ARTICLE IN PRESS

0022-2313/$ - se

doi:10.1016/j.jlu

�CorrespondE-mail addr

Journal of Luminescence 124 (2007) 321–326

www.elsevier.com/locate/jlumin

Slow dephasing and local field effects of exciton–biexciton transition inb�ZnP2 detected by femtosecond degenerate four-wave mixing

S. Nakanishia,�, H. Itoha, H. Miyagawaa, M. Sakamotob, O. Arimotob

aDepartment of Advanced Materials Science, Kagawa University, Takamatsu 761-0396, JapanbDepartment of Physics, Okayama University, Okayama 700-8530, Japan

Received 10 October 2005; received in revised form 10 February 2006; accepted 28 March 2006

Available online 16 May 2006

Abstract

In femtosecond degenerate four-wave mixing (FWM) spectroscopy for the exciton and biexciton system in a semiconductor b�ZnP2,

we have observed that the spectrally resolved FWM signals from the exciton–biexciton transition exhibit a specific time profile consisting

of a growing up and decay for negative and positive delay between two excitation pulses, respectively. The decay profile indicates

unexpectedly much slower dephasing of the exciton–biexciton transition compared to 1s exciton polariton. The time profile is interpreted

as the manifestation of local field renormalization due to many-body Coulomb correlation among carriers in b�ZnP2, which is made

observable due to the narrow homogeneous linewidth of the exciton–biexciton transition.

r 2006 Elsevier B.V. All rights reserved.

PACS: 71.35.�y; 78.47.+p; 42.50.Md

Keywords: b�ZnP2; Exciton–biexciton transition; Local field effects; Four-wave mixing

1. Introduction

The dynamics of the carriers in semiconductor materialshas been of fundamental interest in the pure science andtheir application to the novel functional devices. In last twodecades many studies have been performed on the carrierdynamics in semiconductor on picosecond and femtose-cond time scales [1]. Those studies provided us richknowledge of the dynamics of electrons, holes, excitonsand biexcitons in the bulk [2,3] and quantum mechanicallyconfined systems such as quantum well, quantum wire andquantum dots [4–6]. In the investigation of ultrafastdephasing dynamics in semiconductors, four-wave mixing(FWM) technique has been conventionally employedbecause it is one of the most versatile techniques inultrafast spectroscopy. By using FWM technique, manyworks have found out that the response of carriers insemiconductor to the coherent excitation is significantlydifferent from that of atomic system [1,7–15], and in some

e front matter r 2006 Elsevier B.V. All rights reserved.

min.2006.03.017

ing author. Tel./fax: +81 87 864 2413.

ess: nakanish@eng.kagawa-u.ac.jp (S. Nakanishi).

cases the biexciton state (or two-exciton state) playssignificant role in the FWM signals [16–22]. The differentbehavior of FWM signals from the atomic systemprincipally originates from many-body correlation effectsamong the carriers, such as the local field renormalization(LFR), phase space filling (PSF) and excitation-induceddephasing (EID).In this paper we present the results of time-integrated

(TI) and spectrally resolved (SR) degenerate femtosecondFWM spectroscopy to study the dephasing dynamics ofexcitons and biexcitons in a semiconductor b�ZnP2 with adirect band gap of 1.603 eV at low temperature. In additionto rather conventional results for the 1s exciton polariton,we have found out that the SR-FWM signal from theexciton–biexciton ðx2bÞ transition exhibits a specific timeprofile consisting of a growing up and a decay for negativeand positive time delay between two femtosecond excita-tion pulses, respectively. This time profile makes a markedcontrast to that for 1s exciton polariton. More interest-ingly, the decay profile indicates unexpectedly much slowerdephasing of the x2b transition compared to 1s excitontransition. We interpret these our findings for the x2b

ARTICLE IN PRESSS. Nakanishi et al. / Journal of Luminescence 124 (2007) 321–326322

transition in terms of LFR due to many-body Coulombcorrelations in b�ZnP2. Though the dephasing of x2b

transitions has been studied since two decades, there areonly a few works that report the evident observation of themany-body correlation effects for the x2b transition.Therefore, we think our findings as a valuable example ofmany-body correlation effects detected in the x2b transi-tions. We discuss the dephasing nature of the x2b

transition in b�ZnP2 compared with previous studiesperformed in other semiconductors [20–22]. It should beemphasized that the effect of LFR is made observable dueto the narrow homogeneous linewidth of the x2b

transition in b�ZnP2.

