Conservation Laws for Mass, Momentum, and Energy - American

2 Conservation Laws for Mass, Momentum, and Energy Application to Semiconductor Devices and Technology Bill Baerg

Downloaded by UNIV LAVAL on July 15, 2016 | http://pubs.acs.org Publication Date: October 2, 1985 | doi: 10.1021/bk-1985-0290.ch002

Intel Corporation, Santa Clara, CA 95051

This paper describes the application of three fundamental conservation laws for continuous media, to three aspects of semiconductor technology: device modeling, electromigration, and laser annealing. The response of a semiconducting material to an applied e l e c t r i c f i e l d can be described in terms of the behavior of the conduction and valence electrons that exist within i t . In a nondegenerate semiconductor (_1), these electrons may be considered as a c l a s s i c a l ideal gas mixture, of electrons (in the conduction band) and holes (in the valence band). The goal of t h i s paper is to show that the methods of continuum mechanics, as developed by C. Truesdell (2), can provide a useful description of the transport phenomena whicTi are observed. F i r s t , we w i l l b r i e f l y review the general conservation equations for mass and momentum, for continuous media. Then we w i l l use these equations to describe the transport of electrons and holes in a semiconductor. The results w i l l correspond to those which are used in device modeling, such as in the SEDAN (3) and MINIMOS (4) programs, and w i l l demonstrate the role of momentum conservation. Next, we w i l l apply the general equations to the problem of electromigration in a metal. In t h i s case, the components of the mixture w i l l be conduction electrons and metallic ions. The motion of the metallic ions in the interconnect constitutes a f a i l u r e mechanism, which has received renewed interest as production c i r c u i t dimensions approach one micron (5). In t h i s work, the ions are considered to constitute a l i n e a r l y "elastic material {2). The results w i l l be compared to previous work by approximations, and a set of equations with i n i t i a l and boundary conditions, which could be solved numerically, w i l l be presented for the f i r s t time. In addition to mass and momentum, energy is also conserved. The basic equations w i l l be reviewed, and applied to laser annealing. Laser annealing i s , among other things, a promising method of recrystal1izing the damaged surface layers of a 0097-6156/ 85/ 0290-0012$06.00/ 0 © 1985 American Chemical Society

Stroeve; Integrated Circuits: Chemical and Physical Processing ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

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13

Application of Three Fundamental Conservation Laws

semiconductor following ion implantation (6). Instead of to an applied e l e c t r i c f i e l d in the usual sense, the electron and hole transport w i l l be in response to incident optical r a d i a t i o n . Conservation of Mass


Consider a gas mixture of several components, labeled with the subscript A : A = 1, 2, . . . . Each component is made up of p a r t i c l e s with mass m/\ and has a density of n/\ p a r t i c l e s per unit volume, where n/\ i s a mathematically continuous, d i f f e r e n t i a t e function of space and time. Further, consider an arbitrary volume V of the mixture, enclosed by a surface S. Then the mass of component A is conserved i f l^rriAnAdV ^ m A W A - n d S + JmA(GA-RA)dV

(1)

where VA i s the v e l o c i t y vector of component A, and n is the outer normal unit vector for S. (Vectors are i d e n t i f i e d with boldface l e t t e r s . ) GA and RA are the generation and recombination rates, respectively, for component A. Equation 1 states that, for component A, in unit time, the increase in the mass in V is equal to the flux of mass from outside V, plus the net generation of mass within V. The surface integral has a minus sign because n is directed outward from V. Since rry\ i s constant, and appears in a l l three i n t e g r a l s , i t cancels out. A l s o , we can apply the divergence theorem, converting the surface integral to a volume i n t e g r a l : , J[|Â + v.n VA -(GA-RA)] dV = 0 V Since V is a r b i t r a r y , the integral must vanish, and A

+ V.nAVA = GA-RA:

{

?

(

3

.

