INVESTIGACiÓN REVISTA MEXICANA DE FlslCA '5 (6) ,81-,83 DICIEMBRE 1999

On polar optical vibrational quantum excitations in semiconductornanostructures

A. Malos-AhiagucDepartamento de Física, Universidad de Oriente

Patricio Lumumba s/n, 90500 Salltiago de Cuba, Cuba

Recibido el 29 de enero de 1999; aceptado el 23 de abril de 1999

By applying a phenomenological treatment of long-wavelength polar optical oscillalions in semiconductor nanoslructures we derive a generalapproxímnted cxpression fm the energy spcclrum of the elementary cxcitations that depends essentially on the nanostructure geomclry.

KeyworclJ: Polar aplical oscillations; elcrncntary excitations in nanoslructurcs

Aplicando un modelo fenomenológico de oscilaciones óptico-polares de onda larga en nanocslructuras scrniconductoras, se deriva unaexpresión general aproximada para el cspectro energético de las cxcitacioncs clcmcntales. Dicho espectro depende de manera esencial de lageometría de la nanoestructura.

[k\'cril'ton's: Oscilaciones óptico-polares: excitaciones clementales en nanoeslruduras

PAes: 63.20.Dj: 68.65.+g

2. Brief review of (he model

interesleu in the study of the cnergy spectrum 01' the long-wavelength polar optical vibratiolls in SNS's characterizingIhc lícld represcnled hy ( 1).

(4)

(2)

(3)

n _ DC _ D"- D (~;7) - p DI

Ji = nD" - CDI •where

We assume a material wilh isotropic dielectric response anuvibrational dispersion relatiol1. Thc oscillations are descriheuby the displacemenl vector lield alhat represents the relativedisplaccment of the two ions involved. The electric pOlential</J is reJalcd with Ihe e¡eclrie f¡cld hy E = - \71>. eorrcspond-ing to a quasistalionary trcaUncnt of Maxwell equations inthe unretarJcd !imit (e ~ (0). Thc internal stresses of IheIllcdiulll are incorporatcd in the fifth tcrm 01' (1) through thestrai 11 tensor:

and the tensor >"i]kl closcly related lO the elastic moduli ten-sor [23,2.1). This tefln results relevant in SNS's because itgivcs the possibility to introduce houndary-matching con-Jitions for {¡ at the interfaces. At the same time it leadslo dispersive oscillations. Other physical paramcters 01' themcdilllll, prcsent in (1), are the same that in Ref. 21.

In mder lO investigate the inllllence of the nanostructureshape on the energy of the syslcm wc, firstly. find an expres-sion ror the Hamiltonian dcnsity 'H. By definition

is the momentum dcnsity canonically conjugale to 17.

Polar optical vibrations in semiconductor nanostructures(SNS's) of Ihe Iype of ,!uantum wells [1-8]. ,!uantumwires [DJ. ,!uantum dOls [1O-12J. and superJaniees [13]. havehren intensivcly investigated in the ¡ast years. Severa! ad-vanced Icchniques have permitted the growth 01' such low-Jimcnsional systcms that posscss important physical prop-crlies suggcsting a broad spcctrum 01' devicc applications.Thereforc. the energy spectrum ano optical properties inSNS's are submitred lO extcnsive studies. Important rolein many physical processes play the polar optical oscilla-lions, cspecially in the long-wavelength limil. Several lhc-orcticai treatments for long-wavelength oscillations hascdon phcnomenological continuum approach or involving mi-croscopic numerical calcularions may he met in previousworks 114-20j.

Recently C. Trallero-Giner el al. 121, 22J have proposed aphenomenological approach ro long-wavelength polar opticalphonons in SNS's. They postulated, in the spirit of the c1as-sical lheory of macroscopic media the following Lagrangiandcnsity:

1 2 1+ -8 0_1\71>1 + ->"1,,1<';1<", (1)7r. 2

which takes into account phonon dispersion up to quadraticterms in lhe wave vector, as well as the coupling between(he electrostatic potential 4J anu the mechanical vibration dis-placemcnt fí.

COllsidering lhe Lagrangian density (1), Ihey obtained theequiltions 01" motion and sol ved it within a given homoge-ncous part 01' the structure and applying suitable bounJary-millching conditions at the interfaces. In Ihis work, we have

1. Introduction

582 A. MATOS-AHIAGUE

m

(8)

(9)m

bctwcen clcctrie licld and Illcchanical vibrations has becosymmclril.cd lo cnsure lhe Humiltonian hcrmilicity.

Thc opcrators fr, ti, and ¿ may he written [21] in the form

(5)Wc musl note that lhe momentum density canonically conju-gate lo cP is idcntically zera.

From lhe Lagrangian Jensity (1) wc obtain lhe followingHamiltonian dcnsity:

1 ") 1 "'"JH = -11- + -(1"'-"- + n,,' \7d!

2/) 2

3. Hamiltonian operator

Taking ¡!lIO account (5) ano working in lhe Schr()dingcrpiclurc, lhe Humiltonian opcrator dcscribing lhe 1011g-w<lvclcngth polar aplica) oscillations in Ihe nanostructurc isthen givcn by

Ii=j(6)

m

whcre i'im(i7) and rfJm(f) are spatial solutions of rhe equa-tions of Illotion gencrated by (1), corresponding to lhe m-thIllolle and Wm its eigenfrequcncy. The operators bm and b~lare the annihilation and creation operators respectively for aphol1on in lhe Ill-th state, and Cm is the real constan!:

( 10)

Arter Ledious algebra we obtain

whcre tI, íi. ano ~ are lhe operators corresponding lo lhe re-spective quantizcd neJds and intcgration is extended lo lhewholc nanostruclurc. The tcnn corrcsponding lo lhe coupling

