- freely available
- re-usable

*J. Low Power Electron. Appl.*
**2014**,
*4*(3),
201-213;
doi:10.3390/jlpea4030201

^{1}

^{2}

^{†}

Published: 16 July 2014

## Abstract

**:**The global variability of ultra-thin body and buried oxide (UTBB) MOSFETs in subthreshold and off regimes of operation is analyzed. The variability of the off-state drain current, subthreshold slope, drain-induced barrier lowering (DIBL), gate leakage current, threshold voltage and their correlations are considered. Two threshold voltage extraction techniques were used. It is shown that the transconductance over drain current (g

_{m}/I

_{d}) method is preferable for variability studies. It is demonstrated that the subthreshold drain current variability in short channel devices cannot be described by threshold voltage variability. It is suggested to include the effective body factor incorporating short channel effects in order to properly model the subthreshold drain current variability.

## 1. Introduction

Ultra-thin body and buried oxide (UTBB) technology is promising for future MOSFET nodes due to its short channel effects immunity [1] and low variability [2]. The quantification of global variability improves fabrication process optimization and MOSFET model extraction. Whereas threshold voltage variability has been widely studied [3,4,5], subthreshold variability has received little attention. Characterization of subthreshold variability has become essential nowadays. Design is shifting towards the subthreshold regime for low power applications. Furthermore, the off-state drain current variability is important for stand-by power considerations.

The variability of MOSFET parameters is due to geometry fluctuations, the granularity of materials and doping randomness [6]. Doping randomness affects MOSFET variability through random dopant fluctuations (RDF) in the channel, the source and drain extensions and in the ground plane. Fabrication imperfections cause the variability of device dimensions, such as line edge roughness (LER), as well as Si channel and buried oxide (BOX) thicknesses. The granularity of gate material also contributes to the parameter variability of a device. Typically, the channel of an FD SOI device (including UTBB) is left undoped, thus eradicating RDF in the channel. In general, well-controlled process, small, short channel effects [3] and low series resistance [7] contribute to the low variability of FD SOI [1,2,5,7].

The aim of the study is to perform a qualitative analysis of global subthreshold parameter variability. Such an analysis can be carried out even on devices that are not yet fully optimized from the short channel effects perspective. The parameters of interest for the subthreshold regime are the off-state drain current (I_{d-off}) and gate current (I_{g-off}), threshold voltage (V_{th}), subthreshold slope (S) and drain-induced barrier lowering (DIBL). The off-state currents were extracted at gate voltage V_{g} = 0 V. Absolute values of the correlation coefficients are presented in this work. This paper is an extended version of our previous work [8]. It also incorporates some results from [9]. The data from [9] were improved by doubling the number of characterized devices, thus enhancing statistics.

## 2. Experimental Details

The devices studied in this work are fully-depleted (FD) silicon-on-insulator (SOI) n-channel MOSFETs fabricated on 25 nm-thick BOX. The top Si layer is 7.5 nm-thick. The channel is left undoped allowing the avoidance of RDF. The n-type ground plane was incorporated below BOX. The equivalent oxide thickness of the gate dielectric is 1.2 nm. Devices with gate lengths L_{g} from 28 to 88 nm and a gate width W_{g} of 10 µm are characterized. 106 transistors of each gate length are measured across the wafer. A 3-sigma normality test was carried out to exclude outliers.

Two threshold voltage extraction techniques are used in this work: using the constant current method, V_{th} is obtained as V_{g}, where I_{d}/(W_{g}/L_{g})) = 10^{−7} A; the transconductance over drain current (g_{m}/I_{d}) method is described in [10,11,12]. Both extraction methods are widely used in the linear and saturation regimes of operation.

## 3. Results and Discussion

Figure 1 shows the ratios of standard deviations (σ) to mean values (μ) of I_{d-off}, S, DIBL and V_{th} extracted using the g_{m}/I_{d} technique at drain voltages V_{d} of 1 V and 20 mV. As expected, shorter devices exhibit stronger variation in both regimes of operation. It can be seen that I_{d-off} variability is rather strong and presumably impacted by the variability of other parameters (e.g., V_{th}, S and DIBL).

**Figure 1.**The ratio of standard deviation over the mean value of the off-state drain current (I

_{d-off}), subthreshold slope (S), drain-induced barrier lowering (DIBL) and threshold voltage (V

_{th}) extracted using the g

_{m}/I

_{d}method in devices with gate lengths of 28 and 88 nm.

