RAPID COMMUNICATIONS
PHYSICAL REVIEW B 69, 121409共R兲 共2004兲
Nonlinear screening and percolative transition in a two-dimensional electron liquid
Michael M. Fogler
Department of Physics, University of California, San Diego, La Jolla, California 92093, USA
共Received 11 December 2003; published 24 March 2004兲
A variational method is proposed for calculating the percolation threshold, the real-space structure, and the
ground-state energy of a disordered two-dimensional electron liquid. Its high accuracy is verified against exact
asymptotics and prior numerical results. The inverse thermodynamical density of states is shown to have a
strongly asymmetric minimum at a density that is approximately the triple of the percolation threshold. This
implies that the experimentally observed metal-insulator transition takes place well before the percolation point
is reached.
DOI: 10.1103/PhysRevB.69.121409
PACS number共s兲: 73.21.Fg, 71.30.⫹h, 73.43.Cd
The discovery that a two-dimensional 共2D兲 electron liquid
can be a metal at moderate electron density n e and an insulator at small n e was a major surprise that questioned our
fundamental understanding of the role of disorder in such
systems. Today, almost a decade later, it remains a subject of
an intense debate.1 One important reason why the conventional theory fails could be its flawed basic premise of the
‘‘good’’ metal, i.e., a uniform electron liquid slightly perturbed by impurities and defects. Indeed, modern nanoscale
imaging techniques2–5 unambiguously showed that lowdensity 2D electron systems are strongly inhomogeneous,
‘‘bad’’ metals, where effects of disorder are nonperturbatively strong. In particular, depletion regions 共DRs兲, i.e., regions where n(r) is effectively zero, exist. They appear
when n e is too small to adequately compensate fluctuating
charge density of randomly positioned impurities. As n e is
reduced in the experiment, e.g., in order to approach the
vicinity of the metal-insulator transition 共MIT兲, the DRs are
expected to grow in size and concentration and eventually
merge below some percolation threshold n e ⫽n p . An important and controversial issue is whether or not this percolation
transition plays any role in the observed MIT.1 To resolve it
one needs to have a theory that is able to calculate n p and
that can describe the inhomogeneous structure of the 2D
metal at n e ⬃n p . Great progress in this direction has been
achieved by Efros, Pikus, and Burnett,6 whose paper is intellectually tied to earlier work on nonlinear screening by Efros
and Shklovskii.7 Still, analytical results remained scanty and
numerical simulations6,8 were the only known way to quantitatively study the n e ⬃n p regime. These simulations are
very time consuming and redoing them in order to get any
information beyond what is published6,8 or to study novel
experimental setups seems impracticable. Below I will show
that a variational approach to the problem can be a viable
alternative. Comparing it with the available numerical results
for a typical model of the experimental geometry 共Fig. 1兲, I
establish that it correctly predicts the value of n p and accurately reproduces the energetics of the ground state, in particular, the density dependence of the electrochemical potential ␮ and of the inverse thermodynamical density of states
共ITDOS兲, ␹ ⫺1 ⫽d ␮ /dn e , over a broad range of n e . The
most striking feature of the resultant function ␹ ⫺1 (n e ) is a
strongly asymmetric minimum, which is observed in real
experiments.9–12 I will elaborate on the origin of this feature
0163-1829/2004/69共12兲/121409共4兲/$22.50
and show that it occurs at n e ⬇3n p largely independently of
the parameters of the system. Recently, this minimum attracted much interest when Dultz and Jiang10 reported that in
some samples it virtually coincides with the apparent MIT.
The proposed theory indicates that, at least for these samples,
any connection between the MIT and the percolation of the
DRs can be ruled out. Thus, the explanation of the MIT lies
elsewhere.
The Hamiltonian of the model is adopted from Efros,
Pikus, and Burnett6 共EPB兲 共see also Fig. 1兲,
H⫽
冕 再
d 2r
冎
1
关 n 共 r兲 ⫺n e 兴 ⌽ 共 r兲 ⫹H 0 共 n 兲 ,
2
where ⌽(r) is the electrostatic potential,
⌽⫽
冕
d 2r ⬘
冋
共1兲
册
e 2 n 共 r⬘ 兲 ⫺n e
n d 共 r⬘ 兲 ⫺n i
⫺
,
␬ 兩 r⬘ ⫺r兩
冑共 r⬘ ⫺r兲 2 ⫹s 2
共2兲
␬ is the dielectric constant, and H 0 (n) is the energy density
of the uniform liquid of density n,
H 0 共 n 兲 ⫽⫺ 共 e 2 / ␬ 兲 n 3/2h 0 共 n 兲 .
