Ch-3 1
Chapter 3 p-n junctions
3.1.
Processing and classification of p-n junctions (diodes)
1. Processing techniques
Alloying
Diffusion
Ion implantation
Epitaxial growth using CVD and MBE
2. Junction interfacial profiles
Abrupt (Step function)
Graded (Linear grading; Complementary error function; Gaussian function)
3. Junction depths
Shallow junctions (for high speed and high frequency devices)
can be fabricated using low energy ion implantation, low-temperature CVD, or MBE.
Deep junction (conventional for high reverse bias voltage uses)
3.2.
Electrostatic analysis - potential diagrams
In this section, we will show how to use the Poisson equation and current density equations to calculate the
built-in electric field, ε, the potential band bending, φ, and the width of the depletion layer, xd.
3.2.1.
Depletion approximation
(1) In the space charge region adjacent to the junction interface, there are only uncompensated donor and
acceptor ions, ND+ and NA+, but not free carriers, n and p, i.e., all carriers have been depleted --- a so-called
depletion layer.
(2) At the boundaries of the space charge region, the charge density is abruptly changed.
3.2.2.
Equilibrium conditions - p-n junction under zero bias
Using the depletion approximation, we have n << ND+ and p << NA-. Under an equilibrium condition the
current density passing through a p-n junction is equal to zero, i.e.,
r kT ∂ p
j p = qµ p ( pE −
) = 0.
q ∂x
Hence, we find
Ex =
kT 1 ∂ p
dφ
=−
q p∂x
dx
(3.2.1)
(3.2.2)
Integrating the above equation over the entire junction region,
φn
− ∫ dφ =
φp
kT
q
∫
pn
pp
dp
,
p
(3.2.3)
we obtain
− (φn − φ p ) =
kT pn
ln
q
pp
(3.2.4)
Ch-3 2
Because of ni = nn pn = pp np, and assuming a full ionization, i.e., pp ≈ NA, and nn ≈ ND, we find
2
np = ni2/ ND,
pn = ni2/ NA.
(3.2.5)
(3.2.6)
Therefore,
Vb = (φ n − φ p ) =
kT ⎡ N A ( − x p ) N D ( x n ) ⎤ E g kT ⎡ N A N D ⎤
+
ln ⎢
ln ⎢
⎥=
⎥,
q ⎣
q
q ⎣ Nv Nc ⎦
ni2
⎦
(3.2.7)
and
nn / np = pp / pn = exp(qVb/kT),
(3.2.8)
where Vb is called the built-in voltage.
Fig. 3.1.1 Built-in voltage for one-side abrupt junction in Ge, Si, and GaAs.
3.2.3.
Non-equilibrium conditions - p-n junction under bias
In this case, the minority carrier density should be presented using the quasi-Fermi level. For holes at the nside of a junction we have
p(xn) = Nv exp{-[EFp - Ev(xn)]/kT},
(3.2.9)
and for electrons at the p-side we have
n(xp) = Nc exp{-[ Ec(xp)- EFn]/kT}.
(3.2.10)
The energy difference between the two quasi-Fermi levels for electrons and holes, respectively, is given by
EFn - EFp = Vb - VB ,
(3.2.11)
where VB is the external bias voltage.
3.2.4.
Distribution of the electric field, and potential in the space charge region
In the space charge region, the relationships among the charges, built-in electric field, and potentials are
represented by Poisson’s equation. For an abrupt p-n junction, if we only consider a 1-D case, we have
Ch-3 3
ρ( x)
d 2φ ( x)
=−
.
2
dx
ε ε0
(3.2.12)
In a general case, if all dopants are ionized, we can write the charge neutrality as below,
ρ(x) = q( ND - NA +p - n)
(3.2.13)
By using the depletion approximation, we find
⎧− qN A
⎪
ρ ( x ) = ⎨ qN D
⎪ 0
⎩
− xp ≤ x ≤ 0
0 ≤ x ≤ xn
x ≤ − x p and x ≥ x n
(3.2.14)
(1) Solution for the electric field
Substituting the charge density solution into Poisson’s equation gives the equations to be solved for the
electric field.
⎧ qN A / (ε ε 0 )
d 2φ ⎪
= ⎨− qN d / (ε ε 0 )
dx 2 ⎪
0
⎩
− xP ≤ x ≤ 0
0 ≤ x ≤ xn
x ≤ x n and x ≥ x n
(3.2.15)
Integrating from the depletion region edge to an arbitrary point x, one obtains the p-side of the depletion region
E ( x) = −
qN
dφ ( x )
= − ( A x + A)
dx
ε ε0
-xp ≤ x ≤ 0
(3.2.16)
Using the boundary condition E(-xp) = - dφ(x)/dx = 0, we obtain
A=
qN A
ε ε0
xp
(3.2.17)
Hence
E ( x) = −
qN A
ε ε0
( x p + x)
-xp ≤ x ≤ 0
(3.2.18)
( xn − x)
0 ≤ x ≤ xn
(3.2.19)
Similarly, we obtain
E ( x) = −
Discussion:
qN D
ε ε0
(i) At the junction interface, the continuity of the electric field is found to require
NA xp = ND xn
(3.2.20)
(ii) At the junction interface, E(x=0) = Emax = - qNA xp/(εε0). Therefore, the continuity of
the electric field ensures qNA xp = qND xn, which indicates that the numbers of positive or
negative charges at both sides of the space charge region are equal.
