Inclusion of Tunneling and Size-Quantization Effects in Semi-

Classical Simulators

Outline:

What is Computational Electronics?

Semi-Classical Transport Theory Drift-Diffusion Simulations Hydrodynamic Simulations Particle-Based Device Simulations

Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators Tunneling Effect: WKB Approximation and Transfer Matrix Approach Quantum-Mechanical Size Quantization Effect

Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment Methods

Particle-Based Device Simulations: Effective Potential Approach

Quantum Transport Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical

Basis of the Green’s Functions Approach (NEGF) NEGF: Recursive Green’s Function Technique and CBR Approach Atomistic Simulations – The Future

Prologue

Quantum Mechanical Effects

There are three important manifestations of quantum mechanical effects in nano-scale devices:

- Tunneling

- Size Quantization

- Quantum Interference Effects

Inclusion of Tunneling and Size-Quantization Effects

Tunneling Effect: WKB Approximation and Transfer Matrix Approach

Quantum-Mechanical Size Quantization EffectDrift-Diffusion and Hydrodynamics:

• Quantum Correction and • Quantum Moment Methods

Particle-Based Device Simulations: Effective Potential Approach

Tunneling Currents vs. Technology Nodes and Tunneling Mechanisms

10-16

10-14

10-12

10-10

10-8

10-6

10-4

0 50 100 150 200 250

Cu

rren

t (A

/m

)

Technology Generation (nm)

Ion

IG

Ioff

• For tox 40 Å, Fowler-Nordheim (FN) tunneling dominates• For tox < 40 Å, direct tunneling becomes important• Idir > IFN at a given Vox when direct tunneling active• For given electric field: - IFN independent of oxide thickness

- Idir depends on oxide thickness

B Vox > B

Vox = BVox < B

FN FN/Direct Direct

tox

B Vox > B

Vox = BVox < B

FN FN/Direct Direct

tox

This slide is courtesy of D. K. Schroder.

WKB Approximation to Tunneling Currents Calculation

0

EF

B

0

EF

B

a

No applied bias With applied bias

- eEx

x-axis

The difference between the Fermi level and the top of the barrier is denoted by B

According to WKB approximation, the tunneling coefficient through this triangular barrier equals to:

a

dxxT0

)(2exp where: eExm

x B 2*2

)(

WKB Approximation to Tunneling Currents Calculation

eE

mT B

3*24

exp2/3

Calculated and experimental tunnel current characteristics for ultra-thin oxide layers.

(M. Depas et al., Solid State Electronics, Vol. 38, No. 8, pp. 1465-1471, 1995)

The final expression for the Fowler-Nordheim tunneling coefficient is:

Important notes:

The above expression explains tunneling process only qualitatively because the additional attraction of the electron back to the plate is not included

Due to surface imperfections, the surface field changes and can make large difference in the results

Tunneling Current Calculation in Particle-Based Device Simulators

If the device has a Schottky gate then one must calculate both the thermionic emission and the tunneling current through the gate WKB fails to account for quantum-mechanical

reflections over the barrier Better approach to use in conjunction with

particle-based device simulations is the Transfer Matrix Approach

W. R. Frensley, “Heterostructure and Quantum Well Physics,” ch. 1 in Heterostructure and Quantum Devices, a volume of VLSI Electronics: Microstructure Science, N. G. Einspruch and W. R. Frensley, eds., (Academic Press, San Diego, 1994).

Transfer Matrix Approach

Within the Transfer Matrix approach one can assume to have either

Piece-wise constant potential barrierPiecewise-linear potential barrier

D. K. Ferry, Quantum Mechanics for Electrical Engineers, Prentice Hall, 2000.

Piece-Wise Constant Potential Barrier (PCPBT Tool) installed on the nanoHUB

www.nanoHUB.org

The Approach at a Glance

Slide property of Sozolenko.

The Approach, Continued

Slide property of Sozolenko.

