Second-order optical response in semiconductors
Abstract
We present a new general formalism for investigating the second-order optical response of solids, and illustrate it by deriving expressions for the second-order susceptibility tensor ${\ensuremath{\chi}}_{2}(\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\Sigma}};{\ensuremath{\omega}}_{\ensuremath{\beta}},{\ensuremath{\omega}}_{\ensuremath{\gamma}}),$ where ${\ensuremath{\omega}}_{\ensuremath{\Sigma}}={\ensuremath{\omega}}_{\ensuremath{\beta}}+{\ensuremath{\omega}}_{\ensuremath{\gamma}},$ for clean, cold semiconductors in the independent particle approximation. Based on the identification of a polarization operator $\mathbf{P}$ that would be valid even in a more complicated many-body treatment, the approach avoids apparent, unphysical divergences of the nonlinear optical response at zero frequency that sometimes plague such calculations. As a result, it allows for a careful examination of $\mathit{actual}$ divergences associated with physical phenomena that have been studied before, but not in the context of nonlinear optics. These are (i) a coherent current control effect called ``injection current,'' or ``circular photocurrent,'' and (ii) photocurrent due to the shift of the center of electron charge in noncentrosymmetric materials in the process of optical excitation, called ``shift current.'' The expressions we present are amenable for numerical calculations, and we demonstrate this by performing a full band-structure calculation of the shift current coefficient for GaAs.
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Circular injection currents can occur within 18 of the 21 noncentrosymmetric crystal classes, but are forbidden for zincblende and certain other crystals due to antisymmetry under permutation of the last two coordinates.
“Circular injection currents, for instance, can occur within 18 of the 21 noncentrosymmetric crystal classes, but are forbidden for (zincblende), , and crystals, due to the antisymmetry of under permutation of the last two coordinates42.”
From Ultrafast terahertz emission from emerging symmetry-broken materials (2023) · cites this paper as [42]
Injection currents involve asymmetric carrier excitation in k-space with a net group velocity, due to helicity-dependent quantum interference between different polarization components of a circularly polarized excitation beam.
“Injection currents, on the other hand, involve asymmetric carrier excitation in -space with a net group velocity, due to helicity-dependent quantum interference between different polarization components of a circularly polarized excitation beam42,43.”
From Ultrafast terahertz emission from emerging symmetry-broken materials (2023) · cites this paper as [42]
Cites
Self-Consistent Equations Including Exchange and Correlation Effects
Kohn, W., & Sham, L. J. (1965) · Physical Review · doi:10.1103/physrev.140.a1133 · Cited by 64,135 · see citing works
Hohenberg, P., & Kohn, W. (1964) · Physical Review · doi:10.1103/physrev.136.b864 · Cited by 53,541 · see citing works
Perdew, J. P., Chevary, J. A., Vosko, S. H., Jackson, K. A., Pederson, M. R., Singh, D. J., & Fiolhais, C. (1992) · Physical review. B, Condensed matter · doi:10.1103/physrevb.46.6671 · Cited by 22,217 · see citing works
Maximally localized generalized Wannier functions for composite energy bands
Marzari, N., & Vanderbilt, D. H. (1997) · Physical review. B, Condensed matter · doi:10.1103/physrevb.56.12847 · Cited by 4,715 · see citing works
Macroscopic polarization in crystalline dielectrics: the geometric phase approach
Resta, R. (1994) · Reviews of Modern Physics · doi:10.1103/revmodphys.66.899 · Cited by 2,567 · see citing works
Symmetry and Strain-Induced Effects in Semiconductors
Bir, G. L., & Pikus, G. E. (1974) · Cited by 2,312 · see citing works
Quantum Theory of the Optical and Electronic Properties of Semiconductors
