quantum corrections for monte carlo simulationmcc.illinois.edu/summerschool/2002/umberto...

Quantum Corrections forMonte Carlo Simulation

Brian Winstead and Umberto RavaioliBeckman Institute

University of Illinois at Urbana-Champaign

2002 School on Computational Material Science May 21-31, 2002

Outline

• Quantum corrections for quantization effects– Effective potential– Wigner-based– Schrödinger-based

• Quantum corrections for tunneling• Extending Schrödinger-based correction to device

simulation


20 30 40 50 60 70

0

5

10

15

20

25

30

35

Lateral direction (nm)

Dep

th d

irect

ion

(nm

)

Source

Channel

Drain

Monte Carlo Snapshot of a MOSFET


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 themotion of wave packets centroids in the crystal

• Monte Carlo does not account for interferenceeffects due to rapidly varying applied fields orheterojunctions

• Quantum corrections can capture non-coherentinterference effects

• Coherent transport effects are small for puresilicon devices above 10-nm regime


Corrections in Monte Carlo

Quantum effects can be approximated in MonteCarlo by correcting the classical potential

Tunneling

SizeQuantization

UU + Uc


Snapshot from MOS Capacitor

0 5 10 15−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Depth (nm)

Ene

rgy

(eV

) potentialparticles

E1 = 0 eV

Tox

= 3 nmV

g = 1.0 V

NA = 2 x 1017 cm−3


Effective Potential

• Feynman developed the effective potential in the1960s and applied it to quantum corrections instatistical mechanics

• Particles feel nearby potential due to quantumfluctuations around classical path of least action

• Veff is a non-local function of the nearby potential

mkTadxexVxV a

xx

eff12

,')()(2

2)'( 2

2 ==−

−

∫


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


0 0.5 1 1.5 2 2.5 3

1012

1013

She

et c

harg

e cm

−2

Gate bias (V)

NA = 2 x 1017 cm−3

ND

= 2 x 1017 cm−3

Tox

= 30 Å

Schrödinger−PoissonSchrödinger−correctedWigner−based correctedEffective potential corrected

Effective potential for MOS

• Effective size a can be tuned to match sheetcharge density


0 1 2 3 4 5 610

16

1017

1018

1019

1020

Con

cent

ratio

n (c

m−

3 )

Depth below interface (nm)

Tox

= 30 Å N

A = 2 x 1017 cm−3

Vg = 0.25 V

0.5

1.0

2.0

3.0

Schrödinger−PoissonEffective potential corrected MC

Effective potential for MOS


...192024 5

5

5

54

3

3

3

32

+∂∂

∂∂

+∂∂

∂∂

−

∂∂

∂∂

+∂∂

∂∂

−=∂∂

pf

xH

pf

xH

pf

xH

xf

pH

tf

ww

www

Wigner Formulation

• The Wigner method was developed in the 1930sfor quantum correction to statistical mechanics

• Wigner formulation of quantum mechanicsformally separates the quantum and classicalcontributions to the equation of motion

QuantumContribution

ClassicalBoltzmann equation


• We start from the general Wigner function representation ofquantum transport

where

is the density matrix.

• The Wigner function is the quantum equivalent of thedistribution function in the semi-classical Boltzmannequation.

Quantum corrections for Monte Carlo simulation

( ) ( )31, 2, 2j p y

wf r p dy e x y x yr◊= + -Ú

( ), 'x xr


• The quantum transport equation of the Wigner function hasthe form (parabolic bands, ballistic)

• Direct solution of the Wigner transport equation is still aconsiderable numerical challenge.

• We are interested in determining a truncated expansion ofthe quantum equation, that resembles the standardBoltzmann equation, so that the standard Monte Carlotechnique can be applied with minor modifications.


3

* 21

( , ) ( , , )

( , ) sin( )2 2

ww w

f pf d s dP K s P f r p P t

t m

s sK s P r U r s PU

p

∂+ ◊ — = +

∂

= + - - ◊

Ê ˆÁ ˜Ë ¯

È Ê ˆ Ê ˆ ˘Í ˙Ë ¯ Ë ¯Î ˚

ÚÚ


• The complete Wigner transport equation, inclusive of collisionterms, can be reformulated to resemble Boltzmann equation

• At first order, we truncate considering only α = 1.