Piece-Wise Linear Potential Barrier

This algorithm is implemented in ASU’s code for modeling Schottky junction transistors (SJT)

It approximates real barrier with piece-wise linear segments for which the solution of the 1D Schrodinger equation leads to Airy functions and modified Airy functions

Transfer matrix approach is used to calculate the energy-dependent transmission coefficient

Based on the value of the energy of the particle E, T(E) is looked up and a random number is generated. If r<T(E) than the tunneling process is allowed, otherwise it is rejected.

Tarik Khan, PhD Thesis, December 2004.

The Approach at a Glance

E

ai-1 ai ai+1

Vi

Vi+1

Vi-1 V(x)

1D Schrödinger equation:

Solution for piecewise linear potential:

The total transmission matrix:

T(E):

ExVdx

d

m

)(

2 2

22

)()( )2()1( iiiii BCAC

1 2 1........T FI N BIM M M M M M

12

011

1 N

T

kT

Km

' '1 1

0 0

' '1 1

0 0

' '1 1

' '1 1

1 1[ (0) (0)] [ (0) (0)]

2 2

1 1[ (0) (0)] [ (0) (0)]

2 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

i i i i

FI

i i i i

N i N N i N N i N N i NBI

n N i N N i N N i N N i N

r rA A B B

ik ikM

r rA A B B

ik ik

r B ik B r B ik BM

r r A ik A r A ik A

'

' ''1 1

( ) ( )( ) ( )

( ) ( )( ) ( )

i i i ii i i i ii

i i i i i i ii i i i i

A Br B BM

r r A r Br A A

Simulation Results for Gate Leakage in SJT

10-7

10-6

10-5

10-4

10-3

0.1 0.2 0.3 0.4 0.5 0.6 0.7

Drain current Gate CurrentTunneling Current

Cur

rent

[A/u

m]

Gate Voltage [V]

T. Khan, D. Vasileska and T. J. Thornton, “Quantum-mechanical tunneling phenomena in metal-semiconductor junctions”, NPMS 6-SIMD 4, November 30-December 5, 2003, Wailea Marriot Resort, Maui, Hawaii.

Quantum-Mechanical Size Quantization

Quantum-mechanical size quantization manifests itself as:

- Effective charge set-back from the

interface

- Band-gap increase

- Modification of the Density of States

function

D. Vasileska, D. K. Schroder and D.K. Ferry, “Scaled silicon MOSFET’s: Part II - Degradation of the total gate capacitance”, IEEE Trans. Electron Devices 44, 584-7 (1997).

Effective Charge Set-Back From The Interface

Schrodinger-Poisson SolversQuantum Correction ModelsQuantum Moment Models

Substrate

Gate

ox

oxox t

εC

polyC

invC deplC inv

ox

poly

ox

ox

deplinv

ox

poly

ox

oxtot

C

C

C

C1

C

CC

C

C

C1

CC

inv

ox

poly

ox

ox

deplinv

ox

poly

ox

oxtot

C

C

C

C1

C

CC

C

C

C1

CC

D. Vasileska, and D.K. Ferry, "The influence of poly-silicon gates on the threshold voltage, inversion layer and total gate capacitance in scaled Si-MOSFETs," Nanotechnology Vol. 10, pp.192-197 (1999).

Schrödinger-Poisson Solvers

At ASU we have developed: 1D Schrodinger – Poisson Solvers (inversion

layers and heterointerfaces) 2D Schrodinger – Poisson solvers (Si

nanowires) 3D Schrodinger – Poisson solvers (Si quantum

dots)

S. N. Miličić, F. Badrieh, D. Vasileska, A. Gunther, and S. M. Goodnick, "3D Modeling of Silicon Quantum Dots," Superlattices and Microstructures, Vol. 27, No. 5/6, pp. 377-382 (2000).

