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.,

j q pEkTq

pxp p= −µ

∂∂

(r

) = 0. (3.2.1)

Hence, we find

EkTq p

px

ddxx = =

1−

∂∂

φ (3.2.2)

Integrating the above equation over the entire junction region,

− =∫ ∫dkTq

dppp

n

p

n

p

pφ

φ

φ, (3.2.3)

we obtain

− − =( ) lnφ φn pn

p

kTq

pp

(3.2.4)

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Because of ni

2 = nn pn = pp np, and assuming a full ionization, i.e., pp ≈ NA, and nn ≈ ND, we find np = ni

2/ ND, (3.2.5) pn = ni

2/ NA. (3.2.6) Therefore,

Vb = ( ) ln( ) ( )

lnφ φn pA p D n

i

g A D

v c

kTq

N x N xn

Eq

kTq

N NN N

− =−⎡

⎣⎢

⎤

⎦⎥ = +

⎡

⎣⎢

⎤

⎦⎥2 , (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 n-side 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

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d x

dxx2

20

φ ρε ε

( ) ( )= − . (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

(3.2.14) ρ( )xqN x x

qN x xx x and x x

A p

D

p n

=− − ≤ ≤

≤ ≤≤ − ≥

⎧

⎨⎪

⎩⎪

00

0n

(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.

ddx

qN x xqN x x

x x and x x

A P

d

n n

2

2

0

0

00

0

φε εε ε=

− ≤ ≤− ≤

≤ ≥

⎧

⎨⎪

⎩⎪

/ ( )/ ( ) 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 xd x

dxqN

x AA( )( )

(= − = − + )φ

ε ε0 -xp ≤ x ≤ 0 (3.2.16)

Using the boundary condition E(-xp) = - dφ(x)/dx = 0, we obtain

AqN

xAp=

ε ε0 (3.2.17)

Hence

E xqN

x xAp( ) ( )= − +

ε ε0 -xp ≤ x ≤ 0 (3.2.18)

Similarly, we obtain

E xqN

x xDn( ) ( )= − −

ε ε0 0 ≤ x ≤ xn (3.2.19)

Discussion: (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)

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(2) Solution for the potential Integrating Eq. 3.2.16, once again, the electrostatic potential can be obtained,

φε ε ε ε

( )xqN

xqN

x x CA Ap= +

2 0

2

0+ . -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 xp2/(2εε0). Hence,

φε ε

( ) ( )xqN

x xAp= +

2 0

2 -xp ≤ x ≤ 0 (3.2.23)

Similarly, on the n-side of the junction, using a boundary condition φ(xn) = Vb , we obtain

φε ε

( ) ( )x VqN

x xbD

n= − −0

2 0 ≤ x ≤ xn (3.2.24)

Discussion: (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

x VqN

xAp b

Dnε ε ε ε0

2

0

2= − (3.2.25)

(iii) The potential difference between p- and n sides is

φ[( xn-(-xp)] = φ( xn) - φ(-xp) = q

N x N xA p D n2 0

2

ε ε( + 2 ) = 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) , (3.2.28) xp = xm ND /( NA + ND) . (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 N x

N NA D m

A D

2

02ε ε ( )+. (3.2.30)

Therefore,

xq

V VN NN Nm b B

A D

A D= −

+⎡

⎣⎢

⎤

⎦⎥

2 01 2

ε ε( )

/

. (3.1.31)

For single side p-n junctions, e.g., p+-n junctions where NA >> ND, we thus have

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x xV V

qNm nb B

D≈ =

−⎡

⎣⎢

⎤

⎦⎥

2 01 2

ε ε ( )/

. (3.1.32)

Similarly, for n +- p junctions, we find

x xV V

qNm pb B

A≈ =

−⎡

⎣⎢

⎤

⎦⎥

2 01 2

ε ε ( )/

. (3.1.33)

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.

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(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,

VkTq

xnb

m

i= 2

2ln( )

α. (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.