)

)

which is the continuity equation for component A. When the gas mixture consists of p a r t i c l e s with charge qA, we may define an e l e c t r i c a l current density, j , so that JA = qAnAVA

(4)

Then the continuity Equation 3 becomes the conservation of charge: ft (PA)

+ V.JA = qA(fy\-RA)

(

5

)

For a binary gas mixture of electrons and holes, we define n and p as the electron and hole number d e n s i t i e s , respectively, and also use the same l e t t e r s as subscripts to represent the electron and hole components. Then we can 'write the charge Equation 5 for holes of charge +q, and electrons of charge - q , respectively:


14

C H E M I C A L A N D PHYSICAL PROCESSING O F INTEGRATED CIRCUITS

ft ( w ) ~ |t (-qn)

and

+

-jp

v

V.j

+

q(V p

=

R

(6)

}

= -q(G -R )

n

n

(7)

n

Now l e t ' s consider the total charge density in a semiconductor,


P

= q (p-n

Q

+ N -N D

+ (1-f)

A

n) T

(

g

)

where NQ and NA are the dopant donor and acceptor concentrations, respectively, ny i s the trap density, and f is the f r a c t i o n of traps occupied by electrons. Here we have in mind the Shockley-Read-Hall recombination theory (_7), and the traps are s i t e s within the semiconductor, which have energy levels within the band gap. These s i t e s may be due to crystal d i s l o c a t i o n s , impurity atoms or surface defects. Each s i t e may be occupied by an electron, making the trap neutral, or a hole, making i t p o s i t i v e l y charged. The trap density ny, as well as NQ and NA, is constant in time, but t h i s is not generally true for f: for example, in a time-dependent e l e c t r i c f i e l d , an MOS capacitor with p-type S i l i c o n has surface traps which may be f i l l e d with holes (accumulation) or electrons (inversion) depending on the e l e c t r i c f i e l d . Therefore, taking the time derivative of Equation 8 gives |£Q =

| _ (p_

q

n

+

n T ( 1

.

f ) )

(

g

)

Now we can add Equations 6 and 7, so that q ft (

where

p-n

)

+

V

J is the total current density, Up U

a n d

'

j

- ^ V M

=

(10)

- Jp + Jn and = -(Gp-Rp) = -(G -R )

n

n

(11)

(12)

n

are the net recombination rates for holes and electrons, respectively. Adding Equations 9 and 10, |£0

+

v

.j

=

q

n

Hl|fl _

j

q

(

U p

.

U n

)

(

1

3

)

Since there is no creation of net charge,

If ~ +

and therefore,

V

'J

=

0

(14)

f

-nifl = V n U

(15)

The Shockley-Read-Hall theory assumes a steady-state condition, so that, U =U , since 3f = 0. A recombination theory for the more F general case does not yet e x i s t . n

n

P


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15


Conservation of Momentum In order to derive the momentum equations, we must consider an integral over a volume which i s not fixed in space. Rather, the integral is taken over the set of p a r t i c l e s which define an a r b i t r a r y volume of the gas mixture at some instant in time. As these p a r t i c l e s move in space, so does the volume of integration. Thus the volume of integration changes with time. The force F/\ acting on component A in a volume V(t) of the gas mixture, enclosed by a surface SCt), i s F = fctAds +)fAdV +]pAdV (16) S V V where f/\ i s the external or applied force density exerted on component A, p i s the force density exerted on A by the other components, and t i s the stress vector for component A (2). The stress vector represents the force of the material outside of V(t) acting on the material inside V(t). The assumption, based on the short range of the i n t e r - p a r t i c l e forces, is that t h i s force which the material exerts on i t s e l f acts e n t i r e l y on the surface S ( t ) . Truesdell c a l l s this stress p r i n c i p l e the defining concept of continuum mechanics. The stress vector depends not only on time and space, but also on the surface o r i e n t a t i o n . However, according to Cauchy's fundamental theorem, r

f

A


A

*A

=

T

A-

(17)

n

where n is the unit normal, as before, and T/\ i s the stress tensor. The stress tensor depends only on time and space, and is not a physical quantity, but a concept. A p a r t i c u l a r material is defined in t h i s regard by a c o n s t i t u t i v e equation, r e l a t i n g the stress tensor to i t s motion. The subscript A means that the stress tensor represents only the force of component A acting on i t s e l f . In addition, there is the force exerted on component A by the other components in the system. We c a l l t h i s force density p . The law of conservation of momentum states that the t o t a l force acting on the material is equal to the rate of change of i t s momentum: /• r d f F = j T . n d s + ] ( f + p )dV = ^ P A V d V A

A

A

A

A

A

(

1

8

)

