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CHAPTER 4
CONDUCTION IN SEMICONDUCTORS
Prof. Dr. Beşire GÖNÜL
CHAPTER 4CONDUCTION IN SEMICONDUCTORS
• Carrier drift• Carrier mobility• Saturated drift velocity• Mobility variation with temperature• A derivation of Ohm’s law• A derivation of Ohm’s law• Drift current equations• Semiconductor band diagrams with an electric field present• Carrier diffusion• The flux equation• The Einstein relation• Total current density• Carrier recombination and diffusion length
• We now have some idea of the number densityof charge carriers (electrons and holes) presentin a semiconductor material from the work wecovered in the last chapter. Since current is therate of flow of charge , we shall be able calculate
Drift and DiffusionDrift and Diffusion
covered in the last chapter. Since current is therate of flow of charge , we shall be able calculatecurrents flowing in real devices since we knowthe number of charge carriers. There are twocurrent mechanisms which cause charges tomove in semiconductors. The two mechanismswe shall study in this chapter are driftdrift andanddiffusiondiffusion.
• Electron and holes will move under the influence of anapplied electric field since the field exert a force oncharge carriers (electrons and holes).
• These movements result a current of ;
F qE=
Carrier DriftCarrier Drift
dI• These movements result a current of ;dI
d dI nqV A=:dI
:dV
drift current number of charge carriers per unit volume
charge of the electrondrift velocity of charge carrier
area of the semiconductor:A
:q
:n
Carrier Mobility ,Carrier Mobility , µ
dV Eµ=
:E:µ
applied field
mobility of charge carrier
[ ]
−=
SecV
cm2
µ µ is a proportionality factor
=E
Vdµ
So is a measure how easily charge carriers move under the influence ofan applied field or determines how mobile the charge carriers are.
µµ
n n -- type Sitype Si
+ -V
n – type Si
e-
V
L
VE =
e
dV
Electric field
Electron movement
Current flow
Current carriers are mostly electrons.
+ -V
p – type Si
holeV
p p -- type Sitype Si
L
VE =
hole
dV
Electric field
Hole movement
Current flow
Current carriers are mostly holes.
Carrier MobilityCarrier Mobility
E
Vd=µ
Macroscopic understanding
In a perfect Crystal
Microscopic understanding? (what the carriers themselves are doing?)
*
q
m
τµ =
0ρσ
=
→ ∞
It is a superconductor
*mgeneralinmm he
** ⟨
*
*
;
;
e
h
m n type
m p type
−
−
• A perfect crystal has a perfect periodicity and thereforethe potential seen by a carrier in a perfect crystal iscompletely periodic.
• So the crystal has no resistance to current flow andbehaves as a superconductor. The perfect periodicpotential does not impede the movement of the charge
µ
potential does not impede the movement of the chargecarriers. However, in a real device or specimen, thepresence of impurities, interstitials, subtitionals,temperature , etc. creates a resistance to current flow.
• The presence of all these upsets the periodicity of thepotential seen by a charge carrier.
The mobility has two component
The mobility has two componentsThe mobility has two components
Impurity interaction component
Lattice interaction component
• Assume that s/c crystal is at thermodynamic equilibrium (i.e. there isno applied field). What will be the energy of the electron at a finitetemperature?
• The electron will have a thermal energy of kkTT//22 per degree offreedom. So , in 3D, electron will have a thermal energy of
Thermal velocityThermal velocity
* 23 1 3 3kT kT kT= ⇒ = ⇒ =* 23 1 3 3
2 2 2th th
kT kT kTE m V V
m ∗= ⇒ = ⇒ =
( ) 21
*
2
1
:
−
mV
TV
electronofvelocitythermalV
th
th
th
α
α
• Since there is no applied field, the movement ofthe charge carriers will be completely random.This randomness result no net current flow. As
Random motionRandom motion result result no currentno current..
This randomness result no net current flow. Asa result of thermal energy there are almost anequal number of carriers moving right as left, inas out or up as down.
