Unintentional high density p-type modulation doping of a GaAs/AlAs core-multi-shell nanowire
J. Jadczak, P. Plochocka, A. Mitioglu, I. Breslavetz, M. Royo, A. Bertoni, G. Goldoni, T. Smolenski, P. Kossacki, Hadas Shtrikman, D. K. Maude
UUnintentional high density p-type modulation dopingof a GaAs/AlAs core-multi-shell nanowire
J. Jadczak, † , ∇ P. Plochocka, ∗ , † A. Mitioglu, † I. Breslavetz, ¶ M. Royo, § A. Bertoni, § G. Goldoni, § T. Smolenski,
P. Kossacki,
A. Kretinin, @ Hadas Shtrikman, @ andD. K. Maude † Laboratoire National des Champs Magnétiques Intenses, CNRS-UJF-UPS-INSA, 143, avenue deRangueil, 31400 Toulouse, Institute of Applied Physics, Academiei Str. 5, Chisinau, MD-2028,Republic of Moldova, Laboratoire National des Champs Magnétiques Intenses,CNRS-UJF-UPS-INSA, 25, avenue de Martyrs, 38042 Grenoble, Institute for Nanoscience,CNR-NANO S3, via Campi 213/A, 41125 Modena, Italy, Departament de Química Física IAnalítica, Universitat Jaume I, Av. Sos Baynat s/n, 12080 Castelló, Spain, Department of Physics,Informatics and Mathematics, University of Modena and Reggio Emilia, Italy, Institute ofExperimental Physics, Faculty of Physics, University of Warsaw, Ho˙za 69, 00-681 Warsaw,Poland, Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot76100, Israel, and School of Physics and Astronomy, University of Manchester, UK
E-mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r bstract Achieving significant doping in GaAs/AlAs core/shell nanowires (NWs) is of considerabletechnological importance but remains a challenge due to the amphoteric behavior of the dopantatoms. Here we show that placing a narrow GaAs quantum well in the AlAs shell effectivelygetters residual carbon acceptors leading to an unintentional p-type doping. Magneto-opticalstudies of such a GaAs/AlAs core multi-shell NW reveal quantum confined emission. Theo-retical calculations of NW electronic structure confirm quantum confinement of carriers at thecore/shell interface due to the presence of ionized carbon acceptors in the 1 nm GaAs layer inthe shell. Micro-photoluminescence in high magnetic field shows a clear signature of avoidedcrossings of the n = n = Keywords: GaAs core/shell nanowires, 2D confinement, resonant phonon coupling.Semiconductor nanowires (NWs) represent a rapidly expanding field of research largely due totheir great technological promise.
For example, transistor action has been demonstrated usingcarbon nanotubes and silicon nanowires, it has been suggested that indium phosphide nanowirescan be used as building blocks in nanoscale electronics, and doped radial core-multi-shell NWshave good chances to find industrial applications as high efficiency solar cells. NWs with twodimensional (2D) carriers localized at the core/shell interface offer new perspectives in quantum ∗ To whom correspondence should be addressed † Laboratoire National des Champs Magnétiques Intenses, CNRS-UJF-UPS-INSA, 143, avenue de Rangueil, 31400Toulouse ‡ Institute of Applied Physics, Academiei Str. 5, Chisinau, MD-2028, Republic of Moldova ¶ Laboratoire National des Champs Magnétiques Intenses, CNRS-UJF-UPS-INSA, 25, avenue de Martyrs, 38042Grenoble § Institute for Nanoscience, CNR-NANO S3, via Campi 213/A, 41125 Modena, Italy (cid:107)
Departament de Química Física I Analítica, Universitat Jaume I, Av. Sos Baynat s/n, 12080 Castelló, Spain ⊥ Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Italy
Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Ho˙za 69, 00-681 Warsaw, Poland @ Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel (cid:52)
School of Physics and Astronomy, University of Manchester, UK ∇ Institute of Physics, Wroclaw University of Technology, 50-370 Wroclaw, Poland However, to introduce carriers the control and understanding of the doping mecha-nisms in NWs is crucial.Epitaxial GaAs has been investigated for more than 40 years, a field of research propelled bythe discovery of the quantum Hall effect with its panoply of exotic many body ground states.
