Tunneling Effects on Fine-Structure Splitting in Quantum Dot Molecules
TTunneling Effects on Fine-Structure Splitting in Quantum DotMolecules
Hanz Y. Ram´ırez and Shun-Jen Cheng
Department of Electrophysics, National Chiao Tung University,Hsinchu 300, Taiwan, Republic of China (Dated: October 30, 2018)
Abstract
We theoretically study the effects of bias-controlled interdot tunneling in vertically coupledquantum dots on the emission properties of spin excitons in various bias-controlled tunnelingregimes. As a main result, for strongly coupled dots we predict substantial reduction of opticalfine structure splitting without any drop in the optical oscillator strength. This special reductiondiminishes the distinguibility of polarized decay paths in cascade emission processes suggesting theuse of stacked quantum dot molecules as entangled photon-pair sources.
PACS numbers: 71.45.Gm, 03.67.Bg, 78.67.Hc, 78.55.Cr, 74.50.+r a r X i v : . [ c ond - m a t . m e s - h a ll ] J un unneling is a remarkable quantum property of microscopic particles that has no classicalcounterpart, which allows coupling between two objects spatially separated by a finite po-tential barrier. Currently, extending the analogy between atoms and 0D solid state systems,coupled quantum dots (QDs) are widely studied as artificial molecules where importantproperties of single dots are improved for optimization and scalability of applications. Re-cent examples of interesting and useful tunnel effects in coupled dot systems include thetunability of fluctuations in Kondo currents [1], reduction of electronic spin decoherence byinteraction with nuclear spin [2], conditional dynamics of transitions [3] and bias control of g tensors [4].Currently, a highly desirable feature of QD-based photon emitters is the reduced finestructure splitting (FSS) between the intermediate one-exciton (X) spin states. The FSS iswidely believed to be a consequence of the electron-hole ( e-h ) exchange interaction causedby the intrinsic lack of perfect symmetry of QD structures [5]. The FSSs make the twopossible decay paths in bi-exciton cascade processes energetically distinguishable, and havebecome a main obstacle in the production of polarization-entangled photon pairs from QDs[6, 7, 8, 9]. Researchers have recently demonstrated significant reductions in the FSSs ofsingle QDs using strain and post-annealing techniques, and the application of electric andmagnetic fields [10, 11, 12]. Nevertheless, to solve the “which-path” problem, the FSSsof bright X’s (typically only 10 ∼ µ eV) must be within the intrinsic broadenings oftheir emission lines (typically only 10 ∼ µ eV) [13]. In most experiments, however,it is not clear if the reduction of FSS is caused by the undoing of symmetry breaking orthe reduction of e-h wave function overlap. The latter effect reduces not only the FSS butalso the oscillator strength of e-h recombination, yielding narrow intrinsic broadening inthe corresponding emission lines and actually inhibiting the generation of entangled photonpairs [13, 16].In this letter, we theoretically examine the effects of quantum tunneling in vertical QDmolecules on the optical fine structure properties using the configuration interaction (CI)method. This study is based on a