2. Experimental

The b�ZnP2 sample and experimental setup in thepresent study were almost the same as those used in ourprevious papers [23,24]. Single crystals of monoclinicb�ZnP2 were grown from the vapor phase with the opticalflat surface of a bc face. In b�ZnP2 excitons have a largebinding energy of approximately 40meV, which leads tothe distinct series of singlet and triplet exciton resonances[25]. The triplet exciton resonances were previously studiedby femtosecond coherent pulse propagation [26]. It isknown that the 1s exciton resonance at 1.560 eV shows thepolariton effect with a longitudinal-transverse splitting aslarge as 4.8meV [27,28]. The luminescence study indicatedthe binding energy of the biexciton in b�ZnP2 to beapproximately 10.4meV [29,30]. Whereas in b�ZnP2 it isimpossible to manipulate the contribution of the biexcitonto FWM signal by the polarization selection rule, this largebinding energy of the biexciton is advantageous indiscriminating between the contributions to FWM signalsof the exciton and biexciton, compared to the well-studiedsemiconductors with a relatively small binding energy ofexciton and biexciton. Fig. 1 shows the reflection spectrumof the sample at 10K, together with the energy diagram for

1.54 1.56 1.58Photon Energy (eV)

Ref

lect

ivity

(ar

b. u

nits

)

10 K

E//c

g

x

b

2x

1.56

0 eV

10.4 meV

1.55

0 eV

Fig. 1. Left: reflection spectrum of b�ZnP2 in the vicinity of the 1s exciton

polariton at 10K. The reflection structure around 1.561 eV corresponds to

the 1s exciton polariton resonance. Right: energy level diagram for exciton

and biexciton system in b�ZnP2. g, x and b denote crystal ground state,

exciton and biexciton state, respectively.

exciton and biexciton system. One can see a large reflectionstructure around 1.561 eV arising from the 1s excitonpolariton. There is no distinct reflection structure at1.550 eV, where the sample used in previous study showeda structure due to the A-exciton [23].For the degenerate FWM measurement around 1s

exciton polariton region, we used a cw mode-lockedTi:sapphire laser with the repetition rate of 76MHz, thepulse duration of 100 fs corresponding to the bandwidth(FWHM) of 23meV and the typical power of 1mW. Thecenter energy of the laser was set at 1.561 eV. In themeasurements we employed the two-beam backwarddegenerate FWM geometry because of the large absorptionof the 1s exciton polariton. The two beams, with the wavevectors ~k1 and ~k2, were focused onto the sample, one ofwhich was time delayed by t with respect to the other. Inorder to excite 1s excitons and biexcitons, the electricvector ~E of laser pulse was set parallel to c-axis. Thebackward degenerate FWM signals in the direction of2~k2 �

~k1 were fed either directly into a photomultiplieror a monochromator for the TI and SR measurement,respectively.

3. Results and discussion

Fig. 2(a) represents a typical t-dependence of the TI-FWM signal observed at 10K by exciting b�ZnP2 samplewith a cw mode-locked Ti:sapphire laser at 1.561 eV. Thespectrum of excitation laser covered the wide energy regionaround the 1s exciton polariton resonance as depicted by adashed line in Fig. 2(b). The FWM signal for the positivedelay shows that the fast signal decay is followed by aslower picosecond decay. The latter decay componentprovides the dephasing time T2 of approximately 5.8 ps,assuming the extreme inhomogeneous broadening. Itshould be noted that, in the negative delay, there exists asmall FWM signal, which is attributed to the contributionfrom the x2b transition as mentioned below.In order to elucidate the detailed origin of the FWM

signal, we performed the SR-FWM measurements at afixed time delay. Fig. 2(b) shows the spectrum of FWMsignal at the fixed time delay of t ¼ 0. ET and EL inFig. 2(b) denote the 1s transverse and longitudinal excitonenergy, respectively. It is observable that the majorcontribution to the FWM signal arises from the 1s excitonpolariton between 1.560 and 1.565 eV. Therefore, thedephasing time T2 ¼ 5:8 ps obtained above from Fig. 2(a)probably corresponds to that of the 1s exciton polariton.As one can see the spectrum of the exciton polariton with arelatively broad line shape compared with the homoge-neous linewidth (FWHM) of 0.22meV derived fromT2 ¼ 5:8 ps, the assumption of the extreme inhomogeneousbroadening is justified for the exciton polariton resonance.In addition, we point out that the contribution from the 1sexciton polariton at ET is relatively small compared to thatat EL. It is probably because the contribution at EL isenhanced in the backward FWM geometry due to the low

ARTICLE IN PRESS

1.54 1.56 1.58

0

0.5

1

0 2 410-2

10-1

100

Photon Energy (eV)

FWM

Sig

nal I

nten

sity

Laser

abcd

FWM

Sig

nal I

nten

sity

Delay Time (ps)

10 K

(a)

(b)

ET EL

Fig. 2. (a) TI-FWM signal of b�ZnP2 at 10K as a function of the time

delay t between two femtosecond excitation pulses measured in the

vicinity of the 1s exciton polariton. (b) Spectrum of the FWM signal in (a)

at a fixed time delay t ¼ 0. ET and EL denote the 1s transverse and

longitudinal exciton energy, respectively. The dashed line shows the

spectrum of the femtosecond laser.

