Giant permanent dipole moment of 2D excitons bound to a single stacking fault

3.7 MB·

by  Todd Karin

Giant permanent dipole moment of 2D excitons bound to a single stacking fault
Todd Karin,1, Xiayu Linpeng,1, M.M. Glazov,2 M.V. Durnev,2 E.L. Ivchenko,2
∗ ∗
Sarah Harvey,1 Ashish K. Rai,3 Arne Ludwig,3 Andreas D. Wieck,3 and Kai-Mei C. Fu1,4
1Department of Physics, University of Washington, Seattle, Washington 98195, USA
2Ioffe Institute, 194021 St.-Petersburg, Russia
3Lehrstuhl fu¨r Angewandte Festk¨orperphysik, Ruhr-Universita¨t Bochum, D-44870 Bochum, Germany
4Department of Electrical Engineering, University of Washington, Seattle, Washington 98195, USA
(Dated: June 8, 2016)
Weinvestigatethemagneto-opticalpropertiesofexcitonsboundtosinglestackingfaultsinhigh-
6 purityGaAs. Wefindthatthetwo-dimensionalstackingfaultpotentialbindsanexcitoncomposed
1 of an electron and a heavy-hole, and confirm a vanishing in-plane hole g-factor, consistent with
0 the atomic-scale symmetry of the system. The unprecedented homogeneity of the stacking-fault
2 potentialleadstoultra-narrowphotoluminescenceemissionlines(withfull-widthathalfmaximum
(cid:46)80 µeV) and reveals a large magnetic non-reciprocity effect that originates from the magneto-
n
Stark effect for mobile excitons. These measurements unambiguously determine the direction and
u
magnitude of the giant electric dipole moment ((cid:38)e 10 nm) of the stacking-fault exciton, making
J ·
stacking faults a promising new platform to study interacting excitonic gases.
6

]
l Introduction. The stacking fault (SF), a planar, atomi- by subtracting a layer (intrinsic SF, see Fig. 1c) or adding
l
a cally thin defect, is one of the most common extended de- a layer (extrinsic SF). The intrinsic SF can be viewed as
h fectsinzinc-blende,wurtzite,anddiamondsemiconductors. a monolayer of wurtzite (ABAB stacking) surrounded by
- A fundamental understanding of the SF potential is im- zinc-blende (ABCABC stacking) [3, 24]. Due to the band
s
e portant for determining how the defect affects semiconduc- offset [25–27] and spontaneous polarization at the stacking
m tor device performance [1, 2], engineering heterostructures fault [28], electrons and/or holes are attracted to the SF
. based on crystal phase [3–5], and providing a new two- plane. Whileusefulforphysicalmotivation,thisbulkphase
t
a dimensional (2D) platform for fundamental physics [6, 7]. change model must be taken with caution when applied to
m Herewereportonexcitonsboundtolarge-area,singleSFsin atomically thin SFs, which can deviate from simple the-
high-purityGaAs,auniquesystemwhereSFsareeasilyiso- ory [29]. Here, however, we find that single SFs in bulk
-
d lated with far-field optical techniques. The atomic smooth- GaAsbindexcitons,confirmingthatthepotentialisattrac-
n
nessofthepotentialandextremeperfectionofthesurround- tive for at least one carrier.
o
ing semiconductor result in ultra-high optical homogeneity In the confocal scan in Fig. 1(a), most of the SF defects
c
[ ((cid:46)80 µeV). This enables optical resolution of the SF ex- appear as single triangles, which we identify as a pair of
citon fine-structure and thus direct measurement of the gi- nearby SFs [30, 31]. Because the binding energy of exci-
4 ant built-in dipole moment ((cid:38)e 10 nm) via the magneto- tons to a pair of SFs depends on the distance between the
v ·
1 Stark effect. These results indicate that the extremely- SFs [32], the PL emission energy from excitons bound to
9 homogeneous SF potential may be promising for studies of these structures has a high variability of 10 meV between
9 many-body excitonic physics, including coherent phenom- structures. Strikingly, this inhomogeneity disappears when
3 ena [8–10], spin currents [11], superfluidity [12], long-range four SFs grow in an inverted pyramid structure consisting
0 order [13–17], and large optical nonlinearities [18–20]. of four well-isolated 111 SF planes [Fig. 1(b)], which we
1. Stacking fault photoluminescence. Figure 1(a) shows a refer to as up, down,{left}and right [33]. The full width at
0 spectrallyresolvedconfocalscanofSFstructuresinaGaAs half-maximum (FWHM) of the SF PL line in our sample is
6 epilayer, excited with an above band-gap laser (1.65 eV, (77 19)µeVatzeromagneticfield[21],somewhatnarrower
1 1.5 K) [21]. The image is colored red, green or blue accord- tha±n excitonic lines associated with stacking faults in pre-
:
v ing to three characteristic emission bands shown in Fig. 1e. vious work [22, 34]. In comparison, the narrowest reported
i The narrow-band PL at 1.493 and 1.496 eV originates from linewidthforaGaAs/AlGaAsquantumwellis130µeV[35],
X
excitons, electron-hole pairs, bound to the 2D SF poten- while PL linewidths from analogous zinc-blende/wurtzite
ar tial [22, 23]. The sample consists of a 10 µm GaAs layer quantumdiscsinnanowiresrangefrom0.6 10meV[27,36–
on 100 nm AlAs on a 5 nm/5 nm AlAs/GaAs (10 ) super- 38]. This unprecedented homogeneity allo−ws us to resolve
×
lattice grown directly on a semi-insulating (100) GaAs sub- the SF-bound exciton fine structure
strate. Stackingfaultstructuresnucleatenearthesubstrate-
Nature of hole in SF exciton. Experimentally, we deter-
epilayer interface during epitaxial growth [21].
mine that the SF exciton is composed of an electron and a
The physical origin of the potential can be understood
heavy-hole using polarization resolved PL, consistent with
fromtheatomicstructureoftheSFdefect: thelattice-plane
the atomic-scale symmetry of the system [21]. For linearly
ordering in the [111] direction of zinc-blende is modified
polarized light incident from above (along the [001] axis),
the largest overlap between the light polarization and the
in-SF-planeheavy-holedipoleoccurswhenexcitingandcol-
lectingalongtheHdirectionforthedown SF[Fig.1(d)], in
∗ Theseauthorscontributedequallytothiswork. agreement with our experimental data [Fig. 1(f)]. On the
2
(a) (b) (c) B C 1.491 1.493
L D RU kVH CA AB A.U.) (fH) SF
[111] B B L( V
A A P
[11¯2]
PerfectCrystal StackingFault
(d) V
[010] [100] up H (e) Edge SF SF
)
U.
eft ght (A.
l ri PL x10
10µm 5µm down 1.460 1.470 1.480 1.490 1.500
E(eV)
FIG. 1. (a) Confocal scan of SF structures. The image is formed by coloring emission in different wavelength bands as red, blue or
green,asdepictedine. Excitationat1.53eV,100µW,1.9K,exciteandcollectHpolarization(seeb). (b)DiagramofSFpyramid.
