Simulations Of "Atomistic" Effects In Nanoscale Dopant Profiling

Petru Andrei, Mohit Mehta Department of Electrical and Computer Engineering

Florida State University Tallahassee, USA

Mark J. Hagmann Newpath Research L.L.C.

Salt Lake City, USA

Abstract—The most commonly used method for dopant profiling in semiconductors is scanning capacitance microscopy (SCM). However, the analysis on which SCM is based assumes the dopant atoms are a continuous fluid, and “atomistic” effects caused by the locations of the individual dopant atoms make this assumption invalid at a resolution finer than 10 nm for a dopant concentration of 1018cm-3 where there is an average of only 1 atom per pixel. Simulations are presented which include quantum mechanical effects such as the confinement of carriers near the dopant atoms, as well as the discrete nature of the dopant atoms. These simulations suggest that error bars should be used to show the lack of certainty in dopant profiling as such a fine resolution is approached.

Keywords— Dopant profiling, atomistic effects, scanning capacitance microscopy, SCM, random dopant fluctuations, RDF

I. INTRODUCTION In 2011 Intel announced they had begun volume production

on their 22 nm microprocessor, and they plan to achieve volume production with 14 nm lithography by 2013 and 10 nm by 2015 [1]. The International Technology Roadmap for Semiconductors (ITRS, 2011) [2] states that sub-nanometer spatially resolved 2-D and 3-D dopant profiling is required in sub-16 nm lithography. However, a resolution of 10 nm with a dopant concentration of 1018cm-3 would correspond to an average of only 1 dopant atom per pixel so the meaning of a resolution finer than this may be questioned. Scanning capacitance microscopy (SCM) is the most commonly used method for dopant profiling, but the analysis upon which SCM is based considers the dopant atoms as a continuous fluid. Thus, it appears that even coarser resolution than was just calculated is required with SCM. It is surprising that “atomistic” simulations, allowing for the discrete nature of the dopant atoms, have been used to simulate nanoscale devices for more than 15 years [3-6] but this may be the first time that “atomistic” simulations have been considered for characterizing the materials to be used in these devices.

The article is structured as follows. In section II we summarize the traditional procedure of capacitance-voltage profiling and its extension to SCM. Then, in section III we present the model for simulating the transport properties in semiconductor materials and the technique for the computation of the variability of the differential capacitance. Section IV presents sample numerical results for a 50×50×50 nm MOS capacitor, and conclusions are drawn in section V.

II. TRADITIONAL PROCEDURE FOR DOPING PROFILING The standard procedure for the extraction of doping profiles

in semiconductors is based on using differential capacitance-voltage measurements to extract the distribution of majority carriers according to Eq. (1) and then computing the doping concentration with Eq. (2).

( )13C dCn x

q dVε

− = −

(1)

( ) ( ) ( )( )

2

1 dn xkT dN x n xdx n x dxq

ε = +

(2)

where x is the direction perpendicular to the surface of the semiconductor, n is the concentration of majority carriers, C is the differential capacitance, N is the doping concentration, and V is the bias voltage. Eq. (2) was introduced by Kennedy et al [7] to correct for the charge neutrality approximation in the region below the oxide-semiconductor interface.

Eqs. (1) and (2) can be used to extract the three-dimensional doping profile of semiconductor materials if the electrode is relatively narrow and can be moved across the surface of the semiconductor in the y and z directions. Although apparently sound, this approach deserves a bit more discussion. Equations (1) and (2) are derived using a one-dimensional analysis and based on the assumptions that there is a relatively large number of dopants in the semiconductor and the doping concentration is a continuous function of location. These assumptions are good if the depletion region is big enough so it contains a large number of dopants, however, they fail when the depletion region is too small because the total number of dopants is significantly reduced. For instance, the average number of dopants in the depletion region created by an electrode with cross-sectional area of 30×30 nm depends on the bias voltage and varies between 1 and 10 for an average doping concentration of 1017 cm-3. Any change in the number or location of dopants will result in significant changes of the dimensions and shape of the depletion region. The doping concentration has discontinuities at the locations of the doping atoms and the assumptions under which Eqs. (1) and (2) are derived are violated. Hence, it is natural to ask about the errors that we have if we are still employing these equations to extract

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the doping profile of the semiconductor. In addition, it is important to analyze how we need to modify these equations to include the effects induced by the discrete nature of the doping.

