Quantum Corrections for Monte Carlo Simulation

Quantum Corrections for Monte Carlo Simulation Brian Winstead and Umberto Ravaioli Beckman Institute University of Illinois at Urbana-Champaign

Outline Quantum corrections for quantization effects Effective potential Wigner-based Schrödinger-based Quantum corrections for tunneling Extending Schrödinger-based correction to device simulation

Monte Carlo Snapshot of a MOSFET 0 5 Channel Depth direction (nm) 10 15 20 25 30 Source Drain 35 20 30 40 50 60 70 Lateral direction (nm)

Motivation for Quantum Corrections Full-quantum transport is often impractical Goal is to extend the validity of semi-classical Monte Carlo to the 10-nm regime Quantum corrections can extend the validity of Monte Carlo in a practical way Mixed quantum/classical effects are treated in a unified fashion Little extra computational overhead is added in both 2D and 3D

Role of Quantum Corrections Monte Carlo particles correctly represent the motion of wave packets centroids in the crystal Monte Carlo does not account for interference effects due to rapidly varying applied fields or heterojunctions Quantum corrections can capture non-coherent interference effects Coherent transport effects are small for pure silicon devices above 10-nm regime

Corrections in Monte Carlo Quantum effects can be approximated in Monte Carlo by correcting the classical potential Tunneling Size Quantization U U + U c

Snapshot from MOS Capacitor 0.35 0.3 0.25 0.2 Energy (ev) 0.15 0.1 0.05 0 E 1 = 0 ev potential particles T ox = 3 nm V g = 1.0 V N A = 2 x 10 17 cm 3 0.05 0.1 0.15 0 5 10 15 Depth (nm)

Effective Potential Feynman developed the effective potential in the 1960s and applied it to quantum corrections in statistical mechanics Particles feel nearby potential due to quantum fluctuations around classical path of least action Veff is a non-local function of the nearby potential V eff ( x) = V ( x) e ( x x') 2a 2 dx', a = 2 2 12mkT

Properties of Effective Potential Simple to implement and to calculate Not sensitive to noise from Monte Carlo Works best for smooth, symmetric potentials a can be treated as a fitting parameter describing the size of the particle Detailed solution near large heterojunctions is typically incorrect and cannot be fit

Effective potential for MOS Effective size a can be tuned to match sheet charge density 10 13 N A = 2 x 10 17 cm 3 Sheet charge cm 2 10 12 N D = 2 x 10 17 cm 3 T ox = 30 Å Schrödinger Poisson Schrödinger corrected Wigner based corrected Effective potential corrected 0 0.5 1 1.5 2 2.5 3 Gate bias (V)

Effective potential for MOS Concentration (cm 3 ) 10 20 10 19 10 18 3.0 2.0 1.0 0.5 V = 0.25 V g Schrödinger Poisson Effective potential corrected MC T ox = 30 Å N A = 2 x 10 17 cm 3 10 17 10 16 0 1 2 3 4 5 6 Depth below interface (nm)

Wigner Formulation The Wigner method was developed in the 1930s for quantum correction to statistical mechanics Wigner formulation of quantum mechanics formally separates the quantum and classical contributions to the equation of motion f t w = H p f x w + H x f p w Classical Boltzmann equation Quantum Contribution 2 24 3 H 3 x 3 f p w 3 + 4 1920 5 H 5 x 5 f p w 5 +...

Quantum corrections for Monte Carlo simulation We start from the general Wigner function representation of quantum transport 1 fw r p = Ú dy e x+ y x- y 3 where jpy (, ) r ( 2, 2) is the density matrix. r ( xx, ') The Wigner function is the quantum equivalent of the distribution function in the semi-classical Boltzmann equation.

Quantum corrections for Monte Carlo simulation The quantum transport equation of the Wigner function has the form (parabolic bands, ballistic) 3 fw p Ê 1 ˆ + f (, ) (,, ) * w = dsdp K s P f 2 w r p+ P t t Á ÚÚ m Ë p È s s KsP (, ) = U Ê ˆ Ê ˆ r+ -U r- sin( s P) Í Ë 2 Ë 2 Î Direct solution of the Wigner transport equation is still a considerable numerical challenge. We are interested in determining a truncated expansion of the quantum equation, that resembles the standard Boltzmann equation, so that the standard Monte Carlo technique can be applied with minor modifications.