共3兲
FIG. 1. The geometry of the theoretical model. The 2D layer of
interest is sandwiched between the top and the bottom metallic
gates. The dopants reside in the plane with the dashed border. The
depletion regions 共shown as holes in the probed layer兲 enhance the
penetrating electric field E p .
69 121409-1
©2004 The American Physical Society
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 69, 121409共R兲 共2004兲
MICHAEL M. FOGLER
At low densities, h 0 (n)⬃1 is a slow function13 of n. The
negative sign in Eq. 共3兲 reflects the prevalence of the
exchange-correlation energy over the kinetic one in this regime. The potential created by the random concentration of
dopants n d (r) can be expressed in terms of the effective
in-plane background charge ˜␴ (q)⫽ñ d (q)exp(⫺qs) 共the tildes denote the Fourier transforms兲. With these definitions,
⌽(r) coincides with the potential created by the charge density ␴ (r)⫽n(r)⫺n e ⫺ ␴ (r). I will assume that the dopants
have the average concentration 具 n d 典 ⬅n i Ⰷs ⫺2 and are uncorrelated in space. In this case the one- and two-point distribution functions of ␴ (r) have the Gaussian form,
of such holes, their fields are additive, leading to ␹ ⫺1
⫽(8 ␲ e 2 /3␬ )N h a 3 . From here the exact large-n e asymptotics of ␹ ⫺1 and ␮ can be obtained by substituting the proper
N h and averaging over the distribution of a. This can be done
by noting that these holes appear around the minima of ␴ (r)
whose statistics is fixed by Eqs. 共4兲–共6兲. It is easy to see that
the most probable holes are nearly perfect circles with radii6
a⬃s 冑K 0 /n e . The charge distribution around a single hole at
distances r⬎a is given by the formula
P 1 共 ␴ 兲 ⫽ 共 2 ␲ K 0 兲 ⫺1/2exp共 ⫺ ␴ 2 /2K 0 兲 ,
where the hole is assumed to be centered at r⫽0 and a 2
⫽⫺3 关 n e ⫹ ␴ (0) 兴 / ␴ xx . Equation 共7兲 can be obtained, e.g.,
by generalizing the textbook solution15 for the hole in the
metallic sheet16 and is also the limiting form of Eq. 共11兲 in
Ref. 17. Previously, Eq. 共7兲 was used for study of quantum
dots in Ref. 18.
In the current problem the main factor that determines the
⫺1
of the depletion holes to ␹ ⫺1 is their
net contribution ␹ DR
exponentially small concentration, proportional to P 1 (⫺n e )
⬀exp(⫺n2e /2K 0 ). The final result,
P 2⫽
1
2␲冑
2
K 20 ⫺K r ⬘
冋
exp
2K r ⬘ ␴␴ ⬘ ⫺K 0 共 ␴ 2 ⫹ ␴ ⬘ 2 兲
2
2 共 K 20 ⫺K r ⬘ 兲
册
共4兲
, 共5兲
where ␴ ⬘ ⫽ ␴ (r⫹r⬘ ) and K r is given by
K r ⬅ 具 ␴ 共 r 兲 ␴ 共 0 兲 典 ⫽n i s/ ␲ 共 r 2 ⫹4s 2 兲 3/2.
共6兲
6
The characteristic scales in the problem are as follows. The
typical amplitude of fluctuations in ␴ is 冑n i /s 关see Eqs. 共4兲
and 共6兲兴. Their characteristic spatial scale is the spacer width
s 关cf. Eqs. 共4兲–共6兲 and Fig. 1兴. In the cases studied below,
n e ⲏn p and n e Ⰷn p , s exceeds the average interelectron
. As usual in Coulomb problems, the
separation a 0 ⫽n ⫺1/2
e
energy is dominated by the longest scales, in this case s. The
fluctuations H 0 (n)⫺H 0 (n e ) of the local energy density
come from the interactions on the much shorter scale of a 0
Ⰶs and can be treated as a perturbation.6 For the purpose of
calculating the ground-state density profile n(r) I neglect
H 0 . Once such a ground state is known, I correct the total
energy H by adding to it H 0 averaged over n(r).
