Metal-insulator transition in spatially-correlated random magnetic field system

8 Pages·0.12 MB·English

by  D. N. Sheng

Metal-insulator transition in spatially-correlated random magnetic field system
D.N. Sheng, Z. Y. Weng
Texas Center for Superconductivity, University of Houston, Houston, TX 77204-5506
Wereexaminetheproblemofdelocalization oftwo-dimensionalelectronsinthepresenceofrandom
magnetic field. By introducing spatial correlations among random fluxes, a well-defined metal-
9
insulator transition characterized by a two-branch scaling of conductance has been demonstrated
9 2
numerically. Critical conductance is found non-universal with a value around e /h. Interest-
9
ing connections of this system with the recently observed B = 0 two-dimensional metallic phase
1
(Kravchenkoet al.,Phys. Rev. B 50, 8039 (1994)) are also discussed.
n
a 71.30.+h, 73.20.Fz, 73.20.Jc
J

4 Whether two-dimensional (2D) electrons can become erally increases as the Fermi energy shifts towards the
delocalized in the presence of random magnetic field band center. With much reduced error bar, the present
]
n (RMF) is still controversial. This is a very important numerical algorithm is also applied to an uncorrelated
n issue related to many interesting systems, like half-filled (white noise limit) RMF case and the results unambigu-
- quantum Hall effect (QHE) [1,2], gauge-field descrip- ously show that all states are localizedwithout a critical
s
i tion [3] of high-T superconductor and so on. By us- region at strong strength of RMF. Possible connections
d c
ing the standard transfer-matrix method [4], a number of the present RMF system to the zero-magnetic-field
.
at ofnumericalcalculations[5–7]havebeenperformedfora (B=0) 2D metal [14] are also discussed at the end of the
m non-interacting 2D electron system subject to spatially- paper.
uncorrelatedRMF.Theresultsindicatethatelectronsare We consider a tight-binding lattice model of noninter-
-
d alwayslocalizednearthebandedge,whilethereisadra- actingelectronsunderRMF.The Hamiltonianis defined
n
matic enhancement of localization length as one moves as follows:
o
towards the band center. However, the interpretation
[c of the latter is rather conflicting, ranging from that all H =− X eiaijc+i cj +Xwic+i ci (1)

