Light Absorption near Threshold with Phonon
Partiipation for Impurities in Semiondutors
M. A. Amato
InstitutodeFsia,Universidadede Braslia
70910-900, Braslia,DF,Brazil
Reebidoem7dejunho,2002. Aeitoem3desetembro,2002.
Itispresentedasimplemodelforthealulationofthetransitionrateforimpuritiesin
semiondu-torsinwhiheletron-phononinterationis takenintoaountinaseondordertimedependent
perturbationtheory. Thisresultshowstheexpliitdependeneofthetransitionrateonthephonon
density of statesand that the absorption urve of a semiondutoris modulated by the phonon
struture.
In reent years progress has been made in
inves-tigating the physial properties of impurity entres in
semiondutors either theoretially or experimentally.
Fordeeplevelimpurities theiridentiationand
har-aterisationremainsas adiÆultproblemitisstill an
important and ative eld in semiondutor researh
[1℄. On the other hand, many of the shallow
impuri-ties are by now well understood. The eetive mass
theory of shallowimpurities [2℄ led to the well-known
hydrogenitheoryprovidingbothoneptualguidane
and, inmanyases,quantitativepreditionsaboutthe
bindingenergies[3℄.
Deepimpuritiesaresodierentfromshallow
impu-rities that extension of eetive mass theory has not
provedto besuessfulin understanding their
proper-ties. Theirdistinguishingfeatureisintheharateristi
ofthelongrangepotential. Shallowimpuritiesare
on-trolled by longrange potentials, whereasthe presene
ofshortrangepotentialproduesonlyaslighthemial
shift. For deep impurities this relative importane of
thepotentialworksontheoppositeway.
Sometheoretialapproahesareused totaklethe
impurity problem. There are rst priniples type
of approahes that aim at a truly quantitative
as-sessment of arefully seleted impurity-host systems
[4℄[5℄. They employ self-onsistent pseudopotential
tehniques,whih providesaverydetailed information
abouteletroni andvibronipropertiesof spei
de-fets, suh as harge densities [6℄, lattie relaxation
arounddefets[7℄,andenergylevels. Thesemiempirial
approahesoer an impreisebut global view of deep
impurities in many dierenthosts [8℄ whih employ a
modernversionof tightbindingtheory[9℄, augmented
bytheKoster-SlaterGreen'sfuntiontehnique. These
purity levels energiesrather than to give quantitative
agreementwith existingdata.
WhatmakesdiÆulttoobtainanauratesolution
to the problem is the presene of the interation
be-tween the eletrons and the lattie. The model
gen-erally used for a quantitative treatment of
eletron-phonon interation is the so alled onguration
o-ordinate model. Its formalism has developed a long
timeagoandreviewsexistonthistopi for
semilassi-alaswellasquantumtreatment[10℄[11℄[12℄[13℄. Inits
original form this model is worked out for transitions
within loalised states, and it has been applied
su-essfullyto suh systemsasF-entres in alkali halides
andrare-earthimpuritiesin semiondutors. Itis also
widelyaeptedforstudyingtransitionsrelatedtodeep
entres, mainly due to a bigger Frank-Condon shift.
However,forasmallFrank-Condonshift,whihour
forsomedeepentres,itwouldbeappropriatetomodel
transitionsthroughindiretproesses. Thiswouldalso
allow to study transitions onneted to phonon
side-bands. Indiret proessesanplayasigniantrolein
determiningtheshapeandthepositionofthepeaksin
the phonon side band struture of deep impurities in
semiondutors[14℄.
It is generally aepted that even were afull
the-oryavailable,itwould beneessaryto extrat from it
anaeptable,simpliedmodelwhihouldbeusedto
denethe prinipal measurable quantities and to
pro-vide a useful terminology for desribing experimental
results. Thepreseneofsuhamodelisofsome
impor-taneandweattemptin thisartileto illustratewhat
experimental features lend themselves mostreadily to
modelling.
dis-it is diÆult to go into the theory underlying the
ab-sorptionproessunlessthestudentshaveasound
bak-ground in time dependent perturbation theory.
How-ever,someimportant quantitativefeatures ofthe
pro-essanbeonveyedbyelementarymethodsdesribed
inthis paper.
