Jpn. J. Appl. Phys. Vol. 42 (2003) pp. 4890–4893
Part 1, No. 7B, July 2003
#2003 The Japan Society of Applied Physics

Thermodynamic Analysis of the Interaction between Kr Adsorbateson Si(111)-7  7 Surface
Osamu T AKEUCHI and Hidemi SHIGEKAWA
Institute of Applied Physics and 21st COE, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan(Received March 20, 2003; accepted for publication April 4, 2003)

Intra-half-unit attractive interactions between Kr adsorbates on Si(111) substrate was investigated on the basis of scanningtunneling microscopy observation at 8 K. A simple thermodynamic model was proposed to extract the interaction energies
from the adsorbate distribution. Obtained energies were compared to what is expected from the Lennard-Jones potential andthe existence of a considerable degree of additional interaction besides the Van der Waals interaction was suggested.
[DOI: 10.1143/JJAP.42.4890]
KEYWORDS: scanning tunneling microscopy, adsorption, rare gas, Van der Waals interaction

Rare gas adsorbates on substrates have been accepted as amodel case of two-dimensional ideal gas systems.1,2) It was
expected that a simple Van der Waals (VdW) interaction
between rare gas atoms governs such systems and that acomparison of experimental results with thermodynamic
theory would fortify the fundamental understanding ofvarious important phenomena in complicated realistic
systems, such as clustering of adsorbates, nucleation ofsmall islands, and phase transition between diﬀerent surface
structures. In reality, however, recent studies utilizing
scanning tunneling microscopy (STM) have revealed thatweak VdW interaction between rare gas adsorbates can beeasily aﬀected by other factors such as dipole-dipole
interaction due to a small but ﬁnite amount of chargetransfer between adsorbates and substrate, and a small
deviation of substrate properties due to local structures suchas steps, defects, and previously adsorbed rare gas atoms on
the surface.3–12) In particular, rare gas adsorptions beneathstep edges showed large deviations from those of theterraces on several metallic substrates.3–8,11) Importantly, theeﬀect of the local structure can aﬀect the global manner ofadsorption processes. For instance, in a Kr/Pt(111) system,
it was suggested that islands that grow from step edges havea diﬀerent packing structure from those in the middle ofterraces8) and this was directly conﬁrmed by atomicallyresolved STM observation.11) Here, the ﬁrst step of island
nucleation determines the whole atomic structure of adomain.
When the substrates have a more complicated structure,
like the Si(111)-7  7 surface, the situation becomes further
complicated. Rare gas adsorption onto such surfaces islargely aﬀected by a small diﬀerence in the electric structureof the substrate. So far, adsorption of Xe10) and Kr12) on toSi(111)-7  7 have been investigated at 8 K by STM. These
rare gas atoms formed complex adsorption structures andtheir adsorption processes were diﬀerent in an unfaulted halfunit (UH) and in a faulted half unit (FH) though the atomic
structures of the ﬁrst few layers of the substrate are exactly
the same in both half unit cells. The result suggested theadsorption process was not simply determined by substratecorrugations but reﬂected the small deviation in electronic
states of the substrate due to the underlying stacking fault.
Understanding the origin of these characteristic adsorbateadsorbate/adsorbate-substrate interactions and its relation

http://dora.ims.tsukuba.ac.jp

ship to the global adsorption processes was desired. In thisstudy, we further investigated the attractive interaction
between Kr adsorbates on Si(111) substrate at 8 K. A
theoretical model was proposed to describe the interaction,and the interaction energy was retrieved from the adsorbate
distribution obtained by STM observation. Finally, thestrength of the obtained interaction was compared to the
VdW interaction.
The details of the experimental setup are described
elsewhere.10) The Si strip cut from a commercial wafer
was prebaked at 1000 K for a night, ﬂashed at 1400 K andcooled to 8 K in an ultrahigh-vacuum chamber with a liquid
He cryostat, whose base pressure was 1:0  108 Pa. Kr
dosing was done from a stainless-steel tube at room
temperature whose end was 5 cm from the Si substrate
kept at 8 K on the STM stage. The typical measurement
condition for STM was tunnel current setting It ¼ 0:3 nA
and sample bias voltage Vs ¼ þ2:0 V.
As reported,12) Kr atoms adsorb diﬀerently in a UH and in
an FH of a Si(111)-7  7 unit cell. In a UH, as shown in
Fig. 1, Kr atoms, at ﬁrst, formed an unresolved baseFig. 1. STM topograph of Kr/Si(111) at 8 K (Vs ¼ 2:0 V, It ¼ 0:3 nA,
28  25 nm2 ). The adsorption structure consists of an unresolved basestructure and Kr atoms at the center adatom site in UH, and Kr trimers at
three corners of the FH. Some UHs with the base structure are indicatedby the gray triangles.

