The Dielectric Impact of Layer Distances on Exciton and Trion Binding

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The dielectric impact of layer distances on exciton and trion binding energies in van der Waals heterostructures Matthias Florian, Malte Hartmann, Alexander Steinhoff, Julian Klein, Alexander W. Holleitner, Jonathan J. Finley, Tim Oliver Wehling, Michael Kaniber, and Christopher Gies Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b00840 • Publication Date (Web): 20 Mar 2018 Downloaded from http://pubs.acs.org on March 22, 2018

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Nano Letters

The Dielectric Impact of Layer Distances on Exciton and Trion Binding Energies in van der Waals Heterostructures Matthias Florian,∗ † Malte Hartmann,† Alexander Steinho,† Julian Klein,‡ Alexander W. Holleitner,‡ ¶ Jonathan J. Finley,‡ ¶ Tim O. Wehling,† § Michael Kaniber,‡ ¶ and Christopher Gies∗ † ,

,

,

,

,

,

†Institut für Theoretische Physik, Universität Bremen, P.O. Box 330 440, 28334 Bremen,

Germany ‡Walter Schottky Institut and Physik Department, Technische Universität München, Am Coulombwall 4, 85748 Garching, Germany ¶Nanosystems Initiative Munich (NIM), Schellingstr. 4, 80799 Munich, Germany §Bremen Center for Computational Materials Science, Universität Bremen, 28334 Bremen, Germany E-mail: m[email protected]; [email protected]

Abstract

tures, transition-metal dichalcogenides, dielectric screening, trion binding energy, band gap engineering, 2D materials

The electronic and optical properties of monolayer transition-metal dichalcogenides (TMDs)

Introduction.

and van der Waals heterostructures are strongly subject to their dielectric environment. In each layer the eld lines of the Coulomb interaction

mate-

basic constituents is a relatively new discipline,

are screened by the adjacent material, which re-

driven by seemingly endless possibilities in ma-

duces the single-particle band gap as well as ex-

terial combinations in so-called van der Waals

citon and trion binding energies. By combining

heterostructures (vdWH),

an electrostatic model for a dielectric hetero-

1

and by the manipu-

lation and control of the electronic and optical

multi-layered environment with semiconductor

properties through their dielectric environment.

many-particle methods, we demonstrate that

The Coulomb interaction between charge carri-

the electronic and optical properties are sen-

ers in the atomically thin layer is screened only

sitive to the interlayer distances on the atomic scale.

Quantum-mechanical

rial design with atomically thin layers as the

weakly, which is the reason for the exceptionally

An analytic treatment is used to pro-

large exciton binding energies of hundreds of

vide further insight into how the interlayer gap

meV, and for the importance of GW corrections

inuences dierent excitonic transitions. Spec-

to band gaps calculated from density-functional

troscopical measurements in combination with

theory.

a direct solution of a three-particle Schrödinger

2,3

The eects of electric eld screening

by the surrounding dielectric environment have

equation reveal trion binding energies that cor-

been heavily investigated in the recent years.

rectly predict recently measured interlayer dis-

49

The possibility to use dielectric encapsulation

tances and shed light on the eect of tempera-

to externally control the band gap and the

ture annealing.

binding energy is now recognized as a virtue

Keywords: van der Waals heterostruc-

to tailor new excitonic and optoelectronic de-

ACS Paragon Plus Environment 1

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Figure 1:

(a)

Page 2 of 15

Schematic representation of a vdWH with an ideal plane boundary and

(b)

a re-

alistic interface with a nite interlayer gap. For two point charges in the TMD layer, calculated equipotential (black) and electric eld lines (red) have been superimposed to visualize the eect of changes in the dielectric environment caused by the interlayer gap.

(c) Illustration of the change

of the band-gap and the bound-state energies due to the dierent screening environments: freestanding TMD monolayer (left), TMD monolayer on substrate with no distance between the layers causing strong screening reducing both the band gap and trions

ET

Egap

and the binding energies of excitons

EX

(middle), and non-vanishing gap between TMD and substrate leading to reduced

screening (right).

vices.

10

Further knobs to tune and tailor the

lines.

It is the topic of this letter to provide

optical properties of vdWH include phonon

both an analytic and a quantitative under-

modes at the interfaces with surrounding lay-

standing of the impact of the interlayer gap on

ers.

