PHYSICAL REViEW B VOLUME 46, NUMBER 18 1 NOVEMBER 1992-II

Localization in a one-dixnensional Thue-Morse chain

Danhong Huang

Department of Physics, The University of Lethbridge, Lethbridge, Alberta, Canada T1KSM)

Godfrey Gumbs

Department of Physics and Astronomy, Hunter College and the Graduate School,

City University of New York, 09$ Park Avenue, New York, New York 100M

Miroslav KolarDepartment of Chemistry, The University of Lethbridge, Lethbridge, Alberta Canada T1KSMg

(Received 6 May 1992)

The mean resistance of a one-dimensional wire is calculated with the use of Landauer formula

for three types of arrangements: the random, Thue-Morse, and Fibonacci chains for which thepositions of the atoms and the scattering strengths are modulated according to the prescribed rules.

Comparison of the obtained numerical results shows that for the position modulation, a Thue-Morse

chain is more localized than a Fibonacci chain, while for the scattering strength modulation it is

less localized. It is shown that the Thue-Morse chain can be switched from being localized to beingextended when the ratio of the strength modulation to the position modulation is increased. A

similar change occurs in the generalized Thue-Morse chain.

I. INTRODUCTION "1," can be generated by the following recursion formula:

Technical advances in submicron physics have en-abled experimentalists to fabricate nearly ideal one-dimensional wires (see, for example, Ref. 1 and refer-ences given therein). In these systems, both the magneto-optical absorption and the transport properties have beenstudied recently. 1z The relationship between the elec-trical conductance at zero temperature and the trans-mission coefilcient, given by the well-known Landauerformula, s indicate that some experimentally measurablequantities can be adequately explained when regardinga one-dimensional (1D) wire as an infinite 1D array ofpotentials.

The discovery of quasicrystals4 has stimulated inter-est in exploring the physical nature of quasiperiodicsequencess as well as commensurate-incommensuratesystems. s Work on aperiodic sequences, including theThue-Morse, incommensurate, and others, has also in-cluded studies of direct experimental relevance to semi-conductor multilayers (superlattices) (Ref. 7) and super-conducting networks. s It is interesting to note the con-tradiction between some calculated thermodynamic andspectral properties and the results from the structure fac-tor in a Thue-Morse (TM) sequence as to whether theTM chain is more random or more "periodic" than theFibonacci chain (see Kolar et al in Ref. 5). I. n this paper,we will concentrate on comparing the mean resistance forthe TM and the Fibonacci chains with position or scat-tering strength modulation. Our calculations of the lo-calization length as a function of the chain length haveshown that the TM chain is more localized for positionmodulation but less localized for the strength modula-tion.

The TM sequence composed of two symbols, "0" and

where e„denotes the nth element of the sequence. Theresulting infinite string of digits never repeats itself. Inspite of this periodicity, the TM sequence is still selfsimilar in nature. Both the TM and Fibonacci sequencescan be generated by the simple substitution rules: 0 -+01, 1 ~ 10 for the TM sequence, and 0 -+ 1, 1 ~ 10 forthe Fibonacci one. The finite chain obtained from a single"0" by n times of applications of the respective substitu-tion rule is called the nth generation chain. It has 2" ele-ments for the TM case, and F„elements for the Fibonaccicase, where the Fibonacci numbers (F„)are recursivelydefined by F„+1 = F„+F„1with Fp = F1 = 1. Forthe TM sequence, the ratio of the number of 0's to thenumber of 1's is equal to one in any generation. How-

ever, this is not true for the Fibonacci sequence. Forthe infinite Fibonacci sequence, this ratio has the limit-

ing value v = (~5+ 1)/2 (golden mean), an irrationalnumber. We can choose two fundamental lengths a and

b, and assign them to the numbers "0" and "1" in thesequence, respectively. Then the successive TM chainsare (ab, abbar abbabaab, . . .), and the Fibonacci ones arefb, ba, bab, babba, . . .). The nth generation of these se-quences is obtained in a straightforward way. For theTM case, we have W„+q ——W„W„', where Wo ——a andW„' is obtained from W„by exchanging a and b. Fortile FlboIlaccl sequence, we obtain Sny1 = SnSn 1 withSp = a, S1 = b The lengt. hs a and b are the separationsbetween neighboring scatterers and the whole sequencedetermines where the scatterers are. For example, thethird-generation Fibonacci chain (bab) has scatterers at(zn)n=1, .. .P; = b, (b + a), and 2b + a. The length of

46 11 479 1992 The American Physical Society

ll 480 DANHONG HUANG, GODFREY GUMBS, AND MIROSLAV KOLAR 46

the nth generation (n ) 0) is I~M = 2" (a+ b) andI» = (E„ ib+ F„za) for the TM and Fibonacci se-quences, respectively.

