Condensation of

Quasi-Two-Dimensional

Biexcitons in a Single

Heterojunction Quantum Well

ARL-TR-1934 April 1999

Patrick A. Folkes

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ARL-TR-1934 April 1999

Army Research Laboratory

Adelphi, MD 20783-1197

Condensation of

Quasi-Two-Dimensional

Biexcitons in a Single

Heterojunction Quantum Well

Patrick A. Folkes 

Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.
Excitons that coexist with a degenerate two-dimensional electron gas

in the same quantum well subband have been observed in the

photoluminescence from the recombination of electrons with

localized photoexcited holes. At a critical electron density, an abrupt

decrease in the exciton radiative recombination rate is observed, along

with the formation of biexcitons. With increased excitation intensity,

photoluminescence spectra are observed that verify theory on the

radiative renormalization of biexcitons and strongly indicate the

Abstract

occurrence of a Bose-Einstein condensation of biexcitons.

ii
iii

Contents

Introduction ............................................................................................................................................... 1

Experimental Results ............................................................................................................................... 2

Discussion .................................................................................................................................................. 6

Conclusion ................................................................................................................................................. 8

Acknowledgment ..................................................................................................................................... 8

References .................................................................................................................................................. 9

Distribution ............................................................................................................................................. 11

Report Documentation Page................................................................................................................. 15

Figures

1. Schematic of backgated coupled-well bandstructure for negative gate voltage and

positive gate voltage ........................................................................................................................... 3

2. Observed PL spectra from SHQW with Vg = –38 V at 6, 15, and 25 K ........................................ 3

3. Observed 6 K PL spectra for I = 0.3, 0.46, and 0.75 mW with Vg = –37.3 V; and for

I = 0.75 mW, Vg = –38 V ...................................................................................................................... 4

4. Approximate X and XX 6 K PL lineshapes for I = 0.3, 0.46, and 0.75 mW and

Vg = –37.3 V; and for I = 0.75 mW and Vg = –37.3 V; and Vg = –38 V .......................................... 5
1

Introduction

In certain semiconductors, excitons (electron-hole bound states) and

biexcitons (exciton-exciton bound states) are expected to undergo BoseEinstein condensation (BEC) at relatively high temperatures because of

their small effective masses, which are typically on the order of the free

electron mass [1–3]. Experimental results have shown that excitons in

Cu2O [4] and Ge [5] and biexcitons in CuCl [6] exhibit quantum degenerate Bose-Einstein statistics at high densities, as well as anomalous luminescence and ballistic transport, which have been attributed to the occurrence of BEC [7–12]. The sharp luminescence features observed [7] and an

increased optical phase-conjugate signal have been attributed to the

presence of a resonant two-photon absorption-induced condensate of

biexcitons in CuCl. A recent report [12] describes the condensation of

spatially indirect excitons in coupled AlAs/GaAs quantum wells (QWs),

as evidenced by anomalous exciton transport and the concurrent appearance of a huge low-frequency noise in the photoluminescence (PL) intensity. Theory has shown that BEC of a two-dimensional (2D) ideal Bose gas

in a confining potential can occur [13]; however, weakly interacting 2D

bosons in a confining potential are predicted to undergo a KosterlitzThouless phase transition instead of BEC, because of the absence of

infinitely long-ranged phase coherence [14]. Huang [2] points out, however, that in 2D systems the existence of phase coherence over a large

finite distance could lead to a local BEC.

My colleagues and I have recently observed [15] excitons that coexist with

a degenerate 2D electron gas (2DEG) in the same subband (that is, Mahan

excitons) in the PL spectra from the recombination of quasi-2D electrons

with localized photoexcited holes in a single heterojunction quantum well

(SHQW). When the 2DEG density ns ≈ 1.9 × 1011 cm–2, we observed an

abrupt decrease in the quasi-2D Mahan exciton (X) PL intensity and

linewidth, along with the formation of quasi-2D biexcitons (XX) and a

large discontinuity in the X groundstate energy. These intriguing observations led to a high-resolution study of the PL from this system as the

excitation intensity is increased with ns ≈ 1.9 × 1011 cm2. I report here the

observation of PL spectra that strongly indicate the occurrence of the BEC

of weakly localized quasi-2D biexcitons (XX) and verify recent theory on

the radiative renormalization of XX [16].

