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3.8 Temperature Accelerated Measurements

Typical capture and emission times of oxide defects vary by many orders of magnitude, from µs to weeks, depending on their properties, the temperature, and the bias conditions applied to the MOSFET. However, especially in TDDS measurements the experimental window is limited due to the fact that each measurement contains, e.g., $100$ stress/recovery cycles. In order to capture the characteristics of a defect with a characteristic emission time of one week, the measurement would take approximately two years for one gate and drain voltage combination. If it is taken into account that a thorough characterization of defects requires more than one measurement, such measurement durations are not feasible. However, defects with very large emission times are of special interest because they contribute among others to the permanent component of degradation, which is expected to dominate the device lifetime distribution. Therefore, their thorough characterization would be essential.

Not only defects with large emission times can be a challenge for the experimental characterization. Defects with emission times smaller than $t_\mathrm {rec,min}$ are a challenge as well. One possibility to overcome these challenges is to accelerate or to slow down the charge carrier exchange by changing the temperature during the stress and recovery phases in TDDS measurements independently. The impact of $T$ on $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ is discussed in Subsection 2.1.3. In order to shift $\tau _{\mathrm {c}}$ and $\tau _{\mathrm {e}}$ lying outside experimentally feasible time slots, defined temperature ramps (one example is shown in Figure 3.27) can be applied during stress and/or recovery. For this purpose, an in situ heating technology for temperature accelerated measurements has been introduced recently [119–121].

Figure 3.25: **Poly-heater-device system:** Polycrystalline silicon wires (poly-heater) are processed near the MOSFET and are electrically isolated.

Such temperature accelerated measurements can be based on local heaters realized as polycrystalline silicon wires (poly-heater). They are processed near the MOSFET and electrically isolated as shown in Figure 3.25. In contrast to the experimental setups for the $\Delta V_{\mathrm {th}}$ extraction in eMSM measurements, the poly-heater-device system has two additional contacts for the application of a voltage to the poly-heater (see Figure 3.26). As long as no voltage $V_\mathrm {PH}$ is applied to the wires, the MOSFET and the poly-heater are held at a fixed device temperature ( $T_\mathrm {dev}$ ) and at a fixed poly-heater temperature ( $T_\mathrm {PH}$ ), respectively, both corresponding to the thermo chuck temperature ( $T_\mathrm {chuck}$ ): $T_\mathrm {dev}$   $=$   $T_\mathrm {PH}$   $=$   $T_\mathrm {chuck}$ . When a voltage is applied to the wires resulting in a current flow through the poly-heater ( $I_\mathrm {PH}$ ), the dissociated heat corresponding to the power dissipated in the poly-heater ( $P_\mathrm {PH}$ ) elevates $T_\mathrm {PH}$ first. Immediately afterwards, a temperature gradient forms vertically across the device stack because $T_\mathrm {chuck}$   $<$   $T_\mathrm {PH}$ . As a consequence, $T_\mathrm {dev}$ is elevated as well: $T_\mathrm {chuck}$   $<$   $T_\mathrm {dev}$   $<$   $T_\mathrm {PH}$ .

Figure 3.26: **Experimental setup for temperature accelerated measurements:** The voltages applied to the gate and drain contacts are realized as constant voltage sources and $I_\mathrm {D}$ is measured using a transimpedance am- plifier. The heating with the polyheater is realized with a constant voltage source voltage applied to poly-heater ( $V_\mathrm {PH}$ ). Simultaneously the poly-heater cur- rent $I_\mathrm {PH}$ is measured.

$ I_\mathrm {D} $ — Figure 3.26: **Experimental setup for temperature accelerated measurements:** The voltages applied to the gate and drain contacts are realized as constant voltage sources and $I_\mathrm {D}$ is measured using a transimpedance am- plifier. The heating with the polyheater is realized with a constant voltage source voltage applied to poly-heater ( $V_\mathrm {PH}$ ). Simultaneously the poly-heater cur- rent $I_\mathrm {PH}$ is measured.

Figure 3.27: **Measurement procedure using a poly-heater:** During an MSM sequence, the temperature can be elevated, e.g., during recovery. This would accelerate recovery, which allows for the characterization of effects typically lying outside the measurement window.

