The accurate description of the optical spectra of insulators and semiconductors remains an important challenge for time-dependent density-functional theory (TDDFT). Evidence has been given in the literature that TDDFT can produce bound as well as continuum excitons for specific systems, but there are still many unresolved basic questions concerning the role of dynamical exchange and correlation (xc). In particular, the roles of the long spatial range and the frequency dependence of the xc kernel fxc for excitonic binding are still not very well explored. We present a minimal model for excitons in TDDFT, consisting of two bands from a one-dimensional (1D) Kronig-Penney model and simple approximate xc kernels, providing an easily accessible model system for studying excitonic effects in TDDFT. For the 1D model system, it is found that adiabatic xc kernels can produce at most two bound excitons, confirming that the long spatial range of fxc is not a necessary condition. It is shown how the Wannier model, featuring an effective electron-hole interaction, emerges from TDDFT. The collective, many-body nature of excitons is explicitly demonstrated.
REFERENCES
1.
2.
F.
Ariasetiawan
and O.
Gunnarsson
, Rep. Prog. Phys.
61
, 237
(1998
).3.
W.
Hanke
and L. S.
Sham
, Phys. Rev. B
21
, 4656
(1980
).4.
G.
Onida
, L.
Reining
, and A.
Rubio
, Rev. Mod. Phys.
74
, 601
(2002
).5.
E.
Runge
and E. K. U.
Gross
, Phys. Rev. Lett.
52
, 997
(1984
).6.
Time-Dependent Density Functional Theory
, Lecture Notes in Physics
, edited by M. A. L.
Marques
, C. A.
Ullrich
, F.
Nogueira
, A.
Rubio
, K.
Burke
, and E. K. U.
Gross
(Springer
, Berlin
, 2006
).7.
C. A.
Ullrich
, Time-Dependent Density-Functional Theory: Concepts and Applications
(Oxford University Press
, Oxford, England
, 2012
).8.
P.
Elliott
, F.
Furche
, and K.
Burke
, Reviews in Computational Chemistry
, edited by K. B.
Lipkowitz
and T. R.
Cundari
(Wiley
, Hoboken, NJ
, 2009
), p. 91
.9.
S.
Botti
, A.
Schindlmayr
, R.
Del Sole
, and L.
Reining
, Rep. Prog. Phys.
70
, 357
(2007
).10.
J. P.
Perdew
, R. G.
Parr
, M.
Levy
, and J. L.
Balduz
, Phys. Rev. Lett.
49
, 1691
(1982
).11.
X.
Gonze
and M.
Scheffler
, Phys. Rev. Lett.
82
, 4416
(1999
).12.
R.
Stubner
, I. V.
Tokatly
, and O.
Pankratov
, Phys. Rev. B
70
, 245119
(2004
).13.
M.
Gatti
, V.
Olevano
, L.
Reining
, and I. V.
Tokatly
, Phys. Rev. Lett.
99
, 057401
(2007
).14.
Y.-H.
Kim
and A.
Görling
, Phys. Rev. Lett.
89
, 096402
(2002
).15.
L.
Reining
, V.
Olevano
, A.
Rubio
, and G.
Onida
, Phys. Rev. Lett.
88
, 066404
(2002
).16.
F.
Sottile
, K.
Karlsson
, L.
Reining
, and F.
Aryasetiawan
, Phys. Rev. B
68
, 205112
(2003
).17.
A.
Marini
, R.
Del Sole
, and A.
Rubio
, Phys. Rev. Lett.
91
, 256402
(2003
).18.
R.
Del Sole
, G.
Adragna
, V.
Olevano
, and L.
Reining
, Phys. Rev. B
67
, 045207
(2003
).19.
F.
Sottile
, M.
Marsili
, V.
Olevano
, L.
Reining
, and Olevano
, Phys. Rev. B
76
, 161103
–R
(2007
).20.
V. U.
Nazarov
and G.
Vignale
, Phys. Rev. Lett.
107
, 216402
(2011
).21.
S.
Sharma
, J. K.
Dewhurst
, A.
Sanna
, and E. K. U.
Gross
, Phys. Rev. Lett.
107
, 186401
(2011
).22.
V.
Turkowski
and C. A.
Ullrich
, Phys. Rev. B
77
, 075204
(2008
).23.
V.
Turkowski
, A.
Leonardo
, and C. A.
Ullrich
, Phys. Rev. B
79
, 233201
(2009
).24.
25.
G.
Dresselhaus
, J. Phys. Chem. Solids
1
, 14
(1956
).26.
P.
Yu
and M.
Cardona
, Fundamentals of Semiconductors
, 4th ed. (Springer
, Berlin
, 2010
).27.
S. W.
Koch
, M.
Kira
, G.
Khitrova
, and H. M.
Gibbs
, Nature Mater.
5
, 523
(2006
).28.
G. D.
Scholes
and G.