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Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7B

structure, which is indicated by a hatched round triangle inthe schematic. Note that gaps between adatoms in some UH
units, for instance, those indicated by the gray triangles inthe ﬁgure, look slightly brighter than those in clean UHunits. This base structure looks more prominent whenobserved with Vs ¼ þ3:0 V.12) On the base structure,
successive adsorption occurs at the center adatom sites.
Corner adatom sites can also be occupied by Kr atoms.
However, these sites are much more unstable so that most Kr
atoms on the base structure stay at the center adatom sites.
On the other hand, in an FH, Kr atoms always adsorb as
trimers without the base structure. Other forms of Kr
adsorbate were seldom observed in our experiment. When an
FH is fully occupied by three Kr trimers, the atomic
structure is identical to that of Xe adsorption in a UH and in
an FH, even though the adsorption process is diﬀerent
between them; Xe atoms prefer the dimer form.10) The
reason why Kr atoms prefer the trimer form in an FH
remains to be studied. Consequently, both a UH unit cellwith the base structure and an FH unit cell have threeadsorption sites, each of which can be occupied by one Kr
atom or one Kr trimer, respectively.
In order to analyze the intra-half-unit interaction between
these Kr adsorbates, all half unit cells were classiﬁed by the
number of adsorbates inside, and the occurrence of each typeis plotted by thick solid lines with square markers in Fig. 2.
Note that we neglected UHs without the base structure. In
the same ﬁgure, the theoretically expected random distribu-

O. TAKEUCHI and H. S HIGEKAWA

tion, that corresponds to the case where no interaction exists
between adsorbates, is plotted by dashed lines with circle
markers. The random distribution is calculated purely
stochastically, i.e.
Nk ¼ 3 Ck k ð1  Þ3k N;

Number of occurrences

ð1Þ

with the numbers of half unit cells that have k adsorbate(s)inside fNk g0k3 , the total number of half unit cells N, andthe coverage per adsorption site .
Compared with random distribution, the experimental
results showed larger probabilities for the cases where moreadsorbates aggregate in the same half unit cell. Thus, theexistence of some attractive interaction between adsorbates
is conﬁrmed. However, the magnitude of the interaction can
not be directly derived from these plots. In the remainder ofthis paper, we evaluate the interaction energies by detailed
theoretical analysis and discuss the validity of analysis
procedures and the origin of the interaction.
First, let us assume there are N unit cells in an adsorbate
system at thermal equilibrium with system temperature T.
Each of the cells has three adsorption sites. The attractive
interaction is characterized by the interaction energies E2and E3 . Namely, when two or three adsorbates exist in thesame unit cell, the free energy of the system is lowered by E2
or E3 per adsorbate, respectively, compared to the cases
where each of these adsorbates exists alone in a diﬀerent
unit cell. The situation is illustrated in Fig. 3. Then, thefollowing relationships exist:
M ¼ 3N ¼

3
X

kNk ;

ð2Þ

k¼0

Experimental
Random

N¼

200

3
X

Nk ;

ð3Þ

k¼0

E ¼ 2E2 N2  3E3 N3 ;

100
0
0

1

2

3

Number of Kr atoms in a UH unit(a)

Number of occurrences4891

with the number of adsorbates M and the total energy of thesystem E. To obtain eq. (4), the constant component of ME1
has been dropped.
A macroscopic state of the system is deﬁned by a series ofnumbers, fNk g ¼ fN0 ; N1 ; N2 ; N3 g. Limited by the conditions
eqs. (2) and (3), however, this macroscopic state can be
identiﬁed when only fN2 ; N3 g are speciﬁed. N0 and N1 can bederived from fN2 ; N3 g as
N0 ¼ N  M þ N2 þ 2N3 ;Experimental
Random

1600

ð4Þ

ð5Þ

N1 ¼ M  2N2  3N3 :1500

Next, we calculate the number of microscopic states thatgives the same macroscopic state fN2 ; N3 g, i.e. the partition

300
200
100
0
0

1

2

3

Number of Kr trimers in an FH unit(b)
Fig. 2. Distribution of Kr adsorbates. Kr adsorbates tend to be gathered
together in the experimental results compared to the theoretically
predicted random distribution.

Schematic
Number ofadsorbates

0

1

2

3

Energy0E12(E1-E2)

3(E1-E3)

Fig. 3. Deﬁnition of interaction energies. Attractive interactions between
Kr adsorbates decrease the free energy of the system by E2 and E3 peradsorbate when two/three adsorbates exist in the same half unit cell.

