# Thermomigration of Au-Ge microdroplets on Ge(111): Atomic mechanisms and effects on surface morphology

Nayef Abu Dahech, Frédéric Leroy, Pierre Müller, Fabien Cheynis, Stefano Curiotto

► To cite this version:

Nayef Abu Dahech, Frédéric Leroy, Pierre Müller, Fabien Cheynis, Stefano Curiotto. Thermomigration of Au-Ge microdroplets on Ge(111): Atomic mechanisms and effects on surface morphology. Physical Review Materials, 2025, 9 (10), pp.103403. 10.1103/p11f-r9y9. hal-05332833

HAL Id: hal-05332833

https://hal.science/hal-05332833v1

Submitted on 27 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Thermomigration of Au-Ge microdroplets on Ge(111): Atomic mechanisms and effects on surface morphology

Nayef Abu Dahech,\* Frédéric Leroy,† Pierre Müller,‡ Fabien Cheynis,§ and Stefano Curiotto¶

Aix Marseille University, CNRS, CINAM, Marseille, France

The motion of microdroplets under thermal gradients, also known as thermomigration (TM), plays a critical role in a wide range of surface phenomena and technological applications. In this work, we study the TM behavior of Au-Ge micro-droplets on Ge(111) surfaces and show that the droplet motion leads to the formation of atomically flat terraces, which are essential for thin film epitaxy. Experimental observations are combined with a microscopic model to investigate droplet motion and the formation of flat atomic terraces. Our results show that, at low temperatures, droplet velocity increases with size, but this size dependence diminishes as the temperature rises. The microscopic model we develop is based on thermally activated detachment, diffusion, and attachment of solute (Ge) atoms across the droplet, driven by a temperature-dependent concentration difference between the leading edge and the trailing edge. The model reproduces the experimental trends. Our results provide insights into the fundamental mechanisms of droplet TM and suggest ways to control surface morphology.

## I. INTRODUCTION

The motion of objects under the influence of a thermal gradient - commonly referred to as the Soret effect, thermophoresis, or thermomigration (TM) - has been the subject of scientific study since Ludwig's seminal work in 1856 [1, 2], and further detailed in comprehensive reviews [3–5].

Droplet dynamics on solid surfaces driven by thermal gradients play a critical role in different areas. For example, in self-cleaning coatings where engineered morphologies minimize droplet adhesion, one might consider using thermal gradients to further facilitate droplet removal [6]. For example, Bjelobrk et al. [7] demonstrated that droplets on lubricant-impregnated surfaces can achieve velocities up to 6.5 mm/s under a thermal gradient as low as 2 K/mm. These advances underscore the potential of TM effects in enhancing the functionality of self-cleaning surfaces. On the other hand, in semiconductor fabrication, where precise control of metal droplet positioning is essential for catalyzing nanowire growth [8], uncontrolled TM can lead to unintended droplet displacement or modification of surface morphology, potentially compromising device functionality.

Different investigations have shed light on the mechanisms underlying droplet thermomigration, emphasizing the influence of diffusion, phase transformation, and interfacial kinetics. For example, Ichinokawa et al. [9] demonstrated that metallic islands on Si(100) migrate toward hotter regions with velocities proportional to island size. Yang et al. [10] proposed a dissolution-diffusion-deposition mechanism for Pt-Si droplets thermomigrating on Si(100). More recent investigations, including those by Stenger et al. [11] and Poelsema et al. [12], have expanded our understanding of the thermomigration dynamics on different systems.

In this article we investigate the motion of Au-Ge microdroplets on Ge(111) surfaces. Au-Ge has been chosen because

it is a model system with a low-temperature eutectic, where droplets have already been observed [13–17]. We find that droplet velocity increases linearly with droplet size at low temperatures, but from about 1000 K this dependence weakens, and the velocity becomes independent of size at 1180 K. This phenomenon is analyzed in detail in subsections III A and III B. Our approach integrates experimental observations (section III) with microscopic modeling (section IV). The model extends the atomistic framework introduced in [18], incorporating the temperature-dependent equilibrium concentration of a solute B in a solvent A as predicted by the liquidus lines of the corresponding phase diagram. We also show that the thermomigration of metallic droplets can be used to obtain very flat surfaces. Atomically flat terraces can be used for further studies and are essential in epitaxial growth of thin film [19].

## II. EXPERIMENTAL DETAILS

A Germanium (111) wafer (BT-Electronics) is cut by wire saw into 9 × 10 mm² samples. The samples are then cleaned in three successive ultrasonic baths with acetone, ethanol, and distilled water. In each experiment, a different sample is then chosen and introduced into an ultra-high vacuum (UHV) system with a base pressure of 2 × 10⁻¹⁰ Torr. We have used a sample holder specially conceived to apply a thermal gradient, as described in [18]. The samples are degassed at about 1150 K, roughly 60 K below the Ge melting point, for 18 hours. The temperature is measured using an optical pyrometer, assuming a germanium emissivity of 0.56 [20]. We estimate an error of +/- 30 K on the absolute temperature measurements. The thermal gradient, considered linear, is obtained by focusing the pyrometer and measuring the temperatures of the hot side and of the cold side of the sample. The temperature difference is divided by the distance between the two measured spots (5 mm) to obtain the thermal gradient: $\nabla T = \frac{\Delta T}{\Delta x}$, (see figure 1(a)). Typical thermal gradients vary between 0.24 × 10⁻² K/µm and 2.48 × 10⁻² K/µm from 650 K to 1180 K. After degassing, the samples are cleaned at room temperature by 1 keV Ar⁺ ion bombardment for 15 minutes; then, they are

