Probing excitons and biexcitons in coupled quantum dots by coherent ...

PHYSICAL REVIEW B 79, 155324 共2009兲

Probing excitons and biexcitons in coupled quantum dots by coherent two-dimensional optical spectroscopy E. G. Kavousanaki, O. Roslyak, and S. Mukamel Department of Chemistry, University of California, Irvine, California 92617, USA 共Received 18 December 2008; published 29 April 2009兲 We calculate two-dimensional 共2D兲 photon-echo and double-quantum-coherence spectra of two coupled InGaAs/GaAs quantum dots at various distances, taking into account electron, hole, and exciton hopping. Signatures of direct and indirect excitons in two-exciton resonances are revealed. At short distances, electron delocalization contributes to the creation of new biexcitonic peaks, and dipole-dipole interactions shift the two-exciton energies. DOI: 10.1103/PhysRevB.79.155324

PACS number共s兲: 78.47.Fg, 78.67.Hc

I. INTRODUCTION

The optical response of confined excitons in semiconductor quantum dots 共QDs兲 has been studied extensively.1–3 The ability to control their electronic properties makes them ideal candidates for studying fundamental many-body effects.4–6 In addition, they are promising candidates for numerous applications, including fluorescence labels of biomolecules,7 lasers,8 solar cells,9 and quantum computing.10 Arrays of QDs were proposed as building blocks in quantum information applications, e.g., as a quantum register for noiseless information encoding.11 Biexcitons have been suggested as a source of entangled photons.12 Much effort has been devoted to the investigation of the coupling between the dots, either due to exciton13,14 or carrier migration.15–17 However, both coupling mechanisms may coexist, and separating them is of considerable interest. Two-dimensional 共2D兲 spectroscopy provides a new tool for studying coupled excitons in molecular aggregates,18 photosynthetic complexes,19 semiconductor quantum wells,20 and QDs.17,21 In these experiments, the system is subjected to three temporally well separated femtosecond pulses propagating in the directions k1, k2, and k3 and centered at times ␶1, ␶2, and ␶3 共Fig. 1兲. 2D signals of two coupled quantum wells were simulated recently using a free-carrier model, neglecting Coulomb interactions and many-body effects such as biexcitons.22 In this paper, we shall calculate these signals for a system of two coupled quantum dots. This QD molecule is described by a tight-binding two-band Hamiltonian for electrons and holes,23 taking into account interdot electron and hole hoping along with monopole-monopole and dipole-dipole Coulomb interactions. We use realistic parameters for the system studied in Refs. 16 and 24. The Hamiltonian is recast in terms of e-h pair variables, truncated at the two-exciton manifold.25 We consider two types of heterodyne detected signals,26 SI and SIII, generated in the phasematching directions kI = −k1 + k2 + k3 共photon echo兲, and kIII = + k1 + k2 − k3 共double-quantum coherence兲, respectively. Both are recorded as a function of time delays t1 = ␶2 − ␶1, t2 = ␶3 − ␶2, and t3 = t − ␶3, where t is the detection time. 2D spectra, obtained by a Fourier transform of SI with respect to t1 and t3, SI共⍀3 , t2 , ⍀1兲, and of SIII with respect to t2 and t3, SIII共⍀3 , ⍀2 , t1兲, reveal exciton correlations and two-exciton resonances.27 The signals are obtained by solving the nonlin1098-0121/2009/79共15兲/155324共10兲

ear exciton equations 共NEE兲,27,28 which account for excitonexciton interactions and their quasibosonic nature.17,29 To classify biexciton states in terms of their single-exciton constituents, we analyze both the SI and SIII signals using the corresponding sum-over-states expressions 共SOS兲.27 The roles of different coupling mechanisms at various interdot distances d is discussed. We show that at short distances electron delocalization contributes to the creation of new biexcitonic peaks in the spectra, while exciton hopping shifts the two-exciton peaks. In Sec. II we present our model Hamiltonian for two coupled quantum dots. In Sec. III we discuss the variation in the absorption spectra and single-exciton eigenstates with interdot distance. In Sec. IV we present the photon echo and double-quantum coherence spectra and identify the biexcitonic contributions. Summary and conclusions are given in Sec. V. II. MODEL HAMILTONIAN FOR TWO COUPLED QUANTUM DOTS

We consider a system of two vertically stacked, lensshaped InGaAs/GaAs QDs of height 2 nm and radius 10 nm aligned along the z axis.24 It is described by the tight-binding two-band Hamiltonian,23,30 共1兲

HQD = H0 + HC .

Here H0 represents noninteracting electrons and holes, and HC are the Coulomb interactions. Due to quantum confinek1 1 1 1 1 1 1 1 1 1 1 1 1

k2 k3

1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1

ks

t3 t2 t τ3

1 1 1 1 1 1 1 1 1 1 1 1

t1 τ2

τ1

FIG. 1. The pulse sequence in a four-wave-mixing experiment: the system is excited by three pulses propagating at directions k1, k2, and k3 and centered at times ␶1, ␶2, and ␶3, respectively. The signal is heterodyne detected in the phase-matching direction ks at time t.

