Semiconductor Devices A brief review
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Transcript of Semiconductor Devices A brief review
Semiconductor DevicesA brief review
Dr. K. Fobelets
Purpose of the course
• Study bipolar devices in more detail– Diodes and BJTs– Closer to reality: recombination – What causes the delays in these devices when
switching?
The most frequently used sentence in this course will be:
Excess minority carrier concentration
Structure
• 1. Lectures : 10 hrs– Basic principles based on Q&A session– Recombination and how does it impact the
characteristics– LONG pn diode – correct and approximated
solutions– LONG BJT– Switching of pn diodes and BJTs
• 2. Classes: solving past exam papers
Review
• Electrons and holes• Minority and majority carriers• Energy band diagram
Intrinsic Si
Si Si Si Si
Si Si Si Si
Si Si Si Si
Movement: kT
Si Si Si Si
Si Si Si Si
Si Si Si Si
Thermal energy: kT
Si Si Si Si
Si Si Si Si
Si Si Si Si
Si
Covalent bond
Free charged carriers in Si
Extrinsic Si
Si B Si Si
Si Si Si Si
Si Si Si Si
NA
Extrinsic Si
Si As Si Si
Si Si Si Si
Si Si Si Si
ND
Obtained by dopingB
As
Extrinsic Si
p-type n-type
In semiconductors two types of free charged carriers exist: electrons and holes.
Q1: What are holes?
a) Spherical voids in a semiconductorb) A positively charged Si atom that has lost its electronc) A positively charged particle that is the result of quantum mechanics
SiSi
SiSiSi
+ SiSi
Si
Si
SiSi
SiSiSi
SiSi
Si
Si
CThe two charged particles describe together the conduction in semiconductors.
Electron e- with charge q=-e and mass mn = m0 m*n
Hole h+ with charge q=+e and mass mp = m0 m*p
Intrinsic silicon (Si) has a small number of both free electrons and holes such that n i=pi.In order to increase the free carrier concentration, the semiconductor can be doped. With donors ND more electrons are created, with acceptors NA more holes are generated.
Q2: When intrinsic Si is doped with donor atoms, which of the following statements is correct?
a) n = p = ni = pi
b) n > ni & p < ni
c) n > p > ni
d) p > n > ni
n: electron concentrationp: hole concentrationni: intrinsic electron concentrationpi: intrinsic hole concentration
Bn > ni & p < ni in an n-type semiconductor.
n-type semiconductorn = ND p = ni
2/ND
p-type semiconductorn = ni
2/NA p = NA By heart
The concept of majority carrier and minority carrier is important in semiconductor devices. Majority carrier is the carrier type in a doped semiconductor with the highest concentration. Minority carrier is the carrier type with the lowest concentration.
Q3: True or False? The holes are the majority carriers in a p-type semiconductor (doped with acceptor atoms NA).
TRUEp-type semiconductor
pp
holeconcentration
p-typesemiconductor
np
electronconcentration
p-typesemiconductor
>
n-type semiconductor
nn
electronconcentration
n-typesemiconductor
np
holeconcentration
n-typesemiconductor
>
MAJORITY CARRIERS MINORITY CARRIERS
Drift and diffusion
• Two types of carrier movement– As a result of an electric field → DRIFT– As a result of a carrier gradient → DIFFUSION
Drift of carriers under influence of an electric field: E
E+ -
E+ -
EqJqJ
carriers ofnumber
v carriers ofnumber
Diffusion of carriers due to a carrier gradient
carriers ofnumber D
gradiention concentratconstant diffusion
dxdqJ
qJ
x
The purpose of semiconducting devices is to generate a current/voltage in response to an applied voltage/current. Two different types of current can exist in a semiconductor: drift and diffusion current. The expression of the total current that can flow in a semiconductor is given by the drift-diffusion equation:
Q4: Which statement is true?
a) Term (1) is drift current and (2) diffusion currentb) Term (2) is drift current and (1) diffusion currentc) Only term (1) can exist in a semiconductord) Only term (2) can exist in a semiconductor
dxxdpeDxExpexJ
dxxdneDxExnexJ
ppp
nnn
)()()()(
)()()()(
(1) (2)
ADrift current is proportional to the carrier concentration and the electric fieldDiffusion current is proportional to the carrier gradient.
E(x) Jndrift
Jpdrift
n(x) Jn
diff
p(x) Jpdiff
Motion of free charged carriers in a semiconductor.
