2.1 Poisson's Equati

2 ' phenomena, fem I only (2.1-1).

As the next step i

potential i /J. For remembering th:

B=rot

Note that we as

compared to all

We substitute (2

rot (

lf "rot z = O" hol can be expresse

expressed as:

Some Fundamental Properties

To accurately analyze an arbitrary semiconductor strucll1$ which is intended as a self

contained device under various operating conditions, a mathematical model has to be given. The

equations which form this mathematical model are commonly called the basic semiconductor

equations. They can be derived from Maxwell's equations (2-1), (2-2), (2-3) and (2-4), several

relations obtained from solid-state physics knowledge about semiconductors and various -

sometimes overly simplistic - assumptions. E=-

at5 rotH=l+- (2-1) at aB

rot E= -- (2-2) at

div 15= p (2-3)

div B=O (2-4)

Now we substit

15= -

The first term

homogenous. 1

Poisson's equ

div (s The space char]

elementary che

the negatively

which will be !

E and i5 are the electric field and displacement vector; Hand Bare the magnetic field and induction vector, respectively.] denotes the conduction current density, and pis the electric

charge density.

The next sections will be devoted entirely to an outline of the procedures which have to be

carried out in order to derive the basic semiconductor equations.

2.1 Poisson's Equation

Poisson's eguation is essentially the third Maxwell equation (2-3). However, to make this

equation directly applicable to semiconductor problems, some manipulations have to be

undertaken. We first introduce a relation for the electric displacement vector i5 and the electric field vector E (2.1-1).

p=q

From a purel

without intro!

brought about

Sections 2.3 a1

The permitti

quantity. In p~

materials cum

of the permi

amorphous str

(2.1-1)

e denotes the ~rmittivity tensor. This relation is valid for all materials which have a time

independent permittivity . Furthermore, polarization by mechanical forces is neglected [2.44].

Both assumptions hold relatively well considering the usual applications of semiconductor

devices. However, an investigation of piezoelectric

I

I

2.1 Poisson's Equation 9

ended as a

model has

commonly

Maxwell's

solid-state

ies overly

phenomena, ferroelectric phenomena and nonlinear optics is impossible when using } only

(2.1-1).

As the next step it is desirable to relate the electric field vector E to the electrostatic potential If;. For that purpose we solve (2-4) by introducing a vector field A and remembering that "div rot" applied to any vector is always zero.

B=rotA,j!Y-A-=0-~'.~- -:ffr / rol- s: = -:> (2.1-2) Note that we assume for the applicability of (2.1-2) that the speed of light is large compared

to all velocities which are relevant for a device [2.44].

We substitute (2.1-2) into (2-2) and we obtain readily (2.1-3).

- a.4)

rot E+at =0

(2.1-3)

If "rot z = O" holds for a vector field z we know from basic differential calculus that z can be expressed as a gradient field. Therefore, the electric field vector E can be expressed as:

- a.4 E= ---grad!f; at

(2.l-4)

(2-1

)

Now we substitute (2.l-4) into (2.1-1) and then the result into (2-3).

- a.4 D = - e . - - c; . grad 1/; i3 t

div(:-~~)+ div (e. grad If;)= -p

(2.1-5)

(2-2)

(2-3) (2-4)

gnetic

field ty,

and pis

(2.1-6)

vhich have

The first term in (2.1-6) is zero if the permittivity s can' be considered to be homogenous.

Thus, we finally end up with (2.1-7) which is the well known form of Poisson's equation.

div (e. grad If;)= -p (2.1-7) The space charge density p can be further broken apart (2.1-8) into the product of the

elementary charge q times the sum of the positively charged hole concentration p, the

negatively charged electron concentration n and an additional concentration C which will

be subject of later investigations.

owever, to

ne mani-

he electric

p = q . (p - n + C) (2.1-8)

(2.1-1

)

From a purely mathematical point of view (2.1-8) represents a substitution only, without

introducing any assumptions. However, additional assumptions are brought about by

modeling the quantities n, p etc. as will become clearly apparent in Sections 2.3 and 2.4.

The permittivity s will be treated here in all further investigations as a scalar quantity. In

principle it has to be represented as a tensor of rank two. However, the materials currently

in use for device fabrication do not show a significant anistropy of the permittivity owing

to their special composition, e.g. cubic lattice or amorphous structure. Inhomogeneity

effects of the permittivity have been neglected

rich have a

al forces is

the usual

ezoelectric

(

IO 2 Some Fundamental Properties 2.3 Carrier Transport Equations

material e, [] --

Si 11.7

Si02 3.9

Si3N4 7.2 typical Ga As 12.5 Ge 16.l

This result is interpreted fai

total conduction current an

charge. In order to obtain t

carried out. We first define a

making use of the d~o1

- an divJ -q- -=q n ot

in (2.1-7). There does not exist pronounced experimental evidence for inhomogeneity

effects. For some materials the relative permittivity constants s, = e/e0 are summarized in Table 2.1-1.

Table 2.1-1. Relative permittivity constants

div grad t/J =CJ_. (n-p-C) e

(2.1-9)

. - op div J +q--=-

P 8t

It is obvious that we can not

different ways (2.2-5), (2.2-6;

equation more easily. The 4

the net generation or reco

recombination and negativ

about the structure of R ex.

carefully (cf. Section 4.2) us

doctors. If we have a moc

considered as two equations

no necessity or even evidenc

upon local quantities and

recombination phenomena

considering only the deriva

In particular for Si3N4 the value of e, depends strongly on the individual processing

conditions; it can vary quite significantly.

Ifwe introduce (2.1-8) and the assumption of a homogeneous scalar permittivity into (2.1-

7), we obtain the final form of Poisson's equation to be uSeal'or semiconductor device

modeling.

2.2 Continuity Equations

The continuity equations can be derived in a straightforward manner from the first Maxwell

equation (2-1). If we apply the operator "div" on this equation we obtain:

- - op divrotH=divJ +-=0 (2.2-1) ot

2.3 Carrier Transp

(2.2-4)

The derivation of current

cumbersome task. It is not

wide field of physics behim

relations in detail. Therefo

proof, but with reference 14

Without loss of generality t

the charge constant per par

_{lrift velocity} of the

particl density can be written

as (

JP=q-p-vP 1.=

-q n v.

The major problem is to fin

to the electric field vector

information about the drift

means of a distribution fu

coordinates x=(x,y,z)r, m

~en dimensional space.1

Now we split the conduction current density J into a component JP caused by holes and a component Jn caused by electrons:

(2.2-2)

Furthermore, we assume that all charges in the semiconductor, except the mobile carriers

electrons and holes, are time invariant. Thus we neglect the influence of charged defects,

e.g. vacancies, dislocations, deep recombination traps, which may change their charge state

in time.

oc -=0 ot If we substitute (2.1-8) and (2.2-2) into (2.2-1) and if we make use of (2.2-3) we obtain:

(2.2-3)

)

'

'roperties 2.3 Carrier Transport Equations IL

nhomo

f./e0 are

This result is interpreted fairly trivially. It just means that sources and sinks of the total

conduction current are fully compensated by the time variation of the mobile

charge. In order to obtain two continuity equations a few formal steps have to be carried

out. We first define a quantity R in (2.2-5) and, secondly, we rewrite (2.2-4) by

making use of the definition R.

- an divJ -q -=q - R n at (2.2-5

)

(2.1-9

).

. - ap div J + q . - = - q. R v at

It is obvious that we can not gain informat ion by writing one equation (2.2 -4) in two

different ways (2.2-5), (2.2-6). However, these formal steps enable us to interprete the

equation more easily. The quantity R can be understood as a function describing the net

generation or recombination of electrons and holes. Positive R means recombination

and negative R means generation. So far we have no informat ion about the structure of

R except equations (2.2-5) and (2.2-6). R has to be modeled carefully (cf. Section 4.2)

using knowledge from the solid -state physics of semiconductors. If we have a model

for R, equations (2.2-5) and (2.2-6) can really be considered as two equations. It seems

worthwhile to note explicitly here that there is no necessity or even evidence that R can

be expressed as a function depending only upon local quantities and not upon integral

quantities; non-local generation or recombination phenomena may certainly take place

in semiconductor devices considering only the derivation of the continuity equations.

(2.2-6)

icessmg

/ity into

nducto

r

the

first

obtain:

(2.2-1) 10

'Y holes

(2.2-2)

mobile

ience of

. ch

may

(2.2-3)

2-3)

we

(2.2-4)

2.3 Carrier Transport Equations

The derivation o f current relations for the semiconductor equations is a very

cumbersome task. It is not the intention of this book to cover the extraordinarily wide

field of physics behind all the considerations necessary to derive the current relations

in detail. Therefore, some of the required relations will be given without proof, but

with reference to a text more specialized in that field.

Without loss of generality the current density of charged particles is the product of the

charge constant per particle, the part icle concentration and the average velocity ~drift

velocit}'.} of the particles. So the hole current density and the electron current density

can be written as (2.3-1) and (2.3-2), respectively .

Jv=q-p-vp

Jn= =tl : n. vn (2.3-1

)

(2.3-2

)

The major problem is to fin~ expressions which relate the average carrier velocit ies to

the electric field vector E and to the carrier concentration. In order to obtain

information about the drift velocity we have to describe the carrier concentration by means of a distribution function fv in phase space which is the space of spatial

coordinates x = (x, y, z)", momentum coordinates k = (kx, kY , k,f and time t , thus a seven dimensional space. The distribution function determines the carrier concen-

)

I

1

2

2 Some Fundamental Properties 2.3 Carrier Transport Eqi

tration per unit volume of phase space. By integrating the distribution function over the

entire momentum volume V k we obtain the carrier concentration v (x, t ). v stands for n or p,

denoting electrons or holes.

