D Quantum Corrections

8/12/2019 D Quantum Corrections

1/41

Inclusion of Tunneling and Size-Quantization Effects in Semi-

Classical Simulators


2/41

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 QuantumMoment Methods

Particle-Based Device Simulations: Effective Potential Approach

Quantum Transport

Direct Solution of the Schrodinger Equation (Usuki Method) and TheoreticalBasis of the Greens Functions Approach (NEGF)

NEGF: Recursive Greens Function Technique and CBR Approach

Atomistic SimulationsThe Future

Prologue


3/41

Quantum Mechanical Effects

There are three important manifestations

of quantum mechanical effects in nano-

scale devices:

- Tunneling

- Size Quantization

- Quantum Interference Effects


4/41

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: EffectivePotential Approach


5/41

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

rrent(A/

m)

Technology Generation (nm)

Ion

IG

Iof f

For tox 40 , Fowler-Nordheim (FN) tunneling dominates For t

ox< 40 , 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=

B

Vox<

B

FN FN/Direct Direct

tox

B Vox>

B

Vox=

B

Vox<

B

FN FN/Direct Direct

tox

This slide is courtesy of D. K. Schroder.


6/41

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 isdenoted by B

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

a

dxxT 0 )(2exp where: eExm

x B 2*2

)(


7/41

WKB Approximation to Tunneling

Currents Calculation

eE

mT B

3

*24exp

2/3

Calculated and experimental tunnelcurrent characteristics for ultra-thin oxidelayers.

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

The final expression for theFowler-Nordheim tunnelingcoefficient is:

Important notes:

The above expressionexplains tunneling processonly qualitatively becausethe additional attraction ofthe electron back to the plateis not included

Due to surfaceimperfections, the surface

field changes and can makelarge difference in the results


8/41

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-mechanicalreflections over the barrier

Better approach to use in conjunction with

particle-based device simulations is the


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).


9/41


Within the Transfer Matrix approach one

can assume to have either

Piece-wise constant potential barrier

Piecewise-linear potential barrier

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


10/41

Piece-Wise Constant Potential Barrier

(PCPBT Tool) installed on the nanoHUB

www.nanoHUB.org


11/41

The Approach at a Glance

Slide property of Sozolenko.


12/41

The Approach, Continued

Slide property of Sozolenko.


13/41

Piece-Wise Linear Potential Barrier

This algorithm is implemented in ASUs code formodeling Schottky junction transistors (SJT)

It approximates real barrier with piece-wiselinear segments for which the solution of the 1D

Schrodinger equation leads to Airy functions andmodified Airy functions

Transfer matrix approach is used to calculate theenergy-dependent transmission coefficient

Based on the value of the energy of the particleE, T(E) is looked up and a random number isgenerated. If r


14/41

The Approach at a Glance

E

ai-1 aiai+1

Vi

Vi+1

Vi-1 V(x)

1D Schrdingerequation:

Solution for piecewiselinear potential:

The total transmissionmatrix:

T(E):

ExVdx

d

m

)(

2 2

22

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

1 2 1........T FI N BI M M M M M M

1

2011

1 NT

kT

K m

' '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 N

BI

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

r rA A B B

ik ik M

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 i

i

i i i i i i ii i i i i

A Br B B

Mr r A r Br A A


15/41

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 currentGate CurrentTunneling Current

Current[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.


16/41

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 Statesfunction

D. Vasileska, D. K. Schroder and D.K. Ferry, Scaled silicon MOSFETs: Part II - Degradationof the total gate capacitance, IEEE Trans. Electron Devices44, 584-7 (1997).


17/41

Effective Charge Set-Back From The

Interface

Schrodinger-Poisson Solvers

Quantum Correction Models

Quantum Moment Models

Substrate

Gate

ox

oxox

t

C

polyC

invC deplC inv

ox

poly

ox

ox

deplinv

ox

poly

ox

oxtot

C

C

C

C1

C

CC

C

C

C1

C

C

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,"NanotechnologyVol. 10, pp.192-197 (1999).