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Poisson’s equations:

ddx

q x x x xx x and x x

m m

m m

2

20 2 2

0 2φ α ε ε=

− − ≤ ≤≤ − ≥

⎧⎨⎩

/ ( ) / // / 2

(3.2.36)

The electric filed

E xq

xxm

( ) = −−α

ε ε0

22

22

, -xm/2 ≤ x ≤ xm /2 (3.2.37)

and the maximum electric field at the junction interface

E xq xm

max ( )= = −08

2

0

αε ε

. (3.2.38)

By setting reference zero potential at xp = -xm/2, we find the electrostatic potential distribution

φαε ε

( ) ( ) ( )xq x x

x xm m= +⎡

⎣⎢

⎤

⎦⎥6

22 20

3 2 − 3 . -xm/2 ≤ x ≤ xm /2 (32.39)

The voltage drop across the depletion region is equal to

V Vq x

b Bm− =

23 20

3αε ε

( ) . (3.2.40)

Finally, we obtain the width of the depletion region

xq

V Vm b= −⎡

⎣⎢

⎤

⎦⎥

12 01 3

ε εα

( )/

B . (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

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(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

LkT

q NdnD

2 02

2=

ε ε, (3.2.42)

LkT

q NdpA

2 02

2=

ε ε, (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), p-side (3.2.44a) ρ(x>0) = q( ND + p - n). n-side (3.2.44b) According to Eq. 3.2.2, we have

− = =ddx

kTq p

dpdx

kTq p

d pdx

φ 1 1 ln. (3.2.45)

By integrating the above equation, we find

φ = −kTq

pC

ln . (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] = ni

2exp(-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

ddx

qN N

kTN

V VkTA A D

b B2

20

φε ε

φ φ= − − + − − −

− −⎧⎨⎩

⎫⎬⎭

exp( ) exp( ) (p-side) (3.2.50)

ddx

qN N

kTN

V VkTD A D

b B2

20

φε ε

φ φ= − + − − −

− −⎧⎨⎩

⎫⎬⎭

exp( ) exp( (n-side) (3.2.51)

Using the boundary conditions E(φ=0) = E(φ=Vb-VB) = 0

E xL

kTq

qkT

qkT

NN

qkT

q V VkTp

D p

D

A

b B( ) exp( ) exp exp( ) /

< = − − − −⎡⎣⎢

⎤⎦⎥+ −

⎡⎣⎢

⎤⎦⎥

−−⎡

⎣⎢⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

02

1 11 2

φ φ φ

(3.2.52a)

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E xL

kTq

qkT

V Vq V V

kTNN

qkT

q V VkTn

Dnb B

b B A

D

b B( ) ( ) exp(( )

) exp( ) exp(( )

)/

> = − − − − − −− −⎡

⎣⎢⎤⎦⎥+ − − −

−⎡⎣⎢

⎤⎦⎥

⎧⎨⎩

⎫⎬⎭

02

11 2

φφ φ

(3.2.52b)

where, Ldp and Ldp are the Debye radius, i.e., LkT

q NdnD

2 02

2=

ε ε and L

kTq Ndp

A

2 02

2=

ε ε.

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.

E EL

kTq

qkT

qkT

NN

qkT

q V VkTDp

D

A

b Bmax ( )

( )exp(

( )) exp

( )exp

( )= = − − − −

⎡⎣⎢

⎤⎦⎥+ −

⎡⎣⎢

⎤⎦⎥

−−⎡

⎣⎢⎤

⎦⎥⎧⎨⎩

⎫⎬⎭

02 0

10 0

11/2

φ φ φ

(3.2.53) and

φ( ) ( ) ( ) exp(( )

)0 1=+

− +−+

− −−⎡

⎣⎢⎤⎦⎥

NN N

V VN NN N

kTq

q V VkT

D

A Db B

A D

A D

b B . (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

xqN

VnkT

qVm

Bb= −

2 0ε ε( B− )

]

(3.2.53)

= [L V VD b B2 β( )− − 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 x q p E qDpxp n x p

n( ) = −µ∂∂

, (3.3.1)

