science.sciencemag.org/content/366/6467/870/suppl/DC1

Supplementary Materials for

Electrical control of interlayer exciton dynamics in atomically thin

heterostructures

Luis A. Jauregui, Andrew Y. Joe, Kateryna Pistunova, Dominik S. Wild, Alexander A.

High, You Zhou, Giovanni Scuri, Kristiaan De Greve, Andrey Sushko, Che-Hang Yu,

Takashi Taniguchi, Kenji Watanabe, Daniel J. Needleman, Mikhail D. Lukin, Hongkun

Park, Philip Kim*

*Corresponding author. Email: pkim@physics.harvard.edu

Published 15 November 2019, Science 366, 870 (2019)

DOI: 10.1126/science.aaw4194

This PDF file includes:

Materials and Methods

Supplementary Text

Figs. S1 to S18

References

2

Materials and Methods

Materials

Figure S1 provides a schematic for our multi-terminal and multi-gate device structure. Red and

green gates correspond to contact top-gates for WSe2 and MoSe2 respectively; blue (red) gate

electrodes are the heterostructure top (bottom) gates. Light-blue layers correspond to h-BN. The

transition metal dichalcogenide (TMD) layers are depicted as atomic layers. Electrical contacts for

MoSe2 and WSe2, shown as yellow electrode, are prepatterned 20 nm thick Pt contacts. Thin layers

of h-BN and monolayers of MoSe2 and WSe2 were prepared by mechanical exfoliation from bulk

crystals onto Si substrate coated with 285 nm SiO2. The thickness of h-BN layers was measured

by an atomic force microscope. Monolayers of MoSe2 and WSe2 were identified by optical contrast

and verified by photoluminescence measurements. The bottom h-BN layer was first transferred by

a dry transfer method onto the Cr (1 nm)/AuPd (9 nm) bottom gate defined by electron beam

lithography. Afterwards, electrodes were deposited on top of bottom h-BN by electron-beam

evaporation of Cr (1 nm)/Pt (19 nm), with the pattern defined by electrode beam lithography. The

h-BN/MoSe2/WSe2 heterostructure was then assembled and stamped on top of pre-patterned

electrodes on bottom h-BN. During the transfer, MoSe2 and WSe2 were aligned by their

crystallographic axis that were identified by second harmonic generation. Next, the Cr (1 nm)/Pd

(9 nm) top gate and contact gates were deposited on top of the heterostructure by thermal

evaporation. Finally, electrical contacts were deposited by electron-beam evaporation of Cr (5

nm)/Au (120 nm) to connect thin pre-patterned electrodes and gates to the wire-bonding pads.

Methods

Optical measurements were carried out in a confocal microscope using a 100x objective with a

numerical aperture of 0.75 in a 4K cryostat (Montana Instruments). Two sets of galvo mirrors

were used to scan excitation and collection spots independently on the sample.

Spectral Photoluminescence measurements

For Figs. 1, A and E we measured the respective spectra using a continuous wave 660 nm diode

laser as excitation with power = 6 µW at T = 4 K.

Absorption measurements

For Fig. 1B, we used a supercontinuum laser from NKT, integrating for at least 10 seconds for

each of the spectra to make sure to not saturate the spectrometer. For Fig. 1F, we use a white light

lamp source and integrate 4 times for 5 seconds. We normalize the spectra outside the

heterostructure, but in an area where there are similar conditions.

Diffusion measurements

For Figs. 2B-E, we used a continuous wave excitation from a 660 nm diode laser with the powers

noted in each figure. The PL maps were generated by scanning the collection galvo mirrors and

collecting with a single photon counting module (Excelitas Technologies). All the gates were at

zero to generate interlayer excitons.

Trion diffusion measurements

3

For Figs. 3A-C we used a continuous wave excitation from a 660 nm diode laser and power =

500 µW. Contact gates were employed at +13 V for MoSe2 and -13 V for WSe2, while the top-

gate and bottom-gate is marked in each figure. For the electrical measurements a sourcemeter

Keithley 2400 was used.

Time-resolved PL measurements

Time-resolved photoluminescence measurements in Fig. 1C were performed with a mode-locked

Ti:Sapphire laser (Mira, Coherent, repetition rate 80 Mhz, 6 ps pulse width at half maximum, 20

nW average power) coupled to a pulse-picker (ConOptics) to reduce the repetition rate to 400 kHz.

Time-resolved photoluminescence measurements in Fig. 1G (measured in a separate cooldown of

the sample) were performed with a supercontinuum laser from NKT (1 W average power, 2 Mhz

with pulse-picker module) coupled to an NKT provided tunable band-pass filter around 660 nm.

Time-resolved photoluminescence measurements in Figs. 2F-J were performed by pulsing a diode

laser (excitation wavelength 660 nm, repetition rate 100 kHz, peak power is labeled). All excitation

sources were synchronized to a single photon counting module (Excelitas Technologies) and a

picosecond event timer (Picoharp 300, PicoQuant).

Electroluminescence measurements

Electrical measurements in Fig. 4A were performed in a Janis VTI cryostat with all voltages (drain-

source and gates) supplied by a Keithley 2400 sourcemeter. Electroluminescence measurements

were performed in the above-mentioned optical setup by driving voltage supplied by a Keithley

2400 sourcemeter. For the electroluminescence lifetime measurements, sawtooth wave electrical

pulses (0–5 V, 200 μs pulses) generated by a wavefunction generator were applied to the device.

Supplementary Text

Section 1. Electric field in the TMD

In a parallel plate capacitor with a single slab of material in between, the electric field can be

defined to be the ratio between the voltage difference and the distance between the parallel plates.

When there is a thin sheet of material with a different dielectric constant as in Figure S3a, the

electric field within this sheet will be effectively smaller. Let us define the top and bottom h-BN

thicknesses to be ttotal/2 and the thickness of the TMD to be 𝑑, where 𝑑 ≪ 𝑡𝑡𝑜𝑡𝑎𝑙 . We can start by

drawing a Gaussian pillbox (Figure S3b), where the displacement field into and out of the box will

be equal (there are no free charges):

𝜀ℎ−𝐵𝑁 𝐸𝐵𝑁 = 𝐷𝑖𝑛 = 𝐷𝑜𝑢𝑡 = 𝜀𝑇𝑀𝐷 𝐸ℎ𝑠 (1)

𝐸𝐵𝑁 =𝜀𝑇𝑀𝐷

𝜀ℎ−𝐵𝑁 𝐸ℎ𝑠 (2)

where 𝐸𝐵𝑁 and 𝐸𝐻𝑆 are the electric fields in the h-BN and TMD layer respectively. Now if we

integrate the electric field along a straight line from the top to the bottom plate:

𝑉𝑏𝑔 − 𝑉𝑡𝑔 = −∫ 𝑬𝑭𝒊𝒆𝒍𝒅 ∙ 𝒅𝒍𝑏𝑜𝑡

𝑡𝑜𝑝= −(𝐸𝐵𝑁 ∙ (𝑡𝑡𝑜𝑝 + 𝑡𝑏𝑜𝑡) + 𝐸ℎ𝑠 ∙ 𝑑) (3)

4

We can use Eq. S2 and 𝑡𝑡𝑜𝑡𝑎𝑙 = 𝑡𝑡𝑜𝑝 + 𝑡𝑏𝑜𝑡 to get an expression for 𝐸ℎ𝑠:

𝑉𝑏𝑔 − 𝑉𝑡𝑔 = −((𝜀𝑇𝑀𝐷

𝜀ℎ−𝐵𝑁 𝐸ℎ𝑠) ∙ 𝑡𝑡𝑜𝑡𝑎𝑙 + 𝐸ℎ𝑠 ∙ 𝑑) (4)

𝐸ℎ𝑠 = (𝑉𝑡𝑔 − 𝑉𝑏𝑔)/(𝜀𝑇𝑀𝐷

𝜀ℎ−𝐵𝑁 ∙ 𝑡𝑡𝑜𝑡𝑎𝑙 + 𝑑) =

𝑉𝑡𝑔−𝑉𝑏𝑔

𝑡𝑡𝑜𝑡𝑎𝑙/ (

𝜀𝑇𝑀𝐷

𝜀ℎ−𝐵𝑁 +

𝑑

𝑡𝑡𝑜𝑡𝑎𝑙 ) (5)

Because of our initial assumption, 𝑑 ≪ 𝑡𝑡𝑜𝑡𝑎𝑙 , to first order, we can ignore the second term in the

denominator. Therefore, we can define the electric field felt by the interlayer excitons (𝐸ℎ𝑠) to be:

𝐸ℎ𝑠 =𝑉𝑡𝑔−𝑉𝑏𝑔

𝑡𝑡𝑜𝑡𝑎𝑙∙ (𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷) (6)

where 𝑉𝑡𝑔 is the top gate voltage, 𝑉𝑏𝑔 is the bottom gate voltage, 𝜀ℎ−𝐵𝑁 = 3.9 and 𝜀𝑇𝑀𝐷 = 7.2

(based on reference (38)) are the dielectric constants for h-BN and TMD, and 𝑡𝑡𝑜𝑡𝑎𝑙 = 184 nm is

the total h-BN thickness. This allows us to calculate 𝐸ℎ𝑠and extract the electron hole separation

(𝑑) from the linear Stark shift:

∆𝐸𝑛𝑒𝑟𝑔𝑦 = 𝒑 ∙ 𝑬𝒉𝒔 = 𝑒 𝑑 𝐸ℎ𝑠 (7)

where �⃑� is the dipole moment and 𝑒 is the electron charge.