4.1n3 f fJx,k,t )-dk=v(x,t ) (2.3-3

)

(2.3-8) is termed~

number of carriers sc

time . .fv (:X, k, t) gives [1-fv(x,k',t )] gives

unoccupied and can, t

of the scattering even1

equals the number of

unit time. Thorough i

found in, e.g., [2.21]

The derivative of x. carriers.

Vk

This normalization (2.3-3) defines .fv as a probability. In the literature various different normalizations can be found, e.g. [2.42], [2.49]. The distribution function has the property that its derivative along a particle trajectory xv(t ),

k.(t ) with respect to time vanishes in the entire phase space in compliance with the Liou

ville theorem about the invariance of the phase volume for a system moving along the

phase paths or an account of the conservation of the number of states [2.49]. d +

xv dt=uv

d dt I, (xv (t), kv (t), t) = O (2.3-4

)

We have now to subst

the Boltzmann trans1

of; ftve -+-.g at -n

= - f (fv(x.

Equation (2.3-4) is the Boltzmann transport equation in implicit form. By expanding the total derivative we obtain:

(2.3-5)

Vk'

Here grad, denotes the gradient operator with respect to the momentum coordinates k; grad , is the gradient operator with respect to the spatial coordinates x. Equation (2.3-5tshows that the variation of the distribution fu. nction at each point of phase

space (x, k) with time is caused by the motion of particles in normal space(x) and in mom en

tum space t: The derivative of kv wi~h respect to time multiplied with Planck's constant h equals the sum of all forces F. These forces have to be devided into two classes (2.3-7).

-fv(x,

dkv

dt

t; h h=-

h ' 2 ti (2.3-6)

A fairly accurate appr

carrier concentrations

ficult task to accompl

seven independent v a

rather requires the use

for numerical approac

[2.42].

An alternative approa

the motion of one or n

e.g. [2.43]. However,

[2.64], [2.65] and thei

application. One should be awarer

implies already seven

The scattering pro

The duration of a c particle; collisions

Carrier-carrier inte the right hand sid:

External forces an dimensions of the

F.=Fve+Fvi (2.3- 7)

f ve comprises forces due to macroscopic external fields and ft vi denotes forces due to internal localized crystal attributes like impurity atoms or ions, vacancies, and thermaf)att1ce vibrations. It is quite impossible to calculate the effect of internal j forces F vi upon the distribution function from the laws of dynamics [2.49]. Statistical laws have to be invoked instead. By introducing the quantity S; (k, k'). d k' which is the probability per unit time that a carrier in the state k will be scattered into the momentum volume dk ',we can write the internal collision term as follows:

t vi I ( _ ( _ ) __ grad, .fv J; = fv (.x, k, t) 1-.fv (x, k', t) S; (k, k')-

Vk' (2.3-8) - .fv (x, k', t) (1-.fv (x, t; t)) S. (k', k)) d/('

I I

'

2.3 Carrier Transport Equations 13

al Properties

(2.3-3)

(2.3-8) is termed the collision integral. The first term in the integrand describes the number

of carriers scattered from the state k into the volume element dk' per unit time. j~ (x, k, t) gives the probability that a carrier initially occupies the state k. [1-fv (x , k ', t)] gives the probability that the volume element d k' is initially unoccupied and can, therefore, accept a carrier. S; (k, k') gives an a priori probability of the scattering event. Correspondingly, the

second term in the integrand of (2.3-8) equals the number of electrons scattered from

volume element dk' into state k per unit time. Thorough investigations about the scattering probabilitx S; (k, k') can be found in, e.g., [2.21 ], [2.95]. The derivative of xv with respect to time represents the group velocity of the carriers.

iction over

t). v stands

re vanous

a particle e

space in

volume for

tion of the r

Xv

dt=uv (2.3-9)

(2.3-5)

We have now to substitute the relations (2.3-6) to (2.3-9) into (2.3-5), and we obtain the Boltzmann transport eguation in exnlicit form.

of, f ve -+- gradkfv+u, gradxfv = ot n

= - J tJ: (x,i(, t) (1- I. (x, k', t)) . s. (k, k')- Vk'

- I, (x, k', t) (1- I; (x, k, t)) S, (k', k)) dk'

(2.3-10)

(2.3-4)

expanding

itum coor-

rdinates :X

:. ch point

of 11

space(x)

(2.3-6)

A fairly accurate approach would be to directly solve (2.3-10) in order to calculate carrier

concentrations and drift velocities. However, this is an extraordinarily difficult task to

accomplish. (2.3-10) represents an integro-differential equation with seven independent

variables. This equation does not admit a closed solution. It rather requires the use of

iterative procedures which, moreover, are scarcely suitable for numerical approaches [2.10],

or additionally, invoke very stringent assumptions [2.42].

An alternative approach to solving the Boltzmann equation consists in simulating the

motion of one or more carriers ~t microscopic level with Monte Carlo methods, e.g. [2.43].

However, this category of simulations is very computationally intensive [2.64], [2.65] and

therefore, with a few exceptions only, not suitable for engineering application.

One should be aware of the fact that the validity of the Boltzmann equation (2.3-10) implies

already several assumptions (cf. [2.10], [2.17]).

The scattering probability is independent of external forces.

The duration of a collision is much shorter than the average time of motion of a particle;

collisions are instantaneous.

Carrier-carrier interaction is negligible. This effect would change the integrand of the right hand side integral in (2.3-10) highly nonlinear inj, [2.4].

External forces are almost constant over a length comparable to the physical dimensions of the wave packet describing the motion of a carrier.

nt n equals es (2.3-7).

(2.3- 7)

,rces due to .ncies, and of internal j ics [2.49].

!

quantity .t

e k will be ion term as

(2.3-8)

'

14

2 Some Fundamental Properties 2.3 Carrier Transport Equations

The band theory and the effective mass theorem apply to the semiconductor under consideration [2.76].

However, it is my intention here to outline the derivation of the classical current relat ions

and only to pinpoint the problems associated with much more basic and error-prone

models.

' By assuming that all scattering processes are elastic and by neglecting all effects~

of .egeneracy the scattering integral can be approximated and the Boltzmann equation is reduced to a pure differential equation [2.21], [2.42], [2.76].

ofv t ; d f, ~ d I. J.-fvo - + - gra k + u gra = - ---

o t 1i v v x v . v

~ ofvdk=4 u. at

Vk

Vk

(2.3-11

)

f e. (u. grad, f.)

Vk The physical mot ivation for the right hand side of (2.3-11) is as follows: Suppose that

at some moment of t ime t =0 all external forces are switched off and fv is homogeneous.

ft T grad, fv + u. grad, J~ = 0

Vk

(2.3-12

)

v denotes the carrier concent

mass, T denotes the lattice te

average collision time. The c

Boltzmann constant. The ext

field E if the magnetic inducti also for the validity

of (2.3-

It follows from (2.3-11) that the distribution function will change as a result of

collisions only. (2.3-13) will reduce to:

at r; (2.3-13

)

I, (x, t; t) =L, (x, k)+ (f,. (x,/Z, O)-f.0 (x, k)) v>

(2.3-14

)

In (2.3-18) it has been

assurm compared to the

thermal e carriers (cf. [2.67]).

We obtain ordinary differen

holes using the above given

The solution of this differential equation is quite simple.

J.0 is the equilibrium distribution function, and the quantity r, shows the rate of return to the state of equilibrium from the disturbed state, therefore, it is termed the relaxation

time. Under the very restrictive assumptions stated above the problem of I solving the Boltzmann equation can be eased drastically by modeling the relaxation time as only a

function of energy [2.42].

l In order to obtain the current relations from (2.3-11) we multiply this equation with I the group velocity ii., and then we integrate the equation over momentum space.

a q -(n-v)+-n ot n m:

a q -(p-u)--.p ot p m

p

These equations can be reg;

"closed solution" of these ~

obtain an approximate soluti

Vk Vk Vk

= -f Uv_Jv-fvo. dk T .

v

(2.3-15

)

Vk

For the solution of(2.3-15) we make use of the following four solutions to integrals,

the verificat ion and discussion of which is not necessarily trivial, but well established

in the literature, e.g. [2.13], [2.21], [2.71].

We rewrite (2.3-21) and

(2.3-'. collision times r, and

chargewe end up with:

!

f

l

J

I

2.3 Carrier Transport Equations 15

amental Properties

semiconductor - ofv - 3 a - u .-.dk=4n -(vv) v at at v (2.3-16)

:lassical current

more basic and

Vk

.ng all effects~ of the Boltzmann

[2.76].

. ( F ve ) - F ve J . - 3 - V Uv h.gradkfv dk=----r; Uvgradkfvdk=-4n Fvem:

Vk Vk

(2.3-17)

(2.3-11)

Vk

(2.3-18)

s: Suppose that

i off and fv is

(2.3-19)

Vk Vk

(2.3-12)

v denotes the carrier concentration, D y is the 9.!:ill, velocity, m: represents the effective mass, Tdenotes the lattice temperature and Ty in the right hand sides of (2.3-19) is an average

collision time. The constant k on the right hand side of(2.3-18) denotes the Boltzmann

constant. The external forces F,e can be expressed in terms of the electric field E if the magnetic induction B (Lorentz force) is neglected which is a requirement also for the validity of (2.3-16).

e as a result of

(2.3-13)

(2.3-20)

(2.3-14)

In (2.3-18) it has been assumed that the drift energy of the carriers is negligibly small

compared to the thermal energy. Therefore, relation (2.3-18) is invalid for hot carriers (cf.

