ECEE 302: Electronic Devices

28 October 2002

ECEE 302 ElectronicDevices

Drexel UniversityECE Department

BMF-Lecture 4-102802-Page -1Copyright © 2002 Barry Fell

ECEE 302: Electronic Devices

Lecture 4. Effect of Excess Carriers in Semi-Conductors

28 October 2002

28 October 2002




Outline• Optical Absorption

• Luminescence– Photo-Luminenscence

– Cathodoluminescence

– Electroluminescence

• Carrier Lifetime and Photoconductivity– Direct Re-Combination of electrons and holes

– Indirect Combination; Trapping

– Steady State Carrier Generation: Quasi-Fermi Levels

– Photoconductive Devices

• Diffusion of Carriers– Diffusion Process

– Diffusion and Drift of Carriers, (built in fields)

– Continuity Equation (Diffusion and Recomination)

– Steady State Carrier Injection and Diffusion Length

– Haynes-Shockley Experiment

– Gradients in the Quasi-Fermi levels

28 October 2002




Optical Absorption

• Optical Absorption Process Text, Figure 4-1

• Absorption Experiment Text, Figure 4-2 & 4-3

• Band Gaps of common semi-conductors Text, Figure 4-4

)(

lightincident the of wavelength of function a is

material the oft coefficien absorption the called is

I(0)eI(d)

iprelationsh the by

given is solid the into d distance a radiation of intensity the foreThere

I(0)eI(x)

is equation this to solution The

I(x)dx

dI(x)-

pointthat at intensity remaining the to

alproportion decreases I(x), intensity,that assumes model absorption Optical

d-

x-

28 October 2002




Luminescence

• Luminescence refers to light emission from solids

• Types of Luminescence– Photoluminescence Text, Figures 4-5 & 4-6

• Direct excitation and recombination of an EHP

• Trapping

• Color is determined by impurities that create different energy levels within the solid

– Florescence• fast luminescence process

– Phosphorescence (phosphors)• slow luminescence process

• mulitple trapping process

– Electroluminescence• mechanism for LEDs

• electric current causes injection of minority carriers to regions where they combine with majority carriers to produce light

28 October 2002




Example: Absorption (Example 4-1) (1 of 2)Problem: GaAs with t=.46m. Illumination=monochromatic light

=h=2eV, a=5x104 cm-1. Pincident=10mW

(a) Find the total energy absorbed by the sample per sec (J/s)

(b) Find the rate of excess thermal energy given up to the electrons in the lattice prior to recombination (J/s)

(c) Find the number of photons per second given off from recombination events (assume 100% quantum efficiency)

s/J109mw9mw1mW10Power Absorbed

Watts1010.W10We10

eW1010eILI (a)

:Solution

3

323.22

m/cm10m1046.0cm1053L0

2614

28 October 2002




Example: Absorption (Example 4-1) (2 of 2)

cphotons/se 108.2

eV/photon 2eV/J106.1

s/J109 secper emitted photons ofnumber

have we absorbed, photon eachfor dtransmitte is photon one If (c)

s/J1057.2s/J1090.285 isheat to up given energy total the Thus

0.285eV 2

eV 1.43 -eV 2 is photonper up given

energy percentage The photon.per eV 1.43 is gap energy the of

ntransistio to due up given energy The eV. 2 is energy photon heT (b)

s/J109mw9mw1mW10Power Absorbed

Watts1010.W10We10

eW1010eILI (a)

:Solution

16

19

3

33

3

323.22

m/cm10m1046.0cm1053L0

2614

28 October 2002




Carrier Lifetime and Photoconductivity

• Excess electrons and holes increase conductivity of semi-conductors

• When excess carriers are produced from optical luminescence, the resulting increase in conductivity is called photoconductivity

• This is the primary mechanism in the operation of solar cells

• Mechanisms– Direct Recombination Text, Figure 4- 7– Indirect Recombination, Trapping Text, Figure 4- 8– Impurity Energy Levels Text, Figure 4- 9– Photo-conductive decay Text, Figure 4-10

• Steady State Carrier Generation; Quasi-Fermi Levels Text, Fig 4-11

• Photo-conductive Devices

28 October 2002




Direct Recombination of Electrons and Holes (1 of 2)