Along with the electric field, there will be electrostatic doping due to the gate voltages. We define

the carrier density doping due to the two gates in the intrinsic regime to be:

𝑛 =𝐶𝑡𝑔 𝑉𝑡𝑔

𝑒+𝐶𝑏𝑔 𝑉𝑏𝑔

𝑒− 𝑛0 (8)

where 𝐶𝑡𝑔 = 𝜀ℎ−𝐵𝑁/𝑡𝑡𝑜𝑝 and 𝐶𝑏𝑔 = 𝜀ℎ−𝐵𝑁/𝑡𝑏𝑜𝑡 are the capacitances per unit area for the top and

bottom gates, 𝑒 is electron charge, and 𝑛0 is the density of in-gap states needed to be filled before

filling the conduction band. In order to avoid electrostatic doping of the TMDs, we use

𝑉𝑡𝑔 = −𝛼𝑉𝑏𝑔 (9)

with = 𝑡𝑡𝑜𝑝/𝑡𝑏𝑜𝑡𝑡𝑜𝑚 = 0.614, where 𝑡𝑡𝑜𝑝 = 70 nm and 𝑡𝑏𝑜𝑡𝑡𝑜𝑚 = 114 nm are the top and

bottom h-BN thicknesses respectively. When Eq. S9 is inserted into Eq. S8, we get no change in

doping due to the electrostatic gates. This gate configuration is defined to be only changing the

electric field without changing the doping level in the TMD layers and is confirmed from our

absorption measurements as shown in Fig. 1B.

Section 2. Second device with similar analysis

We observe similar sample illumination and tunable lifetime behavior in an additional sample.

Figure S4a shows an optical image of the second sample and Figure S4b shows the electric field

5

dependence. Note that the WSe2 and MoSe2 layers are inverted with respect to the first device

(inset of Figure S4b). The resulting slope with 𝑑 = 0.596 nm is the same magnitude, but opposite

in sign, indicating the same interlayer electron-hole separation with a downward electric dipole

direction. We perform PL diffusion measurements (Figure S4c), similar to Figs. 2B-D, where see

interlayer exciton emission throughout the sample with P = 1.75 mW excitation. Finally, we

measure the lifetime of the interlayer excitons and show qualitatively similar behavior as a function

of doping (Figure S4d). We note that the measured lifetime of 14 s in this sample at Vbg = 0 V is

an unprecedented several orders of magnitude longer than the main device.

Section 3. Lifetime under electric field

The lifetime of the interlayer excitons has a strong dependence on the overlap of the electron and

hole wavefunctions. The wavefunctions are expected to stretch due to an external electric field,

which can be expected to change the tails of the wavefunctions. We note that even a slight change

of the tails of the electron and hole wavefunctions is sufficient to significantly affect the IE lifetime

owing to the strong localization of the carriers to separate layers. This explains why the IE dipole

moment remains constant over the entire range of electric fields, as evidenced by the linear Stark

shift, despite the large change in the IE lifetime. Our argument further suggests that a quantitative

description of this effect requires an accurate solution of the Bethe-Salpeter equation, which may

be the topic of future theoretical studies.

Section 4. Quantum efficiency

We extract the quantum efficiency of the interlayer excitons. Based on the definition of quantum

efficiency (𝜂) and the total decay rate (1/𝜏𝑡𝑜𝑡𝑎𝑙 = 𝛾𝑡𝑜𝑡𝑎𝑙 = 𝛾𝑟𝑎𝑑 + 𝛾𝑛𝑜𝑛−𝑟𝑎𝑑):

𝜂 =𝛾𝑟𝑎𝑑

𝛾𝑡𝑜𝑡𝑎𝑙= 1 −

𝛾𝑛𝑜𝑛−𝑟𝑎𝑑

𝛾𝑡𝑜𝑡𝑎𝑙= 1 − 𝛾𝑛𝑜𝑛−𝑟𝑎𝑑 𝜏𝑡𝑜𝑡𝑎𝑙 (10)

where 𝛾𝑛𝑜𝑛−𝑟𝑎𝑑 and 𝛾𝑟𝑎𝑑 are the non-radiative and radiative decay rates, respectively. The

quantum efficiency is proportional to the photoluminescence (PL) intensity (𝐼𝑃𝐿) and so we can

replace Eq. S10 with 𝐼𝑃𝐿 to get

𝐼𝑃𝐿 = 𝐴(1 − 𝛾𝑛𝑜𝑛−𝑟𝑎𝑑𝜏𝑡𝑜𝑡𝑎𝑙) (11)

where A is an arbitrary scaling factor and 𝛾𝑛𝑜𝑛−𝑟𝑎𝑑 is the non-radiative decay rate. Now when

applying an electric field, we can tune both the 𝐼𝑃𝐿 and the 𝜏𝑡𝑜𝑡𝑎𝑙. If we plot intensity vs total

lifetime (Figure S5a), we find that the dependence is linear and so the 𝛾𝑛𝑜𝑛−𝑟𝑎𝑑 can be extracted

using Eq. S11 by extrapolating the fit to the limit of 𝐼𝑃𝐿 = 0. We find a non-radiative lifetime of

577 ± 12 ns. Using the non-radiative decay rate and Eq. S10, we extract both the electric field

dependent radiative lifetime (Figure S5b), ranging from 100 ns to 4 s, as well as the electric field

dependent quantum efficiency (Figure S5b inset). We find that for negative electric fields, our

radiative lifetime is shorter than the non-radiative lifetime, allowing us to achieve large quantum

efficiencies up to ~80 %. For large positive electric fields, our total lifetime plateaus near the non-

radiative lifetime (Fig. 1C) and results in a much lower quantum efficiency.

Section 5. Electric field applied with Vtg and Vbg when using only single gates

6

-Theory for double gate electrostatics

When using a single gate for the heterostructure, because of the gate structure and screening from

either the MoSe2 or WSe2 layer, the applied electric field will differ depending on the doping level.

Here we will describe the electrostatics for our heterostructure (MoSe2 on top) while controlling

both gates. The same analysis can be considered with the inverted heterostructure (WSe2 on top).

Let’s begin with the following definitions (Figure S6): 𝐶𝑇𝑀𝐷 =𝜀𝑇𝑀𝐷

𝑑 is the capacitance per unit

area of the TMD layers, 𝐶𝑡𝑔 =𝜀ℎ−𝐵𝑁

𝑡𝑡𝑜𝑝 and 𝐶𝑏𝑔 =

𝜀ℎ−𝐵𝑁

𝑡𝑏𝑜𝑡 are the electrostatic capacitances per unit

area of the top and bottom gates, 𝐶𝑞,𝑡𝑔 and 𝐶𝑞,𝑏𝑔 are the quantum capacitances per unit area

associated with the top and bottom gates, 𝜇𝑚 and 𝜇𝑤 are the chemical potentials of the MoSe2 and

WSe2 layers, 𝜇𝑚0 and 𝜇𝑤

0 are the chemical potentials at which the MoSe2 and WSe2 layers begin to

be doped, 𝜎𝑚 and 𝜎𝑤 are the carrier densities in the MoSe2 and WSe2 layers, and 𝑑 is the interlayer

distance. From here we can set up a system of equations to solve for the chemical potentials:

{ 𝜎𝑚 = 𝜀𝑇𝑀𝐷 𝐸ℎ𝑠 − 𝜀ℎ−𝐵𝑁 𝐸𝑡𝑜𝑝 𝜎𝑤 = 𝜀ℎ−𝐵𝑁 𝐸𝑏𝑜𝑡 − 𝜀𝑇𝑀𝐷 𝐸ℎ𝑠

(12)

{

𝑡𝑡𝑜𝑝 𝐸𝑡𝑜𝑝 = 𝑉𝑡𝑔 − 𝜇𝑚 𝑡𝑏𝑜𝑡 𝐸𝑏𝑜𝑡 = 𝜇𝑤 − 𝑉𝑏𝑔 𝑑 𝐸ℎ𝑠 = 𝜇𝑚 − 𝜇𝑤

(13)

{ 𝜎𝑚 = −𝐶𝑞,𝑡𝑔(𝜇𝑚 − 𝜇𝑚

0 )

𝜎𝑤 = −𝐶𝑞,𝑏𝑔(𝜇𝑤 − 𝜇𝑤0 )