The rest of the paper is organized as follows. In Sec.II, we describe our model and present the theory forcalculating the resistance. This includes a brief reviewof the transfer-matrix and optical methods, along withsome related numerical results. In Sec. III, we calculatethe mean resistance for the random, TM, and Fibonaccichains, and compare the localization length as a functionof the chain length for these three systems. Section IVis devoted to a discussion on the generalized TM and Fi-bonacci chains. A short summary of the key results isalso presented at the end of Sec. IV.

II. MODEL AND THEORETICAL DESCRIPTION

The scaled (5 = 2m = 1) one-dimensional (1D)Schrodinger equation in the presence of potential scat-terers is given by

(2)

The short-range scattering potential for N scatterers canbe expressed as

V(z) = ) q„6(x —x„) .

Reading and Sigels obtained an exact solution of Eq. (2)using the potential in Eq. (3) with finite N and arbitrarystrengths and positions. Gumbsio later solved the 1DDirac equation for arbitrary b-function scatterers. Forsimplicity, in this paper, we let q„have two values tosimulate the randomness in the scattering strength. Thatis, we take q„= qi, qz (qi P q2). The positions {x„}ofthe scat terers are taken as being given by the Thue-Morse(TM) sequence. Specifically, if we know the positionsof the scatterers for the nth generation TM chain, say{yi,yz, . . . , yz },then the positions of the scatterers forthe next generation {zi,xz, . . . , zz +i }are given by

x= fori = 1, . . . , 2",yz + [(b + a)(i —2") —y, z j for i = (2" + 1), . . . , 2"+

with the known first-generation scatterers located at{a,b+a} To defin. e the notation and for comparison, wealso list the results for the Fibonacci chain. In this case,if the positions of the scatterers for the (n —1)th and nthgenerations are {yi,yz, . . . , y~„,}and {zi,zz, . . . , zs„},respectively, we can obtain the positions of the scatterersfor the (n+ 1)th generation at {zi,xz, . . . , z~„+,}from

~

~ ~

z' fori =1, . . . , F„,z~„+y, F„ for i = (F„+1), . . . , I'„+1,

with the first- and second-generation scatterers locatedat {b}and {b,b+a}.

With the use of the 1D single-channel Landauer for-mula, we can calculate the dimensionless conductance gof the system from

2ez/h 1 —T (6)

A. Transfer-matrix method

where T is the transmission coefBcient through the wholequasicrystal chain. For the calculation of transmissioncoefficient T, we shall use two methods, i.e. , the transfer-matrix and optical methods which are described in detailin the following subsections.

where g~ (z) and Q~ l (z) are represented by the spinorsof the coefficients from the plane-wave expansion of theirwave functions. The transfer matrix T„ is given explic-itly by

1 —iq„/2k —iq„/2kiq„/2k 1 + iq„/2k

which is position dependent due to the two different kindsof scatterers in the system. Here, k = ~E is the wavevector along the chain direction and E is the incidentenergy of an electron. The propagation of the right-goingplane wave between the nth and (n + 1)th barriers isrelated by

where D„ is the displacement matrix, given by

~iky O

o —ckp~W

and y„= (z„—z„ 1). The scattering strengths {q„}aredistributed according to

qi«r y~-i =~,gn =

qg for y„g ——b,

Making use of the transfer matrix, we can obtain therelationship between the wave functions Q

' (x) on theleft and right sides of the nth scattering barrier. Astraightforward calculation yields

with yi = xi. The total transfer matrix M for this chainwith N scatterers is obtained from

(7)

