Revealing exciton-exciton couplings in semiconductors...

Revealing exciton-exciton couplings in semiconductors using multidimensional four-wavemixing signals

Lijun Yang and Shaul MukamelChemistry Department, University of California, Irvine, California 92697-2025, USA

Received 20 July 2007; revised manuscript received 5 November 2007; published 28 February 2008

We show that the femtosecond four-wave mixing signal from GaAs quantum wells generated in the phase-matching direction k1+k2−k3 can fully resolve the pure heavy-hole, pure light-hole, and the mixed twoexcitons by directly observing double quantum coherences. Time-dependent Hartree-Fock contributions fordifferent types of two excitons may be completely separated from higher-order Coulomb correlations byspecific pulse polarization configurations. Simulations performed on a one-dimensional three-band tight-binding model for a GaAs quantum well are analyzed using double-sided Feynman diagrams which reveal theLiouville space pathways contributing to the various peaks.

DOI: 10.1103/PhysRevB.77.075335 PACS numbers: 78.47.p, 78.67.De, 42.50.Md, 71.35.Cc

I. INTRODUCTION

Nonlinear four-wave mixing FWM experiments havelong been known to provide direct probes for the many-bodyeffects in the ultrafast dynamics of excitons in semiconductorquantum wells SQWs.1–13 FWM signals are commonly dis-played as a function of a single time or frequency variablewhich provides a one-dimensional 1D projection of the mi-croscopic information. Signatures of complex many-bodydynamics may not be easily identified. It is difficult to re-solve the two-exciton correlations involving both heavy-holeHH and light-hole LH excitons by projecting all informa-tion into a 1D signal. During the past decade, coherent opti-cal two-dimensional correlation spectroscopy14–17 2DCStechniques have been developed and applied to many chemi-cal, biological systems,18–21 and SQWs.22–26 These femtosec-ond analogs of nuclear magnetic resonance techniques27,28

correlate the polarization phase evolution in independenttime periods. Many-body couplings show up as cross peaksbetween different resonances.

2DCS signals depend on three time delays t1 , t2 , t3 be-tween the incoming pulses, k1, k2, k3, and the signal, ks.Three 2DCS techniques, SI, SII, and SIII are possible for anexciton model.15 The signals are, respectively, generatedalong the phase-matching directions kI=−k1+k2+k3, kII=k1−k2+k3, and kIII=k1+k2−k3. Through a Liouvillespace pathway analysis,14,15,24 2DCS can reveal detailedmany-body couplings. Recent simulations of the SI and SIItechniques show their capacity to separate coherences fromdifferent pathways and to monitor two-excitoncorrelations.22–26 However, SI and SII only partially resolvedifferent two-exciton resonances. Pure HH and pure LH twoexcitons can not be completely separated from the mixedLH+HH two excitons.12,29 This was shown for SI in Ref.24 and for SII in Appendix A of this paper.

Most past work on many-body interactions in SQWs hadfocused on the sub-HH excitons space. In a recent study ofHH excitons,32 in GaAs SQW, we demonstrated that SIIIRefs. 15, 30, and 31 is particularly sensitive to two-excitoncorrelations, as demonstrated recently for HH excitons in aSQW. In this work, we extend that study to include HH andLH exciton couplings. Apart from the coexistence of several

types of two excitons in the LH and HH exciton couplings,two-exciton resonances are further complicated by manysources such as time-dependent Hartree-Fock TDHF andhigher many-body correlations HMBC which are neglectedby the TDHF.

SIII, however, can completely separate pure HH, LH, andmixed two excitons. With the proper choice of pulse polar-izations, it can even separate TDHF and HMBC two-excitoncorrelations. Moreover, as shown in Ref. 32, both frequencyaxes in SIII involve two excitons; one of them t2 or its con-jugate frequency 2 is solely related to two excitons andprovides a very clean projection. We can thus combine theinformation from both axes to improve two-exciton reso-lution. In Sec. II, we present the model Hamiltonian and theHeisenberg equations of motion. In Sec. III, we present thelevel scheme for GaAs SQWs, the Feynman diagrams forSIII, and the schematic 2D spectrum. Numerical calculationswithin and beyond TDHF are presented respectively in Secs.IV and V.

II. TIGHT-BINDING HAMILTONIAN AND EQUATIONSOF MOTION

We used a three-band one-dimensional tight-bindingHamiltonian of a linear chain with 10 sites and 40electrons.33–38 This model with various number of sites hasbeen successfully employed to reproduce many experimental1D FWM38 and 2DCS26 signals in GaAs SQWs.

The free band Hamiltonian is given by

HK = ijc

Tijc ci

c†cjc +

ijvTij

vdiv†dj

v, 1

where cic† ci

c are the creation annihilation Fermi operatorsof electrons in site i from the conduction band c and di

v† div

are the corresponding operators of holes in the valence bandv. The diagonal elements Tii

c,v describe the site energies forthe electrons holes in the conduction valence band, whilethe off-diagonal elements Tij

c,v represent the couplings be-tween different sites.

