Applied Mathematics and Mechanics
(English Edition, Vol.3, No.6, Dec.
W.
SHOCKLEY'S
Published by HUST Press,
Wuhan, China
1982)
EQUATION
#/~D ITS L I M I T A T I O N *
Bai Zhe(S
~)
(Huazhong University of Science
and Technology, Wuhan)
(Received Dec. 19,198[)
Abstract
ABSTRACT
The one-dimensional
the motion
a rigid flying
plate under
attack has
The p r o b l e m problem
of p-n of
junction
in of
Ref.[1]
is discussed
in explosive
this
paper. only
We consider
Tsai index
Shu-tang's
view that
W. Shockan analytic solution
when the that
polytropic
of detonation
products
equals to three. In
ley's equation
be In
applied
to all
circumstances
cor-"weak" shock
general, a numerical
analysis iscannot
required.
this paper,
however,
by utilizingisthe
rect, but we cannot say, b e c a u s e of this, that W. Shockley's
behavior of the
reflection shock in the explosive products, and applying the small parameter purmethod of treatment and its conclusion for p-n junction are
terbation method,
an analytic,
first-order approximate solution is obtained for the problem of flying
wrong.
In this p a p e r we d e m o n s t r a t e d that W. Shockley's eplate driven by
various
high
explosives
with ideal
polytropic
othermodel.
than but nearly equal to three.
quation merely describes
P-s indices
junction
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
index)
for estimation of the velocity of flying plate is established.
I. Introduction
Concerning
p-n
the basis of the design of transistor and integrated circuit,the
1.
Introduction
junction theory in semiconductor physics
is very important.
First,this pro-
blemExplosive
was solved
by flying-plate
W. Shockley
and he ffmds
gave its
the
result, use
i.e.,
W. study
Shockley's
equadriven
technique
important
in the
of behavior
of
tion.
Tsai Shu-tang
deducedloading,
a general
equation
for describing
materials
under
intense impulsive
shockcurrent
synthesis density
of diamonds,
and explosive
welding and
cladding
of
metals.
The
method
of
estimation
of
flyor
velocity
and
the
way
of
raising
it
are
p-n
junction according to the continuous equation of fluid d y n a m i c s . T h e questions
purpose
ofofcommon
interest.
this paper is to analyze, compare and discuss both the methods of treatment
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
and the ofconclusion
p-n junction.
approach
solving the about
problemtheof motion
of flyor is to solve the following system of equations
governing the flow field of detonation products behind the flyor (Fig. I):
II. The Errors between Ideal and Practical Cases and the Cause
Experimental
and practical
example,
--ff
=o,
research proves
that there
ap +u_~_xp
+ au are more errors between the ideal
cases, especiallyauSi p-n
and GaAs p-n junction.
au junction
1
a Si p-n
junction
y
=0,
is shown in the following figure.
(i) When the p o s i t i v e l y biased
voltage
aS
as
voltage
its c h a r a c t e r i s t i c s
is high,
a--T
=o,
pare
=p(p,
s), in curve
shown
the r e c o m b i n a t i o n
(i.0
is low, the recombination currert
density is greater than the diffused current density.
junction;
As an
In potential barrier p-n
a .When the p o s i t i v e l y biased
current may be neglected,
the diffused cur-
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
rent is the
main
and its
a r a c t e r ishock
s t i c sofare
shown in
curve
When injecrespectively,
with
the one
trajectory
R ofc hreflected
detonation
wave
D as b.
a boundary
and the
trajectory
of flyor
as another
Bothvoltage
are unknown;
the position
of R is
andthat
the state
paration is Fgreat
(when
p o s i t iboundary.
v e l y biased
is great,
the case
injecmeters
on
it
are
governed
by
the
flow
field
I
of
central
rarefaction
wave
behind
the
detonation
wave
tion n o n e q u i l i b r i u m m i n o r i t y carrier c o n c B n t r a t i o n is close to or outstrips maD and by initial stage of motion of flyor also; the position of F and the state parameters of products
jority carrier concentration),
the p o s i t i v e l y biased voltage drops to the hole
293
* C o m m u n i c a t e d by Li Hao.
828
Bai Zhe
10 ~
....