*
0
2
? 300 1.18
? 1000 / 0.15 /( )
th e
d
V RT K m m
V E V m m V sµ
= = =
= = = −
• Calculate the velocity of an electron in a piece of n-typesilicon due to its thermal energy at RT and due to theapplication of an electric field of 1000 V/m across thepiece of silicon.
CalculationCalculation
5
? 1000 / 0.15 /( )
31.08 10 / sec
150 / sec
d
th th
d d
V E V m m V s
kTV V x m
m
V E V m
µ
µ
∗
= = = −
= ⇒ =
= ⇒ =
How long does a carrier move in time before collision ?
The average time taken between collisions is called as relaxation time, (or mean free time)
Microscopic understanding of mobilityMicroscopic understanding of mobility??
τHow far does a carrier move in space (distance) before a collision?
The average distance taken between collisions is called as mean free path, .l
*d
FV
mτ= ×
CalculationCalculation
Drift velocity=Acceleration x Mean free time
Force is due to the applied field, F=qE
* *d
d
F q EV
m m
qV E
m
τ τ
τµ µ
∗
= × =
= ⇒ =
Force is due to the applied field, F=qE
*
2 2
12 13
? ? 1 .18 0.59
0.15 /( ) 0 .0458 /( )
10 sec 1.54 10 sec
e o h o
e h
e e h h
l m m m m
m V s m V s
m mx
τ
µ µ
µ µτ τ
∗
∗ ∗− −
= = = =
= − = −
= = = =
• Calculate the mean free time and mean free path forelectrons in a piece of n-type silicon and for holes in apiece of p-type silicon.
CalculationCalculation
12 13
5 5
5 12 7
10 sec 1.54 10 sec
1.08 10 / 1 .052 10 /
(1 .08 10 / )(10 ) 10
(1.052 10
elec hole
elec
hole
e e h he h
th th
e th e
h th h
m mx
q q
v x m s v x m s
l v x m s s m
l v x
µ µτ τ
τ
τ
− −
− −
= = = =
= =
= = =
= = 5 13 8/ )(1 .54 10 sec) 2 .34 10m s x x m− −=
Saturated Drift VelocitiesSaturated Drift Velocities
dV Eµ=
dV for efor e--
So one can make a carrier go as fast as we like just by increasing the electric field!!!
dV
E E
for efor e--
for holesfor holes
(a) Implication of above eqn. (b) Saturation drift velocity
• The equation of does not imply that VVddincreases linearly with applied field EE..
•• VVdd increases linearly for low values of EE and then itsaturates at some value of VVdd which is close VVthth at
EVd .µ=
Saturated Drift VelocitiesSaturated Drift Velocities
saturates at some value of VVdd which is close VVthth athigher values of EE.
• Any further increase in EE after saturation point does notincrease VVdd instead warms up the crystal.
Mobility variation with temperatureMobility variation with temperature
µ µ
TTTT
High temperature Low temperature
)ln(µ
ln( T )ln( T )
Peak depends on the density of
impurities
High temperature Low temperature
IµLµ
1 1 1
T L Iµ µ µ= +
This equation is called as Mattheisen’s rule.
Variation of mobility with temperatureVariation of mobility with temperature
At high temperature (as the lattice warms up)
Lµ component becomes significant.
Lµ decreases when temperature increases.
2
3
2
3
1
−−⇒×= TTCLµ
It is called as a power law.
CC11 is a constant.
1.5T −
Carriers are more likely scattered by the lattice atoms.
At low temperatures
Variation of mobility with temperatureVariation of mobility with temperature
Iµ component is significant.
Iµ decreases when temperature decreases.
2
3
2 TCI ×=µ CC22 is a constant.
Carriers are more likely scattered by ionized impurities.
The peak of the mobility curve depends on the number density of ionized impurities.
Variation of mobility with temperatureVariation of mobility with temperature
Highly doped samples will therefore cause more scattering, and have a lower mobility, than low doped
samples.
This fact is used in high speed devices called High Electron Mobility Transistors (HEMTs) where electrons are made to
move in undoped material, with the resulting high carrier mobilities!
HEMTs are high speed devices.
samples.
