8/10/2019 Class (11).pdf

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 1/16

Subject 11: Outline

Rectification inpnJunctions

* The rectifier equation

* Electron and hole currents

* R-G currents

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 2/16

Today we develop a QUANTITATIVE analysis that allows us to compute the CURRENT that

flows through apnjunction in response to the application of an EXTERNAL voltage (Va)

* We begin by noting that apnjunction may be broken up into two QUASI-NEUTRALregions that surround the DEPLETION region

For the sake of the analysis here we will assume that the ELECTRIC FIELD in

these quasi-neutral regions is exactly equal to ZERO

The Rectifier Equation

!

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!" #!"

!" $!#

!" %

x

E(x)

QUASI-NEUTRALREGION

QUASI-NEUTRALREGION

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 3/16

Since the electric field is equal to zero the electron and hole DRIFT currents that flow in

the quasi-neutral regions must ALSO be equal to zero

* Previously we obtained expressions for the MINORITY-CARRIER concentrationsunder the influence of an external bias

The Rectifier Equation

JP =JP drift+JP diffusion = ! qDpd"pn

dx(11.2)

JN =JN drift+JN diffusion = qDnd!np

dx(11.1)

!pn (x)=ni

2

NDexp

qVa

kBT

"

#$

%

&'(1

"

#$

%

&'exp (

x

Lp

"

#$

%

&' , Lp = D)p (10.9)

!np(x)=ni

2

NAexp

qVa

kBT

"

#$

%

&'(1

"

#$

%

&'exp (

x

Ln

"

#$

%

&' , Ln = D)n (10.8)

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 4/16

By combining Eqs. 10.9, 10.10, 11.1, and 11.2 we thus obtain expressions for the

electron and hole CURRENT DENSITIES in the junction

* Note the DIFFERENT way in which the directionxis defined in these two equations

The Rectifier Equation

JN = !qDnd"npdx

= qDn

Ln

ni2

NAexp

qVa

kBT

#

$%

&

'(!1

#

$%

&

'(exp !

x

Ln

#

$%

&

'( (11.3)

JP =! qDp d"pn

dx= qDp

Lp

ni2

NDexp qVa

kBT#$% &

'(!1#

$% &

'(exp ! x

Lp#$% &

'( (11.4)

#

#

#

#

#

#

#

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#

#

#

#

#

#

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#

#

#

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#

#

#

#

#

#

#

x

x = 0

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!

!

!

!

!

!

!

!

x

x = 0

Eq. 11.3

Eq. 11.4

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 5/16

By SUMMINGthe electron and hole currents we obtain the TOTALjunction current

* Due to Kirchhoff's law we need only perform this sum at a SINGLE point within thejunction and we do this here by summing the currents at x = 0 (justified later!)

where have have taken the junction to have cross-sectional areaA

The Rectifier Equation

I =A(JN+JP )=Io exp qVa

kBT

!

"#

$

%&'1

!

"#

$

%& , Io = qA

Dn

Ln

ni2

NA+

Dp

Lp

ni2

ND

!

"#

$

%& (11.5)

#

#

#

#

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x

x = 0

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!

x

x = 0

1. WE EVALUATEJNHERE 2. AND ADD TO THEVALUE OFJPHERE

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 6/16

The IDEAL-DIODE equation (Eq. 11.5) predicts a forward-bias current that grows

dramatically once Va> kBT/q

* Under REVERSEbias the current is expected to be roughly CONSTANTand equal to

the REVERSE SATURATION CURRENT (Io) whose value is NEGLIGIBLYsmall in the

range of pA

The Rectifier Equation

V(V)

1

2

-1

-2

-3

-4

-5

I(mA)

2

-2

AD

A

i

n

no

NNN

n

L

DqAI >>= ,

2

DA

D

i

p

p

o NN

N

n

L

DqAI >>= ,

2

I =A(JN+JP )=Io exp qVa

kBT

!

"#

$

%&'1

!

"#

$

%& , Io = qA

Dn

Ln

ni2

NA+

Dp

Lp

ni2

ND

!

"#

$

%& (11.5)

1

2

-1

-2

-3

-4

-5

I(mA)

2

-2

FORWARDCONDUCTION

REVERSE BIAS:NEGLIGIBLE

CURRENT

IN A JUNCTION WITH HEAVY DOPING ONONE SIDEIoIS DETERMINED BY THE DOPING

DENSITY ON THE LIGHTLY-DOPED SIDE

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 7/16

The rectifier equation gives the TOTAL current that flows through the junction under the

influence of an externally applied voltage

* This total current is INDEPENDENT of position within thepnjunction as required byKirchoffs current law

