Semiconductor Devices A brief review

1
Semiconductor DevicesA brief review

• Dr. K. Fobelets

2
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?

3
The most frequently used sentence in this course
will be

• Excess minority carrier concentration

4
Structure

• 1. Lectures 10 hrs
• Basic principles based on QA 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

5
Review

• Electrons and holes
• Minority and majority carriers
• Energy band diagram

6
Free charged carriers in Si
Covalent bond
Intrinsic Si
Movement kT
Si
Si
Si
Si
Si
Si
Si
p-type
n-type
7
In semiconductors two types of free charged
carriers exist electrons and holes. Q1 What
are holes?

1. Spherical voids in a semiconductor
2. A positively charged Si atom that has lost its
   electron
3. A positively charged particle that is the result
   of quantum mechanics

8
C The two charged particles describe together the
conduction in semiconductors.
Electron e- with charge q-e
and mass mn m0 mn Hole h with charge
qe and mass mp m0 mp
9
Intrinsic silicon (Si) has a small number of both
free electrons and holes such that nipi. 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?

1. n p ni pi
2. n gt ni p lt ni
3. n gt p gt ni
4. p gt n gt ni

n electron concentration p hole
concentration ni intrinsic electron
concentration pi intrinsic hole concentration
10
B n gt ni p lt ni in an n-type semiconductor.
n-type semiconductor n ND p ni2/ND
p-type semiconductor n ni2/NA p NA
11
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).
12
TRUE
p-type semiconductor
p
n
gt
p
p
p-type semiconductor
p-type semiconductor
hole concentration
electron concentration
n-type semiconductor
n
p
gt
n
n
n-type semiconductor
n-type semiconductor
electron concentration
hole concentration
MAJORITY CARRIERS
MINORITY CARRIERS
13
Drift and diffusion

• Two types of carrier movement
• As a result of an electric field ? DRIFT
• As a result of a carrier gradient ? DIFFUSION

14
Drift of carriers under influence of an electric
field E
E
-

E
-

15
Diffusion of carriers due to a carrier gradient
x
16
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?
(1)
(2)

1. Term (1) is drift current and (2) diffusion
   current
2. Term (2) is drift current and (1) diffusion
   current
3. Only term (1) can exist in a semiconductor
4. Only term (2) can exist in a semiconductor

17
A
Drift current is proportional to the carrier
concentration and the electric field Diffusion
current is proportional to the carrier gradient.
E(x) Jndrift
Jpdrift   n(x) Jndiff   p(x)
Jpdiff
18
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?
E
-
E
-


a)
c)
(b)
(d)
-

-

E
E
19
B
When 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.
20
Q6 Solve diffusion processes
n

1. Draw arrows indicating the direction of diffusion
   of carriers.
2. Identify the type of carriers that is diffusing.

21
Solution
p
n
p
Holes
Electrons
22
p
n
p
Q7 Why is there no net current while diffusion
is happening?

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.

23
E
E
-

-

p
n
p
Holes
Holes
diffusion
drift
Electrons
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.
24
Depletion
n-Si
ND
p-Si
NA
Si
Si
Si
B
-

As
B
Capacitive effect
Capacitive effect
-

Si
As
B
-


-
E
E
25
Q8 True - False
Ec
EF
Ev
The position of the Fermi level EF determines the
type of the semiconductor.
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Q9 Multiple choice
Ec
EF
Ev

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.

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Ec
Bottom of conduction band
EF
EG
Bandgap. No energy levels in this energy region.
Intrinsic level. Is the position of the Fermi
level EF when the semiconductor is intrinsic.
Ei
Ev
Top of valence band
Position of Fermi level is determined by the
doping type and density For n-type Si
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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.

29
Energy band diagram

• e.g.
• p-Si n-Si
• p-Si n-Si p-Si
• It is possible to start from the knowledge on
  workfunctions, f and the energy reference the
  vacuum level, Evac. The workfunction is dependent
  on the doping concentration!