Quantum corrections for Monte Carlo simulation The complete Wigner transport equation, inclusive of collision terms, can be reformulated to resemble Boltzmann equation f t 1 + v rf - ru k f a + 1 ( 1) ( ) ( ru k f ) a a = 1 a + - 2a + 1 f + Â = Á 4 2 1! Ë t C Ê ˆ At first order, we truncate considering only α = 1.

Quantum corrections for Monte Carlo simulation The truncated equation has a form resembling Boltzmann equation f t 1 qc + v r f - F k f = Á Ë t C where F qc contains the quantum correction to the forces. With the modified forces, the particles move as if under the influence of a classical potential, but following equivalent quantum trajectories. The quantum correction essentially modifies the potential energy felt by the particles. Ê f ˆ

Quantum corrections for Monte Carlo simulation The corrected forces can be evaluated by making assumptions on distribution function and bandstructure. For f Ï 1 = exp Ì- È E k-k + U (r) - k E BT Î Ó Displaced Maxwellian Ê ( ) 3 U 3 ( ) 2 2 2 2 2 2 U x x x x 2 y y y y 2 qc 1 Ô Fx =- ÁU Íg k k g Íg k k g x Á - Ì - - + - - 24 ÔÍ x Í y Á Ë Ê ( ) 3 U 3 ( ) 2 2 2 2 2 2 U y y y y 2 x x x x 2 qc Á 1 Ô Fy =- U Íg k k g Íg k k g y Á - Ì - - + - - 24 ÔÍ y Í x Á Ë 2 with g x= ; g mkt y= 2 mkt y B x B f ( x, y, z) ( k - k ) 2 2 i i Ek-k = Â i= x,, y z 2mi k = k k k = momentum centroid Parabolic bands ÏÈ È ÓÎ Î ÏÈ È ÓÎ Î ˆ Ô Ô ˆ Ô Ô

Quantum corrections for Monte Carlo simulation The forces obtained have still some practical problems in regions like sharp interfaces, where quantum effects are prominent. To obtain a smooth potential, we can use approximate relations obtained by integrating the displaced Maxwellian distribution with the momentum. We obtain these alternative expressions for the second order derivatives of the potential ( ln n) U ( ln n) 2 2 2 2 U -k T ; -k T x x y y 2 B 2 2 B 2

Quantum corrections for Monte Carlo simulation The smooth version of the quantum corrected forces is 2 2 qc Ê kbt ÏÔ 2 2 ln n 2 2 ln n) ˆ Fx =- U - Èg x( kx -kx) - 3g x + 3 È g 2 y( ky -ky) -g Ô Á Ì y x 24 2 Ë x Í Ô Î Ó Î y Ô 2 2 qc Ê kbt ÏÔ 2 2 ln n 2 2 ln n Fy =- U - È g y( ky -ky) - 3g ˆ y + 3Èg ( ) 24 2 x kx -kx -g Ô Á Ì x y 2 Ë Í Ô y Î Ó Î x Ô This formulation has explicit momentum dependence and improves upon previous results in the literature where the momentum terms were evaluated with the thermal energy 2 ( ) 2 k - k k T = 2m 2 i i B i

Quantum corrections for Monte Carlo simulation Test of the quantum corrections in Monte Carlo: Single GaAs/AlGaAs/GaAs barrier with a fixed potential (1)

Quantum corrections for Monte Carlo simulation Single GaAs/AlGaAs/GaAs barrier with a fixed potential (2)

Quantum corrections for Monte Carlo simulation Single GaAs/AlGaAs/GaAs barrier with a fixed potential (3)

Quantum corrections for Monte Carlo simulation Single GaAs/AlGaAs/GaAs barrier with a fixed potential (4)

Wigner Correction Approximate quantum series Truncate to the first non-classical term Assume displaced maxwellian Parabolic bands Leads to a momentum-dependent correction Possible to avoid last two approximations by making use of Monte Carlo full band

Simplified Wigner Correction Averaging out the momentum-dependence V V 2 12m * 2 ln( n) Feynman showed this can also be derived as an approximation to the effective potential Further approximation leads to the density gradient method in drift-diffusion