Significant efforts have been made to obtain high mobility 2D carriers in GaAs heterostructuresgrown by molecular beam epitaxy (MBE). Notably, the development of remote or modulationdoping, which spatially separates the carriers from the dopant atoms, was a crucial step in thediscovery of the fractional quantum Hall effect. The direct application of the modulation dop-ing techniques to GaAs/AlAs NWs would seem to be a natural evolution. Today high qualityGaAs/AlAs nanowires with a large aspect ratio and typical diameters of a few tens of nanometersare routinely grown by MBE using the vapour-liquid-solid (VLS) method.
While core multishell NWs would seem to be ideally suited for modulation doping, in practice achieving a signifi-cant doping remains a challenge. In the VLS method, depending upon the growth plane, dopantscan act as donors or acceptors and the dopant incorporation can be different for axial and lateral(sidewall) growth resulting in an inhomogeneous dopant distribution, or even compensation andnegligible doping. In NWs the use of the AlGaAs ternary alloy for the shell can lead to the segre-gation of the Ga and Al atoms leading to the formation of the quantum dots.
Moreover, usingAlGaAs for the shell leads to a red shift of the photoluminescence emission from the NW core,which is not understood. On the other hand both experiment and theory suggest that modulationdoping in core-multi-shell NWs can lead to non uniform charge distribution with an accumulationof charge at the facets or corners of the hexagonal NW which can lead to quantum confinement. In this work, we show that incorporating a narrow GaAs quantum well, which can be used toaccommodate dopants, in the AlAs shell of the NW can lead to significant unintentional p-typedoping due to the incorporation of residual carbon acceptors. Experiment and calculations demon-strate that charge transfer at low temperature leads to quantum confinement with the formation ofa high density two dimensional hole gas at the core/shell interface of the NW.The GaAs/AlAs NWs were grown by MBE using the self-assisted VLS method, on ( ) -3igure 1: (a) typical transmission electron microscopy (TEM) image taken from the center of aGaAs/AlAs core-multi-shell NW, (b)pure zinc blende structure confirmed by the electron diffrac-tion taken from the [011] direction, the arrow is aligned along the [111] growth direction, (c)scanning electron microscope (SEM) image of the GaAs/AlAs core-multi-shell NWs ∼ µ mlong.oriented silicon bearing a native oxide layer. After water removal at 200 ◦ C, the Si wafer wasoutgassed in a separate chamber (600 ◦ C), before being transferred into the MBE growth chamber.Growth was initiated by Ga condensation at pin holes in the SiO layer and carried out at 640 ◦ C anda group V/III (As /Ga) ratio of 100. Uniform diameter GaAs NWs were grown with a high aspectratio, no significant tapering and pure zinc-blende structure, as revealed by careful transmissionand scanning electron microscopy (Fig. 1(a) and (c)) and electron diffraction taken from the [011]direction (Fig. 1(b)). For growth of the shell (nominally AlAs 3 nm/GaAs 1 nm/ AlAs 3 nm) and12 nm GaAs capping layer, the temperature was lowered to 520 ◦ C. The hexagonal GaAs NW coreof side length d (cid:39) nm composes a substrate for the multilayer structure, which consists of aAlAs/GaAs/AlAs shell of a nominal thickness of 7 nm and a GaAs capping layer.A typical µ PL spectrum of a single core-multi-shell NW obtained in the absence of magneticfield at a temperature T = . µ PL spectrum measured on a single core-shell NW. Thestructure of each sample is shown schematically in the inset of Fig. 2. We stress that both sampleswere grown in the same MBE chamber under nominally the same growth conditions. Both NWshave a rather large core (80 nm for core-shell and 60 nm for core-multi-shell) so that in a first4igure 2: Typical µ PL spectra of core-shell GaAs/AlAs NW (black line) and core-multi-shellGaAs/AlAs (red line) excited by 660 nm laser line. Inset: schematic cross-section of core-shelland core-multi-shell NWs.approximation quantum confinement can be neglected. The common feature of both emissionspectra is the prominent peak at 1 .
