Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

4.2 Defect Distribution in the Oxide

Using the electrostatics described in the previous section, the defect population can be sampled in the oxide in a subsequent step. Every defect has a certain probability of capturing and emitting a charge, which is determined by the transition rates \( k^\mathrm {CB/VB}_{ij} \) and \( k^\mathrm {CB/VB}_{ji} \) in (3.20) from 2-state NMP theory. Using these transition rates it is possible to compute the occupancy of each trap according to the Master equations

(4.5) \{begin}{align} \begin {split} \frac {\dd p_i(t)}{\dd t}&= - p_i \sum \limits _{r=1}^{N_r}k_{ij,r} + p_j \sum \limits _{r=1}^{N_r}k_{ji,r}\\ \frac {\dd p_j(t)}{\dd t}&= p_i \sum \limits _{r=1}^{N_r}k_{ij,r} - p_j \sum \limits _{r=1}^{N_r}k_{ji,r}, \end {split} \{end}{align}

where \( p_{i,j} \) are the probabilities that a defect is in the initial/final state. These equations include the interaction of each defect with all charge reservoirs \( r \), which are in the case of a MOSFET the valence and conduction band of the channel and the gate. The solution of the differential equation system is

(4.6–4.7) \{begin}{align} \begin {split} p_j(t) &= \frac {k_{ij}}{k_{ij}+k_{ji}}+\Big (p_j(0)-\frac {k_{ij}}{k_{ij}+k_{ji}}\Big )\mathrm {e}^{-t(k_{ij}+k_{ji})} \end {split}\\ p_i(t) &= 1-p_j(t) \{end}{align}

and can be used to weigh the impact on the potential caused by the defects by

(4.8) \{begin}{align} V_\mathrm {traps} = -\varepsilon _0\varepsilon _\mathrm {r}WL\sum \limits _{n=1}^{N_\mathrm {T}}q_{\mathrm {T},n}d_\mathrm {ox}(1-\frac {x_\mathrm {T}}{d_\mathrm {ox}}) \label {eq:Vtraps} \{end}{align}

with

(4.9) \{begin}{align} q_\mathrm {T}(t) = q p_j(t) \{end}{align}

being the charge stored at a defect. Subsequently, the surface potential of the device can be computed using (4.2). The shift of the potential compared to the initial time step then gives the threshold voltage shift. The total threshold voltage shift thus depends on the probability for a charge transition which itself depends on the defect parameters, the applied gate voltage and the temperature as shown in Fig. 4.1. Note that in (4.8) the contribution of a defect to the total \( V_\mathrm {traps} \) depends on its depth and is considered according to the charge sheet approximation (CSA). However, recent studies have shown that the CSA considerably underestimates the impact of a single defect on the device performance, which might give rise to a slight overestimation of the trap densities in the current version of Comphy [210, MJJ3, MJC1, MJC2].

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Figure 4.1: The mean charge state of a pre-existing oxide defect depends on the defect parameters trap level \( E_\mathrm {T} \), relaxation energy \( E_\mathrm {R} \), curvature ratio \( R \) and position in the oxide \( x_\mathrm {T} \), on the applied gate voltage and the temperature. With these parameters it is possible to compute transition rates \( k_{ij} \) and \( k_{ji} \) using NMP theory which gives then the impact on the electrostatics using the charge sheet approximation.

Specified Defect Distributions

Each defect can be described by a set of model parameters: the relaxation energy \( E_\mathrm {R} \), the curvature ratio \( R \), the trap level \( E_\mathrm {T} \), the spatial position inside the oxide \( x_\mathrm {T} \), and the configuration coordinate offset \( \Delta Q \). The latter is only relevant when using a quantum mechanical 2-state NMP transition rate. Typically, in large area devices an ensemble of defects is assumed to be distributed across a so called defect band. For this, it has often been assumed that the trap level of defects is normally distributed, that relaxation energies are normally distributed as well and that the spacial position is uniformly distributed [136]. Since the curvature of the PECs is correlated with the relaxation energy, it is not necessary to treat \( R \) as a stochastically distributed unit. Finally, a uniform spatial defect concentration \( N_\mathrm {T} \) across parts of the dielectric is assumed. The parameters can be seen in Fig. 4.2. The sampling of the defects can either be done by Monte-Carlo sampling or more efficiently, the mean degradation of the defect band can be computed by sampling the parameters on a grid and introducing a weighting scheme.

