13.2 Two-Dimensional Compound Semiconductors
397
(a)
(b)
Fig. 13.23 a Differential reflection contrast spectrum of WS 2 monolayer (inset: reflection contrast). X A denotes the
A-exciton, X A,T at the low energy side the trion. Based on data from [1311]. b Experimental transition energies (circles)
for WS 2 from magneto-optical data [1313] (cf. Fig.15.13) with fits using the modified hydrogen model ∝ 1/(n + δ) 2
(solid line) and a 2D hydrogen model with constant dielectric function fitted to the n = 3, 4, 5 levels (both theories
displayed for continuous n). The (very similar) band gap E g = 2.239 eV for the two models is shown as lines on the left
erostructures [1188] (cmp. Sect. 12.3.2), must be considered. The thin slab is surrounded by vacuum
which gives a huge contrast in . Even when packed, as often done in experiments, between other
layers such as h-BN ( ≈ 4.5) or on a substrate, the contrast in dielectric constant is very large.
6
This problem has been treated theoretically first in [1308, 1309] and recently rigorously in
[1310]. The exciton binding energy is typically increased by this effect to several 100 meV. Also,
the strongly anisotropic dielectric surrounding affects different quantum states differently, leading to
non-hydrogenic terms in the exciton spectrum [1311] (Fig. 13.23). Schematics of the field lines for
the 1s- and 2s-states are depicted in Fig. 13.24c, motivating that the anisotropic dielectric environment
acts differently on these states. The higher the quantum number n is, the lower is the average dielectric
constant ’seen’ by the 2D-exciton. In the example of excitons in WS 2 , the effective dielectric constant changes from about 5 for the 1s state to 2.5 for the 2s state to values close to 1 for the higher
states [1311].
In [1312] it has been discussed that the exciton level can be well described by the formula E n =
E g − R
∗
/(n − δ)
2 , with δ being a fitting parameter. Such parameter δ is known as ’quantum defect’ in
the description of alkali spectra [1314]. For WSe 2 (embedded between h-BN layers), E g = 1.873 eV,
R
∗
= 140.5 meV and δ = 0.083 has been found [1312]. The data from [1313] for WS 2 (also sandwiched
between slabs of exfoliated hexagonal h-BN) can be fitted with E g = 2.239 eV, R
∗
= 140.4 meV and
δ = 0.118 (Fig. 13.23b).
The calculated plane-view of the electron wave function of the exciton ground state (A-exciton) in
a MoS 2 monolayer is depicted in Fig. 13.24a; the contributions to the exciton wave packet stem, from
around the K-points as shown in Fig. 13.24b [1299]. Magneto-optical spectroscopy of 2D excitons and
the Zeeman splittings and diamagnetic shifts up to the 5s state in WS 2 and Mo(S,Se,Te) 2 monolayers
are reported in [1313] and (on Si/SiO 2 substrate) [1315]; the g-factor (cf. Sect. 15.2) of the exciton
is very close to 4 μ B (cf. Fig. 15.13d), a fact that is under theoretical consideration [1316]. Charged
excitons have been discussed in [1317, 1318]. The influence of various surrounding media to the TMDC
on the dielectric screening has been investigated theoretically [1319] and experimentally [1320].
6 In QW heterostructures, the differences in are typically around 1 or smaller [1188].
397
(a)
(b)
Fig. 13.23 a Differential reflection contrast spectrum of WS 2 monolayer (inset: reflection contrast). X A denotes the
A-exciton, X A,T at the low energy side the trion. Based on data from [1311]. b Experimental transition energies (circles)
for WS 2 from magneto-optical data [1313] (cf. Fig.15.13) with fits using the modified hydrogen model ∝ 1/(n + δ) 2
(solid line) and a 2D hydrogen model with constant dielectric function fitted to the n = 3, 4, 5 levels (both theories
displayed for continuous n). The (very similar) band gap E g = 2.239 eV for the two models is shown as lines on the left
erostructures [1188] (cmp. Sect. 12.3.2), must be considered. The thin slab is surrounded by vacuum
which gives a huge contrast in . Even when packed, as often done in experiments, between other
layers such as h-BN ( ≈ 4.5) or on a substrate, the contrast in dielectric constant is very large.
6
This problem has been treated theoretically first in [1308, 1309] and recently rigorously in
[1310]. The exciton binding energy is typically increased by this effect to several 100 meV. Also,
the strongly anisotropic dielectric surrounding affects different quantum states differently, leading to
non-hydrogenic terms in the exciton spectrum [1311] (Fig. 13.23). Schematics of the field lines for
the 1s- and 2s-states are depicted in Fig. 13.24c, motivating that the anisotropic dielectric environment
acts differently on these states. The higher the quantum number n is, the lower is the average dielectric
constant ’seen’ by the 2D-exciton. In the example of excitons in WS 2 , the effective dielectric constant changes from about 5 for the 1s state to 2.5 for the 2s state to values close to 1 for the higher
states [1311].
In [1312] it has been discussed that the exciton level can be well described by the formula E n =
E g − R
∗
/(n − δ)
2 , with δ being a fitting parameter. Such parameter δ is known as ’quantum defect’ in
the description of alkali spectra [1314]. For WSe 2 (embedded between h-BN layers), E g = 1.873 eV,
R
∗
= 140.5 meV and δ = 0.083 has been found [1312]. The data from [1313] for WS 2 (also sandwiched
between slabs of exfoliated hexagonal h-BN) can be fitted with E g = 2.239 eV, R
∗
= 140.4 meV and
δ = 0.118 (Fig. 13.23b).
The calculated plane-view of the electron wave function of the exciton ground state (A-exciton) in
a MoS 2 monolayer is depicted in Fig. 13.24a; the contributions to the exciton wave packet stem, from
around the K-points as shown in Fig. 13.24b [1299]. Magneto-optical spectroscopy of 2D excitons and
the Zeeman splittings and diamagnetic shifts up to the 5s state in WS 2 and Mo(S,Se,Te) 2 monolayers
are reported in [1313] and (on Si/SiO 2 substrate) [1315]; the g-factor (cf. Sect. 15.2) of the exciton
is very close to 4 μ B (cf. Fig. 15.13d), a fact that is under theoretical consideration [1316]. Charged
excitons have been discussed in [1317, 1318]. The influence of various surrounding media to the TMDC
on the dielectric screening has been investigated theoretically [1319] and experimentally [1320].
6 In QW heterostructures, the differences in are typically around 1 or smaller [1188].