Research on coherent effects and condensation of excitons and biexcitons

has grown significantly in recent years. Potential future applications of

this research include the development of a nonlasing coherent source of

emission, the phase-coherent transport of optical signals through

nanometer-scale devices in optical computers, and the efficient transport

of light into near-field optical microscope tips [1].
2

Experimental Results

The experiments were carried out on an AlGaAs/GaAs modulationdoped coupled-well heterostructure [16], shown schematically in figure 1.

Shubnikov-de Haas measurements at 4.2 K show that the 2DEG is confined in only the lowest subband of the SHQW, formed by the transfer of

electrons from the doped AlGaAs into the lower bandgap GaAs. I varied

the 2DEG density ns

 by applying a voltage Vg to a Schottky contact on the

120-μm-thick substrate, as shown in figure 1. Measurements show that ns

varies linearly with Vg with a depletion rate dns/dVg ≈ 6.3 × 108 cm–2V–1.

The samples were mounted in a variable-temperature liquid helium

cryostat and excited with a 5145-Å emission from an argon ion laser, with

intensities up to approximately 2 W/cm2. The PL spectrum was measured

with a 1-m monochromator.

The observed 6 K PL over the range from 1557 to 1545 meV is the superposition of (1) the sharp, resonant PL from the spatially direct recombination of weakly localized quasi-2D Mahan excitons and (2) the relatively

broad PL from the spatially direct recombination of free electrons with

localized heavy holes in the SHQW [16]. The observed PL over the 1545 to

1535 meV energy range is attributed to the spatially indirect recombination of SHQW electrons with photoexcited holes, which are localized 100

to 150 Å from the AlGaAs/GaAs interface [16]. In the presence of a 2DEG,

the X eigenstates will depend on various many-body interactions and

effects [17]. Over the range 1.9 × 1011 cm–2 < ns

 < 2.2 × 1011 cm–2, the

spatially direct X emission results in a strong resonance in the observed

PL intensity at the X groundstate energy Een = 1553 meV. Around ns ≈ 1.9

× 1011 cm–2, an abrupt large decrease in the 1553-meV X PL intensity and

linewidth occurs, along with the appearance of another sharp peak in the

PL spectrum near 1551 meV, as shown in figure 2 for Vg = –38 V.

The observed abrupt decrease in the 1553-meV X PL intensity at ns ≈ 1.9 ×

1011 cm–2 signifies that there is a suppression of the recombination rate

(possibly due to many-body effects), with an attendant increase in the

radiative lifetime of the 1553-meV X. The increased lifetime will result in

increases in the X density and the exciton/exciton scattering rate and

possibly the achievement of thermal equilibrium with the lattice—

conditions that are conducive to the formation of biexcitons [3]. A phenomenological model for the decay of an XX into a photon and an X,

which subsequently radiatively decays, results in a double-peak emission

spectrum consisting of the X peak at Een and an XX spectrum whose

high-energy edge is lower than Een by an amount equal to the XX binding

energy. The high-energy edge of the XX PL spectrum arises from the

emission from biexcitons that occupy the groundstate. The 1551-meV

feature is observed only over the narrow range of ns

 where the 1553-meV

X has a reduced recombination rate; this fact strongly suggests that the

1551-meV feature can be attributed to the formation and subsequent

radiative decay of biexcitons in the SHQW and precludes the possibility

that the 1551-meV peak is the emission from the recombination of

impurity-bound excitons. The observed separation of the X and XX peaks

gives an XX binding energy of approximately 2 meV, which is consistent
3

with recent experimental reports of 1.8 to 2.7 meV for 80 to 100 Å GaAs

QWs [18–21].