Q5: If a p-type semiconductor at room temperature is conducting carriers due to drift, which of the following motion paths would be followed by the holes?
a)
(b)
c)
(d)
E+ - E+ -
E+ - E+ -
BWhen carriers move in a semiconductor they are scattered along the way. This means that they will be accelerated by the electric field (in this case) and then interact with atoms, impurities, other carriers that makes them lose some of their kinetic energy = scattering. Therefore the carriers will travel with an average velocity in amplitude and direction.
me
Ev
Q6: Solve diffusion processes
p+ n p
1. Draw arrows indicating the direction of diffusion of carriers.2. Identify the type of carriers that is diffusing.
Solution
p+ n p
Holes
Electrons
p+ n p
1. Because hole diffusion and electron diffusion cancel each other.2. Because an internal electric field is built up across each junction
causing drift of holes/electrons that cancel the diffusion of .holes/electrons.
3. Because holes and electrons diffuse automatically back to where they came from.
Q7: Why is there no net current while diffusion is happening?
p+ n p
Holes
Electrons
2. Because an internal electric field is built up across each junction causing drift of holes/electrons that cancel the diffusion of .holes/electrons.
Holes
Electronsdiffusion drift
+- E + -E
p-Si
Si B Si Si
Si Si Si Si
Si Si Si Si
NA n-Si
Si As Si Si
Si Si Si Si
As Si Si Si
ND
Depletion
Si
B
As
Si
Si
B
Cap
aciti
ve e
ffec
t
E+ -
--
B- : boron atom ionised
Si
Si
Si
Cap
aciti
ve e
ffec
t
E- +
As+ : arsenic atom ionised
+
+
Q8: True - False
The position of the Fermi level EF determines the type of the semiconductor.
Ec
Ev
EF
Q9: Multiple choice
1. This is the energy band diagram of an n-type semiconductor.2. This is the energy band diagram of a p-type semiconductor.3. This is the energy band diagram of an intrinsic semiconductor.
Ec
Ev
EF
Bottom of conduction bandEc
Top of valence bandEv
EiIntrinsic “level”. Is the position of the Fermi level EF when the semiconductor is intrinsic.
EG Bandgap. No energy levels in this energy region.
Position of Fermi level is determined by the doping type and densityFor n-type Si:
D
CFc
D
CCFc
FcC
NN
kTEE
NN
nN
kTEE
kTEE
Nn
ln
exp
exp
EF
Devices• A combination of n and p type
semiconductors plus ohmic contacts to apply the external voltages/currents makes devices
• When combining a-similar materials diffusion will occur and as a result an internal electric field will be built up to an amount that opposes diffusion current.
Energy band diagram
e.g.p-Si – n-Sip-Si – n-Si – p-Si
It is possible to start from the knowledge on workfunctions, and the energy reference: the vacuum level, Evac. The workfunction is dependent on the doping concentration!
Evac
n-Sie×n-Si
EF
p-Si
e×p-Si
EF
Evac
p-Si
e×p-Si
EF
Evac
n-Sie×n-Si
EF
Evac
p-Si
EF EF
Depleted region on both sides
Ec
Ev
Ec
Ev
e×p-Si
Evac
n-Sie×n-Si
Evac
SinSipeVe 0
Diffusion and drift can occur at the same time.
E
Both also always occur across junctions
A charge packet
A look at the short pn-diode
PN diode I
V
p n
p n
p n
E
Short PN diodeI
V
p n
p n
p n
E
DIFFUSION
Short PN diodeI
V
p n
p n
p n
E
DIFFUSION
Short PN diodeI
V
p n
p n
p n
E
Linear variation of minority carrier concentration
How do we find the current?
DIFFUSION
distanceMin
ority
car
rier c
once
ntra
tion
Apply diffusion current formula to the minority carrier variation
Short PN diodeI
V
p n
p n
Ep n
Only few carriers can contribute to the current
Contents of course this year
• Long pn diode– Introducing the concept of recombination of carriers.– Switching of the pn diode, where does the delay come
from?
• Bipolar junction transistor– Internal functioning– Switching delays
p n
Long
But what happens in a long pn diode?
p n
Ln Lp
Minority carrier diffusion length
Short
In long semiconductors recombination of the minority carriers will occur whilst
diffusing
Loss of both carrier type, but felt most in excess minority carriers. Remember: the amount of majority carriers is much larger than the excess.
