Kondo Excitons in Self-Assembled Quantum Dots

RAPID COMMUNICATIONS

PHYSICAL REVIEW B 67, 241307͑R͒͑2003͒

Kondo excitons in self-assembled quantum dots

A. O. Govorov,1,2 K. Karrai,3 and R. J. Warburton4 1Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA 2Institute of Semiconductor Physics, 630090 Novosibirsk, Russia 3Center for NanoScience and Sektion Physik, Ludwig-Maximilians-Universita¨t, Geschwister-Scholl-Platz 1, 80539 Mu¨nchen, Germany 4Department of Physics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom ͑Received 12 May 2003; published 17 June 2003͒ We describe excitons in quantum dots by allowing for an interaction with a Fermi sea of electrons. We argue that these excitons can be realized very simply with self-assembled quantum dots, using the wetting layer as host for the Fermi sea. We show that a tunnel hybridization of a charged exciton with the Fermi sea leads to two striking effects in the optical spectra. First, the photoluminescence lines become strongly dependent on the vertical bias. Second, if the exciton spin is nonzero, the Kondo effect leads to peculiar photoluminescence line shapes with a linewidth determined by the Kondo temperature.

DOI: 10.1103/PhysRevB.67.241307 PACS number͑s͒: 73.21.La, 78.67.Hc, 78.55.Cr

The Kondo effect concerns the interaction of a localized In voltage-tunable structures, self-assembled QDs are em- spin with a Fermi sea of electrons.1 It is a crucial effect in bedded between two contacts.7,9 This makes it possible to various areas of nanophysics where often a nanosized object control the number of electrons in a QD by the application of 2 ͓ ͑ ͔͒ 9 with spin is in close proximity to a metal. Semiconductor a voltage Ug Fig. 1 a . By generating a hole with optical nanostructures are ideal for investigations of Kondo physics excitation, it has been demonstrated that there are regions of because, in contrast to metals, their properties are voltage gate voltage with constant excitonic charge, with the exci- 3 7 tunable. Characteristic to the Kondo effect are a screening tonic charge changing abruptly at particular values of Ug . of the localized spin and a resonance in the quasiparticle The charged excitons, labeled XnϪ, contain nϩ1 electrons density of states ͑DOS͒ at the Fermi energy. This leads to a and one hole. Excitons with n up to 3 have been observed in maximum in conductance at very low temperatures in the InAs/GaAs quantum dots.7,10 At higher voltages, 1 4 transport through a spin- 2 quantum dot. So far, the Kondo capacitance-voltage spectroscopy shows that electrons fill 9 effect in nanostructures has been studied almost exclusively the 2D wetting layer. In this case, the Fermi energy (EF)in in relation to transport properties. In optics, Kondo effects the wetting layer depends linearly on Ug : we find that EF ϭ Ϫ o have only been discussed theoretically with respect to non- (ao*/4d)(Ug Ug), where ao* is the effective Bohr radius, linear and shake-up processes in a quantum dot.5,6 o d is the distance between the back gate and wetting layer, Ug Here we present a theory of effects which arise from a is the threshold gate voltage at which electrons start to fill the combination of the Kondo effect and the optics of nanostruc- wetting layer, and dӷa* .9 Existing PL results in the regime ͑ ͒ o tures. In our model, a charged exciton in a quantum dot QD of wetting layer filling show broadenings and shifts of the PL interacts with a Fermi sea of free electrons. We call the re- lines,7 which we believe is some evidence of an interaction sulting quasiparticles Kondo excitons. We suggest that such with the Fermi sea, although this has not been analyzed in excitons can exist in self-assembled quantum dots where ex- detail. The excitons studied to date in the regime of a filled citons with an unambiguous charge can be prepared in 3Ϫ 4Ϫ 7 3Ϫ 4Ϫ 7 wetting layer are X and X . X has zero spin and X voltage-tunable structures. Furthermore, the wetting layer suffers from intradot Auger broadening in the final state, and can be filled with electrons and used as a two-dimensional therefore these excitons are unsuitable for Kondo physics. ͑2D͒ electron gas ͓Fig. 1͑a͔͒, offering a simple means of generating a Fermi sea in close proximity to a quantum dot. A significant point is that self-assembled dots are very small, only a few nanometers in diameter, allowing the Kondo tem- ϳ perature TK to be as high as 10 K. The time to form the ϳប ϳ 8 Kondo state is /kBTK 1 ps, which is much less than the typical radiative lifetime of 1 ns, giving the exciton enough time to form the Kondo state. We find that the photoluminescence ͑PL͒ from Kondo excitons is determined by the Kondo DOS at the Fermi level such that the PL line ͒ width is equal to kBTK (kB is the Boltzmann constant . Fur- thermore, the PL energy depends strongly on the Fermi energy, and therefore also on the bias voltage. This behavior FIG. 1. ͑a͒ Schematic of the heterostructure with a quantum dot contrasts with conventional single dot PL where the lines are embedded between front and back gates; dӶl. ͑b͒ The band dia- very sharp depending only weakly through the Stark effect gram of a quantum dot and associated wetting layer showing also on the bias.7 the energies typical to self-assembled quantum dots.