<footnote>\* nayef.abu-dahech@univ-amu.fr</footnote>

<footnote>† frederic.leroy.3@univ-amu.fr</footnote>

<footnote>‡ pierre.muller@univ-amu.fr</footnote>

<footnote>§ fabien.cheynis@univ-amu.fr</footnote>

<footnote>¶ stefano.curiotto@cnrs.fr</footnote>
annealed for 15 minutes at 1160 K.

Gold (Au, 99.999% pure, Goodfellow) was deposited using an MBE-Komponenten effusion cell at a rate of 0.08 monolayers (ML)/minute, calibrated with the formation of the Au-Si(111) $\sqrt{3} \times \sqrt{3}$ surface phase. Different amounts of Au are deposited, ranging from 0.25 to 20 ML. The motion of Au-Ge droplets on the substrates is observed in real time using a low-energy electron microscope (LEEM III, Elmitec GmbH). The images are taken in bright-field mode, with electron energies ranging from 1 to 10 eV. Data analysis was carried out using the Fiji software with the TrackMate plugin [21].

<figure><img src="image_2.png" /><figcaption>FIG. 1. (a) Ge(111) sample under a controlled thermal gradient, with enhanced thermal radiation on the right side corresponding to a higher temperature than on the left. (b) A phase diagram of the Au-Ge system, calculated using the thermodynamic data of Chevalier [22]. The diagram illustrates the relevant phase regions including liquid, liquid+solid Au, and liquid+solid Ge, with the eutectic point around 628 K at approximately 27% Ge. The coexistence of fcc Au and diamond-structured Ge is indicated at lower temperatures.</figcaption></figure>

### III. RESULTS AND DISCUSSION

#### A. Formation of Au-Ge liquid micro-droplets

After depositing Au at room temperature, we gradually increased the temperature close to the melting point of Ge, while observing the sample surface with LEEM. A few kelvin below the Ge melting point (1211 K), we observed circular structures moving across the sample and leaving tracks behind. These circular structures are initiated by a sudden jump from pre-existing surface defects. This jump is a critical prerequisite for the motion to occur. Afterward, the structures can also move at lower temperatures.

We suggest that these structures are liquid droplets: according to the Au-Ge bulk phase diagram [22], a eutectic between solid Ge, solid Au-Ge, and a Au-Ge liquid forms at 636 K (see figure 1(b)). After the first jump, the droplets also move at temperatures much lower than the Ge melting point.

These droplets move on the sample along the thermal gradient axis, towards the hot side (see figure 2), with approximately constant velocity. The exact trajectories are influenced by surface topography. The droplet velocity does not depend on the Au coverage (ML deposited), increases with the temperature, and is not modified by the surface roughness. Along their motion, the droplets interact with the substrate and leave

<figure><img src="image_3.png" /><figcaption>FIG. 2. Images illustrating two micro-droplets of different radii (small and large droplets are highlighted in yellow and red circles, respectively) migrating toward the hotter side of the sample. Panels (a) and (b) show droplets moving at $T = 913$ K, with a thermal gradient given by $1.58 \times 10^{-2}$ K/µm, the time interval between the images is $t = 33$ s, and a starting voltage of 0.42 eV. The blue arrows highlight some anchored droplets that coalesce with the larger droplets upon contact during the motion. Panels (c) and (d) are recorded at $T = 1031$ K, with a thermal gradient given by $2.26 \times 10^{-2}$ K/µm, a time interval of 22 s, and a starting voltage of 0.83 eV. Panels (e) and (f) are recorded at $T = 1180$ K, with a thermal gradient given by $2.48 \times 10^{-2}$ K/µm, a time interval of 3 s, and a starting voltage of 0.42 eV. A defect of the channel plates is highlighted with a blue circle. All the images are obtained after 10 ML of Au deposition (not during deposition). The droplets move from right to left, toward the higher-temperature region, despite the value of the average temperature.</figcaption></figure>
tracks that display flat terraces of micrometer scale. During their motion, droplets can merge with other droplets that are moving more slowly or that remain immobile because they are anchored at defects (see the blue arrows in figure 2(a)). The motion of these droplets is similar to that observed by other authors for large droplets in a thermal gradient, for instance in Pt-Ge [12] and Pt-Si [10]. As expected [23, 24], the terraces have a 1 × 1 surface structure, while at lower temperatures, where the droplets do not move, the surface shows a (√3 × √3)R30° reconstruction (see the LEED patterns in figures 3(a) and 3(b)).