-20 0 20 4010-3

10-2

10-1

100 1.550 eV

Delay Time (ps)

FWM

Sig

nal I

nten

sity

10 K

18 K

Fig. 3. Decay profiles of SR-FWM signal at 1.550 eV in b�ZnP2 at 10K

(solid line) and 18K (dashed line) as a function of t.

S. Nakanishi et al. / Journal of Luminescence 124 (2007) 321–326 323

reflectivity and the large nonlinear Fresnel factor [31,32]. Itshould be noted that a small contribution to the FWMsignal definitely exists at 1.550 eV, where the biexciton isknown to fluoresce with the transition to the 1s exciton (seethe energy level diagram in Fig. 1).

When the t-dependence of SR-FWM signal wasmeasured at three energy points (a–c in Fig. 2(b)) aroundthe L-T gap of 1s exciton polariton, we observed that theSR-FWM signal showed the time profile without appreci-able signal intensity for negative delay and that thedephasing time became longer as the observation energywas varied from EL to ET. The former fact is an evidenceof the inhomogeneous broadening of exciton resonance.The latter fact is consistent with our previous observations[23] and accounted for in terms of phonon scattering forthe exciton polariton. From these measurements, weattribute the fast decay for 0oto0:5 ps in Fig. 2(a) tothe contribution from the exciton polariton at EL. Thedephasing time determined in Fig. 2(a) corresponds to thatof the exciton polariton at ET.

The FWM signal component at 1.550 eV (d in Fig. 2(b))was found to exhibit a distinct t-dependence in theSR measurement, as depicted in Fig. 3. Compared with

the TI-FWM signal in Fig. 2(a) where the contribution of1s exciton polariton dominates, it shows a much slowerdecay both for positive and negative time delay, implyingslow dephasing of the relevant transitions. In particular,the evident FWM signal for negative time delay makes asharp contrast to that for 1s exciton. The time constants ofthe rise and decay for FWM signal at 10K in Fig. 3 arederived to be 3.52 and 7.32 ps, respectively. The FWMsignal for negative time delay in Fig. 2(a) might beattributed to this FWM signal at 1.550 eV, whereas therise time constant in Fig. 2(a) is smaller than that in Fig. 3.The discrepancy in the rise time constant probably resultsfrom insufficient precision in determining the time constantin Fig. 2(a) because of a small contribution of FWM signalat 1.550 eV.The directionality and t-dependence of the FWM signals

in Fig. 3 confirm that they originate from the coherentpolarizations created at 1.550 eV by the present excitationgeometry. The contribution of A-exciton at 1.550 eVobserved in our previous work [23] is excluded based onthe reflection spectrum in Fig. 1. Since the biexciton inb�ZnP2 is known to show the fluorescence at 1.550 eV, weattribute the FWM signal to the coherent polarizationsassociated with the x2b transition. We emphasize that, asseen in Fig. 2(b), the spectral line at 1.550 eV has asymmetric shape and much narrower linewidth ðo0:8meVÞthan that of the biexciton fluorescence ð�2:4meVÞ reportedpreviously [29]. The biexciton fluorescence spectrum wasreported to have an asymmetric line shape with a tail tolower energy side, which is caused by the thermaldistribution of the biexciton in its dispersion curve. Thecharacteristics we observed are consistent with our recentobservation of the very sharp spectrum at 1.550 eV in themeasurement of femtosecond transient reflectivity changesfor the same b�ZnP2 sample reported elsewhere [24].As we excited the sample by the coherent femtosecond

pulses, the evident FWM signal for negative time delay inFig. 3 is not explained in the framework of optical Blochequations (OBE) for the independent two-level systems.Although the OBE for the independent three-level systems

ARTICLE IN PRESSS. Nakanishi et al. / Journal of Luminescence 124 (2007) 321–326324

can explain the FWM signal for negative time delay astwo-photon coherence induced FWM signals [17,33], itcannot account for the FWM signals for negative andpositive time delay simultaneously. Instead, we considerthat the observed FWM signals at 1.550 eV manifestthemselves as the results of many-body correlation amongthe carriers in b�ZnP2. The many-body correlation effects,such as LFR, PSF and EID, are taken into account in theanalysis of nonlinear optical response of semiconductors byusing the semiconductor Bloch [7–15], the extended OBEfor the multi-level scheme [18,19,34], and the theoreticalapproach on a microscopic basis [22,35–37]. Thesetheoretical calculations can explain the FWM signalsobserved for negative time delay in exciton and biexcitonsystem. Especially, the latter two theoretical approachesexplicitly take into account the contribution of thebiexciton and two exciton states.

