Group theoretical analysis of symmetry breaking in two-dimensional quantum dots

Physical Review B - Condensed Matter and Materials Physics

Volumn 68, Issue 3, 2003, Pages 353251-3532516

  Yannouleas, Constantine ^a   Landman, Uzi ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; ELECTRON; ELEMENTARY PARTICLE; FERMION; MAGNETIC FIELD; MATHEMATICAL ANALYSIS; MODEL; QUANTUM CHEMISTRY; QUANTUM DOT; QUANTUM MECHANICS; ROTATION; SEMICONDUCTOR;

EID: 11444249983     PISSN: 10980121     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (79)

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9. See, e.g., exact calculations by M. Eto, Jpn. J. Appl. Phys., Part 1 36, 3924 (1997); Hartree-Fock calculations by A. Natori, Y. Sugimoto, and M. Fujito, ibid. 36, 3960 (1997); H. Tamura, Physica (Amsterdam) 249B-251B, 210 (1998); M. Rontani, F. Rossi, F. Manghi, and E. Molinari, Phys. Rev. B 59, 10 165 (1999); the spin density-functional theory at B = 0 by In-Ho Lee, V. Rao, R.M. Martin, and J.P. Leburton, ibid. 57, 9035 (1998) and Ref. 43; and also at nonzero B by O. Steffen, U. Rössler, and M. Suhrke, Europhys. Lett. 42, 529 (1998).
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11. See, e.g., exact calculations by M. Eto, Jpn. J. Appl. Phys., Part 1 36, 3924 (1997); Hartree-Fock calculations by A. Natori, Y. Sugimoto, and M. Fujito, ibid. 36, 3960 (1997); H. Tamura, Physica (Amsterdam) 249B-251B, 210 (1998); M. Rontani, F. Rossi, F. Manghi, and E. Molinari, Phys. Rev. B 59, 10 165 (1999); the spin density-functional theory at B = 0 by In-Ho Lee, V. Rao, R.M. Martin, and J.P. Leburton, ibid. 57, 9035 (1998) and Ref. 43; and also at nonzero B by O. Steffen, U. Rössler, and M. Suhrke, Europhys. Lett. 42, 529 (1998).
12. See, e.g., exact calculations by M. Eto, Jpn. J. Appl. Phys., Part 1 36, 3924 (1997); Hartree-Fock calculations by A. Natori, Y. Sugimoto, and M. Fujito, ibid. 36, 3960 (1997); H. Tamura, Physica (Amsterdam) 249B-251B, 210 (1998); M. Rontani, F. Rossi, F. Manghi, and E. Molinari, Phys. Rev. B 59, 10 165 (1999); the spin density-functional theory at B = 0 by In-Ho Lee, V. Rao, R.M. Martin, and J.P. Leburton, ibid. 57, 9035 (1998) and Ref. 43; and also at nonzero B by O. Steffen, U. Rössler, and M. Suhrke, Europhys. Lett. 42, 529 (1998).
13. See, e.g., exact calculations by M. Eto, Jpn. J. Appl. Phys., Part 1 36, 3924 (1997); Hartree-Fock calculations by A. Natori, Y. Sugimoto, and M. Fujito, ibid. 36, 3960 (1997); H. Tamura, Physica (Amsterdam) 249B-251B, 210 (1998); M. Rontani, F. Rossi, F. Manghi, and E. Molinari, Phys. Rev. B 59, 10 165 (1999); the spin density-functional theory at B = 0 by In-Ho Lee, V. Rao, R.M. Martin, and J.P. Leburton, ibid. 57, 9035 (1998) and Ref. 43; and also at nonzero B by O. Steffen, U. Rössler, and M. Suhrke, Europhys. Lett. 42, 529 (1998).
14. C. Yannouleas and U. Landman, Phys. Rev. Lett. 82, 5325 (1999).
15. C. Yannouleas and U. Landman, Phys. Rev. B 61, 15 895 (2000).
16. For studies pertaining to the geometrical arrangements of classical point charges in a harmonic confinement, see Yu.E. Lozovik, Usp. Fiz. Nauk 153, 356 (1987) [Sov. Phys. Usp. 30, 912 (1987)]; V.M. Bedanov and F.M. Peeters, Phys. Rev. B 49, 2667 (1994).
17. For studies pertaining to the geometrical arrangements of classical point charges in a harmonic confinement, see Yu.E. Lozovik, Usp. Fiz. Nauk 153, 356 (1987) [Sov. Phys. Usp. 30, 912 (1987)]; V.M. Bedanov and F.M. Peeters, Phys. Rev. B 49, 2667 (1994).
18. For studies pertaining to the geometrical arrangements of classical point charges in a harmonic confinement, see Yu.E. Lozovik, Usp. Fiz. Nauk 153, 356 (1987) [Sov. Phys. Usp. 30, 912 (1987)]; V.M. Bedanov and F.M. Peeters, Phys. Rev. B 49, 2667 (1994).
