Generating gradient germanium nanostructures by shock-induced amorphization and crystallization

Significance

Amorphization and nanocrystallization are two powerful methods to tailor material properties by altering their microstructure without changing the overall chemistry. Using powerful laser-driven shocks, we demonstrate that amorphization and nanocrystallization can be achieved within a time scale that is considerably shorter than other conventional techniques. Our results provide compelling insights into pressure/shear amorphization and propose a route to fabricate gradient semiconducting nanostructures using lasers. Additionally, shear-driven amorphization is demonstrated as the dominant deformation mechanism in this extreme regime.

Abstract

Gradient nanostructures are attracting considerable interest due to their potential to obtain superior structural and functional properties of materials. Applying powerful laser-driven shocks (stresses of up to one-third million atmospheres, or 33 gigapascals) to germanium, we report here a complex gradient nanostructure consisting of, near the surface, nanocrystals with high density of nanotwins. Beyond there, the structure exhibits arrays of amorphous bands which are preceded by planar defects such as stacking faults generated by partial dislocations. At a lower shock stress, the surface region of the recovered target is completely amorphous. We propose that germanium undergoes amorphization above a threshold stress and that the deformation-generated heat leads to nanocrystallization. These experiments are corroborated by molecular dynamics simulations which show that supersonic partial dislocation bursts play a role in triggering the crystalline-to-amorphous transition.

Amorphous and gradient nanostructures are drawing intense attention due to their superior functional and mechanical properties (1, 2). Since they are thermodynamically metastable, amorphous materials can transform into nanocrystalline ones if appropriate treatments are applied (3). One of the most common methods to achieve amorphization is to quench a liquid at ultrafast cooling rates, which is extremely difficult for most pure elements (4). Alternatively, it has been shown that application of pressure leads to amorphization of materials whose melting point displays a negative Clapeyron slope (dT/dP < 0) (5–9); germanium (Ge) falls into this category (10). However, instead of pressure-induced amorphization, numerous studies, under both static (11, 12) and dynamic conditions (13–15), have shown that Ge undergoes a polymorphic transition at elevated pressures. Consequently, amorphization was not unambiguously identified in Ge until Clarke et al. (16) observed the indentation-induced crystalline-to-amorphous transition. More recently, a high-speed nanodroplet test also showed surface amorphization of Ge in an extremely localized manner (17).

Despite being widely studied, the underlying microstructural mechanisms of pressure-induced amorphization remain vague. This is due to the notorious brittleness of germanium at room temperature which renders its recovery from pressurization extremely challenging. The deposition of high-power pulsed-laser energy onto a millimeter-scale target generates transient states of extreme stresses that promptly build up and decay rapidly as the pulse propagates. The short duration of the stress pulse and impedance-matched encapsulation preserves the integrity of the target by suppressing the full development of cracks and enables postshock microstructure characterization. Using this methodology, we have previously reported shock-induced amorphization in silicon (18) and boron carbide (19). Before that, Jeanloz et al. (20) discovered this phenomenon in olivine (iron/magnesium silicate) subjected to shock compression. Amorphous bands formed on compression of crystalline silica were also observed in diamond anvil cell experiments (21, 22), where the role of stress in promoting the transition was also pointed out. The important, albeit often ignored, role of shear in pressure-induced phase transition (21–24) is clearly evidenced by the directional nature of the amorphous bands.