507 eV with shoulders which develop into well resolved emissionlines in a magnetic field. This transition is related to the so called KP series of excitons boundto defect pairs with different separations in extremely high quality epitaxial GaAs. The emissionenergy of 1 .
507 eV corresponds well to the strongest line in the KP series in 1 . − .
511 eVspectral region. The PL spectrum of the core-multi-shell NW is far richer than for the core-shell NW. In additionto the KP series, a large number of emission lines are resolved already in the zero magnetic fieldspectrum. At higher energies but still below the band gap of bulk GaAs ( E ZBg = .
519 eV) wecan distinguish the free exciton (X) at 1 .
515 eV and exciton bound to neutral acceptor (AX) at1 .
513 eV. Below the KP series, in the energy range 1 . − .
493 eV we observe a number ofweak features with emission energies characteristic of the free-electron carbon-acceptor A e andthe donor carbon-acceptor ( D − A ) transitions. This suggests that we have doped regions withinthe NW due to the unintentional incorporation of carbon. The shape of the spectra at low energy5s characteristic for p-doped structures, as recently observed for a single GaAs NW with an axialheterojunction. In particular, it was shown that the emission energy for n-type and p-type materialis quite different; for p-type GaAs NWs the emission is dominated by recombination via acceptorcenters, whereas for n-type NWs emission is blue shifted with respect to the GaAs band gap dueto a band filling effect with increasing doping concentration.
For the core-multi-shell NWs we observe additional emission lines at energies higher than theband gap of GaAs in the range 1 . − .
528 eV. This emission energy is typical for a GaAsquantum well (QW) of width 15 −
20 nm suggesting a quantum confinement of the carriers inthe core. To distinguish this high energy emission from the emission in other spectral regions werefer to it a “2D like” in the rest of the paper. The core-multi-shell structure incorporates a 1 nmGaAs narrow quantum well in the AlAs shell at a distance of 3 nm from the core. No emissionfrom this ultra thin quantum well (QW) is detected at higher energies, presumably due to the rapidthermalization (escape) of photo-created carriers. The 1 nm GaAs layer between the two AlAslayers is expected to getter impurities, notably residual carbon atoms.
It acts as an efficientimpurity trap due to the higher solubility of carbon atoms in GaAs and due to the floating ofcarbon atoms at the AlAs vacuum interface during the MBE growth.
Moreover, for AlAs weexpect the concentration of carbon to be quite high because Al atoms are more reactive with carbonor other impurities. For example, the residual carbon incorporation in AlAs can be two ordersof magnitude higher than for GaAs. In our core-multi-shell NWs the carbon incorporated in theGaAs quantum well can lead to a non-uniform charge distribution with excess holes accumulatedat the core/shell interface.This hypothesis is confirmed by 3D self-consistent simulations of the NW electronic structure.The modeling assumes spatial invariance along the NW axis and includes the nominal multi-shellmaterial modulation over the NW cross section. The interstitial GaAs quantum well is uniformlydoped with a constant density of acceptors δ A . Our results predict that a hole gas starts to accumu-late at the core/shell interface for p-doping densities of the order of 2 . × cm − (correspondingto a linear density of 4 . × cm − in the NW). The hole gas density distribution for a NW with6igure 3: Calculated PL spectra (main panel) and hole gas distributions (small left panels) for core-multi-shell NWs with the p-doping densities ρ A indicated in the left panels, together with the 2Dfree hole density N h . The colormap insets illustrate the squared envelope functions for the electron(left) and hole (right) states whose recombination yields the PL peaks. The vertical dashed lineindicates the position of the GaAs bulk band gap.three selected doping densities is illustrated in the left panels of Fig. 3. The different localizationpatterns are in line with previous results. At low densities the hole gas is distributed near theheterojunction forming six wide channels at the center of the hexagonal facets (bottom panel). Asthe doping density is increased the hole gas moves closer to the heterojunction and it forms, first, aquasi-uniform sixfold bent 2DEG (middle panel), and finally, six tunnel-coupled narrow channelsat the edges (top panel).The calculated PL spectra corresponding to the same doping densities can be seen in the mainpanel of Fig. 3. With increasing doping densities the exciton ground state red shifts and its intensityis substantially reduced. This can be explained as follows: The peak red shifts due to the largerattracting mean field experienced by the electron when the hole density is increased. At the sametime, whereas the hole ground state is further localized near the heterojunction, the electron groundstate remains in the center of the GaAs core (see two top-left insets in Fig. 3) due to the repulsivepotential generated by the high density of acceptors. Therefore, the overlap between the groundelectron and hole states becomes very small and the PL intensity of such transition is reduced. The7L spectra of the NWs with higher acceptor densities show an additional intense peak at 1 .
