Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope
The formation of an adatom adsorption structure in dynamic force microscopy experiment is shown as a result of the spontaneous appearance of shear strain caused by external supercritical heating. This transition is described by the Kelvin-Voigt equation for a viscoelastic medium, the relaxation Land...
Gespeichert in:
| Datum: | 2014 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
2014
|
| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/152897 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope / A.V. Khomenko // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33401: 1–10 — Бібліогр.: 50 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-152897 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1528972019-06-14T01:25:28Z Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope Khomenko, A.V. The formation of an adatom adsorption structure in dynamic force microscopy experiment is shown as a result of the spontaneous appearance of shear strain caused by external supercritical heating. This transition is described by the Kelvin-Voigt equation for a viscoelastic medium, the relaxation Landau-Khalatnikov equation for shear stress, and the relaxation equation for temperature. It is shown that these equations formally coincide with the synergetic Lorenz system, where the shear strain acts as the order parameter, the conjugate field is reduced to the stress, and the temperature is the control parameter. Within the adiabatic approximation, the steady-state values of these quantities are found. Taking into account the sample shear modulus vs strain dependence, the formation of the adatom adsorption configuration is described as the first-order transition. The critical temperature of the tip linearly increases with the growth of the effective value of the sample shear modulus and decreases with the growth of its typical value. Формування структури адсорбованих адатомiв при дослiдженнi в режимi динамiчної силової мiкроскопiї представлено як результат спонтанної появи зсувної деформацiї в результатi зовнiшнього надкритичного нагрiвання. Цей перехiд описується рiвнянням Кельвiна-Фойгта для в’язкопружного середовища, релаксацiйним рiвнянням Ландау-Халатнiкова для зсувних напружень та релаксацiйним рiвнянням для температури. Показано, що цi рiвняння формально збiгаються iз синергетичною системою Лоренца, де зсувна деформацiя вiдiграє роль параметра порядку, спряжене поле зводиться до напружень, та температура є керувальним параметром. В рамках адiабатичного наближення знайденi стацiонарнi значення цих величин. Враховуючи залежнiсть модуля зсуву зразка вiд деформацiї, формування конфiгурацiї адсорбованих адатомiв описано як перехiд першого роду. Критична температура зонда лiнiйно зростає з ростом ефективного значення модуля зсуву зразка i зменшується при зростаннi його характерного значення. 2014 Article Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope / A.V. Khomenko // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33401: 1–10 — Бібліогр.: 50 назв. — англ. 1607-324X DOI:10.5488/CMP.17.33401 PACS: 46.55.+d, 64.60.-i, 61.72.Hh, 62.20.F-, 62.20.Qp, 68.37.Ps arXiv:1301.4379 http://dspace.nbuv.gov.ua/handle/123456789/152897 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The formation of an adatom adsorption structure in dynamic force microscopy experiment is shown as a result of the spontaneous appearance of shear strain caused by external supercritical heating. This transition is described by the Kelvin-Voigt equation for a viscoelastic medium, the relaxation Landau-Khalatnikov equation for shear stress, and the relaxation equation for temperature. It is shown that these equations formally coincide with the synergetic Lorenz system, where the shear strain acts as the order parameter, the conjugate field is reduced to the stress, and the temperature is the control parameter. Within the adiabatic approximation, the steady-state values of these quantities are found. Taking into account the sample shear modulus vs strain dependence, the formation of the adatom adsorption configuration is described as the first-order transition. The critical temperature of the tip linearly increases with the growth of the effective value of the sample shear modulus and decreases with the growth of its typical value. |
| format |
Article |
| author |
Khomenko, A.V. |
| spellingShingle |
Khomenko, A.V. Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope Condensed Matter Physics |
| author_facet |
Khomenko, A.V. |
| author_sort |
Khomenko, A.V. |
| title |
Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope |
| title_short |
Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope |
| title_full |
Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope |
| title_fullStr |
Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope |
| title_full_unstemmed |
Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope |
| title_sort |
self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/152897 |
| citation_txt |
Self-organization of adatom adsorption structure at interaction with tip of dynamic force microscope / A.V. Khomenko // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33401: 1–10 — Бібліогр.: 50 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT khomenkoav selforganizationofadatomadsorptionstructureatinteractionwithtipofdynamicforcemicroscope |
| first_indexed |
2025-07-14T04:21:53Z |
| last_indexed |
2025-07-14T04:21:53Z |
| _version_ |
1837594747057733632 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33401: 1–10
DOI: 10.5488/CMP.17.33401
http://www.icmp.lviv.ua/journal
Self-organization of adatom adsorption structure at
interaction with tip of dynamic force microscope
A.V. Khomenko1,2
1 Department of Complex Systems Modelling, Sumy State University,
2 Rimskii-Korsakov St., 40007 Sumy, Ukraine
2 Peter Grünberg Institut-1, Forschungszentrum-Jülich, 52425 Jülich, Germany
Received April 5, 2014, in final form June 04, 2014
The formation of an adatom adsorption structure in dynamic force microscopy experiment is shown as a re-
sult of the spontaneous appearance of shear strain caused by external supercritical heating. This transition is
described by the Kelvin-Voigt equation for a viscoelastic medium, the relaxation Landau-Khalatnikov equation
for shear stress, and the relaxation equation for temperature. It is shown that these equations formally coin-
cide with the synergetic Lorenz system, where the shear strain acts as the order parameter, the conjugate field
is reduced to the stress, and the temperature is the control parameter. Within the adiabatic approximation,
the steady-state values of these quantities are found. Taking into account the sample shear modulus vs strain
dependence, the formation of the adatom adsorption configuration is described as the first-order transition.