The optical spectrum of GaAs SQWs connects a J= 32 va-

lence band and a J= 12 conduction band the J= 1

2 valence

PHYSICAL REVIEW B 77, 075335 2008

1098-0121/2008/777/07533511 ©2008 The American Physical Society075335-1

http://dx.doi.org/10.1103/PhysRevB.77.075335

band is well separated from the J= 32 valence band due to

spin-orbit interaction and need not be considered. Only theband edge states Bloch vector k near 0 are optically activedue to momentum conservation. The conduction band canthus be accurately represented by two conduction orbitalsJz=

12 and four valence orbitals, Jz=

32 HH and Jz

= 12 LH states.12,29,39,40 The dipole selection rules are pre-

sented in Fig. 1. The allowed transitions are denoted by Rand L arrows, representing right and left circularly polarizedphotons, respectively. The corresponding transition dipolesare vc, where v=1,2 denote heavy holes Jz=

32

and v=3,4 represent light holes Jz=

12

. c=1 and 2 denote elec-trons with different spins in the conduction band Jz=

12

.The dipole interaction with the radiation field has the

form

HI = − Er,t · P , 2

where P is the interband polarization operator

P ijvc

ijvcpij

vc + H.c. . 3

Here, ijvc are interband transition dipoles and pij

vcdivcj

c. Theelectric field Er , t of the three incoming pulses is given by

Er,t = =1

3

Er,t + E*r,t

= =1

3

eE+t − eik·r−t

+ eE−t − e−ik·r−t . 4

Here, E+ E

− = E+* is the envelope of the positive

negative-frequency component of the th pulse centered at, with carrier frequency , polarization unit vector e, andwave vector k.

We next turn to the many-body couplings. We assumemonopole-monopole Coulomb interaction between electronsand holes,33,41,42

HC =1

2 ijcvcv

cic†ci

c − div†di

vVijcjc†cj

c − djv†dj

v , 5

where

Vij = U0d0

i − jd0 + a0.

Here, U0 is the interaction strength, a0 is a spatial cutoff, andd0 is the lattice constant.

The total Hamiltonian is finally given by

H = HK + HI + HC. 6

The Heisenberg equations of motion derived from thisHamiltonian can be truncated by employing the nonlinearexciton equations NEEs formalism which has been widelyapplied to Frenkel43,44 and Wannier4,45,46 excitons. The opti-cal field is treated perturbatively, while Coulomb interactionsare included nonperturbatively. The NEE is most suitable fordescribing Coulomb correlations in the low excitation 3

regime.9,47,48 In this work, we focus on coherent excitationsand thus only retain the two types of variables involvingsingle exciton and two exciton, respectively.4 To third orderin the radiation field, the equations of motion for the firstvariable, pij

vc= pijvc, are given by34,36–38

− i

tpij

vc −i

texpij

vc = − n

Tjnc pin

vc − m

Tmiv pmj

vc + Vijpijvc

+ klvc

Vkj − Vki − Vlj + Vli

plkvc*plj

vcpikvc − plk

vc*plkvcpij

vc

− plkvc*Blkij

vcvc + Er,t · ijvc*

− klvc

ilvc*pkl

vc*pkjvc

+ klvc

ljvc*plk

vc*pikvc , 7

where the exciton dephasing time tex is introduced phenom-

enologically. The correlated two-exciton amplitudes Blkijvcvc

is defined by

Blkijvcvc dl

vckcdi

vcjc + plj

vcpikvc − plk

vcpijvc , 8

and satisfy the equations of motion,

− i

tBlkij

vcvc −i

tbiBlkij

vcvc = − m

Tjmc Blkim

vcvc + Tmiv Blkmj

vcimevc

+ Tkmc Blmij

vcvc + Tmlv Bmkij

vcimevc + Vlk

+ Vlj + Vik + Vij − Vli − VkjBlkijvcvc

− Vlk + Vij − Vli − Vkjpikvcplj

vc

+ Vik + Vlj − Vli − Vkjplkvcpij

vc,

9

where tbi is the two-exciton dephasing time. 2DCS signalscan be calculated by selecting the spatial Fourier componentsof Eqs. 7 and 9 as shown in Appendix B.

2

1:

1−c

2

1:

2c

2

3:

1−v

2

1:

3−v

2

1:

4v

2

3:

2v

11cv

µ14cv

µ

22cv

µ23cv

µR

R

L

L

FIG. 1. Spin orbitals and selection rules for a single site of theten-site tight-binding model.

LIJUN YANG AND SHAUL MUKAMEL PHYSICAL REVIEW B 77, 075335 2008

075335-2

In all calculations, we use the following parameters.12,49

For the site energies and carrier coupling energies, we takeTij

c =8 meV, Tijv=1,2=4.75 meV HH band, and Tij

v=3,4

=2.52 meV LH band to account for the in-plane dispersionof the valence-band structure in the quantum well. The siteenergies Ti=j

c and Ti=jv=1,2 are taken as half of the band gap Eg.

U0=10 meV and a0 /d=0.5. The parameters of our one-dimensional model were optimized for the description of ex-citons and biexcitons in quantum wells. We assumed Gauss-ian pulse envelopes,

Et − t = exp− t − t2/

2 , 10

where =250 fs for the calculations of all 2DCS signalsand =100 fs for linear absorption. The carrier frequencywas detuned by =3.62 meV to the blue of the HH exci-ton energy.

III. SIII TECHNIQUE APPLIED TO GaAs QUANTUMWELLS

To investigate the excitonic structure of our model Eq.6, we transform from the real site space to momentum kspace. Applying periodic boundary conditions gives the fol-lowing k values:38

k = 0, 2

Na,

4

Na, . . . ,

N/2 − 12

Na,

a,

where N is the number of sites. For N=10, there are 10 HHand 10 LH single exciton states. However, because k aredegenerate, we have 12 different energies. We first calculatethe linear absorption using14,24

= j

nc 0ImP · Eopt

* Eopt2 , 11

where n is the average, frequency-independent refractiveindex of the quantum well, 0 is the vacuum permittivity, andc is the speed of light. Eopt is Fourier transform of thepulse envelope E

+. P is the Fourier transform of the linearpolarization obtained by integrating the linear equation Eq.B5 over a time period of 38 ps with 1660 time grid points.

Figure 2 shows the calculated linear absorption. There aresix pairs of peaks one pair is very weak and each paircorresponds to HH and LH band contributions. The smallnegative features in the absorption reflect numerical error.The two strong peaks to the red are, respectively, referred toas HH and LH excitons. These are the dominant peaks in thelinear or nonlinear responses of a quantum well. As N in-creases, the strength of other peaks will be reduced and theircontributions to the linear absorption will form a continuum.In the numerical calculations, we used spectrally narrowpulses to reduce the influence of continuum states. Thus, inthe following analysis, we shall only consider the twomarked HH and LH excitons.

