Charge Trapping and Variability in CMOS Technologiesat Cryogenic Temperatures

6 Time-Zero Characterization

Understanding the time-zero properties of a device under test is the basis for every variabiltiy and reliability study of a certain technology. The reason is that (math image)((math image)) and CV curves show not only a strong dependence on drain bias or device dimensions, but also on the device temperature. The temperature dependence is caused by reduced phonon-scattering, which increases the mobility of the charge carriers. Furthermore, the temperature-dependent shift of the Fermi-levels, the temperature dependence of the carrier distributions and the band gap also seriously affect the device characteristics. All these effects can change the shape of (math image)((math image)) and CV curves dramatically, which has in turn a strong impact on any reliability study. The impact of the temperature on the transition curves is discussed detailed in Section 6.1, the impact on CV curves is discussed in Section 6.2.

Additionally, quantum mechanical effects become dominant towards cryogenic temperatures, which do not occur in the classical limit. An example for this is resonant tunneling caused by quantum dots. This effect has been measured across various MOSFET technologies and is presented in Section 6.3.

After studying the temperature dependent time-zero characteristics on large-area devices, the characteristics are studied on SmartArrays, employing thousands of devices which can be addressed digitally. This enables the study of the temperature dependence of the variability of parameters which are an important measure for the MOSFET performance of the devices. This has been done for two different SmartArray generations with various geometries. The results are presented in Section 6.4.

6.1 Transfer Characteristics

The most widely used method for device characterization is the analysis of the (math image)((math image)) curve of a single MOSFET. This can be done for a pristine or a stressed device and allows the extraction of important quantities which determine the quality and performance of a MOSFET. To record an (math image)((math image)) curve, (math image) is kept constant, while (math image) is swept from deep depletion to inversion. From the recorded (math image)((math image)) curve it is possible to extract multiple quantities like sub-threshold swing \( \mathit {SS} \), threshold voltage (math image), on-state current \( I_\mathrm {ON} \), transconductance \( g_\mathrm {m} \), or leakage current \( I_\mathrm {OFF} \), as shown in Fig. 6.1. All these parameters depend not only on the device dimensions, but also show a strong temperature dependence as can be seen by a comparison of the (math image)((math image))-curve for 4 K and 298 K in Fig. 6.1. Depending on the examined quantity, (math image)((math image)) curves are plotted on a logarithmic scale (particularly for subthreshold quantities) or on linear scales as can be seen in Fig. 6.1 (right). The temperature dependence of the transfer characteristics can be seen for Tech. A in Fig. 6.2 (top) and Tech. B in Fig. 6.2 (bottom) for both, nMOS and pMOS. Towards 4 K the (math image)((math image)) curve becomes steeper and the threshold voltage and on-state current increase.


Figure 6.1: The schematic (math image)((math image)) curves for 298 K and 4 K show a strong temperature dependence which influences various important device parameters, such as the on-current \( I_\mathrm {ON} \), the subthreshold swing \( SS \), the leakage current \( I_\mathrm {OFF} \), the (maximal) transconductance (math image), or the threshold voltage (math image) which can be defined at a constant current \( I_\mathrm {cc} \) or with the maximum transconductance \( \max (\gm ) \).


Figure 6.2: The transfer characteristics of Tech. A (top) and Tech. B (bottom) show qualitatively a similar behavior for both pMOS (left) and nMOS (right). Towards 4 K the (math image)((math image)) gets steeper and \( I_\mathrm {ON} \) increases. All (math image)((math image)) curves intersect close to a zero temperature coefficient point.

6.1.1 Subthreshold Swing

The steepness of the (math image)((math image)) is well-know as sub-threshold slope and is central for the switching between ON-state and OFF-state of a transistor and thus an essential performance criterion. For device characterization often the reciprocal value is used, which is known as subthreshold swing \( SS \). In the classical limit, the \( SS \) is defined in the weak inversion regime and given by [220]