To find n(r) one needs to solve the electrostatic problem
with the following dual boundary conditions: if n(r)⬎0,
then ⌽(r)⫽ ␮ ⫽const; otherwise, if n⫽0 共DR兲, then6,7,14
⌽⬎ ␮ . I start with the analysis of the large-n e case, which
clarifies why the ␹ ⫺1 (n e ) dependence is nonmonotonic and
which provides a formula for the density n m where ␹ ⫺1 has
the minimum.
In the limit n e Ⰷ 冑n i /s, the asymptotically exact treatment
is possible because the DRs appear only in rare places where
␴ (r) dips below ⫺n e . 共In practice, this limit is realized
when n e ⬎K 1/2
0 ⬇0.2冑n i /s.兲 The corresponding electrostatic
problem is analogous to that of the metallic sheet perforated
by small holes 共see Fig. 1兲. The most elegant way to derive
␹ ⫺1 in this regime is to calculate the fraction E p /E 0 of the
electric field that reaches the bottom layer in the geometry of
Fig. 1. Indeed, E p /E 0 is nonzero only if the probe layer is
not a perfect metal, ␹ ⫺1 ⫽0. In the simplest case, where
distances s 1 and s 2 are large, the following formula holds:
␹ ⫺1 ⫽4 ␲ (e 2 s 2 / ␬ )(dE p /dE 0 ) 共cf. Ref. 9兲. This is essentially the formula used to deduce ␹ ⫺1 in the experiment.9–12
It is immediately obvious that holes in the metallic sheet
enhance the penetrated field E p . For example, the field leaking through a round hole of radius a is the field of a dipole15
p⫽(a 3 /3␲ )E0 . If there is a finite but small concentration N h
n共 r 兲⫽
␴ xx a 2
␲
冋冑
r2
a
⫺1⫺
2
册
冉 冊
r2
a
2
⫹
2
a
arccos ,
3
r
⫺1
␹ DR
⯝ 共 3 冑2/8␲ 兲共 e 2 n i / ␬ sn 2e 兲 exp共 ⫺4 ␲ s 2 n 2e /n i 兲 ,
共7兲
共8兲
agrees with that of EPB but has no numerical coefficients left
undetermined. To finalize the calculation, one needs to aug⫺1
2
2
by the local term ␹ ⫺1
ment ␹ DR
0 ⫽d 具 H 0 (n) 典 /dn e . In the
present case, fluctuations around the average density are
small. Hence, 具 H 0 (n) 典 ⯝H 0 (n e ) and
2
␹ ⫺1
0 共 n e 兲 ⯝⫺ 共 e / ␬ 兲 h 1 共 n e 兲 / 冑n e ,
n e Ⰷ 冑n i /s,
共9兲
where h 1 (n e )⫽(3/4)h 0 (n e )⫹3h 0⬘ n e ⫹h 0⬙ n 2e ⬃1.
Formula 共8兲 implies a sharp upturn of the ITDOS as n e
→0 caused by the exponential growth of the DRs. At the
⫺1
boundary of its validity, n e ⬃ 冑n i /s, Eq. 共8兲 gives ␹ DR
⬃e 2 s/ ␬ , which is large and positive. On the other hand, the
关Eq. 共9兲兴 shows a weak dependence on n e ,
local term ␹ ⫺1
0
remaining small and negative. Combined, they produce a
⫺1
⫹ ␹ ⫺1
at
strongly asymmetric minimum in ␹ ⫺1 (n e )⫽ ␹ DR
0
the density
n m⫽
1
冑n i
4 冑␲ s
ln1/2
冉
4096
␲ h 41
冊
n is 2 .
共10兲
The logarithmic factor in Eq. 共10兲 is rather insensitive to n i s 2
and h 1 . For n i s 2 ⫽h 1 ⫽1, one gets n m ⬇0.38冑n i /s. In comparison 共see Ref. 6 and below兲 the percolation threshold is
n p ⬇0.12冑n i /s,
共11兲
so that n m ⬇3n p . In accord with EPB’s heuristic argument,
at such density a very small area fraction is depleted 关see
inset in Fig. 2共a兲兴, and so the use of the asymptotic formula
共8兲 is justified.
Let us now proceed to the case n e ⬃n p . Again, I start with
the electrostatic problem (H 0 ⬅0). One expects DRs to be
abundant and irregularly shaped. For a given n e , the ground-
121409-2
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PHYSICAL REVIEW B 69, 121409共R兲 共2004兲
NONLINEAR SCREENING AND PERCOLATIVE . . .