529 eV.As illustrated in the corresponding insets in Fig. 3, this peak originates from the recombination ofan electron in the ground state and a hole in an excited, but still occupied, state with an azimuthal-like nodal surface. Since this hole state is well spread over the center of the GaAs core it has alarge overlap with the electron ground state leading to an intense peak in the PL spectrum. Thispeak is not observed at lower doping density since the excited hole state is unoccupied.Figure 4: Evolution of the PL spectra (solid curves) of core-multi-shell GaAs/AlAs NW as functionof the excitation power, at T= 1.8 K. The calculated PL spectra for different doping levels areplotted for comparison (dashed curves). The black dot-dashed lines are a guide to the eye tohighlight the evolution of the position of 2D emission lines described in the text. Note that thespectra have been shifted vertically for clarity.The concentration of photo created electron hole pairs, and thus the concentration of holes canalso be tuned by varying the excitation power. A comparison between experimental results andtheoretical simulation is presented in Fig. 4, which shows plots of µ PL spectra for different exci-tation powers. The relative intensity of the KP series of lines, below E ZBg = .
519 eV, decreases asthe excitation power increases. This is due to the saturation of these transitions when all the defectpairs have a bound exciton. In contrast, the 2D like emission intensity increases continuously withexcitation power. For the highest excitation power, the spectra are dominated by emission relatedto the 2D hole gas. This further confirms the very different origins of each emission channel.Theory predicts that the mechanism of the 2D recombination depends on the concentration8f photo-excited carriers. Increasing the number of photo created electron-hole pairs changes theoverlap between electron and ground hole states due to the modified distribution of 2D hole gasin the core. The calculated PL spectra for two different hole concentrations are shown in Fig. 4for comparison, showing qualitative agreement with the experimental data. The transition around1.526 eV shifts toward lower energies with increasing excitation power, while the second peakaround 1 .
