Metallic transport in a monatomic layer of in on a silicon surface.

Abstract

We have succeeded in detecting metallic transport in a monatomic layer of In on an Si(111) surface, Si(111)-sqrt[7]×sqrt[3]-In surface reconstruction, using the micro-four-point probe method. The In layer exhibited conductivity higher than the minimum metallic conductivity (the Ioffe-Regel criterion) and kept the metallic temperature dependence of resistivity down to 10 K. This is the first example of a monatomic layer, with the exception of graphene, showing metallic transport without carrier localization at cryogenic temperatures. By introducing defects on this surface, a metal-insulator transition occurred due to Anderson localization, showing hopping conduction.

Full text

Page 1

Metallic Transport in a Monatomic Layer of In on a Silicon Surface
Shiro Yamazaki,1,*Yoshikazu Hosomura,1Iwao Matsuda,2Rei Hobara,1Toyoaki Eguchi,2
Yukio Hasegawa,2and Shuji Hasegawa1
1Department of Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2The Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, 277-8581, Japan
(Received 3 February 2009; revised manuscript received 13 November 2010; published 14 March 2011)
We have succeeded in detecting metallic transport in a monatomic layer of In on an Si(111) surface,
Sið111Þ ?
exhibited conductivity higher than the minimum metallic conductivity (the Ioffe-Regel criterion) and kept
the metallic temperature dependence of resistivity down to 10 K. This is the first example of a monatomic
layer, with the exception of graphene, showing metallic transport without carrier localization at cryogenic
temperatures. By introducing defects on this surface, a metal-insulator transition occurred due to
Anderson localization, showing hopping conduction.
ffiffiffi
7
p
?
ffiffiffi
3
p
-In surface reconstruction, using the micro-four-point probe method. The In layer
DOI: 10.1103/PhysRevLett.106.116802 PACS numbers: 73.20.?r, 73.25.+i, 73.63.?b
Metallic transport, which means conductivity higher
than the minimum metallic conductivity and a decrease
in resistivity by cooling, is an old but still new issue in
condensed matter physics, especially at nanometer scales.
Since electrical transport in nanoscale low-dimensional
systems is affected by atomic disorder and defects much
more than in three-dimensional bulk materials, it has been
thought thatitisdifficultforsuch low-dimensionalsystems
to exhibit metallic transport at low temperatures due to
defect-induced Anderson localization [1,2]. However, in
recent years, atomic-scale low-dimensional systems show-
ing metallic transport even at low temperatures have been
found, such as high-quality polyaniline one-dimensional
(1D) molecular chains [3] and graphene, a monatomic
layer of carbon [4,5].
Here, we report a transport study on an indium mona-
tomic layer deposited on a Si(111) surface, Sið111Þ ?
ffiffiffi
four-point probe (?4PP) method [6,7]. It was found for the
first time that with the exception of graphene, the mona-
tomic layer showed a higher conductivity than the mini-
mum metallic conductivity as well as metallic temperature
dependence of resistivity. Furthermore, by intentionally
introducing defects on the surface, the resistivity increased
dramatically and a metal-insulator transition occurred due
to Anderson localization.
By depositing In on the Si(111) surface, various kinds of
surface reconstructions are formed [8]. The
[Figs. 1(a) and 1(b)] and the
[Figs. 1(c) and 1(d)] are insulating with a band gap of
?1 eV at the Fermi level [9]. The 4 ? 1 surface
[Figs. 1(e) and 1(f)] exhibits a quasi-1D metallic band
structure [10]. Because of its Peierls instability at low
temperatures,thissurfaceexhibitsametal-to-insulatortran-
sition upon cooling [11]. The
and 1(h)] exhibits a circular Fermi surface and a parabolic
band dispersion as revealed by photoemission spectroscopy
7
p
?
ffiffiffi
3
p
-In surface reconstruction, by using the micro-
ffiffiffi
3
p
?
ffiffiffi
3
p
ffiffiffiffiffiffi
31
p
?
ffiffiffiffiffiffi
31
p
surfaces
ffiffiffi
7
p
?
ffiffiffi
3
p
surface [Figs. 1(g)
(PES), suggesting a free-electron-like two-dimensional
(2D) metallic system [12]. The evaluated effective mass
m?and Fermi wave number kFare 1:1me(me; electron’s
rest mass) and 14 nm?1, respectively. kFis quite large
compared to the values for typical metallic surfaces such
as Sið111Þ ?
