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Zonal theory of a solid and the limits of its applicability
Volkenshtein, F. F.
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MATIOMAL F 2 3 U R C H COUUCIL O F CATlkDA
T i t l e : The z c ~ d tlleorjr o f a solid a,rd the l i m i t s of i t s a p p l i c a b i l i t y . ( ~ o c n a y a t e o r i y a tverdogo t s l a i predely ee p r i m e c i m o s t ~
By :
F.F.
Volkecshteic.Eeferecce : Gsl.e:d~i P i z . ITauk
42:
11-29,1951.
Trmsl zted by: Esther Rabkin.TI92
ZCNAL
THEOIE OF ASOLID
ANU TEE LIMITS OF ITSAPPLICABILITY
*
-
I, I n t r o d u c t i o nThe coctemporary t h e o r y of s o l i d s i s of i n t e r e s t from v a r i o u s view- poic-t;s. Various b r m c h a s of p h y s i c s and e n g i m e r i c g depecd, i n t h e fical
m a l y s i s , on t h e t h e o r y of s o l i d s .
1. The t h e o r y of s o l i d s i s of i c t e r e s t t o e l e c t r i c a l e c g i c e e r s . Ic f a c t , a r i e l e c t r o t e c h n i c a l problem can be reduced t o t h e combication of bodies havicg e i t h e r s m a l l o r l a r g e c o n d u c t i v i t i e s . The problems of t h e e l e c t r o c o n d u c t i v i t y of m e t a l l i c and non-metallic c r y s t a l s , t h e problems of t h e e l e c t r i c d breakdonr, of d i e l e c t r i c s
-
a l l deperd or, t h e cofitemporary t h e o r y of s o l i d s t h e momect we become i n t e r e s t e d i n t h e mechanism of t h e phenome con.2,
The problems of l u m i ~ e s c e c c e a l s o l e a d t o t h e g e c e r a l p r o b l e ~ of t h e t h e o r y of s o l i d s . The mechw-ism of t h e processes which p l a y a p a r ti c
t h e c r y s t a l phosphors u ~ d e r t h e a c t i o n of i l l u m i n a t i o n car, be understood o n l y o c t h e b a s i s of t h e contemporary t h e o r y of s o l i d s .3,
The t h e o r y of s o l i d s i s of i n t e r e s t t o p h y s i c e l chemists who a r e s t u d y i c g a d s o r p t i o n acd h e t e r o g e ~ e o u s cataJysis, C r y s t a l s ( m s t a l l i c ai-don-metallic)
a r e , a s a r u l e , c a t a l y s t s , The chemical y r o c e s s e s developing ir.side a c r y s t a l acd or. i t s s u r f a c e car,r,ot be completely understood i f we i g ~ o r e t h 3 contemporary coccepts of t h eT-
A r e p o r t giver, a t t h e s e v e c t h A l l Ucior, Corference or, t h e Physics of Semi-cocductors.
t h e o r y o f s o l i d s . I n t h e s a p r o c e s s e s tile s o l i d a c t s c o t a s ar, i c e r t medium o r bsckicg, b u t a s ar, eciuivalent p a r t i c i p = t i n -the process.
The t h e r o y of heterogeceous c a t a l y s i s i s based on the t h e o r g o f s o l i d s .
That, t h e e , i s t h e problem of t h e t h e o r y of s s c l i d s ?
The problem c o c s i s t s o f e x p l a i c i c g t h e e l e c t z i c , p h o t c - e l e c t r i c , magcetic, azd i c g e c e r a l t h e macroscopic, p r o p e r t i e s of t h e s o l i d f r o m t h e g e ~ e r a l v i e l y ~ i c t of t h e c r y s t a l l l . c e s t r u c t u r e . The e s s e c t i a l p o i n t i s d i s c o v e r y of t h e microscopic rnecha.ism of t h e macr-oscopic p r o c e s s e s proceeding i c t h e s o l i d ,
What, then, i s t h e posii;ior, i n t h i s f i e l d ?
I c d e s c r i b i c g t l ~ e e 1 e c t r o r . i ~ p r o c e s s e s
ir, the
c r y s t d , use i s g z c e r e l l y macic: of the concepts of t h e so-called z o c d t h e o r y of t h e :jolid, This i s a very co~:vecier,t ar,d d e s c r i p t i v e tshsorx, Gqjerimecters generzlly. speak i c the 1aci;;vage of t h e z o c a l theory.Ir,
t h i s 1cu:g.la.ge they. i c t e r p r e t t h e i r r e s u l t s ,A t tlle same time, t h e o r e t i c i a c s c a t e g o r i c a l l y o b j e c t t o t h e zonal t h e o r y o e t h e premise t h a t d e f e c t s of a p r i n c i p a l c l i e r a c t s r l i e a t t h e v e r y f o u n d a t i o c of i$.
The z o c a l t h e o r y of t h e s o l i d
is
22 approximate theory.Like acy a p p r o x i ~ n a t e th e o r y i t s a p p l i c a t i o n
i s
l i m i t e d . E x p e r i m e ~ t e r s o f t e n exceed t h e s e l i n i t s ar,d make u s e of t h e z o c a l t h e o r y i n c a s e s f o r whichi t
i s d e l i n i t e l y nor,=ay;;licable. Or. t h e c o n t r a r y , numeroust h e o r e t i c i a n s exceed t h e s e l i m i t s i n t h e o p p o s i t e d i r e c t i o r , acd
,
d e f ectii7:g t h e p u r i t y of t h e t h e o r y , s t r i k e o u t t h e z o c a l model aver. ic t h o s e c a ( ? s f o r vihich it i s q u i t e aR a c c e p t a b l e ap;~roximation,Ic P a c t , ir, t h e t h e o r y of s o l i d s , a whole number of problems
e x i s t which can be solved by a completely s a t i s f a c t o r y approximatior. w i t h i r , t h e frame of t h e z o c a l t h e o r y . However, t h e r e a l s o e x i s t problems vrblch by t h e i r v e r y essence $0 b e y o ~ d tlhese bour:daries. The f i r n p o r t a ~ t thirig i s
.to d e c i d e what group o f problems car, be onluraced t y t h e s e l i m i t s and what group of problems l i e s o u t s i d e t h e s a
limits.
The p r e s e c t paper d e a l s w i t h thi: questi.oc o f t h e limits of
a p p l i c a b i l i t y of t h e z o c a l t h e o r y ,
11, Bgsic Assumptiocs of t h e Zocal The,ox
F i r s t c f all,
I
vould l i k e t o dlwel.1 b r i e f l y o e t h 3 mair, assumptions which m e t h e b a s i s o f t h e Z ~ i : u l t h e o r y o f solic!s.m a t i s a c r y s t a l from a p h y s i c a l vievqoint,?
It i s a house i c ivhich l i v e s a whole f~uni1.y of ~ l e c t r o n s , T h i s h o ~ s e h a s a d e f i c i t e a r c h i t e c t u r e , ar,d the' u l a c t r o r : ~ p o p ~ l a t i n g i t obay d e f i c i t & ru1.a~ o f -the i c c e r errar.gemect.
The problem c o c s i s t s i n t h e d e s c r i p t i o n of t h e behaviour of t h e s e e l e c t r o c s
.
The e l e c t r o c s f i l l i n g t h e c r y s t a l i c t e r a c t with each o t h e r . The acoount of t h i s i c t e r a c t i d n c o n s t i t u t e s t h e b a s i c =d p r i c c i p a l d i f f i c u l t y of t h e theory.
It
r e p r e s e c t s a d i f f i c u l t y be cause quantum mechanics c m c o t e x a c t l y solve t h e problem of a system of i c t e r a c t h g p a r t i c l e s , I n the s o l u t i o c of such problems, quacturn mechanics r e s o r t s t o approximate methods.Therefore, t h e t h e o r y of t h e c r y s t a l is, by its very c a t u r e , ar,
approximate theory. &e has t o ecsure only t h a t t h i s theory r e p r e s e n t s a s u f f i c i e c t l y good approximation.
