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Nanostructures Research Group Arizona State University

Modeling Quantum and Coulomb Effects in

Nanoscale Devices

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Discrete Impurities

(Short-Range Interaction)

Efficient 3D Poisson

Equation Solvers

3D Monte Carlo

Transport Kernel

Effective Potential

(Space Quantization)

3DMCDS3DMCDS

Discrete Impurities

(Short-Range Interaction)

Efficient 3D Poisson

Equation Solvers

3D Monte Carlo

Transport Kernel

Effective Potential

(Space Quantization)

3DMCDS3DMCDS

Models Developed and Used

(1) Corrected Coulomb Approach (2) P^3 M Algorithm (3) Fast Multipole Method (FMM)

(1) Ferry’s Effective Potential Method (2) Quantum Field Approach

Statistical Enhancement: Event Biasing Scheme

Short-Range Interactions and Discrete/Unintentional Dopants

Quantum Mechanical Size-quantization Effects

Boltzmann Transport Equations (Particle-Based Monte Carlo Transport Kernel)

Long-range Interactions (3D Poisson Equation Solver)

In principle, the effective role of the potential can be written in terms of the non-local density as:

Ferry’s Effective Potential Approach

where the summation over i is a summation over the carriers themselves. The finite size of the electron has been replaced by smoothing of the real potential.

dxdydz a

z z a

y y a

x x V x y z a a a

V x y z x y z x y z

2 2

2 2

2 eff (^322)

( , , )exp ( 2 )

In three dimensions:

D.K. Ferry, “The onset of quantization in ultra-submicron semiconductor devices”, Superlattices and Microstructures, 27, 61, (2000).

Quantization energy

Charge set-back - quantum capacitance effect

Built-in potential for triangular po- tential approxima- tion.

Effective potential approximation

Derivation of Thermodynamic Quantum Potential Approach:

     

         

3

3

2

( ) , 2 , exp , 2 2

( ) , exp ,

quant eq eq

eq eq eq

eq eq

a f x p x x i p d

b x y F x y

c V m

  

 

     

    

 







 ^    

 ^  ^  

H 

   

2 , exp 2

class eq

p f x p e V x m

      3

2 exp , , 2 3 , exp 2 2 2

p (^) e V Q (^) x p eq x x i p d m

  ^ ^   ^ ^ ^ 

 ^ ^  ^  ^ ^  

   ^ 

Classical:

Quantum Mechanical:

Quantum Field Approach

C. Ringhofer, C. Gardner and D. Vasileska, “Effective potentials and quantum fluid models: A thermodynamic approach,” Inter. J. on High Speed Electronics and Systems 13, 771, 2003.

The Quantum Barrier Field:

     

   

   

2 2

2 2

2 sin , , exp 2 8

1,0, 2

2 sinh 2 , , 1,0,0 exp 2 8

  

  (^)   

 ^   

  ^  

 ^   

   ^   ^  

y

T B x Q x B B x T B x z

y y Q T^ y^ i^ y B y y y

e V x B y p m V x p m V x m p B e V

p m B m e V x p e d m p

The barrier potential VB ( x ) is a step function which takes into account the discontinuity at the Si/SiO 2 interface due to the difference in the semiconductor and the oxide affinities.

The fact that we have to calculate this integral only once is a reason why we should decide to tabulate the result in the very beginning.

Quantum Field Approach

The Hartree Field:

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

  2 2 2 2 2 ( , , , ) 1 24 2 exp 8 ( , ) Q x x H H

p V x p t V x t m m

where:

   

2 2 (^0) , exp , 8

x

VH x t  m  VH x t

(^2 2 ) 0 0 2 , 1

r r 24

Q n n n n n n x H x H j k j k r H j k

V x p t V x t p p x x x V x t n N m

 

Then, the Hartree Field is computed using:

S. Ahmed, C. Ringhofer, D. Vasileska, “Parameter–Free Effective Potential Method for Use in Particle–Based Device Simulations”, IEEE Transactions on Nanotechnology , in press, 2005.

Quantum Field Approach

Model Verification:MOS capacitor

Quantum Field Approach

Quantization energy

Charge set-back - quantum capacitance effect

Built-in potential for triangular po- tential approxima- tion.

Effective potential approximation

2.0E+

4.0E+

6.0E+

8.0E+

1.0E+

1.2E+

1.4E+

1.6E+

1.8E+

2.0E+

0.8 1 1.2 1.4 1.6 1.8 2 2. Gate Voltage [V]

Sheet Electron Density [cm

-2]

Classical SCHRED New Ferry

N (^) A = 10^18 cm- T (^) ox = 1.5 nm

Adjusting a 0 or the form/shape of the wave packet can improve the situation

The Corrected Coulomb Approach

Basic idea:

Short-range force is calculated directly from first principles by the implementation of a MD routine and then combined with the Long-range forces found from the mesh.

To avoid double counting:

The corrected Coulomb force is obtained by pre-computing the mesh force from a single ion and then subtracting it from the Coulomb force. This corrected force is then added to the actual mesh force obtained during the real simulation to get the total force.

ion

1

2

elec

1

2

2

N

k (^) i k

k ik

N

j (^) i j

Coul ij i

r r

q q a

r r

q a

F

mesh i

Coul i

corr

Fi  F  F

mesh i

corr i

total

Fi  F  F

(Precomputed)

D. Vasileska, W. J. Gross and D. K. Ferry, “Monte-Carlo particle- based simulations of deep-submicron n-MOSFETs with real- space treatment of electron-electron and electron-impurity interactions”, Superlatt. Microstruct ., Vol. 27, pp. 147-157 (2000).

