We present a simple approach to Anderson localization in one-dimensional disordered lattices. We introduce the tight-binding model in which one orbital and a single random energy are assigned to each lattice site, and the hopping integrals are constant and restricted to nearest-neighbor sites. The localization of eigenstates is explained by two-parameter scaling arguments. We compare the size scaling of the level spacing in the bare energy spectrum of the quasi-particle (in the ideal lattice) with the size scaling of the renormalized disorder seen by the quasi-particle. The former decreases faster than the latter with increasing system size, giving rise to mixing and to the localization of the bare quasi-particle wave functions in the thermodynamic limit. We also provide a self-consistent calculation of the localization length and show how this length can be obtained from optical absorption spectra for Frenkel excitons.

1.
W. A. Harrison, Solid State Theory (Dover, New York, 1980).
2.
J. M. Ziman, Models of Disorder (Cambridge U.P., Cambridge, 1979).
3.
P. W.
Anderson
, “
Absence of diffusion in certain random lattices
,”
Phys. Rev.
109
,
1492
1505
(
1958
).
4.
For more information on the Nobel prize, see 〈http://www.nobel.se/physics/laureates/1977/〉.
5.
N. F.
Mott
and
W. D.
Twose
, “
The theory of impurity conduction
,”
Adv. Phys.
10
,
107
163
(
1961
).
6.
E.
Abrahams
,
P. W.
Anderson
,
D. C.
Licciardello
, and
T. V.
Ramakrishnan
, “
Scaling theory of localization: Absence of quantum diffusion in two dimensions
,”
Phys. Rev. Lett.
42
,
673
676
(
1979
).
7.
D. J.
Thouless
, “
Electrons in disordered systems and the theory of localization
,”
Rep. Prog. Phys.
13
,
93
142
(
1974
).
8.
P. A.
Lee
and
T. V.
Ramakrishnan
, “
Disordered electronic systems
,”
Rev. Mod. Phys.
57
,
287
337
(
1985
).
9.
B.
Kramer
and
A.
MacKinnon
, “
Localization: Theory and experiment
,”
Rep. Prog. Phys.
56
,
1469
1564
(
1993
).
10.
J.
Frenkel
, “
On the transformation of light into heat in solids. I
,”
Phys. Rev.
37
,
17
44
(
1931
).
11.
A. S. Davydov, Theory of Molecular Excitons (Plenum, New York, 1971).
12.
V. M. Agranovich and M. D. Galanin, in Electronic Excitation Energy Transfer in Condensed Matter, edited by V. M. Agranovich and A. A. Maradudin (North-Holland, Amsterdam, 1982).
13.
F.
Domı́nguez-Adame
,
E.
Maciá
,
A.
Khan
, and
C. L.
Roy
, “
LCAO approach to non-relativistic and relativistic Kronig-Penney models
,”
Physica B
212
,
67
74
(
1995
).
14.
V. A.
Malyshev
, “
Localization length of a 1D exciton and temperature dependence of the radiative lifetime in frozen dye-solutions with J-aggregates
,”
Opt. Spectrosc.
71
,
505
506
(
1991
);
V. A.
Malyshev
, “
Localization length of one-dimensional exciton and low-temperature behaviour of radiative lifetime of J-aggregated dye solutions
,”
J. Lumin.
55
,
225
230
(
1993
).
15.
V.
Malyshev
and
P.
Moreno
, “
Hidden structure of the low-energy spectrum of a one-dimensional localized Frenkel exciton
,”
Phys. Rev. B
51
,
14587
14593
(
1995
).
16.
V. A.
Malyshev
,
A.
Rodrı́guez
, and
F.
Domı́nguez-Adame
, “
Linear optical properties of one-dimensional Frenkel exciton systems with intersite energy correlations
,”
Phys. Rev. B
60
,
14140
14146
(
1999
).
17.
A. V.
Malyshev
and
V. A.
Malyshev
, “
Statistics of low energy levels of a one-dimensional weakly localized Frenkel exciton: A numerical study
,”
Phys. Rev. B
63
,
195111
(
2001
).
18.
E. E.
Jelley
, “
Spectral absorption and fluorescence of dyes in the molecular state
,”
Nature (London)
138
,
1009
1010
(
1936
).
19.
G.
Scheibe
, “
Über die Veränderung der Absorptionspektren in Lösung und die van der Waalss-chen Kräfte als Ihre Ursache
,”
Angew. Chem.
50
,
51
(
1937
).
20.
J-Aggregates, edited by T. Kobayashi (World Scientific, Singapore, 1996).
21.
E. W.
Knapp
, “
Lineshapes of molecular aggregates. Exchange narrowing and intersite correlation
,”
Chem. Phys.
85
,
73
82
(
1984
).
22.
L. D.
Bakalis
and
J.
Knoester
, “
Linear absorption as a tool to measure the exciton delocalization length in molecular assemblies
,”
J. Lumin.
87
,
66
70
(
2000
).
23.
F.
Domı́nguez-Adame
and
V. A.
Malyshev
, “
Frenkel excitons in one-dimensional random systems with correlated disorder
,”
J. Lumin.
83–84
,
61
67
(
1999
).
This content is only available via PDF.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.