Acceptor energy levels in GaAs/AlGaAs quantum wells in the presence of an external magnetic field

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Acceptor energy levels in GaAs/AlGaAs quantum wells in the presence of an external magnetic field

ZHAO, Q.X., et al.

Abstract

Energy levels of ground- and excited shallow acceptor states in quantum wells (QWs) have been calculated in the presence of an external magnetic field within a four-band effective-mass theory, in which the valence-band mixing as well as the mismatch of the band parameters and the dielectric constants between well and barrier materials have been taken into account. The energy separations between the ground and different excited acceptor states are deduced at various magnetic fields. The g-factors of the acceptor 1S3/2 ground states and the 2P3/2 excited states are obtained for QWs with different well widths. The oscillator strengths of the acceptor infrared transitions in QWs corresponding to the G, D, and C lines of acceptors in bulk GaAs have also been calculated vs magnetic field up to 10 T.

ZHAO, Q.X., et al . Acceptor energy levels in GaAs/AlGaAs quantum wells in the presence of an external magnetic field. In: Lockwood, David J. 22nd International Conference on the

Physics of Semiconductors, vol. 3 . Singapore : World Scientific, 1995. p. 2283-2286

Available at:

http://archive-ouverte.unige.ch/unige:120904

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1 / 1

...

\

~/ \.· .. - J •

ACCEPTOR ENERGY LEVELS

IN%aAs/AIGaAs~UANTUM

^{WE°LLS IN THE}

PHESENCE OF AN EXTERNAL MAGNETIC FIELD Q.X. Zhao, P.O. Holtz

Dcpanmcnl of Physics and Measurcmcnl Technology, Linkoping University, S-581 83 Linkl>ping, .Sweden

Alfredo Pasquarello

lnslitut Romand de Rechcrcl1e Num~riquc en Physique des Mar~riaux (IRRMA), PHD- Ecublcns, CH-I 015 Lausanne, Switzerland

D. Moncmar, and M. Willander

Depanmem of Physics and Measurement Technology, Linki.iping Universiiy, S-581 83 Linkoping, Sweden

Energy levels of i;round-and exciled shallow acceptor states in quantum wells (QWs) have been calcul:ucd in the presence of an cxtemal magnetic field within a four-band effective-mass theory, in which lhc valence-band mixing as well as the mismatch of the band parameters and the dielectric constants between well and barrier materials have been taken into account. The energy separations between !he ground and different excited acceptor states arc deduced :11 various magnetic fields. The g-faciors of 1he acceptor 1S312 ground Slates and the 2P:i12 excited sta1cs arc obtainec.J for QWi; with different well widths. The oscillator strcngll1s of the acceptor infrarcd transitions in QWs corresponding to the G, D, and C lines of acceptors in bulk GaAs have also been calculated vs magnetic field up to

10 T

1. I11troduclio11

lk.cau~e of the relative complexity of calculating energy levels of acceptors t:anfincd in quantum well (QW) structures, the number of papers which c.Jcal wi1h 1he presence of an external magnetic ricld

is

very limited. To our lmowlcdgc, the only work which considers the mai;nctic field dependence of ll1c acceptor ground states is reported by Massclink et al I. However, !heir prcdic1cd spliuing (Ref. I) is unrcalis1ically large in comparison \Vith c11pcri111cntal data 2,3.

In 1his work. we have cxicnded 1hc theory by Fra.izzoli and Pasquarcllo4 to cover 1he case of an external magnetic field. TI1e advantage of this theory with respect to others is that ii allows one 10 obtain acceptor ground slates as well as excited slates of any symmetry within the same fran1ework. The impurity s1a1es arc calculated will1i11 a four- band cffec1ive-mass theory, in which ll1c valence-band mixing as well as the mismatch of the band parameters and the dielectric constants between well and barrier materials have been taken into account. The spliuin& of 1he shallow acccplor S-likc ground slates as well as the P-like cxci1cd slates have been calculated with a.n applied magnetic field paraJlcl to the growth direction. Transition energies between the ground and the excited srates arc obtained for varying magnetic field strength. The ca1cula1ed transition energies from tl1e ground IS312(f"6) acceptor stale .to 2S312(r6) excited stale arc found lo be in cxccllcnl agreement with our experimental data derived from Resonant Ran1an Scancring (RRS) and

228·1 Q. X. Zha.o cl a.I.