The problem of t h e c r y s t a l would be q u i t e a simple problem, i f
i t
were p o s s i b l e t o ignore t h e i c t e r a c t i o n between t h e e l e c t r o c s , r e t a i c ocly t h e i c t e r a c t i o r . of t h e e l e c t r o c s w i t h , t h e n u c l e i , Of course t h i s i n c o r r e c t method couldbe
corrected somewhat i f t h e i n t e r a c t i o c of t h e e l e c t r o n s i s s u b s t i t u t e d by t h e i n t r o d u c t i o n of a c e r t a i n e x t e r n a l e f f e y t i v e f o r c e f i e l d , i n which each of the e l e c t r o n s car, move indeperdently of t h e o t h e r s , Suchar.
e f f e c t i v e f i e l d c a l c u l a t e d a s a f i e l d produced by a d i s t r i b u t e d charge of a l l t h e e l e c t r o c s o f our system i s c a l l e d a s e l f - c o n s i s t e n t f i e l d ,Accoucting f o r t h e i n t e r a c t i o n s by t h e s e l f - c o n s i s t e n t f i e l d i s q u i t e
an
incomplete method. However, t h i s methodi s
a t t r a c t i v e becauseit permits us t o transform t h e poly-electronic pro.blem i n t o a s e r i e s of
independect sir:gle e l e c t r o c i c problems.mher, we use t h i s method we car. d e a l with t h e behaviour of each electror, separately; me car. assign t o e ~ c h e l e c t r o n i t s i n d i v i d u a l wave f u ~ c t i s c =d t h e i c d i v i d u a l er,ergy velue. Within t h e frame of t h i s method, each e l e c t r o c l i v e s
i t s
i c d P ~ i u u a l l i f e as though "not c o t i c i n g " t h erernaii,~r.g e l e c t r o ~ s , as i f t h e y did r.ot e x i s t a t all. The o r J y t h i c g 17l;ich renixtrs L,IO e l e c t r o n of t h e i r existence i s the P a u l i p r i c c i p l e
r h i c h forbicis it t o occppy t h e ciuantum s t a t e s already occupied by o t h e r e1'2ctror,s *
It
i s or, t h i s very nethod, tha method of t h e s e l f - c o n s i s t e c t f i e l d , t h a t tho sc- c d l e d zocal theory of tho s o l i d i s based.Thus, the b a s i s
o f
thc zonal theory i s t h e reductioc of t h e poly- e l e c t r o c i c problem t o 8 s i n g l e e l e c t r o c i c problem. The zonal theory i s as i c g l e e l e c t r o n theory,
This i s t h e y r i n a r y , but r.ot t!~e only, c h a r a c t e r i s t i c f e a t u r e of the zocal theory,
Tk.e secccd c h a r a c t e r i s t i c f e aturc i s tha def i c i t e assumptior: col:cercir:g the c h a r a c t e r of the force f i e l d i n which each i n d i v i d u a l e l e c t r o a moves. This f i e l d i s of z p e r i o d i c c a t u r e , r e f l e c t i c g t h e periodic s t r u c t u r e of the l a t t i c e , . For t h e zocal theory t h e concrete form of t h i s f i e l d i s
c ~ t
e s s e c t i a l ; The e s s e c t i a l t h i c g is orJy t h a t t h i s f i e l d should be p e r i o d i c with t h e period of -the l a t t i c e .Thz c h a r a c t e r i s t i c zocEil structu-o of t h a ecergy spectrum of the
e l e c t r o c , t h a t is, a systen of permissible energy bocds g e c e r a l l y sub- divided by forbiddec regions, i s a r e s u l t of p r e c i s e l y t h i s p e r i o d i c i t y .
Thus, t h e zonal p i c t u r e of the spectrum i s a l r e a d y p r o s e ~ t i r , t h e very p r e r e q u i s i t e of t h e thaory; i c t h e assumption c o c c e r c h g the p e r i o d i c c h a r a c t e r of the p o t e c t i a l ,
111, C e r t a i n Short_c_o_micgs of t h e ,Zor.al Theory
17e v~ill cote c e r t a i r . c h a r a c t e r i s t i c shortcomicgs o f t h e zor.al theory
-
c e r t a i c of i t s featui-es hiddec i c t h e p r e s e c t a t i o e of t h e problem i t s e l f .If
we cocsider t h a t t h e e l e c t r o c s "do c o t take c o t i c e " o f each o t h e r , t h e c w e carnot exclude t h e p o s s i b i l i t y of t h e i r accumulation on oce and t h e same atom.It
may happec t h a t or. oce m d t h e same atom ( o r ion) s e v e r a l e l e c t r o c s :vilI. coricectrate a t t h e expecse of s t r i p p i n g.
.?J o t h e ro r s e v e r a l o t h e r atoms ( o r i o n s ) ( l ) . Such s t a t e s , c h a r a c t e r i z e d by an "over c o c c e c t r a t i o c t ' of e l e c t r o c s i c oce s p o t , figure i c t h e zocal theory or, ar, equal f o o t i n g wi.;h t h e o t h e r s t a t e s .
\Thee ar. atom of the l a t t i c e acquires an e x t r a e l e c t r o n (above t h e cormal complement), t h e c such u: atom i s trtxsforrned i n t o a r:egative ioc, This e x t r a e l e c t r o n i s borrowed from another atom which i s thus tramformed i n t o a p o s i t i v e ion,
The e s s e n t i a l t h i n g i n t h i s case i s c o t s o much t h e f a c t t h a t t h e zonal thoory allows t h e t r a n s i t i c n of a c e l e e t r o n from ope atom of t h e l a t t i c e t o some o t h e r atom, as t h e f a c t t h a t such a t r a n s i t i o n does not r e q u i r e an expr,di&ure of ecergy from t h e viewpoint of t h e zocal theory.
A s i m i l a r r e s u l t i s -11 kEov~c from t h e moleculer t L s o r y .
Ir,
t h e molecular t h e o r y a method e x i s t s which i s completely eciuivalect t o t h e 3ioch which v:e u s e i c t h e zonal t h e o r y , T h i s i s t h e s o - c a l l e d( 3 )
rinethod of molecular o r b i t a l s o r , i n o t h a r words, t h e b.ullikec-Eiuxd method , S o l v i n g w i t h t h i s mntl~od, f o r example, t h a problem of t h e hydrogec n o l s c u l e , we o b t a i c , t c z e t h e r
: v i t h
t h e s o - c a l l e d noc-polar s t a t e s , -the p o l a r s t a t e s , i . e . , t h e s t a t e s when both a l e c t r o n s a r e c o c c e ~ t r a t e d o c one o f t h e t v ~ o c u c l e i of t h e hydrogor, molecule acd t h e o t h a r c u c l e u s i s s t r i p p e d . The hjrdrogee molecule t h u s a c q u i r e s a h e t e r o p o l a r c h a r a c t e r . I n t h e I i u l l i k e c - Hued ~ . e t b o l l , such h e t e r o p o l a r s t a t e s have t h e samd s t a t i s t i c a l weight a s t h e homopolm s t a t e s , a d correspoficl t o t h e same valuo o f efiergy,In
t h e c r y s t a l l i c e l a t t i c e , c o ~ s i d e r e d from t h s y o i c t of view of the Bloch method, we have e x a c t l y t h e same s t a t e of afl"airs.Such a r e s u l t i s a cor,sec,uecce of' t h e ~ ~ c o l l a c t i v i s a t i o c t l o f t h e
e l e c t r o c , c h a r a c t e r i s t i c of t h e i:.ullilcen-Iiucd mothod a s w e l l as 01' t h e Bloch ~ e t h o d (acd i c c e c e r a l o f t h e z o c d t h e o r y ) , Each e l e c t r o c oelocgs t o t h e same d e g r e e t o i t s onc c u c l e u s a s ::lei1 a s t o t h o o t h e r c ~ ~ c l e i i n t h a l a t t i c e a d i.t moves a l o c g t h e l a t t i c e i ~ d e p e n d e c t l y of t h e o t h e r e l e c t r o c s . Thc e l e c t r o c i c t h e z o c a l t h e o r y p o s s e s s e s two p r o p e r t i e s :
it
"does c o t c o t i c e t t t h e o t h e r e l e c t r o n s acd a t t h e same timeit
l t f o r g e t s aboutIt t h e atom t o vrhichit
i c i t i a l l y oelocged ( t o g e t h e r with whichit
was i c t r o d u c e d i c t o t h e l a t t i c e ) ,IV,
The Limits o f A p p l i c a b i l i t y of t h e Zocal TheoryThe z o c a l t h e o r y r e p r e s e c t s
an
approximate t h e o r y , ar,d i t t h e r e f o r e h a s i t s l i m i t s of a p p l i c a b i l i t y ,There a r e problems t o which the a p p r o x i m a t i o ~ s on which the zonal theory i s based
are
applicable. On t h e o t h e r hmd, t h e r e a r e problems t o which these approximationsare
c o t applicable.Re w i l l formulate tha cocdit.ioes determining: t h e sphere of a p p l i c a b i l i t y of the zoi:al theory, It i s p o s s i b l e t o s t a t e f o u r such cor,ditiocs,
1) Each atom ( o r i o ~ ) e n t e r i c g i c t c t h e compositioc of t h e c r y s t a l l i ~ e l a t t i c e possesses i t s cormal complemect of e l e c t r o n s , To these l1proper" e l e c t r o n s belocgicg t o the atom o r ion of t h e l a t t i c e , t h e zor.al theory i s d e f i ~ i i e l y c o t a p p l i c a b l e , The behavicur of these e l e c t r c ~ s carxot be c o r r e c t l y described w i t h i n the frame of t h e zonal theory.
