Lecture 16 The pn Junction Diode (III)

Outline • Small-signal equivalent circuit model • Carrier charge storage –Diffusion capacitance

Reading Assignment: Howe and Sodini; Chapter 6, Sections 6.4 - 6.5

6.012 Spring 2007 Lecture 16 1 I-V Characteristics

Diode Current equation:

⎡ V ⎤ I = I ⎢ e(Vth )−1⎥ o ⎢ ⎥ ⎣ ⎦

I lg |I|

0.43 q kT =60 mV/dec @ 300K Io

0 0 V 0 V Io linear scale semilogarithmic scale

6.012 Spring 2007 Lecture 16 2 2. Small-signal equivalent circuit model Examine effect of small signal adding to forward bias: ⎡ ⎛ qV()+v ⎞ ⎤ ⎛ qV()+v ⎞ ⎜ ⎟ ⎜ ⎟ ⎢ ⎝ kT ⎠ ⎥ ⎝ kT ⎠ I + i = Io ⎢ e −1⎥ ≈ Ioe ⎢ ⎥ ⎣ ⎦ If v small enough, linearize exponential characteristics: ⎡ qV qv ⎤ ⎡ qV ⎤ ()kT (kT ) (kT )⎛ qv ⎞ I + i ≈ Io ⎢e e ⎥ ≈ Io ⎢e ⎜ 1 + ⎟ ⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎝ kT⎠ ⎦⎥

qV qV qv = I e()kT + I e(kT ) o o kT

Then: qI i = • v kT

From a small signal point of view. Diode behaves as conductance of value: qI g = d kT

6.012 Spring 2007 Lecture 16 3 Small-signal equivalent circuit model

gd

gd depends on bias. In forward bias:

qI g = d kT

gd is linear in diode current.

6.012 Spring 2007 Lecture 16 4 Capacitance associated with depletion region:

ρ(x)

+ qNd p-side − n-side (a) xp x = xn vD VD − qNa = − QJ qNaxp

ρ(x)

+ qNd p-side −x −x n-side (b) p p x xn xn

= + > > vD VD vd VD-- − qNa x < x |q | < |Q | p p, J J = − qJ qNaxp

= ∆ ∆ρ = ρ − ρ qj qNa xp (x) (x) (x) + qNd X p-side d n-side (c) x n xn − − xp xp x

q = q − Q > j j j 0 − qN = −qN x − −qN a − = − ∆ a p ( axp) qj qNd xn = − qNa (xp xp) ∆ = qNa xp

Depletion or junction capacitance:

dqJ C j = C j (VD ) = dvD VD

qεsNa Nd C j = A 2()Na + Nd ()φB −VD

6.012 Spring 2007 Lecture 16 5 Small-signal equivalent circuit model

gd Cj

can rewrite as:

qεsNa Nd φB C j = A • 2()Na + Nd φB ()φB −VD

C or, C = jo j V 1− D φB φ Under Forward Bias assume V ≈ B D 2 C j = 2C jo

Cjo ≡ zero-voltage junction capacitance

6.012 Spring 2007 Lecture 16 6 3. Charge Carrier Storage: diffusion capacitance What happens to majority carriers?

Carrier picture thus far:

ln p, n

Na Nd po no

p n ni2 ni2 Nd Na 0 x

If QNR minority carrier concentration ↑ but majority carrier concentration unchanged? ⇒ quasi-neutrality is violated.

6.012 Spring 2007 Lecture 16 7 Quasi-neutrality demands that at every point in QNR: excess minority carrier concentration = excess majority carrier concentration

carrier concentrations (cm−3)

(p-type) (n-type)

= + = + pp(x) Na np(x) nn(x) Nd pn(x)

p = N po a

metal n = N metal contact to no d contact to p region n region

= . VD/Vth pn(xn) pno e

pn(x)

− = . VD/Vth np( xp) npo e

np(x)

0 0 − −x x x Wp − = p n = Wn np( Wp) npo pn(Wn) pno Mathematically:

pn (x) − pno = nn (x) − nno

Define integrated carrier charge: 1 q = qA ()p (x ) − p •()W − x Pn 2 n n no n n 2 W − x n ⎡ qV ⎤ = qA n n i e kT −1 =−q ⎣⎢ ⎦⎥ Nn 2 Nd

6.012 Spring 2007 Lecture 16 8 Now examine small increase in V:

carrier concentrations (cm−3)

p-type n-type

nn(xn) − [for V + v ] pp( xp) D d [for V + v ] D d

nn(xn) −dq dq Nn P p [for VD] − pp( xp) [for VD]

p (x ) n n + [for VD vd] metal n (−x ) metal contact to p p contact to + [for VD vd] p region dq n region pn(xn) Pn [for VD] −dq Np − np( xp)

[for V ] D 0 0 − − x Wp xp xn Wn

Small increase in V ⇒ small increase in qPn ⇒ small increase in |qNn |

Behaves as capacitor of capacitance:

2 ⎡ qVD ⎤ dq W − x n q ⎢ ⎥ C = Pn = qA n n i e⎣ kT ⎦ dn dV 2 N kT VD d

6.012 Spring 2007 Lecture 16 9 Similarly for p-QNR:

2 ⎡ qVD ⎤ dq W − x n q ⎢ ⎥ C = Np = qA p p i e⎣ kT ⎦ dp dV 2 N kT VD a

Both capacitors sit in parallel ⇒ total diffusion capacitance:

Cd = Cdn + Cdp

Complete small-signal equivalent circuit model for diode:

gd Cj Cd

6.012 Spring 2007 Lecture 16 10 Bias dependence of Cj and Cd:

C

Cd

C Cj

0 0 V

•Cj dominates in reverse bias and small forward bias 1 ∝ φB − V

•Cd dominates in strong forward bias

⎡ qV ⎤ ⎢ ⎥ ∝ e⎣ kT ⎦

6.012 Spring 2007 Lecture 16 11 What did we learn today?

Summary of Key Concepts

Large and Small-signal behavior of diode: • Diode Current: qV [kT ] I = Io(e − 1) • Conductance: associated with current-voltage characteristics

–gd ∝ I in forward bias,

–gd negligible in reverse bias

• Junction capacitance: associated with charge modulation in depletion region 1 C j ∝ φB − V • Diffusion capacitance: associated with charge storage in QNRs to maintain quasi-neutrality. ⎡ qV ⎤ ⎢ ⎥ ⎣ kT ⎦ Cd ∝ e

6.012 Spring 2007 Lecture 16 12