The many-electron level scheme12,29,39,40 including HHand LH excitons and four bound two-exciton transitions ispresented in Fig. 3. The ground state to exciton dipoles su-perpositions of vc and the exciton to two-exciton dipolesare collectively denoted by ge and ef, respectively. Sub-

scripts g, e, and f represent, respectively, the ground state,the single, and the two-exciton manifold. e can be eithereHeH or eLeL denoting HH and LH excitons. f can beeither fH, fL, or fM representing, respectively, HH, LH, andmixed two excitons. The unbound two-exciton transitionsnot shown for clarity will be included in the numericalcalculations.

We shall consider the 2D projection of the SIIIt3 , t2 , t1signal24 in the 3 ,2 plane for fixed t1,

SIII3,2,t1 dt3dt2SIIIt3,t2,t1ei3t3ei2t2.

12

Within the rotating wave approximation, two basic Feynmandiagrams vii and viii in the notation of Refs. 15 and 24contribute to this signal, each gives six diagrams, as shownin Fig. 4, when all contributions of HH and LH transitionsare spelled out. The contributions of each diagram Liouvillespace pathway to the signal are represented by differentsymbols shown at the bottom. Solid open symbols denoteredshifted blueshifted two excitons. Open circles describethe contributions from bare two excitons whose contributionsvary for fH, fL, and fM. However, we denote them collec-tively by the same symbol circle for clarity.

With the help of Fig. 4, we can schematically sketch theSIII spectrum. This is shown in Fig. 5, where the overlappingsymbols are slightly displaced for clarity. We first consider

0 20 40 60

0.0

0.2

0.4

0.6

0.8

1.0

LH

HH

LinearAbsorption(arb.units)

∆ω (meV)

FIG. 2. Linear Absorption of the ten-site chain model for asemiconductor quantum well. =0 is the HH exciton energywhich is around 1.5 eV it varies for different samples.

'geµ

efµefµ

efµ

efµ

efµefµ

efµ

efµ

Lf

Mf

Hf

Mf

'

Le

Le

He

'

He

Le2

LHee +

''

LHee +

He2

geµ

'geµ

geµ

R

R

R

L

L

L

FIG. 3. Exciton level scheme for GaAs semiconductors.

REVEALING EXCITON-EXCITON COUPLINGS IN… PHYSICAL REVIEW B 77, 075335 2008

075335-3

the HH exciton contributions diagrams viia and viiia.Diagram viiia, when fH is redshifted and equals to 2eH−H1

, shows the resonance between 2eH−H1and the

ground state during the t2 period. During t3, viiia shows aresonance between fH=2eH−H1

and eH with frequency eH

−H1. Therefore, according to viiia and taking fH=2eH

−H1, we have a peak at 3 ,2= eH−H1

,2eH−H1

solid red triangle. The two coordinates correspond, respec-tively, to the resonance energies during the evolution timeperiods t3 and t2. The blueshifted two-exciton energy fH=2eH+H2

in viiia gives a peak at 3 ,2= eH

+H2,2eH+H2

open red triangle. Similarly, using dia-gram viia and by taking, respectively, fH=2eH−H1

and2eH+H2

, we obtain the solid and open red squares. For baretwo excitons, fH=2eH, there are no HMBC two-excitoncoupling50 and both diagrams viia and viiia contribute tothe signal at eH ,2eH open circle. Finally, the analysis ofother diagrams involving LH excitons is similar to the casewith only HH excitons.

The unique capacity of SIII to resolve two excitons withinthe HH region have been demonstrated in Ref. 32. The 2axis provides a clean projection of two excitons which onlycontains two exciton to ground state resonances. By combin-ing the two-exciton information along the 2 and 3 axes,we can achieve high resolution. We note that the peaks re-lated only to HH two excitons around 3 ,2= eH ,2eHand the LH two excitons around eL ,2eL are separated fromthe mixed two-exciton peaks at 2=eH+eL. Thus, the mixed

two excitons do not affect the pure HH LH two excitons, asis the case in SI Ref. 24 and SII see Appendix A tech-niques. Moreover, as we shall show later, by choosing spe-cific pulse polarizations, SIII can also fully separate the cor-related pure or mixed two excitons from their correspondingTDHF contributions. For example, in the vicinity ofeH ,2eH, there will be only contributions from HMBC butno bare two excitons arising from TDHF.

IV. TIME-DEPENDENT HARTREE-FOCK SIMULATIONSOF TWO-DIMENSIONAL CORRELATION

SPECTROSCOPY

SIII is equal to PkIIIt3 , t2 , t1 , t, where PkIII is the inter-band polarization in the phase-matching direction kIII, givenby

PkIIIt3,t2,t1,t ijvc

ijvcpij

vc:kIIIt3,t2,t1,t . 13

To calculate the 2D spectra, we first compute PkIII for fixedt1=150 fs from t3=0 to 28 ps with 660 time grid points bysolving Eqs. B1, B5, and B6. These calculations werethen repeated by varying t2 from t2=0 ps to 14 ps on a 330-point grid.