developed 3D model for coupled QDs that considers theboth of mesoscopic (envelope function) and microscopic (Bloch function) nature of electronsand holes. As a result of quasi-resonant tunneling in stacked double dot systems, FSSs andphotoluminescence (PL) intensities can by tuned by applying external bias fields and/orvarying inter-dot distances. Remarkably, we predict a significant reduction of the optical2SSs in strongly coupled DQDs with small inter-dot distances without any decrease in theoptical oscillator strength.Let us consider a pair of vertically stacked quantum dots along the growth z − axis, sepa-rated by an inter-dot distance d and subject to an applied electric field F , as shown in Fig.1(a) [17]. The e-h Hamiltonian for a single spin exciton in a coupled double QD is writtenas H = (cid:88) j,σ ( ε ej + eF z j ) c † jσ c jσ + (cid:88) n,χ ( ε hn − eF z n ) h † nχ h nχ − (cid:88) j ∈ L,k ∈ R,σ t ejk ( c † jσ c kσ + c † kσ c jσ ) − (cid:88) n ∈ L,m ∈ R,σ t hnm ( h † nχ h mχ + h † mχ h nχ ) − (cid:88) kmnj,σχ V ehkmnj c † jσ h † mχ h nχ c jσ − (cid:88) kmnj,σχχ (cid:48) σ (cid:48) V eh,exkσ,mχ,nχ (cid:48) ,jσ (cid:48) c † kσ h † mχ h nχ (cid:48) c jσ (cid:48) , (1)where the composite indexes j, k ( n, m ) denote the electron (valence hole) orbitals anddot positions ( L / R for the left/right dot), σ = ↑ / ↓ ( χ = ⇑ / ⇓ ) represents electron(hole) spin with s z = / − ( j z = / − ), c † jσ and c jσ ( h † nχ and h nχ ) are the elec-tron (hole) creation and annihilation operators respectively, ε ei ( ε hn ) is the kinetic energyof an electron (a valence hole), e is the unit charge, and z j ∈ L = 0 ( z j ∈ R = d ) is the z − position of the left (right) dot. Here, the valence hole orbitals of the highly quan-tized strained dots are assumed to be purely heavy-hole like. The terms with the hop-ping parameters ( t ejk , t hnm ) describe the (spin-conserved) carrier tunneling between adjacentdots. The matrix elements of conventional e-h Coulomb interaction and the e-h exchange interactions are V ehkmnj ≡ (cid:82) (cid:82) d r d r Φ e ∗ k ( (cid:126)r )Φ h ∗ m ( (cid:126)r ) e π(cid:15)r Φ hn ( (cid:126)r )Φ ej ( (cid:126)r ) and V eh − exkσ,mχ,nχ (cid:48) ,jσ (cid:48) ≡ (cid:82) (cid:82) d r d r Φ e ∗ k ( (cid:126)r ) u ∗ cσ ( (cid:126)r )Φ h ∗ m ( (cid:126)r ) u ∗ vχ ( (cid:126)r ) × e π(cid:15) r Φ hn ( (cid:126)r ) u vχ (cid:48) ( (cid:126)r )Φ ej ( (cid:126)r ) u cσ (cid:48) ( (cid:126)r ), respectively,where Φ α are single-particle envelope wave functions, u cσ ( u vχ ) are the electron (hole) Blochfunctions, (cid:15) is the dielectric constant and r ≡| (cid:126)r − (cid:126)r | . Remarkably, after undergoing an e-h exchange interaction, an electron or a hole could lose its spin conservation. Within thedipole-dipole approximation, the long-range part of the e-h exchange interaction is given by3 eh − ex ( Lr ) kmnj ≡ δ kmnj ≈ e (cid:126) E p (cid:15)m E g (cid:82) (cid:82) d (cid:126)r d (cid:126)r Φ e ∗ k ( (cid:126)r )Φ h ∗ m ( (cid:126)r )Φ hn ( (cid:126)r )Φ ej ( (cid:126)r )[( y − y ) − ( x − x ) +2 i ( x − x )( y − y )] / ( r ) ,where E p is the conduction-valence band interaction energy, E g the band gap energy, and m the mass of a free electron [18].Based on the lowest single-particle orbitals of single