Binding energies of trions and biexcitons in two-dimensional semiconductors from diffusion quantum Monte Carlo calculations

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by  M. Szyniszewski

Binding energies of trions and biexcitons in two-dimensional semiconductors from
diffusion quantum Monte Carlo calculations
M. Szyniszewski,1,2 E. Mostaani,1,3 N. D. Drummond,1 and V. I. Fal’ko1,2
1Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom
2National Graphene Institute, University of Manchester,
Booth St E, Manchester M13 9PL, United Kingdom
3Cambridge Graphene Centre, Engineering Department, University of Cambridge,
7 9 J. J. Thomson Avenue, Cambridge CB3 0FA, United Kingdom
1
0 Excitonic effects play a particularly important role in the optoelectronic behavior of two-
2 dimensional (2D) semiconductors. To facilitate the interpretation of experimental photoabsorp-
tion and photoluminescence spectra we provide statistically exact diffusion quantum Monte Carlo
n
binding-energydataforMott-Wanniermodelsofexcitons,trions,andbiexcitonsin2Dsemiconduc-
a
tors. Wealsoprovidecontactpairdensitiestoallowadescriptionofcontact(exchange)interactions
J
between charge carriers using first-order perturbation theory. Our data indicate that the binding
5
energyofatrionisgenerallylargerthanthatofabiexcitonin2Dsemiconductors. Weprovideinter-
2
polation formulas giving thebindingenergy andcontact densityof 2D semiconductorsas functions

of theelectron and hole effective masses and thein-plane polarizability.
]
l
l
a PACSnumbers: 78.20.Bh,31.15.-p,73.20.Hb,78.55.-m
h
-
s
The optical properties of two-dimensional (2D) semi- two electrons (e and e ) can be found by solving the
e 1 2
m conductorssuchasmonolayerMoS2,MoSe2,WS2,WSe2, Schr¨odinger equation (in Gaussian units)
InSe, and phosphorene have recently attracted a great
.
at deal of interest [1–8]. Numerous observations have ~2 2 ~2 2
-m bspeeecntrmaaodfethoefseth2eDrimchatestrriaulcst,uirne wofhicthhethluemminoesstcepnrcoe- k=Xe1,e2(cid:20)− 2m∇rek −U(rkh)(cid:21)− 2m∇hrh +U(re1e2)Ψ
d nounced features have been interpreted in terms of neu-  =[ T]Ψ, (1)
n −U −E
tral excitons [9–14], charged excitons (trions) [15–20],
o
and biexcitons [21–23], while recent experiments on where m and m are the electron and hole effective
c e h
[ higher-qualitymonolayertransition-metaldichalcogenide masses and rij = ri rj is the position of particle i
−
(TMDC) samples have revealed additional structure in relativeto particle j. The Keldyshpotential U describes
1
their spectra [24–27]. the Coulomb interaction screened by the polarization of
v
7 In this work we study a Mott-Wannier model of exci- the electron orbitals in the 2D lattice [28–30, 34],
0
tons and excitonic complexes in monolayer 2D semicon-
4 e2 d2q 2πeiq·r
7 ductors, taking into account the polarizability of the 2D U(r)=
0 crystal [28–30] and providing data to allow for a pertur- ǫ Z (2π)2q(1+qr∗)
. bative treatment of contact interactions between carri- πe2
01 ers. We use the diffusion quantum Monte Carlo (DMC) = 2ǫr∗ [H0(r/r∗)−Y0(r/r∗)], (2)
7 approach [31–33] to find the energies of trions and biex-
1 citons, and we provide approximate formulas for the ex- where r∗ = 2πκ⊥ is a parameter directly related to the
: citon ( ), trion ( ), and biexciton ( ) binding en- in-plane susceptibility κ⊥ of the material, which has di-
v T XX
i ergies aUs functionsEof the in-plane polaErizability and the mensions of length, and ǫ is the average permittivity of
X electron and hole effective masses, which fit the DMC the media on either side of the 2D semiconductor. The
r data to within 5%. We calculate andreportcontactpair potential is expressed in terms of a Struve function H0
a
densities, enabling the evaluationof perturbative correc- andaBesselfunctionofthesecondkindY0. Equation(1)
tions to the energies of charge-carriercomplexes, as well determines the main contribution T towards the trion
E
asintervalleyscattering,duetocontact(exchange)inter- bindingenergy,whichiscountedfromtheexcitonbinding
actionsbetweenchargecarriers. Thestrengthofthecon- energy . Similar Schr¨odinger equations can be written
U
tact interactions could in principle be determined from for an exciton and a biexciton.
first-principles calculations for different 2D semiconduc- Numerical solution of the Mott-Wannier Schr¨odinger
tors; alternatively, the strengths of the contact interac- equation for an exciton yields the r∗-dependent binding
tions can be regarded as parameters to be determined energy (see the inset in Fig. 1), which agrees with the
upsaiinrgdeexnpsietrieims erenptaolrtdeadtaheinre.conjunction with the contact aansydmUp(trUo∗tic→lim0i)ts=[28−–430R,y∗3,5,a3s6w] Uell(ra∗s→th∞e )co∼ntǫaer2c∗tlnpaari∗B∗r
The energy −U −ET of a trion with one hole (h) and density geXh = hδ(reh)i. Their r∗ dependence was fitted
2
by 0.7(cid:4) 4(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
m(cid:1)/m(cid:2) (cid:1)
U/gReXyh∗ ≈≈((1a8∗B−.0)2x)1[+41−2−01..0x0√xxln,(1−x)]/(cid:2)1+1.31√x(cid:3)((34)) *(cid:5)/RyT0000....3456(cid:1)(cid:3)(cid:2) (cid:2)(cid:1)(cid:3)(cid:4)(cid:2)(cid:3)(cid:1)(cid:3)(cid:4)(cid:2)(cid:4) 0512....5000 *(cid:6)/Ry0312 (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)
(cid:1) 0.0 0.2 0.4 0.6 0.8 1.0
where x = r∗/(a∗B + r∗), a∗B = ǫ~2/(µe2) is the exci- 0.2 (cid:2) (cid:1)(cid:3) (cid:4) x
mtoansics,BaonhdrrRay∗di=us,µµe4=/((2mǫ2e~m2)h)i/s(mthee+emxchi)toinsitcheRryeddbuecregd. 0.1 (cid:2) (cid:1)(cid:3) (cid:4) (cid:2) (cid:1)(cid:3) (cid:4) (cid:2) (cid:1)(cid:3)(cid:2)(cid:4)(cid:1)(cid:3)(cid:2) (cid:4)(cid:1)(cid:3)(cid:4)
Here, the √x term is not a physical singularity; rather, 0.0
0.0 0.2 0.4 0.6 0.8 1.0
itreflectstheenhancedcontactdensityofa2Dhydrogen x=r /(a* +r)
* B *
atom. gX is plotted in Fig. 2.
eh
FIG. 1. (color online). Binding energies of trions at different
Theground-statesolutiontoEq.(1)foratrionwasob-
mass ratios against rescaled in-plane polarizability r∗. The
tained using the DMC approach [31, 32], with the trial
inset shows the binding energies of excitons against rescaled
wave function being optimized using variational Monte r∗. The lines show the fittingformulas of Eqs. (5) and (3).
Carlo (VMC). The trial wave function was of the Jas-
trow form Ψ = exp[J(R)], where the Jastrow exponent
uJ0((Rr)) c=onsics1terd2loofg(ar)p+aircw2ri2se+suc3mr3of/te1r+msc4orf2t,hewfhoerrme 1.5 (cid:2)(cid:1)(cid:3)(cid:4) 2)y 00..1102(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) me/mh
taMftrcmaohha1rnri,inaardztettgeehceeead2emt-rc,nhbi[hai3beeoccl7ry3dpag,i,iyannn(cid:2)y3uartap8ntenRml]iorord.caleegilynTwfcesc.nt-i4ehmpoio[oie3famgain0izhtsrri]ahhetaa)tete.olryiodeortKpptenT-sevtrarrai)amtma[i,t4roanisit1ilsagzcofi]tnawe.ughecbseaadbeptlTvteeh(ehcmhaesp(cid:3)oarreefiaenvuenwridgnit(cid:0)oatmriirchrtmtouthiieuioonzeonfnattcSnwetadsuusirt-o0ofpseso(w-ntpdbrr((cid:1)ael)doeat[rti3demehitffs9yefieew,osrnnalpuaeo4tintcvntga0idheet--l] 2X2T**,()()agagBehBeh01..50 m(cid:1)(cid:3)(cid:4)(cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:1)ge(cid:1)(cid:1)(cid:1)e/T2051(cid:2)m(cid:1)h(cid:1)(cid:1)....(cid:1)(cid:1)(cid:1)0500(cid:3)(cid:1)h(cid:1)(cid:1)(cid:4)(cid:1) (cid:1)(cid:2)(cid:1)(cid:1)(cid:1)g(cid:1)(cid:1)eX(cid:1)(cid:3)(cid:1)h(cid:1)(cid:1)(cid:4)(cid:1)2T*(cid:1)1()/(-ag(cid:2)Bee(cid:1)(cid:1)(cid:1)(cid:1)00000(cid:1).....(cid:3)00000(cid:1)(cid:1)024680(cid:1).0(cid:4)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)0(cid:1)(cid:1)(cid:1).(cid:1)(cid:1)2(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:1)(cid:1)(cid:1)0(cid:1)(cid:1)(cid:1)(cid:1).4(cid:2)(cid:1)00(cid:1)(cid:1)..(cid:1)x(cid:1)25(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:1)(cid:1)(cid:1)0(cid:1)(cid:1)(cid:1)(cid:2)(cid:1).(cid:1)(cid:1)(cid:1)6(cid:4)(cid:1)(cid:1)01(cid:1)(cid:1)(cid:1)(cid:1)..(cid:1)(cid:3)(cid:1)(cid:1)(cid:2)8(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)0(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1).(cid:1)(cid:1)(cid:1)(cid:4)8(cid:1)(cid:1)12(cid:1)(cid:1)(cid:1)(cid:1)..251.0
