Module 2.3: PN Junctions Prof. Ali M. Niknejad Prof...
32
Department of EECS University of California, Berkeley Prof. Ali M. Niknejad Prof. Rikky Muller Module 2.3: PN Junctions
Transcript of Module 2.3: PN Junctions Prof. Ali M. Niknejad Prof...
Department of EECS University of California, Berkeley
Prof. Ali M. NiknejadProf. Rikky Muller
Module 2.3: PN Junctions
EE 105 Fall 2016 Prof. A. M. Niknejad
2
What is a pn-junction?
n-typep-type
NDNA
)(0 xpaNp =0
d
i
Nnp2
0 =
a
i
Nnn2
0 =
Transition Region
dNn =00px- 0nx
AKA: diode
EE 105 Fall 2016 Prof. A. M. Niknejad
3
Module 2.3 Outlinel Part 1: Carrier concentration variation and
potential– Show that any time there’s a variation in carrier
concentration, then at thermal equilibrium there must be a variation in potential
l Part 2: Apply this to a pn-junction at thermal equilibrium– Extend result to a reverse biased junction
l Part 3: Look at a forward biased pn-junction
University of California, Berkeley
EE 105 Fall 2016 Prof. A. M. Niknejad
4
Carrier Concentration and Potential
l In thermal equilibrium, there are no external fields and we thus expect the electron and hole current densities to be zero:
University of California, Berkeley
dxdnqDEqnJ o
nnn +== 000 µ
dxdn
kTqEn
Ddxdn
oon
no 00
fµ÷øö
çèæ=÷÷
ø
öççè
æ-=
0
0
00 n
dnVndn
qkTd th
o =÷÷ø
öççè
æ=f
EE 105 Fall 2016 Prof. A. M. Niknejad
5
Carrier Concentration and Potential (2)
l We have an equation relating the potential to the carrier concentration
l If we integrate the above equation we have
l We define the potential reference to be intrinsic Si:
University of California, Berkeley
)()(ln)()(00
0000 xn
xnVxx th=-ff
inxnx == )(0)( 0000f
0
0
00 n
dnVndn
qkTd th
o =÷÷ø
öççè
æ=f
EE 105 Fall 2016 Prof. A. M. Niknejad
6
Carrier Concentration Versus Potential
l The carrier concentration is thus a function of potential
l Check that for zero potential, we have intrinsic carrier concentration (reference).
l If we do a similar calculation for holes, we arrive at a similar equation
l Note that the law of mass action is upheld
University of California, Berkeley
thVxienxn /)(
00)( f=
thVxienxp /)(
00)( f-=
2/)(/)(200
00)()( iVxVx
i neenxpxn thth == - ff
EE 105 Fall 2016 Prof. A. M. Niknejad
7
The Doping Changes Potentiall Due to the log nature of the potential, the potential changes
linearly for exponential increase in doping:
l Quick calculation aid: For a p-type concentration of 1016
cm-3, the potential is -360 mVl N-type materials have a positive potential with respect to
intrinsic Si
University of California, Berkeley
100
0
0
0
00 10
)(log10lnmV26)()(lnmV26
)()(ln)( xn
xnxn
xnxnVx
iith »==f
100
0 10)(logmV60)( xnx »f
100
0 10)(logmV60)( xpx -»f
Department of EECS University of California, Berkeley
Prof. Ali M. NiknejadProf. Rikky Muller
Revesed Biased PN Junctions
EE 105 Fall 2016 Prof. A. M. Niknejad
9
PN Junctions: Overviewl The most important device is a junction
between a p-type region and an n-type regionl When the junction is first formed, due to the
concentration gradient, mobile charges transfer near junction
l Electrons leave n-type region and holes leave p-type region
l These mobile carriers become minority carriers in new region (can’t penetrate far due to recombination)
l Due to charge transfer, a voltage difference occurs between regions
l This creates a field at the junction that causes drift currents to oppose the diffusion current
