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

Chapter 8 Role of Electron-Electron Scattering

An important aspect in modeling hot-carrier degradation for short-channel devices is electron-electron scattering (EES) [85, 86, 13, 24]. The steps for calculating the carrier DFs with EES are shown in Figure 8.1. EES populates the high-energy fraction of the carrier ensemble, thereby considerably changing the shape of the carrier DF, which is visible in a characteristic hump pronounced at high energies, see Figure 8.2 [11, 15]. Apparently EES also determines the temperature behavior of HCD in scaled devices [11, 88]. To obtain the carrier DF, the BTE should be solved taking this scattering effect into account. However, it might be possible to approximate the DFs utilizing the knowledge of the effects of EES on the DF. In this section, the drift-diffusion based model for the carrier energy distribution

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Figure 8.1: Flow chart for the calculation of the carrier DF.

function, Section 4.3, which was derived to describe hot-carrier degradation in LDMOS transistors, is extended for the case of decananometer devices nMOSFETs with gate lengths of 65, 150, and 300 \( \, \)nm.

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Figure 8.2: A schematic representation of the carrier DFs for the channel area (left panel) and for the drain (right panel) with all characteristic peculiarities marked.

Special attention is paid to the effect of EES on the carrier DFs. To approximately consider the important effect of electron-electron scattering on the shape

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Figure 8.3: Comparison of carrier distribution functions calculated with ViennaSHE and our DD-based method in nMOSFET with \( L_{\mathrm {G}} \) = 65 \( \, \)nm for stress condition \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 1.8 \( \, \)V. DFs are plotted for four different positions in the device ( \( x \) = -32.5 \( \, \)nm corresponds to the source, while \( x \) = 32.5 \( \, \)nm is related to the drain).

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Figure 8.4: Same as Figure 8.3 but for \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 2.0 \( \, \)V.

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Figure 8.5: Same as Figure 8.3 but for \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 2.2 \( \, \)V.

of the distribution function, a balance equation is solved for the in- and out-scattering rates. The DFs obtained from the extended analytic approach are compared with those calculated with the deterministic Boltzmann transport equation solver ViennaSHE. Both sets of DFs are then used with the hot-carrier degradation model, Section 4.4, to calculate changes in the drain current as a function of stress time. The results of the extended DD-based model are compared with the results obtained using the full solution of the Boltzmann transport equation and with experimental data. The accuracy and limits of the applicability of the DD-based model is also studied on devices with a range of gate lengths. This model allows to avoid the computationally expensive solution of the BTE but is expected to still provide excellent accuracy.

8.1 Extended DD-Based Distribution Function Model

The reference electron DFs have been calculated with ViennaSHE for three different devices of the same architecture with SiON gate dielectric but with different gate lengths of 65, 150, and 300 \( \, \)nm and for three different stress conditions ( \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 1.8, 2.0, 2.2 \( \, \)V). The resulting DFs for the 65 \( \, \)nm device are shown in Figures 8.3 \( - \) 8.5. From these DF curves one

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Figure 8.6: The knee energy \( \varepsilon _{\mathrm {k,2}} \), where EES starts to dominate the high-energy tail of the carrier DF, calculated with the analytic model plotted vs. ViennaSHE results for two different stress conditions \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 1.8 \( \, \)V (top) and 2.0 \( \, \)V (bottom).

can see that the DF shape substantially changes from the source to the drain. Indeed, DFs evaluated near the source and in the channel have a maximum visible at low and moderate energies with the corresponding energy labeled as \( \varepsilon _{\mathrm {k,1}} \) in Figure 8.2. As for the drain DFs, they typically feature a Maxwellian tail visible at low energies followed by a plateau. The end of this plateau also corresponds to the knee energy \( \varepsilon _{\mathrm {k,1}} \), see Figure 8.2. Another characteristic energy where the DF changes its curvature is related to the onset of the EES hump and is labeled as \( \varepsilon _{\mathrm {k,2}} \).

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Figure 8.7: The knee energy \( \varepsilon _{\mathrm {k,2}} \), where EES starts to dominate the high-energy tail of the carrier DF, calculated with the analytic model plotted vs. ViennaSHE results for stress condition \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 2.2 \( \, \)V.