A. O. GOVOROV, K. KARRAI, AND R. J. WARBURTON PHYSICAL REVIEW B 67, 241307͑R͒͑2003͒

We argue here that X2Ϫ is the perfect exciton to study as it has a nonzero spin without intradot Auger broadening. This exciton has not yet been studied with an occupied wetting layer. Typically, the QD X0 and X1Ϫ excitons are strongly bound and exist only at gate voltages where the wetting layer is empty. It is therefore unlikely that X0 and X1Ϫ can couple with the Fermi sea. Instead, we focus on excitons X2Ϫ and X3Ϫ for which the electrons on the upper state may couple with extended states of the Fermi sea as we illustrate in Fig. 1͑b͒. Since self-assembled QDs are usually anisotropic, we model a QD with two bound nondegenerate orbital states, FIG. 2. ͑a͒ Calculated energy of the initial state as a function of having indexes s and p. For a p state electron, the electro- the delocalized state energy. The energy EF plays the role of the ϭ static potential consists of the QD short-range confinement Fermi level. Here, Udot 64 meV and Eg is the band gap of the and the long-range Coulomb repulsion from the charges in QD. ͑b͒ Electron configurations contributing to the initial ground the lower s state. This potential will therefore have a barrier state. Electron ͑hole͒ spins are represented with solid ͑open͒ at the edge of the QD ͓Fig. 1͑a͔͒, allowing the p electron͑s͒ triangles. to tunnel in and out of the QD. ͉ ˆ To investigate the interaction between a quantum dot ex- where i͘ is the initial state and the operator Vo ϭ ϩ citon and a Fermi sea of electrons, we employ the Anderson Vopt(bs,Ϫ3/2as,↑ bs,3/2as,↓) describes the strong transi- ˆ Hamiltonian which includes the single-particle energy Hsp , tions between the hole and electron s states. the intra-dot Coulomb interaction, and a hybridization We start with a zero bandwidth model, in which the Fermi term Hˆ : sea is replaced by a single ‘‘delocalized’’ state of energy tun 1 EF . In this simplified model, EF plays the role of the Fermi level, allowing us to predict the main features in the evolu- 1 ˆ ϭ ˆ ϩ ee ϩ ϩ tion of the PL with Fermi energy. We describe the electron H Hsp ͚ U␣ ,␣ ,␣ ,␣ a␣ a␣ a␣ a␣ ␣ ␣ ␣ ␣ 1 2 3 4 1 2 3 4 2 1 , 2 , 3 , 4 wave function with the occupations of the electron states, labelling the delocalized state e. The aim is explore the hy- Ϫ Ϫ Ϫ eh ϩ ϩ ϩ ˆ bridization of the X2 and X3 excitons, so we consider four ͚ U␣ ,␣ ,␣ ,␣ a␣ b␣ b␣ a␣ Htun , ␣ ␣ ␣ ␣ 1 2 3 4 1 2 3 4 1 , 2 , 3 , 4 electrons and one hole. We take the hole angular momentum ϩ 3 ͑ ͑ ͒ as 2 denoted sϩ3/2); equivalent results are obtained also 1 Ϫ 3 with 2 . The electron wave function of the hybridized ini- ϩ ϩ tial state