Results

We performed the experiments at the Omega Laser Facility, Laboratory of Laser Energetics, University of Rochester, using a pulsed laser with a nominal square pulse shape (wavelength = 352 nm; laser duration = 1 ns). The nominal laser energies were E_laser = 20 ∼ 100 J, resulting in intensities of 0.2 ∼ 1.1 TW/cm². Fig. 1A shows schematically the shock-recovery assembly. The high-density laser energy vaporizes the polystyrene (CH) ablator, which drives a compressive wave that eventually propagates into the [001] monocrystalline Ge target. For the 100-J experiment, the peak shock stress, σ₃₃ $\approx$ 33 GPa, can be inferred indirectly from the particle velocity (U_p) measurement by velocity interferometer system for any reflector (VISAR) (Fig. 1 B–D) and impedance matching (Fig. 1E), as detailed in Materials and Methods. Note that the shock wave decays rapidly when traveling across the sample thickness, as evidenced by the significant drop of particle velocity at the rear surface (Ge/LiF interface) of the target (Fig. 1D). The time dependence of shock-wave propagation and decay of the longitudinal shock stress as function of depth can be simulated by 1D hydrodynamic simulation, as shown in Fig. S1.

Fig. 1.

Laser-driven, shock-recovery experimental setup, velocimetry measurement, and determination of shock stresses. (A) Shock-recovery assembly with target package (along the laser path: 20-µm CH ablator, 50-µm Al pusher, and Ø3 × 3-mm Ge crystal) encapsulated in Ti and backed by momentum trap (Ti). The assembly is mounted in a recovery tube. (B) VISAR target package is similar to that of the recovery experiment except that a 100-µm-thick half-cylindrical Ge foil is used so as to capture the information from both front (Al/LiF) and rear (Ge/LiF) surface of the target. The laser-transparent LiF window is glued to the rear surfaces of Ge target on one side and Al pusher on another side. (C) Temporally resolved VISAR fringes showing the shock breakout at front and rear surfaces of the sample. (D) Measured particle velocity, U_p as a function of time. (E) Determination of the peak shock stress on the front surface of Ge target by impedance matching. Note that Al and LiF are closely impedance-matched in the regime involved in our study.

Fig. S1.

HYADES simulation of shock-wave propagation and decay in Ge. (A) Calibration of the simulation with the experimentally determined velocity profile at the Ge/LiF interface (rear surface). (B) Simulated longitudinal shock stress vs. time profile at different depths below the shock surface, showing the decay of the amplitude of the stress wave as it passes through the material.

Transmission electron microscopy (TEM) was used to inspect the postshock deformation micro- and nanostructure. The TEM samples were extracted by focused ion beam cutting from the as-shocked surface; the microstructural hierarchy is displayed in Fig. 2A. Along the direction of shock-wave propagation (left to right), nanocrystalline material can be observed as deep as 3 µm below the shock surface. Statistical analysis (Fig. 2B) shows that the average grain size is 62 ± 31 nm, whereas much smaller (5∼10-nm) grains can be seen toward the end of the nanocrystalline regime. The germanium nanocrystals show the form of diamond-cubic structure which is energetically more favorable than their high-pressure polymorphs.

Fig. 2.

TEM/HRTEM micrographs of laser-shock recovered germanium. (A) TEM image shows the hierarchy of the deformation microstructure. (B) Distribution of the grain size in the nanocrystalline domain and (C) amorphous bandwidth and spacing. (D) Zoomed view of nanocrystal. (E) Zoomed view of partially amorphous band with embedded nanocrystals. (F) Zoomed view of a completely amorphous band showing zero contrast inside the band. (Insets) Corresponding Fourier-transformed diffractions in the boxed regions are shown. (G) Lattice image in a nanocrystal shows nanoscale twins/stacking faults on {111} planes. (Inset, red) The Fourier-filtered image reveals the zigzag feature of these planar defects. (H) Amorphous band with stacking faults in its vicinity. (I) Two {111} stacking fault packets intersect, resulting in the early stage of amorphization and two sets of twin spots on the diffraction pattern.