155324-1

©2009 The American Physical Society


KAVOUSANAKI, ROSLYAK, AND MUKAMEL

ment, the conduction and valence bands split into discrete atomiclike levels. Each dot in InGaAs has two spindegenerate electron states, with angular-momentum quantum numbers 共J , M兲 = 共 21 , ⫾ 21 兲, and two heavy-hole states 共 23 , ⫾ 23 兲. The free-carrier Hamiltonian has the form,17,28 H0 =

兺 tm共1兲n cˆm† cˆn + m兺n tm共2兲n dˆm† dˆn m n 1 1

1

1

2 2

2

2

共2兲

2 2

1 1

where the indices 1 共2兲 correspond to electrons 共holes兲 in the conduction 共valence兲 band. The subscripts, e.g., m1 = 共Rm1 , ␴m1兲, describe the QD position R = 兵T , B其共“T” for top and “B” for bottom兲 and the spin projection ␴ of the carrier 共↑ , ↓ for up or down-spin projection, respectively兲. The creation and annihilation operators of an electron 共hole兲 in site † and cˆm1 Rm1共Rm2兲 with spin z component ␴m1共␴m2兲 are cˆm 1 † ˆ ˆ 共d and d 兲. These satisfy the Fermionic algebra, m2

m2

兵cˆm1,cˆn† 其 ⬅ cˆm1cˆn† + cˆn† cˆm1 = ␦m1n1 ,

共3兲

兵dˆm2,dˆn† 其 = ␦m2n2 .

共4兲

1

1

1

2

All other anticommutators are zero. The diagonal elements of t共1兲 and t共2兲 give the electron and hole energies, while off-diagonal elements represent carrier hopping in the conduction and valence bands, respectively. The Coulomb interaction is HC =

1 1 兺 Vee cˆ† cˆ† cˆn cˆm + 兺 Vhh dˆ† dˆ† dˆn dˆm 2 m1n1 m1n1 m1 n1 1 1 2 m2n2 m2n2 m2 n2 2 2 −

兺 Vmeh n cˆm† dˆn† dˆn cˆm

m1n2

+

1 2

1

兺

m1m2n1n2

2

2

F Vm

1

cˆ† dˆ† dˆ cˆ .

1m2,n1n2 m1 m2 n2 n1

共5兲

TABLE I. Hamiltonian parameters for a system of two vertically stacked quantum dots, labeled as T 共top兲 and B 共bottom兲, obtained from Ref. 24: electron and hole on-site energies 共Ee, Eh兲, tunneling couplings 共te, th兲, as well as e-h Coulomb interaction elements 共Veh兲 vs interdot distance. Parameter 共meV兲

Distance dependence 共d in nm兲

EeT EeB EhT EhB te th Veh BB Veh TT Veh BT Veh TB

1450− 436d−1 + 3586d−2 − 7382d−3 1449− 452d−1 + 3580d−2 − 6473d−3 167+ 129d−1 − 2281d−2 + 6582d−3 163+ 274d−1 − 3780d−2 + 9985d−3 −255 exp共−d / 2.15兲 −4.25 exp共−d / 3.64兲 −29.0+ 7.98/ d −29.6+ 19.6/ d −99.1/ 冑d2 + 3.722 −98.5/ 冑d2 + 4.212

F Vm

1m2,n1n2

=

1 3 ⑀rmn

−

冋

␮ m1m2 · ␮ n1n2

3共␮m1m2 · rmn兲共␮n1n2 · rmn兲 2 rmn

,

共6兲

where ⑀ is the dielectric constant, ␮m1m2 is the interband dipole moment at site Rm1 = Rm2 ⬅ Rm, and rmn = 兩Rm − Rn兩 is the distance between sites m and n. Equation 共6兲 has been shown to be adequate for direct-gap semiconductor quantum dots of radius 0.5–2 nm even when they are almost in contact.13,32 The interaction with the optical field in the rotating wave approximation is described by the Hamiltonian, ˆ † + H.c.兴, Hint共t兲 = − 关E共t兲 · V

Rm =Rm ⫽Rn =Rn 1 2 1 2

The first three terms are monopole-monopole contributions of electron-electron, hole-hole, and electron-hole interactions, respectively. Pseudopotential calculations, including strain and realistic band structure, have been performed for this system.16,24 The on-site energies, hopping parameters, and Coulomb interaction energies have been reported vs interdot distance. Electrons tunnel at short 共ⱗ8 nm兲 distances creating delocalized bonding and antibonding one-particle states. The heavy holes however remain localized, even at short d共ⱗ8 nm兲, but their energies are lowered with decreasing distance. This is due to the high interdot barrier for the heavy holes, which suppresses hole tunneling 共the heavyhole-electron effective-mass ratio is mhh / me ⬇ 0.4/ 0.06⬇ 6兲, as well as to the effect of strain and band structure. We will use the parameters from Ref. 24, as listed in Table I, and assume Vee = Vhh = −Veh. The last term in Eq. 共5兲 describes the electrostatic dipoledipole interactions between the charge distributions in the QDs which induce exciton hopping,13,14,31

册

共7兲

ˆ =兺 ˆ ˆ is the dipole moment annihilawhere V m1m2␮ m1m2d m2c m1 tion operator, and E共t兲 is the negative frequency part of the optical field. The interband dipole moment ␮m1m2 at site Rm1 = Rm2 is given by17,33

␮↑↑ =

␮

冑2 共x + iy兲 ˆ

ˆ

␮↓↓ =

␮

冑2 共x − iy兲, ˆ

ˆ

共8兲

where xˆ and yˆ are the polarization directions of the optical pulses. The measured dipole moments in InGaAs quantum dots are ␮ = 25– 35 D.34 The total Hamiltonian for the QD molecule-light system is H = HQD + Hint共t兲.

