Analytical Study of Drift Velocity in N-type Silicon Nanowires

Amir Hossein Fallahpour, Mohammad Taghi Ahmadi, Razali Ismail Faculty of Electrical Engineering

Universiti Teknologi Malaysia 81310 Skudai, Johor

E-mail:amir_falah@yahoo.com

Abstract The limitations on carrier drift velocity due to high-field

effect and randomly velocity vector in equilibrium is reported. The results are based on asymmetrical distribution function that converts randomness velocity vectors in zero-field to streamlined one in a very high electric field. The ultimate drift velocity is found to be appropriate thermal velocity for a given dimensionality for non-degenerately doped Silicon nanowires. However, the ultimate drift velocity is the Fermi velocity for degenerately doped Silicon nanowires. Other important parameter in carrier transport phenomena, for nanoscale devices is quantum confinement effect that leads to one-dimensional behavior in silicon nanowire .

Key words: drift velocity, Silicon nanowire, degenerate limit

1. Introduction The quest for high-speed devices and circuits in favor of Ultra-Large-Scale-Integration (ULSI) is continuous. The speed is determined by the ease with which the carrier (electron or holes) can propagate through the length of the device. In the earlier designs, the mobility of the carrier was believed to be paramount importance. That was the push for Gallium Arsenide (GaAs) as the mobility of electrons in GaAs is 5-6 times higher than that of electron in silicon. However, as development of the devices to nanoscale dimensions continued, it became clear that the saturation velocity plays a predominant role. The reduction in conducting channel length of the device results reduced transit-time-delay and hence enhanced operational frequency. Until today, there is no clear consensus on the interdependence of saturation velocity on low-field mobility that is scattering-limited. There are a number of theories of high-field transport to answer this interdependence. Among them are Monte Carlo simulations, energy-balance theories, path integral methods, green function and many others. No clear consensus has emerged on the true nature of saturation velocity and its dependence on band structure parameters, doping profiles or ambient temperature. Often high mobility is attributed to higher saturation velocity. This outcome is not supported by experimental observations prompting our careful study of the process controlling the ultimate saturation velocity. In section 2, we present the theories framework for one-dimensional silicon nanowires. Section 3 concludes the important findings and reiterates the most crucial findings that the saturation velocity does not sensitively depend on the low-field mobility.

2. Theory In Silicon nanowire only one Cartesian direction is much larger than the De-Broglie wavelength. Therefore energy spectrum is analog-type in x and are digital in y, z-directions as given by:

2 2

*2x

chh

kE E

m

(2.1)

With Eigen function )(rk

propagated waves in all three

dimensions are:

z

Ly

Lezyx

zy

xkjk

x sinsin

2),,( ).(

(2.2)

Where zyxk

,, the wave vector components with momentum

space are P k

,2*

3

22

2*2

22

1 22 zycoc LmLm

EE

is the

unaltered conduction band edge, m is the carrier effective

mass assumed isotropic, zyx LLL is the volume of the

samples with zyxL

,, the length. The distribution function of

the energy Ek is given by:

1

1( )

1k F

B

k E E

k T

f E

e

(2.3)

where EF1 is the Fermi energy at which the probability of occupation is half and T is the ambient temperature. In non- degenerately doped semiconductors the ‘1’ in the denominator is negligible compared to exponential factor, the distribution is then Maxwellian. In the other extreme for strongly degenerate carriers, the probability of occupation is 1 where Ek < EF and it is zero if Ek > EF. Arora [1] modified the equilibrium distribution function of Equation(2.3) by replacing EF1 (the chemical potential) with the

electrochemical potential 1.

FE q

. Here

is the

applied electric field, q is the electronic charge and

the mean free path during which carriers are collision free (ballistic). Arora’s distribution function is thus given by:

978-1-4244-4952-1/09/$25.00 ©2009 IEEE 252 1st Int'l Symposium on Quality Electronic Design-Asia

1 .

1( )

1k F

B

k E E q

k T

f E

e

(2.4)

This distribution has simpler interpretation as given in the tilted band diagram of figure 1.

“Figure 1: Partial streamlined of electron motion on a tilted band diagram in an electric-field.”

