Relation between the interband dipole and momentum matrix elements in semiconductors

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Abstract

It is shown that a frequently used relation between the interband momentum and dipole matrix elements (shortened to the "p-r relation") in semiconductors acquires an additional correction term if applied to finite-volume crystals treated with periodic boundary conditions. The correction term, which is a generalization of the one obtained by Yafet for infinite crystals, does not vanish in the limit of infinite volume. We illustrate this with numerical examples for bulk GaAs and GaAs superlattices. The persistence of the correction term is traced to the subtle nature of the dipole matrix element with spatially extended wave functions. In contrast, a straightforward application of the findings by Blount and Haug [Theoretical Solid State Physics (Pergamon, Oxford, 1972)] yields the usual p-r relation in the distribution sense, without any corrections, when Bloch wave functions normalized to delta functions in crystal momentum space are used. Our findings therefore show that, for the interband dipole matrix element, using Bloch wave functions under periodic boundary conditions is not the proper way to approach the infinite-volume limit. From our numerical evaluations, we find that the correction term is large in the case of interband transitions in bulk GaAs, and that it can be chosen to be small in the case of intersubband transitions in superlattices, which are important in the context of terahertz (THz) radiation. We also show that one can interpret the infinite-volume p-r relation in terms of a limiting procedure using progressively broadened wave packet states that approach delta-normalized Bloch wave functions. Finally, we discuss the p-r relation for nanostructures in the envelope function approximation and show that the cell-envelope factorization of the nanostructure dipole matrix element into a cell-matrix element and an envelope overlap integral involves the cell gradient-k rather than the cell dipole matrix element.

Original languageEnglish (US)
Article number125301
JournalPhysical Review B - Condensed Matter and Materials Physics
Volume87
Issue number12
DOIs
StatePublished - Mar 4 2013

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Momentum
Semiconductor materials
dipoles
Wave functions
momentum
matrices
wave functions
envelopes
Superlattices
cells
Crystals
superlattices
Nanostructures
Boundary conditions
Solid state physics
boundary conditions
crystals
Delta functions
Wave packets
solid state physics

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Electronic, Optical and Magnetic Materials

Cite this

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title = "Relation between the interband dipole and momentum matrix elements in semiconductors",
abstract = "It is shown that a frequently used relation between the interband momentum and dipole matrix elements (shortened to the {"}p-r relation{"}) in semiconductors acquires an additional correction term if applied to finite-volume crystals treated with periodic boundary conditions. The correction term, which is a generalization of the one obtained by Yafet for infinite crystals, does not vanish in the limit of infinite volume. We illustrate this with numerical examples for bulk GaAs and GaAs superlattices. The persistence of the correction term is traced to the subtle nature of the dipole matrix element with spatially extended wave functions. In contrast, a straightforward application of the findings by Blount and Haug [Theoretical Solid State Physics (Pergamon, Oxford, 1972)] yields the usual p-r relation in the distribution sense, without any corrections, when Bloch wave functions normalized to delta functions in crystal momentum space are used. Our findings therefore show that, for the interband dipole matrix element, using Bloch wave functions under periodic boundary conditions is not the proper way to approach the infinite-volume limit. From our numerical evaluations, we find that the correction term is large in the case of interband transitions in bulk GaAs, and that it can be chosen to be small in the case of intersubband transitions in superlattices, which are important in the context of terahertz (THz) radiation. We also show that one can interpret the infinite-volume p-r relation in terms of a limiting procedure using progressively broadened wave packet states that approach delta-normalized Bloch wave functions. Finally, we discuss the p-r relation for nanostructures in the envelope function approximation and show that the cell-envelope factorization of the nanostructure dipole matrix element into a cell-matrix element and an envelope overlap integral involves the cell gradient-k rather than the cell dipole matrix element.",
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N2 - It is shown that a frequently used relation between the interband momentum and dipole matrix elements (shortened to the "p-r relation") in semiconductors acquires an additional correction term if applied to finite-volume crystals treated with periodic boundary conditions. The correction term, which is a generalization of the one obtained by Yafet for infinite crystals, does not vanish in the limit of infinite volume. We illustrate this with numerical examples for bulk GaAs and GaAs superlattices. The persistence of the correction term is traced to the subtle nature of the dipole matrix element with spatially extended wave functions. In contrast, a straightforward application of the findings by Blount and Haug [Theoretical Solid State Physics (Pergamon, Oxford, 1972)] yields the usual p-r relation in the distribution sense, without any corrections, when Bloch wave functions normalized to delta functions in crystal momentum space are used. Our findings therefore show that, for the interband dipole matrix element, using Bloch wave functions under periodic boundary conditions is not the proper way to approach the infinite-volume limit. From our numerical evaluations, we find that the correction term is large in the case of interband transitions in bulk GaAs, and that it can be chosen to be small in the case of intersubband transitions in superlattices, which are important in the context of terahertz (THz) radiation. We also show that one can interpret the infinite-volume p-r relation in terms of a limiting procedure using progressively broadened wave packet states that approach delta-normalized Bloch wave functions. Finally, we discuss the p-r relation for nanostructures in the envelope function approximation and show that the cell-envelope factorization of the nanostructure dipole matrix element into a cell-matrix element and an envelope overlap integral involves the cell gradient-k rather than the cell dipole matrix element.

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