PHYSICAL ELECTRONICS(ECE3540) CHAPTER 7 –...

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PHYSICAL ELECTRONICS(ECE3540)CHAPTER 7 – THE PN JUNCTION

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Typewritten Text

Brook Abegaz

The PN Junction Chapter 4: we considered the semiconductor in

equilibrium and determined electron and holeconcentrations in the conduction and valencebands, respectively.

The net flow of the electrons and holes in asemiconductor generates current. The process bywhich these charged particles move is calledtransport.

Chapter 5: we considered the two basic transportmechanisms in a semiconductor crystal: drift: themovement of charge due to electric fields, anddiffusion: the flow of charge due to density gradients.


The PN Junction Chapter 6: we discussed the behavior of non-

equilibrium electron and hole concentrations asfunctions of time and space.

We developed the ambi-polar transport equationwhich describes the behavior of the excess electronsand holes. Previous Chapters: we have been considering

the properties of the semiconductor material bycalculating electron and hole concentrations inthermal equilibrium and determined the position ofthe Fermi level.


The PN Junction Previous Chapters: We considered the non-

equilibrium condition in which excess electronsand holes are present in the semiconductor.

Chapter 7: We now wish to consider thesituation in which a p-type and an n-typesemiconductor are brought into contact with oneanother to form a PN junction.


The PN Junction Most semiconductor devices contain at least one

junction between p-type and n-type semiconductorregions.

Semiconductor device characteristics and operation areintimately connected to these PN junctions, thereforeconsiderable attention is devoted initially to this basicdevice.

The PN junction diode provides characteristics thatare used in rectifiers and switching circuits and willalso be applied to other devices.

The electrostatics of the PN junction is considered inthis chapter and the current-voltage characteristics ofthe PN junction diode are developed in the nextchapter.


Basic Structure of the PN Junction The entire semiconductor is a single-crystal material

with one region doped with acceptor impurity atoms“p-region” and the adjacent region doped with donoratoms to form the “n-region”. The interfaceseparating n and p regions is the metallurgicaljunction.

We consider a step junction (the dopingconcentration is uniform in each region and there isan abrupt doping change at the junction.)

Initially, at the metallurgical junction, there is a verylarge density gradient in both the electron and holeconcentrations.


Basic Structure of the PN Junction Majority carrier electrons in the n region will begin

diffusing into the p-region and majority carrier holesin the p-region will begin diffusing into the n region.

As electrons diffuse from the n region, positivelycharged donor atoms are left behind. Similarly, asholes diffuse from the p region, they uncovernegatively charged acceptor atoms.

The net positive and negative charges induce anelectric field in the region near the metallurgicaljunction from the positive to the negative charge, orfrom the n to the p region.


Basic Structure of the PN Junction


Fig. 7.1: Basic Structure of the PN Junction

Basic Structure of the PN Junction The two regions are referred to as the space charge

region or depletion region. Essentially all electrons andholes are swept out of the space charge region by theelectric field.

Density gradients still exist in the majority carrierconcentrations at each edge of the space charge region.We can think of a density gradient as producing a"diffusion force" that acts on the majority carriers.

The electric field in the space charge region producesanother force on the electrons and holes which is inthe opposite direction to the diffusion force for eachtype of particle. In thermal equilibrium, the diffusionforce and the E-field force exactly balance each other.


Flat Band Diagram of a PN Junction


Fig. 7.2: Flat Band Diagram of a PN Junction

Zero applied bias The properties of the step junction in thermal equilibrium, the

space charge region width, the electric field, and the potentialthrough the depletion region where no currents exist and noexternal excitation is applied are studied.

Assuming no voltage is applied across the PN junction, thenthe junction is in thermal equilibrium and the Fermi energylevel is constant.

The conduction and valance band energies must bend as we gothrough the space charge region, since the relative position ofthe conduction and valence bands with respect to the Fermienergy changes between p and n regions.