In this paper we would like to enlighten the
rele-vaneof theeletron-phononinteration intheoptial
absorptionspetraduetothepreseneofimpuritiesin
semiondutors. Aswehavepointed theonguration
oordinatemodelisagenerallyaeptedonetoprovide
theamountof interation with phonons. However,on
adierentbasis we propose amoresimple treatment,
whih extrat from the experimental results the
har-ateristi phonon frequeny that assists the eletroni
transition.
Forsakeofonvenienewerestritourdisussionto
theaseofdonorimpurity,althoughthisouldalsobe
formulatedin termsof aeptors. We shallstart with
theHamiltonian
H=H rys
+H int
(1)
wheretheperturbationH int
isgivenby
H int =H e p +H rad (2) The H rad
ontains the usual desription in terms of
eletroniandphononbandstates,andH e p
takesinto
aountthe eletron-photon interation only. For the
eletron-phononinterationonetakes[9℄
H e p = X ~ k X ~ q V( ! q)a y ! q+ ! k a! k (b! q +b ! q ) (3)
withbeinga!
k (b!
q
)theeletron(phonon)operators.
MakinguseoftheFermiGoldenRule,thetransition
rateisgivenby
W = 2 ~ X f H int f I 2 Æ(E f E I ) (4)
where hfj and hIj are the ondution band and trap
states,respetively. Thewavefuntion of theeletron
in the presene of the impurity potential may be
ex-panded intermsofBlohfuntionsintheform
I = 1 p V X n; ! k A n; ! k n; ! k (5)
This form applies to both trap and ondution band
state, though the oeÆients will be generally
dier-ent in the two ases. Proeeding further and
on-sidering only phonon emission, for a donor impurity
onegetsforthe transitionratein thelongwavelength
approximation[19℄ W 2 ~V X n;n0; X ! k ` ; ! k; ! q A n; ! k A n 0 ; ! k 0 (6) X i n; ! k H rad i; ! k i; ! k H e p n 0 ; ! k 0 E! k E i 2 Æ(E! k 0 E I ~!+~ ! q ) (7) d ~ ! q
isthethbranhphonon energywith wave
ve-tor !
q and the index i refers to intermediate states.
Inanindiretproessmomentumonservationrequires
! k 0 ! k = !
q, andfor n=n 0
equation (6) anbe
fur-ther simplied. The oeÆients in the wave funtion
expansionanbealulatedaordingto theimpurity
model[15℄
Equation(6)maybeseenasomposedoftwomajor
terms. Therstdesribingthestrutureof the
transi-tion rate,and the seond inorporatingthe densityof
equation an befatorised into aterm ontaining the
matrixelementandasumoverthedeltafuntions
lead-ingtothesimpleprodutstruture. Thetransitionrate
anbeexpressedasaprodutofthesquaredmatrix
el-ementsandthedensityofnalstates
jMj 2
= 2
~V X
n;n0 X
!
k `
; !
k A
n; !
k A
n 0
; !
k 0
(9)
X
i
n; !
k
H
rad
i; !
k
i; !
k
H
e p
n 0
; !
k 0
E!
k E
i
2
(10)
d
We may argue that this approximationis justied
by saying that the trap state is very loalised
imply-ingthat suh astatehasFourieromponentsalongall
the dierent k 0
s in the Brillouin zone. It is obvious
that suh analysis fails whenever the matrix element
vanishes due to symmetry of the involved states. In
summary,this expressesthetransition rateasa
prod-ut ofthe squaredmatrixelementsand thedensityof
nal states.
Equation(8)showsinasimplewayhowthephonon
densityofstatesmodulatesthetransitionrate. The
ab-sorptionthresholdinthepreseneofphononsisshifted
bytheamountofphononenergysuppliedtowardlower
energies for phonon absorption or toward higher
en-ergies for phonon emission [17℄[18℄. Any further
al-ulation should take into aount the eets of band
struture. As an exerise,for a paraboliband
stru-ture,andasinglephononfrequeny,onean easily
re-produesthe powerlawfor the absorption oeÆient,
(~!)[20℄.
Inreferringtothevalidityofthepresentmodel,may
be pointed out that it takes into aount only single
phonon proesses,oritisarstordereletron-phonon
model. In the ase of deep impurities, the eletron
relaxation is assisted by many phonons, so that it is
analysed byhigherorder perturbationtheory,and the
presentanalysisfails, althoughforsmalllattie
distor-tion, the model an be applied to the understanding
of the behaviour of a deep impurity near the optial
threshold. Finally,theaboveresultdoesnotrequirene
detailsoftheeletron-phononinteration,but
unfortu-natelyitdoesnotapplytoregionswellabovethreshold.