(14)

where Eqs. S12 are Gauss’s law for MoSe2 and WSe2 layers, Eqs. S13 are the electric field between

each layer, and Eqs. S14 describe the effects of quantum capacitance. When solving these

equations for 𝐸ℎ𝑠, we get the following expression:

𝐸ℎ𝑠 =𝜇𝑚−𝜇𝑤

𝑑

=1

𝑑 (𝐶𝑞,𝑏𝑔 + 𝐶𝑏𝑔)(𝐶𝑡𝑔 𝑉𝑡𝑔 + 𝐶𝑞,𝑡𝑔 𝜇𝑚

0 )−(𝐶𝑞,𝑡𝑔 + 𝐶𝑡𝑔)(𝐶𝑏𝑔 𝑉𝑏𝑔 + 𝐶𝑞,𝑏𝑔 𝜇𝑤0 )

(𝐶𝑞,𝑡𝑔 + 𝐶𝑡𝑔 + 𝐶𝑇𝑀𝐷)(𝐶𝑞,𝑏𝑔 + 𝐶𝑏𝑔+ 𝐶𝑇𝑀𝐷)−𝐶𝑇𝑀𝐷2 (15)

From plugging in the relevant numbers for the capacitances,

𝐶𝑇𝑀𝐷 = 𝜀𝑇𝑀𝐷

𝑑≅

7.2𝜀0

0.6𝑛𝑚 ~ 12 𝜀0 nm-1 (16)

𝐶𝑡𝑔/𝑏𝑔 = 𝜀ℎ−𝐵𝑁

𝑡𝑡𝑜𝑝/𝑏𝑜𝑡≅

3.9𝜀0

100𝑛𝑚 ~ 4 × 10−2 𝜀0 nm-1 (17)

where 𝜀ℎ−𝐵𝑁 = 3.9 and 𝜀𝑇𝑀𝐷 = 7.2 (38) are the dielectric constants for h-BN and TMD, we can

make the approximation that

7

𝐶𝑇𝑀𝐷 ≫ 𝐶𝑡𝑔/𝑏𝑔 (18)

The quantum capacitance will depend on whether we are in an intrinsic or doping regime:

𝐶𝑞 ≅ 0 (intrinsic) (19)

𝐶𝑞 = 𝑒2 𝑚∗

𝜋 ℏ2= 4

𝑚∗ 𝜀0

𝑚0 𝑎0 ~ 102 𝜀0 nm-1 (doped) (20)

where 𝑚∗ is the effective mass of the relevant band, 𝑚0 is the bare electron mass, and 𝑎0 is the

Bohr radius. The expression for the quantum capacitance includes the valley degeneracy, while

assuming that only a single a single spin component per valley is occupied. This is valid at low

doping, as in our experiments, where the Fermi energy is below (above) the second conduction

(valence) band, which is split by spin-orbit coupling. When either layer is doped, then we can also

make the approximation that

𝐶𝑞 ≫ 𝐶𝑡𝑔/𝑏𝑔. (21)

Using the approximations in Eq. S18, we can simplify the electric field expression from Eq. S15

to be

𝐸ℎ𝑠 ≅1

𝑑 (𝐶𝑞,𝑏𝑔 + 𝐶𝑏𝑔)(𝐶𝑡𝑔 𝑉𝑡𝑔 + 𝐶𝑞,𝑡𝑔 𝜇𝑚

0 )−(𝐶𝑞,𝑡𝑔 + 𝐶𝑡𝑔)(𝐶𝑏𝑔 𝑉𝑏𝑔 + 𝐶𝑞,𝑏𝑔 𝜇𝑤0 )

𝐶𝑇𝑀𝐷(𝐶𝑞,𝑡𝑔 + 𝐶𝑞,𝑏𝑔+ 𝐶𝑡𝑔 + 𝐶𝑏𝑔). (22)

In the intrinsic regime, where 𝜇𝑀 = 𝜇𝑊 = 0 and 𝐶𝑞,𝑏𝑔 ≅ 𝐶𝑞,𝑡𝑔 ≅ 0, we recover the electric field

from Section 2, Eq. S6. In the doped regime, our expression becomes more complicated. First, we

define the chemical potential to be

𝜇𝑚 ≅ 𝜇𝑤 ≅1

𝐶𝑏𝑔+𝐶𝑡𝑔(𝐶𝑏𝑔 𝑉𝑏𝑔 + 𝐶𝑡𝑔 𝑉𝑡𝑔 ) (23)

when the given layer is in the doped regime. Let us consider the case for n-doping. If we define

the gate voltages required to reach the conduction band edge when combined as 𝑉𝑡𝑔𝑒 and 𝑉𝑏𝑔

𝑒

(experimentally we can begin doping with an arbitrary combination of top and bottom gates), then

we define this chemical potential as

𝜇𝑚0 ≅

1

𝐶𝑏𝑔+𝐶𝑡𝑔(𝐶𝑏𝑔 𝑉𝑏𝑔

𝑒 + 𝐶𝑡𝑔 𝑉𝑡𝑔𝑒 ). (24)

The corresponding electric field at these gate voltages will be

𝐸𝐹𝑖𝑒𝑙𝑑𝑒 =

𝑉𝑡𝑔𝑒 −𝑉𝑏𝑔

𝑒

𝑡𝑡𝑜𝑡𝑎𝑙∙ (𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷). (25)

We know from our experimental results that for n-doping, only the MoSe2 layer becomes doped.

Thus, we can assume 𝜇𝑤0 ≅ 0 and 𝐶𝑞,𝑏𝑔 ≅ 0. Using this assumption and Eqs. S22 and S24, with

some algebra, we obtain the expression

8

𝐸ℎ𝑠 ≅ 𝐸𝐹𝑖𝑒𝑙𝑑𝑒 +

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷 1

𝑡𝑏𝑜𝑡(∆𝑉𝑏𝑔

𝑒 −𝐶𝑡𝑔

𝐶𝑞,𝑡𝑔∆𝑉𝑡𝑔

𝑒 +𝐶𝑡𝑔

𝐶𝑞,𝑡𝑔𝑉𝑡𝑔𝑒 ) (n-doped) (26)

where ∆𝑉𝑏𝑔𝑒 = 𝑉𝑏𝑔

𝑒 − 𝑉𝑏𝑔 and ∆𝑉𝑡𝑔𝑒 = 𝑉𝑡𝑔

𝑒 − 𝑉𝑡𝑔 are the voltages applied once the layer is doped.

A similar calculation can be done for the p-doping regime to get:

𝐸ℎ𝑠 ≅ 𝐸𝐹𝑖𝑒𝑙𝑑ℎ −

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷 1

𝑡𝑡𝑜𝑝(∆𝑉𝑡𝑔

ℎ −𝐶𝑏𝑔

𝐶𝑞,𝑏𝑔∆𝑉𝑏𝑔

ℎ +𝐶𝑏𝑔

𝐶𝑞,𝑏𝑔𝑉𝑏𝑔ℎ ) (p-doped) (27)

where 𝑒 → ℎ corresponds to discussing hole doping in the WSe2 valence band. Although we see

that the last term in both Eqs. S26 and S27 is a discontinuity in electric field from the intrinsic

regime to a doped regime, this term can be ignored because 𝐶𝑡𝑔

𝐶𝑞,𝑡𝑔𝑉𝑔𝑒 ~ 10−4V in our measurements,

which corresponds to an energy shift of ~0.3 eV, significantly smaller than our interlayer exciton

linewidth ~10 meV. Therefore, we can ignore this final term and any observable jumps in exciton

energy cannot be attributed to an abrupt change in electric field.

-Electrostatics for special cases

When considering special cases of Eqs. S6, S26, and S27, we can obtain expressions for the electric

field when sweeping a single gate. When sweeping the top gate only and fixing 𝑉𝑏𝑔 = 0:

𝐸ℎ𝑠 =

{

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷(𝑉𝑡𝑔ℎ

𝑡𝑡𝑜𝑡𝑎𝑙+∆𝑉𝑡𝑔

ℎ

𝑡𝑡𝑜𝑝) 𝑉𝑡𝑔 < 𝑉𝑡𝑔

ℎ

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷

𝑉𝑡𝑔

𝑡𝑡𝑜𝑡𝑎𝑙 𝑉𝑡𝑔

ℎ < 𝑉𝑡𝑔 < 𝑉𝑡𝑔𝑒

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷

𝑉𝑡𝑔𝑒

𝑡𝑡𝑜𝑡𝑎𝑙 𝑉𝑡𝑔

𝑒 < 𝑉𝑡𝑔

(28)

When sweeping the bottom gate only and fixing 𝑉𝑡𝑔 = 0:

𝐸ℎ𝑠 =

{

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷

𝑉𝑏𝑔ℎ

𝑡𝑡𝑜𝑡𝑎𝑙 𝑉𝑏𝑔 < 𝑉𝑏𝑔

ℎ

−𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷

𝑉𝑏𝑔

𝑡𝑡𝑜𝑡𝑎𝑙 𝑉𝑏𝑔

ℎ < 𝑉𝑏𝑔 < 𝑉𝑏𝑔𝑒

𝜀ℎ−𝐵𝑁

𝜀𝑇𝑀𝐷(𝑉𝑏𝑔𝑒

𝑡𝑡𝑜𝑡𝑎𝑙−∆𝑉𝑏𝑔

𝑒

𝑡𝑏𝑜𝑡) 𝑉𝑏𝑔

𝑒 < 𝑉𝑏𝑔

(29)

The 𝑉𝑡𝑔 (𝑉𝑏𝑔) sweeping configuration allows for n- (p-) doping without changing electric field in

the TMD for one doping direction. This allows us to explore the doping response of the interlayer

exciton peaks without the Stark effect. A visual representation of the electric fields for the two

special cases is shown in Figure S7.

