Tunneling Effects on Fine-Structure Splitting in Quantum-Dot Molecules
Hanz Y. Ramı´rez and Shun-Jen Cheng*Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan, Republic of China (Received 17 March 2009; published 18 May 2010)
We theoretically study the effects of bias-controlled interdot tunneling in vertically coupled quantum dots on the emission properties of spin excitons in various bias-controlled tunneling regimes. As a main result, we predict substantial reduction of optical fine-structure splitting without any drop in the optical oscillator strength for the coupled dots with high tunneling rates. This special reduction diminishes the distinguishability of polarized decay paths in cascade emission processes suggesting the use of stacked quantum-dot molecules as entangled photon-pair sources.
DOI:10.1103/PhysRevLett.104.206402 PACS numbers: 71.45.Gm, 03.67.Bg, 78.55.Cr, 78.67.Hc
Tunneling is a remarkable quantum property of micro-scopic particles that has no classical counterpart, which allows coupling between two objects spatially separated by a finite potential barrier. Recent examples of tunnel effects in coupled quantum-dot (QD) systems include the tunabil-ity of fluctuations in Kondo currents [1], reduction of electronic spin decoherence by interaction with nuclear spin [2], conditional dynamics of transitions [3], and bias control of g tensors [4].
Currently, a highly desirable feature of QD-based pho-ton emitters is the reduced fine-structure splitting (FSS) between the intermediate one-exciton spin states [5]. The FSSs (typically 101-102 eV, greater than the intrinsic broadening of emission line 1 eV) make the two pos-sible decay paths in biexciton cascade processes energeti-cally distinguishable, and have become a main obstacle in the production of polarization-entangled photon pairs from QDs [6–10]. Researchers have recently demonstrated sig-nificant reductions in the FSSs of single QDs using strain and postannealing techniques, and the application of elec-tric and magnetic fields [11–13]. In most experiments, however, it is not clear if the reduction of FSS is caused by the undoing of symmetry breaking or the reduction of e-h wave function overlap. The latter effect reduces not only the FSS but also the oscillator strength of e-h recom-bination, yielding narrow intrinsic radiative broadening and actually inhibiting the generation of entangled photon pairs [14,15].
In this Letter, we theoretically examine the effects of quantum tunneling in vertical QD molecules on the optical fine-structure properties by using the (partial) configura-tion interacconfigura-tion (PCI) method. Remarkably, we predict a significant reduction of the optical FSSs in coupled double quantum dots (DQDs) with high tunneling rates, without any decrease in the optical oscillator strength.
Let us consider a pair of vertically stacked quantum dots along the growth z axis, separated by an interdot distance d and subject to an applied electric field F, as shown in Fig. 1(a) [16]. The e-h Hamiltonian for a single spin exciton in a coupled double QD is written as
H ¼X j; ð"e jþ eFzjÞc y jcjþ X n; ð"h n eFznÞh y nhn X j2L;k2R; te jkðc y jckþ c y kcjÞ X n2L;m2R; th nmðh y nhmþ h y mhnÞ X kmnj; Ve-h kmnjc y kh y mhncj þ X kmnj;00 Ve-hexch k;m;n0;j0c y kh y mhn0cj0; (1)
where the composite indexes j, k (n, m) denote the electron (valence hole) orbitals and dot positions [L ðRÞ for the left (right) dot], ¼" or # ( ¼* or +) represents electron (hole) spin with sz¼12 or 12 (jz¼32 or 32), cyj and cj (hyn and hn) are the electron (hole) creation and
annihilation operators, respectively, "ei ("hn) is the kinetic
energy of an electron (a valence hole), e is the unit charge, and zj2L¼ 0 (zj2R¼ d) is the z position of the left (right)
Energy(meV) 40 30 20 10 5 6 7 8 9 10d(nm) (a) (b) (c)
FIG. 1 (color online). Schematic diagrams of (a) a double QD structure and (b) spin-exciton configurations. (c) The calculated hopping parameters, te(blue) and th(green), vs interdot distance
d. Horizontal dashed lines: the values ofeandhconsidered
throughout this work.