[2.67]).

We obtain ordinary differential equations for the drift velocities of electrons and holes using

the above given integrals and the force relations (2.3-20).

a q - 1 n. vn -(n vn)+-. n E+-. grad(n k T)= ---

3t mn mn Tn

(2.3-21)

ows the rate of ,

it is termed the e the problem of I g the relaxation a _ q _ 1 P. vp

- (p vp)- -. p E + -. . grad (p . k . 1) = - -- e t mp mp 'p

These equations can be regarded also as macroscopic force balance equations. A "closed

solution" of these equations is, unfortunately, not possible. In order to

obtain an approximate solution we introduce effective carrier mobilities n and P.

(2.3-22) ls equation with en tum space.

(2.3-15)

(2.3-23)

(2.3-24)

ms to integrals,

well established

We rewrite (2.3-21) and (2.3-22) after multiplication with the corresponding average

collision times Tv and charge constant q, and - remembering (2.3-1) and (2.3-2) - we end up with:

I

I

l

l

16

2 Some Fundamental Properties 2.3 Carrier Transport Equations

Then we can use the substitutions (2.3-34) and (2.3-35) which by means of physical

interpretation are termed the .Einstein relations

k . T Dn = 11 - (2.3-34)

q

k . T D = .--

P p q (2.3-35)

All scattering processes h: polar optical phonon scat

has been neglected.

The spatial variations oft This implies a slowly var

path.

Effects of degeneracy ha

V( integral.

The spatial variation oft

varying electric field vect

The influence of the Le induction.

The time and spatial

furthermore, lattice and c

the diffusion of hot carrier

overcome this problem b)

[2.67], [2.68], [2.69], [2

Parabolic energy bands

degenerate semiconducto

the realistic band structi

[2.20]. However, for a re

system of Boltzmann equ:

one (cf. [2.96]).

The zero order term of t

collision time only has

conductivity phenomena.

The semiconductor has b distribution function is c

vicinity of boundaries, for

expected that the drift-di

free paths of boundaries.

In the literature we can fine

different form of the current

eguivalent to the drift-diffu

slightly different relations v

statistics that the equilibriu

Fermi functions.

(2.3-25)

a JP _ (- 1 ( k . T)) -r . - + J = q. . p. E - - grad p --

P 8t P p p q

The average collision times -r v are very small, typically in the order of tenth of picoseconds.

Therefore, equations (2.3-25) and (2.3-26) can be understood as being

singular!)'. pertu~ This suggests to expand the solution into powers of the

perturbation parameter which is the collision time.

(2.3-26)

a)

Jn(-rn ) = I t: (-rn Y

i=O

(2.3-27)

00

Jp(,p)= I Jpi (rp)1 (2.3-28) i=O

We have an algebraic equation for the zero order term of the current density.

_ (- 1 ( k. T)) J 110 = q ,, n E + --;:;- grad n . -q-

- (- 1 ( k. T)) J po= q P p E - p grad p -q-

These equations are formally approximations of order rv.

Jn = Jno + 0

(Tn) Jp=]p0+0(rp)

(l We further assume that the lattice temperature is constant; T=const.

(2.3-29)

(2.3-30)

(2.3-31)

(2.3-32)

(2.3-33)

to define the diffusion constants Dv , and, finally, we are able to write down the current

relations in the well known, established form as sums of a drift and a diffusion component.

1,/~:! q n 11 E + q D1 1 grad

n Jr~ q P v E - q DP grad p

(2.3-36)

(2.3-37)

~ k)-

i: x, - (

l+exp

In the following I should like to summarize the most important assumptions which

had to be performed over and above to the ones necessary for the validity of the Boltzmann

equation to obtain the current relations (2.3-36) and (2.3-37).

fp0(X,k)=--l

+exp (

I

(

I

(2.3-31)

(2.3-32)

(2.3-33)

ins of physical

(2.3-34) 3" I

(2.3-35)

-rite down the

a drift and a

(2.3-36)

(2.3-37)

nptions which

validity of the

.3-37).

nental Properties

(2.3-25)

(2.3-26)

. er of tenth of

stood as being

powers of the

(2.3-27)

(2.3-28)

en t density.

(2.3-29)

(2.3-30)

2.3 Carrier Transport Equations

All scattering processes have been assumed to be elastic. Therefore, for instance, polar optical phonon scattering, which is a major scattering mechanism in GaAs, has been neglected.

The spatial variations of the collision time and the band structure are neglected. This implies a slowly varying impurity concentration over a carrier mean free path .

Effects of degeneracy have been neglected in the approximation for the scattering integral.

The spatial variation of the external forces is neglected which implies a slow!)". varying electric field vector.

The influence of the Lorentz force is ignored by assuming zero magnetic induction.

The time and spatial variation of carrier temperature is neglected and, furthermore, lattice and carrier temperature are assumed to be equal. Therefore, the diffusion of hot

carriers is improperly described. Several authors have tried to overcome this problem by

using modified Einstein relations [2.9], [2.52], [2.53], [2.67], [2.68], [2.69], [2.85].

Parabolic energy bands are assumed which is an additional reason why degenerate

semiconductor materials cannot be treated properly. Calculations of the realistic band

structure of various semiconductors can be found in, e.g., [2.20]. However, for a

realistic band structure it can become necessary to use a system of Boltzmann equations

to describe the carrier distribution instead of just one (cf. [2.96]).

The zero order term of the series expansions of Jn and JP into powers of the collision time only has been taken into account. Thus, all time dependent EOnductivity

phenomena, like velocity overshoot, are not included.

The semiconductor has been assumed to be infinitely large. In a real device the distribution function is changed in a complex, highly irregular manner in the vicinity of

boundaries, for instance contacts [2.58] and interfaces [2.36]. It can be I expected that the drift-diffusion approximation fails within a few carrier mean free paths of boundaries.

In the literature we can find quite a few papers and books whose authors use a different

form of the current relations. These are based upon special assumptions equivalent to the

drift-diffusion approximations. The procedure to derive these slightly different relations

will be outlined next. We know from semiconductor statistics that the eguilibrium

distribution functions for electrons and holes are Fermi functions.

1 i: (x, I Z ) = (Ee (x, k)- E Jn (x))

1 +exp k . T(x)

1

(2.3-38)

fpo (x, k) = (E JP (x) - Ev (x, tZ)) 1 -l-exp k. T(x)

(2.3-39)

17

18

2 Some Fundamental Properties 2.3 Carrier Transport Equations

Ee and E,, denote the conduction and the valence band energy, respectively.

- h2.f.f EJx, k) = E,0 - q I/I (x) + (2.3-40)

2 m: _ n2kk E0(:x,k)=E00-q l/J(x)- (2.3-41)

2. m; Eco and E"0 are the conduction band and the valence band edge, respectively; I/I (x) is

the electrostatic potential as defined in Section 2.1. -

E Jn and E fp in (2.3-38) and (2.3-39) determine the Fermi energy for electrons and holes. We

shall try now to calculate a correction term (2.3-42) to the equilibrium distribution function.

By assuming vanishing vai

relations (2.3-47) to (2.3-50) i

for the distribution function

fn ~fno - in fno (1

. The current densities can no velocity and distribution ft

i: 1. I, (x, t; t ) = i.; (x, k) + i.. (x, k, t)

(2.3-42) J - -q f n - 4 . n3 . Un . J,

We recall the Boltzmann equation with the relaxation time approximation:

a I, five d f ~ d f I, - fvo (2 3 43) - + - gra k v + U v gra x v = - --- - ot n 'tv

Vk

By assuming

J - q f : 4 . 7[3 . iip. j

Vk

(2.3-46)

ifJn and

2.3 Carrier Transport Equations 19

ental Properties

(2.3-40)

By assuming vanishing vanat10n of temperature grad, T=O and substituting relations

(2.3-47) to (2.3-50) into (2.3-46) we obtain expressions (2.3-51) and (2.3-52) for the

distribution functions for electrons and holes, respectively.

ectively.

(2.3-41) (2.3-51)

.ively; t/J (x) is ~ . av

fv=fvo+

20

2 Some Fundamental Properties 2.3 Carrier Transport Equations

_ ( k . T ( p )) JP= -q P p grad ijJ +-q-. In n;e

Then we evaluate the "grad" operator and obtain relations (2.3-62) and (2.3-63) for the electron

and hole current, respectively.

(2.3-61) quasi-Fermi potential v

mean free path.

Only to first order is the

Fermi potential. Away

important.

Various authors have publ.

carrier transport in semicor out

a one dimensional si rm [2.31].

They use for the de

OU OU ~ -=-U-+-

Ot ox

_ - (k. T ) J;= n n E +q Dn grad n-q ., n -q- gradln(n;e) (2.3-62)

(2.3-63)

The last term in these relations represents a current which is caused by a possible dependence on

position of the intrinsic concentration. It thus accounts for variations in the balldgap, and it will

describe the bandgap narrowing effect observed in heavily doped semiconductors (see also

[2.91]). If we assume a constant intrinsic concentration we obviously do not have a gradient of

the intrinsic concentration, and then relations (2.3-62) and (2.3-63) are indeed identical to

(2.3-36) and (2.3-37). For practical purpose it is often useful to define effective fields for the

drift current components of the electron and hole current density.