• Direct Recombination of an electron and hole occurs spontaneously

(t)nn(t)pn-

(t)nn(t)pn-n-n

(t)nn(t)pn-pn-n

p(t)pn(t)n-ndt

tnd

dt

tnnd

dt

tdn

Then

equal?) these are (Why ionsconcentratcarrier excess"" the be p(t)n(t)

let and ionconcentratcarrier of values mequilibriu the be p and nLet

rate" ionrecombinat the minus rate genertion thermal

to equal is electrons conduction in change of ratenet " the represents which

tptnndt

tdn

equation the by drepresente be can electrons conduction of changeNet

200r

200r

2ir

2ir

200r00r

2ir

00r2ir

0

00

r2ir

28 October 2002




Direct Recombination of Electrons and Holes (2 of 2)

pn

1

low is carriers of level injection the if general, In

p

1 lifetime" ionrecombinat" the where

nenen(t)

solution the has which

n(t)p-dt

tnd

have we Then

np then type,-p e.g. extrinsic, is material the If 2)

neglected be can and

n(t)pn(t)n

then small is n(t), ion,concentratcarrier excess the 1)

:sassumption following the making by

(t)nn(t)pn-dt

tnd

equation the to solution a down write can We

00r0

0r0

/t-tp-

0r

00

002

200r

00r

28 October 2002




Steady State Carrier Generation

g pn present not is trapping if and

nnpng have we

n neglecting and ),T(gpn ,but

nnpnpnnpgg(T)

have we and ,pn Trapping, no with state steady At

ppnnnpgg(T)

then sample, the ontolight shine we Suppose .generation

band-to-band as well as centersdefect to due be can carriers of generation The

etemperatur absolute is T where , pnnTg

rate theat pairs hole-electron generatesthat with crystal aConsider

noptical

n00roptical

200r

200r00rroptical

00rroptical

00r2ir

28 October 2002




Example 4-2 (Textbook, Page 121)

3-130

3-63-14

2310

0

2i

0

3-13313noptical

pn3-14

0

313

cm102 p is state steady new The

cm1025.2cm 10

cm/carriers105.1

n

n p

is ionconcentratcarrier (p) minority mequilibriu initial The (b)

cm102sec2secEHP/cm 10gpn

by given is

ionconcentratcarrier hole)(or electron excess state steady the (a) :Solution

state? steady new the to K)300(T

mequilibriu from ionconcentratcarrier minority in change the is What (b)

ion?concentrat hole)(or electron excess state steady the is What )a(

sec2 and cm 10n with Si of sample a in

dmicrosecon each optically created are EHP/cm 10 Assume:Problem

28 October 2002




Quasi-Fermi Levels

• Fermi Level is valid only when there are no excess carriers present

• We define the “quasi-Fermi” level for electrons (Fn) and holes (Fp) to describe steady state carrier concentrations

)/kTF-(E

i

)/kTE-(Fi

pn

pi

in

epp and

enn

by given are holes and

electrons of ionsconcentratcarrier excess determine to sexpression The

.F and F

by denoted are holes and electronsfor levels miquasi"-Fer" The

Excess Electrons Excess Holes

ECONDUCTION

EVALANCE

EFERMI

Fn

Fp

28 October 2002




Example, Text page 122

0.186eVF-E

Then

?)Why(cm101.5p and ,cm102p holes,For

.233eV

99.80.0259eV

108ln0.0259eV

cm101.5

cm101.2ln0.0259eV

n

nlnkTEF

equation the from Fn determine can We

K300Tat eV 0.0259kT and K,300Tat Sifor cm101.5n where

enn

have we level miquasi"-Fer" the of terms In

cm101.2cm102cm10nnn

was ionconcentrat electron state steady the example previous the In

pF

3-10i

3-13

3

3-10

3-14

iFn

3-10i

kT

EF

i

3-143-133-140

Fn

28 October 2002




Optical Sensitivity of a Photo conductor

• Photo-conductors are conductors that change their conductivity when illuminated by light

• Applications are electric eyes, exposure meters for photography, solar cells, etc

• Sensitivity to specific light color (frequency) is determined by the energy gap

mobility) (high high and time)

ionrecombinat (mean high requires response ctivephotocondu Maximum

. Trapping is there If . ionrecombinat simpleFor

qg

by given is ctivityphotocondu in change resulting The

gp and gn Then

and by

given be band respective its incarrier eachfor times mean theLet

ctorphotocondu afor g rate generation optical theonsider C

pnpn

ppnnoptical

opticalpopticaln

pn

optical

28 October 2002




Diffusion of Carriers

• Diffusion Process Text, Figure 4-12 & 4-13– motion of carriers from high density to low density states

• Diffusion and Drift - Built in Fields Text, Figure 4-14 & 15• Continuity Equation (Diffusion and Recombination) Text, Fig 4-16• Steady State Carrier Injection (Diffusion Length) Text, Fig 4-17• Haynes-Shockley Experiment Text, Figure 4-18 & 4-19• Gradients in the Quasi-Fermi Levels