The up, down, left and right SFs are labeled, along with the H and V polarizations. (c) Comparison of perfect zinc-blende and
stackingfaultcrystalstructure. (d)DetailofSFpyramidstructure. Excitationat1.53eV,100µW,1.7K.(e)LowpowerPLspectra
at colored dots in d. Polarizations: blue - excite/collect H; green, red - excite/collect V. Broad-band luminescence is observed from
the SF edges (red). 1.53 eV, 2 µW, 1.7 K. (f) PL from down SF (blue dot in d). Polarizations: dark blue - excite/collect H; light
blue - excite/collect V.
otherhand, themaindipolemomentforthelight-holeexci- pending on whether the magnetic field is parallel (positive)
tonisalongtheSFnormal,whichwouldgiverisetoamaxi- or antiparallel (negative) to the [¯110] axis. Interestingly,
mumsignalatVpolarization,contrarytowhatisobserved. the down SF demonstrates the opposite behavior. Such
Further, we also note that no hole Zeeman splitting is ob- an asymmetric behavior of the PL is surprising because in
servedforin-planemagneticfieldsBupto7T(Fig.2). This general, time reversal symmetry makes B and B equiva-
−
observation is fully consistent with our symmetry analysis, lent [40]. The observed non-reciprocal behavior of the PL
which finds that B-linear splitting for in-plane fields is for- spectrumwithrespecttoinversionB Bisonlypossible
→−
bidden for heavy-holes but allowed for light-holes [21]. The ifthePLarisesfrommovingexcitons. Inthiscase, timere-
substantial separation of the heavy- and light-hole states versal changes the direction of both the magnetic field and
prevents their magnetic-field induced mixing, in line with the exciton wavevector K.
experiments on GaAs nanowires [21, 39]. Based on the C point symmetry of the SF and time
3v
reversalinvariance, theeffectiveHamiltonianforanexciton
(a) Up (b) Right moving in the presence of an in-SF-plane magnetic field B
PL 7.0 T Down PL 7.0 T Left is g
d Bd = eµ (σ B +σ B )+βB2+β [K B] , (1)
ze 3.5 T L Uze 3.5 T HKB 2 B x x y y (cid:48) × z
mali 0.0 T D Rmali 0.0 T where ge is the electron g-factor, µB is the Bohr magneton,
or -3.5 T or -3.5 T σx,y are the electron spin Pauli matrices, β is a parame-
N N
ter describing the excitonic diamagnetic shift, and β is a
-7.0 T -7.0 T (cid:48)
constant responsible for the non-reciprocal effect [21]. In
1.492 1.494 1.496 1.498 1.496 1.498 1.500 1.502
Eq. (1) we only retain 1st- and 2nd-order terms in B and
E (eV) E (eV)
use a frame of axes related to the SF plane: z [111] is the
SF normal, x [11¯2] and y [¯110]. Each symm(cid:107)etry-derived
FIG. 2. (a) Spectra from up and down SFs as a function of in- || ||
term in Eq. (1) manifests itself in the energetic shift of the
plane magnetic field. The spectra show a non-reciprocity with
applied magnetic field. Excite at 1.65 eV, 0.5 µW, 1.6 K, excite SF PL lines with magnetic field (Fig. 2). The first term
and collect H. Inset shows the geometry of the stacking fault is the electron Zeeman effect and gives rise to the doublets
pyramid and applied magnetic field. (b) Spectra from left and visible at 7 T, since an electron with a particular spin
±
right SFs as a function of partially out-of-plane magnetic field. projection can recombine with the corresponding hole. The
Thespectraaresimilaratpositiveandnegativefields. Excitation second term is the exciton diamagnetic shift, arising from
at 1.65 eV, 0.5 µW, 1.6 K, excite and collect V. the magnetic-field-induced shrinking of the exciton wave-
function [41]. The last term is the magneto-Stark effect,
Non-reciprocalphotoluminescence. PLfromSFsshowsa which, as we show below, quantitatively explains the non-
remarkable non-reciprocity with in-plane applied magnetic reciprocal PL spectra.
field: Figure 2(a) shows that the PL detected in linear po- The experimental geometry, Fig. 1(b), is such that only
larization from the up SF occurs at a different energy de- light emitted normal to the sample surface is collected. For
3
a high quality 2D potential, in-plane exciton momentum is SF pyramids were used at different angles. This does not
transferredtothephotonduringrecombination,asdepicted introduce artifacts because of the extreme similarity of dif-
in Fig. 3(a). This conservation of momentum implies ferent SFs, which have a standard deviation of line-center
energies of only 57 µeV, less than the linewidth. Spectra
ωn
Kx = c sinθ(cid:48)(cid:48), (2) wuperaenadcqduoiwrendSwFist.hWBeyfiratntghiensgpfercotmra−to6.o5nTeotroa6.s5umToonfttwhoe
Voigt function(s) depending on whether the electron Zee-
where θ is the angle between the SF normal and the emit-
(cid:48)(cid:48)
man splitting is resolved. The singlet or doublet line center
ted photon momentum inside the semiconductor, Fig. 3(a),
is denoted E (B ). The part of the exciton energy
ωisthephotonfrequency,nistherefractiveindexandcthe up/down y
odd with magnetic field is found by computing
speed of light. Thus, the collected SF PL arises only from
excitonswithaspecificcenterofmassmomentum[42]. The
∆E (B )=E (B ) E ( B ) (4)
lastterminEq.(1)provides, forafixedKx (Eq.2), anodd up/down y up/down y − up/down − y
in B contribution to the overall PL energy shift, giving
y It follows from Eq. (3) that the asymmetric shift is
rise to a magnetic non-reciprocity effect. It is worth noting
that the up and down SFs are related by a mirror reflec- ed
∆E (B )= 2n(cid:126)ω he sin(θ θ )B . (5)
tioninthe(110)planeandsuchareflectionisaccompanied up/down y ∓ Mc SF ± (cid:48) y
by B B , resulting in the opposite behavior of up and
y y
→− Thus the proportionality constant of ∆E vs. B
down PL spectra observed in Fig. 2(a). up/down y
provides a measurement of the SF exciton’s built-in dipole
Magneto-Stark effect. The physical origin of the non-
moment. Theexperimentalvaluesandfirst-ordertheoryfor
reciprocal PL is the magneto-Stark effect, the interaction
∆E are shown in Fig. 3(f)-(g). Further, the ratio
of a moving exciton’s electric dipole moment with a mag-
netic field [43, 44]. The effect can be understood with a
∆E ∆E
relativistic argument: motion with velocity v=((cid:126)K /M)xˆ r(θ)= | up|−| down| (6)
x 1(∆E + ∆E )
through a magnetic field B=Byyˆ gives rise to an electric 2 | up| | down|
field E =(cid:126)K B /(Mc)zˆ in the moving frame of refer-
eff x y depends (to first order in B ) only on the experimental ge-
ence, where M is the exciton mass in translational motion y
ometry andthe indexof refraction: r(θ) vanishes forcollec-
and c the speed of light. Since for the SF, zˆ [111] and
∝ tion angle θ =0 and increases as a function of θ [Fig. 3(h)].