OW ow -=-U-+q

ot ox

J =A . n . . E + q . D . grad n n n 11 11 (2.3-66)

(2.3-67)

v denotes the electron drift

electron energy and w rep (2.3-68) is almost equivalei

obtained during the deriva

energy w has been assumer

3. k. T m' W= "+-

2

Equation (2.3-69) is obtair

energy E and then integrat

procedure we had to go thrc of

the order u. (u. u) one ha "displaced" Maxwellian [2

_ _ k . T

En=E--- grad ln(n;J q

- - k. T EP = E + -- -grad In (nie) q

(2.3-64)

(2.3-65)

By rewriting (2.3-62) and (2.3-63) we obtain a form of the current relations which is very similar

to the classical drift-diffusion approximations but which can, as in our example, take into account

positional variations of the band gap.

It is worth noting that Boltzmann statistics for the carrier concentrations have been used in

(2.3-66) and (2.3-67). With Fermi statistics for the carrier concentrations an equally simple form

of the current relations compared to the classical drift-diffusion approximations is not derived so

easily.

In the following I would like to summarize some of the simplifying assumptions which became

clear in the derivation of (2.3-51) and (2.3-52), and which, additionally have been implicitly

used for the derivation of the drift diffusion relations (2.3-34) and (2.3-35) (see also [2.10]).

Higher order derivatives of the quasi-Fermi potentials have been neglected (cf. (2.3-46)).

This means that we transform a non-local solution of the Boltzmann equation into an

approximate one depending only upon the local gradient of the quasi-Fermi potential.

The dependence of the distribution function upon the gradient of the quasi-Fermi potential

has been linearized. That means that the scale of length over which the

- ( E f;.(x,k)~exp --

If one uses such a model

circumvents the assumption

always in equilibrium wit]

required to derive (2.3-69),

[2.42], [2.95]), and the addi

lead to many open question

these types of models becom

such an electron temperat1

principles particle model has

which is that simulation res

upon classical current relati, A

two. dimensional simulat

[2 .16]' [2.22].

1

amental Properties

(2.3-61)

and (2.3-63) for

L (n;,)) (2.3-62) 50

(n;e)) (2.3-63)

:d b~ a possible

ts for variations ect observed in

instant intrinsic

; concentration,

36) and (2.3-3 7).

he drift current

(2.3-64)

(2.3-65)

lations which is

.h can, as in our

(2.3-66)

(2.3-67)

.tions have been

ncentrations an

al drift-diffusion

ng assumptions

2), and which,

: drift diffusion

en neglected (cf.

the Boltzmann

l gradient of the

the quasi-Fermi

i over which the

2.3 Carrier Transport Equations 21

quasi-Fermi potential varies by k . T/q must be large compared to the carrier mean free

path.

Only to first order is the carrier transport driving force the gradient of the quasiFermi potential. Away from equilibrium the electric field vector will become important.

Various authors have published approaches for a more sophisticated treatment of carrier

transport in semiconductors. Froelich and Blakey, for instance, have carried out a one

dimensional simulation using energy and momentum conservation laws [2.31]. They use

for the description of electron transport:

av av q. E 2 a ( ( m*. v2 )) v at= - v. a;:+----;;- - 3. m*. n . ax n. w- --2- - -:;::

(2.3-68)

aw =-V aw +qEV--2-,!_(nV(W- m*.v2))- w-w0 at ax 3. n ax 2

22

2 Some Fundamental Properties

Thornber [2.88] has published a generalized current equation for the simulation of

submicron devices by supplementing the drift and diffusion current components with

so-called gradient,~ and relaxation current comQonents in order to include the most

important features of velocity overshoot.

oE oE) on on J0=q n v(E)+ W(E) - +B(E). - -q D(E) - -q. A(E). - ox ot ox ot

(2.3- 72)

Graphs of the gradient coefficient W(E), the rate coefficient B (E) and the relaxation

coefficient A (E) have been presented by Thornber for electrons in silicon at room

temperature. v (E) and D (E) are the well known terms for the drift velocity and the diffusion

coefficient. In the classical drift-diffusion current approximations all terms except those two

are assumed to be negligibly small. Thornber stated in his article [2.88] that relations of the

form (2.3- 72) are adequate to represent current densities whenever the characteristic

distances over which the particle concentration or electric field changes exceed 20 nm in

silicon (200 nm in GaAs) and the characteristic time intervals of such changes exceed about

0.4 picoseconds in silicon (2 picoseconds in GaAs). However, as far as I know, relations of

the form (2.3- 72) - a generalization of which to higher dimensional form is supposed to be

straightforward - have not been tested for practical applicability, although it can be

speculated that the range of validity of such treatments is greatly extended.

In recent work a new concept of device operation has been brought about, namely ballistic

transport. It was argued that by properly selecting the material, temperature, geometry and

bias, a device could be built which is much smaller than the mean free path between

scattering events [2.41], [2.78]. This question, however, is still open, although quite a few

activities in that field can be observed [2.3], [2.6], [2.7], [2.37], [2.45], [2.74], [2.78], [2.87],

[2.93].

As review papers on which kind of problems have to be faced especially for the simulation

of miniaturized devices, references [2.10], [2.29], [2.30] can be suggested. Investigations on

how the carrier transport equations, i.e. current relations, are changed, particularly by heavy

doping effects, are presented in [2.59], [2.63], [2.70], [2.73], [2.90], [2.91]. In the next

section we shall address these problems with emphasis on adequate models for the carrier

concentrations. However, considering the current relations, throughout this text we shall

favour current relations which have a structural appearing like (2.3-66) and (2.3-67). These

formulations will allow us to best characterize, in a more mathematical sense, the problem

of carrier transport. From a pragmatic physical point of view equations (2.3-66) and

(2.3-67) offer, considering the state of the art in understanding their background, a

sufficiently large set of parameters (effective mobilities "'effective fields E., effective diffusivities D,) to be invoked in order to reach a specific goal for agreement between

results obtained by simulation and measurement. One must keep in mind that all these

equations are just models in any case, which only imitatively simulate a real process, more

or less accurately, in a qualitative and quantitative sense.

2.4 Carrier Concentrations

2.4 Carrier Conce:

Accurate models for the

necessity if qualitatively < obtained. I first shall revi

concentrations, which gii

approaches are certainly

physics. However, I shall

assumptions which are ver:

keeps pace with the currei

Assuming a parabolic and

the conduction band (2.4-1

given in the well known f<

4 n (2 pcCE) h

4 n (2

Pv(E) = h3

I have avoided in (2.4-1), (2

- introducing a reference t

differences are of relevanc

literature because that pre

seems to be very conveni:

arbitrary reference point v Ee

and Ev are the so-called J the width of the forbidden

Eg=Ec-Ev

Numerical values of the ba.

frequently used materials

Table 2.4-1.

Table 2.4-1. Band gaps in undope

material E9 [eV]

Si 1.l2

Ga As 1.35

Ge 0.67

For silicon the temperatu

accurately with (2.4-4) or (

7.0 Eg= 1.17eV- -

I

(

ndamental Properties 2.4 Carrier Concentrations 23

the simulation of

rrent components

n order to include

2.4 Carrier Concentrations

In on --q. A(E)'x . ot

(2.3-

72)

Accurate models for the carrier concentrations in semiconductor devices are .a necessity if

qualitatively and quantitatively correct simulation results are to be obtained. I first shall

review the "classical" approaches of modeling the carrier concentrations, which give fairly

simple, closed form algebraic results. These approaches are certainly well documented in

many books on semiconductor physics. However, I shall place particular emphasis on

properly pointing out assumptions which are very possibly going to be violated when

device sophistication keeps pace with the current trends.

Assuming a parabolic and isotropic band structure the density of possible~ for the

conduction band (2.4-1) and the valence band (2.4-2) as a function of energy E are given in

the well known form with properly averaged effective masses [2.42]:

4 . n (2 m:)312 Pc(E)= ,, VE-Ee

(2.4-1)

and the relaxation

in silicon at room

ft velocity and the

imations all terms

tated in his article

t current densities

concentration or

the characteristic

.on (2 picoseconds

'. m (2.3-72) - a sd to

be straightthough

it can be ly

extended.

;ht about, namely

naterial, tempera-

Iler than the mean

1, however, is still

[2.3], [2.6], [2.7],

(2.4-2)

I have avoided in (2.4-1 ), (2.4-2) - as I shall do for all other formulae in this section -

introducing a reference energy with a specific absolute value, because only energy

differences are of relevance. One should, however, be quite careful in reading the literature

because that problem is absolutely not treated in a unique manner. It seems to be very

convenient, a matter of taste, to some people to introduce an arbitrary reference point with

zero energy on the energy scale. Ee and Ev are the so-called band edges. Their difference E9 denotes the band gap, i.e. the width of the forbid d en band between condu ctio n band and valen ce band.

especially for the I, [2.30] can be ions,

i.e. current

resented in [2.59],

iall address these

trations. However,

tll favour current d

(2.3-67). These

matical sense, the

of view equations

iderstanding their

ilities .,

effective a specific

goal for ement.

One must case,

which only a

qualitative and

(2.4-3)

Numerical values of the band gap and its linear temperature coefficient for the most

frequently used materials semiconductor devices are made of are summarized in Table

2.4-1.