#### 3.1. Off-State Gate Current

Figure 2 shows σI_{d-off} in devices with gate lengths from 28 nm to 88 nm in linear and saturation regimes. The higher variability of I_{d-off} is observed in shorter devices. Different parameters, such as short channel effects, I_{g} and V_{th}, have an impact on I_{d} in the off-state.

First, I_{g-off} is considered. The standard deviation of I_{g-off} is shown in Figure 2. An increase of σI_{g-off} with V_{d} can be related to the amplified generation current, which is a function of V_{d} [13]. There is no pronounced dependence of I_{g-off} variability on the gate length. Furthermore, σI_{g-off} remains small compared with σI_{d-off}, except for the longest devices at V_{d} of 1 V. Absolute values of I_{g-off} are also negligibly small compared with I_{d-off} (except for devices with a gate length of 88 nm at V_{d} of 1 V). Therefore, the effect of I_{g-off} variability on I_{d-off} is expected to be negligible. In order to confirm this, Figure 3 plots coefficients of correlation between I_{g-off} and I_{d-off} for devices with various gate lengths at V_{d} of 1 V and 20 mV and at temperatures of 25 and 125 °C. Only in the case of V_{d} of 1 V and a temperature of 25 °C does the correlation become stronger with a gate length increase as the I_{g-off} component of I_{d-off} increases. I_{g-off} is small at low V_{d}, as the gate-induced drain leakage (GIDL) is alleviated. This is confirmed by weak I_{g-off} and I_{d-off} correlation at V_{d} of 20 mV for all gate lengths. With a temperature rise, due to V_{th} shift and S degradation, I_{d} increases stronger than I_{g}. Thus, the effect of I_{g} on I_{d} decreases, and the correlation between I_{d-off} and I_{g-off} reduces (Figure 3). The above discussion suggests that the effect of I_{g-off} variability on I_{d-off} variability should be accounted for only in relatively long devices at high V_{d} and room temperature, whereas in other cases, it can be neglected.

**Figure 2.**σI

_{d-off}and σI

_{g-off}variations with gate length at drain voltages (V

_{d}) of 20 mV and 1 V.

**Figure 3.**The I

_{d-off}-I

_{g-off}correlation in devices with various gate lengths at V

_{d}of 20 mV and 1 V and temperatures of 25 °C and 125 °C.

#### 3.2. Threshold Voltage

Secondly, the effect of V_{th} variability on σI_{d-off} is considered. Figure 4 shows σV_{th} for devices with gate lengths from 28 to 88 nm at V_{d} of 20 mV and 1 V. The variability in shorter devices is obviously stronger than in long devices. At V_{d} of 1 V, σV_{th} is larger than at V_{d} of 20 mV in short devices. However, in long devices, values of σV_{th} are similar in both regimes of operation. At V_{d} of 1 V in the shortest devices, σV_{th} is considerably different for two V_{th} extraction methods. The g_{m}/I_{d} method results in a lower V_{th} variability than the constant current method. This is due to V_{th} extraction from different parts of current-voltage characteristics and the smaller sensitivity of the g_{m}/I_{d} method to short channel effects (S, DIBL), as discussed in [9]. This difference vanishes in long devices.

**Figure 4.**σV

_{th}obtained using the constant current and g

_{m}/I

_{d}methods at V

_{d}of 20 mV (

**top**); and V

_{d}of 1 V (

**bottom**).

Since in the subthreshold regime, I_{d} exponentially depends on V_{th}, it is generally assumed that σI_{d} in this region is dominated by σV_{th}. This holds true for the relatively long devices studied in this work. Correlation between I_{d-off} and V_{th} at V_{d} of 1 V is plotted in Figure 5 for 28 nm- and 100 nm-long devices. V_{th} was extracted using the constant current and g_{m}/I_{d} methods. To quantify the correlation, Figure 6 plots the coefficients of correlation between I_{d} and V_{th} (extracted using the constant current method) as a function of V_{g} in devices with various gate lengths. In the subthreshold region, correlation is strong for 68 and 88 nm-long devices. In strong inversion, the correlation reduces due to the dominance of mobility and series resistance variability [13]. However, in shorter devices, the I_{d} and V_{th} correlation significantly decreases, not only in strong inversion, but also in the subthreshold regime. This effect is also seen in Figure 7, where variation of the I_{d-off} and V_{th} correlation coefficients with the gate length is plotted. Therefore, the variability of other parameters apart from I_{g-off} and V_{th} has to be taken into account.

**Figure 5.**The variation of I

_{d-off}at V

_{d}= 1.0 V with V

_{th}extracted using the constant current and g

_{m}/I

_{d}methods at V

_{d}= 1 V in devices with gate lengths of 28 nm and 88 nm.

