Ee5502 Presentation 1p6l6f

EE5502 MOS Devices CA Small-Signal Analysis of Silicon Nanowire Transistors based on a [1] Poisson/Schrodinger/Boltzmann Solver a

- Reddy Devarajan Srinivasan (A0165006N) - Vinay Panduranga Gudisagar (A0164992N)

Introduction  Wire like structure made of Silicon with diameter or lateral dimension of nanometer (10-9m).

 The current from the source to the drain is turned on and off by the voltage applied to the gate like a conventional transistor. Gate Gate Insulator Drain

W

Source

HAT HY

Channel

Insulator Channel

 With the under 10nm technology generation being under intensive development, the change in geometry of transistors from fin to wire is expected to further enable device scaling due to better short-channel control as well as high current density.  Because the gate in nanowires is surrounding the channel, it can control the electrostatics of the 2 channel more efficiently than the conventional MOSFET.

SiNW vs Bulk-Si Transistor  With multi-gate structures around the channel (nano-wire), we get the best control with gate voltage (VG).  Improved control over VG directly improves the critical parameters like Subthreshold Slope (SS) and Drain Induced Barrier Lowering (DIBL) to suppress Short Channel Effect (SCE) in ultrashort channels. TYPE

Above Threshold

Sub-Threshold

Bulk-Si

Surface Conduction High E field

Surface conduction Low E field

SiNW

Bulk Conduction Low E field

Bulk Conduction High E field

 Vth depends on doping, width and thickness of nanowire.

Measured ID(VG) of an n-channel device with Weff=25nm and L=1μm

 SiNW can exhibit low leakage currents and excellent short channel behavior . 3

Steady State Analysis  The Poisson equation is solved for the electrostatic potential in the 3D real space. The finite volume method is used to ensure flux conservation in the discretization process.  The 1D Boltzmann transport equation is discretized using H-transformation[2],[3] which removes the derivative with respect to kz and aligns the energy grid with the trajectories of ballistic carriers.  In order to treat strong quantization effects, the time-independent Schrodinger Equation (SE) is numerically solved in x-y slices perpendicular to the transport direction using the FEAST[4] eigensolver package.

 The coupled Boltzmann, Poisson, and Schrodinger equations are solved by the Newton-Raphson scheme, with the unknown variables being the electrostatic potential ϕ(r) and the distribution function f ν(z, H). 4

Small Signal Analysis  Small-signal analysis requires linearization of the system of equations around the stationary values.

 Only additional coefficients required for small signal analysis is from the time-derivative of the Boltzmann Equation (BE).  Since the small-signal computations are often performed for many frequencies, the steady-state solutions and the corresponding Jacobian matrix are saved after the convergence of Newton Raphson system and reused at each frequency value. Hence, the time-consuming evaluation of the Jacobian matrix is avoided.  While moving from the continuous BE to the discretized one might lead to unexpected problems with the reciprocity and ivity of the small-signal parameters in equilibrium conditions.  Factorization[5] of the distribution function into its equilibrium and non-equilibrium parts, and making sure that we have a consistent formulation of BE and PE in small-signal sense, restores the necessary symmetries.  The small-signal terminal currents are calculated using a formulation of the Ramo-Shockley theorem[6] that is consistent with the simulation framework, i.e. one-dimensional BE along the transport direction and two-dimensional SE in the transverse planes. 5

Advantages  As compared to the usual approach to self-consistently solve the coupled system of Poisson, Schrodinger, and Boltzmann equations in Gummel-type iterations[7],[8], semi-classical simulators based on Boltzmann’s equation is used.  Deterministic nature of simulators based on Boltzmann’s equation allows us to easily simulate rare events, deep-subthreshold operating points, and events on completely different time scales which is an important feature that stochastic Monte Carlo simulators lack.