From the viewpoint of perturbation treatment, thecoherent polarizations at 1.550 eV are regarded as the fifthorder signals, as follows. The first femtosecond pulse withthe wave vector ~k1 induces the population of second orderat the exciton energy and then, in the third order, itgenerates the polarizations with ~k1 at the x2b transition.The second pulse with ~k2 interacts twice with the generatedpolarizations and creates the polarizations with 2~k22~k1 atthe x2b transition, which emit the FWM signal in thedirection of 2~k22~k1. Since the recent microscopic approachhas been developed with the description of exciton andx2b transitions up to fifth order [22], we employ it totheoretically understand our FWM signals at 1.550 eV. Inthat approach the optical nonlinear response of single-exciton p and biexciton transitions B to the excitationlight EðtÞ is considered to follow the equations of motiongiven by

qtp ¼ i½oxpþ mx � E � bmxEðp�pþ p�p�pp

þ B�ppþ Bp�p� þ B�BÞ

þ V ðp�Bþ p�p�pBþ B�pBÞ� � gxp, ð1Þ

qtB ¼ i½obB þ mx;bEðpþ p�BÞ� � gbB, ð2Þ

where all indices are neglected and gx and gb are thedephasing rate of the exciton and biexciton transitions withthe energies of ox and ob, respectively. mx and mx;b

represent the dipole matrix element of the excitontransition and x2b transition, respectively. b and V denotethe strength of PSF and LFR due to the Coulombinteraction, and these are primary sources of the opticalnonlinearity. The induced total polarization ~P up to fifthorder is obtained by

~P ¼ mxðpð3Þ þ pð5ÞÞ þ mx;bpð5Þ, (3)

where pðnÞ stands for the nth order solution of p.For the two beam degenerate FWM signals with

the ultrashort-pulse limit, i.e. EðtÞ ¼ E1dðtÞ expði~k1 �~rÞþE2dðt� tÞ expði~k2 �~rÞ, one can obtain the analytical resultsof pð3Þ in a simplified model, as shown in the appendix ofRef. [35]. While the analytical results were derived for the

pump-probe signals, they are regarded as the third orderpolarizations corresponding to the FWM signals byexchanging the roles of first and second pulses. Simplecalculations based on the results lead to the fact that theTI-FWM signal intensity I ð3ÞðtÞ, which we detect in presentstudy, reveals the time profile as a function of delay time tbetween two exciting pulse, as

I ð3ÞðtÞ / YðtÞAe�2gxt þYð�tÞBe4gxt, (4)

where YðtÞ denotes the Heaviside step function and thefactors A and B involve b and V , meaning the nonlinearpolarizations originate from PSF and LFR. This expres-sion is for the homogeneously broadened exciton transitionand consists of the signal rise and decay with the decayrates of 4gx and 2gx for negative and positive time delay,respectively. It is known that, when the exciton transition isinhomogeneously broadened, the FWM signal for negativetime delay is smeared out and the decay rate for positivedelay becomes 4gx [1,7]. Therefore, as seen in Fig. 2(a), theexciton transition in b�ZnP2 is concluded to be inhomo-geneously broadened.It is very difficult to obtain the analytical solution in fifth

