Anomalous optical absorption in a random system with scale-free disorder

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by  E. Diaz

Anomalous optical absorption in a random system with scale-free disorder
E. D´ıaz,1 A. Rodr´ıguez,2,∗ F. Dom´ınguez-Adame,1,∗ and V. A. Malyshev3,†
1Departamento de F´ısica de Materiales, Universidad Complutense, E-28040 Madrid, Spain
2Departamento de Matem´atica Aplicada y Estad´ıstica, Universidad Polit´ecnica, E-28040 Madrid, Spain
3Institute for Theoretical Physics and Materials Science Center,
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
5
(Dated: February 2, 2008)
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0
Wereportonananomalousbehavioroftheabsorptionspectruminaone-dimensionallatticewith
2 long-range-correlateddiagonaldisorderwithapower-likespectrumintheformS(k)∼1/kα. These

n typeof correlations give rise to a phase of extended states at the band center, provided α is larger
a than a critical value αc. We show that for α<αc the absorption spectrum is single-peaked, while
J an additional peak arises when α > αc, signalling the occurrence of the Anderson transition. The

3 peak is located slightly below the low-energy mobility edge, providing a unique spectroscopic tool
1 tomonitor thelatter. Wepresent qualitative arguments explaining thisanomaly.

] PACSnumbers: 78.30.Ly;71.30.+h;71.35.Aa;36.20.Kd
n
n
-
s Quantumdynamics of quasiparticlesin randommedia Recently, we have shown that 1D disordered systems
i has been a subject of extensive studies since the seminal with the above mentioned correlated disorder in the site
d
. paperbyAnderson,whoarguedthatquasiparticlestates energies support Bloch-like oscillations [16]. The am-
t
a become localized for sufficiently large disorder, thus giv- plitude of the oscillations turned out to carry informa-
m ing rise to a localization-delocalizationtransition (LDT) tion about the energy difference between the two mobil-
- in three dimensions (3D) [1]. The hypothesis of single- ity edges. This finding opens the possibility to perform
d
parameter scaling, introduced in Ref. 2, lead to the gen- experimentsoncoherentdc chargetransportformeasur-
n
eralbeliefthatalleigenstatesofnoninteractingquasipar- ing the bandwidth of the delocalizedphase in disordered
o
c ticleswerelocalizedinone-(1D)andtwodimensions,and systems with long-range correlated randomness. In this
[ that the LDT does not exist in low dimensional systems work we report on an anomaly of the linear absorption

(for recentoverviewssee Refs. 3 and4). However,atthe within the underlined model. We show a crossover of
1
v end of the eighties and beginning of the nineties several the absorptionspectrum fromasingle-peakedto double-
1 theoretical works provided clear evidences that short- peaked shape when the exponent α crosses the critical
1 range correlations in disorder may cause delocalization value α . This behavior is not shared by the standard
c
3
evenin1Dsystems[5,6,7,8]. Thisfactwasputforward Anderson model and has never been mentioned before.
1
to explain the high conductivity of doped polyaniline [6] Remarkably, the additional peak of the absorption is lo-
0
5 as well as the transportproperties of GaAs-AlxGa1−xAs catedcloseto the low-energymobility edge,thus provid-
0 superlattices with intentional correlated disorder [9]. ingasimplespectroscopictooltomonitorit. Wepropose
/ a simple explanation of the behavior found.