Space Quantization Literature

Bacarani and Worderman transconductance degradation (Proceedings of the IEDM, pp. 278-281, 1982)

Hartstein and Albert estimate of the inversion layer thickness (Phys. Rev. B, Vol. 38, pp.1235-1240, 1988)

van Dort et al. analytical model for Vth which accounts for QM effects (IEEE TED, Vol. 39, pp. 932-938, 1992)

Takagi and Toriumi physical origins of Cinv(IEEE TED, Vol. 42, pp. 2125-2130, 1995)

Vasileska, Schroder and Ferry influence of many-body effects on Cinv (IEEE TED, Vol. 44, pp. 584-587, 1997)

Hareland et al. modeling of the QM effects in the channel (IEEE TED, Vol. 43, pp. 90-96, 1996)

Krisch et al. poly-gate capacitance attenuation (IEEE EDL, Vol. 17, pp. 521-524, 1996)

1D Schrodinger-Poisson Solver for Si Inversion Layers – SCHRED

• 1D Poisson equation:

• 1D Schrödinger equation:

)()()()()(

1znzpzNzNe

zzz AD

EF

VG>0

z-axis [100](depth)

)()()()(

12

2zEzzV

zzmz ijijiji

2-band :m=ml=0.916m0, m||=mt=0.196m0

4-band:m=mt=0.196m0, m||= (ml mt)

1/2

2-band :m=ml=0.916m0, m||=mt=0.196m0

4-band:m=mt=0.196m0, m||= (ml mt)

1/22-band

4-band

• Electron density:

Tk

EETkmN

zNzn

B

ijFBi

ij

ijji

ij

exp1ln

)()(

2||

2

,

Simulation Results Obtained With SCHRED

0

5 x019

1x1020

1.5x1020

2x1020

0 5 10 15 20 25 30 35 40

n(z)

[cm

-3]

Distance from the SiO2/Si interface [Å]

QM

VG= 2.5 V

SC0

5

10

15

20

25

1011 1012 1013

QMSC

z av [

Å]

Ns [cm-2]

Cinv reduces Ctot by about 10%

Cpoly+ Cinv reduce Ctot by about 20%

With poly-depletion Ctot has pronoun-ced gate-voltage dependence

Cinv reduces Ctot by about 10%

Cpoly+ Cinv reduce Ctot by about 20%

With poly-depletion Ctot has pronoun-ced gate-voltage dependence

0.2

0.4

0.6

0.8

1.0

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

Cto

t [F

/cm

2 ]V

G [V]

Cox

SCNP QMNP

SCWPQMWP

invoxpolytot CCCC1111

The classical charge density peaks right at the SC/oxide interface.

The quantum-mechanically calculated charge density peaks at a finite distance from the SC/oxide interface, which leads to larger average displacement of electrons from that interface.

The classical charge density peaks right at the SC/oxide interface.

The quantum-mechanically calculated charge density peaks at a finite distance from the SC/oxide interface, which leads to larger average displacement of electrons from that interface.

Simulation Results Obtained With SCHRED

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 2 3 4 5 6 7 8 9 10

classical M-B, metal gates

classical F-D, metal gates

quantum, metal gates

quantum, poly-gates ND=6x1019 cm-3

quantum, poly-gates ND=1020 cm-3

quantum, poly-gates ND=2x1020 cm-3

Cto

t/Co

x

Oxide thickness tox

[nm]

T=300 K, NA=1018 cm-3

Degradation of the Total Gate Capacitance Ctot

for Different Device Technologies

Degradation of the Total Gate Capacitance Ctot

for Different Device Technologies

Simulation Results Obtained With SCHRED

0

50

100

150

200

250

300

1016 1017 1018

Vth(SCWP)

-Vth(SCNP)

Vth(QMNP)

-Vth(SCNP)

Vth(QMWP)

-Vth(SCNP)

V

th

[mV

]

NA [cm-3]

T=300 K

ND= 1020 cm-3

tox

= 4 nm

0

50

100

150

200

250

300

1016 1017 1018

Van Dort experimental data

Our simulation results

V

th

[mV

]N

A [cm-3]

