p n

photoresist

through photoresist patternion implant of acceptor impurities

after thermal annealingand stripping off photoresist layer

completed pn junction

(a)

(b)

(c)

(d)Boron implant

(ntype) substrate

p(ntype) substrate

n (= substrate)p

80

PROFESSORSs NOTES

8.1 SEMICONDUCTOR JUNCTIONS

Electronics is a philosophy and technology that is defined in terms of active devices. The terminology active de-vice usually means that it is of a construction that controls the flow of currents by means of special layers, patterns,grids, and terminals. For modern electronic circuits, the majority of active devices are semiconductor devices, andalmost all are constructed in terms of layers or layer patterns. These layers and patterns invariably include semicon-ductor junctions, most of which are intentional, some of which are not. In order to assess the characteristics and perfor-mance of an active device we need to understand the electrical characteristics of the semiconductor junctions em-bedded therein.

Although there are many interesting types of semiconductors and therefore many nifty and fascinating types of junc-tions that we can create, it is best to confine our attention to a single semiconductor material and let the others be anextension of concepts. Our best semiconductor choice is silicon, since it is used for the majority of the present genera-tion of circuits. Silicon is favored as a base material since its fabrication processes are relatively straightforward, itdoes not require special handling, is nontoxic, and the raw material (SiO2) is readily available.Semiconductor junctions are formed when two layers of different doping concentrations are metallurgically joined.If we dope one layer as ptype and fuse it to one which is doped ntype, this junction is called a pn junction. Ifthe two layers are of like impurities, we call isotype junctions, but its electrical properties are not as pronouncedas they are for junctions of opposite gender.We will confine our perspective to the simple pn junction and two representative profiles:

(a) abrupt junction(b) linearlygraded junction.

There are probably as many different junction doping profiles as there are electrical engineers. For the sake of simplic-ity and an assessment of the basic electrical characteristics of the junction, we will focus only on the simpler of thejunction types and identify all other junctions as being either approximately abrupt or approximately linear, orsome combination thereto.

The semiconductor junction is usually a result of either deposition of one layer on another, or, more likely, implantationof a concentration of one type impurity into a substrate of the opposite gender, as indicated by figure 8.11, to forma diffused junction.

Figure 8.11: Planar pn junction construction

81

Naturally, variations in the implant and annealing process can make some very interesting junctions. But for the sakeof simplicity we will focus on the junction as being of the simple crosssectional form as indicated by figure 8.11(d).

In terms of the electrical properties we tend to think of pn junctions as being of characteristic form and electrical prop-erties as identified by figure 8.12

p n (a) As represented by

(b) As represented by

encyclopaedia

circuits textbook

p n (c) As fabricated by planarimplant process, ( figure 8.11)

Figure 8.12 pn junction representations.

Our mission, should we choose to undertake it, is to make a good, complete physical identification of the characteris-tics of this basic pn interface, with the greater view of it serving as a electrical component that exists within a numberof other important semiconductor devices, as well being as an important active component in its own right.

8.2 EQUILIBRIUM POTENTIALSOne of the things that we know about semiconductor materials with dopings of opposite gender is that the equilibriumindex for average electron electron energy, the Fermilevel, is placed very differently for the two types, as representedby figure 8.21(a). For the ptype material, the Fermi level is close to the valence band edge, EV, whereas for thentype material it is close to the conduction band edge, EC. The difference between the Fermi levels represents a differ-ence in the average electron energy. So when the materials are metallurgically joined, thermodynamics forces thematerials to come to equilibrium, as represented by figure 8.21(b).

In this case, the transfer of energy is manifested by a migration of electrons in the vicinity of the metallurgical junction,migrating from the ntype material to the ptype material and creating a difference of potential across the junction.The transfer of energy across the junction also may be represented in terms of a bandbending of the energies ECand EV associated with the crystalline lattice. This difference in the lattice energy also is a means of identifying theelectrical potential that is developed across the junction.

EC

EVEF

EC

EV

ntypeptype

EC

EVEF

EC

EV

transition

EF

(a) separated p and ntype materials. EF represents the average electron energy

(b) Joined p and ntype materials. Energy (electrons) must migrate from n to pon order for thermal equilibrium to occur. Note that this creates a bandbendingenergy change from p to n materials. When the materials are at thermal equilibri-um then EF = same throughout.

(Separated)

ptype

ntype

q

0

82

Figure 8.21: Equilibrium processes in the fusion of n and ptype materials.

