- Email: [email protected]
The Point Defect
The point defect
40 charge-compensation
for
a divalent
cation
sub-lattice
would
be
the
foUowing defect equation: : 2 M 2+ -%
2.1.I.where
M+
+
M 3+
t h e M 3+ a n d M + are s i t u a t e d on n e a r e s t
neighbor
cation sites,
which were originally divalent.
Thus, the charge compensation mechan/sm represents the single most important mechAnlsrn which operates withln the defect solid. B e c a u s e of this, th e n u m b e r a n d t y p e s of defects, w h i c h c a n a p p e a r in t h e solid, are limited. T h i s r e s t r i c t s t h e n u m b e r of d e f e c t t y p e s we n e e d to c o n s i d e r , in b o t h e l e m e n t a l (all th e s a m e k i n d of atom) a n d ionic l a t t i c e s (having b o t h c a t i o n s a n d a n i o n s p r e s e n t ) . We have s h o w n t h a t by s t a c k i n g a t o m s or p r o p a g a t i o n
units
together,
a solid w i t h
specific
synm~etry
r e s u l t s . If w e have d o n e t h i s properly, a p e r f e c t solid s h o u l d r e s u l t w i t h no holes or d e f e c t s in it. Yet, th e 2 n d law of t h e r m o d y n a m i c s d e m a n d s t h a t a c e r t a i n n u m b e r of p o i n t d efects (vacancies) a p p e a r in t h e lattice. It is i m p o s s i b l e to o b t a i n a solid w i t h o u t s o m e sort of defects. A p e r f e c t solid w o u l d violate t h i s law. T h e possible
at
absolute
zero
2nd
law s t a t e s t h a t zero entropy is only
temperature.
Since
most
solids
exist
at
t e m p e r a t u r e s far f r o m a b s o l u t e zero, t h o s e t h a t we e n c o u n t e r eu~e d e f e c t solids. It is n a t u r a l to a s k w h a t th e n a t u r e of t h e s e d e f e c t s m i g h t be, p a r t i c u l a r l y w h e n we a d d a foreign c a t i o n (activator) to a solid to f o r m a phosphor. C o n s i d e r t h e s u r f a c e of a solid. In th e in terior, we see a c e r t a i n s y m m e t r y which
depends
upon the
surface from the interior,
structure
of t h e
solid. As we a p p r o a c h
the
t h e s y m m e t r y b e g i n s to c h a n g e . At the v e r y
surface, t h e s u r f a c e a t o m s see only h a l f t h e s y m m e t r y t h a t the i n t e r i o r a t o m s do. R e a c t i o n s b e t w e e n solids t a k e place at t h e surface. T h u s , t h e s u r f a c e of a solid r e p r e s e n t s
a defect in itself s i n c e it is not like t h e
i n t e r i o r of t h e solid. In a t h r e e - d i m e n s i o n a l solid, we c a n p o s t u l a t e t h a t t h e r e o u g h t to be
2.1 Types of point defects three
major
types
of defects,
having
41
either
one-,
two-
or
three-
d i m e n s i o n s . Indeed, this is exactly t h e case found for defects in solids, as we briefly d e s c r i b e d in the p r e c e d i n g
chapter.
We have a l r e a d y given
n a m e s to e a c h of t h e s e t h r e e t y p e s of defects. T h u s a o n e - d i m e n s i o n a l defect of t h e lattice is called a "point" defect, a t w o - d i m e n s i o n a l defect a "line" or "edge" defect a n d a t h r e e - d i m e n s i o n a l defect is called a "plane" or "volume" defect. We have a l r e a d y d e s c r i b e d , in an e l e m e n t a r y way, l i n e a n d v o l u m e defects a n d will not a d d r e s s t h e m f u r t h e r e x c e p t to point out h o w t h e y m a y arise w h e n c e r t a i n point defects are p r e s e n t . It is sufficient to realize t h a t t h e y exist a n d are i m p o r t a n t
for a n y o n e w h o
studies
h o m o g e n e o u s m a t e r i a l s s u c h as m e t a l s . Point defects
are c h a n g e s at a t o m i s t i c levels, while
line
and volume
defects are c h a n g e s in s t a c k i n g of p l a n e s or g r o u p s of a t o m s ( m o l e c u l e s ) in t h e s t r u c t u r e . The former affect t h e c h e m i c a l p r o p e r t i e s of the solid w h e r e a s the l a t t e r affect t h e p h y s i c a l p r o p e r t i e s of t h e solid. Note t h a t t h e a r r a n g e m e n t ( s t r u c t u r e ) of the individual a t o m s (ions) are not affected, only the m e t h o d in w h i c h the s t r u c t u r e u n i t s are a s s e m b l e d . T h a t is, t h e s t r u c t u r e of t h e solid r e m a i n s i n t a c t in spite of t h e p r e s e n c e of d e f e c t s . Let us n o w e x a m i n e e a c h of t h e s e defects in m o r e detail, s t a r t i n g w i t h t h e one-dimensional
lattice
defect
and
then
with
the
multi-dimensional
defects. We will find t h a t specific types have b e e n found to be a s s o c i a t e d w i t h e a c h type of d i m e n s i o n a l defect w h i c h have specific effects u p o n t h e stability of the solid s t r u c t u r e . It s h o u l d be clear t h a t t h e type of p o i n t defect
prevalent
homogenous
in any given
(same
atoms)
or
solid will d e p e n d heterogeneous
upon whether
(composed
it is
of differing
atoms). I. The Po'mt Defect in H o m o g e n e o u s Solids We begin by identifying t h e various defects w h i c h c a n arise in solids a n d later will s h o w how t h e y c a n be m a n i p u l a t e d to o b t a i n d e s i r a b l e p r o p e r t i e s not found in naturally f o r m e d solids. Let us look first at t h e h o m o g e n e o u s type of solid. We will first r e s t r i c t o u r d i s c u s s i o n to solids w h i c h stoichiometric,
are
a n d later will e x a m i n e solids w h i c h c a n be classified as
" n o n - s t o i c h i o m e t r i c " , or having a n excess of one or a n o t h e r of one of t h e
The point defect
42
b u i l d i n g b l o c k s of t h e solid. T h e s e o c c u r in s e m i - c o n d u c t o r s as well as o t h e r t y p e s of e l e c t r o n i c a l l y or optically active solids. Suppose
you w e r e
given
the
problem
of identifying
defects
in
a
h o m o g e n e o u s solid. Since all of t h e a t o m s in t h i s type of solid are t h e s a m e , t h e p r o b l e m is s o m e w h a t simplified over t h a t of the h e t e r o g e n e o u s solid (that is- a solid c o n t a i n i n g m o r e t h a n one type of a t o m or ion). After s o m e i n t r o s p e c t i o n , you could s p e c u l a t e t h a t t h e h o m o g e n e o u s solid c o u l d have t h e following t y p e s of p o i n t d e f e c t s : 2 . 1 . 2 . - Types of Point Defects E x p e c t e d in a H o m o g e n e o u s S o l i d Vacancies
Substitutional Impurities
Self-interstitial
Interstitial Impurities
On t h e left are t h e two types of p o i n t defects w h i c h involve the lattice itself, while t h e o t h e r s involve i m p u r i t y a t o m s (Note t h a t i n t e r s t i t i a l a t o m s c a n involve e i t h e r a n i m p u r i t y a t o m or the s a m e a t o m t h a t m a k e s up t h e lattice s t r u c t u r e itself). Indeed, t h e r e do n o t s e e m to any m o r e t h a n t h e s e four, a n d indubitably, no o t h e r s have b e e n observed. Note t h a t we are l i m i t i n g o u r defect family to p o i n t defects in t h e lattice a n d are i g n o r i n g line a n d v o l u m e defects
of t h e lattice. T h e s e
four p o i n t defects, given
above, are i l l u s t r a t e d in the following diagram, given as 2.1.3. o n the n e x t page. Note t h a t w h a t we m e a n by an "interstitial" is a n a t o m t h a t c a n fit into t h e s p a c e s b e t w e e n t h e m a i n a t o m s in the c r y s t a l l i n e array. In this case, w e have s h o w n a h e x a g o n a l lattice a n d have labeled e a c h type of p o i n t defect. O b s e r v e t h a t we have s h o w n a v a c a n c y in o u r h e x a g o n a l lattice, as well as a foreign i n t e r s t i t i a l a t o m w h i c h is small e n o u g h to fit into t h e i n t e r s t i c e between
the
atoms
of the
structure.
Also
shown
are
two
types
of
s u b s t i t u t i o n a l a t o m s , one larger a n d the o t h e r s m a l l e r t h a n the a t o m s c o m p o s i n g the p r i n c i p a l h e x a g o n a l lattice. In b o t h cases, t h e h e x a g o n a l p a c k i n g is d i s r u p t e d due to a "non-fit" of t h e s e a t o m s in t h e s t r u c t u r e . Additionally, we have i l l u s t r a t e d a n o t h e r type of defect t h a t c a n arise
2.1 Types of point defects
43
I Defects Which Can Occur in a Homogeneous Hexagonal Lattice I
within
the
homogeneous
lattice
(in
addition
to
the
vacancy
and
s u b s t i t u t i o n a l i m p u r i t i e s t h a t are b o u n d to arise). This is called t h e "selfinterstitial". Note t h a t it h a s a decisive effect on t h e defect.
Since
the
atoms
are
all t h e
same
size,
the
structure
at t h e
self-interstitial
i n t r o d u c e s a l l n e - d e f e c t in the overall s t r u c t u r e . It s h o u l d be e v i d e n t t h a t t h e l i n e - d e f e c t i n t r o d u c e s a difference in p a c k i n g o r d e r since t h e c l o s e p a c k i n g a t t h e a r r o w s h a s c h a n g e d to cubic a n d t h e n r e v e r t s to h e x a g o n a l in b o t h lower a n d u p p e r rows of a t o m s . It m a y be t h a t this type of d e f e c t is a m a j o r c a u s e of t h e line or edge type of defects t h a t a p p e a r in m o s t homogeneous
solids.
In c o n t r a s t ,
the
other
defects
produce
only
a
d i s r u p t i o n in the l o c a l i z e d p a c k i n g o r d e r of t h e h e x a g o n a l lattice, i.e.- t h e defect
does not extend
t h r o u g h o u t t h e lattice,
b u t only close
to t h e
The point defect
44 specific defect.
It s h o u l d be evident
t h a t m e t a l s or solid solutions of
m e t a l s (alloys) show s u c h behavior in c o n t r a s t to h e t e r o g e n e o u s l a t t i c e s w h i c h involve c o m p o u n d s s u c h as ZnS. This a c c o u n t s for the t r e m e n d o u s d i s c r e p a n c y b e t w e e n t h e o r e t i c a l a n d actual s t r e n g t h of c e r t a i n alloys in p r a c t i c a l a p p l i c a t i o n s due to "fatigue" failure w h e n the object is b e i n g used. Now, s u p p o s e t h a t we have a solid solution of two (2) e l e m e n t a l solids. Would t h e point defects be the same, or not? An easy way to visualize s u c h point defects is s h o w n in the following d i a g r a m : 2.1.4.- Defects in the H o m o g e n e o u s Solid C o n t a i n i n g 2 Solids in Solution
..~i~~
vm=
Interstitial 1~
Int e rst it .
.
.
.
Vacancy
,IInterstitial cog e n e o u ~ g o n a/ Structure -- Same Size Atoms
Homogeneous Solid Solution Differing Size Atoms
-
Here, we u s e a h e x a g o n a l l y - p a c k e d r e p r e s e n t a t i o n of a t o m s to depict t h e c l o s e - p a c k e d solid. Both types of h o m o g e n e o u s solids are shown, w h e r e one solid is c o m p o s e d
of the
same
sized
atoms while
the
other
is
c o m p o s e d of two different sized atoms. On the right are the types of p o i n t
2.1 Types of point defects
45
d e f e c t s t h a t c o u l d o c c u r for t h e s a m e sized a t o m s in t h e lattice. T h a t is, given a n a r r a y of a t o m s in a t h r e e d i m e n s i o n a l lattice, only t h e s e two t y p e s of lattice p o i n t d e f e c t s could o c c u r w h e r e t h e size of t h e a t o m s are t h e s a m e . T h e t e r m " v a c a n c y " is s e l f - e x p l a n a t o r y b u t " s e l f - i n t e r s t i t i a l " m e a n s t h a t o n e a t o m h a s s l i p p e d into a s p a c e b e t w e e n t h e rows of a t o m s . In a lattice w h e r e t h e a t o m s are all of t h e s a m e size, s u c h b e h a v i o r is e n e r g e t i c a l l y v e r y difficult u n l e s s a severe d i s r u p t i o n of t h e lattice o c c u r s (usually a " l i n e - d e f e c t " results). T h i s b e h a v i o r is q u i t e c o m m o n in c e r t a i n t y p e s of h o m o g e n e o u s solids. In a like m a n n e r , if t h e m e t a l - a t o m w e r e to have b e c o m e m i s p l a c e d in t h e lattice a n d w e r e to have o c c u p i e d o n e of t h e i n t e r s t i t i a l p o s i t i o n s , as s h o w n in t h e different sized a t o m solid ( s e e 2.1.4.)
t h e n t h e lattice is d i s r u p t e d by its p r e s e n c e
at t h e i n t e r s t i t i a l
position. T h i s type of defect h a s also b e e n o b s e r v e d . Note t h a t t h e a t o m s are usually n o t c h a r g e d in t h e h o m o g e n e o u s l a t t i c e . S u m m a r i z i n g , t h r e e t y p e s of p o i n t defects are e v i d e n t in a h o m o g e n e o u s lattice. In a d d i t i o n to t h e Vacancy, two t y p e s of s u b s t i t u t i o n a l defects c a n also be
delineated.
Both are
direct
substitutions
in t h e
"lattice",
or
a r r a n g e m e n t of t h e a t o m s . One is a s m a l l e r a t o m , w h i l e t h e o t h e r is l a r g e r than
the
atoms
comprising
the
lattice.
In
both
cases,
the
lattice
a r r a n g e m e n t affects t h e h e x a g o n a l o r d e r i n g of t h e lattice a t o m s a r o u n d it. T h e lattice p a c k i n g is s e e n to be affected for m a n y lattice d i s t a n c e s . It is for t h i s r e a s o n t h a t c o m p o u n d s c o n t a i n i n g i m p u r i t i e s s o m e t i m e s have q u i t e different c h e m i c a l r e a c t i v i t i e s t h a n t h e p u r e s t ones. However, t h e i n t e r s t i t i a l i m p u r i t y does not affect t h e lattice o r d e r i n g at all. Now, let us look at t h e h e t e r o g e n e o u s l a t t i c e II. T h e Point Defect in H e t e r o g e n e o u s S o l i d s The
situation concerning
defects
in h e t e r o g e n e o u s
inorganic
solids is
s i m i l a r to t h a t given above, e x c e p t for one v e r y i m p o r t a n t factor, t h a t of c h a r g o o n t h e a t o m s . Covalent i n o r g a n i c solids a r e a r a r i t y while i o n i c i t y or p a r t i a l ionicity s e e m s to be t h e n o r m . T h u s , h e t e r o g e n e o u s solids are usually c o m p o s e d of c h a r g e d m o i e t i e s , half of w h i c h are positive ( c a t i o n s ) a n d half negative (anions). In g e n e r a l , t h e total c h a r g e of t h e c a t i o n s will
The point defect
46
e q u a l t h a t of t h e a n i o n s (Even in the case of s e m i - c o n d u c t o r s , w h e r e t h e total of the c h a r g e s is n o t zero, the excess c h a r g e (n- or p- type) is s p r e a d over t h e w h o l e lattice so t h a t no single atom, or g r o u p of a t o m s , ever has a c h a r g e different from its n e i g h b o r s . Note also t h a t m o s t of the s e m i conductors
that
we
use
are
homogeneous
in
nature,
modified
by
h o m o g e n e o u s a d d i t i o n s to form p- or n - t y p e electrically c h a r g e d areas). In a given s t r u c t u r e , c a t i o n s are usually s u r r o u n d e d by anions, a n d vice-versa. B e c a u s e of this, we c a n r e g a r d the lattice as being c o m p o s e d of a c a t i o n s u b - l a t t i c e a n d an a n i o n mab-lattice. ( R e m e m b e r w h a t w a s stated in Chapter
1
concerning
the
fact
that
most
structures
are
oxygen-
d o m i n a t e d ) . W h a t we m e a n by a "sub-lattice" is i l l u s t r a t e d in the following diagram: 2.1.5.-
IA Cubic LatticeShowing the Cation and Anion Sub-Lattices I
Cation Sub-Lattice
Anion Sub-Lattice
Combined Lattice
In this case, we have s h o w n b o t h the cubic cation a n d a n i o n "sub-lattices s e p a r a t e l y , a n d t h e n t h e c o m b i n a t i o n . It s h o u l d be clear t h a t all positive c h a r g e s in t h e c a t i o n s u b - l a t t i c e will be b a l a n c e d by a like n u m b e r of negative c h a r g e s in the a n i o n sub-lattice, even if excess c h a r g e exists in o n e or the o t h e r of the sub-lattices. If an a t o m is m i s s i n g , t h e n the overall lattice
r e a d j u s t s to c o m p e n s a t e
different
atom
compensation
present,
for this loss of c h a r g e .
having
a
differing
charge,
If t h e r e the
is a
charge-
m e c h a n i s m again m a n i f e s t s itself. T h u s , a cation with an
e x t r a c h a r g e n e e d s to be c o m p e n s a t e d by a like anion, or by a n e a r e s t n e i g h b o r c a t i o n w i t h a l e s s e r charge. An e x a m p l e of this type of c h a r g e -
2.1 Types of point defects compensation mechanism
47
for a divalent c a t i o n s u b - l a t t i c e w o u l d be t h e
following defect equation: 2.1.6.-
2 C a 2+ ~
Li +
+
Sb 3+
w h e r e t h e Sb 3+ a n d Li + are s i t u a t e d on n e a r e s t n e i g h b o r c a t i o n s u b - l a t t i c e sites, in t h e d i v a l e n t Ca 2+ s u b - l a t t i c e . Note t h a t a total c h a r g e of 4+ e x i s t s on b o t h s i d e s of t h e above e q u a t i o n .
Thus, the charge compensation mechR:nlsm represents the single most important mechanism which operates withln the d e f e c t ~ solid. We find t h a t t h e n u m b e r a n d t y p e s of defects, w h i c h c a n a p p e a r in t h e h e t e r o g e n e o u s solid, are l i m i t e d b e c a u s e of two factors: 1) T h e c h a r g e - c o m p e n s a t i o n f a c t o r 2) T h e p r e s e n c e of two s u b - l a t t i c e s in t h e ionic solid. These
factors
restrict
the
number
of p o i n t
defect t y p e s
we
need
to
c o n s i d e r in ionic h e t e r o g e n e o u s l a t t i c e s (having b o t h c a t i o n s a n d a n i o n s p r e s e n t ) . For ionic s o l i d s , t h e following t y p e s of d e f e c t s h a v e b e e n f o u n d to exist: S c h o t t k y d e f e c t s ( a b s e n c e of b o t h c a t i o n a n d a n i o n ) Cation v a c a n c i e s Anion v a c a n c i e s F r e n k e l defects (Cation v a c a n c y p l u s s a m e c a t i o n as interstitial) I n t e r s t i t i a l i m p u r i t y a t o m s (both c a t i o n a n d a n i o n ) Substitutional impurity atoms(both cation and anion) T h e s e d e f e c t s are i l l u s t r a t e d in t h e following d i a g r a m , given as 2.1.7. o n the next page. Note t h a t , in g e n e r a l , a n i o n s are l a r g e r in size t h a n c a t i o n s d u e to t h e e x t r a e l e c t r o n s p r e s e n t in t h e former. A h e x a g o n a l lattice is s h o w n in
The point defect
48
2.1.7.] P o i n t Defects Which Can Occur in the Heterogeneous Ionic Solid]
Schottky Defect
Frenkel Defect
Anion Vacancy
Cation Vacancy
Substitutional Cation InterstitialCation Substitutional Anion
2.1.7. w i t h b o t h F r e n k e l a n d S c h o t t k y defects, as well as s u b s t i t u t i o n a l defects. T h u s , if a c a t i o n is m i s s i n g (cation vacancy) in the cation sublattice, a like a n i o n will be m i s s i n g in the a n i o n sub-lattice. T h i s is k n o w n as a S c h o t t k y
defect
(after
the
first i n v e s t i g a t o r
(1935)
to note
its
where
the
existence). In t h e c a s e of t h e F r e n k e l defect, cation was supposed
to r e s i d e
th e
in th e
"square" r e p r e s e n t s lattice
before
it m o v e d
to its
i n t e r s t i t i a l p o s i t i o n in t h e cation sub-lattice. Additionally, "A nti-Fre nke l" d e f e c t s c a n exist in t h e an io n sub-lattice. T h e s u b s t i t u t i o n a l defects are s h o w n as t h e s a m e size as th e c a t i o n or a n i o n it displaced. Note t h a t if
49
2.2 The plane net
t h e y w e r e not, t h e lattice s t r u c t u r e w o u l d b e d i s r u p t e d f r o m r e g u l a r i t y at t h e p o i n t s of i n s e r t i o n of t h e foreign ion. To s u m m a r i z e , t h e c a t e g o r i e s of p o i n t d e f e c t s p o s s i b l e for t h e s e two t y p e s of l a t t i c e s are i l l u s t r a t e d are: 1). In a n e l e m e n t a l solid, we m a y have: Vacancies Self-interstitial Interstitial Impurities Substitutional Impurities 2). In t h e ionic solid, i.e.- h e t e r o g e n e o u s solid, we m a y have: S c h o t t k y d e f e c t s ( a b s e n c e of b o t h c a t i o n a n d a n i o n ) Cation or a n i o n v a c a n c i e s F r e n k e l d e f e c t s (Cation v a c a n c y p l u s t h e s a m e c a t i o n as interstitial) I n t e r s t i t i a l i m p u r i t y a t o m s (both c a t i o n a n d anion) Substitutional impurity atoms(both cation and anion) All of t h e s e
point
defects
are
intrinsic
to
the
solid.
The
factors
r e s p o n s i b l e for t h e i r f o r m a t i o n are e n t r o p y effects (point d e f e c t faults) a n d i m p u r i t y effects. At t h e p r e s e n t time, t h e h i g h e s t - p u r i t y m a t e r i a l s available still c o n t a i n about 1.0 p a r t p e r billion of various i m p u r i t i e s , y e t are 9 9 . 9 9 9 9 9 9 9
% p u r e . S u c h a solid will c o n t a i n about 1014
impurity
a t o m s p e r mole. So it is safe to say t h a t all solids c o n t a i n i m p u r i t y a t o m s , a n d t h a t it is u n l i k e l y t h a t we shall ever b e able to o b t a i n a solid w h i c h is
c o m p l e t e l y pure and d o e s not contain defects. 2.2. - T H E PLANE N E T Now, let u s c o n s i d e r h o w s u c h d e f e c t s arise in a n y given solid. T h e e a s i e s t w a y to visualize h o w i n t r i n s i c d e f e c t s o c c u r in t h e solid is to s t u d y t h e PLANE N E T . I m a g i n e t h a t we have M as c a t i o n s a n d X as a n i o n s (we shall
The point defect
50
i g n o r e formal c h a r g e for the m o m e n t ) . This is s h o w n in the following diagram:
[The Pl one Net'l
2.2.1.-
X M g M X M xMI~x M M X MXx M X M X M X
x
x
,.(,(~._.~~ M XM M X X M
x
,__
x i i,•
M X M X X M X M X M X M X
M M X M
X M X X M X M M X M X X M X M
It is easy to see t h a t we c a n stack a series of t h e s e
"NETS" to form a
t h r e e - d i m e n s i o n a l solid. Note t h a t we have u s e d t h e labeling: V = vacancy; i = i n t e r s t i t i a l ; m = c a t i o n site; x= a n i o n site a n d s = surface site. One m i g h t t h i n k t h a t p e r h a p s V-m a n d V+x o u g h t to be i n c l u d e d in o u r list of v a c a n c i e s . However, a n e g a t i v e l y - c h a r g e d c a t i o n v a c a n c y alone, p a r t i c u l a r l y when
it is s u r r o u n d e d b y negative a n i o n s , w o u l d
Neither Either
n o t be very stable.
s h o u l d a p o s i t i v e l y - c h a r g e d a n i o n v a c a n c y be any m o r e arrangement
would
require
high
energy
stable.
stabilization to exist.
T h e r e f o r e , we do n o t i n c l u d e t h e m in o u r listing. However,
a Vm
c o u l d c a p t u r e a positive c h a r g e
likewise for t h e Vx w h i c h t h e n b e c o m e s a V'x
to b e c o m e
V+m
and
Both of t h e s e are stable
w h e n s u r r o u n d e d by o p p o s i t e l y c h a r g e d sites. We have a l r e a d y s t a t e d t h a t s u r f a c e sites are special. intrinsic
defects.
The
Hence,
they
same criteria
are i n c l u d e d
in o u r listing
apply concerning
charge
of
on t h e
defect. For t h e p l a n e net, we c a n e x p e c t the following types of i n t r i n s i c defects:
2.2 The plane net 2.2 . 2 . -
Vacancies
51 Charged Particles
VM, V x , V+M , V x
e,
Surface Sites
Interstitials
Ms,Xs,
M i , ~- , M+i, X'i The
positive
hole
requires
e q u i v a l e n t of t h e e l e c t r o n ,
n+
more
explanation,
n + is
the
M+i , X s electronic
e- , in s o l i d s a n d t h e y a n n i h i l a t e e a c h o t h e r
upon reaction2.2.3.-
e-
+
n+
=
energy.
T h e following d i a g r a m s h o w s h o w t h e positive hole c a n exist in t h e solid: 2.2.4-
The Germanium Lattice Containing a Positive Hole [
+
73
- G e a 2 = A r C ore" 4s 2 4 p 2 -- G a a l7o =
Here, w e r e p r e s e n t
A r C o r e " 4s 2 4 p ~
Ge w h i c h h a s 32 e l e c t r o n s .
In t h e solid, it f o r m s
4 ( s p ) 3 h y b r i d b o n d s as t h e s e m i - c o n d u c t o r . Ga h a s only 31 e l e c t r o n s a n d c a n f o r m b u t t h r e e b o n d s . If it is i n c o r p o r a t e d into t h e s t r u c t u r e , t h e
The point defect
52
lattice responds by forming a positive electronic "hole", i.e.- =+. Still another type of electronic defect is the color-center, following diagram: 2.2.5.-
shown in the
] A n E l e c t r o n T r a p p e d a t A n A n i o n V a c a n c y ]'
(E) e (E) e (E) e
~G ~ Q $ Q
@e e@
e
(E) e @ e (E) e ]Absorption as a Function of Cation Present in Lattice] 8500 4000 I
4000
6000
i
I
I
I
I
I
6000 I
I
NaC1
LiC1
4000 I
6000 |
I
4000 I
KCI
6000 8000 A
m
m
Rb
-
,
m m
'
"
4-~
D-. 0 [8
,j
I
4
3
24
3
2
3
, 2
3
2
Energy in Electron Volts In order to compensate for the loss of negative charge, the lattice has captured an electron and charge-compensation has resulted at the anion vacancy. This combination is called an "F-center". This is a special case where an electron is localized at the vacancy and is optically active. That
53
2.2 The plane net
is, it a b s o r b s light w i t h i n a w e l l - d e f i n e d b a n d a n d is called a c o l o r - c e n t e r s i n c e it i m p a r t s a specific color to t h e crystal. Note t h a t t h e p e a k of t h e b a n d c h a n g e s as t h e size of t h e c a t i o n in t h e alkali h a l i d e s i n c r e a s e s . There
a p p e a r s to be an i n v e r s e r e l a t i o n b e t w e e n
(actually, t h e
polarizability of t h e
cation)
t h e size of t h e c a t i o n
and the peak
e n e r g y of t h e
a b s o r p t i o n b a n d of t h e s e F - c e n t e r s . We c a n n o w p r o c e e d to w r i t e a s e r i e s of d e f e c t r e a c t i o n s for o u r p l a n e n e t in t e r m s of specific defects, i n c l u d i n g t h a t of t h e F - C e n t e r , w h e r e we u s e 5 as a s m a l l f r a c t i o n : 2.2.6.-
SCHOTII~
:
MX = M1. s X 1 . 5
FRENKEL :
+ 5Vm
MX = M 1-5 X
+ 5 Mi +
ANTI-FRENKEL:
MX = M X I - 6
+ 6Vx
F - CENTER:
MX = M X I - + { V x / e }
+
+ 6Vx 5Vm 5Xi
+8/2
X2
T h e s e e q u a t i o n s are valid for t h e d e f e c t s in a h e t e r o g e n e o u s lattice w h e r e , for t h e F - C e n t e r , t h e b r a c k e t s e n c l o s e t h e c o m p l e x c o n s i s t i n g of an e l e c t r o n c a p t u r e d a t a n a n i o n v a c a n c y , i.e.- {Vx/e-}. Actually, t h e f o r m a t i o n of a n
F-center
is
more
complicated
than
this.
A more
complete
e x p l a n a t i o n is given as follows. It is well k n o w n t h a t F - c e n t e r s c a n b e f o r m e d by e x p o s i n g t h e NaCI c r y s t a l to s o d i u m m e t a l vapor. T h e following d e f e c t r e a c t i o n s t a k e p l a c e : 2.2.7.-
Na ~
=
Na +
Na +
+
{ N a +,C1-}
VC1
+
e-
=
+ =
e
-
2{Na +[CI-,VCll}
[Vcl/e-]
T h e s e e q u a t i o n s i l l u s t r a t e h o w t h e c r y s t a l r e s p o n d s to t h e p r e s e n c e of s o d i u m v a p o r , i.e.- e x c e s s N a + , by f o r m i n g a n i o n v a c a n c i e s , to f o r m t h e Fc e n t e r . However, if o n e a n i o n v a c a n c y c a n be f o r m e d w h i c h c a p t u r e s an e l e c t r o n w h i c h is optically active, t h e n you m i g h t t h i n k t h a t m o r e t h a n
The point defect
54
one anion-vacancy c o m p l e x m i g h t be possible. Indeed, this is the case. This is illustrated in the following diagram: 2.2.8.-
+-+-+-+-+2-+
- ~ -
!
-I-- -k t -I--
- .i._ .~.~@__i_ -+--l---l-y-l---l--
4--
-I---I--
-I-~-I--
-I---I-
[II0]
F-Center A better diagram:
representation
of the
M-Center M-center
is s h o w n
in
the
following
2.2.9.-
An "M-Center" Shown on T w o Planes of a Cubic LatticeI
(2)
U n d e r c e r t a i n precise e x p e r i m e n t a l conditions, two anion sites can be conjoined (in the s a l t - s t r u c t u r e , along the {i,I,0} plane) to form what is
55
2.2 The plane net
t e r m e d t h e "M-Center" as s h o w n above in t h e two d i a g r a m s , 2.2.8. a n d 2.2.9. The
optical p r o p e r t i e s
of t h e s e two t y p e s of c e n t e r s
are given in t h e
following d i a g r a m : 2.2.10.-
Absorption Spectra at 77 ~ for a KCI Crystal with F-and M-Centers Present i
I
1.0
~
i
F-Center M-Center Absorption
0.8
0-.0.4
0,:2
0
400
500
600 700 800 Wavelength in Nanometers
900
T h e defect e q u a t i o n for f o r m a t i o n of t h e M - c e n t e r is also given as follows: 2.2.11.-
M-center:
KC1 = KCll.5 + ib/2 C12 t" + 5 / 4 [ V-C1 ] VC1 ]
Note t h a t e a c h v a c a n c y in 2.2.9. h a s c a p t u r e d a n e l e c t r o n , in r e s p o n s e to
The point defect
56
t h e c h a r g e - c o m p e n s a t i o n m e c h a n i s m w h i c h is o p e r a t i v e for t h e d e f e c t reactions.
These
associated,
negatively-charged,
vacancies
have
quite
different a b s o r p t i o n p r o p e r t i e s t h a n t h a t of the F - c e n t e r . T h e r e are o t h e r i m p u r i t y s y s t e m s to w h i c h this n o t a t i o n c a n b e applied. For t h e case of an AgCI crystal c o n t a i n i n g the Cd 2+ c a t i o n as an i m p u r i t y , w e have: 2Ag +
2.2.12.-
~
[Cd 2+,vAg
T h i s is an e x a m p l e of a h e t ~ r o ~
]
s y s t e m . A n o t h e r s u c h s y s t e m is CdC12
c o n t a i n i n g Sb 3+ . Here, we c a n w r i t e at least t h r e e
different
e q u a t i o n s involving d e f e c t
equilibria: 2.2.13.-
2 Cd 2+
z..
2 Cd 2+
z.._7
Sb3+ + Sb3+ +
2 Cd 2+
-7~
Sb3+
+
p+
+
VCd
V+Cd Li+
In t h e last e q u a t i o n , c h a r g e - c o m p e n s a t i o n h a s o c c u r r e d due to i n c l u s i o n of a m o n o v a l e n t cation. All of t h e s e
equations
are c a s e s of i m p u r i t y
substitutions. A n o t h e r type is t h e so-called h o m o t y p e i m p u r i t y s y s t e m . The s u b s t a n c e , n i c k e l o u s oxide, is a p a l e - g r e e n
insulator, when
prepared
in an i n e r t
a t m o s p h e r e . If it is r e h e a t e d in air, or if a m i x t u r e of NiO a n d Li+ is reheated
in
an
inert
atmosphere,
the
NiO
becomes
a black
semi-
c o n d u c t o r . T h i s is a classical e x a m p l e of t h e effect of defect r e a c t i o n s u p o n t h e i n t r i n s i c p r o p e r t i e s of a solid: 2.2.14.-
2 Ni 2+ 2 Ni 2+
--7 L_ .-I
[ N i3+ / V N i ] N i 3+ + Li +
+
P+
T h i s b e h a v i o r is typical for t r a n s i t i o n m e t a l s w h i c h easily u n d e r g o c h a n g e s in v a l e n c e in the solid state.
2.3 Defect equation symbolism
57
Up to t h i s point, we have only i n v e s t i g a t e d s t o i c h i o m e t r i c lattices 9 Let us now
examine
non-stoichiometric
Consider the semi-conductor,
lattices
Ge. T h e
in
light
of o u r
symbolism.
defect reactions associated w i t h
t h e f o r m a t i o n of p - t y p e a n d n - t y p e l a t t i c e s are. 2.2.15. -
n-type:
Ge+6As
=
[GelAs~]
+
6e-
p-type:
Ge+6Ga
=
[Ge/Ga~]
+
6p+
T h e e x c e s s c h a r g e s s h o w n are s p r e a d over t h e e n t i r e
lattice,
as s t a t e d
before. 2 . 3 . - D E F E C T EQUATION SYMBOLISM Whether
you realize
it or not,
we
have
already developed
our
own
s y m b o l i s m for d e f e c t s a n d defect r e a c t i o n s b a s e d on t h e P l a n e Net.
It
m i g h t b e well to c o m p a r e o u r s y s t e m to t h o s e of o t h e r a u t h o r s , w h o h a v e also c o n s i d e r e d t h e s a m e p r o b l e m in t h e p a s t . It w a s p r o b a b l y R e e s (1930) w h o w r o t e t h e first m o n o g r a p h on d e f e c t s in solids. Rees u s e d nM to r e p r e s e n t t h e c a t i o n vacancy, as did Libowitz (1974). T h i s h a s c e r t a i n a d v a n t a g e s si n c e we c a n w r i t e t h e first e q u a t i o n in 2 . 2 . 1 2 . as:
K-l }+
+ +
2 3 1 9
.
2 Cd 2+
-
" 7
Col
"
p+
Cd
Likewise, t h e o t h e r e q u a t i o n s b e c o m e :
2.3.2.-
2 C d 2+
~
~-I +c d
and:
K~ 9 .
.-
-7
"
+ Cd
+
"
V ~ +col
+
+ p Cd
A l t h o u g h t h e r e s u l t s are e q u a l as far as utility is c o n c e r n e d ,
we
s ha ll
c o n t i n u e to u s e o u r s y m b o l i s m , for r e a s o n s w h i c h will b e c o m e c le a r l a t e r .
The point defect
58
T h e following c o m p a r e s defect s y m b o l i s m , as u s e d by p r i o r A u t h o r s . N o t e t h a t o u r s y m b o l i s m m o s t r e s e m b l e s t h a t of Krc~ger, b u t n o t in all a s p e c t s . 2.3.4-
Rees[1930]
Kruger [1954]
Libowitz [1974}
C a t i o n Site Vacancy:
I-]JIM
VM
~VI
Anion Site V a c a n c y
D[k
Vx
U] x
Cation Interstitial
AM
M i , M+i
Mi
Anion I n t e r s t i t i a l
Ax
Xi , X~
Xi
Negative Free C h a r g e
e
e ~
e-
Positive Free C h a r g e
p
h+
h+
Interstices
---
a V.
a V.
Unoccupied Interstitial
---
Vi
A
1
I
Anti-structure Occupation---
M M , X x , MX,XM
T h e s e prior a u t h o r s have c o n s i d e r e d s o m e i n t r i n s i c d e f e c t s t h a t we have n o t t o u c h e d , n a m e l y i n t e r s t i c e s a n d t h e so-called " a n t i - s t r u c t u r e " o c c u p a t i o n . T h e l a t t e r deals w i t h a n i m p u r i t y a n i o n o n a cation s i t e c o u p l e d w i t h a n i m p u r i t y c a t i o n on a n a n i o n site, b o t h w i t h t h e p r o p e r charge. We have m e n t i o n e d i n t e r s t i c e s b u t not in detail. T h e y a p p e a r as a f u n c t i o n of s t r u c t u r e . T h e r e is one site in a t e t r a h e d r o n , four in a b o d y - c e n t e r e d cube, a n d six in a s i m p l e cube. T h u s , a in {a Vi} is I, 4 or 6, r e s p e c t i v e l y . We shall n e e d t h i s s y m b o l later, as well as V i , t h e u n o c c u p i e d i n t e r s t i t i a l . 2 . 4 . - S O M E APPLICATIONS FOR D E F E C T CHEMISTRY Before
we
proceed
to
analyze
defect
reactions
by a m a t h e m a t i c a l
a p p r o a c h , let u s c o n s i d e r two a p p l i c a t i o n s of solid s t a t e c h e m i s t r y . We b e g i n w i t h a d e s c r i p t i o n of s o m e p h o s p h o r d e f e c t c h e m i s t r y .
2.4 Some applications for defect chemistry
59
I. - P h o s o h 0 r s In t h e prior l i t e r a t u r e , it w a s f o u n d (Kinney- 1955) t h a t Ca2 P2 07 c o u l d b e a c t i v a t e d b y Sb 3+ to form t h e p h o s p h o r : Ca2P207:Sb.02
(this f o r m a l i s m
actually m e a n s a solid-solution of two p y r o p h o s p h a t e c o m p o u n d s , [(Ca.99,Sb.ol)2P2OT]). T h e b r i g h t n e s s r e s p o n s e of t h i s p h o s p h o r
i.e.was
m o d e r a t e w h e n e x c i t e d by ultraviolet r a d i a t i o n b u t w a s i m p r o v e d four t i m e s by t h e a d d i t i o n of Li + . T h e o p t i m u m a m o u n t p r o v e d to be t h a t e x a c t l y e q u a l to t h e a m o u n t of Sb 3+ p r e s e n t in the p h o s p h o r . The d e f e c t reactions occurring were. 2.4.1.- Defect R e a c t i o n s O c c u r r i n g in C a l c i u m P y r o p h o s p h a t e P h o s p h o r PHOSPHOR BRIGHTNESS 2 Ca 2+ or (2 Ca 2+ 2 Ca 2+
~ Sb3+Ca
"--7
+
Sb3+Ca
z_.
Sb3+Ca
V+Ca +
25 %
VCa + P+ ) +
Li+ca
100
It is well k n o w n t h a t p h o s p h o r b r i g h t n e s s in a p h o s p h o r is p r o p o r t i o n a l to t h e n u m b e r s of activator ions, i.e.- Sb 3+ i o n s , actually i n c o r p o r a t e d i n t o t h e p y r o p h o s p h a t e s t r u c t u r e . P h o s p h o r s are p r e p a r e d b y h e a t i n g t h e i n g r e d i e n t s at h i g h t e m p e r a t u r e (> 1000 ~ C . ) t o o b t a i n a c o m p o u n d h a v i n g high
crystallinity.
