Boundary Conditions, the Critical Conductance Distribution, and One-Parameter Scaling

6 Pages·0.18 MB·English

by  Daniel Braun

Boundary Conditions, the Critical Conductance Distribution, and One–Parameter
Scaling
Daniel Braun(1), Etienne Hofstetter(2), Gilles Montambaux(3), and Angus MacKinnon(4)
(1) IBM T.S. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598
(2) Institute for Quantitative Finance, Imperial College, London SW7 2BX, England
1 (3) Laboratoire de Physique des Solides, associ´e au CNRS
0 Universit´e Paris–Sud, 91405 Orsay, France
0 (4) Blackett Laboratory, Imperial College, London SW7 2BW, England
2
February 1, 2008

n
a
J
Westudytheinfluenceof boundaryconditions transverse tothetransport direction for disordered

9 mesoscopicconductorsbothattheAndersonmetal–insulatortransitionandinthemetallicregime.

We show that the boundary conditions strongly influence the conductance distribution exactly at
]
l themetal–insulatortransitionandwediscussimplicationsforthestandardpictureofone–parameter
l
a scaling. We show in particular that the scaling function that describes the change of conductance
h with system size depends on the boundary conditions from the metallic regime up to the metal–
- insulatortransition. Anexperimentisproposedthatmighttestthecorrectnessoftheone–parameter
s
e scaling theory.
m PACS numbers:
.
t I. INTRODUCTION BCs [12]. Samples with periodic boundary conditions
a
show a much stronger level repulsion than samples with
m
hard walls (Dirichlet boundary conditions).
More than fourty years after its discovery by Ander-
-
d son [1] the disorder–induced metal–insulator transition In this work we show with a numerical analysis of the
n is still the subject of strong theoretical as well as exper- conductance distribution atthe criticalpoint that P∗(g)
o does indeed depend on the BCs applied perpendicular
imental research [2]. One of the major achievements in
c to the transport direction. Choosing the appropriate
the long history of the Anderson metal–insulator tran-
[
sition (MIT) is the renormalization group theory, which boundary conditions, we can reproduce both the results

1 of refs. [8] and [9]. In particular, the average critical
has also become known as one–parameter scaling theory
v conductance g depends on the BCs. This alone already
[3]. Its basic assumption is that close to the transition c
2 implies a dependence of β(g) on the BCs since g is de-
the change of the dimensionless conductance g with the c
2
fined as β(g ) = 0. We confirm the BC–dependence of
1 samplesizeLdependsonlyontheconductanceitselfand c
β(g)analyticallybyreinvestigatingitsforminthemetal-
1 not separately on energy, disorder, the size of the sam-
licregimewiththe helpofa1/g expansion. Muchtoour
0 ple, its shape, the elastic mean free path l , etc.. Many
e
1 predictions, like the lower critical dimension, or the crit- surprisewefindthatearlieranalysesoverlookedtheeffect
0 ofthe BCsbyapproximatingasumoverdiffusionmodes
icalbehavior[4,5]weresuccessfullybasedonthis theory,
/ by an integral. Evaluating the sum more carefully, we
t as well as an enormous amount of numerical work that
a not only find a dependence on the BCs, but also a so
aimed at the direct calculation of the scaling function
m
farunkonwnln(l /L)/g terminβ(g)inthreedimensions
β(g) = dlng/dlnL. Another important consequence of e
- that makes β(g) non–universal in the metallic regime.
the one parameter scaling theory is the prediction of a
d
n universal conductance distribution P∗(g) exactly at the
o MIT[6]. Earliernumericalworkonthethreedimensional
II. NUMERICAL INVESTIGATION AT THE
c Anderson model seemed to confirm the universality of
ANDERSON TRANSITION
:
v theconductancedistribution[7]. Thedependenceonthe
i universality class was stressed in [8].