-Interlayer exciton response for single gate sweeps

We collect single-gate-dependent PL from the heterostructure and observe the general trends agree

with Eqs. S28 and S29 (Figure S8a-b). For 𝑉𝑡𝑔 (𝑉𝑏𝑔), p- (n-) doping the heterostructure has the

largest Stark shift and the intrinsic regime has a smaller Stark shift. However, there are a few

9

differences that indicate a charged interlayer exciton has formed. For 𝑉𝑡𝑔 (𝑉𝑏𝑔), n- (p-) doping the

heterostructure does not result in a constant PL energy. Instead, we observe a slight redshift. It has

been well-accepted that trions in monolayer TMDs exhibit a redshift with increasing doping. In

addition, for 𝑉𝑡𝑔 (𝑉𝑏𝑔), the Stark shift for p- (n-) doping the heterostructure does not match the

expected electric field calculated from Eqs. S28 and S29. When the redshift slope from the 𝑉𝑏𝑔

(𝑉𝑡𝑔) p- (n-) doping is subtracted from the 𝑉𝑡𝑔 (𝑉𝑏𝑔) p- (n-) doping slope, we can match the

expected electric field, indicating that the redshift occurs for both doping regimes with either gate.

Finally, there are discontinuities in the interlayer exciton energy at the voltage required to begin

doping, 𝑉𝑔𝑒 and 𝑉𝑔

ℎ. From the electrostatics described above, we do not expect to see any

discontinuities in interlayer exciton energy. Instead, we attribute the difference in energy to the

binding energy of the charged interlayer exciton. The extracted binding energy of ~10 meV (~15

meV) for n- (p-) doping is the same for both 𝑉𝑡𝑔 (𝑉𝑏𝑔) sweeping configurations. In the main text,

we summarize these findings by including the combination of the 𝑉𝑡𝑔 (𝑉𝑏𝑔) sweeping

configurations (Figs. 1D-G) to show the binding energy and redshift as a function of carrier

density.

We also perform time dependent PL (TDPL) to measure the lifetime as a function of the single

gates (Figure S8c-d). We observe similar results for both gate dependences indicating that

electrostatic doping dominates the electric field in terms of controlling interlayer exciton lifetimes.

Section 6. Extracting interlayer exciton density

Interlayer exciton are permanent dipoles whose dipole-dipole repulsion increases the exciton

energy, resulting in a blueshift of PL energy. In the mean-field approximation, where the average

distance between the excitons is much larger than the separation between the layers, we use the

parallel plate capacitor model (39). Equal number of electrons in one TMD layer and holes in the

other layer will create potential difference ∆𝜙 =𝑛𝑒𝑑

𝜀𝑇𝑀𝐷𝜀0, where 𝑛 is the interlayer exciton density,

𝑒 is the electric charge, 𝑑 is the separation between the layers, 𝜀𝑇𝑀𝐷 is the TMD dielectric constant,

and 𝜀0 is vacuum permittivity. Adding one more interlayer exciton will increase the energy by

𝑒∆𝜙, resulting in a “plate capacitor formula”,

∆𝐸 =𝑛𝑒2𝑑

𝜀𝑇𝑀𝐷𝜀0 (31)

We can now extract interlayer exciton density:

𝑛 =∆𝐸𝜀𝑇𝑀𝐷𝜀0

𝑒2𝑑 (32)

Using 𝑑 = 0.6 nm, 𝜀𝑇𝑀𝐷 = 7.2 and ∆𝐸=10 meV at P = 1 mW we get an interlayer exciton density

𝑛 ≈ 5x1011 cm-2.

The parallel plate capacitor model is known to underestimate the interlayer exciton density because

the model does not account for a reduction in interaction energy due to the rearrangement of the

interlayer excitons. This reduction in interaction energy as the excitons avoid each other means a

10

higher density of excitons will be required to achieve the same energy shift (40). However, it

should be noted that the model also assumes a linear relation of energy shift and interlayer exciton

density, which may break down when the exciton separation approaches the layer separation. The

Mott critical density of interlayer excitons has been predicted to occur around ~ 4x1012 cm-2 based

on the critical ratio of the exciton size and inter-exciton distance of 0.3, taking into account valley

degeneracy (2). These considerations allow us to provide a lower and upper bound for the

interlayer exciton densities.

Section 7. Additional analysis of experimental results for exciton diffusion

Figure S9a-b show two representative diffusion maps at 10 µW and 1 mW when excited at the

center of the sample. The red arrow and circle represent the area averaged over for the diffusion

analysis for Fig. 2E, as described in the main text. To further demonstrate that we are observing

the transport of IEs instead of the tail of a non-gaussian excitation we look at the steady-state

spatial map of IE at different electric fields as depicted in Figure S9d. We used spatial maps, as

Figs. 2B-D, where we obtain a radial average PL intensity (IPL) with rs (the radius averaged over

the region depicted in Figure S9c, marked as red dashed lines and red arrow). We used P = 1 mW,

and we vary 𝐸ℎ𝑠. We observe that the IPL gets higher with increased 𝐸ℎ𝑠 (increased ) for rs > 2 m

away from the excitation spot., while at the excitation spot IPL reduces with increasing 𝐸𝐻𝑆 (as

shown in Figure 1A). The increase of IPL away from the excitation spot confirms that we are

increasing the density of IE at the excitation spot and therefore IE move away from larger density

areas. In contrast, if we would be observing the tail of a non-gaussian shaped laser excitation, the

IPL across the sample should be modulated proportionally with the IPL at the excitation spot.

Section 8. Theoretical description of interaction driven diffusion

-Exciton diffusion

We now derive a simple model for the diffusion of excitons that takes into account electrostatic

interactions. Excitons that are generated with a rate 𝑅(𝐫, 𝑡) per unit area and recombine with rate

𝛾 satisfy the continuity equation

𝝏𝒏(𝒓,𝒕)

𝝏𝒕+ 𝜵 ⋅ 𝑱(𝒓, 𝒕) = 𝑹(𝒓, 𝒕) − 𝜸𝒏(𝒓, 𝒕), (33)

where 𝑛(𝐫, 𝑡) is the exciton density, and 𝐉(𝐫, 𝑡) the exciton current density. We assume that the

excitons are locally in equilibrium such that

𝐉(𝐫, 𝑡) =𝜏

𝑚𝑛(𝐫, 𝑡)[−𝛁𝜇(r, 𝑡) + 𝑭(𝐫, 𝑡)], (34)

where 𝜏 is the momentum relaxation time, 𝑚 the exciton mass, 𝜇(𝐫, 𝑡) the local chemical potential,

and 𝑭(𝐫, 𝑡) a local force. The chemical potential is to be determined self-consistently to match the

local density. For a non-interacting Bose gas in two dimensions, we have

𝜇 = 𝑘𝐵𝑇 ln[1 − 𝑒−𝑛(𝐫,𝑡)/𝑛𝑐 ], (35)

where 𝑛𝑐 = 𝑚𝑘𝐵𝑇/(𝜋ℏ2), having accounted for valley degeneracy.