Two-Hole Transitions (THTs) of acceptor bound excitons observed in selective pholOluminescencc in ll1e presence of a magncl.ic field up to 16 T.5

The infrared transitions between lhe IS-ground state and lhe excited states arc also cakulateu. For instance, lhe magnetic field dependence of ll1e infrarcd transitions, denoleu by G(lS312(rg+)-2P312(r3-)), D(lS312(rg+)-2P512(rs-)) and C(lS312(rs+)-2P512(r1·)) in bulk GaAs, arc calculated for the case of acceplOrs confined in QWs at magnelic fields up lo 16 T.6 The polarization properties of these transitions arc also investigated. The corresponding infrared 1.ransition oscillalor strcnglhs arc also calculaled.

The validity of defining g-values for different acceptor levels is discussed. The g- valucs related to lSJf2 and 2P312 acceptor levels in QWs arc calculated for varyini; well widlhs. The calcula1cd g-valucs for !he 1S312(r6) acccp!Or level show an excellent agreement wi1h the available experimental da1a 2,3.

2. Theory

Consic.Jering a single QW, grown in the [001] direction, which we take as the quantilation axis z, !he acceptor Hamiltonian is given by a 4x4 mauix operator 4,6,7

H

=

Jjkin + H'IW + HC+ 2µ lJ Kll•J+ c1µ I (l.l A l 1¹ +

n

J J 1¹ +

n

&I. 1¹) (1)

I lcrc J Ikin rcprc.scnt.s the kinetic cncri;y of the free holes, I 1•1w the co11fincmc11l po1c11tial due to !he valance-band tliscontinuiiy, and HC the potential of the cenU'al acceptor and of ll1e image-charges due 10 the mismatch of tht cliclcctric constant. The kinetic-energy 1cm1, IJkin, quadratic in k=-iV + lclA/(hc), describes the dispersion of the fs valence band with

!he applied magnelic field U=V x A, and is given by the Luuingcr-Kohn Hamiltonian 7. IC

aml ll an; 1hc Luuinger paramclers uescribini; the magnelic field effects on the

r

a valence band. The QW polential represented by Hqw is a square-well potential of barrier height V and well width L. TI1c Coulomb po1cn1ial of a. point charge in a syslem of three dielectric separated by two infinite planes is contained in He. In order to sal.isfy ll1e Maxwell boundary condiiions, He mus! coniain an inrinite series of image charges 4.

We will restrict ourselves lo an applied magnetic field along tl1c z direction in ll1c following discussion. The vector potential A is defined as A=_!_Il x r

=

_!_(-yl3, xl3, 0),

2 2

where

n

is the magnetic field slrcngll1 along' the z direction.

·nie acccplor wavefunctions can be writlcn as a four-component envelope function fm(p,0,l) = [fm,sJ=[fm,312, pn,1/2, fm,-1/2, pn,-J/2j. When the Hamilionian given in (1)

~cts on this four-component funclion Fm, HFm = EFm, the energy posi1ions of the shallow acceptor s1a1es and corresponding wavefuncrions arc derived. The s componcn1 of an acceplor envelope funclion of definite angul~ momentum m can be expanded into a basis set of functions which arc separable in the coordinalcs' p and z:

Fm~(p, O,z) = c-<m·•l• .2.>lm·•',LA:·'c""•Pg: (z)

(2)

I

Here the function g⁵n is chosen lo be !he s component of the four-component envelope funciion g11 , which describes a QW subband stale at k 11 =0. The exponents CXJ arc chosen

·/

I .

Acceptor Energy Levels in GaAs/ AIGaAs Quantum Wells in . • . 2285 in a geomeuical series covering the relevant physical region, :ind the coefficients An1m.s

:ire t:iken as v:iriation parameters. The convergence of the calculations is quite fast :is long as the magnetic length Lb=

y-;n fh

is not significantly smaller than the in-plane orbital length of the acceptor states. The number of basis functions used can be different for different symmetry states. Generally. for more locnlised stntcs, a larger number of basis functions is needed. The dipole transition rule is assumed for transitions between acceptor states. TI1e oscillator strength of transitions from the ground state to the excited states is given by

fm'mio(e)=2mo(Eirri ~ tbm')!(h2 Y1) 1<Fom'1e · rlFim>12 (3).

Here Eom", Fam· and Eim, Fim arc the energies and envelope functions of the ground and excited states, respectively. c is the pol:u-ization vector of the electromagnetic radiation.

3. Results 20.

I'ig.1 (a)

l,

~ G~·lll_,.:::==:::::---

Gh-1 - - - - -

x 18 l-G!"311

c;l·l/2 1,x

>

¹⁷

"'

s

₁₆

;...,

~ ....

"'

w c:

Fig.I (b)

34

c

¹

t

³¹¹ ---·--

c

^{1-J/2 ...} ~ ...

