Supplementary Information: Optophononics with Coupled Quantum Dots

Transcription

1 Suppleentary Inforation: Optophononics with Coupled Quantu Dots Suppleentary Figures Suppleentary Figure 1: Overview Spectru. (A) Overview of electric field dispersed spectru of the olecular neutral exciton discussed in the ain text. Displayed is the spectral region of the transitions into the optical polaron states, X 0,. Two phonon bands are observed at 29 ev to 31 ev and 34 ev to 35 ev above the X 0 transitions. Fro their energies we attribute the to GaAs-like and InAs-like LO and TO optical phonons. Also seen in the spectru are the transitions into the single QD-like exciton state X 2 and the sequence of interdot exciton states ix 0 to ix 4. (B) Level-Anticrossing sequence of the interdot states, ix 0 to ix 4, with single QD-like exciton ground state X 0 [1]. The resonance ix 2 X 0 (saller dashed square) and the resonances ix 3 X 0 and ix 4 X 0 (dashed rectangle) are shown in higher contrast in panels (C) and (D) to the left, respectively. Vertical and horizontal scale bars in (C) and (D) easure 1 ev and 0.1 V respectively.

2 Suppleentary Figure 2: Quantu states involved in the resonant optical and phonon transitions. The photon energy hits the resonance ix n,, and the exciton ix n becoes coupled with the exciton X 0 via the resonant phonon transition. Here, we introduce notations ore convenient for theory: The state 1= X 0 is an exciton in QD1, 1,= X 0, is the sae state with an additional optical phonon ode, and the state 2= ix n is a spatially indirect exciton, or interdot exciton. 0= cgs is the crystal ground state.

3 Suppleentary Figure 3: Illustration of the phonon-assisted processes: (A) Phonon-assisted optical absorption processes in QD1. (B) Phonon assisted tunneling process of a hole fro QD1 to QD2 (or an electron fro QD2 to QD1).

4 Suppleentary Figure 4: Polaronic Fano-Effect: This graph illustrates the polaronic Fano equation, I(). For this graph, we intentionally chose a sall Fano factor to show the possibility of the optical transparency window (the case V=0). Experientally, sall Fano factors are possible in our syste in the non-linear regie. The nubers for the gate voltage are given with respect to the voltage, V center, at which the indirect-exciton line crosses the center of the phonon band, V=V-V center.

5 Suppleentary Figure 5: Non-linearity of the phonon-induced Fano effect. (A-E) Excitation power dependence of the Fano line shape at power densities in nw -2 as indicated in the top left corner of each panel. The solid lines are fits to the data. Values for (F) the linewidth of the interdot transition,, and (G) the Fano factor, q Fano, obtained fro the fits in (A-E). The data in (F) and (G) was fit according to and, yielding q Fano,0 =3.57, a=7.47, ed =87.8 ev and ph =14.40 ev. (H) Coparison of ix 4 transition line profiles at excitation energies 1314 ev ( 0 0, without phonons) and 1317 ev ( 0 >0, with phonons) for excitation power densities 0.56 nw -2, 2.00 nw -2, and 6.45 nw -2. The spectra were noralized (I ax -I in =1), and vertically offset for clarity. Scale bars in (A and D) indicate the full width at half axiu, 2.

6 Suppleentary Figure 6: Residual Tunneling: (A) Anticrossings calculated fro the eigenvalues of the Hailtonian in Eq. (9) for different values of the resonant tunneling energy, T. The resonant tunneling energy, T, aounts for half the splitting between the upper and lower branch of an anticrossing. The residual tunneling energy, t r, at a detuning, E, fro resonance is here easured as the difference between the sloped dashed line and the upper branch of an anticrossing. (B) Residual tunneling energy, t r, calculated for a detuning E of 36 ev. The tunneling rate of the hole tunneling resonance X 0 ix 4 as deterined by PL spectroscopy easures (50050) ev.

