequently, in all materials of this kind, heterojunctions can be induced by an external manipulation of the Coulomb interaction.