529 eV, originated from recombination of electrons with holes in excited state, remainsat the same energy. Its intensity is significantly enhanced as a straightforward consequence of theincreased occupation of the electron and hole sub-bands involved in the radiative recombination.Figure 5: (a) Evolution of PL spectra of core-multi-shell GaAs/AlAs NW as a function of temper-ature, (b) Temperature dependence of the emission energy (symbols) together with the establishedtemperature dependence of the GaAs band gap (lines) (c) Integrated PL intensity rate ln ( − + I / I ) as function of 1 / T (symbols). The lines are the linear fits used to deduce the activation energies.The temperature dependence of the µ PL spectra presented in Fig. 5 (a) provides further confir-mation of the 2D character of the observed emission. In the µ PL spectra at T=12 K, two relativelybroad peaks are observed corresponding to the recombination of the acceptor bound exciton (AX)and the confined 2D exciton ( X D ) at higher energies. Both shift to lower energy with increas-9ng temperature at a rate which tracks the temperature dependence of the GaAs band gap (seeFig. 5 (b)). The emission intensity of AX, observed below E ZBg , decreases much more rapidly thanthe intensity of the confined exciton peak indicative of a distinctly different dissociation channelfor each transition. The thermal dissociation of excitons leads to a decrease in the normalized inte-grated emission intensity I / I = + α e − E / kT . In order to estimate the activation energies associ-ated with both processes, in Fig. 5 (b) we plot the integrated intensity rate ln ( − + I / I ) as functionof 1 / T . The slope of the linear fit to the data gives the activation energies E ( AX ) = . E ( X D ) = . and is also very close to the value reported earlier for ZBGaAs NWs (4 meV). The twofold higher activation energy of X D is expected for 2D confinedexcitons further confirming the localization of holes on the facets.In Fig. 6 the color plot shows differential µ PL spectra, obtained by subtracting suitably av-eraged spectra, in magnetic fields up to 22 T applied perpendicular to the core-multi-shell NWgrowth axis. The PL emission lines sharpen and greatly increase in strength in a magnetic field,which allows us to resolve many more features. On top of the color plot the symbols (white stars)show the position of peaks manually identified for each spectra. This is useful to show weak fea-tures, particularly at low magnetic fields. At high energies we predominantly observe two linescorresponding to emission from quantum confined 2D carriers. At intermediate energies there is aseries of lines previously identified with the KP series related to excitons bound to closely spaceddefect pairs. At low energies, as we will later show, the observed lines can be identified with theLO and TO phonon replicas of the 2D emission observed at high energies.For all the lines, at low magnetic field, the emission energy increases quadratically due tothe diamagnetic shift. At higher magnetic field the dependence becomes linear in the Landauquantization regime. The magnetic field dependence of the KP series of lines in an undoped NWwas discussed in detail in a previous publication. Here we focus on the lines linked to confinedcarrier emission. The energy of emission as a function of magnetic field can be described by theground state of a 2D harmonic oscillator in perpendicular magnetic field,10igure 6: Color plot showing differential µ PL spectra of core-multi-shell GaAs/AlAs NW mea-sured as function of magnetic field. The excitation power was a few nW and the temperature of themeasurements was 1.7 K. The lines show the calculated evolution of the two high energy 2D emis-sion lines together with their LO (dashed lines) and TO (solid lines) phonon replicas as describedin the text. The observed avoided crossing is the result of resonant polaron coupling. E ( B ) = E + ¯ h (cid:113) ω + ( ω c / ) (1)where ω is the harmonic trap frequency, which controls the diamagnetic shift and ω c = eB / m ∗ is the cyclotron frequency. Here we neglect the Zeeman splitting which is not resolved in ourdata. The LO and TO phonon replicas are given by E ( B ) − ¯ h ω ph where ¯ h ω ph is the LO or TOphonon energy in GaAs with ¯ h ω LO = .
25 meV or ¯ h ω TO = .
29 meV. The 2D emission linesare quite weak at low magnetic field and the lower line shows distinct evidence for an avoided11rossing around 12 T which makes it difficult to fit Eq.(1) to the data. Fortunately, the phononreplicas have reasonable strength over a wide range of magnetic field allowing us to reliably extract¯ h ω = .
