Phenomena in cold exciton gases: Condensation, macroscopic ordering and beyond L.V. Butov University of California San Diego Excitons and Electron-Hole Plasma electron_ exciton – bound pair of electron and hole hole + mexciton = melectron + mhole << matom exciton – light bosonic particle in semiconductor

_ _ at high densities + + excitons dissociate plasma of free electrons and holes exciton condensation

d dilute exciton gas (na <<1) d B dense electron-hole system (na >>1) B excitons are weakly interacting Bose particles excitons are formed at Fermi level like Cooper pairs exciton condensation is analogous to exciton condensate called excitonic insulator is analogous to BCS superconductor state Bose-Einstein condensation

Tc matched electron and hole Fermi surfaces mismatched

d n L.V. Keldysh, Yu.E. Kopaev (1964) naB ~1 L.V. Keldysh, A.N. Kozlov (1968)

why it's interesting? Kelvin for excitons exciton condensate is a new form of matter high T for exciton BEC due to light exciton mass: T exciton ~ 1 K c c microKelvin for atoms possibility to study crossover from BEC to BCS-like state possibility of manipulating condensate in microscopic semiconductor devices

how to get cold exciton gas? T << 1 K in He refrigerators lattice finite lifetime of excitons could result to high exciton temperature: T > T X lattice find excitons with lifetime >> cooling time T ~T X lattice find materials with low e-h recombination rate Condensation in 3D and 2D systems BEC Macroscopic occupation of ground state quasi-condensate - macroscopic occupation of low energy states difference between quasicondensate and BEC is not essential for most experiments

2/ 3 2πh2 n 3D systems: BEC is possible a t finite T T =0.527 c M g2/ 3

2D systems: BEC is possible at T=0 only 2 4πh n 1 / n s 1 Bogoliubov temperature T ~ B 2

n 2Mg ln[ln(g/na )] y t i Onset of nonzero order parameter

= onset of local superfluidity 2 2 - d dens π h n (T=T ) s KT ui f Kosterlitz-Thouless temperature T = KT 2Mg uper s T =0 T T pairing of vortexes = onset of macroscopic BEC KT B superfluidity which is not destoyed by vortexes 4πh2n 1 Finite 2D systems: BEC is possible at finite T T = c 2Mg ln(nS/g) include 2D systems with in-plane (random) potential in this case S - area of local potential minimum V.N. Popov, Theor. Math. Phys. 11, 565 (1972) D.S. Fisher, P.C. Hohenberg, PRB 37, 4936 (1988) W. Ketterle, N.J. van Drutten, PRA 54, 656 (1996) Materials with low e-h recombination rate classical semiconductors obstacles for with low e-h highlights experimental realization recombination rate of cold exciton gases Ge, Si discovery of electron-hole ground state – metallic liquid electron-hole liquid alternative to excitonic ground state

Cu2O a number of high rate of Auger interesting effects in recombination exciton gases

Novel systems indirect excitons in coupled quantum wells polaritons in microcavities excitons in quantum-Hall bilayers Why indirect excitons in CQWs? Al Ga As x 1-x GaAs

e long exciton lifetime due to separation GaAs/AlGaAs between electron and hole layers 103 times shorter exciton cooling time h CQW than that in bulk semiconductors

Γ coldest exciton gas: T << 1K < T X e X c AlAs/GaAs CQW E h potential candidate for realization of exciton condensation z exciton energy relaxation by LA-phonon emission 3D: coupling of E=0 state to single state E=E0 2D: coupling of E=0 state to continuum of energy states E > E 0 E =2M v 2 exciton 0 x s dispersion ~ 0.05 meV effective cooling of 2D excitons by bulk phonons E K How to get cold exciton gas?

excitons are generated hot and cool TX drops down down to Tlattice via phonon emission to 400 mK in 5 ns

< Tc << lifetime ways to overcome the obstacle of hot generation and study cold gases of indirect excitons with TX ~Tlattice

separation in time separation in space study indirect excitons study indirect excitons a few ns after the end of excitons beyond photoexcitation pulse photoexcitation spot Repulsive interaction between indirect excitons

e Dipole-dipole repulsive interaction stabilizes exciton state against formation of metallic electron-hole droplets h D. Yoshioka, A.H. MacDonald, J. Phys. Soc. Jpn. 59, 4211 (1990) + _ X. Zhu, P.B. Littlewood, M. Hybertsen, T. Rice, PRL 74, 1633 (1995)