Table 2.4- l. Band gaps in undoped material at T = 300 K

material E. [eV] dE./d T [eV K - t]

Si 1.12 -2.7. 10-4

Ga As 1.35 -5.0. 10-4

Ge 0.67 -3.7. 10-4

For silicon the temperature dependence of the band gap can be modeled more accurately with (2.4-4) or (2.4-5) [2.32], [2.34].

7.02. 10-4eV. (-fJ E9=1.17eV- (T)

1108+ - K

(2.4-4)

24 2 Some Fundamental Properties

Eg=l.1785eV-9.025-10-5eV- (;)-3.05-10-7eV- (;)2 (2.4-5) (2.4-4) is the older formula; it has also been published with slightly different constants, e.g. [2.5], [2.33].

The temperature dependence of the effective masses m: and m; of electrons and holes for the density of states in silicon can be modeled with polynomials which are fitted to

experimental data [2.32], [2.34].

m~ =m0 (t.045 +4.5 10-4 (;)) (2.4-6)

m;=m0 (o.523+1.4.10-3. (;)-1.4s.10-6. (;)2) (2.4-7)

Values for the effective masses at room temperature are summarized for some of the

relevant semiconductor materials in Table 2.4-2.

Table 2.4-2. Effective mass ratios at T = 300 K

material m:fmo [ ] m;/m0 [ ]

Si 1.18 0.5

Ga As 0.068 0.5

Ge 0.55 0.3

In order to obtain expressions for the carrier concentrations we have to integrate the density

of states function multiplied with the corresponding carrier distribution function over the

energy space.

00

n= J pc(E). f,,(E). dE (2.4-8) E, E

p= f Pv(E)f,,(E)-dE (2.4-9)

- 00

The lower integration bound in (2.4-8) is E e because no possible states for electrons do exist

for energies below the conduction band edge. For a similar reason the upper bound in

(2.4-9) is Ev. The distribution functions f,, (E) and fp (E) are Fermi functions.

fn(E)= (E-Efn) 1 +exp k . T

(2.4-10)

1 (2.4-11)

fp(E) = (E1p-E) 1 -l-exp k. T

2.4 Carrier Concentrations

E Jn and EfP denote the h exact meaning will be dis:

evaluated to

2 n=Nc Vn F112

2 p=Nv. Vn. F1;2

where N c and N" denote th

the valence band, respecti

Nc=2 cn: Nv=2 cn:

F 112 (x) is the Fermi integ

closed form solution.

00

F1;2(X)=J _t l+e

0

The asymptotic behavior'

however, is analytic.

F1;2 (x) ~ ~ x31; 3

The qualitative behavior Fig.

2.4-1. For arguments

approximated with an ext

Vn 2

F1;2(x)~ c(x)+(

Quite a few suggestions ha

very crude approximation

c (x) = 1/4, -1 <

However, due to this simp.

expressions where F 112 (x)

been proposed and used i

2.4 Carrier Concentrations 25 ital Properties

r (2.4-5) E Jn and E I denote the Fermi energies for electrons and holes, respectively. Their exact meaning will be discussed later. The integrals in (2.4-8) and (2.4-9) can be evaluated to

:ly different n=N ,_2_.F (Efn-Ec) c l c 1/2 k V tt . T

(2.4-12)

ectrons and

ls which

are

(2.4-6)

2 (Ev-Efv) p=Nv. -- . F1;2 --~~

}In k.T where N c and N" denote the effective density of states in the conduction band and in the

valence band, respectively.

(2.4-13)

(2.4- 7)

Ne= 2. (2 n :~ T m: )3'2 (2.4-14) some of the

Nv=2 cnI~~ T.m;y12

(2.4-15)

F112(x) is the Fermi integral of order 1L2 which, unfortunately, does not have a closed form

solution.

.ntegrate the

distribution

oo VY F x = -d 1/2 ( ) . - v y

0

The asymptotic behavior of F112 (x) for large negative and large positive argument,

however, is analytic.

(2.4-16)

(2A-17)

(2.4-8) ~ 2 3/2 F 112 (x) = - x , x 1

3 (2.4-18)

(2.4-9)

The qualitative behavior of F112 (x) and its asymptotic expansions are

shown in Fig.2.4-1. For arguments close to zero it has been shown

that F112(x ) can be approximated with an expression of the following type:

(2.4-10)

Vn 2

F (x)~--- 112 - c(x)+e-x Quite a few suggestions have been made in the literature for c (x). A most simple but very

crude approximation reads:

(2.4-19)

for

electrons on

the upper ni

functions.

c(x)=l/4, -l

2

6

2 Some Fundamental Properties 2.4 Carrier Concen tra ti ons

10 l -:j {ri. x I

-e I '7 2 I /' I I I

~ 10. ~

I I

I I

/. I

/, I

LL b I 1....x31

2 I 3 I

. ,

10-1

However, this approach

range of arguments. A

inverse function is pre

formulae with high ac

Chebyshev approximati

To come back to the ca

(2.4-17) for the Fermi ii

holds. The validity of t

electrons is sufficiently ~

energy for holes is suffici

are equivalent to the use

will then simplify to

n=Ncexp(~

p=Nv. exp e

1 o-i

-

5

5 x

Fig. 2.4-1. Fermi-integral of order 1/2 and its asymptotic expansions

0

10

15

I

c(x)=0.31-0.044.x, x

arnental Properties 2.4 Carrier Concentrations 27

However, this approach will only deliver formulae which are valid for a restricted range of

arguments. A review on approximations for Fermi integrals and their inverse function is

presented in [2.12]. For the purpose of implementation of formulae with high accuracy on

large computers it is better to use rational Chebyshev approximations as demonstrated in

[2.19].

To come back to the carrier concentrations, we can use the asymptotic expansion (2.4-17)

for the Fermi integral in the expressions (2.4-12), (2.4-l3)j[

Efn-Ec -1 k . T

(2.4-25)

Ev-Efp

2

8

2 Some Fundamental Properties 2.4 Carrier Concentrations

ental Properties 2.4 Carrier Concen tra ti ons 29

hey describe

ibrium state.

ations to the

y

differences :

relevant for

iitrarily. It is

1e zero if the

etowhichno :

zero for the

um. Thus,

E, .ulated in

the ng the

above

band and it is, therefore, well separated from both band edges. Thus Boltzmann statist ics

for intrinsic semiconductors in equilibrium are usually valid.

For many applications it is convenient to define a so-called intrinsic concentration n; as the

geometric average of the carrier concentrations in a semiconductor in equilibrium.

(2.4-40)

The existence of dopants is allowed in (2.4-40). If Boltzmann statistics are valid for

describing the carrier concentrations, n, is evaluated with small algebraic effort:

n; = VNc N,, -exp (- 2. ~9- T) (2.4-41)

(2.4-33)

We see that n; is position dependent if the band gap E9 is position dependent. The carrier

concentrat10ns can now be rewnften into the well known form with five parameters:

intrinsic concentration n.; electrostatic potential t /; , quasi-Fermi potentials

30

2 Some Fundamental Properties 2.4 Carrier Concentrations

(2.4-48)

o E e of the conduction barn causes a shift o Ev of the va majority electrons also scree

donor ions. As already s

description of the interactio

these subjects can be found i

on the results which are esi

derivation.

Kane [2.48] bas derived a1 conduction and the valeno that the local potential fluct function can be defined as density of states functions states functions over the la

+ _ k. T (NZ) N n NA --> l jJ b ~ -- ln --

q n,

_ + . k T (NA_) NA Nn --> tj;b;;;; ---.In --

q n;

However, it is to note that the validity of Boltzmann statistics becomes a very poor

assumption for high doping concentrations, because, as already mentioned, the Fermi

energies E fm E f P are shifted towards one of the band edges. If the error introduced by the

assumption of Boltzmann statistics is not acceptable, one has to solve (2.4-49) for the

built-in potential.

2 (Ev-Efp) 2 (Efn-Ec) + - Nv,cF112 -Nc,cF112 +Nn-NA=O vn k-T vn k-T

(2.4-49)

(2.4-47)

Again, this can only be done with numerical methods. It is obvious that the sum of the

intrinsic Fermi energy E; and built-in potential t f;b, which is often termed the extrinsic Fermi

energy, can be calculated simultaneously from (2.4-49). Most semiconductor devices contain regions with doping levels above 1018 cm - 3 and the transport of carriers through these heavily doped parts can play an essential role in determining device behavior and performance. Therefore, the models for the carrier concentrations have to properly reflect the underlying physics of heavy doping effects. In the preceding considerations we have only addressed the problem of carrier statistics in this context. All of the possible problems associated with the density of states functions have been ignored, except that shift energies for the conduction and the valence band, which have been assumed to be parabolic, have been allowed. In the following we shall examine more in depth why and how the band structure is changed in heavily doped semiconductors. However, the statements we shall make have to some extent a speculative character, because, as it has to be said, our understanding of the physics of heavily doped semiconductors is fairly limited. The density of states function for electrons and holes is influenced by essentially two categories of phenomena [2.62]. The first category consists of interactions between carriers and between carriers and ionized impurity atoms. The second category comprises the effects of electrostatic potential fluctuations which account for the random distribution of impurities together with the overlap of the electron wave functions at the impurity states causing bandtails [2.48] and impurity bands [2.66]. While the second category of heavy doping phenomena alters the shape of the density of states functions for electrons and holes, the first category produces only rigid shifts of both the conduction and the valence bands towards each other. To give an example we shall discuss the possible carrier interaction phenomena in ntype silicon. For p-type material the facts are analogous. Inn-type semiconductors three phenomena become apparent: electron-donor interaction, electron-electron interaction and electron-hole interaction. The electron-donor interaction does not yield changes in the band edges, but the number of electrons in the semiconductor becomes so large that they screen the donor ions, which effectively reduces the impurity ionization energy so that the donor levels ultimately disappear into the conduction band (see also [2.62]). Electron-electron interaction yields a rigid shift