18/41

Schrdinger-Poisson Solvers

At ASU we have developed:

1D SchrodingerPoisson Solvers (inversion

layers and heterointerfaces)

2D SchrodingerPoisson solvers (Sinanowires)

3D SchrodingerPoisson solvers (Si quantum

dots)

S. N. Milii, 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).


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20/41


21/41

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 []

QM

VG= 2.5 V

SC0

5

10

15

20

25

1011 1012 1013

QMSC

zav

[]

Ns [cm-2]

Cinvreduces Ctotby about 10%

Cpoly+ Cinvreduce Ctotby about 20%

With poly-depletion Ctothas 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

Ctot

[F/cm

2]

VG [V]

Cox

SCNP QMNP

SCWPQMWP

invoxpolytot CCCC

1111

The classical charge density peaks rightat the SC/oxide interface.

The quantum-mechanically calculatedcharge density peaks at a finite distancefrom the SC/oxide interface, which leadsto larger average displacement of

electrons from that interface.


22/41


SCHRED

0.2

0.3

0.4

0.50.6

0.7

0.8

0.91

1 2 3 4 5 6 7 8 9 10

classical M-B, metal gates

classical F-D, metal gatesquantum, metal gates

quantum, poly-gates ND=6x1019cm-3

quantum, poly-gates ND=1020cm-3

quantum, poly-gates ND=2x1020cm-3

Ctot/

Cox

Oxide thickness tox

[nm]

T=300 K, NA

=1018cm-3

Degradation of the Total Gate Capacitance Ctot

for Different Device Technologies


23/41


SCHRED


24/41

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)

EnergyE10[meV] T = 4.2 K, Ndepl=10

11 cm-2

Ns [cm-2 ]

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

Experiment:

far-ir

radiation

LED

SiO2 Al-Gate

Si-Sample

Vg

Transmission-Line Arrangement

Infrared Optical Absorption

Experiment:

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.


25/41

SCHRED Usage on the nanoHUB

SCHRED has 92 citations in Scientific Research Papers,

1481 users and 36916 jobs as of July 2009


26/41

3D Schrodinger-Poisson Solvers

3D SchrodingerARPACK

3D 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


27/41

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 42

s

0.0

0.8

0.6

0.4

0.2

1.0

P(s) 11 12 13

14 15 16

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 ofMicrosystems, San Diego, California, March 27-29, 2000, pp. 520-523 (Computational Publications, 2000).


28/41

Quantum Correction Models

- Hansc and Van Dort Approach -

These quantum-correction models try to incorporate thequantum-mechanical description of the carrier density in aMOSFET device structure via modification of certain deviceparameters:

HANSC model - modifies the effective DOS function

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

2* /exp1 LAMBDAzNN CC

CONVQMn

CONVs

QMs zzzzEq ,/

Accounts for the band-gap wideningeffect because of the upward shift ofthe lowest allowed state

Accounts for the larger displacementof the carriers from the interface andextra bend-bending needed for givenpopulation:

94zqEn


29/41

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 transitionbetween classical and quantum description (pinch-off andinversion regions)

3/23/1

4

,

9

13

E

Tqk

EE

B

SiCONVg

QMg

B.DORT (MODEL)

QMi

CONVii

BCONV

gQM

gCONVi

QMi

nyFyFnn

TkEEnn

)()(1

2/exp

refyyaaayF /,2exp1/exp2)( 22

N.DORT (MODEL)


30/41

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

E0

E1

distance

Energy n(z)

z

z

CONVz

QMz

Classical density

Quantum-mechanicaldensity


31/41

Modification of the DOS Function

The modification of the DOS function affects thescattering rates and must be accounted for inthe adiabatic approximation via solution of the1D/2D Schrodinger equation in slices along the

channel of the device

This is time consuming and for all practicalpurposes only charge set-back and modificationof the band-gap are to a very good accuracyaccounted for using eitherBohm potential approach to continuum modeling

