Theory of internal transitions of charged excitons in quantum wells in magnetic fields

a r X i v :c o n d -m a t /9911128v 1 [c o n d -m a t .m e s -h a l l ] 9 N o v 1999

Theory of internal transitions of charged excitons

in quantum wells in magnetic ?elds

A.B.Dzyubenko

Institut f¨u r Theoretische Physik,J.W.Goethe-Universit¨a t,60054Frankfurt,Germany

General Physics Institute,RAS,Moscow 117942,Russia

A.Yu.Sivachenko

The Weizmann Institute of Science,Rehovot 76100,Israel

(February 1,2008)

For charged semiconductor complexes in magnetic ?elds B ,we discuss an exact classi?cation of states,which is based on magnetic translations.In this scheme,in addition to the total orbital angular momentum projection M z and electron and hole spins S e ,S h ,a new exact quantum number appears.This oscillator quantum number,k ,is related physically to the center of the cyclotron motion of the complex as a whole.In the dipole approximation k is strictly conserved in magneto-optical transitions.We discuss implications of this new exact selection rule for internal intraband magneto-optical transi-tions of charged excitons X ?in quantum wells in B .73.20.Dx,71.70.Di,76.40.+b

Recently,there has been considerable experimental

and theoretical interest in charged excitons X ?and X +in magnetic ?elds B in 2D systems.Experimentally,magneto-optical interband [1]transitions of charged ex-citons have been studied extensively.Theoretically,the binding of charged excitons X ?has been considered in quantum dots [2],in a strictly-2D system in the high-magnetic ?eld limit [3],and in realistic quantum wells at ?nite B [4].In all these theoretical works on charged excitons,the existing exact symmetry —magnetic trans-lations —has not been identi?ed.The aim of the present theoretical work is to describe in some detail this symme-try and its manifestations in intraband magneto-optical transitions (see also [5]).Experimental evidences for in-ternal X ?singlet and triplet transitions have been re-ported very recently in [6],where also comparison with quantitative calculations is presented.

We consider a system of interacting particles of charges e j in a magnetic ?eld B =(0,0,B )described by the Hamiltonian

H =

j

?π2j

2

i =j

U ij (r i ?r j ),

(1)

here ?πj =?i ˉh ?j ?

e j

2B

×r ,there is the axial symmetry about

the z -axis [H,?L z ]=0,where ?L z = j

(r j ×?i ˉh ?j )z .Therefore,the total angular momentum projection M z ,

an eigenvalue of ?L

z ,is a good quantum number.In a uniform B ,the Hamiltonian (1)is also invariant un-der a group of magnetic translations whose generators are the components of the operator ?K = j ?K j ,where ?K

j =?πj ?e j c

Q ,Q ≡

j

e j ,(2)

while [?K p ,?πq ]=0,p,q =x,y .For neutral complexes

(atoms,excitons,biexcitons)Q =0,and classi?cation of states in B are due to the continuous two-component vec-tor —the 2D magnetic momentum K =(K x ,K y ).For

charged systems the components of ?K

cannot be observed simultaneously.This determines the macroscopic Lan-dau degeneracy of exact eigenstates of (1).For a dimen-sionless operator ?k

= 2are Bose rais-ing and lowering ladder operators:[?k

+,?k ?]=?Q/|Q |.It follows then that ?k

2=?k +?k ?+?k ??k +has discrete os-cillator eigenvalues 2k +1,k =0,1,....Since [?k

2,H ]=0and [?k

2,?L z ]=0,the exact charged eigenstates of (1),in addition to the electron S e and hole S h spin quan-tum numbers,can be simultaneously labeled by the dis-crete quantum numbers k and M z .The labelling there-fore is |kM z S e S h ν .Here νis the “principal”quantum number,which can be discrete (bound states)or con-tinuous (unbound states forming a continuum)[5].The k =0states are Parent States (PS’s)within a degen-erate manifold.All other Daughter states in each ν-th family are generated out of the PS iteratively:for Q <0|k,M z ?k,S e S h ν =(?k

?)k |0,M z ,S e S h ν /

√

e-and

π/2e2/?l B,

l B=(ˉh c/eB)1/2.The states within each such

form a continuum corresponding to the extended

of a neutral magnetoexciton(MX)as a whole

the second electron in a scattering state.As an

example,the continuum in the lowest(N e N h)=(00)LL

consists of the MX band of width E0extending down in

energy from the free

(00)LL.This corresponds to the

1s MX(N e=N h=0)[9]plus a scattered electron in the zero LL,labeled X00+e0.The structure of the con-tinuum in the(N e N h)=(10)LL is more complicated:in addition to the X00+e1band of the width E0,there is another MX band of width0.574E0also extending down in energy from the free(N e N h)=(10)LL.This corre-sponds to the2p+exciton(N e=1,N h=0)[9]plus a scattered electron in the N e=0LL,labeled X10+e0.

