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 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.


Introduction
Nowadays due to large scientific and practical importance the phenomena, taking place on the sample surface at interaction with tip of dynamic force microscope, e.g., atomic force microscope (AFM) and friction force microscope, attract more and more attention (see the reviews in [1,2,3,4,5] and the literature cited therein). Particularly, the experimental and theoretical data are obtained on structural instabilities, phase transformations, plastic dislocation, neck and adatoms structures formation [6,7,8,9,10,11,12,13]. These processes are characterized by hysteresis of dependencies of adhesion force and potential energy surface on the tip-surface distance [10,14,15] and by hysteresis of the sample stress vs strain curve [16].
Since the nature of such phenomena remains poorly understood the basic goal of the present study is the construction of a qualitative nonlinear model [17,18,19,20,21,22] describing the hysteresis processes which occur on the germanium surface during interaction with the AFM tip [10]. Here the macroscopic continuum mechanics models [23,24] are supposed to be still applicable to the atomic length-scales, where discrete atomistic interactions become significant [2,25,26]. However, a total explanation of studied macroscopic phenomena requires the consideration of the microscopic processes. 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 the 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 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 a viscoelastic semiconductor approximation subject to the heat conductivity. The adatom structure formation is shown in Section 3 to be supercritical in character (has 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 determined also within adiabatic approximation. Using such limit, the synergetic potential is obtained, that is the analog of thermodynamic potential, from basic evolution equations. Section 5 contains a short conclusions.

Basic equations
Let us start from the supposition that the relaxation behavior of the shear component ε of the strain tensor in a viscoelastic semiconductor is governed by the Kelvin-Voigt equation [23] where τ ε is the Debye relaxation time and η ε 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 component of the stress σ. In the steady state,ε = 0, we obtain the Hooke-type expression σ = G ε ε, G ε ≡ η ε /τ ε . The next assumption of our approach is that the relaxation equation of the sample shear stress σ has a similar form to the Landau-Khalatnikov equation [19,27,21]: Here the first term on the right-hand side describes the relaxation during time τ σ ≡ η/G determined by values of the shear viscosity η and modulus G. In the stationary caseσ = 0 the kinetic equation (2) is transformed into the Hooke's law σ = Gε.
Substituting ∂ε/∂t instead of ε/τ σ in Eq. (2) reduces it to a Maxwell-type equation for a viscoelastic matter [24]. Eqs. (1) and (2) give a new rheological model because they can be transformed to a differential equation of second order with respect to strain or stress. Note that effective values of viscosity η ε ≡ τ ε G ε and modulus G ε ≡ η ε /τ ε do not coincide with the real values η and G. Physically such difference is conditioned by that the Maxwell-type equation (2) is not equivalent to the Kelvin-Voigt equation (1) [23,24,21]. As is known the values G ε , η, η ε depend very weakly on the sample temperature T , while the real shear modulus G vanishes, when the temperature decreases to T c [28,29,30,31]. Further, the simplest approximate temperature dependencies are used: where G 0 ≡ G(T = 2T c ) is the typical value of modulus. According to the synergetic concept [17,20,21,30,31,32,33] for completing the equations 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 § 31 in Ref. [24]. Thus, it is necessary to start from the continuity equation for the heat δQ = T δS: Here the heat current is given by the Onsager equation where κ is the heat conductivity. In the elementary case of the thermoelastic stress the entropy consists of the purely thermodynamic component S 0 and the dilatation: where α is the thermal expansion coefficient, T 0 is the equilibrium temperature, I is the unit tensor and K is the compression modulus (see § 6 in [24]). In the considered situation we should transfer from the dilatational component Kαε 0 to the elastic energy −σε/T of the shear component divided by temperature (here the minus sign takes into account the connection T δS = pδV ⇒ −σδε at S 0 = const, which is caused by the contrary choice of the pressure p and the stress σ signs). As a result, Eq. (5) possesses the form Taking into account the equality (κ/l 2 )(T e − T ) ≈ κ∇ 2 T (l is the scale of heat conductivity, T e is the AFM tip temperature) and the definition of heat capacity c p =T dS 0 /dT , Eq. (9) assumes the form: Substituting here expression for theε from Eq. (1) we obtain the term σ 2 /η ε . It describes the dissipative heating of a viscous liquid, flowing under influence 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 influence of the tip whose value T e is not reduced to the Onsager component and is fixed by external conditions. In view of these circumstances the square contribution of the stress is supposed to be included in T e . The obvious account of this term leads to a significant complication of the subsequent analysis, though it results in a renormalization of quantities. Therefore component T e in Eq. (10) is assumed to be constant for our further consideration. It is convenient to introduce the following measure units: for the variables σ, ε, T , respectively (τ T ≡ l 2 c p /κ is the time of heat conductivity). Then, the basic equations (1), (2), and (10) take the form: where the constant is introduced. Equations (12) -(14) have a similar form that of the Lorenz scheme [17] which allows us to describe the thermodynamic phase and the kinetic transitions [20,21,30,31,32,33].