We first present calculations at the TDHF level. To that

end, we set the correlated two-exciton variables Blkijvcvc:k1+k2

in Eq. B1 to be zero. According to Eq. 8, this is equiva-lent to the following TDHF decoupling:

dlvck

cdivcj

c = plkvcpij

vc − pljvcpik

vc

= dlvck

cdivcj

c − dlvcj

cdivck

c . 14

Using the parameters given in Sec. II, we obtained the 2Dspectra Eq. 12 shown in Fig. 6. In all figures, the origin iseH for 3 and 2eH for 2. For XRRR polarization configu-ration indices are chronologically ordered from right to left,i.e., k1 is R and signal is X, only pure HH two excitonsappear at 0,0 and pure LH two excitons appear at around4, 8, as shown in the top panel. For the XRLR configura-tion middle panel, only mixed two excitons appear at 0,4and 4,4. All possible two excitons are seen for XXXX con-figuration bottom panel. Finally, we note that only bare twoexcitons show up at this TDHF level.50

(viiia) (viiib))(viiia'

)(viiib' (viiic) (viiid)

gg

HHee

He

Le

Hf

gg gg

gggggg

HHee

HHee

LLee

LLee

LLee

1k

3k-

4k

4k

4k

Lf

Mf

Mf M

f

Mf

2k

1k

1k

1k

4k

1k

4k

1k

4k

2k

2k

2k

2k

2k

3k-

3k-

3k-

3k-

3k-

He

He

LeL

e

(viic))(viib'

(viib))(viia'(viia)

gg

He

Le

Hf

gg gg1k

3k-

4k

4k

4k

Lf

Mf

Mf

Mf

Mf

2k

1k

1k

1k

4k

1k

4k

1k

4k

gg gg gg

gg gg gg

gg gg gg

3k-

3k-

3k-

3k-

3k-

2k

2k

2k

2k

2k

He

He

He

He

He

Le

Le

LeL

e

Le

(viid)

2t

1t

3t

t

1τ

2τ

3τ

2t

1t

3t

t

1τ

2τ

3τ

FIG. 4. Color online Feyn-man diagrams for SIII techniquederived from diagrams vii andviii of Fig. 1 in Ref. 24.

2Ω

3ΩH

eLe

He2

Le2

LHee +

2

2LL

e ∆+

1

2LL

e ∆−

2MLH

ee ∆++

1MLH

ee ∆−+

2

2HH

e ∆+

1

2HH

e ∆−

FIG. 5. Color online The SIII3 ,2 , t1 signal derived fromFig. 4.


075335-4

Figure 6 can be interpreted using the wave-vector-selectedequations of motion in Appendix B. In Eq. B1, apart from

the term −Sk3k2k1plk

vc:k3*Blkijvcvc:k1+k2 containing the corre-

lated two excitons, all other terms are within TDHF. Thepermutation symbol Sk3k2k1

and its action on different ex-pressions are defined in Eqs. B2 and B4 in Appendix B.Among the TDHF terms, the terms containing optical fieldEt =1,2 ,3 represent Pauli blocking. Our simulationsshow that their contributions to the signal are very small.

Another nonlinear TDHF term Sk3k2k1plk

vc:k3*plkvc:k2pij

vc:k1

in Eq. B1 does not contribute to the signal for translation-ally invariant systems.38 Retaining only the dominant TDHF

term Sk3k2k1plk

vc:k3*pljvc:k2pik

vc:k1, Eq. B1 becomes

− i

tpij

vc:k1+k2−k3 klvc

Sk3k2k1plk

vc:k3*pljvc:k2pik

vc:k1 .

15

R polarized photons can induce two transitions, v1 ,c1and v3 ,c2. L polarized photons induce two other transi-tions, v2 ,c2 and v4 ,c1, as shown in Fig. 1. We first con-

sider the term plkvc:k3*plj

vc:k2pikvc:k1 in Eq. 15 arising from

one permutation of Sk3k2k1.

For the XRRR configuration, the indices v ,c in

plkvc:k3* can be either v1 ,c1 or v3 ,c2 for the R polarized

pulse k3, as shown in Fig. 1. For v ,c= v1 ,c1, we have

klvc

plkvc:k3*plj

vc:k2pikvc:k1 =

kl

plkv1c1:k3*plj

v1c:k2pikvc1:k1.

16

Since both pulses k2 and k1 are R polarized, the c and vindices in Eq. 16 can only assume, respectively, the valuesc=c1 and v=v1. Therefore, in this case, we have only pureHH two excitons. Similarly, if the indices v ,c in

plkvc:k3* take the other possible value of v3 ,c2, then we

only have pure LH two excitons because pulses k2 and k1can only excite LH excitons through transitions v3 ,c2.Analysis of other terms arising from the permutation ofSk3k2k1

yields the same conclusion: XRRR can only accesspure HH or LH two excitons at the TDHF level. Mixed twoexcitons are not excited.

For the XRLR configuration, we consider the same TDHF

term plkvc:k3*plj

vc:k2pikvc:k1 in Eq. 15 where the R polar-

ized pulse k3 leads to v ,c= v1 ,c1 or v3 ,c2 in

plkvc:k3*. For v ,c= v1 ,c1, we again obtain Eq. 16.

Because pulse k2 is L polarized, pljv1c:k2 in Eq. 16 is always

zero for any c values. Even c=c1, pljv1c:k2 = plj

v1c1:k2 is still zerobecause L polarized photons cannot access v1 ,c1 transition,as shown in Fig. 1. Thus, the TDHF term

plkvc:k3*plj

vc:k2pikvc:k1 does not contribute to the signal when

v ,c= v1 ,c1.We next consider other terms arising from the permuta-

tions of plkvc:k3*plj

vc:k2pikvc:k1. Under the same condition

v ,c= v1 ,c1, we have

Sk3k2k1 klvc

plkvc:k3*plj

vc:k2pikvc:k1

= klvc

plkvc:k3*plj

vc:k1pikvc:k2 + ¯

= kl

plkv1c1:k3*plj

v1c:k1pikvc1:k2 + ¯ . 17

Equation 17 can contribute to the FWM signal by takingc=c1 and v=v4, which gives

− i

tpij

vc:k1+k2−k3 kl

plkv1c1:k3*plj

v1c1:k1pikv4c1:k2 + ¯ ,

18

where R polarized pulse k1 accesses v1 ,c1 transition and Lpolarized pulse k2 accesses v4 ,c1 transition, which are bothpossible, as shown in Fig. 1. Moreover, when c=c1 and v=v4, pij

vc:k1+k2−k3 becomes pijv4c1:k1+k2−k3 and is also acces-

sible by the final X polarized heterodyne pulse according tothe selection rules. Thus, in Eq. 18, pulses k1 and k2, re-spectively, create a HH exciton and a LH exciton, thus form-

