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Since Tamm published his first paper on surface states[1], the investigations on various properties and/or problems of surface states have been a very active and important area in condensed matter physics.

A naturally simple but fundamental question is how many surface states are in a band gap in a one-dimensional crystal of length ( is the potential period, and is a positive integer)? A well-accepted concept proposed by Fowler[2] first in 1933, then written in Seitz's classic book[3] that "in a finite one-dimensional crystal the surface states occur in pairs, one state being associated with each end of the crystal." Such a concept seemly was never doubted since then for nearly a century, as shown, for example, in [4]. However, a recent new investigation[5][6][7][8][9] gives an entirely different answer.

The investigation tries to understand electronic states in ideal crystals of finite size based on the mathematical theory of periodic differential equations.[10] This theory provides some fundamental new understandings of those electronic states, including surface states.

The theory found that a one-dimensional finite crystal with two ends at and always has one and only one state whose energy and properties depend on but not for each band gap. This state is either a band-edge state or a surface state in the band gap. Numerical calculations have confirmed such findings[6][7][8][9]. Further, these behaviors have been seen in different one-dimensional systems, such as in.[11][12][13][14][15][16][17]

Therefore:

  • The fundamental property of a surface state is that its existence and properties depend on the location of the periodicity truncation.
  • Truncation of the lattice's periodic potential may or may not lead to a surface state in a band gap.
  • An ideal one-dimensional crystal of finite length with two ends can have, at most, only one surface state at one end in each band gap.

Further investigations extended to multi-dimensional cases found that

  • An ideal simple three-dimensional finite crystal may have vertex-like, edge-like, surface-like, and bulk-like states.
  • A surface state is always in a band gap is only valid for one-dimensional cases.

References

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  1. ^ Tamm, I. (1932). "On the possible bound states of electrons on a crystal surface". Phys. Z. Sowjetunion. 1: 733.
  2. ^ Fowler, R.H. (1933). "Notes on some electronic properties of conductors and insulators". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 141 (843): 56–71. Bibcode:1933RSPSA.141...56F. doi:10.1098/rspa.1933.0103. S2CID 122900909.
  3. ^ Seitz, F. (1940). The Modern Theory of Solids. New York, McGraw-Hill. p. 323.
  4. ^ Davison, S. D.; Stęślicka, M. (1992). Basic Theory of Surface States. Oxford, Clarendon Press. doi:10.1007/978-3-642-31232-8_3.
  5. ^ Ren, Shang Yuan (2002). "Two Types of Electronic States in One-dimensional Crystals of Finite length". Annals of Physics. 301 (1): 22–30. arXiv:cond-mat/0204211. Bibcode:2002AnPhy.301...22R. doi:10.1006/aphy.2002.6298. S2CID 14490431.
  6. ^ a b Ren, Shang Yuan (2006). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves. New York, Springer. Bibcode:2006escf.book.....R.
  7. ^ a b Ren, Shang Yuan (2006). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Chinese edition. Beijing, Peking University Press.
  8. ^ a b Ren, Shang Yuan (2017). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves (2 ed.). Singapore, Springer.
  9. ^ a b Ren, Shang Yuan (2023). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves, Chinese edition (2 ed.). Beijing, Peking University Press.
  10. ^ Eastham, M.S.P. (1973). The Spectral Theory of Periodic Differential Equations. Edinburgh, Scottish Academic Press.
  11. ^ Hladky-Henniona, Anne-Christine; Allan, Guy (2005). "Localized modes in a one-dimensional diatomic chain of coupled spheres" (PDF). Journal of Applied Physics. 98 (5): 054909 (1-7). Bibcode:2005JAP....98e4909H. doi:10.1063/1.2034082.
  12. ^ Ren, Shang Yuan; Chang, Yia-Chung (2007). "Theory of confinement effects in finite one-dimensional phononic crystals". Physical Review B. 75 (21): 212301(1-4). Bibcode:2007PhRvB..75u2301R. doi:10.1103/PhysRevB.75.212301.
  13. ^ El Boudouti, E. H. (2007). "Two types of modes in finite size one-dimensional coaxial photonic crystals: General rules and experimental evidence" (PDF). Physical Review E. 76 (2): 026607(1-9). Bibcode:2007PhRvE..76b6607E. doi:10.1103/PhysRevE.76.026607. PMID 17930167.
  14. ^ El Boudouti, E. H.; El Hassouani, Y.; Djafari-Rouhani, B.; Aynaou, H. (2007). "Surface and confined acoustic waves in finite size 1D solid-fluid phononic crystals". Journal of Physics: Conference Series. 92 (1): 1–4. Bibcode:2007JPhCS..92a2113E. doi:10.1088/1742-6596/92/1/012113. S2CID 250673169.
  15. ^ El Hassouani, Y.; El Boudouti, E. H.; Djafari-Rouhani, B.; Rais, R (2008). "Sagittal acoustic waves in finite solid-fluid superlattices: Band-gap structure, surface and confined modes, and omnidirectional reflection and selective transmission" (PDF). Physical Review B. 78 (1): 174306（1–23）. Bibcode:2008PhRvB..78q4306E. doi:10.1103/PhysRevB.78.174306.
  16. ^ El Boudouti, E. H.; Djafari-Rouhani, B.; Akjouj, A.; Dobrzynski, L. (2009). "Acoustic waves in solid and fluid layered materials". Surface Science Reports. 64 (1): 471–594. Bibcode:2009SurSR..64..471E. doi:10.1016/j.surfrep.2009.07.005.
  17. ^ El Hassouani, Y.; El Boudouti, E.H.; Djafari-Rouhani, B. (2013). "One-Dimensional Phononic Crystals". In Deymier, P.A. (ed.). Acoustic Metamaterials and Phononic Crystals, Springer Series in Solid-State Sciences 173. Vol. 173. Berlin, Springer-Verlag. pp. 45–93. doi:10.1007/978-3-642-31232-8_3. ISBN 978-3-642-31231-1.

Category:Materials science Category:Electronic band structures Category:Semiconductor structures

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