The critical temperature of the tip linearly increases with the growth of the effective value of the sample shear
modulus and decreases with the growth of its typical value.
Key words: phase transition, rheology, plasticity, strain, stress, atomic force microscopy
PACS: 46.55.+d, 64.60.-i, 61.72.Hh, 62.20.F-, 62.20.Qp, 68.37.Ps
1. Introduction
Nowadays, due to large scientific and practical importance, the phenomena taking place on the sam-
ple surface at interaction with the tip of a dynamic force microscope, e.g., atomic force microscope (AFM)
and friction force microscope, attract more andmore attention (see the reviews in [1–6] and the literature
cited therein). Particularly, the experimental and theoretical data are obtained on structural instabilities,
phase transformations, plastic dislocation, neck and adatom structures formation [7–20]. These processes
are characterized by hysteresis of dependencies of adhesion force and potential energy surface on the tip-
surface distance [11, 16, 21–24] and by hysteresis of the sample stress vs strain curve [25].
Since the nature of such phenomena remains poorly understood, the basic goal of the present study
is the construction of a qualitative nonlinear model [26–31] describing the hysteresis processes which
occur on the germanium surface during interaction with the AFM tip [11]. Here, the macroscopic con-
tinuum mechanics models [32, 33] are supposed to be still applicable to the atomic length-scales, where
discrete atomistic interactions become significant [3, 16, 18, 34, 35]. However, a total explanation of the
studied macroscopic phenomena requires a consideration of microscopic processes. Phenomenological
description used here makes it possible to connect the parameters of microscopic theories with macro-
scopic measurements. However, this is a separate independent problem that is hard to solve just now.
In the presented approach, the formation conditions of the adatom adsorption structure are defined on
the semiconductor surface due to both thermal and deformation effects. The total set of freedom degrees
is considered as equivalent variables. The adatom configuration formation is described analytically as
a result of self-organization caused by the positive feedback of shear strain and temperature on shear
stress on the one hand, as well as the negative feedback of shear strain and stress on temperature on the
© A.V. Khomenko, 2014 33401-1
http://dx.doi.org/10.5488/CMP.17.33401
http://www.icmp.lviv.ua/journal
A.V. Khomenko
other hand. This study is based on the assumption that stress relaxation time diverges because the shear
modulus vanishes at the point of transition.
The paper is organized as follows. In section 2 the self-consistent Lorenz system of the governed
equations is written for approximation of semiconductor characterized by heat conductivity. The adatom
structure formation is shown in section 3 to be supercritical in character (is of the second order) when
the effective shear modulus of the germanium does not depend on the strain value; it then transforms to
a subcritical mode with this dependence appearance (section 4). In these sections, the steady-state values
of shear strain and stress, as well as temperature are also determined within adiabatic approximation.
Using such a limit, a synergetic potential is obtained, that is the analog of a thermodynamic potential,
from basic evolution equations. Section 5 contains short conclusions.
2. Basic equations
Let us start with the supposition that the relaxation behavior of the shear component ε of the strain
tensor in a semiconductor is governed by the Kelvin-Voigt equation [32, 36]
ε̇=−ε/τε+σ/ηε , (1)
where τε is the Debye relaxation time and ηε is the effective shear viscosity coefficient. The second term
on the right-hand side describes the flow of a viscous liquid caused by the corresponding shear compo-
nent of the stress σ. In the steady state, ε̇= 0, we obtain the Hooke-type expression σ=Gεε, Gε ≡ ηε/τε.
The next assumption of our approach is that the relaxation equation of the sample shear stress σ has
a form similar to the Landau-Khalatnikov equation [28, 30, 37]:
τσσ̇=−σ+G(T )ε. (2)
Here, the first term on the right-hand side describes the relaxation during time τσ ≡ η/G(T ) determined
by the values of the shear viscosity η and modulus G(T ) depending on the sample temperature. In the
stationary case σ̇= 0, the kinetic equation (2) is transformed into the Hooke’s law
σ=G(T )ε. (3)
Note that effective values of viscosity ηε ≡ τεGε and modulus Gε ≡ ηε/τε do not coincide with the real
values η and G(T ). Physically, such difference is conditioned by the Landau-Khalatnikov-type equation (2)
being not equivalent to the Kelvin-Voigt equation (1) [30, 32, 33]. As is known, the values Gε, η, ηε very
weakly depend on the sample temperature T , while the real shear modulus G(T ) vanishes, when the
temperature decreases to Tc [38–42]. Further, the simplest approximate temperature dependencies are
used: Gε(T ), η(T ),ηε(T ) = const,
G(T ) =G0 (T /Tc −1) , (4)
where G0 ≡G(T = 2Tc) is the typical value of modulus.