In addition, a high density of {111} nanotwins/stacking faults is identified (Fig. 2 D and G), further subdividing the nanostructure. As one goes deeper, deformation bands dominate (Fig. 2C) with an average width (w_band) of 25 ± 17 nm and interspacing (w_spacing) of 124 ± 63 nm. Fourier-transformed diffraction pattern (Fig. 2E, Inset) in these regions shows a mixture of halo ring and sharp spots, suggesting that these bands are essentially amorphous with embedded nanocrystals. In another contrast-free band (Fig. 2F), the halo-ring diffraction indicates a complete amorphous structure. The calculated structure factor profile is similar to the previous reports of amorphous Ge by Bhat et al. (25) and Solomon et al. (26), and is shown in (Fig. S2). These amorphous bands align roughly with {111} slip planes of the diamond-cubic lattice, which are close to the maximum shear surface which forms a cone with an angle of 45°. Moreover, profuse stacking faults can be observed at the amorphous/crystalline interface (Fig. 2H), suggesting that these planar defects serve as the precursors to amorphization. Note that most of these bands align parallel to each other; however, other variants can be observed which tend to intersect and bifurcate, forming a complex network. This is shown in Fig. 2I; thus, the stacking faults and their intersections are the first stage of amorphous band generation, similar to our previous observations for Si (18). At a much lower shock stress (E_laser = 20 J; σ₃₃ $\approx$ 13 GPa), the recovered sample shows bulk amorphization close to the shock surface. The TEM sample (Fig. S3) exhibits a completely amorphous state up to 4 µm below the surface without undergoing crystallization.

Fig. S2.

(A) Calculated S(Q) from the electron diffraction pattern in B and compared with previous reports by Bhat et al. (25) in C.

Fig. S3.

Micrographs of Ge shocked at a lower laser energy (20 J). (A) SEM image of the shocked surface of [001] Ge single crystal. The rectangle indicates the position of the TEM sample cut by FIB technique. (B) SEM image of the TEM sample cross-section. (C) HRTEM image shows that the deformed structure is indeed amorphous without long-range order. The amorphous feature can be also confirmed by FFT diffractography in the bottom-left corner.

To better understand the atomistic mechanisms of amorphization, we have also carried out large-scale molecular dynamics (MD) simulations to mimic the laser shock experiments. The Tersoff (27) interatomic potential was implemented and executed in the LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) (28) code. MD simulations (Fig. 3A) show that amorphization occurs above a critical particle velocity of 1.2 km/s during compression and that the directional features of the amorphous bands and massive stacking fault/nanotwin formation before the onset of amorphization (Fig. 3B) agree well with the TEM observation; this strongly suggests that these planar faults are the precursors of amorphous banding. Dislocation velocity measurement (Fig. 3G) from MD snapshots (Fig. 3 C–F) shows that these stacking faults can travel supersonically before the crystalline-to-amorphous transition; subsequently their speed drops to the transonic regime. We propose that supersonic dislocations have a much-reduced ability to relax shear stresses because of the stress-field distortion produced by their velocity; an additional effect is that they radiate energy (29). Thus, their speed drops to the trans/subsonic regime after a short excursion. It is also proposed that supersonic dislocations (stacking faults) transport the imposed energy to their surroundings, which helps to trigger amorphization. The pressure dependence of longitudinal (C_L) and transversal (C_T) sound speeds are shown in Fig. S4 and the ultrafast nucleation and motion of dislocation/amorphous band can be viewed in Movie S1.

Fig. 3.

MD simulation of partial dislocation propagation and amorphization: (A) Three-dimensional visualization of shocked germanium colored by coordination number. The amorphous bands are colored red. (B) Only the plastically deformed (defected) atoms are shown, suggesting that the amorphous bands are preceded by partial dislocations (stacking faults). The amorphous band aligns roughly with {111} slip plane. (Inset) Radial pair distribution functions distinguish the amorphous domain from crystalline structure. (C–F) Four snapshots showing the evolution of stacking faults and then amorphous bands. (G) Measurement of dislocation/amorphous band speed during shock compression and supersonic burst of dislocation is notified before the formation of amorphous band.

Fig. S4.

Pressure dependency of longitudinal sound speed, transverse sound speed (split into two different values due to crystal anisotropy) along the <110> slip direction. The intersection of the vertical red dotted line with the black line gives the longitudinal sound speed at the pressure matched with the condition of the MD simulation (roughly agrees with experiment as well).