l In thermal equilibrium, drift current and diffusion must balance
n-type
p-type
ND
NA
− − − − − −
+ + + + ++ + + + ++ + + + +
− − − − − −− − − − − −
−V+
EE 105 Fall 2016 Prof. A. M. Niknejad
10
PN Junction Currentsl Consider the PN junction in thermal equilibriuml Again, the currents have to be zero, so we have
dxdnqDEqnJ o
nnn +== 000 µ
dxdnqDEqn o
nn -=00µ
dxdn
nqkT
ndxdnD
En
on
0
000
1-=
-=
µ
dxdp
pqkT
ndxdpD
Ep
op
0
000
1-==
µ
EE 105 Fall 2016 Prof. A. M. Niknejad
11
PN Junction Fields
n-typep-type
NDNA
)(0 xpaNp =0
d
i
Nnp2
0 =diffJ
0E
a
i
Nnn2
0 =
Transition Region
diffJ
dNn =0
– – + +
0E
0px- 0nx
EE 105 Fall 2016 Prof. A. M. Niknejad
12
Total Charge in Transition Regionl To solve for the electric fields, we need to write
down the charge density in the transition region:
l In the p-side of the junction, there are very few electrons and only acceptors:
l Since the hole concentration is decreasing on the p-side, the net charge is negative:
)()( 000 ad NNnpqx -+-=r
)()( 00 aNpqx -»r
0)(0 <xr0pNa >
00 <<- xxp
EE 105 Fall 2016 Prof. A. M. Niknejad
13
Charge on N-Sidel Analogous to the p-side, the charge on the n-side is
given by:
l The net charge here is positive since:
)()( 00 dNnqx +-»r 00 nxx <<
0)(0 >xr0nNd >
a
i
Nnn2
0 =
Transition Region
diffJ
dNn =0
– – + +
0E
EE 105 Fall 2016 Prof. A. M. Niknejad
14
“Exact” Solution for Fieldsl Given the above approximations, we now have an
expression for the charge density
l We also have the following result from electrostatics
l Notice that the potential appears on both sides of the equation… difficult problem to solve
l A much simpler way to solve the problem…
îíì
<<-<<--
@-
0/)(
/)(
0 0)(0)(
)(0
0
nVx
id
poaVx
i
xxenNqxxNenq
xth
th
f
f
r
s
xdxd
dxdE
erf )(0
2
20 =-=
EE 105 Fall 2016 Prof. A. M. Niknejad
15
Depletion Approximationl Let’s assume that the transition region is
completely depleted of free carriers (only immobile dopants exist)
l Then the charge density is given by
l The solution for electric field is now easyîíì
<<+<<--
@0
0 00
)(nd
poa
xxqNxxqN
xr
s
xdxdE
er )(00 =
)(')'()( 000
00
p
x
xs
xEdxxxEp
-+= ò- er
Field zero outsidetransition region
EE 105 Fall 2016 Prof. A. M. Niknejad
16
Depletion Approximation (2)l Since charge density is a constant
l If we start from the n-side we get the following result
)(')'()(0
00 po
s
ax
xs
xxqNdxxxEp
+-== ò- eer
)()()(')'()( 0000
000 xExxqNxEdxxxE n
s
dx
xs
nn +-=+= ò eer
)()( 00 xxqNxE ns
d --=e
Field zero outsidetransition region
EE 105 Fall 2016 Prof. A. M. Niknejad
17
Plot of Fields In Depletion Region
l E-Field zero outside of depletion regionl Note the asymmetrical depletion widthsl Which region has higher doping?l Slope of E-Field larger in n-region. Why?l Peak E-Field at junction. Why continuous?
n-typep-type
NDNA
– – – – –– – – – –– – – – –– – – – –
+ + + + +
+ + + + +
+ + + + +
+ + + + +
DepletionRegion
)()( 00 xxqNxE ns
d --=e
)()(0 pos
a xxqNxE +-=e
EE 105 Fall 2016 Prof. A. M. Niknejad
18
Continuity of E-Field Across Junction
l Recall that E-Field diverges on charge. For a sheet charge at the interface, the E-field could be discontinuous
l In our case, the depletion region is only populated by a background density of fixed charges so the E-Field is continuous
l What does this imply?
l Total fixed charge in n-region equals fixed charge in p-region! Somewhat obvious result.