According to the synergetic concept [26, 29, 30, 40, 41, 43, 44] to complete the equation system (1)
and (2), which contains the order parameter ε, the conjugate field σ, and the control parameter T , we
should deduce a kinetic equation for the temperature. This equation can be obtained using the basic
relationships of elasticity theory stated in § 32 in [33]. Thus, it is necessary to start with the continuity
equation for the heat δQ = TδS:
T Ṡ =−∇q. (5)
Here, the heat current is given by the Onsager equation
q =−κ∇T, (6)
where κ is the heat conductivity. In the elementary case of the thermoelastic stress, the entropy
S = S0(T )+Kαε0 (7)
33401-2
Self-organization of adatom adsorption structure at interaction with the tip of dynamic force microscope
consists of the purely thermodynamic component S0 and the dilatation:
ε̂0
= ε0 Î , ε0
≡α(T −T0) , (8)
where α is the thermal expansion coefficient, T0 is the equilibrium temperature, Î is the unit tensor and
K is the compression modulus (see § 6 in [33]). In the considered situation, we should transfer from the
dilatational component Kαε0 to the elastic energy −σε/T of the shear component divided by tempera-
ture (here, the minus sign takes into account the connection TδS = pδV ⇒−σδε at S0 = const, which is
caused by the opposite choice of the pressure p and the stress σ signs). As a result, equation (5) has the
form
T Ṡ0(T )−σε̇=κ∇2T. (9)
Taking into account the approximation (κ/l 2)(Te−T ) ≈κ∇2T (l is the scale of heat conductivity, Te is the
AFM tip temperature) and the definition of heat capacity cp=T dS0/dT , equation (9) assumes the form:
cpṪ =
κ
l 2
(Te −T )+σε̇. (10)
Substituting the expression for the ε̇ from equation (1) we obtain the term σ2/ηε. It describes the dis-
sipative heating of a viscous liquid flowing under the effect of the stress σ that can be neglected in the
case under consideration. On the other hand, the process of an AFM tip moving into contact with the
surface has the following peculiarity. It is necessary to consider the thermal effect of the tip whose value
Te is not reduced to the Onsager component and is fixed by external conditions. In view of these circum-
stances, the square contribution of the stress is supposed to be included in Te. The obvious account of this
term leads to a significant complication of the subsequent analysis, though it results in a renormalization
of the quantities. Therefore, component Te in equation (10) is assumed to be constant for our further
consideration.
It is convenient to introduce the following measure units:
σs =
(
cpηεTc/τT
)1/2
, εs =σs /Gε , Tc , (11)
for the variables σ, ε, T , respectively (τT ≡ l 2cp/κ is the time of heat conductivity). Then, the basic equa-
tions (1), (2), and (10) take the form:
τεε̇=−ε+σ, (12)
τσσ̇=−σ+ g (T −1)ε, (13)
τTṪ = (Te −T )−σε, (14)
where the constant
g =
G0
Gε
(15)
is introduced. Equations (12)–(14) have a form similar to the Lorenz scheme [26] which allows us to
describe the thermodynamic phase and the kinetic transitions [29, 30, 40, 41, 43–45].
3. Continuous transition
In general the system (12)–(14) cannot be solved analytically. Therefore, we use the following adia-
batic approximation:
τσ ≪ τε , τT ≪ τε . (16)
This implies that in the course of the matter evolution, the stress σ(t) and the temperature T (t) follow the
variation of the strain ε(t). The first of these inequalities is fulfilled because it contains the macroscopic
time τε and the microscopic Debye time τσ ≈ a/c∼10−12 s, where a ∼ 1 nm is the lattice constant or the
33401-3
A.V. Khomenko
intermolecular distance and c ∼ 103 m/s is the sound velocity. The second condition (16) can be reduced
to the form
l ≪ L, (17)
where the maximal value of the characteristic length of the heat conductivity
L =
√
χνε
c2
ε
, (18)
the thermometric conductivity χ ≡ κ/cp, the effective kinematic viscosity νε ≡ ηε/ρ and the sound ve-
locity cε ≡ (Gε/ρ)1/2 are introduced (ρ is the medium density). Then, we can put the left-hand sides of
equations (13) and (14) to be equal to zero. As a result, the stress σ and the temperature T are expressed
in terms of the strain ε:
σ=
gε(Te −1)
1+ gε2
, (19)
T = 1+
Te −1
1+ gε2
. (20)
In accordance with equation (20), in the important range of values of the parameter Te > 1, the temper-
ature T decreases monotonously with an increasing strain ε from the value Te at ε = 0 to (Te +1)/2 at
ε = εm ≡
√
1/g . Obviously, this decrease is caused by the negative feedback of the stress and the strain
on the temperature in equation (14), which is explained by the Le Chatelier principle for this problem.