-2 0 2 4 6

-4

0

4

8

12

Ω3(meV)

Ω2(m

eV)

1.0

12

1.5E

1.9E

2.3E

2.8E

-4

0

4

8

12

Ω2(m

eV)

1.0

10

1E2

1.1E

1.1E

1.1E

-4

0

4

8

12

Ω2(m

eV)

1.0

14

1.8E

2.5E

3.3E

4.5E

(A) XRRR

h

b

e f

e f

h

b

4.5x105

3.3x104

2.5x103

1.8x102

1.4x101

1.0x100

1.1x105

1.1x104

1.1x103

1.0x102

1.0x101

1.0x100

2.8x105

2.3x104

1.9x103

1.5x102

1.2x101

1.0x100

(A) XRRR(A) XRRR

(B) XRLR

(C) XXXX

FIG. 6. Color online Calculated SIII3 ,2 , t1 using variouspulse polarization configurations at the TDHF level. 3=0 corre-sponds to HH exciton energy eH and 2=0 to 2eH. t1=150 fs andt2=0 fs.


075335-5

ing mixed two excitons, as shown in Fig. 6B. The same

conclusion holds if the indices v ,c in plkvc:k3* assume

the other possible value of v3 ,c2. The above analysis ex-plains why only mixed two excitons are created by theXRLR configuration at the TDHF level.

Finally, the XXXX configuration involves a superpositionof both XRRR and XRLR and is thus nonselective. Bothpure HH LH two excitons and mixed two excitons are gen-erated, as shown in Fig. 6C.

V. HIGHER CORRELATION EFFECTS: BEYOND TIME-DEPENDENT HARTREE-FOCK

The full NEE calculation, beyond TDHF, reveals many-body correlations among HH and LH excitons. By selectingpulse polarizations, one can completely separate correlationeffects from TDHF contributions for different types of twoexcitons. For the XRRR configuration Fig. 6A, any peaksother than b and h would be due to HMBC. For example,peaks around 0, 4 and 4, 4 must be from the correlatedmixed two excitons. This is because around these positionsthere should be no TDHF contributions according to Fig.6A but there should be HMBC contributions from mixedtwo excitons according to Fig. 5. For XRLR Fig. 6B, allpeaks other than e and f are induced by HMBC. For ex-ample, the peaks around 0, 0 and 4, 8 must be from thecorrelated pure HH LH two excitons. This is because thereare no TDHF contributions in their vicinity according to Fig.6B. However, there should be contributions from correlatedpure HH two excitons at 0, 0 and pure LH two excitons at4, 8 according to Fig. 5. Thus, the XRRR and XRLR con-figurations completely separate the correlation effects fromthe TDHF contributions for mixed two excitons and pure HHLH two excitons.

In Fig. 7, we present the full NEE calculation for theXRLR configuration. To show how 2D spectra evolve withtime-delay parameters, in panels A and B, we use differentvalues of t1 and the initial t2, t2. We see new peaks a andg not predicted in Fig. 5. The origin of these peaks is notclear. It may be due to interferences along different axes, oramong LH and HH excitons arising from small LH and HHexciton splitting. For example, peak a in panel A might bethe tail of peak e along 2 but may become an isolatedpeak due to the interference with LH excitons along 3.However, it is easy to identify peaks such as a and g inthe 2D spectra because their 2 values vary with differentexcitation conditions. Thus, we can easily exclude themwhen determining the two-exciton correlation energy. UsingFig. 7A, we can directly deduce the bound correlation en-ergies for various two excitons. From peaks c and d andm and n, we obtain that the two-exciton binding ener-gies for HH and LH two excitons are, respectively, 1.35 and1.45 meV.

The peaks that follow the predicted pattern in Fig. 5 suchas b–f, h, m, and n in panels A and B retain thesame 2 values for different time-delay parameters or pulsepolarization configurations. Therefore, these peaks can beused to determine two-exciton couplings, attractive two-exciton binding energy TBE or repulsive two-exciton scat-

tering energy TSE. To that end, we need only identify thepeaks below and above the lines 2=2eH=0,eH+eL4 meV and 2eL8 meV, respectively. For example, wecan obtain the energy of peak c along 2 and thus get theTBE for LH two excitons. Note that the unresolved peak, ashoulder d, to the red of peak c, as predicted by Fig. 5.Peaks d and c always appear in pairs and thus can helpidentify both peaks even though they are very weak and notwell resolved. Similarly, we obtain the TBE for HH twoexcitons from peaks m and n. Moreover, by using spec-trally narrow pulses and different detunings, one can evenclearly show the blueshifted TSE of correlated same-spinexcitons.32 For current pulse parameters, the TSE feature isnot seen.

The spectra shown in Fig. 7 completely separate the pureHHLH two excitons from mixed two excitons peaks eand f. We can thus accurately determine the correlationenergies of pure HH and LH two-excitons, even when thetwo-exciton features are weak e.g., peaks m and n in Fig.7A. The XRLR configuration excludes the TDHF contri-butions at 0, 0 and 4, 8 which contain correlated HH LHtwo excitons and thus further enhance the resolution. Suchresolution may not be achieved by other techniques SI andSII where features such as d and c or m and n arehidden under the stronger mixed two-exciton peaks, e andf.

