Thermally Induced Crossover from 2D to 1D Behavior in an Array of Atomic Wires:
Silicon Dangling-Bond Solitons in Si(553)-Au
B. Hafke ,1,* C. Brand ,1T. Witte,1B. Sothmann ,1M. Horn-von Hoegen ,1and S. C. Erwin 2
1Faculty of Physics and Center for Nanointegration (CENIDE), University of Duisburg-Essen, 47057 Duisburg, Germany
2Center for Computational Materials Science, Naval Research Laboratory, Washington, D.C. 20375, USA
(Received 12 April 2019; published 10 January 2020)
The self-assembly of submonolayer amounts of Au on the densely stepped Si(553) surface creates an
array of closely spaced “atomic wires”separated by 1.5 nm. At low temperature, charge transfer between
the terraces and the row of silicon dangling bonds at the step edges leads to a charge-ordered state within
the row of dangling bonds with ×3periodicity. Interactions between the dangling bonds lead to their
ordering into a fully two-dimensional (2D) array with centered registry between adjacent steps. We show
that as the temperature is raised, soliton defects are created within each step edge. The concentration of
solitons rises with increasing temperature and eventually destroys the 2D order by decoupling the step
edges, reducing the effective dimensionality of the system to 1D. This crossover from higher to lower
dimensionality is unexpected and, indeed, opposite to the behavior in other systems.
Physical phenomena associated with low dimensionality
are suppressed when the temperature is raised. For exam-
ple, the 2D fractional quantum Hall effect [1,2] and the 1D
Tomonaga-Luttinger liquid [3–5] are observed only at low
temperature. In 1D atomic wire systems at low temper-
atures, Peierls distortions or more general symmetry break-
ings can open a gap at the Fermi level and lower the total
energy by forming a charge density wave [6–8] or spin-
density wave [9–12].
Excitations generally wash out the effects of this
anisotropy and hence suppress low-dimensional behavior.
The resulting crossover to higher dimensionality at
increased temperatures is exhibited by many systems.
Recent examples include the atomic wire systems Pt
(110)-Br and Si(557)-Pb. In these systems, structural
changes are accompanied by a delicate interplay between
charge density wave correlations and short-range inter-
actions of the adsorbate atoms  and by correlated spin-
orbit order that triggers a metal-to-insulator transition,
respectively [14–17]. The resulting dimensional crossover
from 1D to 2D is typical for atomic wire systems.
In this Letter, we demonstrate the opposite case: a system
of coupled atomic wires exhibiting 2D order at low
temperatures in which thermal excitations at higher temper-
atures induce a dimensional crossover to 1D behavior. We
identify the mechanism driving this crossover to be the
creation of phase solitons and antisolitons [18,19], which
leads to a reversible order-disorder transition at higher
temperatures . We track the crossover across its
characteristic temperature (approximately 100 K) using a
combination of a quantitative high resolution spot profile
analyzing low-energy electron diffraction (SPA-LEED)
study, density-functional theory (DFT) calculations,
Monte Carlo statistical simulations, and an exactly solvable
We studied the self-organized Si(553)-Au atomic wire
surface consisting of Au double-atom rows on (111)-
oriented Si terraces separated by bilayer steps [Fig. 1].
FIG. 1. Ground-state structure and low-energy excitation of the
Si(553)-Au atomic wire system. The underlying substrate consists
of Si(111) terraces separated by steps. Each terrace contains a
dimerized double row of Au atoms (gold) and a row of Si dangling
bonds (gray spheres) at the step edge. The electron occupancy of
these dangling bonds creates a ground state with tripled perio-
dicity: for every two saturated dangling bonds (SDBs, small
spheres) there is a third, unsaturated dangling bond (UDB, large
spheres). At finite temperatures, some of these UDBs (blue)
become mobile and hop to adjacent sites (red). This excitation
creates a soliton-antisoliton pair that can subsequently dissociate.
PHYSICAL REVIEW LETTERS 124, 016102 (2020)
0031-9007=20=124(1)=016102(5) 016102-1 © 2020 American Physical Society
Charge transfer from the terraces leads to incomplete filling
of the dangling sp3orbitals at the Si step edge [20–22]. The
low-temperature ground state consists of a charge-ordered
state with ×3periodicity along the step edges, which
is observed in scanning tunneling microscopy (STM)
[20,22–29] and LEED experiments [20,28–31]. The ×3
periodicity along the wires represents the simplest way to
distribute the available electrons among the row of dangling
bonds while maximizing the number of fully saturated
dangling bonds (electron lone pairs) . Angle-resolved
photoemission spectroscopy measurements [21,24,30,33]
and DFT calculations [24,33–35] reveal that the dangling-
bond states do not cross the Fermi level. Hence, all the
dangling-bond orbitals have integer electron occupancies of
0, 1, or 2. We will refer to orbitals with occupancy 2 as
saturated dangling bonds (SDBs) and to those less than 2 as
unsaturated dangling bonds (UDBs). Figure 1depicts the
arrangement of UDBs andSDBs schematically. The ordering
of the Si dangling-bond structure is mediated by Coulomb
interaction of the UDBs (large spheres) with approximately
equal spacing within and across the rows. The SDBs merely
provide a compensating background charge to balance the
reduced electron occupancy of the UDBs.