30 x

26

0h·S x

22 0 5 10 15

Magnetic Field (T)

Fig. I 111e magnetic field tlcpcndcnce of the infr:uctl tr.1nsitions correspontling to Ilic (a) C- line, (b) D· :ind C-lincs for ncccp1or:s confinctl in :i 100 A wide GaAs/AlQ.JG:io.7As QW. The

a ⁽ⁱⁿ xy pl:tnc) anti rc (along t direction) pobrit:ition tr:insitions nre indicated by !he solid and the dashed lines, respectively.

o"'(,.) Fii:.1 (a)

120 _<:'11)

n IOT

>

_.... ⁸⁰

--

^.r: _bD E ^{40 l__,____J}

^v

^{L..../I ST}

c:

..,

or

^JI

.... A OT

- _"'

_E ·

120

0¹?1) Fig.2 (b)

<'2

"5: _c31tl rin c31.tl

0 lro

M n 0 JOT

--<

:r.:

₁₀

.... ^.~

^IUW ₃₀

^...

40 ^{I , ST OT} Energy (mcV)

Fig.2 1l1c oscill:ltor strcngll1s of Ilic C. D :intl G

;ibsorplion lines in the acceptor spectrum versus 1.he transition energy for x polariz:ition in a 100 wide QW. The ground sl:ltc is the (a) ISJf2{r6)

, (b) IS312(r7) :icccpior s1:1tcs. TI1c spcctr.1 :it a finite field have an off-set rcbtivc the zc.m field spectrum for clarity.

228G Q. X. Zhao cl al.

The valence b:ind parameters used in the calculations can be found in our previous papcrs5.6, TI1e valence band off set was taken as ~ V=l244 x 0.35 x x mcV (x is the Aluminium-mole-fraction). 111c energy separations between the I S312(r 6) and 2S312(r 6) acceptor states arc calculated in a 100

A

wide G:v\s/Alo.JGao.7As QWs, and show :in excellent agreement with the data deduced from our RRS d:ita . .5 Figs. I and 2 show 1hc energies and oscillntor strength of rhe infr.:1.rcd acceptor transitions vs magnetic fields. The detailed opiic:il transition rules, symmetry and labels used in the figures arc discussed in Refs. 6 and 8. TI1e <lomin:int infrared tr.insition in QWs is the D-linc as also observed in bulk Ga As. P'ig.3 shows the g-valucs of the I S312 and 2PJ/2 ncceptor states. TI1e open :in<l

~olid points arc from Rcfs. 2 and 3, respectively, for the I S312(r6) :icceptor states. We have calculated g-values for m:ignetic fields strength from I T to 16 T. The results show that the g-value variation is less than

s~~~---~--~, o Rcr.~

"'

"' :::J l

cq 2

;.. .;.

I

0

• Rcr.s

2'lt1 (T'6) l\,i~l

IS,n(T'.,>

.1 ... ~~ ... ~ ... ~~ ... ~~ ... ~....,._,

JOO 150 100 150

Well Width

CA)

Fig.3 The g-v:tlucs vs well width

2% for the 1S312(r6) and 2P312(r G) states, less than 6%

for the 1S312(r7) state, and less than 15% for the 2P312(r7) state with the magnetic field up to 16 T. Accordingly, we can conclude that it is still a satisfactory approximation to

use a constant g-factor to describe the splitting for the ground lS acceptor states, and also for the 2P312(r 6) states.

For the 2P312(r1) state the variation with field is too l:lrgc to define a constant g·factor .

In summary, detailed calculations on the acceptor levels have been presented for GaAs/Alo.3Ga(1.7As QWs. The energies and oscillator strengths of the infr:ired acceptor trnnsitions arc deduced vs magnetic fields. The g-values of the lS312 and 2P312 acceptor states :ire also obtained from the calculations. The results show a excellent agreement with recent experimental data for the IS3fi(f6) acceptor state.

4. . References

I 2 3 4 5 6 7 8

W.T. Masselink, Y.-C. Chang, and H. Morkoc, Plrys. Rev. Il32. 5190 (1985).

D.N. Mirlin and A.A. Sircnko, Sov. Phys. Solid Srau 34, 108 (1992).

V.r:. Sapega, et al, Plrys. Ri!v. Il 45, 4320 ( 1992).

S. fraiz.zoli and A. Pasquarello, Phys. Rt!v. ll42, 5349 ( 1990); ll44, 1118 (1991).

Q.X. Zhao, cl al, Plrys. Rev. ll49, 10794 (1994)

Q.X. Zhao, P.O. Hohz, A. Pasquan:llo, D. Moncmar and M. Willandcr, Phys. Rev.

n,

15 July (1994)

J.M. Luttinger, Phys. Rtv. Il 102, 1030 (1956).

Q.X. Zhao, P.O. Holtz, A. Pasquarello, D. Moncmar and M. Willandcr, (unpublished)