<figure><img src="image_4.png" /><figcaption>FIG. 3. Low-Energy Electron Diffraction (LEED) patterns of a Au-Ge(111) surface after 1.5 ML of Au deposition: (a) at 1100 K, we observe a (1 × 1) surface phase; (b) at 550 K, the surface displays a (√3 × √3)R30° reconstruction. The electron beam energy is 25 eV.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 4. Log-log plot of droplet velocity $v$ versus droplet radius $R$ at four different temperatures: 913 K (dark gray), 1002 K (light gray), 1060 K (purple), and 1180 K (green). Dashed lines represent linear fits to the data. Error bars indicate measurement uncertainties [25]. The slope of the linear fit through the data changes from approximately 1 at lower temperatures, indicating an attachment/detachment-limited regime ($v \propto R$), to near 0 at higher temperatures, consistent with a diffusion-limited regime where $v$ is approximately constant.</figcaption></figure>

## B. Micro-droplets' migration behavior

In this study, we have investigated the migration behavior of Au-Ge droplets as a function of droplet radius and average sample temperature. Notice that in our experiments the thermal gradient depends on the average sample temperature (the importance of these two parameters is discussed in the modeling section). Figure 2 shows droplets of small and large radii (highlighted in yellow and red, respectively), before and after motion, at three temperatures: 913 K, 1031 K, and 1180 K. The thermal gradients are $1.58 \times 10^{-2}$ K/µm, $2.26 \times 10^{-2}$ K/µm, and $2.48 \times 10^{-2}$ K/µm, respectively. At the lowest temperature (913 K), a pronounced difference in migration velocity was observed, with larger droplets exhibiting higher velocities. As the temperature increases, this velocity gap narrows, and at 1180 K, the velocities are the same for all droplet sizes. The shadowing observed around some droplets is an artifact of the sliding-average image acquisition mode: each displayed frame contains 50% of the signal from the current exposure (0.1 s) and 50% from the previously integrated image. This mode improves the signal-to-noise ratio, but for fast-moving droplets it produces a shadowing effect. The concentric rings within the shadow arise from a short lag time in data transmission to the computer between frames (0.1 s), during which no signal is recorded.

Figure 4 shows the droplet velocity as a function of radius in logarithmic scale, for four different temperatures. At low temperature, $v \propto R$. However, the size dependence weakens when the temperature increases and the droplet velocity becomes independent of size at 1180 K. This behavior highlights the dominant role of temperature in modulating the migration kinetics. Overall, our findings show that droplet size and temperature act as key factors governing the migration of Au-Ge micro-droplets under thermal gradients, with the effect of droplet size diminishing as the temperature increases.

## C. Formation of atomically flat terraces

Since all experiments have been carried out above 900 K, i.e. at temperatures higher than the Au-Ge eutectic [22], the droplets dissolve Ge from the substrate to form a Au-Ge liquid. Ge is more easily dissolved from steps and step bunches than from flat (111) terraces [13]. The droplets tend to move along step bunches left behind by previously moving droplets. As Ge is dissolved, the tracks are modified, particularly if the moving droplets are large. The track size depends on the droplet radius: larger droplets make wider tracks, while smaller ones make narrower tracks. After a prolonged period during which the droplets move and etch the sample, flat terraces separated by atomic steps are observed (see Figures 5(a) and 5(b)). The steps are more visible in the low-temperature surface phase (√3 × √3)R30° of Au-Ge(111), and become evident upon decreasing the temperature. For instance, in Figure 5, we observed the transition on a sample with about 2 ML of deposited Au, when cooling from 930 K to 870 K. Note that, according to the detailed studies by Giacomo et al. [24] on the 1 × 1 to (√3 × √3)R30° phase transition, the tran-
sition temperature depends on the Au coverage and is about 913 K for 1 ML Au. The phase transition progresses from the lower-temperature side to the higher-temperature side (see Figures 5(c) and 5(d)). The atomically-flat terraces are separated by domain boundaries and by steps, and have a size of up to 7 µm. If Au is further deposited on the sample at low temperatures, nano-droplets form within the terraces. These droplets behave differently from the large droplets studied in this work and will be investigated in another article. Our findings on the formation of flat terraces are an alternative to the method proposed by other authors [26, 27] and offer potential strategies for controlling surface morphology in nanoscale manufacturing processes.

<figure><img src="image_6.png" /><figcaption>FIG. 5. Two monolayers of Au were deposited on the sample at room temperature, after which it was annealed at high temperature. The figure presents a series of LEEM images illustrating the evolution of the phase transition on a Au-Ge(111) surface during cooling (at constant Au content) from 930 K to 870 K. Panel (a) displays a flat terrace in the track left by Au-Ge moving droplets, where the surface appears largely featureless. Upon decreasing the temperature, the 1 × 1 surface phase transforms into a $\sqrt{3} \times \sqrt{3}$ phase. Panel (b) shows that the onset of the phase transition occurs on the low-temperature side (right), then propagates towards the hotter side (left). Panels (c) and (d) present the same region after further progression of the transition, with the formation of well-defined domains. The circular dark features are probably structures pinned at defects of the substrate, where Au has accumulated.</figcaption></figure>

## IV. MODELING

We now consider the effect of the thermal gradient on the droplet displacement from an atomistic point of view, and propose a microscopic model to explain the motion and the relationship between the velocity and the radius. The model is 1D and can be understood using figure (6). It is based on [18], and expanded to consider the change of equilibrium concentration of a solute B in a solvent A at different temperatures as suggested by liquidus lines in phase diagrams. The motion of a droplet is linked to the displacement of the hole underneath [13]. The holes underneath the droplets are much deeper than one monolayer; however, we consider that every layer behaves in the same way, so that a simple 1D model can approximate

<figure><img src="image_7.png" /><figcaption>FIG. 6. Schematics of the microscopic model discussed in the text. An A-B droplet displaces, along with its underlying hole, toward the hot side. The displacement is governed by detachment, diffusion, and attachment of B atoms. This figure serves as a schematic representation of a general, simplified case. In certain systems, however, the solid substrate B can be an A-B alloy richer in B than the liquid phase, as in the case of Au-Ge.</figcaption></figure>

the whole motion. There are thus (i) a detachment flux of B atoms from the hot side of the liquid-solid interface ($J_{det}$), where the concentration of B in the liquid should be higher (for a eutectic phase diagram of the kind shown in figure 1), (ii) a diffusion flux inside the liquid ($J_{diff}$), and (iii) an attachment flux on the cold side ($J_{det}$), where the B concentration in the liquid should be lower. The solid of figure 6 can also contain A atoms (as in the case of the Au-Ge system), however, in the model, we neglect the change in composition of the solid. The expressions for the three fluxes have been obtained in [18] with a linear approximation development, considering the temperature difference between the droplet edges small with respect to the average temperature. The fluxes are (equations (3), (4), and (5) of [18]):

$$J_{diff} = J = D_0 \frac{n(-L/2) \exp \frac{-Q}{kT(-L/2)} - n(L/2) \exp \frac{-Q}{kT(L/2)}}{\frac{2kT_0^2}{(E_m - Q) \frac{dT}{dx}} \exp \frac{E_m - Q}{kT_0} \sinh \frac{(E_m - Q)\Delta T}{2kT_0^2}} \quad (1)$$

$$J_{att} = J = -k'_{-L/2}(n(-L/2) - n_{eq}(-L/2))$$

$$J_{det} = J = -k'_{L/2}(n_{eq}(+L/2) - n(+L/2))$$

$T_0$ is the temperature in the middle of the droplet ($x = 0$). $\Delta T$ is the temperature difference between the opposite sides of the droplet. $D_B = D_0 \exp(-E_m/kT)$ is the diffusion coefficient of B in the liquid, with $E_m$ the B diffusion energy and $D_0$ a prefactor. Q is a quantity related to the binding energy of B in the liquid [28, 29]. n is the concentration of the B species in the liquid in atoms per unit volume, evaluated at the positions $-L/2$ and $+L/2$ (see the schematics of the micromodel figure), that can be different from the equilibrium concentration $n_{eq}$ predicted by the A-B phase diagram at those positions. $k'_i$, where $i = \pm L/2$ is the attachment/detachment coefficient at different droplet edges. From $dn/dT$, obtained from the inverse
of the slope of the liquidus curve in a phase diagram, we write:

$$ 
\begin{aligned}
n_{eq}(-L/2) &= n_{eq}(0) + \frac{dn_{eq}}{dT} \frac{dT}{dx} \cdot \left(-\frac{L}{2}\right) \\
n_{eq}(+L/2) &= n_{eq}(0) + \frac{dn_{eq}}{dT} \frac{dT}{dx} \cdot \left(+\frac{L}{2}\right)
\end{aligned} 
\quad (2) $$

In steady state, the three fluxes of equation 1 are equal and using equations 2 we obtain:

$$ J = - \frac{\frac{dT}{dx} \left( \frac{dn}{dT} + \frac{n_{eq}(0)Q}{kT_0^2} \right)}{\frac{1}{D_B} + \frac{1}{L} \left[ \frac{1}{k'_{-L/2}} \left( 1 + \frac{Q\Delta T}{2kT_0^2} \right) + \frac{1}{k'_{+L/2}} \left( 1 - \frac{Q\Delta T}{2kT_0^2} \right) \right]} \quad (3) $$

The hole and the droplet move opposite to the atomic fluxes, therefore we write the droplet velocity as $v = -\frac{1}{1-\omega n_{eq}(0)} J\omega$, where $\omega$ is the atomic volume, and $\frac{1}{1-\omega n_{eq}(0)}$ a correction term due to advection (see for instance [30]). We take a same $\omega$ for both A and B. Therefore the velocity is:

$$ 
\begin{aligned}
v &= \frac{1}{1 - \omega n_{eq}(0)} \omega \frac{\frac{dT}{dx} \left( \frac{dn}{dT} + \frac{n_{eq}(0)Q}{kT_0^2} \right)}{\frac{1}{D_B} + \frac{1}{L} \left[ \frac{1}{k'_{-L/2}} \left( 1 + \frac{Q\Delta T}{2kT_0^2} \right) + \frac{1}{k'_{+L/2}} \left( 1 - \frac{Q\Delta T}{2kT_0^2} \right) \right]} \\
v &\approx \frac{1}{1 - \omega n_{eq}(0)} \omega \frac{\frac{dT}{dx} \left( \frac{dn}{dT} + \frac{n_{eq}(0)Q}{kT_0^2} \right)}{\frac{1}{D_B} + \frac{1}{Rk'}}
\end{aligned} 
\quad (4) $$

The approximated version of equation 4, where we have used $L/2 = R$ is valid when $k'_{-L/2} \approx k'_{+L/2} = k'$. From equation 4, we can conclude that, in the framework of this microscopic model, (i) when the displacement is limited by the diffusion of B in the liquid ($D_B$ much smaller than $R \cdot k'$ at the droplet edges), the velocity of a droplet should be independent from the droplet size; (ii) when the droplet motion is limited by attachment/detachment mechanisms ($R \cdot k'$ much smaller than $D_B$), the droplet velocity should increase with the droplet size; (iii) for very large droplets (large $R$ in equation 4) the motion should always tend towards a diffusion-limited mechanism. Notice that the velocity changes linearly with the thermal gradient, but depends exponentially from the average temperature (exponential dependence on the temperature of $D_B$ and $k'$). We have observed that at low temperature the velocity of Au-Ge microdroplets increases linearly with their radius. Within our model, this behavior can be explained by a motion limited by the detachment and attachment of Ge from the droplet edges. We can thus neglect diffusion and evaluate $k'$ at $T_0 = 913K$. Using the data from Fig. 4 for droplet velocity and radius, we determine the linear relation $v = 0.49R$. With $dT/dx = 1.58 \cdot 10^{-5}$ K/nm, values from the phase diagram $\omega n_{eq} = 0.5$ and $\omega dn/dT = 10^{-3}$, and taking $Q = 3$ eV (a binding energy typically on the order of a few eV), we obtain $k' = 7.2 \cdot 10^5$ nm/s.

At high temperatures the Au-Ge droplet velocity is independent from droplet size, suggesting that the motion is no more limited by Ge detachment/attachment, but by diffusion of Ge inside the droplets. The change of regime takes place between 1000 K and 1180 K, with the slope through the data points in figure 4 changing between 1 and 0. The motion of large droplets is probably diffusion limited also for

temperatures lower than 1180 K (L dependence in equation 4). At $T_0 = 1180K$, we estimate $D_B$. We use the velocity of the green data points in figure 4, $dT/dx = 2.5 \cdot 10^{-5}$ K/nm, from the phase diagram we obtain $\omega n_{eq} = 0.85$ and $\omega dn/dT = 3.4 \cdot 10^{-3}$, and we obtain $D_B = 3.5 \cdot 10^9$ nm$^2$/s. This value can be compared with typical diffusion coefficients in liquid metals ($10^9$ nm$^2$/s [31]).

The regime transition from attachment/detachment limited to diffusion limited displacement, observed when the temperature is increased, is probably due to the different temperature dependence of $D_B$ and $k'$. $k'$ depends on the energy needed to detach atoms from an interface. The energy necessary to detach a kink atom is probably the most relevant one [32], and thus we suggest $k' \propto \exp(-E_{kink}/(kT))$. As kink energies are larger than diffusion energies [33], $k'$ decreases faster than $D_B$ when the temperature decreases, and so at low temperature we find that the droplet displacement is limited by attachment/detachment. However, the ratio between $D_B$ and $k'$ decreases when the temperature increases, which explains the change of regime of the droplets from attachment/detachment limited to diffusion limited.

We now estimate the value of R for which droplet motion should be diffusion-limited even at low temperatures. Using an activation energy for the diffusion of Ge in the Au-Ge liquid alloy of 0.5 eV (taken to be equal to the diffusion energy of Si in Au-Si alloys [34]), and the previously calculated value of $D_B$ at 1180 K, we estimate $D_0=5.19 \cdot 10^{11}$ nm$^2$s$^{-1}$. Therefore, at the temperature of 913 K, where we have estimated $k' = 7.2 \cdot 10^5$, $D_B = 9.04 \cdot 10^8$ nm$^2$s$^{-1}$. We consider $Rk'$ much larger than $D_B$ if the ratio between the two values is larger than 100. Therefore, the droplet radius above which the motion is always diffusion-limited is $R > \frac{D_B}{0.01k'} = 125\mu m$.

Our model also explains the results of several other authors that observe, in systems different from Au-Ge [10, 35], droplets moving faster when their size increases. Furthermore, notice that, when the term $\frac{n_{eq}(0)Q}{kT_0^2}$ is negligible, in the diffusion limited regime, equation 4 is similar to the phenomenological expression suggested by Cline and Anthony [35] and used by other authors after them [10, 12]. However, for Au-Ge, that term is much larger than $\frac{dn}{dT}$ and cannot be neglected [36]. A critique of the Cline and Anthony model was already advanced by Poelsema et al., who studied the thermomigration of eutectic droplets in the Pt-Ge system [12], and concluded that droplet motion is determined by the rate of dissolution of the substrate wetting layer at the advancing side of the droplets. Our model incorporates this interpretation of motion limited by attachment/detachment and further predicts that the mechanism controlling droplet motion changes with parameters such as droplet radius and temperature.

We now estimate the effective driving force underlying the droplet displacement. The velocity of an object can be written as the product of a mobility ($\mu$) and a force ($F$), $v = \mu \cdot F$. Using the approximated version of equation 4, multiplying the expression by $kT/(kT)$ we can separate an effective mobility
$\mu$ from a thermo-chemical driving force $F$:

$$ 
\begin{aligned}
\mu &= \frac{1}{1 - \omega n_{eq}(0)} \frac{1}{kT} \frac{1}{\frac{1}{D_B} + \frac{1}{R \cdot k'}} \\
F &= \omega \cdot kT \frac{dT}{dx} \left( \frac{dn}{dT} + \frac{n_{eq}(0)Q}{kT_0^2} \right)
\end{aligned} 
\quad (5) $$

With the values used in the estimation of $D_B$ and $k'$, at 1180 K, in the diffusion limited regime, we find $\mu = 2.3 \cdot 10^{11} \text{nm}^2 \text{s}^{-1} \text{eV}^{-1}$ and $F = 6.3 \cdot 10^{-8} \text{eV} \text{nm}^{-1}$. In the attachment/detachment limited regime, at 913 K, we find $\mu = 4.3 \cdot 10^{10} \text{nm}^2 \text{s}^{-1} \text{eV}^{-1}$ and $6.7 \cdot 10^{10} \text{nm}^2 \text{s}^{-1} \text{eV}^{-1}$ for droplets of 2350 nm and 3650 nm respectively, while $F = 2.7 \cdot 10^{-8} \text{eV} \text{nm}^{-1}$. The force exerted on the droplets is of the same order of magnitude as that obtained for the displacement of ad vacancy islands on Si(111) under a thermal gradient [37]. This force is much smaller than the one reported by Ternes et al. for mechanically displacing an adatom with the tip of an atomic force microscope at low temperatures ($\approx 1.3 \text{ eV/nm}$, [38]). At low temperatures, adatoms diffuse very little and are almost immobile; however, at the temperatures used in our experiments, adatoms exhibit high diffusivity. In this regime, thermomigration introduces a small bias into an otherwise random diffusion process, and this bias is sufficient to induce droplet motion.

Droplet motion is only possible when there is some kind of anisotropy between two opposite edges of the droplet. Using a temperature gradient is an effective way to obtain a strong driving force for droplet motion. Even stronger forces can be achieved with electric fields, where the displacement of objects is due to electromigration [14, 17, 39–42]. However, moving droplets have also been observed in several other contexts, where the underlying driving forces are weak. In these cases, droplets typically move only when they interact with the topmost substrate layers. For instance, Benter et al. (2023) studied InAs surfaces with lithographically deposited AlPd patches and showed that after annealing, In droplets nucleated everywhere except in a depleted zone around the metallic patch [43]. This demonstrates how compositional modifications of the substrate can bias droplet positioning via preferential nucleation. Da Silva et al. (2017) investigated Au droplets on GaAs and observed their spontaneous motion, later exploiting the phenomenon to grow nanowires inside the droplet tracks [44]. They proposed that the driving force may arise either from surface roughness—droplets moving to smooth the substrate and reduce surface energy—or from chemical composition gradients, since Ga and As evaporate at different rates. Tersoff et al. (2009) demonstrated that Ga droplets on GaAs surfaces can move spontaneously: a small fluctuation in droplet position uncovers a substrate region locally enriched

in As [45]. The resulting asymmetry in chemical potential (or equivalently surface free energy) across the droplet then drives its motion. Hilner et al. (2009) reported yet another mechanism in GaP, where sublimation of steps beneath Ga droplets governs their motion: droplets climb step bunches in one regime, while in another they follow evaporating steps, leaving behind flat terraces [46]. Droplets far from chemical equilibrium can also move to react with the substrate surface [47–49], and when substrate dissolution is possible only at defects such as surface steps, their displacement can be spectacular [13, 50, 51]. Together, these studies highlight (i) the diversity of mechanisms that can lead to droplet motion, and (ii) the generality of droplet dynamics across material systems. In contrast to these cases, our present work demonstrates that an externally imposed temperature gradient, with defined strength and direction, provides a means to drive droplet motion, enabling control of both velocity and trajectory.

## V. SUMMARY

In this work we have studied the thermomigration of Au-Ge microdroplets on Ge(111). The droplets move towards the hot side of the sample. The droplet velocity increases linearly with the droplet size at low temperatures, while it does not depend on size at higher temperatures. We have developed a model to explain the mechanism of droplet motion under thermal gradients. Within our model, the change of regime of the dependence between the droplet velocity and the droplet size implies a change in the mechanism limiting the droplet displacement, from attachment/detachment limited (of Ge from the droplet edges) to diffusion limited. When the droplets displace by thermomigration, they dissolve Ge from the substrate and leave tracks with flat regions that display atomically-flat terraces with sizes of the order of 10 microns. These terraces are large defect-free regions that can be used for further investigations in fields like epitaxy and thin-film deposition.

## ACKNOWLEDGMENTS

This work has been supported by the ANR grant Ther-motweez (ANR-22-CE09-0009-01).

## DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.

[1] C. Ludwig, Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse 20, (1856).

[2] C. Soret, Etat d'équilibre des dissolutions dont deux parties sont portées à des températures différentes (On the equilibrium

state of solutions with two parts kept at different temperatures), Archives des Sciences Physiques et Naturelles de Geneve 2, 48 (1879).
[3] M. A. Rahman and M. Z. Saghir, Thermodiffusion or soret effect: Historical review, International Journal of Heat and Mass Transfer **73**, 693 (2014).

