Università degli Studi di Pisa Giuseppe Iannaccone G. Iannaccone Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Pisa Via Diotisalvi 2, I-56122, Pisa, Italy IWCE 2004 Analytical and Numerical Investigation of Noise in nanoscale Ballistic Field Effect Transistors Acknowledgments: Support from EU (SINANO), MIUR (FIRB), Fondazione CRP

Università degli Studi di Pisa Giuseppe Iannaccone Motivation 1. Effects typical of mesoscopic devices can be observed in “mundane” MOSFETs at room temperature as the ballistic component of the drain current increases 2. Suitable noise models are required by circuit designers, especially for analog and mixed signal applications Here we focus on the limit of ballistic transport.

Università degli Studi di Pisa Giuseppe Iannaccone Noise of the drain current Transition from Thermal to shot in as the ballistic limit is approached Shot noise of the gate current Plus contributions due to defects (not considered here) Noise in nanotransistors INTEL - Prototype 20 nm MOSFET NMOS Gate delay 0.6 ps Shot noise of the gate tunnel current Noise of the drain current

Università degli Studi di Pisa Giuseppe Iannaccone Shot Noise Noise is an extremely sensitive probe of electron- electron interaction. No interactions  Poissonian process Interaction introduce “coordination” in the collective motion of electrons, making the process non Poissonian ( S  S full ). Interaction: Pauli Exclusion and Coulomb Repulsion Full shot noise S: power spectral density of the noise current

Università degli Studi di Pisa Giuseppe Iannaccone Ballistic transport in MOSFETs (I) Density of states in the first subband in the channel: Electron density at the subband peak in the channel: y SourceDrain

Università degli Studi di Pisa Giuseppe Iannaccone Charge fluctuations in MOSFETs Fluctuations of n 2D as function of  f S and  f D : fluctuations electrostatics in the contacts Subband maximum E M depends on n 2D via electrostatics Electrostatic effects are included in a single capacitance per unit area C C

Università degli Studi di Pisa Giuseppe Iannaccone Equivalent circuit Add quantum capacitance towards the source and the drain. Equivalent circuit: y C

Università degli Studi di Pisa Giuseppe Iannaccone Barrier modulation Fluctuation of channel barrier Current density is modulated by barrier height ! Total capacitance Longitudinal velocity

Università degli Studi di Pisa Giuseppe Iannaccone Current fluctuations depend on fluctuations of the occupation factors and of the channel barrier: fluctuations electrostatics in the contacts Current fluctuations expressed as a function of contact fluctuations Current Fluctuations

Università degli Studi di Pisa Giuseppe Iannaccone Shot noise power spectral density Power spectral density: Far from equilibrium, if f D = 0, we have C QD = 0, and

Università degli Studi di Pisa Giuseppe Iannaccone Noise suppression factor (Fano factor) Fano factor is always < 1 and If C C is very large (e.g., large gate capacitance) then Coulomb interaction is completely screened: For Maxwell-Boltzmann statistics (e.g. below threshold), f S << 1 Effect of Pauli Exclusion Effect of Coulomb Interaction

Università degli Studi di Pisa Giuseppe Iannaccone 25 nm “Well tempered” MOSFET Doping Profile of the 25 nm “well tempered” MOSFET (D. Antoniadis) Effective channel length 25 nm Super-halo doping in the channel minimizes charge sharing effects Lowest subband profile from 2D PS solver (G. Fiori et al., APL 81, 3672 (2002)) With Vg=1V, Vds=0.1 V, 96.5% of current is carried by the 1st subband

Università degli Studi di Pisa Giuseppe Iannaccone Subband Maximum and Source Quantum Capacitance C QS

Università degli Studi di Pisa Giuseppe Iannaccone Shot noise suppression in well tempered MOSFETs V DS = 0.5 V V DS = 1 V

Università degli Studi di Pisa Giuseppe Iannaccone Noise in the partially ballistic MOSFETs (I) (with G. Mugnaini)  the first N-1 MOSFETs can be aggregated in an equivalent drift- diffusion MOSFET. ( G. Mugnaini et al., submitted to IEEE-TED). The channel of an arbitrary MOSFET is decomposed in a chain of ballistic MOSFETs of length the mean free path.

