empirical observations of v br

President University Erwin Sitompul SDP 7/1

Lecture 7

Semiconductor Device Physics

Dr.-Ing. Erwin SitompulPresident University

http://zitompul.wordpress.com

2 0 1 3


Dominant breakdown mechanism is tunneling

BR 0.75B

1 V

N

Empirical Observations of VBR

Chapter 6 pn Junction Diodes: I-V Characteristics

• VBR : breakdown voltage

VBR decreases with increasing N,

VBR decreases with decreasing EG.


2S CR A D

BR biA D2

N NV V

q N N

E

A DCR bi BR

S A D

2 N NqV V

N N

E

• At breakdown, VA=–VBR

Breakdown Voltage, VBR

Chapter 6 pn Junction Diodes: I-V Characteristics

If the reverse bias voltage (–VA) is so large that the peak electric field exceeds a critical value ECR, then the junction will “break down” and large reverse current will flow.

Thus, the reverse bias at which breakdown occurs is


2 A DCR BR

S A D

2 N NqV

N N

E

Small E-field:

High E-field:

Low energy, causing lattice vibration and localized heating only

High energy, enabling impact ionization which causing avalanche, at doping level N < 1018 cm–3

Breakdown Mechanism: Avalanching Chapter 6 pn Junction Diodes: I-V Characteristics

• ECR : critical electric field in the depletion region

2CR

BR 2sVqN


Breakdown Mechanism: Zener Process Chapter 6 pn Junction Diodes: I-V Characteristics

Zener process is the tunneling mechanism in a reverse-biased diode.

Energy barrier is higher than the kinetic energy of the particle.

The particle energy remains constant during the tunneling process.

Barrier must be thin dominant breakdown mechanism when both junction sides are heavily doped.

Typically, Zener process dominates when VBR < 4.5V in Si at 300 K and N > 1018 cm–3.


2i

thermal p 1 n 1R-G( ) ( )

np nn

t n n p p

n

p

R-G

thermalR G

x

x

nI qA dx

t

T i( )1 i

E E kTn n e

Effect of R–G in Depletion RegionChapter 6 pn Junction Diodes: I-V Characteristics

R–G in the depletion region contributes an additional component of diode current IR–G.

The net generation rate is given by

i T( )1 i E E kTp n e

• ET: trap-state energy level


For reverse bias, with the carrier concentrations n and p being negligible,

n

p

2i

R-Gp 1 n 1( ) ( )

x

x

np nI qA dx

n n p p

iR-G

02

qAnWI

1 10 p n

i i

1

2

n p

n n

where


Continuing,

• Reverse biases with VA< – few kT/q

• Thermal carrier generation in the depletion layer• Carriers swept by electric field and generate

additional current


A 2R-G i

qV kTI qAnWe


n

p

2i

R-Gp 1 n 1( ) ( )

x

x

np nI qA dx

n n p p

Continuing,

For forward bias, the carrier concentrations n and p cannot be neglected,

• High carrier concentrations in the depletion layer• Additional carrier recombination in the region

that decreases current


A2 N PDIFF i

N A P D

( 1)qV kTD DI Aqn e

L N L N

A

A

iR-G

0 n p 2bi A

0

( 1)

21

2

qV kT

qV kT

qAnW eI

V Ve

kT q

DIFF R GI I I

Diffusion, ideal diode



J A SV V IR

Voltage drop, significant for high I

A S

J

( )0

0 A bi,

q V IR kT

qV kT

I I eI e V V

Effect of Series ResistanceChapter 6 pn Junction Diodes: I-V Characteristics

The assumption that applied voltage is dropped only across the depletion region is not fully right.


RS can be determined experimentally

SV IR

Sslope R

Effect of Series ResistanceChapter 6 pn Junction Diodes: I-V Characteristics

As part of the applied voltage is wasted, a larger applied voltage is necessary to achieve the same level of current compared to the ideal.


n n0n n (for a p+n junction)

(for a pn+ junction)p p0p p

Effect of High-Level InjectionChapter 6 pn Junction Diodes: I-V Characteristics

As VA increases and about to reach Vbi, the side of the junction which is more lightly doped will eventually reach high-level injection:

This means that the minority carrier concentration approaches the majority doping concentration.

Then, the majority carrier concentration must increase to maintain the neutrality.

This majority-carrier diffusion current reduces the diode current from the ideal.


Perturbation of both minority and majority carrier

High-Level Injection EffectChapter 6 pn Junction Diodes: I-V Characteristics


Forward-bias current Reverse-bias current

Deviations from ideal I-V

Due to avalanching and Zener process

Due to high-level injection and series resistance in

quasineutral regions

Due to thermal generation in depletion region

Due to thermal recombination in depletion region

SummaryChapter 6 pn Junction Diodes: I-V Characteristics


nP n ( )

xQ qA p x dx

pN p ( )

xQ qA n x dx

Minority-Carrier Charge StorageChapter 6 pn Junction Diodes: I-V Characteristics

When VA>0, excess minority carriers are stored in the quasineutral regions of a pn junction.


Charge Control ApproachChapter 6 pn Junction Diodes: I-V Characteristics

n

P n ( , )x

Q qA p x t dx

Consider a forward-biased pn junction.