7 Suppleentary Notes: Suppleentary Note 1: Theoretical description of the Fano Effect We now develop a odel of the phonon-induced Fano-like effect in coupled QDs. Suppleentary Figure 2 shows a set of quantu states that need to be involved. This set of states is derived fro the band diagra shown in Figure 1A in the ain text. First, we write down Feri s golden rule describing the rate of energy absorption in our syste: I( ) W, 2 2 ˆ opt 0 ( f CGS ), f W f V E where is the photon frequency; E CGS is the energy of the coon crystal ground state and CGS 0 is its wave function. In the following, we will set for convenience: ECGS 0. f are all final states of our syste, which can involve both electrons and phonons. The optical interaction operator prootes the electrons fro the ground state to excited electronic states in both QDs: Vˆ V cˆ cˆ cˆ cˆ V cˆ cˆ cˆ cˆ, opt indirect CB,QD1 VB,QD2 VB,QD2 CB,QD1 direct CB,QD1 VB,QD1 VB,QD1 CB,QD1 V A, V A, direct 0 CB,QD1 VB,QD1 indirect 0 CB,QD1 VB,QD2 (1) (2) where the second quantization operators involve the electrons fro the conduction and valence bands (CB and VB) in both dots. The spatially-direct and indirect optical transition eleents include envelope wave functions,, and the inter-band eleent A E /2, where 0 cv 0 cv is the inter-band atoic dipole and E0 is the aplitude of the laser field. The function W can be written in ters of the optical Green function G( ): 2 W I G ( ), f 0 Vˆ f f Vˆ 0 1 G V V opt opt ˆ ( ) 0 ˆ opt opt 0. ˆ f f ECGS i f ECGS i H In the next step, we need to identify the perturbation operator to su up all interacting final states in the su in G( ). This operator is the electron-phonon interaction energy which includes two iportant ters (3)

8 Vˆ Vˆ Vˆ, e-ph e-ph,qd1-qd1 e-ph,qd1-qd2 Vˆ b b ( r) cˆ cˆ e-ph,qd1-qd1 VB,QD1 VB,QD1 VB,QD1 VB,QD1 b b ( r) cˆ cˆ, CB,QD1 CB,QD1 CB,QD1 CB,QD1 ˆ ˆ ˆ e-ph,qd1-qd2 VB,QD2 ( ) VB,QD1 VB,QD2 VB,QD1 b ˆ ˆ VB,QD1 r VB,QD2 cvb, QD1 cvb, QD2 V b r c c ( ). (4) In the above equations, b and b are the phonon operators of odes and () r are the Fröhlich potentials of the odes. In our case, the phonon odes coe fro localized phonon vibrations in a QDM and we will not specify the odes since the details of such vibrations are not known. The operator Vˆ describes weak electron-phonon (polaron) interaction for the exciton in QD1; we need this e-ph,qd1-qd1 weak interaction to involve phonon-assisted inter-band transitions in QD1 (Suppleentary Fig. 3). In the case of the inter-qd operator V ˆ, we only keep the resonant ters which describe phonon-assisted e-ph,qd1-qd2 tunneling between QDs. Then, the final states of our syste should be calculated including the electronphonon operators (4) and the Hailtonian of our syste should be written as: Hˆ Hˆ Vˆ Hˆ Vˆ Vˆ, (5) 0 e-ph 0 e-ph,qd1-qd1 e-ph,qd1-qd2 where Ĥ 0 is the purely electronic Hailtonian. To ake diagraatic calculations in the following sipler, we first diagonalize partially the total Hailtonian by including only the intra-dot polaronic Vˆ as a sall perturbation. Then, the states becoe operator e-ph,qd1-qd1 0 0 ;1 ;1 0 a 0 ;1 ;1 CB,QD1 VB,QD1 VB,QD2 phonons CB,QD1 VB,QD1 VB,QD2 1 1 ;1 ;0 0 b 1 ;1 ;0 CB,QD1 VB,QD1 VB,QD2 phonons CB,QD1 VB,QD1 VB,QD2, (6) where the state 0 is the coon crystal ground state and 1 is the state of an exciton in QD1. Both states acquire now polaronic wave functions written for the regie of weak electron-phonon interaction. Using the new states, we also can express the optical aplitude for the polaron state X 0; ph, n as: ˆ () r v 1, V 0 A dv. 2 2 opt 0 CB,QD1 VB,QD1 CB,QD1 VB,QD1 (7) In the fraework of the new states (6), the perturbation operator is V ˆ and the total Hailtonian is written as as follows [2]: ' 0 e-ph,qd1-qd2 e-ph,qd1-qd2 Hˆ Hˆ Vˆ. The rewritten operator in Eq. 3 can be easily expanded