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Figure 4.2: A defect is described by the relaxation energy (math image), the trap level (math image), the spatial position in the oxide (math image), the curvature ratio \( R \), and the configuration coordinate offset \( \Delta Q \). While the latter two are correlated with the energies, (math image) is assumed to be uniformly distributed and the energies are normally distributed in defect bands. With the specified distributions, defects can then be sampled in the band using Monte-Carlo sampling.

Effective Single Defect Decomposition

While the approach of using Gaussian trap bands has proven to be very successful [136], it has the disadvantage that large relaxation energies, compared to values obtained from DFT calculations of suitable oxide defect candidates [185], are required eventually [MJJ2]. This can give rise to unphysically large charge transition times, as exemplary shown for the shallow SiO2  trapband of the 28 nm technology examined in [136] in Fig. 4.3, where the majority of all sampled defects shows transition times above \(10^7\) s, which is experimentally inaccessible. The smaller time constants correspond to the tail of the distribution of the relaxation energies. Thus, an alternative approach using an effective single defect decomposition (ESiD) has been developed to avoid artificially high relaxation energies. In this approach no assumptions on the defect parameter distributions are made when the model is calibrated to experimental data [MJJ2].

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Figure 4.3: Gaussian trap bands as used in the 28 nm technology published in [136] show that only a small part of the distribution results in active defects. The majority of the extracted defect parameters of the trap band leads to charge transistion times above \(10^7\) s (left), and is thus experimentally inaccessible. These large charge transition times correspond to relaxation energies \( E_\mathrm {R} \) larger than compared to those derived from DFT calculations (right) and are thus an artifact of the Gaussian distributions. Figure recreated from [MJJ2].

When using the ESiD approach, the total threshold voltage shift \( \Delta \vth \) is expressed as superposition of the threshold voltage shifts caused by each defect \( \delta \vth \)

(4.10) \{begin}{align} \Delta \vth = \sum \limits _{E_\mathrm {T}, E_\mathrm {R}, x_\mathrm {T}, \Delta Q} N(t; E_\mathrm {T}, E_\mathrm {R}, x_\mathrm {T}, \Delta Q) \delta V_\mathrm {th}(E_\mathrm {T}, E_\mathrm {R}, x_\mathrm {T}, \Delta Q) \{end}{align}

where \( N(E_\mathrm {T}, E_\mathrm {R}, x_\mathrm {T}, \Delta Q) \) is the weight of each defect. Instead of assuming Gaussian distributions for \( E_\mathrm {T} \) and \( E_\mathrm {R} \), the ESiD algorithm employs a uniform parameter grid sampled for \( E_\mathrm {T} \), \( E_\mathrm {R} \), \( x_\mathrm {T} \) and \( \Delta Q \). These parameters span a grid that is considered in the response matrix \( \pmb {(\delta V)} \), where at every point \( i \) the response \( \delta V_\mathrm {th} \) is computed. By using the response matrix and defining the observation vector \( \pmb {(\Delta V)_j} = \Delta \vth (t_j) \) a non-negative linear least square (NNLS) algorithm can be used for computing the weights \( \pmb {(N)_i}=N(E_{\mathrm {T},i}, E_{\mathrm {R},i}, x_{\mathrm {T},i}, \Delta Q_i) \) to

(4.11) \{begin}{align} \pmb {N} = \arg \min _{\pmb {\hat {N}}\geq 0}\left \lVert {\pmb {(\delta V)}\cdot \pmb {\hat {N}}-\pmb {\Delta V}}\right \rVert _2^2. \{end}{align}

Since this is mathematically an ill-posed problem that can lead to solutions with physically unrealistic high defect densities, a Tikhonov regularization [211] is added to ensure that the least-square solution results in smoother defect densities

(4.12) \{begin}{align} \pmb {N} = \arg \min _{\pmb {\hat {N}}\geq 0}\left \lVert {\pmb {(\delta V)}\cdot \pmb {\hat {N}}-\pmb {\Delta V}}\right \rVert _2^2+\gamma ^2\left \lVert \pmb {\hat {N}}\right \rVert _2^2 \{end}{align}

where \( \gamma \) is the regularization parameter. The ESiD algorithm allows an efficient extraction of trap parameters from measurement data which can be compared for their agreement with DFT simulations [MJJ2, 212]