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dot. Here, the valence hole orbitals of the highly quantized strained dots are assumed to be purely heavy-hole-like. The terms with the hopping parameters (te
jk, thnm) describe
the (spin-conserved) carrier tunneling between adjacent dots. The matrix elements of conventional e-h Coulomb interaction and the e-h exchange interactions are Ve-h
kmnj RR d3r1d3r2e k ð~r1Þhmð~r2Þðe2=4r12Þhnð~r2Þejð~r1Þ and Vk;m;ne-hexch 0;j0 RR d3r1d3r2ek ð~r1Þucð~r1Þhmð~r1Þuvð~r1Þ ðe2=4r 12Þhn ð~r2Þuv0ð~r2Þ e jð~r2Þuc0ð~r2Þ, respectively,
where are single-particle envelope wave functions,
uc (uv) are the electron (hole) Bloch functions, is
the dielectric constant and r12j ~r1 ~r2 j . Within the dipole-dipole approximation, the long-range part of the e-h exchange interaction is given by Vkmnje-hexchðLRÞ kmnj1 ½3e2@2Ep= ð4Þ4m0E2g RR d3~r1d3~r2e k ð~r1Þ h mð~r1Þhn ð~r2Þejð~r2Þ½ðy1 y2Þ2 ðx1 x2Þ2 þ 2iðx1
x2Þðy1y2Þ=ðr12Þ5, where Ep is the conduction-valence
band interaction energy [17], Egthe band gap energy, and
m0the mass of a free electron [18]. The energy spectrum of an interacting exciton in a DQD can be calculated by direct diagonalization of Eq. (1) in the basis of exciton configu-rations constructed from the all S and P orbitals of indi-vidual dots [19].
Besides the fully numerical approach, the following analysis is performed using a simplified ‘‘rigid orbital’’ model to improve physical understanding. Based on the lowest single-particle orbitals of single dots, eight spin-exciton configurations are constructed, as displayed in Fig. 1(b). To analyze further the (linear) polarization of emitted light, a new basis is defined by the linear trans-formation of the configurations according to the parity symmetry: jLLi p1ffiffi2ðjL " L +i jL # L *iÞ, jRRi
1ffiffi 2 p ðjR " R +i jR # R *iÞ, jLRi 1ffiffi 2 p ðjL" R +i jL# R *iÞ, jRLi 1ffiffi 2 p ðjR " L +i jR # L *iÞ. Notably,
only the configurations with positive (negative) parity are associated with a x- (y-)polarized light emission. In the basis ordered by jLLi, jRRi, jLRi, jRLi, the de-coupled4 4 Hamiltonian matrix is
^H¼ Ve-h DD II th te II Ve-hþ eþ h DD te th th te eFd þ h II II te th II eFd þe II 0 B B @ 1 C C A; (2)
where the kinetic energy offset "e
Lþ "hL is removed for
brevity, ðeÞ h "ðeÞ hR " ðeÞ h
L denotes the difference
be-tween kinetic energies of the two adjacent dots due to the inevitable slight differences in size, shape, or chemical composition, Ve-h VLLLLe-h ¼ VRRRRe-h denotes the direct
Coulomb interaction between an e-h pair in the same single dot, and DD RRRR
1 ¼ LLLL1 (II LRRL1 ¼
RLLR
1 LLRR1 ¼ RRLL1 ) is the long-range e-h
ex-change interaction in a direct exciton (an indirect exci-ton). Within the three-dimensional parabolic model for the confining potentials of single QDs, the single-particle wave functions of the lowest orbitals of single dots can be described by L=Rðx; y; zÞ ¼ ð3=2lxlylzÞ1=2 expf1 2½ðlxxÞ 2þ ðy lyÞ 2 þ ðzzL=R lz Þ 2g, characterized by the
wave function extents l¼x;y;z. Accordingly, we derive
Ve-h e2 41lf½
ffiffiffi 2 p
sin1ðpffiffiffiffiffiffiffiffiffiffiffiffiffi1a2Þ=½ ffiffiffiffipðpffiffiffiffiffiffiffiffiffiffiffiffiffi1a2Þg, DD ¼ fð3pffiffiffiffie2@2EpÞ = ½ð4Þ16pffiffiffi2m0E2gg½ðly lxÞ = l2xl2y eð3pffiffiffi=4Þ2ðlz=lyÞ2erfc½ð3pffiffiffiffi=4Þðl z=lyÞ, [18,20,21] and II ¼ DDeðd 2=2l2
zÞ, for a slightly deformed DQD ( lylx
ly
1 Þ 0, where l ðlxþ lyÞ=2, a lz=l). The parameters te
are evaluated using a standard exponential model [22] and fitted to the results of pseudopotential calculations in Ref. [23]. For the evaluation of th, we follow the
formula-tion in Ref. [24] derived from the four band Luttinger-Kohn model with the consideration of spin-orbit coupling. The energy spectrum fEx;ig (fEy;ig) of the exciton
states jx; ii (jy; ii) as the initial states of the x- (y
)-polarized light emission is calculated by diagonalizing Hþ
(H) in Eq. (2) [25]. In the combined energy spectrum,
fEx;i; Ey;ig, each level is a doublet of the spin-exciton
states, jx; ii and jy; ii, which are split by an FSS Ei
Ey;i Ex;i [inset of Fig. 2(a)]. The x- (y
)-linear-polarized photoluminescence (PL) spectra are ob-tained using Fermi’s golden rule: IxðyÞð!Þ ¼
P
iFðEi; TÞjh0jPxðyÞjxðyÞ; iij2ðExðyÞ;i @!Þ, where
the subscript i denotes initial states of the PL tran-sition, ! is the frequency of the emitted photon, the
F V F V
(a)
(b) (d)
(c)
FIG. 2 (color online). Calculated energy spectra vs bias field F of (a) a DQD with d ¼8:5 nm and (b) a DQD with d ¼ 4:5 nm. Straight dashed lines describe the energy spectrum of a de-coupled DQD. (c) and (d) are magnified fine structures of the corresponding energy spectra (a) and (b), at near resonance, and the (schematic) configuration intermixings of the lowest exciton states.