I varied ns

 slightly around the value ns ≈ 1.9 × 1011 cm–2 to minimize the

1553-meV peak amplitude and, using enhanced resolution, studied the

evolution of the PL lineshape as the excitation power I is increased. The

observed high-resolution 6 K PL data plotted in figure 3 clearly show that,

with increasing I, the XX PL at 1551 meV increases at a faster rate than the

1553-meV X PL, in agreement with previous observations [22–24]. Figure

3 shows that, for Vg = –37.3 V, the XX PL lineshape evolves strikingly: at I

= 0.3 mW, it has a small relatively broad peak, and at I = 0.75 mW, it has a

large, very narrow peak at 1551 meV and a distinctive cusp at 1551.5 meV.

Concurrently (for Vg = –37.3 V, I = 0.75 mW), the figure shows large lowfrequency fluctuations in the PL intensity over the range 1546 meV to

1535 meV. Figure 3 also shows that the PL observed under the same

conditions (Vg = –37.3 V, I = 0.75 mW) is extremely sensitive to ns

, as

evidenced by the abrupt decrease in XX PL intensity and the disappearance of the very narrow 1551-meV line and the low-frequency PL intenFigure 1. Schematic

of backgated

coupled-well

bandstructure for

(a) negative gate

voltage and

(b) positive gate

voltage.

(a) (b)

2

1

0

1.53 1.54 1.55 1.56

Energy (eV)

PL intensity

6 K

25 K

15 K

Figure 2. Observed

PL spectra from

SHQW with Vg =

–38 V at 6, 15, and

25 K.
4

sity fluctuations when Vg is changed to –38 V; this change corresponds to

reducing ns

 by only 4 × 108 cm–2. The intersection of the –37.3 and the

–38 V PL spectra (both 0.75 mW) at 1548 meV gives an approximate

amplitude of the PL from 2D free-electron/localized-hole (e-h) recombination; this amplitude is not sensitive to slight changes in ns

, varies linearly

with I, and can be approximated by a square pulse lineshape. Figure 4

shows approximate lineshapes of the combined PL from X and XX, which

I obtained by subtracting the estimated e-h PL from the data shown in

figure 3.

Figure 4 shows that as the XX density increases, the XX PL narrows over

the range from 1548 to 1552.5 meV. The half-maximum width of the lowenergy side of the XX PL, which is determined by the XX energy distribution, decreases from 1.6 meV (for I = 0.3 mW, Vg = –37.3 V) (bottom

curves, fig. 4a) to 1 and 1.2 meV (for I = 0.75 mW, Vg = –37.3 V and –38 V)

(both curves, fig. 4b). The observed narrowing of the XX energy distribution with increasing XX density confirms that the biexcitons obey BoseEinstein statistics and that, for I = 0.75 mW, the XX are in the quantum

degenerate regime [9], where a significant fraction of the 2D XX have

small wavevectors. In the quantum degenerate regime, polariton (mixed

exciton-photon state) effects are predicted [17] to significantly

renormalize the XX dispersion, resulting in a shift in the location of the

groundstate and an increase in the XX binding energy from the unperturbed value of ~1 meV [25] to 2.2 meV, which agrees with the observed

value of 2 meV.

Figure 3. Observed

6 K PL spectra for I =

0.3, 0.46, and 0.75 mW

with Vg = –37.3 V

(solid lines) and I =

0.75 mW, Vg = –38 V

(dashed lines).

2

1

0

0.75 mW

Intersection

point

0.46 mW

0.30 mW

Energy (eV)

1.54 1.55 1.56

Vg = –38 V

Vg = –37.3 V

PL intensity

6 K
5

Figure 4.