We can identify this difference of potential by use of the relationships in which the chargecarrier densities are relatedto energy by means of thermal statistics:

n ni exp (EF E in) kT (8.21a)

p ni exp (E ip EF) kT (8.21b)

Recognizing that the energies are electron energy levels, then every one can be replaced by an equivalent voltage po-tential, which relates to the electronic charge, q, by:

(8.22) E (

q) or E

q

As indicated by figure 8.22. In the figure, the drop in energy of crystalline lattice across the junction from lefttoright indicates a positive increase in electrical potential, since the electrons are of negative polarity (!).

Using the association identified by equation (8.22), the free electron densities on each side of the junction can beexpressed as:

np ni exp ( F pi ) VT (8.23a)

nn ni exp ( F ni ) VT (8.23b)

Take note of the syntax that we use in equation 8.23. Since we have two sides we must identify a notation for theelectron densities on each side. Density nn represents the electron density on nside of the junction and np representsthe (minoritylevel) electron density on the pside of the junction. We can similarly identify hole densities pp andpn.

If we take the ratio of 8.23a to 8.23bnpnn exp ( pi ni ) VT (8.24)

n (= substrate)p

p n

NA ND

83

where we have, for convenience defined the thermal potential as being associated with the thermal fugacity kT, by

VT kTq thermal potential (8.25)

Now, since the pn junction is in an equilibrium state, thennp

n2i

NA and nn ND

assuming the we are operating at moderate temperatures for which ni

EC

EV

EC

EVEF

ptype

ntype

q

0

NA ND

metallurgical boundary

!"#$% &'()+*,''-,.'')'0/21)'3)45

687:9

';)(

J

x

x

x

"!$#&%(')*# +,-'.#&/$01#201')354-687

9%:%;0

86

As indicated by Figure 8.34, the separation of charge in the junction region forms an electric field and potential there-to. The characteristics of this spacecharge region can be analyzed straightforwardly by the application of Gausslaw, which, in onedimensional form, as is the case represented by figure 8.34, is:

dEdx

s(8.31)

where es is the permittivity of the semiconductor material. For silicon, the permittivity s 1.045 pF/cm[1]. Thedensity of the uncovered doping sites is given by

qNA for xp x 0

qND for 0 x xn

where the dimensions of the spacecharge region (which we will call the SCR), are indicated by figure 8.34. TheSCR does not terminate at xp and xn as abruptly as indicated by the figure. The transition from SCR to the neutralregions are more on the order of a FermiDirac distribution, but the distinction is not sufficiently different from theuniform depletion approximation to merit the extra mathematical complexity.

Integrating equation (8.31) from the left,

E

0

dE qNAs

x

xp

dx

for which we get

E(x) qNAs

(x xp) for xp x o (8.32a)

If we integrate equation (8.31) from the right, using x = x, which admittedly is backward from our usual way ofthinking, but is perfectly OK for the mathematics, the result is

E(x) qNDs

(x xn) for xn x 0 (8.32b)

of course, at x = 0 and x = 0, the electric field is a maximum, and is

E(0) qNAxps

qNDxn

s

EMAX (8.33)

Charge balance offers us some simplification since

qNAxp qNDxn QS (8.34)

[1] The relative permittivity of silicon is r = 11.8. Since the vacuum permittivity 0 = 8.85 1012 F/m =

8.85 pF/cm, then

s 11.8 (8.85 pF m) 104.5 pF m 1.045 pF cm.

We might as well use this value, rather than retracing our computational process every time.

87

We have identified QS as charge/area that is uncovered on each side of the SCR. This equality also gives us a way toidentify the full thickness, W, of the spacecharge region (SCR),

W xp

xn (8.35)

since equation (8.34) allows us to relate the boundaries xp and xn to each other, for which

xnxp

NAnD

so that

W xp

1 xnxp xp 1

NAND

for which

xp W

1 NAND (8.36a)

and similarly,

xn W

1 NDNA (8.36b)

We can apply (8.36) to equation (8.34) to put the charge/area, QS, in terms of depletion layer thickness W,

QS qNAxp qNAW 1 NAND which we can express as,

QS qW NAND(ND NA) qWNB (8.37)

where

1NB

1ND 1

NA(8.38)

We have taken some extra pains to explain the development of equations (8.37) and (8.38) since they provide somenice simplification options.

