[IEEE 2014 IEEE 29th International Conference on Microelectronics (MIEL) - Belgrade, Serbia (2014.5.12-2014.5.14)] 2014 29th International Conference on Microelectronics Proceedings - MIEL 2014 - Variability analysis — Prediction method for nanoscale triple gate FinFETs

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<ul><li><p>99978-1-4799-5296-0/14/$31.00 2014 IEEE</p><p>PROC. 29th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2014), BELGRADE, SERBIA, 12-14 MAY, 2014</p><p>Variability Analysis Prediction Method for Nanoscale Triple Gate FinFETs </p><p> D. Tassis, I. Messaris, N. Fasarakis, S. Nikolaidis, G. Ghibaudo and C. Dimitriadis </p><p>Abstract We expanded our analytical compact model for the drain current of undoped or lightly doped nanoscale FinFETs, in order to predict and decompose variability in the electrical characteristics of FinFETs. The model has been evaluated by comparison to TCAD simulated devices with predefined variability. Successful application to experimental data of FinFETs with fin width Wfin= 15 nm, gate length LG =30 nm, equivalent gate oxide thickness tox = 1.7 nm and fin height Hfin= 65 nm, has attributed their behavior to geometrical variations (of LG, Wfin) and variability in the metal gate work function (m). Furthermore, variability of FinFETs having different number of fins (2-50) and fins pitch (200-1000 nm) has been investigated </p><p>I. INTRODUCTION </p><p>Short channel triple-gate (TG) FinFETs are very promising candidates for the sub-20 nm technology. But nowadays, continuous shrinkage of the device tends to create even on the same chip non identical transistors, having random differences, with a significant impact on the electrical performance of the devices [1, 2]. In a circuit, a huge number of such similar devices can have unpredictable results, degrading the performance of the circuit, unless variability is taken into account. </p><p>Our analytical model [3] describes the TG FinFETs electrical characteristics, namely, the drain current ID, the threshold voltage Vt and the subthreshold swing (SS), in terms of devices parameters such as the geometrical characteristics, doping concentrations etc. and only six model specific parameter, taking into account short channel and quantum effects. Exploiting the analytical expressions of the model, we can estimate the variation in any variable using the error propagation formula: </p><p>22 yy xixi i</p><p>y can be any output variable of the model such, as Vt (threshold voltage roll-off), Vt, Ion (on current), and SS, while x is any variable pertaining the device characteristics or the model. y, x is the standard deviation of y and x respectively. The compact model </p><p>provides the partial derivatives of the output variables in respect to the input variables xi. As sources of variability, we have considered geometrical variations in the fin width (Wfin), gate length (LG), equivalent oxide thickness (tox) and variability in the metal gate work function m [4]. Previous attempts [4] [6] are using also the error propagation formula, but are based in direct measurements of the geometrical variability, while the remaining variability is attributed to the metal gate work function variability. This method is time and effort consuming and needs electron microscopy. This means sample preparation and destruction, thus can not be achieved while testing (electrically) the devices. There is an easier and faster method, exploiting directly the electrical measurements we are interested to, such as the transfer characteristics of the devices. </p><p>II. MODEL VALIDATION A. Comparison to TCAD Simulated Device </p><p>In order to check the validity and accuracy of our method, we compared the results with TCAD devices that exhibit predefined fluctuations around a nominal value in the aforementioned parameters (Wfin, LG, tox and m). We designed 100 similar FinFETs with nominal values Wfin = 12 nm, LG = 24 nm tox = 1 nm, fin height Hfin = 20 nm, m = 4.71 eV and normal distribution in their values. The variation of each one has been chosen to meet the requirements of the ITRS [7], i.e. the