Haug, H., Koch, S. W., & Keldysh, L. V. (1994) · Physics Today · doi:10.1063/1.2808410 · Cited by 1,626 · see citing works
Fuchs, M., & Scheffler, M. (1999) · Computer Physics Communications · doi:10.1016/s0010-4655(98)00201-x · Cited by 1,485 · see citing works
Ab InitioCalculation of Excitonic Effects in the Optical Spectra of Semiconductors
Albrecht, S., Reining, L., Sole, R. D., & Onida, G. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.4510 · Cited by 954 · see citing works
Electron-Hole Excitations in Semiconductors and Insulators
Rohlfing, M., & Louie, S. G. (1998) · Physical Review Letters · doi:10.1103/physrevlett.81.2312 · Cited by 950 · see citing works
Nonlinear optical susceptibilities of semiconductors: Results with a length-gauge analysis
Aversa, C., & Sipe, J. E. (1995) · Physical review. B, Condensed matter · doi:10.1103/physrevb.52.14636 · Cited by 903 · see citing works
Self-citationBockstedte, M., Kley, A., Neugebauer, J., & Scheffler, M. (1997) · Computer Physics Communications · doi:10.1016/s0010-4655(97)00117-3 · Cited by 692 · see citing works
Optical Absorption of Insulators and the Electron-Hole Interaction: AnAb InitioCalculation
Benedict, L. X., Shirley, E. L., & Bohn, R. B. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.4514 · Cited by 499 · see citing works
Quantum Theory of the Optical and Electronic Properties of Semiconductors
Haug, H., & Koch, S. W. (1990) · WORLD SCIENTIFIC eBooks · doi:10.1142/0936 · Cited by 468 · see citing works
Theory of the bulk photovoltaic effect in pure crystals
Baltz, R. V., & Kraut, W. (1981) · Physical review. B, Condensed matter · doi:10.1103/physrevb.23.5590 · Cited by 455 · see citing works
Acceleration of Electrons in a Crystal Lattice
Houston, W. V. (1940) · Physical Review · doi:10.1103/physrev.57.184 · Cited by 407 · see citing works
Linear optical response in silicon and germanium including self-energy effects
Levine, Z. H., & Allan, D. C. (1989) · Physical Review Letters · doi:10.1103/physrevlett.63.1719 · Cited by 392 · see citing works
Symmetry Principles in Solid State and Molecular Physics
Barton, G. (1975) · Physics Bulletin · doi:10.1088/0031-9112/26/7/031 · Cited by 391 · see citing works
Corso, A. D., Pasquarello, A., Baldereschi, A., & Car, R. (1996) · Physical review. B, Condensed matter · doi:10.1103/physrevb.53.1180 · Cited by 273 · see citing works
Quantum Theory of the Solid State
(1974) · Elsevier eBooks · doi:10.1016/c2013-0-10455-2 · Cited by 265 · see citing works
Symmetry Principles in Solid State and Molecular Physics
Lax, M. D., & Herman, F. (1976) · Physics Today · doi:10.1063/1.3023427 · Cited by 194 · see citing works
Aspnes, D. E. (1972) · Physical review. B, Solid state · doi:10.1103/physrevb.6.4648 · Cited by 169 · see citing works
Ghahramani, E., Moss, D., & Sipe, J. E. (1991) · Physical review. B, Condensed matter · doi:10.1103/physrevb.43.8990 · Cited by 134 · see citing works
Self-citationFunctional theory of extended Coulomb systems
Martin, R. M., & Ortíz, G. (1997) · Physical review. B, Condensed matter · doi:10.1103/physrevb.56.1124 · Cited by 119 · see citing works
Quantum Interference in Electron-Hole Generation in Noncentrosymmetric Semiconductors
Fräser, J. M., Shkrebtii, A. I., Sipe, J. E., et al. (1999) · Physical Review Letters · doi:10.1103/physrevlett.83.4192 · Cited by 71 · see citing works
Self-citationQuantum interference control of currents in CdSe with a single optical beam
Laman, N., Shkrebtii, A. I., Sipe, J. E., & van Driel, H. M. (1999) · Applied Physics Letters · doi:10.1063/1.125084 · Cited by 62 · see citing works
Self-citationLevine, Z. H. (1994) · Physical review. B, Condensed matter · doi:10.1103/physrevb.49.4532 · Cited by 59 · see citing works
藤原, 毅. (1975) · 日本物理學會誌 · Cited by 49 · see citing works
On the ω4 law in microscopic theories of inelastic light scattering
Zeyher, R., Bilz, H., & Cardona, M. (1976) · Solid State Communications · doi:10.1016/0038-1098(76)91728-2 · Cited by 34 · see citing works