S(t) V(t) V(t) The l a s t term is proportional to the mass of a p a r t i c l e of • component A, whereas the applied force density for a coulomb force, which we consider below, is proportional to the charge of a p a r t i c l e of component A. The small mass-to-charge r a t i o of an electron has the effect that t h i s acceleration term i s n e g l i g i b l e under ordinary conditions, as w i l l be shown at the end of t h i s section, and we make this assumption throughout t h i s paper. As a r e s u l t , the momentum conservation law reduces to the statement that the total force F on a volume moving with the material must vanish. Applying the divergence theorem, A

F A - ^ -

T

A

+

V P A

,

D

V

" °

VCt)


(19)

16


Since the volume i s a r b i t r a r y , and the integrand is assumed continuous, p + f + V.T = 0 A

A

A

(

2

Q

)

The t o t a l applied force acting on the mixture i s , f = Ef A A (21) and the t o t a l stress is T = IT A (22) where we have neglected a term on the same order of smallness as the acceleration term in Equation 18. Then summing Equation 20 over a l l components in the mixture, Z p + f + V.T = 0 A

A


A

( ) 23

But the total force for the mixture as a whole must also vanish, f + V.T = 0 (

2

4

)

The l a s t equation is v a l i d for any material, regardless of the forces which the components of the mixture may exert on each other (neglecting accelerations). Therefore A

( ) 25

Equation 20 is our momentum, or force balance equation which we may now apply to semiconductors. To do so, we w i l l need c o n s t i t u t i v e equations for the p , subject to condition Equation 25, and for the stress tensors T . A

A

Device Modeling Equations In order to model electron and hole transport in a semiconductor device, such as a t r a n s i s t o r , we need solutions to a set of equations for the concentrations n and p, the e l e c t r i c current densities j and j p , and for the e l e c t r o s t a t i c potential defined by E = -V. Since there are nine unknowns (counting a vector as three, for each component), we must have nine equations. Two of them have already been derived: Equations 6 and 7, for the conservation of charge. Another is Poisson's equation for the e l e c t r i c f i e l d : V.D = p (26) n

Q

with PQ given by Equation (8). D i s the e l e c t r i c displacement which, for a simple m a t e r i a l , is given by (27)

e E

where e is the d i e l e c t r i c constant of the m a t e r i a l . Poisson's equation becomes eV $ = -pg

Therefore,


(28)

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17


The remaining six equations are the (vector) momentum Equations 20, for electrons and holes. The coulomb force density on component A is fA = qAnAE

(

2

9

)


The additional (non-coulomb) force density on component A due to the other components must include the interaction of the fixed ions and atoms in the semiconductor c r y s t a l . This force is modeled as a drag force, analogous to the drag of a f l u i d moving past a s o l i d sphere, and i s proportional to the r e l a t i v e velocity of each component. The drag force of the electrons on the holes, and vice versa, is neglected, which is an assumption of diluteness of the gas mixture r e l a t i v e to the density of the fixed atoms. The drag of the crystal on the electrons is given by Pn

=

Jn/^n

(

3

0

)

where p is the electron m o b i l i t y . The effects of the atoms, or l a t t i c e s c a t t e r i n g , the dopant ions, or ionized impurity s c a t t e r i n g , and defects (which are p a r t i c u l a r l y important at an i n t e r f a c e ) , are a l l lumped into t h i s d e f i n i t i o n of mobility. Even when the diluteness assumption is i n v a l i d (in low-doped, high i n j e c t i o n regions), the mobility is empirically adjusted to f i t the data. S i m i l a r l y , for holes, n

Pp = - J p p /P

(31)

where jjp i s the hole mobility. The difference in sign i s due to the positive and negative charges of electrons and holes. The hole current and v e l o c i t y vectors have the same d i r e c t i o n , or s i g n , and the hole drag force has the opposite d i r e c t i o n . The condition Equation 25, which states that jP/\ = 0, is not of any value here since i t merely t e l l s us that the drag force of the electrons plus holes on the crystal i s equal to the drag force of the crystal on the electrons and holes. It is not t r u e , of course, that p + p = 0. F i n a l l y , we assume the drag force is proportional to rather than just VA, because i t is more plausible to allow the density and v e l o c i t y to vary inversely without affecting the drag, and JA = qA A A convenient variable. The divergence of the stress tensor, TA, represents the force which component A exerts on i t s e l f . For a perfect f l u i d , n

n

v

1 S

p

a

TA = -PA*

(32)

where 1 i s a unit tensor (with components $ i j ) , and PA i s the p a r t i a l pressure of component A. For a mixture of ideal gases, PA = A n

k T

(33)

where k is the Boltzmann constant and T is the temperature (not to be confused with the total stress T, which is used only once,


18


in Equation 22.