To check the convergence of the correlation energies, wehave repeated the calculation of the 2D spectrum with thesame parameters as in Fig. 7A but with different basis sets.These are shown in Figs. 8A 14 sites and 8B 20 sites.We note that the main features such as mixed exciton peakse and f, pure LH two-exciton peaks c and d, and pureHH two-exciton peaks m and n had converged and agreewith Fig. 7A. In fact, the binding energies of pure HH and

-4

0

4

8

12

Ω2(m

eV)

1.0

8.5

73

6.2E

5.3E

4.5E

-2 0 2 4 6

-4

0

4

8

12

Ω2(m

eV)

Ω3(meV)

1.0

9.3

87

8.1E

7.5E

7E4

4.5x104

5.3x103

6.2x102

7.3x101

8.5x100

1.0x100

7.0x104

7.5x103

8.1x102

8.7x101

9.3x100

1.0x100

(A)

e f

cb

d

n m

c

d

e

a

n m

f

g

a

(B)

FIG. 7. Color online Calculated SIII3 ,2 , t1 spectra withXRLR polarization configuration. Exciton correlations are includedEqs. 7 and 9. A t1=300 fs and t2=300 fs. B t1=150 fs andt2=0 fs.


075335-6

LH two excitons can be obtained with very high resolution inthis experiment. For example, although we cannot resolveclearly peaks c and d, by connecting the ridges betweenpeaks c and d, we can obtain a straight line, whose energyalong 2 accurately gives the LH two-exciton binding en-ergy. However, we also note that the XRLR polarization con-figuration does not resolve the correlated mixed two excitonsbecause peaks e and f are dominated by the TDHF con-tribution, as shown in Fig. 6B. The contributions fromHMBC mixed two excitons will overlap with these TDHFcontributions and may not be easily distinguished.

The XRRR spectra are shown in Fig. 9. As in Fig. 7, theorigin of peaks a and g is not clear and might be attrib-uted to the cross interferences among HH and LH excitonsalong different axes. They do not follow the pattern in Fig. 5and their 2 values vary for different configurations. Accord-ing to Fig. 6A, there should be no signals at positions eand f for the XRRR configuration at the TDHF level.Therefore, e and f in Fig. 9 must come from correlatedmixed two excitons, according to Fig. 5. In this way, weachieve a complete separation of correlated mixed two exci-tons from their TDHF counterparts. In other techniques, SIRef. 24 and SII Appendix A, correlated mixed two exci-tons always overlap with other dominant peaks and may notbe easily resolved. Furthermore, we note that the mixed twoexcitons formed by two same-spin excitons do not have ap-preciable red shifts or blueshifts from eH+eL4 meV. Thisindicates that if using other techniques these correlation fea-tures will completely overlap with the stronger single-exciton transitions and may not be detected at all.

The XXXX spectra are shown in Fig. 10. As expected,this configuration cannot completely separate the correlationand TDHF contributions neither for pure HH LH two ex-citons nor for mixed two excitons. This is because the TDHFpeaks occupy all the major peak positions, as shown in Fig.

6C. Panels A and B are calculated with the same param-eters except that t1 and the initial t2, t2, are slightly different.We find strong interferences between the within and beyondTDHF contributions. For example, even though peaks eand f in Fig. 6C show the mixed two excitons from TDHFcontributions, in Fig. 10, peak f is missing due to interfer-ence. However, we can still resolve several peaks such as b,e, and h which have the same 2 values as compared tothe corresponding peaks in Figs. 7 and 9.

-4

0

4

8

12

Ω2(m

eV)

1.000

9.311

86.70

807.3

7518

7E4

(A)

e f

c

d

nm

-2 0 2 4 6

-4

0

4

8

12

Ω2(m

eV)

Ω3(meV)

1.000

9.311

86.70

807.3

7518

7E4

e f

c

d

nm

7.0x104

7.5x103

8.1x102

8.7x101

9.3x100

1.0x100

7.0x104

7.5x103

8.1x102

8.7x101

9.3x100

1.0x100

(B)

FIG. 8. Color online Calculated SIII3 ,2 , t1 spectra withsame parameters as in Fig. 7A but with different basis size: A 14sites and B 20 sites.

-2 0 2 4 6-2 0 2 4 6-2 0 2 4 6-2 0 2 4 6

-4-4-4-4

0000

4444

8888

12121212

ΩΩΩΩ3333(meV)(meV)(meV)(meV)

ΩΩΩΩ2222(m

eV)

(meV)

(meV)

(meV)

1.0

12

1.5E

1.8E

2.1E

2.6E

-4-4-4-4

0000

4444

8888

12121212

ΩΩΩΩ2222(m

eV)

(meV)

(meV)

(meV)

1.0

11

1.3E

1.4E

1.6E

1.8E1.8x101.8x101.8x101.8x105555

1.6x101.6x101.6x101.6x104444

1.4x101.4x101.4x101.4x103333

1.3x101.3x101.3x101.3x102222

1.1x101.1x101.1x101.1x101111

1.0x101.0x101.0x101.0x100000

2.6x102.6x102.6x102.6x105555

2.1x102.1x102.1x102.1x104444

1.8x101.8x101.8x101.8x103333

1.5x101.5x101.5x101.5x102222

1.2x101.2x101.2x101.2x101111

1.0x101.0x101.0x101.0x100000

eeee ffff

hhhh

bbbb

aaaa

ffff

hhhh

bbbb

aaaa

eeee

gggg

gggg

(A)(A)(A)(A)

(B)(B)(B)(B)

FIG. 9. Color online Calculated SIII3 ,2 , t1 spectra withXRRR polarization configuration. Exciton correlations are includedEqs. 7 and 9. A t1=300 fs and t2=300 fs. B t1=150 fs andt2=0 fs.