.,o,I L i g'i' I I ill
x,l
:
}
I
i
i
m:
ideal
negative
direction
n:
ideal
positive
direction
o:
practical
negative
direction
Q:
practical
positive
direction
p:
junction
breakdown
Abstract
o--~--~" 75 '_o ~ 3'dk
Vy!,IA'~,T
The one-dimensional
problem of the motion of a rigid flying plate under explosive attack has
an analytic
only when
thej upolytropic
index of detonation products equals to three. In
The dsolution
-I /
relation
of P-n
nction
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
behavior
the reflection
applying
small
d i f f u s eof zone
and its shock
c h a r a cin
t ethe
r i s texplosive
i c s are products,
shown inand
curve
c . the
When
theparameter
p o s i t i v epurly
terbation method, an analytic, first-order approximate solution is obtained for the problem of flying
c u r r e n t is himher, the v o l t a g e drops in solid r e s i s t a n c e due to the effect of
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
series
c o n n e cof
t iflying
o n resistance.
ontac
t of
e l e c t r oresults
d e of by
both
ends ofThus
p-n
Final
velocities
plate obtainedWhen
agreethe
verycwell
with
numerical
computers.
anjunction
analytic formula
is very with
good,
two its
parameters
v o l t a g eof drop
high explosive
may be neglected.
(i.e. detonation
In this
velocity
case,
and polytropic
the volindex)
for
estimation
of
the
velocity
of
flying
plate
is
established.
tage drop of p o t e n t i a l b a r r i e r of p-n junction is very small; the j_[r relation
is not
shown
the
relation
in curve
(ii)
When
of
index,
but a linear
1.
6 .
the b i a s e d
voltaae
relation.
Its c h a r a c t e r i s t i c s
are
Introduction
is negative,
the t h e o r e t i c a l
value
of the re-
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
verse cunder
u r r e n tintense
is smaller
the shock
e x p e r isynthesis
m e n t a l of
value
and it
to saturation.
materials
impulsivethan
loading,
diamonds,
andtends
explosive
welding and
cladding
The p r oof
d umetals.
c t i v e The
c u r rmethod
e n t is ofthe
estimation
main one
of flyor
for velocity
the silicon
and theclass
way ofmraising
a t e r i aitl sarebquestions
ecause
ofthe
common
interest.
f o r b i d d e n band is wider, the intrinsic carrier c o n c e n t r a t i o n is low, and
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
the p r o d u c t i v e c u r r e n t d e n s i t y of the p o t e n t i a l b a r r i e r zone is g r e a t e r than the
approach of solving the problem of motion of flyor is to solve the following system of equations
n
e g a t i v ethed iflow
f f u sfield
i o n ofc udetonation
r r e n t density.
(This the
total
o d u cI):
t i v e c u r r e n t consists of
governing
products behind
flyorp r(Fig.
the p r o d u c t i v e
tion
current
of a m e t a l l u r g i c a l
zone
--ff
ap +u_~_xp+
and that of a surface
three
components.
of the
field
induced
In addition,
crease
in n e g a t i v e
in n e g a t i v e
The
biased
depletion
surface
junction
the width
biased
electrical
au
and
surface
aS
as
of a field
or it
=o,c o n s i s t s
induced
junc-
of one or a few of the
au
field
affects
1
yd e p l e t=0,
ion
of the p o t e n t i a l
barrier
the p r o d u c t i v e
current
zone).
zone
increases
the p r o d u c t i v e
a--T therefore,
=o,
This is not saturation.
p =p(p, s),
voltage;
voltage.
au
junction,that
current
with
(i.0
in-
the
increases
where
density,
and
particle
III. p,Tp,
h e S, Lui mare
i t apressure,
tion
o f W. specific
S h o c k l entropy
ey's
Equa
t i o n velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
Firstly,
a c cas
o ranother
d i n g toboundary.
q u a s i - F eBoth
r m i are
level
we cantheobtain
n c e nthe
t r astate
t i o nparaof
trajectory
F of flyor
unknown;
positionthe
of Rc oand
meters
on
it
are
governed
by
the
flow
field
I
of
central
rarefaction
wave
behind
the
detonation
wave
the i n j e c t i o n n o n e q u i l i b r i u m m i n o r i t y c a r r i e r at both ends of the p o t e n t i a l barD and by initial stage of motion of flyor also; the position of F and the state parameters of products
rier.