Continuous transition
In general the system (12) - (14) can not be solved analytically, therefore we use the following adiabatic approximation: This implies that in the course of 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 τ ε and the microscopic Debye time τ σ ≈ a/c∼10 −12 s, where a ∼ 1 nm is the lattice constant or the intermolecular distance and c ∼ 10 3 m/s is the sound velocity. The second condition (16) can be reduced to the form l ≪ L, where the maximal value of the characteristic length of the heat conductivity the thermometric conductivity χ ≡ κ/c p , the effective kinematic viscosity ν ε ≡ η ε /ρ and the sound velocity c ε ≡ (G ε /ρ) 1/2 are introduced (ρ is the medium density). Then, we can put the left-hand sides of Eqs. (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: In accordance with Eq. (20), in the important range of values of the parameter T e > 1, the temperature T decreases monotonically with increasing strain ε from the value T e at ε = 0 to (T e +1)/2 at ε = ε m ≡ 1/g. Obviously, this decrease is caused by the negative feedback of the stress and the strain on the temperature in Eq. (14), that is explained by the Le Chatelier principle for this problem. Really the reason for the formation of adatoms adsorption structure is the positive feedback of the strain and the temperature on the stress in Eq. (13). Hence the increase in the temperature should intensify the self-organization effect. However according to Eq. (14) the system behaves so that the consequence of transition, i.e., growth of the strain, leads to a decrease in its reason (temperature). Equation (19), expressing the stress in terms of the strain, has the linear form of the Hooke's law at ε ≪ ε m with the effective shear modulus G ef ≡ g (T e − 1). At ε = ε m the function σ(ε) has a maximum and at ε > ε m it decreases that has no physical meaning. Thus, the constant ε m ≡ 1/g gives the maximal strain. The increase in the typical value of the modulus G 0 leads to a decrease in the maximal strain ε m and an increase in the effective modulus G ef whose value is proportional to the characteristic temperature T e . Substituting Eq. (19) into Eq. (12) we obtain the Landau-Khalatnikov-type equation [19,27] where the synergetic potential has the form At steady state the conditionε = 0 is fulfilled and the potential (22) acquires a minimum. When the temperature T e becomes smaller than the critical value this minimum corresponds to ε = 0, i.e., the adatom adsorption structure is not realized. In the reverse case T e > T c0 , the stationary shear strain has the nonzero value which increases with T e 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: Note that, on the one hand, the steady temperature T 0 coincides with the critical value (23) and, on the other hand, its value differs from the temperature T e . Since T c0 is the minimal temperature at which 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 last term on the right-hand side of Eq. (14)) decreases the sample temperature so much that only in the limit does it ensure the self-organization process. At steady state the value of the shear modulus is The two cases can be marked out by the parameter g = G 0 /G ε . In the situation g ≫ 1, meeting the large value of the modulus G 0 , Eqs. (23) -(25) take the form This corresponds to the "solid (fragile)" limit. The opposite case g ≪ 1 (small modulus G 0 ) meets the "strongly viscous liquid"

