Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

7.2 1/f Noise

Based on the measured drain-source signals or the corresponding (math image) traces in the time domain, the power spectral densities (PSDs) in the frequency domain can be obtained using for example the Welch algorithm [230] or Bartlett’s method [231]. The PSD of a two-state RTN signal follows a Lorentzian, as can be seen in Fig. 7.1 (right). The Lorentzian of defect \( i \) can be described using

(7.1) \{begin}{align} S_i(f) = \frac {(2d_i\tau _{0,i})^2}{(\tau _{\mathrm {c},i}+\tau _{\mathrm {e},i})(1+(2\pi f \tau _{0,i})^2)}, \{end}{align}

where \( d_i \) is the step height and \( \tau _{0,i} = 1/(1/\tau _{\mathrm {c},i}+1/\tau _{\mathrm {e},i}) \). The superposition of multiple electrically active defects results in the superposition of the Lorentzian spectra

(7.2) \{begin}{align} S(f) = \sum \limits _i S_i(f) = \sum \limits _i\frac {(2d_i\tau _{0,i})^2}{(\tau _{\mathrm {c},i}+\tau _{\mathrm {e},i})(1+(2\pi f \tau _{0,i})^2)}.

As shown schematically in Fig. 7.1, on large area devices the Lorentzian spectra are uniformly distributed in \( \log (\tau ) \). When considering only a subset of defects in a small time interval, the approximation \( \taue =\tauc =\tau _i \) holds. Using the equivalent step height \( d_\mathrm {e}=d_i \) for all defects \( i \), allows to simplify the expression to

(7.3) \{begin}{align} S(f) \approx \frac {d_\mathrm {e}^2}{2} \sum \limits _i\frac {\tau _i}{1+(\pi f \tau _i)^2}. \{end}{align}

Due to the characteristic corners of the Lorentzian PSDs, a single defect always dominates at a certain frequency. For the time constant of this dominating defect \( \tau _i=1/(\pi f) \) holds at the corner frequency \( \pi f \). This allows considering only the dominant defect, leading to

(7.4) \{begin}{align} S(f) \approx \frac {d_\mathrm {e}^2}{4\pi f}. \{end}{align}

This final result shows how the superposition of Lorentzian PSDs stemming from single defects results in a 1/f signal which is observed on large area devices.

These general considerations about connections between RTN signals caused by single defects and 1/f noise in large area devices can be related to the empirical relation between the spectral density of the drain-source current \( S_{\id } \) and the number of charge carriers \( N \) in the channel proposed by Hooge [232]

(7.5) \{begin}{align} \frac {S_{\id }(f)}{\id ^2}=\frac {\alpha _\mathrm {H}}{f N} \{end}{align}

with \( \alpha _\mathrm {H} \) being the empirical Hooge parameter. Based on this empirical model, a unified flicker noise model [233] for MOSFETs was developed which describes the drain current PSD for a device with dimensions \( W \) and \( L \) as

(7.6) \{begin}{align} S_{\id }(f) = \frac {k_\mathrm {B}T\id ^2}{\gamma f W L}\left (\frac {1}{N}+\alpha \mu \right )^2N_\mathrm {t}(E_\mathrm {F,n}) \{end}{align}

with \( \gamma \) being a WKB tunneling factor for the interface traps following a trap distribution \( N_\mathrm {t}(E) \), which is evaluated at the channel quasi Fermi level \( E_\mathrm {F,n} \). \( N \) is the charge carrier density, \( \mu \) the carrier mobility and \( \alpha =\pi m_e q^3/16\varepsilon ^2 h k_\mathrm {B}T \) with the average dielectric constant \( \varepsilon = (\varepsilon _\mathrm {Si}+\varepsilon _\mathrm {Ox})/2 \). From this equation the corresponding gate voltage power spectral density can be derived using

(7.7) \{begin}{align} S_{\vg }(f) = \frac {S_{\id }(f)}{\gm ^2}. \{end}{align}

resulting in

(7.8) \{begin}{align} S_{\vg }(f) = \frac {k_\mathrm {B}T q^2}{\gamma f W L C_\mathrm {ox}^2}\big (1+\alpha \mu N \big )^2N_\mathrm {t}(E_\mathrm {F,n}). \{end}{align}

These models are well established at room temperature, however, the linear relation in \( T \), classical mobility models for \( \mu \), and the simplified evaluation of the trap density at the quasi Fermi level fail at cryogenic temperatures and cannot explain the increasing noise levels at 4.2 K compared to room temperature as shown in Fig. 7.2.

(-tikz- diagram)

Figure 7.2: The spectral noise density of (math image) (left) at 4.2 K (blue circles) increases compared to room temperature (red circles), while the spectral noise density of (math image) (right) stays constant. Both noise spectra show the occurrence of Lorentzian corners at cryogenic temperatures, which can be attributed to single defects. The red dashed line indicates 1/f behavior and the black dashed lines show Lorentzian corners in the spectrum. Figures taken from [234].

While the spectral noise density of (math image) increases towards 4.2 K, the spectral noise density of (math image) is rather temperature independent. However, both show the occurrence of Lorentzian features at cryogenic temperatures which can be attributed to single defects. This shows the importance of understanding the role of single defects at cryogenic temperatures and the necessity of a charge transition model being valid in the cryogenic regime.