9 1 i Hˆ Hˆ ' 0 int 1 1 ˆ 1 1 ˆ 1 ˆ 1 H H H i Hˆ i Hˆ i Hˆ i Hˆ i Hˆ i Hˆ..., ' ' int ' ' int ' int ' where the interaction Hailtonian should be taken as H ˆ ˆ int Ve-ph,QW1-QW2. Then, we eploy the ground state fro Eq. 6 and write G( ) as a su of a few infinite series. Then, we su up these geoetrical series and coe to the following equation: X0 2 e vt G I vv, i ix tt tt tt, i vt vt, i vv vv, i, ph, where ix 0 CB,QD1 VB,QD2 A and v is the optical aplitude for the transition fro 0 to the polaron state 1,, given by Eq. 7. The phonon-assisted tunneling aplitude is obtained in the for: (8) t 1, Vˆ 2. (9) e-ph VB,QD2 VB,QD1 The eployed ethod of suation of the diagra ladders with creation and eission of phonons is siilar to the approach used by us previously in Ref. 3 for the case of a Stark ladder interacting with optical phonons in a seiconductor superlattice. We now assue that the phonon band has a flat density of state in the interval D/ 2 D/ 2. This odel corresponds well to the experiental spectra, which show ph,0 ph, ph,0 optical-phonon bands in the for of stripes. Then, the sus in Eqs. 8 are replaced by integrals: ph,0 D/2 (...) (...) 0dph. (10) ph,0 D/2 Assuing that the transition aplitudes are constant, we obtain the Green function and the absorption rate:

10 2 I( ) G( ), F ( ) G v q F F q phon ph 2 2 qfano ( ) I Fano ph ( ), ix i ix ph Fph ( ) ph D 1 1 ( ) d ph, i D ix ix 2 Fano ph 0 vph vt0 QD1 ph,0 ph t, t, X 0 (11) where ph is the broadening of the discrete exciton line ( ix ) due to the phonon-induced tunneling and qfano is the Fano factor which controls an asyetry of the spectru. In Suppleentary Figure 4, we illustrate one particular case of the function I( ) for various applied gate voltages. The spectru has a discrete resonance (indirect exciton) and the phonon band. With the applied voltage, we ove the discrete line through the phonon band. This case shows that the polaronic Fano effect gives an interesting opportunity to produce a strong anti-resonance, which is a narrow optical transparence window, under certain conditions. This transparency window is realized in Suppleentary Figure 4 at the voltage V 0 when the discrete resonance is inside the phonon band. Soe of these conditions for the narrow Fano transparency window are: Sall-in-agnitude Fano factor, a narrow discrete exciton line (i.e. sall ix ), and relatively sall phonon-induced broadening. The non-linear regie can facilitate the appearance of Fano anti-resonances since the non-linearity leads to saller Fano factors (see Ref. 4 and the Suppleentary Note 3 below). In order to siulate the electric field dispersed optical spectra, as seen in Fig 2A, the energy of the (0) interdot state was varied according to. Here is the energy of the interdot state at zero ix ix U applied bias (U=0V), and is the linear Stark shift [refs 1&5], which is deterined by the dot separation and easured as slope of the interdot exciton transitions. The data in the figures is plotted against the bias voltage, U. Here the negative voltage values indicate that the diode was operated under reverse bias. (0) ix Suppleentary Note 2: Validation of the Fano factors In order to validate the assuption of the above Fano schee it is useful to analyze what ratio of the two atrix eleents, and ix, is required for a Fano factor of q Fano 1, and to check if this ratio is reasonable. Besides the ratio of the atrix eleents the optical phonon density of states,, and the tunneling rate,. ix 1 q Fano t enter the Fano factor,. Typical values for the phonon density of states are around 0.1 ev -1 [6&7]. If for the tunneling rate we use values between the residual tunneling as calculated above and the value we can extract fro fitting the power dependence of ph in Suppleentary Fig. 5 below (both fro ix 4 ), i.e., t0.01 ev-0.20 ev, we find 20 ix -300 ix. Further we have to consider that the transition rate into the polaron state, X 0,, is liited by the lifetie of the intradot neutral exciton, X 0, which is on the order of 1 ns. Consequently, it follows that the lifetie of the interdot exciton

