Boundary effects on the plastic flow of amorphous layers during high-energy heavy-ion irradiation

Swift heavy-ion irradiation of amorphous materials causes nonsaturating plastic deformation. Thin samples deform anisotropically as if they were hammered. Thick samples or thin layers on top of thick crystalline substrates flow plastically during nonperpendicular irradiations. In this paper, we demonstrate that the constitutive equation of ion hammering not only describes the homogeneous flow of uniformly irradiated matter but also the deformation effects at boundaries between irradiated and unirradiated matter. An analytic expression is derived which describes the flow of thin surface layers in the vicinity of an abrupt boundary. This solution is complemented by finite-element calculations, which reveal the influence of finite transition widths. The constitutive equation is also solved numerically to unveil the boundary effects for thick samples, where the ion-induced deformation depends on the distance from the specimen surface. The calculations are compared with experiments carried out with thin amorphous silicon surface layers and thick crown glasses. The excellent agreement confirms the validity of the constitutive equation and supports its microscopic basis, namely the idea of an efficient relaxation of thermally induced shear stresses along the ion path during the thermal spike period.

Figure 1

Schematic view of the irradiation geometry with a $y$ boundary in the case of an embedded amorphous silicon $\left(a\text{−}\mathrm{Si}\right)$ layer of thickness $d$ in crystalline silicon $\left(c\text{−}\mathrm{Si}\right)$. The projection of the ion beam onto the specimen surface determines the $x$ axis. The $x$ axis is the borderline between irradiated area $(y<0)$ and unirradiated area $(y>0)$. The surface of the unirradiated specimen part defines $z=0$ and the incidence angle of the ions $ϴ$ is measured from the $z$ axis. The same notation is used for the irradiation of the crown glass. For $y\to -\infty$, the transition from $c\text{−}\mathrm{Si}$ to $a\text{−}\mathrm{Si}$ at $x=a$ forms an $x_a$ boundary and the transition from $a\text{−}\mathrm{Si}$ to $c\text{−}\mathrm{Si}$ at $x=b$ forms an $x_b$ boundary since $c\text{−}\mathrm{Si}$ is almost inert to irradiation with swift heavy ions (Ref. 41).

Figure 2

Electronic energy deposition per unit path length $S_e\left(\xi \right)$ of a penetrating ion for the irradiation conditions used in our experiments in the case of $a\text{−}\mathrm{Si}$ (a) and the crown glass (b). The distributions were calculated with SRIM2003 (Ref. 49).

Figure 3

Shift at the surface $\Delta x(z=0,t)$ in the uniformly irradiated central region versus ion fluence ${\Phi }_{0}t$ during $600\mathrm{MeV}$ Au irradiation of $a\text{−}\mathrm{Si}$ at $T_0=80\mathrm{K}$ and the crown glass at $T_0=300\mathrm{K}$.

Figure 4

DEKTAK surface profile of an $a\text{−}\mathrm{Si}$ layer embedded in $c\text{−}\mathrm{Si}$ after an irradiation with $390\mathrm{MeV}$ Xe ions at $T_0=300\mathrm{K}$, flux ${\Phi }_0=8.3\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, and fluence ${\Phi }_{0}t=9.3\times {10}^{15}{\mathrm{cm}}^{-2}$. The $a\text{−}\mathrm{Si}$ layer extends from $x_a=0$ to $x_b=3000\mu \mathrm{m}$.

Figure 5

DEKTAK and AFM surface profiles of the sample from Fig. 4 at the $x_a$ boundary (a) and $x_b$ boundary (b). The origin of the $x$ scale is set arbitrarily.

Figure 6

Steady-state shear velocity $v_x(y,z)$ of $a\text{−}\mathrm{Si}$ at the $y$ boundary with a transition width $\Gamma =0\mu \mathrm{m}$ (a) and (b), and $\Gamma =30\mu \mathrm{m}$ (c) and (d), respectively. (a) and (c) show $v_x\left(y\right)$ for constant values $z$ and (b) and (d) show $v_x\left(z\right)$ for constant values $y$. The numerical and analytical calculations coincide in (a) and (b). All calculations were carried out for $d=5.7\text{−}\mu \mathrm{m}$-thick $a\text{−}\mathrm{Si}$ layer irradiated with $350\mathrm{MeV}$ Au ions at $T_0=80\mathrm{K}$ and ion flux ${\Phi }_0=4.6\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$.

Figure 7

Shift at the surface, $\Delta x(y,z=0,t)$, of $a\text{−}\mathrm{Si}$ at the $y$ boundary for different transition widths $\Gamma$. The numerical calculations were carried out for a $5.7\text{−}\mu \mathrm{m}$-thick $a\text{−}\mathrm{Si}$ layer irradiated with $350\mathrm{MeV}$ Au ions at $T_0=80\mathrm{K}$, flux ${\Phi }_0=4.6\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, and fluence ${\Phi }_{0}t=1.7\times {10}^{15}{\mathrm{cm}}^{-2}$.

Figure 8

Comparison of the calculated shift at the surface $\Delta x(y,z=0,t)$ of $a\text{−}\mathrm{Si}$ with the experimental result for a transition width $\Gamma =0\mu \mathrm{m}$ (a) and $\Gamma =10\mu \mathrm{m}$ (b). The underlying micrograph of the specimen surface was taken from a $5.7\text{−}\mu \mathrm{m}$-thick $a\text{−}\mathrm{Si}$ layer after an irradiation with $350\mathrm{MeV}$ Au ions at $T_0=80\mathrm{K}$, ${\Phi }_0=4.6\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ and ${\Phi }_{0}t=1.7\times {10}^{15}{\mathrm{cm}}^{-2}$. The calculation results were taken from Fig. 7. The origin $y=0$ is chosen to give optimal agreement with the experiment, $x=0$ is arbitrary.

Figure 9

Steady-state shear velocity $v_x(y,z)$ of the crown glass at the $y$ boundary with a transition width $\Gamma =0\mu \mathrm{m}$ (a) and (b), and $\Gamma =30\mu \mathrm{m}$ (c) and (d), respectively. (a) and (c) show $v_x\left(y\right)$ for constant values $z$ and (b) and (d) show $v_x\left(z\right)$ for constant values $y$. All numerical calculations were carried out for an irradiation with $600\mathrm{MeV}$ Au ions at $T_0=300\mathrm{K}$ and ion flux ${\Phi }_0=4.4\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. The maximum deformation length is ${\xi }_0\sim -40\mu \mathrm{m}$.

Figure 10

Comparison of the calculated shift at the surface for a $y$ boundary, $\Delta x(y,z=0,t)$, of the crown glass with the experimental result. The underlying micrograph of the specimen surface was taken after an irradiation with $600\mathrm{MeV}$ Au ions at $T_0=300\mathrm{K}$, flux ${\Phi }_0=4.4\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ and fluence ${\Phi }_{0}t=1\times {10}^{14}{\mathrm{cm}}^{-2}$. For the calculation, a transition width $\Gamma =0\mu \mathrm{m}$ was used. The location of $y=0$ was chosen to give optimum agreement with the experiment. The origin of the $x$ scale is arbitrary.