Electrons in the conduction band of the n region see apotential barrier in trying to move into the conduction band ofthe p region. This barrier is the built-in potential barrier andis denoted by Vbi.

The potential Vbi maintains equilibrium, therefore no current isproduced by this voltage.


PN Junction in Thermal Equilibrium


Fig. 7.3: PN Junction in Thermal Equilibrium

Zero applied biasThe intrinsic Fermi level is equidistant from the conductionband edge through the junction, thus the built-in potentialbarrier can be determined as the difference between theintrinsic Fermi levels in the p and n regions.

In the n region, the electron concentration in theconduction band is given by:

where ni and EFi are the intrinsic carrier concentration andthe intrinsic Fermi energy respectively.


|||| FpFnbiV

kT

EENn Fcc

)(exp0

kT

EEnn FiFi

)(exp0

Zero applied biasThe potential in the n region can be defined as:

Similarly, in the p region, the hole concentration is given by:

Therefore, the built-in potential voltage is calculated as:


FFiFn EEe ||

kTenn Fn

i)(exp0

i

dFn n

NekT ln

FFiFp EEe ||

kTe

nn Fpi

)(exp0

i

aFp n

Ne

kT ln

22 lnln||||

i

dat

i

daFpFnbi n

NNVn

NNe

kTV

Zero applied bias At this time, we should note a subtle but important

point concerning notation. Previously in the discussion of a semiconductor

material, Nd and Na denoted donor, and acceptorimpurity concentrations in the same region, therebyforming a compensated semiconductor.

From this point on, Nd and Na will denote the netdonor and acceptor concentrations in the individual nand p regions, respectively. If the p region, forexample, is a compensated material, then Na willrepresent the difference between the actual acceptorand donor impurity concentrations.

The parameter Nd is defined in a similar manner forthe n region.


Electric field Electric field is created in the depletion region by

the separation of positive and negative spacecharge densities.

We will assume that the space charge regionabruptly ends in the n region at x = +xn andabruptly ends in the p region at x = -xp (xp is apositive quantity).


PN Junction in Thermal Equilibrium


Fig. 7.4: a) Charge density in a p-n junction, b) Electric Field, c) Potential d) Energy band Diagram

Electric field The electric field is determined from Poisson's equation which, for a one

dimension alanalysis, is:

where (x) is the electric potential, E(x) is the electric field, (x) is the volume chargedensity, and ɛs is the permittivity of the semiconductor. The charge densities are:

The electric field in the p region is found by integrating Poisson’s eqn:

The electric field is zero in the neutral p region for x < -xp. As there areno surface charge densities within the PN junction structure, the electricfield is a continuous function. The constant of integration is determinedby setting E = 0 at x=-xp. For the n-region:


dxxdEx

dxxd

s

)()()(2

2

ndpa xxeNxxxeNx 0:)(0:)(

)()(1 p

s

a

s

a

s

a

s

xxeNCxeNdxeNdxxE

)()(1 xxeNCxeNdxeNdxxE n

s

d

s

d

s

d

s

Electric field The electric field is also continuous at the metallurgical junction, or

at x = 0 therefore:

For the uniformly doped pn junction, the E-field is a linearfunction of distance through the junction, and the maximum(magnitude) electric field occurs at the metallurgical junction. Anelectric field exists in the depletion region even when no voltage isapplied between the p and n regions.

Integrating the electric field to find the built-in potential:


ndpa xNxN

)0(:2

)2

.(2

)( 222

nps

an

s

d xxxeNxxxeNx

)0(:)(2

)( 2 xxxxeNx pps

a

)(2

|)(| 22pand

snbi xNxNexxV

Space Charge Width We can determine the distance that the space

charge region extends into the p and n regionsfrom the metallurgical junction. This distance isknown as the space charge width.