293
W. Shockley's Equation and Its Limitations
829
(3.1)
An=no(x) =u~,c A,~T
qF
w h e r e K 0 is B o l t z m a n n
type
P,~
electronic
constant,
np
concentration;
is n -type
equilibrium
According
is P type
n,~ is p type
electronic
to eqs.(3.1)
and
(3.2)
K,,T-
Ap=p.(x)=p.:
electronic
concentration;
P,
n
is
e q u i l i b r i u m e l e e t r o n i c concentration;
concentration.
(3.2),
we k n o w
that An
and A p are
functions
of
voltage.
applied
Secondly,
Boltzmann
eqs. (3.1)
statistical
and
(3.2)
distribution
are
the b o u n d a r y
Abstract
of
conditions.We
the n o n e q u i l i b r i u m
minority
shall o b t a i n
c a r r i e r by the
of a state
rigid flying
plate under
s o l uThe
t i o n one-dimensional
of c o n t i n u o u s problem
e q u a t i of
o n sthe
. motion
W h e n the
is steady,
the explosive
c o n t i n u oattack
u s e-has
an analytic solution only when the polytropic index of detonation products equals to three. In
q u a t i o n of the n o n e q u i l i b r i u m
m i n o r i t y c a r r i e r in the h o l e d i f f u s i o n zone is
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
behavior of the reflection
products,
the small parameter
purd2p.shock in the
dp. explosive
_:z~p dd~_
_ , , 2 #and
. , = applying
0
(3.3)
terbation method, an analytic, first-order approximate solution is obtained for the problem of flying
plate driven
by various high explosives with
thann e but
three.
A s for the small i n j e c t i o n ,
a vpolytropic
a l u e of indices
dex/dx other
m a y be
g l e cnearly
t e d ; equal
e,=0 to in
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
the n e u t r a l zone of n -type.
So eq. (3.3) m a y be w r i t t e n as
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
index) for estimation of the velocity ofD.flying
is established.
d~po plate
_ p~--P.o
=0
(3.~)
The
solution
of eq.(3.4)
m a 1.
y be Introduction
obtained according
to the b o u n d a r y
condi-
tion
x.--z
Explosive driven flying-plate technique qYffmds its important
use in the study of behavior of
K:?" 1)e L--P-(3.5)
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
a n a l o g yinterest.
w i t h this
ofBycommon
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
llp(E)__t,,o=ttpo( e ~ T
, ) e--L~
(3.6)
approach of solving the problem of motion of flyor is to solve the following system of equations
governing
the flow field of detonation products behind the flyor (Fig. I):
W h e n p-n j u n c t i o n has an a p p l i e d v o l t a g e , eqs. (3.5) and (3.6) show B o l t z -
p.(x)_:,oo=p.o(e
mann
statistical
fusion
zone.
r§ is the
fficient
distribution
In the a b o v e
life
of hole,
of e l e c t r o n ,
lue of P b o u n d a r y
We
extract
x,
of p-n
the
diffusion
equation.
calculate
the
f+
is
is d
z f f u s iau
on
au
x
value
aS
junction.
current
current
of the n o n e q u i l i b r i u m
--ff
e q u aap
t i o+u_~_xp
n , e, +is
If we
of
minority
=o,
the i n t e n s i t y
au
carrier
of the
c o1
efficient
of hole,
n
of p-n
y
=0,
boundary
L_
electrical
field,
is d i f f u s i o n
junction,
x~
coe-
(i.0
is x va-
as
a--T
=o,
of d i f f u s i o n w i t h eqs.(3.5) and
can p o=p(p,
b t a i n s),
the d i f f u s i v e c u r r e n t density,
density
density
in the dif-
of the n o n e q u i l i b r i u m
minority
carrier.
(3.6)
from
then we can
When
the
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
i n j e c t i o n with
is small,
ex=0 R
in ofthe
n e u t r ashock
l zone;
h e n c e thewave
c u rD
r e nas
t a dboundary
e n s i t y of
respectively,
the trajectory
reflected
of detonation
anddifthe
trajectory
F of rflyor
state
f u s i o n can
e p r e sas
e n tanother
the tboundary.
o t a l c u r rBoth
e n t are
d e n sunknown;
ity.
