Transverse Averaging Technique for Depletion Capacitance of Nonuniform PN-Junctions

a r X i v :c o n d -m a t /0701552v 1 [c o n d -m a t .o t h e r ] 23 J a n 2007Transverse Averaging Technique for Depletion Capacitance of Nonuniform

P N -Junctions

Anatoly A.Barybin ?,and Edval J.P.Santos ??

?

Electronics Department,

Saint-Petersburg State Electrotechnical University,

197376,Saint-Petersburg,Russia ??Laboratory for Devices and Nanostructures,

Departamento de Eletr?o nica e Sistemas,

Universidade Federal de Pernambuco,

C.P.7800,50670-000,Recife-PE,Brasil

E-mail:edval@ee.ufpe.br

This article evolves an analytical theory of nonuniform P N -junctions by employing the transverse

averaging technique (TAT)to reduce the three-dimensional semiconductor equations to the quasi-

one-dimensional (quasi-1D)form involving all physical quantities as averaged over the longitudinally-

varying cross section S (z ).Solution of the quasi-1D Poisson’s equation shows that,besides the usual

depletion capacitance C p and C n due to the p -and n -layers,there is an additional capacitance C s

produced by nonuniformity of the cross-section area S (z ).The general expressions derived yield the

particular formulas obtained previously for the abrupt and linearly-graded junctions with uniform

cross-section.The quasi-1D theory of nonuniform structures is demonstrated by applying the general

formulas to the P N -junctions of exponentially-varying cross section S (z )=S 0exp(αz )as most

universal and applicable to any polynomial approximation S (z )?S 0(1+αz )n .

Keywords:I.INTRODUCTION Analytical methods have the advantage of promoting insight into device behavior,guiding the interpretation of numerical simulations.However,the analytical theory of semiconductor devices in terms of rigorous physical equations is basically developed in complete form only within the framework of the one-dimensional models [1].Allowance for nonuniformity in the doping impurity distribution and cross-sectional geometry peculiar to various semiconductor devices requires applying the two-or three-dimensional models.Their application can yield practi-cally useful results only if one resorts to numerical calculations.But the numerical treatment of the sets of partial di?erential equations over the two-or three-dimensional domain often proves to be computationally too intensive even for relatively simple physical models and device structures.In most instances,the multidimensional treatment,despite its comprehensiveness,gives too much detailed and often redundant information,which prevents one from clear physical interpretation of numerical results.Moreover,the pure numerical approach is inferior to analytical methods in predictability since it is bound up with the speci?c parameters and geometrical structure of a device under calculation.To deal with this problem,we shall search a mathematical description of complex semiconductor structures that retains the essential physical properties peculiar to the multidimensional models but enables their computational complexity to be reduced.Direction of our search is determined by the following fairly simple considerations.Any semiconductor device is connected with external electric circuits by its terminals so that a driving contribution into the circuit is caused by the internal electronic processes inside the whole bulk of a device under examination.Hence,the integral character of the internal process contributions into the external circuit allows us to transform the input di?erential equations in such a way as to obtain their output form involving some device characteristics directly related to circuit ones (for example,such as the external circuit current and the voltage applied to a semiconductor diode).Such mathematical transformations will be performed by integrating the three-dimensional equations that govern the

internal electronic processes over the cross section of semiconductor structures and so be referred to as the transverse averaging technique (TAT).

Application of the TAT to the fundamental equations of semiconductor electronics converts them into the one-dimensional form.Such a form contains the physical scalar quantities (potential,charge density,etc.)and the longitudinal components of vector quantities (electric ?eld,current density,etc.)which,being averaged over the cross section S (z ),depend only on the longitudinal coordinate z .Besides,the one-dimensional equations obtained by using the TAT will also include some contour integrals along interface lines to take into account the proper boundary conditions between the di?erent domains of semiconductor structure.Such equations will be referred to as the quasi-one-dimensional (quasi-1D)ones.They are completely equivalent to the initial three-dimensional equations and in this respect are accurate but they deal solely with the physical quantities averaged over the cross section of a

2 FIG.1:Schematic illustration of the nonuniform P N-junction with axial symmetry;the depletion layers are situated between

the cross sections z=?W p of area S p and z=W n of area S n.

semiconductor structure.