However, i n a d d i t i o n t o the "properv e l e c t r o n s , ar. atom o r an ion of t h e l a t t i c e may a l s o possess ar. e x t r a e l e c t r o n . Ic t h e case of t h e atomic l a t t i c e such ar, e x t r a e l e c t r o c , placed on the ~ e u t r a l atom i n a d d i t i o c t o the complemect, makes
i t
a c e g a t i v e ion. I n t h e case of t h e i o n i c l a t t i c e , such e x t r a e l e c t r o n s lead t o the appearance of armmalous i o c s , i , e , i o c s havicg an anomzlous charge, These e x t r a e l e c t r o n s may be i ~ t r o d u c e d i n t o the l a t t i c e from the o u t s i d e , o r they m a y be borrowed from t h e r e s o u r c e s of the l a t t i c e proper a t the expecse of t r m s f e r r i c g e l e c t r o c s from oce cormal atom o r ior, t o acother atom o r ioc.I n g e c e r a l , i f oce can speak of t h e a p p l i c a b i l i t y or" tk,e zonal t h e o r y t o t h e e l e c t r o n s of t h e l a t t i c e , i t i s ocly i n r e l a t i o n t o t h e s e " e x t r a " e l e c t r o e s (1,
4 )
2) The zonal t h e o r y does not always giva a c o r r e c t d e s c r i p t i o n of 2hs behavicur o f t h e s e " e x t r a t t e l e c t r o c s , -but o ~ l y viher, t h e c o c c e c t r a t i o c of these Itextratt e1ectror.s i s s u f f i c i e ~ t l j r small. IR o t h e r words, t h a zocal theory r e q u i r e s t h a t tiia 5:unber c f the ar.omalous atoms ( i o n s ) should be smal.1 ir, r e l a t i o r ; tc the total numile: of atoms (i0r.s) ir, t h e l a t t i c e .
3)
The i;re.vious c o r . d i t i o c i s a r i ? q u i r x l c o r i i i t i c c , but it i s ~ o t s u f f i c i e n t , The 2 0 ~ 2 . 1 k h e o ~ ~ y i s apj~lj.carjle t o t h a ttextratt e l e c t r o n s o n l y Gl- ,lie : corxiizioi: til::t $,,.~:li tlextrti't e l e e t r o r , does c o t chaxge t h e s t a t e oft h e llprocarll ( i r d ~ e r ) e l e c t r o r : ~ of t h e atom ( o r ior.) or, which t h i s e x t r a eiectrot: I r T h i s Lakes place, f o r exanple, r,lrec t h e flpropeitl e l e c t r o r s o c . t h e atom ( o r icc) form a closed s h a l l .
4 )
Ever. when all t h e above c o ~ d i t i o c s a r e observed, t h e zonal t h e o r y w i l l caase t c f u n c t i e ~ ir. problems where t h e r e i sar.
i ~ t e r a c t i o n -mtweox t v o o r s e v e r a l o f t h e ltaxtral' e l i ~ c t r o r , ~ , Ir: o t , , ~ r lsords, the zonal. t h s o r y i s c o t a p p l i c a b l e t o prob1e:ils ~ r , which we are d e a l i c , withthe encoucter of two ar.omalous atoms ( o r i o c s ) , wh3c t h e s e t-uo momalous atoms ( o r i o n s ) are adjaescb t o s a c h o t h e r ar,d car, e c t e r h t o i n t e r a c t i o n s with each o t h e r ,
The c0nditior.s formulatod above def ice, f n g e n e r a l , t h e zocal t h e o r y as a met!lod of d s s c r i b i c g tile bahavicur of' e l e c t r o n s ir, a l a t t i c e , r e g a r d l e s s of whether m e are d e a l i c g with a m e t a l l i c or coc-metallic l a t t i c e .
V,
Conccrnicg t h e A p ~ ~ l i c a . b i l i t y of t h e Zonal Theory t o M e t a l l i c m.d t o-
' . . N o n - b t a l i c C r y 5 t a l s _r___--..,_.-.-.-- .-..---.-.--..---.._-.---...-..--.--._~.- "-...----W w i l l e ~ o n c o c s i d e r t o what e x t e ~ t ; the f o u r ccr.ditiocs formulated car. be satisfied i r , -the case of metals.
Disregarding t h z cp.12 stior: concercir:g -the f u l f i l n e c t ox t h e non- f u l f i l n i e n t of the f i r s t two conditions,
it.
@el: be stated t-hat i n t h ecase of a metal t h e t h i r d cor:diticr, i s d e f i n i . t e l y c o t fulfilled: the proper o l e c t r o ~ s of thr? ::;etc+l.'iic at,on cl;c r:ct form a closed s h e l l , m.d
t h e e x t r a e l e c t r o n adjoir,e=l t c , the !;:etallic atom d i s t o r t s tklo l~ehavi0LW of i t s i n c e r e l e c t r o c , 'These i m e r e l e c t r o ~ s , o r rnore c o r r e c t l y proFer e l e c t r o n s , of t h e a$oir, f e e l t h e aj;Jearar:ce of t h e " e x t r a u e l e c t r o r , ,
Thus, the a p p l i c a t i s r , of tho zor:al theory t c the case of metals, s t r i c t l y spe ;!iir.g, shoalcl be mc.s%c?erod as raa t s u b s t a ~ t i a t e d .
\'hat, then, i s t h e p o s i t i o n i n t h e case of ~01.1-netallic i o n i c c r y s t a l s ?
Tho i o n i c c r y s t a l may c0r.t a i c " e x t r a t t e l o c t r o a s . These e l e c t r o n s ( r e g a r d l e s s of t h e i r o r i g i n ) a r e the c u r r e z t c a r r i e r s ~ r , a nor,-neta1l.i~ c r y s t a l , I n otdi.18r words, anocg t h e corm& i o r ~ s t h e r e e x i s t s a c e r t s t i . ~ ~ u ~ t i t y of momalous ioris with
w.
excessive r,cg-ative or. a~ excessive pljositive charge.The
f i r s t dnter.!sl.r:es t h e el6:ctrcnic c o ~ d u c t i v i t y ; the secor,d deter*mir,es t h ~ h o l e cccduc t.i.v.i:ty,,I!:
tile h s h a v i c r ~ ~ r . of these a:omalous ions which it-.tsrests 11i3 vrher, vrz sp3:d;cf
t h e eic;.ct-.,:)clcc o c d u c t i v i t y of t h e l a t t i c e .
The
concentrat,i.or: of tliase z~omalous ior,,.;i n
noc-metallic i o c i c cryz-t.als i s , under ordiriary conditior,s,quite s m a l l . It i s s o small t h a tt h e i o ~ s m.ay be cocsidered or, t:52 average as h i c g separated by s u f f i c i e ~ t l y l a r g e aistar.ces
-
d i s t e n c e s a t which i t i s completely p o s s i b l e t o i g ~ o r e t h e u i n t c r a c t i c r , with ,3ach other,,F u r t h e r , t h e proper e l e c t r o c s of t h e l a t t i c e i o n s f o r n closed s h e l l s ; t h e r e f o r e t h e e x t r a e l e c t r o c , adjoiced t o t h e ior, i c additior, t o t h e ~ o r m a l conylemect, does c o t d i s t o r t o r almost does c o t d i s t o r t the behaviour of t h e proper e l e c t r o n s .
Thus, t h e f i r s t t h r e e c o c d i t i o c s a r e approximately f u l f i l l e d ,
The c r i t e r i o n of t h e a p p l i c a b i l i t y of t h e zonal t h e o r y i n t h e case of t h e i o n i c c r p s t a J i s t h e r e f o r e t h e l a s t , f o u r t h , c o c d i t i o c .
We car, make use of t h e zocal theory i c t h e d e s c r i p t i o c of the heha- a i o u r of t h e c o c d u c t i v i t y e l e c t r o c s as l o c g ' a s t h e s e e l e c t r o c s car, be cocsidered as being s u f f i c i e c t l y s e p a r a t e d , There e x i s t s a wide group of ~ r o b l e m s f o r which t h i s assumption can b accepted.
However, t h e r e a r e problerns i n which t h e c l o s e n e s s of the ~ l e c t r o c s
m.d the r e s u l t i r . 6 i r - t e r a c t i o c betweec them i s e s s e n t i a l , P r o b l m s of t h i s type c a r i o t be described ir, t h e l m ~ a g e of t h e zonal theory. These
r e q u i r e a d i f f e r e n t approach,
KC,
The Zocal P i c t u r e ir. T r m . s l a t ~ o r .t o the Model
Lm;,ua,?eTe w i l l c o t e t h a t ir, t h e case of t h e i o c i c c r y s t a l t h e z o c a l ( 5 )
>j c t u r e may be e a s i l y t r a x s l a t e d i n t o t h e d e s c r i p t i v e model l a g p a g e
?le will cocsider
an
i d e a l i o n i c l a t t i c e , L e t u s essume f o r t h e sake of s i m p l i c i t y t h a t we a r e d e a l i x g w i t ha
c r y s t a l of t h e " type MmRr (whereLA
i s t h e symbol of t h e metal aEd R is t h e symbol of t h e m e t a l l o i d ) , c o c s i s t i n g of p o s i t i v e m e t a l l i c i o n s ar,d of negative m e t a l l o i d i o c s ,A s long a s all t h e i o c s of t h e l a t t i c e possess t h e i r cormal
charge, ar. e l e c t r o n i c c o c d u c t i v i t y i c the l a t t i c e i s bwnera.lly c o t possible. Ic t h e zocal schemlu t h i s nsacs t h a t t h e lower zone ( t h e so-called ~ o r m a l zone) i s completely f i l l e d with e l e c t r o n s , i . e .
i t
does not contain a s i n g l e hole, ard the upper z o ~ e ( t h e c o c d u c t i v i t y zoce) d o e s c o t eontain a sir.gle e l e c t r o c .L e t u s now imagise t h a t vve have remoaed ar, e l e c t r o c from oce of cho negative ior:s a d have placed
it
or. o m of the s u f f i c i e n t l y rernoved p o s i t i v e i o ~ s . t h e z c c a l scheme t h i s 7 , v i l l meac t h a t ive have t,rar:sferred ar:e l e c t r o n from t h e l o v e r zoce irlto tha upper zoce. Thus, t h e aph-eararxt! of
ar.
e l e c t r o c i n t,he c o c d u c t i v i t y zoce a d l;hu s i n x l t m e o u s appaararxe 01' a hole i c t h e normal zoce mewas t h a t two m.omaious i o c s with opposite a u x i l i s r y charges have beec created i n the l s t t i c e , which are s u f f i c i e n t l y f a r apart. ?'he ecergy expended or. such ar, o p e r a t i o n i s equal t o t h e d i s t m c e betweenthe lower ar.d u p p r zol:es (mo-e e x a c t l y , betwees, t h e c e c t r e s of these z o ~ e s )
.
