Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

3 Charge Transfer Models

Charge trapping is a stochastic process and refers to the exchange of charge carriers between a defect and a charge reservoir. A mathematical description of the trapping and detrapping probabilities is central for making this process available to simulators and to enable reliability studies of MOSFETs. Note that charge trapping is a dynamic process which is determined by the applied biases, temperatures as well as intrinsic properties of the particular defect like its trap level and relaxation energy. As such, the development of a theoretical model which covers all features of charge trapping is a formidable challenge.

The very first model for describing charge transition rates was proposed by Hall [171] and Shockley and Read [172] in the 1950s, and is thus called Shockley-Read-Hall (SRH) model. This model is well suited for describing trapping at the interface with little relaxation, which was one of the biggest fabrication issues in the first generation of transistors as described in Section 1.1. This model will be outlined in Section 3.1. However, the SRH model can not accurately describe the bias and temperature dependence observed in many BTI and RTN experiments, where oxide defects are involved, which typically show a significant relaxation upon charge transfer. In order to account for such relaxation effects, the nonradiative multiphonon (NMP) model is introduced, which describes charge transition events accurately. While for many applications a 2-state NMP model is sufficient for describing the defect kinetics, there are phenomena such as anomalous RTN (aRTN), which prove the existence of additional metastable states and thus require an extension to a 3 or 4-state model.

In Section 3.2 the 2-state NMP model will be introduced. Since the calculation of the full quantum mechanical transition rate is computationally expensive, a classical approximation is typically used for above room temperature applications. However, this classical approximation is not suitable for cryogenic applications, because it does not take nuclear tunneling into account. Therefore, a WKB-based, numerically efficient approximation has been developed, which is presented and benchmarked in Section 3.2.1 and Section 3.2.2, respectively, based on [MJJ1]. An even more efficient approximation can be used when assuming that the potential energy surfaces which represent a defect are undistorted before and after a charge transition, see Section 3.2.3. In Section 3.3 the 4-state NMP model will outlined, which can be necessary for describing single defect measurements.

3.1 Shockley-Read-Hall Model

With the Shockley-Read-Hall model, defects are described by a neutral and a charged configuration state. The defects can either be acceptor- or donor like. Acceptor-like defects accept an electron during charge trapping and thus have a negative charged state, while donor-like defects donate an electron and become positively charged. Additionally, the terminology of electron- and hole-like defects is frequently used, depending on the charged state. This leads to four different processes shown in Fig. 3.1:

• Electron capture: A defect traps an electron from the conduction band, which is represented by $0\rightarrow -$ . This is equivalent to a hole transitioning from the defect to the conduction band, represented by $+\rightarrow 0$ .
• Electron emission: A defect releases an electron to the conduction band, which is represented by $-\rightarrow 0$ . This is equivalent to a hole captured from the conduction band, which is represented by $0\rightarrow +$ .
• Hole capture: A defect traps a hole from the valence band, which is represented by $0\rightarrow +$ . This is equivalent to an electron release from the defect to the valence band represented by $- \rightarrow 0$ .
• Hole emission: A defect emits a hole to the valence band, which is represented by $+\rightarrow 0$ . This is equivalent to an electron capture from the valence band, represented by $0\rightarrow -$ .

Analogously to the derivation in [140], the hole capture process is used in the following to present the mechanics of the SRH model. However, the derivation can be done in a similar fashion for hole emission and electron trapping. In the case of hole capture, the defect can be in the initial neutral configuration 0 and in the charged configuration $+$ . In the SRH model, capturing a hole from the valence band is possible without overcoming a barrier, leading to a hole capture probability $c_p$ which is often written as being directly proportional to the thermal velocity $v_\mathrm {th}=\sqrt {8k_\mathrm {B}T/(\pi m_{p,\mathrm {eff}})}$ of the charge carriers