-2 0 2 4 6

-4

0

4

8

12

Ω2(m

eV)

Ω3(meV)

1.0

11

1.3E

1.4E

1.6E

1.8E

-4

0

4

8

12

Ω2(m

eV)

1.0

10

1.1E

1.1E

1.2E

1.2E1.2x105

1.2x104

1.1x103

1.1x102

1.0x101

1.0x100

1.8x105

1.6x104

1.4x103

1.3x102

1.1x101

1.0x100

g

b

h

a

a

g

e

h

b

(A)

(B)

FIG. 10. Color online Calculated SIII3 ,2 , t1 spectra withXXXX polarization configuration. A t1=300 fs and t2=300 fs.B t1=150 fs and t2=0 fs.


075335-7

In summary, the proposed 2DCS technique is particularlyuseful for resolving different correlated two excitons. Theseparation of two-exciton HMBC effects from the TDHFlevel may be achieved by selecting specific pulse polariza-tions.

ACKNOWLEDGMENT

The support of the Chemical Sciences, Geosciences, andBiosciences Division, Office of Basic Energy Sciences, U.S.Department of Energy is gratefully acknowledged.

APPENDIX A: THE SII TECHNIQUE

The SI technique was analyzed in Ref. 24. For complete-ness and comparison with the SIII technique discussed in thiswork, we present below a Feynman diagram analysis of theSII technique. We denote the heterodyne-detected signal gen-erated by three very short impulsive pulses along thephase-matching direction kII=k1−k2+k3 as SIIt3 , t2 , t1.24

The corresponding 2D signal defined as

SII3,t2,1 dt3dt1SIIt3,t2,t1ei3t3ei1t1

A1

will be displayed in the 3 ,1 plane for a fixed t2. Usingthe notation of Refs. 15 and 24, this signal is described bythree basic Feynman diagrams iv, v, and vi, where ivand v only involve single excitons and vi involves bothsingle and two excitons. The complete set of diagrams de-rived from the three basic diagrams by specifying the various

HH and LH excitons are shown in Fig. 11. fH, fL, and fMdenote the various two excitons. For example, due to Cou-lomb interactions, fH can be either redshifted fH=2eH−H1

,bare fH=2eH, or blueshifted fH=2eH+H2

compared to twicethe single HH exciton energy, 2eH. In diagrams via to vid,each solid symbol describes a type of redshifted two excitonsand the corresponding open symbol describes blueshiftedones. Open circles denote contributions from bare two exci-tons.

The SII signals from iva–ivd, va–vd, and via–vid are plotted schematically in Figs. 12iv, 12v, and12vi, respectively. The total spectrum obtained by sum-ming all diagrams is shown in the panel s of Fig. 12, whereoverlapping peaks are slightly displaced for clarity.

Note that different types of two excitons are only partiallyseparated in panel s. Pure HH LH two excitons e.g., solidred triangle are still overlapped with the mixed two excitonse.g., solid green square. The situation is similar in all other2D projections of SII where pure HH LH two excitons andthe mixed two excitons may not be fully separated.

(va)

gg gg

gggg

gg gg

gggg

)(vb

(vc) (vd)

He

He

Le

Le

3k

4k

1k

2k-

3k

3k

3k

4k

4k4

k

2k-

2k-

2k-

1k

1k

1k

He

He

Le

Le

g

gg

g

(iva)

He

gg gg

gggg

gg gg

gggg

)(ivb

(ivc) (ivd)

He

He

Le

Le

He

Le

Le

1k

2k-

3k

4k

1k

1k

1k

2k-

2k-

2k-

3k

3k

3k

4k

4k

4k

(via) (vib))(via'

gg

HHee

He L

e

Hf

gg gg

gggggg

HHee

HHee

LLee

LLee

LLee

1k

2k-

3k

3k

3k

4k 4

k4

k

He

Lf

Mf

Mf

Mf

)(vib' (vic)

Mf

2k-

2k-

1k

1k

1k

3k

4k

2k-

1k

3k

4k

2k-

1k

3k

4k

2k-

(vid)

He

Le

Le

2t

1t

3t

t

1τ

2τ

3τ

2t

1t

3t

t

1τ

2τ

3τ

FIG. 11. Color online Feyn-man diagrams for SII techniquederived from diagrams iv, v,and vi of Fig. 1 in Ref. 24.

1Ω

He

Le

1Ω

3Ω

He

Le

He

Le

(s)

He

Le

3Ω

(iv) (v)

(vi)

FIG. 12. Color online The SII3 , t2 ,1 signal. iv Contri-butions of pathways iva, ivb, ivc, and ivd. v Contributionsof pathways va, vb, vc, and vd. vi Contributions of path-ways via to vid. x Total spectrum. The dashed line marks thediagonal peaks.


075335-8

APPENDIX B: WAVE-VECTOR SELECTION FOR THE SIII

TECHNIQUE

In our earlier work,24 we presented the equations of mo-tion for calculating SI 2DCS for a two-pulse configuration.Below, we extend them to calculate SIII 2DCS signals in-

duced by three pulses. To that end, we present the equationsof motion for selecting the kIII=k1+k2−k3 component ofthe interband density matrix pij

vc in Eq. 7. Additional termsand equations are necessary in the three-pulse scheme due tothe more permutations among different pulses.