Università degli Studi di Pisa Giuseppe Iannaccone Noise in the partially ballistic MOSFETs (II) (with G. Mugnaini - preliminary) Thermal noise source + shot noise source As the ratio between the device length and the mean free path is reduced, Noise has a transition THERMAL  SHOT Presently including the effect of electrostatics on noise N=L/

Università degli Studi di Pisa Giuseppe Iannaccone Gate currents: Fresh and stressed oxides Experimental results by F. Crupi – From G. Iannaccone et al. IEEE-TED 50, 1363 (2003) Stress voltage 7.8 V(8 V is the breakdown voltage) SILCs should introduce alter also the noise properties Current (A) SILC Fresh Voltage (V) I-V characteristics 6 nm oxide Noise properties of the current through fresh oxides: full shot noise at large currents

Università degli Studi di Pisa Giuseppe Iannaccone Current through fresh oxides (tunneling + native TAT) theory and exp. Trap distribution is a gaussian centered at 1.8 eV below the oxide CB, with 0.1 eV standard deviation Exp. By F. Crupi

Università degli Studi di Pisa Giuseppe Iannaccone TAT model G. Iannaccone et al. IEEE-TED 50, 1363 (2003 Generation and Recombination rates Trap occupation factor FANO Factor

Università degli Studi di Pisa Giuseppe Iannaccone Extraction of SILC trap distribution Comparison with experiments 6 nm oxide Gaussian distribution centered -0.5 eV below Si CB, standard deviation 82 meV

Università degli Studi di Pisa Giuseppe Iannaccone Extraction of SILC trap distribution (V) For thicker oxides shot noise suppression is due to transitions through native traps

Università degli Studi di Pisa Giuseppe Iannaccone Conclusion We have derived an analytical expression of noise in ballistic MOSFETs with two well defined contributions from Pauli exclusion and Coulomb repulsion. Noise properties can be computed from a numerical simulation of DC electrical properties. Numerical results for “well tempered” MOSFETs operating in the ballistic regime have been shown, exhibiting room temperature suppression of shot noise, in typical operating conditions, down to Shot Noise of the gate current: contribution of native traps may be important also for noise properties (experiments here are still missing) For thicker oxides a distribution of traps can be extracted that reproduces both DC and noise characteristcs

Università degli Studi di Pisa Giuseppe Iannaccone Current Fluctuations Current fluctuations depend on fluctuations of the occupation factors and of the channel barrier: fluctuations electrostatics in the contacts We can introduce two average velocities v S and v D :

Università degli Studi di Pisa Giuseppe Iannaccone Equilibrium and far from equilibrium If f S = f D, S reduces to 4KTG, as it must be, where Far from equilibrium, if f D = 0, we have C QD = 0, and The noise suppression factor is a weighted average

Università degli Studi di Pisa Giuseppe Iannaccone Extraction of native and SILC trap distribution Extraction of native and SILC trap distribution (I) Simulations with a distribution uniform in energy do not provide satisfactory results Trap distribution indipendent of position Integral equation withas the unknown Hp:

Università degli Studi di Pisa Giuseppe Iannaccone Extraction of SILC trap distribution (II) Electrons from cathode VB Model A: Riccò, Gozzi, Lanzoni, IEEE TED 45, “mean” quantities: fluxes and capture cross section Model B: Ielmini, Spinelli, Lacaita, IEEE TED 47, Transient SILC components

Università degli Studi di Pisa Giuseppe Iannaccone Extraction of SILC trap distribution (III) 5.9 nm oxide ? Comparison with exp. performed in Pisa Other thicknesses ? Comparison with experiments drawn from the literature (Ricco’ et al.) Effect of surface traps for very low voltages J-V Curves

Università degli Studi di Pisa Giuseppe Iannaccone Extraction of SILC trap distribution (IV) Fano Factor Stronger suppression for E m approaching the silicon gap center

Università degli Studi di Pisa Giuseppe Iannaccone Understanding the nature of SILCs Stress-induced leakage currents (SILCs) are the single most important limit to downscaling of non-volatile memories ? [read disturb, retention degradation] SILCs are due to tunneling assisted by traps generated by electric field stress. The energy distribution of traps is not known We show that detailed modeling, coupled with DC and noise characterization, can provide enough information to extract information about the energy distribution of traps

Università degli Studi di Pisa Giuseppe Iannaccone Tunneling Current: fresh oxides Electron effective mass in the oxide conduction band Determination of the oxide thickness Native traps are required for fitting the current at low fields

Università degli Studi di Pisa Giuseppe Iannaccone Transition probability Capture cross section TAT model (I) Two-Step tunneling Anelastic TAT G. Iannaccone et al., IEEE-TED 2003