The latter ensures that the average density is equal to n e .
Introducing the Lagrange multiplier ␮ v 共variational estimate
of the electrochemical potential兲, one obtains that H v is minimized if, for all ␴ ⬎ f , ␳ ( ␴ ⬘ ) satisfies
冕
FIG. 2. 共a兲 Electrochemical potential vs electron density for the
electrostatic problem, H 0 ⬅0, according to the present theory 共solid
line兲 and EPB’s numerical simulations 共dots兲. Inset: Fraction of the
depleted area vs density. 共b兲 ␹ ⫺1 vs density according to the numerical simulations of Shi and Xie 共Ref. 8兲 共squares兲, the present
theory 共solid line兲, and for the uniform electron liquid 共dashed line兲.
state n(r) is some nonlocal functional of ␴ and there is no
hope to find it exactly. What I wish to report here is that a
variational solution sought within the class of purely local
functionals, n(r)⫽n 关 ␴ (r) 兴 remarkably accurately reproduces the ␮ (n e ) and ␹ ⫺1 (n e ) dependencies found in numerical work.6,8 More interestingly, it predicts the correct value
of n p 关Eq. 共11兲兴.
The system of equation that defines such a variational
state n( ␴ ) follows from the fact that for a Gaussian random
function the averages over the total area L 2 of the system and
over the distribution function are the same. This yields 关cf.
Eqs. 共1兲–共5兲兴
1
⫽
2
2
L
Hv
G共 ␴,␴⬘兲⫽
冕
冕冕
d ␴ d ␴ ⬘␳共 ␴ 兲G共 ␴ , ␴ ⬘ 兲␳共 ␴ ⬘ 兲,
共13兲
where ␳ ( ␴ )⫽n( ␴ )⫺n e ⫺ ␴ and V(r ⬘ )⫽e 2 / ␬ r ⬘ . The energy
H v needs to be minimized with respect to all functions n( ␴ )
that obey the constraints n( ␴ )⭓0 and
冕
d ␴ P 1 共 ␴ 兲 n 共 ␴ 兲 ⫽n e .
共14兲
共15兲
Here f is such that n( f )⫽0. 关I found that n( ␴ ) is always a
monotonically increasing function, so that n⬎0 corresponds
to ␴ ⬎ f ]. The kernel G( ␴ , ␴ ⬘ ) 关Eq. 共13兲兴 is logarithmically
divergent, G⬀⫺ln兩␴⫺␴⬘兩 at ␴ → ␴ ⬘ , and decays exponentially at large ␴ and ␴ ⬘ . There is a certain analogy between
Eq. 共15兲 and the integral equations of 1D electrostatics,
which also have logarithmically divergent kernels.15,18 This
analogy entails that n( ␴ )⬃ 冑␴ ⫺ f at ␴ close to f. Note that
␴ (r)⫺ f is proportional to the distance in the real space between the given point r and the boundary of the nearby DR.
Thus, the variational principle renders correctly the squareroot singularity in n(r) at the edge of the DRs 关cf., e.g., Eq.
共7兲兴. I was not able to establish the analytical form of the
solution beyond this property and resorted to finding n( ␴ )
numerically. To do so the integral in Eq. 共15兲 was converted
into a discrete sum over 101 points on the interval 兩 ␴ 兩
⬍1.5冑n i /s and the resultant system of 101 linear equations
was solved on the computer. The solution can be approximated by a simple analytical ansatz
n a 共 ␴ 兲 ⫽ 关共 n e ⫹ ␴ 兲 2 ⫺ 共 n e ⫹ f 兲 2 兴 1/2␪ 共 ␴ ⫺ f 兲 ,
共16兲
where ␪ (z) is the step function. For example, the energy H v
is nearly the same whether it is calculated using n a or using
the actual solution of Eq. 共15兲. So, in principle, Eq. 共16兲
obviates the need to solve Eq. 共15兲. The only equation that
needs to be solved is Eq. 共14兲 for f.