zˆ directions are not equivalent, the SF-bound exciton has
− We obtain good agreement between r(θ) calculated exper-
a non-zero dipole moment p=ed zˆ, where e= e is the
he | | imentally from the B =0 slope of ∆E without any fit pa-
elementary charge, and d is the average separation be-
he rameters [Fig. 3(h)].
tween the hole and electron along the z-axis. The Stark
Further,byfitting∆E (B )withaB -linearfunc-
effect H = p E in the exciton’s reference frame thus up/down y y
s − · eff tion, we can estimate the dipole moment of the exciton
becomes the magneto-Stark effect:
p=ed =e (10+20) nm. The main uncertainties result
e(cid:126) fromthheeaccu·racy−o1ftheBy-linearfitandthevalueofthein-
S = dheKxBy, (3) (111)-plane heavy-hole mass, which depends on the details
H −Mc
oftheSFpotential[21]. Theexcitonmasscanberoughlyes-
in agreement with Eq. (1) with β = e(cid:126)d /(Mc), see timatedas0.17m ,thesumofthebulk-GaAsin-(111)-plane
(cid:48) he o
−
Ref. [21, 41] for formal derivation. heavy-hole mass and the isotropic electron mass, where m
o
Physically, the dipole moment of a SF bound exciton is a isthefreeelectronmass. Inaddition,wenotethemagneto-
consequence of symmetry breaking and spontaneous polar- Stark induced splitting saturates at high fields [Fig. 3(f,g)],
ization similar to that in zinc-blende/wurtzite heterostruc- possibly due to a decreased exciton dipole moment from
tures [23, 45]. The hole in the exciton is presumably lo- the magnetic-field-induced shrinking of the exciton wave-
calized in the SF plane while the electron is weakly bound function. Future work will investigate exciton confinement
via the Coulomb interaction. The spontaneous polarization potentialsconsistentwiththeobserveddipolemoment,dia-
shifts the electron cloud to one side of the SF, resulting in magnetic shift and saturation of the magneto-Stark effect.
a giant excitonic dipole moment. A microscopic understanding of the confinement potential
Equations (1)-(3) predict that the asymmetric energy may enable predictions for the binding potential and exci-
shift of exciton PL is linearly related to the in-plane tonic dipole moment for SFs in other semiconductors.
wavevector K. Since the angle of light collection deter- Conclusion. WehaveshownthatSFsinGaAsareanal-
mines the exciton momentum [Eq. (2)], we test the ap- most perfect 2D potential which binds heavy-hole excitons.
plicability of the model by recording spectra of the up These excitons freely propagate in the SF plane, a conclu-
and down SFs as a function of the collection angle θ sion confirmed via the magneto-Stark effect. Further, an
and magnetic field B [Fig. 3(b)]. The collection angle asymmetry of the SF potential induces a giant dipole mo-
y
is related to the emission angle from the up/down SF by mentoftheSF-boundexciton. Suchexcitonscouldbeuseful
sinθ =nsinθ = nsin(θ θ ), where θ is the angle for studying the many-body physics of interacting dipoles.
(cid:48) (cid:48)(cid:48) SF SF
± −
the SF normal zˆ [111] makes with [001] [Fig. 3(c)]. Inconventionalexcitonicsystems,typicalelectron-holesep-
||
In this experiment, we modified the collection angle by arations are on the order of several nm [6, 46], whereas the
mounting the sample at different angles. Since the sample SF-bound exciton has a gigantic electron-hole separation
wasremovedfromthecryostattochangetheangle,different of 10 nm and the possibility to modify this value with an
4
(a) B (d) ∆Eup θ =0◦ (f) θ =0◦ 0.3 (h)
y ) 800
θ00 K SF nits p V)
x U U e600
µ
Eeff (Arb. n -6.5∆TEdow6n.5T ∆E(|400 Up 0.2
L w |200
Collect Up P o Down
D 6.5T -6.5T
(b) 0 )
1.4965 1.4970 1.4975 0 2 4 6 θ
θ r(
Bya (e) E∆(EeuVp) θ =43◦ (g) B(T) θ =43◦ 0.1
) 800
s
Down nit p V)
U U e600
µ
θ θ Arb. -6.5T∆Edown6.5T E(|400 Data
(c)D θ0 θSFθ0 F PL( own ∆|200 UDpown 0.0 Theory
own UpS zy D1.4696.50T1.49-665.51T.4970 00 2 4 6 0 20 40
SF x
E(eV) B(T) θ(◦)
FIG. 3. (a) Because of conservation of in plane momentum during exciton recombination, the angle of light emission depends on
the exciton wavevector. Collecting different angles probes different exciton momenta. The SF has a built in potential that creates a
zero-field dipole moment for the SF exciton. In the exciton frame of reference, the in-plane magnetic field becomes an out-of-plane
electricfield,leadingtothemagneto-Starkeffect. (b)Spectraofup anddown SFsasafunctionofθ andin-planemagneticfieldBy.
(c) Light from the up SF originates from excitons with larger Kx than light from the down SF (for θ >0). (d-e) Spectra of up and
down SFatpositiveandnegativeBy forθ=0◦ and43◦. Atθ=0◦,∆Eup and∆Edown havethesamemagnitude,whileforθ=43◦,
the magnitude of ∆Eup is larger than ∆Edown. (f-g) Splitting ∆Eup/down as a function of magnetic field. Data are obtained from
Voigt fits to spectra similar to those shown in d-e. Solid lines are a fit to ∆E =aB for the first three data points. (h) The ratio of
B =0slopes,Eq.(6),dependsonlyongeometricalconstraints. Thetheory(solidline)hasnoadjustableparameters. Dataforother
angles in [21].
applied field. In addition, the ultra-narrow linewidths in Number 1150647 and the National Science Foundation
the SF system will allow the small energy shifts present in Graduate Research Fellowship under grant number DGE-
many-body interactions to be observed. As a rough esti- 1256082, and in part by the State of Washington through
mate, the interaction energy of two such dipoles will ex- the University of Washington Clean Energy Institute. The
ceed the SF FWHM of 77 µeV when the exciton density IoffeteamhasbeenpartiallysupportedbyRFBR,RFPres-
is greater than 230 µm 2. Using a wavefunction size of ap- ident grant No. MD-5726.2015.2 and Dynasty foundation.