t
a Attheendofthelastdecade,itwasdemonstratedthat We consideraregularopenchainofN opticallyactive
m long-range correlations in disorder, with no characteris-
two-levelunitswithparalleltransitiondipoles,whichare
- tic spatial length (scale-free disorder), also acts towards coupled by the resonant dipole-dipole interaction. The
d delocalization of 1D quasiparticle states [10]. Random
n corresponding Hamiltonian reads
sequences of correlated site energies characterized by a
o
power-like spectrum S(k) 1/kα with α > 0, result N N−1
c
∼ = ε n n n n+1 + n+1 n , (1)
v: in extended states provided α is larger than some crit- H n| ih |− | ih | | ih |
i ical value αc [10]. The extended states form a phase nX=1 nX=1(cid:16) (cid:17)
X at the band center, which is separated from localized where n denotesthestateinwhichthen-thunit,having
| i
r states by twomobility edges. This theoreticalprediction the energy εn, is excited, whereas all the other units are
a
was experimentally validated by measuring microwave in the ground state. The intersite dipole-dipole coupling
transmission spectra of a single-mode waveguide with is restricted to nearest-neighbors and set to 1 over the
−
inserted correlated scatterers [11]. The observed trans- entire lattice. ε1,ε2,...,εN is a stochastic long-range
mission spectra were nicely simulated by the theoretical correlated sequence. We generate a realization of the
model [12], confirming the existence of a phase of de- sequence according to the following rule [10]
localized states, in spite of the underlying randomness.
N/2
Peculiarities of this type of disorder had also their trace 1 2πkn
ε =σC cos +φ . (2)
in biophysics, explaining the long-distance charge trans- n α kα/2 N k
k=1 (cid:18) (cid:19)
port in DNA sequences [13, 14] as well as the existence X
of a new class of level statistics [15]. Here, C = √2 N/2k−α −1/2, and φ ,φ ,...,φ
α k=1 1 2 N/2
(cid:0)P (cid:1)
2
are N/2 independent random phases uniformly dis- theband,indicatingstrongandunexpectedeffectsofthe
tributed within the interval [0,2π]. The random distri- long-rangecorrelateddisorderonthespatialdistribution
bution (2) has zero mean ε = 0 and a correlation of the probability amplitude. In contrast to the case
n
h i
function α < 2, the broadening of the peaks drops down on in-
creasing α and then saturates. Finally, notice that the
σ2C2 N/2 1 2πk(n m) low-energytailofthelow-energypeaklosesitscharacter-
ε ε = α cos − , (3)
h n mi 2 kα N istic Gaussianshapeobservedinuncorrelateddisordered
kX=1 (cid:20) (cid:21) systems. All these features pointout arichphenomenol-
where ... indicates an average over the distribution of ogy of the model under study that cannot be accounted
randomhphiasesφ andN isassumedtobeeven. From(3) for within the standard theoretical frameworks.
k
it follows that σ = ε2 1/2 is the standard deviation of
h ni
the distribution (2). This quantity will be referred to
as magnitude of disorder. The long-range nature of the
site potential correlationsresults fromthe power-lawde-
pendenceoftheamplitudesoftheFouriercomponentsin s)
Eq. (2). As was shown in Ref. 10, a phase of extended nit
u
states occurs at the band center provided α > αc = 2 rb.
whTenheσq=ua1n.tity of our primary interest is the absorption A(E) (a
spectrum, which is defined as
N
1
A(E)= F δ(E E ) , (4) 0
ν ν
N*ν=1 − + 1
X
2
where E are the eigenenergies of the normalized eigen-
ν 3
α
functionsψ oftheHamiltonian(1). ThequantityF =
νn ν 4
2 0
Nn=1ψνn is the dimensionless oscillator strength of 5 −4 −3 −2 −1
t(cid:16)he ν-th sta(cid:17)te. E
P
WehavenumericallydiagonalizedtheHamiltonian(1)
andobtainedtheabsorptionspectrumfordifferentvalues FIG.1: Evolutionoftheabsorptionspectrumshapeasafunc-
of the power exponent α, considering open linear chains tionofthepowerexponentαandagivenmagnitudeofdisor-
of size N = 250 and setting the magnitude of disorder der σ = 1. Notice the appearance of a double-peaked struc-
σ = 1. The results comprise an average over 3 104 re- ture of the absorption spectrum when α exceeds the critical
alizations of the disorder for each value of α. ×Figure 1 valueαc =2 for the LDTto occur.
shows the output of the simulations. When α 1 the
≪
absorption spectrum displays a single and asymmetric To get insight into the effects of scale-free disorder on
peak slightly below the lower band edge E = 2 of the the optical properties of the system, it is useful to have
−
periodic lattice (σ = 0), i.e., only the lowest states of a look at the eigenfunctions of the Hamiltonian (1). In
the band contribute to the absorption spectrum. This Fig. 2 we depict a subset of wave functions obtained for
trendisexactlythe sameasthatobservedin1Dsystems a typical random realization of the potential landscape.
