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Journal of Luminescence 112 (2005) 11–16

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Bosonic stimulation of cold excitons in a harmonic potentialtrap in Cu2O

N. Naka�, N. Nagasawa

Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Available online 24 November 2004

Abstract

A high-density cold exciton gas is generated in a harmonic potential trap by two-photon resonance excitation of

Cu2O at 2 K. Strongly localized light signal is observed at the bottom of the potential trap when the exciton density

exceeds 1017 cm�3: The possibility of bosonic stimulation of excitons from the trap rim to the bottom is discussed in

connection with the origin of the light signal.

r 2004 Elsevier B.V. All rights reserved.

PACS: 71.35.Lk; 71.45.�d; 03.75.Kk

Keywords: Exciton; Bosonic stimulation; Two-photon spectroscopy

1. Introduction

Along with the rapid advancement of experi-ments on Bose–Einstein condensation (BEC) inatomic systems, BEC of excitons has been attract-ing considerable attention as a phase transition ina photo-controlled open system composed of finitelifetime particles.

An exciton is a bound pair of photo-excitedelectron and hole in a semiconductor. If thecarriers are generated by band-to-band excitation,the effective temperature of the excitons is

e front matter r 2004 Elsevier B.V. All rights reserve

min.2004.09.035

ng author. Tel.: +81 3 5841 4182, fax:

.

ss: naka@ap.t.u-tokyo.ac.jp (N. Naka).

generally higher than the lattice temperature. Thisusually hinders the system in going beyond theBEC phase boundary. On the other hand, reso-nance excitation creates excitons selectively at themomentum close to zero due to momentum-energyconservation. The initial kinetic energy of excitonsthus generated is as small as 10 meV, and thesystem temperature can be even lower than thelattice temperature at the initial stage.

This study is concerned with resonantly excitedorthoexcitons and paraexcitons in cuprous oxide(Cu2O). This system has long been thought to bethe best candidate to realize excitonic BEC.Because of the strong exchange interaction, whichcauses large (� 12 meV) splitting betweenspin–singlet orthoexcitons and spin–triplet

d.

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N. Naka, N. Nagasawa / Journal of Luminescence 112 (2005) 11–1612

paraexcitons, these excitons are stable against theformation of biexcitons or electron hole dropletsaround the critical density for BEC. Also due tothe unique band structure with positive-paritybands, they have extremely long lifetime that isadvantageous to reach quasi-thermal equilibrium.In order to achieve a cold exciton gas in Cu2O,however, we need to employ two-photon excita-tion instead of one-photon resonance excitation,because the latter process is only quadrupoleallowed for orthoexcitons. The two-photon excita-tion is dipole-allowed for orthoexcitons. Theparaexcitons are optically inactive and producedby down-conversion of orthoexcitons. The time-scale of the conversion and the lifetime ofparaexcitons have been measured as several [1]and 300 [2] nanoseconds, respectively.

In this paper, we overview the experimentalresults obtained by our new approach using two-photon resonance excitation of high-density ex-citons in potential traps. We also discuss therelevance of BEC to the observed phenomenon,and give prospects to prove BEC along thisapproach.

Fig. 1. (a) Schematic illustration showing experimental config-

uration. The stress is applied along z parallel to [1 1 0] axis. (b)

Emission image under two-photon excitation of orthoexcitons

at two potential bottoms of a double trap. The viewing axis is

along x, and is perpendicular to the stress axis. The area of view

is 2.5 mm on a side.

2. Experimental setup

Regulating the spatial extension of an excitongas is important to avoid density lowering causedby diffusion. We utilize a strain trapping method,which is originated from an old problem on strain-distribution between two bodies in contact [3,4].Our strategy is to use one well of a double trap sothat reproducibility of any phenomena observed inone well can be easily checked in the other well. Adouble trap has a larger potential gradient than asingle trap, to make tighter confinement possible.As presented in our recent paper [5], we ‘‘visua-lize’’ the potential trap by mapping the emissionassociated with two-photon excitation. By thismethod, we obtain precise shapes of the traps andconfirm the formation of harmonic traps for theexcitons.