Such sophisticated DFs can be approximated by following the algorithm depicted in Figure 8.1. First, Equation 8.1 allows for a rough evaluation of the electron DF without the effect of electron-electron scattering using the parameters \( A \), \( \varepsilon _{\mathrm {ref}} \), and \( C \) evaluated as earlier in Section 4.3

(8.1) \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-1} \end{equation}

The parameter \( b \) assumes a different value after every knee position, see Figure 8.2. In the channel and source regions \( b \) is assigned a value of -1 before \( \varepsilon _{\mathrm {k,1}} \), 1 between \( \varepsilon _{\mathrm {k,1}} \) and \( \varepsilon _{\mathrm {k,2}} \), and 2 after \( \varepsilon _{\mathrm {k,2}} \). While for the drain region, \( b \) has a value of 0.2 before \( \varepsilon _{\mathrm {k,1}} \), 1 between \( \varepsilon _{\mathrm {k,1}} \) and \( \varepsilon _{\mathrm {k,2}} \), and 2 after \( \varepsilon _{\mathrm {k,2}} \).

The first knee energy in the low energy region is evaluated analytically using the electric field from the DD simulations:

(8.2) \begin{equation} \varepsilon _{\mathrm {k,1}}=\alpha \exp [\beta -\left (\gamma -\delta E\right )]^{\frac {1}{2}}, \end{equation}

where \( E \) is the electric field, while \( \alpha \), \( \beta \), \( \gamma \) , and \( \delta \) are fitting parameters with values 0.4157 \( \, \)eV, 1.3, 11.04, \( 1.51\times 10^{-6} \) \( \, \)cmV \( ^{-1} \), respectively. The second knee energy, where EES starts to dominate the high energy tail, is obtained by considering the balance of in- and out-scattering rates. The in-scattering rate corresponds to EES, while the out-scattering includes such mechanisms as interactions of electrons with phonons and ionized impurities [206, 207].

(8.3) \begin{equation} r_{\mathrm {ees}}=r_{\mathrm {ii}}+r_{\mathrm {op/abs}}+r_{\mathrm {op/emi}}+r_{\mathrm {acc}}\label {eq:ek2} \end{equation}

The collision integral, for a nondegenerate semiconductor, is written as [175]:

(8.4) \begin{equation} \frac {\partial f}{\partial t}=-\sum S(p,p_{2};p',p_{2}')f(p)f(p_{2})+\sum S(p',p_{2}';p,p_{2})f(p')f(p_{2}').\label {eq:scattering-1} \end{equation}

suggesting that carriers with momentum \( p \) out-scatter and carriers with \( p' \) in-scatter due to carriers with \( p_{2} \) and \( p_{2}' \) respectively. Since there is an equal probability for a carrier being in/out scattered, Equation 8.4 can be written as:

(8.5) \begin{equation} \frac {\partial f}{\partial t}=-\sum S(p,p_{2};p',p_{2}')\left (f(p)f(p_{2})-f(p')f(p_{2}')\right ).\label {eq:scattering-2} \end{equation}

Equilibrium is reached when the in- and out-scattering rates balance.

For the scattering rates in Equation 8.3 the same standard expressions [175, 208] were used in both ViennaSHE and the DD-based model. The scattering rate of carriers by accoustic phonons \( r_{\mathrm {acc}} \) can be written as [175]

(8.6) \begin{equation} r_{\mathrm {acc}}=\frac {D_{\mathrm {A}}^{2}k_{\mathrm {B}}T_{\mathrm {L}}m^{*}p}{\pi c_{\mathrm {l}}\hbar ^{4}}, \end{equation}

where \( D_{\mathrm {A}} \) is the accoustic deformation potential, \( c_{\mathrm {l}} \) is the elastic constant ( \( v_{\mathrm {s}}=\sqrt {c_{\mathrm {l}}/\rho } \), \( v_{\mathrm {s}} \) is the velocity of sound and \( \rho \) is the mass density). For a spherical parabolic band the rate becomes

(8.7) \begin{equation} r_{\mathrm {acc}}=\frac {D_{\mathrm {A}}^{2}k_{\mathrm {B}}T_{\mathrm {L}}m^{*}\sqrt {2m^{*}\varepsilon }}{\pi c_{\mathrm {l}}\hbar ^{4}}. \end{equation}

Note that for a fair comparison the same physical parameters are used in both the SHE code and the analytic approximation.

