Many Partile Theory for Luminesene in Quantum Wells
M.F. Pereira Jr. y
InstitutodeFsia,UniversidadeFederaldaBahia,
40210-340, Salvador,BA,Brazil
Reeivedon30June,2002
AGreen'sfuntiontheoryisappliedtothedesription oflumineseneandabsorptionspetraof
lowdimensionalsemiondutors. ProgressinthenumerialsolutionoftheBetheSalpeterequation
foroupledbandquantumwellswithaT-matrixstrutureforthepolarisationfuntionand
arrier-arrierdephasingisgivenwithinanapproahthatsatisfytheKubo-Martin-Shwingersumruleand
eliminatestypialartifatsinomputedoptialspetra.
I Introdution
Photo-luminesene Spetrosopy (PL) is one of the
major non-destrutive haraterisation tehniques of
novelsemiondutor materials. [1℄As thegrowthand
optialmeasurementtehniquesprogress,aparallel
de-velopmenttakesplaeonthetheoretialanalysisof
op-tialspetra,anditisnowawell-establishedfat that
many-body eets are ruial for the understanding
oftheoptialpropertiesofsemiondutormaterials.[2℄
KeldyshGreen'sfuntions arepartiularlywell-suited
todesribebothsteadystateandnon-equilibrium
situ-ations,andtheorrespondingdiagramsallowfora
sys-tematiinlusionofinreasinglyhigher-ordereets.[3℄
If the semiondutor media have low eetive
dimen-sionality, likequantum wells dotsand wires,quantum
onnementofthehargedarriersandnon-paraboli
bandstruturefundamentallyaltertheeletroni
stru-ture. [4℄Reenttheorieshavebeenabletoinorporate
all these eets onsistently and be able to desribe
theirinterplayintheresultingoptialspetra. [5℄
How-ever, there is, so far, no approah free of
approxima-tions. Dierenttehniqueshavebeenusedtoobtain
nu-merialodes,fastenoughforsystematiomputations
allowingdiret omparisonwith experimentsand
on-tainingenoughpreditable power,andseveral
disrep-aniesfoundinomputationsinthepastarebeing
pro-gressivelyeliminated. Oneexampleisthat iftheband
oupation, or equivalently, the Pauli-bloking fator
is written under thequasi-partile approximation, i.e.
with the Fermi funtions evaluated at the dispersion
relations, f
e (e
e (k)), f
h (e
h
(k)), a spurious absorption
develops below the gain region. The diÆulty an be
eliminatedwhennondiagonaldephasingtermsare
on-sidered in the (oherent)polarisation. [6, 7℄ Another
development ofnegativeluminesene onthehigh
en-ergysideofthespetra. Noneoftheseartifatsappear
in analternativeapproahpresentedreently,[5℄ sine
bymaking surethat thepolarisationfuntion satises
the Kubo-Martin-Shwinger (KMS) ondition. [2, 8℄,
highk-valuesontributionsthatultimatelygiveriseto
the artifats are eliminated. Furthermore, the theory
automatiallyguaranteesthat theswith from
absorp-tion to gain ours exatly at the hemial potential,
so that the tehnique proposed in Ref. [9℄ to extrat
the gain spetra from spontaneous emission data an
besafely used withouta disrepany between
"exper-imental" and"omputed"totalhemialpotential
dif-ferenes. The theory has, verysuessfully, predited
severalimportantexperimentalresultsandprovided
ex-planationsforwhyertainpreviousapproahesatually
worked. Speiallyforthegainase,theavailable
ex-perimental datafor asystemati omparisonwere
ob-tainedforlaserstruturesandmaterialsinwhih
high-ordersub-bandtransitionsplayedanegligiblerole,and
,e.g.,stronginhomogeneitiesmaskeddetailsofthe
de-phasingmehanismasinRef. [10℄. So,insteadof
om-puting those eetsin detail,at to theGW
dephas-ing expression (orresponding to arrier- arrier
spe-trum) hasbeenused. [11℄ Theaim ofthe present
pa-per is not to ompare and ontrast the dierent
ap-proahesintheliterature,buttopresentastepforward
towardsthe omputation of Photo-luminesene (PL)
spetrosopy, within the ontext of a T-matrix based
theory. Here,insteadofapartlyphenomenologial
de-phasing,orfrequeny-independentresultsthatrequires
non- diagonaldephasing to removeinonsistenies, an
iteratedGWarrierself-energyisusedgivingrisetoa
k- and frequeny-dependent dephasing, extending the
approah of Ref. [12℄ for bulkmaterials to the MQW
dependentdephasingtoexplainertainfeatures of
op-tial spetra.
