Thermodynamics and Transport Model of Charge ... - IEEE Xplore

BOULAIS et al.: THERMODYNAMICS AND TRANSPORT MODEL OF CHARGE INJECTION IN SILICON 2733
Fig. 7. FVC variations as a function of laser power, (circles) experiments, and
(triangles) simulation.
The physical model presented so far predicts a saturation of the
current entering the junction as a function of the laser power in
the third region. In fact, at those powers, the temperature rise
in silicon is strong enough to induce melting of the substrate.
Although melting of the surface is included in the model,
its direct effect on electronic transport has been neglected.
The next section introduces a model that considers those
effects.
B. Coupled Thermodynamic and Transport Model
With Fusion
Basically, the equations of coupled thermodynamic and
transport model presented in the previous section remains
essentially the same, except that one must include the effect
of the melted silicon, which has become metallic. Inverse
bremsstrahlung then becomes the main light-absorption
mechanism, and direct electron–hole pair creation can be
neglected. However, when fusion temperature is reached,
semiconductor–metal phase transformation induces confined
electrons in covalent bonds to reach conduction bands thus significantly
increasing equilibrium electronic concentration [20].
The density of states of silicon is greatly modified when melted,
and the Fermi level position decreases by approximately 0.6 eV
according to experimental studies [21]. The solid–liquid interface
is assumed to be at equilibrium so that thermionic
charge injection can be neglected. Electron mobility is also
modified [22]. Therefore, at relatively high laser power, the
process will result in a metal–semiconductor heterostructure
whose interface moves with the melted front. Because of the
Fermi level drop in the melted silicon, carriers in liquid silicon
are trapped inside a 0.6-eV well. Therefore, even if the carrier
density within the melted silicon is very high, those carriers
are probably not able to diffuse, and they probably stay trapped
inside the melted region. Melting of the substrate is thus believed
to have a negligible impact on the current perturbation
affecting the sensitive circuit. From simulation results and using
Fermi statistics, one can estimate the number of carriers in the
melted region that is able to pass over the potential barrier to be
approximately 10 19 times lower than the photoinduced charges
injected in the solid semiconductor outside the melted region.
As the laser power increases, reflected light from the impact
point could be scattered to relatively large distances
through multiple reflections in the interdielectrics and reach
the p-n junction, thus creating additional electron–hole pair
in the silicon. A simple calculation can be made to approximate
the magnitude of this scattered light. Neglecting backscattering
from dielectric/dielectric interface since their refraction
indexes are expected to be very similar, the proportion of light
backscattered from the irradiated surface, to the interface top
layer/air, and back to the surface is about 10% of the incident
light at the laser impact point. However, since the target is not
experimentally located exactly at the laser waist, it is expected
that the incoming lightwave will not be perfectly plane at the
surface, yielding to a reflected light over a large area. Moreover,
when melted, the silicon surface experiences different transformations
that increase this light scattering. First, liquid silicon’s
reflection coefficient is about 1.5 times larger than the one of
solid silicon. In addition, besides reflection coefficient modification,
silicon undergoes purely geometrical modification when
melted. Fig. 8 shows an image from a transmission electron
microscopy of a silicon area which has been melted by a laser
irradiation whose parameters are analog to the one used for
trimming processes. Silicon has experienced strong geometrical
modification and has reshaped as a rounded pyramid, similar
to a sombrero shape. Other shapes have been reported in
the literature, including bowl, for silicon and other material
surfaces [23], [24]. These studies showed that irradiating at
low energy tends to create bowllike structures, while at higher
energy, sombrero structures are mostly observed. As stated
in [23], bowllike structure formation results from temperature
gradient inducing thermocapillarity flow that drives material
from the hot center to the cold periphery. However, at higher
laser power, native oxide layer is removed (at least partially),
thus creating a surfactant-concentration gradient driving material
toward the center. The observed sombrerolike structure is
expected to increase the quantity of scattered light reaching the
transistor region. Wave front at the surface is indeed no longer
plane, and reflections will occur at angles far from normal, thus
increasing the proportion of light reaching regions distant from
the laser impact point. Scattered light creates photo-induced
carriers in regions far from impact point and outside the strong
temperature gradient area. Those carriers then diffuse freely
to the circuit area, thus contributing to the total current. Note
that all samples contained those substrate deformation from
the beginning, since measurements have been conducted on a
previously irradiated circuit. Consequently, the laser power at
the edge of region 3 in Fig. 2 does not stand for the point at
which the deformation appears but rather for the point at which
scattered-light induced carrier effect on the oscillator becomes
dominant over the diffusion process. Within this model, lightscattered
induced carrier generation is modeled as a uniform
and constant light background proportional to the incident laser
power. Light-scattered generation have thus to be added to the
optical generation
G = G optical + G scattering . (12)