Optical phonon scattering can be due to absorption or emission of the optical phonons. The corresponding rates are given by Equation 8.8 and 8.9, respectively [175]:

(8.8) \begin{equation} r_{\mathrm {op/abs}}=\frac {D_{\mathrm {o}}^{2}N_{\mathrm {o}}m^{*}\sqrt {2m^{*}\left (\varepsilon +\hbar \omega _{\mathrm {o}}\right )}}{2\pi \rho \omega _{\mathrm {o}}\hbar ^{3}}\label {eq:Optical_abs} \end{equation}

(8.9) \begin{equation} r_{\mathrm {op/emi}}=\frac {D_{\mathrm {o}}^{2}\left (N_{\mathrm {o}}+1\right )m^{*}\sqrt {2m^{*}\left (\varepsilon -\hbar \omega _{\mathrm {o}}\right )}}{2\pi \rho \omega _{\mathrm {o}}\hbar ^{3}}\label {eq:Optical_emi} \end{equation}

where \( D_{\mathrm {o}} \) is the accoustic deformation potential, \( \hbar \omega _{\mathrm {o}} \) is the energy of optical phonon, \( N_{\mathrm {o}} \) is the density of optical phonons ( \( N_{\mathrm {o}}=1/\left (\exp \left [\hbar \omega _{\mathrm {o}}/k_{\mathrm {B}}T_{\mathrm {L}}\right ]-1\right ) \)).

The ionized impurity scattering rates are [175]:

(8.10) \begin{equation} r_{\mathrm {ii}}=\frac {N_{\mathrm {I}}q^{4}}{16\sqrt {2m^{*}}\pi \left (\epsilon _{\mathrm {Si}}\epsilon _{\mathrm {0}}\right )^{2}}\left [\ln \left (1+\gamma _{\mathrm {D}}^{2}\right )-\frac {\gamma _{\mathrm {D}}^{2}}{1+\gamma _{\mathrm {D}}^{2}}\right ]\frac {1}{\varepsilon ^{3/2}} \end{equation}

where

\[ \gamma ^{2}=8m^{*}\varepsilon L_{\mathrm {D}}^{2}/\hbar ^{2} \]

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Figure 8.8: The electron DFs simulated with ViennaSHE and with the DD-based model for nMOSFETs with gate lengths of 150 and 300 \( \, \)nm for \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 1.8 \( \, \)V. These DFs are plotted for different positions in the device. The source is at \( x \) = \( -L_{\mathrm {G}}/2\, \)nm and the drain is at \( x \) = \( +L_{\mathrm {G}}/2\, \)nm.

with \( L_{\mathrm {D}} \) being the Debye length given as:

\[ L_{\mathrm {D}}=\sqrt {\frac {\epsilon _{\mathrm {Si}}\epsilon _{\mathrm {0}}k_{B}T}{q^{2}n_{0}}} \]

As for electron-electron scattering, we use the following expression [208]:

(8.11) \begin{equation} r_{\mathrm {ees}}=\frac {mq^{4}n}{\epsilon _{\mathrm {Si}}^{2}\epsilon _{\mathrm {0}}^{2}\hbar ^{3}}\sum _{\varepsilon }\frac {\sqrt {2m^{*}}/\hbar \left |\sqrt {\varepsilon }-\sqrt {\varepsilon _{0}}\right |}{\beta ^{2}\left (\frac {2m^{*}}{\hbar ^{2}}\left (\varepsilon -\varepsilon _{0}\right )+\beta ^{2}\right )}f\left (\varepsilon \right ) \end{equation}

where \( \mathbf {\beta } \) is the inverse Debye length. It is important to emphasize that calculation of the rate \( r_{\mathrm {EES}} \) requires a DF. For energies \( E<\varepsilon _{\mathrm {k,2}} \) the DF calculated with Equation 4.22 can be used, while for \( E\geq \varepsilon _{\mathrm {k,2}} \) the DF is perturbed due to EES. Once the knee energies are obtained, the DFs are evaluated by changing the value of the parameter \( b \) at the knees in Equation 4.22. Same approach is followed for the 150 and 300 \( \, \)nm devices to obtain the DFs in Figure 8.8.

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Figure 8.9: The knee energy \( \varepsilon _{\mathrm {k,2}}(x) \) for 150 and 300 \( \, \)nm devices calculated with the analytic model plotted vs. ViennaSHE results for stress conditions \( V_{\mathrm {gs}} \) = \( V\mathrm {_{ds}} \) = 1.8 \( \, \)V and 2.0 \( \, \)V.