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operator PðÞx ¼ Pn;jSn;jðhn*cj# þ hn+cj"Þ [P ðÞ y ¼
iPn;jSn;jðhn*cj# hn+cj"Þ] describes the all possible e-h recombinations that produce the x½y linear
polar-ized PL, Sn;j ¼Rd3rh
n ð~rÞejð~rÞ is the e- and h-wave
function overlap, and FðEi; TÞ ¼expðEi=kBTÞ=
½PlexpðEl=kBTÞ is the probability of occupation of
state jii, where kB is the Boltzmann constant and T is temperature.
Figure 2(a) shows the calculated energy spectra of a coupled DQD with d ¼8:5 nm under various applied biases. Under the weak tunneling (WT) condition (defined by te ðhÞ=e ðhÞ 1), the 4 4 Hamiltonian matrix, Eq. (2),
can be decomposed into two2 2 blocks that are coupled only by relatively weak electron hopping. Thus, the Hamiltonian matrix for the two lowest spin-exciton states can be approximated as the following2 2 block:
^HWT
¼ Ve-th DD th
h eFd þ h DD
; (3)
with respect to the basis jLLi and jLRi. Equation (3) is actually equivalent to the widely used solvable three-orbital model for DQDs [26]. The eigenstates of Eq. (3) are superpositions of the optically active exciton configu-ration jLLi and the inactive configuconfigu-ration jLRi, deter-mined by the bias-controlled detuning from resonance (jedF ðhþ Ve-hÞj). Expanding the exciton eigenstates in the used basis for Eq. (2), i.e., jx; ii ¼PnjCx
nj;ijnjþi
and jy; ii ¼PnjCynj;ijnji, the intensities and the FSS associated with the lowest spectral lines are given by I1 FðE1; TÞðCLL;1SDþ CLR;1SIÞ2 and E1 2ðC2LL;1DDþ C2LR;1IIÞ, where CLL;1 Cx
LL;1¼ CyLL;1(CLR;1 CxLR;1 ¼
CyLR;1) are the expansion coefficients associated with the bright (dark) exciton configurations jLLi (jLRi) and SD SLL ¼ SRR 1 (SI SLR¼ SRL ¼ eðd
2=4l2 zÞ) is
the e-h wave function overlap in a direct-exciton (an indirect-exciton) configuration. Accordingly, both the I1 and the E1 of a weakly spatially coupled DQD in the weak tunneling regime (SI SD and II DD) are
mainly proportional to C2LL;1 and should depend similarly on applied bias fields. Figure3presents the calculated F dependences of the I1 andE1, and selected polarized PL spectra of the DQD in the fields near resonance, obtained using the PCI method. The results obtained using the model Eq. (2) are also presented in Fig.3(a)for compari-son. Both sets of results show similar features that are consistent with the analysis presented above. The only remarkable difference is that the magnitude of the reso-nance field obtained using the PCI method is smaller than that from the simple model because of the reduced indirect-exciton energy by the interdot Coulomb attraction. At very low bias (jedF=ðVe-hþ hÞj 1), the ground
states of the exciton are jx; 1i jLLþi and jy; 1i
jLLi. The intensity (FSS) of the corresponding linear polarized emission lines is I1 ðSDÞ2 (E1 2DD),
approaching the value of the intensity (FSS) of the lowest
spectral lines of a single dot, ISD (ESD). At near reso-nance (edF=ðVe-hþhÞ 1), where jx;1i p1ffiffi2ðjLLþi
jLRþiÞ and jy; 1i 1ffiffi 2
p ðjLLi jLRiÞ, only the hole
in the exciton can be transferred between dots while the electron is stably localized in the left dot. The intensity (FSS) of the corresponding polarized emission lines is I1 ðSDþ SIÞ2=2 (E
1 DDþ II), which is only about
50% of that for a single dot. The resonant interdot tunnel-ing of a stunnel-ingle hole significantly reduces the overlap of the electron and hole wave functions, leading to not only the decrease in the optical FSS but also the oscillator strength of an e-h recombination. The decreased oscillator strength of e-h recombination reduces the intrinsic broadening width of the main exciton lines. Thus, such a FSS reduction does not support the feasibility of the dot-based entangled photon-pair source devices [27,28].