The actual number of electrons in the UDBs has been
previously investigated using DFT. The result is sensitive to
the choice of the exchange-correlation functional. The
original prediction , which used the functional of
Perdew, Burke, and Ernzerhof (PBE) , was that five
electrons are shared among three dangling bonds with an
electron configuration (2,2,1) having ×3periodicity and
one unpaired spin . More recent work , based on
the revised PBEsol functional , predicted that only four
electrons are shared among three dangling bonds, implying
the configuration (2,2,0) with no unpaired spins. At present
it is not possible to distinguish between these scenarios on
the basis of experimental data. Here, we consider both
possibilities and show that they lead to very different
estimates of the order-disorder transition temperature.
The experiment was performed under ultrahigh vacuum
conditions at a base pressure lower than 1×10−10 mbar.
The Si substrate was cut from an n-type Si(553) wafer
(phosphorus doped, 0.01 Ωcm). Prior to Au deposition, the
sample was cleaned in several short flash-anneal cycles by
heating via direct current to 1250 °C. Next, 0.48 ML
[monolayer, referred to the atomic density of a Si(111)
surface, i.e., 1ML ¼7.83 ×1014 atoms per cm2]Auwas
deposited from an electron-beam-heated graphite crucible
 at a substrate temperature of 650 °C, followed by a
postannealing step at 850 °C for several seconds  and
subsequent cooling to 60 K on a liquid helium cryostat. The
temperature was measured by an Ohmic sensor (Pt100)
directly clamped to the back of the sample.
At 60 K, the SPA-LEED pattern [Fig. 2(a)] reveals spots
at ×3positions and streaks at ×2positions in the ½1¯
direction. The latter indicates the formation of Au atomic
wires. The spacing of the ×1spots corresponds to the
reciprocal lattice constant a
k¼2π=ð3.84 ÅÞof the Si
substrate along the steps. The ×3spots arise from ordering
of the UDBs within the rows; hence, the UDBs have an
intrarow separation of 3×ak¼11.5Å. The UDBs in
different rows are in registry: recent investigations by
SPA-LEED and STM reveal a centered pð1×3Þarrange-
ment [Figs. 2(a) and 2(c)]. In the ½¯
reciprocal step-to-step distance is a
and thus the separation between the UDB rows is 14.8 Å.
Hence, at low temperatures the UDBs in Si(553)-Au are
arranged in rows in a fully ordered 2D array with
approximately equal spacing within (11.5 Å) and across
(14.8 Å) the rows. The ×2streaks are attributed to the
dimerized double row of Au atoms on the (111) terrace of
the surface [gold spheres in Fig. 2(c); the unit cell is shown
by the blue-shaded areas in Figs. 2(a) and 2(c)][35,40].We
did not detect any ×6periodicity in the ½1¯
indicating that the Au atoms and Si step-edge atoms are
structurally decoupled [27,28,31,37].
At 180 K, the intensity of the ×3spots fades markedly
[Fig. 2(b)], in agreement with an earlier study ,
while the intensity of the ×2streaks is nearly unaffected.
FIG. 2. SPA-LEED pattern of Si(553)-Au at an electron energy
of 150 eV and temperatures (a) 60 K and (b) 180 K. The ×2
streaks between the rows of sharp integer-order spots arise from
dimerized Au double rows on the (111)-oriented terraces. The
rows of elongated spots at ×3positions indicate the tripled
periodicity and long-range order of the UDBs at the Si step edge.
The intensity of the ×2streak is nearly unaffected at higher
temperature, while the ×3features fade away. In (a) the unit cells
of the Au (blue) and Si (green) sublattices as well as the directions
of the line profiles LPkand LP⊥(Fig. 3) are indicated (for more
details see ). (c) Surface structural model showing Si step-
edge atoms (gray) and Au atoms (gold). The unit cells are
depicted with the same color coding as in (a).
PHYSICAL REVIEW LETTERS 124, 016102 (2020)
To analyze the evolution of the diffraction pattern between
60 to 190 K (heating rate 0.13 K=s), we recorded a series of
line profiles [Figs. 3(a) and 3(b)] through the ×3spots, in
two orthogonal directions: ½¯
310(LP⊥, across the steps)
10(LPk, along the steps). The ×3diffraction spots
[insets of Figs. 3(a) and 3(b)] of each of the line profiles
were best fitted by a series of equidistant Lorentzian
functions. No Gaussian-like central spike is found and
the positions of the ×3diffraction spots do not shift with
temperature. Across the steps, the full width at half
maximum (FWHM) κ⊥steadily increases as the temper-
ature is raised from 60 to 130 K. Eventually, the spots
merge into streaks, consistent with the vanishing of the ×3
periodicity reported earlier .
