4. Carrier Transport

4. Carrier transport 2102385

• Why do we want to know about carrier transport in semiconductors?

• Transport mechanisms– Drift (under external electric field)– Diffusion (under concentration gradient)

– Drift-diffusion inter-relationship

1

4. Carrier Transport

Information (I, V)

Information transfer =carrier transport ( I )

Information read-out (I, V)

want large bandwidth must understand transport mechanisms

amout of infotime Mbps

external force

(battery, -field)

Contents

4.1 Drift

4.2 Diffusion

4.3 Current densities

4.4 Einstein relation

4.5 Continuity equations

v.2018.AUG

all about motion ofCB electrons & VB holes

4. Carrier transport 2102385 2

Transport: in a semiconductor

( ) ( ) ( )

x

t > 0, x = 0 t > 0, x > 0 t = 0

Drift Diffusion Current densities Einstein relation Continuity equations

Individual: yes

Overall: no

Individual: yes

Overall: yes

Carriers motion due to thermal energy (equilibrium condition) does not lead to net

displacement of “carriers”. No net charge transport.

External electric fields cause the carriers to drift apart. Net charge transport results.

4.1 Drift

Drift is the underlying mechanism of conduction in MOSFET (C7).

(ข)

(ค)


Transport: in a circuit (semiconductor + metal + battery)


electrons holes

generation recombination(sam

e as

C3

, sl

ide

#6

)

Metal-semiconductor junctions serve as

fast EHP generation-recombination center

All electronic circuits consist of wires (metals, electrons only) and chips

(semiconductors, electrons and holes). How can electrons/holes complete a circuit?

Under electric fields, both types of carriers drift.

Metal-semiconductor junctions serve as

low-resistance contacts


note 1: p = Momentum, hole concentrationnote 2: t = mean free Time (of equilibrium carriers),

recombination lifetime (of escess carriers)reminder: Force = rate of change of Momentum

4

Acceleration (by -field)

• for single e-: net force = - ex

• for n electrons/cm3:

Deceleration (by collisions)

• assume truly random collisions

• initial momentum lost every t (mean free time,

average time between scattering events)

x

field

x nedt

dp

tx

collisions

x p

dt

dp

0x

•

steady-state current flow• average momentum per e-:• average velocity per e-:

• Current density = # e- crossing unit area per unit time n<vx> × e

• Drude model of conduction (1900): Ohm’s law (1827)

• If carriers include both electrons (n) and holes (p):

00 xx

collisions

x

field

x nep

dt

dp

dt

dp

t

x

nn

x

x

xx

x

m

e

m

pv

en

pp

t

t

**

23*

2

cm

A

s

cm

cm

Coulomb

m

nevenJ x

n

xx

electron

electron unit

t

t

nem

ne

n

*

2

pn pene

xxv

xxJ

*

nm

et

drift velocity


4.1.1 Mobility (…) r high

rr

r

JA

I

L

V

A

LIIRV

;mobility

conductivity

current density


102

103

104

p

n

Ge, 300 K

102

103

104

Si, 300 KMo

bil

ity

(cm

2/V

-s)

n

p

1014

1015

1016

1017

1018

1019

102

103

104

GaAs, 300 K

Doping Concentration (cm-3)

n

p

5

(ND, NA) [ND+, Na

-] [ionized] impurity scattering t

(N)22

2*

* ;

dkEdm

m

e

t

)(

)(

Ap

Dn

N

N

(…)

Impurity scattering

The higher the doping conc, the lower the mobility

t = mean free timerecombination lifetime

minority carrier lifetime


carrier typescattering mechanisms

ionized impurity lattice carrier-carrier

majority

minority

1014

1015

1016

1017

1018

1019

1020

10

100

1,000

Si, 300 K

Mo

bil

ity

(cm

2/V

-s)

Doping Concentration (cm-3

)

n,

n,

p,

p,

(…)

)(

)(

Ap

Dn

N

N

)(

)(

Dp

An

N

N

(N,type): majority minority


• Matthieson’s rule: or

mechanism which causes lower mobility dominates

21

111

ttt

21

111

low temperatures limit

(ionized) impurity scattering

high temperatures limit

phonon/lattice scattering

As+

rc

KE > |PE|e

KE = 1/2mev

2

KE |PE|

KE < |PE|

Scattering of electrons by an ionized impurity

h+

e

Thermal vibrations of atoms can break bonds and thereby create electron-hole pairs.