 Multi-subband deterministic framework pre-developed for quadratically converging stable Full Newton-Raphson (FNR)[9] nanowire transistor solver of the combined system of Poisson, Schrodinger and Boltzmann equations is used.  The main advantage of such FNR approach is evident when we notice how such formulation can pave the way towards exact small-signal analysis, i.e. once an FNR solver is available for the stationary problem, our system of equations is already linearized and the same derivatives can be used in investigation of small AC perturbations covering the full frequency range.

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Results  Doping Concentration  Source – Drain ND= 2 X 1019 cm-3  Channel region ND = 1 X 1017 cm-3  Assumption Lch = LG

 SE is solved limiting up to 5 sub-bands in the Boltzmann equation

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Results  ittance Vs Frequency at Equilibrium VDS = 0V & VGS = 0.5V  Similar to Pao-Sah model for MOSFET, it was observed the reciprocity holds good in equilibrium (Y12 = Y21)  Small-signal current gain.

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Results  Cut-Off frequencies shows discontinuities for different gate lengths.  So stability factor is defined as Roller Factor,  Stable at frequencies in which K > 1.

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Conclusion  Nanowire solver suitable for steady state analysis and small-signal analysis.  Robust and stable computations in the complete frequency range.

 Small-signal parameters show reciprocity and ivity in equilibrium conditions similar to Pao-Sah and charge sheet model for MOSFET  Discontinuities observed when changing any quantity that influence the steady-state which was consequences of transforming BE into total energy

 These can be reduced by refining energy grid.  Presented solver of nanowire devices for RF and High Frequency applications.

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Conclusion

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References [1] Maziar Noei, Dino Ruic and Christoph Jungemann, “Small-Signal Analysis of Silicon Nanowire Transistors Based on a Poisson/Schrodinger/Boltzmann Solver,” 2017 International Conference on Simulation of Semiconductor Processes and Devices. [2] A. Gnudi, D. Ventura, G. Baccarani, and F. Odeh, “Two-dimensional MOSFET simulation by means of a multidimensional spherical harmonics expansion of the Boltzmann transport equation,” Solid-State Electron., vol. 36, no. 4, pp. 575 - 581, 1993. [3] S.-M. Hong, A. T. Pham, and C. Jungemann, Deterministic solvers for the Boltzmann transport equation. Computational Microelectronics, Wien, New York: Springer, 2011. [4] E. Polizzi, “Density-matrix-based algorithm for solving eigenvalue problems,” Phys. Rev. B, vol. 79, p. 115112, Mar 2009. [5] D. Ruic and C. Jungemann, “Numerical aspects of noise simulation in MOSFETs by a Langevin-Boltzmann solver,” Journal of Computational Electronics, vol. 14, no. 1, pp. 21–36, 2015. [6] H. Kim, H. S. Min, T. W. Tang, and Y. J. Park, “An extended proof of the Ramo-Shockley theorem,” Solid–State Electron., vol. 34, pp. 1251–1253, 1991. [7] S. Jin, M. V. Fischetti, and T. w. Tang, “Theoretical study of carrier transport in silicon nanowire transistors based on the multisubband Boltzmann transport equation,” Electron Devices, IEEE Transactions on, vol. 55, pp. 2886–2897, 2008. [8] S. Scaldaferri, G. Curatola, and G. Iannaccone, “Direct solution of the boltzmann transport equation and poisson/schrodinger equation for nanoscale mosfets,” vol. 54, no. 11, 2007. [9] M. Noei and C. Jungemann, “Numerical investigation of junctionless nanowire transistors using a boltzmann/schrdinger/poisson full newton- raphson solver,” Proc. SISPAD, 2016. [10] Junctionless Nanowire Transistor (JNT): Properties and Design Guidelines. A. Kranti, R. Yan, C.-W. Lee, I. Ferain, R. Yu, N. Dehdashti Akhavan, P. Razavi, JP Colinge. Tyndall National Institute, University College Cork, Cork, Ireland 12