The
sintering
process
decreases
entropy
and
is
counterproductive to t h e f o r m a t i o n of v a c a n c i e s in t h e p y r o p h o s p h a t e lattice. In t h e a b s e n c e of Li+, l a c k of v a c a n c y - f o r m a t i o n actually d e c r e a s e s t h e a m o u n t of Sb 3+ i n c o r p o r a t e d into activator sites. A p p a r e n t l y , four t i m e s as m a n y activator ions w e r e i n c o r p o r a t e d into t h e lattice w h e n t h e c h a r g e - c o m p e n s a t i n g Li+ ions w e r e p r e s e n t on n e a r e s t n e i g h b o r sites. Note t h a t we have w r i t t e n two defect r e a c t i o n s for t h e c a s e of v a c a n c y f o r m a t i o n in 2.4.1. P y r o p h o s p h a t e is a n i n s u l a t o r a n d t h e f o r m a t i o n of a p o s i t i v e l y - c h a r g e d v a c a n c y is m u c h m o r e ~likely t h a n t h e v a c a n c y p l u s a free positive c h a r g e .
The point defect
60
THUS, ALTHOUGH MORE THAN ONE DEFECT REACTION MAY BE APPLICABLE TO A GIVEN SITUATION, ONLY ONE IS USUALLY FAVORED BY THE PREVAILING THERMO DYNAMIC AND ELECTRICAL CONDITIONS. II.- L i t h i u m Ni0b~tr L i t h i u m niobate, LiNbOs, is a photorefractive material, discovered in 1 9 6 6 (Ashkin et al). T h a t is, it is an electroSptic m a t e r i a l in w h i c h the indices of refraction c a n be c h a n g e d by an applied electric field. It is used in optical devices w h i c h employ its nonlinear optical a n d e l e c t r o S p t i c p r o p e r t i e s . As a single crystal, LiNb03 has high t r a n s p a r e n c y to e l e c t r o m a g n e t i c radiation. If a laser b e a m is d i r e c t e d down the length of s u c h a crystal, its f r e q u e n c y is doubled. YAG:Nd 3§ i.e.- Y3A15OI2:Nds+, is a c o m m o n l y u s e d laser crystal, w h o s e emission lies at 10,600 A. (near-infrar e d radiation). The c o m b i n a t i o n of a LiNbO3 crystal a n d YAG:Nd s§ p r o d u c e s a laser b e a m at 5,300 A (green light). If a second crystal is also i n c o r p o r a t e d in the optical setup, radiation at 2 , 6 5 0 A (ultraviolet radiation) is obtained. Although the radiation f r e q u e n c y c h a n g e s , so does the i n t e n s i t y of light p r o d u c e d . Losses of 100 times or more are c o m m o n . However, this d e t r i m e n t a l factor is overcome by i n c r e a s i n g the power of the laser b e a m . Fortunately, LiNbOs has a high r e s i s t a n c e to d a m a g e by a laser b e a m , unlike m a n y o t h e r similar crystals. It gains its unique c h a r a c t e r i s t i c s b e c a u s e the crystal s t r u c t u r e has "ouflt-in" d e f e c t s . LiNbOs is a ferroelectric s t r u c t u r e related to t h a t of the cubic p e r o v s k i t e (CaTiOs) s t r u c t u r e . T h a t is, electric polarization of the lattice e l e c t r o n s o c c u r s u p o n application of an electric field. Although a voltage will i n d u c e s u c h polarization, so will the electric vectors of a b e a m of light, p a r t i c u l a r l y t h a t of a laser beam. However, LiNbOa consists of d i s t o r t e d o x y g e n - o c t a h e d r a s h a r i n g faces so t h a t a p l a n a r hexagonal a r r a n g e m e n t results. The pile-up of the o c t a h e d r a along the p e r p e n d i c u l a r direction, caxis, follows the cation s e q u e n c e "Li, Nb a n d v a c a n t alte". The p o i n t s y n m l e t r y group is C~ (see 1.4.14. a n d 1.4.16 of the first chapter) w i t h the trigonal axis along the cation rows. It is very close to a C3v (3m) configuration.
2.4 Some applications for defect chemistry
61
The following d i a g r a m s h o w s t h e a r r a n g e m e n t of t h e t h r e e LiNbOa s t r u c t u r e .
ions in t h e
2.4.2.-
Note t h a t we have a s t r u c t u r e w i t h a '~)uilt-in" c r y s t a l defect, a vacancy. B o t h t h e l i t h i u m a n d n i o b i u m c a t i o n s are in a n o c t a h e d r a l c o o r d i n a t i o n . In fact, t h e two ions, Li § a n d Nb 5+, have n e a r l y the s a m e r a d i u s a n d o c c u p y o c t a h e d r a l sites w i t h t h e s a m e Csv s y m m e t r y . The l i t h i u m d e f i c i e n c y in c o n g r u e n t c r y s t a l s is a c c o m m o d a t e d b y m e a n s of Nbu a n t i - s i t e s a n d Nb 5+ v a c a n c i e s in a relative c o n c e n t r a t i o n neutrality.
Note
that
many
t h a t g u a r a n t i e s overall
electrical
physical
properties
depend
upon
s t o i c h i o m e t r y , e.g.- Curie t e m p e r a t u r e , p a r a m e t e r s a n d p h o t o r e f r a c t i v e yield.
absorption
spectra,
lattice
However, it h a s b e e n f o u n d t h a t i m p u r i t i e s play a m a j o r role in t h e o p e r a t i o n of a f r e q u e n c y - d o u b l i n g c r y s t a l like LiNbOs. M a n y i m p u r i t i e s have ionic radii similar to t h a t of l i t h i u m . T h e y s u b s t i t u t e at Li§ s i t e s r a t h e r t h a n Nb s§ sites (possibly b e c a u s e t h e NbO 4 c o o r d i n a t i o n is s t r o n g e r at t h e n i o b i u m site). A m o n g t h e s e are: M n 2+, Fe 3+ a n d N i 2+. As we have a l r e a d y seen, s u b s t i t u t i o n of s u c h m u l t i v a l e n t c a t i o n s on a m o n o v a l e n t s i t e results
in
lattice
compensation
such
as
oxygen
vacancies
and
the
f o r m a t i o n of c o l o r - c e n t e r s at t h e o x y g e n v a c a n c i e s . T h e s e i n t e r f e r e w i t h the photorefractive properties
of s u c h defect c r y s t a l s since t h e e a s e of
electric p o l a r i z a t i o n of t h e l a t t i c e is i m p a i r e d d u r i n g use. A l t h o u g h , as w e
The point defect
62
will see, the crystal-growing p ro ces s is also a purification process, it is not able to exclude all of t h e i m p u r i t i e s as the crystal grows. III. Bubble M e m o r i e s "Bubble m e m o r y " is the t e r m applied to the device w h i c h uses a "soft" m a g n e t i c m a t e r i a l to carry information. If a f e r r o m a g n e t i c film s u c h as e u r o p i u m gallium g a r n e t is grown epitaxially u p o n a suitable s u b s t r a t e s u c h as gadolinium gallium garnet, i.e.- Gd3 Ga5 O11 (= GGG), it f o r m s m a g n e t i c d o m a i n s in w h i c h the electron spins of the cations are aligned in the s a m e direction in the s a m e domain. This is s h o w n in the following: 2.4.3.- A F e r r o m a g n e t i c Film Grown EpitaxiaUy on
r
ARROWS DIRECTION
INDICATE
9 ....
l;~.; ~
IS/~4
r/ss/4
"'""
"
~,,,,
I~.;./I
9 ....
v##/si WsS~'~l v////I
rr // ///Z/ 4/ ~ l "
"
.... "
i;T:I
Ir162 MAGNETIZATION
|V /.... I ///I
IT#~##4 r/s//4 ITSSSS4
.... " "
OF
.... 9 99 I T. s. .s.s / 4 4
4
V/##4
4
r ....4
1
I;~;I Ir r////,l
1.....
I./.~././I
Ws//4 v,,~4 VSSS4 v/l/4
V/ss'4
rf~I.~m r;m;~
1
I~l;l
I:~1~3
v////m
Vss/,T
Magnetic F iIm
I
rd~ir11,F,r1,r162 SubstPate
The following diagram, given as 2.4.4. on the next page, illustrates how t h e s e would look u n d e r polarized light (the Faraday effect) using c r o s s e d Nicol polarizers. (The b lack an d white polarity).
p a r t s are d o m a i n s of o p p o s i t e
W h e n a m a g n e t i c field is applied, with the field vector horizontal to t h e film, the d o m a i n s collapse to form s e p a r a t e d cylinders within the film, as shown. T h e s e a p p e a r to be "bubbles" w h e n viewed from the top, h e n c e the n a m e . The bubbles t h e n b e c o m e mobile u n d e r the influence of a s e p a r a t e electric field a n d will move. Actually, the electric field causes t h e dom a i n - wa l l to collapse by a spin-flip m e c h a n i s m , while the c y l i n d e r volume is m a i n t a i n e d by the m a g n e t i c field.
2.4 Some applicationsfor defect chemistry
2.4.4.[M~gnetic
I ~ b b l e s as V i ~ d
63
f r o m t h e T o p of t h e Film in P o l a r i z e d L i g h t i
00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 ~-- 50~ ---o Magnetic Domains
Magnetic Bubbles
T h i s c a u s e s t h e a p p a r e n t m o v e m e n t of t h e wall. Bubbles d e n s i t i e s as h i g h as 107 p e r c m 2, a n d bubble velocities of u p to 10 3 - 1 0 4 c m . / s e c , h a v e b e e n r e p o r t e d . Obviously, b u b b l e velocity d e p e n d s u p o n E, t h e e l e c t r i c field s t r e n g t h , as well as t h e c o m p o s i t i o n of t h e epitaxially g r o w n film. T h e u s e of a v a p o r - p h a s e - d e p o s i t e d m e t a l grid u p o n t h e s u r f a c e of t h e film s e r v e s to s w i t c h b u b b l e s from site to site. T h e p r e s e n c e (1) or a b s e n c e (0) of a b u b b l e is d e t e c t e d b y p o l a r i z e d light b e a m s . T h u s , i n f o r m a t i o n s t o r e d , a n d r e t r i e v e d , on a "chip" in b i n a r y language. However, t h e epitaxial film m u s t be defect-free. ~
c a n be
is difficult to grow as
a defect-free single crystal. F r e n k e l defects a p p e a r , a n d give rise to l i n e dislocations. W h e n t h e c r y s t a l is c u t into wafers,
t h e line d i s l o c a t i o n s
r e m a i n a n d c a n be r e v e a l e d by c h e m i c a l p o l i s h i n g in h o t p h o s p h o r i c acid. T h e r e is a s t r o n g c h a n c e t h a t t h e s e d i s l o c a t i o n s will be p r o p a g a t e d i n t o t h e epitaxial Film w h e n it is d e p o s i t e d .
The
resulting film-defects
then
"pin" one or m o r e b u b b l e s to one location on t h e film, m a k i n g it ( t h e m )
The point defect
64
unusable. In addition, a lattice m i s m a t c h t e n d s to a c c e n t u a t e the defects of the ~ s u b s t r a t e onto the Film grown on it. The ao lattice p a r a m e t e r for ~ is 12.53 A (this is the length of the side of the cube enclosing t h e 19 a t o m s ) w h i l e t h a t of Eu3 Ga5 O l l is 12.48 A. This m i s m a t c h of lattice p a r a m e t e r is about 50x g r e a t e r t h a n t h a t d e s i r e d so t h a t a c o m p o s i t i o n s u c h as Y2.45 Eu5.5 Fe3.8 Gal.2 O12 is generally m o r e suitable to c o r r e c t l y m a t c h t h e lattice p a r a m e t e r of CW_~. In this way, defects p r e s e n t in t h e s u b s t r a t e c a n be avoided in the epitaxially grown • m , if p r o p e r g r o w t h c o n d i t i o n s are m a i n t a i n e d . Nevertheless, a l t h o u g h the "bubble-memory" a p p r o a c h to h i g h e r d e n s i t y m e m o r i e s in c o m p u t e r s was t h o u g h t to hold m u c h promise, the p r o b l e m s a s s o c i a t e d with intrinsic defects in the film a n d those involving the s u b s t r a t e proved to be too d a u n t i n g a n d today t h e b u b b l e - m e m o r y a p p r o a c h h a s b e e n a b a n d o n e d in favor of m o r e advantageous m e t h o d s . IV. Calcium Sulfide P h o s p h o r L e n a r d in 1928 r e p o r t e d
on a c a l c i u m sulfide p h o s p h o r ,
having
Actually,
a blue
emission.
he
probably
i.e.- CaS:Bi 3+,
prepared:
CaS:Bi3§
initially. As you c a n see, the divalent cation site would have to be o c c u p i e d by a negative vacancy since charge c o m p e n s a t i o n dictates t h a t the vacancy will appear. L e n a r d found t h a t the emission i n t e n s i t y of CaS:Bia§ was very low. It is likely t h a t the V site is a color c e n t e r w h i c h d i s s i p a t e d m o s t of the excitation energy. He found t h a t the use of chloride fluxes greatly i m p r o v e d t h e emission i n t e n s i t y a n d t h a t the use of KCI p r o d u c e d the b r i g h t e s t p h o s p h o r . He c o n c l u d e d t h a t K§ was a coactivator, i.e.- as CaS:Bia§
+ , since he u s e d first u s e d NaCI as a flux d u r i n g its p r e p a r a t i o n
a n d t h e n KCI. Even today, the p h o s p h o r is r e f e r r e d to in t h a t m a n n e r a l t h o u g h it is obvious t h a t the c h a r g e c o m p e n s a t i o n m e c h a n i s m is m o s t likely the c o r r e c t m e c h a n i s m in the formation of the p h o s p h o r . This c o n c l u d e s o u r c o n s i d e r a t i o n of defect applications. Let us now c o n s i d e r a m o r e m a t h e m a t i c a l a p p r o a c h to the d e s c r i p t i o n of p o i n t defects. It h a s b e e n said: "If you c a n n o t calculate the p r o p e r t i e s of any given theory, you really do not u n d e r s t a n d it".
2.5 Thermodynamics of the point defect
65
2.5. -THERMODYNAMICS OF THE POINT D E F E C T We shall u s e two a p p r o a c h e s to derive s o m e w o r k i n g values for t h e p o i n t defect in solids, n a m e l y t h a t of S t a t i s t i c a l M e c h a n i c s Thermodynamics
of Defects. T h e r e
a n d t h a t of t h e
are t h o s e w h o have s o m e f a m i l i a r i t y
w i t h s t a t i s t i c a l m e c h a n i c s . For o t h e r s , s o m e e x p l a n a t i o n is due. S t a t i s t i c a l M e c h a n i c s as a discipline w a s originally d e r i v e d in t h e early 1920's w h e n it w a s realized t h a t one h a d to deal w i t h large p o p u l a t i o n s of a t o m s o r m o l e c u l e s in various e n e r g y s t a t e s ( p a r t i c u l a r l y g a s e s at t h a t time). T h e s e s t a t e s arise b e c a u s e e a c h m o l e c u l e , for e x a m p l e , is v i b r a t i n g in a m a n n e r slightly different t h a n its n e i g h b o r . W h a t we d e s c r i b e
as t h e e n e r g y s t a t e for a given set of c o n d i t i o n s
is
actually t h e average of t h a t of a B o l t z m a n n p o p u l a t i o n . The discipline b e s t s u i t e d for h a n d l i n g s u c h a s y s t e m is s t a t i s t i c s , h e n c e t h e n a m e . T h e approach
used
for
manipulating
molecular
populations
in
Statistical
M e c h a n i c s is quite involved, a n d we shall t o u c h very briefly on t h e m a t h e m a t i c s involved. We will first d e s c r i b e e a c h of t h e s e a p p r o a c h e s s e p a r a t e l y a n d t h e n a c o m b i n e d version. Hopefully, t h i s will aid in y o u r u n d e r s t a n d i n g of t h e two m e t h o d s of d e t e r m i n i n g the effect of the p o i n t defect u p o n t h e p r o p e r t i e s of t h e solid. I. S t a t i s t i c a l M e c h a n i c s A p p r o a c h T h e l a n g u a g e of S t a t i s t i c a l M e c h a n i c s evolved over a c o n s i d e r a b l e p e r i o d of time. For e x a m p l e , t h e t e r m " e n s e m b l e " is u s e d to d e n o t e a s t a t i s t i c a l p o p u l a t i o n of m o l e c u l e s ; "partition function" is t h e integral, over p h a s e s p a c e of a s y s t e m , of t h e e x p o n e n t i a l of {-E I kT} [where E is t h e e n e r g y of t h e s y s t e m , k is B o l t z m a n n ' s c o n s t a n t , a n d T is t h e t e m p e r a t u r e in ~ F r o m t h i s "function", all of t h e t h e r m o d y n a m i c f u n c t i o n s c a n be d e r i v e d . T h e definitions t h a t we shall n e e d are given as follows in 2.5.1. on t h e n e x t page. Here, Nj is t h e s u m of t h e individual a t o m s t i m e s a n e n t r o p y factor a n d 1~I is t h e s u m of all of t h e N j 's.
The point defect
66
2 . 5 . 1 . - S t a t i s t i c a l M e c h a n i c s Definitions N e e d e d W
_
thermodynamic probability
G
----
e n s e m b l e of r e l a t e d a t o m s ( m o l e c u l e s )
Nj
----
~ m
1~I
----
~. Nj
Q
--
Partition Function
U s i n g t h e s e , we c a n derive t h e following e q u a t i o n for t h e total e n e r g y of a n y given s y s t e m : E T = total e n e r g y of s y s t e m = 1~I E(ave.) = ~ Nj Ej
2.5.2.-
where
Nj
is t h e total n u m b e r of a t o m s (molecules)
involved. Using t h e
m e t h o d s of S t a t i s t i c a l M e c h a n i c s , we c a n derive b y p r o b a b i l i t y r e l a t i o n s : 2.5.3.-
In
these
W
=
Q
=
Q
=
equations,
F is
A F/h3 ~ r
Nj !
exp- Ej / k T
{for s t a t e s )
]~ exp- Ej / k T {for levels)
a so-called
"partition
coefficient",
~
is
a
d e g e n e r a c y , k is B o l t z s m a n n ' s c o n s t a n t , a n d T is in d e g r e e s Kelvin. T h e first e q u a t i o n in 2.5.3. is a s t a t i s t i c a l m e c h a n i c a l definition of w o r k , w h e r e a s t h e l a s t two d e s c r i b e total e n e r g y s t a t e s . Having t h e s e d e f i n i t i o n s and
equations
allows
us
to
define
point
defects
from
a Statistical
Mechanical viewpoint. II, S c h o t t k y a n d F r ~ n k e l D e f e c t s C o n s i d e r a p l a n e n e t h a v i n g N sites, of w h i c h N L are lattice sites, Nv are v a c a n c i e s , a n d N i are I n t e r s t i t i a l s . (Note t h a t we do not c o n s i d e r c h a r g e at t h e sites for t h e m o m e n t ) . equations:
U s i n g t h e s e , we have the following two
2.5 Thermodynamics of the point defect 2.5.4.-
NL
=
2.5.5.-
~ Ni
=
N + Nv
67
- Ni
ct N L
E q u a t i o n 2.5.4. h o l d s if s o m e fraction of Ni
is a s s o c i a t e d w i t h N L , t h e
n u m b e r of lattice sites, i.e.- F r e n k e l defects. A c c o r d i n g to t h e B i n o m i a l T h e o r e m , we c a n c o m b i n e p a i r s of t h e s e sites as: 2.5.6.-
W ( N v , NI)
=
Nit /Nv!(NL-NV)
2.5.7.-
W i ( N i , aNl)
=
r
Note
that
these
equations
are
simply the
! / (r
!
[
Combinatorial Equation
as
a p p l i e d to t h e s e two s e t s of defects. We c a n set u p a p a r t i t i o n f u n c t i o n ( s e e t h e above definition), u s i n g e q u a t i o n 2.5.3. We a p p l y t h i s to t h e S c h o t t k y a n d F r e n k e l defects as e x a m p l e s : 2.5.8.- SCHOTTKY:
Qv (T)
=
2.5.9.- FRENKEL:
Qi (T)
=
Wi(Nv, NL) exp -Nv Ev / k T W i ( N i , aNL} exp -Ni Ei ! k T
We n o w solve t h e s e e q u a t i o n s u s i n g c e r t a i n a p p r o x i m a t i o n s e m p l o y e d in Statistical
Mechanics,
including
n u m b e r s . If t h e e n e r g y r e q u i r e d
Stirling's
Approximation
for
large
to f o r m v a c a n c i e s a n d I n t e r s t i t i a l s
is
g r e a t e r t h a n kT, a n d ff t h e defects are only a fraction of t h e total n u m b e r of lattice sites, t h e n we get: 2.5.10.-
SCHO~-
Nv
=
N exp -Ev / k T
2.5.1 I.-
FRENKEL-
Ni
=
r N exp - Ei / k T
Note t h a t we n o w have B o l t z m a n n d i s t r i b u t i o n e q u a t i o n s for e a c h type of defect., a n d t h a t t h e energy, E, to f o r m t h e defect is like a n a c t i v a t i o n energy. T h e fraction of defects p r e s e n t , e i t h e r Nv / N e x p o n e n t i a l f u n c t i o n of t h i s activation e n e r g y .
or
N i / N , is an
The point defect
68
This l e a d s to t h e following c o n c l u s i o n : It is c l e a r w i t h o u t a doubt t h a t b o t h S c h o t t k y a n d F r e n k e l d e f e c t s are t h e r m a l in origin. Although
you
may
have
wondered
why
we
considered
Statistical
M e c h a n i c s at all in r e l a t i o n to the p o i n t defect, the above o b s e r v a t i o n is critical to o u r u n d e r s t a n d i n g of the f o r m a t i o n of p o i n t defects in solids. It is a fact t h a t in m a n y c a s e s the activation e n e r g y r e q u i r e d
to f o r m
v a c a n c i e s a n d / o r i n t e r s t i a l s is a p p r o x i m a t e l y t h a t of r o o m t e m p e r a t u r e : 2.5.12.-
E v , Ei
,,
kT
It is t h i s m e c h a n i s m w h i c h p r o d u c e s defects in the lattice. T e m p e r a t u r e s slightly above r o o m t e m p e r a t u r e will p r o d u c e defects in the solid, e v e n t h o u g h t h e r a t e of p r o d u c t i o n m a y be e x t r e m e l y slow b e l o w about 500 ~ K. T h e c o n c l u s i o n t h a t w e r e a c h is t h a t d e f e c t f o r m a t i o n is favored in t h e solid. It is m u c h m o r e difficult to obtain a ' ~ e r f e c t " solid, so t h a t the defect-solid results. We c a n s u m m a r i z e this statistical m e c h a n i c a l a p p r o a c h to the solid s t a t e in t h a t it gives u s a m e t h o d to evaluate the fraction of i n t r i n s i c d e f e c t s p r e s e n t u n d e r specified c o n d i t i o n s , a n d also gives u s a m e a s u r e of t h e deviation f r o m absolute s t o i c h i o m e t r y . If we c a n evaluate the energy of f o r m a t i o n of v a c a n c i e s a n d i n t e r s t i t i a l s at a given t e m p e r a t u r e for a given c r y s t a l lattice, t h e n we c a n calculate t h e n u m b e r s of defects f o r m e d at any other temperature. III - Defect T h e r m o d y n a m i c s The r e a s o n w h y we first i n v e s t i g a t e d t h e Statistical M e c h a n i c s a p p r o a c h to defect f o r m a t i o n is t h a t it gives u s a good basis for u n d e r s t a n d i n g t h e a p p l i c a t i o n of c h e m i c a l t h e r m o d y n a m i c s to the defect solid state.
2.5 Thermodynamics of the point defect
69
We begin this a p p r o a c h by defining the total n u m b e r s of defects p r e s e n t as N d. In addition, we n e e d t h e following definitions2.5.13.-
Definitions N e e d e d to Define the T h e r m o d y n a m i c A p p r o a c h ENTROPY OF DEFEC'I~ ENTHALPY OF DEFECTS FREE ENERGY OF DEFECTS
= =
Sd
=
Gd
Hd
According to t h e laws of t h e r m o d y n a m i c s , we c a n write2.5.14.-
AG
=
AH
- ATS
Note t h a t we are u s i n g the G ~ b s free e n e r g y r a t h e r t h a n t h e H e l m h o l t z free e n e r g y at this point. For the defect solid, we m u s t define f r e e - e n e r g y in t e r m s of the f r e e - e n e r g y of t h e p e r f e c t solid, G o , as r e l a t e d to t h e f r e e - e n e r g y of t h e d e f e c t solid, vis: 2.5.15.-
AG
=
Gd
-
Go
Note t h a t due to i n c r e a s e d entropy, the e n e r g y of the defect
s t a t e is
h i g h e r t h a n t h a t of the perfect solid. For this reason, we m u s t d i s t i n g u i s h b e t w e e n t h e s o u r c e s w h i c h c o n t r i b u t e to the total entropy, a n d m u l t i p l y by t h e n u m b e r of defects p r e s e n t . The applicable e q u a t i o n is: 2.5.16.-
Gd
= Go +
Nd Hd - T Nd A Svib. - T A Sconfig.
where
ASmb. is t h e e n t r o p y c h a n g e c a u s e d by c h a n g e s in vIlmatlomal f r e q u e n c y w i t h i n t h e solid by t h e p r e s e n c e of d e f e c t s and A Sconfig. is t h e c h a n g e of e n t r o p y c a u s e d b y c o n f l ~ ~ t i o n a l
c h a n g e s in t h e vicqniW
defects. This a s p e c t of defect solid s t a t e c h e m i s t r y h a s b e e n t h o r o u g h l y s t u d i e d a n d the c h a n g e in the various q u a n t i t i e s of 2.5.16., as a function of n u m b e r s of defects, is s h o w n in t h e foUowing d i a g r a m :
The point defect
70
2.5.17.Numbers of Defects as a Function of Gibbs Free Energy
Nd&H
ol
-NdT ~
d
Config.
Gd I I
~ - Hlntr|nsi
I
Nd
c IL
"
You will n o t e t h a t w h e n it is possible to plot G d as a f u n c t i o n of G, t h e Gibbs free energy, a m i n i m u m o c c u r s w h i c h is the n u m b e r of defects at t h e given t e m p e r a t u r e for t h e given solid. Note also t h a t t h e e n t h a l p y of d e f e c t s i n c r e a s e s c o n t i n u o u s l y as a f u n c t i o n of N d while b o t h of t h e e n t r o p y - c o n t r i b u t i o n s d e c r e a s e . The n e t effect is a m i n i m u m value for G d , at w h i c h p o i n t we c a n get t h e n u m b e r of i n t r i n s i c defects p r e s e n t . T h u s , we have s h o w n again t h a t t h e r e is a specific n u m b e r of i n t r i n s i c d e f e c t s present
in
the
solid
at
any
given
temperature,
this
time
by
Thermodynamics. Let u s n o w consider a combined (Statistical Mechanics and T h e r m o d y n a m i c ) a p p r o a c h to t h e p r o b l e m of c a l c u l a t i n g defect n u m b e r s . IV,- C o m b i n e d A o o r o a c h to Defect F o r m a t i o n In t h i s m e t h o d , we will d e l i n e a t e the e x a c t p r o c e d u r e u s e d to d e r i v e
2.5 Thermodynamics of the point defect equations
for
specific
defects
in
the
solid.
To
71 show
how
this
is
a c c o m p l i s h e d , we let N L be t h e t o t a l n u m b e r of lattice sites a n d Nv t h e n u m b e r of v a c a n c i e s . T h e w a y t h a t N v v a c a n c i e s c a n b e a r r a n g e d u p o n N L lattice s i t e s is given b y c o m b i n a t o r i a l s t a t i s t i c s as: 2.5.18.-
W =
~ N L
! ~
Nv!(Nv-Nv)
!
The cortfigurational or m i x i n g e n t r o p y will be d e f i n e d by: 2.5.19.-
ASM= k l n W
U s i n g S t i r l i n g ' s a p p r o x i m a t i o n for large n u m b e r s (this m e t h o d is u s e d e x t e n s i v e l y in S t a t i s t i c a l M e c h a n i c s ) , we c a n get: 2.5.20.-
AS M = k [N L I n N L - N v l n N v - ( N L - N v) In (N L - N v) ]
You will n o t e t h a t we h a v e t h e c h a n g e in e n t r o p y as a f u n c t i o n of t h e d i f f e r e n c e s b e t w e e n n o r m a l lattice s i t e s a n d v a c a n c i e s . S i n c e we k n o w t h a t N L >> N v, we c a n w r i t e for t h e e n t r o p y of m i x i n g : 2.5.21.and
AS M ~ [N L I n N L - N v l n N v ] ASM=- k[N v InNv/NLI=-RXv
InXv
w h e r e X v is t h e f r a c t i o n of defects actually p r e s e n t . As e x p e c t e d , for n o n i n t e r a c t i n g defects, t h e e n t r o p y of m i x i n g is ideal so t h a t t h e free e n e r g y of t h e s y s t e m c a n be w r i t t e n as: 2.5.22.-
AFv = N v (AE v - T ASVv) + RT [N L- N v / N L (In N L- N v / N L ) + N v /N L I n - N v / N O ]
w h e r e AShy is t h e c h a n g e in v i b r a t i o n a l e n e r g y of t h e lattice a r i s i n g f r o m t h e c h a n g e in v i b r a t i o n a l f r e q u e n c y a r o u n d t h e v a c a n t lattice site. If w e m i n i m i z e t h e free energy, i.e.2.5.23.and:
0Fv/ON v = 0 = A E v - T A S v v ) + R T I n ( N v / N L - N v) l n ( N v / N L - N v} = l n X v = A E v - T A S ~ v ) / R T
The point defect
72
T h u s , we c a n w r i t e for t h e atomic fraction of v a c a n c i e s p r e s e n t : 2.5.24.-
Xv = exp - AF~ /RT
You m a y w o n d e r
w h y we
= e ' a E v / RT
have e x a m i n e d
this m e t h o d
for specifying
v a c a n c i e s in g e n e r a l t e r m s . The r e a s o n is t h a t we c a n apply the s a m e m e t h o d to t h e p r o b l e m of p o i n t - d e f e c t pairs. This is i n t e n d e d to help you u n d e r s t a n d h o w t h e s e v a r i o u s types of defects in the solid arise. a. IntErstitial A t o m s The p r o c e s s of c r e a t i n g a n i n t e r s t i t i a l is j u s t the o p p o s i t e of c r e a t i n g a vacancy. If t h e r e are N, possible i n t e r s t i t i a l p o s i t i o n s in t h e lattice, a n d if AFi is t h e free e n e r g y n e e d e d to m o v e the a t o m into its i n t e r s t i t i a l position, t h e n t h e i n t e r s t i t i a l c o n c e n t r a t i o n at e q u i l i b r i u m will be: 2.5.25.-
N~ / N, = X~ = exp (- AF~ / k T )
The s a m e a p p r o a c h applies to F r e n k e l pairs. b. F r e n k e l p a i r s We have, in this case, b o t h a v a c a n c y a s s o c i a t e d with an i n t e r s t i t i a l atom. Using t h e a p p r o a c h s h o w n in 2.5.18., we have: 2.5.26.-
W =
NL~ N v ! ( N L - N v) !
X
NIJ N i ! ( N x - N i) 7
Now, we have two c o m b i n a t o r i a l f u n c t i o n s , one for the v a c a n c i e s and o n e for the i n t e r s t i t i a l s . Since the n u m b e r s of vacancies a n d interstials are equal, i.e.- N F = N L = N i , we c a n use the s a m e s t e p s given in 2.5.19., ,20, 21., a n d 2 . 5 . 2 2 . to get: 2.5.27.-
ASM= RT [(NLIn NL) " (2NF InNv) + (Nxln N~) - { ( N L - N F) l n ( N ~ - N F) ln(N L-NF)}]
2.5 Thermodynamics of the point defect
73
Minimizing t h e free e n e r g y as before, we c a n w r i t e for t h e F r e n k e l pairs: 2.5.28.-
NF = (N L Nx )112 exp (- ttFiv ! 2RT)
Note t h a t ZkF applies to b o t h i n t e r s t i t i a l a n d v a c a n c y sites. Using 2 . 5 . 2 3 . a n d 2 . 5 . 2 4 . , we c a n get the fraction of F r e n k e l defects as: 2.5.29.-
X~ --- exp (AFI v 12RT
since the free e n e r g i e s are a p p r o x i m a t e l y t h e s a m e , i.e.- ttF i v ~- ZXFI + ~ v
c. S c h o t t ~ The
Defects
c a l c u l a t i o n of S c h o t t l ~
defects
follows the
same method
given. We u s e cv for c a t i o n - v a c a n c y a n d av for t h e
associated
already anion
vacancy. The free e n e r g y of this defect is t h e n -
V
2 . 5 . 3 0 . - AFsh = Ncv AEcv + NAV AEAv - T (Ncv ASVcv + NAy AS Av) RT In
N
~
X
N c v [ ( N -Ncv) [
N
!
NAv!(N -N^v) [
Again, we have two c o m b i n a t o r i a l factors, o n e e a c h for t h e a s s o c i a t e d c a t i o n v a c a n c i e s . Here, t h e total n u m b e r of lattices sites, N >> Nsh = Ncv = N^v. Minimizing t h e free e n e r g y gives: 2.5.31.and:
Nsh / (N - Nsh )
= exp (-AFsh I 2 R T )
Xsh = exp (-AFsh I 2 R T )
You wiU n o t e t h a t this a p p r o a c h u s e s Statistical M e c h a n i c s to a p p r o x i m a t e t h e t h e r m o d y n a m i c c o n s t a n t s for t h e n u m b e r of defects p r e s e n t . V. Defect E q u i l i b r i a J u s t as c h e m i c a l r e a c t i o n s c a n be d e s c r i b e d a n d c a l c u l a t e d in t e r m s of
The point defect
74 thermodynamic
constants
and
chemical
equilibria,
so
can
we
also
d e s c r i b e d e f e c t f o r m a t i o n in t e r m s of e q u i l i b r i a . T h i s is g i v e n as follows: 2.5.32-
L a w of M a s s A c t i o n :
bB+cC
K =
-~ dD + e E 9
adD aeE / ~tbB acc
Using this equation, we can calculate the numbers
of d e f e c t s for v a r i o u s
d e f e c t s in t h e MX c r y s t a l as: 2.5.33.-
F r e n k e l D e f e c t s [for t h e MX c r y s t a l ] Mx
2.5.34.-
~
Mi
+
VM
"
KF
-
~tMi a v M [ aM
S c h o t t ~ y D e f e c t s tfor t h e MX c r y s t a l ] MX
~
VM + V x
Note that we have specified
,
KSh
- a v M a v M / aM X
the equilibrium constants
activity, a, of t h e m m o c ~ t e d defe 9
in t e r m s
of t h e
W e c a n also w r i t e t h e r m o d y n a m i c
e q u a t i o n s for t h e s e d e f e c t s 2.5.36.-
Chemical Thermodynamics AG = AH - T AS = - RT In K 9 K =
2.5.37.-
exp
AS / R 9 e x p - AH / R T
Defect Thermodynamics Kd = e x p
ASd I R " e x p - AHd I R T
w h e r e d r e f e r s to t h e specific d e f e c t . We may summarize
the knowledge
we have already developed
for t h e
Schottky and Frenkel defects: I. W e h a v e s h o w n b y S t a t i s t i c a l
Mechanics
that we can calculate
n u m b e r s of d e f e c t s p r e s e n t at a g i v e n t e m p e r a t u r e .
2.6 Defect equilibria in various types of compounds 2. T h e r e
is a n Activation E n e r g y for defect
75
formation.
In m a n y
c a s e s , t h i s e n e r g y is low e n o u g h t h a t defect f o r m a t i o n o c c u r s at, o r slightly above, r o o m t e m p e r a t u r e . 3. Defects m a y be d e s c r i b e d in t e r m s of t h e r m o d y n a m i c
constants
a n d equilibria. T h e
the
presence
vibrational frequencies
of d e f e c t s
changes
both
local
in t h e vicinity of t h e defect a n d t h e local
lattice c o n f i g u r a t i o n a r o u n d t h e d e f e c t . One q u e s t i o n we m a y logically a s k is h o w are we to k n o w w h a t t y p e s of defects will a p p e a r in a given solid? T h e a n s w e r to t h i s q u e s t i o n is g i v e n as follows: IT HAS B E E N FOUND:
"Yhere are t w o a s s o c i a t e d effects on a given solid w h i c h have o p p o s i t e e f f e c t s on s t o i c h i o m e t r y . U ~ R U y , o n e involves t h e cation site and t h e o t h e r t h e Rnlon site. B e c a u s e of t h e d i f f e r e n c e s in d e f e c t - f o r m a t i o n e n e r g i e s , t h e c o n c e n t r a t i o n of o t h e r d e f e c t s is usually n e g l i g ~ l e ' . T h u s , if F r e n k e l Defects p r e d o m i n a t e usually n o t p r e s e n t . applies for a,~ociater types
of
defects
in a given solid, o t h e r d e f e c t s a r e
Likewise, for t h e
Schottky
Defect.
Note
that this
d e f e c t s . If t h e s e are n o t p r e s e n t , t h e r e will still b e 2 present,
each
having
an
opposite
effect
upon
s t o i c h i o m e t r y . Thus, w e c o n c l u d e that i n t r i n s i c d e f e c t s usually o c c u r i n patl~,
This
conclusion
cannot
be
overemphasized.
The
following
d i s c u s s i o n s h o w s h o w t h i s o c c u r s in t h e r e a l w o r l d of d e f e c t s in solids. 2.6.Up
D E F E C T EQUILIBRIA IN VARIOUS TYPES OF COMPOUNDS to now,
we
have b e e n
concerned
with
the
MX c o m p o u n d
h y p o t h e t i c a l e x a m p l e of t h e solid state. We will n o w u n d e r t a k e concrete
examples
as f o u n d in
the
real
world,
using
the
as a more
concepts
d e v e l o p e d for t h e s i m p l e MX c o m p o u n d . For t h e s a k e of simplicity, w e r e s t r i c t o u r s e l v e s to b i n a r y c o m p o u n d s , t h a t is- o n e c a t i o n a n d o n e anion. An e x a m p l e of a t e r n a r y c o m p o u n d is ~ , w h e r e A a n d B are d i f f e r e n t cations, a n d S is a s m a l l w h o l e n u m b e r .
76
The point defect
Our example of a b i n a r y c o m p o u n d will be:
MXs We will d i s t i n g u i s h b e t w e e n four states for this h y p o t h e t i c a l c o m p o u n d , to wit: s t o i c h i o m e t r i c vs: n o n - s t o i c h i o m e t r i c non-ionized vs: ionized I. S t o i c h i o m e t r i c Bini~ry C o m p o u n d s of M ~ In the real world of defect chemistry, we find t h a t in addition to t h e simple defects, o t h e r types of defects appear, d e p e n d i n g u p o n the type of crystal we are dealing with. These m a y be s u m m a r i z e d as s h o w n in t h e following. According to o u r n o m e n c l a t u r e , VM is a vacancy at an M cation site, etc. The first five pairs of defects given above have b e e n o b s e r v e d e x p e r i m e n t a l l y in solids, w h e r e a s the last four have not. 2.6.1-
Defects in the MXs C o m p o u n d Schottky Frenkel Anti-Frenkel Anti-Structure Vacancy-Structure Structure-Vacancy Interstitial Interstitial-Structure Structure-Interstitial
P~$ OF D E F E C T S VM + V x VM + Mi
~q XM
+
+
Vx MX
VM + MX Vx + XM Mi + Xi MX +
Xi
Mi
XM
+
This a n s w e r s the h y p o t h e s i s p o s e d above, namely t h a t d e f e c t s in s o l i d s o c c u r in pairs. S t u d y these defect-pairs carefully so t h a t you b e c o m e familiar with them. T h e y r e p r e s e n t the type of s t r u c t u r e defects found in m o s t solids. We have now i n t r o d u c e d into our n o m e n c l a t u r e a d i s t i n c t i o n b e t w e e n s t r u c t u r e a n d a n t i - s t r u c t u r e defects. What this m e a n s is that s t a c k i n g faults can s o m e t i m e s result in XM a n d Mx defects, which are
]
CHAPTER 2 The Point Defect
There
are two types of defects a s s o c i a t e d w i t h p h o s p h o r s . One involves
controlled
point
defects
in
which
a
foreign
activator
cation
is
i n c o r p o r a t e d in the solid in defined a m o u n t s . The o t h e r involves line a n d point defects i n a d v e r t e n t l y f o r m e d in the solid s t r u c t u r e b e c a u s e of i m p u r i t y a n d e n t r o p y effects. This c h a p t e r will define a n d c h a r a c t e r i z e the n a t u r e of all of t h e s e p o i n t defects in the solid, t h e i r t h e r m o d y n a m i c s a n d equilibria. It will b e c o m e a p p a r e n t t h a t the type of defect p r e s e n t will d e p e n d u p o n the n a t u r e of the solid in w h i c h they are i n c o r p o r a t e d . T h a t is, the c h a r a c t e r i s t i c s
of the p o i n t defects
in a given p h o s p h o r
will
d e p e n d u p o n its c h e m i c a l c o m p o s i t i o n . Of necessity, this c h a p t e r is n o t i n t e n d e d to be exhaustive, a n d the r e a d e r is r e f e r r e d to the m a n y t r e a t i s e s c o n c e r n e d w i t h the p o i n t defect. 2.1. - TYPES OF POINT DEFECTS Let us n o w c o n s i d e r the defect solid from a g e n e r a l p e r s p e c t i v e . C o n s i d e r the case of s e m i - c o n d u c t o r s , w h e r e m o s t of the a t o m s are the same, b u t t h e total of the c h a r g e s is not zero. In t h a t case, the excess charge (n- o r p- type) is s p r e a d over the whole lattice so t h a t no single atom, or g r o u p of atoms, h a s a c h a r g e different from its n e i g h b o r s . However, m o s t inorganic solids are c o m p o s e d of c h a r g e d moieties, half of w h i c h are positive (cations) a n d half negative (anions). The total charge of t h e c a t i o n s equals, in general, t h a t of the anions. If an a t o m is missing, t h e lattice r e a d j u s t s to c o m p e n s a t e for this loss of charge. If t h e r e is an e x t r a a t o m p r e s e n t , the c h a r g e - c o m p e n s a t i o n m e c h a n i s m again m a n i f e s t s itself. A n o t h e r possibility is the p r e s e n c e of an a t o m w i t h a c h a r g e larger or s m a l l e r t h a n t h a t of its neighbors. In a given s t r u c t u r e , c a t i o n s are usually s u r r o u n d e d by anions, a n d vice-versa ( R e m e m b e r w h a t we said in C h a p t e r 1 wherein
it w a s s t a t e d t h a t m o s t s t r u c t u r e s
are o x y g e n - d o m i n a t e d ) .