Approximate X and

XX 6 K PL lineshapes

(a) for I = 0.3, 0.46,

and 0.75 mW and Vg =

–37.3 V; (b) for I =

0.75 mW, Vg = –37.3 V

(solid line); I = 0.75

mW, Vg = –38 V

(dashed line). Label g

marks appearance of

narrow 1551-meV

line; label h marks

cusp at 1551.5 meV.

Label X marks

coherent polarization

mode; label X– marks

negatively charged

exciton at 1552 meV.

2

1

0

X X

X– X–

h h

g g

6 K, Vg = 37.3 V 6 K, I = 0.75 mW

1.55 1.553

Energy (eV)

PL intensity

1.55 1.553

Energy (eV)

(a) (b)

Vg = –38 V

Vg = 37.3 V

I = 0.3 mW

I = 0.46 mW

I = 0.75 mW
6

Discussion

The large increase in the 6 K XX PL intensity and the abrupt appearance

of the very narrow 1551-meV line (labeled g in fig. 4a) in the XX PL

spectrum for Vg = –37.3 V, I = 0.75 mW, together with the extreme sensitivity of this XX PL spectrum to small changes in ns

, suggest that the very

narrow 1551-meV line comes from the recombination of XX that have

undergone BEC into the groundstate. This conclusion is also supported

by the close resemblance of the distinctive lineshape of the observed XX

PL for Vg = –37.3 V, I = 0.75 mW, to the calculated [3] emission spectrum

for three-dimensional Bose-condensed biexcitons. The remarkable disappearance of the very narrow 1551-meV line and the change in the XX PL

lineshape when ns

 is reduced by only 4 × 108 cm–2 (as shown in fig. 4b)

can only be explained by a decrease in XX density: from a density greater

than the critical density for BEC when Vg = –37.3 V, to a density smaller

than the critical density when Vg is changed to –38 V. The calculated

critical density [14] for BEC of the weakly localized quasi-2D XX at 6 K is

approximately 2 × 1010 cm–2, assuming that the localization energy for the

XX is ≈3 meV (the observed value for the hole localization energy [16])

and that the XX effective mass equals the free electron mass. Figure 2

shows that at 15 K the amplitude of the XX PL is strongly reduced compared to the X peak, and the XX PL no longer has a sharp peak near 1551

meV, indicating that the XX are not quantum degenerate at 15 K.

As shown in figure 4b, the cusp at 1551.5 meV (labeled h) observed for

Vg = –37.3 V, I = 0.75 mW, evolves into a sharp resonance at 1551.2 meV

for Vg = –38 V, I = 0.75 mW. This resonance is attributed to the theoretically predicted [17] van Hove singularity in the biexciton-polariton joint

density of states, which is expected to be manifested in the XX PL spectrum as a peak that is ~0.4 meV higher in energy than the XX groundstate

energy [3,17]. The shift in the location of the van Hove singularity when

BEC occurs provides evidence that BEC results in significant modification

of the XX dispersion. Figure 4 shows that when BEC occurs, the observed

X PL narrows, and its peak shifts by 0.2 meV to lower energy; these

changes suggest that the radiative decay of condensed X results in the

formation of a coherent polariton mode (labeled X in fig. 4). The

1552-meV resonance, labeled X– in figure 4 (which is observed only when

the X recombination rate is suppressed near ns ≈ 1.9 × 1011 cm–2), is not

sensitive to small changes in ns

, but increases with increasing excitation

intensity. The observed 1-meV separation between the X and X– peaks

suggests that the X– resonance can be attributed to the emission from

negatively charged excitons, which have been recently observed in

modulation-doped quantum well structures [26]. The X– consists of two

electrons bound to a single hole, with a calculated binding energy for the

second electron of 1 to 1.4 meV for GaAs quantum wells [27].