III. MATHEMATICAL MODELING

A. Transport equations In this subsection we summarize the transport equations

that we use to simulate the device. In order to capture the screening and quantum confinement effects that are caused by the atomistic nature of the dopants at nanoscale we perform "atomistic" simulations by solving the density-gradient equations coupled with the continuity and the Poisson equations. The density-gradient equations are widely used in the computational electronics community to account for quantum mechanical corrections of electron and hole transport [8]. The complete set of equations that we solve is

( ) ( ) 0D Aq p n N Nε ϕ + −∇ ⋅ ∇ + − + − = (3)

( )

( )2

0n

n n

b nT

nϕ φ

∇ ⋅ ∇+ − −Φ = (4)

( )

( )2

0p

n p

b pT

pϕ φ

∇ ⋅ ∇− + +Φ = (5)

( ) 0n nn nt

µ φ∂−∇ ⋅ ∇ =

∂ (6)

( ) 0p pp pt

µ φ∂+∇ ⋅ ∇ =

∂ (7)

where ϕ is the electric potential, n , nµ , nD , nφ and p , pµ ,

pD , pφ are the concentration, mobility, diffusivity, and quasi-Fermi potentials of the electrons and holes, respectively.

( )n TΦ and ( )p TΦ are functions that depend on the nature of electron and hole statistics in the semiconductor. In our simulations we use Boltzmann statistics for which

( ) lnni

kT nTq n

Φ = and ( ) ln ip

nkTTq p

Φ = , where T is the

absolute temperature. The quantum mechanical effects are “controlled” by the parameters:

2

*4nn n

ћbr qm

= and 2

*4pp p

ћbr qm

= (8)

where *nm and *

pm denote the effective masses of the electrons and holes, while nr and pr are dimensionless parameters that account for statistics of electron and holes in semiconductor devices. The values of nr and pr vary asymptotically from 1, when only the lowest energy sub-band is occupied (e.g. at low

temperature), to 3 when other sub-bands become populated as well (e.g. at high temperatures).

The system of Eqs. (3)-(7) is subject to appropriate boundary conditions and should be solved self-consistently to compute the electrostatic potential and carrier concentrations in realistic MOS capacitor structures. More details about the boundary conditions for Eqs (3)-(7) can be found in [9, 10].

Due to the stochastic nature of the ion implantation and diffusion processes, the doping concentration in Eqs. (3)-(7) is a random field. Hence, Eqs. (3)-(7) represent a system of stochastic partial differential equations that is solved to evaluate the electrostatic potential, quasi-Fermi potentials, and the electron and hole concentrations throughout the device. This differential capacitance can be expressed as a functional of the solution of this stochastic system of equations and is also a stochastic quantity that can be characterized by a given average value and variance. In the next section we present a technique that is based on the linearization of the transport equations and the computation of doping sensitivity functions to compute the variance of the differential capacitance. This technique is a generalization of the method of doping superposition coefficients developed in [11-13] for the evaluation of the variability of y-parameters in nanoscale MOSFETs. The doping sensitivity functions are also instrumental in the design of fluctuation-resistant structures and optimization of semiconductor devices [14].

B. Doping Sensitivity Analysis The doping sensitivity functions of a parameter were

initially introduced in [9] and were defined as to how much that parameter changes if we add one dopant atom at some location inside the semiconductor device. In this section, we are interested in computing the doping sensitivity function of the differential capacitance, which we denote by ( )Cγ r and which is a function of the position inside of the device. The differential capacitance is defined as the imaginary part of the small-signal electrode current divided by the angular frequency

( ) ( )Im GG

i vC V

ω∂ ∂

= (9)

It is shown in [4], that if the doping sensitivity function of parameter C is known we can compute the variance of that parameter by using the following equation

( ) ( )2 2C C N dσ γ

Ω= ∫ r r r (10)

where the integration is performed over the whole volume of the semiconductor region, Ω. Hence, for the remainder of this section, we present the technique that we use to derive the doping sensitivity function of the differential capacitance.