So, at constant temperature, V.T

= -Vp

A

= -kT7n

A

(34)

A

Now we can substitute Equations 34, 29, and 30 or 31 in Equation 20. +JA/PA PAÂE -kTVnA = 0 ( ) +

35

For electrons and holes, respectively,

Equation 35 is

j = UqE UnkTVn jp = ypqpE - ypkTVp n

n

n

+

(36)


These equations complete the set required for device modeling, such as SEDAN or MINIMOS. We repeat the others below:

ID. at . lilnQ = -un 3t

and

q

(f>) 1°)

u

/

9

x

x

eV* = -q (p-n + N - N + ( l - f ) n ) D

A

(28)

T

where we have used Equations 8 and 12. The unknowns are n, p, Jn* J p » and c|>, with E = -Vcf>. In addition, the quantities u Up Un, U , and f must be given in terms of the unknowns, and Np, NA and nj must be given. Note that the e l e c t r i c f i e l d , through the potential is related to the net charge density through Poisson's equation, even though the f i e l d is externally applied (to the terminals of the device). To summarize t h i s section, we have seen that device modeling can be performed with a set of equations for mass and momentum conservation, plus Poisson's equation for the e l e c t r i c f i e l d . We have obtained the conventional equations from the general laws for continuous media by neglecting accelerations, by treating the electrons and holes as an ideal gas mixture, and by introducing the concept of the drag force (originated by Truesdell for general mixtures) between the electrons or holes, and the c r y s t a l . n9

n

Acceleration In Equation 18, the acceleration term would be s i g n i f i c a n t i f i t were comparable in magnitude to the coulomb term, or i f qnE * mn £

(

3

?

)

We consider an MOS t r a n s i s t o r with channel length ft, and ask at what value of I does Equation 37 hold? If the t r a n s i t time for an electron to move the distance I i s T , then 3t "

7

T


(

3

8

)

2.

19


BAERG

But, we also have V * pE where y

(39)

is the electron mobility in the channel, so T

~ v ~ pE

(40)

Therefore, dV „ 3t B

bE) !L

2

(41)


Equation 41 gives the electron acceleration in the channel in terms of the channel length, m o b i l i t y , and e l e c t r i c f i e l d . Substituting in Equation 37, and using E - V%

(42)

where V is the potential drop across ft, we obtain (43) Substituting numerical values, u = 500 cm? v o l t - 1 sec-1, V = 5 volt, m = 9.1 x 10-28gm and qV = 5eV = 8.0 x 1 0 - ^ e r g , we find £ = 0.27]jm. So, for t h i s MOS example we find that acceleration w i l l become important when the channel length approaches a quarter-micron. However, from Equation 43, the channel length at which acceleration becomes important is proportional to the m o b i l i t y , so a high-mobility material would have a longer ft. The effect of the acceleration w i l l be to reduce the effective electron mobility in the c r y s t a l . Electromigration Electromigration is the displacement of atoms in a conductor due to an e l e c t r i c current. A metal, such as aluminum, consists of p o s i t i v e l y charged ions, and Z conduction electrons per ion (Z = 3 for aluminum). To simplify the discussions, we w i l l consider an idealized homogenous conductor, rather than the more r e a l i s t i c p o l y c r y s t a l l i n e d e s c r i p t i o n . Our mixture is then made of two components, an electron gas and an ionic body, which we w i l l describe as a l i n e a r l y e l a s t i c s o l i d . There are no generation or recombination mechanisms in a metal, so the continuity Equation 3 becomes §£e + V . n v e

for electrons, and

V.n

i V i

e

= 0

= 0

for the ions.