T h u s , a cation w i t h an e x t r a c h a r g e n e e d s to be c o m p e n s a t e d by a like anion, or by a n e a r e s t n e i g h b o r cation w i t h a lesser charge. An e x a m p l e of
39
The point defect
78 number number
of s i t e s , w h e r e a s t h e l a s t five a r e b a s e d u p o n a n e x c e s s in t h e of s i t e s available. T h i s e x c e s s w e call "5". Note t h a t w e are n o t
s p e a k i n g of t h e r a t i o of c a t i o n s t o a n i o n s , i . e . - s t o i c h i o m e t r y , e x c e s s of c a t i o n s or a n i o n s to t h e n o r m a l
concentration
but0f
an
of c a t i o n s o r
anions. II. D e f e c t C o n c e n t r a t i o n s in M ~ s _ _ C o m p o u n d s It is of i n t e r e s t to b e able to d e t e r m i n e
t h e n u m b e r of i n t r i n s i c d e f e c t s in
a g i v e n solid. As w e h a v e s h o w n , p a i r s of d e f e c t s p r e d o m i n a t e in a n y g i v e n solid. T h u s , t h e n u m b e r of e a c h t y p e of i n t r i n s i c d e f e c t s , N i (M) or N i (X), will e q u a l e a c h o t h e r , F o r S c h o t t k y d e f e c t s in t h e MXs c r y s t a l , w e have: 2.6.2.-
Ni {VM )
=
This makes our mathematics
NI ( S V x ) simpler since we can rewrite
the Schottky
e q u a t i o n of T a b l e 2-1 as: 2.6.3.-
0 ~-~ Ni(VM) + SNi(VM)
Here, we have e x p r e s s e d the c o n c e n t r a t i o n
4- ctVi as t h e ratio of d e f e c t s to t h e
n u m b e r of M- a t o m s i t e s (this h a s c e r t a i n a d v a n t a g e s , as w e will see). We c a n t h a n r e w r i t e t h e d e f e c t e q u i l i b r i a e q u a t i o n s of T a b l e 2-1 in t e r m s of n u m b e r s of i n t r i n s i c d e f e c t c o n c e n t r a t i o n s . T h e s e are given as follows: 2 . 6 . 4 . - Eq~ilibri~tm C o n s t a n t s = F u n c t i o n of N u m b e r s of I n t r i n s i c D e f e c t s SCHO~:
KSh = Ni {S N i ) 2 = S s Ni S+I
FRENKEL:
KF
ANTI-FRENKEL:
KAF = Ni 2 / (S- Ni){cx- N i )
ANTI-STRUCT:
KAS = Ni 2 1 ( l - N i ) ( S - N i ) S KVS = S s (S + I )s+1. Ni2S+l i (S-Ni -S Ni )S
VAC. -STRUCT:
= Ni 2 / (I- Ni){cx- N i )
STRUCT-VAC:
KSV
= ( S + I ) S + I NiS+2 / (S - Ni - S Ni)
INTERSTITIAL:
KI
= S S Ni S+l / (r - N i - SNi) %S+I
S o m e of t h e s e e q u a t i o n s a r e c o m p l i c a t e d a n d w e n e e d to e x a m i n e t h e m in
2.6 Defect equilibria in various types of compounds m o r e d e t a i l so as to d e t e r m i n e
79
h o w t h e y a r e to b e u s e d . E q u a t i o n 2 . 5 . 1 6 .
given above shows that intrinsic defect concentrations
will i n c r e a s e w i t h
increasing temperature
a n d t h a t t h e y wiU b e low for h i g h e n t h a l p i e s
defect
arises because
formation.
This
the
entropy
effect
of
is a p o s i t i v e
e x p o n e n t i a l w h i l e t h e e n t h a l p y effect is a n e g a t i v e e x p o n e n t i a l . C o n s i d e r the following practical e x a m p l e : TI0
is c u b i c w i t h t h e NaCl s t r u c t u r e . A s a m p l e w a s a n n e a l e d at 1 3 0 0 ~
Density and X-ray measurements revealed that the intrinsic defects were S c h o t t k y in n a t u r e (VTi + V 0 ) a n d t h a t t h e i r c o n c e n t r a t i o n w a s 0 . 1 4 0 . I n t h i s c a s e , S = 1 so t h a t : KSh
2.6.5.-
- 0 . 0 1 9 6 = 2 x 1 0 -2
T h i s c r y s t a l is q u i t e d e f e c t i v e s i n c e 1 o u t of 7 Ti- a t o m - s i t e s
( 0 . 1 4 -I) is a
v a c a n c y , a n d l i k e w i s e for t h e o x y g e n - a t o m - s i t e s . A n o t h e r e x a m p l e is: CcI-I2
.
intrinsic
From
thermodynamic
defects
were
measurements,
Anti-Frenkel
it w a s
in n a t u r e ,
i.e.-
{Hi
has
the
found
that
the
+ VH ). A n
e q u i l i b r i u m c o n s t a n t w a s c a l c u l a t e d as: 2.6.6.-
KAF
at a t e m p e r a t u r e
=
3 . 0 X I 0 -4
of 6 0 0
~
This
compound
cubic
s t r u c t u r e w i t h o n e o c t a h e d r a l i n t e r s t i c e p e r Ce a t o m . T h e r e f o r e , a n d S = 2 for C e l l 2 . W e c a n t h e r e f o r e w r i t e : 2.6.7.-
kAF
=
Ni
=
or
Ni 2 /
(2-Ni)(1-Ni
2 . 4 x 10 -2
)
=
3.0
is v a c a n t .
a
= I,
x 1 0 -4
(600 ~
T h i s m e a n s t h a t 1 o u t of 4 2 h y d r i d e a t o m s is i n t e r s t i t i a l , hydride-atom-sites
fluorite
a n d 1 o u t of 8 4
The point defect
80
Let us review w h a t we have covered
concerning
stoichiometric
binary
compounds: I. We have s h o w n t h a t defects occur in pairs. The r e a s o n for this lies in t h e c h a r g e - c o m p e n s a t i o n principle w h i c h o c c u r s in all solids. 2.
Of
the
nine
defect-pairs
possible,
only
5
have
actuaUy
been
e x p e r i m e n t a l l y o b s e r v e d in solids. T h e s e are: Schottky, F r e n k e l , Anti- Frenkel, A n t i - S t r u c t u r e , V a c a n c y - S t r u c t u r e . 3. We have given d e f e c t - e q u a t i o n s for all nine types of defects, a n d t h e E q u i l i b r i u m C o n s t a n t (EC) t h e r e b y associated. However, t h e s e equilibria would r e q u i r e values in t e r m s
calculation of
of e n e r g y at each site,
values w h i c h are s o m e t i m e s difficult to d e t e r m i n e . A better
method
is to c o n v e r t
these
EC e q u a t i o n s to t h o s e involving
n u m b e r s of e a c h t y p e of i n t r i n s i c defect, as a ratio to an intrinsic cation o r anion. This allows us to calculate the actual n u m b e r of intrinsic d e f e c t s p r e s e n t in t h e crystal, at a specified t e m p e r a t u r e . III. Non-StoiChiomctri~ Binary C o m p o u n d ~ we
will now
extend
our treatment
of i n t r i n s i c
defects
to the
non-
s t o i c h i o m e t r i c n o n - i o n i z e d c o m p o u n d s , as r e p r e s e n t e d by: 2.6.9.-
MXs+
6
w h e r e 8 is a small i n c r e m e n t . The q u e s t i o n is: "How do we obtain nons t o i c h i o m e t r y in t h e solid?". Consider a c o m p o u n d g o v e r n e d by either or b o t h t h e following equilibria: 2.6.10.-
Xx
2.6.11.-
M(external phase) ~
One
example
might
~
be
I / 2 X2{gas)~
a halide
+ Vx
MM crystal
+ S V X + VMi which
has
become
non-
s t o i c h i o m e t r i c due to its being h e a t e d to a t e m p e r a t u r e sufficient to cause a small a m o u n t of the halide to b e c o m e volatile.
2.6 Defect equilibria in various types of compounds
81
A n o t h e r c a s e m i g h t be an oxide, h e a t e d in t h e p r e s e n c e of e x c e s s m e t a l , e.g.- ZnO + Zn. For a n o n - s t o i c h i o m e t r i c crystal, the c o n c e n t r a t i o n of e a c h point d e f e c t , in e a c h c o n j u g a t e pair, is no l o n g e r equal. If t h e r e is an e x c e s s of V M , X i , or XX, t h e n the c o m p o u n d will have a s u r p l u s of X (or deficiency of M, w h i c h is the s a m e thing) over t h e ideal s t o i c h i o m e t r i c c o m p o s i t i o n . T h i s is called a positive deviation from s t o i c h i o m e t r y . Conversely, for a n e g a t i v e deviation, t h e r e will be an e x c e s s of V x , M i , or M M . T h i s explains t h e p l u s a n d m i n u s in e q u a t i o n 2.6.9. In t e r m s of the above given defects, m a y be e x p r e s s e d as s h o w n in t h e following Table: TABLE 2-2. N o n - S t o i c h i o m e t r y , 8 , as a F u n c t i o n of Specific T y p e s of D e f e c t s in MXs + 5 Binary C o m p o u n d s Vacancy Formation Vx f r o m e x t e r n a l M
Defect E q u a t i o n M -~ MM + S[Vx] + (~Vi
Equilibrium Constant KV X= MM[Vx] s Vi a /M
VM froIn go~eotlk$ X
1 / 2X 2 --~ S i x + [VM] +(xVi
KVM=XxS[VM]Vi a / %
{if Xx ~ 1 a n d Vi =a,
then
1 / 2 X2 +(~VI ~ [Xi]
K m = [Xi] / ( ~- X4) p x 21/2
X-Interstitials
5 = S[VM]/
1/2S
1- VM}
or: {Px21/2 =1/Kxi- 8/(~-5} M-Interstitlals
M M + S X x + ( I +(~) Vi [Mil + S / 2 X2
KMi= [Ui ]px2S/2/MMXX S [(~ - M~ (t+a) or: Px2 S = KMi ((~(S + 6) + 5) 1+5 /(. ~) (S+6)a) l/S (from g a s e o u s X2)
X-Substituttonals
1 / 2(S+ 1)X2 -% SXx +r
Kx M =[XM] (~a / PX2 (S+1/2)
(from g a s e o u s X2 on
. Vi(l+a)
{or: p x 2(S+1/2) = 1/ KXM 9
( S + 1 ) X x + MM + a Vi
6/(l+S+ti) I/S+I } KMx= PX2((S+I)/2)
((S+I)/2) X2 + Mx
(1-Mx) S+I eta
an M-site) M-Substitutionals (gaseous X2 f o r m e d )
{or:
[Mx] /
Px2((S+l)/2)
(KMx) I/S+I
=
(S+ 1 )(S+d) /
[(-5)(S+ 1+6) s ] I/S+I )
The point defect
82
Note t h e various m e c h a n i s m s w h i c h give rise to the specific c o m b i n a t i o n s of defects. T h e s e m e c h a n i s m s have b e e n t h o r o u g h l y s t u d i e d as a function of specific c o m p o u n d s .
It is sufficient for us, at t h i s point, to observe
w h i c h defect e q u a t i o n s govern b o t h t h e equilibria s t o i c h i o m e t r y of the g e n e r a l c o m p o u n d , MXs + 8.
and
the
non-
The e q u a t i o n u s e d to calculate the n o n - s t o i c h i o m e t r y factor, 8 , in t h e g e n e r a l case is: 2 . 6 . 1 2 . - 8 = (Xi-Vx)+ S(VM-Mi) + ( S + I ) ( X M - M x ) / I + Mi + Mx - V M - XM We c a n e x p r e s s r e l a t i o n s h i p s b e t w e e n defect formation, the influence of various ext 9 factors, a n d the e q u i l i b r i u m c o n s t a n t t h e r e b y related. We do this in t e r m s of 8, the degree of n o n - s t o i c h i o m e t r y , as given in Table 22. Even t h o u g h t h e s e e q u a t i o n s are r a t h e r formidable-looking, we shall be able to u s e t h e m to good advantage. Note t h a t e a c h case c o r r e s p o n d s to the influence of a r e a c t i n g e x t e r n a l factor on a s t o i c h i o m e t r i c solid, w h i c h c o n t a i n s intrinsic defects. T h e s e
additional d e f e c t s b e c a u s e of n o n s t o i c h i o m e t r y a n d c h a r g e - c o m p e n s a t i o n . The d e f e c t p r o d u c e d is e n c l o s e d in b r a c k e t s in Table 2-2. factors
produce
C o n s i d e r this factor carefully by again e x a m i n i n g Table 2-2. Also given is t h e r e a c t i o n p r o d u c i n g t h e defect, with its c o r r e s p o n d i n g e q u i l i b r i u m c o n s t a n t . In m o s t cases, the deviation, 8, is p r e s e n t e d in t e r m s of t h e e q u i l i b r i u m c o n s t a n t a n d the partial p r e s s u r e
of the e x t e r n a l gaseous
reactant. T h u s , if the n u m b e r of d e f e c t s p r o d u c e d can be m e a s u r e d and an e q u i l i b r i u m c o n s t a n t calculated, t h e n 8 c a n be d e t e r m i n e d both as a f u n c t i o n of partial p r e s s u r e , p x 2 , a n d t e m p e r a t u r e (see 2 . 5 . 2 2 . ) . IV. Defect t~oncentration~ in MX,s• We now p r o c e e d as we did for the s t o i c h i o m e t r i c - c a s e , n a m e l y to develop
2.6 Defect equilibria in various types of compounds
83
d e f e c t - c o n c e n t r a t i o n e q u a t i o n s for t h e n o n - s t o i c h i o m e t r i c case, i.e.MXS• C o n s i d e r t h e effect of A n t i - F r e n k e l defect p r o d u c t i o n . F r o m T a b l e 2-1, we get KAF, w i t h its a s s o c i a t e d e q u a t i o n , kAF 9 In T a b l e 2-2, we u s e Kxi for X - i n t e r s t i t i a l s . 2.6.13.-
When
C o m b i n i n g t h e s e , w e get:
KAF = KVx " Kxi
both
Vx
and
Xi
coexist
=
Ni 2 I ( S - N i ) ( a - N i )
in
the
lattice,
the
deviation
from
s t o i c h i o m e t r y (from 2 . 6 . 1 2 . ) b e c o m e s : 2.6.14.-
5
= [Xi] -
[Vx]
U s i n g t h e e q u i l i b r i u m c o n s t a n t of 2 . 6 . 1 3 . , i.e. 2.6.15.-
Kv x _
p x 2112 [Vx ] /Xx
_
p x 2112 [Vx] [ S - [Vx ]
a n d t h e a p p r o p r i a t e o n e f r o m T a b l e 2 - 2 (i.e.- Kxi ), we get ( a s s u m e for simplicity that S = r = 1): 2.6.16.-
5
= a Px21/2 Kxi/Px~2Kxi + 1
-
S K v x / KVx+ Px2 I/2
We c a n r e a r r a n g e t e r m s in 2 . 6 . 1 6 . to o b t a i n : Kxa (1- 8 ) p x 2 - 8( K v x K x i + 1 ) Px2 I/2 a n d i f - KVx 2 (1- 8 ) = 0, w e c a n , b y u s i n g 2 . 6 . 1 3 . a n d Ni ~1, o b t a i n : 2.6.17.-
Ni 2 ( i - 5 ) p x 2 - 5 Kv x Px2 I / 2 -
KVx 2 (I- 8} = 0 .
S o l v i n g for p x 2 yields 9 2.6.18.- px 2 = KVx2(82+
Since
at s t o i c h i o m e t r i c
2Ni(1-
composition,
f o r m i d a b l e e q u a t i o n r e d u c e s to: 2.6.19.-
52 ) • 8 1 5 2 + 4 N i ( 1 - 8 2 ) ] 1 / 2 / 2 N i 4 ( I - 5 ) 2
Px2 o = KVx 2 ! Ni 2
8 must
e q u a l zero,
this
rather
The point defect
84 where
Px2 o
is the p r e s s u r e of X2
gas in equilibrium w i t h the MXs
crystal at the s t o i c h i o m e t r i c composition. This gives us the o p p o r t u n i t y to divide 2.6.18. b y 2 . 6 . 1 9 , to obtain: 2.6.20.- Px2 / Px2 o
=
82 +2 Ni (I- 62 ) •
6[62 + 4 Ni ( I - 6 2 ) 1 2 N i 2 ( I - 6 ) 2
We c a n therefore calculate 8 in t e r m s of the ratio of Px2 to Px2 o a n d N i , s h o w n as follows2.6.21.Effect of E x t e r n a l P r e s s u r e of X 2 G a s o n N o n - S t o i c h i o m e t r y of t h e H y p o t h e t i c a l C o m p o u n d , M X S_+8
N i = 10-4
J
Ni = 10 -3
PX 2
I~
-
0.06
= 1 0 -2
0
+ 0.06
~, C h a n g e in S t o i c h i o m e t r y Although we will not t r e a t the o t h e r types of pairs of defects, it is well to note t h a t similar e q u a t i o n s c a n also be derived for the o t h e r intrinsic defects. What we have shown is t h a t ezlmnml z~Letmats can cause f u r t h e r c h a n g e s in the n o n - s t o i c h i o m e t r y of the solid.
2.6 Defect equilibria in various types of compounds
85
V. Ionization Of D e f e c t s We have a l r e a d y covered,
albeit briefly, n o n - i o n i z e d
stoichiometric
and
n o n - i o n i z e d n o n - s t o i c h i o m e t r i c i n t r i n s i c - d e f e c t c o m p o u n d s . Let u s n o w c o n s i d e r t h e ionization of defects in t h e s e c o m p o u n d s . In t h e MXs c o m p o u n d , if we r e m o v e s o m e of t h e X - a t o m s to f o r m Vx~ t h e e l e c t r o n s f r o m t h e r e m o v e d X - a t o m (or f r o m t h e b o n d h o l d i n g t h e X - a t o m in t h e crystal)
are
left
behind
for
charge
compensation
reasons.
At
low
t e m p e r a t u r e s , t h e s e e l e c t r o n s are localized n e a r t h e v a c a n c y b u t b e c o m e d i s s o c i a t e d f r o m t h e p o i n t defect at h i g h e r t e m p e r a t u r e s . T h e y b e c o m e free to m o v e t h r o u g h t h e crystal, a n d we say t h a t t h e i n t r i n s i c defect h a s b e c o m e ionized. We c a n w r i t e t h e following e q u a t i o n s for t h i s m e c h a n i s m : 2.6.22.-
VACANCIES 8
Vx ~
Vx + + e
KVx
VM --~ VM" + p+ In
the
equilibrium
constant
=
IV x [ +
=
I VM- I I P + I / VM
s
KVM
equations,
each
symbol
le
is
I/Vx
actually
a
c o n c e n t r a t i o n , i.e.- n u m b e r s of specified defects a n d e l e c t r o n s , e t c . In a like m a n n e r , we w r i t e for i n t e r s t i t i a l s a n d a n t i - s t r u c t u r e d e f e c t s 2.6.23.-
INTERSTITIALS Mi
--~ Mi + + e-
Xi-+p +
Xi ~
2.6.24.-
A useful
KMi* = I Mi+l I e l
/ Mi
Kxi* =
IX~l I p + l / X ~
IXM + I I e
ANTI-STRUCTURE XM ~ X M +
+ e -
KXM =
Mx
+ p+
KM X
understanding
the
~
example
Mx
for
ionization of defects,
=
IMx" [ [P + I/ MX
above
is t h a t of c o b a l t o u s oxide,
I/XM
equations,
and
the
CoO. E x t e r n a l o x y g e n
The point defect
86
p r e s s u r e will affect s t o i c h i o m e t r y a n d p r o d u c e c o b a l t v a c a n c i e s , b u t t h e v a c a n c i e s a r e ionized at r o o m t e m p e r a t u r e :
1 / 2 0 2 ~ Oo + V C o
2.6.25.-
+n + +aVi
T h e n + is free to m i g r a t e t h r o u g h o u t t h e lattice. However, at p r e s s u r e s b e l o w 10 .6 ( o b t a i n e d by a p p l i c a t i o n of a v a c u u m ) , t h e Co v a c a n c i e s b e c o m e
doubly-ionized: 2.6.26.-
VCo
~
Vco 2
+~+
T h i s i l l u s t r a t e s h o w t h e ionized defect a r i s e s . We h a v e a l r e a d y i l l u s t r a t e d h o w v a c a n c i e s arise t h r o u g h PLANE ~ .
the us e of a
( S E E 2.2.1.). T h e r e i n , we u s e d MX as t h e m o l e c u l e to b u i l d
t h e NET. Let u s n o w r e t u r n , u s i n g t h e ions, M 2+ a n d X 2-
as the i o n i c
f o r m s w i t h w h i c h to b u i l d t h e NET. T h i s is s h o w n as follows: 2.6.27.-
[A Plane I 2+ •
Net fo r the Ionized MX Compound
X 2 - M 2+ X 2 - i~t 2 . X 2x2-r vl ' ~ + X =- M 2 . X 2 - M 2+ X 2 - M 2 § I M~'x~-r'12"• ~- H 2 " X ~ M2+X2-
I
I Ioo,~d I ! x_~i Mz'IX2- M 2 + X j . ~ ~ ' I ~ . ~ X Hole
. p'r--M2 + X 2 - M 2+ 2 ~ ~
M 2+
22"-- M 2 + X 2 _
JX
I Vacancy Hole Complex
I I x~- M~'x~--~N~_s~ ~- H ~" x ~- H ~*
El
x
2-IMP'Ix =- M 2+ X 2- M 2 "
Electron
I
.~IM " I X 2- M 2+ M 2 - X2-1~t 2~ X 2 X 2 - M 2+ X 2- M 2+ M 2 + X 2- M2+X2 -
•
§ •
X 2- M ""
l.x2M~_IM [M 2+
M 2" X 2- p,12 " X 2- M 2- X
2-]M +
X 2rVl2 . )(;2 + x 2~l~2- M 2" X2- M 2+ M2+ X 2- M Z - X 2 -
H ~§ x 2 - M ~" X 2- M 2.
M 2+ X 2 - M 2 + X 2- M 2 + X 2 - pvl2+ X 2 X 2 - M 2 + X2-. M2+ X2- M 2§ X 2- M 2 §
VacancyTrapped Electron Complex
2.6 Defect equilibria in various types of compounds
87
A v a c a n c y - h o l e c o m p l e x is s h o w n , as weU as a v a c a n c y - t r a p p e d e l e c t r o n complex.
In addition,
an example
of a n ionized
hole
and
an ionized
e l e c t r o n - s i t e is also given. We m u s t t h e r e f o r e a d d to o u r list of i o n i z a t i o n e q u a t i o n s t h e following: 2.6.28.-
MM+Z -.~
MM+Z-1 + p+
MM+Z ~
MM+Z+I + e-
B e c a u s e of t h e w a y t h a t we h a v e d e f i n e d o u r MXs crystal, t h e analogous ionization of t h e anion, X, does n o t o c c u r . Let u s n o w s u m m a r i z e w h a t we h a v e c o v e r e d : 1. We h a v e s h o w n t h a t defect e q u a t i o n s a n d e q u i l i b r i a c a n b e w r i t t e n for t h e MXs c o m p o u n d , b o t h for t h e s t o i c h i o m e t r i c a n d non-stoichiometric cases. 2. In a d d i t i o n , we h a v e s h o w n t h a t f u r t h e r defect f o r m a t i o n c a n b e i n d u c e d b y e x t e r n a l r e a c t i n g s p e c i e s , a n d t h a t t h e s e a c t to f o r m specific t y p e s of defects, d e p e n d i n g u p o n t h e c h e m i c a l n a t u r e of t h e crystal lattice. 3. We have also s h o w n
that the
intrinsic
defects
can
become
ionized. B e c a u s e m o s t of w h a t we have c o v e r e d to t h i s p o i n t h a s a s s u m e d l a c k of ionization, we n e e d to p r o c e e d f u r t h e r so as to develop e q u a t i o n s m o r e suitable for t h e real w o r l d . Most of t h e c a s e s we e n c o u n t e r in t h e real w o r l d involve ionic l a t t i c e s containing
charged
considered
simple hypothetical compounds
various k i n d s
cations and charged
of n o n - i o n i z e d
intrinsic
a n i o n s . Up to now, we
have
s u c h as MX a n d d e l i n e a t e d
defects
that would be p r e s e n t .
S e c t i o n 2 - 6 w a s c o n c e r n e d w i t h various t y p e s of d e f e c t s i n t r i n s i c to t h e s t o i c h i o m e t r i c MXs c o m p o u n d a n d t h e n o n - s t o i c h i o m e t r i c c o m p o u n d ,
The point defect
88 MXs•
Additionally, we e x a m i n e d the ionized MXs c o m p o u n d . The n o n -
ionized types are the easiest to illustrate, including the
mathematics
involved. Nevertheless, even those e q u a t i o n s (see Tables 2-1 and 2-2) b e c a m e s o m e w h a t c u m b e r s o m e at t i m e s . S u c h e q u a t i o n s are applicable to metallic, a n d possible s o m e covalent, crystals. E q u a t i o n s i n c l u d i n g ionization a n d electric charge are n e e d e d for ionic crystals a n d semic o n d u c t o r s . But, the m a t h e m a t i c s involved s t a r t s to b e c o m e uasolvablc. The m a i n p r o b l e m is the need to have a complete description, utilizing e q u a t i o n s d e s c r i b i n g all possible p r o c e s s e s , including: Intrinsic defect f o r m a t i o n Effect of extrinsic factors Ionization of intrinsic d e f e c t s Effects of c h a r g e - c o m p e n s a t i o n . By the time we are finished, we find t h a t we have derived a set of multiv a r i a n t e q u a t i o n s in several u n k n o w n s . Brouwer (1954) c o n s i d e r e d this case a n d was able to formulate a m e t h o d of solution. The following is a s u m m a r y of t h a t m e t h o d . 2.7. - BROUWER'S APPROXIMATION METHOD Let us c o n s i d e r an MXs crystal with S c h o t t l ~ defects (this is one of t h e m o r e easily defined types, mathematically). We define S=1. The types of intrinsic defects was given in 2.6.1., the equilibrium c o n s t a n t s of i n t r i n s i c defects was given in Table 2-1, n u m b e r s of intrinsic defects was given in 2.6.4., ionization of vacancies in 2.6.22, 2.6.23 & 2.6.24, and 2.6.21. gave the effect of an external factor (X2 gas) on the p r o d u c t i o n of n o n s t o i c h i o m e t r y . We can rewrite these equilibrium c o n s t a n t equations for the MX c o m p o u n d in the form given in 2.7.1. on the next page.
This set of equations completely defines the defect concentrations in 8n MX crystal conta/n/ng S c h o t ~ defects.
89
2.7 Brouwer' s approximation method
2.7.1-
a.
In KS = In VM + In VX
b.
InKvx* -Ine+InVx*
C.
In KVM* = In p + In VM" - In VM
d.
In KVx
e.
lnKVM = I n V M
=
-lnVx
1 / 2 In P x9 + In Vx - 1/2inPx 2
The i n t r i n s i c ionization c o n s t a n t is: f.
In Kion = In e + In p ( w h e r e Kion is t h e i o n i z a t i o n equilibrium constant)
and the electroneutrality condition i s . g.
e +VM-
=
p
+ VX+
It is i m p o r t a n t to realize t h a t e is t h e s u m of t h e e l e c t r o n s ,
not a s i n g l e
n e g a t i v e c h a r g e , i.e.2.7.2-
e = E e
and p
= Y p+
This set of equations c o m p l e t e l y defines the d e f e c t c o n c e n t r a t i o n s in an MX crystal cont~Inlng S c h o t t ~ defects. T h e r e a r e staten e q u a t i o n s in 2.7. I. a n d if t h e e q u i l i b r i u m c o n s t a n t s a r e k n o w n , t h e r e r e m a i n s e v e n u n k n o w n s in t h e s e s i m u l t a n e o u s e q u a t i o n s , i.e.- T h e s e involve- VM, Vx
, V M - , VX+ , e , p , a n d px 2. T h e first six
e q u a t i o n s are l i n e a r r e l a t i o n s b e t w e e n l o g a r i t h m s of c o n c e n t r a t i o n s ,
and
l o g a r i t h m s of e q u i l i b r i u m c o n s t a n t s , b u t t h e l a s t e q u a t i o n is not. T h e s o l u t i o n to t h i s set of e q u a t i o n s c a n be a c c o m p l i s h e d , b u t t h e m e t h o d s a r e c o m p l e x . However, if t h e r e is f u r t h e r ionization of d e f e c t s , where
more
than one pair
of d e f e c t s
results,
the
or t h e c a s e
situation becomes
h o p e l e s s . T h e e q u a t i o n s c a n b e w r i t t e n , b u t t h e set of e q u a t i o n s c a n n o t b e easily solved. In
1954,
Brouwer
proposed
a
graphical
method
for
solving
these
e q u a t i o n s . T h e m e t h o d h a s b e e n a d o p t e d b e c a u s e of f u r t h e r d e v e l o p m e n t b y Kroeger a n d Vink (1956).
The method
e n t a i l s dividing t h e r a n g e of
The point defect
90
defect c o n c e n t r a t i o n s into regions, s u c h t h a t c h a r g e - c o m p e n s a t i o n involves only two defects, so t h a t the e l e c t r o - n e u t r a l i t y equation (2.7.1-g.) is simplified. Brouwer's Method h a s b e e n applied to the case of silver chloride w h e r e i m p u r i t i e s like c a d m i u m chloride m a y be p r e s e n t . T h e p h o t o g r a p h i c p r o p e r t i e s can be e n h a n c e d or deteriorated, d e p e n d i n g u p o n the state of the i m p u r i t y added, i n a d v e r t e n t l y or not. For
a large
negative
stoichiometry
deviation,
VM-
and
p+ b e c o m e
negligible c o m p a r e d to Vx + and e - - T h e n , Equation 2.7.1.- g. b e c o m e s : In e = In Vx + . By using this relation, all defect c o n c e n t r a t i o n s can be plotted in t e r m s of X2 p r e s s u r e a n d the equilibrium c o n s t a n t s . This is s h o w n as a Brouwer plot, given as 2.7.3. on the next page. It is m o r e c o n v e n i e n t to use N d , the n u m b e r s of a given defect, and to define a quantity, R, as: 2.7.4.-
R = KVM Px21/2
for the MX c o m p o u n d c o n t a i n i n g S c h o t t l ~ defects. We can t h e n e x p r e s s Nd in p o w e r s of R, so that: 2.7.5.-
InNd
= 01nR
+C
We find t h a t 0, the slope, is either + 1 / 2
or + I. This simplifies m a t t e r s
greatly, as c a n be seen in 2.7.3. Here, the defect e q u a t i o n s for o u r MX c o m p o u n d {containing S c h o t t k y defects) are plotted for the case: Kion > KSh . In obtaining this plot, we have derived the following equations from equations 2.7.1. and 2.7.2.: 2.7.6.-
VM = R Vx = K s / R e=Vx + ={KvxVx}
1/2
= ( K v x.
Ks / R } 1 / 2
p = Ki ( R /KV X 9 KS)1/2 VM
$
= KVx*. VM / p = (R KVx* KS) KVM ) /
Kion
91
2.7 Brouwer' s approximation method 2.7.3.-
B r o u w e r Plot for Schottky Defects in the Hypothetical Compound,
MX.
for the
Case:
[Kion
le
'
KSh
i
i
I,
§ -" "X
"
''= I
i I
,'I
'n...,
I
. . j
I
I I I
^
I I
~,,,
. /
Z
i
11[
"0
%,
~ 9 - ,,v
"~ %, "~
I
.i,II''' .i,r
"Jill/,, .111'
.'~L'
I
I
imam
,,li"'~i,
i ~'l-'i t''" I ~ M I' I .'' i ,#" I .d# I .,d' I .li If'
%
.,nr
_
. ,,i'""'l ..,,,"" I .,,,,,,,'"' !
V M ,,,,,'"'"'
i
I Hi
jlllililliillilllllll
-i-
v;, . V
r
-"-"'-~i, +++"
I I
V
I
-+-'-
_.
I
,t:
+
--++ +4M
i
M "~1111
Jlej~T
V,-, l ++T "%, X I,.§ % +'lz.~ ~ -+ I '%.+-'- I -+-%. I • "T "~i~l ,+~
+" ~. ~r +++ Y M 4-
I I !
§247 "
',
++ ,+++
~
.. +
~-~Vx
i Hi
"',,,,..
a! i
I
I
We can readily see that concentrations and that by taking logarithms we
el
} ', +++":{in R - 1/2 lnl p + in KVM. .~l,.++ iI X 2 i +++
2.7.3.,
~
I
V M
concentrations,
~
.
.,i' ",i%
Yv
I I I
c
>
I
I
V__
]
can
now
II
are now expressed
in powers
o f R,
, w e g e t s l o p e s o f _+ 1 / 2 o r + 1. A s s h o w n
differentiate
between
namely: REGION I
-
VX + > VM
REGIONII
-
VX +
-
R E G I O N III -
VM
> VX +
VM-
3
regions
of
in
defect
The point defect
92
At l a r g e p o s i t i v e d e v i a t i o n s f r o m t h e s t o i c h i o m e t r i c c o m p o s i t i o n ( R E G I O N III), V M >> e a n d p >> V x + . In t h i s R E G I O N , 2 . 7 . 1 . - g b e c o m e s : 2.7.7. -
In V M "
=
In p
In t h e v i c i n i t y of t h e s t o i c h i o m e t r i c c o m p o s i t i o n c o n c e n t r a t i o n of d e f e c t s d e p e n d s u p o n w h e t h e r KS
(REGION II), t h e > Kion , or v i c e -
versa. The
ionic
former
usually holds
for l a r g e b a n d - g a p
compounds.
Then, 2.7.1.- g becomes: 2.7.8.-
VM-
=
+
W e c a n also s h o w t h a t t h e c h a n g e f r o m n e a r s t o i c h i o m e t r y to a n e g a t i v e d e v i a t i o n of s t o i c h i o m e t r y (REGION I) o c c u r s w h e n :
(R KS KVx* /Kion )112
2.7.9.-
=
(Ks
Kion / K v X
R) 1/2
T h i s gives us: 2.7.10.-
RII -- I
=
Kion / KVx*
-
KS
a n d in a l i k e m a n n e r : 2.7.11.-
Because
RII - Ill
the
stoichiometry
is
/ Kv X
rigidly
defined
in
REGION
II
by
the
condition: 2.7.12.-
VM
c h a n g e s in R or Px2
+ VM"
-
VX
+
VX +
do n o t g r e a t l y affect d e v i a t i o n f r o m s t o i c h i o m e t r y .
But, t h e r e a r e l a r g e c h a n g e s in b o t h Z e- a n d Y p+ in t h i s r e g i o n . T h u s , t h e s t o i c h i o m e t r i c c o m p o s i t i o n is p r o b a b l y b e s t d e f i n e d w h e n e = p . T h i s o c c u r s , as s h o w n in 2 . 7 . 3 . , at t h e value:
93
2.7 Brouwer' s approximation method
2.7.13.-
Ro
= {Ks Kion )1/2 / KVx
An e x a m p l e of a c o m p o u n d w h e r e KS o b t a i n e d values (at 6 0 0 ~
>> Kion
is IEBr. Kroeger {I 9 6 4 )
of:
KS = 8 x 1 0
"14
Kion = 3 x 10 -35 If we s u b t r a c t 2.7.9. f r o m 2 . 7 . 1 0 . , we find t h a t t h e Br2 g a s - p r e s s u r e c h a n g e s b y 1043 (since R ~ PBr2 I/2 ) over Region II. Moreover, t h e deviation
from
stoichiometry, 8 , changes
e s s e n t i a l l y c o n s t a n t . However,
only b y
10 -3, or
remains
Z e- (or Z p+ ) c h a n g e s b y a factor of
a p p r o x i m a t e l y 1021 over Region II. In e l e c t r o n i c s e m i - c o n d u c t o r s , t h e c o n d i t i o n : Kion >> KS , usually prevails. We usually get a B r o u w e r analysis like t h a t of 2 . 7 . 3 . Now, c o n s i d e r t h e c a s e w h e r e ionization is t h e n o r m . T h i s case, s h o w n i n 2.7.14.
(next
page),
is for t h e
hypothetical
semi-conductor
alloy MX,
w h e r e M a c t s as a c a t i o n , a n d X a c t s m o r e like a n anion. T h a t is, w h e n w e get i o n i z a t i o n of defects, M loses a n e l e c t r o n a n d X is positively i o n i z e d . An e x a m p l e c o u l d be GaAs. In t h i s case, e l e c t r o n i c insensitive
(Region
II)
to
composition,
whereas
c h a r g e is r e l a t i v e l y the
deviation
from
s t o i c h i o m e t r y , d , v a r i e s c o n s i d e r a b l y . In t h e p o s i t i v e - d e v i a t i o n d i r e c t i o n (Region III), t h e m a j o r d e f e c t s are VM" , p , a n d VM. T h i s gives t h e r e l a t i o n : 2.7.15.-
VM ~
VM"
+
p+
In t h e n e g a t i v e - d e v i a t i o n d i r e c t i o n (Region I), e a n d VM" p r e d o m i n a t e . For t h e m a j o r d e f e c t s of t h i s s y s t e m , we h a v e t h e c o n d i t i o n s 2 . 7 . 1 6 . o n t h e n e x t page.
shown in
The point defect
94 2.7.14.-
Brouwer Plot for S c h o t t k y D e f e c t s in the H y p o t h e t i c a l Compound. M X . W h e n I o n i z a t i o n of D e f e c t s is the N o r m i.e.-
I
[ Kion > K s h ]
I,'
le,-I
I]
le = ~I
le,
-i
k l
-~.e
1-
+ , V X
I m I
I a I
-
,i
,
V
I I
I I
..f'
,' aJmm~
-
I
,p."
N
dRiP! V
I ~ at,. I ,,i" "qirmnmmmnmmmmnnmmmmmmmmmmmmmmmmmmnmmmid,
,,,,' 'm"'"'"',.