2 10 100

105

106

E

lect

ron

Mo

bil

ity

n (

cm2/V

-s)

Temperature (K)

ND = 41013

cm-3

combined

piezoelectric

neutral

impurity

deformation potential

ionized

impurity polar

optical phonon

GaAs

(…) (T)

1t

2t

t = mean free time (s)1/t = scattering rate (Hz)


Example: Si

n(ND,T)

100 200 400 600 800 1000

102

103

104

E

lect

ron M

obil

ity

n (

cm2/V

-s)

Temperature (K)

Si

ND = 1019

cm-3

1016

1017

1018

1014

T3/2

T3/2

impurity

scattering

lattice

scattering

(…) (N,T)


log(n)

INTRINSIC

EXTRINSIC

IONIZATION

log( )

log( )

T-3/2

T3/2

Lattice

scattering

Impurity

scattering

1/TLow TemperatureHigh Temperature

T

Metal

Semiconductor

LO

GA

RIT

HM

ICS

CA

LE

Res

isti

vit

y

Temperature dependence of electrical conductivity for a doped (n-

type) semiconductor.9

note:

log() vs 1/T

log(r) vs T

&

)()()( TeTnT

ne

r

4.1.2 Resistivity & Conductivity


(…) r high

)/1(

)(

T

T

r

Reminder: = C3, #20

= C4, #7


1014

1015

1016

1017

1018

1019

1020

1021

10-4

10-3

10-2

10-1

100

101

102

p-GaAs

n-GaAs

p-Si

n-Si

300 K

Res

isti

vit

y (

-cm

)

Doping Concentration ND, N

A (cm

-3)

)(Nr

1% alloy, solid solubility limits


• (low-field) mobility – default• at high fields: Ohm’s law is not true

• at high fields, vd ~ vth (107 cm/s) carriers are hot added energy transferred to lattice, not to increase vdrift (scattering limits velocity)

• mobility is meaningless• (saturation) current density:

v

J

IRV

Tkvm Bth 2/2*• velocities– drift velocity (vdrift) = (see graph)

• at low fields• vs at high fields

– vth does not contribute to current (random motion)

– vdrift does not contribute to current (directed by )

driftthtotal vvv

velocity saturation (vs) | impact ionization (II)

snevJ

4.1.3 High (electric) field effects

102

103

104

105

106

105

106

107

108

Si (holes)

Si (electrons)

300 K

Dri

ft V

elo

city

(cm

/s)

Electric Field (V/cm)

GaAs (electrons)

GaAs (holes)


(…) r high

vs is the cause of saturation current Id-sat in nanoscale CMOS

! no

Si b

reakd

ow

n (II)

Ebr


A

A

C

b

c

d

x

D

B

EC

EV

A/B

c

velocity saturation (vs) | impact ionization (II)


(…) r high

II is the cause of breakdown of many devices (diodes, transistors), effective mostly while devices are under reverse bias

nMP

nPPnn

1

1)1( 2

cause: energetic carriers break bond (ionize), generating 2 additional carriers upon each collision (impact)

effect: carrier multiplication—by a factor M related to ionization probability P according to


t0

x = 0

x

t1

t2 t

y = 0

x

t0

t1

t2

t

y =

0

y

4.2 DiffusionDrift Diffusion Current densities Einstein relation Continuity equations

Diffusion is a natural process occuring in many branches of science and engineering.

Carrier diffusion results from non-uniform creation of excess carriers.

Diffusion is the underlying mechanism of conduction in diodes (C5) & BJTs (C6).


Diffusion process:

• Spatial variation (gradient) in n and p diffusion (net motion of carriers from regions of high to low concentration)

• equal prob. of moving in both directions• rate of e- flow in +x-direction per unit area (flux density)

n(x) p(x)

n(l)

n(-l) n(0)

x

n p

In

Ip

0

14

Important parameters

: flux (#/area-time)

l: mean free path

t: mean free time

D: diffusion coefficient

I: current

Units

: /cm2-s or (cm-2-s-1)

l: cm

t: s

D: cm2/s

I: A (or C/s)

)()(2

1

2

)(

2

)(0

lnlnlllnlln

xn

ttt

dx

xdnD

dx

xdnl

dx

xdnln

dx

xdnln

lx

n

n

)()(

)()0(

)()0(

2

1)(

2

t

t

How is I related to ?

“first-order approximation”(Taylor series)


• electrons flux:

• holes flux:

• electron current density:

• hole current density:

dx

xdnDx nn

)()(

dx

xdpDx pp

)()(

dx

xdnqD

dx

xdnDqdiffJ nnn

)()()(.)(

dx

xdpqD

dx

xdpDqdiffJ ppp

)()()(.)(

Diffusion current:

Diffusion flux: (#/cm2-s)

(A/cm2)

q = |electronic charge| = 1.6*10-19 C


• Drift and diffusion components of e- and h+:

dx

xdpqDxxpqxJ

dx

xdnqDxxnqxJ

ppp

nnn

)()()()(

)()()()(

Ex

n-Type Semiconductor

x

Hole Drift

Hole Diffusion

Electron Drift

Electron Diffusion

Semitransparent electrode

Light

drifts

diffusions

Total current:)()( xJxJ pn

Agree with the

+/- signs?