(

4

4

)

(45)

20


In the momentum Equation 20, we again neglect accelerations, and use the coulomb force Equation 29 for the applied force on ions and electrons, with the e l e c t r i c f i e l d given by Poisson's Equation 28. That i s , f i = Zqn -E

( )

fe = qneE

(47)

46

n

and eV.E = q(Zni-n )

(48)


e

since the ionic charge is Zq. The d i e l e c t r i c constant of the metal is e. The force densities of the electrons acting on the ions (the electron "wind") and of the ions acting on the electrons must balance, by Equation 25, since there are only two components in the mixture (in the semiconductor model there were three : electrons, holes and fixed ions or atoms). Hence, as in Equation 30, we have Pe

(49)

Je/ê

=

for the electrons, where Ug i s the electron mobility in the metal, and since by Equation 25, 0

(50)

Pi = -Je/ve

( )

PP

+

Pi

51

we are neglecting vi compared to v in the model for the drag force, which is more precisely proportional to (v -v-j). Note that there is no ion mobility c o e f f i c i e n t involved. However, we are not neglecting vi elsewhere, such as in the continuity Equation 45, since vi describes the electromigration. Now we can substitute these results in Equation 20, for the ions e

e

j /u

e

j /y

e

e

= ZqniE + V.Ti

(

5

2

)

and for the electrons, e

= qn - - e e

E

v

(53)

T

The stress tensors are Tj and T for the ions and conduction electrons, respectively. Before we discuss c o n s t i t u t i v e equations for the s t r e s s , we note by subtracting Equation 53 from Equation 52 that e

V.Ti = - q ( Z n i - n ) E - V . T e

e

(54)

This equation strongly suggests space charge plays an important role in electromigration, since i f the divergence of the ionic


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stress tensor vanishes, there isn't any electromigration. So, we w i l l not want to assume that Zni = n i n this problem. The nine Equations 44, 45, 48, 52, and 53 form acomplete set, which we repeat below: e

W

+

J*

+

V

-"e e = v

V.n

(44)

0

=0

i V i

(45)

Je/Pe = qni'E + V.Ti

(52)

j /y

(53)

e

=

e

q e n

E

" « e 7

T

EV.E = q(Znj -

n)

(48)

e

where E = - VcJ> as before. The nine unknowns are n i , n , V i , v and . The c o e f f i c i e n t s \iq and e must be given, as well as c o n s t i t u t i v e equations for Ti and Te in terms of the unknowns. F i n a l l y , boundary and i n i t i a l conditions must be given. The stress tensor models, e s p e c i a l l y the one for the ions, play key roles in t h i s problem. Comparison of these equations with the ones for a-semiconductor shows that the differences are in the absence of source terms in the continuity equations, the absence of impurity ions or traps, and in the stress tensors. We start with the simpler model, for the electron gas. In a metal, the electron density is so high that the Pauli exclusion p r i n c i p l e must be taken into account, i . e . , there can be only one electron in each quantum state. The result is that the electrons obey the Fermi d i s t r i b u t i o n rather than Boltzman's, and may be considered strongly degenerate (8). The equation of state is


e

e

_ ( f2)Z73 ^2

5/3

3

Pe " " 5

m

e

n

( )

e

55

As before, we assume the electron gas is a perfect f l u i d so that Equation 30 applies, and V.T = -VPe e

= 5| (3l2n )2/3 v n (56) This result i s , of course, non-linear in the electron density. Since we have considered only f i r s t order or linear terms throughout t h i s theory, i t seems inconsistent to include a non-linear term here. Therefore, we w i l l revert to the ideal gas model for the electrons, so that e

e

V.T

e

e

= -kTVn

e

(57)

We now turn to the stress tensor for the m e t a l l i c ions, making use of the theory of e l a s t i c i t y : Let x be the vector describing the position of a p a r t i c l e in the body before i t i s deformed, and let x be the vector to this same p a r t i c l e after a deformation. Then 0

u = x-x

0


(58)

21

22


is c a l l e d the displacement vector and, for small deformations, the s t r a i n tensor E has the components E

k£

=

+

V*L

5 ^k)

(59)

9 x

A perfectly e l a s t i c body is one whose stress arises solely in response to the s t r a i n from i t s o r i g i n a l s t a t e , and i f the response is also l i n e a r and i s o t r o p i c , the material obeys Hooke's Law (9): Ti = X(trace E ) l + 2GE

(

6

Q

)

The c o e f f i c i e n t s X and G are c a l l e d Lame's constants, and are related to the Young's modulus E and Poisson's r a t i o v by X

(1 + v M l - 2v)

=


and

(61)

^ G

2TT^T

=

( 6 2 )

Note that by Equation 58, trace E = E|

Conservation Laws for Mass, Momentum, and Energy - American

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