−
proximately10nm,thecriticaldensityforexcitonoverlapin A.K.R., A.L., and A.D.W. acknowledge partial support of
the 2D potential is 10000 µm 2. Therefore, the SF-bound Mercur Pr-2013-0001, DFG-TRR160, BMBF - Q.com-H
−
excitonsystemcouldshowsizabledipole-dipoleinteractions 16KIS0109, and the DFH/UFA CDFA-05-06. We would
andmaydemonstratecoherentphenomenaatreasonableex- like to acknowledge helpful discussions with John Schaib-
citon densities. ley,PasqualRivera,XiaodongXu,DavidCobdenandMatt
Acknowledgements. This material is based upon work McCluskey.
supportedbytheNationalScienceFoundationunderGrant
5
Supplemental Materials: Fundamental properties of 2D excitons bound to single stacking
faults in GaAs
S1. STACKING FAULT FORMATION (Andor).
Stacking fault (SF) structures can grow from the
substrate-epilayerinterfaceduringepitaxialgrowth[33,47]. S3. STACKING FAULT PHOTOLUMINESCENCE
LINEWIDTH
Inthepresentwork,SFsformina10µmGaAslayergrown
bymolecularbeamepitaxywithroomtemperatureelectron
density n 1.9 1014 cm 3 and mobility 7400 cm2/Vs.
−
∼ × ∼
Theentirestructureconsistsofthe10µmGaAslayeron100
s)
nm AlAs on a 5 nm/5 nm AlAs/GaAs (10 ) superlattice nit
× U
growndirectlyonasemi-insulating(100)GaAsverticalgra- b. (94±13)µeVbefore (77±19)µeVafter
dientfreezesubstrate(WaferTechnologyLtd),startedwith Ar deconvolution deconvolution
AlAs. Thesamplewasgrownatapyrometertemperatureof L (
P
600 CwiththerelativelylowAs beamequivalentpressure
◦ 4
of8 10 6 Torr,measuredbyafluxtube. Thegrowthrate 1.492 1.4925 1.493 1.4935 1.494
× − E (eV)
was 0.7 ML/s for the GaAs and 0.35 ML/s for the AlAs.
Oxide removal before growth was performed at 620 C un-
◦ FIG.S5. HighresolutionPLspectrumofup SF.TheFWHMof
der As flux. This growth procedure resulted in oval defects
the line is (77 19) µeV from a weighted Lorentzian fit, taking
in the sample surface, a feature commonly associated with intoaccountth±einstrumentresolution. Beforedeconvolutionthe
stacking faults [22, 33, 47, 48]. We observe two types of FWHMis(94 13)µeV.Excitationat1.65eV,1.44K,1.4nW.
stacking fault defects, a SF pyramid and a SF pair defect, ±
shown in Fig. S4. The size of the pyramid structure (14.1
FigureS5showsahighresolutionPLspectrumoftheSF.
µm top edge in Fig. 1a) is consistent with stacking faults
In order to extract the true PL linewidth, we need to take
thatnucleatenearthesubstrate-epilayerinterfaceandgrow
into account the spectral resolution of our setup. The spec-
along 111 planes through the 10 µm thick epilayer.
{ } trometerinstrumentresolutionisfoundbytakingaspectro-
graph of a narrow band Ti:Sapphire laser and fitting to a
SFPyramid SFPairDefect Voigtfunction,seeFig.S6(a). WefindthatneitheraGaus-
siannoraLorentzianaccuratelydescribethespectralpoint
V
[1¯11] spreadfunction,soweuseaVoigtfit. Forspectrallinesthat
H
arenearlyasnarrowasthespectrometerFWHM(fullwidth
k
athalfmaximum),themeasuredFWHMwillbewiderthan
[001] the true FWHM. Figure S6(b) shows the FWHM of a line
obtained by the convolution of a Lorentzian or Gaussian
spectral lineshape with the spectrometer response function.
[010] For example, a measured linewidth of 94 µeV corresponds
[100]
to a true linewidth of 72 or 82 µeV, depending on whether
thetruelineshapeisassumedtobeLorentzianorGaussian.
FIG. S4. Diagram showing the types of defects visible in the Hence,toevaluatetheintrinsiclinewidthoftheSFemission
sample. TheSFpyramidconsistsoffourstackingfaultsarranged we fit the spectrum in Fig. S5 with a weighted Lorentzian
in a pyramid shape. The SF pair defect is a set of two SFs with and use the deconvolution procedure [Fig. S6(b)] to obtain
a small nm separation. an intrinsic PL linewidth of only (77 19) µeV. Here, the
∼ ±
uncertainty combines the original fit uncertainty and the
uncertainty of the deconvolution procedure.
S2. PHOTOLUMINESCENCE SPECTROSCOPY
SYSTEM S4. MAGNETO-STARK HAMILTONIAN
The sample is excited with a continuous wave Sirah Ma- Microscopically, the magneto-Stark effect can be derived
tisse Ti:Sapphire laser. The laser is focused to a spot size fromtheHamiltonianforanelectronandholeinamagnetic
of 1 µm on the sample using an aspheric lens (numeri- field B =B yˆ:
∼ y
cal aperture 0.77) mounted inside a liquid helium immer-
sion cryostat (Janis). Coarse positioning of the sample was (cid:126)2 (cid:16) e (cid:17)2 (cid:126)2 (cid:16) e (cid:17)2
H = k + A + k A , (S7)
performed with slip-stick positioners (Attocube). For con- 2m e (cid:126)c 2m h− (cid:126)c
e h
focal scanning, spatial selectivity is achieved with a pinhole
in the intermediate image plane and a scanning mirror to where k (k ) is the wavevector of the electron (hole), m
e h e
raster the excitation and collection spot over the sample. (m ) is the effective mass of the electron (hole) and the
h
The photoluminescence (PL) is imaged on a spectrometer vector potential is A=B zxˆ. The Coulomb interaction
y
6
(a)1.5 (b)180
Voigt Lorentzian FWHM = 23.9 µeV
160 Lorentzian
Voigt Gaussian FWHM = 1.93e-06 µeV Gaussian
s) Voigt Total FWHM = 23.9 µeV eV)140
Unit 1 µM (120
Arb. FWH100 Exp. FWHM = 94 µeV
ounts ( 0.5 asured 6800 72 82
C e
M 40 µ µ
e e
Laser V V
Voigt Fit 20
0 0
1.5008 1.501 1.5012 1.5014 1.5016 0 25 50 75 100 125 150
Energy (eV) True FWHM (µeV)
FIG. S6. (a) Spectrum of the narrow band Ti:Sapphire laser used to determine the spectrometer instrument resolution. The best
fit is a Voigt function with a 32.7 µeV FWHM. The Voigt lineshape is the convolution of a Lorentzian and a Gaussian with best fit
widthsprovidedasaninsetinthefigure. (b)ConvolutionofthespectrometerinstrumentresponseinawithaLorentzianorGaussian
lineshape. By interpolating backwards, the deconvoluted FWHM of a spectral line can be found.