Properties of Wigner Correction Requires long run times due to Monte Carlo noise in 2 ln(n) and restricts grid spacing Works well in drift-diffusion where noise is not an issue Unlike effective potential, 2 ln(n) is local For MOS, a single fitting parameter was found to adjust the correction at the oxide interface Requires no fitting in the silicon region

Wigner-corrected MOS Inversion Wigner correction is accurate across range of biases empirical nox = 1e15 cm -3 to fit interface Electron concentration (cm 3 ) 10 20 10 19 10 18 10 17 3.0 2.0 1.0 0.5 V g = 0.25 V Schroedinger Poisson Monte Carlo T ox = 3nm N A = 2 x 10 17 cm 3 0 2 4 6 8 10 Depth below interface (nm)

Wigner-corrected MOS Accumulation nox = 1e15 cm -3 Electron concentration (cm 3 ) 10 20 10 19 10 18 3.0 2.0 1.0 0.5 V = 0.25 V g Schroedinger Poisson Monte Carlo T ox = 3 nm N D = 2 x 10 17 cm 3 10 17 0 2 4 6 8 10 Depth below interface (nm)

Schrödinger-Based Correction Treats quantum effects in the direction perpendicular to transport Accurate No fitting parameters Not sensitive to noise in the Monte Carlo concentration estimator Efficient, additional computation time is small

Applying Schrödinger Correction Schrödinger equation is solved along 1D slices of the 2-D domain Self-consistent Monte Carlo potential is the input to Schrödinger and quantum density, nq, is output Concentration is linked to the correction with a Boltzmann dependence n q ( z) e { Vp( z) + Vqc( z) }/ ktt

Consistent with Non-equilibrium Transport Schrödinger energy levels/wavefunctions are filled on a Boltzmann distribution Within each slice, the correction forces the shape of the quantum density onto Monte Carlo No Fermi level is required Relative concentration between the slices is determined naturally by Monte Carlo transport

Degenerate Statistics in MOS Degeneracy can be important in highly inverted MOS structures 6 5 4 Fermi energy Subband 0 Subbands 1,2 Subband 3 Subband Energy (ev/kt) 3 2 1 0 1 2 T ox = 30 Å N A = 2 x 10 17 cm 3 3 4 0 0.5 1 1.5 2 2.5 3 Gate potential (V)

Correction Satisfies Degeneracy Vqc(z) is only a function of the relative nq(z) and the first three subbands have a similar shape 10 20 Quantum corrected MC Degenerate statistics Non degenerate statistics Concentration (cm 3 ) V g = 3.0 V T ox = 30 Å N A = 2 x 10 17 cm 3 10 19 0 0.5 1 1.5 2 Depth below interface (nm)

Quantum-corrected Simulation Flow Potential Energy levels Quantum density Quantum correction Move Particles in Corrected Potential 2 ε Vp = ρ d 1 dz m 2 * n q dk dt q dψ dz + ( z) ψ ( E, z) E V ψ p = e E / kt = r ( Vp + Vqc) 2 Eψ ( nq( z) ) Vp( z Vo ) kt t log + t

Schrödinger-corrected MOS Inversion Concentration (cm 3 ) 10 20 10 19 10 18 3.0 2.0 1.0 0.5 V = 0.25 V g Schrödinger Poisson Monte Carlo T ox = 30 Å N A = 2 x 10 17 cm 3 10 17 10 16 0 1 2 3 4 5 6 Depth below interface (nm)

Schrödinger-corrected MOS Accumulation Concentration (cm 3 ) 10 20 10 19 10 18 3.0 2.0 1.0 0.5 V = 0.25 V g Schrödinger Poisson Monte Carlo T ox = 30 Å N D = 2 x 10 17 cm 3 10 17 0 1 2 3 4 5 6 Depth below interface (nm)

Schrödinger-corrected Double-gate 10 20 V gs = 0.6 V T ox = 20 Å Schrödinger Poisson Monte Carlo N A = 1 x 10 15 cm 3 T si = 5 nm 10 nm 20 nm 40 nm Concentration (cm 3 ) 10 19 10 18 10 17 20 15 10 5 0 5 10 15 20 Depth direction (nm)