Along the steps, the FWHM κkis relatively constant up
to about 100 K and then steadily increases as the temper-
ature is raised further. This broadening of the ×3diffraction
spots is due to increasing disorder in the arrangement of
UDBs. This type of disorder originates from a simple
microscopic process in which an electron [or two electrons,
for the (2,2,0) configuration] hops from an SDB onto a
neighboring UDB [middle panel of Fig. 3(c)]. As long as
these electron hops do not bring neighboring UDBs closer
than 2ak, the configuration is metastable.
We used DFT to determine the formation energy E0of
this elementary excitation, which can be viewed as a
soliton-antisoliton bound pair. The calculations were per-
formed in a 1×6cell of Si(553)Au with four silicon double
layers plus the reconstructed surface layer and a vacuum
region of 10 Å. All atomic positions were relaxed except
the hydrogen-passivated bottom double layer. Total ener-
gies and forces were calculated using the generalized-
gradient approximation of Perdew, Burke, and Ernzerhof
(PBE) for the (2,2,1) configuration and the PBEsol revision
of that functional for the (2,2,0) configuration, with
projector-augmented wave potentials as implemented in
[38,41,42]. The plane-wave cutoff was 250 eVand the
sampling of the surface Brillouin zone was 6×6.
For the (2,2,1) ground-state configuration we find
E0¼30 meV, suggesting these defects will be numerous
at temperatures above ∼300 K, which is consistent with our
experimental data. For the (2,2,0) configuration we find
E0¼85 meV, implying a much higher temperature scale
of ∼1000 K.
To investigate the concentration and distribution of
defects as a function of temperature, we used the
Metropolis Monte Carlo method to sample the steady-state
arrangement of UDBs in an infinite array of dangling-bond
wires with the Si(553)-Au geometry. We performed 107
trial hops at each temperature and computed the diffraction
intensity from the positions of the UDBs. The spectra were
convolved with a Gaussian to account for instrumental
broadening in the experimental data. For the (2,2,1)
configuration, the resulting FWHM of the ×3peaks is
constant up to ∼100 K and then increases gradually with
temperature, in agreement with experiment but with smaller
values [blue squares in Fig. 3(b)]. For the (2,2,0) configu-
ration the FWHM is flat up to temperatures about 3× higher
(blue triangles), as expected from the q2scaling of the
Coulomb energy. See Supplemental Material  for
FIG. 3. FWHM of the ×3diffraction spots (red data points) as a
function of temperature in (a) the ½¯
310direction and (b) the
10direction, as indicated in Fig. 2(a). Results from
Monte Carlo simulations based on DFT interactions are shown
in (b) by blue squares for the (2,2,1) configuration and triangles
for the (2,2,0) configuration. The result from the analytical model
in Eq. (1) is shown in black. The increase of the FWHM in
(a) indicates the loss of interwire correlation, while in (b) it
indicates a decreasing correlation length along the steps. At low
temperatures the FWHM is constant in the ½¯
310direction up to
∼90 K. Insets in (a),(b): Line profiles for both directions at
various temperatures (shifted for better visibility). (c) Structural
model of creation and separation of a soliton-antisoliton pair.
Charge is transferred from an SDB to a UDB, generating a hop of
the UDB to a neighboring site and creating a soliton-antisoliton
pair. If this pair separates then a phase-shifted domain with ×3
periodicity is formed.
PHYSICAL REVIEW LETTERS 124, 016102 (2020)
Even though the geometry of our model is 2D, the energy
scale of Coulomb interactions across the wires is only
0.1 meV, 3 orders of magnitude smaller than E0.Hence,
the interactions in the Monte Carlo simulations are essen-
tially 1D. Our DFT calculations, however, reveal a much
stronger interaction across the wires of order 1 meV. These
may arise from the interaction of strain fields from the UDBs
but other sources may contribute as well. Regardless of their
origin, we turn now to investigating their role in the order-
disorder transition. We show next that by including these 2D
interactions, the FWHM at all temperatures is brought into
quantitatively excellent agreement with experiment.
We constructed an exactly solvable Potts model
Hamiltonian describing the dynamics of coupled wires
and the resulting steady-state FWHM of the ×3peaks as a
function of temperature:
where δi;j denotes the Kronecker delta. A single UDB can
take three positions within each unit cell i: left, center, and
right, ui¼fl; c; rg. The first term, with parameter b,
describes the energy needed to displace neighbouring
UDBs relative to each other: specifically, the energy needed
to create a soliton-antisoliton pair within one wire is 2b.