The carriers arrive from left or right, a mean-free-path away from either side. It can be series that the Fermi level on

left

.3

qEF

and that on the right

.3

qEF

are the two quasi Fermi levels with EF . The current flow is

due to the gradient of Fermi energy 1( )

FE x is not

constant when electric field is applied. The ballistic motion in a mean-free path can be interrupted by the onset of a

quantum emission of energy 0 .This quantum may be an

optical phonon or any digital (continues) energy difference between the quantum level or photon if an electromagnetic field is present. The mean-free path with the emission of a

quantum of energy is related to 0 (zero-field mean free

path) by an expression [2]

]1[]1[ 0000

QQ

ee q

E

(2.5)

00 )1( NEQ (2.6)

1

100

TkBe

N (2.7)

Here )1( 0 N gives the probability of quantum emission

with N0 being the Bose-Einstein probability factor that determine the probability of emission. The inelastic scattering length during which a quantum is emitted is given by:

q

EQQ

(2.8)

Obviously Q in zero –electric field and will not

modify the traditional scattering describes by mean free

path 0 as 0 Q , However the presence of high electric

field make 0 Q .In that extreme.

0

q

EQQ (2.9)

This itself may be enough to explain mobility degradation

in a high electric field:

Qc

i i i

qq q

m v m v m v

(2.10)

Here c is the mean free time in which, the electron motion

is ballistic. iv Is the mean intrinsic velocity [3]

1

2

2

2

1

12

d d

id thd d

v v

(2.11)

m

Tkv B

th

2 (2.12)

dxe

x

j x

j

j

0)( 1)1(

1)( F (2.13)

Where,1

(3 )2

j D , )( j is a Gamma function whose value

is )1()( jj !, 22

3

, j, is integer .with

degraded by the onset of quantum emission ,the maximum velocity is evaluated :

QmQ

i

Ev

m v

(2.14)

This does not take into account the left–right asymmetry at the distribution function [4,5] . The distribution function of Equation(2.4) transfer random velocity vectors into the streamlined are in a very high electric field as represent in Figure 2.

The maximum average velocity per electron is now 1iv

and is given by iv as a function of temperature and doping

concentration. [6]

11 0 1*

1

2 CBi F

hh

Nk Tv

m p

(2.15)

Where 1CN is carrier density

3

2

1 2

*2

2B

C

m k TN

(2.16)

“Figure 2: random velocity and the streamlined in a very high electric field.”

“Figure 3:Velocity VS temperature for silicon.” Figure 3 demonstrates for small number of carrier concentration, velocity changes with temperature and for high concentration has been observed that velocity only depends on doping concentration. The ultimate drift velocity is found to be appropriate thermal velocity for a given dimensionality for non- degenerately doped Silicon nanowires. [7]

1 1 *

210.564 B

i th th thn

k Tv v v v

m (2.17)

In figure 4 low concentration velocity curve follows degenerate approximation, however in high concentration, velocity curves of different temperatures are closer together. Subsequently, the ultimate drift velocity is the Fermi velocity for degenerately doped Silicon nanowires.

1 1( )4 *iv Deg n

m

(2.18)

“Figure 4: velocity verses doping concentration.”

3. Conclusion Using this model, we have investigated Silicon nanowire device parameters (1D). The distribution function transfers random velocity vectors into the streamlined in a very high electric field. The ultimate drift velocity is found to be appropriate thermal velocity for a given dimensionality for non- degenerately doped Silicon nanowire. However, the ultimate drift velocity is the Fermi velocity for degenerately doped Silicon nanowire.

4. References [1] V. K. Arora, Japanese Journal of Applied Physics, 24

537(1985). [2] Vijay K. Arora, Applied Physics Letters, 80, 3763(2002). [3] V. K. Arora and Michel L.P.Tan,Ismail Saad Razali

Ismail Applied Physics Letters(2007) [4] Vijay K. Arora, proceedings of IEEE international

conference on Microelectronics,Belgrade,Serbia and Montenegro,14-17 May 2006(IEEE,NewYork)pp.17-24

[5] A.M.T.Fairus V.K.Arora Microelectron.J.32,679(2001) [6] Nanoscale Transistors Device Physics, Modeling and

Simulation. Jing Guo.Mark S. Lundestorm Spriger (2006)

[7] Vijay K. Arora.Jpn.J, Applied Physics Letters, part 1 24,537(1985)