_ the ground state of the + system is excitonic indirect excitons results in effective screening of in-plane disorder are oriented dipoles A.L. Ivanov, EPL (2002) R. Zimmermann

2 Wex=4 W/cm W =0.5 W/cm2 ex Repulsive interaction in experiment: y

sit Exciton energy increases with density

ten L.V. Butov, A. Zrenner, G. Bohm, G. Weimann, J. de Physique 3, 167 (1993)

In L.V. Butov, A. Zrenner, G. Abstreiter, G. Bohm, G. Weimann, PRL 73, 304 (1994)... V. Negoita, D.W. Snoke, K. Eberl, PRB 61, 2779 (2000) PL energy shift: δE~4πne2d/ε → estimate for exciton density

1.54 1.55 1.56 1.57 Energy (eV) Experiments on cold exciton gases in CQW nanostructures effects indicating exciton condensate

0.12 superradiance (macroscopic dipole),

0.020 ity

0.10 s

0.015 ) onset of exciton superfluidity, and 1

- )

0.08 1 s -

n s 0.010 ( n

1 ( - 0.06

r 1 fluctuations near phase transition

M M - τ 12

12 PL inten a 0.005 a τ g g 0.04 n 8 n 8 e e t 0.000 t ic 6 ic 0.02 0481216 4 4 F 5 F 6 Butov et al. J. de Physique 3, 167 (1993) i ie 4 e 5 ) l ld 3 e (K d 4 ) B (T) ur 3 (K ( 0 2 rat ( 0 re T pe T 2 ratu ) em ) pe T Tem PRL 73, 304 (1994) PRB 58, 1980 (1998)

bosonic stimulation of exciton shrinkage of spatially

ity scattering - signature of localized exciton cloud s degenerate Bose-gas of excitons with reducing T →

PL inten Butov et al. PRL 86, 5608 (2001) degenerate exciton gas 50 100 150 PRL 87, 216804 (2001) Butov et al. Nature 417, 47 (2002) Time (ns)

exciton rings macroscopically ordered exciton state Butov et al. cond-mat/0204482 [Nature, 418, 751 (2002)] cond-mat/0308117 [PRL 92, 117404 (2004)]

SSC 127, 89 (2003) http://physics.ucsd.edu/~lvbutov/ Bosonic stimulation of exciton scattering exciton E exciton PL kinetics rectangular photon phonon excitation pulse

PL jump occupation kinetics 10 W/cm2 of optically active 0.15 W/cm2 0.1 meV exciton states monoexponential PL kinetics exciton mass 50 100 150 is controlled in situ 10 K T =50 mK 1.2 bath 0 4 exp 0.8

/ m X )

end of excitation 1.6 M 0.4 -1 0.5 theory pulse normalized PL intensity normalized PL

010 (ns B (T)

0.15 Butov et al. rise

τ PRL 87, 26804 (2001) W/cm2

1/ 0 73 74 75 76 77 0100.1 1 10 0816

2 Time (ns) W (W/cm ) T (K) B (T) enhancement of exciton scattering rate to low energy states with increasing exciton concentration reveals bosonic stimulation of exciton scattering scattering rate of bosons signature of degenerate Bose-gas of excitons to a state p is ~(1+Np) L.V. Butov, A.L. Ivanov, A. Imamoglu, P.B. Littlewood, A.A. Shashkin, V.T. Dolgopolov, K.L. Campman, and A.C. Gossard, PRL 86, 5608 (2001) Experiment vs theory

10 10

experiment 1

4 (K) E=0

1

10 theory X N 0.1 T 0.01 0.1 3 10 50 100 150 50 100 150 Time (ns) PL intensity

2 T/0 T X 10 NE=0 = e -1 rectangular _ 2 excitation pulse T0 = πh n / 2gMXkB 50 100 150 temperature of quantum degeneracy Time (ns)

ph ph dNE=0/dt = ΓphNE(1 + NE=0)(1 + nE ) – Γph (1 + NE)NE=0 nE –NE=0 / τ = ph ph = Γph (NE -nE )NE=0 + Γph (1 + nE ) NE –NE=0 / τ

at low Tlattice and in presence Frolich inversion condition of generation of hot excitons counterpart of population ph inversion condition for lasers NE –nE >0 2D image of indirect exciton PL vs Pex