It is a bit easier to develop the algorithm for the computation of Cγ by starting from the discretized version of Eqs. (3)-(7), which can be written in the following form

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( )1 1,..., , ,...,k

n mdI x x N Ndt

+

( )1 1,..., , ,..., , 0k n mF x x N N V+ = , 1,...,k n= (11)

where ix and iD denote the state variables and the doping concentration at the mesh point i. Functions kI and kF come from the discretization of Eqs. (3)-(7), n is the total number of discretized equations, and m is the number of discretization points in the semiconductor material. If voltage V does not depend on time, the steady-state solution of Eq. (11) satisfies

( )1 10 0,..., , ,..., , 0k n m

GF x x N N V = (12)

Now consider a small-signal harmonic variation of parameter V

0 0 0j tV V V V V e ωδ δ= + = + (13)

where Vδ depends on time. After the transient regime has passed, we look for a solution of Eq. (11) with the form

0 0i i i i i j tx x x x x e ωδ δ= + = + (14)

where ixδ is the small-signal change in the state variables that is caused by Vδ . By combining Eqs. (13) and (14) with Eq. (11), expanding in a Taylor series, and keeping only the first-order terms we obtain

0k k k

i ii i

I F Fj x x VVx x

ω δ δ δ∂ ∂ ∂+ + =

∂∂ ∂ (15)

By introducing the functions

k k kG j I Fω= + (16)

equation (15) can be written more compactly as

0k k

ii

G Fx VVx

δ δ∂ ∂+ =∂∂

(17)

which can be inverted to give

i k

ik

x Fx VVG

δ δ∂ ∂= −

∂∂ (18)

The differential capacitance C can be defined as a linear

combination of ix

Vδδ

as

i i k

i i k

x x FC C CV VG

δδ

∂ ∂= = −

∂∂ (19)

where the iC are coefficients that do not depend of the ix or jD (Eq. (19) can be obtained from Eq. (9) by discretizing the

small-signal terminal current).

We are now interested in evaluating how much the differential capacitance changes when the doping variable is modified by an infinitesimally small quantity jN . An infinitesimal change in the doping variable jN results in infinitesimal changes of 0

ix (denoted by 0ix ) that can be

computed from Eq. (12), to give

0 0k k

i ji j

F Fx Nx N

∂ ∂+ =

∂ ∂

(20)

0

i ki j

k j

x Fx NF N∂ ∂

= −∂ ∂

(21)

To compute the variation of C we need to compute the variation of matrix elements i kx G∂ ∂ and kF V∂ ∂ ; these

variations are denoted by i kx G∂ ∂ and kF V∂ ∂ , respectively.

1) Computation of kF V∂ ∂ Matrix elements kF V∂ ∂ are evaluated at point

( )1 10 0,..., , ,..., ,n mx x N N V . Hence, we can use the chain rule to

write

2 2

0

k k ki j

i j

F F Fx NV V x V N

∂ ∂ ∂= + =

∂ ∂ ∂ ∂ ∂

2 2k i l k

ji l j j

F x F F NV x F N V N

∂ ∂ ∂ ∂= − + ∂ ∂ ∂ ∂ ∂ ∂

(22)

2) Computation of k iG x∂ ∂ Matrix elements k iG x∂ ∂ are also evaluated at point

( )1 10 0,..., , ,..., ,n mx x N N V so that

2 2

0

k k ki j

i i i j

G G Gx Nx V x x N

∂ ∂ ∂= + =

∂ ∂ ∂ ∂ ∂

2 2k j m k

ji j m j i j

G x F G Nx x F N x N

∂ ∂ ∂ ∂= − + ∂ ∂ ∂ ∂ ∂ ∂

(23)

3) Computation of i kx G∂ ∂ To compute the variation of the elements of the inverse

matrix of k iG x∂ ∂ we have

i i k k

ijk k j j

x x G GG G x x

δ ∂ ∂ ∂ ∂

+ + = ∂ ∂ ∂ ∂

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0i k j i k j

k j l k j l

x G x x G xG x G G x G∂ ∂ ∂ ∂ ∂ ∂

+ =∂ ∂ ∂ ∂ ∂ ∂

i i l j

k l j k

x x G xG G x G∂ ∂ ∂ ∂

= −∂ ∂ ∂ ∂

(24)

where we have used Eq. (23) and k j

klj l

G xx G

δ∂ ∂=

∂ ∂, where k

lδ is

the Kronecker delta function.