共9兲

Using the method proposed by Chernyak and Mukamel,25 Hamiltonian 共9兲 can be transformed into the excitonic representation by introducing the electron-hole pair operators,

155324-2


III. SINGLE-EXCITON MANIFOLD AND THE ABSORPTION SPECTRUM

The absorption spectrum was calculated by a Fourier transform of the linear response obtained by Eq. 共B7兲. It is given by26

冋兺

A共␻兲 ⬀ Im

e

册

兩␮eg兩2 , ␻ − Ee + i␥eg

共11兲

where e denotes single-exciton states with energy Ee and dephasing rate ␥eg. We set ␥ = 0.05 meV, obtained from the measured linewidths in a similar system.15 The singleexciton block of the Hamiltonian has four fourfold spindegenerate eigenstates. Since spin does not affect the absorption, we will drop it here, but include it in the nonlinear response in Sec. IV, since two-exciton states may be formed by single excitons of different spins. The single-exciton eigenstates 兩␣典, 兩␤典, 兩␥典, and 兩␦典 are expanded in the basis 共Fig. 2兲, 兩a典 = cˆB†dˆB†兩g典

兩c典 = cˆB†dˆT†兩g典

兩b典 = cˆT†dˆT†兩g典

兩d典 = cˆT†dˆB†兩g典

共12兲

where B 共T兲 denotes the bottom 共top兲 QD, and 兩g典 is the ground state. 兩a典 and 兩b典 describe direct excitons, while 兩c典 and 兩d典 are indirect excitons. The variation of the simulated absorption spectra with interdot distance d is shown in Fig. 3. At large distances 共d = 17 nm兲, the two QDs are uncoupled and indirect excitons are optically forbidden. Because of QD slight asymmetry 共see Table I兲, there are two peaks corresponding to direct excitons 兩␣典 = 兩a典 and 兩␤典 = 兩b典. As the distance is decreased, the absorption peaks are blueshifted and their splitting is reduced. This is due to the decreasing hole energies, as well as to the electron tunneling, which leads to carrier delocalization. As shown in Fig. 2, exciton splitting is minimized at d = 8.6 nm, where the single-exciton eigenstates are approximately given by 兩␣典 ⬇ 兩a典 + 兩b典

兩␥典 ⬇ 兩c典,

兩␤典 ⬇ 兩a典 − 兩b典

兩␦典 ⬇ 兩d典,

d=11.3 nm

The main steps of this transformation are given in Appendix A. The transformed Hamiltonian will be used in Secs. III and IV to study the one and two-exciton properties.

d=10.2 nm

共10兲

d=9.8 nm

2

d=8.6 nm

1

d=7.9 nm

Bˆm1m2 = dˆm2cˆm1 .

† ˆ† = cˆm dm ,

d=6.8 nm

1m2

d=5.5 nm

† Bˆm

d=17.0 nm

PROBING EXCITONS AND BIEXCITONS IN COUPLED…

FIG. 2. 共Color online兲 The four single-exciton eigenstates 兩␣典, 兩␤典, 兩␥典, 兩␦典 vs interdot distance, expanded in the e-h basis 兩a典共both particles in the bottom QD兲, 兩b典共both particles in the top QD兲, 兩c典共electron in the bottom and hole in the top QD兲, and 兩d典共electron in the top and hole in the bottom QD兲.

兩␥典 ⬇ 兩b典 − 兩c典,

兩␤典 ⬇ 兩a典 + 兩d典

兩␦典 ⬇ 兩a典 − 兩d典,

共14兲

with energies E␣ ⬍ E␤ ⬍ E␥ ⬍ E␦. Thus, the electron is delocalized but the hole is not. This is attributed to the smaller (α)

(β) (α)

共13兲

d = 17.0 nm

(β)

(α)

with energies E␣ ⬇ E␤ ⬍ E␥ ⬍ E␦. The QD-localized excitons at large d now become strongly entangled bonding and antibonding excitons. At the same time, indirect excitons appear. These are blueshifted since their binding energy is smaller than for the direct excitons, due to the reduced Coulomb attraction. At shorter distances, the two lower excitons anticross, while the contribution of the two upper ones in the absorption spectrum become stronger. The exciton eigenstates now form bonding and antibonding states,

兩␣典 ⬇ 兩b典 + 兩c典

(β)

(α)

(β) (β)

1.24

(δ)

(γ) (β)

(α)

d = 5.5 nm

d = 9.8 nm

(δ)

(γ)

(α)

d = 6.8 nm

d = 10.2 nm

(δ)

(γ)

(α)

d = 7.9 nm

1.23

(β)

d = 11.3 nm

(δ)

(γ)

(α)

d = 8.6 nm

(δ)

(γ)

(γ)

(δ)

(β) 1.25

(γ)

1.26

1.27

1.28

1.29

(δ) 1.3

Energy (eV)

FIG. 3. 共Color online兲 Absorption spectra 共logarithmic scale兲 at various interdot distances d.

155324-3


KAVOUSANAKI, ROSLYAK, AND MUKAMEL (a)

(a)

ks

t t3 τ3 t2

g

t3

g

g

g

k2

g g

−k 1

g

ks

t t3 τ3 1.26

1.27

1.28

1.29

t2

1.3

Energy (eV)

τ2

FIG. 4. 共Color online兲 Absorption spectrum with 共a兲 and with no 共b兲 dipole-dipole interactions for the shortest interdot distance d = 5.5 nm.

