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1 Flutution nd Commensurility Effet of Exiton Density Wve Sen Yng 1, L.V. Butov 1, B.D. Simons 2, K.L. Cmpmn 3, nd A.C. Gossrd 3 1 Deprtment of Physis, University of Cliforni t Sn Diego, L Joll, CA , USA 2 Cvendish Lortory, Mdingley Rod, Cmridge CB3 OHE, United Kingdom nd 3 Mterils Deprtment, University of Cliforni t Snt Brr, Snt Brr, CA , USA (Dted: Otoer 18, 218) rxiv: v1 [ond-mt.mes-hll] 7 Fe 215 At low tempertures, indiret exitons formed t the in-plne eletron-hole interfe in oupled quntum well struture undergo spontneous trnsition into sptilly modulted stte. We report on the ontrol of the instility wvelength, mesurement of the dynmis of the exiton emission pttern, nd oservtion of the flutution nd ommensurility effet of the exiton density wve. We found tht flututions re strongly suppressed when the instility wvelength is ommensurte with defet seprtion long the exiton density wve. The ommensurility effet is lso found in numeril simultions within the model desriing the exiton density wve in terms of n instility due to stimulted proesses. PACS numers: An indiret exiton (IX) is ound pir of n eletron nd hole onfined in sptilly seprted quntum well lyers [1, 2]. Long lifetimes llow indiret exitons to ool down elow the temperture of quntum degenery giving n opportunity to study low-temperture exiton sttes. Remrkle phenomen in old IX gses inluding spontneous trnsition into sptilly modulted exiton stte [3, 4], spontneous oherene nd ondenstion of exitons [4 6], perfet Coulom drg [7], longrnge spin urrents nd spin textures [4, 8], enhned exiton rditive reomintion [9], tunneling reomintion [1, 11], nd sttering [12] rtes, nd orreltion phenomen [13 19] hve een found. Reently, sptilly ordered exitoni stte ws oserved in whih exiton density undergoes modultionl instility [3]. This stte, dued the mrosopilly ordered exiton stte (MOES), exhiits pproximtely periodi sptil modultion ouring within n exiton ring. The MOES forms when the IX gs is ooled elow few Kelvin lose to the temperture of quntum degenery (T db = 2π 2 n/m 3 K for the exiton density per spin stte n = 1 1 m 2 nd exiton mss m =.22m relevnt to the experiments). The MOES is hrterized y high degree of exiton oherene, with the oherene length rehing mirometers [4 6]. The lrge exiton oherene length is n order of mgnitude greter thn in lssil exiton gs showing tht the MOES is ondenste in momentum spe. The ourrene of sptil modultion in n exitoni system [3] initited intensive experimentl [4 6, 2] nd theoretil [21 28] studies. The following properties re importnt for understnding the MOES origin. The MOES forms in the externl ring of the exiton pttern formtion. The externl ring itself forms on the interfe etween the eletron-rih nd hole-rih regions [29 33]. The existne of suh interfe is essentil for the MOES ourrene nd, for instne, no spontneous density modultion is oserved in old IX gs in nother ring of the exiton pttern formtion the inner > <g (2) (x)> =6 =5.5 =5 2) g (2 4 8 x ( m).3.2 =6 = Gte Voltge(V) FIG. 1: (olor online). () The seond order orreltion funtion for the exiton density wve g (2) (x) = I(x )I(x +x) for I(x ) 2 the IX emission intensity profile I(x) long the ring segment etween LBS 1 nd LBS 2 of length L [shown in ()] with verging over 8 frmes in 27 seond dt quisition movie. The ommensurility numers ν = L/λ re 7 (red, light) Fig.4 nd 6.5 (lk, drk). () Imges of the emission pttern of indiret exitons verged over the 8 frmes for different ν. () Stndrd devition of g (2) s funtion of gte voltge, whih ontrols the instility wvelength λ. The peks indite the suppression of phse flututions of the exiton density wve t integer ν. ring, whih forms due to exiton trnsport nd ooling nd does not involve the order etween the eletronrih nd hole-rih regions [3, 34]. The other importnt property is tht the MOES is hrterized y repulsive exiton intertion [2]. This is onsistent with the predited repulsive intertion etween IXs, whih re

