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  • 99978-1-4799-5296-0/14/$31.00 2014 IEEE

    PROC. 29th INTERNATIONAL CONFERENCE ON MICROELECTRONICS (MIEL 2014), BELGRADE, SERBIA, 12-14 MAY, 2014

    Variability Analysis Prediction Method for Nanoscale Triple Gate FinFETs

    D. Tassis, I. Messaris, N. Fasarakis, S. Nikolaidis, G. Ghibaudo and C. Dimitriadis

    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

    I. INTRODUCTION

    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.

    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:

    22 yy xixi i

    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

    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.

    II. MODEL VALIDATION A. Comparison to TCAD Simulated Device

    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.

    The variability of these parameters has an impact on the variability of the transfer characteristics of the devices (Fig. 1).

    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

    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

    G. Ghibaudo is with IMEP, MINATEC, Parvis Louis Neel, 38054 Grenoble Cedex 9, France.

  • 100

    = 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.

    B. Variability decomposition

    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).

    In order to estimate total standard deviations, the components (yi) are summed according to (1) - in a square manner:

    2 2y yii (2)

    For the theoretical estimation, (1) gives: yy xi ixi

    (3)

    We used the standard deviations from the distribution of each variable xi while the partially derivatives are calculated from the model.

    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.

    C. Estimation of variability sources

    For a given set of output parameters and their

    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

    0.0 0.2 0.4 0.6 0.8 1.0 1.210-1410-1210-1010-810-610-4

    0.0 0.2 0.4 0.6 0.8 1.0 1.20

    1x10-5

    2x10-5

    VD= 20 mVD

    rain

    cur

    rent

    , ID (A

    )

    Dra

    in c

    urre

    nt, I

    D (A

    )

    Gate Voltage, VG (V)

    0.0 0.2 0.4 0.6 0.8 1.0 1.210-1210-1010-810-610-410-2

    0.0 0.2 0.4 0.6 0.8 1.0 1.201x10-42x10-43x10-44x10-4

    Dra

    in c

    urre

    nt, I

    D (A

    )

    VD= 1 V

    Dra

    in c

    urre

    nt, I

    D (A

    )Gate Voltage, VG (V)

    VD= 1 V

    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).

    0.00 0.01 0.02 0.03 0.04

    Vt

    VD=1

    VV

    D=2

    0mV

    Threshold Voltage variability (V)

    All m tox Wfin Lg

    0.000 0.001 0.002 0.003 0.004

    VD =

    20

    mV

    VD =

    1 V

    Subthreshold Swing variability, SS (V/dec)

    All m tox Wfin Lg

    0.0 5.0x10-6 1.0x10-5 1.5x10-5 2.0x10-5

    All m tox Wfin LgV

    D =

    20

    mV

    VD =

    1 V

    On-current variability, Ion (A)0 5x10-6 1x10-5

    VD =

    1 V

    Overdrive Current variability, Iov (A)

    All m tox Wfin LgV

    D =

    20

    mV

    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)

    0.00 0.01 0.02 0.03 0.04

    Vt

    VD=1

    VV

    D=2

    0mV

    Threshold Voltage variability (V)

    All m tox Wfin Lg

    0.000 0.001 0.002 0.003 0.004

    VD =

    1 V

    VD =

    20

    mV

    Subthreshold Swing variability, SS (V/dec)

    All m tox Wfin Lg

    0.0 5.0x10-6 1.0x10-5 1.5x10-5 2.0x10-5

    VD =

    20

    mV

    VD =

    1 V

    On-current variability, Ion (A)

    All m tox Wfin Lg

    0.0 5.0x10-6 1.0x10-5

    Overdrive Current variability, Iov (A)

    All m tox Wfin LgV

    D =

    20

    mV

    VD =

    1 V

    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 r