Figure2(b)shows the energy spectra of a coupled DQD with smaller d ¼4:5 nm. Figure3(c)plots the normalized I1andE1of the lowest spectral lines vs F. Generally, the strongly coupled DQDs have smaller FSSE1but larger I1 than single dots or weakly coupled dot molecules, since small interdot distance makes te ðhÞgreater thane ðhÞ and the interdot tunneling more likely. In the strong tunneling (ST) limit (t ), both electrons and holes can be transferred between dots over a very wide range of detun-ing. Thus, Eq. (2) can be approximated to
^HST 0 0 th te 0 0 te th th te 0 0 te th 0 0 0 B B B @ 1 C C C A: (4)
The lowest eigenstates for Eq. (4) are 12ðjLLi þ jRRi þ jLRi þ jRLiÞ, highly intermixing all exciton
edF/(Veh+∆h) edF/(Veh+∆h)
FIG. 3 (color online). (a) Calculated normalized FSS E1=ESD [light (magenta) lines] and intensity I1=ISD [dark
(green) lines] of the main PL spectral lines of the DQD with d ¼ 8:5 nm as functions of F, where ESD(ISD) denotes the FSS (PL
intensity) for a single dot. Solid (dashed) lines show the results calculated by using the PCI method [the simple model of Eq. (2)]. (b) Selected polarized PL spectra for the considered DQD in (a) under the fields Fi around resonance at T ¼10 K.
(c) and (d) The same calculated results as (a) and (b) but for the DQD with short interdot distance d ¼4:5 nm.
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configurations. This indicates that both kinds of particles, electrons and holes, in the eigenstates are likely delocal-ized and can be simultaneously transferred between the coupled dots. Accordingly, we have I1 ðSDþ SIÞ2 and E1 DDþ 2II, i.e., that the FSS is only about
one-half of the magnitude of ESD but the intensity of the polarized emission lines is slightly larger than ISD. In the ST regime, not only valence holes but also electrons are spread over the two coupled dots. The simultaneous e and h resonant transfer between dots enlarges the optically active volume and increases the mean distance hr12i in the long-ranged e-h exchange interactions, resulting in the larger I1 and smallerE1. In Fig.3(c), the PCI results reveal a similar feature but smaller magnitudes of I and FSS than the model Eq. (2). This is because the compo-nents of indirect exciton, which are mixed in the radiative exciton states, are increased by the interdot Coulomb attraction that is associated with both Coulomb direct and correlation interactions.
Figure4plots the normalized I1 and E1 (by ISD and ESD) of DQDs versus the d and F, obtained from the PCI
calculation. In the WT regime, as discussed previously, I1 andE1 depend similarly on F. As a DQD is driven into the ST regime, I1 are markedly increased and the FSS is reduced to only 50% of ESD (see the regions high-lighted by dashed-line boxes) [16]. This finding suggests that in a dot ensemble the number of useful dots with sufficiently small FSSs that are suitable for fabricating devices can be roughly doubled if such devices are made of double QD structures. The increased I1 and reduced E1are robust against the detuning, being almost
insensi-tive to F. The values ofe ¼ 30 meV and h¼ 10 meV, are estimated for a pair of dots with around 1–2 monolayer difference in height. Whether double QDs can be easily prepared in the ST regime, and the strong tunneling effect can be subsequently exploited, depends on the ratios te=e 1 and th=h 1. Fabricating more similar dots
can increase the ratios te=e and th=h [29], and makes the device more efficient.
In summary, this study discusses the effects of quantum tunneling on polarized photon emission from spin excitons in vertically stacked double quantum dots. Results show that an increase in the optically active volume and charge delocalization via quantum tunneling inhibits the optical FSS of coupled QDs in the strong tunneling regime without any decrease in the optical oscillation strength. This tunneling-driven FSS reduction makes strongly coupled vertical quantum-dot molecules better sources of entangled photon pairs than single dots.
The authors would like to thank the NSC of Taiwan for support. Wen-Hao Chang (NCTU) is appreciated for his valuable discussions.
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Lett. 89, 233 110 (2006). FIG. 4 (color online). Normalized intensity I1=ISD (left) and
FSSE1=ESD(right) of the lowest PL spectral lines of coupled DQDs, as functions of d and F, obtained from the PCI calcu-lation. The red dashed line boxes highlight the feature of reduced E1 and increased I1 of the DQDs with short d. The vertical
dotted lines indicate d ¼4:5 nm and d ¼ 8:5 nm for which Figs. 2and 3are calculated. The magenta dash-dotted line in the upper (lower) half plane indicates the hole (electron) reso-nance.
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