before photon emission will have a total electron where a␣ (b␣ ) is the intra-dot creation operator of electrons spin S ϭ0. We therefore express the ground state as ͉i͘ ͑ ͒ ee eh e holes ; U and U are electron-electron and electron-hole ϭA ͉i ͘ϩA ͉i ͘, where Coulomb potential matrix elements, respectively. The index 1 1 2 2 ␣ stands for (␤,␴), where ␤ is the orbital index and ␴ the 1 spin index; ␤ can be s or p, and ␴ϭϮ 1 for electrons ͉ ϭ ͉ Ϫ͉ ͉ 2 i1͘ ͑ s↑ ,s↓ ,p↑ ,e↓͘ s↑ ,s↓ ,p↓ ,e↑͘) sϩ3/2͘, Ϯ 3 ˆ ˆ ͱ2 and 2 for heavy holes. Htun is given by Htun ϭ͚ ͓ ϩ ϩ ϩ ͔ k,␴Vk ck,␴a p,␴ a p,␴ck,␴ , where the operators ck,␴ de- ͉i ͘ϭ͉s↑ ,s↓ ,p↑ ,p↓͉͘sϩ ͘. scribe the delocalized electrons in the Fermi sea; k and Ek 2 3/2 are the 2D momentum and kinetic energy, respectively. For ͉ ͉ 2Ϫ The states i1͘ and i2͘ correspond to the excitons X and the tunnel matrix element V , we assume V ϭV in the in- Ϫ k k X3 , respectively ͑Fig. 2͒, and have energies E and E . terval 0ϽE ϽD, and V ϭ0 elsewhere.11 In the operator i1 i2 k k The above energies E and E should be calculated in the Hˆ , we include only coupling between the p state and the i1 i2 tun ͉ ͘ Fermi sea. In our approach, the quantization in a QD is as- absence of tunnel coupling. In the Kondo function i1 , the sumed to be strong so that the Coulomb interactions can be delocalized electron ‘‘screens’’ the net spin in the QD. By included with perturbation theory.12,13 The tunnel broadening diagonalising the Hamiltonian, we find that the initial ground state has energy E ϭ 1 ͓E ϩE Ϫͱ(E ϪE )2ϩ8V2͔, and is the smallest energy in the problem. By considering the i 2 i1 i2 i1 i2 tunneling through the barrier established by the Coulomb that A ϭϪa/(1ϩa2)1/2, A ϭϪA /a, where aϭ(E 1 2 1 i2 interaction, we estimate that the tunnel energy lies between 0 ϪE )/ͱ2V. and 2 meV depending on the energy of the p state with i As the energy EF increases, the ground state evolves from respect to the bottom of the continuum. Our approach is to the X2Ϫ to the X3Ϫ exciton. The transition occurs when E ͑ ͒ i1 solve Eq. 1 for the initial and final states, and calculate the intra intra intra intra ϷE where E ϷE → ϭE ϪE where E and E optical emission spectrum at zero temperature: i2 F 1 2 2 1 1 2 are the intradot energies. This transition indicates the so- 1 ϱ called mixed-valence regime. To calculate Ei numerically, ͑␻͒ϭ ͵ Ϫi␻t͗ ͉ ˆ ϩ͑ ͒ ˆ ͑ ͉͒ ͘ we represent a QD as an anisotropic harmonic oscillator tak- I Re dte i Vo t Vo 0 i , 0 ing the electron ͑hole͒ oscillator frequencies as 25 and 20

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KONDO EXCITONS IN SELF-ASSEMBLED QUANTUM DOTS PHYSICAL REVIEW B 67, 241307͑R͒͑2003͒