The motion of t h e e l e c t r o n along t h e zoce of c o n d u c t i v i t y memas t h e %rar.spositioc alocg the l a t t i c e of t h e n i e t & l l i c anomalous ior!. The motion of the hole along the r-ormal zone n!ear.s tlie t r a c s p o s i t i o n of t h e metalloid azomalous ion, Here, of course, we have i n micd not t h e traEspoSitior, of the anomalous i o n i t s e l f , but t h e t r a r s p o s i t i o n of t h e snomalous s t a t e which i s t r a n s f e r r e d from oce of t h e i o n s t o a neighbouricg i o n of t h e same kind.
The recombir.atioc of tu~o opposite anomalous s t a t e s , accornprnied by a reductior, of a p a i r of normal ior,s, i s described ir, t h e lwL.uage of the zonal scheme as tile f a l l i . c g out of ar. e l e c t r o n from t h e zoce of c o c d u c t i v i t y i c t o t h e h o l e of the normal zoce.
It should be coted t h a t t h i s p r o c e s s of recombinetion i s o u t s i d e
t h e frme of t h e z o c a l t h e o r y , T h i s i s an example o f a problem which c s c c o t be d e s c r i k d by t h e z o c a l t h e o r y ,
IR
f a c t , t h e reco!nbicatioc of ar, e l e c t r o n a d a h o l e i s a r e s u l t of t h e i r ~ t e r a c t i o n produced 'be-tvieer, them. T h i s me =s t h a t t n o ar.ornalous i o n s n i t h o p p o s i t e a u x i l i a r y cha.;e ss3'e foucd d i r e c t l y a 6 j a c e r . t t o each o t i i e r ,
Ir,
t h i s p r o c e s s of r e c c ~ b i c a -t i o c t.he z o c a l t h e o r y i s capable t o 2i.vi.n~ c n l y tohe i n i t i a l ard t;he J.'ir:d.
T h i s , of c o u r s e , i s ~ i h o t r u e of the r e v e r s e p r o c e s s
-
t h a ;;recessof
i o c i z a t i o c ( t h e formztior, o f tvio m o r n d l ~ ~ u s ior:s f r o n a >air cf I I O ~ ~ E L ! . i c ~ s ) ,2-;
t h i s case t o o , .tile zc;:&l tlleory c a c ,d::. . : : c r i b c n l y t h e i r : i t i a l zr:ci %!it:f i l - a 1 s t ~ t e s ,
Batv~eec th e s e two stat.-es .tilure e x i s t i c t e r n e d i a t z s t a t e s v;llich, ~ a t u r a l l y , c a ~ : iri no nsy bt? f i . t t s d i c t o t h e z m a l scheme, These are t h e so- c a l l e d e x c i t o n s t a t e s , 1% n i l 1 d i s c u s s t h e s e l a t e r ,
?JII, The ~tNorn~allL Zcce ir, t h e Zcr:al Scheme of t h e N c c - b , e t c r l l ~ q = ~ ~ ~ s ~ a ~
-.
Thus, t h e behaviour of t h e c o ~ i i u c t i v i t y e l e c t r o c s , i n - . , t b a l -tl-ocs
ir,
t h e upper z m e , may be d e s c r i b e dby
t h e Bloch method. T h i s prob1.-I;? i s feud w i t h i n t h e frame of t h e z o c a l theory,, Ir, tj3.s c a s e t i i e r eo b j e c t i o c s t o t h e z 0 c . d t h e o r y ,
O b j e c t i c c s t o t h e zocnl t h s o r y apkoar, however, rrhsn cre ~o ov5r t o t h e e l e c t r o c s o f t h e lower (normiil) zosa, t h a t i s , nhen :!a a r e f a c e d v l i t i l
t h e problem of t h e b e h a v i o u ~ of c1ectror.s f i l l i n g ((coiipletely o r alrvost (:ample t e l y ) t h e lower zoce
,
Actually, i.r_ desc:;.il;ing <he i ~ : ~ ~ . v i o u . ~ - o f t h ? e1ectror.s i c the zoce of c o c d u c t i v i t y we remain w i t h i e t h e frame of tile sing1.e e$actror,ic problem. However, i n d o s c r i b i c g th3 b e h ~ v i o u r of t h a holes ir, the cormal zone, we a r e d e a l i c g with a "si>:gle h o l e f t problem, i f oce car, use such a:: e x p r e s s i ~ c , but s t i l l not with a s k g l e e l e c t r o c i c problem,
This m e a s t h a t the e l e c t r o s s f i 1 l i r . g the lower zom cnncot pretacd t o i c d i v i d u a l i t y so e s t o %e considered as being i ~ d e p e n d e n t of esch o t h e r ( i c o t h e r words, t h e y cannot have i n d i v i d u a l wave f ' u ~ c t i o ~ s ) .
A c o e s i d e r a t i o c t h a t t h i s i s so would cceJ be j u s t i f i a b l e , s i c c e ti,o e l e c t r o r , ~ of t h e norrriril zoce are the proper e1ect.ror.s of t h e lat-tiice i o c s , The problem concernicg t h e i r behaviour i s essectia!.ly a p o l y - e l e c t r c c i c
pr:)blsm and
i t
cancot -IX reduced t-o s s e r i e s o f . indepe1:dent s i c g l e eluctror:ic prc 17: ems.The e l e c t r o m of' t h e lower zuce form a c e r t a i r group, t h e h l l a v i o u r of which c m b3 d e s c r i h d by a j o i n t wave f u c c t i o c ::llich d e p e ~ d s or, t h e
co-crdicates of all t h e e l e c t r o c s mid which does c o t break up i n t o products o f t h e separated i n d i v i d u a l i'utlctiocs,
This j o i c t f u c c t i o c , zs has already been s t a t e d , i s d e p e ~ d e c t oe the
co-ordicates of a l l -the e l s c t r o c s , Here we have ir: nir.d dl t h e c a t u r a l e l e c t r o n s of the n e t a l l o i d i o n s of t h e l a t t i c e , with t h e exception of t h e or,e sir,gle e l e c t r o c , t h e a.csecce of vhich means t h a t t'tiere i s a hole. I t s co-ordicates a r e c o t r e p r e s e c t a d i n the j o i c t fur.ctioc and t h e r e f o r e we cac s a y t h c t t h e behavicur of a hole car. b described with the help of t h i s functior:,
-
15-
Thus, tlie lower z o ~ e has a d i f f e r e c t mear.ir,r, t h = t h e upper zoce,
'The u?!.)er zor-e r e p r e s e c t s a syst,ern of e l e c t r o c l e v e l s , ar.d t h e lower zone :cepresentJs a system of h o l e l e v e l s ,
Thz r r e l a t i o c s h i l ~ betweer, thorn i s t h e s=e as t h e r e l - a t i o c s h i p betweer. t h e optical. terms i r , a s l ~ g l e e l e c t r o ~ i c atom ar,d tile X-ray terms of t h e p o l y - e l e c t r o c i c atom which correspocd t o t h e removal of ar, e l e c t r o r , fro11, one of the e l e c t r o ~ i c l a y e r s , The X-ray Le1.m d o e s c o t e x p r e s s t h e e c e r g y of ar, r i r d i v i d u a l electron., buL t h e er,ergy of t h e t o t a l system as a whole,
J's'e o b t a i n f o r t h e atom a wliole s e r i e s of' X-ray terms, sir,ce t h e
'3r,ergy of t h e system w i l l depecd t o some d e g r e e o; t h e f a c t of p r e c i s e l y T ~ b i c h e l e , - t r o c i s d e f i c i e c t
i n
t h i s system, ' This and o r J y t h i s weaningI:= ue giver. t,o t h e lower zoce i n t h e o r d i ~ a r y z o c a l scheme of t h e rclr.-
n ~ a t a l l i c crys+,al,
Ir,
t h e diagrams giver, ir, t e x t books on atomicthe or^^
t h e e l e c t r o c> I. ..-.