2 2 dipoles with uilt-in dipole moment [35]. A serh for mehnism responsile for the formtion of the MOES hd led to model ttriuting n instility to stimulted proesses of exiton formtion t the interfe etween the eletron-rih nd hole-rih regions tht uild up ner quntum degenery [22]. In this work, we report on the oservtion of flututions of the exiton density wve nd finding the ommensurility effet: The flututions vnish when the numer ν of wvelengths of the exiton density wve onfined etween defets is n integer. This new phenomenon in old exiton gses is presented in Figure 1. As detiled further in the text, the suppression of flututions of the exiton density wve t integer ν is reveled y pronouned mxim in the stndrd devition of the seond order orreltion funtion for the exiton density wve g (2) (x) for the IX emission intensity profile I(x) long the ring segment etween defets. We lso nlyzed the stility of the exiton density wve y numeril simultions nd found the ommensurility effet within the model desriing the exiton density wve in terms of n instility due to stimulted proesses. The oupled quntum well (CQW) struture ontins two 8 nm GAs QWs seprted y 4 nm Al.33 G.67 As nd surrounded y 2 nm Al.33 G.67 As rrier lyers (for detils see [3]). IXs in the CQW re formed from eletrons nd holes onfined in the seprted QWs. Photoexittion ws done y w 633 nm HeNe lser with 5 µm spot. The smll disorder in the CQW is indited y the IX emission linewidth of out 1 mev in the ring. The experiments were performed t T = 1.6 K. IX emission imges were quired y CCD mer fter n 8 ± 5 nm interferene filter mthing the IX energy. Previous studies hve shown tht inresing the lser exittion power P leds to the inrese of the externl ring rdius due to the enhnement of hole soure, while inresing the pplied gte voltge V leds to the derese of the ring rdius due to the enhnement of eletron soure [29 33]. Here, we vry P nd V simultneously so tht the ring rdius nd position re kept onstnt. This simultneous inrese of P nd V leds to the enhnement of oth eletron nd hole soures nd, s result, exiton density in the ring. Figure 2 shows tht inresing the exiton density leds to n inrese of the MOES wvelength λ. Note tht MOES eds re essentilly equidistnt forming n ordered rry, while the ed intensities vry from ed to ed (Fig. 2). We refer to suh qusiperiodi rry s the exitoni density wve. λ is ontrolled y P nd V within the rnge 9 24 µm in the experiments presented in Fig. 2. Lrger vlues of λ up to 4 µm were hieved for other ring rdii set y other vlues of P nd V. The exiton pttern formtion lso inludes lolized right spots (LBS), whih re ssoited with defets in the smple eletron urrent filments [29]. LBS eds re lerly distint from MOES eds: the positions MOES Wvelength (μm) 2μm 2μm Lser Power (mw) Gte Voltge (V) pek position (μm) l (μ m ) l (μ m ) pek numer FIG. 2: (olor online). (,) Left: A segment of the externl ring in the exiton emission pttern. Right: The orresponding IX emission intensity profile long the ring. () Gte voltge V = 1.26 V, lser exittion power P = 1.12 mw. () V = 1.13 V, P =.28 mw. () The MOES wvelength λ vs V nd P, whih re vried simultneously so tht the ring rdius is fixed. (d) The MOES ed position vs pek numer for the ring shown in () (irles) nd () (squres). d Fig.1 of LBS eds re fixed while MOES eds move with the ring when the position of the lser spot is djusted, esides LBS eds hve hot ores ssoited with the urrent-indued heting while MOES eds don t [29]. Figure 3 nd movie in supplementry mterils show tht LBS eds re stle while MOES eds flutute with time. Both the oserved flututions of the exiton density wve (Fig. 3) nd its wvelength vrition with density for the fixed ring position (Fig. 2) indite tht the exiton density modultion in the MOES forms spontneously rther thn due to the in-plne disorder in CQW. The stility of LBS eds nd flututions of MOES eds (Fig. 3) show tht the phse of the exiton density wve in the ring is loked t LBS defets nd flututes in etween them. Controlling the exiton density in the ring (y vrying P nd V ) llows to proe the flututions of the exiton density wve for different rtios etween the MOES wvelength λ nd the length L of the ring segment etween two LBS on the ring (suh s LBS 1 nd 2 in Figs. 1 nd 4). Figure 4 shows tht the mplitude of the flututions is smll when the numer of wvelengths of the exiton density wve onfined etween the defets ν = L/λ is n integer. In turn, flututions inrese for non integer ν, ompre in Fig. 4, the medium pnel

3 5μm 5μm 5μm 1μm 1μm d PL Intensity (r.unit) e Bed position (μm) 2 1 2 2 1 LBS Bed 1 1 1μm 2 1 2 3 2 MOES Bed 2 1 Fig.2 FIG. 3: (olor online).