Università degli Studi di Pisa Giuseppe Iannaccone Tunneling Current: Model The electron density n(x) at the Si-SiO 2 interface is computed by solving the Schrödinger equation for the two-fold and four-fold degenerate conduction band minima. 1D Poisson and Schrödinger equations are solved iteratively. Once the band profiles and charge densities are obtained, we can compute the tunneling current

Università degli Studi di Pisa Giuseppe Iannaccone 25 nm “Well tempered” MOSFET Doping Profile of the 25 nm “well tempered” MOSFET (D. Antoniadis) Effective channel length 25 nm Super-halo doping in the channel minimizes charge sharing effects Quantum confinement in the middle of the channel (z = 45 nm) With Vg=1V, Vds=0.1 V, 96.5% of current is carried by the 1st subband

Università degli Studi di Pisa Giuseppe Iannaccone Subband profile and characteristics 2D simulation First subband profile in the longitudinal direction for increasing Vds.

Università degli Studi di Pisa Giuseppe Iannaccone Papers on the subject Y. Naveh, A. N. Korotkov, K. K. Likharev, Shot noise suppression in multimode ballistic Fermi conductors, Phys. Rev. B, 60 (1998), R O. M. Bulashenko and J. M. Rubì, Shot-noise suppression by Fermi and Coulomb correlations in ballistic conductors, Phys. Rev. B, 65 (2001) O. M. Bulashenko and J. M. Rubì, Self-consistent theory of current and voltage noise in multimode ballistic conductors, Phys. Rev. B, 66 (2002), G. Gomila, I. R. Cantalapiedra, T. Gonzalez, L. Reggiani, Semiclassical Theory of shot noise in ballistic n+-i-n+ semiconductor structures: Relevance of Pauli and long-range Coulomb correlations, Phys. Rev. B, 66 (2002)

Università degli Studi di Pisa Giuseppe Iannaccone  Far-from equilibrium transport in each ballistic MOSFET, gives a local shot noise [van der Ziel,1986]: If the chain is long enough, local equilibrium holds in the whole sructure and then local shot noise reduces to conventional thermal noise:  Similarly to the aboveseen current macromodel, a noise macromodel for a device operating in intermediate tranposrt regime, is given by the series of a thermal noise generator with a shot noise generator.  We expect that when the ratio between the device length and the mean free path reduces, a more pronounced far-from equilibrium behavior emerges both in the static current and in the noise. Noise in the partially ballistic MOSFETs (II) (with G. Mugnaini)

Università degli Studi di Pisa Giuseppe Iannaccone Pauli and Coulomb interactions In most cases interactions make the collective motion more regular Limits density in real space Limits density in phase space Reduced fluctuations Sub-poissonian process

Università degli Studi di Pisa Giuseppe Iannaccone Fully ballistic transport regime Electrons with sufficient energy to overcome the barrier near the source reach the drain conserving energy and transversal momentum Electron states originating from the source obey the Fermi-Dirac statistics with source Fermi Energy Efs Electron states originating from the drain obey the Fermi- Dirac statistics with drain Fermi Energy Efd This ensures continuity of current density per unit energy in each subband

Università degli Studi di Pisa Giuseppe Iannaccone Model Poisson equation in 2D The electron density n(  ) in the quantum region is obtained from the solution of the Schrödinger equation with density functional theory p(  ), N D + (  ), N A - (  ) and of n(  ) out of the quantum region are given by the corresponding semiclassical expressions  Discretization with the box integration method  Newton-Raphson method with predictor-corrector iteration scheme

Università degli Studi di Pisa Giuseppe Iannaccone The Schrödinger equation must be solved twice: ? For the 2 minima along the vertical (x) direction ? For the other 4 minima ? The quantum electron density becomes x y z kxkx kzkz kyky kxkx kyky kzkz Mass anisotropy and electron density

Università degli Studi di Pisa Giuseppe Iannaccone Model out of equilibrium When the Poisson-Schrödinger equation is solved, and charge density and potential profiles are known, we compute the current density in the i-th subband The total current density is

Università degli Studi di Pisa Giuseppe Iannaccone Examples of nanotransistors INTEL, in production now oxide thickness 2 nm INTEL test device In production by 2005 (ITRS 2002 update oxide thickness 0.8 nm

Università degli Studi di Pisa Giuseppe Iannaccone Summary Motivation: ? VLSI devices are already nanoelectronic devices !! Mesoscopic Noise in MOSFETs ? Shot Noise of the drain current in ballistic MOSFETs ? Shot Noise of the gate current in fresh oxides and in the case of tunneling assisted by traps Conclusion Acknowlegments: F. Crupi, A. Nannipieri, G. Curatola, G. Fiori