At this point one can compare the predictions of the variational method for ␮ v (n e ) with EPB’s numerical results that
were also obtained for the H 0 ⫽0 problem. As Fig. 2共a兲 illustrates, they are in a good agreement.19
To test the theory further I compare it next with the numerical data of Shi and Xie.8 To this end, one needs to calculate ␹ ⫺1 including the effect of a finite H 0 . The first step is
⫺1
⫽d ␮ v /dn e , which is easily done
to take the derivative ␹ DR
numerically. An accurate fit to the result is provided by the
interpolation formula
⫺1
␹ DR
⬇
共12兲
d 2 r ⬘ V 共 r ⬘ 兲关 P 2 共 ␴ , ␴ ⬘ 兲 ⫺ P 1 共 ␴ 兲 P 1 共 ␴ ⬘ 兲兴 ,
d ␴ ⬘G 共 ␴ , ␴ ⬘ 兲 ␳ 共 ␴ ⬘ 兲 ⫽ ␮ v共 n e 兲 P 1共 ␴ 兲 .
0.30⫹ ␩
e 2 s 3 冑2
exp共 ⫺4 ␲ ␩ 2 兲 ,
␬ 8 ␲ ␩ 0.036⫹0.12␩ ⫹ ␩ 2
共17兲
where ␩ ⬅n e s/ 冑n i . This particular form is devised to match
Eq. 共8兲 at large n e and is consistent up to logarithmic corrections with the behavior expected7,20 at very small n e . The
next step is to evaluate the quantity
␹ ⫺1
0 ⫽
d2
d2
dn e
dn 2e
具 H 0 典 ⫽⫺
2
冕
⬁
f
d ␴ P 1 共 ␴ 兲 H 0 关 n 共 ␴ 兲兴 , 共18兲
which is also easily done on the computer. The total ITDOS,
⫺1
␹ ⫺1 (n e )⫽ ␹ DR
⫹ ␹ ⫺1
0 , calculated from Eqs. 共17兲 and 共18兲,
121409-3
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PHYSICAL REVIEW B 69, 121409共R兲 共2004兲
MICHAEL M. FOGLER
and the theoretical value of n m ⬃0.7⫻10⫺3 a B⫺2 关per Eq.
共10兲兴 compare very well with the simulations 关see Fig. 2共b兲兴.
The parameters I used are n i ⫽6.25⫻10⫺4 a B⫺2 and s
⫽10a B , where a B is the effective Bohr radius.21 In agree⫺1
ment with the exact results presented above, ␹ ⫺1
0 Ⰶ ␹ DR at
⫺1
n e ⬍n m , and so the upturn of ␹
at low n e is driven the
growth of DRs.
Within the variational method the boundaries of the DRs
coincide with the level lines ␴ (r)⫽ f of the zero-mean random function ␴ . Consequently, the DRs percolate at f ⭓0.
Solving Eq. 共15兲 with f ⫽0, one arrives at Eq. 共11兲, which is
in excellent agreement with EPB’s result n p ⬇0.11冑n i /s.
One important quantity not reported in the published numerical works6,8 is the area fraction of the DR. Within the variational method, it is equal to erfc(⫺ f / 冑2K 0 )/2, where
erfc(z) is the complementary error function. It is exactly 1/2
at n e ⫽n p and increases as n e →0 as shown in Fig. 2共a兲 共inset兲.
Let us now compare our results with the experimental
data. Taking a rough number n i ⫽3⫻1011 cm⫺2 and a typical spacer width s⫽40 nm in Eq. 共10兲, one gets n m ⬇5.2
⫻1010 cm⫺2 , in agreement with observed values.10 The estimate for the percolation point is n p ⬇n m /3⬇1.7
⫻1010 cm⫺2 . Despite some small uncertainty in the last
number 共due to the uncertainties in n i and h 1 ), n p and n m
differ substantially, and so they can be easily distinguished
1
For review, see E. Abrahams, S.V. Kravchenko, and M.P. Sarachik, Rev. Mod. Phys. 73, 251 共2001兲.
2
G. Eytan, Y. Yayon, M. Rappaport, H. Shtrikman, and I. BarJoseph, Phys. Rev. Lett. 81, 1666 共1998兲.
3
G. Finkelstein, P.I. Glicofridis, R.C. Ashoori, and M. Shayegan,
Science 289, 90 共2000兲.
4
S. Ilani, A. Yacoby, D. Mahalu, and H. Shtrikman, Phys. Rev.
Lett. 84, 3133 共2000兲; Science 292, 1354 共2001兲.
5
M. Morgenstern, Chr. Wittneven, R. Dombrowski, and R. Wiesendanger, Phys. Rev. Lett. 84, 5588 共2000兲.
6
A.L. Efros, F.G. Pikus, and V.G. Burnett, Phys. Rev. B 47, 2233
共1993兲, and references therein.