T=300 KMetal gatestox

= 14 nm

(IEEE TED, Vol.39, pp. 932-938, 1992)

Vth shows strong substrate doping dependence when poly-gate depletion and QM effects in the channel are included

There is close agreement between the experimentally derived threshold voltage shift and our simulation results

MOS Capacitor with both Metal and Poly-Silicon Gates

Comparison With Experiments

0

10

20

30

40

50

60

0

10

20

30

40

50

60

0 5x1011

1x1012

1.5x1012

2x1012

2.5x1012

3x1012

Exp. data [Kneschaurek et al.]V

eff(z)=V

H(z)+V

im(z)+V

exc(z)

Veff

(z)=VH(z)

Veff

(z)=VH(z)+V

im(z)

Exp. data [Kneschaurek et al.]V

eff(z)=V

H(z)+V

im(z)+V

exc(z)

Veff

(z)=VH(z)

Veff

(z)=VH(z)+V

im(z)

Ene

rgy

E10

[meV

] T = 4.2 K, Ndepl

=1011 cm-2

Ns[cm-2]

Kneschaurek et al., Phys. Rev. B 14, 1610 (1976) Infrared Optical AbsorptionExperiment:

far-ir

radiation

LED

SiO2 Al-Gate

Si-Sample

Vg

Transmission-Line Arrangement

Infrared Optical AbsorptionExperiment:

far-ir

radiation

LED

SiO2 Al-Gate

Si-Sample

Vg

far-ir

radiation

LEDLED

SiO2 Al-Gate

Si-Sample

Vg

Transmission-Line Arrangement

D. Vasileska, PhD Thesis, 1995.

SCHRED Usage on the nanoHUB

SCHRED has 92 citations in Scientific Research Papers, 1481 users and 36916 jobs as of July 2009

3D Schrodinger-Poisson Solvers

3D Schrodinger – ARPACK3D Poisson: BiCGSTAB method

Aluminum

Chrome

PECVD SiO2

Thermal SiO2

p-type bulk silicon

Na = 1016 cm-3

400 nm

30 nm

93 nm20 nm

5 nm

Built-in gates

Aluminum

Chrome

PECVD SiO2

Thermal SiO2

p-type bulk silicon

Na = 1016 cm-3

400 nm

30 nm

93 nm20 nm

5 nm

Built-in gates

3D Schrodinger-Poisson Solvers

Left: The energy level spacing distribution as a function of s =E/(E)avg obtained by combining the results of a number of impurity configurations. Right: The 11th to 16th eigenstates of the silicon quantum dot.

0 1 3 42s

0.0

0.8

0.6

0.4

0.2

1.0

P(s

)

S. N. Milicic, D. Vasileska, R. Akis, A. Gunther, and S. M Goodnick, "Discrete impurity effects in silicon quantum dots," Proceedings of the 3rd International Conference on Modeling and Simulation of Microsystems, San Diego, California, March 27-29, 2000, pp. 520-523 (Computational Publications, 2000).

Quantum Correction Models- Hansc and Van Dort Approach -

These quantum-correction models try to incorporate the quantum-mechanical description of the carrier density in a MOSFET device structure via modification of certain device parameters:

HANSC model - modifies the effective DOS function

Van Dort model - modifies the intrinsic carrier density via modification of the energy band-gap. Within this model, the modification of the surface potential is:

2* /exp1 LAMBDAzNN CC

CONVQMn

CONVs

QMs zzΔzzEq ,/

Accounts for the band-gap widening effect because of the upward shift of the lowest allowed state

Accounts for the larger displacement of the carriers from the interface and extra bend-bending needed for given population:

94

zqEn

With these modifications, the energy band-gap becomes:

This results in modification of the intrinsic carrier density, which now, anywhere through the depth of the device, takes the form:

The function F(y) is introduced to describe smooth transition between classical and quantum description (pinch-off and inversion regions)

3/23/1

4,

913

E

TqkEE

B

SiCONVg

QMg

B.DORT (MODEL)

QM

iCONVii

BCONVg

QMg

CONVi

QMi

nyFyFnn

TkEEnn

)()(1

2/exp

refyyaaayF /,2exp1/exp2)( 22

N.DORT (MODEL)

The Van Dort model is activated by specifying N.DORT on the MODEL statement.

