We propose a new type of hybrid systems formed by conventional semiconductor
nanostructures with the addition of remote insulating layers, where the
electron-hole interaction is enhanced by combining quantum and dielectric
confinement over different length scales. Due to the polarization charges
induced by the dielectric mismatch at the semiconductor/insulator interfaces,
we show that the exciton binding energy can be more than doubled. For
conventional III-V quantum wires such remote dielectric confinement allows
exciton binding at room temperature.Comment: 4 pages, 3 PostScript figures embedded, best printed in color. Uses
  RevTex, multicol, and psfig styles. To appear in Phys. Rev. Let

Goldoni, G.

Molinari, E.

Rossi, F.

English

arXiv

We propose-a new type of hybrid systems formed by conventional semiconductor nanostructures with the addition of remote insulating layers,where the electron-hole interaction is enhanced by combining quantum and dielectric confinement over different length scales. Because of the polarization charges induced by the dielectric mismatch at the semicondcutor/insulator interfaces, we show that the exciton binding energy can be more than doubled. For conventional m-V quantum wires such remote dielectric confinement allows exciton binding at room temperature

GOLDONI, Guido

F. Rossi

MOLINARI, Elisa

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998Strong Exciton Binding in Quantum Structures through Remote Dielectric ConfinementGuido Goldoni, Fausto Rossi, and Elisa MolinariIstituto Nazionale per la Fisica della Materia (INFM), and Dipartimento di Fisica, Università di Modena,Via Campi 213/A, I-41100 Modena, Italy(Received 10 February 1998)We propose a new type of hybrid systems formed by conventional semiconductor nanostructures withthe addition of remote insulating layers, where the electron-hole interaction is enhanced by combiningquantum and dielectric confinement over different length scales. Because of the polarization chargesinduced by the dielectric mismatch at the semiconductor/insulator interfaces, we show that the excitonbinding energy can be more than doubled. For conventional III-V quantum wires such remote dielectricconfinement allows exciton binding at room temperature. [S0031-9007(98)06255-3]PACS numbers: 78.66.Fd, 71.35.–y, 73.20.Dx, 77.55.+fThe electron-hole Coulomb interaction in semiconduc-tors leads to bound excitonic states that could play a cru-cial role in the next generation of optical devices. Forthis purpose, however, the binding energy Eb must exceedthe thermal energy at room temperature, a condition thatis not yet met in conventional III-V materials. In fact, alarge enhancement of Eb with respect to bulk materials hasbeen obtained by confining electron and hole wave func-tions in nanostructures of low dimensionality (quantumconfinement), the most promising type of structures beingquasi-one-dimensional systems (quantum wires); withinGaAs-based materials, however, the observed values of Ebare still well below the room-temperature thermal energy.In this Letter we propose an alternative approach to en-hance Eb that combines quantum confinement with remotedielectric confinement. As first pointed out by Keldysh[1], the electron-hole Coulomb attraction can be greatlyenhanced in layered structures with strong dielectric mis-match, due to the polarization charge induced at the in-terfaces. For conventional semiconductor nanostructuressuch as GaAsyAlGaAs- or GaAsyInGaAs-based samples,however, this is a minor effect due to the small dielec-tric mismatch between the constituents [2]. On the otherhand, interfaces between III-V semiconductors and materi-als with very different dielectric constants, such as oxides,are usually very far from the excellent optical quality of theconventional ones. Our approach is based on the idea thatquantum and dielectric confinement can be spatially sepa-rated, since they are effective over different length scales.We will show that a very large increase in Eb can be ob-tained in GaAs-based quantum wires by adding remote in-sulating layers: these induce strong dielectric confinementwithout degrading the good optical properties ensuing fromquantum confinement.We consider GaAsyAlGaAs quantum wires (QWI),where excitonic properties have been studied extensively[3]. We focus on QWIs with T-shaped [4] and V-shaped[5] cross sections, which rank among the best availablesamples from the point of view of optical properties.Starting from these geometries, we design hybrid struc-0031-9007y98y80(22)y4995(4)$15.00tures adding oxide layers at some distance from theQWI. In the new structures, the electron and hole wavefunctions are confined by the inner, lattice-and-dielectric-matched GaAsyAlGaAs interfaces; the outer AlGaAsyoxide interfaces, owing to the strong dielectric mismatch,provide polarization charges, thus enhancing the electron-hole interaction.To obtain quantitative predictions of excitonic proper-ties in such hybrid structures, an accurate description of thecomplex interplay between quantum and dielectric confine-ment is needed. To this end, we have developed a noveltheoretical scheme that allows us to treat arbitrary dielec-tric configurations, where the low symmetry makes simpleimage-charge methods [2,6] not applicable. Furthermore,we adopt a nonperturbative approach for the self-energyterm, which, in principle, is needed to describe strong di-electric mismatch combined with shallow confining poten-tials, as in some state-of-the-art QWIs. The electron-holeinteraction is treated within the conventional approach ofthe semiconductor Bloch equations, adapted to quasi-one-dimensional (1D) systems [7].More specifically, for a spatially modulated dielectricconstant esrd the Coulomb interaction between twocharges of opposite sign 6e sitting at positions r andr0—our electron-hole pair—is given by V sr, r0d ­2e2Gsr, r0d, where Gsr, r0d is the Green’s function of thePoisson operator, i.e.,=r ? esrd=rGsr, r0d ­ 2dsr 2 r0d . (1)We see that the space dependence of esrd modifies Gsr, r0dwith respect to the homogeneous case, where esrd ­ e0implies G0sr, r0d ­14pe0jr2r0j . This, in turn, gives rise tosignificant modifications in the excitonic spectrum; withinthe conventional Hartree-Fock scheme, such modificationsoriginate from the electron-hole Coulomb matrix elementsentering the evaluation of the absorption spectrum [7],V ehij ­ 2e2ZFepi srdFhpj sr0dGsr, r0dFhi sr0dFej srd dr dr0.