[13]. This fact is crucial to yield low surface resistivity and
metallic transport. The structure of the
ffiffiffi
3
p
?
ffiffiffi
3
p
-Ag (0:8 nm?1) and -Au (1:5 nm?1)
p
ffiffiffi
7
?
ffiffiffi
(b)
3
p
surface is
7 3
31 31
3 3
T(K)
ρ2D(kΩ)
(g)
(i) In/Si(111) surfaces (a)
(c)
(e)
(d)
(f)
(h)
1.2
1.0
0.8
0.6
100 500
41
1
10
101
102
103
300 250200 150100 500
h/2e
2
7 3
FIG. 1 (color online).
images and RHEED patterns of the
size:5nm), the
4 ? 1(V ¼ ?1:0 V,
?0:0091 V, 5 nm) surfaces, respectively. The unit cell of each
surface is indicated in the image. [The lnð001Þ ? ð1 ? 1Þrectunit
cell is also indicated in (g).] (i) Measured sheet resistivities (?2D)
plotted on a logarithmic scale as a function of temperature for
the In/Si(111) surfaces: the
3
?
(crosses), 4 ? 1 (diamonds), and
horizontal broken line indicates the inverse of the minimum
metallic conductivity. The inset shows the measured sheet re-
sistivity for the
7
?
images and resistivity data were obtained in different UHV
systems, both of which were equipped with RHEED.
(a)(b), (c)(d), (e)(f), and (g)(h) are STM
ffiffiffi
31
?
10nm), andthe
3
p
?
ffiffiffi
3
p
(VðSampleÞ¼ 1:6 V,
10 nm),
ffiffiffi
ffiffiffiffiffiffi
p
ffiffiffiffiffiffi
31
p
(V ¼ 1:9 V, the
7
p
?
ffiffiffi
3
p
(V ¼
ffiffiffi
p
ffiffiffi
3
p
p
(filled circles),
ffiffiffi
ffiffiffiffiffiffi
31
p
?
ffiffiffiffiffiffi
31
p
7
?
ffiffiffi
3
p
(open circles). The
ffiffiffi
p
ffiffiffi
3
p
surface at low temperatures. The STM
PRL 106, 116802 (2011)
PHYSICAL REVIEWLETTERS
week ending
18 MARCH 2011
0031-9007=11=106(11)=116802(4) 116802-1
? 2011 American Physical Society

Page 2

approximately understood to be a monatomic layer of In
(001) having quasi-fourfold symmetry [9]. The unit cell
(0:866 ? 0:833?A2) based on the rectangular lattice is also
shown in Fig. 1(g).
The resistivity was measured in situ using a ?4PP chip
with a probe spacing of 20 ?m [6], at temperatures (T)
ranging from room temperature (RT) to 10 K, in an
ultrahigh-vacuum (UHV) chamber [7]. The applied force
to the ?4PP chip was 10–100 nN [6,7]. The damaged area
at the probe contact was several nm2, which was negligible
comparing with the probe spacing (20 ?m). Scanning
tunneling microscope (STM) images were recorded with
a customized STM based on a commercial STM (JEOL) in
a separate UHV chamber. Both UHV chambers were
equipped with reflection-high-energy electron diffraction
(RHEED) to confirm common preparation of samples.