betweentheelectronandtheholeaswellastheSFpotential
[001] z [111]
are omitted in Eq. (S7) for brevity. We use the standard k
definitionofcenterofmass(COM)andrelativecoordinates
m
e
k = K+k
e
M
(S8)
m
h
k = K k
h
M −
y [¯110]
where M = m + m . With these substitutions, Eq. S7 k
e h
becomes
COMKinetic Rel. Kinetic COMMagneto-Stark [010]
(cid:122) (cid:125)(cid:124) (cid:123) (cid:122)(cid:125)(cid:124)(cid:123) (cid:122) (cid:125)(cid:124) (cid:123)
(cid:126)2K2 (cid:126)2k2 e(cid:126)
[100]
H = + (z z )K B
h e x y
2M 2µ −Mc −
(S9)
Orb. Zeeman Diamagnetic
(cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123)
+ ec(cid:126)(cid:18)mzh + mze (cid:19)kxBy+e22cB2y2 (cid:18)mze2 + mzh2 (cid:19) xk[11¯2]
h e e h
where µ=(m−e1+m−h1)−1 [41]. The firstterm is the COM FIG.S7. GaAscrystallinelattice. Stackingfaultplaneisa(111)
kinetic energy of the exciton. The second term is the ki- plane, shown as a triangle. The mirror reflection plane for a SF
netic energy associated with the electron-hole relative mo- sends y y. Note that x x is not a mirror reflection
→ − → −
plane.
tion. The third term describes the magneto-Stark effect for
the exciton COM motion. The fourth term describes the
orbital Zeeman effect [41]. The fifth term is the harmonic
potential created by the magnetic field which produces the light-hole(LH)bandsaredegenerateandtransformaccord-
diamagnetic shift and would produce Landau quantization ingtothefour-dimensionalΓ8 irreduciblespinorrepresenta-
for higher magnetic fields. The COM magneto-Stark term tion of the Td point symmetry group [49], see Refs. [50, 51]
is the same as Eq. (3) in the main text, derived from rela- for notations.
tivistic arguments. A SF oriented in one of the 111 planes possesses the
{ }
lower C point symmetry, as illustrated in Fig. S7. The
3v
C symmetry group contains six symmetry operations: E
3v
S5. HEAVY HOLE – LIGHT HOLE SPLITTING (identity), C3 (rotationby2π/3aboutzˆ), C3−1 (rotationby
−2π/3 about zˆ), σv (reflection y →−y), σv(cid:48) (σv(cid:48) =σvC3−1),
The possible carrier spin states entering into the SF ex- and σv(cid:48)(cid:48) (σv(cid:48)(cid:48) = σvC3). These operations are depicted in
citon can be predicted on general symmetry considerations. Fig. S8. We note that the SF symmetry group is different
Due to spin orbit coupling, the valence band in bulk GaAs from the C symmetry of wurtzite [52].
6v
splitsintotheheavy-hole/light-hole(j = 3)bandsandsplit- When a SF is introduced into a zinc-blende crystal, the
2
off (j = 1) band. At the Γ point, the heavy-hole (HH) and degeneracy of the valence and conduction band edges could
2
7
belifted. Thecompatibilityanalysisshowsthattheconduc- here that HH-LH mixing can affect the in-plane HH effec-
tionbandedgetransformsaccordingtothetwo-dimensional tive mass [58]. Using second-order perturbation theory, we
irreducible representation Γ of the C point group, i.e. obtain:
4 3v
the conduction band does not split. On the other hand,
the degeneracy of the valence band is lifted: the Γ8 irre- mo =γ +γ +(4γ2+2γ2)(cid:126)2 (cid:88)|(cid:104)LH,n|kˆz|HH(cid:105)|2,
ducible representation of T decomposes into Γ Γ Γ m 1 3 2 3 m E E
of C . The two degeneradte states Γ transf4or⊕m a5s⊕th6e hh,|| o n HH − LH,n
3v 4 (S11)
spinors 3/2,1/2 , 3/2, 1/2 (or 1/2,1/2 , 1/2, 1/2 ),
where the summation goes over all the light-hole states n
| (cid:105) | − (cid:105) | (cid:105) | − (cid:105)
where j,m signifies the basic function for angular mo-
j (both bound and continuum), HH denotes the ground
| (cid:105)
mentum j and angular momentum z-component j . The | (cid:105)
m subbandHHenvelopefunctionalongthez axisand LH,n
Γ5 and Γ6 hole states transform as linear combinations the denote the LH envelopes and kˆ = ∂/∂z and γ ,γ|,γ are(cid:105)
spinors 3/2,3/2 and 3/2, 3/2 , see Table I. At B =0 z − 1 2 3
| (cid:105) | − (cid:105) the Luttinger parameters [49]. The sum in Eq. S11 is sen-
and K =0, the two states Γ and Γ are guaranteed to be
5 6 sitive to the details of the HH and LH envelope functions
degenerate by time reversal symmetry [53–55]. It follows
as well as to the energy positions of the size-quantized lev-
that the levels of excitons bound to a SF are split into sub-
els [58, 59].
levels of symmetry Γc Γv and Γc (Γv Γv) where the
4× 4 4× 5⊕ 6 The estimate of the exciton dipole moment involves the
superscripts c,v refer to the conduction and valence bands.
exciton’s effective mass, M =m +m , see Eq. (5), where
For convenience, the former and latter excitonic states are ∗h ∗e
m and m are the in-plane effective masses of the hole
called the light- and heavy-hole excitons, or the LH and ∗h ∗e
and electron. The electron effective mass is isotropic and
HH excitons, irrespective of the relation between the exci-
therefore its in-plane value is m =0.067m . Due to the
ton binding energy and the splitting ∆ between the HH ∗e o
HL anisotropy of the hole effective mass and the unknown ex-
and LH sublevels.