..
c ; v ~ l s are g e r , e s d l y p l o t t e d i r , upward d i r s c t i o r , a d t h e h o l e (X-ray) :!!i?-?.veis i c a dowcv!ar.d d i r e c t i o r , , , ipe hav.e t h e same i r , the zor,al scheme o fi r r y s t t i : ir, i t i h i ~ h ~ lioiva~~i.', tlia e l e c t r o r , a d t h e h o l e l e v e l s a r e combiced cr, oce diagram,
It shoulci be toted t h a t t h i s o r J y a c c e p t a b l e i c t e r p r e t a t i o c of t h e r.oiClfid zoce d i f f e r s from tho o r d i ~ a r y I3loch ior, i r , t e & r e t a t i o c , Ir, f a c t ,
i r * t-:16 Hloc21 method t h e "propert1 e l e c t r o c s tire c o c s i d e r e d or, ar, e q u a l f o o t i c g
w i t h t h ~ "ext,rat' e l e c t r o n s , Ir, t h e Dloch t h e o r y t h e e l e c t r o c s of t h s lower zoce a l e also indepecdent of each o t h e r , t h e same as t h e e l e c t r o c s of t h e upper zone, a d each i s d e s c r i b e d by i t s owr, i c d i v i d u d f'ucctior,, This
i s a l s o i n c o r r e c t , j u s t a s
it would
be
incorrect; t o c o ~ s t r u c t ap l y -
e l e c t r o n i c atom c o c t a i n i ~ g G e l e c t r o n s , havicg taker, t h e system of t h e
hydrogen l e v e l s arcl placed or. these l e v e l s , one a f t e r m o t h e r , a l l of our c e l e c t r o n s , a ~ r d havicg cocsidered only t h e
Peull.
p r i c c i y l e .Objections t o tho application
of
the zonal theory t o t h e e l e c t r o n s of the corplal zoce are, ir. t:ssecce, objec.bions t o the Blochiac i c t e r p r e t a t l o r . of t h i s zcne. Such a concopt of the lowermce
i c
t h enarrow
l i t e r a l Blochim mear.ir.g i s of course i n c o r r e c t . This i s completely obvious,However, whec
an
e x p e r i ~ e n t e r d e p i c t s t h e I.otver cormal zoce (and. he c a m o t g e t aloe, withoutit
ir, t h e bterpretatior, of h i s e q , e r i m e c t a l re:jul.ts), he by no mear.s a s s i s s t oit
a Blochiar. meardng.The lower zoce i s not a system of e l e c t r o n l e v e l s , f i l l e d or almost f i l l e d with electrons. It 'is a system of hole l e v e l s which are c o t a t the p o i c t of being f i l l e d with holes. The lower zoce
i.s
"emptyuic
the same degree with respect t o holes a s the upper i s with r e s p e c t t o e l e c t r o c s .IVhec we speak of the lower zone, we speck not of the behaviour of t h e i r d i v i d u a l e l e c t r o c s i ~ d e p e c d e c t of sach o t h e r , but of the behaviour of the i c d i v i d u a l holes icdependect of each other. , I n t h i s case we do not reduce the p l y - e l e c t r o n i c problem t o a single e l e c t r o n i c problem. Here w e reduce the poly-electrorLc problem t o a s i R g l e hole problem.
Thus, the cormal s t a t e of the system cancot be described with the help of t h e zonal theory, whec i n t h e upper zoce there does c o t e x i s t a s i c g l e e l e c t r o n , =d i n t h e lower zone not a s i n g l e hole. However, with
t l ~ e h c l p of t h e zor,al the.o;.y jt
is
p o s s i b l e t,o d e s c r i b e t i l ~ e x c i t e d s t a t e s . cf t h e eyste1.1, c l ~ i l r a c t e r i z e r i by t h e p r e s e c c e of e l e c t r o n s ir. t h e u , , e ~ -zoce =d by the preser.ce o f h o l e s i r , t h e lomsr zor,e.
VIII
, C e r t air, Approxirnatiocs C o ~ t a i r g t i ir: t h e Zor,r;L S c h ~ m eIt
i s r,ecesss.ry t o ~ o t e t h a t t h e zor,al scherne, ~.epreser,t,ir,g ir, i t s simplest. form two er,ergy 'um.ds suk-clividcri by e forLli(ida1: r ~ g i o n , i s a s s o c i a t 2 d w i t h s t i l l o t h e r s i m p l i f y i c g a s s u r n g t i o ~ s .T h i s scheme r e f e r s o c l y t o i d a a l , acd moreover an i c f i c i t e , c r y s t a l i i r , e l a t t i c e , i,e, a l a t t i c e i n which t h e r e does r,ot e x i s t ally b o m d a r y s u r f a c e s o r any i n c e r s t r u c t u r a l d e f a c t s , T h i s c o c s t i t u t e s t h e f i r s t si.rii!,lifyicg assunpt;icc,
'Thus, a l l t h e p r o p e r t i e s of t h e re,al. c r y s t a l s rrihi.ch s r e determhed by t : h e i r d e v i a t i o r , frorn -t113 ideal s t a t e , i,e, the? wl1ol.e group or t'he
properties secsi-Live t o s t r u c t i ~ c + car:r:ut i c t e r p r e t e d frola t h e v i e s ~ p o i c t of this sirnp1.e z o ~ a l scheme,
lir.ot,her ~ p p r o x i m n t i o c cor.t,air:eri ii-: t h i s sche~ae c o c s i s t s i c t h e assumptioe t h a t t h e apyeara.r,ce of w. w-omelous -ioc i c t l L e l a t t i c e does not produce acy c h a ~ g e s i c t h e i.,osi%ior: of t h ~ cormdl i o c s s u r r o u c d i c g i t , A l l tk i o ~ s romaic a t t h e i r p l a c e s , I n o t h e r -aords, t h e appearacce of nc e l e c t r o c ir. t h e zoce o f c o ~ c l u c t i . v i t y , o r ,of n ho1.e i c t h e cormal zoce
,
i s r_o t accor:lpar.ied by ar, deformatioc of the l a t t i c e ,h t h a s s u m p t i c r . ~ ( t h e ass~~rrq.!tloc cocca:~r.ir.i_; ai~sor,ce c f dc.f@cts i n t h e l a t t i c e sr.6 t h e assuiq;tiul: concer.ni.r.g tkx2 dcf ometic,r. tile a ~ o m a l o u s io c s ) r e p r o e a c t a kr:o~n ~ i ; ~ p l l i ' i c a t . i o r : ~ f ' tbs E I C ~ U U I p i c t u r e ,
If
we i:et 1.56 of L1,e f'irst assuri:ptiol:, -tiioc .:re w i l l a r r i v a fitt h e c o ~ c e p t cccccrcir:g tl1e e o - c d l e r i l f ~ ~ o l ~ o r , s t ~ .
We w i l l t a k e both t h e s e s-teps.
IX,
The "Disordern i n t h e L a t t i c e ar,d its i i e f l e c t i c r . j.r, t h e Zor:al ScherzeW w i l l e pass from
w.
i d e a l c r y s t z h t o a r e d c r y s t d ,The r e a l c r y s t a l d i f f e r s from t h e i d e a l crystal. b j 7 t h e preseccz of
d e f e c t s , i , e , l o c a l d i s t o r t i o c s i c t h e s t r i c t l y p e r i o d i c s t r u c t u r e cf t h e l a t t i c e , Empty l a t t i c e y o i c t s , atoms o f a f o r e i g n i n p u r i t y , as w e l l as t h e p r o p e r atoms o r ion:; of t h e l a t t i c e d l i c h have e c t e r e d i c t o t h s i . r . t e r s t i c e s , may -be used as aE e x a m ~ ~ l e o f snch d e f e c t s
( 5 )
The d e f e c t s i n the l a t t l c e have t h e i r . o;vc r e f l e c t , i c r , ir,
tb
occtrgy spectrum, The preser.ce o f a d e f e c t g e m r a l l y l e a d s t o t h e zppearmce of a d i s c r e t e e n e r g y l e v e l (th;: s o - c a l l e d l o c a l l e v e l ) , which &ecu.rall3: f a l l s i c t o th-. forbidder, regior, ixtirccr, tlle zoces.The l o c a l l e v e l s e r e cheracterizecl by vreve f u c c t i 0 r . s rj!lich are e s s o c t , i a l l y d i f f e r e c t ic form from t h e wave f u n c t i c c s c o r r e s p o c d i c g
t c ;
t h e l e v e l s of t h e er:ergy zor,e.Ir.
