Lecture 11
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Lecture 11
OUTLINE• pn Junction Diodes (cont’d)
– Narrow-base diode– Junction breakdown
Reading: Pierret 6.3.2, 6.2.2; Hu 4.5
Introduction
Lecture 11, Slide 2
• The ideal diode equation was derived assuming that the lengths of the quasi-neutral p-type & n-type regions (WP’ , WN’) are much greater than the minority-carrier diffusion lengths (Ln , Lp) in these regions.
Excess carrier concentrations decay exponentially to 0. Minority carrier diffusion currents decay exponentially to 0.
• In modern IC devices, however, it is common for one side of a pn junction to be shorter than the minority-carrier diffusion length, so that a significant fraction of the “injected” minority carriers reach the end of the quasi-neutral region, at the metal contact.
Recall that p = n = 0 at an ohmic contact
In this lecture we re-derive the diode I-V equation with the boundary condition that p = 0 at a distance xc’ (rather than ) from the edge of the depletion region.
EE130/230A Fall 2013
Excess Carrier Distribution (n side)• From the minority carrier diffusion equation:
• For convenience, let’s use the coordinate system:
• So the solution is of the form:
• We have the following boundary conditions:
)1()0'( / kTqVnon
Aepxp
22
2
p
n
pp
nn
L
p
D
p
dx
pd
pp LxLxn eAeAxp /'
2/'
1)'(
Lecture 11, Slide 3
0)''( cn xxp
xc'
0 x’x’’ 0
EE130/230A Fall 2013
• Applying the boundary conditions, we have:
• Therefore
• Since this can be rewritten as
• We need to take the derivative of pn’ to obtain the hole diffusion current within the quasi-neutral n region:
'
//
/'/'/
0 '0 ,)1()'( ''
''
cLxLx
LxxLxxkTqV
nn xxee
eeepxp
PcPc
PcPc
A
0)(
)1()0(/
2/
1'
/21
''
pcpc
A
LxLxcn
kTqVnon
eAeAxp
epAAp
2sinh ee
''
'/
0 '0 ,/sinh
/'sinh)1()'( c
Pc
PckTqVnn xx
Lx
Lxxepxp A
Lecture 11, Slide 4x
xpqDJ n
pP
)(
EE130/230A Fall 2013
Thus, for a one-sided p+n junction (in which the current is dominated by injection of holes into the n-side) with a short n-side:
)1(0 kTVq AeII Pc
Pc
D
i
P
P
Lx
LxNn
LDqAI
/sinh
/cosh'0 '
'2
where
Pc
PcPkTqV
npp Lx
LxxL
epqDJ A
/sinh
/cosh1
1 /0
2cosh ee where
Lecture 11, Slide 5
Pc
PckTVq
D
i
P
p
xp Lx
Lxe
N
n
L
DqJ A
/sinh
/cosh)1(
2
0
Evaluate Jp at x=xn (x’=0) to find the injected hole current:
EE130/230A Fall 2013
0 as sinh 0 as 1cosh 2
Therefore if xc’ << LP:
For a one-sided p+n junction, then:
and
D
i
c
p
c
P
D
i
P
p
c
P
Pc
Pc
Pc
Pc
N
n
x
DqA
x
L
N
n
L
DqAI
x
L
Lx
Lx
Lx
Lx
22
0
2
/
/1
/sinh
/cosh
Lecture 11, Slide 6EE130/230A Fall 2013
pn is a linear function:
Jp is constant
(No holes are lost due to recombination as they diffuse to the metal contact.)
'/
0'
'/
0
'
'/
0
'1)1(
/
/')1(
/sinh
/'sinh)1()'(
A
A
c
kTqVn
Pc
PckTqVn
Pc
PckTqVnn
x
xep
Lx
Lxxep
Lx
Lxxepxp
A
Lecture 11, Slide 7
Excess Hole Concentration Profile
)1( / kTqVno
Aep
x'0 x'c0
slope is constant
pn(x)
If xc’ << LP:
EE130/230A Fall 2013
General Narrow-Base Diode I-V• Define WP‘ and WN’ to be the widths of the quasi-neutral regions.