When we continue the analysis, to obtain the relationship between the potential across the junction and the extent of theSCR, we use the definition of electric field as a gradient of the potential, which in one dimension, is:

E dVdx

Applying this definition to equation (8.32) and integrating, we get potential drop across the pside of:

Vp qNA

s

0

xp

(xp x)dx qNAx2p

2 s(8.39a)

88

and if we likewise integrate from the right,

Vn qND

s

0

xn

(xn x )dx qNDx2n

2 s(8.39b)

for which the total junction potential will be

J Vp Vn

qNAxp2 s

xp

qNAxp2 s

xn

QS2 s

(xp xn)

since qNAxp = Qs = qNDxn, as given by equation (8.34). Now, since W = xn + xp we can rewrite this equation verysimply as

J

QS2 s

W qW2NB

2 s(8.310)

We are usually interested on just how the thickness of the SCR layer relates to the junction potential, which, from(8.310) will be given by

W 2

s

J

qNB

(8.311)

This equation is complete, but has a much more convenient form. Any time we associate electrostatic fields with anextended spacecharge layer within a material, a characteristic length[2], called the Debye length,

LB

sVTqNB

(8.312)

can be defined. Using this characteristic length in the definition of depletion width, equation (8.311) can be writtenas

W LB 2

J VT

(8.313)

We sometimes call the ratio

J/VT the normalized junction potential. Equation (8.313) is a form that will be useful inthe description of many devices for which the junctions are approximately abrupt and doping densities are approxi-mately uniform.

[2] The Debye length is actually defined in terms of the total charge density level, and emerges any time we

make a Gausslaw analysis of a distributed charge density, whether at the molecular level, as indicated by thistreatment, or at the atomic level. The Debye length is

LB

sVTq(n

p)

but since, for isolated dopings within a semiconductor, (n + p) = either NA or ND. We have taken the libertyof applying it to the junction with NB as defined by equation (8.313).

89

It is also convenient to take note that equation (8.33) will give

EMAX QS

s(8.314)

If we combine this equation with (8.310), we get the nice simplification

EMAX 2 JW (8.315)

To get a feeling for typical layer thicknesses and Efields within the semiconductor junction, consider the followingexample:

*************************************************************************************

EXAMPLE E8.31: An abrupt silicon pn junction is formed by creating an implant of NA = 5 1016 #/cm3into an ntype substrate of doping ND = 1015 #/cm3. Determine: (a) Builtin potential 0 , and (b) W and (c) EMAX at this (zero bias) condition. Assume default temperature 300K.

SOLUTION: (a) According to equation (8.23), the builtin junction potential is

0 VT ln

NANDn2i

.02585 ln 5 1031

2.25 1020 0.675V

(b) As a matter of convenience, we will find the depletion layer thickness W by first applying equation(8.312) to find Debye length. For the doping levels given, we get

L2B

sVTqNB

1.045pF cm .02585V1.6 10 7pC

11015# cm3

15 1016# cm3

1.725 10 10cm2

for which we get Debye length:

LB 1.31 10 5 cm

This can also be expressed as:

LB 131 nm 0.131 m

Note that we need to pay careful attention to units of measure.

You might take note of the choices of the units of measure used in this calculation, for example, s = 1.045pF/cm and q = 1.6 107 pC. This scheme may make the process of keeping track of unit magnitudes a littlesimpler, and it makes sense to designate these quantities in terms of picosize magnitudes.

90

From this measure of Debye length we get SCR layer thickness of

W LB 2

0 VT

0.131 m 1.35.02585

0.9495 m

(c) Equation (8.315) gives us a quick convenient means of determining the electric field at the metallurgicalboundary once we have identified the layer thickness W. Application of this equation gives

EMAX 2 0.675 V0.9495 m

1.42 V

m 1.42 104 V

cm

Take note that the breakdown voltage of air, for which we are always able to see fairly spectacular effects, is E= 104 V/cm. The Efields in the pn junction are formidable fields indeed! In this case, the equilibrium Efield, for which NO external potential has been applied, is greater than the Efields that exist within naturallightning storms!

*************************************************************************************

CASE II. The Gradual (Linearlygraded) Junction

For processes in which the junction is annealed over a long period of time, impurities will migrate and diffuse furtheracross the transition region, which tends to make the junction transition more gradual. To firstorder, this diffusedjunction may be assumed to be approximately linear, for which we may identify charge regions and electric fields be-havior much like that represented by figure 8.36

Often the pn junction may be assessed as a linear profile in the vicinity of the transition from p to n, with the dopinglevels beyond the junction region being relatively uniform, as represented by figure 8.35. Therefore it would be ap-propriate to combine a linearlygraded analysis with an abrupt analysis.

ND

NA

a = (ND NA)/d

linearly graded

Figure 8.34 Representation of gradual junction profile, assuming that the transition is approximately linear.In the vicinity of the metallurgical junction, the transition is approximately linear, and therefore the junction may beanalyzed as a linear distribution of uncovered charge sites, with the junction boundary being defined either by thechange of polarity of uncovered sites or when ND = NA. For convenience of analysis, this point we should let thispoint be the center of coordinates, as represented by the charge analysis represented by figure 8.36.

J

x

x

x

!" !#$%'&)(+*

,-./!0213#/).54678:9

/;$ !?

