Comment on “Condensation of Excitons in a Trap”Dirk Semkat,* Siegfried Sobkowiak, Gunter Manzke, and Heinrich Stolz

Institut fur Physik, Universitat Rostock, D-18051 Rostock, Germany

Condensation of excitons is still a fascinating topic of solidstate physics. In a recent Letter “Condensation of

Excitons in a Trap”,1 High et al. claim to have observed acondensed state in a system of indirect excitons in doublequantum well structures within an electrostatic potential trap.As in every Bose−Einstein condensate, spontaneous coherenceof matter waves should emerge in the exciton system. Thiscoherence is transferred to the decay luminescence and thusshould be observable in the light emission from the excitoncloud. Indeed, the authors have observed in a series ofexperiments a pronounced increase of the coherence of thelight emitted from the excitons either by lowering thetemperature or by increasing the power of the laser excitingthe excitons (compare Figures 3d,e, 4, and 5 of ref 1).The coherence properties of the emitted light were measured

with the well-known technique of shift interferometry.2 Heretwo images of the same object, shifted by a small amount of δ,are superimposed, and by varying the phase delay between thetwo light paths, interference fringes are generated (see Figure 2of ref 1). To determine the interference contrast, the authorsuse the simple formula C = (I12 − I1 − I2)/2(I1I2)

1/2. Furthersupport for the interpretation of the occurrence of a“condensate” is derived by the authors from a simple idealBoson model for the excitons captured in a harmonic trap fromthe critical temperature Tc = (61/2)/(π)ℏω2d with ω2d =(ωxωy)

1/2 being the 2d oscillator frequency. With theassumptions for the trap oscillator frequencies the authorsgive, this indeed would lead to a critical temperature for BECfor N ≃ 3 × 103 excitons in the trap of 2 K.As will be shown in this comment, there are several

objections against the statements in ref 1 based on a rigoroustheory of shift interferometry as well as on the thermodynamicproperties of a dense interacting exciton gas in a potential trap.Theory of Shift Interferometry. We start with a single

point emitter at position xo,yo in the object plane at −d1, whichis imaged by a lens. The amplitude of the light field at a point(xi,yi) in the image plane at d2 is then given by3

λ= − +

× − +

× − + · + +

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

E x yM

dikd M

ikMd

x y E x y

ikd

x y P x Mx y My

( , ) exp[ (1 1/ )]

exp2

( ) ( , )

exp2

( ) ( , )

I i i

i i

d i i

12 1

1

2 2O o o

1o2

o2

2 o o(1)

with EO(x,y) denoting the field amplitude of the emitter andP2d(x,y) the amplitude point spread function (PSF) of the lens.M = d1/d2 is the system magnification.In shift interferometry we superimpose on this image that of

an identical object shifted by δ in, for example, the x directionand with an additional phase Φ. Its field is given by

δλ

δ

δ

′ = − + + Φ

× − +

× − − +

× − + +

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

E x yM

dikd M i

ikMd

x y E x y

ikd

x y

P x Mx y My

( , , ) exp[ (1 1/ ) ]

exp2

( ) ( , )

exp2

(( ) )

( , )

I i i

i i

d i i

12 1

1

2 2O o o

1o

2o2

2 o o (2)

Using the notation of ref 1, the interference pattern of asingle point emitter is given by

δ δ= | + ′ | = + +I x y E x y E x y I I I( , , ) ( , ) ( , , )i i I i i I i i122

1 2 inter

(3)

While I1 and I2 are the images of the two shifted objects, theinterference term is given by

δ

δ

δ

= ′*

∝ | | − − Φ

× + +

× * − + +

Ι

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

I E x y E x y

E x yikd

x i

P x Mx y My

P x Mx y My

2Re[ ( , ) ( , , )]

2 ( , ) Re exp

( , )

( , )

I i i i i

d i i

d i i

inter

O o o2

1o

2 o o

2 o o(4)

In the following, we are interested only in the case of anincoherently emitting cloud of thermal excitons. Then we canapproximate the first-order field correlation functionGO(x, y, x′, y′) = EO(x, y)E*O(x′, y′), which represents themutual coherence function of the emitter4 by

δ δ′ ′ = ′ ′ − ′ − ′G x y x y I x y x y x x y y( , , , ) ( , , , ) ( ) ( )O O (5)

with IO = |EO(xo, yo)|2 representing the intensity distribution of

the emitter. Therefore, we obtain the interference pattern of thetotal emission by integrating eq 4 over the whole emitter. Thisgives the following expression