withuncorrelatedrandomness. Theonlynoticeableeffect The baselines display the corresponding eigenenergies.
upon increasing α within the range α < 2 is an increase The lowest state in Fig. 2 (labelled by 1) is localized
ofthe absorptionbandwidth. In otherwords,long-range in the sense that it has a considerable amplitude within
correlations in disorder cause a stronger localization as a domain of size N∗, which is smaller than the system
compared to the uncorrelated case. A similar effect was size N. It shows a bell-like shape and carries a large
found in 1D disorder-correlated systems with a charac- oscillator strength F N∗. There are several states of
1
∝
teristic spatial length and explained on the basis of the suchtype(notshowninFig.2),whicharecloseinenergy
exchange narrowing concept [17] (see also Ref. 18). to the state 1 and do not overlap with each other. They
The absorption shape changes dramatically when α> contributetothelow-energypeakoftheabsorptionspec-
α , as seen in Fig. 1: A single peak splits into a dou- trum. Onincreasingtheenergy,oneobserveseigenstates,
c
blet. One of the doublet components (at low energy) is liketheonelabelledby2inFigure2,whicharemoreex-
located at the bottom of the band as usual, whereas the tended, as compared to the lowest one, and present sev-
other one (at higher energy) lies deep inside the band. eralnodeswithinthelocalizationsegment. Theoscillator
This means that the oscillator strength does not de- strengthofsuchstatesis smallerthanF . Consequently,
1
crease monotonously from the bottom to the center of they contribute to the absorption spectrum to a lesser
3
extent. Remarkably,goingfurtherupinenergywe again 3 α = 3
find bell-like states, as the one labelled by 3 in Figure 2, 2 α = 4
α = 10
which are characterized by a large oscillator strength. y 1
g
Those states form the high-energy peak of the doublet er 0
n
in the absorption spectrum. Finally, on approaching the E -1
centerofthebandoneexpectstheoccurrenceofextended -2
a)
states. The state 4 in Figure 2 represents an example. -3
2 Site energy 3 Allowed energies
2
4
y 1
g
r 0
0 e
n
E -1
y -2
g
Ener-2 3 -3 b)
00 200 400 600 800 11000000
2
nn
FIG. 3: (a) The site energy landscape εn given by (2) for
-4
1 σ = 1 and three different values of α. The random phases
have been shifted the same amount so that φ1 = −π/2. (b)
A model in which the actual site energy landscape (dashed
0 100 200 300 400 500
n line)isreplaced byastep-likeenergyprofile(solid line). The
shadedregionsindicatetheallowedenergybandsofeachseg-
FIG. 2: A subset of eigenstates for a typical realization of ment,whose width is 4 in the chosen units.
therandomenergypotential(dashedline),forachainofsize
N = 500, magnitude of disorder σ = 1, and power exponent
α=3.0(larger thanthecritical valueαc =2). Thebaselines the sake of clarity,to as subband), ranging from ε¯ 2 to
−
indicate the energies of each eigenstate. The states 1 and 3 ε¯+2 and from ε¯ 2 to ε¯+2 for the left and right
arethosewhichcontributetothelow-andhigh-energypeaks sublattices, resp−ecti−vely [se−e Fig. 3(b)]. The absorption
of the absorption spectrum, respectively.
spectrumofsuchasystemis expectedtohavetwopeaks
caused by the transitions from the ground state to the
Aiming to elucidate the anomaliesfound, wepresenta
bottom state of each subband. For σ = 1 their loca-
simplified model that explains the occurrence of the op-
tions are ε¯ 2 = 1.11 and ε¯ 2 = 2.89. We stress
ticallyactivestatesdeepinside the bandaswellassheds − − − − −
thatthese values areina fairlygoodagreementwith the
lightonthenatureoftheLDTinthemodelunderstudy.