92

E(x) qas

x

W 2

xdx qas

12 x

2 W24 (8.317)

When x = 0, the electric field is at its maximum,

EMAX qa8 s

W2 (8.318)

The relationship between the thickness of the SCR (= W) and the potential across the junction is readily obtained byintegration of equation (8.317) according to the definition of the electric field as a gradient in the electrostatic poten-tial, for which

J

W 2

W 2

E(x)dx qa2 s x3

3

xW24

W 2 W 2

qaW312 s

(8.319)

The width of the depletion (SCR) layer is then

W 12

s

Jqa

1 3

(8.320)

The junction potential itself is still of the form

J

BI VR

where

BI is given by equation (8.26) and VR is the applied reverse bias. The use of equation equation (8.320) isconstrained by the fact that linear profile is not infinite in extent. Doping levels are expected to reach approximatelyuniform limits ND and NA far from the junction, for which equation (8.26) is applicable. Equation (8.320) is a repre-sentation of electrical characteristics for the charge distribution that is uncovered by the builtin plus applied fields.

The relationship between EMAX and

J is defined by taking the ratio of equations (8.318) and (8.319)

EMAX 32

J

W (8.321)

This result might be compared to the analogous result, for the abrupt junction, as given by equation (8.3.15).

*************************************************************************************

EXAMPLE E8.32 A diffused junction has a profile, as shown, that makes an approximately linear transi-tion over a distance d = 5 m from NA = 4

1016#/cm3 to ND = 1

1016 #/cm3. (a) Determine the location ofthe junction boundary by finding XP and XN. (b) Determine the equilibrium value of W. (c) Determine theupper limit of voltage such that the extent of the SCR remains within the linear profile.

NA = 4 1016

XNXP

ND = 1016

93

SOLUTION:

(a) Using similar triangles,XPXN

NAND

Since d 5 m XP XN XP

1

XNXP

XP

1

NDNA

then XP d 1 ND NA 5 m 1 1016 (4 1016) 4 m

similarly, XN d 1 NA ND 5 m 1 (4 1016) 1016 1 m

(b) Using equation (8.26) BI VT ln NAND n2i

BI

.02585 ln (5 1032)

(2.25 1020)

0.729 V

and grading coefficient a |4 1016 ( 1016)|

5 10 4 1.0 1020#

cm3

Then, using equation (8.320)

W

12 s Jqa

1 3

W (12 1.045 pF

cm 0.729V )(1.6 10 7 pC) (1 1020 cm 3)

1 3

0.8295 m

(c) W2 min(XN, XP) 1 m

then from equation (8.319)

J

qaW312 s

(1.6 10

7) (1.0 1020) (2 10

4)312 1.045

10.22V

then VR 10.22V 0.7292 9.492 V

This result tells us that if we apply a VR = 9.492 V to the junction then W/2 = 1 m = XN, which is the limit ofbias for which the bilateral linear junction characteristics remains valid. If we exceed this potential, then wemust evaluate the junction as if it were a linear profile on the NA side and a uniform profile on the ND side.

*************************************************************************************

Example E8.32 Also tells us that, for the onesided junction, most of the linear gradient will lie on the heavilydopedside of the junction boundary. In many cases it is best to assess the junction as if it is half linear and half uniform.Then we must apply the results of both of these type profiles to analyze the junction behavior. For example, if we hadchosen NA = 1.73 1017 #/cm3, then XN would have been 0.273 m and W at J = BI would have been 0.546 m.A number of interesting problems that combine linear and uniformlygraded junction profiles are available at the endof the chapter.

94

8.4 JUNCTION CAPACITANCE

As noted by the previous sections, the pn junction in reverse bias is characterized by a spacecharge region (SCR) inwhich doping sites of opposite polarity are uncovered on both sides of the junction. As long as the junction is kept inreverse bias, no current will flow. This aspect is exactly the same as that for a capacitance. The pn junction happens tohave considerable capacitance, since the thickness of the SCR is small, usually on the order of microns, as was repre-sented by example E8.31.

Figure 8.41: Spacecharge region and separation of charges = pn junction storage capacitance.

As might be expected, however, the pn junction capacitance is voltage dependent, which may make it unsatisfactoryfor some applications but makes it invaluable for others. As a voltage variable capacitance it is usually referred to as avaractor, although it is in fact just another durn pn junction.

To get a handle on the capacitance behavior of the junction in reverse bias, consider the junction slice represented byfigure 8.42. We should acknowledge that the effective capacitance to which timevarying signals will respond isdefined as

C dQdV (8.41)

where, as represented by the figure, dQ represents the increment of charge with dV, which we might note takes place atthe outside boundaries of the SCR as illustrated by figure 8.42.

Figure 8.42: pn junction incremental capacitance.