∫ ∫ δ

δ

∝ − − Φ

× + +

× * − + +

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭

I x y I x yikd

x i

P x Mx y My

P x Mx y My x y

( , ) 2 ( , )Re exp

( , )

( , ) d d

i i

d i i

d i i

inter O o o1

o

2 o o

2 o o o o(6)

Received: July 6, 2012Published: July 23, 2012

Letter

pubs.acs.org/NanoLett

© 2012 American Chemical Society 5055 dx.doi.org/10.1021/nl302504h | Nano Lett. 2012, 12, 5055−5057

Equation 6 is the central relation for shift interferometry. Itshows that the interference pattern depends not only on thePSF but also on the intensity distribution of the emittingsource!To obtain quantitative results, we need the PSF of the actual

imaging setup, which fortunately can be obtained from Figure2S of the supporting material of ref 1. For our purposes, thepoint spread function can be approximated quite well by asimple Gaussian,5

σ= −

+⎛⎝⎜⎜

⎞⎠⎟⎟P x y P

x y( , ) expd

p2 0

2 2

2(7)

Noting that the intensity distribution shown in Figure 2S of ref1 is given by the square of the PSF, eq 7, we obtain σp = 2.0 μm.Representing the intensity distribution of the exciton cloud

also by a Gaussian with different halfwidths in x- and y-directions as

σ σ= − −

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥I x y I

x y( , ) exp

x yO 0

2 2

(8)

the interference pattern can be calculated analytically.Fortunately, due to the choice of the PSF, the final resultfactorizes into a product of x- and y-dependent functions, sothat for a shift in x-direction, the result for the interferencecontrast C(σx, δ) = (I12 − I1 − I2)/2(I1I2)

1/2 depends only on δand σx.As a first result we demonstrate in Figure 1 that, contrary to

the statement in ref 1, the image of a point source (full red line)

is different from the interference visibility function; that is, thecontrast as a function of shift δ of an incoherent source is muchlarger than the optical resolution. Therefore, Figure 1S of ref 1seems to be incorrect, and the scale of upper and lower abscissamust differ by a factor of 1.4. By comparing Figures 1b and 2cof ref 1, one can deduce that it must be the shifts given in ref 1(in the following: δB) that have to be scaled by V = 1.4 to getconsistency (δ = V·δB).The interference contrast at x = 0 is shown in Figure 2 as a

function of the source size for different shifts. Looking moreclosely at the curve for δB = δ/V = 4 μm, our theory predicts acontrast of 0.07 at source sizes above 6 μm HWHM inagreement with the experiment. Reducing the source size, the

contrast increases reaching values of about 0.2−0.3 for sizes of2 μm similar to those measured experimentally (Figure 3e of ref1). This shows that our theory is able to reproduce theexperimental results of ref 1 almost quantitatively without thenecessity of a coherent condensate.

Thermodynamics. To analyze the thermodynamics of theexciton gas in the trap, we use a Hartree−Fock−Bogoliubov−Popov (HFBP) theory (for an overview, see ref 6) which wehave recently applied to excitons in bulk Cu2O.

7 We use a localdensity approximation well-justified by the extension of the trap(Figure 1 in ref 1) which is, even in the narrower y-direction,still large compared to typical length scales, for example, the 2dexcitonic Bohr radius.Applying the HFBP theory in local density approximation to

the 2d case, the densities of thermally excited nT and ofcondensed excitons nc (in Thomas−Fermi approximation) inan external potential Vext form a coupled system of equations,

∫π

= + −

× Θ

⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥n

d k k rE k r

n E k r

E k r

(r)(2 )

( , )( , )

( ( , ))12

12

( ( , ))

T2

2 B

2(9)

μ

μ

= − −

× Θ − −

nU

V r U n r

V r U n r

(r)1

[ ( ) 2 ( )]

( ( ) 2 ( ))

c

0ext 0

T

ext 0T

(10)

Here nB is the Bose distribution, and the quasiparticle energy Eis given by (compare7)

= −E k r k r U n r( , ) ( , ) ( ( ))20

c 2(11)

with

μ= ℏ + − +k r k M V r U n r( , ) /2 ( ) 2 ( )2 2ext 0 (12)

The total density is given by n = nT + nc, μ is the globalchemical potential, and U0 is the strength of the interexcitonicinteraction. We extract the latter parameter from theexperimentally given energy shift of δE = 1.3 meV at n = 3 ×1010 cm−2 by δE = 2U0n leading to U0 = 2.2 μeV·μm2. The trappotential was taken directly from Figure 1c and d of ref 1.In Figure 3, the critical particle number Ncrit for Bose−

Einstein condensation according to eqs 9−12 is presented independence on the temperature by the black curve. The dotted

Figure 1. Comparison of the relative intensity of a point source (fullred line) with the interference visibility function (blue dashed line) fora PSF given by eq 7 with σP = 2.0 μm and a source size σx = 6 μm (seeeq 8).

