Spin-polarized electrons in monolayer MoS

Moore's law has predicted feature sizes in the semiconductor industry for more than several decades: the number of transistors on an integrated circuit doubles every two years. However, as the feature sizes in transistors approach the nanometer scale, the semiconductor industry is hitting a physical limit: the electron flow at these small scales is limited by the quality of the interfaces in all three spatial directions.
Layered van der Waals crystals gained a great interest after K. Novoselov and A. Geim were awarded the Nobel Prize for their work on graphene. As the name indicates it, these crystals consist of weakly bound sheets with a thickness of less than a nanometer, given by the size of the unit cell of the material. The extend of the electron wavefunction in these materials is mostly limited to the plane of the sheets, so that a single sheet of a van der Waals material can be used as an active material and be a "clean" interface at the same time.
Standard semiconductor materials, based on bulk 3D crystals (silicon, germanium, gallium arsenide...) have strong chemical bonds in the three spatial directions, imposing strict rules for hetero-epitaxy such as lattice-matching. The weak binding between the sheets of van der Waals crystals makes it possible to stack different van der Waals materials without compromises within a van der Waals heterostructure, creating functional devices with semiconductor materials, metals and insulators layers within a few nanometers. There is a wide variety of van der Waals materials, and new materials continuously join this growing class of materials. Relevant to this thesis are the monolayers of the optically active transition metal dichalcogenides (TMDCs) and in particular molybdenum disulfide (MoS$_2$), molybdenum diselenide (MoSe$_2$), tungsten disulfide (WS$_2$) and tungsten diselenide (WSe$_2$). These four materials are semiconductors in the monolayer limit with a direct band-gap in the red part of the optical spectrum. It was shown that both the optical and electronic transport properties are significantly improved when the monolayer crystal is placed within two thin layers of the insulating material hexagonal boron nitride (h-BN), forming a Van der Waals heterostructure.
The lack of dangling bonds combined with the atomic thicknesses of van der Waals materials make them interesting as a way to extend Moore's law. There are however a few challenges to overcome before van der Waals materials make it to our private computers. An obvious challenge is the growth: large scale growth of van der Waals materials remains challenging. The best properties are still obtained by mechanical exfoliation (the scotch tape method, discussed in the Methods chapter). This method provides an easy way to fabricate elaborated heterostructures. Another challenge is to understand how the truly two-dimensional nature of these materials impact their physical properties. The reinforced Coulomb interaction in 2D impacts basic properties of the semiconductor, such as the effective mass of the electrons or the ground state for instance.
This thesis focusses mostly on the second challenge: understanding how the 2D nature of van der Waals materials impacts the physical properties of a material. Absorption and photoluminescence spectroscopy is used to study the optical properties of monolayer MoS$_2$ with and without the presence of free electrons.
In Section 2, the band structure of TMDCs and their optical excitation spectrum are disussed, introducing the concept of exciton. Then, Coulomb interaction in a 2D material will be introduced. The most important Coulomb matrix elements for TMDCs are derived in this Section 2. In Section 3, the fabrication techniques used for the fabrication of our sample are explained. In the same Section, the scotch tape technique and how individual few-nanometer-thick layers can be stacked to form a van der Waals heterostructure are discussed. The experimental details on photoluminescence and absorption spectroscopy are also discussed in this Section.
Excitons, the lowest energy optical excitations of an uncharged semiconductor, are formed by an electron-hole pair. When an external electric field is applied, the electron and the hole will tend to separate thus creating a dipole moment aligned with the electric field, lowering the energy of the exciton. This feedback mechanism is named the Stark effect, characterised by the polarisability of the exciton. In Section 4, we will discuss the measurement of the Stark effect in a van der Waals heterostructure formed by a single layer MoS$_2$ encapsulated in the insulator material hexagonal boron nitride. The minute polarisability that we measure unambiguously proves the 2D nature of monolayers TMDCs.
The 2D dimensionality of our sample also has dramatic consequences on the electron-electron Coulomb interaction. We present in Section 5 how we make use of optical absorption to investigate the ground state of free electrons in a directly contacted monolayer of MoS$_2$. The extreme strength of Coulomb interaction in MoS$_2$ allows a regime in which Coulomb interaction dominates over Pauli-blocking to be probed. We find that the electronic ground state is spin-polarized up to a large electron density. This spontaneous symmetry breaking was not expected by standard 2DEG , in which any long-range ferromagnetic order is excluded at finite temperature as a result of the Mermin-Wagner theorem. In MoS$_2$, the small but finite spin-orbit interaction lifts the conditions of the Mermin-Wagner theorem and allows for an Ising-type of ferromagnetic ordering. The roots of the spin-polarisation are to be found in infrared electron-hole pair excitations near the Fermi surface.
In the last part, we will discuss the dramatic effects of the spin-polarised electronic ground state on the optical properties of electron doped MoS$_2$. We will show how photoluminescence spectroscopy can be used to witness that a MoS$_2$ 2DEG undergoes a first-order phase transition between the ferromagnetic phase and the normal paramagnetic phase as the electron density increases. The first order nature of the phase transition is not expected from the standard Gainsburg-Landau theory and relies on non-analycities in the thermodynamic potential.