A mthe
o n g position
the d i fof
f uR
s i and
v e the
curr
e n t paradenmeters
on it are governed by the flow field I of central rarefaction wave behind the detonation wave
sities, the c u r r e n t d e n s i t y of h o l e d i f f u s i o n is
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
830
Bai Zhe
,-,.=(
,-, dp.(~)
The current
density of electron
qV
qD~
diffusion
p,o '~/~w;'r
-1)
is
!_=qD_ dn,(x)
qv
dx I..... =( qD_--~:~ -)(e-K~176
)
Lastly,
if we neglect
barrier
zone,
(3.8)
the action of the p r o d u c t i v e - r e c o m b i n a t i o n
then the total
current density
in a potential
is
qV
J=J'+J'=( qD-nr~L_+ qD'P'~ e-R~ - 1 )
(3.9)
Abstract
Eq. (3.9),
i.e., W. Shockley's
equation,
describes
the relation of current den-
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
of the p-n junction.
an analytic solution only when the polytropic index of detonation products equals to three. In
up, W.analysis
Shockley's
equation
is paper,
deduced
like this;when
the
general,Toa sum
numerical
is required.
In this
however,
by utilizing we
the satisfy
"weak" shock
behavior
of the
shock in the explosive
the small
purconditions
forreflection
the nonequilibrium
minorityproducts,
carrierandin applying
accordance
withparameter
Boltzmann
terbation
method,
an
analytic,
first-order
approximate
solution
is
obtained
for
the
problem
of
flying
statistical distribution; the snap-off layer is depletion(the nonequilibrium
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
minority carrier in P -zone and s -zone are pure motion of diffusion, and the
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
current
diffusion
which
represents
theexplosive
total current;
the injection
small
an
analyticofformula
with two
parameters
of high
(i.e. detonation
velocity and is
polytropic
index)
velocity of
flying plate
is established.of injection is smaller than
(the nfor
o n eestimation
q u i l i b r i uof
m the
minority
carrier
concentration
sity with the applied voltage
the equilibrium
bination
majority
of carrier
conditions
carrier
concentration);
1.
in the depletion
are necessary
conditions
the action of production-recom-
Introduction
layer must be
of constituting
neglected.
However,
the ideal
these
p-n junction mo-
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
del; hence, in fact, W. Shockley's equation merely describes the current densimaterials
under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
ty-voltage
relation
of the ofideal
p-n of
junction
model.
cladding
of metals.
The method
estimation
flyor velocity
and the way of raising it are questions
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
IV. T h e C o m p a r i s o n
approach of solving the problem of motion of flyor is to solve the following system of equations
Making
use field
of of
Einstein's
Tsai the
Shu-tang
governing
the flow
detonation relation,
products behind
flyor (Fig.has
I): given the continuous
equation
making
of electrons
the boundary
the current
density
and holes based on continuous
condition
is:
au
and holes.
au
1
With the two added to together,
y biased
=0, voltage
When the
density.
aS
dynamics.Purposely
=o,
into the equation,he
has Qbtainedsubsequently
ap +u_~_xp+
au
has got the total current
tal current density
--ff
enter
of electrons
equation
is positive,
he
the to-
(i.0
as
a--T
=o,
p =p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R ofD_
reflected
shock of detonation
! -D-7_--T-dx
D. . wave
9 D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para+q.,L
f;_~dx *
l;/dx.
(a.lO)
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
[
When the biased voltage
o o.
is negative,
the total
i:
current density
l
is
293
W. Shockley's Equation and Its Limitations
831
n~
i
n~~
~- ] p-~--f-ax
0..
