Fractal Conductance Fluctuations of Classical Origin
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Fra
tal Condu
tan
e Flu
tuations of Classi
al OriginH. Hennig , , R. Fleis
hmann , , L. Hufnagel , T. Geisel , Max Plan
k Institute for Dynami
s and Self-Organization, 37073 Göttingen, Germany Department of Physi
s, University of Göttingen, Germany and Kavli Institute for Theoreti
al Physi
s, University of California, Santa Barbara, USA(Dated: November 1, 2018)In mesos
opi
systems
ondu
tan
e (cid:29)u
tuations are a sensitive probe of ele
tron dynami
s and
haoti
phenomena. We show that the
ondu
tan
e of a purely
lassi
al
haoti
system with eitherfully
haoti
or mixed phase spa
e generi
ally exhibits fra
tal
ondu
tan
e (cid:29)u
tuations unrelatedto quantum interferen
e. This might explain the unexpe
ted dependen
e of the fra
tal dimensionof the
ondu
tan
e
urves on the (quantum) phase breaking length observed in experiments onsemi
ondu
tor quantum dots.PACS numbers: 73.23.Ad, 05.45Df, 05.60.CdA prominent feature of transport in mesos
opi
sys-tems is that the
ondu
tan
e as a fun
tion of an externalparameter (e.g. a gate voltage or a magneti
(cid:28)eld) showsreprodu
ible (cid:29)u
tuations
aused by quantum interferen
e[1℄. A predi
tion from semi
lassi
al theory that inspired anumber of both theoreti
al and experimental works in the(cid:28)elds of mesos
opi
systems and quantum
haos was thatin
haoti
systems with a mixed phase spa
e these (cid:29)u
-tuations would result in fra
tal
ondu
tan
e
urves [2, 3℄.Su
h fra
tal
ondu
tan
e (cid:29)u
tuations (FCFs) have sin
ebeen
on(cid:28)rmed in gold nanowires and in mesos
opi
semi-
ondu
tor billiards in various experiments [4, 5, 6, 7, 8℄.In addition FCFs have more re
ently been predi
ted too
ur in strongly dynami
ally lo
alized [9℄ and in di(cid:27)u-sive systems [10℄. Due to the quantum nature of the FCFsit
ame as a surprise when re
ent experiments indi
atedthat de
oheren
e does not destroy the fra
tal nature ofthe
ondu
tan
e
urve but only
hanges its fra
tal di-mension [11, 12℄. In the present letter we show, thatthe
ondu
tan
e of purely
lassi
al (i.e. in
oherent) low-dimensional Hamiltonian systems very fundamentally ex-hibits fra
tal (cid:29)u
tuations, as long as transport is at leastpartially
ondu
ted by
haoti
dynami
s. Thus mixedphase spa
e systems and fully
haoti
systems alike gen-erally show FCFs with a fra
tal dimension that is de-termined analyti
ally. We show that it is governed byfundamental properties of
haoti
dynami
s.In a disordered mesos
opi
ondu
tor (cid:21) whi
h issmaller than the phase
oheren
e length of the
harge
arriers but large
ompared to the average impurity spa
-ing (cid:21) the transmission is the result of the interferen
e ofmany di(cid:27)erent, multiply-s
attered and
ompli
ated pathsthrough the system. As these paths are typi
ally verylong
ompared to the wave length of the
harge
arriers,the a
umulated phase along a path
hanges basi
allyrandomly when an external parameter su
h as the en-ergy or the magneti
(cid:28)eld is varied. This results in a ran-dom interferen
e pattern, i.e. reprodu
ible (cid:29)u
tuationsin the
ondu
tan
e of a universal magnitude on the orderof e /h , the so
alled universal
ondu
tan
e (cid:29)u
tuations(UCFs). For a review see [13℄ or [14℄. The role of disorderin providing a distribution of random phases
an as well be taken by
haos. Thus ballisti
mesos
opi
avities likequantum dots in high mobility two-dimensional ele
trongases that form
haoti
billiards show the same universal(cid:29)u
tuations [1, 15, 16℄. If the average of the phase gaina
umulated on the di(cid:27)erent paths traversing the systemexists, the
ondu
tan
e
urves are smooth on parameters
ales that
orrespond to a
hange of the average phasegain on the order of and smaller than the wave length ofthe
arriers. In systems with mixed phase spa
e, where
haoti
and regular motion
oexist, this phase gain, how-ever, is typi
ally algebrai
ally distributed and its aver-age phase gain does not exist (negle
ting the (cid:28)nitenessof the
oheren