Effective potential approach in conjunction withparticle-based device simulators


32/41

Quantum Corrected Approaches

Drift DiffusionDensity Gradient

HydrodynamicQuantum Hydrodynamic

Particle-based device simulations

Effective Potential Approaches due to:

- Ferry, and

- Ringhofer and Vasileska


33/41

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 ,/ t

iVm

22

2

qc ffQVdt

dm

mR

RmtQ

tQtVSmt

tS

S/mtRtSmt

t

v

r

eq.JacobiHamiltonrrr

vrrr

2/122/12

22

2

2

2

1

2),,(

);,,(),(2

1),()2(

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

)1(


34/41

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 thenon-local density as (Ferry et al.1):

Classical densitySmoothed,

effective potential

Buil t-in potential

for tr iangular po-

tent ial approx ima-

t ion.

Effective potential

approximation

Quantization

energy

Set backof ch arge --

quantum capacitance

effects

Buil t-in potential

for tr iangular po-

tent ial approx ima-

t ion.

Effective potential

approximation

Quantization

energy

Set backof ch arge --

quantum capacitance

effects

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

Design, in press.


35/41

Parameter-Free Effective Potential

The basic concept of the thermodynamic approach to effectivequantum 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 betweenthe density matrix and the density function f is given by the Weyl

quantization

, 0it H

12 *

0t x x k mf k f V f

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


36/41

The thermal equilibrium density matrix in the quantum

mechanical setting is given by eq = e-H, 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 Maxwellian

distribution function.

Consequently, to obtain the quantum mechanically correctequilibrium states in the semiclassical Liouville equation with the

effective quantum potential VQ, we set:

22

2 *

( , ) exp [ ] ( / 2, / 2)exp( )k Q eq H

eq m

f 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. 2005Page(s):227236.


37/41

Different forms of the effective quantum potential arise from

different approaches to approximate the matrix exponential e-H.

In the approach presented in this paper, we represent e-Has theGreens 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 thehighest order differential operator (the Laplacian).

This involves successive solution of a heat equation for which

the Greens 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


38/41

The Barrier Potential

The 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 1

22 11

1 1

2 *sinh 2 *( , ) (1, 0, 0) exp2 8 *

Q i xTx B

p

mB me V x p e d m p


39/41

The Hartree Field


40/41

Output Characteristics of DG Device

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

20 cm-3

Nb = 0 g= 1 nm/decade

G = 4.188 VG = 0.4 V

1.E+10

1.E+13

1.E+16

1.E+19

1.E+22

DopingDensity[cm-3]

Source DrainSi Channel

Substrate

BOXBackGate

FrontGate

LG

LT

Tsi

Lsd

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

20 cm-3

Nb = 0 g= 1 nm/decade

G = 4.188 VG = 0.4 V

1.E+10

1.E+13

1.E+16

1.E+19

1.E+22

DopingDensity[cm-3]


Substrate

BOXBackGate

FrontGate

LG

LT

Tsi

Lsd


Substrate

BOXBackGate

FrontGate

LG

LT

Tsi

Lsd


Substrate

BOXBackGate

FrontGate

LG

LT

Tsi

Lsd

0

500

1000

1500

2000

0 0.2 0.4 0.6 0.8 1Drain Voltage [V]

DrainCurrent[uA/um]

0

15

30

45

%C

hangeinC

urrent

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

QM (1nm)%Change

3 nmVG= 0.4 V

1 nm


41/41

Summary

Tunneling that utilizes transfer matrix approach can quiteaccurately be included in conjunction with particle-baseddevice simulators

Quantum-mechanical size-quantization effect can beaccounted in fluid models via quantum potential that is

proportional to the second derivative of the log of thedensity

Effective potential approach has been proven to includesize-quantization effects rather accurately in conjunctionwith particle-based device simulators

Neither the Bohm potential nor the effective potential canaccount for the modification of the density of statesfunction, and, therefore, scattering rates modificationbecause of the low-dimensionality of the system, and,therefore, mobility and drift velocity

D Quantum Corrections

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