4) Computation of C The variation of the differential capacitance C , can be

obtained by expending Eq. (19) in a Taylor series and keeping only the first-order terms to get

i k i k

i ik k

x F x FC C CV VG G

∂ ∂ ∂ ∂= − −

∂ ∂∂ ∂ (25)

Substituting Eqs. (22) and (24) into Eq. (25) we obtain

i l m k

i l m k

x G x FC CVG x G

∂ ∂ ∂ ∂=

∂∂ ∂ ∂

2 2i k i l k

ji k i l j j

x F x F FC NG V x F N V N

∂ ∂ ∂ ∂ ∂− − + = ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2i l n k l m k

ji l m n k j m j k

x G x F G x FC NVG x x F N x N G

∂ ∂ ∂ ∂ ∂ ∂ ∂= − + ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2i k i l k

ji k i l j j

x F x F FC NG V x F N V N

∂ ∂ ∂ ∂ ∂− − + ∂ ∂ ∂ ∂ ∂ ∂ ∂

which can be used to obtain the following expression for the doping sensitivity coefficient at location j.

2 2

j

i l n k l m k

i l m n k j m j kC

x G x F G x FCVG x x F N x N G

γ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= − + ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

2 2i k i l k

i k i l j j

x F x F FCG V x F N V N

∂ ∂ ∂ ∂ ∂− − + ∂ ∂ ∂ ∂ ∂ ∂ ∂

(26)

The last two equations give the discretized values of the doping sensitivity function of C. The tensors that appear on the right-hand side of Eq. (26) can be computed either analytically from the discretized system of the transport equations or by using the Jacobian matrix. It turns out that the last equation can be slightly simplified in the case of doping profiling. Indeed, the doping concentration at location j , jN , and the electrode potential appear in separate equations in the system of transport equations (3)-(7). As a consequence, the mixed derivatives with respect to the two variables are equal to zero

2

0k

j

FV N∂

=∂ ∂

(27)

In addition, the electrode potential V enters linearly in the transport equations and we can write

2

0k

l

FV x∂

=∂ ∂

(28)

Using these observations, the last term in (26) vanishes and doping sensitivity coefficient at location j becomes

2 2

j

i l n k l m k

i l m n k j m j kC

x G x F G x FPVG x x F N x N G

γ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= − + ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (29)

The last equation is used to compute the doping sensitivity function of the differential capacitance.

IV. SIMULATION RESULTS The technique presented in the previous section was

implemented in our mixed-mode electronic device simulator, RandFlux© [15] and used to analyze the effect of discrete dopant fluctuations on the doping profiling of a semiconductor device. For this purpose we considered a 50×50×50 nm MOS capacitor, in which the gate electrode is situated at a distance of 1 nm away from the interface of the semiconductor. The space between the electrode and semiconductor was modeled as SiO2, as in the case of a classical MOS capacitor.

First we use Eq. (26) to compute the doping sensitivity function of the differential capacitance. Then, Eq. (10) is used to compute the variance of C. The results of these simulations are represented in Fig. 1, which shows the differential capacitance and the standard deviation of the differential capacitance (σC) as a function of the potential of the electrode for three average doping concentrations, Na = 1017 cm-3, 1018 cm-3, and 1019 cm-3. The height of the vertical bars is equal to three times σC and the bias voltage is shifted in such a way as to obtain flat bands in the semiconductor when the bias is zero. Notice that there is a strong variability of the differential capacitance, particularly for positive bias voltages where the width of the depletion region is sensitive to the value of the electrode potential. This strong variability is due to the microscopic variation of the dopant placement and concentration inside of the device, and is expected to cause significant errors if the differential capacitance is used in extracting the doping profile as in Eqs. (1) and (2)).