X Themodelstudiedisthethreedimensionaltightbind-
Recently, however, some doubts have been cast on
r whether the conductance distribution is universalwithin ing Anderson Hamiltonian with diagonal disorder on a
a
simple cubic lattice,
the same universality class. Two different numerical
studiesreportedtwodifferentformsofP∗(g)forthesame H = e i i +u i j
i
system [8,9], and it was found that the difference origi- | ih | | ih |
nates in the use of different boundary conditions (BCs) Xi <bXuijl>k
[10]. +u c(e2πiφ i j +h.c.). (1)
The idea that P∗(g) might depend on the BCs ap- | ih |
pears indeed very natural after the discovery that spec- σ<Xyi,jσ>z
tral statistics, and in particular the energy level spacing The e are distributed uniformly and independently be-
i
distribution P(s) exactly at the MIT, do depend on the tween w/2 and w/2. The notation < ij > means next
−
1
nearestneighbors,uisthehoppingmatrixelementwhich thedistributionisalmostindependentofthesystemsize,
wesetequaltounity inthe following,andw isthe disor- as is to be expected from the criticality of the ensemble
der parameter. The last sum in eq.(1) links correspond- at w = 16.5. But the distributions are clearly very dif-
c
ing sites on opposite sides of the cubic sample perpen- ferentforthetwoBCs. Themaximumofthedistribution
dicular to the y and z directions, assuming that trans- is considerably more pronounced for periodic BCs than
port occurs in the x–direction. Hopping between these forhardwalls. Amoredetailedstatisticalanalysisispre-
boundary sites ariseswhen the system is closedto a ring sentedintableIandfortheaveragevalues lng infigure
(c=1) and includes a phase factor ei2πφ, where φ is the 2. h i
magnetic flux in units of h/e inclosed by the ring. Hard
wall(Dirichlet) BCs correspondto c=0. The model (1)
shows a MIT at the critical disorder wc 16.5 [14]. L BC hgi=gc σg hlngi σlng
The numerical calculation of the cond≃uctances uses a 6 P 0.356 0.314 -1.554 1.183
8 P 0.377 0.324 -1.476 1.159
standard Green’s function recursion technique [11] that
10 P 0.392 0.329 -1.412 1.129
yieldsthetransmissionmatrixtofthesample. Thelatter
12 P 0.402 0.334 -1.378 1.118
isconnectedtothetwo–probeconductanceofthesample
16 P 0.413 0.336 -1.329 1.092
by the Landauer–Bu¨ttiker formula
6 HW 0.313 0.306 -1.777 1.281
g =trtt+, (2) 8 HW 0.326 0.310 -1.710 1.252
10 HW 0.331 0.312 -1.685 1.246
where g =G/(e2/h) denotes the conductance G in units 12 HW 0.338 0.311 -1.675 1.211
16 HW 0.348 0.319 -1.614 1.222
oftheinverseofthevonKlitzingconstanth/e2. Whether
the two–probe conductance formula or the four–probe
TABLE I. Statistical analysis of the critical conductance
conductance formula is used, is quite irrelevant at the
distributionfordifferentboundaryconditions(P periodicand
metal insulator transition, since the bulk resistance al-
HW hard wall). Besides the averages of g and lng also the
ways dominates largely over the contact resistance [15].
standarddeviationsofthesequantities,σgandσlngaregiven.
All conductances were calculated at energy E 0. The
≃
number of conductances used for each BC and system
size ranged between 105 for L = 6 and L = 8 to 2 103
·
for L = 16. All system sizes L are measured in units of
the lattice constant.
−1.0
0.4
−1.2
0.3 −1.4
>
g
n
) <l −1.6
g
n 0.2
(l
P
−1.8
0.1
−2.0
6 8 10 12 14 16
L
0.0
−6.0 −4.0 −2.0 0.0
ln g FIG.2. As a function of system size the average hlngi is
plotted for periodic (circles) and hard wall boundary condi-
FIG.1. Criticalconductancedistributionforperiodicand
tions (squares).
hard wall boundary conditions, and different sample sizes.