Prestructured substrates have been pre-

the electronic and optical properties of vdWH.

to induce lateral heterojunctions by

VdWH consist of vertically stacked single lay-

local modication of the Coulomb interaction

ers of two-dimensional materials that can be

on the length scale of a few unit cells, which

semiconducting, such as MoS2 , MoSe2 , WS2

has recently been conrmed experimentally.

and WSe2 , conducting, such as graphene, or in-

11

dicted

12,13

6

1,14

In the literature, the electric eld between two

sulating, such as boron nitride.

opposite charges in the 2D layer is often vi-

fabricated under ambient conditions and with-

sualized by the intuitive picture of eld lines

out lattice matching due to the weak van der

passing the surrounding material where they

Waals interlayer bonding, which has led to an

experience screening. A realistic representation

explosion of research activity on band-structure

of the calculated electrostatic potential and the

and interface engineering in this toolbox of ma-

eld lines that satisfy the boundary conditions

terials. The atomistic modeling of heterostruc-

dictated by Maxwell's equations is shown in

tures that are formed from incommensurate lay-

Fig. 1. While a quantitative assessment of the

ers is strongly limited by computational de-

strength of the Coulomb interaction cannot be

mand. At the same time, there is no particular

inferred from such a picture, it becomes clear

benet from an ab-initio treatment of the en-

that a dierence in the dielectric environment,

vironment. For this reason, dierent multiscale

such as a nite gap at the heterostructure in-

approaches have been developed, in which the

terface (right panel) instead of an ideal plane

active TMD layer is treated atomistically, and

boundary (left panel) has an impact on the eld

macroscopic approaches are used to model the


They can be

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Nano Letters

inuence of the dielectric environment to obtain

strongly inuence the long-range Coulomb in-

independently an eective non-local dielectric

teraction in the active layer and play an impor-

function.

tant role in the characteristics of optoelectronic

5,1518

The results are then successively

used e.g. in Wannier-equation or BSE calcula-

devices.

tions to obtain access to the optical properties, such as excitonic resonances.

5,6

Continuum Electrostatics Approach for Calculating the Non-local Dielectric Function in Stacked Layers. While the

Alternatively,

the optical response is obtained from the solution of semiconductor Bloch equations, which have been used before to evaluate the shift of

ease of fabrication is a particular benet in

excitonic resonances in optically or electrically

creating

excited TMDs.

calculations that are required to determine,

1921

Only recently, cross-sectional STEM

22

vdWH,

material-realistic

ab

initio

and

amongst other things, band osets and band

measurements have provided rst in-

gaps, quickly hit the computational limits, es-

sight into the actual layer separation at the

pecially when supercells are required to repre-

interfaces and into its modication due to the

sent incommensurable multi-layered materials.

anealing step that is often used in sample fabri-

The result of recent eorts in the community

cation. Reported results of 3 to 8 Å imply that

has lead to dierent multi-scale approaches

Coulomb screening is signicantly reduced by

that share a common idea: While the electronic

the gap between the layers.

properties of the active TMD layer are deter-

AFM

23

By applying an

electrostatic approach that builds on the

nier function continuum electrostatics

Wan-

mined from atomistic models, such as density

(WFCE)

functional theory

12,13,25,26

2,3,24

or eective tight-binding

scheme introduced in Ref. 16 to calculate the

models,

non-local dielectric function for an arbitrary

results from adjacent layers of various mate-

number of stacked layers, we demonstrate a sig-

rials is treated in an electrostatic approach

nicant impact of realistic interface conditions

that is oblivious to the atomic resolution of

on the non-local dielectric function that deter-

each layer. This approach is based on the as-

mines the screened Coulomb interaction and

sumption that hybridization of orbitals from

thereby the electronic and optical properties

adjacent layers plays a minor role as compared

of vdWH.

to dielectric screening. We further assume that

To layer

directly gaps

at

evaluate the

the

impact

interfaces

in

of

inter-

vdWH,

the dielectric screening that

adjacent layers excert no strain on the active

we

TMD layer.

As long as we concentrate on

present a combined theoretical and experimen-

observables that emerge from the vicinity of

tal study of trion binding energies in various

the

TMD/substrate combinations.

are well-protected against hybridization eects,

Trion binding

energies are calculated with sucient accuracy

K

and

K'

points in reciprocal space that

this assumption is justied.

to predict TMD-substrate layer separations in

It is our aim to establish the importance of

the experiment, which we nd to be in agree-

layer interfaces in vertical vdWH, in which the

ment with recent cross-sectional STEM mea-

density of a polarizable medium is reduced due

surements.