(2)© 1998 The American Physical Society 4995VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998Here Feshdl denotes the electron (hole) single-particle en-velope function for the l ; kz , n state of the QWI, kz thewave vector along the wire, and n the subband index.To study realistic geometries, we find it convenient tocast the problem in Fourier space, following the theoreticalscheme in Ref. [7]. It is easy to rewrite Eqs. (1) and (2)(the symbol e denotes Fourier transform throughout):Xk00esk 2 k00dk ? k00 eGsk00, k0d ­ dsk 1 k0d , (3)V ehij ­ 2e2XkFeijskdXk0eGsk, k0dFhjisk0d , (4)where Feyhlm skd ­RFeyhpl srdeik?rFeyhm srd dr. We stressthat, in order to evaluate the Coulomb matrix elements in(4), it is not necessary to solve Eq. (3) for each eGsk, k0d.In fact, if we multiply (3) by Fhjisk0d and sum over k0,we get a Poisson equation for the “potential” yjiskd ­Pk0eGsk, k0dFhjisk0d [see Eq. (4)] arising from the “source”Fhjisk0d, which is solved only once for each pair i, j.The Green’s function Gsr, r0d also gives rise to a self-energy termSsrd ­e22limr0!rfGsr, r0d 2 GBsr, r0dg , (5)where GBsr, r0d ­14pesrd jr2r0 j is the (local) bulk solutionof (1). S is a local correction (equal for electrons andholes) which adds to the confining potential in determiningthe single-particle envelope functions Feyhl . This self-energy contribution is evaluated within the same plane-wave approach by solving (3) at a set of k0 [8]: it can beshown that its Fourier transform iseSsGd ­ e22Xg•eGµG 1 g2, G 2g2¶2 eGBsG, gd‚ ,(6)where eGBsG, gd ­ ge21sGdyg2 with the definitions G ­sk 1 k0dy2 and g ­ k 2 k0 [9].We first discuss our findings for QWIs obtained byepitaxial growth on V-grooved substrates (V-QWIs) [5].As a reference sample, we consider a GaAs wire withAlAs barriers [10], and we add two oxide layers, belowand above the QWI [see Fig. 1(a)], at a distance L fromthe GaAsyAlAs interfaces [11]. Note that the oxide layersare characterized by a small dielectric constant that wetake equal to 2 [12].Our results for the V-QWRs are shown in Fig. 1. For thesample shown in Fig. 1(a), we find Eb ­ 29.3 meV, to becompared with 13 meV of the conventional (i.e., with nooxide layers) structure. Figure 1(a) shows that the originof this dramatic enhancement is the large polarizationof the AlAsyoxide interfaces induced by the hole chargedensity [13]; the polarization is larger in the region where49960 2 4 6 8 10 12 141015202530354045(b)Conventional structurekTroom  Full calculation  Eq. (7)Exciton binding energy (meV)L (nm)FIG. 1. (a) (color) Cross section of a hybrid V-QWI, showingthe interface polarization charge (colors, units of nm21) inducedby the charge-density distribution of the lowest-subband hole(grey scale, arbitrary units). The profile of the GaAsyAlAsinterfaces is obtained from Ref. [10]; the oxide layers are atL ­ 5 nm from the GaAsyAlAs interfaces. The polarizationcharge at the GaAsyAlAs interfaces is multiplied by 3. (b) Ebversus distance of the oxide layers from the internal interfaces,L. Solid dots: Full calculation. Solid line: Eq. (7) with L0 ­6.56 nm. Dashed line: Energy E0 of the corresponding con-ventional structure (no oxide layers). Dotted line: thermalenergy at Troom ­ 300 K.the hole is localized, and is more pronounced at the lowerinterface, due to the larger curvature. A small polarizationcharge is also induced at the GaAsyAlAs interface, dueto the small dielectric mismatch. Note that quantumconfinement localizes the wave function well within theinner interfaces; therefore, the AlAsyoxide interface doesnot affect the electron and hole wave functions.In Fig. 1(b) we show the calculated Eb for selectedvalues of L. We also show, for comparison, the calculatedbinding energy for the conventional structure, E0, and theroom-temperature thermal energy. Eb is maximum whenthe oxide layer is at minimum distance [14], L ­ 0: itis enhanced by a factor larger than 3 with respect to E0,and it is well above kTroom [15]. It is important to notethat Eb decreases slowly with L, and is still significantlyVOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998larger than kTroom at L ­ 6 nm. Since Eb is the resultof the Coulomb interaction of, say, the electron with thehole and the polarization charge which is excited at adistance ,L, we intuitively expect Eb to decay as L21,with a typical decay length L0 comparable to the Bohrradius in the QWI [7]; this, in turn, is of the order of theconfinement length. Indeed, Fig. 1(b) shows that Eb isvery well interpolated byEbsLd ­ E0 1Ebs0d1 1 LyL0, (7)with L0 ­ 6.56 nm. Note that Eb crosses kTroom when Lis as large as 9 nm.A second type of structures, which have recently at-tracted considerable attention, are the so-called T-shapedwires (T-QWI), obtained by the cleaved-edge overgrowthmethod [3,4]. The typical sample of our calculations [seeFig. 2(a)] consists of a T-QWI with GaAs parent quantumwells (QWs) of the same width, and AlAs barriers. An0 2 4 6 8 10 12 14161820222426 (b)Conventional structurekTroom  Full calculation  No self-energy  Eq. (7)Exciton binding energy (meV)L (nm)FIG. 2. (a) (color) Same as Fig. 1(a) for a hybrid T-QWI.The QW widths are 5.4 nm; the oxide layer is at L ­ 3 nmfrom the GaAsyAlAs interface. (b) Eb versus L including(solid dots) and neglecting (empty dots) the self-energy contri-bution. Solid line: Eq. (7) with L0 ­ 7.55 nm. Dashed line:Energy E0 of the corresponding conventional structure (no ox-ide layers). Dotted line: Thermal energy at Troom ­ 300 K.oxide layer is added on top of the exposed surface at a dis-tance L from the underlying QW. Note that, in this case,an oxide layer is present only on one side of the QWI. Asin the case of the V-QWIs, a strong polarization chargeforms at the AlAsyoxide interface, with a maximum in theregion of the hole wave function confinement. A smallpolarization charge is also present at the GaAsyAlAs inter-face, peaked around the corners of the intersecting QWs.In Fig. 2(b) we show the calculated Eb vs L. The bindingenergy for the conventional structure, E0, and the room-temperature thermal energy are also shown for comparison.As in the case of the V-QWI, Eb is maximum at L ­ 0,where it is enhanced by a factor of 1.5 with respect to E0,and decreases slowly with L. Although Eb is smaller thanin the case of the V-QWI studied previously, for the small-est L values Eb