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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.

DOI: 10.1103/PhysRevLett.124.016102

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 [13] 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 [20]. 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

analytical model.

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)

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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) [32]. 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 [34], which used the functional of

Perdew, Burke, and Ernzerhof (PBE) [36], was that five

electrons are shared among three dangling bonds with an

electron configuration (2,2,1) having ×3periodicity and

one unpaired spin [28]. More recent work [37], based on

the revised PBEsol functional [38], 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

[39] at a substrate temperature of 650 °C, followed by a

postannealing step at 850 °C for several seconds [25] 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¯

10

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)][28]. In the ½¯

3¯

310direction the

reciprocal step-to-step distance is a

⊥¼2π=ð14.8ÅÞ[21]

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¯

10direction,

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 [20],

while the intensity of the ×2streaks is nearly unaffected.

(a) (b)

(c)

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 [28]). (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)

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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¯

3¯

310(LP⊥, across the steps)

and ½1¯

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 [20].

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

VASP

[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 [43] for

additional details.

(a)

(b)

(c)

FIG. 3. FWHM of the ×3diffraction spots (red data points) as a

function of temperature in (a) the ½¯

3¯

310direction and (b) the

½1¯

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¯

3¯

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)

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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:

H¼X

i½−bδui;uiþ1

−aδui;c;ð1Þ

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 [43]

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 [43]. 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[44]. To characterize the

transition between these two limits, we fit the spot profiles

to a generalized Lorentzian,

LðkkÞ¼ ΓðνÞ

ﬃﬃﬃ

π

pΓðν−1=2Þ

κ2ν−1

k

½ðkk−k0Þ2þκ2

kν;ð2Þ

where kkis the reciprocal space coordinate in the ½1¯

10

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 [43] 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¯

10direction. The

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)

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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-

sional systems.

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.

*Corresponding author.

bernd.hafke@uni-due.de

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