(iii) The electric field is linearly decayed from the junction interface to the edge of the
depletion region.
E(x) = Emax(1 - x/xp ) = [qNA /(εε0)](x - xp).
(3.2.21)
Ch-3 4
(2) Solution for the potential
Integrating Eq. 3.2.16, once again, the electrostatic potential can be obtained,
φ ( x) =
qN A 2 qN A
x +
x x+C.
2ε ε 0
ε ε0 p
-xp ≤ x ≤ 0
(3.2.22)
By setting the arbitrary reference potential equal to 0 at x = xp, i.e., φ(xp) = 0, C = qNA xp /(2εε0). Hence,
2
φ ( x) =
qN A
( x p + x) 2
2ε ε 0
-xp ≤ x ≤ 0
(3.2.23)
Similarly, on the n-side of the junction, using a boundary condition φ(xn) = Vb , we obtain
φ ( x ) = Vb −
Discussion:
qN D
ε ε0
( xn − x) 2
0 ≤ x ≤ xn
(3.2.24)
(i) The φ versus x dependence is quadratic in nature, with a concave curvature on the p-side
of the junction, and a convex curvature on the n-side.
(ii) The continuity of the electrostatic potential at x = 0 is found to require
qN A
ε ε0
x 2p = Vb −
qN D
ε ε0
x n2
(3.2.25)
(iii) The potential difference between p- and n sides is
φ[( xn-(-xp)] = φ( xn) - φ(-xp) =
q
2ε ε0
( N A x 2p + N D x n2 ) = Vb
(3.2.26)
(3) Solution for the width of the depletion layer
The total width of a depletion layer is
xm = xn - (-xp).
(3.2.27)
Following Eq. 3.1.20, we find
xn = xm NA /( NA + ND) ,
xp = xm ND /( NA + ND) .
(3.2.28)
(3.2.29)
Inserting these results into Eq. 3.2.26, while considering an external bias such that Vb → Vb - VB, we obtain
Vb - VB
=
qN A N D x m2
.
2ε ε 0 ( N A + N D )
(3.2.30)
Therefore,
1/ 2
⎡ 2ε ε0
N + ND ⎤
xm = ⎢
(Vb − VB ) A
⎥
N A ND ⎦
⎣ q
.
For single side p-n junctions, e.g., p+-n junctions where NA >> ND, we thus have
(3.1.31)
Ch-3 5
⎡ 2ε ε 0 (Vb − V B ) ⎤
xm ≈ xn = ⎢
⎥
qN D
⎣
⎦
1/ 2
.
(3.1.32)
⎡ 2ε ε 0 (Vb − V B ) ⎤
xm ≈ x p = ⎢
⎥ .
qN A
⎣
⎦
(3.1.33)
Similarly, for n +- p junctions, we find
1/ 2
It indicates that the depletion only occurs at the lowly doped side for the abrupt single side junctions.
The results of the electric field and electrostatic potential as a function of position for an abrupt p-n
junction are shown in Fig. 3.2.1..
Fig. 3.2.1 Step junction solution. Depletion approximation based quantitative solution for the
electrostatic variables in a pn step junction under equilibrium conditions (VB = 0). (a) Step
junction profile, (b) charge density, (c) electric field, and (d) electrostatic potential as a function of
position.
Ch-3 6
(4) In case of a linearly graded doping profile across the junction interface
The linearly graded profile is a more realistic approximation for junction formed by deep diffusion into
moderate or heavily doped substrates. As shown in Fig. 3.2.2, the linearly graded profile is mathematically modeled
by
N(x) = ND - NA = α x .
(3.1.34)
Hence, we have the built-in voltage across a linearly graded p-n junction,
Vb = 2
kT α x m
ln(
).
q
2ni
(3.1.35)
Fig. 3.2.2 Linearly graded solution. Depletion approximation based quantitative solution for the
electrostatic variables in a linearly graded junction under equilibrium conditions (VB = 0). (a)
linear graded profile, (b) charge density, (c) electric field, and (d) electrostatic potential as a
function of position.
Ch-3 7
Poisson’s equations:
d 2 φ ⎧ − qα x / ( ε ε 0 )
=⎨
dx 2 ⎩
0
− xm / 2 ≤ x ≤ xm / 2
x ≤ − x m / 2 and x ≥ x m / 2
(3.2.36)
-xm/2 ≤ x ≤ xm /2
(3.2.37)
The electric filed
x m2
2
qα 2 − x
E ( x) = −
,
2
ε ε0
and the maximum electric field at the junction interface
qα x m2
E max ( x = 0) = −
.