is still of the order of kTroom. It is impor-tant to note that the reduced effect of dielectric confinementin the T-QWI sample with respect to the previous exampleof V-QWI is just due to the presence of a single oxide layer;i.e., geometric effects due to different cross sections playa minor role. In fact, despite the very different geome-try, Eb decays with L in the same way in both cases. Asshown in Fig. 2(b), also in the case of T-QWIs Eb is verywell interpolated by Eq. (7) with L0 ­ 7.55 meV, whichis still of the order of the QWI confinement length. Notethat the above examples of structures are based on stan-dard state-of-the-art QWIs that were previously studied inthe literature [4,5], and no particular optimization of Ebwith respect to sample parameters (constituents and/or ge-ometry) has been attempted.Finally, we discuss the effect of the self-energy termon Eb . We have compared the full calculations discussedabove with calculations performed neglecting Ssrd in thesingle-particle potential. For the V-QWI we have verifiedthat the self-energy contribution tends to increase Eb , butit is so small (,0.2 meV) that the two results cannot bedistinguished on the scale of Fig. 1(b). In the case of theT-QWIs, on the other hand, the self-energy contributionis qualitatively and quantitatively different, as can be seenfrom Fig. 2(b); in this case, in fact, it amounts to ,1 meVat the smallest L, and tends to reduce Eb . This is a conse-quence of the interplay between the dielectric confinementand the shallow quantum confinement of these structures;this is apparent from Fig. 3, where we compare the electronand hole single-particle wave functions calculated neglect-ing [Fig. 3(a)] and including [Fig. 3(b)] the self-energyterm. The self-energy potential [Fig. 3(c)] inside the GaAslayer pushes the electron and hole wave functions awayfrom the oxide layer; because of the different masses andshallow confining potentials, such shift is different forelectrons and holes, and the overlap is diminished, therebyreducing Eb .In summary, we have developed a theoretical schemethat allows us to include dielectric confinement andself-energy effects in a full three-dimensional descriptionof correlated electron-hole pairs [16]. Our calculations4997VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998FIG. 3. Lowest-subband electron and hole charge densities fora T-QWI with L ­ 0 (see inset), neglecting (a) and including(b) the self-energy potential Ssrd shown in (c). All curves arecalculated along the white line shown in the inset.show that a dramatic enhancement of the exciton bindingin GaAs-based quantum structures is made possible byremote insulating layers, bringing Eb in the range of theroom-temperature thermal energy. This enhancementscales slowly with the distance of the insulating layerfrom the quantum confinement region, thus allowing oneto design nanostructures that should be compatible withexcellent optical efficiency. Dielectric confinement by re-mote insulating layers is predicted to be a novel powerfultool for tailoring excitonic confinement in semiconductornanostructures.We are grateful to E. Kapon, L. Sorba, and W. Weg-scheider for useful discussions. This work was supportedin part by the EC through the TMR Network Ultrafastquantum optoelectronics.[1] L. V. Keldysh, Pis’ma Zh. Eksp. Teor. Fiz. 29, 716 (1979)[JETP Lett. 29, 658 (1979)].[2] L. C. Andreani and A. Pasquarello, Phys. Rev. B 42, 8928(1990).[3] For recent reviews see H. Sakaki et al., in Proceedingsof the Fifth International Meeting on Optics of Excitonsin Confined Systems [Phys. Status Solidi (a) 164, 2414998(1997)]; G. Goldoni, F. Rossi, and E. Molinari, ibid. 164,265 (1997).[4] L. Pfeiffer et al., Appl. Phys. Lett. 56, 1697 (1990).[5] E. Kapon et al., Phys. Rev. Lett. 63, 430 (1989).[6] M. Kumagai and T. Takagahara, Phys. Rev. B 40, 12 359(1989).[7] F. Rossi and E. Molinari, Phys. Rev. Lett. 76, 3642 (1996);Phys. Rev. B 53, 16 462 (1996); F. Rossi, G. Goldoni, andE. Molinari, Phys. Rev. Lett. 78, 3527 (1997).[8] It turns out that eGsk, k0d is needed on a coarser gridfor k0 than for k, due to the different length scalesof quantum and dielectric confinements; this makes thenumerical evaluation of (6) feasible.[9] To avoid numerical instabilities we found it convenientto define a reference material with the homogeneousdielectric constant eref ­ esk ­ 0d. It can be shownthat an equation similar to (3) can be obtained for thedifference d eG ­ eG 2 eGref with respect to such reference,where the deltalike term on the right-hand side is canceledexactly. Despite this careful treatment, the evaluation ofS is by far the heaviest part of our calculation.[10] R. Rinaldi et al., Phys. Rev. Lett. 73, 2899 (1994).[11] The GaAsyAlAs band offsets are 1.036 eV (electrons) and0.558 eV (holes). Effective masses are 0.067 (electrons)and 0.38 (holes). Dielectric constants are 12.5 (GaAs),10.0 (AlAs), and 2 (oxide). We neglect the variation ofother band parameters between AlAs and oxide layers.Because of the exponential decay of the wave functions inthe AlAs layers, these effects should be negligible whenL exceeds a few nm.[12] This is just intended as a typical value: oxides that couldbe of practical relevance [see, e.g., A. Fiore et al., Nature(London) 391, 463 (1998)] have similar or smaller values.The value of the dielectric constant could be made evensmaller if one could take advantage of interfaces with air[S. G. Tikhodeev et al., Phys. Status Solidi (a) 164, 179(1997)].[13] Of course, there is no electron-hole asymmetry, and wecould equivalently show the electron charge density andits induced polarization charge; the present choice isconsistent with the choice of Fhji as “source” term, asdiscussed below Eq. (4).[14] For completeness, our calculations have been extendeddown to L ­ 0, although in this limit they should betaken with caution, due to the neglect of the AlAsyoxideinterface effects. See also note [11].[15] In spite of such a dramatic enhancement, Eb is stillmuch smaller than the GaAs band gap, so that the usualapproximation which neglects the interband exchangeinteraction is still valid in the present structures.[16] This approach can be relevant also for applications toother systems, including nanocrystals embedded in glassmatrices [see, e.g., A. P. Alivisatos, Science 271, 933(1996)] where dielectric confinement effects are expectedto be strong.