To gain more insight into the variability of the differential capacitance we have performed Monte-Carlo simulations, in which we have formed a large number of models for devices with macroscopically identical average doping concentration but having different atomic configurations, and then solved the transport equations (3)-(7) for each of these individual devices. In this way, we have computed the differential capacitance as a function of the bias voltage for each device. The technique used to generate the device models is to the same as the one

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-1 0 1 20.0

0.2

0.4

0.6

0.8

C/C ox

Bias voltage (V)

Na = 1018 cm-3

Na = 1017 cm-3

Variability analysis using the doping sensitivity method

Na = 1019 cm-3

Fig. 1. Differential capacitance, C, and its standard deviation for three 50nm x 50 nm x 50nm MOSCs with different doping concentrations.

-1 0 1 20.0

0.2

0.4

0.6

0.8

Na = 1019 cm-3

Na = 1017 cm-3

Na = 1018 cm-3

3 x 200 supposedly identical devices Monte-Carlo simulations

C/C ox

Bias voltage (V) Fig. 2. Differential capacitance as a function of the bias voltage (V) computed for 3 groups of 200 supposedly identical devices.

presented by Asenov [16]. The results of these simulations are presented in Fig. 2 for the same device specifications as in Fig. 1. The doping profiles for each device were generated such that the average doping concentration is equal to the three values that were used in the previous simulations. First, notice the good agreement between the previous technique and the Monte-Carlo simulations, and then, the high variability of the differential capacitance, particularly at low doping concentrations where any variation in the number and position of dopants causes large variations of C.

We will now discuss about the impact of random doping fluctuations on the extracted doping profiles of semiconductors. For this purpose we use Eqs. (1) and (2) to compute the doping profile for each of the devices simulated in Fig. 2, in the direction perpendicular to the surface of the semiconductor. The results of these computations are presented in Figs. 3(a), 3(b), and 3(c) for the same three sets of devices with different average doping concentrations (Na = 1017 cm-3, 1018 cm-3, and 1019 cm-3, respectively). Notice the large discrepancy between the extracted doping concentrations of each device. The discrepancy is larger for lower average

doping concentrations where there are fewer dopant atoms in the depletion region, as may also be seen in the previous simulations.

Figure 4 shows the electron concentration and electrostatic potential in two randomly selected devices in a section of 30×30×30 nm of the MOS capacitor. The average doping concentration is 1019 cm-3 and the bias voltage is indicated below each figure. The three planes on the right show the electron concentration and electrostatic potential in three cross-sections throughout the device. Notice that, although the doping concentration is relatively high, one can clearly distinguish the location of the dopant atoms inside the semiconductor. The electrostatic potential has relatively low values in the regions around these atoms, and the electron concentration is relatively high. During a normal ac impedance measurement the electrons will follow on given paths inside the semiconductor. The conductivity of these paths depends on the bias voltage and on the exact location of doping atoms. The shape and number of these paths varies, which causes a large variability of the small-signal current from device to device. The total number of dopants inside the semiconductor decreases even more for doping concentrations below 1019 cm-3. For instance, for a doping concentration of 1017 cm-3 the average number of dopants in the region shown in Fig. 4 is 2.7 impurities. Any variation in this number and in the location of the dopants results in large variations of the differential capacitance and in the extracted doping concentration, as shown in the previous figures.

V. CONCLUSION Atomistic simulations predict relatively large variations of

the differential capacitance and extracted doping profiles in nanoscale MOS capacitor structures. The discrete and relatively small number of dopant atoms in the depletion region can lead to erroneous results for the dopant concentration during doping profiling. New techniques are required to extract the doping profile information at the nanoscale. These techniques must account not only for the quantum confinement of the carriers at the interface of the semiconductor but also for the confinement of carriers near and around the impurities, and also for the discrete nature of the dopant atoms.

REFERENCES [1] Intel Newsroom, "Intel 22 nm 3-D tri-gate transistor technology,"

http://newsroom.intel.com/docs/zfpv-2032. [2] International Technology Roadmap for Semiconductors, 2012. [3] A. Asenov, "Random dopant induced threshold voltage lowering and

fluctuations in sub-0.1 µm MOSFDT's: A 3-D 'Atomistic' simulation study," IEEE Trans. Electron Devices, vol. 45, pp. 2505-2513, December 1998.