The total depletion, space charge width W is thesum of the two:


a

ndp N

xNx 21

12

dad

abisn NNN

NeVx

21

12

daa

dbisp NNN

NeVx

21

.2

da

dabis

NNNN

eVW

Exercise1. Calculate Vbi in a Silicon PN junction at T =

300K for (a) Nd = 1015 cm‐3 and:i) Na =1015

ii) Na =1016

iii) Na =1017

iv) Na =1018.

(b) Repeat part (a) for Nd = 1018 cm‐3 .


22 lnln||||

i

dat

i

daFpFnbi n

NNVn

NNe

kTV

Solution1. Using the equation:where Vt = 0.0259V and ni = 1.5x1010cm-3,


2ln

i

datbi n

NNVV

For Nd = 1015cm‐3 Vbi(V)

Na = 1015cm‐3 0.575V

Na = 1016cm‐3 0.635

Na = 1017cm‐3 0.695

Na = 1018cm‐3 0.754

For Nd = 1018cm‐3 Vbi(V)

Na = 1015cm‐3 0.754V

Na = 1016cm‐3 0.814

Na = 1017cm‐3 0.874

Na = 1018cm‐3 0.933

Exercise2. An abrupt Silicon PN junction at zero bias has

dopant concentration of Na = 1017 cm-3 and Nd= 2 x 1016 cm-3, T = 300K. (a) Calculate theFermi level on each side of the junction withrespect to the intrinsic Fermi level. (b) Sketch theequilibrium energy band diagram for the junctionand determine Vbi from the diagram and theresults of part (a),(c) Calculate Vbi and comparethe results to part b). d) Determine xn and thepeak electric field for this junction.


22 lnln||||

i

dat

i

daFpFnbi n

NNVn

NNe

kTV

21

.2

da

dabis

NNNN

eVW

Solution (a) n‐side

p-side,

b)

c)


eVx

xEE FiF 3653.0105.1102ln)0259.0( 10

16

eVx

xEE FFi 3653.0105.1102ln)0259.0( 10

16

VVbi 7306.03653.03653.0

Vx

xxnNNVVi

datbi 7305.0

)105.1()102)(102(ln)0259.0(ln 210

1616

2

Solution d)


21

161616

16

19

14

1021021

102102

106.1)7305.0)(1085.8)(7.11(2

xxxx

xxxn

cmVx

xxxxxeNE nd 4

14

41619

max 1076.4)1085.8)(7.11(

)10154.0)(102)(106.1(||

mxx pn 154.0

Reverse Applied Bias If we apply a potential between the p and n regions, we will no

longer be in an equilibrium condition and the Fermi energy levelwill no longer be constant through the system.

If a positive voltage is applied to the n-region with respect to thep-region, as the positive potential is downward, the Fermi level onthe n side is below the Fermi level on the p side. The differencebetween the two is equal to the applied voltage in units of energy.

The total potential barrier, indicated by Vtotal has increased. Theapplied potential is the reverse-bias condition. The total potentialbarrier is now given by:

where VR is the magnitude of the applied reverse-bias voltage andVbi is the same built-in potential barrier defined earlier.


RFpFntotal VV |||| Rbitotal VVV

Reverse and Forward Applied Bias


Fig. 7.5: Energy band diagram of a PN Junction under reverse and forward bias

Space Charge Width and Electric Field The electric fields in the neutral P and N regions are

essentially zero, or at least very small, which means that themagnitude of the electric field in the space charge regionmust increase above the thermal-equilibrium value due tothe applied voltage.

The electric field originates on positive charge andterminates on negative charge; this means that the numberof positive and negative charges must increase if theelectric field increases.

For given impurity doping concentrations, the number ofpositive and negative charges in the depletion region can beincreased only if the space charge width W increases.

The space charge width W increases with an increasingreverse-bias voltage VR.


Space Charge Width and Electric Field In all of the previous equations, the built-in

potential barrier can be replaced by the totalpotential barrier. The total space charge width incase of reverse-bias can be written as:


21

)(2

da

daRbis

NNNN

eVVW

21

max)(2

da

da

s

Rbi

NNNNVVeE

WVVE Rbi )(2

max

Junction Capacitance Since we have a separation of positive and negative

charges in the depletion region, a capacitance isassociated with the PN Junction.