11

Interactions between excitons are taken into account via the force term. For a rigid exciton, the net

electric field is given by the difference of the field acting on the hole and the electron, i.e. E =Eℎ − E𝑒. If no in-plane external field is applied, the only contribution to the electric field is

generated by the excitons themselves. By approximating each exciton as a positive and a negative

point charge separated by the layer separation 𝑑, we can express the potential energy as

𝑼(𝒓, 𝒕) =𝒆𝟐

𝟐𝝅𝜺∫ 𝒅𝟐𝒓′ 𝒏(𝒓′, 𝒕) (

𝟏

|𝒓−𝒓′|−

𝟏

√|𝒓−𝒓′|𝟐+𝒅𝟐). (36)

Together with 𝑭 = −𝛁𝑈, Eqs. S33 – S36 form a set of nonlinear integro-differential equations

for the exciton density 𝑛(r, 𝑡). While these equations may be tractable numerically, it is desirable

to make further simplifications. In particular, we note that the exciton density is expected to vary

only weakly over the length scale 𝑑 ≈ 0.6 nm, which enables us to approximate the integral in

Eq. S35 by 𝑈(𝐫, 𝑡) ≈ 𝑒2𝑑𝑛(𝐫, t)/𝜀. This expression has the simple interpretation of the potential

inside a parallel plate capacitor, where the charge density is allowed to slowly vary in space. In

Section 6, the same expression was used to estimate the exciton density from the observed blue

shift of the PL resonance. We substitute this result back into Eq. S33 to obtain

𝑱(𝒓, 𝒕) = −𝑫 [𝒏(𝒓,𝒕)/𝒏𝒄

𝒆𝒏(𝒓,𝒕)/𝒏𝒄−𝟏+𝒏(𝒓,𝒕)

𝒏∗] 𝜵𝒏(𝒓, 𝒕), (37)

where 𝐷 = 𝑘𝑇𝜏/𝑚 and 𝑛∗ = 𝜀𝑘𝐵𝑇/(𝑒2𝑑). By substituting back into Eq. S34, we thus arrive at

the nonlinear diffusion equation

𝝏𝒏(𝒓,𝒕)

𝝏𝒕−𝑫𝜵 ⋅ [(

𝒏(𝒓,𝒕)/𝒏𝒄

𝒆𝒏(𝒓,𝒕)/𝒏𝒄−𝟏+𝒏(𝒓,𝒕)

𝒏∗)𝜵𝒏(𝒓, 𝒕)] = 𝑹(𝒓, 𝒕) − 𝜸𝒏(𝒓, 𝒕). (38)

In the limit of vanishing density, the above model reduces to conventional single-particle diffusion

with diffusion constant 𝐷. Single particle diffusion is suppressed for densities approaching 𝑛𝑐 as

the occupation of the zero momentum state is enhanced by the Bose occupation factor. At the same

time, repulsion between the excitons leads to increasing diffusion with increasing density. At

temperature 4 K, using 𝑑 = 0.6 nm, 𝜀 = 𝜀0, 𝑚 ≈ 𝑚𝑒, we have 𝑛∗ ≈ 3 × 109 cm−2 and 𝑛𝑐 ≈5 × 1011 cm−2.

We note that the above treatment ignores a number of important effects. We neglect short-range

correlations that arise due to interactions in the exciton fluid, which have been predicted to

significantly affect the interaction induced blue shift (32). We further ignore the role of dielectric

screening, which plays an important role in determining the exciton binding energy (41, 42). While

these effects may modify the value of 𝑛∗, we expect Eq. S37 to remain valid to a good

approximation. Finally, we have ignored the possibility of temperature gradients as well as

interactions between excitons and phonons, which have been proposed to play an important role

in monolayer TMDs (43).

We consider a Gaussian pump profile given by

𝑹(𝒓) = 𝜸𝒏∗𝑨𝒆−𝟐𝒓𝟐/𝒘𝟎

𝟐. (39)

12

The dimensionless constant 𝐴 quantifies the density of excitons in units of 𝑛∗ generated at the

center of the beam per exciton lifetime. By defining the diffusion length 𝑙2 = 𝐷/𝛾, 𝑛 = 𝑛/𝑛∗, and

𝜏 = 𝛾𝑡, Eq. S37 can be concisely written as

𝝏�̃�(𝒓,𝒕)

𝝏𝝉− 𝒍𝟐

𝟏

𝒓

𝝏

𝝏𝒓[𝒓 (

𝜶�̃�(𝒓,𝒕)

𝒆𝜶�̃�(𝒓,𝒕)−𝟏+ �̃� (𝒓, 𝒕))

𝝏�̃�(𝒓,𝒕)

𝝏𝒓] = 𝑨𝒆−𝟐𝒓

𝟐/𝒘𝟎𝟐− �̃� (𝒓, 𝒕), (40)

where 𝛼 = 𝑛∗/𝑛𝑐, and we assumed that the boundary and initial conditions are cylindrically

symmetric such that there is no angular dependence at any time. Based on our estimate above, we

set 𝛼 = 0.01 for all numerical simulations below. To explore the exciton dynamics quantitatively,

we define the width

𝒘(𝒕) = √⟨𝒓𝟐(𝒕)⟩ = [∫ 𝒅𝟐𝒓 |𝒓|𝟐𝒏(𝒓,𝒕)

∫ 𝒅𝟐𝒓 𝒏(𝒓,𝒕)]𝟏/𝟐

. (41)

The steady-state width as a function of the amplitude 𝐴 is shown for three different values of the

diffusion length 𝑙 in Figure S10. For 𝐴 ≪ 1, the diffusion term due to interaction is negligible and

𝛼�̃�(𝑟, 𝑡)/(𝑒𝛼�̃�(𝑟,𝑡) − 1) ≈ 1, such that the width is given by 𝑤 ≈ √𝑤02/2 + 4𝑙2. In this regime,

the steady-state density is given by 𝑛(𝑟) ∝ ∫ d2𝐫 𝑅(|𝐫 − 𝐫′|)𝐾0(𝒓′/𝑙), where 𝐾0 is the modified

Bessel function, which asymptotically approaches 𝑛 ∼ 𝑒−𝑟/𝑙/√𝑟/𝑙 when 𝑟, 𝑙 ≫ 𝑤0. The effect of

interaction driven diffusion becomes apparent when 𝐴 ∼ max(1,𝑤02/𝑙2). For 𝐴 much greater than

this value, the width scales as 𝑤 ∝ 𝐴1/4 until finite size effects start to matter. We can associate

the width with an effective diffusion constant 𝐷eff = (𝑤2/4 − 𝑤0

2/8)𝛾, which corresponds to the

diffusion constant in a linear model that would reproduce the same steady-state width. At high

power, in the absence of finite size effects, we expect 𝐷eff ∝ 𝐴1/2.

This behavior is in qualitative agreement with the experimental observation of increased diffusion

at high power. Nonlinear diffusion also provides a potential explanation for the fact that only little

diffusion is observed after the laser has been switched off: As the exciton population decays, the

effective diffusion constant decreases. This effect is evident in the plots of the exciton density as

a function of time in Figure S11, where the exciton generation rate is set to zero at 𝛾𝑡 = 10. For

𝛾𝑡 < 10, the amplitude 𝐴 was chosen such that the steady state width is 2.2 μm for all three values

of the diffusion length, corresponding to the circles in Figure S10. There is little diffusion at late

times when the steady-state width is dominated by the interaction term (𝑙 = 0.01 μm and 𝑙 =0.1 μm), whereas conventional single particle diffusion is significant for 𝑙 = 1 μm.

-Carrier diffusion

The above model can be extended to also include diffusion of free charge carriers. We neglect

interactions between free carriers and between carriers and exciton to arrive at the set of coupled

equations

13

𝝏𝒏𝒆

𝝏𝒕−𝑫𝒆𝜵

𝟐𝒏𝒆 = 𝑹 − (𝜞 + 𝜞′)𝒏𝒆𝒏𝒉𝝏𝒏𝒉

𝝏𝒕−𝑫𝒉𝜵

𝟐𝒏𝒉 = 𝑹 − (𝜞 + 𝜞′)𝒏𝒆𝒏𝒉𝝏𝒏𝒙

𝝏𝒕− 𝑫𝒙𝜵 ⋅ [(

𝒏𝒙/𝒏𝒄

𝒆𝒏𝒙/𝒏𝒄−𝟏+𝒏𝒙

𝒏∗)𝜵𝒏𝒙] = −𝜸𝒏𝒙 + 𝜞𝒏𝒆𝒏𝒉.

(42)

Here 𝑛𝑒, 𝑛ℎ, and 𝑛𝑥 denote the density of free electrons, free holes, and excitons, and 𝐷𝑒, 𝐷ℎ, and

𝐷𝑥 are their respective diffusion constants. Optical excitation at rate 𝑅 is assumed to generate free

carriers rather than excitons, which is appropriate if the excitation frequency is above gap. The

coefficients 𝛤 and 𝛤′ determine the free carrier recombination rate, which is proportional to both

the electron and hole density. We allow for two recombination pathways: forming an interlayer

exciton (coefficient 𝛤) or direct decay with no intermediate exciton state (coefficient 𝛤′). In the

limit of large generation rate 𝑅, one can eliminate the carrier degree of freedom, and the model

reduces to the one discussed above.

This model offers an alternative way to reproduce the reduced exciton diffusion observed in

experiment after the laser is switched off. If we assume that only free carriers diffuse while

excitons are localized (𝐷𝑥 = 0), the spatial distribution of excitons evolves over a time scale set

by the exciton recombination rate. This effect is illustrated in Figure S12, where we plot both the

free carrier and exciton density as a function of time and radius. We note that as a consequence of

the nonlinear recombination term, the numerical results are rather sensitive to the boundary

conditions used for the charge carriers. We further point out that without the nonlinear exciton

diffusion term, this model does not explain the increased diffusion observed at high powers in the

experiment. However, it is plausible that the carrier diffusion constants themselves increase with

power as a higher carrier density may screen disorder more effectively.