An inhomogeneous strain applied by a sphericalplunger to a body with flat surface yields shearstress maximum inside the body. When a (0 0 1)face of a Cu2O crystal is strained by such a

Hertzian contact, the position of the shear stressmaximum approximately coincides with the posi-tion of the largest energy shift for excitons or thebottom of the potential trap. However, if thepressed surface is selected as a (1 1 0) face, a doublepotential trap is formed along two orthogonal[0 0 1] axes near the shear stress maximum [5]. Thisis because of elastic anisotropy of Cu2O, whichleads to a largest strain-induced energy shift along[001] axes.

The sample we used was a 3-mm cube cut from anatural Cu2O crystal of high quality. The surfaces,oriented as four (1 1 0) and two (0 0 1) crystal faces,were carefully polished by mechanical means. Adouble potential trap was created by pressing aspherical glass lens of curvature 7.78 mm against a(1 1 0) surface of the sample held in a cryostat at2 K. As schematically shown in Fig. 1(a), twopotential wells are created along two [0 0 1]directions. These two wells are aligned along y-axis depicted in the figure.

Fig. 1(b) shows an example of emission imageswhen the infrared laser beam is tuned to the two-photon resonance of orthoexcitons at the trapbottom. The excitation light source was a LiF:color-center laser (Solar, LF151) pumped by a Q-switched YAG laser. The repetition rate and thepulse duration were 400 Hz and 12 ns, respectively.The laser beam was directed along y and scannedalong z, where y and z are defined in Fig. 1(a). Thedetection was made with a high-sensitivity CCDcamera (Wright Instruments) through the zeroth

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Fig. 2. (a) Spectrograms of the emission taken through a high-

resolution spectrometer. Upper (lower) panel is for orthoexci-

tons (paraexcitons). (b) Analysis of (a). The vertical axis is the

potential energy measured from the zero stress values. In

reality, paraexcitons locate lower in energy than orthoexcitons.

The potential gradient is about 200 (100) meV mm�2 for

orthoexcitons (paraexcitons). The fitting functions are shown

by dashed lines, which completely overlap with the line

connecting the data points.

N. Naka, N. Nagasawa / Journal of Luminescence 112 (2005) 11–16 13

order diffraction of a monochromator (JASCO,CT25T).

In Fig. 1(b), two trap bottoms separated by� 0:5 mm are recorded as luminous spots. Theemission is originated from second harmonicscattering mediated by orthoexcitons and fromdirect luminescence of orthoexcitons. The lumines-cence from paraexcitons is forbidden in thisviewing direction.

Fig. 2(a) is a spectrogram of an image similar tothat shown in Fig. 1(b). The spectrogram wasobtained by dispersing the image through a high-resolution spectrometer (Jobin Yvon, THR1500).The horizontal axis holds spatial informationalong y. The region near the trap bottoms isenlarged. The image was taken from the bottomside of the crystal, with the viewing axis parallel tothe stress axis. To this direction, emission ofluminescence light from both the orthoexcitonsand paraexcitons1 is allowed. The vertical axisrepresents the energy of the photon emitted asdirect luminescence of orthoexcitons or paraexci-

1Optically inactive paraexcitons become slightly active for

direct luminescence under finite stresses.

tons. Since direct luminescence occurs with re-combination of excitons at momentum close tozero, the contours represent potential energy of therespective exciton states as a function of position.

Fig. 2(b) is a plot of the potential shapes oforthoexcitons (open circles) and paraexcitons(closed circles) around the respective potentialbottoms. The vertical axis is the energy shiftmeasured from each exciton resonance under zerostress. The potential shapes are well described byparabolic functions. This means that the potentialwells work as harmonic traps for the excitons atlow temperatures. By fitting the data to aparabolic function, we obtained the potentialgradient, a; to be 200 (100) meV mm�2 fororthoexcitons (paraexcitons). The position of thepotential minimum for paraexcitons is about45 mm inside that for orthoexcitons.