f a c t , ar, e l e c t r o r , l o c a t e d ir. t,he zoce of c c ~ d u c t i v i t . j . ma;; be d e s c r i a e d by wave f u c c t i o c s t,h- model o f which i s p e r i o d i c mith t h e p e r i o dof t h e l a t t i c e . This mar.s t h a t such ar. e l e c t r o r ,
i s
f r e e t o t r a v e l over t h e whole c r y s t a l o r , ic o t h e r words, t h e r e i s ar, 'equal p r o b a b i l i t y t h a t such a x e l e c t r o n m a y ba foucd i n m y placo or. t h s c r y s t a l . E i t i ~ r e s p e c t t o such ac e l e c t r c r . , i t car, i ~ e s a i d t h a t it i s uciformly d i s t r i b u t e d over t h e whole c r y s t a l ,The e l e c t r o r . placed oh t h e l o c a l l e v e l i s c i ~ a r a c t e r i z e d b;ir a wave fur.ctioc 1,hich has a sharp maximum Rear the d e f e c t arAd which i s s h a r p l y damped with a departur.2 from
i t ,
This mear.s t h a t such aR e l e c t r o r . i s l o c a l i z e d ir. a def ir.itc? r e g i o n of the c r y s t a l (namely, ir. t h e regior. of t h e d e f e c t ) , It has a mavimun p r o b a b i l i t y of existj.r.g n e r r t h e d e f e c t , .=d t h i s d e c r e a s e s with t h e d e p a r t u r e from t h e clefect, Cor.siderir.g tho e l e c t r o r , as a. d i s t r i b u t e d charge, oce cm s t ~ t e that, t h e d e x s i t y of t h i s charge d e c r e a s e s ~ i i t l i t h e d e p a r t u r e from t.he d e f e c t ,It Should 'be r,oted t h a t ' t h o degree of t h e l o c a l i z a t i o n of tho
electcror. cittperds o c tile positior: of the l o c a l l e v ? l s i c t h e ecergy s p e c t r u n , The c l o s e r the e l e c t r o c i c l o c a l level, i s t o t h e zor.2 of cor.ductivity, t h c l e s s i s the degree of l o c d i z a t i o r , of t h e e l e c t r o n locaixd or: t h i s l e v e l , ad t h e more c i i s t r i h t e d is t h e -r;ve fur,ctior. correspor.dicg t o t h i s l e v e l . A s t h e l o c a l l e v e l approaches t h e lower extrerne of t h e c o ~ d u c t i v i - t y zor.e, t h e l o c d i z e d e l e c t r o n i s &ra:ciually t r a c s f orned i n t o a c o l l e c t i v F z e d
a l c c t r o r : arcl t h e very concept of the l o c a l l e v e l g r a d u a l l y l o s e s mesnir.G(6).
Ir.
g e x e r a l t h e d e f e c t s 1;rt;ser:t ic t h e l a t t i c e arid d e t e r r n i r . 5 ~ ~ t h el o c a l l e v e l s i r , tile er,ergy spectrum p o s s e s s a c e r t a i n rnobil-L'ty, This n o b i l i t y i s a r e s u l t of t h e p s r i o d i c s t r u c t u r e of tlie. l a t t i c e i t s e l f ,
A fiur~ber of f a.ctors substm.tiat,e t h e t r a v e l of tiie d e f e c t s ir.side the c r y s t a i , The ior.ic c o c d u c t i v i t y
ir,
the c r y s t a l is, fr, i t s e l f , such a s ~ l l ; s t ~ ~ t i a t i c r , ,The d e f e c t s t r a v e i l i c g
ix
the l a t t i c e e x t o r i c t o ictderact3.cr, w i t h each o t h e r vlic:c thay 8pysoaci.1, Thus, f o r exengla, two empty n e t a 1 l . i ~ l a t t i c e poir.Ss repulse aach o t h e r , t m d i c g t o break up i n t l ~ opposite d i r e c t r o c , Cr, the cor:t.rary a m e t ~ . l . l i c empty corcar accl 8 n e t a l l i c i n t e r s ' t i c i a l ior.a t t r z c t each ot1:er.
Ther, b i g or s e v ~ r ~ d d e f e c t s are joicad toget,l~dr t h e cevr d ~ f z c t
ah' i s producecl, &et:eral?y s;,uakir,g, 1,onse:ises r,ew ~ r 0 p e r t j - e ~ ~ TIii.1~ we have icsicie a l . n t t i c e a c h a r a c t e r i s . t i c "c1ieni~trjt- of d 2 f e c t s td7)
kimocgst the d i f f e r e c t r e a c t i o c s i n t o which tlle defec-ts car, e c t e r , t h e r e <2xlsl; such rr:actior:s as g e c e r a l l y l e s d t o tiie clisappearmco of the reect:l.r,g ' , ' p a r t i c J e . ~ ~ ~ (reactj.ocs olio recmrnbical;ioc), as v ~ l Z a s reac.t;"lria whicl~, or, the cocti-my, lead to Lk~e a;,p,mance of d e f e c t s ,
Ir:
o t h e r v~ords, .ithe 1.a-ttlce i t s e i f i s caiia'hle of o r i ~ . ~ a t i c g acd of absorbicg d e f e c t s ,A t axy given teniper zture, t h e r e e x i s t d e f i c i t e eq7lililjriur: tor.- cei.:t:xetion.s f c r the d e f e c t s of various kjLr,ds ,,
The: ~:cergjr spectrum o r the crystzl. shou1.d t h u s be col;side.red i c 2er;er.a.l as vury5r.g wit"n t h e t e n p e r a t u r e , Upor, h e a t i c g , l o c a l i.e~i:~is of oce lcind ,-ippe.s,r a t t h e e ~ e n s e of t h s dLsap;;earar,ce of t h e l 9 c a l leve1.s of i:ti.ier kir:G:i.. ,A;,.,., L,,c t ~ t . a ; ~urnLxr of l e v e l s v s r i e s ,
Ir.
0 t h ~ P Y O ~ C ~ S : , theeEergy speztl::gn r e f l e c t s t h e c1:iernicd r e a c t i o c s i:2 zhic'h tha d c f e c t , ~
The t o t a l number of d e f e c t s i c a l a t t i c e c h a r a c t e r i z e t h e t which
mzy
be c a l l e d t h s d i s o r d e r i n t h e l a t t i c e . . T h i s d i s o r l ~ e r h a s a double o r i g i n , VJe cac d i s t i r : g u i s h i n it two p o r t i o n s : t h e h e a t p o r t i o n acd t h e bic2gse;shic p o r t i o n ,The b i o g r a p h i c d i s o r d e r r e p r e s e c t s t h e n o n - r e v e r s i b l e p o r t i o c of t h e d i s o r d e r , T h i s i s t h e p o r t i o r , of t h e d i s o r d e r which i s r e t ~ n e d a t zero temperature, The d e g r e e of t h e b i o g r a p h i c d i s o r d e r i s determiced by t h e p r o d u c t i o n o f . t h e sample and by t h e whole h i s t o r y of i t s l i f e
-
i r . o t h e r words, by i t s biography, It i s a d i s o r d e r which i s g i v e n t o t h e l a t t i c e , s o t o speak, " a t b i r t h " ,On t o p of t h i s b i o p a p h i c d i s o r d e r i s s u p e r h p o s e d t h e r e v e r s i b l e h e a t d i s o r d e r rvhicll i s of a ternpa~aturc.: o r i g i c ,
The r e l a t i o r s h i p betweec t h e h e a t acd tlla u i o g r e p h i c d i s o r d a r s i s c a t u r o l l y depende1:t on t h e tenlpersture ar:d or, t h e biogi-z,phy of t h e s a m p l e .
Thus, t h e t r a c s i t i o n from a c i d e a l c r y s t 6 1 t o a r e a l c r y s t s l i s a s s o c i a t e d w i t h t h e s u p e r i m p o s i t i o n of a c e r t a i c d i s o r d e r on t o p of t h e s t r i c t o r d e r which r u l e s t h e i d e a l c r y s t a l ,
I'Jhile t h e s o - c a l l e d s t a b l e p r o p e r t i e s of a c r y s t a l car: be determined by t h e f a c t o r of o r d e r ,
i c
o t h e r words, by t h e p e r i o d d c s t r u c t u r e of t h e l a t t i c e , , t h e p r o p e r t i e s s e c s i t i v e t o s t r u c t u r e on t h e c o n t r a r y a r e determined by t h e f a c t o r of d i s o r d e r , i,e, by t h e l o c a l d i s t o r t i o R of t h i s p e r & o d i c i t y ,The accour:t of tile d i s o r d e r i:: t h e l . a t t i c e , i,e, t h e t r m s i t i o r . fro^ ar. i d e d t o a r e a l c r y s t t i l , cac, 'k sccoapiislielj r v i t h i c t h e frarrs of the zor,al t h e o r y . Ths z o l ~ a l t h a o r y i s r,ot tlssociated o r g a ~ i c a l l y ir, m.y
way with t h e f a c t t h z t bhc! c r y s t a l i s i d e a i ,
X,
.The:
Idea Concegpin.,: P o l a r o c s .a:d.Ae Zone1 Theory!?e m i l l t u r r , c ~ t o acother simplif'yir,; s . ; s ~ ~ n , ~ t i c c r ~ v!ilch i s four,d i r , t h e c o ~ c l u c t i v i t y e l e c t r o r . , i , e , , moucu t h e " e x t r a " e l e c t r o c ir,tcod:lced i n t o t h e l ~ . t t i c e ,
I n
t b s case of t h e i o n i c l a t t i c e t h i s daforrnation c m be reduced t o t h 6 d i e l e c t z - i c p o l a r i ~ ~ a t i c :)f t h e l a t t i z o by the. f i e l d or!' t h e e l e c t . r o n ,Ic the or.dir,axy z o c a l t h e o r y t h i s effect I s i ~ r , c r i ? d .