The transverse averaging approach was?rst applied in paper[2]to the planar MESFET-structures[3]in order for the e?ective boundary of current channel to be found with taking account of carrier di?usion.In this paper,the TAT is developed,when applied to the mesa-structures of nonuniform P N-junctions.In addition to the geometrical nonuniformity given by the cross section S(z),the theory will also allow for the averaged doping pro?lesˉN D(z)and

ˉN A (z);in so doing the three functions of z should be assumed as known.Section2contains the general integral

relations underlying the transverse averaging technique and their application to deriving the quasi-1D depletion layer equations for nonuniform junctions is demonstrated in Section3.Solution of the quasi-1D Poisson’s equation and general analysis of the depletion-layer capacitance for the nonuniform P N-junctions are set forth in Sections4and 5.Section6completes the quasi-1D analysis by applying the general equations to the abrupt and linearly-graded junctions with uniform and exponentially-varying cross sections.

II.INTEGRAL RELATIONSHIPS OF TRANSVERSE A VERAGING TECHNIQUE

In order to apply the transverse averaging technique to the general three-dimensional equations of semiconductor electronics,the known integral relations[4]

S (?·A)dS=

d

dz S(e z?)dS+

L

(n?)dl,(2)

written for the uniform cross section S bounded by the contour L,should be generalized to the case of nonuniform structures with the longitudinally-varying cross section S(z),as shown in Fig.1.It is easy to see that the desired

3 generalization of Eqs.(1)and(2)gives the following forms:

S(z)(?·A)dS=

d

cosθ

dl,(3)

S(z)(??)dS=

d

cosθ

dl.(4)

In Eqs.(3)–(4),ˉA(z)andˉ?(z)are the quantities averaged over the cross-section S(z)(with r being the cross-sectional radius-vector,see Fig.1):

ˉA(z)=1

S(z) S(z)?(r,z)dS,(6)

n is the outward unit vector normal to the boundary contour L(z),andθis the angle of slope of the boundary line a(z)(characterized by a tangential vectorτ)with respect to the longitudinal unit vector e z,so that(see Fig.1)

cosθ=

1

1+(da/dz)2

.(7)

III.DERIV ATION OF QUASI-ONE-DIMENSIONAL DEPLETION-LAYER EQUATIONS FOR

NONUNIFORM P N-JUNCTIONS

Electric?eld inside the depletion layer obeys the usual quasi-static equations[1]:

?·D=ρwith D=?E,(1)

?×E=0with E=???.(2) Applying the TAT equations(3)and(4)to Eqs.(1)and(2),we obtain the following integral relations for a semiconductor medium:

d

cosθ

dl=ˉρS,(3)

ˉE z S=?

d

cosθ

?dl.(4)

Here the average longitudinal?eld and charge density are de?ned similar to Eqs.(5)and(6),namely,

ˉE z (z)=

1

S(z) S(z)ρ(r,z)dS.(6)

4 Outside semiconductor whereρ≡0,the equations analogous to(3)and(4)have the following form(with marking all outside quantities by superscript o):

d

cosθ

dl=0,(7)

ˉE o z S o=?

d

cosθ

?o dl,(8)

where the outward unit vector for the outside region S o is?n.Here S o means an e?ective localization area of the outside?eld E o where usually?o

ˉD

z

S+ˉD o z S o=?ˉE z S 1+?oˉE z S o

ˉE

z

S o

ˉ?

S o

dz (ˉE z S)=

ˉρS

dz

(ˉ?S).(12)

IV.SOLUTION OF QUASI-ONE-DIMENSIONAL POISSON’S EQUATION Substituting(12)into Eq.(11)yields the quasi-one-dimensional Poisson’s equations:

?for the n-depletion layer(where the excess ionized donors create charge densityˉρ=qˉN D≡q(ˉN+D?ˉN?A)>0)

d2

?ˉN

D

(z)S(z)for0

?for the p-depletion layer(where the excess ionized acceptors create charge densityˉρ=?qˉN A≡?q(ˉN?A?ˉN+D)<0)

d2

?ˉN

A

(z)S(z)for?W p

where the depletion layer widths W n and W p(shown in Fig.1)should be found.