τ1

t1

effective mass of the electron, as well as to the more complicated valence-band structure and the effect of strain, which favor hole localization.24 The effect of dipole-dipole interactions for the shortest distance, d = 5.5 nm, is shown in Fig. 4. The peak positions are the same, but their intensity is changed. The more pronounced effect of dipole-dipole interactions in the nonlinear response will be discussed in Sec. IV. IV. TWO-EXCITON MANIFOLD AND THE 2D SPECTRA

We focus on the two-exciton states and their energies E f . The optical response is calculated using the NEE 共Refs. 27 and 28兲 for the single- and two-particle variables 具Bˆm典 and 具BˆmBˆn典共Appendix B兲. The 2D spectra 关Eqs. 共C2兲 and 共C9兲兴 are calculated in terms of the single-exciton Green’s functions and the exciton scattering matrix27 共Appendix C兲. All simulations were carried out using the SPECTRON package.35 However, it is more convenient to analyze them using the alternative sum-over-states expressions17 SI␯4␯3␯2␯1共⍀3,t2 =

0,⍀1兲 = i 兺 ee⬘

⫻

␯

␮ge1⬘

⍀ 1 + E e⬘ + i ␥ e⬘g

冋兺 f

␯

␯2 ␯3 4 ␮eg ␮ fe ␮e⬘ f

⍀3 − E f + Ee⬘ + i␥ fe⬘ ␯

−

␯

␯2 ␯3 ␯4 ␮e 3g + ␮e 2g␮eg 兲␮ge 共␮eg

⬘

⬘

⍀3 − Ee + i␥eg

册

, 共15兲

␯4␯3␯2␯1 共⍀3,⍀2,t1 = 0兲 = i 兺 SIII

ee⬘ f

⫻

冋

␯1 ␯2 ␮eg ␮ fe ⍀2 − E f + i␥ fg ␯

⍀3 − E f + Ee⬘ + i␥ fe⬘ ␯

−

␯

␮ge4⬘␮e⬘3f ␯

␮ge3⬘␮e⬘4f

⍀ 3 − E e⬘ + i ␥ e⬘g

册

,

共16兲

τ2

g

e

g

e

e’

g

e’

g

g

k3

k3

k2

t1

(c)

1.25

t2

e’

τ1

(b)

g

τ3

τ2 t1

ks

t

e k3

(b)

g

τ1

e’

e’

f

e’

e

e’

g

e’

g

g

−k 1

k2 −k 1

FIG. 5. Feynman diagrams for the kI technique.

where g denotes the ground state, e and e⬘ are single excitons, and f is a doubly excited state. ␥eg, ␥ fe, and ␥ fg are the corresponding dephasing rates. The kI signal shows resonances at single-exciton energies along the ⍀1 axis, at ⍀1 = −Ee⬘, and two types of resonances along ⍀3: at ⍀3 = Ee and ⍀3 = E f − Ee⬘. Thus, the diagonal peaks along ⍀3 = −⍀1 = Ee reveal single-exciton states, similar to the linear absorption, while the off-diagonal cross peaks reveal exciton coherences and biexciton contributions. To see which excitons contribute to the formation of each biexciton we turn to the Feynman diagrams, which represent the sequences of interactions with the optical fields and the state of the excitonic density matrix during the intervals between interactions.35 For the kI technique, there are three diagrams, shown in Fig. 5: ground-state bleaching 关共a兲兴, stimulated emission 关共b兲兴, and excited-state absorption 关共c兲兴. The two-exciton states f formed by excitons e and e⬘ only show up in 共c兲. Thus, a cross peak at 共⍀1 = −Ee⬘ , ⍀3 = E f − Ee⬘兲 indicates that exciton e⬘ contributes to that biexciton. The kIII spectrum provides complementary information. As in kI, the resonances along ⍀3 are at single-exciton energies ⍀3 = Ee⬘ and at ⍀3 = E f − Ee⬘. Along ⍀2 though, the signal directly reveals two-exciton energies ⍀2 = E f . There are two corresponding Feynman diagrams, shown in Fig. 6, which both describe excited-state absorption. The twoexciton state is formed by excitons e and e⬘ and results in cross peaks at 共⍀2 = E f , ⍀3 = Ee⬘兲关共a兲兴 or 共⍀2 = E f , ⍀3 = E f − Ee⬘兲关共b兲兴. We thus obtain information on the contributing single-exciton states. The variation in the kI signals with d is displayed in Fig. 7, and Fig. 8 shows the kIII signals. At d = 17 nm, direct excitons 兩␣典 = 兩a典 and 兩␤典 = 兩b典 result in two diagonal peaks at E␣ and E␤ in the kI spectrum. Biexcitons can be formed either from excitons within the same QD, or from excitons in different QDs. Because of the slight asymmetry between the dots, there are two same-dot biexcitons 共in the top or bottom QD兲, which create two closely spaced peaks, labeled A, in the kIII spectrum 关Fig. 8共a兲兴 at E f 1 = 2360 meV and E f 2 = 2363 meV. In kI 关Fig. 7共a兲兴, they create two cross peaks at

155324-4



(a) t t3 τ3

ks

g

−k 3

e’

t2 τ2 t1 τ1

g

ks

t t3

g

e’

e’

f

e’

f

g

e

g

g

g

τ3

f

t2

g τ2

k2 k1

(b)

e

g

g

g

t1 τ1

−k 3

d = 5.5 nm 关compare with panels 共d兲 and 共e兲 of Fig. 7兴. The lowest excitonic peak on the diagonal becomes stronger. Biexcitons A, formed by bonding orbitals, redshifts while the remaining biexcitonic peaks remain at the same position.

k2 k1

V. CONCLUSIONS

FIG. 6. Feynman diagrams for the kIII technique.