tent of HH-LH coupling, determining the correct value of
The experimental data imply that the in-plane hole g-
m requiresacomplexcalculation,whichwedonotperform
factor in the ground exciton state is negligible. This is ex- ∗h
here. However,wecanestimatetheHHeffectivemassusing
actlytruefortheHHexcitonΓc (Γv Γv)atzeromagnetic
4× 5⊕ 6 somesimplearguments. IfHH-LHcouplingisneglected,the
fieldB andzerowavevectorK . Thisimpliesthatthesplit-
y in-planeheavyholemassisthatofaheavy-holeinthe(111)
ting∆ ishighenoughtopre(cid:107)ventthemixingbetweenthe
HL plane, m =0.10m [59]. Another estimate of the in-plane
LH and HH sublevels induced by the finite value of K re- ∗h o
x heavy hole mass can be made from the spatially averaged
alized in the experiment and by the applied magnetic field,
heavy-hole mass of 0.45m . We will use the lower value to
B 7 T. e
| |≤ estimate the dipole moment, and the higher value to esti-
In order to confirm that the SF splits HH and LH
mate the error in our measurement of the dipole moment.
states, we we can estimate whether the HH-LH splitting is
This conservative procedure yields the minimum possible
muchgreaterthaninteractionswhichmixHHandLH.The
value of the dipole moment.
magnetic-field-induced HH-LH mixing can be estimated by
using the hole Zeeman Hamiltonian
= 2µ κ(J B), (S10) S6. HAMILTONIAN BY SYMMETRY
Z B
H − ·
whereµ istheBohrmagneton,κisthemagneticLuttinger The form of the Hamiltonian describing the SF-bound
B
parameterontheorderofunity,J =(J ,J ,J )arethema- exciton state can be derived based on the symmetry of the
x y z
trices of the angular momentum 3/2 operator, and we ne- SF system. Specifically, we would like to find terms of the
glect a weak cubic anisotropy [56]. According to Eq. (S10), Hamiltonian that are odd with magnetic field and can thus
the magnitude of the Zeeman HH-LH coupling matrix el- explain the magnetic non-reciprocity data.
ement at the maximum field of 7 T is < 1 meV. As for It is first necessary to find how the symmetry operations
the HH-LH mixing caused by the nonzero value of K , we affectavector, suchasapositionvectorr =(x,y,z), anda
x
notethatthemaximumexcitonwavevectormeasuredinour pseudo-vector, such as magnetic field B =(B ,B ,B ). In
x y z
system is 5 µm−1. In this case the terms of the Luttinger- freespace,r andB transformaccordingtotheD1− andD1+
KohnHamiltonianthatcoupletheHHandLHexcitonshave irreduciblerepresentations,where denotesparitywithre-
±
magnitudes less than 2 µeV [49]. Thus, we conclude that spect to space inversion. Making use of the compatibility
no significant mixing occurs if the splitting ∆ is greater tables for C point symmetry, one can readily check that
HL 3v
than a few meV. The lack of significant HH-LH mixing in D1− = Γ1⊕Γ3, while D1+ = Γ2⊕Γ3. Taking into account
our experiments is in agreement with the HH-LH splitting thatunderrotationsthecomponentsofpolarandaxialvec-
of16meVestimatedforinterfaceexcitonsinpolytypiczinc- torstransformidenticallyandmakinguseofFig.S8wefind
blende/wurtzite GaAs nanowires [39]. that z and B transform according to the irreducible repre-
z
In order to determine the dipole moment of the SF ex- sentationsΓ andΓ ,respectively,whilethepairs(x,y)and
1 2
citon using Eq. 5, it is necessary to estimate the in-plane ( B ,B ) form equivalent bases of the two-dimensional Γ
y x 3
−
exciton effective mass, the sum of the electron and hole in- irreducible representation of C , see Table I.
3v
plane effective masses. While the electron mass is isotropic To apply the method of invariants, we need to estab-
in GaAs, the effective in-plane hole mass depends on the lish transformation rules for the basic 2 2 matrices act-
×
detailed nature of the stacking fault potential. We note ing in the spin subspaces of electrons and heavy-holes. We
8
TABLE I. Character table for the double group of C3v. Double group characters given in Ref. [57].
C3v E E C32 C32 3σv 3σv Basis Functions
C3 C3
ΓΓΓΓ2341 1122 −1122 −1111 −−1111 −0011 −0011 Szisnpcc-iaponllmaa12rnp;:eoz(cid:0)nvc(cid:12)(cid:12)eeo12ncm,tt−oporf12o;n(cid:11)p(e,sKne(cid:12)(cid:12)utx12d,o,Kof12v(cid:11)vye(cid:1))ec;ctto(or−r;;BBzyz;,;BxBx2)x3+;−(y−23;σByeBx,Bx2σyx2+e;);Bσ(yz2e−;;σBσylxhhy3h,−;σxlσ3hzBh);hy;(BBσx2yzl2;h−I;Bσx2yh,h2BxBy)
Γ5 1 −1 −1 1 i −i Heavy-hole spin ‘up’; √12(cid:0)(cid:12)(cid:12)32,−32(cid:11)+i(cid:12)(cid:12)32,32(cid:11)(cid:1)
Γ6 1 −1 −1 1 −i i Heavy-hole spin ‘down’; √12(cid:0)(cid:12)(cid:12)32,−32(cid:11)−i(cid:12)(cid:12)32,32(cid:11)(cid:1)
y
x
E C3 C3−1 σv σv0 σv00
Identity 2π rotation 2π rotation Reflection Reflection Reflection
3 − 3
FIG. S8. Symmetry operations of C3v, the point group for a (111) SF in GaAs.
introduce basic electron matrices Ie (the 2 2 unit ma- potentially different:
×
trix) and σe =(σe,σe,σe) (Pauli matrices) acting in the
x y z 1 1
basis of s= 1/2 electron spinors. The decomposition H = ge µ B σe+ geµ (B σe +B σe) (S13)
Γ4⊗Γ∗4 =| Γ1±+Γ2(cid:105)+Γ3 indicatesthewaysinwhichtheba- Z,e 2 zz B z z 2 || B x x y y
sic electron spin matrices transform. By calculating the
For the heavy holes, the Hamiltonian takes the form [56]:
effect of the C symmetry operators on the matrices, one
3v
findsthatIe isinvariant,σze transformsaccordingtoΓ2 and H = 1µ B (cid:0)ghhσhh+ghhσhh(cid:1). (S14)
(σxe,σye)formabasisofthetwo-dimensionalirreduciblerep- Z,hh 2 B z zz z zx x
resentation Γ with σe and σe transforming equivalently to
3 x y
Eqquation S14 implies that the heavy hole has a zero in-
B and B respectively (Tab. I). The basic matrices for the
x y
plane g-factor, and that an out-of-plane component B cre-
light-hole doublet transform in exactly the same way. z
atesatiltedeffectivefieldwithaspinprecessionvectorlying
By contrast, the heavy-hole spin doublet transforms ac-
inthexz plane. Wenotethattheatomic-scalesymmetryof
cording to the reducible representation Γ Γ . The direct
5⊕ 6 the SF makes the two in-plane directions x and y inequiva-
product
lent.