The Si(111) substrate was an n-type (P-doped) with
1–10 ?cm resistivity at RT. Indium was deposited on a
clean Sið111Þ-ð7 ? 7Þ reconstructed surface kept at
appropriate annealing temperatures. The In coverage and
annealing temperature used for the respective In/Si(111)
surface were (0.33 ML, 500?C) for the
(0.50 ML, 450?C) for the
(1.0 ML, 450?C) for the 4 ? 1 surface. The deposition
rate was calibrated using the phase diagram of the In/Si
(111) system [14]. About 1.7 ML In was first deposited on
the 7 ? 7 surface at RT and flash heated at 500?C to
fabricate the
7
?
In atoms are desorbed from the sample, resulting in the
formation of the
7
?
1.2 ML In coverage. Longer heating for a few seconds
caused the formation of the 4 ? 1 surface by further
desorbing In atoms.
We estimated the conductivity of the surface-space-
charge layer of the respective In/Si(111) surfaces by solv-
ing the Poisson equation with parameters derived from the
reported band bending [15–18]. This conductivity was
found to be negligibly small compared to the measured
conductivity. Furthermore, the band bending indicates that
a pn junction is formed between the surface-space charge
layer (p-type) and the underlying bulk (n-type), which
prevents the measuring current from penetrating into the
substrate bulk. In addition, the temperature dependence of
the measured resistivity of the
pletely different from that of the surface-space-charge
layer and the bulk Si crystal at low temperatures.
Therefore, we can state that the measured conductivity
was dominated by the surface states.
Figure 1(i) shows the sheet resistivities ?2D, which were
obtained from the measured 4PP resistance R by ?2D¼
ð?RÞ=ln2, plotted on a logarithmic scale as a function of
temperature for all surfaces. The values of resistivity and
its temperature dependences significantly depend on the
surface. The resistivities of the semiconducting ones, the
ffiffiffiffiffiffi
ffiffiffi
3
p
ffiffiffiffiffiffi
?
ffiffiffi
3
p
surface,
ffiffiffiffiffiffi
31
p
?
31
p
surface, and
ffiffiffi
p
ffiffiffi
ffiffiffi
3
p
surface [12]. In this process, excess
p
ffiffiffi
3
p
surface which ideally has
ffiffiffi
7
p
?
ffiffiffi
3
p
surface was com-
31
p
?
ffiffiffiffiffiffi
31
p
surface and the
ffiffiffi
3
p
?
ffiffiffi
3
p
surface, are much
larger than the inverse of the minimum metallic conduc-
tivity (quantum resistivity: h=2e2¼ 12:9 k?) and in-
crease withcooling (semiconducting
dependence). The activation energy evaluated from the
temperature dependence of the
about 0.2 eV, which is smaller than the reported band gap
?1 eV [9]. A metal-insulator transition by Peierls insta-
bility has been reported on a 4 ? 1 surface [19]. Our 4 ? 1
surface shows a semiconducting temperature dependence
at the high-temperature phase because the resistivity is
higher than the inverse of the minimum metallic conduc-
tivity probably due to defects. An abrupt change in the
temperature dependence at 150 K is indicative of the
metal-to-insulator transition.
Figure 1(i) and the inset show sheet resistivities of the
ffiffiffi
temperature, respectively. Both resistivities showed the
same temperature dependence irrespective of the tempera-
ture ramping direction, demonstrating the reliability of our
measurements. The resistivity of the
much lower than the inverse of the minimum metallic
conductivity, and monotonically decreased with cooling
(metallic temperature dependence) from RT to 10 K. In
other words, this surface satisfies the Ioffe-Regel criterion,
which is needed for metallic conduction [20]. The inset
in Fig. 1(i) shows a linear-scale plot below 100 K. The
solid line is the fitting result by the metallic temperature
dependence:
@e2k2
This equation is derived from the 2D Boltzmann equation,
?2D¼ ð2?m?Þ=ðe2k2
of the phonon-induced inelastic relaxation time ?in¼
temperature
ffiffiffiffiffiffi
31
p
?
ffiffiffiffiffiffi
31
p
surface is
7
p
?
ffiffiffi
3
p
surface recorded with decreasing and increasing
ffiffiffi
7
p
?
ffiffiffi
3
p
surface is
?2D¼4?2m??