e length and assuming the semi
lassi
allimit ~ eff → ; the role of the (cid:28)nite ~ eff is dis
ussed in[3℄). Therefore, as shown in [2℄, the
ondu
tan
e
urveof su
h a system (cid:29)u
tuates on all parameter s
ales andforms a fra
tal. The fra
tal dimension D is
onne
tedto the exponent γ of the algebrai
distribution of phasegains by D = 2 − γ . Experiments on quantum dots that study the depen-den
e of the
ondu
tan
e (cid:29)u
tuations on several systemparameters like size and temperature seem to partly
on-tradi
t the semi
lassi
al theory of fra
tal s
aling [11, 12℄.Namely it was found that with de
reasing
oheren
elength the s
aling region over whi
h the fra
tal was ob-served did not shrink (cid:21) as would be expe
ted from thesemi
lassi
al arguments (cid:21), but that the fra
tal dimen-sion
hanged. An impli
it assumption of the semi
las-si
al theory is that the
lassi
al dynami
s remains un-
hanged as the external parameter is varied and thusonly phase
hanges are relevant. In most experimentalsetups, however, the
lassi
al phase spa
e
hanges withvariation of the
ontrol parameter. In this arti
le we showthat the
lassi
al
haoti
dynami
s itself already leads tofra
tal
ondu
tan
e
urves! Moreover, from this followsthat even on very small parameter s
ales the (cid:29)u
tuationsdue to
hanges in the
lassi
al dynami
s are important.In general the
ondu
tan
e
urve is a superposition oftwo fra
tals: one originating in interferen
e whi
h is sup-pressed by de
oheren
e to reveal the fra
tal (cid:29)u
tuationsre(cid:29)e
ting the
hanges in the
lassi
al phase spa
e stru
-ture. In addition, we predi
t that FCFs are not onlyobservable in systems with a mixed phase spa
e but inpurely
haoti
systems.Figure 1: Classi
al Condu
tan
e g ( B ) through a stadium(left, geometry as in ref. [5℄) and square billiard (right, ge-ometry as in ref. [11℄) versus magneti
(cid:28)eld B . Both (cid:29)u
-tuating
ondu
tan
e
urves are fra
tals (see insets and text).Their respe
tive dimensions are D ≈ . for the stadium and D ≈ . for the square billiard. The fra
tal dimensions arein good agreement with experimental measurements [5, 11℄.As a starting point of our investigations and to
onne
tit to the experiments we numeri
ally study the
lassi
al
ondu
tan
e through a re
tangle (hard-wall) and a sta-dium billiard (soft-wall) as a fun
tion of a magneti
(cid:28)eldas shown in Fig. 1. (Throughout this arti
le, we willstudy the transmission, whi
h, in a
ordan
e with theLandauer theory of
ondu
tan
e, is proportional to the
ondu
tan
e, see e.g. [17℄.) Note that not only the phasespa
e of the stadium but also of the re
tangle billiard ismixed in the presen
e of a perpendi
ular magneti
(cid:28)eld.In both
ases, a modi(cid:28)ed version [18, 19℄ of the box-
ounting analysis
learly reveals the fra
tal nature of the
ondu
tan
e
urves. As the simulation is purely
lassi-
al, the fra
tal s
aling
annot be
aused by interferen
ee(cid:27)e
ts. So what is the underlying me
hanism for thefra
tality of the
ondu
tan
e
urve and how
an we un-derstand its dimension?To study this me
hanism in detail we will, be
auseof its numeri
al and
on
eptual advantages, analyze thetransport in Chirikov's standard map [21, 22, 23℄. Thisparadigmati
system shows all the ri
hness of Hamilto-nian
haos. And sin
e (cid:21) as will be
ome apparent below (cid:21)our theory relies only on very fundamental properties of
haoti
systems, it is a natural
hoi
e as a model system.The standard map is de(cid:28)ned by θ ′ = p + θp ′ = p + K sin θ ′ with momentum p , angle θ and the 'nonlinearity pa-rameter' K , whi
h drives the dynami
s from fully inte-grable ( K = 0 ) to fully
haoti
( K & ). In betweenthe phase spa
e is mixed. The standard map
an be seenas the Poin
aré surfa
e of a
onservative system of twodegrees of freedom. As su
h the map
an by viewed to di-re
tly
orrespond to the Poin
aré map at the boundary Figure 2: How lobes translate into (cid:29)u
tuations: In the lowerrow the entryset of the standard map with absorbing bound-ary
onditions at ± π for K = 7 . and K = 7 . resp.
