Longitudinal dielectric constant for quantum wells

Superlattices

and Microstructures,

Vol. 1, No. 4, 1985

295

Imgitudinal DielectricConstantfor QuantumWalls RaphaelTsu* and LiderioIoriatti Institutode Fisicae mimica de Sao Carlos Universidade de Sao Paul0 13.560Sao Carlos,SP Brazil Ws have calculatedthe longitudinal dielectricconstantfor a G&sAlGaAsguantumhell structure.In particular, the total contribution to the transitionsistakenas a sumof separatecontributionsofthe regionsaroundthe T, X, and L symnetrypointsof the Brillouinzone. Since the electron-and hole-stateare quantizedh a wsll formedby potentialbarriersof the hetrojunction, the staticdielectricconstant is significantly reduced.For a wall width of 2 m and barriers formedby A1_35Ga_65As, tbs reductionis almost20%.

Sincethe

I. Introduction Eversincetheintr~ctionofamn-mde and the observatim andexcitons in a doublebarrierG&s-AlGaAs guantm wells4,many aspectsof the physicalproperties relatedto these have been aggressively explored,such as the quantum Ball effectsin a high mobilitylayer with modulationdopings.The transversedielectric ~nstanthas~measuredbyTsueta16,ina GaAs-AlAssuperlattice showingthat the indexof refractionlies batmen that of a Ga_5Al5As alloy and GaAs. Mre recently,experimentswith zinc diffusion7into GaAs-AlGaAssuperlattice, indicate that the indexof refractionof a superlattice is differentfrm its corresponding alloy having the same averagealloy amposition. The questionof dielectricconstantis importantfor an understandingof impuritylevel,excitcns,as wall as the role of carrierscreeningin general. lhe dielectricamstantof GaAs-AlGaAssuperlatticeshas been amputed using a k-p energy bends and takingthe contributions of the T, X, L symmtrypointsof theband structureseperately8, tiver the guantization was consideredmly in the r point. For polar smiconductorsuchasGaAs, the ioniccontributicm to the staticdielectricconstant representsa significantfactorin amparison with thatpartdueto its covalent nature. Since the ionic contribution invo$veschargetransfer betweentheGaand As atis, it is assmadthat the guantm wzll plays a negligiblerole for the ionicpart. me to the guantization of the high lyingvalenceband statesand the low lying amductionbaud statesby the AlGaAsbarriersseperrated by a well width W = 5 rnn,the electronic contribution to the staticdielectricomstant is reducedprimarilyfrcmthe increasein the energy bethe valence-and conduction-baud.

manentunmatrixelenr2ntbetw2enthein-

itial and final stateshas littleenergyckapendence, themin featuresfrmtheVan Mve singularitiesof the jointdensityof states.In GaAs these peaks are due to transitions at the X-point andL-pointof the B.Z. where the valenceband runs nearlyparallelto ths conductionband along the A and A directions. The zone centertransition is iqm-tantbutrepresents a smllmntribution to the overalldielectricfunction.This is really the centralpoint of Phillipssc&ll~ In this work one seperatesths k-spaceinto three regions:tl-ra r-pointgivingthe initialabsorptionedge bit contriimting littleto the dielectricconstant, the L-pointwhich startsat the Ll-L3 transition and joins into the thirdandmost important region the X5-X1 transition.Similarlyfor the staticdielectricconstant,the transitioninvolvesthe positionmatrixel-t which is relatedto the pmatrix elementvia the equationof motion,* = is identical [x,B] . Again the main contribution to that of the opticalcase.With spatialconfinesent by the energy barrierssuch as a guautmwzll the 2-dimensional naturemanisfestsitselfnear the r-point over a limitedrange of energy.Below Eb, the barrierdue to AlGaAson each side of the well, statesare quantized.Since the transition in the three regionsare additive,we can calculate the differemcebetwsentheunboundedandthe boundeddue to qmntmwzll, leadingto the result we setoutto caqnlte. Under the applicationof a perturbingelectric fieldF, the wavefunctions that satisfythe boundaryconditionsare given by (1) Y*