In order for the quasi-1D Poisson’s equations(1)and(2)to be solved,we shall apply the usual boundary condi-tions[1]:

?for the n-depletion layer(0

ˉE

z

(W n)=0andˉ?(W n)=V n,(3)?for the p-depletion layer(?W p

ˉE

z

(?W p)=0andˉ?(?W p)=V p,(4)

5

where V n and V p are the voltage drop across the n -and p -depletion layers of widths W n and W p ,respectively.

Integration of Eqs.(1)and (2)with allowing for Eq.(12)and the ?rst boundary conditions (3)and (4)imposed on the electric ?eld ˉE

z yields ˉE

z (z )S (z )=?q ?

A p (z )for ?W p

A n (z )=

W n z ˉN D (z ′)S (z ′)dz ′,(7)

A p (z )=

z ?W p ˉN A (z ′)S (z ′)dz ′.(8)

Integration by parts of Eqs.(5)and (6)with allowing for Eq.(12)and the second boundary conditions (3)and (4)imposed on the potential ˉ?yields

ˉ?(z )S (z )=V n S n +q

?

zA p (z )?B p (z ) for ?W p

where we have denoted B n (z )=W n z z ′ˉN

D (z ′)S (z ′)dz ′,(11)

B p (z )=

z ?W p z ′ˉN A (z ′)S (z ′)dz ′,(12)

and (see Fig.1)S n ≡S (W n ),S p ≡S (?W p ).(13)

Electric ?eld continuity at z =0imposed on relations (5)and (6)in the form

ˉE

z (0?)=ˉE z (0+)(14)gives rise to the equality A p (0)=A n (0),which expresses the electrical neutrality condition written as

0 ?W p

ˉN

A (z )S (z )dz =W n 0ˉN D (z )S (z )dz.(15)Potential continuity at z =0imposed on relations (9)and (10)in the form

ˉ?(0?)=ˉ?(0+)=0

(16)gives expressions for the voltage drop across the n -and p -layers of depletion written as

V n =q

?S n W n 0z ˉN

D (z )S (z )dz,(17)

6

V p =q

?S p

?W p

z ˉN

A (z )S (z )dz.(18)

The total voltage drop across the P N -junction consists of the built-in potential V bi and the applied voltage V [1]:

V

n

?

V

p =

V bi ?V.

(19)

Sign of V is so chosen that the positive (V >0)and negative (V <0)values correspond to the forward-bias and

reverse-bias conditions.From Eqs.(17)–(19)it follows that

V bi ?V =

q

S n

W n

z ˉN D (z )S (z )dz ?1?|V |

.(1)

Here Q is the total charge of excess ionized donors equal to the total charge of excess ionized acceptors taken in the

appropriate layers of depletion,namely,

Q (V )=q

W n (V ) 0

ˉN

D (z )S (z )dz =q 0

?W P (V )

ˉN

A (z )S (z )dz.(2)

Di?erentiating expressions (2),as integrals with variable limits,with respect to |V |,we can rede?ne the depletion-layer capacitance (1)in the following form:

C (V )=q ˉN

Dn S n ?W n (V )?|V |

,

(3)

where by analogy with notation (13)we have denoted

ˉN

Dn ≡ˉN D (W n ),ˉN

Ap ≡ˉN A (?W p ).(4)

Di?erentiating equation (20),as integrals with variable limits W n (V )and ?W p (V ),with respect to |V |=?V (for

the reverse-biased junction)and taking into account that,in accordance with notation (13),

S n (V )=S W n (V ) and S p (V )=S ?W p (V )

,

we obtain

1=

q

S n

?W n (V )

?

ˉN

Ap W p ?V p S ′

p

?|V |

,(5)

where we have denoted

S ′n ≡

dS (z )

dW n

,

S ′

p ≡

dS (z )

dW p

.(6)

7 Expressing the derivatives?W n/?|V|and?W p/?|V|that enter into Eq.(5)in terms of the capacitance C de?ned by(3),we?nally arrive at the desired expression for the inverse capacitance C?1of the nonuniform P N-junction in the following form:

1

C p +

1

C s

.(7)

In expression(7),we have introduced the depletion capacitances for p-and n-layers:

C p=?S p

W p

,C n=

?S n

W n

,(8)

and an additional inverse capacitance C?1s allowing for nonuniformity of the cross-section area S(z):