Really, the reason for the formation of adatom adsorption structure is the positive feedback of the strain
and the temperature on the stress in equation (13). Hence, the increase in the temperature should in-
tensify the self-organization effect. However, according to equation (14), the system behaves in such a
way that the consequence of transition, i.e., the growth of the strain, leads to a decrease in its cause (i.e.,
temperature). Equation (19), expressing the stress in terms of the strain, has a linear form of the Hooke’s
law at ε≪ εm with the effective shear modulus Gef ≡ g (Te −1). At ε= εm , the function σ(ε) has a maxi-
mum and at ε> εm it decreases, which has no physical meaning. Thus, the constant εm ≡
√
1/g gives the
maximal strain. An increase in the typical value of the modulus G0 leads to a decrease in the maximal
strain εm and to an increase in the effective modulus Gef whose value is proportional to the characteristic
temperature Te.
Substituting equation (19) into equation (12), we obtain the Landau-Khalatnikov-type equation [28,
37, 46, 47]
τεε̇=−∂V /∂ε, (21)
where the synergetic potential has the form
V =
1
2
[
ε2
+ (1−Te) ln
(
1+ gε2
)]
. (22)
At a steady state, the condition ε̇ = 0 is fulfilled and the potential (22) acquires a minimum. When the
temperature Te becomes smaller than the critical value
Tc0 = 1+ g−1; g ≡G0/Gε < 1, Gε ≡ ηε/τε , (23)
this minimum corresponds to ε = 0, i.e., the adatom adsorption structure is not realized. In the reverse
case Te > Tc0, the stationary shear strain has the nonzero value
ε0 =
[
Te − (1+ g−1)
]1/2
(24)
which increases with Te growth in accordance with the root law. This causes the formation of the adatom
configuration. Equations (19) and (20) give the stationary values of stress and temperature:
σ0 = ε0 , T0 = 1+ g−1. (25)
Note that, on the one hand, the steady temperature T0 coincides with the critical value (23) and, on the
other hand, its value differs from the temperature Te. Since Tc0 is the minimal temperature at which
33401-4
Self-organization of adatom adsorption structure at interaction with the tip of dynamic force microscope
the formation of the adatom adsorption structures can be observed, the above implies that the negative
feedback of the stress σ and the strain ε on the temperature T [see the last term on the right-hand side
of equation (14)] decreases the sample temperature so much that only in the limit does it ensure the
self-organization process. At a steady state, the value of the shear modulus is
Gs =Gε . (26)
The two cases can be marked out by the parameter g = G0/Gε. In the situation g ≫ 1, meeting the
large value of the modulus G0, equations (23)–(25) take the form
ε0 = (Te −1)1/2, T0 = Tc0 = 1. (27)
This corresponds to the “solid (fragile)” limit. The opposite case g ≪ 1 (small modulus G0) meets the
“strongly viscous liquid”
ε0 =
(
Te − g−1
)1/2
, T0 = Tc0 = g−1
=Gε/G0 . (28)
4. Effect of deformational defect of modulus
The Kelvin-Voigt equation (1) assumes the use of the idealized Genki model. For the dependence σ(ε)
of the stress on the strain, this model is described by the Hooke’s expression σ=Gεε at ε< εm and by the
constant σm = Gεεm at εÊ εm (σm, εm are the maximal stress and strain, σ> σm results in viscous flow
with the deformation rate ε̇ = (σ−σm)/ηε). Actually, the σ vs ε dependence curve has two regions: the
first one, Hookean, has a large slope corresponding to the shear modulus Gε, followed by a more gently
sloping section of the plastic deformation whose tilt is defined by the hardening factor Θ<Gε. Obviously,
such a picture means that the shear modulus, introduced in equation (1), depends on the strain value. Let
us use the simplest approximation [48, 49]
Gε(ε) =Θ+
Gε−Θ
1+
(
ε/εp
)2
, (29)
which describes the above mentioned transition of the elastic deformation mode to the plastic one. It
takes place at a characteristic value of the strain εp, which is smaller than εs (otherwise plastic mode
is not realized). Note that an expression of the type equation (29) was originally proposed by Haken
[26] describing the rigid mode of laser radiation. It is used [29, 43, 44] to describe the first-order phase
transition, and equation (29) contained the square of the ratio ε/εp (so, the V vs ε dependencies in [29, 43,
44] and equation (30) have an even form). In the description of structural phase transitions of a liquid, the
third-order invariants, breaking the specified parity, are present [27]. Therefore, in the study [30, 40, 41],
in the approximation (29), we used the linear term ε/εp, instead of the square term
(
ε/εp
)2
. Obviously, in
this case, the V vs ε dependence is already uneven.