() MOES ed positions flutute in time: left nd right imges re mesured t the sme prmeters vs time. () Stndrd devition of the IX emission intensity.

(d) IX emission intensity vs time in the points mrked y rosses in () nd () round LBS ed (left) nd MOES ed (right). (e) Bed position vs time for LBS ed (left) nd MOES ed (right). V = 1.217 V.

3 3 5μm 5μm 5μm 1μm 1μm d PL Intensity (r.unit) e Bed position (μm) LBS Bed 1 1 1μm MOES Bed 2 1 Fig.2 FIG. 3: (olor online). () An imge of the IX emission pttern extrted from rel time movie, whih shows flututions of the exiton density wve. () MOES ed positions flutute in time: left nd right imges re mesured t the sme prmeters vs time. () Stndrd devition of the IX emission intensity. Yellow (light) olor indites high flutution regions nd lue (drk) olor indites low flutution regions. (d) IX emission intensity vs time in the points mrked y rosses in () nd () round LBS ed (left) nd MOES ed (right). (e) Bed position vs time for LBS ed (left) nd MOES ed (right). V = V. MOES eds flutute while LBS eds re stle. presenting lrge flututions t non integer ν with left nd right pnels presenting smller flututions t integer ν. This ommensurility effet is quntified in Fig. 1, whih presents the seond order orreltion funtion for for the IX emission intensity profile I(x) long the ring segment etween LBS 1 nd 2. Apprently, stle periodi wve produes strong osilltions in g (2) (x), while flututions of the wve smer out suh osilltions. Stndrd devition of g (2) gives mesure for the flututions. Figure 1 shows pronouned mxim in g (2) inditing suppression of the flututions of the exiton density wve t integer ν. the exiton density wve g (2) (x) = I(x )I(x +x) I(x ) 2 The oserved flututions of the exiton density wve nd ommensurility effet re disussed elow. The PL Intensity (. u.) Bed position (μm) MOES Bed 2 MOES Bed 2 MOES Bed A FIG. 4: (olor online). () Imges of the IX emission pttern for different MOES wvelengths λ. The ommensurility Fig.3 numer ν = L/λ in the ring segment etween LBS 1 nd LBS 2 of length L is ν = 8 (left) nd 6 (right). () Flututions of the IX emission intensity for integer (left nd right) nd non-integer (middle) ν in the point mrked y ross in (). () Flututions of the ed positions for the sme onditions s in (). V = (left), (middle), nd (right) V. Flututions of MOES eds vnish t integer ν. MOES is stte with spontneously roken symmetry. It involves lrge numer of exitons, for instne the estimted numer of exitons in the ring segment etween LBS 1 nd LBS 2 is 1 6. The ommensurility effet indites tht the flututions of the exiton density wve re olletive. Colletive flututions in sttes with spontneously roken symmetry is generl phenomenon oserved in vriety of systems. Rottionl flututions nd wves in liquid rystls, sound wves in liquids nd solids, nd seond sound wves in superfluids present hrteristi exmples. Here, we ompre the experimentl results for the exiton density wve with the theory ttriuting the development of the MOES to stimulted proesses tht uild up ner quntum degenery [22] nd show tht the experimentlly oserved ommensurility effet is lso found within this model. This instility is enpsulted y simple kineti theory involving the interply of the eletron/hole nd exiton densities. In prtiulr, the dimensionless eletron density, g e De l n e, stisfies the nonliner diffusion equ-