_
".,,,'"
Z
,,,,'
J
,me"
_pm 9,,' ,,,"
m 'II. m ',,. m '%
..+ VX
Ii
-mJ""
'",,,.'%~
i "'mmm.f
I
!
im /v./
i
:/
I i m
~r'
/
_ r '
i/ ,
2.7.16.
m
I
K__ion >> Region I
-
R e g i o n II
-
R e g i o n III -
jr r
'lllm
l
mmmmm
'Im'"",,,,,,,,, V x
1 '
using
'"',,.
m
Immll
l
InlPx i
m
2
+ in Kv_ ,
rl
Ksh
e>
p
e ~ p
p
> e
Note that we have rather well defined the defects present semi-conductor,
~..~e
'1
,-.
I [lnR = 1 / 2
_/ r"-
. % '
~...._r j-,--~
I
X. M
-,m,.
Iv../ i _Fro i./
~r'rm i
v
i m n I
V -..r r'
r
~ M /
~
"'"',,,,. Ai ")b~
m I
~
/
I
rl ..rm
/!------__e
/,,,,,,.
I
I
~
r .F
relationships
defined
for t h i s t y p e o f
by Brouwer's
method.
It
2.8 Analyses of real crystals using Brouwer's method
95
s h o u l d be a p p a r e n t t h a t this m e t h o d a d d s c o n s i d e r a b l e power ability to analyze intrinsic defects w h e r e ionization is the n o r m . .8.-
to our
ANALYSES OF REAL CRYSTAI~ USING BROUWER'S METHODCOMPARISON TO THE THERMODYNAMIC METHOD
The silver halide series of c o m p o u n d s have b e e n extensively s t u d i e d b e c a u s e of their usage in p h o t o g r a p h i c Film. In particular, it is k n o w n t h a t if silver b r o m i d e is i n c o r p o r a t e d into a p h o t o g r a p h i c emulsion, any i n c i d e n t p h o t o n will create a F r e n k e l defect. W h e n the • m is d e v e l o p e d , the Agi + is r e d u c e d to Ag metal. T h e s e localized a t o m s act as nuclei to c a u s e m e t a l .....crystal formation at the p o i n t s "sensitized" by the p h o t o n action. (Note t h a t this d e s c r i p t i o n is an oversimplification of the actual m e c h a n i s m . Nevertheless, it s h o u l d be a p p a r e n t t h a t a k n o w l e d g e of defect c h e m i s t r y of the c o m p o u n d , AgBr, s h o u l d prove to be very i m p o r t a n t in u n d e r s t a n d i n g the c h e m i s t r y of p h o t o g r a p h i c films). I. The AgBr Crystal with a Divalent Impurity, Cd 2+ Consider the crystal, AgBr. Both cation a n d anion are m o n o v a l e n t , i.e.- Ag+ a n d B r - . The addition of a divalent cation s u c h as Cd 2+ s h o u l d i n t r o d u c e v a c a n c i e s , WAg , into the crystal, b e c a u s e of the c h a r g e - c o m p e n s a t i o n mechanism. s y s t e m as: 2.8.1.-
To m a i n t a i n
electro-neutrality,
we
prefer
to
define
the
(1-5) Ag +Br- - 5 C d 2+ S =
Fortunately, AgBr is easy to grow as a single crystal, u s i n g S t o c k b a r g e r T e c h n i q u e s . P o s s e s s i o n a n d m e a s u r e m e n t of a single crystal g r e a t l y facilitates o u r m e a s u r e m e n t of defects. The i m p e r f e c t i o n s we expect to find are: 2.8.2.-
VAg ,Agi , e - , p + , C d A g , a n d
[CdAg,VAg]
The last defect is one involving two n e a r e s t n e i g h b o r cation sites in t h e lattice.
The point defect
96
The following table gives the defect r e a c t i o n s governing this case:
DEFECT REACTIONS IN THE AgBr (~RYSTAL CONTAINING Cd 2 + _ e"
ao
0
~
b. C.
gAg
+ctVi
d. e.
f. g. h.
AgAg lgi
+ (xVi
--, ~
+ -~ ~_
p@
+ hgi
WAg WAg
+ Agi +
Agi + + e
1 / 2 B r 2 ~ BrBr + VBr CdAg--7~-CdAg+ + eCdAg + WAg"
[CdAg, VAgl
CdAg+ +VAg"
[CdAg, VAgl
i. For C o n s t a n t Cd CdT = CdAg +CdAg + + [CdAg ,VAg] = K j. For E l e c t r 0 - n e u t r a l i t y e- + W A g = P + + A g i +
+ CdAg+
We w o u l d normally plot In N d vs: In KAgi + In KVAg 9 However, we find it m o r e c o n v e n i e n t to plot In N d vs: 1 / T . The r e a s o n for this is as follows. E x p e r i m e n t a l l y , we fund t h a t if we fLx the Cd 2+ c o n t e n t at s o m e c o n v e n i e n t level, it is n e c e s s a r y to anneal the AgBr crystals at a fLxed t e m p e r a t u r e for t i m e s long e n o u g h to achieve c o m p l e t e equilibrium. If the t e m p e r a t u r e is c h a n g e d , t h e n b o t h type a n d relative n u m b e r s of defects m a y also change. T h u s , we plot In Nd vs: 1/T, as in the Brouwer d i a g r a m of 2.8.3., given on the next page. At low t e m p e r a t u r e s (Region III), s i n g l y - c h a r g e d d e f e c t s p r e d o m i n a t e , i.e.- 2 WAg = Agi + + CdAg+. At the j u n c t i o n of III II, the c h a r g e d moieties begin to c l u s t e r to form CdAg+ and Vag-. T h e s e in t u r n m a y form the c o m p l e x : 2.8.4.-
CdAg+ +
VAg
~
[CdAg
, VAg]
At the s a m e time, the c o n c e n t r a t i o n of Agl + d r o p s dramatically.
2.8 Analyses of real crystals using Brouwer's method
97
2.8.3.-
,,
I
IBrouwer Diagram for the AgBr Crystal Containing Cd2+I I
I
I
,I
IP,EGION I I
I
I REGION III
,I
I v
[Cd Ag, V.~q] "0
Z
~, t %.l'[Cd Ag,..,V,~g ' 4-
I REGION IIII
!
. Ag
I
,'; A g i
Ecd;,g,v,...,. ............ g] I ......... ...,." . . . . .I- " " ' ......... .,:,-'
~.,'
-'
I [Cd~
] ,,' '~Ag .,'
,'
V
..
l I LIII r I II F I dl
9 CdAg i iiiiii
iii
i iiii
iii
iii
ii i
I
'
I
II II iiI II IIIIII
+ ,,,' Ag~.,
iIII
I
A iiiiII IIIII
[Cd Ag, VAgl
Ol 0
In, P.
.~,t' I i.~ I
IRE0tON I!
I i
-
f[ l..,,'T)
i,
I RE01ONIII
'I I I 1 'I!
,,
| iI |
IREGiON lltl ~LJ / "
,
I ! I
em
I ,
I
I o
/
I I I l9
I
'
I
~ i i.
I'I
J d
!I
J
II1"
At t h e j u n c t u r e of II - I in t h e B r o u w e r d i a g r a m , t h e c o m p l e x c l e a r l y dominates.
The point defect
98 While
the
above r e s u l t s
show
how
intrinsic
defects
are
affected
by
t e m p e r a t u r e , we still do n o t k n o w h o w t h e e l e c t r o n s a n d h o l e s vary as a function
of t e m p e r a t u r e .
(Note
that
temperature,
as specified,
is
a
preparation t e m p e r a t u r e , not measurement t e m p e r a t u r e . M e a s u r e m e n t of i n t r i n s i c c o n d u c t i v i t y , o , is s h o w n at t h e b o t t o m of 2.8.3. At low t e m p e r a t u r e s , c o n d u c t i v i t y d u e to v a c a n c i e s , o v , a p p e a r s to be the m a j o r contributor: 2.8.5.-
WAg --~ WAg-
Conductivity concentrations
decreases decrease.
at
+ p+ higher
temperatures
because
FinaUy, at t h e v e r y high t e m p e r a t u r e
Agi + region
(Region I), c o n d u c t i v i t y is relatively low, a n d a p p r o a c h e s zero, b e c a u s e t h e c o m p l e x [CdAg, WAg] p r e d o m i n a t e s as t h e n u m b e r of c h a r g e d m o i e t i e s , (CdAg + + WAg) d e c r e a s e s . V a r y i n g t h e Cd 2+ c o n t e n t in t h e AgBr c r y s t a l affects t h e relative d e f e c t ratios, as s h o w n in t h e foUowing d i a g r a m , s h o w n as 2.8.6. on the n e x t page. Again, we c a n identify 3 Regions as a f u n c t i o n of C d - c o n c e n t r a t i o n : 2.8.7.-
REGION I -
Agi + > VAg
REGION II -
Agi + @ WAg
REGION III -
WAg > Agi +
R e g i o n I of 2 . 8 . 7 . c o r r e s p o n d s
closely to t h a t of R e g i o n III of 2 . 8 . 5 .
T e l t o w (1949) also s t u d i e d t h i s crystal. He m e a s u r e d c o n d u c t i v i t y of AgBr c r y s t a l c o n t a i n i n g various a m o u n t s of Cd 2+ , as a f u n c t i o n of m e a s u r e m e n t temperature. those
Up to 175 ~
of t h e m i d d l e
h e o b t a i n e d c o n d u c t i v i t y c u r v e s similar to
of 2.8.6.
But, as t h e
measurement
temperature
i n c r e a s e d , t h e p r o n o u n c e d dip s e e n in 2.8.6. t e n d e d t o f l a t t e n out. At t h e h i g h e s t m e a s u r e m e n t t e m p e r a t u r e of 4 1 0 ~ t h e c o n d u c t i v i t y w a s fiat. He c o n c l u d e d t h a t e l e v a t e d t e m p e r a t u r e s p r e c l u d e t h e f o r m a t i o n of c l u s t e r s a n d / o r c o m p l e x e s , so t h a t c o n d u c t i v i t y d u e to Agi + r e m a i n s t h e
2.8 Analyses of real crystals using Brouwer's method
99
2.8.6.-
Brouwer Diagram Showing Effect of Cd Concentration on Defect Formation in AgBr Crystal
[ Cd
I n Cd
I
4-
~g
i! i I
s' J
VJI! ,r=
u 9
Z e~
-A g +
u-
IL~U
lr
Aa.
n
n
~
m
............. ~. . . . . . . . . . . . . . . %. . . . . . . . . . . . . . b , , " -
....~'-
C:d
.I
+ _
Ag .I
I
,,'"
/
I n I
=liP'
I
.r o" .
dr
, V _
CCd.
J/
,,I
/ f
2~ u ^ g
".n~,,
/ A-'J"
g
~
t
n
u I
/
I
i .
I
",,,.Ag_ .,,,.,,,,, I
In
'.,,,
l "'nhn
9
1
]
.4=
, A g .,,i, ......................
...... "
in [ca] i
a i
i
I/
m r .,air
410~ 325 ~ C
I
.am-.
n
In [CA]]
9
Is. hi"
I
I
t~_..__L.,..--..,.-
i
225 ~
n 175
ti
~
n
I
l
...............n
I
J l "mr
/
v~]
ir
D
r
]
I~a~pd m
.,Ir
I
A,!]
2g.,r F I
..~.,,'-
_-,,,--"
./I'"',..,, A"%.g"i ,e I
Iii
n, J-.'" .
f
V
/[c.~ g ' 11
I r,+
~,g '
n-
---~
The point defect
100
m a j o r c o n t r i b u t o r to t h e conductivity, as a c c o r d i n g to t h e defect r e a c t i o n "5" of Table 2-3. LET US NOW SUMMARIZE WHAT WE HAVE COVERED TO DATE: 1. By r e w r i t i n g t h e e q u i l i b r i u m c o n s t a n t s of Table 2-1 a n d 2 . 7 . 2 0 . (ionization of vacancies) as l o g a r i t h m s , we o b t a i n e d linear r e l a t i o n s a m o n g t h e set of defect e q u a t i o n s . 2. By defining s e t s of defects as a ratio, R, we c a n t h e n plot t h e r a t i o s so as to s h o w h o w t h e relative n u m b e r s vary as a f u n c t i o n of t h e type of defect p r e s e n t in t h e c h o s e n c r y s t a l lattice. This is t h e Brouwer Method. 3. We also i11ustrated the m e t h o d for a AgBr c r y s t a l c o n t a i n i n g Cd 2+ . The
set of defect
reactions were
given,
so as to illustrate
the
p o s s i b l e defects p r e s e n t . T h e n , a B r o u w e r d i a g r a m i l l u s t r a t e d t h e n u m b e r s a n d t y p e s of defects actuaUy p r e s e n t as a f u n c t i o n of Cd 2+ c o n t e n t in t h e crystal. II. Defec~ D i s o r d e r in A g B r - A T h e r m o d y n a m i c AoDroach TO i l l u s t r a t e yet a n o t h e r a p p r o a c h to analysis of defect formation, c o n s i d e r the i n f l u e n c e of Br2 - gas u p o n defect f o r m a t i o n in AgBr. The free e n e r g y of f o r m a t i o n , ~G, is r e l a t e d to the r e a c t i o n . 2.8.8.-
Ag ~ + 112 Br2 {g) ~
AgBr{g)
{ A G A g B}r
This c a n be r e w r i t t e n as: 2.8.9.-
gAg p l / 2
Br 2
=
exp AGAgBr / R T
It m a k e s no difference as to w h i c h of t h e activities we use. If we n o w fix PBr 2 at s o m e low value, we find t h a t the i~os~ble defects in o u r AgBr crystal, as i n f l u e n c e d by t h e ~ I f ~ ,
PBr 2, will be:
2.8 Analyses of real crystals using Brouwer's method 2.8.10.-
Agi +
, WAg
,
Bri-
, VBr +
, e
and
101
~+
w h e r e w e u s e ~ for t h e positive c h a r g e to d i f f e r e n t i a t e b e t w e e n p r e s s u r e , p, of t h e e x t e r n a l gas. B e c a u s e of t h e h i g h e l e c t r o s t a t i c e n e r g y r e q u i r e d to m a i n t a i n t h e m in a n ionic c r y s t a l s u c h as AgBr, w e c a n safely i g n o r e t h e foUowing p o s s i b l e d e f e c t s : 2.8.11.-
AgBr + , AgBr ++ ,
BrAg"
, BrAg-- .
If w e h a v e t h e r m a l d i s o r d e r a t r o o m t e m p e r a t u r e
(I do n o t k n o w of a n y
c r y s t a l for w h i c h t h i s is n o t t h e case), t h e n w e c a n e x p e c t t h e f o l l o w i n g defect reaction relations: 2.8.12.-
Agi +
=
WAg"
Bri-
=
Agi +
WAg"
=
VBr +
Bri-
=
VBr +
At e q u i l i b r i u m , t h e following e q u a t i o n s a r i s e : 2.8.13.-
Ag~ Kd
+ 1 / 2 Br2 tg) ~ =
AgBr(s)
+
VAg"
+ g+
V A g - ' ~ + / p 112 Br
2
T h i s gives u s a t o t a l of e i g h t (8) c o n c e n t r a t i o n s
to c a l c u l a t e . T h e y involve
t h e foUowing c r y s t a l d e f e c t s 2.8.14.-
Ag~,
Our procedure i.e.- for AgBr:
BrBr , A g i ,
Bri
, WAg , VBr + , e-
, n+
is to s e t u p a site b a l a n c e in t e r m s of l a t t i c e m o l e c u l e s ,
The point defect
102 2.8.15.-
AgAg , + V a g BrBr
+ e-
+ VBr + + ~+
= 1
( e-
=
(~+ ~
1
--
Ag'Ag) Br+Br)
S i n c e Br2 {gas) is t h e d r i v i n g f o r c e for d e f e c t f o r m a t i o n , w e n e e d also to consider
deviation from stoichiometry,
6 . T h u s , w e also s e t a Agl-8 Br
balance: 2.8.16.-
Ag~
+Agi +
+e-
BrBr + Bri-
=
+u+
1+8
=
1
To m a i n t a i n e l e c t r o n e u t r a l i t y : 2.8.17.-
Agi +
+ VBr +
+ ~+
=
Bri-
+ WAg
+ e-
W e also s e t u p t h e f o l l o w i n g e q u a t i o n s : 2.8.18.-
AgAg
+
~xVl ~
Agi +
2.8.19.-
BrBr
+
r
Bri-
and 2.8.20.-
e-
~
+ p+ ~
+ VAg"
Ke
= Agi + V A g
VBr +
Kg
=
+
0
Kb =
/ Vi a
Bri- VBr + / Vi a
(e)
(~+)
N o t e t h a t w e h a v e d i s t i n g u i s h e d b e t w e e n t h r e e (3) s i t u a t i o n s , to wit. a. E l e c t r o n e u t r a l i t y b. T h e r m a l D i s o r d e r c. N o n - s t o i c h i o m e t r y These
(excess cation)
are the eight equations (2.8.12.
to 2 . 8 . 2 0 . )
required
to c a l c u l a t e
t h e d e f e c t c o n c e n t r a t i o n s a r i s i n g f r o m t h e e f f e c t s of t h e e x t e r n a l f a c t o r , PBr2 9 F r o m measurements of c o n d u c t i v i t i e s , transfer numbers ( e l e c t r o m i g r a t i o n of c h a r g e d s p e c i e s ) , l a t t i c e c o n s t a n t s a n d e x p e r i m e n t a l
2.8 Analyses of real crystals using Brouwer's method
103
d e n s i t i e s , it h a s b e e n s h o w n t h a t F r e n k e l d e f e c t s p r e d o m i n a t e
(Lidiard -
1957). T h i s m e a n s t h a t : 2.8.21.-
Agi + , VAg"
Furthermore,
>>
Bri
,
VBr +
Agi + so t h a t in t e r m s of o u r e q u i l i b r i u m c o n s t a n t s
VAg-
we get: 2.8.22.-
FOR FRENKEL DEFECTS:
Ke >> Kg
a n d Ke
T h u s , w e n e e d o n l y to c o n s i d e r t h e a b o v e two (2) d e f e c t s ,
>> Kd n a m e l y - VAg"
a n d Agi + , s i n c e t h e y a r e t h e m a j o r c o n t r i b u t o r s to n o n - s t o i c h i o m e t r y . c a l c u l a t i n g pO
Br 2
as b e f o r e ( w h e n 6 = 0 , s e e 2 . 7 . 2 2
& 2.7.23.),
By
we can
e x p r e s s o u r o v e r a l l d e f e c t e q u a t i o n as: 2 . 8 . 2 3 . - p I/2Br2 / (P~
1/2 = {6/2g +[( 1 +6/2~)2] I/2 } {6/2~+[ 1 +(6/2~) 2] I/2 }
B e c a u s e of t h e c o n d i t i o n s g i v e n in 2 . 8 . 1 6 . , t h e f i r s t h a l f of t h e e q u a t i o n c a n b e s e t e q u a l to o n e . N o t e t h a t w e a r e u s i n g ~ , ~ , a n d u a s t h e
eqnillhrium c o n s t a n t s 2.8.24.-
. i.e. -
~ ~. Ke 1/2
~
~
K~ 1/2
y -~Z Ky 112
5 >> 13 9 By t a k i n g l o g a r i t h m s , w e
In t h e r e m a i n i n g p a r t of t h e e q u a t i o n , can then obtain: 2.8 25.9
l121n
PBr 2
lpO
Br 2
=
In6
-
In
T h i s r e s u l t t h e n l e a d s u s to a p l o t of t h e effect of p a r t i a l p r e s s u r e
of Br2
o n t h e d e v i a t i o n f r o m s t o i c h i o m e t r y , d , for t h e AgBr c r y s t a l , a s s h o w n i n 2 . 8 . 2 6 . o n t h e n e x t p a g e (this w o r k is d u e to G r e e n w o o d - 1 9 6 8 ) . F o r ~ = 0 , t h e r e is a p o i n t of i n f l e c t i o n w h e r e t h e s l o p e o f t h e line is d e f i n e d b y t h e e q u i l i b r i u m c o n s t a n t , i.e.- [3 = K~ I12 .
The point defect
104 2.8.26.-
Effect of Partial Pressure of BrzGas on Deviation from Stoichiometry of the AgBr Crystal
99 - ~
A
+8
f
log p Br z
T h e larger this value, the flatter is the curve. All r e l a t i o n s r e g a r d i n g t h e defects c a n n o w be d e r i v e d . T h e m a j o r defects t u r n o u t to be: 2.8.27.-
WAg
-7
~+
~--
e-
~
Agi+ K~ {liKe} 1/2
pl/2
Br2
(Ke) I/2 (1/Ke) Kb( 1/ PBr2) I/2
T h i s s h o w s t h a t b o t h P+ a n d e- are m i n o r i t y defects d e p e n d e n t on
PBr 2
The following gives the s t a n d a r d e n t h a l p i e s a n d e n t r o p i e s of these d e f e c t r e a c t i o n s , a c c o r d i n g to Krosger ( 1 9 6 5 ) -
105
2.9 Summary and conclusions 2.8.28.-
D E F E C T REACTION
AS
AH
Cal. / mol I ~ AgAg ~
Agi +
+ WAg" ( F r e n k e l )
-7 AgAg + BrBr ~-
0~..
e
WAg-
25.6
+ VBr +
+ p+
I I 2 Br2 + A-gag-~ It is a p p a r e n t
AgBr + WAg- + P+
that the
Frenkel
process
Kcal. ! mol. 29.3
- 13.3
36
25
78
4.9
25.4
coupled
with
the
electronic
p r o c e s s a r e t h e p r e d o m i n a t i n g m e c h a n i s m s in f o r m i n g d e f e c t s in AgBr t h r o u g h t h e a g e n c y of e x t e r n a l r e a c t i o n w i t h Br2 gas. A final c o m m e n t : we c a n u s e t h e s e t h e r m o d y n a m i c v a l u e s to c a l c u l a t e t h e e q u i l i b r i u m c o n s t a n t s a c c o r d i n g to: 2.8.29.-
Ki
=
exp-
AGi o I R T
a n d c a n also o b t a i n t h e activity of t h e silver a t o m in AgBr f r o m 2 . 8 . 2 9 . By u s i n g e q u a t i o n 2.8.9., w e c a n s h o w 9 For aAg = 1 ,
@ T = 277 ~
For PBr2 = 1 a t m .
@ 277 ~
8
= + 1012
9 8 = - 10 -7
w h e r e t h e p l u s or m i n u s i n d i c a t e a n e x c e s s or deficit of t h e silver a t o m in AgBr. T h i s r e s u l t is d u e to W a g n e r ( 1 9 5 9 ) . 2.9.- SUMMARY AND CONCLUSIONS Let u s n o w s u m m a r i z e t h e m a j o r c o n c l u s i o n s r e a c h e d r e g a r d i n g t h e defect solid. You will n o t e t h a t w e h a v e i n v e s t i g a t e d t h e f o l l o w i n g h y p o t h e t i c a l c o m p o u n d s : MX, MXs a n d MXs+ 8- But, w h e n we i n v e s t i g a t e d c r y s t a l s in t h e r e a l world, w e f o u n d t h a t a c t u a l d e f e c t s in s u c h solids d i d
The point defect
106
not c o n f o r m e n t i r e l y to t h o s e of o u r h y p o t h e t i c a l c o m p o u n d s . Nonetheless, understand
in
order
to
how defects
comprehend
and
affect the p r o p e r t i e s
form
a
foundation
of actual solids,
n e c e s s a r y to s t u d y t h o s e h y p o t h e t i c a l c o m p o u n d s .
The
to
it was
following
is a
s u m m a r y of the c o n c l u s i o n s we r e a c h e d r e g a r d i n g t h e defect solid state: I. The c h a r g e c o m p e n s a t i o n m e c h a n i s m r e p r e s e n t s the single m o s t i m p o r t a n t m e c h a n i s m w h i c h o p e r a t e s w i t h i n the defect solid. 2. We have s h o w n t h a t defect e q u a t i o n s a n d equilibria c a n be w r i t t e n for t h e MXs c o m p o u n d , b o t h for the s t o i c h i o m e t r i c a n d n o n - s t o i c h i o m e t r i c cases. 3. The c o n c l u s i o n t h a t we r e a c h is t h a t defect formation is favored in the solid b e c a u s e of the e n t r o p y factor. difficult
to o b t a i n a "perfect"
solid,
It is m u c h m o r e
so t h a t the
defect-solid
r e s u l t s . We have also s h o w n t h a t the i n t r i n s i c defects c a n b e c o m e ionized. 4. Although m o r e t h a n one defect r e a c t i o n m a y be applicable to a given situation, only one thermodynamic that intrinsic
is usually favored by the
a n d electrical defects
conditions.
usually occur
Thus,
in pairs.
we
This
prevailing conclude conclusion
c a n n o t be o v e r e m p h a s i z e d . 5. We have s h o w n t h a t defects o c c u r in pairs. The r e a s o n for t h i s lies in t h e c h a r g e - c o m p e n s a t i o n p r i n c i p l e w h i c h o c c u r s in all solids. 6. Of the
nine
defect-pairs
possible,
only 5 have actually b e e n
e x p e r i m e n t a l l y observed in solids. T h e s e are: Schottky, F r e n k e l , Anti- Frenkel, A n t i - S t r u c t u r e , V a c a n c y - S t r u c t u r e . 7. T h e r e c a n be no doubt t h a t b o t h S c h o t t k y a n d F r e n k e l d e f e c t s are t h e r m a l in origin. 8. We have s h o w n by Statistical M e c h a n i c s t h a t we c a n calculate n u m b e r s of defects p r e s e n t at a given t e m p e r a t u r e . 9. T h e r e
is an Activation E n e r g y for defect
formation.
In m a n y
cases, this e n e r g y is low e n o u g h t h a t defect formation o c c u r s at, or slightly above, r o o m t e m p e r a t u r e .
2.10 The effects of purity
107
10 .Defects m a y be d e s c r i b e d in t e r m s of t h e r m o d y n a m i c a n d equilibria. T h e p r e s e n c e
constants
of d e f e c t s c h a n g e s b o t h t h e local
v i b r a t i o n a l f r e q u e n c i e s in t h e vicinity of t h e defect a n d t h e local lattice c o n f i g u r a t i o n a r o u n d t h e d e f e c t . 11 .There
are two a s s o c i a t e d effects on a given solid w h i c h
opposite cation
effects on site
differences
and
stoichiometry.
the
other
the
Usually, anion
site.
in d e f e c t - f o r m a t i o n - e n e r g i e s ,
the
one
have
involves
Because
the
of
the
concentration
of
o t h e r d e f e c t s is usually n e g l i g i b l e . 12 9 In e x a m i n i n g t h e defect s t a t e of real c r y s t a l s s u c h as AgBr, w e find t h a t we c a n write, u s i n g e q u i l i b r i u m c o n s t a n t s a n d d e f e c t t h e r m o d y n a m i c s derived from Statistical Mechanics and classical Thermodynamics,
valid e q u a t i o n s for t h e n u m b e r s a n d t y p e s of
various a s s o c i a t e d d e f e c t s p r e s e n t . However, we also find t h a t w e c a n n o t solve for t h e value of t h e u n k n o w n q u a n t i t i e s in a set of simultaneous
equations
since
the
equations
are
not
linearly
solvable. T h e e q u a t i o n s c a n b e w r i t t e n , b u t t h e set of e q u a t i o n s c a n n o t b e easily solved. It is for t h i s r e a s o n t h a t we have r e s o r t e d to g r a p h i c a l m e t h o d like t h a t of B r o u w e r , e v e n t h o u g h it is n o t e n t i r e l y s a t i s f a c t o r y in its s o l u t i o n s to t h e n u m b e r s a n d t y p e s of d e f e c t s p r e s e n t in real c r y s t a l s . T h u s , it s h o u l d b e clear t h a t lattice d e f e c t s in t h e solid s t a t e is t h e n o r m a l s t a t e of affairs a n d t h a t it is t h e d e f e c t s w h i c h affect t h e p h y s i c a l a n d c h e m i c a l p r o p e r t i e s of t h e solid. 2.10. -
T H E E F F E C T S OF PURITY (AND I M P U R I T I E S )
Our s t u d y h a s led u s to t h e p o i n t w h e r e we c a n realize t h a t t h e p r i m a r y effect of i m p u r i t i e s in a solid is t h e f o r m a t i o n of defects, p a r t i c u l a r l y t h e F r e n k e l a n d S c h o t t l ~ t y p e s of a s s o c i a t e d defects. T h u s , t h e p r i m a r y e f f e c t o b t a i n e d in p u r i f y i n g a solid is t h e m d m l m L ~ t i o n of defects.
Impurities,
p a r t i c u l a r l y t h o s e of differing v a l e n c e s t h a n t h o s e of t h e lattice,
cause
c h a r g e d v a c a n c i e s a n d / o r i n t e r s t i t i a l s . We c a n also i n c r e a s e t h e r e a c t i v i t y of a solid to a c e r t a i n e x t e n t b y m a k i n g it m o r e of a d e f e c t c r y s t a l b y t h e a d d i t i o n of s e l e c t e d i m p u r i t i e s .
The point defect
108
It is not so a p p a r e n t as to w h a t h a p p e n s to a solid as we c o n t i n u e to purify it. To u n d e r s t a n d this, we n e e d to examine the various g r a d e s of purity as we n o r m a l l y e n c o u n t e r them. Although we have e m p h a s i z e d inorganic c o m p o u n d s t h u s far (and will c o n t i n u e to do so), the s a m e p r i n c i p l e s apply to organic crystals as well. COMMERCIAL GRADE is usually about 95% p u r i t y (to o r i e n t ourselves, w h a t we m e a n is t h a t 95% of the m a t e r i a l is t h a t specified, with 5% being different (unwanted-?) material. Laboratory or "ACS-REAGENT GRADE" averages about 9 9 . 8 % in purity. 2.11.1.-
GRADES OF PURITY FOR COMMON CHEMICAI~
GRADE
%
opm IMPURITIES
IMPURITY ATOMS PER MOLE OF
Commercial Laboratory Luminescent Semi-conductor
95 99.8 99.99 99.999
50,000 2000 100 10
Crystal G r o w t h Fiber-Optics
99.9999 99.999999
1 0.01
~QMPQUND. 3.0 x 1022 1.2 x 1021 6 x 1019 6 x 1018 6 x 1017 6 x 1015
The GRADES listed above are n a m e d for the usage to w h i c h they are i n t e n d e d , a n d are usually m i n i m u m purities r e q u i r e d for the particular application. Fiber-optic m a t e r i a l s are c u r r e n t l y p r e p a r e d by c h e m i c a l v a p o r deposition t e c h n i q u e s b e c a u s e any h a n d l i n g of m a t e r i a l s i n t r o d u c e s impurities. F u r t h e r m o r e , this is the only way found to date to p r e p a r e t h e r e q u i r e d m a t e r i a l s at this level of purity. The frontiers of p u r i t y a c h i e v e m e n t of solids p r e s e n t l y lie at the fraction of p a r t s per billion level. However, b e c a u s e of E n v i r o n m e n t a l Demands, analytical m e t h o d o l o g y p r e s e n t l y available far exceeds this. We can now analyze m e t a l s a n d anions at the femto level (parts per quadrillion= 10 -15 ) if we w i s h to do so.
Nevertheless, it is becoming apparent that as h/gh purity inorganic solids are being obtained, we observe that their physical properties may be
2.11 Nanotechnology and the solid state
109
different than t h o s e usually a c c e p t e d for the s a m e c o m p o u n d of lower
purity. The h i g h e r - p u r i t y c o m p o u n d m a y u n d e r g o solid state r e a c t i o n s s o m e w h a t differently t h a n t h o s e c o n s i d e r e d "normal" for the c o m p o u n d . If w e reflect b u t a m o m e n t , we realize t h a t this is w h a t we m i g h t expect to o c c u r as we obtain c o m p o u n d s (crystals) c o n t a i n i n g far fewer i n t r i n s i c defects. It is u n d o u b t e d l y true t h a t m a n y of the d e s c r i p t i o n s of physical a n d solid state reaction m e c h a n i s m s now existing in the literature are only partially correct. It s e e m s t h a t p a r t of the frontier of k n o w l e d g e for C h e m i s t r y of The Solid State lies in m e a s u r e m e n t of physical a n d chemical p r o p e r t i e s of inorganic c o m p o u n d s as a function of purity. A case in point is t h a t of the so-called "Nano-Technology", the v a n g u a r d of r e s e a r c h into c h e m i c a l a n d physical p r o p e r t i e s of m a t e r i a l s in the r e s e a r c h c o m m u n i t y today. 2.11.-
N a n o t e c h n o l o g y a n d The Solid S t a t e
In the next c h a p t e r , we shall e x a m i n e the m e t h o d s of c h a r a c t e r i z i n g solids including: the p r o p e r t i e s of individual particles (including single crystals); the solid state r e a c t i o n s t h a t are u s e d to form various solids; a n d m e t h o d s u s e d to describe an a s s e m b l y of particles (particle size). We will find t h a t m o s t solid m a t e r i a l s are c o m p o s e d of particles in the 1- 3 0 0 ~m. range. This is 1-300 x 10 -6 m e t e r s . Most inorganic m a t e r i a l s are p r o d u c e d having particles in this size range. T h e s e are the familiar p o w d e r s s u c h as coal dust, inorganic chemicals, silt a n d fine sand, a n d even bacteria. C u r r e n t r e s e a r c h defines n a n o - t e c h n o l o g y as the use of m a t e r i a l s a n d s y s t e m s w h o s e s t r u c t u r e s and c o m p o n e n t s exhibit novel a n d significantly c h a n g e d p r o p e r t i e s w h e n control is achieved at the atomic a n d / o r m o l e c u l a r level. What this m e a n s is t h a t w h e n a given m a t e r i a l is p r o d u c e d having particle sizes at fractions of a ~rn (micron), it displays novel p r o p e r t i e s not found in the s a m e material w h o s e particles are l a r g e r t h a n 1.0 ~ n (micron). N a n o t e c h n o l o g y involves d i m e n s i o n s w h e r e a t o m s a n d molecules, a n d i n t e r a c t i o n s b e t w e e n them, influence their c h e m i c a l
110
The point defect
a n d p h y s i c a l b e h a v i o r . A u t h e n t i c n a n o - p a r t i c l e s are so s m a l l that~there are m a n y m o r e a t o m s o n t h e surface of e a c h p a r t i c l e t h a n t h e n o r m a l p a r t i c l e of 1.0 ~ma. P a r t i c l e s of 1.0 ~rn, i.e.- 1000 n m or 1000 x 10 -9 m, m a y s e e m small but those
atoms on the
surface of e a c h p a r t i c l e
are only about
0 . 0 0 1 5 % or 15 in a million of t h e a t o m s c o m p o s i n g t h e lattice. A n a n o p a r t i c l e w i t h d i m e n s i o n s of 10 n m . b r i n g s t h e surface a t o m s to about 1 5 % of t h e total a t o m s c o m p o s i n g the particle. At this size range, q u a n t u m p h y s i c s a n d q u a n t u m effects d e t e r m i n e s
t h e p r i m a r y b e h a v i o r of s u c h
particles. C o n s i d e r t h a t a t o m s have a size r a n g e of about 1-2 A. Most i n o r g a n i c solids, w i t h t h e e x c e p t i o n of halides, sulfides (and o t h e r pnictides),
are
D
b a s e d u p o n t h e oxygen a t o m , i.e.- oxide = O- , w h o s e a t o m i c r a d i u s d o e s not change
even w h e n
sulfates, p h o s p h a t e s
a n d silicates
are f o r m e d .
Oxide h a s a n a t o m i c d i a m e t e r of 1.5 A or 0 . 1 5 nm. = 0 . 0 0 0 1 5 ~m. Nanoparticles particle
are c l u m p s of 1000 to 1 0 , 0 0 0 of 0 . 1 5
~n.
in d i a m e t e r .
They
a t o m s . The l a t t e r w o u l d be a c a n be m e t a l
oxides,
semi-
c o n d u c t o r s , or m e t a l s w i t h novel p r o p e r t i e s useful for e l e c t r o n i c , optical, m a g n e t i c a n d / o r catalytic uses. W h e n light m e e t s p a r t i c l e s this small, it b e h a v e s differently. One e x a m p l e is TiO 2 ( t i t a n i u m dioxide), w h i c h h a s b e e n u s e d as an ultra-violet a b s o r b e r for s u n - s c r e e n p r o d u c t s . The usual p r o d u c t is a p p l i e d to t h e skin as a w h i t e - r e f l e c t i n g c r e a m . The p r o c e s s for m a k i n g t i t a n i u m dioxide varies b u t usually e m p l o y s TiC14 a n d its h y d r o l y s i s u n d e r c o n t r o l l e d c o n d i t i o n s . When particles
of 50 n m .
are formed,
the
sun-screen
cream
now is
t r a n s p a r e n t since the p a r t i c l e s absorb a n d s c a t t e r visible light m u c h less t h a n t h e l a r g e r p a r t i c l e s p r e v i o u s l y u s e d . However, t h e ultraviolet light a b s o r p t i o n is n o t c h a n g e d , only the reflection of w h i t e light. As we
shall see
in t h e
next
chapter,
particles
are
formed
first
as
"embryos" w h i c h are m i n u t e p a r t i c l e s of t h e n a n o - p a r t i c l e class. T h e s e t h e n grow into "nuclei" w h i c h t h e n grow into particles. The
science of
p a r t i c l e g r o w t h h a s b e e n a m a j o r s o u r c e of o u r u n d e r s t a n d i n g of p a r t i c l e s . As we have a l r e a d y s h o w n , lattice defects, d u e to t h e r m a l effects, are t h e n o r m w h e n a c r y s t a l g r o w s to sizable p r o p o r t i o n s . However, w h e n n a n o -
111
2.11 Nanotechnology and the solid state
crystals are formed, the n u m b e r s of e m b r y o s a l l o w e d to form, w i t h c o r r e s p o n d i n g nuclei, are controlled. The nuclei growth is t h e n c o n f i n e d to atomic d i m e n s i o n s . Much of this growth forms by " S p o n t a n e o u s Assembly". T h a t is, w h e n n a n o - p a r t i c l e s are formed, a t o m s are a d d e d o n e at a time to form the embryo a n d t h e n the nucleus. It is the size of t h e n u c l e u s t h a t is r e s t r i c t e d . I s u b m i t t h a t the p r e d i c t i o n given in the previous section, i.e.- see p. 107, h a s already b e e n realized. T h a t is, n a n o p a r t i c l e s form by self-assembly of a t o m s (ions) into defect-~ree crystals. It is this lack of intrinsic d e f e c t s t h a t give s u c h particles their u n i q u e c h e m i c a l a n d physical p r o p e r t i e s . Note t h a t if n o r m a l growth were allowed to p r o c e e d further, t h e n w e w o u l d have the n o r m a l d e f e c t - c r y s t a l . Suggested Reading 1. A.C. D a m a s k a n d G.J. Dienes, Point Defects in Metals, Gordon
&
Breach, New York ( 1 9 7 2 ) . 2. G.G. Libowitz, "Defect Equilibria in Solids", Treatice
on Solid State
Chem.- (N.B. Hannay- Ed.), I, 3 3 5 - 3 8 5 , (I 9 7 3 ) . The Chemistry of Imperfect Amsterdam (1964). 3. F.A. Kra~ger,
Crystals, N o r t h - H o l l a n d ,
4. F.A. K r u g e r & H.J. Vink in Solid State Physics, Advances in Research and Applications (F. Seitz & D. Turnbull-Eds.), pp. 3 0 7 - 4 3 5 (I 9 5 6 ) . 5. J.S. A n d e r s o n in Problems of Non-Stotchtometry (A. R a b e n a u - E d . ) , pp, 1-76, N. Holland, A m s t e r d a m ( 1 9 7 0 ) . 6. W. Van Gool, Principles of Defect chemistry Academic Press, New York ( 1 9 6 4 ) .
of Crystalline Solids,
112
The point defect
7. G. Brouwer, "A General A s y m m e t r i c Solution of Reaction Equations C o m m o n in Solid State Chemistry", Philips Res. Rept., 9 , 3 6 6 - 3 7 6 (1954) 8. A. B. Lidiard, "Vacancy Pairs in Ionic Crystals", Phys. Rev., I 1 2 , 5 4 - 5 5 (1958). 9. J.S. A n d e r s o n , "The Conditions of E q u i l i b r i u m of N o n s t o i c h i o m e t r i c C h e m i c a l C o m p o u n d s , Proc. Roy. Soc. ( L o n d o n ) , A185, 6 9 - 8 9 (1946). I0. N.N. Greenwood, Ionic Crystals, Lattice Defects & Non-Stotchiometry, Butterworths, London (1968). 1 I. Hayes a n d S t o n e h a m , "Defects a n d Defect P r o c e s s e s in Non-Metallic Solids"- J. Wiley & Sons, New York ( 1 9 8 5 ) .