The problem cor:csrcir,g t . 1 ~ e l e c t r o r , ir, t h e i o ~ i c l a t . t i c e , ~vl..l.th
ar.
account of t 1 . i ~ p o l a r i z a t i o r , prcduced by t h i s e l e ctroi:, ;.Jas malyzed b y S..I .
~ e k a r ( 9 1 Ar. e l e c t r o c surroucded 5y a p o l a r i z s d l a t t i c e cas ~:amrd a y o l ~ r o n . ,Such
ax
elec+,ron t r a v e l s 8lo;:g t h e c r y s t e l toge:t.hsr x i t h t h e deforrnn- t i o r . which surroul=S~sit.,
A s i t moves t>Josi;; t h e c r y s t , a l , , 5.t t r a i l s p o r t s t h ep o l f i r i z a t i o n which surr*cucds j.t, TIE c o ~ d u c t i v i t y d e t e r n i ~ e d b2i s ~ c h e l e c t r o r , ~ i s c a l l e d t114 p o l a r o ~ i c c o ~ ~ u c t i v i t ~ r The d i f lrerucce batbveeR the e.Leciror,i.c
=d t h o p o l a o c i c co::ductjivities caE be reduced t o t h e d i f f e r . - c c e ic t h e v a l u e of t h e e f f e c t i v e Inass o r t h a value of ',he m o b i l i t y o f t.he c u r r e n t c a r r i a r s ,
The p o l a r o c i c cor,ciucti.vity i s iR e s s e m e t h e o r d i r , s r y elect,rc?r,lc conducLi.vity ir, a i l i c h , hornever, t h e p o l a r i z a t . i o r , oi" t h e ; . a t t i c e ir, t h e regior. surro11cdir.g t h e moving e l e c t r o n 5.s t a k e c i ~ t o esco:lct,
Ilowever, t h e a c c o u ~ t . of t h g p o ? . a r i z a t i o r : :leads t o ar. e s s e c t i a l .
cornpii.catior, of t h e c h s r a c t e r o f t h e elect.ror: motioc; t h e c r y s t a l polari2,ed w i t h t h e electrnoc r e p r e s e c t s f o ~ t h e s l . e c t r c ~ r , A ~ o t ~ e l l t i a l p i t i n s i d e nf 1 ~ 2 1 I ~ l i t h e e l e c t r o r , o s c i i l a t e s , The p o ' t e c t i a l . pi-t i t s e 1 . f i s t r m s p o r t e d
a1 ! ) ~ l r ! O t h e c r y s t a l t o g e t k r n i t h t h e ct1ectrc.n r ~ ~ c i ' l . l a t i r , ; ; i ~ . it (1-0) *
From t h e v i e q j ~ i ~ t o.f qila~tun: m a c h a ~ i c s , tJlo accour,t of tile p o l a r i z a - t.j.3~ mear,s t h e a c c o a c t of t h e i ~ t s r a c t i o r , of t+he e.lect,ror, i ~ i . t h t.hc phcnor. f je:d, This i c t . e r a c t i o r , may be e i t h e r l a r g e r j r ' s m d l ir: cornpatson w i t h
the: etuergy of tihe el:?ctror, ic t h e p e r i o d i c f!..el.(! of aL ideal. 'l.at~ti.@e, H e ~ c e
Ic t.hs c a s e o f t h e "weak borXlt"the a e c o u c t o f t h e p o l a r i z a t i o r , , as skicruid be e x p e c t e d , i n t r o d u s e s o c l y s m a l l
corrections
fr, t h e o r d i c a r y z m a l p i c t u r e ,Tk:.
c o ~ d u c t i v i t y zone i s somewhat, d . + l e ~ ? . a c e d dotv,naasds .which .is c ~ i + e r,atu:ral, It car-; ever, b~ said t h a t i t is evickxt. G, x)'riorf, I c fact.,t h e dov~:icuard cl5~p,Saeement o f t h e a l e c ' r - ~ . c n i c I.evt?l s r;l.e=s a r"lacrea::e ic. the e z e r g y of t h e system, Bat t h e dc?for~aa.Lior, i 7 f t h e I.at.tice also o c c u r s
because i t is e n e r g e t i c ~ l l y p r o f i t s b l c ,,
Ir,
t h e case of -the llst.ror,;; t x ~ r ~ d " ';rrzJ..ize2 ir, r:.umerous s r y s t a l . ~ ) , .the c c s e i s somewhat n-~cre complex, From tole cor,siderat,iocs of ti12 t r m s : i t i o n a l i c v a r i e c c e it is evider,t t h a t t h e er,argjr. sl~ect:rum of the systerii h a s a zor,al. c h a r ~ c t ~ e r ir; t , h i s case too, IIowever,
t h e . strnc-Lure of tliese zoces in decaral* i s qui-ce d i f f e r e r t from t h a t gi'ver, by t h e o r d i c a r y B.och tiieory. They are c h a a c t e r i z e d by d i f f e r e c t parameters aEd have a mori: complex form,It should
be
c o t e d t h a t t o g v t h e r v i t h tlie t t e l e c t r o c i c polarocsfl, i f we caR use such ar. e x p r e s s i c c , i t should be p o s s i b l e t o c o n s i d e r p o l a r o c s of m o t h e r k i c d which could bs caned Ithole p o l a r o c s ~ .IE
f a c t , a holt., i.e.en
axomalous i o c w i t h ar, e x c e s s i v e p o s i t i v e charge, l i k e m e l e c t r o c p o l a r i z e s t h e l a t t i c e ar0ur.d i t s e l f . . Ir: c r y s i t . l s p o s s e s s i c g a h o l e c o c d u c t i v i t y t h e c u r r e n t i s t r m s p o r t e d 'by such hole polarocs.account of t h i s p o l a r i z e t i o r , produces ar, upward d i s p l a c s n e c t of t h e t o t a l hole zone a s a u c i t , vrjlich r e p r e s e c t s a gair, i c t h e ecergy
de termiced by t h e de formatior, ( p o l a r i z a t i o c ) of t h e l a t - t i c e
.
Thus, t a k i n g i c t o c o c s i d e r a t i o c t h e e f f e c t of t h e p o l a r i z a t i o n l e a d s , i c t h e f i n a l aRalysis, t o a narrowir.!~ of t h e forbidder. r e g i o n botween t h o upper m d lower zor.es, .However, from t h e viawpoict of t h e zonal scheme, t h i s c o c d t t i o c on which t h e experimecter o p e r a t e s i s c o t e s s e n t i a l , I n f a c t , such values a s t h e midth of the zone, o r tlie width of t h e forbidder, s e c t i o c betweor, t h e zones, a r e c e v e r taker, frcr: theoret,j.cal fcrmulae but a r e detormiced or, t h e b a s i s of e x p e r i n e c t a 1 d a t a ,
The width of t h e f c r b i d d e c s e c t i o n , i . e . t h e ecergy of i o c i z a t i o ~ , may
be
d e t e r n i c e d , f o r e x a r ~ p l e , from t h e o p t i c ~ l d a t a o r from t h e d a t a on elec.Lror,ic c o c d u c t i v i t y ,It should 'be noted t h a t t h e s e two methods of d e t e r m i n i c g t h e i o c i z a -
t i o n ecergy ger:erally give d i f f e r e c t r e s u l t s , The o p t i c a l i o c i z a t i o c energy i s found t o be l a r g e r thar. t h s thermal,
T h i s c o c d i t i o r , beconies q u i t e ucderstar:3ab,le from t h e -vi.ervpoi.r:t o f t h e "polal-oc" t h e o r y ,
Ic f a c t , t h e absorpti.oc of a
lickit
quz-~t;um r e p r e s a n t s a m o n e ~ t r i r y ijrocess durir.g which th,:re i s cot s u f f i c i e c t time f o r t h e i o c s t o bec;isi;lacecl from t h e i r a q a i l i b r i u m p o s i t i o n , a d t h e . r e f o r e there i s r,ot s u f f i.cier,t t:me f o r t h e F o l a r i z n ~ t , l . ~ c of .the l a i t i c e t o k accomi;.lish~-td.
Taler,. na measire t h e i o ~ i z a t i o r , ar:ergy Sy t h e o p t i c a l method, rle o b t e i r . tht, d i s t a ~ c e betaear, t h e z o m s c o r r e s p o x d i c g t o t h e o l d Blochi* t*lleory,
Ho:vever
,
Sc %he c a s e of: t h e thern.al i - ~ e r u of i o c i z a t i o c we are rl.oal,i:2: wit,h a p o l a r i ~ e d lai;t,iclt':;. ~ c c o u ~ y l , of tile p v l a r i z a t i i > r , , as Ta:; ;.,:[re &y s t ax$ed,
le -r.!; c: !:~~.r~owir,i; of' sk. dist.ance bettwear, t h a io\ve;- ar.dupper z o r . ~ o , Tho f or>hi.dderl r s g i o c ke t,ct.er; t.tie zoces i s .carrtiwed,
The i d e a cc:~r:cr-rr:i.r,g t;lio : ! O ; ; F ~ S : ~ ~ L r<>~iail:;j ~it,ii-i.r, tht? fral.rle of t h e
zal::a 'c.];eol-y af s o l id;:,
..