( )( )

( )

r r k

12 1

r k1

1v

1

4 2 1 ! C

ff U f

t

fU f

t

aa

aa a

+•+

=

∂+ ◊ — - — ◊ —

∂

- ∂+ — ◊ — =

∂+

Ê ˆÁ ˜Ë ¯Â


• The truncated equation has a form resembling Boltzmannequation

where Fqc contains the quantum correction to the forces.

• With the modified forces, the particles move as if under theinfluence of a classical potential, but following equivalentquantum trajectories.

• The quantum correction essentially modifies the potential energyfelt by the particles.


r k1v qc

C

f ff F ft t

∂ ∂Ê ˆ+ ◊— - ◊— = Á ˜Ë ¯∂ ∂


• The corrected forces can be evaluated by making assumptions ondistribution function and bandstructure. For


( ) ( )

( ) ( )

2 22 22 22 2

2 22 22 22 2

2

1 3 324

1 3 324

with ;

qcx x x x y y y yx

qcy y y y x x x xy

x yx B

U UF U k k k kx x y

U UF U k k k ky y x

m k T

g g g g

g g g g

g g

Ê ˆÏ ¸È ˘ È ˘Ô ÔÁ ˜Í ˙ Í ˙Ì ˝Á ˜Í ˙ Í ˙Ô ÔÁ ˜Î ˚ Î ˚Ó ˛Ë ¯

Ê ˆÏ ¸È ˘ È ˘Ô ÔÁ ˜Í ˙ Í ˙Ì ˝Á ˜Í ˙ Í ˙Ô ÔÁ ˜Î ˚ Î ˚Ó ˛Ë ¯

∂ ∂ ∂=- - - - + - -∂ ∂ ∂

∂ ∂ ∂=- - - - + - -∂ ∂ ∂

= =2

y Bm k T

k k1exp (r) fB

f E U Ek T -

Ï ¸È ˘= - + -Ì ˝Î ˚Ó ˛

( )

( )

22

k k, ,

momentum centroid

2, ,

i i

ii x y z

x y z

k kE

mk k k k

-=

-=

= =

Â

Displaced Maxwellian Parabolic bands


• The forces obtained have still some practical problems in regionslike sharp interfaces, where quantum effects are prominent.

• To obtain a smooth potential, we can use approximate relationsobtained by integrating the displaced Maxwellian distribution withthe momentum.

• We obtain these alternative expressions for the second orderderivatives of the potential


( ) ( )2 22 2

2 2 2 2ln ln;B B

n nU Uk T k Tx x y y

∂ ∂∂ ∂- -∂ ∂ ∂ ∂


• The “smooth” version of the quantum corrected forces is

• This formulation has explicit momentum dependence andimproves upon previous results in the literature where themomentum terms were evaluated with the thermal energy


( ) ( )

( ) ( )

2 2222 22 2

2 22 22 22 2

ln ln )3 3

24

ln ln3 3

24

qc Bx x x x x y y y y

qc By y y y y x x x x

k T n nF U k k k k

x x y

k T n nF U k k k k

y y x

g g g g

g g g g

∂ ∂ ∂= - - - - + - -

∂ ∂ ∂

∂ ∂ ∂= - - - - + - -

∂ ∂ ∂

Ê ˆÏ ¸Ô ÔÈ ˘È ˘Ì ˝Á ˜Í ˙Î ˚ Î ˚Ë ¯Ô ÔÓ ˛Ê ˆÏ ¸Ô ÔÈ ˘ È ˘Ì ˝Á ˜Í ˙ Î ˚Î ˚Ë ¯Ô ÔÓ ˛

( )22

2 2i i B

i

k k k Tm-

=


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



• Single GaAs/AlGaAs/GaAs barrier with a fixed potential (2)



