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3 Experimental Characterization

The study of degradation mechanisms prevalent in transistors includes a thorough experimental characterization of time-dependent variation and time dependent drifts of MOSFET parameters. Therefore, different measurement methods and sequences have been developed. In this chapter, an overview is given and the purposes, advantages and challenges of commonly used techniques are introduced. Then, challenges which had to be faced during the measurements for this thesis, are discussed. In this context, the fact that most of the measurement techniques have been developed for the characterization of either NBTI or HCD degradation plays an important role. For the discussion of such challenges, the focus lies on the $V_{\mathrm {th}}$ extraction methods applied using MSM techniques as this method is most relevant for the understanding of the results presented in Chapter 5.

Figure 3.1: **Range of stress conditions:** Schematic illustration of the different stress conditions NBTI, HCD and mixed NBTI/HC. The area of stress conditions defines the boundaries of the 2-dimensional parameter space $($ $V_{\mathrm {D}}^\mathrm {str}$ $,$ $V_{\mathrm {G}}^\mathrm {str}$ $)$ applied in this thesis. Different colors separate the voltage combinations which trigger different degradation mechanisms, while the color intensity indicates the increasing impact of the stress on the parameter shifts.

$ ( $ — Figure 3.1: **Range of stress conditions:** Schematic illustration of the different stress conditions NBTI, HCD and mixed NBTI/HC. The area of stress conditions defines the boundaries of the 2-dimensional parameter space $($ $V_{\mathrm {D}}^\mathrm {str}$ $,$ $V_{\mathrm {G}}^\mathrm {str}$ $)$ applied in this thesis. Different colors separate the voltage combinations which trigger different degradation mechanisms, while the color intensity indicates the increasing impact of the stress on the parameter shifts.

As an introduction to this chapter, Figure 3.1 illustrates the stress conditions for triggering both degradation mechanisms, NBTI and HCD, including the so-called mixed NBTI/HC stress conditions applied for the characterization of the unavoidable interplay of both. The region of stress conditions in this figure shows the 2-dimensional parameter space $($ $V_{\mathrm {D}}^\mathrm {str}$ $,$ $V_{\mathrm {G}}^\mathrm {str}$ $)$ applied in this thesis. The measurements discussed in the current and the following chapters were conducted on 2.2 nm $\ch{SiON}$ pMOSFETs of a 130 nm commercial technology ( $V_{\mathrm {DD}}$   $=$  −1.5 V and $V_{\mathrm {th}}$   $=$  465 mV). One has to distinguish between large-area devices and nano-scale devices. The first have the dimensions $W$   $=$  10 µm and $L$   $=$  120 nm or $L$   $=$  130 nm, and the second $W$   $=$  160 nm and $L$   $=$  120 nm or $L$   $=$  130 nm.

3.1 On-The-Fly (OTF) Measurements

Using the OTF measurement method, MOSFET parameter shifts are probed directly during operation without interruption of the applied voltages. It was introduced in 2004 for the purpose of measuring the device degradation $\Delta V_{\mathrm {th}}$ during stress conditions [106], which typically refer to much higher biases than used at nominal operating conditions. This technique aims for the direct characterization of the $\Delta V_{\mathrm {th}}$ evolution.

The basic measurement procedure is shown in Figure 3.2 for BTI measurements. It consists of a periodic modulation of $V_{\mathrm {G}}^\mathrm {str}$ with the modulation amplitude ( $\Delta V_\mathrm {G}^\mathrm {str}/2$ ) at a certain drain measurement voltage ( $V_\mathrm {D}^\mathrm {meas}$ ) while $I_\mathrm {D}$ is determined at three measurement points for each modulation period $n$ . The modulation of $V_{\mathrm {G}}^\mathrm {str}$ induces a $\Delta I_\mathrm {D}(n)$ , which changes over the modulation periods due to the degradation-induced $\Delta V_{\mathrm {th}}$ during stress. From the modulation amplitude and the corresponding $\Delta I_\mathrm {D}$ the transconductance can be obtained according to

$(3.1) \begin{equation} \label {eq:slopeidvg} g_\mathrm {m}(V_\mathrm {G}^\mathrm {str},V_\mathrm {D}^\mathrm {meas},t_\mathrm {str})= \dfrac {\partial I_\mathrm {D}}{\partial V_\mathrm {G}}\bigg |_{t_\mathrm {str},V_\mathrm {th}}=-\dfrac {\partial I_\mathrm {D}}{\partial V_\mathrm {th}}\bigg |_{t_\mathrm {str},V_\mathrm {G}}\approx \dfrac {\Delta I_\mathrm {D}}{\Delta V_\mathrm {G}^\mathrm {str}} \end{equation}$

with

Figure 3.2: **OTF measurement procedure at NBTI conditions:** $V_{\mathrm {G}}^\mathrm {str}$ is modulated periodically at a certain $V_\mathrm {D}^\mathrm {meas}$ while $I_\mathrm {D}$ is determined. With this, the transcon- ductance $g_\mathrm {m}=\Delta I_\mathrm {D}/\Delta V_\mathrm {G}^\mathrm {str}$ can be calculated for each modulation step.