Additional work is required to unambiguously determine the parameters of these models. It is

indeed possible that both interaction driven exciton diffusion and carrier diffusion contribute to

the exciton dynamics, as well as other effects such as the Seebeck effect and coupling to phonons

(43). The models may be further refined to account for spatial inhomogeneity, the observed power

dependence of the exciton lifetime, the nonlinear increase of the PL intensity with power. The

latter two effects may be caused by Auger-type exciton decay processes, which have been shown

to play an important role in monolayer TMDs (44).

Section 9. Spatial control of charged interlayer excitons

To verify the spatial control of charged interlayer excitons (CIEs), we show spatial maps as a

function of Vds, normalized by dividing the spatial map at Vds = 0V as shown in Figs S14 (a-f) and

S15 (a-f). We used a continuous wave excitation with = 660 nm and power = 500 µW focused at

the center of the sample. We maintain the same carrier density for the different Vds by

compensating with either the Vbg or Vtg for the p-type or n-type CIE, respectively. We observe the

undoped regime shift in the PL spectra with applied Vds vs either Vtg or Vbg (Figures S13a-b) and

compensate in the CIE transport measurements. We choose a starting doping at Vds = 0V and thus,

apply 𝑉𝑏𝑔 = 2.125 𝑉𝑑𝑠 − 5𝑉 and 𝑉𝑡𝑔 = 1.6 𝑉𝑑𝑠 for the p-type and n-type CIE, respectively.

Figures S14 (a-c) show increasing Vds for p-type CIEs, increases the density of CIE excitons at the

right edge of the sample near where the sample is electrically grounded. In Figure S14d-f, we

14

change the direction of the applied bias voltage and show the same analysis of the spatial maps for

increasing Vds. In order to compare the different data sets, we plotted the averaged vertical linecuts

(IPL(Vds)/IPL(Vds = 0V)) vs. X, as shown in Figs. S14g and S14h. These average linecuts are

obtained from the spatial maps of Figs. S14 (a-c) and Figs. S14 (d-f) respectively. When looking

at the averaged linecuts in Figure S14g (S14h), we observe a finite slope in the IPL(Vds)/IPL(Vds =

0V) vs. X of the CIEs, with a higher intensity towards the right (left) side of the sample, the

expected direction of transport. In Figure S15a-f, we performed similar experiments but with n-

type CIEs. We find that the n-type CIEs move in opposite direction to the applied in-plane field as

the p-type CIEs. Figures S15g-h show a similar linecut analysis of the PL maps as performed with

the p-type CIEs, which verifies the spatial control of the CIEs. Our experiments demonstrate that

both negative and positive CIEs can be moved with an in-plane electric field. We see a stronger

effect with an enhancement of the recombination on the edge when the CIEs are moved to the right

of the sample, which could be an indication of the sample inhomogeneity, which can inhibit the

transport of CIE in the left direction. We exclude the effect being due to a contact issue because

the enhancement occurs many microns away from the contacts – at the edge of the heterostructure

gate and the WSe2 contact gate set to -13 V (MoSe2 contact gates are set to +13 V). The top inset

of Fig. 1A shows the contacts are ~ 10 µm away from the heterostructure edge.

We note that the observed experimental evidence can be due to a few alternative mechanisms such

as free carriers aid the transport of the CIEs or the CIEs upconvert to IEs while traveling towards

the edge of the heterostructure. Additionally, thermal gradients and coupling to phonons may also

play an important role for CIE diffusion in TMDs as described in reference (43). Further

experiments such as pump-probe experiment and systematic photo-current measurement might be

required for further clarification in a future study.

Section 10. Spatially tunable electroluminescence

In Figure S16a, we show the same plot as Fig. 4A inset, Ids vs Vtg and Vbg for Vds = -3V from

MoSe2 to WSe2 (grounded) in a forward bias configuration. Here, we label three regimes which

correspond to p-doped (region I), n-doped (region II) and intrinsic (region III). We can tune the

spatial location of the electroluminescence based on the doping of the heterostructure (Figure

S16b). When the heterostructure is p-doped (n-doped) as in region I (II), holes (electrons) are free

to move across the heterostructure where recombination occurs at the edge of the heterostructure

and the n-doped MoSe2 (p-doped WSe2) regime. The electroluminescence energy at this interface

is still at around 1.3eV and is tunable with electric field (Fig. 4D), indicating the recombination

still occurs in the heterostructure and is due to interlayer excitons. This agrees well with the band

structure at the interface as the hole (electron) in the heterostructure would have to overcome an

energy barrier of ~400 meV in order to enter and recombine in the monolayer MoSe2 (WSe2).

Because only one layer in the heterostructure becomes doped at a time (as described in the main

text), we know that the free hole (electron) will exist in the WSe2 (MoSe2). When the

heterostructure is tuned to the intrinsic regime along the compensated 𝐸ℎ𝑠 line as in region III, the

electrons and holes recombine within the heterostructure, with the spatial location determined by

the in-plane electric field of Vds, Vtg, and Vbg. In Fig. 4B (4C), we apply Vds = 7V, Vtg = -1V (1V)

and, Vbg = 12V (8.75V), marked with a green circle (red pentagon) in Fig. 4A inset, to demonstrate

spatial control of the recombination location within the heterostructure.

15

Section 11. Electroluminescence spectra

In Figure S17, we show Vds vs. electroluminescence energy with 𝑉𝑡𝑔 = 𝑉𝑏𝑔 = 0𝑉 with the

corresponding I-V response in the Figure S17 inset. We observe interlayer exciton emission for Ids

> 10nA. We observe a broadening of the peak with larger currents, which we attribute to heating

as we are applying over 1 A of current through the sample.

Section 12. Temperature dependence

In Figure S18, we show exciton diffusion maps for different cryostat temperatures. We excite at

the center of the sample with P = 550 W and at T < 50 K, observe emission from the entire

sample. When the sample is warmed above 50K, we see significant decrease in exciton emission

away from the excitation spot. While the single particle diffusion constant is expected to rise as a

function of temperature, a sharp decrease in lifetime has been shown at temperatures above 50 K

(45), which can explain the change in diffusion behavior.

16

`

Figure S1. Schematic of a multi-terminal and multi-gate device structure. Red and green gates

correspond to contact top-gates for WSe2 and MoSe2 respectively; blue (brown) gate electrodes

are the heterostructure top (bottom) gates. Light-blue layers correspond to h-BN. The transition

metal dichalcogenide (TMD) layers are depicted as atomic layers. Electrical contacts for MoSe2

and WSe2 are shown as yellow electrodes.

17

Figure S2. Photoluminescence (PL) spectra from MoSe2 (green), WSe2 (red), and MoSe2/WSe2

heterostructure region (blue) at Vtg = Vbg = 0 V.

18

Figure S3. (a-b) Schematics of the device structure and definition of variables for electric field

in the TMD.

19

Figure S4. (a) An optical image of the second measured device with a similar device structure

with top and bottom gates with contact gates above the MoSe2 and WSe2 contacts. (b) PL spectra

as a function of electric field shows a similar interlayer distance but opposite slope to the main

sample. The flipped order of WSe2 and MoSe2 flips the dipole orientation. (c) Exciton diffusion

map with excitation power = 1.75 mW shows the entire sample area illuminating. (d) Lifetime as

a function of Vbg shows qualitatively the same behavior. We note that at Vbg = 0 V, we measure

an unprecedented lifetime of 14 s.

20

Figure S5. (a) IPL vs. lifetime, taken from Fig. 1A. (b) Extracted radiative lifetime (rad) and

non-radiative lifetime (non-rad) vs. electric field (Ehs). Inset: extracted quantum efficiency () vs.

Ehs.

21

Figure S6. Schematic of the device structure and definition of variables for electric field with

single gates.

22

Figure S7. Representation of electric field as a function of a single gate as described in Eq. S28

(a) and Eq. S29 (b).

23

Figure S8. (a) - (b) Photoluminescence intensity vs. top (Vtg) and bottom (Vbg) gate respectively,

while the other gate is grounded. (c) – (d) lifetime vs. Vtg and Vbg respectively.

24

Figure S9. CCD image of steady state exciton PL when excited at the center of the sample at 10

W (a) and 1 mW (b). The red circle and dashed line represent the radial averaging used for Fig.

2E. The white arrow shows the line cut and direction used for spatial time resolved measurements

in Figs. 2I-J. (c) APD scanning image of exciton diffusion at 10 W. The red curve and the dashed

lines represent the radial averaging used for Figure S9d. (d) Radially averaged intensity as a

function of electric field.

25

Figure S10. Steady-state width as a function of the pump amplitude 𝐀 for three different values

of the diffusion length. The simulation was performed with Gaussian beam waist 𝐰𝟎 = 𝟏 𝛍𝐦 on

a disk of radius 𝟏𝟎 𝛍𝐦 with hard wall boundary conditions. The dashed line indicates the scaling

𝐰 ∝ 𝐀𝟏/𝟒, corresponding to 𝐃𝐞𝐟𝐟 ∝ 𝐀𝟏/𝟐, which is the behaviour expected at high power in the

absence of finite size effects. The circles indicate the parameters for which time dependence is

plotted in Figure S11.