Since this incremental charge is added at the boundaries, then the separation between +dQ and dQ is the SCR layerthickness W, so that the junction capacitance/area is

+_

p n

W

p n

dQ

dQ

95

CJ

s

W (8.42)

If you dont like this lazy (but accurate) argument, then recognize that junction capacitance can also be defined byexamining the Efield. Since an increment in charge also represents an increment in the Efield, then

dE dQs

The corresponding change in the applied voltage is

dV WdE W dQs

(8.43)

from which we get the same result for junction capacitance CJ as given by equation (8.42).

But since the increment of charge also represents more of the doping sites being uncovered, we might also recognizethat equations (8.42) and (8.43) represent a means for examining this profile, particularly if one side of the junctionis very heavily doped, so that there is relatively little effect on W due to the doping sites uncovered on its side. In thisrespect we may approximate. If, for example, we consider a p+n junction (for which NA >> ND) then

dQ qNBdW qNDdWthen

dV W dQs

W (qNDdW )s

qND

s WdW qND2 s

d(W2) qN(W)2 s d

2s C2J

so that the doping profile can be examined by means of the slope of 1/CJ2 with respect to V.

N(W) 2q s

ddV

1C2J

1

(8.44)

For the abrupt junction the depletion capacitance will beCJ

s

LB 2( 0 VR) VT

(8.45)

For the linearlygraded junction the depletion layer capacitance is

CJ qa 2s

12( 0 VR) 1 3

(8.46)

We might take note that both of these equations may be written as

CJ CJ0

1

VR 0

1 (m 2)(8.47)

96

for which CJ0 is a constant, corresponding to the zerobias (VR = 0) capacitance. This is the form for junction capaci-tance that is used by the SPICE software. It default to m = 0, corresponding to the abrupt, uniform junction profile,for which

CJ0

s

LB

2 0 VT

(8.48)

*************************************************************************************

EXAMPLE 8.41: Consider the circuit for which the capacitances are replaced by reversebiased diodes,as shown by figure E8.41. This circuit is a bandpass circuit and the peak frequencies are defined by

1C R1R2 (E8.41)

When the diodes are reversebiased they will behave as if theyare voltagecontrolled capacitances of the form:

CJ CJ0 1 VR 0 1 (m 2)

where CJ0 is the SPICE diode model parameter CJO.

Figure E8.41: DellyannisFriend Biquad

We execute the DF biquad in SPICE, choosing CJ0 to be 400 pF. Using either the .STEP command or someother SPICE input option, we may apply bias sequence V = {0, 2, 6, 14}V. Since a voltage divider exists at theinput we will will get a diode reverse bias of VR = {0, 1, 3, 7}V applied to the two diodes. Using PROBE tosee all traces concurrently, then we will see a SPICE output something like that shown by figure E8.42.

Measured values, using PROBE cursorf1 = 3.95 kHz (for VR = 0 )f2 = 5.53 kHz (for VR = 1 )f3 = 7.76 kHz (for VR = 3 )f4 = 10.78 kHz (for VR = 7 )

Figure E8.42: Approximate transfer response for DF biquad with biased varicaps.

Using these measurements we can find the values of CJ from equation (E8.41), for which

CJ1 = 403 pFCJ2 = 288 pFCJ3 = 205 pFCJ4 = 148 pF

From the plot we get VBI = 1.2 V

*************************************************************************************

+

2R1 = 20 k R2 = 1 M

vi V+

vo

2R1 = 20 k

|vO|

f(Hz)10k3k1k 30k

f1 f2 f3 f4

VR

1/C2

VBI

97

The grading constant, m, defines the profile, which may be assumed to be of the form:

N N0

xx0

m

(8.49)

When m = 0, the doping profile on each side of the junction is constant. When m = 1, the doping profile is linear.For the special case in which m = 3/2, which we call the hyperabrupt doping profile, we then have a junction capaci-tance for which

CJ CJ0 1 VR 0

2 (8.410)

NA

ND N0 xx0

3 2

Figure 8.43 Typical hyperabrupt junction profileSo that, should this junction be used in a resonant LC circuit,

f 12 LC

and if we use a hyperabrupt junction, the frequency will be directly proportional to the applied bias:f

1 VR 0

giving us a frequency that is linearlycontrolled by applied voltage. Pretty neat, huh?