1

qˉN Ap S p S′p

qˉN Dn S n

S′n

C p+C n =

?S0

2? ˉN D W2n+ˉN A W2p .(2) From Eqs.(1)and(2)it is easy to obtain the following expressions for the partial width of depletion layers:

W p(V)= q(ˉN A+ˉN D)ˉN DˉN A+ˉN D W(V),(3)

8

W n(V)= q(ˉN A+ˉN D)ˉN AˉN A+ˉN D W(V),(4) so that the total depletion width W=W p+W n is

W(V)= qˉN(V bi?V)withˉN=ˉN AˉN D

W(V)

= 2(V bi?V)S0.(6)

Expressions(5)and(6)are fully identical to those previously obtained for two-sided abrupt junctions[1].

For one-sided junctions(e.g.,the P+N-structure with a highly-doped emitter whenˉN A?ˉN D so that W p?W n), expressions(5)and(6)assume the following form:

W(V)?W0n(V)= qˉN D(V bi?V),(7)

C(V)?C0n(V)=

?S0?qˉN D

2

.(9) The voltage relation(20)with allowing for Eq.(9)takes the following form:

V bi?V=q

?

W/2

?W/2

z2dz=

qa

qa

(V bi?V) 1/3.(11)

Because of the symmetry relation(9),the partial depletion capacitances for the p-and n-layers de?ned by formulas

(8)are equal to each other:

C p=C n=

2?S0

W(V)

= qa?2

9 C.Abrupt junction of linearly-varying cross section:ˉN A(z)=constant for z<0,ˉN D(z)=constant for z>0,

S(z)=S0(1+αz).

In this case the electrical neutrality condition(15)takes the following form:

ˉN

A

W p(1?αW p/2)=ˉN D W n(1+αW n/2).(14) The voltage drops across the n-and p-layers of depletion calculated from Eqs.(17)and(18)by using S n= S0(1+αW n)and S p=S0(1?αW p)are equal to

V n=qˉN D W2n

1+αW n

,(15)

V p=?qˉN A W2p

1?αW p

.(16)

Substitution of Eqs.(15)and(16)into equality(19)yields an expression that together with Eq.(14)forms the set of algebraic equations to calculate the dependencies W p(V)and W n(V)and then to?nd the depletion capacitance by using Eqs.(7)–(9).However,there is no point in doing such calculations because the junction of linearly-varying cross section under consideration is a particular case of the more general situation corresponding to the junction of exponentially-varying cross section,which will be proved just below.

D.Abrupt junction of exponentially-varying cross section:ˉN A(z)=constant for z<0,ˉN D(z)=constant for

z>0,S(z)=S0exp(αz).

In this case the electrical neutrality condition(15)takes the following form:

ˉN

A 1?e?αW p =ˉN D eαW n?1 .(17) The voltage drops across the n-and p-layers of depletion calculated from Eqs.(17)and(18)by using S n= S0exp(αW n)and S p=S0exp(?αW p)are equal to

V n=qˉN D W2n

(αW n)2

=

qˉN D

?S p/S0

1?(1+αW p)exp(?αW p)

?α2 eαW p?αW p?1 .(19)

Before applying Eqs.(17)–(19)to the subsequent analysis,let us demonstrate that these equations give rise to the above formulas(14)–(16)for the abrupt junctions of linearly-varying cross section.Indeed,whenαW p?1and αW n?1so that S n=S0exp(αW n)?S0(1+αW n)and S p=S0exp(?αW p)?S0(1?αW p),from Eqs.(17)–(19)it follows that

ˉN

A

W p(1?αW p/2)?ˉN D W n(1+αW n/2),(20)

V n=qˉN D W2n

(αW n)2

?

qˉN D W2n

1+αW n

,(21)

V p=?qˉN A W2p

(αW p)2

??

qˉN A W2p

1?αW p

.(22)

Here the following approximate expansions were used:e x?1+x+x2/2for Eq.(20)and e x?1+x+x2/2+x3/6 for Eqs.(21)–(22),where x=?αW p or x=αW n.