II Main equations
Thetimeevolutionofthequantumstatistialaverages
desribingtheinteratingarriers(G),photons(D)and
plasmons(W),withintheKeldyshformalism,givenby
Dysonequations. [13℄Coulomborretionsto thefree
partilepropagators,G
0 ,D
0
,andW
0
,aregivenbythe
respetiveself-energies,,P,andp. Funtional
deriva-tiveordiagrammatitehniquesallowtheonsideration
ofinreasinglyhigherorderCoulomborretionsinthe
self-energies. Thearrierandphotonself-energies,ina
shematinotation,are(fordetailsseeRefs. [5,12℄).
(1;2) =
GW
(1;2)+
T
(1;2); (1)
P(1;2) = G(1;2)G(2;1)+G(1;3)G(4;1)T(3;4;5;6)G(5;2)G(2;6): (2)
where
GW
(1;2)=G(1;2)W(1;2),
T
(1;2)=T 00
(1;3;2;4)G(4;3). T 00
denotestheT-matrixwithoutthersttwo
iterationsoftheladdersheme,whihgivesrisetotheBethe-Salpeterequation
T(1;2;3;4)=W(1;2)Æ
13 Æ
24
+W(1;2)G(1;5)G(6;2)T(5;6;3;4): (3)
d
Inthispaper,T-matrixorrelationsbeyondGWin
the spetraldensityof photonsareonsidered but the
arrierself-energiesarerestritedtoGW,i.e.,(1;2)=
G(1;2)W(1;2), sinethe T-matrixontributes
signi-antlytoarrierself-energyin quantum wellsand
su-perlattiesonlyatlowtemperatures. [14℄
Under stationary onditions, to be desribe here,
a Fourier-transform with respet to time, yields
r
n
(k;!) = r
n;X
(k;!)+ r
n;
(k;!). Spetral
infor-mation is inluded in the retarded Keldysh
ompo-nents,supersript r
. Thedephasing,responsibleforthe
gain/absorptionbroadening,isgivenbytheimaginary
partof r
n;
(k;!),namely
~
n
(k;!)= X qq 0 ~ n (k;!)F
n (k 0 ;p 0 ;q 0 )W r (q) 2 ~ 2
(!+e
n (k) e n (k 0 ) e (p 0
)+e
(q 0 )) 2 +~ 2 2 n ; (4) wheref <
isaFermidistribution,f >
=1 f < ,p 0 =q 0 +k k 0
,andF
n (k;p
0
;q 0
)=f > (e n (k 0 ))f > (e (p 0 ))f < (e (q 0 ))+ f < (e n (k 0 ))f < (e (p 0 ))f > (e (q 0
)):Theiterateddephasingisevaluatedatk=0onlyandforsimpliity,andwefurther
ignoretherealpartof r
n
(k;!)withoutaeting thegeneralonlusionspresentedhere.
TheemissionI(!)andabsorptionspetra(!)aregivenrespetivelyby
I(!)= ~! 2 =2 2 ImfP r (!)g
1 exp((~! ))
; (!)==(2! p
(1))ImP r
(!); (5)
d
where is the total hemial potential, and I(!) has
been derived from the KMS relation. [5℄ In the
nu-merialdatapresentedhere,theintegralEquation2is
solvedthroughnumerialmatrixinversion.
III Numerial Results and
Dis-ussion
The solid lines of Fig. 1 shows absorption (a,)
and Photoluminesene (b,d) of a 2.5 nm ZnSe/
(Zn,Mg)(S,Se)quantumwellat300Kandexitedwith
with respet to the free arrier band gap. The
line-shapes omputed with the iterated GW dephasing at
k=0 (solid lines) are sharper then several
experimen-tal ndings for 300 K, sine higher order diagrams,
arrier-phononsatteringandinhomogeneous
broaden-ing(presentinmostsamples)havenotbeenonsidered,
as thegoal hereis to illustrate the importane of the
iteratedfrequeny-dependentdephasing. For
ompar-ison, the dephasing is omputed two expressions that
simulate thefull Eq. 4,
=
0
1
1+exp[(~e e
(k)+~e h
(k) ~!)=℄
; (6)
=
0
exp(
h
)+exp[(~e e
(k)+~e h
(k) ~!)=℄
exp(
h )+1
; (7)
d
wheree e
(k),e h
(k)arerenormalisedenergies,
h isthe
holeshemialpotential,=1/K
B
T,withthe
temper-atureTinKelvins,and
h
istheholeshemial
poten-tial. Thet- formulasdepends ontwo
phenomenolog-ialparameters that must beadjusted in omparison,
e.g. withPLEorabsorptionspetra: ,andwhihis
aretail fators,and
0
. Theyan be alternatively(in
someases)beadjustedusingthedephasingalulated
hereandtheurvesgeneratedwithit,assuggestedby
thenumerialresultspresented.
Thistypeofexpressionhashelpedto,very
suess-fully, reprodue a wealth of experimental absorption
data, e.g. asin Refs. [11,10℄(and referenestherein).