8ε ε0
(3.2.38)
By setting reference zero potential at xp = -xm/2, we find the electrostatic potential distribution
φ ( x) =
⎤
xm 2
qα ⎡ x m 3
3
⎢2 ( ) + ( ) x − x ⎥ .
2
6ε ε0 ⎣ 2
⎦
-xm/2 ≤ x ≤ xm /2
(32.39)
The voltage drop across the depletion region is equal to
Vb − VB =
2qα x m 3
( ) .
3ε ε0 2
(3.2.40)
Finally, we obtain the width of the depletion region
⎡12ε ε 0
⎤
xm = ⎢
(Vb − V B ) ⎥
⎣ qα
⎦
1/ 3
.
(3.2.41)
Fig. 3.2.3 Depletion-layer width and depletion capacitance per unit area as a function of doping
for one-side abrupt junction in Si.
(5) Comments on the depletion approximation
Ch-3 8
(i) The depletion approximation is a good description only for a symmetrical p-n junction, i.e., NA ≈ ND,
and under a high reverse bias. It is not suitable for single-side abrupt junctions. In those cases, the space
charge effect due to free carriers should be taken into account
(ii) There are actually no abrupt changes at the boundaries of the space charge region. Thin boundary
layers exist between the depletion region and neutral layers, which are characterized as the Debye
radius
L2dn =
2ε ε0 kT
,
q2 ND
(3.2.42)
L2dp =
2ε ε0 kT
,
q2 N A
(3.2.43)
Ldp and Ldp are in the order of 10-6 cm at room temperature.
3.2.5.*
Space charge effect
For asymmetrical abrupt p-n junctions (n+-p or p+-n), due to accumulation of free carriers there is an
inversion layer at the junction interface close to the lowly doped side. To solve the problem, the free carrier, p and
n, must be taken into account during calculations. In this case the charge densities in the space charge region are
ρ(x<0) = q(- NA + p - n),
ρ(x>0) = q( ND + p - n).
p-side
n-side
(3.2.44a)
(3.2.44b)
According to Eq. 3.2.2, we have
−
dφ kT 1 dp kT 1 d ln p
=
=
.
dx
q p dx
q p dx
(3.2.45)
By integrating the above equation, we find
φ=−
kT p
ln .
q
C
(3.2.46)
C can be determined, if we take a reference zero potential at a place where p = NA. Therefore,
p = NA exp(-qφ/kT).
(3.2.47)
With an external bias, we have
pn = NcNv exp[-(Ec-Ev+Efp’-Efn’)/kT] = ni2exp(-VB/kT)= NDNA exp[-(Vb-VB)/kT] ,
(3.1.48)
Therefore,
n = NDNA exp[-(Vb-VB)/kT] / p = ND exp[-(Vb-VB-φ)/kT].
(3.1.49)
The Poisson equations can thus be written as
Vb − V B − φ ⎫
d 2φ
q ⎧
φ
) ⎬ (p-side)
⎨− N A + N A exp( − ) − N D exp( −
2 = −
kT
kT
dx
ε ε0 ⎩
⎭
2
Vb − V B − φ ⎫
d φ
q ⎧
φ
(n-side)
⎬
⎨ N D + N A exp( − ) − N D exp( −
2 = −
kT
kT
dx
ε ε0 ⎩
⎭
(3.2.50)
(3.2.51)
Using the boundary conditions E(φ=0) = E(φ=Vb-VB) = 0
1/ 2
2 kT ⎧ q φ ⎡
qφ ⎤ N D ⎡
qφ
⎡ q (V b − V B ) ⎤ ⎫
⎤
E p ( x < 0) = −
− 1 − exp( −
) +
exp
− 1⎥ exp ⎢ −
⎨
⎥⎬
L D p q ⎩ kT ⎢⎣
kT ⎥⎦ N A ⎢⎣
kT
kT
⎦
⎣
⎦⎭
(3.2.52a)
Ch-3 9
1/ 2
En ( x > 0) = −
2 kT ⎧ q
q (Vb − VB − φ ) ⎤ N A ⎡
qφ
q (Vb − VB ) ⎤ ⎫
⎡
)⎥ +
exp( − ) − exp( −
)⎥⎬
⎨ (Vb − VB − φ ) − ⎢1 − exp( −
⎢
LDn q ⎩ kT
kT
kT
kT
⎣
⎦ ND ⎣
⎦⎭
(3.2.52b)
where, Ldp and Ldp are the Debye radius, i.e., Ldn =
2
2ε ε0 kT
2ε ε0 kT
2
and Ldp =
.
2
q ND
q2 N A
Fig. 3.2.4 Space charge effect for a strongly asymmetrical junction. (a) Energy band, (b) electric
field distribution, and (c) space charge distribution.
Because the electric field must be continuous at the junction interface (x = 0), we can then obtain the
maximum electric filed and the electrostatic potential at this point.