30
Evac
Evac
n-Si
p-Si
EF
EF
31
Evac
p-Si
EF
32
Depleted region on both sides
Evac
Evac
p-Si
n-Si
Ec
Ec
EF
EF
Ev
Ev
33
Diffusion and drift can occur at the same time.
A charge packet
E
Both also always occur across junctions
34
A look at the short pn-diode

• PN diode

I
p
n
V
35

• Short PN diode

DIFFUSION
I
p
n
V
36

• Short PN diode

DIFFUSION
I
p
n
V
37

• Short PN diode

DIFFUSION
I
How do we find the current?
p
n
V
Linear variation of minority carrier concentration
Apply diffusion current formula to the minority
carrier variation
38

• Short PN diode

I
p
n
V
Only few carriers can contribute to the current
39
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

40
But what happens in a long pn diode?
Short
41
In long semiconductors recombination of the
minority carriers will occur whilst diffusing
42
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 Lp

Injection of carriers
x
Lp
43
Generation-recombination

• Generation of carriers and recombination is
  continuously happening at the same time such that
  the equilibrium carrier concentrations are
  maintained.

RG
Charge neutral
44
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 RgtG

When there is a shortage, then G gt R
45
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
Recombination of h in n-type semiconductor
46
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?
47
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.

48
Mathematical description of diffusion and
recombination
Jp(x)
Jp (xDx)
x
xDx
x
49
Mathematical description of diffusion and
recombination
bulk defined excess concentration
Jp total current drift diffusion
Neglect drift current (no electric field applied)
50
Mathematical description of diffusion and
recombination
bulk defined excess concentration
51
Solve equation in steady state
Boundary conditions
General solution of 2nd order differential
equation
52
Too complicated

• Short approximation

• Long approximation

Xn ltlt Lp
Xn gtgt Lp
EXPONENTIAL
LINEAR
53
Short semiconductor

• Xn Lp carriers do not have time to recombine
  (t8) !
• Taking linear approximation.

pn
NO recombination variation of the excess
carrier concentration linear
54
Diffusion and recombination

• Xn gtgt Lp carriers do have time to recombine (tlt8)
  !
• Taking exponential approximations

dpn(x)
pn(x)
pn
pn(x)pn0
Dp
When recombination occurs and Xn gtgt Lp variation
of the excess carrier concentration is exponential
?pn(x)
pn0
Xn
Lp
0
x
dpn(Xn)0
Contact imposes
55
dpn still too complex for quick calculations

• Take really extreme case
• Xn gtgtgt Lp or Xn ? 8

dpn(x)
dpn(x)
Note I and Q of both expressions of
for the same I for
same as for linear
approximation when XnLp
Xn ? 8
dpn(x)
56
Diffusion and recombination

• Xn gtgtgt Lp carriers do have time to recombine
  (tlt8) !
• Taking exponential approximations

pn(x)
pn
pn(x)pn0Dp e-x/Lp
When recombination occurs and Xn ? 8 variation of
the excess carrier concentration is exponential
Dp
?pn(x)
pn0
8
Lp
Imposes
dpn(Xn)0
0
x
57
SHORT ? LONGapproximation
LpXn200nm
Lp200 nm, Xn20nm
dpn(x)
dpn(x)
Correct solution Exponential solution Linear
solution
Boundary of short
Short
x
x
Lp200 nm, Xn1000nm
Lp200 nm, Xn400nm
dpn(x)
dpn(x)
Long
Intermediate
x
x
58

• 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.

59
Carrier injections forward bias

• Carrier injection across junction

• Creates minority carrier concentration gradients

np0ni2/NA ppNA pn0 ni2/ND nnND
60
Carrier injections reverse bias

• Minority carriers are swept across junction Vlt0

x

• Small amount of minority carriers ? small current

61
Thus
Dnp
Dpn
Dpn pn0 (eeV/kT -1)
Dnp np0 (eeV/kT -1)
62
Two methods to calculate current
-wp
wn
0
I
x
dpn
dnp
Dpn
Dnp
-x
0
x
0

