On the intersubband absorption line width in semiconductor quantum wells

On the intersubband absorption line width in semiconductor quantum wells

~ - I ~ $5.~ + . ~ Perg~on Press L ~ Solid Stare Commu~cations, Vol. 82, No. ~ pp. 565-567, 1992. Printed ~ Groat Britain. ON THE ~ E R S U B B A ~ ...

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~ - I ~ $5.~ + . ~ Perg~on Press L ~

Solid Stare Commu~cations, Vol. 82, No. ~ pp. 565-567, 1992. Printed ~ Groat Britain. ON THE ~ E R S U B B A ~

A B S O R ~ O N HNE ~D~ ~LLS .

~ SEMICO~U~OR

QUA~UM

M. ~ y

Institum ~ Physics, M. C~e-Skl~owska U ~ v e r ~

2~031 Lubfin, ~ M. C ~ S H ~ o w s ~

l, P~and

(Received 29 N o , e m i r 1991 ~ ~ M ~ l e g e O

The influence of the nonparaboficity of the ground subband induced by the exchange ~teraction on the inte~ubband absorption fine shape of a doped GaAs/Al~Ga~_~As quantum well ~ ~ u s s e d . The obtained ms~m ~ c a t e that contrary m the Bandara's suggestion t~s nonparabofid~ Oves rather small cont~bution to the fine width ~ the depolarization and ex~ton effecm are taken ~to accoun~ The similar ~tuation takes ~ace when nonpar~leHsm of the subbands adses from the ~trinsic nonparabolidty of the well m a m ~ . INTERSUBBAND transitions ~ quantum w~Hs (QW'~ have, besides thor ~mmst' from the fundamental point of ~ew, recen0y, aRracted substantial atmntion for their utilization ~ novel ~frared ~R) detector. In such detecto~ the QW ~ doped to den~ties ~ o ~ to 10~2cm -~ where the influence of the man,body effects of energy level structure and ~tersubband absorption spectrum shoed be ~zable. In recent pape~ Bandara a aL [1, ~ have attempted to ~ u d e the exchange ~mraction in an appro~mated way on the subband ~speriom They derived an expfi~t formu~ for in the #ane momentum (k) dependence of the exchange inmraction for the subband eneroes at T ~ • It pm~cm the Mrge variation with ~ of the energy separat~n between the exited and ground subband. From tMs large variation the autho~ of [2] Mentified the exchange interaction as one of the m~or source of fine broade~ng in the ~m~ubband absorpt~n spectra. The purpo~ of tMs nora ~ to show that even ff the exchange ~duced nonparab~i~ty ~ as large as sugge~ Bandara et al. [1] ~s contribution to the fine broade~ng ~ ~ ~rge exmnd compensated by depoMrization effect not con~dered ~ [2]. The ~mi~r compensation takes #ace when k dependence of the ~m~ubband ~paration afi~s ~om the nonparabofici~ of the well and barrier mamfi~s [3]. Assume, Hke ~ [I, ~, that ~ the ab~nce of the dectron--electron ~mraction the subhands am parab~ic. The energy ~paration (AE) between the ground ~ = 0) and exalted subband ~ = I) ~ then independent on k. When the well ~ doped the charge neutr~ity mi~rnizes directCoulomb interactionsand we can expect that exchange interaction can contribus the most impo~ant co~ection to the mnormal-

izafion of the onemlectron energy leve~. Bandara's ms~ts [1, ~ suggest thaL in an electrically neutral doped we~ t~s correction ~ much larger than that ms~ting from &rect Coulomb ~teraction and ~ the, ka&ng order ~ ~dependent on the detailed shape of the ~ectron wave functions. The renormalized ~mrsubhand energy (E~o(k)) can then be approximated by the form~a [1, ~ El0(k) ~ A E + :k/E(k/k/)/2~2~

(1)

where k / ~ ~ N , ) ran, N, ~ the two-dimensional dectron density ~ the quantum we~ ~ ~ the &~ectric consent and E(x) ~ a complem ~fiptic ~mgral. (We assume that o~y ground subband ~ occupied and T = 0.) The nume~cal evaluation of the exchange correction to the ~tersubband energy shows Fig. 1. At this p~nt ~ ~mmsting to nora that t~s correction ~ ~veral times ~rger than that ob~ined wit~n the loc~ densi~ approximation (see for exarapk [4]). Momove~ ~ contrast with the loc~ density approximatio~ we observe the strong va~ation of the co~ection with ~ (k-dependence of the subband ~paration induced by ~t~ns~ nonparabolicity ~ at T ~ 0 ne~io~e [3]). The &ffemnce E~0(0) - E~0(k:) change with the electron concentration fike N: ~ ~ee Fi~ ~. It ~ well known that ~ ad&tion to the m a n , body effects wh~h m n o r m ~ e the subband energy levels, the int~subband absorption process ~ mottled by depolarizaton and ex~ton effect [~. In th~ work we take ~to account above effec~ using the formafism of Ando ~, ~. When an exmrn~ ~ectric fie~ Ee - ~ ~ applied ~ the z ~rection the absorbing power of the QW of u~t area ~ proportion~ to the m~ part of the zz component of the mottled two ~mension~ frequency

565

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SEMICONDUCTOR Q U A N T U M WELLS

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Fig. 3. Absorption ~ne shape G(~) as a function of photon energy h~ for 85-A GaA~AlxGaj_xAs QW's with different carrier concentrations calculated for two values of F: (a) F = 2 meV and (b) F = 6 meV. The solid Hnes contain the depolarization and exdton effect~ dashed Hnes do not. For comparison we show also the results calculated with the hdp of equation (5) where the depolarization, ex~ton and broadening effects are neglected (dotted ~ne~. The barrier height used ~ VB = 240 meV.