Pc(E)= 4-n:-(2-1 h3

Pv(E)= 4-n-(21

with:

x

y(x)= iJ;. f V -0)

A simple approximation fc

Slotboom [2.81].

x:::;0.601 y(x);;;;;

x?:0.601

Results which are fairly sim

same time by Bonch-Bru:

infinite tails for the condu-

states functions are princi

band, but they fall off rapid

edge. As expected (2.4-5

asymptotically equivalent

(2.4-2), respectively. Amor

functions has been carried

are remarkably more comj

ductors I NZ I;;;;; IN.4 I 0 superior to Kane's methoi

{

ndarnental Properties

(2.4-47)

(2.4-48)

omes a very poor

y mentioned, the

dges, If the error

otable, one has to

(2.4-49)

JS that the sum of

often termed the

2.4-49).

above 1018 cm-3 i

play an essential

the models for the

physics of heavy

essed the

problem .sociated

with the ergies for

the conabolic,

have been nd how

the band the

statements we sit

has to be said, is

fairly limited. by

essentially two

eractions between

second category

h account for

the :he electron

wave rity bands

[2.66]. the shape

of the >ry

produces only Is

each other. To

phenomena in ne

semiconductors

electron-electron

eraction does not

he semiconductor

ively reduces the

isappear into the

yields a rigid shift

2.4 Carrier Concentrations

Ee of the conduction band towards the valence band. Electron-hole interaction causes a shift o E; of the valence band towards the conduction band, because the majority electrons also screen the mobile minority holes in addition to the immobile donor ions. As already

said, completely analogous statements hold for the description of the interaction

phenomena in p-type material. An excellent review on these subjects can be found in [2.57].

We shall primarily concentrate in the following on the results which are established without

going very much into details of their derivation.

Kane [2.48] has derived approximations for the density of states function for the

conduction and the valence bands in heavily doped semiconductors by assuming that the

local potential fluctuations are sufficiently slow that a local density of states function can

be defined as if the local potential were constant. The macroscopic density of states

functions which are the statistical average of the local density of states functions over the

lattice can then be expressed as:

_ 4.n.(2m~)312 .'C-. (E-Ec) Pc (E)- , ~ V a.; Y

(Jcv (2.4-50)

(2.4-51)

with:

x

y(x)= J; f V x-u exp(-u2). du - 00

(2.4-52)

A simple approximation for the unwieldy equation (2.4-52) has been suggested by

Slotboom [2.81].

x~0.601 y(x)~

x;;::0.601

1 2. Vn e-x2 (1.225 -0.906. (1-ez"'))

irx-(1--1 ) 16. x2

(2.4-53)

Results which are fairly similar to (2.4-50), (2.4-51) have been presented at almost the

same time by Bonch-Bruevich [2.15]. These density of states functions include infinite tails

for the conduction and the valence bands. That means the density of states functions are

principally different from zero everywhere in the forbidden band, but they fall off rapidly

with increasing distance from the corresponding band edge. As expected (2.4-50) and

(2.4-51) are for small doping concentrations asymptotically equivalent to the parabolic

density of states functions (2.4-1) and (2.4-2), respectively. A more rigorous approach to

the derivation of density of states functions has been carried out by Halperin and Lax

[2.38]. However, their results are remarkably more complex. For strongly compensated,

heavily doped semiconductors IN; I~ I NA. I 0 only, the Halperin and Lax theory is expected to be superior to Kane's method (cf. [2.72]).

31

{

32

2 Some Fundamental Properties 2.4 Carrier Concentrations

a.; is the characteristic standard deviation of the Gaussian tails of the density of states

functions (2.4-50), (2.4-51). The best established model for O"cv has been published by

Morgan [2.66].

O"cv= q1 . v(Nii + N-:i_).). . exp (-_a_) e 4-n 2.J..

(2.4-54)

For the derivation of this fo valence band structures are r: (2.4-50), (2.4-51). Fermi stati accounted for in the calcula screening length in metals [2 that the Fermi energy lies i forbidden band. This does extraordinarily high which appropriate for semiconduct In Fig. 2.4-2 a comparison of given. The solid line correspc Mock, Polsky et al.); the da: infinite (the model of Slotbooi model ofVanOverstraeten et (2.4-56) as a reference.

Jc denotes the screening length, and a is the crystal lattice constant, numerical values of which are summarized in Table 2.4-3.

Table 2.4-3. Crystal lattice constants

material a [10-9 m]

Si GaAs Ge

0.543072

0.565315

0.565754

Kane [2.48] as well as Morgan [2.66] in their original work have used a so-called cutoff

radius instead of a/2 in the exponential term of (2.4-54). However, there is evidence to

relate this quantity to the lattice constant [2.66]. VanOverstraeten et al. [2.90] and Slotboom

[2.81] have in their investigations fully neglected the exponential factor of (2.4-54). For the

screening length Jc two models are most frequently in use. The first one has been proposed

by Stern [2.84].

E 0

~+ V1_an 1+1-ope l+-Ni_+N_-;;. oEfn oE!P k T;00

For non-degenerate material when Boltzmann statistics can be applied this formula reduces

to the well known Debye length.

(2.4-55)

en . . c. . , . .>

O J c C J) 10-'

(2.4-56)

0

)

c

c C J ) C J ) L 0 en

Jc=~ -vt . k T

q n+ P

T;0n in (2.4-55) represents an effective temperature for ion screening. In the original paper

of Stern [2.84] this quantity is treated as adjustable parameter in order to fit experimental

data. Stern has speculated that at room temperature T;011 should be in the range from about 7000 K to 9000 K. Mock [2.63] and Polsky et al . [2.73] have used 9000 K in thei r work; Nakagawa

[2.70] has claimed that 6000 K is more appropriate to obtain quant itatively correct results; and

Slotboom [2.81] has assumed T;011 to be infinite so that the last term in the denominator of (2.4-55) vanishes. In [2.51] and [2.90] a different model for the screening length which has also been suggested

by Stern [2.83] in an early work has been used.

10-s ,--~- 10 17

d

Fig. 2.4-2. Screening length '

(2.4-57)

When the doping concentra

described by a delta function

the electrons of the impurir

impurity band. Morgan [2.6( ~

for the impurity band.

J

il Properties

density of

has been

(2.4-54)

numerical

t

so-called .

r, there is

ieten et al.

iected the

are most

(2.4-55)

is formula

(2.4-56)

re original

irder to fit

ould be in

~.73]

have (is

more

2.81] has

J f (2.4-55)

which has

(2.4-57)

2.4 Carrier Concentrations

For the derivation of this formula it has been assumed that the conduction and valence band

structures are parabolic so that (2.4-57) seems to be inconsistent with (2.4-50), (2.4-51).

Fermi statistics for the carrier distribution functions have been accounted for in the

calculation of (2.4-57) which can also be identified as the screening length in metals [2.50].

A requirement for the applicability of (2.4-57) is that the Fermi energy lies in one of the

carrier bands, and not as usual in the forbidden band. This does not happen unless the

doping concentration is extraordinarily high which should lead to the conclusion that

(2.4-57) is inappropriate for semiconductor device modeling.

In Fig. 2.4-2 a comparison of the models for the screening length inn-type silicon is given.

The solid line corresponds to (2.4-55) with T;0n equal to 9000 K (the model of Mock,

Polsky et al.); the dashed line is also (2.4-55) but with T;0n assumed to be infinite (the

model ofSlotboom); and the dot-dashed line corresponds to (2.4-57) (the model

ofVanOverstraeten et al.). The dotted line denotes the classical Debye length (2.4-56) as a

reference.

silicon

E o

- en L

..., Ol c Q) 1

o-> OJ c ,_ c Q) Q)

L o en

- -- - Mock Slolboom Uan Overslraelen el.al. De bye

10 -9 ,---,--- 10 17

1 0 18 1 0 IS 1 0 ZO 10 21

donor concentration [cm-31

Fig. 2.4-2. Screening length versus donor concentration in silicon at 300 K temperature

When the doping concentration is large, the impurity energy level cannot be described by a

delta function as it is the case in simple theory. The wave function of the electrons of the

impurity atoms overlap, thus causing the formation of an impurity band. Morgan [2.66] has

developed a theory which predicts a Gaussian sfiape for the impurity band.

33

34

2 Some Fundamental Properties

(2.4-59)

2.4 Carrier Concentrations

1C 23

iu= j valence band

- de I I 0 21 ::>

Q)

M I

E

l 0 20

o - (I)

Q)

.., 10 19

cu .., (I) c, 10 18

0

. .., -

(/) I 0 17

c

Q)

'O

I 0 16

I 0 IS

-.7 -.6

Fig. 2.4-3. Band structure f,

(2.4-58)

En and EA are the activation energies for specific donor and acceptor atoms, respectively.

Numerical values for Ev and EA are well documented in the literature, e.g. [2.86]. The

expression for uDA like equation (2.4-54) for ucu has also been proposed by Morgan [2.66].

q2 v(Nj; +NA)). ( 1 ) a DA= - 1.0344 - exp - ~========= s 4.n Vn.3206n(Nj;+NA).).3

(2.4-60)

Some discussion about the models for avA can be found in, e.g., [2.39], [2.72].