**Figure 6.**The variation of the I

_{d}-V

_{th}correlation coefficient with V

_{g}in devices with various gate lengths. V

_{th}was obtained using the constant current method.

It is important to note the importance of the V_{th} extraction method in variability studies. In [9], it was shown that the g_{m}/I_{d} V_{th} extraction method is beneficial over other techniques from the short channel effects perspective. A similar property can be observed in Figure 7. The low correlation of V_{th‑gm/Id} and I_{d-off} in short devices suggests the parasitic effect immunity of V_{th} extraction using the g_{m}/I_{d} technique [11,12] compared with the constant current method, as the latter is obviously sensitive to DIBL [9].

**Figure 7.**The V

_{th}-I

_{d-off}correlation coefficients in devices with various gate lengths. V

_{th}was obtained using the constant current (cc) and g

_{m}/I

_{d}methods.

#### 3.3. Short Channel Effects

Thirdly, an impact of short channel effect variability on subthreshold I_{d} is considered. The data in Figure 8 show strong correlation between I_{d-off} and DIBL in shorter devices. The correlation coefficient decreases to ~0.2 and even more for devices with gate lengths of 68 nm and 88 nm, where the average DIBL is below 60 mV/V. Correlation between I_{d-off} and V_{th} (extracted at V_{d} of 20 mV) featured an opposite trend increasing with the gate length, as seen from Figure 7. Therefore, in devices with gate lengths of 68 and 88 nm, I_{d-off} and V_{th} variability is caused by the same mechanisms, while the variability of DIBL and subthreshold slope dominates in short devices.

**Figure 8.**The DIBL-I

_{d-off}correlation coefficients in devices with various gate lengths.

Figure 9 presents the variation of the subthreshold slope and V_{th} extracted using the constant current and the g_{m}/I_{d} methods, at high V_{d} in saturation. Scatter is much stronger in short devices than in longer ones, as expected. Furthermore, in short devices, the correlation of the subthreshold slope with V_{th-cc} is stronger than the correlation with V_{th-gm/Id}. This correlation is quantified in Figure 10. Figure 10 shows DIBL and V_{th}, as well as the S and V_{th} correlation in 28 nm- and 88 nm-long devices at 25 °C and 125 °C. V_{th} was extracted with the g_{m}/I_{d} and constant current methods. V_{th} extracted using the g_{m}/I_{d} technique shows little dependence on short channel effects, even at high temperatures. V_{th} obtained with the constant current method shows very strong correlation with DIBL at room and elevated temperatures. This comparison confirms that the g_{m}/I_{d} technique enables an evaluation of intrinsic V_{th}, only slightly affected by short channel effects. Evaluation of V_{th} free of short channel effects is of interest for device models that incorporate variability. If variability is imposed on each of the correlating parameters in a model, the total performance variability might be overestimated [9]. This can be avoided either by including correlation factors in the models or using independent (zero correlation) parameters.

**Figure 9.**Variations of V

_{th}obtained using the g

_{m}/I

_{d}and constant current methods as a function of the subthreshold slope.

**Figure 10.**The V

_{th}-DIBL and V

_{th}-S correlation coefficients in devices with gate lengths of 28 nm and 88 nm at 25 °C and 125 °C temperatures. V

_{th}was obtained using the constant current and g

_{m}/I

_{d}methods.

The temperature dependence of the I_{d-off} and DIBL correlation is shown in Figure 11. In the case of short devices (gate lengths of 28 nm and 38 nm), the correlation factor being very strong shows nearly no dependence on temperature. However, in long devices (gate lengths of 68 nm and 88 nm), the correlation between I_{d-off} and DIBL rises with temperature. This can be explained by the DIBL increase with temperature, as the inset in Figure 11 shows.

**Figure 11.**The I

_{d-off}–DIBL correlation in devices with various gate lengths in the 25–125 °C temperature range at V

_{d}of 1 V. The inset shows the DIBL temperature dependence.

#### 3.4. Drain Current Variability Modelling

Following the above discussion, Figure 12 compares σI_{d} and g_{m}·σV_{th} dependences on V_{g} in devices with a gate length of 88 nm at V_{d} of 1 V and 20 mV. σI_{d} agrees well with g_{m}·σV_{th} in around-threshold and subthreshold regions, i.e., moderate and weak inversion. In order to ease the comparison of devices with different gate lengths and at various temperatures, the results are plotted as a function of the gate overdrive V_{g}–V_{th}.

**Figure 12.**Variation of σI

_{d}and its components with gate overdrive at V

_{d}of 20 mV and 1 V at 25 °C in devices with a gate length of 88 nm. V

_{th}was extracted using the g

_{m}/I

_{d}method.