40 charge-compensation
for
a divalent
cation
sub-lattice
would
be
the
foUowing defect equation: : 2 M 2+ -%
2.1.I.where
M+
+
M 3+
t h e M 3+ a n d M + are s i t u a t e d on n e a r e s t
neighbor
cation sites,
which were originally divalent.
Thus, the charge compensation mechan/sm represents the single most important mechAnlsrn which operates withln the defect solid. B e c a u s e of this, th e n u m b e r a n d t y p e s of defects, w h i c h c a n a p p e a r in t h e solid, are limited. T h i s r e s t r i c t s t h e n u m b e r of d e f e c t t y p e s we n e e d to c o n s i d e r , in b o t h e l e m e n t a l (all th e s a m e k i n d of atom) a n d ionic l a t t i c e s (having b o t h c a t i o n s a n d a n i o n s p r e s e n t ) . We have s h o w n t h a t by s t a c k i n g a t o m s or p r o p a g a t i o n
units
together,
a solid w i t h
specific
synm~etry
r e s u l t s . If w e have d o n e t h i s properly, a p e r f e c t solid s h o u l d r e s u l t w i t h no holes or d e f e c t s in it. Yet, th e 2 n d law of t h e r m o d y n a m i c s d e m a n d s t h a t a c e r t a i n n u m b e r of p o i n t d efects (vacancies) a p p e a r in t h e lattice. It is i m p o s s i b l e to o b t a i n a solid w i t h o u t s o m e sort of defects. A p e r f e c t solid w o u l d violate t h i s law. T h e possible
at
absolute
zero
2nd
law s t a t e s t h a t zero entropy is only
temperature.
Since
most
solids
exist
at
t e m p e r a t u r e s far f r o m a b s o l u t e zero, t h o s e t h a t we e n c o u n t e r eu~e d e f e c t solids. It is n a t u r a l to a s k w h a t th e n a t u r e of t h e s e d e f e c t s m i g h t be, p a r t i c u l a r l y w h e n we a d d a foreign c a t i o n (activator) to a solid to f o r m a phosphor. C o n s i d e r t h e s u r f a c e of a solid. In th e in terior, we see a c e r t a i n s y m m e t r y which
depends
upon the
surface from the interior,
structure
of t h e
solid. As we a p p r o a c h
the
t h e s y m m e t r y b e g i n s to c h a n g e . At the v e r y
surface, t h e s u r f a c e a t o m s see only h a l f t h e s y m m e t r y t h a t the i n t e r i o r a t o m s do. R e a c t i o n s b e t w e e n solids t a k e place at t h e surface. T h u s , t h e s u r f a c e of a solid r e p r e s e n t s
a defect in itself s i n c e it is not like t h e
i n t e r i o r of t h e solid. In a t h r e e - d i m e n s i o n a l solid, we c a n p o s t u l a t e t h a t t h e r e o u g h t to be
2.1 Types of point defects three
major
types
of defects,
having
41
either
one-,
two-
or
three-
d i m e n s i o n s . Indeed, this is exactly t h e case found for defects in solids, as we briefly d e s c r i b e d in the p r e c e d i n g
chapter.
We have a l r e a d y given
n a m e s to e a c h of t h e s e t h r e e t y p e s of defects. T h u s a o n e - d i m e n s i o n a l defect of t h e lattice is called a "point" defect, a t w o - d i m e n s i o n a l defect a "line" or "edge" defect a n d a t h r e e - d i m e n s i o n a l defect is called a "plane" or "volume" defect. We have a l r e a d y d e s c r i b e d , in an e l e m e n t a r y way, l i n e a n d v o l u m e defects a n d will not a d d r e s s t h e m f u r t h e r e x c e p t to point out h o w t h e y m a y arise w h e n c e r t a i n point defects are p r e s e n t . It is sufficient to realize t h a t t h e y exist a n d are i m p o r t a n t
for a n y o n e w h o
studies
h o m o g e n e o u s m a t e r i a l s s u c h as m e t a l s . Point defects
are c h a n g e s at a t o m i s t i c levels, while
line
and volume
defects are c h a n g e s in s t a c k i n g of p l a n e s or g r o u p s of a t o m s ( m o l e c u l e s ) in t h e s t r u c t u r e . The former affect t h e c h e m i c a l p r o p e r t i e s of the solid w h e r e a s the l a t t e r affect t h e p h y s i c a l p r o p e r t i e s of t h e solid. Note t h a t t h e a r r a n g e m e n t ( s t r u c t u r e ) of the individual a t o m s (ions) are not affected, only the m e t h o d in w h i c h the s t r u c t u r e u n i t s are a s s e m b l e d . T h a t is, t h e s t r u c t u r e of t h e solid r e m a i n s i n t a c t in spite of t h e p r e s e n c e of d e f e c t s . Let us n o w e x a m i n e e a c h of t h e s e defects in m o r e detail, s t a r t i n g w i t h t h e one-dimensional
lattice
defect
and
then
with
the
multi-dimensional
defects. We will find t h a t specific types have b e e n found to be a s s o c i a t e d w i t h e a c h type of d i m e n s i o n a l defect w h i c h have specific effects u p o n t h e stability of the solid s t r u c t u r e . It s h o u l d be clear t h a t t h e type of p o i n t defect
prevalent
homogenous
in any given
(same
atoms)
or
solid will d e p e n d heterogeneous
upon whether
(composed
it is
of differing
atoms). I. The Po'mt Defect in H o m o g e n e o u s Solids We begin by identifying t h e various defects w h i c h c a n arise in solids a n d later will s h o w how t h e y c a n be m a n i p u l a t e d to o b t a i n d e s i r a b l e p r o p e r t i e s not found in naturally f o r m e d solids. Let us look first at t h e h o m o g e n e o u s type of solid. We will first r e s t r i c t o u r d i s c u s s i o n to solids w h i c h stoichiometric,
are
a n d later will e x a m i n e solids w h i c h c a n be classified as
" n o n - s t o i c h i o m e t r i c " , or having a n excess of one or a n o t h e r of one of t h e
The point defect
42
b u i l d i n g b l o c k s of t h e solid. T h e s e o c c u r in s e m i - c o n d u c t o r s as well as o t h e r t y p e s of e l e c t r o n i c a l l y or optically active solids. Suppose
you w e r e
given
the
problem
of identifying
defects
in
a
h o m o g e n e o u s solid. Since all of t h e a t o m s in t h i s type of solid are t h e s a m e , t h e p r o b l e m is s o m e w h a t simplified over t h a t of the h e t e r o g e n e o u s solid (that is- a solid c o n t a i n i n g m o r e t h a n one type of a t o m or ion). After s o m e i n t r o s p e c t i o n , you could s p e c u l a t e t h a t t h e h o m o g e n e o u s solid c o u l d have t h e following t y p e s of p o i n t d e f e c t s : 2 . 1 . 2 . - Types of Point Defects E x p e c t e d in a H o m o g e n e o u s S o l i d Vacancies
Substitutional Impurities
Self-interstitial
Interstitial Impurities
On t h e left are t h e two types of p o i n t defects w h i c h involve the lattice itself, while t h e o t h e r s involve i m p u r i t y a t o m s (Note t h a t i n t e r s t i t i a l a t o m s c a n involve e i t h e r a n i m p u r i t y a t o m or the s a m e a t o m t h a t m a k e s up t h e lattice s t r u c t u r e itself). Indeed, t h e r e do n o t s e e m to any m o r e t h a n t h e s e four, a n d indubitably, no o t h e r s have b e e n observed. Note t h a t we are l i m i t i n g o u r defect family to p o i n t defects in t h e lattice a n d are i g n o r i n g line a n d v o l u m e defects
of t h e lattice. T h e s e
four p o i n t defects, given
above, are i l l u s t r a t e d in the following diagram, given as 2.1.3. o n the n e x t page. Note t h a t w h a t we m e a n by an "interstitial" is a n a t o m t h a t c a n fit into t h e s p a c e s b e t w e e n t h e m a i n a t o m s in the c r y s t a l l i n e array. In this case, w e have s h o w n a h e x a g o n a l lattice a n d have labeled e a c h type of p o i n t defect. O b s e r v e t h a t we have s h o w n a v a c a n c y in o u r h e x a g o n a l lattice, as well as a foreign i n t e r s t i t i a l a t o m w h i c h is small e n o u g h to fit into t h e i n t e r s t i c e between
the
atoms
of the
structure.
Also
shown
are
two
types
of
s u b s t i t u t i o n a l a t o m s , one larger a n d the o t h e r s m a l l e r t h a n the a t o m s c o m p o s i n g the p r i n c i p a l h e x a g o n a l lattice. In b o t h cases, t h e h e x a g o n a l p a c k i n g is d i s r u p t e d due to a "non-fit" of t h e s e a t o m s in t h e s t r u c t u r e . Additionally, we have i l l u s t r a t e d a n o t h e r type of defect t h a t c a n arise
2.1 Types of point defects
43
I Defects Which Can Occur in a Homogeneous Hexagonal Lattice I
within
the
homogeneous
lattice
(in
addition
to
the
vacancy
and
s u b s t i t u t i o n a l i m p u r i t i e s t h a t are b o u n d to arise). This is called t h e "selfinterstitial". Note t h a t it h a s a decisive effect on t h e defect.
Since
the
atoms
are
all t h e
same
size,
the
structure
at t h e
self-interstitial
i n t r o d u c e s a l l n e - d e f e c t in the overall s t r u c t u r e . It s h o u l d be e v i d e n t t h a t t h e l i n e - d e f e c t i n t r o d u c e s a difference in p a c k i n g o r d e r since t h e c l o s e p a c k i n g a t t h e a r r o w s h a s c h a n g e d to cubic a n d t h e n r e v e r t s to h e x a g o n a l in b o t h lower a n d u p p e r rows of a t o m s . It m a y be t h a t this type of d e f e c t is a m a j o r c a u s e of t h e line or edge type of defects t h a t a p p e a r in m o s t homogeneous
solids.
In c o n t r a s t ,
the
other
defects
produce
only
a
d i s r u p t i o n in the l o c a l i z e d p a c k i n g o r d e r of t h e h e x a g o n a l lattice, i.e.- t h e defect
does not extend
t h r o u g h o u t t h e lattice,
b u t only close
to t h e
The point defect
44 specific defect.
It s h o u l d be evident
t h a t m e t a l s or solid solutions of
m e t a l s (alloys) show s u c h behavior in c o n t r a s t to h e t e r o g e n e o u s l a t t i c e s w h i c h involve c o m p o u n d s s u c h as ZnS. This a c c o u n t s for the t r e m e n d o u s d i s c r e p a n c y b e t w e e n t h e o r e t i c a l a n d actual s t r e n g t h of c e r t a i n alloys in p r a c t i c a l a p p l i c a t i o n s due to "fatigue" failure w h e n the object is b e i n g used. Now, s u p p o s e t h a t we have a solid solution of two (2) e l e m e n t a l solids. Would t h e point defects be the same, or not? An easy way to visualize s u c h point defects is s h o w n in the following d i a g r a m : 2.1.4.- Defects in the H o m o g e n e o u s Solid C o n t a i n i n g 2 Solids in Solution
..~i~~
vm=
Interstitial 1~
Int e rst it .
.
.
.
Vacancy
,IInterstitial cog e n e o u ~ g o n a/ Structure -- Same Size Atoms
Homogeneous Solid Solution Differing Size Atoms
-
Here, we u s e a h e x a g o n a l l y - p a c k e d r e p r e s e n t a t i o n of a t o m s to depict t h e c l o s e - p a c k e d solid. Both types of h o m o g e n e o u s solids are shown, w h e r e one solid is c o m p o s e d
of the
same
sized
atoms while
the
other
is
c o m p o s e d of two different sized atoms. On the right are the types of p o i n t
2.1 Types of point defects
45
d e f e c t s t h a t c o u l d o c c u r for t h e s a m e sized a t o m s in t h e lattice. T h a t is, given a n a r r a y of a t o m s in a t h r e e d i m e n s i o n a l lattice, only t h e s e two t y p e s of lattice p o i n t d e f e c t s could o c c u r w h e r e t h e size of t h e a t o m s are t h e s a m e . T h e t e r m " v a c a n c y " is s e l f - e x p l a n a t o r y b u t " s e l f - i n t e r s t i t i a l " m e a n s t h a t o n e a t o m h a s s l i p p e d into a s p a c e b e t w e e n t h e rows of a t o m s . In a lattice w h e r e t h e a t o m s are all of t h e s a m e size, s u c h b e h a v i o r is e n e r g e t i c a l l y v e r y difficult u n l e s s a severe d i s r u p t i o n of t h e lattice o c c u r s (usually a " l i n e - d e f e c t " results). T h i s b e h a v i o r is q u i t e c o m m o n in c e r t a i n t y p e s of h o m o g e n e o u s solids. In a like m a n n e r , if t h e m e t a l - a t o m w e r e to have b e c o m e m i s p l a c e d in t h e lattice a n d w e r e to have o c c u p i e d o n e of t h e i n t e r s t i t i a l p o s i t i o n s , as s h o w n in t h e different sized a t o m solid ( s e e 2.1.4.)
t h e n t h e lattice is d i s r u p t e d by its p r e s e n c e
at t h e i n t e r s t i t i a l
position. T h i s type of defect h a s also b e e n o b s e r v e d . Note t h a t t h e a t o m s are usually n o t c h a r g e d in t h e h o m o g e n e o u s l a t t i c e . S u m m a r i z i n g , t h r e e t y p e s of p o i n t defects are e v i d e n t in a h o m o g e n e o u s lattice. In a d d i t i o n to t h e Vacancy, two t y p e s of s u b s t i t u t i o n a l defects c a n also be
delineated.
Both are
direct
substitutions
in t h e
"lattice",
or
a r r a n g e m e n t of t h e a t o m s . One is a s m a l l e r a t o m , w h i l e t h e o t h e r is l a r g e r than
the
atoms
comprising
the
lattice.
In
both
cases,
the
lattice
a r r a n g e m e n t affects t h e h e x a g o n a l o r d e r i n g of t h e lattice a t o m s a r o u n d it. T h e lattice p a c k i n g is s e e n to be affected for m a n y lattice d i s t a n c e s . It is for t h i s r e a s o n t h a t c o m p o u n d s c o n t a i n i n g i m p u r i t i e s s o m e t i m e s have q u i t e different c h e m i c a l r e a c t i v i t i e s t h a n t h e p u r e s t ones. However, t h e i n t e r s t i t i a l i m p u r i t y does not affect t h e lattice o r d e r i n g at all. Now, let us look at t h e h e t e r o g e n e o u s l a t t i c e II. T h e Point Defect in H e t e r o g e n e o u s S o l i d s The
situation concerning
defects
in h e t e r o g e n e o u s
inorganic
solids is
s i m i l a r to t h a t given above, e x c e p t for one v e r y i m p o r t a n t factor, t h a t of c h a r g o o n t h e a t o m s . Covalent i n o r g a n i c solids a r e a r a r i t y while i o n i c i t y or p a r t i a l ionicity s e e m s to be t h e n o r m . T h u s , h e t e r o g e n e o u s solids are usually c o m p o s e d of c h a r g e d m o i e t i e s , half of w h i c h are positive ( c a t i o n s ) a n d half negative (anions). In g e n e r a l , t h e total c h a r g e of t h e c a t i o n s will
The point defect
46
e q u a l t h a t of t h e a n i o n s (Even in the case of s e m i - c o n d u c t o r s , w h e r e t h e total of the c h a r g e s is n o t zero, the excess c h a r g e (n- or p- type) is s p r e a d over t h e w h o l e lattice so t h a t no single atom, or g r o u p of a t o m s , ever has a c h a r g e different from its n e i g h b o r s . Note also t h a t m o s t of the s e m i conductors
that
we
use
are
homogeneous
in
nature,
modified
by
h o m o g e n e o u s a d d i t i o n s to form p- or n - t y p e electrically c h a r g e d areas). In a given s t r u c t u r e , c a t i o n s are usually s u r r o u n d e d by anions, a n d vice-versa. B e c a u s e of this, we c a n r e g a r d the lattice as being c o m p o s e d of a c a t i o n s u b - l a t t i c e a n d an a n i o n mab-lattice. ( R e m e m b e r w h a t w a s stated in Chapter
1
concerning
the
fact
that
most
structures
are
oxygen-
d o m i n a t e d ) . W h a t we m e a n by a "sub-lattice" is i l l u s t r a t e d in the following diagram: 2.1.5.-
IA Cubic LatticeShowing the Cation and Anion Sub-Lattices I
Cation Sub-Lattice
Anion Sub-Lattice
Combined Lattice
In this case, we have s h o w n b o t h the cubic cation a n d a n i o n "sub-lattices s e p a r a t e l y , a n d t h e n t h e c o m b i n a t i o n . It s h o u l d be clear t h a t all positive c h a r g e s in t h e c a t i o n s u b - l a t t i c e will be b a l a n c e d by a like n u m b e r of negative c h a r g e s in the a n i o n sub-lattice, even if excess c h a r g e exists in o n e or the o t h e r of the sub-lattices. If an a t o m is m i s s i n g , t h e n the overall lattice
r e a d j u s t s to c o m p e n s a t e
different
atom
compensation
present,
for this loss of c h a r g e .
having
a
differing
charge,
If t h e r e the
is a
charge-
m e c h a n i s m again m a n i f e s t s itself. T h u s , a cation with an
e x t r a c h a r g e n e e d s to be c o m p e n s a t e d by a like anion, or by a n e a r e s t n e i g h b o r c a t i o n w i t h a l e s s e r charge. An e x a m p l e of this type of c h a r g e -
2.1 Types of point defects compensation mechanism
47
for a divalent c a t i o n s u b - l a t t i c e w o u l d be t h e
following defect equation: 2.1.6.-
2 C a 2+ ~
Li +
+
Sb 3+
w h e r e t h e Sb 3+ a n d Li + are s i t u a t e d on n e a r e s t n e i g h b o r c a t i o n s u b - l a t t i c e sites, in t h e d i v a l e n t Ca 2+ s u b - l a t t i c e . Note t h a t a total c h a r g e of 4+ e x i s t s on b o t h s i d e s of t h e above e q u a t i o n .
Thus, the charge compensation mechR:nlsm represents the single most important mechanism which operates withln the d e f e c t ~ solid. We find t h a t t h e n u m b e r a n d t y p e s of defects, w h i c h c a n a p p e a r in t h e h e t e r o g e n e o u s solid, are l i m i t e d b e c a u s e of two factors: 1) T h e c h a r g e - c o m p e n s a t i o n f a c t o r 2) T h e p r e s e n c e of two s u b - l a t t i c e s in t h e ionic solid. These
factors
restrict
the
number
of p o i n t
defect t y p e s
we
need
to
c o n s i d e r in ionic h e t e r o g e n e o u s l a t t i c e s (having b o t h c a t i o n s a n d a n i o n s p r e s e n t ) . For ionic s o l i d s , t h e following t y p e s of d e f e c t s h a v e b e e n f o u n d to exist: S c h o t t k y d e f e c t s ( a b s e n c e of b o t h c a t i o n a n d a n i o n ) Cation v a c a n c i e s Anion v a c a n c i e s F r e n k e l defects (Cation v a c a n c y p l u s s a m e c a t i o n as interstitial) I n t e r s t i t i a l i m p u r i t y a t o m s (both c a t i o n a n d a n i o n ) Substitutional impurity atoms(both cation and anion) T h e s e d e f e c t s are i l l u s t r a t e d in t h e following d i a g r a m , given as 2.1.7. o n the next page. Note t h a t , in g e n e r a l , a n i o n s are l a r g e r in size t h a n c a t i o n s d u e to t h e e x t r a e l e c t r o n s p r e s e n t in t h e former. A h e x a g o n a l lattice is s h o w n in
The point defect
48
2.1.7.] P o i n t Defects Which Can Occur in the Heterogeneous Ionic Solid]
Schottky Defect
Frenkel Defect
Anion Vacancy
Cation Vacancy
Substitutional Cation InterstitialCation Substitutional Anion
2.1.7. w i t h b o t h F r e n k e l a n d S c h o t t k y defects, as well as s u b s t i t u t i o n a l defects. T h u s , if a c a t i o n is m i s s i n g (cation vacancy) in the cation sublattice, a like a n i o n will be m i s s i n g in the a n i o n sub-lattice. T h i s is k n o w n as a S c h o t t k y
defect
(after
the
first i n v e s t i g a t o r
(1935)
to note
its
where
the
existence). In t h e c a s e of t h e F r e n k e l defect, cation was supposed
to r e s i d e
th e
in th e
"square" r e p r e s e n t s lattice
before
it m o v e d
to its
i n t e r s t i t i a l p o s i t i o n in t h e cation sub-lattice. Additionally, "A nti-Fre nke l" d e f e c t s c a n exist in t h e an io n sub-lattice. T h e s u b s t i t u t i o n a l defects are s h o w n as t h e s a m e size as th e c a t i o n or a n i o n it displaced. Note t h a t if
49
2.2 The plane net
t h e y w e r e not, t h e lattice s t r u c t u r e w o u l d b e d i s r u p t e d f r o m r e g u l a r i t y at t h e p o i n t s of i n s e r t i o n of t h e foreign ion. To s u m m a r i z e , t h e c a t e g o r i e s of p o i n t d e f e c t s p o s s i b l e for t h e s e two t y p e s of l a t t i c e s are i l l u s t r a t e d are: 1). In a n e l e m e n t a l solid, we m a y have: Vacancies Self-interstitial Interstitial Impurities Substitutional Impurities 2). In t h e ionic solid, i.e.- h e t e r o g e n e o u s solid, we m a y have: S c h o t t k y d e f e c t s ( a b s e n c e of b o t h c a t i o n a n d a n i o n ) Cation or a n i o n v a c a n c i e s F r e n k e l d e f e c t s (Cation v a c a n c y p l u s t h e s a m e c a t i o n as interstitial) I n t e r s t i t i a l i m p u r i t y a t o m s (both c a t i o n a n d anion) Substitutional impurity atoms(both cation and anion) All of t h e s e
point
defects
are
intrinsic
to
the
solid.
The
factors
r e s p o n s i b l e for t h e i r f o r m a t i o n are e n t r o p y effects (point d e f e c t faults) a n d i m p u r i t y effects. At t h e p r e s e n t time, t h e h i g h e s t - p u r i t y m a t e r i a l s available still c o n t a i n about 1.0 p a r t p e r billion of various i m p u r i t i e s , y e t are 9 9 . 9 9 9 9 9 9 9
% p u r e . S u c h a solid will c o n t a i n about 1014
impurity
a t o m s p e r mole. So it is safe to say t h a t all solids c o n t a i n i m p u r i t y a t o m s , a n d t h a t it is u n l i k e l y t h a t we shall ever b e able to o b t a i n a solid w h i c h is
c o m p l e t e l y pure and d o e s not contain defects. 2.2. - T H E PLANE N E T Now, let u s c o n s i d e r h o w s u c h d e f e c t s arise in a n y given solid. T h e e a s i e s t w a y to visualize h o w i n t r i n s i c d e f e c t s o c c u r in t h e solid is to s t u d y t h e PLANE N E T . I m a g i n e t h a t we have M as c a t i o n s a n d X as a n i o n s (we shall
The point defect
50
i g n o r e formal c h a r g e for the m o m e n t ) . This is s h o w n in the following diagram:
[The Pl one Net'l
2.2.1.-
X M g M X M xMI~x M M X MXx M X M X M X
x
x
,.(,(~._.~~ M XM M X X M
x
,__
x i i,•
M X M X X M X M X M X M X
M M X M
X M X X M X M M X M X X M X M
It is easy to see t h a t we c a n stack a series of t h e s e
"NETS" to form a
t h r e e - d i m e n s i o n a l solid. Note t h a t we have u s e d t h e labeling: V = vacancy; i = i n t e r s t i t i a l ; m = c a t i o n site; x= a n i o n site a n d s = surface site. One m i g h t t h i n k t h a t p e r h a p s V-m a n d V+x o u g h t to be i n c l u d e d in o u r list of v a c a n c i e s . However, a n e g a t i v e l y - c h a r g e d c a t i o n v a c a n c y alone, p a r t i c u l a r l y when
it is s u r r o u n d e d b y negative a n i o n s , w o u l d
Neither Either
n o t be very stable.
s h o u l d a p o s i t i v e l y - c h a r g e d a n i o n v a c a n c y be any m o r e arrangement
would
require
high
energy
stable.
stabilization to exist.
T h e r e f o r e , we do n o t i n c l u d e t h e m in o u r listing. However,
a Vm
c o u l d c a p t u r e a positive c h a r g e
likewise for t h e Vx w h i c h t h e n b e c o m e s a V'x
to b e c o m e
V+m
and
Both of t h e s e are stable
w h e n s u r r o u n d e d by o p p o s i t e l y c h a r g e d sites. We have a l r e a d y s t a t e d t h a t s u r f a c e sites are special. intrinsic
defects.
The
Hence,
they
same criteria
are i n c l u d e d
in o u r listing
apply concerning
charge
of
on t h e
defect. For t h e p l a n e net, we c a n e x p e c t the following types of i n t r i n s i c defects:
2.2 The plane net 2.2 . 2 . -
Vacancies
51 Charged Particles
VM, V x , V+M , V x
e,
Surface Sites
Interstitials
Ms,Xs,
M i , ~- , M+i, X'i The
positive
hole
requires
e q u i v a l e n t of t h e e l e c t r o n ,
n+
more
explanation,
n + is
the
M+i , X s electronic
e- , in s o l i d s a n d t h e y a n n i h i l a t e e a c h o t h e r
upon reaction2.2.3.-
e-
+
n+
=
energy.
T h e following d i a g r a m s h o w s h o w t h e positive hole c a n exist in t h e solid: 2.2.4-
The Germanium Lattice Containing a Positive Hole [
+
73
- G e a 2 = A r C ore" 4s 2 4 p 2 -- G a a l7o =
Here, w e r e p r e s e n t
A r C o r e " 4s 2 4 p ~
Ge w h i c h h a s 32 e l e c t r o n s .
In t h e solid, it f o r m s
4 ( s p ) 3 h y b r i d b o n d s as t h e s e m i - c o n d u c t o r . Ga h a s only 31 e l e c t r o n s a n d c a n f o r m b u t t h r e e b o n d s . If it is i n c o r p o r a t e d into t h e s t r u c t u r e , t h e
The point defect
52
lattice responds by forming a positive electronic "hole", i.e.- =+. Still another type of electronic defect is the color-center, following diagram: 2.2.5.-
shown in the
] A n E l e c t r o n T r a p p e d a t A n A n i o n V a c a n c y ]'
(E) e (E) e (E) e
~G ~ Q $ Q
@e e@
e
(E) e @ e (E) e ]Absorption as a Function of Cation Present in Lattice] 8500 4000 I
4000
6000
i
I
I
I
I
I
6000 I
I
NaC1
LiC1
4000 I
6000 |
I
4000 I
KCI
6000 8000 A
m
m
Rb
-
,
m m
'
"
4-~
D-. 0 [8
,j
I
4
3
24
3
2
3
, 2
3
2
Energy in Electron Volts In order to compensate for the loss of negative charge, the lattice has captured an electron and charge-compensation has resulted at the anion vacancy. This combination is called an "F-center". This is a special case where an electron is localized at the vacancy and is optically active. That
53
2.2 The plane net
is, it a b s o r b s light w i t h i n a w e l l - d e f i n e d b a n d a n d is called a c o l o r - c e n t e r s i n c e it i m p a r t s a specific color to t h e crystal. Note t h a t t h e p e a k of t h e b a n d c h a n g e s as t h e size of t h e c a t i o n in t h e alkali h a l i d e s i n c r e a s e s . There
a p p e a r s to be an i n v e r s e r e l a t i o n b e t w e e n
(actually, t h e
polarizability of t h e
cation)
t h e size of t h e c a t i o n
and the peak
e n e r g y of t h e
a b s o r p t i o n b a n d of t h e s e F - c e n t e r s . We c a n n o w p r o c e e d to w r i t e a s e r i e s of d e f e c t r e a c t i o n s for o u r p l a n e n e t in t e r m s of specific defects, i n c l u d i n g t h a t of t h e F - C e n t e r , w h e r e we u s e 5 as a s m a l l f r a c t i o n : 2.2.6.-
SCHOTII~
:
MX = M1. s X 1 . 5
FRENKEL :
+ 5Vm
MX = M 1-5 X
+ 5 Mi +
ANTI-FRENKEL:
MX = M X I - 6
+ 6Vx
F - CENTER:
MX = M X I - + { V x / e }
+
+ 6Vx 5Vm 5Xi
+8/2
X2
T h e s e e q u a t i o n s are valid for t h e d e f e c t s in a h e t e r o g e n e o u s lattice w h e r e , for t h e F - C e n t e r , t h e b r a c k e t s e n c l o s e t h e c o m p l e x c o n s i s t i n g of an e l e c t r o n c a p t u r e d a t a n a n i o n v a c a n c y , i.e.- {Vx/e-}. Actually, t h e f o r m a t i o n of a n
F-center
is
more
complicated
than
this.
A more
complete
e x p l a n a t i o n is given as follows. It is well k n o w n t h a t F - c e n t e r s c a n b e f o r m e d by e x p o s i n g t h e NaCI c r y s t a l to s o d i u m m e t a l vapor. T h e following d e f e c t r e a c t i o n s t a k e p l a c e : 2.2.7.-
Na ~
=
Na +
Na +
+
{ N a +,C1-}
VC1
+
e-
=
+ =
e
-
2{Na +[CI-,VCll}
[Vcl/e-]
T h e s e e q u a t i o n s i l l u s t r a t e h o w t h e c r y s t a l r e s p o n d s to t h e p r e s e n c e of s o d i u m v a p o r , i.e.- e x c e s s N a + , by f o r m i n g a n i o n v a c a n c i e s , to f o r m t h e Fc e n t e r . However, if o n e a n i o n v a c a n c y c a n be f o r m e d w h i c h c a p t u r e s an e l e c t r o n w h i c h is optically active, t h e n you m i g h t t h i n k t h a t m o r e t h a n
The point defect
54
one anion-vacancy c o m p l e x m i g h t be possible. Indeed, this is the case. This is illustrated in the following diagram: 2.2.8.-
+-+-+-+-+2-+
- ~ -
!
-I-- -k t -I--
- .i._ .~.~@__i_ -+--l---l-y-l---l--
4--
-I---I--
-I-~-I--
-I---I-
[II0]
F-Center A better diagram:
representation
of the
M-Center M-center
is s h o w n
in
the
following
2.2.9.-
An "M-Center" Shown on T w o Planes of a Cubic LatticeI
(2)
U n d e r c e r t a i n precise e x p e r i m e n t a l conditions, two anion sites can be conjoined (in the s a l t - s t r u c t u r e , along the {i,I,0} plane) to form what is
55
2.2 The plane net
t e r m e d t h e "M-Center" as s h o w n above in t h e two d i a g r a m s , 2.2.8. a n d 2.2.9. The
optical p r o p e r t i e s
of t h e s e two t y p e s of c e n t e r s
are given in t h e
following d i a g r a m : 2.2.10.-
Absorption Spectra at 77 ~ for a KCI Crystal with F-and M-Centers Present i
I
1.0
~
i
F-Center M-Center Absorption
0.8
0-.0.4
0,:2
0
400
500
600 700 800 Wavelength in Nanometers
900
T h e defect e q u a t i o n for f o r m a t i o n of t h e M - c e n t e r is also given as follows: 2.2.11.-
M-center:
KC1 = KCll.5 + ib/2 C12 t" + 5 / 4 [ V-C1 ] VC1 ]
Note t h a t e a c h v a c a n c y in 2.2.9. h a s c a p t u r e d a n e l e c t r o n , in r e s p o n s e to
The point defect
56
t h e c h a r g e - c o m p e n s a t i o n m e c h a n i s m w h i c h is o p e r a t i v e for t h e d e f e c t reactions.
These
associated,
negatively-charged,
vacancies
have
quite
different a b s o r p t i o n p r o p e r t i e s t h a n t h a t of the F - c e n t e r . T h e r e are o t h e r i m p u r i t y s y s t e m s to w h i c h this n o t a t i o n c a n b e applied. For t h e case of an AgCI crystal c o n t a i n i n g the Cd 2+ c a t i o n as an i m p u r i t y , w e have: 2Ag +
2.2.12.-
~
[Cd 2+,vAg
T h i s is an e x a m p l e of a h e t ~ r o ~
]
s y s t e m . A n o t h e r s u c h s y s t e m is CdC12
c o n t a i n i n g Sb 3+ . Here, we c a n w r i t e at least t h r e e
different
e q u a t i o n s involving d e f e c t
equilibria: 2.2.13.-
2 Cd 2+
z..
2 Cd 2+
z.._7
Sb3+ + Sb3+ +
2 Cd 2+
-7~
Sb3+
+
p+
+
VCd
V+Cd Li+
In t h e last e q u a t i o n , c h a r g e - c o m p e n s a t i o n h a s o c c u r r e d due to i n c l u s i o n of a m o n o v a l e n t cation. All of t h e s e
equations
are c a s e s of i m p u r i t y
substitutions. A n o t h e r type is t h e so-called h o m o t y p e i m p u r i t y s y s t e m . The s u b s t a n c e , n i c k e l o u s oxide, is a p a l e - g r e e n
insulator, when
prepared
in an i n e r t
a t m o s p h e r e . If it is r e h e a t e d in air, or if a m i x t u r e of NiO a n d Li+ is reheated
in
an
inert
atmosphere,
the
NiO
becomes
a black
semi-
c o n d u c t o r . T h i s is a classical e x a m p l e of t h e effect of defect r e a c t i o n s u p o n t h e i n t r i n s i c p r o p e r t i e s of a solid: 2.2.14.-
2 Ni 2+ 2 Ni 2+
--7 L_ .-I
[ N i3+ / V N i ] N i 3+ + Li +
+
P+
T h i s b e h a v i o r is typical for t r a n s i t i o n m e t a l s w h i c h easily u n d e r g o c h a n g e s in v a l e n c e in the solid state.
2.3 Defect equation symbolism
57
Up to t h i s point, we have only i n v e s t i g a t e d s t o i c h i o m e t r i c lattices 9 Let us now
examine
non-stoichiometric
Consider the semi-conductor,
lattices
Ge. T h e
in
light
of o u r
symbolism.
defect reactions associated w i t h
t h e f o r m a t i o n of p - t y p e a n d n - t y p e l a t t i c e s are. 2.2.15. -
n-type:
Ge+6As
=
[GelAs~]
+
6e-
p-type:
Ge+6Ga
=
[Ge/Ga~]
+
6p+
T h e e x c e s s c h a r g e s s h o w n are s p r e a d over t h e e n t i r e
lattice,
as s t a t e d
before. 2 . 3 . - D E F E C T EQUATION SYMBOLISM Whether
you realize
it or not,
we
have
already developed
our
own
s y m b o l i s m for d e f e c t s a n d defect r e a c t i o n s b a s e d on t h e P l a n e Net.
It
m i g h t b e well to c o m p a r e o u r s y s t e m to t h o s e of o t h e r a u t h o r s , w h o h a v e also c o n s i d e r e d t h e s a m e p r o b l e m in t h e p a s t . It w a s p r o b a b l y R e e s (1930) w h o w r o t e t h e first m o n o g r a p h on d e f e c t s in solids. Rees u s e d nM to r e p r e s e n t t h e c a t i o n vacancy, as did Libowitz (1974). T h i s h a s c e r t a i n a d v a n t a g e s si n c e we c a n w r i t e t h e first e q u a t i o n in 2 . 2 . 1 2 . as:
K-l }+
+ +
2 3 1 9
.
2 Cd 2+
-
" 7
Col
"
p+
Cd
Likewise, t h e o t h e r e q u a t i o n s b e c o m e :
2.3.2.-
2 C d 2+
~
~-I +c d
and:
K~ 9 .
.-
-7
"
+ Cd
+
"
V ~ +col
+
+ p Cd
A l t h o u g h t h e r e s u l t s are e q u a l as far as utility is c o n c e r n e d ,
we
s ha ll
c o n t i n u e to u s e o u r s y m b o l i s m , for r e a s o n s w h i c h will b e c o m e c le a r l a t e r .
The point defect
58
T h e following c o m p a r e s defect s y m b o l i s m , as u s e d by p r i o r A u t h o r s . N o t e t h a t o u r s y m b o l i s m m o s t r e s e m b l e s t h a t of Krc~ger, b u t n o t in all a s p e c t s . 2.3.4-
Rees[1930]
Kruger [1954]
Libowitz [1974}
C a t i o n Site Vacancy:
I-]JIM
VM
~VI
Anion Site V a c a n c y
D[k
Vx
U] x
Cation Interstitial
AM
M i , M+i
Mi
Anion I n t e r s t i t i a l
Ax
Xi , X~
Xi
Negative Free C h a r g e
e
e ~
e-
Positive Free C h a r g e
p
h+
h+
Interstices
---
a V.
a V.
Unoccupied Interstitial
---
Vi
A
1
I
Anti-structure Occupation---
M M , X x , MX,XM
T h e s e prior a u t h o r s have c o n s i d e r e d s o m e i n t r i n s i c d e f e c t s t h a t we have n o t t o u c h e d , n a m e l y i n t e r s t i c e s a n d t h e so-called " a n t i - s t r u c t u r e " o c c u p a t i o n . T h e l a t t e r deals w i t h a n i m p u r i t y a n i o n o n a cation s i t e c o u p l e d w i t h a n i m p u r i t y c a t i o n on a n a n i o n site, b o t h w i t h t h e p r o p e r charge. We have m e n t i o n e d i n t e r s t i c e s b u t not in detail. T h e y a p p e a r as a f u n c t i o n of s t r u c t u r e . T h e r e is one site in a t e t r a h e d r o n , four in a b o d y - c e n t e r e d cube, a n d six in a s i m p l e cube. T h u s , a in {a Vi} is I, 4 or 6, r e s p e c t i v e l y . We shall n e e d t h i s s y m b o l later, as well as V i , t h e u n o c c u p i e d i n t e r s t i t i a l . 2 . 4 . - S O M E APPLICATIONS FOR D E F E C T CHEMISTRY Before
we
proceed
to
analyze
defect
reactions
by a m a t h e m a t i c a l
a p p r o a c h , let u s c o n s i d e r two a p p l i c a t i o n s of solid s t a t e c h e m i s t r y . We b e g i n w i t h a d e s c r i p t i o n of s o m e p h o s p h o r d e f e c t c h e m i s t r y .
2.4 Some applications for defect chemistry
59
I. - P h o s o h 0 r s In t h e prior l i t e r a t u r e , it w a s f o u n d (Kinney- 1955) t h a t Ca2 P2 07 c o u l d b e a c t i v a t e d b y Sb 3+ to form t h e p h o s p h o r : Ca2P207:Sb.02
(this f o r m a l i s m
actually m e a n s a solid-solution of two p y r o p h o s p h a t e c o m p o u n d s , [(Ca.99,Sb.ol)2P2OT]). T h e b r i g h t n e s s r e s p o n s e of t h i s p h o s p h o r
i.e.was
m o d e r a t e w h e n e x c i t e d by ultraviolet r a d i a t i o n b u t w a s i m p r o v e d four t i m e s by t h e a d d i t i o n of Li + . T h e o p t i m u m a m o u n t p r o v e d to be t h a t e x a c t l y e q u a l to t h e a m o u n t of Sb 3+ p r e s e n t in the p h o s p h o r . The d e f e c t reactions occurring were. 2.4.1.- Defect R e a c t i o n s O c c u r r i n g in C a l c i u m P y r o p h o s p h a t e P h o s p h o r PHOSPHOR BRIGHTNESS 2 Ca 2+ or (2 Ca 2+ 2 Ca 2+
~ Sb3+Ca
"--7
+
Sb3+Ca
z_.
Sb3+Ca
V+Ca +
25 %
VCa + P+ ) +
Li+ca
100
It is well k n o w n t h a t p h o s p h o r b r i g h t n e s s in a p h o s p h o r is p r o p o r t i o n a l to t h e n u m b e r s of activator ions, i.e.- Sb 3+ i o n s , actually i n c o r p o r a t e d i n t o t h e p y r o p h o s p h a t e s t r u c t u r e . P h o s p h o r s are p r e p a r e d b y h e a t i n g t h e i n g r e d i e n t s at h i g h t e m p e r a t u r e (> 1000 ~ C . ) t o o b t a i n a c o m p o u n d h a v i n g high
crystallinity.