4 tion, 2 g e = exp [ηδg x ] g e (g e x), where lengths re mesured in terms of the diffusion length, nd flutution of the lol exiton density from its vlue in the unmodulted stedy stte, δg x g x ḡ x, depends non-lolly on the flutution in eletron density δg e g e ḡ e through the reltion, δg x ( x) = δg e ( x)+ d 2 x 2πl 2 x K ( x x l x )δg e ( x ), with K the modified Bessel funtion. Here denotes the totl rrier flux t the interfe, D e denotes the eletron diffusion onstnt, nd the ontrol prmeter, η, involves oth the proximity to the degenery temperture nd exiton density of the unmodulted stte t the eletron-hole interfe (for detils, see [22]). In the ring geometry, n nlysis of the kineti eqution ove shows tht, elow ritil temperture, the ring undergoes type II instility towrd the development of sptilly modulted stte with wvelength, ut onstrined to e ommensurte with the overll ring irumferene (in the relevnt prmeter regime, the diffusion length l x exeeds the rnge of eletron nd hole overlp l). To ssess the potentil for olletive flututions to drive the oserved ommensurility effet, we explored the pttern of instility for fixed vlue of η = 1, where the instility is well-developed, for hnging vlues of l/l x. When the instility is llowed to nnel from the unmodulted stte, the wvenumer, λ, of the modulted stte hnges sequentilly through sequene of plteus set y disrete vlues omptile with the ring irumferene (Fig. 5). However, when the system is llowed to evolve from the fully-modulted stte s funtion of hnging l/l x, one sees oth hysteresis nd the exlusion of n intermedite stle wvenumer. These effets mirror the ommensurility effet seen in experiment the stohsti trnsfer etween stle nd metstle sttes feture of disontinuous trnsitions, nd the exlusion of stle modultions mnifesttion of the nonlinerity of the dynmis. set y the length sle λ l 1/3 l 2/3 x The experiment shows lso tht oth the MOES wvelength λ nd ring width δ r inrese with density (Fig. 2). This dt is lso onsistent with the model where oth λ nd δ r inrese with density [λ l 1/3 l 2/3 x nd δ r l x nd l x nd l inrese with density due to sreening of the in-plne disorder]. In onlusion, we oserved flututions of the exiton density wve nd the ommensurility effet the flutution suppression when the numer of wvelengths onfined etween defets is n integer. We thnk Leonid Levitov for vlule disussions nd ontriutions t the erlier stge of the exiton pttern formtion studies. This work ws supported y NSF. Density (r. unit) Wvelength (r. unit) x (r. unit) d Density (r. unit) x (r. unit) ζ 3 Density (r. unit) x (r. unit) FIG. 5: (olor online). () Wvelength of the exiton density wve s funtion of (l x/l) 3. Squres (irles) show the evolution of λ s l x/l is rmped up progressively from 3 to 4.4 (down progressively from 4.2 to 3.6) using the previous vlue s the seeding density. Tringles show evolution of λ for stedy-stte exiton density wve s l x/l is hnged from 3 to 4.2 using smll rndom perturtion from the uniform solution s seeding density. (-d) Corresponding profiles of the exiton density wve long the eletron-hole interfe. Commensurte sttes with integer ν re found to e roust with respet to the prmeter vrition produing the plteus, while flututions develop in the trnsition region etween integer ν where hysteresis is found. [1] Yu.E. Lozovik, V.I. Yudson, JETP 44, 389 (1976). [2] T. Fukuzw, S.S. Kno, T.K. Gustfson, T. Ogw, Surf. Si. 228, 482 (199). [3] L.V. Butov, A.C. Gossrd, D.S. Cheml, Nture 418, 751 (22). [4] M. Alloing, M. Bein, M. Lewenstein, D. Fuster, Y. González, L. González, R. Comesot, M. Comesot, F. Duin, Europhys. Lett. 17, 112 (214). [5] Sen Yng, A.T. Hmmk, M.M. Fogler, L.V. Butov, A.C. Gossrd, Phys. Rev. Lett. 97, (26). [6] A.A. High, J.R. Leonrd, A.T. Hmmk, M.M. Fogler, L.V. Butov, A.V. Kvokin, K.L. Cmpmn, A.C. Gossrd, Nture 483, 584 (212). [7] D. Nndi, A.D.K. Fink, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Nture 488, 481 (212). [8] A.A. High, A.T. Hmmk, J.R. Leonrd, Sen Yng, L.V. Butov, T. Osttniký, M. Vldimirov, A.V. Kvokin, T.C.H. Liew, K.L. Cmpmn, A.C. Gossrd, Phys. Rev. Lett. 11, (213). [9] L.V. Butov, A.I. Filin, Phys. Rev. B 58, 198 (198). [1] I.B. Spielmn, J.P. Eisenstein, L.N. Pfeiffer, K.W. West, Phys. Rev. Lett. 84, 588 (2). [11] J.P. Eisenstein, A.H. MDonld, Nture 432, Fig.5

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THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny