ATOMISTIC-CONTINUUM MODELING OF
LASER-INDUCED PHASE TRANSITIONS IN
SILICON: MELTING, ABLATION, SOLIDIFICATION,
AND AMORPHIZATION

Dmitry S. Ivanov, Tatiana Itina

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

To cite this version:

Dmitry S. Ivanov, Tatiana Itina. ATOMISTIC-CONTINUUM MODELING OF LASER-INDUCED PHASE TRANSITIONS IN SILICON: MELTING, ABLATION, SOLIDIFICATION, AND AMORPHIZATION. Fundamentals of Laser-assisted Micro and Nanotechnologies, FLAMN-2025, ITMO, Jun 2025, Saint-Peterbourg, Russia. ujm-05133704

HAL Id: ujm-05133704

https://ujm.hal.science/ujm-05133704v1

Submitted on 27 Jun 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Atomistic-Continuum Modeling of Laser-Induced Phase
Transitions in Silicon: Melting, Ablation, Solidification,
and Amorphization

Dmitry S. Ivanov and Tatiana E. Itina

Université Jean Monnet Saint-Etienne, Laboratoire Hubert Curien, Saint-
Etienne, France

<figure><img src="image_3.png" /></figure>
Nanostructuring of Si with laser pulses

<figure><img src="image_4.png" /></figure>

Liangliang Yang et al, Nanomaterials **9**, 1789 (2019)

<figure><img src="image_5.png" /></figure>
Technological advances in amorphization of Si

CMOS Technology:

<figure><img src="image_6.png" /></figure>

(a)

TFT Technology

<figure><img src="image_7.png" /></figure>

(b)

Vishwas Acharya et al, *Mater. Res. Express* **10**, 082002 (2023)

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

A. Engel, Th. Pflug, M. Olbrich, and A. Horn

<figure><img src="image_11.png" /></figure>

Laserinstitut
Hochschule Mittweida

<figure><img src="image_12.png" /></figure>

David Grojo et al, *Laser Photonics Rev.* **18**, 2301327 (2024)

<figure><img src="image_13.png" /></figure>

Feiyu Xu et al, *Chemical Engineering Journal* **430**, 133140 (2022)
# The Model: MD and nTTM

## nTTM:

Temperature dynamics of free carriers

$$ 
\left\{ 
\begin{aligned}
\frac{\partial n}{\partial t} &= \frac{\alpha I}{\hbar \omega} + \frac{\beta I^2}{2 \hbar \omega} + \theta n - \gamma n^3 - \mathrm{div}(\vec{J}) \\
\frac{\partial U}{\partial t} &= (\alpha + \theta n) I + \beta I^2 - \mathrm{div}(\vec{W}) - G(T_e - T_a) \\
C_a \frac{\partial T_a}{\partial t} &= \mathrm{div}(k_L \nabla T_L) + G(T_e - T_a)
\end{aligned}
\right. 
$$

<figure><img src="image_14.png" /></figure>

J.K. Chen, D.Y. Tzou, and J.E. Beraun, Int. J. of Heat and Mass Transfer 48, 501-509 (2005)

<figure><img src="image_15.png" /></figure>

## MD:

Non equilibrium kinetics at atomic level

$$ m_i \frac{d^2 \vec{r}_i}{dt^2} = \vec{F}_i = - \sum_{j, j \neq i} \nabla_{\vec{r}_j} U(|\vec{r}_i - \vec{r}_j|); i = 1, \dots, N $$

<figure><img src="image_16.png" /></figure>

Laser-assised direct imprinting in Si by 308 nm, 20 ns, 1.6 J/cm², S. Y. Chou, C. Keimel, and J. Gu, Nature 417, 835 (2002)
The combined MD-nTTM model

$$
\left\{
\begin{aligned}
\frac{dI(z,t)}{dz} &= -(\alpha + \alpha_{FCA}(\varepsilon(N_e)))I(z,t) - \beta I^2(z,t) \\
\frac{\partial n_e}{\partial t} &= \frac{\alpha I(z,t)}{h\nu} + \frac{\beta I^2(z,t)}{2h\nu} + \delta n_e - \gamma N_e^3 - \nabla \cdot \vec{J}(n_e, T_e, E_g) \\
\frac{\partial U_e(N_e, T_e, E_g)}{\partial t} &= (\alpha + \alpha_{FCA}(\varepsilon(N_e)))I(z,t) + \beta I^2(z,t) \\
&\quad - \nabla \cdot \vec{W}(N_e, T_e, E_g) - \frac{C_{e-h}(N_e, T_e, E_g)}{\tau_e}(T_e - T_l) \\
C_l \frac{\partial T_l}{\partial t} &= \vec{\nabla} \cdot (\kappa \vec{\nabla} T_l) + \frac{C_{e-h}(N_e, T_e, E_g)}{\tau_e}(T_e - T_l) \rightarrow m_i \frac{d\vec{r}_i}{dt^2} = -\sum_{j=1,N} \vec{\nabla} U(\vec{r}_{ij}, T_e) + \xi m_i \vec{V}_i^T
\end{aligned}
\right.
$$

1. TTM + additional eq for the
density of carriers

2. Implicit Crank-Nicolson
algorithm needs parallel.

3. Drude-Lorenz model for
reflectivity and free car abs

nTTM description of free carriers

Tersoff Modified Potential

<figure><img src="image_17.png" /></figure>

nTTM description of lattice

Combined MD + nTTM

<figure><img src="image_18.png" /></figure>
$F = 0.31 \text{ J/cm}^2$

Si <001>
100 fs
800 nm

<figure><img src="image_19.png" /></figure>

25 nm

$CSP_{thr} = 0.97$

CSP

<figure><img src="image_20.png" /></figure>

0.99

0.98

0.97

0.96

0.95

0.94

0.93

0.92

0.91

0.9

<figure><img src="image_21.png" /></figure>

10 ps

<figure><img src="image_22.png" /></figure>

20 ps

<figure><img src="image_23.png" /></figure>

30 ps

<figure><img src="image_24.png" /></figure>

40 ps

<figure><img src="image_25.png" /></figure>

50 ps

<figure><img src="image_26.png" /></figure>

70 ps

<figure><img src="image_27.png" /></figure>

100 ps

<figure><img src="image_28.png" /></figure>

150 ps
Central Symmetry Parameter Dynamics

Laser pulse

<figure><img src="image_29.png" /></figure>

CSP = 0.97 threshold value

1. Homogeneous mechanism in Si is **dominant**

2. Heterogeneous melting mechanism is observed within **few nm only**.
# Melting Kinetics of Si vs Metals

Laser Pulse

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /><figcaption>Ni 1 ps pulse, fluence 129 mJ/cm², 100 nm Ni film</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>Si 270 fs pulse, fluence 400 mJ/cm², 1000 nm Si film</figcaption></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

Ni: face
center cubic

<figure><img src="image_35.png" /></figure>

1. Well distinguished solid-liquid interface propagation

2. T/P interplay results in both heterogeneous and homogeneous melting mechanism

3. The melting start upon the **drop** in pressure

Si: Diamon cubic

<figure><img src="image_36.png" /></figure>

1. Rapid/instant advancement of the melting front

2. T/P interplay results in mainly homogeneous melting mechanism

3. The melting starts upon the **rise** of pressure
Clapeyron equation for metals and Si, the crystal structure effect

<figure><img src="image_37.png" /></figure>

$$ \begin{array}{l} \text{At P = 0 Pa} \\ \text{T}_m = 1439 \text{ K} \end{array} $$

Face center cubic structure: $V_{\text{melt}} = 5\%$

Volume expands results in internal stresses increase

$$ \left(\frac{dT}{dP}\right)_m = \frac{\Delta V_m}{\Delta S_m} = \frac{T_m \Delta V_m}{\Delta H_m} $$

1. For metals **dT/dP > 0**, so increase of P reinforce the crystal stability against of melting.

2. For Si **dT/dP < 0**, so the increase of pressure destabilize the crystal structure and facilitates the onset of melting.

<figure><img src="image_38.png" /></figure>

$$ \begin{array}{l} \text{At P = 0 Pa} \\ \text{T}_m = 1696 \text{ K} \end{array} $$

Open diamond crystal structure: $V_{\text{melt}} = -7.5\%$

Volume contracts results in internal stresses drop
Melting mechanism of Si vs Metals

Laser Pulse

<figure><img src="image_39.png" /><figcaption>Ni
1 ps pulse, fluence 129 mJ/cm², 100 nm Ni film</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>Si
270 fs pulse, fluence 400 mJ/cm², 1000 nm Si film</figcaption></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

Ni: face
center cubic

<figure><img src="image_43.png" /></figure>

1. At the fluences around melting threshold the solid-liquid transition via **heterogeneous mechanism** is more energetically favorable.

Si: Diamon cubic

<figure><img src="image_44.png" /></figure>

2. At the fluences around melting threshold, the **homogeneous melting** mechanism plays a determinant role and makes formation of the solid-liquid interface at a significant depth instantly.

3. **Spallation** mechanism of ablation **is suppressed** in Si
Depth of melting vs. fluence

1. Melting threshold 0.3 J/cm²

2. There some experimental results matching our measurements

3. Two melting mechanisms at low fluence

4. Heterogeneous mechanism is suppressed at high fluences

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>
Theory vs Experiment comparison

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

David Grojo et al, Laser Photonics Rev. 18, 2301327 (2024)

$$d_{\text{melt}} = A \ln(F/F_{\text{th}}) + B$$

A = 600nm and B = -220nm
# Melting dynamics in <001> and <111> crystal orientation and depth of melting

<figure><img src="image_49.png" /></figure>

<100>

<figure><img src="image_50.png" /></figure>

30 ps

<figure><img src="image_51.png" /></figure>

CSP 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

<figure><img src="image_52.png" /></figure>

<111>

<figure><img src="image_53.png" /></figure>

40 nm

<figure><img src="image_54.png" /></figure>

50 ps

<figure><img src="image_55.png" /></figure>

100 ps

<figure><img src="image_56.png" /></figure>

Z
X
Y

<figure><img src="image_57.png" /></figure>

30 ps

<figure><img src="image_58.png" /></figure>

50 ps

<figure><img src="image_59.png" /></figure>

100 ps

Solid > CSP<sub>thr</sub> = 0.97 > Liquid

1. Two identical samples but <001> and <111>

2. Heterogeneous vs homogeneous melting mechanism

3. <001> results 26% more melt

4. Speed of sound in <111> 8% more

5. Optical properties can be refined
<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /></figure>

<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

<figure><img src="image_64.png" /></figure>

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

position, nm

CSP

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
Central Symmetry Parameter Dynamics

<figure><img src="image_67.png" /></figure>

CSP = 0.97 threshold value

1. The effect of crystal structure makes the homogeneous mechanism of melting in Si **dominant and the only** mechanism at fluences above the melting threshold.

2. Due to instant relaxation of the laser-induced stresses at the surface, the heterogeneous melting mechanism is observed within **few presurface nm only**.

<figure><img src="image_68.png" /></figure>

Due to ~ 10% higher speed of sound in <111> the melting threshold is lower, but the optical properties during the laser light absorption must be considered carefully.
Cooling rate due to nTTM and MD

<figure><img src="image_69.png" /><figcaption>100 fs pulse, fluence 0.16 J/cm² at 800nm</figcaption></figure>

The cooling rate given by nTTM model ~ 2.5*10¹² K/s

<figure><img src="image_70.png" /><figcaption>100 fs pulse, fluence 0.18 J/cm² at 800nm</figcaption></figure>

The necessary cooling rate for amorphization ~ 3.5*10¹³ K/s

? Working just above the melting threshold warrant the surface melting ~ (10-20)nm with fastest cooling rate, which makes the amorphization possible ?

<figure><img src="image_71.png" /></figure>
$F = 0.31 \text{ J/cm}^2$

Si <001>
100 fs
800 nm

<figure><img src="image_72.png" /></figure>

CSP < 0.97

CSP

<figure><img src="image_73.png" /></figure>

0.99

0.98

0.97

0.96

0.95

0.94

0.93

0.92

0.91

0.9

<figure><img src="image_74.png" /></figure>

10 ps

<figure><img src="image_75.png" /></figure>

20 ps

<figure><img src="image_76.png" /></figure>

30 ps

<figure><img src="image_77.png" /></figure>

40 ps

<figure><img src="image_78.png" /></figure>

50 ps

<figure><img src="image_79.png" /></figure>

70 ps

<figure><img src="image_80.png" /></figure>

100 ps

<figure><img src="image_81.png" /></figure>

150 ps
T < T<sub>m</sub> and NO mechanical
equilibrium!

F = 0.31 J/cm²

Si <001>
100 fs
800 nm

<figure><img src="image_82.png" /></figure>

CSP < 0.97

CSP

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /></figure>

<figure><img src="image_86.png" /></figure>

<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /></figure>

<figure><img src="image_89.png" /></figure>

Melting dynamics

1. The melting process is over by 150 ps

2. At the time of 300 ps the slow process of recrystallization can be observed

3. The surface oscillations are due to thermal expansion and melting contraction interplay

4. At the time of 250 - 450 ps the strong temperature gradient still exist providing the cooling

5. The carriers' density gradient attributes to the total cooling rate
Melting Dynamics of 1000nm Si film at 800 nm, 100 fs, $F_{inc}$ = 0.31 J/cm², and $F_{abs}$ = 0.16 J/cm²

<figure><img src="image_90.png" /></figure>

a) CSP dynamics emphasizes the homogeneous melting mechanism

<figure><img src="image_91.png" /></figure>

<figure><img src="image_92.png" /></figure>

b) The carriers' density drops 15 times as compared their value upon the pulse deposition, but the gradient is also significant

c) Lattice temperature gradient provides the cooling rate due to phonon thermal conductivity

<figure><img src="image_93.png" /></figure>

d) Free carriers' temperature gradient assists the additional cooling mechanism
**Metals:** Face center cubic structure: $V_{\text{melt}} = 5\%$

Solid-liquid interface advancement is favored

<figure><img src="image_94.png" /></figure>

**Silicon:** Open diamond crystal structure: $V_{\text{melt}} = -7.5\%$

Solid-liquid interface advancement is hampered

<figure><img src="image_95.png" /></figure>

**Proposed mechanism of amorphization of Si due to laser cooling**

<figure><img src="image_96.png" /></figure>
Face center cubic structure: $V_{\text{melt}} = 5\%$

Solid-liquid interface advancement is favored

<figure><img src="image_97.png" /></figure>

Open diamond crystal structure: $V_{\text{melt}} = -7.5\%$

Solid-liquid interface advancement is hampered

<figure><img src="image_98.png" /></figure>

Proposed mechanism of
amorphization of Si due
to laser cooling

<figure><img src="image_99.png" /></figure>

$$G(P,V) = U + PV - TS \Leftrightarrow \Delta G = \Delta H - T\Delta S$$

$$\text{Also diffusion coefficient } D = f(T_{\text{ph}})$$
Conclusion

> The atomistic-continuum MD-nTTM model is proposed for laser-Si interactions

> Homogeneous melting mechanism was found dominant

> The melting depth vs fluence was related to the experimental measurements

> The effect of crystal orientation was investigated

> The amorphization mechanism is attributed with temperature gradient in Tph, Te-h, and Ne-h in the absence of mechanical (acoustic) equilibrium