Thermodynamics and Transport Model of Charge ... - IEEE Xplore
Thermodynamics and Transport Model of Charge ... - IEEE Xplore Thermodynamics and Transport Model of Charge ... - IEEE Xplore
BOULAIS et al.: THERMODYNAMICS AND TRANSPORT MODEL OF CHARGE INJECTION IN SILICON 2731 Fig. 3. Custom electrical model of an inverter of the ring oscillator. The exact position-dependent mobility parameters for both holes and electron are difficult to determine, since integrated circuits are not only composed of pure and crystalline silicon but also of polycrystalline silicon which has suffered many stresses in some regions, including high-power laser impacts and phase changes. Furthermore, the real doping-concentration profile remains unknown, since those parameters are not provided by the foundry. Position-dependent dislocations, defects, and dopants would all decrease the carrier mobility, resulting in a modified position and temperature-dependent mobility given by with μ M n,p(x, y, T )=φ(x, y, T )μ C n,p(x, y, T ) (6) φ(x, y, T ) ∈ [0, 1] (7) where the superscript M and C stands, respectively, for “modified” and “crystalline,” as the subscripts n and p, respectively, stands for electrons and holes. Carrier mobility in crystalline silicon can be found in standard texts [18]. Now, since the exact expression for ϕ(x, y, t) is very difficult to obtain, both theoretically and experimentally, we propose to use in a first approximation an equivalent mobility given by with μ eq n,p(x, y, T )=φμ C n,p(x, y, T ) (8) φ ∈ [0, 1] (9) where μ eq n,p is the equivalent mobility of electrons and holes and φ is an empirical parameter adjusted to fit experimental value. Similarly, the carrier recombination time is also expected to be different from the one for crystalline silicon, and a uniform equivalent recombination has been assumed τ eq n,p(x, y, t) =τ. (10) While the introduction of equivalent recombination times, absorption coefficient, and mobilities are required to fit experimental data, their influence over the general behavior of Fig. 4. FVC variation as a function of laser power, (circles) experiments, and (crosses) simulation. injected current as a function of incident laser power remains limited. Finite-element simulations were performed on a 1.8-V reversed-biased p-n junction located at d = 30 μm from a pulsed focused laser. The calculated current was then inserted as a current source perturbing the electrical model to obtain the simulated variation of the FVC output voltage. Mobility factor and recombination time were then adjusted to fit experimental results. Simulations show that φ = 0.17 and τ = 200 ns reproduces the experimental results with the best accuracy. Fig. 4 shows both the best-fit simulated and measured results of the FVC output variation as a function of P L . Results in Fig. 4 show that, while the model predictions agree quite well with simulation results within the first region (P L < 0.375 W), it does not follow the behavior observed in the second region (0.375 W
2732 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 55, NO. 10, OCTOBER 2008 IV. COUPLED THERMODYNAMICS AND TRANSPORT MODEL A. Coupled Thermodynamic and Transport Model Without Fusion In this improved model based on coupled thermodynamic and transport equations, the carrier spatial distribution is simulated in a two-step process. Temperature is first calculated using a model essentially based on the one introduced by Degorce et al. [19], involving the resolution of coupled Beer–Lambert and heat equations, where the phase transformation is accounted for by using an enthalpic model. Siliconoxide effect on temperature distribution is neglected to reduce the model’s complexity. The carrier distribution is then determined through the resolution of the coupled Boltzmann and Poisson equations. The temperature calculated in the first step is used into these equations without any explicit feedback. However, carrier-distribution effects on temperature are taken into account by using an empirical relation of the temperaturedependent absorption coefficient α(T ). This approximation reduces greatly the model’s complexity. The calculated maximum temperature gradient of 3.5 × 10 7 K/cm remains small enough to justify the use of Boltzmann equation for any laser power considered in this paper. As expected, temperature gradient has a considerable effect on the charge transport in the material which can be easily understood by looking at the Boltzmann equation. Developing (3), for the electrons, one can write Fig. 5. Schematic of the time evolution of the distribution of electronic current in the material. At the beginning of the pulse (A), electrons induced in the material diffuse from the laser impact point to the circuit. The laser heats the substrate, and when temperature gradient gets high enough over a certain area (B), this area experiences an inversion of its electronic current. After the end of the pulse (C), the temperature gradient gets lower, and the inverse current area disappear. ( ) −→ G5/2 J n = −enμ n ∇V + nμ n ∇F − k B nμ n − η ∗ ∇T G 3/2 (11) where the first terms represent the drift and temperatureconstant diffusion current and the last one corresponds to a current generated when a temperature gradient is present within the material. A similar equation can be obtained for holes. Calculations show that the coefficient multiplying the temperature gradient (11) is always positive. In our application, in addition to injecting carriers, the laser acts as a heat source generating a thermal gradient having the same sign as the carrier concentration and the quasi-Fermi level. Therefore, the temperature gradient always opposes the effect of the carrierconcentration gradient, here represented by the quasi-Fermi level gradient, on the diffusion current. Strong temperature gradient could limit and even stop the diffusion of carriers from the laser impact point to the junction. Fig. 5 shows a schematic time profile of the electronic current present in the material. As shown in Fig. 5, when a strong temperature gradient is present in the material, an inversion of the electronic current is experienced over a small area of the substrate. Carriers are then confined in this small region limited by this diffusion barrier where they can recombine. Only carriers created at the very beginning of the laser pulse, which is before the formation of a strong temperature gradient, or created outside the diffusion barrier will eventually get to the junction. Fig. 6 shows a simulation result showing that this inversion of the electronic current indeed occurs for a 0.5-W laser pulse in a region located about 400 nm from the laser impact point. Fig. 7 now shows Fig. 6. Electronic current as a function of distance from laser impact point at the end of a 0.5-W pulse as calculated with (solid line) coupled thermodynamic and transport model and (dashed-dot line) constant-temperature semiconductor constant model. a comparison of the simulated FVC output voltage using this new model to the experimental results. Comparison of Fig. 7 to Fig. 4 shows that introducing temperature effects into the semiconductor transport model permits one to almost reproduce the experimental behavior in the second region which could thus be attributed for a large part to the presence of strong temperature gradient within the substrate. It is of interest that the electrical model used to simulate FVC output is no longer valid beyond 0.7 W. At that point, the electric model predicts that current perturbation required to reproduce experimental values shortcuts the n-well junction. The linear-circuit model used here is then no longer a good approximation to the phenomena taking place. It is thus difficult to correlate simulated and experimental behaviors for incident powers beyond 0.7 W.