Excess holes, in an n-type semiconductor will recombine with the large amount of available electrons.
p
In long semiconductors recombination of the minority carriers will occur whilst
diffusing
• Diffusing minority carriers (e.g. holes) recombine with majority carriers (electrons) within a diffusion length LpIn
ject
ion
of c
arrie
rs
x
Loss of both carrier type, but felt most in excess minority carriers. Remember: the amount of majority carriers is much larger than the excess.
Lp
Excess holes, in an n-type semiconductor will recombine with the large amount of available electrons.
p
Generation-recombination
• Generation of carriers and recombination is continuously happening at the same time such that the equilibrium carrier concentrations are maintained.
Charge neutral
R=G
Recombination - generation
• In case there is an excess carrier concentration then the recombination rate R of the excess, will be larger than its generation rate, G: R>G
When there is a shortage, then G > R
Recombination - generation
• Simple model: Recombination/generation rate is proportional to excess carrier concentration.
• Thus no net recombination/generation takes place if the carrier density equals the thermal equilibrium value.
Recombination of e- in p-type semiconductor
p
n
p
nnppp
n
p
n
ppnnn
pppGRU
nnnGRU
0
0
Recombination of h+ in n-type semiconductor
Diffusion, drift and recombination of carriers
What is the consequence of this recombination on the characteristics of the pn diode with neutral regions
larger than the diffusion lengths of the minority carriers?
In the pn diode the carrier gradient determines the current thus we have
to find the function p(x) of the minority carrier concentration.
• Note, reasoning done for p(x). For n(x) analogous approach.
Mathematical description of diffusion and recombination
xx x+x
Jp(x) Jp (x+x)A
p
pp
xxx
px
xxJxJqt
p
)()(1
Rate of hole variation
Variation of hole concentration in x x A/s
Recombination rate= +
Mathematical description of diffusion and recombination
p
p px
Jqt
p
1
= bulk defined + excess concentration
Jp : total current = drift + diffusion
Neglect drift current (no electric field applied)
p
pp
xxx
px
xxJxJqt
p
)()(1
p
p px
Jqt
txpx
1),(:0
ppp 0
D
in N
npp2
0 0
with
Mathematical description of diffusion and recombination
pp
pp
p
p
px
pDtp
ppp
pxpDp
xJ
qtp
2
2
0
2
21
= bulk defined+ excess concentration
dxxdpeDxJ pp)()(
D
in N
npp2
0 0
with
Solve equation in steady state
22
2
0
ppp Lp
Dp
xp
tp
Diffusion length
Boundary conditions:ppx
pXx n
00
General solution of 2nd order differential equation:
21 sinh)( C
LxCxpp
x
p
Xn0
pcontact
p
n
p
nL
Xx
LX
pxp sinh
sinh
)(
Too complicated
• Short approximation • Long approximation
Xn << Lp
p
n
p
nL
Xx
LX
pxp sinh
sinh
)(
Xn >> Lp
LINEAR EXPONENTIAL
Short semiconductor• Xn ≤ Lp carriers do not have time to recombine (=∞) !• Taking linear approximation.
pn(x)
x0
pn0
p
pn(x
)
Xn
NO recombination : variation of the excess carrier concentration linear
pn(x)= pn0+ p (1–x/Xn)
pn(x)
Contact imposes pn(Xn)=0
p’n
Diffusion and recombination• Xn >> Lp carriers do have time to recombine (t<∞) !• Taking exponential approximations
When recombination occurs and Xn >> Lp variation of the excess carrier concentration is exponential
pn(x)
x0
pn0
p
pn(x
)
pn(x)=pn0+p’n
LpContact imposes pn(Xn)=0
Xn
p
n
p
p
nLX
Lx
LX
p expexp
exp1
pn(x)
p
n
p
p
nLX
Lx
LX
p expexp
exp1pn(x)=
pn still too complex for quick calculations
• Take really extreme case• Xn >>> Lp or Xn → ∞
pLxp exp
Note: I and Q of both expressions of for the same
I for same as for linear approximation when Xn=Lp
pn(x) Xn → ∞
pn(x)=
pLxp exp
Diffusion and recombination
When recombination occurs and Xn → ∞ variation of the excess carrier concentration is exponential
pn(x)
x0
pn0
p
pn(x
)
pn(x)=pn0+p e-x/Lpp’n
Lp
• Xn >>> Lp carriers do have time to recombine (t<∞) !• Taking exponential approximations
Imposes pn(Xn)=0∞
SHORT ↔ LONGapproximation
Short
Boundary of short
LongIntermediate
Correct solutionExponential solutionLinear solution
pn(x)
pn(x)
pn(x)
pn(x)Lp=200 nm, Xn=400nm
Lp=200 nm, Xn=20nm Lp=Xn=200nm
Lp=200 nm, Xn=1000nmx
x
x
x
• Calculation of currents in pn diode with neutral regions larger than the diffusion length, using the long semiconductor approximation
→• Exponential variation of the excess minority
carrier concentration.