SEMICONDUCTOR QUANTUM WELLS

Vd. 82, No. 7

567

N~ = 0.5 x 10hem -2 and assuming ~ r ~m#ici~ that ?. ~ N,. The ~sdts a ~ p ~ e d ~ H ~ 3. If we ne~e~ the depdarization and exoton eff~ts then ~ the a b ~ n ~ of the b~ade~ng of the ~vds the ~ncfion G(w) has the ma~mum at h~ = ~0(0) and is ~ron~y ~ymm~tic. Hgure 3 shows that by con~dedng ~ e ~vel broadening the Re ~ ) = ~e~#(E,o(O)m)-'G(m), (2) peak of G(~) ~ s~fied to ~ e smaller energy, ~s h ~ with width becomes ~rger and the h ~ g ~ d ~ a ~ Indu~on of the depolarization and exoton eff~ts = ~o~) Im ~ 1 + ~,,g(~ O) changes the ~tuafion d~stically. We s ~ that where depolatization effect not o~y s ~ s the peak position to the ~gher energy but ~so tends to compens~e ~ e fine ~ o ~ i o n ~s~fing ~om the k dependence of the ~) = (2IN,) ~)2dk E~o(k) _E~0(0)Om):- ~ F ' (4) intersubband energy ~paration. Absorption spectrum ~a~ic~ady in QW with ~gher d~Won conce~ and ~0 ~ the osdHator ~ n ~ h cor~spon~ng tratio~ has, except ~w energy pa~, nearly Lo~ntzian n = 0 ~ n = 1 intersubband ~an~tion, the hctor shape and ~ c o m ~ wi~ Bandara's ~l~ = ~ ~s~ts ?~ accoums ~ r dep~atization and ex~ton s ~ ~ - ~ the h ~ w i & h of the ~ne ~ determ~ed m ~ y by the and F is the phenomendo#cM p a ~ m a ~ d~ctiMng broade~ng p a ~ m ~ F. the ~ne broadening induced by the electron scattetin~ In condu~on, we have shown that when the k From equation (4) we see ~at absorption ~ne dependence of the subband separat~n ~s~ts ~om pro~e ~ de~rmined by the behafiour of the function exchange interaction the ~tersubband absorption ~ne G(~). (We ne#ect the change of~0 wi~ L) U~ng width ~ ~ron~y aff~ted by depo~dzafion effecL relation ~na.~0 [a/(~ + ~ = 6(x) we find that ~ ~mit ~ y. ~ 0 equation for G(~) ~duces ~ the ~ r m Acknowledgement -This w o ~ was supported by KBN under Grant No. 2 031091 01. 2~e~E,o(0) 1 dependem ~nductifi~ ~nsor ~(~)]. I n ~ c ~ quamum-w~ IR d ~ AE ~ 0.1 eV and ratio I~0(0) - E~o(ky)l]E~o(O)~ small. Thu~ a~(~) ~ n be c ~ c u ~ d ~ a way simil~ ~ that d e ~ d ~ d ~ Ando~ paper [6]. A~uming that h~ ~ d o ~ to the ~bband separation ~0(0) we find

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whe~ E(~ and K(x) are comple~ elliptic h~grals and k ~ the sdufion o f ~ = E~o(k). (The above ~ r m ~ a ~ con~stent with that obt~ned by Bandara [l~. We have calculated the variation of G(m) ~ r cou#e v~ues of F ~ n g the numerical v~ues of the parameters ~E and ~, approptia~ ~ r the uni~rm~ doped 8 ~ GaAs quantum well with 240meV bartiers of A~G~.~As and N, = 10ncm -~ [4]. For comparison we have ~so performed ca~uhfion of G(w) ~ r the p ~ o u ~ y spec~ed QW but ~ n g

REFERENCES 1. 2. 3. 4. 5. 6.

K.M.S.V. Bandara, D.D. Coon, Byungsung O, Y.F. Lin & M.H. Francombe, AppL Phys. Lett. 53, 1931 (1988). J.-W. Cho~ Byungsung O, K.M.S.V. Bandara & D.D. Coon, Superla~ices and Microstructures 10, 1 (1991). M. Zah~n~ Phys. Re~ B43, 5411 (1991~ W.L. Bloss, J. AppL Phys. 66, 3639 (1989). T. Ando, A.B. Fowler & F. Stern, Mo& Re~ Phys. 54, 437 (1982). T. Ando, Z Phys. Jpn. 44, 475 (1978~