Interlayer excitons with tunable dispersion relation

0.9 MB·

by  Brian Skinner

Interlayer excitons with tunable dispersion relation
Brian Skinner
Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(Dated: February 2, 2016)
Interlayerexcitons,comprisinganelectroninonematerialboundbyCoulombattractiontoahole
inanadjacentmaterial,arecompositebosonsthatcanassumeavarietyofmany-bodyphases. The
phasediagramofthebosonicsystemislargelydeterminedbythedispersionrelationofthebosons,
which itself arises as a combination of the dispersion relations of the electron and hole separately.
HereIshowthatinsituationswhereeithertheelectronortheholehasanon-monotonic,“Mexican
hat-shaped” dispersion relation, the exciton dispersion relation can have a range of qualitatively
6
differentforms,eachcorrespondingtoadifferentmany-bodyphaseatlowtemperature. Thisdiver-
1
sity suggests a novel platform for continuously tuning between different quantum phases using an
0
external field.
2

n
a When an electron binds to a hole in a solid state sys-
exciton

J tem,theresultingexcitonhaspropertiesthatarequalita- εe ε
1
tivelydifferentfromeithertheelectronorholeseparately.
3
Forexample, theelectronandholehavefermionicstatis- k k electron

] tics, while the exciton is a boson. Electrons and holes 0 P
ε
l interactviaalong-rangedCoulombinteraction,whileex-
l
a k
citons have only a short-ranged dipolar interaction. But 0
h
in terms of its dispersion relation, an exciton is usually
-
s qualitatively similar to a free electron or hole. In partic- ε P
e ε
m ular, one can usually describe the center of mass coordi- h hole

nate of the exciton as an effectively free particle having
.
at massequaltothesumoftheelectronandholemasses,so k P
m thatthetotalenergyεoftheexcitongrowsasP2,where
P is the center of mass momentum. FIG. 1. (Color online) Schematic illustration of a composite
-
d But what happens when the electron and hole have bosonwithtunabledispersionrelation. Amaterialcontaining
n dispersion relations that are qualitatively different from electrons(bluelayer)isplacedadjacenttoamaterialcontain-
o each other? Which of its constituent particles does the ing holes (yellow layer), with the electron dispersion relation
c following Eq. (1) and the hole dispersion relation following
excitontakeafter, intermsofitsdispersionrelation: the
[
Eq. (2). In this configuration the electron and hole can bind
electron or the hole?
together to form an interlayer exciton, which has a disper-
1
v In particular, consider the case of a two-dimensional sionrelationthatcantakeanumberofqualitativelydifferent
5 (2D) exciton for which the electron has a “Mexican hat- forms, depending on the value of k0.
2 shaped” dispersion,
3
0 (|k|−k )2
.0 εe(k)= 2me0 , (1) qexutaenrtnuaml fipelhda.ses within a single device by changing an
2
0 while the hole has the usual parabolic dispersion The idea of using interlayer excitons to realize a 2D
6 Bose system goes back more than forty years,[2] but its
1 |k|2 experimentalrealizationhasbeenenabledonlyrelatively
: ε (k)= . (2)
v h 2m recentlybythedevelopmentofsufficientlycleanandsuf-
h
i ficiently thin bilayer devices.[3–5] The ongoing develop-
X
Here, m and m are the effective masses of the electron mentofnew2Dmaterialscontinuestoprovidenovelplat-
e h
r
and hole, respectively, k is the momentum, and k is a forms for realizing bilayer exciton physics.[6–8] Of par-
a 0
characteristic momentum that defines the width of the ticular significance is the identification of a number of
“brim”oftheMexicanhat(seeFig.1). Onecannowask different materials that have, or can be made to have,
the question: what is the form adopted by the exciton a Mexican hat shape at low energy ε. Such a disper-
dispersion relation ε(P)? sionrelationisusuallyassociatedwithRashbaspin-orbit
This seemingly quaint problem actually has a range of coupling,[9] which can be significant in semiconductors
experimental implications. As explained below, its solu- like GaAs or InAs,[10–12] at the surface of topological
tion suggests the ability to engineer interlayer excitons insulators like Bi Se ,[13] or at oxide interfaces such as
2 3
with widely tunable dispersion relation.[1] This tunabil- LaAlO /SrTiO .[14] A Mexican hat-shaped dispersion
3 3
ity offers the potential to realize a range of many-body also arises in materials such as bilayer graphene[15, 16]
2
that have an avoided crossing between two intersecting displacement of the electron from the center of the po-
bands with opposite-sign velocity. Importantly, for each tential well, and the corresponding size of the electron
of the preceding examples in this paragraph, the brim of wavefunction is that of the ground state of the 2D har-
the Mexican hat, k , can be widely and continuously ad- monic oscillator — namely, r ∼ (a r3)1/4. (As in the
0 e e h
justed by applying a transverse electric field. As shown previous case, this expression for r is correct even for
e
below,thisadjustabilityofferstheabilitytocontinuously large k .) The Mexican hat shape of the dispersion re-
0
tune the dispersion relation of interlayer excitons. The lation remains only when k (cid:29)1/r , which is equivalent
0 e
remainder of this paper is dedicated to deriving the ex- to the condition
citon dispersion relation, and to discussing the different
(cid:18)m (cid:19)3/4
many-body phases that can be obtained by tuning k0 k a (cid:29) h , (m /m (cid:28)1). (4)
and the exciton density n. For simplicity, the amplitude 0 e me h e
of inter-layer tunneling is considered everywhere to be
Thesequalitativepredictionscanbeverifiedinaquan-
negligibly small.
