Simulation Technique

Threshold voltage is one of the key device parameters in the characteristics of nanoscale metal-oxide-semiconductor field effect transistors. This section presents the characteriza-tion technique for intrinsic parameter fluctuacharacteriza-tions consisting of line edge roughness (LER), oxide thickness fluctuation (OTF), random-dopant-fluctuation (RDF), and an emerging fluctuation source: work-function fluctuation (WKF). The characterization approaches are examined with experiment data. Base upon the independent of random variables, the total threshold voltage fluctuation, σVth,totalis expressed as follows [74]:

σ²Vth,total ≈ σ²Vth,RDF + σ²Vth,LER+ σ²Vth,OT F + σ²Vth,W KF, (2.9)

where σVth,RDF, σVth,LER, σVth,OT F, and σVth,W KF are the threshold voltage fluctuations caused by the random-dopant-fluctuation, line edge roughness, oxide thickness fluctuation, and the workfunction fluctuation, respectively. The statistical addition of individual fluc-tuation sources herein, as shown in Equation (2.9), simplifies the variability analysis of nano-devices and circuits, significantly [74]. In addition, the methodology of LER, OTF and WKF has been proposed and LER- ,OTF- and WKF-induced Vth fluctuations are ex-amined in our previous work [74]. However, the result shows that the RDF dominates the Vthfluctuation in NMOSFETs, as disclosed in Fig. 2.2.

Therefore, in this thesis, we focus on the characteristic fluctuations induced by random

2.3 : Simulation Technique 27

dopant and propose suppression technique to mitigate RDF. The nominal channel dop-ing concentration of the control devices is 1.5×10¹⁸ cm⁻³. They have a 16-nm gate, a TiN/HfSiON gate stack of 0.8-nm EOT, and a workfunction of 4.52 eV. Outside the chan-nel, the doping concentrations in the source/drain and background are 1.0×10²⁰cm⁻³and 1.0×10¹⁵cm⁻³, respectively. For the channel region, to consider the effect of random fluc-tuation of the number and location of discrete channel dopants, 1327 dopants are firstly randomly generated in a large cube (96 nm)³, in which the equivalent doping concentration is 1.5×10¹⁸cm⁻³, as shown in Fig. 2.3(a). The (96 nm)³ cube is then partitioned into 216 sub cubes of 16 nm³. The number of dopants may vary from zero to 14, and the average number is six, as shown in Figs. 2.3(b) and 2.3(c), respectively. In principle, 3D device simulation with the 216 channel structures almost covers cases, shown in Fig. 2.4, and thus will be fairly meaningful to reflect statistical randomness of dopant number. We have no-ticed that in this simulation only dopant within the channel region is treated discretely. The doping concentrations remain continuous in the source/drain region because the volume of source/drain region is two-order magnitude greater than that of channel region. These sub-cubes are equivalently mapped into the device channel for the 3D “atomistic” device simulation with discrete dopants, as shown in Fig. 2.5(a). In “atomistic” device simulation, the resolution of individual charges within a classical drift-diffusion simulation using a fine mesh creates problems associated with singularities in the Coulomb potential [75-77]. The

potential becomes too steep with fine mesh, and therefore, the majority carriers are un-physically trapped by ionized impurities, and the mobile carrier density is reduced [75-77].

Thus, the density-gradient approximation is used to handle discrete charges by properly in-troducing related quantum-mechanical effects, and coupled with Poisson equation as well as electron-hole current continuity equations [42-44,78-82].

Without loss of generality, the dual material gate (DMG) and inverse dual material gate (inDMG) are with 16-nm-gate and 1.5×10¹⁸ cm⁻³ equivalent channel doping concentra-tion. The estimated device with dual material gate has two types, DMG and inverse DMG, as shown in Figs. 2.5(b) and 2.5(c). For DMG device, the workfunction (WK) at the source and drain sides are WK1 and WK2, respectively, and WK1 > WK2. The inverse DMG device are designed accordingly, and WK1 < WK2. The gate materials could be MoN, TiN, and Ta, whose distributions of grain orientation and work-function are summarized in Fig. 2.5(d) [10].

As for the generation of lateral asymmetry doping, the adapted drain-end and near-source-end channel doping profile are shown in Figs. 2.6(a) and 2.6(b). Only half of the channel is doped and 1327 dopants are randomly generated in a large rectangular solid (gate width, source-drain direction, channel depth: 48 nm× 96 nm × 96 nm). Therefore, effective channel doping concentration is still 1.5×10¹⁸cm⁻³. Then the large cube is par-titioned into 216 sub-cubes ((8 nm)× (16 nm) × (16 nm)) and mapped into drain-end of

2.3 : Simulation Technique 29

channel region for discrete dopant simulation. Similarly, the dopants in sub-cubes may vary from zero to 14 (the average number is six) within its sub-cubes, as shown in Figs. 2.7 and 2.8. To estimate the device characteristics on the same basis, the threshold voltage for all devices are calibrated to 250 mV according to ITRS roadmap 2007 [2] for low-operating-power application.