Strong exciton binding in quantum structures through remote dielectric confinement

We propose a new type of hybrid systems formed by conventional semiconductor nanostructures with the addition of remote insulating layers, where the electron-hole interaction is enhanced by combining quantum and dielectric confinement over different length scales. Because of the polarization charges induced by the dielectric mismatch at the semiconductor/insulator interfaces, we show that the exciton binding energy can be more than doubled. For conventional III-V quantum wires such remote dielectric confinement allows exciton binding at room temperature

GOLDONI G.

MOLINARI E.

ROSSI, FAUSTO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEStrong exciton binding in quantum structures through remote dielectric confinement / GOLDONI G.; ROSSI F.;MOLINARI E.. - In: PHYSICAL REVIEW LETTERS. - ISSN 0031-9007. - 80:22(1998), pp. 4995-4998.OriginalStrong exciton binding in quantum structures through remote dielectric confinementPublisher:PublishedDOI:10.1103/PhysRevLett.80.4995Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1405204 since:APS The Americal Physical SocietyVOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998Strong Exciton Binding in Quantum Structures through Remote Dielectric ConfinementGuido Goldoni, Fausto Rossi, and Elisa MolinariIstituto Nazionale per la Fisica della Materia (INFM), and Dipartimento di Fisica, Università di Modena,Via Campi 213/A, I-41100 Modena, Italy(Received 10 February 1998)We propose a new type of hybrid systems formed by conventional semiconductor nanostructures withthe addition of remote insulating layers, where the electron-hole interaction is enhanced by combiningquantum and dielectric confinement over different length scales. Because of the polarization chargesinduced by the dielectric mismatch at the semiconductor/insulator interfaces, we show that the excitonbinding energy can be more than doubled. For conventional III-V quantum wires such remote dielectricconfinement allows exciton binding at room temperature. [S0031-9007(98)06255-3]PACS numbers: 78.66.Fd, 71.35.–y, 73.20.Dx, 77.55.+fThe electron-hole Coulomb interaction in semiconduc-tors leads to bound excitonic states that could play a cru-cial role in the next generation of optical devices. Forthis purpose, however, the binding energy Eb must exceedthe thermal energy at room temperature, a condition thatis not yet met in conventional III-V materials. In fact, alarge enhancement of Eb with respect to bulk materials hasbeen obtained by confining electron and hole wave func-tions in nanostructures of low dimensionality (quantumconfinement), the most promising type of structures beingquasi-one-dimensional systems (quantum wires); withinGaAs-based materials, however, the observed values of Ebare still well below the room-temperature thermal energy.In this Letter we propose an alternative approach to en-hance Eb that combines quantum confinement with remotedielectric confinement. As first pointed out by Keldysh[1], the electron-hole Coulomb attraction can be greatlyenhanced in layered structures with strong dielectric mis-match, due to the polarization charge induced at the in-terfaces. For conventional semiconductor nanostructuressuch as GaAsyAlGaAs- or GaAsyInGaAs-based samples,however, this is a minor effect due to the small dielec-tric mismatch between the constituents [2]. On the otherhand, interfaces between III-V semiconductors and materi-als with very different dielectric constants, such as oxides,are usually very far from the excellent optical quality of theconventional ones. Our approach is based on the idea thatquantum and dielectric confinement can be spatially sepa-rated, since they are effective over different length scales.We will show that a very large increase in Eb can be ob-tained in GaAs-based quantum wires by adding remote in-sulating layers: these induce strong dielectric confinementwithout degrading the good optical properties ensuing fromquantum confinement.We consider GaAsyAlGaAs quantum wires (QWI),where excitonic properties have been studied extensively[3]. We focus on QWIs with T-shaped [4] and V-shaped[5] cross sections, which rank among the best availablesamples from the point of view of optical properties.Starting from these geometries, we design hybrid struc-0031-9007y98y80(22)y4995(4)$15.00tures adding oxide layers at some distance from theQWI. In the new structures, the electron and hole wavefunctions are confined by the inner, lattice-and-dielectric-matched GaAsyAlGaAs interfaces; the outer AlGaAsyoxide interfaces, owing to the strong dielectric mismatch,provide polarization charges, thus enhancing the electron-hole interaction.To obtain quantitative predictions of excitonic proper-ties in such hybrid structures, an accurate description of thecomplex interplay between quantum and dielectric confine-ment is needed. To this end, we have developed a noveltheoretical scheme that allows us to treat arbitrary dielec-tric configurations, where the low symmetry makes simpleimage-charge methods [2,6] not applicable. Furthermore,we adopt a nonperturbative approach for the self-energyterm, which, in principle, is needed to describe strong di-electric mismatch combined with shallow confining poten-tials, as in some state-of-the-art QWIs. The electron-holeinteraction is treated within the conventional approach ofthe semiconductor Bloch equations, adapted to quasi-one-dimensional (1D) systems [7].More specifically, for a spatially modulated dielectricconstant esrd the Coulomb interaction between twocharges of opposite sign 6e sitting at positions r andr0—our electron-hole pair—is given by V sr, r0d ­2e2Gsr, r0d, where Gsr, r0d is the Green’s function of thePoisson operator, i.e.,=r ? esrd=rGsr, r0d ­ 2dsr 2 r0d . (1)We see that the space dependence of esrd modifies Gsr, r0dwith respect to the homogeneous case, where esrd ­ e0implies G0sr, r0d ­14pe0jr2r0j . This, in turn, gives rise tosignificant modifications in the excitonic spectrum; withinthe conventional Hartree-Fock scheme, such modificationsoriginate from the electron-hole Coulomb matrix elementsentering the evaluation of the absorption spectrum [7],V ehij ­ 2e2ZFepi srdFhpj sr0dGsr, r0dFhi sr0dFej srd dr dr0.(2)© 1998 The American Physical Society 4995VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998Here Feshdl denotes the electron (hole) single-particle en-velope function for the l ; kz , n state of the QWI, kz thewave vector along the wire, and n the subband index.To study realistic geometries, we find it convenient tocast the problem in Fourier space, following the theoreticalscheme in Ref. [7]. It is easy to rewrite Eqs. (1) and (2)(the symbol e denotes Fourier transform throughout):Xk00esk 2 k00dk ? k00 eGsk00, k0d ­ dsk 1 k0d , (3)V ehij ­ 2e2XkFeijskdXk0eGsk, k0dFhjisk0d , (4)where Feyhlm skd ­RFeyhpl srdeik?rFeyhm srd dr. We stressthat, in order to evaluate the Coulomb matrix elements in(4), it is not necessary to solve Eq. (3) for each eGsk, k0d.In fact, if we multiply (3) by Fhjisk0d and sum over k0,we get a Poisson equation for the “potential” yjiskd ­Pk0eGsk, k0dFhjisk0d [see Eq. (4)] arising from the “source”Fhjisk0d, which is solved only once for each pair i, j.The Green’s function Gsr, r0d also gives rise to a self-energy termSsrd ­e22limr0!rfGsr, r0d 2 GBsr, r0dg , (5)where GBsr, r0d ­14pesrd jr2r0 j is the (local) bulk solutionof (1). S is a local correction (equal for electrons andholes) which adds to the confining potential in determiningthe single-particle envelope functions Feyhl . This self-energy contribution is evaluated within the same plane-wave approach by solving (3) at a set of k0 [8]: it can beshown that its Fourier transform iseSsGd ­ e22Xg•eGµG 1 g2, G 2g2¶2 eGBsG, gd‚ ,(6)where eGBsG, gd ­ ge21sGdyg2 with the definitions G ­sk 1 k0dy2 and g ­ k 2 k0 [9].We first discuss our findings for QWIs obtained byepitaxial growth on V-grooved substrates (V-QWIs) [5].As a reference sample, we consider a GaAs wire withAlAs barriers [10], and we add two oxide layers, belowand above the QWI [see Fig. 1(a)], at a distance L fromthe GaAsyAlAs interfaces [11]. Note that the oxide layersare characterized by a small dielectric constant that wetake equal to 2 [12].Our results for the V-QWRs are shown in Fig. 1. For thesample shown in Fig. 1(a), we find Eb ­ 29.3 meV, to becompared with 13 meV of the conventional (i.e., with nooxide layers) structure. Figure 1(a) shows that the originof this dramatic enhancement is the large polarizationof the AlAsyoxide interfaces induced by the hole chargedensity [13]; the polarization is larger in the region where49960 2 4 6 8 10 12 141015202530354045(b)Conventional structurekTroom  Full calculation  Eq. (7)Exciton binding energy (meV)L (nm)FIG. 1. (a) (color) Cross section of a hybrid V-QWI, showingthe interface polarization charge (colors, units of nm21) inducedby the charge-density distribution of the lowest-subband hole(grey scale, arbitrary units). The profile of the GaAsyAlAsinterfaces is obtained from Ref. [10]; the oxide layers are atL ­ 5 nm from the GaAsyAlAs interfaces. The polarizationcharge at the GaAsyAlAs interfaces is multiplied by 3. (b) Ebversus distance of the oxide layers from the internal interfaces,L. Solid dots: Full calculation. Solid line: Eq. (7) with L0 ­6.56 nm. Dashed line: Energy E0 of the corresponding con-ventional structure (no oxide layers). Dotted line: thermalenergy at Troom ­ 300 K.the hole is