28 October 2002 ECEE 302 Electronic Devices Drexel University ECE Department BMF-Lecture 4-102802-Page -1 Copyright © 2002 Barry Fell ECEE 302: Electronic Devices Lecture 4. Effect of Excess Carriers in Semi- Conductors 28 October 2002
Jan 24, 2016
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -2Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -3Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -4Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -5Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -6Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -7Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -8Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -9Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -10Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -11Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -12Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -13Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -14Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -15Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -16Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -17Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -18Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -19Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -20Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -21Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -22Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -23Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -24Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -25Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -26Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -27Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -28Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -29Copyright © 2002 Barry Fell
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
ECEE 302 ElectronicDevices
Drexel UniversityECE Department
BMF-Lecture 4-102802-Page -30Copyright © 2002 Barry Fell
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