dots, eight spin-X configurationsare constructed, as displayed in Fig. 1(b). To analyze further the (linear) polarizationof emitted light, a new basis is defined by the linear transformation of the configurationsaccording to the parity symmetry: | LL ±(cid:105) ≡ √ ( | L ↑ L ⇓(cid:105) ± | L ↓ L ⇑(cid:105) ), | RR ±(cid:105) ≡ √ ( | R ↑ R ⇓(cid:105) ± | R ↓ R ⇑(cid:105) ), | LR ±(cid:105) ≡ √ ( | L ↑ R ⇓(cid:105) ± | L ↓ R ⇑(cid:105) ), | RL ±(cid:105) ≡ √ ( | R ↑ L ⇓(cid:105) ± | R ↓ L ⇑(cid:105) ).In the redefined basis, the 8 × × π x -( π y -) polarized light emission. In the basis ordered by | LL ±(cid:105) , | RR ±(cid:105) , | LR ±(cid:105) , | RL ±(cid:105) , thedecoupled 4 × ˆ H ± = − V eh ∓ δ DD ∓ δ II − t h − t e ∓ δ II − V eh + ∆ e + ∆ h ∓ δ DD − t e − t h − t h − t e − eFd + ∆ h ∓ δ II ∓ δ II − t e − t h ∓ δ II eFd + ∆ e ∓ δ II , (2) where the kinetic energy offset ε e/hL + ε e/hL is removed for brevity, ∆ e/h ≡ ε e/hR − ε e/hL denotes the difference between kinetic energies of the two adjacent dots due to the inevitableslight differences in size, shape or chemical composition, V eh ≡ V ehLLLL = V ehRRRR denotesthe direct Coulomb interaction between an e-h pair in the same single dot, and δ DD ≡ δ RRRR = δ LLLL ( δ II ≡ δ LRRL = δ RLLR ≈ δ LLRR = δ RRLL ) is the long range e-h exchangeinteraction in a direct X (an indirect X). Previous studies concerning e-h exchange matrixelements in single and laterally coupled dots use 2D approaches [18, 19, 20]. However, afully three-dimensional formulation, including dot height and interdot distance, is requiredto accurately consider tunneling effects in stacked QD molecules. Within the 3D parabolicmodel for the confining potentials of single QDs, the single-particle wave functions of thelowest orbitals of single dots can be described by Φ L/R ( x, y, z ) = ( π l x l y l z ) − / exp[ − (( xl x ) +( yl y ) +( z − z L/R l z ) )], characterized by the wave function extents l α = x,y,z . Accordingly, we derive V eh ≈ e π(cid:15) l √ − (1 − a ) √ π (1 − a ) , δ DD = e (cid:126) E p √ π(cid:15)m E g ( l x − l y ) l y l y l z , and δ II = δ DD e − d l z , for a slightly deformed DQD ( ξ ≡ l x − l y l y (cid:28) (cid:54) = 0 , where l ≡ ( l x + l y ) / , a ≡ l z /l ). The values of t e and t h , are determined by theformulation presented in Refs. [21] and [22], respectively. Figure 1(c) shows the calculated4 e and the t h as functions of d [23].The energy spectrum { E π x ,i } ( { E π y ,i } ) of the exciton states | π x ; i (cid:105) ( | π y ; i (cid:105) ) for the π x ( π y )-polarized light emission is calculated by diagonalizing H + ( H − ) in Eq.(2) [24]. In the com-bined energy spectrum, { E π x ,i , E π x ,i } , each level is a doublet of the spin X states, | π x ; i (cid:105) and | π y ; i (cid:105) , which are split by an FSS ∆ E i ≡ E π y ,i − E π x ,i [inset of Fig. 2(a)]. The π x ( π y )-linear-polarized photoluminescence (PL) spectra are obtained using Fermi’s golden rule: I x ( y ) ( ω ) = (cid:80) i F ( E i , T ) |(cid:104) | P − x ( y ) | π x ( π y ); i (cid:105)| δ ( E π x ( π y ) ,i − (cid:126) ω ), where the subscript i ( f ) denotes initial(final) states of the PL transition, ω is the frequency of the emitted photon, the operator P ( − ) x = (cid:80) n,j S n,j ( h n ⇑ c j ↓ + h n ⇓ c j ↑ ) [ P ( − ) y = − i (cid:80) n,j S n,j ( h n ⇑ c j ↓ − h n ⇓ c j ↑ )] describes the all pos-sible e-h recombinations that produce the π x [ π y ] linear polarized PL, S n,j = (cid:82) d r Φ h ∗ n ( (cid:126)r )Φ ej ( (cid:126)r )is the e- and h-wave function overlap, and F ( E i , T ) = exp( − E i /k B T ) / [ (cid:80) l exp( − E l /k B T )]is the probability of occupation of state | i (cid:105) , where k B is the Boltzmann constant and T istemperature.Figure 2(a) shows the calculated energy spectra of a weakly coupled DQD with d = 8 . t e = 6 . t h = − . e (cid:29) t e > t h ), the 4 × × t e / ∆ e (cid:28) × H W C ± = − V eh ∓ δ DD − t h − t h − eF d + ∆ h ∓ δ DD (3)with respect to the basis | LL ±(cid:105) and | LR ±(cid:105) . Equation (3) is actually equivalent to thewidely used solvable three-orbital model for DQDs [25]. The eigen states of Eq.(3) arehybridized by the optically active X-configuration | LL ±(cid:105) and the inactive configuration | LR ±(cid:105) , determined by the bias-controlled detuning from resonance ( | edF − (∆ h + V eh ) | ).Expanding the X eigen states in the used basis for Eq.(2), i.e. | π x ; i (cid:105) = (cid:80) nj C xnj,i | nj + (cid:105) and | π y ; i (cid:105) = (cid:80) nj C ynj,i | nj −(cid:105) , the intensities and the FSS associated with the lowest spectrallines are given by I ≈ F ( E , T )( C LL, S D + C LR, S I ) and ∆ E ≈ C LL, δ DD + C LR, δ II ),where C LL, ≡ C xLL, = C yLL, ( C LR, ≡ C xLR, = C yLR, ) are the expansion coefficients asso-ciated with the bright (dark) X configurations | LL ±(cid:105) ( | LR ±(cid:105) ) and S D ≡ S LL = S RR ≈ S I ≡ S LR = S RL = e − d l z ) is the e-h wave function overlap in a direct-X (an indirect-X)5onfiguration. Accordingly, both of the I and the ∆ E of a weakly coupled DQD ( S I (cid:28) S D and δ II (cid:28) δ DD ) are mainly proportional to C LL, and should depend similarly on applied biasfields. Figure 3(a) shows the calculated polarized PL spectra of the weakly coupled DQD atsome bias fields in the near-resonance regime and the inset shows the F -dependences of the I and ∆ E .At very low bias ( | edF/ ( V eh + ∆ h ) | (cid:28) | π x ; 1 (cid:105) ≈| LL + (cid:105) and | π y ; 1 (cid:105) ≈ | LL −(cid:105) . The intensity (FSS) of the corresponding linear polarizedemission lines is I ≈ ( S D ) (∆ E ≈ δ DD ), approaching the value of the intensity (FSS) ofthe lowest spectral lines of a single dot, I SD (∆ E SD ). At near resonance ( edF/ ( V eh + ∆ h ) ≈ | π x ; 1 (cid:105) ≈ √ ( | LL + (cid:105) − | LR + (cid:105) ) and | π y ; 1 (cid:105) ≈ √ ( | LL −(cid:105) − | LR −(cid:105) ), only the hole inthe exciton can be transferred between dots while the electron is stably localized in the leftdot. The intensity (FSS) of the corresponding polarized emission lines is I ≈ ( S D + S I ) / E ≈ δ DD + δ II ), which is only about 50% of that for a single dot. The resonant inter-dottunneling of a single hole significantly reduces the overlap of the electron and hole wavefunctions, leading to not only the decrease in the optical FSS but also the oscillator strengthof an e-h recombination. The decreased oscillator strength of e-h recombination reduces theintrinsic broadening width of the main X lines. Such an FSS