• If both sides of a pn junction are narrow (i.e. much shorter than the minority carrier diffusion lengths in the respective regions):
11 /0
/2
kTqVkTqV
AP
N
DN
Pi
AA eIeNW
D
NW
DqAnI
Lecture 11, Slide 8
xJN
xn-xp
JJP
e.g. if hole injection into the n side is greater than electron injection into the p side:
EE130/230A Fall 2013
Summary: Narrow-Base Diode• If the length of the quasi-neutral region is much shorter than the
minority-carrier diffusion length, then there will be negligible recombination within the quasi-neutral region and hence all of the injected minority carriers will “survive” to reach the metal contact.– The excess carrier concentration is a linear function of distance.
For example, within a narrow n-type quasi-neutral region:
The minority-carrier diffusion current is constant within the narrow quasi-neutral region.
Shorter quasi-neutral region steeper concentration gradient higher diffusion current
)1( / kTqVno
Aep
x
pn(x)
xn0
location of metal contact(pn=0)
WN’
EE130/230A Fall 2013 Lecture 11, Slide 9
pn Junction Breakdown
A Zener diode is designed to operate in the breakdown mode:
Breakdown voltage, VBR VA
Lecture 11, Slide 10EE130/230A Fall 2013
C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-10
Review: Peak E-Field in a pn Junction
DA
DA
Si
Abi
Si
nD
Si
pA
Si
NN
NNVVqxqNxqN
dx
2 )0(
xxn-xp
E(x)
E(0)
Si
Abi NVVq
2
)0(
where N is the dopant concentration on the lightly doped side
For a one-sided junction,
Lecture 11, Slide 11EE130/230A Fall 2013
Breakdown Voltage, VBR
• If the reverse bias voltage (-VA) is so large that the peak electric field exceeds a critical value ECR, then the junction will “break down” (i.e. large reverse current will flow)
• Thus, the reverse bias at which breakdown occurs is
biCRs
BR VqN
V 2
2
s
BRbiCR
VVqN
2
Lecture 11, Slide 12EE130/230A Fall 2013
Avalanche Breakdown Mechanism
if VBR >> VbiHigh E-field:
Low E-field:ECR increases slightly with N:
For 1014 cm-3 < N < 1018 cm-3, 105 V/cm < ECR < 106 V/cm
qNV CRsBR 2
2
Lecture 11, Slide 13EE130/230A Fall 2013
R. F. Pierret, Semiconductor Device Fundamentals, Figure 6.12
Tunneling (Zener) Breakdown Mechanism Dominant breakdown mechanism when both sides of a junction are very heavily doped.
VA = 0 VA < 0
biCRs
BR VqN
V 2
2
V/cm 106CRTypically, VBR < 5 V for Zener breakdown
Ec
Ev
Lecture 11, Slide 14EE130/230A Fall 2013C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-12
Empirical Observations of VBR
• VBR decreases with increasing N
• VBR decreases with decreasing EG
Lecture 11, Slide 15EE130/230A Fall 2013
R. F. Pierret, Semiconductor Device Fundamentals, Figure 6.11
VBR Temperature Dependence
• For the avalanche mechanism: – VBR increases with increasing T, because the mean free
path decreases
• For the tunneling mechanism: – VBR decreases with increasing T, because the flux of
valence-band electrons available for tunneling increases
Lecture 11, Slide 16EE130/230A Fall 2013
Summary: Junction Breakdown• If the peak electric field in the depletion region exceeds a
critical value ECR, then large reverse current will flow.
This occurs at a negative bias voltage called the breakdown voltage, VBR:
where N is the dopant concentration on the more lightly doped side
• The dominant breakdown mechanism isavalanche, if N < ~1018/cm3
tunneling, if N > ~1018/cm3
biCRs
BR VqN
V 2
2
Lecture 11, Slide 17EE130/230A Fall 2013
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