Many-body renormalization of the electron effective mass of InSe

The layered semiconductor InSe has a wide range of attractive electronic and optoelectronic properties, in which the effective mass of the charge carriers plays a key role. Here, we study from first principles the many-body renormalization of the electron effective mass in $\gamma$-InSe, taking into account the effects of both electron-electron and electron-phonon interactions. Electron-electron interaction, treated within the many-body $GW$ approximation, leads to around 15% of the increase in the in-plane effective mass over the result from density functional theory, and a more than threefold increase in the out-of-plane electron effective mass. The surprisingly large directional anisotropy in the mass renormalization is explained in terms of the symmetries of band-edge wave functions. The mass enhancement induced by electron-phonon interactions, which we find to mainly originate from Fröhlich electron-phonon coupling, is less than 10% at room temperature, indicating weak polaronic effect. After including the many-body renormalization effects, the calculated electron effective masses of InSe are 0.12 and 0.09 in the in-plane and out-of-plane directions, respectively.

Figure 1

The atomistic model and electronic band structure of $\gamma$-InSe. (a) Side view of the structural model, showing the layer stacking sequence of the bulk. The rhombohedral unit cell is shown alongside, and the in- and out-of-plane directions are indicated. (b) Top view of InSe monolayer and a trigonal-prismatic structural unit in the monolayer. The latter corresponds to the shaded region on the top view of the monolayer structure. (c) The first Brillouin zone corresponding to the rhombohedral unit cell. (d) The electronic band structures from many-body $G_0{W}_0$ calculation, as well as density functional theory (DFT) calculation within the local density approximation (LDA). The $G_0{W}_0$ and DFT-LDA results are represented by blue and gray lines respectively. The zero of energy corresponds to the energy maximum of the valence band.

Figure 2

Electron effective mass enhancement of $\gamma$-InSe induced by EPI. The in- and out-of-plane effective mass enhancements are calculated as a function of temperature ($T$) using both Rayleigh-Schrödinger and Brillouin-Wigner perturbation theory. (Top) In-plane electron mass enhancement (${\lambda }_{xx}$). (Bottom) Out-of-plane mass enhancement (${\lambda }_{zz}$).

Figure 3

Calculated phonon spectrum of $\gamma$-InSe. The phonons that exhibit LO-TO splitting are indicated by arrows in (a). The representative pattern of atomic displacements corresponding to the in-plane Fröhlich phonons are illustrated in (b).

Figure 4

Decomposition of phonon-mode contribution to the effective mass enhancement of the electron carriers in InSe, calculated within the Rayleigh-Schrödinger perturbation theory at 300 K. (a) and (b) represent the results for in-plane and out-of-plane effective mass enhancement respectively. The size of magenta circles superimposed on the phonon spectra is proportional to the relative contribution of each phonon mode. The results show that the dominant contributions to the effective mass enhancement originate from the Fröhlich optical phonons and long-wavelength acoustic phonons.

Figure 5

The temperature dependence of phonon-branch contributions to the effective mass enhancement of electron carriers in InSe, calculated within the Rayleigh-Schrödinger perturbation theory. We plot the in-plane effective mass enhancement (${\lambda }_{xx}$) as a function of temperature ($T$) and the respective contributions of the Fröhlich and acoustic phonons to the mass enhancement. Similar results are found for the out-of-plane effective mass enhancement.