75 meV and m ∗ = . m e . The second term in the Taylor expansion of Eq.(1) gives thecoefficient for the diamagnetic shift ¯ he / ω m ∗ (cid:39) µ eV/T which is reasonable for a confinedexciton. Avoided crossings in the emission from the lowest ( n =
0) Landau level have previously beenobserved in high density 2DEGs and are due to a resonant polaron coupling which occurs when ∆ n ¯ h ω c = ¯ h ω LO . Occupancy arguments require that in order to observe such an effect the Fermienergy should be similar to or greater than the phonon energy, which requires a high 2D carrierdensity ≥ × cm − . The avoided crossing behavior can be described using a perturbationapproach where the unperturbed energies are replaced by E ± = (cid:16) E + E nph (cid:17) ± (cid:113) ( E − E nph ) − γ ph (2)where E nph is the energy of the n th phonon replica obtained from Eq.(1) with ω c / → ( n + ) ω c and γ ph is the characteristic interaction energy for each phonon. The evolution of the emissionlines, calculated using Eq.(2) and shown by the yellow lines in Fig. 6, is in good agreement withour data, nicely reproducing the phonon replicas and the avoided crossing behavior observed inthe n = m ∗ and ω were extracted from the unperturbed n = γ ph ) which we findto be equal to γ TO (cid:39) γ LO (cid:39) n = n = n = In conclusion, we have carried out optical studies of the single GaAs/AlAs core-multi-shell NW12n high magnetic field. We have compared the obtained results with the typical spectra collected forsingle GaAs/AlAs core-shell NW with comparable diameter. In the PL spectra of the core-multi-shell NW we have observed emission above E ZBg = .
519 eV which is related to 2D confinementof carriers in the core. Our results are in good agreement with theoretical calculations, whichpredict different localization regimes for carriers as a function of doping. Moreover, in magneto-PL spectra from core-multi-shell NWs we have observed avoided crossings of emission lines. Theunderlying coupling is caused by the hole-phonon interaction in a 2D system with a dense gas.This shows that the presence of 1 nm GaAs layer in the shell, which acts as an efficient impuritytrap, can lead to the efficient incorporation of residual acceptors (carbon) and the formation of adense 2D hole gas at the facets of the NW core.
Acknowledgement
We thank Ronit Popovitz-Biro for the careful TEM mesurements. This work was partially sup-ported by the Region Midi-Pyrenées, the Programme Investissements d’Avenir under the programANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT, ANR JCJC project milliPICSand project APOSTD/2013/052 Generalitat Valenciana Vali+d Grant. G.G. and A.B. acknowledgesupport from EU-FP7 Initial Training Network INDEX. G.G. acknowledges support from Univer-sity of Modena and Reggio emilia, through grant ’Nano- and emerging materials and systems forsustainable technologies’. We would also like to acknowledge partial support by the Israeli ScienceFoundation grant
Supporting Information AvailableExperimental techniques
The study of the optical properties of core-multi-shell NW’s was carried out in two experimentalsetups. For measurements of the micro-photoluminescence ( µ PL) in magnetic field the sample wasplaced in a system composed of piezoelectric x-y-z translation stages and a microscope objective.13he µ PL system was cooled to a temperature of T=1 . B =
22 T. The field was applied in the Faraday configuration,perpendicular to the NW (cid:104) (cid:105) growth axis. The sample was illuminated by a diode laser at 660nm. Both the exciting and collected light were transmitted through a monomode fiber coupleddirectly to the microscope objective. The diameter of the excitation beam on the sample was ofthe order of 1 µ m. The emission from the sample was dispersed in a spectrometer equipped witha multichannel CCD camera. For additional µ PL measurements in the absence of magnetic field,the sample was placed in helium flow cryostat with optical access. The cryostat was mounted onmotorized x-y translation stages. The µ PL was measured for temperatures varying from 10 to100 K. Excitation and collection was implemented using a microscope objective with a numericalaperture NA=0.66 and magnification ×
50. The diameter of the excitation spot on the sample wasof the order of 1 µ m and the µ PL spectra have been recorded using a spectrometer equipped witha multichannel CCD camera.