A step-by-step integration of $\partial$ $I_\mathrm {D}$ / $g_\mathrm {m}$ gives the threshold voltage shift during stress:

$(3.2) \begin{equation} \label {eq:deltaVth} \Delta V_\mathrm {th}(t_\mathrm {str})=-\int _{I_\mathrm {D}(0)}^{I_\mathrm {D}(t_\mathrm {str})}\dfrac {\partial I_\mathrm {D}}{g_\mathrm {m}(t_\mathrm {str})}\approx -\sum _{n=1}^N \dfrac {I_\mathrm {D}(n)-I_\mathrm {D}(n-1)}{1/2(g_\mathrm {m}(n)+g_\mathrm {m}(n+1))} \end{equation}$

with

One challenge arises due to the fact that a parameter which is typically measured at a level near the threshold regime, namely $V_{\mathrm {th}}$ , is extracted from a measurement at stress level where the applied voltages are considerably higher than the threshold voltage. A proper separation of the impact of mobility fluctuations at stress level and $V_{\mathrm {th}}$ drifts at a level near the threshold regime on the measurement is, therefore, an issue [107]. The reason is that shifts of the transfer characteristics along the $V_\mathrm {G}$ -axis ( $V_{\mathrm {th}}$ drifts) and “tilts" of the transfer characteristics (see Figure 3.3) are indistinguishable at the stress level. Such “tilts" are caused by defects located near or at the oxide/substrate interface which contribute to surface scattering and lower the carrier mobility. Based on a simple SPICE level 1 model, the following three parameter equation meets this challenge and separates the mobility and the $V_{\mathrm {th}}$ effect from each other:

Figure 3.3: **Transfer characteristic before and after stress:** Both curves are fitted using the SPICE level 1 model (dashed lines), which describes $I_\mathrm {D}$ above $g_\mathrm {m,max}$ very well. The difference between the zero crossing points is $\Delta V_{\mathrm {th}}$ . Figure source: [107].

$ I_\mathrm {D} $ — Figure 3.3: **Transfer characteristic before and after stress:** Both curves are fitted using the SPICE level 1 model (dashed lines), which describes $I_\mathrm {D}$ above $g_\mathrm {m,max}$ very well. The difference between the zero crossing points is $\Delta V_{\mathrm {th}}$ . Figure source: [107].

$(3.3) \begin{equation} \label {eq:OTF} G_\mathrm {SD}= \beta \dfrac {V_\mathrm {G}-V_\mathrm {th}}{1+\Theta (V_\mathrm {G}-\mathrm {V_\mathrm {th}})} \end{equation}$

with

The SPICE model has an empirical background and describes $I_\mathrm {D}$ above $g_\mathrm {m,max}$ in the linear $V_\mathrm {D}$ regime rather well, as shown in Figure 3.3. However, it does not explain the threshold voltage shifts near the regime typically defined as the threshold regime in other measurement methods.

The extraction of $\Delta V_{\mathrm {th}}$ depends on the modulation around $V_{\mathrm {G}}^\mathrm {str}$ . Therefore, two measurement points on the transfer characteristics, namely $I_\mathrm {D}$ ( $V_\mathrm {0}=$ $V_{\mathrm {G}}^\mathrm {str}$ - $\Delta$ $V_{\mathrm {G}}^\mathrm {str}$ /2) and $I_\mathrm {D}$ ( $V_\mathrm {1}=$ $V_{\mathrm {G}}^\mathrm {str}$ + $\Delta$ $V_{\mathrm {G}}^\mathrm {str}$ /2) or $G_\mathrm {0}=G_\mathrm {SD}(V_\mathrm {0})$ and $G_\mathrm {1}=G_\mathrm {SD}(V_\mathrm {1})$ , respectively, are inserted into Equation 3.3. Assuming that $\Theta$ is known from an initial characterization of the device the two-equation system containing only two unknowns, $\beta$ and $V_{\mathrm {th}}$ , can be solved and as a result the threshold voltage can be calculated using