In order to obtain a density of states function for electrons and holes, the density of states

functions of the conduction band (2.4-50), the valence band (2.4-51) and the impurity bands

(2.4-58), (2.4-59) have to be combined. Kleppinger and Lindholm [2.51] have simply added

up the corresponding functions for that purpose. VanOverstraeten et al. [2.30], however,

have assumed that the total density of states function of the mobile carriers is composed of

the envelope of the conduction (valence) density of states and the corresponding impurity

band density of states function. This approach is physically much more sound since adding

up the density of states functions implies that a substitute impurity atom and a silicon atom

that were at that same lattice site both contribute to the density of states (cf. [2.72]). The

concentration of electrons and holes can now, finally, be calculated by:

(2.4-62)

1023

1022

- 7 10 2t ::> Q)

M I E 10 u

- (IJ Q)

.., 10'" cu

.., (IJ

.._ 10 18 0 ::Jl

.., - 10 17 (/) c Q) 'O

I 0"

10 16 ,

valence band

I I 11111111

-.7 -.6

00

(2.4-61)

- 00

00

- 00

The integration bounds are now - co and co in contrast to (2.4-8) and (2.4-9) because of the

infinite tails of the density of states functions. It is obvious that the

integrals (2.4-61), (2.4-62) do not have a closed form algebraic solution; they have to

be solved with numerical methods. Details on how to design efficient algorithms for the

self-consistent solution of the carrier concentrations and the built-in potential are given in,

e.g., [2.46].

Fig. 2.4-3, Fig. 2.4-4 and Fig. 2.4-5 summarize the results we have obtained in a graphical

way. They show the density of states function for electrons max(pc(E),pD(E)) and the density

of states function for holes max(pv(E),pA(E)). The dashed line in the conduction band

denotes the distribution function of electrons, i.e. the integrand of (2.4-61). Fig. 2.4-3

corresponds to a doping of : =1016cm-3, NA =0, i.e. fairly low doping concentration. Fig.2.4-4 has been

Fig. 2.4-4. Band structure f(

l

ne Fundamental Properties 2.4 Carrier Concentrations 35

(2.4-58)

10 2Z

donor concentration

valence band conduction band

(2.4-59) - I ;:) I 0 21 [2.72]).

, be calculated by:

10 ,.

10 ts -.

6 -.

5 -.4 .

4

.5 .6 . 7

energy [et.JJ

Fig. 2.4-3. Band structure for N't, = I016'cm - 3, N;, =0 in silicon at 300 K temperature

- ~ 10" Q)

1J

1022 donor concentration

(2.4-61)

(2.4-62)

I ~ JO"

I 'll I

E o I 0.,

~t to (2.4-8) and (2.4-9) is.

It is obvious that the .ic

solution; they have to 1 1

efficient algorithms for

nd the built-in potential

"' OJ -;;; I 0'" - .>

'- 0 10

:n - .>

10'"

; we have obtained in a

function for electrons

ioles max (P v (E),p A (E)). distribution function of

sponds to a doping of tion.

Fig. 2.4-4 has been

valence band conduction band

I I I I I I

-.6 -.5 -.4 .

4

.

s .6

.7

energy leUJ

Fig. 2.4-4. Band structure for N't, = 1018 cm -3, N;, =0 in silicon at 300 K temperature

Fig. 2.4-5. Band structure for NZ= 1021 cm-3, N;; = 0 in silicon at 300 K temperature

1023

~ acceptor concentration= 10 16cm-3

10221

donor concentration= 1016cm-3

- .... I 10., ::> Q J i" 'l

I E 10 20 0

- (/)

QJ

.,.> 10" C D

.,.> C f )

(/) 10 17

c QJ

1J

10,.

10'" ~ ' ~~~~~~~' ~~~'d' I, l I ,l, 'I'' ~~~~~~~li'D'~ '~~~~ -.? -.6 -.5 -.4 .4 .5 .6 .

7 energy [eUJ

36

10 23

10 22

- .... I ::

> 10 21

QJ i" 'l I

E 0 l 0 20 - [I)

QJ

.,.> l 0 1 9 C D .,.>

[I

)

(0

L

..., c 10 12 (j)

o c ~ 0 T=30( o

~ 10

14 E

o

al (T)= -1.99765 10+1 +2.01814. 10-1. T-1.97040. 10-4. T

2 (2.4-68)

The coefficients a2 ( T) and a3 ( T) which fit the empirical formulae best to Mock's

model in the doping range [1012, 1020]cm-3 are:

a2(T)=9.60563. 10-1-3.94127 .10-3. T+4.41488. 10-6. T2

a3(T)=l.29363 .10-1 +l.10709 .10-3. T-9.56981-10-7. T2

U 10 II (/) c

L

~ 1010

with:

10 s -t--- 10 17

(2.4-69

) whereas for Slotboom's model in the doping range [1012, 3. 1020] cm -3 they read: Fig. 2.4-7. Intrinsic concentr

(

' I

undamental Properties

2.61], have proved

of bipolar devices,

oach with (2.4-63),

t the concept of an

tal impurity con)f

compensation is

erial any

approach . s to note

that for rom

equilibrium is

eason can be found

I) compared to the

concentration has

(2.4-66)

'imental data up to

ons the theoretical

o fit experimental

in [2.11].

:asured band gaps

for heavily doped

inly the rigid shifts

to lattice disorder,

h contributions to

ive electrical band

gations.

omplex models

of ] which have

been ~ structure

for the

:m-3

(2.4-67)

. 10-4. T2 (2.4-68)

ae best to Mack's

~ o-6. T2 .0-1.

T2 (2.4-69

) ] cm -3 they read:

2.4 Carrier Concentrations

a2(T)=7.95811. 10-1-3.20439. 10-3 . T+3.54153. 10-6. T2 a3

(1)=2.97104. 10-1 +6.75707. 10-4. T-4.90892. 10-7. T2

(2.4-70)

and for VanOverstraeten's et al. model in the doping range [ 101 7, 1021] cm - 3 they

evaluate to:

a2(T)=2.38838 .10-1-9.57814. 10-4 . T+ 1.07551. 10-6. T2 a3 ( T) =

5.10190. 10-1+5.75190. 10-4. T- 7.01029. 10-1. T2

(2.4-71)

The temperature T has to be given in Kelvin in (2.4-68) to (2.4- 71 ). The maximum

relative difference of formula (2.4-67) with the above given coefficients and the exactly

evaluated models is always smaller than ten percent (cf. [2.46], [2.47]) in the temperature

range [250, 400] K. Formulae for an effective intrinsic concentration for compensated

material are also given in [2.46], however, they are much more complicated.

In Fig. 2.4-7 the effective intrinsic concentration for Mock's model (solid line), Slotboom's

model (dashed line) and VanOverstraeten's et al. model (dot dashed line) are shown in

conjunction with the experimental values of Mertens et al. [2.61], Slotboom [2.81], Wieder

[2.92] and Wulms [2.94]. Although the agreement between the models and the

experimental data is not overwhelming, it can be considered pragmatically to be

quite.good, because of the fairly pronounced scatter

of the measured data. However, a judgment as to which of the models is to be prefered can

not, therefore, be given.

10 15

~

Moel< I t

40 2 Some Fundamental Properties 2.6 The Basic Semiconductor Eq1

2.5 Heat Flow Equation J =qn- .E+1 II n

For the design of power devices it is often desired to simulate interaction of electrothermal

phenomena. Changes in the temperature and its distribution in the interior of a device can

influence significantly the electrical device behavior. Particularly, two effects usually have

to be considered. Thermal runawa:t is one, a rather common mechanism where the

electrical energy dissipated causes a temperature rise over an extended area of a device

resulting in increased power dissipation. The device temperature increases which leads to

an irrecoverable device failure (burn out), unless an equilibrium situation can occur with a

heat sink removing all of the energy dissipated. The existence of such an equilibrium

situation is the second question which is sometimes quite difficult to answer [2.54]. In order

to account for thermal effects in semiconductor devices tqe heat flow equation (2.5-1) has

to be solved.

JP= q. p - p. EThe

last expression in (2.5-'.

temperature field as the dr

being constant for the deriv.

As can be proved with m

perature in (2.3-29), (2.3-30)

verified these relations witl

solution of the Boltzmann e1

the thermal diffusion coeffi

ar p. c. - -H =div k(T). grad T at (2.5-1)

D T p

DP~ 2. T

material c[m2s-

2K-

1] p [VAs

3m-

5]

Si 703 2328

Si02 782 2650 typical

Si3N4 787 3440 typical

GaAs 351 5316 Ge 322 5323

These coefficients are small

the procedure just sketchec

result owing to the comple

obtain and discrepancies

demonstrated that Stratton

presence of dopands the th e

factor of five. Some more cc

[2.76]. However, one need 1

the con text of semiconduct:

these relatively rough mod

on the current densities, e.

p and c are the specific mass density and specific heat of the material. Numerical values for

p and cat room temperature are summarized in Table 2.5-1 for the most frequently used

materials in device processing.

I Table 2.5-L Specific heat and density constants al T= 300 K

The temperature dependence of p and c can be assumed to be negligibly small in

consideration of practical device applications [2.50]. If one is not interested in thermal

transients one can assume for the simulation that the partial derivative of the temperature

with respect to time vanishes, which eases the problem of solving the heat flow equation by

one dimension. However, one is absolutely incorrect in using this assumption in a

simulation for which an equilibrium condition does not exist. The simulation program will

"blow up" in a manner analogous to the real device.

k ( T) and H denote the thermal conductivity and the locally generated heat. Models for these

quantities will be given and discussed in Section 4.3 and Section 4.4, respectively.