The second term, with parameter a, favors the occupation
of the central position and arises from the coupling of the
wire to neighboring wires. See Supplemental Material 
for additional details.
The model fits best to our experimental data for
a¼2.1meV and b¼21 meV. These fitted values are
also consistent with our DFT results: ashould be equal to
the calculated energy difference per UDB, 2.1 meV,
between (2,2,1) configurations in staggered and centered
alignments, and bcorresponds to E0=2¼15 meV. In the
Supplemental Material, we derive analytical expressions
for the profiles and FWHM of the ×3peaks as a function of
temperature . The resulting FWHM, convolved as
above with a Gaussian, is now in excellent agreement with
our experimental results [black curve in Fig. 2(b)]. This
improved agreement demonstrates that 2D coupling
between neighboring wires indeed plays an important,
central role in the order-disorder transition.
We turn now to the crossover from 2D to 1D behavior. At
temperatures above ∼120 K, the ×3diffraction spots are
well described by a standard Lorentzian. At temperatures
below ∼90 K, the 2D character of the diffraction is more
pronounced and hence the spot profiles are described by a
Lorentzian to the power 3=2. To characterize the
transition between these two limits, we fit the spot profiles
to a generalized Lorentzian,
where kkis the reciprocal space coordinate in the ½1¯
direction, k0is the position of the ×3diffraction spot, and
ΓðxÞis the Gamma function. The parameter ν¼ðdþ1Þ=2
characterizes the dimensionality dof the system: ν¼3=2
describes 2D systems while ν¼1describes 1D systems
[44–46]; we constrained νto lie in this range.
We find that νexhibits a well-defined transition from 1.5
to 1.0 between T−¼93 K and Tþ¼128 K (Fig. 4). The
transition begins at about the temperature for which the
FWHM κkalong the steps begins to increase [Fig. 2(b)].
Fitting the spot profiles without allowing νto vary leads to
significantly worse fits (insets to Fig. 4). The transition is
completed at Tþ, where the FWHM κ⊥across the wires
exceeds the size of the surface Brillouin zone [Fig. 2(a)],
reflecting the complete loss of long-range order across the
wires. The underlying origin of this dimensional crossover
is subtle but simple: the approximate geometrical isotropy
of the 2D array of UDBs is broken by the strong anisotropy
of the energy scales for creating disorder across and within
the UDB wires. At temperatures above T−soliton defects
are still rare, but a rapidly growing fraction of the wire rows
undergoes registry shifts with respect to each other and
hence the 2D low-temperature state begins to behave like a
collection of uncoupled 1D wires. As the temperature
approaches Tþthis crossover becomes nearly complete.
See the Supplemental Material  for additional discus-
sion, modeling, and analysis.
To summarize, we have shown that silicon dangling-bond
solitons in Si(553)-Au are created by thermal excitation.
These defects interact via Coulomb forces within each step-
edge atomic wire and via another mechanism, probably
strain, across the wires. As the temperature is raised, the
resulting disorder destroys the ×3positional long-range
order of the UDBs within each wire and their registry across
the wires. The nature of the interactions and their respective
energy scales create a dimensional crossover of the
FIG. 4. Temperature dependence of the exponent νof the
generalized Lorentzian of Eq. (2) in the ½1¯
exponent drops from 1.5 at T−¼93 K to 1.0 at Tþ¼128 K,
indicating a crossover from 2D to 1D behavior. Insets: Exper-
imental and fitted spot profiles at 60 K and at 128 K. At 60 K the
profile is best fit by ν¼3=2(2D behavior), while at Tþthe best
fit is ν¼1(1D behavior).
PHYSICAL REVIEW LETTERS 124, 016102 (2020)
order-disorder transition from 2D at low temperature to 1D
at high temperature. The generality of this crossover can
readily be investigated—both experimentally and using our
theoretical methods—in other atomic wire systems in the
Ge/Si(hhk)-Au family, where differences in the surface
morphology and ground-state electron configuration may
lead to further expanding our understanding of low-dimen-
We acknowledge fruitful discussions with J. Aulbach,
F. Hucht, J. König and J. Schäfer. This work was funded by
the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation)–Projektnummer 278162697–SFB
1242 and through Projektnummer 194370842–FOR1700.
B. S. acknowledges financial support from the Ministry of
Innovation NRW via the “Programm zur Föderung der
Rückkehr des hochqualifizierten Forschungsnachwuchses
aus dem Ausland.”This work was partly supported by
the Office of Naval Research through the Naval
Research Laboratory’s Basic Research Program (SCE).
Computations were performed at the DoD Major Shared
Resource Center at AFRL.
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PHYSICAL REVIEW LETTERS 124, 016102 (2020)