46 LOCALIZATION INN A ONE-DIMENSIONAL THUE-MORSE CHAIN 11 481

The uq asiperiodic, periodic, and ransimulated in our d 1

(*„jtob dtbtdri u e quasiperiodicalldo 1, ' 1. Th o 1

ficient T and the r fle re ection coefficient}1 11 f

T = I/[M2giR = /Mz, i/Mz

where ItI = T is the transmissio

b i h o d g o the resistance p = 1

y using this transfer-m ', asS dXi" dA' erend calcu atedan vishai and B

modulatio .In Figs. 1(a) and 1(b), we plot the re

'

tio of th i id t'en energy E of ano an electron or t e TM

sition and strength modulation are u'

cu1 ioM. W}1 h

' 'i i

e position-modulationth al of thself-similarity of thet o e spectrum isis contracted, but the'p (opo e spectrum is res

e ones . On thee

11- d 1a ion ratio increases the gy

T11 11i lit of }I 't in s, an the resistan

d dbh od 1

'iomes quite senaitiv

a ion see the right-arro ' 's

()fu io of E fo h TMof ththe position modulati

r e M sequence wit'

h difFerent ratiosu a ion to the strength dmo ulation.

The ratio egect can b 1e c early seen in thesWe emphasize that al

these four 6gures.

on the transmissio ffi'

a though there ar'

n coe cient for the F'e several studies

tie discussion of the ffatio s. Theref

bo i dnce an compare it to

ce. owever, for the Fithe magnitude of th

' ' 'ee resistance eak

'

to th tio of t}1e position modulatiothe strength modul t'

ion. As the ratio of

sltghtly reduced. Cu a ion increases the

onsequentl there peak value is only

11 io of h ffo e e ects of the twocase where both occur simultaneou 1 .

wo modulations in thus y.

e

B. Optical method

The optical method has been use ' ge cient T for a er'

e

."I th'n ss section, we energ'c a ice and then corn

0

io p

s or t e periodic and ipansfer-mat 's o tained when

e egin by consie n arrier which in

mis-

t fl tiMb —i e ec-

ft 1 hfo d 1 1

i io fo 1:SQ

t„ i/(1- p(')p e"--~ forg(&)

t(2)t„ i/(1 (&)p„ ie's"-') for y„ i = 6

THLIE-IVlORSE CHAIN (T&,

Q OoO

O

Z OO

V)

V)M o

OQ O

Po

O

O

O

w

Z OO

OR oIJJ O

0.635 0.600 0.. &0 0.6&5 0.650 0.655i~ ~l

0.635 0.640 0.6850.605 0.650 0.655

FIG. 1. Calculated resistance R as

K= E

chain with positii ion and scatterinsis ance R as a function of th e inci ent ener E

g strength modulationsÃ

o, ti 1 . (= I (scale on right); (b) i ——q2 = 0.5 and a = I, 5 = 4a =, 5 = 4r (scale on left) and

11 482 DANHONG HUANG, GODFREY GUMBS, AND MIROSLAV KOLAR

THEE-VAPORSE CHAIN (TM, }

(D

Clu)

o

Lfl

0C)

C)

nru

a

z}— o

LLI v)

0.635 0.6&0 0.645 0.650 0.655 0.635 0.640 0.6&5 0.650 0.655

FIG. 2. Calculated resistance B as a function of the incident energy E of an electron for the eighth-generation Thue-Morsechain with both position and scattering strength modulations. (a) qq = ~2/4, q2 = 0.5 and a = 1, b = r (scale on left) and

q& = v 2/16, q2 = 0.5 and a = 1, b = ~ (scale on right); (b) q& = v 2/4, q2 = 0.5 and a = 1, b = 4~ (scale on left) and

qq——v2/16, q2 = 0.5 and a = 1, b = 47 (scale on right).

(1,2) iql 2

2k+ iqg, 2

The sequence {q„)is the same as in Eq. (11). Currentconservation gives two additional constraints

t„p„' + p„t„' = 0 .(16)

If the values of (t„q, p„q) are known, we can calculate(t„, g„) from

(Z, 2)

1 —~(~ 2)~

The total transmission coefBcient through N barriers isobtained by T = t~t~.