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Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7B

O. TAKEUCHI and H. S HIGEKAWA

function WN2 ;N3 . For our purpose, we count the ways of
selecting M occupied sites from 3N adsorption sites. The
number of methods of splitting N unit cells into
fN0 ; N1 ; N2 ; N3 g is given by N!=ðN0 !N1 !N2 !N3 !Þ. In addition,
three ways exist to determine occupied sites in a unit cellthat contains one or two adsorbates. As a result, the partition
function is given as
3N1 þN2 N!
:
N0 !N1 !N2 !N3 !

ð6Þ

Assuming a Boltzmann distribution, the probability ofﬁnding the system in a macroscopic state fN2 ; N3 g can becalculated as
WN2 ;N3 expðEN2 ;N3 =kB TÞ
PðN2 ; N3 Þ ¼ P
ð7ÞN ;N WN2 ;N3 expðEN2 ;N3 =kB TÞ
0

0

2

3

0

0

0

0

with the total energy of the system EN2 ;N3 corresponding to
the state. The statistical mechanics predicts that the mostprobable macroscopic state is always realized, where thefollowing conditions hold.
@PðN2 ; N3 Þ=@N2 ¼ 0@PðN2 ; N3 Þ=@N3 ¼ 0:

3.0

E3

2.5
2.0
E2
1.5
0

ð8Þ

2000
4000
6000
8000
Number of trials per adsorbate

10000

(b)

In a speciﬁc system, substitution of eq. (7) into this condition
results in the following formula.
1
N12
E2 ¼  kB T ln
2
3N0 N21
N13
E3 ¼  kB T ln:
3
27N02 N3Interaction energy / kBT

WN2 ;N3 ¼

Fig. 4. Evaluation of statistical error by a Monte Carlo simulation. (a)Fluctuation of adsorbate distribution for the system of Kr trimers in an
FH. (b) Fluctuation of apparent interaction energies calculated using eq.
(9). See text for details.

ð9Þ

Finally, these equations directly give the magnitude of
intraunit interaction energies E2 and E3 from the experimentally obtained distribution of adsorbates fNk g. In thecase of Kr atoms at center adatom sites in a UH, E2 ¼
0:179kB T and E3 ¼ 0:566kB T are obtained fromfNk g ¼ f224; 220; 103; 43g, while in the case of Kr trimers
at the corners of an FH, E2 ¼ 1:79kB T and E3 ¼ 2:81kB T are
obtained from fNk g ¼ f1590; 68; 35; 21g.
Before proceeding to the detailed discussion, let us
consider the statistical error of the analysis. For this purpose,
Monte Carlo simulations were performed. The model
systems consisted of approximately 104 cells in a triangularlattice with a periodic boundary condition. An appropriate
amount of adsorbates were, at ﬁrst, scattered randomly inthis lattice. Note that random distribution corresponds to theinﬁnite temperature of the system. Then, the Monte Carlotrial described below was run repeatedly: 1) picking up anadsorbate randomly, 2) selecting one of the six neighboring
cells, 3) selecting one of the three adsorption sites in the cell,4) quitting the trial if the cell is already occupied, 5)
calculating the energy cost E necessary to move theadsorbate to the selected site and 6) accepting the movementwith a probability of expðE=kB TÞ. The calculation of E
is done by using eq. (4) with the substitution of experimentally obtained E2 and E3 . During repetition of the trials,
a selected area with the same size as the experiment isintermittently examined to obtain snap shots of the adsorbate
distribution.
A typical result of the simulations for Kr trimers in an FH
is shown in Fig. 4. The lateral axes indicate the number ofMonte Carlo trials divided by the total number of adsorbatesin the system, which gives an arbitrary time scale. The
vertical axis of Fig. 4(a) indicates the number of unit cells
that contain a speciﬁed number of adsorbates and the values
obtained by our experiment are indicated by black horizontal
lines. As soon as the trials started, the system temperature
rapidly cooled down from inﬁnity and reached the equilibrium in 100 trials per adsorbate. The ﬂuctuation after
reaching equilibrium gives the statistical error. Averaging
104 trials per adsorbate, we obtained the root mean squareamplitude of the ﬂuctuation fNk g  f11:5; 11:9; 9:2; 6:1g
for a UH and fNk g  f8:44; 8:0; 5:1; 3:9g for an FH. Figure
4b indicates the corresponding change in E2 and E3
calculated from instantaneous fNk g using eq. (9). Theinteraction energy including the statistical error level (2)is summarized in Table I.
Here we discuss the results in detail. First, so far theinteraction energies have been written as multiples ofthermal energy kB T. Then, what are the absolute values? If
we simply substitute the measured system temperature
T ¼ 8 K, E2 ¼ 0:123 meV and E3 ¼ 0:391 meV for Kr