3.8.1 Poly-Heater Calibration

In order to assign the correct $T_\mathrm {dev}$ to $P_\mathrm {PH}$ , the system has to be calibrated prior to the measurements. The calibration procedure is shown schematically in Figure 3.28. The calibration consists of the following steps:

1. $I_\mathrm {D}$ is obtained at different $T_\mathrm {chuck}$   $=$   $T_\mathrm {dev}$ at $P_\mathrm {PH}$   $=$  0 W over a wide range of temperatures. It has to be considered that $I_\mathrm {D}$ $($ $V_\mathrm {G}$ $,$ $V_\mathrm {D}$ $)$ has to be chosen in such a way that no stress is introduced to the device, e.g., in the subthreshold region near $V_{\mathrm {th}}$ .
2. $I_\mathrm {D}$ is obtained at different $P_\mathrm {PH}$ at a fixed $T_\mathrm {chuck}$ , which corresponds to the temperature minimum ( $T_\mathrm {min}$ ) of the setup.
3. $I_\mathrm {D}$ $($ $T_\mathrm {chuck}$ ) and $I_\mathrm {D}$ $($ $P_\mathrm {PH}$ $)$ are fitted with a polynomial fit of first or second order.
4. With the coefficients of the fits, $T_\mathrm {dev}$ $($ $P_\mathrm {PH}$ $)$ is interpolated for arbitrary poly-heater power at a certain $T_\mathrm {chuck}$ .

Figure 3.28: **Schematic poly-heater calibration:** $I_\mathrm {D}$ is obtained at the required $T_\mathrm {chuck}$   $=$   $T_\mathrm {dev}$ for $P_\mathrm {PH}$   $=$  0 W and at different $P_\mathrm {PH}$ at a fixed $T_\mathrm {chuck}$ , which corresponds to the $T_\mathrm {min}$ of the setup. $I_\mathrm {D}$ $($ $T_\mathrm {chuck}$ $)$ and $I_\mathrm {D}$ $($ $P_\mathrm {PH}$ $)$ are fitted with a polynomial fit of first or second order. With the coefficients of the fits, $T_\mathrm {dev}$ $($ $P_\mathrm {PH}$ $)$ is interpolated for arbritrary poly-heater power at a certain $T_\mathrm {chuck}$ .

This calibration method can also be applied to obtain the temperature of the poly-heater $T_\mathrm {PH}$ $($ $P_\mathrm {PH}$ $)$ . In this context, the increase of the poly-heater resistance ( $R_\mathrm {PH}$ ) with $T$ caused by the reduction of the carrier mobility of the polycrystalline silicon wires can be characterized. Therefore, $R_\mathrm {PH}$ $($ $T_\mathrm {chuck}$ $)$ at $P_\mathrm {PH}$   $=$  0 W and $R_\mathrm {PH}$ $($ $P_\mathrm {PH}$ $)$ at $T_\mathrm {chuck}$   $=$   $T_\mathrm {min}$ are measured and fitted with a polynomial fit of first or second order. With the fitted coefficients $T_\mathrm {PH}$ $($ $P_\mathrm {PH}$ $)$ can be interpolated. For such calibrations it has to be considered that the coefficients of the polynomial fits are valid only for a certain $T_\mathrm {chuck}$ . If the poly-heater is used at a different $T_\mathrm {chuck}$ the calibration has to be repeated for each required $T_\mathrm {chuck}$ .

The poly-heater technique is able to reach temperatures far beyond the scope of conventional thermo chuck systems. While the latter are typically used up to 200 °C, poly-heater systems can elevate $T_\mathrm {dev}$ up to 300 °C and more. As a result, the probing temperature range of the poly-heater technique is much wider than the one used for its calibration. In order to make use of this wide range, an analytical expression for $T_\mathrm {dev}$ $($ $P_\mathrm {PH}$ $)$ has been proposed [121]. Unfortunately, $T_\mathrm {dev}$ does not depend linearly on $P_\mathrm {PH}$ but exponentially. A linear dependence is associated with simple Joule heating where the thermal resistivity ( $R^\mathrm {th}$ ) of surrounding materials does not play a role. By contrast, in the case of the poly-heater system $R^\mathrm {th}$ increases simultaneously with the temperature increase. Thus the functional dependence of the device temperature on the power supplied to the heater is an exponential function:

$(3.12) \begin{equation} \label {eq:TvsPPH} T_\mathrm {dev}(P_\mathrm {PH})=T_\mathrm {0}-\dfrac {1}{\alpha }+\left (\dfrac {1}{\alpha } + T_\mathrm {chuck} - T_\mathrm {0} \right ) e^{\alpha R_\mathrm {sub,0}^\mathrm {th}P_\mathrm {PH}} \end{equation}$

with

Figure 3.29: **Heating and cooling characteristics:** For $T_\mathrm {chuck}$   $=$  −60 °C. **(1)** Left: The heater power is abruptly turned on. Right: Within 1 ms the maximum of $P_\mathrm {PH}$ is reached. Afterwards, $P_\mathrm {PH}$ tends to decrease slightly for approxi- mately 1 s until the thermal equilibrium between heater, wafer and chuck is restored. Due to the delayed thermal coupling of poly-heater and MOSFET, $T_\mathrm {dev}$ needs up to 10 s until stabi- lization. **(2)** Left: The heater power is abruptly turned off. Right: $P_\mathrm {PH}$ decreases to zero within 1 ms and $T_\mathrm {dev}$ needs up to 10 s until it reaches $T_\mathrm {chuck}$ . Figure source: [120].