⍀3 = E f 1 − E␣ and E f 2 − E␤. The absence of cross peaks at ⍀3 = E f 1 − E␤ and E f 2 − E␣ implies that biexciton f 1共f 2兲 is formed by two 兩a典-type 共兩b典-type兲 single excitons localized in the bottom 共top兲 QD. The third biexciton, labeled B, has energy E f 3 = 2453 meV, higher than the other two. As shown in the kI spectrum, it creates peaks at both ⍀3 = E f 3 − E␣ and E f 3 − E␤, indicating that it is formed by one exciton on each QD 共兩␣典 and 兩␤典兲. At d = 10.2 nm, the indirect excitons 兩␥典 = 兩c典 and 兩␦典 = 兩d典 show up on the diagonal 关Figs. 7共b兲, 7共c兲, 8共a兲, and 8共b兲兴. In the kIII spectrum, there are three bound biexcitons, labeled A and B, while the remaining cross peaks are due to unbound two-exciton states at energies 2E␣ , E␣ + E␤ , 2E␤. The first two biexcitons 共A兲 are formed from excitons within the same QD, as in d = 17 nm case. The third biexciton 共B兲 is formed mostly from exciton 兩␤典 = 兩b典 and a small contribution from 兩␥典 = 兩c典, which is the first sign of electron delocalization. The 2D spectra at the critical distance of d = 8.6 nm, where the two lower excitons become degenerate, are shown in Figs. 7共c兲 and 8共c兲. The kIII spectrum has four biexciton peaks at energies E f 1 = 2381 meV and E f 2 = 2383 meV 共labeled A兲, E f 3 = 2457 meV 共B兲, and E f 4 = 2523 meV 共C兲. The remaining peaks, at energies Ee + Ee⬘ 共e , e⬘ = ␣ , . . . , ␦兲, are due to unbound two-exciton states. The kI spectrum shows that biexcitons A and B are mostly formed by the first two excitons, which are now delocalized and may not be attributed to a single QD. The excitons are represented by bonding and antibonding orbitals 兩a典 ⫾ 兩b典. The last cross peak C consists mostly of the two higher single-exciton states, the indirect excitons 兩c典 and 兩d典. At shorter distances 共d = 6.8 nm and d = 5.5 nm兲, the electrons become delocalized and yield richer spectra, as shown in panels 共d兲 and 共e兲 of Figs. 7 and 8. Four sets of biexcitonic peaks now appear in kIII. At d = 5.5 nm we observe eight biexcitonic peaks, at E f 1 = 2407 meV, E f 2 = 2411 meV 共A兲, E f 3 = 2424 meV, E f 4 = 2440 meV, E f 5 = 2449 meV 共D兲, E f 6 = 2490 meV 共B兲, and E f 7 = 2560 meV, E f 8 = 2574 meV 共C兲. Looking at region 共I兲 of the kI spectrum 关Fig. 7共b兲兴 one can see that biexcitons f 1 and f 2 of group A and f 3 of group D are formed mostly from the bonding orbitals 兩␣典 = 兩b典 + 兩c典 and 兩␤典 = 兩a典 + 兩d典. Similarly, regions 共I兲 and 共II兲 show that f 4 and f 5 of group D are formed mostly from the antibonding ones 兩␥典 = 兩b典 − 兩c典 and 兩␦典 = 兩a典 − 兩d典. Region 共II兲 also suggests that all single-exciton states contribute significantly to the formation of biexcitons B and C. In Fig. 9 we display the kI spectra calculated by neglecting dipole-dipole interactions at short distances, d = 6.8 and

We have studied the single and double excitons in two coupled quantum dots and their variation with interdot distance. Our calculation takes into account both carrier tunneling and exciton migration via dipole-dipole interactions. The absorption spectra were used to classify the single-exciton states in terms of localized e-h pairs. The 2D spectra in directions kI = −k1 + k2 + k3 and kIII = + k1 + k2 − k3 were calculated by means of NEE. Analysis of these spectra using the corresponding SOS expressions, allowed us to classify the biexcitons according to their single-exciton components. At large distances, only direct excitons are active, and two types of biexcitons are formed, either within or at different QDs. At shorter distances, we also see biexcitons created from indirect excitons. At distances where electron interdot tunneling becomes noticeable, additional biexcitonic peaks appear, providing a clear signature of electron delocalization. Exciton hopping is significant only at short distances, where it affects the intensity of the excitonic peaks and shifts the biexciton energies. ACKNOWLEDGMENTS

This work was supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy and the National Science Foundation Grant No. CHE-0745892. We thank Darius Abramavicius and Steven Cundiff for many useful discussions. Figures of 2D spectra were produced using the program dat2eps-v3.2.pl written by Cyril Falvo. APPENDIX A: RECAST OF THE ELECTRON-HOLE HAMILTONIAN USING EXCITONIC VARIABLES

By introducing the electron-hole pair operators 关Eq. 共10兲兴 Chernyak and Mukamel recasted Hamiltonian 共9兲, as well as the commutation relations of these operators, in terms of an infinite series of normally ordered operators Bˆ† and Bˆ.25,28,29 Since the Hamiltonian conserves the number of particles, each term contains an equal number of creation 共Bˆ†兲 and annihilation 共Bˆ兲 operators. For computing the third-order response, the Hamiltonian can be truncated at fourth order, † ˆ Bn + H = 兺 hmnBˆm mn