Discussion

Germanium melts with a reduction in volume and its melting temperature decreases as pressure increases [dT/dP $\approx$ −35 K/GPa (30)]. Thus, compressive stresses favor amorphization because the disordered phase possesses a smaller specific volume than the original crystalline phase, although the amorphous state is energetically less favorable at ambient pressure. In addition to pressure, the superposed shear stress also facilitates amorphization by inducing large lattice displacements through stacking-fault generation and propagation (18). This is due to the nature of shock waves which generate a uniaxial strain with high-amplitude hydrostatic pressure (P) and shear stresses. The pressure (P) and the maximum shear stress (τ_max) are related through the generalized Hooke’s law, as detailed in SI Text and Fig. S5. Specifically, the pressure dependency of elastic moduli and the ratio of shear stress over hydrostatic pressure is given in Fig. S5.

Fig. S5.

(A) MD prediction of pressure dependency of elastic constant and (B) ratio of maximum shear over pressure (assuming elasticity and no relaxation).

Applying classical nucleation theory we can obtain the pressure and shear-stress dependence of the nucleation barrier, as explained previously by Zhao et al. (31),

$$\mathrm{\Delta }{G}_{c-a}=\mathrm{\Delta }{g}_{c-a}\cdot \frac{4}{3}\pi r^3-\left(P{\epsilon }_v+{\tau }_{\mathit{max}}\gamma \right)\cdot \frac{4}{3}\pi r^3+{\gamma }_{c/a}\cdot 4\pi r^2,$$ [1]

where $\mathrm{\Delta }{G}_{c-a}$ is the energy gain of nucleating a spherical amorphous nucleus, $\mathrm{\Delta }{g}_{c-a}$ = 14.2 kJ/mol (32), and ${\gamma }_{c/a}$ = 0.08 J/m² (32) are the volumetric Gibbs free-energy barrier and crystalline/amorphous interfacial energy, respectively. ${\epsilon }_v$ and $\gamma \approx {\epsilon }_v\left(1+w_{spacing}/w_{band}\right)$ are the volumetric and localized shear strain, which can be obtained from the Ge shock-Hugoniot data (ref. 33, pp. 521–568). The second term on the right-hand side of Eq. 1 represents the work done by pressure and shear, which help to overcome the energy barrier and interfacial energy gain of nucleating an amorphous embryo (expressed by the third term). Such an effect is plotted in Fig. 4A. The monotonically increasing energy as a function of radius curve for zero-stress state (blue) indicates the difficulty of forming amorphous phase at ambient state, whereas under shock the curves (red and yellow) are convex and exhibit a critical condition $d{G}_{c-a}/dr=0$, corresponding to the critical nucleus size of $r_c=2{\gamma }_{c-a}/\left(P{\epsilon }_v+{\tau }_{\mathit{max}}\gamma -\mathrm{\Delta }{g}_{c-a}\right)$.

Fig. 4.

Thermodynamic analysis of amorphization. (A) Gibbs free-energy change associated with nucleation of an amorphous embryo. The presence of shock stress (pressure plus shear) renders it possible to overcome the energy barrier of crystalline-to-amorphous transformation. (B) Pressure-induced homogeneous temperature (T_homo) and shear-induced localized temperature (T_local) compared with the decreasing melting temperature (T_m) as a function of shock stress (negative Clapeyron slope). The intersections represent the critical shock stresses of crystalline-to-amorphous transition, indicating that shear stress lowers the threshold.

Shock-generated heat is another important factor in amorphization and subsequent nanocrystallization. Several effects have to be considered: (i) the temperature rise at the shock front reduces the energy barrier of crystalline-to-amorphous transition; (ii) upon further heat transfer, the newly formed amorphous structure can retransform into energetically more favorable crystalline phase; (iii) if the temperature is sufficiently high, shock-induced melting may occur; and (iv) the transient nature of the shock-induced thermal flux results in a self-quenching mechanism which leads to a hierarchical nanostructure.