8.5 PN JUNCTION IN FORWARD BIAS: LOWLEVEL INJECTION

When the pn junction is forward biased, the inhibiting force of the electric field is reduced in magnitude. Charge carri-ers are more free to emigrate across the junction, and current flow will take place. The emigration of charge carriersis primarily a diffusion process, which is a thermal process and is driven by thermal statistics.Therefore we turn to our thermal statistics to identify the effects that are taking place and define the basic junctionelectrical characteristics. If we look at the density of electrons on both sides of the junction, as defined by the thermalstatistics we see that

nn NC exp (EFn ECn) kT (8.51a)

np NC exp (EFp ECp) kT (8.51b)

where EFn , ECn are the energy levels on the nside of the junction and EFp, ECp are the energy levels on the psideof the junction, respectively. If the junction is at equilibrium, then EFn = EFp, and we are back to equation (8.24),with

0 defined by the difference in the lattice energy levels across the junction or the equivalent electron potentials,for which

EC

EV

EC

EVEF

ptype

ntype

q

0

NA ND

V = 0

ECp

EVpEFp

q

NA ND

V > 0

ECn

EVn

EFn

qV

98

(8.52)nnnp exp (ECp ECn) kT exp( 0 VT)

where

(8.53) 0

(ECp ECn) q

(E ip E in) q

in ip

as defined by equation (8.26).

Solving equation (8.52) for np, for which the equilibrium value can be designated as np0, we get

(8.54)np0

nn exp(

0 VT )

Figure 8.51. Junction potentials (a) at equilibrium and (b) at forward bias

When the junction is at forward bias, for which EFn > EFp , then the junction potential is reduced by an amount,(8.55)V

(EFn EFp) q

as represented by figure 8.51. We also see that

(8.56)ECp ECn

q

q( 0 V)

for which equation (8.52) becomes

(8.57)nnnp exp VT exp ( 0 V ) VT

and equation (8.54) becomes

(8.58)np

nn exp

( 0 V) VT

If we take the ratio of equations (8.57) and (8.54) we get

(8.59)np

np0 exp V VT

99

This result is the level of electrons that are injected across the junction by the thermal processes. The voltage V isthe applied bias that imbalances the thermal equilibrium and allows carriers to migrate across. Since the relationshipis exponential, the forward conduction current is very strong.

Similarly, holes will also emigrate from the pside into the nregion and we will see an injected hole density of

(8.510)pn pn0 exp

V VT

Naturally, these injected carrier densities will upset thermal equilibrium on each side, and the recombination processeswill then begin to act to restore equilibrium.

If we make the approximation that the minoritycarrier populations will be the primary densities that are affected, thenwe can assess the junction current by following the action of the minoritycarrier levels. This assumption is calledlowlevel injection, corresponding roughly to the condition

np < 0.1 pp and pn < 0.1 nn lowlevel injection (8.511)If we have a forward bias V such that the majoritycarrier densities are also affected, then we are in a situation whichwe identify as highlevel injection, and the analysis becomes more complicated. However, in most cases, the normaljunction operation corresponds to lowlevel injection, so we will postpone high level injection to later entertainment.

The zones beyond the space charge region (SCR) include many charge carriers. As the excess enemy carriers of theopposite gender invade these regions, war is declared, and recombination processes take place. A combination of dif-fusion processes and recombination processes make up the factors that drive the carrier levels toward equilibrium.There will be relatively little free charge within these regions, so they would aptly be described as quasineutral re-gions (QNR), and are so represented by figure 8.52.

Figure 8.52: The quasineutral regions.

Since a nonequilibrium situation exists, recombination and diffusion processes will govern the fate of the injectedcharge carriers. In the case of excess electrons injected into the pside, the density will have a survival profile given by

(8.512) np

np(0) exp( x Ln)

where

np(0) is the level of excess carriers above equilibrium at the boundary between SCR and the QNR for the ptype side. This excess level of carriers = np(0) np0. The parameter Ln is the recombination length, also called thediffusion length, and is given by

(8.513)Ln Dn n

p

n

SCRQNR QNR

pn

np

xx

100

We need to note that Dn = nVT is the diffusion constant for ntype carriers in the ptype environment and n is therecombination decay time constant for these carriers.

The invading ntype carriers recombine as result of an encounter with the deadly holes, which results in an annihilationof both, and emission of a photon or phonon as a marker of the encounter. The level np(0) is the level of carriers enteringthe pside (at x = 0) after having migrated across the somewhat diminished SCR barrier.Similarly, the density of minority type carriers injected into the nside will be

(8.514) pn

pn(0) exp( x Lp)

where

pn(0) represents the excess injected carrier density at x = 0. These injected carriers will suffer the same recom-bination fate as their cousins, according to a diffusion and recombination process for holes, as defined by the recom-bination (diffusion) length

(8.515)Lp Dp p

We note that Dp = pVT is the diffusion constant for ptype carriers in the ntype environment and p is the recombina-tion time constant for these carriers. Whether we consider the injected holes or electrons it is essential that we identifythe action charge carriers are the minority carriers, and therefore we must identify the diffusion constants or mobilitiesfor these minority carriers NOT the majoritycarrier diffusion constants.The minoritycarrier levels for the junction in forwardbias are shown by figure (8.53). Typical recombinationlengths are on the order of 10 m.