)0()0(00
==-=-== xExqNxqNxE pno
s
dpo
s
an
ee
nodpoa xqNxqN =
EE 105 Fall 2016 Prof. A. M. Niknejad
19
Potential Across Junctionl From our earlier calculation we know that the
potential in the n-region is higher than p-regionl The potential has to smoothly transition from high
to low in crossing the junctionl Physically, the potential difference is due to the
charge transfer that occurs due to the concentration gradient
l Let’s integrate the field to get the potential:
ò- ++-=x
x pos
apo
p
dxxxqNxx0
')'()()(e
ffx
x
pos
ap
p
xxxqNx0
'2')(2
-÷÷ø
öççè
æ++=
eff
EE 105 Fall 2016 Prof. A. M. Niknejad
20
Potential Across Junctionl We arrive at potential on p-side (parabolic)
l Do integral on n-side
l Potential must be continuous at interface (field finite at interface)
20 )(
2)( p
s
ap
po xxqNx ++=
eff
20 )(
2)( n
s
dnn xxqNx --=
eff
)0(22
)0( 20
20 pp
s
apn
s
dnn xqNxqN f
ef
eff =+=-=
EE 105 Fall 2016 Prof. A. M. Niknejad
21
Solve for Depletion Lengthsl We have two equations and two unknowns. We are
finally in a position to solve for the depletion depths
20
20 22 p
s
apn
s
dn xqNxqN
ef
ef +=-
nodpoa xqNxqN =
(1)
(2)
÷÷ø
öççè
æ+
=da
a
d
bisno NN
NqN
x fe2÷÷ø
öççè
æ+
=ad
d
a
bispo NN
NqN
x fe2
0>-º pnbi fff
EE 105 Fall 2016 Prof. A. M. Niknejad
22
Sanity Checkl Does the above equation make sense?l Let’s say we dope one side very highly. Then
physically we expect the depletion region width for the heavily doped side to approach zero:
l Entire depletion width dropped across p-region
02lim0 =+
=¥®
ad
d
d
bis
Nn NNN
qNx
d
fe þ
a
bis
ad
d
a
bisNp qNNN
NqN
xd
fefe 22lim0 =÷÷ø
öççè
æ+
=¥®
EE 105 Fall 2016 Prof. A. M. Niknejad
23
Total Depletion Widthl The sum of the depletion widths is the “space
charge region”
l This region is essentially depleted of all mobile charge
l Due to high electric field, carriers move across region at velocity saturated speed
÷÷ø
öççè
æ+=+=
da
bisnpd NNqxxX 112000
fe
µ11012150 »÷øö
çèæ=
qX bisd
fecmV10
µ1V1 4=»pnE
EE 105 Fall 2016 Prof. A. M. Niknejad
24
Have we invented a battery?l Can we harness the PN junction and turn it into a
battery?
l Numerical example:
2lnlnlni
ADth
i
A
i
Dthpnbi n
NNVnN
nNV =÷÷
ø
öççè
æ+=-º fff
mV600101010logmV60lnmV26 20
1515
2 =´==i
ADbi n
NNf
?
EE 105 Fall 2016 Prof. A. M. Niknejad
25
Contact Potentiall The contact between a PN junction creates a
potential differencel Likewise, the contact between two dissimilar
metals creates a potential difference (proportional to the difference between the work functions)
l When a metal semiconductor junction is formed, a contact potential forms as well
l If we short a PN junction, the sum of the voltages around the loop must be zero:
mnpmbi fff ++=0
pnmnf
pmf
+
−bif )( mnpmbi fff +-=
EE 105 Fall 2016 Prof. A. M. Niknejad
26
PN Junction Capacitorl Under thermal equilibrium, the PN junction does
not draw any (much) currentl But notice that a PN junction stores charge in the
space charge region (transition region)l Since the device is storing charge, it’s acting like a
capacitorl Positive charge is stored in the n-region, and
negative charge is in the p-region:
nodpoa xqNxqN =
EE 105 Fall 2016 Prof. A. M. Niknejad
27
Reverse Biased PN Junctionl What happens if we “reverse-bias” the PN
junction?