We further present results for the

to the mere van der Waals interlayer bonding.

band-gap reduction and for the increase of ex-

As a consequence, eld lines passing this in-

citon binding energies as function of the in-

terlayer gap, as we will refer to it in the fol-

terlayer gap.

Simple estimates are provided

lowing, are more weakly screened in compari-

that allow for calculating corrections to the

son to those passing the adjacent material. As

bound-state resonance energies and the band-

we will show, the impact of the interlayer gap

gap renormalizations for realistic interface con-

on the band gap and on excitonic binding en-

ditions via Eqs. (9) and (11), respectively.

ergies is large due to the strong Coulomb ef-

22

In

the emerging eld of band-structure and inter-

fects in these materials.

face engineering in vdWH, our results demon-

process to rst provide closed equations for the

strate that layer separations at the interfaces

non-local macroscopic two-dimensional dielec-


We use a two-step

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tric function

ε2D mac (q).

The latter describes a

Page 4 of 15

formed back into the Wannier basis.

To use

TMD encapsulated in a sub- and superstrate

them in equations of motion formulated in mo-

heterostructure that includes additional layers

mentum space, they are subsequently trans-

of air to model the interlayer gaps.

In a sec-

formed into the Bloch basis using expansion

ond step, this dielectric function is transformed

coecients that connect the Wannier and the

into a microsopic basis and used to solve a

Bloch basis on a G0 W0 -level as described in Ref.

generalized two- and three-particle Schrödinger

19.

equation to study the impact of interlayer dis-

The starting point of our derivation of a

tance on the exciton and trion binding ener-

model dielectric function for TMD heterostruc-

gies.

tures is Poisson's equation, which yields the

While free carriers are known to further

φ(r) for a given charge ρ(r) in the presence of a dielectric funcεr (r, r0 ) describing nonlocal screening ef-

modify the electronic and optical properties of

electrostatic potential

the TMD by reducing both, the single-particle

density

band gap and exciton binding energies,

tion

19,27,28

we assume a low density of carrier doping in

fects

5

the TMD layer in accordance with the experi-

Z

In

order

to

d3 r0 εr (r, r0 )∇r0 φ(r0 ) = −

∇r ·

mental situation. calculate

properly

screened

ρ(r) . ε0

(1)

Coulomb matrix elements for the embedded

To nd a unique solution for the potential

TMD, we begin with ab initio calculations

we solve Poisson's equation for each layer of

for the freestanding monolayer to obtain bare

the heterostructure separately assuming an in-

Uαβ (q)

and screened

Vαβ (q)

Coulomb matrix

elements in a Wannier orbital basis

q

α ,

φ,

nite extension of each layer in the x-y plane

where

and a charge density

is a two-dimensional wave vector from the

layer.

ρ

ρ

only in the active TMD

may either describe intrinsic, doped

rst Brillouin zone. Calculations are performed

or optically excited charges.

on the G0 W0 level as outlined in Ref. 19. We

following to formally derive an expression for

stress that calculations of the excitonic proper-

the dielectric response experienced by arbitrary

ties are performed on the the full Brillouin zone

charges in the TMD layer. The response is en-

so that potential eects due to the interplay of

coded in the macroscopic two-dimensional di-

carriers in dierent valleys are accounted for.

electric function

Such eects have been shown to arise e.g. from

later to solve dynamical equations for specic

strain,

charge densities.

27,29

and at elevated carriers densities.

27

To take into account environmental screen-

ε2D mac (q),

It is used in the

which will be used

At the interfaces, boundary

conditions dictated by electrostatics must be

ing eects in vertical heterostructures, we em-

fullled.

ploy and extend the WFCE approach. The cen-

in-plane component to reciprocal space and use

tral idea introduced by Roesner

an ansatz for

only the leading eigenvalue of

et al. 16

U

is that

To solve Eq. (1), we transform the

φ(q)

that takes into account the

is connected

vanishing of the potential at innity and its con-

to long-wavelength charge-density modulations,

tinuity at each interface following from the con-

for which environmental screening is expected

tinuity of the tangential electric eld:

to be strongest.

The remaining eigenvalues

are linked to microscopic details and are well described as constants.

φ(q, z) =

The same argument

holds for the dielectric function, whose leading eigenvalue

N −1 X ρ(q) e−q |z| + Bj e−q |z−zj | . 2 ε0 εTMD (q) q j=1

ε1 (q) = ε2D mac (q) is hence analytically

(2)

modeled by an eective two-dimensional dielec-

Here, the rst term accounts for the inhomo-

tric function using continuum medium electrostatics, as we show below.

geneity due to the charge density in the active

The matrix ele-

TMD layer.