E0

E1

distance

Energy n(z)

z

z

CONVz

QMz

Classical density

Quantum-mechanicaldensity

Modification of the DOS Function

The modification of the DOS function affects the scattering rates and must be accounted for in the adiabatic approximation via solution of the 1D/2D Schrodinger equation in slices along the channel of the device

This is time consuming and for all practical purposes only charge set-back and modification of the band-gap are to a very good accuracy accounted for using either Bohm potential approach to continuum modeling Effective potential approach in conjunction with

particle-based device simulators

Quantum Corrected Approaches

Drift Diffusion Density Gradient

Hydrodynamic Quantum Hydrodynamic

Particle-based device simulations Effective Potential Approaches due to:

- Ferry, and - Ringhofer and Vasileska

Bohm Theory

The hydrodynamic formulation is initiated by substituting the wavefunc-tion into the time-dependent SWE:

The resultant real and imaginary parts give:

nRReiS , /

tiV

m

22

2

qc ffQVdt

dm

mR

RmtQ

tQtVSmt

tS

S/mtRtSmt

t

v

r

eq. Jacobi Hamilton rrr

v rr r

2/122/12

22

2

2

2

1

2),,(

);,,(),(2

1),()2(

;),(),(;01),(

)1(

Effective Potential Approach Due to Ferry

ieffi

ii

ii

ii

Vd

Vdd

dVd

ndrVV

)r()rr(r~

'rrexp)'r('r)rr(r~

)r'r('rr

exp'r)r(r~

)r()r(

2

2

2

2

In principle, the effective role of the potential can be rewritten in terms of the non-local density as (Ferry et al.1):

Classical densitySmoothed,effective potential

Built-in potentialfor triangular po-tential approxima-tion.

Effective potential approximation

Quantization energy

“Set back” of charge --quantum capacitance effects

Built-in potentialfor triangular po-tential approxima-tion.

Effective potential approximation

Quantization energy

“Set back” of charge --quantum capacitance effects

1 D. K. Ferry, Superlatt. Microstruc. 27, 59 (2000); VLSI Design, in press.

Parameter-Free Effective Potential

The basic concept of the thermodynamic approach to effective quantum potentials is that the resulting semiclassical transport picture should yield the correct thermalized equilibrium quantum state. Using quantum potentials, one generally replaces the quantum Liouville equation

for the density matrix (x,y) by the classical Liouville equation

for the classical density function f(x,k). Here, the relation between the density matrix and the density function f is given by the Weylquantization

, 0it H

12 *

0t x x kmf k f V f

( , ) [ ] ( / 2, / 2)exp( )f x k W x y x y ik y dy

The thermal equilibrium density matrix in the quantum mechanical setting is given by eq = e-βH, where =1/kBT is the inverse energy, and the exponential is understood as a matrix exponential.

In the semi-classical transport picture, the thermodynamic equilibrium density function feq is given by the Maxwelliandistribution function.

Consequently, to obtain the quantum mechanically correct equilibrium states in the semiclassical Liouville equation with the effective quantum potential VQ, we set:

22

2 *( , ) exp [ ] ( / 2, / 2)exp( )

k Q eq Heq mf x k V W e x y x y ik y dy

D. Vasileska and S. S. Ahmed, “Modeling of Narrow-Width SOI Devices”, Semicond. Sci. Technol., Vol. 19, pp. S131-S133 (2004).D. Vasileska and S. S. Ahmed, “Narrow-Width SOI Devices: The Role of Quantum Mechanical Size Quantization Effect and the Unintentional Doping on the Device Operation”, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 – 236.