I

f¡ =L ,1'm (b~J'm+ ~) + ¿)1Jmbmbm + !L.c] + V, (11)"1 111

where Xm is thc real quantity

, ,!(,[-n,,- J - é~ 1\7" 1')<t3,. - h,1m = !¿wm + cm. 20 Re Um' V'+'m 87T ,+,m m, (12)

with

"',!f! [2(3~ IOu,'.'," l' + (3; 1Oll~m , OU}m l' + 2((3; - 2(3;)Re (Ollim , OU;m)] <t3,..L. O, 0, 0.",' D,l,¡ Dx}• '1 '"J

(i,J)"S'

Moreover,11m is, in general, a complex quantiLy given by

( 13)

( 14)

with

.'JIII = e;" ¿ ! [2' (Oll,m) 2 + (32 (Oll'm + OIlj '." ) 22 f! (3L O . T O l' O e

.ti "J' 1(i,j)"s'

+2((J2 _ 21J') (DIL'm , OU},'m)] (¡'l'.1. T D.Ci D.rj ( 15)

For more dClails SC!; Rcfs, 21. 22 and 24.

In Eqs. (13) and (14) the indexes i,j indicate Lhe com-ponent of the vector 17m(P) and 11¿ lahels lhe mode. TheSlIllllll<Jtions over (i,j) mn Lhrough thc set of comhinationsS = {(1.2),(2,:J),(:J, I)J, Sinee \Ve consider an iS(llmpiemcdiulll, lo oblain (13) anJ (14) we taken all non-zero COIll-

ponents nf the tensor Aijld given hy

'\úji = IJ¡3~, Aüjj = p(f3Z - 2{3Z),

In Eq. (1 1), lhe operalor fr descrihes the coupling be-twecn dilTerenL modes. It givcs rise to a small broadening(JI' Ihe cllcrgy lcvcls, so iL Illay be treated as a pcrturbation.However in Ihis work, as a preliminary stuJy, we ncglcct thecoupling bctwccll dilTerenl moJes. thal is, we assumc that thelifetilllc 01"the elementary cxcitations is rcasonably long, solhat \Ve dial wilh a simpler Hmniitonian

¡[(o) = L.l'm(btJ)llI +~)+ ¿(Ilmbmbm + "',c,]. (16)\ _ .\ = (".I{:;.,<' i}.ii - ¡jij , (j,j = 1,2,3(; f.j).

m m

He!'. AJeT fII'. 45 ((1) ( 199t.l) SR 1-5R3

S1I3

S. Conclusiolls

( 17)

(1 X)2 2 _ ,1'm

0m + 13m - -=====,1,1;';' - 41'lml"

, I"ml( '1 ., A'+)fJIlI = -- (~mf l1l + (jm 111'

h.II1

ON POLAR OI'TICAL VIHRATIONAL QUANTUM EXCITATlONS IN SEMICONDUCTOR NANOSTRUCTURES

\Ve musl ohserve Ihal, in lhe limil, when we ignorehoth the couplctl clcctric ficld ami the internal stress of theIllcdium, Eq. (21) hccoll1cs ~m = Wm as could be dcsired.

It follows from Eq. (20) Ihal lhe slales 01' Ihe systemucscrihcd by Ihe Lagr<lngian dcnsity (1) corrcspond lo ele-Illcntary cxcitations wilh encrgics hf.m. ILrcsults rcmarkablc.howcvcr. that hccausc ofthc inclusion of the internal stress ofthe mcdiulll. Ihe clcctric ficld ami its coupling with the dis-placcmcnt. the cncrgy of lhe long-wavclcngth polar aplicalphonons in SNS's t1cpcnds nol only on lhe frequcncies Wm•

hut also in a Illore complicatcd way on the nanostructure gc-olllcLry.

where t.:;u = '1m :lI1dO'm, ¡3m are real quanlities salisfying

and

4. Energy spectrum

Once the Hamiltonian (16) is not diagonal in lhe phonoonumhcrs. we will find Ihe eigenvalues 01' (16) hy diagonal-izing the Hamiltonian using a linear transformalion 0'- lheBose-opernlors. Such a linear tranSf!lfI11ation is carried outby changing ro new Bose-operators Arn through the equation

In summary. we llave derived. using a phcr\omenologicaltrcatmenl proposed in previous works, an approximated ex-prcssion ror the energy 01'elcmentary excitations in semicon4ductor nanostructures.

Onc musl stress that rhe study ol' cncrgy spectrum 0'-optical vihr:llional qllantum cxcitations in quanturn wells,qualllum well wires, supcrlattices and scmiconducting het-erostructures in general, as weH as several thermodynamicproperties sllch as heat capacily, may he readily obtained asstraightforward applicalions [251 0'- the above approach onIhe physical properties of hClerostructurcs.

( 19)

(20)

(21 )J.1";1 - 41'/m¡:l

f¡

f¡ = L f¡(",(Á~,Am + 1/2),m

(1m X 13m =

[,"1 =

-111mlV"';, - 41'lml'

In Icrrns 01'the new operators the Hamiltonian (16) resullsdi<lgonal in the cigenvalues of the operators J.Vm = .4~I.timami, il can he wrillen as '-01l0w5:

where

\Ve musl note that 11m • ..l'm musl salisfy the condilion21'/1111 < .l'm rOl"lhe validily 01'our transfonnalions, once weconsidered o.m JJm as real quanlities. h [ollows from (12),(14) that such a condition may he satisfied, ir lhe oscillationsare suHiciently small.

Acknowlcdgmcnts

Thc allthOl"lhanks E. Roca and R.L. Rodríguez for helpfuldisclIssions abollt lhe prcscnt prohlcm.

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