SamplesizesL=8,10,12,16aredenotedbycircles, squares,
diamonds and triangles, respectively; open symbols indicate The average is always over the disorder ensemble.
periodicBCs,fulloneshardwalls. Thefulllinesareaverages Fig.2 shows that the average logarithmic conductance
overthe abovesystem sizes. still depends slightly on the system size in the regime
investigated. But the difference between periodic and
Our main numerical result is shown in Fig.1, where hard wall BCs does not diminish with increasing L, and
we have plotted the distributions of the logarithmof the the dependence on L decreases for larger L. Where we
conductance at the transition for periodic and hard wall have used the same system sizes as in Refs. [8,9] our
(HW) BCs, and different system sizes. For the same BC values for all quantities calculated ( g , lng , and the
h i h i
2
standard deviations of g and lng) coincide within one The argument y is defined as
percent with the values given in these references. For
2
comparisonwith [9] g should be multiplied with a factor Dτe 1 le
y = = . (7)
2, since we consider only one spin direction. Thus, the L2 3 L
(cid:18) (cid:19)
discrepancy between [8] and [9] can indeed be explained
by the influence of the BCs (see also [10]). Previousanalysesintheliteratureproceededbyapproxi-
matingthesumbyanintegral[18],whereuponalldepen-
Our result has important implications for the scaling dence on the boundary conditions is lost. While this is a
theory of the metal–insulator transition, since it shows goodapproximationforg ,importantcorrectionsof
→∞
that the scaling function β(g) must depend on the BCs. theorder(lng)/g ariseforfiniteg,whichwearegoingto
The conductance that enters in this equation has to be derive now, assuming that to this order no further dia-
understoodasanaverageconductance[18],andthe crit- grams beyond the diffuson approximationcontribute. In
ical conductance is given by β(g ) = 0. According to [19] it was shown by field theoretical methods combined
c
our results g depends on the BCs, g = 0.413 for peri- with a renormalizationgroupapproachthat the diffuson
c c
odic BCs and g = 0.348 for hard walls at L = 16 (see approximation gives the leading perturbative contribu-
c
table I), and therefore the β(g) curves must at least be tion to the small energybehavior ofthe spectral correla-
shifted as a function of the BCs. In the next section we tion function to order 1/g2.
show by re–examining the weak localization corrections In order to proceed it is convenient to differentiate
to the conductance that also in the metallic regime β(g) SBC(y). The derivatives for both PBCs and HW BCs
depends on the BCs. can be written with the help of the function
∞
F(y)= e−π2n2y (8)
III. METALLIC REGIME
n=1
X
as
It is well knownthat already in metallic regime g 1
≫
the quantum interference of diffusing electrons reduces ∂ S (y)= π2F(y)(2F(4y))2, (9)
y P
the conductance comparedto the classicalvalue g =σL, −
∂ S (y)= π2F(y)(1+F(y))2. (10)
where σ is bulk conductivity. The weak localization cor- y HW
−
rection δg is given by a sum over diffusion modes as [18]
The function F(y) is related to the complete elliptic in-
e−Dq2τe tegralsK ≡K(k)andK′ ≡K(k′)with k′ =√1−k2 by
δg = 2 . (3) [20]
− q2L2
q
X 1/2 ∞
The sum is limited to the diffusive regime where Dq2 1 2K 1 = e−πn2K′/K. (11)
1/τe. This limitation is taken into account by the exp≪o- 2 (cid:18) π (cid:19) − ! nX=1
nentialcut-off;τe istheelasticcollisiontime,D =vF2τe/3 Since we are interested in y 1, we need K′/K 1
denotes the diffusion coefficient and vF the Fermi veloc- and therefore k 1 (k′ 1).≪For small values of k′≪the
ity. The sum (3) depends on the BCs via the quantiza- → ≪
elliptic integrals behave like
tion condition for the diffusion modes q. For the trans-
port direction the wave vector is quantized according to 4 π
K =ln + (k′2), K′ = + (k′2), (12)
qx =nxπ/L,nx =1,2,.... Periodicboundaryconditions k′ O 2 O
in the y-direction imply q = n 2π/L, n = 1, 2,...
y y y
and correspondingly for the z-direction. Hard±wa±ll BCs and we therefore obtain
on the other hand lead to q = n π/L, n = 0,1,2,...
y y y 1 1 1/2
and qz =nzπ/L, nz =0,1,2,.... Consequently we have F(y) 1 . (13)
≃ 2 (cid:18)πy(cid:19) − !