28 October 2002




Diffusion Process

• Diffusion refers to the process of particles moving from areas of high density to areas of low density

• The diffusion rate is driven by the concentration at a point

••

• •••

•

•

•

•

•

•

BeforeClustered Group of Particles

AfterUniformly Distributed Group of Particles

28 October 2002




Diffusion Equation (1 of 2)

n1 n2

L L

L

x0

n1 >n2

n-nt2

L

At

LAn-n2

1

x

is x time, area/unit flow/unit of rate The

LAn2

1-LAn

2

1 is 2---1 from flownet The

LA.n2

1 is 1 to 2 from moving particles ofnumber The LA.n

2

1

is 2 area to 1 area from moving particles ofnumber net The

particles. of )n(n n and n ionconcentrat a containing regions

adjacent twoconsider we equation diffusion the derive To

21

21

0n

0n

21

21

2121

occurs diffusion whichat rate the describes D

tcoefficien diffusion the called is t2

LD ,where

dx

xdnDL

x

xxn-xn

t2

Ln-n

t2

Lx and

Lx

xxn-xn

x

Lxxn-xnn-n

as ionconcentrat electron in difference the write can eW

n

2

n

n210n

21

28 October 2002




Diffusion Equation (2 of 2)Particles and Current

this?) causes(What holesfor direction same the in and

electronsfor direction opposite the in moves diffusioncurrent gradient.

ionconcentrat the of direction the in move diffusion particle :Note

(holes) dx

xdpDq

dx

xdpDqJ

)(electrons dx

xdnDq

dx

xdnDqJ

sexpression the by holes and electronsfor given is and region a crossing

area)nit (current/u densitycurrent the iscurrent diffusion The

(holes) dx

xdp-Dx

)(electrons dx

xdn-Dx

by given are holes and electronsfor equation diffusion heT

ppp

nnn

pp

nn

28 October 2002




Diffusion and Drift of Carriers

• Forces that can cause electron (hole) drift are– Diffusion - driven by carrier concentration

– Electro-Motive Force - driven by an Electric Field (F=qE)

xJxJxJ

expression the by given is J(x), density,current total The

(holes) dx

xdpqDxExpqxJ

)(electrons dx

xdnqDxExnqxJ

Hence

pn

pxpp

nxnn

28 October 2002




Built in Electric Fields

dx

Ed

q

1

q

E

dx

d

dx

xdVE

energy reference

our be conductor)-semi intrinsic the of energy Fermi (the E Letting

xpoint the to from q,

charge the bring to required charget energy/uni the is xV wheredx

xdVE

dimension one in or

zy,x,V-gradE

potential electric the of definition the Recall

iix

i

x

28 October 2002




Einstein Relationship

Volts 0.026D

e),Temperatur (room K300Tfor that Note

Relation Einstein the called is q

kTD general In

D

q

kTD and

xEqkT

1DxE hence xEq

dx

Ed but

0dx

Ed since ,

dx

Ed

kT

1D

dx

Ed

dx

Ed

kT

1DxE and

xndx

Ed

dx

Ed

kT

1

dx

xdn have we ,enn Since

dx

xdn

xn

1DxE Hence .0xJ mequilibriu At

dx

xdnqDxExnqxJ current, hole) (and electronfor expression an have We

p

p

n

n

xn

nxx

i

Fi

n

niF

n

nx

iFkT

EE

i0

n

nxn

nxnn

iF

28 October 2002




Example, Text page 130

(c)

m

V106.2m10V026.0a

q

kT(x)E (b)

aq

kT(x)na

(x)n

1D

dx

(x)dn

(x)n

1D(x)E have We(a) :Solution

E of direction the indicate and diagram band a Sketch (c)

m)1(a when (x)E Evaluate (b)

nN whichfor range theover mequilibriuat (x)Efor expression an Find (a)

eNNthat such side one from donors with doped is sample Si intrinsic An:oblemPr

416x

n

n

n

nx

x

-1x

idx

-ax0d

n(x)

x

ni

N0

x

E(x)