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 bymaking use of Monte Carlo full band


• Averaging out the momentum-dependence

• Feynman showed this can also be derived as anapproximation to the effective potential

• Further approximation leads to the densitygradient method in drift-diffusion

Simplified Wigner Correction

)ln(12

2*

2

nm

VV ∇−→


Properties of Wigner Correction

• Requires long run times due to Monte Carlonoise in ∇2ln(n) and restricts grid spacing

• Works well in drift-diffusion where noise isnot an issue

• Unlike effective potential, ∇2ln(n) is local

• For MOS, a single fitting parameter was foundto adjust the correction at the oxide interface

• Requires no fitting in the silicon region


0 2 4 6 8 10

1017

1018

1019

1020

Ele

ctro

n co

ncen

trat

ion

(cm

−3 )


Schroedinger−PoissonMonte Carlo

Tox

= 3nmN

A = 2 x 1017 cm −3

Vg = 0.25 V

0.5

2.0

1.0

3.0

Wigner-corrected MOS Inversion

Wigner correction is accurate across range ofbiases—empirical nox = 1e15 cm-3 to fit interface


Wigner-corrected MOS Accumulation

nox = 1e15 cm-3

0 2 4 6 8 1010

17

1018

1019

1020

Ele

ctro

n co

ncen

trat

ion

(cm

−3 )


Schroedinger−PoissonMonte Carlo

Tox

= 3 nmN

D = 2 x 1017 cm−3

Vg = 0.25 V

0.5

1.0

2.0

3.0


Schrödinger-Based Correction

• Treats quantum effects in the directionperpendicular 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 ofthe 2-D domain

• Self-consistent Monte Carlo potential is the inputto Schrödinger and quantum density, nq, is output

• Concentration is linked to the correction with aBoltzmann dependence

{ } tqcp kTzVzVq ezn /)()()( +−∝


Consistent with Non-equilibrium Transport

• Schrödinger energy levels/wavefunctions are filledon a Boltzmann distribution

• Within each slice, the correction forces the shapeof 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 MOSDegeneracy can be important in highly invertedMOS structures

0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

3

4

5

6

Sub

band

Ene

rgy

(eV

/kT

)

Gate potential (V)

Tox

= 30 Å N

A = 2 x 1017 cm−3

Fermi energySubband 0Subbands 1,2Subband 3


0 0.5 1 1.5 2

1019

1020

Vg = 3.0 V

Tox

= 30 Å N

A = 2 x 1017 cm−3

Con

cent

ratio

n (c

m−

3 )


Quantum−corrected MCDegenerate statisticsNon−degenerate statistics

Correction Satisfies DegeneracyVqc(z) is only a function of the relative nq(z) andthe first three subbands have a similar shape


Quantum-corrected Simulation Flow

Potential

Energy levels

Quantum density

Quantum correction

Move Particles inCorrected Potential

ρε −=∇⋅∇ pV

)( qcpr VVqdtkd

+∇=

ψψψ EVdzd

mdzd

p =+

− *

2 12

∑ −∝E

kTEq

tezEzn /2),()( ψ

( ) opqt VzVznkT +−− )()(log


Schrödinger-corrected MOS Inversion

0 1 2 3 4 5 610

16

1017

1018

1019

1020

Con

cent

ratio

n (c

m−

3 )


Tox

= 30 Å N

A = 2 x 1017 cm−3

Vg = 0.25 V

0.5

1.0

2.0

3.0

Schrödinger−PoissonMonte Carlo


Schrödinger-corrected MOS Accumulation

0 1 2 3 4 5 610

17

1018

1019

1020

Con

cent

ratio

n (c

m−

3 )


Tox

= 30 Å N

D = 2 x 1017 cm−3

Vg = 0.25 V

0.5

1.0

2.0

3.0



Schrödinger-corrected Double-gate

−20 −15 −10 −5 0 5 10 15 2010

17

1018

1019

1020

Con

cent

ratio

n (c

m−

3 )

Depth direction (nm)