Target electron (^) Fixed electron

The P^3 M Approach

ij

Coul ij

corr Fij  F  R

 Reference force replaces the mesh force and needed to avoid double counting of the short-range force. The reference force should be equal to the mesh force inside the SR domain and equal to the Coulomb force outside the short-range domain.

 Smoothing of the total interparticle force between the long-and short-range domains can be thought of as ascribing a finite size to particle i. A sphere with uniformly decreasing density profile, S ( r ) is a good choice for smoothing in three dimensions.

2

1 4

( ) r

qq Rij r i j  





 

   

  

 (^)    0 , ,

, / 2 2

48 (r )^4 otherwise

r r r

r S (^) r sr

sr sr

( 224 224 70 48 21 ) 35

1 4

( )  2  ^3  ^4  ^5  ^6 

 sr

i j ij (^) r

qq R r

(^1222489684022470487 ) 35

1 4

( ) 2 2    ^2  ^3  ^4  ^5  ^6 

 

 sr

i j ij (^) r

qq R r

0  r  rsr 2

rsr 2  r  r sr

r  r sr

r sr  ^2 r and

2

1 4

( ) r

qq Rij r i j  





 

   

  

 (^)    0 , ,

, / 2 2

48 (r )^4 otherwise

r r r

r S (^) r sr

sr sr

( 224 224 70 48 21 ) 35

1 4

( )  2  ^3  ^4  ^5  ^6 

 sr

i j ij (^) r

qq R r

(^1222489684022470487 ) 35

1 4

( ) 2 2    ^2  ^3  ^4  ^5  ^6 

 

 sr

i j ij (^) r

qq R r

0  r  rsr 2

rsr 2  r  r sr

r  r sr

r sr  ^2 r and



 

   

  

 (^)    0 , ,

, / 2 2

48 (r )^4 otherwise

r r r

r S (^) r sr

sr sr

( 224 224 70 48 21 ) 35

1 4

( )  2  ^3  ^4  ^5  ^6 

 sr

i j ij (^) r

qq R r

(^1222489684022470487 ) 35

1 4

( ) 2 2    ^2  ^3  ^4  ^5  ^6 

 

 sr

i j ij (^) r

qq R r

0  r  rsr 2

rsr 2  r  r sr

r  r sr

r sr  ^2 r and

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (New York, McGraw-Hill, 1981).

The FMM

Central Strategy of the FMM: How does the FMM speed up the computation?

In FMM one uses multiples moments to represent distant particle groups and introduces a local expansion to evaluate the contribution from distant particles in the form of a series. The multipole moment associated with a distant group can be translated into the coefficient of the local expansion associated with a local group. Interactions with particles which are nearby are handled directly.

distant particles local particles

distant particles local particles

multipole moment local expansion

local point

translation

1

1 1

1 1

1

N i i i N p i k i k i k p (^) N k i k i k i p k k k

q

x

x y

q y x

x q y

x A



 

 



L.Greengard and V. Rokhlin , “ A Fast Algorithm for Particle Simulations”, J. Comp. Phys. V. 135, 280-292 (1997).

The FMM

1. Defining the Kernel and forming Multipole Moments

Consider a two-dimensional (2D) physical model which consists of a set of N charged particles. Suppose that a point charge of unit strength is located at the point

 x 0^ ,^ y 0^  ^ x 0 ^2.^ Then, for any^ x ^  x y^ ,  ^2 with^ x  x 0 , the potential due to this charge

is described by the expressions

 x 0  x y ,    log || x  x 0 || ,

It is well known that  x 0 is harmonic in any region not containing the point x 0. Moreover,

for every harmonic function u, there exists an analytic function w :  such that

u x y  ,  Re  w x y  ,  and w is unique except for an additive constant. In the remainder

of this discussion we will work with analytic functions, to simplify notation, making no

distinction between a point  x y ,  ^2 and a point x  iy  z . We note that

 x 0  x   Re   log  z  z 0 .

The FMM

1. Defining the Kernel and forming Multipole Moments

LEMMA.1. If u x y  ,  Re  w x y  ,  describe the potential field at  x y ,  , then the

corresponding force field is given by

 u   u x , uy    Re  w ^  , Im  w ,

where w  is the derivation of w.

LEMMA 2. Let a point charge of intensity q be located at z 0. Then for any z such that

z  z 0 ,

0 ^ ^ ^0 ^ ^ ^0

1

log log.

k z k

z z q z z q z k z





 ^  

The FMM

LEMMA 3.

Suppose that

0 0 (^1 )

log k k k

a z a z z z z





is a multipole expansion of the potential due to a set of m charges of strengths

q 1 (^) , q 2 ,..., qm , all of which are located inside the circle D of radius R with center at z 0_._

Then for z outside the circle D 1 of radius  R  z 0  and center at the origin,

1

log ll , l

b z a z z





where

0 0 0 1

l l l k l k k

l (^) a z b a z k (^) l

 

  ^  

with

l k

the binomial coefficients.

2. M2M shifting

LEMMA 4. Suppose that m charges of strengths q 1 (^) , q 2 ,..., qm are located inside the circle

D 1 with radius R and center at z 0 , and that z 0   c  1  R with c  1. Then the

corresponding multipole expansion converges inside the circle D 2 of radius R centered

about the origin. Inside D 2 , the potential due to the charges is described by a power

series,

0

l l^ , l

 z b z





where,

0 ^ ^0 ^0 

(^1 )

1 log k^ k k k

a b a z z





and

0 1 0 0

p (^) k k l (^) l k l k

a l^ k a b for l z (^)  z k l z

  ^ ^   

The FMM

3+5. M2L translation and Evaluation