Carrier injections: forward bias
• Carrier injection across junction
-wp wn0
p ne-
diff
h+diff
• Creates minority carrier concentration gradients
np(-x)n’p
pn(x)
p’n
np0=ni2/NA & pp=NA
pn0= ni2/ND & nn=ND
pn0
x
np0
-x
Tnn
Tpp
VVpp
VVnn
exp'
exp'
0
0
Carrier injections: reverse bias
• Minority carriers are swept across junction V<0
-wp wn0
p ne-
drift
h+drift
• Small amount of minority carriers → small current
pn0
x
np0
-x
np(-x)
n’’p
pn(x)
p’’n
Tnn
Tpp
VV
pp
VV
nn
exp''
exp''
0
0
Thus
pn = pn0 (eeV/kT -1)
-wp wn0
p ne-
diff
h+diff
np(-x)n’p
pn(x)
p’n
pn0
x
np0
-x
np = np0 (eeV/kT -1)
nppn
nLx
pp enxn)(
)(
pLx
nn epxp)(
)(
Two methods to calculate current
x-wp wn0 I
nppn
x-x
np pn
Slope
1. Gradient excess carrier concentration2. Re-supply of recombined excess charge
0 0
Qn
Qp
1. Excess carrier concentration gradient
-wp wn
np pn
x-x
np pn
Slope
e-
In = e A Dn dnp/dx = max @ x=0
h+
Ip = -e A Dp dpn/dx = max @ x=0
Maximum diffusion currents at the edges of the transition region
0 0
1. Excess carrier concentration gradient
e- h+
Fill in expression for excess carrier concentration
1exp
1exp
1exp)(
0
0
0
max
0
)(
max
)(
kTeV
LDeAn
I
dx
ekTeVdn
eADI
ekTeVnxn
n
npdiffn
x
Lx
p
ndiffn
Lx
pp
n
n
1exp
1exp
1exp)(
0
0
0
max
0
)(
max
)(
kTeV
LDeAp
I
dx
ekTeVdp
eADI
ekTeVpxp
p
pndiffp
x
Lx
n
pdiffp
Lx
nn
p
p
In Ip
Changing gradient!→
Changing diffusion current density
p n
ItotIp In
Itot=In + Ip
x
diffntotxdriftp
Lx
n
npxdiffn
III
ekTeV
LDeAn
I n
)(
1exp0
InIp
x
diffptotxdriftn
Lx
p
pnxdiffp
III
ekTeV
LDeAp
I p
)(
1exp0
x-wp wn0 I
np pn
x
np0
-x
np pn
pn0
In
Ip
np = np e-(-x)/Ln
pn = pn e-(x)/LpQn
Qp
0 0
2. Re-supply of recombined excess carriers
Excess carrier charge Q recombines every seconds (carrier life time).For steady state Q has to be re-supplied every seconds → current
-wp wn0
np pn
x
np0
-x
np pn
pn0
In
Qn = -e A ∫-∞0np dx
In = Qn/n = e A Ln np /n
Ip
Qp = e A ∫0∞pn dx
Ip = Qp/p = e A Lp pn /p
Charge – minority carrier life time ratio
np = np e-(-x)/Ln
pn = pn e-(x)/LpQn
Qp
0 0
2. Re-supply of recombined excess carriers
Charge = area under excess carrier concentration: integrate-∞ and + ∞ are the contacts: excess charge = 0!