4.3 Current densitiesDrift Diffusion Current densities Einstein relation Continuity equations

In actual devices, current will be dominated by one of the four terms, i.e.drift

diffusion NFET, PFET npn, pnp


EC

EV

EF

x

E n(x)

x

E

0

x = 0

Ei

17

• Device: assumed optical excitation from the left (as in previous slide), or n-Si doped by surface diffusion to attain the profile

• Mechanism: diffusion causes -field which causes drift (oppose diffusion).

• Open circuit:set Jp(x) = 0

dx

dE

qdx

xdVx

dxdE

i

F

1)()(.2

0/.1

q

kTD

dx

xdp

xp

Dx

p

p

...

)(

)(

1)(

diffusiondrift

Conditions:

4.4 Einstein relationDrift Diffusion Current densities Einstein relation Continuity equations

Si, 300 K

1014

1015

1016

1017

1018

1019

1020

0

200

400

600

800

1,000

1,200

1,400

Doping Concentration (cm-3

)

(c

m2/V

-s)

0

5

10

15

20

25

30

35

D (cm

2/s)

(see p.162)

Equilibrium requirement:

(Streetman #3.5, #5.1.1)0/ dxdEF

• Open circuit:set Jp(x) = 0


4.5 Continuity equationsDrift Diffusion Current densities Einstein relation Continuity equations

A dx

G R J (x) J (x+dx) I

Changes in carrier concentrations due to

- generation G and recombination R were considered in C3

- drift Jdrift and diffusion Jdiff (combined J(x)) were considered in 4.1, 4.2 (4.3)

- the 4 mechanisms are now considered together, forming basic equation that can

determine changes in carrier concentration (in space x and time t) of all devices

pp

p

nnn

RGx

J

qt

p

RGx

J

qt

n

1

1

The rates of increase of electrons and holes

in the shaded volume:

… (detailed derivation pp.165-6)

p

ppp

n

nnn

pG

x

p

dx

pdD

t

p

nG

x

n

dx

ndD

t

n

t

t

2

2

2

2

Continuity equations for electrons & holes

… C3, #37,30

… C4, #16

a continuity equation: a partial differential equation which

gives a relation between the amount of the quantity and the

"transport" of that quantity. It states that the amount of the

conserved quantity at a point or within a volume can only

change by the amount of the quantity which flows in or out of

the volume. "Quantity" which can flow or move, such as mass,

energy, electric charge, momentum, number of molecules, etc.

(electrodynamics, hydro-, fluid dynamics...) (source: Wiki)


Solutions: 1. t | 2. x


4.5.1

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

n

(G

opt n

)

t/tn

0 2 4 6 8 10 12 14 16

t/tn

( ) ( )

0.1

0.9

t = t0

nt

nop eGtntt /

1)(A.

ntt

nop eGtntt /)( 0)(B.

A B

A & B (material parameters) can be used to predict rise & fall times (device/circuit parameters).

n

nnn

nG

x

n

dx

ndD

t

n

t

2

2

Case 1. uniform illumination of a semiconductor (no bias): on @ t = 0, off @ t = t0

Determine n(t), p(t)

@ A Gop

@ B 0“uniform”

(in space, x)


Solutions: 1. t | 2. x


4.5.2

The equation can be used to design layer thickness for optical absorption (solar cells, photocells...)

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

t/tn

p

/p

x/Lp

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0x = 0

t/tn

p

/p

( ) ( ) = 2

0.25 = 1

x

x = 0

22

2

22

2

ppp

nnn

L

p

D

p

dx

pd

L

n

D

n

dx

nd

t

t

pp

nn

LxLx

LxLx

eCeCxp

eCeCxn/

2

/

1

/

2

/

1

)(

)(

nnn DL t ppp DL t

pLxpexp

/)(

Diffusion equations

Diffusion length:

Lp = (for the whole) average distance the number of holes reduced to 1/e

Lp = (for a hole) average distance a hole diffuses before recombining

general solutions

Case 2.1 steady illumination of a semiconductor (no bias) at one surface

Case 2.2 steady injection of carriers in biased p-n junctions

Determine carrier distribution: n(x), p(x)

n

nnn

nG

x

n

dx

ndD

t

n

t

2

2

“steady”(in time, t)

no bias- complete absorption @ surface- no carriers generated inside bulk

specific solution

(C1 = 0)



(…) r high 1. t | 2. x

Conclusions

1. Drift of carriers under electric field

• Mobility is a function of doping concentration N and type (min or maj carrier), T

• Resistivity r and conductivity also functions of N, T

• At high fields, velocity saturates at vs and at very high fields material breaks down by impact ionization (II)

2. Diffusion of carriers under concentration gradient (due to doping or uneven excitation)

3. Current density

4. Einstein relation:

5. Continuity equations:

• t-dependent solution (rise/fall time – pulse excitation)

• x-dependent solution (diffusion length – uniform, steady state excitation)

he pene

driftJ

dx

xdpqD

dx

xdnqDqJ pndiff

)(,

)(

diffdrifttotal JJJ

q

kTD

tDL

4. Carrier Transport

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