Figure 8.53: Injected carrier profiles.Since equations (8.512) and (8.514), and figure 8.53 identify that the injected carriers will have a gradient in densi-ty, due to recombination, then also they also imply that we will have two components of diffusion current that occur:

(8.516a)Jn qDn ddx [

n(x)]x 0

qDn

np(0)Ln

(8.516b)Jp qDp ddx [

p(x )]x 0

qDp

pn(0)Lp

Since

(8.517a) np(0) np(0) np0 np0 exp(V VT) np0

(8.517b) pn(0) pn(0) pn0 pn0 exp(V VT) pn0

p n

SCRQNR QNR

pn(x)np(x)

pn0np0

101

then

J Jp Jn

J

qDnnp0

Ln qDp

pn0Lp

exp(V VT) 1

J JS

exp(V VT) 1 (8.518)

This equation is called the Shockley equation, or also the ideal diode equation, and is the effective description of cur-rent in the forward direction.

Or it is ALMOST the description of current in the forward direction. There is more, as we will see in coverage given insection 8.7.

In equation (8.518), JS is called the (reverse) saturation current density, given by

JS

qDnnp0

Ln qDp

pn0Lp

or, simplifying, we get JS qn2i

DnLnNA

DpLpND

(8.519)

since np0 = ni 2/NA and pn0 = ni 2/ND. As we see from the Shockley equation, this is the level of current that results whenV < 0, corresponding to reverse bias. It is small, on the order of fA/cm2. When biased in the forward direction, typicaljunction current density levels are on the order of A/cm2.

8.6 QUASIEQUILIBRIUM STATISTICS AND QUASIFERMI LEVELS

Since, at forward bias, we clearly are in a nonequilibrium situation, and the equilibrium statistics that we used socheerfully with equation (8.21) are ruined. Equation (8.21) assumed that the index, EF, for equilibrium, had to be aconstant, from one type semiconductor, all the way across the junction, to the other type semiconductor, for which wecould readily identify a builtin voltage 0 as a consequence.

But when the semiconductor is in forward bias, the Fermi energies on the opposite sides of the junction are no longerthe same, so we might as well identify them as EFp and EFn. As indicated by figure (8.52b) the difference betweenEFn and EFp is just

qV EFn EFp (8.61)When we are well away from the junction, deep within the quasineutral region, we expect that thermal statistics,such as is represented by equation (8.21), is fine. Equations (8.62) should be fine, and our use of the massactionlaw,

pn0 n2i nn0

as assumed so cheerfully by equations (8.517), should also be fine and reasonable. Far from the junction, conductionis entirely identifiable in terms of majority carrier flow, for which equilibrium thermal statistics is fine.It is only in the vicinity of the junction that the thermal statistics may be compromised, because it is only in the vicinityof the junction for which pn > pn0 and np > np0.We may retain all of the simplicity of the thermal statistics by assuming a state of quasiequilibrium for which equa-tions (8.21) are valid, provided that we define an EFn and EFp everywhere. We therefore define EFn and EFp as quasi

102

Fermi energy levels since they represent a statement of quasiequilibrium. For minoritycarrier levels as well as ma-jority carrier levels, equations (8.21) will apply, for which:

np ni exp

(EFn(x) E ip) kT (8.62a)

pn ni exp

(E in EFp(x)) kT (8.62b)

where an EFn is now assumed to also be defined on the pside of the junction concurrently with Eip, and is distinctlydifferent from EFp. Similarly, we expect an EFp(x), distinctly different from EFn which may be defined on the nsideof the junction concurrently with Enp). The quasiFermi levelsNote that, for lowlevel injection conditions, equations (8.21) are still just fine, and define the majoritycarrier levels

nn ni exp (EFn E in) kT (8.63a)

pp ni exp

(E ip EFp) kT (8.63b)

If we take the product of equation (8.62a) and equation (8.63b), and assume that we are at the injection boundaryof the QNR, for which x = 0, we get

nnpn n2i exp

(EFn EFp) kT (8.64)

Using equation (8.61) and nn ND, thenpn

n2iND

exp qV kT pn0 exp V VT (8.65)

which is exactly the same as equation (8.59). Similarly, we could take the product of equations (8.62b) and (8.63a),for which, in like manner, we would find

np

n2iNA

exp qV kT np0 exp V VT (8.66)which is the same as equation (8.510).We expect that the behavior of the quasiequilibrium levels will be something like that represented by figure 8.61.

Figure 8.61: Nonequilibrium conditions, for which it is convenient to define quasiequilibriumFermi levels. In the SCR and in the QNR region near to the junction, EF splits into EFn and EFp. Far from thejunction where there is little excess minority carrier levels, the quasiequilibrium Fermi levels coincide.