titative way as follows. The exciton dispersion ε(P) is
Before giving a detailed calculation of the exciton dis-
foundbysolvingtheSchrodingerequationHψ =εψwith
persion, it is worth outlining how the dispersion rela-
the Hamiltonian
tion can be understood qualitatively using the following
simple scaling arguments, which for simplicity assume H =ε +ε −V(r ), (5)
(cid:98)e (cid:98)h eh
small interlayer separation d. The key idea is to com-
pare k with the typical internal momentum k of the where ε and ε are the kinetic energy operators for the
0 e (cid:98)e (cid:98)h
electron within a stationary exciton. When k is much electron and hole, respectively, and
0
smaller than k , the (small) Mexican hat feature in the
e e2
electron dispersion is irrelevant, since the electron wave V(r )=− (6)
eh (cid:112)
function uses primarily much larger momentum compo- κ |reh|2+d2
nents. When k (cid:29) k , on the other hand, the Mexican
0 e is their mutual Coulomb energy. Here r = r −r is
hat structure in ε (k ) is reflected in the exciton disper- eh e h
e e the displacement vector between the electron and hole,
sion relation, since the energy of the electron is strongly
withr beingtheelectroncoordinateandr theholeco-
reduced when the exciton acquires enough momentum e h
ordinate. The wavefunction with fixed total momentum
that the electron momentum approaches k .
0 P is written as
Consider first the case where the hole mass is heavy,
mh (cid:29) me. In this case the characteristic radius rh ψ(r ,r )=exp[i(k0·r +k0 ·r )]ϕ(r ), (7)
e h e e h h eh
of the hole wavefunction is much smaller than that of
the electron, r . Consequently, r is similar to the ra- wherek0 andk0 arewavevectorssuchthatk0+k0 =P
e e e h e h
dius of a 2D hydrogen atom with a stationary nucleus, and (dε /dk)| = (dε /dk)| . These two condi-
e k=k0 h k=k0
r ≈ κ/m e2 ≡ a (the electron effective Bohr radius, tions ensure that tehe electron and hhole together have to-
e e e
in Gaussian units with (cid:126) ≡ 1 — here κ is the dielec- tal momentum P, and that they both have the same
tric constant). This expression for r is roughly correct group velocity. (Equivalently, k0 and k0 are the val-
e e h
even when k is large, since large k leads only to a log- ues of momentum that minimize the total kinetic en-
0 0
arithmic renormalization of r .[17] The typical electron ergy ε (ke)+ε (kh) under the constraint k0+k0 =P.)
e e 0 h 0 e h
momentum is therefore k ∼1/a . If k is small enough Onecansaythatthedistributionofelectronmomentais
e e 0
thatk (cid:28)1/a ,thenthenon-monotonicpartoftheelec- centered around k0, while the hole has momenta in the
0 e e
tron dispersion at small k is smeared out by the internal neighborhood of k0. For concreteness, below I take P to
h
motion of the electron around the hole, and the exciton be in the x direction, so that k0 and k0 are also in the x
e h
dispersionisparabolic. Thus,itisonlyatlargerk ,such direction.