EV

EC

Ei

EF

28 October 2002




Continuity Equation

holesfor expressionsimilar a with electronsfor equations diffusion the called is This

n

x

xnD

t

tx,n and

x

nqDJ diffusion, to strickly due iscurrent the If

n

x

xJ

q

1

t

tx,n

becomes electronsfor and

p

x

xJ

q

1

t

tx,p

becomes holesfor equation this ,0x Letting

p

x

q

xxJ

q

xJ

t

p Hence

rate ionrecombinat-ionconcentrat hole of increasebuildup Hole of Rate The

:boundary the

by defined volume the within (sink) destroyedor (source) created isthat charge

the to boundary a through passing charge the relates equation continuity The

n2

2

n

nn

n

n

p

p

p

pp

xxx

28 October 2002




Diffusion Length: Steady State Carrier Injection

holes and electronsfor length diffusion the as DL and DL define We

holes)(for L

p

D

p

x

xp and

electrons)(for L

n

D

n

x

xn

become equations Diffusion the and

0t

tx,p

t

tx,n then state steady reaches ionconcentratcarrier the If

holes)(for p

x

xpD

t

tx,p and

electrons)(for n

x

xnD

t

tx,n showedjust We

pppnnn

2ppp

2

2

2nnn

2

2

p2

2

n

n2

2

n

28 October 2002




Diffusion Length

versa)-vice (and electron an with recombinesit before diffusemust

hole a distance average the is This value.their of 1/e to carriers of

number the reducesthat solid the in distance the is L length diffusion The

pexp so pB and 0 AHence

p0p and ,0p

conditions boundary the from determined are B and A

BeAexp

form the on takes equation Diffusion the to solution The

p

Lx

Lx

Lx

p

pp

28 October 2002




Haynes-Shockley Experiment

• The Haynes-Shockley Experiment results in the independent determination of minority carrier mobility () and the minority carrier diffusion constant (D)

dd

d

2

p

tD4

x

p

2

2

p

x

dp

dd

t

Ltvtx where

t16

xD determine can we this From

etD2

pt,xp

onDistributi Gaussian the called is equation this to solution Thex

t,xpD

t

t,xp is holes thefor equation diffusion The

E

v Hence

t

Lv carriers

minority the of velocitydrift the measuring by determined is mobility Hole

p

2

28 October 2002




Example, Text Page 136-137

)K300T(for q

kTeV026.

secVoltcm

109.1

seccm

4.49D )b(

sec

cm4.49

sec105.216

sec1017.1 cm95.

t16

tL

t16

xD

secVolt

cm109.1

cm1/Volts2sec1025.

cm95.0

LVoltage Battery

timetransit separation probe

E

v (a)

Solution

Relation Einstein theagainst this Check (b)

tCoefficien Diffusion the and mobility hole Calculate (a)

s117 width pulse t

.25ms pulse of timetransit t

volts 2 E Voltage Battery

cm 0.95 separation probe

cm 1sample of lengthL

Experiment Shockley-Haynes

23

2

p

p

2

34

24

3d

2

d

2

p

23

3-

x

dp

d

0

28 October 2002




Gradients in the Quasi-Fermi Levels• Equilibrium implies no gradient in the Fermi level

• Combination of drift (due to Electric Field) and diffusion implies there is a gradient in the “quasi” Fermi Level

holes)(for dx

qF

d

dx

qF

dxpqxJ

electrons)(for dx

qFd

dx

qFd

xnqdx

dFxnxJ

level Fermi quasi"" the of terms in Law" sOhm'" dgeneralize a have we ,xEqdx

dE sincebut

dx

dE

dx

dFxnxExnqxJ

find we

,q

kTD Relation, Einstein the From

dx

dE

dx

dF

kT

xnen

dx

d

dx

xdn

wheredx

xdnqDxExnqxJ

by given is diffusion anddrift with ionconcentrat (hole) electron mequilibriu-non of case general The

p

p

p

pp

n

n

n

nn

nn

xi

innxnn

n

n

inkT

EF

i

nxnn

in

28 October 2002




Summary• We described methods of calculating carrier concentrations under

equilibrium conditions in the previous lecture

• This lecture we discussed carrier concentrations under non-equilibrium conditions

– Mechanisms (Optical Absorption-Direct and Indirect Recombination)

– Quasi-Fermi Levels to describe non-equilibrium carrier concentrations

• Diffusion Process– Current Density Mechanisms

• Diffusion

• Electric Field

– Einstein Relation

– Continuity Equation

– Diffusion Length

– Haynes-Shockley Experiment

– Generalized Ohm’s Law (Quasi-Fermi Levels)

• Photo conductive devices

28 October 2002




Next Time - Semi-conductor Junctions

• Fabrication of p-n junctions

• p-n Junction equilibrium conditions– contact potential

– Fermi Level

– Space Charge

• Forward and Reverse Biased Junctions– Steady State Conditions

– Reverse Bias Breakdown

– A-C conditions

– Diode Operation

– Capacitance of the p-n junction

– Varactor Diode

• Shottky Barriers

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