2
δg = S (y) (4)
−π2 BC Inserting this into eqs.(9), (10) and integrating with re-
spect to y yields
wheretheindexBC standsforaboundaryconditionand
√π 5 π2
exp π2(n2 +4n2+4n2)y SP = + πlny 2π3/2√y+ y αP (14)
S (y)= − x y z , (5) 4√y 8 − 2 −
P n2 +4n2+4n2
nynX,xn>z6=00 (cid:2) x y z (cid:3) SHW = √π 1πlny+ 1π3/2√y+ π2y αHW . (15)
4√y − 8 4 8 −
exp π2(n2 +n2+n2)y
S (y)= − x y z . (6)
HW n2 +n2+n2 whereas,replacingthe sum(3)byanintegral,one would
nynX,xn>z≥00 (cid:2) x y z (cid:3) have found
3
√π It remains to reexpress L˜ by g. To this end we invert
S = α. (16)
4√y − g(L˜) from (18) to order 1/g,
whereαisanintegrationconstantresultingfromthecut- 1
L˜ = (g+alng aln(σ˜ A)+b) , (21)
off at small q 1/L. Thus, the leading term for small σ˜ A − −
y, π/y/4, is≃the same for both boundary conditions. −
TheintegrationconstantsαP andαHW canbeevaluated and insert it in (20). We obtain the final result
p
numerically, by subtracting from the exactly calculated
sums the analytical formulae (14) and (15) without the 1
β(g)=1 (b+a(1+ln(σ˜ A)) alng)+ (1/g2).
constants. At the same time this serves as a sensitive − g − − O
check for the correctness of these formulae. For small y
(22)
the differences converge to
α 6.1509,α 2.3280. (17) It is now obvious that the scaling function does indeed
P HW
≃− ≃
depend on the BCs via the coefficients a and b, and
We have evaluated the sum numerically down to values the dependence arises at order (lng)/g. Furthermore,
y =10−6, where in particular the logarithmic term with β(g)depends toorder1/g aswellonthe materialdepen-
the prefactors given above could be clearly verified. dent dimensionless bulk conductivity σ˜, and is therefore
With Eqs.(14)and(15)the conductanceas a functionof non–universal! Again, the non–universality vanishes for
the dimensionless length L˜ L/l takes the form g (equivalently, on the metallic side of the transi-
e
≡ → ∞
tion: L ), but is important if one is interested in
g =(σ˜ A)L˜ alnL˜+b+ (1/L˜) (18) β(g) at fi→nit∞e values of g. Since HW BCs lead to smaller
− − O
valuesofβ(g)atintermediatevaluesofg thanPBCsbut
for both periodic and hard wall BCs. The dimensionless toasmallercriticalconductance,thereshouldbeapoint
bulk conductivity σ˜ is defined as σ˜ = σleh/e2, and the where the two curves cross, which would imply that in
constantA=√3/(2π3/2)is the same forboth BCs. The that point the change of g with the system size is inde-
coefficients a and b on the other hand do depend on the pendent of the BCs. Due to the dependence of β(g) on
boundary conditions; their values are given in table II. σ˜, this point is not expected to be universal, though.
Note that in the traditional approach the coefficient a Themostinterestingquestionisofcourse,whetheralso
vanishes! theslopeofβ(g)atg =g ischangedbytheBCsand/or
c
Quite surprisingly a < 0 for PBCs, which means that σ˜,asthisslopedeterminesthecriticalexponentν defined
the conductance increases even slightly faster than lin- by ξ(w) g g −ν according to β(g) = 1(g g )/g .
early with the system size. This looks as if there was This que∝sti|on−arcis|es actually already fromν th−e dcepenc-
anti–localization, but it should be noted that the lead- dence of spectral statistics on the BCs, since the scal-
ing behavior due to weak localization is still the usual ingfunctioncanbe determinedalsofrompurely spectral
decreaseofthe (bulk–)conductivity, i.e.the leadingterm statistics [16,17]. Very recently it has been argued that
is linear in the system size and with the expected neg- within the same universalityclassν does atleastnotde-
ative sign. The fact that a < 0 only for PBCs suggests pend on the shape of the sample [29]. Since the critical
a simple physical explanation for the logarithmic term: spectral statistics does depend on the shape of the sam-
Closing the sample to a double torus by imposing PBCs ple muchinthe same wayas onthe BCs[30](indeed, all
allows for additional paths that interfere constructively that has been said above about the dependence on the
and lead to enhanced localization for small system sizes BCs translates one to one to a dependence on the shape
compared to the HW case. When increasing the system of the sample), one might suspect that ν is also inde-
size these additional localizing paths quickly stop con- pendent of the BCs. On the other hand, considering the
tributing and the conductance therefore increases more qualitative behavior of the two scaling curves a critical
rapidly than what would be expected just from the vol- exponentindependentoftheBCswouldappearratheras
ume part of the weak localization. coincidence. However, so far it is an open question and
definitly deserves attention [26].