V(q) in the eigenbasis of the bare interaction U(q) are then −1 obtained via Vi (q) = εi (q) Ui (q) and transments of the screened interaction

The second term stems from the

homogeneous solutions of Poisson's equation in each layer, where


j runs over all interfaces of the

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Nano Letters layers. Its particular form captures the fact

The details of the electrostatic calculation are

Bj

given in the Supporting Information and yield

accumulate at each interface, thereby superim-

a system of coupled linear equations that can

posing the two-dimensional Coulomb potential

easily be solved for any relevant heterostruc-

φ0 (q)/εTMD (q) = ρ(q)/(2 ε0 εTMD (q) q), which is due to the charges ρ(q) in the active layer,

ture size.

with induced potentials.

εTMD (q),

N

that surface charges given by the coecients

In the following, we assume that

the dielectric response of the active layer itself,

Taking into account

is isotropic.

the combined action of the simple Coulomb po-

For the simple yet typical cases displayed

tential and the induced potentials, we can for-

in Fig. 2 compact analytic expressions can be

mulate the two-dimensional dielectric function

found. For a TMD layer of width

ε2D mac (q) that describes the dielectric response to

placed

on a substrate and accounting for the interlayer gap of width

any charge density in the active TMD layer.

ε2D mac (q) =

hTMD

hint

at the interface, we obtain

ε3 (1 + ε˜1 ε˜2 β + ε˜1 ε˜3 α2 β + ε˜2 ε˜3 α2 ) , 1 + ε˜1 αβ + ε˜2 α − ε˜3 α + ε˜1 ε˜2 β − ε˜1 ε˜3 α2 β − ε˜2 ε˜3 α2 − ε˜1 ε˜2 ε˜3 αβ

(3)

ple expression

ε2D mac (q) = ε3

1 − ε˜1 αβ − ε˜2 α + ε˜1 ε˜2 β . 1 + ε˜1 αβ + ε˜2 α + ε˜1 ε˜2 β

(4)

In Fig. 3 the impact of the substrate distance

hint

on the non-local dielectric function

is shown for MoS2 on top of hBN as obtained from Eq. (3). Figure 2:

Schematic respresentation of fre-

tor

quently encountered realizations of heterostructures accounting for an interlayer gap

hint

be-

(b)

In the long-wavelength limit (q

→ 0),

substrate and superstrate dielectric constants in agreement with the Keldysh potential.

ing. The eective non-local dielectric function and

q.

the eective screening is given by the average of

tween the active TMD layer and its surroundfor the supported and encapsulated cases

Dierent regimes of screening

can be identied depending on the wave vec-

(a)

21,30

For

small but nite momenta that are sensitive to the direct vicinity of the active TMD layer, the

is given by Eqs. (3) and (4), respec-

gap weakens the eective substrate screening

tively.

and causes a pronounced dip below the long-

ε˜i =

εi+1 −εi and εi+1 +εi

wavelength value that is absent for

hint = 0.

α = ε1 =

For large momenta

that

responds to charges being very close to each

for a vanishing gap at the interface Eq. (3) re-

other in the TMD layer. The discrepancy be-

produces the result of.

For the symmetric

tween the cases with and without gap become

case of a TMD layer sandwiched between sub-

particularly relevant in light of recent results

and superstrate with equal dielectric constants

that have investigated interlayer gaps in vdWH

εsub ,

that vary in dierent material classes and under

where we have dened

e−q hTMD , β = e−q 2hint with the parameters εsub , ε2 = 1, ε3 = εTMD (q), ε4 = 1. Note 16,19

q,

the eective dielectric

function approaches the bulk limit, as it cor-

the dielectric function is given by the sim-

annealing.

22,23

It further becomes clear that the

material-realistic eective dielectric function is


12

MoS2 / hBN

10 8 6 4 without interlayer gap with 5Å interlayer gap linearized

2 0 0

Figure

1

3:

2

3 4 q in 1/nm

Macroscopic

ε2D mac (q) for MoS2

5

dielectric

6

function

on hBN, comparing results for

an ideal plane (dashed line) interface and for a realistic interlayer gap of

hint = 5 Å (solid

line).

The dotted line represents the linear behavior of the dielectric function if the Coulomb interaction is approximated by a Keldysh potential.

(a)

400

MoS2 / hBN

300

X1s X2s X3s Egap

200 100 0

Exciton transition energy in eV

Dielectric function ε(q)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 15

Exciton binding energy EB in meV

Nano Letters

(b)

2.4 2.2 2 1.8 1.6 0

clearly beyond a linear description that is provided by a Keldysh potential (dotted line), and

(a)

2

4 6 8 Interlayer gap in Å

10

in using the latter one may strongly miscalcu-

Figure 4:

Impact of the interlayer gap on

late the impact of screening.

the binding energies of the exciton series in MoS2 on an hBN substrate.

Eect of Interlayer Distance on the TwoParticle Optical Properties and the Band Gap. The reduced screening in the presence

(b) Absolute exci-

tonic energies take into consideration the renormalized band gap, which is shown as a dotted line together with the energies of the 1s, 2s and 3s exciton transition. Further results for vari-

of non-vanishing interlayer distances in vdWH

ous combinations of TMDs and dielectric em-

signicantly modies the observable optical and

beddings are given in the Supporting Informa-

single-particle properties of vdWH. Before we

tion.

provide a direct assessment in terms of a theoryexperiment comparison of the trion binding energy in the following section, we rst take a

tion

look at the impact on the binding energies of

32

and read in Fourier space

¯ ω)ψkhe (ω) (εek + εhk − h 1 X X eh0 he0 h0 e0 V 0 0 ψ 0 (ω) − A k0 h0 e0 k,k ,k,k k

the bound-state exciton series. Exciton states emerge as solutions of the semiconductor Bloch equations (SBE) for the microscopic interband

h e 19,31 he polarisations ψk = ak ak . In the limit of vanishing excitation density, the SBE become

(5)

∗ = (dhe k ) E(ω) .

formally equivalent to the Bethe-Salpeter equa-

The linear response of the material is given by the macroscopic susceptibility χ(ω) = P P 1 he he k he dk ψk + c.c. /E(ω), which conA


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Nano Letters

tains excitons as discrete resonances below a

theory

continuum of optical interband transitions. The screened Coulomb matrix elements them exciton binding energies

EB

V

Z

and with

1/a

∆EB ≈

depend di-

q∆V (q) dq ,

(7)

0

rectly on the dielectric function via the WFCE sensitive to modications caused by variations

e2 ∆ε−1 (q) is the dierence 4πε0 q between the screened Coulomb potential with

in the interlayer gaps in vdWH discussed in the

and without the interlayer gap.

where

approach discussed above. Therefore, they are

context of Fig. 3.

For

We solve Eq. (5) by direct

∆V (q) =

the

frequently

used

case

of

encapsu-

diagonalization using, in addition to properly

lated TMDs with a dielectic function given by

screened Coulomb matrix elements, material-

In Fig. 4(a) the variation of the binding en-

Eq. (4) an analytic expression can be derived in −1 case of small interlayer gaps where ∆ε (q) ≈ −1 ∂ε (q) hint . Assuming for simplicity that the di∂hint electric response of the TMD layer itself εTMD is

ergy of the 1s to 3s exciton resonances with

momentum-independent and given by its bulk

interlayer gap is shown for the structure con-

value, we obtain as a result:

realistic input for band structures of the TMD slab as explained in detail in Ref. 33.

sidered in Figs. 1(b) and 3. An increasing in-

4(ε2 − 1) e2 hint 2 sub + − × ∆EB ≈ 4πε0 hTMD ε ε hTMD − + − × 1−ε Λ + ln[(ε + ε )Λ] , a

terlayer distance weakens the screening of the Coulomb interaction, which leads to a stronger electron-hole attraction and an increase of the binding energy. A comparison of 1s to 3s exciton binding en-

(8)

ε± = εsub ± εTMD and Λ = [ε− + ε exp (hTMD /a)]−1 . Taking advantage of the

with +

ergies reveals that more tightly bound excitons are more susceptible to this eect. An understanding of this can be obtained by a series of

fact that the thickness of the TMD layer is

approximations that is derived along the lines

small compared to the exciton Bohr radius

of Ref. 34. The central idea is to calculate an

(hTMD /a

eective dielectric constant that is obtained by

simple expression that is valid if the substrate 2 screening is suciently strong (εsub 1):

averaging

ε2D mac (q) over |q| up to 1/a, with a be-

1)

Eq. (8) reduces to a remarkably

ing the exciton Bohr radius:

ε = 2a2

Z 0

Without the

1/a

∆EB ≈ q ε2D mac (q) dq .

q -dependence

of

ε,

(6)

e2 hint . 4πε0 a2

(9)

For the asymmetric case of supported TMDs (cf. Eq. (4)) we nally obtain the same re-

the exciton

sult diering only by a factor of two which re-

problem can be solved analytically by means

ects the missing screening of the capping layer.

of the model of a 2D hydrogen atom. In this case the Bohr radius a = h ¯ 2 ε/(2e2 µex ) is pro-

From Eq. (9) it becomes obvious that the exciton binding energy increases in the presence of

portional to the dielectric constant and denes,

an interlayer gap

together with Eq. (6), a self-consistency prob-

2D lem. Assuming that εmac (q) depends linearly on

hint .