26

Figure S11. Exciton density as a function of time and distance from the beam center for three

different values of the diffusion length. The white line indicates the width as defined by Eq. S40.

The amplitude is chosen such that the steady-state width is 𝟐. 𝟐 𝛍𝐦. All remaining computational

parameters are the same as in Figure S10.

27

Figure S12. Carrier density (left) and exciton density (right) as a function of position and time.

The white lines indicate the width of the distribution. We set 𝐥 = √𝐃𝐞/𝛄 = √𝐃𝐡/𝛄 = 𝟒 𝛍𝐦 and

𝐃𝐱 = 𝟎. The beam waist is 𝐰𝟎 = 𝟏 𝛍𝐦 and a disk of radius 𝟏𝟎 𝛍𝐦 was used for the computation.

The boundary conditions enforce zero carrier density at the edge of the disk. The excitation

amplitude 𝐀 was chosen such that the width of the exciton distribution equals 𝟐. 𝟐 𝛍𝐦 after 𝛄𝐭 =𝟏𝟎.

28

Figure S13. (a) Photoluminescence spectra taken near the center of the sample with P = 500 W

as a function of Vds (applied in the left to right sample direction) and Vbg. Note that the interlayer

exciton regime shifts with Vds, requiring an adjusted Vbg to maintain the same doping in the sample

for the p-type CIEs. (b) Same as (a) but for Vds and Vtg to calibrate the n-type CIEs.

29

Figure S14. (a-c) Normalized spatial dependence of 𝐼𝑃𝐿 under different Vds’s applied on the left

side for p-doped charged interlayer excitons (CIEs). We normalized 𝐼𝑃𝐿 by 𝐼𝑃𝐿(𝑉𝑑𝑠)/𝐼𝑃𝐿(𝑉𝑑𝑠 =0). We observed a larger population of charged IEs near the right WSe2 electrode by increasing

Vds. (d-f) Same as (a-c) except for switched Vds direction. g and h vertical average linecut of the

normalized 𝐼𝑃𝐿 for data from (a-c) and (d-f) respectively, vs. X for different Vds. For (a) and (d)

Vbg = -3.375V, for (b) and (e) Vbg = -1.25V and for (c) and (f) Vbg = 0.875V while Vtg = 0 in all

the cases. The WSe2 contact top-gates were kept at -13V during this measurement.

30

Figure S15. (a-c) Normalized spatial dependence of 𝐼𝑃𝐿 under different Vds’s applied on the left

side for n-doped charged interlayer excitons (CIEs). We normalized 𝐼𝑃𝐿 by 𝐼𝑃𝐿(𝑉𝑑𝑠)/𝐼𝑃𝐿(𝑉𝑑𝑠 =0). (d-f) Same as (a-c) except for switched Vds direction. g and h vertical average linecut of the

normalized 𝐼𝑃𝐿 for data from (a-c) and (d-f) respectively, vs. X for different Vds. For (a) and (d) Vtg

= 1.6 V, for (b) and (e) Vtg = 3.2 V and for (c) and (f) Vtg = 4.8 V while Vbg = 0 in all the cases.

The WSe2 contact top-gates were kept at -13 V and the MoSe2 contact top-gates were kept at 13

V during this measurement.

31

Figure S16. (a) Ids vs. Vtg and Vbg (with Vds = -3V on MoSe2 and grounded WSe2). The white

dashed line represents the compensated electric field line where 𝑉𝑏𝑔 = 10.37𝑉 − 𝑉𝑡𝑔 and is

defined in Section 1. (b) Spatial dependent electroluminescence maps taken in the labeled regimes

(I, II, III). These regimes are in the p-doped, n-doped, and intrinsic regimes, respectively.

32

Figure S17. Electroluminescence spectra as a function of applied Vds. Emission energy is similar

to that of PL. Inset: Electroluminescence intensity as a function of applied Vds.

33

Figure S18. Temperature dependent log PL intensity exciton diffusion maps with continuous wave

excitation P = 550 µW at the center of the sample. The white dashed lines define the heterostructure

region.

References and Notes

1. L. V. Butov, A. L. Ivanov, A. Imamoglu, P. B. Littlewood, A. A. Shashkin, V. T. Dolgopolov,

K. L. Campman, A. C. Gossard, Stimulated scattering of indirect excitons in coupled

quantum wells: Signature of a degenerate Bose-gas of excitons. Phys. Rev. Lett. 86,

5608–5611 (2001). doi:10.1103/PhysRevLett.86.5608 Medline

2. M. M. Fogler, L. V. Butov, K. S. Novoselov, High-temperature superfluidity with indirect

excitons in van der Waals heterostructures. Nat. Commun. 5, 4555 (2014).

doi:10.1038/ncomms5555 Medline

3. M. Combescot, R. Combescot, F. Dubin, Bose-Einstein condensation and indirect excitons: A

review. Rep. Prog. Phys. 80, 066501 (2017). doi:10.1088/1361-6633/aa50e3 Medline

4. I. Schwartz, D. Cogan, E. R. Schmidgall, Y. Don, L. Gantz, O. Kenneth, N. H. Lindner, D.

Gershoni, Deterministic generation of a cluster state of entangled photons. Science 354,

434–437 (2016). doi:10.1126/science.aah4758 Medline

5. E. Poem, Y. Kodriano, C. Tradonsky, N. H. Lindner, B. D. Gerardot, P. M. Petroff, D.

Gershoni, Accessing the dark exciton with light. Nat. Phys. 6, 993–997 (2010).

doi:10.1038/nphys1812

6. A. A. High, J. R. Leonard, M. Remeika, L. V. Butov, M. Hanson, A. C. Gossard,

Condensation of excitons in a trap. Nano Lett. 12, 2605–2609 (2012).

doi:10.1021/nl300983n Medline

7. A. A. High, J. R. Leonard, A. T. Hammack, M. M. Fogler, L. V. Butov, A. V. Kavokin, K. L.

Campman, A. C. Gossard, Spontaneous coherence in a cold exciton gas. Nature 483,

584–588 (2012). doi:10.1038/nature10903 Medline

8. A. A. High, A. K. Thomas, G. Grosso, M. Remeika, A. T. Hammack, A. D. Meyertholen, M.

M. Fogler, L. V. Butov, M. Hanson, A. C. Gossard, Trapping indirect excitons in a GaAs

quantum-well structure with a diamond-shaped electrostatic trap. Phys. Rev. Lett. 103,

087403 (2009). doi:10.1103/PhysRevLett.103.087403 Medline

9. A. A. High, A. T. Hammack, L. V. Butov, L. Mouchliadis, A. L. Ivanov, M. Hanson, A. C.

Gossard, Indirect excitons in elevated traps. Nano Lett. 9, 2094–2098 (2009).

doi:10.1021/nl900605b Medline

10. A. T. Hammack, M. Griswold, L. V. Butov, L. E. Smallwood, A. L. Ivanov, A. C. Gossard,

Trapping of cold excitons in quantum well structures with laser light. Phys. Rev. Lett. 96,

227402 (2006). doi:10.1103/PhysRevLett.96.227402 Medline

11. L. V. Butov, C. W. Lai, A. L. Ivanov, A. C. Gossard, D. S. Chemla, Towards Bose-Einstein

condensation of excitons in potential traps. Nature 417, 47–52 (2002).

doi:10.1038/417047a Medline

12. R. Rapaport, G. Chen, D. Snoke, S. H. Simon, L. Pfeiffer, K. West, Y. Liu, S. Denev, Charge

separation of dense two-dimensional electron-hole gases: Mechanism for exciton ring

pattern formation. Phys. Rev. Lett. 92, 117405 (2004).

doi:10.1103/PhysRevLett.92.117405 Medline

13. X. Hong, J. Kim, S.-F. Shi, Y. Zhang, C. Jin, Y. Sun, S. Tongay, J. Wu, Y. Zhang, F. Wang,

Ultrafast charge transfer in atomically thin MoS2/WS2 heterostructures. Nat.

Nanotechnol. 9, 682–686 (2014). doi:10.1038/nnano.2014.167 Medline

14. P. Rivera, J. R. Schaibley, A. M. Jones, J. S. Ross, S. Wu, G. Aivazian, P. Klement, K.

Seyler, G. Clark, N. J. Ghimire, J. Yan, D. G. Mandrus, W. Yao, X. Xu, Observation of

long-lived interlayer excitons in monolayer MoSe2-WSe2 heterostructures. Nat. Commun.