Within the adiabatic approximation (16), the Lorenz equations (12)–(14), where Gε is replaced by a
dependence Gε(ε), is reduced to the Landau-Khalatnikov equation (21). The synergetic potential has the
form:
V =
1
2
ε2
−
gα2(Te −1)
2
{
1
gα2 −θ
ln
∣∣∣∣
1+ gε2
1+θ(ε/α)2
∣∣∣∣
+
1
θgα2(θ−1 − g−1α−2)
[
θ−1 ln
∣∣θ−1
+ (ε/α)2
∣∣− g−1α−2 ln
∣∣g−1α−2
+ (ε/α)2
∣∣]
}
. (30)
Here, the constant α≡εp/εs < 1 and the parameter θ=Θ/Gε < 1, describing the ratio of the tilts for the de-
formation curve on the plastic and the Hookean sections, are introduced. At a small value of temperature
Te , the dependence (30) has a monotonously increasing shape with its minimum at ε= 0 corresponding
to the steady state of the absence of the adatom adsorption structure (curve 1 in figure 1). As shown in
figure 1, at
T 0
c =1+θg−1
+α2(1−2θ)+2α
√
g−1θ(1−θ)(1−gα2), (31)
33401-5
A.V. Khomenko
Figure 1. Dependence of the synergetic potential on the strain at g=0.2, θ=α=0.25 and various tempera-
tures: (curve 1) Te<T 0
c , (curve 2) Te=T 0
c , (curve 3) T 0
c <Te<Tc , and (curve 4) TeÊTc0.
Figure 2. Dependence of the steady-state values of the strain on the temperature Te at parameters of
figure 1 (the solid curve corresponds to the stable value ε0, the dashed curvemeets the unstable one, εm).
a plateau appears (curve 2), which for Te > T 0
c is transformed into a minimum, meeting the strain ε0 , 0,
and a maximum at εm that separates the minima corresponding to the values ε= 0 and ε = ε0 (curve 3)
[11]. With a further increase in the temperature Te, the “ordered” phase minimum, corresponding to the
adatom adsorption configuration ε= ε0, grows deeper, and the height of the interphase barrier decreases
vanishing at the critical value Tc0 = 1+ g−1 (23). The steady-state values of the strain have the form (see
figures 1 and 2)
(
εm
0
)2
=
(
2gθ
)−1
{
g
(
Te −1−α2
)
−θ∓
√[
g
(
Te−1−α2
)
−θ
]2
−4gα2θ
[
1−g (Te−1)
]}
, (32)
where the lower sign meets the stable adatom structure and the upper sign corresponds to the unstable
one. At Te Ê Tc0, the dependence V (ε) is characteristic of the absence of the modulus defect (see curve 4
in figure 1).
It is worth noting that the potential barrier inherent in the synergetic first-order transition manifests
itself only due to the deformational defect of the modulus. Since the latter is realized always [11], it
follows that the studied adatom structure formation is a synergetic first-order transition. The considered
situation differs from typical thermodynamic phase transitions. Really, in the latter case, the stationary
value of the semiconductor temperature T0 is equal to the thermostat value Te. In this study, T0 is reduced
to the critical value Tc0 for a synergetic second-order transition (see section 3). When the modulus defect
33401-6
Self-organization of adatom adsorption structure at interaction with the tip of dynamic force microscope
is taken into account, the temperature
T0 = 1+
Te −1
1+ gε2
0
, (33)
whose value is defined by a minimum position of the dependence (30), is realized. In accordance with
equations (32) and (33), the quantity T0 monotonously decays from the value
Tm = 1+
T 0
c −1
1+ g
(
εc
0
)2
, εc
0 =
(
2gθ
)−1 [
g
(
T 0
c −1−α2
)
−θ
]
(34)
at Te = T 0
c , to 1 at Te →∞. As shown in figure 3, the stationary temperature T0 = Te in the range from 0 to
Tc0. The jump down occurs at Te = Tc0, following which the value T0 smoothly decreases. If the quantity
Te then decays, the steady-state temperature T0 increases. At the point T 0
c [equation (31)], the T0 vs Te
dependence has a jump from Tm [equation (34)] up to T 0
c . For Te < T 0
c , the steady-state temperature T0 is
also equal to Te.
Figure 3. Dependence of the steady-state value of the sample temperature T0 on the temperature Te at
parameters of figure 1.
Since T 0
c > 1, the maximal sample temperature (34) is lower than the minimal temperature of the
AFM tip (31). As shown in figure 3, at Te > T 0
c , the stationary temperature T0 of the sample is smaller
than Te.
5. Summary
In accordance with the analysis presented above, the formation of an adatom adsorption structure is
caused by self-organization of shear components of the strain and by the stress fields, on the one hand,
and by the sample temperature, on the other hand. Here, the strain ε acts as the order parameter, the
conjugate field is reduced to the stress σ, and the temperature T is the control parameter. The cause
for self-organization is the positive feedback of T and ε on σ [see equation (13)]. According to equa-
tions (2) and (4), it is caused by the temperature dependence of the shear modulus. With an allowance
for the effective shear modulus vs strain dependence, we obtain expressions for temperatures corre-
sponding to the absolute instability of the adatom configuration T 0
c [equation (31)] and its stability limit
Tc0 [equation (23)]. A real thermodynamic transition temperature can be determined from the equality
V (0) =V (ε0) of potentials in different phases and it is in the (T 0
c ,Tc0) region. According to equation (23),
systems predisposed to the formation of adatom structure have large typical G0 and small effective Gε
values of shear modulus.