F
kBT þ ?0:
(1)
F?inÞ. Temperature dependence
@=ð2??kBTÞ is derived using Debye approximation at
temperature higher than Debye temperature [21,22], where
kBis the Boltzmann constant. Here umklapp processes are
neglected. Our fitting result gives an electron-phonon cou-
pling constant of ? ¼ 1:2 and residual sheet resistivity of
?0¼ 820 ?. This value of ? is large, but consistent with
the value estimated by PES, ? ? 1 [12]. In general, the
? value is typically 0.1–0.3 for bulk metals, while it
tends to be large at surfaces, and exceeds unity in some
surfaces [22]. The residual resistivity ratio [RRR ?
?ðRTÞ=?ð4KÞ ? ?ðRTÞ=?0] is found to be 1.3, which is
smaller by 2 orders of magnitude than the RRR of typical
bulk metals. This means that in such atomic-scale low-
dimensional systems, carrier scattering by defects contrib-
utes significantly to the resistivity. The evaluated average
relaxation time (?) (0.92 fs at RT, 1.3 fs at 10 K) and mean
free path (‘ ¼ vF?) (1.4 nm at RT, 1.9 nm at 10 K) do not
change as much upon cooling. This leads to kF‘ ¼ 20–30,
which is much larger than unity, ensuring the Ioffe-Regel
criterion [20]. The mean free path ?2 nm is slightly
shorter than the roughly estimated point defect separation
PRL 106, 116802 (2011)
PHYSICAL REVIEWLETTERS
week ending
18 MARCH 2011
116802-2

Page 3

?10 nm in STM images. The deviation may be due to line
defects such as steps and domain boundaries that scatter
electronsmorestrongly.Recently,the
been reported to be the first one-atomic layer exhibiting
possible superconductivity at T < 3:18 K, by scanning
tunneling spectroscopy measurements [23]. Therefore a
transport measurement at lower temperatures is highly
desired.
Figure 2(a) shows the measured sheet resistivities as a
function of temperaturefor
different densities of defects. Defects were observed by
wide-view STM images [Figs. 2(b), 2(d), and 2(f)]
and in situ RHEED patterns around the 0th Laue zone
[Figs. 2(c), 2(e), and 2(g)]. As shown in Fig. 1(i) and
also in Fig. 2(a), the pristine
metallic transport. The STM image exhibits a small num-
ber of dark spots with a defect density of 0:004 nm?2.
However, the 100L-O2exposed
higher resistivities and a semiconducting temperature de-
pendence (circled crosses), following a thermal activation-
type function with a small activation energy ?E ¼
1:0 meV.The STMimageexhibited additionalbrightspots
indicated by the black arrow in Fig. 2(d). The defect
density was roughly 0:03 nm?2. The RHEED pattern
shows weaker superlattice spots [Fig. 2(e)]. A surface
fabricated by 2 ML In deposition on a Sið111Þ-ð7 ? 7Þ
surface at RT without postannealing exhibits a larger re-
sistivity (triangles) eventhoughthe surface has In coverage
ffiffiffi
7
p
?
ffiffiffi
3
p
surface has
severalsurfaces with
ffiffiffi
ffiffiffi
7
p
?
ffiffiffi
ffiffiffi
3
p
surface shows
7
p
?
3
p
surface shows
nearly 2 times greater than the
image shows a granular In film partially connected to each
other[Fig.2(b)]. Thedefect
0:3–0:5 nm?2. The RHEED pattern shows no spots except
for the specular reflection [Fig. 2(c)]. The temperature
dependence of the resistivity is thermal activation type
with an activation energy of ?E ¼ 32 meV. The conduc-
tion mechanism is thought to be hopping conduction
between granular islands. One of the other samples
of
7
?
ffiffiffi
perature dependence as shown by downward triangles in
Fig. 2(a). The critical resistivity dividing the metallic
T dependence from the insulating one is about 3 k? in
this case. This value is several times smaller than the
inverse of the minimum metallic conductivity. These re-
sults clearly show that the resistivity is significantly
increased by introducing small amounts of defects, and
the temperature dependence also changes drastically.