results of exact calculations presented in Fig. 1, despite
To this end, we recall that the site energy potential (2)
both the simplification involved and neglecting the cou-
is given by a sum of spatial harmonics. The amplitude
pling. Thisisastrongindicationthattheloweststatesof
of each term, σC k−α/2, decreases upon increasing the
α thesubbandsarenotverymuchmodifiedafterswitching
harmonic number k. For sufficiently high values of α,
on the interaction between sublattices.
the first term in the series (2) will be dominant, while
In order to understand the above features, let us con-
the others are considerably smaller. Consequently, the
sider the elements of the coupling matrix between seg-
site potential for a given realization represents a deter-
ments, given by [17]
ministic function of period N (harmonic with k = 1),
perturbed by a colored noise (harmonics with k 2). 4 2πk 2πk
≥ V =( 1)kl sin l sin r . (5)
Figure 3(a) shows the site energy landscape (2) for dif- klkr − N +2 N +2 N +2
(cid:18) (cid:19) (cid:18) (cid:19)
ferent values of α > 2, illustrating the statement above.
Therefore, relevant information can be obtained by ana- Here, k and k , ranging from 1 to N/2, number the
l r
lyzing the first (k =1) term in the series (2). eigenstates of the left and right sublattices, respectively.
Asafurtherstepofsimplification,wereplacethe sine- From (5) we find that the magnitude of coupling of the
like site energy potential by a step-like one, as shown in lowest state of the left sublattice (k = 1) to the closest
r
Fig.3(b). Morespecifically,wetakeε =ε¯sign(N/2 n), state of the left one (k = 1) is V 16π2/(N +2)3,
n l 11
− | | ≈
whereε¯=(2/π)σC =0.63σC(α)istheaveragevalueof thatismuchsmallerthan2ε¯=1.78,theenergydifference
α
the site energies on the left half of the system. In doing betweenthelowestbandedges(thelimitofN 1isim-
≫
so, we map the original lattice onto two uniform sublat- plied). Themagnitudeofthecouplingoftheloweststate
tices, coupled to each other through the hoping between oftheleftsublattice(k =1)tothecentralbandstatesof
l
sites N/2 and N/2 + 1. The allowed energies of each the rightone is about V 8π/(N+2)2, whereas the
| 1kr|≈
decoupled sublattice form a band (referred hereafter, for energy spacing at the center of the band is 4π/(N +2).
4
Again, the coupling is smaller than the energy spacing. model; this signals the occurrence of the localization-
This explains the low sensitivity of the lowest states of delocalizationtransition. Thelow-energypeakislocated
thesubbandstoswitchingontheinteractionbetweenthe atthebottomoftheband,whereasthehigh-energypeak
sublattices. lies deep inside of the band, indicating the presence of
bandstateswithalargeoscillatorstrength. Thelocation
0
of the high-energy peak is slightly below the low-energy
mobility edge of the phase of extended states. This pro-
-100 vides a unique possibility to monitor the mobility edge
0
spectroscopically. The crossoverfoundis acharacteristic
τ feature of the underlying long-range correlated random
g -200
o -0.2 sequence and, thus, can also be used to distinguish it
l
from other types of correlations.
-300
-0.4
-0.2 0.0 0.2
The authors thank A. V. Malyshev, M. L. Lyra and
-400
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 F. A. B. F. de Moura for helpful comments. This work
E wassupportedbyDGI-MCyT(ProjectMAT2003-01533)
and CAM (Project GR/MAT/0039/2004). V. A. M. ac-
FIG. 4: Log-normal plot of the transmission coefficient τ as
knowledges support from ISTS (grant #2679).
a function of the incoming energy when α = 4.0 and N =
3×103. Theinsetrepresentsanenlargedviewofτ closetothe
centerofthebandforN =3×103(solidline)andN =3×104
(dashed line), showing that τ does not change on increasing
thesize. 3×103 disorder realizations were considered.
∗ Also at GrupoInterdisciplinar de Sistemas Complejos.
† On leave from ”S.I. Vavilov State Optical Institute”,
Note the existence of an overlap of the two subbands
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c
α is the critical value for a LDT to occur in the present torial Complutense, Madrid, 2000).
c