0
that The function ϕ(r ) represents the wave function for
eh
the relative motion of the electron and hole around each
k a (cid:29)1, (m /m (cid:29)1), (3) other. Such motion is, in general, anisotropic, corre-
0 e h e
sponding to a hydrogen-like state that is elongated in
that the exciton dispersion has a Mexican-hat shape. one direction. One can see how this anisotropy arises by
Ontheotherhand,whentheholemassislightenough consideringthespecialcaseinwhichk islargeandm is
0 e
that m (cid:28) m , the hole wavefunction acquires a large small. Inthiscasetheelectronkineticenergyhasasharp
h e
size r ≈κ/m e2, while the electron wavefunction is rel- minimum at |k | = k , and k0 is very close to k xˆ. The
h h e 0 e 0
ativelycompact. Thespatialextentoftheelectronwave- dispersionrelationfortheelectronintheneighborhoodof
functioninthiscasecanbefoundbyconsideringthatthe k0 isthereforeparabolicinthexdirectionwithmassm ,
e e
electron sits in the bottom of a parabolic potential well andnearlyflatinthey direction. Consequently,theelec-
createdbytheCoulombpotentialofthehole. Theshape tron wavefunction acquires a shape that is tightly con-
of this potential well is u(ρ) ∼ e2ρ2/κr3, where ρ is the fined in the y direction (extended in momentum space)
h
3
and more extended in the x direction (compact in mo-
4
mentum space).
)
Inordertoaccountforthisanisotropy,onecanusethe ε(0 3
variational wavefunction −
)
(cid:114) P k
ϕ(x,y)= 2βπ2λexp[−β(cid:112)x2+(λy)2] (8) y,ε( 2 0ea = 0.
g 1
er 1
(following, for example, Refs. 2 and 18). Here, β and λ n k
are variational parameters, with β corresponding to the ne 0ae = 1
o 0
iλnv>ers1e bsieziengofththeedwimaveenfsuionnctleiossnainnistohteroxpydiroefcttihoen,waanvde excit k0ae = 2
-1
function. Inthelimitingcasek →0andd→0,Eq.(S2)
0 -4 -2 0 2 4
reproduces the ground state wavefunction of the hydro- center of mass momentum, Pa /¯h
e
gen atom at λ=1. More generally, the energy of the ex-
citon is approximated by the minimum value of (cid:104)ψ|H|ψ(cid:105) FIG. 2. (Color online) The dispersion relation ε(P) of an
over all values of the variational parameters. That is, interlayerexcitonwithm =m =1andd/a →0. Different
h e e
curves are labeled by the corresponding value of k . Units of
0
ε(P)(cid:39)min(εk+εc), (9) energy on the vertical axis are e2/κae.
β,λ
where ε is the expectation value of the kinetic energy,
k
given by
32β4 (cid:90) ε (k0+q/2)+ε (k0 −q/2)
ε = d2q e e h h , (10)
k πλ (cid:2)4β2+q2+q2/λ2(cid:3)
x y
and ε is the Coulomb energy of the exciton, given by
c
−2e2β2λ(cid:90) exp[−2β(cid:112)x2+(λy)2]
ε (cid:39) d2r . (11)
c πκ (cid:112)x2+y2+d2
Equations (10) and (S7) are derived in the Supplemen-
taryInformation, alongwithananalyticalexpressionfor
Eq. (S7) in the limit d/a →0.
e
Figure 2 shows an example calculation of the exciton
FIG. 3. (Color online) The normalized hat height of the ex-
dispersion relation ε(P) for different values of k , taking
0 citondispersion,h=[ε(P =0)−ε ]/[k2/(2(m +m ))],as
thecaseofm =m andd/a →0. Forsmallk (cid:28)1/a , min 0 e h
h e e 0 e a function of the electron “hat brim” k and the mass ratio
0
thedispersionisparabolic,andwelldescribedbyε(P)= m /m . Regionsofh≈1aresuchthattheexcitondispersion
h e
P2/[2(me+mh)]. Atlargek0 (cid:29)1/ae,ontheotherhand, relationisMexicanhat-like,ε(P)(cid:39)(P−k0)2/(2(me+mh)),
the exciton has a pronounced minimum in the disper- whileregionsofh≈0aresuchthatε(P)(cid:39)P2/(2(m +m )).
e h
sion at |P| (cid:39) k , and ε(P) (cid:39) (|P|−k )2/[2(m +m )]. The dashed white lines indicate the approximate boundary
0 0 e h
between these two regions derived in Eqs. (3) and (4).