$ T_\mathrm {chuck} $ — Figure 3.29: **Heating and cooling characteristics:** For $T_\mathrm {chuck}$   $=$  −60 °C. **(1)** Left: The heater power is abruptly turned on. Right: Within 1 ms the maximum of $P_\mathrm {PH}$ is reached. Afterwards, $P_\mathrm {PH}$ tends to decrease slightly for approxi- mately 1 s until the thermal equilibrium between heater, wafer and chuck is restored. Due to the delayed thermal coupling of poly-heater and MOSFET, $T_\mathrm {dev}$ needs up to 10 s until stabi- lization. **(2)** Left: The heater power is abruptly turned off. Right: $P_\mathrm {PH}$ decreases to zero within 1 ms and $T_\mathrm {dev}$ needs up to 10 s until it reaches $T_\mathrm {chuck}$ . Figure source: [120].

As shown in Figure 3.29, this type of temperature elevation is quite fast compared to other heating setups, e.g., the furnace which was used for the temperature control of devices mounted on a ceramic package in this thesis. As soon as a certain voltage is applied to the poly-heater, it takes approximately 1 ms until the maximum $P_\mathrm {PH}$ is reached, a process mainly limited by the finite speed of the voltage source [121]. Afterwards, $P_\mathrm {PH}$ tends to decrease slightly for approximately 1 s because $R_\mathrm {PH}$ increases due to the elevated $T_\mathrm {PH}$ until the thermal equilibrium between heater, wafer and chuck is restored. Due to the delayed thermal coupling of poly-heater and MOSFET, $T_\mathrm {dev}$ increases after turning on the poly-heater and needs up to 10 s for the stabilization. Then, $T_\mathrm {dev}$ remains constant until the heater is turned off again. After turning off the heater power, $P_\mathrm {PH}$ decreases to zero within 1 ms, again mainly limited by the finite speed of the voltage source, and $T_\mathrm {dev}$ needs up to 10 s until it reaches $T_\mathrm {chuck}$ finally. As a comparison, the furnace which was used for the experimental characterizations of MOSFETs mounted on ceramic packages in this thesis needs more than 30 min until the thermal equilibrium of the system is restored after a temperature change. Due to the significant temperature switching speed, the poly-heater system is quite promising for the realization of fast temperature ramps [119].

However, the possibility of an application of defined temperature ramps is not the only advantage of a poly-heater system. It can also overcome limitations typically associated with a thermo chuck. One limitation is that switching of the temperature in conventional setups, where $T_\mathrm {dev}$ is controlled by the thermo chuck and the MOSFET is contacted by probe-needles, the probe-needle contact can get lost. The reason for this is that heating and cooling a wafer on a thermo chuck results in a considerable thermal expansion of the probe-needles. As a consequence, a continuous manual needle adjustment is required when $T_\mathrm {dev}$ is changed. By contrast, heating with the poly-heater is local and causes no thermal expansion of needles and pads. This enables a change of temperature simultaneously with the measurement cycles without introducing additional delays due to manual needle adjustments.

3.8.2 Temperature Accelerated Measurements in Ceramic Packages

A hardware and software application for temperature control of local poly-silicon heater structures based on the setup in Figure 3.26 has been developed within the TDDS framework [122]. This allows for controlled temperature pulses or ramps during device recovery within the TDDS sequence of stress and recovery cycles. The poly-heater setup can be easily realized with standard equipment as well. Unfortunately, during the measurements with this application, difficulties arose, which are discussed in the following.

Figure 3.30: **Dependence of the drain current on the dissipated poly-heater power in packaged large-area devices:** The left panels show $I_\mathrm {D}$ and $P_\mathrm {PH}$ after the poly-heater is turned on and the right panels show $I_\mathrm {D}$ and $P_\mathrm {PH}$ after the poly-heater is turned off. While the stabilitzation of $P_\mathrm {PH}$ needs approximately 10 ms, the stabilization of $T_\mathrm {dev}$ needs 30 min at least (100 s are shown in this figure).

For the experimental characterization only poly-heater MOSFETs mounted on a ceramic package were available, which introduces a number of complications due to two facts. On the one hand, a ceramic package has no defined heat sink as the thermo chuck in the previously described setup. On the other hand the thermal resistance of the materials surrounding the heater/device system is higher in a package than in the poly-heater-device-chuck-system shown in Figure 3.25 and described in [112, 121]. These two facts lead to a completely different thermal coupling and thermal dynamics between the poly-heater and the MOSFET, which can be seen from the characterization of the heating and cooling dynamics in Figure 3.30. $V_\mathrm {PH}$ was applied abruptly while $I_\mathrm {PH}$ and $I_\mathrm {D}$ (proportional to $T_\mathrm {dev}$ ) were measured simultaneously. $P_\mathrm {PH}$ was calculated by multiplication of $V_\mathrm {PH}$ and $I_\mathrm {PH}$ .