[4] W. Köhler and K. I. Morozov, The soret effect in liquid mixtures – a review, Journal of Non-Equilibrium Thermodynamics **41**, 151 (2016).

[5] L. Talbot, Thermophoresis – a review, Progress in Astronautics and Aeronautics **74**, 467 (1981).

[6] T. Xu, Y. Xu, J. Wang, H. Lu, W. Liu, and J. Wang, Sustainable self-cleaning evaporator for long-term solar desalination using gradient structure tailored hydrogel, Chemical Engineering Journal **415**, 128893 (2021).

[7] N. Bjelobrk, H.-L. Girard, S. B. Subramanyam, H.-M. Kwon, D. Quéré, and K. K. Varanasi, Thermocapillary motion on lubricant-impregnated surfaces, Physical Review Fluids **1**, 063902 (2016).

[8] R. S. Wagner and W. C. Ellis, Vapor-liquid-solid mechanism of single crystal growth, Applied Physics Letters **4**, 89 (1964).

[9] T. Ichinokawa, C. Haginoya, D. Inoue, H. Itoh, and J. Kirschner, Electro- and Thermomigration of metallic islands on Si(100) surface, Japanese Journal of Applied Physics **32**, 1379 (1993).

[10] W.-C. Yang, H. Ade, and R. J. Nemanich, Stability and dynamics of Pt-Si liquid microdroplets on Si(001), Phys. Rev. B **69**, 045421 (2004).

[11] B. Stenger, A. Dorsett, J. Miller, E. Russell, C. Gabris, and S. Chiang, Growth and motion of liquid alloy droplets of au on ge (1 1 0), Ultramicroscopy **183**, 72 (2017).

[12] B. Poelsema, Z. Zhang, J. S. Solomon, H. J. W. Zandvliet, and A. van Houselt, Shining new light on the motion of eutectic droplets across surfaces: A PEEM study of PtGe on Ge(110), Physical Review Materials **5**, 125602 (2021).

[13] S. Curiotto, F. Leroy, F. Cheynis, and P. Müller, Surface-dependent scenarios for dissolution-driven motion of growing droplets, Scientific Reports **7**, 902 (2017).

[14] F. Leroy, A. El-Barraj, F. Cheynis, P. Müller, and S. Curiotto, Atomic Transport in Au-Ge Droplets: Brownian and Electromigration Dynamics, Physical Review Letters **123**, 176101 (2019).

[15] A. El-Barraj, S. Curiotto, F. Cheynis, P. Müller, and F. Leroy, Dynamics of Au-Ge liquid droplets on Ge(111) terraces: Nucleation, growth and dynamic coalescence, Applied Surface Science **509**, 144667 (2020).

[16] A. El-Barraj, *Growth and Electro-Thermomigration on semiconductor surfaces by Low Energy Electron Microscopy*, Ph.D. thesis, Aix-Marseille University, Campus de Luminy, 13288 Marseille (2019).

[17] F. Leroy, A. El-Barraj, F. Cheynis, P. Müller, and S. Curiotto, Electromigration of Au on Ge(111): Adatom and island dynamics, Physical Review B **106**, 115402 (2022).

[18] S. Curiotto, N. Combe, P. Müller, A. El Barraj, N. Abu Dahech, F. Cheynis, O. Pierre-Louis, and F. Leroy, Surface thermomigration of 2d voids, Applied Physics Letters **125**, 121601 (2024).

[19] C. Wang, B. Müller, and K. Hofmann, Epitaxy of atomically flat caf2 films on si (1 1 1) substrates, Thin Solid Films **410**, 72 (2002).

[20] F. Allen, Emissivity at 0.65 micron of silicon and germanium at high temperatures, Journal of Applied Physics **28**, 1510 (1957).

[21] Fiji contributors, Trackmate - imagej (2023), accessed: 2025-05-15.

[22] P. Y. Chevalier, A Thermodynamic Evaluation of the Au-Ge and Au-Si systems, Thermochimica Acta **141**, 217 (1989).

[23] G. Le Lay, Surface Science **132**, 169 (1983).

[24] J. A. Giacomo, C. H. Mullet, and S. Chiang, Growth, phase transition, and island motion of Au on Ge(111), The Journal of Chemical Physics **155**, 054701 (2021).

[25] The error of a function $\Delta f(x)$, given an error $\Delta x$ of the input argument, is approximately $\Delta f(x) \approx \frac{df}{dx} \cdot \Delta x$. Using $\log_{10}(x) = \frac{\ln x}{\ln 10}$, $\Delta \log_{10}(x) = 0.434 \cdot \frac{\Delta x}{x}$. In our case $x$ is the droplet velocity.

[26] D. Lee and J. Blakely, Formation and stability of large step-free areas on Si(001) and Si(111), Surface Science **445**, 32 (2000).

[27] S. Tanaka, C. C. Umbach, J. M. Blakely, R. M. Tromp, and M. Mankos, Fabrication of arrays of large step free regions on Si(001), Applied Physics Letters **69**, 1235 (1996).

[28] A. Roux and N. Combe, Thermal mass transport mechanism of an adatom on a crystalline surface, Physical review B **108**, 115410 (2023).