The total excess hole charge in the n quasineutral region is:

2n n n

P 2p

p p pD

t x

The minority carrier diffusion equation is (without GL):

nP P

pqD

x

J

n P n

p

( )q p q p

t x

J

Since the electric field E0,

Therefore (after all terms multiplied by q),


P

n p n n

( )

n P np( )

1J

x J x x

dqA p dx A d qA p dx

dt

J

P

P n

( )

P P P n

( )

( ) ( )J

J x

A d A A x

J J J

P PP n

p

( )dQ Q

A xdt

J

0

0 In steady state

QP QP


Integrating over the n quasineutral region (after all terms multiplied by Adx),

Furthermore, in a p+n junction,

So:

P n( )A x J


PP n P n

p

( ) ( )Q

A x I x

J

NN p

n

( )Q

I x

P PP n

p

( ) 0dQ Q

A xdt

J


In steady state, we can calculate pn junction current in two ways:

From slopes of Δnp(–xp) and Δpn(xn) From steady-state charges QN and QP stored in each

“excess minority charge distribution”

Therefore,

Similarly,


N p( ) 0x J

P PDIFF

p

dQ Qi

dt

DIFF P n( )xJ J

0 In steady state


Moreover, in a p+n junction:


0x 0 x

cxnx

n-side contact

cx

Narrow-Base DiodeChapter 6 pn Junction Diodes: I-V Characteristics

Narrow-base diode: a diode where the width of the quasineutral region on the lightly doped side of the junction is on the order of or less than one diffusion length.

px


An n0( 0) ( 1)qV kTp x p e

P Pn 1 2( ) x L x Lp x Ae A e

A

c P c P

n0 1 2

1 2

( 1)

0

qV kT

x L x L

p e A A

Ae A e

Narrow-Base Diode I–VChapter 6 pn Junction Diodes: I-V Characteristics

We have the following boundary conditions:

n c( ) 0p x x

Then, the solution is of the form:

Applying the boundary conditions, we have:


cc P cc P

A

cc P cc P

( ) ( )/

n n0 c( ) ( 1) , 0x x L x x L

qV kTx L x L

e ep x p e x x

e e

A c Pn n0 c

c P

sinh ( )( ) ( 1) , 0

sinhqV kT x x L

p x p e x xx L


Solving for A1 and A2, and substituting back:

Note that sinh( ) , cosh( )2 2

e e e e

The solution can be written more compactly as


A c Pn n0

c P

( )( ) ( 1)qV kT x x L

p x p ex L

An n0

c

( ) ( 1) 1qV kT xp x p e

x


With decrease base width, xc’0:

02

0

limsinh( )

lim cosh( ) 12

• Δpn is a linear function of x due to negligible thermal R–G in region much shorter than one diffusion length

• JP is constant

• This approximation can be derived using Taylor series approximation


Because , then

A P c PP P n0

c P

1 cosh ( ) ( 1)

sinhqV kT L x x L

qD p ex L

J

nP P

( )p xqD

x

J

A

2c PP i

DIFF PP D c P

cosh( )( 0) ( 1)

sinh( )qV kT x LD n

I A x qA eL N x L

J

2c PP i

0P D c P

cosh( )

sinh( )

x LD nI qA

L N x L

ADIFF 0 ( 1)qV kTI I e


Then, for a p+n junction:


2 2P i P P i

0P D c c D

D n L D nI qA qA

L N x x N

2c P

c P

( )1

2( )

x L

x L


If xc’ << LP,

02

0

limsinh( )

lim cosh( ) 12

c PP

c

( )

2

x LL

x

P

c

L

x

c P

c P

cosh( )

sinh( )

x L

x L

Resulting

Increase of reverse bias means• Increase of reverse current• Increase of depletion width• Decrease of quasineutral region xc’=xc–xn


A c Pn n0

c P

sinh ( ) /( ) ( 1)

sinh /qV kT x x L

p x p ex L

c P c P

A

c P c P

( ) ( )

n n0( ) ( 1)x x L x x L

qV kTx L x L

e ep x p e

e e

A Pn n0( ) ( 1)qV kT x Lp x p e e Back to ideal

diode solution

Wide-Base DiodeChapter 6 pn Junction Diodes: I-V Characteristics

Rewriting the general solution for carrier excess,

For the case of wide-base diode (xc’>> LP),

pc P c PP

A

c P c P

// //

n0 ( 1)x Lx L x Lx L

qV kTx L x L

e e e ep e

e e


A

2c PP i

DIFFP D c P

cosh( )( 1)

sinh( )qV kT x LD n

I qA eL N x L

A

2P i

DIFFP D

( 1)qV kTD nI qA e

L N Back to ideal

diode solution

Wide-Base DiodeChapter 6 pn Junction Diodes: I-V Characteristics

Rewriting the general solution for diffusion current,

For the case of wide-base diode (xc’>> LP),

lim sinh( )2

lim cosh( )2

e

e


Homework 5Chapter 6 pn Junction Diodes: I-V Characteristics

Due: 11.06.2012.

1.(8.14)The cross-sectional area of a silicon pn junction is 10–3 cm2. The temperature of the diode is 300 K, and the doping concentrations are ND = 1016 cm–3 and NA = 8×1015 cm–3. Assume minority carrier lifetimes of τn0 = 10–6 s and τp0 = 10–7 s.

Calculate the total number of excess electrons in the p region and the total number of excess holes in the n region for (a) VA = 0.3 V, (b) VA = 0.4V, and (c) VA = 0.5 V.

2. (7.2)Problem 6.11, Pierret’s “Semiconductor Device Fundamentals”.

empirical observations of v br

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