an beseen. The three pi
tures in the
enter row show the magni(cid:28)-
ation of the
entral se
tions of the entryset for three di(cid:27)erentvalues of K = 7 . , . and . . The transmission T ( K ) for K = 7 . . . . . [20℄ is shown in the top left pi
ture. Note thata small
hange in K shifts the lobes verti
ally, but
onservesthe overall phase spa
e stru
ture, and that the largest (cid:29)u
-tuations are
aused by interse
tion with the apex of lobes.Starting from K = 7 . , a large transmission lobe is
ut bythe horizontal line (see text), i.e. the transmission in
reaseswith K . In the same way, e.g. the (cid:29)u
tuations of T ( K ) near K = 7 . an be understood. The box-
ounting analysis re-veals a fra
tal stru
ture (top right).of a
haoti
ballisti
avity,
onne
ting it
on
eptuallywith the experimental system. We introdu
e absorbingboundary
onditions (see e.g. ref. [24℄), i.e. when p ex-
eeds (drops below) a maximum (minimum) thresholdvalue, the parti
le is transmitted (re(cid:29)e
ted) and leavesthe
avity. As
an be seen right from the de(cid:28)nition ofthe standard map, the envelope of the entryset (whi
h is,the phase spa
e proje
tion of the inje
tion lead) is simplyhalf a period of a sine fun
tion times K .A traje
tory entering the system eventually
on-tributes either to the total transmission or re(cid:29)e
tion, andwe mark the
orresponding point in the entryset by a
olor
ode (transmission: red, re(cid:29)e
tion: blue). Chaoti
dynami
s, through its fundamental property of stret
h-ing and folding in phase spa
e, leads to a lobe stru
ture(see Fig. 2 (bottom)), whi
h is typi
al for
haoti
systemsand not spe
ial to the standard map. The distributionof widths w of lobes exhibits a power law n ( w ) ∝ w − α . (1)The lobe stru
ture is translated into transmission bysumming up the interse
tions of the transmission lobesalong a horizontal line, see Fig. 2. A lobe of thi
kness w gives rise to a maximum
ontribution ∆ T ∝ w β . Varia-tion of the external parameter K leads to a fra
tal trans-mission
urve T ( K ) with D ≈ . .How does the fra
tal dimension depend on the powerlaw distribution of lobe-widths and the
urvature of thelobes? To this aim, we study a random sequen
e of
urvesegments mimi
king the interse
tion of
onse
utive lobesof widths w , distributed algebrai
ally with exponent α and
urved like w β . We de(cid:28)ne X i := P ij =1 w j and T ( X ) = ( − i ( X − X i ) β : X i < X ≤ X i +1 . An example of this
urve of (cid:16)random lobes(cid:17) with α = 1 . and β = is shown in Fig. 3 (top). The box-
ountinganalysis
learly reveals a fra
tal stru
ture.Figure 3: Transmission T ( X ) for lobes (red upper
urve,shifted along the y-axis for
larity) and stripes (bla
k lower
urve) for one and the same random distribution with α = 1 . , β = 0 . . The inset shows the box-
ounting analysis for theupper (red triangles) and lower transmission
urve (bla
ksquares). The regression line is drawn for the upper
urve,whose fra
tal dimension is . .We further simplify the problem by repla
ing the lobesby a sequen
e of stripes of widths x with power law dis-tribution n ( x ) ∝ x α . Dispensing with the sign of the(cid:29)u
tuation, the transmission reads T ( X ) = ( X i +1 − X i ) β . This yields histogrammati
transmission
urves T ( X ) like the bottom
urve of Fig. 3. As shown in the in-set, the fra
tal dimension of the resulting transmission
urve remains un
hanged
ompared to the
orrespond-ing
al
ulation with random lobes within the pre
ision ofthe box-
ounting analysis. Thus, the fra
tal dimension of the
urve does not
hange noti
eably when
onsideringstripes instead of lobes and also when negle
ting the signof ea
h
ontribution,
on(cid:28)rming the intuition, that thefra
tal dimension depends only on the relative s
aling,i.e. α and β , but not on the detailed form of the
urvese
tions.For these
urves like the bottom one of Fig. 3 with α − β > , we
an give an analyti
al expression for thefra
tal dimension and then estimate the fra
tal dimensionof the transmission
urve in the standard map. We ap-ply the box-
ounting method, whi
h we therefore reviewshortly (see e.g. [19℄ for a more detailed introdu
tion). Inthis approa
h the fra
tal
urve lying in a n − dimensionalspa
e is
overed by a n − dimensional grid. Let the grid
onsist of boxes of length s
ale s . The box-
ounting di-mension is then given by D = − lim s → log N ( s ) log ( s ) , (2)where N ( s ) is the number of non-empty boxes. Forour problem, we divide N ( s ) into three