[29] A. Roux and N. Combe, Erratum: Thermal mass transport mechanism of an adatom on a crystalline surface [Phys. Rev. B 108, 115410 (2023)], Physical Review B **110**, 119902 (2024).

[30] F. Hausser, P. Kuhn, J. Krug, and A. Voigt, Morphological stability of electromigration-driven vacancy islands, Physical Review E **75**, 046210 (2007).

[31] B. N. Bhat, NASA technical report: Solute Diffusion in Liquid Metals, Tech. Rep. (National Aeronautics and Space Administration, 1973).

[32] B. Mutaftschiev, The atomistic nature of crystal growth, Vol. 43 (Springer, 2001).

[33] G. Antczak and G. Ehrlich, Surface diffusion: metals, metal atoms, and clusters (Cambridge University Press, 2010).

[34] Y. Wang, A. Santana, and W. Cai, Atomistic mechanisms of orientation and temperature dependence in gold-catalyzed silicon growth, Journal of Applied Physics **122**, 085106 (2017).

[35] H. E. Cline and T. R. Anthony, High-speed droplet migration in silicon, Journal of Applied Physics **47**, 2325 (1976).

[36] at $T_0 = 913$ K, we measure from the phase diagram $\omega dn/dT = 10^{-3} \text{ K}^{-1}$. $\omega n_{eq}=0.5$ and supposing $Q=3$ eV, we find $\frac{\omega n_{eq}Q}{kT_0^2} = 2 \cdot 10^{-2}$. $dn/dT$ is twenty times smaller than $\frac{n_{eq}Q}{kT_0^2}$. With a similar calculation at $T_0 = 1180$ K we find that the difference between the two terms narrows but $dn/dT$ is still six times smaller than $\frac{n_{eq}Q}{kT_0^2}$.

[37] F. Leroy, A. El-Barraj, F. Cheynis, P. Müller, and S. Curiotto, Determination of the thermomigration force on adatoms, Physical Review Letters **131**, 116202 (2023).

[38] M. Ternes, C. P. Lutz, C. F. Hrjibehedin, F. J. Giessibl, and A. J. Heinrich, The force needed to move an atom on a surface, Science **319**, 1066 (2008).

[39] F. Leroy, A. El-Barraj, F. Cheynis, P. Müller, and S. Curiotto, Electric forces on a confined advacancy island, Physical Review B **102**, 235412 (2020).

[40] S. Curiotto, P. Müller, A. El-Barraj, F. Cheynis, O. Pierre-Louis, and F. Leroy, 2D nanostructure motion on anisotropic surfaces controlled by electromigration, Applied Surface Science **469**, 463 (2019).

[41] S. Curiotto, P. Müller, F. Cheynis, and F. Leroy, 2D Manipulation of Nanoobjects by Perpendicular Electric Fields: Implications for Nanofabrication, ACS Applied Nano Materials **3**, 1118 (2020).

[42] S. Curiotto, P. Müller, F. Cheynis, and F. Leroy, Kinetic Monte Carlo simulations of the diffusion and shape evolution of single-layer clusters on a hexagonal lattice with and without external force, Applied Surface Science **552**, 149454 (2021).

[43] S. Benter, A. Jönsson, J. Johansson, L. Zhu, E. Golias, L.-E. Wernersson, and A. Mikkelsen, Geometric control of diffus-
ing elements on inas semiconductor surfaces via metal contacts, Nature Communications **14**, 4541 (2023).

[44] B. C. Da Silva, D. Soares Oliveira, F. Likawa, O. D. D. Couto, J. Bettini, L. F. Zagonel, and M. Alonso Cotta, Exploring au droplet motion in nanowire growth: a simple route towards asymmetric gap morphologies, Nano Letters **17**, 7274 (2017).

[45] J. Tersoff, D. E. Jesson, and W. X. Tang, Running Droplets of Gallium from Evaporation of Gallium Arsenide, Science **324**, 236 (2009).

[46] E. Hilner, A. A. Zakharov, K. Schulte, P. Kratzer, J. N. Ander-
sen, E. Lundgren, and A. Mikkelsen, Ordering of the nanoscale
step morphology as a mechanism for droplet self-propulsion,
Nano Letters **9**, 2710 (2009).

[47] A. K. Schmid, N. C. Bartelt, and R. Q. Hwang, Alloying at Sur-
faces by the Migration of Reactive Two-Dimensional Islands,

Science **290**, 1561 (2000).

[48] F. Leroy, Y. Saito, F. Cheynis, E. Bussmann, O. Pierre Louis, and P. Müller, Nonequilibrium diffusion of reactive solid islands, Physical Review B **89**, 235406 (2014).

[49] F. Leroy, Y. Saito, S. Curiotto, F. Cheynis, O. Pierre-louis, and P. Müller, Shape transition in nano-pits after solid-phase etching of SiO2 by Si islands, Applied Physics Letters **106**, 191601 (2015).

[50] W. Swiech, E. Bauer, and M. Mundschau, Surface Science **253**, 283 (1991).

[51] S. Curiotto, F. Leroy, F. Cheynis, and P. Müller, Surface Science **632**, 1 (2015).