In view of a harmonic oscillator, the separationof the energy ladder is o ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a=mex

p¼ 0:1 meV

with a typical potential gradient ofa ¼ 100 meV mm�2 and effective mass of theexcitons, mex ¼ 3m0 where m0 is the free electronmass at rest. This energy scale is much smallerthan our spectral resolution. The round-trip timeof this harmonic oscillator is 50 ns, which issomewhat longer than the lifetime of theorthoexcitons and shorter than the lifetimeof paraexcitons. On one hand, the spatialextent of the ground-state wavefunction isdgs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_=mexo

p¼ 0:6 mm.

In the following sections, we take up only onewell of the double trap. Sufficiently good reprodu-cibility was found between one well and the other.Time-resolved detections were made by a gatedICCD camera system (LaVision, PicoStar HR12).The gate width was typically 5 ns.

3. Results

Fig. 3(a) shows an emission image viewed fromthe bottom of the crystal when the two-photonenergy is tuned at a rim of the potential well fororthoexcitons (see the inset). The laser beam wasdirected along y and scanned along x shown in theinset. Thus, cross section in a xy plane is cut out inthe image. The laser beam was energetically off the

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Fig. 3. (a) Emission image under two-photon excitation of

orthoexcitons at a potential rim, viewed from the bottom

surface of the crystal along the stress axis. The inset

schematically shows the potential trap and the oval contour

due to resonance emission DE in the inset denotes the two-

photon excitation energy measured from the bottom of the

trap. The area of view is 200 � 300mm. (b) Intensities of the

central peak (closed circles) and the side peaks (open circles,

mean value for two peaks) as a function of time. (c)

Polarization dependence of the intensity profile across the

center of the trap. The right-hand side peak and central peak

are shown.

Table 1

Energy conditions for observing strongly localized signal at the

center of the trap DE is the two-photon excitation energy

measured from the bottom energy of the trap.

Trap depth (meV) Energy difference, DE

5.95 0.67–1.17

5.15 0.87

4.90 1.37

3.68 1.02

N. Naka, N. Nagasawa / Journal of Luminescence 112 (2005) 11–1614

trap bottom but spatially across it. The ovalcontour in the image represents emission fromorthoexcitons at the two-photon resonance. Inspite of the fact that the two-photon energy is offresonant at the center of the trap, a strong signalappeared at the center of the trap. In addition, twoside peaks appeared slightly inside the ovalcontour.

Plotted in Fig. 3(b) are intensities of the centralpeak (closed circles) and the side peaks (opencircles, mean value for two peaks) as a function oftime. With the maximum laser power ofP ¼ 7 mW, which is the case for Fig. 3(b), thecentral peak clearly showed a delayed onset withrespect to the side peaks. The spatial width of thecentral peak was ‘ ¼ 7:7 mm at the time ofmaximum intensity. Although not shown in thefigure, the central and side peaks showed similartemporal behavior with lower incident laser power.

The intensity ratio of the central peak to the sidepeak decreased with lowering the incident power.The central peak was indiscernible with incidentpower less than 1 mW.

The central peak suddenly disappears, when thebeam position is shifted or the two-photon energyis changed. In order to observe the central peak,the laser beam should traverse the center of thetrap (not energetically but spatially). The energycondition for observing peaks is tabulated in Table1, where one can see that manifestation of thecentral peak occurs only when the two-photonenergy is set 1 meV above the trap bottom. InTable 1, the energy difference between the two-photon energy and the bottom energy of the trap isrepresented as DE:

From the spectroscopic observation [6], weknow that the position of the central peakcoincides with the trap bottom of paraexcitons(compare Fig. 2(b)), but that the signal is not dueto direct luminescence of ortho/paraexcitons at thetrap bottom. The photon energy of the centralpeak was found to be the same as the two-photonenergy of the excitation light. Fig. 3(c) showsintensity profile across the center of the trap, withpolarizer inserted before the detector. The emittedlight was polarized along x-direction.