l Since no current is flowing, the entire reverse biased potential is dropped across the transition region
l To accommodate the extra potential, the charge in these regions must increase
l If no current is flowing, the only way for the charge to increase is to grow (shrink) the depletion regions
+
−Dbi V+-f
DV 0<DV
EE 105 Fall 2016 Prof. A. M. Niknejad
28
Voltage Dependence of Depletion Width
l Can redo the math but in the end we realize that the equations are the same except we replace the built-in potential with the effective reverse bias:
÷÷ø
öççè
æ+
-=+=
da
DbisDnDpDd NNq
VVxVxVX 11)(2)()()( fe
bi
Dn
da
a
d
DbisDn
VxNN
NqN
VVxf
fe-=÷÷
ø
öççè
æ+
-= 1)(2)( 0
bi
Dp
da
d
a
DbisDp
VxNN
NqN
VVxf
fe-=÷÷
ø
öççè
æ+
-= 1)(2)( 0
bi
DdDd
VXVXf
-= 1)( 0
EE 105 Fall 2016 Prof. A. M. Niknejad
29
Charge Versus Biasl As we increase the reverse bias, the depletion
region grows to accommodate more charge
l Charge is not a linear function of voltagel This is a non-linear capacitorl We can define a small signal capacitance for small
signals by breaking up the charge into two terms
bi
DaDpaDJ
VqNVxqNVQf
--=-= 1)()(
)()()( DDJDDJ vqVQvVQ +=+
EE 105 Fall 2016 Prof. A. M. Niknejad
30
Derivation of Small Signal Capacitance
l Do a Taylor Series expansion:
l Notice that
!++=+ DV
DDJDDJ v
dVdQVQvVQ
D
)()(
RD VVbipa
VV
jDjj
VxqNdVd
dVdQ
VCC== ú
úû
ù÷÷ø
öççè
æ--===f
1)( 0
bi
D
j
bi
Dbi
paj V
CV
xqNC
fff -
=-
=112
00
da
da
bi
s
da
d
a
bis
bi
a
bi
paj NN
NNqNN
NqN
qNxqNC
+=÷÷
ø
öççè
æ+÷÷
ø
öççè
æ==
fefe
ff 22
220
0
EE 105 Fall 2016 Prof. A. M. Niknejad
31
Physical Interpretation of Depletion Cap
l Notice that the expression on the right-hand-side is just the depletion width in thermal equilibrium
l This looks like a parallel plate capacitor!
da
da
bi
sj NN
NNqC+
=fe20
0
1
011
2 d
s
dabissj XNN
qC efe
e =÷÷ø
öççè
æ+=
-
)()(
Dd
sDj VXVC e
=
EE 105 Fall 2016 Prof. A. M. Niknejad
32
A Variable Capacitor (Varactor)l Capacitance varies versus bias:
l Application: Radio Tuner
0j
j
CC
![%JTDPWFS$IFSSZ#MPTTPNTJOUIF+BQBOFTF$PVOUSZTJEFg ÊEï5T 7]F d&mD* ö±ö¹ö·ö¸ ö ö³ ö²ö ö±ö´ m ùö¹öµ. z ë 0 [®ú ê5Y ó N AøÊÓÿRå](https://static.fdocuments.in/doc/165x107/5e843f8dd7d2c168874f63c6/jtdpwfsifsszmpttpntjouifbqboftfpvousztjef-g-e5t-7f-dmd-.jpg)
![F:8 gb flash 31-05-2012Noorul Huda oor ul hudaTajdar-e ... Karbla.pdf · ÜÜÜnnnûûûÜûuuuuôôônô†††$$$$ÖÖ†Ö]]]Ö]àààôôôôÛÛÛFFFàFuuuuøøøÛø†††$$$$ÖÖ†Ö]]Ö]ääääôôô]ô×××###ÖÖ×Ö]]Ö](https://static.fdocuments.in/doc/165x107/5eab36cff01e01439d645ad5/f8-gb-flash-31-05-2012noorul-huda-oor-ul-hudatajdar-e-karblapdf-oeoeoennnoeuuuunaaaaffffuuuuaaaa.jpg)