A Derivation of Ohm’s LawA Derivation of Ohm’s Law
d d d
dd
I nqV A V E
I qJ
A m
µ
τµ
τ
∗
= =
= =
2
2 1
x d x x
x x
nqJ nqV nq E J E
m
nqJ E
m
τµ
τσ σ ρ
σ
∗
∗
= = =
= = =[ ] [ ][ ] [ ]1 ( )
m
m
ρ
σ
= Ω −
= Ω −
A Derivation of Ohm’s LawA Derivation of Ohm’s Law
V
L
R
VI =
VA VI
L Rρ= =
area
current
This is in fact ohm’s law which is written slightly in a different form.
1x
x xx
I IV VJ E
A L A Lσ σ
ρ= = ⇒ =
Drift Current EquationsDrift Current Equations
For undoped or intrinsic semiconductor ; n=p=ni
For electron
n nJ nqEµ=
For hole
p pJ pqEµ=n nJ nqEµ=
drift current for electrons
number of free electrons per unit volume
mobility of electron
p pJ pqEµ=
drift current for holes
number of free holes per unit volume
mobility of holes
Drift Current EquationsDrift Current Equations
Total current density
i e h
i n P
J J J
J nqE pqE
n p n
µ µ
= +
= +
= =
( )
i
i i n p
n p n
J n q Eµ µ
= =
= +
since
For a pure intrinsic
semiconductor
Drift Current EquationsDrift Current Equations
?=totalJ for doped or extrinsic semiconductor
n-type semiconductor;
T n D nn p J nq E N q Eµ µ>> ⇒ ≅ =T n D nn p J nq E N q Eµ µ>> ⇒ ≅ =where ND is the shallow donor concentration
p-type semiconductor;
T p A pp n J pq E N q Eµ µ>> ⇒ ≅ =where NA is the shallow acceptor concentration
Variation of resistivity with temperatureVariation of resistivity with temperature
Why does the resistivity of a metal increase withincreasing temperature whereas the resistivity of asemiconductor decrease with increasingtemperature?
1 1ρ = =
nqρ
σ µ= =
This fact is used in a real semiconductor device called a
thermistor, which is used as a temperature sensing element.
The thermistor is a temperature – sensitive resistor; that is its terminalresistance is related to its body temperature. It has a negative temperaturecoefficient , indicating that its resistance will decrease with an increase in itsbody temperature.
Semiconductor Band Diagrams with Semiconductor Band Diagrams with Electric Field PresentElectric Field Present
At equilibrium ( with no external field )
EC
EĐ
E
All these energies are horizontal
Pure/undoped semiconductor
EVhorizontal
How these energies will change with an applied field ?
+ -
n – type
Electric field
Electron movementHole flow
EC
EĐ
EV
Efe-
hole
qV
• With an applied bias the band energies slope down for the givensemiconductor. Electrons flow from left to right and holes flow from rightto left to have their minimum energies for a p-type semiconductor biasedas below.
EC
e-+_
p – type
Electric field
Electron movementHole flow
EC
EĐ
EV
Ef
hole
qV
Under drift conditionsUnder drift conditions;;
••UnderUnder driftdrift conditionsconditions;; holes float and electrons sink.Since there is an applied voltage, currents are flowingand this current is called as drift current.
•There is a certain slope in energy diagrams and thedepth of the slope is given by qV, where V is the batteryvoltage.
work qE L
Vq L work qV gain in energyL
qVSlope of the band qE Force on the electron
= − ×
= − × ⇒ = − =
= − = − =
The work on the charge carriersThe work on the charge carriers
Work = Force x distance = electrostatic force x distance
qVSlope of the band qE Force on the electron
L
VE
L
= − = − =
=
Since there is a certain slope in the energies, i.e. theenergies are not horizontal, the currents are able to flow.
where L is the length of the s/c.