† ˆ†ˆ ˆ Umn,klBˆm B nB kB l 兺 mnkl

ⴱ † − 兺关␮m · E共t兲Bˆm + H.c.兴

共A1兲

m

where electron-hole pairs are denoted with Latin indices without subscript m = 共m1 , m2兲, n = 共n1 , n2兲, etc. The parameters hmn and Umn,kl of the effective Hamiltonian can be determined by comparing successively order by order the matrix elements of Hamiltonians 共9兲 and 共A1兲 in the space of

155324-5



FIG. 7. 共Color online兲 Absolute value of the kI signal for xxxx polarization at several interdot distances d. Regions denoted as I and II are magnified in the center and right columns, respectively. Single-exciton energies are marked with dashed lines and biexciton peaks are circled. 155324-6



(a)

2500

B

2400

A

10

共1兲共2兲 eh hmn = tm n ␦ m2n2 + ␦ m1n1t m n − V m m ␦ m1n1␦ m2n2

1

1 1

2350

+

Ω2 (meV)

(b) d=10.2 nm

one, two, three, etc. electron-hole pair excitations. This is possible because normally ordered N creation and N annihilation operators do not contribute in the subspaces of N − 1 and smaller number of excitations. Thus, in the one electronhole pair excitation subspace we obtain

3

102 Ω2 (meV)

d=17.0 nm

2450

10

2300 1100 1120 1140 1160 1180 1200 1220 1240 Ω3 (meV)

10

0

2550

10

4

2500

10

3

2450

B

2350 1100

(c)

10

1

10

0

10

4

C

10

3

B

102

A 1150

1200 Ω3 (meV)

1250

1300

2540

Ω2 (meV)

d=8.6 nm

2480 2460

2400 2380

(d) Ω2 (meV)

d=6.8 nm

2550

1220

1240 1260 Ω3 (meV)

共2兲共1兲 + ␦ m1k1t m k ␦ n1l1␦ n2l2 + ␦ m1k1␦ m2k2t n l ␦ n2l2 2 2

10

1

10

0

共A4兲

Fmn,kl + Fmn,lk − 2 兺 Pmn,pqF pq,kl = 0.

10

4

10

3

P is tetradic matrix defined as 1 1 Pmn,pq = ␦m1q1␦m2p2␦n1p1␦n2q2 + ␦m1p1␦m2q2␦n1q1␦n2p2 . 2 2 共A6兲

102

D

10

1

10

0

10

4

10

3

The U matrix is invariant to the addition of any matrix F that satisfies Eq. 共A5兲 and it is thus not uniquely defined. This freedom arises since Hamiltonians 共9兲 and 共A1兲 are only required to coincide in our physically relevant subspace of one and two e-h pair excitations but may differ in higher manifolds. Similarly, the commutation relation of the electron-hole particle operators can be expanded in a series of normally ordered operators Bˆ† and Bˆ. For the third-order response, this can be truncated at quadratic order,

A 1200 1250 Ω3 (meV)

1300

2600

2550

C 102

2500

2450

D

B

关Bˆm,Bˆ†n兴 = ␦mn − 2 兺 Pmp,nqBˆ†pBˆq ,

A 1150

1200 1250 Ω3 (meV)

1300

1350

共A7兲

pq

101

where ␦mn = ␦m1n1␦m2n2. The P matrix is responsible for the deviation from boson statistics.

2400 1100

共A5兲

pq

B

2400

Ω2 (meV)

11

␦m1k1␦m2k2␦n1l1tn共2兲2l2兴.

We further define the matrix F by the equation,

C

1150

d=5.5 nm

1300

共A3兲

2500

2450

(e)

1280

共A2兲

1 hh + Vm ␦ ␦ ␦ ␦ 兴 − 关tm共1兲1k1␦m2k2␦n1l1␦n2l2 2n2 m1k1 n1l1 m2l2 n2k2 4

A 1200

1

1 ee ¯ U mn,kl = 关Vm1n1␦m1l1␦n1k1␦m2k2␦n2l2 4

+

1180

1

where

2440 2420

− ␦Rm Rn 兲.

¯ Umn,kl = U mn,kl + Fmn,kl ,

2520 2500

1 2

Diagonal elements 共m = n兲 describe the electron-hole pair energy, given as the sum of electron and hole kinetic energies reduced by the electron-hole Coulomb attraction. Offdiagonal elements 共m ⫽ n兲 describe electron, hole, or exciton hopping between adjacent sites. Similarly, to describe the one and the two electron-hole pair subspace we need the quartic term,

102

2400

2 2

F ␦ ␦ 共1 Vm 1m2,n1n2 Rm1Rm2 Rn1Rn2

100

FIG. 8. 共Color online兲 Absolute value of the kIII signal for xxxx polarization at various interdot distances d. Arrows mark biexciton contributions.