localized, and is more pronounced at the lowerinterface, due to the larger curvature. A small polarizationcharge is also induced at the GaAsyAlAs interface, dueto the small dielectric mismatch. Note that quantumconfinement localizes the wave function well within theinner interfaces; therefore, the AlAsyoxide interface doesnot affect the electron and hole wave functions.In Fig. 1(b) we show the calculated Eb for selectedvalues of L. We also show, for comparison, the calculatedbinding energy for the conventional structure, E0, and theroom-temperature thermal energy. Eb is maximum whenthe oxide layer is at minimum distance [14], L ­ 0: itis enhanced by a factor larger than 3 with respect to E0,and it is well above kTroom [15]. It is important to notethat Eb decreases slowly with L, and is still significantlyVOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998larger than kTroom at L ­ 6 nm. Since Eb is the resultof the Coulomb interaction of, say, the electron with thehole and the polarization charge which is excited at adistance ,L, we intuitively expect Eb to decay as L21,with a typical decay length L0 comparable to the Bohrradius in the QWI [7]; this, in turn, is of the order of theconfinement length. Indeed, Fig. 1(b) shows that Eb isvery well interpolated byEbsLd ­ E0 1Ebs0d1 1 LyL0, (7)with L0 ­ 6.56 nm. Note that Eb crosses kTroom when Lis as large as 9 nm.A second type of structures, which have recently at-tracted considerable attention, are the so-called T-shapedwires (T-QWI), obtained by the cleaved-edge overgrowthmethod [3,4]. The typical sample of our calculations [seeFig. 2(a)] consists of a T-QWI with GaAs parent quantumwells (QWs) of the same width, and AlAs barriers. An0 2 4 6 8 10 12 14161820222426 (b)Conventional structurekTroom  Full calculation  No self-energy  Eq. (7)Exciton binding energy (meV)L (nm)FIG. 2. (a) (color) Same as Fig. 1(a) for a hybrid T-QWI.The QW widths are 5.4 nm; the oxide layer is at L ­ 3 nmfrom the GaAsyAlAs interface. (b) Eb versus L including(solid dots) and neglecting (empty dots) the self-energy contri-bution. Solid line: Eq. (7) with L0 ­ 7.55 nm. Dashed line:Energy E0 of the corresponding conventional structure (no ox-ide layers). Dotted line: Thermal energy at Troom ­ 300 K.oxide layer is added on top of the exposed surface at a dis-tance L from the underlying QW. Note that, in this case,an oxide layer is present only on one side of the QWI. Asin the case of the V-QWIs, a strong polarization chargeforms at the AlAsyoxide interface, with a maximum in theregion of the hole wave function confinement. A smallpolarization charge is also present at the GaAsyAlAs inter-face, peaked around the corners of the intersecting QWs.In Fig. 2(b) we show the calculated Eb vs L. The bindingenergy for the conventional structure, E0, and the room-temperature thermal energy are also shown for comparison.As in the case of the V-QWI, Eb is maximum at L ­ 0,where it is enhanced by a factor of 1.5 with respect to E0,and decreases slowly with L. Although Eb is smaller thanin the case of the V-QWI studied previously, for the small-est L values Eb is still of the order of kTroom. It is impor-tant to note that the reduced effect of dielectric confinementin the T-QWI sample with respect to the previous exampleof V-QWI is just due to the presence of a single oxide layer;i.e., geometric effects due to different cross sections playa minor role. In fact, despite the very different geome-try, Eb decays with L in the same way in both cases. Asshown in Fig. 2(b), also in the case of T-QWIs Eb is verywell interpolated by Eq. (7) with L0 ­ 7.55 meV, whichis still of the order of the QWI confinement length. Notethat the above examples of structures are based on stan-dard state-of-the-art QWIs that were previously studied inthe literature [4,5], and no particular optimization of Ebwith respect to sample parameters (constituents and/or ge-ometry) has been attempted.Finally, we discuss the effect of the self-energy termon Eb . We have compared the full calculations discussedabove with calculations performed neglecting Ssrd in thesingle-particle potential. For the V-QWI we have verifiedthat the self-energy contribution tends to increase Eb , butit is so small (,0.2 meV) that the two results cannot bedistinguished on the scale of Fig. 1(b). In the case of theT-QWIs, on the other hand, the self-energy contributionis qualitatively and quantitatively different, as can be seenfrom Fig. 2(b); in this case, in fact, it amounts to ,1 meVat the smallest L, and tends to reduce Eb . This is a conse-quence of the interplay between the dielectric confinementand the shallow quantum confinement of these structures;this is apparent from Fig. 3, where we compare the electronand hole single-particle wave functions calculated neglect-ing [Fig. 3(a)] and including [Fig. 3(b)] the self-energyterm. The self-energy potential [Fig. 3(c)] inside the GaAslayer pushes the electron and hole wave functions awayfrom the oxide layer; because of the different masses andshallow confining potentials, such shift is different forelectrons and holes, and the overlap is diminished, therebyreducing Eb .In summary, we have developed a theoretical schemethat allows us to include dielectric confinement andself-energy effects in a full three-dimensional descriptionof correlated electron-hole pairs [16]. Our calculations4997VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998FIG. 3. Lowest-subband electron and hole charge densities fora T-QWI with L ­ 0 (see inset), neglecting (a) and including(b) the self-energy potential Ssrd shown in (c). All curves arecalculated along the white line shown in the inset.show that a dramatic enhancement of the exciton bindingin GaAs-based quantum structures is made possible byremote insulating layers, bringing Eb in the range of theroom-temperature thermal energy. This enhancementscales slowly with the distance of the insulating layerfrom the quantum confinement region, thus allowing oneto design nanostructures that should be compatible withexcellent optical efficiency. Dielectric confinement by re-mote insulating layers is predicted to be a novel powerfultool for tailoring excitonic confinement in semiconductornanostructures.We are grateful to E. Kapon, L. Sorba, and W. Weg-scheider for useful discussions. This work was supportedin part by the EC through the TMR Network Ultrafastquantum optoelectronics.[1] L. V. Keldysh, Pis’ma Zh. Eksp. Teor. Fiz. 29, 716 (1979)[JETP Lett. 29, 658 (1979)].[2] L. C. Andreani and A. Pasquarello, Phys. Rev. B 42, 8928(1990).[3] For recent reviews see H. Sakaki et al., in Proceedingsof the Fifth International Meeting on Optics of Excitonsin Confined Systems [Phys. Status Solidi (a) 164, 2414998(1997)]; G. Goldoni, F. Rossi, and E. Molinari, ibid. 164,265 (1997).[4] L. Pfeiffer et al., Appl. Phys. Lett. 56, 1697 (1990).[5] E. Kapon et al., Phys. Rev. Lett. 63, 430 (1989).[6] M. Kumagai and T. Takagahara, Phys. Rev. B 40, 12 359(1989).[7] F. Rossi and E. Molinari, Phys. Rev. Lett. 76, 3642 (1996);Phys. Rev. B 53, 16 462 (1996); F. Rossi, G. Goldoni, andE. Molinari, Phys. Rev. Lett. 78, 3527 (1997).[8] It turns out that eGsk, k0d is needed on a coarser gridfor k0 than for k, due to the different length scalesof quantum and dielectric confinements; this makes thenumerical evaluation of (6) feasible.[9] To avoid numerical instabilities we found it convenientto define a reference material with the homogeneousdielectric constant eref ­ esk ­ 0d. It can be shownthat an equation similar to (3) can be obtained for thedifference d eG ­ eG 2 eGref with respect to such reference,where the deltalike term on the right-hand side is canceledexactly. Despite this careful treatment, the evaluation ofS is by far the heaviest part of our calculation.[10] R. Rinaldi et al., Phys. Rev. Lett. 73, 2899 (1994).[11] The GaAsyAlAs band offsets are 1.036 eV (electrons) and0.558 eV (holes). Effective masses are 0.067 (electrons)and 0.38 (holes). Dielectric constants are 12.5 (GaAs),10.0 (AlAs), and 2 (oxide). We neglect the variation ofother band parameters between AlAs and oxide layers.Because of the exponential decay of the wave functions inthe AlAs layers, these effects should be negligible whenL exceeds a few nm.[12] This is just intended as a typical value: oxides that couldbe of practical relevance [see, e.g., A. Fiore et al., Nature(London) 391, 463 (1998)] have similar or smaller values.The value of the dielectric constant could be made evensmaller if one could take advantage of interfaces with air[S. G. Tikhodeev et al., Phys. Status Solidi (a) 164, 179(1997)].[13] Of course, there is no electron-hole asymmetry, and wecould equivalently show the electron charge density andits induced polarization charge; the present choice isconsistent with the choice of Fhji as “source” term, asdiscussed below Eq. (4).[14] For completeness, our calculations have been extendeddown to L ­ 0, although in this limit they should betaken with caution, due to the neglect of the AlAsyoxideinterface effects. See also note [11].[15] In spite of such a dramatic enhancement, Eb is stillmuch smaller than the GaAs band gap, so that the usualapproximation which neglects the interband exchangeinteraction is still valid in the present structures.[16] This approach can be relevant also for applications toother systems, including nanocrystals embedded in glassmatrices [see, e.g., A. P. Alivisatos, Science 271, 933(1996)] where dielectric confinement effects are expectedto be strong.