Schrödinger vs Other Corrections for MOS Typical behavior of different corrections for MOS 1 Potential (V) 0.8 0.6 0.4 0.2 Schrödinger Poisson potential Ideal correction Schrödinger correction Wigner correction, n = 1e15 cm 3 ox Effective potential, a = 5 Å V g = 1.0 V T ox = 30 Å N A = 2 x 10 17 cm 3 0 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 Depth below interface (nm)

Schrödinger vs Other Corrections for MOS 10 19 Concentration (cm 3 ) 10 18 V g = 1.0 V T ox = 30 Å N A = 2 x 10 17 cm 3 Schrödinger Poisson Schrödinger based correction Wigner correction, n = 1e15 cm 3 ox Effective potential, a = 5 Å 10 17 0 0.5 1 1.5 2 2.5 3 3.5 4 Depth below interface (nm)

Adding Tunneling to Monte Carlo Calculation of transfer matrix tunneling probability for each Monte Carlo particle is accurate Implementation of transfer matrix is cumbersome and can be inefficient Effective potential is simple and gives reasonable results for current through a small barrier Useful in current project to extend MOCA to SiGe or could be applied to source/channel tunneling

Effective Potential for Tunneling 10 7 10 6 10 5 T = 300 K a = 14 Å φ b = 0.1 ev Transfer Matrix Effective Potential J (A/cm 2 ) 10 4 10 3 10 2 φ b = 0.2 ev 10 1 10 0 φ b = 0.4 ev 10 3 10 4 10 5 10 6 Field (V/cm)

Extending Schrödinger Correction to Devices Heating occurs in the direction to transport Cannot make use of an electron temperature because it is not well-defined for non-equilibrium Define a transverse temperature Tt to describe the variation of the concentration with potential in the direction Tt is validated if a single temperature at each point along the transport path accurately describes the variation

Validating Transverse Temperature Tt(z) accurately describes potential concentration for highly non-equilibrium transport in 25nm FET 10 20 Source (drain is similar) Beginning of channel Concentration (cm 3 ) 10 19 Middle of channel End of channel Concentration Temperature derived concentration 10 18 0 0.5 1 1.5 2 Depth below interface (nm)

Typical Transverse Temperature Transverse temperature for a 25-nm MOSFET in saturation bias 10 4 Average Energy/kB Transverse Temperature Temperature (K) 10 3 Source Channel Drain 10 2 30 35 40 45 50 55 60 65 Distance along channel (nm)

Effect of Heating in the Direction 10 20 Cut at source/channel transition Cut at channel/drain transition Concentration (cm 3 ) 10 19 10 18 Transverse temperature No transverse temperature No quantum correction 0 0.5 1 1.5 2 2.5 3 Depth below interface (nm)

Surface Scattering Surface scattering model of Yamakawa, using roughness parameters obtained from experiment 10 3 Cooper and Nelson Takagi et. al. Monte Carlo Mobility (cm 2 /Vs) 10 2 10 4 10 5 10 6 Effective Normal Field (V/cm)

25 nm MOSFET concentration Depth direction (nm) 0 1 2 3 4 2e+20 8e+19 4e+19 2e+19 1e+19 5e+18 5e+17 L eff = 25 nm T ox = 15 Å V g = 1 V V d = 1 V 5 6 20 30 40 50 60 70 Lateral direction (nm) 5e+16

25 nm MOSFET corrected potential Depth direction (nm) 0 1 2 3 4 5 6 7 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 L eff = 25 nm T ox = 15 Å V g = 1 V V d = 1 V 8 9 0.7 0.8 0.9 10 20 30 40 50 60 70 Lateral direction (nm) 1

Properties along transport path 2.5 3 x 107 S MC MC only 0.7 0.58 Average velocity (cm/s) 2 1.5 1 0.47 0.35 0.23 Average energy (ev) 0.5 0.12 0 20 30 40 50 60 70 Distance along channel (nm) 0

Conclusions Quantum corrections can be used to extend the validity of Monte Carlo device simulation to the 10-nm regime Wigner-based corrections accurate, and momentum-dependent is interesting somewhat impractical due to noise in Monte Carlo Effective potential simple and fast accurate for small heterojunctions Schrödinger-based correction accurate and efficient with no fitting parameters best choice for size quantization effects