APPENDIX B: THE NONLINEAR EXCITON EQUATIONS

The nonlinear optical response is calculated using the Heisenberg equation of motion for the electron-hole operator

155324-7



102 1200

II

I

A

-1200 Ω1 (meV)

-1250

1150

A

Ω3 (meV) 1

10

1100 -1100

I -1200 -1250 Ω1 (meV)

-1300

-1350

101

B

Region II

D

104

C

1270 1260

A

D

1140

10

103

1250

102

1

1240 1230

102

B

1220 1210

1

10

1200

A

1110

100

B

1280

1120

-1150

1220

1190 -1220 -1230 -1240 -1250 -1260 -1270 -1280 -1290 -1300 Ω1 (meV)

103

1130

1150

102

1230

1200

1160 1150

1240

1210

101

Region I

1250

II

102

D

1140

1170

10

1250

1180

102

3

C

1260

1110 -1220 -1230 -1240 -1250 -1260 -1270 -1280 -1290 -1300 Ω1 (meV)

-1300

103

1200

3

Ω3 (meV)

-1150

1300

Ω3 (meV)

1160

1120

1350

d=5.5 nm

D

1130

101

1150

1170

Region II

1270 10

Ω3 (meV)

3

10

1250

1100 -1100

(e)

Region I

1180 1300

Ω3 (meV)

Ω3 (meV)

d=6.8 nm

(d)

-1230 -1240 -1250 -1260 -1270 -1280 -1290 -1300 -1310 Ω1 (meV)

100

1190

B

D

-1230 -1240 -1250 -1260 -1270 -1280 -1290 -1300 -1310 Ω1 (meV)

100

FIG. 9. 共Color online兲 Absolute value of the kI signal for xxxx polarization without dipole-dipole interactions.

pair operators 关Eq. 共10兲兴 the anticommutation relations 共3兲 and 共4兲, and Eqs. 共A2兲–共A6兲 to recast the V matrix in the form

d Bˆm, i dt 具Bˆm典 = 具关Bˆm , H兴典, obtained by Eqs. 共A1兲 and 共A7兲,

i

d ˆ ⴱ 具Bm典 = 兺 hmn具Bˆn典 + 兺 Vmn,kl具Bˆ†nBˆkBˆl典 − ␮m · E共t兲 dt n nkl + 2 兺 ␮ⴱn · E共t兲Pmk,nl具Bˆ†k Bˆl典.

共B1兲

ee hh Vmn,kl = Vm n ␦ m1l1␦ n1k1␦ m2k2␦ n2l2 + V m n ␦ m1k1␦ n1l1␦ m2l2␦ n2k2 1 1

nkl

1 F F + 共Vm ␦ ␦ + Vm ␦ ␦ 兲 1n2k1l2 n1l1 m2k2 1n2l1k2 n1k1 m2l2 2

The last term in the right-hand side is known in the context of the simpler semiconductor Bloch equations as phase-space filling.30 The second term describes exciton-exciton interactions where V is given by

1 + 共VnF m l k ␦m1k1␦n2l2 + VnF m k l ␦m1l1␦n2k2兲 1 2 12 2 1 21 2

Vmn,kl = Umn,kl + Umn,kl − 2 兺 Pmn,pkh pl − 2 兺 Pmn,pqU pq,kl p

1 eh 1 eh + Vnehm 兲␦m1k1␦n1l1␦m2k2␦n2l2 − 共Vm − 共Vm 1n2 1 2 1n2 2 2

pq

¯ ¯ ¯ =U mn,kl + Umn,kl − 2 兺 Pmn,pkh pl − 2 兺 Pmn,pqU pq,kl . p

+ Vnehm 兲␦m1l1␦n1k1␦m2l2␦n2k2 . 1 2

共B5兲

pq

共B2兲 Note that it is independent of the matrix F, Eq. 共A5兲. Similarly, for the two-particle variable, 具BˆmBˆn典, to second order in the optical field, i

2 2

This expression, which is the same as Eq. 共20兲 in Ref. 25, clearly shows that V describes Coulomb interactions between the particles that constitute the two pairs. Neglecting incoherent exciton transport, we can make the factorization,36

d ˆ ˆ 共2兲 ⴱ 具BmBn典 = 兺 hmn,kl 具BˆkBˆl典 − 关␮m · E共t兲具Bˆn典 + 具Bˆm典␮ⴱn · E共t兲兴 dt kl + 2 兺 ␮ⴱl · E共t兲Pmn,kl具Bˆk典,

and

具Bˆ†nBˆkBˆl典 = 具Bˆ†n典具BˆkBˆl典. 共B6兲

共B3兲

kl

where 共2兲 = ␦mkhnl + ␦nlhmk + Vmn,kl . hmn,kl

具Bˆ†k Bˆl典 = 具Bˆ†k 典具Bˆl典

共B4兲

By expanding the equations of motion in orders of the optical field E共t兲, and defining Bm = 具Bˆm典, Y mn = 具BˆmBˆn典, we finally obtain the nonlinear exciton equations,27,28

The diagonal elements of h共2兲 represent two electron-hole pair energies, while Vmn,mn describes the biexciton binding energy. To see that, we use the definition of the electron-hole 155324-8

i

d 共1兲 ⴱ B = 兺 hmnB共1兲 n − ␮m · E共t兲, dt m n

共B7兲



i

Hamiltonian with energies Ee and dephasing rates ␥e,

d 共2兲 ⴱ 共1兲 ⴱ Y = 兺 h共2兲 Y 共2兲 − 关␮m · E共t兲B共1兲 n + Bm ␮n · E共t兲兴 dt mn n mn,kl kl + 2兺

␮ⴱl

·

E共t兲Pmn,klB共1兲 k ,

共B8兲

kl

ˆ 共t兲兩e典 = − i␪共t兲e−iEet−␥et , Ie共t兲 ⬅ 具e兩G

共C3兲

ˆ 共␻兲兩e典 = 共␻ − E + i␥ 兲−1 , Ie共␻兲 ⬅ 具e兩G e e

共C4兲

and G共t兲 is the single-exciton Green’s function. The Fourier transform is defined as

d 共3兲共1兲ⴱ 共2兲 = 兺 hmnB共3兲 i Bm n + 兺 Vmn,klBn Y kl dt n nkl 共1兲 + 2 兺 ␮ⴱn · E共t兲Pmk,nlB共1兲ⴱ k Bl .