FIG. 4. ͑a͒ Contributions to the initial state ͑2͒. ͑b͒ Calculated emission spectrum of the X2Ϫ exciton with the triplet final state configuration; ⌬ϭ1 meV. The Fermi energies of 2 and 3.4 meV correspond to KT ϭ7 and 14 K, respectively. FIG. 3. ͑a͒ Final intradot states related to the singlet and triplet K states of the X2Ϫ exciton, and to the X3Ϫ exciton. ͑b͒ Calculated We focus now on the PL line shapes in the Kondo regime energies of the optical transitions as a function of the delocalized Ͻ state energy without (Vϭ0) and with (Vϭ0.5 meV) the hybridi- EF E1→2. While the zero bandwidth model is adequate for zation. The thickness of the line represents the intensity of the the general picture of the PL, we have to employ the finite emission. bandwidth model in order to calculate the PL line shapes. The initial Kondo state is the exciton X2Ϫ coupled with the ϭ 1,14 ͑12.5 and 10͒ meV, typical values for a self-assembled quan- Fermi sea. A trial function for the Se 0 ground state is ͑ ͒ tum dot. Figure 2 a shows Ei as a function of EF . ͉ ͘ ͉ ͘ After photon emission from state i , the final state f ͉i͘ϭͫ A ͉␾ ͘ϩ ͚ A ͉␾ ͬ͘*͉sϩ ͘, Ϫ 0 0 k k 3/2 ϭ 1 2 kϾk has Se 2 . The intra dot configuration in the X final state F ϭ ͱ is either a singlet (Sdot 0), with a configuration (1/ 2) ϫ ͉ Ϫ͉ ϭ ͉␾ ͘ϭ͉s↑ ,s↓ ,p↑ ,p↓ ;⍀͘, ͑2͒ ( s↑ ,p↓͘ s↓ ,p↑͘), or a triplet (Sdot 1), with configura- 0 tions ͉s↑ ,p↑͘ and (1/ͱ2)(͉s↑ ,p↓͘ϩ͉s↓ ,p↑͘) ͑Ref. 7͓͒Fig. ͑ ͔͒ 1 3 a . The PL related to these configurations are well sepa- ͉␾ ͘ϭ ͑ ˆ ϩ ͉ ⍀͘Ϫ ˆ ϩ ͉ ⍀͘ ϳ 7,10 k ck,↓ s↑ ,s↓ ,p↑ ; ck,↑ s↑ ,s↓ ,p↓ ; ), rated, typically by 5 meV, due to the exchange interac- ͱ2 tion between the s and p states. Nonzero optical matrix elements exist for the three final states: as represented diagrammatically in Fig. 4͑a͒. The configurations are described with the localized states and with the ⍀ 1 symbol which denotes all states of the Fermi sea with k ͉ ͘ϭ ͉͑ ͘ϩ͉ ͘Ϫ ͉ ͘ Ͻ f 1 ͱ s↑ ,p↓ ,e↑ s↓ ,p↑ ,e↑ 2 s↑ ,p↑ ,e↓ ), kF , where kF is the Fermi momentum. From the Anderson 6 Hamiltonian, we find that1 ͱ 1 2VA0 1 ͉ ͘ϭ ͉͑ ͘Ϫ͉ ͘ A ϭϪ , A ϭ , f 2 s↑ ,p↓ ,e↑ s↓ ,p↑ ,e↑ ), k intraϩ Ϫ 0 ͑ ϩ ⌬ ␲␦͒1/2 ͱ2 E1 Ek Ei 1 2 / where ⌬ϭ␲V2␳ (␳ is the 2D DOS͒. We write the ground ͉ ϭ͉ f 3͘ s↑ ,p↑ ,p↓͘. ϭ intraϩ Ϫ␦ ␦ state energy as Ei E1 EF , where is the lowering ␦Ͻ Ϫ ͉ ϭ of energy due to the Kondo effect. If E1→2 EF ,we In the state f 1͘, the intradot triplet state with Sdot 1is ͉ obtain ‘‘screened’’ by the delocalized electron. f 2͘ has the singlet intradot state, and the function ͉f ͘ plays the main role in the intra intra 3 ␲͑E ϪE ϪE ͒ emission of the X3Ϫ exciton. ␦ϭ͑ Ϫ ͒ ͩ Ϫ 2 1 F ͪ ϭ ͑ ͒ D EF exp ⌬ kBTK . 3 The calculated PL spectrum contains three lines ͑Fig. 3͒, 2 2Ϫ 2Ϫ 3Ϫ labeled as Xt , Xs , and X , with relative intensities Here, the energy ␦ plays the role of the Kondo temperature 3 2 1 2 2 4 A1 : 4 A1 :A2. Without the tunnel interaction, there is an in the initial state, TK . The temperature TK can be as high as 2Ϫ 3Ϫ ϳ ϭ ⌬ abrupt jump in the PL from X to X , but with the tunnel 7 –14 K for realistic parameters EF 2 –3.4 meV, interaction the PL shows hybridization effects very clearly. ϭ1 meV, and Dϳ30 meV. It is worthwhile to note that an Ϸ In particular, in the mixed-valence regime EF E1→2, the estimate for TK becomes even higher if we take into account intensity of the X2Ϫ lines rapidly decrease and the X3Ϫ line the Coulomb charging energy and admix the exciton X1Ϫ to appears but the energies of all three lines depend on the function ͑2͒. Ͻ delocalized state energy EF . Hence, a clear prediction is that In the regime EF E1→2, the final intra-dot configuration in the hybridization regime, the PL depends on the Fermi is either a singlet or a triplet state ͓Fig. 3͑a͔͒, as in the case of energy and therefore also on the gate voltage. the zero bandwidth calculation. The triplet state couples with