Coherent current control in semiconductors: a susceptibility perspective
Aversa, C., & Sipe, J. E. (1996) · IEEE Journal of Quantum Electronics · doi:10.1109/3.535360 · Cited by 26 · see citing works
Self-citationComparison of calculated optical response in cubic and hexagonal II-VI semiconductors
Hughes, J. L. P., & Sipe, J. E. (1998) · Physical review. B, Condensed matter · doi:10.1103/physrevb.58.7761 · Cited by 19 · see citing works
Self-citationFull band structure calculation of the linear electro-optic susceptibility
Ghahramani, E., & Sipe, J. E. (1994) · Applied Physics Letters · doi:10.1063/1.111587 · Cited by 9 · see citing works
Self-citationGeometric phase for electric polarization along `rational' directions in crystals
Sipe, J. E., & Žák, J. (1999) · Physics Letters A · doi:10.1016/s0375-9601(99)00308-4 · Cited by 8 · see citing works
Self-citation
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de Juan, F., Grushin, A. G., Morimoto, T., & Moore, J. E. (2017) · Nature Communications · doi:10.1038/ncomms15995 · Cited by 708 · see citing works
Topological nature of nonlinear optical effects in solids
Morimoto, T., & Nagaosa, N. (2016) · Science Advances · doi:10.1126/sciadv.1501524 · Cited by 569 · see citing works
Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl semimetal
Osterhoudt, G. B., Diebel, L. K., Gray, M. J., Yang, X., Stanco, J., Huang, X., Shen, B., Ni, N., Moll, P. J. W., Ran, Y., et al. (2019) · Nature Materials · doi:10.1038/s41563-019-0297-4 · Cited by 399 · see citing works
Shift current bulk photovoltaic effect in polar materials—hybrid and oxide perovskites and beyond
Tan, L. Z., Zheng, F., Young, S. M., Wang, F., Liu, S., & Rappe, A. M. (2016) · npj Computational Materials · doi:10.1038/npjcompumats.2016.26 · Cited by 393 · see citing works
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Ma, J., Gu, Q., Liu, Y., Lai, J., Yu, P., Zhuo, X., Liu, Z., Chen, J.-H., Feng, J., & Sun, D. (2019) · Nature Materials · doi:10.1038/s41563-019-0296-5 · Cited by 327 · see citing works
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Liu, J., Xia, F., Xiao, D., García de Abajo, F. J., & Sun, D. (2020) · Nature Materials · doi:10.1038/s41563-020-0715-7 · Cited by 326 · see citing works
Riemannian geometry of resonant optical responses
Ahn, J., Guo, G.-Y., Nagaosa, N., & Vishwanath, A. (2022) · Nature Physics · doi:10.1038/s41567-021-01465-z · Cited by 252 · see citing works
“To explain the Hermitian connection, we write it as Cbca = r nmr a mn,c = −ir nmr mnR mn, where R mn = Amm−Ann + i∂c log r a mn is the shift vector describing the shift of the electron position during the optical excitation (note that R mn ∼ r mm−r nn) [25, 34].”
MethodUltrafast photocurrents at the surface of the three-dimensional topological insulator Bi2Se3
Braun, L., Mussler, G., Hruban, A., Konczykowski, M., Schumann, T., Wolf, M., Münzenberg, M., Perfetti, L., & Kampfrath, T. (2016) · Nature Communications · doi:10.1038/ncomms13259 · Cited by 236 · see citing works
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Dai, Z., & Rappe, A. M. (2023) · Chemical Physics Reviews · doi:10.1063/5.0101513 · Cited by 162 · see citing works
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Orenstein, J., Moore, J. E., Morimoto, T., Torchinsky, D. H., Harter, J. W., & Hsieh, D. (2021) · Annual Review of Condensed Matter Physics · doi:10.1146/annurev-conmatphys-031218-013712 · Cited by 145 · see citing works
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Sotome, M., Nakamura, M., Fujioka, J., Ogino, M., Kaneko, Y., Morimoto, T., Zhang, Y., Kawasaki, M., Nagaosa, N., Tokura, Y., et al. (2019) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1802427116 · Cited by 138 · see citing works
Rectification and shift currents in GaAs
Côté, D., Laman, N., & van Driel, H. M. (2002) · Applied Physics Letters · doi:10.1063/1.1436530 · Cited by 123 · see citing works
