1

Ballisticity of the Linear response Transport in Nanometric Silicon Devices

C. Jungemann

Institute for Electronics

Bundeswehr UniversityNeubiberg, Germany

EIT4

2

Outline

• Introduction

• Theory

• Results for 40nm N+NN+ structure– High bias– Zero bias

• Conclusions

3

Introduction

4

Introduction

1D 40nm N+NN+ structure

5 1019cm-3 2 1017cm-3 5 1019cm-3

0 10 50 60[nm]

• Macroscopic models (DD, HD) fail for strong nonequilibrium due to

Ballistic transport!

• Macroscopic models also fail near equilibrium in nanometric devices!

Why?

5

Theory

6

Theory

• Boltzmann and Poisson equations

• Deterministic solver based on an nth-order Spherical harmonics expansion

• Newton-Raphson method to solve the nonlinear system of equations

• Modena electron model (analytical band structure)

7

Theory

3s

r 3k

Boltzmann equation (steady state, neglecting the valley index, spherical bands):

eE(r) f (r, k) v(k) f (r, k) W(r,k | k ')f (r, k ') W(r, k ' | k)f (r, k)d k '

2

Projection of the distributio

3l,m l,m3

l

l,

0,0

m l,ml 0 m l

0

1,1

n function onto spherical harmonics:

2g (r, ) (k) Y (k), (k) f (r, k)d k

2

g (r, )Y ,

Electron density:

n(r) 4 g (r, )d

Current density

g

:

g4

j(r) v( )3

(r, , , )

1

0

, 1

1,0

(r, )

g (r, ) d

g (r, )

8

Results

1D 40nm silicon N+NN+ structure

Transport is in x-direction

9

ResultsBiased at 0.5VBiased at 0.5V

Velocity overshoot is a sign of (quasi-)ballistic transportVelocity overshoot is a sign of (quasi-)ballistic transport

10

ResultsBiased at 0.5VBiased at 0.5V Quasi-ballisticQuasi-ballisticScattering dominatedScattering dominated

11

Results

Distribution function at 0.5VDistribution function at 0.5V

2

l 0

1 l

l,m l,mm l

g(r, , , ) g (r, )Y ,

12

Results

Distribution function at 0.5VDistribution function at 0.5V

2

l 1

1 l

l,m l,mm l

g(r, , , ) g (r, )Y ,

13

Results

Quasi-ballisticQuasi-ballisticScattering dominatedScattering dominated Linear responseLinear responsewithout zero orderwithout zero order

14

Results

21 l

l,ml,m

m ll 1

g (r, )g(r, , , )Y ,

V V

Differential distribution function at equilibriumDifferential distribution function at equilibrium

15

Conclusions

16

Conclusions

• Ballistic transport occurs in nanometric devices at high bias

• The linear response of the distribution function shows ballistic peaks at zero bias in regions with large built-in fields

• The ballistic peaks of the linear response can be negative

• Linear response in nanometric devices with large built-in fields is fundamentally different from the bulk case