+q.,L
From e q s . ( 4 . 1 )
and
biased voltages,
D. I~D-~/Odx~
i;~ dx *
l;_~dx"
(4.2)
we know t h a t
in describing
(3.11)
the positive
and negative
the real constant is
qn,
Abstract
~
D_
(3.12)
/(a)
p/dx*
/dx*
problem of the motion of a rigid flying plate under explosive attack has
,i
P~/(fl) D.+
The one-dimensional
an
analytic
solution
only when
the J~
polytropic
of detonation
equals toand
three. In
In the above
equation,
~ and
are the index
current
densitiesproducts
of electrons
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
holes at the origin; i is c h a r a c t e r i s t i c length, IVul is characteristic voltage;
behavior
of the reflection shock in the explosive products, and applying the small parameter purn.
is intrinsic
holes concentration;
~ is
the potential;
q is ofthe
terbation
method, an electrons
analytic, first-order
approximate solution
is obtained
for the problem
flying
plate
driven
by
various
high
explosives
with
polytropic
indices
other
than
but
nearly
equal
to
three.
absolute value of the electronic charge; p~=p/n., n*=n/n. , ~ * = 4/IVo], x * = x / L ,
Final
velocities of flying plate obtained agree very well with numerical results by computers. Thus
a and ~, are the value of both ends of x* ; n: ,n;,p~,p~, V~ Vp are the values of
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
c o r r e s p o n d i n g physical quantities at the boundary; /~ is the wave vector; T is
index)
for estimation of the velocity of flying plate is established.
the temperature.
Comparing W. Shockley's equation with Tsai Shu-tang's current density equa-
1.
Introduction
tion, we have come to know that
Explosive
driven
technique
its important
use in theequation
study of shows
behavior of
(i) When
the flying-plate
barrier voltage
is ffmds
negative,
W. Shockley's
materials
under
intense
impulsive
loading,
shock
synthesis
of
diamonds,
and
explosive
welding
and
that the density of the reverse current has no relation with the applied voltage
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
and their c h a r a c t e r i s t i c curve is in a state of saturation.
Equation (4.3) shows
of common interest.
that Under
the reverse
currentofdensity
has a relation
with the
barrier
vmltage.
the assumptions
one-dimensional
plane detonation
and reverse
rigid flying
plate, the
normal
approach
of motion
flyor isofto saturation.
solve the following
equations
Their c hof
a r asolving
c t e r i sthe
t i c problem
curve is
not ~n of
a state
The system
latterofis
close
governing
the
flow
field
of
detonation
products
behind
the
flyor
(Fig.
I):
to the Dractical case.
(ii) Equation
and
(4.3) has no relation with
(4.2) contains 8 which
error between theoretical
current,
@ , but the last term of eas.(4.1)
--ff
=o,
marks
recombination
ap the
+u_~_xp
+ au
y
but that is not the only source error.
the surface effect,
=0,
In addition,
the effect aS
of series
a s connection
produce error.
(iii) Tsai Shu-tang's
current.
This shows that the
and practical
results from the recombination
au
au values
1
a--T
=o,
s), Shockley's
viewp =p(p,
that W.
large injection,
resistances,
(i.0
etc., all can
equation cannot be applied to
general cases is correct, but, W. Shockley's method of treatment and his conclu-
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
sion about the ideal p-n junction are also correct.
Moreover, this conclusion
respectively,
with the trajectory R of reflected shock of detonation
wave D as a boundary and the
is of great
importance
forboundary.
the research
in unknown;
basic theories.
trajectory
F of flyor
as another
Both are
the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
832
Bai Zhe
Acknowledgement
The author is grateful
preparation
to Prof.
Li Hao for a great deal of help in the
of this paper.
References
1.
2.
3.
4.
Tsai Shu-tang, Two problems of semiconductor physics discussed in the point
of view of fluid dynamics, Applied Mathematics and Mechanics, Vol.], No.3,
(1980),311.
Shockley, W., BSTJ, 28,(1949),435.
Sze, S. M.
Physics of Semiconductor Devices, Chap.3, John Wiley & Sons,
New York, (1969).
Abstract
Smith, R. A., Semiconductor, second
edition, Chap.7, London, New York,(1978).
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
an analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
index) for estimation of the velocity of flying plate is established.
1.
Introduction
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
approach of solving the problem of motion of flyor is to solve the following system of equations
governing the flow field of detonation products behind the flyor (Fig. I):
--ff
ap +u_~_xp+
au
au
aS
as
au
y1
=o,
=0,
(i.0
a--T
=o,
p =p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293