Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

Chapter C Analytical Expression for the Moments

To calculate the moments as in Equations 4.24 and 4.25 we use the general format for a moment’s equation:

$(C.1) \begin{equation} m_{i}=\int _{0}^{\infty }\varepsilon ^{i}f(\varepsilon )g(\varepsilon )d\varepsilon \label {eq:general-moment} \end{equation}$

for $f(\varepsilon )$ we use the DF expression used in this work:

$f(\varepsilon )=A\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]+C\exp \left [-\frac {\varepsilon }{k_{\mathrm {B}}T_{n}}\right ],$

and a general formulation of the the density of states $g(\varepsilon )=g_{\mathrm {0}}\sqrt {\varepsilon }\left (1+(\eta \varepsilon )^{\varsigma }\right )$ . The moment equation Equation C.1 then becomes:

$(3.2–C.3) \{begin}{align} m_{i}= & \int _{0}^{\infty }\varepsilon ^{i}\left (A\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]+C\exp \left [-\frac {\varepsilon }{k_{\mathrm {B}}T_{n}}\right ]\right )\left (g_{\mathrm {0}}\sqrt {\varepsilon }\left (1+(\eta \varepsilon )^{\varsigma }\right )\right )d\varepsilon \\ = & \int _{0}^{\infty }\varepsilon ^{i}g_{\mathrm {0}}\sqrt {\varepsilon }A\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]d\varepsilon +\int _{0}^{\infty }\varepsilon ^{i}g_{\mathrm {0}}\sqrt {\varepsilon }\left (n\varepsilon \right )^{\varsigma }A\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]d\varepsilon \label {eq:appendixB-3}\\ & \qquad \qquad \qquad \qquad \qquad \qquad \quad +\nonumber \\ & \int _{0}^{\infty }\varepsilon ^{i}g_{\mathrm {0}}\sqrt {\varepsilon }C\exp \left [-\frac {\varepsilon }{k_{\mathrm {B}}T_{n}}\right ]d\varepsilon +\int _{0}^{\infty }\varepsilon ^{i}g_{\mathrm {0}}\sqrt {\varepsilon }\left (n\varepsilon \right )^{\varsigma }C\exp \left [-\frac {\varepsilon }{k_{\mathrm {B}}T_{n}}\right ]d\varepsilon \nonumber \{end}{align}$

Solving the first term in Equation C.3:

$(3.4–C.3) \{begin}{align*} m_{i,1}= & \int _{0}^{\infty }\varepsilon ^{i}g_{\mathrm {0}}\sqrt {\varepsilon }A\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]d\varepsilon \\ = & Ag_{0}\int _{0}^{\infty }\varepsilon ^{i+\frac {1}{2}}\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]d\varepsilon \{end}{align*}$

substituting $\left (\varepsilon /\varepsilon _{\mathrm {ref}}\right )^{b}$ as t gives:

$(3.4–C.3) \{begin}{align*} m_{i,1}= & \frac {Ag_{0}}{b}\left (\varepsilon _{\mathrm {ref}}\right )^{\frac {2i+3}{2}}\int _{0}^{\infty }t^{\frac {2i+3-2b}{2b}}\exp \left [-t\right ]dt\\ = & \frac {Ag_{0}}{b}\left (\varepsilon _{\mathrm {ref}}\right )^{\frac {2i+3}{2}}\Gamma \left (\frac {2i+3}{2b}\right ) \{end}{align*}$

since $\Gamma (x)=\int _{0}^{\infty }t^{x-1}\exp \left [-t\right ]dt$ . Similarly, the other terms in Equation C.3 can be written as:

$(3.4–C.3) \{begin}{align*} m_{i,2}= & \frac {Ag_{0}}{b}\left (\varepsilon _{\mathrm {ref}}\right )^{\frac {2i+3}{2}}\left (\eta \varepsilon _{\mathrm {ref}}\right )^{\varsigma }\Gamma \left (\frac {2i+3+2\varsigma }{2b}\right )\\ m_{i,3}= & Cg_{0}\left (k_{\mathrm {B}}T_{n}\right )^{\frac {2i+3}{2}}\Gamma \left (\frac {2i+3}{2}\right )\\ m_{i,4}= & Cg_{0}\left (k_{\mathrm {B}}T_{n}\right )^{\frac {2i+3}{2}}\left (\eta \varepsilon _{\mathrm {ref}}\right )^{\varsigma }\Gamma \left (\frac {2i+3+2\varsigma }{2}\right ) \{end}{align*}$

Thus the analytical expression for a moment can be written as:

$(C.4) \{begin}{align} m_{i}= & \frac {Ag_{0}}{b}\left (\varepsilon _{\mathrm {ref}}\right )^{\frac {2i+3}{2}}\left (\Gamma \left (\frac {2i+3}{2b}\right )+\left (\eta \varepsilon _{\mathrm {ref}}\right )^{\varsigma }\Gamma \left (\frac {2i+3+2\varsigma }{2b}\right )\right )+\label {eq:analytical-moment}\\ & \qquad \qquad \qquad \qquad \qquad \;\nonumber \\ & Cg_{0}\left (k_{\mathrm {B}}T_{n}\right )^{\frac {2i+3}{2}}\left (\Gamma \left (\frac {2i+3}{2}\right )+\left (\eta \varepsilon _{\mathrm {ref}}\right )^{\varsigma }\Gamma \left (\frac {2i+3+2\varsigma }{2}\right )\right )\nonumber \{end}{align}$