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BOULAIS et al.: THERMODYNAMICS AND TRANSPORT MODEL OF CHARGE INJECTION IN SILICON 2731<br />
Fig. 3.<br />
Custom electrical model <strong>of</strong> an inverter <strong>of</strong> the ring oscillator.<br />
The exact position-dependent mobility parameters for both<br />
holes <strong>and</strong> electron are difficult to determine, since integrated<br />
circuits are not only composed <strong>of</strong> pure <strong>and</strong> crystalline silicon<br />
but also <strong>of</strong> polycrystalline silicon which has suffered many<br />
stresses in some regions, including high-power laser impacts<br />
<strong>and</strong> phase changes. Furthermore, the real doping-concentration<br />
pr<strong>of</strong>ile remains unknown, since those parameters are not provided<br />
by the foundry. Position-dependent dislocations, defects,<br />
<strong>and</strong> dopants would all decrease the carrier mobility, resulting<br />
in a modified position <strong>and</strong> temperature-dependent mobility<br />
given by<br />
with<br />
μ M n,p(x, y, T )=φ(x, y, T )μ C n,p(x, y, T ) (6)<br />
φ(x, y, T ) ∈ [0, 1] (7)<br />
where the superscript M <strong>and</strong> C st<strong>and</strong>s, respectively, for “modified”<br />
<strong>and</strong> “crystalline,” as the subscripts n <strong>and</strong> p, respectively,<br />
st<strong>and</strong>s for electrons <strong>and</strong> holes. Carrier mobility in crystalline<br />
silicon can be found in st<strong>and</strong>ard texts [18]. Now, since the<br />
exact expression for ϕ(x, y, t) is very difficult to obtain, both<br />
theoretically <strong>and</strong> experimentally, we propose to use in a first<br />
approximation an equivalent mobility given by<br />
with<br />
μ eq<br />
n,p(x, y, T )=φμ C n,p(x, y, T ) (8)<br />
φ ∈ [0, 1] (9)<br />
where μ eq<br />
n,p is the equivalent mobility <strong>of</strong> electrons <strong>and</strong> holes <strong>and</strong><br />
φ is an empirical parameter adjusted to fit experimental value.<br />
Similarly, the carrier recombination time is also expected to<br />
be different from the one for crystalline silicon, <strong>and</strong> a uniform<br />
equivalent recombination has been assumed<br />
τ eq<br />
n,p(x, y, t) =τ. (10)<br />
While the introduction <strong>of</strong> equivalent recombination times,<br />
absorption coefficient, <strong>and</strong> mobilities are required to fit experimental<br />
data, their influence over the general behavior <strong>of</strong><br />
Fig. 4. FVC variation as a function <strong>of</strong> laser power, (circles) experiments, <strong>and</strong><br />
(crosses) simulation.<br />
injected current as a function <strong>of</strong> incident laser power remains<br />
limited. Finite-element simulations were performed on a<br />
1.8-V reversed-biased p-n junction located at d = 30 μm from<br />
a pulsed focused laser. The calculated current was then inserted<br />
as a current source perturbing the electrical model to obtain the<br />
simulated variation <strong>of</strong> the FVC output voltage. Mobility factor<br />
<strong>and</strong> recombination time were then adjusted to fit experimental<br />
results. Simulations show that φ = 0.17 <strong>and</strong> τ = 200 ns reproduces<br />
the experimental results with the best accuracy. Fig. 4<br />
shows both the best-fit simulated <strong>and</strong> measured results <strong>of</strong> the<br />
FVC output variation as a function <strong>of</strong> P L . Results in Fig. 4<br />
show that, while the model predictions agree quite well with<br />
simulation results within the first region (P L < 0.375 W), it<br />
does not follow the behavior observed in the second region<br />
(0.375 W
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