(6.1) \{begin}{align} SS = \frac {\partial \vg }{\partial \log _{10}(\id )}=\ln (10)\frac {k_\mathrm {B}T}{q}\frac {C_\mathrm {ox}+C_\mathrm {D}+C_\mathrm {it}}{C_\mathrm {ox}}=m\ln
(10)\frac {k_\mathrm {B}T}{q}, \{end}{align}

where \( C_\mathrm {ox} \) is the capacitance of the oxide, \( C_\mathrm {D} \) the depletion capacitance and \( C_\mathrm {it} \) the capacitance of the interface states. The factor \( m \) which is defined by the capacitances is often called subthreshold-swing factor. The minimal value for \( SS \) at a certain temperature is reached in the limit \( C_\mathrm {D} \rightarrow \SI {0}{F} \), \( C_\mathrm {it} \rightarrow \SI {0}{F} \) and \( C_\mathrm {ox} \rightarrow \infty \ \mathrm {F} \). Experimentally, \( SS \) can be extracted using a linear fit for the subthreshold region of the logarithmic (math image)((math image)) curve. By doing so the temperature dependence of \( SS \) can be extracted from (math image)((math image)) measurements, and is shown in Fig. 6.3.


Figure 6.3: At temperatures above around 150 K, the subthreshold swing follows the Boltzmann relation (black line). Towards cryogenic temperatures it starts to saturate which can be seen for pMOS (left) and nMOS (right). This behavior has been observed for various technologies and can be explained by band tail states [110].

According to the classical Boltzmann thermal limit \( SS \) is defined as \( m \ln (10) k_B T/q \). Here, \( m \) is the subthreshold-slope factor which can be approximated by \( m=\partial \vg /\partial \varphi _\mathrm {S} \) with \( \varphi _\mathrm {S} \) being the surface potential. The classical Boltzmann thermal limit is experimentally not accessible below around 100 K, but starts to saturate towards a constant value. This can be seen in Fig. 6.3 for Tech. A for pMOS (left) and nMOS (right). This saturation effect of the subthreshold swing has been measured across multiple different FET technologies as shown in [110]. The deviation from the Boltzmann thermal limit can be explained by band tail states near the interface between the substrate and oxide. These band tail states occur for example due to static or dynamic disorder from interface defects, impurities or electron-phonon scattering and lead to a blurred band edge. This can be modeled precisely using Gauss hypergeometric functions for the band tails [221]. However, these functions require a large computational effort compared to a simplified model using exponential band tails \( g(E) \), as can be seen in Fig. 6.4, which is derived in detail in [MJJ4].


Figure 6.4: Crystalline disorder leads to blurred band edges. These can be modeled assuming exponentially decaying band tail states depending on a critical temperature \( T_c \). This widened band edge contribute to the energy scan by the Fermi distribution and can explain the saturation of the subthreshold slope. Figure taken from [MJJ4].

As shown by Beckers et al.  [MJJ4] the introduction of a critical temperature \( T_\mathrm {c} \), which is typically around 30 K to 50 K, allows the adaption of the classical \( SS \) to

(6.2) \{begin}{align} SS=m\frac {k_\mathrm {B}T}{q}\ln (10)b_\mathrm {c} \{end}{align}


(6.3) \{begin}{align} b_\mathrm {c}=\frac {1+f(\theta )\cdot y}{1+\theta \cdot f(\theta ) \cdot y} \{end}{align}

(6.4) \{begin}{align} f(\theta )=\frac {\pi \cdot (\theta -1)}{\sin (\pi \theta )} \{end}{align}


(6.5) \{begin}{align} y=\exp \left [(1-\theta )\big (\frac {x}{k_\mathrm {B}T}\big )\right ]. \{end}{align}

In this closed form expression \( x=E_\mathrm {C}-E_\mathrm {F} \) is the energetic offset between the conduction band and the Fermi level and \( \theta =T/T_\mathrm {c} \). This closed-form simplification of the model presented in [221] describes the temperature dependence of \( SS \) and the occurring saturation efficiently and is thus well suited for usage in commercial device simulators.

6.1.2 Transconductance

The transconductance (math image) (short for transfer conductance) relates the drain current with the gate voltage via

(6.6) \{begin}{align} \gm = \frac {\mathrm {d}\id }{\mathrm {d}\vg } \{end}{align}

at a constant (math image). Towards lower temperatures, electron-phonon scattering gets reduced, leading to a higher carrier mobility. This has the effect of an increasing transconductance, as can be seen in Fig. 6.5 for Tech. A for both pMOS (left) and nMOS (right). The overall shape of (math image) gets steeper because the Fermi-Dirac distribution becomes narrower at low temperatures. The point of maximum (math image) which also gets continuously larger for lower temperatures, as can be seen in Fig. 6.5 (bottom), is widely used for the definition of the threshold voltage, as discussed in the upcoming section in detail.