There are also bands(not shown in Fig.1)above each free LL originating from the bound internal motion of two electrons in the absence of a hole[5].Internal transitions to such bands have extremely small oscillator strengths and not discussed here.Bound X?states(?nite inter-nal motions of all three particles)lie outside the continua (Fig.1).In the limit of high B the only bound X?state in the zeroth LL(N e N h)=(00)is the X?-triplet.There are no bound X?-singlet states[3,4]in contrast to the B=0case.The X?-triplet binding energy in zero LL’s (N e N h)=(00)is0.043E0[3,4].In the next electron LL (N e N h)=(10)there are no bound X?-singlets,and only one bound triplet state X?t10,lying below the lower edge 2

of the MX band[5].The X?t10binding energy is0.086E0,

twice that of the X?t00,and similar to the stronger binding of the D?-triplet in the N e=1LL[8].

We focus here on internal transitions in theσ+polar-ization governed by the usual selection rules:spin con-

served,?M z=1.In this case the e-CR–like inter-LL

(?N e=1)transitions are strong and gain strength with B.Both bound-to-bound X?t00→X?t10and photoion-izing X?transitions are possible.For the latter the

?nal three-particle states in the(10)LL belong to the continuum(Fig.1),and calculations show that the FIR absorption spectra re?ect its rich structure[5].Tran-sitions to the X00+e1continuum are dominated by a sharp onset at the edge(transition1)at an energyˉhωce plus the X?t00binding energy.In addition,there is a broader and weaker peak corresponding to the transi-tion to the X01+e0MX band,transition2.The latter may be thought of as the1s→2p+internal transition of the MX[10],which is shifted and broadened by the presence of the second electron.In accordance with this picture,it is visible from Fig.2that with increasing sep-aration d between the e-and h-layers(when the exciton binding and,thus,transition energies are reduced and the X–e interaction is e?ectively diminished),the sec-ond peak is redshifted and sharpened.Thus the X?-triplet behaves physically in the photoionizing bound-to-continuum transitions as an exciton that very loosely binds an electron,and the two“parts”of the complex can absorb the FIR photon,to some extent,independently. The double-peak structure of the bound-to-continuum transitions is a generic feature for transitions from both the singlet and triplet ground X?states in quasi-2D sys-tems in strong B.Such transitions in translationally in-variant systems are discussed theoretically in[6],where also experimental results for bound-to-continuum transi-tions are reported and comparison between theory and experiment is made.

The inter-LL bound-to-bound transition,X?t00→X?t10, has a very speci?c spectral position:since the?nal state is more stronger bound,it lies below the e-CR energy ˉhωce=ˉh eB/m e c.However,it has exactly zero oscillator strength,a manifestation of the magnetic translational invariance:the two selection rules–conservation of k and?M z=1cannot be satis?ed simultaneously.In-deed,e.g.,the X t00PS(with k=0)has M z=?1, while the X t10PS has M z=1,so that the usual se-lection rule?M z=1cannot be satis?ed.Localization of charged excitons breaks translational invariance and relaxes the k-conservation rule.As a result,the bound-to-bound X?t00→X?t10transition,which is prohibited in translationally-invariant systems,develops below the e-CR[5].Such a peak is a tell-tale mark of localization of charged triplet excitons.

In conclusion,we have studied the exact symmetry

—magnetic translations—for charged excitons in B and established its consequences for intraband magneto-optical transitions.In particular,we have shown that in translationally invariant quasi-2D system with a sim-ple valence band the bound-to-bound transition from the triplet ground state X?t to the next electron Landau level is prohibited.In the presence of translationally-breaking e?ects(disorder,impurities etc.)the intraband bound-to-bound triplet transition develops below the electron cyclotron resonance.This suggests a method of studying localization of charged excitons.

ABD is grateful to the Alexander von Humboldt Foun-dation for research support.