A comprehensive study of the velocity, momentum and position matrix elements for Bloch states using a local orbital basis

Abstract

We present a comprehensive study of the velocity operator, $\hat{\mathbf{v}}=\frac{i}{\hslash }[\hat{H},\hat{\mathbf{r}}]$, when used in crystalline solids calculations. The velocity operator is key to the evaluation of a number of physical properties and its computation, both from a practical and fundamental perspective, has been a long-standing debate for decades. Our work summarizes the different approaches found in the literature, connecting them and filling the gaps in the sometimes non-rigorous derivations. In particular we focus on the use of local orbital basis sets where the velocity operator cannot be approximated by the $k$-derivative of the Bloch Hamiltonian matrix. Among other things, we show how the correct expression can be found without unequivocal mathematical steps, how the Berry connection makes its way in this expression, and how to properly deal with the two popular gauge choices that coexist in the literature. Finally, we explore its use in density functional theory calculations by comparing with its real-space evaluation through the identification with the canonical momentum operator. This comparison offers us, in addition, a glimpse of the importance of non-local corrections, which may invalidate the naive momentum-velocity correspondence.

arXiv comment12 pages, 4 figures

References in corpus (14)

• Maximally localized Wannier functions: Theory and applications
• Measurement of the Optical Conductivity of Graphene
• Excitonic Effects on the Optical Response of Graphene and Bilayer Graphene
• Ab initio calculation of the anomalous Hall conductivity by Wannier interpolation
• Seeing many-body effects in single- and few-layer graphene: Observation of two-dimensional saddle-point excitons
• Theory and it ab initio calculation of radiative lifetime of excitons in semiconducting carbon nanotubes
• Gauge covariances and nonlinear optical responses
• Optical properties of correlated materials -- Generalized Peierls approach and its application to VO2
• Excitons in boron nitride single layer
• Thermal Transport for Many Body Tight-Binding Models
• Tight-binding calculations of optical matrix elements for conductivity using non-orthogonal atomic orbitals: Anomalous Hall conductivity in bcc Fe
• Assessing the role of interatomic position matrix elements in tight-binding calculations of optical properties
• Dipole matrix element approach vs. Peierls approximation for optical conductivity
• Comment on arXiv:0709.3700 "Orientation dependence of the optical spectra in graphene at high frequencies"

Cited by in corpus (8)

• Orbital Hall effect in transition metals from first-principles scattering calculations
• Quantum correction to the orbital Hall effect
• Including many-body effects into the Wannier-interpolated quadratic photoresponse tensor
• Optical valley separation in two-dimensional semimetals with tilted Dirac cones
• Enabling the bulk photovoltaic effect in centrosymmetric materials through an external electric field
• Designer gapped and tilted Dirac cones in lateral graphene superlattices
• Decomposing Electronic Structures in Twisted Multilayers: Bridging Spectra and Incommensurate Wave Functions
• Ge as an orbitronic platform: giant in-plane orbital magneto-electric effect in a 2-dimensional hole gas