In Figs. 3(a) and 3(b), we compare the results of ourcalculations for the resistance of a periodic chain by us-

ing the transfer-matrix and optical methods, respectively.These caIcuIations show that there is a good agreementof the values of the height and position of each peak ob-

where &q ——ti ) and pq = pi &. Also, H„q = 2ky„is the phase angle acquired by an electron after two suc-cessive reflections at the (n —1)th and nth barriers. Theamplitudes for transmission and reflection (t, p) througha single b barrier are given by

(y 2) 2k

2k+ q,. '

tained using two methods. If more scatterers are addedto the chain, less of the fine structure will be retainedfrom the optical calculation. There is also less agreementbetween the results of the two calculations. In Figs. 3(c)and 3(d), we plot the calculated results of the resistanceof the Fibonacci chain with the use of transfer-matrixand optical methods, respectively. Clearly, there is stillsome agreement both in the height and in the positionof the peaks obtained with these two methods. If severalmore scatterers are added to the chain, the fine structurewill be gradually eliminated from the optical calculations.However, the overall agreement between the results is notas good as in a periodic chain since there is more of thefine structure in a quasiperiodic chain. We conclude thatthe optical method could be used in both periodic andquasiperiodie chains if the number of scatterers is notvery large.

III. LOCALIZATION EFFECT

In this section, we address the main question of our in-

vestigation, consisting of which of the two chains, ThueMorse or Fibonacci, is more localized. We answer thisquestion by comparing the mean resistance as a functionof the chain length. This will show which chain has theshorter localization length for the same chain length. Wehave computed the mean resistance by averaging the re-sistance over an interval of k c [ko, ko + Nhk] with thestep Ak for different lengths l. or difFerent generations:

p = ) p;(kp+ ib, k) .

From this calculation, we are able to compare the L de-

46 LOCALIZATION IN A ONE-DIMENSIONAL THUE-MORSE CHAIN 11 483

O

Q oO

OD

UJu)z

Cn oO

UJ

OO

PERIODIC CHAIN (N=13)S

Ct

Lxl

ZI—

M

LVCC

tOOO

oo

oo

PERIODIC CHAIN (N=13)

QC)

O

CC

OO

OO

O

FIBONACCI CHAIN (F,)cv

QO

zo

V)

oO

FIBONACCI CHAIN (F,)

10.0 15.0 20.0 2S.O 30.0 10,0

FIG. 3. Calculated resistance R as a function of the incident energy 8 of an electron for the periodic chain wiscatterers and the sixth-generation Fibonacci chain with position modulation by using the transfer-matrix and optical methods,resPectively. (a) and (b) correspond to qi = q2 = 0.5 and a = b = 1 for transfer-matrix and optical methods, respectively; (c)and (d) correspond to qs = q2 = 0.5 and a = 1, b = r for transfer-matrix and optical methods, respectively.

pendence of p for different systems. The so-called o.parameter s can also be obtained from this L dependence:

The inverse of o. defines the localization length (. Inour numerical calculations, we take the interval to be[0.9, 1.0] and b,k = 10 s.

Figures 4(a)—4(c) show plots of the mean resistance asa function of the chain length L for the random, TM,and Fibonacci chains with position modulation. Thesecalculations show that the threshold value of L (indi-cated by a vertical arrow in the figures) is the largest forthe Fibonacci chain. For the random chain, the mean re-sistance increases exponentially with L. However, for theperiodic and quasiperiodic chains, this increase is accord-ing to a power law. This is clearer from Figs. 5(a)—5(c),where the localization length ( is plotted as a functionof L In a random ch. ain, there are oscillations of ( onlyfor a small range of values of L. As L increases, thereis Anderson localization~s where ( is independent of L,i.e., the mean resistance increases exponentially with L.

For the Fibonacci and TM chains, there are substantialoscillations in the entire range of values of L. This meansthat there is a power-law increase of the mean resistancewith L. The mean localization length in the TM chainis smaller than in the Fibonacci chain. This implies thatin the case of position modulation the TM chain is morelocalized than the Fibonacci one.

For the strength modulation in the random, TM, andFibonacci chains, from the L dependence of the meanresistance we find that the threshold values of L do notmutually differ very much in all three arrangements. Themain difference is that the rate of increase of the meanresistance with L is the largest for the random and Fi-bonacci chains. The absolute magnitude of the meanresistance is much larger in the Fibonacci chain. In Figs.6(a)—6(c), the L dependence of the localization length ispresented. In the random and Fibonacci chains, thereare oscillations of the localization length in the shortchain regime only. As L increases, there is localizationfor which ( is independent of L For the TM ch.ain, onthe other hand, there are large oscillations of the local-ization length which is attributed to a power-law increase

&1 484 DANHONG HUANNG, GODFREY GUMBS, AND MIROSLAV KOLAR

C3

X

RANDOM CHAIN THUE MORSE CHAINC3

X

FIBONACCI CHAIN

O

(DCV

Oln

OIO

OIll

n 5 O

O

Q)