=

Ink>+

Z,k<*l eFx In'k" In'k'> Eti

-

Then the static dielectric ox&ant

Enmk, becarnes

(2) E = E. ionic

II. The BPA Calculationand Discussion: where ws have used =cnklxln'k>&, Opticalabsorptionoriginatesin ths transitionwith the applicationof a vectorpotential.

and set the Fermi functionsf =l and f =0 for n=vand n'=c. Let us take Eb%ng the i&s limit *Also at energyConversionDevices,Inc..lYoy, MI 48084 of energy involved,then for EcEb, the amtriimtion

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+ 03 $02.0010

0 1985 Academic Press Inc. (London)

Limited

296

Superlattices

should be subtracted and replacea by transitions between the quantized states. The part in E@.(2) to be subtracted is given by

and Microstructures,

E~J= 3eV , E;=5,

(3)cg~r)=~(ur/mo)3’2~~oE~2)1’2cEg)-2. . $.pg (x-1y2

4

&

Vol. 7, No. 4. 1985

E;=4.95,

E; =1.43

X

where pr

is the reduced mass for whichever transiis appropriate, for example, transition involving the light-hole band to the conduction band, it is the reduced mass between the light-hole mass and the electron mass at the r-point. Similarly,taking the wavefunction of the quantized region to be

2 al

H

UC (4) Y = U (r,k)si.n(mvx/w)exp(iktrt) s 6 which lead to pmn,= p , he arrive at nm'mn'6ktkr_, the term to replace ES(r)

z

Lu

- - - - A!35GaeysAs where II dates electric field parallel to the plane of the ~~11. For sr wa replace pll by pl in eq.(5). !R-Etotal for+ &e three regiZ-!sism" (6)

c,,~=E -

( E;+E+;

-4

r

X

WAVE VECTOR Fig.1

Energy band structure of C&As and A1_35Ga+65As

Fig.2

E,,~

) + kiL + E;~+ E;&

Eguatlb (6) clearly indicates iur a&roacLto the problem. Instead of calculating the dielectric mnstantusing full band structure obtained by pseudopotential Be or k.p12 calculations, ws used the effective mess model which is only applicable near t&se symnatry points. lhis method should give reasonable results because the presence of the guanti wzll only affects a srall region of energy near these symmetry points. Therefore the second term in eq.(6) represents the portion of the transition afected and thus removed and replaced by the third term representing the contribution drze to transitions betwzn quantized states. FigurelshowsthebandstructureofGaAs( solid line) and M_35Gas65As (dashed line) . Note that the energygapbetwaen thevalence-andconduction-band for GaAs is denoted by E and that of We have asti that 68% of the dif=mbyEb.

I

=m ,t + m-1 vtPand = 0.2 mo, and NL= 4, the numm = 0.075 m , m b&of valley: at ?$ wz have 312 (8) EB(L)= w (2mLEg/"' ) 1’2 @+l) (‘-‘) y2 E2 g

Due

to

the

breaking of the -try , p;Wh)=

by thewall,

16

p2/3 , p:(.%h) = 2p2/3 for the

r-point and pf,= 4p2/3, pi =,2p2/3 for the Lpoint, with p /m. = 7.5 eV. With these values, ws have calculated tha contrihtion due to transitions

sl

of the bound states in guantUm ~11.

Superlattices

and Microstructures, I

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I

E GaAs=

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Vol. 1, No. 4, 1985 I

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13.2

a factorof twoover itsperpendicularcounterpart. Thamain reductionames frm the L-point ofthe Brillouinzone.Unlike superlattice whare the dielectricconstantshouldeventual.&,a~ac~c~;ie_ of an alloy of equal canposition lectricconstantapproachthat of the GaAs bulk for largew and 20% reductionfor w 2 2 rm. Therefore for mall well width, it is importantto account for the reductionof E when dealingwith +urity statesand excitons.Ourapproachmaynotapply to walls havingvery high barriersbecausewa have not conserv~thestates in our reimvingand insertion process,and wa have used parabolicmodels.