The increase in temperature due to shock compression can be evaluated by considering both the homogeneous temperature rise (ΔT_homo) resulting from work done by hydrostatic pressure and the localized temperature rise (ΔT_local) resulting from work done by shear stress. The latter assumes a balance between relaxation of deviatoric strain energy and decrease in internal energy with heat loss to its surroundings, and thus gives a rough estimate of the temperature inside the amorphous band (19):

$$\begin{array}{l}{T}_{band}=T_0+\mathrm{\Delta }{T}_{homo}+\mathrm{\Delta }{T}_{local}\\ =T_0+\frac{P\mathrm{\Delta }V}{C_p}+\frac{2\dot{Q}{w}_{band}}{\sqrt{\pi }}\cdot \sqrt{\frac{t}{k\rho C_p}},\end{array}$$ [2]

where T₀ is the preshock temperature, $\mathrm{\Delta }V$ is the volume change, $\dot{Q}\approx \beta {\tau }_{\mathit{max}}\gamma /\mathrm{\Delta }t$ is the rate of heat generation obtained from the conversion of deviatoric strain energy, $\beta$ is the conversion efficiency (usually taken as 0.9), $\mathrm{\Delta }t$ ∼ 1 ns is the duration of the laser pulse. k, ρ, C_p are the heat conductivity, density, and heat capacity of the amorphous band, respectively. It should be noted that the shear stress is assumed to be independent of shock stress after the Hugoniot elastic limit [∼4 GPa for Ge (15)] as it is relaxed by plastic deformation. Fig. 4B shows the plot of shock-induced temperature together with melting temperature as a function of shock stress. The intersections of T_homo and T_band with T_m give the critical shock stresses (17.5 GPa for T_homo and 14 GPa for T_band) for the onset of amorphization. Clearly, the presence of shear stress lowers the threshold for amorphization, in agreement with early static compression experiments (21, 22). These calculations, whose accuracy depends largely on the material parameters, agree qualitatively with our experimental observations.

In summary, we have shown that germanium undergoes amorphization and nanocrystallization under extreme shock deformation. Although it is difficult, at this stage, to conclude whether these processes are solid state or the result of melting and quenching, the presence of shear stress is definitely crucial, and enhances the understanding of pressure-induced amorphization and polyamorphisms (21, 22, 34–36). After its discovery in olivine (20), shock-induced directional amorphization has now been confirmed in Si (18) and B₄C (19). Therefore, shear-induced amorphization should be considered as an important deformation mechanism in this extreme regime. More practically, our results suggest high-power, pulsed laser as a tool to rapidly produce substrate-free, micrometer-scale, gradient nanostructured semiconductors.

Materials and Methods

Laser Shock-Recovery Experiments.

We performed laser-driven shock-recovery experiments at the Omega Laser Facility. The shock wave is created by the following sequence of processes. First, the high-power pulsed-laser energy is deposited on the 20-μm CH ablator of the target package, ionizing it into plasma. Second, as the plasma flows away, the target surface experiences a reaction force equal to the rate at which momentum is carried away due to the rocket effect. The stress pulse promptly builds up and transforms into a shock wave. Third, the shock wave propagates inward and quickly decays as the laser duration is very short (1 ns). The amplitude of the ablation pressure (P_abl) can be estimated by the analytical model put forward by Lindl (37), $P_{abl}=C{\left(I/\lambda \right)}^{\beta }$, where C is a material-dependent constant and λ is the wavelength of the laser. $I=E_{laser}/At$ is the laser irradiance and β is the material-dependent exponential [calibrated to be 0.71 for diamond (38), which is similar to our CH ablator]. The target package consists of a 20-µm CH ablator, 50-µm Al foil, Ø 3 × 3-mm cylindrical Ge target, and Ti momentum trap. The assembly is encapsulated within a Ti cup. The Al foil has two functions: (i) as a heat shield to minimize the preheating induced by laser irradiation; and (ii) as a pulse shaper to render the shock pulse on the target surface planar (uniaxial strain state). The laser pulse is nominally a 1-ns square pulse of 351 nm (3 omega) laser light. The beam was used without phase plates and defocused to a spot size of 3-mm diameter.