1/2
E max
qφ (0) ⎤ N D ⎡
qφ (0) ⎤
2 kT ⎧ qφ (0) ⎡
⎡ q (Vb − V B ) ⎤ ⎫
= E (0) = −
− ⎢1 − exp( −
)⎥ +
⎨
⎥⎬
⎢exp kT − 1⎥ exp ⎢−
L Dp q ⎩ kT
kT
N
kT
⎦
⎦
⎣
⎦⎭
⎣
A ⎣
(3.2.53)
and
φ (0) =
N A − N D kT
ND
(Vb − V B ) + (
)
N A + ND q
NA + ND
q (Vb − V B ) ⎤
⎡
)⎥ .
⎢1 − exp( −
kT
⎣
⎦
(3.2.54)
The fact that at x = 0, φ(0) ≠ 0 indicates that there is a charge accumulation at the junction interface. In an extreme
case, when NA >> ND,
φ(0) ≈ kT/q.
(3.2.55)
Insert this result into Eq.3.2.47, we find
p(0) = NA exp[-qφ(0)/kT] ≈ NA exp(-1) ≈ 0.368 NA,
(3.2.56)
Ch-3 10
indicating that the hole concentration at x = 0 has been very close to the doping concentration (NA) at the heavily
doped side and must be much larger than that (ND) at the lowly doped side. Therefore, a larger number of holes will
be accumulated at the n-side of the junction to form a p-type inversion layer . Such a charge accumulation changes
the value of the built-in voltage, i.e., Vb → Vb + n kT/q (n = 1-2). The resulted change of the width of the space
charge region is
xm =
2ε ε 0
nkT
(Vb −
− VB )
qN B
q
[β (V
= LD 2
b
(3.2.53)
]
− VB ) − n ,
(3.2.54)
where, β = q/(kT), and LD is the Debye radius.
3.3.
DC analysis (I-V characteristics)
In this section, we will show how to use the current density equations and continuity equations to calculate
the I-V characteristics of a p-n junction.
3.3.1.
Ideal junction equations
(1) Assumptions for an ideal p-n junction:
(i) Low resistance in the p- and n- regions, i.e., all voltage can be applied on the space charge region.
(ii) The length of p- or n-region is larger than the diffusion length of minority carriers, i.e., pn(x=W) = pn0.
(iii) No any generation-recombination processes occur within the space charge region.
(iv) np or pn << n or p (low injection condition).
(v) No surface recombination.
(2) Basic equation:
In the section 2.6, we have derived the following equations
j p ( x ) = qµ pn E x − qD p
∂ pn
∂x
∂ p n 1 ∂ j p p n − pn 0
=
−
∂t
q ∂x
τp
,
(3.3.1)
.
(3.3.2)
Inset Eq. 3.3.1 into 3.3.2, we find
∂ pn p n − pn 0
∂ pn
∂ 2 pn
= Dp
−
2 − µp E x
∂x
τp
∂t
∂x
.
(3.3.3)
By applying the first assumption, and also under a forward bias condition the 2nd term at the right side of the above
equation is very small, we can re-writhe Eq. 3.3.3 at the steady state, i.e., ∂p/∂t=0,
∂ 2 p n pn − p n 0
Dp
−
= 0,
τp
∂ x2
(3.3.4)
∂ 2 pn pn − p n 0
=
∂ x2
L2p
(3.3.5)
or
where Lp =
,
D pτ p is the diffusion length.
(3) Under forward bias (VB = VF > 0)
Ch-3 11
The solution of the Eq 3.3.5 is
pn - pn0 = Aexp[(x-xn)/Lp]+Bexp[-(x-xn)/Lp]
(3.3.6)
We have known
p = pn0exp[qVF/(kT)]
P = Pn0, (the second assumption)
at x = xn
at x = Wn
(3.3.7a)
(3.3.7b)
where, VF is the forward bias voltage. Hence, we have
pno{exp[qVF/(kT)]-1} = A + B,
A exp(Wn/Lp) + B exp(-Wn/Lp) = 0.
(3.3.8a)
(33.8b)
Fig. 3.3.1 Energy band diagram, intrinsic Fermi level (ψ), quasi-Fermi level, and carrier
distributions under (a) forward bias, and (b) reverse bias conditions.
We obtain
aV F
W − xn
) exp( − n
)
kT
Lp
,
Wn − x n
2 Sh(
)
Lp
− pn 0 (1 − exp
A=
aV F
W − xn
) exp( n
)
kT
Lp
.
Wn − x n
2 Sh(
)
Lp
(3.3.9a)
pn 0 (1 − exp
B=
Therefore, we find
(3.3.9b)
Ch-3 12
p = pn − pno = pn 0 (1 − exp
[
]
aVF Sh (Wn − x n ) / L p
)
.
kT
Sh(Wn + Ln )
(3.3.10)
The hole current density can then be written as
j p ≈ j pD = − qD p
∂ pn
∂x
x = xn
=
qD p pno
Lp
Cth(
Wn − x n ⎡
qV
⎤
) ⎢exp( F ) − 1⎥ ,
Lp
kT
⎣
⎦
(3.3.11)
Fig. 3.3.2 Ideal I-V characteristics of a diode. (a) Linear plot and (b) semilog plot.