To just calculate the temperature distribution and the associated thermal power dissipation

without taking into account the current induced by gradients of the temperature is a fairly

crude approach which is only appropriate for limited application [2.35]. In a more rigorous

approach the current density equations have to be supplemented by additional terms.

2.6 The Basic Sem

We shall now summarize tr

in order to be able to wri

equations, which we shall 1

sake of transparency and e

and complexity of our mod

major number of engim

Certainly, conditions do ex

doubt. However, as I tried 1

results in semiconductor

applicable and still sufficiei

The basic semiconductor

continuity equations for el

for electrons (2.6-4) and hr

this set the heat flow equ:

nental Properties

interaction of

ibution in the

rice behavior.

iway is one, a

ted causes a

reased power

verable device

1 a heat sink

rium situation

4]. In order to

~(2.5-1)

(2.5-1)

al. Numerical

1 for the most

gibly small in interested in

I derivative of em of solving

y incorrect in

ition does not

us to the real

heat. Models j

Section 4.4,

iermal

power .dients

of the e for

limited

[uations have

2.6 The Basic Semiconductor Equations 41

- - '/' 111 = q n 1 1 E + q D11 grad n + q n D11 grad T

- - T J p = q . p . v . E - q D p grad p - q p . D p . grad T

(2.5-2

)

(2.5-3

)

The last expression in (2.5-2), (2.5-3) represents a drift current component with the

~mperature field as the driving force. In Section 2.3 we did assume temperature being

constant for the derivation of the classical drift-diffusion relations (cf. (2.3-33)). As can be

proved with minor algebraic effort, by assuming non-constant temperature in (2.3-29),

(2.3-30) we obtain equations (2.5-2), (2.5-3). Stratton [2.85] has verified these relations

with a much more rigorous approach, from a perturbation solution of the Boltzmann

equation. He also derived in his paper approximations for the thermal diffusion

coefficientsn;'.

DT~~ - ~ f (2.5-4)

II - 2. T - l..~ ..

D '-

DT~_P -:;: - r-~ (2.5-5) p-2.r 11 These coefficients are smaller by a factor of two compared to those we obtain with the

procedure just sketched above. However, as pointed out in [2.85] a more exact result

owing to the complexity of the problem is cumbersome, if at all possible, to obtain and

discrepancies of that order are not at all surprising. Dorkel [2.27] demonstrated that

Stratton's result is applicable for intrinsic semiconductors; in the presence of dopands the

thermal diffusion coefficient is underestimated by at most a factor offive. Some more

considerations on this subject can be found in, e.g. [2.14], [2.76]. However. one need not

worry as all publications on non-isothermal effects in the context of semiconductor device

modeling certify more or less the applicability of these relatively rough models for

describing the feedback of temperature gradients on the current densities, e.g. [2.1], [2.18].

2.6 The Basic Semiconductor Equations

We shall now summarize the results which we have obtained in the previous sections in order to be able to write down a set of equations, the "basic" semiconductor equations,

which we shall use in all further investigations. It is obvious that for the sake of transparency and efficiency, we shall perform a trade-off between accuracy and complexity of our model. The equations we shall concentrate on are valid for the major number of engineering applications, particularly for silicon devices. Certainly, conditions do exist for which their validity is not guaranteed, or at least in doubt. However, as I tried to express in the previous sections, the more sophisticated results in semiconductor physics are too complex to give a rigorous, generally applicable and still sufficiently simple model for the purpose of device simulation. The basic semiconductor equations consist of Poisson's equatjon (2.6-1), the continuity equations for electrons (2.6-2) and holes (2.6-3) and the current relations for electrons (2.6-4) and holes (2.6-5). For some applications it is desired to add to this set the beat flow equations (2.6-6).

42

2 Some Fundamental Properties

div grad t/J= .!!:__. (n-p-C) (2.6-1)

- op div] +q -= -q. R p at

(2.6-3)

2. 7 References

[2.12] Blakemore, J. S.: Appro Used

to Describe Eleen 1067- 1076

(1982). [2.13] Blatt, F. J.: Physics

of E

[2.14) Bletekjaer, K.: Transpon Electron Devices ED-17, [2.15] Bonch-Bruevich, V. L.: C State4, No.10, 1953-1' [2.16) Buot, F. A., Frey, J.: Et Design Information. Sol; [2.17] Capasso, F., Pearsall, T. Carlo Simulations of Higl EDL-2, 295 (1981). [2.18] Chryssafis, A., Love, W.: in n-p-n Power Transistc [2.19] Cody, W. J., Thacher, H. Orders - 1 /2, I /2 and 3 / [2.20] Cohen, M. L., Bergstres Fourteen

Semiconductor 789- 796 (1966). [2.21] Conwell, E. M.: High F [2.22) Cook, R. K., Frey, J.: 1 in Si and GaAs MES (1982). [2.23] Curtice, W. R.: Direct C.

(Monte-Carlo) Model fo 1942-1943 (1982). [2.24] DeMan, H. J. J.: The J Transistor. IEEE Trans. [2.25] DelA!amo, J. A., Swan Narrowing in Moderate! [2.26] Dhariwal, S. R.,

Ojha, ' Electron. 25, No. 9, 909 [2.27] Dorkel, J. M.: On El( Electron. 26, No. 8, 819 [2.28] Engl, W. L., Dirks, H. K {1983).

[2.29] Frey, J.: Transport Pb

Semiconductor Devices : [2.30]

Frey, J.: Physics Preble

Semiconductor Devices : [2.31]

Froelich, R. K., Blakey, Wave lmpatt Diodes. Pr [2.32) Gaensslen,

F. H., Jaege Depletion Mode Devio

Meeting, pp. 520- 524 ( [2.33)

Gaensslen, F. H., Rideoi

Temperature Operation. [2.34)

Gaensslen, F. H., Jaeger Mode

MOSFET's. Solid [2.35) Gaur, S. P.,

Navon, D.

Nonisothermal Conditio [2.36) Gnadinger, A. P., Talley, and the Thickness of the l 13' 1301 -1309 (1970). [2.37) Grondin, R. 0., Lugli, P Device Lett. EDL-3, No

- on div J -q. - = q. R n ot

(2.6-2)

Jn=q n 11 E11+q D11 gradn

JP=q. p . P. EP-q. DP. gradp ar p . c . - - H =div k ( T) . grad T ot

(2.6-4)

(2.6-5)

(2.6-6)

I

To almost this level of completeness, these equations were first presented by

VanRoosbroeck [2.89].

Models for C, the net doping concentration, for R, the net generation/ recombination,

for 1 1, P, the carrier mobilities, for H, the thermal generation and fork ( T), the thermal

conductivity will be d iscussed in the following chapters. E,, and EP, the effective fields in the current relations are to first order the electric field, however, we may use

supplementary correction terms to account for heavy do12ing (cf. Section 2.3, Section

2.4) o r thermally induced currents (cf. Section 2.5). For such mathemat ical

investigations, relat ively slight perturbations are of only secondary importance. Hence,

for most applicat ions, accounting for some specific effect is possible by properly

modeling the parameters in the basic equations.

J

2. 7 References

[2.1] Adler, M. S.: Accurate Calculations of the Forward Drop and Power Dissipation in Thyristors.

IEEE Trans. Electron Devices ED-25, No. I, 16-22 (1978). [2.2] Adler, M. S.: An Operational Method to Model Carrier Degeneracy and Band Gap Narrowing.

Solid-State Electron. 26, No. 5, 387-396 (1983). [2.3] Ankri, D., Eastman, L. F.: GaA!As-GaAs Ballistic Hetero-Junction Bipolar Transistor.

Electronics Lett. 18, No.17, 750-751 (1982). [2.4] Anselm, A. I.: Einfiihrung in die Halbleitertheorie. Berlin: Akademie-Verlag 1964.

[2.5] Antoniadis, D. A., Dutton, R. W.: Models for Computer Simulation of Complete IC Fabrication Process.

IEEE J. Solid State Circuits SC-14, No. 2, 412-422 (1979).

[2.6] Awano, Y., Tomizawa, K., Hashizume, N., Kawashima, M.: Monte Carlo Particle Simulation of a GaAs

Submicron n+-i-n+ Diode. Electronics Lett. 18, No. 3, 133-135 (1982).

[2. 7] Awano, Y., Tomizawa, K., Hashizume, N., Kawashima, M.: Monte Carlo Particle Simulation of a GaAs

Short-Channel MESFET. Electronics Lett. 19. No. I, 20-21 (1983).

[2.8] Aymerich-Humett, X., Serra-Mestres, F., Millan, J.: An Analytical Approximation for the Fermi-Dirac

Integral F3/2(x). Solid-State Electron. 24, No. 10, 981-982 (1981).

[2.9] Baccarani, G., Mazzone, A. M.: On the Diffusion Current in Heavily Doped Silicon. Solid-State Electron.

18, 469-470 (1975).

[2.10] Baccarani, G.: Physics of Submicron Devices. Proc. VLSI Process and Device Modeling, pp. 1-23.

Katholieke Universiteit Leuven, 1983.

[2.11] Bennett, H. S.: Improved Concepts for Predicting the Electrical Behavior of Bipolar Structures in Silicon.

IEEE Trans. Electron Devices ED-30, No. 8, 920-927 (1983).