OO

Oln

25. 0

I

50.0 75.0.0 100.0 125.0 150.0 175.0

CHAIN LENGTH L

O I I I

100.0 200.0 300.0 400.0 500.0 600.0

CHAIN LENGTH l

OO

500.0 1000.0 1500.0

CHAIN LENGTH L

FIG. 4. CCalculated mean resistance R aswith osi

' gce as a function of the chain len th Ido h: = =05 d0(

; (c) i o u h i : = = 0.5 d = 1 5 =

RANDOM CHAIN THUE-MORSE CHAIN FIBONACCI CHAIN

O

O

in, !

OOO

C)

OOVlP4

O0Pv

4pCI

0

OIll

OOOID

OOIn

OO

OO

O

P4

CIln

OO

25.0 50.0 75.0 100.0 125.0 150.0 175.0

CHAI N LENGTH L

O . I

100.0 200.0 300.0 000.0 500.0 600.0

CHAIN LENGTH L

500.0 1000.0 1500.0 2000.0

CHAIN LENGTH L

FIG.G. 5. Calculated localization len th ~ aschains with

eng ~~ as a function of the chain len th L forg or the random, Thue-Morse, and Fibo'

e y. aj~ andom chain: q& ——q2 ——0.5 and 0 &(), , — . y, , (b)TM h . q —q —0.5)

RANDOM CHAIN THUE-MORSE CHAIN FIBONACCI CHAIN

OO I

OOP4

OCI

OO

OO

OO

OO

oOin

J

OClO

OCIIn

OON

OO

oI I I

I I I I

100.0 200.0 300.0 (00..0 5oo. o coo. o200.0 100.0 600.0 800.0 1000.0

CHAIN LENGTH L

100.0 200.0 300.0 800.0 500.0 600.0

CHAIN LENGT H L

FIG. 6. Calculated localization len th fCHAIN LENGTH L

chains with scattering strength m d 1

' a ion engt as a function of the chain lmo u ation, respectively. (a) Random chain:

ength L for the random Thue-M d Pue- orse, an Fibonacci™; ~c& i onacci chain: qq ——&2~8 —0.

) c ain:

, q2 —— . an a=6=1.

46 LOCALIZATION IN A ONE-DIMENSIONAL THUE-MORSE CHAIN 11 485

THLlE-MORSE CHAIN FIBONACCI CHAIN

tV

Q

Al

oI cr

iOO. O 2OO. O SOO. O ~OO. O SOO. O SOO. O

CHAIN LENGTH L

O

Q

oo

Do

100.0 200.0 300.0 400.0 500.0

CHAIN LENGT H L

C3

C)

C)

C)

PJ

CD

oPVoC)Lfj

C7

C)oo

FIG. 7. Calculated mean resistance R (scale on left) and the n parameter (scale on right) as a function of the chainlength L for the Thue-Morse and Fibonacci chains with both position and scattering strength modulation. (a) TM chain:ql, = ~2/8, q2 = 0.5 and a = 1, b = r; (b) Fibonacci chain: q& = ~2/8, q2 = 0.5 and a = 1, b = r.

of the mean resistance with L. The localization length inthe Fibonacci chain is comparable to that in a randomchain. This indicates that the Fibonacci chain with posi-tion modulation is almost localized. We conclude that inthe case of strength modulation, the TM chain is muchmore extended than the Fibonacci one.

From the results presented above, we expect some can-cellation of the effects of the position and strength mod-ulation in the TM and Fibonacci chains. This is con-firmed in Figs. 7(a) and 7(b), where we have shown theL dependences of the mean resistance (see the left-arrowindicated ones) and the a parameter (see the right-arrowindicated ones) for the TM and Fibonacci chains withboth position and strength modulation. The thresholdvalues of L and the rate of the resistance increase nowbecome comparable to each other. The behavior of o; asa function of L is quite similar in these two systems. Thisgives strong evidence for the large cancellation of the ef-fects of the position and strength modulation in the TMand Fibonacci chains. Based on these studies, we predictthat one can switch the TM sequence from an extendedstate to a localized state by increasing the ratio of am-plitude of the position modulation to amplitude of thestrength modulation.