Similarly, we obtain the electron current density at p-side
jn ≈
qDn n po
Ln
Cth(
Wp − x p ⎡
qV
⎤
) ⎢exp( F ) − 1⎥ .
Ln
kT
⎣
⎦
(3.3.12)
Therefore, the total current density passing through a p-n junction is
⎡ qD p pno
qDn n po
Wp − x p ⎤ ⎡
W − xn
qV
⎤
)+
) ⎥ ⎢exp( F ) − 1⎥
J F = j p + jn = ⎢
Cth(
Cth( n
Lp
Ln
Ln
kT
⎦
⎢⎣ L p
⎥⎦ ⎣
qVF
⎡
⎤
= J s ⎢exp(
) − 1⎥ .
(3.3.13)
kT
⎣
⎦
where, J s =
qD p pno
Discussion:
Lp
Cth(
qDn n po
Wp − x p
Wn − x n
)+
Cth(
) is called the saturation current.
Lp
Ln
Ln
(i) When Wn << Lp, Wp << Ln,
Ch-3 13
JF = (
qD p pno
Wn − x n
+
qDn n po ⎡
qV
qV
⎤
⎡
⎤
) ⎢exp( F ) − 1⎥ = J s* ⎢exp( F ) − 1⎥ . (3.3.14)
Wp − x p ⎣
kT
kT
⎦
⎣
⎦
(ii) When Wn >> Lp, Wp >> Ln,
JF = (
qD p pno
Lp
+
qDn n po ⎡
qV
qV
⎤
⎡
⎤
) ⎢exp( F ) − 1⎥ = J s** ⎢exp( F ) − 1⎥ . (3.3.15)
Ln
kT
kT
⎣
⎦
⎣
⎦
(4) Under reverse bias (VB = VR < 0)
Because VB = VR < 0, exp[qVR/(kT)] → 0. Therefore, we find
JR = - J s
(3.3.16)
The Eqs 3.3.13 and 3.3.16 are called (Shockley) ideal diode equations.
3.3.2
Real I-V characteristics
We will discuss non-ideal effects which influence the measured diode I-V characteristics.
(1) Generation-recombination process
According to the carrier recombination theory, at the steady state the carrier net recombination rate is
np − ni2
U R = R − Gth =
.
τ p (n − nt ) + τn ( p − pt )
(3.3.17)
To facilitate the discussion, we have made the following assumptions:
(i) The analysis is restricted at the boundary where there is an equality of the electron and hole densities,
i.e., n = p, and np = ni2 exp[qVF/(kT)] within the space charge region.
(ii) The life time of electrons is the same as that of holes, i.e., τt = τn = τp.
(iii) The energy level of the recombination center is located close to the middle of the bandgap. We thus
have nt ≈ pt ≈ ni, and nt = ni exp[(Et - Ei)/(kT)].
For the forward bias: Following the assumption (i), we can simplify Eq. 3.1.17 as
qVF
ni exp( kT − 1)
.
UR ≈
qVF
2τt
− 1)
exp(
2 kT
(3.3.18)
When VF > 2kT/q, the above equation can be further simplify as
UR ≈
ni
qV
exp( F ) .
2τt
2 kT
(3.3.19)
Hence, the recombination current density in the space charge region is
J RG = ∫ qU R dx = ∫
xm
0
xm
0
qni
qV
n
qV
qV
0
exp( F )dx = qxm i exp( F ) = J RG
exp( F ) .
2τ t
2 kT
2τ t
2 kT
2 kT
(3.3.20)
The total forward current density is a sum of the diffusion current density and the recombination current
density within the space charge region, i.e.,
Ch-3 14
J F = J D + J RG = J s exp(
qV F
qV
0
) + J RG
exp( F )
2 kT
kT
(3.3.21)
In practice, it is frequently more convenient to use an empirical formula
J F = J s − eff exp(
qV F
),
nkT
(3.3.22)
where n is an ideality factor.
For the reverse bias: VB = VR < 0, there is only generation current,
we thus obtain
0
J R = J s + J RG
= Js +
qx m ni
.
2τ t
(3.3.23)
(2) Surface recombination effects
(i) Surface recombination current, ISG .
ISR = qASp[ pn(x=0) - pn0]
(3.3.24)
(ii) Leakage current due to a surface inversion layer, Ii.
(iii) Surface leakage current due to absorption of water vapor and ions, etc., Is.
(3) Ccurrent due to defect assisted tunneling processes as reverse bias
If there are some defects (in particular dislocations) at the junction interface. Some defect states can be
introduced at energy positions close to the middle of the bandgap. Electrons and holes can thus tunnel through the
junction via these states under reverse bias, forming a tunneling current, ITu.