Different forms of the effective quantum potential arise from different approaches to approximate the matrix exponential e-βH.

In the approach presented in this paper, we represent e-βH as the Green’s function of the semigroup generated by the exponential.

The logarithmic Bloch equation is now solved asymptotically, using the Born approximation, i.e. by iteratively inverting the highest order differential operator (the Laplacian).

This involves successive solution of a heat equation for which the Green’s function is well known, giving

2 2

23 2

1 2 *( , ) sinh exp ( )

2 * 8 *2

i x yQ

Q QB H

m kV x k V y e dyd

m mk

V V

The Barrier PotentialThe total potential is divided into Barrier and Hartree potential, where Barrier is a Heviside step function and Hartree is the solution to the Poisson equation.

The barrier field is then calculated using:

Note: It is evaluated only once at the beginning of the simulation!!!

1 1

1 122

11

1 1

2 *sinh2 *

( , ) (1,0,0) exp2 8 *

Q i xTx B

pm

B me V x p e d

m p

The Hartree Field

Hartree potential is expanded using the assumption that it is slowly varying function in space. In that case, one can write:

where:

Then, the Hartree Field is computed using:

22 22 2

2( , , ) 1 exp ( , ),

8 *24 *xxQ

HHp

V x p t V x tmm

220 ( , ) exp ( , )

8x

H HV x t V x tm

2 2 20 02

, 1( , , ) ( , ) ( , ), 1, ,

24 *r r

Q n n n n n nx x H j k j k r HH

j kV x p t V x t p p x x x V x t n N

m

C. Gardner, C. Ringhofer and D. Vasileska, I nt. J . High Speed Electronics andSystems, Vol. 13, 771 (2003).

Output Characteristics of DG DeviceDG SOI Device:

Tox = 1 nm Tsi = 3 nmLG = 9 nm LT = 17 nmLsd = 10 nm Nsd = 2 x 1020 cm-3

Nb = 0 g = 1 nm/decadeΦG = 4.188 VG = 0.4 V

1.E+10

1.E+13

1.E+16

1.E+19

1.E+22

Dop

ing

Den

sity

[cm

-3]

Source Drain Si Channel

Substrate

BOX

Back Gate

Front Gate

LG

LT

Tsi

Lsd

Tox = 1 nm Tsi = 3 nmLG = 9 nm LT = 17 nmLsd = 10 nm Nsd = 2 x 1020 cm-3

Nb = 0 g = 1 nm/decadeΦG = 4.188 VG = 0.4 V

1.E+10

1.E+13

1.E+16

1.E+19

1.E+22

Dop

ing

Den

sity

[cm

-3]

Source Drain Si Channel

Substrate

BOX

Back Gate

Front Gate

LG

LT

Tsi

Lsd

Source Drain Si Channel

Substrate

BOX

Back Gate

Front Gate

LG

LT

Tsi

Lsd

Source Drain Si Channel

Substrate

BOX

Back Gate

Front Gate

LG

LT

Tsi

Lsd

0

500

1000

1500

2000

0 0.2 0.4 0.6 0.8 1Drain Voltage [V]

Dra

in C

urre

nt [u

A/u

m]

0

15

30

45

% C

hang

e in

Cur

rent

W/o quant. (3nm)QM (3nm)NEGF (3nm)W/o quant. (1nm)QM (1nm)%Change

3 nm V G = 0.4 V

1 nm

Summary

Tunneling that utilizes transfer matrix approach can quite accurately be included in conjunction with particle-based device simulators

Quantum-mechanical size-quantization effect can be accounted in fluid models via quantum potential that is proportional to the second derivative of the log of the density

Effective potential approach has been proven to include size-quantization effects rather accurately in conjunction with particle-based device simulators

Neither the Bohm potential nor the effective potential can account for the modification of the density of states function, and, therefore, scattering rates modification because of the low-dimensionality of the system, and, therefore, mobility and drift velocity