* The INDIVIDUAL electron and hole currents are generally NOT independent of

position however

In the QUASI-NEUTRAL regions Equations 11.3 & 11.4 show that the MINORITY

carrier currents DECAY exponentially with distance due to recombination

Electron and Hole Currents

JN = !qDnd"npdx

= qDn

Ln

ni2

NAexp

qVa

kBT

#

$%

&

'(!1

#

$%

&

'(exp !

x

Ln

#

$%

&

'( (11.3)

JP =! qDpd"pndx

= qDp

Lp

ni2

NDexp

qVa

kBT

#

$%

&

'(!1

#

$%

&

'(exp !

x

Lp

#

$%

&

'( (11.4)

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 8/16

INSIDE the depletion regions the electric field is NOT equal to zero and we must use the

continuity equations to compute the electron and hole currents

* If we consider a diode under STEADY-STATE conditions then the continuityequations (Eqs. 7.4 & 7.5) inside the depletion region become

* In the IDEAL-DIODE APPROXIMATION we assumed NO R-G processes within the

depletion region which requires in turn that the electron and hole currents in this

region must be INDEPENDENTof position

Electron and Hole Currents

!p

!t= 0 = "

1

q

dJP

dx+!p

!t ThermalR"G

(11.7)

!n

!t=

1

q"#J

N+

!n

!t thermalR$G

+

!n

!tOther

(7.4)

!p

!t=$

1

q"#J

P+

!p

!t thermalR$G

+

!p

!tOther

(7.5)

!n

!t= 0 =

1

q

dJN

dx+!n

!tThermalR"G

(11.6)

NO R-G PROCESSES!

NO R-G PROCESSES!

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 9/16

With these considerations in mind the electron and hole currents inside apnjunction

may be sketched as shown below

* The minority-carrier currents decay exponentially away from the edges of thedepletion region

The majority-carrier currents counteract this variation in a manner that keeps

the TOTAL current CONSTANT at any particular point

* Inside the depletion region the electron and hole currents are CONSTANT

Electron and Hole Currents

x-xp xn

JN,P

JP JN

x-xp xn

JN,P

JP JN

1. DECAY ON A FEWMINORITY-CARRIERDIFFUSION LENGTH, Lp

1. DECAY ON A FEWMINORITY-CARRIER

DIFFUSION LENGTHS, Ln

2. MAJORITY CURRENTCOMPENSATES FOR CHANGEOF MINORITY CURRENT

2. MAJORITY CURRENTCOMPENSATES FOR CHANGE

OF MINORITY CURRENT

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 10/16

We have seen that in FORWARDbias the minority-carrier concentrations are INCREASED

around the depletion region

* In REVERSE bias these concentrations are SUPPRESSED due to the associatedsuppression of carrier diffusion across the junction

The depletion region essentially acts as a SINK for minority carriers

* Reverse-bias voltages of only a few kBT/qreduce the minority concentrations toZEROat the edge of the depletion region

SUPPRESSED MINORITY-CARRIER CONCENTRATIONS IN ApnJUNCTION UNDER REVERSE BIAS

ni2/NA

ni2/ND

xxn-xp

n p

ni2/NA

ni2/ND

xxn-xp

n p

np = ni2exp

qVa

kBT

!

"#

$

%&

LAW OF THE JUNCTION (Eq. 10.5): MINORITYCONCENTRATIONSSUPPRESSED NEAR

DEPLETIONREGION DUE TO

INCREASEDBARRIER FORDIFFUSION

Va< 0

Electron and Hole Currents

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 11/16

In deriving the rectifier equation it was assumed that NO R-G processes occur within the

depletion region

* In real junctions typically those fabricated in Si and GaAs it is found that inreverse bias or with a small forward bias the current can noticeably EXCEEDIo

This is due to ADDITIONAL current that is generated by R-G processes taking

place within the depletion region itself

* In REVERSE bias mode GENERATION of electron holes pairs in the depletion region

enhances the DRIFT current above its expected value

R-G Currents

SCHEMATIC ILLUSTRATION OF R-G CURRENTSIN A REVERSE-BIASEDpnJUNCTION

ELECTRONS AND HOLES CREATED IN THEDEPLETION REGION ARE RAPIDLY SWEPT AWAYBY THE HIGHELECTRIC FIELD IN THIS REGION

IN THIS CASE IT CAN BE SEEN THAT R-GPROCESSES ENHANCE THE DRIFT CURRENT INTHE JUNCTION WHICH THEREFORE EXCEEDS THEEXPECTED REVERSE SATURATION CURRENTIo

Ec

Ev

qVeff

ELECTRONDRIFT

HOLEDRIFT

e-h PAIRGENERATION

R-G HOLEDRIFT

R-G ELECTRONDRIFT

EF

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 12/16

In FORWARD bias mode the main effect of R-G processes is to cause an ENHANCEMENT

of the RECOMBINATION rate

* Remember that forward biasing basically INCREASES the carrier concentrations inthe depletion regions above their equilibrium values