Therefore, the current terms for a real p-n junction should be
expressed as
P+
JF = JD + JRG + JSG + Ji + Js
(3.3.25)
JR = Js0 + JRG0 + Jtu
(3.3.26)
n(x)
nn
pn
(4) High current injection condition (pn ≥ nn)
At low injection we have
p(xn)nn = ni2 exp[qVF/(kT)].
n
p(x)
xn
Wn
Fig. 3.3.3 High current injection
(3.3.27)
At high-injection, there is a large hole density injecting from the p+- to n-region. To keep the charge neutrality, the
same amount of electron will also flow from the contact to n-region. We thus have
p(xn) - pn = n(xn) - nn
(3.3.28)
where pn is the equilibrium (minority) hole density in the n-region, which is very low, and can be ignored. Hence,
we find
n(xn) = nn + p(xn).
(3.3.29)
Eq. 3.3.27 has thus to be re-written as
p(xn)[ nn + p(xn)] = ni2 exp[qVF/(kT)],
or
(3.3.30)
Ch-3 15
p(xn)[ 1 + p(xn)/ nn ] =( ni2/ nn )exp[qVF/(kT)].
(3.3.31)
Since p(xn)/ nn >> 1, we thus have
p(xn) = ni exp[qVF/(2kT)].
(3.3.32)
Therefore, we find at high injection
JF ∝ exp[qVF/(2kT)]
(3.3.33)
(4) Series-resistance
In this case, there are voltage drops across the series resistance, VR = IRs. The applied voltage across the
depletion region is thus reduced to
VF = VB - IRs
(3.3.34)
Fig. 3.3.4 I-V characteristics of a practical diode. (a) Generation-recombination current region,
(b) diffusion current region, (c) high injection region, (d) series resistance effect, and (e) reverse
leakage current due to generation-recombination and surface effect.
3.4.
AC analysis
3.4.1
Small signal admittance
For the practical uses, we need to add an AC modulation signal on the DC bias.,
V = V0 + V1 exp(iωt).
(3.4.1)
Therefore, the hole density (in case of a p+-n junction) injecting to the n-region contains both AC and DC
components. Assuming the AC signal is a sine or cosine function, we can write
p(x,t) = p0(x) + p1(x,t) = p0(x) + p1(x)exp(iωt),
and its derivative to the time is
(3.4.2)
Ch-3 16
∂ p
= iω p1
∂t
(3.4.3)
Insert Eqs. 3.4.2 and 3.4.3 into the continuity equation 3.3.3, we can obtain two equations. One is dealing with the
DC component, which has the same form as Eq. 3.3.4, and another is dealing with the AC component,
Dp
∂ 2 p1 p1
−
= iω p1 ,
∂ x2 τ p
(3.4.4)
or
∂ 2 p1 1
−
(1 − iω τ p ) p1 = 0 .
∂ x 2 L2p
(3.4.5)
The solution of the above equation is in the form of
[
]
[
]
p1 ( x ) = A1 exp − ( x − x n )(1 + iω τ p ) 1/ 2 / L p + B1 exp ( x − x p )(1 + iω τ p ) 1/ 2 / L p (3.4.6).
We need to find the boundary conditions. It is evident at x =
Following Eq. 3.4.1, the injecting hole density at x = xn is
∞ , p1 = 0. How about anther one at x = xn.
p(xn,t) = pn exp{[ q( V0 + V1) exp(iωt)]/kT}.
(3.4.7)
For the small signal, the Eq. 3.4.7 can be extended as a series, and we only take the first-order term as an
approximation, we thus find
p(xn,t) = pn exp[ qV0 /(kT)] [(1+q V1)/(kT) exp(iωt)]
= p0(xn) + p1(xn)exp(iωt),
(3.4.8)
i.e.,
p1(xn) = pn [qV1/(kT)] exp[qV0 /(kT)].
(3.4.9)
Using two boundary conditions at x = xn, and x =∞, respectively, we obtain
A1 = pn [qV1/(kT)] exp[qV0 /(kT)],
(3.4.10a)
B1 = 0.
(3.4.10b)
Therefore, the solution 3.4.6 becomes
p1 ( x ) = pn
⎡ ( x − xn )(1 + iω τ p )1/ 2 ⎤
qV1
qV
exp( o ) exp ⎢−
⎥
kT
kT
Lp
⎥⎦
⎢⎣
(3.4.11)
The AC component of the hole (diffusion) current density passing through the junction is thus written as
jF = jp = − qD p
∂ [ p1 ( x ) exp(iω t )]
∂x
Ch-3 17
⎡ ( x − x n )(1 + iω τ p ) 1/ 2 ⎤
⎤
⎡ qD p pno q
qVo
1/ 2
= A⎢
exp(
)(1 + iω τ p ) ⎥ ⋅ exp ⎢−
⎥ ⋅ V1 exp(iω t ) .
kT
Lp
⎥⎦
⎢⎣ L p kT
⎢⎣
⎥⎦
(3.4.12)
Define the small signal admittance, Y, as
Y=
qD p pno q
j F ( x − xn )
qV
q
=A
exp( o )(1 + iω τ p ) 1/ 2 = I F
(1 + iω τ p ) 1/ 2
L p kT
kT
kT
V1 exp(iω t )
(3.4.13)
When (ωτp)2 << 1,
Y≈
τp
qI F
(1 + iω
),
kT
2
(3.4.14)
and define Gd = qIF/(kT). Hence, we find
Y = G d + iω G d (
τp
2
),
(3.4.15)
where, Gd(τp/2) is a value of the diffusion capacitance.