6, 6242 (2015). doi:10.1038/ncomms7242 Medline

15. D. Unuchek, A. Ciarrocchi, A. Avsar, K. Watanabe, T. Taniguchi, A. Kis, Room-temperature

electrical control of exciton flux in a van der Waals heterostructure. Nature 560, 340–344

(2018). doi:10.1038/s41586-018-0357-y Medline

16. X. M. Liu, K. Watanabe, T. Taniguchi, B. I. Halperin, P. Kim, Quantum Hall drag of exciton

condensate in graphene. Nat. Phys. 13, 746–750 (2017). doi:10.1038/nphys4116

17. L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M.

Campos, D. A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, C. R. Dean, One-

dimensional electrical contact to a two-dimensional material. Science 342, 614–617

(2013). doi:10.1126/science.1244358 Medline

18. See supplementary materials.

19. B. Fallahazad, H. C. P. Movva, K. Kim, S. Larentis, T. Taniguchi, K. Watanabe, S. K.

Banerjee, E. Tutuc, Shubnikov-de Haas Oscillations of High-Mobility Holes in

Monolayer and Bilayer WSe2: Landau Level Degeneracy, Effective Mass, and Negative

Compressibility. Phys. Rev. Lett. 116, 086601 (2016).

doi:10.1103/PhysRevLett.116.086601 Medline

20. G. Scuri, Y. Zhou, A. A. High, D. S. Wild, C. Shu, K. De Greve, L. A. Jauregui, T.

Taniguchi, K. Watanabe, P. Kim, M. D. Lukin, H. Park, Large Excitonic Reflectivity of

Monolayer MoSe2 Encapsulated in Hexagonal Boron Nitride. Phys. Rev. Lett. 120,

037402 (2018). doi:10.1103/PhysRevLett.120.037402 Medline

21. Y. Zhou, G. Scuri, D. S. Wild, A. A. High, A. Dibos, L. A. Jauregui, C. Shu, K. De Greve, K.

Pistunova, A. Y. Joe, T. Taniguchi, K. Watanabe, P. Kim, M. D. Lukin, H. Park, Probing

dark excitons in atomically thin semiconductors via near-field coupling to surface

plasmon polaritons. Nat. Nanotechnol. 12, 856–860 (2017). doi:10.1038/nnano.2017.106

Medline

22. Z. Wang, L. Zhao, K. F. Mak, J. Shan, Probing the Spin-Polarized Electronic Band Structure

in Monolayer Transition Metal Dichalcogenides by Optical Spectroscopy. Nano Lett. 17,

740–746 (2017). doi:10.1021/acs.nanolett.6b03855 Medline

23. P. Rivera, K. L. Seyler, H. Yu, J. R. Schaibley, J. Yan, D. G. Mandrus, W. Yao, X. Xu,

Valley-polarized exciton dynamics in a 2D semiconductor heterostructure. Science 351,

688–691 (2016). doi:10.1126/science.aac7820 Medline

24. I. V. Bondarev, M. R. Vladimirova, Complexes of dipolar excitons in layered quasi-two-

dimensional nanostructures. Phys. Rev. B 97, 165419 (2018).

doi:10.1103/PhysRevB.97.165419

25. A. M. Jones, H. Yu, N. J. Ghimire, S. Wu, G. Aivazian, J. S. Ross, B. Zhao, J. Yan, D. G.

Mandrus, D. Xiao, W. Yao, X. Xu, Optical generation of excitonic valley coherence in

monolayer WSe2. Nat. Nanotechnol. 8, 634–638 (2013). doi:10.1038/nnano.2013.151

Medline

26. J. S. Ross, S. Wu, H. Yu, N. J. Ghimire, A. M. Jones, G. Aivazian, J. Yan, D. G. Mandrus, D.

Xiao, W. Yao, X. Xu, Electrical control of neutral and charged excitons in a monolayer

semiconductor. Nat. Commun. 4, 1474 (2013). doi:10.1038/ncomms2498 Medline

27. G. Finkelstein, H. Shtrikman, I. Bar-Joseph, Negatively and positively charged excitons in

GaAs/AlxGa1-xAs quantum wells. Phys. Rev. B 53, R1709–R1712 (1996).

doi:10.1103/PhysRevB.53.R1709 Medline

28. K. Kheng, R. T. Cox, M. Y. d’Aubigné, F. Bassani, K. Saminadayar, S. Tatarenko, S.

Tatarenko, Observation of negatively charged excitons X- in semiconductor quantum

wells. Phys. Rev. Lett. 71, 1752–1755 (1993). doi:10.1103/PhysRevLett.71.1752 Medline

29. M. A. Lampert, Mobile and Immobile Effective-Mass-Particle Complexes in Nonmetallic

Solids. Phys. Rev. Lett. 1, 450–453 (1958). doi:10.1103/PhysRevLett.1.450

30. M. Sidler, P. Back, O. Cotlet, A. Srivastava, T. Fink, M. Kroner, E. Demler, A. Imamoglu,

Fermi polaron-polaritons in charge-tunable atomically thin semiconductors. Nat. Phys.

13, 255–261 (2017). doi:10.1038/nphys3949

31. R. Schmidt, T. Enss, V. Pietila, E. Demler, Fermi polarons in two dimensions. Phys. Rev. A

85, 021602 (2012). doi:10.1103/PhysRevA.85.021602

32. B. Laikhtman, R. Rapaport, Exciton correlations in coupled quantum wells and their

luminescence blue shift. Phys. Rev. B 80, 195313 (2009).

doi:10.1103/PhysRevB.80.195313

33. L. V. Butov, L. S. Levitov, A. V. Mintsev, B. D. Simons, A. C. Gossard, D. S. Chemla,

Formation mechanism and low-temperature instability of exciton rings. Phys. Rev. Lett.

92, 117404 (2004). doi:10.1103/PhysRevLett.92.117404 Medline

34. J. S. Ross, P. Rivera, J. Schaibley, E. Lee-Wong, H. Yu, T. Taniguchi, K. Watanabe, J. Yan,

D. Mandrus, D. Cobden, W. Yao, X. Xu, Interlayer Exciton Optoelectronics in a 2D

Heterostructure p-n Junction. Nano Lett. 17, 638–643 (2017).

doi:10.1021/acs.nanolett.6b03398 Medline

35. C. H. Lee, G.-H. Lee, A. M. van der Zande, W. Chen, Y. Li, M. Han, X. Cui, G. Arefe, C.

Nuckolls, T. F. Heinz, J. Guo, J. Hone, P. Kim, Atomically thin p-n junctions with van

der Waals heterointerfaces. Nat. Nanotechnol. 9, 676–681 (2014).

doi:10.1038/nnano.2014.150 Medline

36. X. D. Xu, W. Yao, D. Xiao, T. F. Heinz, Spin and pseudospins in layered transition metal

dichalcogenides. Nat. Phys. 10, 343–350 (2014). doi:10.1038/nphys2942

37. L. Jauregui, Replication data for: Electrical control of interlayer exciton dynamics in

atomically thin heterostructures. Harvard Dataverse (2019);

https://doi.org/10.7910/DVN/IZQAVH.

38. K. Kim, S. Larentis, B. Fallahazad, K. Lee, J. Xue, D. C. Dillen, C. M. Corbet, E. Tutuc,

Band Alignment in WSe2-Graphene Heterostructures. ACS Nano 9, 4527–4532 (2015).

doi:10.1021/acsnano.5b01114 Medline

39. A. L. Ivanov et al., Quantum diffusion of dipole-oriented indirect excitons in coupled

quantum wells. Europhys. Lett. 59, 586–591 (2002). doi:10.1209/epl/i2002-00144-3

40. S. V. Lobanov, N. A. Gippius, L. V. Butov, Theory of condensation of indirect excitons in a

trap. Phys. Rev. B 94, 245401 (2016). doi:10.1103/PhysRevB.94.245401

41. P. Cudazzo, I. V. Tokatly, A. Rubio, Dielectric screening in two-dimensional insulators:

Implications for excitonic and impurity states in graphane. Phys. Rev. B 84, 085406

(2011). doi:10.1103/PhysRevB.84.085406

42. T. C. Berkelbach, M. S. Hybertsen, D. R. Reichman, Theory of neutral and charged excitons

in monolayer transition metal dichalcogenides. Phys. Rev. B 88, 045318 (2013).

doi:10.1103/PhysRevB.88.045318

43. M. M. Glazov, Phonon wind and drag of excitons in monolayer semiconductors. Phys. Rev. B

100, 045426 (2019). doi:10.1103/PhysRevB.100.045426

44. M. Kulig, J. Zipfel, P. Nagler, S. Blanter, C. Schüller, T. Korn, N. Paradiso, M. M. Glazov,

A. Chernikov, Exciton Diffusion and Halo Effects in Monolayer Semiconductors. Phys.

Rev. Lett. 120, 207401 (2018). doi:10.1103/PhysRevLett.120.207401 Medline

45. P. Nagler, G. Plechinger, M. V. Ballottin, A. Mitioglu, S. Meier, N. Paradiso, C. Strunk, A.

Chernikov, P. C. M. Christianen, C. Schüller, T. Korn, Interlayer exciton dynamics in a

dichalcogenide monolayer heterostructure. 2D Mater. 4, 025112 (2017).

doi:10.1088/2053-1583/aa7352