In between these two extremes, at k a ≈ 1, the exci-
0 e
ton dispersion relation becomes relatively flat at small
momentum, resembling ε(P)(cid:39)P4/(2mk2) for momenta
0
|P|(cid:46)k . One can now consider the implications of this tunable
0
Onecanexploretheevolutionofthedispersionrelation dispersionforalargesystemofexcitonswithfinitenum-
in a more systematic way by evaluating ε(P) for a range ber density n. For such a system, altering the dispersion
of values of k and m /m . The degree of Mexican hat- relation can imply a change to the ground state phase.
0 h e
ness for each case can be quantified by making the ratio Consider, for example, that when k = 0, the system
0
h = [ε(P = 0)−ε ]/[k2/(2(m +m ))], where ε is is equivalent to a 2D collection of dipolar bosons with
min 0 e h min
the minimum energy of the exciton as a function of mo- finite mass m = m +m . Such a system can occupy
e h
mentum P. h can be called the “normalized hat height” either a liquid phase (a Bose-Einsten condensate or su-
of the exciton, with h=0 indicating a parabolic disper- perfluid) or a solid phase (a Wigner crystal), depend-
sion relation and h=1 indicating a Mexican hat-shaped ing on the value of the density.[20, 21] For point-dipoles,
dispersion. The normalized hat height is plotted in Fig. the solid phase exists at n(cid:29)κ2/(m2e4d4). (This condi-
3 for a wide range of the parameters k a and m /m . tioncomesfromtherequirementthatthetypicalnearest-
0 e h e
Its behavior is in close agreement with the qualitative neighbor interaction energy ∼ e2d2n3/2/κ is larger than
derivation presented in Eqs. (3) and (4). the corresponding quantum confinement energy ∼ n/m
4
associated with forming a crystalline state.) However,
1
in a system of interlayer excitons such high densities
are generally not accessible: at much smaller densities η
n ∼ 1/(max{ae,d})2 the exciton wavefunctions overlap tio, 0.8
stronglywitheachother,andtheindividualexcitonsdis- ra 0.6
sociate to form a state that resembles a uniform electron n
n
gas parallel to a uniform hole gas. Thus, at k0 = 0 only ma 0.4 liquid
liquid phases are typically possible — including, for ex- e
d
ample,aBose-Einsteincondensateatsmallenoughd.[22] n 0.2 η
Li c
Consider, however, that as k is increased from zero solid
0
0
by the application of a perpendicular electric field, the
10-1 100 101
dispersion relation of the excitons begins to flatten, as electron hat brim, k0ae
illustratedinFig.2. Thisflatdispersionimpliesthatthe
quantum confinement energy associated with forming a
FIG.4. Lindemannratioη foratrialwavefunctionofdipolar
crystalline state is much reduced. Correspondingly, a bosons (solid line), calculated using the dispersion relation
Wigner crystal state becomes energetically favored over derivednumericallyforinterlayerexcitons. Parametervalues
a uniform state at sufficiently low density as k0ae ap- are me = mh, d = 0.1ae, and n = 0.01/a2e. As the value
proachesunity. Inotherwords,onecandrivealiquid-to- of k0 is increased, the system demonstrates a tendency to-
wardWignercrystallization(smallη). Thedottedlineshows
solidtransitionatfixeddensitybyincreasingatransverse
the value η = η ≈ 0.23 corresponding to the liquid-solid
electric field. c
transition.[20] Large k a (cid:29) 1 (dashed line) corresponds to
0 e
Thefeasibilityofthistransitioncanbecheckednumer-
bosons with a strongly Mexican hat-shaped dispersion, for
ically in a simple way by examining a trial many-body which more exotic states may exist that are not captured by
wavefunction of dipolar bosons arranged on a triangular the simple variational wavefunction employed here.
lattice (as employed, for example, in Ref. [3]). In this
approach, each lattice site is taken to be the locus of a
Gaussian wavepacket of particle density, and the width low density, since its chemical potential µ ∝ n2/mk2 is
0
w of the wave packet is used as a variational parameter. belowthetypicalstrengthofthedipole-dipoleinteraction
For such a wavefunction, the phase of the system can be ∼e2d2n3/2/κ.