In addition to the above striking changes in the XX PL when Vg = –37.3 V,

I = 0.75 mW, figure 3 shows concurrent large low-frequency fluctuations

(peak-to-peak times of 20 to 60 s) in the PL intensity over the range 1546

to 1535 meV, despite the fact that each data point was obtained after
7

averaging over a 10-s interval. The energy range over which the anomalous PL intensity fluctuations are observed shows that they come from the

recombination of XX composed of at least one spatially indirect X. Similar

large low-frequency fluctuations in the PL intensity of spatially indirect

condensed excitons have been observed before [13]. In the absence of

BEC, the PL intensity exhibits the relatively small shot noise associated

with the Poisson distribution of randomly emitted photons from independent recombination of XX. Statistical mechanics shows that BEC of an

open system of bosons is accompanied by the onset of large fluctuations

in the occupancy of the groundstate [28]. Furthermore, these fluctuations

in the condensate population must be phase coherent, as pointed out by

Nozieres [1]. A coherent fluctuation in the steady-state nonequilibrium

groundstate occupancy of condensed XX will regress via coherent radiative recombination with a characteristic fluctuation time τ f

, which determines the spectral density of the PL intensity fluctuations. Since a large

number of excitons must acquire the appropriate spatial configuration for

coherent recombination of condensed XX, one would expect τ f

 to depend

on and be much larger than the X radiative lifetime, which ranges up to

1 μs for spatially indirect X [29]. I conclude that the observed large lowfrequency noise (large τf

) in the PL intensity is a manifestation of XX

coherence; this observation provides strong evidence of the BEC of XX.

The data indicate that for spatially direct XX, τf « 10 s.
8

Conclusion

In summary, excitons that coexist with a degenerate two-dimensional

electron gas in the same quantum well subband have been observed in

the photoluminescence from the recombination of electrons with localized

photoexcited holes. At a critical electron density, an abrupt decrease in the

exciton radiative recombination rate is observed, along with the formation of biexcitons. With increased excitation intensity, I observe photoluminescence spectra that are sensitive to electron density, verify theory on

the radiative renormalization of biexcitons, and strongly indicate the

occurrence of a Bose-Einstein condensation of weakly localized quasitwo-dimensional biexcitons.

Acknowledgment

It is a pleasure to acknowledge discussions with S. Rudin, M. Dutta,

H. Shen, Doran Smith, and W. Zhou and the wafer growth and processing

provided by P. Newman and M. Taysing-Lara.
9

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11

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Attn AMSRL-SE-EP P Folkes (10 copies)

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US Army Research Lab (cont’d)

Attn AMSRL-SE-RL P Shah

Attn AMSRL-SE-RL R Lareau

Attn Chief AMSRL-SE

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Adelphi MD 20783-1197
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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Condensation of Quasi-Two-Dimensional Biexcitons in a

Single Heterojunction Quantum Well

April 1999 Final, Oct 1996 to Jan 1999

Excitons that coexist with a degenerate two-dimensional electron gas in the same quantum well

subband have been observed in the photoluminescence from the recombination of electrons with

localized photoexcited holes. At a critical electron density, an abrupt decrease in the exciton

radiative recombination rate is observed, along with the formation of biexcitons. With increased

excitation intensity, photoluminescence spectra are observed that verify theory on the radiative

renormalization of biexcitons and strongly indicate the occurrence of a Bose-Einstein condensation of

biexcitons.

semiconductor

Unclassified

ARL-TR-1934

9NENF1

611102.H47

AH47

61102A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release; distribution

unlimited.

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Unclassified Unclassified UL

2800 Powder Mill Road

Adelphi, MD 20783-1197

U.S. Army Research Laboratory

Attn: AMSRL-SE-EP pfolkes@arl.mil

U.S. Army Research Laboratory

15

Patrick A. Folkes

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197
DEPARTMENT OF THE ARMY

U.S. Army Research Laboratory2800 Powder Mill RoadAdelphi, MD 20783-1197

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