In estimating current mismatch of current mirror circuit, dynamic characteristics of common source amplifier, ultra-small nanoscale devices, and for capturing the discrete-dopant-position-induced fluctuations, a device-circuit coupled simulation approach [33] is employed. The nodal voltage and loop current in the circuit can be calculated. The for-mulation of circuit equations is mainly base upon the Kirchhoff’s current law. The circuit nodal equation of current mirror, as illustrated in Fig. 2.9, is shown in below:

Node1 : V1= VDD, (2.10)

Node2 : V₂= V_DD−I_REFR_REF, (2.11)

Node3 : V3= 0, (2.12)

Node4 : V4= VDD−IOUTRL, (2.13)

and

Node1 : V5= VDD. (2.14)

The current mismatch of current mirror circuit and dynamic characteristics of common source amplifier is then estimated. Similarly, the circuit nodal equation of common source amplifier, as displayed in Fig. 2.10(b), is shown in below:

Node1 : V₁= V_in, (2.15)

Node2 : V₂= V_DD, (2.16)

Node3 : V3= VDD−IDR1, (2.17)

Node4 :V4

R2+CdV4

dt = ID, (2.18)

and

Node5 : V₅= 0. (2.19)

The operation bias of common-source circuit is VIN = 0.5V with sinusoid input wave, as shown in Fig. 2.10(a), which is used as a tested circuit to explore the fluctuation of dynamic characteristics. The time-domain simulation results are simultaneously used for the calcula-tion of the property of the frequency response, where the frequency is swept from 1×10⁸Hz to 1×10¹³ Hz. The common-source amplifier circuit with control devices is first explored to illustrate the details of random-dopant-fluctuation in high-frequency integrated circuits.

Thus, all device and circuit characteristics are obtained without any devices’ equivalent

2.3 : Simulation Technique 31

circuit models. The flowchart for mix-mode simulation method is shown in Fig. 2.11. The characteristics of devices of tested circuit are first estimated by solving the device trans-port equations and using as initial guesses in the device-circuit coupled simulation. The circuit nodal equations of the test circuit are formulated and then directly coupled to the device transport equations (in the form of a large matrix containing the circuit and device equations), which are solved simultaneously to obtain the circuit characteristics [33]. The flowchart of decoupled PDE is shown in Fig. 2.12. First we solve the nonlinear Poisson equation until it is convergence, and then the current continuity equations of electron and hole are following solved. If the error is less than the tolerance, the program stops and out-put the initial solution of device’s potential and perform the mixed-mode simulation. We solve the device’s equations coupled with circuit nodal equations until it is convergence.

Figure 2.13 shows the flow for solving decoupled PDE. First, the simulation domain has to be discretized. Applying the finite element approximation to the decoupled PDE, we ob-tained the nonlinear algebraic equations corresponding to the discretized grid. The Newton linearization method is then used to linearize the nonlinear equations. Finally, the linear algebraic equations can be solved using either direct or iterative method.



Figure 2.2: The RDF-, LER-, WKF-, OTF-induced Vthfluctuation for

NMOS devices, where the total σVthis calculated with Eq.

2.9.

2.3 : Simulation Technique 33 Figure 2.3: (a) Discrete dopants randomly distributed in the (96 nm)³

cube with the average concentration of 1.5×10¹⁸cm⁻³. There will be 1327 dopants within the cube, but dopants may vary from 0 to 14 ( the average number is 6 ) within its 216 sub cubes of 16 nm× 16 nm × 16 nm ((b) and (c)).

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 Figure 2.4: The histogram of the dopants in 216 sub cubes for 16 nm

devices. The dopants number can be describe by Gaussian Distribution with a mean of six.

2.3 : Simulation Technique 35

Figure 2.5: The sub-cubes are equivalently mapped into channel region for discrete dopant simulation as shown in MOSFET (a) control, (b) DMG, and (c) inverse DMG devices,

respectively. (d) The properties of metal material used in this study.

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Figure 2.6: The sub-cubes are equivalently mapped into channel region for discrete dopant simulation as shown in MOSFET (b) conventional lateral asymmetric channel (LAC) and (b) inverse LAC devices, respectively.

2.3 : Simulation Technique 37

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 Figure 2.7: The histogram of the dopants in 216 sub cubes (16 nm× 16

nm× 8 nm) for near-source-end channel doping profile of conventional LAC device. The dopants number can be describe by Gaussian Distribution with a mean of six.

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 Figure 2.8: The histogram of the dopants in 216 sub cubes (16 nm× 16

nm× 8 nm) for near-drain-end channel doping profile of inverse LAC device. The dopants number can be describe by Gaussian Distribution with a mean of six.

2.3 : Simulation Technique 39

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 Figure 2.9: The current mirror of analog circuit consisted of NMOS for

exploring the variation sources induced current (IREF/IOUT) mismatch.



Figure 2.10: (a) The common-source circuit is used as a tested circuit to explore the fluctuation of high-frequency characteristics.

(b) The input signal is a sinusoid input wave with 0.5 V offset. The frequency is swept from 1×10⁸Hz to 1×10¹³ Hz [33].

2.3 : Simulation Technique 41

 Figure 2.11: The flowchart for the mixed-mode device circuit coupling

simulation. The devices simulation is performed first and get the initial solution of the devices potential. The mixed-mode simulation is then executed until the final step [83].

 Figure 2.12: A flowchart of the decoupling algorithm. First we solve

the nonlinear Poisson equation until it is convergence, and then the current continuity equation of electron and hole is following solved. If the error is less than the tolerance, the program stops.

2.3 : Simulation Technique 43

 Figure 2.13: A flowchart for solving decoupled PDE. First the

simulation domain have to be discretized. Applying the finite element method approximation to the decoupled PDE, we obtained the nonlinear algebraic equations corresponding to the discretized grid. The Newton linearization method is then used to linearized the nonlinear equations. Finally, the linear algebraic equationscan be solved using either direct or iterative method [84].

在文檔中 16奈米場效應電晶體特性擾動抑制暨TFT-LCD驅動電路設計優化之研究 (頁 66-84)