Berry Phase Modification to the Energy Spectrum of Excitons

Abstract

By quantizing the semiclassical motion of excitons, we show that the Berry curvature can cause an energy splitting between exciton states with opposite angular momentum. This splitting is determined by the Berry curvature flux through the k-space area spanned by the relative motion of the electron-hole pair in the exciton wave function. Using the gapped two-dimensional Dirac equation as a model, we show that this splitting can be understood as an effective spin-orbit coupling effect. In addition, there is also an energy shift caused by other "relativistic" terms. Our result reveals the limitation of the venerable hydrogenic model of excitons, and highlights the importance of the Berry curvature in the effective mass approximation.The effective mass approximation provides a simple yet extremely useful tool to understand a wide variety of electronic properties of semiconductors [1]. Within this approximation, electrons behave almost like free particles in response to external fields, provided that one replaces the bare electron mass with an effective mass derived from the band dispersion. Much of our intuition on electron transport is based on this semiclassical picture. However, it has been shown that such a picture is actually incomplete, and one must include the Berry curvature of the Bloch states [2]. Essentially, the Berry curvature modifies the electron dynamics through an anomalous term in the group velocity of the Bloch electrons [3,4], i.e.,where ε n (k) is the band energy, V (r) is the external potential, and Ω n (k) = i ∇ k u nk | × |∇ k u nk is the Berry curvature defined in terms of the periodic part u nk (r) of the Bloch function. The importance of the Berry curvature has been well established in a number of transport phenomena such as the anomalous Hall effect [5][6][7] and the spin Hall effect [8][9][10]. In this Letter we consider another type of problems for which the effective mass approximation must be modified to include the Berry curvature, namely, the bound state problem of Bloch electrons. To be specific, we will consider the energy spectrum of an exciton, even though our result should be equally applicable to other problems such as shallow impurity states. Our motivation is two fold. First, giant exciton binding energies (about a few hundred meV) have recently been observed in monolayers of transition metal dichalcogenides [11][12][13][14][15][16][17][18][19][20], in which the low-energy carriers behave like massive Dirac fermions with nonzero Berry curvature [21]. Thus, the detailed experimental study of excitons in the presence of the Berry curvature appears to be feasible. Secondly, there have been a few calculations of the exciton energy spectrum in these materials [18,[22][23][24][25][26][27], but the role of the Berry curvature is not explicitly discussed. We will show that, at the level of the effective mass approximation, the Berry curvature is essential to understand the exciton energy spectrum.Our main results are summarized below. We show that, quite generally, the Berry curvatu...

Optical selection rules for excitonic Rydberg series in the massive Dirac cones of hexagonal two-dimensional materials

“…We show that the entanglement of the exciton envelope function with the pseudospin texture leads to anomalous selection rules for the one-photon generation of excitons, where the d-states become bright, and with opposite valley selection rule from the s-states. Such anomalous exciton optical selection rules result from the effect on the interband process by the pseudospin texture, complimentary to the intraband correction (Berry phase correction on exciton binding 39,40 ). The latter effect cannot change the selection rule, but can quantitatively modify the transition dipole strength.…”

Influence of the effective layer thickness on the ground-state and excitonic properties of transition-metal dichalcogenide systems

“…For typical TMDC parameters, the effective coupling constant is in the range of α ∝ 3/κ − 5/κ, clearly questioning the WGA. Corrections to the WGA result both from the full relativistic dispersion and from the lifting of the degeneracy between states with opposite orbital angular momentum 31,37 Numerically solving the full DWE (32) we obtain the results shown in Fig. 4.…”

Excitonic effects in the optical properties of 2D materials:an equation of motion approach

“…This has attracted a wealth of scientific research [7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,23]. The signature of excitons appeared first in the optical measurements of monolayer MoS 2 [5], where two peaks in the absorbance, with energies ∼ 1.9 eV and ∼ 2.1 eV, were identified.…”

“…One of the most prominent features in the optical spectra of these materials is its dependence on the Berry phase, which generates a modified Rydberg series [19,20]. The form of the electron-electron interaction potential, which deviates form the Coulomb one, also contributes to a modified Rydberg series [21].…”