L.V. Butov, A.C. Gossard, and D.S. Chemla, cond-mat/0204482 [Nature 418, 751 (2002)] Radial dependence of indirect exciton PL

indirect internal bulk exciton D ring

r=0 external ring D PL peak intensity 0 40 80 120 excitation r (µm) spot profile external ring center m

µ I internal

PL intensity ring center 3.7 D excitation spot center r=0 1.54 1.55 1.56 1.57 PL intensity x4 Energy (eV) 1.54 1.56 1.58 Energy (eV) Ring structure of indirect exciton PL

internal ring

P (µW) ex 220

external ring can be remote from 95 excitation spot by 290 hundreds of microns

390 33 530

PL peak intensity 770

6

0 40 80 120 160 200 r (µm) at low densities: spatial profile of indirect exciton PL intensity follows laser excitation intensity at high densities: spatial profile of indirect exciton PL intensity is characterized by ring structure Temperature dependence of ring-shaped PL structure

T=2.4 K T (K) 2.4 4.4

7

10 L intensity P 14

1.54 1.55 1.56 1.57 PL peak intensity

T=14 K

04080120 r (µm)

ring structure of indirect exciton PL is observed at low T PL intensity with increasing T rings wash out and spatial profile approaches monotonic bell-like shape 1.54 1.55 1.56 1.57 Energy (eV) peak m ity µ

s pass n 440 te in L P

1.54 1.55 Energy (eV) m)

Pex µ 800 ( g

400 the rin n o PL intensity

tion 0 0 si

0 100 200 300 400 500 o 02040 position on the ring (µm) P Peak number external ring is fragmented into circular structures that form a periodic array over macroscopic lengths, up to ~1 mm exciton state with spatial order on macroscopic lengths – macroscopically ordered exciton state Temperature dependence of ring fragmentation into spatially ordered array of beads m T=1.8 K e h t ansfor ude of Tr t r i e i Ampl Four 0 0246 T (K) T (K) 1.8 T=4.7 K

4.7

PL intensity 7.7

0 100 200 300 400 500 Position on the ring (µm) ring fragmentation into spatially ordered array of beads appears abruptly at low T Ordered state

ring onset 6 )

K 4 T ( 2 low-temperature ordered phase state 0 0 400 800 1200 excitation power (µW) Features in exciton PL pattern PL Intensity 04080120 inner rings r (µm) external rings localized bright spots macroscopically ordered exciton phase PL pattern spatial distribution of optically active low energy excitons origin of inner ring flow of excitons out of excitation spot due E to exciton drift, diffusion, phonon wind, etc.

exciton

PL Intensity transport ~ 0.1 meV 0 40 80 120 over tens 1.549 repulsive of microns 1.548 interaction n K 1.547 → drift 1.546 Energy (eV) K K K 04080120 0 drift r (µm)

moving excitons are optically inactive

K > K0 v > vs shock excitons can travel in a dark state after excitation spot inner ring having been excited until slowed down high TX lower TX to a velocity below photon emission exciton drift excitons relax threshold, where they can decay radiatively to radiative zone lower occupation TX drops outside of excitation spot of radiative zone higher occupation fraction of optically active excitons increases of radiative zone L.V. Butov, A.C. Gossard, and D.S. Chemla, cond-mat/0204482 [Nature 418, 751 (2002)] simulations: A.L. Ivanov, unpublished origin of external ring off-resonance laser excitation creates additional number of holes in CQW electrons

electrons and holes have different collection efficiency to CQW holes

excitons excess holes are photogenerated in the laser excitation spot electron source is spread out over the entire plane due to current through the CQW from n-doped GaAs layers holes created at the excitation spot diffuse out this depletes electrons in the vicinity of the laser spot creating electron-free excitons are generated within the external ring and hole rich region same for formed at the interface between the hole rich e ↔ h region and the outer electron rich area L.V. Butov, L.S. Levitov, B.D. Simons, A.V. Mintsev, A.C. Gossard, D.S. Chemla, cond-mat/0308117 [PRL 92, 117404 (2004)] R. Rapaport, G. Chen, D. Snoke, S.H. Simon, L. Pfeiffer, K.West, Y.Liu, S.Denev, cond-mat/0308150 [PRL 92, 117405 (2004)] Expansion of the ring with increasing Pex

an increase of Pex increases hole density in CQW . n = D∆n−γnp+ J(r) .p = D'∆p−γnp+ J'(r) J(r) = I(r)−a(r)n(r)