Eq. (9) reveals further

that excitons with larger Bohr radius

a,

such

as 2s and 3s excitons, are less aected and the 2 binding energy follows a characteristic 1/a de-

|q| as in the case of a Keldysh potential, Eq. (6) can be solved analytically. The exciton binding

pendence that we also obtain in the full calcula-

energy is obtained by using the corresponding hydrogenic binding energy EB = h ¯ 2 /(2a2 µex ).

tion. It becomes clear that Coulomb eects are easily underestimated if material-realistic inter-

In the presence of an interlayer gap the change

layer gaps between 3-8Å are treated as ideal

of the binding energy can be obtained in the

plane boundaries.

same spirit by means of rst-order perturbation

The absolute energy of the optical response


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Page 8 of 15

fkλ

of vdWH requires knowledge of the band gap

tions with

in addition to the bound-state binding ener-

corresponding states. In the last step a full v valence band (f = 1) and empty conduction c band (f = 0) have been been considered. An

gies.

In the SBE (5) the exciton binding en-

ergy is aected by environmental screening via eh0 he0 the Coulomb matrix elements Vk,k0 ,k,k0 , while the corresponding renormalization of the single-

being electron occupancies of the

evaluation of Eq. (11) is easily performed, and the numerical results for band-gap energies as a

particle band gap has to be considered sep-

function of the interlayer gap at the interface is

arately.

shown by the dotted line in Fig. 4(b). In combi-

The band structure of freestanding

G0 W0

calculations

nation, the impact of the screened Coulomb in-

become modied since the long-range Coulomb

teraction on the binding energies and the band

interaction

TMD slabs as obtained from

renormaliza-

gap leads to a signicant shift of the bound-

tions to the single-particle states experiences

state optical transitions already for slight vari-

the very same environmental screening.

causing

many-body

This

ations of the interlayer gaps in vdWH. To facili-

eect can be captured by a GdW self-energy,

tate a direct comparison for various experimen-

which was rst brought up by Rohlng

and

tal realizations, the interlayer-gap dependence

used to describe screening-induced band struc-

of the band gap and the exciton binding en-

ture renormalisation in vdWH.

ergies for further TMD/substrate combinations

36

35

The idea is to

approximately split the self-energy

Σ

GW,Het

are provided in the Supporting Information.

of

the heterostructure into a part describing the

Theory/Experiment Comparison of Trion Binding Energies. The energetic separation

isolated TMD monolayer that is treated on a full ab-initio level, and a correction term containing environmental screening eects via a

between the neutral and charged excitons (tri-

continuum-electrostatics model:

ons), here referred to as the trion binding en-

ΣGW,Het ≈ G V Het

ergy

eect of interlayer separation on Coulomb inter-

= G V ML + G ∆ V = ΣGW,ML + ΣGdW

action. Signatures of tightly bound trion com-

(10) with

∆ V = V Het − V ML .

self-energy

leads

to

a

plexes are frequently observed in experimental spectra in the presence of moderate charge car-

Here, the GdW

correction

of

rier densities.

single-

ET

does

rectly reects the strength of the Coulomb in-

band-gap energy in the presence of dielectric

teraction and its screening, see Fig. 1(c). Fur-

screening in vdWH, we evaluate the correc-

thermore, it is more easily and with higher

tion in static approximation, which leads to the self-energy

As the dierence between the

not depend on the band gap and, therefore, di-

To obtain the change of the

screened-exchange-Coulomb-hole

3739

trion and exciton bound-state energy,

particle energies with respect to the monolayer band structure.

ET , is particularly well suited to study the

accuracy experimentally accessible in compar-

19

ison to other methods that involve determining

for conduction- (c) and valence-band (v ) states

the separation between higher excited excitonic states,

,c ,v ∆ EGap,k = ΣGdW − ΣGdW k k X 1 c cccc − f k0 = ∆ Vkk0 kk0 2 0 k X 1 vvvv v − ∆ Vkk0 kk0 − f k0 2 0 k 1X cccc vvvv = (∆ Vkk 0 kk0 + ∆ Vkk0 kk0 ) . 2 k0

6

or

combining

optical

measurements

with single-particle measurements of the band gap.