order, pð5Þ, since all the source terms in Eqs. (1) and (2)contribute to pð5Þ. Instead, numerical calculations up tofifth order can be performed as in Ref. [37] where in thepump-probe experiments the fifth order contributions tothe differential absorption spectra were shown to beessentially the negative of the third order contributionsfor the resonant excitation. Therefore, we infer that thefifth order FWM signals of the exciton, corresponding tothe second term in Eq. (3), show the same time profile asthat described by Eq. (4). Based on these considerations,we are led to the conclusion that the FWM signalsobserved at the x2b transition, corresponding to the thirdterm in Eq. (3), also follow the similar time profile to Eq.(4) for the homogeneously broadened transition. OurFWM signals at 1.550 eV in Fig. 3 exhibits exactly thebehavior predicted by Eq. (4) with replacing gx by gb: theFWM signal at 10K for positive delay decays with a timeconstant 7.32 ps, corresponding to the dephasing timeT2ð¼ 1=gbÞ ¼ 14:6 ps for the x2b transition, and fornegative delay it grows with a rise time of 3.52 ps,approximately half the time constant for positive delay.When the FWM signals were measured with changing thesample temperature, we observed that the similar relationbetween the two time constants for negative and positivedelay fairly holds up to 50K, as displayed in Fig. 4 wherethe signal decay rates (inverse of the time constant) for to0and t40 are plotted as a function of temperature. Thetemperature dependence of the two time constants ensuresthat both FWM signals for negative and positive time delayreflect the dephasing of same transition. Therefore, weconclude that the time profile of FWM signal at 1.550 eVresults from Coulomb correlation among the carriers forthe homogeneously broadened x2b transitions in b�ZnP2.We consider the FWM signals are predominated by theLFR effect compared to the PSF effect, which was proved

ARTICLE IN PRESS

0 20 400

1

2

Temperature (K)

Dec

ay r

ate

(1/p

s)

τ < 0

τ > 0

Fig. 4. Decay rates of FWM signals at 1.550 eV in b�ZnP2 measured for

negative delay (solid circle) and positive delay (open circle) as a function of

sample temperature.

S. Nakanishi et al. / Journal of Luminescence 124 (2007) 321–326 325

by numerical calculations in Ref. [35]. In addition, thishigher order creation process explains why the signalcomponent at 1.550 eV is so small.

Now we discuss our FWM signals observed at the x2b

transition in b�ZnP2 with comparison to those in othersemiconductors. Early investigations on the dephasing ofthe x2b transition were performed in CdSe [16,17] and theauthors reported a longer dephasing time of x2b transitionthan that of excitons, as we observed in b�ZnP2. However,the effects of Coulomb correlations, expected to appear asan evident FWM signal for to0, were not observed,probably due to the inhomogeneous broadening of therelevant transitions. Recent studies on x2b transition havebeen carried out by FWM technique in InGaAs/GaAs [20]and high-quality GaAs [21] single quantum wells, and theauthors have discussed dephasing behavior of x2b

transition compared to the exciton dephasing. Whereasthe former study shows that the signal component for to0of FWM at x2b transition is negligible due to theinhomogeneous broadening, the latter study detects theevident FWM signals for to0 in x2b transition, which areattributed to the two-photon coherence-induced freeinduction decay. In both studies the dephasing at x2b

transition is measured to be comparable to or faster thanthe exciton dephasing. This fact contradicts our observa-tion in b�ZnP2 that the dephasing at x2b transition isalmost three times slower than the exciton dephasing at10K. We believe that the time profile of the FWM signalsin Fig. 3 reveals the homogeneous broadening of the x2b

transition and the effects of LFR play dominant role.Assumption of the homogeneous broadening is supportedby a report in Ref. [30] that the excitation spectra of thebiexciton fluorescence at 1.550 eV in b�ZnP2 show a verysharp resonance (with the width of 0.3meV) at two-photonresonance energy of 1.555 eV, meaning sharp linewidth ofthe biexcitons generated by two-photon excitation. Broad-ening mechanism is usually supposed to correlate between

the exciton and x2b transitions and then the homogeneousbroadening is expected for the exciton transition. However,as the exciton in b�ZnP2 has large polariton effect due tothe coupling between the exciton and photon, the excitonspectra are significantly modified. This polariton effectwould result in the virtually broadened spectra as in Fig. 1and faster dephasing of the exciton transition. In addition,the ratio and temperature dependence of the two timeconstants in Fig. 4 confirm that the FWM signals for to0demonstrate the effects of LFR in b�ZnP2. The similarFWM signals at x2b transition have rarely been reportedso far, and we think our observations give an uniqueexample of LFR effects detected at x2b transition insemiconductors.

4. Conclusion

We have performed femtosecond TI- and SR-FWMexperiments in b�ZnP2 and found that the x2b transitionshave an extraordinary slow dephasing compared with 1sexcitons. We conclude that the observed time profile ofFWM signals from the x2b transitions, consisting of agrowing and decay for negative and positive delay,respectively, is attributed to the LFR effects due tomany-body correlation effects among carriers in b�ZnP2

and indicate the homogeneous broadening of the x2b

transition. The temperature dependence of two timeconstants associated with the FWM signals supports ourunderstanding of the FWM signals based on the micro-scopic theoretical approach. The faster dephasing ofexciton than that of x2b transition is tentatively ascribedto the large polariton effect of the exciton transition.

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