From a comparison of Eqs.(14)–(16)and(20)–(22)follows the conclusion about their perfect coincidence.Moreover, such a conclusion can be generalized to the polynomial function S(z)=S0(1+αz)n,n=2,3,...because the power

10

series of the exponential e x contains all powers of x .Consequently,we can restrict our attention solely to the junction with exponentially-varying cross section S (z )=S 0exp(αz ),as the most universal structure,which allows one to realize any desired polynomial approximation by making choice of the

exponent

factor

α

.

Now,let us return to the equations (18)and (19)just derived above.Their substitution into equality (19)yields an expression that together with Eq.(17)forms the set of transcendental equations to calculate the dependencies W p (V )and W n (V ).

In order to ?nd the total depletion-layer capacitance,?rst insert Eqs.(18)and (19)into expression (9)for C ?1

s

with taking account of S ′p /S p =S ′

n /S n =α,then

1

C p

exp(αW p )?1

C n

1?exp(?αW n )

W p

=

?S 0

W n

=

?S 0

C

=

1

αW p

+

1

αW n

.(25)

When α→0,the above formulas turn into the corresponding formulas (1)–(6)obtained for the uniform two-sided junction.

For one-sided junctions (the P +N -diode with ˉN

A ?ˉN D ,W p ?W n ,and V p ?V n ),substitution of Eq.(18)into equality (19)yields

V bi ?V ?

q ˉN

D 2

αW 0n

2,(27)

where W 0

n (V )is a width of the one-sided uniform (α=0)junction de?ned by Eq.(7).

After calculating W n (V )from Eq.(27),formula (25)allows one to ?nd the total depletion capacitance:

C (V )?C n

αW n

W n /W 0n ?αW 0n

/2,(28)

where C 0n (V )=?S 0/W 0

n (V )is a capacitance of the one-sided uniform (α=0)junction de?ned by formula (8)for the cross section area S 0=constant.

Numerical solution of the transcendental equation (27)is carried out for the normalized depletion-layer width

W n /W 0n as a function of the nonuniformity parameter αW 0

n and shown in Fig.2.Then the normalized depletion

capacitance C/C 0

n is calculated from expression (28).In Fig.2,there are also the normalized inverse capacitance

curves C 0n /C s and C 0

n /C n (broken lines)that give the contributions (23)and (24)into the total capacitance (28)for

the one-sided P +

N -junction.They are calculated from the following formulas:

C 0n

2

e

?αW n

and

C 0n

W 0n

e ?αW n .

As follows from the curves of Fig.2,the character of cross-sectional nonuniformity (α<0for S p >S n or α>0

for S p

capacitance.Despite the monotonic growth of W n with increasing α,the capacitance C/C 0

n grows exponentially for α>0because of increasing an e?ective cross-section area when S n >S p ?S 0(see the right insert).For α=?|α|<0,

the capacitance C/C 0

n tends to small values with growing |α|because of decreasing an e?ective cross-section area when S n ?S p ?S 0(see the left insert).

11

FIG.2:Dependencies of the normalized depletion width W n/W0n,the depletion capacitance C/C0n(solid curves),and the normalized inverse capacitance contributions C0n/C n and C0n/C s(broken curves)versus the nonuniformity parameterαW0n calculated for the one-sided P+N-junction whose nonuniform geometry are qualitatively shown in the left and right inserts for α<0andα>0.

E.Linearly-graded junction of exponentially-varying cross section:ˉN A(z)=?az for z<0,ˉN D(z)=az for

z>0,S(z)=S0exp(αz).

In this case the electrical neutrality condition(15)results in the following expression:

(1+αW p)e?αW p=(1?αW n)eαW n.(29) The voltage drops across the n-and p-layers of depletion calculated from Eqs.(17)and(18)by using S n= S0exp(αW n)and S p=S0exp(?αW p)are equal to

qa

V n=

?α3 (1?2eαW p)+(1+αW p)2 .(31) Substitution of Eqs.(30)and(31)into the voltage relation(19)gives rise to the following:

2qa

V bi?V=

12 The desired expression for the inverse capacitance C?1given by Eq.(7)is obtained by substitution of Eqs.(30) and(31)into formula(9)for C?1s with taking account of S′p/S p=S′n/S n=αand by the use of expressions(24)for C p and C n,then

1

C p exp(αW p)?αW p?1

C n

exp(?αW n)+αW n?1