7
A.L. Efros and B.I. Shklovskii, Electronic Properties of Doped
Semiconductors 共Springer, New York, 1984兲; B.I. Shklovskii
and A.L. Efros, Pis’ma Zh. Éksp. Teor. Fiz. 44, 520 共1986兲
关JETP Lett. 44, 669 共1986兲兴.
8
J. Shi and X.C. Xie, Phys. Rev. Lett. 88, 086401 共2002兲.
9
J.P. Eisenstein, L.N. Pfeiffer, and K.W. West, Phys. Rev. B 50,
1760 共1994兲.
10
S.C. Dultz and H.W. Jiang, Phys. Rev. Lett. 84, 4689 共2000兲.
11
S.C. Dultz, B. Alavi, and H.W. Jiang, cond-mat/0210584 共unpublished兲.
12
G.D. Allison, A.K. Savchenko, and E.H. Linfield, in Proceedings
of the 26th International Conference on Physics of Semiconductors, Edinburgh, 2002, edited by A.C. Long and J.H. Davies
共Institute of Physics, to be published兲.
experimentally. Therefore, the observed apparent MIT10 at
n⫽n m has nothing to do with the percolation of the DRs and
moreover with breaking of the electron liquid into droplets
共the latter occurs at n e Ⰶn p ). From Fig. 2共a兲, one can estimate that the DRs occupy a mere 6% of the total area at n
⫽n m .
One qualitative prediction that follows from Eqs. 共9兲 and
共17兲 is that the upturn of ␹ ⫺1 (n e ) should be sharper in
samples with larger s, which seems to be the case if the data
of Ref. 9 (s⫽14 nm) are compared with those of Refs.
10–12 (s⭐4 nm). Detailed fits are left for future.
I conclude with mentioning some other theoretical work
on the subject. It was suggested22 that near the MIT, function
␹ ⫺1 (n e ) may contain both regular and singular parts. My
theory can be considered the calculation of the former. Its
detailed comparison with experiment may furnish an estimate of the putative singular term. The effect of disorder on
␹ ⫺1 was also studied in Refs. 23 and 24 but the electron
density inhomogeneity was not accounted for. Finally, a nonmonotonic ␹ ⫺1 (n e ) dependence was found in a model25
without disorder.
This work is supported by the C. & W. Hellman Fund and
A. P. Sloan Foundation. I am grateful to B. I. Shklovskii for
indispensable comments and insights, and also to H. W.
Jiang, A. K. Savchenko, and J. Shi for discussions.
B. Tanatar and D.M. Ceperley, Phys. Rev. B 39, 5005 共1989兲.
M.M. Fogler, cond-mat/0402458 共unpublished兲.
15
L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous
Media 共Pergamon, New York, 1960兲.
16
Note that E 0 ⫽4 ␲␴ (0)/ ␬ ⫹const and that p can be read off the
asymptotic law 2 ␲␴ (r)⬃⫺p/r 3 valid at rⰇa. This leads to the
same formula for ␹ ⫺1 as in the case of a hole in a metallic sheet,
see more details in Ref. 14.
17
V.G. Burnett, A.L. Efros, and F.G. Pikus, Phys. Rev. B 48, 14 365
共1993兲.
18
M.M. Fogler, E.I. Levin, and B.I. Shklovskii, Phys. Rev. B 49,
13 767 共1994兲.
19
This is also true in serveral other geometries; see Ref. 14.
20
V.A. Gergel’ and R.A. Suris, Zh. Éksp. Teor. Fiz. 75, 1991 共1978兲
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21
The data in Ref. 8 must be divided by two to correct a notational
error 关J. Shi 共private communication兲兴; n i is four times smaller
than in Ref. 8 to account for a peculiar choice q⫽e/2 of the
impurity charge made there.
22
S. Chakravarty, S. Kivelson, C. Nayak, and K. Voelker, Philos.
Mag. B 79, 859 共1999兲.
23
Q. Si and C.M. Varma, Phys. Rev. Lett. 81, 4951 共1998兲.
24
R. Asgari and B. Tanatar, Phys. Rev. B 65, 085311 共2002兲.
25
S. Orozco, R.M. Méndez-Moreno, and M. Moreno, Phys. Rev. B
67, 195109 共2003兲.
13
14
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