33401-7
A.V. Khomenko
The present study is principally different from [30]. In particular, the basic parameters and equations
are different. For example, the order parameter is the strain, the shear modulus depends on temperature
(4), the derivation of equation for temperature (10) differs and so on. Therefore, the resultant equations
and figures are different from the ones in [30]. Thus, solid-liquid transition of an ultrathin lubricant film
and self-organization of adatoms on the semiconductor surface in contact with the tip of the dynamic
force microscope are described only by a similar approach but with many different aspects.
At a choice of real parameters, there is a difficulty connected primarily with the following features
which are common with the ultrathin lubricant film [30, 45, 48, 49]. The properties of adatom layers due
to their small thickness do not coincide with the properties of volume materials. They are characterized
by various values of elastic constants, density, heat conductivity, etc. The adatom structure temperature
is also an effective quantity, and it can essentially fluctuate, since the adatoms number is limited, and
these fluctuations lead to transitions between states [31, 50]. Therefore, we are restricted by a description
of the qualitative system behavior and all parameters are transformed into dimensionless form.
Acknowledgements
The basis of this study method was founded in the joint papers with my teacher Prof. A.I. Olemskoi
cited here. I thank Dr. Bo N.J. Persson for the invitation, hospitality, helpful comments and suggestions
during my stay in the Forschungszentrum Jülich (Germany). I am grateful to him and to the organizers
of the conference “Joint ICTP-FANAS Conference on Trends in Nanotribology” (12–16 September 2011,
Miramare, Trieste, Italy) for the invitation and financial support for participation, duringwhich this work
was initiated. The work was supported by the grant of the Ministry of Education and Science of Ukraine
“Modelling of friction of metal nanoparticles and boundary liquid films which interact with atomically
flat surfaces” (No. 0112U001380) and by the grant for a research visit to the Forschungszentrum Jülich
(Germany). I am thankful to Dr. Boris Lorenz for an attentive reading and correction of this article.
References
1. Fundamentals of Friction and Wear on the Nanoscale, Gnecco E., Meyer E. (Eds.), Springer, Berlin, 1 edn., 2007.
2. Garcia R., Amplitude Modulation Atomic Force Microscopy, Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim,
Germany, 2010.
3. Garcia R., Pérez R., Surf. Sci. Rep., 2002, 47, No. 6–8, 197; doi:10.1016/S0167-5729(02)00077-8.
4. Szlufarska I., Chandross M., Carpick R.W., J. Phys. D: Appl Phys, 2008, 41, No. 12, 123001;
doi:10.1088/0022-3727/41/12/123001.
5. Hofer W., Foster A., Shluger A., Rev. Mod. Phys., 2003, 75, No. 4, 1287; doi:10.1103/RevModPhys.75.1287.
6. Giessibl F., Rev. Mod. Phys., 2003, 75, No. 3, 949; doi:10.1103/RevModPhys.75.949.
7. Pogrebnjak A.D., Shpak A.P., Azarenkov N.A., Beresnev V.M., Phys.-Usp., 2009, 52, No. 1, 29;
doi:10.3367/UFNe.0179.200901b.0035.
8. Wang X., Kunc K., Loa I., Schwarz U., Syassen K., Phys. Rev. B, 2006, 74, 134305; doi:10.1103/PhysRevB.74.134305.
9. Guo W., Zhu C.Z., Yu T.X., Woo C.H., Zhang B., Dai Y.T., Phys. Rev. Lett., 2004, 93, 245502;
doi:10.1103/PhysRevLett.93.245502.
10. Yong C., Kendall K., Smith W., Philos. Trans. R. Soc. London, Ser. A, 2004, 362, No. 1822, 1915;
doi:10.1098/rsta.2004.1423.
11. Oyabu N., Pou P., Sugimoto Y., Jelinek P., Abe M., Morita S., Pérez R., Custance O., Phys. Rev. Lett., 2006, 96, 106101;
doi:10.1103/PhysRevLett.96.106101.
12. Khomenko A.V., Prodanov N.V., Carbon, 2010, 48, No. 4, 1234; doi:10.1016/j.carbon.2009.11.046.
13. Prodanov N.V., Khomenko A.V., Surf. Sci., 2010, 604, No. 7–8, 730; doi:10.1016/j.susc.2010.01.024.
14. Khomenko A.V., Prodanov N.V., J. Phys. Chem. C, 2010, 114, 19958; doi:10.1021/jp108981e.
15. Pogrebnyak A.D., Ponomarev A.G., Shpak A.P., Kunitskii Y.A., Phys.-Usp., 2012, 55, No. 3, 270;
doi:10.3367/UFNe.0182.201203d.0287.