∂∂

∂∂ τ

pt q

jx

p pn p n n

p= −

−1 0 . (3.3.2)

Inset Eq. 3.3.1 into 3.3.2, we find

∂∂

∂∂

µ∂∂ τ

pt

Dpx

Epx

p pnp

np x

n n n

p= − −

−2

20 . (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,

Dpx

p pp

n n n

p

∂∂ τ

2

20 0−

−= , (3.3.4)

or

∂∂

2

20

2

px

p pL

n n n

p=

−, (3.3.5)

where Lp = Dp pτ is the diffusion length.

(3) Under forward bias (VB = VF > 0)

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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

at x = xn p = pn0exp[qVF/(kT)] (3.3.7a) at x = Wn P = Pn0, (the second assumption) (3.3.7b) where, VF is the forward bias voltage. Hence, we have pno{exp[qVF/(kT)]-1} = A + B, (3.3.8a) A exp(Wn/Lp) + B exp(-Wn/Lp) = 0. (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

Ap

aVkT

W xL

ShW x

L

nF n

p

n n

p

=

− − − n−

−

0 1

2

( exp ) exp( )

( ), (3.3.9a)

Bp

aVkT

W xL

ShW x

L

nF n

p

n n

p

=

− n−

−

0 1

2

( exp ) exp( )

( ). (3.3.9b)

Therefore, we find

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[ ]

p p p paVkT

Sh W x L

Sh W Ln no nF n n p

n n= − = −

−

+0 1( exp )( ) /

( ). (3.3.10)

The hole current density can then be written as

j j qDpx

qD pL

CthW x

LqVkTp pD p

nx x

p no

p

n n

p

Fn

≈ = − =−

−⎡⎣⎢

⎤⎦⎥=

∂∂

( ) exp( ) 1 , (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

jqD n

LCth

W xL

qVkTn

n po

n

p p

n

F≈−

−⎡⎣⎢

⎤⎦⎥

( ) exp( ) 1 . (3.3.12)

Therefore, the total current density passing through a p-n junction is

J j jqD p

LCth

W xL

qD nL

CthW x

LqVkTF p n

p no

p

n n

p

n po

n

p p

n

F= + =−

+−⎡

⎣⎢⎢

⎤

⎦⎥⎥

−⎡⎣⎢

⎤⎦⎥

( ) ( ) exp( ) 1

= JqVkTs

Fexp( ) −⎡⎣⎢

⎤⎦⎥

1 . (3.3.13)

where, JqD p

LCth

W xL

qD nL

CthW x

Lsp no

p

n n

p

n po

n

p p

n=

−+

−( ) ( ) is called the saturation current.

Discussion: (i) When Wn << Lp, Wp << Ln,

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JqD pW x

qD nW x

qVkT

JqVkTF

p no

n n

n po

p p

Fs

F=−

+−

−⎡⎣⎢

⎤⎦⎥= −

⎡⎣⎢

⎤⎦⎥

( ) exp( ) exp( )*1 1 . (3.3.14)

(ii) When Wn >> Lp, Wp >> Ln,

JqD p

LqD n

LqVkT

JqVkTF

p no

p

n po

n

Fs

F= + −⎡⎣⎢

⎤⎦⎥= −

⎡⎣⎢

⎤⎦⎥

( ) exp( ) exp(**1 1) . (3.3.15)

(4) Under reverse bias (VB = VR < 0) Because VB = VR < 0, exp[qVR/(kT)] → 0. Therefore, we find JR = - Js (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

U R Gnp n

n n p pR thi

p t n= − =

−− + −

2

τ τ( ) ( t ). (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 = ni

2 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

Un

qVkT

qVkT

Ri

t

F

F≈

−

−2

1

21τ

exp( )

exp( ). (3.3.18)

When VF > 2kT/q, the above equation can be further simplify as

Un qV

kTRi

t

F≈2 2τ

exp( ) . (3.3.19)