It:
t h i - F . R . - P .., . L. + ntj are d s a l 5 . ~ 2 ::vi t h t h e aecoul:t ofc e r t a i e e f f s c t ( t h e e f f e c t ;;€ pt:~l.al~iziit..i.c:;n) :.lhic't~, i s r,eglc:cl;t;d ir, t h e
/
c~.-cj.ii:ary !3: o c h i mA) vari.&+:. rllf t.i-:e xc!l;s'l .t,.l~i~ijs.y-. T ~ I C accou.r.'t o f t h i s
e i'fect r e p r e s e c t s ax impor-t d c c ' r ~ ~ - : ~ c ~ i . 0 ~ o f ' tilt: z o c a i t!ieory ~rhici:
th,e s t r u c t u r e of thi: zo1:al. spec.bl.um, but by no nlear,s char,;os t h e p.rir.sfpal p o s i t i o c s or, which *I?e zcr~a'l t l ~ m r y i s 'based, and
it
di;es r,ots o l v e t h e d i f f i c u 1 t i . e ~ ~lrll.~ch ?..re c h o . r l a c t e r i s t i c of t h i s t h e o r y ,
Ir: rack, t h e prublem cor:cc!r.r:ir:g t i ~ f polaror, i s considsreri as a
s i ~ g 1 . e e 1 e c t r o e i . c problem, T h e i c t q e r a c t i o . r , 1 ~ t . u ~ t . e ~ t h e pol.ccr(-~ns i s i g ~ o r e d i . n ' i t , j u s t a s t h e i ~ t e r a c t i o n 'betneer, e l e c t r o c s i s i g ~ o r e d i c t h e o r d i c a r y
zocal theory. Therefore, t h i s theory caases t o hold ir. those cases f o r which t h e zocal theory does c o t holc?. Kecce, t h e i r l i m i t s of a p p l i c a b i l i t y a r e i d e c t i c a l ,
The polaror. theor:r, develokecl by SOX. Pekar
ard
h i s co-workers, r e p r d s e c t s ar, accouEt of ar. h p 0 r t a r . t e f f e c t which i s ceglected i c the o r d i c a r y zonal theory ( t h a e f f e c t of the l a t t i c e deformatioc); h o ~ s v e r , i ts t i l l r e t a i c s t h e b a s i c "aturo of the zt c a l theory: i t d i s r e g a r d s the i c t e r a c t i o c of e l e c t r o c s .
The problem coccercing s e v e r a l encoupteric; polarocs i c t e r a c t i c g
with each o t h e r p r a s e c t s ever, g r e a t e r d i f f i c u l t i e s thar, t h e problem coccercicg tile e ~ c o u c t e r i c g e l a c t r o c s wbich, as was already poicted o u t , c a r s o t be
described ir, tht language of t h e zocal theory.
X I , S x c i t o c s
Problems of t h i s type den& n ciifferellt a!;proach,
rn
~ l l e zocal t h e o r y does c o t r e p r e s e c t t h e ocly approximate method f o r t h e s o l u t i o c of tha problem of a l e c t r o c s ir, a c r y s t a l . There are c t h e r approximate methods.
Ir,
cocclusioc, we v i l l d n e l l or, some of t h e problemswhich
extecd beyord t h e frame of the 2or.d theory. A t y j ) i c a l example of such a problemi s t h e _c.roblerr, cor,cercicd t h e exciton,
The coccept of t h e e x c i t o c was proposed by Ya, I. Frenkel- (12) with r e s p e c t t o atomic l a t t i c e s . Let us imagica t h a t amo1:gst t h e atoms of t h e l a t t i c e lound ir, a c o r n a l s t a t e , oce of the a.torns is excitad. Such ar,
i'rom one a ~ o m t o ar:cth:::r r-8 i.ghbi,urirrg zton;, :!l::j .thus Llay 1 1 1 0 7 ~ ~ ;it ;.txLd c ; ~ ~ sl onk; t the ~ 2 - y s t a l ,
Tile i.den of the e:ccFtor, nns dtive i o p d by TrerJcel, es cilx8e.3dy, s t a t e d 9 wi.t.h respe~ct l;o I~ornopolar (motall.ic> l a t , t i c e s , That, t l i u c , corr.espclr,ds t o t h e l?r.eckslizr. e x s i t o ~ : i i l -tli.e c t i m of th? ioci.c l a t a t i c e ?
Jr, the cas;u of' Lha ion:ic ! at;tir:r tilt? exzitor! miiy bs Irna3.ner-l as u p a i r '0 al:orntiLous i0r.s !-iitfi appu:jite ~ \ X < : F ; R : , ~ V C C ~ I G ~ ~ ~ C ~ S :iil%c:stly uiljacefit ti, p ;lc?i ~ t , h e r w.d linlcttd h j .tl!a f'c
rw:;
r:bf' t h o C U U ~ O : I I ? I ~ C s'ttX'tic ti.s~:,d d t s :,minecl
by
exa? Lly such a l;rtlt:ar't?i.b or' at: el.ect,ro~:. f.rbcr:r tiha cogutl.ve 3 o:t:The excitor, t r s v e i l i n g , along t h e c r y s t a l does n o t t r m s f e r any c u r r e n t , s i n c e t h e e x c i t o n r e p r e s e c t s
nr.
ol ~ c t r i c n l l y c e u t i - 4 formet'ioc. Thus "the appearmce of an excit,or, i s 1:ot s s s o c i a t e d w i t h t h e appeararxe of c o n d u c t i v i t y ,If two oppositte =omalous i o n s . joined i n t h e oxcitor,, are
s u f i ' i c i e r , t l y aiclely s e ~ ~ a r a t a d , t h e y become indepecdent of each ot;her, md from t h a t rr?omar,t oc they may be described by t h e zonal theory, Such a b?eahrp of t h e e x c i t o c , o r ir, o t h e r m r d s i t s dissocitztion i c t o ae e l a c t r c r , aEd a h o l e , r e q u i r e s ar, expenditure of e r . e r u , The e x c i t o c , t h e r e f o r e , i s
e n e r g e t i c a l l y nor* p r o f i t a b l e t h m t h e f r e e e l e c t r o r , a d h o l e which figure i n t h e zoxal theory,
XII, Doublons
The probiem of t h e e x c i t o n r e p r e s e c t s a problem of t h e e l e c t r o n acd t h e h o l e i.i:te~.*ac!;ixg with each o t h e r ,
-.
J
W nil.1 e con c o c s i d e r m o t h a r problem vrhich a l s o e x t e c d s beyond
t h e frame of t h e zocal theory: t h e problem of t h e behaviour of two i c t e r a c t i n g e l e c t r o n s ( l 4 ) ,
\Ye w i l l a l s o approach our problem a s a poly-el e c t r o c i c problem t a d vre w i l l s o l v e i t by t h e IIeisecberg method (15)0 This method i s v ~ 1 1 hovin i n t h e t h e o r y of metals.
It
i s f u l l y e q u i v a l e n t t o t h e Heitler-London rnet-hod i r , t h e molecular tlrlenry ( 1 6 I cBy t h i s method t h e problam i s solved a s a p o l y - e l e c t r o ~ d c problem
S o l v i ~ g t h i s problem, we obtair. two types of' so1utior.s (t;wo types of s t a t e s of the system).
F i r s t l y , we o b t ~ i r s o l u t i o ~ s o f the l3loct1i,w. type' c o r r e s p o ~ d i ~ g t o
the
separate ( i c d e p e c d e ~ t of each o t h e r ) e l e c t r o ~ s , rlrher, the p o s i t i o c ic t h e l a t t i c e of one electror. ir, co way p r e d e t e r r ~ i ~ e s tho positior, of tile other,S e c o ~ d l y , we o b t a i n s o l u t i o r , ~ of ar, esser,tinll.y d i f f e r e ~ t type c o r r e s p o ~ d ~ g t o t h e p a i r e d e l e c t r o l : ~ , whec the two electrozls locete:d or. two neighbouricg ions of the l a t t i c e a r e r i g i d l y bocded with each other,
Ir.
other words, the vjave Tunctior, of the systan i r , this casa has a max.i.murn a t t h e mirimurn d i s t m c e betweer, the e?ectror,s ar.d it decreases as t h i s d i s t a c c e i c c r e ase s oThe ~ o r m a l s t a t e of tht? system i s a s t a t e of the f i r s t t j q ~ e , IIorveve~., mongst. the excited s t a t e s of the sy:;tom t h e m cec be preseilt
states
of oce ac3 the ot.!ler t y p e ,The eon~!~ir,afJlor o r b81.1e two e l e c t r o c s i n t o a p a i r i s determixlzd by tllo scco\ir:t af the i1:teraction betwecc t l m ~ ~ lvilich ir:cludes, together x i t h
the
coulomblc i c t e r a c t i o c ,
ar.
exahacge i n t e ~ . a c t i o ~ : of e x a c t l y the same typa ai; i s~ Tsect A i n t h e hydrogen molecule,