* This in turn gives rise to an enhanced recombination rate within the depletion region

that effectively corresponds to an ENHANCEMENT of the DIFFUSION current

R-G Currents

Ec

Ev

qVeff

EF

HOLEDIFFUSION

ELECTRON

DIFFUSION

ELECTRON

DRIFT

HOLEDRIFT

RECOMBINATION OF ELECTRONS AND HOLESINSIDE THE DEPLETION REGION IN A FORWARDBIASEDpnJUNCTION

A LARGE DIFFUSION CURRENT IS CARRIED BYTHOSE ELECTRONS AND HOLES (THICK ARROWS)IN THE DEPLETION REGION WITH SUFFICIENTENERGY TO OVERCOME THE BARRIER THATEXISTS TO THE DIFFUSION OF CURRENT

THE DIFFUSION CURRENTS ARE MUCH LARGERTHAN THE CORRESPONDING ELECTRON AND HOLEDRIFT CURRENTS (SMALLER ARROWS)

THE DIFFUSION CURRENTS ARE EFFECTIVELYENHANCED WHEN CARRIERS WITH INSUFFICIENTENERGY TO OVERCOME THE DIFFUSION BARRIERRECOMBINE (BLACK ARROWS) AND ARE THUSEFFECTIVELY ELIMINATED

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 13/16

R-G Currents

To compute the current due to R-G processes in the depletion region we note that for

every electron-hole PAIRcreated or destroyed per unit time in the depletion region ONEelectron per unit time flows into or out of the diode contacts

* SUMMINGover either the electrons or holes created within the depletion region per

unit time should therefore give the additional current due to R-G processes

Note here that the minus sign accounts for the POLARITYof the current

* Previously (see Subjects 5 & 6) we have been able to express the rate of change of

carrier concentrations due to R-G processes using expressions of the form

This approach CANNOTbe used here however since the R-G processes takingplace in the depletion region do NOTcorrespond to low-level injection

IR!G

= !qA "n

"t thermalR!G

dx

!xp

xn=0

#

$

%%%

(11.8)

!n

!t thermalR"G

= "#n

$n

(11.9)

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 14/16

IR-Gmust be calculated by properly solving for STEADY-STATE conditions under R-Gprocesses with the depletion region an analysis that is BEYONDthe scope of our present

discussion

* In REVERSE bias it may be shown that the additional drift current due to electron-

hole pair creation may be written as

Where !ois a LIFETIME associated with R-G center generation and depends onthe ENERGY of the R-G centers

* In Si and GaAs diodes at room temperatureIR-Gin Eq. 11.10 is much GREATERthan the ideal-diode current expected at reverse bias and small forward bias

In Ge however the larger value of nimakesIomuch larger thanIR-G

R-G Currents

IR!G =qAni

2"oW (11.10)

Io = qA Dn

Ln

ni2

NA+

Dp

Lp

ni2

ND

!

"#

$

%&>IR'G (11.11)

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 15/16

While the ideal-diode equation predicts a CONSTANTcurrent (-Io) for a wide range ofreverse bias the R-G current GROWS with increasing |V

a|

* The growth of the R-G current follows from the increase in the WIDTHof thedepletion region under reverse biasing

As the width of the depletion region grows the volume involved in the creationof electron-hole pairs INCREASES leading to a larger R-G current

R-G Currents

IR!G =qAni

2"oW (11.10)

REVERSECURREN

T(pA)

REVERSE BIAS, Va(V)

-10-30 -20-40

-100

-200

-300REVERSE

CURREN

T(pA)

REVERSE BIAS, Va(V)

-10-30 -20-40

-100

-200

-300

NOTE UNITS!

REVERSE-BIASR-G CURRENT GROWS WITH

INCREASING

BIASING

REVERSE-BIASBREAKDOWN

DIFFERENT IN

ORIGIN TO R-G

CURRENT

SipnJUNCTION300 K

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Spring 2014 EE 430/530: Fundamentals of Solid-State Devices Subject 11, Slide 16/16

For small FORWARD bias it is found that the R-G current typically varies with voltage as

* In this expression the parameter "is an empirical IDEALITY FACTOR whose valuedepends on the relative magnitude of the R-G and forward-bias diffusion currents

When IDEALdiffusion dominates "= 1 but when R-Gprocesses dominate "= 2

R-G Currents

IR!G" exp qVa

#kBT

$

%&

'

() , Va >

kBT

q(13.5)

FORWARD BIAS, Va(V)

0.4 0.8 1.210-9

10-7

10-5

10-3

10-1

FORWARDCU

RRENT(A)

"= 2

"= 1

"= 2

"= 1

Si

GaAs

300 K