Lecture 16 - The pn Junction Diode (II) Equivalent Circuit Model ...

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-1

Lecture 16 - The pn Junction Diode (II) Equivalent Circuit Model November 3, 2005

Contents: 1. I-V characteristics (cont.) 2. Small-signal equivalent circuit model 3. Carrier charge storage: diffusion capacitance Reading assignment: Howe and Sodini, Ch. 6, §§6.4, 6.5, 6.9 Announcements: Quiz 2: 11/16, 7:30-9:30 PM, open book, must bring calculator; lectures #10-18.

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-2

Key questions • How does a pn diode look like from a small-signal point of view? • What are the leading dependences of the small-signal elements? • In addition to the junction capacitance, are there any other capacitive effects in a pn diode?

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-3

1. I-V characteristics (cont.) Diode current equation: I = Io(exp

qV − 1) kT

Physics of forward bias:

Fp p

n

Fn • potential difference across SCR reduced by V ⇒ minority carrier injection in QNR’s • minority carrier diffusion through QNR’s • minority carrier recombination at surface of QNR’s • large supply of carriers available for injection ⇒ I ∝ eqV /kT

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-4

Fp p

n

Fn Physics of reverse bias: • potential difference across SCR increased by V ⇒ minority carrier extraction from QNR’s • minority carrier diffusion through QNR’s • minority carrier generation at surface of QNR’s • very small supply of carriers available for extraction ⇒ I saturates to small value

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-5

I-V characteristics: I = Io(exp qV − 1) kT I

log |I|

0.43 q kT =60 mV/dec @ 300K

Io 0 0 Io linear scale

V

0 semilogarithmic scale

V

6.012 - Microelectronic Devices and Circuits - Fall 2005

Source/drain-body pn diode of NMOSFET:

Lecture 16-6

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-7

Key dependences of diode current: I = qAn2i (

Dn Dp 1 1 qV − 1) + )(exp Na Wp − xp Nd Wn − xn kT

n2i qV (exp N kT

•I∝ − 1) ≡ excess minority carrier concentration at edges of SCR n2i N

– in forward bias: I ∝ exp qV kT : the more carrier are injected, the more current flows n2i −N :

– in reverse bias: I ∝ the minority carrier concentration drops to negligible values and the current saturates • I ∝ D: faster diffusion ⇒ more current 1 : shorter region to diffuse through ⇒ more • I ∝ WQNR current

• I ∝ A: bigger diode ⇒ more current

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-8

2. Small-signal equivalent circuit model Examine effect of small signal overlapping bias: q(V + v) − 1] I + i = Io[exp kT If v small enough, linearize exponential characteristics: I + i = Io(exp

qv qv qV qV exp − 1) ' Io [exp (1 + ) − 1] kT kT kT kT

= Io(exp

qV qV qv − 1) + Io(exp ) kT kT kT

Then: q(I + Io) i= v kT From small signal point of view, diode behaves as conductance of value: gd =

q(I + Io) kT

6.012 - Microelectronic Devices and Circuits - Fall 2005

Small-signal equivalent circuit model, so far:

gd

gd depends on bias. In forward bias: qI gd ' kT gd is linear in diode current.

Lecture 16-9

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-10

Must add capacitance associated with depletion region:

gd

Cj

Depletion or junction capacitance: Cj =

Cjo 1 − φVB

s

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-11

3. Carrier charge storage: diffusion capacitance What happens to the majority carriers? Carrier picture so far: log p, n Na

po

Nd

no p

n ni2 Nd

ni2 Na 0

x

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

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-12

Quasi-neutrality demands that at every point in QNR: excess minority carrier concentration = excess majority carrier concentration n-QNR

n n(xn)

n(x)

qNn Nd p p(xn) p(x)

qPn

ni2 Nd 0

xn

Wn

x

Mathematically: p0 (x) = p(x) − po ' n0 (x) = n(x) − no Define integrated carrier charge: qP n = qA 12 p0 (xn)(Wn − xn) = =

2 Wn −xn ni qA 2 Nd (exp qV kT

− 1) = −qN n

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-13

Now examine small increase in V : n-QNR

n

V +-

n(xn)

-

I

∆qNn=-∆qPn

p

n(x)

n

Nd p p(xn)

+

∆qPn p(x)

ni2 Nd 0

xn

Wn

x

Small increase in V ⇒ small increase in qP n ⇒ small increase in |qN n| Behaves as capacitor of capacitance: Cdn =

dqP n |V dV

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-14

Can write qP n in terms of Ip (portion of diode current due to holes in n-QNR): qP n

Dp (Wn − xn)2 n2i qV − 1) = qA (exp 2Dp Nd Wn − xn kT (Wn − xn)2 = Ip 2Dp

Define transit time of holes through n-QNR: τT p

(Wn − xn)2 = 2Dp

Transit time is average time for a hole to diffuse through n-QNR [will discuss in more detail in BJT] Then: qP n = τT p Ip and Cdn '

q τT pIp kT

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-15

Similarly for p-QNR: qN p = τT nIn Cdp

q ' τT nIn kT

where τT n is transit time of electrons through p-QNR: τT n

(Wp − xp)2 = 2Dn

Both capacitors sit in parallel ⇒ total diffusion capacitance: Cd = Cdn + Cdp =

q q (τT nIn + τT pIp ) = τT I kT kT

with: τT nIn + τT p Ip τT = I Note that qP n + qN p = τT nIn + τT p Ip = τT I

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-16

Complete small-signal equivalent circuit model for diode:

gd

Cj

Cd

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-17

Bias dependence of Cj and Cd : C

Cd C 2Cjo

0

Cj

φB/2 V

0

• Cd dominates in strong forward bias (∼ eqV /kT ) dominates in reverse bias and small forward bias • Cj √ (∼ 1/ φB − V ) - For strong forward bias, model for Cj invalid (doesn’t blow up) - Common ”hack”, let Cj saturate at value corresponding to V = φ2B √ Cj,max =

2Cjo

6.012 - Microelectronic Devices and Circuits - Fall 2005

Lecture 16-18

Key conclusions Small-signal behavior of diode: • conductance: associated with current-voltage characteristics gd ∼ I in forward bias, negligible in reverse bias • junction capacitance: associated with charge modulation in depletion region r

Cj ∼ 1/ φB − V • diffusion capacitance: associated with charge storage in QNR’s to keep quasi-neutrality Cd ∼ eqV /kT Cd ∼ I