variation proposed by ITRS has been equated to 3 (0.7 nm, 0.7 nm 0.17 nm, and 30 meV respectively). The actual values of the corresponding standard deviations (of the dataset used) are Wfin = 0.652 nm, LG = 0.655 nm tox = 0.0172 nm, m = 31.31 eV, slightly different from the initially chosen ones. </p><p>The variability of these parameters has an impact on the variability of the transfer characteristics of the devices (Fig. 1). </p><p>For each characteristic, we calculated Vt, Ion, Iov (overdrive drain current at VG = Vt + 0.65 V), and SS. Finally, after statistical analysis we estimated their variation (Vt, Ion, Iov, SS) both at VD = 20 mV and 1 V. The variation of the threshold voltage roll-off (Vt) is calculated from the values of Vt in low and high VD. These output variables usually have also a normal distribution. The simulated devices results are: Vt = 31.15 mV, Ion = 6.78x10-5 A, Iov = 4.21x10-7 A, SS = 2.40 mV/dec, at VD </p><p>D. Tassis, I. Messaris, N. Fasarakis, S. Nikolaidis and C. Dimitriadis are with the Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece, e-mail: tassis@physics.auth.gr </p><p>G. Ghibaudo is with IMEP, MINATEC, Parvis Louis Neel, 38054 Grenoble Cedex 9, France. </p></li><li><p>100</p><p>= 20 mV and Vt = 31.13 mV, Ion = 2.23x10-5 A, Iov = 1.23x10-5 A, SS = 3.07 mV/dec, at VD = 1 V, while Vt = 4.35 mV. These values are very close to the values predicted by the model: Vt = 31.41 mV, Ion = 7.11x10-5 A, Iov = 3.65x10-7 A, SS = 2.06 mV/dec at VD = 20 mV and Vt = 30.91 mV, Ion = 2.13x10-5 A, Iov = 1.23x10-5 A, SS = 3.07 mV/dec, at VD = 1 V, and Vt = 6.41 mV. Regression statistic analysis has eliminated the possibility of hidden variables affecting the results. But other sources of uncertainties can affect the results, such as uncertainties in the method and calculation of the output variables (using the transfer characteristics). These uncertainties can add their own behavior to the statistics. At least, we must keep them in a relatively low value in order to be negligible. </p><p> B. Variability decomposition </p><p>We also simulated four sets of (a hundred) similar devices having variation in only one input parameter and keeping all the rest constant (at their nominal values). These additional simulations were necessary for the decomposition of the simulated standard deviations (y) into their components (yi), attributed only to the input variables standard deviation (xi). </p><p>In order to estimate total standard deviations, the components (yi) are summed according to (1) - in a square manner: </p><p>2 2y yii (2) </p><p>For the theoretical estimation, (1) gives: yy xi ixi</p><p> (3) </p><p>We used the standard deviations from the distribution of each variable xi while the partially derivatives are calculated from the model. </p><p>The results of the simulated devices are depicted in Fig. 2, while the results calculated by the model are presented in Fig. 3. There is good agreement in the total variabilities and generally also in their components. </p><p> C. Estimation of variability sources </p><p> For a given set of output parameters and their </p><p>variability, we can reverse the problem and estimate the sources of variability, i.e. the variability (standard deviation) of the input variables which cause variability. According to (1), the problem is not linear. We could use m output parameters and n input parameters. For each set of the n input parameters (standard deviations xi) we can estimate one output parameter (standard deviation y). It is an m by n, nonlinear problem. We used m=9 output parameters: Vt, Ion, Iov, SS both at VD = 20 mV and 1 V, and Vt, and considered n=4 sources of variability: Wfin, LG, tox, m. We can use minimization and non linear fitting to estimate the sources of variability. Alternatively we can choose an equal number of output </p><p>0.0 0.2 0.4 0.6 0.8 1.0 1.210-1410-1210-1010-810-610-4</p><p>0.0 0.2 0.4 0.6 0.8 1.0 1.20</p><p>1x10-5</p><p>2x10-5</p><p>VD= 20 mVD</p><p>rain</p><p> cur</p><p>rent</p><p>, ID (A</p><p>)</p><p>Dra</p><p>in c</p><p>urre</p><p>nt, I</p><p>D (A</p><p>)</p><p>Gate Voltage, VG (V) </p><p>0.0 0.2 0.4 0.6 0.8 1.0 1.210-1210-1010-810-610-410-2</p><p>0.0 0.2 0.4 0.6 0.8 1.0 1.201x10-42x10-43x10-44x10-4</p><p>Dra</p><p>in c</p><p>urre</p><p>nt, I</p><p>D (A</p><p>)</p><p>VD= 1 V</p><p>Dra</p><p>in c</p><p>urre</p><p>nt, I</p><p>D (A</p><p>)Gate Voltage, VG (V)</p><p>VD= 1 V</p><p> Fig. 1 Transfer characteristics of 100 TCAD simulated devices with variability in geometry and the metal gate work function (atVD= 20 mV and 1 V). </p><p>0.00 0.01 0.02 0.03 0.04</p><p>Vt</p><p>VD=1</p><p>VV</p><p>D=2</p><p>0mV</p><p>Threshold Voltage variability (V)</p><p> All m tox Wfin Lg</p><p>0.000 0.001 0.002 0.003 0.004</p><p>VD =</p><p> 20 </p><p>mV</p><p>VD =</p><p> 1 V</p><p>Subthreshold Swing variability, SS (V/dec)</p><p> All m tox Wfin Lg</p><p>0.0 5.0x10-6 1.0x10-5 1.5x10-5 2.0x10-5</p><p> All m tox Wfin LgV</p><p>D =</p><p> 20 </p><p>mV</p><p>VD =</p><p> 1 V</p><p>On-current variability, Ion (A)0 5x10-6 1x10-5</p><p>VD =</p><p> 1 V</p><p>Overdrive Current variability, Iov (A)</p><p> All m tox Wfin LgV</p><p>D =</p><p> 20 </p><p>mV</p><p>Fig. 2 Decomposition of total variability of Vt, Ion , Iov , SS, Vt (at VD= 20 mV and 1 V), due to Wfin, LG, tox, m variation (model simulated results) </p><p>0.00 0.01 0.02 0.03 0.04</p><p>Vt</p><p>VD=1</p><p>VV</p><p>D=2</p><p>0mV</p><p>Threshold Voltage variability (V)</p><p> All m tox Wfin Lg</p><p>0.000 0.001 0.002 0.003 0.004</p><p>VD =</p><p> 1 V</p><p>VD =</p><p> 20 </p><p>mV</p><p>Subthreshold Swing variability, SS (V/dec)</p><p> All m tox Wfin Lg</p><p>0.0 5.0x10-6 1.0x10-5 1.5x10-5 2.0x10-5</p><p>VD =</p><p> 20 </p><p>mV</p><p>VD =</p><p> 1 V</p><p>On-current variability, Ion (A)</p><p> All m tox Wfin Lg</p><p>0.0 5.0x10-6 1.0x10-5</p><p>Overdrive Current variability, Iov (A)</p><p> All m tox Wfin LgV</p><p>D =</p><p> 20 </p><p>mV</p><p>VD =</p><p> 1 V</p><p>Fig. 3 Decomposition of total variability of Vt, Ion , Iov , SS, Vt (at VD= 20 mV and 1 V), due to Wfin, LG, tox, m variation (TCAD simulated results) </p></li><li><p>101</p><p>variables (n=m) and solve for the exact solution, if we can transform the problem to linear one. Indeed, we can use the square values of the partial derivatives and variability quantities: y, xi. Then solving (1) for xi2 is a linear (n by n) problem. We must be careful as to which output variables we choose, otherwise the results can be erroneous, because each output variable is more sensitive to specific input variables. If we do not use at least one output variable that is strongly affected by an input variable, then this specific input variable can not be calculated. Also, some output variabilities are calculated more accurately and give better results. Exploiting all these 9 output parameters means that we can estimate a maximum of 9 input parameters in a linear treatment of the problem. </p><p>In our case, averaging the results of different combinations, we calculated Wfin = 0.855 nm, LG = 0.679 nm, m = 31.12 eV with relative errors (compared to the set values) 23.8 %, 3.53 %, and 0.60 % respectively. The variation of tox is relatively small and has negligible effect almost to all output variables thus can be omitted. </p><p>III. EXPERIMENTAL RESULTS </p><p>We successfully applied this method to experimental data of FinFETs with Wfin = 15 nm, LG = 30 nm tox = 1.7 nm and Hfin = 65 nm, each one having 5 fins and Pitch = 200 nm (distance between two adjacent fins). Two different areas of the wafer (Quad A and C) - provided from IMEC (Leuven, Belgium) - were measured. The measured transfer characteristics of the devices exhibited different statistical behavior in Quad A and C [8]. Following the procedure described in [8], we estimated the variability in a device, starting from the experimental data (ID-VG) of an ensemble of devices. The more devices measured the better results we obtain, since statistics are more reliable. Estimation of the unknown variabilities is relatively easy and quick (just electrical measurements of the actual devices). </p><p>For each device we calculated seven output parameters. We did not calculate the threshold voltage at VD = 1V and subsequently