Rumbles
, Nature Mater.
5
, 683
(2006
).29.
H.
Haug
and S. W.
Koch
, Quantum Theory of the Optical and Electronic Properties of Semiconductors
, 5th ed. (World Scientific
, 2009
).30.
31.
G. F.
Giuliani
and G.
Vignale
, Quantum Theory of the Electron Liquid
(Cambridge University Press
, Cambridge
, 2005
).32.
L. J.
Sham
and T. M.
Rice
, Phys. Rev.
144
, 708
(1966
).33.
M. E.
Casida
, Recent Developments and Applications in Density Functional Theory
, edited by J. M.
Seminario
(Elsevier
, Amsterdam
, 1996
).34.
A. L.
Fetter
and J. D.
Walecka
, Quantum Theory of Many-Particle Systems
(Dover
, Mineola, NY
, 2003
).35.
M. E.
Casida
, F.
Gutierrez
, J.
Guan
, F.-X.
Gadea
, D.
Salahub
, and J.-P.
Daudey
, J. Chem. Phys.
113
, 7062
(2000
).36.
T.
Grabo
, T.
Kreibich
, and E. K. U.
Gross
, Mol. Eng.
7
, 27
(1997
).37.
H.
Appel
, E. K. U.
Gross
, and K.
Burke
, Phys. Rev. Lett.
90
, 043005
(2003
).38.
M.
Grüning
and X.
Gonze
, Phys. Rev. B
76
, 035126
(2007
).39.
A. F.
Izmaylov
and G. E.
Scuseria
, J. Chem. Phys.
129
, 034101
(2008
).40.
G. F.
Bertsch
, J.-I.
Iwata
, A.
Rubio
, and K.
Yabana
, Phys. Rev. B
62
, 7998
(2000
).41.
T.
Otobe
, M.
Yamagiwa
, J.-I.
Iwata
, K.
Yabana
, T.
Nakatsukasa
, and G. F.
Bertsch
, Phys. Rev. B
77
, 165104
(2008
).42.
K.
Yabana
, S.
Sugiyama
, Y.
Shinohara
, T.
Otobe
, and G. F.
Bertsch
, Phys. Rev. B
85
, 045134
(2012
).43.
J. C.
Phillips
and L.
Kleinman
, Phys. Rev.
116
, 287
(1959
).44.
L.
Kleinman
and J. C.
Phillips
, Phys. Rev.
118
, 1153
(1960
).45.
A Primer in Density Functional Theory
, Lecture Notes in Physics
, edited by C.
Fiolhais
, F.
Nogueira
, and M.
Marques
(Springer
, Berlin
, 2003
).46.
N.
Helbig
, J. I.
Fuks
, M.
Casula
, M. J.
Verstraete
, M. A. L.
Marques
, I. V.
Tokatly
, and A.
Rubio
, Phys. Rev. A
83
, 032503
(2011
).47.
G. F.
Bassani
and G. P.
Parravicini
, Electronic States and Optical Transitions in Solids
(Pergamon
, New York
, 1975
).48.
C.
Ambrosch-Draxl
, K.
Hummer
, S.
Sagmeister
, and P.
Puschnig
, Chem. Phys.
325
, 3
(2006
).49.
See supplementary material at http://dx.doi.org/10.1063/1.4730031 for the number of discrete eigenvalues when the matrix has a certain symmetry, and proof of only one bound exciton for separable kernels.
50.
Handbook of Mathematical Functions
, edited by M.
Abramowitz
and I. A.
Stegun
(Dover
, New York
, 1972
).51.
W.
Schäfer
and M.
Wegener
, Semiconductor Optics and Transport Phenomena
(Springer
, Berlin
, 2002
).52.
S.
Glutsch
, Excitons in Low-Dimensional Semiconductors
(Springer
, Berlin
, 2004
).53.
S.
Ismail-Beigi
, Phys. Status Solidi C
10
, 3365
(2006
).54.
M. S.
Dresselhaus
, G.
Dresselhaus
, R.
Saito
, and A.
Jorio
, Annu. Rev. Phys. Chem.
58
, 719
(2007
).55.
C.
Uihlein
, D.
Fröhlich
, and R.
Kenklies
, Phys. Rev. B
23
, 2731
(1981
).56.
R. G.
Ulbrich
, Adv. Solid State Phys.
25
, 299
(1985
).57.
N. T.
Maitra
, F.
Zhang
, R. J.
Cave
, and K.
Burke
, J. Chem. Phys.
120
, 5932
(2004
).58.
N. T.
Maitra
, J. Chem. Phys.
122
, 234104
(2005
).59.
M. S.
Hybertsen
and S. G.
Louie
, Phys. Rev. B
34
, 5390
(1986
).60.
K.
Burke
and J. P.
Perdew
, Int. J. Quantum Chem.
56
, 199
(1995
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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