4. Discussion

In discussing the origin of the strongly localizedpeak, the exciton density established under thepresent excitation condition is an essential para-meter. Therefore, we estimate the exciton densityas follows.

The two-photon absorption coefficient wasmeasured by comparing luminescence intensity

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Fig. 4. Schematic showing elastic scattering of second harmo-

nic light by a small index region at the center of the trap. The

arrows indicate direction of the light propagation.

N. Naka, N. Nagasawa / Journal of Luminescence 112 (2005) 11–16 15

under two-photon excitation with that underband-to-band excitation with green light. Namely,using 30% conversion efficiency from greenphotons to excitons [7], we have determinedthe two-photon absorption coefficient asb ¼ 0:0036 cm MW�1: Since two-photon absorp-tion occurs over z � 25 mm, effective generationrate is G0 ¼ GbzI ¼ 1:4 � 1010 excitons pulse�1;where G ¼ 1:1 � 1014 photons pulse�1 is the gen-eration rate at I ¼ 14 MWcm�2 or P ¼ 7 mW.

On one hand, the volume of the exciton gas atthe trap bottom is ð‘ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip= log 4

pÞ3¼ 5:5�

10�9 cm3:2 Dividing the exciton number by thisvolume, one obtains exciton density of2:5 � 1018 cm�3: This value is about 30 times thecritical density for BEC at 2 K, 7:8 � 1016 cm�3:P=

ffiffiffiffiffi30

p� 1 mW approximately agrees with the

incident power with which the central peak startsto appear.

Based on this density estimation and othercurious properties, bosonic stimulation of excitonsfrom the trap rim to the bottom can be proposedas one possible explanation for the manifestationof the localized peak. Classical drift into the trapbottom takes much longer time, e.g., 300 ns [6],and contradicts with the timescale shown in Fig.2(b). Because, the position of the central peakcoincides with the trap bottom of paraexcitons, itis natural to assume that the stimulation occurredin the paraexciton state.

If paraexcitons are condensed at the groundstate, one would expect relevant signal from thetrap bottom. However, no luminescence suggest-ing accumulation of excitons at the trap bottomwas seen in the spectroscopic data. One plausiblemodel to account for the observation is elasticscattering of second harmonic light by a conden-sate.

A Cu2O crystal contains four copper atoms orfour valence electrons in a unit cell of 0.426 nm ona side. This means electron density of5:2 � 1022 cm�3: Because 10�4 of these electronsare excited into the conduction band as aconstituent of excitons, the refractive index would

2Here we take the observed width of ‘ ¼ 7:7 mm. It is

comparable to the spatial resolution of the detection system,

and the real size of the exciton gas can be smaller.

be reduced to some extent. The size of the smallindex region should be of the order of the size ofthe condensate. Such a small gradient index centercan strongly scatter light passing through it. Fig. 4illustrates this situation. Note that we observedsignals from the bottom of the sample, at 90 angleto the incident direction of the laser light. Since,the second harmonic light generated at theexcitation spot propagates across the center ofthe trap, it can be scattered into 90 direction bythe small index region. In fact, the observedpolarization of the signal is consistent with thismodel.

According to simulation [8] with refractive indexchange of 0.01% in diameter of ‘ ¼ 7:7 mm, theintensity of the scattered light is 10�8 that of theincident light. This seems too small in comparisonwith the observed ones. However, since theabsolute intensity of the second harmonic light inthe trap cannot be measured directly, it is difficultto quantitatively compare the intensity of relevantscattering light with the theoretical value. Conse-quently, our model is valid only qualitatively at thepresent stage of the experiments. Comparison ofthe intensities between luminescence from a con-densate and scattering light by a condensate is aninteresting problem to be solved.