The work on the charge carriersThe work on the charge carriers
1(1)
'
i ix x
dVElectrostatic Force gradiant of potential energy
dx
dE dEqE E
dx q dx
one can define electron s electrostatic potential as
= − = −
− = − ⇒ = ⋅
'
(2)
(1) (2) ,
nx
in n
one can define electron s electrostatic potential as
dVE
dx
comparison of equations and gives
EV is a relation betweenV an
q
= −
= − id E
Carrier DiffusionCarrier Diffusion
Current mechanisms
Drift DiffusionP nkT
dP dnkT
=
=
1
dP dnkT
dx dx
dn dP
dx kT dx
=
=
photons
Contact with a metal
Carrier DiffusionCarrier Diffusion
Diffusion current is due to the movement of the carriers from highconcentration region towards to low concentration region. As thecarriers diffuse, a diffusion current flows. The force behind thediffusion current is the random thermal motion of carriers.
1dn dP
dx kT dx= ⋅
A concentration gradient produces apressure gradient which produces the force
dx kT dx= ⋅ pressure gradient which produces the force
on the charge carriers causing to movethem.
How can we produce a concentration gradient in a semiconductor?
1) By making a semiconductor or metal contact.2) By illuminating a portion of the semiconductor with light.
Illuminating a portion of the semiconductor with light
By means of illumination, electron-hole pairs canbe produced when the photon energy>Eg.
So the increased number of electron-hole pairsmove towards to the lower concentration region untilthey reach to their equilibrium values. So there is athey reach to their equilibrium values. So there is anumber of charge carriers crossing per unit area perunit time, which is called as flux. Flux is proportionalto the concentration gradient, dn/dx.
n
dnFlux D
dx= −
FluxFlux
[ ][ ]
2 1
2,th
Flux m s
D l D m s
The current densities for electrons and holes
dn dnJ q D qD for electrons
ν
− −= −
= =
= − − =
2
,
n n n
p p p
n p
J q D qD for electronsdx dx
dp dpJ q D qD for holes
dx dx
J A m
= − − =
= + − = −
=
Einstein RelationEinstein Relation
Einstein relation relates the two independent currentmechanicms of mobility with diffusion;
pn
n p
DD kT kTand for electrons and holes
q qµ µ= =
Constant value at a fixed temperature
( )( )2
2
/sec
sec
J K Kcm kTvolt volt
cm V q C= = =
−
2 5kT
m V a t ro om tem p era tu req
=
Total Current DensityTotal Current Density
When both electric field (gradient of electric potential) and concentration gradient present, the total current density ;
n n n
dnJ q nE qD
dxµ= +
p p p
total n p
dpJ q pE qD
dx
J J J
µ= −
= +
Carrier recombination and diffusion lengthCarrier recombination and diffusion length
• By means of introducing excess carriers into an intrinsics/c, the number of majority carriers hardly changes, butthe number of minority carriers increases from a low- tohigh-value.
• When we illuminate our sample (n-type silicon with 1015• When we illuminate our sample (n-type silicon with 1015
cm-3 ) with light that produces 1014 cm-3 electron-holepairs.
• The electron concentration (majority carriers) hardlychanges, however hole concentration (minority carriers)goes from 1.96 x 105 to 1014 cm-3.
Recombination rate• Minority carriers find themselves surrounded by very high
concentration of majority carriers and will readily recombine withthem.
• The recombination rate is proportional to excess carrier density, .pδ
( ) (0)expt
p t pδ δτ
= −
1
p
d pp
d t
δδ
τ= −
1
n
d nn
d t
δδ
τ= − ( ) (0)exp
tn t nδ δ
τ = −
pd t τ
Lifetime of holes
Excess hole concentration when t=0
Excess hole concentration decay exponentially with time.
Similarly, for electrons;
Diffusion length,Diffusion length,
When excess carriers are generated in a specimen, the minoritycarriers diffuse a distance, a characteristic length, over whichminority carriers can diffuse before recombining majority carriers.This is called as a diffusion length, L.
Excess minority carriers decay exponentially with diffusion distance.
L
( ) (0)expn
xn x n
Lδ δ
= −
( ) (0)exp
p
xp x p
Lδ δ
= −
n n nL Dτ= p p pL Dτ=
Excess electron concentration when x=0
Diffusion length for electronsDiffusion length for holes
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