The figure shows the approximate behavior of the quasiFermi levels, EFn and EFp across the junction, as continuouslevels extending from one side of the semiconductor to the other. Since we are in a nonequilibrium state of forwardbias, EF on the nside is higher than EF on the pside by bias energy qV, as is given by equation (8.61). Far fromthe junction, the quasiequilibrium levels are coincident, EFn = EFp = EF. We might take note of the fact that, in orderto satisfy equations (8.65) and (8.66), it is necessary that EFn and EFp extend all of the way across the SCR withlittle or no change.

p

n

SCRQNR QNR

EFp

EFn

EFpEFn

qV

ECp

EVp

EFp

ECn

EVn

EFn

qV

SCR QNRQNR

Ei

103

In the vicinity of the junction EFn and EFp will split into two levels according to equation (8.62). The split betweenthe levels is an indication of the levels of injected minoritycarriers, np = np(x) and pn = pn(x), as represented by equa-tions (8.512) and (8.514). Since pn(x) asymptotically approaches pn0, we expect that EFp(x) will asymptoticallyapproach EFn, as represented by the figure. Similarly, on the pside, we expect that EFn(x) will asymptotically ap-proach EFp, as governed by equation (8.512).

8.7 RECOMBINATION OF CHARGECARRIERS IN THE SPACECHARGE REGION.

Yes, brothers and sisters, recombination also take place in the SCR. After all, carriers that dare to attempt a transit ofthis region are in a nomans land, in which chargecarriers of both types are present. The usual warfare takes place, inwhich carriers annihilate each other and constitute a recombination current of the form

Jn Qn

n(8.71a)

for ntype carriers, where n represents the recombination time constant for electrons within the SCR. We should havea similar form for the ptype carriers, given by

Jp Qp

p(8.71b)

for which p represents the recombination time constant for holes within the SCR.Since the densities, n and p, of both type chargecarriers, vary across the spacecharge region by several orders ofmagnitude, the process for determining an appropriate

Qn and Qp can be very mathematical if we so choose. In

order to gain insight without the mathematical encumbrance, it is reasonable to make a number of analytical approxi-mations that will make the process a little more tame.

The behavior of energy levels within the SCR is reflected by figure 8.71. Since a builtin Efield exists, the latticeenergies undergo a bandbending as represented by the figure, while the quasiequilibrium Fermi levels remainapproximately constant across the region, as discussed by section 8.6.

Figure 8.62: Energy level behavior within the SCR

We can analyze the carrier densities by means of equation (8.6.2) with knowledge of Ei(x) as function of position withinthe SCR. But we would find that the mathematics would be a task of unwelcome proportions, since we would beentertaining ourselves with integrals of exponential functions. We can accomplish just as much by use of a few approx-imations gained from applying equations (8.62) to the SCR, for which

104

n ni exp

(EFn E i(x)) kT (8.73a)

p ni exp (E i(x) EFp) kT (8.73b)

If we take the product of these two equations, we get:

np n2i exp(EFp EFn) kT n2i exp(V VT) (8.74)

As one of the expectations of lowlevel injection, we expect that somewhere within the spacecharge region the carrierdensities will be n p. Using equation (8.64), this level corresponds to

np n2 p2 n2i exp(V VT)and therefore, taking the square root, the crossover carrier level will be

n p ni exp(V 2VT) (8.65)

EFp

EFn

SCR

Ei

Ei

105

This concept is reinforced by the fact that the differences (EFn Ei) and (Ei EFp) change from one side of the SCRto the other as Ei = Ei(x) changes, as represented by figure 8.62. The point within the SCR at which equation (8.65)is met will not necessarily be at the metallurgical junction.

The approximate behavior of carrier densities from one side of the SCR to the other, with the junction in forward bias,is represented by figure 8.63. If we also make the approximation that the regions within the SCR for which recom-bination takes place are approximately triangular, as also represented by figure 8.63, then

Figure 8.63: Region of recombination in the SCR for excess ntype carriers.

Jn(SCR) q2W

n(SCR)

n(8.66)

If we assume that the capture crosssections for electrons and holes are about the same, then cn = cp, and equation(7.88) will be somewhat simplified,

(7.810)R cnNt(pn n2i )

n p 2ni cosh

106

for which we make the definition of shortcircuit resistance

(15.516)RSCP V VTP

IDSAT

1KP(V VTP)

(15.35)KR

(VI VT1) V VO VT2

t f dt

CLRSC1 0.1

0.9

dyy(2b

y)

CLRSC1 12b ln y

2b

y

0.1

0.9(15.313)

dI(x, t) C V

t dx

dV(x, t) L

I

t dx

(15.6.3a)

(15.6.3b)