The magneto-Stark Hamiltonian can be derived by sym-
(Γ Γ ) (Γ Γ ) =2Γ 2Γ . (S12)
5⊕ 6 ⊗ 5⊕ 6 ∗ 1⊕ 2 metry using a similar procedure. The combination of the
two Γ irreducible representations (K ,K ) and ( B ,B )
3 x y y x
indicates that among the four basic matrices, Ihh, contains an invariant representation, leading to th−e Hamil-
σhh =(σxhh,σyhh,σzhh) acting in the space |±3/2(cid:105), two are tonian
invariantandtwotransformasB [56]. Takingintoaccount
z
that at the mirror reflection σ (¯110) the matrix σhh does H =β (K B K B )
v|| y magneto-Stark (cid:48) x y− y x
notchangesign,wefindthatIhh andσhh transformaccord-
y
Furthermore, the Hamiltonian describing the diamagnetic
ingtoΓ (notethatσhh changesitssignundertimereversal
1 y shiftisderivedfromtheinvariantsB2 andB2+B2,namely,
whereas Ihh does not), while σhh, σhh transform according z x y
z x
to Γ2, see Tab. I and Ref. [56]. H =β B2+β (B2+B2).
dia 1 z 2 x y
Using the transformation properties of the relevant basis
functions (Tab. I), we can build the effective Hamiltonian. For Eq. (1) of the main text, we take into account only an
Any valid term of the Hamiltonian must transform as the in-plane field effect and set β β.
2
≡
identity representation Γ and be even under time rever- We note for completeness that, besides B-linear, B-
1
sal. Inordertoknowwhichcombinationsofbasisfunctions quadratic and KB-terms, the effective Hamiltonian also in-
transformasΓ ,weusetheproductrulesforirreduciblerep- cludesK linearterms,σeK σeK ,whicharisefromspin-
1 x y− y x
resentations. Fromthisinformation,wecanbuildthelinear orbitcoupling. Ourestimatesshowthatthesetermsarenot
in B Hamiltonian. For the electron, this analysis produces significant for the relevant wavevectors and do not lead to
the result that the in-plane and out-of-plane g-factors are the non-reciprocal emission spectra to first-order in B.
9
S7. SUPPLEMENTAL ANGLE RESOLVED DATA where d = z z is the hole-electron separation in
he h e
(cid:104) − (cid:105)
the exciton, z and z are the hole and electron coordi-
h e
Fortheangleresolvedexperimentthatteststhemagneto- nates and ... denotes the quantum mechanical average
(cid:104) (cid:105)
Stark effect, PL spectra were acquired on multiple different (note CGS units). The numerical value uses the estimated
SF pyramids positioned at various angles. The spectra of in-plane exciton mass described in Sec. S5. The quantity
theupanddown SFat 6.5TareshowninFig.S9. Wenote ∆Eup/down(B)=Eup/down(B) Eup/down( B) is the odd
thatthePLspectrasho±wastrongandaweakdoublet. The part of Eup/down doubled, wh−ich is insens−itive to overall
origin of the weak doublet is unknown, but we tentatively offsets and the diamagnetic shift. When θ =θ(cid:48) =0, PL
attributeittoscatterofPLfromotherexcitonpopulations. from the up and down SFs arises from excitons with the
We note that there are two types of SF pyramids: one same absolute value of momentum and hence ∆Eup and
where the left/right SFs show higher energy PL than the ∆Edown have the same magnitude [Fig. 3(d)]. When the
up/down SFs (type a), and a second type where the emis- angleisincreased, ∆Eup and∆Edown havedifferentmagni-
sionenergiesareswapped(typeb). Forexample, usingthis tudes [Fig. 3(e)].
naming scheme the SF pyramid shown in Fig. 1(d) is of In the linear in B regime, the magnetic field is not ex-
y
type a. Here, the left/right directions refer to the direction pected to significantly perturb the exciton dipole moment,
ofthemajorityoftheintrinsic/extrinsicpairdefects,visible and so to first order ∆E should be proportional to B . Ex-
y
as single triangles in Fig. 1(a). We also define a standard perimentally, we find the energy shift is linear with B for
y
and rotated orientation of the bulk crystal depending on B <1 T, and we extract the proportionality constant us-
y
whether the sample is mounted as it is in Fig. 1(a) (stan- ing a fit to ∆E =aB . The estimate of the dimensionless
y
dard) or rotated by 90 about [001] (rotated). dipole moment α depends somewhat on the fitting method
◦
ThePLemissionenergyasafunctionofmagneticfieldfor used to find the slope ∆E vs. B at B =0 from the ex-
y y
the 6 SFs is shown in Fig. S10. These plots show that d , perimental data. Using different methods, we obtain values
he
i.e., the electron-hole separation, changes sign for different of α ranging from 3.2 to 4.3, and we conservatively use the
SF pyramids. The sign of the splitting ∆E reflects lowest value of α=3.2 0.3 to estimate the exciton dipole
up/down ±
the direction of the exciton dipole moment along the SF moment. Given the uncertainty in the measurement of α
normal. From Fig. S10, we find that the atomic structure and the estimation of the in-plane exciton effective mass
oftheSFpyramidhasareflectionsymmetryinthe 110 set (Sec. S5), the best guess for the dipole moment is 10 nm,
{ }
ofplanes,i.e. wheretheup anddown SFsareinterchanged. with the true value estimated to fall between 9 and 30 nm.
We note that the combination
S8. QUANTITATIVE INTERPRETATION OF
∆E ∆E
ANGLE RESOLVED DATA r(θ)= 1(cid:0)|∆Eup|−+|∆Edown| (cid:1) (S17)
2 | up| | down|
CombiningEq.(2)and(3)ofthemaintextandaveraging
over two spin states of the exciton’s electron, the observed dependsonly on givengeometryand theindex of refraction
magneto-Stark shift is (tofirstorderinB),andisindependentoftheexcitondipole
moment. Using Eq. S15 and expanding about small θ, we
E =+αµ B sin(θ +θ ), can produce a small-θ approximation for r(θ),
up B y SF (cid:48)
(S15)
E = αµ B sin(θ θ ),
down B y SF (cid:48)
− −
2cotθ
SF
r(θ) θ (S18)
where the light emission angles are related by Snell’s law: ≈ n
(cid:0) (cid:1)
nsinθ =sinθ,θ =cos 1 1/√3 =54.7 istheanglethe
(cid:48) SF − ◦
SF normal makes with the [001] axis and µ =e(cid:126)/2m c is
B e correct to within 10% for angles up to 45 and n=3.5. We
◦
the Bohr magneton (CGS units). Here α is a dimensionless
obtain good agreement between r(θ) calculated experimen-
measure of the dipole moment:
tally from the B =0 slope of ∆E with Eq. S17. (Fig. 3h).
en(cid:126)ω 1d Weemphasizethatthismodelagreeswiththedatawithout
he
α= d = − , (S16) any fit parameters.