Therefore, forming a high-quality surface is crucial for
obtaining metallic transport.
The inset in Fig. 2(a) shows the sheet resistivity at RTas
a function of the amount of deposited In. For surfaces
without postannealing (as deposited), the resistivity does
not vary systematically with the amount of deposited In
(triangles). This means that additional In does not contrib-
ute to the electrical conduction due to poor connections
between the In islands. The high-temperature flash heating
(500?C) causes the formation of the
resulting in a drastic decrease in resistivity. The resistivity
of the
7
?
of deposited In from 1.5 to 1.6 ML. However, with more
than 1.6 ML, it becomes constant around 1 k?. This
suggests that more than 1.6 ML, the
which ideally has 1.2 ML In, is fabricated completely by
the heating procedure. However, with less than 1.6 ML, the
formed
7
?
Figure 3(a) shows the log-scale sheet resistivity as a
function of the temperature below 50 K for the defective
ffiffiffi
defective
7
?
shows the STM image with a slightly higher defect density
of 0:009 nm?2than the pristine case where the defect
density is 0:004 nm?2. Unlike the pristine
surface showing the metallic temperature dependence,
the defective
7
?
temperature dependence. Three functions based on differ-
ent theories are used for fitting the data. The solid line in
Fig. 3(b)is the fitting usingthe Anderson weak localization
theory for a 2D metal [1,24], ?2D¼ ?ðe2=2?2hÞlnT þ C.
The evaluated fitting parameters are ? ¼ 0:0471 and
C ¼ ?0:257. The physical meaning of ? is the power
related to the temperature dependence of the phase relaxa-
tion time of the carriers (??/ T??). The range of ? needs
to be between 1 and 2. Since the evaluated value of ? is
ffiffiffi
7
p
?
ffiffiffi
3
p
surface. The STM
densityis roughly
ffiffiffi
p
?
ffiffiffi
3
p
RHEED spots, exhibited a semiconducting tem-
surface, which showed a slightly weak
7
p
ffiffiffi
3
p
ffiffiffi
7
p
?
ffiffiffi
3
p
surface,
ffiffiffi
p
ffiffiffi
3
p
surface decreased with increasing amount
ffiffiffi
7
p
?
ffiffiffi
3
p
surface,
ffiffiffi
p
ffiffiffi
3
p
surface is defective.
7
p
?
ffiffiffi
3
p
surface prepared with 1.5 ML In deposition (the
ffiffiffi
p
ffiffiffi
3
p
surface hereafter). The inset inFig. 3(a)
ffiffiffi
7
p
?
ffiffiffi
3
p
ffiffiffi
p
ffiffiffi
3
p
surface shows a semiconducting
7 3
100
101
102
103
300250 200150
T(K)
100500
ρ2D(kΩ)
(a) (b)
(d)
(f)
(c)
(e)
(g)
2ML as-deposited
100L O2
exposed
20nm
Deposited In : θ (ML)
as-deposited
@RT
annealed
100L O2
exposed
100
101
102
2.01.91.8 1.71.6 1.5
h/2e
2
7 3(weaker RHEED spots)
FIG. 2 (color online).
function of temperature for the pristine
(circles), the
7
?
ward triangles), the 100L-O2exposed
crosses), and the 2 ML-In as-deposited surface (triangles). The
inset shows the measured sheet resistivities at RTas a function of
the amount of deposited In for the as-deposited surfaces (tri-
angles) and annealed surfaces (circles and square). (b)(c), (d)(e),
and (f)(g) are wide-view STM images and RHEED patterns
of the 0th Laue rings of the 2 ML as-deposited surface (V ¼
1:6 V), with the 100L-O2exposed
and the pristine
7
?
The common scale is shown in (f). The superlattice spots are
indicated by fractional numbers and lines in (g).