3.4.2
Diffusion capacitance
As we have discussed in the preceding section, the image part of the admittance actually represents the
change of the minority carrier charge density within the diffusion length as a function of the external bias. The
charge density in the diffusion region is increasing with increasing VB, and it in decreasing when VB is decreasing,
which is similar to the charging and discharging processes of a capacitor. Therefore, we call is diffusion
capacitance.
In this section, we will give further discussion about the physical meaning of the diffusion capacitance.
Following Fig. 3.4.1, the charge density in the diffusion region (x = xn → Ln) is presented as
Q=
qV
1
⎤
⎡
AqL p pn 0 ⎢exp( o ) − 1⎥ .
2
kT
⎦
⎣
(3.4.16)
When changing the bias voltage, dV0, the change of the injecting hole density is
dQ =
qV
1
q
AqL p pn 0 ( ) exp( o )dV0
2
kT
kT
(3.4.17)
Ln xp xn
Lp
p+
n
Therefore,
qV
dQ 1
q
= AqL p pn 0 ( ) exp( o )
dV0 2
kT
kT
qV
q τ p AqD p pn 0
= (
)( )
exp( o )
kT 2
Lp
kT
Cdiff =
= (
τp
q τp
)( ) I F = Gd ( )
2
kT 2
p
dQ
pn0
(3.4.18)
x
Fig. 3.4.1 Diffusion capacitance
of a p+-n junction
Ch-3 18
3.4.3
Depletion capacitance
In the preceding sections, we discussed the capacitance introduced in the carrier diffusion region. On the
other hand, the capacitance effect also exits in the depletion region, which is called depletion capacitance.
dQd ε ε0
A
=
dVB
xm
where, xm = xn + xp .
Cd =
(3.4.19)
Fig. 3.4.2 (a) Voltage-dependence of the differential conductance and (b) capacitance of a Si pn
junction at zero frequency.
For an example, for a p+-n junction,
xm ≈ xn =
Hence, we find
2ε ε 0
(V − VB )
qN D b
(3.4.20)
Ch-3 19
Cd = A
ε ε 0 qN D
2(Vb − VB )
.
(3.4.21)
The measurements of the depletion capacitance as a function of the bias voltage can be used to determine
the doping concentration versus the junction depth. In this case, we shall re-write Eq. 3.4.21 as
Cd2 = A 2
ε ε 0 qN D
2(Vb − V B )
.
(3.4.22)
Differentiating the above equation with respect to V, we find
ND =
Cd3
dCd
qε ε0 A
dV
,
(3.4.23)
2
xm =
ε ε0
Cd
A.
(3.4.24)
The dopant depth profile can then be obtained from the above two equations.
For p-n junction with a linearly graded profile, the depletion capacitance should be present as
⎡ q (ε ε 0 ) 2 α ⎤
Cd = A ⎢
⎥
⎣12(Vb − V B ) ⎦
3.4.4
1/ 3
(3.4.25)
Small-signal equivalent circuit of a p-n junction
Cgeom = εε0A/L
p
n
Fig. 3.4.3 Equivalent circuit of a p-n junction
3.5.
Transient response
The transient response between the switch-on (forward bias) and switch-off (reverse bias) states of a diode
is an important characteristic to show whether the device can be used for the high speed (digital) circuits
3.5.1
Charge storage effect (ts)
According to Fig. 3.5.1, at position “1”, the diode is at the on-state
I1 = I F =
V F − Von V F
≈
.
RF
RF
At position “2”, the diode is at the off-state
(3.5.1)
Ch-3 20
I2 = −I R = −
VR + v A
0< t < t s
RR
≈−
VR
.
RR
(3.5.2)
Where, ts is the charge storage time.
For a p+-n junction, using the continuity equation, we have
Dp
d2p ∂ p p
=
+ .
∂ x2 ∂ t τ p
(3.5.3)
Multiplying qAdx at the both sides of the Eq. 3.5.3, and integrating it over the entire n-region, we obtain
⎡∂ p
qAD p ⎢
⎢⎣ ∂ x
x =Wn
−
∂ p
∂x
⎤ ∂
x =0 ⎥ =
⎥⎦ ∂ t
∫
Wn
0
qApn 0 dx +
1
τp
∫
Wn
0
qApn 0 dx
(3.5.4)
or
[
]
dQ p
A J p (0) − J p (Wn ) =
dt
+
Qp
τp
.