An increase in the reverse-bias voltage dVR willuncover additional positive charges in the n regionand additional negative charges in the p region. Thejunction capacitance is defined as:


pand dxeNdxeNdQ '

RdVdQC

''

Junction Capacitance The differential charge dQ' is in units of C/cm2 so that

the capacitance C' is in units of Farads per squarecentimeter (F/cm2), or capacitance per unit area.

For the total potential barrier:

The junction capacitance is: Exactly the same capacitance expression is obtained by

considering the space charge region extending into thep region xp. The junction capacitance is also referred toas the depletion layer capacitance.


R

nd

R dVdxeN

dVdQC

''

21

1)(2

dad

aRbisn NNN

Ne

VVx 21

'

))((2

daRbi

das

NNVVNNeC

Junction Capacitance Comparing the following two equations:

we find that we can write:

The above equation is the same as the capacitance perunit area of a parallel plate capacitor.

Note that the space charge width is a function of thereverse bias voltage so that the junction capacitance isalso a function of the reverse bias voltage applied tothe PN Junction.


21

)(2

da

daRbis

NNNN

eVVW 2

1

'

))((2

daRbi

das

NNVVNNeC

WC s'

Exercise3. Consider a Silicon PN Junction at T = 300K

with doping concentrations of Na = 1016 cm-3

and Nd = 1015 cm-3 . Assume that ni = 1.5 x1010 cm-3 and let VR = 5 V. Calculate the widthof the space charge region.


21

)(2

da

daRbis

NNNN

eVVW

22 lnln||||

i

dat

i

daFpFnbi n

NNVn

NNe

kTV

Solution3. Using the equation:

We can calculate that W = 2.83m.


21

)(2

da

daRbis

NNNN

eVVW

21

1516

1516

19

14

)10)(10(1010

10*6.1)5635.0)(10*85.8)(7.11(2

W

Non-Uniformly Doped Junctions In the PN Junctions considered so far, we have

assumed that each semiconductor region has beenuniformly doped.

In actual PN Junctions, this isn’t always true. In some electronic applications, specific non-

uniform doping profiles are used to obtain specialPN Junction capacitance characteristics.

Different types of doping profiles are used in: Uniformly Doped Junctions Linearly Graded Junctions Hyper-abrupt Junctions


Linearly Doped Junctions Considering a uniformly doped n-type

semiconductor, if we diffuse acceptor atomsthrough the surface, the impurity concentrationswill tend to be like those shown in the Figure.

The depletion region extends into the p and nregions from the metallurgical junction.

The net p-type doping concentration near themetallurgical junction may be approximated as alinear function of distance from the metallurgicaljunction.


axNN ad

Linearly Doped Junctions


Impurity Concentration

Surface X = X’

Nd

Na

----------------P-region------------------------- ---------------N-region-------------

Fig. 7.6: Impurity concentrations of a pn junction with a non-uniformly doped p region.



X = 0

-X0

X0

+

-

P-region N-region

Fig. 7.7: Space charge density of linearly graded PN Junction.

Similarly, the net n-type doping concentration is also a linearfunction of into the n region from the metallurgical junction. Thiseffective doping profile is referred to as a linearly graded junction.

Linearly Doped Junctions The point x = x' corresponds to the metallurgical junction. The

space charge density can be written as (x) = eax where a is thegradient of the net impurity concentration.

The electric field and potential in the space charge region fromPoisson's equation and the electric field can be found as:

The electric field in the linearly graded junction is a quadraticfunction of distance. The maximum electric field occurs at themetallurgical junction. The electric field is zero at both x = +x0 andat x = -x0, The electric field in a non-uniformly dopedsemiconductor is not exactly zero, but the magnitude of this fieldis small therefore E = 0 in the bulk regions.