Total current
• I = Ip(0) + In(0) = e A (Dp pn0 /Lp + Dn np0/Ln )(eeV/kT -1)
• I = I0 (eeV/kT -1)
• With I0 = e A (Dp pn0/Lp + Dn np0/Ln)
Reverse bias current
Same equation as short diode with length exactly equal to the minority carrier diffusion lengths
SHORT ↔ LONGapproximation error on current calculation:
ratio of currents
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5
Xn/Lp
Ireal/Ia
ppro
x
Ireal/IexpIreal/Ilin
Error on linear and exponential approximation
same when Xn=Lp
• Non-idealities in the pn diodes
Log(I)
V
a)b)
c)
idealreal
(a) Low voltage: low injection of carriers
V
Log(I)
V
a)
idealreal
1nkTeV
stot eII
(c) High voltage: high injection of carriers
n’p ≈ pp
p’n ≈ nn
Log(I)
V
c)
idealreal
a) n=2b) n=1c) n=2
(d) Higher currents
Log(I)
d)
idealreal
V
Current determined by resistance
Switching of p-n diodes• When a p-n diode is forward biased, excess carrier
concentrations exists at both sides of the depletion region edge.
• To switch the diode from forward to off or reverse bias, this excess carrier concentration needs to be removed.
• The transients resulting from the time it takes to remove the excess carriers will lead to the equivalent capacitance.
-wp wn0
p nnp pn
Switching off
on
off
i
t0-wp wn0
p nnp pn
e-
h+
-wp wn0
p n
Steady state snap shots
How do we go from this:p
x
pn
pno
pn
To this?
Off: NO current flows!!!
Excess carrier concentration+pno
Variation of the excess carrier concentration as a function of time.
p(x,t)
p
pp
p
p
ppcontactp
p
contact
p
contactp
contact
tQI
dttdQ
tQJ
eeAJ
eeA
dttdQ
dxpeAdxx
Je
eAdxt
txpeA
)()(
)()(
),(
0
000
Relationship for charge Qp
p
p px
Jqt
p
1
Transient during switching off
i(t)= I + dQ/dt = Q/ + dQ/dt
Excess charge due to charge injection at any instance of timeAverage lifetime of minority carriers
Recombination termCharge depletion term (or buildup)
Since no current in “off”, charge has to disappear byrecombination!
For switch from on to off:
At t<0 → Ion=Ion (Von)At t≥0 → Ioff = 0 (Voff = 0)And at t=-0 Q(0)=Ion At t→∞ Q(∞)=0 Q(t)=Ion e-t/
t > 00 = Q/ + dQ/dt
Transient during switching offvariation of the excess carrier concentration as a function of time
t=0
gradient→ i≠0
p
x
Variation in timepn
i=0→gradient=0
A voltage, vd will exists across the diode as long as charge remains
Qp(t)=eA∫p(x,t)dx=Ippe-t/p
p(x,t)=p(vd(t)) e-x/Lp
Revision
• When a pn diode switches, the excess minority carrier concentration needs to change. The removal of the excess minority carrier concentration causes the delay in the pn diode.
• The variation of the excess carrier concentration as a function of time given by:
dttdQtQ
ti p
p
pp
)()()(
ON-OFF (open circuit)take: p+n → Itot ≈ Ip
dttdQtQ
ti p
p
pp
)()()(
p+ nIp
t=0
ppONp
p
p
p
p
p
pONp
tItQ
dttdQtQ
it
QRVIit
exp)(
)()(0
0)0(;0@
)0()0(;0@
vd
R
V
OFF (open circuit) → ONtake: p+n → Itot ≈ Ip
dttdQtQ
ti p
p
pp
)()()(
p+ n
Ip
t=0
vd
pONp
pONpONpp
pONp
ONpp
pONpONpp
p
tONpp
pONpp
p
p
pONpp
p
p
pON
ONp
pp
tItIItQ
tI
ItQ
tIItQ
tItQ
dtItQ
tdQ
tQIdt
tdQ
dttdQtQ
I
RVIit
Qit
exp1exp)(
)(ln
ln)(ln
)(ln
)()(
)()(
)()(
)0(;0@
0)0(;0)0(;0@
0
V
R integrate
Reverse recovery transientSwitch the diode from forward to reverse bias
on
off
i
t0-wp wn0
p nnp pn
e-
h+
Steady state snap shots
How do we go from this:
Reverse bias current flows!!!
Excess carrier concentration 0-wp wn
e-
h+
x
p
pn
0pn
To this?