Figure 6.5: The shape of the transconductance shows a strong temperature dependence for both pMOS (left) and nMOS (right) of Tech. A. The steeper and sharper (math image) at 4.2 K can be explained by the sharper Fermi-Dirac distribution at cryogenic temperatures; \( \max (\gm ) \) increases at lower temperatures because the mobility increases due to reduced phonon-scattering.

It has to be noted that the (math image) characteristics shows a maximum for each transistor. The point of \( \max (\gm ) \) can become degraded due to higher contact resistance or an increase in the number of interface states. The latter affect (math image), because the mobility of the carriers reduces due to Coulomb scattering at interface defects. This effect is deliberately used to link a decreasing \( \max (\gm ) \) to an increasing concentration of interface defects [180].

6.1.3 Threshold Voltage

The threshold voltage (math image) of a MOSFET is the minimal gate voltage (math image) which needs to be applied to turn the device on, which means creating a conduction path between source and drain [220]. While \( SS \) or transconductance are well defined, there are multiple definitions for the threshold voltage. In this work, two different methods for extracting the threshold voltage from measurements will be used: The constant current method and the \( \mathrm {max}(g_\mathrm {m}) \) method, referred to (math image) and (math image), respectively. For the constant current method, a constant current \( I_\mathrm {cc} \) is defined at which the corresponding threshold voltage (math image) is extracted, as can be seen in Fig. 6.1(a). For the \( \mathrm {max}(g_\mathrm {m}) \) method the point of \( \mathrm {max}(g_\mathrm {m}) \) is extracted. Then the intersection point of the tangent line with \( \id =\SI {0}{\ampere } \) is found. The corresponding voltage is defined as (math image).

An extraction of (math image) and (math image) is shown for Tech. A in Fig. 6.6 for pMOS (left) and nMOS (right). As can be seen, with the two methods not the same threshold voltage values are extracted, in fact, they do not even exhibit the same temperature dependence. In the case of pMOS, (math image) decreases by approximately 350 mV while (math image) decreases only by approximately 200 mV when the temperature is changed from 300 K to 4 K. Even worse is the difference for nMOS. Here not only a difference in the absolute shift, but also a change of the shape for the T-dependence can be observed. While for pMOS the T-dependence is almost perfectly linear, the nMOS device shows a saturation of the threshold voltage, which is stronger for (math image) than for (math image). This rather complex T-dependence of the different (math image) definitions is discussed in detail in [111]. By discussing the impact of the temperature dependence of the bulk Fermi potential, dopant freezeout, field-assisted dopant ionization, bandgap widening and interface traps, Beckers et. al. model (math image) and show that the saturation of (math image) for nMOS towards 4.2 K can be explained by the presence of interface defects in combination with the shift of the bulk Fermi potential  [111].


Figure 6.6: The threshold voltage increases towards cryogenic temperatures for both nMOS and pMOS of Tech. A. Different defini- tions and extraction methods of (math image) lead to a different T-dependence which is discussed in detail in [111].

The threshold voltage plays an important role for the evaluation of the reliability of a certain technology. It is often used as reference before applying stress to a device. During stress the shape of the (math image)((math image)) curve can change, which is often simplified by assuming a parallel shift of the whole (math image)((math image)) curve. The change in (math image) extracted from the initial and stress device is often compared. For this, (math image) needs to be extracted from the (math image)((math image)) which makes it necessary to decide on a certain definition. These methods are discussed in detail in the upcoming sections about reliability characterization.

6.1.4 ON-State Current

The ON-state current \( I_\mathrm {ON} \) is extracted in the saturation region, typically at \( \vg =V_\mathrm {DD} \). \( I_\mathrm {ON} \) for Tech. A pMOS (left) and nMOS (right) can be seen in Fig. 6.7, extracted at \( \vg =\SI {-1}{\volt } \) and \( \vg =\SI {0.85}{\volt } \), respectively. As can be seen, (math image) decreases for pMOS and increases for nMOS with decreasing temperatures, which can be explained by the increasing mobility of the charge carriers. This means that at cryogenic temperatures the efficiency of MOS devices increases, which is beneficial for the performance of circuits at these temperatures.


Figure 6.7: The ON-current of Tech. A decreases for pMOS (left) and increases for nMOS (right) towards 4.2 K. This can be explained by the increasing mobility due to reduced phonon-scattering.