G共␻兲 =

共B9兲

nkl

G共t兲 = APPENDIX C: THE 2D SIGNALS

The NEE equations may be solved using the singleexciton Green’s functions and the exciton scattering matrix.27 The polarization is then expressed in terms of the response function, ␯4

V 共t兲 =

冕

dt1dt2dt3S

␯4␯3␯2␯1

共t3,t2,t1兲

␮e␯1␮e␯2ⴱ␮e␯3ⴱ␮e␯4 1

2

3

4

共C8兲 where V is given in Eq. 共B5兲, and I is the tetradic identity matrix in the two-exciton space. Similarly, the response in the kIII direction is given by17

兺 e ,e ,e ,e

␮e␯1ⴱ␮e␯2ⴱ␮e␯3␮e␯4 1

2

3

4

1 2 3 4

1

⫻ G0e3e2共⍀3 + Ee1 + i␥e1兲,

⫻ Ieⴱ 共t1兲Ie4共⍀3兲Ieⴱ 共⍀2 − ⍀3兲 1

A. D. Yoffe, Adv. Phys. 50, 1 共2001兲. D. Scholes and G. Rumbles, Nature Mater. 5, 683 共2006兲. 3 V. I. Klimov, Annu. Rev. Phys. Chem. 58, 635 共2007兲. 4 X. Peng, L. Manna, W. Yang, J. Wickham, E. Scher, A. Kadavanich, and A. P. Alivisatos, Nature 共London兲 404, 59 共1999兲. 5 B. Fisher, J.-M. Caruge, Y.-T. Chan, J. Halpert, and M. G. Bawendi, Chem. Phys. 318, 71 共2005兲. 6 X. Xu, B. Sun, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, and L. J. Sham, Science 317, 929 共2007兲. 7 M. Bruchez, Jr., M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, Science 281, 2013 共1998兲. 8 V. I. Klimov, A. A. Mikhailovsky, S. Xu, A. Malko, J. A. Hollingsworth, C. A. Leatherdale, H.-J. Eisler, and M. G. Bawendi, Science 290, 314 共2000兲. 9 J.-W. Luo, A. Franceschetti, and A. Zunger, Nano Lett. 8, 3174 共2008兲. 10 A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo,

3

⫻ 关⌫e4e3e2e1共⍀3 + Ee3 + i␥e3兲G0e2e1共⍀3

共C2兲

+ Ee3 + i␥e3兲 − ⌫e4e3e2e1共⍀2兲G0e2e1共⍀2兲兴.

where ␯1 , . . . , ␯4 are the polarizations of the optical pulses, e1 , . . . , e4 label eigenstates of the single-exciton block of the

2 G.

1 ␻ − Ee2 − Ee1 + i共␥e2 + ␥e1兲

−1 ⌫共␻兲 = 关I − VG0共␻兲兴−1VG0共␻兲共I − P兲G−1 0 共␻兲 − PG0 共␻兲,

⫻ ⌫e4e1e2e3共⍀3 + Ee1 + i␥e1兲

1

共C6兲

共C7兲

␯4␯3␯2␯1 SIII 共⍀3,⍀2,t1兲 = 2

⫻ Ieⴱ 共t2兲Ie2共t2兲Ieⴱ 共− ⍀1兲Ie4共⍀3兲 1

d␻ exp共− i␻t兲G共␻兲. 2␲

is the Green’s function representing noninteracting two excitons, and the scattering matrix is given by

The response function in the kI = −k1 + k2 + k3 direction is given by17

兺

共C5兲

G0e2e1共␻兲 ⬅ 具e1e2兩Gˆ 0共␻兲兩e1e2典 =

共C1兲

e1,e2,e3,e4

dt exp共i␻t兲G共t兲,

Finally,

⫻E␯3共t − t3兲E␯2共t − t3 − t2兲E␯1共t − t3 − t2 − t1兲.

SI␯4␯3␯2␯1共⍀3,t2,⍀1兲 = 2i

冕

冕

共C9兲

D. Loss, M. Sherwin, and A. Small, Phys. Rev. Lett. 83, 4204 共1999兲. 11 P. Zanardi and F. Rossi, Phys. Rev. Lett. 81, 4752 共1998兲. 12 O. Gywat, G. Burkard, and D. Loss, Phys. Rev. B 65, 205329 共2002兲. 13 G. Allan and C. Delerue, Phys. Rev. B 75, 195311 共2007兲. 14 A. Sitek and P. Machnikowski, Phys. Rev. B 75, 035328 共2007兲. 15 M. Bayer, P. Hawrylak, K. Hinzer, S. Fafard, M. Korkusinski, Z. R. Wasilewski, O. Stern, and A. Forchel, Science 291, 451 共2001兲. 16 G. Bester, J. Shumway, and A. Zunger, Phys. Rev. Lett. 93, 047401 共2004兲. 17 R. Oszwaldowski, D. Abramavicius, and S. Mukamel, J. Phys.: Condens. Matter 20, 045206 共2008兲. 18 S. Mukamel, Ann. Rev. Phys. Chem. 51, 691 共2000兲. 19 G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mancal, C. Yuan-Chung, R. Blankenship, and G. R. Fleming, Nature 共Lon-

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