Hence, the recombination current density in the space charge region is

J qU dxqn qV

kTdx qx

n qVkT

JqV

kTRG R

x i

t

Fx

mi

t

FRG

Fm m= = = =∫ ∫0 0

0

2 2 2 2 2τ τexp( ) exp( ) exp( ) . (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 J J JqVkT

JqV

kTF D RG sF

RGF= + = +exp( ) exp( )0

2 (3.3.21)

In practice, it is frequently more convenient to use an empirical formula

J JqVnkTF s eff

F= − exp( ) , (3.3.22)

where n is an ideality factor. For the reverse bias: VB = VR < 0, there is only generation current, we thus obtain

J J J Jqx n

R s RG sm i

t= + = +0

2τ. (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+ n JF = JD + JRG + JSG + Ji + Js

(3.3.25) JR = Js

0 + JRG0 + Jtu (3.3.26) n(x)

nn p(x) (4) High current injection condition (pn ≥ nn) pn

At low injection we have xn Wn Fig. 3.3.3 High current injection p(xn)nn = ni

2 exp[qVF/(kT)]. (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)] = ni

2 exp[qVF/(kT)], (3.3.30) or

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), (3.4.2) and its derivative to the time is

Ch-3 16

∂∂

ωpt

i p= 1 (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,

Dpx

pi pp

p

∂∂ τ

ω2

12

11− = , (3.4.4)

or

∂∂

ω τ2

12 2 1

11

px L

i pp

p− − =( ) 0 . (3.4.5)

The solution of the above equation is in the form of

[ ] [ ]p x A x x i L B x x i Ln p p p p1 11 2

11 21 1( ) exp ( )( ) / exp ( )( ) // /= − − + + − +ω τ ω τ p (3.4.6).

We need to find the boundary conditions. It is evident at x = ∞ , p1 = 0. How about anther one at x = xn. Following Eq. 3.4.1, the injecting hole density at x = xn is 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

p x pqVkT

qVkT

x x iLn

o n p

p1

11 21

( ) exp( )exp( )( ) /

= −− +⎡

⎣⎢⎢

⎤

⎦⎥⎥

ω τ (3.4.11)

The AC component of the hole (diffusion) current density passing through the junction is thus written as

jF = jp = [ ]

− qDp x i t

xp

∂ ω∂

1 ( ) exp( )

Ch-3 17

= AqD p

Lq

kTqVkT

ix x i

LV ip no

p

op

n p

pexp( )( ) exp

( )( )exp( )/

/

11

1 21 2

1+⎡

⎣⎢⎢

⎤

⎦⎥⎥⋅ −

− +⎡

⎣⎢⎢

⎤

⎦⎥⎥⋅ω τ

ω τω t .

(3.4.12) Define the small signal admittance, Y, as

Yj x x

V i tA

qD pL

qkT

qVkT

i IqkT

iF n p no

p

op F p=

−= + =

( )exp( )

exp( )( ) ( )/ /

1

1 2 1 21 1ω

ω τ ω τ+

(3.4.13) When (ωτp)2 << 1,

YqIkT

iF p≈ +(1

2ω )

τ, (3.4.14)

and define Gd = qIF/(kT). Hence, we find

Y G i Gd dp

= + ωτ

( )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 AqL pqVkTp n

o=⎡⎣⎢

⎤⎦⎥

12

10 exp( ) − . (3.4.16)

When changing the bias voltage, dV0, the change of the injecting hole density is

dQ AqL pqkT

qVkT

dVp no=

12 0 ( ) exp( ) 0 (3.4.17) Ln xp xn Lp

p+ n Therefore,

CdQdV

AqL pqkT

qVkTdiff p n

o= =0

0

12

( ) exp( ) p dQ

= ( )( ) exp( )qkT

AqD pL

qVkT

p p n

p

oτ2

0 pn0

= ( )( ) ( )qkT

I GpF d

pτ τ2 2

= (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.

CdQdV x

Add

B m= =

ε ε0 (3.4.19)

where, xm = xn + xp .