8,40

In the following, we combine measure-

ments of the trion binding energy with a solution of a generalized three-particle Schrödinger equation over the full BZ. In combination with the electrostatic approach presented in the previous section, our model predicts trion binding energies with sucient accuracy to extract layer separations in agreement with experimental re-

(11)

sults. Here, we assume band-diagonal renormaliza-

To access the trion, the SBE (5) are aug-


Page 9 of 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters in Eq. (12) is linked to the optical response of an − electron trion X that describes the correlated

mented by higher-order expectation values of the kind

t− e1 e2 h3 e4 (k1 , k2 , Q)

=

3 ae2 ae1 aeQ4 † ah−(k 1 +k2 −Q) k2 k1

process of annihilating two electrons and one

,

hole, leaving behind an electron with momentum

Q

in the conduction band. Corresponding + + expressions for t (hole trion X ) can be ob-

(12) which are four-operator trion amplitudes. Via

tained by utilizing the electron-hole symmetry.

these three-particle expectation values, excited

The trion amplitudes obey their own equation

carriers are included beyond the scope of simple

of motion, which we derive in rst order in the

occupation factors and give rise to the positive

carrier populations and in linear response:

41

and negative trions. The particular one shown

¯ ω − iΓ)t− (εek11 + εek22 + εhk33 − εeQ4 − h e1 e2 h3 e4 (k1 , k2 , Q) 1 X X e2 h5 h3 e6 − t− (k1 , k2 − q, Q) V A q h ,e k2 ,k3 −q,k3 ,k2 −q e1 e6 h5 e4 5 6 X X 1 h5 h3 e6 Vke11,k t− (k1 − q, k2 , Q) − 3 −q,k3 ,k1 −q e6 e2 h5 e4 A q h ,e 5 6 1 X X e1 e2 e5 e6 Vk1 ,k2 ,k2 +q,k1 −q t− + e6 e5 h3 e4 (k1 − q, k2 + q, Q) A q e ,e 5 6 e1 he = fQ dhe k2 δk1 ,Q δe,e1 − dk1 δk2 ,Q E(ω) .

The homogeneous part of these equations is

gies.

4248

(13)

Especially the deviation from the lin-

a generalization of a three-particle Schrödinger

ear behavior of the dielectric function displayed

equation in reciprocal space for arbitrary band λ structures εk and Coulomb matrix elements Vkλ1,kλ2 λk3 λ,k4 (q). The three-body problem deter-

in Fig. 3 clearly speaks against casting the

1

2 3

Coulomb interaction into the shape of a simple Keldysh potential.

4

mined by the SBE augmented by the trion am-

To

support

our

results

on

the

sensitivity

plitudes (12), together with Eq. (13), is solved

of Coulomb screening on the interlayer gap

by matrix inversion from which we obtain the

and to further demonstrate the accuracy of

linear absorption of the material by calculat-

the trion binding energies obtained from our

ing the macroscopic susceptibility

χ(ω) as a re-

semiconductor model,

we present joint the-

sponse to the electric eld propagating vertical

ory/experimental results for trion binding ener-

to the heterostructure plane.

gies for various TMD/substrate combinations.

The optical response obtained from this ap-

The samples have been prepared by iteratively

proach contains both the bound-state trion and

stacking hBN and TMD akes by viscoelastic

exciton resonances, and the trion binding en-

stamping onto a SiO2 /Si substrate. The thick-

ergy is easily obtained from their energetic sep-

ness of hBN akes used for stacking is typi-

aration.

We point out that our method is a

cally of the order of 10-50 nm. For the anneal-

material-realistic description on the full band structure and beyond both an eective mass

ing step, samples are kept in a N2 -atmosphere ◦ at 50 mbar while being annealed at 300 C

approximation and a Keldysh potential for the

for 30 minutes. In general, encapsulation with

Coulomb interaction that have been used in

hBN and subsequent annealing results in al-

earlier works to calculate trion binding ener-

most lifetime-limited excitonic linewidths, with


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 15

Figure 5: Trion binding energies determined from experiment for dierent vdWH before (closed symbols) and after (open symbols) annealing are shown together with theoretical results that have been obtained for corresponding structures and accounting for dierent sizes of interlayer gaps.

photostable photoluminescence for MoS2 allow-

ing potential intercalated molecules, it has been

ing to extract trion binding energies from low-

shown that the interlayer separation is typically

temperature (10 K) photoluminescence spec-

reduced by several Å in the process of anneal-

tra.