Structural properties of the GaAs/AlAs/GaAs/AlAs/GaAs core/shell QW nanowires
We do not have a structural characterization of the exact NW investigated in our µ PL set up. Weprovide in this section structural characterization and direct evidence for the presence of a well-defined QW within a very similar core multi shell structure grown in the same system and undersimilar growth conditions and layer thicknesses. This sample was studied intensively by crosssectional HR-TEM (prepared using FIB) as demonstrate in Figure 7. Figure 7(a) shows a HR-TEM image taken on a cross section of the multi shell nanowire showing the two AlAs shell layers(bright stripes) and a few monolayers thick GaAs QW embedded between them (GaAs core andcapping layer on the right and left hand sides, respectively, the scale bar is 5 nm. Figure 7(b) is a fullTEM image of the cross section taken from a core multi shell nanowire with very similar nominalthicknesses as the ones studied in this work. Figure 7(c) shows the intensity profile showing clearlythe splitting of the AlAs layer into two layers, consistent with the presence of a thin GaAs QW inbetween them. 14igure 7: (a) HR-TEM image taken on a cross section of the multi shell nanowire showing the twoAlAs shell layers (bright stripes) and a few monolayers thick GaAs QW embedded between them(GaAs core and capping layer on right and left hand sides, respectively, scale bar is 5 nm. (b) TEMimage of a cross section taken from a core multi shell nanowire. (c) intensity profile of the regionmarked on the cross section.
Self consistent calculation of the charge distribution
Calculations have been conducted within a standard envelope function approach, in a single-bandapproximation. We consider the NW as a 3D system spatially invariant along the NW axis direction z . This allows us to factorize the electron (hole) envelope functions as Ψ e ( h ) nk z ( r , z ) = ϕ e ( h ) n ( r ) e ik z z ,with parabolic energy dispersions, E e ( h ) nk z = ε e ( h ) n ± ¯ h k z m ze ( h ) , in the in-wire momentum k z . Over the r ≡ x , y plane, the NW cross section is hexagonal, and the material and doping modulations aredescribed in a corresponding hexagonal domain using a symmetry-compliant triangular grid. Weuse an isotropic electron effective mass ( m ze = m r e ) but a highly anisotropic hole mass. Over thein-plane direction we use the hole mass along the [110] direction which is much larger than thehole mass that we use along the in-wire direction [111]. The ground state hole density n h ( r ) isobtained through a Kohn-Sham LDA procedure.The self-consistent potential V KS ( r ) = V ( r ) + V H ( r ) + V XC ( r ) includes the effect of the spatial15onfinement V ( r ) determined by the materials band offset, the Hartree potential V H ( r ) generatedby free holes and static acceptors, and an approximate exchange-correlation potential V XC (r). Inour samples, conduction band electrons are generated by the laser pumping alone, thus they areminority carriers with a low density: we solve the corresponding Schrödinger equation includingthe electrostatic potential generated by the self-consistent hole density and the density of staticacceptors. In this way we take into account excitonic effects at a mean-field level. Further detailscan be found in references.
From the conduction and valence band states, we compute the PL spectra neglecting dynamicscreening effects and assuming that the photoexcited carriers relax to the lowest available statebefore the radiative recombination. This means that electrons recombine from the lowest-lyingconduction states with holes in states lying above the Fermi level µ . The PL intensity is obtainedas τ ( ω ) ∝ ∑ im | S im | (cid:90) dk z π f e ( E eik z , T )( − f h ( E hmk z − µ , T )) Γ ( E eik z − E hmk z − ¯ h ω − γ ) , (3)where S im = (cid:90) d r ϕ ei ( r ) ϕ hm ( r ) (4)is the overlap integral between a conduction band state i and a valence band state m , f e ( h ) is theelectron (hole) Fermi occupation function at a given temperature T, and Γ is a Lorentzian functionwith a phenomenological bandwidth γ that we set to 1 meV in order to reproduce the width athalf maximum of the experimental peaks. We use the material-dependent parameters indicated inTable 1, a temperature T = . ρ A . The Fermi level is placed0.4 eV above the GaAs valence band edge following experimental observations in highly p-dopedGaAs. Band edges have been calculated assuming the (60:40) rule. Effective masses are taken along [111]direction for m e , hz and [110] for m e , hr . GaAs AlAsBand gap E g [eV] 1.519 3.02Conduction band edge E C [eV] 0.9114 1.812Valence band edge E V [eV] -0.6076 -1.208Electron effective masses m ez m er m hz m hr References (1) Hu, J.; Odom, T. W.; Lieber, C. M.
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