The potential is again found by integrating the electric field as:


ss

eaxxdxdE

)( )(

220

2 xxeadxeaxEss

Edxx)(

Linearly Doped Junctions If we set = 0 at x = -x0 then the potential through the junction is:

The magnitude of the potential at x = + x0 will equal the built-in potentialbarrier for this function. Another expression for the built-in potential barrieris:

If a reverse-bias voltage is applied to the junction, the potential barrierincreases. The built-in potential barrier Vbi is then replaced by the totalpotential barrier Vbi + VR. Solving for x0 and using the total potential barrier,we obtain:

The junction capacitance per unit area can be determined by the same methodas we used for the uniformly doped junction. The junction capacitance is then:


bisss

Veaxxeaxxxeax

303

020

3

32

3)

3(

2)(

20 )ln(i

tbi naxVV

31

0 )}(.23{ Rbi

s VVea

x

312

' })(12

{Rbi

s

VVeaC



dx0

dx0

X = 0

-X0

X0

+

-

P-region N-region

-dQ’

(C/cm3) +dQ’=(x0)dx0 = eax0dx0

Fig. 7.8: Differential change in space charge width with a differential change in reverse-bias voltage for a linearly graded PN Junction.

Note that C' is proportional to (Vbi + VR)-1/3 for the linearly graded junction as compared toC'(Vbi + VR)-1/2 for the uniformly doped junction. In the linearly graded junction, thecapacitance is less dependent on reverse-bias voltage than in the uniformly doped junction.

Hyper-abrupt Junctions The uniformly doped junction and linearly graded junction

are not the only possible doping profiles.

The case of m=0 corresponds to the uniformly dopedjunction and m = +1 corresponds to the linearly gradedjunction. The cases of m = +2 and m = +3 wouldapproximate a fairly low-doped epitaxial n-type layer grownon a much more heavily doped n+ substrate layer.

When the value of m is negative, we have what is referredto as a hyper-abrupt junction. In this case, the n-type doping islarger near the metallurgical junction than in the bulksemiconductor. The equation above is used to approximatethe n-type doping over a small region near x = x0.


mBxN

Hyper-abrupt Junctions


Bx0m

m=+3

m=-1

m=+2

m=+1

m=-2

m=-3

N-type doping profiles

x=0 x0

m=0

Fig. 7.9: Generalized doping profiles of a one-sided p+n junction.

Hyper-abrupt Junctions The junction capacitance can be derived using the same analysis method as before and

is given as:

when m is negative, the capacitance becomes a very strong function of reverse-biasvoltage, a desired characteristic in Varacter diodes. The term Varactor comes from thewords variable reactor and means a device whose reactance can be varied in acontrolled manner with bias voltage.

If a Varactor diode and an inductance are in parallel, the resonant frequency of theLC circuit and the capacitance of the diode can be written in the form:

In a circuit application, we would, in general, like to have the resonant frequency belinear function of reverse-bias voltage VR so we need:

The parameter m required is found from:

A specific doping profile will yield the desired capacitance characteristic.


)2(1)1(

' }))(2(

{

m

Rbi

ms

VVmeBC

LCfr 2

1 )2(

1

0 )(

mRbi VVCC

2VC

22

1

m 23

m

Picture Credits Semiconductor Physics and Devices, Donald Neaman, 4th

Edition, McGraw Hill Publications. B. Van Zeghbroeck, Principles of Electronic Devices,

Department of ECE, University of Colorado,Boulder, 2011.http://ecee.colorado.edu/~bart/book/book/contents.htm

Animation of the PN Junction formation, Universityof Cambridge, 2013.http://www.doitpoms.ac.uk/tlplib/semiconductors/pn.php

W. U. Boeglin, PN Junction, Florida InternationalUniversity, 2011.http://wanda.fiu.edu/teaching/courses/Modern_lab_manual/pn_junction.html


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