0.0
functions forthese systemsarenodeless;hencethe fixed- 0.0 0.2 0.4 0.6 0.8 1.0
node DMC algorithm is exact. The DMC calculations x=r /(a* +r)
* B *
were performed using the casino code [33] with time
steps in the ratio1:4 andthe correspondingtargetcon- FIG.2. (coloronline). Contactelectron-holepairdensitiesof
figuration populations in the ratio 4 : 1. Afterwards, trionsandexcitonsagainst rescaled in-planepolarizability r∗
the energies were extrapolated linearly to zero time step atdifferentmassratios. Theblackcurveisthefittingformula
foranexciton,Eq.(4),whiletheyellowcurveisthefittingfor-
and hence, simultaneously, to infinite population. The
mulaforatrion,Eq.(6). Theinsetshowsthe(muchsmaller)
resulting DMC trion binding energies, shown in Fig. 1,
contactpairdensitybetweentheelectronsinanegativetrion
agree with the asymptotic binding energies found ear-
and thefitting curve(in blue) of Eq. (7).
lier in the limits of r∗ [30] and r∗ 0 [42].
→ ∞ →
The trion contact pair densities gT = δ(r ) and
ee h e1e2 i
gT = δ(r )+δ(r ) were obtained by binning the
eh h e1h e2h i
radial distances sampled in the VMC and DMC calcu- while the contact pair densities are fitted by
lations, evaluating the extrapolated estimate of the pair
0.35
density [32], and then extrapolating the pair density to gT gX + (1 x)3.5 and (6)
eh ≈ eh (a∗)2 −
zero radial separation. The resulting contact pair densi- B
ties are shown in Fig. 2. The trion binding-energy data gT 0.11 1−√x 1 y2 , (7)
are fitted (to an accuracy within 5%: see Fig. 1) by the ee ≈ (a∗)21+√x −
B (cid:2) (cid:3)
formula
where y = µ/m . The term proportional to √1 y in
h
−
Eq. (5) describes the contribution to the ground-state
ET 1 √x 0.44x2 1.16√x+1.46 (2 y) energy due to the harmonic zero-point vibration of two
Ry∗ ≈(cid:0) − (cid:1)h(cid:0) − (cid:1) − heavyelectronstreatedusingtheBorn-Oppenheimerap-
proximation [42].
0.64x2 2.0√x+2.4 1 y ,
−(cid:0) − (cid:1)p − i Similarly, the binding energies EXX of biexcitons were
(5) calculated using DMC and the results are presented in
3
1.4
1.4
1.2 (cid:1) me/mh 0.06 me/mh
1.2 XXgeh01.8. (cid:1) (cid:1)0.5(cid:1)1.0 XXgee0.04 (cid:1)(cid:1) (cid:1)(cid:1)01..50
1.0(cid:1)*2a()B000...246 (cid:1) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) *2a()B0.02 (cid:1) (cid:1) (cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)2.0
*Ry0.8(cid:3)(cid:4)(cid:2) 00..25 0.5 0.75 1. 00.25 0.5 0.75 1.
/ x x
(cid:5)XX0.6 (cid:4)
(cid:4)
0.4 me/mh
(cid:1) 0.2
(cid:2) 0.5
0.2 (cid:4)(cid:3)(cid:2) (cid:1) (cid:3) 0.7
(cid:4) (cid:3) (cid:2) (cid:1) (cid:4) 1.0
0.0 (cid:4) (cid:3) (cid:2) (cid:1) (cid:4) (cid:3)(cid:2)(cid:4)(cid:1)(cid:3)(cid:4)(cid:2)(cid:3)(cid:1)(cid:2)(cid:1) (cid:4)(cid:3)(cid:2)(cid:1)
0.0 0.2 0.4 0.6 0.8 1.0
x=r /(a* +r)
* B *
FIG.3. (color online). Bindingenergies ofbiexcitonsagainst
rescaled in-plane polarizability r∗ at different mass ratios. FIG.4. (coloronline). Ratioofnegative-trionbindingenergy
The lines show the fitting formula of Eq. (8). The left in- tobiexcitonbindingenergy(ET/EXX)asafunctionofrescaled
set shows the electron-hole contact pair densities for a biex- in-plane polarizability r∗ and rescaled mass ratio. The thick
citon and the approximation formula [black curve, Eq. (9)]. blacklineshowsthecurveET=EXX. Experimentallyrelevant
Electron-electron contact pair densities for a biexciton are points for TMDCs are shown using symbols from Table I.
