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

4.3 Carrier Energy Distribution Function

As suggested by the energy driven paradigm, the carrier distribution function is the most important ingredient of any HCD model. To calculate the DFs, we use the deterministic BTE solver ViennaSHE in this work [13, 15, 14]. The Monte-Carlo approach is not well suited for HCD modeling in power devices due its enormous computational demands related to resolving the high energy tails. Thus, it is not used in this work as we try to model devices of different dimensions and operating voltages. An alternative way, as mentioned before, is to use simplified approaches to the BTE solution such as the drift-diffusion scheme. The DFs simulated with ViennaSHE are used as a benchmark for the DD-based approach.

In the analytical approach, a simplified technique is used to obtain the carrier energy DF which uses the drift-diffusion method. However, as discussed in Section 4.2.2.3, the DD model consists of just two moments of the BTE and the full DF cannot be obtained. Thus, in order to estimate the DF, approximate analytic formulations have to be used. The parameters in the analytic DF expression can be calculated from the BTE moments obtained using DD simulations. In this context, an analytic expression for the DF has been used in this work. The analytical expression, which was suggested in [185], considers both hot and cold carriers:

$(4.22) \begin{equation} f(\varepsilon )=A\exp \left [-\left (\frac {\varepsilon }{\varepsilon _{\mathrm {ref}}}\right )^{b}\right ]+C\exp \left [-\frac {\varepsilon }{k_{\mathrm {B}}T_{\mathrm {n}}}\right ],\label {eq:DF} \end{equation}$

where $\varepsilon$ is the carrier energy, $T_{\mathrm {n}}$ the carrier temperature, $k_{\mathrm {B}}$ the Boltzmann constant. In Equation 4.22 the first term represents the DFs in the sections of the device where the carriers are hot, for example carriers in the channel region, the bird’s beak region in LDMOS devices, as well as the high energetic carriers near the drain. The pool of cold carriers in the drain/source region is modeled by an additional cold Maxwellian term. Since the DD model assumes a thermal equilibrium condition, the carrier temperature cannot be obtained directly. Thus, the carrier temperature is evaluated from post-processing of the drift-diffusion simulation results. The electron concentration $n(x)$ , the electric field profile $E(x)$ , and the carrier mobility $\mu (x)$ are extracted as functions of the lateral coordinate from the DD simulations performed using the device and circuit simulator Minimos-NT [124, 125]. These quantities are then employed to estimate the carrier temperature $T_{\mathrm {n}}(x)$ using the balance equation:

$(4.23) \begin{equation} T_{\mathrm {n/p}}=T_{\mathrm {L}}+\frac {2}{3}\frac {q}{k_{\mathrm {B}}}\tau _{\mathrm {n/p}}\mu _{\mathrm {n/p}}E^{2},\label {eq:carrier_Temp} \end{equation}$

where $T_{\mathrm {L}}$ is the lattice temperature, $q$ the modulus of the electron charge, and $\tau _{\mathrm {n/p}}$ the energy relaxation time with typical values of $\tau _{\mathrm {n}}$ = 0.35 $\,$ ps for electrons and $\tau _{\mathrm {p}}$ = 0.4 $\,$ ps for holes [28].

The DF in Equation 4.22 can represent the carriers in thermal equilibrium as well as the non-equilibrium ones. The first part of Equation 4.22 is used to determine the carrier DF in the device regions with high energy carriers, whereas the full expression is required for the regions where both hot and cold carriers lead to defect generation [147]. For the former case, two parameters $A$ , and $\varepsilon _{\mathrm {ref}}$ need to be calculated which is done using the information on the available BTE moments, i.e., the estimated carrier concentration, and the carrier temperature.

$(4.24–4.25) \{begin}{align} & \int _{0}^{\infty }f(\varepsilon )g(\varepsilon )d\varepsilon =n\label {eq:n}\\ & \int _{0}^{\infty }\varepsilon f(\varepsilon )g(\varepsilon )d\varepsilon =\frac {3}{2}nk_{\mathrm {B}}T_{\mathrm {n}}\label {eq:Tn} \{end}{align}$

The Equations 4.24 and 4.25 are reduced to analytical form (for details see Appendix C) using the expression for distribution function $f(\varepsilon )$ from Equation 4.22 while for the density of states $g(\varepsilon )$ the expression proposed in [177] is used:

$(4.26) \begin{equation} g(\varepsilon )=g_{\mathrm {0}}\sqrt {\varepsilon }\left (1+(\eta \varepsilon )^{\varsigma }\right ).\label {eq:DOS} \end{equation}$

Equation 4.26 very accurately reproduces the conventional non-parabolic Kane relation but can be integrated analytically [177]. The parameters $A$ , and $\varepsilon _{\mathrm {ref}}$ thus obtained are substituted in Equation 4.22 to determine the energy DF.

When using the full Equation 4.22, three parameters $A$ , $\varepsilon _{\mathrm {ref}}$ , and $C$ need to be calculated, while the parameter $b$ is assigned a constant value of 1 near the drain and source regions and 2 otherwise. As before, the reduced forms of Equations 4.24 and 4.25, as shown in Appendix C, are used with $f(\varepsilon )$ from Equation 4.22 and $g(\varepsilon )$ from Equation 4.26. However, an additional Equation 4.28 derived from DF normalization, Equation 4.27, is employed so that the three required parameters can be determined.

$(4.27–4.28) \{begin}{align} & \int _{0}^{\infty }f(\varepsilon )d\varepsilon =1\label {eq:DF-norm}\\ & A\frac {\varepsilon _{\mathrm {ref}}}{b}\Gamma \left (\frac {1}{b}\right )-Ck_{\mathrm {B}}T_{\mathrm {n}}\exp \left [-\frac {\varepsilon }{k_{\mathrm {B}}T_{\mathrm {n}}}\right ]=1,\label {eq:DFnorm_reduced} \{end}{align}$

The Equation 4.28 is obtained from Equation 4.27 by assuming that the cold carriers are present in a small fraction of the energy region, thus the corresponding integral is not evaluated for the entire energy range. Instead, the indefinite form is used for the cold carrier term. The ternary equation system consisting of the reduced forms of Equations 4.24 and 4.25, and Equation 4.28 is thus solved to calculate the parameters $A$ , $\varepsilon _{\mathrm {ref}}$ , and $C$ . This equation system is solved at each energy point $\varepsilon$ and the parameters obtained are substituted back to the Equation 4.22 to construct the DF $f(\varepsilon )$ .