ing. We observe a clear indications for a reduc-

4951

By performing spatially resolved low-

temperature (10 K)

µ-PL

measurements and

tion of the binding energy of the electron trion

statistically analyzing emission spectra in dif-

after annealing (open circles), demonstrating

ferent dielectric environments before and after

that interlayer separation plays a noticable role

annealing, we obtain trion binding energies. In

for the Coulomb interaction strength. In fact,

all our measurements, we used continuous wave

a microscopic calculation of the trion bind-

excitation at 2.33 eV with an excitation power −2 density of 0.3 kWcm .

ing energy without an interlayer gap between

Experimental results are shown as circles in

strongly overestimates the experimentally ob-

the TMD and the sub-/superstrate (squares)

Fig. 5 for intrinsically n-doped MoS2 and MoSe2

served binding energy reduction.

as a function of the long-wavelength limit of

for an interlayer gap within our electrostatic

the dielectric screening induced by the dielec-

model of the dielectric screening using Eqs. (3)

tric embedding. Statistical errors are typically

and (4) we obtain quantitative agreement with

below 1 meV reecting high sample uniformity.

the experimentally determined binding ener-

As expected for both TMDs a reduction of the

gies if a gap size of 3-5 Å is assumed (trian-

trion binding energy is observed if the screening

gles, diamonds).

strength is successively increased by changing

recent cross-sectional STEM measurements re-

the substrate from SiO2 to hBN and, further,

porting an hBN-TMD interlayer distance of

by encapsulating the TMD in hBN. Annealing

5-7 Å

has been demonstrated before to be a crucial

cally at hBN layer and the TMD metal atom.

step in the fabrication of vdWH.

A meaningful estimate for the interlayer gap

23,52

By remov-

This is in accordance with

which is measured between the atomi-


Accounting

Page 11 of 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

is therefore obtained from the center-to-center

estimating the reduction of excitonic binding

interlayer distance by subtracting the metalchalcogen vertical separation, which is of the

energies at the interfaces and that reveals a 1/a2 scaling behavior with the Bohr radius a

order of 1.6 Å.

of bound-state resonances.

42

Deviations are observed for

Our results may

MoS2 on SiO2 , where the experimental binding

help explaining the variation in reported trion

energy is larger than the theoretical prediction.

binding energies in the past

However, it has been argued

that water might

the importance of accounting not only for layer

be present on hydrophilic oxide surfaces. This

thicknesses, but also for realistic conditions at

leads to an additional layer of ice under cryo-

the interfaces in the strongly evolving eld of

genic condition with a dielectric constant below

vdWH.

53

The resulting binding energy

might be compared with the theoretical result recent measurements on CVD grown MoS2 on

43,44

Deutsche

Forschungsgemein-

the

graduate

school

the

TUM

International

Graduate

We gratefully acknowledge nancial support

based on

of the German Excellence Initiative via the

an eective-mass model for the band structure

Nanosystems Initiative Munich and the PhD

and a Keldysh potential shows that previously

program ExQM of the Elite Network of Bavaria.

reported binding energies are underestimated

The authors declare no competing nancial

by several meV (32.0-33.8 meV and 27.7-28.4

interest.

meV for free-standing MoS2 and MoSe2 , respectively). Calculated binding energies of both the

Supporting Information

positively and the negatively charged trions in

is available online

and provides additional results for positively

molybdenum- and tungsten-based TMDs and

and negatively charged trions, additional mate-

their dependence on the dielectric screening and

rial and substrate combinations, band-gap and

the interlayer gap is provided in the Supporting

1s and 2s exciton energy renormalizations as

Information.

Conclusion.

the

via

School of Science and Engineering (IGSSE).

Finally, relating our re-

sults to Monte-Carlo calculations

by

schaft

through

SiO2 substrates, where trion binding energies of

51

ported

Quantum3 Mechanical Materials Modelling (QM ) and

for a freestanding sample. This is supported by

35 meV are reported.

and underlines

Acknowledgement This work has been sup-

2 increasing the distance between TMD and substrate layer.

54

a function of the substrate dielectric constant, and details on the experimental and theoretical

We have investigated the im-

methods.

pact of the distance between individual adja-

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Page 14 of 15

Page 15 of 15

TMD Gap

hBN

MoS2

X-

X

increasing gap

Air

Photolumiescence

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters


The Dielectric Impact of Layer Distances on Exciton and Trion Binding

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