IV. GENERALIZED THUE-MORSE CHAIN

We now extend our discussion to the generalized Thue-Morse (GTM) chain. The GTM sequence is generated bythe substitution 0 ~ 0~1", 1 ~ 1"0~ (see Kolar et at. inRef. 5). The tth generation chain has m(m+ n)' ~ digits"0" and n(m+ n)' ~ digits "1." The ratio of the numberof 0's to the number of 1's is a rational number m/n,independent of the generation index. As an example, wechoose two fundamental lengths a and b, corresponding

to the numbers "0" and "1," and generate the successiveGTM chains (abs, abs(bsa)s, . . .) for rn = 1 and n =3. For the tth generation, the length of the string isL = (m, + n)' ~(ma + nb). Procedures for the resistancecalculations developed for the original TM chains (rn =n = 1) can also be applied to this case. If the ratio m/nis very large or very small, a GTM sequence will be veryclose to a periodic one.

Figures 8(a)—8(c) show plots of the resistance as a func-tion of the incident energy E of an electron for the pe-riodic, GTM, and TM chains, with position modulation.When the ratio m/n is small, say m/n =

&&, the behav-ior of the resistance in a GTM chain looks more like itsperiodic counterpart. In fact, we can consider a GTMchain with very large or very small ratios of m/n as be-ing periodic with a slight disorder in the positions of thescatterers. In this way, we can switch the TM sequencefrom a localized to an extended state by increasing orreducing this ratio from a value of one.

In a similar way, one can obtain the so-called generalized Fibonacci sequence. ~s ~7 The 1th generation of thisgeneralized sequence is given by the recursion relationSt+~ ——S, S," „where So ——a, S& ——b. The correspond-ing generalized Fibonacci numbers, de6ned by the recur-sion formula F~+~ ——mFt + nF~ q with Fo ——F~ ——1, in-crease exponentially. For the golden , silver , and coppe-r-mean sequences, we have m = n = 1; m = 1, n = 2; andm = 2, n = 1; respectively. It has been proved~ ' thatthe matrix trace maps of the golden- and silver-mean re-cursion formulas belong to the same universality class ofdynamical systems, while the map of the copper-meansequence has a very diR'erent dynamical behavior: it istwo dimensional, area nonpreserving, and noninvertableand has a fractal invariant. For a generalized Fibonaccisequence, the ratio of the number of 0's to the numberof 1's depends on the generation index. This is difFer-

11 486 DANHONG HUANG, GODFREY GUMBS, AND MIROSLAV KOLAR

PERIODIC CHAIN (N=256)GENERALIZED THUE-IVlORSE CHAIN

GT~ (m=1, n=15) T HUE-VIOR SE CHAIN (TM, )

~ lfl

o

&oPV

QJ oUZ Ul

lo

V)—l1j aEL

LAoo

0.65 0.70 0.75 0.80 0.85 0.90 0.95

K=/E

oCV ill

Q

oK'

UJ o

I-R oMQ3LLjCL

0.65 0.70 0.75 0.80 0.85 0.90 0.95

K=/E

oo

oo

ooo

tooo

oo

CL 8LLl oU oO

I—

M

UJCC

,R0.65

i R R r0.70 0.75 0.80 0.85 0.90 0.95

FIG. 8. Calculated resistance A as a function of the incident energy E of an electron for the periodic chain with 256scatterers, the generahzed Thue-Morse chain with m/n = ~z, and the eighth-generation Thue-Morse chain with positionmodulation, respectively. (a) Periodic chain: q& = Z2 = 0.25 an« = b = 1' (b) genera»z«TM cha'n: + = q2 =a=1, b=~; (c) TMchain: q& ——q2=0.25anda=l, b=7.

ent from a GTM sequence. For the infinite generalizedFibonacci sequences, the limit of this ratio is equal to(v5 + 1)/2, v2+ 1, and 2 for the golden-, silver-, andcopper-mean sequences, respectively.

In conclusion, compared to the Fibonacci chain, wehave found that the TM chain is less localized for thestrength modulation, and more localized for positionmodulation. We can switch it from an extended one toa localized one by increasing or reducing the ratio rn/n

from one or by increasing the ratio of strength modula-tion to position modulation.

ACKNOWLEDGMENTS

This work was supported in part by a grant from theNatural Sciences and Engineering Research Council ofCanada and the City University of New York PSC-CUNYResearch Award Program.

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