The
sintering
process
decreases
entropy
and
is
counterproductive to t h e f o r m a t i o n of v a c a n c i e s in t h e p y r o p h o s p h a t e lattice. In t h e a b s e n c e of Li+, l a c k of v a c a n c y - f o r m a t i o n actually d e c r e a s e s t h e a m o u n t of Sb 3+ i n c o r p o r a t e d into activator sites. A p p a r e n t l y , four t i m e s as m a n y activator ions w e r e i n c o r p o r a t e d into t h e lattice w h e n t h e c h a r g e - c o m p e n s a t i n g Li+ ions w e r e p r e s e n t on n e a r e s t n e i g h b o r sites. Note t h a t we have w r i t t e n two defect r e a c t i o n s for t h e c a s e of v a c a n c y f o r m a t i o n in 2.4.1. P y r o p h o s p h a t e is a n i n s u l a t o r a n d t h e f o r m a t i o n of a p o s i t i v e l y - c h a r g e d v a c a n c y is m u c h m o r e ~likely t h a n t h e v a c a n c y p l u s a free positive c h a r g e .
The point defect
60
THUS, ALTHOUGH MORE THAN ONE DEFECT REACTION MAY BE APPLICABLE TO A GIVEN SITUATION, ONLY ONE IS USUALLY FAVORED BY THE PREVAILING THERMO DYNAMIC AND ELECTRICAL CONDITIONS. II.- L i t h i u m Ni0b~tr L i t h i u m niobate, LiNbOs, is a photorefractive material, discovered in 1 9 6 6 (Ashkin et al). T h a t is, it is an electroSptic m a t e r i a l in w h i c h the indices of refraction c a n be c h a n g e d by an applied electric field. It is used in optical devices w h i c h employ its nonlinear optical a n d e l e c t r o S p t i c p r o p e r t i e s . As a single crystal, LiNb03 has high t r a n s p a r e n c y to e l e c t r o m a g n e t i c radiation. If a laser b e a m is d i r e c t e d down the length of s u c h a crystal, its f r e q u e n c y is doubled. YAG:Nd 3§ i.e.- Y3A15OI2:Nds+, is a c o m m o n l y u s e d laser crystal, w h o s e emission lies at 10,600 A. (near-infrar e d radiation). The c o m b i n a t i o n of a LiNbO3 crystal a n d YAG:Nd s§ p r o d u c e s a laser b e a m at 5,300 A (green light). If a second crystal is also i n c o r p o r a t e d in the optical setup, radiation at 2 , 6 5 0 A (ultraviolet radiation) is obtained. Although the radiation f r e q u e n c y c h a n g e s , so does the i n t e n s i t y of light p r o d u c e d . Losses of 100 times or more are c o m m o n . However, this d e t r i m e n t a l factor is overcome by i n c r e a s i n g the power of the laser b e a m . Fortunately, LiNbOs has a high r e s i s t a n c e to d a m a g e by a laser b e a m , unlike m a n y o t h e r similar crystals. It gains its unique c h a r a c t e r i s t i c s b e c a u s e the crystal s t r u c t u r e has "ouflt-in" d e f e c t s . LiNbOs is a ferroelectric s t r u c t u r e related to t h a t of the cubic p e r o v s k i t e (CaTiOs) s t r u c t u r e . T h a t is, electric polarization of the lattice e l e c t r o n s o c c u r s u p o n application of an electric field. Although a voltage will i n d u c e s u c h polarization, so will the electric vectors of a b e a m of light, p a r t i c u l a r l y t h a t of a laser beam. However, LiNbOa consists of d i s t o r t e d o x y g e n - o c t a h e d r a s h a r i n g faces so t h a t a p l a n a r hexagonal a r r a n g e m e n t results. The pile-up of the o c t a h e d r a along the p e r p e n d i c u l a r direction, caxis, follows the cation s e q u e n c e "Li, Nb a n d v a c a n t alte". The p o i n t s y n m l e t r y group is C~ (see 1.4.14. a n d 1.4.16 of the first chapter) w i t h the trigonal axis along the cation rows. It is very close to a C3v (3m) configuration.
2.4 Some applications for defect chemistry
61
The following d i a g r a m s h o w s t h e a r r a n g e m e n t of t h e t h r e e LiNbOa s t r u c t u r e .
ions in t h e
2.4.2.-
Note t h a t we have a s t r u c t u r e w i t h a '~)uilt-in" c r y s t a l defect, a vacancy. B o t h t h e l i t h i u m a n d n i o b i u m c a t i o n s are in a n o c t a h e d r a l c o o r d i n a t i o n . In fact, t h e two ions, Li § a n d Nb 5+, have n e a r l y the s a m e r a d i u s a n d o c c u p y o c t a h e d r a l sites w i t h t h e s a m e Csv s y m m e t r y . The l i t h i u m d e f i c i e n c y in c o n g r u e n t c r y s t a l s is a c c o m m o d a t e d b y m e a n s of Nbu a n t i - s i t e s a n d Nb 5+ v a c a n c i e s in a relative c o n c e n t r a t i o n neutrality.
Note
that
many
t h a t g u a r a n t i e s overall
electrical
physical
properties
depend
upon
s t o i c h i o m e t r y , e.g.- Curie t e m p e r a t u r e , p a r a m e t e r s a n d p h o t o r e f r a c t i v e yield.
absorption
spectra,
lattice
However, it h a s b e e n f o u n d t h a t i m p u r i t i e s play a m a j o r role in t h e o p e r a t i o n of a f r e q u e n c y - d o u b l i n g c r y s t a l like LiNbOs. M a n y i m p u r i t i e s have ionic radii similar to t h a t of l i t h i u m . T h e y s u b s t i t u t e at Li§ s i t e s r a t h e r t h a n Nb s§ sites (possibly b e c a u s e t h e NbO 4 c o o r d i n a t i o n is s t r o n g e r at t h e n i o b i u m site). A m o n g t h e s e are: M n 2+, Fe 3+ a n d N i 2+. As we have a l r e a d y seen, s u b s t i t u t i o n of s u c h m u l t i v a l e n t c a t i o n s on a m o n o v a l e n t s i t e results
in
lattice
compensation
such
as
oxygen
vacancies
and
the
f o r m a t i o n of c o l o r - c e n t e r s at t h e o x y g e n v a c a n c i e s . T h e s e i n t e r f e r e w i t h the photorefractive properties
of s u c h defect c r y s t a l s since t h e e a s e of
electric p o l a r i z a t i o n of t h e l a t t i c e is i m p a i r e d d u r i n g use. A l t h o u g h , as w e
The point defect
62
will see, the crystal-growing p ro ces s is also a purification process, it is not able to exclude all of t h e i m p u r i t i e s as the crystal grows. III. Bubble M e m o r i e s "Bubble m e m o r y " is the t e r m applied to the device w h i c h uses a "soft" m a g n e t i c m a t e r i a l to carry information. If a f e r r o m a g n e t i c film s u c h as e u r o p i u m gallium g a r n e t is grown epitaxially u p o n a suitable s u b s t r a t e s u c h as gadolinium gallium garnet, i.e.- Gd3 Ga5 O11 (= GGG), it f o r m s m a g n e t i c d o m a i n s in w h i c h the electron spins of the cations are aligned in the s a m e direction in the s a m e domain. This is s h o w n in the following: 2.4.3.- A F e r r o m a g n e t i c Film Grown EpitaxiaUy on
r
ARROWS DIRECTION
INDICATE
9 ....
l;~.; ~
IS/~4
r/ss/4
"'""
"
~,,,,
I~.;./I
9 ....
v##/si WsS~'~l v////I
rr // ///Z/ 4/ ~ l "
"
.... "
i;T:I
Ir162 MAGNETIZATION
|V /.... I ///I
IT#~##4 r/s//4 ITSSSS4
.... " "
OF
.... 9 99 I T. s. .s.s / 4 4
4
V/##4
4
r ....4
1
I;~;I Ir r////,l
1.....
I./.~././I
Ws//4 v,,~4 VSSS4 v/l/4
V/ss'4
rf~I.~m r;m;~
1
I~l;l
I:~1~3
v////m
Vss/,T
Magnetic F iIm
I
rd~ir11,F,r1,r162 SubstPate
The following diagram, given as 2.4.4. on the next page, illustrates how t h e s e would look u n d e r polarized light (the Faraday effect) using c r o s s e d Nicol polarizers. (The b lack an d white polarity).
p a r t s are d o m a i n s of o p p o s i t e
W h e n a m a g n e t i c field is applied, with the field vector horizontal to t h e film, the d o m a i n s collapse to form s e p a r a t e d cylinders within the film, as shown. T h e s e a p p e a r to be "bubbles" w h e n viewed from the top, h e n c e the n a m e . The bubbles t h e n b e c o m e mobile u n d e r the influence of a s e p a r a t e electric field a n d will move. Actually, the electric field causes t h e dom a i n - wa l l to collapse by a spin-flip m e c h a n i s m , while the c y l i n d e r volume is m a i n t a i n e d by the m a g n e t i c field.
2.4 Some applicationsfor defect chemistry
2.4.4.[M~gnetic
I ~ b b l e s as V i ~ d
63
f r o m t h e T o p of t h e Film in P o l a r i z e d L i g h t i
00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 ~-- 50~ ---o Magnetic Domains
Magnetic Bubbles
T h i s c a u s e s t h e a p p a r e n t m o v e m e n t of t h e wall. Bubbles d e n s i t i e s as h i g h as 107 p e r c m 2, a n d bubble velocities of u p to 10 3 - 1 0 4 c m . / s e c , h a v e b e e n r e p o r t e d . Obviously, b u b b l e velocity d e p e n d s u p o n E, t h e e l e c t r i c field s t r e n g t h , as well as t h e c o m p o s i t i o n of t h e epitaxially g r o w n film. T h e u s e of a v a p o r - p h a s e - d e p o s i t e d m e t a l grid u p o n t h e s u r f a c e of t h e film s e r v e s to s w i t c h b u b b l e s from site to site. T h e p r e s e n c e (1) or a b s e n c e (0) of a b u b b l e is d e t e c t e d b y p o l a r i z e d light b e a m s . T h u s , i n f o r m a t i o n s t o r e d , a n d r e t r i e v e d , on a "chip" in b i n a r y language. However, t h e epitaxial film m u s t be defect-free. ~
c a n be
is difficult to grow as
a defect-free single crystal. F r e n k e l defects a p p e a r , a n d give rise to l i n e dislocations. W h e n t h e c r y s t a l is c u t into wafers,
t h e line d i s l o c a t i o n s
r e m a i n a n d c a n be r e v e a l e d by c h e m i c a l p o l i s h i n g in h o t p h o s p h o r i c acid. T h e r e is a s t r o n g c h a n c e t h a t t h e s e d i s l o c a t i o n s will be p r o p a g a t e d i n t o t h e epitaxial Film w h e n it is d e p o s i t e d .
The
resulting film-defects
then
"pin" one or m o r e b u b b l e s to one location on t h e film, m a k i n g it ( t h e m )
The point defect
64
unusable. In addition, a lattice m i s m a t c h t e n d s to a c c e n t u a t e the defects of the ~ s u b s t r a t e onto the Film grown on it. The ao lattice p a r a m e t e r for ~ is 12.53 A (this is the length of the side of the cube enclosing t h e 19 a t o m s ) w h i l e t h a t of Eu3 Ga5 O l l is 12.48 A. This m i s m a t c h of lattice p a r a m e t e r is about 50x g r e a t e r t h a n t h a t d e s i r e d so t h a t a c o m p o s i t i o n s u c h as Y2.45 Eu5.5 Fe3.8 Gal.2 O12 is generally m o r e suitable to c o r r e c t l y m a t c h t h e lattice p a r a m e t e r of CW_~. In this way, defects p r e s e n t in t h e s u b s t r a t e c a n be avoided in the epitaxially grown • m , if p r o p e r g r o w t h c o n d i t i o n s are m a i n t a i n e d . Nevertheless, a l t h o u g h the "bubble-memory" a p p r o a c h to h i g h e r d e n s i t y m e m o r i e s in c o m p u t e r s was t h o u g h t to hold m u c h promise, the p r o b l e m s a s s o c i a t e d with intrinsic defects in the film a n d those involving the s u b s t r a t e proved to be too d a u n t i n g a n d today t h e b u b b l e - m e m o r y a p p r o a c h h a s b e e n a b a n d o n e d in favor of m o r e advantageous m e t h o d s . IV. Calcium Sulfide P h o s p h o r L e n a r d in 1928 r e p o r t e d
on a c a l c i u m sulfide p h o s p h o r ,
having
Actually,
a blue
emission.
he
probably
i.e.- CaS:Bi 3+,
prepared:
CaS:Bi3§
initially. As you c a n see, the divalent cation site would have to be o c c u p i e d by a negative vacancy since charge c o m p e n s a t i o n dictates t h a t the vacancy will appear. L e n a r d found t h a t the emission i n t e n s i t y of CaS:Bia§ was very low. It is likely t h a t the V site is a color c e n t e r w h i c h d i s s i p a t e d m o s t of the excitation energy. He found t h a t the use of chloride fluxes greatly i m p r o v e d t h e emission i n t e n s i t y a n d t h a t the use of KCI p r o d u c e d the b r i g h t e s t p h o s p h o r . He c o n c l u d e d t h a t K§ was a coactivator, i.e.- as CaS:Bia§
+ , since he u s e d first u s e d NaCI as a flux d u r i n g its p r e p a r a t i o n
a n d t h e n KCI. Even today, the p h o s p h o r is r e f e r r e d to in t h a t m a n n e r a l t h o u g h it is obvious t h a t the c h a r g e c o m p e n s a t i o n m e c h a n i s m is m o s t likely the c o r r e c t m e c h a n i s m in the formation of the p h o s p h o r . This c o n c l u d e s o u r c o n s i d e r a t i o n of defect applications. Let us now c o n s i d e r a m o r e m a t h e m a t i c a l a p p r o a c h to the d e s c r i p t i o n of p o i n t defects. It h a s b e e n said: "If you c a n n o t calculate the p r o p e r t i e s of any given theory, you really do not u n d e r s t a n d it".
2.5 Thermodynamics of the point defect
65
2.5. -THERMODYNAMICS OF THE POINT D E F E C T We shall u s e two a p p r o a c h e s to derive s o m e w o r k i n g values for t h e p o i n t defect in solids, n a m e l y t h a t of S t a t i s t i c a l M e c h a n i c s Thermodynamics
of Defects. T h e r e
a n d t h a t of t h e
are t h o s e w h o have s o m e f a m i l i a r i t y
w i t h s t a t i s t i c a l m e c h a n i c s . For o t h e r s , s o m e e x p l a n a t i o n is due. S t a t i s t i c a l M e c h a n i c s as a discipline w a s originally d e r i v e d in t h e early 1920's w h e n it w a s realized t h a t one h a d to deal w i t h large p o p u l a t i o n s of a t o m s o r m o l e c u l e s in various e n e r g y s t a t e s ( p a r t i c u l a r l y g a s e s at t h a t time). T h e s e s t a t e s arise b e c a u s e e a c h m o l e c u l e , for e x a m p l e , is v i b r a t i n g in a m a n n e r slightly different t h a n its n e i g h b o r . W h a t we d e s c r i b e
as t h e e n e r g y s t a t e for a given set of c o n d i t i o n s
is
actually t h e average of t h a t of a B o l t z m a n n p o p u l a t i o n . The discipline b e s t s u i t e d for h a n d l i n g s u c h a s y s t e m is s t a t i s t i c s , h e n c e t h e n a m e . T h e approach
used
for
manipulating
molecular
populations
in
Statistical
M e c h a n i c s is quite involved, a n d we shall t o u c h very briefly on t h e m a t h e m a t i c s involved. We will first d e s c r i b e e a c h of t h e s e a p p r o a c h e s s e p a r a t e l y a n d t h e n a c o m b i n e d version. Hopefully, t h i s will aid in y o u r u n d e r s t a n d i n g of t h e two m e t h o d s of d e t e r m i n i n g the effect of the p o i n t defect u p o n t h e p r o p e r t i e s of t h e solid. I. S t a t i s t i c a l M e c h a n i c s A p p r o a c h T h e l a n g u a g e of S t a t i s t i c a l M e c h a n i c s evolved over a c o n s i d e r a b l e p e r i o d of time. For e x a m p l e , t h e t e r m " e n s e m b l e " is u s e d to d e n o t e a s t a t i s t i c a l p o p u l a t i o n of m o l e c u l e s ; "partition function" is t h e integral, over p h a s e s p a c e of a s y s t e m , of t h e e x p o n e n t i a l of {-E I kT} [where E is t h e e n e r g y of t h e s y s t e m , k is B o l t z m a n n ' s c o n s t a n t , a n d T is t h e t e m p e r a t u r e in ~ F r o m t h i s "function", all of t h e t h e r m o d y n a m i c f u n c t i o n s c a n be d e r i v e d . T h e definitions t h a t we shall n e e d are given as follows in 2.5.1. on t h e n e x t page. Here, Nj is t h e s u m of t h e individual a t o m s t i m e s a n e n t r o p y factor a n d 1~I is t h e s u m of all of t h e N j 's.
The point defect
66
2 . 5 . 1 . - S t a t i s t i c a l M e c h a n i c s Definitions N e e d e d W
_
thermodynamic probability
G
----
e n s e m b l e of r e l a t e d a t o m s ( m o l e c u l e s )
Nj
----
~ m
1~I
----
~. Nj
Q
--
Partition Function
U s i n g t h e s e , we c a n derive t h e following e q u a t i o n for t h e total e n e r g y of a n y given s y s t e m : E T = total e n e r g y of s y s t e m = 1~I E(ave.) = ~ Nj Ej
2.5.2.-
where
Nj
is t h e total n u m b e r of a t o m s (molecules)
involved. Using t h e
m e t h o d s of S t a t i s t i c a l M e c h a n i c s , we c a n derive b y p r o b a b i l i t y r e l a t i o n s : 2.5.3.-
In
these
W
=
Q
=
Q
=
equations,
F is
A F/h3 ~ r
Nj !
exp- Ej / k T
{for s t a t e s )
]~ exp- Ej / k T {for levels)
a so-called
"partition
coefficient",
~
is
a
d e g e n e r a c y , k is B o l t z s m a n n ' s c o n s t a n t , a n d T is in d e g r e e s Kelvin. T h e first e q u a t i o n in 2.5.3. is a s t a t i s t i c a l m e c h a n i c a l definition of w o r k , w h e r e a s t h e l a s t two d e s c r i b e total e n e r g y s t a t e s . Having t h e s e d e f i n i t i o n s and
equations
allows
us
to
define
point
defects
from
a Statistical
Mechanical viewpoint. II, S c h o t t k y a n d F r ~ n k e l D e f e c t s C o n s i d e r a p l a n e n e t h a v i n g N sites, of w h i c h N L are lattice sites, Nv are v a c a n c i e s , a n d N i are I n t e r s t i t i a l s . (Note t h a t we do not c o n s i d e r c h a r g e at t h e sites for t h e m o m e n t ) . equations:
U s i n g t h e s e , we have the following two
2.5 Thermodynamics of the point defect 2.5.4.-
NL
=
2.5.5.-
~ Ni
=
N + Nv
67
- Ni
ct N L
E q u a t i o n 2.5.4. h o l d s if s o m e fraction of Ni
is a s s o c i a t e d w i t h N L , t h e
n u m b e r of lattice sites, i.e.- F r e n k e l defects. A c c o r d i n g to t h e B i n o m i a l T h e o r e m , we c a n c o m b i n e p a i r s of t h e s e sites as: 2.5.6.-
W ( N v , NI)
=
Nit /Nv!(NL-NV)
2.5.7.-
W i ( N i , aNl)
=
r
Note
that
these
equations
are
simply the
! / (r
!
[
Combinatorial Equation
as
a p p l i e d to t h e s e two s e t s of defects. We c a n set u p a p a r t i t i o n f u n c t i o n ( s e e t h e above definition), u s i n g e q u a t i o n 2.5.3. We a p p l y t h i s to t h e S c h o t t k y a n d F r e n k e l defects as e x a m p l e s : 2.5.8.- SCHOTTKY:
Qv (T)
=
2.5.9.- FRENKEL:
Qi (T)
=
Wi(Nv, NL) exp -Nv Ev / k T W i ( N i , aNL} exp -Ni Ei ! k T
We n o w solve t h e s e e q u a t i o n s u s i n g c e r t a i n a p p r o x i m a t i o n s e m p l o y e d in Statistical
Mechanics,
including
n u m b e r s . If t h e e n e r g y r e q u i r e d
Stirling's
Approximation
for
large
to f o r m v a c a n c i e s a n d I n t e r s t i t i a l s
is
g r e a t e r t h a n kT, a n d ff t h e defects are only a fraction of t h e total n u m b e r of lattice sites, t h e n we get: 2.5.10.-
SCHO~-
Nv
=
N exp -Ev / k T
2.5.1 I.-
FRENKEL-
Ni
=
r N exp - Ei / k T
Note t h a t we n o w have B o l t z m a n n d i s t r i b u t i o n e q u a t i o n s for e a c h type of defect., a n d t h a t t h e energy, E, to f o r m t h e defect is like a n a c t i v a t i o n energy. T h e fraction of defects p r e s e n t , e i t h e r Nv / N e x p o n e n t i a l f u n c t i o n of t h i s activation e n e r g y .
or
N i / N , is an
The point defect
68
This l e a d s to t h e following c o n c l u s i o n : It is c l e a r w i t h o u t a doubt t h a t b o t h S c h o t t k y a n d F r e n k e l d e f e c t s are t h e r m a l in origin. Although
you
may
have
wondered
why
we
considered
Statistical
M e c h a n i c s at all in r e l a t i o n to the p o i n t defect, the above o b s e r v a t i o n is critical to o u r u n d e r s t a n d i n g of the f o r m a t i o n of p o i n t defects in solids. It is a fact t h a t in m a n y c a s e s the activation e n e r g y r e q u i r e d
to f o r m
v a c a n c i e s a n d / o r i n t e r s t i a l s is a p p r o x i m a t e l y t h a t of r o o m t e m p e r a t u r e : 2.5.12.-
E v , Ei
,,
kT
It is t h i s m e c h a n i s m w h i c h p r o d u c e s defects in the lattice. T e m p e r a t u r e s slightly above r o o m t e m p e r a t u r e will p r o d u c e defects in the solid, e v e n t h o u g h t h e r a t e of p r o d u c t i o n m a y be e x t r e m e l y slow b e l o w about 500 ~ K. T h e c o n c l u s i o n t h a t w e r e a c h is t h a t d e f e c t f o r m a t i o n is favored in t h e solid. It is m u c h m o r e difficult to obtain a ' ~ e r f e c t " solid, so t h a t the defect-solid results. We c a n s u m m a r i z e this statistical m e c h a n i c a l a p p r o a c h to the solid s t a t e in t h a t it gives u s a m e t h o d to evaluate the fraction of i n t r i n s i c d e f e c t s p r e s e n t u n d e r specified c o n d i t i o n s , a n d also gives u s a m e a s u r e of t h e deviation f r o m absolute s t o i c h i o m e t r y . If we c a n evaluate the energy of f o r m a t i o n of v a c a n c i e s a n d i n t e r s t i t i a l s at a given t e m p e r a t u r e for a given c r y s t a l lattice, t h e n we c a n calculate t h e n u m b e r s of defects f o r m e d at any other temperature. III - Defect T h e r m o d y n a m i c s The r e a s o n w h y we first i n v e s t i g a t e d t h e Statistical M e c h a n i c s a p p r o a c h to defect f o r m a t i o n is t h a t it gives u s a good basis for u n d e r s t a n d i n g t h e a p p l i c a t i o n of c h e m i c a l t h e r m o d y n a m i c s to the defect solid state.
2.5 Thermodynamics of the point defect
69
We begin this a p p r o a c h by defining the total n u m b e r s of defects p r e s e n t as N d. In addition, we n e e d t h e following definitions2.5.13.-
Definitions N e e d e d to Define the T h e r m o d y n a m i c A p p r o a c h ENTROPY OF DEFEC'I~ ENTHALPY OF DEFECTS FREE ENERGY OF DEFECTS
= =
Sd
=
Gd
Hd
According to t h e laws of t h e r m o d y n a m i c s , we c a n write2.5.14.-
AG
=
AH
- ATS
Note t h a t we are u s i n g the G ~ b s free e n e r g y r a t h e r t h a n t h e H e l m h o l t z free e n e r g y at this point. For the defect solid, we m u s t define f r e e - e n e r g y in t e r m s of the f r e e - e n e r g y of t h e p e r f e c t solid, G o , as r e l a t e d to t h e f r e e - e n e r g y of t h e d e f e c t solid, vis: 2.5.15.-
AG
=
Gd
-
Go
Note t h a t due to i n c r e a s e d entropy, the e n e r g y of the defect
s t a t e is
h i g h e r t h a n t h a t of the perfect solid. For this reason, we m u s t d i s t i n g u i s h b e t w e e n t h e s o u r c e s w h i c h c o n t r i b u t e to the total entropy, a n d m u l t i p l y by t h e n u m b e r of defects p r e s e n t . The applicable e q u a t i o n is: 2.5.16.-
Gd
= Go +
Nd Hd - T Nd A Svib. - T A Sconfig.
where
ASmb. is t h e e n t r o p y c h a n g e c a u s e d by c h a n g e s in vIlmatlomal f r e q u e n c y w i t h i n t h e solid by t h e p r e s e n c e of d e f e c t s and A Sconfig. is t h e c h a n g e of e n t r o p y c a u s e d b y c o n f l ~ ~ t i o n a l
c h a n g e s in t h e vicqniW
defects. This a s p e c t of defect solid s t a t e c h e m i s t r y h a s b e e n t h o r o u g h l y s t u d i e d a n d the c h a n g e in the various q u a n t i t i e s of 2.5.16., as a function of n u m b e r s of defects, is s h o w n in t h e foUowing d i a g r a m :
The point defect
70
2.5.17.Numbers of Defects as a Function of Gibbs Free Energy
Nd&H
ol
-NdT ~
d
Config.
Gd I I
~ - Hlntr|nsi
I
Nd
c IL
"
You will n o t e t h a t w h e n it is possible to plot G d as a f u n c t i o n of G, t h e Gibbs free energy, a m i n i m u m o c c u r s w h i c h is the n u m b e r of defects at t h e given t e m p e r a t u r e for t h e given solid. Note also t h a t t h e e n t h a l p y of d e f e c t s i n c r e a s e s c o n t i n u o u s l y as a f u n c t i o n of N d while b o t h of t h e e n t r o p y - c o n t r i b u t i o n s d e c r e a s e . The n e t effect is a m i n i m u m value for G d , at w h i c h p o i n t we c a n get t h e n u m b e r of i n t r i n s i c defects p r e s e n t . T h u s , we have s h o w n again t h a t t h e r e is a specific n u m b e r of i n t r i n s i c d e f e c t s present
in
the
solid
at
any
given
temperature,
this
time
by
Thermodynamics. Let u s n o w consider a combined (Statistical Mechanics and T h e r m o d y n a m i c ) a p p r o a c h to t h e p r o b l e m of c a l c u l a t i n g defect n u m b e r s . IV,- C o m b i n e d A o o r o a c h to Defect F o r m a t i o n In t h i s m e t h o d , we will d e l i n e a t e the e x a c t p r o c e d u r e u s e d to d e r i v e
2.5 Thermodynamics of the point defect equations
for
specific
defects
in
the
solid.
To
71 show
how
this
is
a c c o m p l i s h e d , we let N L be t h e t o t a l n u m b e r of lattice sites a n d Nv t h e n u m b e r of v a c a n c i e s . T h e w a y t h a t N v v a c a n c i e s c a n b e a r r a n g e d u p o n N L lattice s i t e s is given b y c o m b i n a t o r i a l s t a t i s t i c s as: 2.5.18.-
W =
~ N L
! ~
Nv!(Nv-Nv)
!
The cortfigurational or m i x i n g e n t r o p y will be d e f i n e d by: 2.5.19.-
ASM= k l n W
U s i n g S t i r l i n g ' s a p p r o x i m a t i o n for large n u m b e r s (this m e t h o d is u s e d e x t e n s i v e l y in S t a t i s t i c a l M e c h a n i c s ) , we c a n get: 2.5.20.-
AS M = k [N L I n N L - N v l n N v - ( N L - N v) In (N L - N v) ]
You will n o t e t h a t we h a v e t h e c h a n g e in e n t r o p y as a f u n c t i o n of t h e d i f f e r e n c e s b e t w e e n n o r m a l lattice s i t e s a n d v a c a n c i e s . S i n c e we k n o w t h a t N L >> N v, we c a n w r i t e for t h e e n t r o p y of m i x i n g : 2.5.21.and
AS M ~ [N L I n N L - N v l n N v ] ASM=- k[N v InNv/NLI=-RXv
InXv
w h e r e X v is t h e f r a c t i o n of defects actually p r e s e n t . As e x p e c t e d , for n o n i n t e r a c t i n g defects, t h e e n t r o p y of m i x i n g is ideal so t h a t t h e free e n e r g y of t h e s y s t e m c a n be w r i t t e n as: 2.5.22.-
AFv = N v (AE v - T ASVv) + RT [N L- N v / N L (In N L- N v / N L ) + N v /N L I n - N v / N O ]
w h e r e AShy is t h e c h a n g e in v i b r a t i o n a l e n e r g y of t h e lattice a r i s i n g f r o m t h e c h a n g e in v i b r a t i o n a l f r e q u e n c y a r o u n d t h e v a c a n t lattice site. If w e m i n i m i z e t h e free energy, i.e.2.5.23.and:
0Fv/ON v = 0 = A E v - T A S v v ) + R T I n ( N v / N L - N v) l n ( N v / N L - N v} = l n X v = A E v - T A S ~ v ) / R T
The point defect
72
T h u s , we c a n w r i t e for t h e atomic fraction of v a c a n c i e s p r e s e n t : 2.5.24.-
Xv = exp - AF~ /RT
You m a y w o n d e r
w h y we
= e ' a E v / RT
have e x a m i n e d
this m e t h o d
for specifying
v a c a n c i e s in g e n e r a l t e r m s . The r e a s o n is t h a t we c a n apply the s a m e m e t h o d to t h e p r o b l e m of p o i n t - d e f e c t pairs. This is i n t e n d e d to help you u n d e r s t a n d h o w t h e s e v a r i o u s types of defects in the solid arise. a. IntErstitial A t o m s The p r o c e s s of c r e a t i n g a n i n t e r s t i t i a l is j u s t the o p p o s i t e of c r e a t i n g a vacancy. If t h e r e are N, possible i n t e r s t i t i a l p o s i t i o n s in t h e lattice, a n d if AFi is t h e free e n e r g y n e e d e d to m o v e the a t o m into its i n t e r s t i t i a l position, t h e n t h e i n t e r s t i t i a l c o n c e n t r a t i o n at e q u i l i b r i u m will be: 2.5.25.-
N~ / N, = X~ = exp (- AF~ / k T )
The s a m e a p p r o a c h applies to F r e n k e l pairs. b. F r e n k e l p a i r s We have, in this case, b o t h a v a c a n c y a s s o c i a t e d with an i n t e r s t i t i a l atom. Using t h e a p p r o a c h s h o w n in 2.5.18., we have: 2.5.26.-
W =
NL~ N v ! ( N L - N v) !
X
NIJ N i ! ( N x - N i) 7
Now, we have two c o m b i n a t o r i a l f u n c t i o n s , one for the v a c a n c i e s and o n e for the i n t e r s t i t i a l s . Since the n u m b e r s of vacancies a n d interstials are equal, i.e.- N F = N L = N i , we c a n use the s a m e s t e p s given in 2.5.19., ,20, 21., a n d 2 . 5 . 2 2 . to get: 2.5.27.-
ASM= RT [(NLIn NL) " (2NF InNv) + (Nxln N~) - { ( N L - N F) l n ( N ~ - N F) ln(N L-NF)}]
2.5 Thermodynamics of the point defect
73
Minimizing t h e free e n e r g y as before, we c a n w r i t e for t h e F r e n k e l pairs: 2.5.28.-
NF = (N L Nx )112 exp (- ttFiv ! 2RT)
Note t h a t ZkF applies to b o t h i n t e r s t i t i a l a n d v a c a n c y sites. Using 2 . 5 . 2 3 . a n d 2 . 5 . 2 4 . , we c a n get the fraction of F r e n k e l defects as: 2.5.29.-
X~ --- exp (AFI v 12RT
since the free e n e r g i e s are a p p r o x i m a t e l y t h e s a m e , i.e.- ttF i v ~- ZXFI + ~ v
c. S c h o t t ~ The
Defects
c a l c u l a t i o n of S c h o t t l ~
defects
follows the
same method
given. We u s e cv for c a t i o n - v a c a n c y a n d av for t h e
associated
already anion
vacancy. The free e n e r g y of this defect is t h e n -
V
2 . 5 . 3 0 . - AFsh = Ncv AEcv + NAV AEAv - T (Ncv ASVcv + NAy AS Av) RT In
N
~
X
N c v [ ( N -Ncv) [
N
!
NAv!(N -N^v) [
Again, we have two c o m b i n a t o r i a l factors, o n e e a c h for t h e a s s o c i a t e d c a t i o n v a c a n c i e s . Here, t h e total n u m b e r of lattices sites, N >> Nsh = Ncv = N^v. Minimizing t h e free e n e r g y gives: 2.5.31.and:
Nsh / (N - Nsh )
= exp (-AFsh I 2 R T )
Xsh = exp (-AFsh I 2 R T )
You wiU n o t e t h a t this a p p r o a c h u s e s Statistical M e c h a n i c s to a p p r o x i m a t e t h e t h e r m o d y n a m i c c o n s t a n t s for t h e n u m b e r of defects p r e s e n t . V. Defect E q u i l i b r i a J u s t as c h e m i c a l r e a c t i o n s c a n be d e s c r i b e d a n d c a l c u l a t e d in t e r m s of
The point defect
74 thermodynamic
constants
and
chemical
equilibria,
so
can
we
also
d e s c r i b e d e f e c t f o r m a t i o n in t e r m s of e q u i l i b r i a . T h i s is g i v e n as follows: 2.5.32-
L a w of M a s s A c t i o n :
bB+cC
K =
-~ dD + e E 9
adD aeE / ~tbB acc
Using this equation, we can calculate the numbers
of d e f e c t s for v a r i o u s
d e f e c t s in t h e MX c r y s t a l as: 2.5.33.-
F r e n k e l D e f e c t s [for t h e MX c r y s t a l ] Mx
2.5.34.-
~
Mi
+
VM
"
KF
-
~tMi a v M [ aM
S c h o t t ~ y D e f e c t s tfor t h e MX c r y s t a l ] MX
~
VM + V x
Note that we have specified
,
KSh
- a v M a v M / aM X
the equilibrium constants
activity, a, of t h e m m o c ~ t e d defe 9
in t e r m s
of t h e
W e c a n also w r i t e t h e r m o d y n a m i c
e q u a t i o n s for t h e s e d e f e c t s 2.5.36.-
Chemical Thermodynamics AG = AH - T AS = - RT In K 9 K =
2.5.37.-
exp
AS / R 9 e x p - AH / R T
Defect Thermodynamics Kd = e x p
ASd I R " e x p - AHd I R T
w h e r e d r e f e r s to t h e specific d e f e c t . We may summarize
the knowledge
we have already developed
for t h e
Schottky and Frenkel defects: I. W e h a v e s h o w n b y S t a t i s t i c a l
Mechanics
that we can calculate
n u m b e r s of d e f e c t s p r e s e n t at a g i v e n t e m p e r a t u r e .
2.6 Defect equilibria in various types of compounds 2. T h e r e
is a n Activation E n e r g y for defect
75
formation.
In m a n y
c a s e s , t h i s e n e r g y is low e n o u g h t h a t defect f o r m a t i o n o c c u r s at, o r slightly above, r o o m t e m p e r a t u r e . 3. Defects m a y be d e s c r i b e d in t e r m s of t h e r m o d y n a m i c
constants
a n d equilibria. T h e
the
presence
vibrational frequencies
of d e f e c t s
changes
both
local
in t h e vicinity of t h e defect a n d t h e local
lattice c o n f i g u r a t i o n a r o u n d t h e d e f e c t . One q u e s t i o n we m a y logically a s k is h o w are we to k n o w w h a t t y p e s of defects will a p p e a r in a given solid? T h e a n s w e r to t h i s q u e s t i o n is g i v e n as follows: IT HAS B E E N FOUND:
"Yhere are t w o a s s o c i a t e d effects on a given solid w h i c h have o p p o s i t e e f f e c t s on s t o i c h i o m e t r y . U ~ R U y , o n e involves t h e cation site and t h e o t h e r t h e Rnlon site. B e c a u s e of t h e d i f f e r e n c e s in d e f e c t - f o r m a t i o n e n e r g i e s , t h e c o n c e n t r a t i o n of o t h e r d e f e c t s is usually n e g l i g ~ l e ' . T h u s , if F r e n k e l Defects p r e d o m i n a t e usually n o t p r e s e n t . applies for a,~ociater types
of
defects
in a given solid, o t h e r d e f e c t s a r e
Likewise, for t h e
Schottky
Defect.
Note
that this
d e f e c t s . If t h e s e are n o t p r e s e n t , t h e r e will still b e 2 present,
each
having
an
opposite
effect
upon
s t o i c h i o m e t r y . Thus, w e c o n c l u d e that i n t r i n s i c d e f e c t s usually o c c u r i n patl~,
This
conclusion
cannot
be
overemphasized.
The
following
d i s c u s s i o n s h o w s h o w t h i s o c c u r s in t h e r e a l w o r l d of d e f e c t s in solids. 2.6.Up
D E F E C T EQUILIBRIA IN VARIOUS TYPES OF COMPOUNDS to now,
we
have b e e n
concerned
with
the
MX c o m p o u n d
h y p o t h e t i c a l e x a m p l e of t h e solid state. We will n o w u n d e r t a k e concrete
examples
as f o u n d in
the
real
world,
using
the
as a more
concepts
d e v e l o p e d for t h e s i m p l e MX c o m p o u n d . For t h e s a k e of simplicity, w e r e s t r i c t o u r s e l v e s to b i n a r y c o m p o u n d s , t h a t is- o n e c a t i o n a n d o n e anion. An e x a m p l e of a t e r n a r y c o m p o u n d is ~ , w h e r e A a n d B are d i f f e r e n t cations, a n d S is a s m a l l w h o l e n u m b e r .
76
The point defect
Our example of a b i n a r y c o m p o u n d will be:
MXs We will d i s t i n g u i s h b e t w e e n four states for this h y p o t h e t i c a l c o m p o u n d , to wit: s t o i c h i o m e t r i c vs: n o n - s t o i c h i o m e t r i c non-ionized vs: ionized I. S t o i c h i o m e t r i c Bini~ry C o m p o u n d s of M ~ In the real world of defect chemistry, we find t h a t in addition to t h e simple defects, o t h e r types of defects appear, d e p e n d i n g u p o n the type of crystal we are dealing with. These m a y be s u m m a r i z e d as s h o w n in t h e following. According to o u r n o m e n c l a t u r e , VM is a vacancy at an M cation site, etc. The first five pairs of defects given above have b e e n o b s e r v e d e x p e r i m e n t a l l y in solids, w h e r e a s the last four have not. 2.6.1-
Defects in the MXs C o m p o u n d Schottky Frenkel Anti-Frenkel Anti-Structure Vacancy-Structure Structure-Vacancy Interstitial Interstitial-Structure Structure-Interstitial
P~$ OF D E F E C T S VM + V x VM + Mi
~q XM
+
+
Vx MX
VM + MX Vx + XM Mi + Xi MX +
Xi
Mi
XM
+
This a n s w e r s the h y p o t h e s i s p o s e d above, namely t h a t d e f e c t s in s o l i d s o c c u r in pairs. S t u d y these defect-pairs carefully so t h a t you b e c o m e familiar with them. T h e y r e p r e s e n t the type of s t r u c t u r e defects found in m o s t solids. We have now i n t r o d u c e d into our n o m e n c l a t u r e a d i s t i n c t i o n b e t w e e n s t r u c t u r e a n d a n t i - s t r u c t u r e defects. What this m e a n s is that s t a c k i n g faults can s o m e t i m e s result in XM a n d Mx defects, which are
]
CHAPTER 2 The Point Defect
There
are two types of defects a s s o c i a t e d w i t h p h o s p h o r s . One involves
controlled
point
defects
in
which
a
foreign
activator
cation
is
i n c o r p o r a t e d in the solid in defined a m o u n t s . The o t h e r involves line a n d point defects i n a d v e r t e n t l y f o r m e d in the solid s t r u c t u r e b e c a u s e of i m p u r i t y a n d e n t r o p y effects. This c h a p t e r will define a n d c h a r a c t e r i z e the n a t u r e of all of t h e s e p o i n t defects in the solid, t h e i r t h e r m o d y n a m i c s a n d equilibria. It will b e c o m e a p p a r e n t t h a t the type of defect p r e s e n t will d e p e n d u p o n the n a t u r e of the solid in w h i c h they are i n c o r p o r a t e d . T h a t is, the c h a r a c t e r i s t i c s
of the p o i n t defects
in a given p h o s p h o r
will
d e p e n d u p o n its c h e m i c a l c o m p o s i t i o n . Of necessity, this c h a p t e r is n o t i n t e n d e d to be exhaustive, a n d the r e a d e r is r e f e r r e d to the m a n y t r e a t i s e s c o n c e r n e d w i t h the p o i n t defect. 2.1. - TYPES OF POINT DEFECTS Let us n o w c o n s i d e r the defect solid from a g e n e r a l p e r s p e c t i v e . C o n s i d e r the case of s e m i - c o n d u c t o r s , w h e r e m o s t of the a t o m s are the same, b u t t h e total of the c h a r g e s is not zero. In t h a t case, the excess charge (n- o r p- type) is s p r e a d over the whole lattice so t h a t no single atom, or g r o u p of atoms, h a s a c h a r g e different from its n e i g h b o r s . However, m o s t inorganic solids are c o m p o s e d of c h a r g e d moieties, half of w h i c h are positive (cations) a n d half negative (anions). The total charge of t h e c a t i o n s equals, in general, t h a t of the anions. If an a t o m is missing, t h e lattice r e a d j u s t s to c o m p e n s a t e for this loss of charge. If t h e r e is an e x t r a a t o m p r e s e n t , the c h a r g e - c o m p e n s a t i o n m e c h a n i s m again m a n i f e s t s itself. A n o t h e r possibility is the p r e s e n c e of an a t o m w i t h a c h a r g e larger or s m a l l e r t h a n t h a t of its neighbors. In a given s t r u c t u r e , c a t i o n s are usually s u r r o u n d e d by anions, a n d vice-versa ( R e m e m b e r w h a t we said in C h a p t e r 1 wherein
it w a s s t a t e d t h a t m o s t s t r u c t u r e s
are o x y g e n - d o m i n a t e d ) .