estimated by examining the Lindemann ratio η = w/(cid:96), Nonetheless, even in the presence of a dipole inter-
√
where (cid:96) = (2/ 3)1/2n−1/2 is the lattice spacing. Large action a number of proposed states have energy below
valuesofηcorrespondtoaliquidstate,whilesmallηsug- thatofasimpleBose-Einsteincondensateatlowdensity,
gests a Wigner crystal state. Details of this calculation and so there must be some kind of phase transition as
are presented in the Supplementary Information. a function of k . The nature of this transition may pro-
0
Results from such a calculation are shown in Fig. 4 vide fruitful ground for future studies, both theoretical
for the example case m = m , d = 0.1a , and n = and experimental. While the majority of experimental
e h e
0.01/a2. As the value of k is increased from zero, the proposals so far for realizing Mexican hat bosons have
e 0
Lindemann ratio declines, suggesting a tendency toward involved optically-driven Floquet bands or cold atomic
Wigner crystallization. gases,[9, 28] the present work suggests that solid state
If k is increased even further, so that k a (cid:29) 1, bilayers or interfaces may be used as easily tunable plat-
0 0 e
the dispersion relation of the excitons acquires a deep forms for studying the same physics, without the need
and wide-brimmed Mexican hat shape. The phase of for optical excitation.
bosons with such a dispersion has been the subject of Finally, it is worth mentioning that in principle one
a significant amount of theoretical attention in recent can create a tunable excitonic dispersion relation even if
years.[24–29] Among the proposed candidates for the the value of k is fixed, provided that k is large enough
0 0
ground state are a variety of exotic quantum states, and that the small-k part of the electron spectrum can
including a fragmented Bose condensate,[24] a strongly betunedusingsomeotherparameter. Forexample,con-
anisotropicWignercrystal,[25]anda“fermionized”state siderasysteminwhichtheelectronsexperienceastrong
where bosons effectively exhibit Fermi statistics.[26–28] Rashbaspinorbitcoupling. Ifanexternalmagnetic field
Adjudicatingbetweenthesepossiblestatesisbeyondthe is applied to such a system, the spin-split kinetic energy
scope of the present paper, but it is interesting to note bands hybridize and the sharp maximum in energy in
that the presence of a dipole-dipole interaction between the lower band is reduced, as illustrated in Fig. 5. If this
bosons may lead to unusual transitions between the dif- mechanism is used to tune the electron dispersion rela-
ferent candidate states as a function of the parameters tion, then a similar tunability can be achieved for the
n, d, and k . For example, the fermionized state sug- exciton dispersion relation.[30]
0
gested in Refs. [27, 28] for Mexican hat bosons with a Acknowledgments. I am grateful to S. Gopalakrishnan
delta-function interaction cannot persist at arbitrarily and G.Refael forvaluablediscussions, and toA. Nahum
5
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1
Supplementary Information for “Interlayer excitons with tunable dispersion relation”
Brian Skinner
Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(Dated: January 31, 2016)
KINETIC AND POTENTIAL ENERGY OF A SINGLE EXCITON
In Eqs. (10) and (11) of the main text, expressions are presented for the expectation values of the kinetic and
potential energy of an exciton described by the wave function
ψ(r ,r )=exp[i(k0·r +k0 ·r )]ϕ(r ), (S1)
e h e e h h eh
where
(cid:114)
2β2λ (cid:112)
ϕ(x,y)= exp[−β x2+(λy)2]. (S2)
π
The corresponding wave function ψ(cid:101)(ke,kh) in momentum space, defined by
1 (cid:90)
ψ(re,rh)= (2π)4 d2ked2khexp[ike·re+ikh·rh]ψ(cid:101)(ke,kh), (S3)
is
16(2π)5/2β2 δ2(cid:0)k +k −k0−k0(cid:1)
ψ(cid:101)(ke,kh)= √λ [4β2+(k −k −ek0+hk0)2e+(kh −k )2/λ2]3/2. (S4)
ex hx e h ey hy
Here, k =k xˆ+k yˆis the electron momentum, and k =k xˆ+k yˆis the hole momentum.
e ex ey h hx hy
The expectation value of the kinetic energy is given by
(cid:90)
ε = d2r d2r ψ∗(r ,r )(ε +ε )ψ(r ,r ), (S5)
k e h e h (cid:98)e (cid:98)h e h
where ε and ε are the kinetic energy operators for the electron and hole, respectively. Inserting Eq. (S3) into this
(cid:98)e (cid:98)h
expression, and using the fact that plane waves are eigenstates of the kinetic energy operators — i.e., that
ε exp[ik·r ]=ε (k)exp[ik·r ],
(cid:98)e e e e
ε exp[ik·r ]=ε (k)exp[ik·r ]
(cid:98)h h h h
— gives the following general expression for the kinetic energy:
1 (cid:90) (cid:12) (cid:12)2
εk = (2π)4 d2ked2kh[εe(ke)+εh(kh)](cid:12)(cid:12)ψ(cid:101)(ke,kh)(cid:12)(cid:12) . (S6)
Inserting Eq. (S4) into this expression gives Eq. (10) of the main text.