−Mc2µ he 3.2nm
B
[1] S. Guha, J. M. DePuydt, M. A. Haase, J. Qiu, and tronics 17, 829–846 (2011).
H.Cheng,“DegradationofII-VIbasedblue-greenlightemit- [4] N. Akopian, G. Patriarche, L. Liu, J. C. Harmand, and
ters,” Applied Physics Letters 63, 3107–3109 (1993). V. Zwiller, “Crystal phase quantum dots,” Nano Lett. 10,
[2] A.Colli,E.Pelucchi, andA.Franciosi,“Controllingthena- 1198–1201 (2010).
tive stacking fault density in II-VI/III-V heterostructures,” [5] S.Assali,I.Zardo,S.Plissard,D.Kriegner,M.A.Verheijen,
Applied Physics Letters 83, 81–83 (2003). G.Bauer,A.Meijerink,A.Belabbes,F.Bechstedt,J.E.M.
[3] P. Caroff, J. Bolinsson, and J. Johansson, “Crystal phases Haverkort, and E. P. A. M. Bakkers, “Direct band gap
in III–V nanowires: from random toward engineered poly- wurtzitegalliumphosphidenanowires,”NanoLetters,Nano
typism,”IEEEJournalofSelectedTopicsinQuantumElec- Lett. 13, 1559–1563 (2013).
10
(a) θ =0◦,typea,Standard (b) θ =19◦,typea,Rotated (c) θ =19◦,typeb,Rotated
∆E ∆E ∆E
up up up
)p )p )p
sU sU sU
nit nit nit
U U U
b. -6.5T6.5T b. 6.5T -6.5Tb. 6.5T -6.5T
r r r
A ∆E A ∆E A ∆E
( down ( down ( down
L L L
P P P
n n n
w w w
o o o
D D D
6.5T-6.5T -6.5T6.5T -6.5T6.5T
1.496 1.497 1.498 1.499 1.500 1.501 1.496 1.497 1.498 1.499 1.500 1.501 1.496 1.497 1.498 1.499 1.500 1.501
(d) θ =38◦,typea,Rotated (e) θ =38◦,typeb,Rotated (f) θ =42.8◦,typea,Standard
E(eV) E(eV) E(eV)
∆E ∆E ∆E
up up up
)p )p )p
sU sU sU
nit nit nit
U U U
b. 6.5T -6.5T b.6.5T -6.5T b.-6.5T6.5T
r r r
A ∆E A ∆E A ∆E
( down ( down ( down
L L L
P P P
n n n
w w w
o o o
D D D
-6.5T6.5T -6.5T6.5T 6.5T-6.5T
1.496 1.497 1.498 1.499 1.500 1.501 1.496 1.497 1.498 1.499 1.500 1.501 1.496 1.497 1.498 1.499 1.500 1.501
E(eV) E(eV) E(eV)
FIG.S9. Spectraat-6.5Tand6.5TfortheupanddownSFforsixdifferentSFpyramiddefects. Thedifferencebetween∆Eup and
∆Edown is visible when θ=0. 1 µW, 1.5 K, excite at 1.65 eV, excite and collect H polarization.
(cid:54)
[6] L. V. Butov, A. C. Gossard, and D. S. Chemla, “Macro- currentsinacoherentexcitongas,”PhysicalReviewLetters
scopicallyorderedstateinanexcitonsystem,”Nature418, 110 (2013), 10.1103/physrevlett.110.246403.
751–754 (2002). [12] M. M. Fogler, L. V. Butov, and K. S. Novoselov, “High-
[7] A. A. High, A. K. Thomas, G. Grosso, M. Remeika, A. T. temperature superfluidity with indirect excitons in van der
Hammack, A. D. Meyertholen, M. M. Fogler, L. V. Butov, Waalsheterostructures,”NatureCommunications5(2014),
M.Hanson, andA.C.Gossard,“Trappingindirectexcitons 10.1038/ncomms5555.
in a GaAs quantum-well structure with a diamond-shaped [13] A.V.GorbunovandV.B.Timofeev,“Large-scalecoherence
electrostatic trap,” Phys. Rev. Lett. 103, 087403 (2009). ofthebosecondensateofspatiallyindirectexcitons,”JETP
[8] D.Snoke,S.Denev,Y.Liu,L.Pfeiffer, andK.West,“Long- Letters 84, 329 (2006).
rangetransportinexcitonicdarkstatesincoupledquantum [14] D. W. Snoke, “Coherence and optical emission from bi-
wells,” Nature 418, 754–757 (2002). layer exciton condensates,” Advances in Condensed Matter
[9] L. V. Butov, C. W. Lai, A. L. Ivanov, A. C. Gossard, and Physics 2011, 1–7 (2011).
D. S. Chemla, “Towards Bose–Einstein condensation of ex- [15] B.Nelsen,R.Balili,D.W.Snoke,L.Pfeiffer, andK.West,
citons in potential traps,” Nature 417, 47 (2002). “Lasing and polariton condensation: Two distinct transi-
[10] Yehiel Shilo, Kobi Cohen, Boris Laikhtman, Ken West, tions in GaAs microcavities with stress traps,” Journal of
Loren Pfeiffer, and Ronen Rapaport, “Particle cor- Applied Physics 105, 122414 (2009).
relations and evidence for dark state condensation in [16] R.Balili,B.Nelsen,D.W.Snoke,L.Pfeiffer, andK.West,
a cold dipolar exciton fluid,” Nat Commun 4 (2013), “Role of the stress trap in the polariton quasiequilibrium
10.1038/ncomms3335. condensation in GaAs microcavities,” Phys. Rev. B 79,
[11] A. A. High, A. T. Hammack, J. R. Leonard, Sen Yang, 075319 (2009).
L.V.Butov,T.Ostatnicky´,M.Vladimirova,A.V.Kavokin, [17] A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler,
T. C. H. Liew, K. L. Campman, and A. C. Gossard, “Spin L. V. Butov, A. V. Kavokin, K. L. Campman, and A. C.