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A. O. GOVOROV, K. KARRAI, AND R. J. WARBURTON PHYSICAL REVIEW B 67, 241307͑R͒͑2003͒ the Fermi gas, leading to a state with total electron spin Se The important result is that in the Kondo regime, both the ϭ 1 2 . However, the Kondo temperature for this final state PL peak position and the PL line shape depend on the Fermi turns out to be much less than TK allowing us to use the energy and, hence, on the gate voltage. Furthermore, as the non-interacting final states to calculate the PL spectrum. The temperature increases, the peak in the spectral DOS dimin- 1 singlet final state, and also the X3Ϫ final state, have higher ishes rapidly, and therefore the linewidth of the Kondo ex- 2Ϫ energy and are broadened by energy ⌬ through tunneling citon Xt should also be strongly temperature dependent. into empty delocalized states. From state ͉i͘ in Eq. ͑2͒,we In the general case of arbitrary EF , the finite bandwidth ͑ ͒ 15 write the spectral function I(␻) in the form model leads to a PL spectrum similar to that in Fig. 3 b . 2Ϫ 3Ϫ The Xs and X PL lines are close to Lorentzians, with linewidths of ⌬ due to the finite lifetime of the final states. It ͑␻͒ϭϪ Ϫ I Reͫ i ͚ A␤A␤ЈF␤,␤Јͬ, is important to emphasize that the X2 final state lies at ␤,␤Ј t lower energy and does not suffer from tunnel broadening, ␤ ϭ͗␾ ͉ ˆ ϩ ˆ ˆ ͉␾ ͘ such that the line shape and linewidth are determined by where can be either 0 or k, F␤,␤Ј ␤ Vo RVo ␤Ј , and Ϫ Rˆ ϭ1/(Hˆ ϩប␻ϪE Ϫi0). We find that the X2 PL line has many-body effects. i t In summary, we have described excitons which can exist an asymmetric shape ͑Fig. 4͒: in self-assembled QDs when there is an interaction between 2 Ϫ a charged exciton localized on the dot and delocalized elec- 3⌬A D EF 1 2Ϫ͑␻͒ϭ 2 0 ͵ ⑀ Xt Vopt ␲ d ͑⑀ϩ ͒2 trons in the wetting layer. The charged exciton with nonzero 2 0 kBTK ͓ ϰ Ϫ1 spin is surrounded by a ‘‘cloud’’ of radius (kBTK) ]of Fermi electrons. We predict several striking manifestations Ϫi ϫRe , ͑4͒ of these states in the emission spectra: at low temperature, ប␻Ϫ Ј͑ 2Ϫ͒ϩ⑀Ϫ ␥ E Xt i the optical lines are strongly dependent on vertical bias, and Ј 2Ϫ ϭ 2Ϫ Ϫ their widths are determined by the Kondo temperature. where E (Xt ) E(Xt ) kBTK is the renormalized emission energy and ␥ describes the broadening of the final state. We gratefully acknowledge financial support from the We can expect ␥ to be small since the relaxation of the final CMSS program at Ohio University and from the 7 ӷ␥ state requires a spin-flip, and in this case, kBTK , the PL Volkswagen-Stiftung and the SFB 348. One of us ͑K.K.͒ linewidth is equal to KTK . In other words, the PL reflects the gratefully acknowledges J. von Delft and A. Rosch for spectral DOS near the Fermi level in the initial Kondo state. enlightening discussions on Kondo physics.

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