Rossi, Fausto

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[Article] Strong exciton binding in quantum structures through remote
dielectric confinement
Original Citation:
Goldoni G., Rossi F., Molinari E. (1998). Strong exciton binding in quantum structures through
remote dielectric confinement. In: PHYSICAL REVIEW LETTERS, vol. 80 n. 22, pp. 4995-4998. -
ISSN 0031-9007
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VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998
Strong Exciton Binding in Quantum Structures through Remote Dielectric Confinement
Guido Goldoni, Fausto Rossi, and Elisa Molinari
Istituto Nazionale per la Fisica della Materia (INFM), and Dipartimento di Fisica, Università di Modena,
Via Campi 213/A, I-41100 Modena, Italy
(Received 10 February 1998)
We propose a new type of hybrid systems formed by conventional semiconductor nanostructures with
the addition of remote insulating layers, where the electron-hole interaction is enhanced by combining
quantum and dielectric confinement over different length scales. Because of the polarization charges
induced by the dielectric mismatch at the semiconductor/insulator interfaces, we show that the exciton
binding energy can be more than doubled. For conventional III-V quantum wires such remote dielectric
confinement allows exciton binding at room temperature. [S0031-9007(98)06255-3]
PACS numbers: 78.66.Fd, 71.35.–y, 73.20.Dx, 77.55.+f
The electron-hole Coulomb interaction in semiconduc-
tors leads to bound excitonic states that could play a cru-
cial role in the next generation of optical devices. For
this purpose, however, the binding energy Eb must exceed
the thermal energy at room temperature, a condition that
is not yet met in conventional III-V materials. In fact, a
large enhancement of Eb with respect to bulk materials has
been obtained by confining electron and hole wave func-
tions in nanostructures of low dimensionality (quantum
confinement), the most promising type of structures being
quasi-one-dimensional systems (quantum wires); within
GaAs-based materials, however, the observed values of Eb
are still well below the room-temperature thermal energy.
In this Letter we propose an alternative approach to en-
hance Eb that combines quantum confinement with remote
dielectric confinement. As first pointed out by Keldysh
[1], the electron-hole Coulomb attraction can be greatly
enhanced in layered structures with strong dielectric mis-
match, due to the polarization charge induced at the in-
terfaces. For conventional semiconductor nanostructures
such as GaAsyAlGaAs- or GaAsyInGaAs-based samples,
however, this is a minor effect due to the small dielec-
tric mismatch between the constituents [2]. On the other
hand, interfaces between III-V semiconductors and materi-
als with very different dielectric constants, such as oxides,
are usually very far from the excellent optical quality of the
conventional ones. Our approach is based on the idea that
quantum and dielectric confinement can be spatially sepa-
rated, since they are effective over different length scales.
We will show that a very large increase in Eb can be ob-
tained in GaAs-based quantum wires by adding remote in-
sulating layers: these induce strong dielectric confinement
without degrading the good optical properties ensuing from
quantum confinement.
We consider GaAsyAlGaAs quantum wires (QWI),
where excitonic properties have been studied extensively
[3]. We focus on QWIs with T-shaped [4] and V-shaped
[5] cross sections, which rank among the best available
samples from the point of view of optical properties.
Starting from these geometries, we design hybrid struc-
tures adding oxide layers at some distance from the
QWI. In the new structures, the electron and hole wave
functions are confined by the inner, lattice-and-dielectric-
matched GaAsyAlGaAs interfaces; the outer AlGaAsy
oxide interfaces, owing to the strong dielectric mismatch,
provide polarization charges, thus enhancing the electron-
hole interaction.
To obtain quantitative predictions of excitonic proper-
ties in such hybrid structures, an accurate description of the
complex interplay between quantum and dielectric confine-
ment is needed. To this end, we have developed a novel
theoretical scheme that allows us to treat arbitrary dielec-
tric configurations, where the low symmetry makes simple
image-charge methods [2,6] not applicable. Furthermore,
we adopt a nonperturbative approach for the self-energy
term, which, in principle, is needed to describe strong di-
electric mismatch combined with shallow confining poten-
tials, as in some state-of-the-art QWIs. The electron-hole
interaction is treated within the conventional approach of
the semiconductor Bloch equations, adapted to quasi-one-
dimensional (1D) systems [7].
More specifically, for a spatially modulated dielectric
constant esrd the Coulomb interaction between two
charges of opposite sign 6e sitting at positions r and
r0—our electron-hole pair—is given by V sr, r0d ­
2e2Gsr, r0d, where Gsr, r0d is the Green’s function of the
Poisson operator, i.e.,
=r ? esrd=rGsr, r0d ­ 2dsr 2 r0d . (1)
We see that the space dependence of esrd modifies Gsr, r0d
with respect to the homogeneous case, where esrd ­ e0
implies G0sr, r0d ­
1
4pe0jr2r0j . This, in turn, gives rise to
significant modifications in the excitonic spectrum; within
the conventional Hartree-Fock scheme, such modifications
originate from the electron-hole Coulomb matrix elements
entering the evaluation of the absorption spectrum [7],
V ehij ­ 2e
2
Z
Fepi srdF
hp
j sr
0dGsr, r0dFhi sr
0dFej srd dr dr
0.
(2)
0031-9007y98y80(22)y4995(4)$15.00 © 1998 The American Physical Society 4995
VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998
Here Feshdl denotes the electron (hole) single-particle en-
velope function for the l ; kz , n state of the QWI, kz the
wave vector along the wire, and n the subband index.
To study realistic geometries, we find it convenient to
cast the problem in Fourier space, following the theoretical
scheme in Ref. [7]. It is easy to rewrite Eqs. (1) and (2)
(the symbol e denotes Fourier transform throughout):X
k00
esk 2 k00dk ? k00 eGsk00, k0d ­ dsk 1 k0d , (3)
V ehij ­ 2e
2
X
k
Feijskd
X
k0
eGsk, k0dFhjisk0d , (4)
where Feyhlm skd ­
R
F
eyhp
l srdeik?rF
eyh
m srd dr. We stress
