Polarization-dependent photoluminescence-excitation spectra of one-dimensional exciton and

a r X i v :c o n d -m a t /0303322v 2 [c o n d -m a t .m t r l -s c i ] 9 J u l 2003

Polarization-dependent photoluminescence-excitation spectra of one-dimensional exciton and

continuum states in T-shaped quantum wires ?

Hirotake Itoh,?Yuhei Hayamizu,Masahiro Yoshita,?and Hidefumi Akiyama ?

Institute for Solid State Physics,University of Tokyo,and CREST,JST,

5-1-5Kashiwanoha,Kashiwa,Chiba 277-8581,Japan

Loren N.Pfeiffer and Ken W.West

Bell Laboratories,Lucent Technologies,600Mountain Avenue,Murray Hill,NJ 07974,USA

Marzena H.Szymanska and Peter B.Littlewood ?

TCM group,Cavendish Laboratory,University of Cambridge,Cambridge CB30HE,UK

(Dated:February 2,2008)

We measured polarization-dependent photoluminescence-excitation spectra of highly uniform T-shaped quan-tum wires at 5K.We attribute one peak to the one-dimensional(1D)-exciton ground state and the continuous absorption band to 1D continuum states.These had similar polarization dependences.We also observed some other peaks above the 1D-exciton ground state and attribute them to exciton states consisting of excited hole sub-bands.These results show good agreement with a model calculation of a single electron-hole pair in T-shaped geometry with exact diagonalizations of the Coulomb interaction.

In quantum wires,one-dimensional (1D)ex-citons are expected to have a large binding en-ergy 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15and large oscillator strength in the ground state 1,2,3,4,16,while the oscillator strength of exciton excited states and 1D continuum states should be suppressed 3,4.

The absorption line shape of the 1D continuum states was recently measured 17in a photoluminescence-excitation (PLE)spectrum of T-shaped quantum wires (T-wires)with improved homogeneity,which were fabricated by cleaved-edge over-growth with growth-interrupt annealing 18in molecular-beam epitaxy.The results did indeed show reduced absorption of the continuum states,and qualitatively support a simple model calculation 3,4.On the other hand,some excited-exciton states that were predicted in a more detailed calculation 15for the T-wire geometry were not found in the observed spectrum.This seems to indicate an inconsistency between the experi-ment and the theory concerning the above fundamental effect,which must be solved before this effect can be applied to op-tical devices such as quantum wire lasers 5,19,20,21.

In this work,we demonstrate PLE spectroscopy of highly uniform T-wires in an improved optical con?guration that al-lows us to measure polarization-dependent PLE spectra with less scattering-light noise than before 16,17.The observed spec-tra of the T-wires exhibit small isolated peaks,found to be in good agreement with the excited-exciton states predicted in the calculation 15.The peak assignments are supported by the polarization-dependence of the PLE spectra.

We studied several samples that had a nominally identical T-wire-laser structure containing 20T-wires formed at 20T-shaped intersections of 20periods of 14nm Al 0.07Ga 0.93As quantum wells (stem wells)and a 6nm GaAs quantum well (arm well),as schematically shown in Fig. 1.The

barrier

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Photon Energy [eV]

FIG.1:Schematic of a T-wire-laser sample,optical con?guration,and PLE spectra of T-wires at 5K at an input power of 6.7μW.Per-centages show Al-concentration x in Al x Ga 1?x As.PLE was mea-sured via point spectroscopy with a spot size about 0.8μm in di-ameter.The polarization of the light used for excitation in PLE was perpendicular (⊥,normal line)or parallel (//,broken line)to the T-wires.

ization together with the continuous absorption above 1.586eV .The origins of the small peaks is most likely the exci-ton states formed with the electron ground-subband and hole excited-subbands,for the peaks showing strong polarization dependence.The continuous absorption band is expected to correspond to the 1D continuum states.A detailed discussion will now be given comparing a data with a theoretical calcu-lation.

We performed a model calculation of a single electron-hole pair,or an exciton,in the present 14nm ×6nm T-shaped ge-

ometry using a formulation reported previously 15.Our model assumes a single hole band corresponding to the ground hole subband in the arm well,which has an anisotropic effective mass due to the band-mixing effect 11,24,25,26,27.Additional band-mixing effects due to lateral con?nement is neglected,since the second hole subband in the arm well is deviated over 30meV .Calculated results are accurate in the low en-ergy region of our interest.The calculation is based on ex-act diagonalizations of Coulomb interaction matrix elements,which gives energy eigenvalues E n and electron-hole envelope wavefunctions Ψn (x e ,y e ,x h ,y h ,z =z e ?z h )(n =1,2,3,...)of

Photon Energy [eV]

FIG.2:PLE spectra of T-wires at 5K at an input power of 2.4μW for perpendicular (⊥,normal line)and parallel (//,broken line)po-larizations,with calculated oscillator strength (circles).The circles are labeled by index n of the states.The dashed line represents the calculated onset of the 1D continuum state and the peak of the 2D ground-state exciton.Horizontal lines (solid and broken)represent the zero-lines of the PLE spectra.

not only the ground but also the excited states of the exciton system.Here,(x e ,y e ,z e )and (x h ,y h ,z h )denote the positions of an electron and a hole.The method of the calculation is the conjugate-gradient minimization 28in a coordinate where x e ,y e ,x h and y h move within 42nm length,and z goes from -50nm to 50nm.Since we use a dense grid,the results are well converged.Though results for low-energy con?ned states are accurate,care must be taken over results for high-energy states which are not accurate,because of the extension in the ?nite sized box and the mixing effect of high energy bands.Thus,the energies of 2D exciton states in the arm well and 1D continuum states in the T-wires were calculated by another method.The oscillator strength of the n -th state was evaluated as | dxdy Ψn (x ,y ,x ,y ,0)|2,a square of an integral of an electron-hole wavefunction with respect to the same electron and hole positions.This evaluation is justi?ed for wavefunc-tions spatially smaller than the wavelength of light in con-ventional far-?eld optical detections.In a previous paper 15,

oscillator strength was evaluated as dxdy |Ψn (x ,y ,x ,y ,0)|2,which is applicable for a particular detection via a local probe in a near-?eld scanning optical microscope.Such “local-probe oscillator strength”is not used in the present analysis.