Although the switching of $P_\mathrm {PH}$ shows a very comparable time evolution to the measurements on the fabricated wafers (Figure 3.29), $I_\mathrm {D}$ does not reach a thermal equilibrium within 100 s. While the stabilization of $P_\mathrm {PH}$ needs approximately 10 ms, the stabilization of $T_\mathrm {dev}$ needs 30 min at least. This makes the calibration quite tedious and a pulse-like elevation of the temperature impossible because the whole system needs much longer to reach thermal equilibrium than the previously described poly-heater-device-chuck-system.

One way to overcome the challenge of slow heating and cooling dynamics would be to implement a control loop, e.g., using a proportional-integral-derivative (PID) controller illustrated in Figure 3.31. In this context, it is quite easy to implement one in order to hold $P_\mathrm {PH}$ , which is proportional to $T_\mathrm {PH}$ , at a constant value by adjustment of the control variable $V_\mathrm {PH}$ according to the calculated difference between the setpoint for $T_\mathrm {PH}$ ( $P_\mathrm {PH}$ ) and the measured process variable $P_\mathrm {PH}$ = $V_\mathrm {PH}$   $\cdot$   $I_\mathrm {PH}$ . Unfortunately, this does not change the behavior of $I_\mathrm {D}$ , which is proportional to $T_\mathrm {dev}$ . As discussed in Figure 3.30, $P_\mathrm {PH}$ stabilizes within 10 ms and is constant afterwards while $I_\mathrm {D}$ drifts for a longer time. Thus, the implementation of a controller in order to hold $I_\mathrm {D}$ at a constant value by adjustment of $P_\mathrm {PH}$ seems to be the proper solution. In such a controller, $I_\mathrm {D}$ corresponding to the set $T_\mathrm {dev}$ would be the setpoint, $V_\mathrm {PH}$ would be the control variable and $I_\mathrm {D}$ would be the process variable, which is measured in order to calculate the error.

However, such a controller cannot be realized, since $I_\mathrm {D}$ depends not only on $T_\mathrm {dev}$ . In fact, $I_\mathrm {D}$ changes also due to device degradation. If, for example, $I_\mathrm {D}$ was held constant during the stress phase in eMSM measurements by adjustment of $P_\mathrm {PH}$ , $P_\mathrm {PH}$ would increase continuously because $I_\mathrm {D}$ degrades during stress. As a result, $T_\mathrm {dev}$ would increase, which accelerates again the degradation of the device. This could consequently overheat the device dramatically. During the recovery phase, $I_\mathrm {D}$ recovers, which would lead to a decrease of $P_\mathrm {PH}$ and thus to a cooling effect. As a result, neither during stress nor during recovery, $I_\mathrm {D}$ could be held at a constant value by adjustment of $V_\mathrm {PH}$ without changing the degradation and recovery state considerably.

Figure 3.31: **Control loop:** A controller (in this case a PID controller) calculates an error value as the difference between the setpoint and the process variable. Based on this error, a correction is applied to the system.

Besides the difficulties associated with the thermal dynamics of the system, also difficulties related to the calibration of nano-scale devices made it nearly impossible to use the poly-heater system. Due to the fact that single oxide defects cause RTN signals with steps of several pA up to µA, $I_\mathrm {D}$ switches periodically around the calibration point in any nano-scale device. If steps in the drain current of the same magnitude were caused by temperature changes, the corresponding temperature change would be several °C. Even if a nano-scale device is suitable for calibration (no RTN signals), the elevated temperatures during the calibration often lead to the creation of new defects which change the $I_\mathrm {D}$ level. In other words, the calibration is not reproducible. In this case, a lot of devices have to be calibrated in order to be able to calculate mean values for the calibration parameters.

From these experiences, it can be concluded that temperature accelerated measurements using an in situ poly-heater are advantageous in the term of the application of fast temperature ramps only if a defined temperature gradient can form between the heating wires and the thermo chuck. In the case of devices mounted on a ceramic package, the time until thermal equilibrium is reached is quite comparable for devices heated by a poly-heater and devices heated by a furnace.

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Previous: 3.7 Time-Dependent Defect Spectroscopy (TDDS) Top: 3 Experimental Characterization Next: 3.9 Conclusions

$T_\mathrm {dev}$	device temperatre
$P_\mathrm {PH}$	poly-heater power
$T_\mathrm {chuck}$	chuck temperature
$R_\mathrm {sub,0}^\mathrm {th}$	thermal resistance of the substrate
$\alpha$	constant
$T_\mathrm {0}$	constant.