Wearenowinthepositiontoexploretheconsequences
With the dependence of the critical conductance dis-
oftheBCdependentweaklocalizationcorrectionsforthe
tributionontheBCs,anexperimentaltestofthecorrect-
scaling function β(g). Inserting (18) into the definition
ness of the one–parameter scaling picture seems within
reach. Eventhoughanaccurateabsolutemeasurementof
dlng
β(g) (19) the criticalexponentis ratherdifficult [27,28],onemight
≡ dlnL˜ hope to detect a change with the BCs. To this end it is
notevennecessaryto openandclosethesample. Rather
yields
one can investigate the difference between periodic and
1 anti–periodic boundaryconditions. Atleastinonedirec-
β(g)=1+ alnL˜ b a+ (1/L˜) . (20)
g − − O tion anti–periodic BCs, i.e. a phase factor 1 between
−
(cid:16) (cid:17)
4
two opposite sides of the sample, can be easily produced [5] D. Vollhardt and P. W¨olfle, Phys. Rev. Lett.48 699
by closing the sample to a ring and introducing half a (1982).
magentic flux quantum (φ = 1/2 in (1)). Note that for [6] B. Shapiro, Phys.Rev.B 34, 4394 (1986); B. Shapiro,
φ = 1/2 the system still belongs to the orthogonal uni- Mod. Phys.Lett. B5, 687 (1991).
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Phys.Rev.Lett.82, 668 (1999).
upon inclusionof halfa flux quantum wouldbe as wella
[10] K. Slevin and T. Ohtsuki, Phys. Rev. Lett. 82, 669
most welcome contributionto the long-lastingdebate on
(1999), Phys.Rev.Lett. 84, 3915 (2000)
the limits of validity of one–parameter scaling.
[11] A. MacKinnon, Z. Phys. B 59, 385 (1985).
In summary,we haveshownthat the conductance dis-
[12] D.Braun,G.Montambaux,andM.Pascaud,Phys.Rev.
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metallic regime the dependence of a change of the con-
[15] D. Braun, E. Hofstetter, G. Montambaux, A. MacKin-
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the change of conductance when the size of the sample (1993).
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samplefromthemetallicregimeuptothemetalinsulator mans,G.Montambaux,J.-L.PichardandJ.Zinn-Justin;
transition. North-Holland, Amsterdam (1995).
Acknowledgements: D.B. gratefully acknowledges use- [19] V.E. Kravtsov and A.D. Mirlin, JETP Lett. 60, 656
(1994).
ful discussions with B.L. Altshuler, P.A. Braun and
[20] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series,
Y.V.Fyodorov,andG.M.wouldliketothankB.Shapiro,
and Products, Fifth Edition, A. Jeffrey, ed., Academic
A. Kamenev, V. Kravtsov, and V. Yudson. This work
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was supported by the Sonderforschungsbereich237 “Un-
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Akkermans,G.Montambaux,J.-L.PichardandJ.Zinn- (2000)) reported numerical evidencethat thecritical ex-
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[3] E. Abrahams, P.W. Anderson, D.C. Licciardello, and dered and Doped Semiconductors, ed. D.M. Finlayson,
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5
a b
PBC −5/(2π)≃−0.7958 5(ln3)/(4π)−2·6.1509π2 ≃−0.8093
HW 1/(2π)≃0.1592 2·2.328/π2−(ln3)/(4π)≃0.3843
TABLEII. Coefficients in the1/L˜ expansion of g for periodic boundary conditions (PBC) and hard walls (HW).
6