16. Langewisch G., Kamiński W., Braun D.A., Moller R., Fuchs H., Schirmeisen A., Pérez R., Small, 2012, 8, No. 4, 602;
doi:10.1002/smll.201101919.
17. Campbellová A., Ondrácek M., Pou P., Pérez R., Klapetek P., Jelinek P., Nanotechnology, 2011, 22, No. 29, 295710;
doi:10.1088/0957-4484/22/29/295710.
33401-8
http://dx.doi.org/10.1016/S0167-5729(02)00077-8
http://dx.doi.org/10.1088/0022-3727/41/12/123001
http://dx.doi.org/10.1103/RevModPhys.75.1287
http://dx.doi.org/10.1103/RevModPhys.75.949
http://dx.doi.org/10.3367/UFNe.0179.200901b.0035
http://dx.doi.org/10.1103/PhysRevB.74.134305
http://dx.doi.org/10.1103/PhysRevLett.93.245502
http://dx.doi.org/10.1098/rsta.2004.1423
http://dx.doi.org/10.1103/PhysRevLett.96.106101
http://dx.doi.org/10.1016/j.carbon.2009.11.046
http://dx.doi.org/10.1016/j.susc.2010.01.024
http://dx.doi.org/10.1021/jp108981e
http://dx.doi.org/10.3367/UFNe.0182.201203d.0287
http://dx.doi.org/10.1002/smll.201101919
http://dx.doi.org/10.1088/0957-4484/22/29/295710
Self-organization of adatom adsorption structure at interaction with the tip of dynamic force microscope
18. Barth C., Foster A.S., Henry C.R., Shluger A.L., Adv. Mater., 2011, 23, No. 4, 477; doi:10.1002/adma.201002270.
19. Proksch R., Kalinin S.V., Nanotechnology, 2010, 21, No. 45, 455705; doi:10.1088/0957-4484/21/45/455705.
20. Dieška P., Štich I., Phys. Rev. B, 2009, 79, 125431; doi:10.1103/PhysRevB.79.125431.
21. Pokropivnyi A.V., Tech. Phys. Lett., 2000, 26, No. 11, 967; doi:10.1134/1.1329686.
22. Dieška P., Štich I., Pérez R., Phys. Rev. Lett., 2005, 95, 126103; doi:10.1103/PhysRevLett.95.126103.
23. Sahagún E., Sáenz J.J., Phys. Rev. B, 2012, 85, 235412; doi:10.1103/PhysRevB.85.235412.
24. Lange M., van Vorden D., Moller R., Beilstein J. Nanotech., 2012, 3, 207; doi:10.3762/bjnano.3.23.
25. Barsoum M.W., Murugaiah A., Kalidindi S.R., Zhen T., Phys. Rev. Lett., 2004, 92, 255508;
doi:10.1103/PhysRevLett.92.255508.
26. Haken H., Synergetics. An introduction. Nonequilibrium phase transitions and self-organization in physics,
chemistry, and biology, Springer, Berlin, 3 edn., 1983.
27. Landau L.D., Lifshitz E.M., Course of Theoretical Physics, Vol.5: Statistical Physics, Butterworth, London, 1999.
28. Lifshits E.M., Pitaevskii L.P., Course of Theoretical Physics, Vol.10: Physical Kinetics, Pergamon Press, Oxford, 1
edn., 1981.
29. Olemskoi A.I., Khomenko A.V., J. Exp. Theor. Phys., 1996, 83, No. 6, 1180.
30. Khomenko A.V., Yushchenko O.V., Phys. Rev. E, 2003, 68, 036110; doi:10.1103/PhysRevE.68.036110.
31. Khomenko A.V., Phys. Lett. A, 2004, 329, No. 1–2, 140; doi:10.1016/j.physleta.2004.06.091.
32. Rheology, Eirich F. (Ed.), Academic Press, New York, 1960.
33. Landau L.D., Lifshitz E.M., Course of Theoretical Physics, Vol.7: Theory of Elasticity, Pergamon Press, New York,
3 edn., 1986.
34. Mate C., IBM J. Res. Dev., 1995, 39, No. 6, 617; doi:10.1147/rd.396.0617.
35. Khomenko A.V., Prodanov N.V., Condens. Matter Phys., 2008, 11, No. 4, 615; doi:10.5488/CMP.11.4.615.
36. Gómez C.J., Garcia R., Ultramicroscopy, 2010, 110, No. 6, 626; doi:10.1016/j.ultramic.2010.02.023.
37. Landau L.D., Khalatnikov I.M., Dokl. Akad. Nauk SSSR, 1954, 96, 469, (see also: Collected Papers of L.D. Landau,
edited by D. ter Haar. Pergamon, London (1965)).