ontributions N ( s ) = n a + n b + n c , as s
hemati
ally drawn in Fig. 4(A).The number n a of verti
ally pla
ed boxes (see mark (a))
overing
ontributions from stripes of widths x > s , reads n a ( s ) ∝ s Z ∞ s p ( x ) x β dx ∝ s − ( α − β ) . (3)Se
ondly, the number n b of horizontally pla
ed boxes
overing horizontal
ontributions of stripes of widthslarger than s , see Fig. 4A(b), is given by n b ( s ) = 1 s Z ∞ s p ( x ) xdx < s Z ∞ p ( x ) xdx. (4)Hen
e n b s
ales like s − and
an be negle
ted in
om-parison to n a be
ause of α − β > . Finally, we de-termine an upper estimate for the number n c of verti-
ally pla
ed boxes
overing the
ontribution from stripesof widths x ≤ s . The total length of these widths is L ( s ) = R s p ( x ) xdx, therefor L ( s ) /s boxes are needed to
over the length. In(cid:29)ating all heights of the stripes x ≤ s to the maximum possible size s β , see Fig. 4A(
), we (cid:28)nd n c ( s ) < L ( s ) s s β s ∝ s − α + β . (5)For s ≪ thus the dominant terms is n a ( s ) . With Eq. (2), N ( s ) gives rise to the box-
ounting dimension [25℄ D = − lim s → log s − α + β log s = α − β. (6)To
onne
t the analyti
al result with the
al
ulationsof the transmission of the open standard map, we nu-meri
ally estimate the distribution of lobe-widths in theentryset for K = 8 , (cid:28)nding α ≈ . , as shown in Fig. 4B.Together with β = ,
orresponding to (cid:28)rst order Taylorexpansion of the
osine fun
tion, Eq. (6) predi
ts a fra
talFigure 4: A. S
hemati
transmission a
ording to Fig. 3 (bot-tom),
overed with boxes of size s . There are three
ontribu-tions marked (a-
). B. Total number N int ( w ) = R ∞ w n ( w ′ ) dw ′ of lobes (for the open standard map with | p | < π ) ofwidth larger than w on a double logarithmi
s
ale. Thefour
urves show estimates for in
reasing resolution w min =10 − . . . − . The
urves
learly approa
h a power law
orre-sponding to n ( w ) ∝ w − . . The insets show the transmission
urve T ( K ) for values K = 8 . . . . . al
ulated from × traje
tories and its box-
ounting dimension.dimension D ≈ . . Dire
t analysis of the transmission
urve (see insets of Fig. 4B) yields a fra
tal dimension D ≈ . , in good agreement with the expe
ted value.How
an a power law distribution of lobe widthsemerge in a fully
haoti
open system? One might ratherexpe
t to (cid:28)nd an exponential distribution of lobes in afully
haoti
system. To see why the distribution is al-gebrai
, however, let us examine the simplest
ase of anopen
haoti
area preserving map the dynami
s of whi
his governed by a single, positive homogeneous Lyapunovexponent λ . In ea
h iteration phase spa
e stru
tures arestret
hed in one dire
tion by exp( λ ) , shrunk by exp( − λ ) in the other and then folded ba
k. The entryset of theopen system is thus stret
hed into lobes of de
aying width w ( t i ) ∝ exp( − λt i ) . The phase spa
e volume (cid:29)ux outof the system de
ays exponentially as it is typi
al for afully
haoti
phase spa
e, i.e. Γ( t i ) ∝ exp( − t i /τ ) , with(mean) dwelltime τ . The area Γ( t i )∆ t is the fra
tionof the exitset that leaves the system at time t i . With t i ( w ) ∝ − ln( w ) /λ the number of lobes of width w in theexitset is [26℄ w = Γ( t i ( w ))∆ tw ∝ w exp( ln( w ) λτ ) = w λτ − . This suggests that the power law distribution of lobewidths is a generi
property even for fully
haoti
sys-tems. A quantitative expression for the exponent, how-ever, is not as easy to derive, as e.g. the Lyapunov expo-nent for the standard map is not homogeneous.In
on
lusion, we have shown that transport through
haoti
systems due to the typi
al lobe stru
ture ofthe phase spa
e in general produ
es fra
tal
ondu
tan
e
urves, where the fra
tal dimension re(cid:29)e
ts the distribu-tion of lobes in the exit- /entryset. In
ontrast to thesemi
lassi
al e(cid:27)e
t the size of the (cid:29)u
tuations is not uni-versal but depends on spe
i(cid:28)
system parameters. Dueto the fra
tal nature of the
lassi
al
ondu
tan
e, how-ever, there is no parameter s
ale that separates
oherentand in
oherent (cid:29)u
tuations.[1℄ C. M. Mar
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opi
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opi
Sys-tems (Cambridge University Press, 1995).[18℄ Throughout this arti
le, in order to determine the fra
taldimension of a given 2D
urve T (∆ k ) , we used the so
alled variation method, i.e. we
al
ulated k ) =