Table I. Intra-half-unit interaction energies between Kr adsorbates calculated from experimentally obtained adsorbate distribution.
E2 =kB T

E3 =kB T

Kr atoms in FH

0:18  0:18

0:57  0:19

Kr trimers in UH

1:79  0:30

2:81  0:29

Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 7B

O. TAKEUCHI and H. S HIGEKAWA

atoms in a UH, and E2 ¼ 1:24 meV and E3 ¼ 1:93 meV for
Kr trimers in an FH are obtained. The stability of rare gas
structures on Si(111) is, however, known to depend on thetunnel current setting of STM observations.10,12) The higher
current setting results in less stability. Rare gas adsorbates
are not only excited by thermal energy but also by the tunnel
current and/or electric ﬁeld under STM observation. The
inﬂuence of STM observations on rare gas diﬀusion on
metal substrates has been studied.13–15) For the simplest
interpretation, however, the observation eﬀect increases theeﬀective system temperature. Thus, we should take thevalues obtained above as the lower limit of interactionenergies. In order to improve measurement accuracy,
variable temperature experiments and/or variable tunnel
current experiments are required. On the other hand, therelative magnitudes of interaction energies are expected tobe more reliable. If the interaction between Kr adsorbates ispurely additive, the relationship 2E2 ¼ E3 is expected
because one adsorbate interacts with one/two neighboring
adsorbate(s) when two/three adsorbates are in one half unit
cell. In the case of Kr atoms in a UH, 2E2 is slightly smaller
than E3 , and vice versa in the case of Kr atoms in an FH. The
diﬀerence is larger in the latter case but is still comparable tothe error level. We do not insist, at this moment, on whether
these diﬀerences are considerable.
Even with the uncertainty due to the disturbance by STM
observation, comparison of the obtained interaction energies
with the well known Lennard-Jones (L-J) potential enforces
the argument. Figure 5 illustrates the L-J potential for Kr
atoms, where  ¼ 164 K and  ¼ 0:338 nm were employed.
When two center adatom sites in the same unfaulted half unit
cell are occupied by Kr atoms, the separation between theadsorbates is 0.77 nm and the expected interaction energy
from the L-J potential is E ¼ 0:40 meV as indicated in theﬁgure. This value is comparable to the obtained interactionenergy 2E2 ¼ 0:24 meV for Kr atoms at the center adatomsite in a UH with system temperature T ¼ 8 K. If the Kr
atoms only experience the VdW potential, however, they
will form close packed clusters to maintain the most stabledistance between two atoms, . In reality, the potential
corrugation due to substrate structure prevents clustering so

Potential [meV]

0
-4
-8
-12
-16
0.4

0.6

0.8

1.0

1.2

Distance [nm]Fig. 5. Lennard-Jones potential for Kr atoms. The black ﬁlled circle
indicates the distance between two neighboring center adatom sites in thesame half unit cell.

4893

that the Kr atoms are ﬁxed at speciﬁc adsorption sites. Thus,
this potential corrugation is, at least as large as . What isnotable is that Kr adsorbates easily hop around this large
corrugation under the STM observation. Consequently, theeﬀective system temperature under STM observation should
be considerably higher than the measured temperature.
The higher eﬀective temperature gives higher interactionenergies. If this is the case, there is a considerable amount ofadditional interaction besides the VdW interaction between
Kr adsorbates on Si(111). We suggest that some indirectinteraction via the substrate exists even for a rare gas
adsorption system. Namely, the previously adsorbed Kr
atom or trimer aﬀects the substrate’s properties in thesurrounding region and increases the adsorption energies
slightly. Then, an apparent attractive interaction arises
because successively arriving adsorbates eﬀectively probe
this small change in the substrate’s properties. According toour research for the inter-half-unit interaction between Kr
adsorption structures,12) this indirect interaction reaches
across half unit boundaries and also increases the probabilityof Kr adsorption onto the neighboring half unit cells.
In summary, the attractive intra-half-unit interaction
between Kr adsorbates on a Si(111)-7  7 substrate wasinvestigated using scanning tunneling microscopy at 8 K. A
thermodynamic model was proposed for calculating theinteraction energies from the distribution of adsorbates andthe obtained energy values were compared with that
expected from the Van der Waals interaction. The indirectinteraction via the substrate was suggested to explain the
additional component of attractive interaction.
This work was supported by a Grant-in-Aid for ScientiﬁcResearch from the Ministry of Education, Culture, Sports,
Science and Technology.
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