(a) Measured sheet resistivity (?2D) as a
ffiffiffi
7
p
?
ffiffiffi
3
p
surface
ffiffiffi
p
ffiffiffi
3
p
(weaker RHEED spots) surface (down-
ffiffiffi
7
p
?
ffiffiffi
3
p
surface (circled
ffiffiffi
7
p
?
ffiffiffi
3
p
surface (V ¼ 1:9 V)
ffiffiffi
p
ffiffiffi
3
p
surface (V ¼ ?1:19 V), respectively.
PRL 106, 116802 (2011)
PHYSICALREVIEWLETTERS
week ending
18 MARCH 2011
116802-3

Page 4

much smaller than unity, theweak localization picture does
not workinthis case. The solid lineinFig. 3(c) is the fitting
result using the conventional semiconducting temperature
dependence, ?2D¼ ?0expð??E=kBTÞ with fitting pa-
rameters
?0¼ 2:22 ?S=h
However, the best fitted solid line does not reproduce the
experimental results well. This means that the defective
ffiffiffi
semiconducting bands with a finite energy gap at the
Fermi level.
The solid line in Fig. 3(d) is the fitting result using
Mott’s variable range hopping (VRH) theory for 2D metals
that have spatially localized metallic states [25,26],
?2D¼ ?0expð ? ð?E=kBTÞ1=3Þ with ?0¼ 4:75 ?S=h
and ?E ¼ 2:74 meV. This formula fits the experimental
results well. Therefore, although the temperature range is
limited only in10–50K, we adopt the VRH picture here. In
this picture, the defective
7
density of states (DOS) at around the Fermi level, however,
these states are spatially and strongly localized. Therefore,
although the defective
7
?
DOS, it exhibits a semiconducting temperature depen-
dence due to the Anderson strong localization. The local-
ization length ? is estimated from the DOS at the Fermi
and
?E ¼ 0:657 meV.
7
p
?
ffiffiffi
3
p
surface does not exhibit band conduction in
ffiffiffi
p
?
ffiffiffi
3
p
surface has a constant
ffiffiffi
p
ffiffiffi
3
p
surface still has metallic
surface D2D¼ m?=?@2¼ 4:6 eV?1nm?2and the fitting
result ?E ¼ 2:74 meV using ? ¼
? to be 26 nm. The threshold defect density for the
metallic-VRH conduction is roughly estimated to be
0:004–0:009 nm?2.
In summary, we have succeeded in detecting metallic
transportinthe
7
?
10 K. The
7
?
than the inverse of the minimum metallic conductivity and
metallic temperature dependence of the resistivity. This
system is a rare example of monatomic layers exhibiting
metallic transport at cryogenic temperatures. By introduc-
ing defects, a metal-insulator transition occurred due to
Anderson localization.
We thank F. Komori for fruitful discussions. This work
was supported by Grants-In-Aid and the A3 Foresight
Program from the Japan Society for the Promotion of
Science.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
33=?D2D?E
p
, giving
ffiffiffi
p
ffiffiffi
3
p
surface showed a resistivity smaller
-Insurface reconstructiondown to
ffiffiffi
p
ffiffiffi
3
p
*Shiro.Yamazaki@physik.uni-hamburg.de
Present address: Institute of Applied Physics and
Microstructure Advanced Research Center, University of
Hamburg, Jungiusstraße 11, D-20355 Hamburg, Germany.
[1] P.W. Anderson, Phys. Rev. 109, 1492 (1958).
[2] S. Yamazaki et al., Phys. Rev. B 79, 085317 (2009).
[3] K.H. Lee, S. Cho, S.H. Park, A.J. Heeger, C.-W. Lee, and
S.-H. Lee, Nature (London) 441, 65 (2006).
[4] K.S. Novoselov et al., Nature (London) 438, 197 (2005).
[5] Y. Zhang et al., Nature (London) 438, 201 (2005).
[6] http://www.capres.com.
[7] T. Tanikawa et al., e-J. Surf. Sci. Nanotechnol. 1, 50
(2003).