Forhigh performane deviesimulators, theseformula
may be quite useful by adjusting the parameters
ei-thertoexperimentsoralulateddata,allowingforeasy
andfastprogramming. Manysimulatorsinludemany
othereets orrespondingto allharateristisofthe
deviesandapproximationsforthefullmirosopi
ex-pressionsarefrequentlyneessarywiththepresent
nu-merialapabilities available.
Figure1displaysalulatednormalised
Photolumi-nesene(a,b,) and orrespondingabsorption spetra
(d,e,f)ofa3.0 nmZnSe/(Zn,Mg)(S,Se) quantum well
at 300Kand exitedwithN =110 11
arriers=m 2
asafuntion of detuning withrespet to thefree
ar-rierbandgap.Inallplots,thesolidlinesareomputed
with theiterated GW dephasingof Eq. 4at k=0. In
(a,d) the dotted, dot- dashed, and dashed urves are
respetively for (see Eq. 6)
0
= 0.2, 0.1, 0.025.
e
0
,and=108e
0
,wherethe2Dbindingenergyise
0
=79.6meV.In(b,e)thedot-dashedandlong-dashed
urvesare,respetively,for
0
=0.025e
0
,and=108,
0.65e
0
. In(,f)Thedotted anddot-dashedurves
arefor
0
=0.025,and =0.65e
0
, and=0.3
e
0 .
Thenumerialstudypresentedhereshowsthatthe
tformulasareonsistentwiththeGWalulationfor
aertainhoieofparameters,andfailsforothers. As
spetraastomakethemindistinguishableofthose
om-puted with a onstant dephasing =
0
, or in other
words,withaLorentzianlineshape. Notethat,aslong
asthepolarisationfuntionsatisestheKMSondition,
nospuriousabsorptiondevelopsbelowthegainregion,
[15℄andallplotsin(d)ouldwellbeusedas
represen-tativeabsorptionurves. However,asshownin (a)the
PL rises unrealistially in the low- energy side of the
broader spetra. In other words,a theory that yields
reasonableabsorption and gain spetra, not
neessar-ilygivesrealistiPLorPhotolumineseneExitation
Spetrosopy spetra (PLE). Note that this approah
anbeextendedtoomputedbothsingleandtwobeam
PLE. [5, 10, 16℄ So, aslongas the arrieroupation
funtions anbe desribed by quasi-equilibrium
on-ditions, notonlytheKMSrelationmusthold, but the
dephasingmustbesharplyfrequenydependentforPL
(andPLE)spetrawhih areonsistentwithquantum
statistial mehanis and still realisti. The
ompu-tations inludingthe GWdephasing thusinrease the
preditabilityofthetheoretialapproah. Atthispoint
animportantommentisin order,i.e. weatherornot
anexitedsemiondutoratsteady-stateshouldbe
de-sribed byquasi- equilibrium isawhole other matter,
andfurthertheoretialandexperimentalworkisatthis
point neessarytodetermine ifthis approximation,so
suessful so far, really is orret. In other words, if
the system, even at steady state is far from
equilib-rium, theKMSonditiondoesnothaveneessarilyto
hold anymore. In summary, aT-matrix based
many-body theory for the optial spetra of semiondutor
superlatties hasbeen presented and its relevane for
therealistiomputationofphotoluminesenespetra
hasbeenoutlined. Inthepresentapproahthereis no
need forphenomenologialparametersto obtain
spe-tral line broadening,thusinreasing thepreditability
oftheapproah, whih anbeusedasastartingpoint
for simulationsof awealth of dierent optialspetra
harateris-−60
−20
20
Detuning (meV)
0.2
0.6
1
Luminescence (arb. units)
−60
−20
20
Detuning (meV)
0.2
0.6
1
Absorption (arb. units)
(a)
(b)
(c)
(d)
(e)
(f)
Figure1.NormalisedPhotoluminesene(a,b,)andorrespondingabsorptionspetra(d,e,f)ofa3.0nmZnSe/(Zn,Mg)(S,Se)
quantumwellat300KandexitedwithN=110 11
arriers=m 2
asafuntionofdetuningwithrespettothefreearrier
bandgap. Inall plots,thesolidlinesare omputedwiththeiteratedGWdephasingofEq. 4atk=0. In(a,d)thedotted,
dot-dashed,and dashedurvesarerespetivelyfor (seeEq. 6)
0
=0.2, 0.1,0.025. e
0
,and=108 e
0
,wherethe2D
bindingenergyise
0
=79.6meV.In(b,e)thedot-dashedandlong-dashedurvesare,respetively,for
0
=0.025e
0 ,and
=108,0.65e
0
. In(,f)Thedottedanddot-dashedurvesarefor
0
=0.025,and=0.65 e
0
,and=0.3e
0 .
Aknowledgments: Work supported by Conselho
NaionaldePesquisas(CNPq)ofBrazil.
y
Present Address: Institut fuer
Theoretis-he Physik, Tehnishe Universitaet Berlin,
Harden-bergstrasse36, 10623BerlinGermany.
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