Jean Baptiste PerrinNobel Prize in physics 1926

He demonstrated that the currentin a vacuum tube was due to

electron motion.

Collisions of particles affects the mobility term.

+

cphole mobility pp

e

m *

cnelectron mobility nn

e

m *

n nv E

p pv E

-

Effective mass!

equation of motion for a hole in an electric fieldp

p

dvF m * eE

dt p

p

etv E

m *

Lattice structure will also vibrate due to T > 0

“lattice scattering”

lattice mobility 32

lattice1

T

Electron & ion density and conductivity versus temperature -- silicon

Typical mobility values

Silicon 1350 480

Gallium arsenide 8500 400

Germanium 3900 1900

2

ncm

V - s

2

pcm

V - s

Conductivity

S mp ne p n

p n iSe n m

densities may be different

drift p nj e p n E driftj E

Restrictions in design

IA L

material specifies the maximum current density Jmaterial has a certain conductivity

you specify the length L

I JA EA VA

L

VR

I

L

A p n

L

e p n A

you specify the current and the voltage I V

n de N

1 110

0.1

I

A L

Exercise 4.3 For a particular silicon semiconductor device at room temperature, the required material is to be n type with a resistivity of = 0.1 -cm. (a) determine the required impurity doping concentration and (b) the resulting electron mobility.

From figure 4-416 3

dN 9 10 cm

n

2

19 16d

10 cm695 V seN 1.6 10 9 10

Impurity concentration

resi

stiv

ity

Semiconductors are nonlinear. Velocity depends on electric field.

driftv E

e 2ph

2 2

1m *

E1

k

2

ph2

E2 0

k

2phE k phE

kphE

2kk

*em 0

2ph2

E2 0

k

2phE k phE

kphE

2kk

*em 0

Semiconductors are nonlinear. GaAs Velocity depends on electric field.

e 2ph

2 2

1m *

E1

k

em * 0

em * 0

Diffusion current is due to density gradient of density

flux th th1 1n h v n h v

2 2

h 0 h

th1

n h n h v2

th

1 dn 0 dn 0n 0 h n 0 h v

2 dx dx

th

dn 0hv

dx

flux D n

19q 1.602 10 C

current q flux

Diffusion current is due to density gradient of density

nn

J eDx

Exercise 4.5 Assume that, in an n-type GaAs semiconductor at T = 300°K the electron concentration varies linearly. Calculate the magnitude of the diffusion current density.

1810

177 10 cm0.10

cm /s 2nD 225

18 1719

n10 7 10

J 1.6 10 2250.10

Typical values

Silicon 1350 35 480 12.4

Gallium arsenide 8500 200 400 10.4

Germanium 3900 101 1900 49.2

n pnD pD

& 2 2cm cmDV s s

Diffusion current is due to density gradient of density – nonuniform doping

xL

0n( x ) n e

nelectron flux Ddn( x )

dx nD

L

xL

0n e

nDelectron current density

L

xL

0( e ) n e

Eenergy F FiFn

E E

e

Ev

Ec

EF

EFi

-

Diffusion of electrons

Fne 1Fn Fid dE

dx e dx

electric field

Diffusion of electrons1Fn Fid dE

dx e dx

Maxwell Boltzmann distribution

0 exp F Fii d

B

E En n N

T

ln dF Fi B

i

NE E T

n

Diffusion of electrons1Fn Fid dE

dx e dx

ln dF Fi B

i

NE E T

n

for where

xL15 4n

d nN x 10 e x 0 L 10 cm.

Exercise 4.6 Assume that the donor impurity concentration In a semiconductor is given by

Determine the electric field induced in the material in this impurity concentration.

dB

d

dN xT 1E

e N x dx

x

L15 nxL15 nn

0.0259 110 e

L10 e

4

0.0259 V259 cm10

Total current density– 4 components

n p n pen ep eD n eD p J E E

The electron density changes differently than the ion density

As the electrons diffuse, this sets up an electric field.

n( x )

p( x )

dE p( x ) n( x )

dx

Einstein relationsn p n pen ep eD n eD p J E E

V VFTBe

0n n e

n n0 en eD n J E

V E

V VFV VF TB

eTB0e

0 n n

d n edV0 e n e eD

dx dx

Einstein relationsV VF

V VF TBeTB

0e0 n n

d n edV0 e n e eD

dx dx

V V V VF FT TB Be e

n nB

dV e dV0 e D e

dx T dx

nB

n

DT

e

p

p

D

Due to T > 0, “lattice scattering”This effects mobility & diffusion!

National Public Radio puzzlecombinations of 2 & 7 & j yields 8

1)7 2 1 8

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2

)7 855 2 .

2 )7 82

2

3 )2 (7 ) 83

7 )4(4 )2 8

6 )7 2 6 8

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8 )7 log( 2 ) 88

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