states are still localized [5,7] with an extremely large lo- <ij> i
1
calization length close to the band center to the exis-
v Here c+ is a fermionic creation operator, and < ij >
9 tence of a critical region [6] with divergent localization i
refers to two nearestneighboring sites. w is an uncorre-
1 length. Even if a critical region characterized by wave- i
0 functions with fractional dimensionality [8] could exist lated random potential (white noise limit) with strength
1 |w | ≤ W. A magnetic flux per plaquette is given as
here,ametallicphaseseemsbeingruledoutbythosenu- i
0 φ(k)= a ,wherethesummationrunsoverfourlinks
merical calculations since a two-branchscaling as a hall- P2 ij
9
aroundaplaquettelabeledbyk. Weareinterestedinthe
9 markformetal-insulatortransition(MIT)hasneverbeen
case where φ(k) at different k’s is correlated which can
/ found. Analytically, while the study based on a pertur-
t be generated in the following way:
a bativenonlinearsigmamodelapproachpointed[9]tothe
m
localizationofallstates,the existenceof extendedstates
d- was shown [10] possible in the presence of a long-range φ(k)= λh2/04Xfie−|Rk−Ri|2/λ2f (2)
logarithmicinteractionofthe topologicaldensity (due to f i
n
o fluctuatingHallconductance[11]),whichissupportedby
where R (R ) denotes the spatial position of a given
c directnumericalcalculations[12,13]oftopologicalChern k i
: number for the case of spatially-uncorrelatedRMF with plaquette k(i). h0 and λf are the characteristic strength
v andcorrelationlengthscaleofRMF,respectively. f is a
i reduced field strength. i
X random number distributing uniformly between (-1,+1).
In contrast to spatially-uncorrelated RMF, however,
We employ the following numerical algorithm to cal-
r magnetic flux fluctuations in realistic systems [1–3] may
a culate the longitudinal conductance G . Based on the
be much more smooth with finite-range spatial correla- xx
Landauer formula, G for a square sample N = L×L
tions. Such a smoothness can significantly reduce the xx
can be determined as a summation over contributions
random scattering effects while still retain the delocal-
fromalltheLyapunovexponentsoftheHermitiantrans-
ization effect [10–13] introduced by magnetic fluxes. In
fer matrix product T+T [4,15]. To reduce the bound-
this paper, we demonstrate numerically for the first
ary effect of a finite-size system, we connect M different
time the existence of MIT which is characterized by a
square samples together to form a very long stripe along
two-branch scaling of conductance in the presence of
x direction [of size L×(LM)]. Typically M is chosen
spatially-correlatedRMF.Thecriticalconductanceitself
to be larger than 5000 even for the largest sample size
is non-universal, with its value around e2/h which gen-
(L= 200) in this work. In this way, the statistical error
1
bar is significantly reduced in our results (about 1.5%). at W <W , G can be approximately fitted by the fol-
c xx
Inmostofearliernumericalcalculations,finite-sizelocal- lowingform: Gxx =Gs−c0∗exp(−L/ξ0). HereGs isthe
ization length has been computed where the statistical saturatedconductanceatL→∞,whichisnon-universal
fluctuation is usually quite big (especially near the band and depends on the disorder strength W as well as the
center) as compared to a direct calculation of the finite- correlationlength λ of random fluxes.
f
size longitudinal conductance in the present algorithm. The introduction of spatial correlations in random
As a test, we have first re-studied the case in which fluxesiscrucialforsuchametal-insulatortransition. We
the flux φ(k) is randomly distributed between −π to π also found a well-defined MIT at an even shorter cor-
without spatial correlations – the situation investigated relation length: λ = 2.0. But the larger λ is, the
f f
previously[5–7]asmentionedatthebeginningofthepa- stronger the metallic behavior becomes with a larger
per. We find that G monotonically decreases with the saturated conductance. The previously discussed RMF
xx
sample size L at all strengths of the on-site disorders: in white noise limit may only belong to a very special
from W = 0 to W = 4, and is extrapolated to zero at case in which the localization effect of strong random-
largesample-sizelimitasshowninFig. 1atafixedFermi ness of fluxes overwrites the delocalization effect of the
energyE =−1. Inthe insertofFig. 1,G isshownas same fluxes. We would like to point out that even in
f xx
a function of the disorder strength W at different sam- such an uncorrelated random flux case, the delocaliza-
ple sizes: L = 24, 80, and 200, which shows that even tion may be still enhanced if one reduces the strength of
at W =0 the conductance monotonically decreases with RMF.EarliertopologicalChernnumbercalculations[12]
the increase of L, indicating that the dominant role of clearly indicates a delocalization transition as the max-
the random flux here is similar to the random potential imum strength of φ(k) is reduced to around π/2. We
in causing localization of electrons. The one-parameter have computed the conductance in this case using the
scalingofG canbeobtainedbychoosingascalingvari- present method at much larger sample sizes and indeed
xx
able ξ at each random potential W. As plotted in Fig. found a slight increase of the conductance with sample
2, all data can be then collapsed onto a single curve of size at W < W , which is opposite to strong random
c
L/ξ, in which ξ is given in the insert of Fig. 2. Clearly flux limit where conductance always decreases with the