Ackmwledgmsnts: Wa ackuowledge many helpfuldiscussionswith F.H.Pollak,and the supportof this work by the ONR grant no. N00014-834-0140.

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2 4 6 WELL WIDTH

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References *

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1. L.Esakiand R. Tsu, IEM Res. Note Z-2418(1969) L.Esakiand R. Tsu,I5.l J&as and Dev.g,61(1970) 2. R. Tsu and L.Esaki,Appl.Phys. L&t. 2&562(1973) Fig.3 s,,and eI of the quantm well vs. well width. 3. L.L.Chang,L.Esaki and R.Tsu,Appl.Phys. Iett.24 593,(1974) and C.H.&nry,Phys.Rev. 4. R. Dingle,W.Wieqnanu I&t. 3&827(1974) betsmanquantumstatesfor polarizationin the 5. H.L.Stomr,T.Haavasoja,V.Narayanamrti,A.C. plane of the quantuin ml1 layer denotedby EII and Gossard,and W.Wieqmnn,J.Vac.Sci.Technol.B1 perpendiculartothelayerdenotedby EL shown in 423(1983) Fig.2.We have mot includedthe X-pointcontribu6. R. Tsu,A.Kma and L.Esaki,J.Appl. Phys. 46,842 tion becausethe band-edgedifferis tco mall (1975) to be of significance. Note thatbaloww=2m, the Jr. M.D.Camas.K.Hess.J.J. 7. W.D.Laidiq,N.Holonvak quanta statesare squeezedout of the well, Colman, P;D.Dapkusand J.Bardeen,Appl.Phys: giving rise to zero contribution. The oscillation Iett. 38,776(1981) is due tomore andmre quantumstatesinvolvedin 8. J.P.I_&irton and K.Hess,J.Vac.Sci. Technol.~, transitionsas the width of the ~11 is increased. 415(1983) Rev. E,715(1966) When we sum all the contributions for L and r, eg. 9. M.Ilegemsand G.Pearson,Phys lO.J.C.Phillips,in Bonds and Bands in Semiconductors (6) gives the dielectricconstantfor the parallel and perpendicular polarization versus the wall Acad. Press,N.Y.,1973. width.There is a rapid rise fran a value of 10.6 ll.J.R.Cheliko&ky a&d M.L.Cotnan,Phys.Rev.B~ 556 atawidthof 2 m, beames relativelyconstant (1976) above 3 nm as sham in Fig.3.As W increases, 12.F.H.Pollak and M.Cardona,J.Phys.U&n Solids, danselypadcedstatesappearnear the top of the z,423(1966) barrier.They are neither2D like nor 3D in 13.E.E.Mendez.L.L.Chamhamr.G.Landaren.R.Ludeke and L.Esakiand F.H.Poliak,Phys. P&. I&t. 46, nature.The failureof properlytreatingthese 1230(1981) statesled to the poor convergence to the bulk lQ.D.E.Aspnes,Phys. F&v. Bl&5331(1976) value. 15.O.Beroloand J.C.Woolley, Can J.Phys.49,133s Aswa have shown thattheeffectof a guan(1971) tmwell is to reducethe value of the staticdie16.F.H.Pollak, privatecumunication. lectricconstantdue primarilyto the increasein and K.&ass,Int. Conf. the energyof transitioninvolvingquantizedstates. 17. K.B.Kahen,J.P.Leb Sqerlattice,microstructure andMicrcdevices, The breakdownof symmetryresultsin an increaseof 1984 Champaign-Urbana, Ill.USA. the paralleldielectricconstantby approximately

mm1