that, in order to evaluate the Coulomb matrix elements in
(4), it is not necessary to solve Eq. (3) for each eGsk, k0d.
In fact, if we multiply (3) by Fhjisk0d and sum over k0,
we get a Poisson equation for the “potential” yjiskd ­P
k0
eGsk, k0dFhjisk0d [see Eq. (4)] arising from the “source”
Fhjisk0d, which is solved only once for each pair i, j.
The Green’s function Gsr, r0d also gives rise to a self-
energy term
Ssrd ­
e2
2
lim
r0!r
fGsr, r0d 2 GBsr, r0dg , (5)
where GBsr, r0d ­
1
4pesrd jr2r0 j is the (local) bulk solution
of (1). S is a local correction (equal for electrons and
holes) which adds to the confining potential in determining
the single-particle envelope functions Feyhl . This self-
energy contribution is evaluated within the same plane-
wave approach by solving (3) at a set of k0 [8]: it can be
shown that its Fourier transform is
eSsGd ­ e2
2
X
g
•eGµG 1 g
2
, G 2
g
2
¶
2 eGBsG, gd‚ ,
(6)
where eGBsG, gd ­ ge21sGdyg2 with the definitions G ­
sk 1 k0dy2 and g ­ k 2 k0 [9].
We first discuss our findings for QWIs obtained by
epitaxial growth on V-grooved substrates (V-QWIs) [5].
As a reference sample, we consider a GaAs wire with
AlAs barriers [10], and we add two oxide layers, below
and above the QWI [see Fig. 1(a)], at a distance L from
the GaAsyAlAs interfaces [11]. Note that the oxide layers
are characterized by a small dielectric constant that we
take equal to 2 [12].
Our results for the V-QWRs are shown in Fig. 1. For the
sample shown in Fig. 1(a), we find Eb ­ 29.3 meV, to be
compared with 13 meV of the conventional (i.e., with no
oxide layers) structure. Figure 1(a) shows that the origin
of this dramatic enhancement is the large polarization
of the AlAsyoxide interfaces induced by the hole charge
density [13]; the polarization is larger in the region where
0 2 4 6 8 10 12 14
10
15
20
25
30
35
40
45
(b)
Conventional structure
kTroom
  Full calculation
  Eq. (7)
E
xc
ito
n 
bi
nd
in
g 
en
er
gy
 (m
eV
)
L (nm)
FIG. 1. (a) (color) Cross section of a hybrid V-QWI, showing
the interface polarization charge (colors, units of nm21) induced
by the charge-density distribution of the lowest-subband hole
(grey scale, arbitrary units). The profile of the GaAsyAlAs
interfaces is obtained from Ref. [10]; the oxide layers are at
L ­ 5 nm from the GaAsyAlAs interfaces. The polarization
charge at the GaAsyAlAs interfaces is multiplied by 3. (b) Eb
versus distance of the oxide layers from the internal interfaces,
L. Solid dots: Full calculation. Solid line: Eq. (7) with L0 ­
6.56 nm. Dashed line: Energy E0 of the corresponding con-
ventional structure (no oxide layers). Dotted line: thermal
energy at Troom ­ 300 K.
the hole is localized, and is more pronounced at the lower
interface, due to the larger curvature. A small polarization
charge is also induced at the GaAsyAlAs interface, due
to the small dielectric mismatch. Note that quantum
confinement localizes the wave function well within the
inner interfaces; therefore, the AlAsyoxide interface does
not affect the electron and hole wave functions.
In Fig. 1(b) we show the calculated Eb for selected
values of L. We also show, for comparison, the calculated
binding energy for the conventional structure, E0, and the
room-temperature thermal energy. Eb is maximum when
the oxide layer is at minimum distance [14], L ­ 0: it
is enhanced by a factor larger than 3 with respect to E0,
and it is well above kTroom [15]. It is important to note
that Eb decreases slowly with L, and is still significantly
4996
VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998
larger than kTroom at L ­ 6 nm. Since Eb is the result
of the Coulomb interaction of, say, the electron with the
hole and the polarization charge which is excited at a
distance ,L, we intuitively expect Eb to decay as L21,
with a typical decay length L0 comparable to the Bohr
radius in the QWI [7]; this, in turn, is of the order of the
confinement length. Indeed, Fig. 1(b) shows that Eb is
very well interpolated by
EbsLd ­ E0 1
Ebs0d
1 1 LyL0
, (7)
with L0 ­ 6.56 nm. Note that Eb crosses kTroom when L
is as large as 9 nm.
A second type of structures, which have recently at-
tracted considerable attention, are the so-called T-shaped
wires (T-QWI), obtained by the cleaved-edge overgrowth
method [3,4]. The typical sample of our calculations [see
Fig. 2(a)] consists of a T-QWI with GaAs parent quantum
wells (QWs) of the same width, and AlAs barriers. An
0 2 4 6 8 10 12 14
16
18
20
22
24
26 (b)
Conventional structure
kTroom
  Full calculation
  No self-energy
  Eq. (7)
E
xc
ito
n 
bi
nd
in
g 
en
er
gy
 (m
eV
)
L (nm)
FIG. 2. (a) (color) Same as Fig. 1(a) for a hybrid T-QWI.
The QW widths are 5.4 nm; the oxide layer is at L ­ 3 nm
from the GaAsyAlAs interface. (b) Eb versus L including
(solid dots) and neglecting (empty dots) the self-energy contri-
bution. Solid line: Eq. (7) with L0 ­ 7.55 nm. Dashed line:
Energy E0 of the corresponding conventional structure (no ox-
ide layers). Dotted line: Thermal energy at Troom ­ 300 K.
oxide layer is added on top of the exposed surface at a dis-
tance L from the underlying QW. Note that, in this case,
an oxide layer is present only on one side of the QWI. As
in the case of the V-QWIs, a strong polarization charge
forms at the AlAsyoxide interface, with a maximum in the
region of the hole wave function confinement. A small
polarization charge is also present at the GaAsyAlAs inter-
face, peaked around the corners of the intersecting QWs.
In Fig. 2(b) we show the calculated Eb vs L. The binding
energy for the conventional structure, E0, and the room-
temperature thermal energy are also shown for comparison.
As in the case of the V-QWI, Eb is maximum at L ­ 0,
where it is enhanced by a factor of 1.5 with respect to E0,
and decreases slowly with L. Although Eb is smaller than
in the case of the V-QWI studied previously, for the small-
est L values Eb is still of the order of kTroom. It is impor-
tant to note that the reduced effect of dielectric confinement
in the T-QWI sample with respect to the previous example
of V-QWI is just due to the presence of a single oxide layer;
i.e., geometric effects due to different cross sections play
a minor role. In fact, despite the very different geome-
try, Eb decays with L in the same way in both cases. As
shown in Fig. 2(b), also in the case of T-QWIs Eb is very