Figure 2. Contrast at x = 0 for different effective shifts δB = δ/V as afunction of the HWHM of the exciton cloud, which is given in theinset as a function of the size parameter σx (see eq 8). To allow a directcomparison with the results of ref 1. the contrast was scaled by 0.65,which is the maximum contrast in the experiments.

Nano Letters Letter

dx.doi.org/10.1021/nl302504h | Nano Lett. 2012, 12, 5055−50575056

part of the line denotes the region, where the particle numberexceeds the capacity of the trap. Additionally, correspondingcurves are shown for the weakly interacting case (U0/100;dashed blue line) and the noninteracting case (dash−dotted redline).For a temperature of 2 K, we obtain a critical particle number

of about Ncrit ≃ 6.3 × 105 corresponding to a (homogeneous)density of about 6.3 × 1012 cm−2, much higher than theestimate given in ref 1. This discrepancy is obviously due to therather strong exciton−exciton interaction which has beenneglected there. Even with the much weaker interaction of U0/100, Ncrit is still about 3.7 × 105. Neglecting the interaction, ourtheory yields Ncrit ≃ 2 × 103 consistent with the noninteractingestimate.Due to the finite trap depth, there is a temperature-

dependent maximum exciton number Nmax in the trapdetermined by the condition that the chemical potentialreaches the excitonic band gap just outside the trap. The criticalcurve is only meaningful if Ncrit ≤ Nmax. For the interacting case(black line in Figure 3) this holds only at the lowesttemperatures of about 50 mK; that is, the maximum excitonnumber for T ≳ 50 mK is always undercritical in the given trap.Conclusions. As has been shown in this comment, there are

several objections against the statements in ref 1 which togethermake the claims of the Letter highly questionable. First, basedon the theory of imaging of partially coherent light, we havederived a rigorous expression for the contrast of a shiftinterferometer with an arbitrary source. This equation allows usto explain the experimental observations simply by the alsoobserved reduction in spot size (see Figure 3e), without anynecessity to assume the occurrence of a condensate. Second, wederived from a 2d Hartree−Fock−Bogoliubov−Popov theorythe thermodynamic properties of a dense interacting excitongas in a potential trap. Using the trap parameters as given in ref1 we find at T = 100 mK the critical particle number of Ncrit ≃ 3× 104, which is too high to be captured in the trap.

■ AUTHOR INFORMATIONNotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was supported by the Deutsche Forschungsgemein-schaft (Collaborative Research Center SFB 652).

■ REFERENCES(1) High, A. A.; Leonard, J. R.; Remeika, M.; Butov, L. V.; Hanson,M.; Gossard, A. C. Nano Lett. 2012, 12, 2605.(2) Barrett, H. H.; Myers, K. J. Foundations of Image Science; JohnWiley & Sons: New York, 2004.(3) Gu, M. Advanced Optical Imaging Theory; Springer-Verlag: Berlin,2000.(4) Born, M.; Wolf, E. Principles of Optics, 7th ed.; CambridgeUniversity Press: Cambridge, 1999.(5) Zhang, B.; Zerubia, J.; Olivo-Marin, J.-C. Appl. Opt. 2007, 46,1819.(6) Proukakis, N. P.; Jackson, B. J. Phys. B 2008, 41, 203002.(7) Sobkowiak, S.; Semkat, D.; Stolz, H.; Koch, T.; Fehske, H. Phys.Rev. B 2010, 82, 064505.

Figure 3. Critical particle number for BEC vs temperature in the trap.Black line: interacting case (solid: particle number captured by thetrap, dotted: particle number too large to be captured). Dashed blueline: weakly interacting case (U0/100). Dash−dotted red line:noninteracting case.

Nano Letters Letter

dx.doi.org/10.1021/nl302504h | Nano Lett. 2012, 12, 5055−50575057