For comparison with experimental data,we added an offset energy of 1.521eV (corresponding to the bulk GaAs band-gap energy)to E n ,and plotted the calculated oscillator strengths of the n -th states by circles in Fig. 2.The energies of the 1D continuum states and the arm-well 2D-exciton states are shown by dashed lines.The calculated energy of the n =1state with large oscillator strength and the 2D exciton energy in the arm well both show good agreement with the measured strong PLE peaks of the T-wire excitons at 1.575eV and the arm well excitons at 1.596eV .Though the oscillator strength of the n =3state is zero,those of the n =2and 4states have

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?nite small values.The calculated energy positions as well as the small oscillator strength of the n=2and4states show good agreement with the measured small PLE peaks at1.582 and1.586eV.

Figure3shows wavefunctionsΨn(x e,y e,x h,y h,z=z e?z h) for n=1-5and16,where|Ψn|2for various parameter pairs are plotted in the way described in the previous paper15.The wavefunction for the n=1state has no node against any param-eter axis.It is well con?ned in the T-wire and demonstrates that the n=1state is the ground-state exciton in the T-wire. This con?rms that the observed strong PLE peak at1.575eV is due to the ground-state exciton in the T-wire.

For n=2and4,wavefunctions have no node in the elec-tron(x e,y e)motion and the electron-hole relative z motion, while nodes appear in the hole y h motion,which show that the n=2and4states are exciton states consisting of the elec-tron ground subband and the hole excited subbands.Errors in the calculated energy eigenvalues of these excited states in the?nite-sized-box are small,because of the heavy effective mass of hole.The oscillator strengths of these states are small, but not zero,because wavefunctions for different subbands of electrons and holes are approximately orthogonal.The calcu-lated energy and oscillator strength of these states agree well with the experimentally observed two small peaks at1.582 and1.586eV.

Since the present calculation assumes a single hole band, it does not evaluate absolute values of polarization-dependent absorption intensities.However,different hole subbands in general cause different polarization-dependence.Thus,differ-ent hole subbands contained in the n=1,2,and4states quali-tatively explain the experimental?nding that the two peaks at 1.582and1.586eV show stronger absorption for perpendicu-lar polarization,unlike the ground-state exciton peak.

For n=3,5,and16,wavefunctions have no node for the electron(x e,y e)motion and the hole(x h,y h)motion,while nodes appear in the electron-hole relative z motion,which show these states are1D-exciton excited states consisting of the electron ground subband and the hole ground subband in the T-wire.The n=3wavefunction is an odd function of z and has vanishing oscillator strength,whereas the n=5and16 wavefunctions are even functions of z and have large oscilla-tor strength.We should note here that all these states have extended wavefunctions for z,for which the calculated oscil-lator strength and energy eigenvalues are not accurate because of the?nite-sized-box and the light effective mass of electron, and cannot be compared directly with the experimental spec-trum.In fact,the separately calculated onset of1D continuum states is at1.589eV,and all the exciton bound states including the n=5and16states should be below this energy.

An important point we learn for these states is that these 1D-exciton excited states with even parity consisting of the electron ground subband and the hole ground subband in the T-wire have dominant oscillator strength,while other excited states have only negligible oscillator strength.On the basis of this qualitative point,the experimentally observed continuous absorption band around1.589eV is ascribed to excited exci-ton states and1D continuum states consisting of the electron ground subband and the hole ground subband in the T-wire.The similarity in polarization dependence between the contin-uous absorption band and exciton ground state suggests that the contributing hole subband is common.

In summary,we measured polarization-dependent PLE spectra of highly uniform T-wires,and found good agreements with a model calculation.The lowest energy peak due to the 1D-exciton ground state and a continuous absorption band due to1D continuum states show polarization dependence similar to each other,while two small peaks due to excited hole sub-bands have different polarization anisotropy.The oscillator strength of the exciton ground state in T-wires is very large,in stark contrast with the small oscillator strength of the excited exciton states and the1D continuum states.Inverse-square-root singularity was absent in the absorption line shape of1D continuum states in agreement with theories3,4,15.

This work is partly supported by a Grant-in-Aid from the MEXT,Japan.

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FIG.3:Contour-plots of calculated|Ψn|2for n=1,2,3,4,5and16for various parameter pairs.Parameters(x e,y e,z e)and(x h,y h,z h)denote the positions of an electron and a hole,and z is de?ned by z=z e?z h.The directions of x,y and z correspond to[110],[001],and[ˉ110].The size of the boxes is42nm×42nm for the upper two rows,84nm×84nm for the third row,and84nm×100nm for the bottom row.The three contour lines in each box represent80%,50%,and20%lines of maximum.In the relative xy coordinate system,since|Ψn|2is plotted for z relative=0,it is zero everywhere in the n=3state which has a node at z=0.

?E-mail(H.Itoh):hiroitoh@issp.u-tokyo.ac.jp

?also at Bell Laboratories,Lucent Technologies,600Mountain Avenue,Murray Hill,NJ07974,USA

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