T h u s , a cation w i t h an e x t r a c h a r g e n e e d s to be c o m p e n s a t e d by a like anion, or by a n e a r e s t n e i g h b o r cation w i t h a lesser charge. An e x a m p l e of
39
The point defect
78 number number
of s i t e s , w h e r e a s t h e l a s t five a r e b a s e d u p o n a n e x c e s s in t h e of s i t e s available. T h i s e x c e s s w e call "5". Note t h a t w e are n o t
s p e a k i n g of t h e r a t i o of c a t i o n s t o a n i o n s , i . e . - s t o i c h i o m e t r y , e x c e s s of c a t i o n s or a n i o n s to t h e n o r m a l
concentration
but0f
an
of c a t i o n s o r
anions. II. D e f e c t C o n c e n t r a t i o n s in M ~ s _ _ C o m p o u n d s It is of i n t e r e s t to b e able to d e t e r m i n e
t h e n u m b e r of i n t r i n s i c d e f e c t s in
a g i v e n solid. As w e h a v e s h o w n , p a i r s of d e f e c t s p r e d o m i n a t e in a n y g i v e n solid. T h u s , t h e n u m b e r of e a c h t y p e of i n t r i n s i c d e f e c t s , N i (M) or N i (X), will e q u a l e a c h o t h e r , F o r S c h o t t k y d e f e c t s in t h e MXs c r y s t a l , w e have: 2.6.2.-
Ni {VM )
=
This makes our mathematics
NI ( S V x ) simpler since we can rewrite
the Schottky
e q u a t i o n of T a b l e 2-1 as: 2.6.3.-
0 ~-~ Ni(VM) + SNi(VM)
Here, we have e x p r e s s e d the c o n c e n t r a t i o n
4- ctVi as t h e ratio of d e f e c t s to t h e
n u m b e r of M- a t o m s i t e s (this h a s c e r t a i n a d v a n t a g e s , as w e will see). We c a n t h a n r e w r i t e t h e d e f e c t e q u i l i b r i a e q u a t i o n s of T a b l e 2-1 in t e r m s of n u m b e r s of i n t r i n s i c d e f e c t c o n c e n t r a t i o n s . T h e s e are given as follows: 2 . 6 . 4 . - Eq~ilibri~tm C o n s t a n t s = F u n c t i o n of N u m b e r s of I n t r i n s i c D e f e c t s SCHO~:
KSh = Ni {S N i ) 2 = S s Ni S+I
FRENKEL:
KF
ANTI-FRENKEL:
KAF = Ni 2 / (S- Ni){cx- N i )
ANTI-STRUCT:
KAS = Ni 2 1 ( l - N i ) ( S - N i ) S KVS = S s (S + I )s+1. Ni2S+l i (S-Ni -S Ni )S
VAC. -STRUCT:
= Ni 2 / (I- Ni){cx- N i )
STRUCT-VAC:
KSV
= ( S + I ) S + I NiS+2 / (S - Ni - S Ni)
INTERSTITIAL:
KI
= S S Ni S+l / (r - N i - SNi) %S+I
S o m e of t h e s e e q u a t i o n s a r e c o m p l i c a t e d a n d w e n e e d to e x a m i n e t h e m in
2.6 Defect equilibria in various types of compounds m o r e d e t a i l so as to d e t e r m i n e
79
h o w t h e y a r e to b e u s e d . E q u a t i o n 2 . 5 . 1 6 .
given above shows that intrinsic defect concentrations
will i n c r e a s e w i t h
increasing temperature
a n d t h a t t h e y wiU b e low for h i g h e n t h a l p i e s
defect
arises because
formation.
This
the
entropy
effect
of
is a p o s i t i v e
e x p o n e n t i a l w h i l e t h e e n t h a l p y effect is a n e g a t i v e e x p o n e n t i a l . C o n s i d e r the following practical e x a m p l e : TI0
is c u b i c w i t h t h e NaCl s t r u c t u r e . A s a m p l e w a s a n n e a l e d at 1 3 0 0 ~
Density and X-ray measurements revealed that the intrinsic defects were S c h o t t k y in n a t u r e (VTi + V 0 ) a n d t h a t t h e i r c o n c e n t r a t i o n w a s 0 . 1 4 0 . I n t h i s c a s e , S = 1 so t h a t : KSh
2.6.5.-
- 0 . 0 1 9 6 = 2 x 1 0 -2
T h i s c r y s t a l is q u i t e d e f e c t i v e s i n c e 1 o u t of 7 Ti- a t o m - s i t e s
( 0 . 1 4 -I) is a
v a c a n c y , a n d l i k e w i s e for t h e o x y g e n - a t o m - s i t e s . A n o t h e r e x a m p l e is: CcI-I2
.
intrinsic
From
thermodynamic
defects
were
measurements,
Anti-Frenkel
it w a s
in n a t u r e ,
i.e.-
{Hi
has
the
found
that
the
+ VH ). A n
e q u i l i b r i u m c o n s t a n t w a s c a l c u l a t e d as: 2.6.6.-
KAF
at a t e m p e r a t u r e
=
3 . 0 X I 0 -4
of 6 0 0
~
This
compound
cubic
s t r u c t u r e w i t h o n e o c t a h e d r a l i n t e r s t i c e p e r Ce a t o m . T h e r e f o r e , a n d S = 2 for C e l l 2 . W e c a n t h e r e f o r e w r i t e : 2.6.7.-
kAF
=
Ni
=
or
Ni 2 /
(2-Ni)(1-Ni
2 . 4 x 10 -2
)
=
3.0
is v a c a n t .
a
= I,
x 1 0 -4
(600 ~
T h i s m e a n s t h a t 1 o u t of 4 2 h y d r i d e a t o m s is i n t e r s t i t i a l , hydride-atom-sites
fluorite
a n d 1 o u t of 8 4
The point defect
80
Let us review w h a t we have covered
concerning
stoichiometric
binary
compounds: I. We have s h o w n t h a t defects occur in pairs. The r e a s o n for this lies in t h e c h a r g e - c o m p e n s a t i o n principle w h i c h o c c u r s in all solids. 2.
Of
the
nine
defect-pairs
possible,
only
5
have
actuaUy
been
e x p e r i m e n t a l l y o b s e r v e d in solids. T h e s e are: Schottky, F r e n k e l , Anti- Frenkel, A n t i - S t r u c t u r e , V a c a n c y - S t r u c t u r e . 3. We have given d e f e c t - e q u a t i o n s for all nine types of defects, a n d t h e E q u i l i b r i u m C o n s t a n t (EC) t h e r e b y associated. However, t h e s e equilibria would r e q u i r e values in t e r m s
calculation of
of e n e r g y at each site,
values w h i c h are s o m e t i m e s difficult to d e t e r m i n e . A better
method
is to c o n v e r t
these
EC e q u a t i o n s to t h o s e involving
n u m b e r s of e a c h t y p e of i n t r i n s i c defect, as a ratio to an intrinsic cation o r anion. This allows us to calculate the actual n u m b e r of intrinsic d e f e c t s p r e s e n t in t h e crystal, at a specified t e m p e r a t u r e . III. Non-StoiChiomctri~ Binary C o m p o u n d ~ we
will now
extend
our treatment
of i n t r i n s i c
defects
to the
non-
s t o i c h i o m e t r i c n o n - i o n i z e d c o m p o u n d s , as r e p r e s e n t e d by: 2.6.9.-
MXs+
6
w h e r e 8 is a small i n c r e m e n t . The q u e s t i o n is: "How do we obtain nons t o i c h i o m e t r y in t h e solid?". Consider a c o m p o u n d g o v e r n e d by either or b o t h t h e following equilibria: 2.6.10.-
Xx
2.6.11.-
M(external phase) ~
One
example
might
~
be
I / 2 X2{gas)~
a halide
+ Vx
MM crystal
+ S V X + VMi which
has
become
non-
s t o i c h i o m e t r i c due to its being h e a t e d to a t e m p e r a t u r e sufficient to cause a small a m o u n t of the halide to b e c o m e volatile.
2.6 Defect equilibria in various types of compounds
81
A n o t h e r c a s e m i g h t be an oxide, h e a t e d in t h e p r e s e n c e of e x c e s s m e t a l , e.g.- ZnO + Zn. For a n o n - s t o i c h i o m e t r i c crystal, the c o n c e n t r a t i o n of e a c h point d e f e c t , in e a c h c o n j u g a t e pair, is no l o n g e r equal. If t h e r e is an e x c e s s of V M , X i , or XX, t h e n the c o m p o u n d will have a s u r p l u s of X (or deficiency of M, w h i c h is the s a m e thing) over t h e ideal s t o i c h i o m e t r i c c o m p o s i t i o n . T h i s is called a positive deviation from s t o i c h i o m e t r y . Conversely, for a n e g a t i v e deviation, t h e r e will be an e x c e s s of V x , M i , or M M . T h i s explains t h e p l u s a n d m i n u s in e q u a t i o n 2.6.9. In t e r m s of the above given defects, m a y be e x p r e s s e d as s h o w n in t h e following Table: TABLE 2-2. N o n - S t o i c h i o m e t r y , 8 , as a F u n c t i o n of Specific T y p e s of D e f e c t s in MXs + 5 Binary C o m p o u n d s Vacancy Formation Vx f r o m e x t e r n a l M
Defect E q u a t i o n M -~ MM + S[Vx] + (~Vi
Equilibrium Constant KV X= MM[Vx] s Vi a /M
VM froIn go~eotlk$ X
1 / 2X 2 --~ S i x + [VM] +(xVi
KVM=XxS[VM]Vi a / %
{if Xx ~ 1 a n d Vi =a,
then
1 / 2 X2 +(~VI ~ [Xi]
K m = [Xi] / ( ~- X4) p x 21/2
X-Interstitials
5 = S[VM]/
1/2S
1- VM}
or: {Px21/2 =1/Kxi- 8/(~-5} M-Interstitlals
M M + S X x + ( I +(~) Vi [Mil + S / 2 X2
KMi= [Ui ]px2S/2/MMXX S [(~ - M~ (t+a) or: Px2 S = KMi ((~(S + 6) + 5) 1+5 /(. ~) (S+6)a) l/S (from g a s e o u s X2)
X-Substituttonals
1 / 2(S+ 1)X2 -% SXx +r
Kx M =[XM] (~a / PX2 (S+1/2)
(from g a s e o u s X2 on
. Vi(l+a)
{or: p x 2(S+1/2) = 1/ KXM 9
( S + 1 ) X x + MM + a Vi
6/(l+S+ti) I/S+I } KMx= PX2((S+I)/2)
((S+I)/2) X2 + Mx
(1-Mx) S+I eta
an M-site) M-Substitutionals (gaseous X2 f o r m e d )
{or:
[Mx] /
Px2((S+l)/2)
(KMx) I/S+I
=
(S+ 1 )(S+d) /
[(-5)(S+ 1+6) s ] I/S+I )
The point defect
82
Note t h e various m e c h a n i s m s w h i c h give rise to the specific c o m b i n a t i o n s of defects. T h e s e m e c h a n i s m s have b e e n t h o r o u g h l y s t u d i e d as a function of specific c o m p o u n d s .
It is sufficient for us, at t h i s point, to observe
w h i c h defect e q u a t i o n s govern b o t h t h e equilibria s t o i c h i o m e t r y of the g e n e r a l c o m p o u n d , MXs + 8.
and
the
non-
The e q u a t i o n u s e d to calculate the n o n - s t o i c h i o m e t r y factor, 8 , in t h e g e n e r a l case is: 2 . 6 . 1 2 . - 8 = (Xi-Vx)+ S(VM-Mi) + ( S + I ) ( X M - M x ) / I + Mi + Mx - V M - XM We c a n e x p r e s s r e l a t i o n s h i p s b e t w e e n defect formation, the influence of various ext 9 factors, a n d the e q u i l i b r i u m c o n s t a n t t h e r e b y related. We do this in t e r m s of 8, the degree of n o n - s t o i c h i o m e t r y , as given in Table 22. Even t h o u g h t h e s e e q u a t i o n s are r a t h e r formidable-looking, we shall be able to u s e t h e m to good advantage. Note t h a t e a c h case c o r r e s p o n d s to the influence of a r e a c t i n g e x t e r n a l factor on a s t o i c h i o m e t r i c solid, w h i c h c o n t a i n s intrinsic defects. T h e s e
additional d e f e c t s b e c a u s e of n o n s t o i c h i o m e t r y a n d c h a r g e - c o m p e n s a t i o n . The d e f e c t p r o d u c e d is e n c l o s e d in b r a c k e t s in Table 2-2. factors
produce
C o n s i d e r this factor carefully by again e x a m i n i n g Table 2-2. Also given is t h e r e a c t i o n p r o d u c i n g t h e defect, with its c o r r e s p o n d i n g e q u i l i b r i u m c o n s t a n t . In m o s t cases, the deviation, 8, is p r e s e n t e d in t e r m s of t h e e q u i l i b r i u m c o n s t a n t a n d the partial p r e s s u r e
of the e x t e r n a l gaseous
reactant. T h u s , if the n u m b e r of d e f e c t s p r o d u c e d can be m e a s u r e d and an e q u i l i b r i u m c o n s t a n t calculated, t h e n 8 c a n be d e t e r m i n e d both as a f u n c t i o n of partial p r e s s u r e , p x 2 , a n d t e m p e r a t u r e (see 2 . 5 . 2 2 . ) . IV. Defect t~oncentration~ in MX,s• We now p r o c e e d as we did for the s t o i c h i o m e t r i c - c a s e , n a m e l y to develop
2.6 Defect equilibria in various types of compounds
83
d e f e c t - c o n c e n t r a t i o n e q u a t i o n s for t h e n o n - s t o i c h i o m e t r i c case, i.e.MXS• C o n s i d e r t h e effect of A n t i - F r e n k e l defect p r o d u c t i o n . F r o m T a b l e 2-1, we get KAF, w i t h its a s s o c i a t e d e q u a t i o n , kAF 9 In T a b l e 2-2, we u s e Kxi for X - i n t e r s t i t i a l s . 2.6.13.-
When
C o m b i n i n g t h e s e , w e get:
KAF = KVx " Kxi
both
Vx
and
Xi
coexist
=
Ni 2 I ( S - N i ) ( a - N i )
in
the
lattice,
the
deviation
from
s t o i c h i o m e t r y (from 2 . 6 . 1 2 . ) b e c o m e s : 2.6.14.-
5
= [Xi] -
[Vx]
U s i n g t h e e q u i l i b r i u m c o n s t a n t of 2 . 6 . 1 3 . , i.e. 2.6.15.-
Kv x _
p x 2112 [Vx ] /Xx
_
p x 2112 [Vx] [ S - [Vx ]
a n d t h e a p p r o p r i a t e o n e f r o m T a b l e 2 - 2 (i.e.- Kxi ), we get ( a s s u m e for simplicity that S = r = 1): 2.6.16.-
5
= a Px21/2 Kxi/Px~2Kxi + 1
-
S K v x / KVx+ Px2 I/2
We c a n r e a r r a n g e t e r m s in 2 . 6 . 1 6 . to o b t a i n : Kxa (1- 8 ) p x 2 - 8( K v x K x i + 1 ) Px2 I/2 a n d i f - KVx 2 (1- 8 ) = 0, w e c a n , b y u s i n g 2 . 6 . 1 3 . a n d Ni ~1, o b t a i n : 2.6.17.-
Ni 2 ( i - 5 ) p x 2 - 5 Kv x Px2 I / 2 -
KVx 2 (I- 8} = 0 .
S o l v i n g for p x 2 yields 9 2.6.18.- px 2 = KVx2(82+
Since
at s t o i c h i o m e t r i c
2Ni(1-
composition,
f o r m i d a b l e e q u a t i o n r e d u c e s to: 2.6.19.-
52 ) • 8 1 5 2 + 4 N i ( 1 - 8 2 ) ] 1 / 2 / 2 N i 4 ( I - 5 ) 2
Px2 o = KVx 2 ! Ni 2
8 must
e q u a l zero,
this
rather
The point defect
84 where
Px2 o
is the p r e s s u r e of X2
gas in equilibrium w i t h the MXs
crystal at the s t o i c h i o m e t r i c composition. This gives us the o p p o r t u n i t y to divide 2.6.18. b y 2 . 6 . 1 9 , to obtain: 2.6.20.- Px2 / Px2 o
=
82 +2 Ni (I- 62 ) •
6[62 + 4 Ni ( I - 6 2 ) 1 2 N i 2 ( I - 6 ) 2
We c a n therefore calculate 8 in t e r m s of the ratio of Px2 to Px2 o a n d N i , s h o w n as follows2.6.21.Effect of E x t e r n a l P r e s s u r e of X 2 G a s o n N o n - S t o i c h i o m e t r y of t h e H y p o t h e t i c a l C o m p o u n d , M X S_+8
N i = 10-4
J
Ni = 10 -3
PX 2
I~
-
0.06
= 1 0 -2
0
+ 0.06
~, C h a n g e in S t o i c h i o m e t r y Although we will not t r e a t the o t h e r types of pairs of defects, it is well to note t h a t similar e q u a t i o n s c a n also be derived for the o t h e r intrinsic defects. What we have shown is t h a t ezlmnml z~Letmats can cause f u r t h e r c h a n g e s in the n o n - s t o i c h i o m e t r y of the solid.
2.6 Defect equilibria in various types of compounds
85
V. Ionization Of D e f e c t s We have a l r e a d y covered,
albeit briefly, n o n - i o n i z e d
stoichiometric
and
n o n - i o n i z e d n o n - s t o i c h i o m e t r i c i n t r i n s i c - d e f e c t c o m p o u n d s . Let u s n o w c o n s i d e r t h e ionization of defects in t h e s e c o m p o u n d s . In t h e MXs c o m p o u n d , if we r e m o v e s o m e of t h e X - a t o m s to f o r m Vx~ t h e e l e c t r o n s f r o m t h e r e m o v e d X - a t o m (or f r o m t h e b o n d h o l d i n g t h e X - a t o m in t h e crystal)
are
left
behind
for
charge
compensation
reasons.
At
low
t e m p e r a t u r e s , t h e s e e l e c t r o n s are localized n e a r t h e v a c a n c y b u t b e c o m e d i s s o c i a t e d f r o m t h e p o i n t defect at h i g h e r t e m p e r a t u r e s . T h e y b e c o m e free to m o v e t h r o u g h t h e crystal, a n d we say t h a t t h e i n t r i n s i c defect h a s b e c o m e ionized. We c a n w r i t e t h e following e q u a t i o n s for t h i s m e c h a n i s m : 2.6.22.-
VACANCIES 8
Vx ~
Vx + + e
KVx
VM --~ VM" + p+ In
the
equilibrium
constant
=
IV x [ +
=
I VM- I I P + I / VM
s
KVM
equations,
each
symbol
le
is
I/Vx
actually
a
c o n c e n t r a t i o n , i.e.- n u m b e r s of specified defects a n d e l e c t r o n s , e t c . In a like m a n n e r , we w r i t e for i n t e r s t i t i a l s a n d a n t i - s t r u c t u r e d e f e c t s 2.6.23.-
INTERSTITIALS Mi
--~ Mi + + e-
Xi-+p +
Xi ~
2.6.24.-
A useful
KMi* = I Mi+l I e l
/ Mi
Kxi* =
IX~l I p + l / X ~
IXM + I I e
ANTI-STRUCTURE XM ~ X M +
+ e -
KXM =
Mx
+ p+
KM X
understanding
the
~
example
Mx
for
ionization of defects,
=
IMx" [ [P + I/ MX
above
is t h a t of c o b a l t o u s oxide,
I/XM
equations,
and
the
CoO. E x t e r n a l o x y g e n
The point defect
86
p r e s s u r e will affect s t o i c h i o m e t r y a n d p r o d u c e c o b a l t v a c a n c i e s , b u t t h e v a c a n c i e s a r e ionized at r o o m t e m p e r a t u r e :
1 / 2 0 2 ~ Oo + V C o
2.6.25.-
+n + +aVi
T h e n + is free to m i g r a t e t h r o u g h o u t t h e lattice. However, at p r e s s u r e s b e l o w 10 .6 ( o b t a i n e d by a p p l i c a t i o n of a v a c u u m ) , t h e Co v a c a n c i e s b e c o m e
doubly-ionized: 2.6.26.-
VCo
~
Vco 2
+~+
T h i s i l l u s t r a t e s h o w t h e ionized defect a r i s e s . We h a v e a l r e a d y i l l u s t r a t e d h o w v a c a n c i e s arise t h r o u g h PLANE ~ .
the us e of a
( S E E 2.2.1.). T h e r e i n , we u s e d MX as t h e m o l e c u l e to b u i l d
t h e NET. Let u s n o w r e t u r n , u s i n g t h e ions, M 2+ a n d X 2-
as the i o n i c
f o r m s w i t h w h i c h to b u i l d t h e NET. T h i s is s h o w n as follows: 2.6.27.-
[A Plane I 2+ •
Net fo r the Ionized MX Compound
X 2 - M 2+ X 2 - i~t 2 . X 2x2-r vl ' ~ + X =- M 2 . X 2 - M 2+ X 2 - M 2 § I M~'x~-r'12"• ~- H 2 " X ~ M2+X2-
I
I Ioo,~d I ! x_~i Mz'IX2- M 2 + X j . ~ ~ ' I ~ . ~ X Hole
. p'r--M2 + X 2 - M 2+ 2 ~ ~
M 2+
22"-- M 2 + X 2 _
JX
I Vacancy Hole Complex
I I x~- M~'x~--~N~_s~ ~- H ~" x ~- H ~*
El
x
2-IMP'Ix =- M 2+ X 2- M 2 "
Electron
I
.~IM " I X 2- M 2+ M 2 - X2-1~t 2~ X 2 X 2 - M 2+ X 2- M 2+ M 2 + X 2- M2+X2 -
•
§ •
X 2- M ""
l.x2M~_IM [M 2+
M 2" X 2- p,12 " X 2- M 2- X
2-]M +
X 2rVl2 . )(;2 + x 2~l~2- M 2" X2- M 2+ M2+ X 2- M Z - X 2 -
H ~§ x 2 - M ~" X 2- M 2.
M 2+ X 2 - M 2 + X 2- M 2 + X 2 - pvl2+ X 2 X 2 - M 2 + X2-. M2+ X2- M 2§ X 2- M 2 §
VacancyTrapped Electron Complex
2.6 Defect equilibria in various types of compounds
87
A v a c a n c y - h o l e c o m p l e x is s h o w n , as weU as a v a c a n c y - t r a p p e d e l e c t r o n complex.
In addition,
an example
of a n ionized
hole
and
an ionized
e l e c t r o n - s i t e is also given. We m u s t t h e r e f o r e a d d to o u r list of i o n i z a t i o n e q u a t i o n s t h e following: 2.6.28.-
MM+Z -.~
MM+Z-1 + p+
MM+Z ~
MM+Z+I + e-
B e c a u s e of t h e w a y t h a t we h a v e d e f i n e d o u r MXs crystal, t h e analogous ionization of t h e anion, X, does n o t o c c u r . Let u s n o w s u m m a r i z e w h a t we h a v e c o v e r e d : 1. We h a v e s h o w n t h a t defect e q u a t i o n s a n d e q u i l i b r i a c a n b e w r i t t e n for t h e MXs c o m p o u n d , b o t h for t h e s t o i c h i o m e t r i c a n d non-stoichiometric cases. 2. In a d d i t i o n , we h a v e s h o w n t h a t f u r t h e r defect f o r m a t i o n c a n b e i n d u c e d b y e x t e r n a l r e a c t i n g s p e c i e s , a n d t h a t t h e s e a c t to f o r m specific t y p e s of defects, d e p e n d i n g u p o n t h e c h e m i c a l n a t u r e of t h e crystal lattice. 3. We have also s h o w n
that the
intrinsic
defects
can
become
ionized. B e c a u s e m o s t of w h a t we have c o v e r e d to t h i s p o i n t h a s a s s u m e d l a c k of ionization, we n e e d to p r o c e e d f u r t h e r so as to develop e q u a t i o n s m o r e suitable for t h e real w o r l d . Most of t h e c a s e s we e n c o u n t e r in t h e real w o r l d involve ionic l a t t i c e s containing
charged
considered
simple hypothetical compounds
various k i n d s
cations and charged
of n o n - i o n i z e d
intrinsic
a n i o n s . Up to now, we
have
s u c h as MX a n d d e l i n e a t e d
defects
that would be p r e s e n t .
S e c t i o n 2 - 6 w a s c o n c e r n e d w i t h various t y p e s of d e f e c t s i n t r i n s i c to t h e s t o i c h i o m e t r i c MXs c o m p o u n d a n d t h e n o n - s t o i c h i o m e t r i c c o m p o u n d ,
The point defect
88 MXs•
Additionally, we e x a m i n e d the ionized MXs c o m p o u n d . The n o n -
ionized types are the easiest to illustrate, including the
mathematics
involved. Nevertheless, even those e q u a t i o n s (see Tables 2-1 and 2-2) b e c a m e s o m e w h a t c u m b e r s o m e at t i m e s . S u c h e q u a t i o n s are applicable to metallic, a n d possible s o m e covalent, crystals. E q u a t i o n s i n c l u d i n g ionization a n d electric charge are n e e d e d for ionic crystals a n d semic o n d u c t o r s . But, the m a t h e m a t i c s involved s t a r t s to b e c o m e uasolvablc. The m a i n p r o b l e m is the need to have a complete description, utilizing e q u a t i o n s d e s c r i b i n g all possible p r o c e s s e s , including: Intrinsic defect f o r m a t i o n Effect of extrinsic factors Ionization of intrinsic d e f e c t s Effects of c h a r g e - c o m p e n s a t i o n . By the time we are finished, we find t h a t we have derived a set of multiv a r i a n t e q u a t i o n s in several u n k n o w n s . Brouwer (1954) c o n s i d e r e d this case a n d was able to formulate a m e t h o d of solution. The following is a s u m m a r y of t h a t m e t h o d . 2.7. - BROUWER'S APPROXIMATION METHOD Let us c o n s i d e r an MXs crystal with S c h o t t l ~ defects (this is one of t h e m o r e easily defined types, mathematically). We define S=1. The types of intrinsic defects was given in 2.6.1., the equilibrium c o n s t a n t s of i n t r i n s i c defects was given in Table 2-1, n u m b e r s of intrinsic defects was given in 2.6.4., ionization of vacancies in 2.6.22, 2.6.23 & 2.6.24, and 2.6.21. gave the effect of an external factor (X2 gas) on the p r o d u c t i o n of n o n s t o i c h i o m e t r y . We can rewrite these equilibrium c o n s t a n t equations for the MX c o m p o u n d in the form given in 2.7.1. on the next page.
This set of equations completely defines the defect concentrations in 8n MX crystal conta/n/ng S c h o t ~ defects.
89
2.7 Brouwer' s approximation method
2.7.1-
a.
In KS = In VM + In VX
b.
InKvx* -Ine+InVx*
C.
In KVM* = In p + In VM" - In VM
d.
In KVx
e.
lnKVM = I n V M
=
-lnVx
1 / 2 In P x9 + In Vx - 1/2inPx 2
The i n t r i n s i c ionization c o n s t a n t is: f.
In Kion = In e + In p ( w h e r e Kion is t h e i o n i z a t i o n equilibrium constant)
and the electroneutrality condition i s . g.
e +VM-
=
p
+ VX+
It is i m p o r t a n t to realize t h a t e is t h e s u m of t h e e l e c t r o n s ,
not a s i n g l e
n e g a t i v e c h a r g e , i.e.2.7.2-
e = E e
and p
= Y p+
This set of equations c o m p l e t e l y defines the d e f e c t c o n c e n t r a t i o n s in an MX crystal cont~Inlng S c h o t t ~ defects. T h e r e a r e staten e q u a t i o n s in 2.7. I. a n d if t h e e q u i l i b r i u m c o n s t a n t s a r e k n o w n , t h e r e r e m a i n s e v e n u n k n o w n s in t h e s e s i m u l t a n e o u s e q u a t i o n s , i.e.- T h e s e involve- VM, Vx
, V M - , VX+ , e , p , a n d px 2. T h e first six
e q u a t i o n s are l i n e a r r e l a t i o n s b e t w e e n l o g a r i t h m s of c o n c e n t r a t i o n s ,
and
l o g a r i t h m s of e q u i l i b r i u m c o n s t a n t s , b u t t h e l a s t e q u a t i o n is not. T h e s o l u t i o n to t h i s set of e q u a t i o n s c a n be a c c o m p l i s h e d , b u t t h e m e t h o d s a r e c o m p l e x . However, if t h e r e is f u r t h e r ionization of d e f e c t s , where
more
than one pair
of d e f e c t s
results,
the
or t h e c a s e
situation becomes
h o p e l e s s . T h e e q u a t i o n s c a n b e w r i t t e n , b u t t h e set of e q u a t i o n s c a n n o t b e easily solved. In
1954,
Brouwer
proposed
a
graphical
method
for
solving
these
e q u a t i o n s . T h e m e t h o d h a s b e e n a d o p t e d b e c a u s e of f u r t h e r d e v e l o p m e n t b y Kroeger a n d Vink (1956).
The method
e n t a i l s dividing t h e r a n g e of
The point defect
90
defect c o n c e n t r a t i o n s into regions, s u c h t h a t c h a r g e - c o m p e n s a t i o n involves only two defects, so t h a t the e l e c t r o - n e u t r a l i t y equation (2.7.1-g.) is simplified. Brouwer's Method h a s b e e n applied to the case of silver chloride w h e r e i m p u r i t i e s like c a d m i u m chloride m a y be p r e s e n t . T h e p h o t o g r a p h i c p r o p e r t i e s can be e n h a n c e d or deteriorated, d e p e n d i n g u p o n the state of the i m p u r i t y added, i n a d v e r t e n t l y or not. For
a large
negative
stoichiometry
deviation,
VM-
and
p+ b e c o m e
negligible c o m p a r e d to Vx + and e - - T h e n , Equation 2.7.1.- g. b e c o m e s : In e = In Vx + . By using this relation, all defect c o n c e n t r a t i o n s can be plotted in t e r m s of X2 p r e s s u r e a n d the equilibrium c o n s t a n t s . This is s h o w n as a Brouwer plot, given as 2.7.3. on the next page. It is m o r e c o n v e n i e n t to use N d , the n u m b e r s of a given defect, and to define a quantity, R, as: 2.7.4.-
R = KVM Px21/2
for the MX c o m p o u n d c o n t a i n i n g S c h o t t l ~ defects. We can t h e n e x p r e s s Nd in p o w e r s of R, so that: 2.7.5.-
InNd
= 01nR
+C
We find t h a t 0, the slope, is either + 1 / 2
or + I. This simplifies m a t t e r s
greatly, as c a n be seen in 2.7.3. Here, the defect e q u a t i o n s for o u r MX c o m p o u n d {containing S c h o t t k y defects) are plotted for the case: Kion > KSh . In obtaining this plot, we have derived the following equations from equations 2.7.1. and 2.7.2.: 2.7.6.-
VM = R Vx = K s / R e=Vx + ={KvxVx}
1/2
= ( K v x.
Ks / R } 1 / 2
p = Ki ( R /KV X 9 KS)1/2 VM
$
= KVx*. VM / p = (R KVx* KS) KVM ) /
Kion
91
2.7 Brouwer' s approximation method 2.7.3.-
B r o u w e r Plot for Schottky Defects in the Hypothetical Compound,
MX.
for the
Case:
[Kion
le
'
KSh
i
i
I,
§ -" "X
"
''= I
i I
,'I
'n...,
I
. . j
I
I I I
^
I I
~,,,
. /
Z
i
11[
"0
%,
~ 9 - ,,v
"~ %, "~
I
.i,II''' .i,r
"Jill/,, .111'
.'~L'
I
I
imam
,,li"'~i,
i ~'l-'i t''" I ~ M I' I .'' i ,#" I .d# I .,d' I .li If'
%
.,nr
_
. ,,i'""'l ..,,,"" I .,,,,,,,'"' !
V M ,,,,,'"'"'
i
I Hi
jlllililliillilllllll
-i-
v;, . V
r
-"-"'-~i, +++"
I I
V
I
-+-'-
_.
I
,t:
+
--++ +4M
i
M "~1111
Jlej~T
V,-, l ++T "%, X I,.§ % +'lz.~ ~ -+ I '%.+-'- I -+-%. I • "T "~i~l ,+~
+" ~. ~r +++ Y M 4-
I I !
§247 "
',
++ ,+++
~
.. +
~-~Vx
i Hi
"',,,,..
a! i
I
I
We can readily see that concentrations and that by taking logarithms we
el
} ', +++":{in R - 1/2 lnl p + in KVM. .~l,.++ iI X 2 i +++
2.7.3.,
~
I
V M
concentrations,
~
.
.,i' ",i%
Yv
I I I
c
>
I
I
V__
]
can
now
II
are now expressed
in powers
o f R,
, w e g e t s l o p e s o f _+ 1 / 2 o r + 1. A s s h o w n
differentiate
between
namely: REGION I
-
VX + > VM
REGIONII
-
VX +
-
R E G I O N III -
VM
> VX +
VM-
3
regions
of
in
defect
The point defect
92
At l a r g e p o s i t i v e d e v i a t i o n s f r o m t h e s t o i c h i o m e t r i c c o m p o s i t i o n ( R E G I O N III), V M >> e a n d p >> V x + . In t h i s R E G I O N , 2 . 7 . 1 . - g b e c o m e s : 2.7.7. -
In V M "
=
In p
In t h e v i c i n i t y of t h e s t o i c h i o m e t r i c c o m p o s i t i o n c o n c e n t r a t i o n of d e f e c t s d e p e n d s u p o n w h e t h e r KS
(REGION II), t h e > Kion , or v i c e -
versa. The
ionic
former
usually holds
for l a r g e b a n d - g a p
compounds.
Then, 2.7.1.- g becomes: 2.7.8.-
VM-
=
+
W e c a n also s h o w t h a t t h e c h a n g e f r o m n e a r s t o i c h i o m e t r y to a n e g a t i v e d e v i a t i o n of s t o i c h i o m e t r y (REGION I) o c c u r s w h e n :
(R KS KVx* /Kion )112
2.7.9.-
=
(Ks
Kion / K v X
R) 1/2
T h i s gives us: 2.7.10.-
RII -- I
=
Kion / KVx*
-
KS
a n d in a l i k e m a n n e r : 2.7.11.-
Because
RII - Ill
the
stoichiometry
is
/ Kv X
rigidly
defined
in
REGION
II
by
the
condition: 2.7.12.-
VM
c h a n g e s in R or Px2
+ VM"
-
VX
+
VX +
do n o t g r e a t l y affect d e v i a t i o n f r o m s t o i c h i o m e t r y .
But, t h e r e a r e l a r g e c h a n g e s in b o t h Z e- a n d Y p+ in t h i s r e g i o n . T h u s , t h e s t o i c h i o m e t r i c c o m p o s i t i o n is p r o b a b l y b e s t d e f i n e d w h e n e = p . T h i s o c c u r s , as s h o w n in 2 . 7 . 3 . , at t h e value:
93
2.7 Brouwer' s approximation method
2.7.13.-
Ro
= {Ks Kion )1/2 / KVx
An e x a m p l e of a c o m p o u n d w h e r e KS o b t a i n e d values (at 6 0 0 ~
>> Kion
is IEBr. Kroeger {I 9 6 4 )
of:
KS = 8 x 1 0
"14
Kion = 3 x 10 -35 If we s u b t r a c t 2.7.9. f r o m 2 . 7 . 1 0 . , we find t h a t t h e Br2 g a s - p r e s s u r e c h a n g e s b y 1043 (since R ~ PBr2 I/2 ) over Region II. Moreover, t h e deviation
from
stoichiometry, 8 , changes
e s s e n t i a l l y c o n s t a n t . However,
only b y
10 -3, or
remains
Z e- (or Z p+ ) c h a n g e s b y a factor of
a p p r o x i m a t e l y 1021 over Region II. In e l e c t r o n i c s e m i - c o n d u c t o r s , t h e c o n d i t i o n : Kion >> KS , usually prevails. We usually get a B r o u w e r analysis like t h a t of 2 . 7 . 3 . Now, c o n s i d e r t h e c a s e w h e r e ionization is t h e n o r m . T h i s case, s h o w n i n 2.7.14.
(next
page),
is for t h e
hypothetical
semi-conductor
alloy MX,
w h e r e M a c t s as a c a t i o n , a n d X a c t s m o r e like a n anion. T h a t is, w h e n w e get i o n i z a t i o n of defects, M loses a n e l e c t r o n a n d X is positively i o n i z e d . An e x a m p l e c o u l d be GaAs. In t h i s case, e l e c t r o n i c insensitive
(Region
II)
to
composition,
whereas
c h a r g e is r e l a t i v e l y the
deviation
from
s t o i c h i o m e t r y , d , v a r i e s c o n s i d e r a b l y . In t h e p o s i t i v e - d e v i a t i o n d i r e c t i o n (Region III), t h e m a j o r d e f e c t s are VM" , p , a n d VM. T h i s gives t h e r e l a t i o n : 2.7.15.-
VM ~
VM"
+
p+
In t h e n e g a t i v e - d e v i a t i o n d i r e c t i o n (Region I), e a n d VM" p r e d o m i n a t e . For t h e m a j o r d e f e c t s of t h i s s y s t e m , we h a v e t h e c o n d i t i o n s 2 . 7 . 1 6 . o n t h e n e x t page.
shown in
The point defect
94 2.7.14.-
Brouwer Plot for S c h o t t k y D e f e c t s in the H y p o t h e t i c a l Compound. M X . W h e n I o n i z a t i o n of D e f e c t s is the N o r m i.e.-
I
[ Kion > K s h ]
I,'
le,-I
I]
le = ~I
le,
-i
k l
-~.e
1-
+ , V X
I m I
I a I
-
,i
,
V
I I
I I
..f'
,' aJmm~
-
I
,p."
N
dRiP! V
I ~ at,. I ,,i" "qirmnmmmnmmmmnnmmmmmmmmmmmmmmmmmmnmmmid,
,,,,' 'm"'"'"',.