The potential energy of the exciton is given generally by
(cid:90) e2
ε =− d2r d2r |ψ(r ,r )|2
c e h e h (cid:112)
κ (r −r )2+d2
e h
(cid:90) e2
=− d2r |ϕ(r )|2 . (S7)
eh eh (cid:112)
κ r2 +d2
eh
Inserting Eq. (S2) gives Eq. (11) of the main text. In the special case d = 0, this integral can be done analytically,
giving
e =−2e2β (cid:2)K(1−1/λ2)+λK(1−λ2)(cid:3), (S8)
c πκ
where K(x) is the complete elliptic integral of the first kind. An equivalent expression to Eq. (S8) has been derived
previously, for example in Refs. S1 and S2.
2
ENERGY OF A TRIAL MANY-BODY STATE OF DIPOLAR BOSONS
In the main text, the feasibility of a liquid-to-solid transition driven by increasing k was examined using a simple
0
trial wavefunction. Here I provide some details about this calculation, which largely follows the one presented in Ref.
S3.
In this approach, the interlayer excitons are treated as bosonic particles with a dipolar interaction law and the
dispersion relation ε(k) that results from the procedure described in the main text. The many-body wave function
for the system is taken to be a product of Gaussian wave packets φ (r), each centered around some point r on the
ij ij
triangular lattice. These are given by
1 (cid:20) |r−r |2(cid:21)
φ (r)= √ exp − ij , (S9)
ij πw2 2w2
(cid:112)
so that the corresponding uncertainty in position for each particle is (cid:104)r2(cid:105)=w. The Fourier transform of this wave
packet is given by
√
φ(cid:101)(k)= 4πw2exp[−k2w2/2], (S10)
and the corresponding kinetic energy per particle
1 (cid:90) (cid:12) (cid:12)2
εkin = (2π)2 d2kε(k)(cid:12)(cid:12)φ(cid:101)(k)(cid:12)(cid:12) . (S11)
The value of ε is determined by numerically evaluating this integral for each value of w and each instance of the
kin
dispersion relation ε(k). For the case where ε(k)=(k−k )2/2m, the kinetic energy is given by
0
1 (cid:2) √ (cid:3)
ε = 1+k w(k w− π) .
kin 2mw2 0 0
The interaction energy (Hartree energy) in this state is given by[S4]
n (cid:88) 1 (cid:90)
εint = 2 V(cid:101)(q)exp[−q2w2/2]− 2(2π)2 d2kV(cid:101)(q)exp[−q2w2/2]. (S12)
q∈G
Here, n is the particle density, G labels the set of all reciprocal lattice vectors of the triangular lattice, and
4πe2
V(cid:101)(q)= (1−exp[−qd]) (S13)
κq
is the interaction law between dipoles with a dipole arm d. The second term on the right hand side of Eq. (S12)
removes the self-interaction term from the sum in the first term. The set of reciprocal lattice vectors are defined by
(cid:115)
2n
q =2π √ (i2+ij+j2), (S14)
ij
3
where the indices i and j run over all integers.
(cid:112)
In the variational method, the value of the positional uncertainty (cid:104)r2(cid:105) is estimated to be equal to the value of w
which minimizes the total energy per particle, ε +ε .
kin int
[S1] Arno Schindlmayr, “Excitons with anisotropic effective mass,” European Journal of Physics 18, 374 (1997).
[S2] Elsa Prada, J. V. Alvarez, K. L. Narasimha-Acharya, F. J. Bailen, and J. J. Palacios, “Effective-mass theory for the
anisotropic exciton in two-dimensional crystals: Application to phosphorene,” Physical Review B 91, 245421 (2015).
[S3] Brian Skinner, “Chemical potential and compressibility of quantum Hall bilayer excitons,” arXiv:1511.03287 [cond-mat]
(2015), arXiv: 1511.03287.
[S4] BrianSkinner,G.L.Yu,A.V.Kretinin,A.K.Geim,K.S.Novoselov, andB.I.Shklovskii,“Effectofdielectricresponse
on the quantum capacitance of graphene in a strong magnetic field,” Physical Review B 88, 155417 (2013).