$(3.1) \{begin}{align} c_{p}\approx v_\mathrm {th}\sigma , \{end}{align}$

with the electron temperature $T$ , the effective hole mass $m_{p,\mathrm {eff}}$ and the capture cross section $\sigma$ . By taking into account the hole occupancy $f_p(E)$ which follows a Fermi-Dirac distribution, and the density of states $D_\mathrm {p}$ , the capture rate can be computed by

$(3.2) \{begin}{align} k_{0+}=\int \limits _{-\infty }^{E_\mathrm {v}}c_{p}(E)f_{p}(E)D_{p}(E)\mathrm {d}E. \{end}{align}$

Similarly, the hole emission rate is given by

$(3.3) \{begin}{align} k_{+0}=\int \limits _{-\infty }^{E_\mathrm {v}}e_{p}(E)(1-f_\mathrm {p}(E))D_{p}(E)\mathrm {d}E, \{end}{align}$

by replacing the capture probability with the emission probability $e_{p}$ and considering the Fermi-Dirac distribution of the electrons $f_{n}$ . Using

$(3.4) \{begin}{align} \frac {f(E)}{1-f(E)}=\mathrm {e}^{-\beta (E-E_\mathrm {F})} \{end}{align}$

with $E_\mathrm {F}$ being the Fermi level and $\beta =1/k_\mathrm {B}T$ , the rate can be expressed as

$(3.5) \{begin}{align} k_{+0}=\int \limits _{-\infty }^{E_\mathrm {v}}e_{p}(E)f_{p}(E)\mathrm {e}^{-\beta (E-E_\mathrm {F})}D_{p}(E)\mathrm {d}E. \{end}{align}$

The principle of detailed balance states that for an arbitrary energy $E$ the probability of being in state $0$ , $f_0=f(E_\mathrm {T})$ times the transition rate from state 0 to $+$ is equal to the reversed process, which is the probability of being in state $+$ times the transition rate from $+$ to $0$

$(3.6) \{begin}{align} f_0\mathrm {d}k_{0+}(E) = f_+\mathrm {d}k_{+0}(E). \{end}{align}$

From detailed balance the following relation between emission and capture probability can be derived directly as

$(3.7) \{begin}{align} e_\mathrm {p}(E) = c_\mathrm {p}(E)\mathrm {e}^\frac {E-E_\mathrm {T}}{k_\mathrm {B}T}. \{end}{align}$

This allows expressing the hole emission rate

$(3.8) \{begin}{align} k_{+0}=\int \limits _{-\infty }^{E_\mathrm {v}}c_{p}(E)\mathrm {e}^{\beta (E-E_\mathrm {T})}f_{p}(E)\mathrm {e}^{-\beta (E-E_\mathrm {F})}D_{p}(E)\mathrm {d}E. \label {eq:hole_emission_rate} \{end}{align}$

For non-degenerated semiconductors, the Fermi-Dirac statistics can be approximated by the Boltzmann statistics. If additionally the band edge approximation is used, which allows to assume that all carriers are concentrated at the band edges, the integral in (3.8) can be carried out analytically leading to the simplified transition rates

$(3.9) \{begin}{align} \begin {split} k_{0+}&=pv_\mathrm {th}\sigma \\ k_{+0}&=N_\mathrm {v}v_\mathrm {th}\sigma \mathrm {e}^{\beta (E_\mathrm {v}-E_\mathrm {T})} \end {split} \{end}{align}$

with $p$ being the hole concentration and $N_\mathrm {v}$ the valence band weight. The SRH model is typically used for bulk and interface defects. To account for oxide traps, the capture cross section $\sigma$ can be expressed as $\sigma = \sigma _0\lambda$ with $\lambda$ being a Wentzel-Kramers-Brillouin (WKB) tunneling factor which describes the elastic tunneling from the charge reservoir to the defect, see Fig. 3.1. This WKB-tunneling factor allows in principle to include oxide defects [173]. However, bias and temperature dependence of transition rates of oxide defects can not be modeled accurately with the SRH model, therefore a more precise description is needed [174].