In order to avoid misunderstanding in the subjet
due tothesimpliityoftheabovemodelawordabout
the wavefuntions related to the impurities and
on-dution bandstatesoughttobesaid,sothataproper
alulationofthematrixelementsanbeperformed. In
fat thedetermination ofthewavefuntionsisarather
diÆultproblemduetothepotentialintroduedbythe
foreignatominthehostsystem. Thetranslational
sym-Bloh funtion. For bound states, to irunvent the
probleminthesolutionoftheShroedingerwave
equa-tion onehas to model either the wavefuntion or the
potential. Forthe later, the simplest of these models
isthehydrogenimodel,forshallowimpuritiesandthe
Luovskymodelfordeepimpurities. Intheondution
bandtheeletroni wavefuntion ismoreompliated
by sattering of eletrons by the potential. The
sim-plestsolution hasbeento adopt as the wavefuntion
thatfor aplanewave. Thishoie isobviouslyagood
oneforaneutralentrewhihsatteronlyweaklyand
forspreadoutloalizedstatesathighfreeenergies,but
bynomeansanobvioushoieforhargedstates
Insummary,themodeldesribedinthispaperis
un-doubtedlyover-simplisti, and should notberegarded
asanythingmorethanaoneptualaid. Ontheother
side,itdoesprovideaneasilyunderstoodintrodution
to the subjet, and may enablesome useful
quantita-tiveresultstobeobtainedwithoutadetailed quantum
mehanialalulations.
Referenes
[1℄ S.Pantelides, Perspetives in the Past, Present, and
Future of DeepCenters, pp. 1-85, inDeep Centers in
Semiondutors: AStateof theArtApproah,S.
Pan-telides(ed.),1997, GordonandBreah.
[2℄ J. M. Luttinger and W. Kohn. Phys. Rev. 97, 969
(1955).
[3℄ W.Kohn,ShallowImpurityStatesinSilionand
Ger-manium,inSolidStatePhysis,F.SeitzandD.
Turn-bull(eds.),5,257-320(1957).
[4℄ G.A.Bara,and M. Shluter, Phys.Rev.B19, 4965
(1979).
[5℄ M.Jaros,Adv.Phys.29,409(1980).
[6℄ M.Jaros, C.D. Rodriguez,and S.Brand, Phys.Rev.
B19,3137(1979).
[8℄ J. Bernhol, S. Pantelides, N. D. Lipari, and A.
Balderehi, SolidStateCommun.37,705(1981).
[9℄ H.P.Hjalmarson,P.Vogl,and J.D. Dow,Phys.Rer.
Lett.44,810(1980).
[10℄ P. Vogl, H. P. Hjalmarson, and J. D. Dow, J. Phys.
Chem.Solids44,365(1983).
[11℄ K.Huang, and A.Rhys,Pro.RoyalSo. A204,406
(1950).
[12℄ M.Lax,J.Chem.Phys.20,1752(1952).
[13℄ J.J.Markham,Rer.Mod.Phys.31,956(1959).
[14℄ H.Dai,M.A.Gundersen,C.W.Myles,andP.G.
Sny-der,Phys.Rev.B37,1205(1988).
[15℄ M.A.Amato,andB.K.Ridley,J.Phys.C:SolidState
Phys.13,2027(1980).
[16℄ H. N.Nazareno, and M.A. Amato,J. Phys.C:Solid
StatePhys.15,2165(1982).
[17℄ M. A. Amato, M.C. Arikan, and B. K. Ridley, in
Pro.onIII-IVSemi-insulatingMaterials Conferene,
pp.249-252, G. J. Rees (ed.), 1980, Shiva Publishing
Ltd.
[18℄ B.Monemar, andL. Samuelson, Phys.Rev.B18, 809
(1978);Phys.Rev.B18,830(1978).
[19℄ In the long wavelenght approximation the photon
wavevetor is negligible if ompared to the eletron
wavevetorinmagnitude.
[20℄ Thetransition rate andthe absorption oeÆient are
relatedthroughtheequation(w)= WV
hNi
with
be-ing the refrativeindexof themedium,and the