38. Berthier L., Biroli G., Rev. Mod. Phys., 2011, 83, No. 2, 587; doi:10.1103/RevModPhys.83.587.
39. Havranek A., Marvan M., Ferroelectrics, 1996, 176, 25; doi:10.1080/00150199608223597.
40. Olemskoi A.I., Khomenko A.V., Tech. Phys., 2000, 45, 672; doi:10.1134/1.1259700.
41. Olemskoi A.I., Khomenko A.V., Tech. Phys., 2000, 45, 677; doi:10.1134/1.1259702.
42. Khomenko A.V., Lyashenko I.A., Condens. Matter Phys., 2006, 9, No. 4(48), 695; doi:10.5488/CMP.9.4.695.
43. Olemskoi A.I., Khomenko A.V., Kharchenko D.O., Physica A, 2003, 323, 263; doi:10.1016/S0378-4371(02)01991-X.
44. Olemskoi A.I., Khomenko A.V., Phys. Rev. E, 2001, 63, 036116; doi:10.1103/PhysRevE.63.036116.
45. Khomenko A.V., Lyashenko Y.A., Tech. Phys., 2010, 55, No. 1, 26; doi:10.1134/S1063784210010056.
46. Metlov L.S., Phys. Rev. E, 2010, 81, 051121; doi:10.1103/PhysRevE.81.051121.
47. Metlov L.S., Phys. Rev. Lett., 2011, 106, 165506; doi:10.1103/PhysRevLett.106.165506.
48. Khomenko A.V., Lyashenko I.A., J. Phys. Stud., 2007, 11, No. 3, 268.
49. Khomenko A.V., Lyashenko I.A., Phys. Lett. A, 2007, 366, No. 1–2, 165; doi:10.1016/j.physleta.2007.02.010.
50. Khomenko A.V., Lyashenko I.A., Tech. Phys., 2005, 50, No. 11, 1408; doi:10.1134/1.2131946.
33401-9
http://dx.doi.org/10.1002/adma.201002270
http://dx.doi.org/10.1088/0957-4484/21/45/455705
http://dx.doi.org/10.1103/PhysRevB.79.125431
http://dx.doi.org/10.1134/1.1329686
http://dx.doi.org/10.1103/PhysRevLett.95.126103
http://dx.doi.org/10.1103/PhysRevB.85.235412
http://dx.doi.org/10.3762/bjnano.3.23
http://dx.doi.org/10.1103/PhysRevLett.92.255508
http://dx.doi.org/10.1103/PhysRevE.68.036110
http://dx.doi.org/10.1016/j.physleta.2004.06.091
http://dx.doi.org/10.1147/rd.396.0617
http://dx.doi.org/10.5488/CMP.11.4.615
http://dx.doi.org/10.1016/j.ultramic.2010.02.023
http://dx.doi.org/10.1103/RevModPhys.83.587
http://dx.doi.org/10.1080/00150199608223597
http://dx.doi.org/10.1134/1.1259700
http://dx.doi.org/10.1134/1.1259702
http://dx.doi.org/10.5488/CMP.9.4.695
http://dx.doi.org/10.1016/S0378-4371(02)01991-X
http://dx.doi.org/10.1103/PhysRevE.63.036116
http://dx.doi.org/10.1134/S1063784210010056
http://dx.doi.org/10.1103/PhysRevE.81.051121
http://dx.doi.org/10.1103/PhysRevLett.106.165506
http://dx.doi.org/10.1016/j.physleta.2007.02.010
http://dx.doi.org/10.1134/1.2131946
A.V. Khomenko
Самоорганiзацiя структури адсорбованих адатомiв при
взаємодiї iз зондом динамiчного силового мiкроскопа
О.В. Хоменко1,2
1 Кафедра моделювання складних систем, Сумський державний унiверситет, вул. Римського-Корсакова, 2,
40007 Суми, Україна
2 Iнститут Петера Грюнберга-1, Дослiдницький центр Юлiха, 52425 Юлiх, Нiмеччина
Формування структури адсорбованих адатомiв при дослiдженнi в режимi динамiчної силової мiкроскопiї
представлено як результат спонтанної появи зсувної деформацiї в результатi зовнiшнього надкритич-
ного нагрiвання. Цей перехiд описується рiвнянням Кельвiна-Фойгта для в’язкопружного середовища,
релаксацiйним рiвнянням Ландау-Халатнiкова для зсувних напружень та релаксацiйним рiвнянням для
температури. Показано, що цi рiвняння формально збiгаються iз синергетичною системою Лоренца, де
зсувна деформацiя вiдiграє роль параметра порядку, спряжене поле зводиться до напружень, та темпера-
тура є керувальним параметром. В рамках адiабатичного наближення знайденi стацiонарнi значення цих
величин. Враховуючи залежнiсть модуля зсуву зразка вiд деформацiї, формування конфiгурацiї адсорбо-
ваних адатомiв описано як перехiд першого роду. Критична температура зонда лiнiйно зростає з ростом
ефективного значення модуля зсуву зразка i зменшується при зростаннi його характерного значення.
Ключовi слова: фазовий перехiд, реологiя, пластичнiсть, деформацiя, напруження, атомно-силова
мiкроскопiя
33401-10
Introduction
Basic equations
Continuous transition
Effect of deformational defect of modulus
Summary
|