ξ is always finite although it becomes extremely large at increase of sample size (Fig. 1), although a two-branch
weak disorder limit. This is consistent with the conclu- scaling curves here is not as clear-cutas in the spatially-
sion[5,7]thatelectronsarealllocalizedandexcludesthe correlated RMF case shown in Fig. 4.
possibility of a critical region [6] as the error bar in our Asmentionedabove,thecriticalconductanceG varies
c
calculation is much less than the variation of the con- from 0.5e2/h to around 2e2/h as the Fermi energy shifts
ductanceitself. Noticethatinweak-disorderlimitξ may from the band edge towards band center (the insert of
no longer be interpreted as localization length [7] which Fig. 3). It is interesting to note that G obtained here
c
characterizes an exponential decay of conductance with is in the same range as the experimental data found in
sample size at strong localized region. recent B = 0 2D MIT system [14]. In the following,
Now let us focus on RMF with smooth spatial corre- we would like to point out a possible deeper connection
lations as defined in (2). With the correlation length between the two systems.
λ = 5.0 (the lattice constant as the unit) and flux In a recent experiment [16] in p-type GaAs/AlGaAs
f
strengthh0 =1,GxxasafunctionofdisorderstrengthW heterostructure, the evolution of delocalized states was
iscomputedatagivenFermienergyE =−1asshownin studied continuously from the QHE regime at strong
f
Fig. 3. Curvesatdifferentsamplesizes(L=16–200)all magnetic field to zero field limit where the B = 0 MIT
cross at a fixed-point W = W , which is independent of is recovered. The authors found that the critical den-
c
lattice size L within the statistical error bars. It is qual- sity of the lowest extended level in QHE regime flattens
itatively different from the behavior of G in spatially- out, instead of floating up towards infinity, as magnetic
xx
uncorrelated RMF case discussed above. At W > W , field is reduced and can be extrapolated to the critical
c
G continuouslydecreaseswiththeincreaseofthesam- density of B = 0 MIT in such a material. Similar result
xx
ple size, which can be extrapolated to zero at large L has been also observed in Si-MOSFET samples [17,18].
limit, corresponding to insulating phase. On the other At first sight, it is tempting to think that the lowest ex-
hand,atW <W ,G monotonicallyincreaseswithlat- tended level of QHE somehow survives at B = 0, but
c xx
tice sizes like a typical metallic behavior. The insert of physicallyitdoesnotmakemuchsensebecauseQHEex-
Fig. 3 shows the critical conductance G (corresponding tended states carry quantized Hall conductance known
c
to W = Wc) at different Fermi energies and h0’s. The as Chern number whereas at B = 0 the total Hall con-
dataofG inFig. 3canbecollapsedontoatwo-branch ductance must be zero without time-reversal symmetry-
xx
curve as a function of scaling variable L/ξ as shown in breaking. In fact, experiments indicated [17] that be-
Fig. 4 for W > W and W < W , respectively. The fore B vanishes, extended levels of the QHE may al-
c c
insertofFig. 4showsthe scalingvariableξ vs. W which readymergewithadifferentkindofextendedlevel(called
diverges at the critical point W . In the metallic phase QHE/Insulator boundary in Ref, [17]) which carries an
c
2
opposite sign of Hall conductance. Theoretically, it has fromRobertWelchfoundation,andbytheStateofTexas
been previouslyfound [19]that QHE extended statesin- through the Texas Center for Superconductivity at Uni-
deed can be mixed with some boundary extended level versity of Houston.
movingdownfromhigh-energysideatstrongdisorderor
weak magnetic field limit which carries negative Chern
number in a lattice model. When those extended states
with different signs of Chern numbers mix together at
weak magnetic field limit, there could be two conse-
quences: one is that no states will eventually carry non-
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(1994).
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(1995).
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s
fieldproblembasedonbothexperimentalandtheoretical
Fig. 1 The evolution of conductance G (in units of
considerations. xx
Acknowledgments-Theauthorswouldliketothank e2/h)withsamplewidthLatdiffernentdisorderstrength
W’s for spatially-uncorrelated RMF case. The insert:
C. S. Ting, X. G. Wen, and especially S. V. Kravchenko
G as a function of W at different L’s. Fermi energy is
forstimulatingandhelpfuldiscussions. Thepresentwork xx
fixed at E =−1.
is supported by Texas ARP grant No. 3652707, a grant f
3
Fig. 2. The data of G at different L’s and W’s all
xx
collapse onto a scaling curve as a function of L/ξ. The
insert: ξ versus W.
Fig. 3 G versus W at different sample sizes (L =
xx
16(•),24,32,48,64,80,120,200). W is the critical dis-
c
order. Fermi energy is chosen at E = −1. The insert:
f
criticalconductanceG asafunctionofFermienergyE .
c f
Fig. 4. Two branch-scaling curve of G as a single
xx
function of L/ξ for different L’s and W’s. The insert: ξ
versus W.
4
1.5
10
x
x
G
L = 24 80 200
1
0 0.5 1 1.5
W
W = 0
x 1.5
x
G 1 2.0
2.5
3.0
3.25
0.1
3.5
20 40 60 80 100 120 140 160 180 200
L
Fig. 1
10
1
x 7
10
x
G 0.1
5
10
ξ
3
10
0.01
10
0 1 2 3
W
0.001
1e-06 0.0001 0.01 1
L / ξ
Fig. 2
5
h = 0.5 h = 1.0
0 0
2
4
c
G
1
3
x • • •
x • 0
G • • -4 -3 -2 -1
•
• E
f
2 •
•
•
•
•
•
↑ •
•
•
W
1 c •
•
•
•
•
0
0.5 1 1.5 2 2.5 3 3.5 4
W
Fig. 3
10
1
6
10
x
x
G
4
10
↑
ξ
0.1
W
c
2
10
1
1 2 3 4
W
0.01
0.0001 0.01 1 100
L / ξ
Fig. 4