Recently, some authors have claimed that Augerprocess becomes dominant when the density ofexcitons is as high as a critical density for BEC.Here, Auger process means two-body collision ofexcitons, with one non-radiatively decaying andthe other ionizing into free electron–hole pair.With their measured value of Auger coefficient

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N. Naka, N. Nagasawa / Journal of Luminescence 112 (2005) 11–1616

A ¼ 10�16 cm3 ns�1 [9], excitons decay within 1 nsaround the critical density. On the other hand,other groups reported a much lower value ofA ¼ 10�22 cm�3 ns�1 [10], with which BEC isachievable. The collision process has been con-sidered as s-wave scattering, and the rate shouldnot depend on the velocity or kinetic energy of thecolliding excitons. However, the situation undertwo-photon resonance excitation would be differ-ent, because the effective temperature of theexcitons should be much lower than the latticetemperature at least in the initial stage. In fact, ourexperimental result on the incident power depen-dence of the exciton luminescence implies suppres-sion of the non-radiative recombination in Augerprocess [11].

Our observation of the decay of the central peakwithin 10 ns (see Fig. 3(b)) does not necessarilymean the condensate decays in this timescale. Onthe basis of our model, the source of the scatteredlight is the second harmonics of the incident laserlight. The light source itself decays in the pulseduration of 12 ns. In order to measure the lifetimeof possible condensate, we are planning to carryout pump–probe experiments. By optimizing thepolarization of the incident laser light, we are ableto suppress generation of the second harmonics atexcitation spots. Using the second, probe light thathas been optically delayed and converted into redphotons with a second harmonic crystal before thesample, measurement in wider temporal rangesbecomes possible. Another interesting direction isan approach analogous to that for an atomic gasof polarized hydrogen. Nonlinear increase ofparaexciton density as manifestation of BECwould be observed by monitoring the intensity of1s–2p Lyman transition of the excitons [12].Application of external magnetic field may alsohelp to enhance the paraexciton emission efficiencyto make the detection easier [13].

5. Conclusions

We have proposed that the localized signal atthe center of the trap can be qualitativelyexplained by elastic scattering caused by acondensate of paraexcitons. The estimated densitywith which the signal starts to appear is consistentwith the critical density for BEC. Further quanti-tative examination is necessary to certify experi-mental evidence for excitonic BEC. Experiments inpump–probe scheme to confirm our model are inprogress.

Acknowledgements

This work was partially supported by theGrants-in-Aid for Scientific Research (16740179)from The Ministry of Education, Culture, Sports,Science and Technology, Japan.

References

[1] N. Naka, N. Nagasawa, J. Lumin. 94–95 (2001) 413.

[2] N. Naka, N. Nagasawa, Phys. Rev. B 65 (2002) 075209.

[3] A. Marston, Phys. Rev. 1 (1893) 127.

[4] J.P. Wolfe, A. Mysyrowicz, Sci. Am. 250 (1984) 70.

[5] N. Naka, N. Nagasawa, Phys. Rev. B., Phys. Rev. B 70,

in press

[6] N. Naka, N. Nagasawa, Phys. Stat. Sol. (b) 238 (2002) 397.

[7] K.E. O’Hara, J.R. Gullingsrud, J.P. Wolfe, Phys. Rev. B

60 (1999) 10872.

[8] P. Laven, http://philiplaven.com

[9] K.E. O’Hara, J.P. Wolfe, Phys. Rev. B 62 (2000) 12909.

[10] A. Jolk, M. Jorger, C. Klingshirn, Phys. Rev. B 65 (2002)

245209.

[11] N. Naka, N. Nagasawa, Phys. Rev. B 65 (2002) 245203.

[12] M. Kuwata-Gonokami, M. Kubouchi, R. Shimano, A.

Mysyrowicz, J. Phys. Soc. Japan 73 (2004) 1065.

[13] D. Frohlich, private communication