_
".,,,'"
Z
,,,,'
J
,me"
_pm 9,,' ,,,"
m 'II. m ',,. m '%
..+ VX
Ii
-mJ""
'",,,.'%~
i "'mmm.f
I
!
im /v./
i
:/
I i m
~r'
/
_ r '
i/ ,
2.7.16.
m
I
K__ion >> Region I
-
R e g i o n II
-
R e g i o n III -
jr r
'lllm
l
mmmmm
'Im'"",,,,,,,,, V x
1 '
using
'"',,.
m
Immll
l
InlPx i
m
2
+ in Kv_ ,
rl
Ksh
e>
p
e ~ p
p
> e
Note that we have rather well defined the defects present semi-conductor,
~..~e
'1
,-.
I [lnR = 1 / 2
_/ r"-
. % '
~...._r j-,--~
I
X. M
-,m,.
Iv../ i _Fro i./
~r'rm i
v
i m n I
V -..r r'
r
~ M /
~
"'"',,,,. Ai ")b~
m I
~
/
I
rl ..rm
/!------__e
/,,,,,,.
I
I
~
r .F
relationships
defined
for t h i s t y p e o f
by Brouwer's
method.
It
2.8 Analyses of real crystals using Brouwer's method
95
s h o u l d be a p p a r e n t t h a t this m e t h o d a d d s c o n s i d e r a b l e power ability to analyze intrinsic defects w h e r e ionization is the n o r m . .8.-
to our
ANALYSES OF REAL CRYSTAI~ USING BROUWER'S METHODCOMPARISON TO THE THERMODYNAMIC METHOD
The silver halide series of c o m p o u n d s have b e e n extensively s t u d i e d b e c a u s e of their usage in p h o t o g r a p h i c Film. In particular, it is k n o w n t h a t if silver b r o m i d e is i n c o r p o r a t e d into a p h o t o g r a p h i c emulsion, any i n c i d e n t p h o t o n will create a F r e n k e l defect. W h e n the • m is d e v e l o p e d , the Agi + is r e d u c e d to Ag metal. T h e s e localized a t o m s act as nuclei to c a u s e m e t a l .....crystal formation at the p o i n t s "sensitized" by the p h o t o n action. (Note t h a t this d e s c r i p t i o n is an oversimplification of the actual m e c h a n i s m . Nevertheless, it s h o u l d be a p p a r e n t t h a t a k n o w l e d g e of defect c h e m i s t r y of the c o m p o u n d , AgBr, s h o u l d prove to be very i m p o r t a n t in u n d e r s t a n d i n g the c h e m i s t r y of p h o t o g r a p h i c films). I. The AgBr Crystal with a Divalent Impurity, Cd 2+ Consider the crystal, AgBr. Both cation a n d anion are m o n o v a l e n t , i.e.- Ag+ a n d B r - . The addition of a divalent cation s u c h as Cd 2+ s h o u l d i n t r o d u c e v a c a n c i e s , WAg , into the crystal, b e c a u s e of the c h a r g e - c o m p e n s a t i o n mechanism. s y s t e m as: 2.8.1.-
To m a i n t a i n
electro-neutrality,
we
prefer
to
define
the
(1-5) Ag +Br- - 5 C d 2+ S =
Fortunately, AgBr is easy to grow as a single crystal, u s i n g S t o c k b a r g e r T e c h n i q u e s . P o s s e s s i o n a n d m e a s u r e m e n t of a single crystal g r e a t l y facilitates o u r m e a s u r e m e n t of defects. The i m p e r f e c t i o n s we expect to find are: 2.8.2.-
VAg ,Agi , e - , p + , C d A g , a n d
[CdAg,VAg]
The last defect is one involving two n e a r e s t n e i g h b o r cation sites in t h e lattice.
The point defect
96
The following table gives the defect r e a c t i o n s governing this case:
DEFECT REACTIONS IN THE AgBr (~RYSTAL CONTAINING Cd 2 + _ e"
ao
0
~
b. C.
gAg
+ctVi
d. e.
f. g. h.
AgAg lgi
+ (xVi
--, ~
+ -~ ~_
p@
+ hgi
WAg WAg
+ Agi +
Agi + + e
1 / 2 B r 2 ~ BrBr + VBr CdAg--7~-CdAg+ + eCdAg + WAg"
[CdAg, VAgl
CdAg+ +VAg"
[CdAg, VAgl
i. For C o n s t a n t Cd CdT = CdAg +CdAg + + [CdAg ,VAg] = K j. For E l e c t r 0 - n e u t r a l i t y e- + W A g = P + + A g i +
+ CdAg+
We w o u l d normally plot In N d vs: In KAgi + In KVAg 9 However, we find it m o r e c o n v e n i e n t to plot In N d vs: 1 / T . The r e a s o n for this is as follows. E x p e r i m e n t a l l y , we fund t h a t if we fLx the Cd 2+ c o n t e n t at s o m e c o n v e n i e n t level, it is n e c e s s a r y to anneal the AgBr crystals at a fLxed t e m p e r a t u r e for t i m e s long e n o u g h to achieve c o m p l e t e equilibrium. If the t e m p e r a t u r e is c h a n g e d , t h e n b o t h type a n d relative n u m b e r s of defects m a y also change. T h u s , we plot In Nd vs: 1/T, as in the Brouwer d i a g r a m of 2.8.3., given on the next page. At low t e m p e r a t u r e s (Region III), s i n g l y - c h a r g e d d e f e c t s p r e d o m i n a t e , i.e.- 2 WAg = Agi + + CdAg+. At the j u n c t i o n of III II, the c h a r g e d moieties begin to c l u s t e r to form CdAg+ and Vag-. T h e s e in t u r n m a y form the c o m p l e x : 2.8.4.-
CdAg+ +
VAg
~
[CdAg
, VAg]
At the s a m e time, the c o n c e n t r a t i o n of Agl + d r o p s dramatically.
2.8 Analyses of real crystals using Brouwer's method
97
2.8.3.-
,,
I
IBrouwer Diagram for the AgBr Crystal Containing Cd2+I I
I
I
,I
IP,EGION I I
I
I REGION III
,I
I v
[Cd Ag, V.~q] "0
Z
~, t %.l'[Cd Ag,..,V,~g ' 4-
I REGION IIII
!
. Ag
I
,'; A g i
Ecd;,g,v,...,. ............ g] I ......... ...,." . . . . .I- " " ' ......... .,:,-'
~.,'
-'
I [Cd~
] ,,' '~Ag .,'
,'
V
..
l I LIII r I II F I dl
9 CdAg i iiiiii
iii
i iiii
iii
iii
ii i
I
'
I
II II iiI II IIIIII
+ ,,,' Ag~.,
iIII
I
A iiiiII IIIII
[Cd Ag, VAgl
Ol 0
In, P.
.~,t' I i.~ I
IRE0tON I!
I i
-
f[ l..,,'T)
i,
I RE01ONIII
'I I I 1 'I!
,,
| iI |
IREGiON lltl ~LJ / "
,
I ! I
em
I ,
I
I o
/
I I I l9
I
'
I
~ i i.
I'I
J d
!I
J
II1"
At t h e j u n c t u r e of II - I in t h e B r o u w e r d i a g r a m , t h e c o m p l e x c l e a r l y dominates.
The point defect
98 While
the
above r e s u l t s
show
how
intrinsic
defects
are
affected
by
t e m p e r a t u r e , we still do n o t k n o w h o w t h e e l e c t r o n s a n d h o l e s vary as a function
of t e m p e r a t u r e .
(Note
that
temperature,
as specified,
is
a
preparation t e m p e r a t u r e , not measurement t e m p e r a t u r e . M e a s u r e m e n t of i n t r i n s i c c o n d u c t i v i t y , o , is s h o w n at t h e b o t t o m of 2.8.3. At low t e m p e r a t u r e s , c o n d u c t i v i t y d u e to v a c a n c i e s , o v , a p p e a r s to be the m a j o r contributor: 2.8.5.-
WAg --~ WAg-
Conductivity concentrations
decreases decrease.
at
+ p+ higher
temperatures
because
FinaUy, at t h e v e r y high t e m p e r a t u r e
Agi + region
(Region I), c o n d u c t i v i t y is relatively low, a n d a p p r o a c h e s zero, b e c a u s e t h e c o m p l e x [CdAg, WAg] p r e d o m i n a t e s as t h e n u m b e r of c h a r g e d m o i e t i e s , (CdAg + + WAg) d e c r e a s e s . V a r y i n g t h e Cd 2+ c o n t e n t in t h e AgBr c r y s t a l affects t h e relative d e f e c t ratios, as s h o w n in t h e foUowing d i a g r a m , s h o w n as 2.8.6. on the n e x t page. Again, we c a n identify 3 Regions as a f u n c t i o n of C d - c o n c e n t r a t i o n : 2.8.7.-
REGION I -
Agi + > VAg
REGION II -
Agi + @ WAg
REGION III -
WAg > Agi +
R e g i o n I of 2 . 8 . 7 . c o r r e s p o n d s
closely to t h a t of R e g i o n III of 2 . 8 . 5 .
T e l t o w (1949) also s t u d i e d t h i s crystal. He m e a s u r e d c o n d u c t i v i t y of AgBr c r y s t a l c o n t a i n i n g various a m o u n t s of Cd 2+ , as a f u n c t i o n of m e a s u r e m e n t temperature. those
Up to 175 ~
of t h e m i d d l e
h e o b t a i n e d c o n d u c t i v i t y c u r v e s similar to
of 2.8.6.
But, as t h e
measurement
temperature
i n c r e a s e d , t h e p r o n o u n c e d dip s e e n in 2.8.6. t e n d e d t o f l a t t e n out. At t h e h i g h e s t m e a s u r e m e n t t e m p e r a t u r e of 4 1 0 ~ t h e c o n d u c t i v i t y w a s fiat. He c o n c l u d e d t h a t e l e v a t e d t e m p e r a t u r e s p r e c l u d e t h e f o r m a t i o n of c l u s t e r s a n d / o r c o m p l e x e s , so t h a t c o n d u c t i v i t y d u e to Agi + r e m a i n s t h e
2.8 Analyses of real crystals using Brouwer's method
99
2.8.6.-
Brouwer Diagram Showing Effect of Cd Concentration on Defect Formation in AgBr Crystal
[ Cd
I n Cd
I
4-
~g
i! i I
s' J
VJI! ,r=
u 9
Z e~
-A g +
u-
IL~U
lr
Aa.
n
n
~
m
............. ~. . . . . . . . . . . . . . . %. . . . . . . . . . . . . . b , , " -
....~'-
C:d
.I
+ _
Ag .I
I
,,'"
/
I n I
=liP'
I
.r o" .
dr
, V _
CCd.
J/
,,I
/ f
2~ u ^ g
".n~,,
/ A-'J"
g
~
t
n
u I
/
I
i .
I
",,,.Ag_ .,,,.,,,,, I
In
'.,,,
l "'nhn
9
1
]
.4=
, A g .,,i, ......................
...... "
in [ca] i
a i
i
I/
m r .,air
410~ 325 ~ C
I
.am-.
n
In [CA]]
9
Is. hi"
I
I
t~_..__L.,..--..,.-
i
225 ~
n 175
ti
~
n
I
l
...............n
I
J l "mr
/
v~]
ir
D
r
]
I~a~pd m
.,Ir
I
A,!]
2g.,r F I
..~.,,'-
_-,,,--"
./I'"',..,, A"%.g"i ,e I
Iii
n, J-.'" .
f
V
/[c.~ g ' 11
I r,+
~,g '
n-
---~
The point defect
100
m a j o r c o n t r i b u t o r to t h e conductivity, as a c c o r d i n g to t h e defect r e a c t i o n "5" of Table 2-3. LET US NOW SUMMARIZE WHAT WE HAVE COVERED TO DATE: 1. By r e w r i t i n g t h e e q u i l i b r i u m c o n s t a n t s of Table 2-1 a n d 2 . 7 . 2 0 . (ionization of vacancies) as l o g a r i t h m s , we o b t a i n e d linear r e l a t i o n s a m o n g t h e set of defect e q u a t i o n s . 2. By defining s e t s of defects as a ratio, R, we c a n t h e n plot t h e r a t i o s so as to s h o w h o w t h e relative n u m b e r s vary as a f u n c t i o n of t h e type of defect p r e s e n t in t h e c h o s e n c r y s t a l lattice. This is t h e Brouwer Method. 3. We also i11ustrated the m e t h o d for a AgBr c r y s t a l c o n t a i n i n g Cd 2+ . The
set of defect
reactions were
given,
so as to illustrate
the
p o s s i b l e defects p r e s e n t . T h e n , a B r o u w e r d i a g r a m i l l u s t r a t e d t h e n u m b e r s a n d t y p e s of defects actuaUy p r e s e n t as a f u n c t i o n of Cd 2+ c o n t e n t in t h e crystal. II. Defec~ D i s o r d e r in A g B r - A T h e r m o d y n a m i c AoDroach TO i l l u s t r a t e yet a n o t h e r a p p r o a c h to analysis of defect formation, c o n s i d e r the i n f l u e n c e of Br2 - gas u p o n defect f o r m a t i o n in AgBr. The free e n e r g y of f o r m a t i o n , ~G, is r e l a t e d to the r e a c t i o n . 2.8.8.-
Ag ~ + 112 Br2 {g) ~
AgBr{g)
{ A G A g B}r
This c a n be r e w r i t t e n as: 2.8.9.-
gAg p l / 2
Br 2
=
exp AGAgBr / R T
It m a k e s no difference as to w h i c h of t h e activities we use. If we n o w fix PBr 2 at s o m e low value, we find t h a t the i~os~ble defects in o u r AgBr crystal, as i n f l u e n c e d by t h e ~ I f ~ ,
PBr 2, will be:
2.8 Analyses of real crystals using Brouwer's method 2.8.10.-
Agi +
, WAg
,
Bri-
, VBr +
, e
and
101
~+
w h e r e w e u s e ~ for t h e positive c h a r g e to d i f f e r e n t i a t e b e t w e e n p r e s s u r e , p, of t h e e x t e r n a l gas. B e c a u s e of t h e h i g h e l e c t r o s t a t i c e n e r g y r e q u i r e d to m a i n t a i n t h e m in a n ionic c r y s t a l s u c h as AgBr, w e c a n safely i g n o r e t h e foUowing p o s s i b l e d e f e c t s : 2.8.11.-
AgBr + , AgBr ++ ,
BrAg"
, BrAg-- .
If w e h a v e t h e r m a l d i s o r d e r a t r o o m t e m p e r a t u r e
(I do n o t k n o w of a n y
c r y s t a l for w h i c h t h i s is n o t t h e case), t h e n w e c a n e x p e c t t h e f o l l o w i n g defect reaction relations: 2.8.12.-
Agi +
=
WAg"
Bri-
=
Agi +
WAg"
=
VBr +
Bri-
=
VBr +
At e q u i l i b r i u m , t h e following e q u a t i o n s a r i s e : 2.8.13.-
Ag~ Kd
+ 1 / 2 Br2 tg) ~ =
AgBr(s)
+
VAg"
+ g+
V A g - ' ~ + / p 112 Br
2
T h i s gives u s a t o t a l of e i g h t (8) c o n c e n t r a t i o n s
to c a l c u l a t e . T h e y involve
t h e foUowing c r y s t a l d e f e c t s 2.8.14.-
Ag~,
Our procedure i.e.- for AgBr:
BrBr , A g i ,
Bri
, WAg , VBr + , e-
, n+
is to s e t u p a site b a l a n c e in t e r m s of l a t t i c e m o l e c u l e s ,
The point defect
102 2.8.15.-
AgAg , + V a g BrBr
+ e-
+ VBr + + ~+
= 1
( e-
=
(~+ ~
1
--
Ag'Ag) Br+Br)
S i n c e Br2 {gas) is t h e d r i v i n g f o r c e for d e f e c t f o r m a t i o n , w e n e e d also to consider
deviation from stoichiometry,
6 . T h u s , w e also s e t a Agl-8 Br
balance: 2.8.16.-
Ag~
+Agi +
+e-
BrBr + Bri-
=
+u+
1+8
=
1
To m a i n t a i n e l e c t r o n e u t r a l i t y : 2.8.17.-
Agi +
+ VBr +
+ ~+
=
Bri-
+ WAg
+ e-
W e also s e t u p t h e f o l l o w i n g e q u a t i o n s : 2.8.18.-
AgAg
+
~xVl ~
Agi +
2.8.19.-
BrBr
+
r
Bri-
and 2.8.20.-
e-
~
+ p+ ~
+ VAg"
Ke
= Agi + V A g
VBr +
Kg
=
+
0
Kb =
/ Vi a
Bri- VBr + / Vi a
(e)
(~+)
N o t e t h a t w e h a v e d i s t i n g u i s h e d b e t w e e n t h r e e (3) s i t u a t i o n s , to wit. a. E l e c t r o n e u t r a l i t y b. T h e r m a l D i s o r d e r c. N o n - s t o i c h i o m e t r y These
(excess cation)
are the eight equations (2.8.12.
to 2 . 8 . 2 0 . )
required
to c a l c u l a t e
t h e d e f e c t c o n c e n t r a t i o n s a r i s i n g f r o m t h e e f f e c t s of t h e e x t e r n a l f a c t o r , PBr2 9 F r o m measurements of c o n d u c t i v i t i e s , transfer numbers ( e l e c t r o m i g r a t i o n of c h a r g e d s p e c i e s ) , l a t t i c e c o n s t a n t s a n d e x p e r i m e n t a l
2.8 Analyses of real crystals using Brouwer's method
103
d e n s i t i e s , it h a s b e e n s h o w n t h a t F r e n k e l d e f e c t s p r e d o m i n a t e
(Lidiard -
1957). T h i s m e a n s t h a t : 2.8.21.-
Agi + , VAg"
Furthermore,
>>
Bri
,
VBr +
Agi + so t h a t in t e r m s of o u r e q u i l i b r i u m c o n s t a n t s
VAg-
we get: 2.8.22.-
FOR FRENKEL DEFECTS:
Ke >> Kg
a n d Ke
T h u s , w e n e e d o n l y to c o n s i d e r t h e a b o v e two (2) d e f e c t s ,
>> Kd n a m e l y - VAg"
a n d Agi + , s i n c e t h e y a r e t h e m a j o r c o n t r i b u t o r s to n o n - s t o i c h i o m e t r y . c a l c u l a t i n g pO
Br 2
as b e f o r e ( w h e n 6 = 0 , s e e 2 . 7 . 2 2
& 2.7.23.),
By
we can
e x p r e s s o u r o v e r a l l d e f e c t e q u a t i o n as: 2 . 8 . 2 3 . - p I/2Br2 / (P~
1/2 = {6/2g +[( 1 +6/2~)2] I/2 } {6/2~+[ 1 +(6/2~) 2] I/2 }
B e c a u s e of t h e c o n d i t i o n s g i v e n in 2 . 8 . 1 6 . , t h e f i r s t h a l f of t h e e q u a t i o n c a n b e s e t e q u a l to o n e . N o t e t h a t w e a r e u s i n g ~ , ~ , a n d u a s t h e
eqnillhrium c o n s t a n t s 2.8.24.-
. i.e. -
~ ~. Ke 1/2
~
~
K~ 1/2
y -~Z Ky 112
5 >> 13 9 By t a k i n g l o g a r i t h m s , w e
In t h e r e m a i n i n g p a r t of t h e e q u a t i o n , can then obtain: 2.8 25.9
l121n
PBr 2
lpO
Br 2
=
In6
-
In
T h i s r e s u l t t h e n l e a d s u s to a p l o t of t h e effect of p a r t i a l p r e s s u r e
of Br2
o n t h e d e v i a t i o n f r o m s t o i c h i o m e t r y , d , for t h e AgBr c r y s t a l , a s s h o w n i n 2 . 8 . 2 6 . o n t h e n e x t p a g e (this w o r k is d u e to G r e e n w o o d - 1 9 6 8 ) . F o r ~ = 0 , t h e r e is a p o i n t of i n f l e c t i o n w h e r e t h e s l o p e o f t h e line is d e f i n e d b y t h e e q u i l i b r i u m c o n s t a n t , i.e.- [3 = K~ I12 .
The point defect
104 2.8.26.-
Effect of Partial Pressure of BrzGas on Deviation from Stoichiometry of the AgBr Crystal
99 - ~
A
+8
f
log p Br z
T h e larger this value, the flatter is the curve. All r e l a t i o n s r e g a r d i n g t h e defects c a n n o w be d e r i v e d . T h e m a j o r defects t u r n o u t to be: 2.8.27.-
WAg
-7
~+
~--
e-
~
Agi+ K~ {liKe} 1/2
pl/2
Br2
(Ke) I/2 (1/Ke) Kb( 1/ PBr2) I/2
T h i s s h o w s t h a t b o t h P+ a n d e- are m i n o r i t y defects d e p e n d e n t on
PBr 2
The following gives the s t a n d a r d e n t h a l p i e s a n d e n t r o p i e s of these d e f e c t r e a c t i o n s , a c c o r d i n g to Krosger ( 1 9 6 5 ) -
105
2.9 Summary and conclusions 2.8.28.-
D E F E C T REACTION
AS
AH
Cal. / mol I ~ AgAg ~
Agi +
+ WAg" ( F r e n k e l )
-7 AgAg + BrBr ~-
0~..
e
WAg-
25.6
+ VBr +
+ p+
I I 2 Br2 + A-gag-~ It is a p p a r e n t
AgBr + WAg- + P+
that the
Frenkel
process
Kcal. ! mol. 29.3
- 13.3
36
25
78
4.9
25.4
coupled
with
the
electronic
p r o c e s s a r e t h e p r e d o m i n a t i n g m e c h a n i s m s in f o r m i n g d e f e c t s in AgBr t h r o u g h t h e a g e n c y of e x t e r n a l r e a c t i o n w i t h Br2 gas. A final c o m m e n t : we c a n u s e t h e s e t h e r m o d y n a m i c v a l u e s to c a l c u l a t e t h e e q u i l i b r i u m c o n s t a n t s a c c o r d i n g to: 2.8.29.-
Ki
=
exp-
AGi o I R T
a n d c a n also o b t a i n t h e activity of t h e silver a t o m in AgBr f r o m 2 . 8 . 2 9 . By u s i n g e q u a t i o n 2.8.9., w e c a n s h o w 9 For aAg = 1 ,
@ T = 277 ~
For PBr2 = 1 a t m .
@ 277 ~
8
= + 1012
9 8 = - 10 -7
w h e r e t h e p l u s or m i n u s i n d i c a t e a n e x c e s s or deficit of t h e silver a t o m in AgBr. T h i s r e s u l t is d u e to W a g n e r ( 1 9 5 9 ) . 2.9.- SUMMARY AND CONCLUSIONS Let u s n o w s u m m a r i z e t h e m a j o r c o n c l u s i o n s r e a c h e d r e g a r d i n g t h e defect solid. You will n o t e t h a t w e h a v e i n v e s t i g a t e d t h e f o l l o w i n g h y p o t h e t i c a l c o m p o u n d s : MX, MXs a n d MXs+ 8- But, w h e n we i n v e s t i g a t e d c r y s t a l s in t h e r e a l world, w e f o u n d t h a t a c t u a l d e f e c t s in s u c h solids d i d
The point defect
106
not c o n f o r m e n t i r e l y to t h o s e of o u r h y p o t h e t i c a l c o m p o u n d s . Nonetheless, understand
in
order
to
how defects
comprehend
and
affect the p r o p e r t i e s
form
a
foundation
of actual solids,
n e c e s s a r y to s t u d y t h o s e h y p o t h e t i c a l c o m p o u n d s .
The
to
it was
following
is a
s u m m a r y of the c o n c l u s i o n s we r e a c h e d r e g a r d i n g t h e defect solid state: I. The c h a r g e c o m p e n s a t i o n m e c h a n i s m r e p r e s e n t s the single m o s t i m p o r t a n t m e c h a n i s m w h i c h o p e r a t e s w i t h i n the defect solid. 2. We have s h o w n t h a t defect e q u a t i o n s a n d equilibria c a n be w r i t t e n for t h e MXs c o m p o u n d , b o t h for the s t o i c h i o m e t r i c a n d n o n - s t o i c h i o m e t r i c cases. 3. The c o n c l u s i o n t h a t we r e a c h is t h a t defect formation is favored in the solid b e c a u s e of the e n t r o p y factor. difficult
to o b t a i n a "perfect"
solid,
It is m u c h m o r e
so t h a t the
defect-solid
r e s u l t s . We have also s h o w n t h a t the i n t r i n s i c defects c a n b e c o m e ionized. 4. Although m o r e t h a n one defect r e a c t i o n m a y be applicable to a given situation, only one thermodynamic that intrinsic
is usually favored by the
a n d electrical defects
conditions.
usually occur
Thus,
in pairs.
we
This
prevailing conclude conclusion
c a n n o t be o v e r e m p h a s i z e d . 5. We have s h o w n t h a t defects o c c u r in pairs. The r e a s o n for t h i s lies in t h e c h a r g e - c o m p e n s a t i o n p r i n c i p l e w h i c h o c c u r s in all solids. 6. Of the
nine
defect-pairs
possible,
only 5 have actually b e e n
e x p e r i m e n t a l l y observed in solids. T h e s e are: Schottky, F r e n k e l , Anti- Frenkel, A n t i - S t r u c t u r e , V a c a n c y - S t r u c t u r e . 7. T h e r e c a n be no doubt t h a t b o t h S c h o t t k y a n d F r e n k e l d e f e c t s are t h e r m a l in origin. 8. We have s h o w n by Statistical M e c h a n i c s t h a t we c a n calculate n u m b e r s of defects p r e s e n t at a given t e m p e r a t u r e . 9. T h e r e
is an Activation E n e r g y for defect
formation.
In m a n y
cases, this e n e r g y is low e n o u g h t h a t defect formation o c c u r s at, or slightly above, r o o m t e m p e r a t u r e .
2.10 The effects of purity
107
10 .Defects m a y be d e s c r i b e d in t e r m s of t h e r m o d y n a m i c a n d equilibria. T h e p r e s e n c e
constants
of d e f e c t s c h a n g e s b o t h t h e local
v i b r a t i o n a l f r e q u e n c i e s in t h e vicinity of t h e defect a n d t h e local lattice c o n f i g u r a t i o n a r o u n d t h e d e f e c t . 11 .There
are two a s s o c i a t e d effects on a given solid w h i c h
opposite cation
effects on site
differences
and
stoichiometry.
the
other
the
Usually, anion
site.
in d e f e c t - f o r m a t i o n - e n e r g i e s ,
the
one
have
involves
Because
the
of
the
concentration
of
o t h e r d e f e c t s is usually n e g l i g i b l e . 12 9 In e x a m i n i n g t h e defect s t a t e of real c r y s t a l s s u c h as AgBr, w e find t h a t we c a n write, u s i n g e q u i l i b r i u m c o n s t a n t s a n d d e f e c t t h e r m o d y n a m i c s derived from Statistical Mechanics and classical Thermodynamics,
valid e q u a t i o n s for t h e n u m b e r s a n d t y p e s of
various a s s o c i a t e d d e f e c t s p r e s e n t . However, we also find t h a t w e c a n n o t solve for t h e value of t h e u n k n o w n q u a n t i t i e s in a set of simultaneous
equations
since
the
equations
are
not
linearly
solvable. T h e e q u a t i o n s c a n b e w r i t t e n , b u t t h e set of e q u a t i o n s c a n n o t b e easily solved. It is for t h i s r e a s o n t h a t we have r e s o r t e d to g r a p h i c a l m e t h o d like t h a t of B r o u w e r , e v e n t h o u g h it is n o t e n t i r e l y s a t i s f a c t o r y in its s o l u t i o n s to t h e n u m b e r s a n d t y p e s of d e f e c t s p r e s e n t in real c r y s t a l s . T h u s , it s h o u l d b e clear t h a t lattice d e f e c t s in t h e solid s t a t e is t h e n o r m a l s t a t e of affairs a n d t h a t it is t h e d e f e c t s w h i c h affect t h e p h y s i c a l a n d c h e m i c a l p r o p e r t i e s of t h e solid. 2.10. -
T H E E F F E C T S OF PURITY (AND I M P U R I T I E S )
Our s t u d y h a s led u s to t h e p o i n t w h e r e we c a n realize t h a t t h e p r i m a r y effect of i m p u r i t i e s in a solid is t h e f o r m a t i o n of defects, p a r t i c u l a r l y t h e F r e n k e l a n d S c h o t t l ~ t y p e s of a s s o c i a t e d defects. T h u s , t h e p r i m a r y e f f e c t o b t a i n e d in p u r i f y i n g a solid is t h e m d m l m L ~ t i o n of defects.
Impurities,
p a r t i c u l a r l y t h o s e of differing v a l e n c e s t h a n t h o s e of t h e lattice,
cause
c h a r g e d v a c a n c i e s a n d / o r i n t e r s t i t i a l s . We c a n also i n c r e a s e t h e r e a c t i v i t y of a solid to a c e r t a i n e x t e n t b y m a k i n g it m o r e of a d e f e c t c r y s t a l b y t h e a d d i t i o n of s e l e c t e d i m p u r i t i e s .
The point defect
108
It is not so a p p a r e n t as to w h a t h a p p e n s to a solid as we c o n t i n u e to purify it. To u n d e r s t a n d this, we n e e d to examine the various g r a d e s of purity as we n o r m a l l y e n c o u n t e r them. Although we have e m p h a s i z e d inorganic c o m p o u n d s t h u s far (and will c o n t i n u e to do so), the s a m e p r i n c i p l e s apply to organic crystals as well. COMMERCIAL GRADE is usually about 95% p u r i t y (to o r i e n t ourselves, w h a t we m e a n is t h a t 95% of the m a t e r i a l is t h a t specified, with 5% being different (unwanted-?) material. Laboratory or "ACS-REAGENT GRADE" averages about 9 9 . 8 % in purity. 2.11.1.-
GRADES OF PURITY FOR COMMON CHEMICAI~
GRADE
%
opm IMPURITIES
IMPURITY ATOMS PER MOLE OF
Commercial Laboratory Luminescent Semi-conductor
95 99.8 99.99 99.999
50,000 2000 100 10
Crystal G r o w t h Fiber-Optics
99.9999 99.999999
1 0.01
~QMPQUND. 3.0 x 1022 1.2 x 1021 6 x 1019 6 x 1018 6 x 1017 6 x 1015
The GRADES listed above are n a m e d for the usage to w h i c h they are i n t e n d e d , a n d are usually m i n i m u m purities r e q u i r e d for the particular application. Fiber-optic m a t e r i a l s are c u r r e n t l y p r e p a r e d by c h e m i c a l v a p o r deposition t e c h n i q u e s b e c a u s e any h a n d l i n g of m a t e r i a l s i n t r o d u c e s impurities. F u r t h e r m o r e , this is the only way found to date to p r e p a r e t h e r e q u i r e d m a t e r i a l s at this level of purity. The frontiers of p u r i t y a c h i e v e m e n t of solids p r e s e n t l y lie at the fraction of p a r t s per billion level. However, b e c a u s e of E n v i r o n m e n t a l Demands, analytical m e t h o d o l o g y p r e s e n t l y available far exceeds this. We can now analyze m e t a l s a n d anions at the femto level (parts per quadrillion= 10 -15 ) if we w i s h to do so.
Nevertheless, it is becoming apparent that as h/gh purity inorganic solids are being obtained, we observe that their physical properties may be
2.11 Nanotechnology and the solid state
109
different than t h o s e usually a c c e p t e d for the s a m e c o m p o u n d of lower
purity. The h i g h e r - p u r i t y c o m p o u n d m a y u n d e r g o solid state r e a c t i o n s s o m e w h a t differently t h a n t h o s e c o n s i d e r e d "normal" for the c o m p o u n d . If w e reflect b u t a m o m e n t , we realize t h a t this is w h a t we m i g h t expect to o c c u r as we obtain c o m p o u n d s (crystals) c o n t a i n i n g far fewer i n t r i n s i c defects. It is u n d o u b t e d l y true t h a t m a n y of the d e s c r i p t i o n s of physical a n d solid state reaction m e c h a n i s m s now existing in the literature are only partially correct. It s e e m s t h a t p a r t of the frontier of k n o w l e d g e for C h e m i s t r y of The Solid State lies in m e a s u r e m e n t of physical a n d chemical p r o p e r t i e s of inorganic c o m p o u n d s as a function of purity. A case in point is t h a t of the so-called "Nano-Technology", the v a n g u a r d of r e s e a r c h into c h e m i c a l a n d physical p r o p e r t i e s of m a t e r i a l s in the r e s e a r c h c o m m u n i t y today. 2.11.-
N a n o t e c h n o l o g y a n d The Solid S t a t e
In the next c h a p t e r , we shall e x a m i n e the m e t h o d s of c h a r a c t e r i z i n g solids including: the p r o p e r t i e s of individual particles (including single crystals); the solid state r e a c t i o n s t h a t are u s e d to form various solids; a n d m e t h o d s u s e d to describe an a s s e m b l y of particles (particle size). We will find t h a t m o s t solid m a t e r i a l s are c o m p o s e d of particles in the 1- 3 0 0 ~m. range. This is 1-300 x 10 -6 m e t e r s . Most inorganic m a t e r i a l s are p r o d u c e d having particles in this size range. T h e s e are the familiar p o w d e r s s u c h as coal dust, inorganic chemicals, silt a n d fine sand, a n d even bacteria. C u r r e n t r e s e a r c h defines n a n o - t e c h n o l o g y as the use of m a t e r i a l s a n d s y s t e m s w h o s e s t r u c t u r e s and c o m p o n e n t s exhibit novel a n d significantly c h a n g e d p r o p e r t i e s w h e n control is achieved at the atomic a n d / o r m o l e c u l a r level. What this m e a n s is t h a t w h e n a given m a t e r i a l is p r o d u c e d having particle sizes at fractions of a ~rn (micron), it displays novel p r o p e r t i e s not found in the s a m e material w h o s e particles are l a r g e r t h a n 1.0 ~ n (micron). N a n o t e c h n o l o g y involves d i m e n s i o n s w h e r e a t o m s a n d molecules, a n d i n t e r a c t i o n s b e t w e e n them, influence their c h e m i c a l
110
The point defect
a n d p h y s i c a l b e h a v i o r . A u t h e n t i c n a n o - p a r t i c l e s are so s m a l l that~there are m a n y m o r e a t o m s o n t h e surface of e a c h p a r t i c l e t h a n t h e n o r m a l p a r t i c l e of 1.0 ~ma. P a r t i c l e s of 1.0 ~rn, i.e.- 1000 n m or 1000 x 10 -9 m, m a y s e e m small but those
atoms on the
surface of e a c h p a r t i c l e
are only about
0 . 0 0 1 5 % or 15 in a million of t h e a t o m s c o m p o s i n g t h e lattice. A n a n o p a r t i c l e w i t h d i m e n s i o n s of 10 n m . b r i n g s t h e surface a t o m s to about 1 5 % of t h e total a t o m s c o m p o s i n g the particle. At this size range, q u a n t u m p h y s i c s a n d q u a n t u m effects d e t e r m i n e s
t h e p r i m a r y b e h a v i o r of s u c h
particles. C o n s i d e r t h a t a t o m s have a size r a n g e of about 1-2 A. Most i n o r g a n i c solids, w i t h t h e e x c e p t i o n of halides, sulfides (and o t h e r pnictides),
are
D
b a s e d u p o n t h e oxygen a t o m , i.e.- oxide = O- , w h o s e a t o m i c r a d i u s d o e s not change
even w h e n
sulfates, p h o s p h a t e s
a n d silicates
are f o r m e d .
Oxide h a s a n a t o m i c d i a m e t e r of 1.5 A or 0 . 1 5 nm. = 0 . 0 0 0 1 5 ~m. Nanoparticles particle
are c l u m p s of 1000 to 1 0 , 0 0 0 of 0 . 1 5
~n.
in d i a m e t e r .
They
a t o m s . The l a t t e r w o u l d be a c a n be m e t a l
oxides,
semi-
c o n d u c t o r s , or m e t a l s w i t h novel p r o p e r t i e s useful for e l e c t r o n i c , optical, m a g n e t i c a n d / o r catalytic uses. W h e n light m e e t s p a r t i c l e s this small, it b e h a v e s differently. One e x a m p l e is TiO 2 ( t i t a n i u m dioxide), w h i c h h a s b e e n u s e d as an ultra-violet a b s o r b e r for s u n - s c r e e n p r o d u c t s . The usual p r o d u c t is a p p l i e d to t h e skin as a w h i t e - r e f l e c t i n g c r e a m . The p r o c e s s for m a k i n g t i t a n i u m dioxide varies b u t usually e m p l o y s TiC14 a n d its h y d r o l y s i s u n d e r c o n t r o l l e d c o n d i t i o n s . When particles
of 50 n m .
are formed,
the
sun-screen
cream
now is
t r a n s p a r e n t since the p a r t i c l e s absorb a n d s c a t t e r visible light m u c h less t h a n t h e l a r g e r p a r t i c l e s p r e v i o u s l y u s e d . However, t h e ultraviolet light a b s o r p t i o n is n o t c h a n g e d , only the reflection of w h i t e light. As we
shall see
in t h e
next
chapter,
particles
are
formed
first
as
"embryos" w h i c h are m i n u t e p a r t i c l e s of t h e n a n o - p a r t i c l e class. T h e s e t h e n grow into "nuclei" w h i c h t h e n grow into particles. The
science of
p a r t i c l e g r o w t h h a s b e e n a m a j o r s o u r c e of o u r u n d e r s t a n d i n g of p a r t i c l e s . As we have a l r e a d y s h o w n , lattice defects, d u e to t h e r m a l effects, are t h e n o r m w h e n a c r y s t a l g r o w s to sizable p r o p o r t i o n s . However, w h e n n a n o -
111
2.11 Nanotechnology and the solid state
crystals are formed, the n u m b e r s of e m b r y o s a l l o w e d to form, w i t h c o r r e s p o n d i n g nuclei, are controlled. The nuclei growth is t h e n c o n f i n e d to atomic d i m e n s i o n s . Much of this growth forms by " S p o n t a n e o u s Assembly". T h a t is, w h e n n a n o - p a r t i c l e s are formed, a t o m s are a d d e d o n e at a time to form the embryo a n d t h e n the nucleus. It is the size of t h e n u c l e u s t h a t is r e s t r i c t e d . I s u b m i t t h a t the p r e d i c t i o n given in the previous section, i.e.- see p. 107, h a s already b e e n realized. T h a t is, n a n o p a r t i c l e s form by self-assembly of a t o m s (ions) into defect-~ree crystals. It is this lack of intrinsic d e f e c t s t h a t give s u c h particles their u n i q u e c h e m i c a l a n d physical p r o p e r t i e s . Note t h a t if n o r m a l growth were allowed to p r o c e e d further, t h e n w e w o u l d have the n o r m a l d e f e c t - c r y s t a l . Suggested Reading 1. A.C. D a m a s k a n d G.J. Dienes, Point Defects in Metals, Gordon
&
Breach, New York ( 1 9 7 2 ) . 2. G.G. Libowitz, "Defect Equilibria in Solids", Treatice
on Solid State
Chem.- (N.B. Hannay- Ed.), I, 3 3 5 - 3 8 5 , (I 9 7 3 ) . The Chemistry of Imperfect Amsterdam (1964). 3. F.A. Kra~ger,
Crystals, N o r t h - H o l l a n d ,
4. F.A. K r u g e r & H.J. Vink in Solid State Physics, Advances in Research and Applications (F. Seitz & D. Turnbull-Eds.), pp. 3 0 7 - 4 3 5 (I 9 5 6 ) . 5. J.S. A n d e r s o n in Problems of Non-Stotchtometry (A. R a b e n a u - E d . ) , pp, 1-76, N. Holland, A m s t e r d a m ( 1 9 7 0 ) . 6. W. Van Gool, Principles of Defect chemistry Academic Press, New York ( 1 9 6 4 ) .
of Crystalline Solids,
112
The point defect
7. G. Brouwer, "A General A s y m m e t r i c Solution of Reaction Equations C o m m o n in Solid State Chemistry", Philips Res. Rept., 9 , 3 6 6 - 3 7 6 (1954) 8. A. B. Lidiard, "Vacancy Pairs in Ionic Crystals", Phys. Rev., I 1 2 , 5 4 - 5 5 (1958). 9. J.S. A n d e r s o n , "The Conditions of E q u i l i b r i u m of N o n s t o i c h i o m e t r i c C h e m i c a l C o m p o u n d s , Proc. Roy. Soc. ( L o n d o n ) , A185, 6 9 - 8 9 (1946). I0. N.N. Greenwood, Ionic Crystals, Lattice Defects & Non-Stotchiometry, Butterworths, London (1968). 1 I. Hayes a n d S t o n e h a m , "Defects a n d Defect P r o c e s s e s in Non-Metallic Solids"- J. Wiley & Sons, New York ( 1 9 8 5 ) .