shown in the right inset, together with the approximation
formula of Eq. (10).
materials typically have x > 0.9 and y 0.5 (see Table
≈
I),andhencearestronglyintheregimeinwhichthetrion
Fig. 3. is the energy required to dissociate a biex-
XX
E bindingenergyexceedsthebiexcitonbindingenergy. The
citon into two separate excitons. A fitting formula with
qualitativeformofourpredictedtrionspectrumisshown
up to 5% accuracy,
inFig.5. Thetrionpeakoccursatlowerenergythanthe
EXX 1 √x 1 1.2 y(1 y) biexciton peak, in stark contradiction to the classifica-
Ry∗ ≈(cid:0) − (cid:1)h − p − i tion of experimental peaks reported in Ref. [43]. In fact
several experimental works [21–23] have reported biex-
2.0 17.0x+43.0 x3/2 x2 +15.7x5/2 ,
×h − (cid:16) − (cid:17) i citon binding energies of TMDCs that are about twice
(8) as large as the reported trion binding energies [15–20].
However the physical origins of experimentally observed
incorporates the fact that the biexciton binding en-
peaks in optical spectra are not always clear. Our con-
ergy is symmetric under the exchange of electrons and
clusion that the trion binding energy is larger than the
holes and includes the correct behavior in the Born-
biexciton binding energy is robust against large changes
Oppenheimer/harmonic-approximation limit of extreme
inthevaluesoftheeffectivemassesandthesusceptibility
mass ratio. The biexciton electron-hole and electron-
and, taken at face value, suggests that the experimental
electron contact pair densities can be approximated as
“trion” and “biexciton” peaks may be misclassified.
0.5 InTable I we comparethe trionandbiexcitonbinding
gXX 2gX + (1 x)2 and (9)
eh ≈ eh (a∗)2 − energies obtained using Eqs. (5) and (8) with previous
B
theoretical calculations and experimental results in the
1 x
geXeX ≈ (a−∗)2(1−0.44x)(0.1−0.064y). (10) literature for molybdenum and tungsten dichalcogenide
B materials. The theoretical results are in good agreement
The ratio of the negative-trion to the biexciton bind- with each other, and also with experimental results for
∗
ing energy is plotted against x = r∗/(aB + r∗) and the trion. However, for the biexciton, there is a major
y = µ/m = m /(m + m ) in Fig. 4. Although the disagreement between theory and experiment: the ex-
h e e h
biexciton binding energy is larger than the trion binding perimentalbindingenergiesarearoundthreetimeslarger
energyfortheCoulombinteraction(x=0),thesituation thanthetheoreticalbiexcitonbindingenergies. Sinceour
is generally reversed when the interaction is of logarith- DMC solution of the Mott-Wannier model is exact, the
mic form (x = 1). However, at extreme mass ratios, quantitative disagreement between the positions of the
especially where the hole is heavy,the biexciton is stabi- theoretical and experimental trion and biexciton peaks
lized with respect to the negative trion. In practice 2D must indicate either a serious inaccuracy in the Keldysh
4
X
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port from EPSRC, NOWNANO DTC grant number Kis,andA.Imamog˘lu,Nat.Nanotechnol.10,491(2015).
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5
TABLEI.TrionandbiexcitonbindingenergiesforanimportantgroupofTMDCs. Wecompareourfittingformulaswithpath
integral Monte Carlo (PIMC) [44] and DMC data [45] in the literature. Effective masses (in units of the bare electron mass)
are taken from GW calculations in theliterature [46, 47].
Negative-trion binding energy ET (meV) Biexciton binding energy EXX (meV)
Material r∗ (˚A) me mh
PIMC DMC Eq. (5) Exp. PIMC DMC Eq. (8) Exp.
• MoS2 39 [46] 0.35 0.43 32.0 33.8 32 34 [16], 35 [48] 22.7(3) 22.7(5) 24 60 [49], 70 [21]
(cid:4) MoSe2 40 [50] 0.38 0.44 27.7 28.4 31 30 [51, 52] 19.3(5) 17.7(3) 23 N/A
N WS2 38 [53] 0.27 0.32 33.1 34.0 31 34 [54], 36 [28] 23.9(5) 23.3(3) 23 65 [43]
H WSe2 45 [53] 0.29 0.34 28.5 29.5 27 30 [17, 20] 23.9(5) 23.3(3) 20 52 [23]
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