reduction however does notsupport the feasibility of the dot-based entangled photon pair source devices [26, 27].Figure 2(b) shows the energy spectra of a strongly coupled DQD with small distance d = 4 . t e = 106meV and t h = 18 . I and ∆ E of the lowest spectral lines vs. F . Generally,the strongly coupled DQD have smaller FSS ∆ E but larger I than single dots or weaklycoupled dot molecules.In the strong coupling (SC) limit ( t β (cid:29) ∆ β ), both electrons and holes can be transferredbetween dots over a very wide range of detuning and the Hamiltonian, Eq.(2), can beapproximately written as ˆ H SC ± ≈ − t h − t e − t e − t h − t h − t e − t e − t h . (4)The lowest eigenstates for Eq.(4) are ( | LL ±(cid:105) + | RR ±(cid:105) + | LR ±(cid:105) + | RL ±(cid:105) ), highly inter-6ixing all X-configurations. Accordingly, we have I ≈ ( S D + S I ) and ∆ E ≈ δ DD + 2 δ II , i.e. that the FSS is only about one half of the magnitude of ∆ E SD but the intensity of thepolarized emission lines is slightly larger than I SD . In the strong coupling regime, not onlyvalence holes but also electrons are spread over the two coupled dots. The simultaneous e-h resonance transfers between dots enlarges the optically active volume and increase themean distance (cid:104) r (cid:105) in the long ranged e-h exchange interactions, resulting in the larger I and smaller ∆ E .Figure 4 plots the normalized I and ∆ E (by I SD and ∆ E SD ) of DQDs as functions ofthe inter-dot distance d and applied bias fields F . In the WC regime, as discussed previously, I and ∆ E depend similarly on F . As a DQD is driven into the SC regime, I are markedlyincreased and the FSS is reduced to only ∼
50% of ∆ E SD (see the regions highlighted by indash-line boxes) [17]. The increased I and reduced ∆ E are robust against the detuning,being almost insensitive to F . The lower part of Fig. 4 plots the results obtained for DQDs atnegative F , which drives electron inter-dot transfers. The results for the DQDs at negative F show similar physical features to those at positive F . The only slight difference is that thenear resonance region is wider than that for the DQDs at positive F because of the largermagnitude of tunneling coupling for electrons [28, 29].In summary, this study discusses the effects of tunnel coupling on photon emission fromspin excitons in vertically stacked double quantum dots. Results show that an increase in theoptically active volume and electric charge deconcentration caused by simultaneous electronand hole transfers between dots significantly inhibits the optical fine structure splittingof coupled QDs in the strong coupling regime without any decrease in optical oscillationstrength. This tunneling-driven FSS reduction is robust against the bias-controlled detuningfrom resonance, making strongly coupled vertical quantum dot molecules better cascadedecay sources of entangled photon pairs than single dots.The authors would like to thank the National Science Council of Taiwan for finan-cially supporting this research under Contract No. NSC-95-2112-M-009-033-MY3. Wen-HaoChang (NCTU) is appreciated for his valuable discussions. [1] D. T. McClure et al. , Phys. Rev. Lett. , 056801 (2007).[2] D. J. Reilly et al. , Phys. Rev. Lett. , 236803 (2008).