$(3.4) \begin{equation} \label {eq:VtSPICE} V_\mathrm {th}= V_\mathrm {0} - \dfrac {-(1+\Theta \Delta V_\mathrm {G}^\mathrm {str})+\sqrt {(1+\Theta \Delta V_\mathrm {G}^\mathrm {str})^2+4\Theta G_\mathrm {0}/g_\mathrm {m}}}{2\Theta } \end{equation}$

with

$ V_\mathrm {G}^\mathrm {str}-\Delta V_\mathrm {G}^\mathrm {str}/2 $

Unfortunately, the extraction of $V_{\mathrm {th}}$ from a measurement at stress level and not from a measurement in the threshold regime introduces errors. In general, due to the modulation of the gate bias, the stress level changes, leading to a different degradation state than the state obtained after constant stress over the whole $t_\mathrm {str}$ . For example, it has been shown that $\Delta V_{\mathrm {th}}$ is underestimated with the OTF method [43]. This systematic error could be minimized by a small modulation amplitude $\Delta V_\mathrm {G}^\mathrm {str}$ . However, the error in $g_\mathrm {m}$ in Equation 3.1 is inversely proportional to $\Delta V_\mathrm {G}^\mathrm {str}$ and thus the smaller the amplitude is, the more error is introduced to $g_\mathrm {m}$ . Since the error in $g_\mathrm {m}$ determines the error of the $V_{\mathrm {th}}$ extraction in Equation 3.4, the choice of the modulation amplitude affects the statistical error in $V_{\mathrm {th}}$ [107].

Moreover, the modulation time plays a major role in the obtained degradation state. In order to minimize the change of the degradation state due to the modulation, the time within which the modulation is performed must be as short as possible. As a consequence, the integration time of the measurement has to be as short as possible. However, a decrease of the integration time in measurements leads to an increase of the statistical error of the measured $V_{\mathrm {th}}$ . Integration times in the order of µs or ms lead to a relative accuracy in the measured $I_\mathrm {D}$ of $10^{-4}$ or less. This corresponds to a statistical error of $V_{\mathrm {th}}$ of $\pm$ 12 mV, which is too high. In order to achieve a statistical error of, e.g., $\pm$ 1 mV in $V_{\mathrm {th}}$ , $I_\mathrm {D}$ has to be measured with a relative accuracy of $8\times 10^{-6}$ . With standard equipment, the integration time required to achieve such a relative accuracy would be more than 20 ms which enlarges the measurement time enormously. Unfortunately, 20 ms is way too long if the prevention of recovery during the measurement is required. [107]

The OTF method is quite sensitive to mobility changes induced by stress [108] while it is quite insensitive to $V_{\mathrm {th}}$ changes because the $I_\mathrm {D}$ - $V_\mathrm {G}$ curve flattens out at the stress level [107]. Since $V_{\mathrm {th}}$ is the parameter of interest, an insensitivity to the changes of the threshold voltage shift is a considerable disadvantage. Together with the introduced systematic error due to the voltage modulation, this disadvantage makes the OTF method unfavorable for this thesis.

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$V_\mathrm {G}^\mathrm {str}$	gate stress voltage
$V_\mathrm {D}^\mathrm {meas}$	drain voltage applied during measurement
$I_\mathrm {D}$	drain current
$t_\mathrm {str}$	stress time
$V_\mathrm {th}$	threshold voltage.

$n$	sequential number of measurement
$N+1$	number of $I_\mathrm {D}$ measurements.

$G_\mathrm {SD}$	drain-source conductance
$\beta$	global mobility (parameter)
$V_\mathrm {G}$	gate voltage
$V_\mathrm {th}$	threshold voltage (parameter)
$\Theta$	leads to an asymptotic approach to a maximum value $G_\mathrm {SD}=\beta / \Theta$ describing
	the mobility decrease due to surface roughness for high gate fields (parameter).

$V_\mathrm {0}$	corresponds to $V_\mathrm {G}^\mathrm {str}-\Delta V_\mathrm {G}^\mathrm {str}/2$
$V_\mathrm {G}^\mathrm {str}$	gate stress voltage
$\Delta V_\mathrm {G}^\mathrm {str}$	amplitude of gate stress voltage modulation
$G_\mathrm {0}$	drain-source conductance at $V_\mathrm {G}^\mathrm {str}-\Delta V_\mathrm {G}^\mathrm {str}/2$
$\Theta$	asymptotic approach to a maximum value $G_\mathrm {SD}=\beta / \Theta$ describing
	the mobility decrease due to surface roughness for high gate fields.