Vgs

= 0.6 V T

ox = 20 Å

NA = 1 x 1015 cm−3

Tsi

= 5 nm 10 nm 20 nm 40 nm



Schrödinger vs Other Corrections for MOS

• Typical behavior of different corrections for MOS

0 0.5 1 1.5 2 2.5 3 3.5 4−0.2

0

0.2

0.4

0.6

0.8

1


Pot

entia

l (V

) Vg = 1.0 V

Tox

= 30 Å N

A = 2 x 1017 cm−3

Schrödinger−Poisson potential Ideal correctionSchrödinger correctionWigner correction Effective potential, a = 5 Å

, nox

= 1e15 cm−3


Schrödinger vs Other Corrections for MOS

0 0.5 1 1.5 2 2.5 3 3.5 410

17

1018

1019


Con

cent

ratio

n (c

m−

3 )

Vg = 1.0 V

Tox

= 30 Å N

A = 2 x 1017 cm−3

Schrödinger−Poisson Schrödinger−based correction Wigner correction Effective potential, a = 5 Å

, nox

= 1e15 cm−3


Adding Tunneling to Monte Carlo

• Calculation of transfer matrix tunneling probabilityfor each Monte Carlo particle is accurate

• Implementation of transfer matrix is cumbersomeand can be inefficient

• Effective potential is simple and gives reasonableresults for current through a small barrier

• Useful in current project to extend MOCA to SiGeor could be applied to source/channel tunneling


Effective Potential for Tunneling

103

104

105

106

100

101

102

103

104

105

106

107

J (A

/cm

2 )

Field (V/cm)

T = 300 Ka = 14 Å

φb = 0.1 eV

φb = 0.2 eV

φb = 0.4 eV

Transfer MatrixEffective Potential


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 inthe ⊥ direction

• Tt is validated if a single temperature at each pointalong the transport path accurately describes thevariation


0 0.5 1 1.5 210

18

1019

1020

Con

cent

ratio

n (c

m−

3 )


Source (drain is similar)

Beginning of channel

Middle of channel

End of channel

ConcentrationTemperature derived concentration

Validating Transverse Temperature

Tt(z) accurately describes potential→concentrationfor highly non-equilibrium transport in 25nm FET


Typical Transverse Temperature

• Transverse temperature for a 25-nm MOSFET insaturation bias

30 35 40 45 50 55 60 6510

2

103

104

Distance along channel (nm)

Tem

pera

ture

(K

)

Source Channel Drain

Average Energy/kBTransverse Temperature


Effect of Heating in the ⊥ Direction

0 0.5 1 1.5 2 2.5 3

1018

1019

1020

Con

cent

ratio

n (c

m−

3 )


Cut at source/channel transition

Cut at channel/drain transition

Transverse temperatureNo transverse temperatureNo quantum correction


104

105

106

102

103

Mob

ility

(cm

2 /Vs)

Effective Normal Field (V/cm)

Cooper and NelsonTakagi et. al.Monte Carlo

Surface Scattering

Surface scattering model of Yamakawa, usingroughness parameters obtained from experiment


25 nm MOSFET – concentration

5e+16

5e+17

5e+18

1e+19

2e+19

4e+19

8e+19

2e+20

20 30 40 50 60 70

0

1

2

3

4

5

6


Dep

th d

irect

ion

(nm

)

Leff = 25 nmTox = 15 ÅVg = 1 VVd = 1 V


25 nm MOSFET – corrected potential

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

20 30 40 50 60 70

0

1

2

3

4

5

6

7

8

9

10


Dep

th d

irect

ion

(nm

)

Leff = 25 nmTox = 15 ÅVg = 1 VVd = 1 V


Properties along transport path

20 30 40 50 60 700

0.5

1

1.5

2

2.5

3x 10

7

0

0.12

0.23

0.35

0.47

0.58

0.7A

verage energy (eV)

Ave

rage

vel

ocity

(cm

/s)

Distance along channel (nm)

S−MCMC only


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

quantum corrections for monte carlo simulationmcc.illinois.edu/summerschool/2002/umberto...

Documents