3] L. Robledo et al. , Science , 772 (2008).[4] Till Andlauer et al. , Phys. Rev. B , 045307 (2009).[5] A. S. Bracker et al. , Semicond. Sci. Technol. , 114004 (2008) and references there.[6] C. Santori et al. , Phys. Rev. B , 045308 (2002).[7] N. Akopian et al. , Phys. Rev. Let. , 130501 (2006).[8] L. He et al. , Phys. Rev. Lett. , 157405 (2008).[9] R. J. Young et al. , Phys. Rev. Lett. , 030406 (2009).[10] W. Langbein et al. , Phys. Rev. B , 161301(R) (2004).[11] R. M. Stevenson et al. , Nature , 179 (2006).[12] M. Reimer et al. , Phys. Rev. B , 195301 (2008).[13] A. Greilich et al. , Phys. Rev. B , 045323 (2006).[14] A. I. Tartakovskii et al. , Phys. Rev. B 70, 193303 (2004).[15] B. Gerardot et al. , Appl. Phys. Lett. , 041101 (2007).[16] K. Kowalika et al. , Appl. Phys. Lett. , 183104 (2007).[17] W. H. Chang et al. , Phys. Rev. B , 245314 (2008).[18] E. Poem et al. , Phys. Rev. B , 235304 (2007).[19] M. Glazov et al. , Phys. Rev. B , 193313 (2007).[20] J. L. Zhu et al. , Appl. Phys. Lett. , 261119 (2007).[21] H. Krenner et al. , Phys. Rev. Lett. , 057402 (2005).[22] J. I. Climente et al. , Phys. Rev. B , 115323 (2008).[23] M. F. Doty et al. , Phys. Rev. Lett. , 047401 (2009).[24] Parameters for InGaAs/GaAs QDs were used, following Poem et al. l x = 5 . l y = 5nm and l z = 1 . L x = 20 . L y = 20nm and L z = 3nm, respectively).[25] E. A. Stinaff et al. , Science , 636 (2006).[26] R. Hafenbrak et al. , New J. Phys. , 315 (2007).[27] A. J. Hudson et al. , Phys. Rev. Lett. , 266802 (2007).[28] M. Scheibner et al. , Phys. Rev. Lett. , 197402 (2007).[29] B. Szafran et al. , Phys. Rev. B , 115441 (2008). y L z z d e h edF (a) (b)(c) ∣ L L 〉 ∣ L L 〉 ∣ R R 〉 ∣ R R 〉∣ R L 〉∣ R L 〉∣ L R 〉∣ L R 〉 e h t h t e E F F ≠ F = FIG. 1: (Color online) Schematic diagrams of (a) a double QD structure and (b) spin excitonconfigurations. (c) The calculated hopping parameters, t e blue (dark) and t h green (light), vs.interdot distance d . Horizontal dashed lines: the values of ∆ e and ∆ h considered throughout thiswork. a)(b) E E ∣ x / y ; 〉~ - ∣ x , 〉∣ y , 〉 + E edF / V eh h edF / V eh h ~ t h ∣ x / y ; 〉~ + + ∣ vac 〉 y x E d=8.5nmd=4.5nm FIG. 2: (Color online) Calculated energy spectra vs. bias field F of (a) a weakly coupled DQD with d = 8 . d = 4 . b) E (a) F = kV / cm F = kV / cm F = kV / cm F F x − pol. y − pol. meV meV d=8.5nm d=4.5nm E FIG. 3: (Color online) (a) Calculated polarized PL spectra of a weakly coupled DQD with d =8 . F at near resonance at T = 10 K . The inset: the normalized FSS∆ E / ∆ E SD (orange solid line) and intensity I /I SD (purple dashed line) of the main PL spectrallines as functions of F , where ∆ E SD and I SD denotes the FSS and intensity of the main PL lineof a single dot. The considered biases in the calculated PL spectra are indicated with verticalarrows in the inset. (b) The calculated results same as (a) but for a strongly coupled DQD with d = 4 . C WC SC WC d=4.5nm d=8.5nm d=4.5nm d=8.5nm I /I SD ΔE /ΔE SD FIG. 4: (Color online) Normalized intensity I /I SD (left) and FSS ∆ E / ∆ E SD (right) of the lowestPL spectral lines of coupled DQDs, as functions of inter-dot distance d and bias field F . The dashedline boxes highlight the reduced ∆ E and increased I of the strongly coupled DQDs. The verticaldotted lines indicate d = 4 . d = 8 .5nm for which Figs. 2 and 3 are calculated. The reddashed line in the upper (lower) half plane indicates the hole (electron) resonances.