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,

x xqN

V Vm nD

b B≈ = −2 0ε ε

( ) (3.4.20)

Hence, we find

Ch-3 19

C AqN

V VdD

b B=

−ε ε0

2( ). (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

C AqN

V VdD

b B

2 2 0

2=

−ε ε( )

. (3.4.22)

Differentiating the above equation with respect to V, we find

NC

q AdCdV

Dd

d=

3

02ε ε

, (3.4.23)

xC

Amd

=ε ε0 . (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

C Aq

V Vdb B

=−

⎡

⎣⎢

⎤

⎦⎥

( )( )

/ε ε α0

2 1 3

12 (3.4.25)

3.4.4 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

I IV V

RVRF

F on

F

F

F1 = =

−≈ . (3.5.1)

At position “2”, the diode is at the off-state

Ch-3 20

I IV v

RVRR

R A t t

R

R

R

s

2

0= − = −

+≈ −

< <. (3.5.2)

Where, ts is the charge storage time. For a p+-n junction, using the continuity equation, we have

Dd p

xpt

pp

p

2

2∂∂∂ τ

= + . (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

qADpx

px t

qAp dx qAp dxp x W x n

W

pn

W

n

n n∂∂

∂∂

∂∂ τ= =−

⎡

⎣⎢⎢

⎤

⎦⎥⎥= +∫ ∫0 00 00

1 (3.5.4)

or

[ ]A J J WdQdt

Qp p n

p

p( ) ( )0 − = +

τp

. (3.5.5)

If Wn >> Lp, Jp(Wn) ≈ 0. Therefore,

AJdQdt

Qp

p

p( )0 = +

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

IQp

p1 = τ

. (3.5.7)

(2) 0 < t < ts

i = - I2 = AJdQdt

Qp

p

p( )0 = +

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). (3.5.12) Remarks: 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

erft

II

s

p R

F

( )τ

=+

1

1. (3.5.13)

3.5.2 Reverse recovering time (tr = ts + tf) Define the reverse recovering time as tr = ts + tf, (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

t erfI

I Is p=+

−τ 1 2

1 2

( ) , (3.5.15)

erft

t

tII

f

p

f

p

f

p

ττ

πτ

+

−

= +

exp( ). ( )1 01 2

1. (3.5.16)

For a p+-n junction, when Wn >> Lp

t tIIs f

p+ ≈ −

τ2

2

1

1( ) , (3.5.17)

when Wn << Lp

t tWD

IIs f

n

p+ ≈ −

22

1

1( ) . (3.5.18)

3.6. Junction breakdown (VR > VBR, IR → ∞) The followings are three main breakdown mechanisms of p-n junctions. 1. Breakdown due to thermal dissipation

I TEkTR

g∝ 3 0exp( ) (3.6.1)

2. 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

Im q E V A

h Em q E

h Etpn av

g

pn g

av

=⋅ ⋅

−⎡

⎣⎢⎢

⎤

⎦⎥⎥

( )exp

( )/ /

/

/ / /2 8 2

3

1 2 5 2

2 1 2

1 2 1 2 3 2r

rπ

, (3.6.2)

where, mpn is the reduced effective mass mpn = 2(1/mn + 1/mlhp)-1, (3.6.3) and

r

r

EE q V V N N

N Navb B A D

A D= =

−+

⎡

⎣⎢

⎤

⎦⎥

max/

( )( )2 2 0

1 2

ε ε (3.6.4)

3. 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

VE x E

qNBRBR m BR

D

=⋅

=

r r

2 20

2ε ε

, (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

VE E

qBRBR BR= =

23

43

23 2

0 1 2

r r /

/( )ε εα

. (3.6.6)

It is evident that the junction is broken down , when the maximum electric field in the junction exceeds some critical

value, rEBR . For different semiconductors, the

rEBR values at 300 K are listed below.

Ge 100 kV/cm Si 300 kV/cm GaAs 400 kV/cm SiC 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.