11 states is on the order of ix 20 ns-300 ns. These values are in excellent agreeent with experientally deterined values for the interdot exciton lifeties as reported by Boyer de la Giroday, et al., in ref [8]. Suppleentary Note 3: Origin of the non-linearity of the phonon-induced Fano-effect Insight into the echanis behind the non-linear Fano-effect (Fig. 3 and Suppleentary Fig.5A-E) is gained via the change in linewidth and the change of the Fano factor. The total linewidth, ed + ix + ph, is coposed of the hoogeneous linewidth of the interdot exciton transition, ix, its broadening caused by the response of the large static dipole oent of the interdot exciton to fluctuations in the local electric field, ed, and the phonon-assisted tunneling broadening, ph. The change in line shape with increasing power is quantified by a decrease of the Fano factor (Suppleentary Fig. 5F), and an increase of the total width of the resonance (Suppleentary Fig. 5G), as obtained by fitting the data in Suppleentary Fig. 5(A-E) with Eq. (1). Both follow a power broadening law. As Eq. (1) does not account for the inhoogeneous broadening separately the extracted q-values need to be considered apparent Fano factors, q * Fano. In coparison, exaination of the power dependence of the transition into ix 4 at an excitation energy outside the phonon bands shows no significant power broadening (Suppleentary Fig. 5H). Hence power broadening reains negligible copared to the electric dipole-induced broadening, ed. Consequently, the power dependence observed when ix 4 and X 0, are in resonance can be attributed to an increase of ph and thus an increase of the phonon-ediated tunnel coupling, t, between the two states. One ay think of this process as two oscillators being synchronized by a coon driving field. Here the optical driving field locks the relative phase of both states. As the Rabi oscillation frequency increases with increasing power both states see each other on average longer in the case of in-phase driving, before one of the loses its phase due to an incoherent process. The in-phase optically driven coupling and resulting foration of the olecular polaron quasi causes a destructive interference for the eission of the optical phonon, i.e., the relaxation into the intradot exciton ground state, X 0. The destructive interference of both relaxation transitions effectively reduces the transition rate of both optical transitions, which in turn causes an enhanced power broadening. Suppleentary Note 4: Residual Tunneling The occurrence of the Fano effect is indicative of the existence of a weak coupling between the two involved states. Here, quantu echanical tunneling is the obvious echanis. The coupling strength at resonance of the involved hole levels, T r,n, is directly easurable as the agnitude of the anticrossing splitting, 2T n, in the SQD-like exciton ground state transition PL. The coupling strengths depend on the wave function syetries and thus are different for each pair of states [1]. At the energy of the polaron transitions, 28 ev to 38 ev away fro the level resonances, the coupling strength is significantly reduced. The residual tunnel coupling, t r, can be calculated fro the easured anticrossing splitting, 2T n, at resonance of the n-th blue dot hole state with the red dot hole ground state as ( ) ( ). It follows fro the eigenvalues, ( ) ( ( )), of Hailtonians of the for ( ). (12) Here, the energy of the hole ground state of the SQD-like exciton in QD1 was set to zero and is the energy of the optical phonon by which the n-th hole level in QD2 is detuned fro resonance with the ground state hole level in QD1. For the QDMs in this saple, the tunneling energies at resonance reach values up to about 1eV, which at =36 ev detuning yields residual tunneling energies of up to about 20 ev (Suppleentary Fig. 6).

12 Note, that the residual coherent tunnel coupling between X 0 and ix 2 at an energy detuning of ħ30 ev fro resonance aounts to only 0.07 ev. Such a coupling strength is equivalent to a dot to dot separation of about 35n for hole tunneling or 90 n for electron tunneling in InAs/GaAs QDMs [4]. Suppleentary References: 1. M. Scheibner M. Yakes, A.S. Bracker, I.V. Ponoarev, M.F. Doty, C.S. Hellberg, L.J. Whitan, T.L. Reinecke & D. Gaon, Optically Mapping the Electronic Structure of Coupled Quantu Dots, Nature Phys. 4, (2008). 2. A. C. Hewson, The Kondo Proble to Heavy Ferions (Cabridge University Press, Cabridge, 1993). 3. A.O. Govorov, Interband optical absorption in the Wannier-Stark 'ladder' under the electron-lo-phonon resonance condition. Solid State Coun. 92, (1994). 4. M. Kroner, A. O. Govorov, S. Rei, B. Biederann, S. Seidl, A. Badolato, P. M. Petroff, W. Zhang, R. Barbour, B. D. Gerardot, R. J. Warburton & K. Karrai, The nonlinear Fano effect, Nature 451, (2008). 5. M. Scheibner, A. S. Bracker, D. Ki & D. Gaon, Essential concepts in the optical properties of quantu dot olecules, Solid State Coun. 149, (2009). 6. X. Lia, X. Taoa, R. Lia, H. Chena, Y. Ouyanga, Y. Du, Lattice dynaics properties of XAs (X=Al, Ga and In) with zinc-blende structure fro first-principle calculations. J. Phys. Che. Solids 73, (2012). 7. Debernardi, Phonon linewidth in III-V seiconductors fro density-functional perturbation theory. Phys. Rev. B 57, (1998). 8. A. Boyer de la Giroday, N. Sköld, R. M. Stevenson, I. Farrer, D. A. Ritchie, and A. J. Shields, Exciton-Spin Meory with a Seiconductor Quantu Dot Molecule. Phys. Rev. Lett. 106, (2011).