19. W via path-integral Monte Carlo simulations [R. Egger, W. Häusler, C.H. Mak, and H. Grabert, Phys. Rev. Lett. 82, 3320 (1999)]. Since the doubly humped radial electron densities are compatible with the (1,N -1) polygonal structures of classical point charges in the range 6 ≤ N ≤ 8 (after carrying out an azimuthal averaging), this crossover was interpreted as indicating the formation of WM's.
20. These transitions were further elaborated in A.V. Filinov, M. Bonitz, and Yu.E. Lozovik, Phys. Rev. Lett. 86, 3851 (2001).
21. C. Yannouleas and U. Landman, Phys. Rev. Lett. 85, 1726 (2000).
22. C. Yannouleas and U. Landman, J. Phys.: Condens. Matter 14, L591 (2002).
23. C. Yannouleas and U. Landman, Phys. Rev. B 66, 115315 (2002).
24. C. Yannouleas and U. Landman, Phys. Rev. B 68, 035326 (2003).
25. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
26. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
27. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
28. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
29. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
30. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
31. Unlike the HF approach for which a fully developed theory for the restoration of symmetries has long been established (see Sec. IB), the breaking of symmetries within the spin-dependent density functional theory poses a serious dilemma [J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995)]. This dilemma has not been resolved todate; several remedies (like projection, ensembles, etc.) are being proposed, but none of them appears to be completely devoid of inconsistencies [A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J. M. Seminario (Elsevier, Amsterdam, 1996), p. 327]. In addition, due to the unphysical self-interaction energy (which vigorously and erroneously assists the kinetic energy in orbital delocalization), the density functional theory is more resistant against space symmetry breaking [R. Bauernschmitt and R. Ahlrichs, J. Chem. Phys. 104, 9047 (1996)] than the sS-UHF, and thus it fails to describe a whole class of broken symmetries involving electron localization, e.g., the formation at B = 0 of Wigner molecules in QD's (see footnote 7 in Ref. 9), the hole trapping at A1 impurities in silica [J. Laegsgaard and K. Stokbro, Phys. Rev. Lett. 86, 2834 (2001); G. Pacchioni, F. Frigoli, D. Ricci, and J.A. Weil, Phys. Rev. B 63, 054102 (2001)], or the interaction driven localization-delocalization transition in d- and f-electron systems [see, e.g., Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by V. I. Anisimov (Gordon & Breach, Amsterdam, 2000); S.Y. Savrasov, G. Kotliar, and E. Abrahams, Nature (London) 410, 793 (2001)]. In line with the above, no density functional calculations describing space symmetry breaking and formation of Wigner molecules at B = 0 in circular QD's have been reported to date.
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64. Depending on the spin polarization, a WM may (or may not) be accompanied by a SDW. Unlike the pure SDW case, however, the SDW of a WM exhibits necessarily the same number of humps as the number of electrons (see, e.g., the case of N = 3 electrons in Sec. III B).
65. However, for RW≲1, the formation of a special class of SDW's (often called electron puddles) plays an important role in the coupling and dissociation of quantum dot molecules (see Ref. 32 and Ref. 9).
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77. We note that the discrete rotational (and more generally rovibrational) collective spectra associated with symmetry breaking in a QD may be viewed as finite analogs to the Goldstone modes accompanying symmetry-breaking transitions in extended media [see P. W. Anderson, Basic Notions of Condensed Matter Physics (Addison-Wesley, Reading, MA, 1984)].
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79. Nv point-group symmetries.