[4] P. Andrei and I. Mayergoyz, "Analysis of fluctuations in semiconductor devices through self-consistent Poisson-Schrödinger computations," J. Appl. Phys., vol. 96, pp. 2071-2079, August 2004.

[5] A.T. Putra, A Nishida, S. Kamohara, t. Tsunomura and T. Hiramoto, "Consideration of random dopant fluctuation models for accurate prediction of threshold voltage variation of metal-oxide-semiconductor field-effect transistors in 45 nm technology and beyond," Jpn. J. Appl. Phys., vol. 48, art. no. 044502, April 2009.

[6] S. Markov, B. Cheng and A. Asenov, "Statistical variability in fully depleted SOI MOSFETs due to random dopant fluctuations in the source and drain extensions," IEEE Electron Device Lett. vol. 33, pp. 315-317, March 2012.

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40 50 60 70 80 900.5

1.0

1.5

2.0

2.5(a)200 devices "atomistically" doped

Na = 1017 cm-3

Do

ping

conc

entra

tion

(x10

17 cm

-3)

Distance from interface (nm)

20 30 400.5

1.0

1.5

2.0

2.5(b)

Dopi

ng co

ncen

tratio

n (x

1018

cm-3)

Distance from interface (nm)

200 devices "atomistically" doped Na = 1018 cm-3

8 10 12 140.5

1.0

1.5

2.0

2.5(c)200 devices "atomistically" doped

Na = 1019 cm-3

Dopi

ng co

ncen

tratio

n (x

1019

cm-3)

Distance from interface (nm) Fig. 3. Extracted doping concentrations for each of the devices represented in Fig. 2. Note the large variation in the doping concentration particularly at low concentrations, where there are relatively few dopant atoms in the device. Any variation in the location of these dopants results in large fluctuations of C and, consequently, of the extracted doping concentrations.

Fig. 4. Electron concentration and electrostatic potential in a region of 30 nm x 30 nm x 30 nm region of the MOS-C with average doping concentrations and bias voltages indicated below each figure. The 3 planes on the right show the electron concentration and electrostatic potential in three cross-sections throughout the device.

[7] D. P. Kennedy, et al., "On the Measurement of Impurity Atom

Distributions in Silicon by the Differential Capacitance Technique," IBM J. Res. Develop., vol. 115, pp. 399-409, September 1968.

[8] M. G. Ancona and G. J. Iafrate, "Quantum Correction to the Equation of State of an Electron-Gas in a Semiconductor," Phys. Rev. B, vol. 39, pp. 9536-9540, May 1989.

[9] I. D. Mayergoyz and P. Andrei, "Statistical analysis of semiconductor devices," J. Appl. Phys., vol. 90, pp. 3019-3029, Septempber 2001.

[10] P. Andrei and I. Mayergoyz, "Quantum mechanical effects on random oxide thickness and doping fluctuations in ultrasmall semiconductor devices," J. Appl. Phys., vol. 94, pp. 7163-7172, Dec 1 2003.

[11] P. Andrei and I. Mayergoyz, "Analysis of random-dopant induced fluctuations of frequency characteristics of semiconductor devices," J. Appl. Phys., vol. 93, pp. 4646-4652, April 2003.

[12] P. Andrei and I. Mayergoyz, "Sensitivity of frequency characteristics of semiconductor devices to random doping fluctuations," Solid-State Electronics, vol. 48, pp. 133-141, January 2004.

[13] P. Andrei and I. Mayergoyz, "Random doping-induced fluctuations of subthreshold characteristics in MOSFET devices," Solid-State Electronics, vol. 47, pp. 2055-2061, November 2003.

[14] P. Andrei and L. Oniciuc, "Suppressing random dopant-induced fluctuations of threshold voltages in semiconductor devices," J. Appl. Phys., vol. 104, art.nNo. 104508, November 2008.

[15]RandFlux: User's manual v.0.6, Florida State University, http://www.eng.fsu.edu/ms/RandFlux.

[16] X. Wang, et al., "Simulation Study of Dominant Statistical Variability Sources in 32-nm High-kappa/Metal Gate CMOS," IEEE Electron Device Letters, vol. 33, pp. 643-645, May 2012.

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