VISAR Analysis and Impedance Matching.

Separate VISAR experiments were conducted to measure the particle velocity and further infer the shock pressure. A 532-nm probe laser is reflected from the rear surface of the moving target and then it will pass through collection optics and be routed into two separate streak cameras. Each of them uses a different etalon thickness. The VISAR target comprises a 20-µm CH ablator, 50-µm Al foil, a half-moon Ge sample (100-µm thick), and laser-transparent LiF window. This specific geometry of the target allows the measurement of particle velocity of both front and rear surface of the Ge sample, which shows the rapid decay of the shock pressure as function of the depth. A correction factor of 0.775 is applied to the apparent velocity to account for the pressure dependence of the index of refraction of the LiF window. Conservation of mass and momentum give the relationship between the initial density ${\rho }_0$, particle velocity $U_p$, shock velocity $U_s$ and shock stress ${\sigma }_{33}$,${\sigma }_{33}\approx {\rho }_0{U}_s{U}_p$ (39), where ${\rho }_0{U}_s$ is often termed as shock impedance, which can be obtained from the slope of the shock Hugoniot curves (${\sigma }_{33}$ vs. U_p) in Fig. 1E. At the interface between the Al foil and Ge sample, shock wave is reflected and the shock stress changes. The inverted shock Hugoniot of Al (red dotted line in Fig. 1E) gives the estimate of the reflected shock wave and the intersection of this line with Ge curve (black line in Fig. 1E) yields the shock pressure on the front surface of the Ge sample. Such a process is termed as impedance matching. The shock stress at the rear surface of Ge sample can be read directly from the Ge Hugoniot curve.

Radiation-Hydrodynamic Simulations.

Radiation-hydrodynamics simulations were performed using the HYADES code to aid in the design of the experiment and interpretation of the results. The target was modeled as a 1D stack consisting of 20-µm polystyrene ablator, 3 µm of glue (approximated as polystyrene), 50-µm Al, 3 µm glue (polystyrene), and 125 µm Ge. To simulate VISAR data, an additional layer of glue and LiF was substituted at the corresponding interface (front or back of the Ge). A rate-independent Steinberg–Guinan model was used to model the strength of Al (40). The Ge was modeled as elastic-perfectly plastic using the von Mises yield criterion with Y_VM = 4 GPa, consistent with observations of the Ge Hugoniot elastic limit.

TEM Sample Preparation and Observation.

TEM is the ultimate tool to characterize the postmortem microstructure of the shocked target. To prepare TEM samples cite-specifically, a Hitachi NB5000 scanning electron microscope (SEM) equipped with a focused ion beam (FIB) was used to cut TEM samples directly from the laser-shocked germanium monocrystal surface. The TEM foils were ion milled by 30-kV Ga beam and finally polished at 5 kV to minimize FIB damage. The as-lifted sample is shown in the SEM image in Fig. S3B. A Hitachi HF3300 TEM operated at 300 keV was used to characterize the postshock microstructure. In addition to the results shown in the text (Fig. 2) where the Ge target was laser shocked with a high energy (E_laser = 100 J, σ₃₃ ∼ 33 GPa), Fig. S3C shows the microstructure of the Ge target shocked at a low energy (E_laser = 20 J, σ₃₃ ∼ 10 GPa). The contrast-less feature of the high-resolution TEM (HRTEM) image suggests the random arrangement of the atoms. The Fourier-transformed diffractograph exhibits a halo-shaped ring pattern, confirming the amorphous nature of the materials.

MD Simulations.

