26
2 Entering a Two-Dimensional Materials World
Fig. 2.6 a Schematic drawing of a five-layer TMDC with conduction and valence subbands displayed in the common quantum-well representation (left and right part, respectively) with barrier levels encapsulating the well’s minimum gap. Purple arrows indicate intersubband transitions.
Blueish shaded curves on the subband energy levels display calculated spatial probability functions
for charge carriers in the respective bands, with the envelope Bloch wave shown as dotted curves.
b For various numbers N of layers, the calculated lowest intersubband transitions for holes are
compared with the measured valence band transitions for N = 4 and 5. Adapted with permission.
[148] Copyright 2018 Springer Nature
and valley pseudo-spin degree of freedoms are so-to-say locked [146, 147]. Left
(right) circularly polarised light resonant to the A (B) exciton transition will only
populate a single valley and vice versa for the other valley, whereas decoherence due
to scattering processes may repopulate excitons over time (see for instance [32]).
In most TMDC cases, the bilayer already features a momentum-space indirect
exciton I (cf. [3])—the monolayer being a good direct semiconductor. Recently,
even the common quantum-well picture and eigen-state formation in out-of-plane
direction has been revisited in an investigation that probed (with nanoimaging capabilities) experimentally intersubband transitions for a four- and five-layer TMDC
[148]. While the Bloch wave-function of a periodic system—describing the unit
cell of a crystal multiplied with a plane envelope wave—does not apply to single
layers, for which only standing wave solutions of the envelope function avoid destructive wave-function interference in the out-of-plane direction, the transition from an
“atomistic” to a periodic system naturally exhibits the increase of states (formation
of bands) with increasing number of layers from the monolayer towards the bulk
regime (cf. Fig. 2.6).
Oscillating Polarisation Waves of Coherent Excitons
In fact, due to the lack of out-of-plane dipole–dipole couplings (i.e. hybridisation,
in the sense of long-distance exchange interactions) no delocalised bulk excitons
(3D exciton–polariton states, to be more precise, cf. [149]) with typically weaker
3D binding energies (oscillator strength) are formed in monolayer TMDCs, but 2D
excitons (2D exciton–polaritons, with mere in-plane long-range exchange interactions). This is not only important for the strength of light–matter interactions, but also
relevant for the discussion of the energy–momentum dispersion of optically-active
resonances in 2D materials (see [33, 52]).
2 Entering a Two-Dimensional Materials World
Fig. 2.6 a Schematic drawing of a five-layer TMDC with conduction and valence subbands displayed in the common quantum-well representation (left and right part, respectively) with barrier levels encapsulating the well’s minimum gap. Purple arrows indicate intersubband transitions.
Blueish shaded curves on the subband energy levels display calculated spatial probability functions
for charge carriers in the respective bands, with the envelope Bloch wave shown as dotted curves.
b For various numbers N of layers, the calculated lowest intersubband transitions for holes are
compared with the measured valence band transitions for N = 4 and 5. Adapted with permission.
[148] Copyright 2018 Springer Nature
and valley pseudo-spin degree of freedoms are so-to-say locked [146, 147]. Left
(right) circularly polarised light resonant to the A (B) exciton transition will only
populate a single valley and vice versa for the other valley, whereas decoherence due
to scattering processes may repopulate excitons over time (see for instance [32]).
In most TMDC cases, the bilayer already features a momentum-space indirect
exciton I (cf. [3])—the monolayer being a good direct semiconductor. Recently,
even the common quantum-well picture and eigen-state formation in out-of-plane
direction has been revisited in an investigation that probed (with nanoimaging capabilities) experimentally intersubband transitions for a four- and five-layer TMDC
[148]. While the Bloch wave-function of a periodic system—describing the unit
cell of a crystal multiplied with a plane envelope wave—does not apply to single
layers, for which only standing wave solutions of the envelope function avoid destructive wave-function interference in the out-of-plane direction, the transition from an
“atomistic” to a periodic system naturally exhibits the increase of states (formation
of bands) with increasing number of layers from the monolayer towards the bulk
regime (cf. Fig. 2.6).
Oscillating Polarisation Waves of Coherent Excitons
In fact, due to the lack of out-of-plane dipole–dipole couplings (i.e. hybridisation,
in the sense of long-distance exchange interactions) no delocalised bulk excitons
(3D exciton–polariton states, to be more precise, cf. [149]) with typically weaker
3D binding energies (oscillator strength) are formed in monolayer TMDCs, but 2D
excitons (2D exciton–polaritons, with mere in-plane long-range exchange interactions). This is not only important for the strength of light–matter interactions, but also
relevant for the discussion of the energy–momentum dispersion of optically-active
resonances in 2D materials (see [33, 52]).