Transients when switching to reverse biase(t)
tE
-Ep n
e(t)i(t) R If≈E/R
Ir≈-E/R
I
V
If
-Ir
x
pIf → gradient≠0
Ir → gradient≠0
t
v(t)
ti(t)
t
-E
Storage delay time: tsd
i(t)If
t
-Ir
v(t)
Time required for the stored charge to disappear
tsd = minority carrier ln(1 + If/Ir)
tsd
Calculate storage delay time: tsd
dttdQtQ
ti p
p
pp
)()()(
i(t)IF
t
-IR
v(t)
tsd
0)(;@
)0()0(;0@
)0()0(;0@
sdsd
RpR
FpF
tQtt
IQIit
IQIit
dttdQtQ
I
Ititt
p
p
pR
Rpsd
)()(
)(0
X !
Calculated storage delay time: tsd
i(t)IF
t
-IR
v(t)
tsd
pFpRpRpp
FpRp
pRp
p
pRppRpp
tpRp
p
pRp
p
p
p
p
pRp
p
p
pR
tIIItQ
IItQIt
QItQIt
tQIt
tQItdQdt
dttdQtQI
dttdQtQ
I
exp)(
)(exp
)0(ln)(ln
)(ln
)()(
)()(
)()(
0
integrate
Calculated storage delay time: tsd
i(t)IF
t
-IR
v(t)
tsd
R
FRp
FR
Rpsd
p
sdFpRpRp
sd
pFpRpRpp
III
IIIt
tIII
tt
tIIItQ
lnln
exp0
exp)(
i(t)IF
t
-IR
v(t)
tsd
After: tsd
0
0)(
d
sdp
v
tQ
Evd
Build-up of depletion region
deplbu RCt
Small signal equivalent circuit
• Junction capacitance • Diffusion capacitance
p n
w
• Cj = A/w
• w function of bias→ C voltage variable capacitance
• Important in reverse bias
• Due to charge storage effects
-wp wn0
p nnp pn
• Due to depletion region
• Cd = dQ/dV = d (I )/dV
= e/kT I
• Important in forward bias
• Diffusion capacitance
Equivalent conductances
• Diffusion conductance
• gd = dI/dV = e/kT I0 eeV/kT
≈ e/kT I
• Slope of the current voltage characteristic in forward bias
• Series resistance rs
• Due to n and p region + contact resistance
• Vd = Vappl – rs I
rd
rs
Cj
CdOnly linear circuit elements present
Large signal equivalent circuit
C
Rs
Reverse bias: depletion capacitanceForward bias: diffusion capacitance
Non-linear circuit elements present
Conclusions
• The characteristics in a pn diode are based upon excess minority carrier diffusion.– Excess carrier concentrations are being formed
by injection of carriers across the junction.– The gradient of the excess minority carrier
concentration at the junction determines the magnitude of the current.
– Delay times are due to the storage of excess minority charge in the layers.
Revision
• When recombination is taken into account, the excess minority carrier concentration reduces while diffusing through the neutral regions of the diode.
• The variation of the excess carrier concentration is then given by:
pp
px
pDtp
2
2
Lifetime of minority carrier holes
Revision
• The steady state solution for the excess minority carrier concentration is then:
• This is considered too complex for quick calculations and approximations are used in the case of a short or long neutral region.
p
n
p
nL
Xx
LX
pxp sinh
sinh
)(
Revision
• Short: Xn ≤ Lp
pn(x)
x0
pn0
p
pn(x
)
Xn
linear
pn(x)= pn0+ p (1–x/Xn)
pn(x)
Contact imposes pn(Xn)=0
p’n
Revision
• Long: Xn >>> Lp exponential
pn(x)
x0
pn0
p
pn(x
) pn(x)=pn0+p e-x/Lpp’n
Lp Imposes pn(Xn)=0∞
pn(x)=pn0+
p
n
p
p
nLX
Lx
LX
p expexp
exp1
Revision
• These approximation make some errors in the calculation of the current and the charge stored in the neutral regions.
• However we will see that:
1. I and Q for simplified and non-simplified exponential variation of pn(x) for Xn → ∞ is the same
2. I for is same as for linear approximation when Xn=Lp
pn(x) =
pLxp exp
Errors on current
0200400600800
100012001400
1 2 3 4
Xn (nm)
Cur
rent
(a.u
.)
Series1Series2Series3
Lp=20 nm
1020 40 200
CorrectExponentialLinear
Short = good approximation up to Xn = Lp
Long = good approximation up to Xn > 5 ×Lp