MD simulations of shocked Ge were conducted using LAMMPS using the Tersoff potential under NVE (conservation of number of particles, volume, and total energy of the system) conditions. Shock compression was conducted along the [001] direction in single-crystalline germanium using a nonequilibrium drive piston. The piston was linearly accelerated to 1.2 km/s over 1 ps and held constant for the duration of the simulation. The system employs periodic boundaries perpendicular to the shock direction to create a uniaxial strain state. The lateral dimensions are 40 nm and the dimension along the direction of shock wave propagation is 50 nm to the rear surface. To compare the simulation to the experiments where the wave decays fully as it traverses the sample, analysis is only conducted before the reflection of the shock wave from the rear surface. The imparted particle velocity, U_p = 1.2 km/s, generates a shock pressure of 40.5 GPa and a concomitant deviatoric shear stress of 10.5 GPa. The corresponding shock speed is U_s = 5.2 km/s. Analysis of the shock-induced defect structure is completed using the “Identify Diamond Structure” implemented within OVITO (Open Visualization Tool) (41). To successfully apply this analysis technique to a strained lattice, an affine scale is first applied to artificially revert the strain such that analysis correctly identifies the defects. If this step is not completed the analysis tool identifies the uniaxially compressed region as unidentified non-dc atoms. Upon successful analysis, an affine scale exactly opposite in magnitude is applied to perfectly return the atoms to their original positions. Further analysis is conducted of the pair-distribution function (42) on the crystalline and amorphous regions. It describes the probability g(r) of finding an atom at a given distance r away from any other atom. The shape and peaks of the curve can be used to differentiate between phases and can also provide coordination information.

SI Text

Shear Stresses Under Shock Compression.

The shock stress (${\sigma }_{33}$), hydrostatic pressure (P), and maximum shear stress (τ_max) can be obtained from the generalized Hooke’s law for uniaxial strain, ${\sigma }_{ij}=C_{ij33}{\epsilon }_{33}$, where C_ijkl is the elastic constants, ${\epsilon }_{33}$ is the uniaxial strain, and x₃ is the direction of shock-wave propagation. The ratio ${\tau }_{\max }/P$ can be expressed as a function of elastic stiffness (if the deformation is assumed to be purely elastic) as

$$\frac{{\tau }_{\mathit{max}}}{P}=\frac{3\left(C_{11}-C_{12}\right)}{2\left(C_{11}+2C_{12}\right)}.$$ [S1]

For germanium, at ambient pressure, C₁₁ = 129.2 GPa, C₁₂ = 47.9 GPa, rendering ${\tau }_{\mathit{max}}/P\approx 0.54$. This is a first approximation since the elastic moduli are pressure-dependent as predicted by our MD simulations. This high shear stress is relaxed by directional deformation and subsequent amorphous band formation.

Supersonic Dislocation.

Both our TEM observations and MD simulations show that dislocation (stacking faults) activity occurs before the amorphization, suggesting that dislocations are the trigger to the drastic crystalline-to-amorphous transition. Therefore, it is reasonable to infer that the kinetics of dislocation motion determine the kinetics of amorphization. It has been predicated by the theory of linear elasticity that the dislocation velocity is limited by the transverse wave speed (C_T) at which the energy associated with the screw dislocation approaches infinity, as shown in Eq. S2.

$$\frac{W}{l}=\frac{G{b}^2}{4\pi {\left(1-\frac{v_d^2}{C_T^2}\right)}^{1/2}}\ln \frac{R}{b},$$ [S2]

where W is the elastic energy associated with a moving screw dislocation, l is the length, and b is the Burgers vector. However, MD simulations by Gumbsch and Gao (29) suggested the existence of supersonic dislocation in tungsten, even above longitudinal sound speed. Gumbsch and Gao (29) proposed that a supersonic dislocation can be obtained if an ultrafast dislocation seed is nucleated at a stress concentrator. An applied stress is required to help the dislocation to overcome the sound barrier. Such a condition can be most likely achieved in a strong shock experiment where the high stress state (both hydrodynamic and deviatoric components) builds up quickly at the shock front. In our MD simulations, supersonic dislocation bursts are identified, as shown in Fig. 3, before amorphization. The motion of dislocation is nonuniform at this stage and the velocity should be a range instead of a constant. After such a supersonic event, the dislocation velocity quickly falls off when the amorphization is initiated; the motion of dislocation/amorphous band becomes transonic and uniform. Movie S1 and MD snapshots in Fig. 3 show how the partial dislocations nucleate, catch up with the shock front, and eventually give rise to the formation of amorphous bands.