− i

tpij

vc:k1+k2−k3 −i

expij

vc:k1+k2−k3 = − n

Tjnc pin

vc:k1+k2−k3 − m

Tmiv pmj

vc:k1+k2−k3 + Vijpijvc:k1+k2−k3 +

klvc

Vkj − Vki − Vlj + Vli

· Sk3k2k1plk

vc:k3*pljvc:k2pik

vc:k1 − Sk3k2k1plk

vc:k3*plkvc:k2pij

vc:k1

− Sk3k2k1plk

vc:k3*Blkijvcvc:k1+k2 − E3

*t · klvc

ilvc*Sk2k1

pklvc:k2pkj

vc:k1

+ ljvc*Sk2k1

plkvc:k2pik

vc:k1 − E2t · klvc

ilvc*Sk3k1

pklvc:k3*pkj

vc:k1

+ ljvc*Sk3k1

plkvc:k3*pik

vc:k1 − E1t · klvc

ilvc*Sk3k2

pklvc:k3*pkj

vc:k2

+ ljvc*Sk3k2

plkvc:k3*pik

vc:k2 , B1

where ex describes exciton dephasing time. In Eq. B1, Skikjand Skikjkk

i , j ,k=1,2 ,3 describe all possible ways making therelevant terms to contribute along the direction kIII=k1+k2−k3. For example,

Sk3k2k1plk

vc:k3*pljvc:k2pik

vc:k1 = plkvc:k3*plj

vc:k2pikvc:k1 + plk

vc:k3*pljvc:k1pik

vc:k2 + plkvc:−k2*plj

vc:−k3pikvc:k1

+ plkvc:−k2*plj

vc:k1pikvc:−k3 + plk

vc:−k1*pljvc:−k3pik

vc:k2 + plkvc:−k1*plj

vc:k2pikvc−:k3

= plkvc:k3*plj

vc:k2pikvc:k1 + plk

vc:k3*pljvc:k1pik

vc:k2 + plkvc:k2plj

vc:k3*pikvc:k1

+ plkvc:k2plj

vc:k1pikvc:k3* + plk

vc:k1pljvc:k3*pik

vc:k2 + plkvc:k1plj

vc:k2pikvc:k3*, B2

where each density matrix has either positive or negativewave vector. Two other typical examples showing the opera-tions of Skikj

and Skikjkkare

E2t · klvc

ilvc*Sk3k1

pklvc:k3*pkj

vc:k1

= E2t · klvc

ilvc*pkl

vc:k3*pkjvc:k1

+ ilvc*pkl

vc:−k1*pkjvc:−k3

= E2t · klvc

ilvc*pkl

vc:k3*pkjvc:k1

+ ilvc*pkl

vc:k1pkjvc:k3* B3

and

Sk3k2k1plk

vc:k3*Blkijvcvc:k1+k2 = plk

vc:k3*Blkijvcvc:k1+k2

+ plkvc:−k2*Blkij

vcvc:k1−k3

+ plkvc:−k1*Blkij

vcvc:k2−k3

= plkvc:k3*Blkij

vcvc:k1+k2

+ plkvc:k2Blkij

vcvc:k1−k3

+ plkvc:k1Blkij

vcvc:k2−k3. B4

To solve Eq. B1 for the time-dependent interband coher-ences matrix pij

vc:k1+k2−k3, we need the equations of motionfor the first-order in the optical field density matricespij

vck=1,2 ,3; the three second-order two-exciton matri-

ces Blkijvcvc:k1+k2, Blkij

vcvc:k1−k3, and Blkijvcvc:k2−k3. The first-

order equations are given by

− i

tpij

vck −i

expij

vck = − n

Tjnc pin

vck − m

Tmiv pmj

vck

+ Vijpijvck + Et · ij

vc*,

= 1,2,3 . B5

The equations of motion for the three two-exciton densitymatrices are, respectively,


075335-9

− i

tBlkij

vcvc:k1+k2 −i

2exBlkij

vcvc:k1+k2

= − m

Tjmc Blkim

vcvc:k1+k2 + Tmiv Blkmj

vcvc:k1+k2

+ Tkmc Blmij

vcvc:k1+k2 + Tmlv Bmkij

vcvc:k1+k2

+ Vlk + Vlj + Vik + Vij − Vli − VkjBlkijvcvc:k1+k2

− Vlk + Vij − Vli − VkjSk2k1pik

vck1pljvck2

+ Vik + Vlj − Vli − VkjSk2k1plk

vck1pijvck2, B6

− i

tBlkij

vcvc:k1−k3 −i

2exBlkij

vcvc:k1−k3

= − m

Tjmc Blkim

vcvc:k1−k3 + Tmiv Blkmj

vcvc:k1−k3

+ Tkmc Blmij

vcvc:k1−k3 + Tmlv Bmkij

vcvc:k1−k3

+ Vlk + Vlj + Vik + Vij − Vli − VkjBlkijvcvc:k1−k3


vck1pljvck3*


vck1pijvck3*, B7

and

− i

tBlkij

vcvc:k2−k3 −i

2exBlkij

vcvc:k2−k3

= − m

Tjmc Blkim

vcvc:k2−k3 + Tmiv Blkmj

vcvc:k2−k3

+ Tkmc Blmij

vcvc:k2−k3 + Tmlv Bmkij

vcvc:k2−k3

+ Vlk + Vlj + Vik + Vij − Vli − VkjBlkijvcvc:k2−k3


vck2pljvck3*


vck2pijvck3*, B8

where 2ex describes two-exciton dephasing times.By solving Eqs. B1 and B5–B8, we obtain the kIII

component of interband coherences, pijvc:k1+k2−k3. Then, SIII

2DCS can be obtained through Eq. 12. We found that forwell separated pulses, numerical effort can be reduced sig-nificantly by employing only a subset of the above equations.Equations B7 and B8 can be neglected with small effectson the correlation energies. Therefore, all 2D spectra re-ported are calculated by neglecting Eqs. B7 and B8. Thisis equivalent to neglecting the Sk3k2k1

before the term

plkvc:k3*Blkij

vcvc:k1+k2 in Eq. B1, or including in Eq. B4only the term Blkij

vcvc:k1+k2 but neglecting Blkijvcvc:k1−k3 and

Blkijvcvc:k2−k3. This is also the general practice when calculat-

ing 2DCS for Frenkel excitons.

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