EECS 6.012 Spring 1998Lecture 3

B. Diffusion Current Density- Holes

Current density = (charge) x (# carriers per second per area):

If we assume the mean free path is much smaller than the dimensions of our device, then we can consider

=

dx

and usea Taylor expansion on

p(x -

) and

p(x +

):

, where

D

p

=

2

/

c

is the diffusion coefficent

Holes diffuse down the concentration gradient and carry positive charge.

Jpdiff

q

12-- p x ( ) A 1

2-- p x +( )A

Ac---------------------------------------------------------------=

J pdiff

qDp xddp=

I. Carrier Transport: Drift

A. Drift Velocity

Electrons drifting in an Electric Field move (on average) with a drift velocity which is proportional to the Electric Field, E.

x

x

xxf,3

xi

xi

xi

xf,2

xf,1

* xi = initial position * xf, n = final position of electron n after 7 collisions

x

x

xxi

xi

xi

xf,3

xf,2

xf,1

Electron # 1

Electron # 2

Electron # 3

(a) Thermal Equilibrium, E = 0

Electron # 1

(b) Electric Field E > 0

Electron # 2

Electron # 3

E

E

E

EECS 6.012 Spring 1998Lecture 3

Drift Velocity (Cont.)

mn

is an effective mass to take into account quantum mechanics

Lump it into a quantity we call mobility

n

(units:cm

2

/V-s)

vd

= -

nE

t

Velocityin Directionof E Field

v atqEm-------t= =

vaveqc2mn----------

E=

EECS 6.012 Spring 1998Lecture 3

B. Electron and Hole Mobility

mobilities vary with doping concentration-- plot is for 300K

typical values for bulk silicon -- 300 K

n

= 1000 cm

2

/(Vs)

p

= 400 cm

2

/(Vs)

1013 10151014 1019 10201016 1017 1018

1400

1200

1000

800

600

400

200

0

holes

Nd + Na total dopant concentration (cm3)

electrons

mob

ility

(cm

2 /V

s)

EECS 6.012 Spring 1998Lecture 3

C. Velocity Saturation

At electric fields greater >~ 10

4

V/cm

Drift velocities saturate --> max. out at around 10

7

V/cm.

Velocity saturation is very common in VLSI devices, due to sub-micron dimensions

E(V/cm)

10 102 103 104 105

103

104

105

106

107

108

vdn, vdp(cm/s)

holes

electrons

EECS 6.012 Spring 1998Lecture 3

D. Drift Current Density

Electrons drift against electric field

Electrons carry negative charge

J

n

dr

=

(-q) n v

d

units: Ccm

-2

s

-1

= Acm

-2

J

ndr

= (-

q

)

n (-

n

E

) =

q n

n

E

Holes drift with electric field

v

d

=

p

E

J

p

dr

=

(+q) p v

d

J

p dr

=

q p p E

Caution:: The linear relationship between field and drift velocity breaks down for high electric fields.

EECS 6.012 Spring 1998Lecture 3

II. Carrier Transport: Diffusion

A. Diffusion

Diffusion is a transport process driven by gradients in the concentration of particles in random motion undergoing frequent collisions.

Average carrier velocity = vth = 107 cm/s

Average interval between collisions = c = 10-13 s = 0.1 picoseconds

mean free path = = vth c = 10-6 cm = 10 nm

Since their motion is random, half of the carriers in each volume will pass through the plane before their next collision.

xr xr + xr

reference plane(area = A)

x

p(x)

p(xr )

p(xr + )volume A:holes movingin x direction crossreference plane withint = c.

volume A:holes moving in + x direction crossreference plane within t = c.

hole diffusion

Jpdiff (positive)

EECS 6.012 Spring 1998Lecture 3

B. Diffusion Current Density- Holes

Current density = (charge) x (# carriers per second per area):

:

If we assume the mean free path is much smaller than the dimensions of our device, then we can consider = dx and usea Taylor expansion on p(x - ) and p(x + ):

where Dp = 2 / c is the diffusion coefficent

Holes diffuse down the concentration gradient and carry positive charge.

Jpdiff

q

12-- p x ( )[ ] A 1

2-- p x +( )[ ] A

Ac----------------------------------------------------------------------------=

J pdiff

qDp xddp=

p(x)

x

Jpdiff ( > 0)

Jpdiff ( < 0)

EECS 6.012 Spring 1998Lecture 3

C. Diffusion Current Density - Electrons

Electrons diffuse down the concentration gradient, yet carry negative charge --> electron diffusion current density points in the direction of the gradient

D. Einstein Relation

Both mobility and the diffusion constant are related to the mean time between collisions c. There is a relation between these important quantities called the Einstein Relationship.

n(x)

Jndiff ( < 0)

Jndiff ( > 0)

x

Dpp-------- kT

q------= and

Dnn------- kT

q------=

EECS 6.012 Spring 1998Lecture 3

III. Total Current Density

Fortunately, we will be able to eliminate one or the other component in finding the internal currents in microelectronic devices.

Jn

Jndr

Jndiff+ qn

nE qD

n xddn+= =

Jp

Jpdr

Jpdiff+ qp

pE qD

p xddp= =