the threshold voltage roll-off. The results are tabulated in Table I. </p><p>In order to estimate the partial derivatives used in (1), we had to choose a device of average behavior and extract the models parameters as described in [9]. Then, solving for the sources of variability, we estimated for Quad A: Wfin = 2.00 nm, LG = 3.28 nm tox = 0.17 nm, m = 53.2 eV, for Quad C: Wfin = 1.90 nm, LG = 4.68 nm tox = 0.17 nm, m = 49.0 eV and all devices (both Quad A and C): Wfin = 1.80 nm, LG = 3.85 nm tox = 0.21 nm, m = 48.1 eV. </p><p>We also measured a set of FinFETs with Wfin = 10 nm, LG = 30 nm tox = 1.7 nm, fin height Hfin = 65 nm and Pitch = 200 nm, having 2-50 fins. Their variability was comparable to the previous device measured (with Wfin = 15 nm and N=5). Comparison of their variability in respect </p><p>to N is presented in Fig. 4. For a small number of fins the variability is higher (for Vt, SS), but tends to become constant at higher N (N&gt;10). Variability in Ion and Iov, (Fig. 4) almost follows the trend of the drain current (which is proportional to N). But if we divide Ion and Iov by N, we see that variability in currents (per fin) is getting lower for high N, as we expected, because of the larger sample of equivalent fins. Also, Iov is always lower than Ion, because it neglects excess variability in currents due to Vt (thus m). Unfortunately, the quality and (mainly) the quantity of these measurements do not allow to accurately solve for the sources of variability. </p><p> TABLE I </p><p>VARIABILITY OF THE INPUT VARIABLES OF FinFETs IN TWO AREAS (QUAD A AND C) OF A WAFER </p><p> VD Variability Quad A Quad C Overall 20 mV </p><p>Vt (mV) 57.8 51.1 54.5 Ion (x10-6 A) 2.63 3.45 3.46 Iov (x10-6 A) 1.95 2.49 2.77 SS (mV/dec) 7.49 5.50 7.49 </p><p>1 V Ion (x10-5 A) 5.65 5.98 5.88 Iov (x10-5 A) 3.84 4.42 4.40 SS (mV/dec) 17.4 10.4 16.2 </p><p>Another set of FinFETs with Wfin = 10 nm, LG = 30 nm, tox = 1.7 nm, Hfin = 65 nm and N-5, having Pitch 200, 400 and 1000 nm has been studied. A comparison of their transfer characteristics at VD= 20 mV for Pitch 200 and 1000 nm is shown in Fig. 5. Initially their behavior is not expected. From the dispersion of the characteristics, we expect to have higher Vt and SS for low Pitch (200 nm), while the higher I the higher the Pitch (1000 nm) is. </p><p>We can see this for all three Pitches in Fig. 6. Usually higher variability implies more dispersion in the characteristics, thus higher variability in all variables is expected. This behavior can be explained, if we consider also variation in the series resistance RSD of the device. For </p><p>0 10 20 30 40 500.00</p><p>0.01</p><p>0.02</p><p>0.03</p><p>0.04</p><p>0.05</p><p>0.06</p><p> VD= 20 mV</p><p>Vt (</p><p>V)</p><p>Number of fins, N</p><p>0 10 20 30 40 50</p><p>0</p><p>2</p><p>4 VD= 1 V VD= 20 mV</p><p>Ion</p><p> (x10</p><p>-4 A</p><p>)</p><p>Number of fins, N</p><p>02468101214</p><p>Ion</p><p>/N (x</p><p>10-5 A</p><p>)</p><p>0 10 20 30 40 500123456789</p><p>10 VD= 1 V VD= 20 mV</p><p>SS</p><p> (mV</p><p>/dec</p><p>)</p><p>Number of fins, N</p><p>0 10 20 30 40 500</p><p>1</p><p>2 VD= 1 V VD= 20 mV</p><p>Iov</p><p> (x10</p><p>-4 A</p><p>)</p><p>Number of fins, N</p><p>0</p><p>5</p><p>10</p><p>I ov</p><p> /N (x</p><p>10-6 A</p><p>) </p><p>Fig. 4. Variability of FinFETs with Wfin=10 nm, LG=30 nm, Hfin=65 nm, tox=1.7 nm, Pitch =200 nm and N=2-50. Hollow symbols reffer to variability per fin. </p></li><li><p>102</p><p>the low pitch (as in the former devices) RSD was negligible. But in higher Pitches (400 and 1000 nm) a RSD in the order of 200-300 can be observed. If we want to find the sources of these variabilities, we must develop further our model, as to comprise all the parasitics, which would affect also the series resistance and other electrical characteristics, such as capacitances. </p><p>IV. CONCLUSIONS </p><p>A quick and relatively easy method is presented, which can exploit our drain current compact model as well as the parameter extraction procedure, described elsewhere. </p><p>The method...</p></li></ul>

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