Resonance-enhanced optical nonlinearity in the Weyl semimetal TaAs
Patankar, S., Wu, L., Lu, B., Rai, M., Tran, J. D., Morimoto, T., Parker, D. E., Grushin, A. G., Nair, N. L., Analytis, J. G., et al. (2018) · Physical Review B · doi:10.1103/physrevb.98.165113 · Cited by 116 · see citing works
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Sirica, N., Tobey, R. I., Zhao, L. X., Chen, G. F., Xu, B., Yang, R., Shen, B., Yarotski, D. A., Bowlan, P., Trugman, S. A., et al. (2019) · Physical Review Letters · doi:10.1103/physrevlett.122.197401 · Cited by 115 · see citing works
THz emitters based on the photo-Dember effect
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Ultrafast terahertz emission from emerging symmetry-broken materials
Pettine, J., Padmanabhan, P., Sirica, N., Prasankumar, R. P., Taylor, A. J., & Chen, H.-T. (2023) · Light: Science & Applications · doi:10.1038/s41377-023-01163-w · Cited by 81 · see citing works
Interactions Remove the Quantization of the Chiral Photocurrent at Weyl Points
Avdoshkin, A., Kozii, V., & Moore, J. E. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.196603 · Cited by 78 · see citing works
Photon-drag-induced terahertz emission from graphene
Obraztsov, P. A., Kanda, N., Konishi, K., Kuwata-Gonokami, M., Garnov, S. V., Obraztsov, A. N., & Svirko, Y. P. (2014) · Physical Review B · doi:10.1103/physrevb.90.241416 · Cited by 81 · see citing works
Shift charge and spin photocurrents in Dirac surface states of topological insulator
Kim, K. W., Morimoto, T., & Nagaosa, N. (2017) · Physical Review B · doi:10.1103/physrevb.95.035134 · Cited by 79 · see citing works
Unconventional Photocurrents from Surface Fermi Arcs in Topological Chiral Semimetals
Chang, G., Yin, J.-X., Neupert, T., Sanchez, D. S., Belopolski, I., Zhang, S. S., Cochran, T. A., Chéng, Z., Hsu, M.-C., Huang, S.-M., et al. (2020) · Physical Review Letters · doi:10.1103/physrevlett.124.166404 · Cited by 72 · see citing works
Terahertz Pulse Generation from GaAs Metasurfaces
Hale, L. L., Jung, H., Gennaro, S. D., Briscoe, J., Harris, C. T., Luk, T. S., Addamane, S. J., Reno, J. L., Brener, I., & Mitrofanov, O. (2022) · ACS Photonics · doi:10.1021/acsphotonics.1c01908 · Cited by 50 · see citing works
Light-Induced Currents at Domain Walls in Multiferroic BiFeO<sub>3</sub>
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Sirica, N., Orth, P. P., Scheurer, M. S., Dai, Y. M., Lee, M.-C., Padmanabhan, P., Mix, L. T., Teitelbaum, S. W., Trigo, M., Zhao, L. X., et al. (2022) · Nature Materials · doi:10.1038/s41563-021-01126-9 · Cited by 9 · see citing works
- Authors
- Sipe, J. E., & Shkrebtii, A. I.
- Affiliations
- University of Toronto (CA)
- Venue
- Physical Review B
- Publisher
- American Physical Society (APS)
- Published
- 2000-02-15
- Volume
- 61
- Issue
- 8
- Pages
- 5337-5352
- ISSN
- 0163-1829 (ISSN-L)
- 1095-3795
- OpenAlex
- W2060827848
- Cited by
- 910
- References
- 33
- Citations by year
- 2026: 109 · 2025: 128 · 2024: 97 · 2023: 89 · 2022: 73 · 2021: 102 · 2020: 69 · 2019: 52 · 2018: 30 · 2017: 23 · 2016: 18 · 2015: 17 · 2014: 14 · 2013: 11 · 2012: 8
- FWCI
- 3.53
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical review. B, Condensed matter (publishedVersion)
- Funders
- Natural Sciences and Engineering Research Council of Canada
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Photorefractive and Nonlinear Optics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Topological Materials and Phenomena
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Laser-Matter Interactions and Applications
- Keywords
- second-order nonlinear susceptibility, nonlinear optics, semiconductors, polarization operator, injection current, shift current, noncentrosymmetric materials, GaAs, electronic band structure, independent particle approximation
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