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A number of novel two-dimensional materials and nanostructures demonstratecomplex single-electron energy dispersion, which is called the mexican-hatdispersion. In this paper, we analyze interaction of a pair of electrons withsuch an energy dispersion. We show that relative motion of the electron pair isof a very peculiar character. For example, the real space trajectoriescorresponding to electron-electron scattering can have three reversal points,reversal points at non-zero radial momentum and other unusual features. Despitethe repulsive Coulomb interaction, two electrons can be coupled forming acomposite quasi-particle - the bi-electron. The bi-electron corresponds toexcited states of the two-electron system. Because the bi-electron coupledstates exist in continuum of extended (free) states of the electron pair, thesestates are quasi-resonant and have finite times of life. We found that rotatingbi-electron is a long-living composite quasi-particle. The rotatingbi-electrons can be in motion. For slowly moving bi-electrons, we determinedthe kinetic energy and the effective mass. Due to strongly nonparabolic energydispersion, the translational motion of the bi-electron is coupled to itsinternal motion. This results in effective masses dependent on quantum statesof the bi-electron. In the paper, properties of the bi-electron are illustratedfor the example of bigraphene in a transverse electric field.

Rotating bi-electron in two-dimensional systems with mexican-hat single-electron energy dispersion