[8] V.G. Lifshits, A.A. Saranin, and A.V. Zotov, Surface
Phases on Silicon (John Wiley & Sons, West Sussex, 1994).
[9] J. Kraft, M.G. Ramsey, and F.P. Netzer, Phys. Rev. B 55,
5384 (1997).
[10] T. Abukawa et al., Surf. Sci. 325, 33 (1995).
[11] H.W. Yeom et al., Phys. Rev. Lett. 82, 4898 (1999).
[12] E. Rotenberg et al., Phys. Rev. Lett. 91, 246404 (2003).
[13] H.M. Zhang, T. Balasubramanian, and R.I.G. Uhrberg,
Phys. Rev. B 66, 165402 (2002).
[14] A.V. Zotov et al., Appl. Surf. Sci. 159–160, 237 (2000).
[15] H. Lu ¨th, Surface and Interfaces of Solid Materials
(Springer-Verlag, Berlin, 1995).
[16] S. Hasegawa et al., Prog. Surf. Sci. 60, 89 (1999).
[17] S. Takeda et al., Surf. Sci. 415, 264 (1998).
[18] T. Tanikawa, Ph.D. thesis, Univ. Tokyo, 2003.
[19] T. Tanikawa et al., Phys. Rev. Lett. 93, 016801 (2004).
[20] A.F. Ioffe and A.R. Regel, Prog. Semicond. 4, 237
(1960).
[21] G.D. Mahan, Many-Particle Physics (Kluwer Academic/
Plenum, New York, 2000).
[22] E.W. Plummer et al., Prog. Surf. Sci. 74, 251 (2003).
[23] T. Zhang et al., Nature Phys. 6, 104 (2010).
[24] P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57,
287 (1985).
[25] N.F. Mott, J. Non-Cryst. Solids 1, 1 (1968).
[26] A. Bostwick et al., Phys. Rev. Lett. 103, 056404 (2009).
T (K) (Log Scale)
(d) Mott VRH
T (K)
) K / 1 ( T / 1)K/ 1 ( T / 1
2.0
1.8
1.6
1.4
1.2
10 20 30 40 50
2.0
1.8
1.6
1.4
1.2
0.08 0.06 0.040.02
2.0
1.8
1.6
1.4
1.2
0.45 0.40
1/3
0.35
1/3
0.30
ρ2D(kΩ) (Log Scale)
σ2D(µS/
σ2D(µS/
) (Log Scale)
σ2D(µS/
) (Log Scale)
(a)
(c) act. Hopping
(b) weak localization
e
D
⋅=
2
π
α=0.0471<<1
C=-0.257
σ0=4.75(µS/
∆E=2.74(meV)
)
σ0=2.22(µS/
∆E=0.657(meV)
)
7 3
20
defective
500
600
700
800
900
103
5040 30 10
CT
+⋅
ln
2
2
2
ασ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ∆
k
−=
T
E
b
D
exp
02
σσ
⎟⎟
⎠
⎟
⎞
⎜⎜
⎝
⎜
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ∆
k
−=
3
1
02
exp
T
E
b
D
σσ
20nm
FIG. 3 (color online).
plotted on a logarithmic scale as a function of temperature
ranging from 50 to 10 K for the defective
(b)(c)(d) Sheet conductivities ?2Dfor the same data as in (a);
(b) plotted as a function of T (on a logarithmic scale), (c) plotted
on a logarithmic scale as a function of the inverse of T,
(d) plotted on a logarithmic scale as a function of 1=T1=3. In
(b), (c), and (d), the solid lines indicate the fitting results by the
functions with parameters displayed in each figure. The inset in
(a) shows the STM image (V ¼ ?1:77 V).
(a) Measured sheet resistivities (?2D)
ffiffiffi
7
p
?
ffiffiffi
3
p
surface.
PRL 106, 116802 (2011)
PHYSICAL REVIEWLETTERS
week ending
18 MARCH 2011
116802-4