Estimation of Sound Speeds and Dislocation Velocity.

Travel of acoustic waves in crystal is highly anisotropic and orientation dependent. Germanium is a cubic material and the active slip system is of {111}<110> type. Thus, we need to evaluate the sound speeds along the <110> direction to compare them with the dislocation velocity. The corresponding longitudinal sound speed is

$$C_L={\left(\frac{C_{11}+C_{12}+2C_{44}}{2\rho }\right)}^{1/2}.$$ [S3]

The transversal sound speed is polarized in different <110> directions:

$$\begin{array}{l}{C}_{\mathit{Tmax}}={\left(\frac{C_{44}}{\rho }\right)}^{1/2},\\ C_{\mathit{Tmin}}={\left(\frac{C_{11}-C_{12}}{2\rho }\right)}^{1/2}.\end{array}$$ [S4]

It should be noted that the elastic constants C₁₁, C₁₂, C₄₄ are a function of pressure, as shown in Fig. S5A. Thus, the sound speeds also depend on pressure, as plotted in Fig. S4.

Dislocation velocity (V_d) is estimated by tracking the displacement of the defects in time. Multiple snapshots with 1-ps spacing (in MD time) are used and the effective defect length is measured from each snapshot. It should be noted that the motion/growth of these defects is not strictly limited in one particular direction and we also notice the lateral growth of the stacking faults/amorphous bands, suggesting the activation of stacking faults in the adjacent slip planes. Due to these complexities, the defect velocity was measured several times to minimize the errors caused by the uncertainty of the defect length. Despite the error bars, our measurements in Fig. 3G of the main text show that the partial dislocations nucleate with an ultrafast velocity (V_d ∼ C_T) right after the shock front passes by. Shortly after their nucleation, the dislocations are still under extremely high stresses and they quickly accelerate to the supersonic (V_d > C_L) regime. However, the motion of supersonic dislocations is not steady state, i.e., the velocity should be a range instead of a number. After such an explosive motion/growth, their velocity falls to the transonic regime (C_L > V_d > C_T). The supersonic dislocations seem to trigger the amorphization which helps to relax the concentrated deviatoric stress.

Calculation of the Structure Factor.

To better understand the structure of the amorphous germanium, we have thus calculated the S(Q) profile based on our selected area electron diffraction pattern. The result is given in the following figure, Fig. S2. It should be mentioned that the diffractograph that we show in the article is the FFT of the HRTEM image where we can easily distinguish the pattern of amorphous materials from that of crystalline domain. However, the FFT diffractograph is not suitable for the calculation of the structure factor S(Q). Therefore, additional experiments were conducted and the selected area electron diffraction pattern (in the vicinity of the band) is recorded in Fig. S2B. However, the size of the diffraction aperture is much larger than the characteristic size of the amorphous band (∼20 nm) and hence the pattern is inevitably contaminated by the surrounding crystalline matrix, which affects the calculation of the S(Q). Another factor is that the microscope does not have an energy filter and therefore cannot filter the inelastically scattered electrons. Thus, the tail becomes very noisy and therefore we trimmed it at Q ∼ 6 $\dot{\text{A}}$⁻¹. Nevertheless, the first two broad peaks on the S(Q) vs. Q curve correspond mostly to the amorphous structure and we found that they are very similar to the results shown by ref. 25, albeit with a little shift to the right (higher value of scattering vector, Q), indicating that the amorphous structure may be a low-density amorphous polymorph, which may form under unloading.