The situation is different in short channel devices. As seen from Figure 13, σ_{Id} can be described by g_{m}·σV_{th} only in a very narrow V_{g} range around V_{th}. In the subthreshold, the curves strongly deviate. In [14], it was suggested to consider the variability of the body factor in weak inversion in addition to V_{th}. The impact of the body factor dependences on V_{g} and temperature was emphasized and related to depletion width variation [14]. The UTBB devices studied here remain fully-depleted in the whole weak-to-moderate inversion range at any temperature. Furthermore, Si and oxide thickness fluctuations are not significant, due to the well-controlled fabrication process [2,9]. In [2], reflectometry was used to evaluate top silicon layer thickness variability and correlate it with electrically obtained data. Furthermore, in [9], it was found that global variability does not degrade with temperature through Si layer thickness variation, confirming that the process matches σI_{d} sufficiently well (Figure 12). As this is not the case in the shortest device (Figure 13), we consider the effective body factor n, which includes short channel effects. Thus, in our case, n dependence on V_{g} originates from the gate and the drain counteraction on electrostatics. In this work, n (V_{g}) is derived in the first approximation from S(V_{g}) according to
$S=nkT/q\xb7\text{ln}(10)$
, where q is the elementary charge, k is the Boltzmann constant and T is the temperature. At V_{d} of 20 mV, the combined effect of σV_{th} and σn can fit σI_{d} sufficiently well:

However at V_{d} of 1 V, Equation (1) does not allow one to fit σI_{d}. This is due to amplification of short channel effects at V_{d} of 1 V and, thus, their stronger impact on V_{th}. In order to fit σI_{d}, V_{th} and n, correlation coefficient ρ has to be accounted for:

This approach is shown to work sufficiently well at elevated temperature. In Figure 14, it is shown that σI_{d} at 125 °C is fitted well when the V_{g}-dependent effective body factor and its correlation with the V_{th} are considered (Equation (2)).

**Figure 13.**Variation of σI

_{d}and its components with gate overdrive at V

_{d}of 20 mV and 1 V at 25 °C in devices with a gate length of 28 nm. V

_{th}was extracted using the g

_{m}/I

_{d}method.

The results presented in Figure 12, Figure 13 and Figure 14 were obtained using the g_{m}/I_{d} V_{th} extraction technique. Figure 15 shows σI_{d} fitting in 28 nm-long devices at 25 °C with V_{th} obtained using the constant current method. The fitting is acceptable at V_{d} of 20 mV, as short channel effects are not strongly pronounced. However, at V_{d} of 1 V, the fitting works only in a very narrow V_{g} region close to V_{th}. This can be explained by the strong impact of short channel effects on the constant current extraction of V_{th}. This again confirms the advantages of the g_{m}/I_{d} V_{th} extraction method for variability assessment.

Further σI_{d} modelling improvement can be done by accounting for GIDL in the very low V_{g} region. In strong inversion, mobility and series resistance should be considered [13].

**Figure 14.**Variation of σI

_{d}and its components with gate overdrive at V

_{d}of 20 mV and 1 V at 125 °C in devices with a gate length of 28 nm. V

_{th}was extracted using the g

_{m}/I

_{d}method.

**Figure 15.**Variation of σI

_{d}and its components with gate overdrive at V

_{d}of 20 mV and 1 V at 25 °C in devices with a gate length of 28 nm. V

_{th}was extracted using the constant current method.

## 4. Conclusions

The global variability of UTBB devices in the subthreshold has been analyzed through I_{d-off}, I_{g-off}, S, V_{th}, DIBL and their correlations. A generally used approach to model σI_{d} using σV_{th} is shown to work well for long devices. For short channel devices, an improved procedure that accounts for the V_{g}-dependent effective body factor (incorporating short channel effects) was proposed and validated in the temperature range from 25 °C to 125 °C. This is important for power consumption considerations and compact model parameter extraction in the off-regime. It was shown that the g_{m}/I_{d} V_{th} extraction technique is beneficial for accurate variability assessment and modeling.

## Acknowledgments

The research was partially funded by FNRS (Fonds National de la Recherche Scientifique), Belgium, Catrene “Reaching 22” and FP7 NoE “Nanofunction”.

## Author Contributions

Sergej Makovejev, Jean-Pierre Raskin, Denis Flandre and Valeriya Kilchytska performed in-depth analysis and contributed to paper writing. Babak Kazemi Esfeh performed measurements and the initial results analysis. François Andrieu provided devices for this work.

## Conflicts of Interest

The authors declare no conflict of interest.

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