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 = G ε ε at ε < ε m and the constant σ m = G ε ε m at ε ≥ ε m (σ m , ε m are the maximal stress and strain, σ > σ m results in viscous flow with the deformation rateε = (σ − σ m ) /η ε ). Actually, the σ vs ε dependence curve has two regions: first one, Hookean, has a large slope corresponding to the shear modulus G ε , followed by the more gently sloping section of the plastic deformation whose tilt is defined by the hardening factor Θ < G ε . Obviously, such a picture means that the shear modulus, introduced in Eq. (1), depends on the strain value. Let us use the simplest approximation 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 ε p , which is smaller than ε s (otherwise plastic mode is not realized). Note that an expression of the type Eq. (29) was proposed, for the first time, by Haken [17] describing the rigid mode of laser radiation. It is used [20,32,33] to describe the first-order phase transition, and Eq. (29) had contained the square of the ratio ε/ε p (so the V vs ε dependencies in Ref. [20,32,33] and Eq. (30) have the even form). In description of the structural phase transitions of a liquid the third-order invariants, breaking the specified parity, are present [18]. Therefore in study [21,30,31] in approximation (29) we used the linear term ε/ε p , instead of the square one (ε/ε p ) 2 . Obviously, in such case the V vs ε dependence is not already even.
Within the adiabatic approximation (16) the Lorenz equations (12) - (14), where G ε is replaced by dependence G ε (ε), is reduced to the Landau-Khalatnikov equation (21). The synergetic potential possesses the form: Here the constant α≡ε p /ε s < 1 and the parameter θ=Θ/G ε < 1, describing the ratio of the tilts for the deformation curve on the plastic and the Hookean sections, are introduced. At small value of temperature T e dependence (30) has a monotonically increasing shape with its minimum at ε = 0 corresponding to the steady state of the absence of the adatom adsorption structure (curve 1 in Fig. 1). As shown in Fig. 1 at a plateau appears (curve 2), which for T e > T 0 c is transformed into a minimum, meeting the strain ε 0 = 0, and a maximum at ε m that separates the minima corresponding to the values ε = 0 and ε = ε 0 (curve 3) [10]. With further increase in the temperature T e the "ordered" phase minimum, corresponding to the adatom adsorption configuration ε = ε 0 , grows deeper, and the height of the interphase barrier decreases, vanishing at the critical value T c0 = 1 + g −1 (23). The steady-state values of the strain have the form (see Figs. 1 and 2 where the lower sign meets the stable adatom structure and the upper sign corresponds to the unstable one. At T e ≥ T c0 the dependence V (ε) is characteristic for the absence of the modulus defect (see curve 4 in Fig. 1). It is worth noting that potential barrier inherent in the synergetic first-order transition is manifested only in the presence of the deformational defect of the modulus. Since latter is realized always [10], it follows that a studied adatom structure formation is a synergetic first-order transition. The considered situation is much more complex than typical thermodynamic phase transitions. Really, in the latter case the steady-state value of the semiconductor temperature T 0 is equal to the value T e fixed by a thermostat. In this study T 0 is reduced to the critical value T c0 for the synergetic second-order transition (see section 3). When the modulus defect is taken into account, the temperature whose value is defined by a minimum position of the dependence (30), is realized. According to Eqs. (32) and (33), the quantity T 0 smoothly decreases from the value at T e = T 0 c , to 1 at T e → ∞. As shown in Fig. 3, the stationary temperature T 0 increases linearly from 0 to T c0 , with T e being in the same interval and, after the jump down at T e = T c0 , the magnitude T 0   Since T 0 c > 1, the maximal sample temperature (34) is lower than the minimal temperature of the AFM tip (31). As shown in Fig. 3, at T e > T 0 c the stationary temperature T 0 of the sample is lower than T e .

Summary
In accordance with the analysis presented above the formation of the adatom adsorption structure is caused by the self-organization of the shear components of the strain and the stress fields, on the one hand, and 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 Eq. (13)). According to Eqs. (2) and (4), it is caused by the temperature dependence of the shear modulus. The formation of the adatom adsorption structure due to both heat and deformation effects at contact of solid surfaces can be represented as shear and dynamic disorder-driven melting where thermal fluctuations [34] are absent. With an allowance for the effective shear modulus vs strain dependence, we obtain the expressions for the temperatures corresponding to the absolute instability of the adatom configuration T 0 c [Eq. (31)] and its stability limit T c0 [Eq. (23)]. The real thermodynamic transition temperature can be determined from the equality V (0) = V (ε 0 ) of potentials in different phases and it is in the (T 0 c , T c0 ) region. According to Eq. (23) systems predisposed to formation of adatom structure have large typical G 0 and small effective G ε values of shear modulus.
The kinetics of this transition is described by the Landau-Khalatnikov equation (21), where the synergetic potential has the form (30) inherent in the first-order transition. In supercooled adatom configuration with τ ε = ∞, freezing of the system can occur (ε → 0) even in the nonsteady state ∂V /∂ε = 0.