H o p p i n gc o n d u c t i o n | 3 1
H-x
are supposedto bs
so that both for the
esultingfrom the x
n-So our conclusion
[1't2, the rangeof r
maxima,from the x
nsnear E",
(3.2r)
wolfle (1980c,b).
has the property that, whereasin most of spaceif we normalizety'to one
particleper unit volume,thereexistsa volumein which it divergesas (Z/a)3.
Moreover, if the proportion of localized statesis significant(say a finite
fraction r of the whole),then in any volume (3/4, if we pick the right energy,
we should find that a solution of the Schrodingerequation exists which
decaysexponentiallyto zero and yet, when lry'l2is integratedover all space,
the integral is comparablewith that within a few multiples of < from the
centreof the localizedregion.
While we have no formal proof that such statescannot exist,it seemsso
improbableas to raisedoubts about the possibilityof coexistence.
3.5. Hoppingconduction
If the Fermi energyEF lies below the mobility edgeE", we have seenthat
conductionmay be of two kinds.
1. By excitationto the mobility edge.We may then give oo in eqn (3.14)
the value
Dxist at the
bd statescan coexist
rspecimenfor which
E energy.Srivastava
@d statesresult but
re proved that these
p6 1990)to exPlain
ra negligiblenumber
rlieve this to be true.
dzeZ; we shall allow
nparablenumber of
r samesmall energy
they are
eigenstates
to zero
tends
I states
oo = 0.03e2lhLi.
Ths inelasticdiffusion length may then be the result of collisions with
phonons,or Auger processesin which an electronlosesenergyto another
which has energybelow E".
2. By thermally activatedhopping,if N(E.) is finite. This is a processin
which an electronin an occupiedstatewith energybelow E" receivesenergy
from a phonon, which enablesit to move to a nearby state above EF. A
processof this kind was first describedby Miller and Abrahams(1960)as
an explanation of impurity conduction in doped and compensatedsemiconductors(Chapter4). In this work, the electronwas supposedalwaysto
move to the nearestempty centre.Their analysisresultedin an expression
for the conductivity
o:
03 exp(- %lkBT).
e. is expectedto be of the form
cr - l/N( En)cJ
fies separatedbY an
ir6dinger'sequation,
l It is extended,and
further
wherec is the distancebetweennearestneighbours.This is discussed
in Chapter 4.
Mott (1968)first pointed out that at low temperaturesthe most frequent
hopping processwould ,xotbe to a nearestneighbour.The argumentin its
simplestform is the following. Within a range R of a given site the density
32 I Short meanfree
paths and
localization
of stalesper unit energy
rangeis. nearrhe Fermi
energy.
(41t/3)R3N(E).
Thus for_the hopping process
tJ
enersy.
this.^.,!i au'*irii. ;i:',::[],:j:,:i:i,f
,1!;i;*"'
',, n,o'.,, acri
var
ion
._.Efros.and Shklovskiin9
iii_lfti*i;j
Thus,sofar u, ,r," u",iuul'nl
:/@,/3)R3N(E)' the
hopsrhesmaler-ili';;;:
rurrher
rheelecrron
iliiY:t-:_"t*."d.
lili.tt^1"" 'to', u"o"l oo
tunnerrins,r,'.f,"i,oiiiii
disrance
invorves
;ii :""j,:ilt,?:i": rarse
""0
wherel/a is the decay
",;;;;-""T:f #
exp(_ 2aR)
lensth of
beanoprimum
h;;;il;il#tf,l,'jfl,|i
wave
runction.
Sothere
wiu
exp(_2eR)exp(_AElkBn
rs a maximum.Thiswill
occurwhcn
dR + 1l{(rc/3)R3NiJ)kBr}
has its minimum value,
that is wlcn
.
R = {[email protected]+
(323)
S u b s t i r u tlionrgR i n ( 1 . 2 2 ) .
" " , uthat
" . , "the
^ ' .hopping
'
theconducivityis ;; ;;f";J' * " . see
probabilityand thus
A expl- B17tr+7,
where
:'C:)''"(#u)''.
wheree=€- -.
(3.24a)
(3.24b)
flTit}*j:U{iilitilljti iiti;"r*r1,;:*:-$::
ffi";i
ji:
;":n:l:tl;,.;;i:ffi
:,JJ,
{::*;t$gxt,'[txJ::ff
,"
rz+
rHamirtonijii,,
;ffi iifi :?TjxJ;|liJ,,],r":tlces
o=Aexp(_B/T,)
Experimenrauy,
:3t;H,"#";;r;;,ff behaviour.
however,
it i. .,."j,1rrtl
j,i:'ffi:,"r:
T"i:J,';i':,H,"".
j::n1;;:T"rff*.ff
,ri,f
",1ff
,,,?;il
.r-a,.-
th.r" i, u ,ph.rl"o;
;;;H:r".
' iff
(3.22)
o*""sionsee
S
(4;3t
L:::."
"il:::
ifi,..f
jlitH":!f
,I,.r
trfl
ffiffi
.""i*T'ilu[:,ffi:l
i.-,:n
t
,
wnere
(4;3
n(e) :
This must not
rend to infin;r- ^^
p o w e ro f c a n d
N^rFr ;';;"t
:.
,l'ff il:T'i;,TLl,ii:liq
a = l/(. yielding
-" '!" 4ru, snol
o : ao erp-