1. Gradient excess carrier concentration
2. Re-supply of recombined excess charge

63
1. Excess carrier concentration gradient
Maximum diffusion currents at the edges of the
transition region
dnp
dpn
Slope
Dpn
Dnp
-x
0
x
0
-wp
wn
64
1. Excess carrier concentration gradient
Fill in expression for excess carrier
concentration
e-
h
Ip
In
65
Changing gradient!?Changing diffusion current
density
p
n
Itot
Ip
In
ItotIn Ip
66
2. Re-supply of recombined excess carriers
-wp
wn
0
I
x
np
pn
dnp Dnp e-(-x)/Ln
dpn Dpn e-(x)/Lp
Dpn
Qn
Dnp
Qp
pn0
-x
0
x
0
Excess carrier charge Q recombines every t
seconds (carrier life time). For steady state Q
has to be re-supplied every t seconds ? current
67
2. Re-supply of recombined excess carriers
Charge minority carrier life time ratio
dnp
dpn
Ip
dnp Dnp e-(-x)/Ln
In
dpn Dpn e-(x)/Lp
Dpn
Qn
Dnp
Qp
pn0
-x
x
0
0
-wp
wn
0
Charge area under excess carrier concentration
integrate -8 and 8 are the contacts excess
charge 0!
Qn -e A ?-80dnp dx In Qn/tn e A Ln Dnp
/tn
Qp e A ?08dpn dx Ip Qp/tp e A Lp Dpn /tp
68
Total current
Same equation as short diode with length exactly
equal to the minority carrier diffusion lengths

• 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

69
SHORT ? LONGapproximation error on current
calculationratio of currents
Error on linear and exponential approximation
same when XnLp
70

• Non-idealities in the pn diodes

Log(I)
ideal
c)
b)
a)
V
71

• (a) Low voltage low injection of carriers

Log(I)
ideal
real
a)
V
V
72
(c) High voltage high injection of carriers
Log(I)
ideal
real
np pp
c)
pn nn
V

1. n2
2. n1
3. n2

73
(d) Higher currents
Log(I)
ideal
real
d)
Current determined by resistance
V
74
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.

75
Switching off
Steady state snap shots
How do we go from this
pno
Excess carrier concentration
Off NO current flows!!!
76
Variation of the excess carrier concentration as
a function of time.dp(x,t)
Relationship for charge Qp
77
Transient during switching off
i(t) I dQ/dt Q/t dQ/dt
t gt 0 0 Q/t dQ/dt
Q(t)Ion te-t/t
Since no current in off, charge has to
disappear by recombination!
78
Transient during switching offvariation of the
excess carrier concentration as a function of time
Qp(t)eA?dp(x,t)dxIptpe-t/tp
A voltage, vd will exists across the diode as
long as charge remains
dp(x,t)Dp(vd(t)) e-x/Lp
79
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

80
ON-OFF (open circuit) take pn ? Itot Ip
p
n
Ip
R
t0
V
81
OFF (open circuit) ? ON take pn ? Itot Ip
p
n
Ip
R
t0
V
82
Reverse recovery transient
Switch the diode from forward to reverse bias
Steady state snap shots
How do we go from this
Excess carrier concentration
Reverse bias current flows!!!
83
Transients when switching to reverse bias
-Ir
84
Storage delay time tsd
If
i(t)
v(t)
t
-Ir
Time required for the stored charge to disappear
tsd tminority carrier ln(1 If/Ir)
85
Calculate storage delay time tsd
IF
i(t)
v(t)
!
X
t
-IR
86
Calculated storage delay time tsd
IF
i(t)
v(t)
integrate
t
-IR
87
Calculated storage delay time tsd
IF
i(t)
v(t)
t
-IR
88
After tsd
IF
i(t)
v(t)
t
Build-up of depletion region
-IR
89
Small signal equivalent circuit

• Junction capacitance

• Diffusion capacitance

• Diffusion capacitance

• Due to charge storage effects

• Due to depletion region

• Cj e A/w

• Cd dQ/dV d (I t)/dV
• e/kT I t

• w function of bias
• ? C voltage variable capacitance

• Important in reverse bias

• Important in forward bias

90
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

Only linear circuit elements present
91
Large signal equivalent circuit
Rs
C
Reverse bias depletion capacitance Forward bias
diffusion capacitance
Non-linear circuit elements present
92
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.

93
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

Lifetime of minority carrier holes
94
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.

95
Revision

• Short

linear
Xn Lp
pn
96
Revision

• Long

exponential
Xn gtgtgt Lp
pn(x)pn0
pn(x)
pn
pn(x)pn0Dp e-x/Lp
Dp
?pn(x)
pn0
8
Lp
Imposes
dpn(Xn)0
0
x
97
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 dpn(x) for Xn ? 8 is the
same 2. I for
is same as for linear approximation when XnLp
dpn(x)
98
Errors on current
Correct Exponential Linear
Short good approximation up to Xn Lp
Long good approximation up to Xn gt 5 Lp