well interpolated by Eq. (7) with L0 ­ 7.55 meV, which
is still of the order of the QWI confinement length. Note
that the above examples of structures are based on stan-
dard state-of-the-art QWIs that were previously studied in
the literature [4,5], and no particular optimization of Eb
with respect to sample parameters (constituents and/or ge-
ometry) has been attempted.
Finally, we discuss the effect of the self-energy term
on Eb . We have compared the full calculations discussed
above with calculations performed neglecting Ssrd in the
single-particle potential. For the V-QWI we have verified
that the self-energy contribution tends to increase Eb , but
it is so small (,0.2 meV) that the two results cannot be
distinguished on the scale of Fig. 1(b). In the case of the
T-QWIs, on the other hand, the self-energy contribution
is qualitatively and quantitatively different, as can be seen
from Fig. 2(b); in this case, in fact, it amounts to ,1 meV
at the smallest L, and tends to reduce Eb . This is a conse-
quence of the interplay between the dielectric confinement
and the shallow quantum confinement of these structures;
this is apparent from Fig. 3, where we compare the electron
and hole single-particle wave functions calculated neglect-
ing [Fig. 3(a)] and including [Fig. 3(b)] the self-energy
term. The self-energy potential [Fig. 3(c)] inside the GaAs
layer pushes the electron and hole wave functions away
from the oxide layer; because of the different masses and
shallow confining potentials, such shift is different for
electrons and holes, and the overlap is diminished, thereby
reducing Eb .
In summary, we have developed a theoretical scheme
that allows us to include dielectric confinement and
self-energy effects in a full three-dimensional description
of correlated electron-hole pairs [16]. Our calculations
4997
VOLUME 80, NUMBER 22 P HY S I CA L REV I EW LE T T ER S 1 JUNE 1998
FIG. 3. Lowest-subband electron and hole charge densities for
a T-QWI with L ­ 0 (see inset), neglecting (a) and including
(b) the self-energy potential Ssrd shown in (c). All curves are
calculated along the white line shown in the inset.
show that a dramatic enhancement of the exciton binding
in GaAs-based quantum structures is made possible by
remote insulating layers, bringing Eb in the range of the
room-temperature thermal energy. This enhancement
scales slowly with the distance of the insulating layer
from the quantum confinement region, thus allowing one
to design nanostructures that should be compatible with
excellent optical efficiency. Dielectric confinement by re-
mote insulating layers is predicted to be a novel powerful
tool for tailoring excitonic confinement in semiconductor
nanostructures.
We are grateful to E. Kapon, L. Sorba, and W. Weg-
scheider for useful discussions. This work was supported
in part by the EC through the TMR Network Ultrafast
quantum optoelectronics.
[1] L. V. Keldysh, Pis’ma Zh. Eksp. Teor. Fiz. 29, 716 (1979)
[JETP Lett. 29, 658 (1979)].
[2] L. C. Andreani and A. Pasquarello, Phys. Rev. B 42, 8928
(1990).
[3] For recent reviews see H. Sakaki et al., in Proceedings
of the Fifth International Meeting on Optics of Excitons
in Confined Systems [Phys. Status Solidi (a) 164, 241
(1997)]; G. Goldoni, F. Rossi, and E. Molinari, ibid. 164,
265 (1997).
[4] L. Pfeiffer et al., Appl. Phys. Lett. 56, 1697 (1990).
[5] E. Kapon et al., Phys. Rev. Lett. 63, 430 (1989).
[6] M. Kumagai and T. Takagahara, Phys. Rev. B 40, 12 359
(1989).
[7] F. Rossi and E. Molinari, Phys. Rev. Lett. 76, 3642 (1996);
Phys. Rev. B 53, 16 462 (1996); F. Rossi, G. Goldoni, and
E. Molinari, Phys. Rev. Lett. 78, 3527 (1997).
[8] It turns out that eGsk, k0d is needed on a coarser grid
for k0 than for k, due to the different length scales
of quantum and dielectric confinements; this makes the
numerical evaluation of (6) feasible.
[9] To avoid numerical instabilities we found it convenient
to define a reference material with the homogeneous
dielectric constant eref ­ esk ­ 0d. It can be shown
that an equation similar to (3) can be obtained for the
difference d eG ­ eG 2 eGref with respect to such reference,
where the deltalike term on the right-hand side is canceled
exactly. Despite this careful treatment, the evaluation of
S is by far the heaviest part of our calculation.
[10] R. Rinaldi et al., Phys. Rev. Lett. 73, 2899 (1994).
[11] The GaAsyAlAs band offsets are 1.036 eV (electrons) and
0.558 eV (holes). Effective masses are 0.067 (electrons)
and 0.38 (holes). Dielectric constants are 12.5 (GaAs),
10.0 (AlAs), and 2 (oxide). We neglect the variation of
other band parameters between AlAs and oxide layers.
Because of the exponential decay of the wave functions in
the AlAs layers, these effects should be negligible when
L exceeds a few nm.
[12] This is just intended as a typical value: oxides that could
be of practical relevance [see, e.g., A. Fiore et al., Nature
(London) 391, 463 (1998)] have similar or smaller values.
The value of the dielectric constant could be made even
smaller if one could take advantage of interfaces with air
[S. G. Tikhodeev et al., Phys. Status Solidi (a) 164, 179
(1997)].
[13] Of course, there is no electron-hole asymmetry, and we
could equivalently show the electron charge density and
its induced polarization charge; the present choice is
consistent with the choice of Fhji as “source” term, as
discussed below Eq. (4).
[14] For completeness, our calculations have been extended
down to L ­ 0, although in this limit they should be
taken with caution, due to the neglect of the AlAsyoxide
interface effects. See also note [11].
[15] In spite of such a dramatic enhancement, Eb is still
much smaller than the GaAs band gap, so that the usual
approximation which neglects the interband exchange
interaction is still valid in the present structures.
[16] This approach can be relevant also for applications to
other systems, including nanocrystals embedded in glass
matrices [see, e.g., A. P. Alivisatos, Science 271, 933
(1996)] where dielectric confinement effects are expected
to be strong.
4998


https://iris.polito.it/retrieve/handle/11583/1405204/45608/Goldoni-Rossi-Molinari_PRL_80_4995_1998.pdf

Strong exciton binding in quantum structures through remote dielectric confinement

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