(3.5.5)
If Wn >> Lp, Jp(Wn) ≈ 0. Therefore,
AJ p (0) =
dQ p
dt
+
Qp
τp
.
(3.5.6)
Fig. 3.5.1.The turn-off transient. (a) Idealized representation of the awitching circuit, (b) sketch
and charaterization of the current-time transient, and (c) voltage-time transient.
Ch-3 21
(1) t < 0
the p-n junction is at the steady state, i.e., I = I1, ∂Qp/∂t = 0. Hence, we find
I1 =
Qp
τp
.
(3.5.7)
(2) 0 < t < ts
i = - I2 = AJ p (0) =
dQ p
dt
+
Qp
τp
.
(3.5.8)
The solution of the above differential equation is
Qp(t) = - τpI2 + C exp(-t/τp).
(3.5.9)
Because at t = 0, Qp(0) = τpI1, we obtain
C = τp(I1 + I2).
(3.5.10)
Therefore, we find
Qp(t) = - τpI2 + τp(I1 + I2) exp(-t/τp).
(3.5.11)
At t=ts, the stored charge is Qp(ts). Wishing to err on the conservative side (i.e., obtain an estimate of ts that is too
large), we take Qp(ts) to be approximately zero. Therefore, we have
ts = τp ln(1 + I1/ I2).
Remarks:
(3.5.12)
The charge storage time in a p-n junction is primary determined by (i) the minority carrier life
time, τp; (ii) the forward injection current, I1; and (iii) the reverse extraction current I2.
As a point of information, a more precise analysis, based on a complete ∆pn(x,t) solution and properly
accounting for the residual stored charge at t=ts, gives
erf (
3.5.2
ts
τp
)=
1+
1
IR
.
(3.5.13)
IF
Reverse recovering time (tr = ts + tf)
Define the reverse recovering time as
t r = t s + t f,
(3.5.14)
where, tf is called the fall time. The mathematics treatment for this part is very complicated. Here, only results are
shown:
Ch-3 22
I2
),
I1 + I 2
tf
exp( − )
t s = τ p erf
erf
tf
τp
−1
(
τp
+
tf
π
(3.5.15)
= 1 + 01
.(
I2
).
I1
(3.5.16)
τp
For a p+-n junction,
when Wn >> Lp
ts + t f ≈
τ p I2
(
2 I1
) −1 ,
(3.5.17)
when Wn << Lp
ts + t f ≈
3.6.
Wn I 2 −1
( ) .
2 Dp I1
(3.5.18)
Junction breakdown (VR > VBR, IR → ∞)
The followings are three main breakdown mechanisms of p-n junctions.
1. Breakdown due to thermal dissipation
I R ∝ T 3 exp(
2.
E g0
kT
)
(3.6.1)
Breakdown due to the tunneling effect (Zener breakdown)
The meachanism of breakdown for Si and Ge junctions with breakdown voltages less than about 4Eg/q is
found to be due to tunneling effect. The tunneling current density is given by
It =
r
(2m pn ) 1/ 2 q 5/ 2 E av ⋅ V ⋅ A
h 2 E g1/ 2
⎡ 8π (2m pn ) 1/ 2 q 1/ 2 E g3/ 2 ⎤
⎥,
r
exp ⎢−
3h E av
⎢⎣
⎥⎦
(3.6.2)
where, mpn is the reduced effective mass
mpn = 2(1/mn + 1/mlhp)-1,
and
3.
r
1/ 2
r
E max ⎡ q (Vb − V B ) N A N D ⎤
E av =
=⎢
⎥
2
⎣ 2ε ε 0 ( N A + N D ) ⎦
(3.6.3)
(3.6.4)
Avalanche breakdown (due to impact ionization processes)
Avalanche multiplication (or impact ionization) is the most important mechanism for junction breakdown,
which sets an upper limit on the reverse bias for most diodes (VBR ≥ 4Eg/q). For abrupt (one-side) p-n junctions
Ch-3 23
VBR =
r
E BR ⋅ x m
r
=
2
ε ε 0 E BR
2
2qN D
,
(3.6.5)
(a)
(b)
Fig. 3.6.1 Temperature-dependence of I-V characteristics of tunneling breakdown (a) and
Avalanche braeakdown .
For linearly graded p-n junctions
V BR =
r
2 E BR
3
=
r
4 E BR
3
3/ 2
(
2ε ε 0 1/ 2
) .
qα
(3.6.6)
It is evident that the junction is broken down , when the maximum electric field in the junction exceeds some critical
r
r
value, E BR . For different semiconductors, the E BR values at 300 K are listed below.
Ge
Si
GaAs
SiC
100 kV/cm
300 kV/cm
400 kV/cm
2300 kV/cm.
Ch-3 24
Fig. 3.5.2 Avalanche breakdown voltage vs. doping concetration for one-side junction in Ge, Si <100>oriented GaAs and GaP. The dashed line indicates the maximum doping beyond which the tunneling
mechanism will dominates the breakdown characteristics.