J'(r) = Pexδ(r) holes nX ∝np

P ex excitons le density

electrons partic

T= 380 distance mK optical control of the ring by excitation intensity 410 µm Shrinkage of the ring with increasing gate voltage

a an increase of gate voltage increases electron density in CQW . n. = D∆n−γnp+ J(r) p = D'∆p−γnp+ J'(r) J(r) = I(r)−a(r)n(r) T=4.7 Kb J'(r) = Pexδ(r)

nX ∝np

Vg

electronic control of the ring by external gate voltage

T= 380 mK

380 µm Interaction of two exciton rings m µ

240 do not mix with attractive exciton-exciton interaction!

rings attract one another at large distances

the existence of “dark matter” outside the rings that mediates the interaction

T= electron flow outside each ring which is 380 mK perturbed by the presence of another ring

electrons in the area between the rings are depleted more strongly

attraction of the rings direct excitons indicate hotcores atthe collapsed rings P theory increasing P (“stars”) with bright spots to localized exciton rings collap ex se of Collapse ofrings to localizedbrightspots ex 2D imageofindirectexciton PLvsP ex

~ 600 µm rich illuminatedarea embedded inthehole crossing CQW) (at currentfilaments sources ofelectrons are duetolocalized localized brightspots e h holes excitons ele indirect c trons direct Ordered state indirect exciton PL direct exciton PL

410 µm ring onset 6 aggregates on the ring have no hot cores contrary

to bright spots generated by the pinholes )

K 4

aggregates move in concert with the ring when T ( the position of the source is adjusted showing 2 further that in-plane potential fluctuations are ordered phase not strong enough to destroy the ordering 0 0 400 800 1200 excitation power (µW) excitons in external ring are formed from well-thermalized carriers heating sources in the ring - the binding energy released at exciton formation due to long lifetimes of indirect excitons the heating has little effect on their temperature

the rings represent a source of cold excitons with temperature close to Tlattice in external ring exciton gas is the coldest macroscopically ordered exciton state

macroscopically ordered phases can be both in new state, not predicted quantum (e.g. atom BEC) and classical (e.g. Taylor vortices) systems origin of ordered exciton state - ? the macroscopically ordered phase appears abruptly at low temperatures is observed in the same temperature range as bosonic stimulation of exciton scattering (coincidence?) statistically degenerate Bose-gas of excitons note that the experiments on pattern formation and PL kinetics were done at different geometries direct comparison is not available yet Similarities with known phenomena: Modulational instabilities stationary solutions to 1D nonlinear Schrodinger equation under periodic boundary conditions on a ring stationary soliton trains experimental example: soliton train in atom BEC with attractive interaction K.E. Strecker, G.B. Partridge, A.G. Truscott, R.G. Hulet, Nature 417, 150 (2002) soliton train is observed

below Tc only

intrinsic property of atom BEC repulsion between beads of soliton train is wave interference phenomenon attractive interaction for indirect excitons ? positive feedback ? Similarities in astrophysics as far from BEC S. Chandrasekhar and E. Fermi (1953) as possible gravitational instability of an infinite cylinder: the cylinder is unstable for all modes of deformation with wavelengths exceeding a certain critical value for the mode of infinite cylinder maximum gravitational instability λ ~ πD instability fragmentation of gaseous slabs and fragmentation of D the cylinder filaments λ step in star formation

2 T=0.37 K T=1.8 K ed ng

n ri 1 agment o r t r f

attractive interaction o exci f D for indirect excitons ? / λ 0 positive feedback ? 50 100 150 200 ring radius (µm) origin of macroscopically ordered exciton state is, at present, unclear

low temperature state experiment restriction for interpretations